diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzaixh b/data_all_eng_slimpj/shuffled/split2/finalzzaixh
new file mode 100644
index 0000000000000000000000000000000000000000..684b66c21f53161927a6f0659d2bd1b14954460b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzaixh
@@ -0,0 +1,5 @@
+{"text":"\\subsection*{Acknowledgments}\nWe thank Denny Wu, Xiaoyu Liu, Dongruo Zhou, Vedant Nanda, Ziyan Yang, Xiaoxiao Li, Jiahao Su, Wei Hu, Bo Han, Simon S. Du, Justin Brody, and Don Perlis for helpful feedback and insightful discussions. \nAdditionally, we thank Houze Wang, Qin Yang, Xin Li, Guodong Zhang, Yixuan Ren, and Kai Wang for help with computing resources.\nThis research is partially performed while Jingling Li is a remote research intern at the Vector Institute and the University of Toronto.\nLi and Dickerson were supported by an ARPA-E DIFFERENTIATE Award, NSF CAREER IIS-1846237, NSF CCF-1852352, NSF D-ISN \\#2039862, NIST MSE \\#20126334, NIH R01 NLM-013039-01, DARPA GARD \\#HR00112020007, DoD WHS \\#HQ003420F0035, and a Google Faculty Research Award.\nBa is supported in part by the CIFAR AI Chairs program, LG Electronics, and NSERC.\nXu is supported by NSF CAREER award 1553284 and NSF III 1900933. Xu is also partially supported by JST ERATO JPMJER1201 and JSPS Kakenhi JP18H05291. \nZhang is supported by ODNI, IARPA, via the BETTER Program contract \\#2019-19051600005. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein. \n\n{\n\\small\n\\bibliographystyle{unsrtnat}\n\\typeout{}\n\n\\subsection{Related Work}\n\\label{sec:related}\nA commonly studied type of noisy label is the random label noise, where the noisy labels are drawn i.i.d.~from a uniform distribution.\nWhile neural networks trained with random labels easily overfit~\\citep{zhang2016understanding}, it has been observed that networks learn simple patterns first~\\cite{arpit2017closer}, converge faster on downstream tasks~\\cite{maennel2020neural}, and benefit from memorizing atypical training samples~\\cite{feldman2020neural}.\n\nAccordingly, many recent works on noisy label training are based on the assumption that when trained with noisy labels, neural networks would first fit to clean labels~\\cite{ lyu2019curriculum, han2018co, jiang2018mentornet,li2020dividemix, liu2020early} and learn useful feature patterns~\\cite{hendrycks2018using, lee2019robust, bahri2020deep, wu2020topological}.\nYet, these methods are often more effective on random label noise than on more realistic label noise (i.e., class-dependent and instance-dependent label noise).\n\nMany works on representation learning have investigated the features preferred by a network during training~\\cite{arpit2017closer, hermann2020shapes, shah2020pitfalls, sanyal2020benign}, and how to interpret or control the learned representations on clean data~\\cite{alain2016understanding, hermann2020shapes, hermann2019origins, montavon2018methods, yuan2020clime}. \nOur paper focuses more on the predictive power rather than the explanatory power in the learned representations.\nWe adapt the method in~\\cite{alain2016understanding} to measure the predictive power in representations, and we study learning from noisy labels rather than from a clean distribution.\n\nOn noiseless settings, prior works show that neural networks have the inductive bias to learn simple patterns~\\cite{arpit2017closer, hermann2020shapes, shah2020pitfalls, sanyal2020benign}. \nOur work formalizes what is considered as a simple pattern for a given network via architectural alignments, and we extend the definition of alignment in~\\cite{Xu2020What} to noisy settings.\n\n\\section{Introduction}\n\\label{sec:intro}\nSupervised learning starts with collecting labeled data.\nYet, high-quality labels are often expensive.\nTo reduce annotation cost, we collect labels from non-experts~\\cite{snow2008cheap, welinder2010multidimensional, yan2014learning, yu2018learning} or online queries~\\cite{blum2003noise, jiang2020beyond, liu2011noise}, which are inevitably noisy. \nTo learn from these noisy labels, previous works propose many techniques, including modeling the label noise~\\cite{natarajan2013learning, liu2015classification, yao2020dual}, designing robust losses~\\cite{ghosh2017robust, lyu2019curriculum, wang2019symmetric, zhang2018generalized}, adjusting loss before gradient updates~\\cite{arazo2019unsupervised, chang2017active, han2020sigua, hendrycks2018using, ma2018dimensionality, patrini2017making, reed2014training, song2019selfie, wang2017multiclass}, selecting trust-worthy samples~\\cite{lyu2019curriculum, song2019selfie, chen2019understanding, han2018co, jiang2018mentornet, malach2017decoupling, nguyen2019self, shen2019learning, wang2018iterative, yu2019does},  designing robust architectures~\\cite{bekker2016training, chen2015webly, goldberger2016training, han2018masking, jindal2016learning, li2020understanding, sukhbaatar2014training, yao2018deep}, applying robust regularization in training~\\cite{goodfellow2014explaining, hendrycks2019using, jenni2018deep, pereyra2017regularizing, tanno2019learning, zhang2017mixup}, using meta-learning to avoid over-fitting~\\cite{garcia2016noise, li2019learning}, and applying semi-supervised learning~\\cite{nguyen2019self, ding2018semi, li2020dividemix, liu2020early, yan2016robust} to learn better representations.\n\nWhile these methods improve some networks' robustness to noisy labels, we observe that their effectiveness depends on how well the network's architecture aligns with the target\/noise functions, and they are less effective when encountering more realistic label noise that is class-dependent or instance-dependent.\nThis motivates us to investigate an understudied topic: how the network's architecture impacts its robustness to noisy labels.\n\nWe formally answer this question by analyzing how a network's architecture aligns with the target function and the noise.\nTo start, we measure the robustness of a network via the predictive power in its learned representations (Definition~\\ref{def:predict_power}), as\nmodels with large test errors may still learn useful predictive hidden representations~\\cite{arpit2017closer, maennel2020neural}. \nIntuitively, the predictive power measures how well the representations can predict the target function. \nIn practice, we measure it by training a linear model on top of the learned representations using a small set of clean labels and evaluate the linear model's test performance~\\cite{alain2016understanding}.\n\nWe find that a network having a more aligned architecture with the target function is more robust to noisy labels due to its more predictive representations, whereas a network having an architecture more aligned with the noise function is less robust.\nIntuitively, a \\textit{good} alignment between a network's architecture and a function exists if the architecture can be decomposed into several modules such that each module can simulate one part of the function with a \\textit{small} sample complexity.\nThe formal definition of alignment is in Section~\\ref{subsec:alignment_formal}, adapted from~\\cite{Xu2020What}. \n\nOur proposed framework provides initial theoretical support for our findings on a simplified noisy setting (Theorem~\\ref{thm:main}).\nEmpirically, we validate our findings on synthetic graph algorithmic tasks by designing several variants of Graph Neural Networks (GNNs), whose theoretical properties and alignment with algorithmic functions have been well-studied~\\cite{Xu2020What, du2019graph, xu2020neural}. \nMany noisy label training methods are applied to image classification datasets, so we also validate our findings on image domains using different architectures.\n\nMost of our analysis and experiments use standard neural network training. \nInterestingly, we find similar results when using DivideMix~\\cite{li2020dividemix}, a SOTA method for learning with noisy labels:\nfor networks less aligned with the target function, the SOTA method barely helps and sometimes even hurts test accuracy; whereas for more aligned networks, it helps greatly.\n\nFor well-aligned networks, the predictive power of their learned representation could further improve the test performance of SOTA methods, especially on class-dependent or instance-dependent label noise where current methods on noisy label training are less effective.  \nMoreover, on Clothing1M~\\cite{xiao2015learning}, a large-scale dataset with real-world label noise, the predictive power of a well-aligned network's learned representations could even outperform some sophisticated methods that use clean labels.  \n\nIn summary, we investigate how an architecture's alignments with different (target and noise) functions affect the network's robustness to noisy labels, in which we discover that despite having large test errors, networks well-aligned with the target function can still be robust to noisy labels when evaluating their predictive power in learned representations. \nTo formalize our finding, we provide a theoretical framework to illustrate the above connections.\nAt the same time, we conduct empirical experiments on various datasets with various network architectures to validate this finding.\nBesides, this finding further leads to improvements over SOTA noisy-label-training methods on various datasets and under various kinds of noisy labels (Tables~\\ref{table:cifar10_sym}-\\ref{table:webvision} in Appendix~\\ref{suppsec:add_exp_results}).\n\\section{Theoretical Framework} \n\\label{sec:prelim}\nIn this section, we introduce our problem settings, give formal definitions for ``predictive power'' and ``alignment,'' and present our main hypothesis as well as our main theorem. \n\n\\subsection{Problem Settings}\nLet $\\mathcal{X}$ denote the input domain, which can be vectors, images, or graphs.\nThe task is to learn an underlying target function $f: \\mathcal{X} \\rightarrow \\mathcal{Y}$ on a noisy training dataset $S := \\lbrace (\\x_i, y_i) \\rbrace_{i \\in \\mathcal{I}} \\bigcup \\ \\lbrace (\\x_i, \\hat{y}_i) \\rbrace_{i \\in \\mathcal{I}'}$,  \nwhere $y :=  f(\\x)$ denotes the true label for an input $\\x$, and $\\hat{y}$ denotes the noisy label. \nHere, the set $\\mathcal{I}$ contains indices with clean labels, and $\\mathcal{I}'$ contains indices with noisy labels.\nWe denote $\\frac{|\\mathcal{I}'|}{|S|}$ as the \\textit{noise ratio} in the dataset $S$.\nWe consider both regression and classification problems.\n\n\\textbf{Regression settings.} We consider a label space $\\mathcal{Y} \\subseteq \\mathbb{R}$ and two types of label noise: \na) \\textbf{additive label noise}~\\cite{hu2019simple}: $\\hat{y} := y + \\epsilon$, where $\\epsilon$ is a random variable independent from $\\x$; \nb) \\textbf{instance-dependent label noise}: $\\hat{y} := g(\\x)$ where $g: \\mathcal{X} \\rightarrow \\mathcal{Y}$ is a noise function dependent on the input.\n\n\\textbf{Classification settings.} We consider a discrete label space with $C$ classes: $\\mathcal{Y} = \\{1,2, \\cdots, C\\}$, and three types of label noise: \na) \\textbf{uniform label noise}:  $\\hat{y} \\sim \\text{Unif}(1, C)$, where the noisy label is drawn from a discrete uniform distribution with values between $1$ and $C$, and thus is independent of the true label; \nb) \\textbf{flipped label noise}: $\\hat{y}$ is generated based on the value of the true label $y$ and does not consider other input structures;\nc) \\textbf{instance-dependent label noise}: $\\hat{y} := g(\\x)$ where $g: \\mathcal{X} \\rightarrow \\mathcal{Y}$ is a function dependent on the input $\\x$'s internal structures.\nPrevious works on noisy label learning commonly study uniform and flipped label noise. \nA few recent works~\\cite{cheng2020learning, wang2020proselflc} explore the instance-dependent label noise as it is more realistic.\n\n\\subsection{Predictive Power in Representations}\nA network's robustness is often measured by its test performance after trained with noisy labels. \nYet, since models with large test errors may still learn useful representations, we measure the robustness of a network by how good the learned representations are at predicting the target function --- the predictive power in representations.\nTo formalize this definition, we decompose a neural network $\\mathcal{N}$ into different modules $\\mathcal{N}_1, \\mathcal{N}_2, \\cdots$, where each module can be a single layer (e.g., a convolutional layer) or a block of layers (e.g., a residual block).\n\n\\begin{definition}\n\\label{def:predict_power}\n\\textit{(Predictive power).} Let $f: \\mathcal{X} \\rightarrow \\mathcal{Y}$ denote the underlying target function where the input $\\x \\in \\mathcal{X}$ is drawn from a distribution $\\mathcal{D}$. \nLet $\\mathcal{C} := \\lbrace (\\x_i, y_i) \\rbrace_{i=1}^{m}$ denote a small set of clean data (i.e., $y_i = f(\\x_i)$). \nGiven a network $\\mathcal{N}$ with $n$ modules $\\mathcal{N}_j$, let $h^{(j)}(\\x)$ denote the representation from module $\\mathcal{N}_j$ on the input $\\x$ (i.e., the output of $\\mathcal{N}_j$). \nLet ${L}$ denote the linear model trained with the clean set $\\mathcal{C}$ where we use $h^{(j)}(\\x)$ as the input, and $y_i$ as the target value during training.\nThen the predictive power of representations from the module $\\mathcal{N}_i$ is defined as \n\\begin{align}\n    \\pred_{j}(f, \\mathcal{N}, \\mathcal{C}) := \\mathop{\\mathbb{E}}_{\\x \\sim \\mathcal{D}} \n\\left[l\\left(f(\\x), {L}(h^{(j)}(\\x))\\right) \\right],\n\\end{align}\nwhere $l$ is a loss function used to evaluate the test performance on the learning task. \n\\end{definition}\n\\textbf{Remark.} Notice that smaller $\\pred_{j}(f, \\mathcal{N}, \\mathcal{C})$ indicates better predictive power; i.e., the representations are better at predicting the target function. \nWe empirically evaluate the predictive power using linear regression to obtain a trained linear model ${L}$, which avoids the issue of local minima as we are solving a convex problem; then we evaluate ${L}$ on the test set.\n\n\\subsection{Formalization of Alignment}\n\\label{subsec:alignment_formal}\nOur analysis stems from the intuition that a network would be more robust to noisy labels if it could learn the target function more easily than the noise function.\nThus, we use architectural alignment to formalize what is easy to learn by a given network. \\citet{Xu2020What} define the alignment between a network and a deterministic function via a sample complexity measure (i.e., the number of samples needed to ensure low test error with high probability) in a PAC learning framework (Definition 3.3 in~\\citet{Xu2020What}). \nIntuitively, a network aligns well with a function if each network module can easily learn one part of the function with a small sample complexity.\n\n\\begin{definition}\n\\label{def:alignment}\n\\textit{(Alignment, simplified based on~\\citet{Xu2020What}).} \nLet $\\mathcal{N}$ denote a neural network with $n$ modules $\\mathcal{N}_j$. \nGiven a function $f: \\mathcal{X} \\rightarrow \\mathcal{Y}$ which can be decomposed into $n$ functions $f_j$ (e.g., $f(\\x) = f_1(f_2(...f_n(\\x)))$), the alignment between the network $\\mathcal{N}$ and $f$ is defined via \n\\begin{align}\n    \\textit{Alignment}(\\mathcal{N}, f, \\epsilon, \\delta) := \\max_{j} \\mathcal{M}_{{A}_j}(f_j, \\mathcal{N}_j, \\epsilon, \\delta),\n\\end{align}\nwhere $\\mathcal{M}_{{A}_j}(f_j, \\mathcal{N}_j, \\epsilon, \\delta)$ denotes the sample complexity measure for $\\mathcal{N}_j$ to learn $f_j$ with $\\epsilon$ precision at a failure\nprobability $\\delta$ under a learning algorithm $A_j$.\n\n\\textbf{Remark.} \nNotice that smaller $\\textit{Alignment}(\\mathcal{N}, f, \\epsilon, \\delta)$ indicates better alignment between network $\\mathcal{N}$ and function $f$. \nIf $f$ is obtuse or does not have a structural decomposition, we can choose $n=1$, and the definition of alignment degenerates into the sample complexity measure for $\\mathcal{N}$ to learn $f$.\nAlthough it is sometimes non-trivial to compute the exact alignment for a task without clear algorithmic structures, we could break this complicated task into sub-tasks, and it would be easier to measure the sample complexity of learning each sub-task. \n\n\\end{definition}\n\\citet{Xu2020What} further prove that better alignment implies better sample complexity and vice versa.\n\\begin{theorem} \n\\label{thm:xu2020}\n\\textit{(Informal;~\\cite{Xu2020What})} \nFix $\\epsilon$ and $\\delta$.\nGiven a target function $f: \\mathcal{X} \\rightarrow \\mathcal{Y}$ and a network $\\mathcal{N}$, suppose $\\left\\lbrace x_i \\right\\rbrace_{i=1}^M$ are i.i.d. samples drawn from a distribution $\\mathcal{D}$, and let $y_i := f(x_i)$. \nThen $\\textit{Alignment}(\\mathcal{N}, f, \\epsilon, \\delta) \\leq M$\nif and only if there exists a learning algorithm $A$ such that\n\\begin{align}\n    \\mathbb{P}_{x \\sim \\mathcal{D}} \\left[ \\| f_{\\mathcal{N}, A}(x) - f(x) \\| \\leq \\epsilon \\right] \\geq 1- \\delta,\n\\end{align}\nwhere $f_{\\mathcal{N}, A}$ is the function generated by $A$ on the training data $\\left\\lbrace x_i, y_i \\right\\rbrace_{i=1}^M$.\n\\end{theorem}\n\\textbf{Remark.} Intuitively, a function $f$ (with a decomposition $\\{f_j\\}_j$) can be efficiently learned by a network $\\mathcal{N}$ (with modules $\\{\\mathcal{N}_j\\}_j$) iff each $f_j$ can be efficiently learned by $\\mathcal{N}_j$. \n\nWe further extend Definition~\\ref{def:alignment} to work with a random process $\\mathcal{F}$ (i.e., a set of all possible sample functions that describes the noisy label distribution). \n\\begin{definition}\n\\label{def:alignment_extend}\n\\textit{(Alignment, extension to various noise functions).} \nGiven a neural network $\\mathcal{N}$ and a random process $\\mathcal{F}$,\nfor each $f \\in \\mathcal{F}$, the alignment between $\\mathcal{N}$ and $f$ is measured via $\\max_{j} \\mathcal{M}_{{A}_j}(f_j, \\mathcal{N}_j, \\epsilon, \\delta)$ based on Definition~\\ref{def:alignment}.\nThen the alignment between $\\mathcal{N}$ and $\\mathcal{F}$ is defined as\n\\[\n    \\textit{Alignment}^{*}(\\mathcal{N}, \\mathcal{F}, \\epsilon, \\delta) := \\sup_{f\\in\\mathcal{F}} \\max_{j} \\mathcal{M}_{{A}_j}(f_j, \\mathcal{N}_j, \\epsilon, \\delta),\n    \\vspace{-0.5em}\n\\]\nwhere $\\mathcal{N}$ can be decomposed differently for various $f$.\n\\end{definition}\n\n\\subsection{Better Alignment Implies Better Robustness (Better Predictive Power)}\nBuilding on the definitions of \\textit{predictive power} and \\textit{alignment}, we hypothesize that\na network better-aligned with the target function (smaller $\\textit{Alignment}(\\mathcal{N}, f, \\epsilon, \\delta)$) would learn more predictive representations (smaller $\\pred_{j}(f, \\mathcal{N}, \\mathcal{C})$) when trained on a given noisy dataset.\n\\begin{hypothesis}\n\\label{thm:hypothesis} (Main Hypothesis). \nLet $f: \\mathcal{X} \\rightarrow \\mathcal{Y}$ denote the target function. \nFix $\\epsilon$, $\\delta$, a learning algorithm $A$, a noise ratio, and a noise function $g: \\mathcal{X} \\rightarrow \\mathcal{Y}$ (which may be a drawn from a random process).\nLet S denote a noisy training dataset and $\\mathcal{C}$ denote a small set of clean data.\nThen for a network $\\mathcal{N}$ trained on $S$ with the learning algorithm $A$, \n\\begin{align}\n\\textit{Alignment}(\\mathcal{N}, f, \\epsilon, \\delta)\\downarrow \\ \\Longrightarrow \\pred_{j}(f, \\mathcal{N}, \\mathcal{C})\\downarrow,\n\\end{align}\nwhere $j$ is selected based on the network's architectural alignment with the target function (for simplicity, we consider $j = n-1$ in this work).\n\\end{hypothesis}\n\nWe prove this hypothesis for a simplified case where the target function shares some common structures with the noise function (e.g., class-dependent label noise). \nWe refer the readers to Appendix~\\ref{suppapp:thm} for a full statement of our main theorem with detailed assumptions.    \n\\begin{theorem}\n\\label{thm:main}\n\\textit{(Main Theorem; informal)} \nFor a target function $f: \\mathcal{X} \\rightarrow \\mathcal{Y}$ and a noise function $g: \\mathcal{X} \\rightarrow \\mathcal{Y}$, consider a neural network $\\mathcal{N}$ well-aligned with $f$ such that $\\pred_{j}(f, \\mathcal{N}, \\mathcal{C})$ is small when training $\\mathcal{N}$ on clean data (i.e., $\\pred_{j}(f, \\mathcal{N}, \\mathcal{C}) < c$ for some small constant $c$). If there exists a function $h$ on the input domain $\\mathcal{X}$ such that $f$ and $g$ can be decomposed as follows: $\\forall x \\in \\mathcal{X}$, $f(\\x) = f_r(h(\\x))$ with $f_r$ being a linear function, and $g(\\x) = g_r(h(\\x))$ for some function $g_r$, then the representations learned by $\\mathcal{N}$ on the noisy dataset still have a good predictive power with $\\pred_{j}(f, \\mathcal{N}, \\mathcal{C}) < c$.\n\\end{theorem}\n\nWe further provide empirical support for our hypothesis via systematic experiments on various architectures, target and noise functions across both regression and classification settings.\n\\section{Experiments on Graph Neural Networks}\n\\label{sec:gnn}\nWe first validate our hypothesis on synthetic graph algorithmic tasks by designing GNNs with different levels of alignments to the underlying target\/noise functions. \nWe consider regression tasks.\nThe theoretical  properties  of  GNNs  and  their alignment with algorithmic regression tasks are well-studied~\\cite{Xu2020What, du2019graph, xu2020neural, sato2019approximation}.\nTo start, we conduct experiments on different types of additive label noise and extend our experiments to instance-dependent label noise, which is closer to real-life noisy labels.\n\n\\textbf{Common Experimental Settings.}  \nThe training and validation sets always have the same noise ratio, the percentage of data with noisy labels. \nWe choose mean squared error (MSE) and Mean Absolute Error (MAE) as our loss functions.\nDue to space limit, the results using MAE are in Appendix~\\ref{appsec:mae}.\nAll training details are in Appendix~\\ref{suppsec:gnn_training_details}.\nThe test error is measured by mean absolute percentage error (MAPE), a relative error metric.\n\n\\subsection{Background: Graph Neural Networks}\nGNNs are structured networks operating on graphs with MLP modules~\\cite{battaglia2018relational, scarselli2009graph, xu2018how, xu2018representation, xu2021optimization, liao2021information, cai2021graphnorm}. \nThe input is a graph ${\\mathcal{G}}=(V,E)$ where each node $u \\in V$ has a feature vector $\\x_u$, and we use $\\mathcal{N}(u)$ to denote the set of neighbors of $u$.\nGNNs iteratively compute the node representations via message passing: (1) the node representation $\\bm{h}_u$ is initialized as the node feature: $\\bm{h}_u^{(0)} = \\x_u$; (2) in iteration $k=1..K$, the node representations $\\bm{h}_u^{(k)}$ are updated by aggregating the neighboring nodes' representations with MLP modules~\\cite{gilmer2017neural}. \nWe can optionally compute a graph representation $\\bm{h}_{{\\mathcal{G}}}$ by aggregating the final node representations with another MLP module.\nFormally,\n\\begin{align}\n\\vspace{-0.5em}\n\\bm{h}_u^{(k)} &:= \\sum\\limits_{v \\in \\mathcal{N}(u)} \\text{MLP}^{(k)} \\Big( \\bm{h}_u^{(k - 1)}, \\bm{h}_v^{(k - 1)} \\Big), \\\\\n\\bm{h}_{{\\mathcal{G}}} &:= \\text{MLP}^{(K+1)} \\Big( \\sum_{u \\in {\\mathcal{G}}} \\bm{h}_u^{(K)} \\Big).\n\\vspace{-0.5em}\n\\end{align}\nDepending on the task, the output is either the graph representation $\\bm{h}_{{\\mathcal{G}}}$ or the final node representations $\\bm{h}_u^{(K)}$.\nWe refer to the neighbor aggregation step for $\\bm{h}_u^{(k)} $ as \\emph{aggregation} and the pooling step for $\\bm{h}_{{\\mathcal{G}}}$ as \\emph{readout}. Different tasks require different aggregation and readout functions.\n\n\\subsection{Additive Label Noise}\n\\label{subsubsec:unstructured_noise}\n\n\\begin{figure}\n\\vspace{-1em}\n\\centering\n\\begin{minipage}{.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{max_sum_gnn_alignment.pdf}\n  \\captionof{figure}{\\textbf{Max-sum GNN aligns well with the task maximum degree.} \n     Max-sum GNN $h_{{\\mathcal{G}}}$ can be decomposed into two modules: $\\text{Module}^{(1)}$ and $\\text{Module}^{(2)}$, and the target function $f({\\mathcal{G}})$ can be similarly divided as $f({\\mathcal{G}}) = f_2(f_1({\\mathcal{G}}))$. As the nonlinearities of the target function have been encoded in the GNN's architecture, $f({\\mathcal{G}})$ can be easily learned by $h_{{\\mathcal{G}}}$: $f_1(\\cdot)$ can be easily learned by $\\text{Module}^{(1)}$, and $f_2(\\cdot)$ is the same as $\\text{Module}^{(2)}$. }\n  \\label{fig:maxdeg_illustrate}\n\\end{minipage}\n\\hspace{1.0em}\n\\begin{minipage}{.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{GNN_max_sum_gaussian_mean10_reg_true_label.pdf}\n  \\captionof{figure}{\\textbf{PCA visualization of hidden representations} from a max-sum GNN trained with additive label noise drawn from $\\mathcal{N}(10,\\,15)$ at 100\\% noise ratio. Each dot denotes a single training example and is colored with its true label. The x-axis and y-axis denote the projected values at the first and second principal components. As the colors change gradually from left to right, the largest principal component of the representations have a clear linear relationship with the true labels. }\n  \\label{fig:maxdeg_graph_embed}\n\\end{minipage}\n\\end{figure}\n\n\\citet{hu2019simple} prove that MLPs are robust to additive label noises with zero mean, if the labels are drawn i.i.d.~from a Sub-Gaussian distribution.  \n\\citet{wu2020optimal} also show that linear models are robust to zero-mean additive label noise even in the absence of explicit regularization.\nIn this section, we show that a GNN \\textit{well-aligned} to the target function not only achieves low test errors on additive label noise with zero-mean, but also learns \\textit{predictive} representations on noisy labels that are drawn from non-zero-mean distributions despite having large test error.\n\n\\textbf{Task and Architecture.} The task is to compute the maximum node degree:\n\\begin{align}\n\\label{eq:maxdeg}\nf({\\mathcal{G}}) &:= \\text{max}_{u \\in {\\mathcal{G}}} \\sum\\limits_{v \\in \\mathcal{N}(u)} 1.\n\\end{align}\nWe choose this task as we know which GNN architecture aligns well with this target function---a 2-layer GNN with max-aggregation and sum-readout (max-sum GNN):\n\\begin{align}\n\\label{eq:max_sum_gnn}\n\\bm{h}_{{\\mathcal{G}}} &:= \\text{MLP}^{(2)} \\Big( \\text{max}_{u \\in {\\mathcal{G}}} \\sum\\limits_{v \\in \\mathcal{N}(u)} \\text{MLP}^{(1)} \\Big({\\bm{h}}_u, {\\bm{h}}_v\\Big) \\Big), \\\\\n{\\bm{h}}_u &:=  \\sum\\limits_{v \\in \\mathcal{N}(u)} \\text{MLP}^{(0)} \\Big({\\bm{x}}_u, {\\bm{x}}_v\\Big).\n\\end{align}\nFigure~\\ref{fig:maxdeg_illustrate} demonstrates how exactly the max-sum GNN aligns with $f({\\mathcal{G}})$.\nIntuitively, they are well-aligned as the MLP modules of max-sum GNN only need to learn simple constant functions to simulate $f({\\mathcal{G}})$. \nBased on Figure~\\ref{fig:maxdeg_illustrate}, we take the output of $\\text{Module}^{(2)}$ as the learned representations for max-sum GNNs when evaluating the predictive power.\n\n\\textbf{Label Noise.} We corrupt labels by adding independent noise ${\\epsilon}$ drawn from three distributions: Gaussian distributions with zero mean $\\mathcal{N}(0,\\,40)$ and non-zero mean $\\mathcal{N}(10,\\,15)$, and a long-tailed Gamma distribution with zero-mean $\\Gamma(2,\\,1\/15)-30$.  \nWe also consider more distributions with non-zero mean in Appendix~\\ref{appsec:additive}. \n\n\\textbf{Findings.} In Figure~\\ref{fig:maxdeg_unstructured}, while the max-sum GNN is robust to \\textit{zero-mean} additive label noise (dotted yellow and purple lines), its test error is much higher under non-zero-mean noise $\\mathcal{N}(10,\\,15)$ (dotted red line) as the learned signal may be ``shifted'' by the non-centered label noise. \nYet, max-sum GNNs' learned representations under these three types of label noise all predict the target function well when evaluating their predictive powers with 10\\% clean labels (solid lines in Figure~\\ref{fig:maxdeg_unstructured}).\n\nMoreover, when we plot the representations (using PCA) from a max-sum GNN trained under 100\\% noise ratio with $\\epsilon \\sim \\mathcal{N}(10,\\,15)$, the representations indeed correlate well with true labels (Figure~\\ref{fig:maxdeg_graph_embed}). \nThis explains why the representation learned under noisy labels can recover surprisingly good test performance despite that the original model has large test errors.\n\nThe predictive power of randomly-initialized max-sum GNNs is in Table~\\ref{table:unstructured_noise} (Appendix~\\ref{appsec:random}).\n\n\\begin{figure}\n\\vspace{-1em}\n\\centering\n\\begin{minipage}{.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{maxdeg_unstructured_noise_early_stop_reg_improve.pdf}\n  \\captionof{figure}{\\textbf{Representations are very predictive for a GNN well-aligned with the target function under additive label noise.} On the maximum degree task, the representations' predictive power (solid lines) achieves low test MAPE ($< 5\\%$) across all three types of noise for the max-sum GNN, despite that the model's test MAPE (dotted lines) may be quite large (for non-zero-mean noise). We average the statistics over 3 runs using different random seeds.}\n  \\label{fig:maxdeg_unstructured}\n\\end{minipage}\n\\hspace{1.0em}\n\\begin{minipage}{.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{maxdeg_structured_noise_reg_loss_improve_on_last_epoch.pdf}\n  \\captionof{figure}{\\textbf{Representations are more predictive for GNNs more aligned with the target function, and less predictive for GNNs more aligned with the noise function}. \n  On the maximum node feature task, while all three GNNs have large test errors under high noise ratios (dotted lines), the predictive power (solid lines) in representations from Deepset (yellow) and max-max GNN (red) greatly reduces the test MAPE. \n  In contrast, the representation's predictive power for max-sum GNN barely reduces the model's test MAPE (tiny gap between dotted and solid purple lines).}\n  \\label{fig:gnn_structured}\n\\end{minipage}\n\\vspace{-1em}\n\\end{figure}\n\n\\subsection{Instance-Dependent Label Noise}\n\\label{subsubsec:structured_noise}\nRealistic label noise is often instance-dependent. For example, an option is often incorrectly priced in the market, but its incorrect price (i.e., the noisy label) should depend on properties of the underlying stock.\nSuch instance-dependent label noise is more challenging, as it may contain \\textit{spurious signals} that are easy to learn by certain architectures. \nIn this section, we evaluate the representation' predictive power for three different GNNs trained with instance-dependent label noise.\n\n\\textbf{Task and Label Noise.}\nWe experiment with a new task---computing the maximum node feature: \n\\begin{align}\n\\label{eq:max_node_feature}\nf({\\mathcal{G}}) := \\text{max}_{u\\in {\\mathcal{G}}} ||{\\bm{x}}_u||_{\\infty}.\n\\end{align}\nTo create a instance-dependent noise, we randomly replace the label with the maximum degree:\n\\begin{align}\ng({\\mathcal{G}}) :=\\text{max}_{u \\in {\\mathcal{G}}} \\sum\\limits_{v \\in \\mathcal{N}(u)} 1.\n\\end{align}\n\n\\textbf{Architecture.} We consider three GNNs: DeepSet~\\cite{zaheer2017deep}, max-max GNN, and max-sum GNN.  \nDeepSet can be interpreted as a special GNN that does not use neighborhood information: \n\\begin{align}\n    h_{{\\mathcal{G}}} = \\text{MLP}^{(1)} \\Big( \\text{max}_{u \\in {\\mathcal{G}}} \\text{MLP}^{(0)} \\Big({\\bm{x}}_u\\Big) \\Big).\n\\end{align}\nMax-max GNN is a 2-layer GNN with max-aggregation and max-readout. Max-sum GNN is the same as the one in the previous section.\n\nDeepSet and max-max GNN are well-aligned with the target function $f({\\mathcal{G}})$, as their MLP modules only need to learn simple linear functions. \nIn contrast, max-sum GNN is more aligned with $g({\\mathcal{G}})$ than $f({\\mathcal{G}})$ since neither its MLP modules or sum-aggregation module can efficiently learn the max-operation in $f({\\mathcal{G}})$~\\cite{Xu2020What, xu2020neural}.\n\nMoreover, DeepSet cannot learn $g({\\mathcal{G}})$ as the model ignores \\textit{edge information}.\nWe take the hidden representations before the last MLP modules in all three GNNs and compare their predictive power.\n\n\\textbf{Findings.}\nWhile all three GNNs have large test errors under high noise ratios (dotted lines in Figure~\\ref{fig:gnn_structured}), the predictive power in representations from GNNs more aligned with the target function --- DeepSet (solid yellow line) and max-max GNN (solid red line) --- significantly reduces the original models' test errors by 10  and 1000 times respectively. \nYet, for the max-sum GNN, which is more aligned with the noise function, training with noisy labels indeed destroy the internal representations such that they are no longer to predict the target function --- its representations' predictive power (solid purple line) barely decreases test error. \nWe also evaluate the predictive power of these three types of randomly-initialized GNNs, and the results are in Table~\\ref{table:structured_noise} (Appendix~\\ref{appsec:random}).\n\\section{Experiments on Vision Datasets}\nMany noisy label training methods are benchmarked on image classification; \nthus, we also validate our hypothesis on image domains.\nWe compare the representations' predictive power between MLPs and \\mbox{CNN-based} networks using 10\\% clean labels (all models are trained until they could perfectly fit the noisy labels, a.k.a., achieving close to 100\\% training accuracy). \nWe further evaluate the predictive power in representations learned with SOTA methods.\nPredictive power on networks that aligned well with the target function could further improve SOTA method's test performance (Section~\\ref{sec:eval_sota}).\nThe final model also outperforms some sophisticated methods on noisy label training which also use clean labels (Appendix~\\ref{appsec:compare_sota}). \nAll our experiment details are in Appendix~\\ref{suppsec:vision_training_details}.\n\\vspace{-0.5em}\n\\subsection{MLPs vs. \\mbox{CNN-based} networks}\nTo validate our hypothesis, we consider several target functions with different levels of alignments to MLPs and CNN-based networks. \nAll models in this section are trained with standard procedures without any robust training methods or robust losses.\n\\paragraph{Datasets and Label Noise.} We consider two types of target functions: one aligns better with \\mbox{CNN-based} models than MLPs, and the other aligns better with MLPs than \\mbox{CNN-based} networks. \n\n\\textbf{1).} \\textbf{CIFAR-10} and \\textbf{CIFAR-100}~\\cite{krizhevsky2009learning} come with clean labels.\nTherefore, we generate two types of noisy labels following existing works: (1) \\textbf{uniform label noise} randomly replaces the true labels with all possible labels, and (2) \\textbf{flipped label noise} swaps the labels between similar classes (e.g., deer$\\leftrightarrow$horse, dog$\\leftrightarrow$cat) on CIFAR-10~\\cite{li2020dividemix}, or flips the labels to the next class on CIFAR-100~\\cite{natarajan2013learning}.\n\n\\textbf{2).}  \\textbf{CIFAR-Easy} is a dataset modified on CIFAR-10 with labels generated by procedures in Figure~\\ref{fig:our_cifar10_demo} --- the class\/label of each image depends on the location of a special pixel.\nWe consider three types of noisy labels on CIFAR-Easy: (1) \\textbf{uniform label noise} and (2) \\textbf{flipped label noise} (described as above); and (3) \\textbf{instance-dependent label noise} which takes the original image classification label as the noisy label.\n\n\\begin{figure}\n\\centering\n\\begin{minipage}{.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{cat.pdf}\n  \\captionof{figure}{\\textbf{Synthetic Labels on CIFAR-Easy.} For each image, we mask a pixel at the top left corner with pink color. Then the synthetic label for this image is the location of the pink pixel\/mask (i.e., the cat image in the above example has label 4).}\n  \\label{fig:our_cifar10_demo}\n\\end{minipage}%\n\\hspace{1.0em}\n\\begin{minipage}{.45\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{sample_complexity.pdf}\n  \\captionof{figure}{\\textbf{Sample complexity of MLPs and CNNs on CIFAR-Easy.} Both MLPs and CNNs can achieve 100\\% test accuracy given sufficient training examples, but MLPs need far fewer examples than CNNs and thus are more sample-efficient on CIFAR-Easy.}\n  \\label{fig:sample_complex}\n\\end{minipage}\n\\vspace{-1em}\n\\end{figure}\n\n\\input{cifar_baseline_compare}\n\\input{our_cifar10_compare}\n\\input{baseline}\n\\input{our_cifar10}\n\n\\paragraph{Architectures.}\n\\label{exp:our_cifar10}\nOn CIFAR-10\/100, we evaluate the predictive power in representations for three architectures: 4-layer MLPs, 9-layer CNNs, and 18-layer PreAct ResNets~\\cite{he2016identity}.\nOn CIFAR-Easy, we compare between MLPs and CNNs. \nWe take the representations before the penultimate layer when evaluating the predictive power for these networks.\n\nAs the designs of \\mbox{CNN-based} networks (e.g., CNNs and ResNets) are similar to human perception system because of the receptive fields in convolutional layers and a hierarchical extraction of more and more abstracted features~\\cite{lecun1995convolutional, kheradpisheh2016deep}, \\mbox{CNN-based} networks are expected to \\textit{align better} with the target functions than MLPs on image classification datasets (e.g., \\mbox{CIFAR-10\/100}).\n\nOn the other hand, on CIFAR-Easy, while both CNNs and MLPs can generalize perfectly given sufficient training examples, MLPs have a much smaller sample complexity than CNNs  (Figure~\\ref{fig:sample_complex}).\nThus, both MLP and CNN are \\textit{well-aligned} with the target function on CIFAR-Easy, but MLP is \\textit{better-aligned} than CNN according to Theorem~\\ref{thm:main}.\nMoreover, since the instance-dependent label on CIFAR-Easy \\ is the original image classification label, CNN is also \\textit{aligned} with this instance-dependent noise function on CIFAR-Easy.\n\n\\paragraph{Experimental Results.}\nFirst, we empirically verify our hypothesis that \\emph{networks better-aligned with the target function have more predictive representations}. \nAs expected, across most noise ratios on CIFAR-10\/100, the representations in \\mbox{CNN-based} networks (i.e., CNN and ResNet) are more predictive than those in MLPs (Figure~\\ref{fig:cifar_baseline_compare}) under both types of label noise.\nMoreover, the predictive power in representations learned by less aligned networks (i.e., MLPs) sometimes are even worse than the vanilla-trained models' test performance, suggesting that the noisy representations on less aligned networks may be more corrupted and less linearly separable.\nOn the other hand, across all three types of label noise on CIFAR-Easy, MLPs, which align better with the target function, have more predictive representations than CNNs (Figure~\\ref{fig:our_cifar10_compare}). \n\nWe also observe that \\emph{models with similar test performance could have various levels of predictive powers in their learned representations}. \nFor example, in Figure~\\ref{fig:our_cifar10_compare}, while the test accuracies of MLPs and CNNs are very similar on CIFAR-Easy\\ under flipped label noise (i.e., dotted purple and yellow lines overlap), the predictive power in representations from MLPs is much stronger than the one from CNNs (i.e., solid purple lines are much higher than yellow lines).\nThis also suggests that when trained with noisy labels, if we do not know which architecture is more aligned with the underlying target function, we can evaluate the predictive power in their representations to test alignment.\n\nWe further discover that \\textit{for networks well-aligned with the target function, its learned representations are more predictive when the noise function shares more mutual information with the target function}. \nWe compute the empirical mutual information between the noisy training labels and the original clean labels across different noise ratios on various types of label noise. \nThe predictive power in representations improves as the mutual information increases (Figure~\\ref{fig:mutual_info} in Appendix~\\ref{suppsec:add_exp_results}).\nThis explains why the predictive power for a network is often higher under flipped noise than uniform noise: at the same noise ratio, flipped noise has higher mutual information than uniform noise. \nMoreover, comparing across the three datasets in Figure~\\ref{fig:mutual_info}, we observe the growth rate of a network's predictive power w.r.t. the mutual information depends on both the intrinsic difficulties of the learning task and the alignment between the network and the target function.\n\n\\vspace{-0.5em}\n\\subsection{Predictive Power in Representations for Models Trained with SOTA Methods}\n\\label{sec:eval_sota}\n\\vspace{-0.5em}\nAs previous experiments are on standard training procedures, we also validate our hypothesis on models learned with SOTA methods on noisy label training. \nWe evaluate the representations' predictive power for models trained with the SOTA method, DivideMix~\\cite{li2020dividemix}, which leverages techniques from semi-supervised learning to treat examples with unreliable labels as unlabeled data. \n\nWe compare (1) the test performance for models trained with standard procedures on noisy labels (denoted as \\textbf{Vanilla training}), (2) the SOTA method's test performance (denoted as \\textbf{DivideMix}), and (3) the predictive power in representations from models trained with DivideMix in (2) (denoted as \\textbf{DivideMix's Predictive Power}).\n\nWe discover that \\textit{the effectiveness of DivideMix also depends on the alignment between the network and the target\/noise functions}.\nDivideMix only slightly improves the test accuracy of MLPs on CIFAR-10\/100 (Table~\\ref{table:cifar_baseline}), and DivideMix's predictive power does not improve the test performance of MLPs, either.\nIn Table~\\ref{table:mlp_ourcifar}, DivideMix also barely helps CNNs as they are well-aligned with the instance-dependent noise, where the noisy label is the original image classification label.\n  \nMoreover, we observe that \\textit{even for networks well-aligned with the target function, DivideMix may only slightly improve or do not improve its test performance at all} (e.g., red entries of DivideMix on MLPs in Table~\\ref{table:mlp_ourcifar}).\nYet, the representations learned with DivideMix can still be very predictive: the predictive power can achieve over 50\\% improvements over DivideMix for CNN-based models on CIFAR-10\/100 (e.g., 80\\% flipped noise), and the improvements can be over 80\\% for MLPs on CIFAR-Easy\\ (e.g., 90\\% uniform noise).\n\nTables~\\ref{table:cifar_baseline} and~\\ref{table:mlp_ourcifar} shows that the representations' predictive power on networks well aligned with the target function could further improve SOTA test performance. \nAppendix~\\ref{appsec:compare_sota} further demonstrates that on large-scale datasets with real-world noisy labels, the predictive power in well-aligned networks could outperform sophisticated methods that also use clean labels (Table~\\ref{table:clothing} and Table~\\ref{table:webvision}). \n\\section{Concluding Remarks}\n\\vspace{-0.5em}\nThis paper is an initial step towards formally understanding how a network's architectures impacts its robustness to noisy labels. \nWe formalize our intuitions and hypothesize that a network better-aligned with the target function would learn more predictive representations under noisy label training. \nWe prove our hypothesis on a simplified noisy setting and conduct systematic experiments across various noisy settings to further validate our hypothesis.\n\nOur empirical results along with Theorem~\\ref{thm:main} suggest that knowing more structures of the target function can help design more robust architectures. \nIn practice, although an exact mathematical formula for a decomposition of a given target function is often hard to obtain, a high-level decomposition of the target function often exists for real-world tasks \nand will be helpful in designing robust architectures --- a direction undervalued by existing works on learning with noisy labels. \n\n\n\\section{Additional Experimental Results}\n\\label{suppsec:add_exp_results}\nIn this section, we include additional experimental results for the predictive power in (a) representations from randomly initialized models (Appendix~\\ref{appsec:random}), (b) representations learned under different types off additive label noise (Appendix~\\ref{appsec:additive}) and (c) representations learned with a robust loss function (Appendix~\\ref{appsec:mae}). \nWe further demonstrates that the predictive power in well-aligned networks could even outperform sophisticated methods that also utilize clean labels (Appendix~\\ref{appsec:compare_sota}). \n\n\\subsection{Predictive Power of Randomly Initialized Models}\n\\label{appsec:random}\nWe first evaluate the predictive power of randomly initialized models (a.k.a., untrained models), and we compare their results with GNNs trained on clean data (a.k.a., 0\\% noise ratio).\n\n\\begin{table}[ht]\n    \\vspace{-0.5em}\n\t\\centering\n\t\\small\n\t\\caption{\n\t\tPredictive power in representations from random and trained max-sum GNNs on the maximum degree task (Section~\\ref{subsubsec:unstructured_noise}). Notice that lower test MAPE denotes better test performance.\n\t\t}\n\t\\label{table:unstructured_noise}\n\t\\vskip 0.1in\n\t\\begin{tabular}\t{l |c|c }\n\t\t\\toprule\t \t\n\t\t\t\\multirow{2}{*}{\\bf Model}  & \\multicolumn{2}{c}{\\bf Test MAPE} \\\\\n\t\t\t\\cmidrule{2-3}\n\t\t\t& Random & Trained \\\\\n\t\t\t\\midrule\t\t\t\n\t\t\tMax-sum GNN & 12.74 $\\pm$ 0.57 & 0.37 $\\pm$ 0.08 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\vspace{-0.5em}\n\\end{table}\t\t\n\n\\begin{table}[ht]\n    \\vspace{-0.5em}\n\t\\centering\n\t\\small\n\t\\caption{\n\t\tPredictive power in representations from various types of random and trained GNNs on the maximum node feature task (Section~\\ref{subsubsec:structured_noise}). Notice that lower test MAPE denotes better test performance.\n\t\t}\n\t\\label{table:structured_noise}\n\t\\vskip 0.1in\n\t\\begin{tabular}\t{l |c|c|c|c }\n\t\t\\toprule\t \t\n\t\t\t\\multirow{2}{*}{\\bf Model}  & \\multicolumn{2}{c|}{\\bf Test MAPE} & \\multicolumn{2}{c}{\\bf Test MAPE (log scale)} \\\\\n\t\t\t\\cmidrule{2-5}\n\t\t\t& Random & Trained & Random & Trained \\\\\n\t\t\t\\midrule\t\t\t\n\t\t\tDeepSet & 5.14e-05 & 1.06e-05 & -4.29 & -4.97 \\\\ \\hline\n\t\t\tMax-max GNN & 0.794 & 0.0099 & -0.10 & -2.00 \\\\ \\hline\n\t\t\tMax-sum GNN & 54.28 & 3.08 & 1.73 & 0.488 \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\vspace{-0.5em}\n\\end{table}\t\n\n\\subsection{Additive Label Noise on Graph Algorithmic Datasets}\n\\label{appsec:additive}\n We conduct additional experiments on additive label noise drawn from distributions with larger mean and larger variance. We consider four such distributions: Gaussian distributions $\\mathcal{N}(10,\\,30)$ and $\\mathcal{N}(20,\\,15)$, a long-tailed Gamma distribution with mean equal to $10$: $\\Gamma(2, \\dfrac{1}{15}) - 20$, and another long-tailed t-distribution with mean equal to $10$: $\\mathcal{T}(\\nu=1) + 10$. \n Figure~\\ref{fig:app_maxdeg_unstructured} demonstrates that for a GNN well aligned to the target function, its representations are still very predictive even under non-zero mean distributions with larger mean and large variance. \n\\input{app_gnn_unstructured}\n\n\\vspace{-1em}\n\\subsection{Training with a Robust Loss Function}\n\\label{appsec:mae}\nWe also train the models with a robust loss function--Mean Absolute Error (MAE), and we observe similar trends in the representations' predictive power as training the models using MSE (Figure~\\ref{fig:app_maxdeg_l1}).\n\\begin{align}\n    \\text{loss} = \\sum_{i=1}^n |y_\\text{true} - y_\\text{pred}|.\n\\end{align}\n\n\\begin{figure*}[h!]\n    \\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{maxdeg_unstructured_noise_early_stop_l1_improve.pdf}\n         \\vspace{-1em}\n         \\caption{Test errors of max-sum GNNs on the maximum degree task with additive label noise}\n     \\end{subfigure}\n    \\hfill\n     \\begin{subfigure}[b]{0.44\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{maxdeg_structured_noise_l1_loss_improve_on_last_epoch.pdf}\n         \\caption{Test errors of three different GNNs on the maximum node feature task with instance-dependent label noise}\n     \\end{subfigure}\n     \\caption{\\textbf{Predictive power in representations trained with MAE}. For GNNs trained with MAE, the predictive power in representations exhibits similar trends as models trained with MSE. The robust loss function, MAE, is more helpful in learning more predictive representations under smaller noise ratios. \n     }\n     \\label{fig:app_maxdeg_l1}\n    \\vspace{-0.25em}\n\\end{figure*}\n\n\\input{mutual_information}\n\n\\subsection{Comparing with Sophisticated Methods Using Clean Labels}\n\\label{appsec:compare_sota}\n\\input{cifar10_sym_noise}\n\\input{cifar10_asym_noise}\n\\input{cifar100_sym_noise}\n\\input{cifar100_asym_noise}\n\\input{clothing}\n\\input{webvision}\nIn previous experiments (section~\\ref{sec:eval_sota}), we have shown that the predictive power in well-aligned models could further improve the test performance of SOTA methods on noisy label training. \nAs we use a small set of clean labels to measure the predictive power, we also wonder how the improvements obtained by the predictive power compare with the sophisticated methods that also use clean labels.\n\n\\subsubsection{Sophisticated Methods Using Clean Labels}\nIn our experiments, we consider the following methods which use clean labels: L2R~\\cite{ren2018learning}, MentorNet~\\cite{jiang2018mentornet}, SELF~\\cite{nguyen2019self}, GLC~\\cite{hendrycks2018using}, Meta-Weight-Net~\\cite{shu2019meta}, and IEG~\\cite{zhang2019ieg}.\nBesides, as the SOTA method, DivideMix, keeps dividing the training data into labeled and unlabeled sets during training, we also compare with directly using clean labels in DivideMix: we mark the small set of clean data as labeled data during the semi-supervised learning step in DivideMix.\nWe denote this method as \\textit{DivideMix w\/ Clean Labels (DwC)} and further measure the predictive power in representations learned by DwC.\n\n\\subsubsection{Datasets}\nWe conduct experiments on CIFAR-10\/100 with synthetic noisy labels and on two large-scale datasets with real-world noisy labels: Clothing1M and Webvision.\n\n\\textbf{Clothing1M}~\\cite{xiao2015learning} has real-world noisy labels with an estimated 38.5\\% noise ratio.\nThe dataset has a small human-verified training data, which we use as clean data.\nFollowing recent method~\\cite{li2020dividemix}, we use 1000 mini-batches in each epoch to train models on Clothing1M. \n\n\\textbf{WebVision}~\\cite{li2017webvision} also has real-world noisy labels with an estimated 20\\% noise ratio. \nIt shares the same 1000 classes as ImageNet~\\cite{deng2009imagenet}.\nFor a fair comparison, we follow~\\cite{jiang2018mentornet} to create a mini WebVision dataset with the top 50 classes from the Google image subset of WebVision. \nWe train all models on mini WebVision dataset and evaluate on both the WebVision and ImageNet validation sets.\nWe select 100 images per class from ImageNet training data as clean data.\n\n\\subsubsection{Experimental Settings}\nWe use the same architectures and hyperparameters as DivideMix: an 18-layer {PreAct Resnet~\\cite{he2016identity}} for \\mbox{CIFAR-10\/100}, a ResNet-50 pre-trained on ImageNet for Clothing1M, and Inception-ResNet-V2~\\cite{szegedy2016inception} for WebVision.\nWe use the test accuracy reported in the original papers whenever possible, and the accuracy for L2R~\\cite{ren2018learning} are from~\\cite{zhang2019ieg}. For IEG, we use the reported test accuracy obtained by ResNet-29 rather than WRN28-10, because ResNet-29 has a comparable number of parameters as the \\mbox{PreAct ResNet-18} we use.\n\nAs CIFAR-10\/100 do not have a validation set, we follow previous works to report the averaged test accuracy over the last 10 epochs:  we measure the predictive power in representations for models from these epochs and report the averaged test accuracy. \nFor Clothing1M and Webvision, we use the associated validation set to select the best model and measure the predictive power in its representations.\n\n\\subsubsection{Results}\nTables~\\ref{table:cifar10_sym}-\\ref{table:cifar100_asym} show the results on CIFAR-10 and CIFAR-100 with uniform and flipped label noise, where \\textbf{boldfaced numbers} denote test accuracies better than all methods we compared with.\nWe see that across different noise ratios on CIFAR-10\/100 with flipped label noise, the predictive power in representations remains roughly the same as the test performance of the model trained on clean data for a network well-aligned with the target function, which matches with Lemma~\\ref{thm:main_extend}. For CIFAR-10 with uniform label noise, the predictive power in representations achieves better test performance using only 10 clean labels per class on most noise ratios; for CIFAR-100 with uniform label noise, the predictive power in representations could achieve better test performance using only 50 labels per class.\n\nMoreover, we observe that adding clean data to the labeled set in DivideMix (DwC) may barely improve the model's test performance when the noise ratio is small and under flipped label noise.\nAt 90\\% uniform label noise, DwC can greatly improve the model's test performance, and the predictive power in representations can achieve a even higher test accuracy with the same set of clean data used to train DwC.\n\nOn Clothing1M, we compare the predictive power in representations learned by DivideMix with existing methods that use the small set of human-verified data: CleanNet~\\cite{lee2018cleannet}, F-correction~\\cite{patrini2017making} and Self-learning~\\cite{han2019deep}. \nAs these methods also use the clean subset to fine-tune the whole model, we follow similar procedures to fine-tune the model (trained by DivideMix) for 10 epochs and then select the best model based on the validation accuracy to measure the predictive power in its representations.\nThe predictive power in representations could further improve the test accuracy of DivideMix by around 6\\% and outperform IEG, CleanNet, and F-correction (Table~\\ref{table:clothing}). The improved test accuracy is also competitive to~\\cite{han2019deep}, which uses a much more complicated learning framework. \n\nOn Webvision, the predictive power also improves the model's test performance (Table~\\ref{table:webvision}). \nThe improvement is less significant than on Clothing1M as the estimated noise ratio on Webvision (20\\%) is smaller than Clothing1M (38.5\\%).\n\\section{Experimental Details}\n\\label{suppsec:exp_details}\n\n\\subsection{Computing Resources}\nWe conduct all the experiments on one NVIDIA RTX 2080 Ti GPU, except for the experiment on the WebVision dataset~\\cite{li2017webvision} (Table~\\ref{table:webvision}), which uses 4 GPUs concurrently.\n\n\\subsection{Measuring the Predictive Power}\nWe use linear regression to train the linear model when measuring the predictive power in representations. For representations from all models except MLPs, we use ordinary least squares linear regression (OLS). When the learned representations are from MLPs, we e use ridge regression with \\mbox{penalty = 1} since we find the linear models trained by OLS may easily overfit to the small set of clean labels. \n\n\\subsection{Experimental Details on GNNs}\n\\label{suppsec:gnn_training_details}\n\\paragraph{Common settings.} In the generated datasets, each graph ${\\mathcal{G}}$ is sampled from Erd\\H{o}s-R\\'enyi random graphs with an edge probability uniformly chosen from $\\{0.1, 0.2, \\cdots, 0.9\\}$. This sampling procedure generates diverse graph structures. The training and validation sets contain 10,000 and 2,000 graphs respectively, and the number of nodes in each graph is randomly picked from $\\{20, 21, \\cdots, 40\\}$. The test set contains 10,000 graphs, and the number of nodes in each graph is randomly picked from $\\{50, 51, \\cdots, 70\\}$. \n\n\\subsubsection{Additive Label Noise} \n\\paragraph{Dataset Details.} In each graph, the node feature $\\x_u$ is a scalar randomly drawn from $\\{1, 2, \\cdots, 100\\}$ for all $u \\in {\\mathcal{G}}$.\n\n\\paragraph{Model and hyperparameter settings.} We consider a 2-layer GNN with max-aggregation and sum-readout (max-sum GNN):\n\\[\n\\bm{h}_G = \\text{MLP}^{(2)} \\Big( \\text{max}_{u \\in {\\mathcal{G}}} \\sum\\limits_{v \\in \\mathcal{N}(u)} \\text{MLP}^{(1)} \\Big({\\bm{h}}_u, {\\bm{h}}_v\\Big) \\Big), \\\\\n{\\bm{h}}_u =  \\sum\\limits_{v \\in \\mathcal{N}(u)} \\text{MLP}^{(0)} \\Big({\\bm{x}}_u, {\\bm{x}}_v\\Big).\n\\]\nThe width of all MLP modules are set to $128$. The number of layers are set to $3$ for $\\text{MLP}^{(0)}$ and $\\text{MLP}^{(1)}$. The number of layers are set to $1$ for $\\text{MLP}^{(2)}$.\nWe train the max-sum GNNs with loss function MSE or MAE for 200 epochs. We use the Adam optimizer with default parameters, zero weight decay, and initial learning rate set to 0.001. The batch size is set to 64.  We early-stop based on a noisy validation set. \n\n\\subsubsection{Instance-Dependent Label Noise.} \n\\paragraph{Dataset Details.} \nSince the task is to predict the maximum node feature and we use the maximum degree as the noisy label, the correlation between true labels and noisy labels are very high on large and dense graphs if the node features are uniformly sampled from $\\{1, 2, \\cdots, 100\\}$.\nTo avoid this, we use a two-step method to sample the node features.\nFor each graph ${\\mathcal{G}}$, we first sample a constant upper-bound $M_{\\mathcal{G}}$ uniformly from $\\{20, 21, \\cdots, 100\\}$. For each node $u \\in {\\mathcal{G}}$, the node feature ${\\bm{x}}_u$ is then drawn from $\\{1, 2, \\cdots, M_{\\mathcal{G}}\\}$.\n\n\\paragraph{Model and hyperparameter settings.} We consider a 2-layer GNN with max-aggregation and sum-readout (max-sum GNN), a 2-layer GNN with max-aggregation and max-readout (max-max GNN), and a special GNN (DeepSet) that does not use edge information: \n\\[ h_G = \\text{MLP}^{(1)} \\Big( \\text{max}_{u \\in {\\mathcal{G}}} \\ \\text{MLP}^{(0)} \\Big({\\bm{x}}_u\\Big) \\Big). \\] \nThe width of all MLP modules are set to $128$. The number of layers is set to $3$ for $\\text{MLP}^{(0)}, \\text{MLP}^{(1)}$ in max-max and max-sum GNNs and for $\\text{MLP}^{(0)}$ in DeepSet. The number of layers is set to $1$ for $\\text{MLP}^{(2)}$ in max-max and max-sum GNNs and for $\\text{MLP}^{(1)}$ in DeepSet.\nWe train these GNNs with MSE or MAE as the loss function for 600 epochs. We use the Adam optimizer with zero weight decay. \nWe set the initial learning rate to $0.005$ for DeepSet and $0.001$ for max-max GNNs and max-sum GNNs. \nThe models are selected from the last epoch so that they can overfit the noisy labels more.\n\n\\subsection{Experimental Details on Vision Datasets}\n\\label{suppsec:vision_training_details}\n\\paragraph{Neural Network Architectures.} Table~\\ref{tab:models-cnn} describes the 9-layer CNN~\\cite{miyato2018virtual} used on CIFAR-Easy\\ and \\mbox{CIFAR-10\/100}, which contains 9 convolutional layers and 19 trainable layers in total. Table~\\ref{tab:models-mlp} describes the 4-layer MLP used on CIFAR-Easy\\ and \\mbox{CIFAR-10\/100}, which has 4 linear layers and ReLU as the activation function.\n\n\\begin{table}[ht]\n    \\centering\n    \\begin{minipage}[t]{0.45\\textwidth}\n        \\caption{9-layer CNN on CIFAR-Easy\\ and \\mbox{CIFAR-10\/100}.}\n        \\label{tab:models-cnn}\n         \\vskip 0.1in\n        \\centering\\small\n        \\begin{tabular*}{\\textwidth}{l@{\\extracolsep{\\fill}}c}\n            \\toprule\n            \\multirow{1}*{Input}\n            & 32$\\times$32 Color Image \\\\\n            \\midrule\n            \\multirow{5}*{Block 1}\n            & Conv(3$\\times$3, 128)-BN-LReLU \\\\\n            & Conv(3$\\times$3, 128)-BN-LReLU \\\\\n            & Conv(3$\\times$3, 128)-BN-LReLU \\\\\n            & MaxPool(2$\\times$2, stride = 2) \\\\\n            & Dropout(p = 0.25) \\\\\n            \\midrule\n            \\multirow{5}*{Block 2}\n            & Conv(3$\\times$3, 256)-BN-LReLU \\\\\n            & Conv(3$\\times$3, 256)-BN-LReLU \\\\\n            & Conv(3$\\times$3, 256)-BN-LReLU \\\\\n            & MaxPool(2$\\times$2, stride = 2) \\\\\n            & Dropout(p = 0.25) \\\\\n            \\midrule\n            \\multirow{4}*{Block 3}\n            & Conv(3$\\times$3, 512)-BN-LReLU \\\\\n            & Conv(3$\\times$3, 256)-BN-LReLU \\\\\n            & Conv(3$\\times$3, 128)-BN-LReLU \\\\\n            & GlobalAvgPool(128) \\\\\n            \\midrule\n            Score & Linear(128, 10 or 100) \\\\\n            \\bottomrule\n        \\end{tabular*}\n        \n    \\end{minipage}\n    \\hfill\n    \\begin{minipage}[t]{0.45\\textwidth}\n        \\caption{4-layer FC on CIFAR-Easy\\ and \\mbox{CIFAR-10\/100}.}\n        \\label{tab:models-mlp}\n         \\vskip 0.1in\n        \\begin{tabular*}{\\textwidth}{lc}\n            \\toprule\n            \\multirow{1}*{Input}\n            & 32$\\times$32 Color Image \\\\\n            \\midrule\n            \\multirow{3}*{Block 1}\n            & Linear(32$\\times$32$\\times$3, 512)-ReLU \\\\\n            & Linear(512, 512)-ReLU \\\\\n            & Linear(512, 512-ReLU \\\\\n            \\midrule\n            Score & Linear(512, 10 or 100) \\\\\n            \\bottomrule\n        \\end{tabular*}\n    \\end{minipage}\n    \\vspace{-1ex}\n\\end{table}\n\n\\paragraph{Vanilla Training.} For models trained with standard procedures, we use SGD with a momentum of 0.9, a weight decay of 0.0005, and a batch size of 128. For ResNets and CNNs, the initial learning rate is set to $0.1$ on CIFAR-10\/100 and $0.01$ on CIFAR-Easy. For MLPs, the initial learning rate is set to $0.01$ on CIFAR-10\/100 and $0.001$ on CIFAR-Easy. The initial learning rate is multiplied by 0.99 per epoch on CIFAR-10\/100, and it is decayed by 10 after 150 and 225 epochs on CIFAR-Easy.\n\n\\paragraph{Train Models with SOTA Methods.} We use the same set of hyperparameter settings from DivideMix~\\cite{li2020dividemix} to obtain corresponding trained models and measure the predictive power in representations from these models.\n\nOn CIFAR-10\/100 with flipped noise, we only use the small set of clean labels to train the linear model in our method, and the clean subset is randomly selected from the training data.\nOn CIFAR-10\/100 with uniform noise, the clean labels we use are from examples with highest model uncertainty~\\cite{lewis1994sequential}. \nBesides the clean set, we also use randomly-sampled training examples labeled with the model's original predictions to train the linear model. \nWe use 5,000 such samples under 20\\%, 40\\%, 50\\%, and 80\\% noise ratios, and we use 500 such samples under 90\\% noise ratio. \n\\section{Theoretical Results}\n\\label{suppapp:thm}\nWe first provide a formal version of Theorem~\\ref{thm:xu2020} based on~\\cite{Xu2020What}. Theorem~\\ref{thm:xu2020} connects a network's architectural alignment with the target function to its learned representations' predictive power  \\textit{when trained on clean data}.  \n\\begin{theorem} \n\\label{thm:main_formal}\n{(Better alignment implies better predictive power on clean training data; \\cite{Xu2020What}). } \nFix $\\epsilon$ and $\\delta$.\nGiven a target function $f: \\mathcal{X} \\rightarrow \\mathcal{Y}$ that can be decomposed into functions $f_1, ..., f_n$ and given a network $\\mathcal{N}$, where $\\mathcal{N}_1, ..., \\mathcal{N}_n$ are $\\mathcal{N}$'s modules in sequential order,\nsuppose the training dataset $\\mathcal{S} := \\left\\lbrace \\x_j, y_j \\right\\rbrace_{j=1}^M$ contains $M$ i.i.d. samples drawn from a distribution with clean labels $y_j := f(x_j)$. \nThen under the following assumptions, $\\textit{Alignment}(\\mathcal{N}, f, \\epsilon, \\delta) \\leq M$\nif and only if there exists a learning algorithm $A$ such that the network's last module $\\mathcal{N}_n$'s representations learned by $A$ on the training data $\\mathcal{S}$ have predictive power $\\pred_{n}(f, \\mathcal{N}, \\mathcal{S}) \\leq \\epsilon$ with probability $1-\\delta$.\n  \\vspace{0.05in} \\\\\n  Assumptions: \\vspace{0.05in} \\\\\n  \\textbf{(a)} We train each module $\\mathcal{N}_i$'s sequentially: for each $\\mathcal{N}_i$, the input samples are $\\{h^{(i-1)}(\\x_j),f_i(h^{(i-1)}(\\x_j))\\}_{j=1}^M$ with $h^{(0)}(\\x) =\\x$.\n  Notice that each input $h^{(i-1)}(\\x_j)$ is the output from the previous modules, but its label is generated by the function $f_{i}$ on $h^{(i-1)}(\\x_j)$.   \\\\\n \\textbf{(b)} For the clean training set $\\mathcal{S}$, let $\\mathcal{S}' := \\left\\lbrace \\hat{\\x}_j, y_j \\right\\rbrace_{j=1}^M$ denote the perturbed training data ($\\hat{\\x}_j$ and $\\x_j$ share the same label $y_j$).\n Let $f_{\\mathcal{N}, A}$ and $f'_{\\mathcal{N}, A}$ denote the functions obtained by the learning algorithm $A$ operating on $\\mathcal{S}$ and $\\mathcal{S}'$ respectively.\n Then for any $\\x \\in \\mathcal{X}$, $\\| f_{\\mathcal{N}, A}(\\x) - f'_{\\mathcal{N}, A}(\\x) \\| \\leq L_0 \\cdot \\max_{\\x_j \\in \\mathcal{S}} \\| \\x_j - \\hat{\\x}_j \\|$, for some constant $L_0$.  \\\\\n \\textbf{(c) }  For each module $\\mathcal{N}_i$, let $\\hat{f}_i$ denotes its corresponding function learned by the algorithm $A$. Then for any $\\x, \\hat{\\x} \\in \\mathcal{X}$, $ \\| \\hat{f}_j (\\x) - \\hat{f}_j (\\hat{\\x})  \\| \\leq L_1 \\| \\x - \\hat{\\x} \\|$, for some constant $L_1$. \n\\end{theorem}\n\nWe have empirically shown that Theorem~\\ref{thm:main_formal} also hold when we train the models on noisy data. \nMeanwhile, we prove Theorem~\\ref{thm:main_formal} for a simplified noisy setting where the target function and noise function share a common feature space, but have different prediction rules. \nFor example, the target function and noise function share the same feature space under flipped label noise (in classification setting). Yet, their mappings from the learned features to the associated labels are different.\n\n\\begin{theorem}\n\\label{thm:main_extend}\n{(Better alignment implies better predictive power on noisy training data). } \nFix $\\epsilon$ and $\\delta$. \nLet $\\left\\lbrace \\x_j \\right\\rbrace_{j=1}^M$ be i.i.d. samples drawn from a distribution. \nGiven a target function $f: \\mathcal{X} \\rightarrow \\mathcal{Y}$ and a noise function $g: \\mathcal{X} \\rightarrow \\mathcal{Y}$, let $y := f(\\x)$ denote the true label for an input $\\x$, and $\\hat{y} := g(\\x)$ denote the noisy label of $\\x$. \nLet $\\hat{\\mathcal{S}} := \\lbrace (\\x_j, y_j) \\rbrace_{j=1}^N \\bigcup \\ \\lbrace (\\x_j, \\hat{y}_j) \\rbrace_{j=N+1}^M$ denote a noisy training set with $M-N$ noisy samples for some $N \\in \\{1,2,\\cdots,M\\}$.\nGiven a network $\\mathcal{N}$ with modules $\\mathcal{N}_i$, suppose $\\mathcal{N}$ is well-aligned with the target function $f$ (i.e., the alignment between $\\mathcal{N}$ and $f$ is less than $M$ --- $\\textit{Alignment}(\\mathcal{N}, f, \\epsilon, \\delta) \\leq M$). \nThen under the same assumptions in Theorem~\\ref{thm:main_formal} and the additional assumptions below, there exists a learning algorithm $A$ and a module $\\mathcal{N}_i$ such that when training the network $\\mathcal{N}$ on the noisy data $\\hat{\\mathcal{S}}$ with algorithm $A$, the representations from its $i$-th module have predictive power $\\pred_{i}(f, \\mathcal{N}, \\mathcal{C}) \\leq \\epsilon$ with probability $1-\\delta$, where $\\mathcal{C}$ is a small set of clean data with a size greater than the number of dimensions in the output of module $\\mathcal{N}_i$. \n\n\\vspace{0.05in} \nAdditional assumptions (a simplified noisy setting): \\vspace{0.05in} \\\\\n\\textbf{(a)} There exists a function $h$ on the input domain $\\mathcal{X}$ such that the target function $f: \\mathcal{X} \\rightarrow \\mathcal{Y}$ and the noise function $g: \\mathcal{X} \\rightarrow \\mathcal{Y}$ can be decomposed as: $f(\\x) = f_r(h(\\x))$ with $f_r$ being a linear function and $g(\\x) = g_r(h(\\x))$ for some function $g_r$. \\\\\n\\textbf{(b)} $f_r$ is a linear map from a high-dimensional space to a low-dimensional space. \\\\ \n\\textbf{(c)} The loss function used in measuring the predictive power is mean squared error (denoted as $\\|\\cdot\\|$) .  \n\\end{theorem}\n\n\\textbf{Remark.} Theorem~\\ref{thm:main_extend} suggests that the representations' predictive power for models well aligned with the target function should remain roughly similar across different noise ratios under flipped label noise. Empirically, we observe similar phenomenons in Figures~\\ref{fig:cifar_baseline_compare}-\\ref{fig:our_cifar10_compare}, and in Tables~\\ref{table:cifar10_asym} and~\\ref{table:cifar100_asym}.  \nSome discrepancy between the experimental and theoretical results could exist under vanilla training as Theorem~\\ref{thm:main_extend} assumes sequential training, which is different from standard training procedures.\n\n\\paragraph{Proof of Theorem~\\ref{thm:main_extend}.} According to the definition of alignment in Definition~\\ref{def:alignment}, since $\\textit{Alignment}(\\mathcal{N}, f, \\epsilon, \\delta) \\leq M$ and $f(\\x) = f_r(h(\\x))$, we can find a sub-structure (denoted as $\\mathcal{N}_{sub}$) in the network $\\mathcal{N}$ with sequential modules $\\{\\mathcal{N}_1, \\cdots, \\mathcal{N}_i\\}$ such that $\\mathcal{N}_{sub}$ can efficiently learn the function $h$ (i.e., the sample complexity for $\\mathcal{N}_{sub}$ to learn $h$ is no larger than $M$). \nAccording to Theorem~\\ref{thm:main_formal}, applying sequential learning to train $\\mathcal{N}_{sub}$ with labels $h(\\x)$, the representations of $\\mathcal{N}_{sub}$ will have predictive power $\\pred_{i}(h, \\mathcal{N}_{sub}, \\mathcal{C}) \\leq \\epsilon$ with probability $1-\\delta$. \n\nSince for each input $\\x$ in the noisy training data $\\hat{\\mathcal{S}}$, its label can be written as $f_r(h(\\x))$ (if it is clean) or $g_r(h(\\x))$ (if it is noisy), when the network $\\mathcal{N}$ is trained on $\\hat{\\mathcal{S}}$ using sequential learning, its sub-structure $\\mathcal{N}_{sub}$ can still learn $h$ efficiently (i.e., $\\mathcal{M}_{A} (h, \\mathcal{N}_{sub}, \\epsilon, \\delta) \\leq M$ for some learning algorithm $A$). Thus, the representations learned from the noisy training data $\\hat{\\mathcal{S}}$ can still be very predictive (i.e., $\\pred_{i}(h, \\mathcal{N}_{sub}, \\mathcal{C}) \\leq \\epsilon$ with probability $1-\\delta$). \n\nSince $f_r$ is a linear map from a high-dimensional space to a low-dimensional space, and the clean data $\\mathcal{C}$ has enough samples to learn $f_r$ ($\n|\\mathcal{C}|$ is larger than the input dimension of $f_r$), the linear model $L$ learned by linear regression can also generalize $f_r$ (since linear regression has a closed form solution in this case as the problem is over-complete).\nTherefore, as $\\pred_{i}(h, \\mathcal{N}_{sub}, \\mathcal{C}) \\leq \\epsilon$, $\\pred_{i}(f, \\mathcal{N}_{sub}, \\mathcal{C}) \\leq \\epsilon$ also holds.\nNotice that $\\pred_{i}(f, \\mathcal{N}_{sub}, \\mathcal{C}) = \\pred_{i}(f, \\mathcal{N}, \\mathcal{C})$ as $\\mathcal{N}_{i}$ is also the $i$-th module in $\\mathcal{N}$.\nHence, we have shown that there exist some module $\\mathcal{N}_i$ such that $\\pred_{i}(f, \\mathcal{N}, \\mathcal{C}) \\leq \\epsilon$ with probability $1-\\delta$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\n\nArea and orientation preserving diffeomorphisms of the standard 2-disc, referred to as symplectomorphisms of $\\disc$, allow  decompositions in terms of \\emph{positive} twist diffeomorphisms.\nUsing the latter decomposition we utilize the Conley index theory of discrete braid classes  as introduced in \\cite{BraidConleyIndex,GVV-pre} in order to obtain a\nMorse type forcing theory of periodic points: a priori information about periodic points determines a mapping class which may force additional periodic points.\n\n\n\n{\\let\\thefootnote\\relax\\footnotetext{* \\textit{Institute of Computer Science and Computational Mathematics, Jagiellonian University, Krak\\'ow}}}\n\n{\\let\\thefootnote\\relax\\footnotetext{** \\textit{Department of Mathematics, VU University, Amsterdam}}}\n\n\n\\newpage\n\n\n\\begin{sloppypar}\n\n\\section{Prelude}\nLet $\\disc\\subset\\plane$ be the standard unit 2-disc with coordinates $z=(x,y)\\in \\plane$, and let $\\omega=dx\\wedge dy$ be the standard area 2-form on $\\plane$.\nA diffeomorphism $F\\colon\\disc \\to\\disc$ is said to be symplectic  if $F^*\\omega = \\omega$ --- area and orientation preserving ---\nand is referred to as a \\emph{symplectomorphism} of $\\disc$.\nSymplectomorphisms of the 2-disc form a group which is denoted by $\\Symp(\\disc)$. \nA diffeomorphism $F$\nis \\emph{Hamiltonian} if it is given as the time-1 mapping of a Hamiltonian system\n\\begin{equation}\\label{HE} \n    \\begin{aligned}\n      \\dot{x}&=\\partial_y H(t,x,y); \\\\ \\dot{y}&=-\\partial_xH(t,x,y),\n    \\end{aligned}\n\\end{equation}\nwhere $ H \\in C^{\\infty}(\\rr\\times \\disc)$ a the Hamiltonian function with the additional property that $H(t, \\cdot)|_{\\partial \\disc} = const.$ for all $ t \\in \\rr$. The set of  Hamiltonians satisfying these requirements is denoted by ${\\mathcal{H}}(\\disc)$ and\nthe associated flow of \\eqref{HE} is denoted by $\\psi_{t,H}$. The group $\\Ham(\\disc)$ signifies the group of Hamiltonian diffeomorphisms of $\\disc$.\n Hamiltonian diffeomorphisms are symplectic by construction. For the 2-disc these notions are equivalent, i.e. $\\Symp(\\disc) = \\Ham(\\disc)$ and we may therefore   study Hamiltonian systems in order to prove properties about symplectomorphisms of $\\disc$, cf.\\ \\cite{Boyland2005}, and  Appendix \\ref{sec:sympMCG}.\n\nA subset $B\\subset \\disc$ is a invariant set for $F$ if $F(B) = B$.\n We are interested in finite invariant sets.\n Such invariant sets consist of periodic points, i.e.\npoints $z\\in\\disc$ such that $F^k(z) = z$ for some $k\\ge 1$.\nSince $\\partial \\disc$ is also invariant, periodic point are either in ${\\rm int~}\\disc$, or $\\partial \\disc$.\n\nThe main result of this paper concerns a\n \\emph{forcing} problem. Given a finite invariant set $B\\subset \\inter \\disc$ for $F \\in \\Symp(\\disc)$, do there \nexist additional periodic points? More generally, does there exist a finite   invariant set $A\\subset\\inter \\disc$, with $A\\cap B=\\varnothing$? This is much alike a similar question for the discrete dynamics on an interval, where \nthe famous Sharkovskii theorem establishes a forcing order among periodic points based on their period.\nIn the 2-dimensional situation such an order is much harder to establish, cf.\\ \\cite{Boyland2005}.\nThe main results are based on braiding properties of periodic points and are stated and proved in Section \\ref{sec:main1}.\nThe braid invariants introduced in this paper add additional information to existing invariants in the area-preserving case.\nFor instance,  Example \\ref{exm:exist3} describes a braid class which forces additional invariant sets solely in the area-preserving case\nhence extending the  non-symplectic methods described  in      \\cite{JiangZheng}.\n\n\nThe theory in this paper can be further\ngenelarized to include symplectomorphisms of bounded subsets of $\\rr^2$ with smooth boundary, eg. annuli, and  symplectomorphisms of $\\rr^2$.\n\n\\vskip.3cm\n\n{\\bf Acknowledgements:} AC was supported by the Foundation for Polish Science under the MPD Programme `Geometry\nand Topology in Physical Models', co-financed by the EU European Regional Development Fund,\nOperational Program Innovative Economy 2007-2013.\n\n\\section{Mapping class groups}\n\\label{sec:discinv}\nA priori knowledge of finite invariant sets $B$ for $F$\ncategorize mappings in so-called mapping classes.\nTraditionally   mapping class groups are defined for orientation preserving homeomorphisms,\n cf.\\  \\cite{G3} for an overview. \nDenote by $\\Homeo^+(\\disc)$ the space  of orientation preserving homeomorphisms  and by $\\Homeo^+_0(\\disc)$ the homeomorphisms that leave the boundary point wise invariant.\nTwo homeomorphisms $F,G\\in \\Homeo^+(\\disc)$ are   isotopic if there exists an isotopy $\\phi_t$, with $\\phi_t\\in\\Homeo^+(\\disc)$ for all $t\\in [0,1]$,\nsuch that $\\phi_0=F$ and $\\phi_1 = G$.\nThe equivalence classes in $\\pi_0\\bigl(\\Homeo^+(\\disc)\\bigr) = \\Homeo^+(\\disc)\/\\!\\!\\sim$ are called \\emph{mapping classes} and  form a group under composition. The latter is referred to as the  \\emph{mapping class group}  of the  2-disc and is denoted by $\\Mod(\\disc)$. \nFor homeomorphisms that leave the boundary point wise invariant the mapping class  group is denoted by $\\Mod_0(\\disc) = \\pi_0\\bigl(\\Homeo^+_0(\\disc)\\bigr)$. In  Appendix \\ref{sec:MCGclbr} we provide proofs of the relevant facts about mapping class groups.\n\\begin{prop}\n\\label{prop:MCG11}\nBoth mapping class groups $\\Mod(\\disc)$ and $\\Mod_0(\\disc)$ are trivial.\n\\end{prop}\nThe   mapping class groups $\\Mod(\\disc)$ and $\\Mod_0(\\disc)$ may also be defined using diffeomorphisms, cf.\\ Appendix \\ref{sec:MCGclbr}.  \nIn  Proposition \\ref{prop:mapclass1}, we show that $\\pi_0\\bigl(\\Symp(\\disc)\\bigr) = \\Mod(\\disc)$ and in Propositon\n  \\ref{prop:mapclass3} we show that $\\Ham(\\disc) = \\Symp(\\disc)$, which implies that every homeomorphism, or diffeomorphism is isotopic to\n  a Hamiltonian symplectomorphism.\n\n\\begin{prop}\n\\label{prop:MCG11a}\n$\\pi_0\\bigl(\\Symp(\\disc)\\bigr) = \\pi_0\\bigl(\\Ham(\\disc)\\bigr) =\\Mod(\\disc)\\cong 1$.\n\\end{prop}\n\n\n\nMore refined information about mapping classes is  obtained by considering finite invariant sets $B$.\nThis leads to the notion of the \\emph{relative mapping classes}. \nTwo homeomorphisms $F,G\\in \\Homeo^+(\\disc)$ are of the same mapping class \\emph{relative to} $B$ if there\nexists an isotopy $\\phi_t$,  with $\\phi_t\\in\\Homeo^+(\\disc)$ and $\\phi_t(B) = B$ for all $t\\in [0,1]$, such that\n$\\phi_0=F$ and $\\phi_1=G$. The subgroup of such homeomorphisms is denoted by $\\Homeo^+(\\disc\\rel B)$ and $\\Homeo^+_0(\\disc\\rel B)$\nin case $\\partial\\disc$ is point wise invariant.\nThe associated mapping class groups are denoted by \n$\\Mod(\\disc\\rel B) = \\pi_0\\bigl(\\Homeo^+(\\disc\\rel B) \\bigr)$ and\n$\\Mod_0(\\disc\\rel B) = \\pi_0\\bigl(\\Homeo^+_0(\\disc\\rel B) \\bigr)$ respectively.\n\n\\begin{prop}\n\\label{prop:MCG12}\n$\\Mod(\\disc\\rel B)\\cong {\\mathscr{B}}_m\/Z({\\mathscr{B}}_m)$ and $\\Mod_0(\\disc\\rel B)\\cong {\\mathscr{B}}_m$,\nwhere ${\\mathscr{B}}_m$ is the Artin braid group, with $m = \\# B$ and $Z({\\mathscr{B}}_m)$ is the center of the braid group.\n\\end{prop}\n\nLet $\\C_m\\disc$ be the \\emph{configuration space} of unordered configurations of $m$ points in\n$\\disc$. \n\\emph{Geometric braids}  on $m$ strands on $\\disc$ are closed loops  in $\\C_m\\disc$ based at $B_0 = \\{z_1,\\cdots,z_m\\}$, where the points $z_i$ are defined\nas follows: $z_i = (x_i,0)$, $x_0=-1$, and $x_{i+1}= x_i + 2\/(m+1)$.\nThe \\emph{classical braid group} on $\\disc$ is the fundamental group $\\pi_1\\bigl(\\C_m\\disc,B_0\\bigr)$ and is denote by ${\\mathcal{B}}_m\\disc$.\nThe  (algebraic) \\emph{Artin braid group} ${\\mathscr{B}}_{m}$ is a free group spanned by the $m-1$ generators $\\sigma_{i}$, modulo\nfollowing relations:\n\\begin{align}\\label{eqn:braidrel}\n \\begin{cases}\n  \\sigma_{i} \\sigma_{j} = \\sigma_{j} \\sigma_{i}, & \\ |i-j| \\geq 2,\\ i,j \\in \\{1, \\dots ,m-1\\} \\\\\n  \\sigma_{i} \\sigma_{i+1} \\sigma_{i} = \\sigma_{i+1} \\sigma_{i} \\sigma_{i+1}, & \\ 1\\le i \\le m-2.\n \\end{cases}\n\\end{align} \nFull twists are denoted algebraically by $\\square=  (\\sigma_{1} \\dots \\sigma_{m-1})^{m}$ and generate the center of the braid group ${\\mathscr{B}}_m$.\nPresentation of words consisting only of the $\\sigma_i$'s (not the inverses) and the relations in \\eqref{eqn:braidrel} form a monoid\nwhich is called the \\emph{positive braid monoid} ${\\mathscr{B}}_m^+$.\n\nThere exists a canonical isomorphism ${\\bm{i}}_m\\colon {\\mathscr{B}}_m \\to {\\mathcal{B}}_m\\disc$, cf.\\ \\cite[Sect.\\ 1.4]{Birman}.\nFor closed loops ${\\bm{\\beta}}(t)$ based at $B\\in \\C_m\\disc$ we have a canonical isomorphism   ${\\bm{j}}_B\\colon\\pi_1\\bigl(\\C_m\\disc,B\\bigr)\\to \\pi_1\\bigl(\\C_m\\disc,B_0\\bigr) ={\\mathcal{B}}_m\\disc$.\nLet $p\\colon [0,1]\\to \\C_m\\disc$ be a path connecting $B_0$ to $B$, then define ${\\bm{j}}_B\\bigl([{\\bm{\\beta}}]_B\\bigr) := [(p\\cdot {\\bm{\\beta}})\\cdot p^*]_{B_0}\n= [p\\cdot({\\bm{\\beta}}\\cdot p^*)]_{B_0}$, where $p^*$ is the inverse path connecting $B$ to $B_0$. The definition of ${\\bm{j}}_B$ is independent of the chosen path $p$.\nThis yields the isomorphism \n$\n\\imath_B = {\\bm{i}}_m^{-1}\\circ {\\bm{j}}_B\\colon \\pi_1\\bigl(\\C_m\\disc,B\\bigr) \\to\n{\\mathscr{B}}_m.\n$\n\nThe construction of the isomorphism $\\Mod_0(\\disc\\rel B) \\cong {\\mathcal{B}}_{m}\\disc$ can be  understood as follows, cf.\\ \\cite{Birman}, \\cite{Birman2}. For \n$F\\in \\Homeo^+_0(\\disc\\rel B)$\nchoose an   isotopy $\\phi_t\\in \\Homeo^+_0(\\disc), \\ t \\in [0,1]$, \nsuch that $\\phi_1=F$.\nSuch an isotopy  exists since $\\Homeo^+_0(\\disc)$ is contractible, cf.\\ Propostion \\ref{prop:MCG11}.\nFor $G\\in [F]\\in \\Mod_0(\\disc\\rel B)$, the composition and scaling  of the isotopies defines isotopic braids based at $B\\in \\C_m\\disc$.\nThe isomorphism $\\jmath_B\\colon \\Mod_0(\\disc\\rel B) \\to {\\mathscr{B}}_m$  is given by $\\jmath_B([F]) = \\iota_B\\bigl(d_*^{-1}([F])\\bigr)= \\imath_B([{\\bm{\\beta}}]_B)$, with ${\\bm{\\beta}}(t) = \\phi_t(B)$\nthe geometric braid generated by $\\phi_t$. The isomorphism $d_*$ is given in Appendix \\ref{subsec:braidMCG} and $[{\\bm{\\beta}}]_B$ denotes the homotopy class in $\\pi_1\\bigl(\\C_m\\disc,B\\bigr)$.\nFor $\\Mod(\\disc\\rel B)$ we use the same notation for the isomorphism which is given by\n\\[\n\\jmath_B\\colon\\Mod(\\disc\\rel B) \\cong {\\mathscr{B}}_{m}\/Z({\\mathscr{B}}_m),\\quad [F] \\mapsto \\jmath_B([F])= \\beta\\!\\!\\!\\!\\mod\\square,\n\\]\nwhere $\\beta = \\imath_B\\bigl( [{\\bm{\\beta}}]_B\\bigr)$.\nThe above mapping class groups can also be defined using diffeomorphisms and  symplectomorphisms. \n\n\\begin{prop}\n\\label{prop:MCG12a} \n $ \\pi_0\\bigl(\\Ham(\\disc\\rel B)\\bigr) =\\Mod(\\disc\\rel B)\\cong {\\mathscr{B}}_m\/Z({\\mathscr{B}}_m)$.\n\\end{prop}\nIn Appendix \\ref{sec:sympMCG} we show that $\\pi_0\\bigl(\\Symp(\\disc\\rel B)\\bigr) = \\Mod(\\disc\\rel B)$ and  that $\\Symp(\\disc\\rel B) = \\Ham(\\disc\\rel B)$\nand therefore that every mapping class can be represented by Hamiltonian symplectomorphisms.\n\n\n\n\n\n\\section{Braid classes}\n\\label{subsec:2color}\nConsidering free loops in a configuration space as opposed to based loops leads to classes of closed braids, which are the key tool for studying periodic points.\n\n\n\n\\subsection{Discretized braids}\n\\label{subsec:discbr}\nFrom \\cite{BraidConleyIndex} we recall the notion of positive piecewise linear braid diagrams and discretized braids.\n\\begin{defn}\n\\label{PL}\nThe space of {\\em discretized period $d$ closed braids on $n$ strands},\ndenoted $\\Conf^d_m$, is the space of all pairs $({\\bm{b}},\\tau)$ where\n$\\tau\\in S_m$ is a permutation on $m$ elements, and ${\\bm{b}}$ is an \nunordered set of $m$ {\\em strands}, ${\\bm{b}}=\\{{\\bm{b}}^\\mu\\}_{\\mu=1}^m$,\ndefined as follows:\n\\begin{enumerate}\n\\item[(a)]\n\teach strand  \n\t${\\bm{b}}^\\mu=(x^\\mu_0,x^\\mu_1,\\ldots,x^\\mu_d)\\in\\rr^{d+1}$\n\tconsists   of $d+1$ {\\em anchor points} $x_j^\\mu$;\n\\item[(b)] $x^\\mu_d = x^{\\tau(\\mu)}_0$\n\tfor all $\\mu=1,\\ldots,m$;\n\\item[(c)]\n\tfor any pair of distinct strands ${\\bm{b}}^\\mu$ and ${\\bm{b}}^{\\mu'}$\n\tsuch that $x^\\mu_j=x^{\\mu'}_j$ for some $j$,\n\tthe \\emph{ transversality} condition\n\t $\\bigl(x^\\mu_{j-1}-x^{\\mu'}_{j-1}\\bigr)\n\t\\bigl(x^\\mu_{j+1}-x^{\\mu'}_{j+1}\\bigr) < 0$ holds.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{rem}\nTwo discrete braids $({\\bm{b}},\\tau)$ and $( {\\bm{b}}', \\tau')$ are close if the strands ${\\bm{b}}^{\\zeta(\\mu)}$ and $ {\\bm{b}}'^\\mu$\nare close in $\\rr^{md}$ for some permutation $\\zeta$ such that $ \\tau'=\\zeta\\tau\\zeta^{-1}$.\nWe suppress the permutation $\\tau$ from the notation. \nPresentations via the braid monoid ${\\mathscr{B}}_m^+$ store the permutations.\n\\end{rem}\n\n\\begin{defn}[cf.\\ \\cite{BraidConleyIndex}]\n\\label{defn:closure}\nThe closure $\\bar\\Conf_m^d$ of the space $\\Conf_m^d$ consists of pairs $({\\bm{b}},\\tau)$ for which (a)-(b) in Definition \\ref{PL} are satisfied.\n\\end{defn}\n\nThe path components of $\\Conf_m^d$ are the \\emph{discretized braids classes} $[{\\bm{b}}]$.\nBeing in the same path connected component is an equivalence relation on $\\Conf_m^d$, where the braid classes are the\nequivalence classes expressed by the notation ${\\bm{b}},{\\bm{b}}'\\in [{\\bm{b}}]$, and ${\\bm{b}}\\sim{\\bm{b}}'$. \nThe associated permutations $\\tau$ and $\\tau'$ are conjugate. A path connecting ${\\bm{b}}$ and ${\\bm{b}}'$ is called a \\emph{positive isotopy} and the equivalence relation is   referred to \\emph{positively isotopic}.\n\nTo a configuration ${\\bm{b}}\\in\\Conf_m^d$  one can associate a \n\\emph{piecewise linear braid diagram} $\\Bd({\\bm{b}})$. For \neach strand ${\\bm{b}}^\\mu\\in {\\bm{b}}$, consider the piecewise-linear (PL) \ninterpolation\n\\begin{equation}\\label{interpolate1}\n\\Bd^{\\mu}(t) := x^\\mu_{\\floor{d\\cdot t}}+(d\\cdot t-\\floor{d\\cdot t})\n\t(x^\\mu_{\\ceil{d\\cdot t}}-x^\\mu_{\\floor{d\\cdot t}}),\n\\end{equation}\nfor $t\\in[0,1]$.\nThe braid diagram $\\Bd({\\bm{b}})$ is then defined to be the \nsuperimposed graphs of all the functions $\\Bd^{\\mu}(t)$.\nA braid diagram $\\Bd({\\bm{b}})$ is not only a good bookkeeping tool for keeping track of the strands\nin $\\Bd({\\bm{b}})$, but also plays natural the role of a braid diagram projection with only positive intersections, cf.\\ Section \\ref{subsec:discbrinv12}.\n\nThe set of $t$-coordinates of intersection points in $\\Bd({\\bm{b}})$ is denoted by $\\{t_i\\}$, $i=1,\\cdots,|{\\bm{b}}|$, where $|{\\bm{b}}|$ is the total number of\nintersections in $\\Bd({\\bm{b}})$ counted with multiplicity. The latter is also referred to as the \\emph{word metric} and is an invariant for ${\\bm{b}}$.\nA discrete braid ${\\bm{b}}$ is \\emph{regular} if all points $t_i$ and anchor points $x_j^\\mu$ are distinct.\nThe regular discrete braids in $[{\\bm{b}}]$ form a dense subset and every discrete braid is positively isotopic to a regular discrete braid. \nTo a regular discrete braid ${\\bm{b}}$ one can assign a unique positive word $\\beta = \\beta({\\bm{b}})$ defined as follows:\n\\begin{equation}\n\\label{eqn:word1}\n{\\bm{b}} \\mapsto \\beta({\\bm{b}}) = \\sigma_{k_1} \\cdots \\sigma_{k_\\ell},\n\\end{equation}\n where $k_i$ and $k_i +1$ are the positions that intersect at $t_i$, cf.\\ \\cite[Def.\\ 1.13]{Dehornoy1}. \nOn the positive braid monoid ${\\mathscr{B}}_m^+$ two positive words $\\beta$ and $\\beta'$ are positively equal,\nnotation $\\beta \\doteq\\beta'$, if they represent the same element in ${\\mathscr{B}}_m^+$ using  the relations in \\eqref{eqn:braidrel}.\nOn ${\\mathscr{B}}_m^+$ we define an equivalence relation which acts as an analogue of conjugacy in the braid group, cf.\\ \\cite[Sect.\\ 2.2]{BDV}.\nFor a given word $\\sigma_{i_1}\\cdots\\sigma_{i_n}$, define the relation\n\\[\n\\sigma_{i_1}\\sigma_{i_2}\\cdots\\sigma_{i_n} \\equiv \\sigma_{i_2}\\cdots\\sigma_{i_n}\\sigma_{i_1}.\n\\]\n\n\\begin{defn}\n\\label{defn:equiv12}\nTwo positive words $\\beta,\\beta'\\in {\\mathscr{B}}_m^+$ are \\emph{positively conjugate}, notation\n$\\beta \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} \\beta'$, if there exists a sequence of words $\\beta_0,\\cdots,\\beta_\\ell\\in {\\mathscr{B}}_m^+$, with $\\beta_0=\\beta$ and $\\beta_\\ell =\\beta'$, such\nthat for all $k$, either  $\\beta_k\\doteq \\beta_{k+1}$, or $\\beta_k\\equiv \\beta_{k+1}$\n\\end{defn}\n\nPositive conjugacy is an equivalence relation on ${\\mathscr{B}}_m^+$ and the set of  positive conjugacy classes $\\llbracket\\beta\\rrbracket$ of the braid monoid ${\\mathscr{B}}_m^+$ is  denoted by\n$\\CC {\\mathscr{B}}_m^+$. \n\nThe above defined assignment ${\\bm{b}} \\mapsto \\beta({\\bm{b}})$ can be extended to all discrete braids. A discrete braid ${\\bm{b}}$ is positively isotopic to a regular braid ${\\bm{b}}'$ and the mapping $\\Conf_m^d \\to \\CC{\\mathscr{B}}_m^+$, given by ${\\bm{b}} \\mapsto \\llbracket\\beta({\\bm{b}})\\rrbracket$, is well-defined\nby choosing $\\beta({\\bm{b}})$ to be any representative in the positive conjugacy class $\\llbracket\\beta({\\bm{b}}')\\rrbracket$.\nObserve that for fixed $d$ the mapping $\\Conf_m^d \\to \\CC{\\mathscr{B}}_m^+$ is not surjective.\n\n\\begin{rem}\nThe positive conjugacy relation defined in Definition \\ref{defn:equiv12} is symmetric by construction since it is defined on finite words.\nFor instance, consider $\\sigma_1\\sigma_2\\sigma_3 \\equiv \\sigma_2\\sigma_3\\sigma_1$.\nThe question whether the $\\sigma_2\\sigma_3\\sigma_1 \\equiv \\sigma_1\\sigma_2\\sigma_3$ is answered as follows:\n$\\sigma_2\\sigma_3\\sigma_1 \\equiv \\sigma_3\\sigma_1\\sigma_2 \\equiv \\sigma_1\\sigma_2\\sigma_3$, which, by Definition \\ref{defn:equiv12},\nshows that $\\sigma_2\\sigma_3\\sigma_1 \\equiv \\sigma_1\\sigma_2\\sigma_3$.\n\\end{rem}\n\n\n\nThe presentation of discrete braids via words in ${\\mathscr{B}}_m^+$ yields the \n following alternative equivalence relation.\n\\begin{defn}\n\\label{defn:topeq}\nTwo discretized braids ${\\bm{b}}, {\\bm{b}}'\\in \\Conf_m^d$ are \\emph{topologically equivalent} if $\\beta({\\bm{b}}) \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} \\beta({\\bm{b}}')$ in ${\\mathscr{B}}_m^+$, i.e. \n$\\beta({\\bm{b}})$ and $\\beta({\\bm{b}}')$ are positively conjugate.\nNotation: ${\\bm{b}} \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} {\\bm{b}}'$.\n\\end{defn}\n\nSummarizing, ${\\bm{b}}\\sim{\\bm{b}}'$ implies ${\\bm{b}} \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} {\\bm{b}}'$\nand $\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}$ defines a coarser equivalence relation on $\\Conf_m^d$.\nThe equivalence classes with respect to $\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}$ are denote by $[{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$.\nThe converse   is not true in general, cf.\\ \\cite[Fig.\\ 8]{BraidConleyIndex}.\nFollowing \\cite[Def.\\ 17]{BraidConleyIndex}, a discretized braid class $[{\\bm{b}}]$ is \\emph{free} if\n$[{\\bm{b}}] = [{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$.\n\\begin{prop}[\\cite{BraidConleyIndex}, Prop.\\ 27]\n\\label{prop:free}\nIf $d>|{\\bm{b}}|$, then $[{\\bm{b}}]$ is a free braid class.\n\\end{prop}\n\nTaking $d$ sufficiently large is a sufficient condition to ensure free braid classes, but this condition is not a necessary condition.\n\n\\begin{figure}[tb]\n\\centering\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4, line width = 1.5pt]\n  \\draw[-] (0,0) -- (2,6) -- (4,0); \n  \\draw[fill] (0,0) circle (0.07)\n              (2,6) circle (0.07)\n              (4,0) circle (0.07);\n\n   \\draw[-] (0,2) -- (2,2) -- (4,2);            \n   \\draw[fill] (0,2) circle (0.07)\n               (2,2) circle (0.07)\n               (4,2) circle (0.07);\n\n\n   \\draw[-] (0,4) -- (2,4) -- (4,4);\n   \\draw[fill] (0,4) circle (0.07)\n               (2,4) circle (0.07)\n               (4,4) circle (0.07);\n\\foreach \\x in {0,4}\n \\draw[line width=1.0pt, -] (\\x,-1) -- (\\x,7); \n\n\\end{tikzpicture}\n\\qquad\n\\qquad\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4, line width = 1.5pt]\n  \\draw[-] (0,6) -- (2,0) -- (4,6); \n  \\draw[fill] (0,6) circle (0.07)\n              (2,0) circle (0.07)\n              (4,6) circle (0.07);\n\n   \\draw[-] (0,2) -- (2,2) -- (4,2);            \n   \\draw[fill] (0,2) circle (0.07)\n               (2,2) circle (0.07)\n               (4,2) circle (0.07);\n\n\n   \\draw[-] (0,4) -- (2,4) -- (4,4);\n   \\draw[fill] (0,4) circle (0.07)\n               (2,4) circle (0.07)\n               (4,4) circle (0.07);\n\\foreach \\x in {0,4} \n \\draw[line width=1.0pt, -] (\\x,-1) -- (\\x,7); \n\n\\end{tikzpicture}\n\\qquad\n\\qquad\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4, line width = 1.5pt]\n  \\draw[-] (0,0) -- (2,6) -- (4,3.5) -- (6,0); \n  \\draw[fill] (0,0) circle (0.07)\n              (2,6) circle (0.07)\n              (4,3.5) circle (0.07)\n              (6,0) circle (0.07);\n\n   \\draw[-] (0,2) -- (2,2) -- (4,2) -- (6,2);            \n   \\draw[fill] (0,2) circle (0.07)\n               (2,2) circle (0.07)\n               (4,2) circle (0.07)\n               (6,2) circle (0.07);\n\n\n   \\draw[-] (0,4) -- (2,4) -- (4,4) -- (6,4);\n   \\draw[fill] (0,4) circle (0.07)\n               (2,4) circle (0.07)\n               (4,4) circle (0.07)\n               (6,4) circle (0.07);\n\\foreach \\x in {0,6}\n \\draw[line width=1.0pt, -] (\\x,-1) -- (\\x,7); \n\\end{tikzpicture}\n\\caption{The left and middle diagrams show  representatives  ${\\bm{b}},{\\bm{b}}'\\in \\Conf_3^2$ in Example \\ref{exm:free2}.\nThe right diagram shows a representative the same topological braid class in $\\Conf_3^3$ (free).}\n\\label{fig:free1}\n\\end{figure}\n\n\n\\begin{exm}\\label{exm:free2}\nGiven the braids ${\\bm{b}}\\in \\Conf_3^2$ with  ${\\bm{b}}^1 = (1,4,1)$, ${\\bm{b}}^2 = (2,2,2)$ and ${\\bm{b}}^3 = (3,3,3)$ and consider the braid\nclass $[{\\bm{b}}]$, see Figure \\ref{fig:free1}[left and middle]. Since ${\\bm{b}}$ is regular, $\\beta({\\bm{b}})$ is uniquely defined and $\\beta({\\bm{b}}) = \\sigma_1\\sigma_2^2\\sigma_1$.\nAlso define ${\\bm{b}}'\\in \\Conf_3^2$ with ${\\bm{b}}'^1 = (4,1,4)$, ${\\bm{b}}'^2={\\bm{b}}^2$ and ${\\bm{b}}'^3={\\bm{b}}^3$ and the braid\nclass $[{\\bm{b}}']$. Since ${\\bm{b}}'$ is also regular we have the unique braid word $\\beta({\\bm{b}}') = \\sigma_2\\sigma_1^2\\sigma_2$.\nObserve that $ \\sigma_1\\sigma_2^2\\sigma_1\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} \\sigma_2\\sigma_1^2\\sigma_2$, which implies that ${\\bm{b}}$ and\n${\\bm{b}}'$ are topologically equivalent. However, ${\\bm{b}}$ and ${\\bm{b}}'$ are not positively isotopic in $\\Conf_3^2$ and $[{\\bm{b}}]$ and\n$[{\\bm{b}}']$ are two different path components of $\\Conf_3^2$.\nThe positive conjugacy class of $\\sigma_1\\sigma_2^2\\sigma_1$ is given by $\\llbracket\\sigma_1\\sigma_2^2\\sigma_1\\rrbracket = \\{\\sigma_1\\sigma_2^2\\sigma_1,\\sigma_2^2\\sigma_1^2,\n\\sigma_2\\sigma_1^2\\sigma_2,\\sigma_1^2\\sigma_2^2\\}$. The words $\\sigma_2^2\\sigma_1^2$ and $\\sigma_1^2\\sigma_1^2$\nare not represented in $\\Conf_3^2$.\nIf we consider $ {\\bm{b}}''\\in \\Conf_3^3$ given by ${\\bm{b}}'' = \\bigl\\{ (1,4,1,1), (2,2,2,2),  (3,3,3,3)\\bigr\\}$, then\nthe associated braid class $[{\\bm{b}}'']$ is free, which confirms that the condition in Proposition \\ref{prop:free} is not a necessary\ncondition, see Figure \\ref{fig:free1}[right].\n\\end{exm}\n\n\n\n Let \n$\\beta = \\sigma_{i_1} \\cdots \\sigma_{i_d}\\in {\\mathscr{B}}_m^+$ be a positive braid word, then define\n\\[\n\\ev_q(\\beta) :=  {\\bm{b}} = \\{ {\\bm{b}}^\\mu\\} \\in \\Conf_m^{d+q},\\quad  {\\bm{b}}^\\mu = (x_j^\\mu),\\quad\\mu=1,\\cdots, m,~~q\\ge 0,\n\\]\nwith $x_0^\\mu = \\mu$, $x_j^\\mu = x_0^{\\sigma_{i_1} \\cdots \\sigma_{i_j}(\\mu)}$, $j=1,\\cdots,d$, and $x_{d+q}^\\mu =\\cdots = x_d^\\mu$. \nThe expression $\\sigma_{i_1} \\cdots \\sigma_{i_j}(\\mu)$, $\\mu = 1,\\cdots,m$ describes the permutation of the set $\\{1,\\cdots,m\\}$, where\n$\\sigma_{i_1} \\cdots \\sigma_{i_j}$ is regarded as a concatenation of permutations given by the generators $\\sigma_i$ interpreted as\na basic permutation of $i$ and $i+1$.\nBy Proposition \\ref{prop:free}, $[\\ev_q(\\beta)]$ is free for all $q\\ge 1$, and every\n $\\llbracket\\beta\\rrbracket\\in \\CC{\\mathscr{B}}_m^+$  defines a free discrete braid class $[\\ev_q(\\beta)]$ in $\\Conf_m^{d+q}$\n for all $q\\ge 1$. \n\n\n\\subsection{Discrete 2-colored braid classes}\n\\label{subsec:discbr2}\nOn closed configuration spaces we define the following product:\n\\[ \n\\bar\\Conf_n^d\\times\\bar\\Conf_m^d \\to \\bar\\Conf_{n+m}^d,\\quad ({\\bm{a}},{\\bm{b}}) \\mapsto  {\\bm{a}}\\sqcup{\\bm{b}},\n\\]\nwhere ${\\bm{a}}\\sqcup{\\bm{b}}$ is the disjoint union of the strands in ${\\bm{a}}$ and ${\\bm{b}}$ regarded as an element in $\\bar\\Conf_{n+m}^d$.\nThe definition yields a canonical permutation on the labels in ${\\bm{a}}\\sqcup{\\bm{b}}$.\nDefine the space of \\emph{2-colored discretized braids} as the space of ordered pairs\n\\begin{equation}\n\\label{eqn:2color}\n\\Conf_{n,m}^d := \\bigl\\{ {\\bm{a}}\\rel{\\bm{b}}:=({\\bm{a}},{\\bm{b}})~|~ {\\bm{a}}\\sqcup{\\bm{b}} \\in \\Conf_{n+m}^d\\bigr\\}.\n\\end{equation}\nThe strand labels in ${\\bm{a}}$ range from $\\mu=1,\\cdots,n$ and the strand labels in ${\\bm{b}}$ range from $\\mu=n+1,\\cdots,n+m$.\nThe associated permutation $\\tau_{{\\bm{a}},{\\bm{b}}} = \\tau_{\\bm{a}} \\oplus\\tau_{\\bm{b}} \\in S_{n+m}$, where\n $\\tau_{\\bm{a}}\\in S_n$ and $\\tau_{\\bm{b}} \\in S_m$, and $\\tau_{\\bm{a}}$ acts on the labels\n$\\{1,\\cdots,n\\}$ and $\\tau_{\\bm{b}}$ acts on the labels $\\{{n+1},\\cdots,{n+m}\\}$.\nThe strands ${\\bm{a}} = \\{x_j^{\\mu}\\}$, $\\mu=1,\\cdots,n$ are the \\emph{red}, or \\emph{free} strands and\nthe strands ${\\bm{b}} = \\{x_j^{\\mu}\\}$, $\\mu=n+1,\\cdots,n+m$ are the \\emph{black}, or \\emph{skeletal} strands.\nA path component $[{\\bm{a}}\\rel{\\bm{b}}]$ in $\\Conf_{n,m}^d$ is called a \\emph{2-colored discretized braid class}.\nThe canonical projections are given by\n$\\varpi\\colon\\Conf_{n,m}^d \\to \\Conf_m^d$ with ${\\bm{a}}\\rel{\\bm{b}} \\mapsto {\\bm{b}}$ and by\n$\\varpi^*\\colon\\Conf_{n,m}^d \\to \\Conf_n^d$ with ${\\bm{a}}\\rel{\\bm{b}} \\mapsto {\\bm{a}}$.\nThe mapping $\\varpi$ yields a fibration\n\\begin{equation}\n\\label{eqn:fiberbundle2}\n[{\\bm{a}}]\\rel {\\bm{b}} \\to [{\\bm{a}}\\rel {\\bm{b}}] \\to [{\\bm{b}}].\n\\end{equation}\n The pre-images\n$\\varpi^{-1}({\\bm{b}}) = [{\\bm{a}}]\\rel {\\bm{b}}\\subset  \\Conf_{n}^d$,   are called the \\emph{relative discretized braid class fibers}.\n\nThere exists a natural embedding $\\Conf_{n,m}^d \\hookrightarrow \\Conf_{n+m}^d$, defined by ${\\bm{a}}\\rel{\\bm{b}} \\mapsto {\\bm{a}}\\sqcup{\\bm{b}}$.\nVia the embedding we   define the notion of topological equivalence of two 2-colored discretized braids:\n${\\bm{a}}\\rel{\\bm{b}} \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} {\\bm{a}}'\\rel{\\bm{b}}'$ if ${\\bm{a}}\\sqcup{\\bm{b}} \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} {\\bm{a}}'\\sqcup{\\bm{b}}'$. The associated equivalence classes are denoted by $[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$, which are\nnot necessarily connected sets in $\\Conf_{n,m}^d$. A 2-colored discretized braid class $[{\\bm{a}}\\rel{\\bm{b}}]$ is free if $[{\\bm{a}}\\rel{\\bm{b}}] = [{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$.\nIf $d>|{\\bm{a}}\\sqcup{\\bm{b}}|$, then  $[{\\bm{a}}\\rel{\\bm{b}}]$ is free by Proposition \\ref{prop:free}. \n\nThe set of collapsed singular braids in $\\bar \\Conf_{n,m}^d$ is given by:\n\\[\n\\begin{aligned}\n\\Sigma^- := \\{{\\bm{a}}\\rel{\\bm{b}}\\in \\bar\\Conf_{n,m}^d~|~ {\\bm{a}}^\\mu =~&{\\bm{a}}^{\\mu'}, \\hbox{or~} {\\bm{a}}^\\mu={\\bm{b}}^{\\mu'}\\\\\n&\\hbox{for some~~}\\mu\\not=\\mu',~\\hbox{and~} {\\bm{b}}\\in \\Conf_m^d\\}.\n\\end{aligned}\n\\]\nA 2-colored discretized braid class $[{\\bm{a}}\\rel{\\bm{b}}]$ is \\emph{proper} if \n$\\partial [{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}} \\cap \\Sigma^- = \\varnothing$.\nIf a braid class $[{\\bm{a}}\\rel{\\bm{b}}]$ is not proper it is called \\emph{improper}.\nIn \\cite{BDV} properness is considered in a more general setting. The notion of properness in this paper coincides with weak properness in \\cite{BDV}.\n\nA 2-colored discretized braid class $[{\\bm{a}}\\rel{\\bm{b}}]$\nis called \\emph{bounded} if its fibers are bounded as  sets in $\\rr^{nd}$.\nNote that $[{\\bm{a}}\\rel{\\bm{b}}]$\nis \\emph{not} a bounded set in $\\rr^{(n+m)d}$.\n\n\n\\subsection{Algebraic presentations}\n\\label{subsec:algpres}\nDiscretized braid classes are presented via the positive conjugacy classes of the positive braid monoid ${\\mathscr{B}}_m^+$.\nFor 2-colored discretized braids we seek a similar presentation.\n\nIn order to keep track of colors we define coloring on words in ${\\mathscr{B}}_{n+m}^+$.\nWords in ${\\mathscr{B}}_{n+m}^+$ define associated permutations $\\tau$ and the permutations $\\tau$ yield partitions of the set $\\{1,\\cdots,n+m\\}$.\nLet $\\gamma \\in {\\mathscr{B}}_{n+m}^+$ be a word for which the induced partition   contains a union of equivalence classes $\\aset \\subset \\{1,\\cdots,n+m\\}$\nconsisting of $n$ elements. The set $\\aset$ is the \\emph{red coloring} of length $n$ and the remaining partitions are colored black, denoted by $\\bset$. \nThe pair $(\\gamma,\\aset)$ is \ncalled a 2-colored positive braid word, see Figure \\ref{fig:relative1}.\nFor a given coloring $\\aset \\subset \\{1,\\cdots,n+m\\}$ of length $n$ the set of all words $(\\gamma,\\aset)$ forms a monoid which is denoted by ${\\mathscr{B}}^+_{n,m,\\aset}$ and is referred as\nthe \\emph{2-colored braid monoid} with coloring $\\aset$.\n\n\nTwo pairs $(\\gamma,\\aset)$ and $(\\gamma',\\aset')$ are positively conjugate if $\\gamma\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} \\gamma'$ and\n$\\aset' = \\zeta^{-1}(\\aset)$, where $\\zeta$ is a permutation conjugating the induced permutations $\\tau_\\gamma$ and $\\tau_{\\gamma'}$, i.e.  $\\tau_{\\gamma'} = \\zeta\\tau_\\gamma\\zeta^{-1}$.\nIf $\\xi$ is another permutation such that $\\tau_{\\gamma'} = \\xi\\tau_\\gamma\\xi^{-1}$, then \n$ \\zeta\\tau_\\gamma\\zeta^{-1} =  \\xi\\tau_\\gamma\\xi^{-1}$. This implies that $\\tau_\\gamma =  \\zeta^{-1}\\xi\\tau_\\gamma\\xi^{-1}\\zeta$\nand thus $\\xi^{-1}\\zeta(\\aset) = \\aset$, which is equivalent to \n$\\zeta^{-1}(\\aset) = \\xi^{-1}(\\aset)$. This shows\nthat the conjugacy relation in well-defined.\nPositive conjugacy for 2-colored braid  words is again denoted by $(\\gamma,\\aset) \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} (\\gamma', \\aset')$\nand a conjugacy class is denoted by $\\llbracket\\gamma,\\aset\\rrbracket$.\nThe set of 2-colored positive conjugacy classes \nwith red colorings of length $n$ is denoted by $\\CC{\\mathscr{B}}_{n,m}^+$.\n\nThe words corresponding   to the different colors $(\\gamma,\\aset)$  can be derived from the information in $(\\gamma,\\aset)$.\nLet $\\aset_0\\subset \\aset$ be a cycle of length $\\ell\\le n$ and let $k  \\in \\aset_0$. If $\\gamma = \\sigma_{i_1}\\cdots \\sigma_{i_d}$, then we define\nan $\\ell$-periodic sequence $\\{k_j\\}$, with \n\\[\nk_0 = k,\\quad\\hbox{and}\\quad k_{j} = \\sigma_{i_j}(k_{j-1}), ~~~j=1,\\cdots,\\ell d,\n\\]\nby considering the word $\\gamma^\\ell$. Now use the following rule: if $k_j-k_{j-1} \\not = 0$, remove \n$\\sigma_{i_{j'}}$ from $\\gamma$, for $j=1,\\cdots, \\ell d$, where $j' =j \\!\\!\\mod d \\in \\{1,\\cdots,d\\}$.\nMoreover, $\\sigma_{i_j}$ is replaced by $\\sigma_{i_j-1}$, if $k_j=k_{j-1}<i_j$ and $\\sigma_{i_j}$ remains unchanged otherwise.\nWe repeat this procedure for all cycles in $\\aset$ and we obtain the mapping $(\\gamma,\\aset) \\mapsto \\beta \\in {\\mathscr{B}}_m^+$ denoted by\n\\[\n\\pi_{\\aset}\\colon {\\mathscr{B}}^+_{n,m,\\aset} \\to {\\mathscr{B}}^+_m.\n\\]\nBy considering the complementary color $\\bset$ we construct a mapping $(\\gamma,\\bset) \\mapsto \\alpha\\in {\\mathscr{B}}_n^+$ using the same scheme.\n\nWith the notion of coloring braid words we can encode the information of a 2-colored discretized braid ${\\bm{a}}\\rel{\\bm{b}}$ in a 2-colored word\n$(\\gamma,\\aset)$. Given ${\\bm{a}}\\rel{\\bm{b}}\\in \\Conf_{n,m}^d$ (regular), we define \n\\begin{equation}\n\\label{eqn:colored1}\n{\\bm{a}}\\rel{\\bm{b}} \\mapsto  \\gamma({\\bm{a}}\\rel{\\bm{b}}) := \\beta({\\bm{a}}\\sqcup{\\bm{b}}),\n\\end{equation}\n cf.\\ \\eqref{eqn:word1} and the coloring $\\aset = \\cset^{-1}(\\{1,\\cdots,n\\})$, where the permutation $\\cset$\nis defined as follows. Order the coordinates $x_0^{\\mu_1} < \\cdots < x_0^{\\mu_{n+m}}$ and define\n\\[\n\\cset^{-1} = \\left(\\begin{array}{cccc}1 & 2 & \\cdots & n+m \\\\\\mu_1 & \\mu_2 & \\cdots & \\mu_{n+m}\\end{array}\\right).\n\\]\nThe permutations $\\tau_\\gamma$ and $\\tau_{{\\bm{a}},{\\bm{b}}}$ are conjugated: $\\tau_\\gamma = \\cset \\tau_{{\\bm{a}},{\\bm{b}}} \\cset^{-1}$.\nThe mapping ${\\bm{a}}\\rel{\\bm{b}} \\mapsto (\\gamma,\\aset)$ is well-behaved under positive conjugacy: \n${\\bm{a}}\\rel{\\bm{b}} \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} {\\bm{a}}'\\rel{\\bm{b}}'$ implies $(\\gamma,\\aset) \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} (\\gamma',\\aset')$.\n\n\n\nConversely, every positive conjugacy class $\\llbracket\\gamma,\\aset\\rrbracket$ determines a 2-colored discretized braid class $[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$ via\nthe mapping $\\gamma \\mapsto \\ev_q(\\gamma) \\in \\Conf_{n+m}^{d+q}$, with $d$ the number of generators in $\\gamma$ and $q\\ge 0$, cf.\\ Figure \\ref{fig:relative1}[right]. \nThe representation  ${\\bm{a}} \\rel {\\bm{b}}\\in \\Conf_{n,m}^{d+q}$ is obtained from the coloring $\\aset$.\n\n\\begin{defn}\n\\label{defn:proper2}\nA positive conjugacy class $\\llbracket\\gamma,\\aset\\rrbracket$ is called proper if the associated discretized braid class  $[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$  is  proper, cf.\\ Section \\ref{subsec:discbr2}.\n\\end{defn}\n\n\n\\begin{figure}[htb]\n\\centering\n\\begin{tikzpicture}[xscale=0.4, yscale=0.3, line width = 1.5pt]\n   \\draw[-] (0,0) -- (2,4) -- (4,4);            \n   \\draw[fill] (0,0) circle (0.07)\n               (2,4) circle (0.07)\n               (4,4) circle (0.07);\n\n   \\draw[-] (0,4) -- (2,0) -- (4,0);            \n   \\draw[fill] (0,4) circle (0.07)\n               (2,0) circle (0.07)\n               (4,0) circle (0.07);\n\n   \\draw[-,color=red] (0,2) -- (2,6) -- (4,2);            \n   \\draw[fill, color=red] (0,2) circle (0.07)\n                           (2,6) circle (0.07)\n                           (4,2) circle (0.07);\n\n \\foreach \\x in {0,4}\n \\draw[line width=1.0pt, -] (\\x,-1) -- (\\x,7); \n\\end{tikzpicture}\n\\qquad\n\\qquad\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4, line width = 1.5pt]\n   \\draw[-] (0,0) -- (2,0) -- (4,2) -- (6,4);            \n   \\draw[fill] (0,0) circle (0.07)\n               (2,0) circle (0.07)\n               (4,2) circle (0.07)\n               (6,4) circle (0.07);\n\n   \\draw[-] (0,4) -- (2,2) -- (4,0) -- (6,0);            \n   \\draw[fill] (0,4) circle (0.07)\n               (2,2) circle (0.07)\n               (4,0) circle (0.07)\n               (6,0) circle (0.07);\n\n   \\draw[-,color=red] (0,2) -- (2,4) -- (4,4) -- (6,2);            \n   \\draw[fill, color=red]  (0,2) circle (0.07)\n                           (2,4) circle (0.07)\n                           (4,4) circle (0.07)\n                           (6,2) circle (0.07);\n\n \\foreach \\x in {0,6}\n \\draw[line width=1.0pt, -] (\\x,-1) -- (\\x,5); \n\\end{tikzpicture}\n\\caption{Representations of the relative braid class $[{\\bm{a}}\\rel{\\bm{b}}]$.}\n\\label{fig:relative1}\n\\end{figure}\n\n\\begin{exm}\nConsider ${\\bm{a}}\\rel{\\bm{b}}\\in \\Conf_{1,2}^2$, with  strands ${\\bm{a}} = \\{(2,4,2)\\}$, ${\\bm{b}} = \\{(1,3,3), (3,1,1)\\}$.\nSince the strands in ${\\bm{b}}$ have labels $\\mu=2,3$,\nthe permutation is given by $\\tau_{{\\bm{a}},{\\bm{b}}} = (23)$ and $\\gamma = \\beta({\\bm{a}}\\sqcup {\\bm{b}}) = \\sigma_2\\sigma_1\\sigma_2$.\nThe coloring permutation is given as follows: $x_0^2<x_0^1<x_0^3$ and therefore\n$\\cset^{-1} = (12)$.\nThe red coloring is given by $\\aset = \\cset(\\{1\\}) = \\{2\\}$.\nWe verify that $\\tau_\\gamma = \\cset \\tau_{{\\bm{a}},{\\bm{b}}} \\cset^{-1} = (2)(13) = (13)$, see Figure \\ref{fig:relative1}[left].\nThe topological type of ${\\bm{a}}\\rel {\\bm{b}}$ is given by the $\\llbracket\\sigma_2\\sigma_1\\sigma_2,\\{2\\}\\rrbracket$ and $[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$ is a proper and free 2-colored discrete braid class, see Figure \\ref{fig:relative1}[right].\nIn order to compute the skeletal braid word $\\beta$ we consider the following sequence:\n\\[\nk_0 =2,\\quad k_1 = \\sigma_2(2) = 3,\\quad k_2=\\sigma_1(3) = 3,\\quad k_3 = \\sigma_2(3) =2.\n\\]\nThis yields the differences $k_1-k_0= 1$, $k_2-k_1 = 0$ and $k_3-k_2 = -1$, and therefore both letters $\\sigma_2$ are removed from $\\gamma$,\nwhich gives $\\gamma \\mapsto \\beta = \\sigma_1$.\n\\begin{figure}[hbt]\n\\centering\n\\begin{tikzpicture}[xscale=0.4, yscale=0.25, line width = 1.5pt]\n   \\draw[-] (0,0) -- (2,4) -- (4,4);            \n   \\draw[fill] (0,0) circle (0.07)\n               (2,4) circle (0.07)\n               (4,4) circle (0.07);\n\n   \\draw[-] (0,4) -- (2,0) -- (4,0);            \n   \\draw[fill] (0,4) circle (0.07)\n               (2,0) circle (0.07)\n               (4,0) circle (0.07);\n\n   \\draw[-,color=red] (0,2) -- (2,6) -- (4,2);            \n   \\draw[fill, color=red] (0,2) circle (0.07)\n                           (2,6) circle (0.07)\n                           (4,2) circle (0.07);\n\n   \\draw[-] (0+4,0) -- (2+4,4) -- (4+4,4);            \n   \\draw[fill] (0+4,0) circle (0.07)\n               (2+4,4) circle (0.07)\n               (4+4,4) circle (0.07);\n\n   \\draw[-] (0+4,4) -- (2+4,0) -- (4+4,0);            \n   \\draw[fill] (0+4,4) circle (0.07)\n               (2+4,0) circle (0.07)\n               (4+4,0) circle (0.07);\n\n   \\draw[-,color=red] (0+4,2) -- (2+4,6) -- (4+4,2);            \n   \\draw[fill, color=red] (0+4,2) circle (0.07)\n                           (2+4,6) circle (0.07)\n                           (4+4,2) circle (0.07);\n\n\n\\foreach \\x in {0,4,8}\n \\draw[line width=1.0pt, -] (\\x,-1) -- (\\x,7); \n\n\\foreach \\x in {2,6}\n \\draw[loosely dashed, line width=1.0pt, -] (\\x,-1) -- (\\x,7); \n\\end{tikzpicture}\n\\qquad\n\\begin{tikzpicture}[xscale=0.4, yscale=0.25, line width = 1.5pt]\n   \\draw[-] (0,0) -- (2,4) -- (4,4);            \n   \\draw[fill] (0,0) circle (0.07)\n               (2,4) circle (0.07)\n               (4,4) circle (0.07);\n\n   \\draw[-] (0,4) -- (2,0) -- (4,0);            \n   \\draw[fill] (0,4) circle (0.07)\n               (2,0) circle (0.07)\n               (4,0) circle (0.07);\n\n   \\draw[-,color=red] (0,2) -- (2,3) -- (4,2);            \n   \\draw[fill, color=red] (0,2) circle (0.07)\n                           (2,3) circle (0.07)\n                           (4,2) circle (0.07);\n\n   \\draw[-] (0+4,0) -- (2+4,4) -- (4+4,4);            \n   \\draw[fill] (0+4,0) circle (0.07)\n               (2+4,4) circle (0.07)\n               (4+4,4) circle (0.07);\n\n   \\draw[-] (0+4,4) -- (2+4,0) -- (4+4,0);            \n   \\draw[fill] (0+4,4) circle (0.07)\n               (2+4,0) circle (0.07)\n               (4+4,0) circle (0.07);\n\n   \\draw[-,color=red] (0+4,2) -- (2+4,3) -- (4+4,2);            \n   \\draw[fill, color=red] (0+4,2) circle (0.07)\n                           (2+4,3) circle (0.07)\n                           (4+4,2) circle (0.07);\n\\foreach \\x in {0,4,8}\n \\draw[line width=1.0pt, -] (\\x,-1) -- (\\x,7); \n\n\\foreach \\x in {2,6}\n \\draw[loosely dashed, line width=1.0pt, -] (\\x,-1) -- (\\x,7); \n\\end{tikzpicture}\n\\qquad\n\\begin{tikzpicture}[xscale=0.4, yscale=0.25, line width = 1.5pt]\n   \\draw[-] (0,0) -- (2,4) -- (4,4);            \n   \\draw[fill] (0,0) circle (0.07)\n               (2,4) circle (0.07)\n               (4,4) circle (0.07);\n\n   \\draw[-] (0,4) -- (2,0) -- (4,0);            \n   \\draw[fill] (0,4) circle (0.07)\n               (2,0) circle (0.07)\n               (4,0) circle (0.07);\n\n   \\draw[-,color=red] (0,5) -- (2,3) -- (4,5);            \n   \\draw[fill, color=red] (0,5) circle (0.07)\n                           (2,3) circle (0.07)\n                           (4,5) circle (0.07);\n\n   \\draw[-] (0+4,0) -- (2+4,4) -- (4+4,4);            \n   \\draw[fill] (0+4,0) circle (0.07)\n               (2+4,4) circle (0.07)\n               (4+4,4) circle (0.07);\n\n   \\draw[-] (0+4,4) -- (2+4,0) -- (4+4,0);            \n   \\draw[fill] (0+4,4) circle (0.07)\n               (2+4,0) circle (0.07)\n               (4+4,0) circle (0.07);\n\n   \\draw[-,color=red] (0+4,5) -- (2+4,3) -- (4+4,5);            \n   \\draw[fill, color=red] (0+4,5) circle (0.07)\n                           (2+4,3) circle (0.07)\n                           (4+4,5) circle (0.07);\n\\foreach \\x in {0,4,8}\n \\draw[line width=1.0pt, -] (\\x,-1) -- (\\x,7); \n\n\\foreach \\x in {2,6}\n \\draw[loosely dashed, line width=1.0pt, -] (\\x,-1) -- (\\x,7); \n\\end{tikzpicture}\n\\caption{Positively conjugate representations.}\n\\label{fig:relative2}\n\\end{figure}\n\nFigure \\ref{fig:relative2}[left] shows shows a representation  of $(\\gamma,\\aset)$ and Figure \\ref{fig:relative2}[right] shows a positively conjugate representation $(\\gamma',\\aset')$.\nIn the latter case $\\aset' = \\{3\\}$, $\\cset=(123)$ and $\\aset' = \\cset^{-1}(\\{1\\}) = (132)(1) = \\{3\\}$. It follows that $\\tau_{\\gamma'} = \\cset\\tau_{{\\bm{a}},{\\bm{b}}} \\cset^{-1} = (123)(23)(132) = (12)$. Moreover, since $\\gamma' = \\zeta \\gamma\\zeta^{-1}$, with $\\zeta = (23)$ it follows that  $\\tau_{\\gamma'} = \\zeta\\tau_\\gamma\n\\zeta^{-1} = (23)(13(23) = (12)$.\n\\end{exm}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discrete braid invariants}\n\\label{subsec:discbrinv12}\nWe summarize the construction of a topological invariant for 2-colored relative braid classes as described in \\cite{BraidConleyIndex,GVV-pre}.\nLet  ${\\bm{a}}\\rel {\\bm{b}}\\in \\Conf_{n,m}^d$ represent  a proper, bounded discretized 2-colored braid class $[{\\bm{a}}\\rel {\\bm{b}}]$. Then, the fiber $[{\\bm{a}}]\\rel {\\bm{b}}$ defines\na bounded set in $\\rr^{nd}$.\n\nA sequence ${\\mathcal{R}}=\\{{\\mathcal{R}}_j\\}$ of functions ${\\mathcal{R}}_j\\colon \\rr^3\\to \\rr$,\n which satisfy $\\partial_1\\mathcal{R}_j>0$ and $\\partial_3\\mathcal{R}_j>0$ is called a \\emph{parabolic recurrence relation}.\nFrom   \\cite[Lem.\\ 55-57]{BraidConleyIndex} there exists    a parabolic recurrence relation ${\\mathcal{R}}=\\{{\\mathcal{R}}_j\\}$\nsuch that ${\\bm{b}}$ is a zero for ${\\mathcal{R}}$, i.e. ${\\mathcal{R}}_j(x_{j-1}^{\\mu_\\nu},x_j^{\\mu_\\nu},x_{j+1}^{\\mu_\\nu})=0$ for all $j\\in \\zz$ and for all ${\\nu}=1,\\cdots,m$.\nThe recurrence relation ${\\mathcal{R}}$ may regarded as vector field and is integrated via the equations\n\\begin{equation}\n\\label{parabolicvectorfield}\n \\frac{d}{ds}x^{{\\mu_\\nu}}_{j}=\\mathcal{R}_{j}(x^{{\\mu_\\nu}}_{j-1},x^{{\\mu_\\nu}}_{j},x^{{\\mu_\\nu}}_{j+1}),\\quad \\nu=1,\\cdots,m.\n\\end{equation}\nLet $N$ denoted the closure in $\\rr^{nd}$ of  $[{\\bm{a}}]\\rel {\\bm{b}}$.\nBy   \\cite[ Prop.\\ 11 and Thm.\\ 15]{BraidConleyIndex}, the set $N$ is an isolating neighborhood for the parabolic flow generated by\nEquation \\eqref{parabolicvectorfield}.\nWe define ${\\mathrm{h}}({\\bm{a}}\\rel{\\bm{b}})$ as the homotopy Conley index of $\\Inv(N,{\\mathcal{R}})$, cf.\\ \\cite{BraidConleyIndex}, \\cite{ConleyIndex}.\nThe Conley index is independent of the choice of parabolic recurrence relations ${\\mathcal{R}}$ for which ${\\mathcal{R}}({\\bm{b}})=0$, cf.\\ \\cite[Thm.\\ 15(a)-(b)]{BraidConleyIndex},\nas well as the choice of the fiber, i.e. ${\\bm{a}}\\rel{\\bm{b}} \\sim {\\bm{a}}'\\rel{\\bm{b}}'$, then ${\\mathrm{h}}({\\bm{a}}\\rel{\\bm{b}}) = {\\mathrm{h}}({\\bm{a}}'\\rel{\\bm{b}}')$, cf.\\ \\cite[Thm.\\ 15(c)]{BraidConleyIndex}.\nThis makes ${\\mathrm{h}}({\\bm{a}}\\rel{\\bm{b}})$  an invariant of the discrete 2-colored braid class $[{\\bm{a}}\\rel{\\bm{b}}]$.\n\nThere is an intrinsic way to define ${\\mathrm{h}}({\\bm{a}}\\rel{\\bm{b}})$ without using parabolic recurrence relations.\nWe define $N^-\\subset \\partial N$ to be the set of boundary points for which the word\nmetric is locally maximal.\nThe pair $(N,N^-)$ is an index\npair for any parabolic system ${\\mathcal{R}}$ such that ${\\mathcal{R}}({\\bm{b}})=0$, and\nthus by the independence of Conley index on ${\\mathcal{R}}$,  the pointed homotopy type\nof\n$N\/N^-$ gives the Conley index: \n$\n{\\mathrm{h}}({\\bm{a}}\\rel{\\bm{b}}) = [N\/N^-],\n$\nsee Figure \\ref{fig:conley1} and \\cite[Sect.\\ 4.4]{BraidConleyIndex} for more details on the construction.\n\n\\begin{figure}[hbt]\n\\centering\n\\begin{tikzpicture}[xscale=0.5, yscale=0.4, line width = 1.5pt]\n  \\draw[-] (0,4) -- (2,5) -- (4,4); \n  \\draw[fill] (0,4) circle (0.07)\n              (2,5) circle (0.07)\n              (4,4) circle (0.07);\n  \\draw[-] (0,1) -- (2,0) -- (4,1);\n  \\draw[fill] (0,1) circle (0.07)\n              (2,0) circle (0.07)\n              (4,1) circle (0.07);\n\n  \\draw[-] (0,5) -- (2,1) -- (4,5); \n  \\draw[fill] (0,5) circle (0.07)\n              (2,1) circle (0.07)\n              (4,5) circle (0.07);\n  \\draw[-] (0,0) -- (2,4) -- (4,0);\n  \\draw[fill] (0,0) circle (0.07)\n              (2,4) circle (0.07)\n              (4,0) circle (0.07);\n\n  \\draw[-, color=red] (0,2) -- (2,2.5) -- (4,2);\n  \\draw[color=red,fill] (0,2) circle (0.07)\n                        (2,2.5) circle (0.07)\n                        (4,2) circle (0.07);\n\\foreach \\x in {0,4}\n \\draw[line width=1.0pt, -] (\\x,-1) -- (\\x,6); \n\\end{tikzpicture}\n\\qquad\n\\qquad\n\\begin{tikzpicture}[xscale=0.5, yscale=0.5, line width = 1.5pt]\n  \\draw[-,color=white] (1,1);\n  \\fill[color=red!90] (2,6) -- (6,6) -- (6,2) -- (2,2) -- (2,6);\n  \\draw[-] (2,6) -- (6,6) -- (6,2) -- (2,2) -- (2,6); \n  \\draw[->] (1.5,5) -- (2.5,5);\n  \\draw[->] (1.5,4) -- (2.5,4);\n  \\draw[->] (1.5,3) -- (2.5,3);\n\n  \\draw[<-] (1.5+4,5) -- (2.5+4,5);\n  \\draw[<-] (1.5+4,4) -- (2.5+4,4);\n  \\draw[<-] (1.5+4,3) -- (2.5+4,3);\n \n  \\draw[<-] (5,1.5) -- (5,2.5);\n  \\draw[<-] (4,1.5) -- (4,2.5);\n  \\draw[<-] (3,1.5) -- (3,2.5);\n\n  \\draw[->] (5,1.5+4) -- (5,2.5+4);\n  \\draw[->] (4,1.5+4) -- (4,2.5+4);\n  \\draw[->] (3,1.5+4) -- (3,2.5+4);\n\\end{tikzpicture}\n\\qquad\n\\qquad\n\\begin{tikzpicture}[xscale=0.6, yscale=0.8, line width = 1.5pt]\n  \\draw[-,color=white] (1,1);\n  \\draw[fill=red!90] (1,3.5) ellipse (2.0 and 1.0);\n  \\draw[fill=white] (0.5,3.15) ellipse (1.0 and 0.6);\n  \\draw[fill] (0.07,2.63) circle (0.07);\n\\end{tikzpicture}\n\\caption{The Conley index for the braid in Example \\ref{exm:exist2}.\nThe homotopy of the pionted space in given by $h({\\bm{a}}\\rel{\\bm{b}}) = \\sbb^1$.}\n\\label{fig:conley1}\n\\end{figure}\n\nThe invariant ${\\mathrm{h}}({\\bm{a}}\\rel{\\bm{b}})$ is not necessarily invariant with respect to the number of discretization points $d$.\nIn order to have   invariance also with respect to $d$, another invariant for discrete braid classes was introduced in \\cite{BraidConleyIndex}.\nConsider the equivalence class induced by the relation ${\\bm{a}}\\rel{\\bm{b}} \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} {\\bm{a}}'\\rel{\\bm{b}}'$ on $\\Conf_{n,m}^d$, which defines  the class\n$[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$ of proper discrete 2-colored braids.\nVia the projection $\\varpi\\colon [{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}} \\to [{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$ we obtain fibers $\\varpi^{-1}({\\bm{b}})$.\nSuppose $[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$ is a bounded class, i.e. all fibers $\\varpi^{-1}({\\bm{b}})$ are bounded sets in $\\rr^{nd}$.\nFollowing \\cite[Def.\\ 18]{BraidConleyIndex}\nthe closure $N$ of a fiber\n$\\varpi^{-1}({\\bm{b}})$ is an isolating neighborhood since ${\\bm{a}}\\rel{\\bm{b}}$ is proper.\nDefine ${\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}})$ as the homotopy Conley index of $N$.\nIf $[{\\bm{a}}_k]\\rel {\\bm{b}}$ are the fibers belonging to the components $[{\\bm{a}}_k\\rel{\\bm{b}}]$ of $[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$, then  \n\\begin{equation}\n\\label{eqn:braidindex22}\n{\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}) := \\bigvee_{k} {\\mathrm{h}}([{\\bm{a}}_k]\\rel{\\bm{b}}).\n\\end{equation}\n\nDefine the following extension mapping\n $\\E:\\Conf_m^d\\to\\Conf_m^{d+1}$, cf.\\ \\cite{BraidConleyIndex},\nvia concatenation with the trivial braid of period one: \n\\begin{equation}\n\t(\\E{\\bm{b}})^\\mu := \\left\\{\n\t\\begin{array}{cl}\n\t\tx_j^{\\mu} &\tj=0,\\ldots,d; \t\\\\\n\t\tx_d^{\\mu} &  j=d+1 .\n\t\\end{array}\\right.\n\\end{equation}\nProperness remains unchanged under the extension mapping $\\E$, however boundedness may not be preserved.\nDefine the skeletal augmentation:\n\\[\n\\A\\colon \\Conf_m^d \\to \\Conf_{m+2}^d,\\quad {\\bm{b}} \\mapsto \\A{\\bm{b}} = {\\bm{b}}^* = {\\bm{b}}\\cup {\\bm{b}}^-\\cup{\\bm{b}}^+,\n\\]\nwhere ${\\bm{b}}^- =\\{\\min_{\\mu} \\{x_j^\\mu\\} - 1\\}_j$ and ${\\bm{b}}^+ =\\{\\max_{\\mu} \\{x_j^\\mu\\} + 1\\}_j$.\nIf $[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$ is bounded, then\n ${\\mathrm{h}}([{\\bm{a}}_k]\\rel{\\bm{b}}) = {\\mathrm{h}}([{\\bm{a}}_k]\\rel {\\bm{b}}^*)$ for all $k$ and therefore ${\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}) = {\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}^*)$.\nOne can define second skeletal augmentation:\n\\[\n\\B\\colon \\Conf_m^d \\to \\Conf_{m+2}^d,\\quad {\\bm{b}} \\mapsto \\B{\\bm{b}}={\\bm{b}}^\\# = {\\bm{b}}\\cup {\\bm{b}}^s\\cup{\\bm{b}}^n,\n\\]\nwhere ${\\bm{b}}^s =\\{(-1)^j\\min_{\\mu} \\{x_j^\\mu\\} - (-1)^j\\}_j$ and ${\\bm{b}}^n =\\{(-1)^j\\max_{\\mu} \\{x_j^\\mu\\} + (-1)^j\\}_j$.\nAs before, if $[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$ is bounded, then\n ${\\mathrm{h}}([{\\bm{a}}_k]\\rel{\\bm{b}}) = {\\mathrm{h}}([{\\bm{a}}_k]\\rel {\\bm{b}}^\\#)$ for all $k$ and therefore ${\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}) = {\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}^\\#)$.\n\n\nConsider the proper, bounded  2-colored braid classes $[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$ and $[\\E{\\bm{a}}\\rel\\E{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$.\nThe main result in \\cite[Thm.\\ 20]{BraidConleyIndex} is the Stabilization Theorem which states that\n\\begin{equation}\n\\label{eqn:braidindex13}\n{\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}^*) = {\\mathrm{H}}(\\E{\\bm{a}}\\rel \\E{\\bm{b}}^*).\n\\end{equation}\nThe independence of ${\\mathrm{H}}$ on the skeleton ${\\bm{b}}$ can be derived from the Stabilization Theorem.\nSince a 2-colored discretized braid class is free when $d$ is sufficiently large, we have that $[\\E^p{\\bm{a}}\\rel \\E^p{\\bm{b}}^*]$ is free\nfor some $p>0$ sufficiently large, and by stabilization ${\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}^*) = {\\mathrm{H}}(\\E^p{\\bm{a}}\\rel \\E^p{\\bm{b}}^*)$.\nLet ${\\bm{a}} \\rel {\\bm{b}} \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} {\\bm{a}}'\\rel{\\bm{b}}'$, then $\\E^p{\\bm{a}}\\rel \\E^p{\\bm{b}}^* \\sim \\E^p{\\bm{a}}'\\rel \\E^p{\\bm{b}}'^*$.\nBy \\cite[Thm.\\ 15(c)]{BraidConleyIndex}, a continuation can be constructed which proves that\n${\\mathrm{H}}(\\E^p{\\bm{a}}\\rel \\E^p{\\bm{b}}^*) ={\\mathrm{H}}(\\E^p{\\bm{a}}'\\rel \\E^p{\\bm{b}}'^*)$. Consequently,\n\\[\n{\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}^*) = {\\mathrm{H}}(\\E^p{\\bm{a}}\\rel \\E^p{\\bm{b}}^*) = {\\mathrm{H}}(\\E^p{\\bm{a}}'\\rel \\E^p{\\bm{b}}'^*) = {\\mathrm{H}}({\\bm{a}}'\\rel{\\bm{b}}'^*),\n\\]\nwhich shows that the index ${\\mathrm{H}}$ only depends on the topological type $\\llbracket \\gamma,\\aset\\rrbracket$, with $\\gamma=\\beta({\\bm{a}}\\rel{\\bm{b}})$.\n\\begin{defn}\n\\label{defn:discrbrinv}\nLet $\\llbracket\\gamma,\\aset\\rrbracket$ be proper, positive conjugacy class. Then, the  \\emph{braid Conley index} is defined as\n\\begin{equation}\n\\label{eqn:relbrinv}\n{\\mathrm{H}}\\llbracket\\gamma,\\aset\\rrbracket := {\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}^*).\n\\end{equation}\n\\end{defn}\nThe  braid Conley index  ${\\mathrm{H}}$  may be computed using any\nrepresentative ${\\bm{a}}\\rel{\\bm{b}}^*$ for any sufficiently large $d$ and any associated recurrence relation ${\\mathcal{R}}$.\n\nFinally, we mention that besides the extension  $\\E$, we also have a \\emph{half twist} extension operator $\\T$:\n\\begin{equation}\n\t(\\T{\\bm{b}})^\\mu := \\left\\{\n\t\\begin{array}{cl}\n\t\tx_j^{\\mu} &\tj=0,\\ldots,d \t\\\\\n\t\t-x_d^{\\mu} &  j=d+1 .\n\t\\end{array}\\right.\n\\end{equation}\nEvery discretized braid can be dualized via the mapping $\\{x^\\mu_j\\} \\mapsto \\{(-1)^j x_j^\\mu\\}$. On $\\Conf_m^{2d}$ this yields\na well-defined operator $\\D\\colon\\Conf_m^{2d}\\to \\Conf_m^{2d}$  mapping proper, bounded discretized braid classes $[{\\bm{a}}\\rel{\\bm{b}}]$ to proper, bounded discretized braid classes\n$[\\D{\\bm{a}}\\rel\\D{\\bm{b}}]$.\nFrom \\cite[Cor.\\ 31]{BraidConleyIndex} we recall the following result.\nLet ${\\bm{a}}\\rel{\\bm{b}}\\in \\Conf_{n,m}^{2d}$ be proper, then\n\\begin{equation}\n\\label{eqn:thedual}\n{\\mathrm{H}}\\bigl(\\T^2\\circ \\D({\\bm{a}}\\rel{\\bm{b}}^*)\\bigr) = {\\mathrm{H}}\\bigl(\\D({\\bm{a}}\\rel{\\bm{b}}^*)\\bigr) \\wedge \\sbb^{2n},\n\\end{equation}\nwhere the wedge is the $2n$-suspension of the Conley index.\n\nFrom the singular homology $H_*({\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}^*))$ the Poincar\\'e polynomial \nis denoted by $P_t({\\bm{a}}\\rel{\\bm{b}}^*)$, or $P_t\\llbracket\\gamma,\\aset\\rrbracket$ in terms of the topological type.\nThis yields an important invariant: $|P_t({\\bm{a}}\\rel{\\bm{b}}^*)| = |P_t\\llbracket\\gamma,\\aset\\rrbracket|$, which is the number of monomial term in the Poincar\\'e polynomial.\n\n\n\n\n\\section{The variational formulation}\nFor a given symplectomorphism $F\\in \\Symp(\\disc)$\n the problem of finding periodic points can be reformulated in terms of parabolic recurrence relations.\n\n\n\\subsection{Twist symplectomorphisms}\n\n\nLet  $F(x,y) = \\bigl(f(x,y),g(x,y)\\bigr)$ be a symplectomorphism of $\\plane$, with $f,g$ smooth functions on $\\plane$.\nRecall that $F\\in \\Symp(\\plane)$ is a \\emph{positive} twist symplectomorphism if\n\\[\n\\frac{\\partial f(x,y)}{\\partial y} >0.\n\\]\nFor twist symplectomorphisms there exists a variational principle  for finding periodic points, cf.\\ \\cite{LeCalvez}, \\cite{Moser}.\nSuch a variational principle also applies to symplectomorphisms that are given as a composition:\n\\[\nF= F_d\\circ \\cdots \\circ F_1,\n\\]\nwith $F_j\\in \\Symp(\\plane)$ positive twist symplectomorphisms for all $j$.\nIt is important to point out that $F$ itself is \\emph{not} twist in general.\nAn important question is whether every mapping $F\\in \\Symp(\\plane)$ can be written  as a composition of (positive) twist symplectomorphisms, cf.\\ \\cite{LeCalvez}.\nSuppose $F\\in \\Ham(\\plane)$,\nand $F$ allows a Hamiltonian isotopy $\\psi_{t,H}$ with appropriate asymptotic conditions near infinity,\nsuch that $\\psi_{t_i,H}\\circ\\psi_{t_{i-1},H}^{-1}$ is close to the identity mapping in the $C^1$-norm for sufficiently small time steps $t_i-t_{i-1}$. Then, define $G_i = \\psi_{t_i,H}\\circ\\psi_{t_{i-1},H}^{-1}$, $i=1,\\cdots,k$,\nand $F = G_k\\circ \\cdots \\circ G_1$.\nWe remark that in this construction\n the individual mappings $G_i$ are not twist necessarily.\nThe following observation provides a decomposition consisting solely of positive   twist symplectomorphisms.\nConsider the $90^o$ degree clockwise rotation\n\\[\n\\psi(x,y) = (y,-x), \\quad \\psi^4={\\rm id},\n\\]\nwhich is positive   twist symplectomorphism.\nThis yields the decomposition:\n\\begin{equation}\n\\label{eqn:decomp12}\nF = (G_k\\circ \\psi) \\circ\\psi\\circ\\psi\\circ\\psi\\circ \\cdots \\circ (G_1\\circ\\psi)\\circ\\psi\\circ\\psi\\circ\\psi,\n\\end{equation}\nwhere $F_{4i} = G_{i}\\circ \\psi$ and $F_j=\\psi$ for $j\\not = 4i$ for some $i$ and $d=4k$.\nSince the mappings $G_i$ are close to the identity,  the compositions $G_i\\circ \\psi$ are positive twist symplectomorphisms.\nThe above   procedure intertwines  symplectomorphisms with $k$ full rotations. As we will see later on this results in\npositive braid representations of mapping classes.\nThe choice of $\\psi$ is arbitrary since other rational rotations also yield twist symplectomorphisms.\n\nFor  symplectomorphisms $F \\in \\Symp(\\disc)$ we establish a similar decomposition in terms of positive  twist symplectomorphisms, with the additional property that the decomposition can be extended to symplectomorphisms of $\\plane$,\nwhich is necessary  to  apply the variational techniques in \\cite{BraidConleyIndex}.\n\n\\subsection{Interpolation}\\label{subsec:interpl}\n\nA symplectomorphism $F\\in \\Symp(\\plane)$ satisfies the \\emph{uniform twist condition} if the there exists a $\\delta>0$ such that\n\\begin{equation}\n\\label{eqn:twist}\n\\delta^{-1} \\ge \\frac{\\partial f(x,y)}{\\partial y} \\ge \\delta >0,\\quad \\forall (x,y)\\in \\plane.\n\\end{equation}\nThe subset of such symplectomorphism is denoted by $SV(\\plane)$, cf.\\ \\cite{LeCalvez}.\nA result by Moser implies that all  symplectomorphisms of   $\\plane$ with a uniform twist condition \nare Hamiltonian.\n\\begin{prop}[cf.\\ \\cite{Moser}]\\label{MoserThm}\n Let $F \\in SV(\\plane)$. Then, there exists a Hamiltonian  $H \\in \\mathcal{H}(\\plane)$\n such that $0<\\delta \\le H_{yy} \\le \\delta^{-1}$ and\n $\\psi_{1,H} = F$, where $\\psi_{t,H}$ is  the associated Hamiltonian flow.\nAll orbits of $\\psi_{t,H}$ project to straight lines in $(t,x)$-plane, and $\\psi_{t,H}\\in SV(\\plane)$ for all $t\\in (0,1]$.\n\\end{prop}\n\nFor completeness we give a self-contained proof of Proposition \\ref{MoserThm}, which is the same as the proof\n in \\cite{Moser} modulo a few alterations.\n\n\\begin{proof}\nFollowing \\cite{Moser} we consider action\n integral $\\int_{0}^{1} L(t,x(t),\\dot{x}(t)) dt$\n for functions $x(t)$ with $x(0)=x_0$ and $x(1) = x_1$.\n We require that extremals are affine lines, i.e. $\\ddot x(t)=0$. \nFor extremals the action is given by $S(x_0,x_1) = \\int_{0}^{1} L(t,x(t),\\dot{x}(t)) dt$ and we seek\nLagrangians such that $S={\\bm{h}}$, where ${\\bm{h}}$ is the generating function for $F$.\n %\n For Lagrangians this implies \n\\begin{equation}\n\\tfrac{d}{dt}(\\partial_pL)-\\partial_xL =  ( \\partial_{t}+p \\partial_{x} )\\partial_p L-\\partial_x L=0,\n \\label{straightEL}\n\\end{equation}\nwhere $p=\\dot x$. Solving the first order partial differential equation yields\n $L=L_{0}(t,x,p)+p \\partial_x m + \\partial_t m$,\nwith\n\\begin{equation}\n L_{0}:=-\\int_{0}^{p}(p-p')\\partial^2_{x_0 x_1}{\\bm{h}}(x-p't,x+p'(1-t)) dp' .\n \\label{Fzero}\n\\end{equation}\nand $m= m(t,x)$ to be specified later, cf.\\ see \\cite{Moser} for details.\nThe extremals $x(t)$ are also extremals for $L_0$. Let $S_{0}(x_{0},x_{1}) = \\int_{0}^{1} L_0(t,x(t),\\dot{x}(t)) dt$, then\n\\begin{equation}\n \\int_{0}^{1}p \\partial_x m(t,x(t))+\\partial_t m(t,x(t)) dt = m(1,x_{1})-m(0,x_{0})\n\\end{equation}\nand hence \n %\n\\begin{equation}\n S(x_{0},x_{1})=S_{0}(x_{0},x_{1})+m(1,x_{1})-m(0,x_{0}).\n\\end{equation}\nDifferentiating $S$ yields\n\\begin{equation*}\n\n \\partial_{x_0} S = -\\partial_p L(0,x_{0},x_{1}-x_{0}),\\quad\n\n  \\partial_{x_1} S = \\partial_p L(0,x_{1},x_{1}-x_{0})\n \\label{Sdiff}\n\\end{equation*}\nand for the mixed derivat\n\\begin{equation}\n \\partial^2_{x_0 x_1} S_0(x_0,x_1)=-\\partial^2_{pp}L(0,x_{0},x_{1}-x_{0})=\\partial^2_{x_{0} x_{1}}{\\bm{h}}(x_{0},x_{1}).\n \\label{FppRel}\n\\end{equation}\nThen, $S_{0}(x_{0},x_{1})-h(x_{0},x_{1})=u(x_{0})+v(x_{1})$ and the choice\n\\begin{equation*}\n m(t,x):=(1-t)u(x)-tv(x)\n\\end{equation*}\nimplies $S={\\bm{h}}$. Differentiating the relation $y=-\\partial_x {\\bm{h}}(x,x_{1})$ with respect to $y$ and using the fact that\n$x_1 = f(x,y)$, yields\n\\begin{equation*}\n\\begin{aligned}\n1&= -\\partial^2_{y x}{\\bm{h}} (x,x_1) =  - \\partial^2_{x x_{1}}{\\bm{h}} (x,x_{1})\\partial_yf(x, x_1)\\\\\n&= -\\partial^2_{x x_{1}}{\\bm{h}}(x,x_{1})\\partial_y f(x, x_{1})\n \\end{aligned}\n\\end{equation*}\nand thus $\\delta\\le \\partial_y f \\le \\delta^{-1}$   if and only if $-\\delta^{-1} \\leq \\partial^2_{x x_{1}}{\\bm{h}} \\leq -\\delta$.\nBy relation \\eqref{FppRel} we have $\\partial^2_{pp}L \\in [ \\delta, \\delta^{-1} ]$.\n\nThe Hamiltonian is obtained via the Legendre transform\n\\begin{equation}\n H(t,x,y):=yp-L(t,x,p),\n \\label{Ltransform}\n\\end{equation}\nwhere \n\\begin{equation}\n y=\\partial_p L(t,x,p),\n \\label{Ltransform2}\n\\end{equation}\nand we can solve for $p$, i.e. $p = \\lambda(x,y)$. As before, differentiating \\eqref{Ltransform2} gives\n$1=\\partial^2_{pp}L\\cdot  \\partial_y \\lambda$ and differentiating \\eqref{Ltransform} gives $\\partial_yH = \\lambda$. Combining these two identities yields $\\partial^2_{pp}L\\cdot \\partial^2_{yy}H=1$, from which the desired property $\\partial^2_{yy}H \\in [ \\delta, \\delta^{-1} ]$ follows.\n\nFrom the above analysis we obtain the following expression for the isotopy $\\psi_{t,H}$:\n\\begin{equation}\n\\label{eqn:MoserIso}\n\\psi_{t,H}(x,y) = \\Bigl(x+\\lambda(x,y)t,\\partial_pL\\bigl(t,x+\\lambda(x,y)t,\\lambda(x,y)\\bigr)  \\Bigr).\n\\end{equation}\nLet $\\pi_x$ denote the projection onto the $x$-coordinate.~Then, $\\partial_y \\pi_x \\psi_{t,H}(x,y) =\\partial_y \\lambda(x,y) t = \\partial^2_{yy}H t$, which proves that\n$\\psi_{t,H}$ is positive twist for all $t\\in (0,1]$.\n\\end{proof}\n\n\nUsing Proposition \\ref{MoserThm} we obtain\na decomposition of symplectomorphisms $F\\in \\Symp(\\plane)$ as given in \\eqref{eqn:decomp12}\nand which satisfy additional properties such that the discrete braid invariants in \\cite{BraidConleyIndex} are applicable.\n\n\\begin{prop}\\label{Interpolation2} \nLet  $F \\in \\Symp(\\disc)$. Then, there exists  an  isotopy $\\phi_{t} \\subset \\Symp(\\plane)$ for all $t\\in [0,1]$, an integer $d\\in \\nn$ and \n a sequence $\\{t_{j} \\}_{j=0}^{d} \\subset [0,1]$ with $t_j = j\/d$,\n  such that\n  \\begin{enumerate}\n \\item[(i)] $\\phi_{0}= \\id$, $\\phi_{1}|_{\\disc}=F$;\n \\item[(ii)] $\\phi_{t}$ is smooth with respect to $t$ on the intervals $[t_j,t_{j+1}]$ (piecewise smooth);\n \\item[(iii)] $\\widehat F_j :=\\phi_{t_{j}} \\circ \\phi_{t_{j-1}}^{-1} \\in SV(\\plane)$ for all $1\\le j \\le d$, and $F_j:=\\widehat F_j|_{\\disc}$;\n \\item[(iv)] the projection of the graph of $\\phi_{t}(x,y)$ onto $(t,x)$-plane is linear on the intervals $t \\in (t_{j-1},t_{j})$ for all $1\\le j \\le d$, and for all $(x,y) \\in \\plane$;\n \\item[(v)] $\\phi_{t} (\\disc) \\subset [-1,1] \\times \\rr $ for all $t\\in [0,1]$;\n \\item[(vi)] the points $z_\\pm = (\\pm 2,0)$  \n\n are fixed points of $\\widehat F_j = \\phi_{t_{j}} \\circ \\phi_{t_{j-1}}^{-1}$ for all $1\\le j \\le d$;\n  \\item[(vii)] the points $z'_\\pm = (\\pm 4,0)$  are period-2 points of $\\widehat F_j = \\phi_{t_{j}} \\circ \\phi_{t_{j-1}}^{-1}$ for all $1\\le j \\le d$,\n  i.e. $\\widehat F_j(z'_\\pm) = z'_\\mp = -z'_\\pm$, for all $j$.\n \\end{enumerate}\nThe decomposition\n\\begin{equation}\n\\label{eqn:decomp14}\n\\widehat F = \\widehat F_d\\circ \\cdots \\circ \\widehat F_1,\n\\end{equation}\n is a generalization of the decomposition given in \\eqref{eqn:decomp12}.\n\\end{prop}\n\nThe isotopy constructed in Proposition \\ref{Interpolation2} is called a \\emph{chained Moser isotopy}.\nBefore proving Proposition \\ref{Interpolation2} we construct  analogues of the rotation mapping used in \\eqref{eqn:decomp12}.\n\n\\begin{lem}\\label{AlternatePsi}  \nFor every integer $\\ell\\ge 3$ there exists  \n a positive  Hamiltonian twist diffeomorphism $\\Psi$ of the plane $\\plane$,\nsuch that: \n\\begin{enumerate}\n\\item[(i)] the restriction $\\Psi|_{\\disc}$  is a rotation   over angle $2\\pi\/\\ell$ and $\\Psi|_{\\disc}^\\ell = \\id$;\n\\item[(ii)] the points $z_\\pm=(\\pm 2,0)$ are fixed points for $\\Psi$;\n\\item[(iii)] the points $z'_\\pm=(\\pm 4,0)$ are period-2 points for $\\Psi$, i.e. $\\Psi(z'_\\pm) = z'_\\mp$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nA linear rotation mapping on $\\plane$  is a positive twist mapping for all rotation angles $\\vartheta\\in (0,\\pi)$.\nThe generating function for a rotation is given by\n\\begin{equation}\n\\label{eqn:rot12}\n{\\bm{h}}_\\vartheta(x,x') = \\tfrac{1}{2}\\cot(\\vartheta) x^2 - \\csc(\\vartheta) x x' + \\tfrac{1}{2}\\cot(\\vartheta) x'^2.\n\\end{equation}\nIn order to construct the mappings $\\Psi$ we construct special generating functions.\nLet $\\ell\\ge 3$ be an integer and let $\\vartheta_\\ell = 2\\pi\/\\ell \\in (0,\\pi)$.\nConsider generating functions of the form\n\\begin{equation}\n{\\bm{h}}_\\Psi(x,x') = \\xi_\\ell(x) - \\csc(\\vartheta_\\ell) x x' + \\xi_\\ell(x'),\n\\end{equation}\nwhich generate positive twist mappings for all $\\ell\\ge 3$.\nWe choose $\\xi_\\ell$ as follows: $\\xi_\\ell(x) = \\frac{1}{2}\\cot(\\vartheta_\\ell) x^2$ for all $|x|\\le 1$, $\\xi_\\ell(x) = \\frac{1}{2}\\csc(\\vartheta_\\ell) x^2$ for all $3\/2\\le |x|\\le 5\/2$, and  $\\xi_\\ell(x) = -\\frac{1}{2}\\csc(\\vartheta_\\ell) x^2$ for all $7\/2\\le |x|\\le 9\/2$.\nThe mapping $\\Psi$ is defined by $h_\\Psi$ and \n$y = -\\partial_1 {\\bm{h}}_\\Psi(x,x')$ and $y' = \\partial_2 {\\bm{h}}_\\Psi(x,x')$.\\footnote{To simplify notation we express the derivatives of ${\\bm{h}}$ with respect to its two  coordinates \n  by $\\partial_1{\\bm{h}}$ and $\\partial_2{\\bm{h}}$.} \nFor $|x|,|x'|\\le 1$, the generating function restricts to \\eqref{eqn:rot12} which yields the rotation over $\\vartheta_\\ell$ on $\\disc$ and  establishes (i).\nFor $3\/2\\le x,x'\\le 5\/2$ we have \n\\[\ny = \\csc(\\vartheta_\\ell)(x'-x),\\quad\\hbox{and}\\quad y' =  \\csc(\\vartheta_\\ell)(x'-x),\n\\]\nwhich verifies that $z_+$ are fixed points and same holds for $z_-$, completing the verification of (ii).\nFor $7\/2\\le x \\le 9\/2$ and $-9\/2\\le x'\\le -7\/2$, we have \n\\[\ny = \\csc(\\vartheta_\\ell)(x'+x),\\quad\\hbox{and}\\quad y' =  -\\csc(\\vartheta_\\ell)(x'+x),\n\\]\nthen $z'_+$ is mapped to $z'_-$ and similarly $z'_-$ is mapped to $z'_+$, which completes (iii) and proof of the lemma.\n\\end{proof}\n\n\nIn order to extend chained Moser isotopies yet another type of Hamiltonian twist diffeomorphism is needed.\n\n\\begin{lem}\\label{AlternatePsi3}  \nFor every  integer $\\ell\\ge 3$  there exists  \n a positive  Hamiltonian twist symplectomorphism $\\Upsilon$ of the plane $\\plane$,\nsuch that: \n\\begin{enumerate}\n\\item[(i)] the restriction $\\Upsilon|_{\\disc}$  is a rotation   over angle $2\\pi\/\\ell$, i.e. $\\Upsilon|_{\\disc}^{\\ell} = \\id$;\n\\item[(ii)] the points $z_\\pm=(\\pm 2,0)$ and $z'_\\pm=(\\pm 4,0)$ are period-2 points for $\\Upsilon$, i.e. $\\Upsilon(z'_\\pm) = z'_\\mp$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nAs before consider\n generating functions of the form\n\\begin{equation}\n\\label{eqn:rot14}\n{\\bm{h}}_\\Upsilon(x,x') = \\xi_\\ell(x) - \\csc(\\vartheta_\\ell) x x' + \\xi_\\ell(x'),\n\\end{equation}\nwhich generate positive twist mappings for all   $\\ell\\ge 3$.\nWe choose $\\xi_\\ell$ as follows: $\\xi_\\ell(x) = \\frac{1}{2}\\cot(\\vartheta_\\ell) x^2$ for all $|x|\\le 1$,  and  $\\xi_\\ell(x) = -\\frac{1}{2}\\csc(\\vartheta_\\ell) x^2$ for all $3\/2\\le |x|\\le 9\/2$.\nThe mapping $\\Upsilon$ is defined by $h_\\Upsilon$.\nFor $|x|,|x'|\\le 1$, the generating function restricts to \\eqref{eqn:rot12} which yields the rotation over $\\vartheta_\\ell$ on $\\disc$ and  establishes (i).\nFor $3\/2\\le x,\\le 9\/2$ and $-9\/2\\le x'\\le -7\/2$, we have \n\\[\ny = \\csc(\\vartheta_\\ell)(x'+x),\\quad\\hbox{and}\\quad y' =  -\\csc(\\vartheta_\\ell)(x'+x),\n\\]\nthen $z_+$ and $z'_+$ are mapped to $z_-$ and $z'_-$ respectively and similarly $z_-$ and $z'_-$ are mapped to $z_+$ and $z'_+$ respectively, which completes proof.\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{Interpolation2}]\nConsider the  subgroup $\\Symp_{c}(\\plane)$ formed by compactly supported symplectomorphisms of the plane.\\footnote{A symplectomorphism is compactly supported in $\\plane$ if it is the identity outside a compact subset of $\\plane$\n}\n Recall that due to the uniform twist property the set $SV(\\plane)$ is open in the topology given by $C^{1}$-convergence on compact sets, cf.\\ \\cite{LeCalvez}. \n Let $\\Psi\\in \\Ham(\\plane)$ be given by Lemma \\ref{AlternatePsi} for some $\\ell\\ge 3$.\n Then, there exists\nan open neighborhood ${\\mathscr{W}} \\subset \\Symp_{c}(\\plane)$  of the identity, such that $\\varphi\\circ \\Psi\\in SV(\\plane)$ for all $\\varphi \\in {\\mathscr{W}}$.\n\nFor $F\\in \\Symp(\\disc)$, Proposition \\ref{prop:MCG11a}\nprovides a Hamiltonian\n $H_{} \\in \\mathcal{H}(\\disc)$ such that $F = \\psi_{1,H}$. \n Let $H^\\dagger$ be a smooth extension to $\\rr\\times\\plane$ and ${\\mathscr{U}}_\\epsilon(\\disc) = \\{z\\in \\plane~|~|z|<1+\\epsilon\\}$ and\nlet  $\\alpha \\colon\\plane \\rightarrow \\rr$ \nbe a smooth bump function satisfying $\\alpha|_{\\disc}=1$, $\\alpha = 0$ on $\\plane\\setminus{\\mathscr{U}}_\\epsilon(\\disc)$.\nTake $\\epsilon\\in (0,1\/2)$ and define $\\widetilde H = \\alpha H^\\dagger$ with $\\widetilde H \\in {\\mathcal{H}}(\\plane)$.\nThe associated Hamiltonian isotopy is denoted by $\\psi_{t,\\widetilde H}$ and $\\widetilde F = \\psi_{1,\\widetilde H}\n\\in \\Ham(\\plane)$. Moreover, $\\psi_{t,\\widetilde H}$\nequals the identity on ${\\plane\\setminus{\\mathscr{U}}_\\epsilon(\\disc)}$, i.e. $\\psi_{t,\\widetilde H}$ is supported in ${\\mathscr{U}}_\\epsilon(\\disc)$,\nand $\\widetilde F|_{\\disc} = F$.\n\nFix $\\ell\\ge 3$ and choose $k>0$\n sufficiently large  such that\nthe symplectomorphisms \n\\begin{equation}\n G_{i}=\\psi_{i\/k,\\widetilde H} \\circ \\psi_{(i-1)\/k,\\widetilde H}^{-1}, \\ i \\in \\{1, \\dots,k \\}\n\\end{equation}\nare elements of ${\\mathscr{W}}$. Each $G_{i}$ restricted to $\\disc$ can be decomposed as follows:\n\\begin{equation}\n G_{i}|_\\disc = \\bigl(G_{i}|_\\disc \\circ \\Psi\\bigr) \\circ \\underbrace{\\Psi'\\circ\\cdots\\circ\\Psi'}_{\\kappa(\\ell-1)},\\quad \\ell\\ge3,~\\kappa\\in \\nn,\n\\end{equation}\nwhere $\\Psi$ and $\\Psi'$ are obtained from Lemma \\ref{AlternatePsi} by choosing rotation angles $2\\pi\/\\ell$ and $2\\pi\/\\kappa\\ell$ respectively. \nObserve that $\\Psi \\circ \\Psi'^{\\kappa(\\ell-1)}|_{\\disc} = \\id$.\nFrom $\\widetilde F$ we define the mapping $\\widehat F\\in \\Symp(\\plane)$:\n\\begin{equation} \n  \\widehat{F}=(G_{k} \\circ \\Psi) \\circ  \\underbrace{\\Psi' \\circ \\dots \\circ \\Psi'}_{\\kappa(\\ell-1)}\\circ\\cdots\\circ (G_{1} \\circ \\Psi) \\circ  \\underbrace{\\Psi'\\circ\\cdots\\circ \\Psi'}_{\\kappa(\\ell-1)}.\n\\end{equation}\nBy construction we have $\\widehat F|_\\disc = F$.\nLet $\\ell_\\kappa= \\kappa(\\ell-1) +1$ and  $d = \\ell_\\kappa k$ and put \n\\begin{equation}\n  \\widehat F_{j} = \\begin{cases} G_{j\/\\ell_\\kappa} \\circ \\Psi &\\mbox{for }~~ j \\in \\{\\ell_\\kappa, 2\\ell_\\kappa,\\cdots, d\\}\n    \\\\ \\Psi' &\\mbox{for } j \\in \\{1, \\dots, d \\} \\backslash \\{\\ell_\\kappa, 2\\ell_\\kappa, \\dots, d \\}. \\end{cases}\n\\end{equation}  \nwith $\\widehat F_j \\in SV(\\plane)$ for $j \\in \\{1, \\dots, d \\}$ and $F_j  = \\widehat F_j|_{\\disc}$.\nUsing the latter we obtain a decomposition of $F$ as  given in \\eqref{eqn:decomp12},\nand with the additional property that the mappings $F_j$ extend to twist symplectomorphisms of the $\\plane$,\nwhich proves \\eqref{eqn:decomp14}.\n\nEach symplectomorphism $\\widehat F_j$ can be connected to identity by a Hamiltonian path.\nLet $H^j$ be the Hamiltonian given by Proposition  \\ref{MoserThm}, which connects\n$\\widehat F_j$ to the identity via the Moser isotopy $\\psi_{s,H^j}$, $s\\in [0,1]$.\nLet $t_{j}=j\/d$ for all $j \\in \\{0, \\dots, d \\}$ and define \n\\begin{equation}\n\\label{eqn:theisotopy}\n\\phi_t = \\psi_{s^j(t),H^j} \\circ \\widehat F_{j-1} \\circ\\cdots\\circ \\widehat F_0,\\quad t\\in [t_{j-1},t_{j}], ~~j\\in \\{1,\\cdots,d\\},\n\\end{equation}\nwith $s^j(t) = d(t-t_j)$ and $\\widehat F_0 = \\id$.\nObserve that, by construction, $\\phi_{t_j}\\circ \\phi_{t_{j-1}}^{-1} = \\widehat F_j$, for all\n$j=1,\\cdots,d$ and (i) - (iv) is satisfied.\nCondition (v) follows from (iv) and from the fact that each $\\widehat F_j$ leaves the disc $\\disc$ invariant.\n\n   All the symplectomorphisms in the decomposition are supported in the disc ${\\mathscr{U}}_\\epsilon(\\disc)$, hence Conditions (ii) and (iii) of  Lemma\n   \\ref{AlternatePsi} \n   imply Properties (vi) and (vii).\n\\end{proof}\n\n\\begin{rem}\n\\label{rem:extendedMI}\nThe chained Moser isotopies in Proposition \\ref{MoserThm} can be extended with two more parameters  $r\\ge 0$ and $\\rho\\ge 0$.\nConsider the decoposition\n\\begin{equation}\n\\label{eqn:decomp16}\n\\widehat F = \\widehat F_d\\circ \\cdots \\circ \\widehat F_1 \\circ \\underbrace{\\Psi^{\\ell_r}_r\\circ\\cdots\\circ\\Psi^{\\ell_r}_r}_r \\circ\n\\underbrace{\\Upsilon^{\\ell_\\rho}_\\rho\\circ\\cdots\\circ\\Upsilon^{\\ell_\\rho}_\\rho}_\\rho, \n\\end{equation}\nwhere $\\Psi_r^{\\ell_r}|_\\disc = \\id$ and $\\ell_r\\ge 3$, and $\\Upsilon_\\rho^{\\ell_\\rho}|_\\disc = \\id$ and $\\ell_\\rho\\ge 3$.\nWe can again define an Moser isotopy as in \\eqref{eqn:theisotopy} with $d$ replaced by $d+r\\ell_r+\\rho\\ell_\\rho$. \nThe isotopy is again called a chained  Moser isotopy and denoted by $\\phi_t$, and the extended period will again be denoted by $d$.\nThe strands $\\phi_t(z_\\pm)$ link with the cylinder $[0,1]\\times\\disc$ and with each other with  linking number $2\\rho$.\n\\end{rem}\n\n\n\n\n\n\n\n\\subsection{The discrete action functional}\\label{subsec:genfun}\n\nLet $F \\in \\Symp(\\disc)$ be the given  symplectomorphism of the 2-disc and let $\\{ \\phi_{t} \\}_{t \\in \\rr}$ and $\\{t_{j} \\}_{j \\in \\{ 0, \\dots, d \\} }$ \nbe the associated continuous isotopy and   sequence of discretization times as given in Proposition~\\ref{Interpolation2}\nfor the extension $\\widehat F$.\nThe isotopy is extended periodically, that is $ \\phi_{t+s} = \\phi_{t} \\circ \\phi_{1}^{s} $ and $t_{j+s d} = s + t_{j}$ for all $s \\in \\mathbb{Z}$.\nThe decomposition of $\\widehat F$ given by Proposition \\ref{Interpolation2} yields a periodic sequence of positive twist symplectomorphisms $\\{\\widehat F_j\\}$, with $\\widehat F_j =\n\\phi_{t_{j+1}} \\circ \\phi_{t_{j}}^{-1} \\in SV(\\plane)$ and $\\widehat F_{j+d} = \\widehat F_j$.\n\n\n\\begin{defn}\nA sequence $\\{ (x_{j},y_{j}) \\}_{j \\in \\mathbb{Z}}$ is a \\textit{full orbit} for the system $\\{ \\widehat F_j\\}$ if\n\\[\n(x_{j+1},y_{j+1}) =   \\widehat F_j (x_{j},y_{j}), \\quad j\\in \\zz.\n\\]\nIf $(x_{j+n},y_{j+d}) = (x_j,y_j)$ for all $j$, then $\\{ (x_{j},y_{j}) \\}_{j \\in \\mathbb{Z}}$ is called an \\emph{$d$-periodic sequence}\nfor the system $\\{ \\widehat F_j\\}$.\n\\end{defn}\n\n\n\n  For every twist symplectomorphism $\\widehat F_j  \\in SV(\\plane)$ \nwe assign a generating function\n  ${\\bm{h}}_{j}={\\bm{h}}_{j}(x_{j}, x_{j+1})$ on the $x$-coordinates, which implies that\n  $y_{j}= -\\partial_1 {\\bm{h}}_j$\n  and $y_{j+1}=\\partial_2 {\\bm{h}}_j$.\n  From the twist property it follows that\n  \\begin{equation}\\label{hmonotonicity}\n    \\partial_{1} \\partial_{2} {\\bm{h}}_{j} < 0,\\quad \\forall j\\in \\zz.\n  \\end{equation}\n  Note that the sequence $\\{{\\bm{h}}_{j} \\}$ is $d$-periodic.  \n \n  Define the \\textit{action functional} $W_{d}:\\rr^{\\mathbb{Z}\/d\\mathbb{Z}} \\rightarrow \\rr$ by\n  \\begin{equation}\n  \\label{eqn:action1}\n    W_{d}\\bigl(\\{x_j\\}\\bigr):= \\sum\\limits_{j=0}^{d-1} {\\bm{h}}_{j} (x_j, x_{j+1}).\n  \\end{equation}\nA sequence $\\{x_j\\}$ is a critical point of $W_d$ if and only if\n\\begin{equation}\n\\label{eqn:parabrec}\n \\mathcal{R}_{j}(x_{j-1}, x_{j}, x_{j+1}) :=    -\\partial_{2} {\\bm{h}}_{j-1} \\left(x_{j-1}, x_{j}\\right) - \\partial_{1} {\\bm{h}}_j \\left(x_{j}, x_{j+1}\\right)=0,\n\\end{equation}\n    for all $j \\in \\mathbb{Z}$.\n    The $y$-coordinates satisfy $y_{j}=\\partial_{2} {\\bm{h}}_{j-1}\\left(x_{j-1},x_{j}\\right)$.\n \n  Periodicity and exactness of $\\mathcal{R}_j$ is immediate. The monotonicity follows directly from inequality~\\eqref{hmonotonicity}.\nA periodic point $z$, i.e. $F^d(z) = z$, is equivalent to the periodic sequence $\\{ (x_{j},y_{j}) \\}_{j \\in \\mathbb{Z}}$, with $z=(x_0,y_0)\\in \\disc$.\nSince $z=(x_0,y_0)\\in \\disc$, the invariance of $\\disc$ under $F$ implies that $(x_{j},y_{j})\\in \\disc$ for all $j$.\nThe above considerations yield the following variational principle.\n\\begin{prop}\n\\label{prop:varprin1}\nA $d$-periodic sequence $\\{ (x_{j},y_{j}) \\}_{j \\in \\mathbb{Z}}$ is an $d$-periodic orbit for the system $\\{ \\widehat F_j\\}$ if and only if\nthe sequence of $x$-coordinates $\\{ x_{j} \\}$ is a critical point of $W_d$. \n\\end{prop}\n\nThe idea of periodic sequences can be generalized to periodic configurations.\nLet $\\{B_j\\}_{j\\in \\zz}$, $B_j = \\{(x_j^\\mu,y_j^\\mu)~|~\\mu=1,\\cdots,m\\}\\in \\C_m(\\plane)$ and $B_{j+d} = B_j$ for all $j$. \nSuch a sequence $\\{B_j\\}$ is a $d$-periodic sequence for $\\{\\widehat F_j\\}$ if $\\widehat F_j(B_j) = B_{j+1}$ for all $j\\in \\zz$.\n\nFor a $d$-periodic sequence $\\{B_j\\}$, the $x$-projection yields a discretized braid ${\\bm{b}} = \\{{\\bm{b}}^\\mu\\} = \\{x_j^\\mu\\}$, cf.\\ Definition \\ref{PL}.\nThe above action functional can be extended to the space of discretized braids $\\Conf_m^d$:\n\\begin{equation}\n\\label{eqn:action2}\nW_d({\\bm{b}}) := \\sum_{\\mu=1}^m W_d({\\bm{b}}^\\mu), \n\\end{equation}\nwhere $W_d({\\bm{b}}^\\mu)$ is given by \\eqref{eqn:action1}. This yields the following extension of the variational principle.\n\\begin{prop}\n\\label{prop:varprin2}\nA $d$-periodic sequence $\\{ B_j \\}_{j \\in \\mathbb{Z}}$, $B_j\\in \\C_m(\\plane)$, is a $d$-periodic sequence of configurations for the system $\\{ \\widehat F_j\\}$ if and only if\nthe sequence of $x$-coordinates ${\\bm{b}}=\\{ x^\\mu_{j} \\}$ is a critical point of $W_d$ on $\\Conf_m^d$. \n\\end{prop}\n\nA discretized braid ${\\bm{b}}$ that is stationary for $W_d$ if it satisfies the parabolic recurrence relations in \\eqref{eqn:parabrec} for all $\\mu$\nand   the periodicity condition in Definition \\ref{PL}(b).\nIn Section \\ref{sec:braiding} we show that $d$-periodic sequences of configurations $\\{B_j\\}$ for the system $\\{ \\widehat F_j\\}$\nyields  geometric braids.\n\n\n\n\n\n\n\\section{Braiding of periodic points}\n\\label{sec:braiding}\nFor  symplectomorphisms $F\\in \\Symp(\\disc)$, with a finite invariant set $B\\subset \\inter \\disc$, the mapping class\ncan be identified via a chained Moser isotopy.\n\n\\begin{prop}\n\\label{prop:traceout1}\nLet $B\\subset\\inter\\disc$, with $\\# B=m$,  be a finite invariant set for $F\\in \\Symp(\\disc)$ and let\n$\\phi_t$ be a chained Moser isotopy given in Proposition \\ref{Interpolation2}. Then, ${\\bm{\\beta}}(t) = \\phi_t(B)$ represents a   geometric  braid based at $B\\in \\C_m\\disc$     with only positive crossings and $\\beta = \\imath_B\\bigl([{\\bm{\\beta}}]_B\\bigr)$ is a\npositive word  in\nthe braid monoid ${\\mathscr{B}}_m^+$.\nThe $x$-projection ${\\bm{b}}(t) = \\pi_x{\\bm{\\beta}}(t)$ on the $(t,x)$-plane is a (continuous)   piecewise linear braid diagram.\n\\end{prop}\n\n Proposition \\ref{prop:MCG12} implies that the associated positive braid word $\\beta\\in \\mathcal{B}_m$, derived from the braid diagram $\\pi_x\\phi_t(B)$ determines the mapping class of $F$\nrelative to $B$.\nIf the based path ${\\bm{\\beta}}(t)=\\phi_t(B)$ is regarded as a \\emph{free loop}  $\\sbb^1 \\to \\C_m\\disc$, i.e. discarding the base point, then ${\\bm{\\beta}}$ is referred to as a \\emph{closed} geometric braid in $\\disc$.\n\\begin{defn}\n\\label{defn:acylindrical}\nLet ${\\bm{\\beta}}$ be a geometric braid in $\\disc$. A component of ${\\bm{\\beta}}'\\subset {\\bm{\\beta}}$ is called \\emph{cylindrical in} ${\\bm{\\beta}}$ if ${\\bm{\\beta}}'$ can be deformed onto $\\partial \\disc$ as a closed geometric braid. Otherwise ${\\bm{\\beta}}'$ is called \\emph{acylindrical}. A union of components ${\\bm{\\beta}}'$ is called cylindrical\/acylindrical in ${\\bm{\\beta}}$ if all members are.\n\\end{defn}\n\n\\begin{rem}\n\\label{rmk:acylindrical}\nA  positive conjugacy class $\\llbracket \\gamma,\\aset\\rrbracket$ is associated with braid classes in $[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$ \nin $\\Conf_{n,m}^{d+q}$, cf.\\ Section \\ref{subsec:algpres}. If for a representative ${\\bm{a}}\\rel{\\bm{b}}$ it holds that $\\Bd({\\bm{a}})$ is cylindrical\/acylindrical\nin ${\\bm{\\gamma}} =\\Bd({\\bm{a}})\\rel\\Bd({\\bm{b}})$,\nthen $\\llbracket \\gamma,\\aset\\rrbracket$ is said to  cylindrical\/acylindrical, cf.\\  Definition \\ref{defn:proper2}.\n\\end{rem}\n\n\nLet $z,z'\\in \\plane$ be distinct points with the property that $\\widehat F^n(z) = z$ and $\\widehat F^n(z') = z'$, for some $n\\ge 1$, and where $\\widehat F = \\phi_1$ and\n$\\phi_t$ a chained Moser isotopy constructed in Proposition \\ref{Interpolation2}.\nDefine the continuous functions $z(t) = \\phi_t(z)$ and $z'(t) = \\phi_t(z')$ and let $x(t)$ and $x'(t)$ the $x$-projection of $z(t)$ and $z'(t)$ respectively.\nBy Proposition \\ref{Interpolation2}, $x(t)$ and $x'(t)$ are (continuous) piecewise linear functions that are uniquely determined by the sequence\n$\\{t_j\\}_{j=0}^{nd}$, $t_j = j\/d$.\n\n\\begin{lem}[cf.\\ \\cite{BraidConleyIndex}]\n\\label{lem:propergraphs}\nThe two $x$-projections $x(t)$ and $x'(t)$ form a (piecewise linear) braid diagram, i.e.\\ no tangencies.\nThe intersection number $\\iota\\bigl(x(t),x'(t)\\bigr)$, given as the total number of intersections of the graphs of $x(t)$ and $x'(t)$\non the interval $t\\in [0,n]$,\n is well-defined and even.\n\\end{lem}\n\n\\begin{proof}\nLet $x_j = x(t_j)$ and $x_j' = x'(t_j)$, $j=0,\\cdots,nd$ and by the theory in  Section \\ref{subsec:genfun} the sequences satisfy the parabolic recurrence relations\n$\\mathcal{R}_{j}(x_{j-1}, x_{j}, x_{j+1}) =0$ and $\\mathcal{R}_{j}(x'_{j-1}, x'_{j}, x'_{j+1}) =0$.\nSuppose the sequences $\\{x_j\\}$ and $\\{x'_j\\}$ have a tangency at $x_j = x_j'$ (but are not identically equal). Then, either $x'_{j-1}<x_{j-1}$ and $x'_{j+1} \\le x_{j+1}$, or\n$x'_{j-1}\\le x_{j-1}$ and $x'_{j+1} < x_{j+1}$, and similar with the role of $\\{x_j\\}$ and $\\{x'_j\\}$ reversed.\nSince functions $\\mathcal {R}_j$ are strictly increasing in the first and third variables and since $x_j = x'_j$ both evaluations of $\\mathcal{R}_j$ cannot be zero simultaneously,\nwhich contradicts the existence of tangencies. All intersections of $x(t)$ and $x'(t)$ are therefore transverse in the sense of Definition \\ref{PL}(c) and thus \n$\\iota\\bigl(x(t),x'(t)\\bigr)$ is well-defined and even.\n\\end{proof}\n\n\nThe curves $z(t)$ and $z'(t)$ may be regarded as 3-dimensional (continuous) curves $z,z'\\colon \\rr\/\\zz \\to \\rr^3$, $t\\mapsto (t,z(t))$ and $t\\mapsto (t,z'(t))$.\nDue to the special properties of the chained Moser isotopy $\\phi_t$ we have:\n\\begin{lem}\n\\label{lem:linking1}\nThe graphs $t\\mapsto (t,z(t))$ and $t\\mapsto (t,z'(t))$ form a positive 2-strand braid.  Intersections in the $x,x'$-braid diagram correspond to positive crossings in the $z,z'$-braid. The linking number is given by\n${\\rm Link}\\bigl(z(t),z'(t)\\bigr) =\\frac{1}{2} \\iota\\bigl(x(t),x'(t)\\bigr)$.\n\\end{lem}\n\n\\begin{proof}\nBy Lemma \\ref{lem:propergraphs} the projection graphs $t\\mapsto x(t)$ and $t\\mapsto x'(t)$ form a braid diagram.\nIn order to show that the graphs $t\\mapsto (t,z(t))$ and $t\\mapsto (t,z'(t))$ form a positive 2-strand braid we examine the intersections in\nthe $x,x'$-braid diagram.\n\n(i) Consider an intersection on the interval $[t_j,t_{j+1}]\\subset [0,n]$\nfor which $x_j<x'_j$ and $x_{j+1}>x'_{j+1}$.\nLet $\\tau\\in [t_j,t_{j+1}]$ be the intersection point and $x(\\tau) = x'(\\tau) = x_*$.\nAfter rescaling and shifting to the interval $[0,1]$ we have\n$x(s(t)) = x_j + (x_{j+1}-x_j)s(t)$, $s(t) = d(t-t_j) \\in [0,1]$ and the same for $x'(s(t))$.\nRecall that $\\phi_t$ is given by  \n\\eqref{eqn:theisotopy} and therefore by \\eqref{Ltransform2}, \n\\[\ny(s(\\tau)) = \\partial_p L^j\\bigl(s(\\tau),x_*,x_{j+1}-x_j), \\quad y'(s(\\tau)) = \\partial_pL^j\\bigl(s(\\tau),x_*,x'_{j+1}-x'_j),\n\\]\nwhere $L^j$ are the Lagrangians for the Moser isotopies $\\psi_{t,H^j}$ in Proposition \\ref{MoserThm}.\nSince $\\partial^2_{pp}L^j\\ge \\delta>0$ and $x_{j+1}-x_j > x'_{j+1}-x'_j$, we conclude that $y(s(\\tau)) > y'(s(\\tau))$.\nBy reversing the role of $x$ and $x'$, i.e.\\ $x_j>x'_j$ and $x_{j+1}<x'_{j+1}$, we obtain $y(s(\\tau))<y'(s(\\tau))$,\nwhich shows that an intersection in the $x,x'$-diagram\ncorresponds to a positive crossing in the $z,z'$-braid.\n\n(ii) Consider an intersection at $x_j$, for which   $x_{j-1}<x'_{j-1}$, $x_j = x'_j = x_*$ and $x_{j+1}>x'_{j+1}$.\nAs in the previous case\n\\[\n\\begin{aligned}\ny(s(\\tau)) &= \\partial_pL^{j-1} \\bigl(1,x_*,x_*-x_{j-1}) = \\partial_pL^j\\bigl(0,x_*,x_{j+1}-x_*);\\\\\ny'(s(\\tau)) &= \\partial_pL^{j-1} \\bigl(1,x_*,x_*-x'_{j-1}) = \\partial_pL^j\\bigl(0,x_*,x'_{j+1}-x_*),\n\\end{aligned}\n\\]\nand since $x_*-x_{j-1}>x_*-x'_{j-1}$ (and $x_{j+1}-x_* > x'_{j+1}-x_*$) we conclude that $y(s(\\tau)) >y'(s(\\tau))$. Reversing the role of $x$ and $x'$ yields\n$y(s(\\tau))<y'(s(\\tau))$. In this case we also conclude that an intersection in the $x,x'$-diagram\ncorresponds to a positive crossing in the $z,z'$-braid, which concludes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:traceout1}]\nSince $\\phi_t$ is an isotopy and $F(B) = \\phi_1(B) = B$, the path $t\\mapsto \\phi_t(B)$ in $\\C_m\\disc$ represents a geometric braid ${\\bm{\\beta}}(t)$.\nBy Lemma \\ref{lem:linking1} all crossings in ${\\bm{\\beta}}(t)$ are positive. Indeed, if we consider the $m$-fold cover $\\widehat {\\bm{\\beta}}(t)$ of ${\\bm{\\beta}}$, i.e.\\\n$t\\mapsto \\phi_t(B)$, $t\\in [0,m]$, then all pairs of strands satisfy the hypotheses of Lemma \\ref{lem:linking1}.\nConsequently, the presentation of $\\beta\\in {\\mathscr{B}}_m$ of ${\\bm{\\beta}}$ is unique and consists of only positive letters.\n\\end{proof}\n\nOur decomposition automatically selects a braid word in ${\\mathscr{B}}_m^+$ which is positive and which may be represented as a positive\npiecewise linear braid diagram.\nThis allows us to use the theory of parabolic recurrence relations for finding additional periodic points for $F$. \nIn the following lemma $\\widehat F$ is an extension of $F$ to $\\plane$ given by Proposition \\ref{Interpolation2}.\n\n\\begin{lem}\n\\label{prop:reallyindisc}\nLet $A,B\\subset \\plane$ be finite, disjoint sets  and   $B \\subset\\inter\\disc$. \nLet $F\\in \\Symp(\\disc)$ with $F(B) = B$ and let \n  $\\phi_t$ be a chained Moser isotopy given by Proposition \\ref{Interpolation2} with $\\phi_1 = \\widehat F$. Suppose  \n$A$ is an invariant set for $\\widehat F$ and ${\\bm{\\alpha}}(t) = \\phi_t(A)$ is acylindrical in  ${\\bm{\\alpha}}\\rel{\\bm{\\beta}}$.\n Then, $A$ is an invariant set for $F$ with $A\\subset \\inter\\disc$.\n\\end{lem}\n\n\\begin{proof}\nThe set $A = {\\bm{\\alpha}}(0)$ is an invariant set for $\\widehat F$. Assume without loss of generality that ${\\bm{\\alpha}}(t)$ is a single component braid.\nLet ${\\bm{\\alpha}}^\\dagger(t)$ be the $n$-fold cover of ${\\bm{\\alpha}}$, with $t\\in [0,n]$. If ${\\bm{\\alpha}}^\\dagger(t_j) \\in \\plane \\setminus \\inter\\disc$, $t_j=j\/d$, for some $j$, then\n${\\bm{\\alpha}}^\\dagger(t_j) \\in \\plane \\setminus \\inter\\disc$ for all $j$, since the set $\\phi_t(\\disc)$ separates the points inside and outside $\\partial\\disc$ under the isotopy $\\phi_t$. Therefore, ${\\bm{\\alpha}}^\\dagger(t) \\in \\plane\\setminus\\inter \\phi_t(\\disc)$\nfor all $t\\in [0,n]$ and\nthus ${\\bm{\\alpha}}(t) \\in  \\plane\\setminus \\inter \\phi_t(\\disc)$ for all $t\\in [0,1]$.\nBy assumption ${\\bm{\\beta}}(t) \\in   \\phi_t(\\disc)$ for all $t\\in [0,1]$\nand therefore  that ${\\bm{\\alpha}}$ can be contracted onto $\\partial \\phi_t(\\disc)$, \nwhich contradicts the assumption that ${\\bm{\\alpha}} \\rel{\\bm{\\beta}}$ is acylindrical.\nConsequently, ${\\bm{\\alpha}}(t) \\in   \\phi_t(\\disc)$ for all $t\\in [0,1]$.\nThe latter implies $A\\subset \\inter\\disc$, which completes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{The main theorem}\n\\label{sec:main1}\n\n\n\n\nGiven $F\\in \\Symp(\\disc)$ and  a finite invariant set $B\\subset \\inter \\disc$,\nthen \n\\begin{equation}\n\\label{eqn:detMP}\n\\jmath_B([F]) = \\beta\\!\\! \\!\\!\\mod\\square \\in {\\mathscr{B}}_m\/Z({\\mathscr{B}}_m).\n\\end{equation}\nThe braid word $\\beta$ can chosen positive by adding full twists and can be used to force additional invariant sets as quantified by the  following result.\n\n\n\n\\begin{thm}\n\\label{thm:main1}\nLet $B\\subset \\inter\\disc$ be a finite invariant set for a symplectomorphism $F\\colon \\disc\\to \\disc$\nand  let $\\beta\\in\n{\\mathscr{B}}_m^+$  be  a positive braid word representing the mapping class of $F$ in $\\Mod(\\disc\\rel B)$.\nConsider a coloring $\\aset\\subset \\{1,\\cdots,n+m\\}$ of length $n$ and a 2-colored braid $(\\gamma,\\aset) \\in \\pi_\\aset^{-1}(\\beta) \\subset {\\mathscr{B}}^+_{n,m,\\aset}$\nand assume that $\\llbracket\\gamma,\\aset\\rrbracket$ is proper and acylindrical,\\footnote{See Definition \\ref{defn:proper2} and Remark \\ref{rmk:acylindrical}.}\nand ${\\mathrm{H}}\\llbracket\\gamma,\\aset\\rrbracket\\not = 0$.\nThen,\n\\begin{enumerate}\n\\item [(i)] there exists a finite invariant set $A\\subset \\inter\\disc$ of $F$\nsuch that the mapping class of $F$ satisfies \n$\\jmath_{A\\cup B}([F]) = \\gamma\\!\\!\\mod \\square$;\n\\item [(ii)] the number of distinct invariants sets $A$ in (i) is bounded from below by $|P_t\\llbracket\\gamma,\\aset\\rrbracket|$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nRecall from Section \\ref{subsec:algpres} that \n$\\llbracket\\gamma,\\aset\\rrbracket$ determines a 2-colored discretized braid class $[{\\bm{a}}\\rel{\\bm{b}}]_{\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim}}$ via\nthe mapping $\\gamma \\mapsto \\ev_0(\\gamma) \\in \\Conf_{n+m}^{d_0}$, with $d_0$ the number of generators in $\\gamma$. \nThe representative ${\\bm{a}} \\rel {\\bm{b}}\\in \\Conf_{n,m}^{d_0}$ is obtained from the coloring $\\aset$.\n\nConsider a chained Moser isotopy $\\phi_t$ with $r=\\rho=0$, then \n\\[\n\\imath_B([\\phi_t(B)])\\!\\!\\!\\! \\mod \\square= \\beta,\n\\]\nwhere $\\ell\\ge 3$ and $k$ in Proposition \\ref{Interpolation2} are fixed. \nLet ${\\bm{b}}_B := \\bigl\\{ \\pi_x \\phi_{t_j}(B)\\bigr\\}$, $j=0,\\cdots,d$ and \nchoose $\\kappa$ large enough such that $[{\\bm{b}}_B]$ defines a free discrete braid class.\nThe following\n three cases can be distinguished:\n\\[ \n{\\bm{b}} \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} {\\bm{b}}_B,\\quad {\\bm{b}}\\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} \\T^{2\\lambda} {\\bm{b}}_B,\\quad\\hbox{and}\\quad \\T^{2\\lambda} {\\bm{b}} \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} {\\bm{b}}_B,\n\\] \nfor some $\\lambda\\ge 1$. The choices of $\\ell$, $k$ and $\\kappa$ determine $d$.\n\n\n {\\it Case I:} \n The integers $q\\ge 0$ and  $\\kappa$ are chosen large enough such that $\\E^q{\\bm{b}}^*\\sim {\\bm{b}}_B^*$ and $[\\E^q({\\bm{a}}\\rel{\\bm{b}}^*)]$ is free,\n where ${\\bm{b}}_B^* = {\\bm{b}}_B \\cup {\\bm{b}}_B^-\\cup {\\bm{b}}_B^+$, with ${\\bm{b}}_B^\\pm = \\{\\pm 2\\}$. Note that ${\\bm{b}}_B^* = \\bigl\\{ \\pi_x \\phi_{t_j}(B^*)\\bigr\\}$,\n with $B^* = B\\cup \\{-2,+2\\}$.\n Choose ${\\bm{a}}_B\\rel{\\bm{b}}_B^* \\sim \\E^q({\\bm{a}}\\rel{\\bm{b}}^*)$, cf.\\ Figure \\ref{fig:7C1},  then by the invariance of the discrete braid invariant we have that\n\\[\n{\\mathrm{H}}({\\bm{a}}_B\\rel{\\bm{b}}_B^*) = {\\mathrm{H}}\\bigl(\\E^q({\\bm{a}}\\rel{\\bm{b}}^*)\\bigr) = {\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}^*) \\not = 0,\n\\]\nwhich proves that $P_t({\\bm{a}}_B\\rel{\\bm{b}}_B^*) = P_t({\\bm{a}}\\rel{\\bm{b}}^*) =  P_t\\llbracket\\gamma,\\aset\\rrbracket\\not = 0$.\n\nFrom the Morse Theory in \\cite[Lemma 35]{BraidConleyIndex} we derive that the number of stationary \ndiscrete braid diagrams ${\\bm{a}}_B\\rel{\\bm{b}}_B$ for the action $W_d$ in \\eqref{eqn:action1} induced by the Moser isotopy $\\phi_t$ \nis bounded below by $|P_t\\llbracket\\gamma,\\aset\\rrbracket|$.\nThe associated piecewise linear braid diagrams $\\Bd({\\bm{a}}_B\\rel{\\bm{b}}_B)$\nlift to 2-colored braids ${\\bm{\\alpha}}_B(t)\\rel{\\bm{\\beta}}_B(t)$ by\nconstructing the $y$-component via \\eqref{eqn:MoserIso}.\nThe stationary braids ${\\bm{\\alpha}}_B(t)\\rel{\\bm{\\beta}}_B(t)$ are in fact braids in $\\disc$ by Proposition \\ref{prop:reallyindisc} and\n$A={\\bm{\\alpha}}_B(0) \\subset \\inter\\disc$ is an invariant set for $F$. Moreover, ${\\bm{\\alpha}}_B(t)\\rel{\\bm{\\beta}}_B(t)$\ndetermines the mapping class of $F$, which completes the proof in Case I.\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[xscale=0.23, yscale=0.23, line width = 1.5pt]\n\\foreach \\x in {-2,8,16}\n \\draw[line width=1.0pt, -] (\\x,-3) -- (\\x,7); \n\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (16.0,-3.5) -- (8.1,-3.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $q$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (7.9,-3.5) -- (-2.,-3.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $d_0$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (-2,7.5) -- (16.,7.5) node [black,midway,yshift=0.5cm] {\\footnotesize $\\E^q({\\bm{a}}\\rel{\\bm{b}}^*)$};\n   \n   \\draw[-] (-2,2) -- (0,0) -- (2,0) -- (4,2) -- (6,4) -- (8,4) -- (10,4) -- (12,4) -- (14,4) -- (16,4);            \n   \\draw[fill] (-2,2) circle (0.07)\n               (0,0) circle (0.07)\n               (2,0) circle (0.07)\n               (4,2) circle (0.07)\n               (6,4) circle (0.07);\n\n\\foreach \\x in {8,10,12,14,16}\n   \\draw[fill] (\\x,4) circle (0.07);\n\n\n   \\draw[-] (-2,4) -- (0,4) -- (2,2) -- (4,0) -- (6,0) -- (8,2) -- (10,2) -- (12,2) -- (14,2) -- (16,2);            \n   \\draw[fill] (0,4) circle (0.07)\n               (2,2) circle (0.07)\n               (4,0) circle (0.07)\n               (6,0) circle (0.07)\n               (-2,4) circle (0.07);\n\n\\foreach \\x in {8,10,12,14,16}\n   \\draw[fill] (\\x,2) circle (0.07);\n\n\n   \\draw[-,color=red] (-2,0) -- (0,2) -- (2,4) -- (4,4) -- (6,2) -- (8,0) -- (10,0) -- (12,0) -- (14,0) -- (16,0);            \n   \\draw[fill, color=red]  (0,2) circle (0.07)\n                           (2,4) circle (0.07)\n                           (4,4) circle (0.07)\n                           (6,2) circle (0.07)\n                           (-2,0) circle (0.07);\n\n\\foreach \\x in {8,10,12,14,16}\n   \\draw[fill, color=red] (\\x,0) circle (0.07);\n\n\\foreach \\y in {-2,6}\n \\draw[-] (-2,\\y) -- (16,\\y);\n\n\\foreach \\x in {-1,...,8}\n   \\draw[fill] (2*\\x,-2) circle (0.07);\n\n\\foreach \\x in {-1,...,8}\n   \\draw[fill] (2*\\x,6) circle (0.07);\n\n\\end{tikzpicture}\n\\raisebox{55pt}{\n\\begin{minipage}{0.05\\linewidth}\n  \\centering \\Large $\\sim$\n\\end{minipage}}\n\\begin{tikzpicture}[xscale=0.23, yscale=0.23, line width = 1.5pt]\n\n\\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (16,-3.5) -- (-2.,-3.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $d$};\n\n\\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (-2,7.5) -- (16.,7.5) node [black,midway,yshift=0.5cm] {\\footnotesize ${\\bm{a}}_B\\rel{\\bm{b}}_B^*$};\n\n\\foreach \\x in {-2,16}\n \\draw[line width=1.0pt, -] (\\x,-3) -- (\\x,7); \n\n\\draw[-] (-2,2) -- (0,2.5) -- (2,2.9) -- (4,2.6) -- (6,2.7) -- (8,4) -- (10,4.7) -- (12,4.5) -- (14,3.8) -- (16,4);            \n\\foreach \\point in {(-2,2), (0,2.5), (2,2.9), (4,2.6), (6,2.7), (8,4), (10,4.7), (12,4.5), (14,3.8), (16,4)}\n   \\draw[fill] \\point circle (0.07);\n  \n\\draw[-] (-2,4) -- (0,4.2) -- (2,4) -- (4,3.7) -- (6,3.5) -- (8,3.4) -- (10,3) -- (12,2.8) -- (14,2.2) -- (16,2);            \n\\foreach \\point in {(-2,4), (0,4.2), (2,4), (4,3.7), (6,3.5), (8,3.4), (10,3), (12,2.8), (14,2.2), (16,2)}\n   \\draw[fill] \\point circle (0.07);\n\n\\draw[-,color=red] (-2,0) -- (0,0.5) -- (2,2) -- (4,4) -- (6,5) -- (8,5) -- (10,4) -- (12,2) -- (14,1) -- (16,0);            \n\\foreach \\point in {(-2,0), (0,0.5), (2,2), (4,4), (6,5), (8,5), (10,4), (12,2), (14,1), (16,0)}\n   \\draw[fill, color=red] \\point circle (0.07);\n\n\\foreach \\y in {-2,6}\n \\draw[-] (-2,\\y) -- (16,\\y);\n\n\\foreach \\x in {-1,...,8}\n   \\draw[fill] (2*\\x,-2) circle (0.07);\n\n\\foreach \\x in {-1,...,8}\n   \\draw[fill] (2*\\x,6) circle (0.07);\n\n\\end{tikzpicture}\n\\caption{Two representatives for Case I. Numbers of discretization points are linked by $d=d_0+q$.}\n\\label{fig:7C1}\n\\end{figure}\n\n\n {\\it Case II:}\n By Remark \\ref{rem:extendedMI} we\n use a(n) (extended) chained Moser isotopy,  denoted by $\\widetilde\\phi_t$, with $r=\\lambda$, $\\ell_r\\ge 3$ and $\\rho=0$\n and we denote the associated discrete braid by $\\widetilde {\\bm{b}}_B:= \\bigl\\{ \\pi_x \\widetilde\\phi_{t_j}(B)\\bigr\\}$.\n By construction $\\widetilde {\\bm{b}}_B \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} \\T^{2\\lambda}{\\bm{b}}_B \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} {\\bm{b}}$.\n  The integers $q\\ge 0$ and  $\\kappa$ are chosen large enough such that $\\E^{q+2\\lambda}{\\bm{b}}^* \\sim \\widetilde {\\bm{b}}_B^*$ and $[\\E^{2\\lambda+q}({\\bm{a}}\\rel{\\bm{b}}^*)]$ is free,  where $\\widetilde {\\bm{b}}_B^* = \\widetilde {\\bm{b}}_B \\cup {\\bm{b}}_B^-\\cup {\\bm{b}}_B^+ = \\bigl\\{ \\pi_x \\widetilde \\phi_{t_j}(B^*)\\bigr\\}$.\nAs in Case I we choose ${\\bm{a}}_B\\rel \\widetilde{\\bm{b}}_B^*\\sim \\E^{2\\lambda+q}({\\bm{a}}\\rel{\\bm{b}}^*)$, cf.\\ Figure \\ref{fig:7C2}, such that \n\\[\n{\\mathrm{H}}\\bigl({\\bm{a}}_B\\rel \\widetilde {\\bm{b}}_B^*\\bigr) = {\\mathrm{H}}\\bigl( \\E^{2\\lambda+q}({\\bm{a}}\\rel{\\bm{b}}^*)\\bigr) = {\\mathrm{H}}({\\bm{a}}\\rel{\\bm{b}}^*) \\not = 0.\n\\]\nThis proves the theorem in Case II.\n\\begin{figure}\n  \\centering\n\\begin{tikzpicture}[xscale=0.23, yscale=0.23, line width = 1.5pt]\n\n\\foreach \\x in {-4,10,18}\n \\draw[line width=1.0pt, -] (\\x,-3) -- (\\x,7); \n\n\\draw[-] (-4,4) -- (-2,2) -- (0,0) -- (2,0) -- (4,2) -- (6,4) -- (8,4) -- (10,2) -- (12,2) -- (14,2) -- (16,2) -- (18,2);            \n\\foreach \\point in {(-4,4), (-2,2), (0,0), (2,0), (4,2), (6,4), (8,4), (10,2), (12,2), (14,2), (16,2), (18,2)}\n   \\draw[fill] \\point circle (0.07);\n\n\\draw[-] (-4, 2) -- (-2,4) -- (0,4) -- (2,2) -- (4,0) -- (6,0) -- (8,2) -- (10,4) -- (12,4) -- (14,4) -- (16,4) -- (18,4);            \n\\foreach \\point in {(-4, 2), (-2,4), (0,4), (2,2), (4,0), (6,0), (8,2), (10,4), (12,4), (14,4), (16,4), (18,4)}\n   \\draw[fill] \\point circle (0.07);\n\n\\draw[-,color=red] (-4,0) -- (-2,0) -- (0,2) -- (2,4) -- (4,4) -- (6,2) -- (8,0) -- (10,0) -- (12,0) -- (14,0) -- (16,0) -- (18,0);            \n\\foreach \\point in {(-4,0), (-2,0), (0,2), (2,4), (4,4), (6,2), (8,0), (10,0), (12,0), (14,0), (16,0), (18,0)}\n   \\draw[fill, color=red] \\point circle (0.07);\n\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (18.0,-3.5) -- (10.1,-3.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $q$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (9.9,-3.5) -- (-4.,-3.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $d_0$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (-4,7.5) -- (18.,7.5) node [black,midway,yshift=0.5cm] {\\footnotesize $\\E^{2\\lambda+q}({\\bm{a}}\\rel{\\bm{b}}^*)$};\n   \n\n\\foreach \\y in {-2,6}\n \\draw[-] (-4,\\y) -- (18,\\y);\n\n\\foreach \\x in {-2,...,9}\n   \\draw[fill] (2*\\x,-2) circle (0.07);\n\n\\foreach \\x in {-2,...,9}\n   \\draw[fill] (2*\\x,6) circle (0.07);\n\\end{tikzpicture}\n\\raisebox{55pt}{\n\\begin{minipage}{0.05\\linewidth}\n  \\centering \\Large $\\sim$\n\\end{minipage}}\n\\begin{tikzpicture}[xscale=0.23, yscale=0.23, line width = 1.5pt]\n\n\\foreach \\x in {-4,10,18}\n \\draw[line width=1.0pt, -] (\\x,-3) -- (\\x,7); \n\n\\draw[-] (-4,2) -- (-2,1.6) -- (0,2.) -- (2,2.5) -- (4,3.5) -- (6,4.) -- (8,4.5) -- (10,4.3) -- (12,1.2) -- (14,-0.1) -- (16,1.0) -- (18,4);            \n\\foreach \\point in {(-4,2), (-2,1.6), (0,2.), (2,2.5), (4,3.5), (6,4.), (8,4.5), (10,4.3), (12,1.2), (14,-0.1), (16,1.0), (18,4)}\n   \\draw[fill] \\point circle (0.07);\n  \n\\draw[-] (-4,4) -- (-2,4) -- (0,3.9) -- (2,3.1) -- (4,2.2) -- (6,1.5) -- (8,0.5) -- (10,0) -- (12,3.2) -- (14,4.0) -- (16,2.7) -- (18,2);            \n\\foreach \\point in {(-4,4), (-2,4), (0,3.9), (2,3.1), (4,2.2), (6,1.5), (8,0.5), (10,0), (12,3.2), (14,4.0), (16,2.7), (18,2)}\n   \\draw[fill] \\point circle (0.07);\n\n\\draw[-,color=red] (-4,0) -- (-2,0.2) -- (0,0.9) -- (2,1.3) -- (4,2.4) -- (6,2.8) -- (8,3.4) -- (10,3.4) -- (12,4.1) -- (14,2.5) -- (16,1.8) -- (18,0);            \n\\foreach \\point in {(-4,0), (-2,0.2), (0,0.9), (2,1.3), (4,2.4), (6,2.8), (8,3.4), (10,3.4), (12,4.1), (14,2.5), (16,1.8), (18,0)}\n   \\draw[fill, color=red] \\point circle (0.07);\n\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (18.0,-3.5) -- (10.1,-3.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $r\\cdot \\ell_r$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (9.9,-3.5) -- (-4.,-3.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $d$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (-4,7.5) -- (18.,7.5) node [black,midway,yshift=0.5cm] {\\footnotesize ${\\bm{a}}_B\\rel \\widetilde {\\bm{b}}_B^{*}$};\n   \n\n\n\n\\foreach \\y in {-2,6}\n \\draw[-] (-4,\\y) -- (18,\\y);\n\n\\foreach \\x in {-2,...,9}\n   \\draw[fill] (2*\\x,-2) circle (0.07);\n\n\\foreach \\x in {-2,...,9}\n   \\draw[fill] (2*\\x,6) circle (0.07);\n\n\\end{tikzpicture}\n\\caption{Two representatives for Case II. Numbers of discretization points are linked by $ d+r \\ell_r = d_0+q$.}\n\\label{fig:7C2}\n\\end{figure}\n\n\n\n{\\it Case III:} \n By Remark \\ref{rem:extendedMI} we\nuse a(n) (extended) chained Moser isotopy, denoted by $\\widetilde\\phi_t$, with $r=0$ and $\\rho=1$ and $\\ell_\\rho = 2\\lambda+2\\ge 4$.\nWe denote the associated discrete braid by $\\widetilde {\\bm{b}}_B:= \\bigl\\{ \\pi_x \\widetilde\\phi_{t_j}(B)\\bigr\\}$.\n By construction $\\widetilde {\\bm{b}}_B \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} \\T^{2}{\\bm{b}}_B \\stackrel{\\mbox{\\tiny\\textnormal{\\raisebox{0ex}[0ex][0ex]{$+$}}}}{\\sim} \\T^{2\\lambda+2}{\\bm{b}}$.\nThe integers $q\\ge 0$ and  $\\kappa$ are chosen large enough such that $\\T^{2\\lambda+2}(\\E^q{\\bm{b}}^*)\\sim \\widetilde {\\bm{b}}_B^*$ and $[\\T^{2\\lambda+2}\\E^{q}({\\bm{a}}\\rel{\\bm{b}}^*)]$ is free, and where  ${\\bm{b}}_B^* = \\bigl\\{ \\pi_x \\widetilde \\phi_{t_j}(B^*)\\bigr\\}$.\nDefine $\\widetilde {\\bm{b}}_B^{*\\#} = \\bigl\\{ \\pi_x \\widetilde \\phi_{t_j}(B^{*\\#})\\bigr\\}$, with $B^{*\\#} = B\\cup \\{-4,-2,2,4\\}$, which is equivalent to\naugmentating $\\widetilde {\\bm{b}}_B^*$ with the strands ${\\bm{b}}_B^s = \\{(-1)^{j+1}4\\}_j$ and ${\\bm{b}}_B^n = \\{(-1)^j 4\\}_j$.\nChoose ${\\bm{a}}_B \\rel \\widetilde {\\bm{b}}_B^{*} \\sim \\T^{2\\lambda+2}\\E^q({\\bm{a}}\\rel{\\bm{b}}^*)$.\nFrom \\eqref{eqn:thedual} and the fact that $\\E^q({\\bm{a}}\\rel{\\bm{b}}) = \\E^q{\\bm{a}}\\rel\\E^q{\\bm{b}}$ and \n$\\T^{2\\lambda+2}\\bigl(\\E^q{\\bm{a}}\\rel (\\E^q{\\bm{b}}^*)^\\#\\bigr) \\sim {\\bm{a}}_B \\rel \\widetilde {\\bm{b}}_B^{*\\#}$, cf.\\ Figure \\ref{fig:7C3},\nwe derive that\n\\[\n\\begin{aligned}\n0\\not = {\\mathrm{H}}({\\bm{a}} &~\\rel{\\bm{b}}^*) \\wedge  \\sbb^{2n(\\lambda+1)} = \\\\\n& = {\\mathrm{H}}\\bigl(\\E^q({\\bm{a}}\\rel{\\bm{b}}^*)\\bigr) \\wedge \\sbb^{2n(\\lambda+1)} \n= {\\mathrm{H}}\\bigl(\\E^q{\\bm{a}}\\rel(\\E^q{\\bm{b}}^*)^\\#\\bigr) \\wedge \\sbb^{2n(\\lambda+1)}\\\\\n&= {\\mathrm{H}}\\bigl(\\T^{2\\lambda+2}\\bigl(\\E^q{\\bm{a}}\\rel (\\E^q{\\bm{b}}^*)^\\#\\bigr)\\bigr)  ={\\mathrm{H}}\\bigl({\\bm{a}}_B \\rel \\widetilde {\\bm{b}}_B^{*\\#} \\bigr),\n\\end{aligned}\n\\]\nwhich implies that ${\\mathrm{H}}\\bigl({\\bm{a}}_B \\rel \\widetilde {\\bm{b}}_B^{*\\#} \\bigr) \\not = 0$.\nAs in the previous case the non-triviality of the  braid invariant proves the theorem in Case III\nsince the braid class $[{\\bm{a}}_B \\rel \\widetilde {\\bm{b}}_B^{*\\#}]$ is free, bounded and proper and stationary braids of $W_d$ in $[{\\bm{a}}_B \\rel \\widetilde {\\bm{b}}_B^{*\\#}]$ \nyield invariant sets $A\\subset \\inter \\disc$ by Proposition \\ref{prop:reallyindisc}.\n\\end{proof}\n\n\n\\begin{figure}[hbt]\n\\centering\n\\begin{tikzpicture}[xscale=0.23, yscale=0.23, line width = 1.5pt]\n\n\\foreach \\x in {-2,8,12,16,20}\n \\draw[line width=1.0pt, -] (\\x,-5) -- (\\x,9); \n\n\\draw[-] (-2,2) -- (0,0) -- (2,0) -- (4,2) -- (6,4) -- (8,4) -- (10,4) -- (12,4) -- (14,0) -- (16,4) -- (18,0) -- (20,4);            \n\\foreach \\point in {(-2,2), (0,0), (2,0), (4,2), (6,4), (8,4), (10,4), (12,4), (14,0), (16,4), (18,0), (20,4)}\n   \\draw[fill] \\point circle (0.07);\n\n\\draw[-] (-2,4) -- (0,4) -- (2,2) -- (4,0) -- (6,0) -- (8,2) -- (10,2) -- (12,2) -- (14,2) -- (16,2) -- (18,2) -- (20,2);            \n\\foreach \\point in {(-2,4), (0,4), (2,2), (4,0), (6,0), (8,2), (10,2), (12,2), (14,2), (16,2), (18,2), (20,2)}\n   \\draw[fill] \\point circle (0.07);\n\n\\draw[-,color=red] (-2,0) -- (0,2) -- (2,4) -- (4,4) -- (6,2) -- (8,0) -- (10,0) -- (12,0) -- (14,4) -- (16,0) -- (18,4) -- (20,0);            \n\\foreach \\point in {(-2,0), (0,2), (2,4), (4,4), (6,2), (8,0), (10,0), (12,0), (14,4), (16,0), (18,4), (20,0)}\n   \\draw[fill, color=red] \\point circle (0.07);\n\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (7.9,-5.5) -- (-2.,-5.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $d_0$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (11.9,-5.5) -- (8.1,-5.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $q$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (15.9,-5.5) -- (12.1,-5.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $2\\lambda$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (20,-5.5) -- (16.1,-5.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $2$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (-2,9.5) -- (12.,9.5) node [black,midway,yshift=0.5cm] {\\footnotesize $\\E^q{\\bm{a}}\\rel(\\E^q{\\bm{b}}^*)^\\#$};\n   \n\\foreach \\y in {-2,6}\n \\draw[-] (-2,\\y) -- (12,\\y);\n\n\\foreach \\x in {-1,...,6}\n   \\draw[fill] (2*\\x,-2) circle (0.07);\n\n\\foreach \\x in {-1,...,6}\n   \\draw[fill] (2*\\x,6) circle (0.07);\n\n\\foreach \\y in {-2,6}\n \\draw[-, dashed] (12,\\y) -- (20,\\y);\n\n\\foreach \\y in {-4,8}\n \\draw[-, dashed] (-2,\\y) -- (20,\\y);\n\n\\end{tikzpicture}\n\\raisebox{67pt}{\n\\begin{minipage}{0.05\\linewidth}\n  \\centering \\Large $\\sim$\n\\end{minipage}}\n\\begin{tikzpicture}[xscale=0.23, yscale=0.23, line width = 1.5pt]\n\n\\foreach \\x in {-2,12,20}\n \\draw[line width=1.0pt, -] (\\x,-5) -- (\\x,9); \n\n\\draw[-] (-2,2) -- (0,2.3) -- (2,3.8) -- (4,3.2) -- (6,1.2) -- (8,-0.3) -- (10,2.8) -- (12,4.5) -- (14,2.8) -- (16,0.2) -- (18, 1.3) -- (20,4);            \n\\foreach \\point in {(-2,2), (0,2.3), (2,3.8), (4,3.2), (6,1.2), (8,-0.3), (10,2.8), (12,4.5), (14,2.8), (16,0.2), (18, 1.3), (20,4)}\n   \\draw[fill] \\point circle (0.07);\n  \n\\draw[-] (-2,4) -- (0,3.3) -- (2,0.0) -- (4,1.2) -- (6,3.5) -- (8,4.5) -- (10,3.4) -- (12,1.1) -- (14,0.8) -- (16,3.9) -- (18, 3.3) -- (20, 2);            \n\\foreach \\point in {(-2,4), (0,3.3), (2,0.0), (4,1.2), (6,3.5), (8,4.5), (10,3.4), (12,1.1), (14,0.8), (16,3.9), (18, 3.3), (20, 2)}\n   \\draw[fill] \\point circle (0.07);\n\n\\draw[-,color=red] (-2,0) -- (0,1.0) -- (2,5.2) -- (4,2.2) -- (6,0.1) -- (8,2.2) -- (10,5.2) -- (12,-1.0) -- (14,2.2) -- (16,4.9) -- (18, 1.8) -- (20,0);            \n\\foreach \\point in {(-2,0), (0,1.0), (2,5.2), (4,2.2), (6,0.1), (8,2.2), (10,5.2), (12,-1.0), (14,2.2), (16,4.9), (18, 1.8), (20,0)}\n   \\draw[fill, color=red] \\point circle (0.07);\n\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (11.9,-5.5) -- (-2.,-5.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $d$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (20.0,-5.5) -- (12.1,-5.5) node [black,midway,yshift=-0.5cm] {\\footnotesize $\\ell_\\rho$};\n \\draw[line width=1.0pt,decorate,decoration={brace,amplitude=8pt}] (-2,9.5) -- (20.,9.5) node [black,midway,yshift=0.5cm] {\\footnotesize ${\\bm{a}}_B\\rel \\widetilde {\\bm{b}}_B^{*\\#}$};\n   \n\\foreach \\y in {-2,6}\n \\draw[-] (-2,\\y) -- (12,\\y);\n\n\\foreach \\x in {-1,...,6}\n   \\draw[fill] (2*\\x,-2) circle (0.07);\n\n\\foreach \\x in {-1,...,6}\n   \\draw[fill] (2*\\x,6) circle (0.07);\n\n\\foreach \\y in {-2,6}\n \\draw[-, dashed] (12,\\y) -- (20,\\y);\n\n\\foreach \\y in {-4,8}\n \\draw[-, dashed] (-2,\\y) -- (20,\\y);\n\n\\end{tikzpicture}\n\\caption{Two representatives for Case III. Numbers of discretization points are linked by $ d+ \\ell_\\rho = d_0+q+2\\lambda +2$.\nThe dashed lines indicate the oscillating strands due to  the action of $\\T$ and the augmentation of the skeleta by $^\\#$. }\n\\label{fig:7C3}\n\\end{figure}\n\n\\begin{exm}\n\\label{exm:exist2}\nConsider $F\\in \\Symp(\\disc)$ and assume that $F$ has a four point invariant sets $B$ and  mapping class $[F]$ relative to\n$B$ is given by $\\beta = \\sigma_3\\sigma_1\\sigma_2^2\\sigma_1\\sigma_3$, cf.\\ Figure \\ref{fig:braidex1}.\nConsider the 2-colored braid class $\\llbracket\\gamma,\\aset\\rrbracket \\in \\CC{\\mathscr{B}}_{1,4}^+$, with\n\\[\n\\gamma = \\sigma_4\\sigma_1\\sigma_2\\sigma_3\\sigma_2^2\\sigma_3\\sigma_2\\sigma_1\\sigma_4,\\quad \\aset = \\{3\\}.\n\\]\nThen, $(\\gamma,\\aset) \\mapsto \\beta$ and the positive conjugacy class is proper and acylindrical.\nIn \\cite[Sect.\\ 4.5]{BraidConleyIndex} the braid invariant is given: ${\\mathrm{H}}\\llbracket\\gamma,\\aset\\rrbracket = \\sbb^1$, which by Theorem \\ref{thm:main1}\nyields another fixed point.\nThe index calculations in \\cite{BraidConleyIndex} suggest that we can find more periodic point for $F$. Instead of using multi-strand braids with $n>1$, we\nuse $F^k$ instead.\nConsider three different 2-colored braid words: $\\gamma_0 = \\gamma$ as above, $\\gamma_{-1} = \n \\sigma_4\\sigma_1\\sigma_2\\sigma_3^2\\sigma_2\\sigma_1\\sigma_4$, and\n$\\gamma_1  = \\sigma_4\\sigma_1\\sigma_3\\sigma_2^2\\sigma_3\\sigma_1\\sigma_4$.\nFor all three cases the skeletal word is given by $\\beta$, the coloring is given by $\\aset=\\{3\\}$.\nConsider a symbolic sequence $\\{a_i\\}_{i=1}^k$, $a_i\\in \\{-1,0,1\\}$, then\nthe positive conjugacy class $(\\gamma,\\aset)$, with $\\aset = \\{3\\}$, and\n\\[\n\\gamma = \\gamma_{a_0} \\cdot\\gamma_{a_1} \\cdots \\gamma_{a_{k-1}}\\cdot \\gamma_{a_k},\n\\]\nis proper and acylindrical, except for $a_i=-1$, or $a_i=1$ for all $i$.\nIf follows that $(\\gamma,\\aset) \\mapsto \\beta^k$. In \\cite[Sect.\\ 4.5]{BraidConleyIndex} the braid invariant is given by ${\\mathrm{H}}\\llbracket\\gamma,\\aset\\rrbracket = \\sbb^r$,\nwhere $r$ is the number of zeroes in $\\{a_i\\}$.\nThis procedure produces many $k$-periodic point for $F$ that are forced by the invariant set $B$. As matter of fact one obtains a lower bound on the \ntopological entropy of $F$.\n\\end{exm}\n\n\\begin{exm}\n\\label{exm:exist3}\nLet $F\\in \\Symp(\\disc)$ possess an invariant set $B$ consisting of three points,\nand let the mapping class of $[F]$ relative to\n$B$ be represented by a positive braid word $\\beta = {\\bm{i}}_m^{-1}[{\\bm{\\beta}}]$,\nwith ${\\bm{\\beta}} = \\bigl\\{{\\bm{\\beta}}^1(t), {\\bm{\\beta}}^2(t), {\\bm{\\beta}}^3(t)\\bigr\\}$ being a geometric representative of \n\\[\n\\beta = \\sigma_1\\sigma_2^2\\sigma_1^2\\sigma_2^2\\sigma_1\\sigma_2\\sigma_1\\sigma_2^2\\sigma_1,\n\\]\ncf.\\ Figure \\ref{ex:nontriv1}.\nFor the intersection numbers it holds that  $\\iota\\bigl({\\bm{\\beta}}^1(t),{\\bm{\\beta}}^2(t)\\bigr) = \\iota\\bigl({\\bm{\\beta}}^1(t),{\\bm{\\beta}}^3(t)\\bigr) = 6$ and\n  $\\iota\\bigl({\\bm{\\beta}}^2(t),{\\bm{\\beta}}^3(t)\\bigr) = 1 < 6$.\nFrom considerations in~\\cite[Sect.\\ 9.2]{BraidConleyIndex} it follows that through the black strands ${\\bm{\\beta}}$ one can plot a single red strand ${\\bm{\\alpha}}$, \nsuch that the 2-colored braid class \n$\\llbracket\\gamma,\\aset\\rrbracket$ represented by their union \n\\[\n\\gamma = \\sigma_2 \\sigma_1 \\sigma_2 \\sigma_3 \\sigma_2 \\sigma_3 \\sigma_2 \\sigma_3 \\sigma_1\n\\sigma_2 \\sigma_2 \\sigma_1 \\sigma_2 \\sigma_3 \\sigma_2 \\sigma_3 \\sigma_2 \\sigma_1 \\sigma_2 \\sigma_1 \\sigma_2 \\sigma_3 \\sigma_2 \\sigma_3 \\sigma_2 \\sigma_1 \\sigma_2,\n\\]\nwith $\\aset = \\{2\\}$, cf.\\ Figure \\ref{ex:nontriv1}, \nis proper, acylindrical and of nontrivial index.\nThe intersection numbers for ${\\bm{\\alpha}}$ are\n $\\iota\\bigl({\\bm{\\alpha}}(t),{\\bm{\\beta}}^2(t)\\bigr) = \\iota\\bigl({\\bm{\\alpha}}(t),{\\bm{\\beta}}^3(t)\\bigr) = 4$.\nIf the intersection numbers are chosen more generally, i.e.\\\n$\\iota\\bigl({\\bm{\\beta}}^1(t),{\\bm{\\beta}}^2(t)\\bigr) = \\iota\\bigl({\\bm{\\beta}}^3(t),{\\bm{\\beta}}^3(t)\\bigr) = 2p$ and\n  $\\iota\\bigl({\\bm{\\beta}}^2(t),{\\bm{\\beta}}^3(t)\\bigr) = r < 2p$, and  $\\iota\\bigl({\\bm{\\alpha}}(t),{\\bm{\\beta}}^2(t)\\bigr) = \\iota\\bigl({\\bm{\\alpha}}(t),{\\bm{\\beta}}^3(t)\\bigr) = 2q$,\nwhere $r > 0$ and $p \\geq 2$. If $r < 2q < 2p$, then \nthe singular homology of $\\llbracket\\gamma,\\aset\\rrbracket$ is given by\n$$\nH_{k}\\bigl({\\mathrm{H}}\\llbracket\\gamma,\\aset\\rrbracket\\bigr) = \\begin{cases}\n  \\mathbb{R}: k= 2q,\\ 2q+1,\\\\\n 0: \\text{elsewise}.\\\\\n\\end{cases}\n$$\nBy Theorem~\\ref{thm:main1} we conclude that there are at least two additional distinct fixed points $A_1, A_2$\nwith $\\jmath_{A_i\\cup B}([F]) = \\gamma\\!\\!\\mod \\square, \\ i = 1,2$. \nIn addition,\nvia concatenating braid diagrams one can produce an infinite number of periodic solutions of different periods, cf.\\ \\cite[Lemma 47]{BraidConleyIndex}.\nThe above \nforcing result\nis specific for area-preserving mapping of the 2-disc, or $\\rr^2$ and \\emph{not} true for arbitrary diffeomorphisms of the 2-disc.\n\nFor example consider the time-1 mapping $F\\colon \\disc\\to\\disc$ given by the differential equation $\\dot{r} = r(r-a_1)(r-a_2)(r-1)$ and $\\dot{\\theta} = g(r)>0$,\nwith $g(a_1) = \\pi$ and $g(a_2)  = 6\\pi$, and $0<a_1<a_2<1$.\nThe set $B = \\{-a_1,a_1,a_2\\}$ is invariant and the mapping class of $F$ relative to $B$ is given by\nthe skeleton ${\\bm{\\beta}}$. The mapping $F$ has no invariant set matching the braid $\\gamma$.\nThis implies that the invariants in \\cite{JiangZheng} are void for this example and the braid invariant introduced in this\npaper add additional information to existing invariants,   in particular in the area-preserving case.\n\\end{exm}\n\n\n\n\n\\begin{figure}[hbt]\n  \\centering\n\\scalebox{0.6}{               \n\\begin{tikzpicture}[rotate=90, line width=2.0pt, yscale=0.6]\n\\draw[dashed] (0,0) arc (180:360:2.5 and 1.5);\n\\draw (5,0) arc (0:180:2.5 and 1.5);\n\n  \\braid[color=white,\n line width=6.5pt,\n] s_2 s_1 s_2  s_3 s_2 s_3  s_2 s_3-s_1 s_2  s_2 s_1 s_2  s_3 s_2 s_3  s_2 s_1  s_2 s_1 s_2  s_3 s_2 s_3  s_2 s_1 s_2;\n\n  \\braid[\n line width=1.5pt,\n style strands={2}{red}\n] s_2 s_1 s_2  s_3 s_2 s_3  s_2 s_3-s_1 s_2  s_2 s_1 s_2  s_3 s_2 s_3  s_2 s_1  s_2 s_1 s_2  s_3 s_2 s_3  s_2 s_1 s_2;\n\n\\draw[color=white, line width=8.0pt] (5,-26.5) arc (0:180:2.5 and 1.5);\n \n\\draw (2.5,-26.5) ellipse (2.5cm and 1.5cm);\n\n\\draw[-] (5,0) -- (5,-26.5); \n\\draw[-] (0,0) -- (0,-26.5); \n\n\\foreach \\x in {1,...,4}\n \\draw[fill] (\\x,0) circle (0.07);\n\\foreach \\x in {1,...,4}\n \\draw[fill] (\\x,-26.5) circle (0.07);\n\n\\draw[color=red,fill] (2,0) circle (0.07);\n\\draw[color=red,fill] (2,-26.5) circle (0.07);\n\\end{tikzpicture}\n}\n\\caption{A geometric representative of the braid for Example~\\ref{exm:exist3} with $p=3$, $q=2$, and $r=1$.}\n\\label{ex:nontriv1}\n\\end{figure}\n\n\n\n\\begin{rem}\nOne can also consider 2-colored words $(\\gamma,\\aset)$, $\\gamma\\in {\\mathscr{B}}_{n+m}$. Braid words can be expressed in normal form.\nThe fundamental element, or Garside element in ${\\mathscr{B}}_m^+$ is denoted by \n$\n\\triangle := (\\sigma_{1}\\cdots \\sigma_{m-1}) (\\sigma_{1}\\cdots \\sigma_{m-2})\\cdots  (\\sigma_1 \\sigma_2)\\sigma_1,\n$\nand is also referred to as a \\emph{half twist}.\nThe element $\\triangle$ is a factor of a positive word $\\beta$, if $\\beta = \\beta' \\triangle \\beta''$, $\\beta',\\beta''$ positive (possibly trivial) words.\nIf not, $\\beta$ is said to be \\emph{prime} to $\\triangle$.\nIf we use the lexicographical order on the presentations of a positive word $\\beta$, then the smallest positive word $\\beta'$, positively equal to $\\beta$, is called the \\emph{base}\nof $\\beta$. For positive words $\\beta$, prime to $\\triangle$, the base is denoted by $\\bar\\beta$.\nFollowing \\cite{Birman} and \\cite{Hazewinkel4},\nevery word $\\beta\\in {\\mathscr{B}}_m$ is uniquely presented by a word $\\triangle^\\varrho  \\bar\\delta$, with $\\varrho\\in \\zz$ and $\\bar\\delta\\in {\\mathscr{B}}_m^+$,\nwhich is called \\emph{left Garside normal form}.\nVia the relation $\\triangle \\beta = r(\\beta)\\triangle$ we obtain the \\emph{right Garside normal form} $\\beta = r(\\bar\\delta)\\triangle^\\varrho$.\n\nFor the 2-colored braid words the Garside normal form is not the appropriate normal form, since  \n odd powers of $\\triangle$ do not represent trivial permutations.\nIn the case that the Garside power of $\\gamma$ is even, then $\\triangle^2 = \\square$ defines the identity permutation and therefore the associated base $(\\bar\\delta,\\aset)$, with $\\bar\\delta \\in {\\mathscr{B}}_{n+m}^+$, is a positive 2-colored braid word. In this case the left and right normal form are the same since $\\square$ is at the center of the braid group\n${\\mathscr{B}}_{n+m}$.\nWhen the Garside power  is odd, we argue as follows. Let $\\varrho=2\\uplambda +1$, then\n\\begin{equation}\n\\label{eqn:garside2}\n\\gamma=\\triangle^{2\\uplambda +1} \\bar \\delta = \\square^\\uplambda \\triangle \\bar \\delta\n= \\square^\\uplambda r(\\bar \\delta) \\triangle =   r(\\bar \\delta) \\triangle \\square^\\uplambda =   \\bar \\epsilon \\square^\\uplambda\n= \\square^\\uplambda\\bar \\epsilon,\n\\end{equation}\nwhere $\\bar\\epsilon = \\triangle \\bar\\delta = r(\\bar\\delta)\\triangle$ is the base and $\\bar\\epsilon$ is prime to $\\square$.\nThe power $\\uplambda$ is related to the Garside power via $\\uplambda = \\uplambda(\\gamma) = \\floor{\\varrho\/2}$ and is referred to as the \\emph{symmetric Garside power}\nof $\\gamma$.\nFor 2-colored braid words  $(\\gamma,\\aset)$ this yields the following symmetric normal form: $\\gamma = \\square^\\uplambda  \\bar\\epsilon = \\bar\\epsilon \\square^\\uplambda$, with $\\uplambda \\in \\zz$ and $(\\bar\\epsilon,\\aset)$ a positive 2-colored braid word.\nSince mapping classes of $F$ are given modulo full twists  the latter normal form suggests that we capture all forcing via positive braid words. \nIf conjugacy is incorporated, the power $\\uplambda$ can be optimized with respect to different conjugate braid words.\n\\end{rem}\n\n\\begin{figure}[hbt]\n\\centering\n\\begin{tikzpicture}[rotate=90, line width=2.0pt, scale=0.6]\n\\draw[dashed] (0,0) arc (180:360:3.0 and 1.5);\n\\draw (6,0) arc (0:180:3.0 and 1.5);\n\n  \\braid[color=white,\n line width=6.5pt\n] s_4-s_1 s_2 s_3 s_2 s_2 s_3 s_2 s_1-s_4;\n\n  \\braid[\n line width=1.5pt,\n style strands={3}{red}\n] s_4-s_1 s_2 s_3 s_2 s_2 s_3 s_2 s_1-s_4;\n\n\\draw[color=white, line width=8.0pt] (6,-8.5) arc (0:180:3.0 and 1.5);\n \n\\draw (3,-8.5) ellipse (3cm and 1.5cm);\n\n\\draw[-] (6,0) -- (6,-8.5); \n\\draw[-] (0,0) -- (0,-8.5); \n\n\\foreach \\x in {1,...,5}\n \\draw[fill] (\\x,0) circle (0.07);\n\\foreach \\x in {1,...,5}\n \\draw[fill] (\\x,-8.5) circle (0.07);\n\n\n\\draw[color=red,fill] (3,0) circle (0.07);\n\\draw[color=red,fill] (3,-8.5) circle (0.07);\n\\end{tikzpicture}\n\\caption{The canonical representation of the  2-colored braid class $\\llbracket\\gamma,\\aset\\rrbracket \\in \\CC{\\mathscr{B}}_{1,4}^+$\nwith $\\gamma = \\sigma_4\\sigma_1\\sigma_2\\sigma_3\\sigma_2^2\\sigma_3\\sigma_2\\sigma_1\\sigma_4 $ and $\\aset = \\{3\\}$.}\n\\label{fig:braidex1}\n\\end{figure}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $\\mu$ be the M\\\"obius bundle over $S^1$, regarded as a virtual bundle of dimension $0$.  The mod $2$ Moore spectrum is the Thom spectrum\n$$ M(2) \\simeq (S^1)^\\mu. $$\nThe classifying map for $\\mu$ extends to a double loop map\n$$ \\td{\\mu} : \\Omega^2S^3 \\rightarrow BO. $$\nMahowald proved the following theorem \\cite{Mahowald}:\n\n\\begin{thm}[Mahowald]\nThere is an equivalence of spectra\n$$ (\\Omega^2S^3)^{\\td{\\mu}} \\simeq H\\mathbb{F}_2. $$\n\\end{thm}\n\nThe bundle $\\mu$ may also be regarded as a $C_2$-equivariant virtual bundle over $S^1$, by endowing both $S^1$ and the bundle with the trivial action.  Since $B_{C_2}O$ is an equivariant infinite loop space \\cite{Atiyah}, the classifying map for $\\mu$ extends to an\n$\\Omega^{\\rho}$-map\n$$ \\td{\\mu}: \\Omega^{\\rho}S^{\\rho+1} \\rightarrow B_{C_2}O. $$\nHere, $\\rho$ is the regular representation of $C_2$. \nThe purpose of this paper is to prove the following.\n\n\\begin{thm}\\label{thm:Mahowald}\nThere is an equivalence of $C_2$-spectra\n$$ (\\Omega^\\rho S^{\\rho+1})^{\\td{\\mu}} \\simeq H\\ul{\\mathbb{F}}_2. $$\n\\end{thm} \n\n(Here, $\\ul{\\mathbb{F}}_2$ denotes the constant Mackey functor with value $\\mathbb{F}_2$.)\n\n\\subsection*{Acknowledgements}\n\nMany tricks in this paper have been independently discovered by Doug Ravenel, and the first author's involvement in this project is an outgrowth of mathematical discussions with Agn\\'es Beaudry, Prasit Bhattacharya, Dominic Culver, Doug Ravenel, and Zhouli Xu.  The authors also benefited from valuable input from Mike Hill, and the comments of the referee.  The first author was supported by NSF grant DMS-1611786.\n\n\\subsection*{Conventions}\nEquivariant objects in this paper either live in $\\mathrm{Top}^{C_2}$, the category of $C_2$-spaces, or $\\mathrm{Sp}^{C_2}$, the category of genuine $C_2$-spectra.  In both of these categories, the equivalences are those equivariant maps which induce equivalences on both the $C_2$-fixed points spectrum and the underlying spectrum.\nWe let $\\mbf{H}$ denote the Eilenberg-Maclane spectrum $H\\ul{\\mathbb{F}}_2$, with underlying spectrum $H := H\\mathbb{F}_2$.  We use $\\mbf{H}_\\star$ and $\\pi^{C_2}_\\star$ to denote $RO(C_2)$-graded homology and homotopy \\emph{groups} (i.e. \\emph{not} the Mackey functors) of $C_2$-equivariant spaces and spectra, and $H_*$ and $\\pi_*$ to denote the ordinary homology and homotopy groups of non-equivariant spaces and spectra. We let $\\sigma$ denote the sign representation of $C_2$, and let $\\rho = 1+\\sigma$ denote the regular representation.  For a representation $V$, $S(V)$ denotes the unit sphere in $V$, and $S^V$ denotes its one point compactification, and $\\abs{V}$ denotes its dimension.\n\n\\section{Equivariant preliminaries}\n\n\\subsection*{Euler class}\n\nLet $a$ denote the Euler class in $\\pi^{C_2}_{-\\sigma}S$, given geometrically by the inclusion\n$$ S^0 \\hookrightarrow S^\\sigma. $$\nThere is a cofiber sequence\n\\begin{equation}\\label{eq:cofiber}\n{C_2}_+ \\rightarrow S^0 \\hookrightarrow S^\\sigma\n\\end{equation}\nso the cofiber of $a$ is stably given by\n\\begin{equation}\\label{eq:Ca}\nCa \\simeq \\Sigma^{1-\\sigma} {C_2}_+. \n\\end{equation}\nThe equivalence of underlying spectra\n\\begin{equation}\\label{eq:ulSsig}\n (S^1)^e \\simeq (S^\\sigma)^e\n\\end{equation}\ninduces an equivalence of $C_2$-spectra\n$$ {C_2}_+ \\wedge S^1 \\simeq {C_2}_+ \\wedge S^\\sigma. $$\nTherefore, the equivalence (\\ref{eq:Ca}) can actually be regarded as giving an equivalence\n$$ Ca \\simeq {C_2}_+. $$\nIt follows that $Ca$ is a commutative ring spectrum.\nThe adjoint of the equivalence (\\ref{eq:ulSsig}) gives a $C_2$-equivariant map\n$$ {C_2}_+ \\wedge S^1 \\rightarrow S^\\sigma $$\nwhich, by the self-duality of ${C_2}_+$, gives a map\n$$ u : S^1 \\rightarrow {C_2}_+\\wedge S^\\sigma \\simeq Ca \\wedge S^\\sigma $$   \nwhich serves as a Thom class for the representation $\\sigma$.\nFor $X \\in \\mathrm{Sp}^{C_2}$, we have\n\\begin{align*}\n\\pi^{C_2}_{k}(X) & \\cong \\pi_k(X^{C_2}), \\\\\n\\pi^{C_2}_{V}(X \\wedge Ca) & \\cong \\pi_{\\abs{V}}(X^e).\n\\end{align*}\nSaid differently,\n\\begin{equation}\\label{eq:pistarCa}\n\\pi^{C_2}_{\\star} (X \\wedge Ca) \\cong \\pi_* (X^e) [u^\\pm].\n\\end{equation}\n\n\\subsection*{Tate square}\n\nWe will let\n\\begin{align*}\nX^h & := F({EC_2}_+, X), \\\\\nX^\\Phi & := X \\wedge \\td{EC}_2\n\\end{align*}\ndenote the homotopy completion and geometric localization of $X$, respectively.  The fixed points of $X^h$ are the homotopy fixed points of $X$, and the fixed points of $X^\\Phi$ are the geometric fixed points of $X$.\n$X$ is recovered from these approximations by the pullback (``Tate square'') \\cite{GreenleesMay}\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d] & X^\\Phi \\ar[d] \\\\\nX^h \\ar[r] & X^t\n}\n$$\nwhere the spectrum $X^t$ is the equivariant Tate spectrum\n$$ X^t := (X^h)^{\\Phi}. $$\n\nNote that a generalization of the argument establishing (\\ref{eq:Ca}) yields an equivalence\n$$ \\Sigma^{k\\sigma-1}C(a^k) \\simeq S(k\\sigma)_+. $$\nTaking a colimit, we see that we have\n\\begin{align*}\n\\hocolim_k \\Sigma^{k\\sigma-1}C(a^k) & \\simeq {EC_2}_+, \\\\\n\\hocolim_k S^{k\\sigma} & \\simeq \\td{EC}_2. \n\\end{align*}\nIt follows that homotopy completion and geometric localization can be reinterpreted as $a$-completion and $a$-localization:\n\\begin{align*}\n X^h & \\simeq X^\\wedge_a, \\\\\n X^\\Phi & \\simeq X[a^{-1}].\n\\end{align*}\nIn this manner, the Tate square is equivalent to the ``$a$-arithmetic square''\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d] & X[a^{-1}] \\ar[d] \\\\\nX^\\wedge_a \\ar[r] & X^\\wedge_a[a^{-1}]\n}\n$$\nUsing (\\ref{eq:pistarCa}), the $a$-Bockstein spectral sequence takes the form\n$$ E_1^{*,*} = \\pi_*(X^e)[u^{\\pm}, a] \\Rightarrow \\pi^{C_2}_\\star(X^h). $$\nThe $a$-Bockstein spectral sequence can be regarded as an $RO(C_2)$-graded version of the homotopy fixed point spectral sequence (see \\cite[Lem.~4.8]{HillMeier}).\n\n\\subsection*{The mod $2$ Eilenberg-MacLane spectrum}\n\nWe have \\cite{HuKriz}\n$$ \\pi^{C_2}_\\star \\mbf{H} = \\mathbb{F}_2[a,u] \\oplus \\frac{\\mathbb{F}_2[a,u]}{(a^\\infty, u^\\infty)}\\{\\theta\\} $$\nwhere\n\\begin{align*}\n\\abs{u} & = 1-\\sigma, \\\\\n\\abs{\\theta} & = 2\\sigma - 2.\n\\end{align*}\nThe $a$-$u$ divisible factor in $\\pi_\\star\\mbf{H}$ is best understood from the Tate square, using\n\\begin{align*}\n\\pi^{C_2}_\\star \\mbf{H}^h & \\cong \\mathbb{F}_2[a, u^{\\pm1}], \\\\\n\\pi^{C_2}_\\star \\mbf{H}^\\Phi & \\cong \\mathbb{F}_2[a^{\\pm1}, u].\n\\end{align*}\nActually, the second isomorphism lifts to an equivalence\n$$ \\mbf{H}^{\\Phi C_2} \\simeq H[a^{-1}u] := \\bigvee_{i \\ge 0} \\Sigma^{i} H $$\nso we have\n$$ \\mbf{H}^{\\Phi}_\\star X \\cong H_*(X^{\\Phi C_2})[a^{\\pm1},u] $$\nand, restricting the grading to trivial representations, we get\n\\begin{equation}\\label{eq:HPhi}\n \\mbf{H}^\\Phi_* X \\cong H_*(X^{\\Phi C_2})[a^{-1}u].\n\\end{equation}\n\nBy applying $\\pi_{V}^{C_2}$ to the map\n$$ \\mbf{H} \\wedge X \\rightarrow \\mbf{H} \\wedge X \\wedge Ca $$\nwe get a homomorphism\n\\begin{equation}\\label{eq:Phie}\n\\Phi^e: \\mbf{H}_V(X) \\rightarrow H_{\\abs{V}}(X^e).\n\\end{equation}\nTaking geometric fixed points of a map\n$$ S^V \\rightarrow \\mbf{H} \\wedge X $$\ngives a map\n$$ S^{V^{C_2}} \\rightarrow \\mbf{H}^{\\Phi C_2} \\wedge X^{\\Phi C_2} $$\nUsing (\\ref{eq:HPhi}) and passing to the quotient by the ideal generated by $a^{-1}u$, we get a homomorphism\n\\begin{equation}\\label{eq:PhiC2}\n\\Phi^{C_2}: \\mbf{H}_V(X) \\rightarrow H_{\\abs{V^{C_2}}}(X^{\\Phi C_2}).\n\\end{equation}\n\n\\subsection*{A useful lemma}\n\nOur main computational lemma is the following.\n\n\\begin{lem}\\label{lem:useful}\nSuppose that $X \\in \\mathrm{Sp}^{C_2}$ and suppose that $\\{b_i\\}$ is a set of elements of $\\mbf{H}_\\star (X)$ such that\n\\begin{enumerate}\n\\item $\\{ \\Phi^e(b_i) \\}$ is a basis of $H_*(X^e)$, and\n\\item $\\{ \\Phi^{C_2}(b_i) \\}$ is a basis of $H_*(X^{\\Phi C_2})$.\n\\end{enumerate}\nThen $\\mbf{H}_\\star(X)$ is free over $\\mbf{H}_\\star$, and $\\{b_i\\}$ is a basis.\n\\end{lem}\n\n\\begin{proof}\nThe set $\\{b_i\\}$ corresponds to a map\n\t\\[\n\t\\mbf{H} \\wedge \\bigvee S^{|b_i|} \\to \\mbf{H} \\wedge X.\n\t\\]\nAssumption (1) implies this map is an equivalence upon applying\n$\\Phi^e$, while assumption (2) implies this map is an equivalence\nupon applying $\\Phi^{C_2}$. The result follows.\n\\end{proof}\n\n\\section{Homology of $\\rho$-loop spaces}\n\nWe spell out some specific algebraic structure carried by the equivariant homology of a $\\rho$-loop space.  A more detailed and general study of this algebraic structure can be found in \\cite{Hill}.\n\n\\subsection*{Products}\n\nSuppose $X = \\Omega^\\rho Y \\in \\mathrm{Top}^{C_2}$ is a $\\rho$-loop space.  Then $X$ is in particular a $1$-loop space, and is therefore an equivariant $H$-space with product\n$$ m: X \\times X \\rightarrow X. $$\nHowever, the $\\sigma$-loop space structure also endows $X$ with a twisted product related to the transfer.  Namely, let \n$$ S^\\sigma \\rightarrow S^\\sigma\/S^0 \\approx {C_2}_+\\wedge S^1 $$\nbe the pinch map.  This gives rise to a twisted product\n$$ \\td{m}: N^\\times \\Omega Y \\rightarrow \\Omega^\\sigma Y $$\nwhere\n$$ N^\\times Z := \\Map(C_2, Z) = \\underset{\\substack{ \\curvearrowbotleftright \\\\ C_2}}{Z \\times Z} $$\nis the norm with respect to Cartesian product (i.e. the coinduced space).\nIn particular, there is a map\n\\begin{equation}\\label{eq:mtilde}\n\\td{m}: N^\\times \\Omega^2 Y \\rightarrow X.\n \\end{equation}\nUpon applying fixed points to the map (\\ref{eq:mtilde}), we get an additive transfer\n\\begin{equation}\\label{eq:transfer}\nt: X^e \\rightarrow X^{C_2}.\n\\end{equation}\nIn homology, the $H$-space structure give rise to a product\n$$ m: \\mbf{H}_V X \\otimes \\mbf{H}_W X \\rightarrow \\mbf{H}_{V+W} X. $$\nUsing the equivariant commutative ring spectrum structure of $\\mbf{H}$ \\cite{Ullman}, \nthe twisted product $\\td{m}$ gives rise to a ``norm map''(see \\cite[Thm.~7.2]{BlumbergHill})\n$$ n: H_k X^e \\rightarrow \\mbf{H}_{k\\rho} X. $$\n\n\\subsection*{Dyer-Lashof operations}\n\n$X$ has even more structure:\n$X$ is an $E_\\rho$-algebra \\cite{GuillouMay}.  Specifically, regard \n$S(\\rho)$ as a $C_2 \\times \\Sigma_2$-space where $C_2$ acts on $\\rho$ and $\\Sigma_2$ acts antipodally.  \nThen the $E_\\rho$-structure gives a map\n$$ S(\\rho) \\times_{\\Sigma_2} X^{\\times 2} \\rightarrow X. $$\nNote that $\\mbf{H}$ is itself an $E_\\rho$-ring spectrum, because it is actually an equivariant commutative ring spectrum, so $\\mbf{H} \\wedge X_+$ is an $E_\\rho$-ring in $\\mbf{H}$-modules.\nGiven $x \\in \\mbf{H}_V(X)$, represented by a map\n$$ x: S^V \\rightarrow \\mbf{H} \\wedge X_+, $$\nthere is an induced composite \n\\begin{align*} \n\\mbf{H} \\wedge S(\\rho)_+ \\wedge_{\\Sigma_2} S^{2V} \\xrightarrow{1 \\wedge 1 \\wedge x \\wedge x} & \\mbf{H} \\wedge S(\\rho)_+ \\wedge_{\\Sigma_2} (\\mbf{H} \\wedge X_+)^{\\wedge 2} \\\\\n \\rightarrow & \\mbf{H} \\wedge \\mbf{H} \\wedge X_+ \\\\\n \\rightarrow & \\mbf{H} \\wedge X_+\n \\end{align*}\n(where the unlabeled maps come from the $E_\\rho$-ring and $\\mbf{H}$-module structure of $\\mbf{H} \\wedge X_+$).\nApplying $\\pi_\\star^{C_2}$, we get \na total power operation\n$$ \\mc{P}(x): \\td{\\mbf{H}}_\\star(S(\\rho)_+ \\wedge_{\\Sigma_2} S^{2V}) \\rightarrow \\mbf{H}_\\star X. $$\nFor the purposes of this paper we will be only concerned with the case of $V = k\\rho - \\sigma$ for $k \\in \\mathbb{Z}$.\n\n\n\n\n\n\\begin{prop}\nWe have\n$$ \\td{\\mbf{H}}_\\star \\left( S(\\rho)_+ \\wedge_{\\Sigma_2} S^{2(k\\rho-\\sigma)} \\right) \\cong \\mbf{H}_\\star \\{ e_{2k\\rho-\\sigma-1}, e_{2k\\rho - \\sigma} \\}. $$\n\\end{prop}\n\n\\begin{proof} Consider the following cofiber sequences:\n\t\\begin{align}\n\tS^{2(k-1)\\rho} \\to S(\\rho)_+ \\wedge_{\\Sigma_2} S^{2((k-1)\\rho)} \\to S^{2(k-1)\\rho+\\sigma}\n\t\\label{cofiber1}\n\t\\\\\n\t\\Sigma S(\\rho)_+\\wedge_{\\Sigma_2}S^{2((k-1)\\rho)} \\to \n\tS(\\rho)_+\\wedge_{\\Sigma_2}S^{2(k\\rho - \\sigma)} \\to\n\t\\Sigma^{2(k\\rho -\\sigma)}S(\\rho)_+\\label{cofiber2}\n\t\\end{align}\nThe sequence (\\ref{cofiber1}) arises from Theorem 2.15 of \\cite{Wilson}\nand the second arises from the $(C_2\\times \\Sigma_2)$-equivariant\ninclusion $\\Sigma S^{2((k-1)\\rho)} \\to S^{2(k\\rho -\\sigma)}$,\nwhere both $C_2$ and $\\Sigma_2$ act trivially on the first suspension\ncoordinate.\n\nIn (\\ref{cofiber1}), the boundary map on $\\mbf{H}_{\\star}$ is zero because the group\n\t\\[\n\t\\left[S^{2(k-1)\\rho+\\sigma}, \\Sigma^{2(k-1)\\rho+1}\\mbf{H}\\right] = \\mbf{H}_{-1+\\sigma}\n\t\\]\nis zero. Thus\n\t\\[\n\t\\td{\\mbf{H}}_{\\star}\\left(S(\\rho)_+\\wedge_{\\Sigma_2}S^{2(k-1)\\rho}\\right)\n\t\\cong \\mbf{H}_{\\star}\\{e_{2(k-1)\\rho}, e_{2(k-1)\\rho+\\sigma}\\}\n\t=\\mbf{H}_{\\star}\\{e_{2k\\rho - 2\\sigma -2}, e_{2k\\rho-\\sigma-2}\\}.\n\t\\]\nNow we turn to the second cofiber sequence. Notice that $S(\\rho)_+$\nis $C_2$-equivariantly equivalent to $S^{\\sigma}\\vee S^0$. From the\nprevious computation, the boundary is then determined\nby elements in the following four groups:\n\t\\begin{align*}\n\t\\partial_1\\in\\left[S^{2(k\\rho -\\sigma)}, \\Sigma^{2k\\rho-2\\sigma}\\mbf{H}\\right]\n\t&=\\mbf{H}_0 \\\\\n\t\\partial_2\\in\\left[S^{2(k\\rho -\\sigma)+\\sigma}, \\Sigma^{2k\\rho-\\sigma}\\mbf{H}\\right]\n\t&=\\mbf{H}_{0}\\\\\n\t\\partial_3\\in\\left[S^{2(k\\rho -\\sigma)}, \\Sigma^{2k\\rho-\\sigma}\\mbf{H}\\right]\n\t=\\mbf{H}_{-\\sigma} &= 0\\\\\n\t\\partial_4\\in\\left[S^{2(k\\rho -\\sigma)+\\sigma}, \\Sigma^{2k\\rho-2\\sigma}\\mbf{H}\\right] \n\t=\\mbf{H}_{\\sigma} &= 0\n\t\\end{align*}\nElements of $\\mbf{H}_0$ are determined by their restriction to $H_0$, and\ncomparison with the underlying homology forces $\\partial_1=1$ and $\\partial_2 = 0$.\nThe result follows.\n\\end{proof}\n\nThus we get a pair of Dyer-Lashof operations\n\\begin{align*}\nQ^{k\\rho}: \\mbf{H}_{k\\rho-\\sigma}X \\rightarrow \\mbf{H}_{2k\\rho-\\sigma}X, \\\\\nQ^{k\\rho-1}: \\mbf{H}_{k\\rho-\\sigma}X \\rightarrow \\mbf{H}_{2k\\rho-\\sigma-1}X\n\\end{align*}\ngiven by the formulas\n\\begin{align*}\nQ^{k\\rho}(x) & := \\mc{P}(x)(e_{2k\\rho-\\sigma}), \\\\\nQ^{k\\rho-1}(x) & := \\mc{P}(x)(e_{2k\\rho-\\sigma-1}).\n\\end{align*}\n\n\\begin{rmk}\nIf $X$ is actually an equivariant infinite loop space, then $\\mbf{H}_\\star X$ has an action by equivariant Dyer-Lashof operations \\cite{Wilson}, and these operations agree with those defined in that paper.\n\\end{rmk}\n\n\\subsection*{Compatibility with fixed points}\n\nThe compatibility of all this structure with the maps $\\Phi^e$ and $\\Phi^{C_2}$ of (\\ref{eq:Phie}) and (\\ref{eq:PhiC2}) is summarized as follows.\n\n\\begin{description}\n\\item[Products] Note that $X^e$ is an $E_2$-algebra, and $X^{C_2}$ is an $E_1$-algebra.  The maps $\\Phi^e$ and $\\Phi^{C_2}$ are algebra homomorphisms. \\vspace{10pt}\n\n\\item[Norms]  The following diagram commutes:\n$$\n\\xymatrix@C+1em{\n& H_kX^e \\ar[ld]_{t} \\ar[d]_{n} \\ar[rd]^{\\mr{Fr}} \\\\\nH_k X^{C_2} & \\mbf{H}_{k\\rho} X \\ar[l]^-{\\Phi^{C_2}} \\ar[r]_-{\\Phi^e} & H_{2k} X^e\n}\n$$\nHere $t$ is the transfer (\\ref{eq:transfer}) and $\\mr{Fr}$ is the squaring map (Frobenius).\\vspace{10pt}\n\n\\item[Dyer-Lashof operations]  \nThe following diagrams commute, where $\\epsilon = 0,1$:\n$$\n\\xymatrix@C+1em{\n \\mbf{H}_{k\\rho-\\sigma} X \\ar[r]^{\\Phi^e} \\ar[d]_{Q^{k\\rho - \\epsilon}} &\nH_{2k-1}X^e \\ar[d]^{Q^{2k-\\epsilon}} \\\\\n \\mbf{H}_{2k\\rho-\\sigma-\\epsilon} X \\ar[r]_{\\Phi^e}  &\nH_{4k-1-\\epsilon}X^e\n}\n$$\n$$\n\\xymatrix@C+1em{\n\\mbf{H}_{k\\rho-\\sigma} X \\ar[r]^{\\Phi^{C_2}} \\ar[d]_{Q^{k\\rho}}&\nH_kX^{C_2} \\ar[d]^{\\mr{Fr}}  \n \\\\\n \\mbf{H}_{2k\\rho-\\sigma} X \\ar[r]_{\\Phi^{C_2}}&\nH_{2k} X^{C_2}  \n}\n$$\n\\end{description}\n\n\\section{Homology of $\\Omega^\\rho S^{\\rho+1}$}\n\n\\begin{thm}\nThere is an \\emph{additive} isomorphism (of $\\mbf{H}_\\star$-modules)\n$$ \\mbf{H}_\\star \\Omega^\\rho S^{\\rho+1} \\cong\n\\mbf{H}_\\star \\otimes E[t_0, t_1, \\ldots] \\otimes P[e_1, e_2, \\ldots]\n$$\nwith\n\\begin{align*}\n\\abs{t_i} & = 2^i\\rho - \\sigma, \\\\\n\\abs{e_i} & = (2^i-1)\\rho.\n\\end{align*}\n\\end{thm}\n\n\\begin{proof}\nNote that we have\n$$ H_* \\Omega^2 S^3 = \\mathbb{F}_2[x_1, x_2, \\ldots] $$\nwith\n$$ \\abs{x_i} = 2^i-1. $$\nHere $x_1$ is the fundamental class $\\iota_1$, and \n$$ x_i := Q^{2^i}Q^{2^{i-1}}\\cdots Q^{2}x_1. $$\nDefine $t_0 \\in \\mbf{H}_{1} \\Omega^\\rho S^{\\rho+1}$ to be the fundamental class, and define the other ``generators'' $e_i$ and $t_i$ by\n\\begin{align*}\ne_i & := n(x_i), \\\\\nt_i & := Q^{2^i\\rho}Q^{2^{i-1}\\rho}\\cdots Q^{\\rho} t_0.\n\\end{align*}\nConsider the product\n$$ t^{\\ul{\\epsilon}}e^{\\ul{k}} := t_0^{\\epsilon_0}t_1^{\\epsilon_1} \\cdots e_1^{k_1} e_2^{k_2} \\cdots \\in \\mbf{H}_\\star(\\Omega^\\rho S^{\\rho+1}) $$\nwith $\\epsilon_i \\in \\{0,1\\}$ and $k_i \\ge 0$.  We compute\n$$ \\Phi^e(t^{\\ul{\\epsilon}}e^{\\ul{k}}) = x_1^{2k_1+\\epsilon_0}x_2^{2k_2+\\epsilon_1}\\cdots. $$\nMapping out of the cofiber sequence (\\ref{eq:cofiber}) gives a fiber sequence\n$$ \\Omega N^\\times \\Omega S^{\\rho+1} \\rightarrow \\Omega^\\rho S^{\\rho+1} \\rightarrow \\Omega S^{\\rho+1} \\xrightarrow{\\Delta} N^\\times \\Omega S^{\\rho+1}. $$\nUpon taking fixed points we get a fiber sequence\n$$ \\Omega^2 S^{3} \\xrightarrow{t} (\\Omega^\\rho S^{\\rho+1})^{C_2} \\rightarrow \\Omega S^{2} \\xrightarrow{\\mr{null}} \\Omega S^{3} $$\nIn particular there is an equivalence\n$$ (\\Omega^\\rho S^{\\rho+1})^{C_2} \\simeq \\Omega S^2 \\times \\Omega^2 S^3. $$\nand we have\n$$ H_*(\\Omega^\\rho S^{\\rho+1})^{C_2} \\cong P[y]\\otimes P[t(x_1), t(x_2), \\ldots] $$\nwhere $y$ is the image of the fundamental class under the map\n$$ S^1 \\rightarrow (\\Omega^\\rho S^{\\rho+1})^{C_2}. $$\nIt follows that\n$$ \\Phi^{C_2}(t^{\\ul{\\epsilon}}e^{\\ul{k}}) = y^{\\epsilon_0 + 2\\epsilon_1 + 4\\epsilon_2 + \\cdots } t(x_1)^{k_1} t(x_2)^{k_2}\\cdots.  $$\nThus the set\n$$ \\{ t^{\\ul{\\epsilon}}e^{\\ul{k}} \\} \\subset \\mbf{H}_\\star X $$\nsatisfies the hypotheses of Lemma~\\ref{lem:useful}, and the result follows.\n\\end{proof}\n\n\\section{The equivariant Mahowald theorem}\n\nIn order to prove Theorem~\\ref{thm:Mahowald} we will need\nto establish a Thom isomorphism\n\n$$ \\mbf{H}_\\star (\\Omega^\\rho S^{\\rho+1})^{\\td{\\mu}} \\cong \\mbf{H}_\\star \\Omega^\\rho S^{\\rho+1}. $$\n\nWe will do so in two steps. Recall that an $E_0$-algebra\nis just a spectrum $X$ equipped with a map $S^0 \\to X$.\nLet $\\mathrm{Free}^*_{E_{\\rho}}: \\mathrm{Alg}_{E_0}(\\mathrm{Sp}^{C_2})\n\\to \\mathrm{Alg}_{E_\\rho}(\\mathrm{Sp}^{C_2})$ denote a homotopical left adjoint\nto the forgetful functor. An explicit model for this functor is the homotopy pushout\nof $E_{\\rho}$-algebras:\n\t\\[\n\t\\xymatrix{\n\t\\mathrm{Free}_{E_\\rho}(S^0)\\ar[d]\\ar[r] & \\mathrm{Free}_{E_\\rho}(X)\\ar[d]\\\\\n\tS^0 \\ar[r] & \\mathrm{Free}^*_{E_{\\rho}}(X)\n\t}\n\t\\]\n\nWe will need the following theorem.\n\n\\begin{thm}\\label{thm:freeThom} Let $f: X \\to B_{C_2}O$ classify a virtual bundle of dimension zero\nand denote by $\\tilde{f}: \\Omega^{\\rho}\\Sigma^{\\rho} X \\to B_{C_2}O$ the\nassociated $\\Omega^{\\rho}$-map. Then there is a canonical\nequivalence of $E_{\\rho}$-algebras\nin $\\mathrm{Sp}^{C_2}$\n\t\\[\n\t\\mathrm{Free}^*_{E_{\\rho}}(X^f) \\cong \\left(\\Omega^{\\rho}\\Sigma^{\\rho}X\\right)^{\\tilde{f}}.\n\t\\]\n\\end{thm}\n\\begin{proof} Combine the equivariant approximation theorem \\cite{GuillouMay, RourkeSanderson}\nwith Theorem IX.7.1 and Remark X.6.4 of \\cite{LMS}.\n\\end{proof}\n\n\\begin{rmk} The non-equivariant version of Theorem~\\ref{thm:freeThom} was\nfirst observed by Mark Mahowald, and then proven by Lewis. A nice modern\naccount in the non-equivariant setting via universal properties can be found in\n\\cite{OmarToby}.\n\\end{rmk}\n\n\\begin{prop}\nThere is a Thom isomorphism\n$$ \\mbf{H}_\\star (\\Omega^\\rho S^{\\rho+1})^{\\td{\\mu}} \\cong \\mbf{H}_\\star \\Omega^\\rho S^{\\rho+1}. $$\n\\end{prop}\n\n\\begin{proof}\nLet $\\mathrm{Free}^*_{E_{\\rho}, \\mbf{H}}: \\mathrm{Alg}_{E_0}(\\mathrm{Mod}_{\\mbf{H}})\n\\to \\mathrm{Alg}_{E_{\\rho}}(\\mathrm{Mod}_{\\mbf{H}})$ denote a homotopical left\nadjoint to the forgetful functor. Along with the previous theorem, we will need\ntwo facts:\n\t\\begin{enumerate}\n\t\\item $\\mbf{H} \\wedge (-): \\mathrm{Sp}^{C_2} \\to \\mathrm{Mod}_{\\mbf{H}}$ is\n\tsymmetric monoidal.\n\t\\item There is a Thom isomorphism $\\mbf{H} \\wedge (S^1)^{\\mu} \\cong\n\t\\mbf{H} \\wedge S^1_+$.\n\t\\end{enumerate}\nThe proposition is now proved by the following string of equivalences: \n\t\\begin{align*}\n\t\\mbf{H} \\wedge \\left(\\Omega^{\\rho}\\Sigma^{\\rho}S^1\\right)^{\\tilde{\\mu}}\n\t&\\cong \\mbf{H} \\wedge \\mathrm{Free}^*_{E_{\\rho}}\\left((S^1)^{\\mu}\\right)\n\t&\\text{by Theorem~\\ref{thm:freeThom}}\\\\\n\t&\\cong \\mathrm{Free}^*_{E_{\\rho}, \\mbf{H}}\\left(\\mbf{H} \\wedge (S^1)^{\\mu}\\right)\n\t&\\text{by (1)}\\\\\n\t&\\cong \\mathrm{Free}^*_{E_{\\rho}, \\mbf{H}}\\left(\\mbf{H} \\wedge S^1_+\\right)\n\t&\\text{by (2)}\\\\\n\t&\\cong \\mbf{H} \\wedge \\mathrm{Free}^*_{E_\\rho}\\left( S^1_+\\right)\n\t&\\text{by (1)}\\\\\n\t&\\cong \\mbf{H} \\wedge \\Omega^{\\rho}\\Sigma^{\\rho}S^1_+. &\n\t\\end{align*}\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:Mahowald}]\nThe Thom class is represented by a map\n$$ (\\Omega^\\rho S^{\\rho+1})^{\\td{\\mu}} \\rightarrow \\mbf{H}. $$\nWe wish to show this map is an isomorphism on $\\mbf{H}_\\star$.  The homology of $\\mbf{H}$ is the $C_2$-equivariant Steenrod algebra, computed in \\cite{HuKriz} to be\n$$ \\mbf{H}_\\star \\mbf{H} = \\mbf{H}_\\star[\\tau_0, \\tau_1, \\cdots, \\xi_1, \\xi_2, \\cdots ]\/(\\tau_i^2 = (u+a\\tau_0)\\xi_{i+1} + a\\tau_{i+1}) $$\nwith \n\\begin{align*}\n\\abs{\\tau_i} & = 2^i\\rho - \\sigma, \\\\\n\\abs{\\xi_i} & = (2^i-1)\\rho.\n\\end{align*}\nIt suffices to show it is surjective, since the two homologies are abstractly isomorphic and of finite type.  Observe that the composite\n$$ M(2) \\simeq (S^1)^{\\mu} \\rightarrow (\\Omega^\\rho S^{\\rho+1})^{\\td{\\mu}} \\rightarrow \\mbf{H} $$\nhits $\\tau_0$.  Everything is hit then, by \\cite[Thm.~5.4]{Wilson}.\n\\end{proof}\n\n\n\n\\bibliographystyle{amsalpha}\n\\nocite{*}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nThe fleet maintenance scheduling optimization (VFMSO) problem was initially proposed in \\cite{wang2019vehicle} due to the increasing demand by companies, corporations, and organizations of all sorts which rely on vehicle fleets to deliver products and services and need to maintain vehicles for safety reasons. In the problem, a vehicle fleet, such as a taxi fleet, a bus fleet, etc., can be maintained in multiple separate workshops according to a maintenance schedule. To be specific, each workshop has its own capacity and ability, meaning that on the one hand, each workshop has its own team and each team can work on only one car at the same time; on the other hand, each workshop is limited to the maintenance of the specific component(s) due to restrictions in the equipment or skill level of the staff. The maintenance schedule is optimized for each component based on its remaining useful lifetime (RUL) which has been predicted through predictive approaches or models \\cite{elattar2016prognostics}. Furthermore, the cost and time which are needed by different workshops are considered because it is possible that the maintenance of the same component produces different costs and workloads when the operation is performed in different workshops. The VFMSO problem is essential because it can not only ensure the safety of vehicles for use; at the same time, it can lead to low maintenance costs and longer lives for vehicles as well.\n\nTo enhance the approach in \\cite{wang2019vehicle}, to be specific, to handle the uncertainty in the problem and apply it on new application scenarios, in this paper, we improve it from the following two aspects:\n\n\\begin{itemize}\n\\item[1.] There exists a lot of uncertainty when we use the predicted RUL for each component as its due date, because no matter how accurate the predictive model is, it is still possible that the component will break on other dates: before the due date or later. Therefore, instead of only the RUL, we decide to involve the RUL probability distribution as the foundation to assign the maintenance time in the scheduling optimization.\n\n\\item[2.] The VFMSO problem usually leads to a large and complex solution space, however, finding the most preferred solution is the ultimate goal. To this end, AP-DI-MOEA (Automatic Preference based DI-MOEA) is developed based on the framework of DI-MOEA (Diversity-Indicator based Multi-Objective Evolutionary Algorithm) \\cite{wang2019diversity}. The new algorithm can generate the preference region automatically and find solutions with a more fine-grained resolution in the preference region.\n\\end{itemize}\n\nThis paper is organized as follows. Section \\ref{sec:formulation} formulates the enhanced VFMSO problem. A literature review on preference based optimization is provided in Section \\ref{sec:literature}. The customized multi-objective evolutionary algorithm for the enhanced VFMSO is introduced in Section \\ref{sec:customizedalg}, and in Section \\ref{sec:ap-di-moea}, we explain AP-DI-MOEA. Section \\ref{sec:experiments} presents and discusses experiments and their results. Lastly, Section \\ref{sec:conclusion} concludes the paper and outlines directions for future work.\n\n\n\\section{Problem Formulation}\n\\label{sec:formulation}\nFor a car fleet operated by an operator, the components of cars (e.g., springs, brakes, tires or the engine) can fail and should be maintained regularly. Some separate workshops are available for the maintenance of the car fleet, and the repair time and maintenance cost are known for each component in each workshop. Beside the time and cost for repairing the car component, a fixed set-up cost and set-up time are also considered for each visit of a car to a workshop, which correspond to the cost and time required for the preparation of the maintenance operation. \n\nThe enhanced VFMSO problem addressed in this paper is defined as follows:\n\\begin{enumerate}\n\\item There are $n$ cars $C=\\{{C_{1},C_{2}},\\cdots,C_{n}\\}$ and $m$ workshops $W=\\{{W_{1},W_{2}},\\\\\\cdots,W_{m}\\}$.\n\n\\item Each car $C_i$ comprises $l_i$ components to be maintained for $i=1,\\cdots,n$.\n\n\\item For each component $O_{ij}$ ($j=1,\\cdots,l_i$), i.e., the $j$th component of car $C_i$, there is a set of workshops capable of repairing it. The set of workshops is represented by $W_{ij}$ which is a subset of $W$.\n\n\\item The processing time for maintaining component $O_{ij}$ in workshop $W_k$ is predefined and denoted by $p_{ijk}$.\n\n\\item The cost for maintaining component $O_{ij}$ in workshop $W_k$ is predefined and denoted by $q_{ijk}$.\n\n\\item The set-up time of car $C_i$ in workshop $W_k$ is predefined and denoted by $x_{ik}$.\n\n\\item The set-up cost of car $C_i$ in workshop $W_k$ is predefined and denoted by $y_{ik}$.\n\n\\item The number of teams in workshop $W_k$ is predefined and denoted by $z_k$.\n\n\\item The previous repair time of component $O_{ij}$ is recorded and denoted by $L_{ij}$.\n\\end{enumerate}\n\nAt the same time, the following assumptions are made:\n\\begin{enumerate}\n\\item All workshops and teams are available at the time of the optimization and assumed to be continuously available.\n\n\\item All the components are independent from each other.\n\n\\item Times required for transport of cars from\/to workshops are included in the maintenance time and cost of cars, and the set-up time\n\n\\item Environmental changes (such as car accidents) are not considered here.\n\n\\item There are no precedence constraints among the components of different cars. Cars are maintained on a first-come-first-served basis.\n\n\\item Each team can only work on one operation at a time and an operation, once started, must run to completion.\n\n\\item No operation can start before the completion of the previous operation.\n\\end{enumerate}\n\nTwo constraints are considered in the problem. As mentioned earlier, each workshop can only repair specific components, and this is the first constraint. Another constraint is that the maintenance periods of different operations for the same car should not overlap. It is obviously wrong if two overlapping maintenance operations of a car are assigned to different workshops because one car cannot be in two different workshops at the same time. If two overlapping maintenance operations of a car are assigned to the same workshop, it is not correct either because these two maintenance operations should be grouped together as one operation in this case. The grouping strategy will be explained in Section \\ref{sec:customizedalg}.\n\nThree objectives are assumed to be relevant for the vehicle fleet operator, which are the total workload, total cost and expected number of failures. In a multi-objective optimization problem, the objectives typically are conflicting, i.e., achieving the optimal value for one objective requires some compromise on other objectives. In our problem, the fact that a faster maintenance usually is more expensive leads to the conflict between the first two objectives. The expected number of failures counts the times when the vehicles are broken on the road. Here, the expected value is used because the actual value is unknown at the time of the optimization due to uncertainties in the predictions. When the expected number of failures is large, less maintenance tasks are performed, therefore, the workload and cost can drop. \n\nLet $T_k$ denote the sum of the times spent for all operations that are processed in workshop $W_k$; $M_i$ the sum of all costs spent for all maintenance operations of car $C_i$; $F_{ij}$ the number of failures of component $O_{ij}$. Three objectives can be defined as: \n\\begin{flalign}\n&\\text{Minimize the total workload:}  ~~  f_1 = \\sum_{k=1}^{m}T_k &&\\\\\n&\\text{Minimize the total cost:}    ~~ f_2 = \\sum_{i=1}^{n}M_i &&\\\\\n\\nonumber\n&\\text{Minimize the expected number of failures:} \\\\\n&f_3 = \\sum_{i=1}^{n}\\sum_{j=1}^{l_i} \\mathbb{E}(F_{ij})\n\\end{flalign}\n\n\n\\section{LITERATURE REVIEW}\n\\label{sec:literature}\n\nMulti-objective scheduling optimization is a major topic in the research of manufacturing systems. Its fundamental task is to organize work and workloads to achieve comprehensive optimization in multiple aspects, such as the processing time, processing cost and production safety, by deploying resources, setting maintenance time and processing sequence. In past decades, this issue has received a great deal of interest and research in different fields, such as scheduling of charging\/discharging for electric vehicles \\cite{zakariazadeh2014multi}; scheduling in cloud computing \\cite{ramezani2015evolutionary}; scheduling of crude oil operations \\cite{hou2015pareto}; scheduling in the manufacturing industry to reduce carbon emissions \\cite{ding2016carbon}; scheduling medical treatments for resident patients in a hospital \\cite{jeric2012multi}; scheduling for Internet service providers \\cite{bhamare2017multi}, and so on.\n\n\nAs a typical workshop style, the flexible job shop scheduling problem (FJSP) is an essential branch of production planning problems. The FJSP consists of a set of independent jobs to be processed on multiple machines, and each job contains several operations with a predetermined order. It is assumed that each operation must be processed in specified processing time on a specific machine out of multiple alternatives. The problem has been extensively studied in the literature (for example, \\cite{chiang2013simple}, \\cite{yuan2015multiobjective}, \\cite{gao2019review}). The FJSP is the research basis of the maintenance scheduling optimization problem and many real-world problems extend the standard FJSP by adding specific features. \\cite{ozguven2010mathematical} considers FJSP-PPF (process plan flexibility), where jobs can have alternative process plans. It is assumed that the process plans are known in advance and that they are represented by linear precedence relationships. Because only one of the alternative plans has to be adopted for each job, the FJSP-PPF deals with not only routing and sequencing sub-problems, but also the process plan selection sub-problem. In this paper, a mixed-integer linear programming model is developed for the FJSP-PPF. In \\cite{demir2014effective}, a mathematical model and a genetic algorithm are proposed to handle the feature of overlapping in operations. It is assumed that a lot which contains a batch of identical items is transferred from one machine to the next only when all items in the lot have completed their processing, therefore, sublots are transferred from one machine to the next for processing without waiting for the entire lot to be processed at the predecessor machine, meaning that starting a successor operation of job is not necessary to finish of its predecessor completely. Three features are considered in \\cite{yu2017extended}, which are (1) job priority; (2) parallel operations: some operations can be processed simultaneously; (3) sequence flexibility: the sequence of some operations can be exchanged. A mixed integer liner programming formulation (MILP) model is established to formulate the problem and an improved differential evolution algorithm is designed. Because of unexpected events occurring in most of the real manufacturing systems, there is a new type of scheduling problem known as the dynamic scheduling problem. This type of problem considers random machine breakdowns, adding new machine, new job arrival, job cancellation, changing processing time, rush order, rework or quality problem, due date changing, etc. Corresponding works on the FJSP include  \\cite{fattahi2010dynamic}, \\cite{al2011robust}, \\cite{shen2015mathematical}, \\cite{ahmadi2016multi}. Compared with the standard FJSP, our VFMSO problem has some special properties: (1) flexible sequence: the sequence of the components is not predefined, but mainly influenced by the RUL probability distribution. (2) multiple problem parameters: besides the processing time, other problem parameters like the maintenance cost, set-up time, set-up cost, repair teams, etc, also have impacts on the result.\n\n\n\nOur real-world problem, like many other multi-objective optimization problems, can lead to a large objective space. However, finding a well-distributed set of solutions on the Pareto front requests a large population size and computational effort. Therefore, instead of spreading a limited size of individuals across the entire Pareto front, we decide to only focus on a part of the Pareto front, to be specific, the search for solutions will be only guided towards the preference region which, in our algorithm, is determined by the knee point. It has been argued in the literature that knee points are most interesting solutions, naturally preferred solutions and most likely the optimal choice of the decision maker (DM) \\cite{das1999characterizing, mattson2002minimal, deb2003multi, branke2004finding}.  \n\n\n\n\nThe knee point is a point for which a small improvement in any objective would lead to a large deterioration in at least one other objective. In the last decade, several methods have been presented to identify knee points or knee regions. Das \\cite{das1999characterizing} refers the point where the Pareto surface ``bulges\" the most as the knee point, and this point corresponds to the farthest solution from the convex hull of individual minima which is the minima of the single objective functions. Zitzler \\cite{zitzler2004tutorial} defines $\\epsilon$-dominance: a solution $a$ is said to $\\epsilon$-dominate a solution $b$ if and only if $f_i(a)+\\epsilon \\geq f_i(b) ~\\forall i=1,...,m$ where $m$ is the number of objectives. A solution with a higher $\\epsilon$-dominance value with respect to the other solutions in the Pareto front approximation, is a solution having higher trade-offs and in this definition corresponds to a knee point. The authors of \\cite{yu2018method} propose to calculate the density of solutions projected onto the hyperplane constructed by the extreme points of the non-dominated solutions, then identify the knee regions based on the solution density. \n\nDifferent algorithms of applying knee points in MOEA have also been proposed.\nBranke \\cite{branke2004finding} modifies the second criterion in NSGA-II \\cite{deb2002fast}, and replaces the crowding distance by either an angle-based measure or a utility-based measure. The angle-based method calculates the angle between an individual and its two neighbors in the objective space. The smaller the angle, the more clearly the individual can be classified as a knee point. However, this method can only be used for two objective problems. In the utility-based method, a marginal utility function is suggested to approximate the angle-based measure in the case of more than two objectives. The larger the external angle between a solution and its neighbors, the larger the gain in terms of linear utility obtained from substituting the neighbors with the solution of interest. However, the utility-based measure is not suited for finding knees in concave regions of the Pareto front.\n\nRachmawati \\cite{rachmawati2006multi, rachmawati2006multi2} proposes a knee-based MOEA which computes a transformation of original objective values based on a weighted sum niching approach. The extent and the density of coverage of the knee regions are controllable by the parameters for the niche strength and pool size. The strategy is susceptible to the loss of less pronounced knee regions.\n\nSch{\\\"u}tze \\cite{schutze2008approximating} investigates two strategies for the approximation of knees of bi-objective optimization problems with stochastic search algorithms. Several new definitions for identifying knee points and knee regions for bi-objective optimization problems has been suggested in \\cite{deb2011understanding} and the possibility of applying them has also been discussed.\n\nBesides the knee points, the reference points, which are normally provided by the DM, have also been used to find a set of solutions near reference points. Deb \\cite{deb2006reference} proposes an MOEA, called R-NSGA-II, by which a set of Pareto optimal solutions near a supplied set of reference points can be found. The dominance relation together with a modified crowding distance operator is used in this methodology. For all solutions of the population, the distances to all reference points are calculated and ranked. The lowest rank (over all reference points) of a solution is used as its crowding distance. Besides, a parameter $\\epsilon$ is used to control the spread of obtained solutions. Bechikh proposes KR-NSGA-II \\cite{bechikh2010searching} by extending R-NSGA-II. Instead of obtaining the reference points from the DM, in KR-NSGA-II, the knee points are used as mobile reference points and the search of the algorithm was guided towards these points. The number of knee points of the optimization problem is needed as prior information in KR-NSGA-II.\n\nGaudrie \\cite{gaudrie2019targeting} uses the projection (intersection in case of a continuous front) of the closest non-dominated point on the line connecting the estimated ideal and nadir points as default preference. Conditional Gaussian process simulations are performed to create possible Pareto fronts, each of which defines a sample for the ideal and the nadir point, and the estimated ideal and nadir are the medians of the samples.\n\nRachmawati and Srinivasan \\cite{rachmawati2009multiobjective} evaluate the worthiness of each non-dominated solution in terms of compromise between the objectives. The local maxima is then identified as potential knee solutions and the linear weighted-sums of the original objective functions are optimized to guide solutions toward the knee regions. \n\nAnother idea of incorporating preference information into evolutionary multi-objective optimization is proposed in \\cite{thiele2009preference}. They combine the fitness function and an achievement scalarizing function containing the reference point. In this approach, the preference information is given in the form of a reference point and an indicator-based evolutionary algorithm IBEA \\cite{zitzler2004indicator} is modified by embedding the preference information into the indicator. Various further preference based MOEAs have been suggested, e.g., \\cite{braun2011preference, ramirez2017knee, wang2017new}. \n\nIn our proposed algorithm, i.e., AP-DI-MOEA, we adopt the method from \\cite{das1999characterizing} to identify the knee point, design the preference region based on the knee point, and guide the search towards the preference region. The advantages of our algorithm are: (1) no prior knowledge is used in identifying the knee point and knee region; (2) the preference region is generated automatically and narrowed down step by step to benefit its accuracy; (3) our strategy cannot only handle bi-objective optimization problems, but also tri- and many-objective problems; (4) although we integrate the strategy with DI-MOEA, it may be integrated with any standard MOEAs (such as NSGA-II \\cite{deb2002fast}, SMS-EMOA \\cite{beume2007sms} and others); (5) the proposed algorithm is capable of finding preferred solutions for multi-objective optimization problems with linear, convex, concave Pareto fronts and discrete problems.\n\n\\section{Customized Algorithm for Vehicle Fleet Maintenance Scheduling Optimization}\n\\label{sec:customizedalg}\nFor our real-world VFMSO problem, we first define the execution window for each component based on its predicted RUL probability distribution which is assumed to be a normal distribution. The execution window suggests that the maintenance of the component can only start at a time spot inside the window. The mean ($\\mu$) and standard deviation ($\\sigma$) of the predicted RUL probability distribution determine the interval of the execution window, which is defined as: [$\\mu -2\\times \\sigma$, $\\mu +2\\times \\sigma$]. The interval is chosen relatively long because 95\\% of the values are within two standard deviations of the mean, therefore, maintenance before or after the interval hardly makes sense.\n\nAfter the determination of the execution window, the following two special strategies have been taken to improve the process of scheduling optimization: \n\\begin{itemize}\n\\item Grouping components.\n\\item Obtaining the penalty cost and expected number of failures by Monte Carlo simulation.\n\\end{itemize}\nLastly, evolutionary algorithm (EA) is chosen to solve this real-world application problem due to its powerful characteristics of robustness and flexibility to capture global solutions of complex combinatorial optimization problems. Moreover, EAs are well suited to solve multi-objective optimization problems due to their ability to approximate the entire Pareto front in a single run.\n\n\\subsection{Grouping Components}\nIt would be troublesome and also a waste of time and effort to send a car to workshops repeatedly in a short period of time to repair different components. In our algorithm, since each component has its execution window for its maintenance, it is possible to combine the maintenance of several components to one visit if their execution windows overlap. Especially, by grouping the maintenance of multiple components into one maintenance operation, the set-up cost and set-up time are charged only once for the complete group of components. \n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[height=120pt, width=260pt]{newgroup.png} \n\\caption{Possible groups for a car with eight components.} \n\\label{f2}\n\\end{figure}\n\nFigure~\\ref{f2} represents the execution windows of eight components of a car. The overlap of the execution windows shows the possibility of grouping these components. Combining components can only be effective if there is a common overlap of the execution window for components from the same car, and the starting time of the group operation must lie within the common overlap. In this example, component $c1$ can be grouped with $c2$ and\/or $c3$ due to the overlap between their execution windows. Other possible group structures can be deduced in the same manner. \n\n\\subsection{Monte Carlo Simulation}\nWithin the execution window of a component, an arbitrary time can be chosen as the starting time for maintaining the component. The maintenance time of each component should be as close as possible to its real due date, because:\n\\begin{itemize}\n\\item Performing the maintenance too early results in higher maintenance costs in the long term, because more maintenance tasks have to be done. \n\\item The risk of breaking down on the road will increase if the maintenance date is too late.\n\\end{itemize}\nTherefore, we use Monte Carlo simulation to simulate the ``real'' due dates for each component. In our experiments, stability can be achieved at a few hundred samples, in our case, $1000$ samples of the due date are generated in the execution window of each component according to its predicted RUL probability distribution (see Section IV). Figure~\\ref{f1} shows an example of the execution window evolved from the predicted RUL probability distribution of a component. After 1000 sampled due dates are generated in the execution window, the scheduled maintenance date of the component is compared with these samples one by one, and each comparison can lead to three situations. Let us use $d_{ij}^v$  to denote the $v$th due date sample of component $O_{ij}$; and $D_{ij}$ the scheduled maintenance date of component $O_{ij}$. Three possibilities after the comparison are:\n\\\\\nCase 1) $~D_{ij} < d_{ij}^v$\\\\\nThe scheduled maintenance date is earlier than the sample (or the ``real'' due date) means that the component will be maintained before it is broken. In this case, its useful life between the maintenance date and the due date will be wasted. Therefore, a corresponding penalty cost is imposed to reflect the waste. To calculate the penalty cost, a linear penalty function is suggested based on the following assumptions:\n\\begin{itemize}\n\\item If a component is maintained when it is new or the previous maintenance has just completed, the penalty cost would be the full cost of maintaining it, which is $c+s$: the maintenance cost of the component and the set-up cost of the car;\n\\item If a component is maintained at exactly its due date, the penalty cost would be 0.\n\\end{itemize}\n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[ width=4in]{newpenalty.png} \n\\caption{Execution window of a component.} \n\\label{f1}\n\\end{figure}\n\\vspace{-0.0cm}\n\nAssume $d_{ij}^v$ is ``Sampled Due date'' in Figure~\\ref{f1}, and $D_{ij}$ is ``Maintenance date a'', in this case, $D_{ij}$ is earlier than $d_{ij}^v$. The penalty cost of ``Maintenance date a'' for ``Sampled Due date'' would be the vertical dotted line above ``Maintenance date a''.\n\\\\\nCase 2) $~D_{ij} > d_{ij}^v$\\\\\nThe scheduled maintenance date is later than the sample means that the maintenance date is too late and the defect occurs on the use. Still, $d_{ij}^v$ is ``Sampled Due date'' in Figure~\\ref{f1}, but the scheduled maintenance date $D_{ij}$ is ``Maintenance date b''. In this case, $D_{ij}$ is later than $d_{ij}^v$, and the vehicle will break down on the road. In our algorithm, the number of failures will be increased by one.\n\\\\\nCase 3) $~D_{ij} = d_{ij}^v$\\\\\nThe ideal situation is that the maintenance date is scheduled on the due date. The component can be maintained exactly at the date that the component is broken. In this case, there is no penalty or failure.\n\nThe averages of the penalty costs and the number of failures from $1000$ due date samples will be used as the penalty cost and expected number of failures for the scheduled maintenance date of the component. For each operation (the single-component operation or group operation), its cost consists of three parts: the set-up cost of the car, the maintenance costs and the penalty costs of all components of the operation. The penalty cost of components is a part of the total cost, and the expected number of failures of components is the third objective to be minimized in our multi-objective optimization.\n\n\n\\subsection{Implementation of Evolutionary Algorithm Operators}\nTo solve our application problem with an EA, there are several basic issues we need to deal with, such as, how to represent an individual or solution in the population (Chromosome Encoding); how to take these chromosomes into a process of evolution (Genotype-Phenotype Mapping); how to create variations of solutions in each iteration (Genetic Operators). Details of these topics are given in the following subsections.\n\n\\subsubsection{Chromosome Encoding}\nIn our algorithm, a three-vector chromosome (Figure~\\ref{f0}) is proposed to represent an individual, and the three vectors are:\n\\begin{itemize\n  \\item Group structure vector: the group structures of components.\n  \\item Starting time vector: the starting times of operations.\n  \\item Workshop assignment vector: the workshops for operations.\n\\end{itemize}\n\n\\begin{figure*}[!htbp]\n\\hspace{-0.7cm}\n\\includegraphics[height=0.45in, width=5.2in]{chrom1.png}\n\\caption{Three-vector chromosome.}\n\\label{f0}\n\\end{figure*}\n\nThe group structure vector gives the information of which components are in the same group, it is initialized by randomly picking a feasible group structure for each car (check the details in \\cite{wang2019vehicle}). The generation of the starting time vector should be later than the generation of the group structure vector because the starting time of each operation is determined by the execution window which is the entire execution window of the component for a single-component operation or the execution window intersection for a group operation. A time spot is randomly selected from the execution window or execution window intersection for each operation in order to initialize the starting time vector. \n\nA workshop is considered as ``several workshops\" based on its capacity (the number of teams). By this way, the schedule for each workshop team can be achieved from the solution. For example, consider that two workshops have three and four repairing teams respectively. Then, group operations can be randomly assigned to seven ``workshops'', the former three and the latter four represent the corresponding teams in two workshops.\n\n\\subsubsection{Genotype-Phenotype Mapping}\nTo use the power of EAs to obtain a better population, we need to evaluate each chromosome and give the better ones higher probabilities to produce offspring. This is done by genotype-phenotype mapping or decoding the chromosome. In our problem, it is to convert an individual into a feasible schedule to calculate the objectives and constraints which represent the relative superiority of a chromosome. The genotype-phenotype mapping can be easily achieved in our algorithm because the group structure, the starting time and the workshop team of the operations can be acquired directly from each individual. When converting an individual into a schedule, it is possible that the processing times of two or more operations assigned to the same workshop team are overlapping since the starting time of each operation is decided in the starting time vector. In this situation, the principle of first-come-first-served is followed: the starting time and processing time of the earlier started operation remain the same; the starting time of the later started operation is delayed until the completion of the previous operation; the processing time of the later started operation remains the same; while, an extra waiting time is added to the later started operation as a penalty because the vehicle waits in the workshop for the maintenance.\n\n\\subsubsection{Genetic Operators}\nIn accordance with the problem and its encoding, specific crossover and mutation operators have been designed for our problem (check the details in \\cite{wang2019vehicle}). Both operators are applied separately to the three parts of the chromosome.\n\nFor the group structure vector, multi-point crossover can be used as crossover operator and the number of cutting points depends on the length of the vector. The same cutting points can be applied to the starting time vector when performing crossover. However, the change on the group structure vector as a consequence of the crossover may result in the invalidity of genes in the starting time vector because it is possible that the group members and execution window intersections have changed due to the new group structure. Therefore, when performing the crossover on the starting time vector, the starting times of all operations should be checked based on the new group structure and a new starting time is produced randomly from the correct intersection in the case that the starting time of an operation is invalid. The multi-point crossover can be applied to the workshop assignment vector as well. \n\nThe mutation operator alters one or more gene values in a chromosome. Similarly, the mutation should be operated on the group structure vector first due to its impact on the starting time vector; the starting time of operations should be checked and corrected after the mutation is done on the group structure vector. Afterwards, several gene values can be altered in the staring time vector and workshop assignment vector to generate a new individual.\n\n\n\\section{Proposed Preference based Algorithm}\n\\label{sec:ap-di-moea}\nAs the number of objectives and decision variables increases, the number of non-dominated solutions tends to grow exponentially \\cite{pal2018decor}. This brings more challenges on achieving efficiently a solution set with satisfactory convergence and diversity. At the same time, a huge number of solutions is needed to approximate the entire Pareto front. However, a big population means more computational time and resources. To overcome these difficulties, we propose an automatic preference based MOEA, which can generate the preference region or the region of interest (ROI) automatically and find non-dominated solutions in the preference region instead of the entire Pareto front. The automatic preference based MOEA is developed based on the framework of DI-MOEA (Diversity-Indicator based Multi-Objective Evolutionary Algorithm) \\cite{wang2019diversity}. We call our new algorithm AP-DI-MOEA.\n \nDI-MOEA is an indicator-based MOEA, it has shown to be competitive to other MOEAs on common multi-objective benchmark problems. Moreover, it is invariant to the shape of the Pareto front and can achieve evenly spread Pareto front approximations.\nDI-MOEA adopts a hybrid selection scheme:\n\\begin{itemize\n \\item The ($\\mu$ + $\\mu$) generational selection operator is used when the parent population can be layered into multiple dominance ranks. The intention is to accelerate convergence until all solutions are non-dominated.\n \\item The ($\\mu$ + 1) steady state selection operator is adopted in the case that all solutions in the parent population are mutually non-dominated and the diversity is the main selection criterion to achieve a uniform distribution of the solutions on the Pareto front.\n\\end{itemize}\n\nDI-MOEA employs non-dominated sorting as the first ranking criterion; the diversity indicator, i.e., the Euclidean distance based geometric mean gap indicator, as the second, diversity-based ranking criterion to guide the search. Two variants of DI-MOEA, denoted as DI-1 and DI-2, exist, which use the crowding distance and diversity indicator, respectively, as the second criteria in the ($\\mu$ + $\\mu$) generational selection operator. While, to ensure the uniformity of the final solution set, the diversity indicator is used by both variants in the ($\\mu$ + 1) steady state selection operator. Analogously, two variants of AP-DI-MOEA, i.e., AP-DI-1 and AP-DI-2, are derived from the two variants of DI-MOEA\n\n\n\nThe workings of AP-DI-MOEA are outlined in Algorithm 1. (Exceedance of) $Enum\\_P$ is a predefined condition (In our algorithm, $Enum\\_P$ is the number of evaluations.) to divide the algorithm into two phases: learning phase and decision phase. In the learning phase, the algorithm explores the possible area of Pareto optimal solutions and finds the rough approximations of the Pareto front. In the decision phase, the algorithm identifies the preference region and finds preferred solutions. When the algorithm starts running and satisfies $Enum\\_P$ at some moment, the first preference region will be generated and  $Enum\\_P$ will be updated for determining a new future moment when the preference region needs to be updated. The process of updating $Enum\\_P$ continues until the end. The first $Enum\\_P$ is a boundary line. Before it is satisfied, AP-DI-MOEA runs exactly like DI-MOEA to approximate the whole Pareto front; while, after it is satisfied, the preference region is generated automatically and AP-DI-MOEA finds solutions focusing on the preference region. The subsequent values of $Enum\\_P$ define the later moments to update the preference region step by step, eventually, a precise ROI with a proper size can be achieved.\n\n\\begin{figure*}[!htbp]\n\\vspace{-3.5cm}\n\\hspace{-3cm}\n\\includegraphics[height=10in]{pg_0002.pdf}\n\\end{figure*}\n\n\\begin{figure*}[!htbp]\n\\hspace{-2.5cm}\n\\includegraphics[width=7in]{al2.pdf}\n\\end{figure*}\n\n\n\\iffalse \n\\addtocounter{algorithm}{1}\n\\begin{algorithm}[!htbp]\n\\setstretch{0.8}\n \t\\caption{Finding the knee point and defining the preference region.}\n    \\label{algorithm:2}\n \t\\begin{algorithmic}[1]\n        \\STATE $n \\leftarrow$ the number of objectives;\n        \\STATE $P_t \\leftarrow$ current population;\n        \\STATE $\\epsilon$; \/\/parameter ($>$0) for distinguishing convex\/concave shape;\n        \\STATE $popsize \\leftarrow |P_t|$; \/\/population size\n       \n        \\STATE Declare $Q[n]$;  \/\/upper quartile objective values of $P_t$ \n        \\STATE Declare $L[n]$;  \/\/worst objective values of $P_t$\n        \\STATE Declare $knee[n]$;  \/\/knee point of $P_t$\n        \\STATE Declare $P\\_region[n]$;  \/\/preference region of $P_t$\n        \\STATE Declare $Expoints[n][n]$; \/\/extreme points\n        \\STATE $foundknee \\leftarrow false$;\n       \n       \n        \\FOR{{\\bf each} $i \\in \\{ 1, \\dots, n\\}$}\n        \\STATE sort($P_t$) by the $i$th objective in ascending order;\n        \\STATE $Q[i] \\leftarrow P_t$.get\\_index$(\\frac{3}{4}\\times popsize)$.get\\_obj($i$); \/\/upper quartile value of the $i$th objective \n        \\STATE $L[i] \\leftarrow P_t$.get\\_index$(popsize)$.get\\_obj($i$);\/\/the largest (worst) value of the $i$th objective\n        \\ENDFOR\n        \\FORALL{solution $s \\in P_t$}\n        \\IF{$s$.get\\_obj($i=1,...,n) > Q[i]$ }\n        \\STATE remove $s$ from $P_t$;\n        \\ENDIF\n        \\ENDFOR\n        \\STATE $Expoints[\\centerdot][\\centerdot] \\leftarrow$ extreme points in $P_t$;\n        \\STATE $num_a\\leftarrow$ the number of points in concave region of hyperplane formed by $Expoints[\\centerdot][\\centerdot]$;\n        \\STATE $num_v \\leftarrow |P_t|-num_a$; \/\/the number of points in convex region\n        \\IF{$(num_v - num_a > \\epsilon)$}\n        \\STATE \/\/roughly convex shape\n        \\STATE remove solutions in concave region from $P_t$;\n        \\ELSIF{$(num_a - num_v > \\epsilon)$}\n        \\STATE \/\/roughly concave shape\n        \\STATE remove solutions in convex region from $P_t$;\n        \\ELSE\n        \\FORALL{solution $s\\in P_t$}\n         \\STATE calculate hypervolume of $s$ with reference point $L[\\centerdot]$;\n         \\STATE update the largest hypervolume value ($max\\_h$);\n        \\ENDFOR\n        \\STATE $knee[\\centerdot] \\leftarrow$ solution with $max\\_h$;\n        \\STATE $foundknee \\leftarrow true$;\n        \\ENDIF\n        \\IF{($foundknee == false$)}\n        \\FORALL{solution $s\\in P_t$}\n        \\STATE calculate distance between $s$ and hyperplane formed by $Expoints[\\centerdot][\\centerdot]$;\n        \\STATE update the largest distance ($max\\_d$);\n        \\ENDFOR\n        \\STATE $knee[\\centerdot] \\leftarrow$ solution with $max\\_d$;\n        \\ENDIF\n        \\FOR{{\\bf each} $i \\in \\{ 1, \\dots, n\\}$}\n        \\STATE $P\\_region[i] \\leftarrow knee[i] + (L[i]-knee[i]) \\times 85\\%$\n        \\ENDFOR\n \t\\end{algorithmic}\n\\end{algorithm}\n\\fi\n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=3.3in]{knee-explain.png} \n\\caption{Finding the knee point in bi-dimensional space.} \n\\label{knee}\n\\end{figure}\n\nThe first\/new preference region is formed based on the population at the moment when the condition of $Enum\\_P$ is satisfied, especially the knee point of the population. Algorithm 2 gives the details of line 41 in Algorithm 1, it introduces the steps of finding the knee point of a non-dominated solution set and constituting a hypercube shaped preference region according to the knee point. Figure~\\ref{knee} also gives an illustration of finding the knee point in bi-dimensional space. Firstly, the upper quartile objective values (line 13 in Algorithm 2) in the solution set are used as a boundary to define outliers and solutions outside this boundary are removed  (line 16-20 in Algorithm 2). The extreme solutions (the solutions with the maximum value in one objective) (line 21 in Algorithm 2) are then found inside the boundary and a hyperplane is formed based on the extreme solutions. In a bi-dimensional space (Figure~\\ref{knee}), the hyperplane is only a line connecting two extreme solutions. According to the numbers of points below and above the hyperplane (line 22 - 23 in Algorithm 2), the shape of the solution set can be roughly perceived.\nWe will distinguish between ``convex'' and ``concave'' regions. Points in the \\textit{convex} (\\textit{concave}) \\textit{region} are dominating (dominated by) at least one point in the hyperplane spanned by the extreme points. However, when the number of the points in the convex region and the number of points in the concave region is close enough, it implies that the shape of the current solution set is almost linear. This occurs both when the true Pareto front is linear and when the solution set is converged very well in a small area of the Pareto front. A parameter $\\epsilon$ then is used to represent the closeness and it is a small number decided by the size of the solution set. In the case that the shape of the current solution set is (almost) linear, the solution with the largest hypervolume value with regards to the worst objective vector (line 14 in Algorithm 2) is adopted as the knee point (line 32 - 36 in Algorithm 2). While, under the condition that the shape of the current solution set is convex or concave, the solution in the convex or concave region with the largest Euclidean distance to the hyperplane is chosen as the knee point (line 39 - 42 in Algorithm 2). After the knee point is found, the preference region can be determined based on the knee point by the following formula:\n\\begin{align}\nP\\_region[i] = knee[i] + (L[i]-knee[i]) \\times 85\\%\n\\end{align}\n\nLet $i$ denotes the $i$th objective, as in Algorithm 2, $L[i]$ is the worst value of the $i$th objective in the population, $knee[i]$ is the $i$th objective value of the knee point and $P\\_region[i]$ is the upper bound of the $i$th objective. W.l.o.g. we assume the objectives are to be minimized and the lower bound of preference region is the origin point. According to the formula, we can see that the first preference region is relatively large (roughly 85\\% of the entire Pareto front). With the increase in the number of iteration, the preference region will be updated and becomes smaller and smaller because every preference region picks 85\\% of the current Pareto front. Eventually, we want the preference region can reach a proper range, say, 15\\% of the initial Pareto front. The process of narrowing down the preference region step by step can benefit the accuracy of the preference region.\n\nIn the interest of clarity, Algorithm 1 only shows the workings of AP-DI-1, the workings of AP-DI-2 can be obtained by replacing crowding distance with the diversity indicator contribution.\nIn the ($\\mu$ + $\\mu$) generational selection operator (line 14 - 36 in Algorithm 1), when there is no preference region, the second ranking criteria (the crowding distance for AP-DI-1; the diversity indicator for AP-DI-2) for all solutions on the last front are calculated and the population will be truncated based on non dominated sorting and the second ranking criteria (line 28 - 29 in Algorithm 1). While, if a preference region already exists, both the second ranking criteria and Euclidean distance to the knee point for all solutions on the last front are calculated and the population will be truncated based on first non dominated sorting, then the second ranking criteria, lastly, Euclidean distance to the knee point (line 31 - 32 in Algorithm 1). In the ($\\mu$ + 1) steady state selection operator (line 38 - 59 in Algorithm 1), firstly, the value of $Enum\\_P$ is compared with the current number of evaluations to determine if a (new) preference region should be generated. When it is time to do so, the preference region is generated through Algorithm 2 (line 41 in Algorithm 1), at the same time, the value of $Enum\\_P$ is updated to the next moment when the preference region is to be updated (line 42 in Algorithm 1). There are different strategies to assign the values of $Enum\\_P$. In our algorithm, we divide the whole computing budget into two parts, the first half is used to find an initial entire Pareto front approximation, and the second half is used to update the preference region and find solutions in the preference region. Assume the total computing budget is $Enum\\_T$ (the number of evaluations), then the first value of $Enum\\_P$ is $\\frac{1}{2}\\times Enum\\_T$. Due to the reason that we expect a final preference region with a size of around 15\\% of the initial entire Pareto front and each new preference region takes 85\\% of the current Pareto front, according to the formula: $0.85^{12} \\approx 0.14$, the value of $Enum\\_P$ can be updated by the following formula:\n\\begin{align}\nEnum\\_P = Enum\\_P + (Enum\\_T\/2)\/12\n\\end{align}\n\nAnother half of budget can be divided into $12$ partial-budgets and a new preference region is constituted after each partial-budget. In the end, the final preference region is achieved and solutions focusing on this preference region are obtained. For the rest part of the ($\\mu$ + 1) steady state selection operator, likewise, when there is a preference region, three ranking criteria (1. non-dominated sorting; 2. diversity indicator; 3. the Euclidean distance to the knee point) work together to achieve a well-converged and well-distributed set of Pareto optimal solutions in the preference region.\n\n\\section{Experimental Results}\n\\label{sec:experiments}\n\\subsection{Experimental Design}\nIn this section, simulations are conducted to demonstrate the performance of proposed algorithms on both benchmark problems and our real-world application problems. All experiments are implemented based on the MOEA Framework (\\url{http:\/\/www.moeaframework.org\/}), which is a Java-based framework for multi-objective optimization.\n\nFor the two variants of AP-DI-MOEA: AP-DI-1 and AP-DI-2, their performances have been compared with DI-MOEA: DI-1, DI-2 and NSGA-III \\cite{deb2014evolutionary}. We compare our algorithm with NSGA-III because NSGA-III is a representative state-of-the-art evolutionary multi-objective algorithm and it is very powerful to handle problems with non-linear characteristics. For bi-objective benchmark problems, algorithms are tested on ZDT1 and ZDT2 with 30 variables. For three objective benchmark problems, DTLZ1 with 7 variables and DTLZ2 with 12 variables are tested. For the real-world application problem of VFMSO, experiments have been conducted on two instances with different sizes. The configurations of the two instances, such as the predicted RUL probability distribution, the processing time and maintenance cost of each component, the set-up time and cost of each car, are made available on \\url{http:\/\/moda.liacs.nl}. On every problem, we run each algorithm $30$ times with different seeds, while the same $30$ different seeds are used for all algorithms. All the experiments are performed with a population size of $100$; and for bi-objective problems, experiments are run with a budget of $22000$ (objective function) evaluations, DTLZ three objective problems with a budget of $120000$ evaluations, the VFMSO problems with a budget of $1200000$ evaluations. This setting is chosen to be more realistic in the light of the applications in scheduling that we ultimately want to solve.\n\n\n\n\n\n\n\\subsection{Experiments on bi-objective problems}\n\nBi-objective problems are optimized with a total budget of $22000$ evaluations, when the number of evaluations reaches $10000$ times, the first preference region is generated, then after every $1200$ evaluations, the preference region will be updated. Figure~\\ref{fig:ZDT1} shows the Pareto front approximations from a typical run on ZDT1 (left column) and ZDT2 (right column). The graphs on the upper row are obtained from DI-1 and AP-DI-1, while the graphs on the lower row are from DI-2 and AP-DI-2. In each graph, the entire Pareto front approximation from DI-MOEA and the preferred solutions from AP-DI-MOEA (or \\textit{AP solutions}) are presented, at the same time, the preference region of AP-DI-MOEA is also shown by the gray area.\n\n\\begin{figure}[htbp]\n\\hspace{-0.58cm}\n\\subfigure[ZDT1]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{zdt1-di1-tdi1-w.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{zdt1-di2-tdi2-w.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\\subfigure[ZDT2]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{zdt2-di1-tdi1-w.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{zdt2-di2-tdi2-w.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\n\\caption{Pareto front approximation on ZDT1 and ZDT2.}\n\\label{fig:ZDT1}\n\\end{figure}\n\nBesides the visualization of the Pareto fronts, we also compute the knee point of the entire final Pareto front approximation from DI-MOEA via the strategy described in Algorithm 2. For each run of DI-MOEA and AP-DI-MOEA with the same seed, the following two issues have been checked: \n\\begin{itemize\n\\item If the knee point from DI-MOEA is in the preference region achieved by its derived AP-DI-MOEA;\n\\item If the knee point from DI-MOEA is dominated by or dominating AP solutions; or if it is a non-dominated solution (mutually non-dominated with all AP solutions).\n\\end{itemize}\n\nTable~\\ref{table-kneezdt1} shows the results of 30 runs. For ZDT1 problem, all 30 knee points from DI-1 and DI-2 are in the preference regions from AP-DI-1 and AP-DI-2 respectively; in all these knee points, 10 from DI-1 and 7 from DI-2 are dominated by AP solutions. For ZDT2 problem, most knee points are not in corresponding preference regions, but for those in the preference regions, almost all of them are dominated by AP solutions. Please note that when a knee point from DI-MOEA is outside of the preference region from AP-DI-MOEA, it is not possible that it can dominate any AP solutions because all AP solutions are in the preference region and only solutions in the left side of the gray area can dominate AP solutions. \n\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from DI-MOEA and AP solutions on ZDT problems.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{ZDT1}  & \\multicolumn{2}{|c|}{ZDT2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& DI-1\/ & DI-2\/ & DI-1\/ & DI-2\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 & AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 20 & 23 & 1 & 1\\\\\n\\cline{2-6}\npreference & Dominated & 10 & 7 & 9 & 9 \\\\\n \\cline{2-6}\nregion & Dominating & 0 & 0 & 0 &  0 \\\\\n\\hline\nOutside & Incomparable &  0 & 0 & 20 & 20 \\\\\n\\cline{2-6}\np-region & Dominated &  0 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular}\n\\label{table-kneezdt1}\n\\end{table} \n\nWe also perform the same comparison between AP-DI-MOEA and NSGA-III, the results are shown in Table~\\ref{table-kneezdt1-nsga3}. For ZDT1 problem, all knee points from NSGA-III are in the preference regions from AP-DI-MOEA. Some of these knee points dominate AP solutions. For ZDT2 problem, most knee points from NSGA-III are not in the preference regions and these knee points are incomparable with AP solutions. For the knee points in the preference regions, all three dominating relations with AP solutions appear. For both problems, when the knee point from NSGA-III is dominating AP solutions, it only dominates one AP solution.\n\n\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from NSGA-III and AP solutions on ZDT problems.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{ZDT1}  & \\multicolumn{2}{|c|}{ZDT2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& NSGA-III\/ & NSGA-III\/ & NSGA-III\/ & NSGA-III\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 & AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 14 & 19 & 3 & 1\\\\\n\\cline{2-6}\npreference & Dominated & 0 & 0 & 2 & 3 \\\\\n \\cline{2-6}\nregion & Dominating & 16 & 11 & 4 &  6 \\\\\n\\hline\nOutside & Incomparable &  0 & 0 & 21 & 20 \\\\\n\\cline{2-6}\np-region & Dominated &  0 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular}\n\\label{table-kneezdt1-nsga3}\n\\end{table} \n\nInstead of spreading the population across the entire Pareto front, we only focus on the preference region. To ensure that our algorithm can guide the search towards the preference region and the achieved solution set is distributed across the preference region, we compare the performance of AP-DI-MOEA, DI-MOEA and NSGA-III in the preference region. For each Pareto front approximation from DI-MOEA and NSGA-III, the solutions in the corresponding preference region from AP-DI-MOEA are picked, and we compare these solutions with AP solutions through the hypervolume indicator. The point formed by the largest objective values over all solutions in the preference region is adopted as the reference point when calculating the hypervolume indicator. It has been found that all hypervolume values of new solution sets from DI-MOEA and NSGA-III in the preference region are worse than the hypervolume values of the solution sets from AP-DI-MOEA, which proves that the mechanism indeed works in practice. Figure~\\ref{box:ZDT} shows box plots of the distribution of hypervolume indicators over 30 runs.\n\n\n\n\\begin{figure}[htbp]\n\\hspace{-0.58cm}\n\\subfigure[ZDT1]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{zdt1-di1-box1.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{zdt1-di2-box2.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\\subfigure[ZDT2]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{zdt2-di1-box1.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{zdt2-di2-box2.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\\caption{Boxplots comparing the hypervolume values on ZDT1 and ZDT2.}\n\\label{box:ZDT}\n\\end{figure}\n\n\n\\subsection{Experiments on three objective problems}\nDTLZ1 and DTLZ2 are chosen as three objective benchmark problems to investigate our algorithms. They are performed with a total budget of $120000$ fitness evaluations, when the evaluation reaches $60000$ times, the first preference region is formed, then after every $5000$ evaluations, the preference region is updated. Figure~\\ref{fig:dtlz} shows the Pareto front approximations from a typical run on DTLZ1 (left column) and DTLZ2 (right column). The upper graphs are obtained from DI-1 and AP-DI-1, while the lower graphs are from DI-2 and AP-DI-2. In each graph, the Pareto front approximations from DI-MOEA and corresponding AP-DI-MOEA are given. Since the target region is actually an axis aligned box, the obtained knee region (i.e., the intersection of the axis aligned box with the Pareto front) has an inverted triangle shape for these two benchmark problems.\n\n\\begin{figure*}[htbp]\n\\hspace{-3.5cm}\n\\subfigure[DTLZ1 3 objective problem]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.5in,width=4.4in]{dtlz1-di1-tdi1-w.png}\\\\\n        \\vspace{0.45cm}\n        \\includegraphics[height=2.5in,width=4.4in]{dtlz1-di2-tdi2-w.png}\\\\\n       \n       \n    \\end{minipage}%\n}%\n\\hspace{2cm}\n\\subfigure[DTLZ2 3 objective problem]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4.4in]{dtlz2-di1-tdi1-w.png}\\\\\n       \n        \\includegraphics[height=2.7in,width=4.4in]{dtlz2-di2-tdi2-w.png}\\\\\n       \n       \n    \\end{minipage}%\n}%\n\\caption{Pareto front approximation on DTLZ1 and DTLZ2.}\n\\label{fig:dtlz}\n\\end{figure*}\n\n\n\nTable~\\ref{table-kneedtlz} shows the space and dominance relation of the knee point from DI-MOEA and the solution set from AP-DI-MOEA over 30 runs. For DTLZ1 problem, most knee points from DI-MOEA are in their respective preference regions and all knee points are mutually non-dominated with AP solutions. For DTLZ2 problem, we observed that more knee points are not in the corresponding preference regions. This is because too few solutions from DI-MOEA are in the preference region. For DTLZ1 problem, six solutions from DI-MOEA are in the corresponding preference region on average for each run, while, for DTLZ2 problem, only less than two solutions are in the corresponding preference region on average. Therefore, we can see that on the one side, it is normal that many knee points from the entire Pareto fronts are not in their corresponding preference regions; on the other side, our aim of finding more fine-grained resolution in the preference region has been well achieved because only few solutions can be obtained in the preference region if we spread the population across the entire Pareto front. At the same time, one knee point from DI-1 on DTLZ2 is dominated by solutions from the corresponding AP-DI-1, which proves that AP-DI-MOEA can converge better than DI-MOEA because AP-DI-MOEA focuses on the preference region.\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from DI-MOEA and AP solutions on DTLZ problems.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{DTLZ1}  & \\multicolumn{2}{|c|}{DTLZ2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& DI-1\/ & DI-2\/ & DI-1\/ & DI-2\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 & AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 29 & 27 & 10 & 13\\\\\n\\cline{2-6}\npreference & Dominated & 0 & 0 & 1 & 0 \\\\\n \\cline{2-6}\nregion & Dominating & 0 & 0 & 0 & 0 \\\\\n\\hline\nOutside & Incomparable & 1 & 3 & 19 & 17 \\\\\n\\cline{2-6}\np-region & Dominated &  0 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular}\n\\label{table-kneedtlz}\n\\end{table} \n\n\nAP-DI-1 and AP-DI-2 have also been compared with NSGA-III in the same way. Table~\\ref{table-kneedtlz_nsga3} shows the comparison result. For DTLZ1, the average number of solutions from NSGA-III in the corresponding preference regions from AP-DI-MOEA is six. Still, almost all knee solutions from NSGA-III are in the preference region. For DTLZ2, the average number of solutions from NSGA-III in the corresponding preference region from AP-DI-MOEA is less than one, while, in more than half of 30 runs, the knee points from NSGA-III are still in the preference region. To some extent, it can be concluded that the preference regions from AP-DI-MOEA are accurate. It can also be observed that AP-DI-1 behaves better than AP-DI-2 on DTLZ2, because two knee points from NSGA-III dominate the solutions from AP-DI-2.\n\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from NSGA-III and AP solutions on DTLZ problems.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{DTLZ1}  & \\multicolumn{2}{|c|}{DTLZ2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& NSGA-III\/ & NSGA-III\/ & NSGA-III\/ & NSGA-III\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 &AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 30 & 29 & 14 & 17\\\\\n\\cline{2-6}\npreference & Dominated & 0 & 0 & 1 & 1 \\\\\n \\cline{2-6}\nregion & Dominating & 0 & 0 & 0 & 2 \\\\\n\\hline\nOutside & Incomparable & 0 & 1 & 15 & 10 \\\\\n\\cline{2-6}\np-region & Dominated &  0 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular}\n\\label{table-kneedtlz_nsga3}\n\\end{table} \n\nSimilarly, we pick from DI-MOEA and NSGA-III solutions which are in the corresponding preference region of AP-DI-MOEA, and the hypervolume indicator value is compared between these solutions and AP solutions. It has been found that all hypervolume values of solutions from AP-DI-MOEA are better than those of solutions from DI-MOEA and NSGA-III. The left column of Figure~\\ref{box:dtlz} shows box plots of the distribution of hypervolume values over 30 runs on DTLZ1, and the right column shows the hypervolume comparison on DTLZ2.\n\n\n\\begin{figure}[htbp]\n\\hspace{-0.58cm}\n\\subfigure[DTLZ1]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{dtzl1-di1-box.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{dtlz1-di2-box.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\\subfigure[DTLZ2]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{dtlz2-di1-box.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{dtlz2-di2-box.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\n\\caption{Boxplots comparing the hypervolume values on DTLZ1 and DTLZ2.}\n\\label{box:dtlz}\n\\end{figure}\n\nIn our experiments, we decide half of the total budget is used to find an initial Pareto front because it turned out to be a good compromise: half budget for the initial Pareto front and another half budget for the solutions focusing on the preference region. We also run experiments using 25\\% and 75\\% of the total budget for the initial Pareto front. Figure~\\ref{fig:dtlz-budget} presents the entire Pareto front from DI-MOEA and the Pareto front from AP-DI-MOEA with different budgets for the initial Pareto front. The left two images are on DTLZ1 and the right two images are on DTLZ2. The uppper two images are from DI-1 and AP-DI-1; the lower two images are from DI-2 and AP-DI-2. In the legend labels, 50\\%, 25\\% and 75\\% indicate the budgets which are utilized to find the initial entire Pareto front. It can be observed that the preference region from AP-DI-MOEA with 50\\% of budget are located on a better position than with 25\\% and 75\\% budgets, and the position of the preference region from AP-DI-MOEA with 50\\% of budget is more stable. Therefore, in our algorithm, 50\\% of budget is used before the generation of preference region.\n\n\\begin{figure*}[htbp]\n\\hspace{-3.5cm}\n\\subfigure[DTLZ1 3 objective problem]{\n    \\begin{minipage}[t]{0.50\\linewidth}\n        \\centering\n        \\includegraphics[height=2.6in,width=4.28in]{dtlz1-di1-tdi1-3combine-1.png}\\\\\n       \n        \\includegraphics[height=2.6in,width=4.28in]{dtlz1-di2-tdi2-3combine-3.png}\\\\\n       \n       \n    \\end{minipage}%\n}%\n\\hspace{1.5cm}\n\\subfigure[DTLZ2 3 objective problem]{\n    \\begin{minipage}[t]{0.50\\linewidth}\n        \\centering\n        \\includegraphics[height=2.6in,width=4.2in]{dtlz2-di1-tdi1-3combine-16.png}\\\\\n       \n        \\includegraphics[height=2.6in,width=4.2in]{dtlz2-di2-tdi2-3combine-16.png}\\\\\n       \n       \n    \\end{minipage}%\n}%\n\\caption{Pareto front approximation by different budgets generating initial Pareto front.}\n\\label{fig:dtlz-budget}\n\\end{figure*}\n\n\\subsection{Experiments on Vehicle Fleet Maintenance Scheduling Optimization}\nThe budget of $1200000$ evaluations has been used on the real-world application problems, and $600000$ of them are for the initial Pareto front. After that, the preference region is updated after every $50000$ evaluations.\nThe VFMSO problem has been tested with different sizes. Figure~\\ref{fig:20cars} shows Pareto front approximations of a problem with $20$ cars and $3$ workshops (V1), and each car contains $13$ components: one engine, four springs, four brakes and four tires \\cite{van2019modeling}. It can be observed that AP-DI-1 and AP-DI-2 can zoom in the entire Pareto front and find solutions in the preference region, at the same time, both AP-DI-1 and AP-DI-2 converge better than their corresponding DI-1 and DI-2. A similar conclusion can be drawn from Pareto fronts approximations of the problem with $30$ cars and $5$ workshops (V2) in Figure~\\ref{fig:30cars}.\n\n\nIn Figure~\\ref{fig:2030cars}, We put the Pareto front approximations from DI-MOEA, AP-DI-MOEA and NSGA-III on V1 (left) and V2 (right) together. The behaviours of DI-1, DI-2 and NSGA-III are similar on V1, so are the behaviours of AP-DI-1 and AP-DI-2 on this problem. While, DI-2 and AP-DI-2 converge better than DI-1 and AP-DI-1 on V2 problem. The behaviour of NSGA-III is between that of DI-1 and DI-2.\n\n\n\\begin{figure*}[htbp]\n\\hspace{-3.8cm}\n\\subfigure[DI-1 \\& AP-DI-1]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{20car-di1-tdi1-legend.png}\\\\\n       \n    \\end{minipage}%\n}%\n\\hspace{3.2cm}\n\\subfigure[DI-2 \\& AP-DI-2]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{20car-di2-tdi2-legend.png}\\\\\n       \n    \\end{minipage}%\n}%\n\\caption{Pareto front approximation on VFMSO problem with 20 cars and 3 workshops.}\n\\label{fig:20cars}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\hspace{-3.8cm}\n\\subfigure[DI-1 \\& AP-DI-1]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{30car-di1-tdi1-legend.png}\\\\\n       \n    \\end{minipage}%\n}%\n\\hspace{3.2cm}\n\\subfigure[DI-2 \\& AP-DI-2]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{30car-di2-tdi2-legend.png}\\\\\n       \n    \\end{minipage}%\n}%\n\\caption{Pareto front approximation on VFMSO problem with 30 cars and 5 workshops.}\n\\label{fig:30cars}\n\\end{figure*}\n\n\n\n\\begin{figure*}[htbp]\n\\hspace{-3.5cm}\n\\subfigure[V1]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{20car-di1-tdi1-di2-tdi2-ns3.png}\\\\\n       \n    \\end{minipage}%\n}%\n\\hspace{3cm}\n\\subfigure[V2]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{30car-di1-tdi1-di2-tdi2-ns3.png}\\\\\n       \n    \\end{minipage}%\n}%\n\n\\caption{Pareto front approximation on VFMSO problem by DI-MOEA, AP-DI-MOEA and NSGA-III.}\n\\label{fig:2030cars}\n\\end{figure*}\n\n\nTable~\\ref{table-v1} gives the space and dominance relation of knee points from DI-MOEA and solutions from AP-DI-MOEA on these two VFMSO problems. For both problems, only few knee points from DI-MOEA are in the preference regions of AP-DI-MOEA, and the main reason is that the Pareto front of AP-DI-MOEA converges better than that of DI-MOEA, in some cases, the Pareto front of DI-MOEA cannot even reach the corresponding preference region. More importantly, it can be observed that most knee points from DI-MOEA, no matter whether in the preference region or outside of the preference region, are dominated by the solutions from AP-DI-MOEA. This phenomenon is even more obvious for the application problem with bigger size and run with the same budget as the smaller one: for V2, 90\\% of knee points from DI-MOEA are dominated by the solutions from AP-DI-MOEA.\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from DI-MOEA and AP solutions on V1 and V2.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{V1}  & \\multicolumn{2}{|c|}{V2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& DI-1\/ & DI-2\/ & DI-1\/ & DI-2\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 & AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 0 & 0 & 0 & 0\\\\\n\\cline{2-6}\npreference & Dominated & 9 & 7 & 9 & 6 \\\\\n \\cline{2-6}\nregion & Dominating & 0 & 0 & 0 &  0 \\\\\n\\hline\nOutside & Incomparable &  4 & 9 & 3 & 3 \\\\\n\\cline{2-6}\np-region & Dominated & 17 & 14 & 18 & 21 \\\\\n\\hline\n\\end{tabular}\n\\label{table-v1}\n\\end{table} \n\n\nTable~\\ref{table-v1-nsga3} gives the space and dominance relation of knee points from NSGA-III and AP solutions. For both problems, again, most knee points from NSGA-III are not in the preference regions of AP-DI-MOEA. Some knee points from NSGA-III are dominated by AP solutions and most of them are incomparable with AP solutions. \n\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from NSGA-III and AP solutions on V1 and V2.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{V1}  & \\multicolumn{2}{|c|}{V2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& NSGA-III\/ & NSGA-III\/ & NSGA-III\/ & NSGA-III\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 & AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 0 & 0 & 0 & 1\\\\\n\\cline{2-6}\npreference & Dominated & 0 & 1 & 3& 2 \\\\\n \\cline{2-6}\nregion & Dominating & 0 & 0 & 1 &  1 \\\\\n\\hline\nOutside & Incomparable &  23 & 24 & 21 & 18 \\\\\n\\cline{2-6}\np-region & Dominated & 7 & 5 & 5 & 8 \\\\\n\\hline\n\\end{tabular}\n\\label{table-v1-nsga3}\n\\end{table} \n\n\\iffalse \n The right image of Figure~\\ref{fig:30cars-2} presents the entire Pareto front approximations of V2 from four different MOEAs: DI-1, DI-2, RVEA and NSGA-III. It can be seen that DI-MOEA (both DI-1 and DI-2) and NSGA-III converge to the similar area in the end, while, RVEA reaches another area of the objective space. Table~\\ref{table_hy} provides the average hypervolume value of the four Pareto fronts from 30 runs and the reference point for each run is formed by the largest objective value from all solutions. It can be seen that DI-2 behaves the best and RVEA the worst.\n\n\n\\begin{table}[htbp]\n\\caption{Hypervolume values}\n\\label{table_hy}\n\\begin{center}\n\\begin{tabular}{l|c}\n\\hline\nDI-1 & 0.0525\\\\\n\\hline\nDI-2 & 0.0576\\\\\n\\hline\nRVEA & 0.0202\\\\\n\\hline\nNSGAIII & 0.0534\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\fi\n\n\n\n\n\n\\section{CONCLUSIONS}\n\\label{sec:conclusion}\nIn this paper, a preference based multi-objective evolutionary algorithm, AP-DI-MOEA, is proposed. In the absence of explicitly provided preferences, the knee region is usually treated as the region of interest or preference region. Given this, AP-DI-MOEA can generate the knee region automatically and can find solutions with a more fine-grained resolution in the knee region. This has been demonstrated on the bi-objective problems ZDT1 and ZDT2, and the three objective problems DTLZ1 and DTLZ2. In the benchmark, the new approach was also proven to perform better than NSGA-III which was included in the benchmark as a state-of-the-art reference algorithm.\n\nThe research for the preference based algorithm was originally motivated by a real-world optimization problem, namely, Vehicle Fleet Maintenance Scheduling Optimization (VFMSO), which is described in this paper in a new formulation as a three objective discrete optimization problem. A customized set of operators (initialization, recombination, and mutation) is proposed for a multi-objective evolutionary algorithm with a selection strategy based on DI-MOEA and, respectively, AP-DI-MOEA. The experimental results of AP-DI-MOEA on two real-world application problem instances of different scales show that the newly proposed algorithm can generate preference regions automatically and it (in both cases) finds clearly better and more concentrated solution sets in the preference region than DI-MOEA. For completeness, it was also tested against NSGA-III and a better approximation in the preference region was observed by AP-DI-MOEA .\n\nSince our real-world VFMSO problem is our core issue to be solved, and its Pareto front is convex, we did not consider problems with an irregular shape.\nIt would be an interesting question how to adapt the algorithm to problems with more irregular shapes. Besides, the proposed approach requires a definition of knee points. Future work will provide a more detailed comparison of different variants of methods to generate knee points, as they are briefly introduced in Section \\ref{sec:literature}. In the application of maintenance scheduling, it will also be important to integrate robustness and uncertainty in the problem definition. It is desirable to generate schedules that are robust within a reasonable range of disruptions and uncertainties such as machine breakdowns and processing time variability.\n\n\n\n\\section*{Acknowledgment}\nThis work is part of the research programme Smart Industry SI2016 with project name CIMPLO and project number 15465, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).\n\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\nThe fleet maintenance scheduling optimization (VFMSO) problem was initially proposed in \\cite{wang2019vehicle} due to the increasing demand by companies, corporations, and organizations of all sorts which rely on vehicle fleets to deliver products and services and need to maintain vehicles for safety reasons. In the problem, a vehicle fleet, such as a taxi fleet, a bus fleet, etc., can be maintained in multiple separate workshops according to a maintenance schedule. To be specific, each workshop has its own capacity and ability, meaning that on the one hand, each workshop has its own team and each team can work on only one car at the same time; on the other hand, each workshop is limited to the maintenance of the specific component(s) due to restrictions in the equipment or skill level of the staff. The maintenance schedule is optimized for each component based on its remaining useful lifetime (RUL) which has been predicted through predictive approaches or models \\cite{elattar2016prognostics}. Furthermore, the cost and time which are needed by different workshops are considered because it is possible that the maintenance of the same component produces different costs and workloads when the operation is performed in different workshops. The VFMSO problem is essential because it can not only ensure the safety of vehicles for use; at the same time, it can lead to low maintenance costs and longer lives for vehicles as well.\n\nTo enhance the approach in \\cite{wang2019vehicle}, to be specific, to handle the uncertainty in the problem and apply it on new application scenarios, in this paper, we improve it from the following two aspects:\n\n\\begin{itemize}\n\\item[1.] There exists a lot of uncertainty when we use the predicted RUL for each component as its due date, because no matter how accurate the predictive model is, it is still possible that the component will break on other dates: before the due date or later. Therefore, instead of only the RUL, we decide to involve the RUL probability distribution as the foundation to assign the maintenance time in the scheduling optimization.\n\n\\item[2.] The VFMSO problem usually leads to a large and complex solution space, however, finding the most preferred solution is the ultimate goal. To this end, AP-DI-MOEA (Automatic Preference based DI-MOEA) is developed based on the framework of DI-MOEA (Diversity-Indicator based Multi-Objective Evolutionary Algorithm) \\cite{wang2019diversity}. The new algorithm can generate the preference region automatically and find solutions with a more fine-grained resolution in the preference region.\n\\end{itemize}\n\nThis paper is organized as follows. Section \\ref{sec:formulation} formulates the enhanced VFMSO problem. A literature review on preference based optimization is provided in Section \\ref{sec:literature}. The customized multi-objective evolutionary algorithm for the enhanced VFMSO is introduced in Section \\ref{sec:customizedalg}, and in Section \\ref{sec:ap-di-moea}, we explain AP-DI-MOEA. Section \\ref{sec:experiments} presents and discusses experiments and their results. Lastly, Section \\ref{sec:conclusion} concludes the paper and outlines directions for future work.\n\n\n\\section{Problem Formulation}\n\\label{sec:formulation}\nFor a car fleet operated by an operator, the components of cars (e.g., springs, brakes, tires or the engine) can fail and should be maintained regularly. Some separate workshops are available for the maintenance of the car fleet, and the repair time and maintenance cost are known for each component in each workshop. Beside the time and cost for repairing the car component, a fixed set-up cost and set-up time are also considered for each visit of a car to a workshop, which correspond to the cost and time required for the preparation of the maintenance operation. \n\nThe enhanced VFMSO problem addressed in this paper is defined as follows:\n\\begin{enumerate}\n\\item There are $n$ cars $C=\\{{C_{1},C_{2}},\\cdots,C_{n}\\}$ and $m$ workshops $W=\\{{W_{1},W_{2}},\\\\\\cdots,W_{m}\\}$.\n\n\\item Each car $C_i$ comprises $l_i$ components to be maintained for $i=1,\\cdots,n$.\n\n\\item For each component $O_{ij}$ ($j=1,\\cdots,l_i$), i.e., the $j$th component of car $C_i$, there is a set of workshops capable of repairing it. The set of workshops is represented by $W_{ij}$ which is a subset of $W$.\n\n\\item The processing time for maintaining component $O_{ij}$ in workshop $W_k$ is predefined and denoted by $p_{ijk}$.\n\n\\item The cost for maintaining component $O_{ij}$ in workshop $W_k$ is predefined and denoted by $q_{ijk}$.\n\n\\item The set-up time of car $C_i$ in workshop $W_k$ is predefined and denoted by $x_{ik}$.\n\n\\item The set-up cost of car $C_i$ in workshop $W_k$ is predefined and denoted by $y_{ik}$.\n\n\\item The number of teams in workshop $W_k$ is predefined and denoted by $z_k$.\n\n\\item The previous repair time of component $O_{ij}$ is recorded and denoted by $L_{ij}$.\n\\end{enumerate}\n\nAt the same time, the following assumptions are made:\n\\begin{enumerate}\n\\item All workshops and teams are available at the time of the optimization and assumed to be continuously available.\n\n\\item All the components are independent from each other.\n\n\\item Times required for transport of cars from\/to workshops are included in the maintenance time and cost of cars, and the set-up time\n\n\\item Environmental changes (such as car accidents) are not considered here.\n\n\\item There are no precedence constraints among the components of different cars. Cars are maintained on a first-come-first-served basis.\n\n\\item Each team can only work on one operation at a time and an operation, once started, must run to completion.\n\n\\item No operation can start before the completion of the previous operation.\n\\end{enumerate}\n\nTwo constraints are considered in the problem. As mentioned earlier, each workshop can only repair specific components, and this is the first constraint. Another constraint is that the maintenance periods of different operations for the same car should not overlap. It is obviously wrong if two overlapping maintenance operations of a car are assigned to different workshops because one car cannot be in two different workshops at the same time. If two overlapping maintenance operations of a car are assigned to the same workshop, it is not correct either because these two maintenance operations should be grouped together as one operation in this case. The grouping strategy will be explained in Section \\ref{sec:customizedalg}.\n\nThree objectives are assumed to be relevant for the vehicle fleet operator, which are the total workload, total cost and expected number of failures. In a multi-objective optimization problem, the objectives typically are conflicting, i.e., achieving the optimal value for one objective requires some compromise on other objectives. In our problem, the fact that a faster maintenance usually is more expensive leads to the conflict between the first two objectives. The expected number of failures counts the times when the vehicles are broken on the road. Here, the expected value is used because the actual value is unknown at the time of the optimization due to uncertainties in the predictions. When the expected number of failures is large, less maintenance tasks are performed, therefore, the workload and cost can drop. \n\nLet $T_k$ denote the sum of the times spent for all operations that are processed in workshop $W_k$; $M_i$ the sum of all costs spent for all maintenance operations of car $C_i$; $F_{ij}$ the number of failures of component $O_{ij}$. Three objectives can be defined as: \n\\begin{flalign}\n&\\text{Minimize the total workload:}  ~~  f_1 = \\sum_{k=1}^{m}T_k &&\\\\\n&\\text{Minimize the total cost:}    ~~ f_2 = \\sum_{i=1}^{n}M_i &&\\\\\n\\nonumber\n&\\text{Minimize the expected number of failures:} \\\\\n&f_3 = \\sum_{i=1}^{n}\\sum_{j=1}^{l_i} \\mathbb{E}(F_{ij})\n\\end{flalign}\n\n\n\\section{LITERATURE REVIEW}\n\\label{sec:literature}\n\nMulti-objective scheduling optimization is a major topic in the research of manufacturing systems. Its fundamental task is to organize work and workloads to achieve comprehensive optimization in multiple aspects, such as the processing time, processing cost and production safety, by deploying resources, setting maintenance time and processing sequence. In past decades, this issue has received a great deal of interest and research in different fields, such as scheduling of charging\/discharging for electric vehicles \\cite{zakariazadeh2014multi}; scheduling in cloud computing \\cite{ramezani2015evolutionary}; scheduling of crude oil operations \\cite{hou2015pareto}; scheduling in the manufacturing industry to reduce carbon emissions \\cite{ding2016carbon}; scheduling medical treatments for resident patients in a hospital \\cite{jeric2012multi}; scheduling for Internet service providers \\cite{bhamare2017multi}, and so on.\n\n\nAs a typical workshop style, the flexible job shop scheduling problem (FJSP) is an essential branch of production planning problems. The FJSP consists of a set of independent jobs to be processed on multiple machines, and each job contains several operations with a predetermined order. It is assumed that each operation must be processed in specified processing time on a specific machine out of multiple alternatives. The problem has been extensively studied in the literature (for example, \\cite{chiang2013simple}, \\cite{yuan2015multiobjective}, \\cite{gao2019review}). The FJSP is the research basis of the maintenance scheduling optimization problem and many real-world problems extend the standard FJSP by adding specific features. \\cite{ozguven2010mathematical} considers FJSP-PPF (process plan flexibility), where jobs can have alternative process plans. It is assumed that the process plans are known in advance and that they are represented by linear precedence relationships. Because only one of the alternative plans has to be adopted for each job, the FJSP-PPF deals with not only routing and sequencing sub-problems, but also the process plan selection sub-problem. In this paper, a mixed-integer linear programming model is developed for the FJSP-PPF. In \\cite{demir2014effective}, a mathematical model and a genetic algorithm are proposed to handle the feature of overlapping in operations. It is assumed that a lot which contains a batch of identical items is transferred from one machine to the next only when all items in the lot have completed their processing, therefore, sublots are transferred from one machine to the next for processing without waiting for the entire lot to be processed at the predecessor machine, meaning that starting a successor operation of job is not necessary to finish of its predecessor completely. Three features are considered in \\cite{yu2017extended}, which are (1) job priority; (2) parallel operations: some operations can be processed simultaneously; (3) sequence flexibility: the sequence of some operations can be exchanged. A mixed integer liner programming formulation (MILP) model is established to formulate the problem and an improved differential evolution algorithm is designed. Because of unexpected events occurring in most of the real manufacturing systems, there is a new type of scheduling problem known as the dynamic scheduling problem. This type of problem considers random machine breakdowns, adding new machine, new job arrival, job cancellation, changing processing time, rush order, rework or quality problem, due date changing, etc. Corresponding works on the FJSP include  \\cite{fattahi2010dynamic}, \\cite{al2011robust}, \\cite{shen2015mathematical}, \\cite{ahmadi2016multi}. Compared with the standard FJSP, our VFMSO problem has some special properties: (1) flexible sequence: the sequence of the components is not predefined, but mainly influenced by the RUL probability distribution. (2) multiple problem parameters: besides the processing time, other problem parameters like the maintenance cost, set-up time, set-up cost, repair teams, etc, also have impacts on the result.\n\n\n\nOur real-world problem, like many other multi-objective optimization problems, can lead to a large objective space. However, finding a well-distributed set of solutions on the Pareto front requests a large population size and computational effort. Therefore, instead of spreading a limited size of individuals across the entire Pareto front, we decide to only focus on a part of the Pareto front, to be specific, the search for solutions will be only guided towards the preference region which, in our algorithm, is determined by the knee point. It has been argued in the literature that knee points are most interesting solutions, naturally preferred solutions and most likely the optimal choice of the decision maker (DM) \\cite{das1999characterizing, mattson2002minimal, deb2003multi, branke2004finding}.  \n\n\n\n\nThe knee point is a point for which a small improvement in any objective would lead to a large deterioration in at least one other objective. In the last decade, several methods have been presented to identify knee points or knee regions. Das \\cite{das1999characterizing} refers the point where the Pareto surface ``bulges\" the most as the knee point, and this point corresponds to the farthest solution from the convex hull of individual minima which is the minima of the single objective functions. Zitzler \\cite{zitzler2004tutorial} defines $\\epsilon$-dominance: a solution $a$ is said to $\\epsilon$-dominate a solution $b$ if and only if $f_i(a)+\\epsilon \\geq f_i(b) ~\\forall i=1,...,m$ where $m$ is the number of objectives. A solution with a higher $\\epsilon$-dominance value with respect to the other solutions in the Pareto front approximation, is a solution having higher trade-offs and in this definition corresponds to a knee point. The authors of \\cite{yu2018method} propose to calculate the density of solutions projected onto the hyperplane constructed by the extreme points of the non-dominated solutions, then identify the knee regions based on the solution density. \n\nDifferent algorithms of applying knee points in MOEA have also been proposed.\nBranke \\cite{branke2004finding} modifies the second criterion in NSGA-II \\cite{deb2002fast}, and replaces the crowding distance by either an angle-based measure or a utility-based measure. The angle-based method calculates the angle between an individual and its two neighbors in the objective space. The smaller the angle, the more clearly the individual can be classified as a knee point. However, this method can only be used for two objective problems. In the utility-based method, a marginal utility function is suggested to approximate the angle-based measure in the case of more than two objectives. The larger the external angle between a solution and its neighbors, the larger the gain in terms of linear utility obtained from substituting the neighbors with the solution of interest. However, the utility-based measure is not suited for finding knees in concave regions of the Pareto front.\n\nRachmawati \\cite{rachmawati2006multi, rachmawati2006multi2} proposes a knee-based MOEA which computes a transformation of original objective values based on a weighted sum niching approach. The extent and the density of coverage of the knee regions are controllable by the parameters for the niche strength and pool size. The strategy is susceptible to the loss of less pronounced knee regions.\n\nSch{\\\"u}tze \\cite{schutze2008approximating} investigates two strategies for the approximation of knees of bi-objective optimization problems with stochastic search algorithms. Several new definitions for identifying knee points and knee regions for bi-objective optimization problems has been suggested in \\cite{deb2011understanding} and the possibility of applying them has also been discussed.\n\nBesides the knee points, the reference points, which are normally provided by the DM, have also been used to find a set of solutions near reference points. Deb \\cite{deb2006reference} proposes an MOEA, called R-NSGA-II, by which a set of Pareto optimal solutions near a supplied set of reference points can be found. The dominance relation together with a modified crowding distance operator is used in this methodology. For all solutions of the population, the distances to all reference points are calculated and ranked. The lowest rank (over all reference points) of a solution is used as its crowding distance. Besides, a parameter $\\epsilon$ is used to control the spread of obtained solutions. Bechikh proposes KR-NSGA-II \\cite{bechikh2010searching} by extending R-NSGA-II. Instead of obtaining the reference points from the DM, in KR-NSGA-II, the knee points are used as mobile reference points and the search of the algorithm was guided towards these points. The number of knee points of the optimization problem is needed as prior information in KR-NSGA-II.\n\nGaudrie \\cite{gaudrie2019targeting} uses the projection (intersection in case of a continuous front) of the closest non-dominated point on the line connecting the estimated ideal and nadir points as default preference. Conditional Gaussian process simulations are performed to create possible Pareto fronts, each of which defines a sample for the ideal and the nadir point, and the estimated ideal and nadir are the medians of the samples.\n\nRachmawati and Srinivasan \\cite{rachmawati2009multiobjective} evaluate the worthiness of each non-dominated solution in terms of compromise between the objectives. The local maxima is then identified as potential knee solutions and the linear weighted-sums of the original objective functions are optimized to guide solutions toward the knee regions. \n\nAnother idea of incorporating preference information into evolutionary multi-objective optimization is proposed in \\cite{thiele2009preference}. They combine the fitness function and an achievement scalarizing function containing the reference point. In this approach, the preference information is given in the form of a reference point and an indicator-based evolutionary algorithm IBEA \\cite{zitzler2004indicator} is modified by embedding the preference information into the indicator. Various further preference based MOEAs have been suggested, e.g., \\cite{braun2011preference, ramirez2017knee, wang2017new}. \n\nIn our proposed algorithm, i.e., AP-DI-MOEA, we adopt the method from \\cite{das1999characterizing} to identify the knee point, design the preference region based on the knee point, and guide the search towards the preference region. The advantages of our algorithm are: (1) no prior knowledge is used in identifying the knee point and knee region; (2) the preference region is generated automatically and narrowed down step by step to benefit its accuracy; (3) our strategy cannot only handle bi-objective optimization problems, but also tri- and many-objective problems; (4) although we integrate the strategy with DI-MOEA, it may be integrated with any standard MOEAs (such as NSGA-II \\cite{deb2002fast}, SMS-EMOA \\cite{beume2007sms} and others); (5) the proposed algorithm is capable of finding preferred solutions for multi-objective optimization problems with linear, convex, concave Pareto fronts and discrete problems.\n\n\\section{Customized Algorithm for Vehicle Fleet Maintenance Scheduling Optimization}\n\\label{sec:customizedalg}\nFor our real-world VFMSO problem, we first define the execution window for each component based on its predicted RUL probability distribution which is assumed to be a normal distribution. The execution window suggests that the maintenance of the component can only start at a time spot inside the window. The mean ($\\mu$) and standard deviation ($\\sigma$) of the predicted RUL probability distribution determine the interval of the execution window, which is defined as: [$\\mu -2\\times \\sigma$, $\\mu +2\\times \\sigma$]. The interval is chosen relatively long because 95\\% of the values are within two standard deviations of the mean, therefore, maintenance before or after the interval hardly makes sense.\n\nAfter the determination of the execution window, the following two special strategies have been taken to improve the process of scheduling optimization: \n\\begin{itemize}\n\\item Grouping components.\n\\item Obtaining the penalty cost and expected number of failures by Monte Carlo simulation.\n\\end{itemize}\nLastly, evolutionary algorithm (EA) is chosen to solve this real-world application problem due to its powerful characteristics of robustness and flexibility to capture global solutions of complex combinatorial optimization problems. Moreover, EAs are well suited to solve multi-objective optimization problems due to their ability to approximate the entire Pareto front in a single run.\n\n\\subsection{Grouping Components}\nIt would be troublesome and also a waste of time and effort to send a car to workshops repeatedly in a short period of time to repair different components. In our algorithm, since each component has its execution window for its maintenance, it is possible to combine the maintenance of several components to one visit if their execution windows overlap. Especially, by grouping the maintenance of multiple components into one maintenance operation, the set-up cost and set-up time are charged only once for the complete group of components. \n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[height=120pt, width=260pt]{newgroup.png} \n\\caption{Possible groups for a car with eight components.} \n\\label{f2}\n\\end{figure}\n\nFigure~\\ref{f2} represents the execution windows of eight components of a car. The overlap of the execution windows shows the possibility of grouping these components. Combining components can only be effective if there is a common overlap of the execution window for components from the same car, and the starting time of the group operation must lie within the common overlap. In this example, component $c1$ can be grouped with $c2$ and\/or $c3$ due to the overlap between their execution windows. Other possible group structures can be deduced in the same manner. \n\n\\subsection{Monte Carlo Simulation}\nWithin the execution window of a component, an arbitrary time can be chosen as the starting time for maintaining the component. The maintenance time of each component should be as close as possible to its real due date, because:\n\\begin{itemize}\n\\item Performing the maintenance too early results in higher maintenance costs in the long term, because more maintenance tasks have to be done. \n\\item The risk of breaking down on the road will increase if the maintenance date is too late.\n\\end{itemize}\nTherefore, we use Monte Carlo simulation to simulate the ``real'' due dates for each component. In our experiments, stability can be achieved at a few hundred samples, in our case, $1000$ samples of the due date are generated in the execution window of each component according to its predicted RUL probability distribution (see Section IV). Figure~\\ref{f1} shows an example of the execution window evolved from the predicted RUL probability distribution of a component. After 1000 sampled due dates are generated in the execution window, the scheduled maintenance date of the component is compared with these samples one by one, and each comparison can lead to three situations. Let us use $d_{ij}^v$  to denote the $v$th due date sample of component $O_{ij}$; and $D_{ij}$ the scheduled maintenance date of component $O_{ij}$. Three possibilities after the comparison are:\n\\\\\nCase 1) $~D_{ij} < d_{ij}^v$\\\\\nThe scheduled maintenance date is earlier than the sample (or the ``real'' due date) means that the component will be maintained before it is broken. In this case, its useful life between the maintenance date and the due date will be wasted. Therefore, a corresponding penalty cost is imposed to reflect the waste. To calculate the penalty cost, a linear penalty function is suggested based on the following assumptions:\n\\begin{itemize}\n\\item If a component is maintained when it is new or the previous maintenance has just completed, the penalty cost would be the full cost of maintaining it, which is $c+s$: the maintenance cost of the component and the set-up cost of the car;\n\\item If a component is maintained at exactly its due date, the penalty cost would be 0.\n\\end{itemize}\n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[ width=4in]{newpenalty.png} \n\\caption{Execution window of a component.} \n\\label{f1}\n\\end{figure}\n\\vspace{-0.0cm}\n\nAssume $d_{ij}^v$ is ``Sampled Due date'' in Figure~\\ref{f1}, and $D_{ij}$ is ``Maintenance date a'', in this case, $D_{ij}$ is earlier than $d_{ij}^v$. The penalty cost of ``Maintenance date a'' for ``Sampled Due date'' would be the vertical dotted line above ``Maintenance date a''.\n\\\\\nCase 2) $~D_{ij} > d_{ij}^v$\\\\\nThe scheduled maintenance date is later than the sample means that the maintenance date is too late and the defect occurs on the use. Still, $d_{ij}^v$ is ``Sampled Due date'' in Figure~\\ref{f1}, but the scheduled maintenance date $D_{ij}$ is ``Maintenance date b''. In this case, $D_{ij}$ is later than $d_{ij}^v$, and the vehicle will break down on the road. In our algorithm, the number of failures will be increased by one.\n\\\\\nCase 3) $~D_{ij} = d_{ij}^v$\\\\\nThe ideal situation is that the maintenance date is scheduled on the due date. The component can be maintained exactly at the date that the component is broken. In this case, there is no penalty or failure.\n\nThe averages of the penalty costs and the number of failures from $1000$ due date samples will be used as the penalty cost and expected number of failures for the scheduled maintenance date of the component. For each operation (the single-component operation or group operation), its cost consists of three parts: the set-up cost of the car, the maintenance costs and the penalty costs of all components of the operation. The penalty cost of components is a part of the total cost, and the expected number of failures of components is the third objective to be minimized in our multi-objective optimization.\n\n\n\\subsection{Implementation of Evolutionary Algorithm Operators}\nTo solve our application problem with an EA, there are several basic issues we need to deal with, such as, how to represent an individual or solution in the population (Chromosome Encoding); how to take these chromosomes into a process of evolution (Genotype-Phenotype Mapping); how to create variations of solutions in each iteration (Genetic Operators). Details of these topics are given in the following subsections.\n\n\\subsubsection{Chromosome Encoding}\nIn our algorithm, a three-vector chromosome (Figure~\\ref{f0}) is proposed to represent an individual, and the three vectors are:\n\\begin{itemize\n  \\item Group structure vector: the group structures of components.\n  \\item Starting time vector: the starting times of operations.\n  \\item Workshop assignment vector: the workshops for operations.\n\\end{itemize}\n\n\\begin{figure*}[!htbp]\n\\hspace{-0.7cm}\n\\includegraphics[height=0.45in, width=5.2in]{chrom1.png}\n\\caption{Three-vector chromosome.}\n\\label{f0}\n\\end{figure*}\n\nThe group structure vector gives the information of which components are in the same group, it is initialized by randomly picking a feasible group structure for each car (check the details in \\cite{wang2019vehicle}). The generation of the starting time vector should be later than the generation of the group structure vector because the starting time of each operation is determined by the execution window which is the entire execution window of the component for a single-component operation or the execution window intersection for a group operation. A time spot is randomly selected from the execution window or execution window intersection for each operation in order to initialize the starting time vector. \n\nA workshop is considered as ``several workshops\" based on its capacity (the number of teams). By this way, the schedule for each workshop team can be achieved from the solution. For example, consider that two workshops have three and four repairing teams respectively. Then, group operations can be randomly assigned to seven ``workshops'', the former three and the latter four represent the corresponding teams in two workshops.\n\n\\subsubsection{Genotype-Phenotype Mapping}\nTo use the power of EAs to obtain a better population, we need to evaluate each chromosome and give the better ones higher probabilities to produce offspring. This is done by genotype-phenotype mapping or decoding the chromosome. In our problem, it is to convert an individual into a feasible schedule to calculate the objectives and constraints which represent the relative superiority of a chromosome. The genotype-phenotype mapping can be easily achieved in our algorithm because the group structure, the starting time and the workshop team of the operations can be acquired directly from each individual. When converting an individual into a schedule, it is possible that the processing times of two or more operations assigned to the same workshop team are overlapping since the starting time of each operation is decided in the starting time vector. In this situation, the principle of first-come-first-served is followed: the starting time and processing time of the earlier started operation remain the same; the starting time of the later started operation is delayed until the completion of the previous operation; the processing time of the later started operation remains the same; while, an extra waiting time is added to the later started operation as a penalty because the vehicle waits in the workshop for the maintenance.\n\n\\subsubsection{Genetic Operators}\nIn accordance with the problem and its encoding, specific crossover and mutation operators have been designed for our problem (check the details in \\cite{wang2019vehicle}). Both operators are applied separately to the three parts of the chromosome.\n\nFor the group structure vector, multi-point crossover can be used as crossover operator and the number of cutting points depends on the length of the vector. The same cutting points can be applied to the starting time vector when performing crossover. However, the change on the group structure vector as a consequence of the crossover may result in the invalidity of genes in the starting time vector because it is possible that the group members and execution window intersections have changed due to the new group structure. Therefore, when performing the crossover on the starting time vector, the starting times of all operations should be checked based on the new group structure and a new starting time is produced randomly from the correct intersection in the case that the starting time of an operation is invalid. The multi-point crossover can be applied to the workshop assignment vector as well. \n\nThe mutation operator alters one or more gene values in a chromosome. Similarly, the mutation should be operated on the group structure vector first due to its impact on the starting time vector; the starting time of operations should be checked and corrected after the mutation is done on the group structure vector. Afterwards, several gene values can be altered in the staring time vector and workshop assignment vector to generate a new individual.\n\n\n\\section{Proposed Preference based Algorithm}\n\\label{sec:ap-di-moea}\nAs the number of objectives and decision variables increases, the number of non-dominated solutions tends to grow exponentially \\cite{pal2018decor}. This brings more challenges on achieving efficiently a solution set with satisfactory convergence and diversity. At the same time, a huge number of solutions is needed to approximate the entire Pareto front. However, a big population means more computational time and resources. To overcome these difficulties, we propose an automatic preference based MOEA, which can generate the preference region or the region of interest (ROI) automatically and find non-dominated solutions in the preference region instead of the entire Pareto front. The automatic preference based MOEA is developed based on the framework of DI-MOEA (Diversity-Indicator based Multi-Objective Evolutionary Algorithm) \\cite{wang2019diversity}. We call our new algorithm AP-DI-MOEA.\n \nDI-MOEA is an indicator-based MOEA, it has shown to be competitive to other MOEAs on common multi-objective benchmark problems. Moreover, it is invariant to the shape of the Pareto front and can achieve evenly spread Pareto front approximations.\nDI-MOEA adopts a hybrid selection scheme:\n\\begin{itemize\n \\item The ($\\mu$ + $\\mu$) generational selection operator is used when the parent population can be layered into multiple dominance ranks. The intention is to accelerate convergence until all solutions are non-dominated.\n \\item The ($\\mu$ + 1) steady state selection operator is adopted in the case that all solutions in the parent population are mutually non-dominated and the diversity is the main selection criterion to achieve a uniform distribution of the solutions on the Pareto front.\n\\end{itemize}\n\nDI-MOEA employs non-dominated sorting as the first ranking criterion; the diversity indicator, i.e., the Euclidean distance based geometric mean gap indicator, as the second, diversity-based ranking criterion to guide the search. Two variants of DI-MOEA, denoted as DI-1 and DI-2, exist, which use the crowding distance and diversity indicator, respectively, as the second criteria in the ($\\mu$ + $\\mu$) generational selection operator. While, to ensure the uniformity of the final solution set, the diversity indicator is used by both variants in the ($\\mu$ + 1) steady state selection operator. Analogously, two variants of AP-DI-MOEA, i.e., AP-DI-1 and AP-DI-2, are derived from the two variants of DI-MOEA\n\n\n\nThe workings of AP-DI-MOEA are outlined in Algorithm 1. (Exceedance of) $Enum\\_P$ is a predefined condition (In our algorithm, $Enum\\_P$ is the number of evaluations.) to divide the algorithm into two phases: learning phase and decision phase. In the learning phase, the algorithm explores the possible area of Pareto optimal solutions and finds the rough approximations of the Pareto front. In the decision phase, the algorithm identifies the preference region and finds preferred solutions. When the algorithm starts running and satisfies $Enum\\_P$ at some moment, the first preference region will be generated and  $Enum\\_P$ will be updated for determining a new future moment when the preference region needs to be updated. The process of updating $Enum\\_P$ continues until the end. The first $Enum\\_P$ is a boundary line. Before it is satisfied, AP-DI-MOEA runs exactly like DI-MOEA to approximate the whole Pareto front; while, after it is satisfied, the preference region is generated automatically and AP-DI-MOEA finds solutions focusing on the preference region. The subsequent values of $Enum\\_P$ define the later moments to update the preference region step by step, eventually, a precise ROI with a proper size can be achieved.\n\n\\begin{figure*}[!htbp]\n\\vspace{-3.5cm}\n\\hspace{-3cm}\n\\includegraphics[height=10in]{pg_0002.pdf}\n\\end{figure*}\n\n\\begin{figure*}[!htbp]\n\\hspace{-2.5cm}\n\\includegraphics[width=7in]{al2.pdf}\n\\end{figure*}\n\n\n\\iffalse \n\\addtocounter{algorithm}{1}\n\\begin{algorithm}[!htbp]\n\\setstretch{0.8}\n \t\\caption{Finding the knee point and defining the preference region.}\n    \\label{algorithm:2}\n \t\\begin{algorithmic}[1]\n        \\STATE $n \\leftarrow$ the number of objectives;\n        \\STATE $P_t \\leftarrow$ current population;\n        \\STATE $\\epsilon$; \/\/parameter ($>$0) for distinguishing convex\/concave shape;\n        \\STATE $popsize \\leftarrow |P_t|$; \/\/population size\n       \n        \\STATE Declare $Q[n]$;  \/\/upper quartile objective values of $P_t$ \n        \\STATE Declare $L[n]$;  \/\/worst objective values of $P_t$\n        \\STATE Declare $knee[n]$;  \/\/knee point of $P_t$\n        \\STATE Declare $P\\_region[n]$;  \/\/preference region of $P_t$\n        \\STATE Declare $Expoints[n][n]$; \/\/extreme points\n        \\STATE $foundknee \\leftarrow false$;\n       \n       \n        \\FOR{{\\bf each} $i \\in \\{ 1, \\dots, n\\}$}\n        \\STATE sort($P_t$) by the $i$th objective in ascending order;\n        \\STATE $Q[i] \\leftarrow P_t$.get\\_index$(\\frac{3}{4}\\times popsize)$.get\\_obj($i$); \/\/upper quartile value of the $i$th objective \n        \\STATE $L[i] \\leftarrow P_t$.get\\_index$(popsize)$.get\\_obj($i$);\/\/the largest (worst) value of the $i$th objective\n        \\ENDFOR\n        \\FORALL{solution $s \\in P_t$}\n        \\IF{$s$.get\\_obj($i=1,...,n) > Q[i]$ }\n        \\STATE remove $s$ from $P_t$;\n        \\ENDIF\n        \\ENDFOR\n        \\STATE $Expoints[\\centerdot][\\centerdot] \\leftarrow$ extreme points in $P_t$;\n        \\STATE $num_a\\leftarrow$ the number of points in concave region of hyperplane formed by $Expoints[\\centerdot][\\centerdot]$;\n        \\STATE $num_v \\leftarrow |P_t|-num_a$; \/\/the number of points in convex region\n        \\IF{$(num_v - num_a > \\epsilon)$}\n        \\STATE \/\/roughly convex shape\n        \\STATE remove solutions in concave region from $P_t$;\n        \\ELSIF{$(num_a - num_v > \\epsilon)$}\n        \\STATE \/\/roughly concave shape\n        \\STATE remove solutions in convex region from $P_t$;\n        \\ELSE\n        \\FORALL{solution $s\\in P_t$}\n         \\STATE calculate hypervolume of $s$ with reference point $L[\\centerdot]$;\n         \\STATE update the largest hypervolume value ($max\\_h$);\n        \\ENDFOR\n        \\STATE $knee[\\centerdot] \\leftarrow$ solution with $max\\_h$;\n        \\STATE $foundknee \\leftarrow true$;\n        \\ENDIF\n        \\IF{($foundknee == false$)}\n        \\FORALL{solution $s\\in P_t$}\n        \\STATE calculate distance between $s$ and hyperplane formed by $Expoints[\\centerdot][\\centerdot]$;\n        \\STATE update the largest distance ($max\\_d$);\n        \\ENDFOR\n        \\STATE $knee[\\centerdot] \\leftarrow$ solution with $max\\_d$;\n        \\ENDIF\n        \\FOR{{\\bf each} $i \\in \\{ 1, \\dots, n\\}$}\n        \\STATE $P\\_region[i] \\leftarrow knee[i] + (L[i]-knee[i]) \\times 85\\%$\n        \\ENDFOR\n \t\\end{algorithmic}\n\\end{algorithm}\n\\fi\n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=3.3in]{knee-explain.png} \n\\caption{Finding the knee point in bi-dimensional space.} \n\\label{knee}\n\\end{figure}\n\nThe first\/new preference region is formed based on the population at the moment when the condition of $Enum\\_P$ is satisfied, especially the knee point of the population. Algorithm 2 gives the details of line 41 in Algorithm 1, it introduces the steps of finding the knee point of a non-dominated solution set and constituting a hypercube shaped preference region according to the knee point. Figure~\\ref{knee} also gives an illustration of finding the knee point in bi-dimensional space. Firstly, the upper quartile objective values (line 13 in Algorithm 2) in the solution set are used as a boundary to define outliers and solutions outside this boundary are removed  (line 16-20 in Algorithm 2). The extreme solutions (the solutions with the maximum value in one objective) (line 21 in Algorithm 2) are then found inside the boundary and a hyperplane is formed based on the extreme solutions. In a bi-dimensional space (Figure~\\ref{knee}), the hyperplane is only a line connecting two extreme solutions. According to the numbers of points below and above the hyperplane (line 22 - 23 in Algorithm 2), the shape of the solution set can be roughly perceived.\nWe will distinguish between ``convex'' and ``concave'' regions. Points in the \\textit{convex} (\\textit{concave}) \\textit{region} are dominating (dominated by) at least one point in the hyperplane spanned by the extreme points. However, when the number of the points in the convex region and the number of points in the concave region is close enough, it implies that the shape of the current solution set is almost linear. This occurs both when the true Pareto front is linear and when the solution set is converged very well in a small area of the Pareto front. A parameter $\\epsilon$ then is used to represent the closeness and it is a small number decided by the size of the solution set. In the case that the shape of the current solution set is (almost) linear, the solution with the largest hypervolume value with regards to the worst objective vector (line 14 in Algorithm 2) is adopted as the knee point (line 32 - 36 in Algorithm 2). While, under the condition that the shape of the current solution set is convex or concave, the solution in the convex or concave region with the largest Euclidean distance to the hyperplane is chosen as the knee point (line 39 - 42 in Algorithm 2). After the knee point is found, the preference region can be determined based on the knee point by the following formula:\n\\begin{align}\nP\\_region[i] = knee[i] + (L[i]-knee[i]) \\times 85\\%\n\\end{align}\n\nLet $i$ denotes the $i$th objective, as in Algorithm 2, $L[i]$ is the worst value of the $i$th objective in the population, $knee[i]$ is the $i$th objective value of the knee point and $P\\_region[i]$ is the upper bound of the $i$th objective. W.l.o.g. we assume the objectives are to be minimized and the lower bound of preference region is the origin point. According to the formula, we can see that the first preference region is relatively large (roughly 85\\% of the entire Pareto front). With the increase in the number of iteration, the preference region will be updated and becomes smaller and smaller because every preference region picks 85\\% of the current Pareto front. Eventually, we want the preference region can reach a proper range, say, 15\\% of the initial Pareto front. The process of narrowing down the preference region step by step can benefit the accuracy of the preference region.\n\nIn the interest of clarity, Algorithm 1 only shows the workings of AP-DI-1, the workings of AP-DI-2 can be obtained by replacing crowding distance with the diversity indicator contribution.\nIn the ($\\mu$ + $\\mu$) generational selection operator (line 14 - 36 in Algorithm 1), when there is no preference region, the second ranking criteria (the crowding distance for AP-DI-1; the diversity indicator for AP-DI-2) for all solutions on the last front are calculated and the population will be truncated based on non dominated sorting and the second ranking criteria (line 28 - 29 in Algorithm 1). While, if a preference region already exists, both the second ranking criteria and Euclidean distance to the knee point for all solutions on the last front are calculated and the population will be truncated based on first non dominated sorting, then the second ranking criteria, lastly, Euclidean distance to the knee point (line 31 - 32 in Algorithm 1). In the ($\\mu$ + 1) steady state selection operator (line 38 - 59 in Algorithm 1), firstly, the value of $Enum\\_P$ is compared with the current number of evaluations to determine if a (new) preference region should be generated. When it is time to do so, the preference region is generated through Algorithm 2 (line 41 in Algorithm 1), at the same time, the value of $Enum\\_P$ is updated to the next moment when the preference region is to be updated (line 42 in Algorithm 1). There are different strategies to assign the values of $Enum\\_P$. In our algorithm, we divide the whole computing budget into two parts, the first half is used to find an initial entire Pareto front approximation, and the second half is used to update the preference region and find solutions in the preference region. Assume the total computing budget is $Enum\\_T$ (the number of evaluations), then the first value of $Enum\\_P$ is $\\frac{1}{2}\\times Enum\\_T$. Due to the reason that we expect a final preference region with a size of around 15\\% of the initial entire Pareto front and each new preference region takes 85\\% of the current Pareto front, according to the formula: $0.85^{12} \\approx 0.14$, the value of $Enum\\_P$ can be updated by the following formula:\n\\begin{align}\nEnum\\_P = Enum\\_P + (Enum\\_T\/2)\/12\n\\end{align}\n\nAnother half of budget can be divided into $12$ partial-budgets and a new preference region is constituted after each partial-budget. In the end, the final preference region is achieved and solutions focusing on this preference region are obtained. For the rest part of the ($\\mu$ + 1) steady state selection operator, likewise, when there is a preference region, three ranking criteria (1. non-dominated sorting; 2. diversity indicator; 3. the Euclidean distance to the knee point) work together to achieve a well-converged and well-distributed set of Pareto optimal solutions in the preference region.\n\n\\section{Experimental Results}\n\\label{sec:experiments}\n\\subsection{Experimental Design}\nIn this section, simulations are conducted to demonstrate the performance of proposed algorithms on both benchmark problems and our real-world application problems. All experiments are implemented based on the MOEA Framework (\\url{http:\/\/www.moeaframework.org\/}), which is a Java-based framework for multi-objective optimization.\n\nFor the two variants of AP-DI-MOEA: AP-DI-1 and AP-DI-2, their performances have been compared with DI-MOEA: DI-1, DI-2 and NSGA-III \\cite{deb2014evolutionary}. We compare our algorithm with NSGA-III because NSGA-III is a representative state-of-the-art evolutionary multi-objective algorithm and it is very powerful to handle problems with non-linear characteristics. For bi-objective benchmark problems, algorithms are tested on ZDT1 and ZDT2 with 30 variables. For three objective benchmark problems, DTLZ1 with 7 variables and DTLZ2 with 12 variables are tested. For the real-world application problem of VFMSO, experiments have been conducted on two instances with different sizes. The configurations of the two instances, such as the predicted RUL probability distribution, the processing time and maintenance cost of each component, the set-up time and cost of each car, are made available on \\url{http:\/\/moda.liacs.nl}. On every problem, we run each algorithm $30$ times with different seeds, while the same $30$ different seeds are used for all algorithms. All the experiments are performed with a population size of $100$; and for bi-objective problems, experiments are run with a budget of $22000$ (objective function) evaluations, DTLZ three objective problems with a budget of $120000$ evaluations, the VFMSO problems with a budget of $1200000$ evaluations. This setting is chosen to be more realistic in the light of the applications in scheduling that we ultimately want to solve.\n\n\n\n\n\n\n\\subsection{Experiments on bi-objective problems}\n\nBi-objective problems are optimized with a total budget of $22000$ evaluations, when the number of evaluations reaches $10000$ times, the first preference region is generated, then after every $1200$ evaluations, the preference region will be updated. Figure~\\ref{fig:ZDT1} shows the Pareto front approximations from a typical run on ZDT1 (left column) and ZDT2 (right column). The graphs on the upper row are obtained from DI-1 and AP-DI-1, while the graphs on the lower row are from DI-2 and AP-DI-2. In each graph, the entire Pareto front approximation from DI-MOEA and the preferred solutions from AP-DI-MOEA (or \\textit{AP solutions}) are presented, at the same time, the preference region of AP-DI-MOEA is also shown by the gray area.\n\n\\begin{figure}[htbp]\n\\hspace{-0.58cm}\n\\subfigure[ZDT1]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{zdt1-di1-tdi1-w.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{zdt1-di2-tdi2-w.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\\subfigure[ZDT2]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{zdt2-di1-tdi1-w.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{zdt2-di2-tdi2-w.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\n\\caption{Pareto front approximation on ZDT1 and ZDT2.}\n\\label{fig:ZDT1}\n\\end{figure}\n\nBesides the visualization of the Pareto fronts, we also compute the knee point of the entire final Pareto front approximation from DI-MOEA via the strategy described in Algorithm 2. For each run of DI-MOEA and AP-DI-MOEA with the same seed, the following two issues have been checked: \n\\begin{itemize\n\\item If the knee point from DI-MOEA is in the preference region achieved by its derived AP-DI-MOEA;\n\\item If the knee point from DI-MOEA is dominated by or dominating AP solutions; or if it is a non-dominated solution (mutually non-dominated with all AP solutions).\n\\end{itemize}\n\nTable~\\ref{table-kneezdt1} shows the results of 30 runs. For ZDT1 problem, all 30 knee points from DI-1 and DI-2 are in the preference regions from AP-DI-1 and AP-DI-2 respectively; in all these knee points, 10 from DI-1 and 7 from DI-2 are dominated by AP solutions. For ZDT2 problem, most knee points are not in corresponding preference regions, but for those in the preference regions, almost all of them are dominated by AP solutions. Please note that when a knee point from DI-MOEA is outside of the preference region from AP-DI-MOEA, it is not possible that it can dominate any AP solutions because all AP solutions are in the preference region and only solutions in the left side of the gray area can dominate AP solutions. \n\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from DI-MOEA and AP solutions on ZDT problems.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{ZDT1}  & \\multicolumn{2}{|c|}{ZDT2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& DI-1\/ & DI-2\/ & DI-1\/ & DI-2\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 & AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 20 & 23 & 1 & 1\\\\\n\\cline{2-6}\npreference & Dominated & 10 & 7 & 9 & 9 \\\\\n \\cline{2-6}\nregion & Dominating & 0 & 0 & 0 &  0 \\\\\n\\hline\nOutside & Incomparable &  0 & 0 & 20 & 20 \\\\\n\\cline{2-6}\np-region & Dominated &  0 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular}\n\\label{table-kneezdt1}\n\\end{table} \n\nWe also perform the same comparison between AP-DI-MOEA and NSGA-III, the results are shown in Table~\\ref{table-kneezdt1-nsga3}. For ZDT1 problem, all knee points from NSGA-III are in the preference regions from AP-DI-MOEA. Some of these knee points dominate AP solutions. For ZDT2 problem, most knee points from NSGA-III are not in the preference regions and these knee points are incomparable with AP solutions. For the knee points in the preference regions, all three dominating relations with AP solutions appear. For both problems, when the knee point from NSGA-III is dominating AP solutions, it only dominates one AP solution.\n\n\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from NSGA-III and AP solutions on ZDT problems.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{ZDT1}  & \\multicolumn{2}{|c|}{ZDT2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& NSGA-III\/ & NSGA-III\/ & NSGA-III\/ & NSGA-III\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 & AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 14 & 19 & 3 & 1\\\\\n\\cline{2-6}\npreference & Dominated & 0 & 0 & 2 & 3 \\\\\n \\cline{2-6}\nregion & Dominating & 16 & 11 & 4 &  6 \\\\\n\\hline\nOutside & Incomparable &  0 & 0 & 21 & 20 \\\\\n\\cline{2-6}\np-region & Dominated &  0 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular}\n\\label{table-kneezdt1-nsga3}\n\\end{table} \n\nInstead of spreading the population across the entire Pareto front, we only focus on the preference region. To ensure that our algorithm can guide the search towards the preference region and the achieved solution set is distributed across the preference region, we compare the performance of AP-DI-MOEA, DI-MOEA and NSGA-III in the preference region. For each Pareto front approximation from DI-MOEA and NSGA-III, the solutions in the corresponding preference region from AP-DI-MOEA are picked, and we compare these solutions with AP solutions through the hypervolume indicator. The point formed by the largest objective values over all solutions in the preference region is adopted as the reference point when calculating the hypervolume indicator. It has been found that all hypervolume values of new solution sets from DI-MOEA and NSGA-III in the preference region are worse than the hypervolume values of the solution sets from AP-DI-MOEA, which proves that the mechanism indeed works in practice. Figure~\\ref{box:ZDT} shows box plots of the distribution of hypervolume indicators over 30 runs.\n\n\n\n\\begin{figure}[htbp]\n\\hspace{-0.58cm}\n\\subfigure[ZDT1]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{zdt1-di1-box1.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{zdt1-di2-box2.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\\subfigure[ZDT2]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{zdt2-di1-box1.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{zdt2-di2-box2.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\\caption{Boxplots comparing the hypervolume values on ZDT1 and ZDT2.}\n\\label{box:ZDT}\n\\end{figure}\n\n\n\\subsection{Experiments on three objective problems}\nDTLZ1 and DTLZ2 are chosen as three objective benchmark problems to investigate our algorithms. They are performed with a total budget of $120000$ fitness evaluations, when the evaluation reaches $60000$ times, the first preference region is formed, then after every $5000$ evaluations, the preference region is updated. Figure~\\ref{fig:dtlz} shows the Pareto front approximations from a typical run on DTLZ1 (left column) and DTLZ2 (right column). The upper graphs are obtained from DI-1 and AP-DI-1, while the lower graphs are from DI-2 and AP-DI-2. In each graph, the Pareto front approximations from DI-MOEA and corresponding AP-DI-MOEA are given. Since the target region is actually an axis aligned box, the obtained knee region (i.e., the intersection of the axis aligned box with the Pareto front) has an inverted triangle shape for these two benchmark problems.\n\n\\begin{figure*}[htbp]\n\\hspace{-3.5cm}\n\\subfigure[DTLZ1 3 objective problem]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.5in,width=4.4in]{dtlz1-di1-tdi1-w.png}\\\\\n        \\vspace{0.45cm}\n        \\includegraphics[height=2.5in,width=4.4in]{dtlz1-di2-tdi2-w.png}\\\\\n       \n       \n    \\end{minipage}%\n}%\n\\hspace{2cm}\n\\subfigure[DTLZ2 3 objective problem]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4.4in]{dtlz2-di1-tdi1-w.png}\\\\\n       \n        \\includegraphics[height=2.7in,width=4.4in]{dtlz2-di2-tdi2-w.png}\\\\\n       \n       \n    \\end{minipage}%\n}%\n\\caption{Pareto front approximation on DTLZ1 and DTLZ2.}\n\\label{fig:dtlz}\n\\end{figure*}\n\n\n\nTable~\\ref{table-kneedtlz} shows the space and dominance relation of the knee point from DI-MOEA and the solution set from AP-DI-MOEA over 30 runs. For DTLZ1 problem, most knee points from DI-MOEA are in their respective preference regions and all knee points are mutually non-dominated with AP solutions. For DTLZ2 problem, we observed that more knee points are not in the corresponding preference regions. This is because too few solutions from DI-MOEA are in the preference region. For DTLZ1 problem, six solutions from DI-MOEA are in the corresponding preference region on average for each run, while, for DTLZ2 problem, only less than two solutions are in the corresponding preference region on average. Therefore, we can see that on the one side, it is normal that many knee points from the entire Pareto fronts are not in their corresponding preference regions; on the other side, our aim of finding more fine-grained resolution in the preference region has been well achieved because only few solutions can be obtained in the preference region if we spread the population across the entire Pareto front. At the same time, one knee point from DI-1 on DTLZ2 is dominated by solutions from the corresponding AP-DI-1, which proves that AP-DI-MOEA can converge better than DI-MOEA because AP-DI-MOEA focuses on the preference region.\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from DI-MOEA and AP solutions on DTLZ problems.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{DTLZ1}  & \\multicolumn{2}{|c|}{DTLZ2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& DI-1\/ & DI-2\/ & DI-1\/ & DI-2\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 & AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 29 & 27 & 10 & 13\\\\\n\\cline{2-6}\npreference & Dominated & 0 & 0 & 1 & 0 \\\\\n \\cline{2-6}\nregion & Dominating & 0 & 0 & 0 & 0 \\\\\n\\hline\nOutside & Incomparable & 1 & 3 & 19 & 17 \\\\\n\\cline{2-6}\np-region & Dominated &  0 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular}\n\\label{table-kneedtlz}\n\\end{table} \n\n\nAP-DI-1 and AP-DI-2 have also been compared with NSGA-III in the same way. Table~\\ref{table-kneedtlz_nsga3} shows the comparison result. For DTLZ1, the average number of solutions from NSGA-III in the corresponding preference regions from AP-DI-MOEA is six. Still, almost all knee solutions from NSGA-III are in the preference region. For DTLZ2, the average number of solutions from NSGA-III in the corresponding preference region from AP-DI-MOEA is less than one, while, in more than half of 30 runs, the knee points from NSGA-III are still in the preference region. To some extent, it can be concluded that the preference regions from AP-DI-MOEA are accurate. It can also be observed that AP-DI-1 behaves better than AP-DI-2 on DTLZ2, because two knee points from NSGA-III dominate the solutions from AP-DI-2.\n\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from NSGA-III and AP solutions on DTLZ problems.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{DTLZ1}  & \\multicolumn{2}{|c|}{DTLZ2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& NSGA-III\/ & NSGA-III\/ & NSGA-III\/ & NSGA-III\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 &AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 30 & 29 & 14 & 17\\\\\n\\cline{2-6}\npreference & Dominated & 0 & 0 & 1 & 1 \\\\\n \\cline{2-6}\nregion & Dominating & 0 & 0 & 0 & 2 \\\\\n\\hline\nOutside & Incomparable & 0 & 1 & 15 & 10 \\\\\n\\cline{2-6}\np-region & Dominated &  0 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular}\n\\label{table-kneedtlz_nsga3}\n\\end{table} \n\nSimilarly, we pick from DI-MOEA and NSGA-III solutions which are in the corresponding preference region of AP-DI-MOEA, and the hypervolume indicator value is compared between these solutions and AP solutions. It has been found that all hypervolume values of solutions from AP-DI-MOEA are better than those of solutions from DI-MOEA and NSGA-III. The left column of Figure~\\ref{box:dtlz} shows box plots of the distribution of hypervolume values over 30 runs on DTLZ1, and the right column shows the hypervolume comparison on DTLZ2.\n\n\n\\begin{figure}[htbp]\n\\hspace{-0.58cm}\n\\subfigure[DTLZ1]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{dtzl1-di1-box.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{dtlz1-di2-box.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\\subfigure[DTLZ2]{\n    \\begin{minipage}[t]{0.53\\linewidth}\n        \\centering\n        \\includegraphics[width=2in]{dtlz2-di1-box.png}\\\\\n        \\vspace{0.02cm}\n        \\includegraphics[width=2in]{dtlz2-di2-box.png}\\\\\n        \\vspace{0.02cm}\n       \n    \\end{minipage}%\n}%\n\n\\caption{Boxplots comparing the hypervolume values on DTLZ1 and DTLZ2.}\n\\label{box:dtlz}\n\\end{figure}\n\nIn our experiments, we decide half of the total budget is used to find an initial Pareto front because it turned out to be a good compromise: half budget for the initial Pareto front and another half budget for the solutions focusing on the preference region. We also run experiments using 25\\% and 75\\% of the total budget for the initial Pareto front. Figure~\\ref{fig:dtlz-budget} presents the entire Pareto front from DI-MOEA and the Pareto front from AP-DI-MOEA with different budgets for the initial Pareto front. The left two images are on DTLZ1 and the right two images are on DTLZ2. The uppper two images are from DI-1 and AP-DI-1; the lower two images are from DI-2 and AP-DI-2. In the legend labels, 50\\%, 25\\% and 75\\% indicate the budgets which are utilized to find the initial entire Pareto front. It can be observed that the preference region from AP-DI-MOEA with 50\\% of budget are located on a better position than with 25\\% and 75\\% budgets, and the position of the preference region from AP-DI-MOEA with 50\\% of budget is more stable. Therefore, in our algorithm, 50\\% of budget is used before the generation of preference region.\n\n\\begin{figure*}[htbp]\n\\hspace{-3.5cm}\n\\subfigure[DTLZ1 3 objective problem]{\n    \\begin{minipage}[t]{0.50\\linewidth}\n        \\centering\n        \\includegraphics[height=2.6in,width=4.28in]{dtlz1-di1-tdi1-3combine-1.png}\\\\\n       \n        \\includegraphics[height=2.6in,width=4.28in]{dtlz1-di2-tdi2-3combine-3.png}\\\\\n       \n       \n    \\end{minipage}%\n}%\n\\hspace{1.5cm}\n\\subfigure[DTLZ2 3 objective problem]{\n    \\begin{minipage}[t]{0.50\\linewidth}\n        \\centering\n        \\includegraphics[height=2.6in,width=4.2in]{dtlz2-di1-tdi1-3combine-16.png}\\\\\n       \n        \\includegraphics[height=2.6in,width=4.2in]{dtlz2-di2-tdi2-3combine-16.png}\\\\\n       \n       \n    \\end{minipage}%\n}%\n\\caption{Pareto front approximation by different budgets generating initial Pareto front.}\n\\label{fig:dtlz-budget}\n\\end{figure*}\n\n\\subsection{Experiments on Vehicle Fleet Maintenance Scheduling Optimization}\nThe budget of $1200000$ evaluations has been used on the real-world application problems, and $600000$ of them are for the initial Pareto front. After that, the preference region is updated after every $50000$ evaluations.\nThe VFMSO problem has been tested with different sizes. Figure~\\ref{fig:20cars} shows Pareto front approximations of a problem with $20$ cars and $3$ workshops (V1), and each car contains $13$ components: one engine, four springs, four brakes and four tires \\cite{van2019modeling}. It can be observed that AP-DI-1 and AP-DI-2 can zoom in the entire Pareto front and find solutions in the preference region, at the same time, both AP-DI-1 and AP-DI-2 converge better than their corresponding DI-1 and DI-2. A similar conclusion can be drawn from Pareto fronts approximations of the problem with $30$ cars and $5$ workshops (V2) in Figure~\\ref{fig:30cars}.\n\n\nIn Figure~\\ref{fig:2030cars}, We put the Pareto front approximations from DI-MOEA, AP-DI-MOEA and NSGA-III on V1 (left) and V2 (right) together. The behaviours of DI-1, DI-2 and NSGA-III are similar on V1, so are the behaviours of AP-DI-1 and AP-DI-2 on this problem. While, DI-2 and AP-DI-2 converge better than DI-1 and AP-DI-1 on V2 problem. The behaviour of NSGA-III is between that of DI-1 and DI-2.\n\n\n\\begin{figure*}[htbp]\n\\hspace{-3.8cm}\n\\subfigure[DI-1 \\& AP-DI-1]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{20car-di1-tdi1-legend.png}\\\\\n       \n    \\end{minipage}%\n}%\n\\hspace{3.2cm}\n\\subfigure[DI-2 \\& AP-DI-2]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{20car-di2-tdi2-legend.png}\\\\\n       \n    \\end{minipage}%\n}%\n\\caption{Pareto front approximation on VFMSO problem with 20 cars and 3 workshops.}\n\\label{fig:20cars}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\hspace{-3.8cm}\n\\subfigure[DI-1 \\& AP-DI-1]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{30car-di1-tdi1-legend.png}\\\\\n       \n    \\end{minipage}%\n}%\n\\hspace{3.2cm}\n\\subfigure[DI-2 \\& AP-DI-2]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{30car-di2-tdi2-legend.png}\\\\\n       \n    \\end{minipage}%\n}%\n\\caption{Pareto front approximation on VFMSO problem with 30 cars and 5 workshops.}\n\\label{fig:30cars}\n\\end{figure*}\n\n\n\n\\begin{figure*}[htbp]\n\\hspace{-3.5cm}\n\\subfigure[V1]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{20car-di1-tdi1-di2-tdi2-ns3.png}\\\\\n       \n    \\end{minipage}%\n}%\n\\hspace{3cm}\n\\subfigure[V2]{\n    \\begin{minipage}[t]{0.5\\linewidth}\n        \\centering\n        \\includegraphics[height=2.7in,width=4in]{30car-di1-tdi1-di2-tdi2-ns3.png}\\\\\n       \n    \\end{minipage}%\n}%\n\n\\caption{Pareto front approximation on VFMSO problem by DI-MOEA, AP-DI-MOEA and NSGA-III.}\n\\label{fig:2030cars}\n\\end{figure*}\n\n\nTable~\\ref{table-v1} gives the space and dominance relation of knee points from DI-MOEA and solutions from AP-DI-MOEA on these two VFMSO problems. For both problems, only few knee points from DI-MOEA are in the preference regions of AP-DI-MOEA, and the main reason is that the Pareto front of AP-DI-MOEA converges better than that of DI-MOEA, in some cases, the Pareto front of DI-MOEA cannot even reach the corresponding preference region. More importantly, it can be observed that most knee points from DI-MOEA, no matter whether in the preference region or outside of the preference region, are dominated by the solutions from AP-DI-MOEA. This phenomenon is even more obvious for the application problem with bigger size and run with the same budget as the smaller one: for V2, 90\\% of knee points from DI-MOEA are dominated by the solutions from AP-DI-MOEA.\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from DI-MOEA and AP solutions on V1 and V2.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{V1}  & \\multicolumn{2}{|c|}{V2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& DI-1\/ & DI-2\/ & DI-1\/ & DI-2\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 & AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 0 & 0 & 0 & 0\\\\\n\\cline{2-6}\npreference & Dominated & 9 & 7 & 9 & 6 \\\\\n \\cline{2-6}\nregion & Dominating & 0 & 0 & 0 &  0 \\\\\n\\hline\nOutside & Incomparable &  4 & 9 & 3 & 3 \\\\\n\\cline{2-6}\np-region & Dominated & 17 & 14 & 18 & 21 \\\\\n\\hline\n\\end{tabular}\n\\label{table-v1}\n\\end{table} \n\n\nTable~\\ref{table-v1-nsga3} gives the space and dominance relation of knee points from NSGA-III and AP solutions. For both problems, again, most knee points from NSGA-III are not in the preference regions of AP-DI-MOEA. Some knee points from NSGA-III are dominated by AP solutions and most of them are incomparable with AP solutions. \n\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Space and dominance relation of knee point from NSGA-III and AP solutions on V1 and V2.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Problem} & \\multicolumn{2}{|c|}{V1}  & \\multicolumn{2}{|c|}{V2}\\\\ \n\\cline{1-6}\n\\multicolumn{2}{|c|}{\\multirow{2}*{Algorithm}}& NSGA-III\/ & NSGA-III\/ & NSGA-III\/ & NSGA-III\/ \\\\\n\\multicolumn{2}{|c|}{ }& AP-DI-1 & AP-DI-2 & AP-DI-1 & AP-DI-2 \\\\\n\\hline\nIn & Incomparable & 0 & 0 & 0 & 1\\\\\n\\cline{2-6}\npreference & Dominated & 0 & 1 & 3& 2 \\\\\n \\cline{2-6}\nregion & Dominating & 0 & 0 & 1 &  1 \\\\\n\\hline\nOutside & Incomparable &  23 & 24 & 21 & 18 \\\\\n\\cline{2-6}\np-region & Dominated & 7 & 5 & 5 & 8 \\\\\n\\hline\n\\end{tabular}\n\\label{table-v1-nsga3}\n\\end{table} \n\n\\iffalse \n The right image of Figure~\\ref{fig:30cars-2} presents the entire Pareto front approximations of V2 from four different MOEAs: DI-1, DI-2, RVEA and NSGA-III. It can be seen that DI-MOEA (both DI-1 and DI-2) and NSGA-III converge to the similar area in the end, while, RVEA reaches another area of the objective space. Table~\\ref{table_hy} provides the average hypervolume value of the four Pareto fronts from 30 runs and the reference point for each run is formed by the largest objective value from all solutions. It can be seen that DI-2 behaves the best and RVEA the worst.\n\n\n\\begin{table}[htbp]\n\\caption{Hypervolume values}\n\\label{table_hy}\n\\begin{center}\n\\begin{tabular}{l|c}\n\\hline\nDI-1 & 0.0525\\\\\n\\hline\nDI-2 & 0.0576\\\\\n\\hline\nRVEA & 0.0202\\\\\n\\hline\nNSGAIII & 0.0534\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\fi\n\n\n\n\n\n\\section{CONCLUSIONS}\n\\label{sec:conclusion}\nIn this paper, a preference based multi-objective evolutionary algorithm, AP-DI-MOEA, is proposed. In the absence of explicitly provided preferences, the knee region is usually treated as the region of interest or preference region. Given this, AP-DI-MOEA can generate the knee region automatically and can find solutions with a more fine-grained resolution in the knee region. This has been demonstrated on the bi-objective problems ZDT1 and ZDT2, and the three objective problems DTLZ1 and DTLZ2. In the benchmark, the new approach was also proven to perform better than NSGA-III which was included in the benchmark as a state-of-the-art reference algorithm.\n\nThe research for the preference based algorithm was originally motivated by a real-world optimization problem, namely, Vehicle Fleet Maintenance Scheduling Optimization (VFMSO), which is described in this paper in a new formulation as a three objective discrete optimization problem. A customized set of operators (initialization, recombination, and mutation) is proposed for a multi-objective evolutionary algorithm with a selection strategy based on DI-MOEA and, respectively, AP-DI-MOEA. The experimental results of AP-DI-MOEA on two real-world application problem instances of different scales show that the newly proposed algorithm can generate preference regions automatically and it (in both cases) finds clearly better and more concentrated solution sets in the preference region than DI-MOEA. For completeness, it was also tested against NSGA-III and a better approximation in the preference region was observed by AP-DI-MOEA .\n\nSince our real-world VFMSO problem is our core issue to be solved, and its Pareto front is convex, we did not consider problems with an irregular shape.\nIt would be an interesting question how to adapt the algorithm to problems with more irregular shapes. Besides, the proposed approach requires a definition of knee points. Future work will provide a more detailed comparison of different variants of methods to generate knee points, as they are briefly introduced in Section \\ref{sec:literature}. In the application of maintenance scheduling, it will also be important to integrate robustness and uncertainty in the problem definition. It is desirable to generate schedules that are robust within a reasonable range of disruptions and uncertainties such as machine breakdowns and processing time variability.\n\n\n\n\\section*{Acknowledgment}\nThis work is part of the research programme Smart Industry SI2016 with project name CIMPLO and project number 15465, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\n\nThe properties of physical systems in the vicinity of a critical\npoint, such as critical exponents and amplitude ratios, can be\nextracted by a variety of methods, ranging from exact solutions to\nMonte Carlo simulations.\n\nIn the absence of exact results, one of the most successful approaches\nis based on the investigation of the strong-coupling series expansion,\nwhich enjoys the property of a finite radius of convergence, often\n(but not necessarily) coinciding with the extent of the\nhigh-temperature phase.  More generally, when no singular points occur\non the real axis of the complex coupling plane, it is possible to\nexploit strong-coupling results even beyond the convergence radius by\nanalytic continuations, which are based on appropriate resummation\nmethods.  Extending the length of the strong-coupling series and\nimproving the accuracy of the resummations are therefore the two most\ncompelling tasks within this approach to the study of the behavior of\nsystems in the critical region.\n\nAs part of an extended program of strong-coupling calculations we have\nrecently computed an extended series expansion of all nontrivial\ntwo-point Green's functions\n\\begin{equation}\nG(x) = \\left<{\\vec s}(0)\\cdot{\\vec s}(x)\\right>\n\\end{equation}\nfor the nearest-neighbor lattice formulation of two-dimensional \n${\\rm O}(N)$ $\\sigma$ models on the square, triangular, and honeycomb\nlattices, respectively up to 21st, 15th, and 30th order in the\nstrong-coupling expansion parameter $\\beta$.  A complete presentation\nof our strong-coupling computations for ${\\rm O}(N)$ $\\sigma$ models\nin two and three dimensions will appear in a forthcoming paper.\nA preliminary report of our calculations can be found in \nRef.~\\cite{lattice95}.\n\nThe relevance of a better understanding of 2-$d$ ${\\rm O}(N)$ $\\sigma$\nmodels cannot be overestimated.  They appear in condensed matter\nliterature as prototype models for critical phenomena that are\nessentially restricted to two-dimensional layers, including some\ninstances of high-$T_c$ superconductivity.  Moreover, they can be\nemployed as model field theories sharing some of the most peculiar\nfeatures of four-dimensional gauge theories, such as asymptotic\nfreedom and spontaneous mass generation.  This last statement must\nhowever be qualified, since the above-mentioned properties, according\nto common lore, are possessed only by those 2-$d$ ${\\rm O}(N)$ models\nsuch that $N>2$.\n\nWe focus here on these asymptotically free models, analyzing their\nstrong-coupling expansion in order to extract information that may be\nrelevant to the description of their continuum limit\n($\\beta\\to\\infty$), assuming $\\beta_c=\\infty$ to be the only\nsingularity on the real axis.  This hypothesis is favored by all\nnumerical evidence as well as by the successful application of the\nextrapolation techniques that we shall discuss in the present paper.\nThe analysis of our strong-coupling series for \nmodels with $N\\geq 2$ is presented in Ref.~\\cite{Nm2}.\n\nIt is obviously quite hard to imagine that strong-coupling techniques\nmay be really accurate in describing the divergent behavior of such\nquantities as the correlation length and the magnetic susceptibility.\nNevertheless, as our calculations will explicitly confirm, the\nstrong-coupling analysis may provide quite accurate continuum-limit\nestimates when applied directly to dimensionless,\nrenormalization-group invariant ratios of physical quantities.  Two\nbasic ideas will make this statement more convincing.\n\n(i) For any dimensionless, renormalization-group invariant ratio\n$R(\\beta)$, when $\\beta$ is sufficiently large we may expect a\nbehavior \n\\begin{equation}\nR(\\beta)-R^*\\sim {1\\over \\xi^2(\\beta)},\n\\label{scalR}\n\\end{equation}\nwhere $R^*$ is the fixed point (continuum) value and $\\xi$ is the\n(diverging) correlation length.  Hence a reasonable estimate of $R^*$\nmay be obtained at the values of $\\beta$ corresponding to large but\nfinite correlation lengths, where the function $R(\\beta)$ flattens.\nThis is essentially the same idea underlying Monte Carlo studies of\nasymptotically free theories, based on the identification of the\nso-called scaling region.\n\n(ii) On physical grounds, it is understandable that $\\beta$ is not\nnecessarily the most convenient variable to parameterize phenomena\noccuring around $\\beta=\\infty$.  An interesting alternative is based\non the observation that the strong-coupling series of the internal\nenergy\n\\begin{equation}\nE = \\beta + O(\\beta^3)\n\\end{equation}\nmay be inverted to give $\\beta$ as a series in $E$.  This series may\nbe substituted into other strong-coupling expansions, obtaining\nexpressions for physical quantities as power series in $E$.  It might\nnow be easier to reach the continuum limit, since it now occurs at a\nfinite value of the expansion variable, i.e., $E\\to1$.\n\nWe hope to convince the reader that, by exploiting these ideas,\nstate-of-the-art strong-coupling calculations can be made at least as\naccurate as the best Monte Carlo simulations presently available,\nwhen applied to dimensionless renormalization-group invariant quantities.\n\nWe must stress that the analysis of the strong-coupling series\ncalculated on different lattices offers a possibility of testing\nuniversality, and, on the other side, once universality is assumed, it\nrepresents a further check for possible systematic errors and allows\ntheir quantitative estimate; this estimate is usually a difficult task\nin strong-coupling extrapolation methods such as those based on Pad\\'e\napproximants and their generalizations.\n\nOur physical intuition of the behavior of ${\\rm O}(N)$ models is\nstrongly guided by our knowledge of their large-$N$ behavior, and by\nthe evidence of a very weak dependence on $N$ of the dimensionless\nratios.  In order to extend our understanding to those lattices that\nhave not till now received a systematic treatment, and also in order\nto establish a benchmark for the strong-coupling analysis, we decided\nto start our presentation with a detailed comparative study of the\nlarge-$N$ limit of various lattices, in the nearest-neighbor\nformulation.  To the best of our knowledge, only the large-$N$\nsolution on the square lattice was already known explicitly\n\\cite{sqNi}.\n\nThe paper is organized as follows:\n\nIn Sec.~\\ref{secNi} we present the large-$N$ limit solution of ${\\rm\n  O}(N)$ $\\sigma$ models on the square, triangular and honeycomb\nlattices, in the nearest-neighbor formulation, calculating several\nphysical quantities and showing explicitly the expected universality\nproperties.  The triangular- and honeycomb-lattice results are\noriginal, and possess some intrinsic reasons of interest.  However,\nreaders willing to focus on square-lattice results are advised to jump\nto Sec.~\\ref{SCA} after reading Subs.~\\ref{secse} and \\ref{secsqNi},\nwhere the notation is fixed.\n\nSec.~\\ref{SCA} is devoted to a detailed analysis of the available\nstrong-coupling series of $G(x)$ and other physical quantities on the\nsquare, triangular, and honeycomb lattices.  Most of the results we\nshall show there concern the $N=3$ model.  The basic motivation for\nthis choice lies in the observation that all dependence in $N$ is\nmonotonic between 3 and $\\infty$; hence the discussion of higher-$N$\nresults would be only a boring repetition of the considerations\npresented here.  The reader not interested in the analysis of\ntriangular and honeycomb lattices may skip most of the discussion, by\nfocusing on Subs.~\\ref{scsq}, where further definitions are introduced\nand the square-lattice series are analyzed, and on Subs.~\\ref{concl},\nwhere all conclusions are drawn.\n\nApps.~\\ref{apptr} and \\ref{appex} provide the derivation and the\ntechnical details of the large-$N$ calculations on the triangular and\nhoneycomb lattices respectively.  We present as well the calculation\nof the $\\Lambda$-parameters.\n\nApp.~\\ref{singNinf} is a study of the complex temperature\nsingularities of the $N=\\infty$ partition functions on the \ntriangular and honeycomb lattices.\n\nIn Apps.~\\ref{appscsq}, \\ref{appsctr} and \\ref{appscex} we present,\nfor selected values of $N$, the strong-coupling series of some\nrelevant quantities on the square, triangular, and honeycomb lattice\nrespectively.\n\n\n\\section{The large-$\\protect\\bbox{N}$ limit of lattice \n$\\protect\\bbox{{\\rm O}(N)}$ $\\protect\\bbox{\\sigma}$ models}\n\\label{secNi}\n\n\n\\subsection{The large-$\\protect\\bbox{N}$ saddle point equation}\n\\label{secse}\n\nThe nearest-neighbor lattice formulations on square, triangular and\nhoneycomb lattices are defined by the action\n\\begin{equation}\nS_L= -N\\beta\\sum_{\\rm links} {\\vec s}_{x_l}\\cdot {\\vec s}_{x_r},\n\\qquad {\\vec s}_x\\cdot {\\vec s}_x = 1,\n\\label{lattaction}\n\\end{equation}\nwhere $\\vec s$ is a $N$-component vector, the sum is performed over\nall links of the lattice and $x_l,x_r$ indicate the sites at the ends\nof each link.  The coordination number is $c=4,6,3$ respectively for\nthe square, triangular and honeycomb lattice.  The lattice spacing\n$a$, which represents the length unit, is defined to be the length of\na link.  The volume per site is then $v_s=1,\\sqrt{3}\/2, 3\\sqrt{3}\/4$\n(in unit of $a^2$) respectively for the square, triangular, and\nhoneycomb lattice.\n\nStraightforward calculations show that the correct continuum\nlimit of ${\\rm O}(N)$ $\\sigma$ models, \n\\begin{equation}\nS= {N\\over 2t} \\int d^2x\\, \\partial_\\mu {\\vec s}(x)\\cdot\n\\partial_\\mu {\\vec s}(x),\n\\qquad {\\vec s}(x)\\cdot {\\vec s}(x) = 1,\n\\label{contaction}\n\\end{equation}\nis obtained by identifying  \n\\begin{equation}\nt={1\\over \\beta}, \\  {1\\over \\sqrt{3}\\beta},\\ \n{\\sqrt{3}\\over \\beta},\n\\label{temp}\n\\end{equation}\nrespectively for the square, triangular and honeycomb lattice.\nNotice that \n\\begin{equation}\n\\lambda\\equiv t\\beta = {4v_s\\over c}\n\\label{tbeta}\n\\end{equation}\nis the distance between nearest-neighbor\nsites of the dual lattice in unit of the lattice spacing $a$.\n\nWhen the number of field components $N$ per site goes to infinity,\none can use a saddle point equation to evaluate the partition \nfunction. Replacing the constraint $\\vec s_x^{\\,2}=1$\nby a Fourier integral over \na conjugate variable $\\alpha_x$, we write the partition\nfunction as\n\\begin{eqnarray}\nZ&&\\propto \\int \\prod_x d{\\vec s}_x \\,\\delta( \\vec s_x^{\\,2}-1)\\,\n\\exp N\\beta \\sum_{\\rm links} {\\vec s}_{x_l}\\cdot {\\vec s}_{x_r}\n\\nonumber \\\\\n&&\\propto\\int \\prod_x d\\phi_x d\\alpha_x \\,\\exp \nN\\left[ \\sum_x i{\\alpha_x\\over 2}\\left( 1 - \\phi_x^2\\right)\n-{\\beta\\over 2}\\sum_{\\rm links} \\left( \n\\phi_{x_l}-\\phi_{x_r}\\right)^2\\right].\n\\label{fp}\n\\end{eqnarray}\nIntegrating out the $\\phi$ variables we arrive at\nthe expression \n\\begin{equation}\nZ\\propto \\int d\\alpha_x \\,\\exp {N\\over 2}\n\\left( \\sum_x i\\alpha_x - {\\rm Tr}\\,\\ln R\\right),\n\\label{fp2}\n\\end{equation}\nwhere \n\\begin{equation}\nR_{xy}= -{1\\over t} \\Delta_{xy} + i\\alpha_x\\delta_{xy},\n\\label{R}\n\\end{equation}\nand $\\Delta_{xy}$ is a generalized Laplacian operator, such\nthat \n\\begin{equation}\n\\lambda\\, \\sum_{\\rm links} \\left(\\phi_{x_l}-\\phi_{x_r}\\right)^2\n= -\\sum_{x,y} \\phi_x \\Delta_{xy} \\phi_y.\n\\label{Q}\n\\end{equation}\n\nThe large-$N$ limit solution is obtained from the variational \nequation with respect to $\\alpha_x$.\nLooking for a translation invariant solution we set\n\\begin{equation}\ni\\alpha_x={v_s\\over t}\\,z.\n\\label{costalp}\n\\end{equation}\nThe matrix $R$ then becomes\n\\begin{equation}\nR_{xy}= {1\\over t} \\left[ -\\Delta_{xy}+zv_s\\delta_{xy}\\right],\n\\label{R2}\n\\end{equation}\nand the saddle point equation is written as\n\\begin{equation}\n1=\\lim_{N_s\\rightarrow\\infty}\n{1\\over N_s} {\\rm Tr}\\,R^{-1},\n\\label{spe}\n\\end{equation}\nwhere $N_s$ is the number of sites.\n\nThe large-$N$ fundamental two-point Green's function is\nobtained by\n\\begin{equation}\nG(x-y)= R^{-1}_{xy}.\n\\label{NiGx}\n\\end{equation}\n\nIn order to calculate the trace of $R^{-1}$, the easiest\nprocedure consists in Fourier transforming the operator\n$R$. Such transformation is straightforward on lattices,\nlike square and triangular lattices, whose sites\nare related by a translation group, and in these cases it\nyields the diagonalization of the matrix $R_{xy}$.\nThe honeycomb lattice, not possessing a full translation\nsymmetry, presents some complications. \nIn this case a partial diagonalization of $R_{xy}$ can be \nachieved following the procedure outlined in Ref.~\\cite{SCUN2}.\n\n\n\\subsection{The square lattice}\n\\label{secsqNi}\n\nTurning to the momentum space the variational equation\nbecomes\n\\begin{equation}\n{1\\over t}=\\beta =\\int_{-\\pi}^\\pi {d^2 k\\over (2\\pi)^2}\n{1\\over \\widehat{k}^2+z}={1\\over 2\\pi} \n\\rho_{\\rm s}(z) K\\left( \\rho_{\\rm s}(z) \\right), \n\\label{sesq}\n\\end{equation}\nwhere \n\\begin{equation}\n\\rho_{\\rm s}(z)=\\left(1 + {1\\over 4}z\\right)^{-1},\n\\label{rhos}\n\\end{equation}\nand $K$ is the complete integral of the first kind.\n\nLet's define the moments of $G(x)$\n\\begin{equation}\nm_{2j}\\equiv\\sum_x (x^2)^j \\,G(x).\n\\label{momgx}\n\\end{equation}\nStraightforward calculations lead to the following\nresults\n\\begin{equation}\n\\chi\\equiv m_0 = {t\\over z},\n\\label{chisqin}\n\\end{equation}\n\\begin{equation}\n\\xi_{G}^2 \\equiv M_{G}^{-2} \\equiv \n{m_2\\over 4\\chi}= {1\\over z},\n\\label{xigsqin}\n\\end{equation}\n\\begin{equation}\n u\\equiv {m_2^2\\over \\chi m_4}=\n{1\\over 4}\\left( 1 + {z\\over 16}\\right)^{-1}.\n\\label{omsqin}\n\\end{equation}\nNotice that in the large-$N$ limit the renormalization constant of the\nfundamental field is $Z=t$. $u$ is a renormalization-group invariant\nquantity.\n\nThe mass-gap should be extracted from the long distance\nbehavior of the two-point Green's function, which is also\nrelated to the imaginary momentum singularity of the \nFourier transform of $G(x)$.\nIn the absence of a strict rotation invariance, one actually\nmay define different estimators of the mass-gap having \nthe same continuum limit.\nOn the square lattice one may consider $\\mu_{\\rm s}$ and \n$\\mu_{\\rm d}$ obtained respectively by the equations\n\\begin{eqnarray}\n&&\\tilde{G}^{-1}(p_1=i\\mu_{\\rm s},p_2=0)=0,\\nonumber \\\\\n&&\\tilde{G}^{-1}\\left(p_1=i{\\mu_{\\rm d}\\over\\sqrt{2}},\np_2=i{\\mu_{\\rm d}\\over\\sqrt{2}}\\right) = 0.\n\\label{msmd}\n\\end{eqnarray}\n$\\mu_{\\rm s}$ and $\\mu_{\\rm d}$ \ndetermine respectively the long\ndistance behavior of the side and diagonal wall-wall\ncorrelations constructed with $G(x)$. \nIn generalized Gaussian models, such as the large-$N$ limit\nof ${\\rm O}(N)$ models, it turns out convenient to define\nthe following quantities \n\\begin{eqnarray}\n&&M_{\\rm s}^2= 2\\left( {\\rm cosh} \n\\mu_{\\rm s} - 1\\right),\\nonumber \\\\\n&&M_{\\rm d}^2= 4\\left( {\\rm cosh} \n{\\mu_{\\rm d}\\over \\sqrt{2}} -1\\right).\n\\label{MsMd}\n\\end{eqnarray}\nIn the continuum limit \n\\begin{equation}\n{M_{\\rm s}\\over \\mu_{\\rm s}}\\,,{M_{\\rm d}\n\\over \\mu_{\\rm d}}\\rightarrow 1,\n\\label{msmd2}\n\\end{equation}\ntherefore $M_{\\rm s}$ and $M_{\\rm d}$ may be also used\nas estimators of the mass-gap. \n\nIn the large-$N$ limit \n\\begin{equation}\nM_{\\rm s}^2 = M_{\\rm d}^2 = z=M_{G}^2.\n\\label{msmdmg}\n\\end{equation}\n\nThe rotational invariance of $G(x)$ at large distance,\n$d\\gg\\xi$, is checked by the ratios $\\mu_{\\rm s}\/\\mu_{\\rm d}$.\nUsing the above results one can evaluate the scaling violation terms:\n\\begin{equation}\n{\\mu_{\\rm s}\\over \\mu_{\\rm d}}=\n{ \n\\ln\\left( {1\\over 2}\\sqrt{z} + \\sqrt{ 1 + \n{1\\over 4}z}\\right)\\over\n\\sqrt{2}\\ln\\left( {1\\over 2\\sqrt{2}}\\sqrt{z} + \\sqrt{ 1 + \n{1\\over 8}z}\\right)}\n = 1 - {1\\over 48}z + {71\\over 23040}z^2+\nO\\left(z^3\\right).\n\\label{rotviol}\n\\end{equation}\n \nAnother test of scaling is provided by the ratio\n\\begin{equation}\n{\\mu_{\\rm s}\\over M_{G}}\n= {2\\over \\sqrt{z}}\n\\ln\\left( {\\sqrt{z}\\over 2} + \\sqrt{ 1 + \n{z\\over 4}}\\right)=\n1 - {1\\over 24}z + {3\\over 640}z^2+\nO\\left(z^3\\right).\n\\label{scalviol}\n\\end{equation}\n\nThe internal energy can be easily calculated obtaining\n\\begin{equation}\nE \\equiv \\langle {\\vec s}_x\\cdot {\\vec s}_{x+\\mu} \\rangle =\nR^{-1}_{x,x+\\mu}=\n1\\,-\\,{1\\over 4\\beta}\\,+\\,{z\\over 4}.\n\\label{energysq}\n\\end{equation}\nTherefore\n\\begin{equation}\n{1\\over 2}\\sum_\\mu\n\\langle ({\\vec s}_{x+\\mu}-{\\vec s}_x)^2 \\rangle =\n{1\\over 2\\beta}\\,-\\,{z\\over 2},\n\\label{condlatt}\n\\end{equation}\nwhere the term proportional to $z$ is related to the condensate $T$ of\nthe trace of the energy-momentum tensor~\\cite{CRcond}\n\\begin{equation}\n{\\beta(t)\\over 2t^2}\n\\partial_\\mu {\\vec s}(x)\\cdot\\partial_\\mu {\\vec s}(x).\n\\label{temtr}\n\\end{equation}\nIn the large-$N$ limit\n\\begin{equation}\n\\beta(t)=-{1\\over 2\\pi}t^2,\n\\label{beta}\n\\end{equation}\ntherefore from the expression (\\ref{energysq}) we deduce\n\\begin{equation}\n{T\\over M_{G}^2}={1\\over 4\\pi}.\n\\label{condsq}\n\\end{equation}\n\nAnother interesting quantity which can be evaluated in the large-$N$\nlimit is the zero-momentum four-point renormalized coupling constant,\ndefined by\n\\begin{equation}\ng_r = -{\\chi_4\\over \\chi^2\\xi_{G}^2}\n\\label{gr0}\n\\end{equation}\nwhere\n\\begin{equation}\n\\chi_4 = \\sum_{x,y,z} \\langle  {\\vec s}_0\\cdot {\\vec s}_x \n\\, {\\vec s}_y\\cdot {\\vec s}_z \\rangle_c.\n\\label{chi4}\n\\end{equation}\n$g_r$ is zero in the large-$N$ limit, where the theory is Gaussian-like\nand thus $\\chi_4=0$. \nIts value in the continuum limit\n\\begin{equation}\ng_r^* = {8\\pi\\over N}+ O\\left({1\\over N^2}\\right)\n\\label{gr1}\n\\end{equation}\ncan be also evaluated in the large-$N$ expansion of\nthe continuum formulation of the ${\\rm O}(N)$ models~\\cite{gr}.\nOn the square lattice, by using the saddle point equation we find\n\\begin{equation}\nNg_r = -2 {\\partial \\ln z\\over \\partial \\beta},\n\\label{gr2}\n\\end{equation}\nwhich can be made more explicit by writing\n\\begin{equation}\nNg_r = 4\\pi{1+\\rho_{\\rm s}\\over \\rho_{\\rm s} E(\\rho_{\\rm s})} =\n8\\pi\\left[1 + {z\\over 8}\\left(\\ln {z\\over 32}\\,+\\,2\\right) \n+ O(z^2)\\right],\n\\label{gr3}\n\\end{equation}\nwhere $E$ is an elliptic function.\n\nAll the above results can be expressed as functions of $\\beta$\nby solving the saddle point equation.\nConcerning asymptotic scaling, and therefore \nsolving the saddle point equation at large $\\beta$, one finds\n\\begin{equation}\nM_{G}\\simeq 4\\sqrt{2}\\,\\exp \\left(-{2\\pi\\over t}\\right).\n\\label{asysq}\n\\end{equation}\n\nThe analytic structure of the various observables\nhas been investigated in Ref.~\\cite{BCMO}.\nThe complex $\\beta$-singularities are square-root branch points,\nindeed quantities like $\\chi$ and $\\xi_G^2$\nbehave as \n\\begin{equation}\nA(\\beta)+B(\\beta) \\sqrt{\\beta-\\beta_s}\n\\end{equation}\naround a singular point $\\beta_s$, where $A(\\beta)$ and $B(\\beta)$ are\nregular in the neighborhood of $\\beta_s$.  The singularities closest\nto the origin are located at $\\bar{\\beta}=0.32162\\,(\\pm 1\\pm i)$.\nSuch singularities determine the convergence radius of the\nstrong-coupling expansion, which is therefore $\\beta_r=0.45484$,\ncorresponding to a correlation length $\\xi_{G}=3.17160$.\n\n\n\\subsection{The triangular lattice}\n\\label{sectrNi}\n\nOn the triangular lattice, using the results of App.~\\ref{apptr}, \nthe saddle point equation can be written as\n\\begin{equation}\n{1\\over t}=\\sqrt{3}\\beta = \\int^\\pi_{-\\pi} {dk_1\\over 2\\pi}\n\\int^{2\\pi\/\\sqrt{3}}_{-2\\pi\/\\sqrt{3}}\n{dk_2\\over 2\\pi} {1\\over \\Delta(k)+z}\n\\label{setr}\n\\end{equation}\nwhere \n\\begin{equation}\n\\Delta(k)=4\\left[ 1 - {1\\over 3}\\left(\n\\cos k_1+2\\cos {k_1\\over 2}\\cos {\\sqrt{3}k_2\\over 2}\\right)\\right]\n\\label{deltatr}\n\\end{equation}\nand the momentum integration is performed over the Brillouin\nzone corresponding to a triangular lattice.\nBy rather straightforward calculations (making also use\nof some of the formulas of Ref.~\\cite{Gradshteyn})\nthe saddle point equation can be written as\n\\begin{equation}\n{1\\over t}=\\sqrt{3}\\beta =\n{1\\over 2\\pi} \\left( 1 + {z\\over 6}\\right)^{-1\/4}\\,\\rho_{\\rm t}(z)\n\\,K(\\rho_{\\rm t}(z)),\n\\label{setr2}\n\\end{equation}\nwhere\n\\begin{equation}\n \\rho_{\\rm t}(z)= \\left( 1+{z\\over 6}\\right)^{1\/4}\\,\n\\left[ {1\\over 2} + {z\\over 8} + {1\\over 2}\n\\left( 1 + {z\\over 6}\\right)^{1\/2} \\right]^{-1\/2}\\, \n\\left[ {5\\over 2} + {3z\\over 8} - {3\\over 2}\n\\left( 1 + {z\\over 6}\\right)^{1\/2} \\right]^{-1\/2}. \n\\label{setr3}\n\\end{equation}\n\nUsing the results of App.~\\ref{apptr} one can find\n\\begin{equation}\n\\chi={t\\over v_s z}={2\\over 3\\beta z},\n\\label{chitrin}\n\\end{equation}\n\\begin{equation}\n \\xi_{G}^2 \\equiv M_{G}^{-2} =  {1\\over z}\\, ,\n\\label{xitrin}\n\\end{equation}\n\\begin{equation}\nu\\equiv {m_2^2\\over \\chi m_4}\n={1\\over 4}\\left( 1 + {z\\over 16}\\right)^{-1} .\n\\label{omtrin}\n\\end{equation}\n\nAn estimator of the mass-gap $\\mu_{\\rm t}$ can be extracted from the\nlong distance behavior of the \nwall-wall correlation function defined in\nEq.~(\\ref{walldeftr}), indeed for $x\\gg 1$\n\\begin{equation}\nG_{\\rm t}^{(\\rm w)}(x)\\propto e^{-\\mu_{\\rm t} x}.\n\\label{ldgtr}\n\\end{equation}\nIn the large-$N$ limit one finds\n\\begin{equation}\nM^2_{\\rm t}\\equiv{8\\over 3}\\left( {\\rm cosh} {\\sqrt{3}\\over 2}\n\\mu_{\\rm t} -1\\right)=z=M^2_{G}.\n\\label{Mtr}\n\\end{equation}\nA test of scaling is provided by the ratio \n\\begin{equation}\n{\\mu_{\\rm t}\\over M_{G}}=\n{2\\over \\sqrt{3z}}\n{\\rm Arccosh}\\,\\left[ 1 + {3\\over 8}z \\right]\n =  1-{1\\over 32}z + {9\\over 10240} z^2\n+O\\left( z^3\\right),\n\\label{scalvioltr}\n\\end{equation}\nwhere scaling violations are of the same order as those \nfound on the square lattice for the corresponding\nquantity, cfr.\\ Eq.~(\\ref{scalviol}).\n\nThe internal energy is given by the following\nexpression\n\\begin{equation}\nE=\\langle {\\vec s}_{x_l}\\cdot {\\vec s}_{x_r}\\rangle =\n1 - {1\\over 6\\beta} + {z\\over 4} ,\n\\label{etr}\n\\end{equation}\nleading again to the result (\\ref{condsq}) \nfor the condensate of the trace of the\nenergy-momentum tensor, in agreement with universality. \n\nWe calculated $g_r$ on the triangular lattice, finding \nthe following expression (in the derivation we made use of the\nsaddle point equation (\\ref{setr}))\n\\begin{equation}\nNg_r = - {2\\over \\sqrt{3}} {\\partial \\ln z\\over \\partial \\beta},\n\\label{gr2t}\n\\end{equation}\nwhich can be written in a more explicit form using \nEq.~(\\ref{setr2}): \n\\begin{eqnarray}\n Ng_r&=&4\\pi\\left( 1 + {1\\over 6}z\\right)^{1\/4}\n{1\\over z}\\left[ {E(\\rho_{\\rm t})\\over 1-\\rho_{\\rm t}^2} \n{\\partial \\rho_{\\rm t}\\over \\partial z}- \n{1\\over 24}\\left(1+{1\\over 6}z\\right)^{-1}\\rho_{\\rm t} \n  K(\\rho_{\\rm t})\\right]^{-1}\n\\nonumber \\\\\n&=&\n8\\pi\\left[1 + {z\\over 8}\\left(\\ln {z\\over 48}\\,+\\,{11\\over 6}\\right) \n+ O(z^2)\\right],\n\\label{gr3t}\n\\end{eqnarray}\nwhere the continuum value of $Ng_r$, obtained for $z\\rightarrow 0$, is\nin agreement with the results (\\ref{gr1}) and (\\ref{gr3}).\n\nIn the weak coupling region $t\\rightarrow 0$ the\nsaddle point equation leads to the asymptotic\nscaling formula\n\\begin{equation}\nM_{G}\\simeq 4\\sqrt{3} \\exp \\left( \n-{ 2\\pi\\over t}\\right).\n\\label{asytr}\n\\end{equation}\nThe equations (\\ref{asysq}) and\n(\\ref{asytr}) are in agreement with the large-$N$ limit of the  \nratio of the $\\Lambda$-parameters of the square \nand triangular lattice formulations\ncalculated in App.~\\ref{apptr}, cfr.\\ Eq.~(\\ref{ratioltr2}),\nusing perturbation theory.\n\nWe have investigated the analytic structure in the complex\n$\\beta$-plane. Details of such study are presented in\nApp.~\\ref{singNinf}.  As on the square lattice, the singularities are\nsquare-root branch points. Those closest to the origin are placed at\n$\\bar{\\beta}= 0.206711\\pm \\,i\\,0.181627$, leading to a convergence\nradius for the strong-coupling expansion $\\beta_r=0.275169$, which\ncorresponds to a correlation length $\\xi_{G}=2.98925$.\n\n\n\\subsection{The honeycomb lattice}\n\\label{iNhl}\n\nThe analysis of models defined on the honeycomb lattice presents a few\nsubtleties caused by fact that, unlike square and triangular lattices,\nnot all sites are related by a translation, not allowing a\nstraightforward definition of a Fourier transform. Nevertheless,\nobserving that sites at even distance in the number of lattice links\nform triangular lattices, one can define a Fourier-like transformation\nthat partially diagonalizes the Gaussian propagator (up to $2\\times 2$\nmatrices)~\\cite{SCUN2}.  In this section we present the relevant\nresults, some details of their derivation are reported in\nApp.~\\ref{appex}.\n\nUsing the expression of $R^{-1}$ of Eq.~(\\ref{greengex}) \nwe write the saddle point equation in the following form\n\\begin{equation}\n{1\\over t}={\\beta\\over\\sqrt{3}}=\n\\int^{{2\\over3}\\pi}_{-{2\\over3}\\pi} {dk_1\\over 2\\pi}\n\\int^{\\pi\/\\sqrt{3}}_{-\\pi\/\\sqrt{3}}\n{dk_2\\over 2\\pi} {1+{1\\over 4}z\\over \n\\Delta(k)+z\\left(1+{1\\over 8}z\\right)}\n\\label{seex}\n\\end{equation}\nwhere \n\\begin{equation}\n\\Delta(k)={8\\over 9}\\left[ 2 - \\cos {\\sqrt{3}\\over 2}k_2\n\\left( \\cos {3\\over 2}k_1  + \\cos {\\sqrt{3}\\over 2}k_2\\right)  \n\\right],\n\\label{deltaex}\n\\end{equation}\nand integrating over the momentum  we arrive at\n\\begin{equation}\n{1\\over t}={\\beta\\over\\sqrt{3}}=\n{1\\over 2\\pi} \\left( 1 + {z\\over 4}\\right)^{1\/2}\n\\rho_{\\rm h}(z) \\,K(\\rho_{\\rm h}(z)),\n\\label{seex2}\n\\end{equation}\nwhere\n\\begin{equation}\n\\rho_{\\rm h}(z)= \\left( 1 + {z\\over 4}\\right)^{1\/2} \n\\left( 1 + {3z\\over 8}\\right)^{-3\/2} \n\\left( 1 + {z\\over 8}\\right)^{-1\/2}. \n\\label{seex3}\n\\end{equation}\n\nFrom Eq.~(\\ref{greengex}) we also derive\n\\begin{equation}\n\\chi ={t\\over v_s z}={4\\over 3\\beta z},\n\\label{chihoin}\n\\end{equation}\n\\begin{equation}\n\\xi_{G}^2 \\equiv M^{-2}_{G}=\n{1\\over z},\n\\label{xihoin}\n\\end{equation}\n\\begin{equation}\nu ={1\\over 4}\\left( 1 + {z\\over 16}\\right)^{-1}.\n\\label{omhoin}\n\\end{equation}\n\nThe two orthogonal wall-wall correlation functions \n$G^{(\\rm w)}_{\\rm v}(x)$ and $G^{(\\rm w)}_{\\rm h}(x)$ defined in\nEqs.~(\\ref{g1}) and (\\ref{g2}) allow one to define two estimators of\nthe mass-gap from their long distance behavior\n\\begin{eqnarray}\nG^{(\\rm w)}_{\\rm v}(x)\\propto e^{-\\mu_{\\rm v} x},\\nonumber\\\\ \nG^{(\\rm w)}_{\\rm h}(x)\\propto e^{-\\mu_{\\rm h} x},\n\\label{ldbg}\n\\end{eqnarray}\nwhere $x$ is the distance between the two walls in \nunit of the lattice spacing.\nIn the continuum limit $\\mu_{\\rm v}=\\mu_{\\rm h}$ \nand they both reproduce\nthe physical mass propagating in the fundamental channel.\nAs on the square and triangular lattices,   it is convenient\nto define the quantities\n\\begin{eqnarray}\n&&M_{\\rm v}^2 = {8\\over 9}\\left( {\\rm cosh} {3\n\\mu_{\\rm v}\\over 2} - 1\\right),\\nonumber \\\\\n&&M_{\\rm h}^2 = {8\\over 3}\\left( {\\rm cosh} \n{\\sqrt{3}\\mu_{\\rm h}\\over 2} -1\\right),\n\\label{M1M2}\n\\end{eqnarray}\nwhich, in the continuum limit, \nare also estimators of the mass gap. \nIn the large-$N$ limit one finds\n\\begin{eqnarray}\n&&M_{\\rm v}^2 = z\\left( 1+ {z\\over 8}\\right),\\nonumber \\\\\n&&M_{\\rm h}^2=z.\n\\label{M1M22}\n\\end{eqnarray}\nNotice that in the continuum large-$N$ limit the result  \n\\begin{equation}\n{M\\over M_{G}}=1,\n\\label{momg}\n\\end{equation}\nwhere $M$ is any mass-gap\nestimator, is found for all lattice formulations considered.\n\nOn the honeycomb lattice the maximal violation of full rotational\nsymmetry occurs for directions differing by a $\\pi\/6$ angle, and\ntherefore, taking into account its discrete rotational symmetry, also\nby a $\\pi\/2$ angle. So a good test of rotation invariance of $G(x)$ at\nlarge distance is provided by the ratio $\\mu_{\\rm v}\/\\mu_{\\rm h}$:\n\\begin{equation}\n{\\mu_{\\rm v}\\over \\mu_{\\rm h}}=\n{\n{\\rm Arccosh}\\,\\left[ 1 + {9\\over 8}z \\left( 1 + {1\\over 8}\nz\\right)\\right]\n\\over\n\\sqrt{3}{\\rm Arccosh}\\,\\left[ 1 + {3\\over 8}z \\right]}\n= 1+{1\\over 640} z^2\n+O\\left( z^3\\right).\n\\label{rotscalho}\n\\end{equation}\nAs expected from the better rotational symmetry of the honeycomb\nlattice, rotation invariance is set earlier than for the square\nlattice, indeed the $O(z)$ scaling violation is absent.\n\nA test of scaling is provided by the ratio \n\\begin{equation}\n{\\mu_{\\rm h}\\over M_{G}}=\n{2\\over \\sqrt{3z}}\n{\\rm Arccosh}\\,\\left[ 1 + {3\\over 8}z \\right]\n= 1-{1\\over 32}z + {9\\over 10240} z^2\n+O\\left( z^3\\right),\n\\label{scalviolho}\n\\end{equation}\nwhere scaling violations are of the same order of those \nfound on the square lattice for the corresponding\nquantity, cfr.\\ Eq.~(\\ref{scalviol}).\n\nThe internal energy is given by\n\\begin{equation}\nE=1 - {1\\over 3\\beta} + {z\\over 4}\n=1 - {1\\over 3\\beta} + {M_{G}^2\\over 4},\n\\label{eneex}\n\\end{equation}\nwhere the term proportional to $M_{G}^2$\nverifies again universality.\n\nIn the weak coupling region $t\\rightarrow 0$ the\nsaddle point equation leads to the asymptotic\nscaling formula\n\\begin{equation}\nM_{G}\\simeq 4 \\exp \\left( \n-{ 2\\pi\\over t}\\right).\n\\label{asyex}\n\\end{equation}\nThe equations (\\ref{asysq}) and\n(\\ref{asyex}) are in agreement with the large-$N$ limit of the  \nratio of the $\\Lambda$-parameters of the square \nand triangular lattice formulations\ncalculated in App.~\\ref{appex}, cfr.\\ Eq.~(\\ref{ratiolho2}),\nusing perturbation theory.\n\nIn Fig.~\\ref{asyiN} we compare asymptotic scaling from\nthe various lattices considered, plotting the ratio\nbetween $M_{G}$ and the corresponding\nasymptotic formula (cfr.\\ Eqs.~(\\ref{asysq}),\n(\\ref{asytr}) and (\\ref{asyex})).\nNotice that in the large-$N$ limit corrections to \nasymptotic scaling are $O(M^2_{G})$, in that corrections\n$O(1\/\\ln M_{G})$ are suppressed by a factor $1\/N$.\n\nWe have investigated the analytic structure in the complex \ntemperature-plane of the $N=\\infty$ model on the honeycomb lattice\n(details are reported in App.~\\ref{singNinf}).\nAs on the square and triangular lattices,\nsingularities are square-root branch points, and those closest to the\norigin are placed on the imaginary axis\nat $\\bar{\\beta}=\\pm i 0.362095$. \nThe convergence radius for the strong-coupling expansion\nis associated to a quite small correlation length:\n$\\xi_{G}=1.00002$.\n\n\n\\section{Continuum results from strong coupling}\n\\label{SCA}\n\n\n\\subsection{Analysis of the series}\n\\label{analysis}\n\nIn this section we analyze the strong-coupling series of some of the\nphysical quantities which can be extracted from the two-point\nfundamental Green's function.  We especially consider dimensionless\nrenormalization-group invariant ratios, whose value in the scaling\nregion, i.e., their asymptotic value for $\\beta\\rightarrow \\infty$,\nconcerns the continuum physics.  Some strong-coupling series for\nselected values of $N$ are reported in the Apps.~\\ref{appscsq},\n\\ref{appsctr} and \\ref{appscex} respectively for the square,\ntriangular and honeycomb lattices.  The series in the energy are\nobtained by inverting the strong-coupling series of the energy\n$E=\\beta+O(\\beta^3)$ and substituting into the original series in\n$\\beta$.\n\nOur analysis of the series of dimensionless renormalization \ngroup invariant ratios of physical quantities,\nsuch as those defined in the previous section, is \nbased on Pad\\'e approximant (PA) techniques.\nFor a review on the resummation techniques cfr.\\ \nRef.~\\cite{Guttmann}.\n\nPA's are expected to converge well to meromorphic\nanalytic functions. More flexibility is achieved by applying\nthe PA analysis to the logarithmic derivative  \n(Dlog-PA analysis), and therefore enlarging the class\nof functions which can be reproduced to those having\nbranch-point singularities.\nThe accuracy and the convergence of the PA's  \ndepend on how well the function considered, \nor its logarithmic derivative, can be reproduced by a \nmeromorphic analytic function, and may change when considering\ndifferent representations of the same quantity.\nBy comparing the results from different series representations\nof the same quantity one may check for possible\nsystematic errors in the resummation procedure employed.\n\nIn our analysis we constructed $[l\/m]$ PA's and Dlog-PA's\nof both the series in $\\beta$ and in the energy.\n$l,m$ are the orders of the polynomials\nrespectively at the numerator and at the denominator\nof the ratio forming the $[l\/m]$ PA of the series at hand, \nor of its logarithmic derivative (Dlog-PA).\nWhile $[l\/m]$ PA's provide directly \nthe quantity at hand, in a Dlog-PA analysis one gets \na $[l\/m]$ approximant by reconstructing the original quantity\nfrom the $[l\/m]$ PA of its logarithmic derivative,\ni.e., a $[l\/m]$ Dlog-PA of the series $A(x)=\\sum_{i=0}^\\infty a_i x^i$\nis obtained by\n\\begin{equation}\nA_{l\/m}(x) =  a_0\\exp \\int_0^x \ndx' \\, {\\rm Dlog}_{l\/m} A(x').\n\\label{appA}\n\\end{equation}\nwhere ${\\rm Dlog}_{l\/m} A(x)$ indicates the $[l\/m]$ PA\nof the logarithmic derivative of $A(x)$.\n\nWe recall that a $[l\/m]$ PA uses $n=l+m$ terms of the series,\nwhile a $[l\/m]$ Dlog-PA requires $n=l+m+1$ terms.\nContinuum estimates are then obtained by evaluating the approximants\nof the energy series at $E=1$, and those of the $\\beta$\nseries at a value of $\\beta$ corresponding to a reasonably\nlarge correlation length. \n\nAs final estimates we take the average of the results from the\nquasi-diagonal (i.e., with $l\\simeq m$) PA's using all available terms\nof the series. The errors we will display are just indicative, and\ngive an idea of the spread of the results coming from different PA's.\nThey are the square root of the variance around the estimate of the\nresults, using also quasidiagonal PA's constructed from shorter\nseries.  Such errors do not always provide a reliable estimate of the\nuncertainty, which may be underestimated especially when the structure\nof the function (or of its logarithmic derivative) is not well\napproximated by a meromorphic analytic function. In such cases a more\nreliable estimate of the real uncertainty should come from the\ncomparison of results from the analysis of different series\nrepresenting the same quantity, which in general are not expected to\nhave the same analytic structure.\n\nIn the following of this section we present \nthe main results obtained from our strong-coupling analysis.\nMost of them will concern the $N=3$ case.\n\n\n\\subsection{The square lattice}\n\\label{scsq}\n\nOn the square lattice we have calculated the two-point Green's\nfunction up to $O(\\beta^{21})$, from which we have extracted\nstrong-coupling series of the quantities $E$, $\\chi$, $\\xi_{G}^2$,\n$u$, $M_{\\rm s}^2$, $M_{\\rm d}^2$, already introduced in\nSec.~\\ref{secsqNi}, and of the ratios $r\\equiv M_{\\rm s}^2\/M_{\\rm\n  d}^2$, $s\\equiv M_{\\rm s}^2\/M_{G}^2$.  Some of the above series for\nselected values of $N$ are reported in App.~\\ref{appscsq}.  Our\nstrong-coupling series represent a considerable extension of the 14th\norder calculations of Ref.~\\cite{Luscher}, performed by means of a\nlinked cluster expansion, which have been rielaborated and analyzed in\nRef.~\\cite{Butera}.  We also mention recent works where the linked\ncluster expansion technique has been further developed and\ncalculations of series up to 18th order~\\cite{Reisz} and 19th\norder~\\cite{Butera2} for d=2,3,4 have been announced.\n\nIn order to investigate the analytic structure in the complex\n$\\beta$-plane we have performed a study of the singularities of the\nDlog-PA's of the strong-coupling series of $\\chi$ and $\\xi_{G}^2$.  As\nexpected by asymptotic freedom, no indication of the presence of a\ncritical point at a finite real value of $\\beta$ emerges from the\nstrong-coupling analysis of $N\\geq 3$ models, confirming earlier\nstrong-coupling studies~\\cite{Butera}.  The singularities closest to\nthe origin, emerging from the Dlog-PA analysis of $\\chi$ and\n$\\xi_{G}^2$, are located at a pair of complex conjugate points, rather\nclose to the real axis in the $N=3$ case (where $\\bar{\\beta}\\simeq\n0.59\\pm i 0.16$) and moving, when increasing $N$, toward the\n$N=\\infty$ limiting points $\\bar{\\beta}=0.32162\\,(1\\pm i)$.  In Table\n\\ref{zeroes} such singularities are reported for some values of $N$.\nThe singularity closest to the origin determines the convergence\nradius of the corresponding strong-coupling series. For example for\n$N=3$ the strong-coupling convergence radius turns out to be\n$\\beta_r\\simeq 0.61$, which corresponds to a quite large correlation\nlength $\\xi\\simeq 65$.  We recall that the partition function on the\nsquare lattice has the symmetry $\\beta \\rightarrow -\\beta$, which must\nbe also realized in the locations of its complex singularities.\n\nBy rotation invariance the ratio \n$r\\equiv M_{\\rm s}^2\/M_{\\rm d}^2$ should go to one\nin the continuum limit. Therefore the analysis of such ratio\nshould be considered as a test of the procedure employed \nto estimate continuum physical quantities. \nIn the large-$N$ limit $r=1$ at all values of $\\beta$.\nThis is not anymore true \nat finite $N$, where the strong-coupling series of \n$M_{\\rm s}^2$ and $M_{\\rm d}^2$ differ from each other, \nas shown in App.~\\ref{appscsq}.\nFrom $G(x)$ up to $O(\\beta^{21})$ we could calculate\nthe ratio $r$ up to $O(\\beta^{14})$. \nThe results of our analysis of the series of $r$ for $N=3$\nare summarized in Table \\ref{sqr}. There we report\nthe values of the PA's and Dlog-PA's of the $E$-series at $E=1$, \nand of those of the $\\beta$-series at $\\beta=0.55$,\nwhich corresponds to a reasonably large \ncorrelation length $\\xi\\simeq 25$.\nWe considered PA's and Dlog-PA's with \n$l+m\\geq 11$ and $m\\geq l\\geq 5$.\nThe most precise determinations of $r^*$,\nthe value of $r$ at the continuum limit,  come from Dlog-PA's,\nwhose final estimates are $r^*=1.0000(12)$ from the $E$-approximants,\nand $r^*=1.0002(6)$ from the $\\beta$-approximants (at $\\beta=0.55$).\nThe precision of these results is remarkable.\n\nFor all $N\\geq 3$ the violation of rotation invariance in the large\ndistance behavior of $G(x)$, monitored by the ratio \n$\\mu_{\\rm s}\/\\mu_{\\rm d}$, turns out quantitatively very close to that\nat $N=\\infty$ when considered as function of $\\xi_G$ (in a plot the\n$N=3$ curve of $\\mu_{\\rm s}\/\\mu_{\\rm d}$ versus $\\xi_{G}$ as obtained\nfrom the strong-coupling analysis would be hardly distinguishable from\nthe exact $N=\\infty$ one).  $\\mu_{\\rm s}\/\\mu_{\\rm d}$ is one within\nabout one per mille already at $\\xi\\simeq 4$.\n\nCalculating a few more components of $G(x)$ at larger orders\n(i.e., those involved by the wall-wall correlation\nfunction at distance 6 and 7 respectively up to $O(\\beta^{22})$\nand $O(\\beta^{23})$),\nwe computed the ratio \n\\begin{equation}\ns\\equiv {M_{\\rm s}^2\\over M_{G}^2}\n\\label{sdef}\n\\end{equation}\nup to $O(\\beta^{16})$, by applying the technique described in\nRefs.~\\cite{SCUN1,RV}.  We recall that at $N=\\infty$ we found $s=1$\nindependently of $\\beta$.  No exact results are known about the\ncontinuum limit $s^*$ of the ratio $s$, except for its large-$N$\nlimit: $s^*=1$.  Both large-$N$ and Monte Carlo estimates indicate a\nvalue very close to one.  From a $1\/N$ expansion~\\cite{Flyv,CR}:\n\\begin{equation}\ns^*= 1 - {0.006450\\over N} + O\\left( {1\\over N^2}\\right).\n\\label{largeNs}\n\\end{equation}\nMonte Carlo simulations at $N=3$~\\cite{Meyer}\ngave $s=0.9988(16)$ at $\\beta={1.7\/3}=0.5666...$ ($\\xi\\simeq 35$), and\n$s=0.9982(18)$ at $\\beta=0.6$ ($\\xi\\simeq 65$),\nleading to the estimate $s^*=0.9985(12)$.\n\nIn Table \\ref{sqs} we report, for $N=3$, the values of PA's and\nDlog-PA's of the energy and $\\beta$ series of $s$ respectively at\n$E=1$ and at $\\beta=0.55$.  We considered PA's and Dlog-PA's with\n$l+m\\geq 13$ and $m\\geq l\\geq 5$.  Combining PA and Dlog-PA results,\nour final estimates are $s^*=0.998(3)$ from the $E$-approximants, and\n$s^*=0.998(1)$ from the $\\beta$ approximants evaluated at\n$\\beta=0.55$, in full agreement with the estimates from the $1\/N$\nexpansion and Monte Carlo simulations.  With increasing $N$, the\ncentral estimate of $s^*$ tends to be closer to one.\n\nThe scaling-violation pattern of the quantity $\\mu_{\\rm s}\/M_{G}$ for\n$N=3$ is similar to the pattern for $N=\\infty$ (cfr.\\ \nEq.~(\\ref{scalviol})), i.e., it is stable within a few per\nmille for $\\xi\\gtrsim 5$.\n\nAnother dimensionless renormalization-group invariant quantity we have\nconsidered is $u\\equiv m_2^2\/(\\chi m_4)$, whose large-$N$ limit has\nbeen calculated in the previous section, cfr.\\ Eq.~(\\ref{omsqin}). \nAt finite $N$ its continuum limit $u^*$ is not known. \nFrom the expression of the self-energy calculated up to\n$O(1\/N)$~\\cite{Flyv,CR,CRselfenergy}, one can obtain\n\\begin{equation}\nu^*= {1\\over 4}\\left[ 1 - {0.006198\\over N} \n    + O\\left( {1\\over N^2}\\right) \\right].\n\\label{largeNu}\n\\end{equation}\nIt is interesting to notice that the $O(1\/N)$ correction in\nEqs.~(\\ref{largeNs}) and (\\ref{largeNu}) is very small.\n\nAt $N=3$ the analysis of the $O(\\beta^{21})$ strong-coupling series of\n$u$ detected a simple pole close to the origin at $\\beta_0=-0.085545$\nfor the $\\beta$-series, and at $E_0=-0.086418$ for the energy series,\ncorresponding to $M^2_{G}=-16.000$, which, within the precision of our\nstrong-coupling estimate, is also the location of the pole in the\ncorresponding $N=\\infty$ expression (\\ref{omsqin}).  Being a simple\npole, this singularity can be perfectly reproduced by a standard PA\nanalysis, and indeed we found PA's to be slightly more stable than\nDlog-PA's in the analysis of $u$.  The results concerning $N=3$,\nreported in Table \\ref{sqom} (for PA's with $l+m\\geq 16$ and $m\\geq\nl\\geq 8$), lead to the estimates $u^*=0.2498(6)$ from the energy\nanalysis, and $u^*=0.2499(5)$ form the $\\beta$ analysis (at\n$\\beta=0.55$).  The agreement with the large-$N$ formula (\\ref{largeNu})\nis satisfactory.\nIn Fig.~\\ref{figomsq} the curve $u(E)$ as obtained\nfrom the $[10\/10]$ PA and the exact curve $u(E)$ at $N=\\infty$ (cfr.\\ \nEq.~\\ref{omsqin}) are plotted, showing almost no differences.\n\nIn Table \\ref{sum} we give a\nsummary of the determinations of $r^*$, $s^*$, and $u^*$ \nfrom PA's and Dlog-PA's of the energy and $\\beta$-series.\n\nWe mention that we also tried to analyze series in the variable\n\\begin{equation}\nz={I_{N\/2}(N\\beta)\\over I_{N\/2-1}(N\\beta)},\n\\label{chcoeff}\n\\end{equation}\nwhich is the character coefficient of the fundamental representation.\nAs for the $E$-series, the continuum limit should be reached at a\nfinite value $z\\rightarrow 1$, and estimates of $r^*$,$s^*$ and $u^*$\nmay be obtained evaluating the approximants of the corresponding\n$z$-series at $z=1$.  We obtained results much less precise than those\nfrom the analysis of the $E$-series.  Maybe because of the\nthermodynamical meaning of the internal energy, resummations by PA's\nand Dlog-PA's of the $E$-series turn out much more effective,\nproviding rather precise results even at the continuum limit $E=1$.\n\nThe strong-coupling approach turns out to be less effective to the\npurpose of checking asymptotic scaling.  In Table \\ref{mc}, we\ncompare, for $N=3,4,8$, $\\xi_{G}$ as obtained from the plain \n21st order series of $\\xi_{G}^2$ and from its Dlog-PA's with\nsome Monte Carlo results available in the literature.  Resummation\nby integral approximants~\\cite{IA} provides results substantially\nequivalent to those of Dlog-PA's.  For $N=3$ Dlog-PA's follow\n Monte Carlo data reasonably well up to about the convergence radius\n$\\beta_r\\simeq 0.6$ of the strong-coupling expansion, but they fail\nbeyond $\\beta_r$. On the other hand it is well known that for $N=3$\nthe asymptotic scaling regime is set at larger\n$\\beta$-values~\\cite{CEPS}.  More sophisticated analysis can be found\nin Refs.~\\cite{Butera,Bonnier}, but they do not seem to lead to a\nconclusive result about the asymptotic freedom prediction in the\n$O(3)$ $\\sigma$ model.  At larger $N$, the convergence radius\ndecreases, but on the other hand the asymptotic scaling regime should\nbe reached earlier.  At $N=4$ and $N=8$ the 21st order plain\nseries of $\\xi_{G}^2$ provides already quite good estimates of $\\xi_G$\nwithin the convergence radius when compared with Monte Carlo results.\nAgain Pad\\'e-type resummation fails for $\\beta>\\beta_r$.  We mention\nthat at $N=4$ the convergence radius $\\beta_r\\simeq 0.60$ corresponds\nto $\\xi_G\\simeq 25$, and at $N=8$ $\\beta_r\\simeq 0.55$ corresponds to\n$\\xi_G\\simeq 8$.\n\nIn order to check asymptotic scaling we consider the ratio\n$\\Lambda_{\\rm s}\/\\Lambda_{2l}$, where\n$\\Lambda_{\\rm s}$ is the effective $\\Lambda$-parameter which\ncan be extracted by\n\\begin{equation}\n\\Lambda_{\\rm s}\\equiv\\left( {\\Lambda_{\\rm s}\\over M}\\right)\nM={M\\over R_{\\rm s}},\n\\label{effla}\n\\end{equation}\nwhere $M$ is \nan estimator of the mass-gap,\n$R_{\\rm s}$ is the mass-$\\Lambda$ parameter ratio\nin the square lattice nearest-neighbor formulation~\\cite{Hasenfratz}\n\\begin{equation}\nR_{\\rm s}= R_{\\overline{\\rm MS}}\\times\n\\left( {\\Lambda_{\\overline{\\rm MS}}\\over \n\\Lambda_{\\rm s}}\\right)=\\left( {8\\over e}\\right)^{1\\over N-2}\n{1\\over \\Gamma\\left( 1 + {1\\over N-2}\\right)}\\times\n\\sqrt{32} \\exp\\left[ {\\pi\\over 2(N-2)}\\right],\n\\label{RL}\n\\end{equation}\nand $\\Lambda_{2l}$ is the corresponding two-loop formula\n\\begin{equation}\n\\Lambda_{2l}= \\left( {2\\pi N \\over N-2}\\beta\\right)^{1\\over N-2}\n\\exp \\left( -{2\\pi N\\over N-2}\\beta\\right).\n\\label{twoloopla}\n\\end{equation}\nThe ratio $\\Lambda_{\\rm s}\/\\Lambda_{2l}$ should go to one in the\ncontinuum limit, according to asymptotic scaling.  The available\nseries of $M_G^2$ are longer than any series of the mass-gap\nestimators; therefore, neglecting the very small difference between\n$M_{G}$ and $M$ (we have seen that for $N\\geq 3$ $(M_{G}-M)\/M\\lesssim\n10^{-3}$ in the continuum limit), for which formula (\\ref{RL})\nholds, we use $M_{G}$ as estimator of $M$.  In Fig.~\\ref{asysc} we\nplot $\\Lambda_{\\rm s}\/ \\Lambda_{2l}$ for various values of $N$,\n$N=3,4,8$, and for comparison the exact curve for $N=\\infty$.  As\nalready noted in Ref.~\\cite{Wolff} by a Monte Carlo study, for\n$N=3,4$ at $\\xi\\simeq 10$ the asymptotic scaling regime is still far\n(about 50\\% off at $N=3$ and $15\\%$ at $N=4$), while for $N=8$ it is\nverified for $\\xi\\gtrsim 4$ within a few per cent, and notice that the\nconvergence radius $\\beta_r\\simeq 0.55$ corresponds to $\\xi\\simeq 8$.\nAnyway with increasing $N$ curves of $\\Lambda_{\\rm s}\/ \\Lambda_{2l}$\nclearly approach the exact $N=\\infty$ limit.\n  \n\n\\subsection{The triangular lattice}\n\\label{sctr}\n\nOn the triangular lattice we have calculated the two-point Green's\nfunction up to $O(\\beta^{15})$, from which we have extracted\nstrong-coupling series of the quantities $E$, $\\chi$, $\\xi_{G}^2$,\n$u$, $M_{\\rm t}^2$, already introduced in Sec.~\\ref{sectrNi}, and of\nthe ratios $s\\equiv M_{\\rm t}^2\/M_{G}^2$.  Some of the above series\nfor $N=3$ are reported in App.~\\ref{appsctr}.\n\nLike ${\\rm O}(N)$ $\\sigma$ models on the square lattice, \nno indication of the presence\nof a critical point at a finite real value of $\\beta$ emerges\nfrom the strong-coupling analysis for $N\\geq 3$. \nBy a Dlog-PA analysis of the $O(\\beta^{15})$\nstrong-coupling series of $\\chi$ and $\\xi_{G}^2$ at $N=3$, \nwe found that the singularities closest to the origin is \n$\\bar{\\beta}\\simeq 0.358\\pm i 0.085$, giving rise to a convergence\nradius $\\beta_r\\simeq 0.37$ which should correspond to a rather\nlarge correlation length: $\\xi_{G}\\simeq 70$.\nIncreasing $N$ such singularities move toward their\n$N=\\infty$ limit $\\bar{\\beta}=0.206711\\pm\\,i\\,0.181627$.\nSome details of this analysis are given in Table \\ref{zeroes}.\n\nIn our analysis of dimensionless quantities we considered, as on the\nsquare lattice, both the series in the energy and in $\\beta$.  The\nestimates concerning the continuum limit are obtained by evaluating\nthe approximants of the energy series at $E=1$, and those of the\n$\\beta$-series at a $\\beta$ associated to a reasonably large\ncorrelation length.  For $N=3$ we chose $\\beta=0.33$, whose\ncorresponding correlation length should be $\\xi\\simeq 22$, according\nto a strong-coupling estimate.\n\nCalculating a few more components of $G(x)$ at larger orders (i.e.,\nthose involved by the wall-wall correlation function at distance\n$\\sqrt{3}\/2 \\times 5$ up to $O(\\beta^{16})$), we computed the\nratio $s\\equiv M_{\\rm t}^2\/M_{G}^2$ up to $O(\\beta^{11})$\n\\cite{SCUN1,RV}.  For $N=3$ the analysis of the strong-coupling series\nof $s$ (some details are given in Table \\ref{trs}) leads to the\nestimate $s^*=0.998(3)$ from the energy approach, and $s^*=0.998(1)$\nevaluating the approximants at $\\beta=0.33$ (we considered PA's and\nDlog-PA's with $l+m\\geq 8$ and $m\\geq l\\geq 4$).  Such results are in\nperfect agreement with those found for the square lattice.\n\nPA's and Dlog-PA's (with \n$l+m\\geq 11$ and  $m\\geq l\\geq 5$)\nof the strong-coupling series of $u$ expressed \nin terms of the energy, evaluated at $E=1$, lead to the estimate\n$u^*=0.249(1)$ at $N=3$. The analysis of the series in $\\beta$\ngives $u^*=0.2502(4)$.\nAgain universality is satisfied. \n\nA summary of the results on the triangular lattice can be\nfound in Table \\ref{sum}.\n\nAs on the square lattice we checked asymptotic scaling by looking at\nthe ratio $\\Lambda_{\\rm t}\/ \\Lambda_{2l}$, where $\\Lambda_{\\rm t}$ is\nthe effective $\\Lambda$-parameter on the triangular lattice, defined\nin analogy with Eq.~(\\ref{effla}). Beside\nthe formulas concerning asymptotic scaling given for the square\nlattice case, cfr.\\ Eqs.~(\\ref{effla}-\\ref{twoloopla}), we need here\nthe $\\Lambda$-parameter ratio $\\Lambda_{\\rm t}\/\\Lambda_{\\rm s}$\ncalculated in App.~\\ref{apptr}, cfr.\\ Eq.~(\\ref{ratioltr2}).  We again\nused $M_G$ as approximate estimator of the mass-gap $M$.\nFig.~\\ref{asysctr} shows curves of $\\Lambda_{\\rm t}\/ \\Lambda_{2l}$ for\nvarious values of $N$, $N=3,4,8$, and for comparison the exact curve\nfor $N=\\infty$.  Such results are similar to those found on the square\nlattice: for $N=3,4$ asymptotic scaling regime is still far at\n$\\xi_G\\simeq 10$, but it is verified within a few per cent at $N=8$,\nwhere the correlation length corresponding to the strong-coupling\nconvergence radius is $\\xi\\simeq 8$.\n\n\n\\subsection{The honeycomb lattice}\n\\label{scho}\n\nOn the honeycomb lattice we have calculated \nthe two-point Green's function \nup to $O(\\beta^{30})$, from which we extracted \nstrong-coupling series of the quantities $E$, $\\chi$, \n$\\xi_{G}^2$, $u$, $M_{\\rm v}^2$, $M_{\\rm h}^2$, \nalready introduced in  Sec.~\\ref{iNhl},\nand of the ratios $r\\equiv M_{\\rm v}^2\/M_{\\rm h}^2$,\n$s\\equiv M_{\\rm h}^2\/M_{G}^2$.\nSome of the above series for $N=3$ are reported \nin App.~\\ref{appscex}.\n\nAt $N=3$ a Dlog-PA analysis of the $O(\\beta^{30})$\nstrong-coupling series of $\\chi$ and $\\xi_{G}^2$ \ndetected two couples of complex conjugate singularities,\none on the imaginary axis at $\\bar{\\beta}\\simeq\\pm i0.460$, \nquite close to the origin, and the other \nat $\\bar{\\beta}\\simeq 0.93\\pm i 0.29$.\nThe singularity on the imaginary axis\nleads to a rather small convergence radius in terms\nof correlation length, indeed  at $\\beta\\simeq 0.46$ \nwe estimate $\\xi\\simeq 2.6$. \nAt $N=4$ we found $\\bar{\\beta}\\simeq\\pm i0.444$, \nand $\\bar{\\beta}\\simeq 0.88\\pm i 0.41$.\nAt larger $N$ the singularities closest\nto the origin converge toward the $N=\\infty$ value\n$\\bar{\\beta}=\\pm i \\,0.362095$.\nNotice that, as on the square lattice, the partition function on the\nhoneycomb lattice enjoys the symmetry $\\beta\\rightarrow -\\beta$.\n\nAgain we analyzed both the series in the energy and in $\\beta$.\nThe estimates concerning the continuum limit are obtained by evaluating\nthe approximants of the energy series at $E=1$, and \nthose of the $\\beta$-series at $\\beta=0.85$ for the $N=3$ case,\nwhich should correspond to $\\xi\\simeq 22$.\n\nBy rotation invariance the ratio \n$r\\equiv M_{\\rm h}^2\/M_{\\rm v}^2$ should go to one\nin the continuum limit. \nFrom $G(x)$ up to $O(\\beta^{30})$ we extracted\nthe ratio $r$ up to $O(\\beta^{20})$. \nAgain PA's and Dlog-PA's \nof the energy series evaluated at $E=1$\nand of the $\\beta$-series evaluated at $\\beta=0.85$\n(some details are given in Table \\ref{hor})\ngive the correct result in the continuum limit: \nrespectively $r^*=1.00(1)$ and $r^*=1.001(1)$ at $N=3$\n(we considered PA's and Dlog-PA's with \n$l+m\\geq 16$ and $m\\geq l\\geq 7$).\n\nCalculating a few more components of $G(x)$ at larger orders (i.e.,\nthose involved by $G^{({\\rm w})}_{\\rm h}(x)$, defined in\nEq.~(\\ref{g2}), at distances $x=\\sqrt{3}\/2 \\times 9$ and\n$x=\\sqrt{3}\/2 \\times 10$ respectively at $O(\\beta^{34})$ and\n$O(\\beta^{35})$), we computed the ratio $s\\equiv M_{\\rm h}^2\/M_{G}^2$\nup to $O(\\beta^{25})$ \\cite{SCUN1,RV}.  For $N=3$ the analysis of the\nstrong-coupling series of $s$ gives $s^*=0.999(3)$ from the\n$E$-approximants and $s^*=0.9987(5)$ from the $\\beta$-approximants\nevaluated at $\\beta=0.85$ (some details are given in Table \\ref{hos}),\nin agreement with the result found on the other lattices.  We\nconsidered PA's and Dlog-Pa's with $l+m\\geq 22$, $m\\geq l\\geq 10$.\n\nThe analysis of the energy series of $u$ confirms universality:\nPA's and Dlog-PA's (with $l\\leq m$,\n$l+m\\geq 26$, $l\\geq 12$) of the energy series\nevaluated at $E=1$ give $u^*=0.249(3)$,\nand those of the $\\beta$-series at $\\beta=0.85$ $u^*=0.2491(3)$.\nAs for the square lattice, the curve $u(E)$ obtained \nfrom the PA's at $N=3$ and the exact curve $u(E)$ \nat $N=\\infty$, cfr.\\ Eq.~(\\ref{omhoin}), \nwould be hardly distinguishable if plotted together.\n\nAs noted above, the convergence radius $\\beta_r$ is small in terms of\ncorrelation length for all values of $N$: it goes from $\\xi\\simeq 1.0$\nat $N=\\infty$ to $\\xi\\simeq 2.6$ at $N=3$. Nevertheless in this case\nDlog-PA resummations seem to give reasonable estimates of $\\xi_G$ even\nbeyond $\\beta_r$ (apparently up to about the next singularity closest\nto the origin).  In Fig.~\\ref{asyscho} we show curves of \n$\\Lambda_{\\rm h}\/ \\Lambda_{2l}$, where $\\Lambda_{\\rm h}$ is the\neffective $\\Lambda$-parameter on the honeycomb lattice, for various\nvalues of $N$, $N=3,4,8$, and for comparison the exact curve for\n$N=\\infty$.  The necessary ratio of $\\Lambda$-parameters has been\ncalculated in App.~\\ref{appex}, cfr.\\ Eqs.~(\\ref{ratiolho}) and\n(\\ref{ratiolho2}).\n\n\n\\subsection{Conclusions}\n\\label{concl}\n\nWe have shown that quite accurate continuum limit estimates of\ndimensionless renormalization-group invariant quantities, such as $s$\nand $u$ (cfr.\\ Eqs.~(\\ref{sdef}) and (\\ref{omsqin})), can be obtained\nby analyzing their strong-coupling series and applying resummation\ntechniques both in the inverse temperature variable $\\beta$ and in the\nenergy variable $E$.  In particular, in order to get continuum\nestimates from the analysis of the energy series, we evaluated the\ncorresponding PA's and Dlog-PA's at $E=1$, i.e., at the continuum\nlimit.  This idea was already applied to the calculation of the\ncontinuum limit of the zero-momentum four-point coupling $g_r$,\nobtaining accurate results~\\cite{gr}.  These results look very\npromising in view of a possible application of such strong-coupling\nanalysis to four-dimensional gauge theories.\n\nThe summary in Table \\ref{sum} of our $N=3$ strong-coupling results\nfor the continuum values $r^*$, $s^*$ and $u^*$,\nfor all lattices we have considered, shows that\nuniversality is verified within a precision of few per mille, leading\n to the final estimates $s^*\\simeq 0.9985$ and $u^*\\simeq 0.2495$\nwith an uncertainty of about one per mille.\nThe comparison with the exact $N=\\infty$ results, $s^*=1$ and\n$u^*=1\/4$, shows that quantities like $s^*$ and $u^*$, which describe the\nsmall momentum universal behavior of $\\widetilde{G}(p)$ in the\ncontinuum limit, change very little and apparently monotonically from\n$N=3$ to $N=\\infty$, suggesting that\nat $N=3$ $\\widetilde{G}(p)$ is essentially Gaussian at small momentum.\n\nLet us make this statement more precise.\nIn the critical region\none can expand the dimensionless \nrenormalization-group invariant function\n\\begin{equation}\nL(p^2\/M_G^2)\\equiv {\\widetilde{G}(0)\\over \\widetilde{G}(p)}\n\\label{elle}\n\\end{equation}\naround $y\\equiv p^2\/M_G^2=0$, writing\n\\begin{eqnarray}\nL(y)&=&1 + y + l(y)\\nonumber \\\\\nl(y)&=&\\sum_{i=2}^\\infty c_i y^i.\n\\label{lexp}\n\\end{eqnarray}\n$l(y)$ parameterizes the difference from a generalized Gaussian\npropagator.  One can easily relate the coefficients $c_i$ of the\nexpansion (\\ref{lexp}) to dimensionless renormalization-group\ninvariant ratios involving the moments $m_{2j}$ of $G(x)$.\n\nIt is worth observing that\n\\begin{equation}\nu^* = {1\\over 4 (1 - c_2)}.\n\\label{uc2}\n\\end{equation} \nIn the large-$N$ limit the function $l(y)$ is\ndepressed by a factor $1\/N$.\nMoreover the coefficients of its low-momentum expansion are very small.\nThey can be derived from the $1\/N$ expansion of the \nself-energy~\\cite{Flyv,CR,CRselfenergy}. \nIn the leading order in the $1\/N$ expansion one finds  \n\\begin{eqnarray}\nc_{2}&\\simeq&-{0.00619816...\\over N},\\nonumber \\\\ \nc_{3}&\\simeq&{0.00023845...\\over N},\\nonumber \\\\\nc_{4}&\\simeq&-{0.00001344...\\over N},\\nonumber \\\\ \nc_{5}&\\simeq&{0.00000090...\\over N},\n\\label{c1N}\n\\end{eqnarray}\netc..  For sufficiently large $N$ we then expect\n\\begin{equation}\nc_i\\ll c_2\\ll 1 \\;\\;\\;\\;\\;\\;\\;\\;\\;{\\rm for}\\;\\;\\;\\; i\\geq 3.\n\\label{crel}\n\\end{equation}  \nAs a consequence, since \nthe zero of $L(y)$ closest \nto the origin is $y_0=-s^*$, the value of $s^*$ \nis substantially fixed  by the term proportional to\n$(p^2)^2$ in the inverse propagator, through the approximate relation \n\\begin{equation}\ns^*-1\\simeq c_2 \\simeq 4 u^* - 1 .\n\\label{s4u}\n\\end{equation}\nIndeed in the large-$N$ limit one finds from Eqs.~(\\ref{largeNs}) and \n(\\ref{largeNu})\n\\begin{equation}\ns^*-4u^*={-0.000252\\over N}+O\\left( {1\\over N^2}\\right),\n\\end{equation}\nwhere the coefficient of the $1\/N$ term is much\nsmaller than those of $s^*$ and $u^*$.\n\nFrom this large-$N$ analysis one expects \nthat even at $N=3$ the function $l(y)$ be small in a relatively large\nregion around $y=0$. \nThis is confirmed by the strong-coupling estimate of $u^*$,\nwhich, using Eq.~(\\ref{uc2}), leads to  $c_2 \\simeq -0.002$.\nFurthermore, the comparison of the estimates of $s^*$ and $u^*$\nshows that $s^*-4u^*\\simeq 0$ within the precision of our analysis,\nconsistently with Eq.~(\\ref{crel}).\nIt is interesting to note that similar results have been\nobtained for the models with $N\\leq 2$,\nand in particular for the Ising Model, i.e., for $N=1$, where the\nstrong-coupling analysis turns out to be very precise~\\cite{Nm2}.\n\nWe can conclude that the two-point Green function for all $N\\geq 3$ is\nalmost Gaussian in a large region around $p^2=0$, i.e.,\n$|p^2\/M_G^2|\\lesssim 1$, and the small corrections to Gaussian\nbehavior are essentially determined by the $(p^2)^2$ term in the\nexpansion of the inverse propagator.\n\n\n\nDifferences from Gaussian behavior will become important at\nsufficiently large momenta, as predicted by simple weak coupling\ncalculations supplemented by a renormalization group resummation.\nIndeed the asymptotic behavior of $G(x)$ for $x\\ll1\/M$ (where $M$ is\nthe mass gap) turns out to be\n\\begin{equation}\nG(x) \\sim \\left(\\ln{1\\over xM}\\right)^{\\gamma_1\/b_0}, \\qquad\n{\\gamma_1\\over b_0} = {N-1\\over N-2} \\,;\n\\end{equation}\n$b_0$ and $\\gamma_1$ are the first coefficients of the\n$\\beta$-function and of the anomalous dimension of the fundamental\nfield $\\vec s$ respectively.  Let us remind that a free Gaussian\nGreen's function behaves like $\\ln (1\/x)$.\nImportant differences are\npresent in other Green's functions even at small momentum, as shown in\nthe analysis of the four-point zero-momentum renormalized coupling,\nwhose definition involves the zero-momentum four-point correlation\nfunction (\\ref{chi4})~\\cite{gr}.\nHowever monotonicity in $N$ seems to be a persistent feature.\n\nOur strong-coupling calculations allow also a check of asymptotic\nscaling for a relatively large range of correlation lengths. For all\nlattices considered the ratio between the effective\n$\\Lambda$-parameter extracted from the mass-gap and its two-loop\napproximation, $\\Lambda\/\\Lambda_{2l}$, when considered as function of\n$\\xi_G$, shows similar patterns with changing $N$.  Confirming earlier\nMonte Carlo studies, large discrepancies from asymptotic scaling are\nobserved for $N=3$ in the range of correlation lengths we could\nreliably investigate, i.e.,  $\\xi\\lesssim 50$. At $N=8$ and for all\nlattices considered, asymptotic scaling within a few per cent is\nverified for $\\xi\\gtrsim 4$, and increasing $N$ the ratio\n$\\Lambda\/\\Lambda_{2l}$ approaches smoothly its $N=\\infty$ limit.\n\n\n\\acknowledgments\n\nIt is a pleasure to thank B.~Alles\nfor useful and stimulating discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzavko b/data_all_eng_slimpj/shuffled/split2/finalzzavko
new file mode 100644
index 0000000000000000000000000000000000000000..ba3bba9badd8462f0c781edd5603a2b005df204f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzavko
@@ -0,0 +1,5 @@
+{"text":"\\section{Syst\\`emes int\\'egrables quantiques}\n\nLe {\\bf mod\\`ele \\`a 6 sommets} est un c\\'el\\`ebre mod\\`ele de {\\bf physique statistique} introduit par Pauling en 1935, qui permet notamment \nde d\\'ecrire le cristal de la glace  (voir \\cite{BaxterBook}). Il est r\\'ealis\\'e sur un r\\'eseau dont chaque\nsommet est reli\\'e \\`a 4 autres sommets. Un \\'etat du syst\\`eme est une orientation\ndes ar\\^etes telle qu'\\`a chaque sommet arrivent exactement 2 fl\\^eches (Figure 1). \nLes fl\\^eches repr\\'esentent l'orientation des mol\\'ecules d'eau du cristal\nles unes par rapport aux autres. \nIl y a 6 configurations\npossibles \\`a chaque sommet (Figure 2), ce qui justifie l'appellation de ce mod\\`ele.\n~\\begin{center}\n\\begin{figure}\\label{orientation}\n \\hspace{4.5cm}  \\epsfig{file=orientation.eps,width=0.3\n   \\linewidth}\n      \\caption{Une orientation d'un r\\'eseau (mod\\`ele \\`a 6 sommets).}\n\\end{figure}\n\\end{center}\n\\begin{center}\n\\begin{figure}\\label{6vertex}\n  \\hspace{1cm} \\epsfig{file=6vertex.eps,width=0.8\\linewidth}\n      \\caption{6 configurations possibles \\`a chaque sommet.}\n\\end{figure}\n\\end{center}\n\n\nL'\\'etude du mod\\`ele de la glace est fortement li\\'ee \\`a celle d'un autre mod\\`ele, cette fois-ci en {\\bf physique statistique quantique}, appel\\'e {\\bf mod\\`ele $XXZ$} de Spin $1\/2$, dit de Heisenberg quantique (1928). Il s'agit d'une variante en physique quantique du mod\\`ele d'Ising (1925) (voir \\cite{JM}), qui mod\\'elise des cha\\^ines de spins magn\\'etiques quantiques\nayant deux \\'etats classiques, haut ou bas (Figure 3).\n\\begin{center}\n\\begin{figure}\\label{spin}\n \\hspace{4.5cm}  \\epsfig{file=spin2.eps,width=0.3\n   \\linewidth\n   }\n      \\caption{Etats d'un Spin $1\/2$ (haut ou bas).}\n\\end{figure}\n\\end{center}\n\nCes deux mod\\`eles, mod\\`ele \\`a 6 sommets et mod\\`ele $XXZ$, figurent parmi les plus \\'etudi\\'es\nen physiques statistique et quantique. Les structures math\\'ematiques qui les sous-tendent sont tr\\`es proches.\nEn d\\'epit de leur formulation assez \\'el\\'ementaire,\nils sont extr\\^emement riches et leur analyse a une tr\\`es longue histoire.\n\nEn physique statistique (quantique), le comportement du syst\\`eme est contr\\^ol\\'e par la {\\bf fonction de partition} $\\mathcal{Z}$ \\footnote{En physique statistique, la fonction de partition s'exprime comme la somme $\\sum_j \\text{exp}(-E_j\/(k_BT))$ sur tous les \\'etats $j$ du syst\\`eme, o\\`u $E_j$ est l'\\'energie de l'\\'etat $j$, $T$ est la temp\\'erature du syst\\`eme et $k_B$ la constante de Boltzmann. En physique quantique, la somme est remplac\\'ee par une trace $\\text{Tr}_W(\\text{exp}(-E\/(k_BT)))$ o\\`u $E$ est l'op\\'erateur \\og hamiltonien\\fg{}  qui agit sur l'espace $W$ des \\'etats quantiques du syst\\`eme.}, qui permet d'obtenir les grandeurs mesurables\\footnote{Une grandeur mesurable $Q$ est obtenue comme moyenne pond\\'er\\'ee sur les \\'etats $\\frac{\\sum_j  \\text{exp}(-E_j\/(k_BT)) Q_j}{\\mathcal{Z}}$ des valeurs $Q_j$ sur chaque \\'etat $j$.}. Cette fonction\n$\\mathcal{Z}$ est tr\\`es difficile \\`a calculer en g\\'en\\'eral. La m\\'ethode de la {\\bf matrice de transfert}\nest un proc\\'ed\\'e pour tenter de la d\\'eterminer : il s'agit d'\\'ecrire $\\mathcal{Z}$ comme trace\nd'un op\\'erateur $\\mathcal{T}$ (la matrice de transfert) agissant sur l'{\\bf espace des \\'etats} $W$ : \n$$\\mathcal{Z} = \\on{Tr}_W (\\mathcal{T}^M).$$\nIci $M$ est un entier associ\\'e \\`a la taille du r\\'eseau du mod\\`ele.\nAinsi, pour trouver $\\mathcal{Z}$, il suffit d'obtenir les valeurs propres $\\lambda_j$ de $\\mathcal{T}$ :\n$$\\mathcal{Z} = \\sum_j \\lambda_j^M.$$\nLe spectre $\\{ \\lambda_j \\}_j$ de $\\mathcal{T}$ est appel\\'e le {\\bf spectre du syst\\`eme quantique}.\n\nDans un c\\'el\\`ebre article s\\'eminal de 1971, inspir\\'e notamment par les travaux de Bethe (1931), Baxter \\cite{Baxter} a compl\\`etement r\\'esolu ce probl\\`eme\\footnote{Baxter a introduit la m\\'ethode puissante des \\og Q-op\\'erateurs\\fg{} qui lui a \\'egalement permis de r\\'esoudre le mod\\`ele \\og \\`a 8 sommets\\fg, plus complexe. Le mod\\`ele \\`a 6 sommets avait aussi \\'et\\'e r\\'esolu par d'autres m\\'ethodes, notamment dans les travaux de Lieb et Sutherland (1967).}. Gr\\^ace \\`a une \\'etude\ntr\\`es pr\\'ecise il a notamment montr\\'e\nque les valeurs propres $\\lambda_j$ de $\\mathcal{T}$ ont une structure tout \\`a fait remarquable : elles\ns'expriment sous la forme\n\\begin{equation}    \\label{relB}\n\\lambda_j = A(z) \\frac{Q_j(zq^2)}{Q_j(z)} + D(z)  \\frac{Q_j(zq^{-2})}{Q_j(z)},\n\\end{equation}\no\\`u $z,q\\in\\CC^*$ sont des param\\`etres du mod\\`ele (respectivement spectral et quantique),\n$A(z)$ et $D(z)$ sont des fonctions \\og universelles\\fg{} (au sens o\\`u elles ne d\\'ependent pas de la valeur propre\n$\\lambda_j$). La fonction $Q_j(z)$ d\\'epend de la valeur propre, mais c'est un polyn\\^ome.\nLa relation (\\ref{relB}) est la fameuse {\\bf relation de Baxter} (ou \\og relation $TQ$ de Baxter\\fg).\nLes polyn\\^omes $Q_j$ sont appel\\'es {\\bf polyn\\^omes de Baxter}.\n\n\\medskip\n\nEn r\\'esultent alors naturellement les questions suivantes :\n\n- Y a-t-il une explication pour l'existence de la relation de Baxter  ?\n\n- Une expression analogue avec des polyn\\^omes permet-elle de d\\'ecrire le spectre d'autres syst\\`emes quantiques ?\n\n\\medskip\n\nUne conjecture formul\\'ee en 1998 par Frenkel-Reshetikhin \\cite{Fre} affirme que la deuxi\\`eme question doit avoir une r\\'eponse positive. Comme on ne peut esp\\'erer effectuer en g\\'en\\'eral le calcul d\\'etaill\\'e de Baxter qui est connu pour le mod\\`ele $XXZ$, c'est en r\\'epondant \\`a la premi\\`ere question que nous pouvons d\\'emontrer cette conjecture. Pour ce faire, \\'etudions les structures math\\'ematiques, alg\\'ebriques, sous-jacentes \\`a la th\\'eorie.\n\n\\section{Groupes quantiques et leurs repr\\'esentations}\n\nLes {\\bf groupes quantiques} sont graduellement apparus au cours des ann\\'ees 1970, en particulier dans les travaux de l'\\'ecole de Leningrad, comme le cadre naturel math\\'ematique pour \\'etudier les matrices de transfert. Drinfeld \\cite{Dri} et Jimbo \\cite{J} ont ind\\'ependamment d\\'ecouvert une formulation alg\\'ebrique uniforme sous forme d'{\\bf alg\\`ebres de Hopf}. Il s'agit d'un des r\\'esultats cit\\'es pour la m\\'edaille Fields de Drinfeld en 1990.\n\nPour introduire les groupes quantiques de Drinfeld-Jimbo, consid\\'erons d'abord un objet tr\\`es classique, une {\\bf alg\\`ebre de Lie} (simple) complexe de dimension finie. Il s'agit d'un espace vectoriel de dimension finie $\\mathfrak{g}$ muni d'un {\\bf crochet de Lie}, c'est-\\`a-dire d'une application bilin\\'eaire antisym\\'etrique \n$$[,]:\\mathfrak{g}\\times \\mathfrak{g}\\rightarrow \\mathfrak{g}$$ \nsatisfaisant la formule de Jacobi\n$$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0\\text{ pour tous }x,y,z\\in\\mathfrak{g}.$$\nL'exemple le plus simple, mais n\\'eanmoins non trivial car il correspond au mod\\`ele $XXZ$, est celui de l'alg\\`ebre de Lie $\\mathfrak{g} = sl_2$ : c'est l'espace des matrices complexes $2\\times 2$ de trace nulle, muni du crochet naturel \n$$[M,N] = MN - NM$$ \npour lequel il est clairement stable. Pour les g\\'en\\'erateurs lin\\'eaires\n$$E = \\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}\\text{ , }F = \\begin{pmatrix}0&0\\\\1&0\\end{pmatrix}\n\\text{ , }H = \\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix},$$\non a par exemple la relation\n\\begin{equation}\\label{crochet}[E,F] = H.\\end{equation}\n\nCes alg\\`ebres de Lie ont des analogues naturelles de dimension infinie, les {\\bf alg\\`ebres de lacets} \n$$\\hat{\\Glie} = \\Glie \\otimes \\CC[t^{\\pm 1}],$$\navec le crochet de Lie d\\'efini par\n$$[x\\otimes f(t),y\\otimes g(t)] = [x,y]\\otimes (fg)(t)\\text{ pour $x,y\\in\\Glie$ et $f(t),g(t)\\in \\CC[t^{\\pm 1}]$},$$ \nce qui revient \\`a remplacer le corps $\\CC$ par l'anneau des polyn\\^omes de Laurent complexes \n$$\\CC[t^{\\pm 1}] = \\left\\{\\sum_{N\\leq i\\leq M} a_i t^i| N,M\\in\\ZZ, a_i\\in\\CC\\right\\}.$$ \nCes alg\\`ebres sont des quotients d'{\\bf alg\\`ebres de Kac-Moody affines}, qui ont des propri\\'et\\'es alg\\'ebriques semblables \\`a celles des alg\\`ebres de Lie simples de dimension finie (notamment une pr\\'esentation analogue \\`a celle de Serre pour $\\Glie$, comme l'ont montr\\'e Kac (1968) et Moody (1969), voir \\cite{ka}). Elles ont \\'et\\'e \\'etudi\\'ees intensivement pour leurs diverses applications en math\\'ematiques et physique math\\'ematique.\n\nMaintenant, pour \\'etudier les syst\\`emes quantiques qui nous int\\'eressent, ces alg\\`ebres de Lie classiques doivent \\^etre {\\bf quantifi\\'ees}, c'est-\\`a-dire d\\'eform\\'ees en tenant compte du param\\`etre quantique \n$$q = \\text{exp}(h)\\in\\CC^*,$$ \no\\`u $h$ est un analogue de la grandeur de Planck ($q$ sera bien identifi\\'e au param\\`etre quantique de la relation (\\ref{relB})). \nOn retrouve les structures classiques pour $h\\rightarrow 0$, donc $q\\rightarrow 1$. {\\it On supposera dans la suite que $q$ n'est pas une racine de l'unit\\'e.}\n\nBien qu'une telle quantification des alg\\`ebres de Lie $\\Glie$ ou $\\hat{\\Glie}$ elles-m\\^emes ne soit pas connue, Drinfeld et Jimbo ont d\\'ecouvert qu'il existe une quantification naturelle de leurs {\\bf alg\\`ebres enveloppantes} respectives $\\mathcal{U}(\\Glie)$ et $\\mathcal{U}(\\hat{\\Glie})$ (alg\\`ebres universelles d\\'efinies \\`a partir des alg\\`ebres de Lie, par exemple en rempla\\c cant dans la pr\\'esentation de Serre les crochets $[x,y]$ par des expressions alg\\'ebriques $xy - yx$ dans l'alg\\`ebre). On obtient alors les groupes quantiques $\\mathcal{U}_q(\\Glie)$, $\\mathcal{U}_q(\\hat{\\Glie})$ qui d\\'ependent\\footnote{Elles peuvent \\^etre d\\'efinies comme des alg\\`ebres sur $\\CC[[h]]$.} du param\\`etre quantique $q$, voir \\cite{CP}. \n\nPar exemple dans $\\mathcal{U}_q(sl_2)$ la relation (\\ref{crochet}) devient\n$$[E,F] = \\frac{e^{hH} - e^{-hH}}{q - q^{-1}}$$\nqui tend bien vers $H$ quand $h$ tend vers $0$.\n\nLe cas des {\\bf alg\\`ebres affines quantiques} $\\mathcal{U}_q(\\hat{\\Glie})$ est particuli\\`erement remarquable car Drinfeld \\cite{Dri2} a d\\'emontr\\'e\\footnote{La preuve a \\'et\\'e pr\\'ecis\\'ee par la suite par Beck puis par Damiani.} qu'elles peuvent non seulement \\^etre obtenues comme quantification de $\\mathcal{U}(\\hat{\\Glie})$, mais \\'egalement, par un autre proc\\'ed\\'e, comme {\\bf affinisation} du groupe quantique $\\mathcal{U}_q(\\Glie)$. C'est la {\\bf r\\'ealisation de Drinfeld} des alg\\`ebres affines quantiques. Ceci peut \\^etre \\'enonc\\'e dans le diagramme \\og commutatif\\fg{} suivant :\n$$\\xymatrix{ &\\hat{\\Glie}\\ar@{-->}[dr]^{\\text{Quantification}}&   \n\\\\ \\Glie \\ar@{-->}[ur]^{\\text{Affinisation}}\\ar@{-->}[dr]_{\\text{Quantification}}& &  \\U_q(\\hat{\\Glie})\n\\\\ & \\U_q(\\Glie)\\ar@{-->}[ur]_{\\text{Affinisation quantique}}& }$$\nCe th\\'eor\\`eme, qui revient \\`a donner deux pr\\'esentations isomorphes de $\\mathcal{U}_q(\\hat{\\Glie})$, est un analogue quantique du th\\'eor\\`eme classique de Kac et Moody. Il s'agit d'une bonne indication de l'importance de ces alg\\`ebres d'un point de vue alg\\'ebrique.\n\nLes alg\\`ebres affines quantiques $\\U_q(\\hat{\\Glie})$ ont en fait une structure beaucoup plus riche, ce sont des alg\\`ebres de Hopf. Elles sont notamment munies d'une {\\bf comultiplication} (qui est une op\\'eration duale de la multiplication), c'est-\\`a-dire d'un morphisme d'alg\\`ebre\n\\begin{equation}\\label{coproduit}\\Delta : \\U_q(\\hat{\\Glie}) \\rightarrow \\U_q(\\hat{\\Glie})\\otimes \\U_q(\\hat{\\Glie}).\\end{equation}\nMais surtout, $\\U_q(\\hat{\\Glie})$ poss\\`ede une {\\bf $R$-matrice universelle}, c'est-\\`a-dire un \\'el\\'ement (canonique) dans le carr\\'e tensoriel\\footnote{En fait, dans une l\\'eg\\`ere compl\\'etion du carr\\'e tensoriel.}\n$$\\mathcal{R}(z) \\in (\\U_q(\\hat{\\Glie})\\otimes \\U_q(\\hat{\\Glie}))[[z]]$$\nqui est notamment une solution de l'{\\bf \\'equation de Yang-Baxter quantique} :\n$$\\mathcal{R}_{12}(z)\\mathcal{R}_{13}(zw)\\mathcal{R}_{23}(w) = \\mathcal{R}_{23}(w)\n\\mathcal{R}_{13}(zw) \\mathcal{R}_{12}(z).$$\nLes param\\`etres formels $z$, $w$ sont appel\\'es {\\bf param\\`etres spectraux}. Cette \\'equation est \\`a valeurs dans le cube tensoriel \n$$(\\U_q(\\hat{\\Glie}))^{\\otimes 3}[[z, w]].$$\nLes indices dans les facteurs indiquent l'emplacement des termes de la $R$-matrice universelle : \n$$\\mathcal{R}_{12}(z) = \\mathcal{R}(z)\\otimes 1\\text{ , }\\mathcal{R}_{23}(z) = 1 \\otimes \\mathcal{R}(z)...$$\nIl s'agit d'une \\'equation hautement non triviale, li\\'ee aux mouvements de tresses. En effet, dans la figure 4 on retrouve l'\\'equation en lisant de bas en haut et en multipliant\npar un facteur $\\mathcal{R}_{\\alpha\\beta}$ d'indice $(\\alpha,\\beta)$ lorsque le brin $\\alpha$ croise le brin $\\beta$. C'est pour cette raison que la th\\'eorie des repr\\'esentations des groupes quantiques permet de construire des invariants en topologie de basse dimension (notamment les polyn\\^omes de Jones des n\\oe uds). Il s'agit historiquement, avec la construction par Lusztig et Kashiwara de bases canoniques de repr\\'esentations des alg\\`ebres de Lie classiques, d'un des premiers grands succ\\`es de la th\\'eorie des groupes quantiques. Nous n'aborderons pas ces sujets ici pour nous concentrer sur les applications aux syst\\`emes quantiques.\n \\begin{center}\n\\begin{figure}\\label{tresse}\n \\hspace{4cm}  \n {\\epsfig{file=tresse.eps,width=.4\\linewidth}\n}\n      \\caption{Equation de Yang-Baxter}\n\\end{figure}\n\\end{center}\n\nPour d\\'ecrire des solutions de l'\\'equation de Yang-Baxter quantique, on peut sp\\'ecialiser sur des {\\bf repr\\'esentations} de dimension finie de $\\U_q(\\hat{\\Glie})$.\nUne repr\\'esentation (lin\\'eaire) de $\\U_q(\\hat{\\Glie})$ est un espace vectoriel $V$ (ici complexe) muni d'un morphisme d'alg\\`ebre\n$$\\rho_V : \\U_q(\\hat{\\Glie}) \\rightarrow \\text{End}(V).$$\nAutrement dit, l'alg\\`ebre $\\U_q(\\hat{\\Glie})$ agit sur l'espace $V$ par op\\'erateurs lin\\'eaires. \n\nL'\\'etude des repr\\'esentations est un vaste domaine, central en math\\'ematiques, appel\\'e {\\bf th\\'eorie des repr\\'esentations}. En arithm\\'etique par exemple, les repr\\'esentations \nde groupes de Galois jouent un r\\^ole crucial. Elles sont \\'egalement essentielles dans la formulation m\\^eme des principes de la physique quantique car ils font intervenir des repr\\'esentations\nde l'alg\\`ebre des observables.\n\nOn d\\'efinit naturellement la {\\bf somme directe de repr\\'esentations} $(V,\\rho_V)$ et $(V',\\rho_{V'})$ avec l'application $\\rho_{V\\oplus V'} = \\rho_V + \\rho_{V'}$ \\`a valeurs\ndans $\\text{End}(V \\oplus V')$.\n\nLes {\\bf repr\\'esentations simples}, c'est-\\`a-dire qui n'ont pas de sous-repr\\'esentation (sous-espace stable pour l'action de l'alg\\`ebre) propre, sont particuli\\`erement importantes, comme nous allons le voir dans notre \\'etude.\nElles constituent les \\og briques \\'el\\'ementaires\\fg{} de la th\\'eorie des repr\\'esentations. Par exemple, toute repr\\'esentation de dimension finie de $\\U_q(\\Glie)$ est {\\bf semi-simple}, c'est-\\`a-dire isomorphe \\`a une somme\ndirecte de repr\\'esentations simples\\footnote{Ce r\\'esultat d\\'emontr\\'e par M. Rosso et G. Lusztig est un analogue quantique du th\\'eor\\`eme classique de Weyl qui assure que toute repr\\'esentation de dimension finie de $\\U(\\Glie)$ est semi-simple.}. Ce n'est pas le cas\\footnote{Cependant, toute repr\\'esentation $V$ de dimension finie de $\\U_q(\\hat{\\Glie})$ admet une filtration de Jordan-H\\\"older par des sous-repr\\'esentations $V_0 = V\\supset V_1\\supset V_2 \\cdots \\supset V_N = \\{0\\}$  avec les  $V_i\/V_{i+1}$ simples.} pour l'alg\\`ebre affine quantique $\\U_q(\\hat{\\Glie})$.\n\n\nComme $\\U_q(\\hat{\\Glie})$ est munie d'un coproduit (\\ref{coproduit}), pour deux repr\\'esentations $(V,\\rho_V)$ et $(V',\\rho_{V'})$, le produit tensoriel $V\\otimes V'$\nest aussi une repr\\'esentation en utilisant\n$$\\rho_{V\\otimes V'} = (\\rho_V\\otimes \\rho_{V'})\\circ \\Delta : \\U_q(\\hat{\\Glie})\\rightarrow \\text{End}(V)\\otimes \\text{End}(V') = \\text{End}(V\\otimes V').$$\nCette action sur un {\\bf produit tensoriel de repr\\'esentation} sera utile par la suite. Mais ind\\'ependamment on peut faire aussi agir directement la $R$-matrice\nuniverselle sur un carr\\'e tensoriel $V\\otimes V$ pour $V$ une repr\\'esentation de dimension finie de $\\U_q(\\hat{\\Glie})$ : on peut en effet consid\\'erer l'image de la $R$-matrice universelle dans $\\text{End}(V^{\\otimes 2})(z)$\n$$\\mathcal{R}_{V,V}(z) = (\\rho_V\\otimes \\rho_V)(\\mathcal{R}(z))\\in \\text{End}(V)^{\\otimes 2}[[z]] = \\text{End}(V^{\\otimes 2})[[z]].$$\nOn obtient aussi une solution de l'\\'equation de Yang-Baxter quantique, dite {\\bf $R$-matrice}, mais dans l'alg\\`ebre de dimension finie $\\text{End}(V^{\\otimes 2})[[z]]$.\n\nPar exemple, dans le cas $\\Glie = sl_2$, l'alg\\`ebre affine quantique $\\U_q(\\hat{sl_2})$ poss\\`ede une repr\\'esentation de dimension $2$ dite {\\bf repr\\'esentation fondamentale}\net not\\'ee $V_1$. Par le proc\\'ed\\'e d\\'ecrit ci-dessus, elle produit la $R$-matrice suivante\n\\footnote{La solution explicite de l'\\'equation de Yang-Baxter donn\\'ee ici est la $R$-matrice \\og normalis\\'ee\\fg, obtenue en multipliant $\\mathcal{R}_{V_1, V_1}(z)$ par une certaine fonction scalaire de $z$. On peut constater que ses coefficients sont des fonctions m\\'eromorphes en $z$. C'est un ph\\'enom\\`ene g\\'en\\'eral, voir \\cite{efk}.} dans $\\text{End}(V_1^{\\otimes 2})[[z]]$ avec $V_1^{\\otimes 2}$ qui est de dimension $4$ :\n$$\\begin{pmatrix}1&0&0&0\\\\ 0&\\frac{q^{-1}(z-1)}{z-q^{-2}}&\\frac{z(1 - q^{-2})}{z-q^{-2}}&0\\\\0&\\frac{1-q^{-2}}{z - q^{-2}}&\\frac{q^{-1}(z - 1)}{z - q^{-2}}&0\\\\0&0&0&1\\end{pmatrix}.$$\nC'est la $R$-matrice associ\\'ee au mod\\`ele $XXZ$. Mais la th\\'eorie des groupes quantiques en produit beaucoup d'autres, selon qu'on change l'alg\\`ebre de Lie $\\Glie$ ou la repr\\'esentation $V$. \nElles correspondent \\`a autant de syst\\`emes quantiques. \n\nLa {\\bf matrice de transfert} $\\mathcal{T}_V(z)$ est alors d\\'efinie en prenant la trace partielle sur la repr\\'esentation, c'est-\\`a-dire \n\\begin{equation}\\label{transfer}\n\\mathcal{T}_V(z) = ((\\on{Tr}_V \\circ \\rho_V) \\otimes \\on{id})({\\mathcal{R}(z)})\\in \\U_q(\\hat{\\Glie})[[z]].\n\\end{equation}\nLa repr\\'esentation $V$ qui sert \\`a construire la matrice de transfert $\\mathcal{T}_V(z)$ est appel\\'ee {\\bf espace auxiliaire}.\nComme cons\\'equence de l'\\'equation de Baxter, les matrices de transfert commutent, c'est-\\`a-dire que pour une autre repr\\'esentation $V'$ on a\n$$\\mathcal{T}_V(z)\\mathcal{T}_V(z') = \\mathcal{T}_V(z')\\mathcal{T}_V(z)\\text{ dans }\\U_q(\\hat{\\Glie})[[z,z']].$$\nAinsi, les coefficients $\\mathcal{T}_V[N]$ des matrices de transfert, d\\'efinis par\n$$\\mathcal{T}_V(z) = \\sum_{N\\geq 0}z^N \\mathcal{T}_V[N],$$\nengendrent une sous-alg\\`ebre commutative de $\\U_q(\\hat{\\Glie})$. \n\nDonnons-nous une autre repr\\'esentation de dimension finie $W$ de $\\U_q(\\hat{\\Glie})$, dite {\\bf espace des \\'etats}. Les coefficients $\\mathcal{T}_V[N]$ des matrices de transfert agissent donc sur $W$ en une grande famille commutative d'op\\'erateurs. Ainsi, il fait sens de parler des valeurs propres des matrices de transfert $\\mathcal{T}_V(z)$ sur $W$.\n\nDans le cas particulier du mod\\`ele $XXZ$, on rappelle que ${\\mathfrak g} = sl_2$ et $V = V_1$ est une repr\\'esentation fondamentale de dimension $2$.\nL'espace des \\'etats $W$ est un produit tensoriel de repr\\'esentations fondamentales de dimension $2$ et l'image de l'op\\'erateur $\\mathcal{T}_{V_1}(z)$\ndans $\\text{End}(W)[[z]]$ est bien la matrice de transfert de Baxter. Les r\\'esultats de Baxter donnent donc la structure du spectre de $\\mathcal{T}_{V_1}(z)$\nsur $W$ dans ce cas. \n\n\\medskip\n\nQue dire en g\\'en\\'eral ?\n\n\\section{La conjecture du spectre quantique}\n\nEn 1998 \\cite{Fre}, E. Frenkel et N. Reshetikhin ont propos\\'e une nouvelle approche dans le but de g\\'en\\'eraliser les formules de Baxter.\n\n\\`A cette fin, ils ont introduit le {\\bf $q$-caract\\`ere} $\\chi_q(V)$ d'une repr\\'esentation $V$ de dimension finie de $\\U_q(\\hat{\\Glie})$. Il s'agit d'un polyn\\^ome\nde Laurent \\`a coefficients entiers en des ind\\'etermin\\'ees $Y_{i,a}$ ($1\\leq i\\leq n$, $a\\in\\CC^*$) \n$$\\chi_q(V) \\in \\ZZ[Y_{i,a}^{\\pm 1}]_{1\\leq i\\leq n, a\\in\\CC^*}.$$\nL'entier $n$ est ici le {\\bf rang} de l'alg\\`ebre de Lie $\\Glie$, qui par exemple vaut bien $n$ pour $\\Glie = sl_{n+1}$. La d\\'efinition du $q$-caract\\`ere de $V$ repose\nsur une d\\'ecomposition de $V$ en sous-espaces de Jordan\\footnote{Pour une famille commutative d'op\\'erateurs sur $W$, obtenus \\`a partir de la r\\'ealisation de Drinfeld de $\\U_q(\\hat{\\mathfrak{g}})$ et distincts en g\\'en\\'eral des coefficients des matrices de transfert.} $V_m$ param\\'etr\\'es par des mon\\^ome $m$ en les variables $Y_{i,a}^{\\pm 1}$ :\n$$V = \\bigoplus_m V_m.$$\nLe $q$-caract\\`ere encode les dimensions de cette d\\'ecomposition. Il est d\\'efini par\n$$\\chi_q(V) = \\sum_m \\text{dim}(V_m) m.$$\nAinsi, les coefficients de $\\chi_q(V)$ sont en fait positifs et leur somme est la dimension $V$. \n\nPar exemple, pour $\\Glie = sl_2$ et $V = V_1$ la repr\\'esentation fondamentale de dimension $2$,  \n\\begin{equation}\\label{qcar}\n\\chi_q(V) = Y_{1,q^{-1}} + Y_{1,q}^{-1}.\n\\end{equation}\nOn a donc dans ce cas deux sous-espaces de Jordan de dimension $1$ associ\\'es aux mon\\^omes respectifs $Y_{1,q^{-1}}$ et $Y_{1,q}^{-1}$ :\n$$V = V_{Y_{1,q^{-1}}} \\oplus V_{Y_{1,q}^{-1}}.$$\n\nLa {\\bf conjecture du spectre quantique} de Frenkel et Reshetikhin \\cite{Fre} pr\\'edit\\footnote{Dans des cas particuliers, une conjecture analogue avait \\'et\\'e formul\\'ee par N. Reshetikhin \\cite{R3}; V. Bazhanov et N. Reshetikhin\n\\cite{BR}; et A. Kuniba et J. Suzuki \\cite{KS}.} que pour une repr\\'esentation de dimension finie donn\\'ee $V$,\nles valeurs propres $\\lambda_j$ de $\\mathcal{T}_V(z)$ sur une repr\\'esentation simple\\footnote{\nPlus g\\'en\\'eralement, $W$ peut \\^etre un produit tensoriel de repr\\'esentations simples.} $W$ sont obtenues de la mani\\`ere suivante :\ndans le $q$-caract\\`ere $\\chi_q(V)$ de $V$, on remplace chaque variable formelle $Y_{i,a}$ par\\footnote{Pour simplifier l'exposition, on supposera dans la suite de $\\Glie$\nest simplement lac\\'ee (c'est le cas notamment des alg\\`ebres de Lie $sl_{n+1}$).}\n$$F_{i}(az) q^{\\text{deg}(Q_{i,j})} \\frac{Q_{i,j}(zaq^{-1})}{Q_{i,j}(zaq)},$$\no\\`u $F_{i}(z)$ est une fonction universelle, au sens o\\`u elle ne d\\'epend pas de la valeur propre $\\lambda_j$, et $Q_{i,j}(z)$ d\\'epend de la valeur propre $\\lambda_j$ mais est un polyn\\^ome. C'est l'analogue du polyn\\^ome de Baxter.\n\nNotons que c'est bien le $q$-carat\\`ere {\\it de l'espace auxiliaire} $V$ qui est utilis\\'e pour \\'ecrire la formule du spectre de la matrice de transfert {\\it sur l'espace des \\'etats} $W$.\n\nDans le cas particulier du mod\\`ele $XXZ$, on obtient  \\`a partir de (\\ref{qcar}) la formule\n$$\\lambda_j = F_{1}(zq^{-1}) q^{\\text{deg}(Q_{1,j})} \\frac{Q_{1,j}(zq^{-2})}{Q_{1,j}(z)} + (F_1(zq))^{-1} q^{-\\text{deg}(Q_{1,j})} \\frac{Q_{1,j}(zq^2)}{Q_{1,j}(z)}.$$\nAinsi, la conjecture est bien compatible avec la formule de Baxter (\\ref{relB}) en identifiant \n$$A(z) = (D(zq^2))^{-1} = (F_1(zq))^{-1}q^{-\\text{deg}(Q_{1,j})}.$$ \nOn peut d\\'etailler par exemple le cas o\\`u l'espace des \\'etats $W\\simeq V_1$ est de dimension $2$. On a alors $2$ valeurs propres $\\lambda_0$ et $\\lambda_1$. La fonction universelle est\n$$F_1(z) = q^{1\/2}\\text{exp}\\left(  \\sum_{r > 0}  \\frac{z^r (q^{-r} - q^r)}{r(q^r + q^{-r})}\\right),$$\net les polyn\\^omes de Baxter sont\n$$Q_{1,0}(z) = 1\\text{ et }Q_{1,1}(z) = 1 - z(1 + q + q^2).$$\nOn obtient donc le spectre\n$$\\lambda_0 = F_1(zq^{-1})\\left(1 + q^{-3}\\frac{1 - z^{-1}}{1 - z^{-1}q^{-2}}\\right),$$\n\\begin{equation}\\label{bethe}\\lambda_1 = F_1(zq^{-1}) \\left(q\\frac{1-z(1+q^{-1} + q^{-2})}{1 - z(1 + q + q^2)} + q^{-4}\\frac{(1 - z^{-1})(1 - z(q^2 + q^3+ q^4))}{(1 - z^{-1}q^{-2})(1 - z(1 + q + q^2))}\\right).\\end{equation}\n\nEn g\\'en\\'eral la formule peut avoir plus de deux termes.\nPar exemple, dans le cas d'une certaine repr\\'esentation fondamentale $V$ de dimension $3$ de $\\U_q(\\hat{sl_3})$, le $q$-caract\\`ere est\n\\begin{equation}\\label{sl3}\\chi_q(V) = Y_{1,q^{-1}} + Y_{1,q}^{-1}Y_{2,1} + Y_{2,q^2}^{-1},\\end{equation}\net la formule pour le spectre est\n$$F_1(zq^{-1}) q^{\\text{deg}(Q_{1,j})}\\frac{Q_{1,j}(zq^{-2})}{Q_{1,j}(z)} +\\frac{F_2(z) q^{\\text{deg}(Q_{2,j})}}{F_1(zq)q^{\\text{deg}(Q_{1,j})}} \\frac{Q_{1,j}(zq^2)Q_{2,j}(zq^{-1})}{Q_{1,j}(z)Q_{2,j}(zq)} + \\frac{q^{-\\text{deg}(Q_{2,j})}}{(F_2(zq^2))^{-1}} \\frac{Q_{2,j}(zq^3)}{Q_{2,j}(zq)}.$$\n\nNotons qu'en g\\'en\\'eral les repr\\'esentations simples $V$ de dimension finie peuvent avoir une dimension \\og tr\\`es grande\\fg. Par exemple, H. Nakajima a obtenu (\\`a l'aide d'un super-calculateur et en s'appuyant sur \\cite{Nak}) que dans le cas de l'alg\\`ebre de Lie exceptionelle de type $E_8$, une des repr\\'esentations fondamentales a un $q$-caract\\`ere avec $6899079264$ mon\\^omes qui n\\'ecessite un fichier de taille m\\'emoire $180$ Go pour \\^etre \\'ecrit. Il y a donc autant de termes dans la formule de Baxter correspondante. Et les repr\\'esentations fondamentales sont les repr\\'esentations simples de dimensions les plus basses. \n\nIl est donc hors de question d'aborder cette conjecture par un calcul explicite en g\\'en\\'eral. D'ailleurs, m\\^eme si les repr\\'esentations simples de dimension finie de $\\U_q(\\hat{\\Glie})$ ont \\'et\\'e intensivement \\'etudi\\'ees ces vingt-cinq derni\\`eres ann\\'ees, on ne conna\\^it pas en g\\'en\\'eral de formule pour leur $q$-caract\\`ere, ni m\\^eme en fait pour leur dimension.\n\nAinsi, il faut de nouvelles structures pour aborder la conjecture du spectre quantique.\n\nNotre d\\'emonstration avec E. Frenkel \\cite{FH} de la conjecture du spectre quantique repose ainsi sur \nde nouveaux ingr\\'edients dont nous donnons un bref aper\\c cu dans les sections suivantes.\n\n\\section{Repr\\'esentations pr\\'efondamentales}\n\nL'id\\'ee g\\'en\\'erale de la preuve est d'interpr\\'eter les $Q_i$ eux-m\\^emes comme des valeurs\npropres de nouvelles matrices de transfert, construites non pas \\`a partir de repr\\'esentations de dimension\nfinie $V$, mais de repr\\'esentations de dimension infinie dite {\\bf repr\\'esentations pr\\'efondamentales}\n$L_{i,a}^+$ o\\`u $1\\leq i\\leq n$ et $a\\in\\CC^*$.\n\nNous avions construit pr\\'ealablement ces repr\\'esentations pr\\'efondamentales avec M. Jimbo \\cite{HJ} dans un contexte un peu diff\\'erent. Ce ne sont pas des repr\\'esentations de l'alg\\`ebre enti\\`ere $\\U_q({\\hat{\\mathfrak g}})$, mais d'une certaine sous-alg\\`ebre, la {\\bf sous-alg\\`ebre de Borel} \n$$\\U_q(\\hat{\\mathfrak{b}})\\subset \\U_q({\\hat{\\mathfrak g}}).$$ \nCela ne pose cependant pas de probl\\`eme pour construire la matrice de transfert $\\mathcal{T}_{i,a}(z)$ associ\\'ee \\`a la repr\\'esentation pr\\'efondamentale $L_{i,a}$ par la formule (\\ref{transfer}), car il est justement connu que la partie \\og gauche\\fg{} de la $R$-matrice universelle (celle \\`a qui on applique $\\rho_{L_{i,a}}$) est dans la sous-alg\\`ebre de Borel\\footnote{On ne peut cependant pas appliquer la trace \\`a un espace de dimension infinie. On utilise une graduation naturelle de $L_i$ par des espaces de dimension finie (les espaces de poids). Ainsi dans la suite, les traces, matrices de transfert, etc. sont \\og tordues\\fg{} par cette graduation.} :\n$$\\mathcal{T}_{i,a}(z) =  ((\\on{Tr}_{L_{i,a}} \\circ \\rho_{L_{i,a}}) \\otimes \\on{id})({\\mathcal{R}(z)})\\in \\U_q(\\hat{\\Glie})[[z]].$$\nIl n'est alors pas difficile de montrer qu'en utilisant un certain automorphisme de $\\U_q(\\hat{\\mathfrak{b}})$ on a \n$$\\mathcal{T}_{i,a}(z) = \\mathcal{T}_i(az)\\text{ o\\`u }\\mathcal{T}_i(z) = \\mathcal{T}_{i,1}(z).$$\n\nPour le cas du mod\\`ele $XXZ$, c'est-\\`a-dire pour $\\Glie = sl_2$, V. Bazhanov, S. Lukyanov, et A. Zamolodchikov avaient d\\'ej\\`a construit \\og \\`a la main\\fg{} une repr\\'esentation pr\\'efondamentale (appel\\'ee repr\\'esentation de $q$-oscillation) et la matrice de transfert associ\\'ee (appel\\'ee $Q$-op\\'erateur de Baxter) dans l'article important \\cite{BLZ}. \n\nPour obtenir l'existence des repr\\'esentations pr\\'efondamentales en g\\'en\\'eral \\cite{HJ}, on ne peut encore une fois pas faire de calculs explicites : le point crucial est de consid\\'erer des syst\\`emes inductifs\\footnote{Les inclusions $L_k\\subset L_{k+1}$, construites \\`a l'aide de produits tensoriels de sous-espaces \\cite{h3}, ne sont pas compatibles avec l'action de $\\U_q(\\hat{\\Glie})$ enti\\`ere mais avec celle d'une sous-alg\\`ebre $\\U_q^+(\\hat{\\mathfrak{b}})$ de $\\U_q(\\hat{\\mathfrak{b}})$.} de repr\\'esentations simples $L_k$ (les repr\\'esentations de Kirillov-Reshetikhin) de dimension finie strictement croissante avec $k\\geq 0$ et de d\\'eterminer en quel sens l'action de la sous-alg\\`ebre de Borel $\\U_q(\\hat{\\mathfrak{b}})$ \\og converge\\fg{} sur la limite inductive $L_\\infty$, qui elle est de dimension infinie :\n$$L_0\\subset L_1\\subset L_2\\subset \\cdots \\subset L_k\\subset L_{k+1}\\subset \\cdots  \\subset L_{\\infty}.$$\nIl s'agit ainsi d'une construction asymptotique des repr\\'esentations pr\\'efondamentales.\n\nEn utilisant certaines filtrations de la repr\\'esentation pr\\'efondamentale $L_{i,a}$, nous \\'etablissons qu'effectivement, \\`a un facteur scalaire universel $f_i(z)$ pr\\`es, la matrice de transfert associ\\'ee $\\mathcal{T}_i(z)$ agit sur l'espace des \\'etats $W$ par un op\\'erateur polyn\\^omial :\n$$\\rho_W(\\mathcal{T}_i(z)) \\in f_i(z) \\times (\\text{End}(W))[z].$$\nIl n'est pas difficile d'\\'ecrire une formule explicite pour la fonction universelle scalaire $f_i(z)\\in\\CC[[z]]$ (elle ne d\\'epend que de $V$ et de $W$). Il est beaucoup plus d\\'elicat d'obtenir des informations sur la partie lin\\'eaire polyn\\^omiale\n$$(f_i(z))^{-1}\\rho_W(\\mathcal{T}_i(z)) \\in (\\text{End}(W))[z].$$\n\nDe m\\^eme que les matrices de transfert usuelles commutent, on a \n$$\\mathcal{T}_i(z)\\mathcal{T}_i(z') = \\mathcal{T}_i(z')\\mathcal{T}_i(z),$$\net donc on obtient une famille commutative $\\mathcal{T}_i[m]$ si on \\'ecrit\n$$\\mathcal{T}_i(z) = \\sum_{m\\geq 0}\\mathcal{T}_i[m]z^m.$$\nEn utilisant la trigonalisation simultan\\'ee, cette commutativit\\'e implique que les valeurs propres sur $W$ de $(F_i(z))^{-1}\\mathcal{T}_i(z)$ elles-m\\^emes sont \\'egalement des polyn\\^omes.\n\n\\section{Anneau de Grothendieck et relations de Baxter}\n\nIl faut enfin d\\'emontrer que les valeurs propres de la matrice de transfert $\\mathcal{T}_V(z)$\ns'expriment, comme pr\\'evu dans la conjecture, en terme des valeurs propres des $\\mathcal{T}_i(z)$\nselon le $q$-caract\\`ere de $V$. Autrement dit, en rempla\\c cant dans $\\chi_q(V)$ chaque\nvariable $Y_{i,a}$ par le quotient\\footnote{Ce quotient doit en fait \\^etre multipli\\'e par une matrice de transfert d'une repr\\'esentation de dimension $1$ que nous omettons dans la suite pour simplifier l'exposition.} \n$$\\mathcal{T}_i(azq^{-1})\/\\mathcal{T}_i(azq),$$ \nobtient-on la matrice de transfert $\\mathcal{T}_V(z)$ ?\n\nDans le cas ${\\mathfrak g} = \\wh{sl}_2$ et $V$ de dimension du mod\\`ele $XXZ$, un calcul \\cite{BLZ} donne le r\\'esultat. On a bien : \n$$\\mathcal{T}_V(z) = \\frac{\\mathcal{T}_1(zq^{-1})}{\\mathcal{T}_1(zq)} + \\frac{\\mathcal{T}_1(zq^3)}{\\mathcal{T}_1(zq)}.$$\n\nEn g\\'en\\'eral, nous proposons d'utiliser la {\\bf cat\\'egorie $\\mathcal{O}$} que nous avons d\\'efinie avec M. Jimbo \\cite{HJ}. Il s'agit d'une {\\bf cat\\'egorie mono\\\" idale} (stable par produits tensoriels) de repr\\'esentations de l'alg\\`ebre de Borel $\\U_q(\\hat{\\mathfrak{b}})$, contenant les repr\\'esentations de dimension finie ainsi que les repr\\'esentations pr\\'efondamentales. \nNous {\\bf cat\\'egorifions} les relations de Baxter g\\'en\\'eralis\\'ees, c'est-\\`a-dire que nous les exprimons en termes de la cat\\'egorie $\\mathcal{O}$. Pour ce faire, on peut d\\'efinir l'{\\bf anneau de Grothendieck} $K(\\mathcal{O})$ de cette cat\\'egorie. En tant que groupe, il s'agit du groupe libre engendr\\'e par les classes d'isomorphismes de repr\\'esentations simples :\n$$K(\\mathcal{O}) = \\bigoplus_{[V]\\text{ Classe d'un simple dans }\\mathcal{O}.} \\ZZ [V].$$\nAlors tout objet (non n\\'ecessairement simple) de $\\mathcal{O}$ a une image dans $K(\\mathcal{O})$ en imposant la relation\n$$[V''] = [V] + [V']$$\nsi on a une suite exacte dans la cat\\'egorie\n$$0\\rightarrow V \\rightarrow V''\\rightarrow V'\\rightarrow 0.$$\nOn peut alors munir $K(\\mathcal{O})$ d'une structure d'anneau par la relation\n$$[V\\otimes V'] = [V][V']$$\npour des objets $V$, $V'$ de la cat\\'egorie $\\mathcal{O}$.\n\nUn des th\\'eor\\`emes principaux de \\cite{FH} est qu'en rempla\\c cant dans $\\chi_q(V)$ chaque\nvariable $Y_{i,a}$ par le quotient \n$$\\frac{[L_{i,aq^{-1}}]}{[L_{i,aq}]},$$ \nen repla\\c cant $\\chi_q(V)$ par $[V]$ puis en \\og chassant\\fg{} les d\\'enominateurs, on obtient une relation dans l'anneau de Grothendieck $K(\\mathcal{O})$.\n\nPar exemple, dans notre cas favori du mod\\`ele $XXZ$, on obtient\n$$[V] = \\frac{[L_{1,q^{-1}}]}{[L_{1,q}]} + \\frac{[L_{1,q^3}]}{[L_{1,q}]}$$\nqui donne la relation de Baxter cat\\'egorifi\\'ee dans l'anneau de Grothendieck\n$$[V][L_{1,q}] = [V\\otimes L_{1,q}] = [L_{1,q^{-1}}] + [L_{1,q^3}].$$\nEn g\\'en\\'eral on obtient des relations avec plus de termes, comme dans l'exemple pour $\\Glie = sl_3$ ci-dessus pour lequel la formule (\\ref{sl3}) donne\n$$[V\\otimes L_{1,1}\\otimes L_{2,q}] = [L_{1,q^{-2}}\\otimes L_{2,q}] + [L_{1,q^2}\\otimes L_{2,q^{-1}}] + [L_{2,q^3}\\otimes L_{1,1}].$$\nMaintenant, \\og prendre la matrice de transfert\\fg{} est additif et multiplicatif, c'est \\`a dire qu'on a un morphisme d'anneau\\footnote{On peut montrer que ce morphisme d'anneau est injectif et donc que l'anneau de Grothendieck $K(\\mathcal{O})$ est commutatif (bien que la cat\\'egorie ne soit pas tress\\'ee, c'est-\\`a-dire que $V\\otimes V'$ et $V'\\otimes V$ ne sont pas isomorphes en g\\'en\\'eral). En fait, l'application de $q$-caract\\`eres $[V]\\mapsto \\chi_q(V)$ elle-m\\^eme peut \\^etre prolong\\'ee en un morphisme d'anneau injectif sur $K(\\mathcal{O})$.}\n$$\\mathcal{T} : K(\\mathcal{O})\\rightarrow \\mathcal{U}_q(\\hat{\\Glie})[[z]]\\text{ , }[V]\\mapsto \\mathcal{T}_V(z).$$\nAinsi, les relations de Baxter g\\'en\\'eralis\\'ees dans l'anneau de Grothendieck $K(\\mathcal{O})$ impliquent les relations voulues entre les matrices de transfert. La conjecture du spectre quantique est donc d\\'emontr\\'ee.\n\n\\begin{center}*\\end{center}\n\nPour conclure, les formules pour les valeurs propres des matrices de transfert en terme\ndes polyn\\^omes $Q_{i,j}$ impliquent des \\'equations entre les racines de ces polyn\\^omes pour garantir que les p\\^oles\napparents se simplifient (par exemple dans l'\\'equation (\\ref{bethe}), $(1 + q + q^2)^{-1}$ n'est en fait pas un p\\^ole de $\\lambda_1$). Dans le cas du mod\\`ele $XXZ$ ce sont les fameuses \\'equations de l'Ansatz de Bethe.\nCes consid\\'erations ont men\\'e N. Reshetikhin \\cite{R3} \\`a formuler ces \\'equations dans le cas g\\'en\\'eral\n  (voir aussi \\cite{BR,KS, F:icmp}). La preuve de la conjecture du spectre quantique permet de donner une explication et une approche uniforme \\`a ces formules. On a maintenant une autre conjecture importante et ouverte : l'existence d'une bijection entre toutes les valeurs propres et les solutions des \\'equations de l'Ansatz de Bethe (conjecture de compl\\'etude).\n\n\\medskip\n\n{\\bf Remerciements} : je souhaite adresser mes remerciements \\`a E. Ghys pour m'avoir encourag\\'e \\`a \\'ecrire cet article, \\`a E. Frenkel et M. Jimbo pour notre collaboration et enfin \\`a J. Dumont, C. Hernandez, P. Zinn-Justin et l'\\'equipe d'Images de Math\\'ematiques pour leurs remarques sur une version pr\\'eliminaire de ce texte.\n\n\n\\backmatter\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nBuilding large-scale quantum computers is still a challenging task due to a plethora of engineering obstacles \\cite{eng}. One prominent challenge is the intrinsic noise. In fact, implementing scalable and reliable quantum computers requires implementing quantum gates with sufficiently low error rates. There has been substantial progress in characterizing noise in a quantum system \\cite{noise3,noise2,noise1} and in building error correcting schemes that can detect and correct certain types of errors \\cite{qec1,qec2,qec3}.\\\\\nNumerous protocols have been constructed to characterize the noise in quantum devices. Many of these protocols fail in achieving one of the following desirables: scalability to large-scale quantum computers and efficient characterization of the noise. Quantum Process Tomography \\cite{QPT} is a protocol that can give a complete description of the dynamics of a quantum black box, however, it's not scalable to large-scale quantum systems. Randomized Benchmarking (RB) is another protocol that's typically used to estimate the error rate of some set of quantum gates \\cite{ ScalableNoise, RBQG}. Although RB is a scalable protocol in principle, it can only measure a single error rate that's used to approximate the average gate infidelity thus providing an incomplete description of noise. Various other protocols based on RB protocol are able to characterize the correlations of noise between the different qubits, however, these protocols lack scalability \\cite{ScalableNoise,SimultaneousRB,ThreeRB}.\nQuantum Error Mitigation \\cite{QEM} (QEM) is a recently emerging field that aims to improve the accuracy of near-term quantum computational tasks. Whereas Quantum Error Correction (QEC) \\cite{AQEC1,AQEC2} necessitates additional qubits to encode a quantum state in a multiqubit entangled state, QEM does not demand any additional quantum resources. It is considered an excellent alternative for enhancing the performance of Noisy Intermediate-Scale Quantum (NISQ) computing \\cite{nisq}. QEM protocols include zero-noise Richardson extrapolation of results from a collection of experiments of varying noise \\cite{Extrapolation}, probabilistic error cancellation through sampling from a set of quantum circuits generated from a target circuit mixed with Pauli gates from an optimized distribution \n\\cite{pem,ProbabilisticMitigation}, and exploiting state-dependent bias through invert-and-measure techniques to map the predicted state to the strongest state \\cite{bias}.\nMeasurement Error Mitigation (MEM) is another QEM protocol that models the noise in a quantum circuit as a measurement noise matrix $\\bm{E}_{meas}$ applied to the ideal output of the circuit. The columns of $\\bm{E}_{meas}$ are the probability distributions obtained through preparing and immediately measuring all possible $2^n$ basis input states \\cite{QiskitTextbook}.\\\\\nRecently, the authors in \\cite{Nature} developed a protocol based on the RB that relies on the concept of a Gibbs Random Field (GRF) to completely and efficiently estimate the error rates of the Pauli Channel and detect correlated errors between the qubits in a quantum computer. Their effort paves the way to enable quantum error correction and\/or mitigation schemes. Herein, we refer to their efficient learning protocol as the \\{EL protocol\\}. \nIn this paper, we build upon the EL protocol and decompose the average noise of a quantum circuit of specific depth into State Preparation and Measurement (SPAM) error and average gate error.\nWe propose a linear algebraic based protocol and proof to efficiently construct and model the average behavior of noise in a quantum system for any desired circuit depth without having to run a large number of quantum circuits on the quantum computer or simulator. We then rely on this model to mitigate the noisy output of the quantum device. For an n-qubit quantum system, the average behavior of the noise can be well approximated as a special form of a Pauli Channel \\cite{Knill2005,Wallman_2016,Ware_2021}. \nA Pauli channel $\\varepsilon$ acts on a qubit state $\\boldsymbol{\\rho}$ to produce\n\\begin{equation}\n\\varepsilon(\\bm{\\rho})=\\sum_{i}p_i\\bm{P}_{i}\\bm{\\rho}\\bm{P}_{i}\n\\label{equation 1}\n\\end{equation}\nwhere $p_i$ is an error rate associated with the Pauli operator $\\bm{P}_{i}$. The $p_i$'s form a probability distribution $(\\sum_{i}p_{i}=1)$, and are related to the eigenvalues, $\\boldsymbol{\\lambda}$, of the Pauli Channel defined as \n\\begin{equation}\n\\lambda_i=2^{-n}Tr(\\bm{P}_{i}\\varepsilon(\\bm{P}_{i}))\n\\label{Equation 2}\n\\end{equation}\nThus, when a state $\\bm{\\rho}$ is subjected to the noisy\nchannel $\\varepsilon$, $p_i$ describes the probability of a multiqubit Pauli error $\\bm{P}_{i}$ affecting the system, while $\\lambda_i$ describes how faithfully a given multispin Pauli operator is transmitted. $\\bm{p}$ and $\\bm{\\lambda}$ are related by Walsh-Hadamard transform where \n\n\n\\begin{equation}\n\\bm{\\lambda}=\\bm{Wp}\n\\label{Equation 3}\n\\end{equation}\n\nWhile RB only estimates the average value of all $\\lambda_i$ of the Pauli Channel, the EL protocol estimates the individual $\\lambda_i$. A complete characterization of the Pauli channel requires learning more than the eigenvalues or error rates associated with single-qubit Pauli operators such as $\\bm{\\sigma}_{z}^{(1)}$ or  $\\bm{\\sigma}_{x}^{(3)}$; it requires learning all of the noise correlations in the system, that is, also learning the eigenvalues and error rates associated with multiqubit Pauli operators such as $\\bm{\\sigma}_{z}^{(1)} \\otimes \\mathbf{1}^{(2)} \\otimes \\bm{\\sigma}_{x}^{(3)}$ and how they vary compared to the ones obtained under independent local noise. Estimating these correlations is essential for performing optimal QEC and\/or QEM. However, these  correlations increase exponentially as the number of qubits increases, so having an efficient noise characterization protocol is crucial to direct the error mitigation efforts to capture the critical noise correlations.\\\\\nOur method relies on the error rates vector $\\bm{p}$ of the Pauli-Channel to decompose the average behavior of noise for circuits of depth $m$ into two noise components: a SPAM error matrix denoted by the matrix $\\bm{N}$ and a depth dependent component comprising an average gate error matrix denoted by the matrix $\\bm{M}$. We evaluate our model for the average noise by predicting the average probability distribution for circuits of depth $m$ and computing the distance between this predicted distribution and the empirically obtained one. Finally, we use our proposed decomposition to mitigate noisy outputs of random circuits and compare our mitigation protocol with the MEM protocol \\cite{QiskitTextbook}. We applied our noise characterization and mitigation protocols on the following IBM Q 5-qubit quantum computers: Manila, Lima, and Belem\\cite{IBM}.\n\\section*{Results}\n\\subsection*{Proposed Protocol Theory}\nThe ideal output probability distribution of an $n$-qubit quantum circuit with depth $m$ is perturbed by the SPAM and the average gate errors. Our aim is to construct a comprehensive linear algebraic model that takes into account both these errors for an arbitrary depth $m$. Matrix algebra can then be employed to mitigate the noise as follows: \n\\begin{equation}\n    \\bm{C}_{ideal}= \\bm{Q}_{m}^{-1}\\bm{C}_{noisy}\n\\end{equation}\nwhere $\\bm{Q}_{m}$ is the characterized noise matrix for circuits of depth $m$, $\\bm{C}_{ideal}$ and $\\bm{C}_{noisy}$ are the ideal and noisy outputs of a given circuit of depth $m$, respectively.\nThe straight-forward approach would be to construct $\\bm{Q}_m$ from empirical simulations in a similar fashion to the $\\bm{E}_{meas}$ noise matrix that was characterized in the MEM scheme. The columns of $\\bm{Q}_{m}$ comprise the emperical average probability distributions for basis input states $\\ket{\\bm{in}}\\in \\{ \\ket{\\bm{0}},\\,\\ket{\\bm{1}},\\,\\dots,\\,\\ket{\\bm{2^n-1}}\\}$, denoted by $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$, where $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$ are obtained through sampling a number of depth $m$ circuits to incorporate the average gate and SPAM errors.\n\\begin{equation}\n\\bm{Q}_m=\n\\begin{bmatrix}\n \\hat{\\bm{q}}(m,\\ket{\\bm{0}}) & \\hat{\\bm{q}}(m,\\ket{\\bm{1}}) & \\hdots &\\hat{\\bm{q}}(m,\\ket{\\bm{2^n-1}})\n\\end{bmatrix}\n\\label{eq:Q}\n\\end{equation}\nBuilding $\\bm{Q}_m$, however, through empirical simulations can be expensive especially when the circuit depth is large. Herein, we propose a method for an efficient estimation of $\\bm{Q}_m$ where the individual probability distributions $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$ are estimated as follows:\n\\begin{equation}\n    \\bm{q}'(m, \\ket{\\bm{in}}) = \\bm{N}_{in}\\bm{M}_{in}^m\\ket{\\bm{in}}\n    \\label{eq:qhat}\n\\end{equation}\nwhere $\\bm{N}_{in}$ and $\\bm{M}_{in}$ are input-specific matrices that represent the SPAM error matrix and average gate error for input $\\ket{\\bm{in}}$, respectively. Both $\\bm{M}_{in}$ and $\\bm{N}_{in}$ are extracted empirically using random circuits from a set of small circuit depths $T$ and then used in mitigating the outputs for circuits with higher depths. We first show the construction of $\\bm{N}_{0}$ and $\\bm{M}_{0}$.\\\\\nThe construction of $\\bm{N}_{0}$ and $\\bm{M}_{0}$ proceeds by estimating the error rates vector $\\bm{p}$ associated with the Pauli Channel based on the assumption in Equation \\ref{equation 1} for the average behavior of the noisy quantum device at hand using the EL protocol. The protocol proceeds by constructing $K$ random identity circuits of depth $m \\in T$ \\cite{SimultaneousRB, Nature}. Each circuit is constructed by initializing the qubits to the all-zeros state $\\ket{\\bm{0}}$ followed by choosing a random sequence $s \\in S_{m}$, the set of all length $m$ sequences of one-qubit Clifford gates applied independently on each qubit, followed by an inverse gate for the chosen sequence to ensure an identity circuit. It then estimates the resulting empirical probability distribution $\\hat{\\bm{q}}(m,\\ket{\\bm{0}})$ by averaging over all the empirical probability distributions $\\hat{\\bm{q}}(m,s,\\ket{\\bm{0}})$ for the constructed random identity circuits of depth $m$, that is, \n\\begin{equation}\n    \\hat{\\bm{q}}(m,\\ket{\\bm{0}})= \\frac{1}{K}\\sum \\hat{\\bm{q}}(m,s,\\ket{\\bm{0}})\n    \\label{Equation 4}\n\\end{equation}\n$\\hat{\\bm{q}}(m,\\ket{\\bm{0}})$ is a vector with $2^n$ entries each corresponding to the possible observed outcome. A Walsh-Hadamard transform is then applied on each $\\hat{\\bm{q}}(m,\\ket{\\bm{0}})$ to obtain\n\\begin{equation}\n    \\bm{\\Lambda}(m)=\\bm{W}\\hat{\\bm{q}}(m,\\ket{\\bm{0})}\n    \\label{Equation 5}\n\\end{equation}\nEach parameter $\\Lambda_{i}(m)$ in $\\bm{\\Lambda}(m)$ is fitted to the model \n\\begin{equation}\n    \\Lambda_{i}(m)=A_{i}\\lambda_{i}^{m}\n    \\label{Equation 6}\n\\end{equation}\nwhere $A_i$ is a constant that absorbs SPAM errors and \nthe vector $\\bm{\\lambda}$ of all fitted parameters $\\lambda_i$ is a SPAM-free estimate to the eigenvalues of the Pauli Channel defined in Equation \\ref{Equation 2}. Notice that we can rewrite Equation \\ref{Equation 6} as\n\\begin{equation}\n    \\bm{\\Lambda}(m)=\\bm{A\\lambda}^{m}\n    \\label{Equation 7}\n\\end{equation}\nwhere $\\bm{A}$ is a diagonal matrix where the diagonal entries are $A_i$ and $\\bm{\\lambda}^m$ is an element-wise exponentiation of a vector. An inverse Walsh-Hadamard Transform is then applied on $\\bm{\\lambda}$ to get the error rate vector $\\bm{p}$ of the Pauli Channel as\n\\begin{equation}\n    \\bm{p}=\\bm{W}^{-1}\\bm{\\lambda}\n    \\label{Equation 8}\n\\end{equation}\n$\\bm{p}$ is then projected onto a probability simplex to ensure $\\sum_{i}p_i=1$. Introducing the GRF model by the EL protocol allows the scalability of estimating $\\bm{p}$ with the increase in the number of qubits. The GRF model assumes the noise correlations are bounded between a number of neighboring qubits depending on the architecture of the quantum computer at hand. Thus, decreasing the number of noise correlations to be estimated.\\\\\nThe final outcome $\\bm{p}$ of the EL protocol represents the SPAM-free probability distribution of the average noise in the quantum computer. Each element $p_i \\in \\bm{p}$ corresponds to the probability of an error of the form $binary(i)$ on an input state $\\ket{\\bm{0}}$. For example, for a 5-qubit quantum computer, $p_0$ corresponds to the probability of no bit flips on the input state, i.e., error of the form  $IIIII$, $p_1$ to the error of the form $IIIIX$, $p_2$ to the error of the form $IIIXI$, etc\u2026\\\\\nIn order to proceed with the proof for our proposed decomposition of Equation (\\ref{eq:qhat}) for input state $\\ket{\\bm{0}}$, we first state the following lemma (the detailed proof of the lemma can be found Section I in the supplementary):\n\\begin{lemma} \nLet $\\bm{\\lambda}$ and $\\bm{p}$ be the respective eigenvalues and error rates of a Pauli Channel with $n$ qubits, then $\\bm{\\lambda}^m=\\bm{WM}^m\\ket{\\bm{0}}$\nwhere $\\bm{M}$ is a $2^n \\times 2^n$ matrix such that $M_{ij}=p_{i\\oplus j}$ ($i \\oplus j$ is the bitwise exclusive-OR operator).\n\\label{Lemma}\n\\end{lemma}\n\\noindent Using Lemma \\ref{Lemma} and Equations \\ref{Equation 5} and \\ref{Equation 7}, $\\hat{q}(m,\\ket{0})$ can be estimated as\n\\begin{equation}\n    \\bm{q}'(m,\\ket{\\bm{0}})=\\bm{W}^{-1}\\bm{AWM}^{m}\\ket{\\bm{0}}\n    \\label{Equation 11}\n\\end{equation}\nThe transition matrix $\\bm{M}=\\bm{M}_{0}$ represents the average error per gate while the $\\bm{N}=\\bm{W}^{-1}\\bm{AW}=\\bm{N}_{0}$ matrix represents the SPAM errors for an input state $\\ket{\\bm{0}}$. Notice that the average noise for depth $m$ circuits on an input state $\\ket{\\bm{0}}$ behaves as a sequence of $m$ average noise gates $\\bm{M}_{0}$ followed by SPAM errors $\\bm{N}_{0}$.\\\\\nThe construction of $\\bm{N}_{in}$ and $\\bm{M}_{in}$ for input state $\\ket{\\bm{in}}$ proceeds similar to the procedure of constructing $\\bm{N}_{0}$ and $\\bm{M}_{0}$, however, a permutation of $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$ is required before applying a Walsh-Hadamard transform to ensure that each element $p_{i}(\\ket{\\bm{in}})$ in the input-specific error rate vector $\\bm{p}(\\ket{\\bm{in}})$ corresponds to the probability of an error of the form $binary(i)$ on an input state $\\ket{\\bm{in}}$. This permutation is done by applying an input-specific permutation matrix $\\bm{\\pi}_{in}$ on $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$ $\\forall m$ where $\\pi_{in_{ij}}=1$ if $i\\oplus j=in$ and $0$ otherwise. \n\n\\subsection*{Experiments}\nIn this section, we evaluate the accuracy of the model in Equation $\\ref{Equation 11}$ in predicting the average probability output, $\\hat{\\bm{q}}(m,\\ket{\\bm{0}})$, for identity circuits of higher depths by estimating $\\bm{A}_0$ and $\\bm{p}(\\ket{\\bm{0}})$ using only simulations of lower depths identity circuits. Denote by $\\bm{q}'(m,\\ket{\\bm{0}})$ the predicted average probability distribution obtained using Equation \\ref{Equation 11}. We select a \\textit{training set of depths} $T=\\{1,\\,2,\\,\\dots,\\,m_{max}\\}$ to estimate $\\bm{A}_0$ and $\\bm{p}$ using the EL protocol followed by the construction of the average gate error matrix $\\bm{M}_{0}$ and SPAM error matrix $\\bm{N}_{0}$ where $M_{0_{ij}}=p_{i \\oplus j}(\\ket{\\bm{0}})$ and $\\bm{N}_{0}=\\bm{W}^{-1}\\bm{A}_{0}\\bm{W}$. A new \\textit{testing set of depths} $T'=\\{m_{max}+1,\\,m_{max}+2,\\,\\dots,\\,100\\}$ is then selected where we compute the \\textit{Jensen-Shannon Divergence} ($JSD$) between $\\hat{\\bm{q}}(m',\\ket{\\bm{0}})$ and $\\bm{q}'(m',\\ket{\\bm{0}})$ $\\forall m'\\in T'$. The $JSD$ measures the similarity between the two probability distributions \\cite{JSD}. The lower the $JSD$, the closer the two distributions are. More information about the $JSD$ can be found in Section II in the supplementary. Figure \\ref{VaryingTrainingDepth} presents the computed $JSD$ for different quantum computers while varying $m_{max}$. Figure \\ref{TestErrorBars} presents the average and standard deviation for the test $JSD$ values for the different quantum computers. The average test $JSD$ varies between $0.024$ and $0.056$ for the different $m_{max}$ values with lower average $JSD$ values noted for high $m$ for $m_{max}=80$ as indicated in Figure \\ref{TestErrorBars}b.\n\\begin{figure}[h]\n\\begin{subfigure}[t]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Manila}\n  \\caption{IBM Q Manila}\n  \\label{Manila}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Lima}\n  \\caption{IBM Q Lima}\n  \\label{Lima}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Belem}\n  \\caption{IBM Q Belem}\n  \\label{Belem}\n\\end{subfigure}\n\\caption{$JSD(\\hat{\\bm{q}}(m,\\ket{\\bm{0}}),\\,\\bm{q}'(m,\\ket{\\bm{0}}))$ for training sets of depths $T$ and testing sets of depths $T'$ with variable maximum training depth $m_{max}\\in\\{20,\\,50,\\,80\\}$ on different IBM Q 5-qubit quantum computers.}\n\\label{VaryingTrainingDepth}\n\\end{figure}\n\\FloatBarrier\n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}[t]{0.48\\textwidth}\n      \\centering\n      \\includegraphics[width=\\columnwidth]{TestErrorBars.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.48\\textwidth}\n      \\centering\n      \\includegraphics[width=\\columnwidth]{TestErrorBars80_100.png}\n      \\caption{}\n    \\end{subfigure}\n    \\caption{The average and standard deviation of $JSD(\\hat{\\bm{q}}(m,\\ket{\\bm{0}}),\\,\\bm{q}'(m,\\ket{\\bm{0}}))$; (a) over all depths $m \\in [m_{max}+1,100]$ and  (b) over depths $m \\in [80,100]$ while varying the maximum training depth $m_{max}$ on different IBM Q 5-qubit quantum computers.}\n    \\label{TestErrorBars}\n\\end{figure}\n\\FloatBarrier\n\\noindent We rely on $\\bm{q}'(m, \\ket{\\bm{in}})$ to construct and evaluate the mitigation power of $\\bm{Q}_m$ for different depths. We first  select a \\textit{training set of depths} $T=\\{1,\\,20,\\,40,\\,60,\\,80,\\,100\\}$ to estimate $\\bm{A}_{in}$ and $\\bm{p}(\\ket{\\bm{in}})$ for each input state $\\ket{\\bm{in}}$ using the EL protocol followed by the construction of $\\bm{M}_{in}$ using $\\bm{p}(\\ket{\\bm{in}})$ and $\\bm{N}_{in}=\\bm{W}^{-1}\\bm{A}_{in}\\bm{W}$. We then estimate $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$ as $\\bm{q}'(m,\\ket{\\bm{in}})$ for all inputs using Equation \\ref{eq:qhat} in order to construct $\\bm{Q}_{m}$ using Equation \\ref{eq:Q}. We then choose a new \\textit{testing set of depths} $T'=\\{10,\\,30,\\,50,\\,70,\\,90\\}$ so that $\\bm{Q}_m$ is used in mitigating the outputs for circuits of depth $m \\in T'$ where for a given identity circuit of depth $m$ with input $\\ket{\\bm{in}}$ and sequence $s$ of gates, the mitigated output $\\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})_{mit}$ in obtained as \n\\begin{equation}\n    \\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})_{mit}=\\bm{Q}_{m}^{-1}\\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})\n\\end{equation}\n$\\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})_{mit}$ is projected onto a probability simplex to ensure a probability distribution. The $JSD$ between $\\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})_{mit}$ and the ideal output $\\ket{\\bm{in}}$ is computed and then averaged over all input states and all random circuits of depth $m$. We also compare our proposed mitigation protocol using $\\bm{Q}_m$ with the MEM scheme (Figure \\ref{Mitigation}). We report upto 88\\% improvement in the $JSD$ value for the proposed approach compared to the unmitigated approach, and upto 69\\% improvement compared to MEM approach. Note that for the results presented here, we rely on the average SPAM free error rate, $\\bm{p}_{avg}=\\frac{1}{2^n}\\sum_{in=0}^{2^n-1}{\\bm{p}(\\ket{\\bm{in}})}$ to construct $\\bm{M}_{in}=\\bm{M}_{avg}$ for all inputs. We compare the results using $\\bm{p}_{avg}$ and $\\bm{p}(\\ket{\\bm{in}})$ in the supplementary Section V. $\\bm{N}_{in}$ remains input specific. Further elaborations on the results are presented in supplementary Section VI. \n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[h]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{AveManilaAllInputs.png}\n  \\caption{IBM Q Manila}\n  \\label{LimaMitigation}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{AveLimaAllInputs.png}\n  \\caption{IBM Q Lima}\n  \\label{AthensMitigation}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{AveBelemAllInputs.png}\n  \\caption{IBM Q Belem}\n  \\label{belem Mitigation}\n\\end{subfigure}\n\\caption{Average $JSD$ between the ideal output $\\ket{\\bm{in}}$ and each of the unmitigated output $\\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})$, mitigated output by the MEM protocol, and mitigated output by our proposed noise model for each depth $m$ on IBM Q 5-qubit quantum computers.}\n\\label{Mitigation}\n\\end{figure}\n\\FloatBarrier\n\\subsection*{Complexity}\nSo far in the estimation of $\\bm{M}_{in}$ and $\\bm{N}_{in}$ for each input state $\\ket{\\bm{in}}$ using the EL protocol, $K$ random circuits are generated for each depth $1\\le m \\le m_{max}$ where the EL protocol requires $O(2^{2n})$ for the Walsh-Hadamard transform which can be reduced into $O(n^2)$ using fast Walsh-Hadamard transform. Thus, the overall complexity of the construction of $\\bm{M}_{in}$ and $\\bm{N}_{in}$ for all input states is $O(m_{max}Kn^22^{n})$. Furthermore,  the GRF model factors the error rates vector into a product of $f\\sim O(n)$ factors, depending on the architecture of the quantum computer, where each factor depends on a subset of adjacent qubits of cardinality $N<<n$ (typically $N=4$). Thus, the complexity is reduced further into $O(nm_{max}KN^{2}2^{n})$. The construction of $\\bm{Q}_m$ would be based on Equation \\ref{eq:qhat} for each input state $\\ket{\\bm{in}}$ where $\\bm{M}_{in}^m$ can be computed efficiently using the Singular Value Decomposition (SVD) of $\\bm{M}_{in}$, thus the construction of $\\bm{Q}_m$ is $O(2^{3n})$. For the MEM scheme, the construction of $\\bm{E}_{meas}$ requires only generating $K$ circuits with no gates for each input state $\\ket{\\bm{in}}$, thus the complexity is $O(K2^{n})$. For mitigation, both protocols are based on matrix inversion, thus the complexity for mitigation is $O(2^{3n})$.\n\n\n\n\\section*{Discussion}\nThe proposed mitigation protocol builds upon the SPAM-free noise characterization protocols for low circuit depths to generate a SPAM-error matrix $\\bm{N}_{in}$ and an average gate error matrix $\\bm{M}_{in}$ for each input state $\\ket{\\bm{in}}$. It then constructs a noise mitigation matrix $\\bm{Q}_{m}$ for arbitrary circuit depths $m$ where the columns of $\\bm{Q}_{m}$ are the estimated average probability distributions $\\bm{q'}(m,\\ket{\\bm{in}})=\\bm{N}_{in}\\bm{M}_{in}^m$.\nThe mitigated output $\\hat{\\bm{q}}(m,\\,s)_{mit}$ of a given circuit of depth $m$ with sequence $s$ of gates is obtained by applying ${\\bm{Q}_{m}}^{-1}$ on the empirical circuit output $\\hat{\\bm{q}}(m,\\,s)$.\n\n\n\nWe evaluated the accuracy of our model in estimating the average probability distributions for high depth circuits and evaluated our mitigation protocol on the IBM Q 5-qubit quantum devices: Belem, Lima, and Manila.\nFor the model accuracy evaluations, for the different $m_{max}$ values, we reported on average a test  $JSD(\\hat{\\bm{q}}(m,\\ket{\\bm{0}}),\\bm{q}'(m,\\ket{\\bm{0}}))$ value around 0.022-0.028 for Manilla, 0.03-0.055 for Lima, and 0.028-0.048 for Belem. For $m_{max}=20$, the test JSD values varied between 0.005 and 0.05 for Lima computer, 0.01 and 0.06 for Manila computer, and 0.02 and 0.09 for Belem computer. We note that for $m_{max}=20$ the test spans $m \\in [21-100]$. For higher depths $m \\in [80-100]$, on average $m_{max}=80$ resulted in better model error than $m_{max}=50$ and $m_{max}=20$. Results for IBM Q Athens are presented in the supplementary.\n\nFinally, we report upto 88\\% JSD improvement for the proposed approach compared to the unmitigated approach with significant mitigation improvement compared to MEM at mid to higher depths. Specifically, for $m=90$, we reported 58\\%, 66\\% and 85\\% JSD improvement for the proposed approach compared to the unmitigated on Belem, Manilla, and Lima respectively. This is compared 12\\%, 17\\% and 51\\% respectively for the MEM. On average for all the test depths across the different machines, we report 68.4\\% JSD improvement for the proposed versus versus 38.2\\% for the MEM improvement compared to the unmitigated approach.\n\\section*{Methods}\nIn evaluating the accuracy of the model, we run $K=1000$ random identity circuits with each submission requesting 1024 shots for each depth $m \\in \\{1,\\,2,\\,\\dots,\\,100\\}$. In evaluating the mitigation power of $\\bm{Q}_m$, we run $K=1000$ random identity circuits for depths $m \\in \\{1,\\,10,\\,20,\\,30,\\,40,\\,50,\\,60,\\,70,\\,80,\\,90,\\,100\\}$ for each basis input state $\\ket{\\bm{in}}$ with each circuit requesting 1024 shots. The constructed circuits contain single qubit Clifford gates only. We run the circuits on the following IBM Q 5-qubit quantum computers: Manila, Lima, and Belem. Theoretical derivations and numerical details essential to the study are presented in the Results section. More details can be found in the Supplementary Information. For the configurations and noise profiles of the IBM quantum machines, please go to IBM Quantum Experience at \\url{http:\/\/www.research.ibm.com\/quantum}.\n\n\n\\subsection*{Data availability}\nThe data that supports the findings of this study are available from the corresponding author upon reasonable request.\n\n\n\n\\section{Introduction}\n\\label{intro}\nBuilding large-scale quantum computers is still an unachievable task due to a plethora of engineering challenges \\cite{eng}. One prominent challenge is that of noise. There has been substantial progress in analyzing the noise in a quantum system \\cite{noise3,noise2,noise1} and in building error correcting schemes that can detect and correct some types of errors \\cite{qec1,qec2,qec3}. However, no known protocol can completely and efficiently characterize quantum noise. Implementing scalable and reliable quantum computers can be done by implementing quantum gates with sufficiently low error rates. Most protocols for determining the error rates are not scalable. These protocols also suffer from state preparation and measurement errors (SPAM).\n\nWhile numerous protocols have been constructed to characterize the noise in quantum devices, many of these protocols fail in achieving one of the following desirables: scalability to large-scale quantum computers and efficient characterization of the noise.\n\nQuantum Process Tomography \\cite{QPT} is a protocol that can give a complete description of the dynamics of a quantum black box, however, it's not scalable to large-scale quantum systems. \n\nRandomized Benchmarking (RB) is another protocol that's typically used to estimate the error rate of some set of quantum gates \\cite{RB,ScalableNoise,RBQG}. Although RB is a scalable protocol in principle, it can only measure a single error rate that's used to approximate the average gate infidelity thus providing an incomplete description of noise. Recent work has improved the scalability of the RB protocol and has broadened its applicability. Various other protocols extend RB protocol to be able to characterize the correlations of noise between different qubits, however, these protocols are no longer scalable \\cite{RB,ScalableNoise}. \n\nFurthermore, \\cite{Nature} develops a protocol based on the RB that can completely and efficiently estimate the error rates of noise and detect all correlated errors between the qubits in a quantum computer.\n\nHere we extend the protocol developed in \\cite{Nature} to model the average behavior of noise in a quantum system and predict the average output for any desired circuit depth without having to run a large number of quantum circuits on the quantum computer or simulator. \n\n\\section{Background Review}\n\\subsection{Depolarizing Channel}\nA depolarizing channel $\\varepsilon$ is a simple model that describes noise in quantum system \\cite{nielson} . For a single qubit, the depolarizing channel replaces the qubit state with the completely mixed state $\\frac{I}{2}$ with probability $\\alpha$ and keeps it untouched with probability $(1-\\alpha)$. Thus, the state of the qubit after being submitted to depolarizing noise becomes\n\n\\begin{equation}\n    \\varepsilon(\\rho)=(1-\\alpha)\\rho+\\alpha\\frac{I}{2}\n\\end{equation}\n\nThe depolarizing channel is thus characterized by a single parameter $\\alpha$. RB and other protocols aim to estimate this $\\alpha$ for multi-qubit quantum computers.\n\n\\subsection{Quantum Process Tomography (QPT)}\nQuantum Process Tomography (QPT) is a procedure that can characterize the complete characteristics of a quantum black box \\cite{QPT}.\n\nA quantum channel is a Complete Positive Trace Preserving (CTPT) linear map $\\varepsilon$ that describes the dynamics of a quantum system.\n\n\\begin{equation}\n    \\rho \\rightarrow \\rho'=\\varepsilon(\\rho)\n\\end{equation}\n\n\nKraus theorem indicates that $\\varepsilon(\\rho)$ can be described \\begin{equation*}\n    \\varepsilon(\\rho)=\\sum_{i} A_i \\rho A_{i}^\\dagger\n\\end{equation*} where $A_i$ are called Kraus Operators and they satisfy the completeness relation \n\\begin{equation*}\n    \\sum_{i}A_i A_{i}^{\\dagger} = \\mathbb{I}\n\\end{equation*}\n\nThe $A_i$ describes the dynamics of the system alone. These dynamics include quantum unitary operations, measurements, and any noise affecting the system (decoherence).\n\nTypically, $A_i$ operators are derived from the theoretical model of the system and its environment. QPT can be used to experimentally derive the $A_i$ operators thus completely describing the dynamics of a quantum black box. \n\nFor a system of $n$ qubits, QPT proceeds by experimentally preparing $N=2^{2n}$ orthonormal pure states $\\rho_j = \\ket{\\psi_j}\\bra{\\psi_j},j=1,...,N$ and then measuring the output $\\rho^{'}=\\varepsilon(\\rho)$ for each input. The inputs and their corresponding outputs are then used to determine the unknown operators $A_i$. The quantum operation $\\varepsilon$ on any input state $\\rho$ can thus be determined by linearly extending $\\varepsilon$ to all states.\n\n\\cite{QPT} shows the mathematical steps for determining the Kraus operators for a quantum black box of one and two qubit quantum system. However, it becomes cumbersome for larger systems.\n\nOnce the operator-sum representation is determined, many important quantities including entanglement fidelity (measure how closely the dynamics of the quantum system under consideration approximates that of some ideal quantum system), the minimum fidelity, and the channel capacity.\n\n\\subsection{Randomized Benchmarking (RB)}\nRB protocol is used to obtain information about the average error rate over the Clifford group \\cite{RB}.The RB protocol proceeds as follows. \n\n\\begin{algorithm}\n\\caption{Randomized Benchmarking Protocol}\n\\begin{algorithmic}[1]\n\\State Choose a circuit depth $m$.\n\\State Choose a random sequence $s\\in S_m$ of Clifford Gates followed by an inverse gate for this sequence.\n\\State Obtain an estimate $\\hat{q}(m,s)$ of the expectation value of an observable $E$ after preparing state $\\rho$ and applying the gates in $s$.\n\\State Repeat steps 2-3 $K_m$ times to obtain an estimate $\\hat{q}(m)$ of \\begin{equation*}\n    \\bar{q}(m)=\\frac{1}{|S_m|}\\sum_{s \\in S_m}q(m,s)\n\\end{equation*}\n\n\\State Repeat steps 1-4 to fit the model \n\\begin{equation*}\n    \\hat{q}(m)=A\\alpha^m+B\n\\end{equation*}\n\n\\end{algorithmic}\n\\end{algorithm}\n\nThe average gate error $r$ can be extracted from $\\alpha$ as follows:\n\n\\begin{equation*}\n    r=\\frac{(2^n-1)(1-\\alpha)}{2^n}  \n\\end{equation*}\nwhere $n$ is the number of qubits.\n\nWhile RB is a scalable protocol and robust to SPAM errors, it only estimates a single parameter of interest; hence,  it doesn't give a complete description of the dynamics of the quantum system (including correlation of errors between qubits).\n\n\\subsection{Gibbs Random Field Protocol}\n\\cite{Nature} improves the RB protocol to an efficient and reliable protocol that can give a complete description of the noise in the quantum computer.\n\nAlthough not every noisy channel is a Pauli Channel, the average noise in any noisy quantum system can be well approximated by a Pauli Channel. A Pauli Channel $\\varepsilon$ acting on a state $\\rho$ is of the form \\begin{equation*}\n    \\varepsilon(\\rho)=\\sum_{j}p(j)P_{j}\\rho P_{j}\n\\end{equation*} where $p(j)$ is the error rate associated with the Pauli operator $P_{j}$. They are closely related to, but distinct from, the eigenvalues of the Pauli channel, which are defined to be \n\\begin{equation*}\n\\lambda(j)=2^{-n}Tr(P_{j}\\varepsilon(P_{j}))\n\\end{equation*}\n\nThus, when a state $\\rho$ is subjected to the noisy channel $\\varepsilon$, $p(j)$ describes the probability of a multiqubit Pauli error $P_{j}$ affecting the system, while the respective eigenvalue describes how faithfully a given multispin Pauli operator is transmitted. The $p(j)$ and $\\lambda(j)$ are related by a Walsh\u2013Hadamard transform where\n\\begin{equation*}\n    \\mathbf{p}=W\\mathbf{\\lambda}\n\\end{equation*}\n\nThus estimating $p(j)$ would give a complete description of the average noise in the quantum system.\n\nThe protocol in \\cite{Nature} proceeds as follows:\n\\begin{algorithm}\n\\caption{Gibbs Random Field Protocol}\n\\begin{algorithmic}[1]\n\\State Choose a circuit depth $m$.\n\\State Choose a random sequence $s\\in S_m$ of Clifford Gates followed by an inverse gate for this sequence.\n\\State Obtain an estimate $\\mathbf{\\hat{q}(m,s)}$ of the probability distribution over the $2^n$ different measurement outcomes.\n\\State Repeat steps 2-3 $K_m$ times to obtain an estimate $\\mathbf{\\hat{q}(m)}$ of \n\\begin{equation*}\n    \\mathbf{\\bar{q}(m)}=\\frac{1}{|S_m|}\\sum_{s\\in S_m}\\mathbf{q(m,s)}\n\\end{equation*}\n\n\\State Apply Walsh-Hadamard transform $W$ on $\\mathbf{\\hat{q}(m)}$ to obtain\n\\begin{equation*}\n    \\mathbf{\\lambda(m)}=W\\mathbf{\\hat{q}(m)}\n\\end{equation*}\n\n\\State Repeat steps 1-5 for different circuit depths.\n\n\\State For each parameter in $\\mathbf{\\lambda(m)}$, fit to the model\n\\begin{equation*}\n    \\lambda_{i}(m)=A\\lambda_{i}^m\n\\end{equation*}\n\n\\State Apply an Inverse Walsh-Hadamard transform on $\\mathbf{\\lambda}$ to obtain\n\\begin{equation*}\n    \\mathbf{p}=W^{-1}\\mathbf{\\lambda}\n\\end{equation*}\n\n\\end{algorithmic}\n\\end{algorithm}\n\nThe effective error rates $p$ can still describe important body correlations between different sets of qubits.\n\nWhile the nature protocol is not scalable, it introduces the Gibbs Random Field (GRF) model for a scalable estimation of $p(j)$ on a quantum computer. GRF is a strictly positive probability distribution that obeys certain conditional independence properties known as Markov conditions (Methods). These conditions restrict the range of possible correlations enough to make the problem of noise characterization tractable, but they allow sufficient expressive power that a GRF can accurately model noise in real devices. The underlying (but testable) assumption is that realistic devices will only have correlations between a bounded number of qubits. \nHowever, this model depends on the architecture of the quantum processor.\n\n\\section{Proposed Approach}\n\\subsection{Backward Propagation Approach (BPA) for Efficient Noise Mitigation}\nAlthough the noise in a given quantum system behaves randomly, we can analyze the average behavior of the noise in the system. We investigate the protocol used in \\cite{Nature} to develop a model of the average noise in a quantum system.\n\nThe final outcome $p$ of the protocol used in \\cite{Nature} is the SPAM free probability distribution of the average noise in the quantum computer. Each element $p_i$ in $p$ corresponds to the probability of an error of the form $binary(i)$ on an input state $\\ket{0}^{\\otimes n}$(the state where all qubits are initialized to $\\ket{0}$). For example, for a 5-qubit quantum computer, $p_0$ corresponds to the probability of no \"bit flips\" on the input state, $p_1$ to the error of the form 00001 (last qubit is flipped), etc\u2026\n\nUsing $p$, we construct a Markovian transition matrix $M$ that can be used to estimate the average noisy output probability distribution of an identity circuit of depth $m$. $M$ is constructed as follows: $M_{ij}=p_{i\\oplus j}$ where $\\oplus$ is the bitwise $xor$ operator. In other words, each column of $M$ corresponds to a different shuffling of $p$. We show here the construction $M$ for a 5-qubit quantum computer.\n\n\\begin{center}\n$M=\n\\begin{pmatrix}\n &p_0     &p_1     &p_2    &\\hdots &p_{31}\\\\ \n &p_1     &p_0     &p_3    &\\hdots &p_{30}\\\\ \n &p_2     &p_3     &p_0    &\\hdots &p_{29} \\\\ \n &\\vdots  &\\vdots  &\\vdots &\\ddots &\\vdots\\\\ \n &p_{31}  &p_{30}  &p_{29}    &\\hdots &p_0\n\\end{pmatrix}\n$\n\\end{center}\n\nAfter each \"average\" gate, the output for the identity circuit would be multiplied by this transition matrix. Thus, the average \"noisy\" probability distribution of an identity circuit of sequence length $m$ would be equal to $M^m\\ket{0}^{\\otimes n}$where $\\ket{0}^{\\otimes n}=[1,0, ... ,0]^T$. Notice for $m=1$, the probability distribution of the average output:\n\n\\begin{center}\n$\nM\\ket{0}^{\\otimes n}=\n\\begin{pmatrix}\np_0\\\\ \np_1\\\\ \np_2\\\\ \n\\vdots\\\\ \np_{31}\n\\end{pmatrix}\n=p\n$\n\\end{center}\n\nThe transition matrix $M$ doesn't include the SPAM errors, thus it doesn't give a good description of the behavior of the quantum circuit on average. However, we know that\n\\begin{equation}\n    M^m\\ket{0}^{\\otimes n}=W^{-1}\\lambda^m\n\\end{equation}\nAnd we also know that after fitting, we get \n\\begin{equation}\n    \\lambda^m=A^{-1}W\\hat{q}(m)\n\\end{equation}\nwhere $A$ is a diagonal matrix obtained from fitting.\nUsing (1) and (2), we get:\n\\begin{equation}\n    M^m\\ket{0}^{\\otimes n}=W^{-1}A^{-1}W\\hat{q}(m)\n\\end{equation}\nOr in other words:\n\\begin{equation}\n    W^{-1}AWM^m\\ket{0}^{\\otimes n}=\\hat{q}(m)\n\\end{equation}\n\nThe detailed proof of equation 6 can be found in the appendix.\n\n\\begin{algorithm}\n\\caption{Proposed Protocol}\n\\begin{algorithmic}[1]\n\\State Choose a training set of sequence lengths $T=[m_{1},m_{2},...]$.\n\\State Use the protocol in \\cite{Nature} to construct $A$ and $\\lambda$.\n\\State Construct $p=W^{-1}\\lambda$.\n\\State Construct the matrix $M$ where $M_{ij}=p_{i \\oplus j}$ ($\\oplus$ is the bitwise $xor$).\n\\State Choose a new testing set of sequence lengths $T'=[m'_{1},m'_{2}...]$.\n\\State for each $m'\\in T'$, construct the average probability distribution $\\hat{q}(m')=W^{-1}AWM^{m'}\\ket{0}^{\\otimes n}$\n\\end{algorithmic}\n\\end{algorithm}\n\nThus, the average noise of random circuits of fixed length $m$ can be described as a Markovian matrix applied to itself $m$ times followed by a matrix $B=W^{-1}AW$ that takes into account the SPAM errors.\n\nUsing this model, we can predict the average output for an input of $\\ket{0}^{\\otimes n}$ for any sequence length $m'$ not included in our fitting set of sequence lengths. Moreover, we can predict the average output for any input state\n\nOur modeling allows the extraction of data in a short period of time without having to run circuits on a real quantum computer.\n\n\\section{Results}\n\\subsection{Data Generation}\nFor each depth $m \\in$ [1,...,100], we run $1000$ random identity circuits with each submission requesting 1024  (Figure \\ref{DataGeneration}). The constructed circuits contain single qubit gates only. We run the circuits on the following 5-qubit IBM quantum computers: Athens, Lima, and Belem.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\columnwidth]{DataGeneration}\n\\caption{Generating data for circuit depth $m$, $G_1,G_2,...,G_m$ are random gates selected from the Clifford Group, $G^{-1}$ ensures we have an identity circuit}\n\\label{DataGeneration}\n\\end{figure}\n\\FloatBarrier\n\n\\subsection{Results of our Proposed Protocol}\nAfter generating the data, the protocol developed in \\cite{Nature} was used to obtain the vector $p$ that we used to estimate the average noisy probability output $\\hat{q}(m)$ for depth $m$ (Figure \\ref{Protocol}).\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=0.3\\columnwidth,width=\\columnwidth]{Protocol}\n\\caption{Generating data for circuit depth $m$, $G_1,G_2,...,G_m$ are random gates selected from the Clifford Group, $G^{-1}$ ensures we have an identity circuit}\n\\label{Protocol}\n\\end{figure}\n\\FloatBarrier\n\nWe fix the maximum depth of out training set of depths to $m_{max}$, and then try to predict the average noisy output for our testing depths $m_{max}+1,...,100$. We computed the Jenson-Shannon divergence (JSD) between $BM^m\\ket{0}^{\\otimes n}$ and $\\hat{q}(m)$ for each quantum computer with $m_{max}=50$ fixed. The JSD is a measure of the similarity be two probability distributions defined as: \n\n\\begin{equation*}\n    JSD(p,q)=\\frac{1}{2}D(p||m)+\\frac{1}{2}D(q||m) \n\\end{equation*}\nwhere $m=\\frac{1}{2}(p+q)$ and \n\\begin{equation*}\n    D(P,Q)=\\sum_{x}P(x) log(\\frac{P(x)}{Q(x)})\n\\end{equation*}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\columnwidth]{JSDQuantumComputers}\n\\caption{$JSD$ between $BM^m \\ket{0}^{\\otimes n}$ and $\\hat{q}(m)$ for different quantum computers. The dashed lines correspond to the $JSD$ for the training depths with $m_{max}=50$ and the solid lines correspond to the $JSD$ for the testing depths}\n\\label{JSDQuantumComputers}\n\\end{figure}\n\\FloatBarrier\n\nFigure \\ref{JSDQuantumComputers} shows that the JSD fluctuates between 0 and reaches a maximum of 0.10 on the Lima quantum computer, indicating a higher similarity between the constructed average $BM^m \\ket{0}^{\\otimes n}$ and the experimentally obtained average $\\hat{q}(m)$ for each sequence length. Ofcourse, the maximum $JSD$ is reduced if we remove the outliers for each quantum computer.\n\nFurthermore, for each quantum computer, we vary $m_{max}$ and measure the corresponding $JSD$ fore each quantum computer (Figures \\ref{VaryingTrainingDepth}).\n\n\\begin{figure}[h]\n\\begin{subfigure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Belem}\n  \\caption{Belem Quantum Computer}\n\\end{subfigure}\n\\begin{subfigure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Lima}\n  \\caption{Lima Quantum Computer}\n\\end{subfigure}\n\\begin{subfigure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Athens}\n  \\caption{Athens Quantum Computer}\n\\end{subfigure}\n\\caption{$JSD(\\hat{q}(m),BM^{m}\\ket{0})$ for different maximum training depth $m_{max}$}\n\\label{VaryingTrainingDepth}\n\\end{figure}\n\\FloatBarrier\n\nFigure \\ref{VaryingTrainingDepth} shows that fitting with a smaller training depth would result in the same testing pattern as fitting with a larger one. It also shows that a $m_{max}=50$ gives the lowest $JSD$ thus results in the best prediction.\n\nWe also tried the same protocol for different basis inputs state, i.e. $\\ket{in} \\in \\{\\ket{0},\\ket{1},...,\\ket{2^n-1}\\}$ where we estimated $\\hat{q}_{in}=W^{-1}A_{in}WM_{in}|in>$ where $A_{in}$ and $M_{in}$ corresponds to the matrices constructed when running the Gibbs Random Field Protocol on each input. Notice that the protocol can be done with some permutations performed on any $\\ket{in}$ to the $\\ket{0}$.\n\nFigure \\ref{lambdas} and \\ref{As} shows the variation of the entries of the $\\lambda$ and $A$ vector as function of the inputs, respectively.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=10cm,width=\\columnwidth]{lambdas}\n\\caption{Variation of the entries of the $\\lambda$ vector as function of the inputs on Athens Quantum Computer}\n\\label{lambdas}\n\\end{figure}\n\\FloatBarrier\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=10cm,width=\\columnwidth]{As}\n\\caption{Variation of the entries of the $A$ vector as function of the inputs on Athens Quantum Compute}\n\\label{As}\n\\end{figure}\n\\FloatBarrier\n\n\\section{Conclusion}\nabc.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe understanding of the turbulence is one of the main unsolved problems\nof classical physics, in spite of the more than 250 years of\nstrong investigations initiated by D.Bernoulli and L.Euler.\n\nIn the stochastic approach to turbulence~\\cite{Monin},~\\cite{Frish}\nthe turbulent cascade is\nconsidered as a stochastic process, described by the probability\ndistribution $P(\\lambda,v)$, where $\\lambda$ and $v$ are the\nappropriate scaled length and velocity increment respectively.\nRecently~\\cite{Friedrich} R.Friedrich and J.Peinke presented\nexperimental evidence that\nthe probability\ndensity function  $P(\\lambda,v)$ obeys a Fokker-Planck\nequation (FPE)~\\cite{Risken} (see fig.1 and fig.2\nin~\\cite{Friedrich}):\n\\begin{equation}\n\\frac{\\partial P(\\lambda,v)}{\\partial \\lambda} =\n\\left[ -\\frac{\\partial}{\\partial v} D^1(\\lambda,v)\n+ \\frac{\\partial^2}{\\partial v^2} D^2(\\lambda,v) \\right]\nP(\\lambda,v),\n\\label{FPP}\n\\end{equation}\nwhere the drift and duffusion coefficients\n$D^1$ and $D^2$ respectively are derived by analysis of experimental\ndata of a fluid dynamical experiment (see fig.3 in~\\cite{Friedrich}).\n\nIn their paper Friedrich and Peinke consider the application\nof the FPE to obtain the Kolmogorov scaling  with simplified\nassumptions that $D^1$ and $D^2$ are $\\lambda$-independent,\n$D^1$ is linear in $v$ and $D^2$ is quadratic in $v$:\n$$\nD^1= -a\\, v, \\qquad a>0; \\qquad\nD^2 = c\\, v^2,\\qquad c>0 \\,.\n$$\n( In the notations of~\\cite{Friedrich} :\n$ a \\equiv \\gamma $ and $ c \\equiv Q $.)\n\nHere we will consider a more realistic situation\n(see fig.3 in~\\cite{Friedrich})\nof $\\lambda$-dependent $D^1$ and $D^2$ :\n\\begin{equation}\nD^1= -a(\\lambda)\\, v, \\qquad a(\\lambda)>0 ;\\qquad\nD^2 = c(\\lambda)\\, v^2,\\qquad c(\\lambda)>0 .\n\\label{D}\n\\end{equation}\nThus the FPE ~(\\ref{FPP}) will take the form:\n\\begin{equation}\n\\frac{\\partial P}{\\partial \\lambda} = b_0(\\lambda)P(\\lambda,v)+\nb_1(\\lambda) v \\frac{\\partial P}{\\partial v} +\nc(\\lambda)\\left( v \\frac{\\partial}{\\partial v}\\right)^2 P(\\lambda,v),\n\\label{P}\n\\end{equation}\nwhere\n\\begin{equation}\nb_0(\\lambda)= a(\\lambda)+2 c(\\lambda),\n\\quad b_1(\\lambda) = a(\\lambda) + 3 c(\\lambda) .\n\\label{b}\n\\end{equation}\n\n\\begin{center}\n\\section{Exact Solution of the Cauchy Problem for the\nEq.~(\\ref{P})}\n\\end{center}\nIn this section we will find the solution\n$P(\\lambda,v)$\n of the Cauchy problem\nfor the Eq.~(\\ref{P}) with the initial condition\n\\begin{equation}\nP(0,v) = \\varphi (v).\n\\label{In}\n\\end{equation}\nAccording to \\cite{Monin}$-\\!$~\\cite{Friedrich}, when the probability\ndensity function is known, one may derive all properties\nof the turbulent cascade considered as a stochastic process.\n\nFor the solution of the problem (\\ref{P}),~(\\ref{In}) we\nshall use the approach\nof M.Suzuki~\\cite{Suzuki} to the FPE (see also ~\\cite {Donkov} ),\nbased on the disentangling techniques of R.Feynman~\\cite{Feynman}\n and the operational methods developed in the functional\nanalysis, in particular in the theory of pseudodifferential equations\nwith partial derivatives ~\\cite{Hoermander}$-\\!$~\\cite{Maslov}\n\nIn the spirit of the operational methods using the\npseudodifferential operators  we can write the solution\nof the Cauchy problem (\\ref{P}),~(\\ref{In}) in the form\n\\begin{equation}\nP(\\lambda, v) =\n\\left(exp_+ \\int_0^{\\lambda} \\left[ b_0(s)+b_1(s)v\n\\frac{\\partial}{\\partial v}\n+ c(s)\\left(v \\frac{\\partial}{\\partial v}\\right)^2 \\right] {\\rm d}s\n\\right) \\varphi(v) ,\n\\label{form}\n\\end{equation}\nwhere the symbol $\\;\\;exp_+\\;\\;$ designates the V.Volterra ordered exponential\n\\begin{equation}\nexp_+ \\int_0^{\\lambda} \\hat C(s) {\\rm d}s =\n\\hat 1 + \\lim_{n\\to\\infty} \\sum_{k=1}^n\\int_0^{\\lambda}{\\rm d}\\lambda_1\n\\int_0^{\\lambda_1}{\\rm d}\\lambda_2 \\dots \\int_0^{\\lambda_{k-1}}\n{\\rm d}\\lambda_{k}\n\\hat C(\\lambda_1) \\hat C(\\lambda_2) \\dots \\hat C(\\lambda_{k}).\n\\label{exp}\n\\end{equation}\n\nThe linearity of the integral and the explicit form of the operators\nin~(\\ref{form}) permit to write the solution $P(\\lambda,v)$ in terms\nof usual, not ordered, operator valued exponent\n\\begin{equation}\nP(\\lambda,v) = {\\rm e}^{\\beta_0 (\\lambda)}\\,\n{\\rm e}^{\\beta_1(\\lambda)v\\frac{\\partial}{\\partial v} +\n\\gamma (\\lambda)\\left(v\\frac{\\partial}{\\partial v}\\right)^2}\n\\varphi(v) ,\n\\label{expP}\n\\end{equation}\nwhere for convenience we have denoted\n\\begin{equation}\n\\beta_j(\\lambda) = \\int_0^{\\lambda}b_j(s){\\rm d}s,\\;\\; (j=0,1);\n\\qquad \\gamma(\\lambda) = \\int_0^{\\lambda}c(s) {\\rm d}s .\n\\label{beta}\n\\end{equation}\nConsequently (from now on \"$'$\" means $\\frac{\\rm d}{{\\rm d}t} $ ) :\n\\begin{equation}\n\\beta_j(0)=0,\\;\\;\\; {\\beta}'_j(\\lambda) =b_j(\\lambda),\\;\\;\\; (j=0,1);\n\\qquad \\gamma(0)=0,\\;\\;\\; {\\gamma}'(\\lambda)=c(\\lambda).\n\\label{betaPR}\n\\end{equation}\n\nSince the operators \\qquad\n$\\hat A \\equiv \\beta_1(\\lambda) v \\frac{\\partial}{\\partial v}$ \\quad and \\quad\n$\\hat B \\equiv \\gamma (\\lambda)\\left(v \\frac{\\partial}{\\partial v}\\right)^2$\\qquad\ncommute : $[\\hat A , \\,\\hat B] = 0$ ,\nfrom Eq.~(\\ref{expP}) we have\n\n\\begin{equation}\nP(\\lambda,v)=\n{\\rm e}^{\\beta_0(\\lambda)}\\,\n{\\rm e}^{\\beta_1(\\lambda) v\\frac{\\partial}{\\partial v}} \\,\n{\\rm e}^{\\gamma(\\lambda)\\left(v\\frac{\\partial}{\\partial v}\\right)^2}\n\\varphi(v).\n\\label{Form}\n\\end{equation}\n\nTherefore, taking into account the formulae\n\\begin{equation}\n{\\rm e}^{\\beta_1(\\lambda)v\\frac{\\partial}{\\partial v}}f(v)\n= f\\left(v{\\rm e}^{\\beta_1(\\lambda)}\\right)\n\\label{F1}\n\\end{equation}\nand\n$\n\\,\\,{\\rm e}^{\\gamma(\\lambda)\\left(v\\frac{\\partial}{\\partial v}\\right)^2}g(v)\n=\\frac{1}{\\sqrt{4\\pi\\gamma(\\lambda)}}\\int_{-\\infty}^{\\infty}\n{\\rm e}^{-\\frac{s^2}{4\\gamma(\\lambda)}} g\\left(v{\\rm e}^\n{-s}\\right){\\rm d}s\n$$\n\\begin{equation}\n\\qquad\\qquad\\qquad=\\frac{1}{\\sqrt{4\\pi\\gamma(\\lambda)}}\n\\int_{-\\infty}^{\\infty}\n{\\rm e}^{-\\frac{\\left(\\ln v-y\\right)^2}{4\\gamma(\\lambda)}}\ng\\left({\\rm e}^\n{y}\\right){\\rm d}y,\n\\label{F2}\n\\end{equation}\nwe obtain the following expression for the exact solution of the\nCauchy problem (\\ref{P}),(\\ref{In})\n$$\nP(\\lambda,v)\n=\\frac{{\\rm e}^{\\beta_0(\\lambda)}}\n{\\sqrt{4\\pi\\gamma(\\lambda)}}\\int_{-\\infty}^{\\infty}\n{\\rm e}^{-\\frac{s^2}{4\\gamma(\\lambda)}} \\varphi\\left(v{\\rm e}^\n{\\beta_1(\\lambda)-s}\\right){\\rm d}s\n$$\n\\begin{equation}\n\\label{end}\n\\qquad\\qquad=\\frac{{\\rm e}^{\\beta_0(\\lambda)}}\n{\\sqrt{4\\pi\\gamma(\\lambda)}}\\int_{-\\infty}^{\\infty}\n{\\rm e}^{-\\frac{\\left(\\ln v+\\beta_1(\\lambda)-y\\right)^2}\n{4\\gamma(\\lambda)}}g\\left({\\rm e}^{y}\\right){\\rm d}y,\n\\end{equation}\nwhere $\\beta_0(\\lambda), \\beta_1(\\lambda)$  and $\\gamma(\\lambda)$\nare defined in~(\\ref{beta}).\n\nSubstituting the expression~(\\ref{end})\nin the Eqs.~(\\ref{P}) and~(\\ref{In}) and using the Eq.~(\\ref{betaPR})\none can see immediately that $P(\\lambda,v)$ is a solution of the\nproblem (\\ref{P}),~(\\ref{In}) and, according to the Cauchy theorem,\nit is the only classical solution of this problem.\n\n\\section{Concluding remarks}\n\\begin{itemize}\n\\item The exact solution of the Cauchy problem (\\ref{P}),~(\\ref{In})\nis obtained using the algebraic method we have described.\nThe Eq.~(\\ref{P}) is a generalization\nof the equation used by R.Friedrich and J.Peinke\n( see section 1)\nin their description of a turbulent cascade by a\nFokker-Planck equation\nwith coefficients derived by a detailed analysis of\nexperimental data\nof a fluid dynamical experiment.\n\\item If\nthe probability distribution function\n$P(\\lambda,v)$ is known, then\none may derive the properties of\na given stochastic process, in our case -\n the turbulent cascade \\cite{Monin}~$-\\!$~\\cite{Friedrich}.\n\\item For more realistic description of the turbulent cascade\nby a FPE\nit should be desirable to use for\n$D^1(\\lambda, v)$ and $D^2(\\lambda ,v)$ in the Eq.~(\\ref{FPP})\nmore general expressions than these in Eq.~(\\ref{D}),\nfor instance:\\\\\n $ D^1(\\lambda,v) = a_1(\\lambda) - a(\\lambda)v,\\,\\,\\, a(\\lambda)>0$\\quad\nand \\quad\n$D^2(\\lambda,v) = c_1(\\lambda) + c(\\lambda) v^2 $.\n\n\\end{itemize}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{section1}\n\nPathology analysis based on microscopic images is a critical task in medical image computing. In recent years, deep learning of digitalized pathology slide has facilitated the progress of automating many diagnostic tasks, offering the potential to increase accuracy and improve review efficiency. Limited by computation resources, deep learning-based approaches on whole slide pathology images (WSIs) usually train convolutional neural networks (CNNs) on patches extracted from WSIs and aggregate the patch-level predictions to obtain a slide-level representation, which is further used to identify cancer metastases and stage cancer \\cite{Wang2016DeepLF}. Such a patch-based CNN approach has been shown to surpass pathologists in various diagnostic tasks \\cite{Liu2017DetectingCM}\n\nOff-the-shelf CNNs have been shown to be able to accurately classify or segment pathology images into different diagnostic types in recent studies \\cite{huang2018improving, veeling2018rotation}. However, \nmost of these methods are weak in interpretability especially for clinicians, due to a lack of evidence supporting for the decision of interest. During diagnosis, a pathologist often inspects abnormal structures (e.g., large nucleus, hypercellularity) as the evidence for determining whether the glimpsed patch is cancerous. For CAD systems, learning to pinpoint the discriminative evidence can provide precise visual assistance for clinicians. Strong supervision-based feature localization methods require a large number of pathology images annotated in pixel-level or object-level, which are very costly and time-consuming and can be biased by the experiences of the observers. In this paper, we propose a weakly supervised learning (WSL) method that can learn to localize the discriminative evidence for the class-of-interest on pathology images from weakly labeled (i.e. image-level) training data. Our contributions include: \ni) proposing a new CNN architecture with multi-branch attention modules and deep supervision mechanism, to address the difficulty of localizing discrete and small objects in pathology images, \nii) formulating a generalizable approach that leverages gradient-weighted class activation map and saliency map in a complementary way to provide accurate evidence localization, \niii) designing a new attention module which allows capturing spatial attention from various context, \niv) quantitatively and visually evaluating WSL methods on large scale histopathology datasets, and \nv) constructing a new dataset (HPLOC) based on Camelyon16 for effectively evaluating evidence localization performance on histopathology images.\n\n\n\n\\textbf{Related Work.} Recent studies have demonstrated that CNN can learn to localize discriminative features even when it is trained on image-level annotations \\cite{zhou2016learning}. However, these methods are evaluated on natural image datasets (e.g., PASCAL), where the objects of interest are usually large and distinct in color and shape. In contrast, objects in pathology images are usually small and less distinct in morphology between different classes. A few recent studies investigated WSL approaches on medical images, including lung nodule detection and placental ultrasound images \\cite{feng2017discriminative}. These methods employ GAP-based class activation map and require CNNs ending with global average pooling, which degrades the performance of CNNs as a side effect \\cite{zhou2016learning}. \n\n\\section{Methods}\nThe overview of the framework is shown in Fig.{~\\ref{fig:fig1}}. \nThe model is trained to predict the cancer score for a given image, indicating the presence of cancer metastasis.\nIn the test phase, besides giving a binary classification, the model generates a cancerous evidence localization map and performs localization. \n\n \\afterpage{\n \\begin{figure}[t]\n\n \t\\centering\n\n \t\\begin{tabular}{cc}\n \t\t\\includegraphics[scale=0.5]{.\/fw.pdf} &\n \n \t\n\n \t\t\\includegraphics[scale=0.5]{.\/blk.pdf} \\\\%width=\\linewidth\n\n \t\n \t\\end{tabular}\n \t\\caption{ \\textbf{Left:} Framework overview of the proposed WSL method. The under line (e.g., \/1, 16) denotes stride and number of channels.  \\textbf{Right:} A building block for the multi-branch attention based residual module (MA-ResModule). \n \t\n \t} \n \t\\label{fig:fig1}\n \\end{figure}\n}\n \n \\subsection{Cancerous Evidence Localization Networks (CELNet)}\n \\label{subsection_2_3}\n\n\n Given the object of interest is relatively small and discrete, a moderate number of convolutional layers is sufficient for encoding locally discriminative features. As discussed in Section \\ref{section1}, instances on pathology images are similar in morphology and can be densely distributed, the model should avoid over-downsampling in order to pinpoint the cancerous evidence from the densely distributed instances. The proposed CELNet starts with a $3\\times 3$ convolution head followed by 3 Multi-branch Attention-based Residual Modules (MA-ResModule)\n \\footnote{Densely connected module is not employed considering it is comparatively speed-inefficient for WSIs application due to its dense tensor concatenation. } \n \\cite{he2016deep}. Each MA-ResModule is composed of 3 consecutive building blocks integrated with the proposed attention module (MAM) as  shown in Fig.{~\\ref{fig:fig1}} (Right). We use $3\\times 3$ convolution with stride of 2 for downsampling in residual connections instead of $1\\times 1$ convolution to reduce information loss. Batch normalization and ReLU are applied after each convolution layer for regularization and non-linearity\n \n \n \\subsubsection{Multi-branch Attention Module (MAM)}\n To eliminate the effect of background contents and focus on representing the cancerous evidence (which can be sparse), we employ attention mechanism. \n Improved on Convolutional Block Attention Module (CBAM) \n\n , which extracts channel attention and spatial attention of an input feature map in a squeeze and excitation manner, we propose a multi-branch attention module. MAM can better approximate the importance of each location on the feature map by looking at its context at different scales. \n Given a squeezed feature map $F_{sq} $ generated by the channel attention module, we compute and derive a 2D spatial attention map $A_s$ by $ A_s = \\sigma(\\sum_{k'} f^{k' \\times k'} (F_{sq}) ),$\n where $f^{k' \\times k'}$ represents a convolution operation with kernel size of $k' \\times k'$, and $\\sigma$ denotes the sigmoid function. We set $k' \\in \\{3, 5, 7\\}$ in our experiments, corresponding to 3 branches. Hereby, the feature map  $F_{sq} $ is refined by element-wise multiplication with the spatial attention map $A_s$.\n \nMAM is conceptually simple but effective in improving detection and localization performance as demonstrated in our experiments.\n \n\n\n\n \\subsubsection{Deep Supervision}\n Deep supervision \\cite{lee2015deeply} is employed to empower the intermediate layers to learn class-discriminative representations, for building the cancer activation map in a higher resolution.  We achieve this by adding two companion output layers to the last two MA-ResModules, as shown in Fig. \\ref{fig:fig1}. Global max pooling (GMP) is applied to search for the best discriminative features spatially, while global average pooling (GAP) is applied to encourage the network to identify all discriminative parts on the image. Each companion output layer applies GAP and GMP on the input feature map and concatenates the resulting vectors. The cancer score of the input image is derived by concatenating the outputs of the two companion layers followed by a fully convolutional layer (i.e., kernel size $1 \\times 1$) with a sigmoid activation. \n CELNet enjoys efficient inference when applied to test WSIs, as it is fully convolutional and avoids repetitive computation for the overlapping part between neighboring patches. \n \n \n\\subsection{Cancerous Evidence Localization Map (CELM) }\n\\subsubsection{Cancer Activation Map (CAM)}\nGiven an image $I \\in \\mathbb{R}^{H \\times W \\times 3}$, let $y^c = S_c(I)$ represent the cancer score function governed by the trained CELNet (before sigmoid layer). \nA cancer-class activation map $M^c$ shows the importance of each region on the image to the diagnostic value. For a target layer $l$, the CAM $M^c_l$ is derived by taking the weighted sum of feature maps $F_l$  with the weights \\{$\\alpha_{k,l}^c$ \\}, where $\\alpha_{k,l}^c$ represents the importance of $k^{th}$ feature plane. The weights $\\alpha_{k,l}^c$ are computed as $\\alpha_{k,l}^c = Avg_{i,j}( \\frac{\\partial y^c}{\\partial F_l^k(i,j)} )$, i.e., spatially averaging the gradients of  cancer score $y^c$ with respect to the $k^{th}$ feature plane $F_l^k$, which is achieved by back propagation (see Fig.\\ref{fig:fig1}). Thus, the CAM of layer $l$ can be derived by $ M_l^c = ReLU(\\sum_k \\alpha_{k,l}^c F_l^k)$, where ReLU is applied to exclude the features with negative influence on the class of interest \\cite{selvaraju2017grad}. \n\nWe derive two CAMs, $M^c_2$ and $M^c_3$ from the last layer of the second and the third residual module on CELNet respectively (i.e., CAM2 and CAM3 in Fig.\\ref{fig:fig1}). CAM3 can represent discriminative regions for identifying a cancer class  in a relatively low resolution while CAM2 enjoys higher resolution and still class-discriminative under deep supervision. \n\n\\subsubsection{Cancer Saliency Map (CSM)}\nIn contrast with CAM, the cancer-class saliency map shows the contribution of each pixel site to the cancer score $y^c$. This can be approximated by the derivate of a linear function $S^c(I) \\approx w^TI + b$. Thus the pixel contribution is computed as $ w = \\frac{\\partial S^c(I)}{\\partial I} $. Different from \\cite{Simonyan2013DeepIC}, we derive $w$  by the guided back-propagation \\cite{springenberg2014striving} to prevent backward flow of negative gradients. \nFor a RGB image, to obtain its cancer saliency map $M^s \\in \\mathbb{R}^{H \\times W\\times 1} $ from $w  \\in \\mathbb{R}^{H \\times W \\times 3} $, we first normalize $w$ to $[0,1]$ range, followed by greyscale conversion and Gaussian smoothing, instead of simply taking the maximum magnitude of $w$ as proposed in \\cite{Simonyan2013DeepIC}. Thus, the resulting cancer saliency map (see Fig.{~\\ref{fig:vis}} (b)) is far less noisy and more focus on class-related objects than the original one proposed in \\cite{Simonyan2013DeepIC}.\n\n\n\n\\subsubsection{Complementary Fusion}\nThe generated CAMs coarsely display discriminative regions for identifying a cancer class (see Fig.\\ref{fig:vis} (c)), while the CSM is fine-grained, sensitive and represents pixelated contributions for the identification (see Fig.\\ref{fig:vis} (b)). To combine the merits of them for precise cancerous evidence localization, we propose a complementary fusion method. First, CAM3 and CAM2 are combined to obtain a unified cancer activation map $M^c \\in \\mathbb{R}^{H \\times W\\times 1}$   as $M^c = \\alpha f_u(M^c_3) + (1- \\alpha) f_u(M^c_2)$, where $f_u$ denotes a upsampling function by bilinear interpolation, and  the coefficient $\\alpha$ in range [0,1] is confirmed by validation.\nThe CELM is derived by complementarily fusing CSM and CAM as $M = \\beta (M^c \\odot M^s) + (1 - \\beta) M^c$,\nwhere $\\odot$ denotes element-wise product, and  the coefficient $\\beta$ captures the reliability of the point-wise multiplication of CAM and CSM, and the value of $\\beta$ is estimated by cross-validation in experiments. \n\n\n\n\n\\section{Experiments \\& Results}\nWe first evaluate the detection performance of the proposed model as for clinical requirements, followed by evidence localization evaluations.  \n\n\\subsection{Datasets and Experimental Setup}\nThe detection performance of the proposed method is validated on two benchmark datasets, PCam\\cite{veeling2018rotation} and Camelyon16 \\footnote{https:\/\/camelyon16.grand-challenge.org}. \n\n\\textbf{PCam:} The PCam dataset contains 327,680 lymph node histopathology images of size $96 \\times 96$ with binary class labels indicating the presence of cancer metastasis, split into 75\\% for training, 12.5\\% for validation, and 12.5\\% for testing as originally proposed. The class distribution in each split is balanced (1:1). For a fair comparison, following \\cite{veeling2018rotation}, we perform image augmentation by random 90-degree rotations and horizontal flipping during training.\n\n\\textbf{Camelyon16:} The Camelyon16 dataset includes 270 H\\&E stained WSIs (160 normal and 110 cancerous cases) for training and 129 WSIs held out for testing (80 normal and 49 cancerous cases) with average image size about $65000 \\times 45000$, where regions with cancer metastasis are delineated in cancerous slides. To apply our CELNet on WSIs, we follow the pipeline proposed in \\cite{Liu2017DetectingCM}, including WSI pre-processing, patch sampling and augmentation, heatmap generation, and slide-level detection tasks. For slide-level classification, we take the maximum tumor score among all patches as the final slide-level prediction. For tumor region localization, we apply non-suppression maximum algorithm on the tumor probability map aggregated from patch predictions to iteratively extract tumor region coordinates. We work on the WSI data at 10$\\times$ resolution instead of  40$\\times$ with the available computation resources.\n\n\tIn our experiments, all models are trained using binary cross-entropy loss with L2 regularization of $10^{-5}$ to improve model generalizability, and optimized by SGD with Nesterov momentum of 0.9 with a batch size of 64 for 100 epochs. The learning rate is initialized with $10^{-4}$ and is halved at 50 and 75 epochs. We select model weights with minimum validation loss for test evaluation.  \n\n\\subsection{Classification Results}\t\n\nAs Tbl.{~\\ref{table1}} shows, CELNet consistently outperforms ResNet, DenseNet, and P4M-DenseNet \\cite{veeling2018rotation} in histopathologic cancer detection on the PCam dataset. \n\n P4M-DenseNet uses less parameters due to parameter sharing in the p4m-equivariance. \n\n For auxiliary experiments, we perform ablation studies and visual analysis. From Tbl.{~\\ref{table1}} , we observe that our attention module brings 1.77\\% accuracy gain, which is larger than the gain brought by CBAM \\cite{woo2018cbam}. Both the CAM and CELM on CELNet are mainly activated for the cancerous regions (see Fig.\\ref{fig:vis} (c) and (d)). These subfigures indicate that CELNet is effective in extracting discriminative evidence for histopathologic classification. \n \n\\begin{table}[h]\n\t\\centering\n\t\\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={left,top} }}]{table}[\\FBwidth]\n\t\t{\n\t\t\\caption{Quantitative comparisons on the PCam test set. P4M-DenseNet  \\cite{veeling2018rotation}:  current SoTA method for  the PCam benchmark, CELNet: our method, $^{-}$: removal of the proposed multi-branch attention module, +CBAM: integration with convolutional block attention module \\cite{woo2018cbam}.  \n\t\t}}\n\t\t{\n\t\t\t\\begin{tabular}{l  c c c }\n\t\t\t\t\\toprule\n\t\t\t\tMethods\t\t\t\t\t\t\t&   Acc \t\t& AUC \t& \\#Params \\\\\n\t\t\t\t\\midrule    \n\t\t\t\tResNet18 \\cite{he2016deep}\t\t\t& \t  88.73\t\t\t&\t95.36\t\t& 11.2M\t\\\\\n\t\t\t\tDenseNet \\cite{veeling2018rotation}\t\t\t\t&\t 87.20  & 94.60\t& 902K   \\\\\n\t\t\t\tP4M-DenseNet    & 89.80 \t&\t96.30\t& 119K \\\\\n\t\t\t\t\\midrule  \n\t\t\t\tCELNet\t\t  \t\t\t    &\t \\textbf{91.87}\t& \\textbf{97.72} & 297K\t \\\\\n\t\t\t\tCELNet$^{-}$ \t \t\t\t&\t90.10 \t\t& 96.45\t& 292K\t\t\\\\\n\t\t\t\tCELNet$^{-}$ +CBAM &\t  90.86 \t& 97.17 & \t  296K\t\\\\\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular}\n\t\t\\label{table1}\n\t\t}\n\\end{table}\n\n\nOn slide-level detection tasks, as shown in Tbl.\\ref{table2}, our CELNet based approach achieves higher classification performance (1.7\\%) in terms of AUC than the baseline method \\cite{Liu2017DetectingCM}, and  outperforms previous state-of-the-art methods in slide-level tumor localization performance in terms of FROC score. The results illustrate that instead of using off-the-shelve CNNs as the core patch-level model for histopathologic slide detection, adopting CELNet can potentially bring larger performance gain. CELNet is more parameter-efficient as shown in Tbl.\\ref{table1} and testing a slide on Camelyon16 takes about 2 minutes on a Nvidia 1080Ti GPU. \n\\begin{table}[h]\n\t\\centering\n\t\\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={left,top} }}]{table}[\\FBwidth]\n\n\t{\\caption{Quantitative comparisons of slide-level classification performance (AUC) and slide-level tumor localization performance (FROC) on the Camelyon16 test set. *: The Challenge Winner uses $40\\times$ resolution while results of other methods are based on $10\\times$. \n\t}}\n\t{\n\t\t\\begin{tabular}{l  c c  }\n\t\t\t\\toprule\n\t\t\tMethods\t\t\t\t\t\t\t&   AUC \t\t& FROC \t \\\\\n\t\t\t\\midrule    \n\t\t\tP4M-DenseNet\t\t\t& \t  -\t\t\t\t\t\t\t\t\t\t\t\t&\t84.0 \t\t\t\\\\\n\t\t\tLiu \\cite{Liu2017DetectingCM}    & 96.5 \t\t\t\t\t\t&\t79.3  \t \\\\\n\t\t\tChallenge Winner$^*$ \\cite{Wang2016DeepLF}    & \\textbf{99.4} \t&\t80.7\t \\\\\n\t\t\tPathologist \t\t\t\t&\t  \t\t\t96.6 \t\t\t\t\t\t\t& 73.3\t\\\\\n\t\t\tCELNet\t\t  \t\t\t    &\t  97.2 & \\textbf{84.8} \t \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{table2}\n\t}\n\\end{table}\n\n\n\\subsection{Weakly Supervised Localization and Results}\nGiven that the trained CELNet can precisely classify a pathology image, here we aim to investigate its performance in localizing the supporting evidence based on the proposed CELM. To achieve this, based on Camelyon16, we first construct a dataset with region-level annotations for cancer metastasis, namely HPLOC, \nand develop the metrics for measuring localization performance on HPLOC. \n\n\\textbf{HPLOC:} The HPLOC dataset contains 20,000 images of size $96 \\times 96$ with segmentation masks for cancerous region. Each image is sampled from the test set of Camelyon16 and contains both cancerous regions and normal tissue in the glimpse, which harbors the high quality of the Camelyon16 dataset.\n\n\n\\textbf{Metrics:} To perform localization, we generate segmentation masks from CELM\/CAM\/CSM by thresholding and smoothing (see Fig.\\ref{fig:vis} (e)). If a segmentation mask intersects with the cancerous region by at least 75\\%\n\\footnote{The annotated contour in Camelyon16 is usually enlarged to surround all tumors. }\n, it is defined as a true positive. Otherwise, if a segmentation mask intersects with the normal region by at least 75\\%, it is considered as a false positive. Thus, we can use precision and recall score to quantitatively assess the localization performance of different WSL methods, where the results are summarized in Tbl.\\ref{table3}. \n\n\\begin{table}[h]\n\t\\centering\n\t\\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={left,top} }}]{table}[\\FBwidth]\n\t{\\caption{Quantitative comparisons for different weakly supervised localization methods on the HPLOC dataset. Ours: CELNet + CELM. MAM and DS are short for multi-branch attention module and deep supervision respectively. \n\t}}\n\t{\n\t\t\\begin{tabular}{l  c c }\n\t\t\t\\toprule\n\t\t\tMethods\t\t\t\t& Precision & Recall \\\\\n\t\t\t\\midrule   \n\t\t\tResNet18 + Backprop \\cite{Simonyan2013DeepIC}     \t\t& 79.8\t& 85.5 \\\\ \n\t\t\tResNet18 + GradCAM \\cite{selvaraju2017grad}\t\t  \t& \t 85.6\t& 82.4   \\\\\n\t\t\t\\midrule  \n\t\t\tOurs\t\t  \t\t&\t \\textbf{91.6}\t& 87.3  \t \\\\\n\t\t\tOurs w\/o MAM\t \t\t&\t88.1 \t\t\t\t& 85.6\t\\\\\n\t\t\tOurs w\/o DS\t\t&\t  90.5 \t\t& \t\\textbf{87.7}\t\\\\\n\t\t\tCELNet + GradCAM  &\t  91.0\t\t& \t85.4\t\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{table3}\n\t}\n\\end{table}\n\n\n\\begin{figure}[t\n\t\\centering\n\n\t\\begin{tabular}{cccccc}\n\n\n\n\n\n\n\n\n\n\t\t\\includegraphics[scale=0.5]{2a.png}  &\n\t\t\\includegraphics[scale=0.5]{2e_sm.png} &\n\t\t\\includegraphics[scale=0.5]{2f_cam.png} &\n\t\t\\includegraphics[scale=0.5]{2b_celm.png} &\n\n\t\t\\includegraphics[scale=0.5]{2c_loc_on_img.png} &\n\t\t\\includegraphics[scale=0.5]{2d_mask.png}\n \\\\\n\t\t(a) Input  & (b) CSM  & (c) CAM & (d) CELM  & (e) Localization  & (f) GT   \\\\\n\t\\end{tabular}\n\n\t\\caption{ Evidence localization results of our WSL method on the HPLOC dataset.\n\t\t (a) Input glimpse, (b) Cancer Saliency Map, (c) Cancer Activation Map, (d) CELM: Cancerous Evidence Localization Map, (e) Localization results based on CELM,  where the localized evidence is highlighted for providing visual assistance, (f) GT: ground truth, white masks represent tumor regions and the black represents normal tissue} \n\n\t\\label{fig:vis}\n\n\\end{figure}\n\n\nWe observe that our WSL method based on CELNet and CELM consistently performs better than the back propagation-based approach \\cite{Simonyan2013DeepIC} and the class activation map-based approach \\cite{selvaraju2017grad}. Note that we used ResNet18 \\cite{he2016deep} as the backbone for the compared methods because it achieves better classification performance and provides higher resolution for GradCAM (12$\\times$12) as compared to DenseNet (3 $\\times$ 3) \\cite{veeling2018rotation}. \nWe perform ablation studies to further evaluate the key components of our method in Tbl.\\ref{table3}. We observe the effectiveness of the proposed multi-branch attention module in increasing the localization accuracy. \nThe deep supervision mechanism effectively improves the precision in localization despite slightly lower recall score, which can be caused by the regularization effect on the intermediate layers, that is, encouraging the learning of discriminative features for classification but also potentially discouraging the learning of some low-level histological patterns. \nWe observe that using CELM can improve the recall score and precision, which indicates that CELM allows better discovery of cancerous evidence than using GradCAM. \nWe present the visualization results in Fig. {\\ref{fig:vis}}, the cancerous evidence \nis represented as large nucleus and hypercellularity in the images, which are precisely captured by the CELM. Fig.\\ref{fig:vis}(e) visualizes the localization results by overlaying the segmentation mask generated from CELM onto the input image, which demonstrates the effectiveness of our WSL method in localizing cancerous evidence.\n\n\n\\section{Discussion \\& Conclusions}\nIn this paper, we have proposed a generalizable method for localizing cancerous evidence on histopathology images. \nUnlike the conventional feature-based approaches, the proposed method does not rely on specific feature descriptors but learn discriminative features for localization from the data. \nTo the best of our knowledge, investigating weakly supervised CNNs for cancerous evidence localization and quantitatively evaluating them on large datasets have not been performed on histopathology images. \nExperimental results show that our proposed method can achieve competitive classification performance on histopathologic cancer detection, and more importantly, provide reliable and accurate cancerous evidence localization using weakly training data, which reduces the burden of annotations. We believe that such an extendable method can have a great impact in detection-based studies in microscopy images and help improve the accuracy and interpretability for current deep learning-based pathology analysis systems. \n\n\\bibliographystyle{splncs04}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn continuum thermodynamics, the constitutive theories are based, besides some general invariance principles, on the second law of thermodynamics, which states that in every admissible process the entropy production has to be non-negative \\cite{Truesdell}.\n\nA rigorous procedure for the exploitation of the entropy principle has been developed for the first time in 1963 by Coleman and Noll \\cite{Coleman-Noll}, and later by Coleman and Mizel \\cite{Coleman-Mizel}. In both papers, the authors assumed the unbalance of entropy \nin the classical form, say the Clausius-Duhem inequality; in this inequality, the entropy flux  is taken as the ratio between the heat flux and the absolute temperature. Later on, M\\\"uller \\cite{Muller} proposed an extension of the entropy inequality, allowing a more general expression for the entropy flux, thus obtaining the thermodynamic compatibility for wider classes of materials. A slightly different approach has been applied by other authors who accepted the classical Clausius-Duhem inequality, but proposed a more general form of the local balance of energy \\cite{Gurtin,Dunn-Serrin,Dunn}. Furthermore,  in 1972, Liu  \\cite{Liu} developed a different procedure for the analysis of the entropy principle, based on the method of Lagrange multipliers.\n\nIn all these papers, the basic assumption is that the second law of thermodynamics restricts the constitutive equations and not the thermodynamic processes. Hence, the constitutive relations are required to be such that the entropy inequality be satisfied for all solutions of the thermodynamic field equations. This assumption is purely mathematical and from a physical point of view may have two\ndifferent interpretations \\cite{Muschik-Ehrentraut}:\n\\begin{itemize}\n\\item all solutions of the balance equations have to satisfy the second law;\n\\item there are solutions of the balance equations which satisfy the second law, and other ones which do not. \n\\end{itemize}\n\nThe first interpretation requires that the constitutive equations must be assigned in such a way the entropy inequality is satisfied along arbitrary processes, whereas the second one means that we have to exclude from the set of solutions of the balance equations those which are not physically achievable, since they do not satisfy the second law of thermodynamics. In \\cite{Muschik-Ehrentraut},  the authors\nproposed a way to choose between the two statements through an \\emph{amendment to the second law}, by expliciting the nearly self-evident, but never precisely formulated, postulate that there are no reversible process directed towards non-equilibrium. \nBy means of this amendment, they were able to prove that, necessarily, the second law of thermodynamics restricts the constitutive equations and not the processes. Such a result justifies, from the physical point of view, the approach to the exploitation of second law \nthrough  Coleman-Noll and Liu procedures (see also \\cite{Muschik,Muschik-Papenfuss-Ehrentraut}).\n\nIn a series of papers \\cite{Cimmelli-2007,CST-JMP-2009,CST-PRSA-2010,COT-JMP-2011},\nthe two classical Coleman-Noll and Liu procedures have been extended in order to use as constraints in\nthe entropy inequality both the balance equations and their gradient extensions up to a suitable finite order. \nThis approach, successfully used in many applications of physical interest (see, for instance, \n\\cite{CST-JNET-2010,COP-Elasticity-2011,COP-IJNLM-2013,COP-CMT-2015,OPR-2016,CGOP-miscele-2020}), \nrevealed essential in order to ensure the compatibility of non-local constitutive relations with second law of thermodynamics \nwithout modifying \\emph{a priori} the entropy inequality or the energy balance through the introduction of extra-terms.\nIn particular, the extended Liu technique requires to add to the entropy inequality a linear combination of the field equations and of the spatial gradients of the latter (in the following they are called extended equations), up to the order of the gradients entering the state space. The coefficients of this linear combination are the Lagrange multipliers and depend upon the state variables only. Thus,  the number of independent constraints to be taken into account is never less than that of the unknown elements entering the constitutive equations as independent state variables.  \n\nIn this paper,  we start discussing the Liu extended procedure from an abstract mathematical point of view, \nand then apply the results to some\nphysical instances of continua with non-local constitutive equations. In fact, we consider a system of first order balance laws in one space dimension sufficiently general to contain the equations governing the thermodynamical processes occurring in a continuous medium. We assume that this system involves some functions whose constitutive equations are allowed to be non-local, \\emph{i.e.}, we admit the possibility that the \\emph{state space} includes the gradients of (not necessarily all) the field variables up to the order $r\\ge 1$. The compatibility of the constitutive equations with an entropy-like inequality  is then discussed. Once we expand the derivatives in the\nentropy-like inequality and impose as constraints the balance equations and some of their gradient extensions,\na set of sufficient conditions, such that the entropy-like inequality is not violated, are  derived. \nFurthermore, the results are applied to two physical instances of continua: \nthe first example is concerned with a fluid with a scalar internal variable and constitutive equations with first order \nnon-localities, the second one to a fluid whose  state space includes also the second order spatial derivative of the mass density (Korteweg fluid \\cite{Korteweg}).\n\nThe plan of the paper is the following. In Section~\\ref{sec:balance}, we define  mathematically the problem, and introduce the notation that\nwill be used throughout the paper. In Section~\\ref{sec:liu}, we discuss the extended Liu procedure and state the theorem providing the sufficient conditions in order the entropy-like inequality be satisfied for all thermodynamical processes. In Section~\\ref{sec:applications},\nwe give two non-trivial applications of the procedure in meaningful physical situations, and solve the thermodynamical conditions providing explicitly a solution for the constitutive equations. Finally, Section~\\ref{sec:conclusions} contains some concluding remarks.\n\n\\section{Balance equations for a continuous medium}\n\\label{sec:balance}\nPhysical laws describing the mechanical as well as the thermodynamical \nproperties of continuous media are usually expressed in terms of balances of some physical quantities (mass, linear and angular momentum, energy, etc.). Here, for simplicity, we consider the case of one-dimensional continuous media and postpone to a forthcoming paper the multi-dimensional case. \n\nLet \n\\[\n\\mathbf{u}\\equiv(u_1(t,x),\\ldots,u_n(t,x))\n\\] \nbe the vector of $n$ field variables, depending on time $t$ and space $x$, describing a one-dimensional continuum. \n\nIt is well known that the general governing equations of a continuum are underdetermined since they involve some constitutive functions that specify the particular continuum we are dealing with. \nThe various constitutive quantities (for instance, the Cauchy stress tensor or the heat flux) depend on the so called \\emph{state variables}: the state variables include (some of) the field variables when a local constitutive theory is adopted, or,  when non-local constitutive theories are considered, also the spatial derivatives up to a finite order $r$ of some field variables. In the following, we shall be concerned with\na non-local constitutive theory. \nIn fact, in our framework, the state variables will be the elements of the set \n\\[\n\\displaystyle \\mathcal{Z}=\\bigcup_{k=0}^r \\mathcal{Z}^{(k)},\n\\]\nwhere\n\\[\n\\begin{aligned}\n&\\mathcal{Z}^{(0)}\\subseteq \\{u_1,\\ldots,u_n\\}\\equiv \\mathcal{U}^{(0)},\\\\\n&\\mathcal{Z}^{(k)}\\subseteq \\left\\{\\frac{\\partial^k u_1}{\\partial x^k},\\ldots,\\frac{\\partial^k u_n}{\\partial x^k}\\right\\}\\equiv \\mathcal{U}^{(k)}, \\qquad k=1,\\ldots,r,\n\\end{aligned}\n\\]\nwhere $r\\ge 1$.\nFinally, let us denote with $\\mathbf{z}$ the vector whose $N$ components belong to the set $\\mathcal{Z}$ of state variables, and $\\mathbf{z}^\\star$ the vector whose $N^\\star$ components belong to the set $\\mathcal{Z}^\\star=\\mathcal{U}^{(0)}\\bigcup \\mathcal{Z}$.\n\nIn local form, the thermomechanical description of a continuum leads to consider a system of partial differential equations having the form\n\\begin{equation}\n\\label{generalbalance}\n\\mathcal{E}_i\\equiv\n\\frac{D\\Phi_i(\\mathbf{u})}{D t}+\\frac{D \\left(\\Psi_i(\\mathbf{u})+\\chi_i(\\mathbf{z}^\\star)\\right)}{D x}-\\Gamma_i(\\mathbf{z}^\\star)=0, \\qquad i=1,\\ldots, n,\n\\end{equation}\nwhere the differential operators $D\/Dt$ and $D\/Dx$, acting on a composite function $F$ depending on $t$ and $x$ through some quantities $w_1(t,x),\\ldots, w_n(t,x)$\n(in the paper, these variables are the field variables or the elements of the state space), by chain rule, are defined as follows\n\\[\n\\frac{D F}{Dt}=\\sum_{j=1}^n\\frac{\\partial F}{\\partial w_j}\\frac{\\partial w_j}{\\partial t},\\qquad\n\\frac{D F}{Dx}=\\sum_{j=1}^n\\frac{\\partial F}{\\partial w_j}\\frac{\\partial w_j}{\\partial x}.\n\\]\nThese operators do not correspond to total derivatives, and we introduce them in order to avoid confusion when stating Theorem 1, see below.\nIn the equations (\\ref{generalbalance}),\n\\begin{itemize}\n\\item $\\Phi_i(\\mathbf{u})$ are some densities, depending at most on the field variables;\n\\item the fluxes are split as sums of functions $\\Psi_i(\\mathbf{u})$ depending at most on the field variables, and \n$\\chi_i(\\mathbf{z}^\\star)$  depending at most on the field variables and the state variables;\n\\item $\\Gamma_i(\\mathbf{z}^\\star)$ are the production terms depending at most on the field variables and the state variables. \n\\end{itemize}\n\nIt is well known that in general the constitutive functions have to satisfy some universal principles  \n(invariance with respect to rigid motions, time translation, \nscale changes of fundamental quantities, Galilei or Lorentz transformations, etc.). In such a framework, general representation theorems for isotropic scalar, vectorial or tensorial constitutive equations have to be taken into account \\cite{Wang1,Wang2,Smith1,Smith2,Smith3}. \nAdditional constraints are imposed by the second law of thermodynamics, which requires that the admissible processes must be such that the entropy production is non-negative.\n\nTherefore, in our general framework, we have to exploit the compatibility of constitutive relations with an entropy-like inequality, assumed in the form\n\\begin{equation}\\label{entropyineq1}\n\\frac{D s(\\mathbf{z})}{D t}+\\frac{D(v s(\\mathbf{z}) +J_s(\\mathbf{z}))}{D x}\\ge 0,\n\\end{equation}\nwhere  $s$ and $J_s$, which are functions depending on the state variables, represent the entropy and the entropy flux, respectively, and $v$ the velocity. In the case where the velocity does not appear among the field variables (for instance, in the case of a model of rigid heat conductors), \nthe entropy-like inequality is assumed to be\n\\begin{equation}\\label{entropyineq2}\n\\frac{D s(\\mathbf{z})}{D t}+\\frac{D J_s(\\mathbf{z})}{D x}\\ge 0.\n\\end{equation}\n\nIn what follows, we will use the entropy-like inequality in the form \\eqref{entropyineq1};   \nin the cases where the velocity does not belong to the field variables, the corresponding results are obtained simply by setting $v=0$.\n\nFrom the analysis of the entropy inequality, some restrictions on the form of constitutive \nequations can be obtained.\nClassically, the  restrictions placed by the entropy principle on the constitutive functions are found by using the Coleman-Noll  procedure \\cite{Coleman-Noll,Coleman-Mizel}, or the Liu  one \\cite{Liu}.\nBoth procedures have to be extended in order to manage non-local constitutive equations. \nThe entropy principle imposes that the inequality \\eqref{entropyineq1} must be satisfied for arbitrary \nthermodynamic processes \\cite{CJRV-2014,JCL-2010}. To find a set of conditions which are at least sufficient for the \nfulfilment of such a constraint, we apply an extended Liu procedure recently developed in a series of papers \n\\cite{Cimmelli-2007,CST-JMP-2009,CST-PRSA-2010,COT-JMP-2011}, \nincorporating new restrictions consistent with higher order non-local constitutive \ntheories. In fact, in order to exploit the second law, we use as constraints both the balance equations for the unknown fields and their extended equations up to the order of the \nderivatives entering the state space.\n\nSimple mathematical considerations may clarify the necessity of imposing as additional constraints in the \nentropy inequality the gradients of the balance equations when dealing with non-local constitutive equations. \n\nThe thermodynamic processes are solutions of the balance equations, and, if these solutions are smooth enough, \nare trivially solutions of their differential consequences  (see also \\cite{Rogolino-Cimmelli-2019}). \nSince the entropy inequality \\eqref{entropyineq1} has to be satisfied in arbitrary smooth processes,  then it is \nnatural, from a mathematical point of view, to use the differential consequences of the equations governing those \nprocesses as constraints for such an inequality. \nNext Section will be devoted to the description of the extended Liu procedure in this general framework.\n \n\\section{Extended Liu procedure}\n\\label{sec:liu}\nIn this Section, we introduce a general scheme in order to apply the extended Liu technique in the \ncase of $r$-th order ($r\\ge 1$) non-local constitutive equations.\n\nWe consider this rather general case essentially for two reasons. The first reason is to have a unified framework good enough to be applied to different models with non-local constitutive equations of arbitrary order. The second one is of computational nature. In fact, in dealing with applied problems, it is relevant both the derivation of the thermodynamic restrictions arising from the entropy inequality, and, whenever this is possible, a more or less explicit \ncharacterization of the constitutive equations. Since the thermodynamic restrictions may have  a lengthy expression, it is convenient to use a computer algebra system for their possible solution. In order to be able to have a flexible computer algebra package that can be used in many different cases, a general approach reveals useful if not necessary. In fact, we developed some general routines in the CAS Reduce \\cite{Reduce} that implement the algorithm at the core of extended Liu approach. \n\nFirst of all, we need to compute the spatial derivatives of the fundamental balance equations. \nFrom  the system \\eqref{generalbalance}, developing the first order time and space derivatives, one has:\n\\begin{equation}\n\\mathcal{E}_i\\equiv \\frac{\\partial \\Phi_i}{\\partial u_j}\\frac{\\partial u_j}{\\partial t}+\\frac{\\partial \\Psi_i}{\\partial u_j}\\frac{\\partial u_j}{\\partial x}+\n\\frac{\\partial \\chi_i}{\\partial z^\\star_{\\alpha}}\\frac{\\partial z^\\star_\\alpha}{\\partial x}-\\Gamma_i=0,\n\\end{equation}\nwhere the Einstein summation convention over repeated indices is used. \nIn order to write in general the $m$-th order spatial derivative of the balance laws, let us recall \n\\cite{Mishkov} a formula giving an expression for the $m$-th derivative of a composite function when the argument is a vector with an arbitrary number of components. This formula is a generalization of the well known Fa\\`a di Bruno's formula \\cite{FaadiBruno,Roman}.\n\nThe following theorem is no more than a simple rewriting of the main result contained in \\cite{Mishkov}.\n\\begin{theorem}\nLet $\\mathbf{w}=(w_1(t,x),\\ldots,w_s(t,x))$ be a vector, and $F(\\mathbf{w}(t,x))$ a composite function for which all the needed derivatives are defined, then\n\\begin{equation}\n\\label{faabruno}\n\\begin{aligned}\n&\\frac{D^m F(\\mathbf{w}(t,x))}{D x^m}=\n\\sum_{J_0}\\sum_{J_1}\\cdots\\sum_{J_m}\n\\frac{m!}{\\prod_{i=1}^m(i!)^{k_i}\\prod_{i=1}^m\\prod_{j=1}^s q_{ij}!}\\times\\\\\n&\\qquad\\times \\frac{\\partial^k F}{\\partial w_1^{p_1}\\partial w_2^{p_2}\\cdots\\partial w_s^{p_s}}\n\\prod_{i=1}^m \\left(\\frac{\\partial^i w_1}{\\partial x^i}\\right)^{q_{i1}}\n\\left(\\frac{\\partial^i w_2}{\\partial x^i}\\right)^{q_{i2}}\\cdots \n\\left(\\frac{\\partial^i w_s}{\\partial x^i}\\right)^{q_{is}},\n\\end{aligned}\n\\end{equation}\nwhere the various sums are over all nonnegative integer solutions of the Diophantine equations\n\\begin{equation*}\n\\begin{aligned}\n&\\sum_{J_0} \\rightarrow k_1+2k_2+\\ldots +mk_m = m,\\\\\n&\\sum_{J_1} \\rightarrow q_{11}+q_{12}+\\ldots +q_{1s} = k_1,\\\\\n&\\sum_{J_2} \\rightarrow q_{21}+q_{22}+\\ldots +q_{2s} = k_2,\\\\\n&\\ldots\\\\\n&\\sum_{J_m} \\rightarrow q_{m1}+q_{m2}+\\ldots +q_{ms} = k_m,\n\\end{aligned}\n\\end{equation*}\nand it is\n\\begin{equation*}\n\\begin{aligned}\n&p_j=q_{1j}+q_{2j}+\\ldots+q_{mj}, \\qquad j=1,\\ldots,s,\\\\\n&k=p_1+p_2+\\ldots+p_s=k_1+k_2+\\ldots+k_m. \\qquad \\square\n\\end{aligned}\n\\end{equation*}\n\\end{theorem}\n\nBy introducing the $m$-th order $(m=1,\\ldots,r)$ spatial derivatives of the balance laws, we get\n\\begin{equation}\n\\label{estese}\n\\begin{aligned}\n&\\frac{D^m \\mathcal{E}_i}{D x^m}\\equiv\\sum_{h=0}^k\\binom{m}{h}\\left[\\frac{D^{h}}{D x^{h}}\\left(\\frac{\\partial \\Phi_i(\\mathbf{u})}{\\partial u_j}\\right)\\frac{\\partial^{m-h+1}u_j}{\\partial t \\partial x^{m-h}} \\right.\\\\\n&\\left.+\\frac{D^{h}}{D x^{h}}\\left(\\frac{\\partial \\Psi_i(\\mathbf{u})}{\\partial u_j}\\right)\\frac{\\partial^{m-h+1}u_j}{\\partial x^{m-h+1}}+\\frac{D^h}{D x^h}\n\\left(\\frac{\\partial \\chi_i(\\mathbf{z}^\\star)}{\\partial z^\\star_\\alpha}\\right)\\frac{\\partial^{m-h+1}z^\\star_\\alpha}{\\partial x^{m-h+1}}\\right]\\\\\n&-\\frac{D^m \\Gamma_i(\\mathbf{z}^\\star)}{D x^{m}}=0.\n\\end{aligned}\n\\end{equation}\nIn order to take into account in the entropy inequality the restrictions determined by the field equations and their spatial derivatives, let us introduce the Lagrange multipliers $\\Lambda^{(k)}_i$\n$(i=1,\\ldots,n,\\; k=0,\\ldots,r)$ associated to \n$\\displaystyle\\frac{D^k\\mathcal{E}_i}{D x^k}$. Therefore, the entropy inequality \\eqref{entropyineq1} becomes\n\\begin{equation}\\label{dis}\n\\frac{\\partial s(\\mathbf{z})}{\\partial z_\\alpha}\\frac{\\partial z_\\alpha}{\\partial t}+v\\frac{\\partial s(\\mathbf{z})}{\\partial z_\\alpha}\\frac{\\partial z_\\alpha}{\\partial x}+s(\\mathbf{z})\\frac{\\partial v}{\\partial x}+\\frac{\\partial J_s(\\mathbf{z})}{\\partial z_\\alpha}\\frac{\\partial z_\\alpha}{\\partial x}-\\sum_{i=1}^n\\sum_{k=0}^r\n\\Lambda_i^{(k)}\\frac{D^k \\mathcal{E}_i}{D x^k}\\geq 0.\n\\end{equation}\n\nBy expanding the derivatives in \\eqref{dis}, and using formulae \\eqref{faabruno} and \\eqref{estese}, a straightforward though tedious computation provides a very long expression that turns out to be a polynomial in some derivatives of field variables not belonging to the state space with coefficients depending at most on the field and state variables. This polynomial must be non-negative! Once \\eqref{dis} has been expanded, we may distinguish the derivatives of field variables therein appearing,\nand not entering the state space, in two different classes:\n\\begin{itemize}\n\\item \\emph{highest derivatives}: time derivatives of the field variables, time derivatives of the spatial derivatives (up to the order $r$) of the field variables, and spatial derivatives of highest order: it is easily ascertained that these highest derivatives appear linearly with coefficients depending on the field and state variables;\n\\item \\emph{higher derivatives}: spatial derivatives whose order is not maximal but higher than that of the derivatives entering the state space: it is easily recognized that these higher derivatives appear in powers of degree up to $r+1$, with coefficients depending on the field and state variables.\n\\end{itemize}\n\nLet us define the following sets:\n\\[\n\\begin{aligned}\n&\\widehat{\\mathcal{Z}}^{(k)}=\\left\\{ w \\in \\mathcal{Z}^{(k)} \n\\;:\\; \\frac{\\partial^h w}{\\partial x^h}\\notin \\mathcal{Z},\\; h=1,\\ldots,r-k\\right\\},\\quad k=0,\\ldots,r-1,\\\\\n&\\widehat{\\mathcal{Z}}^{(r)}\\equiv \\mathcal{Z}^{(r)},\n\\end{aligned}\n\\]\nand\n\\[\n\\begin{aligned}\n&\\widehat{\\mathcal{Z}}=\\bigcup_{k=0}^r \\widehat{\\mathcal{Z}}^{(k)}.\n\\end{aligned}\n\\]\nThe highest derivatives are the time derivatives of the field variables and of their spatial derivatives up to the order $r$, together with the $(r+1)$th order spatial derivatives of the elements belonging to the set $\\widehat{\\mathcal{Z}}$.\n\nTherefore, denoting with  $\\boldsymbol\\zeta$ the vector whose components $\\zeta_i$ are the highest derivatives, and with \n$\\boldsymbol\\eta$ the vector whose components $\\eta_j$ are the higher derivatives, \nthe entropy inequality \\eqref{dis} can be cast in the following compact form:\n\\begin{equation}\n\\begin{aligned}\nA_i(\\mathbf{z}^\\star) \\zeta_i&+B^{(r+1)}_{j_1\\ldots j_{r+1}}(\\mathbf{z}^\\star)\\eta_{j_1}\\cdots \\eta_{j_{r+1}}+\nB^{(r)}_{j_1\\ldots j_{r}}(\\mathbf{z}^\\star)\\eta_{j_1}\\cdots \\eta_{j_{r}}\\\\\n&+\\ldots+B^{(2)}_{j_1j_{2}}(\\mathbf{z}^\\star)\\eta_{j_1}\\eta_{j_{2}}+B^{(1)}_{j_1}(\\mathbf{z}^\\star)\\eta_{j_1}+B^{(0)}(\\mathbf{z}^\\star)\\ge 0,\n\\end{aligned}\n\\end{equation}\nwhere the coefficients $A_i$, $B^{(r+1)}_{j_1\\ldots j_{r+1}}$, $B^{(r)}_{j_1\\ldots j_{r}}$,\\ldots,\n$B^{(2)}_{j_1j_{2}}$,  $B^{(1)}_{j_1}$ and $B^{(0)}$ may depend upon the field variables and the elements entering the state space.\nThis inequality must be satisfied for every thermodynamical process. \n\nFirst, let us observe that nothing prevents to have a thermodynamic process where $B^{(0)}=0$.\nMoreover, since we used in the entropy inequality all the constraints imposed by\nthe field equations together with their spatial derivatives, the highest and higher derivatives may assume arbitrary values. \nConsequently, we may give a set of conditions that are sufficient in order the inequality \\eqref{dis} be fulfilled for every thermodynamical process. These sufficient conditions provide constraints on the constitutive equations.\n\n\\begin{theorem}\\label{theorem2}\nLet $\\boldsymbol\\zeta=(\\zeta_1,\\ldots,\\zeta_p)$ be the vector of highest derivatives, and $\\boldsymbol\\eta=(\\eta_1,\\ldots,\\eta_q)$ the vector of higher derivatives.  \nLet \n\\begin{itemize}\n\\item $A_i(\\mathbf{\\mathbf{z}^\\star})$ are $p$ functions of $\\mathbf{z}^\\star$;\n\\item $B^{(k)}_{j_1\\ldots j_{k}}(\\mathbf{z}^\\star)$, with $k=1,\\ldots,r+1$, are $\\binom{q-k+1}{k}$ functions of $\\mathbf{z}^\\star$; \n\\item $B^{(0)}(\\mathbf{z}^\\star)$ is a function of $\\mathbf{z}^\\star$.\n\\end{itemize}\nThe inequality\n\\begin{equation}\n\\begin{aligned}\nA_i(\\mathbf{z}^\\star) \\zeta_i&+B^{(r+1)}_{j_1\\ldots j_{r+1}}(\\mathbf{z}^\\star)\\eta_{j_1}\\cdots \\eta_{j_{r+1}}+\nB^{(r)}_{j_1\\ldots j_{r}}(\\mathbf{z}^\\star)\\eta_{j_1}\\cdots \\eta_{j_{r}}\\\\\n&+\\ldots+B^{(2)}_{j_1j_{2}}(\\mathbf{z}^\\star)\\eta_{j_1}\\eta_{j_{2}}+B^{(1)}_{j_1}(\\mathbf{z}^\\star)\\eta_{j_1}+B^{(0)}(\\mathbf{z}^\\star)\\ge 0,\n\\end{aligned}\n\\end{equation}\nholds for arbitrary vectors $\\boldsymbol\\zeta$ and $\\boldsymbol\\eta$ if \n\\begin{enumerate}\n\\item $A_i=0$;\n\\item $B^{(2k-1)}_{j_1\\ldots j_{2k-1}}=0$, $k=1,\\ldots,\\lfloor{\\frac{r+2}{2}}\\rfloor$\\footnote{$\\lfloor{x}\\rfloor$ denotes the greatest integer less than $x$.};\n\\item $B^{(2k)}_{j_1\\ldots j_{2k}}\\eta_{j_1}\\cdots \\eta_{j_{2k}}$, $k=1,\\ldots,\\lfloor{\\frac{r+1}{2}}\\rfloor$,\n nonnegative for all $\\boldsymbol\\eta$;\n\\item $B^{(0)}\\ge 0$. $\\qquad \\square$\n\\end{enumerate}\n\n\\end{theorem}\n\nDue to Theorem~\\ref{theorem2}, by imposing that the coefficients of $u_{j,t}$, $u_{j,t x}$, \\ldots, $u_{j,t,\\underbrace{x\\ldots x}_{r}}$, where the indices ${(\\cdot)}_{,t}$ and ${(\\cdot)}_{,x}$ denote the\npartial derivatives with respect to time and space, respectively,\nare vanishing, we obtain:\n\\begin{equation}\n\\begin{aligned}\n&\\frac{\\partial s}{\\partial u_j}-\\sum_{i=1}^n\\sum_{k=0}^r\\Lambda_i^{(k)}\\frac{D^k}{Dx^k}\\left(\\frac{\\partial\\Phi_i}{\\partial u_j}\\right)=0,\\\\\n&\\frac{\\partial s}{\\partial u_{j,x}}-\\sum_{i=1}^n\\sum_{k=1}^r \\binom{k}{k-1}\\Lambda_i^{(k)}\\frac{D^{k-1}}{Dx^{k-1}}\\left(\\frac{\\partial\\Phi_i}{\\partial u_j}\\right)=0,\\\\\n&\\frac{\\partial s}{\\partial u_{j,xx}}-\\sum_{i=1}^n\\sum_{k=2}^r \\binom{k}{k-2}\\Lambda_i^{(k)}\\frac{D^{k-2}}{Dx^{k-2}}\\left(\\frac{\\partial\\Phi_i}{\\partial u_j}\\right)=0,\\\\\n&\\ldots\\\\\n&\\frac{\\partial s}{\\partial u_{j,\\underbrace{x\\ldots x}_{r}}}-\\sum_{i=1}^n\\Lambda_i^{(r)}\\frac{\\partial\\Phi_i}{\\partial u_j}=0,\n\\end{aligned}\n\\end{equation}\nthat allow us the determination of the Lagrange multipliers. The coefficients of the remaining highest derivatives, \n\\begin{equation}\n\\sum_{i=1}^n\\Lambda_i^{(r)}\\left(\\frac{\\partial\\Psi_i}{\\partial \\widehat{z}_\\alpha}+\\frac{\\partial\\chi_i}{\\partial \\widehat{z}_\\alpha}\\right)=0,\\qquad \\widehat{z}_\\alpha\\in\\widehat{\\mathcal{Z}},\n\\end{equation}\nprovide conditions to be used together with the constraints coming from \nthe arbitrariness of the higher derivatives to restrict the constitutive equations. After all these restrictions have been derived, the residual entropy inequality, say\n\\begin{equation} \nB^{(0)}\\ge 0,\n\\end{equation}\nremains providing further constraints.\n\nIt is evident that the general restrictions here derived can not be discussed\nfrom a physical point of view, but they are essential in writing the computer algebra program that almost automatically computes the restrictions placed by second law.\n\nIn the next Section, we provide some examples of physical interest where the procedure here described can be applied, and we discuss the physical meaning of the results.\n\n\\section{Applications}\n\\label{sec:applications}\nHere, we consider two physical examples of fluids whose constitutive equations involve first or second order non-localities, \\emph{i.e.}, special instances of higher grade fluids. \nIn modern terminology, a fluid is said to be of grade $(r+1)$ \n\\cite{Dunn-Serrin,Truesdell_Noll,Dunn-Rajagopal,Gouin2019} if the constitutive quantities are allowed to depend on gradients of order  $r$.  In recent years, these higher grade \nfluids have been employed, for instance, to model capillarity effects \n\\cite{Gouin1985a,Gouin1985b}, or to analyze the structure of liquid-vapor phase transitions \nunder both static\n\\cite{Aifantis-Serrin-1,Aifantis-Serrin-2} and dynamic \\cite{Slemrod-1,Slemrod-2} conditions. \n\nAs observed in \\cite{Dunn-Serrin,Dunn}, these fluids are, in general, incompatible with the restrictions placed by the second law of thermodynamics. In order to find a remedy to such an incompatibility, these authors proposed a generalization of the classical local balance of energy by postulating the existence of a rate of supply of mechanical energy, the so called interstitial working; in such a framework, the entropy flux has the classical form as the ratio between the heat flux and the absolute temperature. Nevertheless, the same authors \\cite{Dunn-Serrin} remarked that the interstitial working can be removed but at the cost of introducing an entropy extra-flux \\cite{Muller} in order to satisfy the second law of thermodynamics. \n\nIn the applications we consider below, the assumptions we make consist in taking the local energy balance in the classical form without including any extra-term; moreover, we write the entropy inequality without specializing the form of the entropy flux: as will be seen, the expression of entropy flux will arise as a consequence of the extended procedure when solving the constraints placed by the second law.\n\n\\subsection{Fluid of grade 2 with a scalar internal variable}\nLet us consider a fluid of grade 2 whose description involves, in addition to the basic fields of mass density, velocity and internal energy, an internal variable. The latter may describe an additional internal degree of freedom of the material, for instance representative of a suitable scalar microstructure \\cite{Capriz,OS-2008} or another extensive property.\n\nThe governing equations we consider read\n\\begin{equation}\n\\label{fluid-grade-2}\n\\begin{aligned}\n&\\mathcal{E}_1\\equiv\\frac{D\\rho}{D t} + \\frac{D (\\rho v)}{D x}=0, \\\\\n&\\mathcal{E}_2\\equiv\\frac{D(\\rho v)}{D t} + \\frac{D (\\rho v^2  - T)}{D x}=0,\\\\\n&\\mathcal{E}_3\\equiv\\frac{D}{D t}\\left(\\rho\\varepsilon+\\rho \\frac{v^2}{2}\\right) +\\frac{D }{D x}\\left(\\rho v\\varepsilon+\\rho \\frac{v^3}{2}-Tv+ q\\right)=0,\\\\\n&\\mathcal{E}_4\\equiv\\frac{D(\\rho\\gamma)}{Dt}+\\frac{D(\\rho v\\gamma+\\phi)}{Dx}=0,\n\\end{aligned}\n\\end{equation}\nwhere $\\rho$ is the mass density, $v$ the velocity, $\\varepsilon$ the internal energy per unit mass, and $\\gamma$ an internal state variable;\nmoreover, the Cauchy stress $T$, the heat flux $q$, and the flux $\\phi$  of internal variable must be assigned by \nmeans of suitable constitutive equations such that for every admissible process the entropy inequality \n\\begin{equation}\n\\rho\\left(\\frac{Ds}{D t}+v\\frac{Ds}{Dx}\\right)+ \\frac{DJ_s}{Dx} \\ge 0\n\\end{equation}\nbe satisfied, being $s$ and $J_s$ (to be assigned as constitutive quantities) the specific entropy and  the entropy flux, respectively.\n\nWe assume the state space spanned by\n\\begin{equation}\n\\mathcal{Z}=\\{\\rho,\\varepsilon,\\gamma,\\rho_{,x},v_{,x},\\varepsilon_{,x},\\gamma_{,x}\\}.\n\\end{equation}\n\nAs shown in the previous Section, the exploitation of second law of thermodynamics is here performed by taking into \naccount the constraints \nimposed on the thermodynamic processes by the balance equations and their first order extensions; these \nconstraints are imposed by introducing some  Lagrange multipliers.\nTherefore, the entropy inequality becomes\n\\begin{equation}\n\\label{entropyconstrained}\n\\begin{aligned}\n&\\rho\\left(\\frac{Ds}{D t}+v\\frac{Ds}{Dx}\\right)+ \\frac{DJ_s}{Dx} \\\\\n&\\quad- \\Lambda^{(0)}_1 \\mathcal{E}_1- \\Lambda^{(0)}_2 \\mathcal{E}_2- \\Lambda^{(0)}_3 \\mathcal{E}_3\n- \\Lambda^{(0)}_4 \\mathcal{E}_4\\\\\n&\\quad-\\Lambda^{(1)}_1\\frac{D\\mathcal{E}_1}{Dx}-\\Lambda^{(1)}_2\\frac{D\\mathcal{E}_2}{Dx}\n-\\Lambda^{(1)}_3\\frac{D\\mathcal{E}_3}{Dx}-\\Lambda^{(1)}_4\\frac{D\\mathcal{E}_4}{Dx} \\geq 0.\n\\end{aligned}\n\\end{equation}\nFor the sake of clarity, we present the details of the computation we are required to do in applying the extended Liu procedure. \n\nExpanding the derivatives in the entropy inequality \\eqref{entropyconstrained}, we obtain a very long expression, say\n\\begin{align*}\n&\\left(\\rho\\frac{\\partial s}{\\partial\\rho}-\\Lambda^{(0)}_1\\right)\\rho_{,t}-\\left(\\rho\\Lambda^{(0)}_2+\\rho_{,x}\\Lambda^{(1)}_2\\right)v_{,t}+\\left(\\rho\\frac{\\partial s}{\\partial\\varepsilon}-\\rho\\Lambda^{(0)}_3-\\rho_{,x}\\Lambda^{(1)}_3\\right)\\varepsilon_{,t}\\allowdisplaybreaks\\\\\n&\\quad +\\left(\\rho\\frac{\\partial s}{\\partial\\gamma}-\\rho\\Lambda^{(0)}_4-\\rho_{,x}\\Lambda^{(1)}_4\\right)\\gamma_{,t}\n+\\left(\\rho\\frac{\\partial s}{\\partial\\rho_{,x}}-\\Lambda^{(1)}_1\\right)\\rho_{,tx}\\allowdisplaybreaks\\\\\n&\\quad+\\rho\\left(\\frac{\\partial s}{\\partial v_{,x}}-\\Lambda^{(1)}_2\\right)v_{,tx}\n+\\rho\\left(\\frac{\\partial s}{\\partial \\varepsilon_{,x}}-\\Lambda^{(1)}_3\\right)\\varepsilon_{,tx}\n+\\rho\\left(\\frac{\\partial s}{\\partial \\gamma_{,x}}-\\Lambda^{(1)}_4\\right)\\gamma_{,tx}\\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial T}{\\partial\\rho_{,x}}\\Lambda^{(1)}_2- \\frac{\\partial q}{\\partial\\rho_{,x}}\\Lambda^{(1)}_3 \n- \\frac{\\partial\\phi}{\\partial\\rho_{,x}}\\Lambda^{(1)}_4 \\right)\\rho_{,xxx}\\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial T}{\\partial v_{,x}}\\Lambda^{(1)}_2- \\frac{\\partial q}{\\partial v_{,x}}\\Lambda^{(1)}_3 \n- \\frac{\\partial\\phi}{\\partial v_{,x}}\\Lambda^{(1)}_4 \\right)v_{,xxx}\\allowdisplaybreaks\\\\\n&\\quad +\\left(\\frac{\\partial T}{\\partial \\varepsilon_{,x}}\\Lambda^{(1)}_2 - \\frac{\\partial q}{\\partial \\varepsilon_{,x}}\\Lambda^{(1)}_3 \n- \\frac{\\partial\\phi}{\\partial \\varepsilon_{,x}}\\Lambda^{(1)}_4\\right)\\varepsilon_{,xxx}\\allowdisplaybreaks\\\\\n&\\quad +\\left(\\frac{\\partial T}{\\partial \\gamma_{,x}}\\Lambda^{(1)}_2- \\frac{\\partial q}{\\partial \\gamma_{,x}}\\Lambda^{(1)}_3 \n- \\frac{\\partial\\phi}{\\partial \\gamma_{,x}}\\Lambda^{(1)}_4 \\right)\\gamma_{,xxx}\\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial ^{2}T}{\\partial \\rho_{,x}^{2}}  \\Lambda^{(1)}_2 \n- \\frac{\\partial ^{2}q}{\\partial \\rho_{,x}^{2}}  \\Lambda^{(1)}_3\n-\\frac{\\partial ^{2}\\phi }{\\partial \\rho_{,x}^{2}}  \\Lambda^{(1)}_4 \n\\right)  \\rho_{,xx}^{2}\\allowdisplaybreaks\\\\\n&\\quad+2 \\left(\\frac{\\partial ^{2}T}{\\partial \\rho_{,x}\\partial v_{,x}}  \\Lambda^{(1)}_2   \n- \\frac{\\partial ^{2}q}{\\partial \\rho_{,x}\\partial v_{,x}}  \\Lambda^{(1)}_3\n-\\frac{\\partial ^{2}\\phi }{\\partial \\rho_{,x}\\partial v_{,x}}  \\Lambda^{(1)}_4\\right)  \\rho_{,xx}  v_{,xx}\\allowdisplaybreaks\\\\  \n&\\quad+2\\left( \\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}\\partial \\rho_{,x}}    \\Lambda^{(1)}_2 \n- \\frac{\\partial ^{2}q}{\\partial \\varepsilon _{,x}\\partial \\rho_{,x}} \\Lambda^{(1)}_3\n- \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}\\partial \\rho_{,x}}    \\Lambda^{(1)}_4  \\right) \\rho_{,xx} \\varepsilon _{,xx}    \\allowdisplaybreaks\\\\\n&\\quad+2 \\left( \\frac{\\partial ^{2}T}{\\partial \\gamma_{,x}\\partial \\rho_{,x}} \\Lambda^{(1)}_2 \n-\\frac{\\partial ^{2}q}{\\partial \\gamma_{,x}\\partial \\rho_{,x}} \\Lambda^{(1)}_3 \n- \\frac{\\partial ^{2}\\phi }{\\partial \\gamma_{,x}\\partial \\rho_{,x}} \\Lambda^{(1)}_4\\right)  \\rho_{,xx}\\gamma_{,xx}  \\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial ^{2}T}{\\partial v_{,x}^{2}}  \\Lambda^{(1)}_2 \n- \\frac{\\partial ^{2}q}{\\partial v_{,x}^{2}}  \\Lambda^{(1)}_3 \n-\\frac{\\partial ^{2}\\phi }{\\partial v_{,x}^{2}}  \\Lambda^{(1)}_4\\right)  v_{,xx}^{2}\\allowdisplaybreaks\\\\\n&\\quad+2\\left( \\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}\\partial v_{,x}}  \\Lambda^{(1)}_2 \n- \\frac{\\partial ^{2}q}{\\partial \\varepsilon _{,x}\\partial v_{,x}} \\Lambda^{(1)}_3 \n- \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}\\partial v_{,x}} \\Lambda^{(1)}_4\\right) v_{,xx} \\varepsilon _{,xx}   \\allowdisplaybreaks\\\\\n&\\quad+2\\left(  \\frac{\\partial ^{2}T}{\\partial \\gamma_{,x}\\partial v_{,x}}  \\Lambda^{(1)}_2  \n- \\frac{\\partial ^{2}q}{\\partial \\gamma_{,x}\\partial v_{,x}}   \\Lambda^{(1)}_3  \n- \\frac{\\partial ^{2}\\phi }{\\partial \\gamma_{,x}\\partial v_{,x}}\\Lambda^{(1)}_4\\right)  v_{,xx}\\gamma_{,xx} \\allowdisplaybreaks\\\\\n&\\quad+ \\left(\\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}^{2}} \\Lambda^{(1)}_2\n- \\frac{\\partial ^{2}q}{ \\partial \\varepsilon _{,x}^{2}} \\Lambda^{(1)}_3 \n-\\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}^{2}} \\Lambda^{(1)}_4\\right) \\varepsilon _{,xx}^{2}  \\allowdisplaybreaks\\\\\n&\\quad+2  \\left(\\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}\\partial \\gamma_{,x}}\\Lambda^{(1)}_2  \n- \\frac{\\partial ^{2}q}{\\partial \\varepsilon _{,x}\\partial \\gamma_{,x}}  \\Lambda^{(1)}_3 \n- \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}\\partial \\gamma_{,x}}\\Lambda^{(1)}_4 \\right) \\varepsilon _{,xx}  \\gamma_{,xx}\n  \\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial ^{2}T}{\\partial \\gamma_{,x}^{2}}   \\Lambda^{(1)}_2\n- \\frac{\\partial ^{2}q}{\\partial \\gamma_{,x}^{2}}   \\Lambda^{(1)}_3\n-\\frac{\\partial ^{2}\\phi }{\\partial \\gamma_{,x}^{2}}  \\Lambda^{(1)}_4\\right) \\gamma_{,xx}^{2} \\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial s}{\\partial \\rho_{,x}}  \\rho  v+\\frac{\\partial J_s}{\\partial \\rho_{,x}} +\\frac{\\partial T}{\\partial \\rho_{,x}}  \\Lambda^{(0)}_2- \\frac{\\partial q}{\\partial \\rho_{,x}}  \\Lambda^{(0)}_3-\\frac{\\partial \\phi }{\\partial \\rho_{,x}}  \\Lambda^{(0)}_4\\right.\\allowdisplaybreaks\\\\\n&\\qquad-\\Lambda^{(1)}_1 v+\\Lambda^{(1)}_2\\left(\\frac{\\partial T}{\\partial \\rho }  +2 \\frac{\\partial ^{2}T}{\\partial \\rho \\partial \\rho_{,x}}  \\rho_{,x}  +2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon \\partial \\rho_{,x}}  \\varepsilon _{,x}    \n+2 \\frac{\\partial ^{2}T}{\\partial \\gamma\\partial \\rho_{,x}}  \\gamma_{,x}\\right) \\allowdisplaybreaks \\\\\n&\\qquad +\\Lambda^{(1)}_3\\left(\\frac{\\partial T}{\\partial \\rho_{,x}}  v_{,x}-2 \\frac{\\partial ^{2}q}{\\partial \\varepsilon \\partial \\rho_{,x}}  \\varepsilon _{,x}  -2  \\frac{\\partial ^{2}q}{\\partial \\gamma\\partial \\rho_{,x}}  \\gamma_{,x}  \n-2 \\frac{\\partial ^{2}q}{\\partial \\rho \\partial \\rho_{,x}}   \\rho_{,x}  \n-\\frac{\\partial q}{\\partial \\rho }  \\right)    \\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\Lambda^{(1)}_4\\left(  -2  \\frac{\\partial ^{2}\\phi }{ \\partial \\varepsilon \\partial \\rho_{,x}}  \\varepsilon _{,x}    \n-2\\frac{\\partial ^{2}\\phi }{\\partial \\gamma\\partial \\rho_{,x}}  \\gamma_{,x}   \n-2\\frac{\\partial ^{2}\\phi }{\\partial \\rho \\partial \\rho_{,x}}  \\rho_{,x} -\\frac{\\partial \\phi }{\\partial \\rho } \\right)\n\\right) \\rho_{,xx} \\allowdisplaybreaks\\\\\n&\\quad+ \\left(\\frac{\\partial s}{\\partial v_{,x}}  \\rho   v  \n+\\frac{\\partial J_s}{\\partial v_{,x}}  \n+\\frac{\\partial T}{\\partial v_{,x}}  \\Lambda^{(0)}_2  \n- \\frac{\\partial q}{\\partial v_{,x}}  \\Lambda^{(0)}_3  \n-\\frac{\\partial \\phi }{\\partial v_{,x}}  \\Lambda^{(0)}_4\\right.\\allowdisplaybreaks\\\\\n&\\qquad+\\left(-\\Lambda^{(1)}_1  \\rho   \n-\\Lambda^{(1)}_2  \\left(\\rho   v  \n+2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon \\partial v_{,x}}  \\varepsilon _{,x}  \n+2 \\frac{\\partial ^{2}T}{\\partial \\gamma\\partial v_{,x}}  \\gamma_{,x}  \n+2 \\frac{\\partial ^{2}T}{\\partial \\rho \\partial v_{,x}}    \\rho_{,x}\\right)\\right.\\allowdisplaybreaks\\\\\n&\\qquad+\\Lambda^{(1)}_3  \\left(T +\\frac{\\partial T}{\\partial v_{,x}}   v_{,x}\n-2 \\frac{\\partial ^{2}q}{\\partial \\rho \\partial v_{,x}}   \\rho_{,x}  \n-2 \\frac{\\partial ^{2}q}{\\partial \\varepsilon \\partial v_{,x}}  \\varepsilon _{,x} \n-2 \\frac{\\partial ^{2}q}{\\partial \\gamma\\partial v_{,x}}  \\gamma_{,x} \\right) \\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\Lambda^{(1)}_4\\left(-2\\frac{\\partial ^{2}\\phi }{\\partial \\rho \\partial v_{,x}}\\rho_{,x}\n-2  \\frac{\\partial ^{2}\\phi }{ \\partial \\varepsilon \\partial v_{,x}}  \\varepsilon _{,x}  \n-2\\frac{\\partial ^{2}\\phi }{\\partial \\gamma\\partial v_{,x}}  \\gamma_{,x}  \\right) \\right) v_{,xx}\\allowdisplaybreaks\\\\\n&\\quad+ \\left(\\frac{\\partial s}{\\partial \\varepsilon _{,x}}   \\rho   v\n+\\frac{\\partial J_s}{\\partial \\varepsilon _{,x}} \n+\\frac{\\partial T}{\\partial \\varepsilon _{,x}}  \\Lambda^{(0)}_2\n -\\frac{\\partial q}{\\partial \\varepsilon _{,x}}  \\Lambda^{(0)}_3\n-\\frac{\\partial \\phi }{\\partial \\varepsilon _{,x}} \\Lambda^{(0)}_4\\right.\\allowdisplaybreaks\\\\\n&\\qquad+\\Lambda^{(1)}_2\\left(2  \\frac{\\partial ^{2}T}{\\partial \\varepsilon \\partial \\varepsilon _{,x}}  \\varepsilon _{,x} \n+ \\frac{\\partial T}{\\partial \\varepsilon }  \n+2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}\\partial \\gamma}  \\gamma_{,x} \n+2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}\\partial \\rho } \\rho_{,x}\\right)\\allowdisplaybreaks\\\\\n&\\qquad+\\Lambda^{(1)}_3\\left(-2\\frac{\\partial ^{2}q}{\\partial \\varepsilon \\partial \\varepsilon _{,x}}  \\varepsilon _{,x}  \n+\\frac{\\partial T}{\\partial \\varepsilon _{,x}}  v_{,x}\n-  \\rho   v\n-\\frac{\\partial q}{\\partial \\varepsilon } \n-2 \\frac{\\partial ^{2}q}{\\partial \\varepsilon _{,x}\\partial \\gamma} \\gamma_{,x}  \n-2 \\frac{\\partial ^{2}q}{\\partial \\varepsilon _{,x}\\partial \\rho } \\rho_{,x}\\right) \\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\Lambda^{(1)}_4\\left(-2  \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon \\partial \\varepsilon _{,x}}  \\varepsilon _{,x} \n-\\frac{\\partial \\phi }{\\partial \\varepsilon } \n-2  \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}\\partial \\gamma}   \\gamma_{,x} \n-2 \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}\\partial \\rho } \\rho_{,x}\\right)\\right) \\varepsilon _{,xx}   \\allowdisplaybreaks\\\\  \n&\\quad+ \\left(\\frac{\\partial s}{\\partial \\gamma_{,x}}  \\rho   v\n+\\frac{\\partial J_s}{\\partial \\gamma_{,x}}  \n+\\frac{\\partial T}{\\partial \\gamma_{,x}}  \\Lambda^{(0)}_2\n- \\frac{\\partial q}{\\partial \\gamma_{,x}}  \\Lambda^{(0)}_3\n-\\frac{\\partial \\phi }{\\partial \\gamma_{,x}}  \\Lambda^{(0)}_4\\right.\\allowdisplaybreaks\\\\\n&\\qquad+\\Lambda^{(1)}_2\\left(2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon \\partial \\gamma_{,x}}  \\varepsilon _{,x} \n+2  \\frac{\\partial ^{2}T}{\\partial \\gamma\\partial \\gamma_{,x}}  \\gamma_{,x} \n+\\frac{\\partial T}{ \\partial \\gamma}  \n+2  \\frac{\\partial ^{2}T}{\\partial \\gamma_{,x}\\partial \\rho }  \\rho_{,x}\\right)  \\allowdisplaybreaks\\\\\n&\\qquad+\\Lambda^{(1)}_3\\left(-2 \\frac{\\partial ^{2}q}{\\partial \\varepsilon \\partial \\gamma_{,x}}  \\varepsilon _{,x}  \n-2 \\frac{\\partial ^{2}q}{\\partial \\gamma\\partial \\gamma_{,x}}  \\gamma_{,x}  \n-\\frac{\\partial q}{\\partial \\gamma}  \n-2  \\frac{\\partial ^{2}q}{\\partial \\gamma_{,x}\\partial \\rho }  \\rho_{,x}\n+\\frac{\\partial T}{\\partial \\gamma_{,x}} v_{,x}\\right)\\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\Lambda^{(1)}_4 \\left(-2  \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon \\partial \\gamma_{,x}}  \\varepsilon _{,x} \n-2\\frac{\\partial ^{2}\\phi }{\\partial \\gamma\\partial \\gamma_{,x}}  \\gamma_{,x}  \n-\\frac{\\partial \\phi }{\\partial \\gamma} \n-2 \\frac{\\partial ^{2}\\phi }{\\partial \\gamma_{,x}\\partial \\rho }  \\rho_{,x}\n-\\rho  v\\right)\\right)\\gamma_{,xx}\\allowdisplaybreaks\\\\\n& \\rho v\\left(\\frac{\\partial s}{\\partial \\rho }\\rho_{,x} \n+ \\frac{\\partial s}{\\partial \\varepsilon }  \\varepsilon _{,x} \n+ \\frac{\\partial s}{\\partial \\gamma}  \\gamma_{,x}\\right)\n+\\frac{\\partial J_s}{\\partial \\rho }  \\rho_{,x}\n+\\frac{\\partial J_s}{ \\partial \\varepsilon }  \\varepsilon _{,x}\n+\\frac{\\partial J_s}{ \\partial \\gamma}  \\gamma_{,x}\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(0)}_1  \\left(v\\rho_{,x}+\\rho   v_{,x}\\right)\n-\\Lambda^{(0)}_2 \\left(\\rho   v  v_{,x}-\\frac{\\partial T}{\\partial \\rho } \\rho_{,x}\n- \\frac{\\partial T}{\\partial \\varepsilon }  \\varepsilon _{,x} \n-\\frac{\\partial T}{ \\partial \\gamma}  \\gamma_{,x} \\right)\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(0)}_3\\left(\\rho   v\\varepsilon _{,x}  -T  v_{,x}+ \\frac{\\partial q}{\\partial \\rho }  \\rho_{,x}\n+\\frac{\\partial q}{ \\partial \\varepsilon }  \\varepsilon _{,x}  \n+\\frac{\\partial q}{\\partial \\gamma}  \\gamma_{,x}\\right)\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(0)}_4 \\left(\\rho v\\gamma_{,x} +\\frac{\\partial \\phi }{\\partial \\rho }   \\rho_{,x} \n+\\frac{\\partial \\phi }{\\partial \\varepsilon }  \\varepsilon _{,x}  \n+\\frac{\\partial \\phi }{\\partial \\gamma}  \\gamma_{,x}  \\right)\n-2  \\Lambda^{(1)}_1  \\rho_{,x}  v_{,x}\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(1)}_2\\left(v\\rho_{,x} v_{,x}+\\rho   v_{,x}^{2}\n-\\frac{\\partial ^{2}T}{\\partial \\rho ^{2}}   \\rho_{,x}^{2}\n-2 \\frac{\\partial ^{2}T}{ \\partial \\rho \\partial \\varepsilon} \\rho_{,x}  \\varepsilon _{,x}   \n-2  \\frac{\\partial ^{2}T}{\\partial \\rho \\partial \\gamma} \\rho_{,x}  \\gamma_{,x} \\right.\\allowdisplaybreaks\\\\\n&\\qquad\\left.- \\frac{\\partial ^{2}T}{\\partial \\varepsilon ^{2}}  \\varepsilon _{,x}^{2}  \n-2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon \\partial \\gamma}  \\varepsilon _{,x}  \\gamma_{,x}  \n-\\frac{\\partial ^{2}T}{ \\partial \\gamma^{2}}  \\gamma_{,x}^{2} \\right)\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(1)}_3\\left(\n( v\\rho_{,x}  +\\rho   v_{,x})\\varepsilon _{,x}   \n-\\left(\\frac{\\partial T}{\\partial \\rho }   \\rho_{,x}  \n+ \\frac{\\partial T}{\\partial \\varepsilon }  \\varepsilon _{,x}    \n+\\frac{\\partial T}{\\partial \\gamma}  \\gamma_{,x} \\right)   v_{,x}\\right.\\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\frac{\\partial ^{2}q}{\\partial \\rho ^{2}}  \\rho_{,x}^{2}\n+2  \\frac{\\partial ^{2}q}{\\partial \\rho\\partial \\varepsilon} \\rho_{,x} \\varepsilon _{,x}  \n+2  \\frac{\\partial ^{2}q}{\\partial \\rho\\partial\\gamma}  \\rho_{,x}\\gamma_{,x} \\right.\\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\frac{\\partial ^{2}q}{\\partial \\varepsilon ^{2}}  \\varepsilon _{,x}^{2}\n+2\\frac{\\partial ^{2}q}{\\partial \\varepsilon \\partial \\gamma}  \\varepsilon _{,x}  \\gamma_{,x} \n+\\frac{\\partial ^{2}q}{\\partial \\gamma^{2}}  \\gamma_{,x}^{2}  \\right)\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(1)}_4\\left((v \\rho_{,x}+\\rho   v_{,x})\\gamma_{,x}    \n+\\frac{\\partial ^{2}\\phi }{\\partial \\rho ^{2}}   \\rho_{,x}^{2}\n+2  \\frac{\\partial ^{2}\\phi }{\\partial \\rho \\partial \\varepsilon}   \\rho_{,x} \\varepsilon _{,x} \\right.\\allowdisplaybreaks\\\\\n&\\qquad\\left.+2\\frac{\\partial ^{2}\\phi }{\\partial \\rho \\partial \\gamma}  \\rho_{,x} \\gamma_{,x}    \n+\\frac{\\partial ^{2}\\phi }{ \\partial \\varepsilon ^{2}}  \\varepsilon _{,x}^{2} \n+2  \\frac{\\partial ^{2}\\phi }{ \\partial \\varepsilon \\partial \\gamma}  \\varepsilon _{,x}  \\gamma_{,x}  \n+\\frac{\\partial ^{2}\\phi }{\\partial \\gamma^{2}}  \\gamma_{,x}^{2} \\right)\\ge 0,\n\\end{align*}\nwhere we can distinguish the \\emph{highest derivatives}, say\n\\[\n\\{\\rho_{,t},v_{,t},\\varepsilon_{,t},\\gamma_{,t},\\rho_{,tx},v_{,tx},\\varepsilon_{,tx},\\gamma_{,tx},\\rho_{,xxx},v_{,xxx},\\varepsilon_{,xxx},\\gamma_{,xxx}\\},\n\\]\nand the \\emph{higher derivatives}, say\n\\[\n\\{\\rho_{,xx},v_{,xx},\\varepsilon_{,xx},\\gamma_{,xx}\\}.\n\\]\nAs expected, the entropy inequality is linear in the highest derivatives and quadratic in the higher ones; \nthe coefficients are at most functions of the field and the state variables.\nBy vanishing the coefficients of the highest derivatives, we determine the Lagrange multipliers, say\n\\begin{equation}\n\\begin{aligned}\n&\\Lambda_1^{(0)}=\\rho\\frac{\\partial s}{\\partial\\rho}, \\qquad \n&&\\Lambda_2^{(0)}=-\\frac{\\rho_{,x}}{\\rho}\\frac{\\partial s}{\\partial v_x},\\\\\n&\\Lambda_3^{(0)}=\\frac{\\partial s}{\\partial\\varepsilon}-\\frac{\\rho_{,x}}{\\rho}\\frac{\\partial s}{\\partial\\varepsilon_{,x}}, \\qquad \n&&\\Lambda_4^{(0)}=\\frac{\\partial s}{\\partial\\gamma}-\\frac{\\rho_{,x}}{\\rho}\\frac{\\partial s}{\\partial\\gamma_{,x}},\\\\\n&\\Lambda_1^{(1)}=\\rho\\frac{\\partial s}{\\partial \\rho_{,x}},\\qquad\n&&\\Lambda_2^{(1)}=\\frac{\\partial s}{\\partial v_{,x}},\\\\\n&\\Lambda_3^{(1)}=\\frac{\\partial s}{\\partial \\varepsilon_{,x}},\\qquad\n&&\\Lambda_4^{(1)}=\\frac{\\partial s}{\\partial \\gamma_{,x}},\n\\end{aligned}\n\\end{equation}\nas well as the following restrictions involving the entropy, the Cauchy stress tensor, the heat flux and the flux of internal variable:\n\\begin{equation}\n\\begin{aligned}\n&\\frac{\\partial s}{\\partial v_{,x}}\\frac{\\partial T}{\\partial \\rho_{,x}}-\n\\frac{\\partial s}{\\partial \\varepsilon_{,x}}\\frac{\\partial q}{\\partial \\rho_{,x}}-\n\\frac{\\partial s}{\\partial \\gamma_{,x}}\\frac{\\partial \\phi}{\\partial \\rho_{,x}}=0,\\\\\n&\\frac{\\partial s}{\\partial v_{,x}}\\frac{\\partial T}{\\partial v_{,x}}-\n\\frac{\\partial s}{\\partial \\varepsilon_{,x}}\\frac{\\partial q}{\\partial v_{,x}}-\n\\frac{\\partial s}{\\partial \\gamma_{,x}}\\frac{\\partial \\phi}{\\partial v_{,x}}=0,\\\\\n&\\frac{\\partial s}{\\partial v_{,x}}\\frac{\\partial T}{\\partial \\varepsilon_{,x}}-\n\\frac{\\partial s}{\\partial \\varepsilon_{,x}}\\frac{\\partial q}{\\partial \\varepsilon_{,x}}-\n\\frac{\\partial s}{\\partial \\gamma_{,x}}\\frac{\\partial \\phi}{\\partial \\varepsilon_{,x}}=0,\\\\\n&\\frac{\\partial s}{\\partial v_{,x}}\\frac{\\partial T}{\\partial \\gamma_{,x}}-\n\\frac{\\partial s}{\\partial \\varepsilon_{,x}}\\frac{\\partial q}{\\partial \\gamma_{,x}}-\n\\frac{\\partial s}{\\partial \\gamma_{,x}}\\frac{\\partial \\phi}{\\partial \\gamma_{,x}}=0.\n\\end{aligned}\n\\end{equation}\nThe latter, joined with the conditions obtained by annihilating the coefficients of the linear terms  in the higher derivatives, provide the restrictions on the constitutive functions. These conditions can be solved so that we are able to provide an explicit solution that is proved to satisfy the remaining restrictions expressed as inequalities.\nTo proceed further, we start by taking the specific entropy as the sum of an equilibrium term and a non-equilibrium part expressed as a  \nquadratic form in the gradients entering the state space; this quadratic part must be semidefinite negative in order to verify the principle of maximum entropy at equilibrium. By using some routines written in the Computer Algebra System Reduce \\cite{Reduce}, we obtain the following solution to all the  thermodynamic restrictions:\n\\begin{equation}\\label{entr_1}\n\\begin{aligned}\ns&=s_0(\\rho,\\varepsilon)+s_1(\\rho,\\gamma)\\rho_{,x}^2,\\\\\nT&=\\rho^2\\frac{\\partial s_0}{\\partial \\rho}\\left(\\frac{\\partial s_0}{\\partial\\varepsilon}\\right)^{-1}\n+\\tau_1(\\rho,\\varepsilon,\\gamma)v_{,x}\\\\\n&+\\rho^2\\left(\\frac{\\partial s_0}{\\partial\\varepsilon}\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{-1}\n\\left(s_1\\frac{\\partial^2 s_1}{\\partial\\rho\\partial\\gamma}-\\frac{\\partial s_1}{\\partial\\rho}\\frac{\\partial s_1}{\\partial\\gamma}\\right)\\rho_{,x}^2\\\\\n&+\\rho^2\\left(\\frac{\\partial s_0}{\\partial\\varepsilon}\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{-1}\n\\left(s_1\\frac{\\partial^2 s_1}{\\partial\\gamma^2}-2\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^2\\right)\\rho_{,x}\\gamma_{,x},\\\\\nq&= q_1(\\rho,\\varepsilon,\\gamma)\\varepsilon_{,x}+q_2(\\rho,\\varepsilon,\\gamma)\\rho_{,x}+q_3(\\rho,\\varepsilon,\\gamma)v_{,x},\\\\\n\\phi&=\\left(2\\frac{\\partial s_1}{\\partial\\gamma}\\rho_{,x}\\right)^{-1}\n\\left(q_2\\frac{\\partial s_0}{\\partial \\varepsilon}+f(\\rho,\\varepsilon,\\gamma)+2k\\frac{\\partial s_1}{\\partial\\gamma}\\rho_{,x} -2\\rho^2s_1v_{,x}\\right),\\\\\nJ_s&=q\\frac{\\partial s_0}{\\partial\\varepsilon}-\\frac{1}{2}\\left(q_2\\frac{\\partial s_0}{\\partial \\varepsilon}+f(\\rho,\\varepsilon,\\gamma)-2\\rho^2s_1 v_{,x}\\right)\\rho_{,x},\n\\end{aligned}\n\\end{equation}\nwhere the function $s_1(\\rho,\\gamma)$ must be a negative function in order the principle of maximum entropy at the equilibrium be satisfied.\nMoreover, the reduced entropy inequality becomes \n\\begin{equation}\n\\label{reduced1}\n\\begin{aligned}\n&(q_1\\varepsilon_{,x}+q_2\\rho_{,x}+q_3v_{,x})\\left(\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}\\rho_{,x}+\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\varepsilon_{,x}\\right)+\\tau_1\\frac{\\partial s_0}{\\partial\\varepsilon} v_{,x}^2\\\\\n&-\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\left(\\frac{\\partial g(\\rho,\\varepsilon)}{\\partial \\rho}\\rho_{,x}+\\frac{\\partial g(\\rho,\\varepsilon)}{\\partial \\varepsilon}\\varepsilon_{,x}\\right)\\rho_{,x}\\geq 0,\n\\end{aligned}\n\\end{equation}\nalong with the constraint\n\\begin{equation}\nq_2\\frac{\\partial s_0}{\\partial\\varepsilon}+f(\\rho,\\varepsilon,\\gamma)-\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}g(\\rho,\\varepsilon)=0,\n\\end{equation}\nwhere $f(\\rho,\\varepsilon,\\gamma)$ and $g(\\rho,\\varepsilon)$ are arbitrary functions of their arguments.\n\nThe inequality \\eqref{reduced1} is satisfied if and only if the following conditions hold true:\n\\begin{equation}\n\\label{diffconstrgen}\n\\begin{aligned}\n&q_1\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\geq 0,\\qquad\\tau_1\\frac{\\partial s_0}{\\partial\\varepsilon} \\geq 0,\\\\\n&q_2\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}-\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\frac{\\partial g}{\\partial \\rho}\\geq 0,\\qquad q_3^2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_1\\geq 0,\\\\\n&\\left(q_3^2\\frac{\\partial^2 s_0}{\\partial\\rho \\partial\\varepsilon}-4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_2\\right)\\frac{\\partial^2 s_0}{\\partial\\rho \\partial\\varepsilon}+4\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\frac{\\partial s_0}{\\partial\\varepsilon}\\frac{\\partial g}{\\partial \\rho}\\tau_1\\leq 0,\\\\\n&\\left(q_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-q_1\\frac{\\partial^2 s_0}{\\partial\\rho \\partial\\varepsilon}\\right)^2+4\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\frac{\\partial g}{\\partial\\rho} q_1+\\frac{\\partial s_1}{\\partial\\gamma}\\left(\\frac{\\partial g}{\\partial\\varepsilon}\\right)^2\\\\\n&-2\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\left(q_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}+q_1\\frac{\\partial^2 s_0}{\\partial\\rho \\partial\\varepsilon}\\right)\n\\frac{\\partial g}{\\partial\\varepsilon}\\leq 0,\\\\\n&\\left(q_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-q_1\\frac{\\partial^2 s_0}{\\partial\\rho \\partial\\varepsilon}\\right)^2\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1-\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\left(q_3^2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_1\\right)\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\frac{\\partial g}{\\partial\\rho}\\\\\n&-\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\left(q_3^2\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-2\\left(q_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}+q_1\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}\\right)\\frac{\\partial s_0}{\\partial \\varepsilon}\\tau_1\\right)\\frac{\\partial g}{\\partial \\varepsilon}\\\\\n&+\\frac{\\partial s_1}{\\partial \\gamma}\\frac{\\partial s_0}{\\partial\\varepsilon}\\left(\\frac{\\partial g}{\\partial\\varepsilon}\\right)^2 \\tau_1\\leq 0.\n\\end{aligned}\n\\end{equation}\n\nThe solution so recovered contains some degrees of freedom that can be fixed in order to model specific physical situations. The constitutive equation \\eqref{entr_1}$_1$ can be interpreted as an extension of the equilibrium constitutive equation to non-equilibrium  situations \\cite{Muschik-Ehrentraut}.\nIn what follows, we limit ourselves to discuss in detail the case where $g(\\rho,\\varepsilon)=0$, whereupon the constraints \n\\eqref{diffconstrgen} simplify as\n\\begin{equation}\n\\label{lastrestr1}\n\\begin{aligned}\n&q_2\\frac{\\partial^2 s_0}{\\partial \\rho \\partial\\varepsilon}\\geq 0,\\qquad q_1\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\geq 0,\\qquad \\tau_1\\frac{\\partial s_0}{\\partial\\varepsilon}\\geq 0,\\\\\n&\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}\\left(4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_2-q_3^2\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}\\right)\\geq 0,\\\\\n&\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\left(4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_1-q_3^2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\right)\\geq 0,\n\\end{aligned}\n\\end{equation}\ntogether with \n\\begin{equation}\n\\label{relationq1q2}\nq_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-q_1\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}=0.\n\\end{equation}\n\nSome comments about the constitutive relations so characterized are in order. \nIn equilibrium situations, in which the gradients of the field variables (except at most the density $\\rho$) vanish, let us define the absolute temperature $\\theta$ by the classical thermodynamical relation $\\displaystyle \\frac{1}{\\theta}=\\frac{\\partial s_0}{\\partial\\varepsilon}$. \nUnder the hypothesis of invertibility of \n$\\theta$ with respect to $\\varepsilon$, which is allowed by the positivity of the specific heat \n$\\displaystyle c = \\frac{\\partial \\varepsilon}{\\partial\\theta}$, the internal energy $\\varepsilon$ can be\nexpressed as function of the arguments $\\rho$ and $\\theta$. Thus,  differentiating with respect to $\\rho$ the\ncondition\n\\[\n\\frac{\\partial s_0(\\rho,\\varepsilon(\\rho,\\theta))}{\\partial \\varepsilon}-\\frac{1}{\\theta}=0,\n\\]\nwe get\n\\begin{equation}\n\\label{sign}\n\\frac{\\partial^2 s_0}{\\partial\\rho\\partial\\varepsilon}+\\frac{\\partial^2 s_0}{\\partial \\varepsilon^2}\\frac{\\partial\\varepsilon}{\\partial\\rho}=0,\n\\end{equation}\nthat, used in \\eqref{relationq1q2}, provides\n\\begin{equation}\nq_2=-q_1\\frac{\\partial\\varepsilon}{\\partial\\rho}.\n\\end{equation}\n\nThus, the heat flux reduces to\n\\begin{equation}\nq = q_1\\frac{\\partial\\varepsilon}{\\partial\\theta}\\theta_{,x}+q_3v_x,\n\\end{equation}\n\\emph{i.e.}, when $q_3=0$, we have the classical Fourier law of heat conduction. \n\nSince \n\\[\n\\frac{\\partial s_0}{\\partial\\varepsilon}=\\frac{1}{\\theta}>0, \\qquad\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}=-\\frac{1}{\\theta^2}\\frac{\\partial\\theta}{\\partial\\varepsilon}< 0,\n\\]\nit is $\\tau_1\\geq 0$, and, as physics prescribes,  $q_1\\leq 0$.\nAs far as $q_2$ is concerned, its sign is the same as that of $\\displaystyle\\frac{\\partial\\varepsilon}{\\partial\\rho}$, which in turn, from \\eqref{sign}, has the same sign as  \n$\\displaystyle \\frac{\\partial^2 s_0}{\\partial\\rho\\partial\\varepsilon}$; thus, due to $\\displaystyle\\frac{\\partial\\varepsilon}{\\partial\\rho}\\geq 0$, it is $q_2\\geq 0$. \nFinally, the last two inequalities in \\eqref{lastrestr1} provide\n\\begin{equation}\n\\label{onlyone}\nq_3^2\\leq 4 \\left(\\frac{\\partial^2 s_0}{\\partial \\varepsilon^2}\\right)^{-1}\\frac{\\partial s_0}{\\partial\\varepsilon}q_1\\tau_1.\n\\end{equation}\nLast, but not the least, it is worth of being observed that the entropy flux contains the classical term $\\displaystyle \\frac{q}{\\theta}$  and an additional term (extra-flux) coming from the application of the procedure  without the need of postulating it at the beginning. Finally, at equilibrium the flux $\\phi$ of the internal variable $\\gamma$ reduces to a constant, and\nthe stress tensor $T$ assumes the classical local form if the mass density is constant. \n\n\\subsection{Korteweg fluids}\nHere, we consider the case of a fluid of grade 3 \\cite{Truesdell_Noll}; in fact,  we include in the state space the second spatial derivative of the mass density \\cite{Dunn-Serrin}. For this class of fluids, Korteweg \n\\cite{Korteweg} proposed the Cauchy stress tensor to be given by a constitutive equation like\n\\begin{equation} \n\\label{kort}\nT_{ij}=\\left(-p+\\sum_{k=1}^3\\left(\\alpha_1\\frac{\\partial^2\\rho}{\\partial x_k^2}+\\alpha_2\\frac{\\partial\\rho}{\\partial x_k}\n\\frac{\\partial\\rho}{\\partial x_k}\\right)\\right)\\delta_{ij}+\\alpha_3\\frac{\\partial\\rho}{\\partial x_i}\n\\frac{\\partial\\rho}{\\partial x_j}+\\alpha_4\\frac{\\partial^2\\rho}{\\partial x_i\\partial x_j},\n\\end{equation}\nwhere $\\rho$ denotes the mass density, $p$  the pressure of the fluid, and $\\alpha_i$, $i=1,\\ldots,4$,  \nsuitable material functions depending on density and temperature. These fluids received a moderate attention in the \nliterature after the pioneering paper by Dunn and Serrin \\cite{Dunn-Serrin}, where the compatibility with the basic tenets \nof rational continuum thermodynamics \\cite{Truesdell} has been extensively studied. They have been studied also in \n\\cite{CST-JMP-2009,COT-JMP-2011,CST-JNET-2010,COP-Elasticity-2011} \nby means of an extended Liu procedure, and by \nHeida and M\\'alek \\cite{Heida-Malek} following a different methodology. \n\nBy limiting to the one-dimensional case, the balance equations read \n\\begin{equation}\n\\label{korteweg}\n\\begin{aligned}\n&\\mathcal{E}_1\\equiv\\frac{D\\rho}{D t} + \\frac{D (\\rho v)}{D x}=0, \\\\\n&\\mathcal{E}_2\\equiv\\frac{D(\\rho v)}{D t} + \\frac{D (\\rho v^2  - T)}{D x}=0,\\\\\n&\\mathcal{E}_3\\equiv\\frac{D}{D t}\\left(\\rho\\varepsilon+\\rho \\frac{v^2}{2}\\right) +\\frac{D }{D x}\\left(\\rho v\\varepsilon+\\rho \\frac{v^3}{2}-Tv+ q\\right)=0,\n\\end{aligned}\n\\end{equation}\nwhere $\\rho$ is the mass density, $v$ the velocity, and $\\varepsilon$ the internal energy per unit mass;\nmoreover, the stress $T$ and the heat flux $q$ must be assigned by means of constitutive equations.\nThe constitutive equations must be such that for every admissible process the entropy inequality \n\\begin{equation}\n\\rho\\left(\\frac{Ds}{D t}+v\\frac{Ds}{Dx}\\right)+ \\frac{DJ_s}{Dx} \\ge 0,\n\\end{equation}\nbeing $s$ the specific entropy and $J_s$ the entropy flux, is satisfied.\n\nLet us assume the state space spanned by\n\\begin{equation}\n\\mathcal{Z}\\equiv\\{\\rho, \\varepsilon, \\rho_{,x}, \\varepsilon_{,x}, v_{,x}, \\rho_{,xx}\\}.\n\\end{equation}\n\nIn this case, the exploitation of second law of thermodynamics requires that we take into account the constraints \nimposed on the thermodynamic processes by the balance equations together with their first and second order spatial derivatives; \nnevertheless, we observe that, since the unique second order spatial derivative belonging to the state space is $\\rho_{,xx}$, the unique second order extension we need to use as constraint is $\\displaystyle\\frac{D^2\\mathcal{E}_1}{Dx^2}$.\nThus, the entropy inequality writes:\n\\begin{equation}\n\\label{entropyconstrainedkorteweg}\n\\begin{aligned}\n&\\rho\\left(\\frac{Ds}{D t}+v\\frac{Ds}{Dx}\\right)+ \\frac{DJ_s}{Dx} \\\\\n&\\quad- \\Lambda^{(0)}_1 \\mathcal{E}_1- \\Lambda^{(0)}_2 \\mathcal{E}_2- \\Lambda^{(0)}_3 \\mathcal{E}_3\\\\\n&\\quad-\\Lambda^{(1)}_1\\frac{D\\mathcal{E}_1}{Dx}-\\Lambda^{(1)}_2\\frac{D\\mathcal{E}_2}{Dx}\n-\\Lambda^{(1)}_3\\frac{D\\mathcal{E}_3}{Dx}-\\Lambda^{(2)}_1\\frac{D^2\\mathcal{E}_1}{Dx^2} \\geq 0.\n\\end{aligned}\n\\end{equation}\n\nBy expanding the derivatives in \\eqref{entropyconstrainedkorteweg}, we obtain a huge expression that we omit to report here;  \nit results linear in the  highest derivatives, say\n\\[\n\\{\\rho_{,t},v_{,t},\\varepsilon_{,t},\\rho_{,tx},v_{,tx},\\varepsilon_{,tx},\\rho_{,txx},v_{,txx},\\varepsilon_{,txx},v_{,xxxx},\\varepsilon_{,xxxx},\\rho_{,xxxxx}\\},\n\\]\nand cubic in  the higher derivatives, say\n\\[\n\\{v_{,xx},\\varepsilon_{,xx},\\rho_{,xxx},v_{,xxx},\\varepsilon_{,xxx},\\rho_{,xxxx}\\}.\n\\]\n\nTo proceed further, let us write the specific entropy as the sum of the equilibrium part  defined for homogeneous states and\na semidefinite negative quadratic form (in order to satisfy the principle of maximum entropy at the equilibrium) in the gradients appearing in the state space.  At this stage we do not specify the form of the entropy flux.\n\nThe restrictions imposed by entropy inequality can be solved and provide the following solution:\n\\begin{equation}\n\\begin{aligned}\\label{entr_gen}\n&s = s_0(\\rho,\\varepsilon)+s_1(\\rho)\\rho_{,x}^2,\\\\\n&q = q_1(\\rho,\\varepsilon)\\varepsilon_{,x}+q_2(\\rho,\\varepsilon)\\rho_{,x}+q_3(\\rho,\\varepsilon)v_{,x},\\\\\n&T = \\rho^2\\left(\\frac{\\partial s_0}{\\partial \\varepsilon}\\right)^{-1}\\left(\\frac{\\partial s_0}{\\partial \\rho}-\\frac{\\partial s_1}{\\partial \\rho}\\rho_{,x}^2-2 s_1\\rho_{,xx}\\right)+\\tau_1(\\rho,\\varepsilon)v_{,x},\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{constrkdv}\n\\begin{aligned}\n&q_1\\le 0, \\qquad q_2\\ge 0, \\qquad \\tau_1\\ge 0,\\\\\n&q_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-q_1\\frac{\\partial^2 s_0}{\\partial\\rho\\partial\\varepsilon}=0,\\\\\n&4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_2-q_3^2\\frac{\\partial^2 s_0}{\\partial\\rho\\partial\\varepsilon}\\geq 0,\\\\\n&4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_1-q_3^2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\leq 0,\n\\end{aligned} \n\\end{equation}\nand $s_1\\leq 0$ in order the principle of maximal entropy at the equilibrium be satisfied.\nFinally, the entropy flux turns out to be\n\\begin{equation}\n\\begin{aligned}\\label{flux_entr}\nJ_s&=\\frac{\\partial s_0}{\\partial\\varepsilon}\\left(q_1\\varepsilon_{,x}+q_2\\rho_{,x}+q_3v_{,x}\\right)+2\\rho^2 s_1\\rho_{,x}v_{,x}=\\\\\n&=\\frac{q}{\\theta}+2\\rho^2 s_1\\rho_{,x}v_{,x},\n\\end{aligned}\n\\end{equation}\nwhere $\\theta$, defined as usual as\n\\begin{equation}\n\\frac{1}{\\theta}=\\frac{\\partial s_0}{\\partial\\varepsilon},\n\\end{equation}\nis the absolute temperature.\n\nLet us observe that, in the expression \\eqref{flux_entr}  of entropy flux, the second contribution  represents  the  entropy extra-flux \\cite{Muller}, related to the gradients of density and velocity.\nMoreover, a similar reasoning as in previous subsection shows that it is $\\displaystyle q_2=-q_1\\frac{\\partial\\varepsilon}{\\partial\\rho}$ so that, choosing $q_3=0$, we recover the classical Fourier law for heat flux. Also, the last two inequalities \nin \\eqref{constrkdv} provide relation \\eqref{onlyone}.\nFinally, if we consider the classical case, \\emph{i.e.}, $s = s_0(\\rho,\\varepsilon)$, the stress tensor $T$ is expressed in local form, and the classical constitutive equation for the entropy flux $\\displaystyle J_s=\\frac{q}{\\theta}$ is recovered.\n\nTherefore,  in order to obtain a constitutive equation for the stress tensor containing first and second order derivatives of mass density we need to assume that $s$ depends on the gradient of $\\rho$, and the entropy flux involves an extra-flux.\nAs a last remark, a stress tensor depending on the gradients of the density can be obtained only if we use the extended\nprocedure for the exploitation of entropy inequality \\cite{CGOP-miscele-2020}.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this paper, we discussed the extended Liu procedure in order to investigate the restrictions placed by an entropy inequality\non the constitutive equations of a continuum whose state space contains spatial derivatives of the unknown fields. The analysis is\nperformed first from a purely mathematical viewpoint by considering a system of balance laws sufficiently general to contain\nthe governing equations of continua, and sufficient conditions for the fulfilment of an entropy-like inequality have been derived. Then, we specialized the results to two physical cases with first or second order  non-local \nconstitutive equations. Remarkably, in the application of the procedure we do not modify the energy balance equation with the  inclusion of extra-terms (like the interstitial  working), and we do not prescribe \\emph{a priori} the form of entropy flux, whose expression arises as a result of\nthe method; in the applications we considered, the procedure provides an entropy flux decomposed in the classical term and an extra-flux.\n\nWe limited ourselves to one-dimensional models, and in the considered physical applications we were able to solve the\nconstraints imposed by the exploitation of the entropy inequality, so determining an explicit expression of the constitutive functions\nby assuming  an expansion of the specific entropy at first order in the squared gradients of the field variables entering the state space.\n\nThe procedure in principle allows us to fix the constitutive equations (for stress tensor, heat flux, \\ldots), according to experiments, and determine the form of the entropy flux algorithmically. \nIt is trivial to observe that the procedure requires a huge amount of computation, increasing with the order of non-localities;\nhowever, such computations can be almost automatically carried out by using a Computer Algebra System.\n\nWork is in progress about the investigation of the extended Liu procedure for multi-dimensional continuous media, where\nother general principles of representation theory \\cite{Smith3} of vectorial and tensorial quantities need to be considered. \n\n\\section*{Acknowledgments}\nThe authors acknowledge partial supported by G.N.F.M. of ``Istituto Nazionale di Alta Matematica'' and University of Messina. The authors gratefully thank dr. Luca Amata for drawing their attention to the paper \\cite{Mishkov}.\n \\medskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzbhhd b/data_all_eng_slimpj/shuffled/split2/finalzzbhhd
new file mode 100644
index 0000000000000000000000000000000000000000..76ed727034a35cf0491c3fc2df5cf94cde9464c1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzbhhd
@@ -0,0 +1,5 @@
+{"text":"\\chapter{Introduction} \n\n\n\n\nLet $(M, \\omega)$ be a closed symplectic manifold. The group $\\Symp(M, \\omega)$ of symplectomorphisms, equipped with the standard $C^\\infty$-topology, is an infinite dimensional Fr\u00e9chet Lie group. A fundamental theorem of Banyaga~\\cite{Banyaga-Isomorphism} states that the isomorphism type of $\\Symp(M,\\omega)$ as a discrete group determines the conformal symplectomorphism type of $(M,\\omega)$. However, the ways topological properties of $\\Symp(M,\\omega)$ relate to the geometry of $(M,\\omega)$ remain largely unknown. To measure our level of ignorance, it suffices to observe that the following two questions are almost completely open in all dimensions $2n \\geq 4$:\n\n\\begin{enumerate}[label=\\textbf{Q\\arabic*.}]\n\\item At the most basic level, how does the homotopy type of $\\Symp(M,\\omega)$ depend on $\\omega\\,$? \n\\item Similarly, suppose $(M,\\omega)$ admits a symplectic or Hamiltonian actions by a compact Lie group $G$ (possibly finite), viewed here as a Lie subgroup $G\\subset\\Symp(M,\\omega)$. What are the homotopy types of the centralizer and of the normalizer $N(G)$ in $C(G)\\subset\\Symp(M,\\omega)\\,$? Can we understand the inclusions \n\\[G\\hookrightarrow \\Symp(M,\\omega)\\quad \\text{~and~}\\quad C(G)\\hookrightarrow N(G)\\hookrightarrow \\Symp(M,\\omega)\\]\nfrom a homotopy theoretic point of view?\n\n\\end{enumerate}\n\n\n\nRegarding the first question, most efforts as been devoted to the study of symplectomorphism groups of rational $4$-manifolds. Following the seminal work of M. Gromov~\\cite{Gr} who showed that the group of compactly supported symplectomorphisms of $\\mathbb{R}^4$ is contractible, the homotopical properties of the group of symplectomorphisms of $\\mathbb{C}P^2$, $S^2 \\times S^2$ and of the $k$-fold symplectic blow-ups $\\mathbb{C}P^2\\#k\\overline{\\mathbb{C}P}^2$, $k\\leq 5$, were studied in several papers such as \\cite{abreu}, \\cite{AGK}, \\cite{MR1775741}, \\cite{P-com}, \\cite{AG}, \\cite{AP}, \\cite{AE}, and~\\cite{LiLiWu2022}. In particular, for $\\mathbb{C}P^2$, $S^2 \\times S^2$, and $\\mathbb{C}P^2\\#k\\overline{\\mathbb{C}P^2}$, $k\\leq 3$, the rational homotopy type of $\\Symp(M,\\omega)$ can be described precisely in terms of the cohomology class $[\\omega]$. For $k\\geq 4$, partial results are known, mostly for $\\pi_0(\\Symp(M,\\omega))$ and $\\pi_1(\\Symp(M,\\omega))$.\nIt is worth pointing out that all these results rely on  special properties of $J$-holomorphic curves in symplectic $4$-manifolds and, as such, do not generalize readily to higher dimensions.\\\\ \n\nThe second question can be partially answered in the case of Hamiltonian toric actions using moment map techniques, see~\\cite{P-MaxTori}. If a torus $\\mathbb{T}^n\\subset\\Ham(M^{2n},\\omega)$ acts effectively on $(M^{2n},\\omega)$ with moment map $\\mu:M^{2n}\\to\\mathbb{R}^n$, then \n\n\\begin{enumerate}\n\\item the centralizer $C(\\mathbb{T}^n)$ is equal to the group of all symplectomorphisms $\\phi$ that preserve the moment map, that is, such that $\\mu \\circ \\phi=\\mu$.\n\\item $C(\\mathbb{T}^n)$ is a maximal torus in $\\Symp(M^{2n},\\omega)$, that is, a maximal, connected, and abelian subgroup of $\\Symp(M^{2n},\\omega)$. In particular, since toric manifolds are simply-connected, $C(\\mathbb{T}^n) \\subset\\Ham(M^{2n},\\omega)$. \n\\item $C(\\mathbb{T}^n)$ deformation retracts onto $\\mathbb{T}^{n}$. In particular, the homotopy type of the centraliser of a toric action is independent of the action.\n\\item Moreover, the Weyl group $W(\\mathbb{T}^n):=N(\\mathbb{T}^n)\/C(\\mathbb{T}^n)$ is always finite\\footnote{Futhermore, as for maximal tori in compact Lie groups, it can be shown that the number of conjugacy classes of toric centralizers is finite, and that each   $C(\\mathbb{T}^n)$ is flat and totally geodesic in $\\Symp(M^{2n},\\omega)$ for the $L^2$  metric}.\n\\end{enumerate}\nOn the other hand, even for toric actions, very little is known about the homotopy theoretic properties of the inclusions $G\\hookrightarrow\\Symp(M,\\omega)$ or $C(G)\\hookrightarrow N(G)\\hookrightarrow\\Symp(M,\\omega)$. Two noteworthy exceptions are i) the works of McDuff-Slimowitz~\\cite{McDS} and McDuff-Tolman~\\cite{McDT} who show that under some rather mild conditions, Hamiltonian $G$ actions induce injective maps $\\pi_1(G)\\hookrightarrow\\pi_1(\\Ham(M^{2n},\\omega))$, and ii) the results on symplectomorphism groups of rational surface mentionned above that, as a byproduct, allow one to understand the subrings of $\\pi_*(\\Symp(M,\\omega))$ or $H_*(\\Symp(M,\\omega))$ that the various Hamiltonian actions on $(M,\\omega)$ generate.\\\\\n\n\n\nIn this paper we combine pseudo-holomorphic curve techniques with moment map techniques to determine the homotopy type of equivariant symplectomorphisms of $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ under the presence of a Hamiltonian circle action. Our main result are Theorem~\\ref{full_homo} and Theorem~\\ref{full_homo_CCC} that give the full homotopy type of the centralizer $C(S^1)\\subset\\Symp(M,\\omega)$ for all choices of symplectic forms and of Hamiltonian circle actions on these two $4$-manifolds. Contrary to the toric case, the homotopy type of the stabilizer $C(S^1)$ is not constant and depends, essentially, on whether the circle action extends to a single toric action or to two distinct toric actions. In the former case, apart from a few exceptional circle actions that must be treated separately, $C(S^1)$ retracts onto the unique torus $\\mathbb{T}^2$ the circle extends to. In the latter case, the two toric actions $\\mathbb{T}^2_1$, $\\mathbb{T}^2_2$ that extend the circle action to do not commute, even up to homotopy. We show that the Pontryagin products of the generators of $H_1(\\mathbb{T}^2_1)$ and $H_1(\\mathbb{T}^2_2)$ generate a subalgebra $P^{alg}\\subset H_*(C(S^1))$ that contains classes of arbitrary large degrees. In particular, $C(S^1)$ does not have the homotopy type of a finite dimensional $H$-space. Moreover, looking at the action of the centralizer $C(S^1)$ on the space of invariant almost complex structures, we prove that there is an equality of Pontryagin rings $P^{alg}= H_*(C(S^1))$. This homology equivalence implies that $C(S^1)$ has the homotopy type of a pushout $\\mathbb{T}^2_1 \\leftarrow S^1\\to \\mathbb{T}^2_2$ in the category of topological groups. Finally, when viewed as a bare topological space, we show that $C(S^1)$ is homotopy equivalent to the product $\\Omega S^3 \\times S^1 \\times S^1 \\times S^1$.\\\\\n\nIt seems likely that our results on centralizers of Hamiltonian circle actions could be proven using moment map techniques alone. The main advantage of introducing pseudo-holomorphic curve techniques is that our setting generalises to actions of any compact, possibly finite, abelian group $A\\subset\\Ham(M,\\omega)$. For instance, in the companion paper~\\cite{Zn_symp}, we use the same framework to determine the homotopy type of the centralizers of most finite cyclic subgroups $\\mathbb{Z}_n$ acting on $(S^2 \\times S^2,\\omega_\\lambda)$ and $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ by Hamiltonian diffeomorphisms.\\\\\n\nIn an effort to make the paper understandable to both equivariant geometers and symplectic topologists, we include more details than would be needed in a document aimed at either audience alone. The paper is structured as follows:\\\\\n\nIn Chapter 2, we recall T. Delzant's equivariant classification of toric actions in terms of moment polytopes, and Y. Karshon's equivariant classifications of Hamiltonian circle actions on $4$-manifolds in terms of labelled graphs. We then determine all possible toric extensions, up to equivalences, of an arbitrary Hamiltonian circle actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$.\\\\ \n\nThe crux of the paper lies in Chapters 3 and 4 in which we adapt the framework of \\cite{MR1775741} to study groups of equivariant symplectomorphisms in the presence of an $S^1$ action. In particular we show that the space of invariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ decomposes into disjoint strata, each of them being homotopy equivalent to an orbit of the centralizer, with stabilizer homotopy equivalent to $S^1$ equivariant K\\\"ahler isometries (see Theorems~\\ref{homogenous} and \\ref{homog}).  In Chapter 3, we show that the number of invariant strata in the decomposition corresponds to the number of toric extensions of the given circle action (Proposition~\\ref{prop:ToricExtensionCorrespondance}). In particular we prove that $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ decomposes into either one or two strata, and that the later case occurs only for an exceptional family of circle actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ (Corollaries~\\ref{cor:lambda=1_IntersectingOnlyOneStratum},\\ref{cor:IntersectingTwoStrata} and \\ref{cor:IntersectingOnlyOneStratum}). In Chapter 4, using techniques similar to the ones developed in~\\cite{AG} we compute the homotopy type of $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ for all Hamiltonian circle actions on $(S^2 \\times S^2,\\omega_\\lambda)$. \n\\\\\n\nIn Chapter 5, we prove $S^1$ equivariant analogues of some technical lemmas of~\\cite{AGK} involving deformation theory. We use these results to prove that in the case the circle admits two distinct toric extensions, the stratum with positive codimension in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is always of codimension two. \n\\\\\n\nFinally in Chapter 6, we carry out a similar analysis on the manifold $\\CP^2\\# \\overline{\\CP^2}$ and obtain the homotopy type of $\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ for all Hamiltonian circle actions on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. \n\\\\\n\n\n\n\n\n\n\\chapter{Hamiltonian torus actions on ruled $4$-manifolds}\n\n\\section{Preliminaries on Hamiltonian actions}\nLet $G$ be a compact Lie group, possibly finite, acting effectively and symplectically on a symplectic manifold $(M,\\omega)$. Every such action $\\rho:G\\times M\\to M$ induces an injective homomorphism $G\\hookrightarrow\\Symp(M,\\omega)$ and, in particular, defines a subgroup $G\\subset\\Symp(M,\\omega)$. Let $\\mathfrak{g}$ denote the Lie algebra of $G$ and $\\mathfrak{g}^*$ be it's dual. Given $Y\\in \\mathfrak{g}$, we denote by $\\overline{Y}$ the corresponding fundamental vector field on~$M$.  \n\n\\begin{defn}\\label{def:HamiltonianActions} The action of $G$ is called Hamiltonian if\n\\begin{itemize}\n\\item $G$ is finite and the induced homomorphism $G\\to\\Symp(M,\\omega)$ takes values in the subgroup of Hamiltonian diffeomorphisms $\\Ham(M,\\omega)$, or\n\\item $\\dim G\\geq 1$ and there exists a moment map, that is, a smooth map $\\mu:M \\to \\mathfrak{g}^*$ such that \n\\begin{enumerate}\n    \\item $d\\mu_p(X_p)(Y) = \\omega(X,\\overline{Y})$ for all $X \\in T_pM$ and $Y \\in \\mathfrak{g}$, and \n    \\item the map $\\mu$ is equivariant with respect to the $G$ action on $M$ and the coadjoint action on~$\\mathfrak{g}^*$.\n\\end{enumerate}\nIn particular, $G\\subset\\Ham(M,\\omega)\\subset\\Symp(M,\\omega)$. \n\\end{itemize} \n\\end{defn}\n\n\\begin{remark}\\label{rmk:UniquenessMomentMap}\nWe note that in this definition, the moment map $\\mu:M\\to\\mathfrak{g}^{*}$ is an auxiliary structure whose sole existence is required. There is no claim about its unicity. For instance, the moment map of a Hamiltonian torus action is only defined up to adding a constant $c\\in\\mathfrak{t}^*$. However, there are situations where the choice of a specific moment map is needed. For this reason, we will always distinguish between the structure $(M,\\omega,\\rho)$ given by a Hamiltonian action alone, from the structure $(M,\\omega,\\rho,\\mu)$ in which a moment map $\\mu$ is chosen.\n\\end{remark}\n\n\\begin{defn}\\label{defn:EquivalenceActions}\nWe will say that two actions $\\rho_{i}:G\\to\\Symp(M,\\omega)$ are symplectically equivalent if the corresponding subgroups $\\rho_{i}(G)$ belong to the same conjugacy class with respect to the action of $\\Symp(M,\\omega)$.\n\\end{defn}\n\nGiven a symplectic action $\\rho:G\\times M\\to M$, let $C(G)$ be the centralizer of the corresponding subgroup $G\\subset\\Symp(M,\\omega)$, that is,\n\\[C(G)=\\{\\phi\\in\\Symp(M,\\omega)~|~\\phi\\circ g\\circ \\phi^{-1} = g,~\\forall g\\in G\\}\\]\nand let $N(G)$ be its normalizer, namely,\n\\[N(G)=\\{\\phi\\in\\Symp(M,\\omega)~|~\\phi\\circ G\\circ \\phi^{-1} = G\\}\\]\nClearly, equivalent actions have isomorphic centralizers and normalizers. \n\nIn this paper, we will investigate the homotopy type of the centralizers $C(T)$ of Hamiltonian torus actions on some $4$-manifolds. We start by giving a simple characterization of symplectomorphisms commuting with a Hamiltonian torus action.\n\\begin{lemma}\\label{lemma:CharacterizationCentralizer}\nLet $(M, \\omega)$ be a compact symplectic manifold equipped with a Hamiltonian torus action $\\rho:T\\times M\\to M$ with moment map $\\mu$. Then $\\phi \\in C(T)$ if, and only if, $\\mu \\circ \\phi = \\mu$ and $\\phi \\in \\Symp(M,\\omega)$.\n\\end{lemma}\n\\begin{proof}\n$(\\Leftarrow)$ Suppose $\\phi \\in \\Symp(M,\\omega)$ satisfies $\\mu \\circ \\phi = \\mu$. Let $X \\in\\mathfrak{t}$ and let $\\overline{X}$ denote the associated fundamental vector field. We then have\n\\[\\omega(d\\phi^{-1}(\\overline{X}),Y) = \\phi^*\\omega(\\overline{X}, d\\phi(Y)) = \\omega(\\overline{X}, d\\phi(Y) =  d\\mu(d\\phi(Y))=d\\mu(Y) = \\omega(\\overline{X}, Y)\\]\nfor any vector field $Y$, which implies $d\\phi(\\overline{X}) = \\overline{X}$ for all $X \\in \\mathfrak{t}$. Consequently, $\\phi$ commutes with the action.\n\\\\\n\n$(\\Rightarrow)$ If the action $\\rho$ has moment map $\\mu$, then $\\mu \\circ \\phi^{-1}$ is a moment map for the conjugate action $\\phi^{-1} \\circ \\rho \\circ \\phi$. If $\\phi^{-1} \\circ \\rho \\circ \\phi = \\rho$, this implies $\\mu \\circ \\phi^{-1} = \\mu + C$ for some constant $C\\in\\mathfrak{t}$. Since the moment images $\\mu(M)$ and $\\mu\\circ \\phi(M)$ are compact and coincide, this constant must be $0$. \n\\end{proof}\n\nThe image of the moment map of a torus action have special convexity properties.\n\n\\begin{thm}(Atiyah-Guillemin-Sternberg)\nLet $(M,\\omega)$ be a symplectic manifold on which a torus $\\mathbb{T}^d$ is acting with moment map $\\mu$. Then the image of $\\mu$ is a convex polytope of $\\mathfrak{t}^*$ whose vertices are images of the fixed points of the torus action.\n\\end{thm}\n\nAs usual, we call an effective Hamiltonian torus action \\emph{toric} if the torus acting is half the dimension of the manifold $M$. By a theorem of Delzant, the moment map image $\\Delta:=\\mu(M)$ of a toric action determines the symplectic manifold $(M,\\omega)$, the action $\\rho$, and the moment map $\\mu$ up to equivariant symplectomorphisms. \n\\begin{thm}(Delzant \\cite{Delzant}) \nLet $(M_{i},\\omega_{i},\\rho_{i},\\mu_{i})$, $i=1,2$, be two toric manifolds of dimension $2n$ with moment maps $\\mu_{i}:M_{i}\\to\\mathfrak{t}^{*}$. If the two moment polytopes $\\mu_{i}(M_{i})$ coincide, then there exists a $\\mathbb{T}^{n}$-equivariant symplectomorphism $\\phi:(M_{1},\\omega_{1})\\to (M_{2},\\omega_{2})$ such that $\\phi^{*}\\mu_{2}=\\mu_{1}$. Conversely, the moment map images of two equivariantly symplectomorphic toric structures $(M_{i},\\omega_{i},\\rho_{i},\\mu_{i})$, $i=1,2$, coincide.\n\\end{thm}\nChoose an identification $\\mathfrak{t}\\simeq\\mathbb{R}^{n}$ such that $\\ker(\\exp:\\mathfrak{t}\\to \\mathbb{T}^{n})\\simeq \\mathbb{Z}^{n}$. The moment polytopes of toric actions on manifolds of dimension $2n$ -- called Delzant polytopes of dimension $n$ -- are completely characterized by the following three properties, see~\\cite{Delzant}:  \n\\begin{itemize}\n\\item (simplicity) there are exactly $n$ edges meeting at each vertex; \n\\item (rationality) the edges meeting at any vertex $p$ are of the form $ p + t u_{i} $, $ t \\in [0,\\ell_i] $, where $ \\ell_i \\in \\mathbb{R}$ and $ u_{i} \\in \\mathbb{Z}^{n} $;\n\\item (smoothness) for each vertex $p$, the corresponding vectors $u_{1},\\ldots,u_{n}$ can be chosen to be a basis of $\\mathbb{Z}^{n}$ over $\\mathbb{Z}$.\n\\end{itemize}\nThis provides a purely combinatorial description of toric structures. Finally, if we are only interested in the subgroup $\\mathbb{T}^n\\subset\\Ham(M^{2n},\\omega)$ associated to a toric action, that is, if we disregard both the pa\\-ra\\-me\\-tri\\-za\\-tion $\\mathbb{T}^n\\hookrightarrow\\Ham(M^{2n},\\omega)$ and the moment map, Delzant's classification theorem yields a bijection\n\\begin{gather*}\n\\{\\text{Inequivalent toric actions on~} 2n\\text{-manifolds}\\}\\\\\n\\updownarrow\\\\\n\\{\\text{Delzant polytopes in~}\\mathbb{R}^n \\text{~up to~} \\AGL(n;\\mathbb{Z}) \\text{~action}\\}\n\\end{gather*}\nAs the next proposition shows, the centralizer and normalizer of symplectic toric actions are easy to describe in terms of moment maps. In particular, the homotopy type of $C(T)$ does not depend on the toric structure.\n\\begin{prop}\\label{prop:CentralizersToricActions}\nConsider an effective toric action $\\rho:\\mathbb{T}^{n}\\to\\Symp(M^{2n},\\omega)$\nwith moment map $\\mu:M\\to\\mathfrak{t}$. Denote by $T$ the corresponding torus\nin $\\Symp(M,\\omega)$, by $C(T)$ its centralizer, by $N(T)$ its normalizer, and by $W(T) =N(T)\/C(T)$ its Weyl group.\n\\begin{enumerate}\n\\item The normalizer $C(T)$ deformation retracts onto $\\mathbb{T}^{n}$. In particular, $C(T)\\subset\\Ham(M,\\omega)$.\n\\item The normalizer $N(T)$ is equal to the group of all symplectomorphisms $\\psi$ such that $\\mu\\circ\\psi = \\Lambda\\circ\\mu$ for some $\\Lambda\\in\\AGL(n;\\mathbb{Z})$. In particular, the Weyl group $W(T)$ is finite.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nThis is a slightly stronger version of Proposition 3.21 in~\\cite{P-MaxTori}, and only the first statement requires some additional justification. By Lemma~\\ref{lemma:CharacterizationCentralizer}, $\\phi\\in C(T)$ iff $\\phi\\in\\Symp(M,\\omega)$ and $\\mu=\\mu\\circ\\phi$. Since the action is toric, each fiber of the moment map consists in a single orbit. It follows that there is a unique map $g_{\\phi}:\\mu(M)=\\Delta\\to\\mathbb{T}^{n}$ such that $\\phi$ can be written as\n\\[\\phi(m) = g_{\\phi}(\\mu(m))\\cdot m\\]\nPick a base point $b\\in\\Delta$ and consider the evaluation fibration\n\\begin{align*}\nC_{b}(T)\\to C(T)&\\to \\mathbb{T}^{n}\\\\\n\\phantom{C_{b}(T)\\to} \\phi&\\mapsto g_{\\phi}(b) \n\\end{align*} \nwhose fiber is the subgroup\n\\[C_{b}(T)=\\{\\phi\\in C(T)~|~\\phi=\\id\\text{~on~}\\mu^{-1}(b)\\}\\]\nConsider $\\phi\\in C_{b}(T)$. Since $\\Delta$ is contractible, the function $g_{\\phi}$ has a unique lift $\\tilde{g}_{\\phi}:\\Delta\\to\\mathfrak{t}$ such that $\\tilde{g}_{\\phi}(b)=0$, and $g_{\\phi}=\\exp\\circ\\tilde{g}_{\\phi}$. Setting $\\tilde{g}_{\\phi,s}=s\\tilde{g}_{\\phi}$ for $s\\in[0,1]$, we obtain a retraction of $C_{b}(T)$ onto $\\{\\id\\}$ through diffeomorphisms $\\phi_{s}$ leaving $\\mu$ invariant and such that $\\phi_{s}=\\id$ on $\\mu^{-1}(b)$. Applying an equivariant Moser isotopy to the family $\\phi_{s}$, this retraction is seen to be homotopic to a retraction in $C_{b}(T)$. This shows that $C_{b}(T)$ is contractible, which in turns implies the result.\n\\end{proof}\nFor Hamiltonian torus actions $T^{d}\\to\\Ham(M^{2n},\\omega)$ for which the torus $T^{d}$ has dimension $d<n$, there is no simple description of the homotopy type of the symplectic centralizer $C(T)$. As we will see, this already happens in the simplest case, namely, for Hamiltonian circle actions on $4$-manifolds. \n\nIn \\cite{Karshon}, Y. Karshon established an equivariant classification of Hamiltonian circle actions on 4-dimensional symplectic manifolds, similar to Delzant classification, in which the moment map image is replaced by labelled graphs. More precisely, given any Hamiltonian $S^1$ action on a 4-manifold $(M,\\omega)$ with moment map $\\mu$, one can construct a labelled graph as follows:\n\\begin{itemize}\n\\item Each component of the fixed point set corresponds to a unique vertex of the graph.\n\\item Each vertex is labeled by the value of the moment map on the corresponding\nfixed point component. \n\\item An extremal vertex that corresponds to a fixed symplectic\nsurface S is said to be \"fat\". Two additional labels are attached to fat vertices: the genus of that surface,\nand its normalized symplectic area. \n\\item Two vertices are connected by an edge if and only if the corresponding isolated\nfixed points are connected by a $\\mathbb{Z}_k$-sphere i.e by a $S^1$ invariant sphere on which  $S^1$ acts with a global stabilizer $\\mathbb{Z}_k$, $k \\geq 2$.\n\\item  Each edge is labelled by the isotropy weight $k \\geq 2$ of the corresponding $\\mathbb{Z}_k$ sphere.\n\\end{itemize}\nLabeled graphs associated to effective Hamiltonian circle actions are characterized by the following properties. If we order the vertices according to their moment map labels, then\n\\begin{itemize}\n\\item there are exactly two extremal vertices;\n\\item fat vertices are extremal, and if the graph contains two fat vertices, then their genus label must coincide;\n\\item the area label of any fat vertex must be strictly positive;\n\\item a vertex is connected to no more than two edges, and no edge is connected to a fat vertex;\n\\item the moment map labels must be strictly monotone along each chain of edges;\n\\item if $e_{1},\\ldots,e_{\\ell}$ is a chain of edges, and if $k_{1},\\ldots,k_{\\ell}$ are the orders of their stabilizers, then $\\gcd(k_{i}, k_{i+1}) = 1$ for $i = 1,\\ldots\\ell-1$, and $(k_{i-1} + k_{i+1})\/k_{i}$ is an integer for $i= 2,\\ldots,\\ell-1$.\n\\end{itemize}\nWe call such graphs \\emph{admissible}. Just as Delzant polytopes classify toric structures up to symplectomorphisms, admissible labelled graphs classify Hamiltonian $S^1$ actions with moment maps.\n\n\\begin{thm}(Karshon \\cite{Karshon})\\label{graphiso}\nEach admissible labelled graph determines a Hamiltonian circle action $\\rho$ with moment map $\\mu$ on a symplectic $4$ manifold $(M,\\omega)$. The tuple $(M,\\omega,\\rho,\\mu)$ is unique up to $S^1$-equivariant symplectomorphisms preserving the moment map. Conversely, the labelled graphs of two equivariantly symplectomorphic Hamiltonian circle actions with moment maps are isomorphic. \n\\end{thm}\nAgain, if one is only interested in conjugacy classes of circles in Hamiltonian groups, Karshon's classification implies that there is a bijection\n\\begin{gather*}\n\\{\\text{Inequivalent Hamiltonian circle actions on~} 4\\text{-manifolds}\\}\\\\\n\\updownarrow\\\\\n\\{\\text{Admissible labelled graphs up to~} \\AGL(1;\\mathbb{Z}) \\text{~action on moment map labels}\\}\n\\end{gather*}\n\n\\begin{remark}\\label{rmk:NormalizationMomentMap}~\n\\begin{itemize}\n\\item Note that when $M$ is compact, it is always possible to canonically assign a moment map to an Hamiltonian circle action by imposing one of the normalizations $\\min \\mu = 0$ or $\\int_{M}\\mu = 0$. \n\\item Note also that in Karshon's classification, it is not necessary to keep track of invariant spheres on which the restricted action is effective. In particular, these spheres do not appear in the labelled graphs. This point will be important in our analysis.\n\\end{itemize}\n\\end{remark}\n\nThe symplectic slice theorem implies that a Hamiltonian circle action on a $4$-manifold has a simple local description near any fixed point.\n\n\\begin{prop}\\label{prop:DefinitionOfWeights}\n[see~\\cite{Karshon} Corollary A.7 or~\\cite{Lerman-Tolman} Lemma A.1] Let $p$ be a fixed point of a Hamiltonian circle action on a $4$-manifold $(M,\\omega)$. Then there exist complex coordinates $(z,w)$ on a neighborhood of $p$ in $M$, and unique integers $\\alpha$ and $\\beta$, called the isotropy weights at $p$, such that\n\\begin{enumerate}\n\\item the circle action is  $t\\cdot (z, w) = (t^{\\alpha}z, t^{\\beta}w)$,\n\\item the symplectic form is $\\omega = \\frac{i}{2}(dz\\wedge d\\bar z + dw \\wedge d\\bar w)$,\n\\item the moment map is $\\mu(z,w) = \\mu(p) + \\frac{\\alpha}{2}|z|^{2} + \\frac{\\beta}{2}|w|^{2}$.\n\\end{enumerate}\n\\end{prop}\nAlthough complex coordinates are used in the definition of weights, the unordered set $\\{\\alpha,\\beta\\}$ is a symplectic invariant of the circle action at the fixed point $p$. This already imposes some restrictions on equivariant symplectomorphisms.\n\n\\begin{cor}\\label{cor:ActionPreservesWeights}\nLet $\\phi\\in C(S^{1})$ be an $S^1$-equivariant symplectomorphism. Then $\\phi$ acts on the fixed point set preserving their weights as unordered sets. In particular, equivalent actions have the same set of weights.\\qed\n\\end{cor}\n\nAs explained in \\cite{Karshon} pages 6-7, the isotropy weights of a Hamiltonian circle action can be read from its labelled graph as follows:\n\\begin{itemize}\n    \\item for $k \\geq 2$, a fixed point has an isotropy weight $-k$ if and only if it is the north pole of a $\\mathbb{Z}_{k}$ -sphere, and it has an isotropy weight $k$ if and only if it is the south pole of a $\\mathbb{Z}_{k}$ -sphere;\n    \\item an interior fixed point has one positive weight and one negative weight, a maximal fixed point has both weights non-positive, a minimal fixed point has both weights non-negative;\n    \\item a fixed point has a weight $0$ if and only if it lies on a fixed surface.\n\\end{itemize}\nWe end this section by stating two simple lemmata that will be useful in computing homology classes of invariant spheres directly from labelled graphs.\n\\begin{lemma}[Lemma 5.4 in~\\cite{Karshon}]\\label{weight}\nLet $S^1$  act on $S^2$ by rotating it $k$ times while fixing the\nnorth and south poles. Suppose that the action lifts to a complex line\nbundle $E$ over $S^2$. Then $S^1$ acts linearly on the fibers over the north and south\npoles; let $\\alpha$ and $\\beta$ be the weights for these actions. Then\n$$\\alpha - \\beta = -ek$$\nwhere $e$ is the self intersection of the zero section.\\qed\n\\end{lemma}\n\n\\begin{lemma}\\label{Lemma:SymplecticArea}\nLet $S^1$  act on $S^2$ by rotating it $k$ times while fixing the\nnorth and south poles. Let $\\mu(n)$ denote the value of the momentum map at the north pole and $\\mu(s)$ the value of the momentum map at the south pole. Then the symplectic area of the $S^2$ is given by $\\frac{\\mu(n) - \\mu(s)}{k}$.\\qed\n\\end{lemma}\n\n\n\n\n\\section{Torus actions on \\texorpdfstring{$S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$}{S2xS2 and CP2\\#CP2}}\nWe now combine the \\emph{equivariant} classification theorems of Delzant and Karshon, together with the Lalonde-McDuff classification of symplectic forms on ruled surfaces, to describe all inequivalent Hamiltonian circle actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ and their relations to toric actions. \n\n\\subsection{Symplectic forms on \\texorpdfstring{$S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$}{S2xS2 and CP2\\#CP2}}\nWe equip $S^2 \\times S^2$ with the product symplectic form $\\omega_\\lambda=\\lambda \\sigma \\oplus \\sigma$, where $\\sigma$ is the standard area form on $S^2$ normalized such that $\\sigma(S^2)=1$. If we think of $S^2 \\times S^2$ as a trivial fiber bundle, $\\omega_\\lambda$ gives area $1$ to the fibers, while the area of horizontal sections is $\\lambda$. Similarly, if we view $\\CP^2\\# \\overline{\\CP^2}$ as the non-trivial $S^2$ bundle over $S^2$, we define an analogous form $\\omega_\\lambda$ which gives area $1$ to the fibers and area $\\lambda-1$ to symplectic sections of self-intersection $-1$, that is, to sections homologous to the exceptional divisor. From an homological point of view, if $F$ denotes the homology class of a fiber in either $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$, if $B$ denotes the class of a section of self-intersection $0$ in $S^2 \\times S^2$, if $E$ denotes the class of the exceptional divisor in $\\CP^2\\# \\overline{\\CP^2}$, and if $L$ denotes the class of a line in $\\CP^2\\# \\overline{\\CP^2}$, then $[\\omega_\\lambda]F=1$, $[\\omega_\\lambda]B=\\lambda$, $[\\omega_\\lambda]L=\\lambda$ and $[\\omega_\\lambda]E=\\lambda-1$. Note that with our conventions on $\\omega_\\lambda$ for the non-trivial bundle $\\CP^2\\# \\overline{\\CP^2}$, $\\lambda$ must be strictly greater than 1 for the symplectic curve in class $E$ to have positive area. We can now state the Lalonde-McDuff classification theorem.\n\n\\begin{thm}[Lalonde-McDuff~\\cite{MR1426534}]\\label{L-McD-Classification} Any symplectic form on $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$ is conformally diffeomorphic to a form $\\omega_\\lambda$ with $\\lambda\\geq 1$. Moreover, any two cohomologous forms are diffeomorphic.\n\\end{thm}\n\n\\subsection{Toric actions on \\texorpdfstring{$S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$}{S2xS2 and CP2\\#CP2}} It is well known that, up to symplectic equivalence, the only toric actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ are given by the standard torus actions on the Hirzebruch surfaces. We now review the main points of the argument. \n \nRecall that the $m^{\\text{th}}$ Hirzebruch surface $W_m$ is the complex submanifold of $\\mathbb{C}P^1 \\times \\mathbb{C}P^2$ defined by the equation\n\\[\nW_m:=  \\left\\{ \\left(\\left[x_1,x_2\\right],\\left[y_1,y_2, y_3\\right]\\right) \\in \\mathbb{C}P^1 \\times \\mathbb{C}P^2 ~|~  x^m_1y_2 - x_2^my_1 = 0 \\right\\}\n\\]\nThe projection map $\\mathbb{C}P^1 \\times \\mathbb{C}P^2 \\longrightarrow \\mathbb{C}P^1$ gives $W_m$ the structure of a $\\mathbb{C}P^1$ bundle over $\\mathbb{C}P^1$ which is diffeomorphic (but not complex isomorphic) to $S^2 \\times S^2$ if $m$ is even and diffeomorphic to the non-trivial $S^2$ bundle over $S^2$ i.e $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$ if $m$ is odd. Thus, for each $m$ we have an integrable complex structure $J_m$ induced on $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$.  When $m=2k$ even, choose $\\lambda > k$, then we can endow  $\\mathbb{C}P^1 \\times \\mathbb{C}P^2$ with the symplectic form $(\\lambda -k) \\sigma_1 \\oplus \\sigma_2$, where $\\sigma_1$ is   the standard symplectic form on  $\\mathbb{C}P^1$ with area 1 and $\\sigma_2$ is the standard symplectic form on $\\mathbb{C}P^2$ such that $\\sigma_2(L) =1$, where $L$ is the class of the line in $\\mathbb{C}P^2$.  Restricting this symplectic form to $W_m$ makes it a symplectic manifold. Similarly when $m=2k+1$,  choose $\\lambda > k+1$, then we can analogously define the form $(\\lambda - (k+1)) \\sigma_1 \\oplus \\sigma_2$ when $m$ is odd. With these choices of symplectic forms, the Lalonde-McDuff classification theorem~\\ref{L-McD-Classification} implies that $W_m$ is symplectomorphic  to $(S^2 \\times S^2,\\omega_\\lambda)$ if $m$ is even and to $(\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2},\\omega_\\lambda)$ when $m$ is odd.\\\\\n\nGiven an integer $m\\geq 0$, the torus $\\mathbb{T}^2$ acts on $\\mathbb{C}P^1 \\times \\mathbb{C}P^2$ by setting\n\\[\n \\left(u,v\\right) \\cdot  \\left(\\left[x_1,x_2\\right],\\left[y_1,y_2, y_3\\right]\\right) = \\left(\\left[ux_1,x_2\\right],\\left[u^my_1,y_2,vy_3\\right]\\right)\n\\]\nThis action leaves $W_m$ invariant and preserves both the complex and the symplectic structures. Its restriction to $W_m$ defines a toric action that we denote $\\mathbb{T}^2_m$. \n\nWhen $m$ is even, the image of the moment map is the polytope of Figure~\\ref{hirz}\n\\begin{figure}[H]\n\\centering       \n\\begin{tikzpicture}\n\\node[left] at (0,2) {$Q=(0,1)$};\n\\node[left] at (0,0) {$P=(0,0)$};\n\\node[right] at (4,2) {$R= (\\lambda - \\frac{m}{2} ,1)$};\n\\node[right] at (6,0) {$S=(\\lambda + \\frac{m}{2} ,0)$};\n\\node[above] at (2,2) {$D_m=B-\\frac{m}{2}F$};\n\\node[right] at (5.15,1) {$F$};\n\\node[left] at (0,1) {$F$};\n\\node[below] at (3,0) {$B+ \\frac{m}{2}F$};\n\\draw (0,2) -- (4,2) ;\n\\draw (0,0) -- (0,2) ;\n\\draw (0,0) -- (6,0) ;\n\\draw (4,2) -- (6,0) ;\n\\end{tikzpicture}\n\\caption{Even Hirzebruch polygon}\n\\label{hirz}\n\\end{figure}\n\n\\noindent where the labels along the edges refer to the homology classes of the  $\\mathbb{T}^2$ invariant spheres in $S^2 \\times S^2$, and where the vertices $P$,$Q$,$R$,$S$ are the fixed points of the torus action. \nSimilarly, when $m$ is odd, the moment map image is given in Figure~\\ref{fig:OddHirzebruch}\n\\begin{figure}[H]\n\\centering   \n\\label{hirz0}\n\\begin{tikzpicture}\n\\node[left] at (0,2) {$Q=(0,1)$};\n\\node[left] at (0,0) {$P=(0,0)$};\n\\node[right] at (4,2) {$R= (\\lambda - \\frac{m+1}{2} ,1)$};\n\\node[right] at (6,0) {$S=(\\lambda + \\frac{m-1}{2} ,0)$};\n\\node[above] at (2,2) {$D_m=B-\\frac{m+1}{2}F$};\n\\node[right] at (5.15,1) {$F$};\n\\node[left] at (0,1) {$F$};\n\\node[below] at (3,0) {$B+ \\frac{m-1}{2}F$};\n\\draw (0,2) -- (4,2) ;\n\\draw (0,0) -- (0,2) ;\n\\draw (0,0) -- (6,0) ;\n\\draw (4,2) -- (6,0) ;\n\\end{tikzpicture}\n \\caption{Odd Hirzebruch polygon}\\label{fig:OddHirzebruch}\n\\end{figure}\n\\noindent where $B$ now refers to the homology class of a line $L$ in $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$, $E$ is the class of the exceptional divisor, and $F$ is the fiber class $L - E$.\\\\\n\nWe define the zero-section $s_0$ to be\n\\begin{align*}\n   s_0: \\mathbb{C}P^1 &\\to W_m \\\\\n    [x_1;x_2]  &\\mapsto \\{[x_1,x_2], [0;0;1]\\}\n\\end{align*}\nand the section at infinity $s_\\infty$ to be \n\\begin{align*}\n     s_\\infty: \\mathbb{C}P^1 &\\to W_m \\\\\n    [x_1;x_2]  &\\mapsto \\{[x_1,x_2], [x^m_1;x^m_2;0]\\}\n\\end{align*}\nThe image of $s_0$ is an invariant and holomorphic sphere homologous to either $D_m:=B-\\frac{m}{2}F$ in $S^2 \\times S^2$ or to $D_m:=B- \\frac{m+1}{2}F$ in $\\CP^2\\# \\overline{\\CP^2}$, depending on the parity of $m$. Similarly, $s_\\infty$ defines an invariant and holomorphic sphere that represents either $B + \\frac{m}{2}F$ in $S^2 \\times S^2$ or $B+\\frac{m+1}{2}F$ in $\\CP^2\\# \\overline{\\CP^2}$. Finally, the homology class $F$ can be represented by an invariant fibre such as $\\{[1,0], [y_1,0,y_3]\\}$.\\\\\n\nIt is well known (see~\\cite{Audin}) that a Delzant polygon $\\Delta\\subset\\mathbb{R}^{2}$ with $e\\geq 3$ edges defines a toric $4$-manifold $M_{\\Delta}$ whose second Betti number is $b_{2}(M_{\\Delta})=e-2$. It follows that the moment polytope of any toric action on either $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$ is a quadrilateral. It is easy to see that any Delzant quadrilateral is equivalent to one of the above Hirzebruch trapezoids up to $\\AGL(2,\\mathbb{Z})$ action and up to rescaling. It follows from Delzant's classification that any toric action on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ is equivalent to an action of the above form. In particular, the equivalence classes of toric actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ are completely characterized by the existence of invariant spheres in specific homology classes.\n\n\n\n\\begin{lemma}\\label{lemma_torus_action_-vecurve}\nUp to equivalence, the toric action $\\mathbb{T}^2_m$, $m\\geq 1$, is characterised by the existence of an invariant, embedded, symplectic sphere $C_m$ with self intersection $-m$. The toric action $\\mathbb{T}^2_{0}$ is characterized by the existence of invariant, embedded, symplectic spheres representing the classes $B$ and~$F$.\\qed\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lemma_number_of_torus_actions}\nWrite $\\lambda=\\ell+\\delta$ with $\\ell$ an integer and $0<\\delta\\leq 1$. Then, up to symplectomorphisms and reparametrizations, \n\\begin{itemize}\n\\item if $\\lambda \\geq 1$, there are exactly $\\ell+1$ inequivalent toric actions on $(S^2 \\times S^2,\\omega_\\lambda)$ given by the even Hirzebruch actions $\\mathbb{T}^2_{2k}$ with $0\\leq k\\leq \\ell$, and\n\\item if $\\lambda >1$, there are exactly $\\ell$ inequivalent toric actions on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ given by the odd Hirzebruch actions $\\mathbb{T}^2_{2k+1}$ with $0\\leq k\\leq \\ell-1$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nWrite $m=2k$ or $m=2k+1$ with $k\\geq 0$. As seen above, any toric action on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ is $\\mathbb{T}^2$-equivariantly symplectomorphic to one of the actions $\\mathbb{T}^2_m$. The invariant symplectic sphere $D_k$ in class $B-kF$ on $S^2 \\times S^2$ or $L - (k+1)F$ in $\\CP^2\\# \\overline{\\CP^2}$ must have positive area, that is, \n\\[0<\\omega_\\lambda(B-kF) = \\lambda - k = \\ell+\\delta-k\\quad\\text{~or~}\\quad 0<\\omega_\\lambda(L-(k+1)F) =\\lambda-(k+1)F = \\ell+\\delta-(k+1)  \\]\nThe result follows.\n\\end{proof}\n\n\\begin{prop}\\label{prop:NormalizersHirzebruch}\nThe Weyl group $W(\\mathbb{T}^2_{m})=N(\\mathbb{T}^2_{m})\/C(\\mathbb{T}^2_{m})$ is isomorphic to\n\\[W(\\mathbb{T}^2_{m})\\simeq\n\\begin{cases}\n\\text{the dihedral group~} D_{8}, & \\text{~when~} m=0 \\text{~and~} \\lambda=1;\\\\\n\\text{the dihedral group~} D_{2}, & \\text{~when~} m=0 \\text{~and~} \\lambda>1;\\\\\n\\text{the dihedral group~} D_{1}\\simeq\\mathbb{Z}_{2},& \\text{~when~} m\\geq 1.\n\\end{cases}\\]\n\\end{prop}\n\\begin{proof}\nThis follows directly from Proposition~\\ref{prop:CentralizersToricActions} and from the description of Hirzebruch trapezoids. Indeed, for $m=0$ and $\\lambda=1$, the moment map polygon is a unit square whose symmetries can be realized by elements of $\\AGL(2,\\mathbb{Z})$. The same holds for $m=0$ and $\\lambda>1$, with the only difference that the moment polygone is now a rectangle with sides of lengths $1$ and $\\lambda$. Finaly, for $m\\geq 1$, if we write $m=2k$ or $m=2k+1$, the only non-trivial element of  $\\AGL(2,\\mathbb{Z})$ that leaves the standard Hirzebruch trapezoid invariant is\n\\[\\Lambda \n=\\begin{pmatrix}-1 & -m\\\\ 0 & 1\\end{pmatrix}\n+\\begin{pmatrix}\\lambda+k,0\\end{pmatrix}\\]\nwhich is an element of order 2.\n\\end{proof}\n\n\\subsection{Hamiltonian circle actions on \\texorpdfstring{$S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$}{S2xS2 and CP2\\#CP2}}\nA list of all possible Hamiltonian circle action on $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$ comes easily from an extension theorem due to Y. Karshon.\n\\begin{thm}[Karshon~\\cite{finTori}, Theorem 1]\nAny symplectic $S^1$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ and $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ extends to an Hamiltonian toric action. \n\\end{thm} \n\nConsequently, every symplectic $S^1$ action on $S^2 \\times S^2$ and $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$ is given by an embedding \n\\begin{align*}\n    S^1 &\\hookrightarrow \\mathbb{T}^2_m \\\\\n    t &\\mapsto (t^a, t^b)\n\\end{align*}\nwhich is itself characterized by a unique triple of numbers $(a,b;m)\\in \\mathbb{Z} \\times \\mathbb{Z} \\times \\mathbb{Z}_{\\geq 0}$. Since we are only interested in effective actions, this translates numerically into the condition  $\\gcd(a,b) = 1$. We shall always assume this unless otherwise stated. \n\n\n\nIn order to construct the graphs of the circle action $S^1(a,b;m)$, we claim that it is enough to compute the isotropy weights at every fixed points. Indeed, given an Hirzebruch trapezoid, the pre-image of a vertex under the moment map is a toric fixed point, and the pre-image of an edge is an invariant embedded two-sphere connecting two fixed points. When we view the space as a Hamiltonian $S^1$-space, such a two-sphere is either fixed by the action, or is a $\\mathbb{Z}_k$ sphere, or has trivial global isotropy. These three possibilities are completely determined by the weights of the $S^{1}(a,b;m)$ action at its two fixed points. We can thus construct the graphs starting from the fixed points and adding edges according to the weights. If there is a fixed surface, its area label can be read from the Hirzebruch trapezoid. Finally, the moment map labels can be added starting with the minimal vertex (characterised by having two positive weights) that we label $\\mu=0$, and then using Lemma~\\ref{Lemma:SymplecticArea} to compute the moment labels for the remaining interior fixed points, and for the maximal fixed point (characterised by having two negative weights).\n\nNow, for the Hirzebruch actions $\\mathbb{T}^2_{m}=(S^{1}\\times S^{1})_{m}$, each $S^{1}$ factor defines two weights at each of the four fixed points $P,Q,R,S$. These weights are listed in table~\\ref{table_weights}. Further, if the the $\\mathbb{T}^2_m$ action has weights $\\{\\alpha_1,\\beta_{1}\\}$ and $\\{\\alpha_{2},\\beta_2\\}$ at a fixed point $x$, then the restricted $S^1(a,b;m)$ action has weights\n\\[\\left\\{a\\alpha_1 + b\\alpha_{2}, a\\beta_{1} + b\\beta_2 \\right\\} \\]\nat $x$. This gives the weights of the $S^{1}(a,b;m)$ actions at the fixed points $P,Q,R,S$.\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{ |p{2cm}||p{4cm}|p{6cm}|  }\n \\hline\n Vertex & Weights for $\\mathbb{T}^2_m$ action & Weights for the $S^1(a,b;m)$ action \\\\\n \\hline\n P & $\\big\\{\\{1,0\\},\\{0,1\\}\\big\\}$   & $\\{a,b\\}$ \\rule{0pt}{10pt}\\\\\n Q & $\\big\\{\\{1,0\\},\\{0,-1\\}\\big\\}$  & $\\{a,-b\\}$ \\rule{0pt}{10pt}\\\\\n R & $\\big\\{\\{-1,m\\},\\{0,-1\\}\\big\\}$ & $\\{-a,am-b\\}$ \\rule{0pt}{10pt}\\\\\n S & $\\big\\{\\{-1,-m\\},\\{0,1\\}\\big\\}$ & $\\{-a,-am+b\\}$ \\rule{0pt}{10pt}\\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Weights of $\\mathbb{T}^2_{m}$ and $S^{1}(a,b;m)$ actions}\n\\label{table_weights}\n\\end{table}\n\n\n\nIn Figures~\\ref{fig:GraphsWithFixedSurfaces} and~\\ref{fig:GraphsIsolatedFixedPoints}, we present the graphs of circles actions $S^{1}(a,b;m)$ on $(S^2 \\times S^2,\\omega_\\lambda)$ and $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. As before, we write $m=2k$ or $m=2k+1$, and in order to deal with even and odd Hirzebruch surfaces simultaneously, we introduce the symbol $\\epsilon_{m} := m\\mod 2$. Each label $\\mu$ represents the value of the moment map normalized by setting $\\min \\mu=0$, while $A$ is the area label of a fixed surface. All fixed surfaces have genus 0. Note that when the isotropy label on an edge is 1, then the circle action on the corresponding invariant sphere has no isotropy, and we erase that edge from the graph. Note also that the identification of the fixed points with the vertices $P,Q,R,S$ of the Hirzebruch trapezoid is there for convenience only and is not part of the data.  \n\n\n\n\nFor actions with fixed surfaces, one of the weights $a$, $b$, or $am-b$ must be zero. Since $\\gcd(a,b)=1$, the possible triples are $(\\pm1,0;m)$, $(0,\\pm1;m)$, and $(\\pm1,\\pm m;m)$. \n\\begin{figure}[H]\n\\centering\n\\subcaptionbox{Graph of $S^{1}(1,0;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm);\n\\draw [fill] (0,1.2) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,2.8) ellipse (0.1cm and 0.1cm);\n\\draw (0,1.2) -- (0,2.8);\n\\node[right] at (0,2) {m};\n\\node[right] at (0,1.2) {R};\n\\node[right] at (0,2.8) {S};\n\\node[right] at (0.5,2.8){$\\mu= \\lambda +k$};\n\\node[right] at (0.5,0){$\\mu=0$};\n\\node[left] at (-0.4,0) {$A= 1$};\n\\node[right] at (0.5,1.2) {$\\mu=\\lambda - k - \\epsilon_{m}$};\n\\end{tikzpicture}}\n\n\\subcaptionbox{Graph of $S^{1}(-1,0;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm);\n\\draw [fill] (0,-1.2) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-2.8) ellipse (0.1cm and 0.1cm);\n\\draw (0,-1.2) -- (0,-2.8);\n\\node[right] at (0,-2.2) {m};\n\\node[right] at (0,-2.8) {S};\n\\node[right] at (0,-1.2) {R};\n\\node[right] at (0.5,0){$\\mu= \\lambda +k$};\n\\node[right] at (0.5,-2.8){$\\mu=0$};\n\\node[left] at (-0.5,0) {$A= 1$};\n\\node[right] at (0.5,-1.2) {$\\mu=m$};\n\\end{tikzpicture}}\n\n\\subcaptionbox{Graph of $S^{1}(0,1;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm); \n\\draw [fill] (0,2.8) ellipse (0.3cm and 0.1cm); \n\\node[left] at (-0.3,2.8) {$\\mu = 1$};\n\\node[left] at (-0.3,0){$\\mu=0$};\n\\node[right] at (0.3,2.8){$A= \\lambda-k-\\epsilon_{m}$};\n\\node[right] at (0.3,0){$A= \\lambda+k$};\n\\node[above] at (0,2.8) {\\rule{0em}{3em}}; \n\\end{tikzpicture}}\n\n\\subcaptionbox{Graph of $S^{1}(0,-1;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm); \n\\draw [fill] (0,2.8) ellipse (0.3cm and 0.1cm); \n\\node[left] at (-0.3,2.8) {$\\mu = 1$};\n\\node[left] at (-0.3,0){$\\mu=0$};\n\\node[right] at (0.3,2.8){$A= \\lambda+k$};\n\\node[right] at (0.3,0){$A= \\lambda-k-\\epsilon_{m}$};\n\\end{tikzpicture}}\n\\subcaptionbox{Graph of $S^{1}(1,m;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm);\n\\draw [fill] (0,-2) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-3.5) ellipse (0.1cm and 0.1cm);\n\\draw (0,-2) -- (0,-3.5);\n\\node[above] at (0,0) {\\rule{0em}{3em}}; \n\\node[right] at (0,-2.75) {m};\n\\node[right] at (0,-3.5) {P};\n\\node[right] at (0,-2) {Q};\n\\node[right] at (0.5,0){$\\mu= \\lambda +k$};\n\\node[right] at (0.5,-3.5){$\\mu=0$};\n\\node[left] at (-0.5,0) {$A= 1$};\n\\node[right] at (0.5,-2) {$\\mu=m$};\n\\end{tikzpicture}}\n\\subcaptionbox{Graph of $S^{1}(-1,-m;m)$\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,-3.5) ellipse (0.3cm and 0.1cm);\n\\draw [fill] (0,-2) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (0,-2);\n\\node[right] at (0,-1) {m};\n\\node[right] at (0,0) {P};\n\\node[right] at (0,-2) {Q};\n\\node[right] at (0.5,0){$\\mu= \\lambda +k$};\n\\node[right] at (0.5,-3.5){$\\mu=0$};\n\\node[left] at (-0.5,-3.5) {$A= 1$};\n\\node[right] at (0.5,-2) {$\\mu=\\lambda - k-\\epsilon_{m}$};\n\\end{tikzpicture}}\n\\caption{Graphs of circle actions with non-isolated fixed points}\\label{fig:GraphsWithFixedSurfaces} \n\\end{figure}\n\n\nFor actions whose fixed points are isolated, none of the weights are zero. The graphs then depend on the signs of $a$, $b$, and $am-b$, as these signs determine which fixed point is minimal and which one is maximal.\n\n\\begin{figure}[H]\n\\centering\n~\n\n\\subcaptionbox*{}{} \n\\setcounter{subfigure}{6} \n\\subcaptionbox{When $a>0,~b>0$, and $am-b>0$\\label{fig:a>0|b>0|am-b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-1.5) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2.5) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-1.5);\n\\draw (1,-1.5) -- (-1,-2.5);\n\\draw (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw (-1,-2.5) -- (0,-4);\n\\node[right] at (-0.35,-3.2) {$b$};\n\\node[right] at (- 0.4,-1.8) {$a$};\n\\node[left] at (-2.2,-1.5) {$a$};\n\\node[right] at (0.7,-0.5) {$am-b$};\n\\node[above] at (0,0.2) {$\\mu = a(\\lambda + k)$};\n\\node[right] at (-0.6,-2.5) {$\\mu = b$};\n\\node[right] at (1,-1.8) {$\\mu = b + a(\\lambda - k-\\epsilon_{m}) $};\n\\node[below] at (0, -4.2) {$\\mu = 0$};\n\\node[left] at (-1,-2.5) {Q};\n\\node[left] at (0.9,-1.4) {R};\n\\node[right] at (0,0) {S};\n\\node[right] at (0,-4) {P};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a>0,~b>0$ and $am-b<0\n}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (1,-2) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (-1,-3) ellipse (0.1cm and 0.1cm); \n\\draw (0,0) -- (1,-2);\n\\draw (-1,-3) -- (0,-4);\n\\draw (0,0) -- (-1,-3) ;\n\\draw (1,-2) -- (0,-4);\n\\node[right] at (-0.9,-3) {Q};\n\\node[left] at (0.9,-2) {S};\n\\node[right] at (0,0) {R};\n\\node[above] at (0,0.2) {$\\mu = b + a (\\lambda -k-\\epsilon_{m})$};\n\\node[right] at (1.3,-2) {$\\mu = a(\\lambda + k) $};\n\\node[left] at (-1.4,-3) {$\\mu = b$};\n\\node[right] at (0,-4) {P};\n\\node[right] at (-1,-3.7) {$b$};\n\\node[right] at (0.5, -3) {$a$};\n\\node[left] at (-0.5,-1.5) {$a$};\n\\node[right] at (0.7,-1) {$b-am$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a>0,~b<0$\\label{fig:a<0|b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-3) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-3);\n\\draw (-1,-2) -- (0,-4);\n\\draw (0,0) -- (-1,-2) ;\n\\draw (1,-3) -- (0,-4);\n\\node[above] at (0,1) {\\rule{0em}{1em}};\n\\node[above] at (0,0.2) {$\\mu = a(\\lambda +k)-b$};\n\\node[right] at (1.4,-3) {$\\mu = -b $};\n\\node[left] at (-1.4,-2) {$\\mu = a(\\lambda-k-\\epsilon_{m})$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\node[right] at (-1,-3.2) {$a$};\n\\node[right] at (0.5, -3.6) {$-b$};\n\\node[left] at (-0.7,-1.0) {$am-b$};\n\\node[right] at (0.6,-1.3) {$a$};\n\\node[left] at (0.9,-3) {P};\n\\node[right] at (-1,-2) {R};\n\\node[right] at (0,0) {S};\n\\node[right] at (0,-4) {Q};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a<0,~b>0$\\label{fig:a<0|b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-3) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-3);\n\\draw (-1,-2) -- (0,-4);\n\\draw (0,0) -- (-1,-2) ;\n\\draw (1,-3) -- (0,-4);\n\\node[above] at (0,1) {\\rule{0em}{1em}}; \n\\node[above] at (0,0.2) {$\\mu = b - a (\\lambda +k)$};\n\\node[right] at (1.4,-3) {$\\mu = b-am $};\n\\node[left] at (-1.4,-2) {$\\mu = -a(\\lambda+k)$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\node[right] at (-1.2,-3.2) {$-a$};\n\\node[right] at (0.5, -3.6) {$b-am$};\n\\node[left] at (-0.7,-1.0) {$b$};\n\\node[right] at (0.6,-1.3) {$-a$};\n\\node[left] at (0.9,-3) {R};\n\\node[right] at (-1,-2) {P};\n\\node[right] at (0,0) {Q};\n\\node[right] at (0,-4) {S};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a<0,~b<0$, and $am-b>0$\\label{fig:a<0|b<0|am-b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-3) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-3);\n\\draw (-1,-2) -- (0,-4);\n\\draw (0,0) -- (-1,-2) ;\n\\draw (1,-3) -- (0,-4);\n\\node[above] at (0,1) {\\rule{0em}{1em}}; \n\\node[above] at (0,0.2) {$\\mu = -b - a (\\lambda -k-\\epsilon_{m})$};\n\\node[right] at (1.4,-3) {$\\mu = am-b $};\n\\node[left] at (-1.4,-2) {$\\mu = -a(\\lambda-k-\\epsilon_{m})$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\node[right] at (-1.2,-3.2) {$-a$};\n\\node[right] at (0.5, -3.6) {$am-b$};\n\\node[left] at (-0.7,-1.0) {$-b$};\n\\node[right] at (0.6,-1.3) {$-a$};\n\\node[left] at (0.9,-3) {S};\n\\node[right] at (-1,-2) {Q};\n\\node[right] at (0,0) {P};\n\\node[right] at (0,-4) {R};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a<0,~b<0$, and $am-b<0$\\label{fig:a<0|b<0|am-b<0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-1.5) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2.5) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-1.5);\n\\draw (1,-1.5) -- (-1,-2.5);\n\\draw (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw (-1,-2.5) -- (0,-4);\n\\node[right] at (-0.35,-3.2) {$b-am$};\n\\node[right] at (- 0.4,-1.8) {$-a$};\n\\node[left] at (-2.2,-1.5) {$-a$};\n\\node[right] at (0.7,-0.5) {$-b$};\n\\node[above] at (0,0.2) {$\\mu = -a(\\lambda + k)$};\n\\node[right] at (-0.6,-2.5) {$\\mu = b-am$};\n\\node[right] at (1,-1.8) {$\\mu = b-a(\\lambda + k-\\epsilon_{m}) $};\n\\node[below] at (0, -4.2) {$\\mu = 0$};\n\\node[right] at (0,0) {P};\n\\node[left] at (0.9,-1.4) {Q};\n\\node[left] at (-1,-2.5) {R};\n\\node[right] at (0,-4) {S};\n\\end{tikzpicture}}\n\\caption{Graphs of circle actions whose fixed points are all isolated}\\label{fig:GraphsIsolatedFixedPoints}\n\\end{figure}\n\n\n\\section{Equivalent circle actions}\n\nA single equivalence class of Hamiltonian circle actions on $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$ may be represented by more than one triple $(a,b;m)$, and this happens whenever the associated labelled graphs are the same up to the action of $\\AGL(1,\\mathbb{Z})$ on the moment map labels. \n\n\\subsection{Equivalent circle actions in \\texorpdfstring{$\\mathbb{T}^2_{m}$}{Tm}}\nThe action $S^{1}(-a,-b;m)$ is equivalent to $S^{1}(a,b;m)$ since they only differ by a reparametrization of the circle. It follows that in \nFigure~\\ref{fig:GraphsWithFixedSurfaces}, the graphs (A), (C), and (E) are $\\AGL(1,\\mathbb{Z})$-equivalent, respectively, to the graphs (B), (D), and (F). Similarly, the graphs (G), (H), and (I) in Figure~\\ref{fig:GraphsIsolatedFixedPoints} are $\\AGL(1,\\mathbb{Z})$-equivalent, respectively, to the graphs (L), (K), and (J). Other examples of equivalent subcircles in $\\mathbb{T}^2_{m}$ are obtained by letting the normalizer $N(\\mathbb{T}^2_{m})$ act on the circle subgroups of $\\mathbb{T}^2_{m}$. For instance, $S^{1}(a,b;m)$ is conjugated to $S^{1}(-a,b-am;m)$ by the adjoint of the element $\\Lambda\\in N(\\mathbb{T}^2_{m})$ of Proposition~\\ref{prop:NormalizersHirzebruch}.\n\n\\begin{prop}\\label{prop:EquivalentSubcircles}\nThe only subcircles of $\\mathbb{T}^2_{m}$ that are equivalent to $S^{1}(a,b;m)$ are the elements of the orbit of $S^{1}(a,b;m)$ under the action of the Weyl group $W(\\mathbb{T}^2_{m})$ and their reparametrizations. \n\\end{prop}\n\\begin{proof}\nWe have to show that if $S^{1}(a,b;m)$ and $S^{1}(c,d;m)$ are in the same conjugacy class, then there is an element of $N(\\mathbb{T}^2_{m})$ taking one circle to the other. As conjugation of actions corresponds to  uniform translation of the moment map labels, this will follow from a systematic inspection of the possible labelled graphs.\n\nThe first cases to consider are actions $S^{1}(a,b;m)$ whose fixed points are all isolated, which means that the weights $a$, $b$, and $am-b$ are all non-zero. Two equivalent graphs arising from normalized moment maps must have the same moment map values at the maximum, and their respective weights at the maximum and at the minimum must also coincide. \n\n\\begin{table}[H]\n\\begin{tabular}{|l||c|c|c|}\n\\hline\nType of graph & Weights at max & Weights at min & Moment label at max\\\\\n\\hline\n\\hline\nG: $a>0$, $b>0$, $am-b>0$ & $-a$, $b-am$ & $a$, $b$ & $a(\\lambda+k)$\\\\\n\\hline\nH: $a>0$, $b>0$, $am-b<0$ & $-a$, $am-b$ & $a$, $b$ & $b+a(\\lambda-k-\\epsilon_{m})$ \\\\\n\\hline\nI: $a>0$, $b<0$ & $-a$, $b-am$ & $a$, $-b$ & $a(\\lambda+k)-b$\\\\\n\\hline\nJ: $a<0$, $b>0$ & $a$, $-b$ & $-a$, $b-am$ & $b-a(\\lambda+k)$\\\\\n\\hline\nK: $a<0$, $b<0$, $am-b>0$ & $a$, $b$ & $-a$, $am-b$ & $-b-a(\\lambda-k-\\epsilon_{m})$\\\\\n\\hline\nL: $a<0$, $b<0$, $am-b<0$ & $a$, $b$ & $-a$, $b-am$ & $-a(\\lambda+k)$\\\\\n\\hline\n\\end{tabular}\n\\caption{Weights and moment map values for graphs (G) -- (L)}\\label{table:WeightsMomentMapValuesGL}\n\\end{table}\nWe start by assuming $a>0$, $b>0$, and $m>0$, which implies that the graph of $S^{1}(a,b;m)$ is of type (G) or (H).\n\nAssume the graph of $S^{1}(a,b;m)$ is of type (G), that is, $am-b>0$, and suppose $S^{1}(c,d;m)$, $(c,d)\\neq (a,b)$, is in the same conjugacy class.\n\n\\begin{itemize}\n\\item Suppose the graph of $S^{1}(c,d;m)$ is also of type (G). Looking at the weights at the minimum, we see that $(c,d)=(b,a)$ is the only non-trivial possibility. Looking at the moment map values at the maximum, we conclude that $a(\\lambda+k)=b(\\lambda+k)$, that is, $a=b$, contradicting the fact that $(a,b)\\neq(c,d)$.\n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (H). Looking again at the weights at the minimum, we see that $c=b$, and $d=a$, and that $a\\neq b$ as before. The sets of weights at the maximum must also be equal, that is, $\\{-a,b-am\\}=\\{-c, cm-d\\}=\\{-b,bm-a\\}$, which implies that $a=a-bm$, contradicting the fact that $bm\\neq 0$.\n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (I), which implies $c>0$, $d<0$. Looking at the weights at the minimum, we must have either $(c,d)=(a,-b)$ or $(c,d)=(b,-a)$. In the former case, looking at the moment map values at the maximum, we conclude that $a(\\lambda+k)=c(\\lambda+k)-d=a(\\lambda+k)+b$, that is, $b=0$, which is a contradiction. In the later case, the weights at the maximum become $\\{-c, d-cm\\} = \\{-b, -a-bm\\}$, and we must have $\\{-b, -a-bm\\}=\\{-a, b-am\\}$ as sets. Since $a+bm\\neq a$, we must have $b=a$, which implies $a=b=1$, $c=1$, and $d=-1$. The moment map values at the maximum are then $a(\\lambda+k)=(\\lambda+k)$ and $c(\\lambda+k)-d=(\\lambda+k)+1$, which are not equal. \n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (J), which implies $c<0$, $d>0$. Looking at the weights at the minimum, we must have $\\{a,b\\}=\\{-c,d-cm\\}$ as sets. So, either $(c,d)=(-a,b-am)$ or $(c,d)=(-b,a-bm)$. In the first case, $b-am<0$, which is impossible at the minimum. In the second case, looking at the moment map values at the maximum, we conclude that $a(\\lambda+k)=d-c(\\lambda+k)=a-bm+b(\\lambda+k)=a+b(\\lambda-k)$. This implies $a(\\lambda+k-1)\/(\\lambda+k)=b$. As $a$ and $b$ are non-zero integers, this is impossible.\n\n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (K), which implies $c<0$, $d<0$, and $cm-d>0$. Comparing the weights at the minimum, we see that either $(c,d)=(-a, -am-b)$ or $(c,d)=(-b,-bm-a)$ with $a\\neq b$. In the former case, comparing the weights at the maximum yields $\\{-a,b-am\\}=\\{c,d\\}=\\{-a,-am-b\\}$. This implies $b=0$, which is excluded. In the former case, we must have $a\\neq b$ and $\\{-a,b-am\\}=\\{c,d\\}=\\{-b,-bm-a\\}$, which again implies $b=0$.\n\\item Finally, suppose the graph of $S^{1}(c,d;m)$ is of type (K), which implies $c<0$, $d<0$, and $cm-d<0$. Comparing the weights at the minimum, we see that either $(c,d)=(-a, b-am)$ or $(c,d)=(-b,a-bm)$. In the former case, we get the conjugate circle $S^{1}(-a,b-am;m)$. In the later case, the moment map values at the maximum gives $a(\\lambda+k)=-c(\\lambda+k)=b(\\lambda+k)$, that is, $a=b$. Then $(c,d)=(-a, b-am)$ and $S^{1}(c,d;m)=S^{1}(-a,b-am;m)$ as before.\n\\end{itemize}\nWe conclude that the only circle action $S^{1}(c,d;m)$ conjugated to an action $S^{1}(a,b;m)$ of type (G) is $S^{1}(-a,b-am;m)$.\n\nAssume now that the graph of $S^{1}(a,b;m)$ is of type (H), that is, $a>0$, $b>0$, and $am-b<0$. By transitivity, we already know that an action $S^{1}(c,d;m)$ in the same conjugacy class cannot be of type (G) or (L).\n\n\\begin{itemize}\n\\item Suppose the graph of $S^{1}(c,d;m)$ is also of type (H). Comparing the weights at the minimum, we see that we must have $(c,d)=(b,a)$. Looking at the weights at the maximum, we must have $\\{-a,am-b\\}=\\{-c,cm-d\\}$ as sets. Since $(c,d)\\neq(a,b)$, the only possibility is $a=d-cm=a-bm$, which implies $b=0$.\n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (I). Comparing the weights at the minimum, we see that we must have either $(c,d)=(a,-b)$ or $(c,d)=(b,-a)$ with $a\\neq b$. In the first case, comparing the weights at the maximum, we should have $\\{-a,am-b\\}=\\{-c,d-cm\\}=\\{-a,-b-am\\}$. This implies $a=0$, which is excluded. Similarly, in the second case, we get $\\{-a,am-b\\}=\\{-c,d-cm\\}=\\{-b,-b-am\\}$ with $a\\neq b$. This again implies  $a=0$.\n\\item Suppose the graph of $S^{1}(c,d;m)$ is of type (J). Looking at the weights at the minimum, we must have $\\{a,b\\}=\\{-c,d-cm\\}$ as sets. So, either $(c,d)=(-a,b-am)$ or $(c,d)=(-b,a-bm)$. In the first case, we obtain the conjugate action $S^{1}(-a,b-am;m)$. In the second case, looking at the weights at the maximum, we must have $\\{-a,am-b\\}=\\{c,-d\\}=\\{-b, bm-a\\}$. If $a=b=1$, this gives\nus the conjugate action $S^{1}(-1,1-m;m)$. If $a=a-bm$, then $b=0$, which is excluded.\n\\item Finally, suppose the graph of $S^{1}(c,d;m)$ is of type (K). Looking at the weights at the minimum, we must have $\\{a,b\\}=\\{-c,cm-d\\}$ as sets. So, either $(c,d)=(-a,-b-am)$ or $(c,d)=(-b,-a-bm)$. Looking at the weights at the maximum, we must also have $\\{-a,am-b\\}=\\{c,d\\}$. If $(c,d)=(-a,-b-am)$, then $am-b=-am-b$, which implies $a=0$. Instead, if $(c,d)=(-b,-a-bm)$, then either $a=b=1$, which is equivalent to the previous case, or $a=a+bm$ which forces $b=0$.\n\\end{itemize}\nWe conclude that $S^{1}(-a,b-am;m)$ is the only subcircle of $\\mathbb{T}^2_{m}$ conjugated to an action $S^{1}(a,b;m)$ of type (H).\n\n\nAssume now that the graph of $S^{1}(a,b;m)$ is of type (I), that is, $a>0$, and $b<0$. By transitivity, we already know that an action $S^{1}(c,d;m)$ in the same conjugacy class cannot be of type (G), (H), (J), or (L). By symmetry of Table~\\ref{table:WeightsMomentMapValuesGL} under a change of sign of the pair $(a,b)$, and comparing with actions of types (H) and (J), we see that the only action $S^{1}(c,d;m)$ of type (K) conjugated to $S^{1}(a,b;m)$ is $S^{1}(-a,b-am;m)$. \n\n\\begin{itemize}\n\\item Suppose the graph of $S^{1}(c,d;m)$ is also of type (I). Comparing the weights at the minimum, we see that we must have $(c,d)=(-b,-a)$. Looking at the weights at the maximum, we must also have $\\{-a,b-am\\}=\\{-c, d-cm\\}=\\{b,-a+bm\\}$. If $b=-a$, then $(a,b)=(1,-1)=(c,d)$. If $b=b-am$, then we must have $a=0$, which is impossible.\n\n\\end{itemize}\nThis shows that for an action $S^{1}(a,b;m)$ of type (I), the only other action $S^{1}(c,d;m)$ in the same conjugacy class is $S^{1}(-a,b-am;m)$.\n\nThis concludes the proof of the proposition for actions $S^{1}(a,b;m)$ whose fixed points are all isolated, in the case $m>0$. When $m=0$ and $\\lambda>1$, the graphs (G) and (L) no longer exist, and the four remaining graphs (H)--(K) only differ by the action of $D_{2}$ on the pair $(a,b)$, that is, by a change of sign of $a$ or $b$. When $\\lambda=1$, we can also interchange $a$ and $b$ to obtain an equivalent graph, which defines an action of $D_{4}$. Finally, the case of actions with non-isolated surfaces involves the graphs (A)--(F) and is simpler. The details are left to the reader. \n\\end{proof}\n\n\n\n\\subsection{Toric extensions of circle actions}\n\\begin{defn}\nWe shall say a circle action $S^1(a,b; m)$ \\emph{extends} to a toric action $\\mathbb{T}^2_{n}$ if it is $S^1$-equivariantly symplectomorphic to a circle action of the form $S^1(c,d; n)$.\n\\end{defn}\n\nIn this section, we determine all possible toric extensions of $S^{1}(a,b;m)$. We begin with the exceptional case $\\lambda=1$.\n\\begin{prop}\nConsider $(S^2 \\times S^2,\\omega_\\lambda)$ with $\\lambda=1$. Then the only Hamiltonian circle actions are of the form $S^1(a,b; 0)$. In particular, they can only extend to the torus $\\mathbb{T}^2_0$. \n\\end{prop}\n\\begin{proof}\nThis follows directly from Theorem~\\ref{lemma_number_of_torus_actions}.\n\\end{proof}\n\n\n\nBy Lemma~\\ref{lemma_torus_action_-vecurve}, a toric extension of $S^{1}(a,b;m)$ to $\\mathbb{T}^2_{n}$, $n\\geq 1$, implies the existence of an invariant sphere of self-intersection $-n$ and of positive symplectic area. These two numerical invariants can be read from labelled graphs using Lemma~\\ref{weight} and Lemma~\\ref{Lemma:SymplecticArea}. As we will see, this imposes enough conditions on the triple $(a,b;m)$ to determine all possible embeddings $S^{1}(a,b;m)\\hookrightarrow \\mathbb{T}^2_{n}$, $n\\neq m$.\n\n\n\n\\begin{prop}\\label{prop:AtMostTwoExtensions}\nConsider a Hamiltonian circle action $S^1(a,b;m)$ with $\\lambda >1$. Under the following numerical conditions on $a,b,m,\\lambda$, the circle action only extends to the toric action~$\\mathbb{T}^2_m$:\n\\begin{itemize}\n    \\item when $a\\neq\\pm 1$;\n    \\item when $b=0$ or $b=am$\n    \\item when $a = \\pm 1$ and $2 \\lambda \\leq |2b-am|+\\epsilon_{m}$.\n\\end{itemize}\nIn all other cases, the circle action may extend to at most two inequivalent toric actions, namely,\n\\begin{itemize}\n    \\item when $a=\\pm 1$, $2\\lambda > |2b-am|+\\epsilon_{m}$, and $b \\not\\in \\{0,am\\}$, the circle action $S^1(a,b;m)$ only extends to the toric action $\\mathbb{T}^2_{m}$ and, possibly, to $\\mathbb{T}^2_{|2b-am|}$.\n\n\\end{itemize}\n\\end{prop}\n\\begin{proof}\nSuppose $S^{1}(a,b;m)$ is equivariantly symplectomorphic to $S^{1}(c,d;n)$ for some $n\\neq m$. Note that we necessarily have $m=n\\mod 2$. By assumption, the two actions share the same normalized labelled graph\n\nWe first consider an action $S^{1}(a,b;m)$ whose fixed points are all isolated. For the moment, let's also assume that $m\\neq 0$ and $n\\neq 0$. We observe that if the $\\mathbb{T}^2_{n}$ invariant curve $C_{n}$ of self-intersection $-n$ has non-trivial isotropy, then it must appear as some edge in the graph of $S^{1}(a,b;m)$. As the edge connecting the vertices $Q$ and $R$ is the only one that corresponds to an invariant sphere of negative self-intersection, namely $-m$, we conclude that $m=n$. This contradiction shows that $C_{n}$ must have trivial isotropy. By symmetry, the same is true of the invariant curve $C_{m}$ of self-intersection $-m$. It follows that $a=\\pm 1$ and $c=\\pm 1$. \n\nAssume $a=1$. Because there are no fixed surfaces, we know that $b\\neq 0$ and $m-b\\neq 0$. Figure~\\ref{fig:PossibleGraphsToricExtensions} shows the three possible graphs for $S^{1}(1,b;m)$, which are then of types (G), (H), or (I). The dashed edges represent the possible locations of the $\\mathbb{T}^2_{n}$ invariant curves $C_{n}$ and $C_{-n}$. We can compute the self-intersection of $C_{n}$ and $C_{-n}$ by applying Lemma~\\ref{weight} to the normal bundle of these invariant spheres, and using the configurations of weights shown in Figure~\\ref{fig:Self-Intersection}. The self-intersection of the curve between $P$ and $R$ is $2b-m$, while the self-intersection of the curve between $Q$ and $S$ is $m-2b$. Set $n=|2b-m|$. For the toric action $\\mathbb{T}^2_{n}$ to exist, we must also have $0<2\\omega_\\lambda (C_{n}) = 2\\lambda-|2b-m|-\\epsilon_{m}$. We conclude that the action $S^{1}(1,b;m)$, $m\\neq 0$, may extend to $\\mathbb{T}^2_{|2b-m|}$ whenever $2\\lambda>|2b-m|+\\epsilon_{m}$, and that this is the only other possible toric extension.\n\n\n\\begin{figure}\n\\centering\n~\n\\subcaptionbox*{}{}\n\\setcounter{subfigure}{6}\n\\subcaptionbox{When $a=1,~b>0$, and $m-b>0$\\label{fig:a=1|b>0|m-b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-1.5) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2.5) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-1.5);\n\\draw [dashed, blue] (0,0) -- (-1,-2.5);\n\\draw [dashed, blue] (1,-1.5) -- (0,-4);\n\\draw (-1,-2.5) -- (0,-4);\n\\node[left] at (-0.6,-3.2) {$b$};\n\\node[right] at (0.7,-0.5) {$m-b$};\n\\node[above] at (0,0.2) {$\\mu = (\\lambda + k)$};\n\\node[right] at (-0.6,-2.5) {$\\mu = b$};\n\\node[right] at (1,-1.8) {$\\mu = b + (\\lambda - k-\\epsilon_{m}) $};\n\\node[below] at (0, -4.2) {$\\mu = 0$};\n\\node[left] at (-1,-2.5) {Q};\n\\node[left] at (0.9,-1.4) {R};\n\\node[right] at (0,0) {S};\n\\node[right] at (0,-4) {P};\n\\end{tikzpicture}}\n\\subcaptionbox{When $a=1,~b>0$ and $m-b<0$\n}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (1,-2) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (-1,-3) ellipse (0.1cm and 0.1cm); \n\\draw [dashed, blue] (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw [dashed, blue] (1,-2) -- (-1,-3);\n\\draw (0,0) -- (1,-2);\n\\draw (-1,-3) -- (0,-4);\n\\node[right] at (0,-4) {P};\n\\node[left] at (-1,-3) {Q};\n\\node[left] at (0.9,-2) {S};\n\\node[right] at (0,0) {R};\n\\node[above] at (0,0.2) {$\\mu = b +  (\\lambda -k-\\epsilon_{m})$};\n\\node[right] at (1.2,-2) {$\\mu = (\\lambda + k) $};\n\\node[right] at (-0.7,-3) {$\\mu = b$};\n\\node[right] at (-0.4,-3.5) {$b$};\n\\node[right] at (0.7,-1) {$b-m$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\end{tikzpicture}\n}\n\\subcaptionbox{When $a=1,~b<0$\\label{fig:a=1|b>0}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-3) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2) ellipse (0.1cm and 0.1cm);\n\\draw [dashed, blue] (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw [dashed, blue] (-1,-2) -- (1,-3);\n\\draw (0,0) -- (-1,-2) ;\n\\draw (1,-3) -- (0,-4);\n\\node[above] at (0,1) {\\rule{0em}{1em}};\n\\node[above] at (0,0.2) {$\\mu = (\\lambda +k)-b$};\n\\node[right] at (1.4,-3) {$\\mu = -b $};\n\\node[right] at (-0.8,-1.9) {$\\mu = (\\lambda-k-\\epsilon_{m})$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\node[right] at (0.5, -3.6) {$-b$};\n\\node[right] at (-0.4,-1.0) {$m-b$};\n\\node[right] at (1,-3) {P};\n\\node[left] at (-1,-2) {R};\n\\node[right] at (0,0) {S};\n\\node[right] at (0,-4) {Q};\n\\end{tikzpicture}}\n\\caption{Graphs of $S^{1}(1,b;m)$ with possible locations of $C_{n}$ and $C_{-n}$}\n\\label{fig:PossibleGraphsToricExtensions}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n   \n\\begin{tikzpicture}\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (4,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (7,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (11,0) ellipse (0.1cm and 0.1cm);\n\\draw [dashed, blue] (0,0) -- (4,0); \n\\draw [dashed, blue] (7,0) -- (11,0);\n\\draw (0,-0.5) -- (0,0.5);\n\\draw (4,-0.5) -- (4,0.5);\n\\draw (7,-0.5) -- (7,0.5);\n\\draw (11,-0.5) -- (11,0.5);\n\\node[left] at (0,0) {$1\\,$}; \n\\node[above] at (0,0.5) {$b$}; \n\\node[right] at (4,0) {$\\,-1$}; \n\\node[above] at (4,0.5) {$m-b$}; \n\\node[right] at (0,-0.25) {$P$};\n\\node[left] at (4,-0.25) {$R$};\n\\node[above,blue] at (2,0) {$1$};\n\\node[left] at (7,0) {$1\\,$}; \n\\node[above] at (7,0.5) {$-b$}; \n\\node[right] at (11,0) {$\\,-1$};\n\\node[above] at (11,0.5) {$b-m$}; \n\\node[right] at (7,-0.25) {$Q$};\n\\node[left] at (11,-0.25) {$S$};\n\\node[above,blue] at (9,0) {$1$};\n\\end{tikzpicture}\n\\caption{Configurations of weights along $C_{\\pm n}$ when $a=1$}\n\\label{fig:Self-Intersection}\n\\end{figure}\n\nWhen $a=-1$, the same arguments apply to $S^{1}(-1,b;m)$ whose possible graphs are now of types (J), (K), and (L). The self-intersections of the curves $C_{\\pm 1}$ are now $\\pm |2b+m|$. We conclude that the action $S^{1}(-1,b;m)$ may extend to $\\mathbb{T}^2_{|2b+m|}$ whenever $m\\neq 0$ and $2\\lambda>|2b+m|+\\epsilon_{m}$.\n\nIn the special case $m=0$, any invariant sphere appearing in the graph of $S^{1}(a,b;0)$ has zero self-intersection. If $S^{1}(a,b;0)$ extends to $\\mathbb{T}^2_{n}$ with $n\\geq 2$, it again follows that the invariant curve $C_{n}$ of self-intersection $-n$ must have trivial isotropy. Consequently, $c=\\pm 1$, and at least one of the weights $a$ or $b$ must be $\\pm1$. If $a=1$, the possible graphs of $S^{1}(1,b;0)$ are the graphs (H) and (I) of Figure~\\ref{fig:PossibleGraphsToricExtensions}. Then, the same argument as before shows that the self-intersection of invariant curves are $0$ and $\\pm 2b$. Consequently, the action $S^{1}(1,b;0)$ may extend to $\\mathbb{T}^2_{|2b|}$ provided $2\\lambda>|2b|$. Similarly, when $a=-1$, the action $S^{1}(-1,b;0)$ may extend to $\\mathbb{T}^2_{|2b|}$ whenever $2\\lambda>|2b|$.\n\nWhen $m=0$ and $b=1$, then the possible self-intersections of invariant curves are $0$ and $\\pm 2a$. However, the two possible graphs are now of type (H) and (J). In both cases, the area of the tentative invariant curve $C_{\\pm n}$ connecting the two interior fixed points is negative. It follows that it is impossible to have $m=0$ and $b=1$. Similarly, if $b=-1$, the possible graphs are of type (I) and (K) and, as before, the area of the tentative invariant curve $C_{\\pm n}$ connecting the two interior fixed points is negative. Consequently, we cannot have $m=0$ and $b=\\pm 1$. This concludes the proof of the statement for actions whose fixed points are all isolated.\n\n\\begin{figure}\n\\centering\n~\n\\subcaptionbox*{}{}\n\\setcounter{subfigure}{7}\n\\subcaptionbox{When $m=0$, $a>0,~b=1$}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (1,-2) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (-1,-3) ellipse (0.1cm and 0.1cm); \n\\draw [dashed, blue] (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw[dashed, blue] (-1,-3) -- (1,-2);\n\\draw (0,0) -- (-1,-3) ;\n\\draw (1,-2) -- (0,-4);\n\\node[left] at (-1.1,-3) {Q};\n\\node[left] at (0.9,-1.9) {S};\n\\node[right] at (0,0) {R};\n\\node[above] at (0,0.2) {$\\mu = 1 + a (\\lambda -k-\\epsilon_{m})$};\n\\node[right] at (1.2,-2) {$\\mu = a(\\lambda + k) $};\n\\node[right] at (-0.9,-3) {$\\mu = 1$};\n\\node[right] at (0,-4) {P};\n\\node[right] at (0.5, -3) {$a$};\n\\node[left] at (-0.5,-1.5) {$a$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\end{tikzpicture}}\n\\setcounter{subfigure}{9}\n\\subcaptionbox{When $m=0$, $a<0,~b=1$\\label{fig:a<0|b=1}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-3) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-3);\n\\draw (-1,-2) -- (0,-4);\n\\draw [dashed, blue] (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw[dashed, blue] (-1,-2) -- (1,-3);\n\\node[above] at (0,1) {\\rule{0em}{1em}};\n\\node[above] at (0,0.2) {$\\mu = 1 - a (\\lambda +k)$};\n\\node[right] at (1.4,-3) {$\\mu = 1 $};\n\\node[above] at (-1,-2) {$\\mu = -a(\\lambda+k)$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\node[right] at (-1.2,-3.2) {$-a$};\n\\node[right] at (0.6,-1.3) {$-a$};\n\\node[left] at (0.9,-3) {R};\n\\node[left] at (-1,-2.2) {P};\n\\node[right] at (0,0) {Q};\n\\node[right] at (0,-4) {S};\n\\end{tikzpicture}}\n\\caption{Graphs of $S^{1}(a,1;0)$ with impossible locations of $C_{n}$ and $C_{-n}$}\n\\label{fig:ImpossibleGraphsToricExtensions}\n\\end{figure}\n\nNow suppose $S^{1}(a,b;m)$ has non-isolated fixed points, that is, $a=0$, or $b=0$, or $b=am$. As the moment map values of graphs (A), (B), (E), and (F) depend only on $m$, it is impossible to get the same graphs from another action $S^{1}(c,d;n)$ with $n\\neq m$. The same remark applies to the area labels of graphs (C) and (D), showing that $S^{1}(a,b;m)$ only extends to $\\mathbb{T}^2_{m}$.\n\\end{proof}\n\n\nIt remains to investigate whether the action $S^{1}(\\pm 1,b;m)$ extends to $\\mathbb{T}^2_{|2b-m|}$ when $2\\lambda > |2b-am|+\\epsilon_{m}$ and $b \\not\\in \\{0,am\\}$. A straightforward but lengthy comparison of the possible graphs of $S^{1}(\\pm1,b;m)$ and $S^{1}(\\pm,d;|2b-am|)$ shows that this is always the case and yields the equivalences stated in the following two corollaries. The graphs giving the first equivalence are shown in Figure~\\ref{fig:ExampleTwoToricExtensions}. The other cases are left to the reader.\n\n\\begin{cor} \\label{cor:CircleExtensionsWith_a=1}\nConsider the action $S^1(1,b;m)$ and suppose $2\\lambda > |2b-m|+\\epsilon_{m}$. Then under the following numerical conditions on $b$ and $m$, the  $S^1(1,b;m)$ action extends to the toric action $\\mathbb{T}^2_{|2b-m|}$ and is equivariantly symplectomorphic to the following subcircle in $\\mathbb{T}^2_{|2b-m|}$\\,:\n\\begin{enumerate}\n    \\item if $b>0$ and $b>m$, then $S^1(1,b; m)$ is equivalent to $S^1(1,b; |2b-m|)$;\n    \\item if $b>0$, $m>b$, and $2b-m < 0$, then $S^1(1,b; m)$ is equivalent to $S^1(1,-b; |2b-m|)$;\n    \\item if $b>0$, $m>b$, and $2b-m > 0$, then $S^1(1,b; m)$ is equivalent to $S^1(1,b;|2b-m|)$;\n    \\item finally, if $b<0$, then $S^1(1,b; m)$ is equivalent to $S^1(1,-b;|2b-m|)$.\\qed\n\\end{enumerate}\n\\end{cor}\n\n\n\\begin{cor}\\label{cor:CircleExtensionsWith_a=-1}\nConsider the $S^1$ actions $S^1(-1,b;m)$ on $(S^2 \\times S^2,\\omega_\\lambda)$  and suppose $2\\lambda > |2b+m|$. Then under the following numerical conditions on $b$ and $m$, the  $S^1(-1,b;m)$ action extends to the toric action $\\mathbb{T}^2_{|2b+m|}$ and is equivariantly symplectomorphic to the following subcircle in $\\mathbb{T}^2_{|2b+m|}$\\,:\n\\begin{enumerate}\n\\item if $b<0$ and $m>-2b$, then $S^1(-1,b;m)$ is equivalent to  $S^1(-1,-b; |2b+m|)$;\n\\item if $b<0$, $m>-b$, and $-2b>m$, then $S^1(-1,b;m)$ is equivalent to \n$S^1(-1,b; |2b+m|)$;\n\\item if $b<0$ and $-b>m$, then $S^1(-1,b;m)$ is equivalent to $S^1(-1,b; |2b+m|)$;\n\\item if $b>0$, then $S^1(-1,b; m)$ is equivalent to $S^1(-1,-b; |2b+m|)$.\\qed\n\\end{enumerate}\n\\end{cor}\n\n\\begin{figure}[H]\n\\centering\n\\subcaptionbox*{$S^1(1,b;m)$ of type (H)\\label{fig:ExtendedGraph1}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (1,-2) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm); \n\\draw [fill] (-1,-3) ellipse (0.1cm and 0.1cm); \n\\draw (0,0) -- (1,-2);\n\\draw (-1,-3) -- (0,-4);\n\\draw[dashed, blue] (0,0) -- (-1,-3) ;\n\\draw[dashed, blue] (1,-2) -- (0,-4);\n\\node[right] at (-0.9,-3) {Q};\n\\node[left] at (0.9,-2) {S};\n\\node[right] at (0,0) {R};\n\\node[above] at (0,0.2) {$\\mu = b +  (\\lambda-k-\\epsilon_{k})$};\n\\node[right] at (1.3,-2) {$\\mu = (\\lambda+k) $};\n\\node[left] at (-1.4,-3) {$\\mu = b$};\n\\node[right] at (0,-4) {P};\n\\node[right] at (-1,-3.7) {$b$};\n\\node[right] at (0.5, -3) {$a=1$};\n\\node[left] at (-0.5,-1.5) {$a=1$};\n\\node[right] at (0.7,-1) {$b-m$};\n\\node[below] at (0,-4.2) {$\\mu = 0$};\n\\end{tikzpicture}\n}\n~\n\\subcaptionbox*{$S^1(1,b;2b-m)$ of type (G)\\label{sndstrata}}\n[.45\\linewidth]\n{\\begin{tikzpicture}[scale=0.9, every node\/.style={scale=0.9}]\n\\draw [fill] (0,0) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (1,-1.5) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (0,-4) ellipse (0.1cm and 0.1cm);\n\\draw [fill] (-1,-2.5) ellipse (0.1cm and 0.1cm);\n\\draw (0,0) -- (1,-1.5);\n\\draw[dashed, blue] (1,-1.5) -- (-1,-2.5);\n\\draw[dashed, blue] (0,0) .. controls (-4,-1.5) and (-2,-3.5) .. (0,-4);\n\\draw (-1,-2.5) -- (0,-4);\n\\node[right] at (-0.35,-3.2) {$b$};\n\\node[right] at (- 1,-1.8) {$a=1$};\n\\node[left] at (-2.2,-1.5) {$a=1$};\n\\node[right] at (0.7,-0.5) {$(2b-m)-b=b-m$};\n\\node[above] at (0,0.2) {$\\mu = \\lambda+\\frac{(2b-m)-\\epsilon_{k}}{2}= b+\\lambda - k-\\epsilon_{k}$};\n\\node[right] at (-0.6,-2.5) {$\\mu = b$};\n\\node[right] at (0.6,-1.8) {$\\mu=b+\\lambda-\\frac{(2b-m)-\\epsilon_{k}}{2}-\\epsilon_{k}$};\n\\node[right] at (0.91,-2.25) {$=\\lambda+k$};\n\\node[below] at (0, -4.2) {$\\mu = 0$};\n\\node[left] at (-1,-2.5) {Q};\n\\node[left] at (0.9,-1.4) {R};\n\\node[right] at (0,0) {S};\n\\node[right] at (0,-4) {P};\n\\end{tikzpicture}}\n\\caption{The equivalence $S^{1}(1,b;m)\\sim S^{1}(1,b;2b-m)$ when $b>m$}\n\\label{fig:ExampleTwoToricExtensions}\n\\end{figure}\n\n\n\n\n\n\\chapter{Action of \\texorpdfstring{$\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$}{Symp(S2xS2)} on \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda}$}{J\\hat{}S1}}\\label{ChapterActionOfSymp}\nIn this chapter we show that the space of $S^1$ invariant compatible almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ can be decomposed into strata each of which being homotopy equivalent to a homogeneous space under the action of the equivariant symplectomorphism group. \n\n\n\n\n\n\n\\section{\\texorpdfstring{$J$}{J}-Holomorphic Preliminaries}\n\nWe first recall a few facts about compatible almost complex structures and associated $J$-holomorphic curves.\n\n\n\\begin{defn}[Compatible almost complex structures] \nAn almost complex structure $J$ on a symplectic manifold $(M,\\omega)$ is said to be compatible with $\\omega$ if $\\omega(u,Ju)>0$ and $\\omega(Ju, Jv)=\\omega(u,v)$ for all non-zero $u,v\\in T_* M$.\n\\end{defn}\n\n\\begin{lemma}\nThe space $\\mathcal{J}_\\omega = \\mathcal{J}(M,\\omega)$ of all compatible almost complex structures on a symplectic manifold $(M,\\omega)$ is non-empty and contractible.\n\\end{lemma}\n\n\\begin{defn}{\\textit{$J$-holomorphic spheres}:}\nLet $(M,\\omega)$ be a symplectic manifold endowed with a compatible almost complex structure $J$. A rational $J$-holomorphic map, also called a parametrized $J$-holomorphic sphere, is a $C^\\infty$ map\n\\[\nu: ({S}^2, j) \\longrightarrow (M,\\omega,J)\n\\]\nsatisfying the Cauchy-Riemann equation\n\\[\n\\bar\\partial_J(u)=\\frac{1}{2}(du\\circ j - J\\circ du) = 0\n\\]\nwhere $j$ is the usual complex structure on the sphere. The image of a $J$-holomorphic rational map is called a rational $J$-holomorphic curve or simply a $J$-curve.\n\\end{defn}\n\n\n\n\\begin{defn}[Multi-covered and simple maps]\nWe say that a $J$-holomorphic map $u:\\mathbb{C}P^1 \\longrightarrow (M,J)$ is multi-covered if $u = {u}^\\prime \\circ f$, where $f:\\mathbb{C}P^1 \\to \\mathbb{C}P^1$ is a holomorphic map of degree greater than one and where $u':\\mathbb{C}P^1 \\to (M,J)$ is a $J$-holomorphic map. We call a $J$-holomorphic map simple if it is not multi-covered.\n\\end{defn}\n\n\\begin{remark}\nWe usually assume that a $J$-holomorphic map is somewhere injective, meaning that $\\exists z\\in {S}^2$ such that $du_z \\neq 0$ and $u^{-1}u(z) = z$. In particular, somewhere injective maps do not factor through multiple covers $h:S^2\\to S^2$. \n\\end{remark}\n\n\\begin{defn}[Moduli spaces of $J$-holomorphic maps or curves]\nLet $(M,\\omega)$ be a symplectic manifold and let $J \\in \\mathcal{J}_\\omega$. Given $A \\in H_2(M, \\mathbb{Z})$ we denote by $\\widetilde{\\mathcal{M}}(A,J)$ the space of all $J$-holomorphic, somewhere injective maps representing the homology class $A$. The Mobius group $G =\\PSL(2,\\mathbb{C})$ acts freely on this space by reparametrization and the quotient space $\\mathcal{M}(A,J):=\\widetilde{\\mathcal{M}}(A,J)\/G$ is called the moduli space of (unparametrised) $J$-curves in class $A$.\n\\end{defn}\n\nIn dimension 4, the geometric properties of $J$-holomorphic curves are, to a large extend, controlled by homological data. As a result, many properties of complex algebraic curves in complex algebraic surfaces extend to $J$-holomorphic curves in $4$-dimensional symplectic manifolds. Below we list the key properties of $J$-holomorphic curves we will be relying on.\n\n\\begin{thm}[Positivity]\\label{thm_Positivity}Let $(M,\\omega)$ be a 4-dimensional symplectic manifold. If a homology class A $\\in H_2(M,\\mathbb{Z})$ is represented by a nonconstant J-curves for some $J \\in \\mathcal{J_\\omega}$ then $\\omega(A) > 0$.\n\\end{thm}\n\n\\begin{thm}[Fredholm property and automatic regularity.]\\label{thm_Regularity}Let $(M,\\omega)$ be a 4-dimensional symplectic manifold.  \nThen the universal moduli space\n\\[\n\\widetilde{\\mathcal{M}}(A,\\mathcal{J}_{\\omega}) := \\bigcup_{J \\in {\\mathcal{J}_{\\omega}}} \\widetilde{\\mathcal{M}}(A,J)\n\\]\nwith $C^l$-topology ($l \\geq 2$) is a smooth Banach manifold and the projection map\n\\[\n\\pi_A: \\widetilde{\\mathcal{M}}(A,\\mathcal{J}_{\\omega}) \\longrightarrow \\mathcal{J}_{\\omega}\n\\]\nis a Fredholm map of index $2(c_1(A) + 2)$ where $c_1 \\in H^2(M,\\mathbb{Z})$ is the first chern class of $(TM, J)$ (note that the Chern class is independent of choice of $J \\in \\mathcal{J}_{\\omega}$).  An almost complex structure is said to be regular for the class $A$ if it is a regular value for the projection $\\pi_A$.  If this is the case then the moduli spaces $\\widetilde{\\mathcal{M}}(A,J)$ and $\\mathcal{M}(A,J)$ are smooth manifolds of dimensions $2(c_1(A)+2)$ and $2(c_1(A)-1)$ respectively.  The set of regular values $\\mathcal{J} \\in \\mathcal{J}_\\omega$ is a subset of second category and is denoted by $\\mathcal{J}_{\\omega}^{\\textrm{reg}}(A)$. If $J \\in  J_\\omega$ is integrable and $S$ is an embedded $J$-holomorphic sphere with self-intersection number $[S]\\cdot [S] \\geq -1$, then $J$ is regular for the class $[S]$. In dimension $4$, the same conclusion holds without the integrability assumption.\n\\end{thm}\n\n\\begin{defn}[Cusp Curves] \\label{defn_cusp}\nLet $(M,\\omega)$ be a symplectic manifold. Let $J \\in J_\\omega$. A $J$-holomorphic cusp curve $C$ is a connected finite union of $J$-holomorphic curves \n\\[C = C_1 \\cup C_2 \\ldots \\cup C_k\\] where $C_i = u_i(\\mathbb{C}P^1)$ and $u_i: \\mathbb{C}P^1 \\to (M,J)$ is a (possibly multi-covered) $J$-holomorphic map.\n\\end{defn}\n\n\\begin{thm}[Gromov's compactness theorem]\\label{thm_Compactness} Let $(M,\\omega)$ be a compact symplectic manifold. Let $J_n \\in \\mathcal{J}_{\\omega}$ be a sequence converging to $J$ in the $C^\\infty$ topology and let $S_i$ be $J_i$-holomorphic spheres of bounded symplectic area $\\omega(S_i)$. Then there is a subsequence of the $S_i$ which converges weakly to a $J$-holomorphic curve or cusp-curve $S$. In particular if all the $S_i$'s belong to the class $A$, then $S$ also belongs the class $A$, and any cusp curve defines a homological decomposition of $A=\\sum_i A_i$ such that $\\omega(A_i)>0$.\n\\end{thm}\n\n\\begin{thm}[Positivity of intersections]\\label{thm_PositivityIntersections} Let $J \\in \\J_{\\om_\\lambda}$ and $A$, $B$ be two distinct $J$-holomorphic curves in a $4$-dimensional manifold. Then they intersect at only finitely many points and each point contributes positively to the intersection multiplicity $[A]\\cdot[B]$. Moreover,  $[A]\\cdot[B]=1$ iff the curves intersect transversally at exactly one point, while $[A]\\cdot[B]= 0$ iff the curves are disjoint.\n\\end{thm}\n\nAs a corollary of Positivity of intersections we have the following result under the presence of a group action. \n\\begin{cor}\\label{cor_pos}\nLet $(M,\\omega)$ be a symplectic 4-manifold and let $G$ be a compact Lie group acting symplectically on $M$. Suppose that $G$ acts trivially on homology. Let $\\mathcal{J}^{G}$ denote the space of $\\omega$ tame (or compatible) almost complex structures and let $C$ be a $J$ holomorphic curve for some $J \\in \\mathcal{J}^G$. \nThen,\n\\begin{enumerate}\n    \\item if C has  negative self intersection, then $g \\cdot C = C$ for all $g \\in G$.\n    \\item If C has zero self intersection, then $g \\cdot C = C$ or $g\\cdot C \\cap C = \\emptyset$ for all $g \\in G$.\n\\end{enumerate} \n\\end{cor}\n\n\\begin{thm}[Adjunction formula]\\label{thm_Adjunction} Let $u: ({S}^2, j) \\longrightarrow (M^4,J)$ be a somewhere injective $J$-holomorphic map representing the homology class $A$ in a $4$-dimensional manifold. Define the virtual genus of $A$ as\n\\[\ng_v(A) = 1+\\frac{1}{2}([A]\\cdot[A]- c_1(A))\n\\]\nwhere $c_1(A) = \\langle c_1(TM,J), A \\rangle$. Then $g_v(A)\\geq 0$ with equality if, and only if, the map $u$ is an embedding.\n\\end{thm}\n\n\n\\section{\\texorpdfstring{$J$}{J}-holomorphic spheres in \\texorpdfstring{$S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$}{S2xS2 and CP2\\#CP2}}\n\nIn this section, will show how the existence of certain $J$-holomorphic spheres in ruled $4$-manifolds induces a natural partition of the space $\\mathcal{J}_\\omega$. We shall state all the relevant results, and refer the reader to Chapter 9 of \\cite{McD} for more details.\n\n\\begin{cor}\\label{FSimple}\nLet $\\lambda = l + \\delta$ where $l \\in \\mathbb{N}$ and $0<\\delta\\leq 1$.\nThen we have \n\\begin{enumerate}\n    \\item Any $J$-holomorphic representative of the class $F$ is a simple curve.\n    \\item The only $J$-holomorphic decomposition of the class $B$ are of the form $B = (B-kF) + kF$, where $0\\leq k\\leq\\ell$. In this decomposition, the $J$-holomorphic representative of the class $(B-kF)$ is an embedded sphere, while the class $kF$ may be represented by a collection of (possibly multiply covered) spheres representing multiples of the class $F$.\\qed\n\\end{enumerate}\n\\end{cor}\n\n\n\n\\begin{prop}\\label{prop_FIndecomposable}\nThe moduli space $\\widetilde{\\mathcal{M}}(F,J) \\neq \\emptyset$ for all $J \\in  J_\\omega$. In particular, for every compatible almost complex structure $J\\in\\mathcal{J}_{\\omega_\\lambda}$, and for any given point $p\\in S^2\\times S^2$, there is a unique embedded $J$-holomorphic sphere representing the class $F$ passing through~$p$.\\qed\n\\end{prop}\n\n\nWe also have an analogous result for $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$.\n\\begin{thm}\nGiven any point $p \\in \\CP^2\\# \\overline{\\CP^2}$, and any $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}$, there is a $J$-holomorphic curve in the class $F:=L-E$ passing through $p$. \\qed\n\\end{thm}\n\nThe following  Theorem due to Abreu and McDuff~\\cite{MR1775741} tells us about the decomposition of the space of compatible almost complex structures on $(S^2 \\times S^2,\\omega_\\lambda)$ into finitely many strata. \n\\begin{thm} \\label{strata}\nLet $\\mathcal{J}_{\\omega_\\lambda}$ denote the space of all compatible almost complex structures (not necessarily invariant) for the form $\\omega_\\lambda$, on $S^2 \\times S^2$ then the space $\\mathcal{J}_{\\omega_\\lambda}$ admits a finite decomposition into disjoint Fr\u00e9chet manifolds of finite codimensions\n\\[\n\\mathcal{J}_{\\omega_\\lambda} = U_0 \\sqcup U_2 \\sqcup U_4 \\ldots \\sqcup U_{2n}\n\\]\nwhere $2n= \\lceil 2\\lambda \\rceil -1$ and $\\lceil \\lambda \\rceil$ is the unique integer $l$ such that $l < \\lambda \\leq l+1$ and where\n\\[\nU_{2i} := \\left\\{ J \\in \\mathcal{J}_{\\omega_\\lambda}~|~ D_{2i}=B-iF \\in H_2(S^2 \\times S^2,\\mathbb{Z})\\text{~is represented by a $J$-holomorphic sphere}\\right\\}\n\\]\n\\end{thm}\n\nA completely analogous description holds true for $\\CP^2\\# \\overline{\\CP^2}$.\n\n\\begin{thm}\\label{strata-CCC}\nLet $\\mathcal{J}_{\\omega_\\lambda}$ denote the space of all compatible almost complex structures (not necessarily invariant) for the form $\\omega_\\lambda$, then the space $\\mathcal{J}_{\\omega_\\lambda}$ admits a finite decomposition into disjoint Fr\u00e9chet manifolds of finite codimensions\n\\[\n\\mathcal{J}_{\\omega_\\lambda} = U_1 \\sqcup U_3 \\sqcup U_5\\ldots \\sqcup U_{2n-1}\n\\]\nwhere $2n  = \\lceil 2\\lambda \\rceil -1$, and $\\lceil \\lambda \\rceil$ is the unique integer $l$ such that $l < \\lambda \\leq l+1$ and where\n\\[ \nU_{2i-1} := \\left\\{ J \\in \\mathcal{J}_{\\omega_\\lambda}~|~ D_{2i-1}=B-iF \\in H_2(\\CP^2\\# \\overline{\\CP^2},\\mathbb{Z})\\text{~is represented by a $J$-holomorphic sphere}\\right\\}\n\\]\n\\end{thm}\n\n\\begin{remark}\nWe label the strata in $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ by the homological self-intersection of the classes $D_s$.\n\n\\end{remark}\n\n\\begin{remark}\\label{remark-starta-toric}\nWe note that for both $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$,  there is a canonical integrable almost complex structure $J_s$ in the strata $U_s$ coming from realizing $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ as the $s^{\\text{th}}$- Hirzebruch surface $W_s$ of section 1.2. Further recall that associated to each $J_s$ we have a unique $J_s$-holomorphic Hamiltonian toric action $\\mathbb{T}_s$. Thus the set of possible equivalence classes of toric actions on $S^2 \\times S^2$ and $\\CP^2\\# \\overline{\\CP^2}$ are in one-to-one correspondence with the strata in the decomposition of $\\mathcal{J}_{\\omega_\\lambda}$. This fact will be crucial in our later analysis of centralizer subgroups. \n\\end{remark}\n\n\n\n\\section{Intersection of \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda}$}{J\\hat{}S1} with the strata}\\label{section:intersection}\nIn this section we shall answer the question as to which strata the space of invariant almost complex structures for a given action $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2}, \\omega_\\lambda)$ intersects.\\\\\n\nIn what follows, we will use the following simple observation several times. Let $(M,\\omega)$ be a simply connected symplectic $4$-manifold. There is a left-exact sequence\n\\begin{equation}\n1 \\to \\Symp_h(M,\\omega) \\to \\Symp(M,\\omega) \\to \\Aut_{c_1,\\omega}\\left(H_2(M,\\mathbb{Z})\\right)\\label{Sequence:ActionOnHomology}\n\\end{equation}\nwhere $\\Symp_h(M,\\omega)$ is the subgroup of symplectomorphisms acting trivially on homology, and where $\\Aut_{c_1,\\omega_\\lambda}\\left(H_2(M,\\mathbb{Z})\\right)$ is the group of automorphisms of $H_2(M,\\mathbb{Z})$ that preserve the intersection product and the Poincar\u00e9 duals of the cohomology classes $c_1(TM)$ and $[\\omega_\\lambda]$. This later group is the identity for $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ with $\\lambda \\geq 1$ and for $(S^2 \\times S^2,\\omega_\\lambda)$ with $\\lambda > 1$. In the case of $(S^2 \\times S^2,\\omega_\\lambda)$ with $\\lambda = 1$, the group $\\Aut_{c_1,\\omega_\\lambda}\\left(H_2(M,\\mathbb{Z})\\right)$ is equal to $\\mathbb{Z}_2$ and is generated by the symplectomorphism that swaps the two $S^2$ factors. Consequently, for any symplectically ruled rational surface, the above sequence is also right-exact and splits.\n\n \n\n\\begin{lemma}\\label{lemma:ActionOnHomology} We have the following equalities among symplectomorphism groups:\n\\begin{itemize}\n\\item $\\Symp(S^2 \\times S^2,\\omega_\\lambda) = \\Symp_h(S^2 \\times S^2,\\omega_\\lambda)\\ltimes\\mathbb{Z}_2$ when $\\lambda=1$,\n\\item $\\Symp(S^2 \\times S^2,\\omega_\\lambda) = \\Symp_h(S^2 \\times S^2,\\omega_\\lambda)$ when $\\lambda>1$, and\n\\item $\\Symp_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) = \\Symp(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ for all $\\lambda\\geq 1$.\n\\end{itemize}\\qed\n\\end{lemma}\n\n\n\\begin{prop}\\label{prop:ToricExtensionCorrespondance}\nFix a circle action $S^1(a,b;m)$. Then the space of invariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the stratum $U_n$ iff the circle action $S^1(a,b; m)$ extends to the toric action $\\mathbb{T}^2_n$. \n\\end{prop}\n\\begin{proof}\nIf $\\lambda = 1$, the result is immediate as the only toric action is $\\mathbb{T}^2_0$. \n\nLet $\\lambda > 1$ and suppose the action $S^1(a,b;m)$ extends to $\\mathbb{T}^2_n$. This means that there exists a circle action $S^1(c,d; n) \\subset \\mathbb{T}^2_n$ which is equivariantly symplectomorphic to the action $S^1(a,b;m)$ via some  symplectomorphism $\\phi$. Let $J_{n}$ denote the standard complex structure in the stratum $U_n$. By Lemma \\ref{lemma:ActionOnHomology}, when $\\lambda > 1$, the group $\\Symp_h(S^2 \\times S^2,\\omega_\\lambda)$ is equal to $\\Symp(S^2 \\times S^2,\\omega_\\lambda)$ so that $\\phi$ preserves homology. Consequently, $\\phi^*J_{n}$  belongs to the stratum $U_n$ and is invariant with respect to the $S^1(a,b;m)$ action, showing that the space $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the stratum $U_n$.\n\nConversely, suppose that $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_n$. If $n\\geq 1$, then there exists an invariant, embedded, symplectic sphere $C_{n}$ of self-intersection $-n$. Arguing as in Proposition~\\ref{prop:AtMostTwoExtensions} and Corollaries~\\ref{cor:CircleExtensionsWith_a=1} and~\\ref{cor:CircleExtensionsWith_a=-1}, we conclude that $S^1(a,b;m)$ must extend to the torus $\\mathbb{T}^2_n$. If $n=0$, there there exist invariant spheres representing the homology classes $B$ and $F$ passing to a common fixed point. Again, this implies that $S^1(a,b;m)$ extends to the torus $\\mathbb{T}^2_0$. Alternatively, we can adapt the proofs of Lemma~\\ref{lemma:EvaluationFibrationConfigurations} in the present document and of Proposition~4.7 in~\\cite{Liat} to show that if an invariant $J$ is in the stratum $U_n$ the circle action $S^1(a,b; m)$ extends to the toric action $\\mathbb{T}^2_n$.\n\\end{proof}\n\n\nThe following corollaries immediately follow from Proposition~\\ref{prop:AtMostTwoExtensions} and Corollaries~\\ref{cor:CircleExtensionsWith_a=1} and~\\ref{cor:CircleExtensionsWith_a=-1}.\n\n\\begin{cor}\\label{cor:lambda=1_IntersectingOnlyOneStratum}\nSuppose $\\lambda=1$. Then the space $\\mathcal{J}_{\\omega_1}$ of compatible, almost-complex structures on $(S^2 \\times S^2,\\omega_1)$ is made of only one stratum. In particular, any Hamiltonian circle action extends to the toric action $\\mathbb{T}^2_0$ and the subspace $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ of $S^1$ invariant almost-complex structures is contractible. \\qed\n\\end{cor}\n\n\\begin{cor}\\label{cor:IntersectingOnlyOneStratum}\nSuppose $\\lambda >1$. Consider an $S^1(a,b;m)$ action $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ depending on whether $m$ is even or odd. Under the following numerical conditions on $a,b,m,\\lambda$, the space $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects only the stratum $U_{m}$:\n\\begin{itemize}\n    \\item when $a\\neq\\pm 1$;\n    \\item when $b=0$ or  $b=am$;\n    \\item when $a = \\pm 1$ and $2 \\lambda \\leq |2b-m|+\\epsilon_{m}$.\\qed\n\\end{itemize}\n\\end{cor}\n\n\\begin{cor} \\label{cor:IntersectingTwoStrata}\nConsider the $S^1(\\pm 1,b,m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$, then $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects exactly two strata. More precisely,\n\\begin{enumerate}\n    \\item if $a=1$, $b \\neq\\{0,m\\}$, $\\lambda >1$, and $2\\lambda > |2b-m|+\\epsilon_{m}$, then the space of $S^1(1,b;m)$-equivariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the two strata $U_m$ and $U_{|2b-m|}$.\n    \\item If $a=-1$, $b \\neq\\{0,-m\\}$, $\\lambda >1$, and $2\\lambda > |2b+m|+\\epsilon_{m}$, then the space of $S^1(-1,b;m)$-equivariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the two strata $U_m$ and $U_{|2b+m|}$.\\qed\n\\end{enumerate} \n\\end{cor}\n\n\n\n\n\n\n\\section{Symplectic actions of compact abelian groups on~\\texorpdfstring{$\\mathbb{R}^4$}{R4}}\n\nIn order to study the action of the the equivariant symplectomorphism group $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ on each invariant stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_k$, we will need to understand the equivariant topology of linearised symplectic  actions. The main result is Theorem~\\ref{thm:EqGr} which is an equivariant version of Gromov's Theorem on the contractibility of the group of compactly supported symplectomorphisms of star-shaped domains of $\\mathbb{R}^{4}$. This relies on Lemma~\\ref{lemma:remove} and Proposition~\\ref{prop:linearize} that were proven by W. Chen in the unpublished manuscript \\cite{ChenUnpub}. For completeness, we shall reproduce their proof here.\n\nLet $G$ be a compact group acting effectively and symplectically  on $ \\mathbb{C}^2 = \\mathbb{R}^4$ with the symplectic form $\\omega_0:= dx_1 \\wedge dy_1 +  (dx_2 \\wedge dy_2)$. We say it acts linearly if it acts as a subgroup of $\\U(2) \\subset \\Sp(4)$.\n\n\\begin{lemma}[Chen]\\label{lemma:remove}\nLet $G$ be a compact group acting linearly on $(\\mathbb{R}^4, \\omega_0)$. Suppose $V$ is a $G$ invariant, compact, and star-shaped neighbourhood of $0$. Let $f: \\mathbb{R}^4 \\setminus V \\to \\mathbb{R}^4$ be a $G$-equivariant symplectic embedding which is the identity near infinity. Then, for every $G$-invariant neighbourhood $W$ of $V$, there exists a $G$-equivariant symplectomorphism $g:\\mathbb{R}^4 \\to \\mathbb{R}^4$ such that $g|_{\\mathbb{R}^4\\setminus W} = f$.  \n\\end{lemma}\n\n\\begin{proof}\nAs $0 \\in int(V)$ and as $f$ is the identity near infinity, there exist $T>0$ such that $f(Tx) = Tx$ for all $x \\in \\mathbb{R}^4 \\setminus~ V$. Define $f_t(x) = \\frac{f(tx)}{t}$ for $1 \\leq t \\leq T$. As the $G$ action is linear, we observe that $f_t$ is equivariant for all $t \\in [1,T]$. By construction, $f_1 = f$, $f_T = id$  and $f_t^*\\omega_0 = \\omega_0$ for all $t$. Thus there are compact sets $V_t = f_t(V)$ and open neighbourhoods $W_t= f_t(W)$ of $V_t$ such that the restriction  $f_t: \\mathbb{R}^4 \\setminus V \\to \\mathbb{R}^4 \\setminus V_t$ and $f_t: \\mathbb{R}^4 \\setminus W \\to \\mathbb{R}^4 \\setminus W_t$  are diffeomorphic. As $G$ acts linearly, each of the sets $W_t$ and $V_t$ are $G$-invariant. \n\nDefine $X_t$ as the vector field that satisfies $\\frac{d}{dt} f_t = X_t \\circ f_t$ and consider the one form $\\alpha_t = i_{X_t}\\omega_0$. As $f_t$ is $G$ equivariant and symplectic, both $X_t$ and $\\alpha_t$ are $G$-invariant. Let $H_t: \\mathbb{R}^4 \\setminus V_t \\to \\mathbb{R}$ be a one parameter family of Hamiltonians that are $G$-invariant and that satisfy $\\alpha_t= dH_t$. Since $f_t$ is the identity near infinity, this implies that $H_t$ is constant near infinity and we can take this constant to be 0. \n\nFinally, we can take a family of $G$-invariant bump functions $\\rho_t: \\mathbb{R}^4 \\to [0,1]$ such that $\\rho_t \\equiv 0$ in a neighbourhood of $V_t$ and $\\rho_t \\equiv 1$ on $\\mathbb{R}^4 \\setminus W_t$. Then the Hamiltonian $\\rho_t H_t: \\mathbb{R}^4 \\to \\mathbb{R}$ is defined on the whole $\\mathbb{R}^4$ and is also $G$-invariant. The Hamiltonian isotopy $g_t$ generated by $\\rho_t H_t$ is $G$ equivariant for all $1\\leq t\\leq T$ and satisfies the properties $g_T = \\id$ and $g_1|_{\\mathbb{R}^4 \\setminus W} = f$. Consequently, $g_1$ is the symplectomorphism $g$ we were looking for.\n\\end{proof}\n\nAny unitary representation of a compact abelian group $G$ on $\\mathbb{C}^2$ induces a splitting into eigenspaces $\\mathbb{C}^2 = \\mathbb{C}_1 \\bigoplus \\mathbb{C}_2$. This simple fact turns out to be essential in our treatment of the equivariant Gromov's Theorem. Consequently, from now on, we assume that the group $G$ is abelian.\n\n\\begin{prop}[Chen]\\label{prop:linearize}\nLet $V\\subset\\mathbb{R}^{4}$ be a compact star-shaped neighbourhood of $\\{0\\}$ and let $\\omega$ be a symplectic form on $\\mathbb{R}^{4}$ such that $\\omega = \\omega_0$ outside some smaller open neighbourhood of the origin $U\\subset V$. Let $G$ be a compact abelian group acting on $(V,\\omega)$ via symplectomorphisms that are linear near the boundary of V. Then the G action is conjugate to a linear symplectic action of $G$ on $(V,\\omega_0)$ by a diffeomorphism $\\Phi$ which is the identity near the boundary and which satisfies $\\Phi^*\\omega = \\omega_0$.\n\\end{prop}\n\n\\begin{proof}\nIdentify $\\mathbb{R}^4$ with $\\mathbb{C}^2$. The linear action near $\\partial V$ extends to a unitary action on $\\mathbb{R}^4$. As $G$ is abelian this linear action splits into two eigenspaces  $\\mathbb{C}_{1} \\oplus \\mathbb{C}_{2}$. We can then compactify $\\mathbb{C}^{2}$ along these eigenspaces to obtain $\\mathbb{C}P^{1}\\times \\mathbb{C}P^{1}$ (diffeomorphic to $S^2 \\times S^2$) equipped with two symplectic forms $\\tilde\\omega$ and $\\tilde\\omega_{0}$ induced from $\\omega$ and $\\omega_{0}$. The compactified space inherits two actions of $G$, namely, $\\rho: G \\hookrightarrow \\Symp(S^2 \\times S^2,\\tilde\\omega)$ coming from extending the $G$ action on $\\mathbb{R}^{4}$ and $\\rho_{\\lin}:G \\hookrightarrow \\Symp(S^2 \\times S^2,\\tilde\\omega)$ which extends the linear action near $\\partial V$.  \n\\\\\n\n\n\nBy construction, there exists a star-shaped subset $V_1 \\subset V$ such that both the action $\\rho$ and $\\rho_{\\lin}$ agree on $\\mathbb{C}^2 \\setminus V_1$. Note that the point at infinity $p:=(\\infty,\\infty)$ is a fixed point for both the action $\\rho$ and $\\rho_{lin}$. Choose a $\\tilde\\omega$ compatible $G$-invariant almost complex structure $J$ on $S^2 \\times S^2$ which is standard in some neighborhood $X_\\epsilon:= (S^2 \\times D_\\epsilon) \\cup (S^2 \\times D_\\epsilon)$ of the wedge $(S^2\\times\\{\\infty\\})\\vee (\\{\\infty\\}\\times S^2)$ (where $D_\\epsilon$ is a small disk of radius $\\epsilon$ at $\\infty$). As $(S^2\\times\\{\\infty\\})$ and $(\\{\\infty\\}\\times S^2)$ are $J$-holomorphic spheres representing the classes $B$ and $F$ respectively, we conclude that $J$ belongs to the stratum $U_{0}$ so that there exists foliations $\\mathcal{B}_J$ and $\\mathcal{F}_J$ by embedded $J$-holomorphic spheres in the classes $B$ and $F$. Given any $q=(z,w) \\in S^2 \\times S^2$, let $u_w$ denote the unique curve in the class B  passing through $(0,w)$ and, similarly, let $v_z$ be the curve in class F  passing through $(z,0)$. We can define a self-map of $S^2 \\times S^2$ by setting\n\\begin{align*}\n\\Psi_J: S^2 \\times S^2 &\\longrightarrow S^2 \\times S^2  \\\\\n(z,w) &\\longmapsto u_w \\cap v_z\n\\end{align*}\nAs explain in~\\cite{McD} Chapter 9, this map $\\Psi_J$ is a diffeomorphism. Moreover, $\\Psi_J$ is equivariant with respect to the linear action $\\rho_{\\lin}$ on the domain  and the action $\\rho$ on the codomain. Furthermore, as $J$ is the standard complex structure in a neighbourhood $X_\\epsilon$ of the wedge, $\\Psi_J$ is the identity near the base point $p$. We now modify $\\Psi_J$ in order to make it the identity in the neighbourhood $X_\\epsilon$. As $J$ is the standard complex structure on $X_\\epsilon$ we can write\n\\[\n\\Psi_J (z,w)= \n\\begin{cases} \n(z,\\phi_2(z,w)) & \\text{~if~} z \\in    D_\\epsilon \\subset S^2 \\times S^2\\\\\n(\\phi_1(z,w),w) & \\text{~if~} w \\in   D_\\epsilon \\subset S^2 \\times S^2\n\\end{cases}\n\\]\nwhere $\\phi_1(z,0)=z$ and $\\phi_2(0,w)=w$ for all $z,w \\in S^2$. Choose a $G$ equivariant (for the $G$ action on $\\{\\infty\\}\\times S^2)$) smooth map $\\beta_1:S^2 \\longrightarrow S^2$ such that $\\beta_1(z) = z$ for all $z \\in D_\\epsilon$ and $\\beta_1 =\\infty$ in a neighbourhood $D_\\delta$ contained in $D_\\epsilon$ and such that $\\det(d\\beta_{1}(z)) \\geq 0 ~\\forall ~z \\in S^2$.\nSimilarly define a $G$ equivariant map (for the $G$ action on $S^2\\times\\{\\infty\\}$) $\\beta_2: S^2 \\to S^2$ satisfying analogous conditions as $\\beta_1$. Then we define a modification of $\\Psi_{J}$ by setting\n\\[\n\\Psi^{\\prime}_J(z,w) = \n\\begin{cases}\n\\Psi_J   &\\text{~if~}  z \\in  (S^2 \\times S^2) \\setminus X_\\epsilon \\\\\n(z,(\\phi_2(\\beta_1(z),w)) &\\text{~if~} z \\in D_\\epsilon \\\\\n(\\phi_1(z,\\beta_2(w)),w) &\\text{~if~}  w \\in D_\\epsilon\n\\end{cases}\n\\]\nThis modification $\\Psi^{\\prime}_J$ is the identity in a smaller neighbourhood $X_\\delta:=(S^2 \\times D_\\delta) \\cup (S^2 \\times D_\\delta)\\subset X_{\\epsilon} $.\\\\\n\nThe submanifolds $\\{z\\} \\times S^2$ and $S^2 \\times \\{w\\}$ for all $z,w \\in S^2$ are symplectic for the form ${\\Psi'_{\\!J}}^* \\tilde\\omega $ and hence $\\tilde\\omega_0 \\wedge {\\Psi'_{\\!J}}^* \\tilde\\omega>0$. Thus the path $\\omega_t:= t\\tilde\\omega +(1-t){\\Psi'_{\\!J}}^* \\tilde\\omega$ is a path of non-degenerate symplectic forms for all $t \\in [0,1]$. We can then apply an equivariant Moser isotopy to get an equivariant diffeomorphism $\\alpha$ of $S^2 \\times S^2$ such that $\\alpha^* {\\Psi'_{\\!J}}^* \\tilde\\omega = \\tilde\\omega_0$. Further, as ${\\Psi'_{\\!J}}^* \\tilde\\omega= \\tilde \\omega_0$ on $X_\\delta$, the restriction of $\\alpha$ to $X_\\delta$ is the identity. We then set $\\tilde \\Psi_J:= \\Psi'_{\\!J} \\circ \\alpha$.\n\\\\\n\nThe restriction of $\\tilde\\Psi_J: \\mathbb{C}^2 \\to \\mathbb{C}^2$ gives us a map which is $G$-equivariant with respect to the action $\\rho_{\\lin}$ on the domain and $\\rho$ on the codomain. As noted before there exists a star-shaped subset $V_1 \\subset V$ such that both the action $\\rho$ and $\\rho_{\\lin}$ agree on $\\mathbb{C}^2 \\setminus V_1$. We can choose $V_1$ such that $0 \\in int(V_1)$. We now apply Theorem \\ref{lemma:remove} to the map $f:={\\tilde\\Psi_J}^{-1}|_{\\mathbb{C}^2 \\setminus V_1}$ and we choose W in Theorem \\ref{lemma:remove} to be a $G$-invariant open subset of $V$ which contains $V_1$. Let $g:\\mathbb{C}^2 \\to \\mathbb{C}^2$ be an equivariant symplectomorphism as in Theorem \\ref{lemma:remove} such that $g|_{\\mathbb{R}^4\\setminus W} = f$, then the map $\\Phi:= \\left(\\tilde\\Psi_J \\circ g\\right)\\Big|_V: V \\to V$ is identity near the boundary and satisfies $\\Phi^*\\omega = \\omega_0$ and is\n$G$-equivariant where the action of G on the domain of $\\Phi$ is the linear action $\\rho_{\\lin}$ while on the range of $\\Phi$ it is the $G$ action $\\rho$ on V that we started out with. \nThus $\\Phi$ is the required equivariant symplectomorphism that linearizes the given $G$ action and takes the form $\\omega$ to $\\omega_0$.\n\\end{proof}\n\nConsider a polydisk $D^2 \\times D^2$ whose product structure is compatible with the eigenspace decomposition $\\mathbb{R}^{4}=\\mathbb{C}^{2}=\\mathbb{C}_{1}\\oplus \\mathbb{C}_{2}$. Consider the symplectic form $\\omega$ such that $\\omega = \\omega_0$ outside of some smaller polydisk of the form  $D_r \\times D_r \\subset D^2 \\times D^2$  for some radius $r$. \n\n\\begin{thm}[Equivariant Gromov's Theorem for Polydisks]\\label{thm:EquivariantGromovForPolydisks}\nLet $G$ be an abelian group. Let $\\omega$ be a symplectic form on $D^2 \\times D^2$ which is equal to $\\omega_0$ near the boundary. Let $G$ act symplectically on $(D^2 \\times D^2 ,\\omega)$ and suppose the action is linear near the boundary. Then the group $\\Symp_c^{G}(D^2\\times D^2, \\omega)$ of equivariant symplectomorphisms that are equal to the identity near the boundary of $D^2 \\times D^2$ is non-empty and contractible.\n\\end{thm}\n\\begin{proof}\nAs the $G$ action outside of $D_r \\times D_r \\subset \\mathbb{R}^4$  is linear, we can extend this $G$ action to the whole of $\\mathbb{R}^4$. Then we can compactify each eigenspace of $\\mathbb{C}$ to an $S^2$ and hence this $G$ action extends to a symplectic action $S^2 \\times S^2$ with respect to the form $\\tilde\\omega$ induced by $\\omega$. \n\\\\\n\nBy Proposition~\\ref{prop:linearize} we can conjugate our $G$ action by a symplectomorphism which is identity near the boundary to get a linear $G$ action on the whole of $V$. As any two conjugate topological subgroups are  homeomorphic, we shall just study the homotopy type of the compactly supported equivariant symplectomorphism group $\\Symp_{c,\\lin}^G(D^2 \\times D^2,\\omega)$ for the linear $G$ action on $V$.\n\\\\\n\nLet $\\mathcal{J}^G_{\\omega}$  be the non-empty and contractible space of all equivariant almost complex structures on $D^2 \\times D^2$ that are compatible with $\\omega$ and are the standard split almost complex structure $J_0$ outside of $D_r \\times D_r$. As they are the standard almost complex structures outside of a neighbourhood these almost complex structure extend to $S^2 \\times S^2$ and are compatible with $\\tilde\\omega_0$. Further once we pick a base point $p=(\\infty,\\infty)\\in S^2\\times S^2$ and identify $D^2\\times D^2$ with the complement of a standard neighborhood $X_\\epsilon:= (S^2 \\times D_\\epsilon) \\cup (S^2 \\times D_\\epsilon)$  of the wedge $(S^2\\times\\{\\infty\\})\\vee (\\{\\infty\\}\\times S^2)\\subset S^2\\times S^2$ (note that the wedge point $(\\infty,\\infty)$ is a fixed point for the extended action of G on $S^2 \\times S^2$),  then any element $J\\in  \\mathcal{J}^G_{\\sigma}$ extends to a equivariant almost complex structure of $S^2 \\times S^2$ which is the standard product complex structure on $S^2 \\times S^2$ on a neighbourhood $X_\\epsilon$ of the wedge $(S^2\\times\\{y\\})\\vee (\\{x\\}\\times S^2)\\subset S^2\\times S^2$. Conversely any such equivariant almost complex structure compatible with $\\tilde\\omega$ that is  standard in some neighbourhood of the wedge $(S^2\\times\\{\\infty\\})\\vee (\\{\\infty\\}\\times S^2)\\subset S^2\\times S^2$ gives us an element of $\\mathcal{J}^G_{\\omega}$ .\n\\\\\n\nIn order to show that $\\Symp_{c,\\lin}^G(D^2 \\times D^2,\\omega)$ is contractible, we shall prove that it is homotopy equivalent to the contractible space $\\mathcal{J}^G_{\\omega}$. \n\\\\\n\n\n\n\n\nDefine the map $\\tilde \\Psi_J$ as in the proof of Proposition~\\ref{prop:linearize}. Thus we have a map \n\\begin{align*}\n\\tau: \\mathcal{J}^G_\\omega  &\\longrightarrow \\Symp_{c,\\lin}^G(D^2 \\times D^2,\\omega) \\\\ \nJ &\\longmapsto \\tilde\\Psi_J \n\\end{align*}\n\nTo prove that $\\tau$ is a\nhomotopy equivalence we construct a homotopy inverse as follows.  Fix a $J' \\in \\mathcal{J}^G_\\omega$, then the homotopy inverse $\\beta$ is defined as,\n\n\\begin{align*}\n\\beta: \\Symp_{c,\\lin}^G(D^2 \\times D^2,\\omega)  &\\longrightarrow \\mathcal{J}^G_\\omega \\\\ \n\\phi &\\longmapsto \\phi_{*}J'\n\\end{align*}\n\nBy construction we see that $\\tau(\\beta(\\phi)) = \\id$ and the other direction is homotopic to the identity as $J^G_\\omega$ is contractible.\n\\end{proof}\n\nWe shall repeatedly use  the following theorem in our analysis of the homotopy type of the equivariant symplectomorphism groups of $S^2 \\times S^2$.\n\n\n\\begin{thm}(Equivariant Gromov's Theorem)\\label{thm:EqGr}\nLet $(V,\\omega)$ be an  compact star-shaped symplectic domain of $\\mathbb{R}^4$ such that $0 \\in int(V)$ and let $\\omega$ be such that $\\omega = \\omega_0$ near the boundary of $V$. Let $G$ be a compact abelian group that acts symplectically and linearly near the boundary and that sends the boundary to itself, then the space of equivariant symplectomorphisms that act as identity near the boundary (denoted by $\\Symp_c^G(V,\\omega)$) is non-empty and contractible.\n\\end{thm}\n\n\\begin{proof}\nBy Proposition~\\ref{prop:linearize} we can conjugate our $G$ action by a symplectomorphism which is identity near the boundary to get a linear $G$ action on the whole of $V$ and such that it takes the form $\\omega$ to $\\omega_0$. As the homotopy type of the two conjugate equivariant symplectomorphism group is the same (they are in fact homeomorphic), we shall just study the homotopy type of the compactly supported equivariant symplectomorphism group  for the linear $G$ action on $(V,\\omega_0)$. We denote this group by $\\Symp_{c,\\lin}^G(V,\\omega_0)$.\n\\\\\n\nChoose real numbers $r>0$ and $T>1$ , $D_r \\times D_r$ is a polydisk of radius $r$, such that $\\frac{1}{T}V \\subset D_r \\times D_r \\subset int(V)$,\nand consider the family of maps $F_t : \\Symp_{c,\\lin}^G(V,\\omega_0) \\to \\Symp_{c,\\lin}^G(V,\\omega_0)$ for $1 \\leq t \\leq T$ \ndefined by $F_t(\\phi)(x) = \\frac{\\phi(tx)}{t} $ for all $x \\in V$.\n\\\\\n\nThen we have that $F_t$ is $G$ equivariant for all $1 \\leq t \\leq T$, $F_1(\\phi) = \\phi$ for all $\\phi \\in \\Symp_{c,\\lin}^G(V,\\omega_0)$,  $F_t(id) = id$ for all t, and $F_T \\left(\\Symp_{c,\\lin}^G(V,\\omega_0)\\right) \\subset \\Symp_{c,\\lin}^G(D_r \\times D_r,\\omega)$.\n\\\\\n\nThe proof of Theorem \\ref{thm:EquivariantGromovForPolydisks} tells us that the inclusion $i:\\Symp_{c,\\lin}^G(D_r \\times D_r,\\omega) \\hookrightarrow \\Symp_{c,\\lin}^G(V,\\omega_0)$ is contractible. Hence we can fix a contraction $\\alpha_t$ for $T \\leq t \\leq T+1$ such that $\\alpha_T = i$ and $\\alpha_{T+1}(\\phi) = id$ for all $\\phi \\in \\Symp_{c,\\lin}^G(D_r \\times D_r,\\omega)$. Then the concatenation \n$$\\Tilde{F_t}:=\\begin{cases}\nF_t ~~1 \\leq t\\leq T\\\\\nF_T \\circ \\alpha_t ~~ T\\leq t \\leq T+1\n\\end{cases}$$ \ngives  us a retraction of  $\\Symp_{c,\\lin}^G(V,\\omega_0)$  to $id$ and hence $\\Symp_{c}^G(V,\\omega)$ is contractible.\n\\end{proof}\n\n\\begin{remark}\nTo our knowledge, it is not known whether an equivariant version of Gromov's Theorem on holds true for non-abelian compact groups.\n\\end{remark}\n\n\\section{Homotopical description of \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_k$}{J\\hat{}S1\\_k}} \nWe now consider the action of the group of equivariant symplectomorphisms $\\Symp_{h}^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ on the contractible space $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ of invariant, compatible, almost-complex structures, and we investigate the orbit-type stratification of this action up to homotopy. In the case of the nontrivial bundle $\\CP^2\\# \\overline{\\CP^2}$, the analysis of the action of the centralizer on the stratification will be postponed to Section~\\ref{Chapter-CCC}.\n\n\n\\subsection{Notation}\\label{not}\n\n\nWe shall use the following notation in the rest of the document. Let $M$ denote the manifolds $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$. Let $G$ be a compact abelian group acting symplectically on $(M,\\omega_\\lambda)$. Let $p_0$ be a fixed point for the group action. Given a $G$ invariant symplectic  curve C, and a $\\omega_\\lambda$ orthogonal $G$ invariant sphere $\\overline{F}$ in the homology class $F$ that intersects C at a point $p_0$, we define the following spaces: \n\n\\begin{itemize}\n\\item $N(C)$:= The symplectic normal bundle to a symplectic submanifold $C$. \n\\item $\\Symp^{G}_h(M,\\omega_\\lambda)$ := The group of $G$ equivariant symplectomorphisms on $(M,\\omega_\\lambda)$ that acts trivially on homology.\n    \\item $\\Stab^{G}(C)$ := The group of all  $\\phi \\in \\Symp^{G}_h(M,\\omega_\\lambda)$ such that $\\phi(C) = C$, that is, such that $\\phi$ \\emph{stabilises} $C$ but does not necessarily act as the identity on $C$.\n    \\item $\\Fix^{G}(C)$ := The group of all  $\\phi \\in \\Symp^{G}_h(M,\\omega_\\lambda)$ such that $\\phi|_C = id$, that is, such that \\emph{fixes $C$ pointwise}. \n    \\item $\\Fix^{G}(N(C))$:= The group of all $\\phi \\in \\Symp^{G}_h(M,\\omega_\\lambda)$ such that $\\phi|_C = \\id$ and  $d\\phi|_{N(C)}: N(C) \\to N(C)$ is the identity on $N(C)$.\n    \\item $\\Gauge1^{G}(N(C))$:= The group of $G$-equivariant symplectic gauge automorphisms of the symplectic normal bundle of $C$.\n    \\item $\\Gauge1^{G}(N(C \\vee \\overline{F}))$ := The group of $G$-equivariant symplectic gauge automorphism of the symplectic normal bundle of the crossing divisor $C \\vee \\overline{F}$ that are identity in a neighbourhood of the wedge point. \n    \\item $\\mathcal{S}^{G}_{K}$ := The space of unparametrized $G$-invariant symplectic embedded spheres in the homology class $K$. \n    \\item $\\mathcal{S}^{G}_{K,p_0}$:= The space of unparametrized $G$-invariant symplectic embedded spheres in the homology class $K$ passing through $p_0$.\n    \\item $\\mathcal{J}_{\\omega_\\lambda}^{G}(C)$ := The space of $G$-equivariant $\\omega_\\lambda$ compatible almost complex structures s.t the curve $C$ is holomorphic.\n    \\item $\\Symp^{G}(C)$:= The space of all $G$-equivariant symplectomorphisms of the curve C.\n    \\item $\\Fix^{G}(N(C \\vee {\\overline{F}}))$ := The space of all $G$-equivariant symplectomorphisms that are the identity in the neighbourhood of $C \\vee {\\overline{F}}$.\n    \\item $\\Symp^{S^1}({\\overline{F}}, N(p_0))$ := equivariant symplectomorphism of the sphere $\\overline{F}$ that are the identity in an open set of $\\overline{F}$ around $p_0$. \n    \\item $\\overline{\\mathcal{S}^{G}_{F,p_0}}$:= The space of unparametrized $G$-invariant symplectic spheres in the homology class $F$ that are equal to a fixed curve ${\\overline{F}}$ in a neighbourhood of $p_0$.\n    \\item $\\Symp^{G}_{p_0,h}(M,\\omega_\\lambda)$:=  The group  of all  $\\phi \\in \\Symp^{G}_{h}(M,\\omega_\\lambda)$ fixing $p_0$.\n    \\item $\\Stab^{G}_{p_0}(C)$:=  The group of all  $\\phi \\in \\Stab^G(C)$ such that $\\phi(p_0) = p_0$.  \n\\end{itemize}\n\nAll the above spaces are equipped with the $C^\\infty$ topology.\n\n\\subsection{Case 1: \\texorpdfstring{$\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$}{Symp(S2xS2)} action on \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$}{J\\hat{}S1\\_2k} with \\texorpdfstring{$s \\neq 0$}{s>0}}\\label{section:ActionOnU_2k}\n\nLet $\\lambda > 1$ and consider a $S^1$-action $S^1(a,b;m)$ on $(S^2 \\times S^2,\\omega_\\lambda)$ with $m=2k$. Let  ${\\mathcal{S}^{S^1}_{D_{2s}}}$ denote the space of all $S^1$ invariant symplectic embedded spheres in the class $D_{2s}=B-sF$. We shall assume that ${\\mathcal{S}^{S^1}_{D_{2s}}}$ is non-empty which, by Theorems~\\ref{cor:IntersectingOnlyOneStratum} and~\\ref{cor:IntersectingTwoStrata}, means that $2s=m$ or $2s=|2b\\pm m|$ depending on $a$. \n\nWe first show that ${\\mathcal{S}^{S^1}_{D_{2s}}}$ is a homogeneous space under the natural action of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$. To this end, we observe that among the two fixed points on the curve in class $D_{2s}$, Table~\\ref{table_weights} shows that at least one fixed point, say $p$, has weights $(w_1,w_2)$ with $w_1\\neq w_2$. It follows that if an invariant curve in the class of the fiber $F$ intersects a curve in the class $D_{2s}$ at $p$, then the two curves intersect $\\omega_\\lambda$-orthogonally. Let $\\mathcal{C}(D_{2s}\\vee F,p)^{S^1}$ be the space of invariant configurations made of curves in classes $D_{2s}$ and $F$ intersecting orthogonally at $p$. \n\n\\begin{lemma}\\label{lemma:EvaluationFibrationConfigurations} \nThe evaluation map at a standard configuration $\\overline{D}_{2s}\\vee \\overline{F}$ through $p$\n\\[\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) \\twoheadrightarrow \\mathcal{C}(D_{2s}\\vee F,p)^{S^1}\\]\nis a serre fibration.\n\\end{lemma}\n\\begin{proof}\nWe first show that the action is transitive. Given an invariant configuration $C\\vee A \\in \\overline{D}_{2s}\\vee \\overline{F}$, \nthe equivariant symplectic neighbourhood theorem implies that we can find an invariant neighbourhood V of $C \\vee A$, an invariant neighbourhood $V'$ of $\\overline{D} \\cup \\overline{F}$, and an equivariant symplectomorphism $\\alpha:V \\to V'$. We claim that $\\alpha$ can be extended to an ambient diffeomorphism $\\beta$ of $S^2 \\times S^2$. Assume this for the moment. By construction, the pullback form $\\omega_\\beta:=\\beta^*\\omega_\\lambda$ is invariant under the the conjugate action $\\beta^{-1}\\rho\\beta$.\n\nWe observe that the complement of the standard configuration $\\overline{D} \\cup \\overline{F}$ in $W_{2s}$ is symplectomorphic to $\\mathbb{C}^2$ with the symplectic form $\\omega_f:=\\frac{1}{2\\pi}\\partial\\bar\\partial f$ where $f=\\log\\left(\\left(1+||w||^2\\right)^{\\lambda}\\left(1+||w||^{4k}+||z||^2\\right)\\right)$, see~\\cite[Lemma~3.5]{abreu}. Under this identification, the standard $\\mathbb{T}^2_{2s}$ action on $W_{2s}$ become linear on $\\mathbb{C}^2$. It follows that near infinity, the form $\\omega_\\beta$ is equal to $\\omega_f$ and that the action $\\beta^{-1}\\rho\\beta$ is linear. By Proposition~\\ref{prop:linearize} we get that there is an equivariant symplectomorphism $\\gamma$ that is equal to the identity near infinity, and that identifies $(\\mathbb{C}^2,\\omega_f, \\rho)$ with $(\\mathbb{C}^2,\\beta^*\\omega_\\lambda,\\beta^{-1}\\rho\\beta)$. By construction, the equivariant symplectomorphism $\\phi := \\gamma \\circ \\beta$ takes the configuration $C\\vee A$ to $\\overline{D}\\vee\\overline{F}$. \n\nIt remains to see that the local diffeomorphism $\\alpha$ can be extended to an ambient diffeomorphism $\\beta$ of $S^2 \\times S^2$. By the Isotopy Extension Theorem (see~\\cite[Theorem~1.4, p.180]{Hi}), it suffices to show that any two configurations of embedded spheres in classes $\\overline{D}\\cup F$ intersecting transversely and positively are isotopic. In turns, this follows from the fact that any two $F$-foliations corresponding to two almost-complex structures $J$ and $J'$ are diffeotopic, and that any two sections of the product $S^2 \\times S^2$ are diffeotopic through sections iff they are homotopic. This shows that $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ acts transitively on $\\mathcal{C}(D_{2s}\\vee F,p)^{S^1}$.\n\nTo prove the homotopy lifting property, consider any family of maps $\\gamma: D^n \\times [0,1] \\rightarrow \\mathcal{C}(D_{2s}\\vee F,p)^{S^1}$ from a $n$ dimensional disk $D^n$ to $\\mathcal{C}(D_{2s}\\vee F,p)^{S^1}$, and choose a lift $\\overline{\\gamma_0}:D^n \\rightarrow \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ of $\\gamma_0$. Since the complement of a configuration is contractible, the equivariant version of Banyaga's Extension Theorem for families implies that there exists a lift $\\overline{\\gamma}: D^n \\times [0,1] \\rightarrow \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) $ extending $\\overline{\\gamma_0}$. \n\\[\n\\begin{tikzcd}\nD^n \\times \\{0\\} \\arrow[d,hookrightarrow] \\arrow[r,\"\\overline{\\gamma_0}\"]    &\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) \\arrow[d,\"\\theta\"] \\\\\n    D^n \\times [0,1]\\arrow[r,\"\\gamma\"] \\arrow[ur,dashrightarrow,\"\\exists ~ \\overline{\\gamma}\"] &\\mathcal{C}(D_{2s}\\vee F,p)^{S^1}\n\\end{tikzcd}\n\\]\nAlternatively, one can apply the equivariant Gromov-Auroux Lemma~\\ref{Au} to show the existence of the lift  $\\overline{\\gamma}$. In both cases, this concludes the proof.\n\\end{proof}\n\n\\begin{cor}\\label{trans} \nFix the action $S^1(a,b;m)$. Then $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ acts transitively on ${\\mathcal{S}^{S^1}_{D_{2s}}}$.\\qed\n\\end{cor}\n\n\n\n\n\\begin{lemma}\\label{first}\nLet $\\overline{D}$ be an invariant symplectic sphere in the class $B - sF$ for which $\\mathcal{S}^{S^1}_{D_{2s}}$ is nonempty. Then the evaluation map\n\\begin{align*}\n    \\theta: \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) &\\twoheadrightarrow {\\mathcal{S}^{S^1}_{D_{2s}}} \\\\\n    \\phi &\\mapsto \\phi(\\overline{D})\n\\end{align*}\nis a Serre fibration with fibre over $\\overline{D}$ given by\n\\[\\Stab(\\overline{D}):= \\left\\{ \\phi \\in \\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)~|~ \\phi(\\overline{D}) = \\overline{D}\\right\\}\\]\n\\end{lemma}\n\\begin{proof}\nThe evaluation map is transitive by Corollary~\\ref{trans}. The homotopy lifting property follows from Lemma~\\ref{Au} as in the proof of Lemma~\\ref{lemma:EvaluationFibrationConfigurations}. Alternatively, one can also note that the action map factors through the restriction map \n\\[\\mathcal{C}(D_{2s}\\vee F,p)^{S^1}\\to\\mathcal{S}^{S^1}_{D_{2s}}\\]\nwhich is itself a fibration. To see this, note that the restriction maps fits into a commuting diagram\n\\[\n\\begin{tikzcd}\n\\mathcal{J}^{S^1}_{\\om_\\lambda}\\arrow[r,\"f_1\"]\\arrow[rd,\"f_2\"] & \\mathcal{C}(D_{2s}\\vee F,p)^{S^1} \\arrow[d]\\\\\n& \\mathcal{S}^{S^1}_{D_{2s}}\n\\end{tikzcd}\n\\]\nwhere the maps $f_1$ and $f_2$ are fibrations. Observe that the map $f_1$ is well defined because the weights at the chosen fixed point $p$ are not equal. Hence, for any choice of invariant almost complex structure $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}$, the unique invariant $J$-holomorphic curve $C$ in class $D_{2s}$ intersects the unique invariant $J$-holomorphic fiber through $\\omega_\\lambda$-orthogonally at~$p$. \n\\end{proof}\n\n\n\n\n\n\\begin{remark}\nAs both $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ and $\\mathcal{S}^{S^1}_{D_{2s}}$ can be shown to be CW-complexes, we see from Theorem 1 in \\cite{Serrefib} (with proof corrected in \\cite{error}), that a Serre fibration in which the total space and base space are both CW complexes is necessarily a Hurewicz fibration. Thus the map $\\theta: \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) \\twoheadrightarrow {\\mathcal{S}^{S^1}_{D_{2s}}}$ is in fact a Hurewicz fibration and hence the fibre over any arbitrary $D \\in \\mathcal{S}^{S^1}_{D_{2s}}$ is homotopy equivalent to $\\Stab(\\overline{D})$. \n\\end{remark}\n\nThe homotopy type of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ is related to the strata $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$ through the following sequence of fibrations. We use the symbol ``$\\simeq$\" to mean ``weakly homotopy equivalent\" throughout the rest of the document. In the fibrations below, we use the notation established in Section~\\ref{not}.\n\n\n\\[\\Stab^{S^1}(\\overline{D}) \\longrightarrow \\Symp^{S^1}_{h,p_0}(S^2 \\times S^2,\\omega_\\lambda) \\longtwoheadrightarrow {\\mathcal{S}^{S^1}_{D_{2s}}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}\\rule{0em}{2em}\\]\n\n\\[\\Fix^{S^1}(\\overline{D}) \\longrightarrow \\Stab^{S^1}(\\overline{D}) \\longtwoheadrightarrow  \\Symp^{S^1}(\\overline{D}) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1 ~\\text{or}~ \\SO(3)\\rule{0em}{2em}\\]\n\n\\[\\Fix^{S^1} (N(\\overline{D})) \\longrightarrow \\Fix^{S^1}(\\overline{D}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(\\overline{D})) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1\\rule{0em}{2em}\\]\n \n\\[\\Stab^{S^1}(\\overline{F}) \\cap \\Fix^{S^1}(N(\\overline{D})) \\longrightarrow \\Fix^{S^1}(N(\\overline{D})) \\longtwoheadrightarrow  \\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1}(\\overline{D})\\simeq \\{*\\}\\rule{0em}{2em}\\]\n \n\\[\\Fix^{S^1}(\\overline{F})  \\longrightarrow \\Stab^{S^1}(\\overline{F}) \\cap \\Fix^{S^1}(N(\\overline{D})) \\longtwoheadrightarrow  \\Symp^{S^1}(\\overline{F}, N(p_0))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\n\n\\[\\left\\{*\\right\\} \\mathbin{\\textcolor{blue}{\\xleftarrow{\\text{~~~$\\simeq$~~~}}}} \\Fix^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\longrightarrow \\Fix^{S^1}(\\overline{F}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F}))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\nHere $\\overline{F}$ and $\\overline{D}$ are the $\\omega_\\lambda$-orthogonally intersecting invariant curves in the $2s^{\\,\\text{th}}$-Hirzebruch surface $W_{2s}$ and whose moment map images are depicted below in red. We denote by $\\overline{F}$ the unique curve in class $F$ intersecting the given curve $\\overline{D} \\in {\\mathcal{S}^{S^1}_{D_{2s}}}$ $\\omega_\\lambda$-orthogonally at $p_0$. In the second fibration, the group $\\Symp^{S^1}(\\overline{D})$ is homotopy equivalent to $\\SO(3)$ when the $S^1$ action fixes the curve $\\overline{D}$ pointwise. Otherwise, it is homotopy equivalent to~$S^1$. \n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}\n\\draw[red] (0,1) -- (3,1) ;\n\\draw (0,1) -- (0,0) ;\n\\draw (0,0) -- (4,0) ;\n\\draw[red] (3,1) -- (4,0) ;\n\\node[above] at (1.5, 1.0) {$\\overline{D}$};\n\\node[right] at (3.55, 0.75) {$\\overline{F}$};\n\\node[above] at (3,1) {$p_0$};\n\\end{tikzpicture}\n\\caption{The isolated fixed point $p_0$}\\label{fig:IsolatedFixedPoint}\n\\end{figure}\n\n\nAssuming  the homotopy equivalence in the first fibration, we immediately get\n\\begin{thm}\nConsider the $S^1(a,b;m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ with $\\lambda >1$. If $ \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s} \\neq \\phi$, then $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\Stab^{S^1}(\\overline{D})  \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$.\n\\end{thm}\nFurthermore, tracking down the various homotopy equivalences in the other fibrations, we will prove that the equivariant stabilizer of the curve $\\overline{D}$, namely  $\\Stab^{S^1}(\\overline{D})$, is homotopy equivalent to the equivariant stabilizer of the corresponding complex structure under the natural action of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$. More precisely, \n\\begin{itemize}\n    \\item $\\Stab^{S^1}(\\overline{D}) \\simeq \\mathbb{T}^2_{2s}$ when $(a,b) \\neq (0,\\pm1)$;\n    \\item  $\\Stab^{S^1}(\\overline{D}) \\simeq SO(3) \\times S^1$ when $(a,b) = (0,\\pm1)$.\n\\end{itemize}\n \n\n\nWe shall now justify each of the homotopy equivalences in the above fibrations. \n\n\n\n\nNow that we know that the action map\n\\begin{align*}\n \\Stab^{S^1}(\\overline{D}) \\to  \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda) &\\twoheadrightarrow {\\mathcal{S}^{S^1}_{D_{2s}}}\n\\end{align*}\nis a fibration, we show that ${\\mathcal{S}^{S^1}_{D_{2s}}}$ is weakly homotopic to $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$. \n\\begin{lemma}\\label{first2}\nThe natural map $\\alpha: \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s} \\to {\\mathcal{S}^{S^1}_{D_{2s}}}$ defined by sending an almost complex structure $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$ to the unique $J$-holomorphic curve in class $D_{2s}$ is a weak homotopic equivalence. \n\\end{lemma}\n\\begin{proof}\nTo show that $\\alpha$ is a weak homotopy equivalence, we first show that the map is Serre fibration.  To do so, consider an arbitrary element ${D} \\in {\\mathcal{S}^{S^1}_{D_{2s}}}$. As in the proof of Lemma \\ref{first}, it suffices to show that given a family of map $\\gamma_t$ from a n-dimensional disk $D^n$, such that $\\gamma_0(0) = {D}$, and a lift $g_0^\\prime:D^n: \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$ lifting $\\gamma_0$, then there exists a lift $\\gamma_t^\\prime$ to $ \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$.\n\\[\n\\begin{tikzcd}\n    D^n \\times \\{0\\} \\arrow[d,hookrightarrow] \\arrow[r,\"\\gamma_0^\\prime\"] &\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s} \\arrow[d,\"\\alpha\"] \\\\\n    D^n \\times [0,1] \\arrow[r,\"\\gamma\"] \\arrow[ur,dashrightarrow,\"\\exists ~ \\gamma^\\prime\"] &\\mathcal{S}^{S^1}_{D_{2s}}\n\\end{tikzcd}\n\\]\nAs in the proof of Lemma \\ref{first}, we have that there exists a lift $\\overline{\\gamma}: D^n \\times [0,1] \\to \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ of $\\gamma$. Pick an element $ J \\in \\alpha^{-1}({D})$ and define $\\gamma^\\prime(s) := \\overline{\\gamma}^*J$. This defines a lift $\\gamma^\\prime$ of $\\gamma$. Hence $\\alpha$ is a fibration, with fibre begin contractible. Thus we get the required result that $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s} \\simeq {\\mathcal{S}^{S^1}_{D_{2s}}}$.\n\\end{proof}\n\n\n\n\n\n\n\\begin{lemma}{\\label{stab}}\nThe restriction map $\\Stab^{S^1}(\\overline{D}) \\twoheadrightarrow \\Symp^{S^1}(\\overline{D})$ is a fibration.  \n\\end{lemma}\n\n\\begin{proof}\nTo show that the restriction map is a fibration we use Theorem \\ref{palais}, in which we set $X= \\Symp^{S^1}(\\overline{D})$, $G=\\Stab^{S^1}(\\overline{D})$ and the action  is given by \n\\begin{align*}\n    G \\times X &\\to X \\\\\n    (\\phi, \\psi) &\\to \\phi|_{\\overline{D}} \\circ \\psi\n\\end{align*}\nHence in order to show that the restriction map $r :\\Stab^{S^1}(\\overline{D}) \\to \\Symp^{S^1}(\\overline{D})$ is a fibration, we only need to show that the action described above  admits local cross sections. Suppose we only  show that a neighbourhood of identity admits local cross sections and that  $\\Stab^{S^1}(\\overline{D})$ acts transitively on $\\Symp^{S^1}(\\overline{D})$ this would suffice to show that $r$ is a fibration as by Theorem \\ref{palais}, its a local fibration near the identity and the map $r$ is equivariant with respect to the action of $\\Stab^{S^1}(\\overline{D})$, thus completing the proof.\n\\\\\n\nConsider the identity $\\id \\in \\Symp^{S^1}(\\overline{D})$. As Let $\\alpha:N(\\overline{D}) \\to U$ be a equivariant diffeomorphism between the symplectic normal bundle $N(\\overline{D})$ and a neighbourhood $U$ of $\\overline{D}$. As $\\Symp^{S^1}(\\overline{D})$ is locally contractible (this can be seen for example by noticing that the proof of Prop 3.3.14 in \\cite{MS} can be made equivariant) we can find a neighbourhood V of $\\id$, and a fixed retraction $\\beta_t$ of the neighbourhood $V$ onto the identity. Hence given any $\\psi \\in \\Symp^{S^1}(\\overline{D})$, we get a one parameter family $\\beta_t(\\psi)$ of symplectomorphisms.  As $\\pi_1(\\overline{D}) = 0$, $\\beta_t(\\psi)$ is Hamiltonian and is generated by  a function $H_t$. Let $\\pi:N(\\overline{D}) \\to \\overline{D}$ be the projection of the normal bundle. Define $\\Tilde{H_t}:= \\alpha \\circ \\pi^*H_t$. Thus $\\Tilde{H_t}$  defines an invariant function on a U. Fix an invariant bump function $\\rho$ with support in $U$ and is 1 in a small neighbourhood around $\\overline{D}$, then $\\rho \\Tilde{H_t}$ is an invariant function and the corresponding symplectomorphism it generates $\\Tilde{\\psi}$ belong to $\\Stab^{S^1}(\\overline{D})$ and extends $\\psi$.  Note that if we fix the neighbourhood $U$, the bump function and the retraction of the neighbourhood in $\\Symp^{S^1}(\\overline{D})$ then this  procedure gives us a lift of $\\psi$ near id in $\\Symp^{S^1}(\\overline{D}) $ to $\\Stab^{S^1}(\\overline{D})$. By Theorem \\ref{palais} this shows $r$ is fibration. \n\\end{proof}\n\n\nThe above proof also shows that the action is transitive. As given $\\gamma \\in \\Symp^{S^1}(\\overline{D})$ we give a procedure to construct an element $\\tilde\\gamma \\in \\Stab^{S^1}(\\overline{D})$ such that the restriction to $\\overline{D}$ is $\\gamma$, thus showing the action is transitive.\n\n\n\n\\vspace{3mm}\n\n\\begin{lemma}\n$\\Symp^{S^1}(\\overline{D})$ is homotopic to $\\SO(3)$ for the circle action $S^1(0,\\pm 1,m)$  and $\\Symp^{S^1}(\\overline{D})$ is homotopic to $S^1$ for all other circle actions.\n\\end{lemma}\n\n\\begin{proof}\nConsider the circle action induced on $\\overline{D}$. The action $S^1(0,\\pm 1,m)$ fixes $\\overline{D}$ pointwise. Hence $\\Symp^{S^1}(\\overline{D}) = \\Symp(\\overline{D})$. By Smale's theorem we know that $\\Symp(\\overline{D})$ is homotopy equivalent to $\\SO(3)$. \n\\\\\n\nThe restriction for all other actions not equal to $S^1(0,\\pm 1,m)$, do not point wise fix the curve $\\overline{D}$. For all other actions, we have the following two subcases. Assume that the action is effective. Let $\\mu: \\overline{D} \\to \\mathbb{R}$ be it's moment map. Then as explained in the proof of Proposition \\ref{prop:CentralizersToricActions} we have that  $\\Symp^{S^1}(\\overline{D}) \\simeq C^\\infty(\\mu(\\overline{D}),S^1)$, where $C^\\infty(\\mu(\\overline{D}),S^1)$ denotes the space of smooth maps from the image of the moment map to $S^1$. As the image of the moment map is an interval, and as the space of smooth maps from an interval to $S^1$ is homotopy equivalent to $S^1$, we have the required result that $ \\Symp^{S^1}(\\overline{D}) \\simeq S^1$.\n\\\\\n\n\nFinally if the induced symplectic $S^1$ action on $\\overline{D}$ is not effective and has $\\mathbb{Z}_k$ stabilizer, the action of $S^1\/\\mathbb{Z}_k \\cong S^1$, is effective and the space of symplectomorphisms equivariant with respect to this quotient  effective action is the same as space of symplectomorphisms equivariant with respect to the non-effective $S^1$ action. Thus the homotopy type of $\\Symp^{S^1}(\\overline{D}) \\simeq S^1$.\n\\end{proof}\n\n\n\n\n\n\n\\begin{lemma}\\label{gauge}\nThe map\n\\begin{align*} \n   \\alpha: \\Fix^{S^1}(\\overline{D}) &\\twoheadrightarrow \\Gauge1^{S^1}(N(\\overline{D})) \\\\\n     \\phi &\\mapsto d\\phi|_{N(\\overline{D})}\n\\end{align*}\n is a Serre fibration with fibre homotopic to $\\Fix^{S^1}(N(\\overline{D}))$. The base space  $\\Gauge1^{S^1}(N(\\overline{D}))$  is homotopy equivalent to $S^1$. \n\\end{lemma}\n\n\\begin{proof}\nThe fact that $\\Gauge1^{S^1}(N(\\overline{D})) \\simeq S^1$ is explained in Appendix A. Thus we only need to prove that the restriction of the derivative indeed is a  fibration and the fibre is homotopic to $\\Fix^{S^1}(N(\\overline{D}))$.\n\\\\\n\nConsider the action \n \n\\begin{align*}\n     \\Fix^{S^1}(\\overline{D}) \\times \\Gauge1^{S^1}(N(\\overline{D})) &\\to\\Gauge1^{S^1}(N(\\overline{D})) \\\\\n     (\\phi, \\psi) &\\to d\\phi|_{N(D)} \\circ \\psi\n\\end{align*}\n\n\nAgain by Theorem \\ref{palais} it suffices to show that the there is a local section to above action. Such a local section is produced by Lemma \\ref{EqSymN}.\n\nThe fibre is apriori given by all equivariant symplectomorphisms that act as identity on the normal bundle of $\\overline{D}$. The claim is that this is in fact homotopy equivalent to the space $\\Fix^{S^1}(N(\\overline{D}))$.  This follows from lemma \\ref{germ}.\n\\end{proof}\n\nLet $\\overline{\\mathcal{S}^{S^1}_{F,p_0}}$ be the space of unparametrized $S^1$-invariant symplectic spheres in the homology class $F$ that are equal to a fixed invariant curve ${\\overline{F}}$ in a neighbourhood of $p_0$.\n\n\\begin{lemma}\nThe map \\begin{align*}\n    \\Fix^{S^1}(N(\\overline{D})) &\\to \\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\\\\n    \\phi \\mapsto \\phi(\\overline{F})\n    \\end{align*} \nis a fibration and $\\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\simeq \\{*\\}$\n\\end{lemma}\n\n\\begin{proof}\n The proof for this is exactly the same as the proof of Corollary \\ref{trans}. We only note that if $F' \\in \\overline{\\mathcal{S}^{S^1}_{F,p_0}}$ then the map $\\phi$ constructed in the proof of Corollary~\\ref{trans} belongs to $\\Fix^{S^1}(N(\\overline{D}))$. Thus we  have that $\\Fix^{S^1}(N(\\overline{D}))$ acts transitively on $\\overline{\\mathcal{S}^{S^1}_{F,p_0}}$, and thus the action map induces a fibration. \n \\\\\n \n\n \n \n \\textit{Proof that $\\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\simeq \\{*\\}$}:\n This follows from the equivariant version of the standard proof of considering the homeomorphic space of equivariant compatible metrics and noting that this space of metrics is contractible. \n \\\\\n \n\\textit{Proof that $\\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\simeq \\{*\\}$}:\n Let $\\mathcal{S}^\\perp_{F,p_0}$ denote the space of all $S^1$ invariant symplectically embedded spheres S in class F such that $S \\cap \\overline{D} =p_0$ and $S$ and $\\overline{D}$ intersect $\\omega_\\lambda$-orthogonally at $p_0$. By Lemma \\ref{transverse} we see that for every $S \\in  \\mathcal{S}^\\perp_{F,p_0}$ there exists a $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D})$ such that the configuration $S \\vee \\overline{D}$ is $J$-holomorphic.  We now have  the following fibration\n \n \\begin{equation*}\n     \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\longtwoheadrightarrow  \\mathcal{S}^\\perp_{F,p_0}\n \\end{equation*}\n\nwhere the map $\\gamma: \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\to  \\mathcal{S}^\\perp_{F,p_0}$ is just sending $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D})$ to the corresponding curve in class $F$ passing through $p_0$. Note that the above map is well defined because the fixed $p_0$ was chosen such that the weights at $p_0$ were distinct. Hence any $S^1$ invariant $F$ curve passing through $p_0$ must be $\\omega_\\lambda$-orthogonal to $\\overline{D}$. Now we show that $\\gamma$ is a homotopy equivalence. To do that we consider the following commutative diagram\n \n\\[ \n\\begin{tikzcd}\n    &T \\arrow[d,\"\\pi_2\"] \\arrow[r,\"\\pi_1\"] & \\mathcal{S}^\\perp_{F,p_0} \\\\\n    &\\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D}) \\arrow[ur, \"\\gamma\"]\n\\end{tikzcd} \n \\]\n\nwhere $T:= \\left\\{ (A,J) \\in  \\mathcal{S}^\\perp_{F,p_0} \\times \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{D})~|~  A ~~\\text{is J-holomorphic} \\right\\}$. Both the maps $\\pi_1$ and $\\pi_2$ are fibrations (this can be argued as in Lemma \\ref{first}) with contractible fibres. As the diagram commutes, the map $\\gamma$ must be a homotopy equivalence. The proof then follows by showing that $\\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\simeq \\mathcal{S}^\\perp_{F,p_0}$, which is a consequence of Theorem~\\ref{gmpf}.\n\\end{proof}\n\n\n\\begin{lemma}\n      \\begin{align*}\n          \\Stab^{S^1}(\\overline{F}) \\cap \\Fix^{S^1}(N(\\overline{D})) &\\to \\Symp^{S^1}(\\overline{F}, N(p_0)) \\\\\n          \\phi &\\mapsto  \\phi|_{\\overline{F}}\n      \\end{align*}\n      \n      is a fibration and $\\Symp^{S^1}(\\overline{F}, N(p_0)) \\simeq \\{*\\}$\n \\end{lemma}\n \n \\begin{proof}\n The fact that this is a fibration follows from applying the proof of  Lemma \\ref{stab} mutatis mutandis. The proof that $\\Symp^{S^1}(\\overline{F}, N(p_0)) \\simeq \\{*\\}$, is similar to Lemma \\ref{stab}.  $\\Symp^{S^1}(\\overline{F}, N(p_0))$ is homotopy equivalent to maps from the interval $[0,1]$ to $S^1$ that is identity near 0. The space of such maps is contractible thus giving the result. \n \\end{proof}\n\n\n\n\\begin{lemma}\\label{ngauge}\n\n\n\\begin{align*}\n    \\Fix^{S^1}(\\overline{F}) &\\to \\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\\\\n    \\phi &\\mapsto d\\phi|_{N(\\overline{D} \\vee \\overline{F})}\n\\end{align*}\nis a fibration and $\\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\{*\\}$ and the fibre $\\Fix^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\{*\\}$\n\\end{lemma}\n\n\\begin{proof}\nThe proof that this is a fibration is similar to the proof of Lemma \\ref{gauge}. The fact that $\\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\{*\\}$ follows from by Lemma \\ref{Gauge(N(D))}. The fact that $\\Fix_{S^1}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\{*\\}$ follows from Theorem \\ref{thm:EqGr}.\n\\end{proof}\n\n\nPutting all the fibrations together gives the following theorem.\n\\begin{thm}\\label{homogenous}\nConsider the $S^1(a,b;m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ with $\\lambda >1$. If $ \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s} \\neq \\phi$, then we have the following homotopy equivalences:\n\\begin{enumerate}\n    \\item when $(a,b) \\neq (0,\\pm1)$, we have $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_{2s} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$;\n    \\item when $(a,b) = (0,\\pm1)$, we have $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/(SO(3) \\times S^1) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$.\n\\end{enumerate} \n\\end{thm} \n\\begin{proof}\n\n\nWhen $(a,b;m) \\neq (0,\\pm 1;0)$ we have a commutative diagram of fibrations\n\\[\n\\begin{tikzcd}\n     &\\Fix^{S^1}(\\overline{D}) \\arrow[r] &\\Stab^{S^1}(\\overline{D}) \\arrow[r,twoheadrightarrow] &\\Symp^{S^1}(\\overline{D}) \\\\\n     &S^1 \\arrow[u,hookrightarrow] \\arrow[r] &\\mathbb{T}^2_{2s} \\arrow[u,hookrightarrow] \\arrow[r] &S^1 \\arrow[u,hookrightarrow]\n\\end{tikzcd}\n\\]\nwhile in the case $(a,b) = (0,\\pm1)$, we have the diagram\n\\[\n\\begin{tikzcd}\n     &\\Fix^{S^1}(\\overline{D}) \\arrow[r] &\\Stab^{S^1}(\\overline{D}) \\arrow[r,twoheadrightarrow] &\\Symp^{S^1}(\\overline{D}) \\\\\n     &S^1 \\arrow[u,hookrightarrow] \\arrow[r] & S^1 \\times SO(3) \\arrow[u,hookrightarrow] \\arrow[r] &SO(3) \\arrow[u,hookrightarrow]\n\\end{tikzcd}\n\\]\nFrom the discussion above, in both the diagrams the leftmost and the rightmost arrows are homotopy equivalences.  As the diagram commutes, the 5 lemma implies that the middle inclusion $\\mathbb{T}^2 \\hookrightarrow \\Stab^{S^1}(\\overline{D})$  or $\\left(S^1 \\times SO(3)\\right) \\hookrightarrow \\Stab^{S^1}(\\overline{D})$ are also homotopy equivalences. This gives us the required result.\n\\end{proof}\n\n\\begin{remark}\\label{ActionOnJsom}\nLet $J_{2s}$ be the standard complex structure on $W_{2s}$. We note that for the  action $S^1(0,\\pm1,;m)$ the  stabiliser of $J_{2s}$ under the natural action of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ on $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s}$ is the group of K\\\"ahler isometries $S^1 \\times \\SO(3)$. For all other circle actions $S^1(a,b;m)$ with $(a,b) \\neq (0,\\pm1)$, the stabiliser of $J_{2s}$ is the maximal torus $\\mathbb{T}^2_{2s}\\subset S^1 \\times \\SO(3)$.\n\\end{remark}\n\n\n\n\n\n\\subsection{Case 2: \\texorpdfstring{$\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$}{Symp(S2xS2)} action on \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$}{J\\hat{}S1\\_0}}\\label{IntwithU0}\nIn order to describe the action of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ on the open stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$, we need to  modify slightly the setting introduced in the previous section. The main difference comes from the fact that for an almost-complex structure $J\\in\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_0$, there is no invariant curve with negative self-intersection  representing a class $B-kF$, $k\\geq 1$. Instead, each such $J$ determines a regular 2-dimensional foliation of $J$-holomorphic curves in the class $B$. Consequently, there is no natural map between the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ and the space $\\mathcal{S}^{S^1}_{B}$ of invariant curves in the class $B$. However, once we choose a fixed point $p_0$, given any $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$, there is a unique invariant $J$-holomorphic curve in the class $B$ passing through $p_0$. This defines a map $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0 \\to \\mathcal{S}^{S^1}_{B,p_0}$ that can be used to prove that the space $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ is homotopy equivalent to an orbit of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$. To do so, because the fixed point $p_0$ is not unique, we must also investigate how the group $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ acts on the fixed point set of the circle action. This is done in Lemma~\\ref{lemma:SymphPreservesAnIsolatedFixedPoint}. Before we proceed to prove this lemma we first describe the action of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ on $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$. Note that by Theorem~\\ref{cor:IntersectingOnlyOneStratum}, Corollary~\\ref{cor:CircleExtensionsWith_a=1} and Corollary~\\ref{cor:CircleExtensionsWith_a=-1}, the space $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ is non-empty only for the following circle actions:\n\\begin{itemize}\n    \\item $S^1(a,b;0)$, or\n    \\item  $S^1(1,b;m)$ with $|2b-m|=0$ and $2\\lambda > |2b-m|$, or\n    \\item $S^1(-1,b;m)$ with $|2b+m|=0$ and $2\\lambda > |2b+m|$.\n\\end{itemize}\nSecondly, we observe that all these actions have at least one isolated fixed point \\emph{except} the actions of the forms\n\\begin{itemize}\n    \\item $S^1(\\pm 1,0;0)$ and\n    \\item $S^1(0,\\pm 1;0)$\n\\end{itemize}\n\\subsubsection{Actions with an isolated fixed point} We now consider actions $S^1(a,b;m)$ with an isolated fixed point $p_0$. We can choose $p_0$ to correspond to the vertex $R$ in the Hirzerbruch surface $W_m$ shown in Figure~\\ref{hirz}. Given $J\\in\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_0$, there is a unique $J$-holomorphic curve $B_{p_0,J}$ in class $B$ that passes through $p_0$. Because $p_0$ is fixed, $J$ is invariant, and $B\\cdot B=0$, positivity of intersection implies that $B_{p_0,J}$ is $S^1$-invariant. We thus get a well-defined map \n\\[\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0 \\to \\mathcal{S}^{S^1}_{B,p_0}\\]\nwhere $\\mathcal{S}^{S^1}_{B,p_0}$ denotes the space of invariant, embedded, symplectic spheres representing the class $B$ and containing the point $p_0$. \n\\begin{lemma}\\label{projection_surjective}\nConsider any $S^1(a,b; m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$. Let $p_0$ and $p_1$ be two fixed points such that there exists an invariant fibre $\\{*\\} \\times S^2$ passing through them . Then there exists no $S^1$ invariant curve in the class $B-kF$ for $k \\geq 0$ passing through $p_0$ and $p_1$.\n\\end{lemma}\n\\begin{proof}\nSuppose not, let $\\overline{D_{2s}}$ be a $S^1$ invariant curve in the class $B-kF$ with $k \\geq 0$ passing through $p_0$ and $p_1$. Then the projection onto the first factor\n$$ \\pi_1 : \\overline{D_{2s}} \\rightarrow S^2 \\times \\{0\\} \\subset S^2 \\times S^2$$\nis surjective. Hence the curve $\\overline{D_{2s}}$ passes through a third  fixed point $p_2$. As the symplectic $S^1$ action on $\\overline{D_{2s}}$ has three fixed points, it has to fix $\\overline{D_{2s}}$ pointwise. This is a contradiction as all fixed surfaces for $S^1$ actions must be either a maximum or minimum for the moment map, but the fixed points $p_2$, $p_1$ and $p_0$ cannot have the same moment map value.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lemma:SymphPreservesAnIsolatedFixedPoint}\nLet $S^1(a,b;m)$ be a circle action for which the space $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ is non-empty. Assume there is an isolated fixed point $p_0$ corresponding to the vertex $R$ in Figure~\\ref{hirz}.  Then any equivariant symplectomorphism that preserves homology $\\phi\\in \\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda)$ fixes~$p_0$.\n\\end{lemma}\n\\begin{proof}\n\\emph{Case 1: $\\lambda >1$:}\nBy Lemma~\\ref{lemma:CharacterizationCentralizer} and Corollary~\\ref{cor:ActionPreservesWeights} any such $\\phi$ must preserve the moment values and the weights of the fixed points (up to change of order of the tuples). These weights are given in Table~\\ref{table_weights} and  the moment map values are given in the graphs~\\ref{fig:GraphsWithFixedSurfaces} and \\ref{fig:GraphsIsolatedFixedPoints}. The two conditions on the circle action imply that either $m=0$, $|2b-m|=0$, or $|2b+m|=0$. It is now easy to see that under any of these three numerical conditions, the weights and moment map values at $R$ differ from the weights at all other fixed points. The result follows.\n\\\\\n\n\\emph{Case 2: $\\lambda = 1$:}\nIf the actions are \\emph{not} of  the form $S^1(1,1; 0)$ or $S^1(-1,-1;0)$ with  $\\lambda = 1$, then an argument similar to Case 1 holds. \nThe only case left are the actions of the form $S^1(1,1; 0)$ or $S^1(-1,-1;0)$ with  $\\lambda = 1$. In this case, the  homology classes $F$, $B$ have the same area and the fixed points $R$ and $Q$ have the same weights (up to change of order of tuples) and the same moment map values. We again argue by contradiction in this case. Let $\\overline{B}$ denote a fixed curve in class $B$ passing through $R$ and $P$. Suppose $\\phi \\in \\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda)$ doesn't fix the point ~$p_0 = R$. Then $\\phi$ has to take the point $R$ to the point $Q$. Further by Lemma~\\ref{lemma:CharacterizationCentralizer}, $\\phi$ fixes the maximum and minimum and hence $\\phi(P) = P$. As $\\phi$ preserves homology, the curve $\\phi(\\overline{B})$ has homology class $B$ and has as to pass through $Q$ and $P$ which contradicts  Lemma~\\ref{projection_surjective}. \n\\end{proof}\n\nLet $J_0\\in U_0$ be the complex structure of the Hirzebruch surface $W_0$ and let $B_{p_0}$ be the unique $J_0$-holomorphic curve containing $p_0$ and representing the homology class $B$.\n\\begin{cor}\nLet $S^1(a,b;m)$ be a circle action with an isolated fixed point and for which the structure $J_0\\in U_0$ is invariant. Then the group $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ acts transitively on the space $\\mathcal{S}^{S^1}_{B,p_0}$, and the action map \n\\begin{align*}\n\\Symp^{S^1}_{h}(S^2 \\times S^2,\\omega_\\lambda) &\\longtwoheadrightarrow \\mathcal{S}^{S^1}_{B,p_0}\\\\\n\\phi & \\mapsto \\phi(B_{p_0})\n\\end{align*}\nis a Serre fibration.\n\\end{cor}\n\\begin{proof}\n\nSince any element of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ fixes $p_0$, it follows that this group acts on $\\mathcal{S}^{S^1}_{B,p_0}$. Let $p_1$ be the other fixed point corresponding to the point $Q$ in Figure~\\ref{hirz}. All curves in  $\\mathcal{S}^{S^1}_{B,p_0}$ pass through $p_1$ and $p_0$. Since the weights at one of $p_0$ or $p_1$ are always distinct, we can use the fixed point with distinct weights and proceed as in the proofs of Corollary ~\\ref{trans} and Lemma~\\ref{first} to show that the action defines a fibration.\n\\end{proof}\nAs before, we can now show that the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}$ is homotopy equivalent to a space of invariant curves.\n\\begin{lemma}\nThe natural map $\\alpha: \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0} \\to {\\mathcal{S}^{S^1}_{B,p_0}}$ defined by sending an almost complex structure $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}$ to the unique $J$-holomorphic curve in class $B$ passing through $p_0$ is a weak homotopic equivalence. \n\\end{lemma}\n\\begin{proof}\nThe argument is identical to the proof of Lemma~\\ref{first2}\n\\end{proof}\n\nFrom now on, we can determine the homotopy type of $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}$ by going through a similar sequence of fibrations and homotopy equivalences as in Section~\\ref{section:ActionOnU_2k}, namely,\n\\[\\Stab^{S^1}(B_{p_0}) \\to \\Symp^{S^1}_{h}(S^2 \\times S^2,\\omega_\\lambda) \\longtwoheadrightarrow \\mathcal{S}^{S^1}_{B,p_0} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}}  \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}\\rule{0em}{2em}\\]\n    \n\\[\\Fix^{S^1}(B_{p_0}) \\to \\Stab^{S^1}_{p_0}(B_{p_0}) \\longtwoheadrightarrow  \\Symp^{S^1}(B_{p_0}) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1\\rule{0em}{2em}\\]\n\n\\[\\Fix^{S^1} (N(B_{p_0})) \\to \\Fix^{S^1}(B_{p_0}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(B_{p_0})) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1\\rule{0em}{2em} \\]\n\n\\[\\Stab^{S^1}(\\overline{F}) \\cap \\Fix^{S^1}(N(B_{p_0})) \\to \\Fix^{S^1}(N(B_{p_0})) \\longtwoheadrightarrow  \\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1}(B_{p_0})\\rule{0em}{2em}\\]\n   \n\\[\\Fix^{S^1}(\\overline{F}) \\to \\Stab^{S^1}(\\overline{F}) \\cap \\Fix^{S^1}(N(B_{p_0})) \\longtwoheadrightarrow  \\Symp^{S^1}(\\overline{F}, N(p_0))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\n\n\\[\\left\\{*\\right\\} \\mathbin{\\textcolor{blue}{\\xleftarrow{\\text{~~~$\\simeq$~~~}}}} \\Fix^{S^1}(N(B_{p_0} \\vee \\overline{F})) \\to \\Fix^{S^1}(\\overline{F}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(B_{p_0} \\vee \\overline{F}))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\nwhere $\\overline{\\mathcal{S}^{S^1}_{F,p_0}}$ denotes the space of all symplectically embedded curve in the class $F$ that pass through $p_0$ and agree with a standard curve $F_{p_0}$ in a neighbourhood of $p_0$. The proofs that these maps are fibrations, and the proofs of the homotopy equivalences are exactly the same as before.\nConsequently, we obtain the following homotopical description of $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}$.\n\\begin{thm}\\label{Thm_isolatedfixedpt}\nConsider one of the following circle actions on $(S^2 \\times S^2,\\omega_\\lambda)$\n\\begin{itemize}\n    \\item $S^1(a,b;0)$ with $(a,b)\\neq (\\pm1,0)$ and $(a,b)\\neq(0,\\pm1)$, or\n    \\item  $S^1(1,b;m)$ with $|2b-m|=0$ and $2\\lambda > |2b-m|$, or\n    \\item $S^1(-1,b;m)$ with $|2b+m|=0$ and $2\\lambda > |2b+m|$.\n\\end{itemize}\nThen the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ is non-empty and \n\\[\\Symp^{S^1}_{h,p_0}(S^2 \\times S^2,\\omega_\\lambda)\/ \\mathbb{T}^2_0 \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}\\]\n\\end{thm}\\qed\n\n\n\\subsubsection{Actions without isolated fixed points} \n\nWe now turn our attention to the action of the group of equivariant symplectomorphisms $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ on the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}$ when the circle action is either\n\\begin{enumerate}\n    \\item $S^1(\\pm 1,0;0)$ or\n    \\item $S^1(0,\\pm 1;0)$.\n\\end{enumerate}\n\n\nThese actions has no isolated fixed points and the associated graphs are of the form\n\\begin{figure}[H]\n    \\centering\n   \n\n\n\\subcaptionbox{Subcase 1: $S^1(\\pm 1,0;0)$  }\n[.45\\linewidth]\n{\\begin{tikzpicture}\n[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm); \n\\draw [fill] (0,2.8) ellipse (0.3cm and 0.1cm); \n\\node[above] at (0,2.9) {$F_{max}$}; \n\\node[below] at (0,-0.1) {$F_{min}$}; \n\\node[left] at (-0.3,2.8) {$\\mu = \\lambda$};\n\\node[left] at (-0.3,0){$\\mu=0$};\n\\node[right] at (0.3,2.8){$A= 1$};\n\\node[right] at (0.3,0){$A= 1$};\n\\end{tikzpicture}\n}\n\\quad\n\\subcaptionbox{Subcase 2: $S^1(0,\\pm 1;0)$}\n[.45\\linewidth]\n{\\begin{tikzpicture}\n[scale=0.85, every node\/.style={scale=0.85}]\n\\draw [fill] (0,0) ellipse (0.3cm and 0.1cm); \n\\draw [fill] (0,2.8) ellipse (0.3cm and 0.1cm); \n\\node[above] at (0,2.9) {$B_{max}$}; \n\\node[below] at (0,-0.1) {$B_{min}$}; \n\\node[left] at (-0.3,2.8) {$\\mu = 1$};\n\\node[left] at (-0.3,0){$\\mu=0$};\n\\node[right] at (0.3,2.8){$A= \\lambda$};\n\\node[right] at (0.3,0){$A= \\lambda$};\n\\end{tikzpicture}\n}\n\\end{figure}\\label{subcase2:Fixedsurface}\n\\noindent where $\\mu$ denotes the value of the moment map and $A$ denotes the area of the fixed surface. We notice that there are pointwise fixed curves in the class $F$ for the circle action $S^1(\\pm 1,0;0)$ and pointwise fixed curves in class $B$ for the action $S^1(0,\\pm 1;0)$. We denote the fixed surface which is a minimum for the moment map as $F_{min}$, $B_{min}$ respectively and the maximum by $F_{max}$, $B_{max}$.\n\\\\\n\nConsider the action $S^1(0,\\pm 1;0)$. By Lemma~\\ref{lemma:CharacterizationCentralizer} we note that any $\\phi \\in \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ must send $B_{max}$ to itself. Then, given $p_0 \\in B_{max}$, we define the following sequence of fibrations and homotopy equivalences:\n\\[\\Fix^{S^1}(B_{max}) \\longrightarrow \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)  \\longtwoheadrightarrow \\Symp(B_{max}) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} SO(3)\\rule{0em}{2em}\\]\n\\[\\Stab^{S^1}(F_{p_0}) \\longrightarrow \\Fix^{S^1}(B_{max}) \\longtwoheadrightarrow \\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1}_{\\om_\\lambda} \\simeq \\{*\\}\\rule{0em}{2em}\\]\n\\[\\Fix^{S^1} (F_{p_0}) \\longrightarrow \\Stab^{S^1}(F_{p_0}) \\longtwoheadrightarrow \\Symp^{S^1}(F_{p_0}) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1\\rule{0em}{2em}\\]\n\\[\\left\\{*\\right\\}\\mathbin{\\textcolor{blue}{\\xleftarrow{\\text{~~~$\\simeq$~~~}}}} \\Fix^{S^1}(N(B_{max} \\vee F_{p_0})) \\longrightarrow  \\Fix^{S^1}(F_{p_0})\\longtwoheadrightarrow \\Gauge1^{S^1}(N(B_{max} \\vee F_{p_0}))\n\\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}}  \\left\\{*\\right\\}\\rule{0em}{2em}\\]\n\\\\\n\nFor the other circle action $S^1(\\pm 1,0;0)$, we obtain a similar sequence of fibrations and homotopy equivalences in which $B_{\\max}$ is replaced by the curve $F_{\\max}$. As before, putting all the homotopy equivalences together, we obtain the following theorem:\n\\begin{thm}\\label{Thm_fixedsurfaces}\nConsider the following two circle actions on $(S^2 \\times S^2,\\omega_\\lambda)$  \n\\begin{itemize}\n\\item $S^1(\\pm 1,0;0)$ or\n\\item  $S^1(0,\\pm 1;0)$\n\\end{itemize}\nThen there is a homotopy equivalence\n\\[\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/(S^1 \\times \\SO(3)) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0\\]\n\\end{thm}\\qed\n\nFor convenience, we collect together the two main results of this section in the theorem below. \n\\begin{thm}{\\label{homog}}\nConsider the action $S^1(a,b;m)$ on $(S^2 \\times S^2,\\omega_\\lambda)$ such that  one of the following  hold:\n\\begin{itemize}\n    \\item $S^1(a,b;0)$ with $(a,b)\\neq (\\pm1,0)$ and $(a,b)\\neq(0,\\pm1)$, or\n    \\item  $S^1(1,b;m)$ with $|2b-m|=0$ and $2\\lambda > |2b-m|$, or\n    \\item $S^1(-1,b;m)$ with $|2b+m|=0$ and $2\\lambda > |2b+m|$.\n\\end{itemize}\nThen the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0$ is non-empty and \n\\[\\Symp^{S^1}_{h,p_0}(S^2 \\times S^2,\\omega_\\lambda)\/ \\mathbb{T}^2_0 \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{0}\\]\nIf instead the $S^1(a,b;m)$ action satisfies \n\\begin{itemize}\n\\item $(a,b;m) = (\\pm 1,0;0)$ or\n\\item  $(a,b;m) = (0,\\pm 1;0)$\n\\end{itemize} then we have that \\[\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/(S^1 \\times \\SO(3)) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_0\\]\nand $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects only the strata $U_0$.\n\\end{thm}\\qed \n\n\n\n\n\n\n\\chapter{The homotopy type of the symplectic centralisers of \\texorpdfstring{$S^1(a,b;m)$}{S1(a,b;m)}}\nGiven any Hamiltonian circle action on $(S^2 \\times S^2,\\omega_\\lambda)$, the two Theorems~\\ref{cor:IntersectingOnlyOneStratum} and~\\ref{cor:IntersectingTwoStrata} give us a complete understanding of which strata the space $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects. Together with Theorems~\\ref{homogenous}, and~\\ref{homog} describing the strata as homogeneous spaces, this allows us to compute the homotopy type of the group of equivariant symplectomorphisms.\n\n\\section{When \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda}$}{J\\hat S1} is homotopy equivalent to a single symplectic orbit}\n\n\n\\begin{thm}\\label{table}\nConsider the  circle action $S^1(a,b;m)$ on $(S^2 \\times S^2,\\omega_\\lambda)$. Under the following numerical conditions on $a,b,m,\\lambda$, the homotopy type of $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ is given by the  table below.\n\\\\\n\n\\noindent{\n\\begin{tabular}{|p{5cm}||p{3.5cm}|p{2cm}|p{3.3cm}|}\n\\hline\n $S^1$ action $(a,b;m)$ & $\\lambda$ &Number of strata $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects &Homotopy type of $\\Symp^{S^1}(S^2 \\times S^2)$\\\\\n\\hline\n {$(0,\\pm 1;m)$ ~~ $m\\neq 0$}  &$\\lambda > 1$ & 1 & $S^1 \\times SO(3)$ \\\\\n\\hline\n\\multirow{2}{10em}{$(0,\\pm 1;0)$ or $(\\pm 1,0;0)$} &$\\lambda = 1$  &1  &$S^1 \\times SO(3)$ \\\\\n&$\\lambda >1$ &1  &$S^1 \\times SO(3)$ \\\\\n\\hline\n$(\\pm 1,\\pm1,0)$&$\\lambda = 1$ &1 & $\\mathbb{T}^2 \\times \\mathbb{Z}_2$ \\\\\n\\hline\n $(\\pm 1,0;m) ~ m\\neq0$ &$\\lambda >1$ &1 &$\\mathbb{T}^2$\\\\\n \\hline\n$(\\pm 1,\\pm m;m)$ $m \\neq 0$ & $\\lambda > 1$   &1 & $\\mathbb{T}^2$\\\\\n\\hline\n$( 1,b;m)$ $b \\neq \\{ m,0\\}$ &$|2b-m| \\geq2 \\lambda \\geq 1$ &1 &$\\mathbb{T}^2$ \\\\\n\\hline\n$(-1,b;m), b \\neq \\{ -m,0\\}$ &$|2b+m| \\geq2 \\lambda \\geq 1$ &1 &$\\mathbb{T}^2$ \\\\\\cline{2-4}\n\\hline\nAll other values of $(a,b;m)$ except $(\\pm 1,b;m)$ &$\\forall \\lambda$  &1   &$\\mathbb{T}^2$ \\\\\n\\hline\n\\end{tabular}\n}\n\\end{thm}\n \n \\begin{proof}\n By Theorem~\\ref{cor:IntersectingOnlyOneStratum}, in each of the above $S^1(a,b;m)$ actions, the space of $S^1$ invariant compatible almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects only the stratum $U_m$. Consequently,\n \\[\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)\/\\Stab(J_m) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m = \\mathcal{J}^{S^1}_{\\om_\\lambda} \\simeq \\{*\\}\\]\n where $\\Stab(J_m)$ denotes the stabiliser of the standard complex structure $J_m \\in U_m$. Thus, for all the actions in the table, we have that $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\simeq \\Stab(J_m)$. For the $S^1$ action given by the triples $(0,\\pm 1,m)$, $(\\pm1,0,0)$ or the circle action $S^1(0,\\pm1,0)$ when $\\lambda =1$, Theorems~\\ref{homogenous} and~\\ref{homog} imply that $\\Stab(J_m) \\simeq S^1 \\times \\SO(3)$. For all other $S^1$ actions in the table, the stabilizers are homotopy equivalent to $\\mathbb{T}^2$. \n\\\\\n\nWe now show how to recover the homotopy type of the full group $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda) $ from the homotopy type of the subgroup $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$. When $\\lambda > 1$, we have the equality $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda) = \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ as stated in Lemma~\\ref{lemma:ActionOnHomology}.\n\\\\\n\n\n\nWhen $\\lambda = 1$ and $a\\neq b$, \nthere exists standard $S^1(a,b;m)$ invariant curves in classes $B$ and $F$ such that the isotropy weight of the action on the curve in class $B$ is $a$ and the isotropy weight of the $S^1$ action on the curve in the class $F$ is $b$. Hence, as $\\phi$ is an equivariant symplectomorphism, Lemma~\\ref{lemma:ActionOnHomology} implies that must have $\\phi_*[F] = [F]$ and $\\phi_*[B] = [B]$. Consequently, $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) = \\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$.\n\\\\\n\nIn the special case when $\\lambda =1$ and $a=b = \\pm 1$, then we have an equivariant version of the exact sequence~\\ref{Sequence:ActionOnHomology}\n\\begin{equation*}\n   1 \\longrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\longrightarrow  \\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\longrightarrow \\Aut_{c_1,\\omega_\\lambda}(H^2(S^2 \\times S^2)) \\longrightarrow 1\n\\end{equation*}\n\nwhere $\\Aut_{c_1,\\omega_\\lambda}(H^2(S^2 \\times S^2)) \\cong \\mathbb{Z}_2$. The map \\begin{align*}\n    \\phi: S^2 \\times S^2 \\to S^2 \\times S^2 \\\\\n    (z,w) \\mapsto (w,z)\n\\end{align*} \nis a $S^1$ equivariant symplectomorphism (for the action $S^1(1,1,0)$ or $ (-1,-1,0)$) and gives a section from $\\mathbb{Z}_2 \\cong \\Aut_{c_1,\\omega_\\lambda}(H^2(S^2 \\times S^2))$ to $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$. Thus we have $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\cong  \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\rtimes \\mathbb{Z}_2$. As the semidirect product of two topological groups is homotopy equivalent(as a topological space) to the direct product of the groups,  we have that  $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\cong  \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\rtimes \\mathbb{Z}_2 \\simeq  \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda) \\times \\mathbb{Z}_2 \\simeq \\mathbb{T}^2 \\times \\mathbb{Z}_2$. This completes the proof.\n\n\\end{proof}\n\n\\section[The two orbits case]{When \\texorpdfstring{$\\mathcal{J}^{S^1}_{\\om_\\lambda}$}{J\\hat{}S1} is homotopy equivalent to the union of two symplectic orbits}\\label{section:TwoOrbits}\n\n\nTheorem~\\ref{table} gives the homotopy type of the group of equivariant symplectomorphisms for all circle actions on $S^2 \\times S^2$ apart from the following two families of actions:\n\\begin{itemize}\n\\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|$.\n\\end{itemize}\nFor convenience, we will write $m'$ for either $|2b-m|$ or $|2b+m|$ depending on which of the above families we consider. Up to swapping $m$ and $m'$, we will also assume $m'>m$. The goal of this section is to show that the symplectic stabilizers of any of these circle actions is homotopy equivalent to the pushout of the two tori $\\mathbb{T}^2_m$ and $\\mathbb{T}^2_{m'}$ along the common $S^1$ in the category of topological groups.\\\\ \n\nBefore delving into the technicalities, it may be useful to outline the proof, which is an adaptation of the Anjos-Granja argument used in~\\cite{AG} to compute the homotopy type of the full group of symplectomorphisms of $S^2 \\times S^2$ for $1<\\lambda\\leq 2$. The first step is to show that the two inclusions \\[\\mathbb{T}^2_{m} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)\\quad \\text{~and~}\\quad \\mathbb{T}^2_{m'} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)\\]\ninduce injective maps in homology. By the Leray-Hirsch theorem, it follows that the cohomology modules of the total space of the fibrations\n\\[\\mathbb{T}^2_m\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\to \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_m \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_m\\]\n\\[\\mathbb{T}^2_{m'}\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_{m'} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'}\\]\nsplit (with coefficients in an arbitrary field $k$). Using the fact that the contractible space of invariant compatible almost-complex structures decomposes as the disjoint union\n\\[\\mathcal{J}^{S^1}_{\\om_\\lambda}=(\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_m)\\sqcup (\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'})\\]\nthe rank of $H^i(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda);k)$ can be computed inductively from Alexander-Eells duality. We then compute the cohomology algebra and the Pontryagin algebra of the pushout\n\\[P = \\pushout(\\mathbb{T}^2_m\\leftarrow S^1\\to \\mathbb{T}^2_{m'})\\]\nin the category of topological groups. We then show that the natural map\n\\[\\Upsilon:P\\to \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\]\nis a homotopy equivalence in the category of topological groups. We further prove that $P$ is weakly homotopy equivalent, as a topological space, to the product $\\Omega S^{3}\\times S^{1}\\times S^{1}\\times S^{1}$.\n\n\n\n\n\\subsection{Homological injectivity}\nWe first show that the two inclusions $\\mathbb{T}^2_{m} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ and $\\mathbb{T}^2_{m'} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ induce injective maps in homology. As the argument does not depend on $m$, we shall only provide the details for the inclusion $\\mathbb{T}^2_{m} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$.\\\\\n\nFix a symplectomorphism $\\phi_m: W_m\\to (S^2 \\times S^2,\\omega_\\lambda)$ compatible with the fibration structures. Let $\\{*\\}$ be the $S^1(1,b,m)$ fixed point $\\left([0,1][0,0,1]\\right)$ in $W_m$, and let $\\mathcal{E}(W_m, *)$ denote the space of orientation preserving, pointed, homotopy self-equivalences of $(W_m,*)$. Similarly, define $\\mathcal{E}(S^2, *)$ to be the space of all orientation preserving homotopy self-equivalences of the sphere preserving a base point $\\{*\\}$.\n\\\\\n\nWe now observe that for the above two families of circle actions (i) and (ii), the same argument as in Lemma~\\ref{lemma:SymphPreservesAnIsolatedFixedPoint} shows that any $\\phi\\in\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)= \\Symp_h^{S^1}(W_{m})$ fixes the base point $\\{*\\}$.\\\\\n\nNow, recall that the zero section $s_0$ of $W_m$ is given by\n\\begin{align*}\n    s_{0}: S^2 &\\to W_m \\\\\n    \\left[z_{0}, z_{1}\\right] &\\mapsto\\left(\\left[z_{0}, z_{1}\\right],[0,0,1]\\right)\n\\end{align*} \nand the projection to the first factor is\n\\begin{align*}\n    \\pi_1: W_m &\\to S^2 \\\\\n    \\left(\\left[z_{0}, z_{1}\\right],\\left[w_{0}, w_{1}, w_{2}\\right]\\right) &\\mapsto\\left[z_{0}, z_{1}\\right]\n\\end{align*}\nWe define a continuous map $h_1:\\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda) \\to \\mathcal{E}\\left(S^{2}, *\\right)$ by setting\n\\begin{equation*}\n  \\begin{aligned}\nh_1:\\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda) &\\to \\mathcal{E}\\left(S^{2}, *\\right) \\\\\n\\psi &\\mapsto \\psi_1:= \\pi_{1} \\circ \\psi \\circ s_{0}\n\\end{aligned}\n\\end{equation*}\nSimilarly, using the inclusion of $S^{2}$ as the fiber \n\\begin{align*}\n    f: S^2 &\\to W_m \\\\\n    \\left[z_{0}, z_{1}\\right] &\\mapsto\\left([0,1],\\left[ 0,z_{0}, z_{1}\\right]\\right)\n\\end{align*}\nand the projection to the second factor $\\pi_{2}: S^2 \\times S^2 \\to S^{2}$, we can define a map \n\\begin{align*}\n   h_2: \\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda) &\\to \\mathcal{E}\\left(S^{2}, *\\right) \\\\\n \\psi &\\mapsto \\psi_2:= \\pi_{2} \\circ \\psi \\circ f\n\\end{align*}\n\nWe thus get a continuous map \n\\begin{align*}\n    h: \\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda) &\\to \\mathcal{E}(S^2, *) \\times \\mathcal{E}(S^2,*) \\\\\n    \\psi &\\mapsto \\left(h_1(\\psi),h_2(\\psi)\\right) \\\\\n\\end{align*}\n\\begin{lemma} \\label{inj}\nThe inclusion $i_m:\\mathbb{T}^2_m \\hookrightarrow \\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda) $ induces a map which is injective in homology with coefficients in any field $k$.\n\\end{lemma}\n\\begin{proof}\nAs $\\mathbb{T}^2$ is connected, $i_m: H_0(\\mathbb{T}^2_m; k) \\to H_0(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda); k)$ is injective. To show that the inclusion map induces an injection at the $H_1$ level, we consider the composition $\\alpha: \\mathbb{T}^2_m \\to \\mathcal{E}(S^2,*) \\times \\mathcal{E}(S^2,*)$ given by\n\\[\n\\begin{tikzcd}\n    &\\mathbb{T}^2_m \\arrow[r,hookrightarrow]  &\\Symp_h^{S^1} (S^2 \\times S^2,\\omega_\\lambda)  \\arrow[r,rightarrow,\"h\"]  &\\mathcal{E}(S^2, *) \\times \\mathcal{E}(S^2, *) \n\\end{tikzcd}\n\\]\nand show that $\\alpha$ induces a map which is injective in homology. \n\nWe claim that $H_1(\\mathcal{E}(S^2,*);\\mathbb{Z})\\simeq \\mathbb{Z}$. Indeed, the standard action of $\\SO(3)$ on $S^2$ gives rise to a diagram of fibrations\n\\[\n\\begin{tikzcd}\n        & \\mathcal{E}(S^2,*)\\arrow[r] &\\mathcal{E}(S^2) \\arrow[r,twoheadrightarrow,\"\\ev\"] &S^2 \\\\\n     &S^1= \\SO(2) \\arrow[u,hookrightarrow] \\arrow[r] &\\SO(3) \\arrow[u,hookrightarrow] \\arrow[r,\"\\ev\",twoheadrightarrow] &S^2 \\arrow[u,equal] \\\\\n\\end{tikzcd}\n\\]\n\nwhere the maps $\\ev$ are evaluations at the base point $\\{*\\}$. This induces a long exact ladder of homotopy groups\n\\[\n\\begin{tikzcd}\n     &\\cdots \\arrow[r] &\\cancelto{\\mathbb{Z}}{\\pi_2(S^2)} \\arrow[r] &\\pi_1(\\mathcal{E}(S^2,*)) \\arrow[r] &\\pi_1(SO(3)) \\times \\cancelto{0}\\pi_1(\\Omega) \\arrow[r] &\\cancelto{0}{\\pi_1(S^2)} \\\\\n     &\\cdots \\arrow[r] &\\mathbb{Z} \\arrow[u,equal] \\arrow[r]&\\cancelto{\\mathbb{Z}}{\\pi_1(S^1)} \\arrow[r] \\arrow[u,\"\\beta\"] &{\\pi_1(SO(3))}\\arrow[u] \\arrow[r] &\\cancelto{0}{\\pi_1(S^2)} \\arrow[u,equal]\n\\end{tikzcd}\n\\]\nwhere we have used the fact, proven by Hansen in~\\cite{Hans}, that $\\mathcal{E}(S^2) \\simeq SO(3) \\times \\widetilde{\\Omega^2}$, where $\\widetilde{\\Omega^2}$ denotes the universal covering space for the connected component of the double loop space of $S^2$ containing the constant based map, and where the $\\SO(3)$ component is just the inclusion. Consequently, $\\pi_1(\\widetilde{\\Omega^2})= 0 $ and the map $\\pi_1(SO(3)) \\to \\pi_1(SO(3)) \\times \\pi_1(\\widetilde{\\Omega^2})$ is an isomorphism. From the commutativity of the middle square, it follows that $\\beta:\\pi_1(S^1) \\to \\pi_1(\\mathcal{E}(S^2, *))$ is also an isomorphism. As the spaces we consider are topological groups, $\\pi_1$ is abelian and hence $\\pi_1 = H_1$, proving the claim.\\\\\n\nNow, the classes $a$, $b$, of the subcircles $(0,1)$ and $(1,0)$ form a basis for $H_1(\\mathbb{T}^2_m; k)$. We claim that $\\alpha_*[0,1]$ and $\\alpha_*[1,0]$ generate a subgroup of rank 2. To see this, let write $\\alpha_*^1$ and $\\alpha_*^2$ for the components of $\\alpha_*$. Then, $\\alpha^1_*(0,1) = 0$ as the circle $(0,1)$ fixes the zero section $\\left([x_1,x_2],[0,0,1]\\right) \\subset W_m$ pointwise, while $\\alpha^2_*[0,1] \\neq 0$ by the reasoning in the previous paragraph. Similarly, $\\alpha^1_*[1,0] \\neq 0$ and $\\alpha^2_*[1,0]= 0$, proving our claim. We conclude that $\\alpha$ is injective on $H_1(\\mathbb{T}^2_m; k)$. \n\\\\\n\nFinally, to show that $i_*$ is injective on $H_2(\\mathbb{T}^2_m;k)$, we will prove the dual statement, namely, that the map $i^*:H^2(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda);k) \\to H^2(\\mathbb{T}^2_m;k)$ is surjective. A generator of $H^2(\\mathbb{T}^2_m;k) \\cong k$ is given by $a \\cup b$. Because $i_*$ is injective at the $H_1$ level, $i^*: H^1(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda);k) \\to H^1(\\mathbb{T}^2;k)$ is surjective, hence there exists elements $a^\\prime$, $b^\\prime \\in H^1(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda);k)$ such that $i^*(a^\\prime) = a$ and $i^*(b^\\prime) = b$. Since $i^*(a^\\prime) \\cup i^*(b^\\prime) = a \\cup b$, it follows that $i^*:H^2(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda);k) \\to H^2(\\mathbb{T}^2_m;k)$ is surjective.\n\\end{proof}\n\\subsection[Cohomology module of the centralizer]{Cohomology module of the centralizer of $S^1(\\pm1,b;m)$}\\label{subsection:CohomologyModule}\nWe are now ready to compute the cohomology module of the centralizer of $S^1(\\pm1,b;m)$ with coefficients in in a field $k$. By duality, this is equivalent to determining the homology module.\n\nRecall that the contractible space of invariant compatible almost-complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ decomposes as the disjoint union\n\\[\\mathcal{J}^{S^1}_{\\om_\\lambda}=(\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_m)\\sqcup (\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'})=:U_{m}^{S^1} \\sqcup U_{m'}^{S^1}\\]\nwhere, for convenience, we set $U_m^{S^1}=\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_m$ and $U_{m'}^{S^1}=\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'}$. We will show in Chapter~\\ref{Chapter-codimension} the following two important facts:\n\\begin{itemize}\n\\item the strata $U_{m}^{S^1}$ and $U_{m'}^{S^1}$ are submanifolds of $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ (see Corollary~\\ref{cor:StrataAreSubmanifolds}), and \n\\item the stratum $U_{m}^{S^1}$ is open in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$, while $U_{m'}^{S^1}$ is of codimension $2$ (see Theorem~\\ref{codimension_calc}). \n\\end{itemize}\nIn particular, it follows that $U_{m}^{S^1}=\\mathcal{J}^{S^1}_{\\om_\\lambda}-U_{m'}^{S^1}$ is connected. As explained in Appendix~\\ref{Appendix-Alexander-Eells},  Proposition~\\ref{prop:AlexanderEellsGeometric},\n\nthe Alexander-Eells duality induces an isomorphism of homology groups\n\\begin{equation}\\label{eq:AlexanderEellsIsomorphismHomology}\n\\lambda_{*}:H_{p}(U_{m'}^{S^1};k)\\to H_{p+1}(U_{m}^{S^1};k)\n\\end{equation}\n\nNow recall that we also have fibrations\n\\begin{align}\\label{eq:TheTwoMainFibrations}\n\\mathbb{T}^2_m\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\xrightarrow{p_m} \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_m \\simeq U_{m}^{S^1}\\\\\n\\mathbb{T}^2_{m'}\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\xrightarrow{p_{m'}}\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_{m'} \\simeq U_{m'}^{S^1}\\notag\n\\end{align}\nFrom the first fibration, the connectedness of the open stratum $U_{m}^{S^1}$ implies that the group $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ is connected. In turns, the second fibration implies that the codimension 2 stratum $U_{m'}^{S^1}$ is also connected.\nBecause the two inclusions \\[\\mathbb{T}^2_{m} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)\\quad \\text{~and~}\\quad \\mathbb{T}^2_{m'} \\hookrightarrow \\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)\\]\ninduce surjective maps in cohomology, the Leray-Hirsch theorem implies that the cohomology module of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ splits as\n\\begin{align}\\label{eq:SplittingCohomology}\nH^*(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda),k) \\cong H^*(U_{m}^{S^1};k) \\otimes H^*(\\mathbb{T}^2_m;k)\\\\\nH^*(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda),k) \\cong H^*(U_{m'}^{S^1};k) \\otimes H^*(\\mathbb{T}^2_{m'};k)\\notag\n\\end{align}\nBy duality, we have corresponding splittings in homology, namely,\n\\begin{align}\\label{eq:SplittingHomology}\nH_*(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda),k) \\cong H_*(U_{m}^{S^1};k) \\otimes H_*(\\mathbb{T}^2_m;k)\\\\\nH_*(\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda),k) \\cong H_*(U_{m'}^{S^1};k) \\otimes H_*(\\mathbb{T}^2_{m'};k)\\notag\n\\end{align}\nIt follows that \n\\[H_{p}(U_{m};k)\\simeq H_{p}(U_{m'};k)\\text{~for all~}p\\geq 0\\]\nTogether with the Alexander-Eells isomorphism~(\\ref{eq:AlexanderEellsIsomorphismHomology}) and the connectedness of $U_{m'}$, this implies that\n\\[H_{p}(U_{m};k)\\simeq k\\text{~for all~}p\\geq 0\\]\nUsing the splitting~\\ref{eq:SplittingHomology} and dualizing, we can finally compute the cohomology module of $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$.\n\n\\begin{thm}{\\label{cohom}} Consider any of the following circle actions:\n\\begin{itemize}\n\\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|$.\n\\end{itemize} Then, the cohomology groups of the symplectic centralizer are\n$$H^p\\left(\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda); k\\right) \\simeq \\begin{cases}\nk^4 ~~p \\geq 2\\\\\nk^3 ~~p =1 \\\\\nk ~~ p=0\\\\\n\\end{cases}$$\nfor any field $k$. In particular, the topological group $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ is of finite type.\n\\end{thm}\n\n\n\n\\subsection{The homotopy pushout \\texorpdfstring{$T_m\\leftarrow S^1(\\pm1,b;m)\\to T_{m'}$}{the two inclusions}}\nAs explained in Corollary~\\ref{cor:CircleExtensionsWith_a=1} and Corollary~\\ref{cor:CircleExtensionsWith_a=-1}, the circle actions $S^1(\\pm1,b;m)$ we are considering in this section extend to exactly two toric actions $\\mathbb{T}^2_m$ and $\\mathbb{T}^2_{m'}$. Geometrically, this means that the two tori $\\mathbb{T}^2_m$ and $\\mathbb{T}^2_{m'}$ intersect in $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$ along the circle $S^1(\\pm1,b;m)$ and, in particular, that we have two inclusions of Lie groups\n\\[\n\\begin{tikzcd}\nS^{1} \\arrow{r}{(1,b')} \\arrow[swap]{d}{(1,b)} & T^{2}_{m'} \\\\\nT^{2}_{m}  & \n\\end{tikzcd}\n\\]\nIn this section we consider the homotopy pushout of these two inclusions, namely,\n\\[P:=\\pushout(T_m\\leftarrow S^1\\to T_{m'})\\]\nThis pushout is to be understood in the category of topological groups. As we will show later, the topological group $P$ turns out to be a model for the homotopy type of the centralizer $\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)$.\n\n\\subsubsection*{The Pontryagin algebra of the pushout}\nIn what follows, all k algebras are graded, and the commutator of two elements is given by\n\\[[a,b] = ab - (-1)^{|a|\\cdot|b|}ba\\]\nFor any field $k$, and for any abelian group $A$, the Pontryagin algebra $H_{*}(A;k)$ is isomorphic to the cohomology algebra $H^{*}(A;k)$. It follows that $H_{*}(S^{1})$ is isomorphic to $\\Lambda(t)$, where $t$ is of degree $1$. Similarly, the Pontryagin algebra $H_{*}(T^{2};k)$ is isomorphic to the to an exterior algebra $\\Lambda(z_{1},z_{2})$ generated by two elements of degree one. The pushout diagram of topological groups \n\\[\n\\begin{tikzcd}\nS^{1} \\arrow{r}{(1,b')} \\arrow[swap]{d}{(1,b)} & T^{2}_{m'} \\arrow{d}{} \\\\\nT^{2}_{m} \\arrow{r}{} & P\n\\end{tikzcd}\n\\]\nis homologically free (see Definition~3.1 in~\\cite{AG}). As before, $P$ denotes the pushout in the category of topological groups. By Theorem~3.8 of~\\cite{AG}, the Pontryagin algebra of $P$ is the pushout of $k$ algebras\n\\[\n\\begin{tikzcd}\nH_{*}(S^{1};k) \\arrow{r}{H(1,b')} \\arrow[swap]{d}{H(1,b)} & H_{*}(T^{2}_{m'};k) \\arrow{d}{} \\\\\nH_{*}(T^{2}_{m};k) \\arrow{r}{} & H_{*}(P;k)\n\\end{tikzcd}\n\\]\nwhich is isomorphic to\n\\[\n\\begin{tikzcd}\n\\Lambda(t) \\arrow{r}{(1,b')} \\arrow[swap]{d}{(1,b)} & \\Lambda(y_{1},y_{2}) \\arrow{d}{} \\\\\n\\Lambda(x_{1},x_{2}) \\arrow{r}{} & P^{alg}_{*}\n\\end{tikzcd}\n\\]\nwhere $P^{alg}_{*}\\simeq H_{*}(P;k)$. By the description of the pushout of $k$ algebras as amalgamated products (see \\cite{AG} for more details), the $k$ algebra $P^{alg}_{*}$ can be identified with equivalence classes of finite linear combinations of words in the letters $\\{x_{1},x_{2},y_{1},y_{2}\\}$ under the relations $x_{i}x_{i}=0$, $y_{i}y_{i}=0$, $[x_{1},x_{2}]=0$, $[y_{1},y_{2}]=0$, and $x_{1}+bx_{2}=y_{1}+b'y_{2}$.  From the last equality, we can write $y_{1}=(x_{1}+bx_{2})-b'y_{2}$, which means that we can choose, as generators, the elements \\[\\{t=x_{1}+bx_{2}, ~x_{2}, ~y_{2}\\}\\]\nwith the relations $t^{2}=x_{2}^{2}=y_{2}^{2}=0$, $[t,x_{2}]=[t,y_{2}]=0$. The remaining commutator $w=[x_{2},y_{2}]$ is nonzero and commutes with $t$, $x_{2}$ and $y_{2}$. It follows that any word in $t,x_{2},y_{2}$ is equivalent to a linear combination of words of the form\n\\[w^{\\alpha}x_{2}^{\\beta}y_{2}^{\\gamma}t^{\\delta}\\]\nwith $\\alpha\\in\\mathbb{N}\\cup\\{0\\}$, and $\\beta,\\gamma,\\delta\\in\\{0,1\\}$. Hence, there is an isomorphism of graded algebras\n\\[P^{alg}_{*}\\cong \\frac{F(x_{2},y_{2})}{\\langle x_{2}^{2},y_{2}^{2}\\rangle}\\otimes \\Lambda(t)\\]\nwhere $F(x_{2},y_{2})$ denotes the free graded algebra over $k$ generated by the elements $x_{2}$ and $y_{2}$, and where $x_{2},y_{2},t$ are of degree one. In particular,\n\\[P^{alg}_{n}\\simeq\n\\begin{cases}\nk & n=0\\\\\nk^{3} & n=1\\\\\nk^{4} & n\\geq 2\n\\end{cases}\\]\nand the words $w^{\\alpha}x_{2}^{\\beta}y_{2}^{\\gamma}t^{\\delta}$ form an additive basis of the homology module $P^{alg}_{*}$.\n\nBy duality, the cohomology modules $P^{alg,*}$ are $P^{alg,0} \\simeq k$, $P^{alg,1}\\simeq k^3$, and $P^{alg,n}\\simeq k^4$ for all $n\\geq 2$. The algebra structure of $P^{alg,*}$ can be determined as follows. Let $\\hat t$, $\\hat x_{2}$, and $\\hat y_{2}$ be the duals of the generators of degree $1$, and let $\\hat w$ be the dual of the generator $w=[x_2,y_2]$ of degree $2$.\n\\\\\n\nLet us now recall the Hopf-Borel theorem (see~\\cite{McCleary} Theorem 6.36).\n\\begin{thm}(Hopf-Borel)\nLet k be a field of characteristic p where p may be zero or a prime. A connected Hopf algebra $H$ over $k$ is said to be monogenic if $H$ is generated as an algebra by 1 and one homogeneous element $x$ of degree strictly greater than 0. If $H$ is a monogenic Hopf algebra, then\n\\begin{enumerate}\n    \\item if $p \\neq 2$ and degree $x$ is odd, then $H \\cong \\Lambda(x)$,\n    \\item if $p \\neq 2$ and degree $x$ is even, then $H \\cong k[x] \/\\left\\langle x^{s}\\right\\rangle$ where $s$ is a power of p or is infinite i.e $H \\cong k[x]$,\n    \\item if $p=2$, then $H \\cong k[x] \/\\left\\langle x^{s}\\right\\rangle$ where $s$ is a power of 2 or is infinite.\n\\end{enumerate}\n\\end{thm}\n\nAs $P^{alg,*}$ is an associative, graded commutative Hopf algebra of finite type, the Hopf-Borel theorem (see~\\cite{McCleary} Theorem 6.36) implies that $P^{alg,*}$ is a tensor product of monogenic Hopf algebras. For a field $k$ of characteristic $p$ different from $2$, including $p=0$, $P^{alg,*}$ contains a subalgebra of the form \n\\[A^*=\\Lambda(\\hat t,\\hat x_2,\\hat y_2)\\otimes k[\\hat w]\/\\langle \\hat w^s\\rangle\\]\nwhere $s$ is a power of $p$ or is infinite. Suppose $s=p^n\\geq 3$ is finite. Then, the rank of $A^i$ would coincide with the rank of $P^{alg,i}$ up to degree $i=2s-1$, and we would have $A^i=0$ for $i\\geq 2s$. Therefore, we would need $4$ more generators of degree $2s$ to account for the rank of $P^{alg,2s}$, and their pairwise products would imply that $\\rk P^{alg,4s}>4$. This contradiction shows that $s$ must be infinite and that the rank of $A^i$ equals the rank of $P^{alg,i}$ for all $i\\geq 0$. Consequently, for a field $k$ of characteristic $p\\neq 2$, the $k$-algebra $P^{alg,*}$ is isomorphic to\n\\[P^{alg,*}\\cong \\Lambda(\\hat t,\\hat x_{2},\\hat y_{2})\\otimes S(\\hat w)\\]\nIn characteristic $p=2$, $P^{alg,*}$ is the tensor product of truncated polynomial algebras $k[z_i]\/z^{s_i}_{i}$ where $s_i$ is a power of $2$. As before, it contains a subalgebra of the form\n\\[A^*=k[\\hat t,\\hat x_2,\\hat y_2]\/ \\langle \\hat t^2,\\hat x_2^2,\\hat y_2^2 \\rangle \\otimes k[\\hat w]\/\\langle \\hat w^s\\rangle\\]\nAgain, assuming $s$ is finite forces the existence of $4$ new generators in degree $2s$ whose products would yield too many generators in degree $4s$. Therefore, in characteristic $p=2$, the cohomology algebra of $P$ is isomorphic to\n\\[P^{alg,*}\\cong k[\\hat t,\\hat x_{2},\\hat y_{2}]\/ \\langle \\hat t^2,\\hat x_2^2,\\hat y_2^2 \\rangle \\otimes k[\\hat w]\\]\nIn characteristic zero, the computation of the cohomology ring yields the minimal model of $H^{*}(P)\\otimes\\mathbb{Q}$. As $P$ is a H-space, it is a nilpotent space (see Exercise~1.13 in \\cite{dgcRationalHomotopy}), so that the main theorem of dgc rational homotopy theory applies (see \\cite{dgcRationalHomotopy}, Theorem~2.50) namely, the dimension $\\pi_{p}(P)\\otimes\\mathbb{Q}$ for $p\\geq 2$ is equal to the number of generators of degree $p$ in the minimal model. For $p=1$, as $P$ is a topological group, the dimension of $\\pi_1(P) \\otimes \\mathbb{Q}$ is same as the rank of $H_1(P,\\mathbb{Q})$. Consequently,\n\\[\\pi_{p}(P)\\otimes\\mathbb{Q}\\simeq\n\\begin{cases}\n\\mathbb{Q} & p=0\\\\\n\\mathbb{Q}^{3} & p=1\\\\\n\\mathbb{Q} & p= 2\\\\\n0 & p\\geq 3\n\\end{cases}\\]\n\n\\subsubsection*{The homotopy type of $P$}\nWe want to better understand the homotopy type of the space $P$. To this end, consider the embeddings\n\\begin{align}\nf_{m}:T^{2}_{m}&\\to S^{1}\\times S^{1}\\times S^{1}\\\\\n(x_{1},x_{2})&\\mapsto (x_{1},x_{2},b'x_{1})\\notag\\\\\n&\\notag\\\\\nf_{m'}:T^{2}_{m'}&\\to S^{1}\\times S^{1}\\times S^{1}\\\\\n(y_{1},y_{2})&\\mapsto (y_{1},by_{1},y_{2})\\notag\n\\end{align}\nThe universal property of pushouts implies that there is a unique map $f_{P}:P\\to S^{1}\\times S^{1}\\times S^{1}$ making the following diagram commutative\n\\[\n\\begin{tikzcd}\nBS^{1} \\arrow{r}{B(1,b')} \\arrow[swap]{d}{B(1,b)} & BT^{2}_{m'} \\arrow{d}{} \\arrow[bend left=10]{ddr}{Bf_{m'}} & \\\\\nBT^{2}_{m} \\arrow{r}{}\\arrow[bend right=10,swap]{drr}{Bf_{m}} & BP \\arrow[dotted]{rd}{Bf_{P}} & \\\\\n & & BS^{1}\\times BS^{1}\\times BS^{1}\n\\end{tikzcd}\n\\]\nBy Theorem~3.9 of~\\cite{AG}, the homotopy fiber of $Bf_{P}$ is the pushout of the homotopy fibers of the other maps in the diagram. To determine this fiber, we first replace the maps in the diagram of groups by homotopy equivalent fibrations\n\\[\n\\begin{tikzcd}[column sep=huge]\n\\mathbb{Z}\\ar[swap]{d}{(1,1,a_{1})} & \\mathbb{Z}\\times\\mathbb{Z} \\ar{d}{(1,a_{1},a_{2})}\\ar[swap]{l}{a_{2}} \\ar{r}{a_{1}}& \\mathbb{Z} \\ar{d}{(1,1,a_{1})}\\\\\nT^{2}_{m}\\times\\mathbb{R} \\ar[swap]{d}{(a_{1},a_{2},b'a_{1}e({a_{3}}))} & S^{1}\\times\\mathbb{R}\\times\\mathbb{R} \\ar[swap]{l}{(a_{1},ba_{1}e({a_{2}}),a_{3})} \\ar{r}{(a_{1},b'a_{1}e({a_{3}}),a_{2})} \\ar{d}{(a_{1},ba_{1}e({a_{2}}),b'a_{1} e({a_{3}}))}& T^{2}_{m'}\\times\\mathbb{R}\\ar{d}{(a_{1},ba_{1}e({a_{3}}),a_{2})}\\\\\nS^{1}\\times S^{1}\\times S^{1}\\ar{r}{=} & S^{1}\\times S^{1}\\times S^{1} &S^{1}\\times S^{1}\\times S^{1}\\ar[swap]{l}{=}\n\\end{tikzcd}\n\\]\n\nwhere $a_{i}$ denote the $i^{\\text{th}}$ coordinate function and $e(a_j) = e^{2\\pi i a_j}$. Applying the classifying space functor, this gives\n\\[\n\\begin{tikzcd}[column sep=normal]\nS^{1}\\ar[swap]{d}{} & S^{1}\\times S^{1}\\ar{d}{}\\ar[swap]{l}{\\text{pr}_{2}} \\ar{r}{\\text{pr}_{1}}& S^{1} \\ar{d}{}\\\\\nBT^{2}_{m} \\ar[swap]{d}{} & BS^{1} \\ar[swap]{l}{} \\ar{r}{} \\ar{d}{}& BT^{2}_{m'}\\ar{d}{}\\\\\nBS^{1}\\times BS^{1}\\times BS^{1}\\ar{r}{=} & BS^{1}\\times BS^{1}\\times BS^{1} &BS^{1}\\times BS^{1}\\times BS^{1}\\ar[swap]{l}{=}\n\\end{tikzcd}\n\\]\nwhich shows that the homotopy fiber of the canonical map $BP\\to BS^{1}\\times BS^{1}\\times BS^{1}$ is homotopy equivalent to\n\\[\\hocolim\\{S^{1}\\xleftarrow{\\text{pr}_{2}}S^{1}\\times S^{1}\\xrightarrow{\\text{pr}_{1}}S^{1}\\}\\simeq S^{1}*S^{1}\\simeq S^{3}\\]\nConsequently, $BP$ is the total space of a fibration\n\\[S^{3}\\to BP\\to BS^{1}\\times BS^{1}\\times BS^{1}\\]\nthat, after looping, becomes\n\\[\n\\begin{tikzcd}[column sep=normal]\n & T^{2}_{m'}\\ar[swap]{d}{j_{m'}} \\ar{rd}{f_{m'}=(a_{1},ba_{1},a_{2})}& \\\\\n\\Omega S^{3}\\ar{r} & P\\ar{r}{f_{P}} & S^{1}\\times S^{1}\\times S^{1}\\\\\n & T^{2}_{m}\\ar{u}{j_{m}} \\ar[swap]{ru}{f_{m}=(a_{1},a_{2},b'a_{1})}& \n\\end{tikzcd}\n\\]\nThe map $f_{P}$ admits a section given by \n\\[s(a_{1},a_{2},a_{3})= j_{m'}(a_1, {b'}^{-1}a_3)j_{m}(1,{b}^{-1}a_1^{-1}a_{2})\\]\nIt follows that, as a space, $P$ is weakly homotopically equivalent to the product\n\\[P\\simeq \\Omega S^{3}\\times S^{1}\\times S^{1}\\times S^{1}\\]\nwhich is consistent with the algebraic computations of the previous section.\n\n\\subsection{Homotopy type of \\texorpdfstring{$S^1(\\pm1,b;m)$}{S1(1,b;m)} equivariant symplectomorphisms}\n\nWe are now able to determine the homotopy type of the group $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ for the circle actions\n\\begin{itemize}\n\\item $S^1(1,b,m)$ when $2\\lambda > |2b-m|$, and\n\\item $S^1(-1,b,m)$ when $2\\lambda > |2b+m|$.\n\\end{itemize}\nSince the arguments are identical in the two cases, we will only discuss the first one. Again, in order to keep the notation simple, we write $\\mathbb{T}^2_m$ and $\\mathbb{T}^2_{m'}$ for the two tori the circle extends to, assuming $m'>m$, and we write $(1,b):S^1\\to\\mathbb{T}^2_m$ and  $(1,b'):S^1\\to\\mathbb{T}^2_{m'}$ for the two inclusions.\\\\\n \n\nFrom the universal property of pushouts, there is a canonical map \n\\[\\Upsilon:P^{alg}_{*}\\to H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)\\]\nmaking the following diagram commutative\n\\[\n\\begin{tikzcd}\n\\Lambda(t) \\arrow{r}{(1,b')} \\arrow[swap]{d}{(1,b)} & \\Lambda(y_{1},y_{2}) \\arrow{d}{} \\arrow[bend left=10]{ddr}{i_{m'}} & \\\\\n\\Lambda(x_{1},x_{2}) \\arrow{r}{}\\arrow[bend right=10,swap]{drr}{i_{m}} & P^{alg}_{*} \\arrow[dotted]{rd}{\\Upsilon} & \\\\\n & & H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)\n\\end{tikzcd}\n\\]\n\n\\begin{prop}\nFor every field $k$, the map $\\Upsilon:P^{alg}_{*}\\to H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)$ is an isomorphism of $k$-algebras.\n\\end{prop}\n\\begin{proof}\nBy definition, the map $\\Upsilon$ is an homomorphism of $k$-algebras. Since $P^{alg}_{i}\\cong H_{i}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)$ for each $i$, it is sufficient to show that $\\Upsilon$ is surjective.\\\\\n\nLet $R$ be the image of $\\Upsilon$. Since the maps $i_{m}$ and $i_{m'}$ are injective, $R$ is the subring generated by the classes $t,x_{2},y_{2}$ viewed as elements in $H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)$. Consider the two fibrations induced by the action maps\n\\begin{align*\n\\mathbb{T}^2_m\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\xrightarrow{p_m} \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_m \\simeq U_{m}^{S^1}\\\\\n\\mathbb{T}^2_{m'}\\to\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\xrightarrow{p_{m'}}\\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_{m'} \\simeq U_{m'}^{S^1}\n\\end{align*}\nObserve that $p_{m}(t)=0$, $p_{m}(x_{2})=0$, $p_{m'}(t)=0$, and $p_{m'}(y_{2})=0$. Now suppose there is an element $z\\in H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)$, not in $R$, and of minimal degree $d$. Since\n\\begin{multline}\nH_{d}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)\\cong\\\\\nH_{d}(U_{m}^{S^1};k)\\otimes H_{0}(T^{2}_{m};k)\n~\\oplus~ H_{d-1}(U_{m}^{S^1};k)\\otimes H_{1}(T^{2}_{m};k)\n~\\oplus~ H_{d-2}(U_{m}^{S^1};k)\\otimes H_{2}(T^{2}_{m};k)\n\\end{multline}\nwe would have a decomposition\n\\[z=c_{1}\\otimes\\mathbf{1}~\\oplus~ c_{t}\\otimes t~\\oplus~ c_{x_{2}}\\otimes x_{2}+c_{T}\\otimes [T^{2}_{m}]\\]\nwith at least one coefficient $c_{j}$ which is not a polynomial in the classes $p_{m}(w)$ and $p_{m}(y_{2})$. Let $c_{\\ell}$ be such coefficient of minimal degree $d-2\\leq\\ell\\leq d$. The inverse of the Alexander-Eells isomorphism of Proposition~\\ref{prop:AlexanderEellsGeometric}\n\\[\\lambda_{*}^{-1}:H_{p+1}(U_{m}^{S^1})\\to H_{p}(U_{m'}^{S^1})\\]\nwould map $c_{\\ell}$ to a class $c_{\\ell-1}'\\in H_{\\ell-1}(U_{m'}^{S^1};k)$. This class could not be a polynomial in $p_{m'}(w)$ and $p_{m'}(x_{2})$ since, otherwise,  \n\\[c_{\\ell} = \\lambda_{*}(c_{\\ell-1}')=p_{m}\\big([y_{2}\\otimes c_{\\ell-1}']\\big)\\]\nwould be a polynomial in the classes $p_{m}(w)$ and $p_{m}(y_{2})$. \nIn turn, this class $c_{\\ell-1}'$ would have to be the image of some element in $H_{\\ell-1}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);k)$ not in $R$, contradicting the minimality of~$z$.\n\\end{proof}\n\n\\begin{cor}\nThe map $\\Upsilon:P^{alg}_{*}\\to H_{*}(\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda);\\mathbb{Z})$ is an isomorphism of Pontryagin algebras over the ring of integers.\n\\end{cor}\n\\begin{proof}\nThis follows from the well known fact that a map induces isomorphisms on homology with $\\mathbb{Z}$ coefficients iff it induces isomorphisms on homology with $\\mathbb{Q}$ and $\\mathbb{Z}_{p}$ coefficients for all primes $p$, see~\\cite{Ha}, Corollary~3A.7~(b).\n\\end{proof}\n\n\\begin{thm}\\label{full_homo}\nThe map $\\Upsilon:P\\to\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ is an homotopy equivalence.\n\\end{thm}\n\\begin{proof}\nThe map $\\Upsilon$ is a homology equivalence on integral homology. Because $P$ and $\\Symp_h^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ are topological groups, it follows that it is a weak equivalence, see~\\cite{Dror-WhiteheadTheorem}, Example~4.2. Because both spaces are homotopy equivalent to CW-complexes, this weak equivalence is a homotopy equivalence (See~\\cite{Ha}, Proposition~4.74).\n\\end{proof}\n\n\\section{Centralizers of Hamiltonian \\texorpdfstring{$S^1$}{circle} actions on \\texorpdfstring{$S^2 \\times S^2$}{the product}}\n\nWe summarise all the results we have obtained in this chapter in the following theorem. \n\\begin{thm} \\label{circle}\nConsider any Hamiltonian circle action $S^1(a,b;m)$ on $(S^2 \\times S^2, \\omega_\\lambda)$. The homotopy type of the symplectic stabilizer $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ is given in the table below:\n\\begin{center}\n\\begin{tabular}{|p{4.5cm}|p{3.5cm}|p{2cm}|p{4cm}|}\n \\hline\n Values of $(a,b ;m)$ & $\\lambda$ &Number of strata $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ intersects &Homotopy type of $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$\\\\\n\\hline\n {$(0,\\pm 1;m)$, $m\\neq 0$}  &$\\lambda > 1$ & 1 & $S^1 \\times SO(3)$ \\\\\n\\hline\n\\multirow{2}{10em}{$(0,\\pm 1;0)$ or $(\\pm 1,0;0)$} &$\\lambda = 1$  &1  &$S^1 \\times SO(3)$ \\\\\\cline{2-4}\n&$\\lambda >1$ &1  &$S^1 \\times SO(3)$ \\\\\n\\hline\n$(\\pm 1,\\pm1;0)$&$\\lambda = 1$ &1 & $\\mathbb{T}^2 \\times \\mathbb{Z}_2$ \\\\\n\\hline\n $(\\pm 1,0;m), m\\neq0$ &$\\lambda >1$ &1 &$\\mathbb{T}^2$\\\\\n \\hline\n$(\\pm1,\\pm m;m), m \\neq 0$ &$\\lambda > 1$   &1 & $\\mathbb{T}^2$\\\\\n\\hline\n\\multirow{2}{10em}{$(1,b;m), b \\neq \\{ m,0\\}$} \n&$|2b-m| \\geq2 \\lambda \\geq 1$ &1 &$\\mathbb{T}^2$ \\\\\\cline{2-4}\n&$2 \\lambda >|2b-m| \\geq 0$  &2 &$\\Omega S^3 \\times S^1 \\times S^1 \\times S^1$ \\\\\n\\hline\n\\multirow{2}{10em}{$(-1,b;m), b \\neq \\{ -m,0\\}$} \n&$|2b+m| \\geq2 \\lambda \\geq 1$ &1 &$\\mathbb{T}^2$ \\\\\\cline{2-4}\n&$2 \\lambda >|2b+m| \\geq 0$  &2 &$\\Omega S^3 \\times S^1 \\times S^1 \\times S^1$ \\\\\n \\hline\nAll other values of $(a,b;m)$ &$\\forall \\lambda$  &1   &$\\mathbb{T}^2$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\nwhere $\\Omega S^3$ denotes the based loop space of $S^3$. \\qed\n\\end{thm}\n\n\n\n\\chapter{Partition of the space of invariant almost-complex structures}\\label{Chapter-codimension}\nIn the previous section, we calculated the homotopy type of the group of $S^1(\\pm1,b;m)$ equivariant symplectomorphisms assuming that the codimension of the the invariant strata $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{m^\\prime}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ was 2. In this section, we use deformation theory to show that the invariant strata $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{m^\\prime}$ is a submanifold of $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ and to characterize its normal bundle. We then calculate its codimension. \\\\\n\nWe mimic the techniques in \\cite{AGK} in the equivariant setting. Fix a K\\\"ahler 4-manifold $(M,\\omega,J)$ and an $S^1$ action on $(M,\\omega,J)$ such that $g^*\\omega=\\omega$ and $g^*J= J$ $\\forall g \\in S^1$.\nThe holomorphic $S^1$ action on the base manifold $M$ induces a natural action on the various tensor spaces such as $T^{1,0}M$ or $\\Omega^{0,k}_{J}(M, TM)$. We write $T^{1,0}M^{S^1}$, $\\Omega^{0,k}_{J}(M, TM)^{S^1}$, to denote the $S^1$ invariant elements of these tensor spaces.\n\n\n\\section{Space of invariant complex structures}\nLet $\\mathcal{J}_l$ be the space of almost complex structures of regularity $C^l$ on $M$, endowed with $C^l$ topology. Being a space of sections, $\\mathcal{J}_l$ is a smooth Banach manifold. An explicit atlas can be constructed using the Cayley transform, see for instance~\\cite{Smolentsev}. Given $J\\in\\mathcal{J}_l$, let $\\Omega^{0,1}_{J,l}(M,TM) \\subset \\End_l(T M)$ be the space of  endomorphisms of the tangent bundle of regularity $C^l$ that anticommute with $J$, that is,\n$$\n\\Omega^{0,1}_{J,l}(M,TM)=\\left\\{A  \\in \\End_l(T M) \\mid A J+J A=0\\right\\} \n$$\nThe map $\\phi_{J}: \\Omega^{0,1}_{J,l}(M,TM) \\rightarrow \\mathcal{J}_l$ given by\n$$\n\\phi_{J}(A)= J e^{A}\n$$\nis a local diffeomorphism sending $C^k$ endomorphisms ($k \\geq l$) to $C^k$ almost complex structures. If $J$ is $S^1$ invariant, then $\\phi$ gives a bijection between invariant endomorphisms near $0$ in $\\Omega^{0,1}_{J,l}(M,TM)$ and invariant almost complex structures in a neighborhood of $J$. This shows that the space $\\mathcal{J}^{S^1}_l$ of invariant almost complex structures is a Banach submanifold of $\\mathcal{J}_l$ whose tangent space $T_J\\mathcal{J}^{S^1}_l$ at $J$ is naturally identified with the linear subspace $\\Omega^{0,1}_{J,l}(M,TM)^{S^1}$.\\\\\n\n\n\n\nLet $I^{S^1}_l$ denote the space of invariant and integrable almost complex structures of $M$ with regularity $C^l$. We now show  that $I^{S^1}_l$is a Banach submanifold of $\\mathcal{J}^{S^1}_l$. To this end, let  $N_J(X,Y) = [X,Y] + J\\left([JX,Y] + [X,JY]\\right) - [JX,JY]$ denote the Nijenhuis tensor with respect to $J$. By the Newlander-Nirenberg theorem, we know that $J \\in \\mathcal{J}^{S^1}_l$ is integrable iff $N(J)=0$.\n\\\\\n\nConsider the vector bundle $\\Omega^{0,2}_{l-1}(M,TM)^{S^1}$ over $\\mathcal{J}^{S^1}_l$ whose fibre over $J$ is the space $\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$ of $S^1$-invariant $(0,2)$ forms of regularity $C^{l-1}$ with values in the holomorphic tangent bundle $TM$.  \nThe Nijenhuis tensor can be interpreted as a section $N:\\mathcal{J}_l\\to\\Omega^{0,2}_{l-1}(M,TM)$. This section is equivariant since, for all $g \\in S^1$,\n\\begin{align*}\n\\begin{split}\ng \\cdot N_J(X,Y) &:= g \\cdot [X,Y] + g \\cdot J\\left([JX,Y] + [X,JY]\\right) - g \\cdot [JX,JY] \\\\ \n&= g_* [g_*^{-1}X,g_*^{-1} Y]  + g_*J\\left([Jg_*^{-1}X,g_*^{-1}Y] + [g_*^{-1}X,Jg_*^{-1}Y]\\right) - g_* \\cdot [Jg_*^{-1}X,Jg_*^{-1}Y]  \\\\ \n&= [X,Y] + J\\left([JX,Y] + [X,JY]\\right) - [JX,JY] \n\\end{split}\n\\end{align*}\nwhere the last equality follows from the facts that $g_*[X,Y] = [g_*X,g_*Y]$ and that $J$ is invariant. In particular, $N$ takes invariant tensors to invariant tensors, that is,\n\\begin{align*}\n    N: \\mathcal{J}^{S^1}_l &\\to \\Omega^{0,2}_{l-1}(M,TM)^{S^1} \\\\\n      J &\\mapsto N_J\n\\end{align*}\nTo show that $I^{S^1}_l$ is a Banach submanifold, it suffices to show that Nijenhuis tensor intersects the 0-section of the bundle transversally. This is equivalent to showing that, for an integrable $J$, the projection of the derivative to the vertical tangent bundle is surjective. We denote this projection of the derivative of $N$ to the vertical tangent bundle as $\\nabla N$. A priori, $\\nabla N$ depends on a choice of connection on $\\Omega^{0,2}_{l-1}(M,TM)^{S^1}$. However, as shown in Appendix A of \\cite{AGK}, given an arbitrary almost complex structure $J$, we can extend the usual $\\bar\\partial_J$ operator to an operator $\\overline{\\partial}_J:\\Omega^{0,1}_{J,l}(M, TM) \\to \\Omega^{0,2}_{J,l-1}(M, TM)$ so that $\\nabla N_J:=\\nabla N(J)$ is given by the following composition. \n\\[\n\\begin{tikzcd}\n     &\\Omega^{0,1}_{J,l}(M, TM)^{S^1} \\arrow[r,\"dN_J\"] \\arrow[rr,bend right=15,\"\\nabla N_J\"]&\\left(\\Omega^{2}_{l-1}(M, TM \\otimes \\mathbb{C})\\right)^{S^1} \\arrow[r,\"\\pi\"] &\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}\n\\end{tikzcd} \n\\]\n\\noindent where $\\pi$ is the canonical projection of \n\\[\n\\Omega^{2}_{J,l-1}(M, TM \\otimes \\mathbb{C})^{S^1} \n\\cong \\Omega^{2,0}_{J,l-1}(M,TM)^{S^1} \\oplus \\Omega^{1,1}_{J,l-1}(M,TM)^{S^1} \\oplus \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}\n\\]\nonto the last summand. \n\n\\begin{thm}[\\cite{AGK}, Corollary A.9]\n$\\nabla N(J) = -2 J \\overline \\partial_J$. \\qed\n\\end{thm}\nWe are lead to show that $\\overline \\partial_J: \\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$ is surjective. This is trivially true  whenever the manifold  as $M$ is 4 dimensional and $H_J^{0,2}(M,TM)^{S^1} = 0$. \n\n\\begin{lemma}{\\label{Lemma:averaging_surj}}\nConsider a complex manifold $(M,J)$ with a holomorphic $S^1$ action. Then the averaging map \n\\begin{align*}\n    \\rho: H_J^{0,2}(M,TM) &\\rightarrow H_J^{0,2}(M,TM)^{S^1} \\\\\n             [\\beta] &\\mapsto \\left[\\int_{S^1}g^*\\beta ~dg\\right]\n\\end{align*}\nis surjective.\n\\end{lemma}\n\\begin{proof}\nThe fact that the above map is well defined follows by noting that the $\\overline \\partial_J$ operator commutes with the averaging operator. The surjectivity follows as the averaging operator is the identity on invariant forms.\n\\end{proof}\n\nNote that the above theorem works for any compact group. By the discussion in the previous paragraph and Lemma~\\ref{Lemma:averaging_surj} we can conclude that $\\overline \\partial_J: \\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$ is surjective for any holomorphic $S^1$ action on a complex 4-manifold satisfying $H_J^{0,2}(M,TM)=0$.\n\n\n\n\\begin{thm}\nLet $(M,J)$ be a $4$-manifold endowed with an integrable complex structure $J$, and with a holomorphic $S^1$ action. Suppose $H_J^{0,2}(M,TM) = 0$. Then the space $I^{S^1}_l$ of invariant complex structures is a Banach submanifold of $\\mathcal{J}^{S^1}_l$ in a neighbourhood of $J$ with tangent space at $J$ identified with $ \\ker \\overline \\partial_J:\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1} $. Equivalently,\n\\[\nT_J I_l \\cong \\left(\\im  \\overline \\partial_J:\\left(\\Omega^{0,0}_{J,l}(M,TM)\\right)^{S^1} \\to \\Omega^{0,1}_{J,l}(M,TM)^{S^1}\\right) \\oplus H^{0,1}_J(M,TM)^{S^1}\n\\]\n\\end{thm}\n\nLet us now assume that $M$ is symplectic. Let $\\mathcal{J}^{S^1}_{\\omega,l}$ denote the space of all $S^1$ equivariant \\emph{compatible} almost complex structures of regularity $C^l$ endowed with the $C^l$-topology. Our next goal is to show that under some cohomological restrictions, the space of equivariant integrable \\emph{compatible} almost complex structures of regularity $C^l$ denoted by $I^{S^1}_{\\omega,l}$ is a Banach submanifold of $\\mathcal{J}^{S^1}_{\\omega,l}$. We first note that given $J \\in \\mathcal{J}^{S^1}_{\\omega,l}$, the equivariant metric \n$h_J (\\cdot, \\cdot) := \\omega(\\cdot, J\\cdot) - i \\omega(\\cdot, \\cdot)$\n induced by the pair $(\\omega, J)$ identifies \n$T_J \\mathcal{J}^{S^1}_{l} = \\Omega^{0,1}_l(M,TM)^{S^1}$ \nwith the space  $\\left(T^{0,2}\\right)^{S^1}:= \\left(\\Omega^{0,2}(M)\\right)^{S^1} \\otimes \\left(\\Omega^{0,2}(M)\\right)^{S^1}$ of complex equivariant $(0,2)$-tensors via the map\n\\begin{align*}\n\\theta:\\left(T^{0,2}\\right)^{S^1} &\\to \\Omega^{0,1}_l(M,TM)^{S^1} \\\\\nA &\\mapsto \\theta(A):= h_J(A \\cdot, \\cdot)\n\\end{align*}\n\nLet us denote by $S\\Omega^{0,1}_{J,l}(M,TM)^{S^1}$ the tangent space of $T_J \\mathcal{J}^{S^1}_{\\omega,l} \\subset T_J \\mathcal{J}^{S^1}_{l}$ of all equivariant compatible almost complex structures. More explicitly, the tangent space consists of elements $A \\in \\Omega^{0,1}_{J,l}(M,TM)^{S^1} $ such that $\\omega(A\\cdot, \\cdot) = - \\omega(\\cdot, A\\cdot)$. Under the above identification, we can check that $S\\Omega^{0,1}_{J,l}(M,TM)^{S^1}$ gets mapped to the space of symmetric $S^1$ invariant $(0,2)$-tensors which we denote by $\\left(S^{0,2}\\right)^{S^1}$. \n\n\nFurther, the quotient $T_{J} \\mathcal{J}_l^{S^1} \/ T_{J} \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ may be identified with the space of invariant $(0,2)$ forms on $M$ since\n\\[T_{J} \\mathcal{J}_l^{S^1} \/ T_{J} \\mathcal{J}^{S^1}_{\\om_\\lambda, l} = \\Omega^{0,1}_{J,l}(M,TM)^{S^1} \/ S \\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\cong T_{J}^{0,2}(M) \/ \\left(S^{0,2}\\right)^{S^1} = \\Omega_{J}^{0,2}(M)^{S^1}.\\]\nAs before, the Nijenhuis tensor defines a map \n\\[N:\\mathcal{J}^{S^1}_{\\omega,l} \\to \\Omega^{0,2}_{l-1}(M,TM)^{S^1}\\]\nwhose kernel is precisely the subspace $I^{S^1}_{\\omega,l}$. We want to show that the derivative $\\nabla N$ is surjective at all $J\\in I^{S^1}_{\\omega,l}$. As we know that $\\nabla N(J) = -2 J \\overline \\partial_J $, we would need to show that $\\overline\\partial_J:S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to  \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$ is surjective. As $M$ is a 4-manifold, all forms in $\\Omega^{0,2}_{l-1}(M,TM)^{S^1}$ are closed, hence to show that the restriction of $\\overline\\partial_J$ to $S\\Omega^{0,1}_{J,l}(M,TM)^{S^1}$ is surjective, it would suffice to show that the vector space $SH_{J}^{0,2}(TM)^{S^1}$ defined below is 0.\n\\begin{align*}\nSH_{J}^{0,2}(TM)^{S^1}\n&:= \\frac{\\ker \\overline\\partial:\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1} \\to \\Omega^{0,3}_{J,l-2}(M,TM)^{S^1}}{\\im \\overline\\partial:S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to  \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}}\\\\\n&=\\frac{\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}}{\\im \\overline\\partial:S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to  \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}}\n\\end{align*}\nAs the above condition is not easy to check directly, we consider the following commutative diagram \n\\[\n\\begin{tikzcd}\n     &0 \\arrow[d] \\arrow[r] &S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\arrow[r, \"\\overline\\partial\"] \\arrow[d] &\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1} \\arrow[d] \\arrow[r] &0 \\\\\n     &\\Omega^{0,0}_{J,l+1}(M,TM)^{S^1} \\arrow[r,\"\\overline\\partial\"] \\arrow[d,\"\\alpha\"] &\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\arrow[r,\"\\overline\\partial\"] \\arrow[d] &\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1} \\arrow[d] \\arrow[r] &0 \\\\\n     &\\Omega^{0,1}_{J,l+1}(M)^{S^1} \\arrow[r, \"\\overline\\partial\"] &\\Omega^{0,2}_{J,l}(M)^{S^1} \\arrow[r, \"\\overline\\partial\"] &0 \\arrow[r] &0\n\\end{tikzcd}  \n\\]\nwhere the map $S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to \\Omega^{0,1}_{J,l}(M,TM)^{S^1}$ is just the inclusion and the where the map $\\Omega^{0,1}_{J,l}(TM)^{S^1} \\to \\Omega^{0,2}_{J,l}(M)^{S^1}$ is the quotient \\[\\Omega^{0,1}_{J,l}(TM)^{S^1} \\to \\Omega^{0,1}_{J,l}(TM)^{S^1}\/S\\Omega^{0,1}_{J,l}(M,TM)^{S^1}\\]\nfollowed by identifying $\\Omega^{0,1}_{J,l}(TM)^{S^1}\/S\\Omega^{0,1}_{J,l}(M,TM)^{S^1}$ with\n$\\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$ (see\\cite{AGK} p. 548 for more details about the identification). The map $\\alpha$ is defined as\n\\begin{align*}\n    \\alpha: \\Omega^{0,0}_{J,l+1}(M,TM)^{S^1} &\\to \\Omega^{0,1}_{J,l+1}(M)^{S^1} \\\\\n    X &\\mapsto \\alpha(X)(Y):= \\omega(X,JY)-i\\omega(X,Y)\n\\end{align*} \nwhere $J \\in I^{S^1}_{\\omega,l}$ and $X,Y \\in \\Omega^{0,0}_{J,l+1}(M,TM)^{S^1}$. We refer the reader to  Appendix B in \\cite{AGK} for  the proof of commutativity of the diagram in the non-equivariant case, and we note that it still holds in the equivariant setting due to the fact that $\\overline\\partial_J$ is equivariant. The above diagram gives rise to a long exact sequence is cohomology\n\\\\\n\n\n\\begin{equation}\\label{les}\n\\begin{aligned} 0 \\longrightarrow H_{J}^{0}(T M)^{S^1} & \\longrightarrow cl\\Omega_{J}^{0,1}(M)^{S^l} \\stackrel{\\delta}{\\longrightarrow} cl S\\Omega_{J}^{0,1}(M,TM)^{S^1} \\stackrel{q}\\longrightarrow H_{J}^{0,1}(T M)^{S^1} \\longrightarrow \\\\ & \\longrightarrow H_{J}^{0,2}(M)^{S^1} \\longrightarrow SH_{J}^{0,2}(TM)^{S^1} \\longrightarrow H_{J}^{0,2}(T M)^{S^1} \\longrightarrow 0 \n\\end{aligned}\n\\end{equation}\n\n\n\n\\noindent where $cl\\Omega_{J}^{0,1}(M)^{S^l}$ denotes the kernel of $\\overline\\partial_J$ in $\\Omega_{J}^{0,1}(M)^{S^l}$ and similarly  $cl S\\Omega_{J}^{0,1}(M,TM)^{S^1}$ is the kernel of $\\overline\\partial_J: S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\to  \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}$. Thus if we had a 4-manifold $M$ with an $S^1$ invariant compatible integrable almost complex structure $J$ such that  $H_J^{0,2}(M) =0$ and $H_J^{0,2}(TM)=0$, noting that $\\overline{\\partial}_J$ takes $S^1$ invariant elements to $S^1$ invariant elements we can conclude that $H_J^{0,2}(M)^{S^1} =0$ and $H_J^{0,2}(M,TM)^{S^1}=0$. Further, it follows from equation \\ref{les} for such a manifold $(M,\\omega,J)$ as above, that $SH_{J}^{0,2}(TM)^{S^1}=0$ and hence $I^{S^1}_{\\omega,l}$ would indeed be a manifold in a neighbourhood of such a $J$. Thus $H_J^{0,2}(M) =0$ and $H^{0,2}_J(TM)=0$ gives us a simpler condition for when $I^{S^1}_{\\omega,l}$ would indeed be a manifold in a neighbourhood of $J$ as required.\n\\\\\n\nAdditionally as  the averaging operator commutes with the $\\overline{\\partial}_J$ operator,  $H_J^{0,2}(M) =0$ implies that $H_J^{0,2}(M)^{S^1} =0$. This tells us that $q:cl S\\Omega_{J}^{0,1}(M,TM)^{S^1} \\to H_{J}^{0,1}(T M)^{S^1} $ is surjective and hence by the first isomorphism theorem we have $\\frac{cl S\\Omega_{J}^{0,1}(M,TM)^{S^1}}{ker ~q:cl S\\Omega_{J}^{0,1}(M,TM)^{S^1} \\to H_{J}^{0,1}(T M)^{S^1}}$ is isomorphic to $H_{J}^{0,1}(T M)^{S^1}$.\nThen the above long exact sequence gives us \n\\[\\frac{cl S\\Omega_{J}^{0,1}(M,TM)^{S^1}}{ker~ q} \\cong \\frac{cl S\\Omega_{J}^{0,1}(M,TM)^{S^1}}{\\im \\delta} \\cong H_{J}^{0,1}(M,TM)^{S^1}\\]\nPutting all this together we obtain the following local description of $I^{S^1}_{\\omega,l}$.\n\\begin{thm}\\label{integrable}\nLet $(M,\\omega,J)$ be a  K\\\"ahler 4-manifold with a K\\\"ahler $S^1$ action. Suppose that $H_J^{0,2}(M)=0$ and $H_J^{0,2}(TM)=0$. Then  $I^{S^1}_{\\omega,l}$ is a Banach submanifold of $\\mathcal{J}^{S^1}_{\\omega,l}$ in a neighbourhood of $J$ with tangent space at $J \\in I^{S^1}_{\\omega,l}$ identified with\n\\[\nT_J I^{S^1}_{\\omega,l} = cl S\\Omega_{J}^{0,1}(M,TM)^{S^1} = \\ker \\overline\\partial_J: S\\Omega^{0,1}_{J,l}(M,TM)^{S^1} \\longrightarrow  \\Omega^{0,2}_{J,l-1}(M,TM)^{S^1}.\n\\]\nEquivalently,\n\\[T_J I^{S^1}_{\\omega,l} \\cong \\im \\delta \\bigoplus H_{J}^{0,1}(T M)^{S^1}.\\]\n\\end{thm}\n\n\\begin{prop}\nThe conditions $H_J^{0,2}(M) =0$ and $H^{0,2}_J(M,TM)=0$ are satisfied for all the Hirzebruch surfaces.\n\\end{prop}\n\\begin{proof}\nTo prove $H^{0,2}_J(M,TM)=0$ for all Hirzebruch surfaces see the computation in Example 6.2b) p.312 in \\cite{Ko}. To prove, $H_J^{0,2}(M) =0$ we note that the the rank of $H_J^{0,2}(M) =0$ (usually called the geometric genus $p_g$) is a birational invariant. As all Hirzebruch surfaces are birationally equivalent, the result follows from the computation on p.220 in \\cite{Ko}. \n\\end{proof}\nFinally, we would like to show that the strata $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\omega,l}$\nis  a Banach submanifold of $\\mathcal{J}^{S^1}_{\\omega,l}$. The most naive method to try to prove this would be to consider the universal moduli space $\\mathcal{M}(D_s,\\J_{\\om_\\lambda})$ of curves in the class $D_s$ (where $D_s$ is defined to be the class $B -\\frac{s}{2}F$ if $M =S^2 \\times S^2$ or the class $B- \\frac{s+1}{2}F$ if $M= \\CP^2\\# \\overline{\\CP^2}$) and try to prove that the inclusion of $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is transverse to the projection of $\\mathcal{M}(D_s,\\J_{\\om_\\lambda})$ to the space of all compatible almost complex structures of regularity $C^l$:\n\\[\n\\begin{tikzcd}\n&\\mathcal{M}(D_s,\\J_{\\om_\\lambda})\\arrow[d,\"\\pi\"]\\\\\n\\mathcal{J}^{S^1}_{\\om_\\lambda, l} \\arrow[r,\"i\"] &\\J_{\\om_\\lambda}\n\\end{tikzcd} \n\\] \nHowever, this approach is flawed as the two maps are never transversal. An alternative method is to try to define an equivariant universal moduli space $\\mathcal{M}^{S^1}(D_s,\\mathcal{J}^{S^1}_{\\om_\\lambda, l})$ and argue that the image under the projection to $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is a Banach submanifold of $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$. This is the approach we implement in the following section.\n\n\\section{Construction of Equivariant moduli spaces}\n\nIn this section we construct moduli spaces of $S^1$ invariant $J$-holomorphic maps into $S^2 \\times S^2$ or $\\CP^2\\# \\overline{\\CP^2}$. Recall that $J_m$ is the standard complex structure on the $m^\\text{th}$ Hizerbruch surface $W_m$, where $m=2k$ or $m=2k+1$. Let $D_s$ denote the homology class $B-\\frac{s}{2}F$ in $S^2 \\times S^2$ and let it denote the class $B -\\frac{s+1}{2}F$ in $\\CP^2\\# \\overline{\\CP^2}$. As seen in Chapter~2, there is a $\\mathbb{T}^2_m$ invariant, $J_m$-holomorphic curve $\\overline{D}$ in $W_m$ in the homology class $D_s$. Consider the $S^1(a,b;m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. From the graph for the circle action $S^1(a,b;m)$ we see that $S^1$ acts on $\\overline{D}$ in a non-effective manner with global stabilizer $\\mathbb{Z}_{a}$. The following theorem is useful in our analysis. \n\n\\begin{lemma}\\label{WellDefinedModuli}\nConsider the $S^1(a,b;m)$ action on $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. Let $S$ be any $S^1(a,b;m)$-invariant symplectic embedded sphere in the homology class $D_s$ with $s>0$. Then the $S^1$ action on $S$ has global stabilizer isomorphic to~$\\mathbb{Z}_a$. \n\\end{lemma}\n\n\\begin{proof}\nThis follows \\ref{weight} from noting that any other $S^1$ invariant curve passes through the same set of fixed points as $\\overline{D}$ , and hence by \\ref{weight} we that the global stabilizer is the same. \n\\end{proof}\n\nThus we can fix an action on a base sphere, namely the standard $S^1$ action that agrees with the action of $S^1$ on $\\overline{D}$ and consider the moduli space of all equivariant maps $u: (S^2,j_0) \\to (M,J)$ for some $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$. We define $\\mathcal{M}^{S^1}(D_s,\\mathcal{J}^{S^1}_{\\om_\\lambda, l})$ as follows\n\\[\n\\begin{aligned}\n\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}): = \\{(&u,J) ~|~ \\text{$u:S^2 \\to M$ is  equivariant, somewhere injective, J-holomorphic and}  \\\\\n&\\text{represents the class $D_s$}\\}\n\\end{aligned}\n\\]\n\n\n\n\\begin{remark}\nAs we are only interested in the case when $s>0$, the curves in class $D_s$ have negative self intersection and the adjunction formula tells us that these curves are embedded. Thus all somewhere injective curves in class $D_s$ for $s>0$ are embedded. \n\\end{remark}\nAs in the non-equivariant case we now wish to prove that this moduli space is a smooth Banach manifold. To prove this we recall the non-equivariant set up as in Chapter 3 in \\cite{McD} and reformulate it in the equivariant setting.\n\\\\\n\nWe have a bundle \n\\[\n\\begin{tikzcd}\n     &\\mathscr{E}^{S^1}_{q-1,p} \\arrow[d, \"\\pi\"] \\\\\n     & B^{S^1}_{q,p}\\times \\mathcal{J}^{S^1}_l \n\\end{tikzcd}\n\\]\n\\noindent where $\\mathscr{E}^{S^1}_{q-1,p}$ is a vector bundle over $B^{S^1}_{q,p}\\times \\mathcal{J}^{S^1}_l $ with fibre over $(u,J)$ consisting of $S^1$ invariant sections of $\\Omega^{0,1}_J(S^2, u^*TM)$ of Sobolev  regularity $W^{q-1,p}$ i.e \n\\[ \\pi^{-1}(u,J) = \\Gamma(S^2, \\Omega^{0,1}_J(S^2, u^*TM)^{S^1}) \\] and the space $B^{S^1}_{q,p}$ is defined as follows: \\[B^{S^1}_{q,p}:= \\{ u \\in \\left(W^{q,p}(S^2,M)\\right)^{S^1} ~|~ [u]=D_s\\}\\]\nwhere $W^{q,p}(S^2,M))^{S^1}$ denotes the space of equivariant maps  of  Sobolev regularity $W^{q,p}$ from $S^2$ to $M$. \n\\\\\n\nWe would like to show that the  section $\\mathscr{F}^{S^1}(u,J):= (u,\\overline \\partial_J u) :B_{q,p}^{S^1} \\times \\mathcal{J}^{S^1}_l \\to \\mathscr{E}^{S^1}_{q-1,p}$  (where $\\overline \\partial_J u = \\frac{1}{2}(du + J \\circ du \\circ j_{S^2})$) is transversal to the zero section. Note that   $\\left({\\mathscr{F}^{S^1}}\\right)^{-1}(0) = \\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})$, thus giving it a smooth structure as in the non-equivariant case (See \\cite{McD} Lemma 3.2.1) \n\\\\\n\n In order to show trasnversality,  we consider the projection  of the derivative map $d\\mathscr{F}^{S^1}$ to the vertical tangent bundle and show that this map is surjective at $(u,J)$ when $u$ is a simple equivariant curve. We shall denote this projection by $D\\mathscr{F}^{S^1}$. More explicitly, we need to show that the map \n\n\\[\\begin{aligned}\nD\\mathscr{F}_{u,J}^{S^1}: \\mathscr{W}^{q,p}(S^2, u^*TM)^{S^1} \\times &C^l(M, \\End(TM, J, \\omega))^{S^1} \\\\\n\\to &\\mathscr{W}^{q,p}(S^2, \\Omega^{0,1}_J(S^2, u^*TM))^{S^1}\n\\end{aligned}\n\\]\nis surjective. But by Lemma 3.2.1 in \\cite{McD} we know that the analogously defined linearized derivative $D\\mathscr{F}_{u,J}$ in the non-equivariant case \n\\[\\begin{aligned}\nD\\mathscr{F}_{u,J}: \\mathscr{W}^{q,p}(S^2, u^*TM) \\times &C^l(M, \\End(TM, J, \\omega)) \\\\\n\\to &\\mathscr{W}^{q,p}\\left(S^2, \\Omega^{0,1}_J\\left(S^2, u^*TM\\right)\\right)\n\\end{aligned}\n\\]\nis surjective. As $J \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}$, the $\\bar\\partial_J$ operator commutes with the averaging operator with respect to the $S^1$ action.  Averaging the above non-equivariant derivative $D\\mathscr{F}_{u,J}$ by the $S^1$ action then shows that $D\\mathscr{F}_{u,J}^{S^1}$ is surjective as well.\n\n\\begin{thm}\n $\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})$ is a smooth Banach manifold. \\qed\n\\end{thm}\n\n\n\nWe now consider the projection map \n\\[\n\\begin{tikzcd}\n     &\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}) \\arrow[d,\"\\pi\"] \\\\\n     &\\mathcal{J}^{S^1}_{\\om_\\lambda, l}\n\\end{tikzcd}\n\\]\nTo conclude that the image of $\\pi$ is a submanifold of $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ we need the following theorem whose proof can be found in  \\cite{mardsen}.\n\\\\\n\n\n\n\n\\begin{thm}\\label{mrdsn}(Theorem 3.5.18 in \\cite{mardsen})\nLet $f:M \\to N$ be a smooth map between Banach manifolds such that \n\\begin{enumerate}\n    \\item $\\ker Tf$ is a sub-bundle of $TM$\n    \\item For each $m \\in M$, $f_*(T_m M)$ is closed and splits in $T_{f(m)}N$ \n    \\item f is open or closed onto it's image\n\\end{enumerate}    \nThen $f(M)$ is a smooth Banach submanifold of N.\n\\end{thm}\n\nA map that satisfies the above conditions is called a sub-immersion. \n\n\\begin{lemma}\\label{subimmersion}\nThe projection map $\\pi: \\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}) \\to \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is a sub-immersion.\n\\end{lemma}\n\\begin{proof}\n\nNote that the $\\ker  d\\pi$ is of constant rank and is the tangent space to the reparametrization group $\\mathbb{C}^*$ is which freely on $\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})$. Hence $\\ker d\\pi$ is a sub-bundle of $T\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})$.\n\\\\\n\nNow we show that the image of $d\\pi$ is closed  $T_J\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$. Note first that $T_J\\mathcal{J}^{S^1}_{\\om_\\lambda, l} = S\\Omega_J^{0,1}(M,TM)^{S^1}$, hence $\\pi_* T_{(u,J)} \\mathcal{M}^{S^1}\\left(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}\\right)$ as a subspace of $S\\Omega_J^{0,1}(M,TM)^{S^1}$ can be described as follows: \n\\begin{multline}\\label{proj_tangent}\n \\pi_*T_{(u,J)} \\mathcal{M}^{S^1}\\left(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}\\right)=\n \\left\\{ \\alpha \\in S\\Omega_J^{0,1}(M,TM)^{S^1}~| ~ [\\alpha \\circ du \\circ j_{S^2}]= 0 \\in H_J^{0,1}(S^2, u^*TM)^{S^1}\n\\right\\}\n\\end{multline}\n\n(This follows from noting that the proof of proposition 2.8 in \\cite{AGK} goes through under the presence of a compact group action.)\nLet $\\gamma_n \\in \\pi_* T_{(u,J)} \\mathcal{M}^{S^1}\\left(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}\\right)$ i.e $\\gamma_n \\in \\Omega^{0,1}(M,TM)^{S^1}= T\\mathcal{J}^{S^1}_{\\om_\\lambda, l} $  and satisfies  $[\\gamma_n \\circ du \\circ J_{S^2}]= 0 \\in H_J^{0,1}(S^2, u^*TM)^{S^1}$.  Further assume the sequence $\\gamma_n$ converges to $\\gamma$ in $\\Omega^{0,1}(M,TM)^{S^1}$. Then $[\\gamma \\circ du \\circ j_{S^2}] = 0 \\in H_{j_{S^2}}^{0,1}(S^2, u^*TM)^{S^1}$, thus showing image of $d\\pi$ is closed  $T\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$.\n\\\\\n\nNext to that the image of $d\\pi$ splits in $T\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$, we proceed as follows. We firstly show that the codimension of the image of $d\\pi$ in $T\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is finite and hence it follows that the image of $d\\pi$ splits. Consider the map \n\\begin{align*}\n    L: S\\Omega_J^{0,1}(M,TM)^{S^1} &\\rightarrow H_{j_{S^2}}^{0,1}\\left(S^2, u^*TM\\right)^{S^1} \\\\\n    \\alpha &\\mapsto [\\alpha \\circ du \\circ j_{S^2}]\n\\end{align*}\n\nBy equation~\\ref{proj_tangent} we see that the kernel of this map is precisely the image of the map $d\\pi$. As $H_{j_{S^2}}^{0,1}\\left(S^2, u^*TM\\right)^{S^1}$ is finite dimensional it follows that the codimension of $d\\pi$ is finite and hence the image of $d\\pi$ in $T\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ splits.\n\\\\\n\nFinally to show that $\\pi$ is open onto it's image we note that, \n\\[\\begin{tikzcd}\n      &\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}) \\arrow[dr,\"\\pi\"] \\arrow[d, \"q\"] \\\\\n      &\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})\/\\mathbb{C}^* \\arrow[r,\"h\", \"\\cong\"']  & \\text{im} ~\\pi\n\\end{tikzcd}\n\\]\nwhere the map $h:\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})\/\\mathbb{C}^* \\to \\text{im} \\pi$ is a homeomorphism. As $q$ is a quotient map for a group action, we have that $q$ is an open map and as $\\pi = h \\circ q$, we have the $\\pi$ too is an open map, thus showing that $\\pi$ satisfies all the conditions in the lemma and hence $\\pi$ is a sub-immersion.\n\\end{proof}\n\n\\begin{cor}\\label{cor:StrataAreSubmanifolds}\n$U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is a Banach submanifold of $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$.\n\\end{cor}\n\\begin{proof}\nThis follows from Lemma~\\ref{subimmersion} and from observing that the image of $\\pi$ is $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$.\n\\end{proof}\n\nWe will now describe the normal bundle of $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ (when $s > 0$) in terms of infinitesimal deformations of complex structures. To this end, we first find a cohomological condition ensuring that the inclusion of complex integrable structures into $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is transverse to the stratum $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$, in other words, that we have transverse maps\n\\[\n\\begin{tikzcd}\n      &\\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}) \\arrow[d,\"\\pi\"] \\\\\n     I^{S^1}_{\\omega,l} \\arrow[r,\"i\"]  &\\mathcal{J}^{S^1}_{\\om_\\lambda, l} \n\\end{tikzcd}\n\\\\\n\\]\n\\begin{lemma} \\label{compliment}\nLet $(M,\\omega_\\lambda,J_s)$  denote any of the Hirzebruch surfaces $(S^2 \\times S^2,\\omega_\\lambda,J_s)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda,J_s)$, and let $(u,J_s)\\in \\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l})$. Then the induced map $u^*:H_{J_s}^{0,1}\\left(M,TM\\right)^{S^1} \\rightarrow H^{0,1}_{j_{S^2}}\\left(S^2, u^*TM\\right)^{S^1}$ is an isomorphism. \n\\end{lemma}\n\n\\begin{proof}\nFrom  Proposition 3.4 in \\cite{AGK} we know that $u^*:H_{J_s}^{0,1}(M,TM) \\rightarrow H_{j_{S^2}}^{0,1}(S^2, u^*TM) $ is an isomorphism. As $u$ is equivariant this indeed gives us that \\[u^*:H_{J_s}^{0,1}\\left(M,TM\\right)^{S^1} \\longrightarrow H_{j_{S^2}}^{0,1}\\left(S^2, u^*TM\\right)^{S^1}\\]\nis also an isomorphism.\n\\end{proof}\n\n\\begin{lemma}\\label{trans_strata}\nLet $(M,\\omega_\\lambda)$ denote either $(S^2 \\times S^2,\\omega_\\lambda)$ or $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. Further let $i: I^{S^1}_{\\omega,l} \\hookrightarrow  \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ denote the inclusion and  $\\pi: \\mathcal{M}^{S^1}(D_s , \\mathcal{J}^{S^1}_{\\om_\\lambda, l}) \\to \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ denote the projection. Then,\n\\begin{itemize}\n    \\item $i \\pitchfork \\pi$,\n    \\item the infinitesimal complement (i.e the fibre of the normal bundle)  of $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ at $J_s \\in I^{S^1}_{\\omega,l}$ can be identified with $H^{0,1}_{J_s}(M,TM)^{S^1}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}Recall by Theorem~\\ref{integrable}, that the tangent space of $I^{S^1}_{\\omega,l}$ was given by \n\\[T_{J_s} I^{S^1}_{\\omega,l} = cl S\\Omega_{J_s}^{0,1}(M,TM)^{S^1} := \\ker \\overline\\partial_{J_s}: S\\Omega^{0,1}_{{J_s},l}\\left(M,TM\\right)^{S^1} \\longrightarrow  \\Omega^{0,2}_{{J_s},l-1}\\left(M,TM\\right)^{S^1}\\]\n\n\nLet $\\gamma \\in T_{J_s} \\mathcal{J}^{S^1}_{\\om_\\lambda, l} = \\left(S\\Omega^{0,1}_{J_s}(M,TM)\\right)^{S^1}$ and define $\\left[\\gamma \\circ du \\circ j_{S^2}\\right] := \\eta \\in H_{j_{S^2}}^{0,1}(S^2, u^*TM)^{S^1}$. To show that $i \\pitchfork \\pi$, we need to produce $\\beta \\in T_{J_s}I^{S^1}_{\\omega,l} = cl S\\Omega_{J_s}^{0,1}(M,TM)^{S^1}$ such that $[\\left(\\gamma - \\beta\\right) \\circ du \\circ j_{S^2}] = 0$. To do so, we consider the following commutative diagram.\n\n\n\\[ \n\\stackinset{l}{13ex}{b}{6ex}{%\n\\scalebox{.8}\n{%\n\\begin{tikzcd}[row sep=9ex, column sep = 13ex, ampersand replacement=\\&]\n[\\alpha] \n    \\arrow[mapsto]{r}\n    \\arrow[mapsto]{d}\n\\& \\left[\\alpha \\circ du \\right]\n    \\arrow[mapsto]{d}  \\\\\n\\left[\\alpha \\circ J_m\\right]\n    \\arrow[mapsto]{r} \n\\& \\begin{array}{c}[\\alpha \\circ du \\circ j_{S^2}] =\\\\ \\left[\\alpha \\circ J_s \\circ du\\right]\\end{array}\n\\end{tikzcd}\n} \n}{%\n\\begin{tikzcd}[row sep = 20ex, column sep = 20ex, ampersand replacement=\\&]\nH_{J_s}^{0,1}(M,TM)^{S^1} \n    \\arrow{r}{u^*} \n    \\arrow[swap]{d}{J_s^*}\n\\& H_{j_{S^2}}^{0,1}(S^2, u^*TM)^{S^1}\n    \\arrow{d}{j_{S^2}^*}  \\\\\nH_{J_s}^{0,1}(M,TM)^{S^1} \n    \\arrow[swap]{r}{u^*} \n\\& H_{j_{S^2}}^{0,1}(S^2, u^*TM)^{S^1}\n\\end{tikzcd}\n}\n\\]\nwhere all the maps $u^*$,$J^*$ and $j_{S^2}^*$ are isomorphisms. Further we have the equality $\\left[\\alpha \\circ du \\circ j_{S^2}\\right] = \\left[\\alpha \\circ J \\circ du\\right]$ as $u$ is $j_{S^2}$-$J_s$ holomorphic. As we know that $H_{J_s}^{0,2}(M)^{S^1} =0$,  from the long exact sequence equation \\ref{les} we see that the quotient map \n\\[ cl S\\Omega_{J_s}^{0,1}(M,TM)^{S^1} \\to H_{J_s}^{0,1}(M,TM)^{S^1}\\] \nis surjective. As both $u^*$ and $J^*$ are isomorphisms, there exists $\\beta \\in cl S\\Omega_{J_s}^{0,1}(M,TM)^{S^1} = T_{J_s} I_{\\omega_\\lambda,l}^{S^1}$ such that $\\left[\\beta \\circ J \\circ du\\right]  =  \\left[\\beta \\circ du \\circ j_{S^2}\\right]= \\eta:= \\left[\\gamma \\circ du \\circ j_{S^2}\\right] $. Hence we indeed have $\\left[\\left(\\gamma -\\beta\\right) \\circ du \\circ j_{S^2} \\right] = 0 $ as required.\n\\\\\n\n\n\nWe now show that the fibre of the normal bundle of $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ at $J_s \\in I^{S^1}_{\\omega,l}$ can be identified with $H^{0,1}_{J_s}(M,TM)^{S^1}$. As seen in the proof of Lemma~\\ref{subimmersion}, we know that there is a map \n\\begin{align*}\n    L: S\\Omega_{J_s}^{0,1}(M,TM)^{S^1} &\\rightarrow H_{J_s}^{0,1}\\left(S^2, u^*TM\\right)^{S^1} \\\\\n    \\alpha &\\mapsto [\\alpha \\circ du \\circ j_{S^2}]\n\\end{align*}\n\nAs the quotient map $ cl S\\Omega_{J_s}^{0,1}(M,TM)^{S^1} \\to H_{J_s}^{0,1}(M,TM)^{S^1}$ is surjective and the maps  $u^*$ and $j_{S^2}$ are isomorphisms, we have that the map $L$ is surjective. As the kernel of L  is the image of $d\\pi$, the cokernel can be identified with $H_{j_{S^2}}^{0,1}\\left(S^2, u^*TM\\right)^{S^1} \\cong H_{J_s}^{0,1}(M,TM)^{S^1}$  Hence the the fibre of the normal bundle of $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ at $J_s \\in I^{S^1}_{\\omega,l}$ can be identified with $H^{0,1}_{J_s}(M,TM)^{S^1}$.\n\\end{proof}\n\n\n\n\\section{Isotropy representations}\n\nAs shown in the previous section, the codimension of $U_{s,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ inside $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ is equal to the dimension of $H_{J_s}^{0,1}(M,TM)^{S^1}$. This can be calculated using  Lemma~\\ref{trans_strata}. In this section we only perform the calculation for $S^2 \\times S^2$ and we postpone the discussion of $\\CP^2\\# \\overline{\\CP^2}$ to Section~\\ref{Isom_CCC}.\n\\\\\n\n\\subsection{Even Hirzebruch surfaces and their isometry groups}\n\n\n\nLet $2k=m$. By Theorem 4.2 in \\cite{AGK}, the action of the isometry group $K(2k)\\simeq S^{1}\\times \\SO(3)$ on the space $H_{J}^{0,1}(M,TM)$ of infinitesimal deformations is isomorphic to $\\Det\\otimes \\Sym^{2k-2}$, where $\\Det$ is the standard action of $S^{1}=U(1)$ on $\\mathbb{C}^{2}$, and where $\\Sym(\\mathbb{C}^{2})$ is the representation $\\mathscr{W}_{k-1}$ of $\\SO(3)$ induced by the $(2k-2)$-fold symmetric product of the standard representation of $\\SU(2)$ on $\\mathbb{C}^{2}$. We use this fact to calculate the dimension of the $S^1$ invariant subspace of $H_{J}^{0,1}(M,TM)$ and thus obtain the codimension $U_{m,l} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda, l}$ inside $\\mathcal{J}^{S^1}_{\\om_\\lambda, l}$. \\\\\n\nFollowing \\cite{AGK}, we construct the Hirzebruch surface $\\mathbb{F}_{2k}$ by K\\\"ahler reduction of $\\mathbb{C}^{4}$ under the action of the torus $T^{2}_{2k}$ defined by\n\\[(s,t)\\cdot z = (s^{2k}tz_{1},tz_{2},sz_{3},sz_{4})\\]\nThe moment map is $\\phi(z)=(2k|z_{1}|^{2}+|z_{3}|^{2}+|z_{4}|^{2}, |z_{1}|^{2}+|z_{2}|^{2})$ and the reduced manifold at level $(\\lambda+k,1)$ is symplectomorphic to $(S^{2}\\times S^{2},\\omega_{\\lambda})$ and biholomorphic to the Hirzebruch surface $\\mathbb{F}_{2k}$. In this model, the projection to the base is given by $[(z_{1},\\ldots,z_{4})]\\mapsto [z_{3}:z_{4}]$, the zero section is $[w_{0}:w_{1}]\\mapsto [(w_{0}^{2k},0,w_{0},w_{1})]$, and a fiber is $[w_{0}:w_{1}]\\mapsto [(w_{0}w_{1}^{2k},w_{0}w_{1},0,w_{1})]$. The torus $T^{2}(2k)=T^{4}\/T^{2}_{2k}$ acts on $\\mathbb{F}_{2k}$. This torus is generated by the elements $[(1,e^{it},1,1)]$ and $[(1,1,e^{is},1)]$, and its moment map is $[(z_{1},z_{2},z_{3},z_{4})]\\mapsto(|z_{2}|^{2},|z_{3}|^{2})$. The moment polytope $\\Delta(2k)$ is the convex hull of the vertices $(0,0)$, $(1,0)$, $(1,\\lambda+k)$, and $(0,\\lambda-k)$.\n\n\\[\n\\begin{tikzpicture}\n\\draw (0,0) -- (1,0) ;\n\\draw (0,0) -- (0,3) ;\n\\draw (1,0) -- (1,4) ;\n\\draw (0,3) -- (1,4) ;\n\\node[right]{};\n\\end{tikzpicture}\n\\]\n\nThe isometry group of $\\mathbb{F}_{2k}$ is \n\\[K(2k) = Z_{\\U(4)}(T^{2}_{2k})\/T^{2}_{2k}=(T^{2}\\times \\U(2))\/T^{2}_{2k}\\simeq S^{1}\\times\\PU(2)\\simeq S^{1}\\times\\SO(3)\\]\nwhere the middle isomorphism is given by\n\\[[(s,t), A]\\mapsto (s^{-1}t\\det A^{k}, [A])\\]\nUnder this isomorphism, an element $[(1,a,b,1)]$ of the torus $T(2k)$ is taken to\n\\[\\left(ab^{k},\\begin{bmatrix}b&0\\\\0&1\\end{bmatrix}\\right)=\\left(b^{k}a,\\begin{bmatrix}b^{1\/2}&0\\\\0&b^{-1\/2}\\end{bmatrix}\\right)\\]\nConsequently, at the Lie algebra level of the maximal tori, the map identifying the maximal torus of $K(2k)$ whose lie algebra is denoted by $\\lie{t}^{2}(2k)$ with the maximal torus $S^1 \\times SO(2) \\subset S^1 \\times SO(3)$ whose lie algebra is denoted by $\\lie{t}^{2}$ (where $SO(2)$ is identified with the rotations around the z-axis)  is given by\n\\[\\begin{pmatrix}\n1&k\\\\\n0&1\n\\end{pmatrix}\\]\nThe moment polytope associated to the maximal torus $T^{2}\\subset K(2k)$ is thus  the balanced polytope obtained from $\\Delta(2k)$ by applying the inverse transpose $\\begin{pmatrix}1&0\\\\-k&1\\end{pmatrix}$\nand has the following shape\n\\[\n\\begin{tikzpicture}\n\\draw (0,0) -- (1,-1) ;\n\\draw (0,0) -- (0,1) ;\n\\draw (0,1) -- (1,2) ;\n\\draw (1,2) -- (1,-1) ;\n\\end{tikzpicture}\n\\]\n\n\\subsection{Even isotropy representations and codimension calculation}\n\nLet $J_m$ be the standard $S^1$ invariant integrable almost complex structure in the strata $U_m$, coming from the Hirzebruch surface $W_m$. The action of the isometry group $K(2k)\\simeq S^{1}\\times \\SO(3)$ on the space $H_{J_m}^{0,1}(S^2 \\times S^2,T(S^2 \\times S^2)) \\cong \\mathbb{C}^{m-1}$ (see \\cite{Ko} Example 6.2(b)(4), p.309 for more details about how the isomorphism is obtained) of infinitesimal deformations is isomorphic to $\\Det\\otimes \\Sym^{2k-2}$ , where  $\\Det$ is the standard action of $S^{1}=U(1)$ on $\\mathbb{C}^{2}$, and where $\\Sym(\\mathbb{C}^{2})$ is the representation $\\mathscr{W}_{k-1}$ of $\\SO(3)$ induced by the $(2k-2)$-fold symmetric product of the standard representation  of $\\SU(2)$ on $\\mathbb{C}^{2}$ (see Theorem 4.2 in\\cite{AGK}). We shall denote this $(2k-2)$-fold symmetric product of the standard representation  of $\\SU(2)$ on $\\mathbb{C}^{2}$ as $\\mathscr{V}_{2k-2}$. See \\cite{B-tD} for more details about the representation theory of $\\SO(3)$ and $\\SU(2)$  \\\\\n\n\nThe circle of $\\SO(3)=\\PU(2)=\\U(2)\/\\Delta(S^{1})$\n\\[R(t)=\\begin{pmatrix}1 & 0 & 0\\\\ 0 & \\cos(t) & -\\sin(t)\\\\ 0 & \\sin(t) & \\cos(t)\\end{pmatrix}, ~t\\in[0,2\\pi)\\]\nlifts to \n\\[e(t\/2):=\\begin{pmatrix}e^{it\/2} & 0\\\\ 0 & e^{-it\/2}\\end{pmatrix}\\in\\SU(2)\\]\nAs explained above, the action of $K(2k)$ on  $H^{0,1}_{J_m}(S^2 \\times S^2,T^{1.0}_{J_m}(S^2 \\times S^2)) \\cong \\mathbb{C}^{m-1}$ is isomorphic to $\\Det\\otimes\\Sym^{2k-2}$.Hence to calculate the the codimension we only need to calculate the dimension of the invariant subspace of $H^{0,1}(S^2 \\times S^2,T^{1.0}_{J_m}(S^2 \\times S^2)) \\cong \\mathbb{C}^{m-1}$ under this action. To do so we note that a basis of $\\Sym^{2k-2}$ is given by the homogeneous polynomials $P_{j}=z_{1}^{2k-2-j}z_{2}^{j}$ for $j\\in\\{0,\\ldots,2k-2\\}$. The action of $R(t)$ on $P_{j}$ is\n\\[R(t)\\cdot P_{j}=e(t\/2)\\cdot P_{j}=e^{i\\big(2k-2-2j\\big)t\/2}P_{j}=e^{it(k-1-j)}P_{j}\\]\nso that the action of $(e^{is},R(t))\\subset S^{1}\\times\\SO(3)$ on $P_{j}$ is\n\\[\\left(e^{is},R(t)\\right)\\cdot P_{j}=e^{i\\big(s+t(k-1-j)\\big)}P_{j}\\]\nEach $P_{j}$ generates an eigenspace for the action of the maximal torus $T(2k)$. In particular, the circle $S^{1}(a,b;2k)$ acts trivially on $P_{j}$ if, and only if, \n\\[a+b(k-1-j)=(a,b)\\cdot(1,k-1-j)=0\\]\nfor $j\\in\\{0,2k-2\\}$. Equivalently, we must have\n\\[a+bj=(a,b)\\cdot(1,j)=0\\]\nfor $j\\in\\{1-k,\\ldots,k-1\\}$\n\nHence the dimension of the invariant subspace is given by the number of $j \\in \\{1-k,\\ldots,k-1\\}$ such that $a+bj=0$.\n\\\\\n\n\n\nNote that the above codimension calculation was with respect to the basis of the maximal torus in $K(2n)$. Hence to calculate the codimension for the $S^1(1,b,m) \\subset \\mathbb{T}^2_m$ as in our case, we need to transform the basis by multiplication by the matrix $\\begin{pmatrix}\\frac{m}{2}& -1\\\\1&0\\end{pmatrix}$. Thus it takes the vector $\\begin{pmatrix}1\\\\b\\end{pmatrix}$ in the basis for the standard moment polytope \n\n\\[\n\\begin{tikzpicture}\n\\draw (0,1) -- (3,1) ;\n\\draw (0,1) -- (0,0) ;\n\\draw (0,0) -- (4,0) ;\n\\draw (3,1) -- (4,0) ;\n\n\\end{tikzpicture}\n\\]\n\nto the vector $\\begin{pmatrix}\\frac{m}{2}-b\\\\1\\end{pmatrix} $ in the basis for the balanced polytope (for which we did the above calculations).\n\n\\[\n\\begin{tikzpicture}\n\\draw (0,0) -- (1,-1) ;\n\\draw (0,0) -- (0,1) ;\n\\draw (0,1) -- (1,2) ;\n\\draw (1,2) -- (1,-1) ;\n\n\\end{tikzpicture}\n\\]\n\nTherefore the codimension of $S^1(1,b;m)$ is given by the number of $j \\in \\{1-\\frac{m}{2}, \\cdots, \\frac{m}{2}-1\\}$ such that $(\\frac{m}{2} - b) + j = 0$. Relabelling $j^\\prime$ as $\\frac{m}{2}+j$, we have that the codimension is given by \nthe number of $j^\\prime \\in \\{1, \\cdots , m-1\\}$ such that $j^\\prime=b$.\n \n\\begin{thm}\\label{codimension_calc}\nGiven the circle action $S^1(1,b,m)$ with $2\\lambda > |2b-m|$ and $b \\neq \\{0, m\\}$, the complex codimension of of the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ in given by the number of $j \\in \\{1, \\cdots , m-1\\}$ such that $j=b$.\n\\\\\nSimilarly for the action $S^1(-1,b,m)$ with $2\\lambda > |2b+m|$ and $b \\neq \\{0, -m\\}$, the complex codimension of of the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ in given by the number of $j \\in \\{1, \\cdots , m-1\\}$ such that $j=-b$.\n\\end{thm}\n\\begin{cor}\nFor the circle actions \\begin{itemize}\n  \\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|$.\n\\end{itemize}The complex codimension of the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is either 0 or 1.\n\\end{cor} \n\\begin{proof}\nFollows from the calculation and discussion above.\n\\end{proof}\n\n\\begin{remark}\nIn the beginning of the section, we only show that the space $\\mathcal{J}^{S^1}_{\\om_\\lambda, l} \\cap U_{2k,l}$ was a Banach submanifold.  But in order to obtain the topology of the space of $\\Symp^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$ with $\\mathbb{C}^\\infty$-topology, we require that the space  $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2k}$ with the $C^\\infty$ topology is a Fr\\'echet manifold and that the codimension of $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2k}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is given by the same formula as in Theorem \\ref{codimension_calc}. As this discrepancy exists in the literature even in the non-equivariant case, and as a resolution of this issue is well beyond the scope of this document we do not attempt to resolve this here. \n\\end{remark}\n\n\n\n\\chapter{Odd Hirzebruch surfaces}\\label{Chapter-CCC}\n\n\\section{Homotopy type of \\texorpdfstring{$\\Symp(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$}{Symp(CP2\\#CP2}}\nWe now compute the centralisers for the $S^1$ actions on the odd Hirzebruch surfaces. The theory is extremely analogous to the even Hirzebruch case i.e $S^2 \\times S^2$, hence we shall only point out the key differences. \n\\\\\n\nRecall that  the  odd Hirzebruch surface $W_m$ (where m is odd) is defined as a complex submanifold of $ \\mathbb{C}P^1 \\times \\mathbb{C}P^2$ defined by setting\n\\[\nW_m:=  \\left\\{ \\left(\\left[x_1,x_2\\right],\\left[y_1,y_2, y_3\\right]\\right) \\in \\mathbb{C}P^1 \\times \\mathbb{C}P^2 ~|~  {x^m}_1y_2 - x_2{y^m}_1 = 0 \\right\\}\n\\]\n\nThis manifold is diffeomorphic to $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$. The Torus $\\mathbb{T}^2$ acts on $\\mathbb{C}P^1 \\times \\mathbb{C}P^2$ in the following manner. \n\n\\[\n \\left(u,v\\right) \\cdot  \\left(\\left[x_1,x_2\\right],\\left[y_1,y_2, y_3\\right]\\right) = \\left(\\left[ux_1,x_2\\right],\\left[u^my_1,y_2,vy_3\\right]\\right)\n\\]\n\n(again with m being odd) and the  moment map image looks like\n\n\\[\n\\begin{tikzpicture}\n\\node[left] at (0,2) {$Q=(0,1)$};\n\\node[left] at (0,0) {$P=(0,0)$};\n\\node[right] at (4,2) {$R= (\\lambda - \\frac{m+1}{2} ,1)$};\n\\node[right] at (6,0) {$S=(\\lambda + \\frac{m-1}{2} ,0)$};\n\\node[above] at (2,2) {$B-\\frac{m+1}{2}F$};\n\\node[right] at (5,1) {$F$};\n\\node[left] at (0,1) {$F$};\n\\node[below] at (3,0) {$B+ \\frac{m-1}{2}F$};\n\\draw (0,2) -- (4,2) ;\n\\draw (0,0) -- (0,2) ;\n\\draw (0,0) -- (6,0) ;\n\\draw (4,2) -- (6,0) ;\n\\end{tikzpicture}\\label{oddHirz}\n\\]\nwhere $B$ now refers to the homology class of a line $L$ in $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$ and $F$ refers to the class $L - E$ where $L$ is the class of the line and $E$ is the class of the exceptional divisor. There is a canonical form which we also call $\\omega_\\lambda$ on $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$, which has weight $\\lambda$ on B and 1 on $F$. Note that with our convention, $\\lambda$ must be strictly greater than 1 for the curve in class $E$ to have positive symplectic area.\nAs before all symplectic $S^1$ action on $\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}P^2}$ extend to toric actions. Hence only need to consider sub-circles of the above family of torus actions. The graphs for the different circles are given in Figures \\ref{fig:GraphsWithFixedSurfaces} and \\ref{fig:GraphsIsolatedFixedPoints}. As explained in Theorem~\\ref{strata-CCC}, we have a stratification of the space of almost complex structures i.e the space $\\mathcal{J}_{\\omega_\\lambda}$  of all compatible almost complex structures for the form $\\omega_\\lambda$, decomposes into disjoint Fr\u00e9chet manifolds of finite codimensions\n\\[\n\\mathcal{J}_{\\omega_\\lambda} = U_1 \\sqcup U_3 \\sqcup U_5\\ldots \\sqcup U_{2n-1}\n\\]\nwhere \n\\[ \nU_{2i-1} := \\left\\{ J \\in \\mathcal{J}_{\\omega_\\lambda}~|~ D_{2i-1}:= B-iF \\in H_2(S^2 \\times S^2,\\mathbb{Z})\\text{~is represented by a $J$-holomorphic sphere}\\right\\}.\n\\]\n\n\n\nWe shall now use this stratification to construct fibrations for the action of equivariant symplectomorphism group $\\Symp^{S^1}_{h}(S^2 \\times S^2,\\omega_\\lambda)$ on the space $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s+1}$ of invariant almost complex structures in each stratum. We follow the same notation as in \\ref{not}. The proofs that the following maps are in fact fibrations is exactly the same as the proof given in Section~\\ref{section:ActionOnU_2k}.\n\n\\[\\Stab^{S^1}(\\overline{D}) \\longrightarrow \\Symp^{S^1}_{h}(S^2 \\times S^2,\\omega_\\lambda) \\longtwoheadrightarrow {\\mathcal{S}^{S^1}_{D_{s}}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1} \\cap U_{2s+1}\\rule{0em}{2em}\\]\n\n\\[\\Fix^{S^1}(\\overline{D}) \\longrightarrow \\Stab^{S^1}(\\overline{D}) \\longtwoheadrightarrow  \\Symp^{S^1}(\\overline{D}) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1 ~\\text{or}~ SO(3)\\rule{0em}{2em}\\]\n   \n\\[\\Fix^{S^1} (N(\\overline{D})) \\longrightarrow \\Fix^{S^1}(\\overline{D}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(\\overline{D})) \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} S^1 \\rule{0em}{2em}\\]\n   \n\\[\\Stab^{S^1}(\\overline{F}) \\cap  \\Fix^{S^1}(N(\\overline{D})) \\longrightarrow \\Fix^{S^1}(N(\\overline{D})) \\longtwoheadrightarrow  \\overline{\\mathcal{S}^{S^1}_{F,p_0}} \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\mathcal{J}^{S^1}(\\overline{D})\\simeq \\{*\\}\\rule{0em}{2em} \\]\n    \n\\[\\Fix^{S^1}(\\overline{F}) \\longrightarrow \\Stab^{S^1}(\\overline{F}) \\cap  \\Fix^{S^1}(N(\\overline{D})) \\longtwoheadrightarrow  \\Symp^{S^1}(\\overline{F}, N(p_0))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\n    \n\\[\\left\\{*\\right\\} \\mathbin{\\textcolor{blue}{\\xleftarrow{\\text{~~~$\\simeq$~~~}}}} \\Fix^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\longrightarrow \\Fix^{S^1}(\\overline{F}) \\longtwoheadrightarrow  \\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F}))  \\mathbin{\\textcolor{blue}{\\xrightarrow{\\text{~~~$\\simeq$~~~}}}} \\left\\{*\\right\\}\\rule{0em}{2em}\\]\n\nWhen the $S^1(a,b;m) \\subset \\mathbb{T}^2_m$ action is of the form $(a,b)= (0,\\pm1)$\n\\[\n\\begin{tikzcd}\n     &\\Fix^{S^1}(\\overline{D}) \\arrow[r] &\\Stab^{S^1}(\\overline{D}) \\arrow[r,twoheadrightarrow] &\\Symp^{S^1}(\\overline{D}) \\\\\n     &S^1 \\arrow[u,hookrightarrow] \\arrow[r] &U(2) \\arrow[u,hookrightarrow] \\arrow[r] &SO(3) \\arrow[u,hookrightarrow] \\\\\n\\end{tikzcd}\n\\]\nFor all other actions with $(a,b) \\neq (0,\\pm1)$ we have \n\\[\n\\begin{tikzcd}\n     &\\Fix^{S^1}(\\overline{D}) \\arrow[r] &\\Stab^{S^1}(\\overline{D}) \\arrow[r,twoheadrightarrow] &\\Symp^{S^1}(\\overline{D}) \\\\\n     &S^1 \\arrow[u,hookrightarrow] \\arrow[r] &\\mathbb{T}^2_{2s+1} \\arrow[u,hookrightarrow] \\arrow[r] &S^1 \\arrow[u,hookrightarrow] \\\\\n\\end{tikzcd}\n\\]\nIn both diagrams, the leftmost and the rightmost arrows are homotopy equivalences. As the diagram above commutes, the 5 lemma implies that the middle inclusions $\\mathbb{T}^2 \\hookrightarrow \\Stab^{S^1}(\\overline{D})$  or $U(2) \\hookrightarrow \\Stab^{S^1}(\\overline{D})$ are also weak homotopy equivalences.\n\n\\begin{thm}\\label{homogenous_CCC}\nConsider the $S^1(a,b;m)$ action on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ with $\\lambda >1$. If $ \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s+1} \\neq \\phi$, then we have the following homotopy equivalences:\n\\begin{enumerate}\n    \\item when $(a,b) \\neq (0,\\pm1)$, we have $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)\/\\mathbb{T}^2_{2s+1} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s+1}$;\n    \\item when $(a,b) = (0,\\pm1)$, we have $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)\/U(2) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{2s+1}$.\n\\end{enumerate} \\qed\n\\end{thm} \n\n\n\nAs in the $S^2 \\times S^2$ case, to understand the homotopy type of the equivariant symplectomorphism we need to next understand which strata the space of invariant almost complex structures intersect. Consider the circle action $S^1(a,b;m)$ on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ and $\\lambda >1$, Corollaries~\\ref{cor:IntersectingOnlyOneStratum} and \\ref{cor:IntersectingTwoStrata} imply that\n\\begin{enumerate}\n    \\item if $a=1$, $b \\neq\\{0,m\\}$ and $2\\lambda > |2b-m|+1$, then the space of $S^1(1,b;m)$-equivariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the two strata $U_m$ and $U_{|2b-m|}$.\n    \\item If $a=-1$,$b \\neq\\{0,-m\\}$ and $2\\lambda > |2b+m|+1$, then the space of $S^1(-1,b;m)$-equivariant almost complex structures $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects the two strata $U_m$ and $U_{|2b+m|}$.\n    \\item for all other cases $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects only the stratum $U_m$.\n\\end{enumerate}  \n\n\n\n\n\nAs before, we the homotopy type of the equivariant symplectomorphism group can be easily described whenever $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects only one strata.\n\n\\begin{thm}\\label{CCC1strata}\nConsider the  circle action $S^1(a,b;m)$ on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$. Under the following numerical conditions on $a,b,m,\\lambda$, the homotopy type of $\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ is given by the  table below.\n\\noindent{\\begin{center}\n \\begin{tabular}{|p{4.5cm}|p{3.5cm}|p{2cm}|p{4cm}|}\n\n \\hline\n Values of $(a,b ;m)$ & $\\lambda >1$ &Number of strata $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects &Homotopy type of $\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$\\\\\n\\hline\n {$(0,\\pm 1;m)$, $m\\neq 0$}  &$\\forall\\lambda $ & 1 & $\\U(2)$ \\\\\n\\hline\n{$(0,\\pm 1;0)$ or $(\\pm 1,0;0)$} \n&$\\forall\\lambda$ &1  &$\\U(2)$ \\\\\n\\hline\n $(\\pm 1,0;m), m\\neq0$ &$\\forall \\lambda $ &1 &$\\mathbb{T}^2$\\\\\n \\hline\n$(\\pm 1,\\pm m;m), m \\neq 0$ &$\\forall \\lambda$   &1 & $\\mathbb{T}^2$\\\\\n\\hline\n{$(1,b;m), b \\neq \\{m,0\\}$} \n&$|2b-m|+1 \\geq2 \\lambda$ &1 &$\\mathbb{T}^2$ \\\\\n\\hline\n{$(-1,b;m), b \\neq \\{ -m,0\\}$} \n&$|2b+m|+1 \\geq2 \\lambda$ &1 &$\\mathbb{T}^2$ \\\\\n\n\\hline\nAll other values of $(a,b;m)$ except $(\\pm 1,b;m)$&$\\forall \\lambda$  &1   &$\\mathbb{T}^2$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}}\n\\qed\n\\end{thm}\n\n\n\nTheorem~\\ref{CCC1strata} gives the homotopy type of the group of equivariant symplectomorphisms for all circle actions apart from the following two exceptional families:\n\\begin{itemize}\n\\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|+1$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|+1$.\n\\end{itemize}\n\nAt this point, we proceed as in Chapter 4. Firstly, we show that the map $\\mathbb{T}^2_m \\hookrightarrow \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ induces a map that is injective in homology. \n\\\\\n\nFix a curve $\\overline{F}$ in the homology class $F$ and passing through the fixed points $Q$ and $P$ in figure~\\ref{oddHirz}. Let $\\mathcal{S}^{S^1}_{F,Q}$ denote the space of $S^1$-invariant curves in the class $F$ passing through $Q$ (defined in figure~\\ref{oddHirz}) and let $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda)$ denote the space of $S^1$-equivariant symplectomorphisms of $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ that pointwise fix the curve $\\overline{F}$.\n\\\\\n\nWithout loss of generality, assume that $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|}$ is the strata of positive codimension in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$. As $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is contractible, $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m = \\mathcal{J}^{S^1}_{\\om_\\lambda} \\setminus \\left(\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|}\\right)$, and the real codimension of $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is 2 (See Corollary~\\ref{odd_codim}), it follows that $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ is connected. Further we have that $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m\/\\mathbb{T}^2_m$ is connected. As the fixed points for the $S^1(\\pm 1,b,m)$ actions are isolated and as $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ is connected, any element $\\phi \\in \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ takes a fixed point for the $S^1$ action to itself. Thus the action of $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ on $\\mathcal{S}^{S^1}_{F,Q}$ is well defined.\n\n\\begin{lemma}\\label{odd_inj}\nThe inclusion $i: \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda) \\hookrightarrow \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ is a homotopy equivalence. \n\\end{lemma}\n\\begin{proof}\nConsider the fibration\n\\begin{equation*}\n    \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda) \\hookrightarrow \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) \\longtwoheadrightarrow \\mathcal{S}^{S^1}_{F,Q}\n\\end{equation*}\nTo show that the action $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ on $\\mathcal{S}^{S^1}_{F,Q}$ is transitive we note that given $F^\\prime \\in \\mathcal{S}^{S^1}_{F,Q}$, there exists a $J^\\prime \\in \\mathcal{J}^{S^1}_{\\om_\\lambda}$ such that $F^\\prime$ is $J^\\prime$-holomorphic. As $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is connected, consider a path $J_t$ such that $J_0=J^\\prime$ and $J_1=J_m$ where $J_m$ is the standard complex structure on the $m^\\text{th}$-Hirzebruch surface for which the curve $\\overline{F}$ is holomorphic.\n\\\\\n\nBy Theorem~\\ref{prop_FIndecomposable}, for every $J_t$ we have a family of curves $F_t$ (with $F_0= F^\\prime$ and $F_1 =\\overline{F}$) in class $F$ passing through $Q$  and this curve is $S^1$ invariant as $J_t$ are $S^1$ invariant. By Lemma~\\ref{Au} we have a one parameter family of Hamiltonian symplectomorphisms $\\phi_t \\in \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$  such that $\\phi_t(F_0) = F_t$ for all $t$.\n\nThus it suffices to show that $\\mathcal{S}^{S^1}_{F,Q}$ is contractible to complete the proof. To do this note that,\n\\begin{equation*}\n  \\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{F})  \\to \\mathcal{J}^{S^1}_{\\om_\\lambda} \\to \\mathcal{S}^{S^1}_{F,Q}\n\\end{equation*}\n\nwhere $\\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{F})$ denotes the space of $S^1$ invariant almost complex structures for which the curve $\\overline{F}$ is $J$-holomorphic. As both $\\mathcal{J}^{S^1}_{\\om_\\lambda}(\\overline{F})$ and $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ are contractible, $\\mathcal{S}^{S^1}_{F,Q}$ is contractible as well completing the proof.\n\\end{proof}\n\n Define the following projections just as in the exposition above Theorem~\\ref{inj}. In our case we take the point $\\{*\\}$ to be the point $Q$\nin Figure~\\ref{oddHirz}.\n\n \\begin{align*}\n    s_{0}: S^2 &\\to W_m \\\\\n    \\left[z_{0}, z_{1}\\right] &\\mapsto\\left(\\left[z_{0}, z_{1}\\right],[0,0,1]\\right)\n\\end{align*} \nand the projection to the first factor of $\\mathbb{C}P^1 \\times \\mathbb{C}P^2$ is\n\\begin{align*}\n    \\pi_1: W_m &\\to S^2 \\\\\n    \\left(\\left[z_{0}, z_{1}\\right],\\left[w_{0}, w_{1}, w_{2}\\right]\\right) &\\mapsto\\left[z_{0}, z_{1}\\right]\n\\end{align*}\nWe define a continuous map $h_1:\\Symp_h^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) \\to \\mathcal{E}\\left(S^{2}, *\\right)$ by setting\n\\begin{equation*}\n  \\begin{aligned}\nh_1:\\Symp_h^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) &\\to \\mathcal{E}\\left(S^{2}, *\\right) \\\\\n\\psi &\\mapsto \\psi_1:= \\pi_{1} \\circ \\psi \\circ s_{0}\n\\end{aligned}\n\\end{equation*}\n\nFurther define the restriction map $r: \\Symp_h^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda) \\to \\mathcal{E}\\left(S^{2}, *\\right)$ by just restricting $\\phi \\in \\Symp_h^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda)$ to the fibre $\\overline{F}$.\n\nThus we have a well defined map \n\\begin{align*}\n    h: \\Symp_h^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda) &\\to \\mathcal{E}\\left(S^{2}, *\\right) \\times \\mathcal{E}\\left(S^{2}, *\\right) \\\\\n    \\phi &\\mapsto (h_1(\\phi), r(\\phi))\n\\end{align*}\n\n\\begin{thm}\nThe inclusion map $i:\\mathbb{T}^2_m \\hookrightarrow \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ induces a map that is injective in homology.\n\\end{thm}\n\n\\begin{proof} By Lemma~\\ref{odd_inj}, it suffices to prove that the inclusion $i:\\mathbb{T}^2_m \\hookrightarrow \\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\overline{F},\\omega_\\lambda) $ induces a map that is injective in homology. \n\\\\\n\nComposing with $h$ we have an inclusion of $h \\circ i:\\mathbb{T}^2_m \\hookrightarrow \\mathcal{E}\\left(S^{2}, *\\right) \\times \\mathcal{E}\\left(S^{2}, *\\right)$ and it suffices to show that this map induces a map that is injective in homology. The proof of this claim in analogous to the proof of Theorem~\\ref{inj}.\n\n\\begin{remark}\nAs the argument above doesn't depend on $m$, the same proof as above also shows that for the family of circle actions given by $S^1(1,b,m)$ with $2\\lambda > |2b-m|+1$, the inclusion $\\mathbb{T}^2_{|2b-m|}$ into $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ also induces a map that is injective in homology. Similarly we can also show that for the $S^1(-1,b;m)$ actions with $2\\lambda > |2b+m|+1$, the inclusion $\\mathbb{T}^2_{|2b+m|}$ and $T_m$ into $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ also induces a map that is injective in homology.\n\\end{remark}\n\n\n\\end{proof}\n\nAs in the $S^2 \\times S^2$ case, we have that $i:\\mathbb{T}^2_m \\hookrightarrow \\Symp^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) $ induces a map which is injective in homology, From our discussion above we also had that $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)\/\\mathbb{T}^2_m \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ and $\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)\/\\mathbb{T}^2_{|2b-m|} \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|}$. Let $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m := P$ and $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|} ;= Q$, as  $i:\\mathbb{T}^2_m \\hookrightarrow \\Symp^{S^1} (\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda) $ induces a map which is injective in homology, further  by Leray-Hirsch theorem we have that \n\\begin{align*}\n    H^*(\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)) \\cong H^*(P) \\bigotimes H^*(\\mathbb{T}^2) \\\\\n     H^*(\\Symp^{S^1}_h(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)) \\cong H^*(Q) \\bigotimes H^*(\\mathbb{T}^2)\n\\end{align*}\nAs before we need to compute the codimension of the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{|2b-m|} $in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$. The computation in the section~\\ref{Isom_CCC} shows it to be two (see Corollary~\\ref{odd_codim}). \n\\\\\n\nThus we have the following theorem on the ranks of the homology groups of the space of equivariant symplectomorphisms.\n\n\\begin{thm} Consider the following circle actions on  $\\mathbb{C}P^2 \\# \\overline \\mathbb{C}P^2$ \\begin{itemize}\n\\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|+1$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|+1$.\n\\end{itemize} \nThen we have \n$$H^p\\left(\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda), k\\right) = \\begin{cases}\nk^4 ~~p \\geq 2\\\\\nk^3 ~~p =1 \\\\\nk ~~ p=0\\\\\n\\end{cases}$$\nfor any field $k$.\n\\end{thm}\n\nAs the proof of Theorem~\\ref{full_homo} holds verbatim for the $S^1(a,b;m)$ actions on $\\CP^2\\# \\overline{\\CP^2}$ satisfying the conditions\n\\begin{itemize}\n\\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|+1$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|+1$.\n\\end{itemize} \nThe above results give us the homotopy type of the centralizer for all circle actions on $\\CP^2\\# \\overline{\\CP^2}$ which we summarise in the table below.\n\n\\begin{thm}\\label{full_homo_CCC}\nFor the $S^1$ action given by the integers $(a,b;m)$, acting on $(\\CP^2\\# \\overline{\\CP^2}, \\omega_\\lambda)$. Under the following numerical conditions on $a,b,m,\\lambda$, the homotopy type of $\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ is given by the  table below.\n\\begin{center}\n \\begin{tabular}{|p{4.5cm}|p{3.5cm}|p{2cm}|p{3.75cm}|}\n\n \\hline\n Values of $(a,b ;m)$ & $\\lambda>1$ &Number of strata $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ intersects &Homotopy type of $\\Symp^{S^1}(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$\\\\\n\\hline\n {$(0,\\pm 1;m)$, $m\\neq 0$}  &$\\forall\\lambda$ & 1 & $\\U(2)$ \\\\\n\\hline\n{$(0,\\pm 1;0)$ or $(\\pm 1,0;0)$}\n&$\\forall\\lambda$ &1  &$\\U(2)$ \\\\\n\n\\hline\n $(\\pm 1,0;m), m\\neq0$ &$\\forall\\lambda$ &1 &$\\mathbb{T}^2$\\\\\n \\hline\n$(\\pm 1,\\pm m;m), m \\neq 0$ &$\\forall\\lambda$   &1 & $\\mathbb{T}^2$\\\\\n\\hline\n\\multirow{2}{10em}{$(1,b;m), b \\neq \\{m,0\\}$} \n&$|2b-m|+1 \\geq2 \\lambda$ &1 &$\\mathbb{T}^2$ \\\\\\cline{2-4}\n&$2 \\lambda >|2b-m|+1 $  &2 &$\\Omega S^3 \\times S^1 \\times S^1 \\times S^1$ \\\\\n \\hline\n \\multirow{2}{10em}{$(-1,b;m), b \\neq \\{ -m,0\\}$} \n&$|2b+m|+1 \\geq2 \\lambda $ &1 &$\\mathbb{T}^2$ \\\\\\cline{2-4}\n&$2 \\lambda >|2b+m|+1$  &2 &$\\Omega S^3 \\times S^1 \\times S^1 \\times S^1$ \\\\\n\\hline\nAll other values of $(a,b;m)$ &$\\forall \\lambda$  &1   &$\\mathbb{T}^2$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\qed\n\\end{thm}\n\n\n\\section{Isometry groups of odd Hirzebruch surfaces}\\label{Isom_CCC}\n\nIn this section we calculate the codimension of the smaller strata in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$. Let $M$ denote the manifold $\\CP^2\\# \\overline{\\CP^2}$. By Theorem~\\ref{trans_strata}, we know that the normal bundle to the strata $U_{2s+1} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda}$ can be identified with $H^{0,1}_{J_{2s+1}}(M,T^{1,0}_{J_{2s+1}}M)^{S^1}$. Thus to calculate the codimension of $U_{2s+1} \\cap \\mathcal{J}^{S^1}_{\\om_\\lambda}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$, we need to understand how $S^1$ acts on $H^{0,1}_{J_{2s+1}}(M,T^{1,0}_{J_{2s+1}}M)$ and calculate the dimension of the invariant subspace. We use the convention $m = 2k+1$.\n\\\\\n\n\nThe Hirzebruch surface $\\mathbb{F}_{2k+1}$ is obtained by K\\\"ahler reduction of $\\mathbb{C}^{4}$ under the action of the torus $T^{2}_{2k+1}$ defined by\n\\[(s,t)\\cdot z = (s^{2k+1}tz_{1},tz_{2},sz_{3},sz_{4})\\]\nThe moment map is $\\phi(z)=((2k+1)|z_{1}|^{2}+|z_{3}|^{2}+|z_{4}|^{2}, |z_{1}|^{2}+|z_{2}|^{2})$ and the reduced manifold at level $(\\lambda +k,1)$ is symplectomorphic to $(\\mathbb{C}P^2 \\# \\overline{\\mathbb{C}}P^2,\\omega_{\\lambda})$ and biholomorphic to the Hirzebruch surface $\\mathbb{F}_{2k+1}$. In this model, the projection to the base is given by $[(z_{1},\\ldots,z_{4})]\\mapsto [z_{3}:z_{4}]$, the zero section is $[w_{0}:w_{1}]\\mapsto [(w_{0}^{2k+1},0,w_{0},w_{1})]$, and a fiber is $[w_{0}:w_{1}]\\mapsto [(w_{0}w_{1}^{2k+1},w_{0}w_{1},0,w_{1})]$. The torus $T^{2}(2k+1)=T^{4}\/T^{2}_{2k+1}$ acts on $\\mathbb{F}_{2k+1}$. This torus is generated by the elements $[(1,e^{it},1,1)]$ and $[(1,1,e^{is},1)]$, and its moment map is $[(z_{1},z_{2},z_{3},z_{4})]\\mapsto(|z_{2}|^{2},|z_{3}|^{2})$. The moment polytope $\\Delta(2k+1)$ is the convex hull of the vertices $(0,0)$, $(1,0)$, $(1,\\lambda+k)$, and~$(0,\\lambda-k -1)$.\n\\\\\n\nThe isometry group of $\\mathbb{F}_{2k+1}$ is \n\\[K(2k+1) = Z_{\\U(4)}(T^{2}_{2k+1})\/T^{2}_{2k+1}=(T^{2}\\times \\U(2))\/T^{2}_{2k+1}\\simeq \\U(2)\\]\nwhere the last isomorphism is given by\n\\[[(s,t), A]\\mapsto (s^{-1}t\\det A^{k}) A\\]\nUnder this isomorphism, an element $[(1,a,b,1)]$ of the torus $T(2k+1)$ is taken to\n\\[ab^{k}\\begin{bmatrix}b&0\\\\0&1\\end{bmatrix}\\]\nConsequently, at the Lie algebra level of the maximal tori, the map $\\lie{t}^{2}(2k+1)\\to \\lie{t}^{2}$ is given by\n\\[\\begin{pmatrix}\n1&k+1\\\\\n1&k\n\\end{pmatrix}\\]\nThe moment polytope associated to the maximal torus $T^{2}\\subset K(2k+1)$ is thus  the balanced polytope obtained from $\\Delta(2k+1)$ by applying the inverse transpose $\\begin{pmatrix}-k&1\\\\k+1&-1\\end{pmatrix}$.\n\n\\subsection{Odd isotropy representations and codimension calculation}\n\nThe action of the isometry group $K(2k+1)\\simeq \\U(2)$ on the space $H_{J}^{0,1}(M,TM)$ of infinitesimal deformations is isomorphic to $\\Det^{1-k}\\otimes \\Sym^{2k-1}$, where $\\Det$ is the determinant representation of $\\U(2)$ on $\\mathbb{C}$, and where $\\Sym^{k}(\\mathbb{C}^{2})$ is the $k$-fold symmetric product of the standard representation of $\\U(2)$ on $\\mathbb{C}^{2}$. Using the double covering $S^{1}\\times\\SU(2)\\to\\U(2)$, we see that irreducible representations of $\\U(2)$ correspond to irreducible representations of $S^{1}\\times\\SU(2)$ for which $(-1,-\\id)$ acts trivially. If $A_{n}$ denotes the representation $t\\cdot z=t^{n}z$ of $S^{1}$ on $\\mathbb{C}$, and if $V_{k}$ is the $k$-fold symmetric product of the defining representation of $\\SU(2)$ on $\\mathbb{C}^{2}$, then the irreducible representations of $\\U(2)$ are $A_{n}\\otimes V_{k}$ with $n+k$ even. In this notation, we have the identifications $\\Det = A_{2}$, while $\\Sym=A_{1}\\otimes V_{1}$. Consequently, $\\Det^{1-k}\\otimes \\Sym^{2k-1} = A_{1}\\otimes V_{2k-1}$.\n\nWith respect to the double covering $S^{1}\\times\\SU(2)\\to\\U(2)$, the maximal torus $T^{2}\\subset\\U(2)$ of diagonal matrices $D_{s,t}:=\\begin{pmatrix}e^{is} & 0\\\\ 0 & e^{it}\\end{pmatrix}$ lifts to\n\\[\\left(|D_{s,t}|^{1\/2},\\frac{D}{|D|^{1\/2}}\\right)\n=\\left(e^{i(s+t)\/2},\\begin{pmatrix}e^{i(s-t)\/2} & 0\\\\ 0 & e^{i(t-s)\/2}\\end{pmatrix}\\right)\\]\nAs explained above, the action of $K(2k+1)$ on $H^{0,1}(\\CP^2\\# \\overline{\\CP^2},T^{1.0}_{J_m}(\\CP^2\\# \\overline{\\CP^2})) \\cong \\mathbb{C}^{m-1}$ is isomorphic to  $\\Det^{1-k}\\otimes\\Sym^{2k-1}$. Hence to calculate the the codimension we only need to calculate the dimension of the invariant subspace of the vector space $ H^{0,1}(\\CP^2\\# \\overline{\\CP^2},T^{1.0}_{J_m}(\\CP^2\\# \\overline{\\CP^2})) \\cong \\mathbb{C}^{m-1}$ under the $S^1(1,b;m)$ action. To do so we note that a basis of $\\Sym^{2k-1}$ is given by the homogeneous polynomials $P_{j}=z_{1}^{2k-1-j}z_{2}^{j}$ for $j\\in\\{0,\\ldots,2k-1\\}$. The action of $D_{s,t}$ on $P_{j}$ is\n\\[D_{s,t}\\cdot P_{j}=e^{i\\big((s+t)(1-k)+s(2k-1-j)+tj\\big)}P_{j}\\]\nso that each $P_{j}$ generates an eigenspace for the action of the maximal torus $T(2k+1)$ generated by $D_{s,t}$. In particular, the circle $S^{1}(a,b;2k+1)$ acts trivially on $P_{j}$ if, and only if, \n\\[(a-b)(k-j)+b=(a,b)\\cdot(k-j,j-k+1)=0\\]\n\\\\\nThus the codimension (in the balanced basis of the maximal torus of K(2k+1) is given by the number of $j \\in \\{0,\\ldots,2k-1\\}$ such that \\begin{equation}\\label{formula_CCC}\n    (a-b)(k-j)+b=(a,b)\\cdot(k-j,j-k+1)=0.\n\\end{equation}\n\\\\\n\nNote that just as in the $S^2 \\times S^2$ case, the above codimension calculation was with respect to the basis of the maximal torus in $K(2k+1)$. Hence to calculate the codimension for the $S^1(1,b,m) \\subset \\mathbb{T}^2_m$ as in our case, we need to transform the basis by multiplication by the matrix $\\begin{pmatrix}\\frac{m-1}{2}+1& -1\\\\\\frac{m-1}{2}&-1\\end{pmatrix}$. Thus it takes the vector $\\begin{pmatrix}1\\\\b\\end{pmatrix}$ in the basis for the standard moment polytope \n\n\n\\[\n\\begin{tikzpicture}\n\\draw (0,1) -- (3,1) ;\n\\draw (0,1) -- (0,0) ;\n\\draw (0,0) -- (4,0) ;\n\\draw (3,1) -- (4,0) ;\n\n\\end{tikzpicture}\n\\]\n\nto the vector $\\begin{pmatrix}\\frac{m+1}{2}-b\\\\\\frac{m-1}{2}-b\\end{pmatrix}$ in the basis for the balanced polytope. Hence $a$ and $b$ in equation~\\ref{formula_CCC} above need to be replaced by $\\frac{m+1}{2}-b$ and $\\frac{m-1}{2}-b$ respectively to get the correct codimension for the $S^1(1,b,m)$ action. \n\\\\\n\nThus we have the following theorem. \n\n\n\n\\begin{thm}\\label{codimension_calc2}\nGiven the circle action $S^1(1,b,m)$ on $(\\CP^2\\# \\overline{\\CP^2},\\omega_\\lambda)$ with $2\\lambda > |2b-m|+1$ and $b \\neq \\{0, m\\}$, the complex codimension of of the strata $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ in given by the number of $j \\in \\{1, \\cdots , m-1\\}$ such that $j=b$.\n\\\\\n\nSimilarly for the action $S^1(-1,b,m)$ with $2\\lambda > |2b+m|+1$ and $b \\neq \\{0, -m\\}$, the complex codimension of of the strata $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ in given by the number of $j \\in \\{1, \\cdots , m-1\\}$ such that $j=-b$.\n\\qed\n\\end{thm}\n\n\\begin{cor}\\label{odd_codim}\nFor the circle actions \\begin{itemize}\n  \\item (i) $a=1$, $b \\neq \\{0, m\\}$,  and $2\\lambda > |2b-m|+1$; or\n\\item (ii) $a=-1$, $b \\neq \\{0, -m\\}$, and $2 \\lambda > |2b+m|+1$.\n\\end{itemize}The complex codimension of the stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_m$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}$ is either 0 or 1.\n\\qed\n\\end{cor}\n\n\n\\chapter{Equivariant Gauge Groups}\n\nIn this section we basically show how to calculate the homotopy type of the equivariant gauge groups that arise in  lemma \\ref{gauge} and lemma \\ref{ngauge}. \n\\\\\n\nLet $P$ be a principle $G$- bundle over $B$, where $G$ is an abelian group that acts on the right. Let  $H$ be a lie group that act on the base space $B$ and this action lifts to an action on the bundle $P$. We shall denote this action by a left action. Note that $H$ need not act effectively on the space $B$. \n\\\\\n\nLet $\\Gauge1^H(P)$ denote all equivariant (with respect to the action of H) bundle automorphisms i.e equivariant maps $u$ such that the following diagram commutes.\n\\[ \\begin{tikzcd}\n     &P  \\arrow[d,\"\\pi\"] \\arrow[r, \"u\"]  &P \\arrow[dl,\"\\pi\"] \\\\\n     &B \n\\end{tikzcd}\n\\]\n\nGiven $u \\in \\Gauge1^H(P)$, define the map \n\\begin{align*}\n    \\phi_u : P &\\rightarrow G \\\\\nx &\\mapsto \\phi_u(x)\n\\end{align*}\n\nwhere $\\phi_u(x)$ is defined such that \n$$x \\cdot \\phi_u(x) = u(x)$$\n\nLet us now see how the map $\\phi_u$ behaves under the left action of $H$. We have \n\n$$ u(h \\cdot x) = h\\cdot x \\cdot \\phi_u(h \\cdot x) $$ \n\nBut we already have that \n\n$$u(h \\cdot x) = h \\cdot u(x) = h \\cdot x  \\cdot \\phi_u(x)$$\n\nPutting the equalities together and noticing that the $G$ -action is free we get that \n\n$$ \\phi_u(h \\cdot x) = \\phi_u(x) $$ \n\nThat is that the map $\\phi_u$ is invariant under the action of $H$. \n\\\\\n\nAlso from the defintion we can see that $$\\phi_u( x \\cdot g) = g^{-1} \\cdot \\phi_u(x) \\cdot g = \\phi_u(x)$$ \n\nwhere the last equality follows from $G$ being abelian. \n\\\\\n\nDenote by $\\Maps_{H,G}(P,G)$ the space of all $H$ and $G$-invariant smooth functions from P to G equipped with the $C^\\infty$ topology.  Note that as $G$ is abelian, this space of maps is homeomorphic with the space $\\Inv(B,G)$,  of $H$-invariant maps from $B$ to $G$ endowed with the $C^\\infty$-topology. Further we have that the map\n\n\n\\begin{align*}\n    \\rho: \\Gauge1^H (P) &\\rightarrow \\Maps_{H,G}(P,G) \\\\\n    u &\\mapsto \\phi_u\n\\end{align*}\nis an an homeomorphism (with both spaces endowed with the $C^\\infty$-topology) with the inverse being constructed using the definition  $x \\cdot \\phi_u(x) = u(x)$. \n\\\\\n\nThus $\\Gauge1^H(P)$ is homeomorphic to $\\Inv(B,G)$.\n\\\\\n\nLet us now use this to calculate the Homotopy type of the Gauge groups in the fibrations in Chapter 2. \n\\\\\n\nConsider a rank 2 symplectic normal bundle of $\\overline{D}$. Let us fix an equivariant arbitrary compatible fibre wise almost complex structure $J$ on $N(\\overline{D})$.  As this is a rank two bundle, the structure group is $Sp(2)$, and this now can be reduced to $U(1)= S^1$, and  the two bundles are isomorphic. Thus the space of symplectic automorphisms of the original bundle is homeomorphic to the space of symplectic automorphisms of the reduced bundle.\n\\\\\n\nIn our case we have a right group action of $S^1$ on this bundle and we are interested  in the equivariant symplectic automorphisms of this bundle.This is homotopic to the space of Equivariant symplectic automorphisms of the $U(1)$ bundle.(As the reduction of the structure group can be done equivariantly) And as $U(1) = S^1$ the space of  Equivariant symplectic automorphisms of the $U(1)$ bundle is the same as $\\Gauge1^{S^1}(P)$ where $P$ is the associated principal bundle. This is homeomorphic to $\\Inv(S^2,S^1)$ from the above discussion.\n\\\\\n\nFinally note that for any non-trivial $S^1$ action on $S^2$ (possibly non-effective) the quotient space under this action is just the interval. Hence the space $\\Inv(S^2,S^1)$ is just the space of smooth maps from the interval to $S^1$.\n\\\\\n\nBefore we embark on trying to calculate the homotopy type of  $\\Gauge1^{S^1}(N(\\overline{D}))$ (See Section~\\ref{not} for notation), we would need the following technical lemma. Note that in all our calculations above we have used the $C^\\infty$-topology on $\\Gauge1^{S^1}(N(\\overline{D}))$ and $\\Inv_{S^1}(S^2,S^1)$. Let $\\Gauge1_c^{S^1}(N(\\overline{D}))$ denote the space of continuous $S^1$-equivariant gauge transformations of the bundle $N(\\overline{D})$ equipped the $C^0$-topology. Using the same argument at the beginning of the section we can show that $\\Gauge1_c^{S^1}(N(\\overline{D}))$ is homotopic to the space $\\Inv_{S^1,c}(S^2,S^1)$ of continuous $S^1$-invariant maps from $S^2$ to $S^1$. Then we have that,\n\n\n\\begin{lemma}\\label{wolken}\nThe space $\\Gauge1^{S^1}(N(\\overline{D}))$ with the $C^\\infty$-topology is homotopic to the space $\\Gauge1_c^{S^1}(N(\\overline{D}))$ equipped with the $C^0$-topology.\n\\end{lemma}\n\nThe proof of this lemma follows from an equivariant version of the arguments used to prove Theorem 3.2.13 in \\cite{homgauge}. \n\n\\begin{lemma} \\label{Gauge(D)}\n$\\Gauge1^{S^1}(N(\\overline{D})) \\simeq \\Gauge1_c^{S^1}(N(\\overline{D})) \\simeq \\Inv_{S^1,c}(S^2,S^1) \\simeq S^1$\n\\end{lemma}\n\n\\begin{proof}\n$\\Gauge1^{S^1}(N(\\overline{D})) \\simeq \\Gauge1_c^{S^1}(N(\\overline{D})) \\simeq \\Inv_{S^1,c}(S^2,S^1)$ follows from Lemma \\ref{wolken} and the discussion above. \n\\\\\n\nLet $\\Maps(S^2\/S^1,S^1)$ denote the space of continuous maps from $S^2\/S^1$ to $S^1$.Then the space $\\Inv_{S^1,c}(S^2,S^1)$ is homeomorphic to the space $\\Maps(S^2\/S^1,S^1)$. Further we note that as $S^2\/S^1$ is homeomorphic to an interval $[0,1]$, the space $\\Maps(S^2\/S^1,S^1)$  can be identified  the space of continuous maps from the interval $[0,1]$ to $S^1$ which we denote by $\\Maps([0,1],S^1)$ . Let $p_0$ be a fixed point for the $S^1$ action on $S^2$. Then we consider the following fibration, \n\n\n\\[\n\\begin{tikzcd}\n     &\\left\\{*\\right\\} \\simeq \\Inv_{S^1,c}\\left(\\left(S^2,p_0\\right), \\left(S^1,id\\right)\\right) \\arrow[r,hookrightarrow] &\\Inv(S^2,S^1) \\arrow[r,\"ev\"] &S^1 \\\\ \n\\end{tikzcd}\n\\]\n\nWhere the map $ev:\\Inv(S^2,S^1) \\rightarrow S^1 $ is just the evaluation map at the fixed point (for the $S^1$ action on $S^2$) $p_0$ and the space $\\Inv_{S^1,c}\\left(\\left(S^2,p_0\\right), \\left(S^1,id\\right)\\right)$ is the space of all continuous maps from $S^2$ to $S^1$, invariant under the $S^1$ action and send the point $p_0$ to the identity on $S^1$. As above, the space $\\Inv\\left(\\left(S^2,p_0\\right), \\left(S^1,id\\right)\\right)$ can be identified with the space of continuous maps from the the interval $[0,1]$ to $S^1$ that send $0$ to the the identity on $S^1$. This space of pointed maps from $[0,1]$ to $S^1$ is contractible, thus completing the proof.\n\\end{proof}\n\n\\begin{remark}\nWe need the point $p_0$ to be a fixed point as the evaluation map has to be surjective.\n\\end{remark}\n\n\\begin{lemma} \\label{Gauge(N(D))}\n$\\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\left\\{*\\right\\}$\n\\end{lemma}\n\n\\begin{proof}\nAnalogous to our method above, we get that $\\Gauge1^{S^1}(N(\\overline{D} \\vee \\overline{F}))$ is just the space of continuous maps from the following configuration to $S^1$ that send some  neighbourhood of the crossing to the identity in $S^1$\n\\begin{figure}[H]\n    \\centering\n   \n   \n   \n\\begin{tikzpicture}[thick, scale=0.6]\n\\draw (0,-3) -- (0,-1);\n\\draw (0,1) --(0,3);\n\\draw[red] (0,1) --(0,-1);\n\\draw[red] (1,0) --(-1,0);\n\\draw (-1,0) -- (-3,0);\n\\draw (1,0) --(3,0);\n\\draw (10,0) circle (2cm);\n\\draw[->] (4,0.5) -- (7,0.5);\n\\node [right] at (12.2,0) {id};\n\\node[below] at (1.3,0) {$\\overline{D}$};\n\\node[right] at (0,2) {$\\overline{F}$};\n\\draw [fill,red] (12,0) circle (0.1cm);\n\\end{tikzpicture}\n\\end{figure}\n\nAnd the space of such maps is indeed contractible thus completing the proof. \n\\end{proof}\n\\begin{comment}\n\n\nWe now want to carry out similar computation but for action of finite abelian groups $\\mathbb{Z}_n$ on the bundle. As discussed before, we have that $\\Gauge1^{Z_n}(N(\\overline{D})) \\simeq \\Inv_{Z_n}(S^2,S^1)$ where  $\\Inv_{Z_n}(S^2,S^1)$ denotes the space of $Z_n$ invariant maps from $S^2$ to $S^1$. Further let $\\Gauge1_c^{Z_n}(N(\\overline{D}))$ denote the space of continuous $\\mathbb{Z}_n$ equivariant gauge transformations. As in Lemma \\ref{wolken} we have that \n$\\Gauge1^{Z_n}(N(\\overline{D})) \\simeq \\Gauge1_c^{Z_n}(N(\\overline{D}))$. Further, just as in the $S^1$ case we may identify  $\\Gauge1_c^{Z_n}(N(\\overline{D}))$ with the space $\\Inv_{Z_n, c}(S^2,S^1)$ of continuous $\\mathbb{Z}_n$ invariant maps from $S^2$ to $S^1$.\n\\\\\n\nPutting all this together we have, \n\n\\begin{lemma}\\label{finitegauge}\n$\\Gauge1^{Z_n}(N(\\overline{D})) \\simeq \\Gauge1_c^{Z_n}(N(\\overline{D})) \\simeq  \\Inv_{Z_n, c}(S^2,S^1) \\simeq S^1$\n\\end{lemma}\n\n\\begin{proof}\nThe homotopy equivalences $\\Gauge1^{Z_n}(N(\\overline{D})) \\simeq \\Gauge1_c^{Z_n}(N(\\overline{D})) \\simeq  \\Inv_{Z_n, c}(S^2,S^1)$ are all explained above. Thus we only need to show that $\\Inv_{Z_n, c}(S^2,S^1) \\simeq S^1$. In our case we know that the $Z_n$ action on $S^2$ are in fact restrictions of $S^1$ actions on $S^2$, hence they are rotations about  fixed points of $S^2$. Note that $\\Inv_{Z_n,c}(S^2,S^1)$ is homeomorphic to the space $\\Maps(S^2\/Z_n, S^1)$ of continuous maps from $S^2\/\\mathbb{Z}_n$ to $S^1$. Further,  $S^2\/Z_n$ is homeomorphic to $S^2$ and hence $\\Maps(S^2\/Z_n, S^1) \\cong  \\Maps(S^2, S^1)$. Finally, we note that $\\Maps(S^2, S^1) \\simeq S^1$ thus completing the result.\n\\end{proof}\n\n\\begin{lemma}\\label{wedgegauge}\n$\\Gauge1^{Z_n}(N(\\overline{D} \\vee \\overline{F})) \\simeq \\left\\{*\\right\\}$\n\\end{lemma}\n\n\\begin{proof}\nAnalogous to the proof of Lemmas \\ref{Gauge(N(D))} and \\ref{finitegauge},  we can identify the group $\\Gauge1_{Z_n}(N(\\overline{D} \\vee \\overline{F}))$ with maps from $S^2 \\vee S^2$ that send a neighbourhood of the wedge point to the identity in $S^1$. The space of such maps is contractible. \n\\end{proof}\n\nFinally, we need to understand the homotopy type of $\\mathbb{Z}_n$ equivariant symplectomorphisms $\\Symp^{\\mathbb{Z}_n}(S^2)$, in our analysis in Chapter 6. \n\\begin{lemma}\\label{Wang}\nConsider a symplectic action of $\\mathbb{Z}_n$ on $S^2$, then the space $\\Symp^{\\mathbb{Z}_n}(S^2)$ is homotopic  to  $\\SO(3)$, if $\\mathbb{Z}_n$ fixes $S^2$ pointwise, and is homotopic to $S^1$ otherwise.\n\\end{lemma}\n\\begin{proof}\nLet $\\psi \\in \\Symp^{\\mathbb{Z}_n}(S^2)$, consider the graph $\\tilde \\psi$ of $\\psi$ i.e\n\n\\begin{align*}\n    \\tilde \\psi: S^2 &\\rightarrow S^2 \\times S^2 \\\\\n    z &\\mapsto (z,\\psi(z))\n\\end{align*}\n\nLet $\\SO(3)^{\\mathbb{Z}_n}$ denote the centraliser of $Z_n$ inside $\\SO(3)$. Choose a $\\mathbb{Z}_n$ equivariant metric for the product $\\mathbb{Z}_n$ action on $S^2 \\times S^2$ coming from the $\\mathbb{Z}_n$ action on $S^2$. Then by Theorem C, Corollary C and Corollary 4.1 in \\cite{wang}, the mean curvature flow with respect to this equivariant metric gives us a canonical homotopy of $\\psi$ to an element inside $\\SO(3)^{\\mathbb{Z}_n}$. Further this homotopy is identity on all the elements of $\\SO(3)^{\\mathbb{Z}_n}$. Thus the map we get is in fact a deformation retract of $\\Symp^{\\mathbb{Z}_n}(S^2)$ and $\\SO(3)^{\\mathbb{Z}_n}$. Note $\\SO(3)^{\\mathbb{Z}_n} = \\SO(3)~ or~ S^1$ depending on whether $\\mathbb{Z}_n$ is in the centre of $SO(3)$ or not respectively, thus proving the claim.\n\\end{proof}\n\\end{comment}\n\\chapter[Equivariant Gompf argument]{Holomorphic configurations and equivariant Gompf argument}\n\n\n\\begin{thm}\\label{transverse}\nLet $G$ be a compact group. Let A and B be two $G$-invariant symplectic spheres in a 4-dimensional symplectic manifold $(M,\\omega)$ intersecting $\\omega$-orthogonally at a unique fixed point $p$ for the $G$ action. Then there exists an invariant $J \\in \\mathcal{J}_\\omega^{G}$ such that both A and B are $J$-holomorphic. Here $\\mathcal{J}_\\omega^{G}$ denotes the space of $G$ invariant compatible almost complex structures on $M$. \n\\end{thm}\n\n\\begin{proof}\nThe proof follows from mimicking the proof of Lemma A.1 in \\cite{Evans} under the presence of a group action.\n\\iffalse\nAs $A$ and $B$ are symplectic submanifolds of $M$. There exists a compatible(with respect to the restricted form on $A$ and $B$) $G$- invariant  almost complex $J_A$ and $J_B$ such that $A$ is $J_A$-holomorphic  and $B$ is $J_B$-holomorphic. This gives us  invariant metrics $g_A$ and $g_B$ defined on the tangent bundles $TA$ and $TB$ respectively. \n\\\\\n\nIn a neighbourhood of $p$, choose a chart $U_p$ such that $A$ and $B$ are $\\mathbb{R}^2 \\times \\{0\\}$ and $\\{0\\}\\times \\mathbb{R}^2$ respectively. This is possible as $A$ and $B$ are symplectically orthogonal. Define the metric $\\Tilde{g}:= g_A \\oplus g_B$ on $T_p M$. We note that \n\nNow extend $g_A$ and $g_B$ in a neighbourhood of  the point $p$  as follows. Pick a smooth chart $U$ such that in that chart the submanifold $A$ is $\\{0\\}\\times \\mathbb{R}^2$ and the submanifold $B$ is $\\mathbb{R}^2 \\times \\{0\\}$. We think of a metric as a section of $TM \\otimes T^*M$ that is positive definite. Further we demand that $U$ is a trivialising chart for the bundle $TM \\otimes T^*M$. Then in the neighbourhood $U$ both $g_A$ and $g_B$ are function from $\\mathbb{R}^4$ to $\\mathbb{R}^4$.  We define a smooth extension of $g_A$ and $g_B$ to the whole neighbourhood as follows. We denote the extension by $\\Tilde{g}$.\n\n$$ \\Tilde{g}\\left((v_1,v_2,v_3,v_4)\\right):= g_A\\left((0,0,v_3,v_4\\right) + g_B\\left(v_1,v_2,0,0)\\right) - g_A(0,0,0,0)$$\n\\\\\n\nThe restriction $\\Tilde{g}$ to $\\{0\\} \\times \\mathbb{R}^2$ is $g_A(0,0,v_1,v_2) + {g_B(0,0,0,0)} - g_A(0,0,0,0)$. As $g_A(0,0,0,0)= g_B(0,0,0,0)$, we have $g_A(0,0,v_1,v_2) + {g_B(0,0,0,0)} - g_A(0,0,0,0) = g_A(0,0,v_1,v_2) = g_A$. Similarly we can show that the restriction of $\\Tilde{g}$ to $\\mathbb{R}^2 \\times \\{0\\}$ is $g_B$. Thus showing that we can extend $\\Tilde{g}$ smoothly to a neighbourhood $U$ of $p$. Apriori this extension need not be equivariant, we further choose a smaller neighbourhood $\\Tilde{U} \\subset U$ which is invariant under the $G$ action. And we average $\\Tilde{g}$ on $\\Tilde{U}$ under the $G$ action to get a function $g^\\prime$ which is equivariant and extends $g_A$ and $g_B$. \n\\\\\n\nFinally, given any other point $x \\in M$, $x \\neq p$ we can find a $G$ invariant open set $U_x$, such that either $U_x \\cap A \\neq \\emptyset$ or $U_x \\cap B \\neq \\emptyset$ and a $G$-equivariant metric $\\tilde{g}_{U_x}$ such that $\\tilde{g}_{U_x}(a) = g_A$ for all $a \\in U_x cap A$ and $\\tilde{g}_{U_x}(b) = g_B(b)$ for all $b \\in U_x \\cap B$. We then patch together all the above metrics using a $G$-invariant partition of unity for the open cover $\\bigcup_{x\\in M\\setminus p} U_x \\cup \\Tilde{U}$, thus giving us a metric with the required properties.\n\\fi\n\\iffalse\nwhere the co-ordinates is gotten by realizing $T_pM \\cong T_p A \\oplus T_p B$ i.e $(v_1,v_2)$ and $(u_1,u_2) \\in T_p A $ and similarly $(v_3,v_4)$ and $(u_3,u_4) \\in T_p B $. Further we note that $\\Tilde{g}_p$ is $S^1$-invariant. \n\\\\\n\nNow consider the equivariant symplectic normal bundles $N(A)$ to $A$ and $N(B)$ which is normal to $B$. At $p$ we see that the fibre symplectic normal bundle $N(A)$ agrees with $T_p B$. We already have a metric defined on $N_p(A) = T_p B$, we define an $S^1$-invariant fibrewise metric $\\Tilde{g}_A$ on $N(A)$ such that $\\Tilde{g}_A$ at the point $p$ agree with $g_B$ at the point $p$. Similarly define $\\Tilde{g}_B$ on $N(B)$ such that $\\Tilde{g}_B$ at the point $p$ agree with $g_A$ at the point $p$.\n\\\\\n\nDefine $h_A:= g_A \\oplus \\Tilde{g}_A$ and $h_B := g_B \\oplus \\Tilde{g}_B$ and define the metric on $TM|_{A \\cup B}$\n\\[\nh_x := \\begin{cases}\n{h_A} ~~ x \\in A \\\\\n{h_B} ~~ x \\in B \\\\\n\\end{cases}\n\\]\n\nFinally extend this metric $h$ to the whole of M, thus giving us an invariant $S^1$-metric $\\Tilde{h}$. Finally consider the almost complex structure corresponding to this metric, thus solving the problem.\n\\vspace{5mm}\n\\fi\n\n\\end{proof}\n\n        \n\\begin{thm}{(Equivariant Gompf Argument)} \\label{gmpf}\nLet $G$ be a compact group. Let $p$ be a fixed point for the action. Let $A$,$B$ and $\\overline{A}$ be $G$-invariant symplectic spheres in a 4-dimensional symplectic manifold $(M,\\omega)$ such that \n\\begin{itemize}\n    \\item $A \\cap B = \\{p\\}$ and the intersection at $p$ is transverse\n    \\item $\\overline{A} \\cap B = \\{p\\}$ and the intersection at $p$ is $\\omega$-orthogonal.\n\\end{itemize} \nThen there exists an $S^1$-equivariant isotopy $A_t$ of $A$  such that $A_t$ intersects $B$ transversely at $p$ for all $t$,  $A_1$ = $\\overline{A}$ in a small neighbourhood of $p$ and the curve $A_1$ agrees with $A$ outside some neighbourhood of $p$.\n\\end{thm}\n\\iffalse\nBefore we embark on the proof of this statement, we make a few observations. The first being that due to the symplectic neighbourhood theorem we can in fact work on the symplectic normal bundle and as   this is a local problem,  we can simplify our hardships by working in a trivialising chart i.e in $\\mathbb{R}^4$. The problem thus reformulated translates to the following one:\n\\\\\n\nGiven an $S^1$ action on $\\mathbb{R}^4$ and two $S^1$ invariant symplectic planes $A$ and $B$. There is an isotopy $\\psi_t \\in \\Ham_c(\\mathbb{R}^4)$ such that $\\psi_t(A)$ intersects $B$ transversely at $0$ for all $t$,  and $\\psi_1(A) = B^{\\bot_\\omega}$ in a small neighbourhood around the origin, where $\\omega_0$ is the standard symplectic form on $\\mathbb{R}^4$ and $B^{\\bot_\\omega}$ is the symplectic orthogonal to $B$. Further we may as well assume $B$ to be the two plane in $\\mathbb{R}^4$ given by $(0,0,x,y)$, and $B^{\\bot_\\omega}$ to the given by the plane $(x,y,0,0)$.\n\\\\\n\nNext we observe that given a function $A:=(f,g): \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$, the graph of $A$ is a symplectic (for the standard form) submanifold of $\\mathbb{R}^4$ iff $\\left\\{f,g\\right\\} > -1$. This can be proven from  a direct computation. We now have enough background to embark on the proof of the above theorem.\n\\fi\n\\begin{proof}Since this is a local problem, we can work in a trivialising chart in $\\mathbb{R}^4$ in which the action is linear. Let $B^{\\bot_\\omega}$ be the symplectic orthogonal to $B$. We can assume the image of  $B$ to be the two plane in $\\mathbb{R}^4$ given by $(0,0,x,y)$, and $B^{\\bot_\\omega}$ to the given by the plane $(x,y,0,0)$. As $\\overline{A}$ is $\\omega$-orthogonal to $B$, we can choose the neighbourhood such that $\\overline{A} = B^{\\bot_\\omega}$ near $p$. As $A$ is transverse to $B$ at $p$, we can assume its image is given by the graph of function (which we also call A) $A:(f,g): \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$.\n\\\\\n\nNext we observe that given a function $A:=(f,g): \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$, the graph of $A$ is a symplectic (for the standard form) submanifold of $\\mathbb{R}^4$ iff $\\left\\{f,g\\right\\} > -1$. This can be proven from  a direct computation.  We will construct an isotopy of graphs of function of the form $A_t:= \\alpha_t(r^2) A$ where $\\alpha_t$ is a bump function depending only on the radius squared (for a fixed $G$ invariant  metric) in $\\mathbb{R}^2$, and such that \\begin{itemize}\n    \\item $A_0 = A$,\n    \\item $A_1 = 0$  near (0,0),\n    \\item $A_t = A$ outside of some neighbourhood of the origin,\n    \\item $A_t$ is symplectic for all $t$.\n\\end{itemize}  \nNote that as $\\alpha_t$ is depends on the radius for a fixed $G$ invariant metric, $A_t$ is also $G$ invariant.\n\\\\\n\n\nDefine $E = g\\left\\{f,r^2\\right\\} + f\\left\\{r^2,g\\right\\}$. Using the fact that $r^2(0,0) = 0$ and $(r^2)^\\prime (0,0) = 0$ we see that $E(0,0) = 0$ and $\\frac{\\partial}{\\partial r}E(0,0) = 0$. By the intermediate value theorem, there exists $c > 0$, $\\epsilon > 0$ and $u > 0$ such that on the ball of radius $u + \\epsilon$ around the origin $B(0,u+\\epsilon)$  we have $E(x) \\geq -c r^2(x)$. Choose $\\delta$ such that  on $B(0,u+\\epsilon)$, $1 + \\left\\{f,g\\right\\} > \\delta > 0$\n\\\\\n\n\n\nPick $\\alpha: \\mathbb{R} \\rightarrow \\mathbb{R}$ satisfying the following properties\n\n\n\n   \n         \\begin{itemize}\n             \\item  $\\alpha(r^2) = 1$ for $r^2 \\geq u$ .\n             \\item $\\alpha(r) = 0$ for r near $0$.\n            \n             \\item $\\alpha^\\prime(r^2) \\leq \\frac{\\delta}{2cr^2} < \\frac{1 + \\{f,g\\}}{2cr^2}$ \n         \\end{itemize} \n         \n\nDefine $\\alpha_t:= (1-t) + t \\alpha(r^2)$ and $A_t := \\alpha_t A$. To show that $A_t$ is symplectic for all $0 \\leq t \\leq 1$ we need to check that  $\\left\\{\\alpha_tf,\\alpha_tg\\right\\} > -1$ for all $0 \\leq t \\leq 1$. \nIn the neighbourhood $B(u)$ we have\n\\begin{equation*}\n    1 + \\left\\{\\alpha_t f,\\alpha_t g\\right\\} = \\underbrace{1 + \\alpha_t^2 \\{f,g\\}}_{\\geq \\delta} + \\underbrace{\\alpha_t\\alpha_t^\\prime E}_{\\geq \\frac{-\\delta}{2}} \\geq 0\n\\end{equation*}\n\nThe inequality $1 + \\alpha_t^2 \\{f,g\\} \\geq \\delta$ follows from the definition of $\\delta$ and from noting that $0 \\geq \\alpha_t \\geq 1$.  $\\alpha_t\\alpha_t^\\prime E \\geq \\frac{-\\delta}{2}$ follows from the  inequality \n\n\\begin{align*}\n    \\alpha_t\\alpha_t^\\prime E &\\geq \\alpha_t\\alpha_t^\\prime (-cr^2) \\\\\n    &\\geq -\\alpha_t\\frac{\\delta}{2cr^2} (cr^2) \\\\\n    &\\geq -\\alpha_t \\frac{\\delta}{2} \\\\\n    &\\geq \\frac{-\\delta}{2}\n\\end{align*}\n\nThus in the neighbourhood $B(u)$ we have the inequality $1 + \\left\\{\\alpha_t f,\\alpha_t g\\right\\} > 0$ for all  t. Outside of $B(u)$, the derivative $\\alpha_t^\\prime$ is identically 0 and $\\alpha_t \\equiv 1$. Hence $\\alpha_t\\alpha_t^\\prime E \\equiv 0$ outside $B(u)$ and $1 + \\left\\{\\alpha_t f,\\alpha_t g\\right\\} = 1 + \\alpha_t^2 \\{f,g\\} + \\cancelto{0}{\\alpha_t\\alpha_t^\\prime E} = 1 + \\{f,g\\} > 0$ outside of $B(u)$.\n\\\\\n\nFinally we note that $A_1 =0$ in a neighbourhood of $(0,0)$ and it equals $A$ outside the ball of radius $u$ around the origin, thus proving the claim.\n\\end{proof}\n\n\n\n\\chapter[Equivariant Differential Topology]{Equivariant versions of classical results from Differential Topology}\n\n\\begin{comment}\n\n\n\\begin{lemma}[Relative Poincare Lemma](see~\\cite{KM}, Lemma~43.10)\\label{RelativePoincareLemma} Let $M$ be a smooth finite dimensional manifold and let $S\\subset M$ be a closed submanifold. Let $\\omega$ be a closed $(k+1)$-form on $M$ which vanishes on $S$. Then there exists a $k$-form $\\sigma$ on an open neighborhood $U$ of $S$ in $M$ such that $d\\sigma=\\omega$ on $U$ and $\\sigma=0$ along $S$. If moreover $\\omega=0$ along $S$, then we may choose $\\sigma$ such that the first derivatives of $\\sigma$ vanish on $S$.\n\\end{lemma}\n\n\n\n\\begin{proof}\nBy restricting to a tubular neighborhood of $S$ in $M$, we may assume that\n$M$ is a smooth vector bundle $p:E\\to S$ and that $i:S\\to E$ is the zero section. We consider $\\mu:\\mathbb{R}\\times E\\to E$, given by $\\mu(t,x)=\\mu_{t}(x)=tx$, then\n$\\mu_{1}=\\id_{E}$ and $\\mu_{0}=i\\circ p:E\\to S\\to E$. Let $V\\in\\mathfrak{X}(E)$ be the vertical vector field $V(x)=vl(x,x)= \\frac{d}{dt}(x+tx)$ whose flow is $\\text{Fl}_{t}^{V}=\\mu_{e^{t}}$. Locally, for $t$ in $(0,1]$ we have\n\\[\\frac{d}{dt}\\mu_{t}^{*}\\omega = \\frac{d}{dt}(\\text{Fl}_{\\log t}^{V})^{*}\\omega =\\frac{1}{t}(\\text{Fl}_{\\log t}^{V})^{*}\\mathcal{L}_{V}\\omega =  \\frac{1}{t}\\mu_{t}^{*}(i_{V}d\\omega+di_{V}\\omega)=\\frac{1}{t}d\\mu_{t}^{*}i_{V}\\omega\\]\nFor $x\\in E$ and $X_{1},\\ldots,X_{k}\\in T_{x}E$ we have\n\\begin{align*}\n(\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega)_{x}(X_{1},\\ldots,X_{k}) &= \\frac{1}{t}(i_{V}\\omega_{tx}(T_{x}\\mu_{t}\\cdot X_{1},\\ldots,T_{x}\\mu_{t}\\cdot X_{k})\\\\\n&= \\frac{1}{t}\\omega_{tx}(V(tx),T_{x}\\mu_{t}\\cdot X_{1},\\ldots,T_{x}\\mu_{t}\\cdot X_{k})\\\\\n&= \\omega_{tx}(vl(tx, tx), T_{x}\\mu_{t}\\cdot X_{1},\\ldots,T_{x}\\mu_{t}\\cdot X_{k})\n\\end{align*}\nSo the $k$-form $\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega$ is defined and smooth in $(t,x)$ for all $t\\in[0,1]$ and describes a smooth curve in $\\Omega^{k}(E)$. Note that for $x\\in S = 0_{E}$ we have $\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega=0$, and if $\\omega = 0$ on $T_{S}M$, we also have $0=\\frac{d}{dt}\\mu_{t}^{*}\\omega=\\frac{1}{t}d\\mu_{t}^{*}i_{V}\\omega$, so that all first derivatives of $\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega$ vanish along $S$. \nSince $\\mu_{0}^{*}\\omega = p^{*}i^{*}\\omega = 0$ and $\\mu_{1}^{*}\\omega=\\omega$, we have\n\\begin{align*}\n\\omega \n&=\\mu_{1}^{*}\\omega-\\mu_{0}^{*}\\omega\\\\\n&=\\int_{0}^{1}\\frac{d}{dt}\\mu_{t}^{*}\\omega\\,dt\\\\\n&=\\int_{0}^{1}d(\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega)\\,dt\\\\\n&=d\\left(\\int_{0}^{1}(\\frac{1}{t}\\mu_{t}^{*}i_{V}\\omega)\\,dt\\right)\\\\\t\n&=d\\sigma\n\\end{align*}\nIf $x\\in S$, we have $\\sigma= 0$, and all first derivatives of $\\sigma$ vanish along $S$ whenever $\\omega = 0$ on $T_{S}M$.\n\\end{proof}\n\n\\begin{remark} \nIf there is a symplectic action of a compact group $G$ acting on $M$ such that $\\omega$ is $G$ invariant and $S$ is $G$-invariant, then we can constuct $\\sigma$ as above such that in addition to the above conditions $\\sigma$ also is $G$-invariant. This is gotten by noting that \n\\begin{align*}\n    \\omega = \\int_G \\omega = \\int_G d\\sigma = d \\int_G \\sigma\n\\end{align*}\n\nLet $\\tilde\\sigma:= \\int_G \\sigma$ and hence $d\\tilde\\sigma = \\omega$ and $\\tilde\\sigma$ satisfies all the conditions.\n\\end{remark}\n\n\\begin{lemma}[Moser isotopy] Let $(M,\\omega)$ be a symplectic manifold and let $S\\subset M$ be a submanifold. Suppose that $\\omega_{i}$, $i=0,1$, are closed $2$-forms such that at each point $x\\in S$, the forms $\\omega_{0}$ and $\\omega_{1}$ are equal and non-degenerate on $T_{x}S$. Then there exist open neighborhoods $N_{0}$ and $N_{1}$ of $S$ and a diffeomorphism $\\phi:N_{0}\\to N_{1}$ such that $\\phi^{*}\\omega_{1}=\\omega_{0}$, $\\phi|_{S}=\\id$, and $d\\phi|_{S}=\\id$.\n\\end{lemma}\n\n\n\n\\begin{proof}\nConsider the convex linear combination $\\omega_{t}=\\omega_{0}+t(\\omega_{1}-\\omega_{0})$. Since $\\omega_{0}$ and $\\omega_{1}$ are equal along $S$, there exists a neighborhood $U_{1}$ of $S$ on which $\\omega_{t}$ is non-degenerate for all $t\\in[0,1]$. By restricting $U_{1}$ to a possibly smaller neighborhood $U_{2}$, the Relative Poincar\u00e9 Lemma~\\ref{RelativePoincareLemma} implies that there exists a $1$-form $\\sigma$ such that $d\\sigma=(\\omega_{1}-\\omega_{0})$, $\\sigma=0$ on $S$, and all first derivatives of $\\sigma$ vanish along $S$. Define the time-dependent vector field $X_{t}$ on $U_{2}$ by setting\n\\[\\sigma = -i_{X_{t}}\\omega_{t}\\]\nSince $X_{t}=0$ on $S$, by restricting $U_{2}$ to a smaller neighborhood $U_{3}$, we can ensure that the flow $\\psi_{t}$ of $X_{t}$ exists for $t\\in[0,1]$. We then have\n\\[\\frac{d}{dt}\\psi_{t}^{*}\\omega_{t} = \\psi_{t}^{*}\\left(\\frac{d}{dt}\\omega_{t}+\\mathcal{L}_{X_{t}}\\omega_{t}\\right) = \\psi_{t}^{*}\\left(\\frac{d}{dt}\\omega_{t}+di_{X_{t}}\\omega_{t}\\right) = \\psi^{*}(\\omega_{1}-\\omega_{0}-d\\sigma)=0\\]\nso that $\\psi^{*}\\omega_{t}=\\omega_{0}$. Finally, since $\\sigma=0$ on $S$, $\\psi=\\id$ on $S$, and since all first derivatives of $\\sigma$ vanish on $T_{S}M$, $d\\psi=\\id$ on $T_{S}M$.\n\\end{proof}\n\n\n\\begin{remark}\nAs the remark above, when both $\\omega_1$ and $\\omega_2$ are both invariant under a compact group action $G$, and $S$ is an $G$-invariant submanifold, then there is a $G$-equivariant diffeomorphism $\\phi$ that satisfies the conditions as above. \n\\end{remark}\n\\end{comment}\n\n\\begin{lemma}\\label{EqSymN}[Equivariant Symplectic neighborhoods theorem] \nLet $G$ be a compact group, and let $(M_{i},\\omega_{i})$, $i=0,1$, be two symplectic $G$-manifolds. Let $S_{i}\\subset M_{i}$ be two invariant symplectic submanifolds with invariant symplectic normal bundles $N_{i}$. Suppose that there is an equivariant isomorphism $A:N_{0}\\to N_{1}$ covering an equivariant symplectomorphism $\\phi:S_{0}\\to S_{1}$. Then $\\phi$ extends to a equivariant symplectomorphism of neighborhoods $\\Phi:U_{0}\\to U_{1}$ whose derivative along $S_{0}$ is equal to $A$.\n\\end{lemma}\n\\begin{proof}\nWe can extend the automorphism $A$ to a diffeomorphism of neighborhoods $\\psi:U_{0}\\to U_{1}$ by setting\n\\[\\psi = \\exp\\circ A\\circ \\exp^{-1}\\]\nBy construction, $d\\psi = A$ along $S_{0}$, so that $\\omega_{0}$ and $\\psi^{*}\\omega_{1}$ coincides along $S_{0}$. Applying the $G$- equivariant Moser isotopy lemma gives the result.\n\\end{proof}\n\nLet $(M,\\omega)$ be a symplectic manifold. Let $G$ be a compact lie group acting symplectically on $M$. Let $S$ be an invariant submanifold under the $G$ action. Let $\\Op(S)$ be an invariant open neighbourhood of $S$, Further define\n\n\\[\\Symp^G_{\\id,N}(M,S)=\\{\\phi\\in\\Symp^G_{0}(M)~|~\\phi|_{S}=\\id,~d\\phi|_{T_{S}M}=\\id\\}\\]\n\\[\\Symp^G_{\\id,\\Op(S)}(M,S)=\\{\\phi\\in\\Symp^G_{0}(M)~|~\\phi=\\id~\\text{near}~S\\}\\]\n\nthen we would like to show that  $\\Symp^G_{\\id,N}(M,S) =\\Symp^G_{\\id,\\Op(S)}(M,S) $. But before we do that we would need the following lemmas.\n\nFollowing~\\cite{Hi}, we define a invariant tubular neighborhood of a invariant submanifold $\\iota:S\\hookrightarrow M$ as a smooth equivariant embeddings $f:E\\hookrightarrow M$ of a vector bundle $\\pi:E\\to S$ such that \n\\begin{enumerate}\n\\item $f|_{S}=\\iota$ after identifying $S$ with the zero section of $\\pi:E\\to S$.\n\\item $f(E)$ is an open neighborhood of $S$.\n\\end{enumerate}\nIn practice, it is often enough to work with the normal bundle $N\\subset T_{S}M$ defined as the orthogonal of $T_{S}M$ relative to a equivariant Riemannian structure. (See \\cite{Bredon} p. 306 for existence of such invariant tubular neighbourhood.)\n\n\\begin{lemma}[Unicity of tubular neighborhoods]\\label{UnicityTubularNeighborhoods} (See~\\cite{Hi}, Theorem 4.5.3) Let $M$ be a $G$-manifold, let $\\iota:S\\hookrightarrow M$ be a invariant submanifold with normal bundle $N$. Then, \n\\begin{enumerate}\n\\item given any two invariant tubular neighborhoods $f_{i}:N_i \\hookrightarrow M$, $i=0,1$, there is a equivariant gauge transformation $A\\in\\mathcal{G}(N)$ such that $f_{0}$ and $f_{1}\\circ A$ are equivariant isotopic rel. to $S$. \n\\item The space $\\mathcal{T}_{S}$ of all invariant tubular neighborhoods $f:N\\hookrightarrow M$ is homotopy equivalent to the group of equivariant gauge transformations $\\mathcal{G}(N)$.\n\\item The space $\\mathcal{T}_{S,d\\iota}$ of invariant tubular neighborhoods $f:N\\hookrightarrow M$ such that $df|_{S}=d\\iota$ is contractible.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n(1) We construct an equivariant isotopy $F_{t}$ in two steps. Firstly, given an $G$-invariant smooth function $\\delta:S\\to(0,1]$, let $U_{\\delta}\\subset N$ be the invariant disc bundle\n\\[U_{\\delta}=\\{x\\in N~|~|x|<\\delta(\\pi(x))\\}\\]\nLet k is the rank of the bundle and $C(G)$ denote the centraliser of G in $\\SO(k)$. Note that $N$ smoothly retracts onto $U_{\\delta}$ through embeddings $G_{t}:N\\to N$ of the form\n\\[G_{t} = (1-t)\\id + th_{\\delta(x)}\\]\nwhere $h_{r}:\\mathbb{R}^{k}\\to D^{k}(r)$ is a equivariant one-parameter family of contracting, $C(G) \\subset \\SO(k)$-invariant diffeomorphisms, restricting to the identity on $D^{k}(r\/2)$ and varying smoothly with $r$. Then, choosing an appropriate invariant function $\\delta$, and composing $f_{1}$ with $G_{t}$, we can isotope $f_{1}$ to an embedding $f_{\\delta}=f_{1}G_{1}$ satisfying\n\\begin{equation}\\label{inclusion-assumption}\nf_{\\delta}(N)\\subset f_{0}(N)~\\text{and}~f_{\\delta}=f_{1}~\\text{on}~U_{\\delta\/2}\n\\end{equation}\nso that the map $g=f_{0}^{-1}f_{\\delta}:N\\to N$ is well-defined. Secondly, observe that the map $g$ is equivariantly isotopic to its vertical derivative $A_{f_{\\delta}}=dg^{\\text{vert}}\\in\\mathcal{G}(N)$ along $S$ via the canonical smooth isotopy\n\\[H_{0}(x)=\\phi(x),\\quad H_{t}(x)=g(tx)\/t,~0<t\\leq 1\\]\nNote that H is indeed equivariant. An isotopy from $f_{\\delta}$ to $f_{0}\\circ A_{f_{\\delta}}$ is then given by $f_{0}H_{1-t}$. The sought-for equivariant isotopy $F_{t}$ is the concatenation of $f_{1}G_{t}$ with $f_{0}H_{1-t}$.\\\\\n\n(2) Fix a invariant tubular neighborhood $f_{0}:N\\hookrightarrow M$ and choose once for all a smooth family of equivariant diffeomorphisms $h_{r}:\\mathbb{R}^{k}\\to D^{k}(r)$ as in the proof of (1). Given any other invariant tubular neighborhood $f:N\\hookrightarrow M$, the isotopy constructed in (1) only depends on an auxiliary invariant function $\\delta:S\\to\\mathbb{R}_{+}$. Although the choice of this function is not canonical, the set $\\Delta^G(f)$ of all invariant $\\delta$ for which $h_{\\delta}(N)\\subset f^{-1}f_{0}(N)$ is a convex subset of $C^{\\infty,G}(S,\\mathbb{R}_{+})$, where $C^{\\infty,G}(S,\\mathbb{R}_{+})$ denotes the space of $G$ invariant smooth functions to $\\mathbb{R}_+$. Consequently, the projection of the fibre product\n\\[\\mathcal{T}_{\\Delta}=\\{(f,\\delta)~|~f\\in\\mathcal{T}_{S},~\\delta\\in\\Delta^G(f)\\}\\to\\mathcal{T}_{S}\\]\nis a homotopy equivalence. Consider the embedding\n\\begin{align}\n\\phi:\\mathcal{G}(N)\\times C^{\\infty,G}(S,\\mathbb{R}^{+})&\\hookrightarrow \\mathcal{T}_{\\Delta}\\\\\n(A,\\delta)&\\mapsto (f_{0}\\circ A, \\delta)\n\\end{align}\nand the continuous map\n\\begin{align}\n\\psi:\\mathcal{T}_{\\Delta}&\\to \\mathcal{G}(N)\\times C^{\\infty,G}(S,\\mathbb{R}_{+})\\\\\n(f,\\delta)&\\mapsto (A_{f_{\\delta}},\\delta)\n\\end{align}\nThen we have $\\psi\\phi=\\id_{\\mathcal{G}(N)\\times C^{\\infty,G}(S,\\mathbb{R}_{+})}$, while $\\phi\\psi(f,\\delta)=(f_{0}\\circ A_{f_{\\delta}},\\delta)$, the map $f_{0}\\circ A_{f_{\\delta}}$ being the terminal point of the isotopy defined in (1). This shows that $\\phi$ and $\\psi$ are homotopy inverses.\\\\\n\n(3) Choosing $f_{0}$ such that $df_{0}=d\\iota$ along $S$, this immediately follows from the fact that the space $\\mathcal{T}_{S,d\\iota}$ is homotopy equivalent to the subspace of $\\mathcal{T}_{\\Delta}$ that retracts to the contractible subspace $\\{f_{0}\\}\\times C^{\\infty,G}(S,\\mathbb{R}_{+})$ under the isotopy defined in~(1).\n\\end{proof}\n\n\\begin{lemma}{\\label{germ}}\nThe inclusion $\\Symp^G_{\\id,\\Op(S)}(M,S)\\hookrightarrow\\Symp^G_{\\id,N}(M,S)$ is a homotopy equivalence.\n\\end{lemma}\n\\begin{proof}\nWe follow the same ideas as in the non-equivariant case. Consider the short exact sequence\n\\[\\Symp^G_{\\id,\\Op(S)}(M,S)\\hookrightarrow\\Symp^G_{\\id,N}(M,S)\\to\\mathcal{G}_{S,\\omega}\\]\nwhere the group $\\mathcal{G}_{S,\\omega}:=\\Symp^G_{\\id,N}(M,S)\/\\Symp^G_{\\id,\\Op(S)}(M,S)$ is the group of germs along $S$ of equivariant symplectomorphisms $\\phi\\in\\Symp_{\\id,N}(M,S)$. We wish to show that $\\mathcal{G}_{S,\\omega}$ is contractible.\n\\\\\n\nChoose a compatible $G$-invariant almost-complex structure $J$ and let $g$ be the associated equivariant metric. Let $N$ be the symplectic orthogonal complement of $TS$ in $TM$. $N$ is invariant under the $G$ action. Equip $N$ with the minimal coupling form $\\Omega$ and choose $\\epsilon>0$ so that the $\\epsilon$-disk subbundle $V_{\\epsilon}\\subset N$ is symplectomorphic to a tubular neighborhood $U$ of~$S$. Let $\\iota$ denote both inclusions $U\\hookrightarrow M$ and $V\\hookrightarrow N$.\n\\\\\n\nLet $\\Omega_{S}^{\\mathrm{loc},G}$ be the space of germs of $G$ invariant symplectic forms defined near $S$ and agreeing with $\\omega$ along $T_{S}M$. Given any two germs $[\\omega_{0}]$ and $[\\omega_{1}]$, their linear convex combination $\\omega_{t}=(1-t)\\omega_{0}+t\\omega_{1}$ is non-degenerate in some neighborhood of $S$. Consequently, $\\Omega_{S}^{\\mathrm{loc},G}$ is convex, hence contractible. By the Symplectic neighborhood theorem, the group $\\mathcal{G}_{S,\\omega}$ acts transitively on $\\Omega_{S}^{\\mathrm{loc},G}$, giving rise to a fibration\n\\[\\mathcal{G}_{S,\\omega}^{\\mathrm{loc},G}\\stackrel{\\simeq}{\\to}\\mathcal{G}_{S,\\omega}\\to\\Omega_{S}^{\\mathrm{loc},G}\\]\nwhose fiber $\\mathcal{G}_{S,\\omega}^{\\mathrm{loc},G}$ is the group of germs of equivariant diffeomorphisms that are symplectic near $S$. This space is homeomorphic to the space $\\mathcal{E}_{S,\\omega}$ of germs along $S$ of equivariant symplectic embeddings $f:\\Op(S)\\to M$ such that $f|_{S}=\\iota$ and $df|_{S}=d\\iota$. By Lemma~\\ref{UnicityTubularNeighborhoods} (3), we know that $\\mathcal{E}_{S,\\omega}$   is contractible, so that $\\mathcal{G}_{S,\\omega}$ and $\\mathcal{G}_{S,\\omega}^{\\mathrm{loc},G}$   are also contractible, thus completing the proof. \n\\end{proof}\n\n\\begin{lemma}\\label{Au} Let $G$ be a compact group acting symplectically on a compact manifold $(M,\\omega)$. Let $W_t$ be a smooth $k$-parameter family of symplectic submanifolds ($t \\in [0,1]^k$), which are invariant under the $G$ action. Then there exists a  $k$-parameter family of equivariant Hamiltonian symplectomorphisms $\\phi_{t}: M \\rightarrow M$ such that $\\phi_t(W_0) = W_t$\n\\end{lemma}\n\n\\begin{proof}\nThe proof follows by mimicking the proof of Proposition 4 in \\cite{Auroux} under the presence of a group action.\n\\end{proof}\n\nFinally, we present the following theorem due to Palais, that we repeatedly use in Chapters 3 and 6 to justify that our maps between infinite dimensional spaces is a fibration. \n\nLet $X$ be a topological space with an action of a topological group $G$. We say X admits local cross sections at $x_0 \\in X$ if for there is a neighbourhood $U$ containing $x_0$ and a map $\\chi: X \\rightarrow G$ such that $\\chi(u) \\cdot x_0 = u$ for all $u \\in U$.  We say X admits local cross sections if this is true for all $x_0 \\in X$. \n\n\\begin{thm}{(Palais)}\\label{palais}\nLet $X$, $Y$ be a topological spaces with a action of a topological group $G$. Let the $G$ action on $X$ admit local cross sections. Then any equiariant map $f$ from another space $Y$ to $X$ is locally trivial.\n\\end{thm}\n\n\\begin{proof}\nSuppose for every point $x_0 \\in X$ there is a local section $\\chi: U \\rightarrow G$ where $U$ is an open neighbourhood of $x_0$. Then we define a local trivialisation of $f$ as follows. \n\n\\begin{align*}\n    \\rho: U \\times f^{-1}(x_0) &\\rightarrow f^{-1}(U) \\\\\n     (u, \\gamma) &\\mapsto \\chi(u) \\cdot \\gamma\n\\end{align*}\n\nAs $f$ is equivariant we indeed have $f(\\rho(u,\\gamma)) = f( \\chi(u) \\cdot \\gamma) = \\chi(u) \\cdot f(\\gamma) = \\chi(u) \\cdot x_0 = u$, where the last equality follows from the definition of being a local section. Thus $\\rho$ maps $U \\times  f^{-1}(x_0) $ into $ f^{-1}(U)$.\\\\\n\nConversely there is map \n\n\\begin{align*}\n   \\beta: f^{-1}(U) &\\rightarrow U \\times f^{-1}(x_0) \\\\\n   y &\\mapsto \\left(f(y), {\\chi(f(y))}^{-1} \\cdot y\\right)\n\\end{align*}\n\nWe can indeed check that the two maps are inverses of each other. \n\n$\\beta \\circ \\rho (u, \\gamma) = \\beta(\\chi(u) \\cdot \\gamma) = \\left(\\chi(u) \\cdot \\gamma, {\\chi(\\chi(u) \\cdot f(\\gamma))}^{-1} \\cdot \\chi(u) \\cdot \\gamma \\right) = (u, {\\chi(u)}^{-1} \\chi(u) \\cdot \\gamma) = (u,\\gamma)$.\n\nSimilarly we can check that $\\rho \\circ \\beta = id$, thus completing the proof. \n\\end{proof}\n\n\\chapter{Alexander-Eells isomorphism}\\label{Appendix-Alexander-Eells}\n\n\nIn this appendix, we prove the Alexander-Eells isomorphism used in Section~\\ref{subsection:CohomologyModule}. We first recall an isomorphism between the homology of a submanifold $Y\\subset X$ and the homology of its complement $X-Y$ that is reminiscent of the Alexander-Pontryagin duality in the category of oriented, finite dimensional manifolds. This isomorphism, due to J. Eells, exists whenever the submanifold $Y$ is co-oriented and holds, in particular, for infinite dimensional Fr\u00e9chet manifolds. We then give a geometric realization of this isomorphism in the special case $Y$ and $X-Y$ are orbits of a continuous action $G\\times X\\to X$ satisfying some mild assumptions. We closely follow Eells~\\cite{Eells} p. 125--126.\\\\\n\nLet $X$ be a manifold, possibly infinite dimensional, and let $Y$ be a co-oriented submanifold of positive codimension $p$. As explained in~\\cite{Eells}, there exists an isomorphism of singular cohomology groups\n\\[\\phi:H^{i}(Y)\\to H^{i+p}(X, X-Y)\\]\ncalled the Alexander-Eells isomorphism. We define the fundamental class (Thom class) of the pair $(X,Y)$ as $u=\\phi(1)\\in H^{p}(X, X-Y)$.\n\\begin{prop}[Eells, p. 113]\nThe pairing\n\\[\\begin{aligned}\nH^{*}(Y)\\otimes H^{*}(X,X-Y)&\\to H^{*}(X,X-Y)\\\\\ny\\otimes x &\\mapsto y\\cup x\n\\end{aligned}\\]\nmakes $H^{*}(X,X-Y)$ into a free $H^{*}(Y)$-module of rank one, generated by $u$.\n\\end{prop}\n\n \n\nLet $\\phi_{*}:H_{i+p}(X,X-Y)\\to H_{i}(Y)$ be the dual of the Alexander-Eells isomorphism $\\phi^{*}=\\phi$. By definition, we have\n\\[\\phi_{*}(a)=u\\cap a\\]\n\nSuppose a topological group $G$ acts continuously on $X$ (on the left), leaving $Y$ invariant, and in such a way that both $X-Y$ and $Y$ are homotopy equivalent to orbits. We have continuous maps $\\mu:G\\times (X,X-Y)\\to (X,X-Y)$, ~$\\mu:G\\times (X-Y)\\to (X-Y)$, and $\\mu:G\\times Y\\to Y$ inducing $H_{*}(G)$-module structures on $H_{*}(X,X-Y)$, $H_{*}(X-Y)$ and $H_{*}(Y)$. We write $\\mu_{*}(c\\otimes a)=c*a$ for the action of $c\\in H_{i}(G)$.\n\n\\begin{lemma}\\label{lemma:AlexanderEellsGmodule}\nIn this situation, the Alexander-Eells isomorphism preserves the $H_{*}(G)$-module structure, that is, the following diagram is commutative:\n\\[\n\\begin{tikzcd}\nH_{*}(G)\\otimes H_{*}(X,X-Y) \\arrow{r}{\\mu_{*}} \\arrow[swap]{d}{1\\otimes\\phi_{*}} & H_{*}(X,X-Y) \\arrow{d}{\\phi_{*}} \\\\\nH_{*}(G)\\otimes H_{*}(Y) \\arrow{r}{\\mu_{*}} & H_{*}(Y)\n\\end{tikzcd}\n\\]\nThus for any $a\\in H_{i+p}(X,X-Y)$, $c\\in H_{i}(G)$, we have $\\phi_{*}(c*a) = c*\\phi_{*}(a)$.\n\\end{lemma}\n\\begin{proof}\nWe first note that if $u$ is the fundamental class of the pair $(X,Y)$, then $\\mu^{*}(u)=1\\otimes u\\in H^{0}(G)\\otimes H^{p}(X,X-Y)$, because $H^{i}(X,X-Y)=0$ for all $i<p$. The cap product is a bilinear pairing\n\\[\n\\cap: H^{*}\\big(G\\times (X,X-Y)\\big)\\otimes H_{*}\\big(G\\times (X,X-Y)\\big) \\to H_{*}\\big(G\\times X\\big)\n\\]\nthat is adjoint to the cup product. Using the relative form of K\\\"unneth theorem, this bilinear map defines a pairing\n\\[\n\\cap:\\big(H^{*}(G)\\otimes H^{*}(X,X-Y)\\big)\\otimes\\big(H_{*}(G)\\otimes H_{*}(X,X-Y)\\big)\\to H_{*}(G)\\otimes H_{*}(X)\n\\]\nLet's write $j^{*}:H^{*}(X,X-Y)\\to H^{*}(X)$.  Then, for any $c\\otimes x\\in H^{*}(G)\\otimes H^{*}(X,X-Y)$, and any $b\\otimes a\\in H_{*}(G)\\otimes H_{*}(X,X-Y)$ we have\n\\[\\begin{aligned}\n\\big\\langle c\\otimes j^{*}x, \\mu^{*}(u)\\cap (b\\otimes a)\\big\\rangle \n&= \\big\\langle (c\\otimes x)\\cup\\mu^{*}(u),b\\otimes a\\big\\rangle\\\\\n&= \\big\\langle (c\\otimes x)\\cup(1\\otimes u),b\\otimes a\\big\\rangle\\\\\n&= \\big\\langle (c\\cup1)\\otimes (x\\cup u),b\\otimes a\\big\\rangle\\\\\n&= \\big\\langle c, b\\big\\rangle \\big\\langle (x\\cup u),a\\big\\rangle\\\\\n&= \\big\\langle c,b\\big\\rangle \\big\\langle j^{*}x, u\\cap a\\big\\rangle\\\\\n&=\\big\\langle c\\otimes j^{*}x, b\\otimes (u\\cap a)\\big\\rangle\n\\end{aligned}\\]\nIt follows that $\\mu^{*}(u)\\cap (b\\otimes a) = b\\otimes (u\\cap a) = b\\otimes \\phi_{*}(a)$. We then compute\n\\[\\phi_{*}(\\mu_{*}(c\\otimes x)) \n=u\\cap\\mu_{*}(c\\otimes x) \n=\\mu_{*}\\big(\\mu^{*}(u)\\cap(c\\otimes x)\\big)\n=\\mu_{*}\\big(c\\otimes\\phi_{*}(x)\\big)\n=\\mu_{*}\\big((1\\otimes \\phi_{*})(c\\otimes x)\\big)\\]\nwhich is the desired relation.\n\\end{proof}\nSince $H_{i}(X,X-Y)=0$ for $i<p$, the Universal Coefficient Theorem yields a canonical isomorphism $\\beta:H^{p}(X,X-Y)\\to \\Hom(H_{p}(X,X-Y);\\mathbb{Z})$. If $Y$ is connected, $H_{p}(X,X-Y)$ is of rank one. In this case, define $a_{u}\\in H_{p}(X,X-Y)$ as the unique class such that $\\beta(u)a_{u}=1$. Suppose the Leray-Hirsch theorem applies to the evaluation fibration $G\\to Y$. Then, $H_{*}(X,X-Y)$ becomes a $H_{*}(Y)$-module by identifying $H_{*}(Y)$ with $1\\otimes H_{*}(Y)\\subset H_{*}(G)$ and by setting\n\\[b*x := [1\\otimes b]*x\\]\n\\begin{thm}\\label{thm:AlexanderEells}\nUnder the above assumptions, the isomorphism $\\psi_{*}=\\phi_{*}^{-1}:H_{i}(Y)\\to H_{i+p}(X,X-Y)$ is given by $\\psi_{*}(y)=y*a_{u}$. Thus $H_{*}(X,X-Y)$ is generated by $a_{u}$ as a $H_{*}(Y)$-module.\n\\end{thm}\n\\begin{proof}\nSince\n\\[\\langle 1, \\phi_{*}(a_{u})\\rangle\n=\\langle 1, u\\cap a_{u}\\rangle \n=\\langle 1\\cup u, a_{u}\\rangle\n=\\langle u,a_{u}\\rangle=1\\]\nit follows that $\\phi_{*}(a_{u})=1$, which is equivalent to $\\psi_{*}(1)=a_{u}$. Since $\\phi_{*}$ is an isomorphism of $H_{*}(G)$-modules, its inverse is also an isomorphism of $H_{*}(G)$-modules. We can then write\n\\[\\psi_{*}(y) = \\psi_{*}(y*1) = y*\\psi_{*}(1) = y*a_{u}\\]\n\\end{proof}\nFrom the naturality of the connecting homomorphism $\\partial$ in the long exact sequence of the pair $(X, X-Y)$, we get\n\\begin{cor}\\label{cor:AlexanderEells}\nSuppose in addition to the above hypotheses that $X$ is contractible. Then the isomorphism \n\\[\\lambda_{*}=\\partial\\circ\\psi_{*}: H_{i}(Y)\\to H_{i+p-1}(X-Y)\\]\nis given by $\\lambda_{*}(y) = y*x_{u}$, where $x_{u}=\\partial a_{u}$.\n\\end{cor}\n\nWe now apply the Alexander-Eells isomorphism to the situation of Section~\\ref{section:TwoOrbits}. Let $G$ be the centralizer $\\Symp_{h}^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$, let $X=\\mathcal{J}^{S^1}_{\\om_\\lambda}$ be the contractible space of invariant, compatible,  almost-complex structures, and let $Y$ be the codimension 2 stratum $\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'}$. Let $p_m$ denote the map \n\\[p_m: \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\\rightarrow \\Symp^{S^1}_h(S^2 \\times S^2,\\omega_\\lambda)\/\\mathbb{T}^2_m \\simeq \\mathcal{J}^{S^1}_{\\om_\\lambda} \\cap U_{m}\\]\n\nFurther let $y_1$ and $y_2$ denote the generators of the Pontryagin algebra $H_{*}(T^{2}_{m'};k)$.  Note that under the inclusion $i_m: \\mathbb{T}^2_m \\hookrightarrow \\Symp_{h}^{S^1}(S^2 \\times S^2,\\omega_\\lambda)$, the elements $y_1$ and $y_2$ correspond to the circle actions $(1,0;m)$ and $(0,1;m)$ respectively. \nThe connecting isomorphism $\\partial:H_{2}(\\mathcal{J}^{S^1}_{\\om_\\lambda}, \\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m})\\to H_{1}(\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m})$ maps the generator $a_{u}$ to the link of $\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'}$ in $\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m}$, that is, to the loop $p_{m}(y_{2})$ generated by the action of $y_{2}\\subset G$. Consequently, Corollary~\\ref{cor:AlexanderEells} immediately implies the following geometric description of the Alexander-Eells isomorphism.\n\\begin{prop}\\label{prop:AlexanderEellsGeometric}\nThe Alexander-Eells isomorphism\n\\[\\lambda_{*}:H_{p}(\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m'})\\to H_{p+1}(\\mathcal{J}^{S^1}_{\\om_\\lambda}\\cap U_{m})\\]\nis given by\n\\[\\lambda_{*}(y)= y*p_{m}(y_{2})=y*y_{2}*1=[y_{2}\\otimes y]*1=\\mu_{m}\\big([y_{2}\\otimes y]\\otimes 1\\big)=p_{m}\\big([y_{2}\\otimes y]\\big)\\]\nIn particular, if $p_{m'}(\\tilde y) = y\\in H_{*}(Y)$, then $\\lambda_{*}(y)=\\tilde y*y_{2}*1=p_{m}(\\tilde y* y_{2})$. \n\\end{prop}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\nQuantum chromodynamics (QCD) predicts that a new form of matter, consisting of\ndeconfined quarks and gluons, is formed at very high temperatures. Calculations\nin lattice QCD~{\\cite{Karsch:2003jg} indicate that the transition to this\nso-called quark-gluon plasma should occur at a critical temperature of around 150--175\\MeV, corresponding to an energy density of about 1~GeV\/fm$^3$.\nExperiments have provided evidence that dense matter at high temperature can be\ncreated in relativistic heavy ion\ncollisions~\\cite{Adcox:2004mh,Adams:2005dq,Back:2004je,Arsene:2004fa}.\n\nStudies of particle production at high transverse momentum (\\pt) are a well-established way to\nprobe the properties of the medium formed in heavy-ion collisions. The yields and correlations of high momentum particles are modified through the ``jet-quenching'' effect,\nresulting from the energy loss suffered\nby hard-scattered partons passing through the medium~\\cite{Bjorken:1982tu}.\nFast parton energy loss provides key information on the thermodynamic and transport properties of the medium traversed~\\cite{CasalderreySolana:2007zz,dEnterria:2009am}. Evidence for jet quenching was first observed in the suppression of inclusive high-$\\pt$ hadron production and the modification of high-$\\pt$ dihadron angular correlations in nucleus-nucleus collisions at the Relativistic Heavy Ion Collider, in comparison to proton-proton collisions ~\\cite{Adams:2005dq,Adcox:2004mh,Arsene:2004fa,Back:2004je}.\n\nRecent results at the LHC~\\cite{Chatrchyan:2011sx,Aad:2010bu,Alice:dihadron,HIN-10-005,Aamodt:2010jd}, using\nfully reconstructed jets, correlations between jets and single particles, and charged particle measurements,\nprovide detailed information on the jet-quenching effect. For central collisions,\na large broadening of the dijet momentum asymmetry distributions is observed, consistent with\ntheoretical calculations that involve differential energy loss of back-to-back hard-scattered\npartons as they traverse the medium~\\cite{He:2011pd,Young:2011qx,Qin:2010mn}.\nAt the same time, angular correlations between the jets are found to be almost unchanged, ruling\nout single-hard-gluon radiation as the leading energy loss mechanism. Studies of jet-hadron correlations,\ninvolving vector summation of charged hadron momenta,\nfind that the energy balance in events with large dijet asymmetry is recovered on average\nby an excess of low-momentum particles in the hemisphere of the away-side jet,\nat large angles relative to the jet axes~\\cite{Chatrchyan:2011sx}. These results constrain the mechanism of parton energy loss~\\cite{CasalderreySolana:2010eh,Chesler:2011nc}.\nFurther understanding of this mechanism requires the measurement of the \\pt dependence of the\nobserved effects.\n\nThe dijet analysis presented in this paper uses a large dataset of PbPb collisions\nat a nucleon-nucleon center-of-mass energy of $\\rootsNN=2.76\\TeV$, collected with the Compact Muon Solenoid\n(CMS) detector in 2011.\nThe CMS detector has a solid-angle acceptance of nearly 4$\\pi$ and is designed to measure jets and energy flow, an excellent feature for studying heavy ion collisions.\nWith a total integrated luminosity of 150$\\pm$8 $\\mu$b$^{-1}$ this\ndataset provides a significantly larger data sample than the 6.8$\\,\\mu$b$^{-1}$ analyzed in 2010, allowing an extension of previous studies to more peripheral collision\nevents and to jets of transverse momenta in excess of 350\\GeVc.\nAdditionally, this paper also presents  a dijet analysis of pp collisions recorded at $\\sqrt{s}=2.76$\\TeV by CMS in 2011, with a total integrated\nluminosity of 231\\nbinv.\n\n\n\n\\section{Experimental method}\n\\label{sec:experimental_method}\n\\subsection{The CMS detector}\n\\label{sec:cms}\n\nThe CMS detector is described in detail\nelsewhere~\\cite{bib_CMS}.  The calorimeters provide hermetic coverage over a large\nrange of pseudorapidity, $|\\eta| < 5.2$, where $\\eta = -\\ln [\n\\tan(\\theta\/2)]$ and $\\theta$ is the polar angle relative to the counterclockwise\nion beam (the $z$ axis).\nA steel and quartz-fiber Cherenkov\ncalorimeter, called hadron forward (HF), covers the high pseudorapidity range\n$3<|\\eta|<5.2$ and is used to determine the centrality of the PbPb collision.\nHadron calorimeter (HCAL) cells are grouped in projective\ntowers of granularity\n$\\Delta \\eta \\times \\Delta \\phi =0.087\\times0.087$ (where $\\phi$ is the azimuthal angle) at central pseudorapidities, having a segmentation about twice as large at forward\npseudorapidities. The electromagnetic calorimeter (ECAL) has a further segmentation of 5$\\times$5 within a tower, and the signals in these cells are clustered together to reconstruct photons. The central calorimeters are embedded in a 3.8\\unit{T} axial magnetic field produced by a superconducting solenoid.\nThe CMS tracking system, located inside the calorimeter, consists of silicon-pixel and silicon-strip layers covering  $|\\eta| <\n2.5$. This analysis uses tracks reconstructed down to transverse momenta of $900$\\MeVc in PbPb collisions, with\na track momentum resolution of about 1\\% at $\\pt = 100$\\GeVc. At high \\pt, the efficiency of the tracking is not strongly \\pt\\ dependent, and for 100\\GeVc tracks it varies from 65\\% in peripheral collisions to 60\\% in central collisions. A set of scintillator tiles, the beam scintillator counters (BSC), used for triggering and beam-halo rejection, is mounted on the inner side of the HF calorimeters.\nThe detailed Monte Carlo (MC) simulation of the CMS detector response is based on {\\GEANTfour}\n\\cite{bib_geant}.\n\n\\subsection{Jet reconstruction in PbPb collisions}\n\\label{sec:jet_reconstruction}\n\nJets are reconstructed using the CMS ``particle-flow\" algorithm~\\cite{CMS-PAS-PFT-10-002,MattPFlow}.\nThis algorithm attempts to identify all stable particles in an\nevent (electrons, muons, photons, charged and neutral hadrons)\nby combining information from all sub-detector systems.\nThe \\ak\\ sequential recombination algorithm, as encoded in the\n{\\textsc{FastJet}} framework, is used to combine the particle-flow candidates\ninto jets using a distance parameter R = 0.3~\\cite{Cacciari:2008gp}.\n\nThe small value of R helps to reduce the deterioration of the jet energy\nresolution in PbPb collisions due to\nfluctuations of the background from soft interactions.\nThe underlying\nbackground from soft collisions is subtracted using the same method\nas employed in~\\cite{Chatrchyan:2011sx} and originally described in~\\cite{Kodolova:2007hd}.\nThis algorithm is a variant of an iterative ``noise\/pedestal subtraction''\ntechnique, where the mean and dispersion of the energies detected in rings of constant $\\eta$ are subtracted from the jet. The jet and background energies are formed from the total energy, determined by particle-flow, within projective towers with the same segmentation as the HCAL, as described in Section~\\ref{sec:cms}.\n\nThe jets reconstructed with the procedure above are then corrected to final state particle jets.\nCMS uses a factorized multi-step approach to correct the jet energies~\\cite{Chatrchyan:2011ds}. For this analysis, jet energy\ncorrections are derived from \\PYTHIA~\\cite{bib_pythia} simulations without PbPb underlying events. Jets reconstructed\nwith the particle-flow algorithm using heavy-ion tracking~\\cite{HIN-10-005} require different corrections\nthan those derived with the tracking algorithm optimized for pp data, due to the difference of tracking efficiencies.\n\nThe pp sample is reconstructed with the same tracking, particle flow and jet algorithms as those used in the analysis of the PbPb data. Although the underlying event and pile-up in pp collisions at $\\sqrt{s}=2.76$\\TeV is small enough not to require a subtraction method, the jet algorithm works successfully in both types of environment.\n\n\n\\subsection{Data samples and triggers}\n\nFor online event selection,\nCMS uses a two-level trigger system:\na hardware-based level-1 trigger and a software-based high level trigger (HLT). The events used in this analysis are\nselected by an inclusive single-jet trigger that required\na calorimeter based jet reconstructed in HLT with  $\\pt > 80$\\GeVc , where the jet $\\pt$ value is\ncorrected for the $\\pt$-dependent calorimeter energy response.\nThe trigger efficiency is defined as the fraction\nof triggered events out of a sample of minimum bias events (described below)\nin bins of offline reconstructed leading-jet $\\pt$.\nThe trigger becomes fully efficient for collisions with a leading particle-flow jet\nwith corrected $\\pt$ greater than 100\\GeVc.\n\nIn addition to the jet data sample, a minimum bias event sample was collected using\ncoincidences between the trigger signals from both the $+z$ and $-z$ sides of either the BSC\nor the HF, which was pre-scaled to record only about 0.1--0.2\\% of the collisions delivered by the LHC.\nIn order to suppress non-collision-related noise, cosmic-ray muons,\nout-of-time triggers, and beam backgrounds, the minimum bias and jet triggers used in this analysis\nwere required to arrive in time with the presence of both colliding ion bunches in the interaction region.\nThe events selected by the jet trigger described above also satisfy all triggers and selections\nimposed for minimum bias events.\n\n\\subsection{Event selection and centrality determination}\n\\label{sec:event_selection}\nA sample of inelastic hadronic collisions is selected offline from the triggered events. Contamination from beam-halo events is removed based upon the timing of the $+z$ and $-z$ BSC signals. A requirement of a reconstructed primary collision vertex based on at least two tracks with transverse momenta above $75$~\\MeVc is imposed. This requirement removes other beam related background events (e.g., beam-gas, ultraperipheral collisions) with large HF energy deposits but very few pixel detector hits. The vertex is required to be compatible with the length of the pixel clusters reconstructed in the event, as a standard method in CMS~\\cite{Khachatryan:2010us}. Finally, an offline HF coincidence is applied, which requires at least three towers on each side of the interaction point in the HF with at least 3~GeV total deposited energy per tower. This event selection, including the minimum bias trigger, has an efficiency of 97\\% with an uncertainty of 3\\% for hadronic inelastic PbPb collisions. This efficiency is\ntaken into account in the centrality determination, and the uncertainty of the efficiency has a negligible effect\non the results of this study.\n\nTable~\\ref{evselcuts} shows the number of events remaining after the various selection criteria are applied. Events with a jet trigger of $\\pt > 80$\\GeVc are selected, followed by the offline event selection for inelastic hadronic collisions (described above). Prior to jet finding on\nthe selected events, a small contamination of\nnoise events from the electromagnetic calorimeter and hadron calorimeter is removed using signal\ntiming, energy distribution, and pulse-shape information \\cite{ref:EGM-10-002,Chatrchyan:2009hy}. The leading and subleading jets are determined among the jets with pseudorapidity $|\\eta| < 2$, which are reconstructed as described in Section~\\ref{sec:jet_reconstruction}.\nEvents are then selected if the corrected jet \\pt is larger than  $120$\\GeVc (corrected\nfor the $\\pt$- and $\\eta$-dependent detector energy response). The subleading jet in the\nevent is required to have a corrected jet $\\pt > 30$\\GeVc.\nThe azimuthal angle between the leading and the subleading jets is required to be at least $2\\pi\/3$.\nFurther jets found in the event, beyond the leading and the subleading ones, are not considered in this analysis.\nIn order to remove events with residual HCAL noise that are missed by the noise-rejection algorithms,\neither the\nleading or subleading jet is required to have at least one track of $\\pt > 4$\\GeVc. For high-\\pt jet events\nthis selection does not introduce any significant bias on the sample and removes only 2\\% of the\nselected dijet events.\n\nThe centrality of the collisions is represented by the number of participating nucleons (\\npart) in a collision, which is correlated with the total transverse energy measured in HF. The minimum bias event sample is divided into constant fractions of total inelastic cross section and for each fraction the average value of \\npart\\ is determined using a Glauber calculation~\\cite{Miller:2007ri}. The dispersion of the \\npart\\ values due to reconstruction effects is based on {\\GEANTfour} simulations of events generated with a multi-phase transport {\\textsc{ampt}} simulation~\\cite{Lin:2004en}.\n\n\n\\begin{table*}[htbp]\n\\begin{center}\n\\caption{The effects of various selections applied to the data sample. In the third column, the\nfractional values are with respect to the line above and in the fourth column they are with respect to the triggered sample. The selections are applied in\nsequence.}\n\\label{evselcuts} \\begin{tabular}{|l|r|r|r|}\n\\hline\nSelections & Events remaining & \\% of previous & \\% of triggered \\\\\n\\hline\nJet triggered events ($\\pt^{\\text{corr}}>80$\\GeVc) & 369\\,938 & 100.00 & 100.00  \\\\\nOffline collision selection  & 310\\,792 & 84.01 & 84.01  \\\\\nHCAL and ECAL noise rejection   & 308\\,453 & 99.25 & 83.38  \\\\\nLeading jet $\\ptlead>120$\\GeVc  & 55\\,911 & 18.13 & 15.11  \\\\\nSubleading jet $\\ptsub>30$\\GeVc    & 52\\,694 & 94.25 & 14.24  \\\\\n\\dphi $>2\\pi\/3$ &  49\\,993 & 94.87 & 13.51  \\\\\nTrack within a jet &   49\\,054 & 98.12 & 13.26  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\subsection{Simulated data samples}\n\\label{sec:pythia_samples}\n\n\nIn PbPb collisions there is a high multiplicity of soft particles produced, the PbPb underlying event. It is essential to understand how the jet reconstruction is modified in PbPb collisions at different centralities. This is studied with simulations of dijet events in pp collisions with the \\PYTHIA event generator (version 6.423, tune Z2)~\\cite{bib_pythia},\nmodified for the isospin content of the colliding nuclei. A minimum hard-interaction scale ($\\hat{p}_\\mathrm{T}$) selection of 80\\GeVc\\ is used to increase the number of dijet events produced in the momentum range studied. \\PYTHIA simulations at lower $\\hat{p}_\\mathrm{T}$ (discussed in~\\cite{Cacciari:2011tm}) are also investigated and found to agree with the $\\hat{p}_\\mathrm{T} > 80$\\GeVc\\ results within the uncertainties. To model the PbPb background, minimum bias \\PbPb\\ events are simulated with the \\HYDJET event generator~\\cite{Lokhtin:2005px}, version 1.8 (denoted \\PYTHYD in this paper).\nThe parameters of \\HYDJET are tuned to reproduce the total particle multiplicities, charged hadron spectra, and elliptic flow at all centralities, and to approximate\nthe underlying event fluctuations seen in data, differences being within the underlying event systematic uncertainty.\n\nThe full detector simulation and analysis chain is used to process both \\PYTHIA dijet events and \\PYTHIA dijet events embedded into \\HYDJET events. The reconstruction of particle flow jets is studied by using the \\PYTHIA generator jet information in comparison to the same fully reconstructed jet in \\PYTHYD, matched in momentum space. The effects of the PbPb underlying event on jet \\pt\\ and position resolution, jet \\pt\\ scale, and jet-finding efficiency are determined as a function of collision centrality and jet \\pt. These effects do not require corrections on the results but contribute to the systematic uncertainties.\n\\section{Results}\n\\label{sec:results}\n\nThe goal of this analysis is to characterize possible modifications of dijet event properties as a function\nof centrality and leading jet transverse momentum in \\PbPb\\ collisions.\nThe analysis is performed in six bins of collision centrality: 0--10\\%, 10--20\\%, 20--30\\%, 30--50\\%, 50--70\\%, and 70--100\\%, the latter being the most peripheral bin. The 0--20\\% most central events are further analyzed in bins of leading jet $\\pt$: 120--150, 150--180, 180--220, 220--260, 260--300, 300--500\\GeVc.\nThroughout the paper, the results obtained\nfrom \\PbPb\\ data are compared to references based on the \\PYTHYD\nsamples described in Section~\\ref{sec:pythia_samples}. The subscripts $1$ and $2$ in the kinematical quantities always refer to the leading and subleading jets, respectively.\n\n\n\\subsection{Dijet azimuthal correlations}\n\\label{sec:dphi}\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{dijet_dphi_all_pt_0to20_20120104}\n\\end{center}\n\\caption{Distribution of the angle $\\dphi$ between the leading and subleading jets\nin bins of leading\njet transverse momentum from\n120 $ < \\ptlead < 150$\\GeVc\\ to $\\ptlead > 300$\\GeVc\\ for\n  subleading jets of $\\ptsub> 30$\\GeVc.\nResults for 0--20\\% central PbPb events are shown as points while the histogram\nshows the results for\n{\\sc {pythia}} dijets embedded into \\HYDJET PbPb simulated events. The error bars represent the statistical uncertainties.}\n\\label{fig:dphiPt}\n\\end{figure*}\n\nEarlier studies of the dijet events in heavy-ion collisions~\\cite{Chatrchyan:2011sx,Aad:2010bu} have shown persistency in dijet azimuthal correlations despite the asymmetry in dijet momenta. This aspect is crucial\nin the interpretation of energy loss observations~\\cite{CasalderreySolana:2011rq}.\nTo understand the momentum dependence of the quenching effects, this study investigates the angular correlation, \\ie, the opening azimuthal angle, \\dphi, between the leading and subleading jets of the events, in bins of leading jet $\\ptlead$.\n\nFor events with 0--20\\% centrality, two features are visible in the \\dphi\\ distributions shown in Fig.~\\ref{fig:dphiPt}: a peaking structure at $\\dphi = \\pi$, and a constant offset from zero in the overall distribution. The distribution around the $\\dphi = \\pi$ peak reflects the back-to-back dijet production and although this distribution changes across the various leading-jet \\pt\\ bins, there is no significant difference between PbPb data and the \\PYTHYD sample.\nThis observation confirms the conclusions of earlier studies~\\cite{Chatrchyan:2011sx,Aad:2010bu}, extending the analysis to differential leading-jet \\pt bins.\nThe event fraction that extends to small \\dphi\\ values is likely due to the matching of the leading jet with a random underlying event fluctuation instead of the true subleading jet partner. The difference in the rate of such events between the PbPb data and the \\PYTHYD sample is compatible with the effect of quenching, which makes it easier for a background fluctuation to supersede a genuine low \\pt jet.\nThe fraction of these background events strongly depends on the centrality and leading jet \\pt. For the purposes of the study presented in this paper, the contribution of these background events to the results is subtracted by using the events at small \\dphi.\n\n\\subsection{Dijet momentum balance}\n\\label{sec:asymmetry}\nTo characterize the dijet momentum balance (or imbalance) quantitatively, we use the asymmetry ratio\n\\begin{equation}\n\\label{eq:aj}\nA_J = \\frac{\\ptlead-\\ptsub}{\\ptlead+\\ptsub}~.\n\\end{equation}\nDijets are selected with $\\dphi > 2\\pi\/3$.\nIt is important to note that the subleading jet $\\ptsub > 30$\\GeVc\\ selection imposes\na $\\ptlead$-dependent limit on the magnitude of \\AJ. The distributions are normalized to\nthe number of selected dijet events.\n\nAs discussed in Section~\\ref{sec:dphi}, the contribution of background fluctuations is estimated from\nthe events with dijets of $\\dphi < \\pi\/3$, and the distributions obtained from these events are subtracted from the results.\nThe estimated fraction of background events, as a function of both leading jet \\pt\\ and centrality, is shown in the bottom row of Fig.~\\ref{fig:SubRate}.\nThe fraction of dijet events in which the subleading jet is found within the acceptance, after the subtraction of\nbackground events, is shown in the top row of Fig.~\\ref{fig:SubRate}. The events in which the subleading jet is not found should be taken into account when comparing the asymmetry distributions, although the bias is negligible for bins of leading jet $\\pt > 180$\\GeVc.\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{SubLeadingRate_d20120103}\n\\caption{\nFraction of events with a genuine subleading jet with $\\dphi > 2\\pi\/3$, as a function of leading jet $\\ptlead$ (left)\nand \\npart\\ (right). The background due to underlying event\nfluctuations is estimated from $\\dphi < \\pi\/3$ events and subtracted from the number of dijets.\nThe fraction of the estimated background is shown in the bottom panels.\nThe error bars represent the statistical uncertainties.}\n\\label{fig:SubRate}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{dijet_imbalance3_lead120_sub30_all_cent_20120103_subt}\n\\caption{Dijet asymmetry ratio, $A_{J}$, for leading jets of $\\ptlead> 120$\\GeVc\\ and\n  subleading jets of $\\ptsub> 30$\\GeVc\\ with a selection of $\\dphi>2\\pi\/3$ between the two jets.\nResults are shown for six bins of collision centrality, corresponding to selections of 70--100\\% to 0--10\\%  of the total inelastic cross section.\nResults from data are shown as points, while the histogram shows the results\nfor\n\\PYTHIA dijets embedded into \\HYDJET PbPb simulated events.\nData from pp collisions at 2.76\\TeV are shown as open points in comparison to PbPb results of 70--100\\% centrality.\nThe error bars represent the statistical uncertainties.}\n\\label{fig:JetAsymm}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{dijet_imbalance3_0to20_pt_20120103_subt}\n\\caption{Dijet asymmetry ratio, $A_{J}$, in bins of leading jet transverse momentum from\n120 $ < \\ptlead < 150$\\GeVc\\ to $\\ptlead > 300$\\GeVc\\ for\n  subleading jets of $\\ptsub> 30$\\GeVc\\\nand $\\dphi>2\\pi\/3$ between leading and subleading jets.\nResults for 0--20\\% central PbPb events are shown as points, while the histogram\nshows the results for\n\\PYTHIA dijets embedded into \\HYDJET PbPb simulated events. The error bars represent the statistical uncertainties.}\n\\label{fig:JetAsymmPt}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{dijet_imbalance5_0to20_pt_20120103_subt}\n\\caption{Subleading jet transverse momentum fraction ($\\ptsub\/\\ptlead$), in bins of leading\njet transverse momentum from\n120 $ < \\ptlead < 150$\\GeVc\\ to $\\ptlead > 300$\\GeVc\\ for\n  subleading jets of $\\ptsub> 30$\\GeVc\\\nand $\\dphi>2\\pi\/3$ between leading and subleading jets.\nResults for 0--20\\% central PbPb events are shown as points, while the histogram\nshows the results for\n\\PYTHIA dijets embedded into \\HYDJET PbPb simulated events.\nThe arrows show the mean values of the distributions and the error bars represent the statistical uncertainties.}\n\\label{fig:JetFractionPt}\n\\end{center}\n\\end{figure*}\n\nThe centrality dependence of $A_J$ for \\PbPb\\ collisions is shown in\nFig.~\\ref{fig:JetAsymm}, in comparison to results from \\PYTHYD simulations.\nThe most peripheral events are also compared\nto results from \\pp\\ collisions at $\\sqrt{s} = 2.76\\TeV$, where the same jet algorithm is used.\nThis comparison supports the use of the\n\\PYTHYD sample as a reference for the dijet asymmetry, which also takes into account underlying event\neffects when comparing with PbPb data.\nThe shape of the dijet momentum balance distribution experiences a gradual change with collision centrality,\ntowards more imbalance. In contrast, the \\PYTHIA simulations only\nexhibit a modest broadening, even when embedded in the highest multiplicity\n\\PbPb\\ events.\n\nTo study the momentum dependence of the amount of energy loss,\nFig.~\\ref{fig:JetAsymmPt} presents the distributions of $A_J$ in different bins of leading jet \\pt,\nfor 0--20\\% central events. One observes a strong evolution in the shape of the distribution across the\nvarious \\pt\\ bins, while a significant difference between PbPb data and \\PYTHYD simulations persists in\neach \\pt\\ bin. The distributions of the \\ptrat\\ ratio, shown in Fig.~\\ref{fig:JetFractionPt}, provide a more intuitive way of quantifying the energy loss.\nBoth the $A_J$ and \\ptrat\\ distributions are affected by the cut on the subleading jet $\\pt$, which should be taken into account in the interpretation of the average value. However, in the bins with leading jet $\\pt> 180\\GeVc$, more than 95\\% of the leading jets are correlated with a subleading jet, indicating that the bias due to dijet selection is very small.\n\n\\subsection{The dependence of dijet momentum imbalance on the \\texorpdfstring{\\pt}{pt} of the leading jet }\n\nThe dependence of the energy loss on the leading jet momentum can be studied using the jet transverse momentum ratio\n$\\ptsub\/\\ptlead$.\nThe mean value of this ratio is presented as a function of $\\ptlead$\nin Fig.~\\ref{deltaPt} for three bins of collision centrality, 50--100\\%, 20--50\\%, and 0--20\\%.\nThe \\PYTHYD simulations are shown as squares and the\nPbPb data are shown as points. Statistical and systematic uncertainties are plotted as error bars and brackets, respectively.\nThe main contributions to the systematic uncertainty in $\\ptsub\/\\ptlead$\nare the uncertainties in the $\\pt$-dependent residual energy scale and the effects of the underlying event on the jet energy resolution.\nEarlier studies of jet-track correlations~\\cite{Chatrchyan:2011sx} have shown that the energy composition of the quenched jets was not significantly different, which\nputs a constraint on the energy scale uncertainty. The uncertainty on the energy scale is derived from\nthree sources: the uncertainty evaluated in the pp studies \\cite{Chatrchyan:2011ds},\nthe energy scale difference in pp data and MC, and the energy scale and its parton type dependence~\\cite{MattPFlow} in simulations of PbPb events (see Section~\\ref{sec:pythia_samples}). These contributions are added in quadrature to assign the total uncertainty on the jet energy scale.  Using this value as a boundary, the uncertainty in the $\\ptsub\/\\ptlead$\nresults is then estimated by varying the jet response at low \\pt\\ and at high \\pt\\ independently.\nThe uncertainty on the underlying event effects is estimated from the full\ndifference between \\pp and \\PYTHYD.\nThese effects add up to 6\\% in the most central events.\nFor the low leading-jet \\pt\\ bins, jet reconstruction efficiency also introduces a minor uncertainty on the order of 1\\%.\nUncertainties due to additional misreconstructed jets, calorimeter noise, and the track requirement are negligible compared\nto the dominating sources of uncertainty.\nFor the centrality bins of 50--100\\%, 20--50\\% and 0--20\\%, the sources of systematic uncertainty are summarized in Table~\\ref{RatioSystematics}.\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=\\cmsFigWidth]{deltaPtOverPt5_lead120_sub30_diff_20120103}\n\\caption{\nAverage dijet momentum ratio $\\ptsub\/\\ptlead$ as a function of\nleading jet \\pt for three bins of collision centrality, from peripheral to central collisions,\ncorresponding to selections of 50--100\\%,  30--50\\% and 0--20\\%  of the total inelastic cross section.\nResults for \\PbPb\\ data are shown as points with vertical bars and brackets indicating\nthe statistical and systematic uncertainties, respectively.  Results for \\PYTHYD are shown as squares. In the 50--100\\% centrality bin,\nresults are also compared with pp data, which is shown as the open circles.\nThe difference between the \\PbPb\\ measurement and the \\PYTHYD expectations is shown in the bottom panels. }\n\\label{deltaPt}\n\\end{center}\n\\end{figure*}\n\nAs shown in Fig.~\\ref{deltaPt}, both the PbPb data and the \\PYTHYD  samples reveal an increasing trend for the mean value of the\n jet transverse momentum ratio, as a function of the leading jet $\\ptlead$. This can be understood\nby the reduction in the effects of jet splitting and energy resolution as one goes to higher jet momenta.\nHowever, the central \\PbPb\\ data points lie consistently below the \\PYTHYD trend. The difference between the pp data and the \\PYTHYD reference is of the order of the systematic uncertainty of the measurement, whereas the difference between \\PbPb\\ data and the reference is more than twice larger. This difference is related to the parton energy loss and for central PbPb collisions it is of significant magnitude across the whole \\pt\\ range explored in this study.\n\n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Summary of the \\ptrat\\ systematic uncertainties. The range of values represent the\nvariation from low ($\\ptlead<140\\GeVc$) to high\n($\\ptlead>300\\GeVc$) leading jet \\pt. }\n\\label{RatioSystematics} \\begin{tabular}{|l|c|c|c|c|}\n\\hline\nSource &  50--100\\% & 20--50\\% & 0--20\\% \\\\\n\\hline\nUnderlying event & 1\\% &  3\\% & 5\\% \\\\\nJet energy scale & 3\\% &3\\% & 3\\%  \\\\\nJet efficiency  & 1--0.1\\% & 1--0.1\\% & 1--0.1\\% \\\\\nJet misidentification & $<0.1$\\%  & $<0.1$\\% & 1--0.1\\%\\\\\nCalorimeter noise & $<0.1$\\% & $<0.1$\\% & $<0.1$\\%  \\\\\nJet identification & $<0.1$\\% & $<0.1$\\% & $<0.1$\\%\\\\\n\\hline\nTotal   & 3.5\\% & 4.5\\% & 6\\% \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\n\\section{Summary}\n\\label{sec:summary}\n\nDijet production in PbPb collisions at\n$\\rootsNN=2.76\\TeV$ was studied with the CMS detector in a data sample\ncorresponding to an integrated luminosity of \\lum. The \\ak\\ algorithm was\nused to reconstruct jets based on combined tracker and calorimeter\ninformation. Events containing a leading jet with $\\ptlead > 120$\\GeVc\\ and\na subleading jet with $\\ptsub > 30$\\GeVc\\ in the pseudorapidity range\n$|\\eta| < 2$ were analyzed.\nData were compared to \\PYTHYD dijet simulations,\ntuned to reproduce the observed underlying event fluctuations.\nFor the most peripheral collisions, good agreement between data and simulations is\nobserved. For more central collisions, the dijet momentum imbalance in the data\nis significantly larger than seen in the simulation. Across the entire range of jet momenta studied, no significant broadening of the dijet angular correlations is observed with respect to the reference distributions.\n\nThe dijet momentum imbalance was studied as a function of the leading jet $\\ptlead$\nfor different centrality ranges in comparison to the \\PYTHYD simulation.\nFor leading jet momenta\n$\\ptlead > 180$\\GeVc\\ the dijet balance distributions are found to be essentially\nunbiased by the subleading jet threshold of $\\ptsub> 30 $\\GeVc.\nFor mid-central (30--50\\%) and more central PbPb event selections, a significantly\nlower average dijet momentum ratio $\\langle \\ptsub\/\\ptlead \\rangle$\nis observed than in the pp data and in the dijet embedded simulations. The downward shift in\n$\\langle \\ptsub\/\\ptlead \\rangle$, with respect to the \\PYTHYD reference, is seen to increase\nmonotonically with  increasing collision centrality,\nand to be largely independent of the leading jet $\\ptlead$,\nup to $\\ptlead$ values in excess of 350\\GeVc.\n\nIn summary, the results presented in this paper confirm previous observations based on\na smaller dataset and extend the measurements of jet-quenching effects to wider centrality and\nleading jet transverse momentum ranges, as well as to lower subleading jet transverse momentum.\n\n\n\\section*{Acknowledgments}\n\\hyphenation{Bundes-ministerium Forschungs-gemeinschaft Forschungs-zentren} We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC machine. We thank the technical and administrative staff at CERN and other CMS institutes. This work was supported by the Austrian Federal Ministry of Science and Research; the Belgium Fonds de la Recherche Scientifique, and Fonds voor Wetenschappelijk Onderzoek; the Brazilian Funding Agencies (CNPq, CAPES, FAPERJ, and FAPESP); the Bulgarian Ministry of Education and Science; CERN; the Chinese Academy of Sciences, Ministry of Science and Technology, and National Natural Science Foundation of China; the Colombian Funding Agency (COLCIENCIAS); the Croatian Ministry of Science, Education and Sport; the Research Promotion Foundation, Cyprus; the Ministry of Education and Research, Recurrent financing contract SF0690030s09 and European Regional Development Fund, Estonia; the Academy of Finland, Finnish Ministry of Education and Culture, and Helsinki Institute of Physics; the Institut National de Physique Nucl\\'eaire et de Physique des Particules~\/~CNRS, and Commissariat \\`a l'\\'Energie Atomique et aux \\'Energies Alternatives~\/~CEA, France; the Bundesministerium f\\\"ur Bildung und Forschung, Deutsche Forschungsgemeinschaft, and Helmholtz-Gemeinschaft Deutscher Forschungszentren, Germany; the General Secretariat for Research and Technology, Greece; the National Scientific Research Foundation, and National Office for Research and Technology, Hungary; the Department of Atomic Energy and the Department of Science and Technology, India; the Institute for Studies in Theoretical Physics and Mathematics, Iran; the Science Foundation, Ireland; the Istituto Nazionale di Fisica Nucleare, Italy; the Korean Ministry of Education, Science and Technology and the World Class University program of NRF, Korea; the Lithuanian Academy of Sciences; the Mexican Funding Agencies (CINVESTAV, CONACYT, SEP, and UASLP-FAI); the Ministry of Science and Innovation, New Zealand; the Pakistan Atomic Energy Commission; the Ministry of Science and Higher Education and the National Science Centre, Poland; the Funda\\c{c}\\~ao para a Ci\\^encia e a Tecnologia, Portugal; JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); the Ministry of Education and Science of the Russian Federation, the Federal Agency of Atomic Energy of the Russian Federation, Russian Academy of Sciences, and the Russian Foundation for Basic Research; the Ministry of Science and Technological Development of Serbia; the Ministerio de Ciencia e Innovaci\\'on, and Programa Consolider-Ingenio 2010, Spain; the Swiss Funding Agencies (ETH Board, ETH Zurich, PSI, SNF, UniZH, Canton Zurich, and SER); the National Science Council, Taipei; the Scientific and Technical Research Council of Turkey, and Turkish Atomic Energy Authority; the Science and Technology Facilities Council, UK; the US Department of Energy, and the US National Science Foundation.\n\nIndividuals have received support from the Marie-Curie programme and the European Research Council (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation \\`a la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Council of Science and Industrial Research, India; and the HOMING PLUS programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecently, assistive robotics became an increasingly popular research field as it provides use-case applications for solutions from a wide range of disciplines. Whereas special-purposed service robots such as autonomous cleaning systems or lawn mowers already found their way into many households, more general-task directed robotic assistants still entail challenges for the safe and reliable use in unconstrained environments. In general, these systems are aimed at providing various services in the household and workspace of humans that are assigned in direct and human-oriented interactions. This requires the assistive robot to enhance the mostly geometric perception as solely needed for cleaning tasks to a more general semantic understanding of its environment. In particular, a common task for a service robot is to find and bring back a specific kind of object. In order to execute such  a command the robot must be able to interact with the human by understanding its needs and subsequently having the capabilities to navigate and find the dedicated object autonomously.\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{info3.png}\n\t%\n\t\\caption{Example image of the TIAGo robot in front of a mapped office environment. Multiple objects on the desk, such as screen, cup, book, mouse and keyboard, are additionally incorporated as projected spheres into the point cloud map.}\n\t\t\\label{info}\n\\end{figure}\nIn this paper we focus on the perceptual semantics for processing meaning-oriented information for objects and humans.\nMultiple approaches \\cite{hobbit, martinez, lee} have been presented targeting the detection and recognition of semantic data for mobile robots. Common methods harness model-based solution approaches where each object is identified by its shape or color using handcrafted descriptors. Given that the target object has an unique appearance, these methods can be very performant for recognition tasks. On the other side, low image resolution, small objects, an ordinary appearance and multiple instances of similar looking objects can impair the detection and recognition of such systems. In addition, the target models have to be taught beforehand with training images of the object showing it from multiple different perspectives.\n\nWe tackle this challenge by using neural networks and combine them with geometric constraints provided by the robot. Our system can be divided in two separate modules. The first one targets the enrichment of geometric maps with semantic information by detecting and localizing objects in a robust manner (Fig. \\ref{info}). Additionally, we make sure to constantly correct and extend currently mapped objects based on updated perception data from the robot. This makes our system also suitable for localization methods that provide retrospective optimization capabilities. The procedure is carried out on the fly in real-time, when the robot is exploring the surroundings making it a well benefiting add-on for geometric-based mapping algorithms.  \nThe second module detects and analyzes potential human interaction partners where we aim to additionally predict cooperation willingness. This provides a first step of interactions with humans in a proactive manner while at the same time semantic objects can be adhoc incorporated in resulting tasks.\nWe implemented our system on the TIAGo, a mobile humanoid robot platform.\nThe rest of this paper is organized as follows: Section \\ref{sec:system} describes our proposed system. In section \\ref{implementation}, we give details of the current state of implementation. Section \\ref{conclusion} concludes and gives an insight about our future work.\n\\begin{figure*}[th]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{system3.pdf}\n\t\n\n\t\\caption{Overview of the proposed method. We use RGB-D images to feed our process lines for object registration and human condition estimation. The object registration consist of 2D object detections that are mapped and associated to 3D objects. The human condition estimation starts with multi task detection process for prediction persons and faces that are in a subsequent step further analyzed for interaction willingness. }\n\t\\label{system}\n\\end{figure*}\n\n\\section{system components} \n\\label{sec:system}\nOur system (Fig. \\ref{system}) consists of two main pipelines for extracting and evaluating semantic information from visual data. The first one is dedicated to detect and register volumetric objects. The second one classifies and estimates the attention related behavior of humans. In the following subsections, we describe each component in detail.\n\n\\subsection{Object registration}       \nThe first step of the object registration is the classification of image regions in the image stream provided by the camera in the head of the robot. For this matter deep neural networks have been proven to reliably detect wide ranges of different types of objects. Popular architectures are the RCNN model family \\cite{rcnn, FasterRCNN} and the YOLO model family \\cite{yolo, yolo9000}. Due to it outstanding speed we chose YOLOv4 \\cite{yolov4} for our framework. The network is pretrained on the COCO database\\cite{coco} and able to predict up to 80 different classes. However, when testing it outside of the database we experienced a quite decreased accuracy rate and repeating occurrences of false positive detections. Accordingly, the unfiltered processing of the provided predictions will result with false labeled map parts in later stages. To overcome this problem, we apply an adapted Intersection over Union (IOU) tracker\\cite{iou} that associates similarly located and equally labeled bounding boxes from consecutive images. Associated detections form a so called \\textit{track}, where the confidence of a true positive object classification increases with the tracks length.\nAfter reaching a specific track length threshold the 2D detection is projected into the 3D space using the intrinsic camera parameters and the extrinsics provided by the robots localization algorithm. For this purpose the depth estimation from the camera active stereo module is used together with the RGB images to generate metric scaled point clouds. Depending on the bounding box sizes we then cut out object cuboids that are afterwards projected into the robots 3D point cloud map. While this method for registering of semantic objects can be performed in a very efficient way, it is combined with supportive solutions to overcome the following challenges: \n \\linebreak\n\\subsubsection{Object Recognition}\nWhen the robot is moving around it may detect objects in the image stream that already has been registered earlier to the map. Processing it a second time will lead to multiple mapped instances of the same physical objects and must be therefore prevented. Consequently, the object mapping must be able to recognize objects that has been seen and registered before. In general, 3D object recognition has been extensively researched in the recent time, but it commonly requires huge computational efforts. In addition, objects from the same class can resemble each other in a way that taking unique visual fingerprints fails. We approach the issue by comparing the projected localization of a new object \\textit{candidate} with the localization of earlier registered objects of the same class. Is the candidate significantly overlaying with its counterpart it is likely that both belong to the same origin. This approach is implemented using a \\textit{nearest neighbor} association process, where we compare the average distance between the entire point clouds instead of their centroids. This way, we incorporate the objects size in the association decision as point clouds of large objects can contain centroids that are wide apart, while the clouds themselves heavily overlap. \n \\linebreak\n\\subsubsection{Object Optimization}\nThe probabilistic localization of the robot involves uncertainties and will only be a convergence of its real pose. This effect is further enhanced when navigating in an unknown environment and, thus, a simultaneous mapping process is required. For localization methods with optimization capabilities the detection of distinctive or familiar map areas is exploited to re-evaluate the past trajectory and, where appropriate, to correct drifts. Concurrently, we use this mechanism to recalculate the poses of our semantic objects derived from the corrected robot trajectory. At the same time, we analyze if this leads to any strong overlaps between their updated point clouds. This provides us the opportunity to even correct mapping errors induced from the erroneous trajectory. Objects with salient overlap are assumed to belong to the same physical instance and are therefore merged whereby the points clouds are concatenated and the localization and object dimensions are recalculated accordingly. This way, we can not only maintain and correct the perception of semantic objects around us, but with the aid of point cloud merging we are able to build up more complete point cloud appearance models of our objects.\n\\subsection{Human Behavior Estimation}\nTo create optimal preconditions for a successful human-robot interaction we apply a dedicated process pipeline to search for and analyze human interaction partners. Instead of using one network for person detection and one for face detection, we utilize a custom neural network that is able to predict person and face bounding boxes simultaneously.\n \\linebreak \n\\subsubsection{Face-Person Detection}\nThe network architecture is based on the SSD approach~\\cite{ssd} that provides a higher inference rate than two-stages detectors. Moreover, both detectors for persons and faces are combined using multi-task learning which enables them to share their first layers of feature extraction. This equally boost the efficiency and the generalization capabilities.  \n\nThe main difficulty for the combination of the two tasks face and person detection in a single neural network is the fact that publicly available databases contain only ground truths for one of the two tasks. For this purpose, we developed a custom multi-task loss function and designed an architecture consisting of a shared backbone and separate detection layers for each detection task. During training, we alternate between batches of person annotations and batches of face annotations, which are taken from different databases.\nThe prediction of the respective class with non-existing ground truth is assumed to be correctly determined and only the gradients of the detection layers with existing ground truth information are adjusted. Thus, a completely end-to-end trainable framework could be created.\n  \nWhile the predicted bounding boxes for persons give us a good estimation of its localization, the face predictions are further used for behavior examination. In particular, we want to know if the human is interested in an interaction with the robot. A good indication for general interest is the gaze or head pose pointing towards the robot.\nAs the gaze estimation is error-prone at low image resolutions, we focus on the head pose to evaluate general interaction willingness.\n \\linebreak\n\\subsubsection{Interaction Willingness Estimation}\nTo estimate the head pose we first determine the position of facial landmarks in the face images provided by the face predictor. These facial keypoints are distinctive spots in the face (e.g. the corners of the eyes, the sides and top of the nose) that provide us information about the current formation of the face. We estimate the landmark positions by applying an ensemble of regression trees~\\cite{faciallm} that has proven to provide very fast and reliable predictions. In the following step we project these 2D landmarks into the 3D space. This is achieved by using a default 3D model that contains the same facial landmarks and aligning it with the previously predicted 2D counterpart. As this represents a minimization problem we apply the commonly used Levenberg-Marquardt optimization to estimate a matching 3D mask. The rotation and translation for the projection that provides the smallest re-projection error implies the head pose.\n\nThe prediction is performed for every single image that contains a detected face. However, the pose information from a single frame is not sufficient to derive assumptions about an underlying general interaction willingness. Brief views in the direction of the robot can be caused by arbitrary intentions including behavior estimation (e.g. when the robot is moving) or even causal glances without deeper conscious intentions. We therefore track a humans head pose over multiple images and gain confidence about interaction intentions the longer the persons focus is directed at the robot. We assume that a viewing direction towards the robot of a duration of 3~ms represents a reliable threshold to determine interaction willingness. However, short distractions are common that result in brief interruptions of the attention towards the robot. We therefore use a solution based on our previous idea \\cite{hempel} and calculate the interaction willingness in a dynamic manner. We apply a progress bar that loads faster when the attention is directed to the robot and unloads slower in case of distractions. In this way, showing interaction willingness can be resumed in natural way even though it has been abandoned for a short time.   \n\n\\section{Implementation}\n\\label{implementation}\nWe implemented our system in form of multiple Robot Operating System (ROS) compatible modules for seamless intercommunication with other (ROS) components and deployed it on the TIAGo robot. For mapping and localization we use the RTAB-Map~\\cite{rtabmap} as its graph-based approach suits our semantic data update and correction process. The robots IMU is used as guess to perform a laser-based ICP-SLAM that runs along with other ROS nodes for path planning, collision avoidance and motion planning on the robots onboard i7 computer. Our image processing focused methods are deployed on an external mobile system that is placed on the robots shoulders. It contains a Quadro RTX 5000 that is able to process the neural networks more efficiently than CPUs.\n\n\n\\section{Conclusion}\n\\label{conclusion}\nIn this work, we addressed the problem of semantic meaningful perception for mobile assistive robots which constitutes a fundamental requirement for solving complex tasks. \n\nWe propose a neural network enhanced approach for successively mapping and maintaining 3D objects of the robots environment that can run alongside other geometrical mapping modules. \nSimilarly, we process humans by utilizing a custom single-shot detector that simultaneously provides person and face predictions in the image stream. The latter are furtherly used to estimates the persons interaction willingness to enable proactive collaboration behavior on the robots side. All modules are implemented on a mobile humanoid robot platform and are compatible for intercommunication with other ROS modules such as path and motion planning.    \nIn future works we will exploit this advantage to incorporate additional text-to-speech and speech recognition modules. First, we will search for potential interaction partners and proactively call for tasks when interaction willingness is predicted. Afterwards the tasks can be provided orally, processed and executed. Typical tasks will be the localization and bringing of specific objects that have either already been mapped or have to be found in a dedicated exploration drive.\nThe behavior will be evaluated in real world scenarios.  \n\n\\bibliographystyle{IEEEtranDOI}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAccording to core accretion models \\citep{mizuno-78}, giant gas\nplanets such as Jupiter and Saturn formed via the accumulation of an\nprotocore of rock and ice which gained solid material until it reached\nsufficient size to begin accreting the gaseous component of the\nprotosolar nebula. The existence of solid ice in the outer solar\nsystem promotes the rapid growth of the more massive protocores\nallowing the accretion of large quantities of gas necessary for\nJupiter-sized planets. Giant planets thus have a dense core of rock\nand ice surrounded by a H-He envelope. It is not known, however,\nwhether the initial dense core remains stable following the accretion\nof the H-He outer layer or whether the core erodes into the fluid\nhydrogen-rich layers above \\citep{stevenson-pss-82,guillot-book}.\n\nThe gravitational moments of Jupiter and Saturn, which have been\nmeasured by prior planetary missions and will be determined for\nJupiter with unprecedented accuracy by the upcoming Juno mission, may\nbe used in combination with interior models\n\\citep{militzer-apj-08,guillot-book,hubbard-ass-05,saumon-apj-04,nettelmann} to\nestimate the mass of the present-day core, but it is unclear whether\nthese masses correspond to the primordial core mass. It has been\nsuggested \\citep{guillot-book,saumon-apj-04} that the present-day core\nmass of Jupiter may be too small to explain its formation by core\naccretion within the relatively short lifetime of the protosolar\nnebula \\citep{pollack-icarus-96}, although a more recent Jupiter model\n\\citep{militzer-apj-08} predicted a larger core of 14--18 Earth masses\nwhich is consistent with core accretion. Furthermore, direct\nmeasurements of Jupiter's atmosphere suggest a significant enhancement\nin the concentration of heavy $(Z > 3)$ elements\n\\citep{niemann-science-96}, but it is unknown to what extent this\nshould be attributed to a large flux of late-arriving planetesimals\nversus the upwelling of core material. Determining the extent of core\nerosion is thus a major priority for understanding the interiors of\ngiant planets and the process by which they were formed.\n\nIn this work we focus on water ice, presumed to be a major constituent\nof the core, and consider the question of whether it has significant\nsolubility in fluid metallic hydrogen at conditions corresponding to\nthe core-mantle boundaries of giant gas planets. Water ice is the most\nprevalent of the planetary ices (water, methane and ammonia) which may\nbe assumed to make up the outermost layers of a differentiated\nrock-ice core \\citep{hubbard-science-81}. At the conditions of\ntemperature and pressure prevalent at giant planet cores, water ice is\npredicted \\citep{cavazzoni,french-prb-09} to be in either in a fully\natomic fluid phase in which oxygen and hydrogen migrate freely and\nindependently, or in a superionic phase in which oxygen atoms vibrate\naround defined lattice sites while hydrogen atoms migrate\nfreely. Assuming the existence of a core-mantle boundary at which\nwater ice and the fluid H-He phase are in direct contact, the relevant\nquestion is the extent to which the system may lower its Gibbs free\nenergy by the redistribution of the atoms of the ice phase into the\nfluid hydrogen. The extreme pressure and temperature conditions\nprevalent at giant planet core-mantle boundaries (8000--12000K and\n8--18 Mbar for Saturn, 18000--21000K and 35--45 Mbar for Jupiter) are\nnot yet obtainable in the laboratory, thus \\emph{ab initio}\nsimulations provide the best available guide to determining the extent\nof core solubility.\n\n\\section{Theory and Methodology}\n\nWe used density functional molecular dynamics (DFT-MD) calculations\nand coupling constant integration (CCI) techniques to compute the\nGibbs free energy of solvation, $\\Delta G_{sol}$, of H$_2$O in fluid\nmetallic hydrogen, i.e. the change in Gibbs free energy when an H$_2$O\nmolecule is removed from the pure ice phase and dissolved in H.  The\nfree energy of solubility is computed from the free energies of three\nsystems: pure ice, pure fluid H, and a mixed system in which the atoms\nof one water molecule are dissolved in $n$ atoms of hydrogen,\n\n\\begin{equation}\n\\Delta G_{sol} = G \\left(\\mbox{O} \\mbox{H}_{n+2}\\right) - \\left[G \\left( {\\mbox{H}_2 \\mbox{O}} \\right) + G\\left( \\mbox{H}_{n} \\right) \\right]\n\\label{deltag}\n\\end{equation}\n\nwhere $G \\left( {\\mbox{H}_2 \\mbox{O}} \\right)$ is the energy per\n$\\mbox{H}_2\\mbox{O}$ stoichiometric unit of the ice phase, and\n$G\\left(\\mbox{H}_{n}\\right)$ is obtained from an appropriately-scaled\nsimulation of 128 H atoms. This quantity becomes more negative as\nsolubility increases. A $\\Delta G_{sol}$ of zero implies a saturation\nconcentration of exactly one H$_2$O to $n$ H. In order to span the\nrange of likely conditions for the core-mantle boundary of Jupiter and\nSaturn, we considered pressures of 10, 20 and 40 Mbar, at a range of\ntemperatures from 2000 to 20000~K.\n\nComputation of free energies from MD simulations is difficult since\nthe entropy term is not directly accessible. Here we use a two-step\nCCI approach as previously applied by several authors\n\\citep{alfe-nature-99,morales-pnas-09,wilson-prl-10} to compute free\nenergies. The CCI method provides a general scheme for computing\n$\\Delta F$ between systems governed by potential energy functions\n$U_1$ and $U_2$. We construct an artificial system $U_\\lambda$ whose\nforces are derived from a linear combination of the potential energies\nof the two systems $U_\\lambda = (1-\\lambda)U_1 + \\lambda U_2$. The\ndifference in Helmholtz free energy between the two systems is then\n\n\\begin{equation}\n\\Delta F = \\int_0^1 \\langle U_2 - U_1 \\rangle_\\lambda \\: d \\lambda,\n\\end{equation}\n\nwhere the average is taken over the trajectories governed by the\npotential $U_\\lambda$.  We perform two CCIs for each $G$ calculation:\nfirst from the DFT-MD system to a system governed by a classical pair\npotential which we fit to the DFT dynamics of the system via a\nforce-matching approach \\citep{izvekov-jcp-04}, and then from the\nclassical system to a reference system whose free energy is known\nanalytically.\n\n\\subsection{Material phases}\n\nThe material phases in question must be established prior to the Gibbs\nfree energy calculations. At the pressure and temperature conditions\nof interest, hydrogen is a metallic fluid of H atoms in which\nmolecular bonds are not stable. Water dissolved within hydrogen\nlikewise is non-molecular, with a free O atom in an atomic H\nfluid. For water ice, the phase diagram at giant planet core pressures\nis divided into three regimes\n\\citep{cavazzoni,goldman,mattsson-prl-06,french-prb-09}: a\nlow-temperature ($<$~2000~K) crystalline regime, an intermediate\nsuperionic regime in which oxygen atoms vibrate around fixed lattice\nsites while hydrogen atoms migrate freely, and a higher-temperature\nfully fluid regime in which both hydrogen and oxygen atoms are\nmobile. The transition between the crystalline and superionic regimes\nhas not yet been studied in detail at high pressures but our\nsimulations find that it occurs below 2000~K. The transition from\nsuperionic to fully fluid is found to occur in simulation at\ntemperatures ranging from 8000~K for 10 Mbar and 13000~K for 50 Mbar\n\\citep{french-prb-09}. Consequently our study includes the superionic\nand fully fluid ice phases.\n\nPrevious studies of superionic ice \\citep{cavazzoni,french-prb-09}\nhave used an \\emph{bcc} arrangement of atoms for the oxygen\nsublattice. We found that such a lattice was stable at all points\nstudied in the superionic regime except for 10 Mbar at 2000 and\n3000~K. At these conditions we found that the oxygen sublattice was\nstable in the \\emph{Pbca} geometry which we recently reported to be\nthe most stable zero-temperature structure for ice at 10 Mbar\n\\citep{militzer-prl-10}. At 10 Mbar and 5000~K, superionic ice with a\n\\emph{bcc} sublattice was stable but the \\emph{Pbca} was not. The\n\\emph{bcc} oxygen sublattice was found to be stable for 20 and 40 Mbar\npressures at all temperatures studied in the superionic regime.\nAttempts to perform a superionic simulation in the \\emph{Cmcm}\ngeometry reported by \\cite{militzer-prl-10} to be the stable\nzero-temperature structure at these pressures resulted in an unstable\nsystem with a large anisotropic strain. We thus used a \\emph{bcc}\noxygen sublattice for all superionic ice simulations, except the\n2000~K and 3000~K simulations at 10 Mbar which used the oxygen\nsublattice from the \\emph{Pbca} phase, as indicated in Figure 1. While\nwe cannot yet exclude the possibility of the existence of yet another\nsuperionic ice structure, we note that the differences between\ndifferent ice phase energies are found to be on the order 0.1~eV per\nH$_2$O, and thus will not significantly affect our results about the\nstability of ice in giant planet cores.\n\n\\subsection{Computation of Gibbs free energies}\n\nOur goal in this paper is to study the solubility of water ice in\nfluid hydrogen. Due to considerations arising from the entropy of\nmixing, the solubility of one material in another is never zero,\nhowever solubility at trace quantities is not sufficient for core\nerosion. In particular, we wish to know whether solubility is\nthermodynamically favored at concentrations significantly greater\nthan the background concentration of oxygen in the fluid envelope of\nJupiter or Saturn -- this is equal to approximately one O atom to 1000\nH atoms if we assume solar concentrations for the Jovian envelope and\napproximately one part in 300 if we assume a threefold enrichment for\noxygen as observed for most other heavy elements \\citep{mahaffy}. We\nbegin by computing the Gibbs free energies of solubility for\ndissolving H$_2$O in pure H at one part in 125, and generalize later.\n\nThe coupling constant integration approach requires, as an integration\nend point, a reference system whose free energy may be computed\nanalytically. It is important to ensure that the system does not\nundergo a phase change along the integration pathway since this may\ncause numerical difficulties in the integration. For the fluid\nsystems, being pure hydrogen, hydrogen with oxygen, and ice at\ntemperatures above the superionic-to-fluid transition, we used an\nideal atomic gas as the reference system\n\\citep{alfe-nature-99,morales-pnas-09,wilson-prl-10}. For superionic\nice, we use as a reference system a combination of an ideal gas system\nfor the hydrogen atoms with an Einstein crystal of oxygen atoms each\noxygen tethered to its ideal lattice site with a harmonic potential of\nspring constant 30~eV\/$\\mbox{\\AA}^2$.\n\nThe DFT-MD simulations in this work used the Vienna Ab Initio\nSimulation Package (VASP) \\citep{vasp} with pseudopotentials of the\nprojector augmented wave type \\citep{paw} and the exchange-correlation\nfunctional of \\citet{pbe}. The pseudopotentials used had a core radius of 0.8~{\\AA} for hydrogen and 1.1~{\\AA} for oxygen. Wavefunctions were expanded in a basis set\nof plane waves with a 900~eV cutoff and the\nBrillouin Zone was sampled with $2 \\times 2 \\times 2$ $k$-points. The\nelectronic temperature effects were taken into account via Fermi-Dirac\nsmearing. A new set of force-matched potentials was fitted for each\npressure-temperature conditions for each stoichiometry. All MD\nsimulations used a 0.2~fs timestep. In the classical potential under\nsuperionic conditions, an additional harmonic potential term was added\nto ensure that oxygen atoms remained in the appropriate lattice sites,\nhowever, we found that in most cases the fitted pair potential alone\nwas sufficient to stabilize the superionic state.\n\nThe first step of the free energy calculations was the determination\nof the appropriate supercell volumes for each system for each set of\npressure and temperature conditions. This was accomplished via\nconstant-pressure MD simulations\n\\citep{hernandez-jcp-01,hernandez-prl-10} with a duration of 1.6~ps\n(0.6~ps for ice). DFT-MD trajectories were then computed in a fixed\ncell geometry in order to fit classical potentials. A run of 0.4~ps\nwas found to be sufficient for fitting suitable potential. We then\nperformed molecular dynamics runs 600 fs long (400 fs for ice) at five\n$\\lambda$ values to integrate between the DFT and classical\nsystems. Finally, we performed classical Metropolis Monte Carlo at 24\n$\\lambda$ values to integrate from the classical to the reference\nsystem.\n\n\\section{Simulation results}\n\nTable I lists the total Gibbs free energies for each simulated system\n(H$_{128}$, OH$_{127}$ and ice) for each set of temperature and\npressure conditions. Ice was confirmed to remain superionic except at\nthe 20~Mbar\/12000~K and 40~Mbar\/20000~K conditions where it was fully\nfluid. The error bars on the computed $G$ values are dominated primarily by the uncertainty in the computed volume at the desired pressure, and secondarily by uncertainty in the $\\langle U_{DFT} - U_{classical} \\rangle_\\lambda$ term in the coupling constant integration. These free energies are combined using Equation \\ref{deltag} to\ngive $\\Delta G_{sol}$ representing the free energy change associated\nwith removing an H$_2$O from the ice phase and dissolving it in the\nhydrogen fluid at a concentration of molecule per 125 solute H\natoms. $\\Delta G_{sol}$ increases strongly with temperature, but shows\nonly a weak dependence on pressure within the 10--40 Mbar range under\nconsideration. The $\\Delta G$ values were found to be well converged with respect to wavefunction cutoff, k-point sampling to within the available error bars. The effect of the electronic entropy term on $\\Delta\nG_{sol}$ was found to be less than 0.1 eV in all cases.  From a linear\ninterpolation through the adjacent data points we estimated the\ntemperature at which $\\Delta G_{sol}$ passes through zero as 2400~K\n$\\pm$ 200~K at 40 Mbar, 2800~K $\\pm$ 200~K at 20 Mbar, and 3400~K\n$\\pm$ 600~K at 10 Mbar.  As shown in Figure 1, this is clearly far\nlower than any reasonable estimate for the core-mantle boundaries for\neither Jupiter or Saturn. The onset of high solubility occurs within\nthe portion of the phase diagram where ice is superionic, and does not\ncoincide with either the superionic-to-fluid transition or the\ncrystalline-to-superionic transition in ice.\n\nThe total $\\Delta G$ of solubility may be broken down into three\ncomponents: a $\\Delta U$ term from the potential energy, a $P \\Delta\nV$ term from the volume difference, and a $-T \\Delta S$ entropic\nterm. Figure 2 shows this breakdown as a function of temperature for\n20~Mbar.  The breakdown for other pressures looks similar. The $\\Delta\nU$ term, representing the difference in chemical binding energy,\nprovides approximately a 2 eV per molecule preference for ice\nformation at all temperatures where ice is superionic, but a much\nsmaller preference for the 12000~K case where ice is fluid.  The $PV$\nterm is indistinguishable from zero within the error bars, suggesting\nthat this is not a pressure-driven transition, in contrast to recent\nresults on the partitioning of noble gases between hydrogen and helium\nin giant planet interiors in which volume effects were found to be the\ndominant term \\citep{wilson-prl-10}. The $-T \\Delta S$ term dominates\nthe temperature-dependent behavior, underlining that ice dissolution\nis indeed an entropy-driven process.\n\nGiven the Gibbs free energy of solubility for the insertion of one\nH$_2$O into 125 H atoms we can determine an approximate Gibbs free\nenergy of solubility at other concentrations, by neglecting the\ncontribution of the oxygen-oxygen interaction and including only the\nentropic term arising from the mixing.  Under these approximations, we\nobtain the expression\n\n\\begin{eqnarray}\n\\frac{\\Delta G_{sol}[m] - \\Delta G_{sol}[n]}{k_BT} &=& (m+2) \\log \\left( \\frac{(m+2)V_H + V_O}{(m+2)V_H}\\right)\\nonumber \\\\ \n&-& (n+2) \\log \\left( \\frac{(n+2)V_H + V_O}{(n+2)V_H} \\right)\\nonumber \\\\\n&-& \\log \\left( \\frac{m+2}{n+2} \\right) ,\n\\label{conc}\n\\end{eqnarray}\n\nwhere $V_H$ and $V_O$ are the effective volumes of each H and O atom\nin the fluid. This approximation becomes invalid as the oxygen-oxygen\ninteraction term in the fluid oxygen phase becomes significant,\nhowever for our purposes it is sufficient to know that the saturation\nconcentration is significantly higher than the background\nconcentration. If a saturation concentration of oxygen in hydrogen\ndoes indeed exist then this may be expected to have a retarding effect\non the erosion of the core.\n\nWe must also consider possibility that hydrogen and oxygen could\ndissolve separately from the H$_2$O mixture, leaving behind a\ncondensed phase with stoichiometry other than H$_2$O. We tested two\ncases explicitly, computing the free energies of pure oxygen and\none-to-one HO phases at the Jupiter-like 40 Mbar, 20,000~K set of\nconditions. Both O and HO were found to be in a fully fluid state at these temperature\/pressure conditions. We found that HO had a free energy of solubility of -11.2\neV per oxygen, while pure oxygen had -22.8 eV per oxygen. Comparing to\nthe -8.9 eV per oxygen solubility of H$_2$O, this suggests oxygen-rich\ncondensed phases are less thermodynamically stable than the H$_2$O\nphase, and certainly far less favourable than dissolution of the dense\nphase into metallic hydrogen. We have also neglected the possibility\nof hydrogen-enriched dense phases such as H$_3$O. While it possible\nsuch phases may be somewhat energetically preferred to H$_2$O, it is\nextremely unlikely that the preference will be strong compared to the\n$8-12$ eV per O unit preference for solubility.\n\nIn Figure 3 we plot the estimated $\\Delta G_{sol}$ as a function of\nconcentration for various computed temperatures at a pressure of\n40~Mbar. A negative value of $\\Delta G_{sol}$ means that it is\nthermodynamically favorable to dissolve further ice into the hydrogen\nphase at a given hydrogen-phase ice concentration, and the point at\nwhich $\\Delta G_{sol}$ is zero is the saturation concentration.  For\n2000~K and 3000~K the saturation concentrations are estimated to be on\nthe order of 1:500 and 1:20 respectively, while for higher\ntemperatures the saturation concentration is much higher. Given that\nwe neglect O--O interactions in the fluid hydrogen phase, it is\ndifficult to precisely determine the saturation concentrations for\nhigher temperatures. However, we may safely say that the saturation\nconcentration for temperatures in excess of 3000~K is very much\ngreater than the background oxygen concentration, and hence that\nsolubility of ice into hydrogen at the core-mantle boundaries of\nJupiter and Saturn is expected to be strongly thermodynamically\nfavored.\n\n\\section{Discussion}\n\nThe consequences of core erosion for planetary evolution models have\nbeen previously considered by \\citet{stevenson-pss-82} and later\n\\citet{guillot-book}. The effects of core erosion can potentially be\ndetected either by orbital probes such as Juno or by atmospheric entry\nprobes, since the redistribution of core material throughout the\nplanet will manifest itself both by a smaller core (detectable from\ngravitational moments) and a higher concentration of heavy elements in\nthe atmosphere than would be expected in a planet without core\nerosion. Once core material has dissolved into the metallic H layers,\nthe rate at which core material can be redistributed throughout the\nplanet is expected to be limited by double diffusive convection\n\\citep{turner,huppert}. Since the higher density due to compositional\ngradients of the lower material interferes with the convection\nprocess, convection may be slowed significantly. \\citet{guillot-book}\nmodelled the effect of core erosion under the assumptions of fully\nsoluble 30 Mbar core in each planet. Under their assumptions up to 19\nEarth masses could have been redistributed from Jupiter's core but\nonly 2 Earth masses from Saturn's, the difference being Jupiter's\nhigher temperatures. While this prediction is subject to significant\nuncertainty in many aspects of the model, it does suggest that a\nredistribution of a significant fraction of the initial protocore is\npossible, at least in Jupiter.  Further refinement of models for the\nupconvection of core material and its observable consequences may be\nfruitful. The effect of core erosion on the heat transport and mass\ndistribution properties of Jupiter and Saturn should also be taken\ninto account in future static models of these planets' interiors.\n\nThese calculations can be expanded in several ways. We have neglected\nthe presence of helium in the hydrogen-rich mantle, however due to the\nlarge magnitude of $\\Delta G_{sol}$ and the chemical inertness of\nhelium we do not expect the presence of helium in the mantle to\nsignificantly affect the solubility behavior. We have explicitly made\nthe assumption that ice and hydrogen are in direct contact, an\nassumption which might fail in one of two ways. If hydrogen and helium\nare immiscible at the base of the atmosphere then the core may make\ndirect contact with a helium-rich layer. This, however, is unlikely in\nthe context of the calculations of \\citet{morales-pnas-09} who predict\nhydrogen-helium immiscibility only far away from the cores of Jupiter\nand Saturn.  The other possibility is that the ice layers of the core\nmay be gravitationally differentiated, leaving ice beneath layers of\nthe less dense planetary ices methane and ammonia. Since the bonding\nin these is similar to that in water ice one could assume that they\nshow a similar solubility behavior, but this analysis is the subject\nof future work.\n\nWe have also considered only the structure of the present-day\nplanet. As suggested by \\citet{slav}, a proper consideration of core\nsolubility must also include solubility during the formation process,\nas dissolution of the icy parts of the core into the accreting\nhydrogen during the formation may result in the amount of ice on the\ncore itself being small by the time the planet reaches its final\nsize. A treatment of the formation processes for Jupiter and Saturn\nusing ice solubilities derived from \\emph{ab initio} calculations may\nbe valuable.\n\nOur calculations strongly suggest that icy core components are highly\nsoluble in the fluid mantle under the conditions prevalent at the\ncore-mantle boundaries of Jupiter and Saturn. Since many\nrecently-discovered exoplanets are more massive and hence internally\nhotter than Jupiter, it can be expected that any initial icy cores in\nthese exoplanets will also dissolve. The presence of core erosion may allow models predicting a small present-day Jovian core to be made consistent with the large initial core required by core erosion, however models of the interior mass distribution of the planet will need to be revised to take the inhomogeneous composition of the lower layers implied by convection-limited core redistribution into account. Improved models which\ninclude core redistribution processes, combined with the\ndata from the Juno probe, may assist in understanding the history and\npresent structure of Jupiter and other planets in our own and in other\nsolar systems.\n\n\n\n\\acknowledgments\n\n This work was supported by NASA and NSF. Computational resources were supplied in part by TAC, NCCS and NERSC. We thank D. Stevenson for discussions.\n\n\n\\clearpage\n\n\n\\bibliographystyle{apj}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nBoth long and short GRB jets have to cross a significant amount of matter (the stellar atmosphere for long GRBs and the merger's ejecta in short ones) before producing the observed $\\gamma$-rays.  \nThis understanding has lead to great interest in jet propagation within surrounding matter and the question was explored both analytically \\citep[e.g.,][]{Blandford_Rees1974, Begelman_Cioffi1989, Meszaros_Waxman2001, Matzner2003, Lazzati_Begelman2005, Bromberg2011} and numerically \\citep[e.g.,][]{Marti+1995, Marti+1997, Aloy+2000, macfadyen_supernovae_2001, Reynolds+2001, Zhang+2004, Mizuta+2006, Morsony+2007, Wang+2008, Lazzati+2009, Mizuta+2009, Morsony+2010, Nagakura+2011, Lopez+2013, Ito+2015, Lopez+2016, Harrison2018}.\nThis, naturally, raises the possibility that some jets are ``choked\" during their propagation and are unable to break out of the surrounding dense medium. \nThe observed temporal distributions of both long \\citep{Bromberg+2012} and short \\citep{Moharan_Piran2017} GRBs suggest that this happens in both types of events and there are indications that this also happens in some Supernovae \\citep{Piran2019}.\n\nIn cases that the jet does not emerge we may still observed the signature of the cocoon that forms. \nFirst, the breakout of the shock driven by the cocoon produces a bright flash. \nFor example, a cocoon breakout is most likely the origin of low-luminosity GRBs ({\\it ll}GRBs) \\citep{Kulkarni1998, macfadyen_supernovae_2001, Tan+2001, campana_association_2006, Wang+2007, waxman_grb_2007, katz_fast_2010, Nakar_Sari2012,Nakar2015}. \nThese type of GRBs are rarely observed, however, when their low luminosity is taken into account it was realized that they are more numerous than regular LGRBs \\citep{Soderberg+2006}. \nAnother signature arises from the fast cocoon material that engulfs the star once the hot cocoon material breaks out and spreads. \nSpecifically, this material leads to very broad absorption lines that are visible as long as it is optically thick \\citep{Piran2019}. \nSuch lines have been observed in several SNe some accompanied by {\\it ll}GRBs  \\citep{Galama+1998, Iwamoto+1998,   Modjaz+2006, Mazzali+2008, Bufano+2012, Xu+2013, Ashall+2019,Izzo_et_al_2019} and others without \\citep{Mazzali+2000, Mazzali+2002, Mazzali+2009}. \nFinally, the cooling emission of the cocoon will also generate a potentially detectable UV-optical transient on time scale of hours to days \\citep{nakar_piran2017}.\n\nThe important signature that helps determining the origin of the broad absorption lines is the energy-velocity distribution of the fast moving material. \nRegular spherical explosion result in a very steep  distribution  with roughly $\\mathrm{d} E(v)\/\\mathrm{d} \\ln v \\propto v^{-5}$ \\citep[e.g.,][]{nakar_sari2010}. \nHowever, when a jet is involved in the explosion this distribution is expected to be much shallower with much more energy at high velocities.\nRecently, \\cite{eisenberg2022} have shown that when the jet is successful the cocoon generates a unique energy-velocity flat distribution with $\\mathrm{d} E\/\\mathrm{d} \\ln \\Gamma\\beta \\propto {\\rm const.}$ over a wide range of velocities from sub to mildly relativistic, where $\\beta=v\/c$ and $\\Gamma$ is the corresponding Lorentz factor. \nThey also found that, when the jet is choked, it leaves a unique signature of a flat energy-velocity distribution. \nHowever, in the case of choked jets the flat distribution covers a range of velocities that is narrower than that of outflows driven by successful jets. \nMotivated by these results we examine here in detail the energy-velocity distributions of different chocked jet, focusing on the relation between the properties of the choked jet and the final energy-velocity distribution of the outflow after it becomes homologous.\n\nFor our study we use a large set of 2D relativistic hydrodynamical simulations. \nWe consider explosions that are driven by choked jets in which we vary the opening angle and the engine working time of the jet as well as the structure of the progenitor. \nWe follow the simulations until the entire outflow becomes homologous and examine what is the relation between these properties and the outflow energy-velocity distribution.\n\nThe paper is structured as follows. \nIn Section \\ref{sec: methodology} we describe the numerical procedure adopted for the simulations. \nIn Section \\ref{subsec: sim_setup} we describe the code choice and the composite mesh structure adopted, while in Section \\ref{subsec: ics} we report in detail the setup for the stellar and interstellar environment and the initial conditions for the relativistic jet. \nNumerical aspects are discussed in two appendixes: a resolution study is described in Appendix \\ref{sec: appendix A} and in Appendix~\\ref{sec: appendix B} we explore the different use of a numerical smoothing function for the stellar density profile. \nIn Section \\ref{sec: results} we explore the results of our set of simulations.\nWe summarize our findings and consider the implications to  observations in Section \\ref{sec: conclusions}. \n\n\n\n\\section{Methodology}\n\\label{sec: methodology}\n\n\\subsection{Simulation Setup}\n\\label{subsec: sim_setup}\n\nOur simulations are performed using  the open source  massively parallel multidimensional relativistic magneto-hydrodynamic code {\\textsc{pluto}} (v4.3) \\citep{Mignone2007}. \nThe code uses a finite-volume, shock-capturing scheme designed to integrate a system of conservation laws where the flow quantities are discretized on a logically rectangular computational grid enclosed by a boundary. \nWe use the special relativistic hydrodynamics module in 2D cylindrical coordinates. \nWe perform our calculations using a parabolic reconstruction scheme combined with a third-order Runge-Kutta time stepping. \nWe also force the code to reconstruct the 4-velocity vectors at each time step. \n\nThe  2D  simulations enables us to reach  high resolution  with reasonable computational resources. \n3D simulations carried by \\citet{Harrison2018} suggest a similar generic evolution of the jet for the same parameters. \nThe main difference arises in the morphology of the  jet head in 2D simulations that is affected by a plug at the head front. \nThis plug  diverts some of the jet elements sideways to dissipated their energy in oblique shocks. \nThis difference is not significant for our purposes.\n\nWe chose the equation of state of the fluid to be ideal and with a constant relativistic polytropic index of $4\/3$. \nThis equation of state is applicable for a relativistic gas (as in the jet) as well to a radiation dominated Newtonian gas, such as the shocked stellar envelope.\n\nTo study the long term evolution of the jet and the cocoon from the star, we use a large grid spanning for several orders of magnitude. \nThis  allows us to track the evolution of the system for at least two minutes after the breakout.  \nAt that time the entire stellar envelope is shocked by the cocoon and it expands enough to become homologous. \nWe use a grid of size $4736 \\times 4636$ cells, with the radial cylindrical coordinate\\footnote{Throughout the paper $r$ is used for the 2D cylindrical radius while $R$ stands for the 3D radius.} \nextending within the range $r = [0,350] \\times 10^{10} \\cm$ and the vertical coordinate extending within the range $ z= [0.1 , 360] \\times 10^{10} \\cm$. \n\nWe use a combination of a  uniform and two  non-uniform mesh grids in $r-z$ coordinates with a decreasing resolution from the inner region of the simulation box to the outer boundaries. \nThe grid mesh is uniform in the inner part to maintain a high resolution of the jet injection and the formation of the resulting high pressure cocoon.\nThe uniform mesh has  $1000 \\times 900$ grid points extending in the ranges $r = [0, 1] \\times 10^{10} \\cm$ and $ z=[0.1, 1] \\times 10^{10} \\cm$ with a resolution along both coordinates of $\\Delta (r,z)_\\mathrm{unif.} = 10^{7} \\cm $.\nNext to the uniform mesh we placed a stretched mesh with $1278^2$ grid points extending along both coordinates within the range $(r,z) = [1,6] \\times 10^{10} \\cm$ with a stretching ratio of $\\sim1.0018$ \nThe number of grid points for this mesh is chosen such that its initial grid spacing is the same of to the adjacent uniform mesh $\\Delta (r,z)_\\mathrm{s, init} = \\Delta(r,z)_\\mathrm{unif.} = 10^7 \\cm$ and its final grid spacing is $\\Delta(r,z)_\\mathrm{s, final} = 10^8 \\cm $. \nWe cover the remaining grid  at larger distances with a logarithmic spaced mesh with $2458^2$ grid points extending within the range $(r,z) = [6, 360] \\times 10^{10}\\cm$. \nThe number of grid points is chosen such that the  grid spacing of the mesh at $(r,z) = 6 \\times 10^{10} \\cm$  coincides to the resolution of the stretched mesh, such that $\\Delta(r,z)_\\mathrm{log, init} = \\Delta(r,z)_\\mathrm{s, final} = 10^8 \\cm $. \nIn this way we ensure a smooth increase of the resolution without jumps for the entire simulation grid.\nA detailed resolution study for these simulations is reported in Appendix \\ref{sec: appendix A}.\n\nWe inject the jet along  the inner  lower $z$ boundary, denoted $z_0$ (see \\ref{sec:jet}). \nOtherwise, we impose a reflective  boundary condition at this boundary as it approximates the equatorial plane of the system. \nWe impose  axial-symmetric conditions for the inner vertical boundary. \nBoth outer boundaries are set to outflow.\n\n\n\\subsection{Initial conditions}\n\\label{subsec: ics}\n\n\\subsubsection{The star}\n\nWe approximate the stellar density profile  as a continuous power law that mimics the sharp decline of density in radius near the stellar edge\\footnote{This profile diverges at the origin but this region does not influence the jet propagation and it is not included in our computational domain.}\n\\begin{equation}\n\\label{eq: rho_profile}\n \\rho(R) = \\begin{cases}\n \\rho_* \\left( \\dfrac{R_*}{R} - 1\\right)^2 + \\rho_0 ,& \\mathrm{for} ~  R \\leq  R_* \\ , \\\\ \n \\rho_0, & \\mathrm{for} ~ ~ R > R_* \\ .\n \\end{cases}\n\\end{equation}\nHere we choose $\\rho_* = 100 ~ \\g ~ \\cm^{-3}$ and $R_* = 3\\times 10^{10} \\cm$. \nThe total integrated mass of the star is $M_* = (9 \\pi \/ 5) ~ M_\\odot$ (see \\ref{sec:scale} for scaling of these parameters to other values.)\n\nFor this  density profile  the local slope, $\\alpha \\equiv \\mathrm{d} \\log \\rho(R)\/\\mathrm{d} \\log R = {-2}\/{(1-R\/R_*)} $. \nThe slope, $\\alpha$, reaches the critical value of 3 for $R = R_*\/3$. Beyond this value a spherical blast wave  accelerates and eventually looses causality. \nWe present our results for this specific density profile. \nHowever, in Sec.~\\ref{sec: diff_profiles} we show that the results for different stellar density profiles (both inner and outer) are qualitatively similar.\n\nSurrounding the star we have an external CSM density of $\\rho_0 = 1.67 \\times 10^{-21} ~ \\g ~\\cm^{-3}$.\nThis exact value is unimportant as it is added just to avoid a numerical vacuum. \nThe interaction of the jet or the cocoon outflow with this CSM is insignificant. \nTo avoid numerical artifacts arising from the sudden drop in density at the edge of the star  we smooth the density of the outer edge of the star with a power law: \n\\begin{equation}\n\\label{eq: smooth}\n    \\rho_\\mathrm{smooth} (R) = \\rho_\\mathrm{s} \\left( \\dfrac{R}{R_\\mathrm{s}} + 1\\right)^{-8} \\ , \n\\end{equation}\nwith $ \\rho_\\mathrm{s} = 0.05 ~ \\g ~ \\cm^{-3}$ and a gradient scale of $R_\\mathrm{s} = 5 \\times 10^8 \\cm$. \nWe verified that this arbitrary choice of the smoothing function does not affect our results (see Appendix ~\\ref{sec: appendix B}).\nIn order to avoid any initial random motion we set a uniform and low ambient pressure of $P = 3.5 ~ \\keV ~ \\cm^{-3}$ within the simulation grid. \n\n\\subsubsection{The jet}\n\\label{sec:jet}\n\nWe inject a collimated jet with a constant luminosity $L_\\mathrm{j}$, operating for $t_\\mathrm{e}$ so that the total injected energy is $E_{0} = L_\\mathrm{j} \\times t_\\mathrm{e} =  10^{51} \\erg$. \nA uniform jet is injected through a nozzle with a velocity in the $z$ direction with an initial bulk Lorentz factor $\\Gamma_{0,\\mathrm{j}}$ a  density $\\rho_\\mathrm{j} $ and a specific enthalpy $h_\\mathrm{j} \\gg 1$. \nBeing relativistically hot the jet spreads quickly to form an initial opening angle  $\\theta_\\mathrm{j} \\simeq 1\/(1.4 \\Gamma_{0,\\mathrm{j}})$ (see details on this injection method at \\citealt{Mizuta2013} and \\citealt{Harrison2018}).\n\nThe jet is numerically initialized by the injection of density, pressure, and momentum along the $z-$direction through a nozzle parallel to the $z-$axis with a radius $r_\\mathrm{j}$ at an initial height $z = z_0$. \nThe head cross section is then  $\\Sigma_\\mathrm{j} = \\pi r_\\mathrm{j}^2$. \nFor an initial opening angle $\\theta_\\mathrm{j} > 0.1~\\rad$, we set up $r_\\mathrm{j} = 10^8~ \\cm$, allowing a sufficient mesh coverage over the nozzle and we set the initial injection height at $z_0 = 10^9~\\cm$. \n\nWe consider a constant jet luminosity $L_\\mathrm{j}$.\nThis determines the product $\\rho_\\mathrm{j} h_\\mathrm{j}$ as:\n\\begin{equation}\n    \\label{eq: rho_j}\n    \\rho_\\mathrm{j} h_\\mathrm{j} = \\dfrac{L_\\mathrm{j}}{\\Sigma_\\mathrm{j}  \\Gamma_{0,\\mathrm{j}}^2 c^3} \\ .\n\\end{equation}\nWe choose $h_\\mathrm{j} = 100$.  \nThis choice of the enthalpy is arbitrary, as long as $h_\\mathrm{j} \\gg 1$.\nThe jet's pressure is given by $  P_\\mathrm{j} = (h_\\mathrm{j} - 1) {\\rho_\\mathrm{j} c^2}\/{4}$.\n\nWe explored the parameter space running simulations for different initial values of $L_\\mathrm{j} $ at steps of $2.5 \\times 10^{50}~\\erg~\\s^{-1}$ from $2.5 \\times 10^{50}~\\erg~\\s^{-1}$ to $2 \\times 10^{51}~\\erg~\\s^{-1}$ for a total of 9 different luminosities. \nFor each value of the luminosity, we run simulations for a set of different opening angles $\\theta_\\mathrm{j} = [0.05, 0.1, 0.2, 0.4, 0.6]~ \\rad$. \nAs we keep the total jet energy fixed these conditions translate to different engine working times $t_\\mathrm{e} = 10^{51}\\erg \/ L_\\mathrm{j}$. \nFor each of the 9 values of the luminosity we run 5 different values of the opening angle for a total of 45 simulations.\n\n\n\\subsubsection{Scaling  relations}\n\\label{sec:scale}\nWhile we consider specific numerical values for the stellar and jet parameters, our solutions can be scaled to other values. \nThe equation of motion of the forward shock speed $\\beta_\\head$, is regulated by  $\\Tilde{L}$ \\citep{Matzner2003,Bromberg2011}:\n\\begin{equation}\n\\label{eq: tilde_L2}\n    \\Tilde{L} \\simeq  \\dfrac{L_\\mathrm{j}}{ \\Sigma_\\mathrm{j} \\rho(R) c^3} \\ ,\n\\end{equation}  \nwith\n\\begin{equation}\n\\label{eq: beta_h}\n    \\beta_\\head = \\dfrac{1}{1+{\\Tilde{L}}^{-1\/2}} \\ .  \n\\end{equation}\nThe stellar size $R_*$, is the  scale  length of t the system. \nThe scalings $\\Sigma_\\mathrm{j} \\propto R_*^2$ and $\\rho(R\/R_*) \\propto \\rho_*$ we can express $\\tilde{L}$ as\n\\begin{equation} \n\\Tilde{L} \\propto \\dfrac{E_0}{t_\\mathrm{e} \\rho_* R_*^2} \\ . \n\\end{equation}\nIf we scale the  stellar  radius as $R_* = \\lambda R_*'$ we have to scale the density and the jet luminosity accordingly in order to maintain $\\Tilde{L}$ and $\\beta_\\head$ unchanged. \n\nAs we show later the location where the jet is choked (i.e. where the last element launched by the jet reaches the head) with respect to the stellar radius has also to be constant. \nThe choking location $z_\\mathrm{ch}$ is roughly proportional to the engine time $t_\\mathrm{e}$ \\citep{Nakar2015}:\n\\begin{equation}\n\\label{eq: zchoke}\n    z_\\mathrm{ch} = \\int_0^{t_\\mathrm{ch}} \\beta_\\head c \\mathrm{d} t \\simeq \\beta_\\head c t_\\mathrm{ch} = \\dfrac{\\beta_\\head c }{1-\\beta_\\head} t_\\mathrm{e}  \\ .  \n\\end{equation}\nwhere $  t_\\mathrm{ch} =  {t_\\mathrm{e}}\/{(1 - \\beta_\\head)}$ is the choking time. \nIf $\\beta_\\head$ is  kept constant than any transformation on $R_*$ will leave $z_\\mathrm{ch} \/ R_*$ unchanged if $t_\\mathrm{e} = \\lambda t_\\mathrm{e}' \\propto R_*$. \nThus, scaling $E_0 = \\eta E_0'$ and $R_* = \\lambda R_*'$ we require that and $\\rho_* = \\eta \\lambda^{-3} \\rho_*'$.\n\nBecause $M_* \\propto \\rho_* R_*^3$, when keeping this scaling of $t_\\mathrm{e}$ and $R_*$, we can rewrite Eq.~\\ref{eq: tilde_L2} as $ \\Tilde{L} \\propto E_0\/M_*$, which means that since $\\Tilde{L}$ is kept  constant the typical  velocity of the system $v_0 = (2E_0\/M_*)^{1\/2}$ is also conserved under these transformations.\n\nTurning to the jet parameters we recall that the only relevant quantities are $L_\\mathrm{j}$, $t_\\mathrm{e}$ and $\\theta_\\mathrm{j}$.\nThe first two determine $E_0$ and the latter determined $\\Gamma_{0,\\mathrm{j}}$. \nThe luminosity, together with the stellar parameters determine the produce $\\rho_\\mathrm{j} h_\\mathrm{j}$ with the condition that $h_\\mathrm{j}$ while arbitrary should be much larger than one. \n\nIn summary, given the physical scales $R_*$, $\\rho_*$, $E_0$, $v_0$, $t_\\mathrm{e}$ and the scalings $R_* = \\lambda R_*'$, $E_0 = \\eta E_0'$, the parameters defining the physics of our system, i.e. $\\tilde{L}$, $z_\\mathrm{ch}\/R_*$, will not change if  $t_\\mathrm{e} = \\lambda t'_\\mathrm{e}$ and $\\rho_* = \\eta \\lambda^{-3} \\rho'_*$.\n\n\\section{Results}\n\\label{sec: results}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.352]{figures\/frame_different_times.pdf}\n    \\caption{A canonical jet ($\\theta = 0.2$ rad; $L_\\mathrm{j} = 10^{51} \\erg \\ \\s^{-1}$ and $t_e=1 \\s$)  launched in a test star with a density profile given by Eq.~\\ref{eq: rho_profile}. \n    The four panels show, from left to right, the relativistic $\\Gamma\\beta$ factor, the density $\\rho$, the pressure $P$ and a scalar tracer of the ejected jet material (an unitary value implies pure jet material with no mixing). \n    All the quantities but the $\\Gamma\\beta$ factor are normalized to the respective maximum values in order to increase the color contrast. The scale for the velocity of our system is dictated by $v_0$. With the above parameters  $\\beta_0 =v_0\/c =  0.014$ and the relativistic regime begins when $\\beta \\Gamma\/\\beta_0\\approx 50$.\n    The different rows show the evolution of the jet at $t = t_\\mathrm{e} = 1~\\s$ (when the engine stops, first row) - the jet is clearly seen here surrounded by a cocoon, $t=1.3~\\s$ (choking time, second row) - the jet has disappeared as its tail reached the head, $t=14.2~\\s$ (shortly after the breakout, third row), and $t=16.0~\\s$ (sideways spreading outside the star, fourth row). \n    To enhance the color contrast we change the normalization scale of $\\rho\/\\rho_\\max$ and $P\/P_\\max$ in the third and fourth row in order to capture the tenuous material and pressure spilling outside the star. }\n    \\label{fig: jet_t01}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.35]{figures\/frame_040.pdf}\n    \\caption{4-panel figure of the breakout of an unchoked jet at $t=t_\\mathrm{e} = $4 s with an opening angle of $\\theta_\\mathrm{j} = 0.1$ rad with a luminosity of $L_\\mathrm{j}=2.5 \\times 10^{50}~\\erg~\\s^{-1}$. \n    In this case the jet tail did not catch up the jet head, resulting in an unchoked breakout and with a head propagation operating as the jet engine were still active. We use the same color scale of the fourth row of Fig.~\\ref{fig: jet_t01} for comparison.}\n    \\label{fig: jet_t4}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.38]{figures\/spreading_angle_evolution_theta_02_alternate_ratio.pdf}\n    \\caption{\\emph{Left}: A schematic figure of how the aspect ratio $\\theta(t)$ is defined (Eq.~\\ref{eq: theta_of_t}) superposed on the density map of the jet cocoon while it is still inside the star. \n    The dashed line represents the maximal width of the cocoon, while the dashed-dotted line represents the maximal height. \n    \\emph{Centre and right}: Evolution of the aspect ratio of the jet as a function of time (central panel) and of the cocoon\/jet head location $z_\\head$ (right panel). \n    The square dots represent the choking time of the jet while the triangular dots represent the breakout time of the cocoon. \n    In both frames the thick dashed horizontal line represents the condition for an isotropic blast wave ($r_\\mathrm{c} = z_\\head$), while the horizontal dotted line represent the original opening angle of the launched jet ($0.2~\\rad$ for the figure). \n    On the right-hand side the dashed-dotted vertical line represents the star edge, located at $R_* = 3 \\times 10^{10}~\\cm$.}\n    \\label{fig: spreading_angle}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.35]{figures\/curves_divided_by_volume_bo_final.pdf}\n    \\caption{Classification of the energy-velocity distribution grouped according to their different cocoon volume $V_\\bo$ at $t=t_\\bo$. \n    Each curve is marked by a different shade of color and a triplet of numbers indicating, in order: the opening angle in radians, $t_\\mathrm{e}$ in seconds, and then $\\sqrt{V_*\/V_\\bo}$. The dashed black line represents the isotropic case.}\n    \\label{fig: choking_height}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.34]{figures\/correlation_volume_cutoff.pdf}\n    \\caption{Correlation between $\\sqrt{V_*\/V_\\bo}$ and the cutoff of the energy-velocity distribution for the simulated set of jets for a  cutoff value of  $0.25$ of the maximum of each differential energy distribution in the plots of Fig.~\\ref{fig: choking_height}. \n    The red-colored area represents 1-standard deviation error of the red fit curve. \n    From the fitting formula we see that the distribution cutoff corresponds to 4 times the the square root of the volume ratio at the breakout for values of $(\\Gamma\\beta)_\\mathrm{cut}$ above 3. \n    The black dashed line represents the linear limit for high cutoffs.}\n    \\label{fig: volume_cutoff}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.34]{figures\/V_bo_z_ch.pdf}\n    \\caption{Correlation between the cocoon volume at the breakout and the choking height of the jet for different values of the initial opening angle $\\theta_\\mathrm{j}$. \n    The gray-shaded region represents the 1$\\sigma$ error for the fits. }\n    \\label{fig: v_bo_z_ch}\n\\end{figure}\n\n\\subsection{The jet-cocoon system}\n\\label{subsec: analysis}\n\nWe start analyzing our simulation set considering a jet with our \\emph{canonical} parameters of 1-sided luminosity of $L_\\mathrm{j} = 10^{51} \\erg\/\\s$ and $\\theta_\\mathrm{j} = 0.2~\\rad \\simeq 10^\\circ$.\n\nWhile advancing through the stellar atmosphere the interaction of the relativistic jet with the stellar material results in a forward-reverse shock structure that is called the head of the jet \\citep{Blandford_Rees1974, Begelman_Cioffi1989, Meszaros_Waxman2001, Matzner2003, Lazzati_Begelman2005, Bromberg2011}. \nThe jet head propagation  velocity, $\\beta_\\head$, is much lower than the jet velocity before it reaches the head and for typical GRB jets it is Newtonian. \nThe shock-heated jet and stellar material that enters the head flow sideways because of the high head pressure and form a pressurized  cocoon which enshrines the jet.  \nThe contact discontinuity between the material shocked in forward and the reverse shocks divides the cocoon to  inner and outer parts. \nThe inner cocoon is composed of tenuous jet material which has crossed the reverse shock while the outer cocoon is composed of denser shocked stellar material. \nThe cocoon exerts a pressure on the jet such that, if sufficiently high,  collimates it, thus reducing its opening angle and consequently reducing the jet cross section compared to the uncollimated jet. \n\nWithin our chosen stellar structure model the jet head moves at a constant velocity at the inner region of the core where the local density slope $\\alpha = 2$.  \nIf the jet reaches outer regions where $\\alpha > 2$  it starts accelerating.\n\nThe jet is \\emph{choked} if the engine stops while the jet is propagating within the stellar envelope and the last jet element launched by the engine (\\emph{tail}, hereafter) catches up with the jet head before the latter breaks out of the star.\nIn this case all the engine's energy goes into the cocoon. \nClearly, the choking height  (Eq.~\\ref{eq: zchoke}) satisfies $z_\\mathrm{ch} < R_*$. \nOtherwise we define the jet as unchoked or successful. \nThroughout the following analysis we focus mostly on jets choked at various depths inside the star. \nFor comparison we show also the case of an unchoked jet breaking out of the star.\nFor a detailed study on the energy-velocity distribution of stellar explosions which are driven by successful jets see \\cite{eisenberg2022}.\n\nWe divide the evolution of the jet to three different phases: 1) the injection phase and choking phase $t < t_\\mathrm{ch}$, 2) the cocoon expansion phase $t_\\mathrm{ch} < t < t_\\bo$ and 3) the cocoon breakout phase $t > t_\\bo$.  \nThe different phases of a choked jet are shown in Fig.~\\ref{fig: jet_t01}.\n\n\\subsubsection{Injection and Choking: $t \\leq t_\\mathrm{ch}$}\nThe engine operates for $t_\\mathrm{e}$ producing a jet. \nThis is clearly seen in the first row of Fig.~\\ref{fig: jet_t01} in both the rightmost panel showing $\\beta \\Gamma$ and the leftmost one showing that the tracer of the jet material is concentrated mostly within a narrow cylinder along the symmetry axis $z$ with radius $r \\simeq r_\\mathrm{j}$ (color dark red).\nThis behaviour is typical of the collimated regime \\citep{Bromberg2011}.\n\nAfter the jet engine stops the last jet material launched at the injection nozzle propagates upwards. \nAt $t_\\mathrm{e}$ the jet head is still unaware that the engine stopped and the jet head continues to propagate with $\\beta_\\head$ (in this specific simulation $\\beta_\\head \\simeq 0.2)$ after the central jet engine is switched off.\nHowever, as $\\beta_\\head < 1$ while the jet material moves at $\\beta_\\mathrm{t} \\simeq 1$ the jet tail catches up with the head.\nOnly at that time the information that the engine stopped reaches the head and the reverse shock within the jet disappears. \nThis is the time where the jet is choked.  \n\nAs the head propagates with a velocity of $\\beta_\\mathrm{h} c$ and the tail propagates at $\\beta_\\mathrm{t} \\simeq 1$, we can estimate the the time that the jet tail will catch up with the head  at  $t_\\mathrm{ch}$, as defined in Eq.~\\ref{eq: zchoke}.\nUntil $t < t_\\mathrm{ch}$ the jet continues to drive the head forward through the stellar atmosphere. \nThe second row of Fig.~\\ref{fig: jet_t01} shows the  system  at $t = t_\\mathrm{ch}$, roughly 0.3 seconds after the end of the engine activity, in this specific simulation. \nOne can see that the very fast jet material around the core disappeared. At this stage all the jet's energy has been dissipated and given to the surrounding cocoon. \n\n\\subsubsection{Cocoon expansion: $ t_\\mathrm{ch} < t < t_\\bo$}\n\nAfter the jet choking, the cocoon becomes less and less collimated  and proceeds spreading sideways while the forward shock decelerates when it is deep within the envelope and accelerates as it reaches the steep density gradient near the stellar edge \\citep{Irwin_Nakar_Piran2019}.\nDuring the propagation the inner cocoon transfers energy to the freshly shocked material (via PdV work).\n\n\\subsubsection{Brekout: $t> t_\\bo$}\nAfter the breakout the cocoon material spreads both radially and tangentially to engulfs the stellar surface, quickly shrouding the breakout point from most observers (see  the last row of Fig.~\\ref{fig: jet_t01}). \nThe star is blanketed by the ejecta in a time equal to $t_\\mathrm{wrap} \\simeq \\pi R_* \/2 v_\\mathrm{bo}$, where $v_\\mathrm{bo}$ is the breakout velocity of the cocoon near the pole.\nThe shock driven by the cocoon also moves tangentially towards the equator at a slower pace until the entire stellar envelope is shocked at $t_\\mathrm{shock} \\simeq \\pi R_* \/(2 v_\\mathrm{p})$ where $v_\\mathrm{p}$ is the pattern velocity at which the spilled material travels along the stellar surface \\citep{Irwin_et_al2021}. \nShortly after reaching  the equator the shocked material propagates almost radially and outwards and it becomes homologous once the outflow reaches $\\sim 2R_*$. \n\n\\subsubsection{Successful jets} \nJets whose engine operates long enough break out from the stellar envelope before the end of the activity of the central engine. \nThese jets are not choked and can preserve an ultra-relativistic velocity once they get out of the star. \nWe show an example of a successful jet in Fig.~\\ref{fig: jet_t4}. \nBecause the jet broke out without being choked the cocoon structure inside the star is mostly collimated along the vertical axis. \nFrom the first and fourth panel which show $\\Gamma\\beta$ and the jet tracer respectively, it is evident how the innermost region is still dominated  by tenuous, highly relativistic jet material. \nComparing the last row of Fig.~\\ref{fig: jet_t01} and Fig.~\\ref{fig: jet_t4} we notice that for the same normalized density and normalized pressure scale, the longer duration of the unchoked jet results in expulsion of denser and faster stellar material with respect to the choked jet. \n\n\\subsection{The spreading angle and the cocoon volume}\n\nTo describe quantitatively the geometry of the jet-cocoon during its propagation within the stellar envelope we use the aspect ratio, defined as\n\\begin{equation}\n\\label{eq: theta_of_t}\n    \\theta(t) = \\dfrac{\\max(r_\\mathrm{c}(t))}{z_\\head(t)}\\ ,\n\\end{equation}\nwhere $r_\\mathrm{c}$ is the cocoon cylindrical radius and $z_\\mathrm{h}$ is the head position. \nFor $\\theta \\ll 1$ the aspect ratio is a good approximation of the cocoon spreading angle.\nThe expanded cocoon at the moment of the breakout is shown in the third row of  Fig.~\\ref{fig: jet_t01}. \nThe steep density transition results in an elongation and acceleration of the cocoon and the ejection of low-density material from the star, which rapidly engulfs the star's external layers. \nWe define the breakout angle $\\theta_\\bo$ as the geometric opening angle measured at the breakout time $t=t_\\bo$, namely \n\\begin{equation}\n\\label{eq: theta_bo}\n    \\theta_\\bo = \\theta (t_\\bo) = \\dfrac{\\max(r_\\mathrm{c}(t_\\bo))}{R_*}\\ .\n\\end{equation}\n\nThe evolution of $\\theta(t)$ for $\\theta_\\mathrm{j} = 0.2~\\rad$ and several values of $t_\\mathrm{e}$ is reported in Fig.~\\ref{fig: spreading_angle}. \nAt first, immediately after injection, the aspect ratio starts growing. \nThe growth continues until $z_\\head$ is roughly twice the injection radius, $z_0$, at which point the aspect ratio starts decreasing, approaching the point where the cocoon opening angle is comparable to $\\theta_\\mathrm{j}$. \nThis evolution reflects the time it takes the pressure in the cocoon to build up to the point that it starts collimating the jet effectively (see \\citealt{Harrison2018} for details). \nThe evolution of the aspect ratio changes dramatically immediately as the jet is fully choked. \nSince there is no more fresh jet material to drive the head its velocity drops sharply. \nAt the same time the cocoon pressure, and thus its sideways expansion, is not affected. \nThe result is that the aspect ratio grows continuously after $t_{\\mathrm{ch}}$. \nThere is a short episode, just before and after the breakout when the aspect ratio decreases, as the head accelerates near the edge of the star and after the breakout. \nSoon after that the aspect ratio starts increasing rapidly as some of the material that broke out of the star spreads sideways at speed that is close to the speed of light. \nOne clear property that is seen in the figure is that jet that are choked more deeply have longer time to expand before they breakout and therefore a deeper choking results in a wider cocoon with a larger volume at the time of breakout. \nAs we show next this fact has important implications for the energy-velocity distribution of the outflow.  \n\nThe volume of the cocoon at breakout, $V_\\bo$  is another parameter that describes the  properties of the jet-cocoon system. \nAs the energy of a choked jet is given to the cocoon, for a given energy, the cocoon mass (and hence volume) at breakout, will corresponds to a typical expansion velocity of the cocoon material. \nAs the volume-averaged density in the shocked cocoon material and the volume-averaged density of the star are roughly the same, we can define a characteristic velocity at the breakout linked with the breakout volume, namely:\n\\begin{equation}\n\\label{eq: volume_bo}\n    \\beta_\\bo \\simeq \\beta_0 \\sqrt{\\dfrac{V_*}{V_\\bo}} \\ . \n\\end{equation}\n\n\\subsection{The Energy-velocity distribution}\n\nFig.~\\ref{fig: choking_height} depicts the energy-velocity distribution of the entire set of simulations for different values of the engine working time $t_\\mathrm{e}$ and different initial opening angles $\\theta_\\mathrm{j}$ at $t=120 \\s$ (when the outflow is homologous and kinetic energy dominates). \nThe $x$-axis is normalized by $\\beta_0$ and the $y$-axis by $E_0$. \nEach curve is differentiated by color and labeled by a triplet of numbers describing, respectively, $\\theta_\\mathrm{j}$, $t_\\mathrm{e}$, and $\\sqrt{V_*\/V_\\bo}$.  \nWe  grouped the different curves according to $\\sqrt{V_*\/V_\\bo}$. \nFor a comparison we superposed the energy-velocity distribution of an isotropic spherically symmetric explosion (black-dashed line) for all panels. \n\nFig.~\\ref{fig: choking_height} shows, first, that in all cases the energy-velocity distribution exhibits a roughly constant energy per logarithmic scale of $\\Gamma\\beta$ over a range of velocities. \nThe distribution rises quickly before this rough plateau starts and decays sharply after it ends. \nThe rough plateau always starts at $\\beta_0$ and its highest velocity is determined almost entirely by $V_{\\bo}$, with a weak dependence on the jet opening angle. \nTo estimate the highest velocity of the flat part of the distribution we define $\\beta_{\\rm cut}$ as the velocity obtained when the energy-velocity distribution drops to 1\/4 of its maximum value. \nThis arbitrary definition provides a velocity that is slightly larger than the end of the plateau (e.g., in the spherical case $\\beta_{\\rm cut}=2\\beta_0$).\nFig~\\ref{fig: volume_cutoff} shows that there is a strong positive correlation between $\\beta_{\\rm cut}$  and $\\sqrt{V_*\/V_{\\bo}}$. \nFor small values of $\\sqrt{V_*\/V_{\\bo}}<3$, where typically $z_{\\mathrm{ch}} \\ll R_*$, we see that $\\beta_{\\rm cut}\\approx \\beta_{\\bo}$. \nHowever, for larger values of $\\sqrt{V_*\/V_{\\bo}}$ where the choking takes place not very deep within the stellar envelope, $\\beta_{\\rm cut} > \\beta_{\\bo}$. \nThe origin of the material faster than $\\beta_{\\bo}$ in these cases is the inner cocoon, which retain a significant fraction of its energy at the time of the breakout and outer cocoon material that is close to the edge of the star, where the forward shock is faster than $\\beta_{\\bo}$.\n\nThe value of $V_*\/V_{\\bo}$ is expected to depend  on the jet opening angle and the choking depth. The jet opening angle determines the aspect ratio of the cocoon as long as the head that is pushed ahead by the jet is feeding the cocoon ($\\theta\\approx \\theta_\\mathrm{j}$), while the choking depth determines by how much this aspect ratio increases until the breakout.  \nFig.~\\ref{fig: v_bo_z_ch}  depicts the correlation between $V_\\bo$ and $z_\\mathrm{ch}$ for different values of the initial $\\theta_\\mathrm{j}$. \nAs expected, $V_\\bo$ is a function of $z_\\mathrm{ch}$ and $\\theta_\\mathrm{j}$. \nA deeper choking height and a wider jet correspond to a larger  cocoon volume upon breakout.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.64]{figures\/scheme.pdf}\n    \\caption{A sketch of the division of the stellar volume for the analysis of the distribution of the stellar material. \n    The cocoon shape taken at $t=t_\\bo$ is overlaid. We associate a scalar tracer to each of the four sectors: I) internal-axis, II) external-axis, III) internal-equatorial, and IV) external-equatorial. }\n    \\label{fig: scheme}\n\\end{figure}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.34]{figures\/frame_195.pdf}\n    \\caption{Maps of the four tracers I, II, III and IV (from left to right) associated with the four sectors of the stellar material (see Fig.~ \\ref{fig: scheme}). \n    The maps show the distribution of the stellar material at $t=60~\\s$ resulting  from  a jet with canonical parameters (see Sec~\\ref{subsec: analysis}). }\n    \\label{fig: 4_tracers}\n\\end{figure*}\n\n\\subsubsection{The origin of ejecta with different final velocities}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.51]{figures\/energy_vs_velocity_distribution_tracer3_195_single.pdf}\n    \\caption{The energy-velocity distribution of the four different matter tracers associated with the sectors depicted in Fig.~\\ref{fig: 4_tracers}.\n    Matter from region II (external along the axis) dominates the highest velocity region. \n    Matter from region III (internal equatorial) dominates the low velocity regime. \n    Matter from region I (internal along the axis) dominates the intermediate region and the plateau. \n    Matter from IV (external equatorial) is always subdominant.}\n    \\label{fig: e_vs_v_tracers}\n\\end{figure}\n\nTo understand the origin of the various components of the outflow, we tracked the distribution of the ejected material using four different scalar tracers associated with four distinct regions of the star. \nThe division to the different regions is determined at the time of the breakout and it is shown in Fig.~\\ref{fig: scheme}. \nThe tracers follow the mass in each of this region (at the time of breakout): I) internal-axis, II) external-axis, III) internal-equatorial, IV) external-equatorial.\n\nFig.~\\ref{fig: 4_tracers} shows the distribution of the stellar material from each of the regions at $t=60~\\s$, roughly $46~\\s$ after the breakout.\nFig.~\\ref{fig: e_vs_v_tracers} shows the energy-velocity distributions of the four sectors.\n\nWe see that the  quasi-spherical outflow that leads the ejecta is made only of tenuous material coming from the on-axis, external layers of the progenitor directly above the expanding jet cocoon (region II). \nThis component contains only around $2\\%$ of the total stellar mass but it contains 11\\% of the total ejecta energy. \nEvidently, the fastest ejecta is dominated by this sector. \nThe  material associated with the stellar core part that is along the axis (region I; first panel in Fig.~\\ref{fig: 4_tracers}) is much more concentrated than that of the external-axis region (II) but much more extended than the two equatorial sectors. \nIt contains $30\\%$ of the total stellar mass and 46\\% of the outflow energy. \nThis section dominates the energy distribution over a wide range of velocities. \nAlmost all the rest of the mass and the energy are contained in the internal-equatorial sector (III) which carries $60\\%$ of the mass and 32\\% of the energy. \nIt dominates the energy at low velocities $\\lesssim \\beta_0$. \nFinally, the outer-equatorial section carries  $5\\%$ of the ejecta mass and 11\\% of its energy. \nAll its material is moving at intermediate velocities and it is subdominant at all velocities. \n\n\\subsection{The effect of the stellar density profile}\n\\label{sec: diff_profiles}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.43]{figures\/different_profile.pdf}\n    \\caption{\\emph{Left}: Energy-velocity distributions of  jet simulations with canonical parameters for four different stellar density profiles. \n    The blue line ($\\rho \\propto R^{-2} (R_*-R)^2 $) represents the profile used in most of the previous simulations. \\emph{Right}: A comparison of the  energy distributions of the cases $\\alpha=2,n=3$ and $\\alpha=2.5,n=3$ with those arising from from two jets choked in a star with a canonical density profile at the same heights $z_\\mathrm{ch}\/R_*$, respectively.}\n    \\label{fig: density_profiles_s}\n\\end{figure*}\n\n\\begin{center}\n\\begin{table*}\n\\label{tab: different_profiles}\n\\begin{tabular}{|c||ccccc|}\n\\hline \nJets & $t_\\mathrm{e}$ [s] & $\\theta_\\mathrm{j}$ [rad] & $\\rho(r)$ & $z_\\mathrm{ch}\/R_*$ & $t_\\bo$ [s] \\\\ \n\\hline \nCanonical & 1 & 0.2 & $\\propto R^{-2} (R_*-R)^2 $ & 0.21 & 13.9 \\\\ \n$\\alpha2$\\_$n2.5$ & 1 & 0.2 & $\\propto R^{-2} (R_*-R)^{2.5} $ & 0.25 & 11.2 \\\\ \n$\\alpha2$\\_$n3$ & 1 & 0.2 & $\\propto R^{-2} (R_*-R)^3 $ & 0.25 & 9.5 \\\\ \n$\\alpha2.5$\\_$n2$ & 1 & 0.2 & $\\propto R^{-2.5} (R_*-R)^2 $ & 0.28 & 6.8 \\\\ \n$\\alpha2.5$\\_$n3$ & 1 & 0.2 & $\\propto R^{-2.5} (R_*-R)^3 $ & 0.37 & 4.0 \\\\ \n\\hline \nCanonical\\_t1.33 & 1.33 & 0.2 & $\\propto R^{-2} (R_*-R)^2 $ & 0.25 & 11.5 \\\\ \nCanonical\\_t2 & 2 & 0.2 & $\\propto R^{-2} (R_*-R)^2 $ & 0.37 & 8.3 \\\\ \n\\hline\n\\end{tabular}\n\\caption{Properties of the jets injected in different density profiles. The table lists the engine working time $t_\\mathrm{e}$, the initial opening angle $\\theta_\\mathrm{j}$, the density profile $\\rho(r)$ used in the run, the choking height relative to the star radius $z_\\mathrm{ch}\/R_*$, and the breakout time $t_\\bo$.}\n\\end{table*}\n\\end{center}\n\nTo study the effect of different stellar density profiles we consider stellar density profiles that can be written as:\n\\begin{equation}\n\\label{eq: rhogen}\n    \\rho (R) = \\rho_*\\left(\\dfrac{R_*}{R}\\right)^\\alpha  \\left(1-\\dfrac{R}{R_*}\\right)^n \\ ,\n\\end{equation}\nwhere $n$ is the outer slope at the edge, and $\\alpha$ is the inner slope, with $\\alpha <3$. \nThe density profile described by Eq.~\\ref{eq: rho_profile}, which is used through the rest of the paper, is roughly equivalent to the case of $\\alpha=2,~ n=2$ and it will be referred as the \\emph{canonical profile} hereafter. \nThe profiles that we consider are listed in Table.~\\ref{tab: different_profiles}. For each profile we run a simulation with our canonical jet parameters, $\\theta_\\mathrm{j} = 0.2~\\rad$, $L_\\mathrm{j} = 10^{51}~\\erg~\\s^{-1}$, and we inject the jets from the same initial height ($z_0 = 10^9~\\cm$). \n\nFig.~\\ref{fig: density_profiles_s} shows a comparison of the energy-velocity distributions from different stellar profiles. \nFirst, it shows that the distributions are all flat over a range of velocities, implying that them main feature of the outflow from an explosion driven by a choked jet is independent of exact stellar profile (a similar result was found by \\citealt{eisenberg2022} in the case of explosions that are driven by successful jets). \nWhen looking in more detail, the right-hand side shows two pairs of simulation. \nEach pair shows the results of different stellar profiles with similar $\\theta_\\mathrm{j}$ and $z_\\mathrm{ch}$ (which dominates $V_{\\bo}$). \nThe distributions found in the two simulations of each pair are very similar, implying that when the cocoon properties are similar the stellar profile has a minor effect on the outflow energy-velocity distribution. \n\nOn the left-hand side of Fig.~\\ref{fig: density_profiles_s} we compare the energy-velocity distributions of jets with the exact same parameters (including $t_\\mathrm{e}$) but different envelope density profiles. \nIt shows that the stellar profile affects the velocity of the head (as was found previously by \\citealt{Bromberg2011,Harrison2018}) and therefore jets with the same properties are choked at different heights when propagating in different density profiles. \nSince the energy-velocity profile depends strongly on $z_\\mathrm{ch}$ two jets with the same properties that propagate at different stellar profiles will result in outflows with different energy-velocity distributions, as shown on the right-hand panel of this figure.   \n\n\\subsection{The energy velocity distribution at different viewing angles}\n\\label{sec: profiles_different_angles}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.39]{figures\/comparison_same_choking_height_255_four_plots.pdf}\n    \\caption{Energy-velocity distributions for six different viewing angles (chosen so that the corresponding wedges have the same volumes). \\emph{Top}: Two jets with the same choking height $ (V_* \/ V_\\bo)^{1\/2}= 1.95 $  but different initial opening angles. \n    \\emph{Bottom}: Two jets choked at $ (V_* \/ V_\\bo)^{1\/2} \\sim 4 $ (bottom panels) and with the same opening angles of the jets on the top row, respectively. \n    The black dashed line represents the isotropic energy-velocity distribution for $1\/6$ of the total volume. The profiles are taken at $t=120~\\s$.}\n    \\label{fig: energy_profiles_volumes}\n\\end{figure*}\n\nSince jet driven explosions are aspherical, one expects that the outflow will not be isotropic. \nFig.~\\ref{fig: energy_profiles_volumes} depicts the energy-velocity distribution of four simulations. \nFor each simulation we show the distributions at six different sections, where each section is the sum of the ejecta within a range of polar direction. \nTo see the dependence on the initial conditions we show simulations with two different jet opening angles ($0.2$ and $0.6$ rad) and two different values of breakout volume $\\sqrt{V_*\/V_\\bo}$. \nAs expected, the outflow is aspherical. \nA common, also expected, property of all four simulations is that the maximal velocity of the outflow is around the jet axis at lower polar angles. \nThis result was found also for jet-driven explosions of successful jets \\citep{eisenberg2022}. \nIn the two simulations with the large value of $\\sqrt{V_*\/V_\\bo}$ (i.e., low $z_\\mathrm{ch}$ and\/or wide $\\theta_\\mathrm{j}$; top panels) the energy-velocity distribution of the equatorial outflow ($\\theta \\gtrsim 60^\\circ$) is similar to that of a spherical explosion, with a typical velocity $\\beta_0$. The faster outflow is confined to lower angles. \nIn the two simulations with the small value of $\\sqrt{V_*\/V_\\bo}$ (i.e., high $z_\\mathrm{ch}$ and narrow $\\theta_\\mathrm{j}$; bottom panels) a large range of velocities was seen in all directions, but still faster velocities are observed closer to the jet axis.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.42]{figures\/density_profile_175e49_255_with_fit_four_eqvol.pdf}\n    \\caption{Density profiles at different viewing angles of two jets with a similar breakout volume $ (V_* \/ V_\\bo)^{1\/2}= 1.95 $ (top row) but different initial opening angle compared to the density profile of jets choked at $ (V_* \/ V_\\bo)^{1\/2} \\sim 4 $ (bottom row) with the same opening angles, respectively. \n    The solid thick black line in each panel represent a power-law fit of $\\rho(r) \\propto v^{-5}$, corresponding to a flat distribution of energy per logarithmic bin of the velocity. \n    The profiles are taken at $t=120~\\s$.}\n    \\label{fig: density_profiles_1}\n\\end{figure*}\n\nFor clarity we present in Fig.~\\ref{fig: density_profiles_1} the radial density distribution profiles, $\\rho(R)$, for different viewing angles of four different simulations. \nThese are the same simulations and the same divisions to angular sections as in Fig.~\\ref{fig: energy_profiles_volumes}. \nThis presentation is often used in studies of SN ejecta and at sub-relativistic velocity  $\\rho(R) \\propto \\beta^{-5} \\frac{\\mathrm{d} E}{\\mathrm{d} \\log\\beta}$.\n\n\n\\section{Conclusions and implications to observations}\n\\label{sec: conclusions}\n\nWe carried relativistic hydrodynamical simulations in 2D cylindrical coordinates of stellar explosions driven by jets, focusing on configurations of choked relativistic jets and exploring  how those can lead to different realizations of the velocity distribution of the outflow in its homologous expansion phase.  \nWe followed the evolution of a relativistic jet from the injection deep inside the star to the point where it is choked and then continued to follow the cocoon as it emerges from the envelope and ultimately unbinds it up to the point that the outflow becomes homologous. \nWe scrutinized the various stages of the jet inside the star and analyzed what happens during the choking process and the adiabatic cocoon expansion.  \nWhile the results are given for a specific set of parameters we provided scaling relation for the physical parameters of the jet and the star in order to facilitate a dimensionless treatment of the problem.\n\nWe summarize our findings as follows:\n\n\\begin{itemize}\n    \\item All jet driven explosions in which the jet is not choked too deep within the star generate an outflow with a unique feature: a significant range of velocities over which the outflow carries a roughly constant amount of energy per logarithmic scale of the proper velocity ($\\Gamma\\beta$). \n    This is a universal property of jet driven explosions. \n    The main difference between different setups is the range of velocities over which the energy is constant.\n    \n    \\item The plateau of the energy-velocity distribution starts in all cases at $v_0=\\sqrt{E_0\/M_*}$. \n    The maximal velocity of the plateau depends mostly on the cocoon volume upon breakout and the corresponding velocity is  $\\beta_\\bo=\\beta_0\\sqrt{V_*\/V_\\bo}$. \n    For  $\\sqrt{V_*\/V_\\bo} < 3$ the maximal velocity is comparable to $\\beta_\\bo$, while for larger values of $\\sqrt{V_*\/V_\\bo}$ the maximal velocity is larger than $\\beta_\\bo$ and it can become mildly relativistic. \n    \n    \\item The volume of the cocoon upon breakout, $V_\\bo$, depends on the choking height, $z_\\mathrm{ch}$, and on opening angle of the jet upon launching. \n    A higher $z_\\mathrm{ch}$ and narrower opening angle leads to a smaller $V_\\bo$ and thus to an outflow that extends to higher velocities.\n    \n    \\item The outflow from an explosion driven by a choked jet is not isotropic.\n    In general, the material along the poles (that is along the jet direction) is faster while the material along the equator is slower.\n\\end{itemize}\n\nA spherical explosions accelerate only a negligible fraction of the stellar mass to very high velocities. \nWe have shown here that the situation is drastically different when there is a jet that breaks the symmetry. \nSuch a jet can deposit a significant amount of energy at high velocity matter, even in case that the jet is choked within the envelope. \nThis excess in high velocity outflow (compared to a spherical explosion) is certainly expected when the entire stellar explosion is driven by a jet, but it is also expected if the jet is accompanied by a simultaneous more spherical explosion  (see e.g., \\citealt{eisenberg2022}).\nIf sufficiently optically thick such a high velocity material that surrounds a SN would produce a very broad  absorption lines (with typical width corresponding to 0.1-0.2c) in the observed spectrum. \nIt will be observed in the early spectra but will disappear later when this outer envelope that is rapidly expanding becomes optically thin. \n\nLines that show an excess of high velocity material have been observed in several SNe \\citep{Galama+1998, Iwamoto+1998, Mazzali+2000,  Mazzali+2002, Modjaz+2006, Mazzali+2008, Bufano+2012, Xu+2013, Ashall+2019,Izzo_et_al_2019}.\nOur result show that, as suggested by \\cite{Piran2019},  a chocked jet can lead to that high velocity material. \nHowever, we have found that some conditions are needed to observe the corresponding broad absorption lines. \nFirst, the jet must be chocked at sufficiently large distance at the stellar atmosphere. \nThe signature of jets that are chocked too deep will not be so significant.\nSecond, as there is less fast moving material in directions far from the jet direction, the fast moving matter will become optically thin earlier in these directions. \nAs the broad absorption line will fade faster, this implies that observers at such viewing angles are less likely to observe the broad absorption line signature. \nThese last two facts imply that we may not observer broad emission line in all SNe that harbour relativistic jets. \n\nThe excess in fast material was observed in various types of stripped envelope SNe. \nThis include SNe that are associated with long GRBs, SNE that are associated with {\\it ll}GRBs and SNe that are not associated with GRBs at all. \nLong GRBs must contain successful relativistic jets. \n{\\it ll}GRBs contain jets which may very well be choked \\citep{Kulkarni1998, macfadyen_supernovae_2001, Tan+2001, campana_association_2006, Wang+2007, waxman_grb_2007, katz_fast_2010, Nakar_Sari2012,Nakar2015}. \nWe do not know if SNe that are not associated with GRBs harbour jets, but if they are then these jets must be choked ones. \nA previous study by \\cite{eisenberg2022} have shown that successful jets can generate the energy-velocity distribution which is observed in SNe that are associated with long GRBs. \nOur finding here show that choked jets can explain the energy-velocity distribution seen in SNe that are associated with {\\it ll}GRBs and in SNe that are not associated with any type of GRBs. \nThis provides further support for the interpretation of the ``disappearing'' early very broad absorption lines in some SNe as arising from choked jets. \nThese findings also show that such lines may not be detected in all SNe that harbor choked jets. \nFurther exploration of this model, including estimates of the observed spectra and the fraction of events in which these lines will be observed will be carried out in future work. \n\n\n\\section*{Acknowledgments}\nWe kindly thank Christopher Irwin for the stimulating discussions and suggestions.\nThis work is supported by the ERC grants TReX (TP and MP) and JetNS and an ISF grant 1995\/21 (EN).\n\\section*{Data Availability}\nThe data underlying this article will be shared on reasonable\nrequest to the corresponding author.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzbmjn b/data_all_eng_slimpj/shuffled/split2/finalzzbmjn
new file mode 100644
index 0000000000000000000000000000000000000000..004ead1a83afb64d5f558709dd4eadac7f2284bf
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzbmjn
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nTopological order\\cite{W8987,W9039,WN9077} is a new kind of order beyond the\nsymmetry breaking orders\\cite{L3726} in gapped quantum systems.  Topological\norders are patterns of \\emph{long-range entanglement}\\cite{CGW1038} in\n\\emph{gapped quantum liquids} (GQL)\\cite{ZW1490}.  Based on the unitary modular\ntensor category (UMTC) theory for non-abelian\nstatistics\\cite{MS8977,BakK01,K062}, in \\Ref{KW1458,W150605768}, it is proposed\nthat 2+1D bosonic topological orders are classified by $\\{\\text{UMTC}\\}\\times\n\\{\\text{iTO}_B\\}$, where $\\{\\text{UMTC}\\}$ is the set of UMTCs and\n$\\{\\text{iTO}_B\\}$ is the set of invertible topological orders\n(iTO)\\cite{KW1458,F1478} for 2+1D boson systems.  In fact $\\{\\text{iTO}_B\\}=\\Z$\nwhich is generated by the $E_8$ bosonic quantum Hall (QH) state, and a table of\nUMTCs was obtained in \\Ref{RSW0777,W150605768}.  Thus, we have a table (and a\nclassification) of 2+1D bosonic topological orders.\n\nIn a recent work\\cite{LW150704673}, we show that 2+1D fermionic topological\norders are classified by $\\{\\mce{\\sRp(Z_2^f)}\\}\\times \\{\\text{iTO}_F\\}$,\nwhere $\\{\\mce{\\sRp(Z_2^f)}\\}$ is the set of non-degenerate unitary\nbraided fusion categories (UBFC) over the symmetric fusion category (SFC)\n$\\sRp(Z_2^f)$ (see Definition \\ref{defMTC}). We also require\n$\\mce{\\sRp(Z_2^f)}$s to have modular extensions.\n$\\{\\text{iTO}_F\\}$ is the set of invertible topological orders for 2+1D fermion\nsystems.  In fact $\\{\\text{iTO}_F\\}=\\Z$ which is generated by the $p+\\ii p$\nsuperconductor. In \\Ref{LW150704673} we computed the table for\n$\\mce{\\sRp(Z_2^f)}$s, and  obtained a table (and a classification) of\n2+1D fermionic topological orders.\n\nIn \\Ref{LW150704673}, we also point out the importance of modular extensions.\nIf a $\\mce{\\sRp(Z_2^f)}$ does not have a modular extension, it means\nthat the fermion-number-parity symmetry is not on-site (\\ie\nanomalous\\cite{W1313}).  On the other hand, if a $\\mce{\\sRp(Z_2^f)}$\ndoes have modular extensions, then the $\\mce{\\sRp(Z_2^f)}$ is\nrealizable by a lattice model of fermions. In this case, a given $\\mce{\\sRp(Z_2^f)}$ may have several\nmodular extensions. We found that different modular extensions of\n$\\mce{\\sRp(Z_2^f)}$ contain information of iTO$_F$s.\n\nOur result on fermionic topological orders can be easily generalized to\ndescribe bosonic\/fermionic topological orders with symmetry.  This will be the\nmain topic of this paper. (Some of the results are announced in\n\\Ref{LW150704673}).  In this paper, we will consider symmetric GQL phases for\n2+1D bosonic\/fermionic systems.  The notion of GQL was defined in \\Ref{ZW1490}.\nThe symmetry group of GQL is $G$ (for bosonic systems) or $G^f$ (for fermionic\nsystems).  If a symmetric GQL has long-range entanglement (as defined in\n\\Ref{CGW1038,ZW1490}), it corresponds to a symmetry enriched topological\n(SET) order\\cite{CGW1038}.  If a symmetric GQL has short-range entanglement, it\ncorresponds to a symmetry protected trivial (SPT) order [which is also\nknown as symmetry protected topological (SPT)\norder]\\cite{GW0931,PBT1225,CLW1141,CGL1314,CGL1204}.\n\nIn this paper, we are going to show that, 2+1D symmetric GQLs are classified by\n$\\mce{\\cE}$ plus their modular extensions and chiral central charge.  In other\nwords, GQLs are labeled by triples $(\\cC,\\cM,c)$, where $\\cC$ is a\n$\\mce{\\cE}$, $\\cM$ a modular extension of $\\cC$, and $c$ the chiral central\ncharge of the edge state. (To be more precise, a modular extension of $\\cC$,\n$\\cM$, is a UMTC with a fully faithful embedding $\\cC \\to \\cM$.  In particular,\neven if the UMTC $\\cM$ is fixed, different embeddings correspond to\ndifferent modular extensions.) Here the SFC $\\cE$ is given by $\\cE=\\Rp(G)$ for\nbosonic cases, or $\\cE=\\sRp(G^f)$ for fermionic cases.  In yet another way to\nphrase our result: we find that the structure $\\cE \\hookrightarrow \\cC\n\\hookrightarrow \\cM$ classifies\nthe 2+1D GQLs with symmetry $\\cE$, where $\\hookrightarrow$ represents the\nembeddings and\n$\\cen{\\cE}{\\cM}=\\cC$ (see Definition \\ref{cendef}).\n\nAs a special case of the above result, we find that bosonic 2+1D SPT phase with\nsymmetry $G$ are classified by the modular extensions of $\\Rp(G)$, while\nfermionic 2+1D SPT phase with symmetry $G^f$ are classified by the modular\nextensions of $\\sRp(G^f)$ that have central charge $c=0$.\n\nWe like to mention that \\Ref{BBC1440} has classified bosonic GQLs with symmetry\n$G$, using $G$-crossed UMTCs. This paper uses a different approach so that we\ncan classify both bosonic and fermionic  GQLs with symmetry.  We also like to\nmention that there is a mathematical companion \\Ref{LW160205936} of this paper, where\n one can find detailed proof and explanations for related mathematical results. \n\n\nThe paper is organized as the following.  In Section \\ref{GQLsymm}, we review\nthe notion of topological order and introduce category theory as a theory of\nquasiparticle excitations in a GQL.  We will introduce a categorical way to\nview the symmetry as well.  In Section \\ref{inv}, we discuss invertible GQLs\nand their classification based on modular extensions.  In Sections \\ref{clGQL} and\n\\ref{clGQL2}, we generalize the above results and propose a classification of\nall GQLs.  Section \\ref{stack} investigates the stacking operation from\nphysical and mathematical points of view.  Section \\ref{howto} describes how to\nnumerically calculate the modular extensions and Section \\ref{examples}\ndiscusses some simple examples.  For people with physics background, one way to\nread this paper is to start with the Sections \\ref{GQLsymm} and \\ref{clGQL2},\nand then go to Section \\ref{examples} for the examples.\n\n\n\n\\section{Gapped quantum liquids, topological order and symmetry}\n\\label{GQLsymm}\n\n\\subsection{The finite on-site symmetry and symmetric fusion category}\n\nIn this paper, we consider physical systems with an on-site symmetry described\nby a finite group $G$.  For fermionic systems, we further require that $G$\ncontains a central $Z_2$ fermion-number-parity subgroup.  More precisely,\nfermionic symmetry group is a pair $(G,f)$, where $G$ is a finite group, $f\\neq\n1$ is an element of $G$ satisfying $f^2=1,fg=gf,\\forall g\\in G$. We denote the\npair $(G,f)$ as $G^f$.\n\nThere is another way to view the on-site symmetries, which is nicer because\nbosonic and fermionic symmetries can be formulated in the same manner.\nConsider a bosonic\/fermionic product state $|\\psi\\ket$ that does not break the\nsymmetry $G$: $U_g|\\psi\\ket=|\\psi\\ket,\\ g\\in G$. Then the new way to view the\nsymmetry is to use the properties of the excitations above the product state to\nencode the information of the symmetry $G$.\n\nThe product state contain only local excitations that can be created by acting\nlocal operators $O$ on the ground state $O|\\psi\\ket$.  For any group action\n$U_g$, $U_g O|\\psi\\ket=U_g O U_g^\\dag U_g|\\psi\\ket=U_g O U_g^\\dag |\\psi\\ket$ is\nan excited state with the same energy as $O|\\psi\\ket$. Since we assume the\nsymmetry to be on-site, $U_g OU_g^\\dag$ is also a local operator.  Therefore,\n$U_g OU_g^\\dag|\\psi\\ket$ and $O|\\psi\\ket$ correspond to the degenerate local\nexcitations. We see that  local excitations ``locally'' carry group\nrepresentations.  In other words, different types of local excitations are\nlabeled by irreducible representations of the symmetry group. \n\nBy looking at how the local excitations (more precisely, their group\nrepresentations) fuse and braid with each other, we arrive at the mathematical\nstructure called symmetric fusion categories (SFC).  By definition a SFC is a\nbraided fusion category where all the objects (the excitations) have trivial\nmutal statistics (\\ie centralize each other, see next section). \nA SFC is automatically a unitary braided fusion category. \n\nIn fact, there are only two kinds of SFCs: one is representation category of\n$G$: $\\Rp(G)$, with the usual braiding (all representations are bosonic); the\nother is $\\sRp(G^f)$ where an irreducible representation is bosonic if $f$ is\nrepresented trivially ($+1$), and fermionic if $f$ is represented\nnon-trivially($-1$).\n\nIt turns out SFC (or the fusion and braiding properties of the local\nexcitations) fully characterize the symmetry group.  Therefore, it is\nequivalent to say finite on-site symmetry is given by a SFC $\\cE$. By Tannaka\nduality $\\cE$ gives rises to a unique finite group $G$ and by checking the\nbraiding in $\\cE$ we know whether it is bosonic or fermionic.  This is the new\nway, the categorical way, to view the symmetry.  Such a categorical view of\nbosonic\/fermionic symmetry allows us to obtain a classification of symmetric\ntopological\/SPT orders.\n\n\\subsection{Categorical description of topological excitations with symmetry}\n\nIn symmetric GQLs with topological order (\\ie with long range entanglement),\nthere can be particle-like excitations with local energy density, but they\ncannot be created by local operators. They are known as (non-trivial)\ntopological excitations. Topological excitations do not necessarily carry group\nrepresentations. Nevertheless, we can still study how they fuse and braid with\neach other; so we have a unitary braided fusion category (UBFC) to describe the\nparticle-like excitations. To proceed, we need the following key definition on\n``centralizers.''\n\\begin{dfn}\n  The objects $X,Y$ in a UBFC $\\cC$ are said to \\emph{centralize} (mutually\n  local to) each other if \n  \\begin{align}\n   c_{Y,X}\\circ c_{X,Y}=\\mathrm{id}_{X\\otimes Y},\n  \\end{align}\n  where $c_{X,Y}: X\\otimes Y\\cong Y\\otimes X$ is the braiding in $\\cC$.\n\\end{dfn}\n\n  Physically, we say that $X$ and $Y$ have trivial mutual statistics.\n\n\\begin{dfn}\n\\label{cendef}\n  Given a subcategory $\\cD\\subset \\cC$, its \\emph{centralizer}\n  $\\cen{\\cD}{\\cC}$ in $\\cC$\n  is the full subcategory of objects in $\\cC$ that centralize all the objects in\n  $\\cD$.  \n\\end{dfn}\n\nWe may roughly view a category as a ``set'' of particle-like excitations.\nSo the centralizer $\\cen{\\cD}{\\cC}$ is the ``subset'' of particles in $\\cC$\nthat have trivial mutual statistics with all the particles in $\\cD$.\n\n\\begin{dfn}\n\\label{defMTC}\n  A UBFC $\\cE$ is a \\emph{symmetric} fusion category if $\\cen{\\cE}{\\cE}=\\cE$.\nA UBFC $\\cC$ with a fully faithful embedding $\\cE\\inj \\cen{\\cC}{\\cC}$ is called\na UBFC over $\\cE$. Moreover, $\\cC$ is called a non-degenerate UBFC over $\\cE$, or\n$\\mce{\\cE}$, if $\\cen{\\cC}{\\cC}=\\cE$.\n\\end{dfn}\n\n\\begin{dfn} \\label{def:hom-bfce}\nTwo UBFCs over $\\cE$, $\\cC$ and $\\cC'$ are equivalent if there is a unitary braided\nequivalence $F:\\cC\\to\\cC'$ such that it preserves the embeddings, i.e.,\nthe following diagram commute.\n  \\newdir^{ (}{{}*!\/-5pt\/@^{(}}\n\\begin{align}\n\\label{Ceq}\n\\xymatrix{\n  \\cE\\ar@{^{ (}->}[r]\\ar@{=}[d]&\\cC\\ar[d]^{F}\n  \\\\\n  \\cE\\ar@{^{ (}->}[r]&\\cC'\n}\n\\end{align}\nWe denote the category of unitary braided auto-equivalences of $\\cC$ by $\\mathcal{A}\\mathrm{ut}(\\cC)$ and its underlining group by $\\mathrm{Aut}(\\cC)$. \n\\end{dfn}\n\nWe recover the usual definition of\nUMTC when $\\cE$ is trivial, \\ie the category of Hilbert spaces, denoted by\n$\\mathrm{Vec}=\\Rp(\\{1\\})$.  In this case the subscript is omitted.\n\nPhysically, a UBFC $\\cC$ is the collection of all bulk topological excitations\nplus their fusion and braiding data. Requiring $\\cC$ to be a \n$\\mce{\\cE}$ means: (1) the set of local excitations, $\\cE$ (which is the set of\nall the irreducible representations of the symmetry group), is included in\n$\\cC$ as a subcategory; (2) $\\cC$ is anomaly-free, \\ie all the topological\nexcitations (the ones not in $\\cE$) can be detected by mutual\nbraiding\\cite{KW1458}. In other words, every topological excitation must have\nnon-trivial mutual statistics with some excitations.  Those excitations that\ncannot be detected by mutual braiding (i.e., $\\cen{\\cC}{\\cC}$) are exactly the\nlocal excitations in $\\cE$. Moreover, we want the symmetry to be on-site\n(gaugeable), which requires the existence of modular extensions (see Definition \\ref{mextdef}). Such an understanding leads to the following\nconjecture:\n\\begin{conj}\n  Bulk topological excitations of topological orders with symmetry $\\cE$ are\nclassified by $\\mce{\\cE}$'s that have modular extensions.\n\\end{conj}\n\\noindent\n\nWe like to remark that $\\mce{\\cE}$'s fail to classify\ntopological orders.  This is because two different topologically ordered phases\nmay have bulk topological excitations with the same non-abelian statistics (\\ie\ndescribed by the same $\\mce{\\cE}$).  However, $\\mce{\\cE}$'s, with modular\nextensions, do classify topological orders up to invertible ones.  See next\nsection for details.  The relation between anomaly and modular extension will\nalso be discussed later.\n\n\n\\section{Invertible GQLs and modular extension}\n\\label{inv}\n\n\\subsection{Invertible GQLs}\n\nThere exist non-trivial topological ordered states that have only trivial\ntopological excitations in the bulk (but non-trivial edge states). They are\n``invertible'' under the stacking operation\\cite{KW1458,F1478}\n(see Section \\ref{stack} for details). More generally,\nwe define  \n\\begin{dfn}\nA GQL is invertible if its bulk topological excitations are all trivial\n(\\ie can all be created by local operators).\n\\end{dfn}\nConsider some invertible GQLs with the same symmetry $\\cE$.  The bulk\nexcitations of those invertible GQLs are the same which are described by the\nsame SFC $\\cE$. Now the question is: How to distinguish those invertible GQLs?\n\n\nFirst, we believe that invertible bosonic topological orders with no symmetry\nare generated by the $E_8$ QH state (with central charge $c=8$) via\ntime-reversal and stacking, and form a $\\Z$ group. Stacking with an $E_8$ QH state only\nchanges the central charge by $8$, and does not change the bulk excitations or\nthe symmetry. So the only data we need to know to determine the invertible\nbosonic topological order with no symmetry is the central charge $c$.  The\nstory is parallel for invertible fermionic topological orders with no symmetry,\nwhich are believed to be generated by the $p+\\ii p$ superconductor state with\ncentral charge $c=1\/2$.\n\nSecond, invertible bosonic GQLs with symmetry are generated by bosonic SPT\nstates and invertible bosonic topological orders (\\ie $E_8$ states) via\nstacking.  We know that the bosonic  SPT states with symmetry $G$ are\nclassified by the 3-cocycles in $H^3[G,U(1)]$.  Therefore, bosonic invertible\nGQLs with symmetry $G$ are classified by $H^3[G,U(1)]\\times \\Z$ (where $\\Z$\ncorresponds to layers of $E_8$ states).\n\nHowever, this result and this point of view is not natural to generalize to\nfermionic cases or non-invertible GQLs. Thus, we introduce an equivalent point\nof view, which can cover boson, fermion,  and non-invertible GQLs in the same\nfashion.\n\n\\subsection{Modular extension}\n\nFirst, we introduce the notion of modular extension of a $\\mce{\\cE}$:\n\\begin{dfn}\n\\label{mextdef}\n  Given a $\\mce{\\cE}$ $\\cC$, its \\emph{modular extension} is a UMTC\n  $\\cM$, together with a fully faithful embedding $\\iota_\\cM:\n  \\cC\\hookrightarrow\\cM$, such that $\\cen{\\cE}{\\cM}=\\cC$, equivalently\n  $\\dim(\\cM)=\\dim(\\cC)\\dim(\\cE)$.\n\n  Two modular extensions $\\cM$ and $\\cM'$ are equivalent if\n  there is an equivalence between the UMTCs ${F:\\cM\\to\\cM'}$ that preserves the\n  embeddings, i.e., the following diagram commute.\n  \\newdir^{ (}{{}*!\/-5pt\/@^{(}}\n\\begin{align}\n\\label{MEeq}\n\\xymatrix{\n  \\cC\\ar@{^{ (}->}[r]\\ar@{=}[d]&\\cM\\ar[d]^{F}\\\\\n  \\cC\\ar@{^{ (}->}[r]&\\cM'\n}\n\\end{align}\n  We\n  denote the set of equivalent classes of modular extensions of $\\cC$ by $\\mathcal{M}_{ext}(\\cC)$.\n\\end{dfn}\n\\begin{rmk}\n  Since the total quantum dimension of modular extensions of a given $\\cC$ is\nfixed, there are only finitely many different modular extensions, due to\n\\Ref{BNRW13}. In principle we can always perform a finite search to exhaust all\nthe modular extensions.\n\\end{rmk}\n\nRemember that $\\cC$ describes the particle-like excitations in our topological\nstate. Some of those excitations are local that have trivial mutual statistics\nwith all other excitations.  Those local excitation form $\\cE \\subset \\cC$.\nThe modular extension $\\cM$ of $\\cC$ is obtained as adding particles that have\nnon-trivial mutual statistics with the local excitations in $\\cE$, so that\nevery particle in $\\cM$ will always have non-trivial mutual statistics with\nsome particles in $\\cM$.  Since the particles in $\\cE$ carry ``charges'' (\\ie\nthe irreducible representations of $G$), the added particles correspond to\n``flux'' (\\ie the symmetry twists of $G$).  So the modular extension correspond\nto gauging\\cite{LG1209} the on-site symmetry $G$. Since we can use the gauged\nsymmetry to detect SPT orders\\cite{HW1339}, we like to propose the following\nconjecture\n\\begin{conj}\\label{invclassB}\n  Invertible bosonic GQLs with symmetry $\\cE=\\Rp(G)$ are classified by\n$(\\cM,c)$ where $\\cM$ is a  modular extension of $\\cE$ and $c=0$ mod 8.\n\\end{conj}\n\n\n\\subsection{Classify 2+1D bosonic SPT states}\n\nInvertible bosonic GQLs described by $(\\cM,c)$ include both bosonic SPT states\nand bosonic topological orders. Among those, $(\\cM,c=0)$ classify bosonic SPT\nstates.  In other words:\n\\begin{cor}\n2+1D bosonic SPT states with symmetry $G$ are classified by the modular\nextensions of $\\Rp(G)$ (which always have $c=0$).\n\\end{cor} \n\nIn \\Ref{CLW1141,CGL1314,CGL1204}, it was shown that 2+1D bosonic SPT states are\nclassified by $H^3[G,U(1)]$.  Such a result agrees with our conjecture, due to\nthe following theorem, which follows immediately from results in \\Ref{dgno2007}. \n\n\\begin{thm} \\label{thm:1to1-repG}\nThe modular extensions of $\\Rp(G)$ 1-to-1 correspond to 3-cocycles in $H^3[G,U(1)]$. The central charge of these modular extensions are $c=0$ mod 8.\n\\end{thm}\n\n\\begin{rmk}\nIn Sec.\\,\\ref{sec:mext-repG}, we give more detailed explanation of the 1-to-1 correspondence in Theorem\\,\\ref{thm:1to1-repG}. Moreover, we will prove a stronger result in Theorem\\,\\ref{thm:spt}. It turns out that the set $\\cM_{ext}(\\Rp(G))$ of modular extensions of $\\Rp(G)$ is naturally equipped with a physical stacking operation such that $\\cM_{ext}(\\Rp(G))$ forms an abelian group, which is isomorphic to the group $H^3[G,U(1)]$. \n\\end{rmk}\n\n\n\n\\begin{rmk}\n$c\/8$ determines the number of layers of the $E_8$ QH states, which is the\ntopological order part of invertible bosonic symmetric GQLs.  In other words\n\\begin{align}\n&\\ \\ \\ \\\n\\{ \\text{invertible bosonic symmetric GQLs} \\} \n\\nonumber\\\\\n&=\n\\{ \\text{bosonic SPT states} \\}\\times \\{ \\text{layers of $E_8$ states} \\}.\n\\end{align}\n\\end{rmk}\n\n\n\n\n\\subsection{Classify 2+1D fermionic SPT states}\n\nThe above approach also apply to fermionic case.  Note that, the invertible\nfermionic GQLs with symmetry $G^f$\nhave bulk excitations described by SFC $\\cE=\\sRp(G^f)$.\nSo we would like to conjecture that\n\\begin{conj}\\label{invclassF}\n  Invertible fermionic GQLs with symmetry $G^f$  are classified by\n$(\\cM,c)$, where $\\cM$ is a  modular extension of $\\cE=\\sRp(G^f)$,\nand $c$ is the central charge determining the layers of $\\nu=8$ IQH states.\n\\end{conj}\n\\begin{rmk}\nNote that, the central charge $c$ mod 8 is determined by $\\cM$, while\n$ (c - \\text{mod}(c,8))\/8$ determines the number of layers of the\n$\\nu=8$ IQH states.\n\\end{rmk}\n\\begin{rmk}\nInvertible fermionic symmetric GQLs include both fermion SPT states and\nfermionic topological orders. $(\\cM,c)$ with $c=0$ classify fermionic SPT\nstates.  \n\\end{rmk} \nIn other words, \n\\begin{cor}\n2+1D fermionic SPT states with symmetry $G$ are classified by the  $c=0$ modular extensions of\n$\\sRp(G^f)$. \n\\end{cor} \n\\begin{rmk}\nUnlike the bosonic case, in general\n\\begin{align}\n&\\ \\ \\ \\\n\\{ \\text{invertible fermionic symmetric GQLs} \\} \n\\\\\n& \\neq\n\\{ \\text{fermionic SPT states} \\}\\times \\{ \\text{layers of $p+\\ii p$ states} \\}.\n\\nonumber \n\\end{align}\n\\end{rmk}\n\n\nWhen there is no symmetry, the invertible fermionic GQLs become the invertible\nfermionic topological order, which have bulk excitations described by\n$\\cE=\\sRp(Z_2^f)$.  $\\sRp(Z_2^f)$ has 16  modular extensions, with central\ncharges $c=n\/2, n=0,1,2,\\dots,15$.  There is only one modular extension with\n$c=0$, which correspond to trivial product state. Thus there is no non-trivial\nfermionic SPT state when there is no symmetry, as expected.\n\nThe modular extensions with $c=n\/2$ correspond to invertible fermionic\ntopological order formed by $n$ layers of $p+\\ii p$ states.\nSince the modular extensions can only determine $c$ mod 8,\nin order for the above picture to be consistent, we need to show the following\n\\begin{thm}\\label{pE8}\nThe stacking of 16 layers $c=1\/2$ $p+\\ii p$ states is equivalent to a $\\nu=8$\nIQH state, which is in turn equivalent to a $E_8$ bosonic QH state stacked with\na trivial fermionic product state.\n\\end{thm}\n\\begin{proof}\nFirst, two layers of $p+\\ii p$ states is equal to one layer of $\\nu=1$ IQH\nstate.  Thus, 16 layers $c=1\/2$ $p+\\ii p$ states is equivalent to a $\\nu=8$ IQH\nstate.  To show $\\nu=8$ IQH state is equivalent to $E_8$ bosonic QH state\nstacked with a trivial fermionic product state, we note that the  $\\nu=8$ IQH\nstate is described by $K$-matrix $K_{\\nu=8}=I_{8\\times 8}$ which is a 8-by-8\nidentity matrix. While the $E_8$ bosonic QH state stacked with a trivial\nfermionic product state is described by $K$-matrix $K_{E_8\\boxtimes\n\\cF_0}=K_{E_8}\\oplus \\bpm 1 &0 \\\\ 0& -1 \\epm $, where $K_{E_8}$ is the matrix\nthat describe the $E_8$ root lattice. We also know that two odd\\footnote{An odd\nmatrix is a symmetric integer matrix with at least one of its diagonal elements\nbeing odd.} $K$-matrices $K_1$ and $K_2$ describe the same fermionic topological\norder if after direct summing with proper number of $\\bpm 1 &0 \\\\ 0& -1 \\epm\n$'s:\n\\begin{align}\nK_1'&=K_1\\oplus  \\bpm 1 &0 \\\\ 0& -1 \\epm \\oplus \\cdots\n\\nonumber\\\\\nK_2'&=K_2\\oplus  \\bpm 1 &0 \\\\ 0& -1 \\epm \\oplus \\cdots,\n\\end{align}\n$K_1'$ and $K_2'$ become equivalent, \\ie\n\\begin{align}\n K_1' = U K_2' U^T,\\ \\ \\ \\ U \\in SL(N,\\Z).\n\\end{align}\nNotice that $K_{\\nu=8}\\oplus \\bpm 1 &0 \\\\ 0& -1 \\epm$ and $K_{E_8\\boxtimes\n\\cF_0}$ have the same determinant $-1$ and the same signature. Using the result\nthat odd matrices with $\\pm 1$ determinants are equivalent if they have\nthe same signature, we find that $K_{\\nu=8}\\oplus \\bpm 1 &0 \\\\ 0& -1 \\epm$\nand $K_{E_8\\boxtimes \\cF_0}$ are equivalent. Therefore $\\nu=8$ IQH state is\nequivalent to $E_8$ bosonic QH state stacked with a trivial fermionic product\nstate.\n\\end{proof}\n\n\n\\section{A full classification of 2+1D GQLs with symmetry}\n\\label{clGQL}\n\nWe have seen that all invertible GQLs with symmetry $G$ (or $G^f$) have the\nsame kind of bulk excitations, described by $\\Rp(G)$ (or $\\sRp(G^f)$).  To\nclassify distinct invertible GQLs that shared the same kind of bulk\nexcitations, we need to compute the modular extensions of $\\Rp(G)$ (or\n$\\sRp(G^f)$).  This result can be generalized to non-invertible topological\norders.\n\nIn general, the bulk excitations of a 2+1D bosonic\/fermionic SET are described\nby a $\\mce{\\cE}$ $\\cC$.  However, there can be many distinct SET orders that\nhave the same kind of bulk excitations described by the same $\\cC$.  To\nclassify distinct invertible SET orders that shared the same kind of bulk\nexcitations $\\cC$, we need to compute the modular extensions of $\\cC$.  This\nleads to the following\n\\begin{conj}\n\\label{classSET}\n  2+1D GQLs with symmetry $\\cE$ (\\ie the 2+1D SET orders) are classified by $(\\cC,\\cM,c)$,\n  where $\\cC$ is a $\\mce{\\cE}$ describing the bulk topological excitations,\n  $\\cM$ is a  modular extension of $\\cC$ describing the edge state up to\n  $E_8$ states, and $c$ is the central charge determining the layers of $E_8$\n  states.\n\\end{conj}\n\nLet $\\cM$ be a modular extension of a $\\mce{\\cE}$ $\\cC$. We note that\nall the simple objects (particles) in $\\cC$ are contained in $\\cM$ as\nsimple objects.  Assume that the particle labels of $\\cM$ are\n$\\{i,j,\\dots, x, y,\\dots\\}$, where $i,j,\\cdots $ correspond to the particles in\n$\\cC$ and $x,y,\\cdots $ the additional particles (not in $\\cC$).  Physically,\nthe additional particles $x,y,\\cdots $ correspond to the symmetry twists of the\non-site symmetry\\cite{W1447}.  The  modular extension $\\cM$ describes the\nfusion and the braiding of original particles $i,j,\\cdots $ with the symmetry\ntwists.  In other words, the modular extension $\\cM$ is the resulting\ntopological order after we gauge the on-site symmetry\\cite{LG1209}.\n\nNow, it is clear that the existence of modular extension is closely related to\nthe on-site symmetry (\\ie anomaly-free symmetry) which is gaugable (\\ie allows\nsymmetry twists).  For non-on-site symmetry (\\ie anomalous\nsymmetry\\cite{W1313}), the  modular extension does not exist since the symmetry\nis not gaugable (\\ie does not allow symmetry twists). We also have\n\\begin{conj}\n\\label{classSETA}\n  2+1D GQLs with anomalous symmetry\\cite{W1313} $\\cE$ are\nclassified by $\\mce{\\cE}$'s that have no modular extensions.\n\\end{conj}\n\n\nIt is also important to clarify the equivalence relation between the triples\n$(\\cC,\\cM,c)$. Two triples $(\\cC,\\cM,c)$ and $(\\cC',\\cM',c')$ are equivalent\nif: (1) $c=c'$; (2) there exists braided equivalences $F_\\cC:\\cC\\to\\cC'$ and\n$F_\\cM:\\cM\\to\\cM'$ such that all the embeddings are preserved, i.e., the\nfollowing diagram commutes.\n  \\newdir^{ (}{{}*!\/-5pt\/@^{(}}\n\\begin{align}\n\\label{TOeq}\n\\xymatrix{\n  \\cE\\ar@{^{ (}->}[r]\\ar@{=}[d]&\\cC\\ar@{^{ (}->}[r]\\ar[d]^{F_\\cC}\n  &\\cM\\ar[d]^{F_\\cM}\\\\\n  \\cE\\ar@{^{ (}->}[r]&\\cC'\\ar@{^{ (}->}[r]&\\cM'\n}\n\\end{align}\nThe equivalence classes will be in one-to-one\ncorrespondence with GQLs (\\ie SET orders and SPT orders).\n\nNote that the group of the automorphisms of a $\\mce{\\cE}$ $\\cC$, denoted by\n$\\mathrm{Aut}(\\cC)$ (recall Definition\\,\\ref{def:hom-bfce}), naturally acts on the\nmodular extensions $\\mathcal{M}_{ext}(\\cC)$ by changing the embeddings, i.e. $F\\in\\mathrm{Aut}(\\cC)$ acts as follows: \n$$\n(\\cC\\hookrightarrow\\cM)\\mapsto (\\cC\\xrightarrow{F}\\cC\\hookrightarrow \\cM)\n$$ \nFor a fixed $\\cC$, the above equivalence relation amounts to say that GQLs with bulk excitations described by a fixed $\\cC$ are in one-to-one correspondence with the quotient\n$\\mathcal{M}_{ext}(\\cC)\/\\mathrm{Aut}(\\cC)$ plus a central charge $c$. When $\\cC=\\cE$, the GQLs\nwith bulk excitations described by $\\cE$ and central charge $c=0$ are SPT phases. In this case, the group $\\mathrm{Aut}(\\cE)$, where $\\cE$ is viewed as the trivial $\\mce{\\cE}$, is trivial. Thus, SPT phases are classified by the modular extensions of\n$\\cE$ with $c=0$.\n\n\n\\section{Another description of 2+1D GQLs with symmetry}\n\\label{clGQL2}\n\n\nAlthough the above result has a nice mathematical structure, it is hard to\nimplement numerically to produce a table of GQLs.  To fix this problem, we\npropose a different description of 2+1D GQLs.  The second description is\nmotivated by a conjecture that the fusion and the spins of the particles,\n$(\\cN^{IJ}_K,\\cS_I)$, completely characterize a UMTC. We conjecture that\n\\begin{conj}\n\\label{NsNsNs}\nThe data $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$,  up\nto some equivalence relations, gives a one-to-one classification of\n2+1D GQLs with symmetry $G$ (for boson) or $G^f$ (for fermion), with a\nrestriction that the symmetry group can be fully characterized by the fusion\nring of its irreducible representations.  The data $( \\tilde N^{ab}_c,\\tilde\ns_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$ satisfies the conditions described in\nAppendix \\ref{cnds} (see \\Ref{W150605768} for UMTCs).  \n\\end{conj}\n\nHere $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$\nis closely related to $(\\cE;\\cC;\\cM;c)$ discussed above.  The data $(\\tilde\nN^{ab}_c,\\tilde s_a)$ describes the symmetry (\\ie the SFC $\\cE$):\n$a=1,\\cdots,\\tilde N$ label the irreducible representations and $\\tilde\nN^{ab}_c$ are the fusion coefficients of irreducible representations.  $\\tilde\ns_a =0$ or $1\/2$ depending on if the fermion-number-parity transformation $f$\nis represented trivially or non-trivially in the representation $a$.  The data\n$(N^{ij}_k,s_i)$ describes fusion and the spins of the bulk particles\n$i=1,\\cdots,N$ in the GQL. The data $(N^{ij}_k,s_i)$ contains $(\\tilde\nN^{ab}_c,\\tilde s_a)$ as a subset, where $a$ is identified with the first\n$\\tilde N$ particles of the GQL.  The data $(\\cN^{IJ}_K,\\cS_I)$  describes\nfusion and the spins of a UMTC, and it includes $(N^{ij}_k,s_i)$ as a subset,\nwhere $i$ is identified with the first $N$ particles of the UMTC.  Also among\nall the particles in UMTC, only the first $N$ (\\ie $I=1,\\cdots,N$) have trivial\nmutual statistics with first $\\tilde N$ particles (\\ie $I=1,\\cdots,\\tilde N$).\nLast, $c$ is the chiral central charge of the edge state.\n\nIf the data $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$ fully characterized\nthe $\\mce{\\cE}$, then the Conjecture \\ref{NsNsNs} would be equivalent to the\nConjecture \\ref{classSET}.\nHowever, for non-modular tensor category, $( \\tilde\nN^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$ fails to to fully characterize a\n$\\mce{\\cE}$. In other words, there are different $\\mce{\\cE}$'s that have the\nsame data $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$.  We need to include\nthe extra data, such as the $F$-tensor and the $R$-tensor, to fully\ncharacterize the $\\mce{\\cE}$.\n\nIn Appendix \\ref{SETtbl}, we list the data $( \\tilde N^{ab}_c,\\tilde s_a;\nN^{ij}_k,s_i)$ that satisfy the conditions in Appendix \\ref{cnds} (without the\nmodular extension condition) in many tables. Those tables include all the\n$\\mce{\\cE}$'s (up to certain total quantum dimensions), but the tables are not\nperfect: (1) some entries in the tables may be fake and do not correspond to\nany $\\mce{\\cE}$ (for the conditions are only necessary); (2) some entries in\nthe tables may correspond to more then one $\\mce{\\cE}$ (since $( \\tilde\nN^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$ does not fully characterize a $\\mce{\\cE}$).\n\nWe then continue to compute $(\\cN^{IJ}_K,\\cS_I;c)$, the modular extensions of\n$( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$.  We find that the modular\nextensions can fix the imperfectness mentioned above.  First, we find that the\nfake entries do not have modular extensions, and are ruled out.  Second, as we\nwill show in Section \\ref{stack}, all $\\mce{\\cE}$'s have the same numbers of\nmodular extensions (if they exist); therefore, the entry that corresponds to\nmore $\\mce{\\cE}$'s has more modular extensions. The modular extensions can tell\nus which entries correspond to multiple $\\mce{\\cE}$'s.  This leads to the\nconjecture that the full data $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i;\n\\cN^{IJ}_K,\\cS_I;c)$ gives rise to an one-to-one classification of 2+1D GQLs, and allows us to calculate the\ntables of 2+1D GQLs, which include 2+1D SET states and 2+1D SPT states.  Those\nare given in Section \\ref{examples}.\n\nAs for the equivalence relation, we only need to consider\n$(\\cN^{IJ}_K,\\cS_I;c)$, since the data  $(\\tilde N^{ab}_c,\\tilde s_a;\nN^{ij}_k,s_i)$ is included in $(\\cN^{IJ}_K,\\cS_I;c)$.  Two such data\n$(\\cN^{IJ}_K,\\cS_I;c)$ and $(\\bar \\cN^{IJ}_K,\\bar \\cS_I;\\bar c)$ are called\nequivalent if $c=\\bar c$, and $(\\cN^{IJ}_K,\\cS_I)$ and $(\\bar \\cN^{IJ}_K,\\bar\n\\cS_I)$ are related by two permutations of indices in the range $N_{\\cM} \\geq I\n> N$ and in the range $N\\geq I > \\tilde N$, where $N_{\\cM}$ is the range of\n$I$.  Such an equivalence relation corresponds to the one in eqn. (\\ref{TOeq})\nand will be called the TO-equivalence relation.  We use the TO-equivalence\nrelation to count the number of GQL phases (\\ie the number of SET orders and\nSPT orders).\n\nWe can also define another equivalence relation, called ME-equivalence\nrelation: we say $(\\cN^{IJ}_K,\\cS_I;c)$ and $(\\bar \\cN^{IJ}_K,\\bar \\cS_I;\\bar\nc)$ to be ME-equivalent if $c=\\bar c$ and they only differ by a permutation\nof indices in range $I > N$.  The ME-equivalence relation is closely related to\nthe one defined in eqn. (\\ref{MEeq}). We use the ME-equivalence relation to\ncount the number of modular extensions of a \\emph{fixed} $\\cC$.  \n\nLast, let us explain the restriction on the symmetry group.  In the Conjecture\n\\ref{NsNsNs}, we try to use the fusion $\\tilde N^{ab}_c$ of the irreducible\nrepresentations to characterize the symmetry group.  However, it is known that\ncertain different groups may have identical fusion ring for their irreducible\nrepresentations.  So we need to restrict the symmetry group to be the group\nthat can be fully characterized by its fusion ring.  Those groups include\nsimple groups and abelian groups\\cite{Yuan16}.  If we do not impose such a\nrestriction, then the Conjecture \\ref{NsNsNs} give rise to GQLs with a given\nsymmetry fusion ring, instead of a given symmetry group.\n\n\n\\section{The stacking operation of GQLs}\n\\label{stack}\n\n\\subsection{Stacking operation}\n\nConsider two GQLs $\\cC_1$ and $\\cC_2$. If we stack them together (without\nintroducing interactions between them), we obtain another GQL, which is denoted\nby $\\cC_1\\boxtimes \\cC_2$.  The stacking operation $\\boxtimes$ makes the set of\nGQLs into a monoid.  $ \\boxtimes $ does not makes the set of GQLs into a group,\nbecause in general, a GQL $\\cC$ may not have an inverse under $\\boxtimes$.  \\ie\nthere is no GQL $\\cD$ such that $\\cC\\boxtimes \\cD$ becomes a trivial product\nstate.  This is because when a GQL have non-trivial topological excitations,\nstacking it with another GQL can never cancel out those topological\nexcitations.\n\nWhen we are considering GQLs with symmetry $\\cE$, the simple stacking\n$\\boxtimes$ will ``double'' the symmetry, leads to a GQL with symmetry\n$\\cE\\bt\\cE$ ($\\Rp(G\\times G)$ or $\\sRp(G^f\\times G^f)$). In general we allow\nlocal interactions between the two layers to break some symmetry such that the\nresulting system only has the original symmetry $\\cE$ (In terms of the symmetry group, keep only the\nsubgroup $G\\hookrightarrow G\\times G$ with the diagonal embedding $g\\mapsto\n(g,g)$). This leads to the stacking between GQLs with symmetry $\\cE$,\ndenoted by $\\bt_\\cE$. Similarly, $\\bt_\\cE$ makes GQLs with symmetry $\\cE$ a\nmonoid, but in general not all GQLs are invertible.\n\nHowever, if the bulk excitations of $\\cC$ are all local (\\ie all described by\nSFC $\\cE$), then $\\cC$ will have an inverse under the stacking operation\n$\\boxtimes_\\cE$, and this is why we call such GQL invertible.  Those invertible\nGQLs include invertible topological orders and SPT states.\n \n\n\\subsection{The group structure of bosonic SPT states}\n\\label{grpSPT}\n\nWe have proposed that 2+1D SPT states are classified by $c=0$ modular\nextensions of the SFC $\\cE$ that describes the symmetry.  Since SPT states are\ninvertible, they form a group under the stacking operation $ \\boxtimes_\\cE $.\nThis implies that the modular extensions of the SFC should also form a group\nunder the  stacking operation.  So checking if the  modular extensions of the\nSFC have a group structure is a way to find support for our conjecture.\n\nHowever, in this section, we will first discuss such stacking\noperation and group structure from a physical point of view.\nWe will only consider bosonic SPT states.\n\nIt has been proposed that the bosonic SPT states are described by group\ncohomology $\\cH^{d+1}[G,U(1)]$\\cite{CLW1141,CGL1314,CGL1204}.  However, it has\nnot been shown that those bosonic SPT states form a group under stacking\noperation. Here we will fill this gap.  An ideal bosonic SPT state of symmetry\n$G$ in $d+1$D is described the following path integral\n\\begin{align}\n Z =\\sum_{\\{g_i\\}} \\prod_{\\{i,j,\\cdots \\}} \\nu_{d+1}(g_i,g_j,\\cdots )\n\\end{align}\nwhere $\\nu_{d+1}(g_i,g_j,\\cdots )$ is a function $G^{d+1} \\to U(1)$, which is a\ncocycle $\\nu_{d+1}\\in \\cH^{d+1}[G,U(1)]$. Here the space-time is a complex whose\nvertices are labeled by $i,j,\\cdots $, and $\\prod_{\\{i,j,\\cdots \\}}$ is the\nproduct over all the simplices of the space-time complex.\nAlso $\\sum_{\\{g_i\\}}$ is a sum over all $g_i$ on each vertex.\n\nNow consider the stacking of two SPT states described by cocycle\n$\\nu_{d+1}'$ and \n$\\nu_{d+1}''$: \n\\begin{align}\n Z =\\sum_{\\{g'_i,g''_i\\}} \\prod_{\\{i,j,\\cdots \\}} \n\\nu'_{d+1}(g'_i,g'_j,\\cdots )\n\\nu''_{d+1}(g''_i,g''_j,\\cdots ) .\n\\end{align}\nSuch a stacked state has a symmetry\n$G \\times G$ and is a $G \\times G$ SPT state.\n\nNow let us add a term to break the $G \\times G$-symmetry to $G$-symmetry\nand consider\n\\begin{align}\n\\label{ZU}\n Z =\\sum_{\\{g'_i,g''_i\\}}  \\prod_{\\{i,j,\\cdots \\}} &\n\\nu'_{d+1}(g'_i,g'_j,\\cdots )\n\\nu''_{d+1}(g''_i,g''_j,\\cdots ) \\times \n\\nonumber\\\\\n \\prod_i & \\ee^{-U|g_i'-g_i''|^2}\n,\n\\end{align}\nwhere $|g'-g''|$ is an invariant distance between group elements.  As we change\n$U=0$ to $U=+ \\infty $, the stacked system changes into the system for an ideal\nSPT state described by the cocycle\n$\\nu_{d+1}(g_i,g_j,\\cdots)=\\nu_{d+1}'(g_i,g_j,\\cdots )\n\\nu_{d+1}''(g_i,g_j,\\cdots )$.  If such a deformation does not cause any phase\ntransition, then we can show that the stacking of a $\\nu_{d+1}'$-SPT state with\na $\\nu_{d+1}''$-SPT state give rise to a $\\nu_{d+1}=\\nu_{d+1}'\\nu_{d+1}''$-SPT\nstate.  Thus, the key to show the stacking operation to give rise to the group\nstructure for the SPT states, is to show the theory \\eqn{ZU} has no phase\ntransition as we change $U=0$ to $U= +\\infty $.\n\nTo show there is no phase transition, we put the system on a closed space-time\nwith no boundary, say $S^{d+1}$.  In this case, $\\prod_{\\{i,j,\\cdots \\}} \n\\nu'_{d+1}(g'_i,g'_j,\\cdots ) \\nu''_{d+1}(g''_i,g''_j,\\cdots )=1$, since\n$\\nu_{d+1}'$ and $\\nu_{d+1}''$ are cocycles.\nThus the path integral \\eq{ZU} is reduced to\n\\begin{align}\n Z =\\sum_{\\{g'_i,g''_i\\}}  \\prod_i  \\ee^{-U|g_i'-g_i''|^2} \n= \\Big(|G| \\sum_g \\ee^{-U|1-g|^2}\\Big)^{N_v} ,\n\\end{align}\nwhere $N_v$ is the number of vertices and $|G|$ the order of the symmetry\ngroup.  We see that the free energy density\n\\begin{align}\n f = -\\lim_{N_v\\to\\infty}\\ln Z\/N_v\n\\end{align}\nis a smooth function of $U$ for $U\\in [0, \\infty )$. There is indeed no phase\ntransition.\n\nThe above result is highly non trivial from a categorical point of view.\nConsider two 2+1D bosonic SPT states described by two modular extensions $\\cM'$\nand $\\cM''$ of $\\Rp(G)$.  The natural tensor product $\\cM' \\boxtimes  \\cM''$ is\nnot a modular extension of $\\Rp(G)$,  but a modular extension of $\\Rp(G)\n\\boxtimes \\Rp(G)=\\Rp(G \\times G)$.  So, $\\cM' \\boxtimes  \\cM''$ describes a $G\n\\times G$-SPT state.  According to the above discussion, we need to break the\n$G \\times G$-symmetry down to the $G$-symmetry to obtain the $G$-SPT state.\nSuch a symmetry breaking process correspond to the so call ``anyon\ncondensation'' in category theory. We will discuss such anyon condensation\nlater.  The stacking operation $\\bt_\\cE$, with such a symmetry breaking process\nincluded, is the correct stacking operation that maintains the symmetry $G$.\n\n\n\\subsection{Mathematical construction of the stacking operation}\n\n\nWe have conjectured that a 2+1D topological order with symmetry $\\cE$ is\nclassified by $(\\cC,\\cM_\\cC,c)$, where $\\cC$ is a $\\mce{\\cE}$ , $\\cM_\\cC$ is a\nmodular extension of $\\cC$, and $c$ is the central charge.  If we have another\ntopological order of the same symmetry $\\cE$ described by $(\\cC',\\cM_{\\cC'},c')$,\nstacking $(\\cC,\\cM_\\cC,c)$ and $(\\cC',\\cM_{\\cC'},c')$ should give a third\ntopological order described by similar data $(\\cC'',\\cM_{\\cC''},c'')$:\n\\begin{align}\n (\\cC,\\cM_\\cC,c) \\boxtimes_\\cE (\\cC',\\cM_{\\cC'},c') =\n(\\cC'',\\cM_{\\cC''},c'')\n\\end{align}\n\nIn this section, we will show that such a stacking operation can be defined\nmathematically. This is an evidence supporting our Conjecture \\ref{classSET}.\nWe like to point out that a special case of the above result for\n$\\cC=\\cC'=\\cC''=\\cE=\\Rp(G)$ was discussed in section \\ref{grpSPT}.\n\nTo define $\\boxtimes_\\cE$ mathematically, first, we like to introduce\n\\begin{dfn}\\label{alg}\n  A \\emph{condensable algebra} in a UBFC $\\cC$ is a\n  triple $(A,m,\\eta)$, $A\\in\\cC$,\n  $m:A\\ot A\\to A$, $\\eta:\\one\\to A$ satisfying\n  \\begin{itemize}\n    \\item Associative: $m(\\id_A\\ot m)=m(m\\ot \\id_A)$\n    \\item Unit: $m(\\eta\\ot\\id_A)=m(\\id_A\\ot\\eta)=\\id_A$\n    \\item Isometric: $m m^\\dag=\\id_A$\n    \\item Connected: $\\Hom(\\one,A)=\\C$\n    \\item Commutative: $m c_{A,A}=m$\n  \\end{itemize}\n\\end{dfn}\nPhysically, such an condensable algebra $A$ is a composite self-bosonic anyon\nsatisfies additional conditions such that one can condense $A$ to obtain\nanother topological phase.\n\n\\begin{dfn}\n  A (left) \\emph{module} over a condensable algebra $(A,m,\\eta)$ in $\\cC$ is a\n  pair $(X,\\rho)$, $X\\in\\cC$, $\\rho:A\\ot X\\to X$ satisfying\n  \\begin{gather}\n    \\rho(\\id_A\\ot\\rho)=\\rho(m\\ot \\id_M),\\nonumber\\\\\n    \\rho(\\eta\\ot\\id_M)=\\id_M.\n  \\end{gather}\n  It is further a \\emph{local} module if\n  \\begin{align*}\n    \\rho c_{M,A} c_{A,M}=\\rho.\n  \\end{align*}\n\\end{dfn}\n  We denote the category of left $A$ modules by $\\cC_A$.\n  A left module $(X,\\rho)$ is turned into a right module via the braiding,\n  $(X,\\rho c_{X,A})$ or $(X,\\rho c_{A,X}^{-1})$, and thus an $A$-$A$ bimodule.\n  The relative tensor functor $\\ot_A$ of bimodules then turns $\\cC_A$ into a fusion category.\n  (This is known as $\\alpha$-induction in subfactor context.)\n  In general there can be two monoidal structures on $\\cC_A$, since there are\n  two ways to turn a left module into a bimodule (usually we pick one for\n  definiteness\n  when considering $\\cC_A$ as a fusion category).\n  The two monoidal structures coincide for the fusion subcategory $\\cC_A^0$ of\n  local $A$ modules. Moreover, $\\cC_A^0$ inherited the braiding from $\\cC$ and\n  is also a UBFC. The local modules are nothing but the anyons in the\n  topological phases after condensing $A$.\n\\begin{lem}[DMNO\\cite{dmno}]\n  \\[\\dim(\\cC_A)=\\frac{\\dim(\\cC)}{\\dim(A)}.\\]\n  If $\\cC$ is a UMTC, then so is $\\cC_A^0$, and\n  \\[\\dim(\\cC_A^0)=\\frac{\\dim(\\cC)}{\\dim(A)^2}.\\]\n\\end{lem}\n  A non-commutative algebra $A$ is also of interest. We have the left center\n  $A_l$ of $A$, the maximal subalgebra such that $m c_{A_l,A}=m$, and the right\n  center $A_r$, the maximal subalgebra such that $m c_{A,A_r}=m$. $A_l$ and\n  $A_r$ are commutative subalgebras, thus condensable.\n\\begin{thm}[FFRS\\cite{FFRS03}]\n  There is a canonical equivalence between the categories of local modules\n  over the left and right centers, $\\cC_{A_l}^0=\\cC_{A_r}^0$.\n\\end{thm}\n\\begin{dfn}\n  The Drinfeld center $Z(\\cA)$ of a monoidal category $\\cA$ is a monoidal category with\n  objects as pairs $(X\\in\\cA,b_{X,-})$, where $b_{X,-}: X\\ot -\\to -\\ot X$ are\n  half-braidings that satisfy similar conditions as braidings. Morphisms and\n  the tensor product are naturally defined.\n\\end{dfn}\n  $Z(\\cA)$ is a braided monoidal category. There is a forgetful tensor functor\n  $for_\\cA:Z(\\cA)\\to \\cA$, $(X,b_{X,-})\\mapsto X$ that forgets the half-braidings.\n\\begin{thm}[M{\\\"u}ger\\cite{Mue01}]\n  $Z(\\cA)$ is a UMTC if $\\cA$ is a fusion category and\n  $\\dim(Z(\\cA))=\\dim(\\cA)^2$.\n\\end{thm}\n\\begin{dfn}\n  Let $\\cC$ be a braided fusion category and $\\cA$ a fusion category, a tensor\n  functor $F:\\cC\\to \\cA$ is called a central functor if it factorizes through\n  $Z(\\cA)$, i.e., there exists a braided tensor functor $F':\\cC\\to Z(\\cA)$ such\n  that $F=F'for_\\cA$.\n\\end{dfn}\n\n\\begin{lem}\n  [DMNO\\cite{dmno}]\n  Let $F:\\cC\\to\\cA$ be a central functor, and $R:\\cA\\to\\cC$ the right adjoint\nfunctor of $F$.\nThen the object $A=R(\\one) \\in\\cC$ has a canonical structure of condensable algebra.\n$\\cC_A$ is monoidally equivalent to the image of\n$F$, i.e. the smallest fusion subcategory of $\\cA$ containing $F(\\cC)$.\n\\end{lem}\n\n\\begin{exa}\n  If $\\cC$ is a UBFC, it is naturally embedded into\n  $Z(\\cC)$, so is $\\overline\\cC$. Therefore, $\\cC\\bt\\overline\\cC\\hookrightarrow Z(\\cC)$.\n  Compose this embedding with the forgetful functor $for_\\cC:Z(\\cC)\\to\\cC$ we\n  get a central functor\n\\begin{align*}\n  \\cC\\bt\\overline\\cC &\\to \\cC\\\\\n  X\\bt Y&\\mapsto X\\ot Y.\n\\end{align*}\nLet $R$ be its right adjoint functor, we obtain a condensable algebra\n$L_\\cC:=R(\\one)\\cong \\oplus_i ( i\\bt \\bar i) \\in \\cC\\bt\\overline\\cC$ ($\\bar i$\ndenotes the dual object, or anti-particle of $i$) and $\\cC=(\n\\cC\\bt\\overline\\cC)_{L_\\cC}$, $\\dim(L_\\cC)=\\dim(\\cC)$.\nIn particular, for a symmetric category $\\cE$, $L_\\cE$ is a condensable algebra\nin $\\cE\\bt\\cE$, and $\\cE=(\\cE\\bt\\cE)_{L_\\cE}=(\\cE\\bt\\cE)_{L_\\cE}^0$ for $\\cE$\nis symmetric, all $L_\\cE$-modules are local.\nCondensing $L_\\cE$ is nothing but breaking the symmetry from $\\cE\\bt\\cE$ to\n$\\cE$.\n\\end{exa}\n\n\nNow, we are ready to define the stacking operation for $\\mce{\\cE}$'s as well\nas their  modular extensions.\n\\begin{dfn}\\label{stacking}\n  Let $\\cC,\\cD$ be $\\mce{\\cE}$'s, and $\\cM_\\cC,\\cM_\\cD$ their \n  modular extensions. The stacking is defined by:\n  \\begin{align*}\n    \\cC\\bt_\\cE\\cD:=(\\cC\\bt\\cD)^0_{L_\\cE},\\quad \\cM_\\cC\\bt_\\cE \\cM_\\cD:=(\\cM_\\cC\\bt\n    \\cM_\\cD)_{L_\\cE}^0\n  \\end{align*}\n\\end{dfn}\nNote that in Ref.~\\onlinecite{DNO11}, the tensor product $\\bt_\\cE$ for\n$\\mce{\\cE}$'s is defined as $(\\cC\\bt\\cD)_{L_\\cE}$. For $\\mce{\\cE}$'s the two\ndefinitions coincide $(\\cC\\bt\\cD)^0_{L_\\cE}=(\\cC\\bt\\cD)_{L_\\cE}$, for $L_\\cE$ lies in\nthe centralizer of $\\cC\\bt\\cD$ which is $\\cE\\bt\\cE$. But for the  modular extensions we have to take\nthe unusual definition above.\n\n\\begin{thm}\n  $\\cC\\bt_\\cE\\cD$ is a $\\mce{\\cE}$, and\n  $\\cM_\\cC\\bt_\\cE\\cM_\\cD$ is a  modular extension of $\\cC\\bt_\\cE\\cD$.\n\\end{thm}\n\\begin{proof}\n  The embeddings\n$\\cE=(\\cE\\bt\\cE)_{L_\\cE}^0\\hookrightarrow (\\cC\\bt\\cD)^0_{L_\\cE}=\\cC\\bt_\\cE\\cD\n\\hookrightarrow (\\cM_\\cC\\bt\\cM_\\cD)^0_{L_\\cE}=\\cM_\\cC\\bt_\\cE\\cM_\\cD$\nare obvious.\nSo $\\cC\\bt_\\cE\\cD$ is a UBFC over $\\cE$. Also\n\\begin{align*}\n  \\dim(\\cC\\bt_\\cE\\cD)=\\frac{\\dim(\\cC\\bt\\cD)}{\\dim(L_\\cE)}\n  =\\frac{\\dim(\\cC)\\dim(\\cD)}{\\dim(\\cE)},\n\\end{align*}\nand $\\cM_\\cC\\bt_\\cE\\cM_\\cD$ is a UMTC,\n\\begin{align*}\n  \\dim(\\cM_\\cC\\bt_\\cE\\cM_\\cD)\n  =\\frac{\\dim(\\cM_\\cC\\bt\\cM_\\cD)}{\\dim(L_\\cE)^2}=\\dim(\\cC)\\dim(\\cD).\n\\end{align*}\n  Thus, $\\cM_\\cC\\bt_\\cE\\cM_\\cD$ is a\n  modular extension of $\\cC\\bt_\\cE\\cD$.\n\\end{proof}\n\n  Take $\\cD=\\cE$. Note that $\\cC\\bt_\\cE\\cE=\\cC$. Therefore,  for any\n  modular extension $\\cM_\\cE$ of $\\cE$, $\\cM_\\cC\\bt_\\cE\\cM_\\cE$ is still a\n  modular extension of $\\cC$. In the following we want to show the inverse,\n  that one can extract the ``difference'', a modular extension of $\\cE$,\n  between two modular extensions of $\\cC$.\n\n\\begin{lem}\\label{Lag}\n  We have $(\\cC\\bt\\overline\\cC)_{L_\\cC}^0=\\cen{\\cC}{\\cC}$.\n\\end{lem}\n\\begin{proof}\n  $(\\cC\\bt\\overline\\cC)_{L_\\cC}$ is equivalent to $\\cC$ (as a fusion\ncategory). Moreover, for $X\\in\\cC$ the equivalence gives the free module $\nL_\\cC\\ot(X\\bt\n\\one )\\cong L_\\cC\\ot(\\one\\bt X)$. $L_\\cC\\ot(X\\bt\\one )$ is a local $L_\\cC$ module if and only\nif $X\\bt \\one$ centralize $L_\\cC$. This is the same as $X\\in\n\\cen{\\cC}{\\cC}$. Therefore we have $(\\cC\\bt \\overline\\cC)_{L_\\cC}^0=\\cen{\\cC}{\\cC}$.\n\\end{proof}\n\n\n\\begin{thm}\\label{main}\n let $\\cM$ and $\\cM'$ be two modular extensions of the $\\mce{\\cE}$ $\\cC$. There\n exists a unique $\\cK\\in\\mathcal{M}_{ext}(\\cE)$ such that $\\cK\\bt_\\cE\\cM=\\cM'$. Such $\\cK$\n is given by\n   \\begin{align*}\n     \\cK=(\\cM'\\bt \\overline\\cM)_{L_\\cC}^0.\n   \\end{align*}\n\\end{thm}\n\\begin{proof}\n  $\\cK$ is a modular extension of $\\cE$. This follows\n  Lemma \\ref{Lag}, that $\\cE=\\cen{\\cC}{\\cC}=(\\cC\\bt\\overline \\cC)^0_{L_\\cC}$ is a full\n  subcategory of $\\cK$. $\\cK$ is a UMTC by construction, and\n  $\\dim(\\cK)=\\frac{\\dim(\\cM)\\dim(\\cM')}{\\dim(L_\\cC)^2}=\\dim(\\cE)^2$.\n\n  To show that $\\cK=(\\cM'\\bt\\overline\\cM)_{L_\\cC}$ satisfies\n  $\\cM'=\\cK\\bt_\\cE\\cM$, note that\n  $\\cM'=\\cM'\\bt\\mathrm{Vec}=\\cM'\\bt(\\overline\\cM\\bt\\cM)_{L_{\\overline\\cM}}^0$. It suffies that\n  \\begin{align*}\n    (\\cM'\\bt\\overline\\cM\\bt\\cM)_{\\one\\bt\n      L_{\\overline\\cM}}^0=[(\\cM'\\bt\\overline\\cM)_{L_\\cC}^0\\bt\n      \\cM]_{L_\\cE}^0\\\\\n      =(\\cM'\\bt\\overline\\cM\\bt\\cM)_{(L_\\cC\\bt\\one)\\ot(\\one\\bt\n      L_\\cE)}^0.\n  \\end{align*}\n  This follows that $\\one\\bt L_{\\overline\\cM} $ and $(L_\\cC\\bt\\one)\\ot(\\one\\bt\n  L_\\cE)$ are left and right centers of the algebra $(L_\\cC\\bt\\one)\\ot(\\one\\bt\n  L_{\\overline\\cM})$.\n\n  If $\\cM'=\\cK\\bt_\\cE\\cM=(\\cK\\bt\\cM)_{L_\\cE}^0$,\n  then\n  \\begin{align*}\n    \\cK= (\\cK\\bt\\cM\\bt\\overline\\cM)_{\\one\\bt\n    L_\\cM}^0=\n    (\\cK\\bt\\cM\\bt\\overline\\cM)_{(L_\\cE\\bt\\one)\\ot(\\one\\bt \n    L_\\cC)}^0\\\\\n    =[(\\cK\\bt_\\cE\\cM)\\bt\\overline\\cM]_{L_\\cC}^0\n    =(\\cM'\\bt\\overline\\cM)_{L_\\cC}^0.\n  \\end{align*}\n  It is similar here that $\\one\\bt L_{\\cM} $ and\n  $(L_\\cE\\bt\\one)\\ot(\\one\\bt\n  L_\\cC)$ are the left and right centers of the algebra\n  $(L_\\cE\\bt\\one)\\ot(\\one\\bt\n  L_{\\cM})$. This proves the uniqueness of $\\cK$.\n\n\\end{proof}\n\n\nLet us list several consequences of Theorem \\ref{main}.\n\\begin{thm}\\label{hegroup}\n  $\\mathcal{M}_{ext}(\\cE)$ forms an finite abelian group.\n \\end{thm}\n \\begin{proof}\n   Firstly, there exists at least one modular extension of a symmetric fusion\n   category $\\cE$,\n   the Drinfeld center $Z(\\cE)$. So the set $\\mathcal{M}_{ext}(\\cE)$ is not empty.\n   The multiplication is given by the stacking $\\bt_\\cE$.\n   It is easy to verify that the stacking $\\bt_\\cE$ for modular extensions\n   is associative and commutative. To show that they form a group we only need\n   to find out the identity and inverse.\n   In this case $\\cK=(\\cM'\\bt \\overline \\cM)^0_{L_\\cE}=\\cM'\\bt_\\cE\\overline \\cM$,\n   Theorem \\ref{main} becomes $\\cM'\\bt_\\cE\\overline\\cM\\bt_\\cE\\cM=\\cM'$, for any\n   modular extensions $\\cM,\\cM'$ of $\\cE$.\n   Thus, $\\overline{\\cM'}\\bt_\\cE\n   \\cM'=\\overline{\\cM'}\\bt_\\cE\n   \\cM'\\bt_\\cE\\overline\\cM\\bt_\\cE\\cM\n   =\\overline\\cM\\bt_\\cE\\cM$, i.e. $\\overline\\cM\\bt_\\cE\\cM$, is the same category\n   for any extension $\\cM$, which turns out to be $Z(\\cE)$. It is exactly the identity element. It is then\n   obvious that the inverse of $\\cM$ is $\\overline\\cM$.\n   The finiteness follows from \\Ref{BNRW13}.\n \\end{proof}\n\n  \\begin{exa}\n    For bosonic case we find that $\\mathcal{M}_{ext}(\\Rp(G))=H^3(G,U(1))$, which is\n    discussed in more detail in the next subsection. For fermionic case a\n    general group cohomological classification is still lacking. We know some\n    simple ones such as $\\mathcal{M}_{ext}(\\sRp(\\Z_2^f))=\\Z_{16}$, which agrees with\n    Kitaev's\n    16-fold way\\cite{K062}.\n  \\end{exa}\n\n \\begin{thm}\\label{hetorsor}\n   For a $\\mce{\\cE}$ $\\cC$, if the modular extensions exist, $\\mathcal{M}_{ext}(\\cC)$ form\n   a $\\mathcal{M}_{ext}(\\cE)$-torsor. In particular, $|\\mathcal{M}_{ext}(\\cC)|=|\\mathcal{M}_{ext}(\\cE)|$.\n \\end{thm}\n \\begin{proof}\n   The action is given by the stacking $\\bt_\\cE$.\n   For any two extensions $\\cM,\\cM'$, there is a unique extension $\\cK$ of\n   $\\cE$, such that $\\cM\\bt_\\cE\\cK=\\cM'$. To see $Z(\\cE)$ acts trivially, note\n   that $\\cM'\\bt_\\cE Z(\\cE)=\\cM\\bt_\\cE \\cK\\bt_\\cE Z(\\cE)=\\cM\\bt_\\cE\\cK=\\cM'$\n   holds for any $\\cM'$. Due to uniqueness we also know that only $\\cZ_\\cE$\n   acts trivially. Thus, the action is free and transitive.\n \\end{proof}\n This means that for any modular extension of $\\cC$,\n   stacking with a nontrivial modular extensions of $\\cE$, one always obtains\n   a different modular extension of $\\cC$; on the other hand, starting with a\n   particular modular extension of $\\cC$, all the other modular extensions can\n   be generated by staking modular extensions of $\\cE$ (in other words, there\n   is only on orbit). However, in general, there is no preferred choice of the\n   starting modular extension, unless $\\cC$ is the form $\\cC_0\\bt \\cE$ where\n   $\\cC_0$ is a UMTC. \n\n \n\\subsection{Modular extensions of $\\Rp(G)$} \\label{sec:mext-repG}\n\n\\begin{figure}[tb]\n$$\n\\setlength{\\unitlength}{.5pt}\n\\begin{picture}(100, 185)\n   \\put(-80,){\\scalebox{1}{\\includegraphics{pic-half-braiding}}}\n   \\put(0,0){\n     \\put(-18,-40){\n    \n     \\put(20, 265)     { $e\\in \\Rp(G) \\subset \\cM$}\n     \\put(-30, 170)    { $-\\otimes A$}\n     \\put(75, 81)     { $x$}\n     \\put(75,30)        { $\\gamma_2$}\n     \\put(75,132)        { $\\gamma_1$}\n     \\put(100, 210)    {$\\cM$}\n     \\put(230, 81)    {$\\cM_A$}\n     \\put(160,45)     {$\\mathrm{Vec}$}\n     \\put(0, 60)        {$F(e)$}\n     }\\setlength{\\unitlength}{1pt}}\n  \\end{picture}\n \n$$\n\\caption{Consider a physical situation in which the excitations in the $2+1$D\n  bulk are given by a modular extension $\\cM$ of $\\Rp(G)$, and those on the\n  gapped boundary by the UFC $\\cM_A$. Consider a simple particle $e\\in \\Rp(G)$\n  in the bulk moving toward the boundary. The bulk-to-boundary map is given by\n  the central functor $-\\otimes A: \\cM \\to \\cM_A$, which restricted to $\\Rp(G)$\n  is nothing but the forgetful functor $F:\\Rp(G) \\to \\mathrm{Vec}$. Let $x$ be a\n  simple excitation in $\\cM_A$ sitting next to $F(e)$. We move $F(e)$ along the\n  semicircle $\\gamma_1$ (defined by the half-braiding), then move along the\n  semicircle $\\gamma_2$ (defined by the symmetric braiding in the trivial phase\n  $\\mathrm{Vec}$).}\n\\label{fig:G-grading}\n\\end{figure}\n\nWe set $\\cE=\\Rp(G)$ throughout this subsection. Let $(\\cM,\\iota_\\cM)$ be a modular\nextension of $\\Rp(G)$. $\\iota_\\cM$ is the embedding\n$\\iota_\\cM:\\cE\\hookrightarrow\\cM$ that we need to consider explicitly in this\nsubsection. The algebra $A=\\mathrm{Fun}(G)$ is a condensable algebra in $\\Rp(G)$ and\nalso a condensable algebra in $\\cM$. Moreover, $A$ is a Lagrangian algebra in\n$\\cM$ because $(\\dim A)^2 = |G|^2=(\\dim \\Rp(G))^2 = \\dim \\cM$. Therefore, $\\cM\\simeq Z(\\cM_A)$, where $\\cM_A$ is the category of right $A$-modules in $\\cM$. In other words, $\\cM$ describes the bulk excitations in a 2+1D topological phase with a gapped boundary (see Fig.\\,\\ref{fig:G-grading}). Moreover, the fusion category $\\cM_A$ is pointed and equipped with a canonical fully faithful $G$-grading\\cite{dgno2007}, which means that \n$$\n\\cM_A=\\oplus_{g\\in G} (\\cM_A)_g, \\quad (\\cM_A)_g\\simeq \\mathrm{Vec}, \\,\\, \\forall g\\in G, \n$$ \n$$\n\\quad \\mbox{and} \\quad \\otimes: (\\cM_A)_g \\boxtimes (\\cM_A)_h \\xrightarrow{\\simeq} (\\cM_A)_{gh}.\n$$\nLet us recall the construction of this $G$-grading. The physical meaning of\nacquiring a $G$-grading on $\\cM_A$ after condensing the algebra $A=\\mathrm{Fun}(G)$ in\n$\\cM$ is depicted in Figure\\,\\ref{fig:G-grading}.  The process in\nFigure\\,\\ref{fig:G-grading} defines the isomorphism $F(e) \\otimes_A x\n  \\xrightarrow{z_{e,x}} x \\otimes_A F(e) = F(e) \\otimes_A x$,\n  which further gives a monoidal automorphism $\\phi(x)\\in \\mathrm{Aut}(F)=G$ of\nthe fiber functor $F: \\Rp(G) \\to \\mathrm{Vec}$.  \n\nSince $\\phi$ is an isomorphism, the associator of the monoidal category $\\cM_A$ determines a unique\n$\\omega_{(\\cM,\\iota_M)}\\in H^3(G, U(1))$ such that $\\cM_A \\simeq \\mathrm{Vec}_G^\\omega$ as $G$-graded fusion categories. \n\n\n\\begin{thm} \\label{thm:spt}\nThe map $(\\cM, \\iota_\\cM) \\mapsto \\omega_{(\\cM, \\iota_\\cM)}$ defines a group isomorphism $\\cM_{ext}(\\Rp(G)) \\simeq H^3(G, U(1))$. In particular, we have \n$$ \n(Z(\\mathrm{Vec}_G^{\\omega_1}),\\iota_{\\omega_1}) \\boxtimes_\\cE\n(Z(\\mathrm{Vec}_G^{\\omega_2}),\\iota_{\\omega_2}) \\simeq (Z(\\mathrm{Vec}_G^{\\omega_1+\\omega_2}), \\iota_{\\omega_1+\\omega_2}).\n$$\n\\end{thm}\n\nFor the proof and more related details, see also \\Ref{LW160205936}.\n\n\n\\subsection{Relation to numerical calculations}\nIn Section \\ref{clGQL2} we proposed another way to characterise GQLs, using\nthe data $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$\nwhich is more friendly in numerical calculations. We would like to investigate how\nto calculate the stacking operation in terms of these data.\n\nAssuming that $\\cC$ and $\\cC'$ can be characterized by data\n$(N^{ij}_k,s_i)$ and $( N^{\\prime ij}_k,s^\\prime_i)$. Let $(\nN^{\\cD, ij}_k, s^\\cD_i)$ be the data that characterizes the stacked $\\mce{\\cE}$ $\\cD\n=\\cC\\bt_\\cE \\cC'$.\n\nTo calculate $(N^{\\cD, ij}_k, s^\\cD_i)$, let us first construct\n\\begin{align}\n N^{ii',jj'}_{kk'} = N^{ij}_{k} N^{\\prime i'j'}_{k'} \n,\\ \\ \\ \\ \\\ns_{ii'}=s_i+s'_{i'}.\n\\end{align}\nNote that, the above data describes a $\\mce{\\cE\\boxtimes \\cE}$ $\\cD' =\\cC\\bt\n\\cC'$ (\\ie with centralizer $\\cE\\bt \\cE$), which is not what we want.  We need\nreduce centralizer from $\\cE\\bt \\cE$ to $\\cE$. This is the $G\\times G$ to $G$\nprocess and $\\cC$-$\\cC'$ coupling, or condensing the $L_\\cE$\nalgebra, as discussed above\n\nTo do the $\\cE\\bt \\cE$ to $\\cE$ reduction (\\ie to obtain the real stacking\noperation $\\bt_\\cE$), we can introduce an equivalence relation. Noting that the\nexcitations in $\\cD'=\\cC\\bt \\cC'$ are labeled by $ii'=i\\bt i'$, the  equivalence\nrelation is \n\\begin{align}\n  ii'\\sim jj', \\quad{\\text{if }} ii'\\ot L_\\cE=jj'\\ot L_\\cE.\n\\end{align}\nwhere $L_\\cE=\\oplus_a a\\bar a, a\\in\\cE$. In the simple case of abelian groups,\nwhere all the $a$'s are abelian particles, the equivalence relation reduces to\n\\begin{align}\n (a\\ot i)i'\\sim i(a\\ot i'),\\ \\ \\ \\\n\\forall \\ \\ i\\in \\cC,\\ i'\\in \\cC',\\  a\\in \\cE.\n\\end{align}\nMathematically, this amounts to consider only the free local $L_\\cE$ modules.\nThe equivalent classes $[ii']$ are then some composite anyons in\n$\\cD=\\cC\\bt_\\cE\\cC'$\n\\begin{align}\n  [ii']=k\\oplus l \\oplus \\cdots , \\quad\\text{ for some }k,l,\\dots\\in\\cD.\n\\end{align}\nIn other words, they form a fusion sub ring of $\\cD$.\nMoreover, the spin of $ii'$ is the same as the direct summands\n\\begin{align}\n  s_{ii'}=s_k^\\cD=s_l^\\cD=\\cdots\n\\end{align}\nSince it is\nlimited to a subset of data of $\\mce{\\cE}$'s, we can only give these necessary\nconditions. However, as we already give a large list of GQLs in terms of these data,\nthey are usually enough to pick the resulting $\\cC\\bt_\\cE\\cC'$ from the list.\n\n\\section{How to calculate the modular extension of a $\\mce{\\cE}$}\n\\label{howto}\n\n\n\\subsection{A naive calculation}\n\nHow do we calculate the modular extension $\\cM$ of $\\mce{\\cE}$ $\\cC$ from\nthe data of $\\cC$?  Actually, we do not know how to do that.  So here, we will\nfollow a closely related Conjecture \\ref{NsNsNs}, and calculate instead\n$(\\cN^{IJ}_K,\\cS_I,c)$ (that fully characterize $\\cM$) from the data $(\n\\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$ (that partially characterize $\\cC$).\nIn this section, we will describe such a calculation.\n\nWe note that all the simple objects (particles) in $\\cC$ are contained in $\\cM$\nas simple objects, and $\\cM$ may contain some extra simple objects.  Assume\nthat the particle labels of $\\cM$ are $\\{I,J,\\cdots \\}=\\{i,j,\\dots, x,\ny,\\dots\\}$, where we use $i,j,\\cdots $ to label the particles in $\\cC$ and\n$x,y,\\cdots $ to label the additional particles (not in $\\cC$).  Also let us\nuse $a,b,\\cdots $ to label the simple objects in the centralizer of $\\cC$:\n$\\cE=\\cC_\\cC^\\text{cen}$.  Let $\\cN^{IJ}_K$, $\\cS_{I}$ be the fusion\ncoefficients and the spins for $\\cM$, and $N^{ij}_k,\\ s_i$ be the fusion\ncoefficients and the spins for $\\cC$.  The idea is to find as many conditions on\n$(\\cN^{IJ}_K, \\cS_{I})$ as possible, and use those conditions to solve for\n$(\\cN^{IJ}_K, \\cS_{I})$.  Since the data $(\\cN^{IJ}_K, \\cS_{I})$ describe the\nUMTC $\\cM$, they should satisfy all the conditions discussed in\n\\Ref{W150605768}.  On the other hand, as a modular extension of $\\cC$,\n$(\\cN^{IJ}_K, \\cS_{I})$ also satisfy some additional conditions. Here, we will\ndiscuss those additional conditions.\n\nFirst, the modular extension $\\cM$ has a fixed total quantum dimension.\n\\begin{align}\n\\label{dimMCE}\n \\dim(\\cM)=\\dim(\\cE)\\dim(\\cC).\n\\end{align}\nIn other words\n\\begin{align}\n \\sum_{I\\in \\cM} d_I^2 = \\sum_{a\\in \\cE} d_a^2 \\sum_{i\\in \\cC} d_i^2.\n\\end{align}\n\nPhysically, the modular extension $\\cM$ is obtained by ``gauging'' the\nsymmetry $\\cE$ in $\\cC$ (\\ie adding the symmetry twists of $\\cE$).  So the\nadditional particles $x,y,\\cdots$ correspond to the symmetry twists.  Fusing an\noriginal particle $i\\in \\cC$ to a symmetry twist $x\\notin \\cC$ still give us a\nsymmetry twist.  Thus\n\\begin{align}\n \\cN^{ix}_j = \\cN^{xi}_j = \\cN^{ij}_x =0.\n\\end{align}\n\nTherefore, $\\cN_i$ for $i\\in \\cC$ is block diagonal: \n\\begin{align}\n\\cN_i= N_i \\oplus \\hat N_i, \n\\end{align}\nwhere $( N_i)_{jk}=\\cN^{ij}_{k}=N^{ij}_{k}$ and $(\\hat N_i)_{x \ny}=\\cN^{iy}_{x}$.  \n\nIf we pick a charge conjugation for the additional particles $x\\mapsto\n\\bar x$, the conditions for fusion rules reduce to\n\\begin{align}\n  \\cN^{i x}_{y}=\\cN^{x i}_{y}=\\cN^{\\bar{x} y}_{i}=\\cN^{i \\bar{y}}_{\\bar{x}},\n\\nonumber \\\\\n  \\sum_{k\\in\\cC} N^{ij}_k \\cN^{kx}_{y}= \\sum_{z \\notin\\cC} \\cN^{i z}_{ x} \\cN^{j y}_{z}.\n  \\label{extN}\n\\end{align}\nWith a choice of charge conjugation, it is enough to construct (or search for)\nthe matrices $\\hat N_i$ and $\\cN^{xy}_z$ to determine all the extended fusion\nrules $\\cN^{IJ}_K$.  \n\nBesides the general condition \\eqref{extN}, there are also some simple\nconstraints on $\\hat N_i$ that may speed up the numerical search.\nFirstly, observe that \\eqref{extN} is the same as\n\\begin{align}\n\\hat N_i \\hat N_j = \\sum_{k\\in \\cC} N^{ij}_k \\hat N_k,\n\\end{align}\nwhere $i,j,k \\in \\cF$.  This means that $\\hat N_i$ satisfy the same fusion\nalgebra as $ N_i$, and $N^{ij}_k=\\cN^{ij}_k$ is the structure constant;\ntherefore, the eigenvalues of $\\hat N_i$ must be a subset of the eigenvalues\nof $ N_i$. \n\nSecondly, since $\\sum_{y\\notin\\cC} \\cN^{i x}_{y}d_{y}= d_i d_{x}$, by\nPerron-Frobenius theorem, we know that $d_i$ is the largest eigenvalue of\n$\\hat N_i$, with eigenvector $v, v_{x}=d_{x}$. ($d_i$ is also the largest\nabsolute values of the eigenvalues of $\\hat N_i$.) Note that ${\\hat N_{\\bar\ni} \\hat N_{i}= \\hat N_i \\hat N_{\\bar i},}$ ${ \\hat N_{\\bar i}=\\hat\nN_i^\\dag}$.  Thus, $d_i^2$ is the largest eigenvalue of the positive\nsemi-definite Hermitian matrix $\\hat N_{i}^\\dag\\hat N_{i}$. For any unit\nvector $v$ we have $v^\\dag \\hat N_i^\\dag \\hat N_i v\\leq d_i^2$, in\nparticular,\n\\begin{align}\n  (\\hat N_i^\\dag \\hat N_i)_{xx}=\\sum_{y} (\\cN^{ix}_{y})^2\\leq\n  d_i^2.\n  \\label{extentry}\n\\end{align}\nThe above result is very helpful to reduce the scope of numerical search.\n\nOnce we find the fusion rules, $\\cN^{IJ}_K$, we can then use the rational\nconditions and other conditions to determine the spins $\\cS_I$ (for details, see\n\\Ref{W150605768}).  The set of data $(\\cN^{IJ}_K,\\cS_I)$ that satisfy all the\nconditions give us the set of modular extensions.\n\n\nThe above proposed calculation for modular extensions is quite expensive.  If\nthe quantum dimensions of the particles in $\\cC$ are all equal to 1: $d_i=1$,\nthen there is another much cheaper way to calculate the fusion coefficient\n$\\cN^{IJ}_K$ of the modular extension $\\cM$.  Such an approach is explained in\nAppendix \\ref{FRgroup}.  We will also use such an approach in our calculation.\n\nLast, we would like to mention that two sets of data $(\\cN^{IJ}_K,\\cS_I)$ and\n$(\\bar \\cN^{IJ}_K,\\bar \\cS_I)$ describe the same modular extension of\n$\\cC$, if they only differ by a permutation of indices $x \\in \\cM$ but\n$x \\notin \\cC$.  So some times, two sets of data $(\\cN^{IJ}_K,\\cS_I)$ and\n$(\\bar \\cN^{IJ}_K,\\bar \\cS_I)$ can describe different modular extensions,\neven through they describe the same UMTC.  (Two sets of data\n$(\\cN^{IJ}_K,\\cS_I)$ and $(\\bar \\cN^{IJ}_K,\\bar \\cS_I)$ describe the same\nUMTC, if they are only different by a permutation of indices $I \\in \\cM$.)\n\nWhy we use such a permutation in the calculation of modular extensions.  (which\nis the ME-equivalence relation discussed before)?  This is because when we\nconsidering modular extensions, the particle $x \\in \\cM$ but $x \\notin \\cC$\ncorrespond to symmetry twists. They are extrinsic excitations that do not\nappear in the finite energy spectrum of the Hamiltonian.  While the particle\n$i\\in \\cC$ are intrinsic excitations that do appear in the finite energy\nspectrum of the Hamiltonian.  So $x \\notin \\cC$ and $i\\in \\cC$ are physically\ndistinct and we do not allow permutations that mix them.  Also we should not\npermute the particles $a\\in \\cE$, because they correspond to symmetries. We\nshould not mix, for example, the $Z_2$ symmetry of exchange layers and the\n$Z_2$ symmetry of 180$^\\circ$ spin rotation.\n\n\n\\subsection{The limitations of the naive calculation}\n\nSince a $\\mce{\\cE}$ $\\cC$ is not modular, the data $(\\tilde N^{ab}_c,\\tilde\ns_a;N^{ij}_k,s_i)$ may not fully characterize $\\cC$.  To fully characterize $\\cC$, we need to use additional data, such\nas the $F$-tensor and the $R$-tensor\\cite{K062,W150605768}.  \n\nIn this paper, we will not use those additional data. As a result, the data\n$(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ may correspond to several different\n$\\mce{\\cE}$ $\\cC$'s.  In other words, $(\\tilde N^{ab}_c,\\tilde\ns_a;N^{ij}_k,s_i)$ is a one-to-many labeling of  $\\mce{\\cE}$'s.\n\nSo in our naive calculation, when we calculate the modular extensions of\n$(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$, we may actually calculate the\nmodular extension of several different $\\cC$'s that are described by the same\ndata $(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$.  But for $\\mce{\\cE}$'s that\ncan be fully characterized by the data $(\\tilde N^{ab}_c,\\tilde\ns_a;N^{ij}_k,s_i)$, our calculation produce the modular extensions of a single\n$\\cC$.  For example, the naive calculation can obtain the correct modular\nextensions of $\\cC=\\Rp(G)$ and $\\cC=\\sRp(G^f)$, when $G$ and $G^f$ are abelian\ngroups, or simple finite groups\\cite{Yuan16}.\n\nIf the $(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ happen to describe two\ndifferent $\\mce{\\cE}$'s, we find that our naive calculation will produce the\nmodular extensions for both of $\\mce{\\cE}$'s (see Section \\ref{Z2N5}).  So by\ncomputing the modular extensions  of $(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$,\nwe can tell if $(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ corresponds to none,\none, two, \\etc $\\mce{\\cE}$'s.  This leads to the Conjecture \\ref{NsNsNs} that\n$(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i,\\cN^{IJ}_K,\\cS_I;c)$ can fully and\none-to-one classify GQLs in 2+1D.\n\n\\section{Examples of 2+1D SET orders and SPT orders}\n\\label{examples}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{ The bottom two rows correspond to the two modular extensions of\n$\\Rp(Z_2)$ (denoted by $N_c^{|\\Th|}=2^{\\zeta^1_2}_0$).  Thus we have two\ndifferent trivial topological orders with $Z_2$ symmetry in 2+1D (\\ie two $Z_2$\nSPT states). \n} \n\\label{mextZ2} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n \\hline \n$2^{\\zeta_{2}^{1}}_{ 0}$ & $2$ & $1, 1$ & $0, 0$ & $\\Rp(Z_2)$ \\\\\n\\hline\n$4^{ B}_{ 0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, 0, \\frac{1}{2}$ & $Z_2$ gauge\\\\\n$4^{ B}_{ 0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{4}, \\frac{3}{4}$ & double semion\\\\\n \\hline \n\\end{tabular} \n\\end{table}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{The two modular extensions of $N^{|\\Th|}_{c}=3^{\\zeta_{2}^{1}}_{ 2}$.\n$3^{\\zeta_{2}^{1}}_{ 2}$ has a centralizer $\\Rp(Z_2)$.  Thus we have two\ntopological orders with $Z_2$ symmetry in 2+1D which has only one type of\nspin-$1\/3$ topological excitations.  We use $N^{|\\Th|}_{c}$ to label\n$\\mce{\\cE}$'s, where $\\Theta ={D}^{-1}\\sum_{i}\\ee^{2\\pi\\ii s_i} d_i^2=\n|\\Th|\\ee^{2\\pi \\ii c\/8}$ and $D^2=\\sum_id_i^2$.\n} \n\\label{mextZ2a} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$3^{\\zeta_{2}^{1}}_{ 2}$ & $6$ & $1, 1, 2$ & $0, 0, \\frac{1}{3}$ & \n\\tiny $K=\\begin{pmatrix}\n 2 & -1 \\\\\n -1 & 2 \\\\\n\\end{pmatrix}\n$\n\\\\\n\\hline\n$5^{ B}_{ 2}$ & $12$ & $1, 1, 2,\\zeta_{4}^{1},\\zeta_{4}^{1}=\\sqrt{3}$ & $0, 0, \\frac{1}{3}, \\frac{1}{8}, \\frac{5}{8}$ & $(A_1,4)$ \\\\\n$5^{ B}_{ 2}$ & $12$ & $1, 1, 2,\\zeta_{4}^{1},\\zeta_{4}^{1}$ & $0, 0, \\frac{1}{3}, \\frac{3}{8}, \\frac{7}{8}$ & \\\\\n \\hline \n\\end{tabular} \n\\end{table}\n\nIn this section, we will discuss simple examples of $\\mce{\\cE}$ $\\cC$'s, and\ntheir modular extensions $\\cM$. The triple $(\\cC, \\cM,c)$ describe a\ntopologically ordered or SPT phase.  A single $\\mce{\\cE}$ $\\cC$ only describes\nthe set of bulk topological excitations, which correspond to topologically\nordered states up to invertible ones.\n\nHowever, in this section we will not discuss examples of $\\mce{\\cE}$ $\\cC$.\nWhat we really do is to discuss examples of the solutions $(\\tilde\nN^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ (which are not really $\\mce{\\cE}$'s, but\nclosely related).  We will also discuss the modular extensions\n$(\\cN^{IJ}_K,\\cS_I;c)$ of $(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$.\n$(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ will correspond to $\\mce{\\cE}$\n$\\cC$ if it has modular extensions $(\\cN^{IJ}_K,\\cS_I;c)$.  This allows us to\nclassify GQLs in terms of the data $(\\tilde N^{ab}_c,\\tilde\ns_a;N^{ij}_k,s_i,\\cN^{IJ}_K,\\cS_I;c)$.\n\n\\subsection{$Z_2$ bosonic SPT states}\n\nTables \\ref{SETZ2-34}, \\ref{SETZ2-5}, and \\ref{SETZ2-6} list the solutions\n$(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ when  $(\\tilde N^{ab}_c,\\tilde\ns_a)$ describes a SFC $\\Rp(Z_2)$.  The table contains all $\\mce{\\Rp(Z_2)}$'s\nbut may contain extra fake entries.  Physically, they describe possible\nsets of bulk excitations for $Z_2$-SET orders of bosonic systems.  The sets of\nbulk excitations are listed by their quantum dimensions $d_i$ and spins $s_i$.\n\nFor example, let us consider the entry $N_c^{|\\Th|}=2_0^{\\zeta_2^1}$ in Table\n\\ref{SETZ2-34}.  Such an entry has a central charge $c=0$.  Also $N=2$, hence\nthe $Z_2$-SET state has two types of bulk excitations both with $d_i=1$ and\n$s_i=0$.  Both types of excitations are local excitations; one is the trivial type\nand the other carries an $Z_2$ charge.\n\nThe first question that we like to ask is that ``is such an entry a fake entry,\nor it corresponds to some $Z_2$-symmetric GQL's?'' If it corresponds to\nsome $Z_2$-symmetric GQL's, how many distinct $Z_2$-symmetric GQL\nphases that it corresponds to?  In other word, how many distinct\n$Z_2$-symmetric GQL phases are there, that share the same set of bulk\ntopological excitations described by the entry $2_0^{\\zeta_2^1}$?\n\nBoth questions can be answered by computing the modular extensions of\n$2_0^{\\zeta_2^1}$ (which is also denoted as $\\Rp(Z_2)$).  We find that the\nmodular extensions exist, and thus $\\Rp(Z_2)$ does correspond to some \n$Z_2$-symmetric GQL's.  In fact, one of the $Z_2$-symmetric GQL's is the\ntrivial product state with $Z_2$ symmetry.  Other $Z_2$-symmetric GQL's are\n$Z_2$ SPT states.\n\nAfter a numerical calculation, we find that there are only two different\nmodular extensions of $\\Rp(Z_2)$ (see Table \\ref{mextZ2}).  Thus there are two\ndistinct $Z_2$-symmetric GQL phases whose bulk excitations are described by the\n$\\Rp(Z_2)$.  The first one corresponds to the trivial product states whose\nmodular extension is the $Z_2$ gauge theory which has four types of particles\nwith $(d_i,s_i)=(1,0), (1,0),(1,0),(1,\\frac12)$.  (Gauging the $Z_2$ symmetry\nof the trivial product state gives rise to a $Z_2$ gauge theory.)  The second\none corresponds to the only non-trivial $Z_2$ bosonic SPT state in 2+1D, whose\nmodular extension is the double-semion theory which has four types of particles\nwith $(d_i,s_i)=(1,0), (1,0),(1,\\frac14),(1,-\\frac14)$. (Gauging the $Z_2$\nsymmetry of the $Z_2$-SPT state gives rise to a double-semion theory\n\\cite{LG1209}.)  So the $Z_2$-SPT phases are classified by $\\Z_2$, reproducing\nthe group cohomology result\\cite{CLW1141,CGL1314,CGL1204}.  In general, the\nmodular extensions of $\\Rp(G)$ correspond to the bosonic SPT states in 2+1D\nwith symmetry $G$.\n\n\\subsection{$Z_2$-SET orders for bosonic systems}\n\n\\begin{table}[t] \n\\caption{\nThe fusion rule of the $N_c^{|\\Th|}=3_2^{\\zeta_2^1}$ $Z_2$-SET order.\nThe particle $\\textbf{1}$\ncarries the $Z_2$-charge $0$, and the particle $s$ carries the $Z_2$-charge\n$1$.  From the table, we see that $\\sigma\\otimes\\sigma=\\textbf{1} \\oplus s\n\\oplus \\sigma$.\n} \n\\label{SET32} \n\\centering\n\\begin{tabular}{ |c|ccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{3}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 2$\\\\\n\\hline\n $3^{\\zeta_{2}^{1}}_{ 2}$ & $\\textbf{1}$  & $s$  & $\\sigma$ \\\\\n\\hline\n$\\textbf{1}$  & $ \\textbf{1}$  & $ s$  & $ \\sigma$  \\\\\n$s$  & $ s$  & $ \\textbf{1}$  & $ \\sigma$  \\\\\n$\\sigma$  & $ \\sigma$  & $ \\sigma$  & $ \\textbf{1} \\oplus s \\oplus \\sigma$  \\\\\n\\hline\n\\end{tabular}\n\\end{table} \n\n\\begin{table}[t] \n\\caption{\nThe fusion rules of the two $N_c^{|\\Th|}=4_1^{\\zeta_2^1}$ $Z_2$ symmetry\nenriched topological orders with identical $d_i$ and $s_i$. \nWe see that one has a $Z_2\\times Z_2$ fusion rule and\nthe other has a $Z_4$ fusion rule.\n} \n\\label{SETZ2-45} \n\\centering\n\\begin{tabular}{ |c|cccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{4}$ & $ \\frac{1}{4}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$\\\\\n\\hline\n $4^{\\zeta_{2}^{1}}_{ 1}$ & $\\textbf{00}$  & $\\textbf{01}$  & $\\textbf{10}$  & $\\textbf{11}$ \\\\\n\\hline\n$\\textbf{00}$  & $ \\textbf{00}$  & $ \\textbf{01}$  & $ \\textbf{10}$  & $ \\textbf{11}$  \\\\\n$\\textbf{01}$  & $ \\textbf{01}$  & $ \\textbf{00}$  & $ \\textbf{11}$  & $ \\textbf{10}$  \\\\\n$\\textbf{10}$  & $ \\textbf{10}$  & $ \\textbf{11}$  & $ \\textbf{00}$  & $ \\textbf{01}$  \\\\\n$\\textbf{11}$  & $ \\textbf{11}$  & $ \\textbf{10}$  & $ \\textbf{01}$  & $ \\textbf{00}$  \\\\\n\\hline\n\\end{tabular}\n~~~~~~\n\\begin{tabular}{ |c|cccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{4}$ & $ \\frac{1}{4}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$\\\\\n\\hline\n $4^{\\zeta_{2}^{1}}_{ 1}$ & $\\textbf{0}$  & $\\textbf{2}$  & $\\textbf{1}$  & $\\textbf{3}$ \\\\\n\\hline\n$\\textbf{0}$  & $ \\textbf{0}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{3}$  \\\\\n$\\textbf{2}$  & $ \\textbf{2}$  & $ \\textbf{0}$  & $ \\textbf{3}$  & $ \\textbf{1}$  \\\\\n$\\textbf{1}$  & $ \\textbf{1}$  & $ \\textbf{3}$  & $ \\textbf{2}$  & $ \\textbf{0}$  \\\\\n$\\textbf{3}$  & $ \\textbf{3}$  & $ \\textbf{1}$  & $ \\textbf{0}$  & $ \\textbf{2}$  \\\\\n\\hline\n\\end{tabular}\n\\end{table} \n\nThe entry $N_c^{|\\Th|}=3_2^{\\zeta_2^1}$ in Table \\ref{SETZ2-34} corresponds to\nmore non-trivial $\\mce{\\Rp(Z_2)}$. It describes the bulk excitations of\n$Z_2$-SET orders which has only one type of non-trivial topological\nexcitation(with quantum dimension $d=2$ and spin $s=1\/3$, see Table\n\\ref{SET32}).  The other two types of excitations are local excitations with\n$Z_2$-charge $0$ and $1$.  We find that $3_2^{\\zeta_2^1}$ has modular\nextensions and hence is not a fake entry.\n\nTo see how many SET orders that have such set of bulk excitations, we need to\ncompute how many modular extensions are there for $3_2^{\\zeta_2^1}$.  We find\nthat $3_2^{\\zeta_2^1}$ has two modular extensions (see Table \\ref{mextZ2a}).\nThus there are two $Z_2$-SET orders with the above mentioned bulk excitations.\nIt is not an accident that the number of $Z_2$-SET orders with the same set of\nbulk excitations is the same as the number of $Z_2$ SPT states.  This is\nbecause the different $Z_2$-SET orders with a fixed set of bulk excitations are\ngenerated by stacking with  $Z_2$ SPT states.\n\nWe would like to point out that for any $G$-SET state, if we break the\nsymmetry, the $G$-SET state will reduce to a topologically ordered state\ndescribed by a UMTC.  In fact,  the different $G$-SET states described by the\nsame $\\mce{\\cE}$ (\\ie with the same set of bulk excitations) will reduce to the\nsame topologically ordered state (\\ie the same UMTC).  In Appendix \\ref{SB}, we\ndiscussed such a symmetry breaking process and how to compute UMTC from\n$\\mce{\\cE}$.  We found that the two $Z_2$-SET orders from $3_2^{\\zeta_2^1}$\nreduce to an abelian topological order described by a $K$-matrix $\\bpm 2& -1\\\\\n-1& 2 \\epm$.  This is indicated by SB:$K=\\bpm 2& -1\\\\ -1& 2 \\epm$ in the\ncomment column of Table \\ref{SETZ2-34}.  In other place, we use SB:$N^B_c$ or\nSB:$N^F_c({a \\atop b})$ to indicate the reduced topological order after the\nsymmetry breaking (for bosonic or fermionic cases).  (The topological orders\ndescribed by $N^B_c$ or $N^F_c({a \\atop b})$ are given by the tables in\n\\Ref{W150605768} or \\Ref{LW150704673}.)\n\nAs we have mentioned, there are two $Z_2$-SET orders with the same bulk\nexcitations.  But how to realize those $Z_2$-SET orders? We find that one of\nthe $Z_2$-SET orders is the double layer FQH state with $K$-matrix $\\bpm 2 &\n-1\\\\ -1 & 2\\\\ \\epm$ (same as the reduced topological order after symmetry\nbreaking), where the $Z_2$ symmetry is the layer-exchange symmetry.  The\nquasiparticles are labeled by the $l$-vectors $l=\\bpm l_1\\\\l_2\\epm$.  The two\nnon-trivial quasiparticles are given by \n\\begin{align} \nl\n&= \\bpm 1 \\\\ 0\\epm, \\ \\ \\bpm 0 \\\\ 1\\epm, \\ \\ \n\\end{align} \nwhose spins are all equal to $\\frac13$.\n\nSince the layer-exchange $Z_2$ symmetry exchanges $l_1$ and $l_2$, we see that\nthe two excitations $ \\bpm 1 \\\\ 0\\epm, \\ \\ \\bpm 0 \\\\ 1\\epm$ always have the\nsame energy. Despite the $Z_2$ symmetry has no 2-dim irreducible\nrepresentations, the above spin-1\/3 topological excitations has an exact\ntwo-fold degeneracy due to the $Z_2$ layer-exchange symmetry.   This effect is\nan interplay between the long-range entanglement and the symmetry:\n\\emph{degeneracy in excitations may not always arise from high dimensional\nirreducible representations of the symmetry.}\n\nSuch two degenerate excitations are viewed as one type of topological\nexcitations with quantum dimension $d=2$ (for the two-fold degeneracy) and spin\n$s=\\frac13$  (see Table \\ref{SETZ2-34}).  The $Z_2$ symmetry twist in such a\ndouble-layer state carry a non-abelian statistics with quantum dimension\n$d=\\sqrt{3}$.  In fact, there are two such $Z_2$ symmetry twists whose spin\ndiffer by 1\/2.\n\nThe other $Z_2$-SET order can be viewed as the above\ndouble layer FQH state $K=\\bpm 2 & -1\\\\ -1 & 2\\\\ \\epm$ stacked with a $Z_2$ SPT\nstate.\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{ The four modular extensions of $N^{|\\Th|}_{c}=5^{\\zeta_{2}^{1}}_{ 0}$ with $Z_2\\times Z_2$ fusion.\n$5^{\\zeta_{2}^{1}}_{ 0}$  has a centralizer $\\Rp(Z_2)$. The first pair and the\nsecond pair turns out to be equivalent.\n} \n\\label{mextZ2b} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$5^{\\zeta_{2}^{1}}_{ 0}$ & $8$ & $1\\times 4, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0$ & \\\\\n\\hline\n$9^{ B}_{ 0}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0,\n\\frac{1}{2}, \\frac{1}{2}, 0, \\frac{15}{16}, \\frac{1}{16}, \\frac{7}{16},\n\\frac{9}{16}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 0}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0,\n\\frac{1}{2}, \\frac{1}{2}, 0, \\frac{3}{16}, \\frac{13}{16}, \\frac{11}{16},\n\\frac{5}{16}$ & $3^{ B}_{ 3\/2}\\boxtimes 3^{ B}_{-3\/2}$\\\\\n\\hline\n$9^{ B}_{ 0}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0,\n\\frac{1}{2}, \\frac{1}{2}, 0, \\frac{1}{16}, \\frac{15}{16}, \\frac{9}{16},\n\\frac{7}{16}$ & $3^{ B}_{ 1\/2}\\boxtimes 3^{ B}_{-1\/2}$\\\\\n$9^{ B}_{ 0}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0,\n\\frac{1}{2}, \\frac{1}{2}, 0, \\frac{13}{16}, \\frac{3}{16}, \\frac{5}{16},\n\\frac{11}{16}$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n\\hline \n\\end{tabular} \n\\end{table}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{ The four modular extensions of $N^{|\\Th|}_{c}=5^{\\zeta_{2}^{1}}_{ 1}$ with $Z_2\\times Z_2$ fusion.\n$5^{\\zeta_{2}^{1}}_{ 1}$  has a centralizer $\\Rp(Z_2)$.\n} \n\\label{mextZ2c} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$5^{\\zeta_{2}^{1}}_{ 1}$ & $8$ & $1\\times 4, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}$ & \\\\\n\\hline\n$9^{ B}_{ 1}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{16}, \\frac{1}{16}, \\frac{9}{16}, \\frac{9}{16}$ & $3^{ B}_{ 1\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 1}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{13}{16}, \\frac{13}{16}, \\frac{5}{16}, \\frac{5}{16}$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n\\hline\n$9^{ B}_{ 1}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{15}{16}, \\frac{3}{16}, \\frac{7}{16}, \\frac{11}{16}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 1}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0,\n\\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{3}{16}, \\frac{15}{16},\n\\frac{11}{16}, \\frac{7}{16}$ & $3^{ B}_{ 3\/2}\\boxtimes 3^{ B}_{-1\/2}$\\\\\n\\hline \n\\end{tabular} \n\\end{table}\n\n\n\\subsection{Two other $Z_2$-SET orders for bosonic\nsystems}\n\nThe fourth and fifth entries in Table \\ref{SETZ2-34} describe the bulk\nexcitations of two other $Z_2$-SET  orders.  Those bulk excitations have\nidentical $s_i$ and $d_i$, but they have different fusion rules $N^{ij}_k$ (see\nTable \\ref{SETZ2-45}).  \n\nBoth entries have two modular extensions, and correspond to two SET orders.\nAmong the two SET orders for the $Z_2\\times Z_2$ fusion rule,  one of them is\nobtained by stacking a $Z_2$ \\emph{neutral} $\\nu=1\/2$ Laughlin state with a\ntrivial $Z_2$ product state.  The other is obtained by stacking a $Z_2$ neutral\n$\\nu=1\/2$ Laughlin state with a non-trivial $Z_2$ SPT state.  \n\nThe entry with $Z_4$ fusion rule also correspond to two SET orders.  They are\nobtained by stacking a $Z_2$ \\emph{charged} $\\nu=1\/2$ Laughlin state with a\ntrivial or a non-trivial $Z_2$ SPT state.  Here, \\emph{charged} means that the\nparticles forming the $\\nu=1\/2$ Laughlin state carry $Z_2$-charge 1.  In this\ncase, the anyon in the $\\nu=1\/2$ Laughlin state carries a fractional\n$Z_2$-charge $1\/2$.  So the fusion of two such anyons give us a $Z_2$-charge 1\nexcitation instead of a trivial neutral excitation.  This leads to the $Z_4$\nfusion rule.\n\n\n\\subsection{The rank $N=5$ $Z_2$-SET  orders for bosonic systems}\n\\label{Z2N5}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe first and the third entries in Table \\ref{mextZ2b} have different\nfusion rules, despite they have the same $(d_i,s_i)$.\n} \n\\label{frZ2_5_1} \n\\centering\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ 0$ & $ \\frac{1}{16}$ & $ \\frac{7}{16}$ & $ \\frac{9}{16}$ & $ \\frac{15}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 0}$ & $\\textbf{1}$  & $\\textbf{2}$  & $\\textbf{3}$  & $\\textbf{4}$  & $\\textbf{5}$  & $\\textbf{6}$  & $\\textbf{7}$  & $\\textbf{8}$  & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{7}$  & $ \\textbf{8}$  & $ \\textbf{9}$  \\\\\n$\\textbf{2}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{9}$  & $ \\textbf{6}$  & $ \\textbf{7}$  \\\\\n$\\textbf{3}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{7}$  & $ \\textbf{6}$  & $ \\textbf{9}$  \\\\\n$\\textbf{4}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{9}$  & $ \\textbf{8}$  & $ \\textbf{7}$  \\\\\n$\\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  \\\\\n$\\textbf{6}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{1} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{3}$  & $ \\textbf{5}$  \\\\\n$\\textbf{7}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{4}$  \\\\\n$\\textbf{8}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{2} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{4}$  & $ \\textbf{5}$  \\\\\n$\\textbf{9}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{3}$  \\\\\n\\hline\n\\end{tabular}\n\\\\[3mm]\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ 0$ & $ \\frac{1}{16}$ & $ \\frac{7}{16}$ & $ \\frac{9}{16}$ & $ \\frac{15}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 0}$ & $\\textbf{1}$  & $\\textbf{2}$  & $\\textbf{3}$  & $\\textbf{4}$  & $\\textbf{5}$  & $\\textbf{6}$  & $\\textbf{7}$  & $\\textbf{8}$  & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{7}$  & $ \\textbf{8}$  & $ \\textbf{9}$  \\\\\n$\\textbf{2}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{9}$  & $ \\textbf{6}$  & $ \\textbf{7}$  \\\\\n$\\textbf{3}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{9}$  & $ \\textbf{8}$  & $ \\textbf{7}$  \\\\\n$\\textbf{4}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{7}$  & $ \\textbf{6}$  & $ \\textbf{9}$  \\\\\n$\\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  \\\\\n$\\textbf{6}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{1} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{4}$  & $ \\textbf{5}$  \\\\\n$\\textbf{7}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{3}$  \\\\\n$\\textbf{8}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{2} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{3}$  & $ \\textbf{5}$  \\\\\n$\\textbf{9}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{4}$  \\\\\n\\hline\n\\end{tabular} \n\\end{table}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe third and the fourth entries in Table \\ref{mextZ2c} have different\nfusion rules, despite they have the same $(d_i,s_i)$.\n} \n\\label{frZ2_5_3} \n\\centering\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ \\frac{1}{8}$ & $ \\frac{3}{16}$ & $ \\frac{7}{16}$ & $ \\frac{11}{16}$ & $ \\frac{15}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 1}$ & $\\textbf{1}$  & $\\textbf{2}$  & $\\textbf{3}$  & $\\textbf{4}$  & $\\textbf{5}$  & $\\textbf{6}$  & $\\textbf{7}$  & $\\textbf{8}$  & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{7}$  & $ \\textbf{8}$  & $ \\textbf{9}$  \\\\\n$\\textbf{2}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{9}$  & $ \\textbf{6}$  & $ \\textbf{7}$  \\\\\n$\\textbf{3}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{7}$  & $ \\textbf{6}$  & $ \\textbf{9}$  \\\\\n$\\textbf{4}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{9}$  & $ \\textbf{8}$  & $ \\textbf{7}$  \\\\\n$\\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  \\\\\n$\\textbf{6}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{1} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{3}$  & $ \\textbf{5}$  \\\\\n$\\textbf{7}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{4}$  \\\\\n$\\textbf{8}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{2} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{4}$  & $ \\textbf{5}$  \\\\\n$\\textbf{9}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{3}$  \\\\\n\\hline\n\\end{tabular}\n\\\\[3mm]\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ \\frac{1}{8}$ & $ \\frac{3}{16}$ & $ \\frac{7}{16}$ & $ \\frac{11}{16}$ & $ \\frac{15}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 1}$ & $\\textbf{1}$  & $\\textbf{2}$  & $\\textbf{3}$  & $\\textbf{4}$  & $\\textbf{5}$  & $\\textbf{6}$  & $\\textbf{7}$  & $\\textbf{8}$  & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{7}$  & $ \\textbf{8}$  & $ \\textbf{9}$  \\\\\n$\\textbf{2}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{9}$  & $ \\textbf{6}$  & $ \\textbf{7}$  \\\\\n$\\textbf{3}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{9}$  & $ \\textbf{8}$  & $ \\textbf{7}$  \\\\\n$\\textbf{4}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{7}$  & $ \\textbf{6}$  & $ \\textbf{9}$  \\\\\n$\\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  \\\\\n$\\textbf{6}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{1} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{4}$  & $ \\textbf{5}$  \\\\\n$\\textbf{7}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{3}$  \\\\\n$\\textbf{8}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{2} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{3}$  & $ \\textbf{5}$  \\\\\n$\\textbf{9}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{4}$  \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe fusion rules of the first and the second entries in Table \\ref{mextZ2c}.\n} \n\\label{frZ2_5_3a} \n\\centering\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ \\frac{1}{8}$ & $ \\frac{1}{16}$ & $ \\frac{1}{16}$ & $ \\frac{9}{16}$ & $ \\frac{9}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 1}$ & $\\textbf{1}$  & $\\textbf{2}$  & $\\textbf{3}$  & $\\textbf{4}$  & $\\textbf{5}$  & $\\textbf{6}$  & $\\textbf{7}$  & $\\textbf{8}$  & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{7}$  & $ \\textbf{8}$  & $ \\textbf{9}$  \\\\\n$\\textbf{2}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{9}$  & $ \\textbf{6}$  & $ \\textbf{7}$  \\\\\n$\\textbf{3}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{7}$  & $ \\textbf{6}$  & $ \\textbf{9}$  \\\\\n$\\textbf{4}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{9}$  & $ \\textbf{8}$  & $ \\textbf{7}$  \\\\\n$\\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  \\\\\n$\\textbf{6}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{1} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{3}$  & $ \\textbf{5}$  \\\\\n$\\textbf{7}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{4}$  \\\\\n$\\textbf{8}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{2} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{4}$  & $ \\textbf{5}$  \\\\\n$\\textbf{9}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{3}$  \\\\\n\\hline\n\\end{tabular}\n\\\\[3mm]\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ \\frac{1}{8}$ & $ \\frac{5}{16}$ & $ \\frac{5}{16}$ & $ \\frac{13}{16}$ & $ \\frac{13}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 1}$ & $\\textbf{1}$  & $\\textbf{2}$  & $\\textbf{3}$  & $\\textbf{4}$  & $\\textbf{5}$  & $\\textbf{6}$  & $\\textbf{7}$  & $\\textbf{8}$  & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{7}$  & $ \\textbf{8}$  & $ \\textbf{9}$  \\\\\n$\\textbf{2}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{9}$  & $ \\textbf{6}$  & $ \\textbf{7}$  \\\\\n$\\textbf{3}$  & $ \\textbf{3}$  & $ \\textbf{4}$  & $ \\textbf{1}$  & $ \\textbf{2}$  & $ \\textbf{5}$  & $ \\textbf{8}$  & $ \\textbf{7}$  & $ \\textbf{6}$  & $ \\textbf{9}$  \\\\\n$\\textbf{4}$  & $ \\textbf{4}$  & $ \\textbf{3}$  & $ \\textbf{2}$  & $ \\textbf{1}$  & $ \\textbf{5}$  & $ \\textbf{6}$  & $ \\textbf{9}$  & $ \\textbf{8}$  & $ \\textbf{7}$  \\\\\n$\\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  \\\\\n$\\textbf{6}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{1} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{3}$  & $ \\textbf{5}$  \\\\\n$\\textbf{7}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{4}$  \\\\\n$\\textbf{8}$  & $ \\textbf{8}$  & $ \\textbf{6}$  & $ \\textbf{6}$  & $ \\textbf{8}$  & $ \\textbf{7} \\oplus \\textbf{9}$  & $ \\textbf{2} \\oplus \\textbf{3}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{4}$  & $ \\textbf{5}$  \\\\\n$\\textbf{9}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{9}$  & $ \\textbf{7}$  & $ \\textbf{6} \\oplus \\textbf{8}$  & $ \\textbf{5}$  & $ \\textbf{2} \\oplus \\textbf{4}$  & $ \\textbf{5}$  & $ \\textbf{1} \\oplus \\textbf{3}$  \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\nThe first and the second entries in Table \\ref{SETZ2-5} describe two $N=5$\n$\\mce{\\Rp(Z_2)}$'s.  They describe two different sets of bulk excitations for\n$Z_2$-SET  orders.  Those bulk excitations have identical $s_i$ and $d_i$,\nbut they have different fusion rules $N^{ij}_k$: the 4 $d=1$ particles have a\n$Z_2\\times Z_2$ fusion rule for the first entry, and they have a $Z_4$ fusion\nrule for the second entry (as indicated by F:$Z_2\\times Z_2$ or F:$Z_4$ in the\ncomment column of Table \\ref{SETZ2-5}).\n\n\\subsubsection{The first entry in Table \\ref{SETZ2-5}}\n\nLet us compute the modular extensions of the first entry (\\ie\n$5^{\\zeta_2^1}_{0}$ with $Z_2\\times Z_2$ fusion).  Since the total quantum\ndimension of the modular extensions is $D^2=16$, the modular extensions must\nhave rank $N=13$ or less (since quantum dimension $d \\geq 1$).\n\n\nNow we would like to show $N=13$ is not possible.  If a modular extension has\n$N=13$, then it must have 12 particles (labeled by $a=1,\\cdots,12$) with\nquantum dimension $d_a=1$, and one particle (labeled by $x$) with quantum\ndimension $d_x=2$, so that $12\\times 1^2+2^2=D^2=16$. In this case,\nwe must have the fusion rule\n\\begin{align}\n a\\otimes x=x,\\ \\ \\ \\ x\\otimes x= 1\\oplus 2\\oplus 3\\oplus 4.\n\\end{align}\nwhere $x\\otimes x$ is determined by the fusion rule of the $\\mce{\\Rp(Z_2)}$.\nThe above determines the fusion matrix $N_x$ defined as $(N_x)_{ij} \\equiv\nN^{xi}_{j}$.  The largest eigenvalue of $N_x$ should be $2$, the quantum\ndimension of $x$. Indeed, we find that the largest eigenvalue of $N_x$ is $2$.\nBut we also require that $N_x$ can be diagonalized by a unitary matrix (which\nhappens to be the $S$-matrix).  $N_x$ fails such a test.  So $N$ cannot be 13.\n\n$N$ also cannot be 12.  If $N=12$, then the modular extension will have 10\nparticles (labeled by $a=1,\\cdots,10$) with quantum dimension $d_a=1$, one\nparticle (labeled by $x$) with quantum dimension $d_x=2$, and one particle\n(labeled by $y$) with quantum dimension $d_y=\\sqrt 2$.  The fusion of 10\n$d_a=1$ particles is described by an abelian group $Z_{10}$ or $Z_2\\times Z_5$.\nNone of them contain $Z_2\\times Z_2$ as subgroup. Thus $N=12$ is incompatible\nwith the $Z_2\\times Z_2$ fusion of the first four $d_a=1$ particles.\n\nWe searched the modular extensions with $N$ up to 11.  We find four $N=9$\nmodular extensions (see Table \\ref{mextZ2b}), and thus the first entry\ncorresponds to valid $Z_2$-SET states.  \n\n\nIn fact one of the $Z_2$-SET states is the $Z_2$ gauge theory with a $Z_2$\nglobal symmetry, where the $Z_2$ symmetry action exchange the $Z_2$-charge $e$\nand the $Z_2$-vortex $m$.  The degenerate $e$ and $m$ give rise to the\n$(d,s)=(2,0)$ particle (the fifth particle in the table).  The bound state of\n$e$ and $m$ is a fermion $f$.  It may carry the $Z_2$-charge 0 or 1, which\ncorrespond to the third and the fourth particle with $(d,s)=(1,1\/2)$ in the\ntable. \n\nHowever, from the discussion in the last few sections, we know that a\n$\\mce{\\Rp(Z_2)}$ always has 2 modular extensions, corresponding to the 2\nbosonic $Z_2$-SPT states in 2+1D.  This seems contradictory with the above result\nthat the $Z_2$-SET state, $5^{\\zeta_2^1}_{0}$ with $Z_2\\times Z_2$ fusion, has\nfour different modular extensions. \n\nIn fact, there is no  contradiction.  Here, we only use $(N^{ij}_k,s_i)$ to\nlabel different entries. However, a $\\mce{\\cE}$ is fully characterized by\n$(N^{ij}_k,s_i)$ plus the $F$-tensors and the $R$-tensors. \nTo see this point, we note that the Ising-like UMTC $N^B_c=3^B_{m\/2}$,\n$m=1,3,\\cdots,15$ (with central charge $c=m\/2$) has three particles: $1$, $f$\nwith $(d_f,s_f)=(1,1\/2)$, and $ \\sigma $ with $(d_\\sigma ,s_\\sigma\n)=(\\sqrt{2},m\/16)$. Its $R$-tensor is given by\\cite{K062}\n\\begin{align}\n R^{ff}_1 &=-1, &\n R^{ \\sigma f }_\\sigma  &= \n R^{ f \\sigma }_\\sigma  = -\\ii^m,  \n\\\\\n R^{ \\sigma \\sigma }_1 &= (-1)^{\\frac{m^2-1}{8}} \\ee^{-\\ii \\frac{ \\pi }{8} m},  &\n R^{ \\sigma \\sigma }_f &= (-1)^{\\frac{m^2-1}{8}} \\ee^{\\ii \\frac{3 \\pi }{8} m},\n\\nonumber \n\\end{align}\nand some components of the $F$-tensor are given by\n\\begin{align}\n F^{f \\sigma \\sigma; \\sigma   }_{ f;1 }=\n F^{ \\sigma \\sigma f; \\sigma   }_{ f;1 }=1.\n\\end{align}\nThe values of $R^{ \\sigma f }_\\sigma$ and $R^{ f \\sigma }_\\sigma$ are not gauge\ninvariant. But if we fix the values of the $F$-tensor to be the ones given\nabove, this will fix the gauge, and we can treat $R^{ \\sigma f }_\\sigma$ and\n$R^{ f \\sigma }_\\sigma$ as if they are gauge invariant quantities.\n\nIf we stack $N^B_c=3^B_{m\/2}$ and  $N^B_c=3^B_{m'\/2}$ together, the induced\nUMTC $ 3^B_{m\/2}\\boxtimes 3^B_{m'\/2}$ contains particles\n$\\textbf{1}=(1,1)$, $\\textbf{2}=(f,f')$, $\\textbf{3}=(f,1)$,\n$\\textbf{4}=(1,f')$, $\\textbf{5}=( \\sigma , \\sigma' )$.  Those 5 particles are\nclosed under the fusion, and correspond to the 5 particles in $\\mce{\\Rp(Z_2)}$\n$5^{\\zeta_2^1}_{m+m'}$.  We note that some components of the $R$-tensor of $\n3^B_{m\/2}\\boxtimes 3^B_{m'\/2}$ are given by\n\\begin{align}\n R^{(f,1),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (f,1)}_{( \\sigma , \\sigma')} =-\\ii^m,\n\\nonumber\\\\\n R^{(1,f'),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (1,f')}_{( \\sigma , \\sigma')} =-\\ii^{m'}.\n\\end{align}\n\nTaking $(m,m')=(-1,1)$ and $(1,-1)$, it is clear the $ 3^B_{-1\/2}\\boxtimes\n3^B_{ 1\/2}$ and $ 3^B_{ 1\/2}\\boxtimes 3^B_{-1\/2}$ give rise to two different\n$R$-tensors that have identical $(N^{ij}_k,s_i)$.  So the first entry in\nTable \\ref{SETZ2-5} (\\ie $5^{\\zeta_2^1}_{0}$ with $Z_2\\times Z_2$ fusion) split\ninto two different entries if we include the $R$-tensors. Each give rise to two\nmodular extensions, and this is why we got four modular extensions.  In\nTable \\ref{mextZ2b}, the first two modular extensions have the same\n$(N^{ij}_k,s_i)$, $F$-tensor and $R$-tensors when restricted to the first 5\nparticles.  \nThe  second pair of modular extensions also have the same $(N^{ij}_k,s_i)$,\n$F$-tensor and $R$-tensor when restricted to the first 5 particles, but their\n$R$-tensor is different from that of the first pair.\nHowever, note that under the exchange of the two fermions, the $R$-tensor of\nthe first pair becomes that of the second pair.\n\n\nWe like to stress that Table \\ref{mextZ2b} is obtained using the\nME-equivalence relation, \\ie the different entries are different under the\nME-equivalence relation (see Section \\ref{clGQL2}).  We see that for each fixed\n$\\mce{\\Rp(Z_2)}$ (\\ie for each fixed set of $(N^{ij}_k,s_i)$, $F$-tensor and\n$R$-tensor), there are two modular extensions, which agrees with our general result\nfor modular extensions. However, if we ignore  $F$-tensor and $R$-tensor, then\nfor each fixed set of $(N^{ij}_k,s_i)$, we get four modular extensions.  This\nis because $(N^{ij}_k,s_i)$ is only a partial description of a\n$\\mce{\\Rp(Z_2)}$, and\nas discussed above, in this case there are two ways to assign $F$-tensor and $R$-tensor to\nthem.\nThis is why each fixed $(N^{ij}_k,s_i)$ has four modular extensions, while\neach fixed $(N^{ij}_k,s_i,F,R)$ has only two  modular extensions.\n\nOn the other hand, under the TO-equivalence relation (see\nSection \\ref{clGQL2}),\nthe two ways to assign $F$-tensor and $R$-tensor are actually equivalent\n(related by exchanging the two fermions), and the first entry in\nTable \\ref{SETZ2-5} corresponds to only one $\\mce{\\Rp(Z_2)}$. Thus,\nthe first entry is equivalent to the third entry, and\nthe second entry is equivalent to the fourth entry in Table \\ref{mextZ2b}.\nSo the four entries of Table \\ref{mextZ2b} in fact represent only two distinct\n$Z_2$-SET orders.\n\nOne of the two $Z_2$-SET orders have been studied extensively.  It corresponds\nto $Z_2$ gauge theory with a $\\Z_2$ global symmetry that exchanges the\n$Z_2$-gauge-charge $e$ and the $Z_2$-gauge-vortex $m$\\cite{W0303,KLW0834}.  \n\n\\subsubsection{The second entry in Table \\ref{SETZ2-5}}\n\nNext, we compute the modular extensions of the second entry in Table\n\\ref{SETZ2-5} (\\ie $5^{\\zeta_2^1}_{0}$ with\n$Z_4$ fusion).  Again, we can use the same argument to show that modular\nextensions of rank 12 and above do not exist.  We searched the modular\nextensions with $N$ up to 11,  and find that there is no modular extensions.\nSo the second entry is not realizable and does not correspond to any valid\nbosonic $Z_2$-SET in 2+1D.  This is indicated by NR in the comment column of\nTable \\ref{SETZ2-5}.\n\nNaively, the (none existing) state from the second entry is very similar to\nthat from the first entry.  It is also a $Z_2$ gauge theory with a $Z_2$ global\nsymmetry that exchange $e$ and $m$. However, for the second entry, the $f$\nparticles (the third and the fourth particles) are assigned fraction\n$Z_2$-charge of $\\pm 1\/2$. This leads to the $Z_4$ fusion rule.  Our result\nimplies that such an assignment is not realizable (or is illegal).\nIt turns out that all the $5^{\\zeta_2^1}_{c}$'s with $Z_4$ fusion do not have\nmodular extensions. They are not realizable, and do not correspond to any 2+1D\nbosonic $Z_2$-SET orders.\n\n\\subsubsection{The third entry in Table \\ref{SETZ2-5}}\n\nThird, let us compute the modular extensions of the third entry in Table\n\\ref{SETZ2-5} (\\ie $5^{\\zeta_2^1}_{1}$ with $Z_2\\times Z_2$ fusion).  We find\nthat the entry has four modular extensions.  In fact, the entry corresponds to\ntwo different $\\mce{\\Rp(Z_2)}$s, each with two modular extensions, as implied\nby the two $Z_2$-SPT states.  The two $\\mce{\\Rp(Z_2)}$s have identical\n$(N^{ij}_k,s_i,c)$, but different $F$-tensors and $R$-tensors.  Sometimes two\ndifferent $\\mce{\\cE}$'s (with different $F$-tensors and the $R$-tensors) can\nhave the same $(N^{ij}_k,s_i)$'s. The third, seventh,\\dots, entries of Table\n\\ref{SETZ2-5} provide such examples.  We like to stress that this is different\nfrom the first entry in Table \\ref{SETZ2-5} which corresponds to one\n$\\mce{\\Rp(Z_2)}$.\n\nTo see those different $F$-tensors and $R$-tensors, we note that one of the two\n$5^{\\zeta_2^1}_{1}$ with $Z_2\\times Z_2$ fusion has modular extensions given by\n$3^B_{1\/2}\\boxtimes 3^B_{1\/2}$ and $3^B_{-3\/2}\\boxtimes 3^B_{5\/2}$.  We find the\n$R$-tensor for this first $5^{\\zeta_2^1}_{1}$ with $Z_2\\times Z_2$ fusion is\ngiven by\n\\begin{align}\n R^{(f,1),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (f,1)}_{( \\sigma , \\sigma')} =-\\ii,\n\\nonumber\\\\\n R^{(1,f'),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (1,f')}_{( \\sigma , \\sigma')} =-\\ii.\n\\end{align}\nThe second $5^{\\zeta_2^1}_{1}$ with $Z_2\\times Z_2$ fusion has modular\nextensions given by $3^B_{-1\/2}\\boxtimes 3^B_{3\/2}$ and $3^B_{3\/2}\\boxtimes\n3^B_{-1\/2}$.  We find the $R$-tensor for the second $5^{\\zeta_2^1}_{1}$ with\n$Z_2\\times Z_2$ fusion is given by\n\\begin{align}\n R^{(f,1),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (f,1)}_{( \\sigma , \\sigma')} =\\ii,\n\\nonumber\\\\\n R^{(1,f'),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (1,f')}_{( \\sigma , \\sigma')} =\\ii.\n\\end{align}\nWe see that the two $5^{\\zeta_2^1}_{1}$'s with $Z_2\\times Z_2$ fusion are\nreally different $\\mce{\\Rp(Z_2)}$.  Each $5^{\\zeta_2^1}_{1}$ has two modular\nextensions, and that is why we have four entries in Table \\ref{mextZ2c}.\n\nAgain, Table \\ref{mextZ2c} is obtained using the ME-equivalence relation,\nand is not a table of GQLs.  Under the  TO-equivalence relation, the third\nentry is equivalent to the fourth entry of Table \\ref{mextZ2c}.  So the four\nentries in  Table \\ref{mextZ2c} actually describe \\emph{three} different\n$Z_2$-SET orders.  This has a very interesting consequence: \\emph{The $Z_2$-SET\nstate described by the third (or fourth) entry in \\ref{mextZ2c}, after stacked\nwith an $Z_2$-SPT state, still remains in the same phase.} This is an example\nof the following general statement made previously: \\emph{The GQLs with bulk\nexcitations described by $\\cC$ are in one-to-one correspondence with the\nquotient $\\mathcal{M}_{ext}(\\cC)\/\\mathrm{Aut}(\\cC)$ plus a central charge $c$.} \nIn such an example $\\mathrm{Aut}(\\cC)$ is non-trivial.\n\nIt is worth noting here that for the second $5^{\\zeta_2^1}_{1}$, two modular\nextensions $3^B_{-1\/2}\\boxtimes 3^B_{3\/2}$ and $3^B_{3\/2}\\boxtimes 3^B_{-1\/2}$\nare actually equivalent UMTCs. This is an example that different embedings\nleads to different modular extensions. For $3^B_{-1\/2}\\boxtimes 3^B_{3\/2}$ the\nfirst fermion in $5^{\\zeta_2^1}_{1}$ is embedded into $3^B_{-1\/2}$ and the\nsecond fermion is embedded into $3^B_{3\/2}$, while for $3^B_{3\/2}\\boxtimes\n3^B_{-1\/2}$ the first fermion is embedded into $3^B_{3\/2}$ and the second\nfermion is embedded into $3^B_{-1\/2}$.  The equivalence between\n$3^B_{-1\/2}\\boxtimes 3^B_{3\/2}$ and $3^B_{3\/2}\\boxtimes 3^B_{-1\/2}$ that\nexchanges both fermions and symmetry twists fails to relate the two\nembeddings, as they differ by a non-trivial automorphism of $5^{\\zeta_2^1}_{1}$ that exchanges only the two fermions. This is an\nexample that the $\\mathrm{Aut}(\\cC)$ action permutes the modular extensions, as discussed in Section \\ref{clGQL}.\n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe three modular extensions of $\\Rp(Z_3)$.\n} \n\\label{mextZ3} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n \\hline \n$3^{\\zeta_{4}^{1}}_{ 0}$ & $3$ & $1, 1, 1$ & $0, 0, 0$ & $\\Rp(Z_3)$ \\\\\n\\hline\n$9^{ B}_{ 0}$ & $9$ & $1\\times 9$ & $0, 0, 0, 0, 0, \\frac{1}{3}, \\frac{1}{3}, \\frac{2}{3}, \\frac{2}{3}$ & $Z_3$ gauge\\\\\n$9^{ B}_{ 0}$ & $9$ & $1\\times 9$ & $0, 0, 0, \\frac{1}{9}, \\frac{1}{9}, \\frac{4}{9}, \\frac{4}{9}, \\frac{7}{9}, \\frac{7}{9}$ & \\\\\n$9^{ B}_{ 0}$ & $9$ & $1\\times 9$ & $0, 0, 0, \\frac{2}{9}, \\frac{2}{9}, \\frac{5}{9}, \\frac{5}{9}, \\frac{8}{9}, \\frac{8}{9}$ & \\\\\n \\hline \n\\end{tabular} \n\\end{table}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe six modular extensions of $\\Rp(S_3)$.\n} \n\\label{mextS3} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$3^{\\sqrt{6}}_{ 0}$ & $6$ & $1, 1, 2$ & $0, 0, 0$ & $\\Rp(S_3)$ \\\\\n\\hline\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, 0, \\frac{1}{3}, \\frac{2}{3}, 0, \\frac{1}{2}$ & $S_3$ gauge\\\\\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, 0, \\frac{1}{3}, \\frac{2}{3}, \\frac{1}{4}, \\frac{3}{4}$ & \\\\\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, \\frac{1}{9}, \\frac{4}{9}, \\frac{7}{9}, 0, \\frac{1}{2}$ & $(B_4,2)$ \\\\\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, \\frac{1}{9}, \\frac{4}{9}, \\frac{7}{9}, \\frac{1}{4}, \\frac{3}{4}$ & \\\\\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, \\frac{2}{9}, \\frac{5}{9}, \\frac{8}{9}, 0, \\frac{1}{2}$ & $(B_4,-2)$ \\\\\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, \\frac{2}{9}, \\frac{5}{9}, \\frac{8}{9}, \\frac{1}{4}, \\frac{3}{4}$ & \\\\\n \\hline \n\\end{tabular} \n\\end{table}\n\n\\subsection{$Z_3$, $Z_5$, and $S_3$ SPT orders for bosonic systems}\n\nWe also find that $\\Rp(Z_3)$ has 3 modular extensions (see Table \\ref{mextZ3}),\n$\\Rp(Z_5)$ has 5 modular extensions (see Table \\ref{mextZ5}), and $\\Rp(S_3)$\nhas 6 modular extensions (see Table \\ref{mextS3}).  They correspond to the 3\n$Z_3$-SPT states, the 5 $Z_5$-SPT states and the 6 $S_3$-SPT states\nrespectively.  These results agree with those from group cohomology\ntheory\\cite{CGL1314}.\n\nWe note that for $\\Rp(Z_2)$, $\\Rp(Z_3)$, and $\\Rp(S_3)$, their modular\nextensions all correspond to distinct UMTCs.  However, for $\\Rp(Z_5)$, its 5\nmodular extensions only correspond to 3 distinct UMTCs.  $\\Rp(Z_5)$ has 5\nmodular extensions because $\\Rp(Z_5)$ can be embedded into the same UMTC in\ndifferent ways. The different embeddings correspond to different modular\nextensions.\n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe 16 modular extensions of $\\sRp(Z_2^f)$. \n} \n\\label{mextZ2f} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n \\hline \n$2^{ 0}_{0}$ & $2$ & $1, 1$ & $0, \\frac{1}{2}$ & $\\sRp(Z_2^f)$ \\\\\n\\hline\n$4^{ B}_{ 0}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, 0, 0$ & $Z_2$ gauge\\\\\n\\hline\n$4^{ B}_{ 1}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}$ & F:$Z_4$ \\\\\n$4^{ B}_{ 2}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}$ & F:$Z_2\\times Z_2$ \\\\\n$4^{ B}_{ 3}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}$ & F:$Z_4$ \\\\\n$4^{ B}_{ 4}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}$ & F:$Z_2\\times Z_2$ \\\\\n$4^{ B}_{-3}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}$ & F:$Z_4$ \\\\\n$4^{ B}_{-2}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}$ & F:$Z_2\\times Z_2$ \\\\\n$4^{ B}_{-1}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{7}{8}, \\frac{7}{8}$ & F:$Z_4$ \\\\\n$3^{ B}_{ 1\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{1}{16}$ & $p+\\ii p$ SC\\\\\n$3^{ B}_{ 3\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{3}{16}$ & \\\\\n$3^{ B}_{ 5\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{5}{16}$ & \\\\\n$3^{ B}_{ 7\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{7}{16}$ & \\\\\n$3^{ B}_{-7\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{9}{16}$ & \\\\\n$3^{ B}_{-5\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{11}{16}$ & \\\\\n$3^{ B}_{-3\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{13}{16}$ & \\\\\n$3^{ B}_{-1\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{15}{16}$ & \\\\\n \\hline \n\\end{tabular} \n\\end{table}\n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nThe five modular extensions of $\\Rp(Z_5)$.\n} \n\\label{mextZ5} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n \\hline \n$5^{\\sqrt{5}}_{ 0}$ & $5$ & $1\\times 5$ & $0, 0, 0, 0, 0$ & \\\\\n\\hline\n$25^{ B}_{ 0}$ & $25$ & $1\\times 25$ & $0, 0, 0, 0, 0, 0, 0, 0, 0, \\frac{1}{5}, \\frac{1}{5}, \\frac{1}{5}, \\frac{1}{5}, \\frac{2}{5}, \\frac{2}{5}, \\frac{2}{5}, \\frac{2}{5}, \\frac{3}{5}, \\frac{3}{5}, \\frac{3}{5}, \\frac{3}{5}, \\frac{4}{5}, \\frac{4}{5}, \\frac{4}{5}, \\frac{4}{5}$ & $5^{ B}_{ 0}\\boxtimes 5^{ B}_{ 0}$\\\\\n$25^{ B}_{ 0}$ & $25$ & $1\\times 25$ & $0, 0, 0, 0, 0, \\frac{1}{25}, \\frac{1}{25}, \\frac{4}{25}, \\frac{4}{25}, \\frac{6}{25}, \\frac{6}{25}, \\frac{9}{25}, \\frac{9}{25}, \\frac{11}{25}, \\frac{11}{25}, \\frac{14}{25}, \\frac{14}{25}, \\frac{16}{25}, \\frac{16}{25}, \\frac{19}{25}, \\frac{19}{25}, \\frac{21}{25}, \\frac{21}{25}, \\frac{24}{25}, \\frac{24}{25}$ & \\\\\n$25^{ B}_{ 0}$ & $25$ & $1\\times 25$ & $0, 0, 0, 0, 0, \\frac{1}{25}, \\frac{1}{25}, \\frac{4}{25}, \\frac{4}{25}, \\frac{6}{25}, \\frac{6}{25}, \\frac{9}{25}, \\frac{9}{25}, \\frac{11}{25}, \\frac{11}{25}, \\frac{14}{25}, \\frac{14}{25}, \\frac{16}{25}, \\frac{16}{25}, \\frac{19}{25}, \\frac{19}{25}, \\frac{21}{25}, \\frac{21}{25}, \\frac{24}{25}, \\frac{24}{25}$ & \\\\\n$25^{ B}_{ 0}$ & $25$ & $1\\times 25$ & $0, 0, 0, 0, 0, \\frac{2}{25}, \\frac{2}{25}, \\frac{3}{25}, \\frac{3}{25}, \\frac{7}{25}, \\frac{7}{25}, \\frac{8}{25}, \\frac{8}{25}, \\frac{12}{25}, \\frac{12}{25}, \\frac{13}{25}, \\frac{13}{25}, \\frac{17}{25}, \\frac{17}{25}, \\frac{18}{25}, \\frac{18}{25}, \\frac{22}{25}, \\frac{22}{25}, \\frac{23}{25}, \\frac{23}{25}$ & \\\\\n$25^{ B}_{ 0}$ & $25$ & $1\\times 25$ & $0, 0, 0, 0, 0, \\frac{2}{25}, \\frac{2}{25}, \\frac{3}{25}, \\frac{3}{25}, \\frac{7}{25}, \\frac{7}{25}, \\frac{8}{25}, \\frac{8}{25}, \\frac{12}{25}, \\frac{12}{25}, \\frac{13}{25}, \\frac{13}{25}, \\frac{17}{25}, \\frac{17}{25}, \\frac{18}{25}, \\frac{18}{25}, \\frac{22}{25}, \\frac{22}{25}, \\frac{23}{25}, \\frac{23}{25}$ & \\\\\n\\hline\n\\end{tabular} \n\\end{table*}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nAll the 8 modular extensions of $\\sRp(Z_4^f)$.\n} \n\\label{mextZ4f} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$4^{ 0}_{0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}$ & \n$\\sRp(Z_4^f)$ \\\\\n\\hline\n$16^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}$ & \\\\\n\\hline \n$16^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{32}, \\frac{1}{32}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{9}{32}, \\frac{9}{32}, \\frac{17}{32}, \\frac{17}{32}, \\frac{25}{32}, \\frac{25}{32}$ & \\\\\n$16^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{16}, \\frac{5}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $8^{ B}_{ 1}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{32}, \\frac{3}{32}, \\frac{11}{32}, \\frac{11}{32}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{19}{32}, \\frac{19}{32}, \\frac{27}{32}, \\frac{27}{32}$ & \\\\\n$16^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{-3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{32}, \\frac{5}{32}, \\frac{13}{32}, \\frac{13}{32}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{21}{32}, \\frac{21}{32}, \\frac{29}{32}, \\frac{29}{32}$ & \\\\\n$16^{ B}_{-2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{3}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{15}{16}, \\frac{15}{16}$ & $8^{ B}_{-1}\\boxtimes 2^{ B}_{-1}$\\\\\n$16^{ B}_{-1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{32}, \\frac{7}{32}, \\frac{15}{32}, \\frac{15}{32}, \\frac{23}{32}, \\frac{23}{32}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{31}{32}, \\frac{31}{32}$ & \\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nThe two $c=0$ modular extensions of $\\sRp(Z_8^f)$ imply that\nthe $Z_8^f$ fermionic SPT phases are described by $\\Z_2$.\nAll other  modular extensions only appear for integer $c$ and are all abelian (two modular extensions\nfor each integer $c$).\n} \n\\label{mextZ8f} \n\\centering\n\\begin{tabular}{ |c|c|l|p{5.3in}|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$8^{ 0}_{0}$ & $8$ & $1\\times 8$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}$ & \\\\\n\\hline\n$64^{ B}_{ 0}$ & $64$ & $1\\times 64$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}$, $\\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$& \\\\\n$64^{ B}_{ 0}$ & $64$ & $1\\times 64$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}$, $\\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$ & \\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\n\n\n\\subsection{Invertible fermionic topological orders}\n\nWe find that $\\sRp(Z_2^f)$ has 16 modular extensions (see Table \\ref{mextZ2f})\nwhich correspond to invertible fermionic topological orders in 2+1D.  One might\nthought that the invertible fermionic topological orders are classified by\n$\\Z_{16}$.  But in fact, the invertible fermionic topological orders are\nclassified by $\\Z$, obtained by stacking the $c=1\/2$ $p+\\ii p$ states.  The\ndiscrepancy is due to the fact that the modular extensions cannot see the $c=8$\n$E_8$ states. The 16 modular extensions exactly correspond to the invertible\nfermionic topological orders modulo the $E_8$ states.  \n\nWe also find that the modular extensions with $c=$ even have a\n$Z_2\\times Z_2$ fusion rule, while the modular extensions with $c=$ odd have a\n$Z_4$ fusion rule (indicated by F:$Z_2\\times Z_2$ or F:$Z_4$ in the comment\ncolumn of Table).\n\nThe $Z_2^f$-SPT states for fermions is given by the modular extensions with\nzero central charge.  We see that there is only one modular extension with\ncentral charge $c=0$.  Thus there is no non-trivial 2+1D fermionic SPT states\nwith $Z_2^f$ symmetry.  In general, the  modular extensions of $\\sRp(G^f)$ with\nzero central charge correspond to the fermionic SPT states in 2+1D with\nsymmetry $G^f$.\n\n\n\nTo calculate the $Z_2\\times Z_2^f$ SPT orders for fermionic systems, we first\ncompute the modular extensions for $\\sRp(Z_2\\times Z_2^f)$.  We note that\n$\\sRp(Z_2\\times Z_2^f)=\\sRp(Z_2^f\\times \\tilde Z_2^f)$.  Thus, the  modular\nextensions for $\\sRp(Z_2\\times Z_2^f)$ is the  modular extensions of\n$\\sRp(Z_2^f\\times \\tilde Z_2^f)$.  Some of the modular extensions\nof $\\sRp(Z_2^f\\times \\tilde Z_2^f)$ are given by the modular extensions of\n$\\sRp(Z_2^f)$ stacked (under $\\boxtimes$) with the modular extensions of\n$\\sRp(\\tilde Z_2^f)$.  Some of the modular extensions of $\\sRp(Z_2\\times\nZ_2^f)$ are given by the modular extensions for $\\Rp(Z_2)$ stacked (under\n$\\boxtimes$) with the modular extensions of $\\sRp(Z_2^f)$.\n\nThe above mathematical statements correspond to the following physical picture:\nSome fermionic GQLs with $Z_2\\times Z_2^f$ symmetry can be viewed as bosonic\nGQLs with $Z_2$ symmetry stacked with fermionic GQLs with $Z_2^f$ symmetry.\nAlso some fermionic GQLs with $Z_2^f\\times \\tilde Z_2^f$ symmetry can be viewed\nas fermionic GQLs with $Z_2^f$ symmetry stacked with fermionic GQLs with\n$\\tilde Z_2^f$ symmetry.\n\nUsing \\eqn{Hgcnd}, we find that the modular extensions for $Z_2\\times Z_2^f$\nsymmetry must have ranks $7, 9, 10, 12, 16$.  By direct search for those ranks,\nwe find that the modular extensions of $\\sRp(Z_2\\times Z_2^f)$ are given by\nTables \\ref{mextZ2Z2f}, \\ref{mextZ2Z2f12}, \\ref{mextZ2Z2f12b} and\n\\ref{mextZ2Z2f16}.  The $N=9$ modular extensions of $\\sRp(Z_2\\times Z_2^f)$ in\nTable \\ref{mextZ2Z2f} are given by the stacking of the $N=3$ modular extensions\nof $\\sRp(Z_2^f)$ and the $N=3$ modular extensions of $\\sRp(\\tilde Z_2^f)$.  The\n$N=16$ modular extensions of $\\sRp(Z_2\\times Z_2^f)$ in Table \\ref{mextZ2Z2f16}\nare given by the stacking of the $N=4$ modular extensions of $\\sRp(Z_2^f)$ and\nthe $N=4$ modular extensions of $\\sRp(\\tilde Z_2^f)$.  There are also 64 $N=12$\nmodular extensions of $\\sRp(Z_2\\times Z_2^f)$ given by the stacking of the\n$N=4$ ($N=3$) modular extensions of $\\sRp(Z_2^f)$ and the $N=3$ ($N=4$) modular\nextensions of $\\sRp(\\tilde Z_2^f)$.\n\n\nMany of the modular extensions have non-trivial topological orders since the\ncentral charge $c$ is non-zero. There are eight modular extensions for each central\ncharge $c=0,1\/2,1,3\/2,\\dots,15\/2$, and in total $8\\times 16=128$ modular\nextensions.  Those eight with $c=0$  correspond\nto the $Z_2\\times Z_2^f$ fermionic SPT states.  Those are all the $Z_2\\times\nZ_2^f$ fermionic SPT states\\cite{GL1369}.\n\n\n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nAll the 32 modular extensions of $\\sRp(Z_2\\times Z_2^f)$ with $N = 9$.\n} \n\\label{mextZ2Z2f} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$4^{ 0}_{0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}$ & $\\sRp(Z_2\\times Z_2^f)$ \\\\\n\\hline\n$9^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{7}{16}, \\frac{9}{16}, \\frac{15}{16}, 0$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{7}{16}, \\frac{9}{16}, \\frac{15}{16}, 0$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{5}{16}, \\frac{11}{16}, \\frac{13}{16}, 0$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{5}{16}, \\frac{11}{16}, \\frac{13}{16}, 0$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n\\hline \n$9^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{1}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{1}{8}$ & $3^{ B}_{ 1\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{7}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{1}{8}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{7}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{1}{8}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{5}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{1}{8}$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{3}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{1}{4}$ & $3^{ B}_{ 3\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{3}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{1}{4}$ & $3^{ B}_{ 3\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{7}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{1}{4}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{7}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{1}{4}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{5}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{3}{8}$ & $3^{ B}_{ 5\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{5}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{3}{8}$ & $3^{ B}_{ 5\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{3}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{3}{8}$ & $3^{ B}_{ 3\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{7}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{3}{8}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$9^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{7}{16}, \\frac{9}{16}, \\frac{15}{16}, \\frac{1}{2}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{7}{16}, \\frac{9}{16}, \\frac{15}{16}, \\frac{1}{2}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{5}{16}, \\frac{11}{16}, \\frac{13}{16}, \\frac{1}{2}$ & $3^{ B}_{ 5\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{5}{16}, \\frac{11}{16}, \\frac{13}{16}, \\frac{1}{2}$ & $3^{ B}_{ 5\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{-3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{1}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{5}{8}$ & $3^{ B}_{-7\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{-3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{7}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{5}{8}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{-3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{7}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{5}{8}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{-3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{5}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{5}{8}$ & $3^{ B}_{ 5\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{-2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{3}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{3}{4}$ & $3^{ B}_{-5\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{-2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{3}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{3}{4}$ & $3^{ B}_{-5\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{-2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{7}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{3}{4}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{-2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{7}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{3}{4}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{-1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{5}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{7}{8}$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{-1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{5}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{7}{8}$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{-1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{3}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{7}{8}$ & $3^{ B}_{-5\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{-1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{7}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{7}{8}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\n\n\n\\subsection{$Z_{2n}^f$ SPT orders for fermionic systems}\n\nWe also find the modular extensions for $\\sRp(Z_4^f)$, $\\sRp(Z_6^f)$, and\n$\\sRp(Z_8^f)$ (see Tables \\ref{mextZ4f}, \\ref{mextZ6f}, and \\ref{mextZ8f}).\nAgain, many of them has non-trivial topological orders since the central charge\n$c$ is non-zero.  \n\nFor $Z_4^f$ group, only one of them have $c=0$.  So there is no non-trivial\n$Z_4^f$ fermionic SPT states.  For $Z_6^f$ group, only three of them have\n$c=0$.  So, the $Z_6^f$ fermionic SPT states are described by $\\Z_3$.  For\n$Z_8^f$ group, only two of them have $c=0$.  So, the $Z_8^f$ fermionic SPT\nstates are described by $\\Z_2$.  Those results are consistent with the results\nin \\Ref{KTT1429,CJWang}.  However, the calculation present here is more\ncomplete.\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nThe first 32 modular extensions of $\\sRp(Z_2\\times Z_2^f)$ with $N =12$.\n} \n\\label{mextZ2Z2f12} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$4^{ 0}_{0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}$ & $\\sRp(Z_2\\times Z_2^f)$ \\\\\n\\hline\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{9}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{9}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{1}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{1}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{1}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $6^{ B}_{-1\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{1}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $6^{ B}_{-1\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{1}{16}, \\frac{3}{16}, \\frac{11}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{1}{16}, \\frac{3}{16}, \\frac{11}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{11}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{11}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{9}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{9}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{3}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $6^{ B}_{ 1\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{3}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $6^{ B}_{ 1\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{3}{16}, \\frac{3}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{3}{16}, \\frac{3}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{11}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{11}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{9}{16}$ & $6^{ B}_{ 3\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{9}{16}$ & $6^{ B}_{ 3\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{5}{16}, \\frac{5}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{5}{16}, \\frac{5}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{13}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{13}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{11}{16}$ & $6^{ B}_{ 5\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{11}{16}$ & $6^{ B}_{ 5\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{9}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{9}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\section{Summary}\n\\label{sum}\n\nGQLs contain both topologically ordered states and SPT states.  In this paper,\nwe present a theory that classify GQLs in 2+1D for bosonic\/fermionic systems\nwith symmetry.\n\nWe propose that the possible non-abelian statistics (or sets of bulk\nquasiparticles excitations) in 2+1D GQLs are classified by \n$\\mce{\\cE}$, where $\\cE=\\Rp(G)$ or $\\sRp(G^f)$ describing the symmetry in\nbosonic or fermionic systems.  However, $\\mce{\\cE}$'s fail to\nclassify GQLs, since different GQL phases can have identical non-abelian\nstatistics, which correspond to identical $\\mce{\\cE}$.  \n\nTo fix this problem, we introduce the notion of modular extensions for a\n$\\mce{\\cE}$.  We propose to use the triple $(\\cC,\\cM,c)$ to\nclassify 2+1D GQLs with symmetry $G$ (for boson) or $G^f$ (for fermion).  Here\n$\\cC$ is a $\\mce{\\cE}$ with $\\cE=\\Rp(G)$ or $\\sRp(G^f)$,  $\\cM$ is a modular\nextension of $\\cC$ and $c$ is the chiral central charge of the edge state.  We\nshow that the modular extensions of a $\\mce{\\cE}$ has a one-to-one\ncorrespondence  with the modular extensions of  $\\cE$.  So the number of the\nmodular extensions is solely determined by the symmetry $\\cE$.  Also, the $c=0$\nmodular extensions of a $\\cE$ ($\\cE=\\Rp(G)$ or $\\sRp(G^f)$) classify the 2+1D\nSPT states for bosons or fermions with symmetry $G$ or $G^f$.\n\nAlthough the above result has a nice mathematical structure, it is hard to\nimplement numerically to produce a table of GQLs.  To fix this problem, we\npropose a different description of 2+1D GQLs.  We propose to use the data $(\n\\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$,  up to some\npermutations of the indices, to describe 2+1D GQLs with symmetry $G$\n(for boson) or $G^f$ (for fermion), with a restriction that the symmetry group\n$G$ can be fully characterized by the fusion ring of its irreducible\nrepresentations (for example, for simple groups or abelian groups).  Here the\ndata $(\\tilde N^{ab}_c,\\tilde s_a)$ describe the symmetry and the data\n$(N^{ij}_k,s_i)$ describes fusion and the spins of the bulk particles in the\nGQL.  The modular extensions are obtained by ``gauging'' the symmetry $G$ or\n$G^f$.  The data $(\\cN^{IJ}_K,\\cS_I)$  describes fusion and the spins of the\nbulk particles in the ``gauged'' theory.  Last, $c$ is the chiral central\ncharge of the edge state.\n\nIn this paper (see Appendix \\ref{cnds}) and in \\Ref{W150605768}, we list the\nnecessary and the sufficient conditions on the data $(\\tilde N^{ab}_c,\\tilde\ns_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$, which allow us to obtain a\nlist of GQLs.  However, in this paper, we did not give the list\nof GQLs directly.  We first give a list of $(\\tilde N^{ab}_c,\\tilde s_a;\nN^{ij}_k,s_i)$, which is an imperfect list of $\\mce{\\cE}$'s.  We then compute\nthe modular extensions $(\\cN^{IJ}_K,\\cS_I;c)$ for each entry $(\\tilde\nN^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$, which allows us to obtain a perfect list\nof GQLs (for certain symmetry groups).  As a special case, we calculated the\nbosonic\/fermionic SPT states for some groups in 2+1D.\n\nIn \\Ref{LW160205936}, we will give a more mathematical description of our\ntheory.  Certainly we hope to generalize the above framework to higher\ndimensions.  We also hope to develop more efficient numerical codes to obtain\nbigger tables of GQLs.\n\n\\bigskip\n\n\\noindent {\\bf Acknowledgement}: \nWe like to thank Pavel Etingof, Dmitri Nikshych, Chenjie Wang, and Zhenghan\nWang for many helpful discussions.  This research is supported by NSF Grant No.\nDMR-1506475, and NSFC 11274192. It is also supported by the John Templeton\nFoundation No.  39901.  Research at Perimeter Institute is supported by the\nGovernment of Canada through Industry Canada and by the Province of Ontario\nthrough the Ministry of Research.  LK is supported by the Center of\nMathematical Sciences and Applications at Harvard University.\n\n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nThe second 32 modular extensions of $\\sRp(Z_2\\times Z_2^f)$ with $N =12$.\n} \n\\label{mextZ2Z2f12b} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$4^{ 0}_{0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}$ & $\\sRp(Z_2\\times Z_2^f)$ \\\\\n\\hline\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{15}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{15}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{5}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{13}{16}$ & $6^{ B}_{ 7\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{5}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{13}{16}$ & $6^{ B}_{ 7\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{3}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{11}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{3}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{11}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{11}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{11}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{15}{16}$ & $6^{ B}_{-7\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{15}{16}$ & $6^{ B}_{-7\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{5}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{13}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{5}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{13}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{16}, \\frac{11}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{16}, \\frac{11}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $6^{ B}_{-5\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $6^{ B}_{-5\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{15}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{15}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $6^{ B}_{-3\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $6^{ B}_{-3\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{9}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{9}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nAll the 32 modular extensions of $\\sRp(Z_2\\times Z_2^f)$ with $N =16$.\n} \n\\label{mextZ2Z2f16} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$4^{ 0}_{0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}$ & $\\sRp(Z_2\\times Z_2^f)$ \\\\\n\\hline\n$16^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}$ & $4^{ B}_{ 0}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{-1}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{-1}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{-1}\\boxtimes 2^{ B}_{ 1}$\\\\\n\\hline\n$16^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}$ & $4^{ B}_{ 1}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}$ & $4^{ B}_{ 1}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{ 0}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{ 0}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{ 1}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{ 1}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}$ & $4^{ B}_{ 1}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{-1}\\boxtimes 4^{ B}_{ 3}$\\\\\n$16^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{ 2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{ 2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}$ & $4^{ B}_{ 4}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{ 3}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}$ & $4^{ B}_{-3}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{-3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}$ & $4^{ B}_{-3}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{-3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{ 4}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{ 4}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{-3}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{-3}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}$ & $4^{ B}_{-3}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{-2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 3}$\\\\\n$16^{ B}_{-1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{-1}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{-1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{-1}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{-1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{-2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{-2}\\boxtimes 2^{ B}_{ 1}$\\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nAll the modular extensions of $\\sRp(Z_6^f)=\\sRp(Z_3\\times Z_2^f)$.\n} \n\\label{mextZ6f} \n\\centering\n\\begin{tabular}{ |c|c|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ \\\\\n\\hline \n$6^{ 0}_{0}$ & $6$ & $1, 1, 1, 1, 1, 1$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}$  \\hfill $\\sRp(Z_6^f)$\\\\\n\\hline\n$36^{ B}_{ 0}$ & $36$ & $1\\times 36$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{2}{3}, \\frac{2}{3}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}$  \\\\\n$36^{ B}_{ 0}$ & $36$ & $1\\times 36$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{2}{9}, \\frac{2}{9}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{5}{9}, \\frac{5}{9}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{8}{9}, \\frac{8}{9}, \\frac{8}{9}, \\frac{8}{9}$  \\\\\n$36^{ B}_{ 0}$ & $36$ & $1\\times 36$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{4}{9}, \\frac{4}{9}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{7}{9}, \\frac{7}{9}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}$  \\\\\n\\hline\n$36^{ B}_{ 1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{11}{24}, \\frac{11}{24}, \\frac{11}{24}, \\frac{11}{24}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{19}{24}, \\frac{19}{24}, \\frac{19}{24}, \\frac{19}{24}, \\frac{5}{6}, \\frac{5}{6}$  \\\\\n$36^{ B}_{ 1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{72}, \\frac{1}{72}, \\frac{1}{72}, \\frac{1}{72}, \\frac{1}{18}, \\frac{1}{18}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{2}{9}, \\frac{2}{9}, \\frac{25}{72}, \\frac{25}{72}, \\frac{25}{72}, \\frac{25}{72}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{49}{72}, \\frac{49}{72}, \\frac{49}{72}, \\frac{49}{72}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}$  \\\\\n$36^{ B}_{ 1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{17}{72}, \\frac{17}{72}, \\frac{17}{72}, \\frac{17}{72}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{41}{72}, \\frac{41}{72}, \\frac{41}{72}, \\frac{41}{72}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{65}{72}, \\frac{65}{72}, \\frac{65}{72}, \\frac{65}{72}, \\frac{17}{18}, \\frac{17}{18}$  \\\\\n$36^{ B}_{ 2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{12}, \\frac{7}{12}, \\frac{7}{12}, \\frac{7}{12}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{11}{12}, \\frac{11}{12}, \\frac{11}{12}, \\frac{11}{12}$  \\\\\n$36^{ B}_{ 2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{36}, \\frac{1}{36}, \\frac{1}{36}, \\frac{1}{36}, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{18}, \\frac{5}{18}, \\frac{13}{36}, \\frac{13}{36}, \\frac{13}{36}, \\frac{13}{36}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{25}{36}, \\frac{25}{36}, \\frac{25}{36}, \\frac{25}{36}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}$  \\\\\n$36^{ B}_{ 2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{5}{36}, \\frac{5}{36}, \\frac{5}{36}, \\frac{5}{36}, \\frac{2}{9}, \\frac{2}{9}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{7}{18}, \\frac{7}{18}, \\frac{17}{36}, \\frac{17}{36}, \\frac{17}{36}, \\frac{17}{36}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{29}{36}, \\frac{29}{36}, \\frac{29}{36}, \\frac{29}{36}, \\frac{8}{9}, \\frac{8}{9}$  \\\\\n$36^{ B}_{ 3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{24}, \\frac{1}{24}, \\frac{1}{24}, \\frac{1}{24}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{17}{24}, \\frac{17}{24}, \\frac{17}{24}, \\frac{17}{24}, \\frac{5}{6}, \\frac{5}{6}$  \\\\\n$36^{ B}_{ 3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{19}{72}, \\frac{19}{72}, \\frac{19}{72}, \\frac{19}{72}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{43}{72}, \\frac{43}{72}, \\frac{43}{72}, \\frac{43}{72}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{67}{72}, \\frac{67}{72}, \\frac{67}{72}, \\frac{67}{72}$  \\\\\n$36^{ B}_{ 3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{11}{72}, \\frac{11}{72}, \\frac{11}{72}, \\frac{11}{72}, \\frac{5}{18}, \\frac{5}{18}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{4}{9}, \\frac{4}{9}, \\frac{35}{72}, \\frac{35}{72}, \\frac{35}{72}, \\frac{35}{72}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{59}{72}, \\frac{59}{72}, \\frac{59}{72}, \\frac{59}{72}, \\frac{17}{18}, \\frac{17}{18}$  \\\\\n$36^{ B}_{ 4}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{6}$  \\\\\n$36^{ B}_{ 4}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{1}{18}, \\frac{1}{18}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{7}{18}, \\frac{7}{18}, \\frac{7}{18}, \\frac{7}{18}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{13}{18}, \\frac{13}{18}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}$  \\\\\n$36^{ B}_{ 4}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{5}{18}, \\frac{5}{18}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{11}{18}, \\frac{11}{18}, \\frac{11}{18}, \\frac{11}{18}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{17}{18}, \\frac{17}{18}, \\frac{17}{18}, \\frac{17}{18}$  \\\\\n$36^{ B}_{-3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{7}{24}, \\frac{7}{24}, \\frac{7}{24}, \\frac{7}{24}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{23}{24}, \\frac{23}{24}, \\frac{23}{24}, \\frac{23}{24}$  \\\\\n$36^{ B}_{-3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{13}{72}, \\frac{13}{72}, \\frac{13}{72}, \\frac{13}{72}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{37}{72}, \\frac{37}{72}, \\frac{37}{72}, \\frac{37}{72}, \\frac{5}{9}, \\frac{5}{9}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{13}{18}, \\frac{13}{18}, \\frac{61}{72}, \\frac{61}{72}, \\frac{61}{72}, \\frac{61}{72}, \\frac{8}{9}, \\frac{8}{9}$  \\\\\n$36^{ B}_{-3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{5}{72}, \\frac{5}{72}, \\frac{5}{72}, \\frac{5}{72}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{29}{72}, \\frac{29}{72}, \\frac{29}{72}, \\frac{29}{72}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{53}{72}, \\frac{53}{72}, \\frac{53}{72}, \\frac{53}{72}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}$  \\\\\n$36^{ B}_{-2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{12}, \\frac{1}{12}, \\frac{1}{12}, \\frac{1}{12}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{5}{12}, \\frac{5}{12}, \\frac{5}{12}, \\frac{5}{12}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{5}{6}, \\frac{5}{6}$  \\\\\n$36^{ B}_{-2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{11}{36}, \\frac{11}{36}, \\frac{11}{36}, \\frac{11}{36}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{23}{36}, \\frac{23}{36}, \\frac{23}{36}, \\frac{23}{36}, \\frac{13}{18}, \\frac{13}{18}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{8}{9}, \\frac{8}{9}, \\frac{35}{36}, \\frac{35}{36}, \\frac{35}{36}, \\frac{35}{36}$  \\\\\n$36^{ B}_{-2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{7}{36}, \\frac{7}{36}, \\frac{7}{36}, \\frac{7}{36}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{19}{36}, \\frac{19}{36}, \\frac{19}{36}, \\frac{19}{36}, \\frac{11}{18}, \\frac{11}{18}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{9}, \\frac{7}{9}, \\frac{31}{36}, \\frac{31}{36}, \\frac{31}{36}, \\frac{31}{36}, \\frac{17}{18}, \\frac{17}{18}$  \\\\\n$36^{ B}_{-1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{5}{24}, \\frac{5}{24}, \\frac{5}{24}, \\frac{5}{24}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{13}{24}, \\frac{13}{24}, \\frac{13}{24}, \\frac{13}{24}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$  \\\\\n$36^{ B}_{-1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{7}{72}, \\frac{7}{72}, \\frac{7}{72}, \\frac{7}{72}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{31}{72}, \\frac{31}{72}, \\frac{31}{72}, \\frac{31}{72}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{55}{72}, \\frac{55}{72}, \\frac{55}{72}, \\frac{55}{72}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{8}{9}, \\frac{8}{9}$  \\\\\n$36^{ B}_{-1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{23}{72}, \\frac{23}{72}, \\frac{23}{72}, \\frac{23}{72}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{47}{72}, \\frac{47}{72}, \\frac{47}{72}, \\frac{47}{72}, \\frac{7}{9}, \\frac{7}{9}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{17}{18}, \\frac{17}{18}, \\frac{71}{72}, \\frac{71}{72}, \\frac{71}{72}, \\frac{71}{72}$  \\\\\n\\hline\n$27^{ B}_{ 1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{19}{48}, \\frac{19}{48}, \\frac{35}{48}, \\frac{35}{48}$  \\\\\n$27^{ B}_{ 1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{41}{144}, \\frac{41}{144}, \\frac{89}{144}, \\frac{89}{144}, \\frac{137}{144}, \\frac{137}{144}$  \\\\\n$27^{ B}_{ 1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{25}{144}, \\frac{25}{144}, \\frac{73}{144}, \\frac{73}{144}, \\frac{121}{144}, \\frac{121}{144}$  \\\\\n$27^{ B}_{ 3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{25}{48}, \\frac{25}{48}, \\frac{41}{48}, \\frac{41}{48}$  \\\\\n$27^{ B}_{ 3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{11}{144}, \\frac{11}{144}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{59}{144}, \\frac{59}{144}, \\frac{107}{144}, \\frac{107}{144}$  \\\\\n$27^{ B}_{ 3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{43}{144}, \\frac{43}{144}, \\frac{91}{144}, \\frac{91}{144}, \\frac{139}{144}, \\frac{139}{144}$  \\\\\n$27^{ B}_{ 5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{31}{48}, \\frac{31}{48}, \\frac{47}{48}, \\frac{47}{48}$  \\\\\n$27^{ B}_{ 5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{29}{144}, \\frac{29}{144}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{77}{144}, \\frac{77}{144}, \\frac{125}{144}, \\frac{125}{144}$  \\\\\n$27^{ B}_{ 5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{13}{144}, \\frac{13}{144}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{61}{144}, \\frac{61}{144}, \\frac{109}{144}, \\frac{109}{144}$  \\\\\n$27^{ B}_{ 7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{48}, \\frac{5}{48}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{37}{48}, \\frac{37}{48}$  \\\\\n$27^{ B}_{ 7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{47}{144}, \\frac{47}{144}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{95}{144}, \\frac{95}{144}, \\frac{143}{144}, \\frac{143}{144}$  \\\\\n$27^{ B}_{ 7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{31}{144}, \\frac{31}{144}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{79}{144}, \\frac{79}{144}, \\frac{127}{144}, \\frac{127}{144}$  \\\\\n$27^{ B}_{-7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{11}{48}, \\frac{11}{48}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{43}{48}, \\frac{43}{48}$  \\\\\n$27^{ B}_{-7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{17}{144}, \\frac{17}{144}, \\frac{65}{144}, \\frac{65}{144}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{113}{144}, \\frac{113}{144}$  \\\\\n$27^{ B}_{-7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{1}{144}, \\frac{1}{144}, \\frac{49}{144}, \\frac{49}{144}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{97}{144}, \\frac{97}{144}$  \\\\\n$27^{ B}_{-5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{1}{48}, \\frac{1}{48}, \\frac{17}{48}, \\frac{17}{48}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}$  \\\\\n$27^{ B}_{-5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{35}{144}, \\frac{35}{144}, \\frac{83}{144}, \\frac{83}{144}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{131}{144}, \\frac{131}{144}$  \\\\\n$27^{ B}_{-5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{19}{144}, \\frac{19}{144}, \\frac{67}{144}, \\frac{67}{144}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{115}{144}, \\frac{115}{144}$  \\\\\n$27^{ B}_{-3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{7}{48}, \\frac{7}{48}, \\frac{23}{48}, \\frac{23}{48}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}$  \\\\\n$27^{ B}_{-3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{5}{144}, \\frac{5}{144}, \\frac{53}{144}, \\frac{53}{144}, \\frac{101}{144}, \\frac{101}{144}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}$  \\\\\n$27^{ B}_{-3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{37}{144}, \\frac{37}{144}, \\frac{85}{144}, \\frac{85}{144}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{133}{144}, \\frac{133}{144}$  \\\\\n$27^{ B}_{-1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{13}{48}, \\frac{13}{48}, \\frac{29}{48}, \\frac{29}{48}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$  \\\\\n$27^{ B}_{-1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{23}{144}, \\frac{23}{144}, \\frac{71}{144}, \\frac{71}{144}, \\frac{119}{144}, \\frac{119}{144}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$  \\\\\n$27^{ B}_{-1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{7}{144}, \\frac{7}{144}, \\frac{55}{144}, \\frac{55}{144}, \\frac{103}{144}, \\frac{103}{144}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$  \\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\\vfill\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\section{Introduction}\n\\input{introduction.tex}\n\n\\section{Preliminaries} \\label{sec:preliminary}\n\\input{preliminary.tex}\n\n\n\n\\section{Problem Formulation} \\label{sec:problem}\n\\input{problem.tex}\n\n\n\n\n\\section{Formula Synthesis Method}\\label{sec:formula_synthesis}\n\\input{synthesis.tex}\n\n\n\n\\section{Case Study}\\label{sec:case_studies}\n\\input{case.tex}\n\n\n\\section{Conclusion}\\label{sec:conclusion}\n\\input{conclusion.tex}\n\n\n\n\n\\subsection{Signals}\nWe define an n-dimensional discrete  signal  $\\mathbf{x}$ as a mapping from time domain  $\\mathbb{N}^+$ to the real numbers $\\mathbb{R}^n$.  A finite signal of length $K+1$ is shown as a sequence $\\mathbf{x} = x_0, x_1, ...,x_K$. We use $x_t^i$ to denote the projection of the state on the $i$th dimension at time $t$.\n\n\nA dataset of labeled signals is defined as \n\n\\begin{align}\\label{eq:dataset}\n \\mathcal{D} = \\{  (\\mathbf{x},\\mathbf{l}) \\mid  \\mathbf{x} = & x_0, x_1, \\ldots,x_K,  \\\\\n  \\mathbf{l} = & l_0, l_1, \\ldots, l_K,  \\text{ and }  \\nonumber \\\\% K \\in \\mathbb{N} \\nonumber \\\\ \n  x_t \\in &\\mathbb{R}^n, l_t \\in \\{0,1\\}, t=0,\\ldots,K\\}, \\nonumber\n\\end{align}\nwhere $l_t = 1$, means that at time $t$, an event of interest is occurred on signal $\\mathbf{x}$.\n\n\n\\subsection{Past Time Signal Temporal Logic}\nA Past Time Signal Temporal Logic (ptSTL) formula is defined with grammar:\n\\begin{align}\n\\phi = \\mathbf{T} | x^i\\sim c | \\neg\\phi | \\phi_1 \\wedge \\phi_2 | \\phi_1 \\vee \\phi_2 | \\phi_1 \\mathbf{S}_{[a,b]} \\phi_2 | \\mathbf{P}_{[a,b]} \\phi | \\mathbf{A}_{[a,b]}  \\phi \\label{eq:ptstl} \n\\end{align}\nwhere $x^i$ is a signal variable, $\\sim \\in \\{>,<\\}$,  and $c$ is a constant, $\\mathbf{T}$ is the Boolean constant $true$, $\\neg, \\wedge$ and $\\vee$ are the standard Boolean operators, $\\mathbf{S}_{[a,b]}$ $(since)$, $\\mathbf{P}_{[a,b]}$ $(previously)$, and $\\mathbf{A}_{[a,b]}$ $(always)$ are the temporal operators with time interval $[a,b]$. The semantics of a ptSTL formula is defined over a signal for a given time point. \n\nInformally, for signal $\\mathbf{x}$, at time $t$, formula $\\mathbf{P}_{[a,b]} \\phi $ is satisfied if $\\phi$ holds at some time in $[t-b,t-a]$, formula $\\mathbf{A}_{[a,b]} \\phi$ is satisfied if $\\phi$ holds everywhere in $[t-b,t-a]$, and $\\phi_1 \\mathbf{S}_{[a,b]} \\phi_2$ is satisfied if $\\phi_2$ holds at some time $t' \\in [t-b,t-a]$ and $\\phi_1$ holds since then. $(\\mathbf{x},t) \\models \\phi $ denotes that signal $\\mathbf{x}$ satisfies formula $\\phi$ at time $t$. Formally, the semantics are given as follows:\n\\begin{align}\n& (\\mathbf{x}, t) \\models  \\mathbf{T}  & & \\nonumber \\\\\n& (\\mathbf{x}, t) \\models x^i \\sim c  &\\text{ iff } &  x^i_t \\sim c , \\sim \\in \\{>,<\\}\\nonumber \\\\\n& (\\mathbf{x}, t) \\models \\phi_1 \\wedge \\phi_2  & \\text{ iff }  & (\\mathbf{x}, t) \\models \\phi_1  \\text{ and } (\\mathbf{x}, t) \\models \\phi_2 \\nonumber \\\\\n& (\\mathbf{x}, t) \\models \\phi_1 \\vee \\phi_2  & \\text{ iff }  & (\\mathbf{x}, t) \\models \\phi_1  \\text{ or } (\\mathbf{x}, t) \\models \\phi_2 \\nonumber \\\\\n& (\\mathbf{x}, t) \\models \\mathbf{P}_{[a,b]} \\phi & \\text{ iff }  & \\exists t' \\in I(t,[a,b]),  (\\mathbf{x}, t') \\models \\phi \\label{eq:semantics} \\\\\n& (\\mathbf{x}, t) \\models \\mathbf{A}_{[a,b]} \\phi & \\text{ iff }  & \\forall t' \\in I(t,[a,b]),  (\\mathbf{x}, t') \\models \\phi  \\nonumber \\\\\n& (\\mathbf{x}, t) \\models \\phi_1 \\mathbf{S}_{[a,b]} \\phi_2 & \\text{ iff }  & \\exists  t' \\in I(t,[a,b]),  (\\mathbf{x}, t') \\models \\phi_2, \\nonumber \\\\\n& & & \\forall t'' \\in [t', t]  (\\mathbf{x}, t'') \\models \\phi_1 \\nonumber, \n\\end{align}\n\\[ \\text{ where } I(t,[a,b]) = [t-b, t-a] \\cap [0,t]  \\]\n\n\n\nNote that the previously ($\\mathbf{P}$) and always ($\\mathbf{A}$) operators are the special cases of the since operator, $\\mathbf{P}_{[a,b]} \\phi := \\mathbf{T}\\ \\mathbf{S}_{[a,b]} \\phi$ and  $\\mathbf{A}_{[a,b]} \\phi := \\neg \\mathbf{P}_{[a,b]} \\neg \\phi$. We include them as they are used in the proposed methods.\n\n\n$Parametric$ $Past$ $Time$ $Signal$ $Temporal$ $Logic$ is an extension of ptSTL~\\citep{Asarin:2011}. In a  parametric ptSTL formula, instead of numerical values in time interval bounds and predicates, parameters can be used. A parametric formula can be converted to a ptSTL formula by assigning a value to each parameter.\nAs an example consider the parametric formula $\\phi = \\mathbf{P}_{[p_1,p_2]} x < p_3$ with parameters $p_1, p_2$ and $p_3$.  ptSTL formula $\\phi (v)  = \\mathbf{P}_{[3,5]} x < 10.2$ is obtained with valuation $v = [3,5,10.2]$.\n \n\\subsection{Monotonicity of Parametric Signal Temporal Logic}\n\n\n\nMonotonicity properties for parametric STL is introduced by~\\cite{miningjournal}. A parametric STL formula $\\phi$ with parameters $[p_1, \\ldots, p_m]$ is  $monotonically$ $increasing$ with parameter $p_i$ if (\\ref{eq:mon_inc}) holds along any signal $\\mathbf{x}$. Similarly,  it is $monotonically$ $decreasing$ with parameter $p_i$ if (\\ref{eq:mon_dec}) holds.\n\\begin{align}\n&\\text{for all }  v,v' \\text{ with } v(p_i) < v'(p_i), v(p_j) = v(p_j) \\text{ for each } i\\neq j, \\nonumber \\\\\n& \\quad  \\quad  \\quad (\\mathbf{x},t) \\models \\phi(v) \\implies  (\\mathbf{x},t) \\models \\phi(v')   \\label{eq:mon_inc} \\\\\n&\\text{for all }  v,v' \\text{ with }   v(p_i) > v'(p_i),  v(p_j) = v(p_j) \\text{ for each } i\\neq j,  \\nonumber \\\\\n& \\quad  \\quad  \\quad  (\\mathbf{x},t) \\models \\phi(v) \\implies  (\\mathbf{x},t) \\models \\phi(v')   \\label{eq:mon_dec}\n\\end{align}\n\nEssentially, $\\phi$ is monotonically increasing with $p_i$ if the valuation can not change from satisfying to violating when only the value of the parameter $p_i$ is increased. \n\n \nOur aim in this work is to generate a ptSTL formula that represents the labels in a dataset~\\eqref{eq:dataset}. For this purpose, we generate label $\\mathbf{l}^\\phi= l^\\phi_0, l^\\phi_1, \\ldots, l^\\phi_N$ from a given signal $\\mathbf{x} = x_0, \\ldots, x_K$ using a given ptSTL formula $\\phi$ as follows:\n\\begin{align}\\label{eq:label_set}\n\tl^\\phi_t = &\\begin{cases}  1 \\text{ if } (\\mathbf{x}, t) \\models \\phi \\\\\n\t 0 \\text{ otherwise}\n\t\\end{cases}\n\\end{align}\n\nWe define number of positive labels $P^{\\#}(\\phi, \\mathbf{x} )$ (\\ref{eq:positive}) and number of negative labels $N^{\\#}(\\phi, \\mathbf{x} )$ (\\ref{eq:negative}) where $\\mathbf{l}^\\phi$ is generated by evaluating  formula $\\phi$ along signal $\\mathbf{x}$ as defined in~\\eqref{eq:label_set}:\n\\begin{align}\\label{eq:positive}\n P^{\\#}(\\phi, \\mathbf{x}) = \\sum_{i=0}^{K} l_i^\\phi\n\n \\end{align}\n\\begin{align}\\label{eq:negative}\n N^{\\#}(\\phi, \\mathbf{x}) = \\sum_{i=0}^{K} \\neg l_i^\\phi\n \\end{align}\n \n \n\n \n\n\nAlso note that\n\\begin{equation}\\label{eq:totalformula}\n    P^{\\#}(\\phi, \\mathbf{x}) +  N^{\\#}(\\phi, \\mathbf{x}) = K + 1\n\\end{equation}\n\n\n\nWe derive monotonicity properties of $P^{\\#}(\\phi, \\cdot)$ for a parametric ptSTL formula $\\phi$ with respect to the monotonicity of $\\phi$.  $P^{\\#}(\\phi,\\cdot)$ is monotonically increasing with $p_i$ if and only if the satisfaction value of $\\phi$ is monotonically increasing with $p_i$, i.e.,if~\\eqref{eq:mon_inc} holds along any signal $\\mathbf{x}$, then~\\eqref{eq:monoton_inc} holds:\n\\begin{align}\n&\\text{for all }  v,v' \\text{ with } v(p_i) < v'(p_i), v(p_j) = v(p_j) \\text{ for each } i\\neq j, \\nonumber \\\\\n& \\quad  \\quad  \\quad P^{\\#}(\\phi(v),\\mathbf{x}) \\leq P^{\\#}(\\phi(v'),\\mathbf{x})   \\label{eq:monoton_inc}\n\\end{align}\nSimilarly, if $\\phi$ is monotonically decreasing with $p_i$, then  $P^{\\#}(\\phi, \\cdot)$ is also monotonically decreasing with $p_i$.  Specifically, for any signal $\\mathbf{x}$,~\\eqref{eq:monoton_dec} holds when  \\eqref{eq:mon_dec} holds:\n\\begin{align}\n&\\text{for all }  v,v' \\text{ with } v(p_i) > v'(p_i), v(p_j) = v(p_j) \\text{ for each } i\\neq j, \\nonumber \\\\\n& \\quad  \\quad  \\quad P^{\\#}(\\phi(v),\\mathbf{x}) \\leq P^{\\#}(\\phi(v'),\\mathbf{x})   \\label{eq:monoton_dec}\n\\end{align}\n\n\nNote that, by~\\eqref{eq:totalformula}, $P^{\\#}(\\phi, \\cdot)$ and $N^{\\#}(\\phi, \\cdot)$ have the opposite monotonicity property, e.g., if $P^{\\#}(\\phi, \\cdot)$ is monotonically increasing with $p_i$ than $N^{\\#}(\\phi, \\cdot)$ is monotonically decreasing with $p_i$.\n\nIn our work, a parameter appears only once in a parametric ptSTL formula. Therefore, the considered formulas are monotonic in each parameter, i.e., either monotonically increasing or monotonically decreasing.\n\n\n\\subsection{Monotonicity for Temporal Parameters}\nThe number of positive labels $P^{\\#}(\\phi, \\mathcal{D})$ of a ptSTL formula $\\phi$ over a dataset $\\mathcal{D}$ is simply defined as the total number of positive labels, and derived from~\\eqref{eq:positive}. $N^{\\#}(\\phi, \\mathcal{D})$ is defined similarly.\n\\[ P^{\\#}(\\phi, \\mathcal{D}) = \\sum_{(\\mathbf{x}, \\mathbf{l}) \\in \\mathcal{D}}^{K} P^{\\#}(\\phi, \\mathbf{x})  \\quad N^{\\#}(\\phi, \\mathcal{D}) = \\sum_{(\\mathbf{x}, \\mathbf{l}) \\in \\mathcal{D}}^{K} N^{\\#}(\\phi, \\mathbf{x})\\]\n\nAs either a positive ($1$) or a negative ($0$) label is assigned to each data point, the equality $|\\mathcal{D}| \\times (K+1) = P^{\\#}(\\phi, \\mathcal{D})  + N^{\\#}(\\phi, \\mathcal{D})$ trivially holds. We define the number of correctly identified positive instances (\\textit{true positives}) with respect to the labels generated by the formula $\\phi$ using~\\eqref{eq:label_set} and the dataset labels as:\n\\begin{align}\\label{eq:tp}\n\n  TP^{\\#}(\\phi, \\mathcal{D}) = \\sum_{(\\mathbf{x},\\mathbf{l}) \\in \\mathcal{D}}  \\sum_{i=0}^{K} l_i \\wedge l^\\phi _i\n\n \\end{align}\n \nSimilarly, the total number of incorrect positive results, i.e., the data points that have label $0$ in the given dataset and label $1$ according to the ptSTL formula $\\phi$ ($l_i^\\phi=1$) is defined as:\n\\begin{align}\\label{eq:fp}\n\n FP^{\\#}(\\phi, \\mathcal{D}) = \\sum_{(\\mathbf{x},\\mathbf{l})\\in \\mathcal{D}}  \\sum_{i=1}^{K}  \\neg l _i \\wedge l^\\phi_i \n \\end{align}\n \nThe derivations of $TP^{\\#}(\\cdot, \\cdot)$ and $FP^{\\#}(\\cdot, \\cdot)$ preserves monotonicity properties~\\eqref{eq:monoton_inc} and~\\eqref{eq:monoton_dec}. Therefore, if a parametric ptSTL formula $\\phi$ is  increasing (or decreasing) with a parameter $p$, then both $TP^{\\#}(\\cdot, \\cdot)$ and $FP^{\\#}(\\cdot, \\cdot)$ are increasing (or decreasing) with $p$. \n\nWe use $\\mathcal{M}(p,\\phi)$ to denote the monotonicity property of parameter $p$ in $\\phi$ for the number of positives ($TP^{\\#}(\\cdot, \\cdot)$, $FP^{\\#}(\\cdot, \\cdot)$ or $P^{\\#}(\\cdot,\\cdot)$ ):\n\\begin{align}\\label{eq:monotonicity}\n\t\\mathcal{M}(p,\\phi) = &\\begin{cases}\n\t \\mathbf{I}  \\text{ if } p \\text{ is monotonically increasing in } \\phi  \\\\\n\t \\mathbf{D} \\text{ if } p \\text{ is monotonically decreasing in } \\phi\n\t\\end{cases}\n\\end{align}\n\nMonotonicity property, $\\mathcal{M}(\\cdot,\\cdot) $, for each parameter in a basic formula is given in Table~\\ref{tb:monotonicity_table}. \n \n\\begin{table}[h]\n\\begin{center}\n\\caption{\\textsc{Monotonicity Table}}\\label{tb:monotonicity_table}\n\\begin{tabular}{cc|cc}\n$\\phi$   & $\\mathcal{M}(p,\\phi)$ &   $\\phi$  & $\\mathcal{M}(p,\\phi)$ \\\\ \\hline\\hline\n$x > p$  &   $\\mathbf{D}$ & $x < p$  &   $\\mathbf{I}$\\\\ \\hline\n$\\mathbf{A}_{[c, p]} \\varphi$  & $\\mathbf{D}$ & $\\mathbf{A}_{[p, c]} \\varphi$  & $\\mathbf{I}$\\\\ \\hline\n$\\mathbf{P}_{[c, p]}\\varphi$  & $\\mathbf{I}$ & $\\mathbf{P}_{[ p, c]}\\varphi$  & $\\mathbf{D}$  \\\\ \\hline\n$\\varphi_1 \\mathbf{S}_{[c, p]}\\varphi_2$  & $\\mathbf{I}$ & $\\varphi_1 \\mathbf{S}_{[p,c]}\\varphi_2$   & $\\mathbf{D}$  \\\\ \\hline\n\\end{tabular}\n\\end {center}\n\\end{table}\n\nNote that the preceding derivations are based on the number of positive labels. The number of correctly identified negative labels, $TN^{\\#}(\\phi, \\mathcal{D})$, and the number of incorrectly identified negative labels $FN^{\\#}(\\phi, \\mathcal{D})$ are defined similarly. These show the opposite monotonicity property, i.e., if $TP^{\\#}(\\phi, \\mathcal{D})$ is monotonically increasing in parameter $p$, then $TN^{\\#}(\\phi, \\mathcal{D})$ is monotonically decreasing in $p$. \nFurthermore, the negation operator ($\\neg$) inverts the monotonicity property. For example, while $\\mathcal{M}(p,\\mathbf{P}_{[a,b]} x < p)$ is $\\mathbf{I}$, $\\mathcal{M}(p,\\neg \\mathbf{P}_{[a,b]} x < p)$ is $\\mathbf{D}$. A parameter's monotonicity is determined by checking the syntax tree of the formula: each negation that appears from the root node to the parameter inverts the monotonicity of the parameter shown in Table~\\ref{tb:monotonicity_table}.\n\n\n\\begin{example}\n\\label{ex:pptstl_formula}\n\nConsider parametric ptSTL formula\n\\begin{equation}\\label{eq:exformula}\n\\phi =  ( \\mathbf{P}_{[p_1, p_2]} ( qGust < p_3 ) )  \\wedge  ( wGust < p_4 )\n \\end{equation} \nMonotonicity properties of $p_1, p_2, p_3$ and  $p_4$ are $\n\\mathcal{M}(p_1,\\phi) = \\mathbf{D}$, $\n\\mathcal{M}(p_2,\\phi) = \\mathbf{I}$, $\n\\mathcal{M}(p_3,\\phi) = \\mathbf{I}$, $\n\\mathcal{M}(p_4,\\phi) = \\mathbf{I}.$\n\\end{example}\n\n\n\n\n\\subsection{Parameter Optimization Using Monotonicity}\\label{sec:parametersynthesis}\n\nWe now present an efficient method based on monotonicity to find parameters of a parametric ptSTL formula $\\phi$ from a given dataset $\\mathcal{D}$~\\eqref{eq:dataset} such that the number of correctly identified positives of the resulting formula is maximized while the number false positives is below a given threshold:  \n\n\\begin{problem}\\label{prob:optimization} \nGiven a labeled dataset $\\mathcal{D}$~\\eqref{eq:dataset}, a parametric ptSTL formula $\\phi$ with $n$ parameters $p_1, p_2, \\ldots, p_n$, lower and upper bounds $l_i, u_i$ for each parameter $p_i$, an error bound $B \\in \\mathbb{N}$, find the valuation $v$  within the given limits that maximizes $TP^{\\#}(\\phi(v), \\mathcal{D})$ while guaranteeing that $ FP^{\\#}(\\phi(v), \\mathcal{D}) \\leq B$.\n\n\\end{problem}\n\nTo solve this problem, we first present an algorithm for parametric ptSTL formulas with two parameters, and then discuss how this approach is adapted for parametric ptSTL formulas with more than two parameters.\n\nWe present a diagonal search method to solve Prob.~\\ref{prob:optimization} in an efficient way when $n$ is 2, which adapts search problem of the product of an m element chain and an n element chain~\\citep{linial1985searching} for ptSTL parameter optimization. The diagonal search algorithm starts with a valuation $v$ with $v(p_1)$ is the bound on $p_1$ that maximizes $TP^{\\#}(\\phi, \\mathcal{D})$ (i.e. either $l_1$ or $u_1$) and $v(p_2)$ is the bound on $p_2$ that minimizes $TP^{\\#}(\\phi, \\mathcal{D})$. Given step sizes $\\delta_1$ and $\\delta_2$ for both parameters, the algorithm iteratively changes the value of a parameter according to the following rule: change $v(p_1)$ by $\\delta_1$ in the direction decreasing $P^{\\#}(\\phi, \\mathcal{D})$ if the error constraint does not hold at $v$, otherwise change $v(p_2)$ by $\\delta_2$ in the direction increasing $P^{\\#}(\\phi, \\mathcal{D})$. Thus, the algorithm moves along a diagonal of the product of the discretized parameter domains with the objective of satisfying the error bound or improving the optimization criteria. This diagonal method is summarized in Alg.~\\ref{alg:diagonal}\n\n\n\n\\begin{algorithm}\n\\caption{$DiagonalSearch(\\phi,B,\\mathcal{D}, l_1, u_1, \\delta_1, l_2, u_2, \\delta_2)$}\\label{alg:diagonal}\n\\begin{flushleft}\n\\begin{algorithmic}[1]\n\\Require{$\\phi$: A parametric ptSTL formula with parameters $p_1$ and $p_2$, $B$ : bound on $FP^{\\#}(\\phi(v), \\mathcal{D})$, $\\mathcal{D}$: dataset as in~\\eqref{eq:dataset},  $l_i, u_i, \\delta_i$: lower bound, upper bound and step size for parameter $p_i$, $i \\in \\{1,2\\}$}\n\\Ensure{$v_{best} = \\arg\\max_{  v } \\{TP^{\\#}(\\phi(v),\\mathcal{D}) \\mid FP^{\\#}(\\phi(v),\\mathcal{D}) < B\\}$}\n\\If {$\\mathcal{M}(p_1,\\phi) == \\mathbf{I}$} \\label{line:startinitialize}\n\t\\State $v(p_1) = u_1, \\bar \\delta_1 = -\\delta_1$\n\\Else\n\\State  $v(p_1) = l_1, \\bar \\delta_1 = \\delta_1$\n\\EndIf\n\\If {$\\mathcal{M}(p_2,\\phi) == \\mathbf{I}$}\n\t\\State $v(p_2) = l_1, \\bar \\delta_2 = \\delta_2$\n\\Else\n\\State  $v(p_2) = u_1, \\bar \\delta_2 = -\\delta_2$\n\\EndIf  \\label{line:endinitialize}\n\\State $v_{best} = []$,  $TP_{best} = 0$\n\\While{$l_1 \\leq v(p_1) \\leq u_1 \\wedge l_2 \\leq v(p_2) \\leq u_2$}\\label{line:loopstart}\n    \\If{$B < FP^{\\#}(\\phi(v),\\mathcal{D})$} \\label{line:errorconstraint}\n        \\State $v(p_1) = v(p_1) + \\bar \\delta_1$\n    \\Else\n       \n        \\If{$TP^{\\#}(\\phi(v),\\mathcal{D}) \\geq TP_{best}$}\\label{line:bestknown}\n                \\State $TP_{best} = TP^{\\#}(\\phi(v),\\mathcal{D})$, $v_{best} = v$\n        \\EndIf\n        \\State $v(p_2) = v(p_2) + \\bar \\delta_2$\n    \\EndIf\n\\EndWhile\\label{line:loopend}\n\\State \\Return $v_{best}$\n\\end{algorithmic}\n\\end{flushleft}\n\\end{algorithm}\n\nIn lines~\\ref{line:startinitialize}-\\ref{line:endinitialize} of Alg.~\\ref{alg:diagonal}, the initial value and the update direction is defined for each parameter with respect to its monotonicity property. At each iteration of the main loop (lines~\\ref{line:loopstart}-\\ref{line:loopend}), exactly one parameter value is updated. If the error constraint (line~\\ref{line:errorconstraint}) is violated, the parameter that initialized to maximize $TP^{\\#}(\\phi,\\mathcal{D})$, $p_1$, is changed by $\\bar \\delta_1$ to reduce $FP^{\\#}(\\phi(v),\\mathcal{D})$. Otherwise, the current parameter assignment is a candidate solution, and it is checked against the best known solution (line~\\ref{line:bestknown}). Then, the parameter that initialized to minimize $FP^{\\#}(\\phi,\\mathcal{D})$, $p_2$, is changed by $\\bar \\delta_2$ to increase $TP^{\\#}(\\phi(v),\\mathcal{D})$. The iterations end when a parameter is out of the given bounds. Consequently, $O(m_1 + m_2)$ formula evaluations are performed over the given dataset, where $m_1 = \\frac{u_1 - l_1}{\\delta_1},  m_2 = \\frac{u_2 - l_2}{\\delta_2}$.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{8.4 cm}{4.2 cm}{%\n\\begin{tikzpicture}[scale=.6]\n\\begin{scope}\n\n\\fill[black!30!red,opacity=0.6]    (3,0) rectangle (5,1);\n\\fill[black!30!red,opacity=0.6]    (5,0) rectangle (7,5);\n\\fill[black!30!red,opacity=0.6]    (6,5) rectangle (7,6);\n\n\\fill[black!30!green,opacity=0.6]    (1,0) rectangle (3,6);\n\\fill[black!30!green,opacity=0.6]    (3,1) rectangle (5,6);\n\\fill[black!30!green,opacity=0.6]    (5,5) rectangle (6,6);\n\\draw (1, 0) grid (7, 6);\n\n\\draw[very thick, scale=1] (1, 0) grid (3, 0);\n\\draw[very thick, scale=1] (3, 0) grid (3, 1);\n\\draw[very thick, scale=1] (3, 1) grid (5, 1);\n\\draw[very thick, scale=1] (5, 1) grid (5, 5);\n\\draw[very thick, scale=1] (5, 5) grid (6, 5);\n\\draw[very thick, scale=1] (6, 5) grid (6, 6);\n\\draw[very thick, scale=1] (6, 6) grid (7, 6);\n\\tikzset{anchor=west}\n\n\\setcounter{row7}{1}\n\\setrowseven {}{\\begin{turn}{90}\\tiny $\\: \\: \\:\\: \\:l_1$\\end{turn}}{\\begin{turn}{90} \\tiny $\\: \\: \\:\\: \\:   l_1 + \\delta_1$\\end{turn}}{\\tiny ..}{\\tiny .}{\\begin{turn}{90}\\tiny $ \\: \\: \\:\\: \\:u_1 - \\delta_1$\\end{turn}}{\\begin{turn}{90}\\tiny $\\: \\: \\:\\: \\:u_1$\\end{turn}}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 $\\end{turn}}{1}{2}{2}{2}{3}{5}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 1\\delta_2$\\end{turn}}{1}{2}{2}{3}{4}{5}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 2\\delta_2$\\end{turn}}{1}{2}{3}{3}{4}{5}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 3\\delta_2$\\end{turn}}{2}{2}{3}{3}{4}{5}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 4\\delta_2$\\end{turn}}{2}{2}{3}{3}{4}{5}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 5\\delta_2$\\end{turn}}{3}{3}{4}{4}{4}{5}\n\n\\node[anchor=center] at (4.0, -0.5) {$FP^{\\#}(\\phi(v), \\mathcal{D})$};\n\\end{scope}\n\n\\begin{scope}[xshift=7.5cm]\n\n\\fill[black!30!red,opacity=0.6]    (3,0) rectangle (5,1);\n\\fill[black!30!red,opacity=0.6]    (5,0) rectangle (7,5);\n\\fill[black!30!red,opacity=0.6]    (6,5) rectangle (7,6);\n\n\\fill[black!30!green,opacity=0.6]    (1,0) rectangle (3,6);\n\\fill[black!30!green,opacity=0.6]    (3,1) rectangle (5,6);\n\\fill[black!30!green,opacity=0.6]    (5,5) rectangle (6,6);\n\\draw (1, 0) grid (7, 6);\n\n\\draw[very thick, scale=1] (1, 0) grid (3, 0);\n\\draw[very thick, scale=1] (3, 0) grid (3, 1);\n\\draw[very thick, scale=1] (3, 1) grid (5, 1);\n\\draw[very thick, scale=1] (5, 1) grid (5, 5);\n\\draw[very thick, scale=1] (5, 5) grid (6, 5);\n\\draw[very thick, scale=1] (6, 5) grid (6, 6);\n\\draw[very thick, scale=1] (6, 6) grid (7, 6);\n\n\\draw[line width=2pt,<-] (5.5,5.5)--(6.5,5.5) node[right]{};\n\\draw[line width=2pt,<-] (5.5,4.5)--(5.5,5.5) node[right]{};\n\\draw[line width=2pt,<-] (4.5,4.5)--(5.5,4.5) node[right]{};\n\\draw[line width=2pt,<-] (4.5,0.5)--(4.5,4.5) node[right]{};\n\\draw[line width=2pt,<-] (2.5,0.5)--(4.5,0.5) node[right]{};\n\\draw[line width=2pt,<-] (2.5,0)--(2.5,0.5) node[right]{};\n\\tikzset{anchor=west}\n\n\\setcounter{row7}{1}\n\n\\setrowseven {}{\\begin{turn}{90}\\tiny $\\: \\: \\:\\: \\:l_1$\\end{turn}}{\\begin{turn}{90} \\tiny $\\: \\: \\:\\: \\:   l_1 + \\delta_1$\\end{turn}}{\\tiny ..}{\\tiny .}{\\begin{turn}{90}\\tiny $ \\: \\: \\:\\: \\:u_1 - \\delta_1$\\end{turn}}{\\begin{turn}{90}\\tiny $\\: \\: \\:\\: \\:u_1$\\end{turn}}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2$\\end{turn}}{10}{11}{14} {15}{17} {18}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + \\delta_2$\\end{turn}}{10}{12}{15} {16}{17} {19}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 2\\delta_2$\\end{turn}}{11}{12}{15} {17}{18} {21}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 3\\delta_2$\\end{turn}}{13}{13}{17} {19}{21} {22}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 4\\delta_2$\\end{turn}}{14}{14}{18} {20}{22} {24}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 5\\delta_2$\\end{turn}}{15}{17}{22} {24}{28} {30}\n\n\n\\node[anchor=center] at (4.0, -0.5) {$TP^{\\#}(\\phi(v), \\mathcal{D})$};\n\\end{scope}\n\n\n\\end{tikzpicture}\n}\n\\end{center}\n\\caption{An example run of Alg.~\\ref{alg:diagonal}.} \\label{fig:grid}\n\\end{figure}\n\nAn example run of Alg.~\\ref{alg:diagonal} is shown in Fig.~\\ref{fig:grid} for illustration, where $B$ is $3$, both $\\mathcal{M}(p_1,\\phi)$ and $\\mathcal{M}(p_2,\\phi)$ are  $\\mathbf{I}$. In Fig. \\ref{fig:grid}, each cell contains  $FP^{\\#}(\\phi(v), \\mathcal{D})$ and $TP^{\\#}(\\phi(v), \\mathcal{D})$ on the left and right arrays, respectively. Cells with $ FP^{\\#}(\\phi(v), {\\mathcal{D}}) > B$ are marked with red (infeasible parameters) and the rest of the cells are marked with green. The algorithm takes a step to the left when it encounters a red cell and takes a step to the down when it encounters a green cell. The path the algorithm follows is shown on the grid. The algorithm returns $[u_1 - 2\\delta_1, l_2 + 4\\delta_2]$. Note that the algorithm evaluates the optimal in each row it finds the overall optimal for the given step sizes.\n\n\nWe now describe the proposed method to solve Prob.~\\ref{prob:optimization}. Let $\\phi$ be the given parametric ptSTL formula with n parameters $p_1, \\ldots, p_n$. If $n=1$, the optimal value is found with a binary search. If $n=2$, the optimal value is found with $DiagonalSearch$ method described in Alg.~\\ref{alg:diagonal}. Finally, if $n > 2$, $DiagonalSearch$ is run for $p_1$ and $p_2$ for all possible combinations of the last $n-2$ parameters, and the optimal parameters are returned. The whole process is referred as $ParameterSynthesis(\\phi, B, \\mathcal{D})$.\n\n\n\\begin{example}\n\\label{ex:optimization}\nConsider the parametric ptSTL formula~\\eqref{eq:exformula} from Ex. \\ref{ex:pptstl_formula} with parameter ranges: $l_1, l_2 = 0$, $u_1,u_2 = 30$, $l_3=-0.4$, $u_3= 0.3$, $l_4=-240$, $u_4=210$.   \n$2,0.05,0.05$ and $30$ are set as the step sizes $\\delta_1\\delta_2,\\delta_3$ and $\\delta_4$, respectively. Since $n>2$, two of the parameters, namely $p_2, p_3$, are selected for $DiagonalSearch$ based on the size of the parameter domains. \nFor the remaining parameters $p_1$ and $p_4$, $\\phi^{i,j}$ is created as follows:\n\\begin{align}\n\\phi^{i,j} =& \\phi(v) \\text{ where } v = [i, p_2, p_3 , j ],  \\\\ \\nonumber\n & i \\in \\{0,2, \\ldots 30\\}, j \\in \\{-240, -210, \\ldots 210\\\n\\end{align}\nAlg. \\ref{alg:diagonal} is run for each $\\phi^{i,j}$ with $B=5$ over the dataset $\\mathcal{D}$  defined in  Ex.~\\ref{ex:formulation}. The maximum $TP^{\\#}(\\phi^{i,j}(v),\\mathcal{D})$ is attained when $i=4$ and $j=-20$ with $v_{best} = [10, 0]$, which corresponds to ptSTL formula \n$\\phi(v) = (\\mathbf{P}_{[4, 10]}  qGust < 0 )  \\wedge  ( wGust < -120 ))$ with $TP^{\\#}(\\phi,\\mathcal{D}) = 235$ and $FP^{\\#}(\\phi(v),\\mathcal{D}) = 4$. This formula explains approximately $50\\%$ of the large deviations from the normal behavior (label 1). Thus a possible approach would be adjusting internal commands generated from $pilot$ with respect to the wind behavior characterized by $\\phi(v)$.\nNote that while grid search requires 61440 valuations~\\citep{codit2018}, $ParameterSynthesis(\\phi, B, \\mathcal{D})$ can find the optimal solution with $4721$ valuations.\n \n\n\\end{example}\n\n\\subsection{Formula Synthesis}\nIn this section, we present the solution to the main problem (Prob.~\\ref{prob:main}) considered in this paper: find a ptSTL formula that represents labeled events in  a dataset.\nIn general, an unexpected behavior\/fault can occur due to a number of different reasons. To utilize this property, we iteratively construct a formula for the given dataset $\\mathcal{D}$ as a disjunction of ptSTL formulas each representing a different reason. The goal in each iteration is to find a formula for a subset of the labeled instances, while limiting incorrectly labeled instances (FP) as this type of error propagates with disjunction operator. To find such a formula, we define the set of all parametric formulas as in~\\citep{codit2018}, and perform parameter optimization on each of them using $ParameterSynthesis$ method described in Sec.~\\ref{sec:parametersynthesis}. \n\nGiven the set of system variables, $\\{x^1, \\ldots,x^n\\}$, and a bound on the number of operators $N$, the set of all parametric ptSTL formulas with up to $N$ operators $\\mathcal{F}^{\\leq N}$ is recursively defined as:\n\\begin{align}\\label{eq:formula_space}\n\\mathcal{F}^{0} &= \\{  x^i \\sim p_i \\mid  \\sim \\in \\{<,>\\}, i = 1,\\ldots,n\\} \\cup \\{\\mathbf{T}\\} \\\\\n\\mathcal{F}^{N} &= \\{\\neg  \\phi , \\mathbf{P}_{[a,b]} \\phi, \\mathbf{A}_{[a,b]} \\phi \\mid \\phi \\in \\mathcal{F}^{N-1} \\} \\cup \\nonumber \\\\ \n&\\bigcup\\limits_{i=1}^{n-1} \\{ \\phi_{1} \\wedge \\phi_{2} ,  \\phi_{1} \\vee \\phi_{2} , \\phi_{1}\\mathbf{S}_{[a,b]}\\phi_{2}  \\mid \\nonumber \\\\\n& \\quad\\quad\\quad\\quad \\phi_{1} \\in \\mathcal{F}^{i}, \\phi_{2} \\in \\mathcal{F}^{N-i-1}\\}  \\nonumber \\\\\n\\mathcal{F}^{\\leq N} &= \\cup_{i=0}^{N} \\mathcal{F}^{i}\\nonumber\n\\end{align}\n\n\n\n\\begin{algorithm}\n\\caption{$FormulaSynthesis(\\mathcal{F}, B, \\mathcal{D}, p)$}\\label{alg:synthesize}  \n\n\\begin{algorithmic}[1]\n\\Require{$\\mathcal{F}$: a set of parametric ptSTL formulas, $B$: bound on the number of false positives, $\\mathcal{D}$: a dataset as in~\\eqref{eq:dataset}, $p$: upper bound on the number of formulas concatenated with disjunction.} \n\n \\State $\\mathcal{F}^v= \\{ \\phi(v) = ParameterSynthesis( \\phi, B, \\mathcal{D}) \\mid \\phi \\in \\mathcal{F}\\}$ \\label{line:parametersyn}\n \\State $i=0, TP_{prev} = 0, TP=1$, $\\Phi = false$\n \\While{$TP > TP_{prev}$\\text{ and } $i < p$}\\label{line:loopstart2}\n \\State $\\phi(v)^* = \\arg \\max_{\\phi(v) \\in \\mathcal{F}^v} TP^{\\#}(\\Phi \\vee \\phi(v), \\mathcal{D})$\n \\State $\\Phi = \\Phi \\vee \\phi(v)^*$\n \\State $i=i+1$\n\n  \\State $TP_{prev} = TP$, $TP = TP^{\\#}(\\Phi, \\mathcal{D})$\n  \\EndWhile\\label{line:loopend2}\n\n \\State \\Return $\\Phi$\n\\end{algorithmic}\n\\end{algorithm}\n\nThe proposed formula synthesis approach is summarized in Alg.~\\ref{alg:synthesize}. The method takes a set of parametric formulas $\\mathcal{F}$, a bound on the number of false positives $B$, a labeled dataset $\\mathcal{D}$ and a bound $p$ on the number of ptSTL formulas, and generates a ptSTL formula $\\phi^\\star$ in the form of~\\eqref{eq:endformula} with at most $p$ sub-formulas, such that $FP^{\\#}(\\phi^\\star, \\mathcal{D}) < B p$ and $TP^{\\#}(\\phi^\\star, \\mathcal{D})$ is optimized. The set of parametric formulas can be defined as in~\\eqref{eq:formula_space}, or alternatively, an expert of the considered system can write a set of parametric formulas. In the algorithm, first, parameters are optimized for each parametric ptSTL formula $\\phi \\in \\mathcal{F}$ (line~\\ref{line:parametersyn}). Then, starting from $\\Phi = false$, iteratively, the formula $\\phi(v)^\\star$ maximizing the valuation of the combined formula $\\Phi \\vee \\phi(v)^\\star$ is selected from the set of ptSTL formulas $\\mathcal{F}^v$ until the sub-formula limit $p$ is reached, or concatenating new formulas does not improve the result (lines~\\ref{line:loopstart2}-\\ref{line:loopend2}). Note that at each iteration a formula $\\phi(v)^\\star$ is added to $\\Phi$ with disjunction.\n\nIn Alg.~\\ref{alg:synthesize}, $ParameterSynthesis( \\phi, B, \\mathcal{D}) $ is run only once for each parametric ptSTL formula. At every iteration of the algorithm, $TP^{\\#}(\\Phi \\vee \\phi(v), \\mathcal{D})$ is computed for each $\\phi(v) \\in \\mathcal{F}^v$ to select the formula $\\phi(v)^\\star$ that generate the highest increment in $TP$. Note that the resulting formula $\\phi^\\star$ might not be the optimal formula due to the iterative synthesis approach. Essentially, the fitness of the formula is upper bounded by the formula that would be obtained by performing parameter optimization on parametric formulas in the form of~\\eqref{eq:endformula} with $N \\times p$ parameters (as in~\\citep{codit2018}). \nHowever, due to the complexity of the parameter synthesis algorithm, this computation is not feasible for large formulas.\n\n\n\\begin{example}\n\\label{ex:synthesis}\nThe set of all parametric ptSTL formulas $\\mathcal{F}^{\\leq 2}$ with  at most $2$ parameters over the system variables $\\{alpha, pilot, wGust, qGust\\}$ is generated according to~\\eqref{eq:formula_space}. The parameter domains are defined as:\n$p_a, p_b \\in \\{2i \\mid i=0, \\ldots,15   \\}$  for $\\mathbf{A}_{[p_a, p_b]}$,$\\mathbf{P}_{[p_a, p_b]}$,  \\\\\n$p_{alpha} \\in \\{-0.5 + 0.05i \\mid i=0, \\ldots,20   \\}$, \\\\%$\\mathcal{V} = \\{x_0, x_1, x_2\\}$ \n$p_{pilot} \\in \\{-0.5 +05i \\mid i=0, \\ldots,20   \\}$, \\\\%$\\mathcal{V} = \\{x_0, x_1, x_2\\}$ \n$p_{wGust} \\in \\{-240 +30i \\mid i=0, \\ldots,15   \\}$, \\\\\n$p_{qGust} \\in \\{-0.4 - 0.05i \\mid i=0, \\ldots,14   \\}$. \n \nWe run Alg.~\\ref{alg:synthesize} with the parametric formula set $\\mathcal{F}^{\\leq 2}$, the dataset from  Ex.~\\ref{ex:formulation}, bound $B=5$ and subformula limit $p=4$.  The resulting formula is:\n\n\\begin{align}\n& \\phi = \\phi_1 \\vee \\phi_2 \\vee \\phi_3  \\vee \\phi_4 \\\\  \\nonumber\n& \\phi_1 = ( \\mathbf{P} _{[4, 10]} ( qGust < 0 ) ) \\wedge ( wGust < -120 )   \\\\   \\nonumber \n& \\phi_2 = ( wGust > 120 ) \\wedge ( \\mathbf{A} _{[14, 14]} ( pilot > -40 ) )  \\\\  \\nonumber \n&  \\phi_3 =\\mathbf{P} _{[2, 2]} ( ( alpha < 30 ) \\wedge ( wGust < -120 ) )\\\\  \\nonumber\n& \\phi_4 =( \\mathbf{A} _{[4, 16]} ( qGust > 10 ) ) \\wedge ( pilot < -40 )  \\nonumber \n  \\end{align}\n  Each sub-formula  $\\phi_1, \\phi_2 $, $\\phi_3$ and $ \\phi_4$ explains a condition that led to a disturbance in the pitch angle of the aircraft. \n  The first formula $\\phi_1$ shows that a disturbance occurs when $wGust$ is less than $-120$ and $qGust$ was lower than $0$ for some time within the last 10 time steps to last 4 time steps. Formulas $\\phi_2 $, $\\phi_3$ and $ \\phi_4$ show that a disturbance occurs 2) when $wGust$ is greater than $120$ and $pilot$ was greater then $-40$ $14$ time steps ago, or 3) if $alpha$ was less than $30$ and $wGust$ was less than -120   two steps ago, or 4) if $qGust$ was higher than 10 for each step between last 4 and 16  steps and $pilot$ is less than -40 in the current step.\n\n $477$ out of $3000$ data points in $\\mathcal{D}$ are labeled with $1$.\n $TP^{\\#}(\\phi,\\mathcal{D}) $ and $FP^{\\#}(\\phi,\\mathcal{D})$ valuations are 419 and 18 respectively. Total mismatch count of 3000 points is computed as 76 which leads to an accuracy of 97.46\\%.\n\n\nThis result is found in 3350 seconds on a PowerEdge T430 machine with Intel Xeon E5-2650 12C\/24T processor. It is important to note that $\\phi$ includes $11$ operators and $15$ parameters and it is defined over $4$ system variables. This example shows that the proposed method can generate complex formulas from labeled datasets in an efficient way, since, due to the computational complexity, existing formula synthesis algorithms are validated on simpler formulas.\n\n\\end{example}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sect_introduction}\n\nIsotope abundance ratios provide a powerful tool for tracing stellar nucleosynthesis, evaluating the composition of stellar ejecta, and constraining the chemical evolution of the Milky Way \\citep{1994ARA&A..32..191W}. In particular, the $^{12}$C\/$^{13}$C ratio is one of the most useful tracers of the relative degree of primary to secondary processing. $^{12}$C is a predominantly primary nucleus formed by He burning in massive stars on short timescales \\citep[e.g.,][]{1995ApJS...98..617T}. $^{13}$C is produced on a longer timescale via CNO processing of $^{12}$C seeds from earlier stellar generations during the red giant phase in low- and intermediate-mass stars or novae \\citep[e.g.,][]{1994LNP...439...72H,1994ARA&A..32..153M,1994ARA&A..32..191W}. $^{12}$C\/$^{13}$C ratios are expected to be low in the central molecular zone (CMZ) and high in the Galactic outskirts, because the Galaxy formed from the inside out (e.g., \\citealt{2001ApJ...554.1044C}; \\citealt{2012A&A...540A..56P}).\n\nObservations indeed indicate a gradient of $^{12}$C\/$^{13}$C ratios across the Galaxy. \\citet{1976A&A....51..303W} and \\citet{1979MNRAS.188..445W} measured the $J_{Ka,Kc}=1_{1,0}-1_{1,1}$ lines of H$_2^{12}$CO and H$_2^{13}$CO near 5 GHz toward 11 and 24 Galactic continuum sources, respectively. While ignoring effects of photon trapping, the results suggested that the $^{12}$C\/$^{13}$C ratios may vary with galactocentric distance ($R_{\\rm GC}$). With the additional measurement of the $J_{Ka,Kc}=2_{1,1}-2_{1,2}$ line of H$_2$CO at 14.5 GHz, \\citet{1980A&A....82...41H, 1982A&A...109..344H, 1983A&A...127..388H, 1985A&A...143..148H} also reported a gradient after correcting for effects of optical depth and photon trapping. \\citet{1990ApJ...357..477L} used the optically thin lines of C$^{18}$O and $^{13}$C$^{18}$O to trace the carbon isotope ratios. They also found a systematic $^{12}$C\/$^{13}$C gradient across the Galaxy, ranging from about 20--25 near the Galactic center, to 30--50 in the inner Galactic disk, to $\\sim$70 in the local interstellar medium (ISM). \\citet{1996A&AS..119..439W}, complementing these investigations by also including the far outer Galaxy, encountered ratios in excess of 100 and demonstrated that the gradient found in the inner Galaxy continues farther out. \\citet{2005ApJ...634.1126M} obtained $^{12}$C\/$^{13}$C = 6.01$R_{\\rm GC}$ + 12.28 based on the CN measurements of \\citet{2002ApJ...578..211S}. Here and elsewhere, $R_{\\rm GC}$ denotes the galactocentric distance in units of kiloparsecs (kpc). By combining previously obtained H$_2$CO and C$^{18}$O results with these CN data, \\citet{2005ApJ...634.1126M} obtained $^{12}$C\/$^{13}$C = 6.21$R_{\\rm GC}$ + 18.71.\n\nMore recently, \\citet{2017ApJ...845..158H} reported observations of a variety of molecules (e.g., H$_2$CS, CH$_3$CCH, NH$_2$CHO, CH$_2$CHCN, and CH$_3$CH$_2$CN) and their $^{13}$C-substituted species toward Sgr B2(N). These authors obtained an average $^{12}$C\/$^{13}$C value of 24 $\\pm$ 7 in the Galactic center region, which is close to results using $^{12}$CH\/$^{13}$CH (15.8 $\\pm$ 2.4, Sgr B2(M)) by \\citet{2020A&A...640A.125J} and the particularly solid $^{12}$C$^{34}$S\/$^{13}$C$^{34}$S ratio (22.1$^{+3.3}_{-2.4}$, $+$50 km s$^{-1}$ Cloud) from \\citet{2020A&A...642A.222H} who use a variety of CS isotopologs and rotational transitions. \\citet{2019ApJ...877..154Y} proposed a linear fit of $^{12}$C\/$^{13}$C = (5.08 $\\pm$ 1.10)$R_{\\rm GC}$ + (11.86 $\\pm$ 6.60) based on a large survey of H$_2$CO. The latter includes data from the center to the outskirts of the Milky Way well beyond the Perseus Arm. However, data from the CMZ are not similar to those thoroughly traced by \\citet{2020A&A...642A.222H}, not many sources from the innermost Galactic disk could be included in this survey, and also the number of sources beyond the Perseus arm was small, meaning that there is still space for improvement.\n\nWhile the carbon isotope ratio has drawn much attention in the past, it is not the only isotope ratio that can be studied at radio wavelengths and that has a significant impact on our understanding of the chemical evolution of the Galaxy. The isotope ratios of sulfur are providing complementary information on stellar nucleosynthesis that is not traced by the carbon isotope ratio. Sulfur is special in that it provides a total of four stable isotopes, $^{32}$S, $^{34}$S, $^{33}$S, and $^{36}$S. In the Solar System, abundance ratios are 95.02 : 4.21 : 0.75 : 0.021, respectively \\citep{1989GeCoA..53..197A}. $^{32}$S and $^{34}$S are synthesized during stages of hydrostatic oxygen-burning preceding a type II supernova event or during stages of explosive oxygen-burning in a supernova of type Ia; $^{33}$S is synthesized in explosive oxygen- and neon-burning, which is also related to massive stars; and $^{36}$S may be an s-process nucleus. The comprehensive calculations of \\citet{1995ApJS..101..181W} indicate that $^{32}$S and $^{33}$S are primary (in the sense that the stellar yields do not strongly depend on the initial metallicity of the stellar model), while $^{34}$S is not a clean primary isotope; its yield decreases with decreasing metallicity. According to \\citet{1985ApJ...295..604T} and \\citet{1989A&A...210...93L}, $^{36}$S is produced as a purely secondary isotope in massive stars, with a possible (also secondary) contribution from asymptotic giant branch (AGB) stars. Only a small fraction of $^{36}$S is destroyed during supernova explosions (Woosley, priv. comm.). Comparing ``primary'' and ``secondary'' nuclei, we might therefore expect the presence of weak $^{32}$S\/$^{34}$S and $^{34}$S\/$^{33}$S gradients and a stronger $^{32}$S\/$^{36}$S gradient as a function of galactocentric radius.\n\nThere is a strong and widespread molecular species that allows us to measure carbon and sulfur isotope ratios simultaneously, namely carbon monosulfide (CS). CS is unique in that it is a simple diatomic molecule exhibiting strong line emission and possessing eight stable isotopologs, which allows us to determine the above-mentioned carbon and sulfur isotope ratios. Six isotopologs have been detected so far in the ISM (e.g., \\citealt{1996A&A...305..960C}; \\citealt{1996A&A...313L...1M}; \\citealt{2020A&A...642A.222H}; \\citealt{2020ApJ...899..145Y}). \n\nMaking use of the CS species, \\citet{1996A&A...305..960C} and \\citet{1996A&A...313L...1M} obtained average abundance ratios of 24.4 $\\pm$ 5.0, 6.3 $\\pm$ 1.0, and 115 $\\pm$ 17 for $^{32}$S\/$^{34}$S, $^{34}$S\/$^{33}$S, and $^{34}$S\/$^{36}$S for the ISM, respectively. The latter is approximately half the solar value, but similar to the value found in IRC+10216 \\citep{2004A&A...426..219M}. Recently, \\citet{2020A&A...642A.222H} published $^{32}$S\/$^{34}$S ratios of 16.3$^{+2.1}_{-1.7}$ and 17.9 $\\pm$ 5.0 for the $+$50 km s$^{-1}$ cloud and Sgr B2(N) near the Galactic center, respectively. These are only slightly lower than the value of 22 in the Solar System. There is an obvious and confirmed $^{32}$S\/$^{34}$S gradient \\citep{1996A&A...305..960C, 2020ApJ...899..145Y} from the inner Galaxy out to a galactocentric distance of 12.0 kpc. Nevertheless, there is a lack of data at small and large galactocentric distances.\n\nWe are performing systematic observational studies on isotope ratios in the Milky Way, including $^{12}$C\/$^{13}$C \\citep{2019ApJ...877..154Y}, $^{14}$N\/$^{15}$N \\citep{2021ApJS..257...39C},$^{18}$O\/$^{17}$O (\\citealt{2015ApJS..219...28Z}, \\citeyear{2020ApJS..249....6Z}, \\citeyear{2020IAUGA..30..278Z}; \\citealt{2016RAA....16...47L}), and $^{32}$S\/$^{34}$S \\citep{2020ApJ...899..145Y}. We have thus performed a more systematic study on CS and its isotopologs toward 110 high-mass star-forming regions (HMSFRs). $^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S ratios can be directly derived from integrated $^{12}$C$^{34}$S\/$^{13}$C$^{34}$S (hereafter C$^{34}$S\/$^{13}$C$^{34}$S) and $^{13}$C$^{32}$S\/$^{13}$C$^{34}$S (hereafter $^{13}$CS\/$^{13}$C$^{34}$S, see Section \\ref{ratios_13c34s}) intensities, respectively. Also, $^{34}$S\/$^{33}$S and $^{34}$S\/$^{36}$S values could be obtained with measurements of C$^{34}$S, $^{12}$C$^{33}$S (hereafter C$^{33}$S), and $^{12}$C$^{36}$S (hereafter C$^{36}$S). Furthermore, $^{32}$S\/$^{33}$S and $^{32}$S\/$^{36}$S ratios can then be derived with the resulting $^{34}$S\/$^{33}$S and $^{34}$S\/$^{36}$S values combined with the $^{32}$S\/$^{34}$S ratios (see Sections \\ref{section_34s33s} to \\ref{section_32s36s}). In Section \\ref{sou_selection}, we describe the source selection and observations for our large sample. Section \\ref{results} presents our results on $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{34}$S\/$^{33}$S, $^{32}$S\/$^{33}$S, $^{34}$S\/$^{36}$S, and $^{32}$S\/$^{36}$S ratios. Section \\ref{discussion} discusses potential processes that could contaminate and affect the isotope ratios derived in the previous section and provides a detailed comparison with results from earlier studies. Our main results are summarized in Section \\ref{summary}. \n\n\n\n\\section{Source selection and observations}\n\\label{sou_selection}\n\n\\subsection{Sample selection and distance}\n\\label{section_distance}\n\nIn 2019, we selected 18 HMSFRs from the Galactic center region to the outer Galaxy beyond the Perseus arm. To enlarge this sample, we chose 92 sources from the Bar and Spiral Structure Legacy (BeSSeL) Survey\\footnote{http:\/\/bessel.vlbi-astrometry.org} in 2020. These 92 targets were recently released by the BeSSeL project \\citep{2019ApJ...885..131R} and not observed by \\citet{2020ApJ...899..145Y}. In total, 110 objects in the Galaxy are part of our survey. The coordinates of our sample sources are listed in Table \\ref{table_sources}. Determining trigonometric parallaxes is a very direct and accurate method to measure the distance of sources from the Sun \\citep{2009ApJ...700..137R, 2014ApJ...783..130R, 2019ApJ...885..131R}. Over the past decade, mainly thanks to the BeSSeL project, the trigonometric parallaxes of approximately 200 HMSFRs have been determined across the Milky Way through dedicated high-resolution observations of molecular maser lines. Therefore, this is a good opportunity to investigate carbon and sulfur isotope ratios with well-determined distances across the Galaxy. The galactocentric distance ($R_{\\rm GC}$) can be obtained with the heliocentric distance $d$ from the trigonometric parallax data base of the BeSSeL project using \\begin{equation}\nR_{\\rm GC} = \\sqrt{( R_0 cos(l) - d )^2 + R_0^2 sin^2(l)}\n,\\end{equation}\n\\citep{2009ApJ...699.1153R}. $R_0$ = 8.178 $\\pm$ 0.013$\\rm _{stat.}$ $\\pm$ 0.022$\\rm _{sys.}$ kpc \\citep{2019A&A...625L..10G} describes the distance from the Sun to the Galactic center, $l$ is the Galactic longitude of the source, and $d$ is the distance either directly derived from the trigonometric parallax data based on the BeSSeL project or a kinematic distance in cases where no such distance is yet available. Because the uncertainty in $R_0$ is very small, it will be neglected in the following analysis. For 12 of our targets without trigonometric parallax data, we estimated their kinematic distances from the Revised Kinematic Distance calculator\\footnote{http:\/\/bessel.vlbi-astrometry.org\/revised\\_kd\\_2014} \\citep{2014ApJ...783..130R}. The resulting distances indicate that 6 of these 12 sources are located in the CMZ, namely SgrC, the $+$20 km~s$^{-1}$ cloud, the $+$50 km~s$^{-1}$ cloud, G0.25, G1.28$+$0.07, and SgrD. Four targets belong to the inner Galactic disk, namely PointC1, CloudD, Clump2, and PointD1. Two sources, W89-380 and WB89-391, are in the outer regions beyond the Perseus arm. The heliocentric distances ($d$) and the galactocentric distances ($R_{\\rm GC}$) for our sample are listed in Columns 6 and 7 of Table~\\ref{table_sources}.\n\n\n\n\\subsection{Observations}\nWe observed the $J$ = 2-1 transitions of CS, C$^{33}$S, C$^{34}$S, C$^{36}$S, $^{13}$CS, $^{13}$C$^{33}$S, and $^{13}$C$^{34}$S as well as the $J$ = 3-2 transitions of C$^{33}$S, C$^{34}$S, C$^{36}$S, and $^{13}$CS toward 110 HMSFRs with the IRAM 30 meter telescope\\footnote{IRAM is supported by INSU\/CNRS (France), MPG (Germany) and IGN (Spain).} in 2019 June, July, and October under project 045-19 (PI, Christian Henkel) as well as in 2020 August within project 022-20 (PI, Hongzhi Yu). The on$+$off source integration times for our sources range from 4.0 minutes to 10.8 hours. These values are given in Appendix~\\ref{appendix_table} (Table \\ref{fitting_all}). The EMIR receiver with two bands, E090 and E150, was used to cover a bandwidth of $\\sim$16 GHz (from 90.8 to 98.2 GHz and 138.4 to 146.0 GHz) simultaneously in dual polarisation. We used the wide-mode FTS backend with a resolution of 195 kHz, corresponding to $\\sim$0.6 km s$^{-1}$ and $\\sim$0.4 km s$^{-1}$ at 96 GHz and 145 GHz, respectively. The observations were performed in total-power position-switching mode and the off position was set at 30$\\arcmin$ in azimuth. Pointing was checked every 2 hours using nearby quasars. Focus calibrations were done at the beginning of the observations and during sunset and sunrise toward strong quasars. The main beam brightness temperature, $T_{\\rm MB}$, was obtained from the antenna temperature $T_{\\rm A}^*$ via the relation $T_{\\rm MB}$ = $T_{\\rm A}^*\\times F_{\\rm eff}$\/$B_{\\rm eff}$ ($F_{\\rm eff}$: forward hemisphere efficiency; $B_{\\rm eff}$: main beam efficiency) with corresponding telescope efficiencies\\footnote{https:\/\/publicwiki.iram.es\/Iram30mEfficiencies}: $F_{\\rm eff}$\/$B_{\\rm eff}$ are 0.95\/0.81 and 0.93\/0.73 in the frequency ranges of 90.8-98.2 GHz and 138.4-146.0 GHz, respectively. The system temperatures were 100-160 K and 170-300 K on a $T_{\\rm A}^*$ scale for the E090 and E150 band observations. The half power beam width (HPBW) for each transition was calculated as HPBW($\\arcsec$)=2460\/$\\nu$(GHz). Rest frequencies, excitations of the upper levels above the ground state, Einstein coefficients for spontaneous emission, and respective beam sizes are listed in Table~\\ref{table_linelist}.\n\n\n\n\n\\begin{table*}[h]\n\\caption{Observed spectral line parameters$^a$.}\n\\centering\n\\begin{tabular}{cccccc}\n\\hline\\hline\nIsotopolog &  Transition  &  $\\nu_0$\\tablefootmark{b}  &  $E_{\\rm up}$\\tablefootmark{c} & $A_{\\rm u,l}$\\tablefootmark{d} & HPBW\\tablefootmark{e}  \\\\\n  & \\    & (MHz)   &  (K) &  (s$^{-1}$) &  ($\\arcsec$) \\\\\n\\hline\n\\label{table_linelist}\nCS               & 2-1 & 97980.953   &7.1  & 1.68 $\\times$ 10$^{-5}$ & 25.1 \\\\\nC$^{33}$S        & 2-1 & 97172.064   &7.0  & 1.64 $\\times$ 10$^{-5}$ & 25.3 \\\\\nC$^{34}$S        & 2-1 & 96412.95    &6.9  & 1.60 $\\times$ 10$^{-5}$ & 25.5 \\\\\nC$^{36}$S        & 2-1 & 95016.722   &6.8  & 1.53 $\\times$ 10$^{-5}$ & 25.9 \\\\\n$^{13}$CS        & 2-1 & 92494.308   &6.7  & 1.41 $\\times$ 10$^{-5}$ & 26.6 \\\\\n$^{13}$C$^{33}$S & 2-1 & 91685.241   &6.6  & 1.38 $\\times$ 10$^{-5}$ & 26.8 \\\\\n$^{13}$C$^{34}$S & 2-1 & 90926.026   &6.5  & 1.34 $\\times$ 10$^{-5}$ & 27.1 \\\\\nC$^{33}$S        & 3-2 & 145755.732  &14.0 & 5.92 $\\times$ 10$^{-5}$ & 16.9 \\\\\nC$^{34}$S        & 3-2 & 144617.101  &13.9 & 5.78 $\\times$ 10$^{-5}$ & 17.0 \\\\\nC$^{36}$S        & 3-2 & 142522.785  &13.7 & 5.54 $\\times$ 10$^{-5}$ & 17.3 \\\\\n$^{13}$CS        & 3-2 & 138739.335  &13.3 & 5.11 $\\times$ 10$^{-5}$ & 17.7 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{\n\\tablefoottext{a}{From the Cologne Database for Molecular Spectroscopy \\citep[CDMS,][]{2005JMoSt.742..215M,2016JMoSp.327...95E}.}\n\\tablefoottext{b}{Rest frequency.}\n\\tablefoottext{c}{Upper energy level.}\n\\tablefoottext{d}{Einstein coefficient for spontaneous emission from upper $u$ to lower $l$ level.}\n\\tablefoottext{e}{Half power beam width.}}\n\\end{table*}\n\n\\subsection{Data reduction}\n\\label{datareduction}\n\nWe used the GILDAS\/CLASS\\footnote{https:\/\/www.iram.fr\/IRAMFR\/GILDAS\/} package to analyze the spectral line data. The spectra of the $J$ = 2-1 transitions of CS, C$^{33}$S, C$^{34}$S, C$^{36}$S, $^{13}$CS, $^{13}$C$^{33}$S, and $^{13}$C$^{34}$S as well as the $J$ = 3-2 transitions of C$^{33}$S, C$^{34}$S, C$^{36}$S, and $^{13}$CS toward one of our targets, DR21, are shown in Fig.~\\ref{fig_dr21}, after subtracting first-order polynomial baselines and applying Hanning smoothing. The spectra of all 110 targets, also after first-order polynomial-baseline removal and Hanning smoothing, are presented in Appendix~\\ref{appendix_spectra} (Fig.~\\ref{spectra_all}). \n\n\n\n\nAmong our sample of 110 targets, we detected the $J$ = 2-1 line of CS toward 106 sources, which yields a detection rate of 96\\%. The $J$ = 2-1 transitions of C$^{34}$S, $^{13}$CS, C$^{33}$S, and $^{13}$C$^{34}$S were successfully detected in 90, 82, 46, and  17 of our sources with signal-to-noise (S\/N) ratios of greater than 3, respectively. The $J$ = 3-2 lines of C$^{34}$S, $^{13}$CS, and C$^{33}$S were detected in 87, 71, and 42 objects  with S\/Ns  of $\\ge$3.0. Line parameters from Gaussian fitting are listed in Appendix~\\ref{appendix_table} (Table \\ref{fitting_all}). Relevant for the evaluation of isotope ratios is the fact that  for 17 sources with 19 velocity components, the S\/Ns of the $J$ = 2-1 transition of $^{13}$C$^{34}$S are greater than 3, which allows us to determine the $^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S ratios directly with the $J$ = 2-1 lines of C$^{34}$S, $^{13}$CS, and $^{13}$C$^{34}$S.  Toward 82 targets with 90 radial velocity components, the $J$ = 2-1 transitions of C$^{34}$S and $^{13}$CS were both detected with S\/Ns of $\\ge$3.0. The $J$ = 3-2 lines of C$^{34}$S and $^{13}$CS were both found in 71 objects with 73 radial velocity components and S\/Ns of $\\ge$3.0. Furthermore, the $J$ = 2-1 and $J$ = 3-2 transitions of C$^{34}$S and C$^{33}$S were detected  with S\/Ns of $\\ge$3.0 toward 46 and 42 sources, respectively.\n\n\\begin{figure}[h]\n\\centering\n   \\includegraphics[width=0.3\\textwidth]{spectra\/DR21.eps}\n\\caption{Line profiles of the $J$ = 2-1 transitions of CS, C$^{33}$S, C$^{34}$S, C$^{36}$S, $^{13}$CS, $^{13}$C$^{33}$S, and $^{13}$C$^{34}$S as well as the $J$ = 3-2 transitions of C$^{33}$S, C$^{34}$S, C$^{36}$S, and $^{13}$CS toward one typical target (DR21) of our large sample of 110 sources, after subtracting first-order polynomial baselines. The main beam brightness temperature scales are presented on the left hand side of the profiles. The spectra of all 110 objects in our sample are shown in Appendix~\\ref{appendix_spectra} (Fig.~\\ref{spectra_all}).}\n  \\label{fig_dr21}\n\\end{figure}\n\nThe C$^{36}$S $J$ = 2-1 line was successfully detected with S\/Ns of $\\ge$3.0 toward three targets, namely W3OH, the $+$50 km~s$^{-1}$ cloud near the Galactic center, and DR21. As C$^{36}$S and $^{13}$C$^{33}$S are the least abundant among the CS isotopologs, tentative detection with S\/Ns of $\\sim$2.0 are also presented here but not included in further analyses. In another five objects, the C$^{36}$S $J$ = 2-1 line was tentatively detected. For the C$^{36}$S $J$ = 3-2 transitions, we report one detection with an S\/N larger than 3.0 toward Orion-KL and five tentative detections. The $J$ = 2-1 lines of $^{13}$C$^{33}$S were tentatively detected toward three sources, namely Orion-KL, W51-IRS2, and DR21.\nIntegration times and 1$\\sigma$ noise levels of the observed transitions are listed in Columns 3 and 4 of Table~\\ref{fitting_all} for each target.\n\n\n\n\n\\section{Results}\n\\label{results}\nIn the following, we first estimate the optical depths of the various lines to avoid problems with line saturation that might affect our results. We then present the carbon and sulfur isotope ratios derived from different detected CS isotopologs.\n\n\\subsection{Optical depth}\n\\label{section_opacities}\n\nThe main isotopolog, CS, is usually optically thick in massive star-forming regions (e.g. \\citealt{1980ApJ...235..437L}; \\citealt{2020ApJ...899..145Y}). Therefore, the $^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S ratios cannot be determined from the line intensity ratios of $I$(CS)\/$I$($^{13}$CS) and $I$(CS)\/$I$(C$^{34}$S). However, assuming that the $J$ = 2-1 transitions of CS, C$^{34}$S, and $^{13}$CS share the same beam filling factor and excitation temperature, we can estimate the maximum optical depth of the $^{13}$CS $J$ = 2-1 line from:\n\\begin{equation}\n\\frac{T_{\\rm {mb}}(^{12}\\rm C\\rm S)}{T_{\\rm mb}(^{13}\\rm C\\rm S)} \\sim \\frac{1 - e^{-\\tau(^{13}\\rm C\\rm S)R_C}}{1 - e^{-\\tau(^{13}\\rm C\\rm S)}},\\,R_C = \\frac{^{12}\\rm C}{^{13}\\rm C},\n\\end{equation}\nwhere $T_{\\rm {mb}}$ is the peak main beam brightness temperature derived from the best Gaussian-fitting result and listed in Column 8 of Table~\\ref{fitting_all}. In this case, the $^{12}$C\/$^{13}$C ratios can be derived from the integrated line intensities of C$^{34}$S and $^{13}$C$^{34}$S with the assumption of $\\tau(\\rm C^{34}\\rm S) \\textless 1.0$, which then also implies $\\tau(^{13}\\rm C^{34}\\rm S) \\textless 1.0$ (see details in Section \\ref{ratios_13c34s}). Multiplying $\\tau(^{13}\\rm CS)$ by $R_C$ = $^{12}$C\/$^{13}$C, we can get the peak opacity $\\tau(\\rm CS)$ = $\\tau(^{13}{\\rm CS})R_C$. The maximum optical depth of C$^{34}$S can be obtained from:\n\\begin{equation}\n\\frac{T_{\\rm mb}(\\rm C\\rm ^{34}S)}{T_{\\rm mb}(^{13}\\rm C\\rm S)} = \\frac{1 - e^{-\\tau(\\rm C\\rm ^{34}S)}}{1 - e^{-\\tau(^{13}\\rm C\\rm S)}},\n\\end{equation}\nwhere $T_{\\rm {mb}}$ is the peak main beam brightness temperature derived from the best Gaussian-fitting result and listed in Column 8 of Table~\\ref{fitting_all}. As shown in Table \\ref{table_13c34sresults}, the peak optical depths of the $J$ = 2-1 lines of CS, C$^{34}$S, and $^{13}$CS for our 17 targets with detections of $^{13}$C$^{34}$S range from 1.29 to 8.79, 0.12 to 0.55, and 0.05 to 0.34, respectively. Therefore, C$^{34}$S and $^{13}$CS in these 17 objects are optically thin, even though they belong, on average, to the more opaque ones,  being successfully detected\nin $^{13}$C$^{34}$S (see below). Nevertheless, the corrections for optical depth are applied to C$^{34}$S and $^{13}$CS with factors of $f_1$ and $f_2$, respectively.\n\\begin{equation}\nf_1=\\frac{\\tau(\\rm C\\rm ^{34}S)}{1-e^{-\\tau(\\rm C\\rm ^{34}S)}} {\\quad\\rm and}\n\\end{equation}\n\\begin{equation}\nf_2=\\frac{\\tau(^{13}\\rm C\\rm S)}{1-e^{-\\tau(^{13}\\rm C\\rm S)}}\n\\end{equation}\nare listed in Columns 7 and 8 of Table~\\ref{table_13c34sresults}, respectively.\n\nFor those 82 sources with detections of $J$ = 2-1 CS, C$^{34}$S, and $^{13}$CS, the optical depths were calculated based on the $^{12}$C\/$^{13}$C gradient that we derived from our C$^{34}$S and $^{13}$C$^{34}$S measurements (for details, see Section \\ref{ratios_13c34s}). In Table \\ref{table_doubleisotope}, the peak opacities of the $J$ = 2-1 lines of CS, C$^{34}$S, and $^{13}$CS for these 82 targets range from 0.34 to 14.48, 0.02 to 0.74, and 0.01 to 0.39, respectively. The CS $J$ = 2-1 lines are optically thick with $\\tau(\\rm CS) \\textgreater 1.0$ in most sources (89\\%) of our sample, while they tend to be optically thin in seven objects, namely Point C1 ($\\tau(\\rm CS) \\leq 0.59$), Sgr C ($\\tau(\\rm CS) \\leq 0.54$), Cloud D ($\\tau(\\rm CS) \\leq 0.64$), G1.28$+$0.07 ($\\tau(\\rm CS) \\leq 0.82$), Sgr D ($\\tau(\\rm CS) \\leq 0.45$), and Point D1 ($\\tau(\\rm CS) \\leq 0.34$). In contrast, the transitions from rare isotopologs, the C$^{34}$S and $^{13}$CS $J$ = 2-1 lines in our sample, are all optically thin, as their maximum optical depths are less than 0.8 and 0.4, respectively. In the following, we are therefore motivated to consider all CS isotopologs as optically thin, except CS itself. This allows us to use ratios of integrated intensity of all the rare CS isotopologs ---but not CS itself--- to derive the carbon and sulfur isotope ratios we intend to study. Small corrections accounting for the optical depths are applied to the C$^{34}$S and $^{13}$CS $J$ = 2-1 lines with factors of $f_1$ and $f_2$, respectively, and are listed in Columns 7 and 8 of Table~\\ref{table_doubleisotope}. The optical depths of the $J$ = 3-2 transitions cannot be estimated, as the CS $J$ = 3-2 line was not covered by our observations because of bandwidth limitations. However, the $^{32}$S\/$^{34}$S ratios for a given source obtained through the double isotope method from the $J$ = 2-1 and $J$ = 3-2 transitions are in good agreement, indicating that the C$^{34}$S and $^{13}$CS $J$ = 3-2 lines in our sample are also optically thin (see details in Section~\\ref{section_double_32s34s}). \n\nThe RADEX non Local Thermodynamic Equilibrium (LTE)  model \\citep{2007A&A...468..627V} was used to calculate the variation of excitation temperature, $T_{ex}$, with optical depth. Frequencies, energy levels, and Einstein A coefficients for spontaneous emission were taken from the Cologne Database for Molecular Spectroscopy \\citep[CDMS;][]{2005JMoSt.742..215M,2016JMoSp.327...95E}. Recent collision rates for CS with para- and ortho-H$_2$ \\citep{2018MNRAS.478.1811D} were used. Figure~\\ref{fig_c34stex} shows the excitation temperatures and opacities of C$^{34}$S $J$ = 2-1 for a kinetic temperature of 30 K and a molecular hydrogen density of 10$^5$ cm$^{-3}$. Variations of $T_{ex}$ within about 2 K for our sample targets with optical depths of 0.02 $\\leq \\tau(\\rm C^{34}S) \\leq 0.74$ can barely affect our results.\n\n\\begin{figure}[h]\n\\centering\n   \\includegraphics[width=0.5\\textwidth]{newFigures\/c34s21-tau-tex.png}\n\\caption{Excitation temperature, $T_{ex}$, as a function of optical depth for the $J$ = 2-1 transition of C$^{34}$S. The gray dashed lines indicate the range of opacities for our sample sources.}\n  \\label{fig_c34stex}\n\\end{figure}\n\n\n\n\n\\begin{figure*}[h]\n\\centering\n   \\includegraphics[width=460pt]{newFigures\/1allratio12c_13c-lowerlimits.pdf}\n     \\caption{$^{12}$C\/$^{13}$C isotope ratios from C$^{34}$S\/$^{13}$C$^{34}$S, CN\/$^{13}$CN, C$^{18}$O\/$^{13}$C$^{18}$O, H$_2$CO\/H$_2^{13}$CO, CH$^+$\/$^{13}$CH$^+$, and CH\/$^{13}$CH are plotted as functions of the distance from the Galactic center. The red symbol $\\odot$ indicates the $^{12}$C\/$^{13}$C isotope ratio of the Sun. The filled black circles are the results obtained from C$^{34}$S with corrections of opacity in the current work, and the resulting first-order polynomial fit is plotted as a solid line, with the gray-shaded area showing the 1$\\sigma$ interval of the fit. The open black circles are the 3$\\sigma$ lower limits obtained from nondetections of $^{13}$C$^{34}$S in the current work. The blue triangles, orange pentagons, yellow stars, green squares, and green diamonds are values determined from CN \\citep{2002ApJ...578..211S,2005ApJ...634.1126M}, C$^{18}$O \\citep{1990ApJ...357..477L,1996A&AS..119..439W,1998ApJ...494L.107K}, H$_2$CO \\citep{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H,2019ApJ...877..154Y}, CH$^+$ \\citep{2011ApJ...728...36R}, and CH \\citep{2020A&A...640A.125J}, respectively, using the most up-to-date distances. The red crosses visualize the results from the GCE model of \\citet[][see also Section~\\ref{section_discussion_model}]{2011MNRAS.414.3231K,2020ApJ...900..179K}. }\n  \\label{fig_gradient_12C13C}\n\\end{figure*}\n\n\n\\begin{table*}[h]\n\\caption{Isotope ratios derived with the $J$ = 2-1 transitions of C$^{34}$S, $^{13}$CS, and $^{13}$C$^{34}$S.}\n\\centering\n\\small\n\\begin{tabular}{lcc|ccc|cc|cc}\n\\hline\\hline\nSource &  $V_{\\rm LSR}$ & $R_{GC}$  &   \\multicolumn{3}{c}{Optical depth}  &   \\multicolumn{2}{c}{Corrections for} &  $^{12}$C\/$^{13}$C  &  $^{32}$S\/$^{34}$S  \\\\\n  &                &         &     &           &           &  \\multicolumn{2}{c}{optical depth} &   &    \\\\\n  &  (km s$^{-1}$) & (kpc)   &  CS & C$^{34}$S & $^{13}$CS &  $f_1$  &  $f_2$ &   &    \\\\\n\\hline\n\\label{table_13c34sresults}\nW3OH                    & -46.96 &  9.64 $\\pm$ 0.03   & 8.289 $\\pm$ 0.083 & 0.292 $\\pm$ 0.003 & 0.127 $\\pm$ 0.001 &  1.2  &  1.1  &  75.30 $\\pm$ 4.27  &    29.80 $\\pm$ 1.68 \\\\\nOrion-KL                &   8.17 &  8.54 $\\pm$ 0.00   & 3.722 $\\pm$ 0.037 & 0.180 $\\pm$ 0.002 & 0.074 $\\pm$ 0.001 &  1.1  &  1.0  &  54.89 $\\pm$ 12.90 &    21.24 $\\pm$ 4.67 \\\\\nG359.61$-$00.24         &  19.33 &  5.51 $\\pm$ 0.15   & 3.531 $\\pm$ 0.035 & 0.204 $\\pm$ 0.002 & 0.089 $\\pm$ 0.001 &  1.1  &  1.0  &  43.85 $\\pm$ 11.95 &    17.31 $\\pm$ 4.79 \\\\\n$+$50~km~s$^{-1}$~cloud &  46.79 &  0.02 $\\pm$ 0.04   & 4.278 $\\pm$ 0.043 & 0.192 $\\pm$ 0.002 & 0.151 $\\pm$ 0.002 &  1.1  &  1.1  &  31.22 $\\pm$ 2.06  &    21.71 $\\pm$ 1.40 \\\\\nSgrB2                   &  53.18 &  0.55 $\\pm$ 0.05   & 1.290 $\\pm$ 0.013 & 0.140 $\\pm$ 0.001 & 0.113 $\\pm$ 0.001 &  1.1  &  1.1  &  12.18 $\\pm$ 0.73  &    9.77  $\\pm$ 0.58 \\\\\nSgrB2                   &  66.54 &  0.44 $\\pm$ 0.70   & 8.119 $\\pm$ 0.081 & 0.523 $\\pm$ 0.005 & 0.341 $\\pm$ 0.003 &  1.3  &  1.2  &  30.62 $\\pm$ 2.04  &    18.99 $\\pm$ 1.31 \\\\\nSgrB2                   &  83.38 &  0.41 $\\pm$ 0.02   & 2.417 $\\pm$ 0.024 & 0.097 $\\pm$ 0.001 & 0.077 $\\pm$ 0.001 &  1.0  &  1.0  &  32.90 $\\pm$ 5.65  &    26.24 $\\pm$ 4.49 \\\\\nG006.79$-$00.25         &  20.87 &  4.75 $\\pm$ 0.25   & 8.789 $\\pm$ 0.088 & 0.484 $\\pm$ 0.005 & 0.242 $\\pm$ 0.002 &  1.3  &  1.1  &  45.82 $\\pm$ 8.39  &    19.46 $\\pm$ 3.63 \\\\\nG010.32$-$00.15         &  11.99 &  5.34 $\\pm$ 0.29   & 5.039 $\\pm$ 0.050 & 0.237 $\\pm$ 0.002 & 0.111 $\\pm$ 0.001 &  1.1  &  1.1  &  50.95 $\\pm$ 11.58 &    21.55 $\\pm$ 5.05 \\\\\nG019.36$-$00.03         &  26.40 &  5.58 $\\pm$ 0.49   & 6.789 $\\pm$ 0.068 & 0.332 $\\pm$ 0.003 & 0.141 $\\pm$ 0.001 &  1.2  &  1.1  &  56.41 $\\pm$ 14.37 &    21.19 $\\pm$ 5.53 \\\\\nG024.78$+$00.08         & 110.72 &  3.51 $\\pm$ 0.15   & 8.038 $\\pm$ 0.080 & 0.554 $\\pm$ 0.006 & 0.322 $\\pm$ 0.003 &  1.3  &  1.2  &  32.56 $\\pm$ 4.25  &    16.08 $\\pm$ 2.12 \\\\\nG028.39$+$00.08         &  78.01 &  4.83 $\\pm$ 0.17   & 8.705 $\\pm$ 0.087 & 0.443 $\\pm$ 0.004 & 0.291 $\\pm$ 0.003 &  1.2  &  1.2  &  37.04 $\\pm$ 10.05 &    18.74 $\\pm$ 5.12 \\\\\nG028.83$-$00.25         &  87.19 &  4.50 $\\pm$ 0.48   & 7.283 $\\pm$ 0.073 & 0.388 $\\pm$ 0.004 & 0.183 $\\pm$ 0.002 &  1.2  &  1.1  &  47.93 $\\pm$ 10.00 &    19.73 $\\pm$ 4.16 \\\\\nG030.70$-$00.06         &  89.94 &  4.20 $\\pm$ 0.10   & 7.088 $\\pm$ 0.071 & 0.448 $\\pm$ 0.004 & 0.263 $\\pm$ 0.003 &  1.2  &  1.1  &  33.39 $\\pm$ 2.77  &    18.07 $\\pm$ 1.52 \\\\\nG030.74$-$00.04         &  91.82 &  5.76 $\\pm$ 0.36   & 5.924 $\\pm$ 0.059 & 0.404 $\\pm$ 0.004 & 0.191 $\\pm$ 0.002 &  1.2  &  1.1  &  37.63 $\\pm$ 10.87 &    15.61 $\\pm$ 4.59 \\\\\nG030.81$-$00.05         &  98.89 &  5.73 $\\pm$ 0.24   & 4.691 $\\pm$ 0.047 & 0.306 $\\pm$ 0.003 & 0.187 $\\pm$ 0.002 &  1.2  &  1.1  &  29.07 $\\pm$ 2.99  &    14.97 $\\pm$ 1.44 \\\\\nW51-IRS2                &  61.07 &  6.22 $\\pm$ 0.06   & 1.883 $\\pm$ 0.019 & 0.119 $\\pm$ 0.001 & 0.052 $\\pm$ 0.001 &  1.1  &  1.0  &  38.26 $\\pm$ 4.35  &    16.70 $\\pm$ 1.92 \\\\\nDR21                    &  -2.49 &  8.10 $\\pm$ 0.00   & 7.079 $\\pm$ 0.071 & 0.270 $\\pm$ 0.003 & 0.106 $\\pm$ 0.001 &  1.1  &  1.1  &  76.22 $\\pm$ 7.57  &    27.48 $\\pm$ 2.84 \\\\\nNGC7538                 & -57.12 &  9.47 $\\pm$ 0.07   & 3.433 $\\pm$ 0.034 & 0.131 $\\pm$ 0.001 & 0.050 $\\pm$ 0.001 &  1.1  &  1.0  &  73.19 $\\pm$ 12.54 &    26.72 $\\pm$ 4.57 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{ Velocities were obtained from measurements of C$^{34}$S, see Table \\ref{fitting_all} in Appendix \\ref{appendix_table}.}\n\\end{table*}\n\n\\subsection{$^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S ratios derived directly from $^{13}$C$^{34}$S}\n\\label{ratios_13c34s}\n\n\\subsubsection{$^{12}$C\/$^{13}$C ratios}\n\\label{section_results_12c13c}\n\nThe $^{12}$C\/$^{13}$C ratios derived from the integrated intensity ratios of C$^{34}$S and $^{13}$C$^{34}$S with corrections of optical depth are listed in Table~\\ref{table_13c34sresults}. Figure~\\ref{fig_gradient_12C13C} shows our results as filled black circles. A gradient of $^{12}$C\/$^{13}$C is obtained with an unweighted least-squares fit:\n\\begin{equation}\n^{12}{\\rm C}\/^{13}{\\rm C} = (4.77 \\pm 0.81)R_{\\rm GC}+(20.76 \\pm 4.61).\n\\end{equation}\nThe correlation coefficient is 0.82. Around the CMZ toward the $+$50 km~s$^{-1}$ cloud and SgrB2, four velocity components of C$^{34}$S and $^{13}$C$^{34}$S were detected and then an average $^{12}$C\/$^{13}$C value of 27~$\\pm$~3 is derived. The uncertainties given here and below are standard deviations of the mean. Eleven objects within a range of 3.50 kpc < $R_{\\rm GC}$ < 6.50 kpc in the inner Galactic disk lead to an average $^{12}$C\/$^{13}$C value of 41~$\\pm$~9. In the Local arm near the Sun, the $^{13}$C$^{34}$S lines were detected toward two sources, Orion-KL and DR21. These provide an average $^{12}$C\/$^{13}$C value of 66~$\\pm$~10, which is lower than the Solar System ratio. The other two targets beyond the solar neighborhood belong to the Perseus arm and show a slightly higher value of 74~$\\pm$~8.\n\nFor sources with detections of C$^{34}$S and nondetections of $^{13}$C$^{34}$S, 3$\\sigma$ lower limits of the $^{12}$C\/$^{13}$C ratio have been derived and are shown as open black circles in Fig.~\\ref{fig_gradient_12C13C}. All these lower limits are below the $^{12}$C\/$^{13}$C gradient we describe above. \n\n\n\n\n\n\\subsubsection{$^{32}$S\/$^{34}$S ratios}\n\\label{section_ratios3234_13c34s}\n\nThe $^{32}$S\/$^{34}$S ratios directly derived from the integrated intensity ratios of $^{13}$CS\/$^{13}$C$^{34}$S from the $J$ = 2-1 lines with corrections of optical depth are listed in Table~\\ref{table_13c34sresults} and are plotted as a function of galactocentric distance in Fig.~\\ref{fig_gradient_32S34S}. With an unweighted least-squares fit, a gradient with a correlation coefficient of 0.47 can be obtained:\n\\begin{equation}\n^{32}{\\rm S}\/^{34}{\\rm S} =(0.73 \\pm 0.36)R_{\\rm GC}+(16.50 \\pm 2.07).\n\\end{equation}\nAn average $^{32}$S\/$^{34}$S ratio of 19~$\\pm$~2 is obtained in the CMZ, which is based on the measurements from two sources, namely the $+$50 km~s$^{-1}$ cloud next to the Galactic center and Sgr B2. In the inner Galactic disk at a range of 3.50 kpc < $R_{\\rm GC}$ < 6.50 kpc, $^{13}$C$^{34}$S was detected toward 11 objects, leading to an average $^{32}$S\/$^{34}$S value of 18~$\\pm$~4. For sources in the Local and the Perseus arm beyond the Sun, the $^{32}$S\/$^{34}$S ratios are 24~$\\pm$~4 and 28~$\\pm$~3, respectively. This reveals a gradient from the inner Galactic disk to the outer Galaxy, but none from the CMZ to the inner disk.\n\nFor sources with detections of $^{13}$CS and nondetections of $^{13}$C$^{34}$S, we determined 3$\\sigma$ lower limits to the $^{32}$S\/$^{34}$S ratio, which are shown as open black circles in Fig.~\\ref{fig_gradient_32S34S}. All these lower limits are below the $^{32}$S\/$^{34}$S gradient we describe above. \n\n\n\n\\begin{figure*}[h]\n\\centering\n   \\includegraphics[height=600pt]{newFigures\/1Allratio32s_34s_new-lower.pdf}\n     \\caption{$^{32}$S\/$^{34}$S isotope ratios as functions of the distance from the Galactic center. The symbol $\\odot$ indicates the $^{32}$S\/$^{34}$S isotope ratio in the Solar System. In the upper panel, the $^{32}$S\/$^{34}$S ratios directly derived from $^{13}$CS\/$^{13}$C$^{34}$S in the $J$ = 2-1 transition and obtained from the double isotope method in the $J$ = 2-1 transition with corrections for optical depth are plotted as black and green dots, respectively. The $^{32}$S\/$^{34}$S ratios without corrections of opacity in the $J$ = 3-2 transition are plotted as light blue dots. The 3$\\sigma$ lower limits of $^{32}$S\/$^{34}$S ratios obtained from nondetections of $^{13}$C$^{34}$S in the current work are shown as open black circles. The $^{32}$S\/$^{34}$S ratios in \\citet{2020ApJ...899..145Y} derived from the double isotope method in the $J$ = 2-1 transitions are shown as blue dots in the lower panel. The $^{32}$S\/$^{34}$S values in the CMZ obtained from $^{13}$CS\/$^{13}$C$^{34}$S in \\citet{2020A&A...642A.222H} are plotted as red stars in both panels. The resulting first-order polynomial fits to $^{32}$S\/$^{34}$S ratios with the direct method and the double isotope method from the $J$ = 2-1 transition in this work are plotted as black and red solid lines in the two panels, respectively, with the gray and yellow shaded areas showing the 1~$\\sigma$ intervals of the fits. The magenta and cyan dashed-dotted lines show the $^{32}$S\/$^{34}$S gradients from \\citet{1996A&A...305..960C} and \\citet{2020ApJ...899..145Y}. The red crosses visualize the results from the GCE model of \\citet[][see also Section~\\ref{section_discussion_model}]{2011MNRAS.414.3231K,2020ApJ...900..179K}. }\n  \\label{fig_gradient_32S34S}\n\\end{figure*}\n\n\n\n\\subsection{$^{32}$S\/$^{34}$S ratios obtained through the double isotope method}\n\\label{section_double_32s34s}\n\nThe $^{32}$S\/$^{34}$S values can also be derived from measurements of C$^{34}$S and $^{13}$CS  using the carbon gradient obtained from our $^{13}$C$^{34}$S measurements above by applying the following equation:\n\\begin{equation}\n\\frac{^{32}{\\rm S}}{^{34}{\\rm S}} = R_{\\rm C} \\frac{I(^{13}{\\rm CS})}{I(\\rm C^{34}{\\rm S})},\n\\end{equation}\nwhere $R_{\\rm C}$ is the $^{12}$C\/$^{13}$C ratio derived from equation (6). The uncertainty on this latter is also included in our error budget. The $^{32}$S\/$^{34}$S ratios in the $J$ = 2-1 transitions were calculated with corrections of optical depth for 83 targets with 90 radial velocity components, in which the C$^{34}$S and $^{13}$CS $J$ = 2-1 lines were both detected, and are listed in Column 7 of Table \\ref{table_doubleisotope}. An unweighted least-squares fit to these values yields  \n\\begin{equation}\n^{32}{\\rm S}\/^{34}{\\rm S} (2-1) = (0.75 \\pm 0.13)R_{\\rm GC}+(15.52 \\pm 0.78), \n\\end{equation}\nwith a correlation coefficient of 0.54. The C$^{34}$S and $^{13}$CS $J$ = 3-2 lines were both detected in 71 objects with 73 radial velocity components. The $^{32}$S\/$^{34}$S ratios derived with equation (8) from the $J$ = 3-2 transition are shown in Column 8 of Table \\ref{table_doubleisotope}. An unweighted least-squares fit to the $J$ = 3-2 transition yields \n\\begin{equation}\n^{32}{\\rm S}\/^{34}{\\rm S} (3-2) = (0.99 \\pm 0.14)R_{\\rm GC}+(16.05 \\pm 0.95),  \n\\end{equation}\nwhich is within the errors, and is consistent with the trend obtained from the $J$ = 2-1 transition (equation (9)). However, we note that we do not have CS $J$ = 3-2 data, and therefore no opacity corrections could be applied to our C$^{34}$S $J$ = 3-2 spectra (see also Sect.~\\ref{section_34s33s}).\n\n\n\n\n\n\n\\subsection{$^{34}$S\/$^{33}$S ratios}\n\\label{section_34s33s}\n\nThe $^{34}$S\/$^{33}$S ratios can be determined directly from the intensity ratios of C$^{34}$S\/C$^{33}$S. The $^{34}$S\/$^{33}$S ratios from the $J$ = 2-1 lines were then corrected for optical depths derived in Section~\\ref{section_opacities}. However, both the $J$ = 2-1 and $J$ = 3-2 transitions of C$^{33}$S are split by hyperfine structure (HFS) interactions \\citep{1981CPL....81..256B}, which may affect the deduced values of $^{34}$S\/$^{33}$S. \n\nThe C$^{33}$S $J$ = 2-1 line consists of eight hyperfine components distributed over about 9.0 MHz \\citep{2005JMoSt.742..215M,2016JMoSp.327...95E}, which corresponds to a velocity range of about 28~km~s$^{-1}$. Following the method introduced in Appendix D in \\citet{2021A&A...646A.170G}, and assuming that the intrinsic width of each HFS line is 1~km~s$^{-1}$, the $J$ = 2-1 line profile can be reproduced by four components (see Fig.~\\ref{fig_c33s_hfs}, upper left panel). In this case, the main component ($I_{main}$) consists of four HFS lines ($F$=7\/2-5\/2, $F$=5\/2-3\/2, $F$=1\/2-1\/2, $F$=3\/2-5\/2), which account for 70\\% of the total intensity. Among the 46 sources with detections of the C$^{33}$S $J$ = 2-1 line, all of the four components were detected in 10 targets. Toward 16 objects, only the three components with the lowest velocities were detected, accounting for 98\\% of the total intensity. For the remaining 20 sources, only the main component was detected. Based on the above assumptions, 30\\% of the total intensity would be missed. The situation is different when the main component becomes broader. If the line width of the main component is larger than 10~km~s$^{-1}$, 87\\% of the total intensity is covered by the main spectral feature. When the line width of the main component is larger than 19.0~km~s$^{-1}$, then almost all HFS lines are included. In Fig.~\\ref{fig_model_hfs}, we show the dependence of the HFS factor for the $J$ = 2-1 line, $f_{\\rm 21HFS}$, on the line width of the main component. Depending on the specific condition of each target, we derived $f_{\\rm 21HFS}$ for each source. The values are listed in Table~\\ref{table_3433}. \n\nThe C$^{33}$S $J$ = 3-2 line consists of nine hyperfine components covering about 8.0 MHz \\citep{2005JMoSt.742..215M,2016JMoSp.327...95E}, corresponding to a velocity range of about 16~km~s$^{-1}$. Assuming that the intrinsic width of each HFS line is 1~km~s$^{-1}$, the $J$ = 3-2 line profile can be characterized by three components (see also Fig.~\\ref{fig_c33s_hfs}). All these three components are detected in only two sources, Orion-KL and DR21. The main component consists of four HFS lines ($F$=5\/2-3\/2, $F$=3\/2-1\/2, $F$=7\/2-5\/2, $F$=9\/2-7\/2), which account for 86\\% of the total intensity. When the line width becomes larger than 9.2~km~s$^{-1}$, almost all of the HFS lines overlap. The HFS factors ($f_{\\rm 32HFS}$) of the $J$ = 3-2 transition obtained individually for each source are listed in Table~\\ref{table_3433}.\n\nWe calculated the $^{34}$S\/$^{33}$S intensity ratios and present them in Table~\\ref{table_3433}. Applying corrections accounting for the effect of hyperfine splitting, the $^{34}$S\/$^{33}$S ratios are derived and are \nalso listed in Table~\\ref{table_3433}. The $^{34}$S\/$^{33}$S values obtained from the $J$ = 2-1 transitions are always higher than the ones derived from the $J$ = 3-2 lines toward the same source, with the exception of G097.53$+$03.18. This difference could be caused by the lack of corrections of optical depth in the $J$ = 3-2 transition. The average values of $^{34}$S\/$^{33}$S toward our sample are 4.35~$\\pm$~0.44 and 3.49~$\\pm$~0.26 in the $J$ = 2-1 and $J$ = 3-2 transitions, respectively. The $^{34}$S\/$^{33}$S ratios were found to be independent of galactocentric distance (Fig.~\\ref{fig_all33S}). \n\nAfter applying the opacity correction of the $J$ = 2-1 transition to the $J$ = 3-2 line in the same source, $^{34}$S\/$^{33}$S ratios in the $J$ = 3-2 transition are higher than the $^{34}$S\/$^{33}$S values in the $J$ = 2-1 transition in seven targets, suggesting that the C$^{34}$S $J$ = 3-2 lines in these seven sources are less opaque than the C$^{34}$S $J$ = 2-1 lines. The seven targets are the $+$20~km~s$^{-1}$~cloud, G023.43$-$00.18, G028.39$+$00.08, G028.83$-$00.25, G073.65$+$00.19, G097.53$+$03.18, and G109.87$+$02.11. A comparison of the corrected $^{34}$S\/$^{33}$S ratios in these two transitions is shown in Fig.~\\ref{fig_33S_compared}. On the other hand, toward the other 32 objects of the whole sample of 39 sources with detections in these two transitions, the $^{34}$S\/$^{33}$S ratios in the $J$ = 3-2 transition are still lower than the $^{34}$S\/$^{33}$S values in the $J$ = 2-1 transitions. This suggests that the C$^{34}$S $J$ = 3-2 lines may be more opaque than the C$^{34}$S $J$ = 2-1 lines. The ratios of $^{34}$S\/$^{33}$S values without corrections of opacity in the $J$ = 3-2 transition and the corrected $^{34}$S\/$^{33}$S values from the $J$ = 2-1 lines in 31 targets of these 32 sources are within the range of 1.02 to 1.71, which suggest that the optical depths of C$^{34}$S in the $J$ = 3-2 transition in these objects range from  0.05 to 1.20 based on equation (4). The maximum optical depth of the C$^{34}$S $J$ = 3-2 line toward the additional source, G024.85$+$00.08, which has not considered until now, is estimated to be 2.6.\n\n\n\n\\begin{figure*}[h]\n\\centering\n   \\includegraphics[width=500pt]{figure\/hfs_c33s.pdf}\n     \\caption{Synthetic C$^{33}$S (2-1) and C$^{33}$S (3-2) spectra for two intrinsic line widths, 1.0~km~s$^{-1}$ (left panels) and 4.0~km~s$^{-1}$ (right panels).}\n  \\label{fig_c33s_hfs}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\\centering\n   \\includegraphics[width=230pt]{figure\/plot_fHFS21.pdf}\n   \\includegraphics[width=230pt]{figure\/plot_fHFS32.pdf}\n     \\caption{Blue lines are curves showing the theoretical dependencies of the HFS factors, $f_{\\rm 21HFS}$ (left) and $f_{\\rm 32HFS}$ (right), on the line width of the C$^{33}$S main component for sources in which only the main component was detected. The red dotted vertical and horizontal lines indicate the values of the minimal line widths where the factors are reaching almost 1.0, 19.0~km~s$^{-1}$ for the $J$ = 2-1 transition ($f_{\\rm 21HFS} \\sim$ 0.99) and 9.2~km~s$^{-1}$ for the $J$ = 3-2 transition ($f_{\\rm 32HFS} \\sim$ 0.99). }\n  \\label{fig_model_hfs}\n\\end{figure*}\n\n\n\n\n\n\n\n\\subsection{$^{32}$S\/$^{33}$S ratios}\n\\label{section_32s33s}\n\nThe $^{32}$S\/$^{33}$S values can also be derived from the $^{34}$S\/$^{33}$S ratios in Section~\\ref{section_34s33s} using the $^{32}$S\/$^{34}$S ratios ---which we directly obtained from $^{13}$CS\/$^{13}$C$^{34}$S and present in Section~\\ref{section_ratios3234_13c34s}--- by applying the following equation:\n\\begin{equation}\n\\frac{^{32}{\\rm S}}{^{33}{\\rm S}} = \\frac{^{34}\\rm S}{^{33}\\rm S} \\times \\frac{^{32}\\rm S}{^{34}\\rm S} .\n\\end{equation}\nFor sources where we did not detect $^{13}$C$^{34}$S, the $^{32}$S\/$^{34}$S ratios derived from the double isotope method in Section~\\ref{section_double_32s34s} are used. Their uncertainty is also included in our error budget. The resulting $^{32}$S\/$^{33}$S ratios are listed in Table~\\ref{table_3233}. As in the case of the $^{34}$S\/$^{33}$S ratios, the $^{32}$S\/$^{33}$S values obtained from the $J$ = 2-1 transitions with corrections for optical depth are slightly larger than the ones from the $J$ = 3-2 transitions without opacity corrections toward the same source. \n\nIn the CMZ, $^{32}$S\/$^{33}$S $J$ = 2-1 ratios toward four targets, namely the $+$20 km~s$^{-1}$ cloud, the $+$50 km~s$^{-1}$ cloud, Sgr B2, and Sgr D, lead to an average value of 70 $\\pm$ 16. In the inner Galaxy, in a galactocentric distance range of 2.0~kpc~$\\le R_{\\rm GC} \\le$~6.0~kpc, an average $^{32}$S\/$^{33}$S ratio of 82 $\\pm$ 19 was derived from values in 20 sources. Near the solar neighborhood, in a galactocentric distance range of 7.5~kpc~$\\le R_{\\rm GC} \\le$~8.5~kpc, $^{32}$S\/$^{33}$S ratios were obtained toward four objects, Orion-KL, G071.31$+$00.82, DR21, and G109.87$+$02.11, resulting in an average value of 88 $\\pm$ 21. For the outer Galaxy, beyond the local arm, we were able to deduce an average $^{32}$S\/$^{33}$S ratio of 105 $\\pm$ 19  from $^{32}$S\/$^{33}$S values in four sources. All these average values provide us with an indication of the existence of a $^{32}$S\/$^{33}$S gradient in the Galaxy. An unweighted least-squares fit to the $J$ = 2-1 transition data from 46 sources yields\n\\begin{equation}\n^{32}{\\rm S}\/^{33}{\\rm S} (2-1) = (2.64 \\pm 0.77)R_{\\rm GC}+(70.80 \\pm 5.57).  \n\\end{equation}\nIn the $J$ = 3-2 transition with data from 42 targets, an unweighted least-squares fit can be obtained with\n\\begin{equation}\n^{32}{\\rm S}\/^{33}{\\rm S} (3-2) = (2.80 \\pm 0.59)R_{\\rm GC}+(59.30 \\pm 4.22).  \n\\end{equation}\nThe $^{32}$S\/$^{33}$S gradients derived from these two transitions have similar slopes but obviously different intercepts. The lower intercept in the $J$ = 3-2 transition could be due to the fact that we could not correct the values for optical-depth effects. If this difference is caused only by the opacity, then an average optical depth in the C$^{34}$S $J$ = 3-2 transition of about 0.4 can be derived with equation (4). \n\n\\setcounter{table}{3}\n\n\\begin{table*}[h]\n\\centering\n\\caption{\\label{table_3233}  $^{32}$S\/$^{33}$S isotope ratios}\n\\begin{tabular}{lccc}\n\\hline\\hline\nSource &  $R_{GC}$  &   \\multicolumn{2}{c}{$^{32}$S\/$^{33}$S}  \\\\\n       &  (kpc)     &  $J$ = 2-1    &     $J$ = 3-2       \\\\\n\\hline\nWB89-380 & 14.19 $\\pm$ 0.92                & 90.74  $\\pm$ 19.53  &   83.93  $\\pm$ 17.90  \\\\\nWB89-391 & 14.28 $\\pm$ 0.94                & 127.07 $\\pm$ 40.26  &   65.46  $\\pm$ 14.69  \\\\\nW3OH & 9.64 $\\pm$ 0.03                     & 105.49 $\\pm$ 7.26   &   74.20  $\\pm$ 4.35   \\\\\nOrion-KL & 8.54 $\\pm$ 0.00                 & 74.50  $\\pm$ 16.85  &   69.12  $\\pm$ 13.51  \\\\\n$+$20 km~s$^{-1}$ cloud & 0.03 $\\pm$ 0.03  & 82.63  $\\pm$ 18.20  &   78.98  $\\pm$ 17.86  \\\\\nG359.61$-$00.24 & 5.51 $\\pm$ 0.15          & 75.75  $\\pm$ 14.99  &   71.69  $\\pm$ 16.86  \\\\\n$+$50 km~s$^{-1}$ cloud & 0.02 $\\pm$ 0.04  & 76.49  $\\pm$ 16.96  &   66.68  $\\pm$ 14.92  \\\\\nG000.31$-$00.20 & 5.25 $\\pm$ 0.36          & 55.61  $\\pm$ 14.84  &   $\\cdots$           \\\\\nSgrB2 & 0.44 $\\pm$ 0.70                    & 42.98  $\\pm$ 9.62   &   35.66  $\\pm$ 7.84   \\\\\nSgrD & 0.45 $\\pm$ 0.07                     & 77.48  $\\pm$ 20.88  &   50.61  $\\pm$ 13.66  \\\\\nG006.79$-$00.25 & 4.75 $\\pm$ 0.25          & 86.65  $\\pm$ 17.01  &   70.08  $\\pm$ 13.79  \\\\\nG007.47$+$00.05 & 12.35 $\\pm$ 2.49         & 107.98 $\\pm$ 30.91  &   $\\cdots$           \\\\\nG010.32$-$00.15 & 5.34 $\\pm$ 0.29          & 88.27  $\\pm$ 17.82  &   79.56  $\\pm$ 15.99  \\\\\nG010.62$-$00.33 & 4.26 $\\pm$ 0.21          & 106.56 $\\pm$ 33.99  &   $\\cdots$           \\\\\nG011.10$-$00.11 & 5.49 $\\pm$ 0.56          & 72.13  $\\pm$ 29.12  &   $\\cdots$           \\\\\nG016.86$-$02.15 & 5.97 $\\pm$ 0.47          & 86.63  $\\pm$ 17.00  &   76.23  $\\pm$ 15.01  \\\\\nG017.02$-$02.40 & 6.40 $\\pm$ 0.36          & 100.03 $\\pm$ 20.54  &   86.36  $\\pm$ 17.47  \\\\\nG017.63$+$00.15 & 6.77 $\\pm$ 0.04          & 73.12  $\\pm$ 23.04  &   $\\cdots$           \\\\\nG018.34$+$01.76 & 6.31 $\\pm$ 0.07          & 92.66  $\\pm$ 18.57  &   86.44  $\\pm$ 16.87  \\\\\nG019.36$-$00.03 & 5.58 $\\pm$ 0.49          & 94.55  $\\pm$ 18.78  &   80.16  $\\pm$ 15.89  \\\\\nG023.43$-$00.18 & 3.63 $\\pm$ 0.49          & 72.63  $\\pm$ 16.26  &   80.58  $\\pm$ 16.63  \\\\\nG024.78$+$00.08 & 3.51 $\\pm$ 0.15          & 70.88  $\\pm$ 14.22  &   55.64  $\\pm$ 11.12  \\\\\nG024.85$+$00.08 & 3.85 $\\pm$ 0.23          & 134.87 $\\pm$ 43.15  &   48.25  $\\pm$ 11.88  \\\\\nG028.30$-$00.38 & 4.71 $\\pm$ 0.26          & $\\cdots$            &   76.82  $\\pm$ 20.06  \\\\\nG028.39$+$00.08 & 4.83 $\\pm$ 0.17          & 85.30  $\\pm$ 17.17  &   76.11  $\\pm$ 15.02  \\\\\nG028.83$-$00.25 & 4.50 $\\pm$ 0.48          & 78.57  $\\pm$ 15.87  &   80.37  $\\pm$ 15.83  \\\\\nG030.70$-$00.06 & 4.20 $\\pm$ 0.10          & 76.54  $\\pm$ 15.18  &   62.68  $\\pm$ 12.57  \\\\\nG030.74$-$00.04 & 5.76 $\\pm$ 0.36          & 86.28  $\\pm$ 17.07  &   72.73  $\\pm$ 14.12  \\\\\nG030.78$+$00.20 & 4.19 $\\pm$ 0.05          & $\\cdots$            &   72.50  $\\pm$ 17.46  \\\\\nG030.81$-$00.05 & 5.73 $\\pm$ 0.24          & 88.21  $\\pm$ 17.23  &   72.72  $\\pm$ 14.52  \\\\\nG031.24$-$00.11 & 7.48 $\\pm$ 2.00          & $\\cdots$            &   90.57  $\\pm$ 21.01  \\\\\nG032.74$-$00.07 & 4.55 $\\pm$ 0.23          & 75.67  $\\pm$ 14.99  &   68.63  $\\pm$ 13.88  \\\\\nG032.79$+$00.19 & 5.26 $\\pm$ 1.57          & 79.87  $\\pm$ 16.07  &   76.72  $\\pm$ 14.93  \\\\\nG034.41$+$00.23 & 5.99 $\\pm$ 0.06          & 80.54  $\\pm$ 15.82  &   72.63  $\\pm$ 14.63  \\\\\nG034.79$-$01.38 & 6.21 $\\pm$ 0.09          & 103.60 $\\pm$ 22.03  &   82.58  $\\pm$ 16.21  \\\\\nG036.11$+$00.55 & 5.45 $\\pm$ 0.43          & 39.43  $\\pm$ 13.64  &   $\\cdots$           \\\\\nG037.42$+$01.51 & 6.78 $\\pm$ 0.05          & 110.92 $\\pm$ 21.76  &   76.06  $\\pm$ 17.77  \\\\\nG040.28$-$00.21 & 6.02 $\\pm$ 0.10          & 110.40 $\\pm$ 31.44  &   71.86  $\\pm$ 17.32  \\\\\nG045.45$+$00.06 & 6.41 $\\pm$ 0.50          & 83.04  $\\pm$ 21.44  &   76.44  $\\pm$ 16.33  \\\\\nG048.99$-$00.29 & 6.18 $\\pm$ 0.02          & 107.90 $\\pm$ 27.60  &   87.21  $\\pm$ 20.50  \\\\\nW51-IRS2 & 6.22 $\\pm$ 0.06                 & 74.69  $\\pm$ 14.46  &   66.43  $\\pm$ 12.72  \\\\\nG071.31$+$00.82 & 8.02 $\\pm$ 0.16          & 95.37  $\\pm$ 31.90  &   106.87 $\\pm$ 30.60  \\\\\nG073.65$+$00.19 & 13.54 $\\pm$ 2.90         & 102.16 $\\pm$ 32.61  &   128.66 $\\pm$ 31.17  \\\\\nG075.29$+$01.32 & 10.69 $\\pm$ 0.58         & 122.06 $\\pm$ 29.02  &   92.08  $\\pm$ 19.53  \\\\\nDR21 & 8.10 $\\pm$ 0.00                     & 86.80  $\\pm$ 16.37  &   80.51  $\\pm$ 15.38  \\\\\nG090.92$+$01.48 & 10.13 $\\pm$ 0.63         & 87.88  $\\pm$ 19.68  &   95.73  $\\pm$ 20.14  \\\\\nG097.53$+$03.18 & 11.81 $\\pm$ 0.70         & 74.99  $\\pm$ 14.50  &   87.92  $\\pm$ 16.65  \\\\\nG109.87$+$02.11 & 8.49 $\\pm$ 0.01          & 94.41  $\\pm$ 18.40  &   98.52  $\\pm$ 23.17  \\\\\nNGC7538 & 9.47 $\\pm$ 0.07                  & 104.53 $\\pm$ 19.54  &   $\\cdots$           \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{The $^{32}$S\/$^{33}$S isotope ratios from the $J$ = 2-1 transition are corrected for the line saturation effects, while the ones in the 3-2 line are not corrected because of the missing main isotopic species.}\n\\end{table*}\n\n\n\n\\subsection{$^{34}$S\/$^{36}$S ratios}\n\\label{section_34s36s}\n\nThe detection of C$^{36}$S in the $J$ = 2-1 and $J$ = 3-2 transitions allows us to also calculate $^{34}$S\/$^{36}$S ratios. Around the CMZ, the C$^{36}$S $J$ = 2-1 line in the $+$50 km~s$^{-1}$ cloud was detected, resulting in a $^{34}$S\/$^{36}$S ratio of 41~$\\pm$~4. In the local arm toward DR21 and Orion-KL \\citep{2007A&A...474..515M,2013ApJ...769...15X}, the C$^{36}$S $J$ = 2-1 and $J$ = 3-2 transitions were detected, respectively, leading to $^{34}$S\/$^{36}$S values of 117~$\\pm$~24 and 83~$\\pm$~7. This yields an average $^{34}$S\/$^{36}$S ratio of 100~$\\pm$~16 in the ISM near the Sun. In the Perseus arm beyond the Solar System \\citep{2006Sci...311...54X}, we detected the C$^{36}$S $J$ = 2-1 line toward W3OH and obtained a $^{34}$S\/$^{36}$S value of 140~$\\pm$~16. These results reveal the possibility of the existence of a $^{34}$S\/$^{36}$S gradient from the Galactic center region to the outer Galaxy. Five tentative detections in the C$^{36}$S $J$ = 2-1 line and also five tentative detections in the $J$ = 3-2 line provide additional $^{34}$S\/$^{36}$S ratios but with large uncertainties. All of the $^{34}$S\/$^{36}$S values are listed in Table~\\ref{table_all36s} and are plotted as a function of the distance to the Galactic center in Fig.~\\ref{fig_all36S}. \n\n\n\\subsection{$^{32}$S\/$^{36}$S ratios}\n\\label{section_32s36s}\n\nAs in the case of the $^{32}$S\/$^{33}$S ratios, the $^{32}$S\/$^{36}$S values could also be obtained from the resulting $^{34}$S\/$^{36}$S ratios in Section~\\ref{section_34s36s} and the $^{32}$S\/$^{34}$S ratios in Section~\\ref{section_ratios3234_13c34s}  using the following equation:\n\\begin{equation}\n\\frac{^{32}{\\rm S}}{^{36}{\\rm S}} = \\frac{^{34}\\rm S}{^{36}\\rm S} \\times \\frac{^{32}\\rm S}{^{34}\\rm S}.\n\\end{equation}\nThe uncertainties of both isotope ratios in the product on the right-hand side of the equation are included in our error budget. The resulting $^{32}$S\/$^{36}$S ratios are listed in Table~\\ref{table_all36s}. In the CMZ, a $^{32}$S\/$^{36}$S ratio of 884~$\\pm$~104 is obtained toward the $+$50 km~s$^{-1}$ cloud. $^{32}$S\/$^{36}$S values of 1765~$\\pm$~414 and 3223~$\\pm$~742 are derived toward Orion-KL and DR21, leading to an average $^{32}$S\/$^{36}$S value of 2494 $\\pm$ 578 in the local regions near the Solar System. In the Perseus arm beyond the solar neighborhood toward W3OH, we obtain the highest $^{32}$S\/$^{36}$S value, 4181~$\\pm$~531. All these results indicate that there could be a positive $^{32}$S\/$^{36}$S gradient from the Galactic center region to the outer Galaxy. Figure~\\ref{fig_all36S} shows the $^{32}$S\/$^{36}$S ratios plotted as a function of the distance to the Galactic center.\n\n\n\\begin{figure*}[h]\n\\centering\n   \\includegraphics[height=620pt]{newFigures\/0Allratio33s-corrected.pdf}\n     \\caption{ $^{34}$S\/$^{33}$S and $^{32}$S\/$^{33}$S isotope ratios (for the latter, see equations (12) and (13)) plotted as functions of the distance from the Galactic center. \\textbf{Top:}  $^{34}$S\/$^{33}$S ratios derived from C$^{34}$S\/C$^{33}$S in the $J$ = 2-1 and $J$ = 3-2 transitions plotted as black and green dots, respectively. The red solid and the two dashed lines show the average value and its standard deviation, 4.35~$\\pm$~0.44, of $^{34}$S\/$^{33}$S with corrections of optical depth toward our sample in the $J$ = 2-1 transition. The yellow solid and the two dashed lines show the average value and its standard deviation, 3.49~$\\pm$~0.26, of $^{34}$S\/$^{33}$S toward our sample in the $J$ = 3-2 transition. The red symbol $\\odot$ indicates the $^{34}$S\/$^{33}$S isotope ratio in the Solar System. \\textbf{Bottom:} Black and green dots show the $^{32}$S\/$^{33}$S ratios in the $J$ = 2-1 and $J$ = 3-2 transitions, respectively. The red symbol $\\odot$ indicates the $^{32}$S\/$^{33}$S value in the Solar System. The resulting first-order polynomial fits to the $^{32}$S\/$^{33}$S ratios in the $J$ = 2-1 and $J$ = 3-2 transitions in this work are plotted as red and green solid lines, respectively, with the pink and yellow shaded area showing the 1~$\\sigma$ standard deviations. The red crosses are the results from the GCE model of \\citet[][see also Section~\\ref{section_discussion_model}]{2011MNRAS.414.3231K,2020ApJ...900..179K}. }\n  \\label{fig_all33S}\n\\end{figure*}\n\n\\begin{figure}[h]\n\\centering\n   \\includegraphics[height=270pt]{newFigures\/0Allratio33sCompared2.pdf}\n     \\caption{Comparison of $^{34}$S\/$^{33}$S ratios between the $J$ = 2-1 and 3-2 data. The $J$ = 2-1 ratios are opacity corrected, while the same correction factors were also applied to the $J$ = 3-2 data. The red solid line indicates that the $^{34}$S\/$^{33}$S ratios are equal in these two transitions.}\n  \\label{fig_33S_compared}\n\\end{figure}\n\n\\begin{figure*}[h]\n\\centering\n   \\includegraphics[height=550pt]{newFigures\/0Allratio36s-corrected.pdf}\n     \\caption{ $J$ = 2-1 (opacity corrected) and $J$ = 3-2 (no opacity corrections) $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S isotope ratios (see equations (18) and (19)) plotted as functions of the distance from the Galactic center. \\textbf{Top:} Filled black circles and filled black triangle present the $^{34}$S\/$^{36}$S ratios in the $J$ = 2-1 and $J$ = 3-2 transitions derived from C$^{34}$S\/C$^{36}$S in this work with detections of C$^{36}$S, respectively. The open gray circles and open gray triangles present the $^{34}$S\/$^{36}$S ratios in the $J$ = 2-1 and $J$ = 3-2 transitions derived from C$^{34}$S\/C$^{36}$S in this work with tentative detections of C$^{36}$S, respectively. The blue diamonds show the $^{34}$S\/$^{36}$S ratios in \\citet{1996A&A...313L...1M}. The red symbol $\\odot$ indicates the $^{34}$S\/$^{36}$S isotope ratio in the Solar System. The resulting first-order polynomial fit to the $^{34}$S\/$^{36}$S ratios in \\citet{1996A&A...313L...1M} and this work, excluding the values from tentative detections, is plotted as a black solid line, with the green shaded area showing the 1~$\\sigma$ interval of the fit. \\textbf{Bottom:}  $^{32}$S\/$^{36}$S ratios obtained from $^{34}$S\/$^{36}$S ratios combined with the $^{32}$S\/$^{34}$S ratios derived in this work. The filled black circles and filled black triangle present the values in the $J$ = 2-1 and $J$ = 3-2 transitions from this work with detections of C$^{36}$S, respectively. The open gray circles and open gray triangles present ratios in the $J$ = 2-1 and $J$ = 3-2 transitions from this work with tentative detections of C$^{36}$S, respectively. The $^{32}$S\/$^{36}$S ratios derived with $^{34}$S\/$^{36}$S values from \\citet{1996A&A...313L...1M} are plotted as blue diamonds. The red symbol $\\odot$ indicates the $^{32}$S\/$^{36}$S isotope ratio in the Solar System. The $^{32}$S\/$^{36}$S gradient, excluding tentative detections, is plotted as a black solid line, with the pink shaded area showing the 1~$\\sigma$ interval of the fit. The red crosses visualize results from the GCE model of \\citet[][see Section~\\ref{section_discussion_model}]{2011MNRAS.414.3231K,2020ApJ...900..179K}. }\n  \\label{fig_all36S}\n\\end{figure*}\n\n\\begin{table*}[h]\n\\centering\n\\caption{\\label{table_all36s}Isotope ratios of $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S}\n\\begin{tabular}{lc|cc|cc}\n\\hline\\hline\nSource  &   $R_{\\rm GC}$   &  \\multicolumn{2}{c}{$^{34}$S\/$^{36}$S}   &  \\multicolumn{2}{c}{$^{32}$S\/$^{36}$S}    \\\\\n        &    (kpc)           &    $J$ = 2-1    &   $J$ = 3-2          &    $J$ = 2-1    &   $J$ = 3-2            \\\\\n\\hline\nW3OH            & 9.64 $\\pm$ 0.03   &  140 $\\pm$ 16  &  174 $\\pm$ 29\\tablefootmark{*}                   &  4181 $\\pm$ 531  &  5195 $\\pm$ 919\\tablefootmark{*} \\\\\nOrion-KL        & 8.54 $\\pm$ 0.00   &  92  $\\pm$ 53\\tablefootmark{*}  &  83  $\\pm$ 7                    &  1954 $\\pm$ 1207\\tablefootmark{*}  &  1765 $\\pm$ 414  \\\\\n$+$20 km~s$^{-1}$ cloud         & 0.03 $\\pm$ 0.03   &  69  $\\pm$ 11\\tablefootmark{*}   &        $\\cdots$                       &  1015 $\\pm$ 278\\tablefootmark{*}   &        $\\cdots$     \\\\\n$+$50 km~s$^{-1}$ cloud     & 0.02 $\\pm$ 0.04   &  41  $\\pm$ 4   &        $\\cdots$                                        &  884 $\\pm$ 104   &        $\\cdots$     \\\\\nG024.78$+$00.08 & 3.51 $\\pm$ 0.15   &  151 $\\pm$ 39\\tablefootmark{*}  &  226 $\\pm$ 66\\tablefootmark{*}  &  2424 $\\pm$ 699\\tablefootmark{*}  &  3640 $\\pm$ 1164\\tablefootmark{*} \\\\\nG030.81$-$00.05 & 5.73 $\\pm$ 0.24   &  55  $\\pm$ 15\\tablefootmark{*}  &       $\\cdots$                        &  829 $\\pm$ 233\\tablefootmark{*}  &       $\\cdots$      \\\\\nW51-IRS2        & 6.22 $\\pm$ 0.06   &  109 $\\pm$ 27\\tablefootmark{*}  &  203 $\\pm$ 45\\tablefootmark{*}  &  1814 $\\pm$ 506\\tablefootmark{*}  &  3397 $\\pm$ 855\\tablefootmark{*} \\\\\nDR21            & 8.10  $\\pm$ 0.00   &  117 $\\pm$ 24  &  159 $\\pm$ 64\\tablefootmark{*}                  &  3223 $\\pm$ 742  &  4384 $\\pm$ 1829\\tablefootmark{*} \\\\\nNGC7538         & 9.47 $\\pm$ 0.07   &       $\\cdots$       &  109 $\\pm$ 28\\tablefootmark{*}                   &       $\\cdots$       &  2912 $\\pm$ 897\\tablefootmark{*} \\\\\n\\hline\n\\multicolumn{6}{c}{Below are the isotope ratios of $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S from \\citet{1996A&A...313L...1M}} \\\\\n\\hline\nW3OH                         & 9.64 $\\pm$ 0.03   &   $\\cdots$       &  181 $\\pm$ 49  &  $\\cdots$       &  4118 $\\pm$ 1124  \\\\\nOrion-KL                     & 8.54 $\\pm$ 0.00   &   $\\cdots$       &  104 $\\pm$ 7   &  $\\cdots$       &  2281 $\\pm$ 174 \\\\\nIRAS15491\\tablefootmark{**}  & 5.48 $\\pm$ 0.31   &   $\\cdots$       &  108 $\\pm$ 15  &  $\\cdots$       &  2120 $\\pm$ 307 \\\\\nIRAS15520\\tablefootmark{**}  & 5.67 $\\pm$ 0.31   &   128 $\\pm$ 32   &  133 $\\pm$ 11  & 2531 $\\pm$ 641  &  2629 $\\pm$ 243 \\\\\nIRAS16172\\tablefootmark{**}  & 5.03 $\\pm$ 0.29   &   121 $\\pm$ 18   &  119 $\\pm$ 14  & 2334 $\\pm$ 361  &  2296 $\\pm$ 287 \\\\\nNGC6334A                     & 6.87 $\\pm$ 0.12   &   108 $\\pm$ 15   &  154 $\\pm$ 20  & 2232 $\\pm$ 322  &  3183 $\\pm$ 431 \\\\\nNGC6334B                     & 6.87 $\\pm$ 0.12   &   163 $\\pm$ 46   &  156 $\\pm$ 26  & 3369 $\\pm$ 960  &  3225 $\\pm$ 552 \\\\\nW51(M)                       & 6.22 $\\pm$ 0.004  &   105 $\\pm$ 11   &  104 $\\pm$ 15  & 2119 $\\pm$ 237  &  2099 $\\pm$ 313 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{\\tablefoottext{*}{Values with large uncertainties are derived from the tentative detection of C$^{36}$S lines.} \\tablefoottext{**}{For these three sources without parallax data, their kinematic distances were estimated (for details, see Section~\\ref{section_distance}).} The $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S isotope ratios in this work from the $J$ = 2-1 transition are corrected for the optical depth effects, while the ones for the 3-2 line could not be corrected.}\n\\end{table*}\n\n\\section{Discussion}\n\\label{discussion}\n\nWith the measurements of optically thin lines of the rare CS isotopologs, C$^{34}$S, $^{13}$CS, C$^{33}$S, $^{13}$C$^{34}$S, and C$^{36}$S, we derived $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{34}$S\/$^{33}$S, $^{32}$S\/$^{33}$S, $^{34}$S\/$^{36}$S, and $^{32}$S\/$^{36}$S isotope ratios. Combined with accurate galactocentric distances, we established a $^{32}$S\/$^{33}$S gradient for the first time and confirmed the existing gradients of $^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S, as well as uniform $^{34}$S\/$^{33}$S ratios across the Milky Way, which are lower than previously reported. Furthermore, we may have detected  $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S gradients  for the first time. In Section~\\ref{section_comparisons_12c13c}, we compare the $^{12}$C\/$^{13}$C gradient obtained in this work with previous published ones derived from a variety of molecular species. A comparison between the $^{32}$S\/$^{34}$S gradients we obtained and previously published ones is presented in Section~\\ref{section_comparisons_32s34s}. The condition of LTE for C$^{33}$S with its HFS line ratios is discussed in Section~\\ref{section_hfs_c33s}. We then also compare our results on $^{34}$S\/$^{33}$S ratios with previously published values and discuss the $^{32}$S\/$^{33}$S gradient. In Section~\\ref{section_discussion_all36} we evaluate whether or not $^{34}$S\/$^{36}$S, $^{33}$S\/$^{36}$S, and $^{32}$S\/$^{36}$S ratios may show gradients with galactocentric distance. Observational bias due to distance effects, beam size effects, and chemical fractionation are discussed. A comparison of several isotopes with respect to primary or secondary synthesis is provided in Section~\\ref{section_discussion_all}. Results from a Galactic chemical evolution (GCE) model, trying to simulate the observational data, are presented in Section~\\ref{section_discussion_model}.\n\n\n\n\n\n\\subsection{Comparisons of $^{12}$C\/$^{13}$C ratios determined with different species}\n\\label{section_comparisons_12c13c}\n\n$^{12}$C\/$^{13}$C ratios have been well studied in the CMZ where the value is about 20--25 \\citep[e.g.,][]{1983A&A...127..388H,1985A&A...149..195G,1990ApJ...357..477L,2005ApJ...634.1126M,2010A&A...523A..51R,2013A&A...559A..47B,2017ApJ...845..158H,2020A&A...642A.222H}, similar to the results that we obtained from C$^{34}$S in the current work. In the inner Galaxy, the $^{12}$C\/$^{13}$C ratios are higher than the values in the CMZ, which were $\\sim$50 as derived from H$_2^{12}$CO\/H$_2^{13}$CO in \\citet{1985A&A...143..148H} and 41~$\\pm$~9 from C$^{34}$S\/$^{13}$C$^{34}$S in this work. As can be inferred from Fig.~\\ref{fig_gradient_12C13C}, $^{12}$C\/$^{13}$C ratios at 3.0 kpc $\\le R_{\\rm GC} \\le$ 4.0 kpc might be as low as in the CMZ, but this is so far tentative and uncertain, as we only have one detection (G024.78$+$00.08) in this region and data from other groups referring to this small galactocentric interval are also relatively few. In the local regions near the Solar System, an average $^{12}$C\/$^{13}$C ratio of 66~$\\pm$~10 was obtained from C$^{34}$S\/$^{13}$C$^{34}$S in this work, which is consistent with $^{12}$C\/$^{13}$C values derived from other molecular species and their $^{13}$C isotopes, that are 75~$\\pm$~9 from C$^{18}$O \\citep{1998ApJ...494L.107K}, 60~$\\pm$~19 from CN \\citep{2005ApJ...634.1126M}, 74~$\\pm$~8 from CH$^+$ \\citep{2011ApJ...728...36R}, and 53~$\\pm$~16 from H$_2$CO \\citep{2019ApJ...877..154Y}. All these $^{12}$C\/$^{13}$C values for the solar neighborhood are well below the value for the Sun \\citep[89,][]{1989GeCoA..53..197A,2007ApJ...656L..33M}. This indicates that $^{13}$C has been enriched in the local ISM during the last 4.6 billion years following the  formation of the Solar System. Beyond the Sun, at a galactocentric distance of about 10~kpc, our results from C$^{34}$S\/$^{13}$C$^{34}$S show a slightly higher value of 67~$\\pm$~8, which is similar to $^{12}$C\/$^{13}$C ratios from CN (66~$\\pm$~20), C$^{18}$O (69~$\\pm$~10), and H$_2$CO (64~$\\pm$~10). These values are still below the value for the Sun. In the far outer Galaxy at 13.8~kpc toward WB89~437, \\citet{1996A&AS..119..439W} found a 3$\\sigma$ lower limit of 201~$\\pm$~15 from C$^{18}$O\/$^{13}$C$^{18}$O. This suggests that the $^{12}$C\/$^{13}$C gradient  extends well beyond the solar neighborhood to the outer Galaxy. Additional sources with large galactocentric distances have to be measured to further improve the statistical significance of this result.\n\nPreviously published $^{12}$C\/$^{13}$C ratios derived from CN \\citep{2002ApJ...578..211S,2005ApJ...634.1126M}, C$^{18}$O \\citep{1990ApJ...357..477L,1996A&AS..119..439W,1998ApJ...494L.107K}, H$_2$CO \\citep{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H,2019ApJ...877..154Y}, CH$^+$ \\citep{2011ApJ...728...36R}, and CH \\citep{2020A&A...640A.125J} are shown in Fig.~\\ref{fig_gradient_12C13C}, but with respect to the new distance values (see details in Section~\\ref{section_distance}). In Fig.~\\ref{distribution_12C13C}, the $^{12}$C\/$^{13}$C values from different molecular species are projected onto the Galactic plane. This also  visualizes the gradient from the Galactic center to the Galactic outer regions beyond the Solar System. In Table~\\ref{table_all12C13Cgradients}, the fitting results for the old and new distances are presented. The comparison shows that the adoption of the new distances has indeed an effect on the fitting results, such as for the $^{12}$CN\/$^{13}$CN gradient. In \\citet{2002ApJ...578..211S} and \\citet{2005ApJ...634.1126M}, the slope and intercept become (6.75~$\\pm$~1.44) and (5.77~$\\pm$~11.29) from (6.01~$\\pm$~1.19) and (12.28~$\\pm$~9.33), respectively. The fitting for H$_2^{12}$CO\/H$_2^{13}$CO from  \\citet{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H} and \\citet{2019ApJ...877..154Y} becomes (5.43~$\\pm$~1.04) and (13.87~$\\pm$~6.38) from (5.08~$\\pm$~1.10) and (11.86~$\\pm$~6.60), respectively. The Galactic $^{12}$C\/$^{13}$C gradients derived based on measurements of CN, C$^{18}$O, and  H$_2$CO are in agreement with our results from C$^{34}$S and therefore indicate that chemical fractionation has little effect on the $^{12}$C\/$^{13}$C ratios. It is noteworthy that all these fits show a significant discrepancy with observations from the Galactic center. Indeed, they suggest values of 5--17 at $R_{\\rm GC}$=0, substantially below observed values of 20--25 (see also Tables~\\ref{table_all12C13Cgradients}, \\ref{table_allratios}). While the values in the CMZ are clearly lower than in the inner Galactic disk (with the potential exception at galactocentric distances of 2.0--4.0 kpc), they are larger than suggested by a linear fit encompassing the entire inner 12.0 kpc of the Galaxy.\n\n\n\n\\subsection{The $^{32}$S\/$^{34}$S gradient across the Milky Way}\n\\label{section_comparisons_32s34s}\n\nThe existence of a $^{32}$S\/$^{34}$S gradient was first proposed by \\citet{1996A&A...305..960C} based on observations of $^{13}$CS and C$^{34}$S $J$ = 2-1 lines toward 20 mostly southern HMSFRs with galactocentric distances of between 3.0 and 9.0 kpc. Very recently, \\citet{2020ApJ...899..145Y} confirmed the existence of this $^{32}$S\/$^{34}$S gradient and enlarged the sample of measurements of $^{13}$CS and C$^{34}$S $J$ = 2-1 lines to a total of 61 HMSFRs from the inner Galaxy out to a galactocentric distance of 12.0 kpc. In the CMZ, \\citet{2020A&A...642A.222H} found $^{32}$S\/$^{34}$S ratios of 16.3$^{+2.1}_{-1.7}$ and 17.9~$\\pm$~5.0 for the $+$50 km s$^{-1}$ cloud and Sgr B2(N), which is consistent with our values derived from $^{13}$C$^{34}$S and also with our results using the double isotope method. In the inner disk at 2.0~kpc $\\le R_{\\rm GC} \\le$ 6.0~kpc, a similar $^{32}$S\/$^{34}$S value of 18~$\\pm$~4 was derived based on our results in Sections~\\ref{section_ratios3234_13c34s} and \\ref{section_double_32s34s}. While $^{12}$C\/$^{13}$C ratios in the inner disk at $R_{\\rm GC} \\ge$ 4.0~kpc are clearly higher than in the CMZ (see details in Section~\\ref{section_comparisons_12c13c}), this is not the case for $^{32}$S\/$^{34}$S. On the contrary, $^{32}$S\/$^{34}$S ratios in the CMZ and inner disk are similar, as already suggested for the first time by \\citet{2020A&A...642A.222H}. In the local ISM, our results lead to an average $^{32}$S\/$^{34}$S ratio of 24~$\\pm$~4, which is close to the value in the Solar System \\citep[22.57,][]{1989GeCoA..53..197A}. That is also differing from the $^{12}$C\/$^{13}$C ratio and its clearly subsolar value in the local ISM. A more detailed discussion is given in Section~\\ref{section_discussion_all}. \n\n\nFor the first time, we established a $^{32}$S\/$^{34}$S gradient directly from measurements of $^{13}$CS and $^{13}$C$^{34}$S (see Section~\\ref{section_ratios3234_13c34s} for details). Similar $^{32}$S\/$^{34}$S gradients were also found in the $^{32}$S\/$^{34}$S values derived by the double isotope method with the $J$ = 2-1 and $J$ = 3-2 transitions (for details, see Section~\\ref{section_double_32s34s}). A gradient of $^{32}$S\/$^{34}$S = (0.75 $\\pm$ 0.13)$R_{\\rm GC}$+(15.52 $\\pm$ 0.78) was obtained based on a large dataset of 90 values from our detections of $^{13}$CS and C$^{34}$S $J$ = 2-1 lines with corrections of opacity, which is flatter than previous ones presented by \\citet{1996A&A...305..960C} and \\citet{2020ApJ...899..145Y}. Following \\citet{2020ApJ...899..145Y}, the gradient does not significantly change when the ratios in the CMZ or in the outer regions are excluded, indicating that the $^{32}$S\/$^{34}$S gradient is robust. \n\n\n\n\n\n\n\n\\begin{table*}[h]\n\\caption{Measurements of the $^{12}$C\/$^{13}$C gradient.}\n\\centering\n\\begin{tabular}{c|cc|cc}\n\\hline\\hline\n  &\\multicolumn{2}{c}{Previous fitting results} & \\multicolumn{2}{c}{This work} \\\\\n  &  slope  & intercept  &  slope &  intercept \\\\\n\\hline\n\\label{table_all12C13Cgradients}\nCN\\tablefootmark{a}        & 6.01 $\\pm$ 1.19 & 12.28 $\\pm$ 9.33 & 6.75 $\\pm$ 1.44 & 5.57 $\\pm$ 11.29 \\\\ \nC$^{18}$O\\tablefootmark{b} & 5.41 $\\pm$ 1.07 & 19.03 $\\pm$ 7.90 & 5.72 $\\pm$ 1.20 & 14.56 $\\pm$ 9.25 \\\\ \nH$_2$CO\\tablefootmark{c}   & 5.08 $\\pm$ 1.10 & 11.86 $\\pm$ 6.60 & 5.43 $\\pm$ 1.04 & 13.87 $\\pm$ 6.38 \\\\ \nC$^{34}$S (this work)      &     $\\cdots$    &     $\\cdots$     & 4.77 $\\pm$ 0.81 & 20.76 $\\pm$ 4.61 \\\\ \n\\hline\n\\end{tabular}\n\\tablefoot{Fitting results for the old (left) and new (right) distances, respectively. \n\\tablefoottext{a}{From \\citet{2002ApJ...578..211S} and \\citet{2005ApJ...634.1126M}.}\n\\tablefoottext{b}{From \\citet{1990ApJ...357..477L}, \\citet{1996A&AS..119..439W}, and \\citet{1998ApJ...494L.107K}.}\n\\tablefoottext{c}{From \\citet{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H} and \\citet{2019ApJ...877..154Y}.}}\n\\end{table*}\n\n\n\\begin{figure*}[h]\n\\centering\n   \\includegraphics[width=520pt]{newFigures\/MW_12C13C_corrected.png}\n     \\caption{Distribution of $^{12}$C\/$^{13}$C ratios from 93 sources in the Milky Way. The background image is the structure of the Milky Way from the artist's impression [Credit:\nNASA\/JPL-Caltech\/ESO\/R. Hurt]. The $^{12}$C\/$^{13}$C isotope ratios with corrections for optical depth from C$^{34}$S\/$^{13}$C$^{34}$S in this work are plotted as circles. The triangles, pentagons, stars, squares, and diamonds indicate the $^{12}$C\/$^{13}$C ratios derived from CN\/$^{13}$CN \\citep{2002ApJ...578..211S,2005ApJ...634.1126M}, C$^{18}$O\/$^{13}$C$^{18}$O \\citep{1990ApJ...357..477L,1996A&AS..119..439W,1998ApJ...494L.107K}, H$_2$CO\/H$_2^{13}$CO \\citep{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H,2019ApJ...877..154Y}, CH$^+$\/$^{13}$CH$^+$ \\citep{2011ApJ...728...36R}, and CH\/$^{13}$CH \\citep{2020A&A...640A.125J}, respectively. The red symbol $\\odot$ indicates the position of the Sun. The color bar on the right-hand side indicates the range of the $^{12}$C\/$^{13}$C ratios. }\n  \\label{distribution_12C13C}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\\centering\n   \\includegraphics[width=520pt]{newFigures\/MW_32S34S_corrected.png}\n     \\caption{Distribution of $^{32}$S\/$^{34}$S ratios from 112 sources in the Milky Way. The background image is the structure of the Milky Way from the artist's impression [Credit:\nNASA\/JPL-Caltech\/ESO\/R. Hurt]. The $^{32}$S\/$^{34}$S isotope ratios with corrections of opacity derived in this work from $^{13}$CS\/$^{13}$C$^{34}$S and the double isotope method in the $J$ = 2-1 transition are plotted as circles and stars, respectively. The triangles indicate the results from the CMZ obtained by \\citet{2020A&A...642A.222H}. The red symbol $\\odot$ indicates the position of the Sun. The color bar on the right-hand side indicates the range of the $^{32}$S\/$^{34}$S ratios.}\n  \\label{distribution_32S34S}\n\\end{figure*}\n\n\n\n\n\n\\subsection{C$^{33}$S}\n\\label{section_hfs_c33s}\n\n We detected at least three components of the C$^{33}$S $J$ = 2-1 line toward 26 sources. This will guide us to obtain information with respect to LTE or nonLTE conditions. As mentioned in Section~\\ref{section_34s33s}, the main component ($I_{main}$) consists of four HFS lines ($F$=7\/2-5\/2, $F$=5\/2-3\/2, $F$=1\/2-1\/2, $F$=3\/2-5\/2). Under LTE conditions in the optically thin case, the ratio between the other identified components (not belonging to the main one) and the main component can be obtained:\n\\begin{equation}\n\\begin{aligned}\nR_{211} &=  \\frac{I(F=3\/2-1\/2 + F=5\/2-5\/2)}{I_{main}}\\\\ &= 0.25,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\nR_{212} &=  \\frac{I(F=3\/2-3\/2)}{I_{main}}\\\\ &= 0.15,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\nR_{213} &=  \\frac{I(F=1\/2-3\/2)}{I_{main}}\\\\ &= 0.02.\n\\end{aligned}\n\\end{equation}\nA detailed examination of our 26 objects is presented in Table~\\ref{table_c33s_hfs}. Except for Orion-KL there is no evidence for nonLTE effects. All the remaining sources are found to be compatible with LTE conditions. The spectra of CS, C$^{34}$S, and $^{13}$CS toward Orion-KL show broad wings (see Fig.~\\ref{fig_orion}), which might lead to a highly complex C$^{33}$S $J$ = 2-1 line shape, which may be what is causing apparent LTE deviations in Orion-KL.  \n\n\n\n\n\\begin{figure}[h]\n\\centering\n   \\includegraphics[width=220pt]{figure\/Orion-KL-c33s.eps}\n     \\caption{Line profiles of the $J$ = 2-1 transitions of CS, C$^{33}$S, C$^{34}$S, and $^{13}$CS toward Orion-KL.}\n  \\label{fig_orion}\n\\end{figure}\n\n\nNo systematic dependence of the $^{34}$S\/$^{33}$S ratios on galactocentric distance was found in previous studies. The average $^{34}$S\/$^{33}$S values were 6.3~$\\pm$~1.0 and 5.9~$\\pm$~1.5 in \\citet{1996A&A...305..960C} and \\citet{2020ApJ...899..145Y}, respectively. A particularly low $^{34}$S\/$^{33}$S ratio of 4.3~$\\pm$~0.2 was found in the Galactic center region by \\citet{2020A&A...642A.222H}. These authors speculated about the possible presence of a gradient with low values at the center. However, in view of the correction for HFS in Section~\\ref{section_34s33s},  it appears that this is simply an effect of different HFS correction factors, with the one at the Galactic center with its wide spectral lines being 1.0, thus (exceptionally) not requiring a downward correction. \\citet{2020ApJ...899..145Y} considered the effect of HFS, while they overestimated the ratio of the main component to the total intensity of the C$^{33}$S $J$ = 2-1 line (for details, see their Section~4.2). This resulted in higher $^{34}$S\/$^{33}$S values. However, the $^{34}$S\/$^{33}$S ratios appear  to be independent of galactocentric distance based on our results (see details in Section~\\ref{section_34s33s}). The average $^{34}$S\/$^{33}$S value with corrections of optical depth toward our sample in the $J$ = 2-1 transition is 4.35~$\\pm$~0.44, which is similar to the value in the Galactic center region derived by \\citet{2020A&A...642A.222H} and lower than the value of 5.61 in the Solar System \\citep{1989GeCoA..53..197A}. This indicates that no systematic variation exists in the $^{34}$S\/$^{33}$S ratios in our Galaxy, that $^{33}$S is (with respect to stellar nucleosynthesis) similar to $^{34}$S, and that the Solar System (see Fig.~\\ref{fig_all33S}) must be peculiar. An approximately constant $^{34}$S\/$^{33}$S ratio with opacity correction from the $J$ = 2-1 transition across the Galactic plane leads to a $^{32}$S\/$^{33}$S gradient in our Galaxy as we already mentioned in Section~\\ref{section_32s33s}: $^{32}{\\rm S}\/^{33}{\\rm S}$ = $(2.64 \\pm 0.77)R_{\\rm GC}+(70.80 \\pm 5.57)$, with a correlation coefficient of 0.46.\n\n\n\n\n\\begin{table*}[h]\n\\caption{Line intensity ratios of C$^{33}$S $J$ = 2-1 hyperfine components.}\n\\centering\n\\begin{tabular}{lcccc}\n\\hline\\hline\nSource  &     FWHM  & $R_{211}$ & $R_{212}$ & $R_{213}$     \\\\\n        &    (km s$^{-1}$)           &           &           &                \\\\\n\\hline\n\\label{table_c33s_hfs}\nW3OH            &  4.41  & 0.29 $\\pm$ 0.04  &  0.15 $\\pm$ 0.03  &  0.05 $\\pm$ 0.03   \\\\ \nOrion-KL        &  4.25  & 0.60 $\\pm$ 0.02  &  0.30 $\\pm$ 0.01  &  0.44 $\\pm$ 0.02   \\\\ \nG359.61$-$00.24 &  3.17  & 0.40 $\\pm$ 0.05  &  0.62 $\\pm$ 0.05  &  0.14 $\\pm$ 0.04   \\\\ \nG006.79$-$00.25 &  2.73  & 0.28 $\\pm$ 0.02  &  0.17 $\\pm$ 0.02  &     $\\cdots$       \\\\ \nG010.32$-$00.15 &  2.66  & 0.27 $\\pm$ 0.04  &  0.23 $\\pm$ 0.06  &     $\\cdots$       \\\\ \nG016.86$-$02.15 &  3.32  & 0.30 $\\pm$ 0.05  &  0.18 $\\pm$ 0.03  &     $\\cdots$       \\\\ \nG017.02$-$02.40 &  3.74  & 0.17 $\\pm$ 0.03  &  0.17 $\\pm$ 0.08  &     $\\cdots$       \\\\ \nG018.34$+$01.76 &  2.30  & 0.24 $\\pm$ 0.07  &  0.13 $\\pm$ 0.06  &     $\\cdots$       \\\\ \nG019.36$-$00.03 &  3.02  & 0.29 $\\pm$ 0.06  &  0.14 $\\pm$ 0.04  &     $\\cdots$       \\\\ \nG023.43$-$00.18 &  3.81  & 0.40 $\\pm$ 0.14  &  1.15 $\\pm$ 0.34  &     $\\cdots$       \\\\ \nG024.78$+$00.08 &  4.29  & 0.34 $\\pm$ 0.03  &  0.21 $\\pm$ 0.03  &  0.08 $\\pm$ 0.03   \\\\ \nG028.39$+$00.08 &  3.04  & 0.26 $\\pm$ 0.04  &  0.24 $\\pm$ 0.06  &     $\\cdots$       \\\\ \nG028.83$-$00.25 &  2.27  & 0.25 $\\pm$ 0.04  &  0.18 $\\pm$ 0.03  &     $\\cdots$       \\\\ \nG030.70$-$00.06 &  4.85  & 0.33 $\\pm$ 0.02  &  0.23 $\\pm$ 0.03  &  0.11 $\\pm$ 0.02   \\\\ \nG030.74$-$00.04 &  3.55  & 0.19 $\\pm$ 0.03  &  0.25 $\\pm$ 0.06  &     $\\cdots$       \\\\ \nG030.81$-$00.05 &  6.15  & 0.27 $\\pm$ 0.03  &  0.16 $\\pm$ 0.02  &  0.14 $\\pm$ 0.02   \\\\ \nG032.74$-$00.07 &  4.54  & 0.36 $\\pm$ 0.04  &  0.16 $\\pm$ 0.03  &     $\\cdots$       \\\\ \nG032.79$+$00.19 &  4.78  & 0.09 $\\pm$ 0.05  &  1.72 $\\pm$ 0.28  &     $\\cdots$       \\\\ \nG034.41$+$00.23 &  4.56  & 0.32 $\\pm$ 0.04  &  0.28 $\\pm$ 0.04  &  0.12 $\\pm$ 0.03   \\\\ \nG034.79$-$01.38 &  2.40  & 0.44 $\\pm$ 0.15  &  0.14 $\\pm$ 0.06  &     $\\cdots$       \\\\ \nG037.42$+$01.51 &  3.18  & 0.20 $\\pm$ 0.05  &  0.10 $\\pm$ 0.02  &     $\\cdots$       \\\\ \nW51-IRS2        &  8.55  & 0.13 $\\pm$ 0.09  &  3.63 $\\pm$ 0.36  &  0.71 $\\pm$ 0.13   \\\\ \nDR21            &  2.71  & 0.28 $\\pm$ 0.01  &  0.20 $\\pm$ 0.01  &  0.14 $\\pm$ 0.01   \\\\ \nG097.53$+$03.18 &  5.22  & 0.30 $\\pm$ 0.05  &  0.13 $\\pm$ 0.04  &  0.14 $\\pm$ 0.04   \\\\ \nG109.87$+$02.11 &  3.19  & 0.23 $\\pm$ 0.06  &  0.38 $\\pm$ 0.07  &     $\\cdots$       \\\\ \nNGC7538         &  3.71  & 0.22 $\\pm$ 0.01  &  0.21 $\\pm$ 0.01  &     $\\cdots$       \\\\ \n\\hline\n\\end{tabular}\n\\tablefoot{ Full width at half maximum values of the main component were obtained from measurements of C$^{33}$S; see Table \\ref{fitting_all}. The errors provided here are 1$\\sigma$.\n}\n\\end{table*}\n\n\\subsection{C$^{36}$S}\n\\label{section_discussion_all36}\n\nAs mentioned in Section~\\ref{section_34s36s}, we find novel potential indications for a positive $^{34}$S\/$^{36}$S gradient with galactocentric radius. The $^{36}$S-bearing molecule C$^{36}$S was first detected by \\citet{1996A&A...313L...1M}. These authors observed the $J$ = 2-1 and 3-2 transitions toward eight Galactic molecular hot cores at galactocentric distances of between 5.0 kpc and 10.0 kpc.  \\citet{1996A&A...313L...1M} reported an average $^{34}$S\/$^{36}$S ratio of 115~$\\pm$~17, which is smaller than the value in the Solar System \\citep[200.5,][]{1989GeCoA..53..197A}. This is consistent with this nucleus being of  a purely secondary nature. Combining the ratios of \\citet{1996A&A...313L...1M} --- after applying new distances (see details in Table~\\ref{table_all36s})--- with our results in the $J$ = 2-1 transition, the following fit could be achieved:\n\\begin{equation}\n^{34}{\\rm S}\/^{36}{\\rm S} = (10.34 \\pm 2.74)R_{\\rm GC}+(57.45 \\pm 18.59), \n\\end{equation}\nwith a correlation coefficient of 0.71. As the $^{34}$S\/$^{33}$S ratios show a uniform distribution across our Galaxy (see details in Section~\\ref{section_hfs_c33s}), a $^{33}$S\/$^{36}$S gradient is also expected. We obtain (2.38~$\\pm$~0.67)$R_{\\rm GC}$+(13.21~$\\pm$~4.48). After applying our $^{32}$S\/$^{34}$S gradient to the $^{34}$S\/$^{36}$S ratios in \\citet{1996A&A...313L...1M} with equation (9), $^{32}$S\/$^{36}$S ratios were then derived and listed in Table~\\ref{table_all36s}. Combined with our results in the $J$ = 2-1 transition, a linear fit to the $^{32}$S\/$^{36}$S ratios is obtained:\n\\begin{equation}\n ^{32}{\\rm S}\/^{36}{\\rm S} = (314 \\pm 55)R_{\\rm GC}+(659 \\pm 374), \n \\end{equation}\nwith a correlation coefficient of 0.84. The $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S ratios are plotted as functions of galactocentric distances in Fig.~\\ref{fig_all36S}. Measurements of $^{34}$S\/$^{36}$S, $^{33}$S\/$^{36}$S, and $^{32}$S\/$^{36}$S are still not numerous. More sources with detected C$^{36}$S lines would be highly\ndesirable, especially in the CMZ and the inner disk within $R_{\\rm GC}$~=~5.0~kpc.\n\n\n\\subsection{Observational bias due to distance effects}\n\\label{section_discussion_distance}\n\nWhile we have so far analyzed isotope ratios as a function of galactocentric distances, there might be a bias in the sense that the ratios could at least in part also depend on the distance from Earth, a bias caused by different linear resolutions. In Appendix~\\ref{appendix_spectra}, the $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{34}$S\/$^{33}$S, and $^{32}$S\/$^{33}$S, as well as the $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S isotope ratios are plotted as functions of the distance from the Sun and shown in Figs.~\\ref{fig_12C13C_2Sun} to \\ref{fig_all36S_2Sun}, respectively. No apparent gradients can be found, which indicates that any observational bias on account of distance-dependent effects is not significant for $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{32}$S\/$^{33}$S, $^{34}$S\/$^{36}$S, and $^{32}$S\/$^{36}$S. This agrees with the findings of \\citet[][see their Section~4.5]{2020ApJ...899..145Y}.\n\n\n\\subsection{Beam size effects}\n\\label{section_beam}\n\nA good way to check whether the different beam sizes for different lines could affect our results is to compare the isotope ratios derived from different transitions at different frequencies observed with different beam sizes. As shown in Section~\\ref{section_double_32s34s}, $^{32}$S\/$^{34}$S ratios obtained from the double isotope method in the $J$ = 2-1 and 3-2 transitions are in good agreement, suggesting that the effect of beam size is negligible. Furthermore, \\citet{2020A&A...642A.222H} found an average $^{32}$S\/$^{34}$S ratio of 17.9~$\\pm$~5.0 in the envelope of Sgr B2(N) with the Atacama Large Millimetre\/submillimetre Array (ALMA) with 1$\\farcs$6 beam size, which is consistent with our results in the CMZ from the IRAM 30 meter telescope with beam sizes of about 27$\\arcsec$. \\citet{2020ApJ...899..145Y} derived similar $^{32}$S\/$^{34}$S and $^{34}$S\/$^{33}$S ratios from different telescopes ---that is, the IRAM 30 meter and the ARO 12 meter--- toward six HMSFRs and concluded that beam-size effects are insignificant (see details in their Section~4.1). All this suggests that beam-size effects could not obviously affect our results.  \n\n\n\n\n\\subsection{Chemical fractionation}\n\\label{section_fractionation}\n\nIsotopic fractionation could possibly affect the isotope ratios derived from the molecules in the interstellar medium. \\citet{1976ApJ...205L.165W} firstly proposed that gas-phase CO should have a tendency to be enriched in $^{13}$CO because of the charge-exchange reaction of CO with $^{13}$C$^+$. Several theoretical studies also support this mechanism \\citep[e.g.,][]{1984ApJ...277..581L,2020MNRAS.497.4333V}, which was then extended to other carbon bearing species \\citep{2020MNRAS.498.4663L}. Formaldehyde forming in the gas phase is suggested to be depleted in the $^{13}$C bearing isotopolog \\citep[e.g.,][]{1984ApJ...277..581L}. However, if H$_2$CO originates from dust grain mantles, then the $^{13}$C bearing isotopolog might be enhanced relative to species like methanol and CO \\citep{2012LPI....43.1611W,2019ApJ...877..154Y}. Recently, \\citet{2020A&A...640A..51C} performed a new gas-grain chemical model and proposed that molecules formed starting from atomic carbon could also show $^{13}$C enhancements through the reaction $^{13}$C + C$_{3}$ $\\rightarrow$ $^{12}$C + $^{13}$CC$_{2}$. As already mentioned in Section~\\ref{section_comparisons_12c13c}, the Galactic $^{12}$C\/$^{13}$C gradient derived from C$^{34}$S in this work is in good agreement with previous results based on measurements of CN, C$^{18}$O, and H$_2$CO. Therefore, chemical fractionation cannot greatly affect the carbon isotope ratios. \n\nTo date, little is known about sulfur fractionation. \\citet{2019MNRAS.485.5777L} proposed a low $^{34}$S enrichment through the reaction of $^{34}$S$^+$ + CS $\\rightarrow$ S$^+$ + C$^{34}$S in dense clouds. A slight enrichment in $^{13}$C was predicted for CS with the $^{13}$C$^{+}$ + CS $\\rightarrow$ C$^+$ + $^{13}$CS reaction \\citep{2020MNRAS.498.4663L}. $^{32}$S\/$^{34}$S ratios derived directly from $^{13}$CS and $^{13}$C$^{34}$S and the double isotope method involving $^{12}$C\/$^{13}$C ratios (equation 8) turn out to agree very well, suggesting that sulfur fractionation is negligible as previously suggested by \\citet{2020A&A...642A.222H} and in this work. \n\n\\subsection{Interstellar C, N, O, and S isotope ratios}\n\\label{section_discussion_all}\n\nThe data collected so far allow us to evaluate the status of several isotopes with respect to primary or secondary synthesis in stellar objects. From the data presented here in Table~\\ref{table_allratios}, we choose the $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{32}$S\/$^{33}$S, and $^{32}$S\/$^{36}$S ratios because $^{12}$C is mostly primary \\citep{1995ApJS...98..617T} while $^{32}$S is definitely a primary nucleus \\citep{1995ApJS..101..181W}, against which the other isotopes can be evaluated. A question arises as to whether all these ratios, as well as those from nitrogen and oxygen, can be part of the same scheme.\n\n Comparing the CMZ with the ratios in the inner disk, the CMZ with the outer Galaxy, the inner disk with the ratios in the outer Galaxy, and the local ISM values with those of the Solar System, we obtain increases in values for $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{32}$S\/$^{33}$S, and $^{32}$S\/$^{36}$S ratios. All of these values are listed in Table~\\ref{table_allratios_com}. Percentages are clearly highest between $^{32}$S and $^{36}$S. These indicate that $^{36}$S is, as opposed to $^{32}$S, definitely secondary. Percentages between $^{12}$C and $^{13}$C are also high but not as extreme, presumably because $^{12}$C is also synthesized in longer lived stars of intermediate mass (e.g., \\citealt{2020ApJ...900..179K}). More difficult to interpret are the $^{32}$S\/$^{33}$S and $^{32}$S\/$^{34}$S ratios, where percentages are smaller, indicating that $^{33}$S and $^{34}$S are, as already mentioned in Section~\\ref{section_hfs_c33s}, neither fully primary nor secondary. However, percentages in the case of the $^{32}$S\/$^{33}$S ratio systematically surpass those of the $^{32}$S\/$^{34}$S ratio, suggesting a more secondary origin of $^{33}$S with respect to $^{34}$S, even though $^{34}$S\/$^{33}$S appears to be constant across the Galaxy. Finally, local interstellar $^{32}$S\/$^{34}$S and $^{32}$S\/$^{33}$S ratios behave strikingly differently with respect to solar values. While $^{32}$S\/$^{34}$S is (almost) solar, $^{32}$S\/$^{33}$S is far below the solar value. Peculiar Solar System abundance ratios may be the easiest way to explain this puzzling situation. Most likely there is an overabundance of $^{34}$S in the gas and dust that formed the Solar System.\n\n\nAnother clearly primary isotope is $^{16}$O, which allows us to look for $^{16}$O\/$^{18}$O and $^{16}$O\/$^{17}$O ratios \\citep{1993A&A...274..730H,1994LNP...439...72H,1999RPPh...62..143W,2008A&A...487..237W,2020ApJS..249....6Z}. The high percentages in the $^{16}$O\/$^{17}$O ratios show that $^{17}$O is more secondary than $^{18}$O, which is consistent with models of stellar nucleosynthesis, because $^{17}$O is a main product of the CNO cycle while $^{18}$O can also be synthesized by helium burning in massive stars.\n\n$^{14}$N\/$^{15}$N can also be measured; both nuclei can be synthesized in rotating massive stars and AGB stars as primary products \\citep[e.g.,][]{2002A&A...390..561M,2011MNRAS.414.3231K,2020ApJ...900..179K,2018ApJS..237...13L}. However, most of the $^{14}$N is produced through CNO cycling, and is therefore secondary \\citep[e.g.,][]{2011MNRAS.414.3231K,2014PASA...31...30K}. The production of $^{15}$N remains to be understood and may be related to novae \\citep[e.g.,][]{2020ApJ...900..179K,2022arXiv221004350R}. None of the stable nitrogen isotopes are purely primary. While $^{14}$N appears to be less secondary than $^{15}$N \\citep{1975A&A....43...71A,1994LNP...439...72H,1999RPPh...62..143W,2012ApJ...744..194A,2015ApJ...804L...3R,2018A&A...609A.129C,2021ApJS..257...39C}, in this case we do not have a clear calibration against an isotope that can be considered to be mainly primary. Remarkably, \\citet{2022arXiv220910620C} reported a rising $^{14}$N\/$^{15}$N gradient that peaks at $R_{\\rm GC}$ = 11 kpc and then decreases, and suggested that $^{15}$N could be mainly produced by novae on long timescales.\n\n\n\\begin{table*}[h]\n\\caption{Interstellar C, N, O, and S isotope ratios.}\n\\centering\n\\begin{tabular}{ccccccc}\n\\hline\\hline\n  & Molecule & CMZ & Inner disk &Local ISM & Outer Galaxy & Solar System\\tablefootmark{*} \\\\\n\\hline\n\\label{table_allratios}\n$^{12}$C\/$^{13}$C & C$^{34}$S\\tablefootmark{a}&  27 $\\pm$ 3     &   41 $\\pm$ 9  &   66 $\\pm$ 10  &   74 $\\pm$ 8   & 89     \\\\ \n          &        CN\\tablefootmark{b}        & $\\cdots$        &  44 $\\pm$ 12  &   41 $\\pm$ 11  &   66 $\\pm$ 19  &      \\\\ \n          &      C$^{18}$O\\tablefootmark{c}   &  24 $\\pm$ 1     &   41 $\\pm$ 2  &   60 $\\pm$ 5   &   70 $\\pm$ 10  &      \\\\ \n          &      H$_2$CO\\tablefootmark{d}     & $\\cdots$        &   40 $\\pm$ 7  &   50 $\\pm$ 13  &   64 $\\pm$ 10  &      \\\\ \n          &            average                &   25 $\\pm$ 2    &   42 $\\pm$ 9  &   54 $\\pm$ 10  &   69 $\\pm$ 12  &       \\\\\n\\hline\n$^{14}$N\/$^{15}$N  & CN\\tablefootmark{e}      & $\\cdots$        &   269 $\\pm$ 59   &   314 $\\pm$ 104  &   289 $\\pm$ 85  &  270 \\\\ \n          &   HCN\\tablefootmark{f}            & $\\cdots$        &    284 $\\pm$ 63  &   398 $\\pm$ 48   &   388 $\\pm$ 32  &      \\\\ \n          &   HNC\\tablefootmark{f}            & $\\cdots$        &   363 $\\pm$ 100  &   378 $\\pm$ 79   &   395 $\\pm$ 74  &      \\\\ \n          &    N$_2$H$^{+}$\\tablefootmark{g}  & $\\cdots$        &   900 $\\pm$ 250  &   496 $\\pm$ 65   &   581 $\\pm$ 140 &      \\\\ \n          &   NH$_3$\\tablefootmark{h}         &  40 $\\pm$ 13    &   175 $\\pm$ 46   &   297 $\\pm$ 99   &   96 $\\pm$ 44   &      \\\\\n\\hline\n$^{16}$O\/$^{18}$O & H$_2$CO\\tablefootmark{i}  &   263 $\\pm$ 45  &  327 $\\pm$ 32    &  560 $\\pm$ 25    &  625 $\\pm$ 144  &  490 \\\\\n$^{18}$O\/$^{17}$O & CO\\tablefootmark{j}       &  3.4 $\\pm$ 0.1  &   3.6 $\\pm$ 0.2  &   3.9 $\\pm$ 0.4  &   4.8 $\\pm$ 0.6 &  5.5 \\\\\n$^{16}$O\/$^{17}$O\\tablefootmark{**}  &        &  894 $\\pm$ 155  &  1177 $\\pm$ 132  &  2184 $\\pm$ 244  & 3000 $\\pm$ 786  &  2625 \\\\\n\\hline\n$^{32}$S\/$^{34}$S\\tablefootmark{a}  &         &  19 $\\pm$ 2     &   18 $\\pm$ 4      &   24 $\\pm$ 4     &   28 $\\pm$ 3     & 23   \\\\\n$^{34}$S\/$^{33}$S\\tablefootmark{a}  &         &  4.2 $\\pm$ 0.2  &   4.3 $\\pm$ 0.4   &   4.2 $\\pm$ 0.5  &   4.1 $\\pm$ 0.3  & 5.6  \\\\\n$^{32}$S\/$^{33}$S\\tablefootmark{a}  &         &   70 $\\pm$ 16   &   82 $\\pm$ 19     &   88 $\\pm$ 21    &   105 $\\pm$ 19   & 127  \\\\\n$^{34}$S\/$^{36}$S\\tablefootmark{a}  &         &  41 $\\pm$ 4     &   122 $\\pm$ 18    &   111 $\\pm$ 16   &   161 $\\pm$ 32   & 200  \\\\\n$^{32}$S\/$^{36}$S\\tablefootmark{a}  &         &  884 $\\pm$ 104  &   2382 $\\pm$ 368  &   2752 $\\pm$ 458 &   4150 $\\pm$ 828 & 4525 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{The inner disk values refer to the mean values at galactocentric distances of 2.0 kpc~$\\le R_{\\rm GC} \\le$~6.0 kpc. The local ISM values refer to 7.5 kpc~$\\le R_{\\rm GC} \\le$~8.5 kpc. The outer Galaxy values point to 9.0 kpc~$\\le R_{\\rm GC} \\le$~11.0 kpc. \n\\tablefoottext{*}{From \\citet{1989GeCoA..53..197A}.} \n\\tablefoottext{a}{This work.}\n\\tablefoottext{b}{From \\citet{2002ApJ...578..211S} and \\citet{2005ApJ...634.1126M}.}\n\\tablefoottext{c}{From \\citet{1990ApJ...357..477L,1993ApJ...408..539L}, \\citet{1996A&AS..119..439W}, and \\citet{1998ApJ...494L.107K}.}\n\\tablefoottext{d}{From \\citet{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H} and \\citet{2019ApJ...877..154Y}.}  \n\\tablefoottext{e}{From \\citet{2012ApJ...744..194A}, \\citet{2015ApJ...804L...3R}, and \\citet{2015ApJ...808L..46F}. }  \n\\tablefoottext{f}{From \\citet{2012ApJ...744..194A} and \\citet{2018A&A...609A.129C}}  \n\\tablefoottext{g}{From \\citet{2015ApJ...804L...3R}. }  \n\\tablefoottext{h}{From \\citet{2021ApJS..257...39C}. }  \n\\tablefoottext{i}{From \\citet{1981MNRAS.194P..37G} and \\citet{1994ARA&A..32..191W}. }  \n\\tablefoottext{j}{From \\citet{2020ApJS..249....6Z}. } \n\\tablefoottext{**}{The $^{16}$O\/$^{17}$O ratios are derived with values of $^{16}$O\/$^{18}$O and $^{18}$O\/$^{17}$O.}    }\n\\end{table*}\n\n\\begin{table*}[h]\n\\caption{Comparison of isotope ratios at different galactocentric distances. Given are percentage enhancements.}\n\\centering\n\\begin{tabular}{ccccc}\n\\hline\\hline\n   & CMZ          &     CMZ      & Inner disk   &  Local ISM  \\\\\n   & $\\downarrow$ & $\\downarrow$ & $\\downarrow$ & $\\downarrow$ \\\\\n   &  Inner disk  & Outer Galaxy & Outer Galaxy & Solar System \\\\\n\\hline\n\\label{table_allratios_com}\n$^{12}$C\/$^{13}$C & 68 $\\pm$ 33  &  176 $\\pm$ 54  &  64 $\\pm$ 21  &  65 $\\pm$ 31  \\\\\n\\hline\n$^{16}$O\/$^{18}$O  & 24 $\\pm$ 9  &  138 $\\pm$ 61  &  91 $\\pm$ 43   &  -12 $\\pm$ 5   \\\\\n$^{16}$O\/$^{17}$O  & 32 $\\pm$ 8  &  236 $\\pm$ 111  &  155 $\\pm$ 73  &  20 $\\pm$ 13    \\\\\n\\hline\n$^{32}$S\/$^{34}$S  & -5 $\\pm$ 11  &  47 $\\pm$ 10  &  56 $\\pm$ 18  &  -4 $\\pm$ 17   \\\\\n$^{32}$S\/$^{33}$S  & 17 $\\pm$ 8   &  50 $\\pm$ 16  &  28 $\\pm$ 6  &  44 $\\pm$ 34 \\\\\n$^{32}$S\/$^{36}$S  & 169 $\\pm$ 50  &  369 $\\pm$ 125  &  74 $\\pm$ 31  &  64 $\\pm$ 27   \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Galactic chemical environment}\n\\label{section_discussion_model}\n\n\n\\citet{2020ApJ...900..179K} established a Galactic chemical evolution (GCE) model based on the GCE model in \\citet{2011MNRAS.414.3231K} with updates with respect to new solar abundances and also accounting for failed supernovae, super-AGB stars, the s-process from AGB stars, and various r-process sites. Based on this GCE model, the predicted $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{32}$S\/$^{33}$S, and $^{32}$S\/$^{36}$S ratios at $R_{\\rm GC}$~=~2.0, 4.0, 6.0, 8.5, 12.0, and 17.0~kpc are obtained and plotted in Figs.~\\ref{fig_gradient_12C13C}, \\ref{fig_gradient_32S34S}, \\ref{fig_all33S}, and \\ref{fig_all36S}. The initial mass function and nucleosynthesis yields are the same for different galactic radii but star formation and inflow timescales ($\\tau_{\\rm s}$ and $\\tau_{\\rm i}$) depend on the Galactic radius (see \\citealt{2000ApJ...539...26K} for the definition of the timescales). Adopted values are $\\tau_{\\rm s}$~=~1.0, 2.0, 3.0, 4.6, 6.5, and 8.8 Gyr as well as $\\tau_{\\rm i}$~=~4.0, 5.0, 5.0, 5.0, 7.0, and 50.0 Gyr for $R_{\\rm GC}$~=~2.0, 4.0, 6.0, 8.5, 12.0, and 17.0~kpc, respectively. The predicted $^{12}$C\/$^{13}$C ratios are in good agreement with our results, while $^{32}$S\/$^{34}$S and $^{32}$S\/$^{36}$S ratios show significant deviations at larger galactocentric distances. $^{32}$S\/$^{33}$S ratios show an offset along the entire inner 12 kpc of the Milky Way. This indicates that current models of Galactic chemical evolution are still far from perfect. In this context, our data will serve as a useful guideline for further even more refined GCE models.\n\nVery recently, \\citet{2022arXiv220910620C} predicted $^{12}$C\/$^{13}$C gradients with four different models addressing nova systems (see details in their Table~2 and Sect.~4), following \\citet{2017MNRAS.470..401R,2019MNRAS.490.2838R,2021A&A...653A..72R}. The gradients from these four models are shown in Fig.~\\ref{fig_12C13C_GCEmodels}. The results from model 1 show a large deviation with respect to the observed values. The other three models could reproduce the ratios within the dispersion at galactocentric radii beyond the solar neighborhood, while the inner Galaxy is not as well reproduced.\n\n\\begin{figure}[h]\n\\centering\n   \\includegraphics[width=280pt]{newFigures\/1allratio12c_13c-mosels.pdf}\n     \\caption{$^{12}$C\/$^{13}$C isotope ratios from observations in this work and GCE models. The red symbol $\\odot$ indicates the $^{12}$C\/$^{13}$C isotope ratio of the Sun. The $^{12}$C\/$^{13}$C gradient obtained from C$^{34}$S in the current work is plotted as a black solid line, with the gray shaded area showing the 1$\\sigma$ interval of the fit. The red crosses visualize the results from the GCE model of \\citet[][see also Section~\\ref{section_discussion_model}]{2011MNRAS.414.3231K,2020ApJ...900..179K}. The dark green, light green, magenta, and pink lines refer to the predicted gradients from models in Table 2 from \\citet{2022arXiv220910620C}.  }\n  \\label{fig_12C13C_GCEmodels}\n\\end{figure}\n\n\\section{Summary}\n\\label{summary}\n\nWe used the IRAM 30 meter telescope to perform observations of the $J$ = 2-1 transitions of CS, C$^{33}$S, C$^{34}$S, C$^{36}$S, $^{13}$CS, $^{13}$C$^{33}$S, and $^{13}$C$^{34}$S as well as the $J$ = 3-2 transitions of C$^{33}$S, C$^{34}$S, C$^{36}$S, and $^{13}$CS toward a large sample of 110 HMSFRs. The CS $J$ = 2-1 line was detected toward 106 sources, with a detection rate of 96\\%. The $J$ = 2-1 transitions of C$^{34}$S, $^{13}$CS, C$^{33}$S, $^{13}$C$^{34}$S, and C$^{36}$S were successfully detected in 90, 82, 46, 17, and 3 of our sources, respectively. The $J$ = 3-2 lines of C$^{34}$S, $^{13}$CS, C$^{33}$S, and C$^{36}$S were detected in 87, 71, 42, and 1 object(s). All the detected rare CS isotopologs exhibit optically thin lines and allow us to measure the isotope ratios of $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{32}$S\/$^{33}$S, $^{32}$S\/$^{36}$S, $^{34}$S\/$^{33}$S, $^{34}$S\/$^{36}$S, and $^{33}$S\/$^{36}$S with only minor saturation corrections. Our main results are as follows:\n\\begin{itemize}\n    \\item Based on the measurements of C$^{34}$S and $^{13}$C$^{34}$S $J$ = 2-1 transitions, we directly measured the $^{12}$C\/$^{13}$C ratios with corrections of opacity. With accurate distances obtained from parallax data \\citep{2009ApJ...700..137R, 2014ApJ...783..130R, 2019ApJ...885..131R}, we confirm the previously determined $^{12}$C\/$^{13}$C gradient. A least-squares fit to our data results in $^{12}\\rm C$\/$^{13}\\rm C$ = (4.77~$\\pm$~0.81)$R_{\\rm GC}$+(20.76~$\\pm$~4.61), with a correlation coefficient of 0.82. \n    \\item The Galactic $^{12}$C\/$^{13}$C gradients derived based on measurements of CN \\citep{2002ApJ...578..211S,2005ApJ...634.1126M}, C$^{18}$O \\citep{1990ApJ...357..477L,1996A&AS..119..439W,1998ApJ...494L.107K}, and H$_2$CO \\citep{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H,2019ApJ...877..154Y} are in agreement with our results from C$^{34}$S and emphasize that chemical fractionation has little effect on $^{12}$C\/$^{13}$C ratios.\n    \\item While previously it had been assumed that a linear fit would provide a good simulation of carbon isotope ratios as a function of galactocentric distance, our analysis reveals that this does not hold for the Galactic center region. While $^{12}$C\/$^{13}$C ratios are lowest in this part of the Milky Way, they clearly surpass values expected from a linear fit to the Galactic disk sources. This indicates that there is no strict linear correlation of carbon isotope ratios across the Galaxy.\n    \\item We confirm the previously determined $^{32}$S\/$^{34}$S gradients \\citep{1996A&A...305..960C,2020ApJ...899..145Y,2020A&A...642A.222H} with the direct method from $^{13}$CS and $^{13}$C$^{34}$S, as well as the double isotope method also using $^{12}$C\/$^{13}$C ratios in the $J$ = 2-1 and $J$ = 3-2 transitions. Opacity corrections could be applied to the $J$ = 2-1 transitions, but not to the $J$ = 3-2 lines that may show, on average, slightly higher opacities. A $^{32}$S\/$^{34}$S gradient of (0.75 $\\pm$ 0.13)$R_{\\rm GC}$+(15.52 $\\pm$ 0.78) was obtained based on a large dataset of 90 values from our double isotope method in the $J$ = 2-1 transition. The 19 sources permitting the direct determination of this ratio with $^{13}$CS\/$^{13}$C$^{34}$S yield $^{32}$S\/$^{34}$S=(0.73 $\\pm$ 0.36)$R_{\\rm GC}$+(16.50 $\\pm$ 2.07). \n    \\item Differences between the behavior of the $^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S ratios as a function of galactocentric distance are reported and should be used as input for further chemical models: (a) In the inner disk the $^{12}$C\/$^{13}$C ratios at $R_{\\rm GC} \\ge$ 4.0 kpc are clearly higher than the value in the CMZ, while the $^{32}$S\/$^{34}$S ratios in the CMZ and inner disk are similar, as already suggested for the first time by \\citet{2020A&A...642A.222H}. (b) In the local ISM, the $^{12}$C\/$^{13}$C ratio is well below the Solar System value but $^{32}$S\/$^{34}$S is still quite close to it. All of this indicates that, unlike $^{13}$C, $^{34}$S is not a clean secondary isotope.\n    \\item There is no notable $^{34}$S\/$^{33}$S gradient across the Galaxy. Ratios are well below the values commonly reported in earlier publications. This is a consequence of accounting for the full hyperfine structure splitting of the C$^{33}$S lines. The average value of $^{34}$S\/$^{33}$S derived from the $J$ = 2-1 transition lines after corrections for opacity toward our sample is 4.35~$\\pm$~0.44.\n    \\item While there is no $^{34}$S\/$^{33}$S gradient with galactocentric radius, interstellar\n    $^{34}$S\/$^{33}$S values near the solar neighborhood are well below the Solar System ratio, most likely suggesting the Solar System ratio is peculiar, and perhaps also the $^{18}$O\/$^{17}$O ratio. A comparison of local interstellar and Solar System $^{32}$S\/$^{34}$S and $^{34}$S\/$^{33}$S ratios suggests that the Solar System may have been formed from gas and dust with a peculiarly high $^{34}$S abundance. The data also indicate that $^{33}$S is not a clean primary or secondary product of nucleosynthesis, similarly\nto $^{34}$S .\n    \\item For the first time, we report a $^{32}$S\/$^{33}$S gradient in our Galaxy: $^{32}{\\rm S}\/^{33}{\\rm S}$ = $(2.64 \\pm 0.77)R_{\\rm GC}+(70.80 \\pm 5.57)$, with a correlation coefficient of 0.46. \n    \\item We find first potential indications for a positive $^{34}$S\/$^{36}$S gradient with galactocentric radius. Combined $^{34}$S\/$^{36}$S ratios from \\citet{1996A&A...313L...1M} and our new data with corrections of opacity in the $J$ = 2-1 transition and applying new up-to-date distances yield a linear fit of $^{34}{\\rm S}\/^{36}{\\rm S}$ = $(10.34 \\pm 2.74)R_{\\rm GC}+(57.45 \\pm 18.59)$, with a correlation coefficient of 0.71. Considering the uniform $^{34}$S\/$^{33}$S ratios in our Galaxy, a $^{33}$S\/$^{36}$S gradient of (2.38~$\\pm$~0.67)$R_{\\rm GC}$+(13.21~$\\pm$~4.48) is also obtained. \n    \\item For the first time, we report a tentative $^{32}$S\/$^{36}$S gradient with galactocentric radius: $^{32}{\\rm S}\/^{36}{\\rm S}$ = $(314 \\pm 55)R_{\\rm GC}+(659 \\pm 374)$, with a correlation coefficient of 0.84. Our measurements are consistent with $^{36}$S being a purely secondary nucleus. However, observations of $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S isotope ratios are still relatively few, especially in the CMZ and the inner disk within $R_{\\rm GC}$~=~5.0~kpc.\n    \\item The predicted $^{12}$C\/$^{13}$C ratios from the latest Galactic chemical evolution models \\citep[e.g.,][]{2020ApJ...900..179K,2021A&A...653A..72R,2022arXiv220910620C} are in good agreement with our results, while $^{32}$S\/$^{34}$S and $^{32}$S\/$^{36}$S ratios show significant differences at larger galactocentric distances. $^{32}$S\/$^{33}$S ratios even show clear offsets along the entire inner 12 kpc of the Milky Way. Taken together, these findings provide useful guidelines for further refinements of models of the chemical evolution of the Galaxy.\n\\end{itemize}\n\n  \n\\begin{acknowledgements}\nWe wish to thank the referee for useful comments. Y.T.Y. is a member of the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne. Y.T.Y. would like to thank the China Scholarship Council (CSC) and the Max-Planck-Institut f\\\"{u}r Radioastronomie (MPIfR) for the financial support. Y.T.Y. also thanks his fiancee, Siqi Guo, for her support during this pandemic period. C.K. acknowledges funding from the UK Science and Technology Facility Council through grant ST\/R000905\/1 and ST\/V000632\/1. We thank the IRAM staff for help provided during the observations.\n\\end{acknowledgements}\n\n\n\n\\bibliographystyle{aa}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSimple organisms like fungi and slime moulds are able to display complex behaviours. This is surprising given that their network-like body plan lacks any central organizing centre. The slime mould \\emph{Physarum polycephalum}\\ has emerged as a model system to study the complex dynamics these organisms use to adapt to their environment. The organism has been shown to find the shortest path through a maze \\cite{Nakagaki:2000} and connect food sources in an efficient and at the same time robust network comparable to man-made transport networks \\cite{Tero:2010}. Furthermore, the slime mould distributes its body mass among several resources to obtain an optimal diet \\cite{Dussutour:2010} and is able to anticipate recurring stimuli \\cite{Saigusa:2008}.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\textwidth]{cutsite2.pdf}\n\\caption{Wound healing process in \\emph{P. polycephalum}\\ illustrated at four time points using bright field images. The cut occurred at \\SI{18}{\\minute} and the fan grown at cut site reached its maximal size at \\SI{60}{\\minute}. The network morphology was restored after \\SI{85}{\\minute}.}\n\\label{img:cutsite}\n\\end{figure*}\n\n\\emph{P. polycephalum}\\ is a true slime mould that forms a plasmodial network. Nuclei keep on dividing without forming cell walls, which results in a syncytial web-like network. The cytoplasm within this tubular network flows back and forth in a shuttle flow \\cite{Kamiya:1981}. These cytoplasmic flows are driven by cross-sectional contractions of the actin-myosin meshwork lining the gel-like tube walls \\cite{WohlfarthBottermann:1979}. Flows are organized across the entire network in a peristaltic wave of contractions that matches organism size \\cite{Alim:2013}. Flows generated in the organism are optimized for transport as contractions increase the effective dispersion of particles way beyond molecular diffusivity by a mechanism called Taylor dispersion \\cite{Marbach:2016}.\n\n\\emph{P. polycephalum}\\ adapts its network-like morphology to its environment by chemotaxis \\cite{Ueda:1976,DURHAM:1976,Chet:1977}. Here, stimulants are classified by being an attractant or a repellant depending on the organism's response to migrate toward or away from the stimulant. Stimulants have also been shown to affect cross-sectional contractions organism-wide by an increase in their frequency and amplitude for an attractant or a decrease for a repellant \\cite{Miyake:1994,Hejnowicz:1980}. A variety of chemical stimuli have been discussed for \\emph{P. polycephalum}, with glucose being a prominent attractant and salts like NaCl being effective repellants \\cite{Kincaid:1978,HIROSE:1982,MCCLORY:1985}. Temperature \\cite{Matsumoto:1988,Takamatsu:2004} and light \\cite{WohlfarthBottermann:1982,Nakagaki:1999} have also been found to act as stimulants that trigger organism-wide restructuring of the transport networks' morphology. In fact, the cytoplasmic flows themselves serve as the medium by which stimuli pervade the organism \\cite{Alim:2017}. \n\nA lot less is known about the impact of mechanical perturbations on the organism. In its natural habitat the slime mould suffers predation from grazing invertebrates causing severing that disrupts the transport network and its cytoplasmic flows. In experiments it has been found that quickly stretching a strand to 10-20\\% of its length while keeping it intact increases the amplitude of oscillations \\cite{Kamiya:1972}. Excising a single strand from a plasmodial network has been observed to lead to a roughly 20 minute cessation of contractions in the strand until recovery \\cite{Yoshimoto:1978}. This phenomenon was not observed for strands excised from the growing fan region of the slime mould resulting in speculations about the motive force being limited to the fan only. Yet, the cessation of contractions turned out to be hard to reproduce, see \\cite{Cieslawska:1984} and references therein. Among these discordant observations what remains established is local gelation of cytoplasmic flows upon touch without severing the organism \\cite{Achenbach:1981}. Despite the limited knowledge, wounding the organism by severing the network is part of daily laboratory routines and an eminent perturbation in natural habitat. \n\nHere we investigate \\emph{P. polycephalum} 's dynamics during wound healing following the quick and complete severing of a tube within the organism's network. We follow the process of wound healing across the individual's entire body, over the course of one hour after severing. The exemplary quantitative analysis of organism-wide contractions reveals a stepwise response spanning four different states. Briefly after severing, the contractions are often marked by an increase in amplitude and frequency, followed by a several minutes long cessation of contractions and stalling of cytoplasmic flows. This resting state is terminated by a sudden restart of vigorous contractions as the severed tube re-fuses. The vigorous state then transitions into a state of network-spanning contractions and continuous fan growth at the wounding site until the organism reverts back to pre-stimulus dynamics. Timing and significance of individual steps varies with the severity of cutting and cutting site location within the network. For example, stalling is found to be less pronounced when the network is cut in fan-like region. Overall, quick and complete severing triggers a response pattern with characteristics of the response to an attractive stimulus, including an increase in amplitude and frequency and net movement to stimulus site, see Fig.~\\ref{img:cutsite}. The reproducibility of stalling clarifies earlier contradictions and at the same time opens new avenues to investigate the biochemical dynamics behind the highly coordinated acto-myosin contractions underlying \\emph{P. polycephalum}'s arguably fascinating dynamics. \n\\section{Methods}\n\\subsection{Culturing and data acquisition}\nThe plasmodium is prepared from microplasmodia grown in liquid medium. The recipe for the medium is inspired by \\cite{Fessel:2012}, see Sec. S1. The advantage of this method over growing the plasmodium on oat flakes or bacteria is the ability to precisely control the nutritional state and amount of the organism. Also, plasmodia grown this way are free from oat flake residues or vacuoles containing food, which provides a cleaner sample for imaging. To prepare the plate for imaging, 0.2-0.5 mL of the microplasmodia grown in a shaking culture at $30^{\\circ}$C are transferred to an 1.5\\% agar plate and stored in a closed, but not sealed dish in the dark. After 12-24 hours, the microplasmodia fuse into a single plasmodium. The plasmodium is ready for imaging when there are no visible traces of liquid medium and the organism assumed its characteristic network shape, which usually occurs up to 36 hours after plating.\n\nThe imaging is performed with a Zeiss Axio Zoom V.16 microscope, equipped with a Zeiss PlanNeoFluar 1x\/0.25 objective and a Hamamatsu ORCA-Flash 4.0 digital camera. A green filter (550\/50nm) is placed over the transmission light source of the microscope to diminish \\emph{P. polycephalum}'s response to the  light, and a humidity chamber prevents the sample from drying out. The acquisition of the images is done in Zeiss ZEN 2 (Blue Edition) software with bright-field setting. During the acquisition, the illumination of the sample is kept constant, and an image is taken every 3 seconds. The plasmodium is imaged for $\\sim$1 hour before the application of the mechanical stimulus to allow for the accommodation to the light \\cite{DURHAM:1976}. The stimulus is applied manually, using a microinjection needle with a blunt tip. The needle tip is held above the surface of the agar at a small angle and quickly dragged across the chosen plasmodial tube. The cut is severe and complete if the two parts of the tube separate completely. The plasmodium is then further imaged for more than 1 hour.\n\nUsing microplasmodia is so far the optimal way of obtaining non-severed networks, where the size and nutritional state are reproducible. However, there are challenges during the imaging that decrease the reproducibility of the experiment. In particular, plasmodia are highly motile and change their morphology accordingly. Furthermore, the organism tends to develop very large foraging fronts, which are not a suitable input for the presented comprehensive data analysis as they lack network characteristics. Lastly, the microscope light can act as stimulus \\cite{WohlfarthBottermann:1982,Nakagaki:1999,Tero:2010}, and even the green-filtered low-intensity illumination may cause the network to respond and change its behaviour to escape the imaging region. These challenges combined make the reproducibility and required stability of the network morphology over time challenging.\n\n\\subsection{Comprehensive network-based contraction analysis}\nTo quantify contraction dynamics we analyse bright field recordings in two different ways: for two morphologically static networks  (see E2 and E3 in the experiment list) we perform an exhaustive network-based analysis as outlined in the following (see Fig.~\\ref{img:05_lineplots} and Fig. S4). For the additional 19 specimen which alter their network morphology dramatically over the course of the experiment, we analyse kymographs along static parts of the network as described in detail in Sec. S3 (see exemplary E1 and Mov. S5).\n\nImages recorded as a time series are processed as 8-bit uncompressed TIFs. At first every image is processed separately, then the results are stitched together, largely following Ref.~\\cite{Alim:2013}, and lastly the collective is analysed. On every single image, background is removed with the rolling-ball method. Then the image is used to create a mask, a binary image, with an intensity threshold that separates the network from the background. The mask is enhanced further, i.e.~only the biggest structure is considered, small holes are filled and single-pixel edges are smoothed. Subsequently, the resulting mask is used as a template for extracting the network's skeleton with a thinning method. In the skeletonized mask each pixel can be understood as a data point representing local intensity and diameter (see Fig.~\\ref{img:02_skel}). Local diameter is calculated as the largest fitting disk radius around the point within the mask. Within this disk the average intensity is computed and saved as intensity at the considered data point. Intensity and diameter anti-correlate due to the optical density of the slime mould and can therefore be used interchangeably considering Beer-Lambert law. Individual data points are attributed to a specific network branch of the network skeleton. To represent network topology, the network is broken down into vertices and edges where vertices describe pixel positions of branching points and edges represent two connected vertices. Each edge then acts as a parent for one specific branch. In this sense edges are abstracted simple connections and branches represent pixel-based resolution of a tube.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{02_skeleton_170220.pdf}\n\\caption{Scheme of intensity and diameter data extraction based on \\emph{P. polycephalum}\\ bright field images. The light grey area depicts the network mask based on the bright field images. Dark grey lines represent the network skeleton and the corresponding topology is shown in blue. Each pixel of the skeleton acts as a reference point for data derived during the analysis. The diameter is set as the distance from the reference point to the next non-mask pixel. The intensity is calculated by averaging individual pixel intensities over a corresponding disk (red).} \\label{img:02_skel}\n\\end{figure}\n\nAfter the network is extracted in space, the edges, vertices, diameters, and intensities are concatenated in time. To map intensity and diameter over time, a reference image is used, usually from an early time point. For every data point the shortest distance to any pixel in the reference image is calculated. This gives a quasi-static (x, y, t) $\\rightarrow$ (intensity, diameter) dataset, i.e.~the topology and vertex positions stay the same, but  intensity and diameter can vary. This is justified as long as growth of the organism and vertex movement is minimal. The oscillatory behaviour of tubes in a certain time window can be described by four time dependent variables, namely amplitude $A$, frequency $f$ (or period $P$), phase $\\varphi$ and trend (base diameter) $d$. Each can be calculated from the time-evolution of the diameter or the intensity data, but if not stated otherwise the following results are only derived from intensity analysis. \n\nThe trend $d(t)$ is obtained with a moving-average filter with a kernel width of \\SI{200}{s} on each time trace (see Fig.~\\ref{img:03_linecomp}). The dataset is detrended with the calculated trend and smoothed with a Gaussian using a kernel width of \\SI{39}{s}. The kernel widths were chosen to extract the characteristic contraction pattern which usually has a frequency of \\SI{\\sim90}{\\second}. The values at every data point are stored as a complex valued time array, with the detrended and smoothed intensity representing the real part and the corresponding Hilbert transform representing the complex part, see S2 for more details. This time array, denoted analytic signal, serves as a basis to get instantaneous phase, frequency and amplitude by computing the angle or absolute value of the complex time series. Finally, the results are mapped back onto the network structure for each time point. In this fashion one can follow oscillatory behaviour resolved in time and space. Furthermore, the maps can be clustered in sub-networks and averaged separately to pinpoint local events in time. It should be mentioned that averaging of results for line plots, i.e.~Fig.~\\ref{img:05_lineplots}, is always done after the data-point based analysis took place. In this way for example, the apparent amplitude of the averaged intensity (Fig.~\\ref{img:05_lineplots}D) can be lower than the amplitude of each data point averaged (Fig.~\\ref{img:05_lineplots}B).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{03_linecomp_170306.pdf}\n\\caption{Derivation of oscillation specific parameters, i.e.~amplitude $A$(t), frequency $f$(t) and trend $d$(t), from single pixel time series. The trend is calculated using a moving average with a kernel width of \\SI{200}{s}. Intensity is filtered with a Gaussian of width \\SI{39}{s}.  Amplitude and frequency are calculated from the absolute value and angle of the complex-valued analytic signal, respectively.} \\label{img:03_linecomp}\n\\end{figure}\n\\section{Results}\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=\\textwidth]{04_maps_170303.pdf}\n\\caption{Time evolution of an exemplary network and its spatially mapped oscillation parameters at \\SI{13}{\\minute}, \\SI{27}{\\minute}, \\SI{33}{\\minute} and \\SI{55}{\\minute}. The network was cut in the centre at \\SI{17.3}{\\minute} (\\emph{scissor icon}). Top row depicts the raw bright field data, middle row  the local amplitude, and bottom row the local frequency. Amplitude and frequency decrease locally, first at the lower sub-network (\\emph{small dotted arc}) at \\SI{27}{\\minute}, subsequently at upper sub-network (\\emph{large dotted arc}) at \\SI{33}{\\minute}.  At \\SI{38}{\\minute} cytoplasmic flows are re-established at the wounding site. Finally, amplitude and frequency values recover.} \\label{img:04_maps}\n\\end{figure*}\n\\subsection{Wounding induces fan growth at cut site}\nWe observe specimens before and after a quick and complete severing of a tube to follow the response of \\emph{P. polycephalum}\\ to wounding (see Fig.~\\ref{img:04_maps}A, Mov. S1 and Mov. S5). Bright field movies reveal that cutting of main tubes distal to fans triggers cessation of contractions followed by stalling of cytoplasmic flow (n=15 out of 21). After contractions resume the severed tube fuses back together (n=21 out of 21), i.e.~flow is re-established, and a fan starts to grow at the cut site.  Furthermore, we observe accumulation of body mass close to the cut site which is most prominent in peripheral cuts (Fig. S2). However, the growth is transient and after a given time the initial morphology is restored and the organism returns to typical behaviour comparable to before wounding.\\\\\n\nIn consideration of previously mentioned technical limits, we selected one representative dataset with prominent discernible features for network-based analysis. The following findings are derived from this dataset and later compared with other experiments. The specific timing of events in the representative data set is as follows (see Fig.~\\ref{img:04_maps}). Two tubes are severed at \\SI{17.3}{\\minute} effectively dividing the network into two parts. In both sub-networks, the size-wise bigger and smaller part, flows stall transiently around \\SI{30}{\\minute}. At \\SI{38}{\\minute} a connecting tube is reinstated and starts to re-establish cytoplasmic flows across the cut site. Until about \\SI{63}{\\minute} a transient fan is created at the cut site. At \\SI{90}{\\minute} the initial morphology is restored and fans are grown elsewhere.\n\n\\subsection{Spatial mapping reveals localized stalling}\nWe perform network-based analysis on the wounded specimen to extract the interplay of contractions during the healing response. In particular, we map out the amplitude and frequency of contractions spatially (see Fig.~\\ref{img:04_maps}, Mov. S2 and Mov. S3). This allows us to exactly localize the onset of stalling as it goes hand in hand with low values of amplitude and frequency. Likewise, patterns in contraction dynamics in a region of interest are identified by spatially averaging amplitude and frequency in this region (see Fig.~\\ref{img:05_lineplots}).\n\nIn the representative dataset, wounding separates the network into two sub-networks. Spatial mapping reveals that oscillations cease on different time-scales in the two sub-networks. By identifying the two sub-networks as separate regions of interest, we quantify the patterns in contraction taking the spatial average of the respective contraction variables in each region. The small sub-network shows a drop in amplitude at \\SI{21.5}{\\minute} by \\SI{63}{\\percent} and only recovers eight and a half minutes later to comparable values. Here, the percentage is given as ratio of time averages before, during and after stalling. In detail, the averages of the first \\si{21.5} minutes, the \\si{9.5} minutes during stalling and \\si{15} minutes after stalling were considered. The bigger sub-network drops significantly later at \\SI{28}{\\minute} by \\SI{51}{\\percent} and recovers to \\SI{29}{\\percent} below the initial value nine minutes later. In the same time frames the frequency drops by \\SI{32}{\\percent} and \\SI{45}{\\percent} for the small and big sub-network, respectively. Yet, neither sub-network recovers its frequency fully right after the stalling phase. Only the small sub-network recovers 35 minutes later to initial frequencies whereas the bigger region levels off \\SI{35}{\\percent} below the initial value.\\\\\nFurthermore, the phase patterns over time (see Mov. S4) reveal changes in the travelling waves upon cutting. Initially (0 to \\SI{17.3}{\\minute}) one can observe peristaltic waves from the tail (right-hand side) to the front (left-hand side) which finally merge into concentric patterns in the fan regions. Then, at 18 to \\SI{30}{\\minute}, the small sub-network slows down noticeably (see change in frequency) and the big sub-network contracts with less apparent spatial correlation, i.e.~the peristaltic wave pattern is temporarily lost.\n\n\\subsection{Fan growth phase coincides with stable network-spanning contractions}\nAfter re-fusing of the two sub-networks, another distinct phase characterized by stable network-spanning contraction dynamics can be observed. In Fig.~\\ref{img:05_lineplots}D contractions appear uniform from \\SI{44}{\\minute} until \\SI{63}{\\minute}. During this phase, amplitude and frequency level off to a stable value with little fluctuations. The small sub-network shows a slight increase in frequency over this period and has more fluctuations in the average intensity data than the big sub-network. Note, that the time frame of these contractions coincide with fan formation at the cut site. Furthermore, the end of this phase also coincides with the largest fan in respect to area.\\\\\nNetwork-spanning contractions are further supported by the phase time series. When considering the phase development one can already observe a peristaltic wave travelling towards the cut site in the small sub-network as early as \\SI{30}{\\minute}. A spanning pattern in the large sub-network is reinstated around the \\SI{35}{\\minute} mark and a global pattern (small and large sub-network) appears roughly three minutes after re-fusing (\\SI{40}{\\minute}). Then a standing wave pattern appears between the central region including the cut site and the periphery. It is stable and network-spanning until \\SI{63}{\\minute}. Subsequently the phase pattern breaks into a peristaltic wave similar to pre-cut and propagates from the tail and the small sub-network into fan regions in the large sub-network.\n\n\\subsection{Stalling and fan growth periods are bridged by distinct transition periods}\nCloser analysis of contraction dynamics over time reveals that the time point of the cut, the stalling phase and the fan growth phase are transitioned by phases of high fluctuations. Particularly in the presented case, before stalling occurs, amplitude and frequency peak shortly in both sub-networks (see arrows in Fig.~\\ref{img:05_lineplots}). In the small sub-network this peak coincides with the cut, whereas another ten minutes pass for the big sub-network before the amplitude reaches its maximum. Surprisingly, here the frequency decline occurs three minutes before the amplitude drops. After stalling the amplitude increases sharply in both sub-networks, yet stays below previous values in the big sub-network. The small network undergoes a phase of roughly \\SI{13}{\\minute} where the amplitude oscillates vigorously. This also coincides with a second frequency drop even though there is no apparent drop in amplitude at this time point. After the fan growth phase, amplitude and frequency show slight gradients once more. Here behaviour becomes comparable to the pre-cut state as the slime mould develops a preferred growth direction in the periphery and continues foraging.\n\n\\begin{figure}[hpt]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{05_lineplots_170303.pdf}\n\\caption{Comparison of oscillation parameters in the big- and small-sub-network depicted in (A) which result from the cut. Grey area (cut site) is not considered in the analysis. Time series of amplitude (B), frequency (C) and intensity (D) are averaged in the respective domains and compared; top : big sub-network, bottom: small sub-network. In each of these plots the black dashed line indicates the moment of the cut. The first grey dashed line marks the time point of fusion and the second the moment of maximal fan size. Black bars underline periods of stalling in (B) and fan growth in (D). In (D) the solid line represents the Gaussian filtered intensity (kernel width = 39s) and markers show raw averaged data. Black arrows indicate respective extremal peaks in the transition periods. Four black dots on the time line correspond to the four time points chosen in Fig.\\ref{img:04_maps}.\n}\\label{img:05_lineplots}\n\\end{figure}\n\n\\subsection{Fan creation and stalling is reproducible for complete severing}\nFor comparison we analysed a second dataset with the same network-based method (see Fig. S4). The key features, i.e.~cut repair, stalling, a transition phase, stable network-spanning contractions and return to pre-cut behaviour are found likewise, but the succession and timing of the specific events vary. This dataset has a weaker fan growth at the cut site and the time point of maximal fan size follows immediately after fusion. Given the short period of fan growth global network-spanning contractions are not observed. However, standing phase wave patterns  are visible in the larger sub-network before fusion. Lastly, the transition phase shows peaking amplitude and fluctuating frequencies and reverberates for more than 30 minutes. At \\SI{70}{\\minute} the network reinstates a peristaltic wave toward peripheral fan regions resuming pre-cut dynamics.\\\\\n\nIn further experiments analysed with a kymograph based approach, we confirmed stalling to be a common response after a cut (see Fig. E1, n=15 out of 21). However, the degree and duration of stalling is varying between experiments and is most reproducible for a severe cut close to the centre of the network.\\\\\nIn detail, we observe that both the degree and duration of stalling, depend on the network size and morphology, cut location, possibilities of re-routing the flow through neighbouring tubes and presence of large fans. Also, a network undergoing quick changes in morphology due to a presumed light shock is less likely to show stalling. Varying cut location shows that complete severing of a tube, with a diameter comparably large in size and few neighbouring tubes, results in strong stalling, see experiments E[2, 3, 5, 6, 8, 9, 12, 13, and 18]. The effect is even more pronounced in smaller networks and on tubes close to the centre of the network (E[2, 3, 5, 8, and 18]). Stalling is less pronounced, as measured by relative change in amplitude and frequency as well as visual inspection of bright field data, if severing was applied to fan-like regions or peripheral tubes (E[10, 11, 14, 15, 16, 17, 19, 20, and 21]). If a severed tube had alternative routes with a comparable flow direction, neighbouring tubes inflated shortly after the cut, indicating a re-routing of flow. Yet, in this case stalling severity ranged from non-existent (E19) to full-stop (E1). In all data sets fan growth is observed around the cut site, yet duration and fan sized varied greatly (see E2 and E9 as maximal and minimal examples). \\\\\nIn all 15 experiments that show stalling, the period lasted for a minimum of three minutes. The exact time point of stalling onset and its duration varied. Duration of transition periods also varied from complete omission up to \\num{22} minutes between cut and stalling. In 7 out of 15 experiments, a vigorous phase of increase in frequency or amplitude fluctuations could be observed in the transition phases.\n\n\\section{Discussion}\nWe investigate \\emph{P. polycephalum}'s response to wounding in the form of a quick and complete severing of tubes using bright field microscopy and quantitative analysis of contraction patterns. Mapping out the contractions amplitude and frequency in space and time allows us to uncover a multi-step pattern of wound healing in \\emph{P. polycephalum}. \n\nThe key of our network-based analysis is mapping contraction variables onto a few pixels serving as the skeletonized backbone of the complete network. This representation allows us to capture contraction dynamics across the entire network over the course of several hours with handleable amount of data. Furthermore, spatial mapping visualizes abstract variables in an approachable way which outlines region of interests or patterns in space. For example, in the representative data set the time-shift in the response pattern between the two sub-domains of the network would have been lost when averaging contraction dynamics across the entire network (see Fig. S1).\n\nAmong the multiple steps in the response to wounding the cessation of oscillations and stalling of the cytoplasmic streaming is most striking. The phenomenon of stalling of cytoplasmic flows has been observed previously ~\\cite{Kamiya:1972,Yoshimoto:1978}, but its reproducibility was deemed questionable \\cite{Cieslawska:1984}. Our work shows that cut location and severity are crucial parameters for inducing reproducible stalling. The stalling period is omitted when a tube is not completely severed, or cut in a way that allows the cut ends to rejoin quickly. In addition, the specific body plan affects the impact of a cutting stimulus. For example, severed fan-like regions show less pronounced stalling. However, we find reproducible strong stalling in networks where the affected tubes are crucial connections that cannot be re-routed easily - thereby clarifying previously discordant observations.\\\\\n\nStimuli are commonly classified into attractants or repellants. The response of \\emph{P. polycephalum}\\ to an attractive stimulus includes fan growth and mass transport towards the stimulus site, often accompanied with an increase of oscillation frequency and amplitude. When we apply a wounding stimulus resulting in complete cutting of a tube, we observe a multi-step response pattern where only two out of four steps show a noticeable increase in amplitude and frequency. Yet, wounding implies that the network architecture is perturbed. Taken into account that contraction frequency decreases as organism size decreases \\cite{Kuroda:2015} the impairment of network architecture itself might counteract any increase in frequency. Despite the weak indication from contraction frequency and amplitude, we always observe fan growth and movement of mass toward the cut site regardless of the tube hierarchy, plasmodium size or the severity of the cut. Fan growth is a lot bigger than initial spillage of cytoplasm due to cutting. Furthermore, we often identify a specific fan growth phase of network-spanning contractions well separated in time from the cutting event by the stalling phase. We therefore identify wounding as an attractive stimulus. The observation of network-spanning oscillations during fan outgrowth adds to our confidence about cutting being an attractive stimulus since the observed phase patterns resemble contraction patterns found in earlier work with attractive stimuli using glucose as a stimulant \\cite{Alim:2017}.\\\\\n\nEmploying spatial data analysis we uncovered that wounding triggers a choreography of multiple successive steps to heal the severed tube. The mere duration of the healing response now defines a suggested minimal wait time after trimming for \\emph{P. polycephalum}\\ experiments. The complexity of the response hints at an intricate signalling pattern underlying the coordination of contractions. It is likely that also the response to classical attractants and repellants, when scrutinized, reveal multiple steps. Unravelling the workings behind \\emph{P. polycephalum}'s ability to adapt, is arguably a fascinating albeit challenging question. Here, the reproducible cessation of contractions arising during this wound-healing response may open up new avenues to investigate the biochemical wiring underlying \\emph{P. polycephalum}'s complex behaviours. Furthermore, it is fascinating that the impact of wounding can be weakened by network architecture. This suggests that \\emph{P. polycephalum}'s body plan itself could be part of the organisms strategy to not only adapt to its environment, but also specifically prevent severe consequences of wounding. \n\n\\section*{Acknowledgements}\nWe thank Christian Westendorf for instructions on growing microplasmodia, as well as for invaluable discussions and advice. M.K. and F.B. acknowledge support by IMPRS for Physics of Biological and Complex Systems.\n\n\\section*{Bibliography}\n\\bibliographystyle{iopart-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAmong AGNs, blazars are the most luminous and violent objects in the\nuniverse and emit $\\gamma$-rays at energies higher than 100 MeV. They are\ndivided into two main subgroups: Highly\nvariable quasars, sometimes called optically violent variable (OVV) quasars,\nand BL Lacertae objects (BL Lac). Since one of the jets of blazars is\npointed toward the Earth, we see the jet emission strongly Doppler enhanced and\nhighly variable. According to the unification scheme \\citep{up95}, radio\ngalaxies are the mis-aligned parent population of blazars. Synchrotron peak\nfrequencies of BL Lac objects cover a large range from IR to X-rays, and based\non its location they are called low-frequency peaked BL Lac objects (LBL;\nsynchrotron peak in the IR), intermediate BL Lac objects (IBL; synchrotron\npeak in the  optical\/UV), or high-frequency peaked BL Lac objects (HBL;\nsynchrotron peak in the X-rays).\n\nWhile there is little evidence for dense radiation environments\nin the nuclear regions of BL~Lac objects --- in particular, HBLs\n---, strong line emission in Flat Spectrum Radio Quasars (FSRQs)\nas well as the occasional detection of emission lines in the spectra of some\nBL~Lac objects \\citep[e.g.,][]{vermeulen95} indicates dense nuclear\nradiation fields in those objects. This is supported by spectral modeling\nof the SEDs of blazars using leptonic models which prefer scenarios\nbased on external radiation fields as sources for Compton scattering\nto produce the high-energy radiation in FSRQs, LBLs and also some\nIBLs \\citep[e.g.,][]{ghisellini98,madejski99,bb00,acciari08,abdo11}. If the\nVHE $\\gamma$-ray emission is indeed produced in the high-radiation-density\nenvironment of the broad line region (BLR) and\/or the dust torus of an\nAGN, it is expected to be strongly attenuated by $\\gamma\\gamma$ pair\nproduction \\citep[e.g.][]{pb97,donea03,reimer07,liu08,sb08}.\n \\cite{akc08} have suggested that such intrinsic $\\gamma\\gamma$ absorption may be\nresponsible for producing the unexpectedly hard intrinsic (i.e., after\ncorrection for $\\gamma\\gamma$ absorption by the extragalactic background\nlight) VHE $\\gamma$-ray spectra of some blazars at relatively high redshift.\nA similar effect has been invoked by \\cite{ps10} to explain the spectral\nbreaks in the {\\it Fermi} spectra of $\\gamma$-ray blazars.\nThis absorption process will lead to the development of Compton-supported\npair cascades in the circumnuclear environment \\citep[e.g.,][]{bk95,sb10,rb10,rb11}.\n\nIn \\cite{rb10,rb11}, we considered the full 3-dimensional development of\nCompton-supported VHE $\\gamma$-ray induced cascades in the external radiation\nfields in AGN environments.\n In those works, we have left the origin (leptonic SSC or IC, or hadronic) \nof the primary VHE $\\gamma$-ray emission deliberately unspecified in order to \ninvestigate the cascade development in as model-independent a way as possible.\nWe have shown that even very weak magnetic fields $(B\\lesssim\n\\mu$G) may be sufficient for efficient quasi-isotropization of the cascade emission.\nWe applied this idea to fit the {\\it Fermi}\n$\\gamma$-ray emission of the radio galaxies NGC~1275 and Cen~A. In\n\\cite{rb10,rb11}, parameters were chosen such that the synchrotron emission\nfrom the cascades was negligible.\n\nIn this paper, we present a generalization of the Monte-Carlo cascade\ncode developed in \\cite{rb11} to non-negligible magnetic fields and consider\nthe angle dependent synchrotron emission from the cascades. In section\n\\ref{setup}, we will outline the general model setup and assumptions\nand describe the modified Monte-Carlo code. Numerical results for generic\nparameters will be presented in section \\ref{parameterstudy}. We confirm\nthat for the objects Cen~A and NGC~1275 the synchrotron radiation from the\ncascades is negligible for those parameters used in \\cite{rb10,rb11}. In section\n\\ref{degeneracy}, we investigate the effect of the magnetic field and its\ndegeneracy. We show that by studying only the high energy emission from\nthe cascades, the magnetic field can not be determined, and additional\nconstraints are needed from the synchrotron emission component. This\nis illustrated for the case of NGC~1275. In section \\ref{3C279}, we will\ndemonstrate that for moderately strong magnetic fields the synchrotron\nemission from the cascades can produce a signature resembling the big blue\nbump (BBB) observed in several blazars and demonstrate \nthat this may make a non-negligible contribution to UV -- soft X-ray\nSED. We illustrate this for the case of 3C~279. We summarize in Section \n\\ref{summary}.\n\n\n\\section{\\label{setup}Model Setup and Code Description}\n\nThe general model setup used for this work is described in \\cite{rb10,rb11}.\nThe primary VHE $\\gamma$-ray emission is represented as a mono-directional\nbeam of $\\gamma$-rays propagating along the X axis, described by a power-law \nwith photon spectral index $\\alpha$ and a high-energy cut-off at $E_{\\gamma, max}$. \nWe assume that the primary $\\gamma$-rays interact via $\\gamma\\gamma$\nabsorption and pair production with an isotropic radiation field with \narbitrary spectrum within a fixed boundary, given by a radius $R_{\\rm ext}$.\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f1.eps}\n\\caption{\\label{diskabs}$\\gamma\\gamma$ opacity due to accretion\ndisk photons as a function of height $z$ (in units of gravitational\nradii $r_g = GM\/c^2$) of the emission region above the black hole,\nfor three different VHE $\\gamma$-ray photon energies, and different\nblack-hole masses and disk luminosities. }\n\\end{figure}\n\nThe assumption of an isotropic external radiation field is appropriate\nfor line emission from the BLR, for distances from the central engine comparable\nto the size of the BLR ($\\sim 10^{17}$ -- $10^{18}$~cm), and for infrared emission\nfrom cold dust in the nuclear environment, on typical scales of $\\sim$~parsec.\nClose to the central black hole and accretion disk, direct emission from the\naccretion disk may dominate the radiation energy density. However, for moderate\ndistances from the disk, the primary $\\gamma$-ray beam will interact with the\naccretion disk emission under an unfavorable angle for $\\gamma\\gamma$ pair\nproduction. To illustrate this point, we plot in Figure \\ref{diskabs} the opacity\nto $\\gamma\\gamma$ absorption in the radiation field of an optically thick,\ngeometrically thin \\citep{ss73} accretion disk as a function of height\n$z$ of the emission region from the black hole. The figure shows that\ntypically, the accretion-disk $\\gamma\\gamma$ opacity drops below one\nat distances of a few 100 -- $10^3 \\, r_g$ from the black hole. This\nis of the order of the characteristic height of the emission region in\nleptonic models of blazar emission, and much smaller than the size of\nthe BLR or the dust torus. We therefore conclude that the neglect of\nthe accretion-disk emission in our simulations is a good approximation\nthroughout almost all of our simulation volume.\nFor the case of thermal blackbody radiation fields considered below, we\nchoose energy densities and blackbody temperatures characteristic of the\nobserved properties (temperatures and total luminosities) of thermal \ninfrared emission seen in AGN.\nIn order to keep our treatment as model independent as possible, we do not \nspecify the physical origin of the primary VHE $\\gamma$-ray spectrum. The\nsimplest assumption consistent with most models of $\\gamma$-ray emission\nin blazars is a simple power-law, which we use as input spectrum in our\nsimulations. The shape of the resulting pair cascade emission is only \nvery weakly dependent on the exact shape of the incident VHE $\\gamma$-ray \nspectrum. This is illustrated in Figure \\ref{PLExpcomparison}, in which\nwe run a cascade simulation, once with a straight power-law, once with\na powerlaw + exponential cut-off (as a more realistic representation of\na physical blazar high-energy spectrum), with identical environmental\nparameters, as listed in the figure caption. While the overall normalization\nof the cascade spectrum, obviously, depends on the flux of absorbed\nVHE $\\gamma$-rays (which is higher in the pure power-law case), the\ncascade spectra at energies below the $\\gamma\\gamma$ absorption\ntrough are virtually identical. In the cases relevant for this study,\nonly a small fraction of the $\\gamma$-ray power is absorbed and re-processed\ninto cascades. Therefore, feedback between the primary $\\gamma$-ray production\nregion and the cascade emission may be neglected, and the cascade development\ncan be treated as a process separate from the (unspecified) VHE $\\gamma$-ray\nproduction mechanism.\n\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f2.eps}\n\\caption{\\label{PLExpcomparison}Comparison of the Compton emission\nfrom cascades for two different input VHE $\\gamma$-ray spectral\nshapes: Blue (thin) lines indicate the emission for a pure power-law\ninput spectrum; red (thick) lines for a power-law + exponential cut-off.\nEnvironmental parameters are the same as in Figure \\ref{standardfig}\n(see below). Different line styles correspond to different viewing\nangles, $\\mu = \\cos\\theta_{\\rm obs}$, with respect to the jet axis,\nas indicated in the legend. Parameters: $B_x = B_y = 1$~$\\mu$G, \n$\\theta_B = 45^o$; $u_{\\rm ext} = 10^{-6}$~erg~cm$^{-3}$, \n$R_{\\rm ext} = 10^{18}$~cm, $T = 1000$~K. \nThe angular bin $0.8 \\le \\mu \\le 1$ contains the forward direction. \nThe (unabsorbed) primary $\\gamma$-ray input spectra are shown by the \ndot-dot-dashed lines.\n}\n\\end{figure}\n\nOur code evaluates $\\gamma\\gamma$ absorption and pair production using the\nfull analytical solution to the pair production spectrum of \\cite{bs97} under\nthe assumption that the produced electron and positron travel initially along\nthe direction\nof propagation of the incoming $\\gamma$-ray. The trajectories of the particles are\nfollowed in full 3-D geometry. Compton scattering is evaluated using the head-on\napproximation, assuming that the scattered photon travels along the direction of\nmotion of the electron\/positron at the time of scattering. The Compton\nenergy loss to the electron is properly accounted for at the time of each scattering.\n\nFor simplicity, the magnetic field in our simulations is treated as \nhomogeneous, oriented at an angle $\\theta_B$ with respect to the jet axis.\nThis may be considered an appropriate proxy for a helical magnetic field\nwith the ratio of toroidal ($B_{\\rm tor}$) and poloidal ($B_x$) magnetic\nfields given by $\\tan\\theta_B = B_{\\rm tor}\/B_x$.\nThe code calculates the synchrotron energy loss of cascade particles in the\nfollowing way: The energy of electrons\/positrons is decreased by $\\Delta E_{\\rm sy}\n= \\dot E_{\\rm sy} \\frac{l_c}{c}$ between successive Compton scatterings, where\n$l_c$ is the Monte Carlo generated distance traveled to the next scattering\nand $ \\dot E_{\\rm sy} = - 2 c \\sigma_{T} u_B \\gamma^2 \\sin^2 \\psi$ with\n$\\psi$ being a pitch angle between the particle momentum and the magnetic field.\nWe assume that the trajectory of the particles between two Compton scatterings\nis not affected by synchrotron radiation which is valid for $ u_B \\lesssim u_{\\rm ext}$.\nThen for $10$ random points between two successive Compton scatterings, we determine\nthe position and direction of motion of the particles at these points and write\nthe spectral power in synchrotron radiation $P_{\\nu}$ into a synchrotron output\nfile for the angular bin corresponding to the electron's\/positron's direction of\nmotion. The synchrotron power $P_{\\nu}$ of a single $e^{\\pm}$ is approximated as:\n\n\\begin{equation}\nP_{\\nu} = 2 \\frac{c \\sigma_T}{\\Gamma(\\frac{4}{3})} u_B \\beta^2\\gamma^2\n\\frac{\\nu^{1\/3}}{\\nu_c^{4\/3}} e^{-\\nu\/\\nu_c}\n\\label{sy_asymptotic}\n\\end{equation}\n\n\\citep{boettcher12} where the critical frequency \n$\\nu_c = \\frac{3 q B}{4\\pi m_e c}\\gamma^2 \\sin\\psi = \n4.2\\times10^6 \\sin\\psi B_G \\gamma^2$~Hz with $B_G$ being the magnetic \nfield in units of Gauss. \nThe approximation (\\ref{sy_asymptotic}) represents the synchrotron\nspectrum to whithin a few \\% at all frequencies.\n\n\n\\section{\\label{parameterstudy}Numerical Results}\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f3.eps}\n\\caption{\\label{standardfig}\nCascade emission at different viewing angles ($\\mu = \\cos\\theta_{\\rm obs}$). \nParameters of the target photon field are the same as for Figure 1. The \ninput photon spectrum is a pure power-law with $\\alpha = 2.5$, \n{$E_{\\gamma, {\\rm max}} = 5$~TeV}. The green solid line represents \nthe target photon field.\n}\n\\end{figure}\n\nWe have used the cascade Monte-Carlo code described in the previous section\nto evaluate the angle-dependent Compton and synchrotron spectra from VHE\n$\\gamma$-ray induced pair cascades for a variety of generic\nparameter choices. Figure \\ref{standardfig} illustrates the viewing angle\ndependence of the cascade emission. For this simulation, we assumed a\nmagnetic field of $B = \\sqrt{2} \\mu$G, oriented at an angle $\\theta_B = 45^o$\nwith respect to the X axis ($B_x = 1 \\, \\mu$G, $B_y = 1 \\, \\mu$G). The\nexternal radiation field is a thermal blackbody with $u_{\\rm ext} =\n10^{-6}$~erg~cm$^{-3}$, extended over a region of radius $R_{\\rm ext} = \n10^{18}$~cm, with a blackbody temperature of $ T = 10^3$~K (corresponding \nto a peak of the blackbody spectrum at a photon energy of $E_s^{\\rm pk}= \n0.25$~eV). This leads to a $\\gamma\\gamma$ absorption cut-off at an energy\n$E_c = (m_e c^2)^2 \/ E_s \\sim 2$~TeV. The incident $\\gamma$-ray spectrum \nhas a photon index of $\\alpha = 2.5$ and extends out to $E_{\\gamma, {\\rm max}} \n= 5$~TeV.\n\nFor any given viewing angle $\\theta$ with respect to the direction of \npropagation of the primary $\\gamma$-rays a critical electron energy for \nwhich the deflection angle over a Compton length equals the observing \nangle, i.e., $\\theta \\sim \\lambda_{\\rm IC} \/ r_g$ can be defined. This \nyields the characteristic electron energy $E_{\\rm e, br} = \\gamma _c \nm_e c^2$ corresponding to a given observing angle $\\theta$:\n\n\\begin{equation}\n\\gamma_c =\\sqrt{\\frac {3  e B}{4 \\sigma_T  u_{\\rm ext}\\theta}}\\sim 7.2\\times10^5\nB_{-6}^{1\/2} u_{-3}^{-1\/2} \\theta^{-1\/2}\n\\end{equation}\nwhere $B_{-6} = B\/\\mu$G and $u_{-3} = u_{\\rm ext}\/(10^{-3}$~erg~cm$^{-3}$).\n\nThis expression has been derived assuming that the Compton cooling\nlength can be calculated in the Thomson regime, which is valid for\n$\\gamma \\lesssim 2 \\times 10^6 T_3^{-1}$ for a thermal target photon\nfield with temperature $T = 10^3 \\, T_3$~K, or $\\gamma \\lesssim 5 \\times \n10^4$ for a Ly$\\alpha$-dominated target photon field.\nIf these electrons radiate their energy by synchrotron radiation and Compton\nupscattering with the soft photon field in the Thomson regime, we can find the\ncorresponding spectral breaks for synchrotron radiation and Compton scattering\nas a function of viewing angle:\n\n\\begin{equation}\nE_{\\rm sy,br}\\cong \\gamma_c^2 B m_e c^2\/B_{cr}= \\frac{ 3m_e c^2  e B^2}{4\n\\sigma_T  u_{\\rm ext}\n\\, \\theta \\ B_{\\rm cr}}\\sim 6.15 B_{-6}^2 u_{-3}^{-1}\n\\theta^{-1}{\\rm meV}\n\\label{ViewingangleSy}\n\\end{equation}\n\n\\begin{equation}\nE_{\\rm IC, br}\\cong\\gamma_c^2 E_s = {3 \\, e \\, B \\over 4 \\, \\sigma_T \\, u_{ext}\n\\, \\theta}\n\\, E_s \\sim 5.4\\times 10^2 \\, E_{s,1}B_{-6} u_{-3}^{-1} \\theta^{-1}\\; {\\rm GeV}.\n\\label{ICbreak}\n\\end{equation}\nwhere the $B_{\\rm cr}= \\frac{m_e^2 c^3}{e \\hbar} = 4.4 \\times10^{13}~G$ and\n$E_{s,1} = E_s\/(1$~eV).\n\nTherefore, the ratio of the Compton to the synchrotron peak frequency is given by:\n\\begin{equation}\n\\frac{E_{\\rm IC, br}}{ E_{\\rm sy,br}} = \\epsilon_s \\frac{B_{\\rm cr}}{B}\n\\label{RatioComToSyn}\n\\end{equation}\nwhere $\\epsilon_s = \\frac{E_s}{m_e c^2}$.\n\nFigure \\ref{standardfig} shows that with increasing viewing angle, the\nspectral peaks of both the synchrotron and Compton emission shift to\nlower energy. This is because the Compton cooling length of the high\nenergy particles is much smaller than their Larmor radius $\\lambda_{IC}\\ll\nr_g$, so they are emitting while traveling in the forward direction. Instead,\nfor low energy particles, $\\lambda_{IC}\\geq r_g$, so that they are deflected\nbefore they are emitting. For the Compton emission this effect was already\ndiscussed in \\cite{rb10,rb11}.\n\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f4.eps}\n\\caption{\\label{ufig}The effect of a varying external radiation energy density.\nParameters: $B_x = B_y = 10^{-6}$~G; $R_{\\rm ext} = 10^{16}$~cm, $T = 10^3$~K;\n$\\alpha = 2.5$, $E_{\\gamma, {\\rm max}} = 5$~TeV. The cascade emission in the \nangular bin $0.2 \\leq \\mu \\leq 0.4$ is shown.}\n\\end{figure}\n\nFigure \\ref{ufig} shows the cascade spectra for different values\nof the external radiation field energy density $u_{\\rm ext}$. In accordance\nwith equations \\ref{ViewingangleSy} and \\ref{ICbreak}, for higher values of\nthe external radiation field, the spectral breaks of both radiation components\nshift to lower energies. Figure \\ref{ufig} also shows that the synchrotron\nluminosities of the cascades decrease with increasing $u_{\\rm ext}$ while\nthe Compton luminosities of the cascades increase. For a larger value of\n$u_{\\rm ext}$ and fixed blackbody temperature the soft target photon number\ndensity increases and $\\tau_{\\gamma\\gamma}$ becomes larger so that the number\nof VHE photons which will be absorbed increases and the photon flux of Compton\nemission from the cascades becomes larger. For very large values of $u_{\\rm ext}$,\n$\\tau_{\\gamma\\gamma}\\gg 1$ for photons above the pair production threshold so\nthat essentially all VHE photons will be absorbed and the Compton flux from\nthe cascade becomes independent of $u_{\\rm ext}$ \\cite[]{rb10,rb11}. The ratio\nof emitted power in Compton to synchrotron radiation in the linear regime\n$(\\tau_{\\gamma\\gamma} \\lesssim 1)$ is given by:\n\\begin{equation}\n\\frac{P_{sy}}{P_{IC}}=\\frac{B^2 \/ 8 \\pi}{u_{ext}}\n\\label{RatioCom}\n\\end{equation}\nif Compton scattering occurs in the Thomson regime.\nThe flux ratio $\\frac{F_{sy}}{F_{IC}} \\varpropto\nu_{\\rm ext}^{-1}$, so that by increasing the $u_{\\rm ext}$ the synchrotron\nflux decreases.\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f5.eps}\n\\caption{\\label{Bfig}The effect of a varying magnetic field strength for a\nfixed angle of $\\theta_B = 45^o$ between jet axis and magnetic field. Parameters:\n$u_{\\rm ext} = 10^{-6}$~erg~cm$^{-3}$, $R_{\\rm ext} = 10^{18}$~cm, $T = 1000$~K,\n$\\alpha = 2.5$, {$E_{\\gamma, {\\rm max}} = 5$~TeV}. The cascade emission in the\nangular bin $0.2\\leq\\mu\\leq0.4$ is shown. The solid green line represents \nthe target photon field.\n}\n\\end{figure}\n\nFigure \\ref{Bfig} illustrates the effect of a varying magnetic field strength\nfor fixed magnetic field\norientation ($\\theta_B = 45^o$). We see that the synchrotron peak energy\nincreases proportional to the square of the magnetic field strength as expected\nfrom Eq. \\ref{ViewingangleSy}. As already discussed in \\cite{rb10,rb11} the\nCompton energy break is proportional to the magnetic field strength. as long\nas it occurs below the $\\gamma\\gamma$ absorption cut-off energy. The synchrotron\nflux is proportional to the square of the magnetic field strength.\nThe flux ratio is $\\frac{F_{sy}}{F_{IC}} \\varpropto B^2$ until the fluxes become\ncomparable, at which point our treatment of synchrotron losses breaks down.\n\n\\begin{figure}[ht]\n\\vskip 1cm\n   \\centerline{\n        \\includegraphics[width=8cm,height=8cm]{f6a.eps}\n        \\includegraphics[width=8cm,height=8cm]{f6b.eps}}\n        \\caption{\\label{BfigF}The effect of a varying magnetic field orientation\nfor a fixed magnetic field strength of $B = 1 \\, \\mu$G, $u_{\\rm ext} =\n10^{-6}$~erg~cm$^{-3}$, $R_{\\rm ext} = 10^{18}$~cm, $T = 1000$~K, $\\alpha = 2.5$,\n{$E_{\\gamma, {\\rm max}} = 5$~TeV}. Left figure: angular bin $0.8 \\leq\\mu\\leq 1.0$\n(dominated by the forward direction, i.e., the blazar case). Right figure: angular \nbin $0.2 \\le \\mu \\le 0.4$, representative of radio galaxies.\nThe green solid lines represent the target photon fields.\n}\n    \\end{figure}\n\nFigure \\ref{BfigF} illustrates the effects of a varying magnetic-field orientation\nwith respect to the jet axis, for fixed magnetic-field strength $B = 1 \\, \\mu$G\nfor different angular bins. The results for the Compton component have been\ndiscussed \\cite{rb11}. The figure illustrates that primarily the perpendicular\n($B_y$) component of the magnetic field is responsible for synchrotron radiation.\n\n\n\\begin{figure}[ht]\n\\vskip 1cm\n   \\centerline{\n        \\includegraphics[width=8cm,height=8cm]{f7a.eps}\n        \\hskip 1cm\n        \\includegraphics[width=8cm,height=8cm]{f7b.eps}}\n        \\caption{\\label{fitCenAandNGC1275}Compton and synchrotron radiation form\n        the cascades. Left figure: (Fit to the SED of Cen~A.); Right figure:\n        (spectrum of NGC~1275 with a simulated cascade spectrum from a mis-aligned\n        blazar, along with the cascade spectra at larger viewing angles)}\n     \\end{figure}\n\nFigures \\ref{fitCenAandNGC1275} illustrate that the cascades emissions from\nthe synchrotron radiation for parameters used in \\cite[]{rb10,rb11} are\nnegligible compared to the cascade Compton emission and much smaller than\nthe synchrotron radiation from the jet itself. This confirms that neglecting\nsynchrotron radiation in our previous works was justified.\n\n\\section{\\label{degeneracy}Magnetic Field degeneracy}\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f8.eps}\n\\caption{\\label{Degeneracy1}\nSynchrotron and Compton emission form the cascades for NGC~1275\n($0.6 \\leq\\mu\\leq 0.8 $). Parameters: $\\theta_B = 11^o$;\n$u_{\\rm ext} = 5\\times10^{-2}$~erg~cm$^{-3}$, $R_{\\rm ext} = 10^{16}$~cm,\n$E_s=E_{L\\alpha}$,\n$\\alpha = 2.5$, {$E_{\\gamma, {\\rm max}} = 5$~TeV}.\n}\n\\end{figure}\n\nIn \\cite{rb10}, we presented a fit to the \\emph{Fermi} spectrum of the radio\ngalaxy NGC~1275. We now show that there is a degeneracy of the magnetic field,\nboth orientation and strength, if only the high energy output from the cascades\nis considered. Figure \\ref{Degeneracy1} shows this effect for NGC~1275. In this\nplot, the external radiation field is\nparameterized through $u_{\\rm ext} = 5 \\times 10^{-2}$~erg~cm$^{-3}$ with photon\nenergy $E_s = E_{Ly\\alpha}$ and $R_{\\rm ext} = 10^{16}$~cm. This size scale is\nappropriate for low-luminosity AGN as observed in NGC~1275 \\citep[e.g.][]{kaspi07},\nand the parameters combine\nto a BLR luminosity of $L_{\\rm BLR} = 4 \\pi R_{\\rm ext}^2 \\, c \\, u_{\\rm ext} =\n1.9\\times 10^{42}$~erg~s$^{-1}$, in agreement with the observed value for NGC~1275.\nThe magnetic field orientation is at an angle of $\\theta_B = 11^o$. \nThe mass of the black hole in NGC~1275 is uncertain, and estimates \nrange from a few times $10^6 \\, M_{\\odot}$ \\citep{levinson95} to \n$\\sim 10^8 \\, M_{\\odot}$ \\citep{wilman05}. Assuming a characteristic\nfraction of $0.1$ of the accretion-disk luminosity to be re-processed\nin the BLR, the accretion-disk luminosity may be estimated to be $L_D\n\\sim 10^{43}$~erg~s$^{-1}$. Figure \\ref{diskabs} shows the $\\gamma\\gamma$\nabsorption depth due to the disk radiation field for $L_D = 10^{43}$~erg~s$^{-1}$\nfor the two possible extreme values of the black-hole mass, as a function\nof height $z$ of the emission region above the accretion disk. It \nshows that for $M_{\\rm BH} = 10^6 \\, M_{\\odot}$, the $\\gamma\\gamma$\nopacity drops below one at $\\sim 10^3 \\, r_g \\sim 10^{14}$~cm from \nthe black hole, while for $M_{\\rm BH} = 10^8 \\, M_{\\odot}$ $\\gamma\\gamma$\nabsorption becomes negligible at $\\sim 10^2 \\, r_g \\sim \\sim 10^{15}$~cm \nfor primary $\\gamma$-rays of $E_{\\gamma} = 1$~TeV, and much earlier for\nlower-energy photons. Therefore, throughout most of our simulation \nvolume ($R_{\\rm ext} = 10^{16}$~cm), $\\gamma\\gamma$ absorption in\nthe disk radiation field can be safely neglected.\n\nThe cascade spectrum shown in Figure \\ref{Degeneracy1} pertains\nto the angular bin $0.6 < \\mu < 0.8$ (corresponding to $37^o \\lesssim\n\\theta \\lesssim 53^o$), appropriate for the known orientation of NGC~1275. In\n\\cite[]{rb10,rb11}, we have shown that for magnetic field values of $B\\geq 1$~nG\nand for energy density $u_{ext} \\geq 10^{-3}$~erg~cm$^{-3}$, there is no pronounced\nbreak in the cascade spectrum and the cascade is independent of magnetic field.\nIn general we expect no break in the cascade Compton emission if $E_{\\rm IC,br}\n\\gtrsim \\frac{(mc^2)^2}{E_s}$, which leads to the condition:\n\n\\begin{equation}\nB \\gtrsim \\frac{{(m_e c^2)}^2 4\\sigma_T u_{\\rm ext} \\theta}{3 e (E_s)^2} \\sim 5 \\,\nu_{ext,-3} E_{s,1}^{-2} \\theta \\; {\\rm nG}\n\\label{relation}\n\\end{equation}\n\nFigure \\ref{Degeneracy1} shows that while the high energy emission due to\ndeflection of the cascade up to the $\\gamma\\gamma$ absorption trough remains\nthe same for the different magnetic fields, the synchrotron emission from the\ncascade changes. Therefore, determining the B field requires knowledge of the\nsynchrotron emission.\n\nIn the regime where $E_{\\rm IC, br}$ is independent of the magnetic field,\n$\\nu_{\\rm sy}\\varpropto B$ according to Eq. \\ref{RatioComToSyn} and the synchrotron\npower is proportional to the square of magnetic field in agreement with figure\n\\ref{Degeneracy1}.\n\nSince the synchrotron\/Compton flux ratio $\\frac{F_{\\rm sy}}{F_{\\rm IC}}\n\\varpropto B^2$, we expect that for sufficiently high magnetic fields,\nwe will reach the regime where the Compton flux from the cascades is\nequal to or smaller than the synchrotron flux in which case our numerical\nscheme is no longer applicable.\n\n\n\\section{\\label{3C279}The Big Blue Bump}\n\nThe spectral Energy distribution of AGN in the ultraviolet (UV) to soft X-ray\nband ($ \\sim 10$~eV-$1$~keV) is notoriously difficult to observe because of\ndust and gas in our galaxy and the AGN environment. The SEDs of many blazars\nexhibits a UV soft X-ray excess, called the big blue bump (BBB)\n\\citep[]{pian99,palma11,raiteri05,raiteri06,raiteri07}. It is often\nattributed to the thermal emission from the accretion disk. In blazars,\nits signature is often particularly hard to detect because of dominant\nnon-thermal emission from the jet. Understanding the origin of the BBB\nis important since this provides information on the central engine of\nthe AGN.\n\n3C~279 was among the first blazars discovered as a $\\gamma$-ray source with\nthe Compton Gamma-Ray Observatory \\citep[]{hartman92}. In 2007 it was detected\nas a VHE $\\gamma$-ray source with the MAGIC I telescope, making it the most\ndistant known VHE $\\gamma$-ray source at a redshift of $0.536$ \\citep[]{HB93}.\nIts relativistic jet is oriented at a small angle to the line of the sight\nof $< 0.5^0$ \\citep[]{J04}. It is also detected by \\emph{Fermi} \\citep[]{abdo09c}\nwith photon spectral index $2.23$. There is evidence of a spectral break of\naround a few GeV to a photon spectral index of $2.50$. It is strongly believed\nthat the radio to optical emission is due to synchrotron radiation by relativistic\nparticles in the jet. However, the origin of the high energy emission is still not\nwell understood \\citep[see, e.g.,][]{br09}.\n\n\\cite{pian99} monitored 3C~279 in the ultraviolet, using IUE, and combined\ntheir data with higher-energy observations from ROSAT and EGRET from 1992 December\nto 1993 January. During this period, the source was in a very low state, allowing\nfor the detection of a UV excess (the BBB), which is typically hidden below a\ndominant power-law continuum attributed to non-thermal emission from the jet.\n\\cite{pian99} proposed that the $\\gamma$-ray emission in the SED of 3C~279\nis produced by the external Compton mechanism, and suggested that the observed\nUV excess might be due to thermal emission from an accretion disk.\n\nAs an alternative to thermal emission from the accretion disk, \\cite{S97} proposed\nthe bulk Compton mechanism as a possible explanation of a UV\/X-ray excess in quasar\nSEDs. If the jet contains a substantial population of cold (i.e., thermal,\nnon-relativistic or mildly relativistic) electrons, they could scatter\nexternal optical\/UV photons with the bulk Lorentz factor of $\\Gamma \\thicksim 10$,\nresulting in bulk Compton radiation in the far UV or soft X-ray range.\n\nHere we suggest an alternative contribution to the BBB feature from\ncascade synchrotron emission. Figures 2 -- 4\nillustrate that the synchrotron emission from cascades may peak in the UV\/X-ray\nrange, thus mimicking a BBB for sufficiently strong magnetic fields\n($B \\gtrsim 1$~mG). Figure \\ref{fit3C279} illustrates \nthe contribution that synchrotron emission from VHE $\\gamma$-ray\ninduced pair cascades can make to the BBB in 3C~279.\nThe primary HE $ \\gamma$-ray spectrum with a photon spectral index of\n$\\alpha = 2.5$ matcheds the Fermi spectrum of 3C~279.\nThe external radiation field is parameterized through $u_{\\rm ext} =\n10^{-4}$~erg~cm$^{-3}$ and $R_{\\rm ext} = 5\\times10^{17}$~cm,\nand the parameters combine to the luminosity of $L = 4 \\pi R_{\\rm ext}^2 \\, c\n\\, u_{\\rm ext}\n\\sim 10^{43}$~erg~s$^{-1}$ corresponding to a $\\nu F_{\\nu}$ peak\nflux of $\\thicksim 10^9$~JyHz, about $2$ orders of magnitude below the observed\nIR\/optical -- UV flux level. The magnetic field is $B = 10^{-2}$~G, oriented at\nan angle of $\\theta_B = 85^o$. The incident $\\gamma$-ray spectrum extends out\nto $E_{\\gamma, {\\rm max}} = 5$~TeV, and the external radiation field is modeled\nas a blackbody with a temperature of $ T= 2000$~K (corresponding to a peak of the\nblackbody spectrum at a photon energy of $E_s^{\\rm pk}= 0.5$~eV). This leads\nto a $\\gamma\\gamma$ absorption cut-off at an energy $E_c = (m_e c^2)^2 \/ E_s\n\\sim 1$~TeV.\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=10cm]{f9.eps}\n\\caption{\\label{fit3C279}Illustration of a possible BBB in 3C~279 from\ncascade synchrotron emission.\nParameters: $B= 10^{-2}$~G, $\\theta_B = 85^0$;\n $R_{\\rm ext} = 5\\times10^{17}$~cm, $T = 2000$~K,\n$\\alpha = 2.37$, $u_{\\rm ext} = 10^{-4}$~erg~cm$^{-3}$, {$E_{\\gamma, {\\rm max}}\n= 5$~TeV}. Data from \\cite{abdo10}.\n}\n\\end{figure}\n\nWe suggest that synchrotron emission from VHE $\\gamma$-ray induced pair\ncascades can enhance the BBB feature in the SEDs of several\nblazars such as 3C~279. An observational test of this hypothesis may be\nprovided through spectropolarimetry. A BBB due to (unpolarized) thermal\nemission from an accretion disk will produce a decreasing percentage of\npolarization with increasing frequency throughout the optical\/UV range.\nIn contrast, if the BBB is produced as synchrotron emission from cascade\npairs in globally ordered magnetic fields, it is also expected to be polarized.\nTherefore, we predict that a BBB due to cascade synchrotron emission would result\nin a degree of polarization showing only a weak dependence on frequency over the\noptical\/UV range. As an example, in recent observations of the high-redshift\n$\\gamma$-ray loud quasar PKS~0528+134, \\cite{palma11} found a decreasing\ndegree of polarization with increasing frequency throughout the optical range,\narguing for an increasing contribution from thermal emission towards the blue\nend of the optical spectrum.\n\n\n\\section{\\label{summary}Summary}\n\nWe investigated the magnetic-field dependence and synchrotron emission\nsignatures of Compton-supported pair cascades initiated by the interaction\nof nuclear VHE $\\gamma$-rays with arbitrary external radiation fields, \nfor a model-independent, generic power-law shape of the primary\nVHE $\\gamma$-ray emission.\nWe\nfollow the spatial development of the cascade in full 3-dimensional geometry\nand study the dependence of the radiative\noutput on various parameters pertaining to the external radiation field and\nthe magnetic field in the cascade region.\nWe confirm that synchrotron radiation from the cascades is negligible in\nNGC~1275 and Cen~A for the parameters we used in our previous works. We\ndemonstrated that the magnetic field can not be well constrained by considering\nthe high-energy (Compton) output from the cascade emission alone, without\nobservational signatures from their synchrotron emission. This was illustrated\nfor the case of NGC~1275, for which we could produce equally acceptable fits\nto the Fermi spectrum for a variety of magnetic-field values, which resulted\nin substantially different synchrotron signatures.\n\nWe have shown that synchrotron emission from VHE $\\gamma$-ray induced pair\ncascades may produce UV\/X-ray signatures resembling the BBB observed in the\nSEDs of several blazars, in particular in their low states. \nWe used the example of 3C~279 to illustrate that cascade synchrotron\nemission may make a substantial contribution to the BBB feature.\nWe point out that spectropolarimetry may serve\nas a possible observational test to distinguish a thermal from a non-thermal\n(cascade) origin of the BBB.\n\n\\acknowledgements{ This work was supported by NASA through Fermi Guest\nInvestigator Grants NNX09AT81G and NNX10AO49G. We thank the anonymous \nreferee for valuable suggestions. }\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzcduh b/data_all_eng_slimpj/shuffled/split2/finalzzcduh
new file mode 100644
index 0000000000000000000000000000000000000000..729e1e13a819fbd3a466cdcf15c534755cc26ae6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzcduh
@@ -0,0 +1,5 @@
+{"text":"\\section{The new cosmological context}\n\n\\subsection{Model expansion versus universe expansion}\n\nIt is currently assumed that matter does not expand itself during an\neventual universe expansion. {\\it If this were true} some standard\nmodel could be used as the base for a non expanding theoretical\nreference framework. In it, ${dr(ik)}\/{r(ik)}=Hdt $. Then, from (1.1),\n(1.2), and NL mass-energy conservation,\n\\begin{equation}\n\\label{2.1}\nd\\phi (i)=-dw(i)=-\\,d\\sum_{k=1}^\\infty \\frac{Gm(k)f(ik)}{r(ik)}%\n=\\left[ \\sum_{k=1}^\\infty \\frac{Gm(k)f(ik)}{r(ik)}\\right] Hdt,\n\\end{equation}\n\\begin{equation}\n\\label{2.2}\nd\\phi \\left( i\\right) =w(U)Hdt=Hdt=dr(ik)\/r(ik)=d\\lambda\n(i)\/\\lambda (i).\n\\end{equation}\n\nThis means that {\\it every wave and particle would expand itself in\nthe same proportion as the intergalactic distances}. An eventual\nuniverse expansion would not change the relative values of all of\nthem: the distances, the velocities, the temperatures, the WRS, the\nHRS, and therefore, the local physical laws\\footnote {This\ngeneralizes the relativity postulates for eventual universe\nexpansions.}. {\\it Then the universe would have not a well-defined\nage and it may last indefinitely}. Of course, this would invalid the\ncurrent deductions normally made for the universe age, which anyway\nseem to be not consistent with the last measurements made with the\nHubble telescope.\n\n\\subsection{The new kind of black hole (BH)}\n\nThe new exponential G relations have not a singularity at $r=2GM$.\nThen, the new kind of BH is different to that of GR \\cite{V81b}. Its\nnucleus would be just a neutron star (NS) with a strong external {\\it\ngradient of the NL refraction index } that would act as a mirror for\nmost of the internal radiation's. Its outcoming {\\it critical\nreflection angle}, given\nby $sin^{-1}[\\left( 2eGM\/r\\right) e^{-2GM\/r}]$, would be rather\nnegligible. Thus the escape probabilities would be not strictly null.\nThen the BH would absorb and store for long time most of the\nradiation's traveling within the impact parameter $2eGM$.\n\n\\subsection{Relativistic particle generation}\n\nIn a way similar to the earth {\\it auroras}, most of the positively\ncharged nuclei would be driven by the magnetic fields towards the\nBH polar regions. Since the neutron binding energy in a BH is of a\nhigher order of magnitude compared with that in atoms, then one the\nmost probable reactions between them is {\\it nuclear stripping} \\cite\n{V77}, \\cite{V81b}. In it, some of the atomic neutrons\nwould be captured by the BH while the remaining nucleus (proton or\nproton rich nucleus) would be rejected by the NS. The last one\nwould take away the NL mass-energy difference between the\noriginal and final states of the captured neutrons. They could only\nescape from the magnetic fields, in axial directions, within the small\nescape angle given above. They would form {\\it narrow jets of\nrelativistic particles} richer in protons and with higher energies for\nhigher $p\/n$ ratios. They are consistent with the composition and\nenergies of {\\it cosmic ray particles} \\cite{V81b}, \\cite\n{R81}. They are also consistent with the {\\it radio sources\nand jets} going away from central regions of galaxies, most of them\nin just the expected orientations.\n\nThis process would be most important because it would convert G\nwork into mechanical and nuclear latent energies. This would\nregenerate new gas of high nuclear and kinetic energies at the cost\nof rather burnt out materials like He or heavier elements. Such low\nentropy materials can in principle extend the luminous lifetimes of\ngalaxies beyond the limits estimated from the current models.\n\n\\subsection{The entropy switch}\n\n{}From the BH surface, just to the contrary of the outside regions, the\nexternal universe would look as a source of {\\it blue shifted\nradiation's} that would increase both the local temperature and the\nprobabilities for filling up the local SW levels up to the highest NL\nfrequencies. This is equivalent to a decrease of the local entropy. In\nthis way the average NL mass and kinetic energy of the nucleons\nwould increase with the time, with the radiation energy coming from\nthe rest of the universe, up to some unstable state in which any\ndecrease of the NL refraction index gradient generated by external\nbodies would produce {\\it frustrated reflections} that can trigger the\nmass outflow. Thus the BH can {\\it explode} producing low density\ngas flowing away through older stellar remnants orbiting around it.\nThis would transform a fraction of the kinetic energy into rotational\none associated with randomly oriented angular momentum's. This is\nalso consistent with the fronts of H rich matter diverging from very\nsmall regions in the universe.\n\n\\section{The new astrophysical context}\n\n{}From above the universe would last, indefinitely, in a kind of\nconservative and {\\it isentropic steady state}.  In it, {\\it matter and\nradiation's would evolve, indefinitely, in rather closed cycles,\nbetween the states of gas and BHs, and vice versa}.\n\n\\subsection{Matter cycles}\n\nSingle and chain of BH explosions would produce {\\it rather\nspherical stellar clusters and elliptical galaxies,  rather free of\nmetals}. They would regenerate randomly oriented angular\nmomentum's that, in the long run, would be canceled out at faster\nrates compared with those parallel to the galactic axis. Thus an {\\it\nelliptical galaxy} would progressively get {\\it disc and spiral shapes}\nof smaller volumes. Finally it would become reduced to a small\ncentral luminous volume (AGN and {\\it quasar}) with massive stars\nand high density black bodies (black holes, neutron stars)\nsurrounded by a halo of dead stars and planetesimals [{\\it black\ngalaxy}]. The explosive events as supernovas would produce large\nchanges of luminosity, within relatively short periods, that are\nconsistent with those of quasars \\cite {N90}.\n\nDue to the low $\\phi (r)$ in the black galaxy center, their atoms\nwould emit strongly red shifted light rather scattered and reflected by\nthe external bodies. This accounts for the fact that quasar\ncorrelation's improve under the assumption that most of the\nobserved red shift is intrinsic \\cite {B73}. The detection of\nmetal lines would also prove the existence of highly evolved (old)\nmatter.\n\nThe {\\it black galaxy} (BG) resulting from a luminous one would be\ncooled down by its BHs. It would also capture and store radiation\ncoming from the external universe, in a way similar to a huge BH.\nAfter a long period, the explosion of some central BH can trigger a\nchain of BH explosions that would regenerate a luminous galaxy.\n\nWithin a larger time scale, the galaxy regeneration would look like a\nBG explosion that can trigger the virtual explosions of the next BGs,\nand so on. They would produce {\\it clusters}. Superclusters would also\nbe due to similar mechanisms. Thus the {\\it fronts} of galaxies in\nluminous stages would also account for the large scale structure of\nthe universe.\n\n\\subsection{High energy step down in stellar objects}\n\nMechanisms of nuclear stripping similar to those occurring BHs\ncould also occur during the matter fall over neutron stars (NSs),\neither steadily or in pulsed ways. They may also occur rather hidden\ninside some stars or gas clouds. They would transform heavy (burnt\nout) elements into protons of higher kinetic and nuclear latent\nenergies that would promote convection currents. This would\nprevent overheating and stellar collapse after neutrino cooling.\n\nThis kind of stellar model, \\cite{V93}, is consistent with all of\nthem: the low neutrino luminosity's, the higher densities and\ntemperatures, the better defined mass-luminosity's relations and the\nmagnetic structures of {\\it main sequence stars}.\n\n\\subsection{Density and isotropy of the universe}\n\nDue to the higher rates of energy emitted by the luminous galaxies\ncompared with those absorbed by the BGs, it is inferred (after a\nmass-energy balance) that {\\it most of the universe should be in the\nstate of low temperature BGs}, cooled down by their own BHs. This\nis consistent with the high average density of the universe derived\nfrom (1.1), and assuming $H=75 km\/\\sec $ per mps\\footnote {When\nthe common mass and energy unit is the joule, $G = G_{newt} c^\n4$.}.This one is $ \\simeq [4\\pi GR^2]^{-1}$, i. e.,  about $ 10^{-29}\ngm\/cm^3 $. This is of a higher order of magnitude than that of the\nluminous fractions of the universe. This is also consistent with the\ncurrent mass excesses detected from dynamic methods in galaxies\nand clusters.\n\nAfter integration of (2), the space properties are fixed, mostly, by\nmatter existing between $R$ and $3R$. The contribution of\nrelatively local matter is extremely small compared with that of the\nrather uniform universe. This is consistent with {\\it the weakness of\nordinary G interactions, and with  the high isotropy of both the space\nproperties and of the cosmological radiation background}.\n\nThe low temperature black-body radiation coming from BGs, red\nshifted during its long average trip up to the observer, ($2R $),\nwould fix {\\it a rather uniform low temperature cosmic radiation\nbackground}. Thus {\\it the universe would always look like a perfect\nradiation absorber}\\footnote {Only steady state cosmologies can\naccount for the {\\it arrows} in nature \\cite {N90}.}.\n\n\\section{Conclusions}\n\nThe theoretical properties of the SW particle model fix a new kind of\n{\\it conservative and isentropic steady state} in which matter and\nradiation's evolve, indefinitely, in rather closed cycles. These cycles\nare fairly consistent with the luminous bodies ranging between\nelliptical galaxies and quasars, and also with larger scale structures\nof the universe.\n\nThis theory opens the way for new stellar models and non\nconventional interpretations of many celestial phenomena. The new\nuniverse would have not the narrow limits of time fixed by the rather\nconventional theories. In this way, also,  astrophysics could do\nwithout the relatively large number of non testable hypotheses that\ncan be advanced on the universe origin.\n\nThere is simultaneous consistency of the theoretical properties of\nthe SW model with fundamental physics, and of the new\ncosmological context with a wide range of astronomical\nobservations. This seems to be a fair reliability test for all of them,\nthe SW particle model, for the relationships derived from it, and for the\nnew cosmological and astrophysical contexts. This unified way may\ncontribute to understand nature in terms of the most elemental\nproperties of radiation's (or vice versa), thus depending on the\nminimum number of parameters, postulates, and arbitrary\nassumptions normally made on relations between matter and its G field.\n\nDue to the large amount of subjects and materials accumulated\nfrom 1976 up to day, the author intends to complilate all of this\nwork into a single book for that may be useful to those that may\nlike to go in this way for undestanding nature from a self-consistent\nand unified viewpoint\\cite{V95c}.\n\nAcknowledgement. I appreciate very much the help of TW Andrews, after\nsending me some helpful literature, including his own ideas. I would also\nappreciate some encouragement and colaboration for finding\nfurther astronomical tests for this theory.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\tMany measurement operations in signal and image processing as well as in communication follow a bilinear model. Namely, in addition to the measurements depending linearly on the unknown signal, also certain parameters of the measurement procedure enter in a linear fashion. Hence one cannot employ a linear model (for example, in connection compressed sensing techniques \\cite{candes2006robust}) unless one has an accurate estimate of these parameters.\n\n\tWhen such estimates are not available or too expensive to obtain, there are certain asymmetric scenarios when one of the inputs can be recovered even though the other one is out of reach (e.g., \\cite{xu1995least,lee2017spectral}, this scenario is sometimes referred to as passive imaging). In most cases, however, the natural aim will be to recover both the signal and the parameters, that is, to solve the associated bilinear inverse problem. Even when some estimates of the parameters are available, such a unified approach will be preferred in many situations, especially when information is limited. Consequently, the study of bilinear inverse problems, including but not limited to the important problem of blind deconvolution, has been an active area of research for many years \\cite{Haykin1994}.\n\t\n\tObserving that bilinear maps admit a representation as a linear map in the rank one outer product of the unknown signal and the parameter vector, one can approach such problems using tools from the theory of low-rank recovery (see, e.g., \\cite{ahmedrechtromberg,lingstrohmer,jung2017blind}). Under sparsity assumptions, that is, when the signals and\/or parameter vectors admit an approximate representation using just a small (but unknown) subset of an appropriate basis (for more details regarding when such assumptions appear in bilinear inverse problems, see  \\cite{strohmer}), however, the direct applicability of these approaches is limited, as two competing objectives arise: one aims to simultaneously minimize rank and sparsity. As a consequence, the problem becomes considerably more difficult; Oymak et al., for example, have demonstrated that minimizing linear combinations of the nuclear norm (a standard convex proxy for the rank) and the $\\ell_1$ norm (the corresponding quantity for sparsity) exhibits suboptimal scaling \\cite{oymak2015simultaneously}.\tIn fact it is not even clear if without additional assumptions efficient recovery is at all possible for a near-linear number of measurements (as it would be predicted identifiability considerations \\cite{doi:10.1137\/16M1067469}).\n\t\n\tRecently, a number of nonconvex algorithms for bilinear inverse problems have been proposed. For example, for such problems without sparsity constraints several such algorithms have been analyzed for blind deconvolution and related problems\\cite{li2016rapid,ling2017regularized} with near-optimal recovery guarantees. In contrast, our understanding of bilinear inverse problems with sparsity constraints is only in its beginning. Recently, several algorithms have been analyzed for sparse phase retrieval  \\cite{bahmani2017anchored,soltanolkotabi2017structured} or blind deconvolution with sparsity constraints \\cite{qu2017convolutional}. The recovery guarantees for these algorithms, however, are either suboptimal in the number of necessary measurements or only local convergence guarantees are available, i.e., one relies on the existence of a good initialization. (A noteworthy exception are the two related papers \\cite{bahmani2016near,iwen2017robust}, where a two-stage approach for (sparsity) constrained bilinear inverse problems is proposed, which achieves recovery at near-optimal rate. However, the algorithm relies on a special nested structure of the measurements, which is not feasible for many practical applications.)\n\t\n\tIn \\cite{paper2} Lee, Wu, and Bresler  introduced the {\\em sparse power factorization} (SPF) method together with a tractable initialization procedure based on  alternating minimization. They also provide a first performance analysis of their method for random bilinear measurements in the sense that their lifted representation is a matrix with independent Gaussian entries.\n\tThat is, they work with linear operators $\\mathcal{A}\\colon \\mathbb{C}^{n_1 \\times n_2} \\longrightarrow \\mathbb{C}^{m} $ that admit a representation as\n\t\\begin{equation*}\n\t\\left(  \\mathcal{A} \\left( X \\right)   \\right) \\left( \\ell \\right)= \\text{trace} \\left( A_{\\ell}^\\ast X  \\right)\n\t\\end{equation*}\n\tfor i.\\,i.\\,d.\\ Gaussian matrices $ A_{\\ell}  \\in \\mathbb{C}^{n_1 \\times n_2} $.\n\t\n\t\n\tFor such measurements they show that with high probability, SPF converges locally to the right solution, i.e., one has convergence for initializations not too far from the signal to be recovered. \n\t\n\t\n\tFor signals that have a very large entry, they also devise a tractable initialization procedure -- they call it thresholding initialization -- such that one has global convergence to the right solution. Local convergence has also been shown for the multi-penalty approach {\\em A-T-LAS$_{1,2}$} \\cite{fornasier2018}, but to our knowledge, comparable global recovery guarantees are not available to date. This is why we focus on SPF in this paper, using the results of \\cite{paper2} as our starting point.\n\t\n\tThe precise condition for their guarantee to hold is that both (normalized) input signals need to be larger than some $c>\\tfrac{1}{2}$ in supremum norm -- more than one quarter of its mass needs to be located in just one entry, that is, the signals must have a very high peak to average power ratio.\n\t\n\t In this paper, we considerably weaken this rather strong restriction in two ways. Firstly, we show that similar results hold for smaller lower bounds $c$ at the expense of a moderately increased number of measurements. Secondly, we show that similar results can be obtained when the mass of one of the signals is concentrated in more than one, but still a small number of entries.\n\t\n\t\n\tThe SPF algorithm, the thresholding initialization, and the resulting recovery guarantees are reviewed in \\Cref{spfsection} before we discuss and prove our results in \\Cref{resultsection} and Section \\Cref{sectionproof}.\n\t\n\t\\subsection*{Notation}\n\tThroughout the paper we will use the following notation. By $ \\left[n\\right] $ we will denote the set $ \\left\\{1; \\ldots; n \\right\\} $. For any set $J$ we will denote its cardinality by $ \\vert J \\vert $. For a vector $v\\in \\mathbb{C}^m$ we will denote by $ \\Vert v \\Vert $ its $ \\ell_2$-norm and by $ \\Vert v \\Vert_{\\infty}$ the modulus of its largest entry. If $J \\subset \\left[n\\right] $ we will by $v_J$ denote the restriction of $v$ to elements indexed by $J$. For matrices $ A \\in \\mathbb{C}^{n_1 \\times n_2} $ we will denote by $ \\Vert A \\Vert_F$ its Frobenius norm and by $\\Vert A \\Vert $ its spectral norm, i.e., the largest singular value of $A$. \t\n\t\n\t\\section{Sparse Power Factorization: Algorithm and Initialization}\\label{spfsection}\n\t\n\t\\subsection{Problem formulation}\n\t\n\n\n\n\n\n\t\n\tLet $ b \\in \\mathbb{C}^m $ be given by\n\t\\begin{equation*}\n\tb:= B(u,v)+z,\n\t\\end{equation*}\n\twhere $ B \\colon \\mathbb{C}^{n_1} \\times \\mathbb{C}^{n_2} \\rightarrow \\mathbb{C}^m $ is a bilinear map and $z\\in \\mathbb{C}^m$ is noise. Recall that one can represent the bilinear map $B \\colon \\mathbb{C}^{n_1}\\times \\mathbb{C}^{n_2} \\rightarrow \\mathbb{C}^{m}$ by a linear map\n\t$\\mathcal{A}\\colon\\mathbb{C}^{n_1\\times n_2}\\longrightarrow\\mathbb{C}^m$, which satisfies\n\t\\[\n\tB(u,v)= \\mathcal{A}(uv^\\ast).\n\t\\]\n\tfor all vectors $u \\in \\mathbb{C}^{n_1}$ and all $ v \\in \\mathbb{C}^{n_2}$. Note that such a linear map $ \\mathcal{A}$ is characterized by a (unique) set of matrices $ \\left\\{ A_{\\ell}  \\right\\}^m_{\\ell =1}  \\subset \\mathbb{C}^{n_1 \\times n_2} $ such that the $\\ell$th entry of $ \\mathcal{A}\\left( X \\right)$ is given by\n\t\\begin{equation}\\label{operatorrepresentation}\n\t\\left(  \\mathcal{A} \\left( X \\right)   \\right) \\left( \\ell \\right)= \\text{trace} \\left( A_{\\ell}^\\ast X  \\right).\n\t\\end{equation}\n\tIn this notation, our goal will be to reconstruct $u$ and $v$ from linear measurements given by\n\t\\begin{equation*}\n\tb_{\\ell} = \\text{trace} \\left( A^*_{\\ell} uv^* \\right)\n\t\\end{equation*}\n\tAt the core of the Sparse Power Factorization Algorithm, as introduced in \\cite{paper2}, are the linear operators $F \\colon \\mathbb{C}^{n_2} \\longrightarrow \\mathbb{C}^{m \\times n_1} $ and $ G \\colon \\mathbb{C}^{n_1} \\longrightarrow \\mathbb{C}^{m \\times n_2} $ defined by\n\t\n\t\t$$F(y) := \n\t\t\\begin{pmatrix}\n\t\ty^\\ast A_1^\\ast\\\\\n\t\t\\vdots\\\\\n\t\ty^\\ast A_m^\\ast\n\t\t\\end{pmatrix},\n\t\t\\quad\n\t\tG(x) := \n\t\t\\begin{pmatrix}\n\t\tx^\\ast A_1\\\\\n\t\t\\vdots\\\\\n\t\tx^\\ast A_m\n\t\t\\end{pmatrix}.$$\n\t\tA direct consequence of this definition is that $$\\mathcal{A}(xy^\\ast) = [F(y)]x = \\overline{[G(x)]y}$$ for all $  x \\in \\mathbb{C}^{n_1} $ and all $ y \\in \\mathbb{C}^{n_2} $.\n\n\n\t\\subsection{Sparse Power Factorization}\n\t\nThe idea of Sparse Power Factorization is to iteratively update estimates $u_t$ and $v_t$ for $u$ and $v$ in an alternating fashion.\nThat is, in each iteration one keeps one of $v_t$ and $u_t$ fixed and updates the respective other one by solving an (underdetermined) linear system. Solving each of these linear systems then amounts to solving a linear inverse problem with sparsity constraints. Hence, many pursuit algorithms proposed in the context of compressed sensing can be applied such as CoSaMP \\cite{needell2009cosamp}, Hard Thresholding Pursuit \\cite{foucart2011hard} or Basis Pursuit. In \\cite{paper2} the authors used Hard Thresholding Pursuit (HTP) for their analysis and in this paper, we will also restrict ourselves to HTP. With this, the Sparse Power Factorization Algorithm reads as follows.\n\t\n\t\n\\begin{alg}[Algorithm 1 in \\cite{paper2}]\\label{SPF.alg}~\\\\\\vspace*{-4mm}\n\t\\begin{algorithmic}[1]\n\t\t\\sffamily\\small\n\t\t\\Require  Operator $\\mathcal{A}$, Measurement $b$, Sparsity Constraints $s_1, s_2$, Initialisation $v_0$.\n\t\t\\Ensure Estimate $\\widehat{X}$.\n\t\t\\State $t \\gets 0$\n\t\t\\While{stop condition not satisfied}\n\t\t\\State $t \\gets t + 1$\n\t\t\\State $v_{t - 1} \\gets \\frac{v_{t - 1}}{\\big\\|v_{t - 1}\\big\\|}$\\label{vt1norm}\n\t\t\\If{$s_1 < n_1$}\n\t\t\\State $u_t \\gets \\mathrm{HTP}(\\mathrm{F}(v_{t - 1}), b, s_1)$\n\t\t\\Else\n\t\t\\State $u_t \\gets \\operatorname*{{arg\\,min}}\\limits_x \\big\\|b - [\\mathrm{F}(v_{t - 1})]x\\big\\|^2$\n\t\t\\EndIf\n\t\t\\State $u_t \\gets \\frac{u_{t}}{\\big\\|u_{t}\\big\\|}$\n\t\t\\If{$s_2 < n_2$}\n\t\t\\State $v_t \\gets \\mathrm{HTP}(\\mathrm{G}(u_{t}), \\bar{b}, s_2)$\n\t\t\\Else\n\t\t\\State $v_t \\gets \\operatorname*{{arg\\,min}}\\limits_b \\big\\|\\bar{b}- [\\mathrm{G}(u_{t})]b\\big\\|^2$\\label{sparse-else}\n\t\t\\EndIf\n\t\t\\EndWhile \\\\\n\t\t\\Return $\\widehat{X} \\gets u_tv_t^{\\ast}$\n\t\\end{algorithmic}\n\\end{alg}\t\n\\noindent The Hard Thresholding Pursuit Algorithm is defined as follows:\n\\begin{alg}HTP(A, b, s)\\label{HTP.alg}~\\\\\\vspace*{-4mm}\n\t\\begin{algorithmic}[1]\n\t\t\\sffamily\\small\n\t\t\\Require  Measurement matrix $A \\in \\mathbb{C}^{m \\times n}$, measurement $b \\in \\mathbb{C}^m$, sparsity constraint $s \\in \\mathbb{N}$.\n\t\t\\Ensure $ \\hat{x} \\in \\mathbb{C}^n $.\n\t\t\\State $t \\gets 0$\n\t\t\\While{stop condition not satisfied}\n\t\t\\State $t \\gets t+1 $\n\t\t\\State $ w= x_{t-1} + A^* \\left(  b-Ax_{t-1} \\right) $\n\t\t\\State $J \\gets  \\underset{J \\subset \\left[n\\right], \\ \\vert J \\vert =s}{\\arg \\max} \\ \\Vert w_J \\Vert  $\n\t\t\\State $x_t \\gets \\underset{x: \\text{supp} \\left(x\\right) \\subset J }{\\arg \\min}  \\Vert Ax-b \\Vert$\n\t\t\\EndWhile \\\\\n\t\t\\Return $ \\hat{x} \\gets x $\n\t\\end{algorithmic}\n\\end{alg}\t\t\n\t\n\t\n\t\n\t\\subsection{Initialization}\\label{Initialization}\n\tAs for many other non-convex algorithms (e.g., \\cite{Jain,candes2015phase}), the convergence properties of Sparse Power Factorization depend crucially on the choice of the starting point. In \\cite{Jain,candes2015phase} the starting point is chosen via a spectral initialization. That is, one chooses the leading left- and right-singular vectors of $ \\mathcal{A}^* \\left(y\\right) $ as the starting point. However, in order to work this approach requires that the number of measurements is at the order of $ \\max \\left\\{n_1, n_2\\right\\} $, which will in general not be optimal as it does not take into account the sparsity of the vectors $u$ and $v$. One way to incorporate the sparsity assumption would be to solve the  Sparse Principal Component Analysis (SparsePCA) problem.\n\t\\begin{equation}\\label{equ:sparsePCA}\n\t\\begin{split}\n\t\\max \\quad &  \\text{Re} \\left( \\tilde{u}^* \\mathcal{A}^* \\left(y \\right) \\tilde{v} \\right) \\\\\n\t\\text{subject to} \\quad &\\Vert \\tilde{u} \\Vert_0 \\le s_1, \\Vert \\tilde{u} \\Vert =1\\\\\n\t&\\Vert \\tilde{v} \\Vert_0 \\le s_2, \\Vert \\tilde{v} \\Vert =1,\n\t\\end{split}\n\t\\end{equation}\n\twhere $\\Vert \\cdot \\Vert_0  $ denotes the number of non-zero entries. As it was shown in \\cite[Proposition III.4]{paper2}, Algorithm \\ref{SPF.alg}, if initialized by a solution of \\eqref{equ:sparsePCA} is able to recover the solution $u$ and $v$ from a number of measurements at the order of $ \\left( s_1 + s_2 \\right) \\max \\left\\{ \\frac{s_1}{n_1}, \\frac{s_2}{n_2}  \\right\\} $. However, the SparsePCA problem has been shown to be NP-hard \\cite{tillmann2014computational}. Nevertheless, in the last fifteen years there has been a lot of research on the SparsePCA problem and, in particular, on tractable (i.e., polynomial-time) algorithms, which yield good approximations to the true solution. Several computationally tractable algorithms have been proposed for solving (\\ref{equ:sparsePCA}), e.g., thresholdings algorithms \\cite{ma2013sparse}, a general version of the power method  \\cite{journee2010generalized} and semidefinite programs \\cite{d2008optimal}. From the statistical perspective, a particular emphasis has been put for computationally efficient or at least tractable algorithms on the analysis of the single spike model\\cite{amini2009high,krauthgamer2015semidefinite,deshpande2014sparse}. These approaches, however, require that the number of samples scales with the square of the number of non-zero entries of the signal to estimate (up to $\\log$-factors). This raised the question whether there are fundamental barriers preventing the SparsePCA problem to be solved in polynomial time at a sampling rate close to the information theoretic limit. Indeed, it has been shown that an algorithm, that achieves this, would also allow for an algorithm which solves the $k$-clique problem in polynomial time \\cite{berthet2013optimal,wang2016statistical}. However, a widely believed conjecture in theoretical computer science states, that this is not the case, which indicates that this approach will not be suited for initializing bilinear recovery problems either.\\\\\n\n\t\\noindent In this manuscript we will analyse the following initialization algorithm, which is the one proposed in \\cite{paper2}. For a set $ J_1 \\subset \\left[n\\right] $, respectively $ J_2 \\subset \\left[n_2\\right] $ in the following we will denote by $ \\Pi_{J_1} $, respectively $ \\Pi_{J_2} $ the matrix, which projects a vector onto the components which belong to $ J_1$, respectively $J_2$.\n\t\n\\begin{alg}[Algorithm 3 in \\cite{paper2}]\\label{SPF_alginit}~\\\\\\vspace*{-4mm}\n\t\\begin{algorithmic}[1]\n\t\t\\sffamily\\small\n\t\t\\Require  Operator $\\mathcal{A}$, Measurement $b$, Sparsity Constraints $s_1, s_2$, \n\t\t\\Ensure Initial guess $v_0$ for $ v \\in \\mathbb{C}^{n_2} $.\n\t\n\t\n\t\n\t\n\t\n\t\t\\State For all $ i \\in \\left[ n_1 \\right] $ let $ \\xi_i  $ be the $ \\ell_2$-norm of the best $s_2$-sparse approximation of the $i$th row of the matrix $ \\mathcal{A}^* \\left(b\\right)  \\in \\mathbb{C}^{n_1 \\times n_2} $.\n\t\t\\State Let $ \\widehat{J_1} \\subset \\left[n_1\\right] $ be the set of the $ s_2 $ largest elements in $ \\left\\{\\xi_1; \\xi_2; \\ldots ; \\xi_{n_1}  \\right\\} $\n\t\t\\State Choose $\\widehat J_2$ to contain the indices of the $s_2$ columns of $\\Pi_{\\widehat J_1} \\mathcal{A}^* \\left( b \\right)$ largest in $\\ell_2$ norm, i.e.,\n\t\t\\begin{equation}\\label{equ:defwidetilde}\n\t\t\\widehat J_2  := \\underset{ J \\subset \\left[ n_2 \\right], \\ \\vert J \\vert = s_2 }{\\operatorname*{arg\\,max}}    \\big\\|\\Pi_{\\widehat J_1}[\\mathcal{A}^\\ast(b)]\\Pi_{  J }\\big\\|_\\mathrm{F}.\n\t\t\\end{equation}\\\\\n\t\t\\Return  $v_0$, the leading right singular vector of $\\Pi_{\\widehat{J_1}}[\\mathcal{A}^{\\ast}(b)]\\Pi_{\\widehat{J_2}}$.\n\t\n\t\n\t\\end{algorithmic}\n\\end{alg}\t\n\t\n\n\n\t\n\t\n\t\\section{Previous results}\n\t\n\tIn the following we will work with the that the model (\\ref{operatorrepresentation}), i.e., we observe\n\t\\begin{equation*}\n\t\\text{trace} \\left( A_{\\ell}^* uv^* \\right) + z_{\\ell}\n\t\\end{equation*}\n\twhere $u \\in \\mathbb{C}^{n_1}$ is $s_1$-sparse, $ v \\in \\mathbb{C}^{n_2}$ is $s_2$-sparse, and $z \\in \\mathbb{C}^m $ is noise. As in \\cite{paper2}, $ \\nu \\left(z\\right) $ will quantify the Noise-to-Signal Ratio by \n\t\\begin{equation}\\label{def:noiselevel}\n\t\\nu \\left(z\\right) := \\frac{\\Vert z \\Vert}{\\Vert \\mathcal{A} \\left(uv^*\\right) \\Vert }.\n\t\\end{equation}\n\t\\noindent For our analysis, $ \\mathcal{A} $ will be a Gaussian linear operator, that is, all the entries of the matrices $ A_1, \\ldots, A_{m}$ are independent with distribution $ \\mathcal{CN} \\left(0, \\frac{1}{m} \\right) $. (Here a complex-valued random variable $X$ has distribution $ \\mathcal{CN} \\left(0,\\frac{1}{m}\\right) $ if its real and complex part are independent Gaussians with expectation $0$ and variance $\\sqrt{\\frac{\\sigma}{2}} $.)\n\t\n\t\n\t\n\t\n\t\n\n\t\n\t\\noindent In \\cite{paper2}, the authors derived that Algorithm \\ref{SPF.alg}, if initialized by Algorithm \\ref{SPF_alginit}, is able to recover both $u$ and $v$ (up to scale ambiguity), if both $u$ and $v$ belong to a certain restricted class of signals. More precisely, they proved the following result.\n\n\t\\begin{thm}[{\\cite[see Theorems III.7 and Theorem III.10]{paper2}}]\\label{th1}\n\t\tAssume that $\\mathcal{A} \\colon \\mathbb{C}^{n_1 \\times n_2 } \\longrightarrow \\mathbb{C}^{m} $ is a Gaussian linear operator as described above. Let $ b= \\mathcal{A} \\left( uv^* \\right) + z$, where $u$ is $s_1$-sparse and $v$ is $s_2$-sparse. Suppose that $ \\Vert u \\Vert_{\\infty} \\ge 0.78 \\Vert u \\Vert  $, $  \\Vert v \\Vert_{\\infty} \\ge 0.78 \\Vert v \\Vert $, and that the noise level satisfies $ \\nu \\left(z\\right) \\le 0.04 $. Then, with probability exceeding $ 1- \\exp \\left(-c_1 m\\right) $, the output of the Algorithm \\ref{SPF.alg}, initialized by Algorithm \\ref{SPF_alginit}, converges linearly to $uv^*$ provided that\n\t\\begin{equation*}\n\t\tm \\ge c_2 \\left(s_1 + s_2 \\right) \\log \\left(\\max\\left\\{\\frac{n_1}{s_1},\\frac{n_2}{s_2}\\right\\}\\right),\n\t\t\\end{equation*}\t\n\twhere $c_1,c_2>0$ are absolute constants.\n\t\\end{thm}\t\n\t\\noindent Note that in order to apply Theorem \\ref{th1} to signals $u$ and $v$ one needs to require that more than half of the mass of $u$ and $v$ are located in one single entry, which is a severe restriction, which can be prohibitive for many applications. Our goal in the following will be to considerably relax this assumption by slightly increasing the amount of required measurements. We will relax this assumption in two different ways: On the one hand we will show that one can replace $ 0.78$ by an arbitrary small constant that will then show up in the number of measurements. On the other hand we generalize the result to the case that a significant portion of mass of $u$ is concentrated on a small number of entries $k$, rather than just one of them.\n\t\\section{Main Result}\\label{resultsection}\n\tIn this section we will state the main result of this article, Theorem \\ref{thm:mainresultreadable}.\n\tFor that, we need to define the norm\n\t\\begin{equation*}\n\t\\Vert x \\Vert_{\\left[k\\right]} := \\underset{I \\subset \\left[n\\right], \\ \\vert I \\vert = k}{\\max}  \\left(  \\sum_{i \\in I} \\vert x_i \\vert^2   \\right)^{1\/2} = \\left(  \\sum_{i=1}^{k}  \\left( x^*_i \\right)^2 \\right)^{1\/2},\n\t\\end{equation*}\n\tfor any $x\\in \\mathbb{C}^{n_1} $, where $ \\left( x^*_i \\right)^{n_1}_{i=1} $ denotes the non-increasing rearrangement of $ \\left( \\vert x_i \\vert \\right)^{n_1}_{i=1} $. Our main requirement on the vector $u$ will be that a significant amount of its mass is located in the largest $k$ entries, i.e., that $ \\frac{\\Vert u \\Vert_{\\left[k\\right]}}{\\Vert u \\Vert} $ is large enough.\n\t\\begin{thm}\\label{thm:mainresultreadable}\n\t\n\t\tLet $ k \\in \\left[ n_1 \\right] $ and $ 0<\\xi<1, 0<\\mu<1$. Then, there are absolute constants $C_1, C_2, C_3 >0$ such that if\n\t\t\\begin{equation}\n\t\tm \\ge C_1  \\max \\left\\{   \\frac{1}{\\xi^4 \\mu^4}, \\frac{k}{\\xi^2}  \\right\\}  \\left(s_1 + s_2 \\right) \\log \\left(\\max\\left\\{\\frac{n_1}{s_1},\\frac{n_2}{s_2}\\right\\}\\right),\n\t\t\\end{equation}\n\t\tthen with probability at least $ 1-\\exp \\left( - C_2   m \\right)  $ the following holds.\\\\\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\tFor all $s_1$-sparse $u\\in \\mathbb{C}^{n_1}$ with $ \\Vert u \\Vert_{\\left[k\\right]} \\ge \\xi \\Vert u \\Vert $, all $s_2$-sparse $u\\in \\mathbb{C}^{n_2}$ with $ \\Vert v \\Vert_{\\infty} \\ge \\mu \\Vert v \\Vert $, and all $ z \\in \\mathbb{C}^m $ with  $ \\nu\\left(z\\right) \\le C_3 \\min \\left\\{   \\xi^2 \\mu^2  ; \\frac{\\xi}{ \\sqrt{k}}  \\right\\}   $ the iterates $\\{X_t\\}_{t\\in\\mathbb{N}}$ generated by applying Algorithm \\ref{SPF.alg}, initialized by Algorithm \\ref{SPF_alginit},  satisfy\n\t\t$$\\limsup_{t\\to\\infty} \\frac{\\|X_t - uv^*  \\|_\\mathrm{F}}{\\| uv^* \\|_\\mathrm{F}} \\leq 8.3 \\nu.$$\n\t\tFurthermore, the convergence is linear, i.e., for all $ t \\gtrsim   \\log \\left( \\frac{1}{\\varepsilon}\\right) $ we have that\n\t\t\\begin{equation}\\label{equ:linearconvergence}\n\t\t\\frac{\\| X_{t} - uv^* \\|_\\mathrm{F}}{\\| uv^*  \\|_\\mathrm{F}} \\leq 8.3 \\nu + \\varepsilon.\n\t\t\\end{equation}\n\t\\end{thm}\n\t\n\n\n\n\n\n\n\t\n\t\n\t\n\t\n\nIn the following we will discuss some important special cases of Theorem \\ref{thm:mainresultreadable}.\n\\begin{itemize}\n\\item \\textbf{Peaky signals: } In \\cite{paper2} the authors discuss recovery guarantees for signals $u$ and $v$ with $\\tfrac{\\Vert u \\Vert_{\\infty} }{\\Vert u \\Vert} $ and  $\\tfrac{\\Vert v \\Vert_{\\infty} }{\\Vert v \\Vert}  $, both bounded below by an absolute constant $\\mu  \\approx 0.78$. The case $k=1$ of our theorem yields a direct improvement of this result in the sense that $\\mu$ can be chosen arbitrarily small with the number of required measurements only increasing by a factor of order $ \\mu^{-8} $. Hence, even when this constant decays logarithmically in the dimension, the required number of measurements will only increase by logarithmic factors.\n\\item \\textbf{Signals with multiple large entries: } When one of the input signals has multiple large entries, using the $\\Vert \\cdot \\Vert_{[k]} $ norm improves upon the resulting guarantee as compared to the scenario just discussed. As an example, assume that $s_1=s_2=s $, that $u$ and $v$ are normalized with $\\|v\\|_\\infty \\geq c_1 s^{-1\/8}$, and that $k=c_2 s^{1\/2}$ of the entries of $u$ are of absolute value at least $ c_3s ^{-1\/4}$. Then $ \\Vert u \\Vert_{\\left[k\\right]} \\ge \\sqrt{ c_2 } c_3 $. Using Theorem \\ref{thm:mainresultreadable} we obtain that the vectors $u$ and $v$ can be recovered if the number of measurements is on the order of $ s^{3\/2}$, thus below the order of $s^2$ that has been established for arbitrary sparse signals in \\cite{strohmer} (cf. next item). In contrast, applying Theorem \\ref{thm:mainresultreadable} with $k=1$ would yield that the number of measurements would have to be on the order of $   s^{5\/2} $, which is worse than the state-of-the-art.\n\\item \\textbf{Arbitrary sparse signals:}\tApplying Theorem \\ref{thm:mainresultreadable} to non-peaky signals yields suboptimal results. Indeed, let $u \\in \\mathbb{C}^{n_1}$ $s_1$-sparse and $v \\in \\mathbb{C}^{n_2} $ $s_2$-sparse be generic vectors. Observe that $ \\Vert v \\Vert_{\\infty} \\asymp \\frac{1}{\\sqrt{s_2}} \\Vert v \\Vert $.\nConsequently, Theorem \\ref{thm:mainresultreadable} applied with $ \\xi =1 $, $ k= s_1 $, and $ \\mu = \\frac{1}{\\sqrt{s_2}} $ yields that with high probability a generic $s_1$-sparse $u$ and a generic $s_2$-sparse $v$ can be recovered from $ y = \\mathcal{A} \\left( uv^* \\right) +z $, if the number of measurements satisfies\n\\begin{equation*}\nm \\ge C \\max \\left\\{ s_1; s_2^2 \\right\\} \\left(s_1 + s_2 \\right) \\log \\left(\\max\\left\\{\\frac{n_1}{s_1},\\frac{n_2}{s_2}\\right\\}\\right),\n\\end{equation*}\t\nand if the noise level $\\nu$ is on the order of $ \\mathcal{O} \\left( \\max \\left\\{ \\frac{1}{s_2}; \\frac{1}{\\sqrt{s_1}}  \\right\\} \\right) $. Previous results (see, e.g., \\cite{strohmer}), in contrast, require $ m \\ge C \\max \\left\\{ s^2_1; s^2_2 \\right\\}  \\log \\left(\\max\\left\\{\\frac{n_1}{s_1},\\frac{n_2}{s_2}\\right\\}\\right) $ samples.\n\\end{itemize}\t\n\n\n\\begin{remark}\nThe peakiness assumptions in Theorem \\ref{thm:mainresultreadable} may seem arbitrary at first sight but in certain applications they are reasonable. Namely, when $u$ is the signal transmitted via a wireless channel and $v$ is the unknown vector of channel parameters it is natural to assume that $v$ has a large entry, as the direct path will always carry most of the energy. The signal $u$ can be modified by the sender, so some large entries can be artificially introduced. In this regard, being able to consider multiple entries of comparable size is of advantage as adding a single very large entry will result in a dramatic increase of the peak-to-average power ratio.\n\\end{remark}\n\t\t\n\n\t\n\t\n\t\n\t\t\n\n\n\t\n\\section{Proofs}\\label{sectionproof}\n\t\\subsection{Technical tools}\n\tThe goal of this section is to prove Theorem \\ref{thm:mainresultreadable}.\n\tWe will start by recalling the following variant of the well-known restricted isometry property.\n\t\\begin{defi}[see \\cite{paper2}]\n\tA linear operator $ \\mathcal{A}$ has the $(s_1, s_2, r )$-restricted isometry property with constant $\\delta$ if \n\t\\begin{equation}\n\t\\left(1-\\delta\\right) \\Vert X \\Vert_F^2 \\le \\Vert \\mathcal{A} \\left( X \\right)  \\Vert^2 \\le \\left( 1+\\delta \\right) \\Vert  X  \\Vert^2_F\n\t\\end{equation}\n\tfor all matrices $ X \\in \\mathbb{C}^{n_1 \\times n_2}$ of rank at most $r$ with at most $s_1$ non-zero rows and at most $s_2$ non-zero columns.\n\t\\end{defi}\n\t\\noindent The following lemma tells us that this property holds with high probability for a number of measurements close to the information-theoretic limit.\n\t\\begin{lemma}[See, e.g., Theorem III.7 in \\cite{paper2}]\\label{thm37}\n\tThere are absolute constants $c_1, c_2>0 $, such that if\n\t\\begin{equation}\\label{necessarymeasurementsrip}\n\tm \\ge \\frac{c_1}{\\delta^2}  r \\left(s_1 + s_2  \\right) \\log \\left(  \\max \\left\\{ \\frac{n_1}{s_1}, \\frac{n_2}{s_2}  \\right\\}   \\right),\n\t\\end{equation}\n\tfor some $\\delta >0$, then with probability at least $1-\\exp \\left( -  c_2 m \\right) $  $ \\mathcal{A}$ has the $(s_1, s_2, r)$-restricted isometry property with restricted isometry constant $\\delta$.\n\\end{lemma}\n\\noindent As in \\cite[Lemma VIII.7]{paper2} we will need the following quantity, which depends on $ \\delta$ and $ \\nu $.\n\t\\begin{align}\\label{def:omegasup}\n\t\t\t&\\omega_\\mathrm{sup} :=\\sup\\left\\{\\omega\\in[0,\\tfrac{\\pi}{2}):\\omega\\geq\\arcsin\\left(C_\\delta[\\delta\\tan(\\omega)+(1+\\delta)\\nu\\sec(\\omega)]\\right)\\right\\}\n\t\\end{align}\n\tHere, the constant $C_\\delta $ is given by the expression\n\t\\begin{equation*}\n\tC_{\\delta} =  1.1 \\frac{ \\sqrt{ \\frac{2}{1-\\delta^2} }  + \\frac{1}{1-\\delta}}{1- \\sqrt{ \\frac{2}{1-\\delta^2} } \\delta },\n\t\\end{equation*}\n\tas it can be seen by an inspection of the proof of Lemma VIII.1 in \\cite{paper2}. The precise value of $ C_{\\delta}$ will not be important in the following, we will only use that $ 2 \\le C_{\\delta} \\le 5 $ for $\\delta \\le 0.04 $. \\\\\n\t\n\\noindent A simple estimate for $ \\omega_{\\sup}$ is given by the following lemma.\n\\begin{lemma}\\label{sin05}\n\tAssume that $ \\delta \\le 0.04 $ and $\\nu \\le 0.04 $. Then it holds that \n\t\\begin{equation*}\n\t\t\\tfrac{1}{2} \\leq \\sin(\\omega_\\mathrm{sup}) \\leq 1.\n\t\\end{equation*}\n\\end{lemma}\n\t\n\\begin{proof}\n\n\tWe observe that in order to show the claim it is enough to verify that $ \\omega= \\arcsin \\frac{1}{2}  $ fulfills the inequality in (\\ref{def:omegasup}). Indeed, using $ \\cos \\omega = \\sqrt{\\frac{3}{4}} $ and $ C_{\\delta} \\le 5 $ we obtain that\n\t\\begin{align*}\n\t\tC_{\\delta} \\left[    \\delta \\tan \\left(  \\arcsin \\frac{1}{2}   \\right) + \\left(1+ \\delta\\right) \\nu   \\sec \\left( \\arcsin \\frac{1}{2}  \\right)  \\right] &= C_{\\delta} \\left[ 0.04 \\frac{1\/2}{\\sqrt{3\/4}} + \\frac{ 1.04 \\cdot 0.04 }{\\sqrt{3\/4}}   \\right]  \\\\\t\t\n\t\t&\\le \\frac{1}{2}.\n\t\\end{align*}\n\n\\end{proof}\n\t\n\t\\noindent The quantity $ \\omega_{\\sup} $ controls the maximal angle between the initialization $ v_0$ and the ground truth $v$ such that the Sparse Power Factorization is guaranteed to converge as captured by the following theorem.\n\t\t\\begin{thm}[Theorem III.9 in \\cite{paper2}]\\label{thm39}\n\t\tAssume that\n\t\t\\begin{enumerate}[1)]\n\t\t\n\t\t\t\\item $\\mathcal{A}$ has the $(3s_1,3s_2,2)$-RIP with isometry constant $\\delta\\leq0.08$,\n\t\t\t\\item $\\nu \\leq 0.08$,\n\t\t\t\\item the initialization $v_0$ satisfies $\\sin(\\angle(v_0,v)) < \\sin \\left( \\omega_{\\sup}  \\right)$.\n\t\t\\end{enumerate}\n\t\tThen the iterates $\\{X_t\\}_{t\\in\\mathbb{N}} $ generated by Algorithm \\ref{SPF.alg}, initialized via Algorithm \\ref{SPF_alginit}, satisfy\n\t\t$$\\limsup_{t\\to\\infty} \\frac{\\|X_t - uv^*\\|_\\mathrm{F}}{\\|uv^*\\|_\\mathrm{F}} \\leq 8.3 \\nu.$$\n\t\tFurthermore, the convergence is linear in the sense of (\\ref{equ:linearconvergence}).\n\t\\end{thm}\n\t\\noindent Thus, it remains to verify that the initialization satisfies $\\sin(\\angle(v_0,v)) < \\sin \\left( \\omega_{\\sup}  \\right)$. The following lemma gives an upper bound on $\\sin(\\angle(v_0,v))$.\t\n\t\t\\begin{lemma}[Lemma 8 in \\cite{paper2}]\\label{lemma8.10}\n\t\tAssume that the $(3s_1, 3s_2, 2 )$-restricted isometry property holds for some constant $ \\delta >0 $. Furthermore, assume that $ \\Vert u \\Vert = \\Vert v \\Vert =1 $. Let $\\widehat{J_1} \\subseteq \\left[n_1\\right] $ and $\\widehat{J_2} \\subseteq \\left[n_2\\right]$ denote the output resulting from Algorithm \\ref{SPF_alginit}.\n\t\t\n\t\t Denote by $v_0$ the leading right singular vector of $\\Pi_{\\widehat{J_1}}[\\mathcal{A}^\\ast(b)]\\Pi_{\\widehat{J_2}}$. Then it holds that\n\t\t\\begin{equation}\\label{ineq:sufficientcondition2}\n\t\t\\sin(\\angle(v_0,v)) \\leq \\frac{\\big\\|\\Pi_{\\widehat J_1}u\\big\\|\\big\\|\\Pi_{\\widehat J_2}^\\perp v\\big\\| + (\\delta + \\nu+\\delta\\nu)}{\\big\\|\\Pi_{\\widehat J_1}u\\big\\|-(\\delta + \\nu+\\delta\\nu)}.\n\t\t\\end{equation}\n\t\\end{lemma}\n\n\t\\noindent  Furthermore, we will need the following two lemmas for our proof.\n\\begin{lemma}\\label{lemma:lastlemma}[Lemma VIII.12 in \\cite{paper2}]\nLet $u$ and $v$ be as in Lemma \\ref{lemma:supportlowerbound} and assume that the measurement operator $ \\mathcal{A}$ satisfies the $ \\left( 3s_1, 3s_2, 2 \\right) $-restricted isometry property with constant $ \\delta $. Recall that $\\widehat J_1 \\subset \\left[n_1\\right] $ is the support estimate for $v_0$ given by the initialization algorithm \\ref{SPF_alginit}. Define\n\\begin{equation}\\label{equ:definitionJ1}\n\t\\widetilde{J_1} :=  \\left\\{  j \\in \\left[ n_1 \\right]: \\ \\vert u_j \\vert \\ge 2 \\left( \\delta + \\nu + \\delta \\nu \\right)    \\right\\}.\n\\end{equation}\t\nThen we have that $ \\widetilde{J_1}  \\subset \\widehat J_1  $. \n\\end{lemma}\t\n\t\n\\begin{lemma}\\label{lemma:supportlowerbound}\n\tAssume that $ \\mathcal{A}$ has the $(3s_1, 3s_2, 2)$-restricted isometry property with isometry constant $ \\delta >0 $ and assume that $u$, respectively $v$, are $s_1$-sparse, respectively $s_2$-sparse, and satisfy $ \\Vert u \\Vert = \\Vert v \\Vert =1 $. Let $ \\widetilde{J_1} $  be defined as in (\\ref{equ:definitionJ1}).\n\tThen, it holds that\n\t\\begin{equation*}\n\t\\big\\|\\Pi_{ \\widehat J_1 } u   \\big\\|  \\big\\|  \\Pi_{ \\widehat J_2 } v   \\big\\| \\ge   \\big\\|\\Pi_{ \\widetilde{J_1} } u   \\big\\|  \\Vert v \\Vert_{\\infty} - 2   \\left( \\delta + \\nu +\\delta \\nu  \\right).\n\t\\end{equation*}\n\\end{lemma}\n\n\n\\noindent Lemma \\ref{lemma:supportlowerbound} is actually a slight generalization of what has been shown in \\cite[p. 1685]{paper2}. For completeness we have included a proof in Section \\ref{section:supportlowerbound}, which closely follows the proof in \\cite{paper2}.\\\\\n\n\n\t\\subsection{Proof of our main result}\n\t\tWe will now piece together these ingredients to obtain a sufficient condition; in the remainder of the section we will then show that the condition holds in our measurement setup. First note that in order to apply Theorem \\ref{thm39} we need to check that \t$\\sin(\\angle(v_0,v)) < \\sin \\left( \\omega_{\\sup}  \\right)$ is satisfied. By Lemma \\ref{lemma8.10} it is sufficient to show that the right-hand side of inequality (\\ref{ineq:sufficientcondition2}) is strictly smaller than $ \\sin \\left( \\omega_{\\sup} \\right) $. Combining this with the equality $ \\big\\|\\Pi_{\\widehat J_2}^{\\perp}v\\big\\| =  \\sqrt{ 1 - \\big\\|\\Pi_{\\widehat J_2}v\\big\\|^2 }  $ we obtain the sufficient condition\n\t\t\\begin{equation*}\n\t\t\\big\\|\\Pi_{\\widehat J_1}u\\big\\|   \\sqrt{ 1 - \\big\\|\\Pi_{\\widehat J_2}v\\big\\|^2 }  <  \\sin \\left( \\omega_{\\sup} \\right)  \\left(  \\big\\|\\Pi_{\\widehat J_1}u\\big\\|   - \\left( \\delta + \\nu+\\delta\\nu\\right)   \\right) -  \\left( \\delta + \\nu+\\delta\\nu  \\right) \n\t\t\\end{equation*}\n\t\tFurther manipulations yield that this is equivalent to\n\t\t\\begin{equation}\\label{ineq:sufficientcondition}\n\t\t\\begin{split}\n\t\t\\big\\|\\Pi_{\\widehat J_1}u\\big\\|^2 < &\\left( \\sin \\left( \\omega_{\\sup}  \\right) \\big\\|\\Pi_{\\widehat J_1}u\\big\\|   - \\left( 1+ \\sin \\left( \\omega_{\\sup} \\right) \\right) \\left( \\delta + \\nu+ \\delta \\nu \\right)    \\right)^2\\\\\n\t\t+ & \\big\\|\\Pi_{\\widehat J_1}u\\big\\|^2 \\big\\|\\Pi_{\\widehat J_2} v\\big\\|^2.\n\t\t\\end{split} \n\t\t\\end{equation}\t\t\n  \t\tHence, in the following our goal will be to verify (\\ref{ineq:sufficientcondition}).\n\t\t\\noindent We already noticed that the angle $ \\omega_{\\sup} $ measures how much the vector $v_0$ given by the initializiation has to be aligned with the ground truth $v$ in order for the Sparse Power Factorization to converge. Consequently, it is natural to expect that the smaller the constant $ \\delta$ and the noise-to-signal ratio $\\nu$, the less the initializiation vector has to be aligned with the ground truth, i.e., the larger $ \\omega_{\\sup} $ can be. This fact is captured by the following lemma.\n\t\\begin{lemma}\\label{sin2}\n\t\tLet $\\delta \\leq 0.04$ and $\\nu \\leq 0.04$. Then it holds that\n\t\t$$\\sin(\\omega_\\mathrm{sup}) \\geq  1 -C_{\\delta}^2\\left(\\delta + 2\\delta\\nu+2\\nu\\right)^2.$$\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tIt follows directly from \\eqref{def:omegasup} that\n\t\t\\begin{align*}\n\t\t \\omega_{\\sup}   &= \\arcsin \\left(  C_{\\delta}  \\left[ \\delta \\tan \\left( \\omega_{\\sup} \\right) + \\left( 1+ \\delta \\right) \\nu \\sec \\left(  \\omega_{\\sup} \\right)    \\right] \\right).\n\t\t\\end{align*}\n\t\tUsing trigonometric identities we obtain that\n\t\t\\begin{align*}\n\t\t \\sin \\left( \\omega_{\\sup} \\right)  &= C_{\\delta} \\left[ \\delta \\frac{\\sin \\left(  \\omega_{\\sup} \\right)}{\\sqrt{1-\\sin \\left(  \\omega_{\\sup} \\right)^2 }}  + \\left(1+\\delta \\right) \\nu \\frac{1}{\\sqrt{ 1- \\sin \\left( \\omega_{\\sup} \\right)^2  }}    \\right].\n\t\t\\end{align*}\n\t\tLemma \\ref{sin05} implies that\n\t\t\\begin{equation*}\n\t\t\\sin \\left( \\omega_{\\sup}  \\right) \\le \\frac{ \\sin \\left( \\omega_{\\sup} \\right) }{\\sqrt{1-\\sin \\left(  \\omega_{\\sup} \\right)^2}} C_{\\delta} \\left( \\delta + 2 \\left( 1+ \\delta \\right) \\nu  \\right).\n\t\t\\end{equation*}\n\tRearranging terms yields that\n\t\t\\begin{equation*}\n\t\t\\sin \\left( \\omega_{\\sup}  \\right) \\ge \\sqrt{1 - C^2_{\\delta}  \\left( \\delta +2\\delta \\nu +2\\nu   \\right)^2 }.\n\t\t\\end{equation*}\n\t\tThe claim follows then using the fact that $ \\sqrt{x} \\ge x $ for all $ x \\in \\left[ 0,1\\right] $.\n\t\t\\end{proof}\t\n\t\\noindent With these preliminary lemmas, we can now prove the following proposition, which is a slightly more general form of Theorem \\ref{thm:mainresultreadable}.\n\t\t\\begin{prop}\\label{prop:mainproposition}\n\tThere are absolute constants $c_1, c_2, c_3 >0$ such that if\n\t\\begin{equation}\\label{equ:numbermeasurements}\n\tm \\ge c_1 \\delta^{-2} \\left(s_1 + s_2 \\right) \\log \\left(\\max\\left\\{\\frac{n_1}{s_1},\\frac{n_2}{s_2}\\right\\}\\right),\n\t\\end{equation}\t\t\n\tfor some $  0 < \\delta <0.01$, then with probability at least $ 1-\\exp \\left( - c_2 m \\right) $ the following statement holds uniformly for all $s_1$-sparse $u \\in \\mathbb{C}^{n_1} $, $s_2$-sparse $v \\in \\mathbb{C}^{n_2} $ and $ z \\in \\mathbb{C}^m $ such that $\\Vert u \\Vert = \\Vert v \\Vert =1 $ and $ \\nu \\left(z\\right) \\le 0.01 $:\\\\\t\n\t\\noindent Let the measurements be given by $ b= \\mathcal{A} \\left( uv^* \\right) +z $ for $ \\mathcal{A} $ Gaussian as above and let $ \\widetilde J_1 $ be defined by\n\t\\begin{equation}\\label{equ:definitionJ2}\n\t\\widetilde J_1 :=  \\left\\{  j \\in \\left[ n_1 \\right]: \\ \\vert u_j \\vert \\ge  M_{\\delta,\\nu} \\right\\},\n\t\\end{equation}\n\twhere\n\t\\begin{equation*}\n\tM_{\\delta, \\nu} :=  2 \\left( \\delta + \\nu + \\delta \\nu \\right).\n\t\\end{equation*}\n\n\n\n\n\n\tThen, whenever\n\t\\begin{equation}\\label{ineq:peakinessassumption}\n\n\t\\big\\|\\Pi_{ \\widetilde J_1 } u   \\big\\|  \\Vert v \\Vert_{\\infty} > c_3 \\sqrt{    M_{\\delta,\\nu} } ,\n\t\\end{equation}\n\tthe iterates $\\{X_t\\}_{t\\in\\mathbb{N}}$ generated by Algorithm \\ref{SPF.alg} initialized via Algorithm \\ref{SPF_alginit}, satisfy\n\t$$\\limsup_{t\\to\\infty} \\|X_t - uv^*\\|_\\mathrm{F} \\leq 8.3 \\nu.$$ \n\tFurthermore, the convergence is linear in the sense of (\\ref{equ:linearconvergence}).\n\\end{prop}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:mainproposition}]\t\n\n\n\tAssumption (\\ref{equ:numbermeasurements}) and \\Cref{thm37} yield that with probability at least $1 - \\exp \\left(-c  m \\right) $ the ($3s_1$,$3s_2$,$2$)-restricted isometry property holds with constant $\\delta $.\n\tFor the remainder of the proof, we will consider the event that the restricted isometry property holds for such $\\delta$. \t\n\n\n\n\n\n\n\n\n\tWe obtain\n\t\\begin{equation*}\n\t\\big\\|\\Pi_{ \\widetilde J_1 } u   \\big\\|  \\Vert v \\Vert_{\\infty}  \\ge  \\left(    \\sqrt{  C^2_{\\delta} +1 }  + 1       \\right)  \\sqrt{M_{\\delta,\\nu}}\n\t\\end{equation*}\n\tfrom $ 2\\le C_{\\delta} \\le 5 $ and by choosing the constant $c_3$ in assumption (\\ref{ineq:peakinessassumption}) large enough. Combining this with Lemma \\ref{lemma:supportlowerbound}  we obtain that\n\t\\begin{equation}\\label{ineq:chain4}\n\t\\begin{split}\n\t\\big\\|\\Pi_{ \\widehat J_1 } u   \\big\\|  \\big\\|  \\Pi_{ \\widehat J_2 } v   \\big\\| &\\ge   \\big\\|\\Pi_{ \\widetilde{ J_1} } u   \\big\\|    \\Vert v \\Vert_{\\infty}  - M_{\\delta, \\nu}.\\\\\n\t&>  \\sqrt{  \\left( C^2_{\\delta} + 1  \\right)        M_{\\delta, \\nu} },\n\t\\end{split}\n\t\\end{equation}\n\twhere we used that $\\sqrt{x} \\ge x $ for all $ x \\in \\left[0,1\\right] $. This yields a lower bound for the second summand of the right-hand side of (\\ref{ineq:sufficientcondition}). To bound the first summand we estimate\n\t\\begin{equation}\\label{ineq:chain6}\n\t\\begin{split}\n\t&\\sin(\\omega_\\mathrm{sup}) \\|\\Pi_{\\widehat J_1}u\\| - \\left( \\sin(\\omega_\\mathrm{sup}) +1 \\right)  \\left( \\delta+\\nu + \\delta \\nu \\right)\\\\\n\t\\ge&   \\left( 1- C^2_{\\delta} \\left( \\delta + 2\\nu + 2\\delta \\nu  \\right)^2 \\right) \\|\\Pi_{\\widehat J_1}u\\|  -2 \\left( \\delta + \\nu +\\delta \\nu \\right)   \\\\\n\t\\ge&   \\|\\Pi_{\\widehat J_1}u\\| - C^2_{\\delta} \\left( \\delta +2 \\nu + 2\\delta \\nu  \\right)^2 -2 \\left( \\delta + \\nu + \\delta \\nu \\right)   \\\\\n\t\\ge &   \\|\\Pi_{\\widehat J_1}u\\| - \\left( C^2_{\\delta}+1 \\right)   M_{\\delta,\\nu}  \\\\\n\t\\ge & 0.\n\t\\end{split}\n\t\\end{equation}\n\tIn the first line we used \\Cref{sin2} and the fact that $ \\sin(\\omega_\\mathrm{sup}) \\le 1 $. The second line is due to $ \\|\\Pi_{\\widehat J_1}u\\| \\le 1  $ and the third inequality is due to $ \\delta \\ge 0 $, $ \\nu \\ge 0 $. In order to verify the last inequality it is enough to observe that due to Lemma \\ref{lemma:lastlemma} and due to assumption (\\ref{ineq:peakinessassumption}) with $c_3$ large enough\n\t\\begin{align*}\n\t\\|\\Pi_{\\widehat J_1}u\\|  &\\ge \\Vert \\Pi_{\\widetilde{J_1}}  u \\Vert \\ge \\Vert \\Pi_{\\widetilde{J_1}}  u \\Vert   \\big\\|  \\Vert v \\Vert_{\\infty}  \\ge \\left( C^2_{\\delta} + 1  \\right) M_{\\delta,\\nu},\n\t\\end{align*}\n\twhere the last inequality uses that $ C_{\\delta} \\le 5 $ and $ 0 \\le \\delta, \\nu \\le 0.01 $. Hence, by squaring (\\ref{ineq:chain6}) we obtain that\n\t\\begin{equation}\\label{ineq:chain5}\n\t\\begin{split}\n\t&\\left( \\sin(\\omega_\\mathrm{sup}) \\|\\Pi_{\\widehat J_1}u\\| - \\left( \\sin(\\omega_\\mathrm{sup}) +1 \\right) \\left( \\delta + \\nu + \\delta \\nu \\right)  \\right)^2\\\\\n\t\\ge & \\left(    \\|\\Pi_{\\widehat J_1}u\\| -  \\frac{1}{2}\\left( C^2_{\\delta}+1 \\right)  M_{\\delta, \\nu}    \\right)^2\\\\\n\t\\ge& \\|\\Pi_{\\widehat J_1}u\\|^2 -  \\left( C^2_{\\delta}+1 \\right) M_{\\delta, \\nu}  \\|\\Pi_{\\widehat J_1}u\\|  \\\\\n\t\\ge & \\|\\Pi_{\\widehat J_1}u\\|^2 -  \\left( C^2_{\\delta}+1 \\right)  M_{\\delta, \\nu},\n\t\\end{split}\n\t\\end{equation}\n\twhere in the last line we again used that $ \\|\\Pi_{\\widehat J_1}u\\| \\le 1 $. Together with (\\ref{ineq:chain4}) this yields (\\ref{ineq:sufficientcondition}), as desired.\n\t\n\\end{proof}\n\\noindent Finally, we will deduce Theorem \\ref{thm:mainresultreadable} from Proposition \\ref{prop:mainproposition}.\n\\begin{proof}[Proof of Theorem \\ref{thm:mainresultreadable}]\n\tWe will prove this result by applying Proposition \\ref{prop:mainproposition} with \n\t\\begin{equation}\n\t\\delta = \\min  \\left\\{  \\frac{\\xi}{6 \\sqrt{2k}}     ;      \\frac{\\xi^2 \\mu^2}{8c_3^2}     \\right\\}.\n\t\\end{equation}\n     Let $ u \\in \\mathbb{C}^{n_1}$ $s_1$-sparse, $ v \\in \\mathbb{C}^{n_2} $ $s_2$-sparse and $z \\in \\mathbb{C}^m $ such that the assumptions of Theorem \\ref{thm:mainresultreadable} are satisfied. Without loss of generality we may assume in the following that $\\Vert  u \\Vert = \\Vert v \\Vert =1  $.\n\tFirst, we note that invoking $ \\delta, \\nu < 0.01 $ and potentially decreasing the size of $C_3$ we have that\n\t\\begin{align*}\n\t2 \\left( \\delta + \\nu \\left(z\\right) + \\delta \\nu \\left(z\\right)   \\right) < 2 \\left( \\delta + 2 \\nu \\left(z\\right) \\right) \\le \\frac{\\xi}{\\sqrt{2k}}.\n\t\\end{align*}\n\tHence, we obtain that\n\t\\begin{equation}\\label{equ:Jinclusion}\n\t\\breve{J}_1:= \\left\\{  j \\in \\left[ n_1 \\right]:  \\ \\vert u_j \\vert \\ge \\frac{\\xi}{\\sqrt{2k}}   \\right\\}  \\subset \\widetilde{J}_1,\n\t\\end{equation}\n\twhere $ \\widetilde{J}_1 $ is the set defined in (\\ref{equ:definitionJ2}). \n\n\tNote that\n\t\\begin{equation*}\n\t\\sum_{i \\in \\left[k\\right] \\backslash \\breve{J}_1 } \\left( u^*_i \\right)^2 < \\sum_{i \\in \\left[k\\right] \\backslash \\breve{J}_1 } \\frac{\\xi^2}{2k} \\le  \\frac{\\xi^2}{2},\n\t\\end{equation*}\n\twhere in the first inequality we have used that $ u^*_i < \\frac{\\xi}{\\sqrt{2k}} $ for all $ i \\in \\left[k\\right] \\backslash \\breve{J}_1 $. By the assumption $ \\Vert u \\Vert_{\\left[k\\right]} \\ge \\xi $ this yields that $ \\sum_{i \\in \\left[k\\right] \\cap \\breve{J}_1  }  \\left( u^*_i \\right)^2  \\ge \\frac{\\xi^2}{2} $, which in turn implies that $  \\Vert \\Pi_{\\breve{J}_1}  u  \\Vert \\ge \\frac{\\xi}{\\sqrt{2}} $. By the inclusion (\\ref{equ:Jinclusion}) we obtain that $   \\Vert \\Pi_{ \\widetilde J_1 }  u  \\Vert \\ge \\frac{\\xi}{\\sqrt{2}} $. Hence,  using the assumption $ \\Vert v \\Vert_{\\infty} \\ge \\mu $, our choice of $\\delta$, the assumption on the noise level $ \\nu \\left( z \\right) $ and potentially again decreasing the value of the constant $C_3$ we obtain that\n\t\\begin{equation*}\n\t\\Vert \\Pi_{\\widetilde{J}_1}  u   \\Vert \\Vert v \\Vert_{\\infty}  \\ge \\frac{\\xi \\mu }{ \\sqrt{2} } \\ge c_3 \\sqrt{ M_{\\delta, \\nu} }.\n\t\\end{equation*}\n\tThis shows that (\\ref{ineq:peakinessassumption}) is satisfied. Hence, we can apply Proposition \\ref{prop:mainproposition} and by inserting our choice of $\\delta$ into (\\ref{equ:numbermeasurements}), so choosing the constant $C_1$ large enough, we obtain the main result.\n\t\n\\end{proof}\n\n\n\n\t\\section{Outlook}\n\tWe see many interesting directions for follow-up work. Most importantly, it remains to explore whether additional constraints on the signals to be recovered are truly necessary (cf. our discussion on to SparsePCA in Section \\ref{Initialization}). Even if this is the case, there is substantial room for improvement with respect to the noise-dependence of the recovery results. A direction to proceed could be to consider stochastic noise models instead of deterministic noise. Also in this work we exclusively considered operators $\\mathcal{A}$ constructed using Gaussian matrices. However, in many applications of interest, the measurement matrices possess a significantly reduced amount of randomness. For example, in blind deconvolution one typically encounters rank-one measurements. That is, the restricted isometry property as used in this paper does not hold. Thus, one needs additional insight to study whether  there exists a computationally tractable initialization procedure at a near-optimal sampling rate. First steps in this direction were taken in \\cite{lee2015rip,lee2017blind}, but a lot of questions remain open.\n\t\n\t\n\t\n\t\n\t\\section*{Acknowledgements}\n\tJakob Geppert is supported by the  German Science Foundation (DFG) in the Collaborative Research Centre ``SFB 755: Nanoscale Photonic Imaging'' and partially in the framework of the Research Training Group ``GRK 2088: Discovering Structure in Complex Data: Statistics meets Optimization and Inverse Problems''. Felix Krahmer and Dominik St\\\"oger have been supported by the German Science Foundation (DFG) in the context of the joint project ``SPP 1798: Bilinear Compressed Sensing'' (KR 4512\/2-1). Furthermore, the authors want to thank Yoram Bresler and Kiryung Lee for helpful discussions.\n\t\n\t\n\n\t\n\t\n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=1]{f1.eps}\n\\caption{A cartoon picture of our MHD model of BH jets, \nwhere magnetic field lines (solid lines) cross the ergosphere (dashed line) and the event horizon. \nThere is an extending plasma loading zone (shaded region) near the central BH, where particles are injected. \nThe inflow and outflow pattern naturally forms under the mutual influence of the central BH and the EM fields. \\label{fig:cartoon}}\n\\end{figure}\n\n\nRelativistic jets launched by accreting black holes (BHs) play an essential role in several energetic astrophysical phenomena,\nincluding stellar-mass BH X-ray binaries, active galactic nuclei and possibly gamma-ray bursts.\nAfter decades of debates among  astrophysical communities, many open questions concerning the nature of the BH jets still remain to be answered \\citep[see e.g.,][for reviews]{Meier01, Blandford18}. To name a few of the most fundamental ones: what are the central engines of the jets; how the fluid within the jets is accelerated to the relativistic speed; how the jets are collimated.\n\nThe Blandford-Znajek (BZ) mechanism \\citep{BZ77,Znajek77}, which describes an electromagnetic (EM) process of extracting the rotation energy of the central BH in the form of Poynting flux, is believed to be the most promising candidate for the central engines\nof the BH jets. For understanding the jets powered by the BZ mechanism, one needs to study magnetohydrodynamic (MHD) process in the Kerr spacetime, where the EM fields and the fluid motion are coupled in a complicate way. Therefore people treat some components of the MHD process as dynamical variables with other components prescribed in many previous studies on this subject.\nFor studying the EM fields of the jet, force-free electromagnetodynamics (FFE) is a convenient assumption, where\nthe fluid energy is ignored and the EM fields are self-contained \\citep[e.g.,][]{Tanabe08, Kom01, Kom02, Kom04a, \nKom04b, Kom05, Kom07,Beskin13, Contop13,Nathan14, Gralla14,Gralla15,Gralla16,Pan14,Pan15,Pan15b,Pan16,Pan17,Pan18, \nYang14, Yang15, East18, Mahlmann18}.\nFor studying the fluid motion within the jet, people usually treat the fluid as test fluid in prescribed EM fields \\citep[e.g.,][]{Takahashi90, Beskin06, Globus13,Globus14,Pu12, Pu15}. There have also been some full MHD attempts where only\noutflow pattern and jet structure in weak gravity regime are addressed \\citep[see e.g.,][for self-similar outflow solutions in pseudo-potential]{Polko13, Polko14,Cecco18} \nand \\citep[see e.g.,][for outflow solutions in Minkowski spacetime]{Beskin98, Beskin00,Lyub09, Tchek09, Beskin10,Beskin17}.\nFor understanding the physics of BH accretion systems, general relativistic MHD (GRMHD) simulation has been another powerful tool \nin the past two decades, in which full MHD equations in curved spacetime are solved. \nNevertheless,  GRMHD codes tend to become unstable to a vacuum and therefore a matter density floor is usually introduced \n\\citep[e.g.,][]{Gammie03, Shibata05, Porth17}, which may obscure our understanding of plasma loading and the flow within the jet. \n\nBesides all the theoretical explorations summarized above,  substantial progress\nin spatial resolution has been made on the observation side. Especially the Event Horizon Telescope (EHT) \n\\citep[e.g.,][]{Doel08,Doel12,Ricarte15,EHT19I,EHT19V} is expected \nto resolve the structure of supermassive BHs nearby (Sgr A$^*$ and  M87) down to horizon scales.\nIt is promising to unveil the physical nature of the jets in these systems if the coming EHT observations can \nbe correctly deciphered.  This motivates us to construct a full GRMHD jet model, considering that \nall the previous studies are confined by different limitations.\n\nIn this paper, we aim to construct a GRMHD framework for investigating the structure of steady and axisymmetric jets \nof spinning BHs, in which the EM fields and fluid motion are self-consistently taken care of. \nA cartoon picture in Fig.~\\ref{fig:cartoon} is shown to illustrate the major elements of our jet model:\na central BH, EM fields, a plasma loading zone, inflow and outflow. \nThe magnetic field lines penetrate the event horizon of a spinning BH and \nextract the rotation energy from the BH in the form of Poynting flux.\nQuantifying plasma loading within the BH jet is also a complicate problem considering \nthe rich plasma sources, including the accretion flow centered on the equatorial plane, \npair production inside the jet \\citep{Lev11, Brod2015, Hiro16, Chen18}\nand neutrino pair annihilation from an extremely hot accretion flow \\citep[see e.g.][]{Pop99, Narayan01}. \nIn our jet model, we do not deal with these detailed processes. For convenience, we \nintroduce a plasma loading zone where plasma is injected and prescribe the loading function, i.e., particle number flux per magnetic flux  $\\eta(r,\\theta)$.  Under the mutual influence of the central BH and the EM fields, the inflow and outflow pattern naturally forms.\nIn  summary: we aim to construct a framework for investigating MHD jet structure of spinning BHs, in which the EM fields and the fluid motion are self-consistently obtained given  proper boundary conditions and a proper plasma loading function $\\eta(r,\\theta)$.\n\n\nThis paper is organized as follows. In Section~\\ref{sec:setup}, we summarize some basic equations and assumptions to be used in this paper.  We derive the two governing equations: the Bernoulli equation and the MHD Grad-Shafranov (GS) equation in Section~\\ref{sec:Bern} and Section~\\ref{sec:GS}, respectively. We detail the numerical techniques\nfor solving the governing equations in Section~\\ref{sec:eg}. \nThe numerical solutions of MHD jet structure with split monopole magnetic field configuration are presented in Section~\\ref{sec:results}. \nSummary and discussion are given in Section~\\ref{sec:summary}. \nFor reference, we place some  details for deriving the governing equations in Appendix \\ref{sec:D_der} and \\ref{sec:GS_der}.\nThroughout this paper, we use the geometrical units $c=G=M=1$, where $M$ is the mass of the central BH. \n\n\n\n\n\\section{Basic Setting Up} \\label{sec:setup}\n\nThe background  Kerr metric is written in the Boyer-Lindquist coordinates as follows,\n\\begin{eqnarray}\n\t{\\rm d}s^2&=& g_{tt} {\\rm d}t^2 + 2g_{t\\phi}{\\rm d}t {\\rm d}\\phi + g_{\\phi\\phi} {\\rm d}\\phi^2 + g_{rr} {\\rm d}r^2 + g_{\\theta\\theta} {\\rm d}\\theta^2  \\nonumber\\\\\n\t&\\ &  \\nonumber\\\\\n\t&=& \\left( \\frac{2Mr}{\\Sigma}-1 \\right) {\\rm d}t^2 - 2\\ \\frac{2Mar\\sin^2\\theta}{\\Sigma} {\\rm d}t {\\rm d}\\phi  \\nonumber\\\\\n\t&\\ & + \\frac{\\beta\\sin^2\\theta}{\\Sigma} {\\rm d}\\phi^2 + \\frac{\\Sigma}{\\Delta} {\\rm d}r^2 + \\Sigma {\\rm d}\\theta^2\\ ,\n\\end{eqnarray}\nwhere $a$ and $M$ are the BH spin and mass, respectively, $\\Sigma=r^2+a^2\\cos^2\\theta$, $\\Delta=r^2-2Mr+a^2$,  $\\beta =(r^2+a^2)^2-a^2\\Delta\\sin^2\\theta$ and the square root of the determinant $\\sqrt{-g}=\\Sigma\\sin\\theta$.\n\n\nWe investigate the structure of a steady and axisymmetric BH jet and  we assume the plasma within the jet is perfectly conducting, i.e., $\\partial_t = \\partial_\\phi = 0$ and $\\mathbf{E} \\cdot \\mathbf{B}=0$, where $\\mathbf{E}$ and \n$\\mathbf{B}$ are the electric and the magnetic fields, respectively. Then all the non-vanishing components of Maxwell tensor are expressed as follows \\citep[see e.g.,][]{Pan14}\n\\begin{equation}\n\\label{eq:Maxwell}\n\\begin{aligned}\n    F_{r\\phi} &= -F_{\\phi r} =\\Psi_{,r}  \\ ,   &F_{\\theta\\phi} &= -F_{\\phi\\theta} = \\Psi_{,\\theta} \\ , \\\\\n    F_{tr} &= -F_{rt} =\\Omega\\Psi_{,r}  \\ ,    &F_{t\\theta} &= -F_{\\theta t} = \\Omega  \\Psi_{,\\theta} \\ , \\\\\n    F_{r\\theta} &= -F_{\\theta r} = - \\frac{\\Sigma }{\\Delta\\sin\\theta} I \\ ,\n\\end{aligned}\n\\end{equation}\nwhere $\\Psi = \\Psi(r,\\theta)$ is the magnetic flux and $\\Omega = \\Omega(\\Psi)$ is the angular velocity of magnetic field lines. \nFor convenience, we have defined poloidal electric current $I(r,\\theta) \\equiv \\sqrt{-g} F^{\\theta r}$. Therefore,\nthe EM  fields are completely determined by three quantities: $\\{\\Psi(r,\\theta), \\Omega(\\Psi), I(r,\\theta)\\}$.\n\nBefore proceeding on, it is useful to define a few conserved quantities.\nFrom the perfectly conducting condition $F_{\\mu\\nu} u^\\nu = 0$, we find different components\nof fluid velocity are related by\n\\begin{equation}\n \\frac{ u^r}{\\Psi_{,\\theta}} = - \\frac{u^\\theta}{\\Psi_{,r}}\n\t= \\frac{(u^\\phi-\\Omega u^t)}{F_{r\\theta}}\\ ,\n\\end{equation}\nfrom which we can define the particle number flux per magnetic flux\n\\begin{equation}\\label{eq:eta}\n\\begin{aligned}\n \\eta&\\equiv\\frac{\\sqrt{-g} n u^r}{\\Psi_{,\\theta}} = - \\frac{\\sqrt{-g} n u^\\theta}{\\Psi_{,r}} \\\\\n\t&= \\frac{\\sqrt{-g} n(u^\\phi-\\Omega u^t)}{F_{r\\theta}} \\ .\n\\end{aligned}\n\\end{equation}\nFrom the energy-momentum tensor $T^{\\mu\\nu} = T^{\\mu\\nu}_{\\rm EM} + \tT^{\\mu\\nu}_{\\rm MT}$, \nwhere the EM part and the matter (MT) part are \n\\begin{equation}\n\\label{eq:energy_tensor}\n\\begin{aligned}\n\tT^{\\mu\\nu}_{\\rm EM}&= \\frac{1}{4\\pi} \\left( F^{\\mu\\rho}F_{\\ \\ \\rho}^{\\nu} - \\frac{1}{4}g^{\\mu\\nu}F_{\\alpha\\beta}F^{\\alpha\\beta} \\right)\\ , \\\\\n\tT^{\\mu\\nu}_{\\rm MT}&= \\rho u^\\mu u^\\nu = nm  u^\\mu u^\\nu \\ ,\n\\end{aligned}\n\\end{equation}\nwe define total energy per particle $E$ and total angular moment $L$ per particle as follows,\n\\begin{equation}\\label{eq:EandL}\n\\begin{aligned}\n\tE&\\equiv E_{\\rm MT}+E_{\\rm EM}= -mu_t +\\frac{\\Omega I}{4\\pi\\eta}\\ ,  \\\\\n\tL&\\equiv L_{\\rm MT}+L_{\\rm EM} = mu_\\phi +\\frac{I}{4\\pi\\eta}\\ ,\n\\end{aligned}\n\\end{equation}\nwhere  $\\rho$, $n$ and $m$ are the  proper energy density, the proper number density and the particle rest mass, respectively;\nand we have assumed cold plasma.\n\nNow let us examine the conservation property of these quantities along magnetic field lines.\nFor this purpose, we define derivative along field lines\n\\begin{equation}\n    D^\\parallel_\\Psi \\equiv \\frac{1}{\\sqrt{-g}}(\\Psi_{,\\theta}\\partial_r -  \\Psi_{,r}\\partial_\\theta)\\ ,\n\\end{equation}\nand it is straightforward to obtain (see Appendix \\ref{sec:D_der})\n\\begin{equation} \\label{eq:D_eta}\n\tD^\\parallel_\\Psi\\eta = (nu^\\mu)_{;\\mu} \\ ,\n\\end{equation}\ni.e., $D^\\parallel_\\Psi\\eta$ quantifies the plasma loading rate. In general, we can write the\nenergy-momentum conservation as \n\\begin{equation}\\label{eq:smu}\n    T^{\\mu\\nu}_{\\phantom{xy};\\nu} = S^\\mu,\n\\end{equation}\nwhere the source term $S^\\mu$ comes from plasma loading. As a simple example, we assume  \n $S^\\mu = (D^\\parallel_\\Psi\\eta) mu^\\mu$ in this paper,\ni.e., the source term is contributed by the kinetic energy of newly loaded plasma.\nWith a few steps of calculation as detailed in Appendix \\ref{sec:D_der}, we obtain\n\\begin{equation}\\label{eq:D_etaEL}\n\\begin{aligned}\n\tD^\\parallel_\\Psi(\\eta E)&= (D^\\parallel_\\Psi\\eta)(-mu_t)\\ ,\\\\\n\tD^\\parallel_\\Psi(\\eta L)&= (D^\\parallel_\\Psi\\eta)(mu_\\phi)\\ ,\n\\end{aligned}\n\\end{equation}\nwhere $\\eta E$ and $\\eta L$ are the energy flux per magnetic flux and angular momentum flux per magnetic flux, respectively.\n\\emph{Outside} the plasma loading zone where there is no particle injection,\nthe particle number conservation reads as\n\\begin{eqnarray}\n\t\\left(n u^\\mu\\right)_{;\\mu}&=&0\\ ,\n\\end{eqnarray}\ntherefore $\\eta, E, L$ are conserved along field lines,\ni.e., $\\eta=\\eta(\\Psi), E = E(\\Psi), L=L(\\Psi)$.\n\nIn summary: with assumptions of steady and axisymmetric jet structure and perfectly conducting plasma within the jet, we have obtained one conserved quantity $\\Omega(\\Psi)$\nand three ``quasi-conserved\"  quantities $\\{\\eta(\\Psi), E(\\Psi), L(\\Psi)\\}$\nwhich are only conserved \\emph{outside} the plasma loading zone.\n\n\\section{Bernoulli equation} \\label{sec:Bern}\n\nFrom the normalization condition $u^\\mu u_\\mu=-1$ and Eqs.~(\\ref{eq:eta},\\ref{eq:EandL}), we obtain the relativistic Bernoulli equation\n\\begin{eqnarray}\\label{eq:Bern}\n\t\\mathcal{F}(u) = u_p^2+1-\\left(\\frac{E}{m}\\right)^2 U_g(r,\\theta) = 0\\ ,\\end{eqnarray}\nwhere $u_p^2 \\equiv u^Au_A$ with the dummy index $A=\\{r,\\theta\\}$.\nIn the Kerr space-time,  the characteristic function $U_g$ is writen as \\citep{Camen86a,Camen86b,Camen87,Takahashi90,Fendt01,Fendt04,Levinson06,Pu15}\n\\begin{eqnarray}\n\\label{Ug}\n\tU_g(r,\\theta)&=& \\frac{k_0k_2-2k_2\\mathcal{M}^2-k_4\\mathcal{M}^4}{(\\mathcal{M}^2-k_0)^2}\\ ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{Ugk}\n\tk_0&=& -[g_{tt}+2g_{t\\phi}\\Omega+g_{\\phi\\phi}\\Omega^2]\\ ,\\nonumber\\\\\n\tk_2&=& \\left[ 1-\\Omega(E\/L)^{-1} \\right]^2\\ ,\\nonumber\\\\\n\tk_4&=& \\frac{\\left[ g_{tt}(E\/L)^{-2} + 2g_{t\\phi}(E\/L)^{-1} +g_{\\phi\\phi} \\right]}{g_{tt}g_{\\phi\\phi}-g_{t\\phi}^2}\\ ,\n\\end{eqnarray}\nand the \\Alfven Mach number $\\mathcal{M}$ is given by\n\\begin{equation}\\label{eq:mach}\n    \\mathcal{M}^2=\\frac{4\\pi m\\eta^2}{n} = 4\\pi mn\\frac{u_p^2}{B_p^2} = 4\\pi m\\eta\\frac{u_p}{B_p},\n\\end{equation}\nwith the poloidal magnetic field $B_p$ defined by\n\\begin{equation}\n(\\sqrt{-g} B_p)^2 = g_{rr} (\\Psi_{,\\theta})^2 + g_{\\theta\\theta} (\\Psi_{,r})^2  \\ .\n\\end{equation}\n\nSeveral characteristic surfaces can be defined according to the critical points of the flow velocity\n\\citep[see e.g., ][for details]{Michel69,Michel82,Camen86a,Camen86b, Takahashi90, Beskin09}.\nThe light surface (LS) is defined by where the rotation velocity of field lines approaches light speed and where\nparticles are forbidden to corotate with the field lines,\n\\begin{equation}\nk_0  \\big|_{r=r_{\\rm LS}} = 0 \\ .\n\\end{equation}\nThe \\Alfven surface is defined by where the denominator of characteristic function $U_g(r,\\theta)$ vanishes, i.e.,\n\\begin{equation}\n    - k_0 + \\mathcal{M}^2 \\big|_{r=r_A} = 0 \\ .\n\\end{equation}\nOn the \\Alfven surface, we find \n\\begin{equation} \n\\frac{E}{L}= - \\frac{g_{tt} + g_{t\\phi}\\Omega }{g_{t\\phi} + g_{\\phi\\phi}\\Omega}\\Bigg|_{r=r_A}\\ , \n\\end{equation} \nwhere we have used Eqs.(\\ref{eq:Bern}-\\ref{Ugk}).\nThe stagnation surface where $u_p = 0$ is determined by\n\\begin{equation}\\label{eq:stag}\nD^\\parallel_\\Psi k_0 \\big|_{r=r_*}=0 \\ .\n\\end{equation}\nThe fast magnetosonic (FM) surface and  slow magnetosonic (SM) surface are defined by where the denominator of $D_\\Psi^\\parallel u_p$ vanishes. In the cold plasma limit, the SM surface coincides with the stagnation surface.\nOn the stagnation surface, where both $u_p$ and $\\mathcal{M}$ vanish, we find\n\\begin{equation} \\label{eq:Estag}\n\\left(\\frac{E}{m} \\right)^2 = \\frac{k_0}{k_2}\\Bigg|_{r=r_*}\\ ,\n\\end{equation}\nwhere we have used Eqs.(\\ref{eq:Bern}, \\ref{Ug}). \n\nPlugging Eq.(\\ref{eq:eta}) into Eq.(\\ref{eq:Bern}), we find that the Bernoulli equation is a polynomial equation of\nfourth order in $u_p$ with to-be-determined eigenvalue $E\/L$ (or equivalently the location of the \\Alfven surface $r_A$), given prescribed angular velocity $\\Omega$ and particle number flux per field line $\\eta(r,\\theta)$ \\citep[see e.g.,][]{Camen86a,Camen86b, Takahashi90, Fendt01, Pu12, Pu15}.\n\n\\subsection{Single Loading Surface}\n\nAs a first step towards a full MHD jet solution, we mathematically idealize the plasma loading zone as a single surface, and we choose the stagnation surface (Eq.~(\\ref{eq:stag})) as the plasma loading surface \\citep[see e.g.,][ for a detailed gap model]{Brod2015} in this paper. \nTo define the plasma loading for both inflow and outflow, \nwe introduce a pair of dimensionless magnetization parameters on the loading surface,\n\\begin{equation}\\label{sigM}\n\t\\sigma_*^{\\rm in;out} =\\frac{B_{p,*}}{4\\pi m |\\eta|_{\\rm in;out}}\\ ,\n\\end{equation}\nwhere $B_{p,*}$ is the poloidal field on the loading surface.\nIn this way, the particle number flux per magnetic flux $\\eta$ is completely determined by $\\sigma_*^{\\rm in;out} $, recalling that\n$\\eta$ is a conserved quantity along field lines outside the loading zone. Note that $\\eta_{\\rm in} < 0$ and\n$\\eta_{\\rm out} > 0$, therefore there is a jump in $\\eta$ at the loading surface, i.e., \n$D_\\Psi^\\parallel \\eta \\propto \\delta(r-r_*)$.\n\nUsing Eq.(\\ref{eq:Estag}), the Bernoulli equation (\\ref{eq:Bern}) can be rewritten into a fourth-order polynomial equation as\n\\begin{eqnarray} \\label{Bern2}\n\t\\sum_{i=0}^4A_i u_p^i&=&0\\ ,\n\\end{eqnarray}\nwhere the coefficients $A_i$ are functions expressed by\n\\begin{equation}\n\\begin{aligned}\n\tA_4&= \\frac{1}{\\sigma_*^2}\\frac{B_{p,*}^2}{B_p^2}\\ ,\\\\\n\tA_3&= -\\frac{2k_0}{\\sigma_*}\\frac{B_{p,*}}{B_p}\\ ,\\\\\n\tA_2&= k_0^2 + \\left(1 + \\frac{k_{0,*}}{k_{2,*}} k_4\\right) \\frac{1}{\\sigma_*^2}\\frac{B_{p,*}^2}{B_p^2}\\ ,\\\\\n\tA_1&= \\left(-k_0 + \\frac{k_{0,*}}{k_{2,*}} k_2\\right) \\frac{2}{\\sigma_*}\\frac{B_{p,*}}{B_p}\\ ,\\\\\n\tA_0&= k_0^2 - \\frac{k_{0,*}}{k_{2,*}} k_0k_2\\ .\\\\\n\\end{aligned}\n\\end{equation}\n\nAs explored in several previous studies \\citep[e.g.,][]{Takahashi90, Pu15},\nsolving the Bernoulli equation above is in fact an eigenvalue problem, where $(E\/L)_{\\rm in}$ is the to-be-determined eigenvalue ensuring inflow smoothly cross the FM surface, while $(E\/L)_{\\rm out}$ is given by the match condition on the loading surface. Eq.(\\ref{eq:D_etaEL}) provides  conditions connecting the inflow and the outflow for the single surface loading,\n\\begin{eqnarray}\\label{eq:deltaEL}\n\t\\delta(\\eta E)&=& m(\\delta\\eta)(-u_t)_*\\ ,\\nonumber\\\\\n\t\\delta(\\eta L)&=& m(\\delta\\eta)(u_\\phi)_*\\ ,\n\\end{eqnarray}\nwhich give the match condition\n\\begin{eqnarray}\\label{eq:EOLout}\n\t(E\/L)_{\\rm out}&=& \\frac{(\\eta E)_{\\rm out}}{(\\eta L)_{\\rm out}} =\\frac{(\\eta E)_{\\rm in} + m(\\delta\\eta)(-u_t)_*}{(\\eta L)_{\\rm in} + m(\\delta\\eta)(u_\\phi)_*}\\ ,\n\\end{eqnarray}\nwhere $\\delta\\eta\\equiv\\eta_{\\rm out}-\\eta_{\\rm in}$, and we have used the fact that $D_\\Psi^\\parallel\\eta$ is \na $\\delta$-function centered on the loading surface in deriving Eq.~(\\ref{eq:deltaEL}).\nIt is straightforward to see Eq.~(\\ref{eq:deltaEL}) guarantees a same jump in the total energy\nflux and its matter component, therefore the Poynting flux (and all the EM field components) is continuous across the loading surface. \n\nAs along as the Bernoulli equation is solved, i.e., both the eigenvalues $(E\/L)_{\\rm in, out}$ and the poloidal velocity field $u_p$ are obtained,\n$u^r$ and $u^\\theta$ is obtained via Eq.(\\ref{eq:eta}) and $u_p^2=u^Au_A$,\nwhile $u_t$ and $u_\\phi$ are obtained via relation $m(u_t+\\Omega u_\\phi) = -(E-\\Omega L)$ and the normalization condition $u\\cdot u=-1$.\n\nBefore delving into the details of numerically solving the Bernoulli equation, we can now give an estimate of the eigenvalues. Combining the definitions of $E$ and $L$ Eqs.(\\ref{eq:EandL}) with Eq.(\\ref{eq:Bern}), we find\n\\begin{equation}\n(u_t+\\Omega u_\\phi)_*=-\\sqrt{k_{0,*}},\n\\end{equation}\nplugging which back into Eqs.(\\ref{eq:EandL}), we obtain\n\\begin{eqnarray}\n\\label{EOL}\n\t(E\/L)_{\\rm in}&=&\\Omega + \\frac{m\\eta_{\\rm in}\\sqrt{k_{0,*}}}{(\\eta L)_{\\rm in}}\\ <\\Omega\\ ,\\nonumber\\\\\n\t(E\/L)_{\\rm out}&=&\\Omega + \\frac{m\\eta_{\\rm out}\\sqrt{k_{0,*}}}{(\\eta L)_{\\rm out}}\\ >\\Omega\\ ,\n\\end{eqnarray}\nwhich imply $E\/L = \\Omega [1 + O(\\sigma_*^{-1})]$ and we have used the fact $\\eta_{\\rm in}<0$ and \n$\\eta_{\\rm out}>0$. \n\n\\section{MHD Grad-Shafranov equation} \\label{sec:GS}\nWith the aid of the Maxwell's equation\n\\begin{equation}\n\t {F^{\\mu\\nu}}_{;\\nu} = 4\\pi j^\\mu \\ ,\n\\end{equation}\nthe trans-field component of the energy conservation equation (\\ref{eq:smu}) is written as \\footnote{Eq.~(\\ref{eqn_MHD}) \nonly holds for the specific choice of source function $S^\\mu = (nu^\\nu)_{;\\nu} mu^\\mu$.} \n\\begin{equation}\\label{eqn_MHD}\n\t\\frac{F^A_{\\ \\phi}}{F_{C\\phi}F^C_{\\ \\phi}} (mn u^\\nu u_{A;\\nu}-F_{A\\nu}j^\\nu)=0\\ ,\n\\end{equation}\nwhere we have used Eq.~(\\ref{eq:D_eta}) and the source function $S^\\mu$. \nThe repeated Latin letters $A$ and $C$ run over the poloidal coordinates $r$ and $\\theta$ only \n\\citep{Nitta91,Beskin93,Beskin97}.\nThis is known as the MHD GS equation, with the $1^{\\rm st}$ and $2^{\\rm nd}$ terms\nin the bracket being the fluid acceleration and the electromagnetic force, respectively.\n\nAfter some tedious derivation (see Appendix \\ref{sec:GS_der}), we write the full MHD GS equation in a compact form\n\\begin{eqnarray}\n\\label{GS_MHD}\n\t&\\ &\\mathcal{L}\\Psi=\\mathcal{S}_{\\rm EM}+\\mathcal{S}_{\\rm MT}\\ .\n\\end{eqnarray}\nHere $\\mathcal{L}$ is a differential operator defined by\n\\begin{eqnarray}\n\\label{GS_MHD_L}\n\t\\mathcal{L}\\Psi&& = \\left[\\Psi_{,rr} + \\frac{\\sin^2\\theta}{\\Delta} \\Psi_{,\\mu\\mu} \\right]\\ \\mathcal{A}(r,\\theta;\\Omega)   \\nonumber\\\\\n\t&\\ & + \\left[ \\Psi_{,r} \\partial^\\Omega_r  + \\frac{\\sin^2\\theta}{\\Delta} \\Psi_{,\\mu} \\partial^\\Omega_\\mu \\right] \\ \\mathcal{A}(r,\\theta;\\Omega)  \\nonumber\\\\\n\t&\\ & + \\frac{1}{2} \\left[ (\\Psi_{,r})^2 + \\frac{\\sin^2\\theta}{\\Delta} (\\Psi_{,\\mu})^2 \\right] D_\\Psi^\\perp\\Omega\\ \\partial_\\Omega \\mathcal{A}(r,\\theta;\\Omega)  \\nonumber\\\\\n\t&\\ & - \\left[ (\\Psi_{,r})^2 + \\frac{\\sin^2\\theta}{\\Delta} (\\Psi_{,\\mu})^2 \\right] \\frac{D^\\perp_\\Psi\\eta}{\\eta}\\ \\mathcal{M}^2(r,\\theta)\\ ,\\nonumber\\\\\n\\end{eqnarray}\nwhere  $\\mu=\\cos\\theta$, $\\mathcal{A}(r,\\theta;\\Omega)=-k_0(r,\\theta;\\Omega)+\\mathcal{M}^2(r,\\theta)$,\nand we have defined $\\partial^\\Omega_A(A=r,\\mu)$ as the partial derivative with respect to coordinate $A$ with $\\Omega$ fixed, $\\partial_\\Omega$ as the derivative with respect to $\\Omega$, $D_\\Psi^\\perp$ as the derivative perpendicular to field lines\n\\begin{eqnarray}\n\tD_\\Psi^\\perp&\\equiv& \\frac{F^A_{\\ \\phi}\\partial_A}{F_{C\\phi}F^C_{\\ \\phi}}\\ ,\n\\end{eqnarray}\nwhich is equivalent to the ordinary derivative $d\/d\\Psi$ when acting on functions of $\\Psi$.\nThe two source terms are\n\\begin{equation}\n\\label{GS_MHD_S}\n\\begin{aligned}\n\t \\mathcal{S}_{\\rm EM} &=\\frac{\\Sigma}{\\Delta} I D^\\perp_\\Psi I\\ , \\\\\n     \\mathcal{S}_{\\rm MT} &=-4\\pi\\Sigma\\sin^2\\theta mn(u^tD_\\Psi^\\perp u_t + u^\\phi D_\\Psi^\\perp u_\\phi)\\ ,\n\\end{aligned}\n\\end{equation}\nwhere $I=4\\pi(\\eta L-\\eta m u_\\phi)$ [see Eq.(\\ref{eq:EandL})]. \n\nIn the FFE limit, $\\mathcal{M}^2=0$, $\\mathcal{S}_{\\rm MT}=0$, and the  GS equation reduces to \\citep{Pan17}\n\\begin{eqnarray}\n\\label{GS_FFE}\n\t&\\ &\\mathcal{L}\\Psi = \\mathcal{S}_{\\rm EM}\\ \\ .\n\\end{eqnarray}\nThe FFE solutions $\\{\\Psi|_{\\rm FFE}, \\Omega|_{\\rm FFE}, (\\eta L)|_{\\rm FFE} \\}$ have been well explored\nboth analytically and numerically in many previous studies \\citep[see e.g.,][]{BZ77,Tanabe08, Contop13, Pan15, Pan15b}.\nSimilar to the FFE case, solving the MHD GS equation (\\ref{GS_MHD}) is also eigenvalue problem, where $\\Omega$ and $4\\pi\\eta L$ are the to-be-determined eigenvalues ensuring field lines smoothly cross the two Alfven surfaces \\citep{Contop13, Nathan14, Pan17,Mahlmann18}.\n\n\n\n\n\n\\section{A split monopole example} \\label{sec:eg}\n\nAs previewed in the Introduction, we aim to construct a framework for investigating MHD jet structure of spinning BHs, \nin which the EM fields $(F_{\\mu\\nu})$ and the fluid motion $(n, u^\\mu)$ are self-consistently obtained \ngiven a proper plasma loading function $\\eta(r,\\theta)$ and proper boundary conditions.\n\nIn this section, we detail the procedure of consistently solving the two governing equations for an example of\nthe split monopole magnetic field configuration around a rapidly spinning central BH with a dimensionless spin $a=0.95$. For simplicity, we explore two different scenarios\nwith magnetization parameters $\\sigma_*^{\\rm out}=2\\sigma_*^{\\rm in}$ and $\\sigma_*^{\\rm out}=\\sigma_*^{\\rm in}$,\nrespectively. Remember that the loading function $\\eta(r,\\theta)$ is completely determined by the \nmagnetization parameters via the definition (\\ref{sigM}).\n\nBoundary conditions used here are similar to those of force-free solutions. \nExplicitly, we choose $\\Psi|_{\\mu=0}=\\Psi_{\\rm max}$  on the equatorial plane, \n$\\Psi|_{\\mu=1}=0$ in the polar direction,  $\\Psi_{,r}|_{r=r_{\\rm H}}=0$ and $\\Psi_{,r}|_{r=\\infty}=0$\nfor the inner and outer boundaries, respectively. Here $r_{\\rm H}$ is the radius of the event horizon.\n\n\\subsection{Numerical Techniques} \\label{sec:tech}\nWe define a new radial coordinate $R = r\/(r+1)$,  confine our \ncomputation domain $R\\times\\mu$ in the region $[R(r_{\\rm H}), R_{\\rm max}]\\times[0,1]$,\nand implement a uniform $256\\times64$ grid. \nIn practice, we choose $R_{\\rm max}=0.995$, i.e., $r_{\\rm max}\\approx 200 M$.\n\nThe Bernoulli equation (\\ref{eq:Bern}) and the MHD GS equation (\\ref{GS_MHD}), governing the flow along the field lines\nand field line configuration, respectively, are coupled. So we solve them one by one in an iterative way:\n\\begin{eqnarray} \\label{Eq_set}\n\\left\\{\\begin{array}{c}\n    \\mathcal{L}\\Psi^{(l)}=(\\mathcal{S}_{\\rm EM}+\\mathcal{S}_{\\rm MT})\\{(\\eta L)^{(l)}, n^{(l-1)}, u^{(l-1)}\\}\\ , \\\\\n    \\mathcal{F}\\{u^{(l)}; (E\/L)^{(l)}, \\Omega^{(l)}, \\Psi^{(l)}\\} =0 \\ ,\n\\end{array}\n\\right.\n\\end{eqnarray}\nwith $l=1,2,3,\\cdots$ . In a given loop $l$, we solve the GS equation updating $\\Psi$ and $\\{\\Omega, (\\eta L)\\}$ (with $\\{n, u^\\mu\\}$ inherited from the previous loop  $l-1$), ensuring field lines smoothly cross the two \\Alfven surfaces; \nin a similar way, we solve the Bernoulli equation updating $u^\\mu$ and $(E\/L)$ (with freshly updated $\\Omega$ and $\\Psi$ from solving the GS equation), ensuring a super-sonic inflow solution and an outflow solution satisfying the match condition \n(\\ref{eq:EOLout}). Combing solutions to both equations and definitions of $\\{\\eta, E, L\\}$, we finally obtain all the desired quantities $\\{F_{\\mu\\nu}, n, u^\\mu\\}$ as functions of coordinates $r$ and $\\theta$. \n\nWe activate the iteration with an initial guess\n\\begin{eqnarray} \\label{init}\n\\left\\{\\begin{array}{cccc}\n    \\Psi^{(0)}(r,\\theta)&=&\\Psi_{\\rm max}(1-\\cos\\theta)\\ ,  \\\\\n    \\Omega^{(0)}(\\Psi)&=&0.5\\Omega_{\\rm H}\\ ,  \\\\\n    (\\eta L)^{(0)}(\\Psi)&=&\\Omega_{\\rm H} \\Psi[2-(\\Psi\/\\Psi_{\\rm max})]\/(8\\pi)\\ ,\\\\\n    n^{(0)}(r,\\theta) &=& u^{(0)}(r,\\theta) = 0\\ ,\n\\end{array}\n\\right.\n\\end{eqnarray}\nwhere $\\Omega_{\\rm H}\\equiv a\/(r_{\\rm H}^2+a^2)$ is the BH angular velocity.\n\nThe numerical techniques for tackling the two eigenvalue problems are detailed as follows:\n\n\\begin{itemize}\n\\item[\\it Step 1] \nThe MHD GS equation is a second-order differential\nequation which degrades to first order on the\n\\Alfven surfaces where $\\mathcal{A}(r,\\theta)=0$. \nNumerical techniques for dealing this problem have been well developed\nin previous force-free studies \\citep{Contop13, Nathan14, Huang16,Huang18, Pan17,Mahlmann18}, and we briefly recap them here.\n\n\nIn each loop $l$, we solve the GS equation (\\ref{GS_MHD}) with\nthe approximate solution obtained from the previous loop $\\left\\{\\Omega^{(l-1)},(\\eta L)^{(l-1)},\\Psi^{(l-1)}\\right\\}$\nas the initial guess. \nWe evolve the flux function $\\Psi^{(l)}$ using the overrelaxation technique with Chebyshev acceleration \\citep{Press86}, and $\\Psi^{(l)}(r,\\theta)$ is updated on grid points except those in the vicinity of the two \\Alfven surfaces. The  flux function $\\Psi^{(l)}(r,\\theta)$ on the \\Alfven surfaces are obtained via interpolation from neighborhood grid points and the directional derivatives on the \\Alfven surfaces \\citep{Pan17}. \nUsually we obtain two different flux function $\\Psi(r_{\\rm A}^-)$ versus $\\Psi(r_{\\rm A}^+)$ \non the \\Alfven surface via interpolations from grid points inside and outside, respectively. \nTo decrease this discontinuity, we adjust $\\Omega^{(l)}(\\Psi)$ at the outer Alfv\\'en (OA) surface:\n\\begin{eqnarray}\n\t\\Omega^{(l)}_{\\rm new}(\\Psi_{\\rm new})&=& \\Omega^{(l)}_{\\rm old}(\\Psi_{\\rm old}) \\nonumber\\\\\n\t&\\ & + 0.05 [\\Psi(r_{\\rm OA}^+)-\\Psi(r_{\\rm OA}^-)],\n\\end{eqnarray}\nwith $\\Psi_{\\rm new}=0.5[\\Psi(r_{\\rm OA}^+)+\\Psi(r_{\\rm OA}^-)]$, \nwhere the subscript old\/new represents quantities before\/after the above adjustment;\nand adjust both $\\Omega^{(l)}(\\Psi)$ and $(\\eta L)^{(l)}(\\Psi)$ at the inner Alfv\\'en surface (IA):\n\\begin{eqnarray}\n\t\\Omega^{(l)}_{\\rm new}(\\Psi_{\\rm new})&=& \\Omega_{\\rm old}(\\Psi_{\\rm old}) \\nonumber\\\\\n\t&\\ & + 0.05[\\Psi(r_{\\rm IA}^+)-\\Psi(r_{\\rm IA}^-)],\\nonumber\\\\\n\t(\\eta L)^{(l)}_{\\rm new}(\\Psi_{\\rm new})&=& (\\eta L)^{(l)}_{\\rm old}(\\Psi_{\\rm old}) \\nonumber\\\\\n\t&\\ & - 0.05[\\Psi(r_{\\rm IA}^+)-\\Psi(r_{\\rm IA}^-)],\n\\end{eqnarray}\nwith $\\Psi_{\\rm new}= 0.5[\\Psi(r_{\\rm IA}^+)+\\Psi(r_{\\rm IA}^-)]$.\n\nAfter sufficient evolution, we obtain a converged solution $\\{\\Omega^{(l)}, (\\eta L)^{(l)}, \\Psi^{(l)}\\}$ \n which ensures field lines smoothly cross the two \\Alfven surfaces.\n\n\\item[\\it Step 2] The Bernoulli equation in the form of Eq.(\\ref{Bern2}) is  a fourth-order polynomial equation in $u_p$\n\\citep{Camen86a,Camen86b,Camen87,Takahashi90,Fendt01,Fendt04,Levinson06,Pu15},\nwhere the FM point is a standard `X'-type singularity, while the Alfv\\'en point turns out to be a higher-order\nsingularity \\citep{Weber67}.\nMathematically, a FM point is the location of a multiple root to the Bernoulli equation. \nThe existence of FM point is very sensitive to the value of $(E\/L)_{\\rm in}^{(l)}$. \nFor a slightly small value, there exists only sub-sonic solutions in the region $r<r_A^{\\rm in}$, \ni.e., no multiple root. For a slightly larger value, on the other hand, there exists no global solution\nextending from $r_{\\rm H}$ to $r_A^{\\rm in}$. Only for some specific choice, there exist a global super-sonic solution that crossing the FM point (see Fig.~\\ref{fig:up} for numerical examples). \nWe adjust $(E\/L)_{\\rm in}^{(l)}$ on each field line until an inflow solution that smoothly crosses the FM point is found. \n\nWith the inflow solution in hand, $(E\/L)_{\\rm out}^{(l)}$ on each field line is uniquely determined by the match condition Eq.(\\ref{eq:EOLout}), then it is straightforward to compute the outflow velocity.\n\n\\item[\\it Step 3] Combining the inflow and the outflow solutions from Step 2,\nthe global fluid velocity $u^{(l)}$ and therefore the number density $n^{(l)},$ and the  Mach number $\\mathcal{M}^{(l)}$ are obtained along each field line. We feed these quantities into the GS equation (\\ref{GS_MHD})  for the next loop.\nWe iterate Step 1 to Step 3 until all quantities converge to a given precision.\n\\end{itemize}\n\nWe should point out that there is  an unphysical divergence \narising from the idealized plasma loading on a single surface adopted in this paper. \nThe particle number flux function $\\eta$ is negative\/positive for inflow\/outflow and is not continous\non the loading surface. According to the \ndefinition of $\\eta$ in Eq.(\\ref{eq:eta}),\nthe particle number density $n$ is proportional to $\\eta\/u_p$ and therefore\ndiverges on the stagnation surface where $u_p=0$. The particle number density $n$ show up in \nthe source terms of the MHD GS equation (\\ref{GS_MHD}). To overcome the unphysical infinity,\nwe smooth them in the vicinity of the stagnation surface before feeding into Eq.(\\ref{GS_MHD}).\\footnote{\nIn addition, we usually obtain two different angular momentum flux per magnetic flux $(\\eta L)_{\\rm Bern}$ from the Bernoulli equation and $(\\eta L)_{\\rm GS}$ from the GS equation, respectively. The former is obtained from $E\/L$ and Eq.(\\ref{eq:Estag}), while the latter is one of the eigenvalues of the GS equation (\\ref{GS_MHD}). In general, the two do not match exactly,  where $(\\eta L)_{\\rm GS}$ does approach $(\\eta L)|_{\\rm FFE}$ in the limit of $\\sigma_*\\rightarrow \\infty$, while  $(\\eta L)_{\\rm Bern}$ does not. For the cases investigated in \nSection \\ref{sec:results}, we find $(\\eta L)_{\\rm GS}$ is different from $(\\eta L)_{\\rm Bern}$ by \n$\\lesssim 25\\%$. This mismatch indicates that the particle number flux per magnetic \nflux $\\eta(r,\\theta)$ cannot be arbitrarily given, or perfectly conducting fluid is not a sufficient description here. } \n\n\n\\section{Numerical Results} \\label{sec:results}\n\n\\subsection{Case Study}\n\nIn this subsection, we explore two nearly force-free cases with $\\sigma_*\\gg 1$: {\\it Case 1} with magnetization parameters $\\sigma_*^{\\rm out} =\\sigma_*^{\\rm in}=75$  and {\\it Case 2} with $\\sigma_*^{\\rm out} =2\\sigma_*^{\\rm in}=60$.\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[scale=0.9]{f2a.eps}\\\\\n\\includegraphics[scale=0.9]{f2b.eps}\n\\caption{{\\it Top-left:} Comparison of the poloidal field line configuration of an MHD jet of {\\it Case 1} (black solid lines)\nwith parameters $\\sigma_*^{\\rm out} = \\sigma_*^{\\rm in}=75$\nversus its FFE counterpart (black dotted lines).\nThe loading surface (loading), Alfv\\'en surfaces (A), light surfaces (LS),\nand inner fast magnetosonic (FM) are presented in blue, dashed purple,\nmagenta, and aqua line, respectively.\nThe configuration of a force-free jet is also shown for comparison (dotted line).\n{\\it Top-right:} A zoom-out configuration of the left panel.\n{\\it Bottom:} The poloidal field line configuration of an MHD jet solution of {\\it Case 2}\nwith parameters $\\sigma_*^{\\rm out}=2\\sigma_*^{\\rm in}=60$. \\label{fig:fl}}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[scale=0.9]{f3a.eps}\\\\\n\\includegraphics[scale=0.9]{f3b.eps}\n\\caption{{\\it Top-left:} The poloidal fluid velocity $u_p$ of {\\it Case 1}.\n{\\it Top-right:} $|u_p|$ as a function of $r$ on the field line with foot-point on the event horizon $\\mu(r_{\\rm H})=0.508$, where the solutions with correct eigenvalue $E\/L$ are shown in solid lines and those with $E\/L$ of slightly larger\/smaller values are shown in dotted\/long-dashed lines.\nThe Alfv\\'en points and the FM point are marked by open circles.\n{\\it Bottom:} The fluid motion configuration of {\\it Case 2},\nsimilar to top panels.  \\label{fig:up}}\n\\end{figure*}\n\n\n\nIn Fig.~\\ref{fig:fl},  we compare the magnetic field line configurations of MHD solutions with that of FFE version. For both cases, which are nearly force-free, the deviation from the FFE version is as expected small, though the deviation of {\\it Case 1} from the FFE solution is more obvious than that of {\\it Case 2},\ndue to more efficient outflow acceleration in the former case (see Fig.~\\ref{fig:ut}).\n\nIn Fig.~\\ref{fig:up}, we show the poloidal velocity $u_p$. To clarify the different singularity types of FM points and \\Alfven points, we explicitly show all the solutions (both physical one and unphysical ones) to the Bernoulli equation\n(\\ref{Bern2}) for the field line with foot point  $\\mu(r_{\\rm H})=0.508$, where the solutions with correct eigenvalue $E\/L$ are shown in solid lines and the solutions with $E\/L$ slightly off the correct value are shown in dotted and dashed lines. \nAt the FM point which is a `X'-type singularity, there exists a multiple root of the Bernoulli equation for correct eigenvalue $E\/L$, while no global solution (dotted lines) or no multiple root exists (long-dashed lines)  for $E\/L$ of slightly off value.\nAt the \\Alfven point which is a higher-order singularity, there exists multiple root  no matter $E\/L$ takes the correct eigenvalue or not. For the physical solution, the inflow passes along $r_*\\to r_{\\rm A}\\to r_{\\rm FM}\\to r_{\\rm H}$ and the outflow passes along $r_*\\to r_{\\rm A}\\to\\infty$.\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[scale=0.9]{f4a.eps}\\\\\n\\includegraphics[scale=0.9]{f4b.eps}\n\\caption{{\\it Top-left:} The fluid angular velocity $\\Omega_{\\rm MT}$ of {\\it Case 1}.\n{\\it Top-right:} Comparison of a few angular velocity like quantities $\\{\\Omega|_{\\rm FFE}, \\Omega|_{\\rm MHD}, (E\/L)_{\\rm in;out}\\}$ of {\\it Case 1} .\n{\\it Bottom Panels:}  same as the top ones except for {\\it Case 2}.  \\label{fig:Om}}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[scale=0.9]{f5a.eps}\n\\includegraphics[scale=0.9]{f5b.eps}\n\\caption{The configuration of particle energy $-u_t$ of {\\it Case 1} ({\\it left}) \nand {\\it Case 2} ({\\it right}). \\label{fig:ut}}\n\\end{figure*}\n\n\nIn Fig.~\\ref{fig:Om}, we show the fluid angular velocity $\\Omega_{\\rm MT}$, the angular velocity of magnetic\nfield lines $\\Omega$ (solid lines), and the eigvenvalues $(E\/L)_{\\rm in;out}$ (dashed grey lines). \nConsistent with our intuition, we find the rotation of magnetic field lines is dragged down by the fluid inertia compared with the FFE case, i.e., $\\Omega|_{\\rm MHD} < \\Omega|_{\\rm FFE}$. Due to nonzero poloidal velocity of both\ninflow and outflow, the fluid does not corotate with the field lines, i.e., $\\Omega_{\\rm MT}<\\Omega$ for inflow  and $\\Omega_{\\rm MT}>\\Omega$ for outflow. Specifically, the fluid angular velocity on the\nevent horizon $\\Omega_{\\rm MT}(r=r_{\\rm H})$ slightly exceeds the BH angular velocity $\\Omega_{\\rm H}$,\nwhich guarantees the fluid energy to be positive on the horizon (see Fig.~\\ref{fig:ut}). \n\nIn Fig.~\\ref{fig:ut}, we show the specific particle energy $-u_t$ for the two cases.\nBoth of them are positive everywhere, while the outflow of \\emph{Case 1} gains more efficient\nacceleration.\n\n\\subsection{Energy Extraction Rates}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=1]{f6.eps}\n\\caption{Results of the energy extraction rates in relation to $\\sigma_*^{\\rm in}$ with $\\sigma_*^{\\rm out}=2\\sigma_*^{\\rm in}$ assumed.\nThe energy rates measured at the event horizon\n$\\{ \\dot{E}_{\\rm tot}^{\\rm H}, \\dot{E}_{\\rm Poynting}^{\\rm H}, \\dot{E}_{\\rm MT}^{\\rm H}\\}$, are presented in filled squares, small filled squares, and filled circles, respectively. The solid black line in top panel, the dotted black line in top panel, and the solid black line in bottom panel are the corresponding fitting curves.\nSimilarly, the energy rates measured at infinity\n$\\{ \\dot{E}_{\\rm tot}^{\\infty}, \\dot{E}_{\\rm Poynting}^{\\infty}, \\dot{E}_{\\rm MT}^{\\infty}\\}$, are presented in open symbols and the solid grey lines are the corresponding fitting curves. \\label{fig:Edot}}\n\\end{figure}\n\nIn this subsection, we investigate the energy extraction rate from the central BH via the MHD jet,\nwhich is defined as \n\\begin{equation} \n\\begin{aligned}\n      \\dot{E}_{\\rm tot}(r)\n      &= -2\\pi \\int_0^{\\pi} \\sqrt{-g} T^r_{\\ t}(r) {\\rm d} \\theta\\ , \\\\ \n      &= 4\\pi\\int_0^{\\Psi_{\\rm max}} (\\eta E)(r) {\\rm d}\\Psi \\ , \\\\ \n      &= 4\\pi\\int_0^{\\Psi_{\\rm max}} [(E\/L)\\times(\\eta L)](r) {\\rm d}\\Psi \\ ,\n\\end{aligned}\n\\end{equation} \nwhere we have used Eqs.(\\ref{eq:eta}-\\ref{eq:EandL}) in the second line. In the third line, $E\/L$ and $\\eta L$ are the eigenvalues of the Bernoulli equation (\\ref{eq:Bern}) and of the GS equation (\\ref{GS_MHD}), respectively. In the similar way, we can define its matter\/electromagnetic component as\n\\begin{equation} \\label{eq:Edot}\n\\begin{aligned}\n\t\\dot{E}_{\\rm MT}(r)&=  4\\pi\\int_0^{\\Psi_{\\rm max}}(-  \\eta m u_t)(r) {\\rm d}\\Psi\\ , \\\\\n\t\\dot{E}_{\\rm Poynting}(r)\n\t&= 4\\pi \\int_0^{\\Psi_{\\rm max}}(\\Omega I\/4\\pi)(r) {\\rm d}\\Psi \\ .\\\\\n\t&= \\dot{E}_{\\rm tot}(r) - \\dot{E}_{\\rm MT}(r)\\ . \n\\end{aligned}\n\\end{equation}\n\nWe measure these energy extraction rates at $r=r_{\\rm H}$\/ $r=\\infty$, and quantify their dependence\non the magnetization parameter $\\sigma_*$. In practice, we find that these energy extraction rates are not sensitive to the value of $\\sigma_*^{\\rm out}$,\nexcept the matter component of energy rate at infinity $\\dot{E}_{\\rm MT}^\\infty$. \nWithout loss of generality, we only show the rates in relation to $\\sigma_*^{\\rm in}$\nfor the $\\sigma_*^{\\rm out}=2\\sigma_*^{\\rm in}$ scenario in Fig.~\\ref{fig:Edot}, where\nall the rates are displayed in unit of the energy extraction rate in the force-free limit \n$\\dot E_{\\rm FFE}\\approx 0.4 (\\Psi_{\\rm max}^2\/4\\pi)$.\n\nAs we see in Fig.~\\ref{fig:Om}, the rotation of magnetic field lines is dragged down by\nthe loaded plasma, i.e., $\\Omega|_{\\rm MHD} < \\Omega|_{\\rm FFE}$, while the fluid that does not corotate with\nthe field lines with angular velocity $\\Omega_{\\rm MT}|_{\\rm inflow} > \\Omega > \\Omega_{\\rm MT}|_{\\rm outflow}$, tends to bend the field lines and induce a stronger $\\phi$-component of magnetic field, i.e., $I|_{\\rm MHD} > I|_{\\rm FFE}$. The net result is that the Poynting energy extraction rate on the event horizon $\\dot E_{\\rm Poynting}^{\\rm H}$ has little dependence on the magnetization. Going outward along the field lines, part of the Poynting flux is converted into the fluid kinetic energy. For the case with magnetization parameter $\\sigma_*^{\\rm in} = 30$, the matter component makes up about $13\\%$ of the total energy flux at infinity.\n\n\\subsection{Penrose Process}\\label{subsec:Penrose}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=0.68]{f7.eps}\n\\caption{The positron\/electron component of energy extraction rates in relation to $\\sigma_*^{\\rm in}$. \nThe energy rates  $\\{\\dot{E}^{\\rm H}_{e^+}, \\dot{E}^{\\rm H}_{e^-}\\}$ \nare presented in open and filled circles, respectively. \nThe solid and dashed lines are the corresponding fitting curves.\nThe shaded regime denotes where the Penrose process is \nworking (for the electron component). \\label{fig:EdMT}}\n\\end{figure}\n\nAn implicit assumption in our MHD jet model is the two-fluid description, since \nthe electric current density $j^\\mu$ is \\emph{not} proportional to the fluid velocity $u^\\mu$.\nTherefore we can decompose the charged fluid as two oppositely charged components, \npositron ($e^+$) and electron ($e^-$). \\footnote{Though there is a degree of freedom in doing \nthis decomposition, e.g., we can also decompose the fluid as electrons and ions, it does not change\nour conclusion qualitatively.} We denote the number densities and the velocity fields as $n_\\pm$ \nand $u^\\mu_\\pm$, respectively, which are related to $j^\\mu$ and $nu^\\mu$ via relations\n\\begin{eqnarray}\n    j^\\mu&=& e(n_+u^\\mu_+ - n_-u^\\mu_-)\\nonumber\\\\\n    mnu^\\mu&=& m(n_+u^\\mu_+ + n_-u^\\mu_-).\n\\end{eqnarray}\nConsequently, we obtain \n\\begin{equation} \n m(n u^\\mu)_\\pm =  \\frac{1}{2}[\\pm j^\\mu(m\/e) + nmu^\\mu] \\ ,\n\\end{equation} \nand we can decompose the matter energy flux into two components $\\dot{E}_{e^\\pm}$.\nHere we are only interested in the energy extraction rates on the event horizon\n\\begin{eqnarray}\n    \\dot{E}^{\\rm H}_{e^+}&=& 4\\pi \\int_0^{\\Psi_{\\rm max}} \\frac{(-\\eta m n_+u_{t+})(r_{\\rm H})}{n(r_{\\rm H})} {\\rm d}\\Psi\\ ,\\nonumber\\\\\n    \\dot{E}^{\\rm H}_{e^-}&=& 4\\pi \\int_0^{\\Psi_{\\rm max}} \\frac{(-\\eta m n_-u_{t-})(r_{\\rm H})}{n(r_{\\rm H})} {\\rm d}\\Psi \\ .\n\\end{eqnarray}\nAs an example, we choose the horizon enclosed magnetic flux $\\Psi_{\\rm max}=1000(m\/e)$, and show  $\\dot{E}^{\\rm H}_{e^+}\/\\dot{E}_{\\rm FFE}$ \nand $\\dot{E}^{\\rm H}_{e^-}\/\\dot{E}_{\\rm FFE}$ in relation to $\\sigma_*^{\\rm in}$ in Fig.~\\ref{fig:EdMT}. \nThe energy extraction rate from positrons is always negative, while \nthe energy extraction rate from electrons, become positive \nwhen the plasma loading is low enough. In this regime, denoted in shades in Fig.~\\ref{fig:EdMT}, the magnetic Penrose process is working, though, only for one of the two charged component. \\footnote{We should not do any quantitative interpretation for the results of this subsection, because the two-fluid decomposition done here is not accurate,  e.g., there is no guarantee for the velocity of each component $u^\\mu_\\pm$ to be timelike and normalized.\nWe will leave a more accurate two-fluid description of MHD jet structure \\citep{Koide09, Liu18} to future work .} This finding is in good agreement with recent particle-in-cell simulations \\citep{Parfrey18}.\n\n\n\\section{Summary and Discussion}\\label{sec:summary}\n\\subsection{Summary}\nTo describe the MHD structure of BH jets, we need a minimum set of quantities as functions of spacetime: \nMaxwell tensor $F_{\\mu\\nu}$, fluid rest mass density $\\rho$ (or equivalently particle number density $n$), and \nfluid four-velocity $u^\\mu$. For determining all these quantities self-consistently, we constructed a full MHD framework, in which EM fields and fluid motion are governed by the MHD GS equation (\\ref{GS_MHD}) and the Bernoulli equation (\\ref{eq:Bern}), respectively. From these two governing equations, we can completely determine $\\{F_{\\mu\\nu}, \\rho ,u^\\mu\\}$ given proper boundary conditions and a proper plasma loading function $\\eta(r,\\theta)$ (see Eq.(\\ref{eq:eta})). As an example, we consider a split monopole field configuration and idealized plasma loading on the stagnation surface.\n\nAssuming steady and axisymmetric jet structure, and perfectly conductive plasma within the jet, the EM fields are \ncompletely determined by three functions: the magnetic flux $\\Psi(r, \\theta)$, the angular velocity of magnetic field \nlines $\\Omega(\\Psi)$ and the poloidal electric current $I(r,\\theta)$ (see Eq.(\\ref{eq:Maxwell})). \nGiven fluid energy density $\\rho$ and velocity $u^\\mu$, the MHD GS equation (\\ref{GS_MHD}) turns out to be a second-order differential equation with respect to $\\Psi(r,\\theta)$ which degrades to be first-order on the two \\Alfven surfaces. Solving the GS equation is an eigenvalue problem, with eigenvalues $\\Omega(\\Psi)$\nand $I(r,\\theta)$ (or more precisely, the conserved quantity $4\\pi\\eta L(\\Psi)$ defined in Eq.(\\ref{eq:EandL})) to be determined ensuring field lines smoothly cross the \\Alfven surfaces.\n\nGiven EM fields $F_{\\mu\\nu}$, the Bernoulli equation turns out to be a fourth-order polynomial equations in the poloidal fluid velocity $u_p$. Solving the Bernoulli equation is also an eigenvalue problem,  with the eigenvalue $(E\/L)_{\\rm in}$ to be determined ensuring the inflow smoothly cross the FM surface, and $(E\/L)_{\\rm out}$ to be determined by the match condition (\\ref{eq:EOLout}) on the loading surface. With both $E\/L$ and $u_p$ obtained, it is straightforward to obtain $n$ and $u^\\mu$ via Eqs.(\\ref{eq:eta},\\ref{eq:EandL}) and the  normalization \ncondition $u\\cdot u=-1$. \n\nThe two governing equations are coupled, therefore we numerically solved them in an iterative way (see Sec.~\\ref{sec:tech}). \nAs a result, we find the rotation of magnetic field lines is dragged down by the plasma loaded, i.e., $\\Omega|_{\\rm MHD} < \\Omega|_{\\rm FFE}$; for the fluid angular velocity, we find  $\\Omega_{\\rm MT}|_{\\rm outflow}<\\Omega<\\Omega_{\\rm MT}|_{\\rm inflow}$ ; \nthe non-corotating fluid tends to bend the field lines and induce a stronger $\\phi$-component of magnetic field, therefore \na stronger poloidal electric current\ni.e., $I|_{\\rm MHD} > I|_{\\rm FFE}$. The net result is that the Poynting energy extraction on the horizon is insensitive to the\nmagnetization, i.e., $\\dot E_{\\rm Poynting}^{\\rm H} |_{\\rm MHD} \\approx \\dot E_{\\rm Poynting}^{\\rm H}|_{\\rm FFE}$ (see Fig.~\\ref{fig:Edot}).\nGoing outward along the field lines, part of the Poynting flux is converted to the fluid kinetic energy. For the case we explored\nwith $\\sigma_*^{\\rm in} = 30$, the matter component makes up $\\sim 13\\%$ of the total energy flux at infinity.\n\nFinally, we examined the MHD Penrose process for the cases we numerically solved. We found the specific fluid energy $-m u_t$ is always positive on the event horizon, i.e., the MHD Penrose process is not working and therefore the BZ mechanism defines fully the jet energetics. However, if we decompose the charged fluid as two oppositely charged components ($e^\\pm)$, we found the magnetic Penrose process does work for one of the two components when the plasma loading is low enough (see Fig.~\\ref{fig:EdMT}).\n\n\\subsection{Discussion}\n\nAs a first step towards a full MHD jet model, we have investigated the MHD jet structure\nof split monopole geometry assuming an idealized plasma loading on the stagnation  surface. \nThis simplified plasma loading gives rise to a few unphysical problems in the vicinity of the loading surface, \nincluding divergence of particle number density $n(r_*)$, which shows up in the source terms of the MHD\nGS equation (\\ref{GS_MHD_S}). To avoid the singularity arising from the unphysical divergence, we smoothed the\nfunction $n(r,\\theta)$ in the vicinity of the loading surface. Another consequence of the simplified \nplasma loading is that we must impose the continuity equation (\\ref{eq:EOLout}) to ensure the EM fields to be continuous across the loading surface. As a result,  $(E\/L)_{\\rm out}$ is specified by $(E\/L)_{\\rm in}$, i.e., $r_{A,\\rm out}$ is specified by $r_{A, \\rm in}$. Therefore, we lose the freedom to adjust $(E\/L)_{\\rm out}$ until a supersonic outflow solution is found\nas we did for the inflow solution. Consequently, all the outflow solutions obtained in this paper are subsonic (see Fig.~\\ref{fig:up}). \n\nIn future work, we aim to investigate a full MHD jet model with a more realistic extending loading zone \nwhere the plasma injection is described by a continuous function $\\eta(r,\\theta)$. Then all the unphysical\ndiscontinuity and divergence described above would be avoided. For the extending plasma loading, the smooth EM fields would be naturally preserved, and the continuity requirement would not be a constraint. As a result,\nwe can adjust $(E\/L)_{\\rm out}$ for finding a supersonic outflow solution, which is more consistent with recent observations \\citep{Hada16,Mertens16}.\nIn addition to the plasma loading, the BH surroundings also play an important role in shaping the jet structure \\citep[e.g.][]{Tchek10,Beskin17}. The role of more realistic BH environment, including accretion flows and hot plasma with non-zero pressure will also be considered in future work. \n\n\n\n\n\\section*{Acknowledgements}\nWe thank the referee for his\/her careful reading of this manuscript and giving insightful suggestions.\nL.H. thanks the support by the National\nNatural Science Foundation of China (grants 11590784 and 11773054),\nand Key Research Program of Frontier Sciences, CAS (grant No. QYZDJ-SSW-SLH057).\nZ.P. thanks Hung-Yi Pu for his invaluable help throughout this research.  \nZ.P. is  supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute\nis supported  by the Government of Canada through the Department of Innovation, \nScience and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.\nC.Y. has been supported by the National Natural Science Foundation of China (grants 11521303, 11733010 and 11873103).\nThis work made extensive use of the NASA Astrophysics Data System and\nof the {\\tt astro-ph} preprint archive at {\\tt arXiv.org}.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGravitational wave (GW) sources~\\cite{DI:2016,secondBBH:2016,thirddetection,fourth:2017,GW170608,o1o2catalog} are now routinely detected by the advanced LIGO~\\cite{DII:2016,LSC:2015} and Virgo~\\cite{Virgo:2015} detectors. The last two observing runs of these GW detectors indicate that, on average, one GW source has been detected for every fifteen days of analyzed data. It is expected that this number will be superseded in the upcoming third observing run, since the advanced LIGO and Virgo detectors have been undergoing commissioning since August 2017. \\new{In their enhanced sensitivity configuration, they will be able to probe a larger volume of space, thereby boosting the expected detection rate} for binary black hole (BBH) mergers and binary neutron star (BNS), and may yield the first observations of neutron star-black hole (NSBH) mergers~\\cite{o1o2catalog}.\n\nGiven the expected scale of GW discovery in upcoming observing runs, it is in order to explore the use of efficient signal-processing algorithms for low-latency GW detection and parameter estimation. This work is motivated by the need to probe a deeper parameter space that is available to GW detectors, in real-time, and using minimal computational resources to maximize the number of studies that can be conducted with GW data. This combination of constraints is a common theme for large-scale astronomical facilities, which will be producing large datasets in low-latency within the next decade, e.g., the Large Synoptic Survey Telescope~\\cite{lsstbook}. Scenarios in which both LSST, among other electromagnetic observatories, and advanced LIGO and Virgo work in unison, analyzing disparate datasets in real-time to realize the science goals of Multi-Messenger Astrophysics make this work timely and relevant~\\cite{whitepaper:SCIMMA,eliuMMA:2019}. \n\nAmong a number of recent developments in signal-processing, deep learning exhibits great promise to increase the speed and depth of real-time GW searches. The first deep learning algorithms to do classification and regression of GWs emitted by non-spinning BBHs on quasi-circular orbits were presented in~\\cite{geodf:2017a} in the context of simulated LIGO noise. The extension of that study to realistic detection scenarios using real advanced LIGO noise was introduced in~\\cite{geodf:2017b}. Even though these algorithms were trained to do real-time classification and regression of GWs in realistic detection scenarios for a 2-D signal manifold (non-spinning BBHs on quasi-circular orbits), the studies presented in~\\cite{geodf:2017a,geodf:2017b,geodf:2017c,Rebei:2018R} have demonstrated that deep learning algorithms generalize to new types of sources, enabling the identification of moderately eccentric BBH mergers, spin precessing BBH mergers, and moderately eccentric BBH signals that include higher-order modes, respectively. These studies also indicate that while the detection of these new types of GW sources is possible, it is necessary to use higher-dimensional signal manifolds to train these algorithms to improve parameter estimation results, and to go beyond point-parameter estimation analysis. This work has sparked the interest of the GW community, leading to a variety of studies including the classification of simulated BBH waveforms in Gaussian noise, GW source modeling and GW denoising of BBH mergers~\\cite{geodf:2017c,hshen:2017,positionML:2018,wei:2019W,Rebei:2018R,AlvinC:2018,2018GN,Fan:2018,Gonza:2018,2018GN,Fuji:2018,LiYu:2017,Nakano:2018}.\n\nWhile detection and parameter estimation are the key goals for the development of deep learning for GW astrophysics, in this article we focus on the application of deep learning for parameter estimation. At present, GW parameter estimation is done using Bayesian inference~\\cite{bambi:2012MNRAS,bambiann:2015PhRvD,Singer_Price_2016}, which is a well tested and extensively used method, though computationally-intensive. On the other hand, given the scalability of deep learning models in training mode (i.e., the ability to combine distributed training and large datasets to enhance the performance of deep learning algorithms in realistic data analysis scenarios), and their computational efficiency in inference mode, it is natural to explore their applicability for GW parameter estimation, the theme of this article. \n\n\\noindent \\textbf{Previous Work} The first exploration of deep learning for the detection and point-parameter estimation of a 2-D signal manifold was presented in~\\cite{geodf:2017a,geodf:2017b}. For waveform signals with matched-filtering signal-to-noise ratio (SNR) \\(\\textrm{SNR}\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}} 10\\), these neural network models measure the masses of quasi-circular BBH mergers with a mean percentage absolute error \\(\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 15\\%\\), and with errors \\(\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 35\\%\\) for moderately eccentric BBH mergers. These results provided a glimpse of the robustness and scalability of deep neural network models, and the motivation to take these prototypical applications into a production run toolkit for GW parameter estimation. \n\n\\noindent \\textbf{Highlights of This Work} \n\n\\begin{cititemize2}\n\\item We have designed new architectures and training schemes to demonstrate that deep learning provides the means to reconstruct the parameters of BBH mergers in more realistic astrophysical settings, i.e., BHs whose spins are aligned or anti-aligned, and which evolve on quasi-circular orbits. \\new{This 4-D signal manifold marks the first time deep learning models \\textit{at scale} are used for GW data analysis, i.e., models trained using datasets with tens of millions of waveforms, and 1,024 nodes (64 processor per node) to significantly reduce the training stage.} Once fully trained, these deep learning models can reconstruct in real-time the parameters of the BBH catalog presented by the LIGO and Virgo Scientific Collaboration in~\\cite{o1o2catalog}. \n\\item The neural network models we introduce in this article have two different architectures. The first one is tailored for the measurement of the masses of the binary components, whereas the second is used to quantify the final spin and the quasi-normal modes (QNMs) of the BH remnant. Once both neural networks are fully trained, we use them in parallel for inferences studies, finding that we can reconstruct the parameters of BBH mergers within 2 milliseconds using  a single Tesla V100 GPU. \n\\item \\new{We introduce a novel scheme to train Bayesian Neural Network (BNN) models at scale using 1,024 nodes on a High Performance Computing platform while keeping optimal performance for inference. We then adapted this framework to introduce for the first time the use of BNNs for GW parameter estimation. With this approach we can estimate the astrophysical parameters of the existing catalog of detected BBH mergers~\\cite{o1o2catalog}, and their posterior distributions, reporting inference times in the order of milliseconds.}\n\n\\item \\new{We use variational inference to approximate the posterior distribution of model parameters in the probabilistic layers of our neural networks. In the inference stage, we sample the network parameters to evaluate the posterior distribution of the physical parameters. Details of the model and training are in Sections~\\ref{sec:prob_model} and~\\ref{bnn_scale}.}\n\\end{cititemize2}\n\n\nThis article is structured as follows. Section~\\ref{method} introduces the model architectures used in these analyses, it describes the construction and curation of the datasets used to train, validate and test our neural network models. It also includes a revised curriculum learning for neural network training. We quantify the accuracy of these neural network models in realistic detection scenarios using real advanced LIGO noise in Section~\\ref{experiments}. We put at work our deep learning algorithms in Section~\\ref{discussion} to estimate the astrophysical parameters of the BBH mergers reported in~\\cite{o1o2catalog}. We summarize our findings and future directions of work in Section~\\ref{conclusion}.\n\n\\begin{figure*}[t!]\n\t\\centerline{\n\t\t\\raisebox{1cm}{\n\t\t\t\\includegraphics[width=90mm]{spin_omegas_diagram.png}}\n\t\t\\hspace{10mm}\n\t\t\\includegraphics[width=65mm]{masses_diagram.png}\n\t}\n\t\\caption{The \\new{left}\n\tarchitecture is used to estimate the final spin and quasi-normal modes of the black hole remnant. The \\new{right}\n\tarchitecture is used to estimate the masses of the binary black hole components.  }\n\t\\label{model_diagram_spin_omegas}\n\\end{figure*}\n\n\\section{Methods}\n\\label{method}\n\nIn this section, we introduce the neural network models used for parameter estimation, and describe a novel curriculum learning scheme to accurately measure the masses of the binary components, and the final spin and QNMs of the BH remnant. We have used \\texttt{TensorFlow}~\\cite{abadi2016tensorflow,abadi2015tensorflow} to design, train, validate and test the neural network models presented in this section. \n\nThe rationale to use two neural network models stems from the fact that the masses, spins and QNMs span rather different scales. Therefore, to improve the accuracy with which deep learning can measure these parameters we have designed one neural network that is tailored to measure the masses of the binary components, and one to measure the final spin and QNMs of the remnant. The astute reader may have noticed that the final spin of the BH remnant and its QNMs have a similar range of values when the QNMs are cast in dimensionless units, and this is the approach we have followed. In practice, we train the second neural network model using the fact that the QNMs are determined by the final spin \\(a_f\\) using the relation~\\cite{Berti:2006b}\n\n\\begin{equation}\n\\omega_{220}\\left(a_f\\right)= \\omega_R + i\\, \\omega_{I}\\,,\n\\label{qnms}\n\\end{equation}\n\n\\noindent where \\((\\omega_R,\\,\\omega_{I})\\) correspond to the frequency and damping time of the ringdown oscillations for the fundamental \\(\\ell=m=2\\) bar mode, and the first overtone \\(n=0\\). We have computed the QNMs following~\\cite{Berti:2006b}. One can readily translate \\(\\omega_R\\) into the ringdown frequency (in units of Hertz) and \\(\\omega_I\\) into the corresponding (inverse) damping time (in units of seconds) by computing \\(M_f\\,\\omega_{220}\\). \\(M_f\\)  represents the final mass of the remnant, and can be determined using Eq. (1) in~\\cite{HealyLous:2017PRDH}.\n\nAs we describe below, we have found that to accurately reconstruct the masses of the binary components, it is necessary to use a more complex and deeper neural network architecture. It is worth mentioning that once these models are fully trained, a single GPU is sufficient to perform regression analyses in milliseconds using both neural network models. \n\n\n\\subsection{Neural network model to measure the properties of the black hole remnant \\label{subsec:properties_model}}\n\nThe neural network model consists of two main parts: a shared root component for all physical parameters, and three leaf components for individual parameters ($a_f$, $\\omega_R$, and $\\omega_I$), as illustrated in the left panel of Figure~\\ref{model_diagram_spin_omegas}, and Table~\\ref{spin_omega_model_config}. The model architecture looks like a rooted tree. The root is composed of seven convolutional layers, and its output is shared by the leaves. Each leaf component has the same network architecture with three fully connected layers. This approach is inspired by the hierarchical self decomposing of convolutional neural networks described in~\\cite{DBLP:journals\/corr\/abs-1811-04406,hu2018squeeze}. The key idea behind this approach is that the neural network structures are composed of a general feature extractor for the first seven layers, which is then followed up by sub-networks that take values from the output of the general feature extractor. \n\nThe rationale to have splits after the universal structure is to use sub-structures that focus on different sub-groups of the data. As a simile: even though the human body has multiple limb locations (``leaves\"), human motion is controlled by the overall motion of the body (``the root\"). In practice this means that the tree structure of our models leverages the hierarchical structure of the data. It first extracts the universal features through the root, and then passes the information to the different sub-networks (``leaves\") to learn specialized features for different physical parameters. Notice that the root will also prevent overfitting in the ``leaves\", since each leaf is optimized through the root. \n\nAnother change to the conventional architecture is that we remove the nonlinear activation in the second to last layer in the leaf component, i.e., it is a linear layer with identity activation function (see Table~\\ref{spin_omega_model_config}). This allows more neurons to be activated and passed to the final layer.  As discussed in~\\cite{DBLP:journals\/corr\/abs-1709-07634}, removing the nonlinear activation in some intermediate layers smooths the gradients and maintains the correlation of the gradients in the neural network weights, which, in turn, allows more information to be passed through the network as the depth increases.\n\n\\begin{table}[t]\n\t\\setlength{\\tabcolsep}{1.5pt}\n\t\\caption{Architecture of the neural network model used to measure the final spin and QNMs of the black hole remnant. For the root convolutional layers, the setup indicates: (kernel size, \\# of output channels, stride, dilation rate, max pooling kernel, max pooling stride). All convolutional layers have ReLU activation function and the padding is set to ``VALID'' mode. There is no max pooling layer if the last two entries in the configuration are 0's. The leaf fully connected layers setup: (\\# of output neurons, dropout rate). For the last layer, we use \\(\\tanh\\) activation function. However, the activation function in the second last layer is removed.}\n\t\\label{spin_omega_model_config}\n\t\\vskip -0.25in\n\t\\begin{center}\n\t\t\\begin{small}\n\t\t\t\\begin{tabular}{c|c|c}\n\t\t\t\t\\hline\n\t\t\t\t\\specialcell{Layer \\\\ Component} & \\specialcell{Layer \\\\ Configurations}  & \\specialcell{Activation \\\\ Functions}\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\specialcell{Root Layer: \\\\ Convolutional}  & $\\begin{array}{c}\n\t\t\t\t(16, 64, 1, 1, 4, 4) \\\\\n\t\t\t\t(16, 128, 1, 2, 4, 4) \\\\\n\t\t\t\t(16, 256, 1, 2, 4, 4) \\\\\n\t\t\t\t(32, 256, 1, 2, 4, 4) \\\\\n\t\t\t\t(4, 128, 1, 2, 0, 0) \\\\\n\t\t\t\t(4, 128, 1, 2, 0, 0) \\\\\n\t\t\t\t(2, 64, 1, 1, 0, 0) \\\\\n\t\t\t\t\\end{array}$ & ReLU \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\specialcell{Leaf Layer: \\\\ Fully Connected}  & $\\begin{array}{c}\n\t\t\t\t(128, 0.0) \\\\\n\t\t\t\t(128, 0.0) \\\\\n\t\t\t\t(1, 0.0)\n\t\t\t\t\\end{array}$ & $\\begin{tabular}{c}\n\t\t\t\tReLU \\\\\n\t\t\t\tIdentity \\\\\n\t\t\t\tTanh \\\\\n\t\t\t\t\\end{tabular}$ \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\\end{small}\n\t\\end{center}\n\t\\vskip -0.1in\n\\end{table}\n\n\\subsection{Neural network model to measure the masses of the binary components} \n\\label{subsec:mass_model}\n\nThe tree-like network model used for this study is described in the right panel of Figure~\\ref{model_diagram_spin_omegas} and Table~\\ref{mass_model_config}. With respect to the architecture described in the previous section, we reduce the number of convolutional layers in the root from seven to three. We have done this because we are now using more layers in the leaves, which in turn  makes the gradient back-propagation harder. Reducing the number of root layers improves gradient updates to the front layers.\n\nEach leaf component uses a squeeze-and-excitation (SE) structure~\\cite{hu2018squeeze}. The SE block is a sub-structure between two layers (squeeze step). It applies a global pooling, and assigns weights to each of the channels in the convolutional layers (excitation step). Compared to conventional convolutional structures with universal weights, the SE components adjust the importance of each channel with an adaptively learned weight, which, as described in~\\cite{hu2018squeeze}, effectively results in 25\\% improvement in image classification. For images, channels are usually represented in RGB. Since we are using 1-D time-series signals, we treat channels of the original input signals to be 1. The SE block adaptively recalibrates channel-wise feature responses. Furthermore, the weights are optimally learned through a constraint introduced by the global pooling. This ensures that the weights encode both spatial and channel-wise information. Furthermore, the weights help the channels represent group specific features at deeper layers, which is consistent with our objective of using ``leaves\" for different parameters. \n\nFollowing the SE components, the neural networks have two highway blocks~\\cite{srivas:2015S}. The structures are a variant of the residual structure, as proposed in~\\cite{he2016deep}. In the residual block, instead of directly learning the feature, it learns the residual components by an identity shortcut connection, which resolves the gradients vanishing when the model goes deeper. The highway block only introduces weights to the components in the residual block, which is similar to the application of importance weights on channels in SE components. Finally, we apply three fully connected layers with dropouts after the highway blocks to prevent overfitting~\\cite{JMLR:v15:srivastava14a}. The same nonlinearity reduction is also applied in the second last layer. \n\n\n\\begin{table}[t]\n\t\\setlength{\\tabcolsep}{1.5pt}\n\t\\caption{Architecture of the neural network model used to measure the masses of the binary components. For the root convolutional layers, the setup indicates: (kernel size, \\# of output channels, stride, dilation rate, max pooling kernel, max pooling stride). All convolutional layers have ReLU activation function and the padding is set to ``VALID'' mode. For the Leaf SE layer, the setup is: (\\# of output channels, \\# of residual blocks). The general structure for the SE layer follows the configuration described in~\\cite{hu2018squeeze}. Leaf highway layer setup: (kernel size, \\# of channels, stride, \\# of highway blocks). The configuration for the highway is described in~\\cite{srivas:2015S}. The leaf fully connected layers setup is: (\\# of output neurons, dropout rate). For the last layer we use ReLU activation. However, the activation function in the second last layer is removed.}\n\t\\label{mass_model_config}\n\t\\vskip -0.25in\n\t\\begin{center}\n\t\t\\begin{small}\n\t\t\t\\begin{tabular}{c|c|c}\n\t\t\t\t\\hline\n\t\t\t\t\\specialcell{Layer \\\\ Component} & \\specialcell{Layer \\\\ Configurations}  & \\specialcell{Activation \\\\ Functions}\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\specialcell{Root Layer: \\\\ Convolutional} & $\\begin{array}{c}\n\t\t\t\t(16, 64, 1, 2, 4, 4) \\\\\n\t\t\t\t(16, 128, 1, 2, 4, 4) \\\\\n\t\t\t\t(16, 128, 1, 2, 4, 4)\n\t\t\t\t\\end{array}$ & ReLU\\\\\n\t\t\t\t\\hline\n\t\t\t\tLeaf Layer: SE &  $\\begin{array}{c}\n\t\t\t\t(128, 3) \\\\\n\t\t\t\t(128, 3) \n\t\t\t\t\\end{array}$& ReLU \\\\  \n\t\t\t\t\\hline\n\t\t\t\tLeaf Layer: Highway& $\\begin{array}{c}\n\t\t\t\t(4, 128, 2, 30)\n\t\t\t\t\\end{array}$ & ReLU\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\specialcell{Leaf Layer: \\\\ Fully Connected} & $\\begin{array}{c}\n\t\t\t\t(512, 0.1) \\\\\n\t\t\t\t(256, 0.1) \\\\\n\t\t\t\t(1, 0.0)\n\t\t\t\t\\end{array}$ & $\\begin{tabular}{c}\n\t\t\t\tReLU \\\\\n\t\t\t\tIdentity \\\\\n\t\t\t\tReLU \\\\\n\t\t\t\t\\end{tabular}$\\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\\end{small}\n\t\\end{center}\n\t\\vskip -0.1in\n\\end{table}\n\n\\subsection{Probabilistic Model}\n\\label{sec:prob_model}\n\\new{In this section we present the probabilistic framework based on Bayesian inference, which we have applied to the neural networks outlined in Sections \\ref{subsec:mass_model} and \\ref{subsec:properties_model}. We use Bayesian neural networks (BNNs)~\\cite{neal2012bayesian,mackay1992practical}, which are neural networks with uncertainty over their weights, to provide estimates of the BBH masses and properties of the BH remnant  posterior distributions. This is in contrast to standard neural networks which provide point estimates of parameters. We use prior and posterior distribution functions on the last two layers of each leaf. With this approach, each of the leaves becomes an independent probabilistic model that regresses the physical parameters. The root layers, on the other hand, can be viewed as feature extractors for each probabilistic leaf.}\n\n\\new{A BNN can be viewed as a probabilistic model for the posterior distribution, $p(\\boldsymbol{w}|{\\mathcal{D}})$, where $\\boldsymbol{w}$ are the model weights and $\\mathcal{D} = \\{\\boldsymbol{x}_j,\\boldsymbol{y}_j\\}_{j= 1}^n$ is the training dataset. Here, $\\boldsymbol{x}_j $ are the input noisy waveforms and $\\boldsymbol{y}_j$ are the continuous parameters of interest, i.e., the BBH masses and the properties of the BH remnant.} \n\n\\new{According to Bayes theorem, $p({\\boldsymbol{w}} | \\mathcal{D}) \\propto p(\\mathcal{D}|{\\boldsymbol{w}})p({\\boldsymbol{w}}),$ where $p({\\boldsymbol{w}})$ is the prior distribution for the weights and \n$p(\\mathcal{D}|{\\boldsymbol{w}})$ is the likelihood. We assume that the likelihood function for each pair of the training data is, }\n\n\\new{\n\\begin{equation}\n    \\label{eq:likelihood}\n    p\\left(\\boldsymbol{y} \\vert \\boldsymbol{x}, \\boldsymbol{w} \\right) =  \\frac{1}{\\sqrt{2\\pi} \\epsilon }\\exp{ \\left(- \\frac{\\| \\boldsymbol{y} - f_{\\boldsymbol{w}}(\\boldsymbol{x}) \\|^2}{2 \\epsilon^2} \\right)},\n\\end{equation}\n}\n\n\\noindent \\new{where $f_{\\boldsymbol{w}}$ represents the neural network function with weights $\\boldsymbol{w}$ and $\\epsilon$ is the standard deviation. The aleatoric uncertainty is covered by the likelihood distribution. A BNN allows a stochastic sampling of the weight parameters during a forward pass through the network while also encoding prior knowledge through the use of prior distributions. \nWe use a variational inference (VI) algorithm to approximate the weight posterior distribution $p(\\boldsymbol{w} | \\mathcal{D})$ using a Gaussian distribution for the weights assuming a mean field approximation, denoted by $q_{\\boldsymbol{\\theta}}(\\boldsymbol{w})$.   \nIt is parameterized by $\\boldsymbol{\\theta}  = (\\boldsymbol{\\mu}, \\boldsymbol{\\sigma})$, representing the mean vector and the standard deviation vector of the distribution respectively. }\n\n\n\\new{The corresponding cost function can be written as}\n\n\\new{\\begin{equation}\n    \\label{eq:loss}\n    \\mathcal{L} = \\mathrm{KL}\\left ( q_{\\boldsymbol{\\theta}}(\\boldsymbol{w} ) \\| p(\\boldsymbol{w}) \\right) - \\mathbb{E}_{q_{\\boldsymbol{\\theta}}(\\boldsymbol{w})} \\log p(\\mathcal{D} \\vert \\boldsymbol{w} ),\n\\end{equation}}\n\n\\noindent \\new{which is known as the variational free energy. The prior distribution is chosen to be a standard normal distribution. Since the probabilistic layers are parameterized by the mean and variance of the weight distributions, the number of parameters which need to be optimized is doubled compared to a standard neural network. The cost function can be approximated by drawing $N$ samples $\\boldsymbol{w}^{(i)}$ from $q_{\\boldsymbol{\\theta}}(\\boldsymbol{w})$,}\n\n\\new{\\begin{align}\n    \\mathcal{L} & \\approx \\frac{1}{N} \\sum_{i = 1}^N \\left[ -\\log q_{\\boldsymbol{\\theta}} \\left(\\boldsymbol{w} \\right)  - \\log p\\left(\\boldsymbol{w}^{(i)} \\right)  \\right. \\nonumber \\\\\n    & \\quad \\quad \\left . - \\log p\\left(\\mathcal{D} \\vert \\boldsymbol{w}^{(i)} \\right) \\right] \n    \\label{eq:loss2}    \n\\end{align}}\n\n\\noindent \\new{During training, for every forward model pass, the variational posterior  distribution  for  the  model  parameters is estimated. Specifically, we use stochastic gradient descent to estimate $\\boldsymbol \\theta$ of $q_{\\boldsymbol{\\theta}}(\\boldsymbol{w} )$ by minimizing Eq.~\\eqref{eq:loss2}. In testing or inference mode, for input waveform $\\boldsymbol{x}^*$, our approximate predictive distribution is given by,}\n\\new{\n\\begin{align}\n\\label{uncertainty_approx_fun}\n  q( \\boldsymbol{y}^* | \\boldsymbol{x}^* ) & = \\int p(\\boldsymbol{y}^*| \\boldsymbol{x}^*, \\boldsymbol{w}) q_{\\boldsymbol{\\theta}}(\\boldsymbol{w}) \\, d{\\boldsymbol{w}}\\,.\n\\end{align}\n}\n\n\\noindent \\new{ We use sampling to compute the statistics of the corresponding estimated physical parameters, e.g., median and 90\\% confidence interval. In addition to the aleatoric uncertainty, the uncertainty in the predictions arises from uncertainty in the weights or so called `epistemic uncertainty.'}\n\n\n\\new{In this probabilistic modeling, we apply the following simplifications: (1) the likelihood function is assumed to be Gaussian, and (2) neural network weight distributions are assumed to be independent Gaussians. Under these assumptions, the loss in Eq.~\\eqref{eq:loss2} is simplified and tractable. The statistical models and VI method are implemented using the computing framework TensorFlow Probability (TFP) \\cite{2017arXiv171110604D,2018arXiv181203973T} using a modified sampling scheme and distributed across nodes in a data parallel fashion using Horovod~\\cite{2018arXiv180205799S}. Details of the model training at scale are discussed in Section~\\ref{bnn_scale}.}\n\n\\subsection{Dataset Preparation}\n\nTo demonstrate the use of deep learning for parameter estimation, we consider the catalog of BBH mergers presented in~\\cite{o1o2catalog}. Based on the Bayesian analyses presented in that study, we consider the following parameter space to produce our training dataset: \\(m_1\\in[9{\\rm M}_{\\odot},\\,65{\\rm M}_{\\odot}]\\), \\(m_2\\in[5.2{\\rm M}_{\\odot},\\, 42{\\rm M}_{\\odot}]\\). The spin of the binary components span a range \\(a_{\\{1,\\,2\\}}\\in[-0.8,\\,0.8]\\). By uniformly sampling this parameter space we produce a dataset with 300,180 waveforms. These waveforms are produced with the surrogate waveform family~\\cite{blackman:2015}, considering the last second of the evolution which includes the late inspiral, merger and ringdown. The waveforms are produced using a sample rate of 8192Hz.\n\n\nFor training purposes, we label the waveforms using the masses and spins of the binary components, and then use this information to also enable the neural net to estimate the final spin of the BH remnant using the formulae provided in~\\cite{Hofmann:2016yih}, and the QNMs of the ringdown following~\\cite{Berti:2006b}. In essence, we are training our neural network models to identify the key features that determine the properties of the BBHs before and after merger using a unified framework.\n\nIn order to encapsulate the true properties of advanced LIGO noise, we whiten all the training templates using real LIGO noise from the Hanford and Livingstone detectors gathered during the first and second observing runs~\\cite{losc}.\n\nWe use 70\\% of these waveform samples for training, 15\\% for validation, and 15\\% for testing. The training samples are randomly and uniformly chosen. Throughout the training, we use ADAM optimizer to minimize the mean squared error of the predicted parameters with default hyper-parameter setups~\\cite{journals\/corr\/KingmaB14}. We choose the batch size to be 64, the learning rate to be 0.0008, the total number of iterations to be 120,000 (maximum). We use a dropout rate 0.1 for training and no dropout is applied for testing and validation. To simulate the environment where the true GWs are embedded, we use real advanced LIGO noise to compute power spectral density, which is then used to whiten the templates. In addition, we apply a random 0\\% to 6\\% left or right shifts. This endows the neural networks with time-invariance, and improves their performance to estimate the parameters of the signal irrespective of their position in the data stream. On the other hand, this technique also prevents overfitting of the data. Since the locations are randomly shifted with independent noise injected, the training data are different at each epoch. \n\n\\begin{figure*}\n\t\\centerline{\n\t\t\\includegraphics[width=0.48\\linewidth]{masses_plot.png}\n\t\t\\hspace{5mm}\n\t\t\\includegraphics[width=0.48\\linewidth]{spin_omegas_plot.png}\n\t}\n\t\\caption{Relative error with which our deep learning algorithm can measure the masses, final spin, \\(a_f\\), and quasi-normal modes (QNMs), \\((\\omega_R\\,,\\omega_I)\\) of the binary black hole components as a function of optimal matched-filtering signal-to-noise ration (SNR). \\textit{Left panel:} For waveform with \\(\\textrm{SNR}\\geq15\\), the primary and secondary masses can be constrained with relative errors less than \\((7\\%,\\,12\\%)\\), respectively. \\textit{Right panel:} For signals with \\(\\textrm{SNR}\\geq15\\), \\((a_f\\,,\\omega_R\\,,\\omega_I)\\) can be recovered with relative errors less than \\((13\\%,\\, 5\\%,\\,3\\%)\\), respectively.}\n\t\\label{relative_errors_mass}\n\\end{figure*}\n\n\n\\subsection{Curriculum learning with decreasing signal-to-noise ratio}\n\\label{dec_snr}\n\n\n\n\\begin{table}[t]\n\t\\setlength{\\tabcolsep}{7pt}\n\t\\caption{Decreasing peak SNR (pSNR) setup. The pSNR is uniformly chosen within the indicated range. Notice that the early stopping criterion is also applied if the number of iterations is greater than 60,000 and the relative error threshold is met. The relation between match-filtering SNR to pSNR is: 1.0 pSNR $\\approx$ 13.0 SNR.}\n\t\\label{CL_BNN}\n\t\\vskip -0.25in\n\t\\begin{center}\n\t\t\\begin{small}\n\t\t\t\\begin{sc}\n\t\t\t\t\\begin{tabular}{c|c}\n\t\t\t\t\t\\hline\n\t\t\t\t\tIterations & pSNRs  \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t1-12000 & 2.0-3.0\\\\\n\t\t\t\t\t12001-24000 & 1.5-3.0\\\\  \n\t\t\t\t\t24001-36000 & 1.0-3.0 \\\\\n\t\t\t\t\t36001-60000 & 0.5-3.0\\\\\n\t\t\t\t\t60001-90000 & 0.3-3.0\\\\\n\t\t\t\t\t90001-120000 & 0.2-3.0\\\\\n\t\t\t\t\t120001- & 0.1-3.0 \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\\end{tabular}\n\t\t\t\\end{sc}\n\t\t\\end{small}\n\t\\end{center}\n\t\\vskip -0.1in\n\\end{table}\n\n\nIn realistic detection scenarios, GWs have moderate SNRs, and are contaminated by non-Gaussian and non-stationary noise. In order to ensure that neural networks identify GWs over a broad range of astrophysically motivated SNRs, we start training them with large SNRs,  and gradually reduce the SNRs to a lower level. This is an idea taken from curriculum learning literature~\\cite{Bengio:2009:CL:1553374.1553380}, which allows the network to distill more accurate information of the underlying signals with larger SNRs to signals with lower SNRs. This approach has been demonstrated for classification, regression and denoising of GW signals~\\cite{geodf:2017a,geodf:2017b,geodf:2017c,Rebei:2018R,hshen:2017,dgNIPS,wei:2019W,George:2017qtr}. Specifically, each waveform is normalized to have maximum amplitude 1, and then we use curriculum learning  with the decreasing SNR scheme detailed in Table~\\ref{CL_BNN} (The strategy for BNN models is the same).  The noisy data is then normalized to have variance one. We normalize the data to ensure that the trained model can characterize true BBH signals in realistic detection scenarios, covering a broad range of SNRs.\n\nThe different steps followed in our curriculum learning scheme are presented in Table~\\ref{CL_BNN}. In addition, we use an early stopping criterion with the relative error threshold 0.026 for \\((m_1\\,,m_2)\\) and 0.0016 for \\((a_f,\\,\\omega_R\\,,\\omega_I)\\). One additional change to the mass model is we rescale the masses by 1\/20, to make the optimization converge faster. In the evaluation, we just scale the data back to its original amplitude.\n\n\\subsection{Training of the Bayesian Neural Network Model}\n\\label{bnn_scale}\n\n\\new{For the probabilistic layers, as the effective number of parameters to be optimized is double that of a standard layers, we examine the impact of scaling the BNN code across nodes on the pre-exascale Cray XC40 system, Theta, at Argonne National Laboratory. Using an optimized build of both Tensorflow and Horovod for the Intel Xeon-Phi [coded name Knights Landing (KNL)] architecture we distribute the code using one MPI rank per node and 128 hardware threads per node and scale up to 1024 nodes. Results for the number of samples processed per second during training is shown in Figure~\\ref{bnn_nn_scaling}. We achieve $\\sim 75$\\% efficiency up to 1024 nodes on Theta. As the number of nodes is increased, there is increased communication of the gradients at each iteration which causes an expected decrease in performance away from the ideal scaling. As the BNN layers have in effect twice the  parameters of the standard layers, the communication cost is slightly higher which can be seen as a decrease in the number of samples processed per second.}\n\n\\new{In addition to evaluating the efficiency on Theta, we fully trained the two BNN models on Hardware-Accelerated Learning (HAL) cluster at the National Center for Supercomputing Applications. Each model was trained on 4 NVIDIA V100 GPUs with batch size of 64. The parameter $\\epsilon$ in the likelihood function Eq.~\\eqref{eq:likelihood} is chosen to be 0.1 for the mass model and $10^{-3}$ for the final spin and QNMs model. We draw $N = 100$ samples and $M = 1600$ samples from $q_{\\boldsymbol{\\theta}}(\\boldsymbol{w})$ at training and testing respectively. The learning rate for the two BNN models is $8 \\times 10^{-6}$. The total number of iterations is 200,000 to guarantee convergence.}\n\n\\begin{figure}[t!]\n\\includegraphics[width=90mm]{bnn_nn_scaling.pdf}\n\t\\caption{Samples processed per second with increasing number of nodes during training of the neural network. The results for the BNN are shown in cyan and standard neural network in blue. Ideal scaling is shown as a dashed black bar at each node count. Error bars are the variance from all iterations during training.}\n\t\\label{bnn_nn_scaling}\n\\end{figure}\n\n\n\\section{Experimental Results}\n\\label{experiments}\nUsing the signal manifold described in the previous section, we present results of the accuracy with which our neural network models can measure the masses of the binary components, and the properties of the corresponding remnant. \n\n\\indent Figure~\\ref{relative_errors_mass} presents the accuracy with which the binary components \\((m_1,\\,m_2)\\) can be recovered over a a broad range of SNRs. We notice that for signals with \\(\\textrm{SNR}\\geq15\\), the primary and secondary masses can be constrained with relative errors~\\cite{relerror:1965} less than \\((7\\%,\\,12\\%)\\), respectively. These results represent a major improvement to the analysis we reported in the context of a 2-D signal manifold in~\\cite{geodf:2017a,geodf:2017b}. Furthermore, we can also see from the same figure that for signals with \\(\\textrm{SNR}\\geq15\\) our neural network models can measure the  triplet \\((a_f\\,,\\omega_R\\,,\\omega_I)\\) with relative errors less than \\((13\\%,\\, 5\\%,\\,3\\%)\\), respectively. To the best of our knowledge, this is the first time deep learning is used to infer the properties of BH remnants directly from GW signals.\n\n\n\\section{Deep learning parameter estimation of detected binary black hole mergers}\n\\label{discussion}\n\n\n\\begin{table*}[!htp]\n\t\\setlength{\\tabcolsep}{4.5pt}\n\t\\renewcommand{\\arraystretch}{2.0}\n\t\\caption{\\new{Parameter estimation results for the catalog of binary black hole mergers reported in~\\cite{o1o2catalog} using our deterministic deep learning models. We report median values with the the 90\\% confidence interval, which was computed by whitening gravitational wave strain data that contain real gravitational wave signals with up to 240 different power spectral densities.}}\n\t\\label{tab_real_event_results}\n\t\\vskip -0.25in\n\t\\begin{center}\n\t\t\\begin{small}\n\t\t\t\\begin{sc}\n\t\t\t\t\\begin{tabular}{c|ccccc}\n\t\t\t\t\t\\hline\n\t\t\t\t\tEvent Name & $m_1\\, [{\\rm M}_{\\odot}]$ & $m_2\\, [{\\rm M}_{\\odot}]$ &  $a_f$ & $\\omega_{R}$ & $\\omega_{I}$ \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\tGW150914 & $35.64_{-5.55}^{+5.19}$ & $29.74_{-3.90}^{+2.12}$ & $0.658_{-0.006}^{+0.039}$ & $0.5253_{-0.0026}^{+0.0186}$ & $0.0820_{-0.0009}^{+0.0002}$ \\\\\n\t\t\t\t\tGW151012 & $25.01_{-9.09}^{+12.00}$ & $16.45_{-6.01}^{+4.50}$ & $0.637_{-0.015}^{+0.011}$ & $0.5155_{-0.0086}^{+0.0028}$ & $0.0824_{-0.0002}^{+0.0002}$ \\\\  \n\t\t\t\t\tGW151226 & $12.39_{-0.25}^{+3.57}$ & $7.70_{-0.48}^{+5.77}$ & $0.725_{-0.140}^{+0.051}$ &  $0.5558_{-0.0611}^{+0.0241}$ & $0.0776_{-0.0002}^{+0.0055}$ \\\\ \n\t\t\t\t\tGW170104 & $32.28_{-6.33}^{+4.31}$ & $22.31_{-3.06}^{+7.01}$ & $0.684_{-0.035}^{+0.014}$ &  $0.5157_{-0.0068}^{+0.0071}$ & $0.0854_{-0.0015}^{+0.0004}$ \\\\\n\t\t\t\t\tGW170608 & $12.90_{-0.31}^{+3.27}$ & $9.93_{-0.09}^{+2.08}$ & $0.716_{-0.077}^{+0.017}$ & $0.5385_{-0.0154}^{+0.0057}$ & $0.0827_{-0.0004}^{+0.0006}$ \\\\\n\t\t\t\t\tGW170729 & $45.32_{-0.98}^{+2.23}$ & $24.41_{-02.32}^{+03.16}$ & $0.737_{-0.058}^{+0.036}$ &  $0.5682_{-0.0303}^{+0.0038}$ & $0.0739_{-0.0016}^{+0.0054}$ \\\\\n\t\t\t\t\tGW170809 & $35.71_{-8.46}^{+7.53}$ & $24.09_{-2.44}^{+5.80}$ & $0.632_{-0.010}^{+0.008}$ &  $0.5123_{-0.0041}^{+0.0034}$ & $0.0826_{-0.0002}^{+0.0001}$ \\\\\n\t\t\t\t\tGW170814 & $30.54_{-8.78}^{+2.01}$ & $22.33_{-7.96}^{+0.07}$ & $0.679_{-0.003}^{+0.002}$ & $0.5364_{-0.0030}^{+0.0009}$ & $0.0812_{-0.0001}^{+0.0003}$ \\\\\n\t\t\t\t\tGW170818 & $31.52_{-1.95}^{+2.15}$ & $25.97_{-0.87}^{+1.21}$ & $0.716_{-0.021}^{+0.015}$ &  $0.5474_{-0.0104}^{+0.0062}$ & $0.0786_{-0.0013}^{+0.0013}$ \\\\\n\t\t\t\t\tGW170823 & $46.98_{-3.89}^{+0.58}$ & $33.01_{-5.92}^{+2.03}$ & $0.626_{-0.023}^{+0.014}$ & $0.5067_{-0.0057}^{+0.0070}$ & $0.0827_{-0.0003}^{+0.0006}$ \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\\end{tabular}\n\t\t\t\\end{sc}\n\t\t\\end{small}\n\t\\end{center}\n\t\\vskip -0.1in\n\\end{table*}\n\n\\begin{table*}[!htp]\n\t\\setlength{\\tabcolsep}{4.5pt}\n\t\\renewcommand{\\arraystretch}{2.0}\n\t\\caption{\\new{As Table~\\ref{tab_real_event_results}, but now using our  probabilistic deep learning models. The uncertainty for these models is captured by randomness in the network weights, not from various noisy realizations of the signals.}}\n\t\\label{tab_real_event_results_BNN}\n\t\\vskip -0.25in\n\t\\begin{center}\n\t\t\\begin{small}\n\t\t\t\\begin{sc}\n\t\t\t\t\\begin{tabular}{c|ccccc}\n\t\t\t\t\t\\hline\n\t\t\t\t\tEvent Name & $m_1 [{\\rm M}_{\\odot}]$ & $m_2 [{\\rm M}_{\\odot}]$ &  $a_f$ & $\\omega_{R}$ & $\\omega_{I}$ \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\tGW150914 & $36.08_{-4.45}^{+4.77}$ & $27.42_{-3.92}^{+3.49}$ & $0.689_{-0.032}^{+0.017}$ & $0.5390_{-0.0269}^{+0.0124}$ & $0.0797_{-0.0022}^{+0.0011}$ \\\\\n\t\t\t\t\tGW151012 & $21.56_{-2.12}^{+3.07}$ & $15.46_{-2.32}^{+2.44}$ & $0.681_{-0.032}^{+0.016}$ &  $0.5365_{-0.0266}^{+0.0130}$ & $0.0804_{-0.0018}^{+0.0008}$ \\\\  \n\t\t\t\t\tGW151226 & $18.04_{-2.49}^{+1.98}$ & $11.96_{-2.89}^{+1.67}$ & $0.715_{-0.035}^{+0.017}$ &  $0.5533_{-0.0280}^{+0.0142}$ & $0.0763_{-0.0036}^{+0.0017}$ \\\\ \t\n\t\t\t\t\tGW170104 & $31.23_{-3.26}^{+4.09}$ & $23.27_{-3.25}^{+3.62}$ & $0.692_{-0.033}^{+0.016}$ &  $0.5358_{-0.0302}^{+0.0052}$ & $0.0796_{-0.0052}^{+0.0026}$ \\\\\n\t\t\t\t\tGW170608 & $16.73_{-2.19}^{+2.38}$ & $12.44_{-2.21}^{+2.03}$ & $0.673_{-0.036}^{+0.019}$ &  $0.5235_{-0.0297}^{+0.0149}$ & $0.0818_{-0.0012}^{+0.0005}$ \\\\\n\t\t\t\t\tGW170729 & $45.28_{-6.42}^{+6.63}$ & $32.34_{-5.48}^{+3.91}$ & $0.751_{-0.038}^{+0.019}$ &  $0.5776_{-0.0309}^{+0.0151}$ & $0.0756_{-0.0048}^{+0.0023}$ \\\\\n\t\t\t\t\tGW170809 & $32.88_{-3.45}^{+4.49}$ & $26.56_{-3.91}^{+3.54}$ & $0.714_{-0.034}^{+0.016}$ &  $0.5492_{-0.0271}^{+0.0060}$ & $0.0760_{-0.0060}^{+0.0030}$ \\\\\n\t\t\t\t\tGW170814 & $32.40_{-3.60}^{+4.77}$ & $25.22_{-4.34}^{+4.38}$ & $0.675_{-0.033}^{+0.016}$ &  $0.5329_{-0.0272}^{+0.0140}$ & $0.0794_{-0.0024}^{+0.0011}$ \\\\\n\t\t\t\t\tGW170818 & $33.49_{-3.19}^{+4.51}$ & $29.71_{-4.63}^{+4.59}$ & $0.631_{-0.032}^{+0.015}$ &  $0.5159_{-0.0277}^{+0.0135}$ & $0.0829_{-0.0016}^{+0.0008}$ \\\\\n\t\t\t\t\tGW170823 & $38.24_{-5.38}^{+4.78}$ & $28.16_{-3.67}^{+4.63}$ & $0.664_{-0.036}^{+0.018}$ &  $0.5321_{-0.0302}^{+0.0156}$ & $0.0757_{-0.0088}^{+0.0043}$ \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\\end{tabular}\n\t\t\t\\end{sc}\n\t\t\\end{small}\n\t\\end{center}\n\t\\vskip -0.1in\n\\end{table*}\n\n\nIn this section we use our neural network models to measure \\((m_1,\\,m_2,\\,a_f,\\,\\omega_R,\\,\\omega_I)\\) from all the BBH mergers detected to date by the advanced LIGO and Virgo observatories~\\cite{o1o2catalog}. We present results for two types of neural network models, namely, deterministic and probabilistic. \n\n\\subsection{Parameter estimation with deterministic neural networks}\n\n\\new{To get insights into the performance of our deterministic neural network models to infer the astrophysical parameters of BBH mergers, we begin by evaluating them for a given BBH system whose ground truth parameters are \\((m_1,\\,m_2,\\,a_f,\\,\\omega_{R},\\,\\omega_{I})= (31.10M_{\\odot},\\,20.46M_{\\odot},\\,0.718,\\,0.5412,\\, 0.0800)\\). Using 1,600 different noise realizations, we have constructed the model predictions for two different SNR cases, as shown in Figure~\\ref{fig_multiple_noise_check}. We notice that these distributions capture the ground-truth values of the BBH system under consideration, and that the reconstruction of the actual parameters of the system improves for larger SNR values, which is in agreement with the analysis presented with traditional Bayesian analysis for GW parameter estimation~\\cite{bambiann:2015PhRvD}. Having conducted similar experiments for other BBH systems, we then went on to using these deep learning models for the parameter reconstruction of real BBH mergers.}\n\n\nIn Table~\\ref{tab_real_event_results} we present the median and 90\\% confidence level for the astrophysical parameters \\((m_1,\\,m_2,\\,a_f,\\,\\omega_{R},\\,\\omega_{I})\\) of all the BBH mergers presented in~\\cite{o1o2catalog}. These values are computed by whitening the data containing a putative signal with 240 different Power Spectral Densities (PSDs), half of them are constructed using LIGO Hanford noise and the rest with LIGO Livingstone noise. Through this approach we are effectively measuring the impact of PSD variations in the measurements of the astrophysical parameters of BBH mergers. We find that these estimates are in very good agreement with the results obtained with the Bayesian analyses presented in Table III of~\\cite{o1o2catalog}. \n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfigure[SNR = 13.0]{\n\t\t\\includegraphics[width=0.49\\linewidth]{mass_seaborn_det_multi_noise_real_1_9.png}}\n\t\t\\subfigure[SNR = 19.5]{\n\t\t\\includegraphics[width=0.49\\linewidth]{mass_seaborn_det_multi_noise_real_1_14.png}}\n        \\subfigure[SNR = 13.0]{\n\t\t\\includegraphics[width=0.49\\linewidth]{spin_ome_test_GW170104_sim_3_9.png}}\n\t\t\\subfigure[SNR = 19.5]{\n\t\t\\includegraphics[width=0.49\\linewidth]{spin_ome_test_GW170104_sim_3_14.png}}\n\t\n\t\\caption{\\new{Model predictions produced by our deterministic models by evaluating them with 1,600 different noise realizations for a binary black hole system with ground truth parameters \\((m_1,\\,m_2,\\,a_f,\\,\\omega_{R},\\,\\omega_{I})= (31.10M_{\\odot},\\,20.46M_{\\odot},\\,0.718,\\,0.5412,\\, 0.0800)\\). The panels show results for the distribution of the estimates for  (\\(m_1,\\,m_2,\\,a_f,\\,w_R,\\,w_I\\)) assuming \\(\\textrm{SNR}=\\{13,\\,19.5\\}\\).}}\n\t\\label{fig_multiple_noise_check}\n\\end{figure*}\n\n\\subsection{Bayesian neural network parameter estimation}\n\n\\new{In addition to parameter estimation results obtained with our deterministic models, based on varying the noise realization with different PSDs, we also evaluated our BNN models for two types of signals. First, on simulated signals to quantify the performance of our probabilistic models. Results of this exercise are presented in Figure~\\ref{bnn_dist}. We carried out an exhaustive study to confirm that our BNN models provide consistent results for different random initializations, and that the results exhibit strong convergence for the optimal choice of hyperparameters.} \n\n\\new{Upon confirming that our probabilistic models perform well, we used them to estimate the astrophysical parameters of the entire catalog of BBH signals reported in~\\cite{o1o2catalog}. These results, which provide the median and the 90\\% confidence intervals, are summarized in Table~\\ref{tab_real_event_results_BNN}.}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfigure[SNR = 13.0]{\n\t\t\\includegraphics[width=0.49\\linewidth]{mass_seaborn_BNN_newloss_newdata_multi_noise_real_1_9.png}}\n\t\t\\subfigure[SNR = 19.5]{\n\t\t\\includegraphics[width=0.49\\linewidth]{mass_seaborn_BNN_newloss_newdata_multi_noise_real_1_14.png}}\n        \\subfigure[SNR = 13.0]{\n\t\t\\includegraphics[width=0.49\\linewidth]{spin_ome_BNN_test_GW170104_sim_3_9.png}}\n\t\t\\subfigure[SNR = 19.5]{\n\t\t\\includegraphics[width=0.49\\linewidth]{spin_ome_BNN_test_GW170104_sim_2_14.png}}\n\t\n\t\\caption{\\new{Variation inference distributions produced by our Bayesian Neural Network models for a binary black hole system with ground truth parameters \\((m_1,\\,m_2,\\,a_f,\\,\\omega_{R},\\,\\omega_{I})= (31.10M_{\\odot},\\,20.46M_{\\odot},\\,0.718,\\, 0.5412,\\, 0.0800)\\). The panels show results for the distribution of the estimates for  (\\(m_1,\\,m_2,\\,a_f,\\,w_R,\\,w_I\\)). As in Figure~\\ref{fig_multiple_noise_check}, we consider \\(\\textrm{SNR}=\\{13,\\,19.5\\}\\).}}\n\t\\label{bnn_dist}\n\\end{figure*}\n\n\n The deep learning parameter estimation results presented in Tables~\\ref{tab_real_event_results_BNN} are consistent with those obtained with established, Bayesian parameter estimation pipelines~\\cite{o1o2catalog}. \\new{The reliable astrophysical information inferred in low-latency by these deep learning algorithms for each BBH signal (less than 2 milliseconds) warrants the extension of this framework to characterize other GW sources, including eccentric compact binary mergers, and sources such as BBH systems with significant spin and asymmetric mass-ratios that require the inclusion of higher-order modes for accurate GW source modeling. This work is under earnest development and will be presented shortly.} \n\n\n\nHaving demonstrated the application of deep learning at scale for the characterization of BBH mergers, it is now in order to design deep neural networks for real-time detection and characterization of GW sources that are expected to have electromagnetic and astro-particle counterparts, i.e., BNS and NSBH systems. For that study, we expect no additional computational challenges to the ones we have already addressed in this analysis. The central development for such an effort, however, will consist of designing a clever algorithm to readily identify BNS or NSBH in a hierarchical manner, i.e., in principle it is not needed to train neural networks using minute long waveforms. Rather, we need to figure out how much information is needed to accurately reconstruct the astrophysical parameters of one of these events in real-time. These studies should be pursued in the future. \n\n\\section{Conclusion}\n\\label{conclusion}\n\nWe have presented the first application of deep learning at scale to characterize the astrophysical properties of BHs whose spins are aligned or anti-aligned, and which evolve on quasi-circular orbits. Using over \\(10^7\\) waveforms to densely sample this parameter space, and encoding time- and scale-invariance, we have demonstrated that deep learning enables real-time GW parameter estimation. These studies mark the first time BNNs are trained using 1,024 nodes on a supercomputer platform tuned for deep learning research, and when applied for the analysis of real advanced LIGO data, they maintain similar accuracy to models trained on 4 V100 GPUs. Our results are consistent with established, compute-intensive, Bayesian methods that are routinely used for GW parameter estimation. \n\nThe approach we have presented herein provides the means to constrain the parameters of BBHs before and after the merger event. We have shown that deep learning can directly infer the final spin and QNMs of BH remnants, thereby paving the way to directly use QNMs to assess whether BH remnants are accurately described by general relativity. In future work, we will study how accurately these neural network models can tell apart ringdown waveforms described by astrophysically motivated alternative theories of gravity in realistic detection scenarios. The extension of this work to enable real-time detection and parameter estimation of GW sources that are central for Multi-Messenger Astrophysics discovery campaigns, and other astrophysically motivated sources, such as eccentric BBH mergers, should also be investigated. \n\n\n\n\\section{Acknowledgements}\nDue to the size of the data, the datasets utilized in this study are available from the corresponding author on reasonable request. Codes will be available before the official publication. \n\n\nThis work utilized the Hardware-Accelerated Learning (HAL)  cluster, supported by NSF Major Research Instrumentation program, grant \\#1725729, as well as the University of Illinois at Urbana-Champaign. \n\nThis research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357.  \n\nThis research is part of the Blue Waters sustained-petascale computing project, \nwhich is supported by the NSF (awards OCI-0725070 and ACI-1238993) \nand the State of Illinois. Blue Waters is a joint effort of the University of Illinois at \nUrbana-Champaign and its National Center for Supercomputing Applications (NCSA). We acknowledge support from the NCSA, and thank the \\href{http:\/\/gravity.ncsa.illinois.edu}{NCSA Gravity Group} for useful feedback. Tesla P100 and V100 GPUs used for this project were donated by NVIDIA to the \\href{http:\/\/gravity.ncsa.illinois.edu}{NCSA Gravity Group}, and are hosted by Vlad Kindratenko at the Innovative Systems Lab at NCSA.  This work also made used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562. Specifically, it used the Bridges system, which is supported by NSF award ACI-1445606, at the Pittsburgh Supercomputing Center (PSC); TG-PHY160053 grant is gratefully acknowledged. \n\nThis paper was reviewed and approved by the LIGO P\\&P committee.\n\n\\vspace{-0.0cm}\n\\section{Contribution}\n\nEAH envisioned this study, and directed the construction of the data sets used to train\/validate\/test the neural network models. H~Shen developed the neural network structure and carried out the training and evaluation. ZZ supervised on the evaluation of the neural network performance. EJ created the BNN, implemented this to run at scale and advised on its use for parameter predictions. H~Shen created the BNN code with a new sampling approach and training objective, trained and evaluated the BNN model on HAL machine for parameter predictions on real GW events. H~Sharma developed the BNN code and carried out extensive scaling tests on Theta. All co-authors contributed to drafting and editing the manuscript. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\nThe possible existence of a liquid-liquid critical point in deeply supercooled water has been a subject of debate in part due to the challenges associated with providing definitive experimental evidence.\nPioneering work by Mishima and Stanley [Nature 392, 164 (1998) and Phys.~Rev.~Lett. 85, 334 (2000)] sought to shed light on this problem by studying the melting curves of different ice polymorphs and their metastable continuation in the vicinity of the expected location of the liquid-liquid transition and its associated critical point.\nBased on the continuous or discontinuous changes in slope of the melting curves, Mishima suggested that the liquid-liquid critical point lies between the melting curves of ice III and ice V.\nHere, we explore this conjecture using molecular dynamics simulations with a purely-predictive machine learning model based on \\textit{ab initio} quantum-mechanical calculations.\nWe study the melting curves of ices III, IV, V, VI, and XIII using this model and find that the melting lines of all the studied ice polymorphs are supercritical and do not intersect the liquid-liquid transition locus.\nWe also find a pronounced, yet continuous, change in slope of the melting lines upon crossing of the locus of maximum compressibility of the liquid.\nFinally, we analyze critically the literature in light of our findings, and conclude that the scenario in which melting curves are supercritical is favored by the most recent computational and experimental evidence.\nThus, although the preponderance of experimental and computational evidence is consistent with the existence of a second critical point in water, the behavior of the melting lines of ice polymorphs does not provide strong evidence in support of this viewpoint, according to our calculations.\n\n                             \n\n\n\\newpage\n\\parindent 1em\n\n\\section*{Introduction}\n\\label{sec:introduction}\nWater continues to be the focus of intense scientific inquiry, not only because of its importance in the biological and physical sciences, but also on account of its distinctive thermophysical properties and phase behavior. Water exhibits at least 17 different crystalline phases (with new ones continuing to be uncovered) \\cite{Salzmann19,Hansen21}, multiple glassy states \\cite{Loerting11}, and possibly also a liquid-liquid phase transition (LLT) between high-density and low-density liquids (HDL and LDL, respectively) under supercooled conditions \\cite{Poole92,Gallo16}. As such, water provides a rich proving ground to stretch our understanding of diverse thermophysical phenomena including complex phase equilibria, metastable phase transitions, and glass physics \\cite{DebenedettiBook}, as well as the possible relationships between them \\cite{Handle17,Debenedetti03}. \n\nThe possibility of an LLT in water has been the focus of numerous studies \\cite{Gallo16}, and a preponderance of both experimental and computational evidence points to the existence of water's LLT at positive pressures ($P$) and supercooled temperatures ($T$) (i.e., below the melting $T$ of the stable ice I phase) \\cite{Kim20,NILSSON22,Palmer14,Debenedetti20,Palmer18,Gartner22,weis2022liquid}. However, there remain many unresolved questions around the LLT and its relationship to water's properties and various solid phases. A set of observations instrumental to the development of the argument in favor of the LLT came about when Mishima and Stanley characterized the melting of various ice polymorphs to liquid water upon decompression at different $T$ \\cite{Mishima98,Mishima00}. They observed that the melting curve of ice III exhibited a notable but continuous change in slope in the $T-P$ plane, while ice V and ice IV exhibited sharp and seemingly discontinuous changes in slope. Recall that, by the Clausius-Clapeyron equation \\cite{callen1998thermodynamics},\n\\begin{equation}\n\\frac{dP}{dT} = \\frac{\\Delta H}{T_m \\Delta V},\n\\label{eq:Clausius-Clapeyron}\n\\end{equation}\nthe slope $dP \/dT$ of a line of phase coexistence $T_m(P)$ is related to the change in enthalpy $\\Delta H$ and volume $\\Delta V$ across the transition. This idea suggests that if a melting curve exhibits a discontinuous change in slope, it correspondingly reflects a discontinuous change in the properties of ice and\/or liquid water at that point. Given that the enthalpy and volume of crystalline solids is only weakly dependent on $T$ and $P$, Mishima and Stanley concluded that the properties of the liquid phase were changing discontinuously (i.e., evidence of an LLT). This argument, if correct, would place the liquid-liquid critical point (LLCP) somewhere in between the ice V and ice III melting lines, with the LLT coexistence line intersecting the ice V and ice IV melting curves at the point of discontinuous change in slope. Mishima also probed the melting lines of ices VI and XIII, but was unable to extend those curves far enough to intersect with the possible LLT line.\n\nThis rationalization for the observed trends, while plausible, remains difficult to definitively explore experimentally due to rapid crystallization of the stable ice I phase upon melting of the other polymorphs. Similar practical challenges also hamper direct experimental demonstration of the LLT. Thus, open questions remain about the true relationship between a possible LLT and the metastable melting of the ice phases. Molecular modeling represents an attractive route to probe these ideas, as one can design simulation methodologies free from unwanted crystallization, which allow us to directly study the relationship between the LLT and the various ices. In parallel, advances in machine learning (ML)-based interaction potentials \\cite{Noe20,Wen22} allow us to develop predictive intermolecular potential models that describe water's interactions at the level of an \\textit{ab initio} reference calculation (e.g., density functional theory), thus enabling purely-predictive simulations of complex collective properties and phase behavior at tractable computational cost \\cite{Gartner20,reinhardt2021quantum,Zhang21,Piaggi21,Schran21,piaggi2022homog}. In this study, we coupled one such ML-potential method (Deep Potential Molecular Dynamics, DPMD) \\cite{Zhang18,Wang18} with several advanced simulation techniques to shed further light on the possible relationship between the LLT and water's liquid-solid phase behavior. \n\n\\section*{Potential scenarios}\n\\label{sec:potential_scenarios}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{Figure1.pdf}\n\\caption{\\label{fig:Fig1} Hypothetical scenarios describing the possible relationship between ice polymorph melting curves and the LLT. The upper plots show the melting curve of a hypothetical ice polymorph (red solid line), the LLT line (gray solid line), the LLCP (gray circle), and the Widom line (gray dashed line). The lower plots show hypothetical free energy surfaces for the liquid density along the melting curves at the three points marked by \\textbf{+} signs. Scenario 1 (left) shows a case where the melting curve is significantly supercritical, Scenario 2 (center) shows a case where the melting curve is slightly supercritical, and Scenario 3 (right) shows a case where the melting curve is subcritical.}\n\\end{figure*}\n\nBefore describing the details of our approach and results, we illustrate schematically the possible classes of behavior in FIG.~\\ref{fig:Fig1}.\nIn this discussion, we assume the existence of an LLT.\nThe elements that we consider in our analysis are the melting curve of an ice polymorph, the liquid-liquid critical point, the liquid-liquid coexistence line (or binodal), and the Widom line. The Widom line can be regarded as an extension of the liquid-liquid coexistence line to supercritical conditions and is defined by the locus of maxima of the correlation length.\nResponse functions, such as the heat capacity at constant pressure $C_P$ and the isothermal compressibility $\\kappa_T$, also have pronounced maxima at supercritical conditions even far from the critical point, and the values of the response functions diverge as the critical point is approached \\cite{xu2005relation}.\nFurthermore, the lines of maxima of the response functions in the $T-P$ plane asymptotically converge to the Widom line as the critical point is approached from supercritical conditions \\cite{xu2005relation}.\n$C_P=(\\partial H\/ \\partial T)_P$ and $\\kappa_T=-(1\/V)(\\partial V\/\\partial P)_T$ are derivatives of the enthalpy $H$ and volume $V$, and thus we expect the fastest change in these liquid-state properties in the immediate vicinity the Widom line.\nIn turn, a pronounced change in the enthalpy and volume of the liquid at the Widom line will lead to correspondingly pronounced changes in slope of the ice melting line as predicted by Eq.~\\eqref{eq:Clausius-Clapeyron}.\n\nWe now analyze three possible scenarios.\nIf the melting curve of a particular polymorph were to be significantly supercritical (Scenario 1, FIG.~\\ref{fig:Fig1} left), the impact of the critical point would be minimal.\nTherefore, we would expect to observe a modest change in slope of the melting curve and the free energy surface of the liquid state would have a single basin that smoothly moves from high to low density as temperature decreases along the melting curve. If the melting curve passed near to the critical point but still at supercritical conditions (Scenario 2, FIG.~\\ref{fig:Fig1} center), a more significant but still continuous change in slope might be observed as the liquid properties change swiftly but continuously upon crossing the Widom line. In this case, the free energy surfaces would still only show one single minimum at a given state point yet they can show significant asymmetry \\cite{Gartner22}, and broadening at the intersection of the melting curve with the Widom line.\nThe broadening of the free energy surface of the liquid as a function of density at the Widom line follows from the fact that density fluctuations $\\sigma_{\\rho}$ are related to $\\kappa_T$ via $\\sigma_{\\rho}^2=\\rho^2 k_B T \\kappa_T\/ V$ where $\\rho$ is the density and $k_B$ the Boltzmann constant \\cite{pathria2016statistical}.\nFinally, if the melting curve was subcritical (Scenario 3, FIG.~\\ref{fig:Fig1} right), a discontinuous change in liquid properties across the LLT would result in a discontinuous change in the slope of the melting curve, and a free energy surface with two basins of equal depth would develop at the point of liquid-liquid phase coexistence (i.e., where the ice melting line meets the LLT line). Moving forward, we will situate our simulation results in the context of these three potential scenarios.\n\n\\section*{Calculation of melting curves}\n\\label{sec:methods1}\n\nOur molecular dynamics simulations were driven by a deep potential model\\cite{Zhang18} of water developed by Zhang et al. \\cite{Zhang21}\nThe model has been carefully trained to reproduce with high fidelity the potential energy surface of water based on density functional theory (DFT) calculations with the Strongly Constrained and Appropriately Normed (SCAN) exchange and correlation functional \\cite{Sun15}.\nSCAN is one of the best semilocal functionals available and describes with good accuracy many properties of water and ice, and their anomalies \\cite{Sun16,Chen17,Piaggi21}.\nEven though the model is short-ranged with a cutoff of 6 \\AA, it can capture subtle physical effects, such as polarization \\cite{piaggi2022homog} and many-body correlations \\cite{Zhang18}.\nFurthermore, this model describes qualitatively the behavior of water and ice polymorphs in a region of the phase diagram spanning temperatures 0-500 K and pressures 0-50 GPa \\cite{Zhang21}.\nIt is thus suitable to represent ice III, IV, V, VI, and XIII at the conditions of interest for this work.\nAnother aspect of critical importance is whether the model has a liquid-liquid transition at deeply supercooled conditions.\nWe recently proved rigorously using free energy calculations that this model has a liquid-liquid transition with a critical point at $T_c = 242 \\pm 5$ K and $P_c = 0.295 \\pm 0.015$ GPa \\cite{Gartner22}.\nIt is important to note that SCAN also has limitations.\nLargely due to the self-interaction error in semilocal functionals \\cite{sharkas2020self}, the strength of the hydrogen bond is overestimated, resulting in an upward displacement of melting temperatures of about 40 K with respect to experiments \\cite{Piaggi21}. Additionally, the solid polymorphs ice III and ice XV are incorrectly predicted by SCAN to be metastable at all ($T$, $P$) \\cite{Zhang21}. However, given the complexity of water's phase diagram, SCAN predicts the relative location of the various phase boundaries in good agreement with experiment \\cite{Zhang21}.\n\n\\begin{figure*}\n\\includegraphics[width=0.95\\textwidth]{Figure2.pdf}\n\\caption{\\label{fig:Fig2} Overview of the methodology to calculate melting curves of ice polymorphs. The procedure is illustrated using the case of ice III. (A) Number of ice III-like molecules as a function of time in the biased coexistence simulations at various $T$ and $P$. The colors of the curves correspond to the $T$, as labeled to the right of the figure. Empty plots denote that no simulations were run at that ($T$, $P$). The range (324,378) that is reversibly sampled corresponds to one layer of ice III. (B) Free energy surfaces as a function of number of ice III-like molecules, where the dashed line is a linear fit to the free energy surface and the shaded region denotes the uncertainty. Colors match the same $T$ reported in panel (A) above. (C) Chemical potential difference between ice III and liquid at various $T$ and $P$. The gray dashed line is a linear fit to the data, and the shaded region represents one standard deviation of uncertainty in the fit parameters. (D) Melting curve obtained by this procedure, where the blue points represent the $T$ and $P$ of zero chemical potential difference between ice and liquid obtained in panel (C). Error bars represent one standard deviation errors in the fit parameters as shown in (C). The dashed line is the melting curve obtained from the integration of the Clausius-Clapeyron equation. (E,F) Simulation snapshots illustrating ice III and the molecular environments used to generate the order parameter \\cite{Piaggi19,Bore22} to drive the biased coexistence (E), and the ice III-liquid coexistence geometry (F).}\n\\end{figure*}\n\nHerein, we computed the melting lines of the ice polymorphs in two stages.\nIn the first stage, we calculated a few points along the liquid-solid coexistence lines using a biased coexistence approach \\cite{Bore22} in which we simulate a particular ice polymorph and liquid water in direct coexistence (FIG.~\\ref{fig:Fig2}F), and use a bias potential to reversibly crystallize and melt a layer of solid (FIG.~\\ref{fig:Fig2}A).\nThis approach was used in a recent work to calculate the phase diagram of the state-of-the-art empirical model of water TIP4P\/Ice \\cite{Abascal05}, and can be regarded as a generalization of the interface pinning approach \\cite{Pedersen13}.\nFrom biased coexistence simulations carried out at different temperatures and pressures, we extract the difference in chemical potential between the liquid and ice from the slope of the free energy surfaces\\cite{Pedersen13,Bore22} (FIG.~\\ref{fig:Fig2}B), and locate the liquid-ice coexistence temperature at a given pressure as the temperature at which this difference is zero (FIG.~\\ref{fig:Fig2}C-D).\nWe applied this procedure to ice III, IV, V, and XIII to obtain a few coexistence points for each polymorph.\nSee FIG.~\\ref{fig:Fig2} for an overview of this procedure for the case of ice III.\nWe show the results for ice IV, V, and XIII in the Supplementary Material \\cite{SI}.\nWe also validated the coexistence points obtained via the biased coexistence method for ice IV and V using standard direct-coexistence simulations (see the Supplementary Material \\cite{SI}).\nWe subsequently obtained continuous and smooth coexistence lines by integrating the Clausius-Clapeyron equation as first proposed by Kofke \\cite{Kofke93}.\nThis technique is based on the numerical integration of Eq.~\\eqref{eq:Clausius-Clapeyron} using the enthalpy and volume obtained from constant temperature and pressure simulations of each phase (see Methods section and the Supplementary Material\\cite{SI} for further details).\n\n\\section*{Results}\n\nUsing the techniques described above, we calculated the coexistence points and lines shown in FIG.~\\ref{fig:Fig3}A for ice III, IV, V, and XIII.\nThe circles and error bars correspond to biased coexistence simulations, and the lines were computed by integrating the Clausius-Clapeyron equation.\nWe also show in FIG.~\\ref{fig:Fig3}A the data for the liquid-liquid critical point, liquid-liquid coexistence line, and Widom line reported recently by us \\cite{Gartner22}.\nAccording to these calculations, the melting curves of all ice polymorphs as predicted by the SCAN functional are supercritical, \\textit{i.e.}, they pass above the liquid-liquid critical point.\nThe melting line of ice VI is also supercritical and is shown in the Supplementary Material \\cite{SI}.\nThus, all of them intersect the Widom line rather than the LLT line.\nOur simulations result in melting curves that show a pronounced, yet continuous, change of slope upon crossing the Widom line.\nThis behavior is compatible with the expected change of properties of liquid water from HDL-like to LDL-like as the Widom line is traversed from high to low pressures.\nMoreover, the change in slope is smoother for ice III than for the other polymorphs, consistent with an increasingly abrupt change in the properties of the liquid closer to the critical point.\nThe smoother change in slope of the melting curve of ice III resembles the behavior hypothesized in Scenario 1 described in FIG.~\\ref{fig:Fig1} while the more abrupt change shown by ice V, IV, and XIII is reminiscent of Scenario 2.  \n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{Figure3.pdf}\n\\caption{\\label{fig:Fig3} Melting curves of ice polymorphs III, IV, V, and XIII, and their location relative to the liquid-liquid critical point. A) Results obtained using a machine learning model based on the SCAN DFT functional. Circles represent melting points calculated using biased coexistence simulations \\cite{Bore22}, crosses were obtained by integrating the Clausius-Clapeyron equation, and lines are spline interpolations of the latter results. We also show the location of the critical point, the liquid-liquid coexistence line, and the Widom line (line of maxima of $\\kappa_T$) as calculated in our previous work \\cite{Gartner22}. B) Melting curves reported by Mishima \\cite{Mishima00} for heavy water based on decompression-induced melting experiments. The approximate location of the discontinuous change in slope in the melting curves of ice IV and V is marked with an X. The shaded region is the location of the critical point estimated by Bachler et al.~\\cite{Bachler21}. We also show the location of the critical point obtained by Shi and Tanaka using experimental measurements \\cite{Shi20}, by Debenedetti et al.\\ using molecular simulations with the empirical water models TIP4P\/2005 and TIP4P\/Ice \\cite{Debenedetti20}, and by Mishima and Sumita\\cite{mishima2023equation} using an extrapolation based on polynomial fits to equation of state data. On the left, we show atomic configurations representative of ices III, IV, V, and XIII.}\n\\end{figure*}\n\nOur results also show good agreement between the biased coexistence simulations and the integration of the Clausius-Clapeyron equation in the HDL-like region.\nOn the other hand, it was not possible to perform biased coexistence simulations in the LDL-like region due to the long relaxation times of the LDL-like liquid at those thermodynamic conditions.\nIndeed, even for the comparatively less expensive bulk liquid simulations for the Clausius-Clapeyron integration procedure, we needed long simulations (100 ns) of the bulk liquid in the LDL-like region for robust statistical certainty.\n\nThe analysis of the melting curves shown in FIG.~\\ref{fig:Fig3}A does not constitute proof of a continuous change in slope since the curves are obtained from a set of points interpolated with a spline, which is by construction smooth and differentiable.\nIn order to provide evidence for the continuous change in slope, we now analyze in detail the properties of liquid water along the melting curves of ice polymorphs.\nIn FIG.~\\ref{fig:Fig4} we show the enthalpy and density of liquid water as a function of pressure.\nBoth properties exhibit a swift change upon crossing of the Widom line and the change is more abrupt as the melting curves approach the critical point, with sequence ice III $\\rightarrow$ V $\\rightarrow$ IV $\\rightarrow$ XIII.\nWe ruled out that this behavior is a result of ice crystallization by analyzing configurations at regular intervals of 5 ps.\nWe calculated the structural fingerprints CHILL+ \\cite{nguyen2015identification} and Identify Diamond Structure \\cite{Larsen16}, as implemented in Ovito\\cite{Stukowski09}, and we did not find atomic environments compatible with ice I in any of our simulations.\nWe also show in FIG.~\\ref{fig:Fig4} the free energy surfaces (FES) as a function of the liquid water density for selected points along the coexistence lines.\nThe FES of the liquid along the melting curves of all studied ice polymorphs show a behavior reminiscent of Scenario 2 of FIG.~\\ref{fig:Fig1}.\nFor all ices, the FES at the state point closest to the Widom line shows clear broadening.\nFurthermore, the FES in the vicinity of the Widom line exhibits deviations from a quadratic form with significant asymmetry and a shoulder suggestive of the metastable free energy minimum that would appear below the critical point.\nTaken together, this behavior provides strong evidence of a continuous crossover from HDL-like to LDL-like liquids as the melting curves of ice III, V, IV, and XIII are traversed towards lower pressures.\nWe remark that none of the melting lines analyzed here have properties of the liquid compatible with subcritical Scenario 3 of FIG.~\\ref{fig:Fig1} that would lead to a discontinuous change in the slope of the melting line.\nBased on the analysis of the liquid properties described above, we conclude that the changes in slope of the melting curves shown in FIG.~\\ref{fig:Fig3} are indeed continuous.\n\n\\begin{figure*}\n\\includegraphics[width=0.9\\textwidth]{Figure4.pdf}\n\\caption{\\label{fig:Fig4} Properties of liquid water along melting curves of several ice polymorphs. Panels A, B, C, and D correspond to ice III, V, IV, and XIII, respectively. For each ice polymorph, we show the enthalpy of liquid water $H_L$, the density of liquid water $\\rho_L$, and the melting temperature $T$ as a function of pressure $P$. The locus of maxima of isothermal compressibility \\cite{Gartner22} is shown in the $T-P$ pane with a dashed line. We also show the free energy surfaces $F$ as a function of the density of liquid water $\\rho_L$. The free energy surfaces are color-coded to match the color of points along the $T$ vs $P$ coexistence line to specify the thermodynamic conditions at which they were calculated.}\n\\end{figure*}\n\nWe have so far focused on the properties of the liquid phase.\nHowever, according to Eq.~\\eqref{eq:Clausius-Clapeyron}, the properties of ice can also affect the slope of melting curves.\nIn the Supplementary Material \\cite{SI}, we show the change in enthalpy and density of ice polymorphs along the melting lines.\nThe data show that the changes experienced by the bulk ice polymorphs are much more subtle than the corresponding changes in the properties of the liquid phase.\nIn the pressure range shown in FIG.~\\ref{fig:Fig4}, the densities of ice polymorphs change by less than 1\\% while the density of liquid water changes by 10\\%.\nFurthermore, the enthalpy of ices varies by around 8\\% while the enthalpy of liquid water has a significantly larger variation of around 17\\%.\nThis analysis indicates that the changes of the properties of the liquid phase are the main factor driving the sharp changes in slope observed in FIG.~\\ref{fig:Fig3}.\n\nThe results described above correspond to a purely-predictive model derived from first principles calculations.\nAn alternative approach is to evaluate the melting lines of ice polymorphs using semi-empirical water models that are fit to experimental information.\nFor this reason, we calculated the melting line of ice V in the TIP4P\/Ice model \\cite{Abascal05}, which is a state-of-the-art semi-empirical model for the study of ice polymorphs.\nThe location of the liquid-liquid critical point for this model has been determined accurately by Debenedetti et al.~\\cite{Debenedetti20}.\nWe find that the melting curve of ice V within the TIP4P\/Ice model (shown in the Supplementary Material \\cite{SI}) is also supercritical, in agreement with the SCAN calculations reported above.\n\n\\section*{Discussion}\n\nThe picture that emerges from our present results is in contrast with Mishima and Stanley's interpretation \\cite{Mishima98,Mishima00}.\nAs described above, Mishima's interpretation of the experiments considers that the melting curve of ice III is supercritical, and the melting lines of ice IV, V, and XIII are subcritical \\cite{Mishima00}.\nOn the other hand, our calculations based on an \\textit{ab initio} model predict supercritical behavior for all the studied ice polymorphs.\nTo evaluate this discrepancy, we analyze the consistency of each of these two interpretations in the light of the most recent evidence for the location of the critical point.\nThe decompression-induced melting curves measured by Mishima \\cite{Mishima00} are shown in FIG.~\\ref{fig:Fig3}B together with recent estimates of the location of the liquid-liquid critical point.\nThe estimates include an extrapolation by Bachler et al. based on experimental data for the high- and low-density spinodals obtained from compression\/decompression experiments on glassy water \\cite{Bachler21}, an analysis by Shi and Tanaka using experimental measurements \\cite{Shi20}, calculations based on molecular simulations with the two realistic empirical water models TIP4P\/Ice and TIP4P\/2005 \\cite{Debenedetti20}, and a very recent extrapolation based on polynomial fits to equation of state data by Mishima and Sumita\\cite{mishima2023equation}.\nIt follows from FIG.~\\ref{fig:Fig3}B that, if such estimates are correct, all melting curves would be supercritical in experiments.\nFurthermore, the relative positions of the ice polymorph melting curves and the critical point provided by SCAN in FIG.~\\ref{fig:Fig3}A seems to be in excellent qualitative agreement with the experimental results shown in FIG.~\\ref{fig:Fig3}B, i.e., the relative stability of all phases is captured qualitatively.\nHowever, the quantitative positions of the melting curves and critical point in the $T-P$ plane differ significantly from experiments, which we attribute to the known limitations of SCAN \\cite{Gartner20,Piaggi21}. We note that it is possible that SCAN somehow shifts the location of the critical point relative to the ice melting curves, however, given the qualitative correspondence between FIG.~\\ref{fig:Fig3}A and FIG.~\\ref{fig:Fig3}B, we do not expect this to be the case.\nMoreover, the calculations described above based on a semi-empirical model also show that the melting line of ice V is supercritical, in disagreement with the original interpretation of the experiments and supporting the picture provided by the SCAN functional. \n\nIn FIG.~\\ref{fig:Fig3}B we have combined experimental melting curves for heavy water \\cite{Mishima00} with estimates of the critical point based on experiments carried out using light water \\cite{Bachler21,Shi20} and simulations that ignore nuclear quantum effects \\cite{Debenedetti20}.\nA figure equivalent to FIG.~\\ref{fig:Fig3}B, replacing the melting curves of heavy water ice polymorphs with melting curves of light water \n ices \\cite{mishima2021liquid} is shown in the Supplementary Material \\cite{SI}.\nThe isotopic effect in the melting lines is rather small, with melting temperatures of heavy water around 5 K higher than in light water \\cite{mishima2021liquid}.\nOn the other hand, the isotopic effect on the location of the critical point has recently been estimated by Eltareb et al.~\\cite{eltareb2022evidence} using path integral molecular dynamics and a semiempirical model of water.\nThey found a critical point location for heavy water 18 K and 9 MPa higher than in light water.\nThe combined isotopic effect on the melting curves and the location of the critical point may lead to a relative shift of around 12 K in light water compared to heavy water.\nTherefore, isotopic effects are unlikely to affect the picture shown in FIG.~\\ref{fig:Fig3}.\nWe also stress that our simulation results shown in FIG.~\\ref{fig:Fig3}A ignore nuclear quantum effects.\nThey are thus more representative of heavy water than of light water.\n\nThe discrepancy between our simulation results and Mishima's experiments lead to the question of why a sharp discontinuity in slope was observed in the experimental melting curves for ice V and ice IV. Such behavior could perhaps be explained by immediate crystallization of ice I rather than melting to a metastable (relaxed) liquid state, which of course is not an issue in the simulations due to the separation of time scales of ice nucleation and liquid-like equilibration\/relaxation. In this context, it should be noted that Mishima's hypothesized liquid-liquid phase transition is located very close to the homogeneous nucleation locus. Furthermore, the behavior reported by Mishima for the melting curves past the hypothesized LLT\\cite{Mishima00} is remarkably noisy on the low-pressure side. Experimental studies explicitly targeted towards this issue are needed to definitively evaluate this hypothesis.\n\n\\section*{Conclusions}\n\nOur results suggest that experiments reported by Mishima and Stanley that pointed to the existence of a liquid-liquid critical point at $\\sim$0.1 GPa and $\\sim$220 K \\cite{Mishima98}, and subcritical melting curves for ice IV, V, and XIII \\cite{Mishima00}, might call for a different interpretation.\nWhile our first principles calculations do support the existence of a liquid-liquid critical point \\cite{Gartner22}, they suggest its location to occur at lower temperatures than had been hitherto assumed, such that the melting curves of ice III, IV, V, VI, and XIII are in reality supercritical.\nThe relative stability of phases reported here is in excellent agreement with experiments, yet from a quantitative point of view our simulations are limited by the accuracy of our chosen semilocal DFT functional.\nFuture work could test our findings using more sophisticated DFT functionals or higher levels of electronic-structure theory.\nConsidering the plethora of known ice polymorphs, and the ones that continue to be discovered and characterized \\cite{gasser2021structural}, the search for ices with subcritical melting curves may be a fruitful endeavor.\nWe also hope that our work will stimulate further experimental efforts to elucidate the behavior of melting curves in the vicinity of the liquid-liquid critical point and definitively explain the discrepancies between the experimental and computational results.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdloi b/data_all_eng_slimpj/shuffled/split2/finalzzdloi
new file mode 100644
index 0000000000000000000000000000000000000000..3546eb7b943898f4ef838d8069e947bde694f267
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzdloi
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction and Preliminary Discussion}\nAs for standard terminology and other terminology used in this paper, we refer to the book by Bondy and Murty, \\cite{Bon}, and to the papers quoted in the references. Let $G$ be a connected graph. A \\emph{2-block} is a 2-connected graph or a block of $G$ containing more than two vertices. The square of a graph $G$, denoted $G^2$, is the graph obtained from $G$ by joining any two nonadjacent vertices which have a common neighbor, by an edge.\n\nIt was shown in 1970 and published in 1974 that the square of every 2-block contains a       hamiltonian cycle, \\cite{Fle2}. Key in proving this was the existence of EPS-graphs $S$ in connected bridgeless graphs $G$, where $S$ is the edge-disjoint union of a not necessarily connected eulerian subgraph $E$ and a linear forest $P$, and $S$ is connected and spans $G$, \\cite{Fle1}. In subsequent papers \\cite{Fle}, \\cite{FleHob} the existence of various types of EPS-graphs was established. Their relevance was based on the fact that the total graph $T(G)$ of any connected graph $G$ other than $K_1$ is hamiltonian if and only if $G$ has an EPS-graph, \\cite{FleHob}. This and the theory of EPS-graphs led to a description of the most general block-cutvertex graph $\\mbox{bc}(G)$ of a graph $G$ may have such that $T(G)$ is hamiltonian and if $\\mbox{bc}(G)$ does not have the corresponding structure, then exchanging certain 2-blocks in $G$ with some special 2-blocks yields a graph $G^*$ such that $\\mbox{bc}(G)$ and $\\mbox{bc}(G^*)$ are isomorphic but $T(G^*)$ is not hamiltonian, \\cite{FleHob}. In dealing with hamiltonian cycles and hamiltonian paths by methods developed up to that point, it was shown in \\cite{Fle} that in the square of graphs hamiltonicity and vertex-pancyclicity are equivalent concepts, and so are hamiltonian connectedness and panconnectedness. In this context Theorem~\\ref{2-blockcycle} stated below was established as a tool needed to prove the equivalences just mentioned.\n\nHowever, in the course of time much shorter proofs of Fleischner's Theorem were developed \\cite{Geo}, \\cite{Riha}; the same applies to Theorem \\ref{2-blockcycle} below, \\cite{MutRau}. More recently, an algorithm yielding a hamiltonian cycle in the square of a 2-block in linear time, was developed, \\cite{AlsGeo}. The methods developed in these much shorter proofs (including the algorithm just mentioned) do not seem to yield short proofs of Theorems \\ref{H_4} and \\ref{F_4} below, \\cite{EkFle}, \\cite{FleChia}. These latter theorems are, on the other hand, instrumental in proving the central results of this paper, i.e., Theorems \\ref{hamiltonian} and \\ref{hamconnected}, and related algorithms. \n\nLet $\\mbox{bc}(G)$ denote the \\emph{block-cutvertex graph} of $G$. Blocks corresponding to leaves of $\\mbox{bc}(G)$ are called \\emph{endblocks}, otherwise \\emph{innerblocks}. Note that a block in a graph $G$ is either a 2-block or a bridge of $G$. For each cutvertex $i$ of $G$, let $k_{i}$ be the number of 2-blocks of $G$ which include vertex $i$ and let $\\mbox{bn}(i)$ be the number of nontrivial bridges of $G$ which are incident with vertex $i$. In what follows a bridge is called nontrivial if it is not incident to a leaf.\n  \nIn Theorem \\ref{hamiltonian}, we introduce an array $m_{i}(B)$ of numbers with an entry for each pair consisting of a cutvertex $i$ and a 2-block $B$ of $G$. We may think of this number $m_{i}(B)$ as the number of edges of $B$ incident with $i$ which are possibly contained in a hamiltonian cycle in $G^{2}$. \n\nStatement of Theorem \\ref{hamiltonian} describes the most general block-cutvertex structure a graph $G$ may have in order to guarantee that $G^2$ is hamiltonian using parameters $m_{i}(B)$ as in \\cite{FleHob}.\n\n\\begin{theorem}\n \\label{hamiltonian}\n Let $G$ be a connected graph with at least three vertices. Let the 2-blocks of $G$ be labelled $B_{1},B_{2},...,B_{n}$. Let the cutvertices of $G$ be labelled $1,2,...,s$. Suppose there is a labelling $m_{i}(B_{t})$ for each $i\\in\\{1,2,...,s\\}$ and each $t\\in\\{1,2,...,n\\}$ such that the following conditions are fulfilled.\n \\begin{mathitem}\n  \\item[1)] $0\\leq m_{i}(B_t)\\leq2$ for all $i$ and all 2-blocks $B_t$;\n  \\item[2)] for 2-block $B_t$ $m_{i}(B_t)=0$ if and only if cutvertex $i$ is not in $V(B_t)$;\n  \\item[3)] for 2-block $B_t$, $m_{i}(B_t)\\geq\\mbox{bn}(i)$, if cutvertex $i\\in V(B_t)$;\n  \\item[4)] $\\mbox{bn}(i)\\leq2$ for all $i\\in\\{1,2,...,s\\}$; \n  \\item[5)] $\\sum_{i=1}^{s}m_{i}(B_t)\\leq4$ for each 2-block $B_t$ of $G$ and, if $m_i(B_t)=2$ \n            for some $i$, then $\\sum_{i=1}^sm_{i}(B_t)\\leq3$; and\n  \\item[6)] $\\sum_{t=1}^{n}m_i(B_t)\\geq2k_{i}+\\mbox{bn}(i)-2$ for each $i\\in\\{1,2,...,s\\}$.\n \\end{mathitem}\n Then $G^{2}$ is hamiltonian. \n\nMoreover, if the labelling $m_{i}(B_{t})$  satisfying conditions 1), 2) and 3) is given and at least one of conditions  4), 5), 6) is violated by some $G$, then there exists a class of graphs $G'$ with non-hamiltonian square but $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic.\n\\end{theorem}\n\nAlso, we obtain a similar result for hamiltonian connectedness (Theorem~\\ref{hamconnected}). Quite surprisingly, its formulation is much simpler than that of Theorem \\ref{hamiltonian}.\n\n\\begin{theorem}\n \\label{hamconnected}\n Let $G$ be a connected graph such that the following conditions are fulfilled:\n \\begin{itemize}\n  \\item[1)] there is no nontrivial bridge of $G$;\n  \\item[2)] every block contains at most 2 cutvertices.\n \\end{itemize}\n Then $G^2$ is hamiltonian connected.\n\nMoreover,\n\\begin{itemize}\n  \\item[$\\cdot$] if a graph $G$ contains a nontrivial bridge, then $G^2$ is not hamiltonian connected;  \n  \\item[$\\cdot$] if $G$ contains a block containing more than 2 cutvertices, then there is a graph $G'$ such that $\\mbox{bc}(G)$ and $\\mbox{bc}(G')$ are isomorphic but $(G')^2$ is not hamiltonian connected. \n\\end{itemize}\n\\end{theorem}\n\nA fundamental result regarding hamiltonicity in the square of a 2-block is the following theorem.\n\n\\begin{theorem}\\emph{\\textbf{\\cite{Fle}}}\n\\label{2-blockcycle}\nSuppose $v$ and $w$ are two arbitrarily chosen vertices of a $2$-block $G$.\nThen $G^2$ contains a hamiltonian cycle $C$ such that the edges of $C$ incident to $v$ are in $G$ and at least one of the edges of $C$ incident to $w$ is in $G$. Furthermore, if $v$ and $w$ are adjacent in $G$, then these are three different edges.  \n\\end{theorem}\n\nThe hamiltonian theme in the square of a 2-block has been recently revisited (\\cite{EkChiaFle}, \\cite{EkFle}, \\cite{FleChia}), yielding the following results which are essential for this paper. \n\nA graph $G$ is said to have the \\emph{$\\mathcal{H}_{k}$ property} if for any given vertices $x_1,...,x_k$ there is a hamiltonian cycle in $G^2$ containing distinct edges $x_1y_1,...,x_ky_k$ of~$G$.\n \n\\begin{theorem}\\emph{\\textbf{\\cite{EkFle}}}\n \\label{H_4}\n Given a 2-block $G$ on at least 4 vertices, then $G$ has the $\\mathcal{H}_{4}$ property, and there are 2-blocks of arbitrary order greater than 4 without the $\\mathcal{H}_{5}$ property.\n\\end{theorem}\n\nBy a \\emph{$uv$-path} we mean a path from $u$ to $v$ in $G$. If a $uv$-path is hamiltonian, we call it a \\emph{$uv$-hamiltonian path}. Let $A=\\{x_{1},x_{2},...,x_{k}\\}$ be a set of $k\\geq 3$ distinct vertices in $G$. An $x_{1}x_{2}$-hamiltonian path in $G^{2}$ which contains $k-2$ distinct edges $x_{i}y_{i}\\in E(G), i=3,...,k$, is said to be $\\mathcal{F}_{k}$. A graph $G$ is said to have the \\emph{$\\mathcal{F}_{k}$ property} if, for any set $A=\\{x_{1},x_{2},...,x_{k}\\}\\subseteq V(G)$, there is an $\\mathcal{F}_{k}$ $x_{1}x_{2}$-hamiltonian path in $G^{2}$.\n\n\\begin{theorem}\\emph{\\textbf{\\cite{FleChia}}}\n \\label{F_4}\n  Every 2-block on at least 4 vertices has the $\\mathcal{F}_{4}$ property. \n\\end{theorem}\n\nA graph $G$ is said to have the \\emph{strong $\\mathcal{F}_{3}$ property} if, for any set of 3 vertices $\\{x_{1},x_{2},x_{3}\\}$ in $G$, there is an $x_{1}x_{2}$-hamiltonian path in $G^{2}$ containing distinct edges $x_{3}z_{3},x_{i}z_{i}\\in E(G)$ for a given $i\\in\\{1,2\\}$. Such an $x_{1}x_{2}$-hamiltonian path in $G^{2}$ is called a strong          $\\mathcal{F}_{3}$ $x_{1}x_{2}$-hamiltonian path.\n\n\\begin{theorem}\\emph{\\textbf{\\cite{FleChia}}}\n \\label{strongF_3}\n  Every 2-block has the strong $\\mathcal{F}_{3}$ property.\n\\end{theorem}\n\n\\begin{theorem}\\emph{\\textbf{\\cite{FleChia}}}\n\t\\label{strongF_3ends}\nLet $G$ be a $2$-connected graph and let $x, y$ be two vertices in $G$. Then $G^2$ has an $xy$-hamiltonian path $P(x, y)$ such that\n\n(i) $xz \\in E(G) \\cap E(P(x,y))$  for some $z \\in V(G)$, \nand\n\n(ii) either $yw \\in  E(G) \\cap E(P(x,y))$ for some $w\\in V(G)$,  or  else $P(x, y)$ contains an edge $uv$ for some vertices $u, v \\in N(y)$.\n\\end{theorem} \n\n\\section{Proofs and algorithms}\n\nPROOF OF THEOREM \\ref{hamiltonian}\n\n\\begin{proof}\n Set $P_0=G-\\cup_{t=1}^{n}B_t$. Then every component of $P_0$ is a tree. Since by 4) $\\mbox{bn}(i)\\leq 2$ every component of $P_0$ is even a caterpillar. \n \n For every caterpillar $T$ of $P_0$ except $T=K_2$ we have the following observation which can be proved easily. \n\n\\medskip\n\n\\noindent\n\\emph{Observation: Let $T$ be a caterpillar with at least three vertices and $P=x_1x_2...x_m$ be some longest path in $T$. Then $T^2$ contains a hamiltonian cycle containing edges $x_1x_2, x_{m-1}x_m$  and different edges $u_jv_j$, where $u_j,v_j\\in N_G(x_j)$ for $j=2,3,...m-1$.}\n\nSee Figure \\ref{Catter} for illustration in which for $x_3$ we have $u_3=x_2$ and $v_3=x_4$.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\end{center}\\caption{Hamiltonian cycle in a caterpillar for $m=7$ (bold edges)}\\label{Catter}\n\\end{figure} \n\n\\medskip\n\n Every 2-block $B_t$ contains a hamitonian cycle in $(B_t)^2$ which is one of two types depending on labellings $m_{i}(B_{t})$:\n \nIf $m_i(B_t)\\neq 2$ for every $i=1,2,...,s$, then for at most 4 cutvertices $a,b,c,d$ it holds that $m_j(B_t)=1$ for $j=a,b,c,d$ by condition 5). By Theorem \\ref{H_4}, $(B_t)^2$ has a hamiltonian cycle $C_t$ containing 4 different edges $aa',bb',cc',dd'$ of~$B_t$. \n \nIf $m_i(B_t)=2$ for some $i\\in\\{1,2,...,s\\}$, then at most one cutvertex $a$ has $m_a(B_t)=1$ by condition 5). By Theorem \\ref{2-blockcycle}, $(B_t)^2$ has a hamiltonian cycle $C_t$ containing 3 different edges $ii',ii'',aa'$ of $B_t$.   \n\nThe union of hamiltonian cycles $C_t$ in $(B_t)^2$, for $t=1,2,...n$, hamiltonian cycles in the square of each catepillar (nontrivial component of $P_0$) and trivial components of $P_0$  is a connected spanning subgraph $S$ of $G^2$. \n\nWe construct a hamiltonian cycle $C$ in $G^2$ from $S$ repeating step by step the following procedure for every cutvertex $i$ of $G$ with $m_{i}(B)\\geq 1$ for some 2-block $B$. \n\nIf $i$ does not exist, then $n=0$ and $G=P_0$ is a caterpillar. Hence $S$ is a hamiltonian cycle in $G^2$. Otherwise we join all hamiltonian cycles from $S$ containing $i$ together with trivial components of $P_0$ containing $i$ to one cycle in the following way.\n\n\\medskip\n\n\\noindent\nFirst assume that $\\mbox{bn}(i)=0$.\n\nBy condition 6) we have  $\\sum_{t=1}^{n}m_i(B_t)\\geq 2k_i-2$. Without loss of ge\\-nerality for $k_i>1$ we may assume that $m_i(B_1)\\geq 1$, $m_i(B_2)\\geq 1$ and $m_i(B_3)=m_i(B_4)=...=m_i(B_{k_i})=2$, where $m_i(B_t)$  corresponds to the number of edges of $B_t$ incident to $i$ in $C_t$. If $k_i=1$, then by condition 2) we have $m_i(B_1)\\geq 1$.\n\nWe find a cycle $C^i$ on $\\cup_{r=1}^{k_i}V(C_r)\\cup L$, where $L$ is the set of all leaves incident to $i$, by appropriately replacing edges of $C_r\\cap B_r$, $r=1,2,...,k_i$, incident to $i$ (guaranteed by definition of $m_i(B_t)$) with edges of $G^2$ joining vertices in different $C_r$ adjacent to $i$ and leaves adjacent to $i$. Note that we preserve properties given by $m_j(B_t)$ for all $j\\neq i$.\n\n\\noindent\nNow assume that $\\mbox{bn}(i)=1$.\n\nBy condition 6) we have  $\\sum_{t=1}^{n}m_i(B_t)\\geq 2k_i+1-2=2k_i-1$. Without loss of generality we may assume that $m_i(B_1)\\geq 1$ and $m_i(B_2)=m_i(B_3)=...=m_i(B_{k_i})=2$, where $m_i(B_t)$  corresponds to the number of edges of $B_t$ incident to $i$ in $C_t$.\nLet $T$ be the component of $P_0$ containing $i$.\n\nIf $T=K_2=ii'$, where $i'$ is also a cutvertex of $G$ with $m_{i'}(B)\\geq 1$  (T is a trivial component of $P_0$), then we find a cycle $C^i$ on $\\cup_{r=1}^{k_i}V(C_r)\\cup T$ containing the edge $ii'$ by appropriately replacing edges of $C_r\\cap B_r$, $r=1,2,...,k_i$, incident to $i$ (guaranteed by definition of $m_i(B_t)$) with edges of $G^2$ joining $i'$ and vertices in different $C_r$ adjacent to $i$. Also here we preserve properties given by $m_j(B_t)$ for all $j\\neq i$.\n\nIf $T$ is a nontrivial component of $P_0$, then $T^2$ contains a hamiltonian cycle $C_T$ containing end-edges of any fixed longest paths $P$ in $T$ (we choose end-edges containing cutvertices of $G$ with $m_{i}(B_t)\\geq 1$) - see Observation above. Again we find a cycle $C^i$ on $\\cup_{r=1}^{k_i}V(C_r)\\cup V(C_T)$ by appropriately replacing edges of $C_r\\cap B_r$, $r=1,2,...,k_i$, incident to $i$ (guaranteed by definition of $m_i(B_t)$) and the end-edge $ii^*$ of $P$ with edges of $G^2$ joining $i^*$ and vertices in different $C_r$ adjacent to $i$. Again we preserve properties given by $m_j(B_t)$ for all $j\\neq i$ and by $C_T$.\n\n\\noindent\nFinally assume that $\\mbox{bn}(i)=2$.\n\nBy condition 6) we have  $\\sum_{t=1}^{n}m_i(B_t)\\geq 2k_i+2-2=2k_i$. It follows necessarily that  $m_i(B_1)=m_i(B_2)=...=m_i(B_{k_i})=2$, where $m_i(B_t)$  corresponds to the number of edges of $B_t$ incident to $i$ in $C_t$.\n\nLet $T$ be the nontrivial component of $P_0$ containing $i$. Note that $i$ is not an endvertex of $T$ because of $\\mbox{bn}(i)=2$. Then $T^2$ contains a hamiltonian cycle $C_T$ containing end-edges of any fixed longest paths in $T$ (we choose end-edges containing cutvertices of $G$ with $m_{i}(B_t)\\geq 1$) and an edge $u_iv_i$ of $G^2$ where $u_i,v_i\\in N_G(i)$  (see Observation above). We find a cycle $C^i$ on $\\cup_{r=1}^{k_i}V(C_r)\\cup V(C_T)$ by appropriately replacing edges of $C_r\\cap B_r$, $r=1,2,...,k_i$, incident to $i$ (guaranteed by definition of $m_i(B_t)$) and the edge $u_iv_i$ of $P$ with edges of $G^2$ joining $u_i, v_i$ and vertices in different $C_r$ adjacent to $i$ if $k_i>1$. If, however, $k_i=1$, then $u_i$ and $v_i$ are joined to the neighbors of $C_r\\cap B_r$ in $N_G(i)$. Also here we preserve properties given by $m_j(B_t)$ for all $j\\neq i$ and by $C_T$.\n\nNow we choose next cutvertex $i$ with $m_{i}(B)\\geq 1$ for some 2-block $B$ successively and we use all cycles  formed in the previous steps instead of previously formed cycles. Note that we preserve all properties given by $m_j(B)$ for all $j\\neq i$ in every case. We stop with the hamiltonian cycle in $G^2$ as required.\n\n\\vskip 8mm\n\nNow assume that there is no labelling satisfying conditions 1) - 6), that is, the labelling $m_i(B_t)$ satisfying conditions 1), 2) and 3) is given and at least one of conditions  4), 5), 6) is violated. We show that there exists a class of graphs $G'$ with non-hamiltonian square but $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic.\n\n\\bigskip\n\n\\begin{figure}[ht]\n\\begin{center}\n\\end{center}\\caption{Graphs without hamiltonian square}\\label{Counter}\n\\end{figure} \n\n\\noindent\n\\emph{Condition 4) does not hold.} \n\nHence $\\mbox{bn}(i)\\geq 3$ for at least one $i\\in\\{1,2,...,s\\}$. Clearly this is a class of graphs $G'$ such that the square of every such graph $G'$ does not contain a hamiltonian cycle (if we try to construct a hamiltonian cycle in the square, then the degree of the cutvertex $i$ is at least 3, a contradiction), e.g. see graphs in Figure \\ref{Counter} a), where $H_1$ is arbitrary connected graphs, $H_2$, $H_3$, $H_4$ are arbitrary connected graphs with at least one edge each and $\\mbox{bn}(i)=3$. Note that conditions 5) and 6) may hold.\n\n\\vskip 0.9cm\n\n\\noindent\n\\emph{Condition 5) does not hold.}\n\nHence $\\sum_{i=1}^{s}m_{i}(B)\\geq 5$ for some 2-block $B$ and $m_i(B)<2$ for all $i$ or $\\sum_{j=1}^{s}m_{j}(B)\\geq 4$ for some 2-block $B$ and $m_i(B)=2$ for some $i\\in\\{1,2,...,s\\}$.\n\nFirst suppose that $k=\\sum_{i=1}^{s}m_{i}(B)\\geq 5$ for some 2-block $B$ of $G$ and $m_i(B)<2$ for all $i$. Clearly $B$ has exactly $k$ cutvertices by condition 2). Then we exhange $B$ with $K_{2,k}$ where $k$ 2-valent vertices are cutvertices of $G$ and all other blocks with arbitrary blocks to get a class of graphs $G'$ such that $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic. The square of every such graph $G'$ does not contain a hamiltonian cycle (if we try to construct a hamiltonian cycle in the square, then the degree of at least one of the two $k$-valent vertices of $K_{2,k}$ is at least 3, a contradiction), e.g. see graphs in Figure \\ref{Counter} b), where $k=5$ and $H_1,...,H_5$ are arbitrary connected graphs with at least one edge each. Note that conditions 4), 6) and the second part of condition 5) may hold.\n\nNow suppose that $\\sum_{j=1}^{s}m_{j}(B)\\geq 4$ for some 2-block $B$ and $m_i(B)=2$ for some $i$. If $B$ contains at least 5 cutvertices of $G$, then we continue similarly as above. If $B$ contains $k$ cutvertices of $G$ where $2\\leq k \\leq 4$, then without loss of generality we may assume that we tried to set the labelling $m_i(B_t)$ satisfying firstly conditions 5) and subsequently condition 6). Hence $\\mbox{bn}(i)\\geq 2$ and $\\mbox{bn}(j)\\geq 2$ where $j$ is the second cutvertex of $G$ in $B$ if $k=2$, otherwise we find a labelling $m_i(B_t)$ satisfying condition 5), a contradiction (see Algorithm~1 cases e) and f) below).\n\nFor $k=3,4$ we exhange $B$ with a cycle $C_k$ to get a class of graphs $G'$ such that $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic. The square of every such graph $G'$ does not contain a hamiltonian cycle (if we try to construct a hamiltonian cycle in the square, then the degree of the cutvertex $i$ is at least 3, a contradiction), e.g. see graphs in Figure \\ref{Counter} c), where $k=3$ and $H_1,...,H_4$ are arbitrary connected graphs with at least one edge each. Note that conditions 4), 6) and the first part of condition 5) may hold.\n\nFor $k=2$, we exchange $B$ with $K_{2,3}$, where two of the three 2-valent vertices are $i$ and $j$, to get a class of graphs $G'$ such that $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic. The square of every such graph $G'$ does not contain a hamiltonian cycle (it is not possible to find a hamiltonian cycle in the square containing the third 2-valent vertex different from $i$, $j$, a contradiction), e.g. see graphs in Figure \\ref{Counter} d), where $H_1,...,H_4$ are arbitrary connected graphs with at least one edge each. Note that conditions 4), 6) and the first part of condition 5) may hold.\n\n\\medskip\n\n\\noindent\n\\emph{Condition 6) does not hold.}\n  \nHence $\\sum_{t=1}^{n}m_{i}(B_t)<2k_i+\\mbox{bn}(i)-2$ for some $i$ and consequently $m_i(B_t)=1$ for at least $3-\\mbox{bn}(i)$ 2-blocks containing $i$. Note that, clearly, $\\mbox{bn}(i)<2$ with respect to condition 3).\n\nLet $r$ be the number of 2-blocks with $m_i(B_t)=1$. Each of these 2-blocks contains either exactly 2 cutvertices of $G$ or at least 3 cutvertices of $G$. Note that for 2-blocks containing only cutvertex $i$ we have $m_i(B_t)=2$ (see Algorithm 1 case d) below). We exchange every 2-block containing exactly 2 cutvertices of $G$ with a cycle $C_3$ and every 2-blocks containing $k$ cutvertices of $G$, $k\\geq 3$, with a cycle $C_k$. In the first case note that we assume without loss of generality that there is no labelling such that we switch values 1 and 2 for both cutvertices of this 2-block to get a permissible labelling (again see Algorithm~1 case e) below).\n\nSince $r\\geq 3-\\mbox{bn}(i)$, by the exchanging 2-block mentioned above we get a class of graphs $G'$ such that $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic. The square of every such graph $G'$ does not contain a hamiltonian cycle (if we try to construct a hamiltonian cycle in the square, then the degree of the cutvertex $i$ is at least 3, a contradiction), e.g. see graphs in Figure \\ref{Counter} $e_1$) and $e_2$). For the graph in Figure \\ref{Counter} $e_1$) it holds that $r=3-\\mbox{bn}(i)=3-1=2$, the 2-block $B_1$ has exactly 2 cutvertices of $G$, the 2-block $B_2$ has $k=3$ cutvertices of $G$ (and hence $B_1$, $B_2$ are isomorphic to $C_3$) and $H_1,...,H_5$ are arbitrary connected graphs with at least one edge. For the graph in Figure \\ref{Counter} $e_2$) it holds that $r=3-\\mbox{bn}(i)=3-0=3$, the 2-block $B_1$ has exactly 2 cutvertices of $G$, the 2-block $B_2$ has $k=3$ cutvertices of $G$, the 2-block $B_3$ has $k=4$ cutvertices of $G$ (hence $B_1$, $B_2$ are isomorphic to $C_3$ and $B_3$ is isomorphic to $C_4$) and $H_1,...,H_7$ are arbitrary connected graphs with at least one edge. Note that conditions 4) and 5) may hold.\n\nThis finished the proof of Theorem \\ref{hamiltonian}.\n\\end{proof}\n\nIf there is a graph $G$ such that every labelling $m_i(B_t)$ violates at least one of the conditions 4) - 6) of Theorem \\ref{hamiltonian}, then there is a graph $G'$ with $\\mbox{bc}(G')=\\mbox{bc}(G)$ such that $(G')^2$ is not hamiltonian  as it has been shown in the proof of Theorem~\\ref{hamiltonian}. On the other hand, if we are able to construct a labelling $m_i(B_t)$ satisfying conditions 1) - 6) using the following algorithm, then $G^2$ is hamiltonian as it has been shown in the proof of Theorem~\\ref{hamiltonian}.\n\n\\bigskip\n\n\\noindent\n\\emph{ALGORITHM 1:} \n\nSet $P_0=G-\\cup_{t=1}^nB_t$. If any component of $P_0$ is not a caterpillar, then $\\mbox{bn}(i)\\geq 3$ for some $i\\in\\{1,2,...,s\\}$ contradicting condition 4) in Theorem \\ref{hamiltonian} and $G^2$ is not hamiltonian (e.g. see Figure \\ref{Counter} a)). STOP.\n\nIf $G=P_0$, then $G$ is a caterpillar,  $n=0$ and $G^2$ is hamiltonian (see Observation in the proof of Theorem \\ref{hamiltonian}) and $m_i(B_t)$ is not defined ($n=0$). STOP.\n\nIf $G$ is a 2-block, $G^2$ is hamiltonian by Theorem \\ref{2-blockcycle} and $m_i(B_t)$ is not defined ($s=0$ and $n=1$). STOP.\n\nWe set $G_0=G-P_0$ and $m_i(B_t)=0$ if $i\\notin V(B_t)$ for $i\\in\\{1,2,...,s\\}$ and $t\\in\\{1,2,...,n\\}$.\n\n\\bigskip\n\n\\noindent\nSTART\n\nWe choose a 2-block $B$ containing at most 1 cutvertex of $G_0$. Note that $B$ is either a component of $G_0$ or an endblock of some component of $G_0$. If such endblock does not exist, we choose 2-block $B$ as a component of $G_0-H$ or an endblock of $G_0-H$ where $H$ is the union of all 2-blocks for which the labelling $m_i(B_t)$ is already set.  Let $c_1,c_2,...,c_k$ be all cutvertices of $G$ contained in $B$, $k\\geq 1$. \n\n\\medskip\n\n\\begin{itemize}\n \\item[a)] If $k\\geq 5$, then by condition 2) $m_{c_i}(B)\\geq 1$ for $i=1,2,...,k$. Hence condition 5) in Theorem \\ref{hamiltonian} does not hold and $G^2$ may not be hamiltonian (e.g. see Figure \\ref{Counter} b)). STOP.\n\n \\item[b)] If $k\\geq 3$ and $\\mbox{bn}(c_i)=2$ for some $i\\in\\{1,2,...,k\\}$ , then by condition 3) $m_{c_i}(B)=2$ and by 2) $m_{c_j}(B)\\geq 1$ for $j=1,2,...,k$. Hence condition~5) in Theorem \\ref{hamiltonian} does not hold and $G^2$ may not be hamiltonian (e.g. see Figure \\ref{Counter} c)). STOP.\n\n \\item[c)] If $k=2$ and $\\mbox{bn}(c_1)=\\mbox{bn}(c_2)=2$, then by condition 3) $m_{c_1}(B)=2$ and $m_{c_2}(B)=2$. Hence condition 5) in Theorem \\ref{hamiltonian} does not hold and $G^2$ may not be hamiltonian (e.g. see Figure \\ref{Counter} d)). STOP.\n\n \\item[d)] If $k=1$, then we set $m_{c_1}(B)=2$ (we maximize values $m_i(B_t)$ with respect to condition 6) in Theorem \\ref{hamiltonian}). Note that, if the labelling $m_{i}(B_t)$ is set for all 2-blocks incident with $c_1$, then condition 6) holds for cutvertex $c_1$ with respect to the choice of $B$. \n \n If the labelling $m_{i}(B_t)$ is set for all 2-blocks of $G$, then the labelling $m_i(B_t)$ satisfies the conditions of Theorem \\ref{hamiltonian} and $G^2$ is hamiltonian. STOP.\n \n Otherwise we go to START. \n\n \\item[e)] If $k=2$ and $\\mbox{bn}(c_i)\\leq 1$ for $i\\in\\{1,2\\}$, then we set $m_{c_1}(B)$ and $m_{c_2}(B)$ in the following way (without loss of generality $i=1$).\n \n Let $\\mbox{bn}(c_2)=2$. Then we set $m_{c_1}(B)=1$ and $m_{c_2}(B)=2$ with respect to conditions 2), 3) and 5). \n \n Let $\\mbox{bn}(c_2)\\leq 1$. Then for at least one of $c_1$, $c_2$ it holds that $m_{c_j}(B_t)$ for $j\\in\\{1,2\\}$ is set for all 2-blocks $B_t$ except $B$ with respect to the choice of $B$ (again without loss of generality $j=1$). We set $m_{c_1}(B)=1$ and we verify condition 6) for $c_1$. If it holds, then we set $m_{c_2}(B)=2$ (again we maximize values $m_i(B_t)$ with respect to condition 6)). If condition~6) for $c_1$ does not hold for $m_{c_1}(B)=1$, then we set $m_{c_1}(B)=2$ and $m_{c_2}(B)=1$. \n \n Now in both cases we verify condition 6) for $c_1$ and $c_2$ if the labelling $m_{c_1}(B_t)$ and $m_{c_2}(B_t)$ is set for all 2-blocks $B_t$. \n \n If condition 6) does not hold in at least one case, then $G^2$ may not be hamiltonian (e.g. see Figure \\ref{Counter} $e_1)$). STOP. \n \n Hence suppose that condition 6) holds for $c_1$, $c_2$ if $m_{c_1}(B_t)$, $m_{c_2}(B_t)$ is set for all $B_t$, respectively.\n \n If the labelling $m_{i}(B_t)$ is set for all 2-blocks, then the labelling $m_i(B_t)$ satisfies the conditions of Theorem \\ref{hamiltonian} and $G^2$ is hamiltonian. STOP.\n  \n Otherwise we go to START. \n \n  \\item[f)] If $k\\in\\{3,4\\}$ and $\\mbox{bn}(c_i)\\leq 1$, then we set $m_{c_i}(B)=1$ for $i=1,2,...,k$. We verify condition 6) for all $c_i$ if the labelling $m_{c_i}(B_t)$ is set for all 2-blocks $B_t$. \n \n If condition 6) does not hold in at least one case, then $G^2$ may not be hamiltonian (e.g. see Figure \\ref{Counter} $e_2)$). STOP.\n \n Hence suppose that condition 6) holds for all $c_i$, $i=1,2,...,k$, for which $m_{c_i}(B_t)$ is set for all $B_t$.\n \n If the labelling $m_{i}(B_t)$ is set for all 2-blocks, then the labelling $m_i(B_t)$ satisfies the conditions of Theorem \\ref{hamiltonian} and $G^2$ is hamiltonian. STOP.\n \n Otherwise we go to START. \n\\end{itemize}\n\n\\noindent\nPROOF OF THEOREM \\ref{hamconnected}\n\n\\begin{proof}\n Let $x,y\\in V(G)$. First we prove that there exists an $xy$-hamiltonian path $P$ in $G^2$ if there is no nontrivial bridge of $G$ and every block contains at most 2 cutvertices.\n\n\\bigskip\n \n (A) Suppose that $x$ and $y$ are in the same block $B$ of $G$. We proceed by induction on $n$, where $n$ is the number of blocks of $G$, $n\\geq 1$.  \n \n For $n=1$, clearly $G=B$. If $B=K_2=xy$, then $G$ is also the $xy$-hamiltonian path in $G^2$ as required. If $B$ is a 2-block, then by Theorem \\ref{strongF_3}, $G^2=B^2$ contains an $xy$-hamiltonian path $P$ as required.\n\n\n\\bigskip\n\nNow suppose that the statement of Theorem \\ref{hamconnected} is true for every graph with $n$ blocks and $G$ is a graph with $n+1$ blocks, $n\\geq 1$. We distinguish 2 cases.\n\n\\begin{itemize}\n \\item $B$ has exactly one cutvertex $c$.\n \n Without loss of generality we assume that $x\\neq c$. If $B$ is a 2-block, then by Theorem \\ref{strongF_3}, $B^2$ contains an $xy$-hamiltonian path $P_B$ containing an edge $cy'$ where $y'$ is a neighbor of $c$ in $B$. Note that $y'=x$ or $c=y$ is possible. If $B=K_2$, then $B=xy=y'c$ and $P_B=xy$ is an $xc$-hamiltonian path in $B^2$. By the induction hypothesis $(G-B)^2$ contains a $cc'$-hamiltonian path $P_G$ where $c'$ is a neighbor of $c$ in $G-B$. Then $P=P_B\\cup P_G-cy'+y'c'$ is an $xy$-hamiltonian path in $G^2$ as required.\n \n \\item $B$ has two cutvertices $c_1$, $c_2$.\n \n We denote by $G_1$, $G_2$ the two components of $G-B$ such that $c_i\\in V(G_i)$ and let $c'_i$ be a neighbor of $c_i$ in $G_i$, $i=1,2$. By the induction hypothesis $(G_i)^2$ contains a $c_ic'_i$-hamiltonian path $P_{G_i}$, $i=1,2$.\n \n \\begin{itemize}\n  \\item[a)] $c_i\\notin \\{x,y\\}$ ($x$ and $y$ are not cutvertices).\n\n By Theorem \\ref{F_4}, $B^2$ contains an $xy$-hamiltonian path $P_B$ containing the edges $c_iz_i$ where $z_i$ is a neighbor of $c_i$ in $B$, $i=1,2$. Note that $z_i\\in\\{x,y\\}$ is possible. \n \n  \\item[b)] Up to symmetry $c_1=x$ and $c_2\\neq y$ (either $x$ or $y$ is a cutvertex of $G$).\n  \n  By Theorem \\ref{strongF_3}, $B^2$ contains an $xy$-hamiltonian path $P_B$ containing the edges $c_iz_i$ where $z_i$ is a neighbor of $c_i$ in $B$, $i=1,2$. Note that $z_1=c_2$ or $z_2=y$ is possible. \n \n \\item[c)] $c_1=x$ and $c_2=y$ (similarly $c_1=y$ and $c_2=x$).\n \n By Theorem \\ref{strongF_3ends}, $B^2$ contains an $xy$-hamiltonian path $P_B$ containing either the edges $c_iz_i$ where $z_i$ is a neighbor of $c_i$ in $B$, $i=1,2$, or the edges $c_1z_1$, $uv$ where $z_1$ is a neighbor of $c_1$ in $B$ and $u,v$ are neighbors of $c_2$ in $B$. \n \\end{itemize}\n \n In all cases except the case c), if $uv$ is the edge of $P_B$, $$P=P_{G_1}\\cup P_B\\cup P_{G_2}-\\{c_1z_1,c_2z_2\\}\\cup\\{c'_1z_1,c'_2z_2\\}$$ is an $xy$-hamiltonian path in $G^2$ as required. \n \n It remains to find an $xy$-hamiltonian path in $G^2$ if $uv$ is the edge of $P_B$. \n \n If $G_2=K_2=c_2u_2$, then $$P=P_{G_1}\\cup P_B\\cup -\\{c_1z_1,uv,c_2u_2\\}\\cup\\{c'_1z_1,u_2u, u_2v\\}$$ is an $xy$-hamiltonian path in $G^2$ as required.    \n \n If $G_2\\neq K_2$, then we prove that $(G_2)^2$ contains a hamiltonian cycle $C$ containing edges $c_2u_2$, $c_2v_2$ of $G_2$. Let $B_1,B_2,...,B_k$ be all 2-blocks of $G_2$ containing $c_2$. By Theorem \\ref{2-blockcycle}, for  $i=1,2,...,k$, $(B_i)^2$ contains a hamiltonian cycle $C'_i$ containing three different edges $c_2u_2^i$, $c_2v_2^i$, $y_iy'_i$ of $B_i$ where $y_i$ is the second cutvertex of $G_2$ in $B_i$ if it exists.\n \n If $y_i$ exists, then we denote by $H_i$ a component of $G_2-(B_i-y_i)$ containing $y_i$. By the induction hypothesis $(H_i)^2$ contains a $y_id_i$-hamiltonian path $P_i$ where $d_i$ is a neighbor of $y_i$ in $H_i$. Then we set $C_i=C'_i\\cup P_i-y_iy'_i+y'_id_i$. If $y_i$ does not exist, then we set $C_i=C'_i$.\n \n Let $T$ be the set of all leaves of $G_2$ adjacent to $c_2$. Then we find a cycle $C$ on $\\cup_{i=1}^{k}V(C_i)\\cup T$ by appropriately replacing edges $c_2u_2^i$, $c_2v_2^i$ with edges of $G^2$ joining $u_2^i$, $v_2^i$ in different $C_i$ and leaves adjacent to $c_2$ (similarly as in the proof of Theorem \\ref{hamiltonian}) such that we preserve two edges ($c_2u_2^i$, $c_2v_2^i$ or $c_2l_1$, $c_2,l_2$ where $l_1,l_2$ are two leaves of $G_2$ adjacent to $c_2$) as $c_2u_2$, $c_2v_2$. \n  \n Now $$P=P_{G_1}\\cup P_B\\cup C-\\{c_1z_1,uv,c_2u_2,c_2v_2\\}\\cup\\{c'_1z_1,u_2u, v_2v\\}$$ is an $xy$-hamiltonian path in $G^2$ as required.    \n\n\n\\end{itemize}\n\n\\bigskip \n \n (B) Suppose that $x$ and $y$ are in different blocks of $G$.\n\n Let $P_G$ be any $xy$-path in $G$ and $c\\in V(P_G)\\setminus\\{x,y\\}$ be a cutvertex of $G$. Let $K$ be the component of $G-c$ containing $x$, $G_y=G-V(K)$ and $G_x=G-G_y$. Clearly $G_x\\cup G_y=G$ and $G_x\\cap G_y=c$.  If $G_x$, $G_y$ are isomorphic to $K_2$, then we set $P_x=G_x$, $P_y=G_y$, respectively. If $G_x$, $G_y$ are 2-blocks, then $(G_x)^2$, $(G_y)^2$ contains an $xc$-hamiltonian path $P_x$, a $cy$-hamiltonian path $P_y$ by Theorem \\ref{strongF_3}, respectively. We proceed by induction on $n$, where $n$ is the number of blocks of $G$, $n\\geq 2$.\n  \n First assume that $G$ has exactly 2 blocks. Hence $G_x$, $G_y$ are isomorphic to $K_2$ or 2-blocks and $P=P_x\\cup P_y$ is an $xy$-hamiltonian path in $G^2$ as required.\n \n Now suppose that the statement of Theorem \\ref{hamconnected} is true for every graph with $n$ blocks and $G$ is a graph with $n+1$ blocks, $n\\geq 2$. If $G_x$, $G_y$ is not a block, then by the induction hypothesis $(G_x)^2$, $(G_y)^2$ contains an $xc$-hamiltonian path $P_x$, a $cy$-hamiltonian path $P_y$, respectively. Then $P=P_x\\cup P_y$ is an $xy$-hamiltonian path in $G^2$ as required.\n  \n\\bigskip \n\n Now it remains to prove that if there is a nontrivial bridge of $G$, then  $G^2$ is not hamiltonian connected and if $G$ contains a block containing more than 2 cutvertices, then there is a graph $G'$ such that $\\mbox{bc}(G)$ and $\\mbox{bc}(G')$ are isomorphic but $(G')^2$ is not hamiltonian connected. \n \n Clearly, if there exists a nontrivial bridge $xy$ in $G$, then there is no $xy$-hamiltonian path in $G^2$ and $G^2$ is not hamiltonian connected.\n \n \\begin{figure}[ht]\n\\begin{center}\n\\end{center}\\caption{Graphs without $xy$-hamiltonian path in the square}\\label{Counter2}\n\\end{figure} \n\n Finally assume that $G$ contains a block $B$ containing $r$ cutvertices, where $r>2$. Then we exhange $B$ with a cycle $C_r$ and all other blocks with arbitrary blocks to get a class of graphs $G'$ such that $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic. Clearly the square of every such graph $G'$ does not contain a hamiltonian path between arbitrary two cutvertices of $G'$ in $C_r$ and hence $(G')^2$ is not hamiltonian connected, e.g. with Figure \\ref{Counter2}, where $r=3$ and $H_1,...,H_3$ are arbitrary connected graphs with at least one edge.\n\\end{proof}\n\nSimilarly as for Theorem \\ref{hamiltonian} we state the following algorithm to verify conditions of Theorem \\ref{hamconnected}.\n\n\\bigskip\n\n\\noindent\n\\emph{ALGORITHM 2:} \n\nLet $G'=G-S$ where $S$ is the set of all endblocks of $G$. Let $\\mbox{cvn}_G(B)$ be the number of cutvertices of $G$ in $B$.\n\n\\noindent\nSTART\n\nFind an endblock $B$ of $G'$.\n\n\\begin{itemize}\n \\item If $B$ is a bridge of $G'$, then $B$ is a nontrivial bridge of $G$ and $G^2$ is not hamiltonian connected. STOP.\n \\item Let $B$ be a 2-block.  \n   \\begin{itemize}\n     \\item if $\\mbox{cvn}_G(B)>2$, then $G^2$ may not be hamiltonian connected (e.g. see Figure \\ref{Counter2}). STOP.\n     \\item if $\\mbox{cvn}_G(B)\\leq 2$, then $G'=G'-B$.\n        \\begin{itemize}\n         \\item if $G'=\\emptyset$, then $G^2$ is hamiltonian connected. STOP.\n         \\item if $G'\\neq\\emptyset$, then go to START.\n        \\end{itemize}\n   \\end{itemize}\n\\end{itemize}\n\nIn both algorithms in this paper, determining blocks and especially endblocks and bridges, cutvertices, block-cutvertex graphs, and the parameters $\\mbox{bn}(i)$, $\\mbox{cvn}_G(B)$ can be determined in polynomial time. \n\nAs a consequence, polynomial running time in Algorithm 2 is guaranteed. For, determining (potentially) not being Hamiltonian connected, can be determined instantly once a nontrivial bridge, a block with more than 2 cutvertices has been found. And deleting an endblock reduces the size of $G'$ linearly. \n\nNow consider the running time of Algorithm 1. The first decision to be made is whether $P_0$  is a forest of caterpillars \u2013 this can be done in linear time. After that, at every step 'one chooses a 2-block $B$ as a component of $G_0-H$ or an endblock of $G_0-H$ where $H$ is the union of all 2-blocks for which the labelling $m_i(B_t)$ is already set'. Clearly, identifying such $B$ can be done in linear time. The same applies to working through the cases for defining the various values of $m_i(B)$. \n\nSummarizing, it follows that both algorithms run in polynomial time. We note however, that these algorithms can only decide the existence or potential non-existence of hamiltonian cycles or hamiltonian paths in the square of graphs under consideration; they do not construct any such cycle or path.\n\n\n\\section{Conclusion}\nThe main results of this paper are Theorem \\ref{hamiltonian} and Theorem \\ref{hamconnected}.\nAs we mention in Introduction Fleischner in \\cite{Fle} proved that in the square of graphs hamiltonicity and vertex-pancyclicity are equivalent concepts, and so are hamiltonian connectedness and panconnectedness. Hence we proved in fact that for graphs satisfying assumptions of Theorem \\ref{hamiltonian}, Theorem \\ref{hamconnected} the square of these graphs is vertex-pancyclic, panconnected, respectively.\n\\medskip\n\n As easy corollary of Theorem \\ref{hamconnected}  we get the following result.\n\n\\begin{corollary}\n\\label{Block-chain}\n Let $G$ be a block-chain. Then $G^2$ is panconnected if and only if every innerblock of $G$ is a 2-block. \n\\end{corollary}\n\nMoreover Corollary \\ref{Block-chain} is also the answer to Problem 1 stated by Chia et al. in \\cite{ChiaOngTan} that for a graph $G$ with only two cutvertices it is true that $G^2$ is pannconnected if and only if the unique block containing the two cutvertices is not the complete graph on two vertices.\n\n\\medskip\n\n\\noindent {\\bf Acknowledgements}.\nThis work was partly supported by the European Regional Development Fund (ERDF), project NTIS - New Technologies for Information Society, European Centre of Excellence, CZ.1.05\/1.1.00\/02.0090.\n\nThe first author was partly supported by project GA20-09525S of the Czech Science Foundation. The second author was supported in part by FWF grant P27615.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Basic Motivation and Structure for the Generalized Uncertainty~Principle}\n\n In this opening section, we review the ideas and motivations underlying the generalized uncertainty principle (GUP) approach to quantum gravity. GUP is a phenomenological approach to quantum gravity which introduces an absolute minimal length in the theory. Many different approaches to combining quantum mechanics and gravity are thought to require a minimal length~\\cite{vene,amati,amati2,gross,maggiore,garay,KMM,adler,scardigli}. There is a simple physical argument for this. From~the Heisenberg uncertainty relationship (i.e.,~$\\Delta x \\Delta p \\ge \\frac{\\hbar}{2}$), one sees that quantum mechanics gives the following relationship between uncertainty in position and momentum $\\Delta x \\sim \\frac{Const.}{\\Delta p}$. From~the gravity side, one argues that as one tries to probe smaller distances, one needs to go to higher center of mass energies\/momenta. At~some point, the energy\/momentum will be large enough that one will form a micro-black hole whose event horizon size can be estimated by the Schwarzschild radius $r_{Sch} = 2 G \\Delta E \/ c^2$, whereas in this expression, $\\Delta E$ has replaced the conventional mass of the black hole, $M$. Now, further setting $c=1$, replacing $r_{Sch}$ with $\\Delta x$ and $\\Delta E$ with $\\Delta p$, this relationship becomes $\\Delta x = 2 G \\Delta p$, i.e.,~there is now a linear relationship between $\\Delta x$ and $\\Delta p$. It should be noted that since we have a set $c=1$ mass, the energy and momentum are interchangeable. If~one combines this linear relationship from gravity with the inverse relationship from quantum mechanics, one finds $\\Delta x \\sim \\frac{\\hbar}{\\Delta p} + G \\Delta p$, (ignoring the factors of 2). The~interplay between the linear term and the inverse term lead to a minimum in $\\Delta x$ at $\\Delta p_m \\sim \\sqrt {\\hbar \/G}$ of $\\Delta x_m \\sim \\sqrt{\\hbar G}$.\n \n The minimal $\\Delta x$ that comes from the GUP, as~described above, is one way to avoid the point singularities that occur in certain solutions in general relativity such as black hole spacetimes. If~one cannot resolve distances smaller than $\\Delta x_m$, this may lead to the avoidance of the singularities of general relativity. This is the hope for theories of quantum gravity---that they will allow one to avoid the singularities of classical general relativity. Another approach to avoid these singularities of classical black hole solutions is non-commutative geometry~\\cite{nicolini}. In~this approach, one proposes that coordinates do not commute with one another. For~example, $[X,Y] \\ne 0$ or $[Y,Z]\\ne 0$. We will later show that the types of GUP models favored by our analysis are also connected to non-commutative geometry~theories. \n \n One of the strengths of the phenomenological GUP approach to quantum gravity is that it offers the possibility to  make experimental tests of quantum gravity---to experimentally check whether there is a minimal distance resolution, $\\Delta x_m$, as~implied by the above arguments, and if so, what is the size of this minimal distance resolution? Some tests of the GUP scenario rely on astrophysical phenomena. For~example, reference~\\cite{AC-nature} proposed a test of minimal lengths based on the dispersion of high-energy photons coming from short gamma ray bursts. The~idea of~\\cite{AC-nature} was that having a minimal distance scale in ones' theory would alter the standard energy--momentum relationship of special relativity, $E^2 = p^2 c^2 + m^2 c^4$. This altered energy--momentum relationship would then lead to an energy-dependent speed of light in the vacuum, which in turn would lead to photons of different energies dispersing or spreading out as they travel long distances through the vacuum. In~2009~\\cite{abdo}, the Fermi gamma ray satellite detected high energy photons coming from a distant gamma ray burst. Using the analysis of~\\cite{AC-nature}, the observation by the Fermi satellite was able to place bounds on the deviations from $E^2 = p^2 c^2 + m^2 c^4$ due to a minimal distance scale. Surprisingly, if the deviations from the special relativistic photon energy and momentum relationship were linear in energy (i.e.,~$p^2 c^2= E^2 [1 + \\zeta (E\/E_{QG})+\\ldots]$ with $E_{QG}$ being the quantum gravity scale and $\\zeta$ is a parameter of order $1$) then the observations of~\\cite{abdo} implied the bound $E_{QG} > E_{Planck}$, i.e.,~that there was no deviation to energies beyond the Planck energy scale. Or~putting this in terms of length $l_{QG} < l_{Planck}$, which is counter to the expectation that hints of quantum gravity should occur before reaching the~Planck-scale.\n \n There are also tabletop and small-scale laboratory test proposals for testing for effects connected with the minimal distance scale coming from GUP. In~the works~\\cite{vagenas,vagenas2}, the proposal was made to use the Lamb shift, Landau levels, quantum tunneling in scanning tunneling microscopes. This work showed than using these tabletop experiments, one could put a bound on the parameter $\\beta$ which were, in~units of Planck momentum squared, of~$\\beta < 10^{36}$, $\\beta < 10^{50}$ and $\\beta < 10^{21}$ for the Lamb shift, Landau levels and tunneling , respectively. There is also a proposal~\\cite{bekenstein} to test Planck scale physics with a tabletop cryogenic optical step-up where the optical photon's momentum is coupled to the center of mass motion of a macroscopic transparent block in such a way that the block is displaced in space by approximately a Planck length. In~the works~\\cite{sujoy,sujoy2,sujoy3}, it was shown that one could use detailed studies of wavefunctions of large molecule to probe Planck-scale physics. Finally there is recent work~\\cite{bosso} which looked at using gravitational waves to put bounds on the parameters coming from GUP models. All of these various experimental approaches are welcome since they hold out the hope that one may experimentally probe Planck-scale~physics.\n\nWe now briefly review some of the basic background behind GUP models and particularly focus on the role that modified operators have in determining whether or not there is a minimal length. The~uncertainty relationship between two physical quantities is closely tied to the commutation relationship between the operators which represent this quantities. In~general, for two operators ${\\hat A}$ and ${\\hat B}$, one has the following relationship between the uncertainties and the commutator:\n\\begin{equation}\n    \\label{dAdB}\n    \\Delta A \\Delta B \\ge \\frac{1}{2i} \\langle [ {\\hat A} , {\\hat B}] \\rangle ~.\n\\end{equation}\nwhere the uncertainties are defined as $\\Delta A = \\sqrt{\\langle {\\hat A} ^2 \\rangle - \\langle {\\hat A} \\rangle ^2 }$ and similarly for $\\langle {\\hat B} \\rangle $.\nFor standard position and momentum operators in position space ${\\hat x} = x$ and ${\\hat p} = -i \\hbar \\partial _x$,  one obtains the usual commutator $[{\\hat x} , {\\hat p}] = i \\hbar$ which then implies the standard uncertainty~principle, \n$$\\Delta x \\Delta p \\ge \\frac{\\hbar}{2}$$ \nor the operators are ${\\hat x} = i \\hbar \\partial _p$ and ${\\hat p} = p$ in momentum~space. \n\nTo obtain a GUP characterized by $\\Delta x \\sim \\frac{\\hbar}{\\Delta p} + \\beta \\Delta p$, \\cite{KMM} proposed the following modified commutator:\n\\begin{equation}\n    \\label{KMMxp}\n    [ {\\hat X} , {\\hat p}] = i \\hbar (1 + \\beta {\\hat p}^2)  ~.\n    \\end{equation}\n \nNote that following~\\cite{KMM}, we replace $G$ by a phenomenological parameter $\\beta$ which characterizes the scale where quantum gravity is important. Naively, the~quantum gravity scale would be set by $G, \\hbar$ and $c$, but~in large extra dimension models~\\cite{ADD,ADD2} or brane world models~\\cite{RS,RS2,Gog,Gog2,Gog3}, the quantum gravity scale can be lower than the typical Planck scale. In~these brane world models, $\\beta$ would be set by the value of the higher dimensional Newton's constant and the size of the extra dimensions. Furthermore, in \\eqref{KMMxp}, the position operator is capitalized while the momentum operators, on~both the left and right sides of \\eqref{KMMxp}, are not. This is because in reference~\\cite{KMM}, the choice was made that in order to obtain the modified commutator in \\eqref{KMMxp}, they would modify the position and momentum operators as\n\\begin{equation}\n\\label{xp1}\n{\\hat X} =i \\hbar (1 + \\beta p^2) \\partial_p ~~~{\\rm and}~~~ {\\hat p} = p  \n\\end{equation}\nwhere only the position operator is modified. This choice of operators in \\eqref{xp1}\nis absolutely crucial to obtaining a minimal length. Using \\eqref{KMMxp} in  \\eqref{dAdB} gives:\n\\begin{equation}\n    \\label{min-Xp}\n    \\Delta X \\Delta p \\ge \\frac{\\hbar}{2}(1+ \\beta \\Delta p ^2)\n\\end{equation}\nIn arriving at \\eqref{min-Xp}, we are taking the center of mass coordinates with $\\langle {\\hat p} \\rangle =0$. Dividing both sides by $\\Delta p$ gives:\n\\begin{equation}\n    \\label{min-x}\n    \\Delta X \\ge \\frac{\\hbar}{2}\\left( \\frac{1}{\\Delta p} + \\beta \\Delta p \\right) ~,\n\\end{equation}\nso that one arrives at the type of GUP described in the opening paragraph which definitely leads to a minimal length. Minimizing $\\Delta X$ in \\eqref{min-x} shows it has a minimal length at \\mbox{$\\Delta p = \\sqrt{1\/\\beta}$} which then yields $\\Delta X_{min} = \\hbar \\sqrt{\\beta}$.\n\n\nIn contrast to the work on GUP from reference~\\cite{KMM}, as outlined above, most recent works have followed a different approach, such as that found in  \\cite{pedram} where the {\\it same} modified commutator of \\eqref{KMMxp} was obtained by modifying the momentum operator {\\it but} not the position operator. By~taking the operators to be of the form:\n\\begin{equation}\n\\label{xp2}\n    {\\hat x}= i \\hbar \\partial _p ~~~{\\rm and}~~~ {\\hat P} = p\\left( 1 + \\frac{\\beta}{3}p^2 \\right)~, \n\\end{equation}\none can see that plugging these into the commutator immediately leads to the same right hand side of the commutator as in \\eqref{KMMxp} where capitals indicate the operator is modified. However, the~uncertainty between the two operators in \\eqref{xp2} is subtly different from the two operators in \\eqref{xp1}. Using the operators in \\eqref{xp2} to calculate the uncertainty relationship via \\eqref{dAdB}~gives:\n\\begin{equation}\n\\label{min-xP}    \n\\Delta x \\Delta P \\ge \\frac{\\hbar}{2}(1+ \\beta \\Delta p ^2).\n\\end{equation}\nAlthough  this relationship superficially appears to be the same as before, now the left hand side has $\\Delta P$ for the modified momentum, while the right hand side has $\\Delta p$ for the standard momentum. Thus, instead of \\eqref{min-x} one obtains:\n\\begin{equation}\n    \\label{min-x2}\n    \\Delta x \\ge \\frac{\\hbar}{2} \\left( \\frac{1}{\\Delta P} + \\frac{\\beta \\Delta p ^2}{ \\Delta P} \\right). \n\\end{equation}\nIn contrast to \\eqref{min-x}, it is not clear whether the interplay between $\\Delta p$ and $\\Delta P$ on the right hand side of \\eqref{min-x2} will lead to a minimum. If~the second term  $\\beta \\frac{\\Delta p^2}{\\Delta P}$ is increasing with $\\Delta p$, then there is a minimum; but if this term decreases with $\\Delta p$, then there is no minimum. Since the highest power of $p$ in ${\\hat P} =p (1 + \\frac{1}{3} \\beta p^2)$ is $p^3$, one can show that $\\Delta P \\approx \\beta \\Delta p ^3$, which implies that the second term $\\beta \\frac{\\Delta p^2}{\\Delta P}$ will go like $\\frac{1}{\\Delta p}$, i.e.,~decreasing with $\\Delta p$ and thus there is no minimum in~position. \n\nTo explicitly see the difference between the GUP from \\eqref{min-x} versus the GUP from \\eqref{min-x2}, one can plot the uncertainty in position versus the uncertainty in momentum in a family of test functions for each operator pair. In~Figure~\\ref{fig1}, we plot $\\Delta X$ versus $\\Delta p$ from \\eqref{min-x} for $\\beta = 0.01$. One can see that there is a minimum in $\\Delta X$ around $\\Delta p = 1\/\\sqrt{\\beta}$ as~expected.\n\\begin{figure}[H]\n    \\includegraphics[scale=0.8]{kmm.jpg}\n    \\caption{The relationship between $\\Delta X$ and $\\Delta p$ for the GUP from \\eqref{min-x} using $\\beta =0.01$. As~expected, a minimum length occurs at approximately $\\Delta p = 1\\sqrt{\\beta}$, and~after this minimum, $\\Delta X$ increases with $\\Delta p$.}\\label{fig1}\n\\end{figure}\nIn Figure~\\ref{fig2}, we plot $\\Delta x$ versus $\\Delta p$ from \\eqref{min-x2} again with $\\beta = 0.01$. In~contrast to Figure~\\ref{fig1}, Figure~\\ref{fig2} has no minimum $\\Delta x$ and in fact looks like the standard relationship between $\\Delta x$ and $\\Delta p$ from quantum~mechanics.  \n\\begin{figure}[H]\n    \\includegraphics[scale=0.8]{pendram.jpg}\n    \\caption{The relationship between $\\Delta x$ and $\\Delta p$ for the GUP from \\eqref{min-x2} using $\\beta =0.01$. This GUP has no minimal length and essentially behaves in a manner resembling the standard uncertainty~principle. }\\label{fig2}\n\\end{figure}\nThere are some subtleties in how Figures~\\ref{fig1} and \\ref{fig2} were obtained. First, for~both figures, we used the lower bound (i.e.,~the equal sign) of Equations \\eqref{min-x} and \\eqref{min-x2}. Second, in~calculating $\\Delta P$ for use with \\eqref{min-x2}, we used a test Gaussian wavefunction in momentum space, $\\Psi (p) \\propto e^{p^2\/2 \\sigma}$. The~reason for using a specific test wave function, rather than making a general argument that would apply regardless of the form of the wavefunction, was the complicated relationship between ${\\hat P}$ and ${\\hat p}$. For~a GUP like that in \\eqref{min-Xp}, one has the same $\\Delta p$ on both the left and right hand side of the equation, regardless of the wavefunction. This is the result of not modifying the momentum operator in this GUP model. On~the other hand for a GUP such as \\eqref{min-xP}, the momentum uncertainty is different between the left and right hand sides of the equation, and~there is not a simple relationship between $\\Delta P$ and $\\Delta p$. Thus, for this case, we picked a specific wavefunction to calculate $\\Delta P$ and $\\Delta p$ and check how the uncertainty principle worked out. One could choose a different test wavefunction instead of the Gaussian, but~the results would be qualitatively similar to that  shown in Figure~\\ref{fig2}. Note that the effects for the type of GUP shown in Figure~\\ref{fig2} are most pronounced in the $\\Delta p \\to 0$ limit. The~above calculations are a prelude and check for the examples of GUPs given in the following~section.  \n\nFinally, even though the momentum operator associated with \\eqref{min-x} is the standard one from quantum mechanics, the~change in the position operator forces one to change the measure of integration in $p$ in order to ensure that ${\\hat X}$ and ${\\hat p}$ are symmetric~\\cite{KMM}. This change in the measure of the momentum integration amounts to $\\int \\ldots dp \\to \\int \\ldots \\frac{dp}{1+ \\beta p^2}$. This modifies the normalization constants but does not qualitatively affect the behavior $\\Delta X$ in the limit as $\\Delta p \\to \\infty$.  All of these issues are more fully examined in~\\cite{BLS}.\n\nThe main point of this section is to show that many works on GUP following the seminal work of KMM~\\cite{KMM} and make a different choice for how the operators are modified, which does not necessarily result in a minimum length scale. KMM~\\cite{KMM} modified the position operator, but~left the momentum operator unchanged, whereas many other works chose to modify the momentum but leave the position operator unchanged. Thus, certain choices of modifying the operators are wrong in the sense that they do not lead to a minimal length. From~the above, we conclude that, when it comes to determining whether a theory has a minimum length scale, {\\bf how the operators are modified is more important than how the commutator is modified}. In~short, a modified commutator is insufficient to guarantee a minimum length scale; in fact, a~modified commutator is not necessary at all, as shown in the next~section.\n \n\n\\section{A GUP with an  Unmodified~Commutator}\n\nWe now move on to give details of how a modified commutator is not necessary to achieve a minimum length scale. If~we do not modify the commutator nor modify the operators, we would obviously end up with ordinary quantum mechanics, which does not have a minimal length. Thus, we want modified position and momentum operators ${\\hat X}$ and ${\\hat P}$ which give rise to the usual commutator $[{\\hat X} , {\\hat P}] = i \\hbar$,  \\cite{BJLS,mlake}. This in turn leads to a standard looking uncertainty relationship but now in terms of the modified operators $\\Delta X \\Delta P \\ge \\frac{\\hbar}{2}$. In~order to have a minimum $\\Delta X$, one needs to modify ${\\hat P}$ so that $\\Delta P$ is either capped at some constant value or decreases. In~reference~\\cite{BJLS}, a~GUP of this kind was constructed by defining a modified momentum by\n\\begin{equation}\n    \\label{p-tanh}\n    {\\hat P} = p_M \\tanh \\left( \\frac{{\\hat p}}{p_M} \\right)~,\n\\end{equation}\nwhere $p_M$ is the maximum cap on the modified momentum. This maximum momentum, $p_M$, is an upper limit to the momentum in the relationship $E^2 = p^2 + m^2$. This cap in momentum also implies a cap in the energy $E$. This connection between a cap on momentum and a cap on the energy of a single particle, as~well as how this relates to modified position and time operators, is discussed in greater detail in~\\cite{BJLS}. Picking the modified position to have the form:\n\\begin{equation}\n\\label{x-tanh}\n   {\\hat X} = i\\hbar\\cosh^2{\\left(\\frac{{\\hat p}}{p_M}\\right)}\\partial_p \n\\end{equation}\nthen leads to the standard looking commutator $[{\\hat X} , {\\hat P}] = i \\hbar$ as can be verified by the substitution of \\eqref{p-tanh} and \\eqref{x-tanh} into the commutator. Another variant with a capped momentum is given in~\\cite{BJLS} by\n\\begin{equation}\n    \\label{p-tan}\n    {\\hat P}  = \\frac{2 p_M}{\\pi } \\arctan \\left( \\frac{\\pi p}{2 p_M} \\right)~,\n\\end{equation}\nand for the modified position operator, one has:\n\\begin{equation}\n    \\label{x-tan}\n    {\\hat X} = i \\hbar \\left[ 1+ \\left( \\frac{\\pi p}{2 p_M}\\right)^2 \\right] \\partial _p ~.\n\\end{equation}\nBoth forms of the modified momenta given in \\eqref{p-tanh} and \\eqref{p-tan} lead to the result $\\Delta P \\le p_M$ which then gives $\\Delta X \\ge \\frac{\\hbar}{2 p_M}$. Despite modified operators \\eqref{p-tanh} and \\eqref{x-tanh} or \\eqref{p-tan} and \\eqref{x-tan} leading to a minimum $\\Delta X$, there is a difference with how this is achieved compared to the KMM modified operators of \\eqref{xp1}. Plotting $\\Delta X$ versus $\\Delta p$ coming from \\eqref{p-tanh}, \\eqref{x-tanh}, the~associated uncertainty principle gives the result shown in Figure~\\ref{fig3}; the graph of $\\Delta X$ versus $\\Delta p$ for the operators \\eqref{p-tan} and \\eqref{x-tan} looks very similar. In~Figure~\\ref{fig1}, which shows the plot of the Kempf--Mangano--Mann GUP~\\cite{KMM}, the~minimum in $\\Delta X$ is reached at some $\\Delta p$ and thereafter $\\Delta X$ linearly increases with $\\Delta p$. In~contrast, the~GUP shown in Figure~\\ref{fig3} asymptotically approaches the minimum $\\Delta X$. \n\\begin{figure}[H]\n    \\includegraphics[scale=0.8]{mup.jpg}\n    \\caption{The relationship between $\\Delta X$ and $\\Delta p$ for the modified operators \\eqref{p-tanh} and \\eqref{x-tanh} and the associated GUP using $p_M =0.25$}\\label{fig3}\n\\end{figure}\nThe GUP model given by Figure~\\ref{fig1} has a physical motivation for its form: $\\Delta x$ inversely proportional to $\\Delta p$ for small momentum where quantum mechanics dominates and has $\\Delta x$ proportional to $\\Delta p$ for a large momentum where quantum gravity is thought to dominate. The~motivation for this behavior was laid out in the introduction and relied on the formation of micro black holes at large momentum\/small distances. In~contrast the GUP model given by Figure~\\ref{fig3}, which has  $\\Delta x$ inversely proportional to $\\Delta p$ for small momentum, but~at a large momentum has $\\Delta x$ approach a constant value asymptotically, does not have a simple, physical~motivation. \n\nThe main take-away message of this section was to emphasize that the most important factor in whether or not a given GUP leads to a minimum length is how the operators are~modified.  \n\n\\section{Connection to Non-Commutative Geometry and Running of \\boldmath$G$}\n\nIn this section, we tie the above considerations about GUPs to another approach to quantizing gravity and a potential physical consequence of quantum gravity. The~other approach to quantizing gravity is non-commutative geometry and the physical consequence is the running of the coupling constant of a theory---in this case, the running of Newton's $G$.\n\nIn general, for non-commutative spacetimes, one has a non-trivial commutation relationship between coordinates of the form $[X_i, X_j] = i \\theta _{ij}$ where $\\theta _{ij}$ is an anti-symmetric matrix~\\cite{nicolini}. This non-commutativity between coordinates implies an uncertainty relationship $\\Delta X_i \\Delta X_j \\ge \\frac{1}{2} |\\theta _{ij}|$, which in turn implies that there is a minimal area and volume---one cannot make $\\Delta X \\Delta Y$ or $\\Delta X \\Delta Y \\Delta Z$ (for example) arbitrarily small due to the non-commutativity between the coordinates. This has the effect, similar to GUP models, of~preventing the formation of point singularities that occur in a black hole and other solutions of classical general relativity. It has previously been noted~\\cite{KMM} that certain GUP models lead to modified position operators in three dimensions (3D)  which naturally result in the non-commutativity of the modified position operators. In~one spatial dimension (1D), one does not encounter this non-commutativity since a single operator will always commute with itself. However, in~three dimensions, different coordinates may not commute with one another, e.g.,~$[{\\hat X} , {\\hat Z}] \\ne 0$. This is the reason behind the matrix $\\theta _{ij}$ being~antisymmetric. \n\nAs a specific example of the GUP-non-commutative geometry connection, one can look at the 3D version of the modified position operators of~\\cite{KMM}. Letting the position and momentum in \\eqref{xp1} go from 1D to 3D (i.e.,~${\\hat X} \\to {\\hat X}_i$, ${\\hat p} \\to {\\hat p}_i$ and $\\partial_p \\to \\partial _{p_i}$ gives a 3D version of \\eqref{xp1} of the form:\n\\begin{equation}\n\\label{x13d}\n{\\hat X}_i =i \\hbar (1 + \\beta |{\\vec p}|^2) \\partial_{p_i} ~.\n\\end{equation}\nWith this 3D version of \\eqref{xp1}, the coordinate commutator becomes:\n\\begin{eqnarray}\n\\label{x13d-a}\n[{\\hat X}_i , {\\hat X}_j ] &=& -2 \\beta \\hbar ^2 (1 + \\beta |{\\vec p}|^2 \\partial_{p_j} ) (p_i \\partial _{p_j} - p_j \\partial _{p_i} )\\nonumber \\\\\n&=& 2 i \\hbar \\beta ({\\hat p}_i {\\hat X}_j  -{\\hat p}_j {\\hat X}_i )  ~,\n\\end{eqnarray}\nwhich implies a connection to the non-commutative parameter of $\\theta_{ij} =  2 \\hbar \\beta ({\\hat p}_i {\\hat X}_j  -{\\hat p}_j {\\hat X}_i )$. As~required, this $\\theta_{ij}$ is antisymmetric. In~the second line in \\eqref{x13d-a}, we used \\eqref{x13d} to turn this back into an expression in terms of the (modified) position operator and (unmodified) momentum operator. The~result in \\eqref{x13d-a} was previously derived in~\\cite{KMM}.\n\nOne can also create 3D versions of the modified position operators from \\eqref{x-tanh} and \\eqref{x-tan} which take the form:\n\\begin{equation}\n\\label{x-tanh3d}\n   {\\hat X}_i = i\\hbar\\cosh^2{\\left(\\frac{|{\\vec p}|}{p_M}\\right)}\\partial_{p_i} ~,\n\\end{equation}\nand:\n\\begin{equation}\n    \\label{x-tan3d}\n    {\\hat X}_i = i \\hbar \\left[ 1+ \\left( \\frac{\\pi |{\\vec p}|}{2 p_M}\\right)^2 \\right] \\partial _{p_i} ~.\n\\end{equation}\nJust as in the case of the 3D modified position operators in \\eqref{x13d}, the~modified 3D position operators in \\eqref{x-tanh3d} and \\eqref{x-tan3d} also lead to a non-commutativity of the coordinates,  $[{\\hat X}_i, {\\hat X}_j] \\ne 0$. We do not give the specific form of $\\theta_{ij}$ for the modified 3D position operators from \\eqref{x-tanh3d} and \\eqref{x-tan3d} since we only want to make the point here that some GUPs (particularly those studied in this work) lead to non-commutative~geometry.\n\nIn quantum field theory, when interactions are quantized, one encounters the phenomenon that the coupling ``constants'' of the interaction become dependent on the energy\/momentum scale at which the effects of the interaction are measure. Colloquially one talks about  coupling constant ``running'' with the energy\/momentum scale. For~example, in~quantum electrodynamics, the fine structure constant, $\\alpha = \\frac{e^2}{\\hbar c}$, which measures the strength of the electromagnetic interaction, depends on the energy\/momentum scale, $E\/p$, at~which it is measured. Similarly, the weak and strong nuclear interactions have couplings which scale with energy\/momentum. \nOne can view the GUP approach to quantum gravity to be as running from Newton's constant $G$.  The~commutator from \\eqref{KMMxp} implies that the GUP parameter  $\\beta$ depends on the momentum of the interaction as $\\beta (p) = \\beta p^2$, i.e.,~the parameter $\\beta$ scales quadratically with momentum in this~case. \n\nTo translate this running of the phenomenological parameter $\\beta$ into a running of Newton's constant $G$. We simply recall that from the heuristic arguments $\\Delta x_m \\sim \\sqrt{\\hbar G}$ as well as in terms of $\\beta$, one has $\\Delta x_m \\sim \\hbar \\sqrt{\\beta}$. Thus, we have the connection between $G$ and $\\beta$ of $\\beta \\sim \\frac{G}{\\hbar}$. Thus, a $\\beta$ which ``runs'' with the momentum directly implies a running $G$. There are two things to note: (i) the usual running of a coupling like the fine structure parameter $\\alpha = \\frac{e^2}{\\hbar c}$ is usually logarithmic in perturbative quantum fields theory; (ii) there are differences between gravity and the other interactions that make it unclear that one can consistently define a running gravitational coupling, at~least in the usual approach of quantum fields theory, as detailed in the arguments of reference~\\cite{anber}. \n\n\\section{Summary and Conclusions}\n\nIn this short note, we examined the role that modifying the position and momentum operators plays in determining a minimum length and that focusing on modifying the commutator is insufficient. We examined models that had modified commutators with the same right hand sides, as~in \\eqref{KMMxp}, but~where the operators on the left hand side were different. We considered the case where the position was modified and the momentum remained unchanged (see \\eqref{xp1}) and we considered the case where the position remained the same and the momentum was modified (see \\eqref{xp2}). The~former case led to a minimum length while the latter case did not. This can be explicitly  seen by comparing \\mbox{Figures~\\ref{fig1} and \\ref{fig2}}. \nFinally, we presented a case where the position and momentum were modified, but~the commutator remained the same as the standard one from quantum mechanics---\\mbox{Equations \\eqref{p-tanh} and \\eqref{x-tanh}}. This system show that there can be a minimum length scale without modifying the commutation~relation.\n\nA general conclusion that can be distilled from the work in this paper is the following. {\\bf In order for a GUP to have a minimal length, the key factor is the modification of the operators rather than the modification of the commutator.} \n\nThe thrust of this paper was to argue that there are constraints on the specifics of how one can formulate a GUP to obtain a minimal length. This is welcome since while the phenomenological approach of GUPs has strengths (e.g.,~the possibility to confront ideas about quantum gravity with experiments and observations), one would also like a way to constrain and focus on the range of variations in the operators and commutator. Other recent works~\\cite{mlake1,mlake2} have also looked at ways to constrain the form of~GUPs. \n\nFinally, in the last section, we tied the type of GUP models discussed here with other models of quantum gravity and with other potential consequences of quantum gravity, namely non-commutative geometry and the ``running'' of the coupling strengths of interactions. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum computers have attracted much attention recently, mainly due to the rapid development of actual hardware \\cite{Barends2014, Bernien2017, Wright2019}.\nThe quantum computer that is to appear shortly is called noisy intermediate scale quantum devices, or in short, NISQ devices \\cite{Preskill2018}.\nWe expect NISQ devices to have $\\sim$100 of qubits with non-negligible noise in the near future.\nSuch devices are believed to be not simulatable by classical computers when the control precision of the qubits is sufficiently high \\cite{Harrow2017, Boixo2018, Neill2018, Bravyi2018}.\nIn this sense, NISQ devices have computational power that exceeds classical computers.\nMany researchers are actively developing ways to exploit their power for practical applications \\cite{Peruzzo2014, Kandala2017, Nam2019, Farhi2014, Otterbach2017, Mitarai2018, Havlicek2019}.\nHowever, we still suffer from the limited number of qubits available on actual devices and the limited depth of circuits that can be run while maintaining the resultant quantum state meaningful.\n\nIf techniques to decompose a quantum circuit to a smaller one are developed, they can extend the applicability of such devices.\nSmaller quantum circuits may refer to ones with the smaller number of qubits or gates.\nPeng \\textit{et al.} recently proposed a clustering approach based on a tensor network representation of a quantum circuit~\\cite{Peng2019}, which greatly progressed the technical development.\nThey showed that we can ``cut'' an identity gate, by sampling measure-and-prepare channels on a qubit according to a certain quasi-probability distribution.\nIn Ref.~\\cite{Mitarai2019}, we proposed methods to construct quantum circuits equivalent to the Hadamard test, which successfully reduces the depth of certain quantum circuits.\nThese techniques share a same idea in that they reconstruct a result of a coherent quantum operation from certain incoherent operations by combining the results obtained from them.\n\nAn approach which has the same flavor as the above have been utilized in the context of memory-efficient classical simulation of quantum circuits.\nSince the direct simulation of a quantum circuit with over 50 qubits breaks down due to the need of storing $2^{50}$ complex numbers in memory, the classical simulator must decompose the given quantum circuit to smaller ones, especially in the number of qubits.\nRefs. \\cite{Chen2018, Pednault2017} have provided one way for such decomposition, which ``cuts'' controlled-Z gates by separately simulating two cases where the control qubit is $\\ket{0}$ or $\\ket{1}$ and then combining them, and they performed classical simulation of over 50-qubit quantum circuits.\nA similar technique has been utilized by Bravyi \\textit{et al.} in Ref. \\cite{Bravyi2016} to remove a relatively small number of qubits from a large quantum circuit by replacing the qubits with a classical simulator.\nTheir approach can be viewed as ``space-like'' cut rather than the ``time-like'' cut proposed by Peng \\textit{et al} \\cite{Peng2019}.\nHowever, their techniques are intended to run on a classical computer and cannot be utilized for simulating a large quantum circuit with a small quantum computer.\n\nIn this work, we present a technique to perform ``space-like'' cut on a quantum computer.\nMore specifically, we present a way to decompose a controlled gate into a sequence of single-qubit operations which consists of projective measurements of Pauli $X$, $Y$, and $Z$ operators, and single-qubit rotations around x, y, and z-axes.\nWe note that our method does not generate any entanglement between the qubits as it is impossible to do so with such single-qubit operations.\nOur method only ``simulates'' effects of entanglement using classical post-processing and sampling.\nMore concretely, although entangling gates cannot be performed with local operations and classical communications in single-shot experiments as widely known \\cite{PhysRevA.52.3457}, we show that it is possible to perform a computational task of evaluating expectation values of the output of entangling circuits by sampling certain sets of gates and applying classical post-processing.\nThe overhead required for our proposed technique, which scales exponentially to the number of decomposition performed, gives a characterization of the entangling gates from a computational viewpoint, which is different from the existing theories of entanglement quantification in e.g. \\cite{Vidal_2000}.\n\nThe method proposed here can be considered as a generalization of our previous work \\cite{Mitarai2019} and a variant of the quantum circuit decomposition presented in Ref.~\\cite{Peng2019}.\nIt can also be viewed as a fully quantum version of the technique utilized in efficient classical simulation schemes \\cite{Chen2018, Pednault2017, Bravyi2016}. \nIn some cases, our method provides a better scaling against Ref. \\cite{Peng2019} when simulating a large quantum circuit with smaller ones.\nThe proposed technique is also useful when we want to apply two-qubit gates between a distant pair of qubits, which otherwise would require many swap operations to perform.\nThis work extends the applicability of NISQ devices whose circuit depth and connectivity are limited.\n\n\\section{Gate decomposition}\n\\subsection{Tensor network representation of quantum circuits}\nQuantum computation is completely specified with a quantum circuit, $U$, an initial state with its density matrix representation, $\\rho$, and an observable, $O$, measured at the output.\nGiven $U$, $\\rho$, and $O$, Any quantum computation can be represented by a tensor network \\cite{Shi2006,Markov2008,Vidal2003}.\nWe define the tensor representation of $U$, $\\rho$, and $O$ in the following manner.\n\nSuppose that our quantum computer has $n$ qubits.\nWe define a complete set of basis in the space of $2 \\times 2$ complex matrix and its dual as $\\{\\kket{e_i}\\}_{i=1}^{4}$ and $\\{\\bbra{e_i}\\}_{i=1}^{4}$ respectively, and assume orthonormality under the trace inner product; $\\bbraket{e_i|e_j}=\\delta_{ij}$.\nWe use the trace inner product, that is, for matrices $A$ and $B$, $\\bbraket{A|B} = \\mathrm{Tr}(A^\\dagger B)$.\nA density matrix $\\rho$ can be decomposed into the sum of $\\kket{e_{j_1}}\\otimes \\kket{e_{j_2}} \\otimes \\cdots \\otimes \\kket{e_{j_n}} = \\kket{e_{j_1}e_{j_2}\\cdots e_{j_n}}$ as \n\\begin{align}\n    \\kket{\\rho} = \\sum_{j_1,\\cdots, j_n} \\rho_{\\bm{j}} \\kket{e_{j_1}e_{j_2}\\cdots e_{j_n}},\n\\end{align}\nwhere $\\bm{j} = (j_1,j_2,\\cdots,j_n)$. \nWe refer to the elements $\\rho_{\\bm{j}} = \\bbraket{e_{j_1}e_{j_2}\\cdots e_{j_n}|\\rho}$ as the tensor representation of $\\rho$.\nAn observable $O$ can also be decomposed into the same form.\nNote that we can naturally assume tensor representations of observables and density matrices consist of real numbers because they are always Hermitian and we can choose the basis $\\{\\kket{e_i}\\}_{i=1}^{4}$ as Hermitian, e.g., we can use the Pauli matrices $\\{I, X, Y, Z\\}$ as the basis.\nTherefore, we assume $\\rho_i$ and $O_i$ are real henceforth.\nThe quantum circuit, $U$, transforms $\\rho$ into $U\\rho U^\\dagger$.\nWe define a corresponding superoperator $\\mathcal{S}(U)$ whose action is defined by $\\mathcal{S}(U) \\rho  = U\\rho U^\\dagger$.\nSuperoperator can be decomposed as,\n\\begin{align}\n    \\mathcal{S}(U) = \\sum_{j_1,\\cdots, j_n}\\sum_{k_1,\\cdots, k_n} \\mathcal{S}(U)_{\\bm{j}, \\bm{k}} \\kket{e_{j_1}\\cdots e_{j_n}}\\bbra{e_{k_1}\\cdots e_{k_n}}.\n\\end{align}\nNote that this decomposition is not limited to superoperators of unitary matrices, but also is applicable for any linear operator that acts on a density matrix.\nWe call $\\mathcal{S}(U)_{\\bm{j}, \\bm{k}} = \\bbra{e_{j_1}\\cdots e_{j_n}}\\mathcal{S}(U)\\kket{e_{k_1}\\cdots e_{k_n}}$ tensor representation of $\\mathcal{S}(U)$. \nWhen we use the Pauli operators as basis set, $\\mathcal{S}(U)_{\\bm{j}, \\bm{k}}$ is refered as Pauli transfer matrix.\n\nQuantum computation ends with measuring the observable $O$.\nThis output can be written down as,\n\\begin{align}\n    \\bbra{O}\\mathcal{S}(U)\\kket{\\rho} &= \\mathrm{Tr}(O U\\rho U^\\dagger) \\\\\n    &= \\sum_{j_1,\\cdots, j_n}\\sum_{k_1,\\cdots, k_n} O_{\\bm{j}}\\mathcal{S}(U)_{\\bm{j}, \\bm{k}} \\rho_{\\bm{k}},\n\\end{align}\nIn many cases, $U$ is a product of elementary gates $\\{U_i\\}_{i=1}^L$, that is, $U=U_L\\cdots U_1$.\nThe tensor representation of the overall gate, $\\mathcal{S}(U)$, is also a product of $\\mathcal{S}(U_i)$; $\\mathcal{S}(U) = \\mathcal{S}(U_L)\\cdots\\mathcal{S}(U_1)$.\nAn important note is that as long as the tensor representation of each element is unchanged, the result of the overall computation is also unchanged.\nIf $\\mathcal{S}(U)$ can be represented by a sum of some simple operations as $\\mathcal{S}(U) = \\sum_i c_i \\mathcal{S}(V_i)$ with coefficients $\\{c_i\\}$, the expectation value of an observable $O$ can be computed with the following equality,\n\\begin{equation}\\label{eq:superop_sum_decomposition}\n    \\bbra{O}\\mathcal{S}(U)\\kket{\\rho} = \\sum_i c_i \\bbra{O}\\mathcal{S}(V_i)\\kket{\\rho}.\n\\end{equation}\nNote that $c_i$ can, in general, depend on the state $\\kket{\\rho}$.\nWe use this scheme to perform the ``decomposition'' of a circuit in this work.\n\nIt is noteworthy that as we perform decompositions of a superoperator rather than an operator such as $U$ itself, the method becomes friendly for a realistic quantum device.\nA direct decomposition of $U$ into some simple operators $\\{V_i\\}$, i.e. $U=\\sum_{i} c_i V_i$, can also be utilized for the same task; however, as expectation values are calculated as $\\bra{0}U^\\dagger O U\\ket{0}$ where $\\ket{0}$ is an initial state, this approach requires us to evaluate $\\sum_{i,j} c_ic_j^* \\bra{0}V_j^\\dagger O V_i\\ket{0}$ which are rather hard for the NISQ devices.\nThis fact demonstrates the advantage of using the above formalism.\nThe tensor network representation of the superoperator formalism allows us to graphically understand the decompositions.\n\n\n\\subsection{Virtual two-qubit gate}\nWe can show the following, which can then be utilized to decompose any two-qubit gate into a sequence of single-qubit operations.\n\\begin{lemma}\\label{thm:two_to_single}\n    For operators $A_1$ and $A_2$ such that $A_1^2=I$ and $A_2^2=I$,\n    \\begin{align}\n        &\\mathcal{S}(e^{i\\theta A_1\\otimes A_2}) = \\cos^2\\theta \\mathcal{S}(I\\otimes I) + \\sin^2\\theta \\mathcal{S}(A_1\\otimes A_2)+ \\nonumber\\\\\n        &\\frac{1}{8}\\cos\\theta\\sin\\theta\\sum_{(\\alpha_1,\\alpha_2)\\in\\{\\pm 1\\}^2} \\alpha_1 \\alpha_2\\left[\\mathcal{S}((I+ \\alpha_1 A_1)\\otimes (I+ i\\alpha_2  A_2)) \\right. \\nonumber\\\\\n        &\\qquad\\qquad\\qquad\\qquad\\qquad \\left.+ \\mathcal{S}((I+ i\\alpha_1 A_1)\\otimes (I+ \\alpha_2  A_2))\\right]\n    \\end{align}\n\\end{lemma}\n\n\\begin{figure*}\n    \\includegraphics[width=\\linewidth]{two_qubit_operations_cut_wide_crop.pdf}\n    \\caption{\\label{fig:two_qubit_gate_cut} Decomposition of (a) a non-local gate and (b) a non-local non-destructive measurement into a sequence of local operations. $A_1$ and $A_2$ are operators such that $A_1^2 = I$ and $A_2^2 = I$.\n    }\n\\end{figure*}\n\nTo prove this, we can directly check the tensor representation of both hand side is equivalent. For detailed calculation, see Appendix \\ref{app:two_to_single}.\nThis theorem is schematically depicted in Fig. \\ref{fig:two_qubit_gate_cut} (a).\nNotice that the operation that is proportional to $I\\pm A$ and $I \\pm iA$ for $A \\in \\{X, Y, Z\\}$ can respectively be performed by a projective measurement and a single-qubit rotation.\n\nThe correspondence with a single-qubit rotation is clear from the formula, $e^{\\pm i\\pi A\/4} = \\frac{1}{\\sqrt{2}}(I\\pm iA)$, which is the rotation of angle $\\pi\/2$ around the $A$ axis.\nLet $\\mathcal{M}_A$ be the projective measurement on the $A$ basis ($A\\in \\{X,Y,Z\\}$), that is, $\\mathcal{M}_A$ acts on a density matrix $\\rho$ as,\n\\begin{align}\n    \\mathcal{M}_A\\rho &= \\frac{1}{\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right)} \\left(\\frac{I+\\alpha A}{2}\\right)\\rho \\left(\\frac{I+\\alpha A}{2}\\right),\n\\end{align}\ndepending on the result of the measurement $\\alpha \\in \\{1,-1\\}$.\nThis is equivalent to $\\mathcal{S}(I\\pm A)$ up to the factor of  $4\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right)$, that is,\n\\begin{align}\\label{eq:corresp_measurement}\n    \\mathcal{S}(I + \\alpha A) &= 4\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right) \\mathcal{M}_{A,\\alpha},\n\\end{align}\nwhere $\\mathcal{M}_{A,\\alpha}$ is a measurement operation postselected with the measurement outcome $\\alpha$.\n$\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right)$ is the probability of getting the result $\\alpha$ by measuring $\\rho$ on the $A$ basis.\nLemma \\ref{thm:two_to_single} with this fact implies that the gate $e^{i\\theta A_1\\otimes A_2}$ can be decomposed, in a sense of Eq.~(\\ref{eq:superop_sum_decomposition}), into a sum of $I\\otimes I$, $A_1\\otimes A_2$, $\\mathcal{M}_{A_1}\\otimes e^{\\pm i\\pi A_2\/4}$, and $e^{\\pm i\\pi A_1\/4}\\otimes \\mathcal{M}_{A_2}$, which can be stated as Lemma below.\nNotably, this technique can be applied for any $\\theta$, which enables us to perform continuous two-qubit gates.\n\\begin{lemma}\\label{thm:two_qubit_gate_decomposition}\n    A quantum gate $e^{i\\theta A_1\\otimes A_2}$ with operators $A_1$ and $A_2$ such that $A_1^2=I$ and $A_2^2=I$ can be decomposed into 6 single-qubit operations.\n    For any quantum state $\\kket{\\rho}$, to achieve the error $\\epsilon$ of the decomposition with respect to the trace distance with probability at least $1-\\delta$, the required number of circuit runs is $O(\\log (1\/\\delta)\/\\epsilon^2)$.\n\\end{lemma}\nThe detailed proof is given in Appendix \\ref{appsec:proof_of_two_qubit_gate_decomp}.\nIntuitively, since the error comes from the probabilistic part of the decomposition, that is the renomalization factor in Eq. (\\ref{eq:corresp_measurement}) $\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right)$,  if we want to estimate $\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right)$ within error $\\epsilon$, $O(1\/\\epsilon^2)$ repetition would suffice.\n\nLet us finally mention the case of the controlled-Z gate, which we denote by $CZ$.\n$CZ$ can be decomposed into\n\\begin{equation}\n    CZ = e^{i\\pi I\\otimes Z\/4}e^{i\\pi Z\\otimes I\/4}e^{-i\\pi Z\\otimes Z\/4},\n\\end{equation}\nignoring the global phase.\nThis means we can decompose a CZ gate using Lemma \\ref{thm:two_qubit_gate_decomposition}.\nThe decomposition is shown in Fig.~\\ref{fig:cz_gate_cut}.\nSimilar decompositions can be performed on some basic two-qubit gates such as CNOT.\nEndo \\textit{et al.} \\cite{Endo2018} also provides such decomposition (Ref. \\cite{Endo2018}, Appendix B).\nHowever, our protocol above is slightly advantageous in that the number of single-qubit operations required is 6 compared to theirs which requires 9 of them.\n\n\\begin{figure*}\n    \\includegraphics[width=0.8\\linewidth]{cz_gate_cut_wide_crop.pdf}\n    \\caption{\\label{fig:cz_gate_cut} Decomposition of controlled-Z gate into a sequence of single-qubit operations.\n    }\n\\end{figure*}\n\n\n\n\n\\subsection{Virtual non-destructive measurement of two-qubit operators}\nIn the previous subsection, we showed that any two-qubit rotation can be decomposed into a sum of single-qubit operations.\nHere, we extend the strategy to construct virtual non-destructive measurement\nof two-qubit operators.\nSimilar to the previous section, we can show the following.\nThis theorem is schematically shown in Fig.~\\ref{fig:two_qubit_gate_cut}~(b).\n\\begin{lemma}\\label{thm:two_to_single_meas}\n    For operators $A_1$ and $A_2$ such that $A_1^2=I$ and $A_2^2=I$,\n    \\begin{align}\n        &\\mathcal{S}(I + A_1\\otimes A_2) = \\mathcal{S}(I\\otimes I) + \\mathcal{S}(A_1\\otimes A_2)+ \\nonumber\\\\\n        &\\frac{1}{8}\\sum_{(\\alpha_1, \\alpha_2)\\in\\{\\pm 1\\}^2} \\alpha_1 \\alpha_2\\left[\\mathcal{S}((I+ \\alpha_1 A_1)\\otimes (I+ \\alpha_2  A_2)) \\right. \\nonumber\\\\\n        &\\qquad\\qquad\\qquad \\left.- \\mathcal{S}((I+ i\\alpha_1 A_1)\\otimes (I+ i\\alpha_2  A_2))\\right]\n    \\end{align}\n\\end{lemma}\nThis can also be shown by the direct calculation of both hand side. See Appendix \\ref{app:two_to_single_meas} for detailed calculation.\n\nThe above Lemma can be utilized to show the following.\n\\begin{lemma}\\label{thm:twoq_meas_decomp}\n    A non-local projection $\\frac{\\red{I}+A_1\\otimes A_2}{2}$ with operators $A_1$ and $A_2$ such that $A_1^2=1$ and $A_2^2=2$ can be decomposed into 6 single-qubit operations.\n    For any quantum state $\\kket{\\rho}$, to achieve the error $\\epsilon$ of the decomposition with respect to the trace distance with probability at least $1-\\delta$, the required number of circuit runs is $O(\\log (1\/\\delta)\/\\epsilon^2)$.\n\\end{lemma}\nThis can be shown with exactly the same approach taken to prove Lemma \\ref{thm:two_qubit_gate_decomposition}, which is provided in Appendix \\ref{appsec:proof_of_two_qubit_gate_decomp}.\n\n\n\n\n\\section{Application}\n\\subsection{Simulation of large quantum circuits}\\label{sec:simulation}\nThe idea of simulating a large quantum circuit by a small quantum computer has been put forward in Ref.~\\cite{Peng2019}.\nPeng \\textit{et al.} utilized the equivalence shown in Fig. \\ref{fig:identitygatecut}.\nIn the figure,\n\\begin{equation}\\label{eq:Pengcut_pairs}\n\\begin{array}{lll}\n    O_1=I, & \\rho_1 = \\ket{0}\\bra{0}, & c_1 = +1\/2, \\\\\n    O_2=I, & \\rho_2 = \\ket{1}\\bra{1}, & c_2 = +1\/2, \\\\\n    O_3=X, & \\rho_3 = \\ket{+}\\bra{+}, & c_3 = +1\/2, \\\\\n    O_4=X, & \\rho_4 = \\ket{-}\\bra{-}, & c_4 = -1\/2, \\\\\n    O_5=Y, & \\rho_5 = \\ket{+i}\\bra{+i}, & c_5 = +1\/2, \\\\\n    O_6=Y, & \\rho_6 = \\ket{-i}\\bra{-i}, & c_6 = -1\/2, \\\\\n    O_7=Z, & \\rho_7 = \\ket{0}\\bra{0}, & c_7 = +1\/2, \\\\\n    O_8=Z, & \\rho_5 = \\ket{1}\\bra{1}, & c_8 = -1\/2, \n\\end{array}\n\\end{equation}\nwhere $\\ket{\\pm}=(\\ket{0}\\pm \\ket{1})\/\\sqrt{2}$ and $\\ket{\\pm i}=(\\ket{0}\\pm i\\ket{1})\/\\sqrt{2}$. \nThe symbols $\\triangleright$ and $\\triangleleft$ denotes the measurement of a certain observable and the preparation of a certain state, respectively.\nContrasting this technique and ours, we refer to the former and the latter as ``time-like'' and ``space-like'' cut, respectively.\nMore concretely, \\textit{a time-like cut} of a quantum channel can be defined as a decomposition  of the channel in the sense of Eq. (\\ref{eq:superop_sum_decomposition}) using measure-and-prepare channels only.\nIn contrast, \\textit{a space-like cut} of a non-local quantum channel is a decomposition of the channel using local quantum channels only.\n\n\n\\begin{figure}\n    \\includegraphics[width=0.7\\linewidth]{identitygatecut_crop.pdf}\n    \\caption{\\label{fig:identitygatecut} Time-like cut employed in Ref.~\\cite{Peng2019}.}\n\\end{figure}\n\n\\begin{figure}\n    \\includegraphics[width=\\linewidth]{simpleclustering_crop.pdf}\n    \\caption{\\label{fig:simpleclustering} Two decomposition approach compared in main text. The top-right approach is the presented, and the bottom-right approach is of Ref.~\\cite{Peng2019}.}\n\\end{figure}\n\nThe decomposition presented in the previous section can also be used in this direction.\nLet us compare the scaling of cost of our decomposition scheme and that of Peng \\textit{et al.} by a simple example.\nWe consider the case where we have an $n$-qubit quantum computer to simulate a $2n$-qubit quantum circuit of Fig.~\\ref{fig:simpleclustering}, which has only one CZ gate between n-qubit ``cluster''.\nThe task is to estimate the expectation value of a final observable $O_f$ by measuring it in the computational basis.\nTo simplify the discussion, we assume $O_f$ is a string of Pauli $Z$'s.\n\nLet $v$ be a desired variance of the estimation of the expectation value of $O_f$.\nWe can show a naive algorithm, which runs the equal number of circuits for each terms appearing in the decomposition, to perform the decomposition with time-like cuts, in the worst case, requires $2048\/v$ runs of $n$-qubit circuit, while the space-like cut approach takes $\\frac{15}{2v}$ runs.\nThe analysis of this simple example is given in Appendix \\ref{app:simpleexample}.\nAlthough the analysis given here is based on a naive algorithm and there are possibilities to improve it, this analysis somewhat shows the enhancement provided by our space-like cut protocol.\n\n\n\n\\subsubsection*{General case}\nWe can consider a general case where we perform the time-like and space-like cuts simultaneously to make a given $m$-qubit quantum circuit runnable on an $n$-qubit quantum computer.\nLet the number of time-like and space-like cuts be $M_t$ and $M_s$, respectively.\nFor space-like cuts, we assume they are performed only on CZ gates.\nThe input state $\\rho$ is initialized in $\\ket{0}\\bra{0}^{\\otimes m}$ and $O_f$ is an output (diagonal) observable calculated from some output function $f:\\{0,1\\}^m\\to [-1,1]$.\nOur task here is to estimate the expectation $\\mathbb{E}[f(y)]$ for a random bitstring $y\\in\\{0,1\\}^m$ sampled from the original circuit.\nThis model is adopted from Ref. \\cite{Peng2019} which originates in Ref. \\cite{Bravyi2016}.\nWith this definition, we can get the following.\n\n\\begin{theorem}\\label{thm:multiple_twoqgate_cut}\n    The number of $n$-qubit circuit runs required to estimate $\\mathbb{E}[f(y)]$ within accuracy $\\epsilon$ with some high probability $1-\\delta$ is $O\\left(\\frac{9^{M_s} 16^{M_t}}{\\epsilon^2}\\log\\left(\\frac{1}{2\\delta}\\right)\\right)$.\n\\end{theorem}\nThis implies that the decomposition of the circuit should be performed to minimize $9^{M_s} 16^{M_t}$.\nA detailed proof is given in Appendix \\ref{appsec:proof_of_multiple_twoqgate_cut}, however, the above can roughly be explained as follows.\nAt each space-like cut, we get 6 different sets of single-qubit operations, so $M_s$ cuts induce $6^{M_s}$ terms.\nLikewise, $M_t$ time-like cuts induce $8^{M_t}$ terms, which makes the total number of circuits in decomposition $6^{M_s}8^{M_t}$.\nWith this decomposition, we can take a Monte-Carlo approach to estimate the sum, that is, we randomly choose circuits to run and average them.\nHoeffding's inequality can be used to bound the error of such protocol, which states that if a magnitude of a random variable is always bounded by some constant $a$, then $O(a^2\/\\epsilon^2)$ samples would suffice to obtain an accuracy of $\\epsilon$.\nIn this case, we are to estimate $\\mathbb{E}[f(y)] = \\sum_{i=1}^{6^{M_s}8^{M_t}} c_i \\bbra{O_f}\\mathcal{S}(V_i)\\kket{\\rho}$ with $i$ randomly drawn from $\\{1,\\cdots,6^{M_s}8^{M_t}\\}$ and $|c_i|=1\/2^{M_s+M_t}$, that is, $\\mathbb{E}[f(y)]$ is estimated by $\\mathbb{E}_i[6^{M_s}8^{M_t} c_i\\bbra{O_f}\\mathcal{S}(V_i)\\kket{\\rho}]$.\nThe magnitude of random variable $6^{M_s}8^{M_t} c_i\\bbra{O_f}\\mathcal{S}(V_i)\\kket{\\rho}$ is roughly $3^{M_s}4^{M_t}$, thus we can apply the Hoeffding bound to get the result.\n\n\\begin{figure}\n    \\includegraphics[width=\\linewidth]{combination_crop.pdf}\n    \\caption{\\label{fig:combination}Schematic illustration of performing the space-like cut and the time-like cut simultaneously.}\n\\end{figure}\n\n\\subsection{Distant two-qubit gates}\nThe theorem introduced above can be utilized to ``virtually'' perform a two-qubit gate between qubits at distance.\nFigure \\ref{fig:distant_2qubit_gate} shows an example of such a virtual two-qubit gate.\nNotice that this protocol works irrespective of the distance between the qubits.\nMany swap gates are otherwise necessary for performing such gates, which makes them impractical on NISQ devices due to the non-negligible amount of decoherence and gate error of such devices.\n\n\\begin{figure}\n    \\includegraphics[width=\\linewidth]{distant_2qubit_gate_crop.pdf}\n    \\caption{\\label{fig:distant_2qubit_gate} Decomposition of distant two-qubit gate on a square lattice. \n    Each vertex of the graph represents a qubit and the edge represents the connectivity of the qubits.\n    $S$ is the set of pairs of single-qubit operations which appears in the formula in Lemma \\ref{thm:two_to_single}, and $c_s$ is the corresponding coefficient for each pair.}\n\\end{figure}\n\nThis protocol might be useful for the variational algorithms such as the variational quantum eigensolver (VQE) \\cite{Peruzzo2014} and the quantum approximate optimization algorithms (QAOA) \\cite{Farhi2014}.\nHere, we describe an example in the QAOA.\nIn the QAOA, we seek to find a ground state of a Hamiltonian $H$ on $n$-qubit which is a sum of Pauli $Z$'s and its products.\nFor example, a Hamiltonian may have the form of,\n\\begin{align}\n    H = \\sum_{ij} J_{ij} Z_iZ_j.\n\\end{align}\nThe QAOA tries to solve the problem by converting it to a optimization problem of a continuous variable $\\bm{\\beta}$ and $\\bm{\\gamma}$.\nThe optimization of $\\bm{\\beta}$ and $\\bm{\\gamma}$ are performed so as to minimize the function,\n\\begin{align}\n    \\expect{H(\\bm{\\beta}, \\bm{\\gamma})} =\\bra{+}^{\\otimes n} U^\\dagger(\\bm{\\beta}, \\bm{\\gamma}) H U(\\bm{\\beta}, \\bm{\\gamma})\\ket{+}^{\\otimes n},\n\\end{align}\nwhere,\n\\begin{align}\\label{eq:qaoa-circuit}\n    U(\\bm{\\beta}, \\bm{\\gamma}) = e^{i\\beta_p \\sum_i X_i}e^{i\\gamma_p H} \\cdots e^{i\\gamma_2 H}e^{i\\beta_1 \\sum_i X_i}e^{i\\gamma_1 H}.\n\\end{align}\nThis algorithm has been experimentally demonstrated \\cite{Otterbach2017} with the connectivity of the target Hamiltonian being equivalent to the connectivity of the actual device.\n\nThe equivalence of the connectivity is almost necessary from the requirement to perform $e^{i\\gamma H}$.\nThis requirement can somewhat be relaxed by our protocol which enables qubits to virtually interact irrespective of the distance between them.\nLet us now assume that an available device has a square-lattice connectivity of Fig. \\ref{fig:distant_2qubit_gate}, and a Hamiltonian of the QAOA which we aim to solve has a interaction between one pair of qubits that is not included in the hardware connectivity graph.\nIn this case, to execute the QAOA circuit (Eq. (\\ref{eq:qaoa-circuit})), we can use our space-like technique $p$ times to virtually apply the unitary.\nThe scaling of the cost can be bounded by setting $M_t=0$ and $M_s=p$ in Theorem \\ref{thm:multiple_twoqgate_cut} which gives us a scaling of $9^p\\epsilon^{-2}\\log[1\/(2\\delta)]$.\nThe time-like cut approach of Peng \\textit{et al.} \\cite{Peng2019} can also be utilized in this direction.\nHowever, as this approach would require 4 cuts per gate, the cost scaling is bounded by $16^{4p}\\epsilon^{-2}\\log[1\/(2\\delta)]$ by setting $M_t=4p$ and $M_s=0$ in Theorem \\ref{thm:multiple_twoqgate_cut}.\nThis demonstrates an advantage, albeit in this special settings, of our technique over the previous result.\n\nIn the context of the VQE, which is also an algorithm to find a ground state of a Hamiltonian but mainly targets a concrete physical system such as molecules, it has been proposed to use the same kind of quantum circuits as the QAOA \\cite{Wecker2015, Mitarai2019g}.\nOur result may also be applicable in constructing such circuits.\n\n\\section{discussion and conclusion}\nWe described a technique to decompose a non-local operations into a sequence of local operations.\n\\red{As the single-qubit operations are generally more accurate on NISQ devices, the proposed technique can be used to enhance their capability. We believe intrinsic noise on single-qubit operations can be compensated by recent sophisticated error mitigation techniques \\cite{Endo2018}.}\nIn particular, our technique of the space-like cut of two-qubit gates can improve the simulation of a large quantum circuit with a small quantum computer in some cases.\nIt would be interesting to investigate the best strategy to perform ``cuts'' to reduce the number of qubits compatible with an available device.\nAlso, the algorithm we have given to bound the cost scaling is rather straight forward and we believe it can be improved with a more sophisticated strategy.\n\n\\red{The proposed algorithm can also be compared to the classical simulation strategy that splits a large circuit by decomposing two-qubit gates. For example, a controlled-NOT gate can be splitted using a tensor network based technique \\cite{biamonte2017tensor}. However, such techniques generally does not focus on decompositions of $\\mathcal{S}(U)$ considered in this work but rather the two-qubit unitary $U$ itself, which takes makes them difficult to be used on NISQ devices as Eq. (\\ref{eq:superop_sum_decomposition}) cannot be utilized anymore. }\n\nOur technique can induce a entanglement-like effect without performing any two-qubit gate with the cost mentioned in Lemmas \\ref{thm:two_qubit_gate_decomposition} and \\ref{thm:twoq_meas_decomp}.\nThis connects this work to areas like quantum communication.\nThis ``virtual'' entanglement creation could be done with the time-like cut proposed by Peng \\textit{et al.}, but our work lowered the cost to perform the task.\nIt is interesting to know whether ours is the optimal protocol or there is a more efficient way.\n\nTo summarize, our technique allows qubits to virtually interact irrespective of physical distances between them.\nThe result is useful for applying a two-qubit gate to a distant pair of qubits.\nIn particular, when applied to the NISQ devices, this may be employed to enhance the power of them.\nFuture direction can be to explore if we can lower the resource to perform such virtual operations.\n\n\\begin{acknowledgements}\n    KM thanks the METI and IPA for their support through the MITOU Target program.\n    KM is also supported by JSPS KAKENHI No. 19J10978 and No. 20K22330, and JST PRESTO JPMJPR2019.\n\tKF is supported by KAKENHI No.16H02211, JST PRESTO JPMJPR1668, JST ERATO JPMJER1601, and JST CREST JPMJCR1673.\n\tThe authors thank Suguru Endo for fruitful discussions and letting us become aware of Ref.  \\cite{Endo2018}.\n    This work is supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0118067394.\n\\end{acknowledgements}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sect:intro} \nThe exclusion process is a paradigm for non-equilibrium behaviour \\cite{bib:M1}. \nIn the one-dimensional totally asymmetric simple exclusion process (TASEP),\neach site $i$ $ ( i=1,2,\\dots, L) $ is either occupied by one particle ($ \\tau_i =1 $) or empty ($ \\tau_i =0$),\nand a particle at site $ i$ hops to $i+1 $ with rate 1, if the target site is empty. Investigation of shocks is one of central issues in the TASEP\\cite{bib:BCFG,bib:FF,bib:DJLS,bib:DEM,bib:M2,bib:KSKS,bib:CHA}. Let us recall a known result in the TASEP with open boundaries (shortly, open TASEP), where a particle is injected at site $i=1$ with rate $\\alpha$, and extracted at site $i=L$ with rate $ \\beta $ \\cite{bib:DEHP}. These rates are regarded as reservoir densities $\\alpha$ and $ 1-\\beta $, respectively. The case $ \\alpha = \\beta < 1\/2 $ is called the co-existence line. Two plateaus with densities $ \\alpha$ and $ 1-\\alpha$ co-exist in the domains\n $ 1 < x < S(t) $ and $ S(t) < x < L $, respectively. The position $ S (t) $ of the shock (domain wall) is \\textit{dynamical}, and its  behaviour is diffusive \\cite{bib:DEM,bib:KSKS}, \\textit{i.e.}  \n\\begin{align} \\label{eq:D=} \n    \\big\\langle \\big( S (t) - S (0) \\big)^2 \\big\\rangle_{\\mathrm E} \\simeq   2 D (\\alpha ) t,   \\    \n     D (\\alpha ) = \\frac{ \\alpha(1-\\alpha) }{ 1 - 2\\alpha } , \n\\end{align} \nas $t\\to \\infty$. Here $ \\big\\langle \\cdot \\big\\rangle_{\\mathrm E} $ denotes the ensemble average. \nThis asymptotic diffusion coefficient is also true on $ \\mathbb Z $ \\cite{bib:FF}.\n\n\nThe so-called second-class particle ($ \\tau_i =2 $)  microscopically \\textit{defines}  \n the positions of shocks \\cite{bib:BCFG}. It behaves as a hole for particles, and as a particle for holes, \\textit{i.e.}  the  hops $ 20  \\to  02 $ and  $ 12 \\to  21 $ occur between sites $i$ and $ i+1 $, with the same rate   as for $ 10 \\to 01 $. We confine only one second-class particle into the system in the initial state. It is not extracted from the system and we do not inject another second-class particle, \\textit{i.e.}  \n``semi-permeable boundary condition'' \\cite{bib:A1,bib:A2,bib:U,bib:ALS}. Figure \\ref{fig:openTASEP} (a,b) shows   simulation results of the mean-squared displacements (MSD) of the second-class particle, which agree with the formula  \\eqref{eq:D=}. We notice that finite-time effect becomes strong, when $\\alpha$ approaches $1\/2$. The density profile averaged over a large time interval \n\\begin{align}\n  \\rho_i := \\langle \\tau_i  ( 2-\\tau_i ) \\rangle_{\\mathrm T}    \n  \\label{eq:rho_i:=}\n\\end{align}\nlooks different from snapshots in simulations. \nSince the shock position moves evenly in the chain,  $ \\rho_i $ is given by \n\\begin{align} \\rho_i \\simeq ( 1 - 2\\alpha ) \\frac i L  + \\alpha, \\label{eq:rho_i=open} \\end{align}\nwhich was shown by the exact stationary state  \\cite{bib:DEHP}.\n\n\n\\begin{figure}\\begin{center}\n     \\includegraphics[width=0.24\\textwidth]{MSD-open.pdf} \\  \n     \\includegraphics[width=0.24\\textwidth]{D-vs-rho.pdf} \\end{center}\n\\caption{ \n  (a) MSD of the second-class particle vs time, and\n  (b) diffusion coefficient vs boundary rate \n      on the co-existence line of the open TASEP with $ L=10^4 $. \n The markers in (a) are simulation results averaged over  $ 10^3 $ runs.\n  We used simulation data  $  ( S(t) -S(0) )^2  \/ (2t ) $ in $  5\\times 10^3 \\le  t \\le 10^4  $ to plot the markers in (b).   The lines in (a,b) correspond to eq.~\\eqref{eq:D=}.   \n \\label{fig:openTASEP}} \n\\end{figure}\n\n\nOn the other hand,  static shocks are realized in the TASEP, by imposing  attachment and detachment of particles in the bulk of the chain (the Langmuir kinetics \\cite{bib:PFF}).   Static shocks  were also found in  TASEPs with inhomogeneous hopping rates on a ring.     One of the simplest cases    is  the Janowsky-Lebowitz (JL) model \\cite{bib:JL}: \n \\begin{align}    p_i =     1  \\  ( 1 \\le i < L )   , \\  r<1 \\  ( i=L )  ,  \\end{align} \n where  $ p_i $  denotes the hopping  rate from site $i$ to $i+1$ ($ L+1 := 1 $). \n Due the inhomogeneity of the bond between sites $ L $ and 1,  the JL model exhibits a shock profile in a certain parameter region.  A mean-field theory qualitatively explains a phase transition between shock and flat density profiles, which  is still a fascinating problem \\cite{bib:CLST,bib:SPS}. \n \nIn \\cite{bib:TB},  another inhomogeneous TASEP   was introduced: \n\\begin{align}\\label{eq:two_pi=}  p_i =   1  \\  ( 1 \\le i \\le \\ell    )   , \\  r  \\  ( \\ell <  i \\le 2 \\ell=L   )  .\\end{align}\n We refer to this model as 2-segment TASEP.   In a certain region of the parameter space $ (r,\\rho) $ (where $ \\rho$ is the global density \\textit{i.e.} the number of particles $\/L$), this model also exhibits a static shock.  Recently  the authors of  \\cite{bib:BSB} investigated TASEPs with three and four parts, generalizing \\eqref{eq:two_pi=}.  Here, we mainly study a specific  4-segment TASEP \n\\begin{align}\\label{eq:four_pi=}\n  p_i = \n  \\begin{cases}  \n     1 & ( 1 \\le i \\le \\ell   \\ \\vee \\   2\\ell  < i \\le  3\\ell  )   , \\\\ \n     r & (  \\ell < i \\le 2\\ell \\  \\vee\\   3\\ell < i \\le 4\\ell =L) .    \\\\ \n  \\end{cases}\n\\end{align}\nThere can exist two shocks in  the 1st and 3rd segments. The positions of the shocks cannot be fixed even in the macroscopic level, but they are related to each other by an equation derived by the particle number conservation  \\cite{bib:BSB}. \n The purpose of this work is performing more detailed Monte Carlo simulations (in continuous time),  in order to deeply understand this synchronization phenomenon. \n\nBefore investigating the 4-segment TASEP, we reconsider the case of 2 segments via the second-class particle.  As an evidence that the second-class particle indicates the shock position, we check that its spatial distribution becomes gaussian, corresponding to the density profile written in the error function.  We also examine properties of the standard deviation of the shock position. \nThen we turn to  the 4-segment TASEP. We find that the MSD exhibits various behaviours, depending on the time scale that we focus. The open boundary condition that we have already reviewed is the reference case. We quantify the interaction between the  shocks by a correlation function.\n Furthermore we introduce a crossover time distinguishing between time scales  of   independency and synchronization of shocks. We also  numerically  estimate the dynamical exponent of the crossover time. Finally we give the conclusions of this work and some remarks, including possible future studies.  Overall in this work, we use the following  definition  for  macroscopic density profiles  with  \\textit{mesoscopic} length  of the lattice  $ 2 \\delta +1 $ ($ 1 \\ll \\delta \\ll  \\ell $),  which is in general different from  the microscopic density profile \\eqref{eq:rho_i:=}:\n\\begin{align}\n   \\rho (x)  = \\rho ( i\/\\ell   )  = \n    \\frac{1}{ 2 \\delta + 1 } \\sum_{ i' = i - \\delta }^{ i + \\delta }  \\tau_{i'}  ( 2 - \\tau_{i'}  )  .   \\label{eq:macro-density}\n\\end{align}\n \n\n\n\n\n\n\\section{2-segment TASEP}\\label{sect:2-seg}\n Let us consider the model \\eqref{eq:two_pi=}.  \n We begin with the assumption that the global density $ \\rho $ is enough small, and \n each segment has a flat density profile       \n\\begin{align}\\label{eq:rho(x)=12}\n   \\rho(x) =      \\alpha_1   (  0  < x < 1   )   , \\       \\alpha_2   (  1 < x < 2  )   .    \n\\end{align}\nThe conservation laws of the number of  particles  and the stationary current $ J $ are written as \n\\begin{align}\n  \\label{eq:2rho=a1+a2}\n  2 \\rho &=     \\alpha_1   +     \\alpha_2  ,  \\\\  \n        J &=  \\alpha_1 (1-\\alpha_1) = r  \\alpha_2 (1-\\alpha_2)  , \n  \\label{eq:J=J1=J2}\n\\end{align}\nrespectively. \nThese are easy to solve \\cite{bib:TB}: \n\\begin{align}\n\\label{eq:alpha1=,alpha2=}\n  \\alpha_1  &= \\frac{ 1 + r -4r\\rho - R }{  2 (1-r)  } , \\ \n  \\alpha_2  = \\frac{ 4\\rho - (1+r)  + R}{  2 (1-r)  } , \\\\\n  J &=          \\frac{ r(1-2\\rho)(R-(1+r)(1-2\\rho)  ) }{   (1-r)^2  } \n    \\label{eq:J:LDLD}  \n\\end{align}  \n with $ R  = \\sqrt{(1+r)^2-16r\\rho (1-\\rho)}  $. \nThe density profile \\eqref{eq:rho(x)=12} is realized as long as  $ \\alpha_2 <  1\/2  $\nwith \\eqref{eq:alpha1=,alpha2=}, \\textit{i.e.}  \n\\begin{align}  \\label{eq:rho_c=} \\rho <   \\rho_c : =   \\frac { 2-\\sqrt{1- r} }{4}     .\n   \\end{align}\nSince both $ \\alpha_1 $ and $  \\alpha_2 $ are less than $ 1\/2 $, \n the case $ \\rho < \\rho_c $ is called LD-LD phase (LD$=$low density). \n\n\n\\begin{figure}\\begin{center}\n  \\includegraphics[width=0.24\\textwidth]{Phase-diagram.pdf} \\ \n  \\includegraphics[width=0.24\\textwidth]{Fundamental-diagram.pdf} \\end{center} \n\\caption{  \n (a) Phase diagram  and\n (b) fundamental diagram \n    of the 2- and 4-segment TASEPs \\cite{bib:TB, bib:BSB}. \nThe phase boundaries in (a) are given by $ \\rho = \\rho_c ,1 - \\rho_c$  \n\\textit{i.e.}  the parabola  $ r =  1  -  1 6  ( \\rho - 1\/2   )^2    $.  \nFor  (b),  we have set    $ \\ell = 10^3 $ and $ r =1\/2$.  To obtain each marker, we counted the number of flowing particles in a single simulation run, and took average over $ 10^6 \\le t \\le 10^7 $. The line in (b)  corresponds to the   predictions  \\eqref{eq:J:LDLD}, \\eqref{eq:J:SMC}. \n  \\label{fig:diagrams}} \n\\end{figure}\n\n\n\n\nWhen  the total density $  \\rho $ exceeds the critical density, \nthe 2nd segment maintains  the density $1\/2 $ and\na shock appears on the 1st segment: \n\\begin{align}\\label{eq:rho(x)=1lh2}\n   \\rho(x) = \n  \\begin{cases}  \n     \\alpha_1  \\  (  0  < x < s )   , \\  \n     1- \\alpha_1 \\  (  s < x < 1   )   , \\\\\n     \\alpha_2 =   \\frac{1}{2}   \\  (  1 < x < 2  )  .    \n  \\end{cases}\n\\end{align}\nWe refer to this case as S-MC (shock-maximal current) phase. The shock position $s$ is macroscopically static, \\textit{i.e.}  localized.  By solving the conservation laws of the number of particles and the stationary current \n\\begin{align}\n 2 \\rho =&  s \\alpha_1  + (1-s)(1 - \\alpha_1 ) +  \\  \\alpha_2  ,  \\\\\n J  ( \\alpha_1 ) =& \\alpha_1 (1-\\alpha_1) =  r \\alpha_2 (1-\\alpha_2 )  =   r \/ 4   \\label{eq:J:SMC} , \n\\end{align}\nthe densities and the shock position in the 1st segment  are determined as \n\\begin{align}\n  \\alpha_1 = \\frac{1-\\sqrt{1-r}}{2}  ,  \\    s  = \\frac{1}{2} + \\frac{1-2\\rho}{ \\sqrt{1-r}}  . \n  \\label{eq:alpha1=,s=}\n\\end{align}\nThe parameters $ ( r , \\rho ) $ are inversely specified by $ (\\alpha_1,s) $ in the S-MC phase. We have emphasized the current as a function of the density $ \\alpha_1 $. \n \nWhen the global density $\\rho$ exceeds  $1-\\rho_c $, the shock position reaches site $i=1 $. The densities in both segments are flat, and larger than $ 1\/2 $, \\textit{i.e.}  the HD-HD phase (HD$=$high density). Figure \\ref{fig:diagrams} (a) summarizes the three phases. In fig.~\\ref{fig:diagrams} (b), we check that the predicted currents \\eqref{eq:J:LDLD} and \\eqref{eq:J:SMC} are realized by simulations. Because of the particle-hole symmetry, we restrict our consideration to $ \\rho \\le 1\/2 $.  \n\n\nLet us investigate properties of the shock in the S-MC phase in more detail. There is a microscopic deviation of the density profile near the shock position. It obeys a gaussian distribution, \\textit{i.e.}  the probability distribution $ P ( S ) := \\frac 1 2 \\langle \\tau_S (\\tau_S-1) \\rangle_{\\mathrm T} $ of the position $S$ of the second-class particle is given as \n\\begin{align}\n\\label{eq:P(S)=Gauss}\n P (S)  =   \\frac{1}{ \\sqrt{2 \\pi  \\sigma^2  }   }  \n  \\exp  \\bigg[  - \\frac{1}{ 2\\sigma^2 }  ( S - \\langle S \\rangle_{\\mathrm T}  ) ^2  \\bigg]     \n\\end{align}\nwith $ \\langle S \\rangle_{\\mathrm T}  \\simeq  \\ell s $ $ ( \\ell \\to \\infty ) $. We see good agreement of simulations to this distribution in fig.~\\ref{fig:2-seg} (a).  This corresponds to the fact that the (microscopic) density profile \\eqref{eq:rho_i:=} in the 1st segment  is well described in terms of  the error function  $\\mathrm{erf}[\\cdot]$  \\cite{bib:BSB}, see  fig.~\\ref{fig:2-seg} (b): \n\\begin{align}\n  \\label{eq:<tau>=erf}\n    \\rho_i =  \\frac{\\sqrt{ 1-r } }{2}  \\ \\mathrm{erf} \n   \\bigg[  \\frac{ i -  ( \\langle S \\rangle_{\\mathrm T}  +  \\frac 1  2  )  }{ \\sqrt{2   \\sigma^2  } } \\bigg] \n       +     \\frac{1}{2}   .\n\\end{align} \nWe chose the values of  parameters, such that corresponding  densities and shock positions are given as  \n\\begin{align}\n  \\nonumber \n &    ( r , \\rho   ) = (0.96,  0.5  ) , ( 0.84 , 0.48 ) , ( 0.64,0.44 ) , (0.36,0.38  )   \\Leftrightarrow  \\\\  \n &    ( \\alpha_1 , s  ) =  ( 0.4 , 0.5 ) , (0.3, 0.6   ) ,  ( 0.2 , 0.7  ) ,  ( 0.1, 0.8  ) , \n\\end{align}\nrespectively,  according to eq.~\\eqref{eq:alpha1=,s=}. \n(In the 2nd segment, we expect that the density profile, which is deviated from $  1\/2 $,  can be well written by the exact finite-size effect in the maximal current phase of the open TASEP \\cite{bib:DEHP}.)\n\n\n\\begin{figure}\\begin{center}\n     \\includegraphics[width=0.24\\textwidth]{gauss.pdf}  \\ \n     \\includegraphics[width=0.24\\textwidth]{erf.pdf} \\  \n     \\includegraphics[width=0.24\\textwidth]{sigma-vs-ell.pdf} \\ \n     \\includegraphics[width=0.24\\textwidth]{sigma-vs-r.pdf} \n     \\end{center}  \n\\caption{  \n (a) Distributions of the second-class particle, \n (b) density profiles,\n (c) standard deviation $ \\sigma $  vs segment length   $ \\ell $,  and  \n (d) $\\sigma$ vs hopping rate $r$.   \n For (a) and (b) we  have set  the system length as $ \\ell =200$, and for (d) $ \\ell =10^3 $.   \n  Each  plot marker was obtained by averaging  data of a single simulation run  over \n   $ 10^6  \\le t \\le 10^7$ \n  [exceptionally $ 10^6  \\le t \\le 5\\times 10^7$ for the cases $ \\ell > 10^3 $ in  (c)]. \n   The curves  in  (a) and (b) correspond to  \n    eqs.~\\eqref{eq:P(S)=Gauss}  and  \\eqref{eq:<tau>=erf},  \n    respectively, with $ \\langle S \\rangle  $   and $ \\sigma$ obtained from simulation data. Each line  in (c)  corresponds to  $ ( \\ell  \/10^4 )^{1\/2}    \\times  \\sigma |_{\\ell=10^4} $ with $ \\sigma |_{\\ell=10^4} $ obtained by simulations. \n\\label{fig:2-seg} } \n\\end{figure}\n\n In fig.~\\ref{fig:2-seg}  (c), we observe $ \\sigma \\propto \\sqrt \\ell $  \\cite{bib:BSB}.  Furthermore  $ \\sigma \/ \\sqrt \\ell   $ vs $r $ is shown in fig.~\\ref{fig:2-seg}  (d) with different global densities  $ \\rho $.  So far we have not  found an explicit formula of $ \\sigma \/ \\sqrt \\ell $, but it seems independent of $ \\rho $. \n\n\n\n \n  \n\\section{4-segment TASEP}\\label{sect:4-seg}\nNow we turn to the 4-segment TASEP \\eqref{eq:four_pi=} \\cite{bib:BSB}. By the same argument  as for the 2-segment case,  the density $\\alpha_j  $ of each segment $j$ is flat, when  $ \\rho < \\rho_c  $. (The phase transition line $ \\rho = \\rho_c $ is identical to that of the 2-segment case, fig.~\\ref{fig:diagrams} (a).)\n From the translational invariance,  we have  $ \\alpha_1 = \\alpha_3  $ and  $ \\alpha_2 = \\alpha_4  $.  Furthermore these densities have  the same form  \\eqref{eq:alpha1=,alpha2=}  as for the 2-segment case \\cite{bib:BSB}. \n After the global density exceeds $ \\rho_c  $, the 2nd and 4th segments maintain the density $\\alpha_2 =\\alpha_4 = 1\/2 $, and shocks appear in the 1st and 3rd segments \\cite{bib:BSB}:\n  with   $ \\alpha_1 = \\alpha_3   $,  \n\\begin{align}\\label{eq:rho(x)_SMCSMC}\n   \\rho(x) =   \n   \\begin{cases}  \n     \\alpha_1  \\ (  0  < x < s_1 )   ,    \\    1- \\alpha_1 \\  (  s_1 < x < 1   )   , \\\\ \n        \\frac{1}{2} \\   (  1 < x < 2  )  ,  \\\\ \n     \\alpha_3  \\ (  2  < x < s_3'   )   ,    \\    1- \\alpha_3 \\  (  s_3'  < x < 3    )   , \\\\ \n        \\frac{1}{2} \\   (  3 < x < 4   )    . \n  \\end{cases}\n\\end{align}\nThe shock positions are denoted by  $ s_1 $ and $ s_3'   = s_3 +2$  [$ s_3=0 $ (resp. $ s_3=1  $) corresponds to the boundary between 2nd and 3rd  (resp.  3rd and 4th) segments]. \n The conservation of the number of particles is written as \n\\begin{align}\n 4  \\rho = \n  s_1 \\alpha_1  + (1-s_1) (1-\\alpha_1 )  +   \\alpha_2 \n   +   s_3 \\alpha_3  + (1-s_3)(1- \\alpha_3 ) +   \\alpha_4   .\n\\end{align}\nSolving this together with the current conservation  \\eqref{eq:J=J1=J2}, \nwe find a restriction on the shock positions  \\cite{bib:BSB} \n \\begin{align}  s_1 +  s_3 = 2 s  ,  \\label{eq:s1+s3=s} \\end{align} with $s$ \\eqref{eq:alpha1=,s=}. \nThe form of the density $ \\alpha_1  $ is the same as  for the 2-segment case \\eqref{eq:alpha1=,s=}.\nThe current $J$ is also unchanged \\cite{bib:BSB}, see fig.~\\ref{fig:diagrams}(b). One of the interesting findings in \\cite{bib:BSB} is that  we cannot fix  the shock positions $s_1$ and $s_3$ even in the macroscopic level, but they are synchronized  by the restriction \\eqref{eq:s1+s3=s}, see fig.~\\ref{fig:4-seg} (a).    The right bound of $ s_j $ is 1, and the left  $ \\lambda $ is given by solving $   2 s - \\lambda  = 1  $: \n\\begin{align}   \\label{eq:lambda<sj<1}\n  \\lambda      =   \\frac{ 2 ( 1 - 2\\rho ) }{ \\sqrt{ 1 - r }  }\n  <  s_j <  1 .  \\end{align} \nThe shock positions are uniformly distributed in this range except for the boundaries, see fig.~\\ref{fig:4-seg} (b).  Therefore the density profile   by averaging configurations over a large time interval becomes \n\\begin{align}\\label{eq:rho(x)_SMCSMC_over-time}\n\\langle  \\rho(x) \\rangle_{\\mathrm T}=\n \\begin{cases}  \n     \\alpha_1  \\ (  0  < x < \\lambda )   ,    \\    \\alpha_1    +    (x-\\lambda)  g \\  (  \\lambda  < x < 1   )   , \\\\ \n        \\frac{1}{2} \\   (  1 < x < 2  )  ,  \\\\ \n     \\alpha_1  \\ (  2  < x < \\lambda'   )   ,   \\    \\alpha_1  +   (x-\\lambda')  g \\  (  \\lambda'  < x < 3    )   , \\\\ \n        \\frac{1}{2} \\   (  3 < x < 4   )   ,  \n  \\end{cases}\n\\end{align}\nwhere $ \\lambda' = \\lambda+2 $ and $ g = \\frac{ 1 - 2 \\alpha_1 }{ 1 - \\lambda  } $.  In  fig.~\\ref{fig:4-seg} (c), we compare   simulation results of the density profile  $ \\rho_i $ \\eqref{eq:rho_i:=} with the prediction \\eqref{eq:rho(x)_SMCSMC_over-time} with $ x=i \/\\ell  $. We chose the values of parameters, such that the corresponding densities and left boundaries become \n\\begin{align}\n  \\nonumber \n &    ( r , \\rho   ) = (0.96,  0.5  ) , ( 0.84 , 0.48 ) , ( 0.64,0.44 ) , (0.36,0.38  )   \\Leftrightarrow   \\\\  \n &   ( \\alpha_1 , \\lambda  ) =  ( 0.4 , 0 ) , (0.3, 0.2   ) ,  ( 0.2 , 0.4  ) ,  ( 0.1, 0.6  ) , \n\\end{align}\nrespectively, according to eqs.~\\eqref{eq:alpha1=,s=} and \\eqref{eq:lambda<sj<1}. We expect that the discrepancies between the prediction and simulations are  due to the finite-size effect. In the half-filling case $\\rho = 1\/2$, the profiles of time average in the 1st and 3rd segments are identical to that of the well-studied open TASEP \\eqref{eq:rho_i=open}. Therefore one may naively expect that  the 1st and 3rd segments are effectively equivalent to the co-existence line of the open TASEP; e.g. for the 3rd segment, the 2nd and 4th segments play the role of reservoir of densities $\\alpha_1$ and $1-\\alpha_1$, respectively. Let us observe simulation results of MSD in the 4-segment TASEP, to verify  whether this guess is correct or not. \n\n\n\n\\begin{figure}\\begin{center} \n    \\includegraphics[width=0.24\\textwidth]{kymo.pdf} \\ \n    \\includegraphics[width=0.24\\textwidth]{unif.pdf} \n    \\includegraphics[width=0.48\\textwidth]{density-profiles-4.pdf} \n    \\end{center} \n\\caption{  \n (a)  Kymograph probing  \n   second-class particles and macroscopic density profiles\n    [eq.~\\eqref{eq:macro-density}   with   $\\delta=5$]  \n    in the 1st and 3rd segments \n    for $ (r,\\rho ,\\ell ) = (0.64, 0.5,200 )  $.   \n (b) Distribution  of the second-class particles, and (c) density profiles\n     for $ \\ell =200$, averaged over      $ 10^6 \\le  t \\le 10^8 $.\n The lines in (b)  are the uniform distribution over the range \\eqref{eq:lambda<sj<1}, and in (c) the prediction \\eqref{eq:rho(x)_SMCSMC_over-time} with $ x = i \/\\ell    $. \n\\label{fig:4-seg} } \n\\end{figure}\n\nDenote by $ S_j (t) + (j-1) \\ell  $  the microscopic  position  of the second-class particle  in segment  $ j \\in \\{1,3 \\} $ at time $t$, so   $ 1 \\le    S_j (t)\\le \\ell$.   Figure \\ref{fig:4-MSD}  (a)  shows the simulation results of the MSDs \n\\begin{align} M (t) = \\big\\langle \\big(  S_j (t) - S_j (0) \\big)^2 \\big\\rangle_{\\mathrm E} . \\end{align}\nThanks to the translational invariance, we omit the subscript $j$, and practically  we obtain $  M(t)  $ by averaging simulation data of $j=1$ and $j=3 $. In some time interval, indeed we see that the diffusion coefficient   \\eqref{eq:D=}     with  $\\alpha =  \\alpha_1 $ \\eqref{eq:alpha1=,s=} provides a good estimation of the MSD's behaviour: \n\\begin{align} M(t) \\approx 2 D ( \\alpha_1 ) t . \\end{align} \n Note that the observed  $   M(t) \/ (2t) $ in this \\textit{intermediate diffusive regime} is slightly lower than $ D $, which is expected to be a finite-size effect. \n\n\nFigure \\ref{fig:4-MSD} (a) indicates that the initial and asymptotic behaviours are also diffusive. The current of particles \\eqref{eq:J:SMC} gives a good fitting curve for the diffusion coefficient in the \\textit{initial diffusive regime}, see fig.~\\ref{fig:4-MSD} (b):   \n\\begin{align} \\label{eq:M=2Jt}  M(t) \\simeq 2 J  ( \\alpha_1  ) \\,  t \\quad ( t\\to +0)  , \\end{align} \nwhich implies that the second-class particles obey the symmetric random walk with jump rate $ J ( \\alpha_1  ) $ in a very short time scale.  The same finite-time effect is observed in the open TASEP [fig.~\\ref{fig:openTASEP} (a)], where one can prove that  both probabilities of finding configurations $20$ and $12$ are given by $ J(\\alpha) $ by using matrices \\cite{bib:A1,bib:A2,bib:U,bib:ALS}. \n\n \n\nOn the other hand,  for the  \\textit{asymptotic diffusive regime},\n  \\begin{align}  M(t) \\simeq 2 \\mathcal D ( \\alpha_1  ) \\, t \\quad ( t\\to \\infty) ,    \\end{align} \nwe expect the inequality   $ \\mathcal D ( \\alpha_1  ) < D ( \\alpha_1  )$,  see fig.~\\ref{fig:4-MSD} (a). Note that the conclusion on the asymptotic behaviour is based on our simulations of  up to $t = 10^6$.  Intuitively, no further transition is expected in longer-time simulations. (We considered  the ``asymptotic'' behaviour  in   $ t \\ll \\ell^2 \/ \\mathcal D $, so that the shocks do not reach  boundaries $ S_j(t) =\\lambda \\ell , \\ell  $. When we take the limit $  t\\to \\infty $ with $ \\ell$ fixed,  the MSD converges to a constant.)  The notation $ \\mathcal D (\\alpha_1) $ emphasizes that the diffusion coefficient is a function of $\\alpha_1$, but the dependence on $ \\ell $ and $ \\lambda $ may also be nontrivial. \n \n \nIn the  time interval between the intermediate and asymptotic diffusion regimes, the MSD exhibits sub-diffusion.  The bold lines (slope $-1\/2 $) are drawn in the log-log graph, fig.~\\ref{fig:4-MSD} (a), which agree well with simulations. We also estimate its exponent $\\varepsilon$ $(\\ln  M(t) \\approx  \\varepsilon \\ln t )$ by \n\\begin{align}\\label{eq:eps=}\n\\varepsilon = \\frac{ 1 }{ 3t\/2 + 1 }\n \\sum_{t\/2 \\le  t' \\le 2t } \\frac{ \\ln M ( 10t' ) - \\ln M ( t' ) }{ \\ln 10 } , \n\\end{align}\nin this \\textit{transient regime.}  \nThe exponent by this formula takes the minimum, which is  $\\approx 1\/2$, see  fig.~\\ref{fig:4-MSD} (c).  \n\n \n\n\n\\begin{figure}\\begin{center}  \n    \\includegraphics[width=0.48\\textwidth]{MSD-4.pdf}  \\ \n    \\includegraphics[width=0.24\\textwidth]{initial-diffusion.pdf}\n    \\includegraphics[width=0.24\\textwidth]{epsilon.pdf}\n     \\end{center} \n\\caption{  \n (a) MSD of the shock positions,\n (b) initial diffusion coefficient, and\n (c) exponent $ \\varepsilon$   \\eqref{eq:eps=}.        \n In (a), we divided $M(t)$ by $2t$, so as to see its property more easily.\n In (a,b), the solid, dashed and dotted lines correspond to $J$ [eq.~\\eqref{eq:J:SMC}], $D$ [eq.~\\eqref{eq:D=}] and $\\mathcal D$ [simply $M(10^6)  \/ (2\\times 10^6 )  $ from simulations], respectively. \n To obtain numerical data in (a,c), we took averages over  $ 10^6 $ and $ 10^3 $ simulation runs \n for $ t\\le 2 $ and $ t>2  $, respectively.  For each marker in (b),   we averaged  $  M(t)  \/ (2t)  $   of   $ 10^{-3} \\le t\\le  10^{-2}  $  over $ 10^6 $ simulation runs. \n\\label{fig:4-MSD} }  \n\\end{figure}\n\nLet us investigate the correlation of the shocks.  We expect that, in a very short time,  the two shocks move independently, since they are far from each other. On the other hand,   they are  synchronized in the frame of  a larger time scale, e.g. as  fig.~\\ref{fig:4-seg} (a). In order to  quantify  (in)dependency of the two shocks, we introduce a correlation function \n\\begin{align}\n\\label{eq:C(t)=}\n C(t)  = - \\big\\langle   \\big( S_1 (t) - S_1  (0) \\big) \\big( S_3  (t) - S_3  (0)   \\big)  \\big\\rangle_{\\mathrm E}   . \n\\end{align}\nDue the synchronization $ S_1 (t) + S_3 (t) \\approx \\ell s $, we have  \n\\begin{align} C(t) \\simeq M (t) \\quad ( t \\to + \\infty), \\end{align}  \n   which is observed in fig.~\\ref{fig:4-Correlation} (a).\nOn the other hand, their initial behaviours are completely different, see fig.~\\ref{fig:4-Correlation} (b,c).  In the vicinity of $t= 0$, we conjecture that $C(t)$ increases more slowly than any power function $t^a (a>0)$. \n \n\n\\begin{figure}\\begin{center} \n     \\includegraphics[width=0.24\\textwidth]{long.pdf} \\ \n     \\includegraphics[width=0.24\\textwidth]{medium.pdf} \\ \n     \\includegraphics[width=0.24\\textwidth]{short.pdf} \\ \n     \\includegraphics[width=0.24\\textwidth]{t-vs-ell.pdf}  \\end{center} \n \\caption{  \n (a,b,c) Comparison between the MSD and the correlation function  \n  in different time windows, and\n  (d) crossover time vs segment length. \n   For  (a,b,c),  we have set the values of the parameters as $ (\\ell,r, \\rho) = (1000,0.64,0.44) $,  \n   and averaged over $ 10^3,10^4 $ and $ 10^6 $ simulation runs,  respectively.\n In (d), each marker was plotted by averaging over  $ 10^3 $ simulation runs, \n  and the lines are fitting curves $ c \\, \\ell^z$.  \n\\label{fig:4-Correlation} } \n\\end{figure}\n\nLet us define the time $T$ as \n \\begin{align} T  = \\inf \\big\\{ t > 0 \\big|  2  C(t)   > M(t)  \\big\\} ,  \\end{align}  \ncharacterizing the crossover between the time scales of independency and synchronization.  \nFigure \\ref{fig:4-Correlation} (d) shows  $T$ vs segment length $\\ell$. By assuming the power law  $ T \\simeq c\\, \\ell^z $, we perform fitting from the simulation data that we have. We write the results directly in fig.~\\ref{fig:4-Correlation} (d), and draw corresponding straight lines as well. \n Under the further assumption that the exponent is independent from parameters, \nwe simply average the four obtained exponents:  $z \\approx 1.71$.  \n\n\n\n\\section{Discussions}\\label{sect:disc}\nIn this work,  we investigated synchronization of shocks in the 4-segment  TASEP, by means of  the two second-class particles.  We found that the behaviour of the MSD $ M(t) $ of the shocks is not so simple. In the initial regime (very short time scale), it is diffusive and the coefficient is nothing but the formula for the particle current. In the intermediate diffusive regime, the coefficient $D$ known in the open TASEP provides a good estimation. This fact indicates that,  up to the intermediate diffusion regime,  the shocks' motions are determined by the values of low and high densities, and independent from details of boundary conditions.  Via sub-diffusive regime, the MSD achieves the asymptotic diffusive regime, where the diffusion coefficient is different from $D$.  From the log-log graph and the numerical estimation, fig.~\\ref{fig:4-MSD} (a,c), it seems that $ M(t)\\propto \\sqrt t$ in the sub-diffusive regime.   In some random walks,   similar changes of the diffusivity have been found \\cite{bib:PM,bib:TSJS}.    In our case, the \\textit{regime change} is spontaneously induced by the particle number conservation, without directly imposing any interaction between the two second-class particles in defining the model. \n\n\nWe also investigated the correlation function of the shocks, and the crossover time between independency and synchronization of the shocks. The correlation function $C(t)$ \\eqref{eq:C(t)=} becomes identical to $M(t)$ as $t\\to +\\infty$.  In the vicinity of $  t=0 $, $ C(t) $ increases very slowly, whereas  $ M(t) $ is linear.  We defined the crossover time $T$ as the time when the ratio $ C\/M $ exceeds $ 1\/2 $.   We estimated the dynamical exponent $z$ for $ T $ by using our simulation data, which was found between  the KPZ $ z=3\/2  $ \\cite{bib:GS} and the normal diffusion $ z=2 $.  The  exponent $z$ as well as $ \\mathcal D  $ and $ \\epsilon $ should be  more precisely estimated  in larger systems   with longer simulation time and by more simulation runs. \n\n\n\nThe generalization to the $2n $-segment case is straightforward: the $j$th segment has the rate 1 (resp. $r$) when $j$ is odd (resp. even). When $ \\rho < \\rho_c $, the density profile $ \\rho (x)  \\ ( j - 1 < x < j ) $ of $j$th segment is give as $\\rho (x)  = \\alpha_1$ for odd $j$, and $\\rho (x)  = \\alpha_2$ for even $j$, with the same forms as in the 2-segment case \\eqref{eq:alpha1=,alpha2=}. When the global density $ \\rho_c < \\rho < 1- \\rho_c $, the density profiles are flat with density $ 1\/2 $ in the segments with even numbers.  On the other hand, $n$ shocks appear in segments $j=1,3, \\dots,2n-1 $. Denoting their positions by $ s_j + j - 1 $, we find \n $ s_1 + s_3 + \\cdots + s_{2n-1 } = n s ,  $ from the conservation of the number of particles. As an analogue to the $n=2$ case, we expect that they are not static but synchronized. Since only one equation governs the synchronization of the $ n $ shocks, we naturally expect that the asymptotic diffusion constant $ \\mathcal D_n( \\alpha )$ enjoys \n $   \\mathcal D_1 (  \\alpha ) < \\mathcal  D_2 (  \\alpha ) < \\mathcal  D_3 (  \\alpha ) <  \\cdots $\n with $ \\mathcal D_1 \\equiv 0 $ and $ \\mathcal D_2 ( \\alpha ) = \\mathcal D ( \\alpha ) $. \nA further intuitive conjecture is that the sub-diffusive regime vanishes as $ n \\to + \\infty $, and \n$\\lim_{ n\\to\\infty } \\mathcal  D_n (  \\alpha )   = D(\\alpha)$. \n \n It is known that two shocks are synchronized in the model \\cite{bib:CCB}. A simple generalization of the JL model \\cite{bib:SB}, e.g. $ p_i = r ( i \\in \\{\\ell , 2\\ell\\} ) , 1 ( i \\notin \\{\\ell , 2\\ell\\} ) $ with $L =2 \\ell $, also exhibits the same type of synchronization.  One of important questions is whether the MSDs of shocks in these models behave like fig.~\\ref{fig:4-MSD} (a), and if yes,  whether  the asymptotic  diffusion coefficient is the same as for the 4-segment TASEP.   Note that the positions of the two synchronized shocks are, in general, far from each other in these models as well as the 4-segment TASEP.  (See e.g.  \\cite{bib:MH,bib:JWW,bib:J,bib:DG} for other types of  synchronization.) While the motions of the  second-class particles  are locally defined, the shocks  move as if they \\textit{knew} each other's position.    We believe that this viewpoint gives hints to study self-organization phenomena,  e.g. in biological cells,  as the exclusion process is one of basic models also in biophysics \\cite{bib:CMZ}.  Application to traffic flows would be also an interesting problem. The local inhomogeneities in the JL and generalized JL models are very similar to traffic lights, and combinations of segments that we studied here evoke different limit speeds of cars.\n\n\n\n\n\\section*{Acknowledgements}\n The author thanks Ludger Santen and M Reza Shaebani for useful discussions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nComposition and behavior of ecological communities are shaped by direct and indirect interactions between the species of these communities, such as the competition for the physical space and the intrinsic and the extrinsic resources. \nThe examples of such  competitive ecosystems are microbial communities in various biomes such as the soil~\\cite{ratzke2020strength}, the ocean~\\cite{tilman1977resource,strom2008microbial} and the human body~\\cite{foster2017evolution} - in particular the human gut which  hosts a diverse microbiome  whose dynamics are important for human health~\\cite{coyte2015ecology,gorter2020understanding}. \nIn the context of cellular populations within organisms, the evolution of neoplasms and tumor cells \\cite{merlo2006cancer,kareva2015cancer,smart2021roles}, interactions within the immune system ~\\cite{tauber2008immune,schmid2003evolutionary}, as well as the appearance of dominant clones during cell reprogramming~\\cite{shakiba2019cell}, exhibit phenomenology akin to ecological competition. \nBeyond biology \\cite{tilman1982resource,morin2009community,tuljapurkar2013population}, competitive interactions shape behaviors in a vast array of systems such as competition economics \\cite{budzinski2007monoculture} and social networks \\cite{koura2017competitive}.\n\nA classical example of the effects of inter-species competition - which inspired important ecological competition paradigms  - is the differentiation in beak forms of finches in the Gal\\'apagos islands \\cite{lewin1983finches,lack1983darwin}. \nOn these islands, dissimilar finch species possess beaks of varying shapes and sizes allowing them to consume different food sources and thus occupy distinct niches; this type of ecosystem structure is commonly referred to as an ecological niche model \\cite{grant1979darwin,pocheville2015ecological}. \nVarious niche models have been used to describe the community structures observed in diverse ecosystems such as plant grassland communities \\cite{zuppinger2014selection,silvertown2004plant}, marine plankton~\\cite{cullen1998behavior} and conservation ecology~\\cite{melo2020ecological,aguirre2015similar}.\nCommonly, niche specialization results in weaker competition for resources between individuals occupying separate niches (inter-species competition) compared to the competition between individuals of the same kind residing in the same niche (intra-species competition). The competition strength (determining the niche overlap \\cite{badali2020effects,capitan2015similar})  defined as the ratio between the inter-specific and intra-specific competition strengths. \n\nAnother paradigmatic class of ecological models comprises neutral models that are often used to describe noisy ecosystems wherein individuals from distinct species are functionally equivalent. \nIn contrast to niche models, interactions between all individuals in neutral models are identical regardless of their species  \\cite{bell2001neutral,hubbell2001unified,chave2004neutral}.\nThus, neutral models have commonly served as the null hypotheses for the exploration of ecological processes in various settings where the differences between inter-specific and intra-specific interaction are functionally negligible \\cite{bell2001neutral,gotelli2006null,blythe2012neutral}.\nNeutral theories can be viewed as a limit of niche theories where inter-specific and intra-specific interactions are equal: in other words, all species reside in completely overlapping niches ~\\cite{grover1997resource,begon2006ecology,pocheville2015ecological}.\n\nIn multi-species communities, the intra- and inter-species interactions as well as interactions with the environment, can lead to complex community composition and population dynamics; some species survive in the long term, while others are driven to extinction.\nHowever, in large communities with high numbers of competing species, it is often impractical or impossible to characterize the entire system composition by the assemblage of abundances for each species.\nHence, coarse-grained paradigmatic descriptions are often used to provide general insights into the common behavior of these ecological communities.\n\nTwo variables commonly used to characterize complex ecological communities are 1) the richness, reflecting the number of co-occurring species \\cite{adams2009species,kery2020applied}, and 2) the species abundance distributions (SAD) - the number of species present at a given abundance. The latter is  closely related to the species rank abundance (SRA) - the species ranked in terms of their abundance \\cite{nias1968clone, rulands2018universality, de2020naive, mcgill2007species, matthews2015species}.\nThese aggregate variables are observable experimentally and serve as the reporters on the underlying community structure, dynamics and the interaction network \\cite{rahbek2001multiscale,hong2006predicting,adler2011productivity,valencia2020synchrony}. \nRichness, for example,  is commonly considered to be an indicator of the competition strength and stability of the ecosystem \\cite{pimm1984complexity, ives2000stability, jousset2011intraspecific,mallon2015microbial,capitan2017stochastic}.\n\n\nThe shape of the SAD is also used as a proxy for the structure of the underlying interactions' network.  For high immigration or weak inter-species competition, the SAD commonly has a peak at high species abundance, away from extinction.\nThis community structure is closely related to the niche models whereby different species co-exist: most species inhabit their own niches with their species abundance fluctuating around the peak of the SAD. Conversely, other ecosystems, such as many microbial communities and T-cell repertoires, exhibit few high-abundance species  alongside highly diverse populations of low-abundance species \\cite{lynch2015ecology, de2020naive}.\nThis `rare biosphere' or `hollow-curved distribution' is described by a unimodal, monotonically decreasing SAD. Interestingly, this unimodal behaviour is empirically observed in many different ecosystems and is often considered universal (see \\cite{leidinger2017biodiversity} and references therein). Neutral models have been championed to describe the emergence of this universality, although other theoretical explanations for the `rare biosphere' SAD in competitive ecosystems have been suggested \\cite{mcgill2007species,magurran2013measuring}.\n\n\n\n\n\n\n\n\n\n\nMany theoretical studies that have aimed to quantify the competitive dynamics, the richness and the abundance distributions in ecological populations applied to  various systems, commonly employ a small number of paragidmatic models.\nOne common model of ecological competition is the deterministic, competitive Lotka-Voltera (LV) model, which has been especially useful in characterizing the niche regime by describing stable species coexistence as stable fixed points of the model. \nDepending on the ratios of inter- and intra-species competition strengths, deterministic LV models provide examples of both the `niche-like' regimes of multiple species coexistence, and the competitive exclusion where species with weaker intra-species interactions drive others to extinction \\cite{hardin1960competitive,macarthur1967limiting,MacArthur1969species, gause2019struggle}.\nIn complex scenarios, such as when the strengths of inter-specific interactions are randomly distributed among different species pairs, multi-species deterministic LV models can exhibit not only deterministic fixed point coexistence but also chaotic behavior reflected in the SAD shapes and richness \\cite{scheffer2006self,vergnon2012emergent,kessler2015generalized,bunin2016interaction,roy2020complex}. \nBeyond disorder in the interaction network, dynamical noise from various sources - both extrinsic and intrinsic - can have important effects on the system composition and dynamics, especially in the neutral regime.\nIn order to capture experimentally observed stochastic fluctuations of population abundances,  environmental noise is often introduced into the mathematical models \\cite{fisher2014transition,lynch2015ecology,verberk2011explaining,fowler2013colonization,barabas2016effect}.\nIn particular, by tuning the strength of environmental noise the shape of the SAD can change from unimodal to bimodal \\cite{fisher2014transition}, indicating a transition between `niche-like' and `neutral-like' regimes.\n\nRegardless of the presence of the external environmental noise or randomness in the interaction network, the demographic noise - the inherent randomness of birth and death events - is ever-present and has fundamental impact on the community structure and stochastic population dynamics \\cite{hubbell2001unified,alonso2006merits,haegeman2011mathematical}. \n\nIn particular, demographic noise has been suggested to be responsible for the SAD shape in neutral systems; these are characterized by the power law decay with an exponential cutoff that may account for `rare biosphere' abundance and distributions observed in many experimental systems \\cite{hubbell2001unified,baxter2007exact,mckane2004analytic}. On the other hand, birth-death-immigration processes with demographic noise have also been shown to exhibit bimodal SADs at very low immigration rates \\cite{xu2018immigration} breaking from the paradigm wherein neutrality is synonymous to an SAD of `rare biosphere' type.\nAlthough demographic noise models have been shown to reproduce observed features of a number of ecological systems \\cite{haegeman2011mathematical,capitan2015similar,capitan2017stochastic,capitan2020competitive}, a complete picture of the different regimes of community structures, is still missing. \nIn particular, it remains to be fully understood how the interplay of the competition strength, the immigration rate, demographic noise and the resulting dynamics of species turnover shape transitions between these different community structure regimes.\n\n\n\n\\begin{figure}[t!]\n   \\begin{flushleft}\n        A\n   \\end{flushleft}\n    \\includegraphics[width=\\columnwidth]{figures\/neutral-vs-niche.pdf}\n    \\begin{flushleft}\n        B\n   \\end{flushleft}\n    \\includegraphics[width=\\columnwidth]{figures\/island-2.pdf}\n    \\caption{Island model.  Panel A: Conventionally, weak competition is associated with `niche-like' bimodal SAD, while strong competition is linked to `neutral-like' monotonously decreasing SAD. However, this paradigm is not complete, since the dependence on other parameters, such as immigration rate $\\mu$ or diversity $S$, is not fully investigated. Thus, the entire phase space, e.g. $(\\mu, \\rho)$ or $(S, \\rho)$,  remains unexplored. Panel B: The model illustration. An island with $J$ individuals from $S^*$ species.  Each individual may proliferate and die with some rate corresponding to inter- and intraspecific interactions within the island. Here we consider deterministic, symmetric, fully-connected interspecific interactions network, governed by single parameter; the competitive overlap $\\rho$. Additionally, individuals may migrate from a cloud\/mainland, contains $S$ species, into the island with a constant rate $\\mu$.       }\n    \\label{fig:fig1}\n\\end{figure}\n\nIn this paper, we systematically study the full parameter space of the community composition and structure using a competitive LV model with the demographic noise and an interaction network of minimal complexity; more complex scenarios may be examined by building on this paradigmatic null model. We show that, beyond the perception of dichotomous neutral-niche regimes, many different regimes of richness and shape of the abundance distribution emerge from the interplay between the competition strength and immigration in the presence of stochasticity as illustrated in Fig.~\\ref{fig:fig1}).\nThese regimes exhibit contrasting dynamics that underpin the differences in the community structures in different regimes, and the transitions between them. \n\nThe paper is structured as follows. In Sec.\\ref{sec:model} we introduce the minimal model. In Sec.~\\ref{sec:results} we present our main results, including the regimes boundaries, their richness and the abundance distributions, as well as their associated underlying dynamics, and the species correlation structure. Lastly, in Sec.~\\ref{sec:discussion}, we discuss our results in the context of experimental observations. \n\n\n\n\n\\section{Mathematical models and methods}\n\\label{sec:model}\n\n\n\nThe minimal model studied in this paper incorporates three essential features of the ecological processes: competitive interactions, immigration and intrinsic demographic noise~\\cite{black2012stochastic,haegeman2011mathematical}.\nIn the model, illustrated  in Fig.~\\ref{fig:fig1}B, the community composition is characterized by the  species abundances, $\\vec{n}=(n_1,\\dots n_i\\dots n_S)$ where the discrete random variable $n_i$ represents the  number of individuals of the $i$-th species, and $S$ is the total number of species.\nThe dynamics of the system are described by a birth-death process with interactions, whereby the abundance (number of individuals) of any species can increase by one with the birth rate $q^+$ or decrease by one with the death rate $q^-$ defined as \n\\begin{align}\nq_i^+(\\vec{n})&=r^+ n_i +\\mu,  \\\\\nq_i^-(\\vec{n})&=r^- n_i + \\frac{r}{K} n_i \\left(n_i +\\sum_{j\\neq i} \\rho _{j,i} n_j\\right) \\nonumber\n\\end{align}\nfor each species $i\\in \\{1,2,\\dots,S\\}$.\n\nThe birth rate incorporates two factors: the per-capita birth rate $r^+$ corresponding to procreation, and the constant and positive immigration rate $\\mu$ from an external basin which ensures that the system possesses no global absorbing extinction state \\cite{capitan2015similar}. \nThe death rates include the `bare' per-capita death rate of the organisms $r^-$ and the competitive interactions effects that increase the mortality at high population numbers, incorporated through a quadratic term in the death rates; Parameter $\\rho_{j,i}$ quantifies the competition strength  between species $i$ and $j$.\nThe carrying capacity for each species is represented by $K$.\nThe per-capita turnover rate is $r=r^+-r^-$.\n\nThese aggregate coarse-grained parameters are determined by a variety of system factors such as the efficiency of resource consumption, interactions with the environment and external forces. Although it is possible to derive these rates from explicit resource competition models in several special cases, the expressions are highly model-dependent and are not explicitly modeled here\n~\\cite{macArthur1970species,chesson1990macarthur,o2018whence}.\nFor biological reasons,  $K$, $r^+$, $r^- > 0$ are all positive, which results in strictly positive transition rates for all $ n_i\\geq 0$. In this paper, we focus on the homogeneous case where the parameters ($\\mu$, $K$, $\\rho$, $r^+$, and $r^-$) are identical for all species and the competitive interactions  $\\forall i,j :\\rho_{j,i}=\\rho$ for all species pairs. \nThis symmetric and homogeneous interaction network \nhas been used in \\cite{badali2020effects,capitan2017stochastic,capitan2020competitive,haegeman2011mathematical} in contrast to the models   wherein the competition strengths are inhomogeneous and drawn from a distribution \\cite{fisher2014transition,allesina2012stability}. \nThis  minimal complexity model allows us to investigate the full phase space of the system to examine the underlying principle without impractical multi-parameter sweeps.\n\nThe stochastic evolution of the system is described by the master equation\n\\iffalse\n\\begin{multline}\n\\label{master-eq}\n\\partial_t  {\\rm \\mathcal{P}}(\\vec{n};t)= \\sum_{i}\\left\\{ \\vphantom{\\left[ q^+_i\\right]} q^+_i (\\vec{n}-\\vec{e}_i){\\rm \\mathcal{P}}(\\vec{n}-\\vec{e}_i;t) \\right.\\\\\n+q^-_i (\\vec{n}+\\vec{e}_i){\\rm \\mathcal{P}}(\\vec{n}+\\vec{e}_i;t)\\\\ \n-\\left. \\left[q^+_i(\\vec{n})+q^-_i(\\vec{n})\\right]{\\rm \\mathcal{P}}(\\vec{n};t)\n\\right\\}\n\\end{multline}\n\\fi\n\\begin{multline}\n\\label{master-eq}\n\\partial_t  {\\rm \\mathcal{P}}(\\vec{n};t)= \\sum_{i}\\left\\{ -\\left[q^+_i(\\vec{n})+q^-_i(\\vec{n})\\right]{\\rm \\mathcal{P}}(\\vec{n};t) \\vphantom{\\left[ \\sum q^+_i\\right]} \\right.\\\\\n\\left. \\vphantom{\\left[ \\sum q^+_i\\right]} +q^+_i (\\vec{n}-\\vec{e}_i){\\rm \\mathcal{P}}(\\vec{n}-\\vec{e}_i;t)+q^-_i (\\vec{n}+\\vec{e}_i){\\rm \\mathcal{P}}(\\vec{n}+\\vec{e}_i;t)\\right\\},\n\\end{multline}\nwhere $\\vec{e}_i$ is the standard basis vector and ${\\rm \\mathcal{P}}(\\vec{n},t)$ is the joint probability density function for the system to exhibit the species composition $\\vec{n}$ at time $t$ \\cite{gardiner1985handbook}. In the long time limit, the system reaches a stationary state where $\\partial_t \\mathcal{P}=0$.\n\nThe species abundance distribution (SAD) describing the mean fractions of species with $n$ individuals, can be related to the marginal single species probability distribution $P(n)$:\n\\begin{align}\n    {\\rm SAD}&(n) = \\frac{1}{S}\\left\\langle \\sum_{i=1}^{S}\\delta (n_i -n) \\right\\rangle\\\\\n   \\nonumber &=\\frac{1}{S}\\sum_{i=1}^S\\left[ \\sum_{n_1=0}^{\\infty}\\cdots \\sum_{n_{i-1}=0}^{\\infty}\\sum_{n_{i+1}=0}^{\\infty}\\cdots \\sum_{n_S=0}^{\\infty}{\\rm \\mathcal{P}}(\\vec{n})|_{n_i=n}\\right] \\\\  &=\\nonumber  P_i(n) \\equiv  P(n),\n\\end{align}\nwhere $\\delta$ is the Kronecker delta function, and using the fact that in this homogeneous system the marginal distributions $P_i(n)=P(n)$ of population abundance are identical for all species. \n\n\\iffalse\nThe full master of the Equation...  can be reduced to the one dimensional master equation for the marginal distribution $P(n)$ with effective birth-death rates (see SM for derivation)[AZ: is this an exact reduction, or there is an approximation involved?]\n\n\\begin{eqnarray}\nq^+(n)&=&r^+ n +\\mu,  \\\\\nq^-(n)&=&r^- n + \\frac{r}{K} n \\left((1-\\rho)n + \\rho \\langle J | n_i = n \\rangle \\right). \\nonumber\n\\end{eqnarray}\n\\fi\n\nIn the Fokker-Planck approximation, the continuous deterministic limit of the master equation (\\ref{master-eq}) recovers the well-known competitive Lotka-Volterra (LV) equations\n\\begin{align}\n    \\frac{\\partial x_i}{\\partial t}&= q_i^+(\\vec{x}) - q_i^-(\\vec{x})\\nonumber\\\\\n    &=r x_i \\left( 1 - \\frac{x_i}{K} + \\sum_{j\\neq i} \\rho \\frac{x_j}{K} \\right) + \\mu \n    \\label{eq:LV}\n\\end{align}\nfor the variable $x_i$, which corresponds to the continuous deterministic limit\nof the discrete variable $n_i$  \\cite{gardiner1985handbook}; see Supplementary Materials (SM) for further details. \n\nThe deterministic steady state is given by\n\\begin{equation}\n    \\tilde{x}(S) = \\frac{K}{2[1+\\rho(S-1)]}\\left\\{1 +  \\sqrt{1+\\frac{4\\mu[1+\\rho(S -1)]}{r K}}\\right\\}.\n    \\label{eq:fixed-LV}\n\\end{equation}\nNote that in the deterministic LV process all species survive with abundance $\\tilde{x}$ as long as $\\rho\\leq 1$ with $\\mu>0$ \\cite{capitan2015similar}.\nConversely, in the stochastic competitive environment the numbers of individuals of each species fluctuate, occasionally reaching extinction. \nThus, the number of (co-)existing species $S^*$ is a stochastic variable as well, and may be smaller than the overall number species $S$ in the immigration flux from the larger basin, with $S^*\\leq S$.\nThe richness, denoted as $\\langle S^* \\rangle $, is defined as the average number of the (co-)existing species, and is related to the SAD via\n\\begin{equation}\n\\label{eq:richness}\n\\langle S^* \\rangle = S(1-P(0))\n\\end{equation}\nwhich, intuitively, is determined by the probability that a species is present in the system,  $1-P(0)$.\n\nNo exact analytical solution for the high-dimensional master equation  \\eqref{master-eq} is known for a general competition strength $\\rho$. To understand the principles of the community organization and the impact of competition, immigration and demographic noise, we developed approximate analytical solutions to the master equation verified by Gillespie simulations (see SM for details).\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Mean-Field Approximation}\nThe full master equation \\eqref{master-eq} can be reduced to a one dimensional approximation for the marginal distribution $P(n)$ with effective birth-death rates (see SM). The SAD, $P(n)$, is obtained as a self-consistent stationary solution of this equation as\n\\begin{multline}\n   P(n) \\equiv P_i(n_i=n)\\\\\n   =P(0)\\frac{(r^+)^{n}(\\mu\/r^+)_{n}}{n!\\prod_{n_i=1}^{n}\\left(r^-+r n_i\/K+r\\rho \\sum_{j\\neq i}^S\\langle n_j |n_i \\rangle \/K\\right)}.\\\\\n   \\label{eq:mean-field}\n\\end{multline}\nTo obtain an analytical approximation to $P(n)$ we use a mean field closure for the unknown conditional averages $\\langle n_j |n_i\\rangle$ as\n$\\left\\langle \\sum_{j\\neq i} n_j |n_i\\right\\rangle\\approx  (S-1) \\langle n \\rangle$.\nThus, \\eqref{eq:mean-field} becomes a closed-form implicit equation for the probability distribution $P(n)$ which can be solved numerically.\nWe have found a good agreement between exact stochastic simulation results and this mean-field approximation for most of the parameter space examined (see SM). \n\nFollowing \\eqref{eq:richness}, the average richness in the mean-field approximation is (See SM)\n\\begin{eqnarray}\n\\langle S^*\\rangle = S \\left( 1 - \\frac{1}{{_1}F_1[a,b+1;c]}\\right ) , \n\\label{eq:mf-richness}\n\\end{eqnarray}\nwhere $P(0)=1\/{_1F_1}[a,b+1+1;c]$ is the normalization constant of $P(n)$ where ${_1F_1}[a,b;c] $ is the hypergeometric Kummer confluent function, with $a=\\mu\/r^+$, ${b}=\n[r^-K+r\\rho (S-1)\\langle n\\rangle  ]\/r$, and ${c}\n{r^+ K}\/{r}$.\n\n\n\n\n \n  \n\n\n\n\\subsection{The system exhibits rich behavior with distinct regimes of population structures controlled by competition strength, immigration rate and the species number} \n\\label{sec:Phases}\n\nDepending on the values of the competitive strength and the immigration rate, the number of species and the system size, the population can exhibit a number of different regimes of behavior, which can be categorized by their richness and the shape of their SAD, as visualized in Fig.~\\ref{fig:phases_sim} and described below.\n.\n\n\\subsubsection{Richness regimes}\nIn the classical deterministic LV model, the systems exhibits either an interior fixed-point with full coexistence of all species at abundances given by Eq.~\\ref{eq:fixed-LV}, or mass extinction with a single surviving species, in agreement with the well-known Gause's law of deterministic competitive exclusion \\cite{capitan2015similar}. By contrast, the stochastic model may also exhibit partial coexistence due to the\nabundance fluctuations arising from the demographic noise whereby a subset of the species are driven to temporary extinction. Overall, the number of co-existing species and their abundances are determined by the balance between the immigration and stochastic competitive exclusion events.\nThree distinct richness regimes can be discerned as shown in Fig.~\\ref{fig:phases_sim}, based on the variations of the richness of the system {$\\langle S^* \\rangle$} in different regimes in the ($\\rho$,$\\mu$,$S$) parameter space.\n\nAt low competition strength - region (a) in Fig.~\\ref{fig:phases_sim}A - all species co-exist so that the richness of the system is equal to the number of species, same as in the deterministic regime $\\langle S^*\\rangle \\approx S$. \nIn this regime, each species effectively inhabits its own niche because the inter-species competition is not sufficiently strong  to drive any of the species to extinction even in the presence of abundance fluctuations arising from the demographic noise.\nThe probability for a species to be present is determined by the balance of its immigration rate and the death rate.\nAt higher immigration rates this regime extends into regions with higher competition strength $\\rho$: high immigration rates stabilize full richness populations even with a relatively high competition strength.\n\n\n\nIn the second regime - region (b) in Fig.~\\ref{fig:phases_sim}A - only a fraction of the species are simultaneously present on average, which we denote as the partial coexistence regime.\nIn this regime, the immigration influx is not high enough to prevent temporary stochastic extinctions of some species resulting from the competition.\n\nAt the very high competition strengths a complete exclusion regime - region (c) in Fig.~\\ref{fig:phases_sim}A - is found. \nHigh competition along with the very low immigration rates act in unison such that the richness is less than two species.\nAlthough regime (c) may appear similar to regime (b) since both present partial coexistence, they are distinguished by key behavioral features as explained below. \n\nNote that the stochasticity is central to the  effect of the competition on the observed richness.\nStochastic fluctuations increase the risk of extinction with increasing competitive overlap, unlike in the deterministic case where the richness is independent of the interaction strength for $\\rho < 1$ \\cite{capitan2015similar}.\n\n\n\n\n\n\n\n\n\\subsubsection{SAD shape and modality regimes}\n\\begin{figure*}[ht!]\n    \\begin{minipage}{.30\\linewidth}\n   \\begin{flushleft}\n        A\n   \\end{flushleft}\n   \\includegraphics[width=\\textwidth]{figures\/Fig3-A.pdf}\n   \\end{minipage}\n    \\begin{minipage}{.65\\linewidth}\n   \\begin{flushleft}\n        B\n   \\end{flushleft}\n   \\includegraphics[width=\\textwidth]{figures\/Fig3-B.pdf}\n   \\end{minipage}\n    \\begin{minipage}[t]{.48\\textwidth}\n   \\begin{flushleft}\n    C\n    \\end{flushleft}\n    \\vspace{0pt}\n    \\includegraphics[width=\\textwidth]{figures\/Fig3-C.pdf}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[t]{.48\\textwidth}\n    \\begin{flushright}\n        \\begin{flushleft}\n        D\n        \\end{flushleft}\n    \\vspace{8pt}\n    \\includegraphics[width=\\textwidth]{figures\/Fig3-D.pdf}\n    \\end{flushright} \n  \\end{minipage}\n    \\caption{Phenomenology of the population structures. Panel \\textbf{A}: The system possesses three distinct richness phases. (a): full coexistence of all the species $\\langle S^* \\rangle \\approx S$; (b): partial coexistence with $\\langle S^* \\rangle < S$; (c): a single species exists on average. Panel \\textbf{B}: Different population regimes are distinguished by different SAD modalities. (I): immigration dominated regime with unimodal SAD at a typical abundance given by the positive root of $\\tilde{n}$; (II): bimodal regime with species at non-zero abundance $\\tilde{n}$ and a rapid species turnover peak a zero abundance; (III): `rare biosphere' regime of a unimodal SAD with peak at zero abundance resulting from the rapid turnover of the temporarily extinct species; (IV) multimodal regime.\n    Panels \\textbf{C} and \\textbf{D}: Intersection of the modality and richness regimes in the $(\\mu,\\rho)$ plane (\\textbf{ C}) and $(S,\\rho)$ plane (\\textbf{D}); see text for discussion. In panels \\textbf{ A}, \\textbf{B} and \\textbf{C} the number of  species $S=30$.  In panel \\textbf{D} the immigration rate is $\\mu=10^{-1}$. For all panels; Colored regions represent data from simulation (see Methods), whereas boundaries from the mean-field approximation are represented by solid black lines.\n   \n    The solution for the master equation \\eqref{master-eq} is simulated using the Gillespie algorithm with $6 \\cdot 10^8$ time steps, $r^+=2$, $r^-=1$, and $K=100$.\n   \n    }\n    \\label{fig:phases_sim}\n\\end{figure*}\nBesides the richness, the balance between immigration and stochastic competitive extinctions  also dictates the mean abundances of individual species and the species abundance distribution (SAD).\nWhen the immigration influx of individuals into the system is higher than the average out-flux due to the transient extinctions, shown in Fig.~\\ref{fig:phases_sim}B as region (I), most species are forced away from extinction.\nIn this regime, the SAD is unimodal with a peak at relatively high species abundances $\\tilde{n}$  which is approximately located at \n\\begin{equation}\n    \\label{eq:dom-level}\n    \\tilde{n} =\n    \\frac{K-\\rho(S-1)\\langle n\\rangle}{2}\\left\\{1 \\pm \\sqrt{1+4\\frac{(\\mu-r^+) K}{r(K-\\rho(S-1)\\langle n\\rangle)^2}}\\right\\},\n\\end{equation}\nwhich agrees with the simulation results, see Fig.~\\ref{fig:Fig3}; see  SM.\n\n\n\nAt lower immigration rates - regime (II) in Fig.~\\ref{fig:phases_sim}B - the immigration rate is insufficiently strong to overcome the competition driven temporary extinctions of some of the species, and the SAD develops an additional peak around $n=0$ corresponding to the temporarily extinct species. \nThe subset of the `quasi-stably' co-existing species whose abundances fluctuate around $\\tilde{n}$ within the high `niche-like' abundance peak dominate the population number, punctuated by rare  fluctuation-driven extinctions and the occasional invasion of a temporarily extinct species into the dominant population.\nBy contrast, the dynamics of species in the $n=0$ zero peak is characterized by the rapid turnover of the remaining species close to extinction.\n\n\n\n\nAt low immigration rates, the peak at \\eqref{eq:dom-level} coincides with the deterministic stable solution in \\eqref{eq:fixed-LV}  (see SM)\n\\begin{equation}\n\\lim_{\\mu\\rightarrow 0}\\tilde{n}=\\lim_{\\mu\\rightarrow 0}\\tilde{x}\\left(\\langle S^* \\rangle \\right) = \\frac{K}{1+\\rho(\\langle S^*\\rangle -1)}.\n\\label{eq:Lim_ss}\n\\end{equation}\nNamely, in the bimodal regime the coexisting dominant species are fluctuating around $\\tilde{n}$ which, at low immigration, is the deterministic fixed point with $\\langle S^* \\rangle$ species. \nThus, the dynamics of abundant species around the $\\tilde{n}$ can be heuristically understood as spatially dependent diffusion in an effective potential well of the Fokker-Plank Equation (See Sec.\\ref{sec:model} and SM). \n\nSomewhat unexpectedly, at low immigration rate $\\mu \\lesssim .05$, the bimodal regime extends onto the neutral line at $\\rho = 1$  where the SAD has been commonly believed to have the monotonically decreasing `rare biosphere' shape \\cite{hubbell2001unified,baxter2007exact}.\nHowever, in this regime the competition is so strong that most of the time either no species are present at high abundance, or only one species survives in a kinetically `frozen' long lived quasi-stable state with the abundance $\\tilde{n}\\simeq K$, as observed previously \\cite{xu2018immigration}. \n\nFurthermore, at the intermediate immigration rates and relatively high competition strengths we observe a unimodal behaviour with a peak at zero rather than at finite $\\tilde{n}$ - region (III) in Fig.~\\ref{fig:phases_sim}B.\nIn this regime, the competition is strong enough so that the fluctuations competitively drive populations\nto temporary extinction before any species is able to establish a `quasi-stable' state at a high abundance.\nAll species undergo rapid turnover around zero resulting from the balance between random immigration and extinction events.\nThis regime corresponds to what was previously described as the `rare biosphere': fewer number of species are found at higher abundances resulting in a monotonically decreasing SAD. \nThis SAD shape is classically recognized as a hallmark of a `neutral-like' regime. However, as shown in Figure~\\ref{fig:phases_sim} the unimodal regime (III) unexpectedly extends substantially beyond the neutral manifold $\\rho=1$, into the non-neutral regions with $\\rho < 1$, and the monotonic-decreasing SAD persists even for competitions strengths as low as $\\rho \\approx 0.1$ - an order of magnitude weaker than the classical neutral regime.\nThis challenges the common perception that the SAD of the `rare biosphere' is necessarily closely related to neutrality.\n\nFinally, we have found an entirely novel multimodal regime with more than two peaks - regime (IV) in Fig.~\\ref{fig:phases_sim} -  which possesses one rapid turnover peak around extinction and multiple peaks at non-zero abundances. \nSimilar to region (IIc), the peak at $n=0$ comprises species which rapidly turnover around extinction. However, in addition to the peak at positive abundances $K$ formed by one surviving species ($S^*=1$) in a meta-stable frozen state, this regime possesses a second peak  at $\\sim K\/(1+\\rho)$ with two simultaneously surviving quasi-stable species ($S^*=2$), following \\eqref{eq:Lim_ss}. The slow fluctuations between the states with $S^*=1$ and $S^*=2$ result in the appearance of the SAD with two non-zero modes at quasi-stable dominance abundance, $\\tilde{n} \\sim K$ and $\\tilde{n}\\sim K\/(1+\\rho)$ observed in the region (IV); these two peaks are only visibly separated when the richness is low and carrying capacity is high, as follows from \\eqref{eq:Lim_ss}.\n  \n\n\n\n\n\nThe transitions between the different regimes and the corresponding changes in the SAD shapes are illustrated in Fig. ~\\ref{fig:Fig3}. Generally, at low competition strength $\\rho$, where the species are practically independent of each other and reside in largely non-overlapping niches, their typical abundance $\\tilde{n}$ is close to the carrying capacity $K$. Increasing competition strength $\\rho$ makes it harder to sustain co-existing species at high abundances, and accordingly $\\tilde{n}$ decreases, as illustrated in the top panels of Fig.~\\ref{fig:Fig3}A) and  Fig.~\\ref{fig:Fig3}B). With further increase in $\\rho$ the system behavior bifurcates depending on the immigration rats $\\mu$. \nAt high immigration rates, $\\mu \\gtrsim 0.05$, the competition-driven decrease in $\\tilde{n}$ continues up to the critical competition strength (calculated in the next section) where the peak around $\\tilde{n}$ disappears (top right panel of Fig.~\\ref{fig:Fig3}A) and  Fig.~\\ref{fig:Fig3}B), as the system is not able to sustain `quasi-stable' niche-like species co-existence. This corresponds to the transition from bimodal region (II) to the `rare biosphere' neutral-like region (III) in Fig.~\\ref{fig:phases_sim}).\nAt lower immigration rates (top left panel of Fig.~\\ref{fig:Fig3}A) and  Fig.~\\ref{fig:Fig3}B)), further increases in the competition strength eventually cause mass species extinctions which allows the remaining few dominant species to maintain higher abundances (region (III) Fig.~\\ref{fig:phases_sim}). As $\\rho \\rightarrow 1$, the system transitions to the region (IIc) of the Fig.~\\ref{fig:phases_sim}: only one dominant species remains, as described in \\cite{xu2018immigration}, with abundance $K$. \n\n\n\\subsubsection{Global Phase Diagram and Regime Boundaries}\n\\label{sec:regime-bound}\n\n\\begin{figure}[t!]\n   \\begin{flushleft}\n        A\n   \\end{flushleft}\n  \n   \n    \\includegraphics{figures\/Fig2-A.pdf}\n    \\begin{flushleft}\n        B\n   \\end{flushleft}\n   \n    \\includegraphics[width=\\columnwidth, trim= 0 0 0 15,clip]{figures\/Fig2-B.pdf}\n    \\caption{SAD changes between different regimes. Panel \\textbf{A}: (upper left) Simulation results for species abundance distributions (SADs) for fixed $\\mu=10^{-3}$ as a function of $\\rho$. (upper right) same for $\\mu=1$. Different values of the interaction strength  $\\rho$ are emphasized with different colors indicated in the color-bar. (lower left) Simulation results for SADs as a function of $\\mu$ for fixed $\\rho=0.5$  (lower right) same for $\\rho=1$. Different immigration rates $\\mu$ are emphasised with different color shown in the color-bar. \n    \\textbf{B}: The non-zero mode of the SAD  given by the positive solution of  $\\tilde{n}$ representing the dominant species abundance as a function of $\\rho$ for different values of $\\mu$. Markers and dotted lines represent simulation results, while solid lines are given from analytic analysis, \\eqref{eq:dom-level}.}\n \\label{fig:Fig3}\n\\end{figure}\n\nIn this section we describe the complete phase diagram of the system defined by the intersection of the different richness and the SAD shape\/modality regimes, derive the regime boundaries and discuss the transitions between them, as shown in the ($\\mu,\\rho$) space in Fig.~\\ref{fig:phases_sim}C, and in ($S,\\rho$) space in Fig.~\\ref{fig:phases_sim}D.\nWe show that the boundaries between different regimes observed in simulations can be understood within simple mean field theories, and discuss the underlying physical factors responsible for the transitions between different regimes.\n\n\\iffalse\n\\begin{center}\n\\begin{tabular}{lccc}\n\\hline\n\\multirow{2}{4em}{\\textcolor{black}{modality:}} & &\n      \\textcolor{black}{$R_n(\\tilde{x} \\rightarrow \\tilde{x} +1)=R_n(\\tilde{x}+1 \\rightarrow \\tilde{x})$} \\\\ &\n     & \\textcolor{black}{$R_n(0 \\rightarrow 1)=R_n(1 \\rightarrow 0)$}   \\\\\n    \\hline \n    \\multirow{2}{4em}{\\textcolor{black}{richness:}}  & & \\textcolor{black}{$R_{S^*}(1 \\rightarrow 2)=R_{S^*}(2 \\rightarrow 1)$} \\\\ &\n     & \\textcolor{black}{$R_{S^*}(S \\rightarrow S-1)=R_{S^*}(S-1 \\rightarrow S)$}\n\\\\\n\\hline\n\\end{tabular}    \n\\end{center}\n\n\\fi\n\nWe define the boundary between the full coexistence (a) and partial coexistence (b) regimes to be at $\\langle S^* \\rangle=S-1\/2$: the midpoint between full richness $S^*=S$ and the loss of 1 species on average.\nSimilarly, the boundary between the partial coexistence (b) and exclusion (c) regimes is located at $\\langle S^* \\rangle=3\/2$, that is to say where the richness is between one and two species such that on average only 1 species is present in regime (c).\n\nTo derive the boundaries corresponding to the transitions of the SAD modality regimes, we use discrete derivatives of the approximated SAD to determine the existence of peaks and their location (See SM).\nThe immigration dominated regime (I) is characterized by a unimodal SAD with a peak at the positive root of $\\tilde{n}$ given in \\eqref{eq:dom-level}.\nCompared to this immigration dominated regime, the neighboring bimodal and monotonically-decreasing unimodal regimes - regions (II) and (III) respectively - differ by the emergence of a new mode at zero abundance. \n\nThus, the boundary that defines transitions to either regime (II) or (III) from the immigration dominated regime (I) is described by a flattening of SAD at $n=0$: $\\partial P(n)\/ \\partial n|_{n=0} = 0$ which, in the discrete case, heuristically corresponds to $P(0)=P(1)$. Combining this condition for the boundary with the global-balance of the master equation \\eqref{master-eq}  results in the rate balance equation,\n$\\langle q_i^+(\\vec{n})|n_i=0 \\rangle=\\langle q_i^-(\\vec{n})|n_i=1 \\rangle$.\n\nIn the mean-field approximation, this boundary is found at\n\\begin{equation}\n    \\label{eq:boundary-I}\n    \\mu = r^- +\\frac{r}{K}[1+\\rho(S-1)\\langle n \\rangle ].\n\\end{equation}\nThis equation recovers the similar transition for $\\rho=1$ derived independently in \\cite{xu2018immigration}. \n\nThe boundary between the bimodal regime (II) and the `rare biosphere' regime (III) is characterized by the disappearance of the peak at high abundance $\\tilde{n}$ in \\eqref{eq:dom-level}.\nIn the bimodal regime at least one solution to $\\tilde{n}$ is real and positive, then a maximal, real peak-location exists. \nConversely, in the `rare biosphere' regime, both solutions of $\\tilde{n}$ are negative or imaginary.  \nWe find that the boundary between the real and imaginary $\\tilde{n}$ is\n\\begin{equation}\n\\label{eq:boundary-IIIa}\nr(K-\\rho (S-1) \\langle n \\rangle )^2=4(r^+-\\mu){K}\n\\end{equation}\nand the transition line between positive and negative solutions, $\\tilde{n}=0$, is\n\\begin{equation}\n    \\frac{(K- \\rho (S-1) \\langle n \\rangle )^4}{16}=1+ \\frac{K(\\mu-r^+)}{r}.\n    \\label{eq:boundary-IIIb}\n\\end{equation}\nThe intersection of these two conditions defines the `rare biosphere' regime and is shown as the blue line in Fig.~\\ref{fig:phases_sim}B,C.\n\nThe modality and the richness of the system are also affected by the number of species $S$ as shown in  Fig.~\\ref{fig:phases_sim}D.\nIn brief, the frequency of the immigration events rises as more species are present in the immigration flux.\nIncreased immigration causes the total population to rise without providing more room for each species in the system; this  increases the stochastic competition, driving more species to extinction.\nHence, as $S$ increases, the transition from the bimodal regime (II) to the unimodal regime (III) occurs at lower values of competition strength $\\rho$, and the fraction of the concurrently surviving species decreases. This effect has been qualitatively observed experimentally \\cite{gore2021}, and we return to it in the Discussion.\n\nThese analytical expressions for the regime boundaries - confirmed by stochastic simulations - provide insights into the effects of different control parameters on the regime boundaries. In particular, using the low $\\mu$ deterministic approximation for $\\langle n \\rangle \\approx K \/ \\left[ 1+\\rho (S-1) \\right]$, shows that the location of the boundary of the `rare biosphere' regime grows proportionally to the carrying capacity and is a decreasing function of the number of species $S$. Thus, the size of the `rare biopshere' neutral-like regime increases with the number of species $S$ as shown in Fig.~\\ref{fig:Fig3}D,\nwhereas increasing the carrying capacity shrinks this regime (See SM).\n\n\n\n\n\n\n\n\n\n\\subsection{Kinetics of the species turnover, extinction and recovery underlie the transitions between different regimes}\n\\label{sec:Dynamics}\n\nSo far we have focused on the steady-state properties of the system, such as the dominant species abundance, SAD modality, and richness to categorize the different regimes. \nHowever, the transitions between different regimes are closely related to the underlying kinetics of species turnover and fluctuations, which are investigated in this section.\n\nThe kinetics of an individual species change drastically between the unimodal `rare biosphere' regime (III) and the regimes that exhibit a peak in SAD at non-zero abundance as shown in Fig.~\\ref{fig:turnover}A  that contrasts the kinetics in these cases. \nIn regime (III), all species undergoes rapid turnover in the relatively broad range of abundances around extinction.\nIn contrast, in the `niche-like' regimes (I, II, and IV) the `quasi-stable' dominant species undergo fast fluctuations around the co-existence peak at $\\tilde{n}$ in addition to fast turnover of the remaining species near extinction. The `niche-like' regimes also possess slow timescales corresponding to individual species leaving the high abundance peak as they are temporarily driven towards extinction, and the reverse invasions of temporarily extinct species into the dominant `niche-like' peak.\n\nTo characterize the differences in the kinetics in different regimes, we calculate the mean first-passage times (MFPT) of the  transitions between different abundance levels characterizing different regimes, denoting  the MFPT for a species transition from an abundance $a$ to another abundance $b$ as  $T(a \\rightarrow b)$.  Similarly, $T(a\\rightarrow a)$ refers to the mean time of return to an abundance $a$ having left that same abundance.\nThe MFPT is calculated from the one-dimensional backward Master equation (see SM).\n\nAlthough these times are important indicators of the system dynamics, the MFPT ratios of two processes\/events is more informative than the MFPT of each event separately; this ratio measures the discrepancy between the timescales at which these events occur. \n\nTo understand the intrinsic kinetics that give rise to the different regimes in Fig.~\\ref{fig:phases_sim}, we first focus on the ratio of the MFPTs of the transitions from dominance to exclusion over the mean time of return to the dominant abundance level (starting from the dominant abundance level), $T(\\tilde{x}(\\langle S^* \\rangle )\\rightarrow 0) \/ T(\\tilde{x}(\\langle S^* \\rangle )\\rightarrow \\tilde{x}(\\langle S^* \\rangle ))$, shown in (Fig.~\\ref{fig:turnover}B).\nRecall that $\\tilde{x}$ is the solution given in ~\\eqref{eq:fixed-LV}; this deterministic solution is a natural extension of the peak $\\tilde{n}$ in regimes where positive modes are non-existent.\nAs explained below, this ratio underlies the changes in the modality of the SAD as a function of the immigration rates and the competition strength.\n\nLarge values of the ratio $T(\\tilde{x}(\\langle S^* \\rangle )\\rightarrow 0)\/ T(\\tilde{x}(\\langle S^* \\rangle )\\rightarrow \\tilde{x}(\\langle S^* \\rangle ))$\nimply that the extinction rate from $\\tilde{x}(\\langle S^* \\rangle )$ is much slower than the rate of local fluctuations in the effective potential well around $\\tilde{x}(\\langle S^* \\rangle )$. Accordingly, Fig.~\\ref{fig:turnover}B shows that this ratio is high in the bimodal and immigration-dominated regimes. Conversely, this ratio is lower within the `rare biosphere' regime that does not possess a high abundance peak with `quasi-stable' co-existing species.\nThis ratio approximately delineates the `rare biosphere' neutral-like  regime from the `niche-like' regimes, as shown in Fig.~\\ref{fig:turnover}B: the contour lines qualitatively recover the boundaries of region IIIb in Fig.~\\ref{fig:phases_sim}C.\n\nThe second ratio, which underlies the richness transitions in the system, \n$T(0\\rightarrow \\tilde{x}(\\langle S^* \\rangle ))\/T(0\\rightarrow0)$ (Figure.~\\ref{fig:turnover} panel C) relates the mean return time to extinction to the invasion time from extinction at zero abundance  to dominance at $\\tilde{x}$. \nHigh values of this ratio indicate that, for an extinct species, the mean invasion times are longer than return times back to extinction.\nThis MFPT ratio approximates the ratio of the average number of temporarily extinct species, $S - \\langle S^* \\rangle$ to the average number of existing species, $\\langle S^* \\rangle$, see Fig.~\\ref{fig:turnover}C. \nConsequently, the ratio quantitatively recovers the boundaries of richness regimes in Fig.~\\ref{fig:phases_sim} in most regions of the parameter space.\nThese MFPT ratios can be understood as the reciprocal of the rates ratios which describe how much more frequently an event occurs than the other.\nFurther discussion on the dynamical features are presented in the SM.\n\n\\begin{figure*}[t!]\n    \\centering\n    \\begin{minipage}{11cm}\n    \\begin{flushleft}\n        A\n   \\end{flushleft}\\includegraphics{figures\/Fig4-A.pdf}\n   \\begin{flushleft}\n        B\n   \\end{flushleft}\n   \\includegraphics{figures\/Fig4-B.pdf}\n    \\begin{flushleft}\n        C\n   \\end{flushleft}\n        \\includegraphics{figures\/Fig4-C.pdf}\n    \\end{minipage}\n    \\caption{Kinetics of species extinction, invasion and turnover. Panel \\textbf{A}:  Sample of the trajectories of the species abundances (grey lines).  We highlight five species' trajectories for visibility.\n    (Upper panel): stable `niche-like' dynamics, where the abundances of the dominant species mostly fluctuate in the vicinity of $\\tilde{n}$, with occasional transitions from  the dominance to nearly-extinct states. The red curve represents the corresponding bimodal SAD. Lower panel shows the erratic dynamics obtained in the `rare biosphere' regime, where all species are rapidly fluctuating close to extinction. The SAD in this case is unimodal monotonically decreasing function.  Panel \\textbf{B}: The MFPT ratio $ T(\\tilde{x}\\rightarrow 0) \/ T(\\tilde{x}\\rightarrow \\tilde{x})$ as a function of $\\rho$ (left). For very weak immigration rates $\\mu\\approx 10^{-3}$ the ratio is non-monotonic in competition strength, revealing regime (c) in Fig.~\\ref{fig:phases_sim}. The MFPT ratios as a function of both $\\mu$ and $\\rho$ is represented with a color-map (right). \\textbf{C}:  The MFPT ratio $ T(0\\rightarrow \\tilde{x}) \/ T(0\\rightarrow 0)$ as a function of the competition overlap (left) and as a function of both $\\rho$ and $\\mu$ (right). This ratio qualitatively captures the richness behaviour in Fig.~\\ref{fig:phases_sim}. Both panels (B) and (C): the values are represented with the logarithmic color scale.}\n    \\label{fig:turnover}\n   \n\\end{figure*}\n\nOverall, the `niche-like' Regimes I, II and IV  are characterized by a relatively stable behavior; generally species stay longer about the\ndominant species abundances\npunctuated by the occasional crossings between dominance to nearly-extinct states and the reverse invasions from extinction into the dominance.\nThe transition between partial coexistence (b) to regime of competitive exclusion (c) is captured by the non-monotonic behaviour of the ratio as shown in Fig.~\\ref{fig:turnover}C.\nSurprisingly, unlike the other regimes, increasing the competition strength in the competitive exclusion regime (c) increases the stability of the dominant species abundance: return times to the dominant abundance are much shorter than the time to extinction for the single species in the frozen `quasi-stable' state.\nConversely, the `rare biosphere' regime (III) features rapid dynamics where species cycle rapidly between extinction and a broad range of abundances without establishing `quasi-stable' states with slow turnover. \nThese different dynamic types are illustrated by illustrative trajectory plots in Fig.~\\ref{fig:turnover}A. \n \n\n\n\n\n\n\n\n\n\n\\subsection{The abundances of different species are weakly anti-correlated}\n\\label{sec:Correlation}\n\nSo far, we have investigated the single species marginal abundance distribution $P(n)$ and the average richness $\\langle S^*\\rangle $. However,\nthe species are not independent of each other due to the inter-species competitive interactions. It has been suggested that inter-species correlations reflect on the underlying community structure and the phase space \\cite{carr2019use,chance2019native}\nTo investigate the connection between the population structure and the cross-species correlations, we calculated the cross-species abundances correlations, quantified via the Pearson correlation coefficient, as shown in Fig.~\\ref{fig:correlation}A. The population  exhibits weak cross-species anti-correlation that increases with the competition strength $\\rho$. This is expected given that the death rate of each species increases with the abundance of the other species and, consequently, these cross-species influences are more pronounced at high competition strengths.\nConversely, higher immigration rates ensure that the abundance fluctuations of different species are less likely to be correlated with those of other species.\nThus, the anti-correlation is most pronounced in the high competition and low immigration regime\n\n\nFurthermore, the impact of individual species on the total population size varies between community structures, which can be quantified by the correlation between the total population size $J$ and individual abundances.\nWe found that the individual species abundances are positively correlated with the total abundance $J=\\sum_i n_i$, which also fluctuates as the individuals of all species undergo birth and death events.\nInterestingly, as shown in Fig.~\\ref{fig:correlation}B, the magnitude of this correlation $\\text{cov}(J,n_i)\/\\sigma_n \\sigma_J$ exhibits inverse \ntrends compared to the inter-species anti-correlation: the correlation  $\\text{cov}(J,n_i)\/\\sigma_n \\sigma_J$ is weaker when the cross-species anti-correlation is stronger.\nThe magnitude of the correlation between the total population size $J$ and a species abundance $n$ exhibits similar behavior to the average richness: $\\text{cov}(J,n_i)\/\\sigma_n \\sigma_J$ is high in the high immigration, low competitive overlap regime and is low otherwise.\nThis behaviour may be understood heuristically:\nwhereas each species in a system with $S^*$ dominant species only contributes $\\sim J\/S^*$ to the total population size. \n\nSomewhat unexpectedly, neither the inter-species correlations nor the correlations between the species abundance and the total abundance distinguish between the different modality regimes but rather both increase with the richness. As expected, our mean-field approximation works best at very low ${\\rm cov}(n_i,n_j)\/\\sigma_{n_i}\\sigma_{n_j}$, whereas our mean-field deviates from the solution at anti-correlation gets stronger (see SM). \n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{flushleft}\n        A\n    \\end{flushleft}\n    \\includegraphics[width=0.75\\columnwidth,trim= 10 10 10 10, clip]{figures\/Fig5-A.pdf}\n     \\begin{flushleft}\n        B\n    \\end{flushleft}\n    \\includegraphics[width=0.75\\columnwidth,trim= 10 10 10 10, clip]{figures\/Fig5-B.pdf}\n    \\caption{Abundance correlations. Panel \\textbf{A}: Pearson correlation coefficient between the abundances of any two different species. Panel \\textbf{B}: Pearson correlation coefficient between  the total population size $J=\\sum n_j$ and an abundance of any species. Correlations were calculated from Gillespie simulation time course data with $6 \\cdot 10^8$ time steps\n    }\n    \\label{fig:correlation}\n\\end{figure}\n\n\\section{Summary and Discussion}\n\\label{sec:discussion}\n\nEcological systems display a wide variety of different behavior regimes that have been commonly analysed through a limited number of paradigmatic models such as the `niche' and `neutral' theories. However, it remains incompletely understood what features of ecological population structure and dynamics are universal and which are system specific, how different models relate to  each other, and what behavior is expected in the full range of the parameter space. Using a minimal model of the competitive population dynamics with demographic noise, we have investigated\nthe different regimes of the population structures and dynamics as a function of the immigration rate $\\mu$, competitive overlap $\\rho$, and the number of species $S$.\nAlthough this minimal model may not fully capture the more complex interaction structures of many ecological communities, it already exhibiting rich and unexpected behaviours paralleling many experimentally observed ones (see Table \\ref{tab:my_label} below and Table S1 in SM), and illuminates the underlying mechanisms that shape population structures in different ecosystems.\n\nWe have focused on the system richness reflecting the number of the co-existing species, and the SAD shape as the characteristics of the different population regimes, using a combination of simulations and analytical mean-field approaches. Our analysis shows that the ecosystem behaviors can be partitioned into different regimes of richness and SAD shape\/modality, parameterized by the immigration rate and the competitive overlap\n\nOur model recovers the expected limits of  the well known `neutral-like' and the `niche-like' regimes. In particular, at $\\rho = 1$ and intermediate values of $\\mu$, the SAD has the `rare-biosphere' monotonously decreasing shape characteristic of the classical neutral regime. On the other hand, at low competition strength, the system SAD exhibits a peak at high species abundance where all species co-exist, effectively occupying distinct ecological niches. \n\nNote that even independent species with no inter-species competition with $\\rho=0$ may present either a unimodal or bimodal SAD depending on the immigration rate, see Fig.~\\ref{fig:phases_sim}B\\&C. Unlike the immigration dominated high abundance peak at high immigration rates, at the very low immigration rates the SAD is peaked around zero due to high extinction probability solely from the intra-species competition\n\nChanges in the SAD between different regimes, reflected in the changes in the locations and the heights of the SAD peaks as a function of the immigration and the competition strength, occur through different routes.\nFor instance, starting at the bimodal regime (II) at $\\rho=1$, as the immigration rate increases,  the SAD peak height gradually decreases without changing significantly its location, until it finally disappears at the boundary of the `rare biosphere' regime (III). \nBy contrast, at lower competition strengths $\\rho<1$, the transition from the bi-modality to the `rare biosphere' regime occurs via   simultaneous changes in the peak's height and location. This is discussed in Sec.~\\ref{sec:results}B.\n\nWe found that for the intermediate immigration rates the system maintains the monotonically decaying `neutral-like' SAD in regime (III) even at rather low competition strength ($\\rho \\approx 0.1$) contrary to the common expectation that different species inhabit separate niches away from neutrality. Conversely, at low immigration rates, the system SAD unexpectedly maintains the peak at non-zero abundance characteristic of `niche-like' regimes even for the high values of the competition strength $\\rho$ usually considered to be in the `neutral-like' domain (regime (IIc)); see Sec.~\\ref{sec:results}B and Figure \\ref{fig:phases_sim}.\n\nWe have also uncovered an unusual - and to the best of our knowledge hitherto not described - regime characterized by the multi-modal SAD with more than one positive, `quasi-stable' abundance peak (Regime (IV) in Fig~\\ref{fig:phases_sim}). This multi-modality arises from the richness fluctuations in this regime: the number of co-existing species is switching randomly between two relatively long-lasting states with  $S^*=1$ and $S^*=2$. Thus, one peak of the SAD is found around $\\sim K$ and the other one in the vicinity of $\\sim K\/2$, as explained in Sec.~\\ref{sec:results}B. We observe that for low $K$, the multimodal regime is non-existent and appears as $K$ increases; see the corresponding phase diagrams in SM.\n\nWe show that the population structures in different regimes stem from  the underlying dynamics of species fluctuations, extinctions and invasions. In the `rare biosphere' neutral-like regimes, all species undergo relatively fast turnover around extinction. This is reflected in the low ratio of the turnover to the extinction mean first-passage times. Conversely, in the `niche-like' regimes the system develops two additional time scales: relatively fast fluctuations about the high abundance peak, and the long waiting times for the transitions from the `quasi-stable' co-existence at high abundance to extinction.  Accordingly, the ratio of the mean extinction time to the mean time of return to dominance is higher in the `niche-like' regime, as discussed in Sec.~\\ref{sec:results}C. \n\nInterestingly, ecological regimes  akin to those predicted by our demographic noise model (except for the multimodal SAD regime) have been also found using deterministic, noiseless LV models with a random  matrix of inter-species competitive interaction strengths  \\cite{may1972will,allesina2008network,allesina2012stability,kessler2015generalized,gore2021}.\nHowever, the underlying mechanisms that give rise to the apparently similar regimes in the two model types are very different. In the demographic noise model, the partial richness `niche-like' regime is formed by the coexistence of some species at a positive abundance and temporary extinctions of other species induced by the stochastic abundance fluctuations. By contrast, in the deterministic LV models with random asymmetric interactions, this regime is formed from a large number of fixed points where different sets of species are deterministically excluded. At higher interaction strengths, the system transitions to the deterministically chaotic behavior that resembles the `neutral-like' regimes, however the nature of the species turnover and the SAD shape are very different to the behaviour we described above \\cite{bunin2017ecological,kessler2015generalized,gore2021}\n\nThe existence of the predicted regimes and the transitions between them can be tested experimentally by measuring the SAD and the dynamics of the species abundances in ecosystems with varying immigration and competition strengths.\nLong-term observations may provide measurements of the stationary species abundance distributions \\cite{weigelt2010jena} although the steady-state SAD may be difficult to estimate due to the limited amount of data.\nIt may be  difficult to experimentally determine and control the immigration rate, the competition strength, and carrying capacity, but practically useful proxies for these parameters exist. For example, the flow rate carrying bacteria into a chamber of a microfluidic device is a well controlled quantity that approximates well the immigration rate for populations encased in the chamber~\\cite{duran2021slipstreamin}.\nFurthermore,   measurements of the SADs and the community compositions have become more attainable due to the advances in single cell gene sequencing techniques \\cite{ratzke2020strength, gore2021,shakiba2019cell}. Another commonly used and robustly  estimated experimental observable is the species rank abundance (SRA), which can be used to infer the SAD to which it is closely mathematically related, although in practice the conversion might be constrained by limitations of noise and quantity of the experimental data. \n\nDespite these difficulties, the asymptotic behaviour of the SADs may provide indication of qualitative dissimilarities between the various regimes to discern different regimes of behavior among the experimental observations. In the mean-field approximation in our model the asymptotic behaviour of the SAD on the neutral line $\\rho=1$ approximates a power law with an exponential cutoff (see SM) - similar to the commonly used neutral birth-death models with the fixed total population size \\cite{baxter2007exact,mckane2004analytic}.\nNotably, the Yule process that is often used to model neutral processes also results in the SAD of a similar form. However, the Yule process is substantially different from the model of this paper because it does not include inter-species interactions and reaches the steady state SAD only if $r^+ < r^-$\n\nIn Table 1, we qualitatively compare the family of the regimes predicted by our model to the various behaviors inferred from experimental findings based on the SAD measurements and population abundance time series.\nThe notable abundance of the neutral ecosystems observed experimentally may pertain to our finding (Section~\\ref{sec:results}B) that the `neutral-like' `rare biosphere' regime extends substantially beyond the neutral line $\\rho =1$: non-neutral communities appear neutral as they exhibit SAD's characteristic of neutral communities, such as gastrointestinal microbiomes \n\\cite{jeraldo2012quantification}. Furthermore, multimodal SAD's predicted by our model that are related to the richness fluctuations may provide an explanation for the multimodal SADs observed in some ecological data,  complementary to the current explanations such as spatial heterogeneity or emergent neutrality \\cite{dornelas2008multiple,vergnon2012emergent}\n\n\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|l|l|\n   \n       System (Ref.) & Regimes  \\\\ \\hline \nmicrobial competition\\cite{gore2021}  & stable full coexistence (IIa) \\\\   &  stable partial coexistence (IIb)\\\\ & persistent fluctuation (IIIb) \\\\ \nglobal birds species \\cite{callaghan2021global} &\tunimodal - log skew (I) \\\\ \nplankton \\cite{ser2018ubiquitous}  &\tpower-law decay (III)  \\\\ \ncoral \\cite{dornelas2008multiple}\t& multimodal (IV)\t\n\\\\ \narthropods \\cite{matthews2014multimodal} &\tmultimodal (IV)\n\\\\\nT-cell receptors \\cite{oakes2017quantitative} & bimodal (II)\\\\\n& unimodal (III)\n\\\\\nmicrobial competition  \\cite{descheemaeker2020stochastic} & `neutral-like' (III) \\\\ & `niche-like' (I \\& II)\n\\\\ \ngastrointestinal  &\t`neutral-like' (III)\n\\\\ \nmicrobiomes \\cite{jeraldo2012quantification} & \\\\\n    \\end{tabular}\n    \\caption{Qualitative classification of observed population regimes in various ecological systems.  }\n    \\label{tab:my_label}\n\\end{table}\n\n\n\n\nOne quantity that is experimentally relatively easy to control is the total number of species $S$. The regimes predicted by the model and the transitions between them shown in Fig.~\\ref{fig:phases_sim}D show similarities with the experimentally observed ones - which were previously explained within the deterministic LV models with random interaction matrix~\\cite{gore2021}. As shown in Fig.~\\ref{fig:phases_sim}D; our model yields `neutral-like' regimes for high $S$ and $\\rho$, which are characterized by erratic dynamics, and `niche-like' behavior with more stable behavior for either low $S$ or $\\rho$. This qualitatively agrees with observed phase-space the in \\cite{gore2021}, which for  strong competition and large pool of species the system presents `persistent fluctuation' regime, while for small species pool or weak competition exhibit `stable full\/partial coexistence'. \nThe fact that both the deterministic LV model with random interaction matrix and the homogeneous LV model with demographic noise are in qualitative agreement with the experimental data raises interesting and important questions concerning the role of stochastic and deterministic dynamics on community composition.\n\nAnother quantity that may enable qualitative and quantitative testing of different models is the carrying capacity $K$ which may be controllable experimentally in some systems.\nAs shown in SM, `rare biosphere' regime shrinks in size with increasing $K$ because a higher carrying capacity sustains higher average abundance,  and larger (less likely) fluctuations are needed for the extinction events to occur.\nHigher average abundance together with insufficiently\nstrong fluctuations,\nresult in longer MFPTs from\ndominance to extinction abundances and vice-versa.\nThis might be captured by the dependence of $K$ in \\eqref{eq:boundary-I}, but further work on the mean-field approximations is needed.\nUnfortunately, rarer turnover events imply longer times to reach the steady state, thus comparing our analytical prediction of the dependence on very large $K$ becomes unfeasible using simulations.\n\nWe expect that the minimal model of this paper can be used for more complicated scenarios, including incorporating speciation  to probe the interaction of the natural selection, inter-species interactions and population diversity and structure.\nFurthermore, the deterministic models inspire further extension\nof our framework to more complex  distributions of the interaction network $\\rho_{i,j}$.\nFinally, our examination of the local ecosystem within an island in the mainland-island model (see Fig.~\\ref{fig:fig1}),\ncan be expanded to a many-island model which allows studying differences in dynamics between the local community and metacommunity, a prominent topic for conservation ecologists and the study of the human microbiome, amongst others.\n \n\n\n\n \n \\iffalse\n\\section{Deterministic Resilience}\n\nIn this section we examine how fast the deterministic system approaches its fixed point. The time for an ecosystem to return to its steady-state, also known as the system resilience, is one of the properties associated with the system stability.   As was mentioned, when $0\\leq \\rho\\leq 1$ the deterministic fixed point, given in ~\\eqref{eq:solstat}, is stable. The direction and pace\/rate\\textcolor{red}{\/another term instead of pace\/rate? } of the flow toward the fixed point are determined by the eigenvalues and eigenvectors of the considered system.  The eigenvalues are obtained as \n\\begin{equation}\n\\lambda_i =\n\\begin{cases}\n\\frac{r}{K}\\left( K - 2 \\tilde{x}(1+\\rho(S-1)) \\right) & \\text{, if } i=1 \\\\\n\\frac{r}{K}\\left( K - \\tilde{x}(2+\\rho(S-2)) \\right) & \\text{, otherwise}.\n\\end{cases}\n\\end{equation}\nand the eigenvectors are \n\\begin{equation}\n\\vec{v}_i =\n\\begin{cases}\n(1,1,\\cdots,1,1)^T & \\text{, if } i=1 \\\\\n(-1,\\delta_{2,i},\\delta_{3,i},\\cdots,\\delta_{S-1,i},\\delta_{S,i})^T & \\text{, otherwise}.\n\\end{cases}\n\\end{equation}\nFor analyzing $\\{\\lambda_i,\\vec{v}_i\\}|_{\\tilde{x}}$ we can deduce the following behaviour of the flow toward the fixed point.  \n\nFirst, we found that $\\lambda_1|_{\\tilde{x}} \\leq \\lambda_{i\\neq 1}|_{\\tilde{x}}$ (the equality is for $\\rho=0$), which means that the system approach faster to constant $J$. Then, in the vicinity of the circle $||\\vec{n}||_1=J$ in $\\ell_1$, it approaches slower toward its fixed point. \n\nSecond, $|\\lambda_1|$ increases with $S$. It means that for highly diverse system, the process reaches faster to the vicinity of its stable community size. In addition, increasing the immigration rate $\\mu$ increase the pace to approaching constant $J$.   \n\nThird, the other (degenerated) eigenvalues describe the behaviour of the flow in the vicinity of constant $J$ (note that it is not necessarily on the {\\em exact} constant J surface). Similarly to $|\\lambda_1|$, the flow in the neighborhood of $||\\vec{n}||_1=J$ toward the fixed point is faster for higher immigration rate. \n\nForth, we have found that $\\lambda_{i\\neq 1}|_{\\tilde{x}}(\\rho,S)$ is not a necessarily a monotone function. For example, when $\\mu=0.01, K=100$ and $r=1$, we find that for $\\rho=0.1$ higher $S$ gives slower approach on the direction of $\\vec{v}_{i\\neq 1}$. However, for $\\rho=1$ an opposite effect is found; high diversity gives faster flow to the fixed point.   \n\n\n\n\\section{Extinction and Invasion Rates}\n\n\\subsection{Stochastic Stability}\n\nHow to define stability? In stochastic models it is not well defined (depends on the paper you read). Note that the boundaries (or any other point) are not absorbing ($\\mu>0$). Stability might mean: recurrence (with probability 1) for any point, $\\rho$-stability, ergodicity, absorbing region, Lyapanov stability for the mean.   \n\n\\textcolor{red}{Q:can we state something about, the bimodality of the abundance distribution,  the level of dominate species, and the mean time of extinction?  }\n\n\\subsubsection{Mean First Passage Time}\nOne of the properties associated with stability is the first passage time. One may ask how long would it take for a species with low abundance to become dominant (or vice versa). \n\nAssume that a species almost surely reaches $x=a$. In addition, we assume that the species level is described by the one-dimensional probability. Then, the mean first passage time from point $x$ to $a$ (where $x>a$) is given by \n\\begin{equation}\n   \\langle T_a(x) \\rangle =   \\sum_{y=a}^{x-1}\\frac{1}{F_{\\rm right}(y)}{\\rm Prob}\\left[z\\geq y+1\\right] \\label{eq:MFPT}\n\\end{equation}\nwhere $F_{\\rm right}(y)\\equiv q^+(y)P(y)$ is the flux right from point $y$ and ${\\rm Prob}[z\\geq y+1]\\equiv \\sum_{z=y+1}^{\\infty}P(z)$ is the probability to be found on larger (or equal) level than $y+1$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth,trim= 130 300 120 300,clip]{figures\/MFPT_differentMu.pdf}\n    \\caption{Mean first passage time from level $x_0$ to extinction. Here the total number of species is $S=20$. We choose $r^+=2$, $r^-=1$ and $\\rho=0.5$. The immigration rate is $\\mu=1$ (blue circles), $\\mu=0.1$ (green rectangles) and $\\mu=0.01$ (purple diamonds). The results are generated from $10^5$ statistically similar systems. The initial position is uniformly distributed in $[1,60]$, i.e. $x_0\\in U[1,60]$.      }\n    \\label{fig:MFPT_differentMu}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth,trim= 130 300 120 300,clip]{figures\/MFPT_rho.pdf}\n    \\caption{Mean first passage time from level $x_0$ to extinction. Here the total number of species is $S=20$. We choose $r^+=2$, $r^-=1$,  $\\mu=1$, and $\\rho=1$ (blue circles), $\\rho=0.5$ (green rectangles) and $\\rho=0.1$ (purple diamonds). The results are generated from $10^5$ statistically similar systems. The initial position is uniformly distributed in $[1,60]$, i.e. $x_0\\in U[1,60]$.  \\textcolor{red}{no agreement with $\\rho=1$}    }\n    \\label{fig:MFPT_rho}\n\\end{figure}\n\n\\subsubsection{Mean Extinction Time from  Level $n_0$}\n\nIn fig.~\\ref{fig:MFPT_differentMu} we present the mean time to extinction (i.e. $a=0$) where the species has started at the level $x_0$. We find that higher immigration rate $\\mu$ give lower time to extinction. This is due to the fact that the influx is proportional to $\\mu$, and the process is stationary (i.e. inbound flux, from zero to positive numbers, is equal to outbound flux, from positive number to extinction). Moreover, In Fig.~\\ref{fig:MFPT_rho} we fixed the immigration rate, and we have found that for small $\\rho$ (weak mutual competition) the MFPT is higher than high $\\rho$. \n\nImportantly, is some cases, the abundance (one  dimension) distribution is not sufficient to evaluate the mean first passage time (see SM). \n\n\\subsubsection{Mean Extinction Time for the Core Species}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth,trim= 130 280 130 300,clip]{figures\/DominantSpecies_differentS1.pdf}\n    \\caption{The mean extinction time of the core species \\textcolor{red}{(approximated analytic solution, no simulation yet)} versus competition strength $\\rho$.  The number of species varies between $S=10$ (blue circles), $S=50$ (green rectangles), $S=100$ (pink crosses) and $S=200$ (yellow starts).    }\n    \\label{fig:DeterminsticVsStochastic}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth,trim= 130 270 110 300]{figures\/BimodalUnimodalRegionAnalytic_3regions_sim_K50_S30}\n    \\includegraphics[trim= 150 270 130 300,width=\\columnwidth]{figures\/BimodalUnimodalRegionAnalytic_3regions_conv_K50_S30.pdf}\n\\includegraphics[trim= 150 270 130 300,width=\\columnwidth]{figures\/BimodalUnimodalRegionAnalytic_3regions_MeanField_K50_S30.pdf}\n    \\caption{Unimodality and bimidality of SAD depending in immigration rate $\\mu$ and competition $\\rho$. The upper panel is obtain from simulation, and the middle and button panels are given from the approximations. This results are obtained from the approximated abundance distribution given using \\textcolor{red}{ 1st approximation (upper panel) and mean-field approximation (lower panel)}. Here we choose the following parameters $S=30$, $r^+=2$, $r^-=1$, $K=50$. The yellow, turquoise and blue regions represent the values of $(\\mu, \\rho)$ where $P(n)$ is a unimoal distribution with maximum at zero, a bimodal distribution with two maxima, or unimodal distribution with maximum at an existence level, respectively. \\textcolor{red}{More or less all have similar map, accept $\\rho=1$ where 1st approx cannot find bimodality there.  }   }\n    \\label{fig:BimodalUnimodal}\n\\end{figure}\n\n \\begin{figure}\n    \\centering\n    \\includegraphics[trim=150 270 150 270,width=\\columnwidth]{figures\/14Oct2.pdf}\n    \\caption{Simulation results for the location and height of the `most-right' peak (2nd peak in bimodality and the only peak at the unimodal phases) .  Upper:Location of the peak. Lower:Height of the peak. }\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n        \\includegraphics[width=\\columnwidth,trim= 130 270 120 300,clip]{figures\/Richness.pdf} \n    \\caption{Richness vs competition.\n    The total number of species change between $S=50$ (red stars), $S=30$ (green squares) and $S=10$ (blue circles). The lines correspond to the approximated analytic solution: 1st and 2nd methods are represented with black and pink curves (respectively). The immigrating rate is $\\mu=1$,\n    $r^+=50$, $r^-=0.1$, and $K=50$.\n    \\textcolor{red}{To run the same figures for $r^+=2$ and $r^-=1$. $K=50$\n    }\n    }\n    \\label{fig:Ricness}\n\\end{figure}\n\n   \\begin{figure}\n        \\centering\n        \\includegraphics[width=\\columnwidth,trim= 130 270 120 280,clip]{figures\/Richness_10to7_mu.pdf}\n        \\includegraphics[width=\\columnwidth,trim= 130 270 120 280,clip]{figures\/Richness_10to7_rho.pdf}\n        \\caption{Upper: The richness versus $\\rho$ for different $\\mu$. Lower: The richness vs $\\mu$ for different $\\rho$.  Here $K=50$, $S=30$, $r^+=2$ and $r^-=1$. The simulation results are given from $10^7$ reactions.}\n        \\label{fig:Richness_mu}\n    \\end{figure}\n    \n    \\begin{figure}\n        \\centering\n        \\includegraphics[width=\\columnwidth,trim= 140 280 140 300,clip]{figures\/Richness_10to7_sim.pdf}\n        \\caption{Simulation results for the richness for different $\\mu$ and $\\rho$. The values of the richness are represented with colors corresponding the colorbar; from high richness (yellow) to low richness (dark blue).  Here $K=50$, $S=30$, $r^+=2$ and $r^-=1$. The simulation results are given from $10^7$ reactions.   }\n        \\label{fig:Richness_heatMap_sim}\n    \\end{figure}\n    \n    \\fi\n    \n\\section*{Methods}\nThe solution for the master equation \\eqref{master-eq} is simulated using the Gillespie algorithm with $10^8$ time steps. We use $r^+=2$, $r^-=1$, $K=100$. \nModalities' classification   is numerically executed after smoothing the simulated SAD. The MFPT is evaluated via the simulated SAD ($\\tilde{x}(S^*)$ is rounded), where a uni-dimensional approximation of the process is considered, see details in SM.\n\n\n\\begin{acknowledgments}\nThe authors acknowledge helpful discussions and comments from all the members of the Goyal and Zilman Groups. AZ acknowledges the support from the National Science and Engineering Research Council of Canada (NSERC) through the Discovery Grant Program. SG acknowledges the support from the National Science and Engineering Research Council of Canada (NSERC) through the Discovery Grant Program and from the Medicine by DEsign Program at the University of Toronto.\n\\end{acknowledgments}\n\n\n\\section*{Approximations of Species Abundance Distribution }\n\n\\subsection*{Derivation of zero flux in $n_i$ - Global balance equation - derivation for the exact SAD}\n\\label{App:ZeroFlux}\n\n\nConsider the multi-dimensional  master equation \n\\begin{eqnarray}\n    \\partial_tP(n_1,n_2\\dots,n_S) &=& \\sum_{i}\\left\\{ q_{i}^+(\\vec{n}-\\vec{e_i})P(\\vec{n}-\\vec{e_i})+q^-_{i}(\\vec{n_i}+\\vec{e_i}) P(\\vec{n}+\\vec{e_i})-\\left[q_{i}^+(\\vec{n})+q_{i}^-(\\vec{n}) \\right]P(\\vec{n}) \\right\\}\n\\end{eqnarray}\n where $q^+_{n_i}(\\vec{n})$ and $q^-_{n_i}(\\vec{n)}$ represents the birth and death rate of species $i$ (respectively), which are generally depends in $\\vec{n}=(n_1,\\dots, n_s)$. Here, $e_i=\\{0, \\dots, 1, \\dots , 0\\}$ (the one is located in the $i$-th component). \n To find Master equation for $n_1$ we sum over all other components; i.e. \n\\begin{eqnarray}\n  &&\\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty}  \\partial_tP(n_1,n_2\\dots,n_S) = \\\\ \\nonumber &&=  \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} \\left\\{\\sum_{i}\\left\\{ q_{i}^+(\\vec{n}-\\vec{e_i})P(\\vec{n}-\\vec{e_i})+q^-_{i}(\\vec{n_i}+\\vec{e_i}) P(\\vec{n}+\\vec{e_i})-\\left[q_{i}^+(\\vec{n})+q_{i}(\\vec{n}) \\right]P(\\vec{n}) \\right\\}\\right\\} \n  \\end{eqnarray}\n  thus\n  \\begin{equation}\n     \\partial_t P_1(n_1) = \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} \\left\\{\\sum_{i}\\left\\{ q_{i}^+(\\vec{n}-\\vec{e_i})P(\\vec{n}-\\vec{e_i})+q^-_{i}(\\vec{n_i}+\\vec{e_i}) P(\\vec{n}+\\vec{e_i})-\\left[q_{i}^+(\\vec{n})+q_{i}(\\vec{n}) \\right]P(\\vec{n}) \\right\\}\\right\\} .\n  \\end{equation}\n  We can now use the fact that for every $n_i$:\n \\begin{eqnarray}\n      \\sum_{n_i=0}^{\\infty}  q_{i}^+(n_1, \\dots n_i-1, \\dots, n_S)P(n_1, \\dots n_i-1, \\dots, n_S) &=& \\sum_{n_i=0}^{\\infty}  q_{i}^+(n_1, \\dots n_i, \\dots, n_S)P(n_1, \\dots n_i, \\dots, n_S)\n,  {\\rm \\ \\ and} \\\\ \\nonumber\n \\sum_{n_i=0}^{\\infty}  q_{n_i}^-(n_1, \\dots n_n+1, \\dots, n_S)P(n_1, \\dots n_i+1, \\dots, n_S)&=& \\sum_{n_i=0}^{\\infty}  q_{n_i}^-(n_1, \\dots n_i, \\dots, n_S)P(n_1, \\dots n_i, \\dots, n_S)\n \\end{eqnarray}\n  [note that   $q_{n_i}^+(n_1, \\dots -1, \\dots, n_S)P(n_1, \\dots, -1, \\dots, n_S)=q_{n_i}^-(n_1, \\dots 0, \\dots, n_S)P(n_1, \\dots 0, \\dots, n_S)=0$]. Thus, the above equation is given by\n\\begin{eqnarray}\n  \\partial_t P_1(n_1) &=& \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} \\left\\{ q_{1}^+(\\vec{n}-\\vec{e_1})P(\\vec{n}-\\vec{e_1})+q^-_{1}(\\vec{n}+\\vec{e_1}) P(\\vec{n}+\\vec{e_1})-\\left[q_{1}^+(\\vec{n})+q^-_{1}(\\vec{n}) \\right]P(\\vec{n})\\right\\}.\n\\end{eqnarray}\nFor simplicity, we define $F^{+}(\\vec{n})\\equiv q_1^+(\\vec{n})P(\\vec{n})$ and $F^{-}(\\vec{n})\\equiv q_1^-(\\vec{n})P(\\vec{n})$, thus Eq. \ncan be written as \n\\begin{eqnarray}\n    \\partial_t P(n_1) =&& \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} \\left\\{ F^{+}(n_1-1,n_2,\\dots)-F^{+}(n_1,n_2,\\dots)\n     +F^{-}(n_1+1,n_2,\\dots)-F^{-}(n_1,n_2,\\dots) \\right\\}.\n\\end{eqnarray}\nBy using z-transform ($n_1\\rightarrow z$), which is defined for a function $k(n_1)$ as $K(z)=\\sum_{n_1=0}^{\\infty} k(n_1) z^{-n_1} $, we obtain\n\\begin{eqnarray}\n    \\partial_t P(z) = \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} F_{\\rm right}(z,n_2,\\dots)(1-z^{-1})+F_{\\rm left}(z,n_2,n_3\\dots)(1-z) \n\\end{eqnarray}\n\\{used ${\\cal Z}[g(n)-g(n-1)]=[1-z^{-1}]\\hat{G}(z)$, and ${\\cal Z}[g(n+1)-g(n)]]=[1-z]\\hat{G}(z)-zg(0)$ \\}. Stationary solution; $\\partial_t P(z)=0 $ and re-organize the equation yields \n\\begin{eqnarray}\n     \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty}F_{\\rm right}(z,n_2,n_3,\\dots)=\\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} F_{\\rm left}(z,n_2,n_3,\\dots)\\frac{1-z}{z^{-1}-1}=\\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty}F_{\\rm left}(z,n_2,n_3\\dots)z .\n\\end{eqnarray}\nThen, we use the inverse z-transform ($z\\rightarrow n_1$), and find\n\\begin{eqnarray}\n      \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty}q^+_{1}(\\vec{n})P(\\vec{n})= \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty}q^-_{1}(\\vec{n}+\\vec{e_1})P(\\vec{n}+\\vec{e_1}). \n\\end{eqnarray}\nWe use Bayes formula; $P(n_1,n_2,n_3,\\dots,n_S)=P(n_2,n_3,\\dots, n_S|n_1)P(n_1)$ and obtain\n\\begin{eqnarray}\n     \\langle q_{n_1}^+(\\vec{n})|n_1\\rangle_{n_2,n_3,\\dots,n_S}P_1(n_1) = \\langle q_{n_1}^-(\\vec{n}+\\vec{e_1}) |n_1+1\\rangle_{n_2,\\dots n_S} P_1(n_1+1), \n\\end{eqnarray}\nwhere $ \\langle *|n_1\\rangle_{n_2, \\dots, n_S}\\equiv\\langle *|n_1\\rangle \\equiv\\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_S}^{\\infty} (*)P(n_2, \\dots, n_S|n_1) $. \nUp to now there are no assumption in the derivation, and the above is general. For our case, as specified in the main text,  $\\langle q_{1}^+(\\vec{n})|n_1\\rangle=\\mu + r^+ n_1$ (note that $q^+_{i}$ depends solely on $n_i$) and $\\langle q_{1}^-(\\vec{n})|n_1\\rangle=n_1\\left(r^-+r n_1\/K + r \\rho \\sum_{j\\neq 1 }\\langle n_j |n_1\\rangle \/K \\right)$.  Additionally, from symmetry,  $P_i(n_i)=P_j(n_j) = P(n)$ for every $i,j$.\nThus, solving the recursive equation and obtain\n\\begin{eqnarray}\n    P(n)&=&P(0)\\prod_{n'=1}^{n}\\frac{q^{+}(n'-1)}{\\langle q^-(\\vec{n})|n'\\rangle}= \\label{eq:exact_appendix}\n    \\\\ \\nonumber\n    &=&P(0)\\prod_{n'=1}^{n}\\frac{r^+(n'+a)}{n\\left(r^-+r n'\/K + r\\rho \\sum_{j\\neq 1 }\\langle n_j |n'\\rangle \/K \\right)}= P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r n'\/K + r\\rho \\sum_{j\\neq 1 }\\langle n_j |n'\\rangle \/K \\right)}.\n\\end{eqnarray}\nwhere\n $a=\\mu\/r^+$ and $(a)_{n} \\equiv a(a+1)\\dots (a+n-1)$, in the Pochhammer symbol.  Here, $P(0)$ is given from normalization.\nWe emphasize that the above abundance distribution $P(n)$ in \\eqref{eq:exact_appendix} is exact, means no approximations have been taken so far.\n\nNote, that the denominator in the exact solution depends on the effect of interactions from all other species over $n_1$,  through the term $\\sum_{j\\neq 1}\\langle n_j |n_1 \\rangle $. Therefore, in order to provide an explicit expression to $P_1(n_1)$, we need to use some approximations. Three approximation approaches, and a discussion about their limitations are given in the following subsections. We note that all the presented approximations below provide decent results. However, we find that  none of them manage to provide adequate fit for every set of parameters, see Figures.  \n\n\\subsection*{Approximation Approach I: Estimating $\\sum_{j\\neq i}\\langle n_j|n_i\\rangle $ Using Mean-Field Approximation }\nWe assume $\\langle n_j |n_i \\rangle = \\langle n_j \\rangle $. Thus, \n\\begin{equation}\n    P(n)\\approx P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r n'\/K + r\\rho \\sum_{j\\neq 1 }\\langle n_j\\rangle \/K \\right)} = P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r n'\/K + r\\rho (S-1)\\langle n\\rangle \/K \\right)},\n    \\label{eq:MF1}\n\\end{equation}\nwhere the last equality is given from symmetry;  $\\langle n_j \\rangle = \\langle n_i \\rangle = \\langle n\\rangle $ for every species $i,j$. In addition, by definition, $\\langle n \\rangle = \\sum_{n=0}^{\\infty }n P(n)$, hence\n\\begin{equation}\n    \\langle n \\rangle \\approx P(0)\\sum_{n=0}^{\\infty}n\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r n'\/K + r\\rho (S-1)\\langle n\\rangle \/K \\right)}\n    \\label{eq:MF1_closure}\n\\end{equation}\nwhere $P(0)=1\/{_1}F_1[a,b;c]$ is the normalization coefficient,  with ${_1}F_1[a,b;c]$ is the Kummer confluent hypergeometric function, $a=\\frac{\\mu}{r^+}$, $b=\\frac{r^- K + r \\rho (S-1) \\langle n \\rangle }{r}+1$ and $c=\\frac{r^+ K}{r}$. Solving numerically the above implicit equation and evaluate $\\langle n\\rangle$. The last step is to substitute the numerical solution of $\\langle n\\rangle$, obtained from \\eqref{eq:MF1_closure}, into \\eqref{eq:MF1}.    \n\n\\subsection*{Approximation Approach II: Estimating $\\langle J|n\\rangle $ using Convolution }\nThe exact solution can be written as \n\\begin{eqnarray}\n    P_1(n_1)=P(n)= P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r (1-\\rho)n'\/K + r\\rho \\langle J |n_1\\rangle \/K \\right)},\n\\end{eqnarray}\nwhere $J=\\sum_{i=1}^{\\infty} n_i $ is the total population size. Here we assume that the total number of individuals in the system, $J$, is weekly depends on $n_1$. Thus $J$ is an independent random variable. Hence, \n\\begin{equation} \n    P_1(n_1)=P(n_1|\\langle J |n_1 \\rangle ) \\approx P(n_1|J)=P(0)\n    \\frac{(a)_{n_1} \\Tilde{c}^{n_1}}{n_1 ! (\\Tilde{b}+1)_{n_1} } \n\\end{equation}\nwith $a=\\frac{\\mu}{r^+}$, $\\tilde{b}= \\frac{r^-K+r\\rho J}{r(1-\\rho)}$, and $\\tilde{c}=\\frac{r^+ K}{r(1-\\rho)}$ [note that both $\\tilde{b}$ and $\\tilde{c}$ differ from $b$ and $c$ defined in previous subsection].\nMoreover, we assume that the species levels are mutually independent, means ${\\cal P}(n_1,\\dots n_S) \\approx \\prod_i P_i(n_i)$. Thus, the PDF of $\\sum_i n_i$ reads \n\\begin{equation}\n    P\\left(\\left.\\sum_i n_i\\right|J\\right)=\\underbrace{P_1(n_1|J)*P_2(n_2|J)* \\dots * P_S(n_S|J)}_{S {\\rm \\ times}}\n\\end{equation}\nwhere $A*B$ means the convolution of $A$ with $B$. $P\\left(\\sum_i n_i|J\\right)$ is the `analytical' PDF to have $\\sum_i n_i$ individuals where we assume that a single species PDF is $P_1(n_1|J)$ with a given $J$. \nTo capture the fact that $J$ has a meaning of number of individuals as well, we consider\n\\begin{equation}\n    P(J)\\approx \\frac{{\\rm Prob}\\left(\\left.\\sum_i n_1=J\\right|J\\right)}{\\sum_J {\\rm Prob}\\left(\\left.\\sum_i n_1=J\\right|J\\right)},\n\\end{equation}\nwhere $P(J)$ is the approximated distribution of $J$. \nThen\n\\begin{equation}\n    P_1(n_1) = \\sum_{J}P_1(n_1|J) P(J)\n\\end{equation}\nis the approximated PDF. \n\nNote that when $S$ is large, we find\n\\begin{equation}\n    P\\left(\\left.\\sum_i n_i\\right|J\\right) \\sim  {\\cal N}\\left(S\\langle n_1 |J \\rangle, S \\cdot Var(n_i) \\right),\n\\end{equation}\nthus $P(J)\\approx {\\rm Prob}(\\sum_i n_i =J|J)$ reaches its maximum in the vicinity of $J$ which satisfies $J\\approx S \\langle n_i |J \\rangle = \\left\\langle \\sum_i n_i |J \\right\\rangle $. Furthermore, for the approximation $P(J)\\approx {\\rm Prob}(\\sum_i n_i =J|J)$, the values of $J$ where $J\\ll S\\langle n_i |J \\rangle  $ or $J\\gg S\\langle n_i |J \\rangle $ are highly improbable, due to the Gaussian nature of $P(\\sum_i n_i|J)$ for large $S$.\n \n\\subsection*{Approximation Approach III: Estimating $\\langle J|n\\rangle $ using Mean-Field Approximation } In a similar fashion to previous approximation approaches, we assume $\\langle J|n\\rangle \\approx \\langle J\\rangle  $, thus\n\\begin{eqnarray}\n    P(n) \\approx P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r (1-\\rho)n'\/K + r\\rho \\langle J \\rangle \/K \\right)}=  P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r (1-\\rho)n'\/K + r\\rho S\\langle n\\rangle  \/K \\right)}.\n\\end{eqnarray}\nThen, $\\langle n \\rangle$ is given by the numerical solution of\n\\begin{eqnarray}\n    \\langle n \\rangle = \\sum_{n=0}^{\\infty} n P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r (1-\\rho)n'\/K + r\\rho S\\langle n\\rangle  \/K \\right)}.\n\\end{eqnarray}\nwhere here the normalization factor is $P(0)=1\/{_1}F_1[a,b;c]$ with $a=\\frac{\\mu}{r^+}$, $\\tilde{\\tilde{b}}= \\frac{r^-K+r\\rho \\langle J\\rangle }{r(1-\\rho)}$, and $\\tilde{c}=\\frac{r^+ K}{r(1-\\rho)}$.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=0.45\\columnwidth,trim=100 100 100 100]{figures\/Quality.pdf}\n    \\includegraphics[width=0.45\\columnwidth,trim=100 100 100 100]{figures\/examples_approx.pdf}\n    \\caption{Quality of Approximation}\n    \\label{fig:quality}\n\\end{figure}\n\n\\subsection*{Interspecies Correlations and Limitations of the Approximations Approaches}\nFor approximation approaches described above, we assume the species are mutually independent, meaning $P(\\vec{n})=\\prod_{i=1}^S P(n_i)$.\nOf course, this mutual independence cannot exactly be obtained, except for $\\rho=0$, since for every $\\rho>0$ the dynamics of a species, i.e its birth and death rates, depends on other species levels.  Therefore, we expect deviations of our approximation from the true simulated SAD. In particular,  we expect a relation between the interspecies correlation and the quality of approximation. \n\nWe quantify the quality of approximation, using two metrics. The first one, Kolmogorov-Smirnov (KS), is defined with $KS(P,Q)\\equiv\\max \\left|{\\rm CDF}(P)-{\\rm CDF}(Q)\\right|$ for two PDFs, $P$ and $Q$. The second metric we use is the Kullback\u2013Leibler divergence, which is defined by $KL(P,Q)\\equiv \\sum_x (P-Q)\\ln\\left(P\/Q\\right)$. \nIntuitively, the KS metric captures the difference between the approximation and the simulation results, and the KL divergence capture the ratio between the distributions. \nFig.~\\ref{fig:quality} shows the KS and KL metrics comparing the three approximations presented above and the simulation results.\n\nIn the main text, we choose to present the results for the regime boundaries obtained from the Approximation I above. Even so, all approximations present reasonable agreement with the simulation. \nHowever, in some regimes, some approximations work better than the others. \nAdditionally, one approximation may better capture  some features of the system, while other approximation show better agreement with other features. \nFor example, only approximation (I) captures the bi-modality at very low $\\mu$ and $\\rho=1$, where the other approximations seem to align with the simulated SAD slightly better.\n\n\\subsection*{Tail-end of distributions}\n\nThe above model does not describe an ecological zero-sum game as the total population size $J=\\sum_i n_i$ may fluctuate in this proposed formulation.\nConversely, classical Moran models describe stochastic processes in which the finite population is fixed. \nThe species abundance distribution can be solved for exactly in certain cases where this assumption of a fixed total population size holds.\n\nIn this limit,\n\\begin{equation}\n    P(n) = P(0)\\frac{(a)_{n}}{n!}\\approx P(0)\\frac{n^{a-1}}{\\Gamma[a]}\n\\end{equation}\nIn the case where $\\langle J|n_1\\rangle \\approx \\langle J \\rangle $:  \\begin{equation}\n    P(n)= P(0) \\left(\\frac{r^+}{r^- + r \\langle J \\rangle \/K}\\right)^{n}\\frac{ (a)_{n}}{n!} \\approx  P(0) \\left(\\frac{r^+}{r^- + r \\langle J \\rangle \/K}\\right)^{n_1}\\frac{ n_1^{a-1}}{\\Gamma[a]}.\n\\end{equation}\nNote that this is valid only when $n \\gg a$, that is to say for the tail-end of the distributions.\n\n\nMoran models are generally neutral $\\rho = 1$), however using our demographic noise model allows for observing non-neutral systems. \nWe can approximate the tail of the distribution for non-neutral models ($0 < \\rho < 1$) using similar arguments as above, that is to say approximating $\\langle J |n \\rangle \\approx J$.\nUsing our previous solution and this approximation, we find that\n\\begin{equation}\n    P(n) = P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}(r^-+r(1-\\rho)n'\/K+\\rho \\langle J |n' \\rangle \/K)}\\approx P(0)\\frac{(a)_n \\tilde{c}^n}{n!(\\tilde{b}+1)_n}\n\\end{equation}\nwhere $a=\\frac{\\mu}{r^+}$, $\\tilde{b}= \\frac{r^-K+r\\rho J}{r(1-\\rho)}$, and $\\tilde{c}=\\frac{r^+ K}{r(1-\\rho)}$.\nLooking at the tail of the distribution ($n \\gg a, \\tilde{b}, \\tilde{c}$) we find that \n\\begin{equation}\n    P(n) \\xrightarrow{n \\rightarrow \\infty} P(0)\\frac{\\Gamma[\\tilde{b}+1]}{\\sqrt{2 \\pi} \\Gamma[a]} n^{a-\\tilde{b}-\\frac{3}{2}}(\\tilde{c}\/n)^{n}e^n\n\\end{equation} \nwhere we have used Stirling's approximation that $n! \\approx \\sqrt{2\\pi n}n^n e^{-n}$.\nIn the high $n$ limit, we can further approximate $J\\sim n$ such that $P(n)\\sim n^{-n\/(1-\\rho)}e^n$\n\nIn the case where $\\rho \\rightarrow 0$, there is no need for approximating $\\langle J |n\\rangle$, as that term disappears in the exact solution.\nWe are left with the solution\n\\begin{equation}\n    P(n) \\xrightarrow{\\rho\\rightarrow 0}P(0) \n    \\frac{(\\mu\/r^+)_{n} (r^+ K\/r)^{n}}{n ! (r^-K\/r+1)_{n} }\n\\end{equation}\nwhich is exact.\nThe tail of this distribution goes as\n\\begin{equation}\n    P(n) \\xrightarrow{\\rho\\rightarrow 0,n\\rightarrow \\infty} P(0)\\frac{\\Gamma[\\tilde{r^-K\/r}+1]}{\\sqrt{2 \\pi} \\Gamma[\\mu\/r^+]} n^{\\frac{\\mu}{r^+}-\\frac{r^-K}{r}-\\frac{3}{2}}(r^-K\/rn)^{n}e^n.\n\\end{equation}\n\nNote that the denominator of $\\tilde{b}$ and $\\tilde{c}$ goes to 0 as $\\rho \\rightarrow 1$, as such the limit must be taken carefully: $\\tilde{b} \\xrightarrow{\\rho \\rightarrow 1} \\infty $.\nWe use the fact that $(x+1)_n \\xrightarrow{ x \\rightarrow\\infty} x^{n}$ to write\n\\begin{equation}\n     P(n) \\xrightarrow{\\rho\\rightarrow 1}P(0) \n    \\frac{(a)_{n} \\{r^+ K\/[r(1-\\rho)]\\}^{n}}{n ! \\{(r^-K+rJ)\/[r(1-\\rho)]\\}^{n} }= P(0) \n    \\frac{(a)_{n} (r^+ K)^{n}}{n ! (r^-K+rJ)^{n} }  \\xrightarrow{n \\rightarrow \\infty } P(0) \\left( \\frac{r^+K}{r^-K + rJ} \\right)^n \\frac{n^{a-1}}{\\Gamma[a]}\n\\end{equation}\nwhich agrees with what we found earlier for $\\rho=1$ and constant $J$. For $\\rho = 1$, we know that if any one species abundance gets large it should dominate the system. Therefore, we can approximate $J\\approx n$ for large $n$ in the neutral regime.\n\nThis asymptotic behaviour may be compared to analytical solutions for which $J$ is held constant.\nThese Moran type models are often solvable exactly, we choose to show their results wherein $J=S n_{det}$ where $n_{det}$ is the solution to the mean deterministic equation Lotka-Voltera equation. \nIn~\\cite{mckane2004analytic}, an analytical solution to the Hubbel model with immigration is found such that\n\\begin{equation}\n    P(n)={J \\choose n}\\frac{\\beta (n+p,n^*-n)}{\\beta (p^*,n^*-J)} \n\\end{equation}\nwhere $p=1\/S$, $n^*=(J-m)\/(1-m) - p$, and $p^*=m p (J-1)\/(1-m)$ .\nIn this model, $m$ is defined as the probability of immigration at any step.\nThis is different from our immigration rate, however we find a suitable transformation to be $m \\approx \\mu \/ \\langle r^+_n + r^-_n \\rangle$: the probability of immigration is the rate of immigration divided by the mean rate of a reaction.\nNote that the function $\\beta(a,b)=\\Gamma (a) \\Gamma (b)\/ \\Gamma (a+b)$.\n\nIn~\\cite{baxter2007exact}, a continuum Fokker-Planck equation is solved to evaluate a similar multi-allelic diffusion model abundance. \nHowever, in this formalism, immigration is replaced by mutations wherein $u_i$ is the rate of mutation of cell allele $i$.\nAssuming all the mutation rates are equivalent, $u_i=u$\nThe steady state joint probability distribution is \n\\begin{equation}\n    P(\\vec{x})=\\Gamma (2 S u ) \\delta (1-\\sum_i x_i)\\prod^S_{i=0} \\frac{x_i^{2u-1}}{\\Gamma (2 u )}\n\\end{equation}\nwhich may be integrated to find the SAD\n\\begin{equation}\n    SAD(n) = \\langle \\sum_j \\delta (x_j - n\/J) \\rangle_{P(\\vec{x})} \\approx \\left( \\frac{n}{J} \\right)^{2 u - 1 } e^{- (2 u (S-1) -1 ) n \/ J }  \n\\end{equation}\nAlthough mutations and immigration are not completely equivalent, mutations may take on a heuristic role similar to immigration that allows for no species to be truly extinct.\nAs such, we assume $u=\\mu\/r$.\nComparisons of these different asymptotic behaviours are found in Fig. ~\\ref{fig:asymptotic}.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=0.7\\columnwidth]{figures\/asymptotic_neutral.pdf}\n   \\caption{Asymptotic behaviour of various neutral models compared to simulations. (top panels) Using $\\langle J |n \\rangle \\approx n $, our approximation does not recover a bimodality, however the analytical approximation clearly follows the simulation's power law with exponential cutoff. (bottom panels) Moran-like models in the literature of power-law SADs with exponential cutoff. Here, the total population size used is the total population size of the steady-state Lotka-Voltera equation, $J=S n_{LV}$. The continuous Fokker-Planck diffusion model of Baxter, Blythe \\& McKane\\cite{baxter2007exact} shows the immigration dominated peak. However, the Hubbel community model solved by Alonso, McKane \\& Sole\\cite{mckane2004analytic} shows a bimodality in the low immigation regime. Both have power laws with exponential cutoff in different regimes.}\n    \\label{fig:asymptotic}\n\\end{figure}\n\n\\section*{Derivations of Boundary Equations}\n\n\\subsection*{Boundaries for Richness Regimes} For the boundaries defined from richness, we use\n$    \\langle S^* \\rangle = S \\left(1-P(0)\\right) \n$, where $P(0)$ obtained numerically from the approximated SAD. Note that in the mean-filed approximations $P(0)$ is explicitly given as Kummer confluent hypergeometric function. Then, the transition between full richness to partial coexistence is given with $S P(0) = 1\/2$ (as the arithmetic mean between the two boundaries). Similarly, the transition boundary between partial coexistence and excluded regime is drawn where $SP(0)=S-3\/2$.  \n\n\\subsection*{Derivation of $\\tilde{n}$ } The boundaries defined by the modalities can be given directly from the above approximations, see Figures. However, we have found that using the mean-field approximation allows us to derived a closed expression for the boundaries.     \n\nThe transition between neutral-like to bimodal regimes, is defined by the presence  or absence of a local maximum in a real positive level. In another words, in the neutral-like regime $P(n)>P(n+1)$, since the SAD is monotony decreasing, while in the bimodal regime, there is $n>0$ where $P(n)<P(n+1)$. Thus the boundary between the regimes occurs where\n  \\begin{eqnarray}\n      P(n)=P(n+1).\n    \\end{eqnarray}\nUsing the $P(n)$ from the mean-field approximation, we find\n    \\begin{eqnarray}\n      P(0)\\frac{(a)_n c^n}{n! (b+1)_n }&=&P(0) \\frac{(a)_{n+1} c^{n+1}}{(n+1)! (b+1)_{n+1} } \n      \\\\\n      \\frac{(b+1)_{n+1}(n+1)!}{n! (b+1)_n }&=& \\frac{(a)_{n+1} c^{n+1}}{(a)_n c^n } \n      \\\\\n      n(b+n)&=& c (a+n-1)\n      \\\\\n      n&=& \\frac{(c-b)\\pm \\sqrt{(c-b)^2+4(a-1)c}}{2}\n  \\end{eqnarray}\nSubstitute $a=\\frac{\\mu}{r^+}$, $b=\\frac{r^- K + r \\rho (S-1) \\langle n \\rangle }{r}+1$ and $c=\\frac{r^+ K}{r}$ yields\n\\begin{eqnarray}\n   \\tilde{n}\\approx\n    \\frac{K-\\rho(S-1)\\langle n\\rangle}{2}\\left\\{1 \\pm \\sqrt{1+4\\frac{(\\mu-r^+) K}{r(K-\\rho(S-1)\\langle n\\rangle)^2}}\\right\\}.\n    \\end{eqnarray}\nIf we use $S \\langle n \\rangle = \\langle S^* \\rangle \\tilde{n}$, we find\n\\begin{equation}\n    \\tilde{n}=\\frac{K}{2(1-\\rho+\\rho \\langle S^* \\rangle)}\\left\\{1+ \\sqrt{1+\\frac{4(\\mu-r^+)(1-\\rho+\\rho \\langle S^* \\rangle)}{rK}}\\right\\} \\approx \\frac{K}{1-\\rho+\\rho \\langle S^* \\rangle}- \\frac{\\mu-r^+}{r}.\n\\end{equation}\n   \n\\subsection*{Boundaries for Modalities Regimes}\nIn the neutral-like regime the SAD is monotonously decreasing. Therefore, the neutral-like regime is sufficiently define by either imaginary $\\tilde{n}$, or real but negative $\\tilde{n}$. The transition line between real and imaginary $\\tilde{n}$, i.e. where $\\Im(\\tilde{n})=0$, is given by  \n\\begin{eqnarray}\n    {r\\left[K-\\rho  (S-1)\\langle n\\rangle\\right]^2}=4{(r^+-\\mu) K}.\n\\end{eqnarray}\nThe transition line between the negative to positive $\\tilde{n}$, where $\\tilde{n}=0$, is drawn where\n\\begin{equation}\n    \\frac{(K-\\rho(S-1)  \\langle n\\rangle)^2}{4}=1+ \\frac{4K(\\mu-r^+)}{r(K-\\rho(S-1)  \\langle n\\rangle)^2}.\n\\end{equation}\nTherefore, the neutral-like regime is defines as the union of the regions defined by both two equations above.\n\nThe second boundary we derive is the border of uni-modality region with positive probable abundance. At zero abundance, the abundance distribution poses either local maximum ($P(0)>P(1)$; correspond to bimodality or neutral-like regimes) or local minimum ($P(0)<P(1)$; unimodality). Consequently, the boundary between these two possibilities is given by $P(0)=P(1)$.  Thus, following the zero-flux equation, we obtain\n\\begin{equation}\n    q^+(0) = \\langle q^-(\\vec{n})|1\\rangle  \\Longrightarrow  \\mu = r^- +\\frac{r}{K} \\left[1-\\rho + \\rho \\langle J\\rangle\\right]. \n\\end{equation}\n\n\n\n\\section*{Dependence on Carrying Capacity $K$}\n\nThe regime boundaries depend on the carrying capacity, as can be seen in Fig.~\\ref{fig:carry_capacity}; as the carrying capacity increases, the `neutral-like' regime shrinks in size.\nWe also find that the full richness regime extends to larger competitive overlap, intuitively suggesting that interspecies competition is decreased in the larger carrying capacity regimes.\nThe larger carrying capacity allows for more individual species to immigrate into the system and coexist.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=0.45\\columnwidth]{figures\/K50.pdf}\n    \\includegraphics[width=0.45\\columnwidth]{figures\/K200.pdf}\n    \\caption{Varying carry capacity. Left panel: $K=50$. Right panel: $K=200$. Note that the the erroneous classification in certain regimes is due to numerical difficulty in determining optima of the SAD.}\n    \\label{fig:carry_capacity}\n\\end{figure}\n\n\\section*{Kinetics}\n\nAs was mentioned in the main text, we concentrate on time scales associated with mean-first-passage-time (MFPT) from some initial abundance $i$, to a final one $f$. This MFPT, denoted as $\\langle T(i\\rightarrow f)\\rangle$, is inversely proportional to the transition rate from $i$ to $j$, where the first-passage times are exponentially distributed (see Figures). \nIn a unidimensional process in an interval $[0,\\infty)$, the MFPTs we consider are  \n\\begin{eqnarray}\n      \\langle T (\\tilde{x} \\rightarrow 0) \\rangle =   \\sum_{y=0}^{\\tilde{x}-1}\\frac{1}{q^+(y)P(y)}\\sum_{z=y+1}^{\\infty}{P(z)}, & &\n  \\langle T (0 \\rightarrow 0) \\rangle =  \\frac{1}{q^+(0)P(0)}=  \\frac{1}{\\mu P(0)}, \\\\ \\nonumber \n  \\langle T (0 \\rightarrow \\tilde{x}) \\rangle =  \\sum_{y=0}^{\\tilde{x}}\\frac{1}{q^+(y)P(y)}\\sum_{z=0}^{y}{P(z)}, & &\n  \\langle T (\\tilde{x} \\rightarrow \\tilde{x}) \\rangle =    \\frac{1}{P(\\tilde{x})[q^+(\\tilde{x})+q^-(\\tilde{x})]} .\n\\end{eqnarray}\nThese expressions are exact for processes in one dimension \\cite{gardiner1985handbook}, i.e. for a single species were no other species affect its dynamics.  Here we have found, using simulation, that these MFPTs agree with the multi-dimensional scenario, where many species evolve (see Figures). Hence, we substitute the SAD obtained from simulation,  as $P(n)$ in the expressions above and estimated the rates with $ R(a \\rightarrow b)=1\/\\langle T(a \\rightarrow b)\\rangle $. A similar approach is used to estimate the other rate ratio. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.7\\columnwidth]{figures\/supp-MFPT.pdf}\n\\end{center}\n\\includegraphics[trim=140 260 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho1_mu03162new.pdf} \n\\includegraphics[trim=140 260 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho1_mu1_new.pdf} \n\\includegraphics[trim=140 260 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho01_mu03162.pdf} \n\\includegraphics[trim=140 260 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho03162_mu3162.pdf}\n\\\\\n\\includegraphics[trim=140 230 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho1_mu03162_new.pdf}  \n\\includegraphics[trim=140 230 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho1_mu1_new_part2.pdf}\n\\includegraphics[trim=140 230 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho01_mu03162_part2.pdf}\n\\includegraphics[trim=140 230 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho03162_mu3162_part2.pdf}\n\\caption{Turnover time statistics}\n\\end{figure}\n\n\\section*{Fokker-Planck formulation}\n\nIn the analysis so far we have used $n_i$ to represent a discrete variable, however we may approximate the master equation in a continuum should some large characteristic system size exist.\nAssuming here that $K >> 1$ is the system size, we define $x_i$ to be the corresponding continuous limit of $n_i$.\nThe variable may be rescaled by the characteristic system size, $y_i = x_i \/ K$\nThis continuous approximation has as probability density $ P(\\vec{n},t) = p(\\vec{y},t)\/K$ and we may write the the scaled rates as $q_i^{+\/-}(\\vec{n})=K Q_i^{+\/-}(\\vec{y})$ where\n\\begin{align*}\n    Q_i^{+}(\\vec{y}) &= r^+ y_i + \\mu \/ K \\\\\n    Q_i^{+}(\\vec{y}) &= r^- y_i + r y_i \\left( y_i + \\sum_{j\\neq i}\\rho_{j,i} y_j \\right)\n\\end{align*}\nis defined on the continuum.\nThen we can write the corresponding master equation as\n\\begin{equation}\n    \\label{eq:master-eq-cont}\n    \\partial_t  p(\\vec{x};t) =\n    K \\sum_{i}\\left\\{Q^+_i(\\vec{y}-\\vec{e}_i)p(\\vec{y}-\\vec{e}_i,t)+ \n    Q^-_i (\\vec{y}+\\vec{e}_i)p(\\vec{y}+\\vec{e}_i,t) - \\left[Q^+_i(\\vec{y})+Q^-_i(\\vec{y})\\right]p(\\vec{y},t)\\vphantom{\\vec{e}_i} \\right\\}\n\\end{equation}\nwherein $\\vec{e}_i$ is the change in abundance $\\vec{y}$ from the respective event, which in our single birth-death process will be $\\vec{e}_i=(\\delta_{1i}\/K, \\delta_{2i}\/K, ..., \\delta_{Si}\/K)$, in other words a vector of zeros except for $1\/K$ located at species $i$.\nTo go from the master equation to the Fokker-Planck equation, we Taylor expand each of the expressions from the right-hand side of \\ref{eq:master-eq-cont}.\nAs such,\n\\begin{multline*}\n    Q^+_i(\\vec{y}-\\vec{e}_i)p(\\vec{y}-\\vec{e}_i,t)= Q^+_i(\\vec{y})p(\\vec{y},t)+\\sum_j (-(\\vec{e}_i)_j) \\frac{\\partial}{\\partial y_j}\\left( Q^+_i(\\vec{y})p(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right)\n    + \\frac{1}{2!}\\sum_j \\sum_k (\\vec{e}_i)_j(\\vec{e}_i)_k \\frac{\\partial^2}{\\partial y_j \\partial y_k}\\left( Q^+_i(\\vec{y})p(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right)+...\n\\end{multline*}\n\\begin{multline*}\n    Q^-_i(\\vec{y}+\\vec{e}_i)p(\\vec{y}+\\vec{e}_i,t)= Q^-_i(\\vec{y})p(\\vec{y},t)+\\sum_j (\\vec{e}_i)_j \\frac{\\partial}{\\partial x_j}\\left( Q^-_i(\\vec{y})p(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right)\n    + \\frac{1}{2!}\\sum_j \\sum_k (\\vec{e}_i)_j (\\vec{e}_i)_k \\frac{\\partial^2}{\\partial y_j \\partial y_k}\\left( Q^-_i(\\vec{y})P(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right)+...\n\\end{multline*}\nNote that $(\\vec{e}_i)_j=\\delta_{ij}\/K$ in our single birth-death event process, which simplifies these equations considerably. \nNow, replacing these expressions in our master equation \\ref{eq:master-eq-cont}, we note that we can write our equation in orders of $1\/K$.\nThus, we obtain\n\\begin{equation}\n\\label{eq:fokker-planck}\n    \\partial_t  p(\\vec{y};t) =\n    -\\sum_j  \\frac{\\partial}{\\partial y_j}\\left[\\left( Q^+_i(\\vec{y})-Q^-_i(\\vec{y})\\right)p(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right]\n    + \\frac{1}{2 K}\\sum_j  \\frac{\\partial^2}{\\partial y_j \\partial y_k}\\left[\\left( Q^+_i(\\vec{y})+Q^-_i(\\vec{y})\\right)p(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right] + \\mathcal{O}(1\/K^2).\n\\end{equation}\nUp to order $1\/K$, \\ref{eq:fokker-planck} is the Fokker-Planck Equation (FPE) of the process.\nUsing Ito's prescription for SDEs, this corresponds to the Langevin equation\n\\begin{equation}\n    \\label{eq:langevin}\n    d y_i = \\left( Q^+_i(\\vec{y})-Q^-_i(\\vec{y}) \\right)dt + \\sqrt{ \\frac{Q^+_i(\\vec{y})+Q^-_i(\\vec{y})}{K} } dW_i\n\\end{equation}\nwhere $W_i$ is a standard Wiener process. By multiplying both sides of this equation by the characteristic size $K dy_i = dx_i$,\n\\begin{equation}\n    \\label{eq:langevin-LV}\n    d x_i = \\left( q^+_i(\\vec{x})-q^-_i(\\vec{x}) \\right)dt + \\sqrt{ K \\left( q^+_i(\\vec{x})+q^-_i(\\vec{x})\\right) } dW_i\n\\end{equation}\nwhere the force term in the Langevin equation recovers the Lotka-Voltera equation.\n\nIn particular, a common choice is for the diffusion term to be proportional to the square root of the abundance, such that the noise is independent of other species abundances, as in  \n\\begin{equation}\n    \\label{eq:langevin-LV-sqrt-noise}\n    d x_i = \\left( q^+_i(\\vec{x})-q^-_i(\\vec{x}) \\right)dt + \\sqrt{ K r x_i } dW_i.\n\\end{equation}\nHowever not all choice of birth and death rate appropriately recovers this form.\n\nUsing an Euler integration method, simulations of the Langevin equation assess how well the Fokker-Planck approximates the SAD. \nWe find that the Fokker-Planck approximation does not reproduce the complete phase space of the modality regimes for either noise specified in \\eqref{eq:langevin-LV} and \\eqref{eq:langevin-LV-sqrt-noise}, see Fig.~\\ref{fig:langevin}.\nIn both cases, the `neutral-like' regime at high competitive overlap ($\\rho > 2\\cdot10^-1$) is present even at low immigration  such that no bimodality is observed on the neutral manifold ($\\rho = 1$).\nThe boundary between the immigration dominated unimodal regime and other regimes is recovered; in this regime, few fluctuations arise that are sufficiently strong enough to bring any species close to the excluded state, $x_i=0$.\nNote that the force term is always positive for some $x_i>0$, which implies that species will assuredly be deterministically pushed back from exclusion.\nAdditionally, the multimodal regime is absent in all Langevin results.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=0.7\\columnwidth]{figures\/supp-langevin.pdf}\n    \\caption{Langevin numerical simulations. Left: Modality regimes with noise \\eqref{eq:langevin-LV} from the rates defined in the main text. Right: Modality regimes with $\\sqrt{n}$ noise as in \\eqref{eq:langevin-LV-sqrt-noise}. Colours correspond to the modality regime presented in the main text: yellow is the `neutral-like' regime, teal is the bimodal regime and purple is the immigration dominated unimodal regime.}\n    \\label{fig:langevin}\n\\end{figure}\n\n\\section*{Species Rank Abundances vs Species Abundance Distribution }\nIn the vast majority of the manuscript we use species abundance distributions (SADs), together with some dynamical properties, in order to examine and classify processes into deferment regimes.  However, in many experimental studies, the species rank abundances (SRAs) are frequently reported instead, e.g see \\cite{ser2018ubiquitous,mora2018quantifying,hubbell1979tree,descheemaeker2020stochastic,jeraldo2012quantification}. The SRA is closely related to the cumulative distribution correspond to SAD, as is described in the following. First, the cumulative abundance distribution is computed with ${\\rm CAD}(n)\\equiv\\sum_0^n P(n')$. Then, we note that the most abundance species, namely species with rank 1, has abundance between ${\\rm CAD}^{-1}(1-1\/S)$ to  ${\\rm CAD}^{-1}(1)$. The second most abundance species, i.e. the ranked 2,  has abundance between ${\\rm CAD}^{-1}(1-2\/S)$ to  ${\\rm CAD}^{-1}(1-1\/S)$, and so on. Therefore, the x-axis in Fig.~\\ref{fig:SRAvsSAD_rho1} is computed with $1+S(1-{\\rm CAD}(n))$ and the y-axis are the abundances $n$. Using this approach, we generated the SRAs correspond to SAD, see   Fig.~\\ref{fig:SRAvsSAD_rho1} for the results for $\\rho=1$.  However, as is shown in Fig.~\\ref{fig:SRAvsSAD_rho1}, classification through the SRAs is less significant.\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[trim=140 220 100 230, width=0.49\\columnwidth]{figures\/SAD_rho1.pdf}\n    \\includegraphics[trim=140 220 100 230, width=0.49\\columnwidth]{figures\/SRA_rho1.pdf}\n    \\caption{A comparison between the species abundance distribution (left panel) and its corresponding species rank abundances (right panel). Here $\\rho=1$ is fixed, and the immigration rate $\\mu$ varies. Solid lines represent $\\mu\\in [10^{-3}, 10^{-1}, 10]$ following the legend. The dashed, dotted and dashed-dotted lines are in-between $\\mu$-s. The color scheme corresponds the modality classification of SAD; teal (bimodal, very low $\\mu$), yellow (rare-biosphere, intermediate $mu$) and blue (unimodal, high $\\mu$).       }\n    \\label{fig:SRAvsSAD_rho1}\n\\end{figure}\n\n\\section*{Experimental Studies}\n\n\\begin{table}[h!]\n    \\centering\n    \\begin{tabular}{|p{3cm}|p{4.5cm}|p{5cm}|p{3cm}|}\n       System (Ref.) & Regimes &\tConclusions & Observations  \\\\ \\hline \nMicrobial competition \\cite{gore2021} & Stable full coexsitence (IIa), stable partial coexistnce (IIb), persisnt fluctuation (IIIb) &\tPositive correlation between species and instability (rapid fluctuations)& \tCommunity composition\/ richness\/  fluctuating communities   \n\\\\\nGlobal birds species \\cite{callaghan2021global} &\tUnimodal - log skew &\t&\tSAD  \\\\\nPlankton \\cite{ser2018ubiquitous}  &\tAbundance is power-law decay , neutral-like &\t& SAD and SRA \\\\\nCoral \\cite{dornelas2008multiple}\t& Multimodal distribution &\tSAD Not from habitat preferences, most likely due to spatial effects.  Common of large samples  &\tSAD\t\n\\\\\nLymphocyte repertoire \\cite{mora2018quantifying} &\tPower law distributions&\tFunctional repertoire is more relevant than actual repertoire, overlap in antigen coverage reduces size for repertoire &\tSRA \n\\\\\nArthropods \\cite{matthews2014multimodal} &\tMulti-modal distribution\t& Propose that multiple modes come from ecologically distinct communities &\tSAD \n\\\\\nTrees \\cite{hubbell1979tree} &\tPower law distributions &\tDifferent forest types show curves with different slopes, explained by random walk model &\tSRA \n\\\\\nBacteria \\cite{bell2005larger}\t& Increasing K, related to Species Area Relation (SAR) &\tLarger islands have more bacteria taxa on them (increased diversity)\t& Diversity per islands size\t\n\\\\\nT-cell receptors \\cite{oakes2017quantitative}\t& Bimodal and unimodal (exponential) &\tAlthough total population (CD8 naive cells) has bimodal distirbution, subpopulations have exponential\tclonotype & SAD\t\n\\\\\nMicrobial competition  \\cite{descheemaeker2020stochastic} &\tNeutral-like and niche-like & Heavy tailed rank abundance (microbiome tongue seems to have 2 slopes). Model with linear (extrinsic noise) reproduces this & SRA and time series \n\\\\ \nCompetition in gastrointestinal microbiomes \\cite{jeraldo2012quantification} &\tNon-neutral (niche) &\tThe species abundance patterns are seemingly well fit by the neutral theory, however the operational taxonomic units (OTUs)  classify it as niche   &\tSRA and OTUs\n\\\\ \n    \\end{tabular}\n    \\caption{Experimental studies consider competitive species and their reported results.  }\n    \\label{tab:my_label}\n\\end{table}\n\n\n\\iffalse\n\\begin{table}\\centering\n\\caption{This is a table}\n\n\\begin{tabular}{lrrr}\nSpecies & CBS & CV & G3 \\\\\n\\midrule\n1. Acetaldehyde & 0.0 & 0.0 & 0.0 \\\\\n2. Vinyl alcohol & 9.1 & 9.6 & 13.5 \\\\\n3. Hydroxyethylidene & 50.8 & 51.2 & 54.0\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\\fi\n\n\\iffalse\n\\movie{Type legend for the movie here.}\n\n\\movie{Type legend for the other movie here. Adding longer text to show what happens, to decide on alignment and\/or indentations.}\n\n\\movie{A third movie, just for kicks.}\n\n\\dataset{dataset_one.txt}{Type or paste legend here.}\n\n\\dataset{dataset_two.txt}{Type or paste legend here. Adding longer text to show what happens, to decide on alignment and\/or indentations for multi-line or paragraph captions.}\n\\fi\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzewcm b/data_all_eng_slimpj/shuffled/split2/finalzzewcm
new file mode 100644
index 0000000000000000000000000000000000000000..5852e83b5a893ed5324b1aa19a4f9d8cab2568c7
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzewcm
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\\label{sec:introduction}\nLet $q\\geq 2$ be a positive integer. Then every real $\\theta\\in[0,1)$\nadmits a unique expansion of the form\n\\[\\theta=\\sum_{k\\geq1}a_kq^k\\quad(a_k\\in\\{0,\\ldots,q-1\\})\\]\ncalled the $q$-ary expansion. We denote by\n$\\mathcal{N}(\\theta,d_1\\cdots d_\\ell,N)$ the number of occurrences of the block\n$d_1\\cdots d_\\ell$ amongst the first $N$ digits, \\textit{i.e.}\n\\[\\mathcal{N}(\\theta,d_1\\cdots d_\\ell,N):=\\#\\{0\\leq i< n\\colon\na_{i+1}=d_1,\\ldots,a_{i+\\ell}=d_\\ell\\}.\\] Then we call a number normal\nof order $\\ell$ in base $q$ if for each block of length $\\ell$ the\nfrequency of occurrences tends to $q^{-\\ell}$. As a qualitative\nmeasure of the distance of a number from being normal we introduce for\nintegers $N$ and $\\ell$ the discrepancy of $\\theta$\nby \\[\\mathcal{R}_{N,\\ell}(\\theta)=\\sup_{d_1\\ldots\n  d_\\ell}\\lvert\\frac{\\mathcal{N}(\\theta,d_1\\cdots\n  d_\\ell,N)}{N}-q^{-k}\\rvert,\\] where the supremum is over all blocks\nof length $\\ell$. Then a number $\\theta$ is normal to base $q$ if for\neach $\\ell\\geq1$ we have that $\\mathcal{R}_{N,\\ell}(\\theta)=o(1)$ for\n$N\\to\\infty$. Furthermore we call a number absolutely normal if it is\nnormal in all bases $q\\geq2$.\n\nBorel \\cite{borel1909:les_probabilites_denombrables} used a slightly\ndifferent, but equivalent (\\textit{cf.} Chapter 4 of \\cite{bugeaud2012:distribution_modulo_one}), definition of normality to show that almost\nall real numbers are normal with respect to the Lebesgue\nmeasure. Despite their omnipresence it is not known whether numbers\nsuch as $\\log2$, $\\pi$, $e$ or $\\sqrt2$ are normal to any base. The\nfirst construction of a normal number is due to Champernowne\n\\cite{champernowne1933:construction_decimals_normal} who showed that\nthe number\n\\begin{align*}\n0.1\\,2\\,3\\,4\\,5\\,6\\,7\\,8\\,9\\,10\\,11\\,12\\,13\\,14\\,15\\,16\\,17\\,18\\,19\\,20\\dots\n\\end{align*}\nis normal in base $10$.\n\nThe construction of Champernowne laid the base for a class of\nnormal numbers which are of the form\n\\begin{gather*}\n\\sigma_q=\\sigma_q(f)=\n  0.\\left\\lfloor f(1)\\right\\rfloor_q\\left\\lfloor f(2)\\right\\rfloor_q\\left\\lfloor f(3)\\right\\rfloor_q \\left\\lfloor f(4)\\right\\rfloor_q \\left\\lfloor f(5)\\right\\rfloor_q \\left\\lfloor f(6)\\right\\rfloor_q \\dots,\n\\end{gather*}\nwhere $\\left\\lfloor\\cdot\\right\\rfloor_q$ denotes the expansion in base $q$ of the integer\npart. Davenport and Erd{\\H o}s\n\\cite{davenport_erdoes1952:note_on_normal} showed that $\\sigma(f)$ is\nnormal for $f$ being a polynomial such that $f(\\mathbb{N})\\subset\\mathbb{N}$. This\nconstruction was extended by Schiffer\n\\cite{schiffer1986:discrepancy_normal_numbers} to polynomials with\nrational coefficients. Furthermore he showed that for these\npolynomials the discrepancy $\\mathcal{R}_{N,\\ell}(\\sigma(f))\\ll (\\log\nN)^{-1}$ and that this is best possible. These results where extended\nby Nakai and Shiokawa\n\\cite{nakai_shiokawa1992:discrepancy_estimates_class} to polynomials\nhaving real coefficients. Madritsch, Thuswaldner and Tichy\n\\cite{madritsch_thuswaldner_tichy2008:normality_numbers_generated}\nconsidered transcendental entire functions of bounded logarithmic\norder. Nakai and Shiokawa\n\\cite{nakai_shiokawa1990:class_normal_numbers} used\npseudo-polynomial functions, \\textit{i.e.} these are function of the\nform\n\\begin{gather}\\label{mani:pseudopoly}\n  f(x)=\\alpha_0 x^{\\beta_0}+\\alpha_1x^{\\beta_1}+\\cdots+\\alpha_dx^{\\beta_d}\n\\end{gather}\nwith $\\alpha_0,\\beta_0,\\alpha_1,\\beta_1,\\ldots,\\alpha_d,\\beta_d\\in\\mathbb{R}$,\n$\\alpha_0>0$, $\\beta_0>\\beta_1>\\cdots>\\beta_d>0$ and at least one\n$\\beta_i\\not\\in\\mathbb{Z}$. Since we often only need the leading term we write\n$\\alpha=\\alpha_0$ and $\\beta=\\beta_0$ for short. They were also able\nto show that the discrepancy is $\\mathcal{O}((\\log N)^{-1})$. We refer\nthe interested reader to the books of Kuipers and Niederreiter\n\\cite{kuipers_niederreiter1974:uniform_distribution_sequences}, Drmota\nand Tichy \\cite{drmota_tichy1997:sequences_discrepancies_and} or\nBugeaud \\cite{bugeaud2012:distribution_modulo_one} for a more complete\naccount on the construction of normal numbers.\n\nThe present method of construction by concatenating function values is in\nstrong connection with properties of $q$-additive functions. We call a\nfunction $f$ strictly $q$-additive, if $f(0)=0$ and the function\noperates only on the digits of the $q$-ary representation, i.e.,\n\\[\n  f(n)=\\sum_{h=0}^\\ell f(d_h)\\quad\\text{ for }\\quad n=\\sum_{h=0}^\\ell d_hq^h.\n\\]\nA very simple example of a strictly $q$-additive function is the sum of digits\nfunction $s_q$, defined by\n\\[\n  s_q(n)=\\sum_{h=0}^\\ell d_h\\quad\\text{ for }\\quad n=\\sum_{h=0}^\\ell d_hq^h.\n\\]\n\nRefining the methods of Nakai and Shiokawa\n\\cite{nakai_shiokawa1990:class_normal_numbers} the author obtained the\nfollowing result.\n\\begin{thm}[{\\cite[Theorem 1.1]{madritsch2012:summatory_function_q}}]\n  Let $q\\geq2$ be an integer and $f$ be a strictly $q$-additive\n  function. If $p$ is a pseudo-polynomial as defined in\n  (\\ref{mani:pseudopoly}), then there exists $\\eta>0$ such that\n\\begin{gather*}\n  \\sum_{n\\leq N}f\\left(\\left\\lfloor p(n)\\right\\rfloor\\right)\n  =\\mu_fN\\log_q(p(N))\n  +NF\\left(\\log_q(p(N))\\right)\n  +\\mathcal{O}\\left(N^{1-\\eta}\\right),\n\\end{gather*}\nwhere\n\\[\n\\mu_f=\\frac1q\\sum_{d=0}^{q-1}f(d)\n\\]\nand $F$ is a $1$-periodic function depending only on $f$ and $p$.\n\\end{thm}\n\nIn the present paper, however, we are interested in a variant of\n$\\sigma_q(f)$ involving primes. As a first example, Champernowne\n\\cite{champernowne1933:construction_decimals_normal} conjectured and\nlater Copeland and Erd{\\H o}s\n\\cite{copeland_erdoes1946:note_on_normal} proved that the number\n\\begin{align*}\n0.2\\,3\\,5\\,7\\,11\\,13\\,17\\,19\\,23\\,29\\,31\\,37\\,41\\,43\\,47\\,53\\,59\\,61\\,67\\dots\n\\end{align*}\nis normal in base $10$. Similar to the construction above we want to\nconsider the number\n\\begin{gather*}\n\\tau_q=\\tau_q(f)=0.\\left\\lfloor f(2)\\right\\rfloor_q \\left\\lfloor f(3)\\right\\rfloor_q \\left\\lfloor f(5)\\right\\rfloor_q \\left\\lfloor f(7)\\right\\rfloor_q \\left\\lfloor f(11)\\right\\rfloor_q \\left\\lfloor f(13)\\right\\rfloor_q \\dots,\n\\end{gather*}\nwhere the arguments of $f$ run through the sequence of primes.\n\nThen the paper of Copeland and Erd{\\H o}s corresponds to the function\n$f(x)=x$. Nakai and Shiokawa\n\\cite{nakai_shiokawa1997:normality_numbers_generated} showed that the\ndiscrepancy for polynomials having rational coefficients is\n$\\mathcal{O}((\\log N)^{-1})$. Furthermore Madritsch, Thuswaldner and\nTichy\n\\cite{madritsch_thuswaldner_tichy2008:normality_numbers_generated}\nshowed, that transcendental entire functions of bounded logarithmic\norder yield normal numbers. Finally in a recent paper Madritsch and\nTichy \\cite{madritsch_tichy2013:construction_normal_numbers}\nconsidered pseudo-polynomials of the special form $\\alpha x^\\beta$\nwith $\\alpha>0$, $\\beta>1$ and $\\beta\\not\\in\\mathbb{Z}$.\n\nThe aim of the present paper is to extend this last construction to\narbitrary pseudo-polynomials. Our first main result is the following\n\\begin{thm}\\label{thm:normal}\nLet $f$ be a pseudo-polynomial as in (\\ref{mani:pseudopoly}). Then\n\\[\n\\mathcal{R}_N(\\tau_q(f))\\ll(\\log N)^{-1}.\n\\]\n\\end{thm}\n\n\nIn our second main result we use the connection of this construction\nof normal numbers with the arithmetic mean of $q$-additive functions\nas described above. Known results are due to Shiokawa\n\\cite{shiokawa1974:sum_digits_prime} and Madritsch and Tichy\n\\cite{madritsch_tichy2013:construction_normal_numbers}. Similar\nresults concerning the moments of the sum of digits function over\nprimes have been established by K\\'atai\n\\cite{katai1977:sum_digits_primes}.\n\nLet $\\pi(x)$ stand for the number of primes less than or equal to\n$x$. Then adapting these ideas to our method we obtain the following\n\\begin{thm}\\label{thm:summatoryfun}\nLet $f$ be a pseudo-polynomial as in (\\ref{mani:pseudopoly}). Then \n\\[\n\\sum_{p\\leq P}s_q(\\left\\lfloor f(p)\\right\\rfloor)=\\frac{q-1}2\\pi(P)\\log_qP^\\beta+\\mathcal{O}(\\pi(P)),\n\\]\nwhere the sum runs over the primes and the implicit $\\mathcal{O}$-constant may\ndepend on $q$ and $\\beta$.\n\\end{thm}\n\n\\begin{rem}\nWith simple modifications Theorem \\ref{thm:summatoryfun} can be extended to\ncompletely $q$-additive functions replacing $s_q$.\n\\end{rem}\n\n\nThe proof of the two theorems is divided into four parts. In the\nfollowing section we rewrite both statements in order to obtain as a\ncommon base the central theorem -- Theorem \\ref{mani:centralthm}. In\nSection~\\ref{sec:proof-prop-refm1} we start with the proof of this\ncentral theorem by using an indicator function and its Fourier\nseries. These series contain exponential sums which we treat by\ndifferent methods (with respect to the position in the expansion) in\nSection \\ref{sec:expon-sum-estim}. Finally, in\nSection~\\ref{sec:proof-prop-refm2} we put the estimates together in\norder to proof the central theorem and therefore our two statements.\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\nThroughout the rest $p$ will always denote a prime. The implicit\nconstant of $\\ll$ and $\\mathcal{O}$ may depend on the\npseudo-polynomial $f$ and on the parameter\n$\\varepsilon>0$. Furthermore we fix a block $d_1\\cdots d_\\ell$ of\nlength $\\ell$ and $N$, the number of digits we consider.\n\nIn the first step we want to know in the\nexpansion of which prime the $N$-th digit occurs. This can be seen as\nthe translation from the digital world to the world of blocks. To this\nend let $\\ell(m)$ denote the length of the $q$-ary\nexpansion of an integer $m$. Then we define an integer $P$ by\n\\begin{gather*}\n\\sum_{p\\leq P-1}\\ell\\left(\\lfloor f(p)\\rfloor\\right) <N\\leq\n\\sum_{p\\leq P}\\ell\\left(\\lfloor f(p)\\rfloor\\right),\n\\end{gather*}\nwhere the sum runs over all primes. Thus we get the following relation\nbetween $N$ and $P$\n\\begin{equation}\\label{mani:NtoP}\n\\begin{split}\nN&=\\sum_{p\\leq P}\\ell(\\left\\lfloor f(p)\\right\\rfloor)+\\mathcal{O}(\\pi(P))+\\mathcal{O}(\\beta \\log_q(P))\\\\\n&=\\frac{\\beta}{\\log q}P+\\mathcal{O}\\left(\\frac{P}{\\log P}\\right).\n\\end{split}\\end{equation}\nHere we have used the prime number theorem in the form (\\textit{cf.}\n\\cite[Th\\'eor\\`eme 4.1]{tenenbaum1995:introduction_la_theorie}) \n\\begin{gather}\\label{pnt}\n  \\pi(x)=\\mathrm{Li}\\, x+\\mathcal{O}\\left(\\frac x{(\\log x)^G}\\right),\n\\end{gather}\nwhere $G$ is an arbitrary positive constant and\n\\[\n  \\mathrm{Li}\\,x=\\int_2^x\\frac{\\mathrm{d}t}{\\log t}.\n\\]\n\nNow we show that we may neglect the occurrences of the block\n$d_1\\cdots d_\\ell$ between two expansions. We write\n$\\mathcal{N}(f(p))$ for the number of occurrences of this block in the\n$q$-ary expansion of $\\lfloor f(p)\\rfloor$. Then \\eqref{mani:NtoP}\nimplies that\n\\begin{gather}\\label{mani:Ntrunc}\n  \\lvert\\mathcal{N}(\\tau_q(f);d_1\\cdots d_\\ell;N)-\\sum_{p\\leq\n    P}\\mathcal{N}(f(p))\\rvert\\ll\\frac N{\\log N}.\n\\end{gather}\n\nIn the next step we use the polynomial-like behavior of $f$. In\nparticular, we collect all the values having the same length of\nexpansion. Let $j_0$ be a sufficiently large integer. Then for\neach integer $j\\geq j_0$ there exists a $P_j$ such that\n\\[\n  q^{j-2}\\leq f(P_j)<q^{j-1}\\leq f(P_j+1)<q^j\n\\]\nwith\n\\[\n  P_j\\asymp q^{\\frac j\\beta}.\n\\]\nFurthermore we set $J$ to be the greatest length of the $q$-ary\nexpansions of $f(p)$ over the primes $p\\leq P$, i.e.,\n\\begin{gather*}\nJ:=\\max_{p\\leq P}\\ell(\\lfloor f(p)\\rfloor)=\\log_q(f(P))+\\mathcal{O}(1)\\asymp\\log\nP.\n\\end{gather*}\n\nNow we show that we may suppose that each expansion has the same\nlength (which we reach by adding leading zeroes). For $P_{j-1}<p\\leq\nP_j$ we may write $f(p)$ in $q$-ary expansion, i.e.,\n\\begin{gather}\\label{mani:expansionoffp}\nf(p)=b_{j-1}q^{j-1}+b_{j-2}q^{j-2}+\\dots+b_{1}q+b_{0}+b_{-1}q^{-1}+\\dots.\n\\end{gather}\nThen we denote by $\\mathcal{N}^*(f(p))$ the number of occurrences of the block\n$d_1\\cdots d_\\ell$ in the string $0\\cdots0b_{j-1}b_{j-2}\\cdots b_1b_0$, where we\nfilled up the expansion with leading zeroes such that it has length $J$. The error of\ndoing so can be estimated by \n\\begin{align*}\n0&\\leq\\sum_{p\\leq P}\\mathcal{N}^*(f(p))-\\sum_{p\\leq P}\\mathcal{N}(f(p))\\\\\n&\\leq\\sum_{j=j_0+1}^{J-1}(J-j)\\left(\\pi(P_{j+1})-\\pi(P_{j})\\right)+\\mathcal{O}(1)\\\\\n&\\leq\\sum_{j=j_0+2}^{J}\\pi(P_{j})+\\mathcal{O}(1)\\ll\\sum_{j=j_0+2}^{J}\\frac{q^{j\/\\beta}}j\n\\ll\\frac P{\\log P}\\ll\\frac N{\\log N}.\n\\end{align*}\n\nIn the following three sections we will estimate this sum of indicator\nfunctions $\\mathcal{N}^*$ in order to prove the following theorem.\n\\begin{thm}\\label{mani:centralthm}\nLet $f$ be a pseudo polynomial as in \\eqref{mani:pseudopoly}. Then\n\\begin{gather}\\label{mani:centralthm:statement}\n\\sum_{p\\leq\n  P}\\mathcal{N}^*\\left(\\left\\lfloor f(p)\\right\\rfloor\\right)=q^{-\\ell}\\pi(P)\\log_qP^\\beta+\\mathcal{O}\\left(\\frac{P}{\\log\n  P}\\right)\n\\end{gather}\n\\end{thm}\n\nUsing this theorem we can simply deduce our two main results.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:normal}]\nWe insert \\eqref{mani:centralthm:statement} into \\eqref{mani:Ntrunc}\nand get the desired result.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:summatoryfun}]\nFor this proof we have to rewrite the statement. In particular, we use that the\nsum of digits function counts the number of $1$s, $2$s, etc. and\nassigns weights to them, i.e.,\n\\[\ns_q(n)=\\sum_{d=0}^{q-1}d\\cdot\\mathcal{N}(n;d).\n\\]\nThus\n\\begin{align*}\n\\sum_{p\\leq P}s_q(\\left\\lfloor p^\\beta\\right\\rfloor)\n&=\\sum_{p\\leq P}\\sum_{d=0}^{q-1}d\\cdot\\mathcal{N}(p^\\beta)\n=\\sum_{p\\leq\n  P}\\sum_{d=0}^{q-1}d\\cdot\\mathcal{N}^*(p^\\beta)+\\mathcal{O}\\left(\\frac{P}{\\log\n    P}\\right)\\\\\n&=\\frac{q-1}2\\pi(P)\\log_q(P^\\beta)+\\mathcal{O}\\left(\\frac{P}{\\log\n    P}\\right)\n\\end{align*}\nand the theorem follows.\n\\end{proof}\n\n\nIn the following sections we will prove Theorem \\ref{mani:centralthm}\nin several steps. First we use the ``method of little glasses'' in\norder to approximate the indicator function by a Fourier series having\nsmooth coefficients. Then we will apply different methods (depending\non the position in the expansion) for the estimation of the\nexponential sums that appear in the Fourier series. Finally we put\neverything together and get the desired estimate.\n\n\\section{Proof of Theorem \\ref{mani:centralthm}, Part I}\\label{sec:proof-prop-refm1}\nWe want to ease notation by splitting the pseudo-polynomial $f$ into a polynomial and the rest. Then\nthere exists a unique decomposition of the following form:\n\\begin{gather}\\label{pseudo:poly:split}\nf(x)=g(x)+h(x)\n\\end{gather}\nwhere $h\\in\\mathbb{R}[X]$ is a polynomial of degree $k$ (where we set\n$k=0$ if $h$ is the zero polynomial) and\n$$g(x)=\\sum_{j=1}^r\\alpha_jx^{\\theta_j}$$\nwith $r\\geq1$, $\\alpha_r\\neq0$, $\\alpha_j$ real, $0<\\theta_1<\\cdots<\\theta_r$\nand $\\theta_j\\not\\in\\mathbb{Z}$ for $1\\leq j\\leq r$.\n\nLet $\\gamma$ and $\\rho$ be two parameter which we will frequently\nuse in the sequel. We suppose that\n\\begin{gather*}\n0<\\gamma<\\rho<\\min\\left(\\frac1{4(k+1)},\\frac{\\theta_r}{2}\\right).\n\\end{gather*}\n\nThe aim of this section is to calculate the Fourier transform of\n$\\mathcal{N}^*$. In order to count the occurrences of the block\n$d_1\\cdots d_\\ell$ in the $q$-ary expansion of $\\lfloor f(p) \\rfloor$\n($2\\le p \\le P$) we define the indicator function\n\\begin{align*}\n\\mathcal{I}(t)=\\begin{cases}\n  1, &\\text{if }\\sum_{i=1}^\\ell d_iq^{-i}\\leq t-\\lfloor t\\rfloor\n     <\\sum_{i=1}^\\ell d_iq^{-i}+q^{-\\ell};\\\\\n  0, &\\text{otherwise;}\n     \\end{cases}\n\\end{align*}\nwhich is a $1$-periodic function. Indeed, we have\n\\begin{gather}\\label{position}\n\\mathcal{I}(q^{-j}f(p)) = 1 \\Longleftrightarrow d_1\\cdots d_\\ell =\nb_{j-1}\\cdots b_{j-\\ell},\n\\end{gather}\nwhere $f(p)$ has an expansion as in (\\ref{mani:expansionoffp}). Thus\nwe may write our block counting function as follows\n\\begin{gather}\\label{mani:NthetatoNstar}\n\\mathcal{N}^*(f(p))=\\sum_{j=\\ell}^J\\mathcal{I}\\left(q^{-j}f(p)\\right).\n\\end{gather}\n\n\nIn the following we will use Vinogradov's ``method of little glasses''\n(\\textit{cf.} \\cite{vinogradov2004:method_trigonometrical_sums}). We want to approximate\n$\\mathcal{I}$ from above and from below by two $1$-periodic functions\nhaving small Fourier coefficients. To this end we will use the\nfollowing\n\\begin{lem}[{\\cite[Lemma\n    12]{vinogradov2004:method_trigonometrical_sums}}]\\label{vin:lem12}\nLet $\\alpha$, $\\beta$, $\\Delta$ be real numbers satisfying\n\\begin{gather*}\n0<\\Delta<\\frac12,\\quad\\Delta\\leq\\beta-\\alpha\\leq1-\\Delta.\n\\end{gather*}\nThen there exists a periodic function $\\psi(x)$ with period $1$,\nsatisfying\n\\begin{enumerate}\n\\item $\\psi(x)=1$ in the interval $\\alpha+\\frac12\\Delta\\leq x\n  \\leq\\beta-\\frac12\\Delta$,\n\\item $\\psi(x)=0$ in the interval $\\beta+\\frac12\\Delta\\leq x\n  \\leq1+\\alpha-\\frac12\\Delta$,\n\\item $0\\leq\\psi(x)\\leq1$ in the remainder of the interval\n  $\\alpha-\\frac12\\Delta\\leq x\\leq1+\\alpha-\\frac12\\Delta$,\n\\item $\\psi(x)$ has a Fourier series expansion of the form\n  $$\n  \\psi(x)=\\beta-\\alpha+\\sum_{\\substack{\\nu=-\\infty\\\\\\nu\\neq0}}^\\infty\n   A(\\nu) e(\\nu x),\n  $$\n  where\n  \\begin{gather}\\label{mani:A}\n  \\lvert A(\\nu)\\rvert \\ll \\min \\left( \\frac 1\\nu,\n  \\beta-\\alpha,\\frac{1}{\\nu^2\\Delta} \\right).\n  \\end{gather}\n\\end{enumerate}\n\\end{lem}\n\nWe note that we could have used Vaaler polynomials\n\\cite{vaaler1985:some_extremal_functions}, however, we do not gain\nanything by doing so as the estimates we get are already best\npossible. Setting\n\\begin{equation}\\label{mani:abd}\n\\begin{split}\n\\delta=P^{-\\gamma},\\quad\n\\begin{aligned}\n\\alpha_-&=\\sum_{\\lambda=1}^\\ell d_\\lambda q^{-\\lambda}+(2\\delta)^{-1},&\n\\beta_-&=\\sum_{\\lambda=1}^\\ell d_\\lambda q^{-\\lambda}+q^{-\\ell}-(2\\delta)^{-1},\\\\\n\\alpha_+&=\\sum_{\\lambda=1}^\\ell d_\\lambda q^{-\\lambda}-(2\\delta)^{-1},&\n\\beta_+&=\\sum_{\\lambda=1}^\\ell d_\\lambda q^{-\\lambda}+q^{-\\ell}+(2\\delta)^{-1}.\n\\end{aligned}\n\\end{split}\n\\end{equation}\nand an application of Lemma \\ref{vin:lem12} with\n$(\\alpha,\\beta,\\delta)=(\\alpha_-,\\beta_-,\\delta)$ and\n$(\\alpha,\\beta,\\delta)=(\\alpha_+,\\beta_+, \\delta)$,\nrespectively, provides us with two functions $\\mathcal{I}_-$ and\n$\\mathcal{I}_+$. By our choice of\n$(\\alpha_\\pm,\\beta_\\pm,\\delta)$ it is immediate that\n\\begin{equation}\\label{uglI}\n\\mathcal{I}_-(t)\\leq\\mathcal{I}(t)\\leq\\mathcal{I}_+(t) \\qquad\n(t\\in\\mathbb{R}).\n\\end{equation}\nLemma \\ref{vin:lem12} also implies that these two functions have\nFourier expansions\n\\begin{align}\\label{mani:Ifourier}\n\\mathcal{I}_\\pm(t)=q^{-\\ell}\\pm P^{-\\gamma}+\n  \\sum_{\\substack{\\nu=-\\infty\\\\\\nu\\neq0}}^\\infty A_\\pm(\\nu)e(\\nu t)\n\\end{align}\nsatisfying\n\\begin{gather*\n\\lvert A_\\pm(\\nu)\\rvert\n\\ll\\min(\\lvert\\nu\\rvert^{-1},P^{\\gamma}\\lvert\\nu\\rvert^{-2}).\n\\end{gather*}\nIn a next step we want to replace $\\mathcal{I}$ by $\\mathcal{I}_+$\nin (\\ref{mani:NthetatoNstar}). For this purpose we observe, using \\eqref{uglI},\nand \\eqref{mani:Ifourier} that\n\\begin{gather*}\n\\lvert\\mathcal{I}(t)-q^{-\\ell}\\rvert\n  \\ll P^{-\\gamma} + \\sum_{\\substack{\\nu=-\\infty\\\\\\nu\\neq0}}^\\infty\n  A_\\pm(\\nu)e(\\nu t).\n\\end{gather*}\nThus setting $t=q^{-j}f(p)$ and summing over $p\\leq P$ yields\n\\begin{gather}\\label{mani:0.5}\n\\lvert\\sum_{p\\leq P}\\mathcal{I}(q^{-j}f(p))-\\frac{\\pi(P)}{q^{\\ell}}\\rvert\n\\ll\\pi(P)P^{-\\gamma}+\\sum_{\\substack{\\nu=-\\infty\\\\\\nu\\neq0}}^\\infty\nA_{\\pm}(\\nu)\\sum_{p\\leq P}e\\left(\\frac{\\nu}{q^j}f(p)\\right).\n\\end{gather}\n\nNow we consider the coefficients $A_\\pm(\\nu)$. Noting\n\\eqref{mani:A} one observes that\n\\begin{gather*}\nA_\\pm(\\nu)\\ll\\begin{cases}\n  \\nu^{-1},       &\\text{for }\\lvert\\nu\\rvert\\leq P^{\\gamma};\\\\\n  P^{\\gamma}\\nu^{-2}, &\\text{for }\\lvert\\nu\\rvert>P^{\\gamma}.\n          \\end{cases}\n\\end{gather*}\nEstimating all summands with $\\lvert\\nu\\rvert>P^{\\gamma}$ trivially we get\n\\begin{gather*\n\\sum_{\\substack{\\nu=-\\infty\\\\\\nu\\neq0}}^\\infty\n  A_\\pm(\\nu)e\\left(\\frac{\\nu}{q^j}f(p)\\right)\n\\ll\\sum_{\\nu=1}^{P^{\\gamma}}\\nu^{-1}e\\left(\\frac{\\nu}{q^j}f(p)\\right)+P^{-\\gamma}.\n\\end{gather*}\nUsing this in \\eqref{mani:0.5} yields\n\\begin{gather*}\n\\lvert\\sum_{p\\leq P}\\mathcal{I}(q^{-j}f(p))-\\frac{\\pi(P)}{q^{\\ell}}\\rvert\n\\ll\\pi(P)P^{-\\gamma}+\\sum_{\\nu=1}^{P^{\\gamma}}\n\\nu^{-1}S(P,j,\\nu),\n\\end{gather*}\nwhere we have set \n\\begin{gather}\\label{S_Pjnu}\nS(P,j,\\nu):=\\sum_{p\\leq P}e\\left(\\frac{\\nu}{q^j}f(p)\\right).\n\\end{gather}\n\n\\section{Exponential sum estimates}\\label{sec:expon-sum-estim}\nIn the present section we will focus on the estimation of the sum\n$S(P,j,\\nu)$ for different ranges of $j$. Since $j$ describes the\nposition within the $q$-ary expansion of $f(p)$ we will call these\nranges the ``most significant digits'', the ``least significant\ndigits'' and the ``digits in the middle'', respectively.\n\nNow, if $\\theta_r>k\\geq0$, \\textit{i.e} the leading coefficient of $f$ origins\nfrom the pseudo polynomial part $g$, then we consider the two ranges\n$$1\\leq q^j\\leq P^{\\theta_r-\\rho}\\quad\\text{and}\\quad\nP^{\\theta_r-\\rho}<q^j\\leq P^{\\theta_r}.$$\nFor the first one we will apply Proposition \\ref{prop:least_significant} and\nfor the second one Proposition \\ref{prop:most_significant}.\n\nOn the other hand, if $k>\\theta_r>0$, meaning that the leading coefficient of\n$f$ origins from the polynomial part $h$, then we have an additional\npart. In particular, in this case we will consider the three ranges\n$$1\\leq q^j\\leq P^{\\theta_r-\\rho},\\quad\nP^{\\theta_r-\\rho}<q^j\\leq P^{k-1+\\rho},\\quad\\text{and}\\quad\nP^{k-1+\\rho}<q^j\\leq P^{k}.$$ We will, similar to above, treat the\nfirst and last range by Proposition~\\ref{prop:least_significant} and\nProposition \\ref{prop:most_significant}, respectively. For the middle\nrange we will apply Proposition \\ref{prop:middle_digits}. Since\n$2\\rho<\\theta_r$, we note that the middle range is empty if $k=1$.\n\nSince the size of $j$ represents the position of the digit in the\nexpansion (\\textit{cf.} \\eqref{position}), we will deal in the\nfollowing subsection with the ``most significant digits'', the ``least\nsignificant digits'' and the ``digits in the middle'', respectively.\n\n\\subsection{Most significant digits}\nWe start our series of estimates for the exponential sum $S(P,j,\\nu)$\nfor $j$ being in the highest range. In particular, we want to show the\nfollowing\n\\begin{prop}\\label{prop:most_significant}\nSuppose that for some $k\\geq1$ we have $\\lvert\nf^{(k)}(x)\\rvert\\geq\\Lambda$ for any $x$ on $[a,b]$ with\n$\\Lambda>0$. Then\n$$S(P,j,\\nu)\\ll\\frac1{\\log P}\\Lambda^{-\\frac1k}+\\frac{P}{(\\log P)^G}.$$\n\\end{prop}\n\nThe main idea of the proof is to use Riemann-Stieltjes integration\ntogether with \n\\begin{lem}[{\\cite[Lemma 8.10]{iwaniec_kowalski2004:analytic_number_theory}}]\n\\label{ik:lem8.10}\nLet $F\\colon[a,b]\\to\\mathbb{R}$ and suppose that for some $k\\geq1$\nwe have $\\lvert F^{(k)}(x)\\rvert\\geq\\Lambda$ for any $x$ on $[a,b]$\nwith $\\Lambda>0$. Then\n\\[\n\\lvert\\int_a^be(F(x))\\mathrm{d}x\\rvert\n\\leq k2^k\\Lambda^{-1\/k}.\n\\]\n\\end{lem}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:most_significant}]\nWe rewrite the sum into a Riemann-Stieltjes integral:\n\\begin{align*}\nS(P,j,\\nu)=\\sum_{p\\leq P}e\\left(\\frac{\\nu}{q^j}f(p)\\right)\n=\\int_{2}^{P}e\\left(\\frac{\\nu}{q^j}f(t)\\right)\\mathrm{d}\\pi(t)+\\mathcal{O}(1).\n\\end{align*}\nThen we apply the prime number theorem in the form\n\\eqref{pnt} to gain the usual integral back. Thus\n\\begin{align*}\nS(P,j,\\nu)\n=\\int_{P(\\log P)^{-G}}^{P}\ne\\left(\\frac{\\nu}{q^j}f(t)\\right)\n\\frac{\\mathrm{d}t}{\\log t}\n+\\mathcal{O}\\left(\\frac{P}{(\\log P)^G}\\right).\n\\end{align*}\nNow we use the second mean-value theorem to get\n\\begin{equation}\\label{mani:res:most}\n\\begin{split}\nS(P,j,\\nu)\\ll\\frac1{\\log P}\\sup_{\\xi}\n  \\lvert\\int_{P(\\log P)^{-G}}^{\\xi}e\\left(\\frac{\\nu}{q^j}f(t)\\right)\\mathrm{d}t\\rvert\n  +\\frac{P}{(\\log P)^G}.\n\\end{split}\n\\end{equation}\nFinally an application of Lemma \\ref{ik:lem8.10} proves the lemma.\n\\end{proof}\n\n\\subsection{Least significant digits} Now we turn our attention to the\nlowest range of $j$. In particular, the goal is the proof of the\nfollowing\n\\begin{prop}\\label{prop:least_significant}\nLet $P$ and $\\rho$ be positive reals and $f$ be a pseudo-polynomial as in\n\\eqref{pseudo:poly:split}. If $j$ is such that\n\\begin{gather}\\label{mani:gammarange}\n  1\\leq q^j\\leq P^{\\theta_r-\\rho}\n\\end{gather}\nholds, then for $1\\leq\\nu\\leq P^\\gamma$ there exists $\\eta>0$\n(depending only on $f$ and $\\rho$) such that\n\\begin{gather*}\n  S(P,j,\\nu)=(\\log P)^8P^{1-\\eta}.\n\\end{gather*}\n\\end{prop}\n\n\nBefore we launch into the proof we collect some tools that will be\nnecessary in the sequel. A standard idea for estimating exponential\nsums over the primes is to rewrite them into ordinary exponential sums\nover the integers having von Mangoldt's function as weights and then\nto apply Vaughan's identity. We denote by\n\\[\n\\Lambda(n)=\\begin{cases}\n\\log p,&\\text{if $n=p^k$ for some prime $p$ and an integer $k\\geq1$;}\\\\\n0,&\\text{otherwise}.\n\\end{cases}\n\\]\nvon Mangoldt's function. For the rewriting process we use the following\n\\begin{lem}\n\\label{mr:lem11}\nLet $g$ be a function such that $\\lvert g(n)\\rvert\\leq 1$ for all\nintegers $n$. Then\n\\[\n\\lvert\\sum_{p\\leq P}g(p)\\rvert\\ll\\frac1{\\log P}\\max_{t\\leq P}\n\\lvert\\sum_{n\\leq t}\\Lambda(n)g(n)\\rvert+\\mathcal{O}(\\sqrt{P}).\n\\]\n\\end{lem}\n\n\\begin{proof}\nThis is Lemma 11 of \\cite{mauduit_rivat2010:sur_un_probleme}. However,\nthe proof is short and we need some piece later.\n\nWe start with a summation by parts yielding\n\\[\\sum_{p\\leq P}g(p)=\\frac1{\\log P}\\sum_{p\\leq\n  x}\\log(p)g(p)+\\int_2^P\\left(\\sum_{p\\leq\n    t}\\log(p)g(p)\\right)\\frac{\\mathrm{d}t}{t(\\log t)^2}.\\]\n\nNow we cut the integral at $\\sqrt{P}$ and use Chebyshev's inequality\n(\\textit{cf.} \\cite[Th\\'eor\\`eme 1.3]{tenenbaum1995:introduction_la_theorie})\nin the form $\\sum_{p\\leq\n  t}\\log(p)\\leq\\log(t)\\pi(t)\\ll t$ for the lower part. Thus\n\\begin{align*}\n\\lvert\\sum_{p\\leq P}g(p)\\rvert\n&\\leq\\left(\\frac1{\\log\n    P}+\\int_{\\sqrt{P}}^P\\frac{\\mathrm{d}t}{t(\\log t)^2}\\right)\n  \\max_{\\sqrt{P}<t\\leq P}\\lvert\\sum_{p\\leq\n    P}\\log(p)g(p)\\rvert+\\mathcal{O}(\\sqrt{P})\\\\\n&=\\frac2{\\log P}\\max_{\\sqrt{P}<t\\leq P}\\lvert\\sum_{p\\leq t}\\log(p)g(p)\\rvert+\\mathcal{O}(\\sqrt{P}).\n\\end{align*}\n\nFinally we again use Chebyshev's inequality $\\pi(t)\\ll t\/\\log(t)$ to obtain\n\\begin{gather}\\label{mani:log_Mangoldt_equivalence}\n\\lvert\\sum_{n\\leq t}\\Lambda(n)g(n)-\\sum_{p\\leq t}\\log(p)g(p)\\rvert\n\\leq\\sum_{p\\leq\\sqrt{t}}\\log(p)\\sum_{a=2}^{\\left\\lfloor\\frac{\\log(t)}{\\log(p)}\\right\\rfloor}1\n\\leq\\pi(\\sqrt{t})\\log(t)\\ll\\sqrt{t}.\n\\end{gather}\n\\end{proof}\n\nIn the next step we use Vaughan's identity to subdivide this weighted\nexponential sum into several sums of Type I and II.\n\\begin{lem}[{\\cite[Lemma\n    2.3]{bergelson_kolesnik_madritsch+2014:uniform_distribution_prime}}]\n\\label{bkmst:lem23}\nAssume $F(x)$ to be any function defined on the real line, supported on $[P\/2, P]$\nand bounded by $F_0$. Let further $U,V,Z$ be any parameters satisfying $3\n\\leq U < V < Z < P$, $Z \\geq 4U^2$, $P \\geq 64 Z^2 U$, $V^3 \\geq 32 P$ and\n$Z-\\frac12\\in\\mathbb{N}$. Then\n$$\\left| \\sum_{P\/2< n\\leq P} \\Lambda(n) F(n)  \\right| \\ll K\n\\log P + F_0 +  L (\\log P)^8 ,$$\nwhere $K$ and $L$ are defined by\n\\begin{align*}\nK&=\\max_M\\sum_{m=1}^\\infty d_3(m)\\left\\vert\\sum\\limits_{Z<n\\leq M} F(mn)\\right\\vert,\\\\\nL&=\\sup\\sum_{m=1}^\\infty d_4(m)\\left\\vert\\sum\\limits_{U < n < V} b(n) F(mn)\\right\\vert,\n\\end{align*}\nwhere the supremum is taken over all arithmetic functions $b(n)$ satisfying $|b(n)| \\leq d_3(n).$\n\\end{lem}\n\nAfter subdividing the weighted exponential sum with Vaughan's identity\nwe will use the following lemma in order to estimate the occurring\nexponential sums.\n\\begin{lem}[{\\cite[Lemma\n     2.5]{bergelson_kolesnik_madritsch+2014:uniform_distribution_prime}}]\n\\label{bkmst:lem25}\n   Let $X,k,q\\in \\mathbb{N}$ with $k,q\\geq 0$ and set $K=2^k$ and $Q=\n   2^q$. Let $h(x)$ be a polynomial of degree $k$ with real\n   coefficients. Let $g(x)$ be a real $(q+k+2)$ times continuously\n   differentiable function on $[X\/2 , X]$ such that $\\left| f^{(r)}(x)\n   \\right| \\asymp F X^{-r}$ $( r = 1, \\dots, q+k+2) $.  Then, if $F =\n   o (X^{q+2})$ for $F$ and $X$ large enough, we have\n $$\\left| \\sum_{X\/2 < x \\leq X} e(g(x) + h(x)) \\right| \\ll X^{1 - \\frac{1}{K}} + X \\left( \\frac{\\log^k X}{F} \\right)^{\\frac{1}{K}} + X \\left( \\frac{F}{X^{q+2}} \\right)^{\\frac{1}{(4KQ-2K)}}.$$\n\\end{lem}\n\nNow we have the necessary tools to state the\n\\begin{proof}[Proof of Proposition \\ref{prop:least_significant}]\nAn application of Lemma \\ref{mr:lem11} yields\n\\[S(P,j,\\nu)\\ll\\frac1{\\log P}\\max\\lvert\\sum_{n\\leq\n  P}\\Lambda(n)e\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\\rvert+P^{\\frac12}.\\]\nWe split the inner sum into $\\leq \\log\nP$ sub sums of the form $$\\lvert \\sum\\limits_{X< n \\leq\n  2X}\\Lambda(n)e\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\\rvert$$ with\n$2X \\leq P$ and let $S$ be a typical one of them. We may\nassume that $X \\geq P^{1-\\rho}$.\n\nUsing Vaughan's identity (Lemma~\\ref{bkmst:lem23}) with $U = \\frac{1}{4} X^{1\/5}$, $V= 4 X^{1\/3}$ and $Z$ the\nunique number in $\\frac12+\\mathbb{N}$, which is closest to $\\frac{1}{4}\nX^{2\/5}$, we obtain\n\\begin{gather}\\label{mani:S}\nS \\ll 1+(\\log X)S_1+(\\log X)^8S_2,\n\\end{gather}\nwhere\n\\begin{align*}\nS_1&=\\sum_{x < \\frac{2X}{Z}} d_3(x) \\sum_{y > Z, \\frac{X}{x} < y < \\frac{2X}{x}} e\\left(\\frac{\\nu}{q^j}(g(xy)+h(xy))\\right)\\\\\nS_2&=\\sum_{\\frac XV<x\\leq\\frac{2X}U} d_4(x) \\sum_{U < y < V, \\frac{X}{x} < y \\leq \\frac{2X}{x}} b(y) e\\left(\\frac{\\nu}{q^j}(g(xy)+h(xy))\\right)\\notag\n\\end{align*}\n\nWe start with the estimation of $S_1$. Since $d_3(x)\\ll\nx^{\\varepsilon}$ we have for\n\\begin{align*}\n\\lvert S_1\\rvert\\ll X^\\varepsilon\\sum_{x\\leq\\frac{2X}Z}\n  \\lvert\\sum_{\\substack{\\frac Xx<y\\frac{2X}x\\\\y>Z}}e\\left(\\frac{\\nu}{q^j}(g(xy)+h(xy))\\right)\\rvert.\n\\end{align*}\nFor estimating the inner sum we fix $x$ and denote $Y=\\frac Xx$. Since\n$\\theta_r\\not\\in\\mathbb{Z}$ and $\\theta_r>k\\geq0$, we have that\n\\[\\lvert\\frac{\\partial^\\ell g(xy)}{\\partial y^\\ell}\\rvert\n\\asymp X^{\\theta_r}Y^{-\\ell}.\\]\n\nNow on the one hand, since $q^j\\leq P^{\\theta_r-\\rho}$, we have $\\nu\nq^{-j}X^{\\theta_r}\\gg X^{\\rho}$. On the other hand for\n$\\ell\\geq5(\\lfloor\\theta_r\\rfloor+1)$ we get\n\\[\\frac{\\nu}{q^j}X^{\\theta_r}Y^{-\\ell}\\leq P^\\gamma X^{\\theta_r-\\frac25\\ell}\\ll X^{-\\frac12}.\\] \n\nThus an application of Lemma \\ref{bkmst:lem25} yields the following\nestimate:\n\\begin{equation}\\label{mani:estim:S1}\n\\begin{split}\n\\lvert S_1\\rvert &\\ll X^{\\varepsilon}\\sum_{x \\leq 2X\/Z} Y \\left[\n  Y^{-\\frac{1}{K}} + (\\log Y)^kX^{-\\frac{\\rho}{K}} + X^{-\\frac{1}{2} \\frac{1}{4K \\cdot 8L^5 - 2K}} \\right] \\\\\n&\\ll X^{1+\\varepsilon}(\\log X)\\left(X^{-\\rho} + X^{-\\frac{1}{64L^5-4} } \\right)^{\\frac1K}, \n\\end{split}\\end{equation}\nwhere we have used that $\\frac kK<1$ and $\\rho<\\frac13$.\n\nFor the second sum $S_2$ we start by splitting the interval $(\n\\frac{X}{V} , \\frac{2X}{U} ]$ into $\\leq \\log X$ subintervals of the\nform $(X_1, 2X_1]$. Thus\n\\begin{align*}\n\\lvert S_2\\rvert\n&\\leq (\\log X)X^{\\varepsilon}\\sum_{X_1<x\\leq\n  2X_1}\\lvert\\sum_{\\substack{U<y<V\\\\\\frac\n    Xx<y\\leq\\frac{2X}x}}b(y)e\\left(\\frac{\\nu}{q^j}(g(xy)+h(xy))\\right)\\rvert\n\\end{align*}\n\nNow an application of Cauchy's inequality together with $\\lvert\nb(y)\\rvert\\ll X^\\varepsilon$ yields\n\\begin{align*}\n\\lvert S_2\\rvert^2\n&\\leq (\\log X)^2X^{2\\varepsilon}X_1\\sum_{X_1<x\\leq\n  2X_1}\\lvert\\sum_{\\substack{U<y<V\\\\\\frac\n    Xx<y\\leq\\frac{2X}x}}b(y)e\\left(\\frac{\\nu}{q^j}(g(xy)+h(xy))\\right)\\rvert^2\\\\\n&\\ll (\\log X)^2X^{4\\varepsilon}X_1\\\\\n&\\quad\\times\\left(X_1\\frac{X}{X_1}+\\lvert \\sum_{X_1<x\\leq2X_1} \\sum_{A < y_1 < y_2 \\leq B}e\\left(\\frac{\\nu}{q^j} (g(xy_1)-g(xy_2) + h(xy_1)-h(xy_2))\\right)  \\rvert\\right)\n\\end{align*}\nwhere $A = \\max \\{U, \\frac{X}{x} \\} $ and\n$B = \\min \\{U, \\frac{2X}{x} \\}$. Changing the order of summation, we\nget\n\\begin{multline*}\n|S_2|^2 \\ll (\\log X)^2X^{4\\varepsilon}X_1\\\\\n\\times\\left(X+\n  \\sum_{A < y_1 < y_2 \\leq B}\\lvert \\sum_{X_1<x\\leq2X_1} e\\left(\\frac{\\nu}{q^j} (g(xy_1)-g(xy_2) + h(xy_1)-h(xy_2))\\right)  \\rvert\\right)\n\\end{multline*}\n\nAs above we want to apply Lemma \\ref{bkmst:lem25}. To this end we fix\n$y_1$ and $y_2 \\ne y_1$. Similarly to above we get that\n$$\\lvert\\frac{\\partial^\\ell\\left(g(xy_1)-g(xy_2)+h(xy_1)-h(xy_2)\\right)}{\\partial x^\\ell}\\rvert\\asymp\\frac{\\lvert\n  y_1-y_2\\rvert}{y_1}X^{\\theta_r}X_1^{-\\ell}.$$\nNow, on the one hand we have $\\frac{\\nu}{q^j}\\frac{\\lvert\n  y_1-y_2\\rvert}{y_1}X^{\\theta_r}\\gg X^{\\rho}$ and on the other hand\n\\[\\frac{\\nu}{q^j}\\frac{\\lvert\n  y_1-y_2\\rvert}{y_1}X^{\\theta_r}X_1^{-\\ell}\n\\ll X^{\\gamma+\\theta_r}\\left(\\frac{X}{V}\\right)^{-\\ell}\n\\ll X^{\\gamma+\\theta_r-\\frac23\\ell}\n\\ll X^{-\\frac12}\\]\nif $\\ell\\geq2\\lfloor\\theta_r\\rfloor+3$. Thus again an application of Lemma \\ref{bkmst:lem25} yields\n\\begin{equation}\\label{mani:estim:S2}\n\\begin{split}\n\\lvert S_2\\rvert^2 &\\ll (\\log X)^2X^{4\\varepsilon}X_1\\left(X +\n  \\sum_{A<y_1<y_2\\leq B} X_1\\left( X_1^{-\\frac{1}{K}} + X^{-\\frac{\\rho}{K}} + X^{-\\frac{1}{2} \\frac{1}{4K \\cdot 2L^2 - 2K}}\\right)\\right) \\\\\n&\\ll (\\log X)^2X^{4\\varepsilon}\\left(X^{\\frac53} + X^{2-\\frac{\\rho}{K}} + X^{2- \\frac{1}{16KL^2 - 4K}}\\right).\n\\end{split}\n\\end{equation}\n\nPlugging the two estimates \\eqref{mani:estim:S1} and\n\\eqref{mani:estim:S2} into \\eqref{mani:S} proves the proposition.\n\\end{proof}\n\n\\subsection{The digits in the middle}\n\nNow we are getting more involved in order to consider those $j$\nleading to a position between $\\theta_r$ and $k$. These sums\ncorrespond to the ``digits in the middle'' in the proof of Theorem\n\\ref{mani:centralthm}. We want to prove the following\n\\begin{prop}\\label{prop:middle_digits}\n  Let $P$ and $\\rho$ be positive reals and $f$ be a pseudo-polynomial as in\n  \\eqref{pseudo:poly:split}. If $2\\rho<\\theta_r<k$ and $j$ is such that\n\\begin{gather}\\label{mani:middle_range}\n  P^{\\theta_r-\\rho}< q^j\\leq P^{k-1+\\rho}\n\\end{gather}\nholds, then for $1\\leq\\nu\\leq P^\\gamma$ we have\n\\begin{gather*}\n  S(P,j,\\nu)=\\sum_{p\\leq P}e\\left(\\frac{\\nu f(p)}{q^j}\\right)\\ll P^{1-\\frac{\\rho}{4^k}}.\n\\end{gather*}\n\\end{prop}\n\nThe main idea in this range is to use that the dominant part of $f$\ncomes from the polynomial $h$. Therefore after getting rid of the\nfunction $g$ we will estimate the sum over the polynomial by the\nfollowing\n\\begin{lem}\\label{lem:exponential_sum_primes_poly}\nLet $h\\in\\mathbb{R}[X]$ be a\npolynomial of degree $k\\geq2$. Suppose $\\alpha$ is the leading\ncoefficient of $h$ and that there are integers $a$, $q$ such that\n$$\\lvert q\\alpha-a\\rvert<\\frac1q\\quad\\text{with}\\quad\n(a,q)=1.$$\nThen we have for any $\\varepsilon>0$ and $H\\leq X$\n$$\\sum_{X<p\\leq X+H}\\log(p)e(h(p))\\ll\nH^{1+\\varepsilon}\\left(\\frac1q+\\frac1{H^{\\frac12}}+\\frac{q}{H^k}\\right)^{4^{1-k}}.$$\n\\end{lem}\n\n\\begin{proof}\n  This is a slight variant of \\cite[Theorem\n  1]{harman1981:trigonometric_sums_over}, where we sum over an\n  interval of the form $]X,X+H]$ instead of one of the form\n  $]0,X]$.\n\\end{proof}\n\nNow we have enough tools to state the \n\\begin{proof}[Proof of Proposition \\ref{prop:middle_digits}]\n  As in the Proof of Proposition \\ref{prop:least_significant} we start\n  by an application of Lemma~\\ref{mr:lem11} yielding\n\\[S(P,j,\\nu)\\ll\\frac1{\\log P}\\max\\lvert\\sum_{n\\leq\n  P}\\Lambda(n)e\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\\rvert+P^{\\frac12}.\\]\n\nWe split the inner sum into $\\leq \\log P$ sub sums of the form\n\\[S:=\\sum_{X<n\\leq X+H}\\Lambda(n)e\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\\]\nwith $P^{1-2\\rho}\\leq X\\leq P$ and \\[H=\\min\\left(P^{1-\\theta_r}\\lvert\\nu\\rvert^{-1}q^j,X\\right).\\]\nNow we want to separate the function parts $g$ and $h$. Therefore we define two\nfunctions $T$ and $\\varphi$ by\n$$T(x)=\\sum_{X< n\\leq\n  X+x}\\Lambda(n)e\\left(\\frac{\\nu}{q^j}h(n)\\right)\n\\quad\\text{and}\\quad\n\\varphi(x):=e\\left(\\frac{\\nu}{q^j}g\\left(X+x\\right)\\right)$$\nThen an application of summation by parts yields\n\\begin{equation}\\label{mani:eq_1}\n\\begin{split}\n\\sum_{X< n\\leq X+H}\\Lambda(n)\ne\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\n&=\\sum_{n=1}^{H} \\varphi(n)(T(n)-T(n-1))\\\\\n&=\\sum_{n=1}^{H}T(n)\\left(\\varphi(n)-\\varphi(n+1)\\right)+\\varphi(H-1)T(H)\\\\\n&\\ll\\lvert T(H)\\rvert+\\sum_{n=1}^{H-1}\\left|\\varphi(n)-\\varphi(n+1)\\right|\\lvert T(n)\\rvert\n\\end{split}\n\\end{equation}\n\nLet $\\alpha_k$ be the leading coefficient of $P$. Then by Diophantine\napproximation there always exists a rational $a\/b$ with $b>0$,\n$(a,b)=1$,\n\\[1\\leq b\\leq H^{k-\\rho}\n\\quad\\text{and}\\quad\n\\lvert\\frac{\\nu\\alpha_k}{q^j}-\\frac ab\\rvert\\leq\n\\frac{H^{\\rho-k}}{b}.\\]\nWe distinguish three cases according to the size of $b$.\n\\begin{itemize}\n\\item[] \\textbf{Case 1.} $H^\\rho<b$. In this case we may apply Lemma\n  \\ref{lem:exponential_sum_primes_poly} together with\n  \\eqref{mani:log_Mangoldt_equivalence} to get $$T(h)\\ll\n  H^{1-\\frac{\\rho}{4^{k-1}}+\\varepsilon}.$$\n\\item[] \\textbf{Case 2.} $2\\leq b<H^\\rho$. In this case we get\n  that $$\\lvert\\frac{\\nu\\alpha_k}{q^j}\\rvert\\geq\\lvert\\frac\n  ab\\rvert-\\frac1{b^2}\\geq\\frac1{2b}\\geq\\frac12H^{-\\rho}\\geq\\frac12P^{-\\rho}.$$\n  Since $2\\rho<\\theta_r$, this contradicts our lower bound $q^j\\geq P^{\\theta_r-\\rho}$.\n\\item[] \\textbf{Case 3.} $b=1$. This case requires a further\n  distinction according to whether $a=0$ or not.\n  \\begin{itemize}\n  \\item[] \\textbf{Case 3.1.}\n    $\\lvert\\frac{\\nu\\alpha_k}{q^j}\\rvert\\geq\\frac12$. It follows\n    that $$q^j\\leq2\\lvert\\nu\\alpha_k\\rvert$$ again contradicting our lower\n    bound $q^j\\geq P^{\\theta_r-\\rho}$.\n  \\item[] \\textbf{Case 3.2.}\n    $\\lvert\\frac{\\nu\\alpha_k}{q^j}\\rvert<\\frac12$. This implies that\n    $a=0$ which yields \n    \\begin{gather}\\label{case3.2}\n    q^j\\geq\\lvert\\nu\\alpha_k\\rvert H^{k-\\rho}.\n    \\end{gather}\n    We distinguish two further cases according to\n    whether $P^{1-\\theta_r}\\lvert\\nu\\rvert^{-1}q^j\\leq X$ or not.\n    \\begin{itemize}\n    \\item[] \\textbf{Case 3.2.1}\n      $P^{1-\\theta_r}\\lvert\\nu\\rvert^{-1}q^j\\leq X$. This implies that\n      $q^j\\leq P^{\\theta_r}\\lvert\\nu\\rvert$\n      and \\[H=P^{1-\\theta_r}\\lvert\\nu\\rvert^{-1}q^j\\geq\n      P^{1-\\rho}\\lvert\\nu\\rvert^{-1}\\geq P^{1-2\\rho}.\\] Plugging these\n      estimates into \\eqref{case3.2} gives \\[P^{\\theta_r}\\geq\n      \\lvert\\alpha_k\\rvert P^{(1-2\\rho)(k-\\rho)}.\\] However, since\n      $4(k+1)\\rho<1$, we\n      have \\[(1-2\\rho)(k-\\rho)>k-1+2\\rho\\geq\\theta_r\\] yielding a\n      contradiction.\n    \\item[] \\textbf{Case 3.2.2}\n      $P^{1-\\theta_r}\\lvert\\nu\\rvert^{-1}q^j>X$. Then $H=X\\geq\n      P^{1-2\\rho}$ and \\eqref{case3.2}\n      becomes \\[P^{k-1+\\rho}\\geq\\lvert\\nu\\alpha_k\\rvert\n      P^{(1-2\\rho)(k-\\rho)}\\] yielding a similar contradiction as in\n      \\textbf{Case 3.2.1}.\n    \\end{itemize}\n  \\end{itemize}\n\\end{itemize}\nTherefore \\textbf{Case 1} is the only possible and we may always apply\nLemma \\ref{lem:exponential_sum_primes_poly} together with\n\\eqref{mani:log_Mangoldt_equivalence}. Plugging this\ninto~\\eqref{mani:eq_1} yields\n\\begin{align*}\n\\sum_{X< n\\leq X+H}\\Lambda(n)\ne\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\n&\\ll H^{1-\\frac{\\rho}{4^{k-1}}+\\varepsilon}\\left(1+\\sum_{X< n\\leq X+H}\\left|\\varphi(n)-\\varphi(n+1)\\right|\\right)\n\\end{align*}\n\nNow by our choice of $H$ together with an application of the mean\nvalue theorem we have that\n$$\\sum_{X\\leq n\\leq X+H}\\lvert \\varphi(n)-\\varphi(n+1)\\rvert\n\\ll H\\frac{\\nu}{q^j}P^{\\theta-1}\\ll 1.$$\nThus \n\\begin{align*}\n\\sum_{X\\leq n\\leq X+H} \\Lambda(n) \ne\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\n\\ll H^{1-\\frac{\\rho}{4^{k-1}}+\\varepsilon}.\n\\end{align*}\n\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{mani:centralthm}, Part II}\\label{sec:proof-prop-refm2}\nNow we use all the tools from the section above in order to estimate\n\n\\begin{gather}\\label{distance_from_mean}\n\\sum_{j=\\ell}^J\\lvert\\sum_{p\\leq P}\\mathcal{I}(q^{-j}f(p))-\\frac{\\pi(P)}{q^{\\ell}}\\rvert\n\\ll\\pi(P)H^{-1}J+\\sum_{\\nu=1}^{H}\n\\nu^{-1}\\sum_{j=\\ell}^JS(P,j,\\nu).\n\\end{gather}\n\nAs indicated in the section above, we split the sum over $j$ into two\nor three parts according to whether $\\theta_r>k$ or not. In any case\nan application of Proposition \\ref{prop:least_significant} yields for the\nleast significant digits that\n\\begin{gather}\\label{estimate:least}\n\\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}\\sum_{1\\leq q^{j}\\leq\n  P^{\\theta_r-\\rho}} S(P,j,\\nu)\n\\ll (\\log P)^9JP^{1-\\eta}.\n\\end{gather}\n\nNow let us suppose that $\\theta_r>k$. Then an application of Proposition\n\\ref{prop:most_significant} yields\n\\begin{equation}\\label{estimate:most_non_integer}\n\\begin{split}\n\\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}&\\sum_{P^{\\theta_r-\\rho}< q^{j}\\leq\n  P^{\\theta_r}}S(P,j,\\nu)\\\\\n&\\ll \\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}\\sum_{P^{\\theta_r-\\rho}< q^{j}\\leq\n  P^{\\theta_r}}\\frac1{\\log\n  P}\\left(\\frac{\\nu}{q^j}\\right)^{-\\frac1{\\left\\lfloor\\theta_r\\right\\rfloor}}+\\frac{P}{(\\log P)^{G-2}}\\\\\n&\\ll \\frac{P}{\\log P}.\n\\end{split}\n\\end{equation}\n\nPlugging the estimates \\eqref{estimate:least} and\n\\eqref{estimate:most_non_integer} into~\\eqref{distance_from_mean} we\nget that\n$$\\sum_{j=\\ell}^J\\lvert\\sum_{p\\leq P}\\mathcal{I}(q^{-j}f(p))-\\frac{\\pi(P)}{q^{\\ell}}\\rvert\n\\ll\\frac{P}{\\log P},$$\nwhich together with \\eqref{mani:NthetatoNstar} proves Theorem\n\\ref{mani:centralthm} in the case that $\\theta_r>k$.\n\nOn the other side if $\\theta_r<k$, then we consider the two ranges\n$$P^{\\theta_r-\\rho}<q^j\\leq P^{k-1+\\rho}\n\\quad\\text{and}\\quad\nP^{k-1+\\rho}<q^j\\leq P^k.$$\nFor the ``digits in the middle'' an application of Proposition\n\\ref{prop:middle_digits} yields\n\\begin{equation}\\label{estimate:middle}\n\\begin{split}\n\\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}\\sum_{P^{\\theta_r-\\rho}< q^{j}\\leq\n  P^{k-1+\\rho}}S(P,j,\\nu)\n&\\ll \\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}\\sum_{P^{\\theta_r-\\rho}< q^{j}\\leq\n  P^{k-1+\\rho}}P^{1-\\frac{\\rho}{4^k}}\\\\\n&\\ll(\\log P)JP^{1-\\frac{\\rho}{4^k}}.\n\\end{split}\n\\end{equation}\n\nFinally we consider the most significant digits. By an application of\nProposition~\\ref{prop:most_significant} we have\n\\begin{equation}\\label{estimate:most_integer}\n\\begin{split}\n\\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}&\\sum_{P^{k-1+\\rho}< q^{j}\\leq\n  P^{k}}S(P,j,\\nu)\\\\\n&\\ll \\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}\\sum_{P^{k-1+\\rho}< q^{j}\\leq\n  P^{k}}\\frac1{\\log\n  P}\\left(\\frac{\\nu}{q^j}\\right)^{-\\frac1k}+\\frac{P}{(\\log P)^{G-2}}\\\\\n&\\ll \\frac{P}{\\log P}.\n\\end{split}\n\\end{equation}\n\nPlugging the estimates \\eqref{estimate:least}, \\eqref{estimate:middle} and\n\\eqref{estimate:most_integer} into~\\eqref{distance_from_mean} we\nget that\n$$\\sum_{j=\\ell}^J\\lvert\\sum_{p\\leq P}\\mathcal{I}(q^{-j}f(p))-\\frac{\\pi(P)}{q^{\\ell}}\\rvert\n\\ll\\frac{P}{\\log P},$$\nwhich together with \\eqref{mani:NthetatoNstar} proves Theorem\n\\ref{mani:centralthm} in the case that $\\theta_r<k$.\n\n\\section*{Acknowledgment}\nThe author wants to thank G{\\'e}rald Tenenbaum for many fruitful\ndiscussions and suggestions in connection with the proof of\nProposition \\ref{prop:middle_digits}.\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}\\setcounter{equation}{0}}\n\n\n\\newcommand{h_{\\rm gpi}}{h_{\\rm gpi}}\n\n\\newcommand{{\\rm diag}\\,}{{\\rm diag}\\,}\n\\newcommand{{\\rm sgn}\\,}{{\\rm sgn}\\,}\n\\newcommand{{\\rm sig}\\,}{{\\rm sig}\\,}\n\\newcommand{{\\rm Ran}\\,}{{\\rm Ran}\\,}\n\\newcommand{{\\rm rank}\\,}{{\\rm rank}\\,}\n\\newcommand{{\\rm span}\\,}{{\\rm span}\\,}\n\n\n\n\\begin{titlepage}\n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\n\\rightline{DAMTP-R94\/11}\n\\vspace{0.1in}\n\\LARGE\n\\center{Generalised Point Interactions for the \\\\ Radial\nSchr\\\"{o}dinger Equation\\\\ via Unitary Dilations}\n\\Large\n\\vspace{0.2in}\n\\center{C.J. Fewster\\footnote{E-mail address:\nC.J.Fewster@amtp.cam.ac.uk}}\n\\vspace{0.2in}\n\\large\n\\center{\\em Department of Applied Mathematics and Theoretical Physics,\n\\\\ University of Cambridge,\n\\\\  Silver Street, Cambridge CB3 9EW, U.K.}\n\\vspace{0.2in}\n\\center{July 8, 1994}\n\\small\\center{Revised November 28, 1994}\n\\vspace{0.2in}\n\n\\begin{abstract}\nWe present an inverse scattering construction of generalised point\ninteractions (GPI) -- point-like objects with non-trivial scattering\nbehaviour. The construction is developed for single centre $S$-wave\nGPI models with rational $S$-matrices, and starts from an integral\ntransform suggested by the scattering data. The theory of unitary\ndilations is then applied to construct a unitary mapping between\nPontryagin spaces which extend the usual position and momentum\nHilbert spaces. The GPI Hamiltonian is defined as a multiplication\noperator on the momentum Pontryagin space and its free parameters are\nfixed by a physical locality requirement. We determine the spectral\nproperties and domain of the Hamiltonian in general, and construct\nthe resolvent and M{\\o}ller wave operators thus verifying that the\nHamiltonian exhibits the required scattering behaviour. The physical\nHilbert space is identified. The construction is illustrated by GPI\nmodels representing the effective range approximation. For negative\neffective range we recover a known class of GPI models, whilst the\npositive effective range models appear to be new. We discuss the\ninterpretation of these models, along with possible extensions to our\nconstruction.\n\n\\vspace{0.3truecm}\n{\\noindent {\\bf PACS Numbers} 03.65.Nk, 03.65.Db}\n\\end{abstract}\n\n\\setcounter{footnote}{0}\n\\renewcommand{\\thefootnote}{\\arabic{footnote}}\n\\end{titlepage}\n\n\n\\sect{Introduction and Main Ideas}\n\nGeneralised point interactions (GPI) are solvable models in quantum\nmechanics representing point objects with non-trivial scattering\nbehaviour. The prototype for such models is the class of point\ninteractions (PI), corresponding to Hamiltonians with\n$\\delta$-function potentials essentially introduced by Fermi\n\\cite{Fermi} and rigorously defined as self-adjoint extensions of\n$-\\triangle$ on the domain of smooth functions compactly supported\naway from the interaction centre \\cite{BF}. This construction leads\nto 1-parameter families of PI Hamiltonians in dimensions 2 and 3\nwhich provide the leading order (scattering length) approximation to\nthe scattering behaviour of Schr\\\"{o}dinger operators with short\nrange potentials in the sector of zero angular momentum (see, for\nexample \\cite{KF}). We refer to \\cite{Alb} for an extensive\nbibliography on PI models.\n\nGPI models are employed to treat more general scattering behaviour,\nsuch as higher order corrections to $S$-wave scattering, non-trivial\nscattering in non-zero angular momentum sectors or point objects in\ndimensions $d\\ge 4$. Such models have  been studied for a long time\nfrom the pseudo-potential viewpoint in many body physics -- see\n\\cite{Huang,GWu}. The mathematical study of such generalised point\ninteraction (GPI) models began in the Russian literature with the\nwork of Shirokov \\cite{Shiro}, and more rigorous formulations were\nlater developed by Pavlov \\cite{Pav1,Pav2,Pav3} and Shondin\n\\cite{Shond1,Shond2} (see also \\cite{Diejen}). In contrast to PI\nmodels, GPI Hamiltonians are not defined on the usual Hilbert space\n$L^2({\\rm I\\! R}^d)$, but on an extended space whose inner product may be\nindefinite, in which case one must identify a physical Hilbert space\nof states in order to recover the probability interpretation of\nquantum mechanics.\n\nIn this paper, we introduce and develop a new inverse scattering\nconstruction for single centre GPI models with non-trivial $S$-wave\nscattering. It is currently an open problem to extend this\nconstruction to higher angular momenta and dimensions $d\\ge 4$; we\ndiscuss this further in the Conclusion. Our method is based on the\ntechnique of {\\em unitary dilations} due in origin to Sz.-Nagy\n\\cite{Nagy} and extended by Davis \\cite{Davis}.\n\nThe $S$-wave inverse scattering problem has been partially studied by\nShondin \\cite{Shond1}, as we describe below. First, let us briefly\nmention the two principal non-inverse GPI constructions, which we\ncall the {\\em auxiliary space} and {\\em distributional} methods. In\nthe auxiliary space method, developed by Pavlov and co-workers\n\\cite{Pav2,Pav3}, one starts with a given extended Hilbert space and\nthen seeks the class of GPI Hamiltonians which `live' on this space.\nOn the other hand, the distributional method of Shondin \\cite{Shond2}\nis a direct attempt to define $H=-\\triangle+\\ket{\\omega}\\bra{\\omega}$,\nwhere $\\omega$ is somewhere in the $j<0$ portion of the scale of\nSobolev spaces ${\\cal H}_j=(-\\triangle+1)^{-j\/2}L^2({\\rm I\\! R}^d)$. The\nconstruction leads to a Pontryagin space\\footnote{A Pontryagin space\nis an indefinite (Krein) inner product space with a finite rank of\nindefiniteness -- see Section~2.1.} $\\Pi_m={\\cal H}_0\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}^{2m}$,\nwhere $m$ is the unique integer such that\n$\\omega\\in{\\cal H}_{-m-2}\\backslash{\\cal H}_{-m-1}$ and the inner product has\nsignature $(m,m)$ on the finite dimensional part. The GPI Hamiltonian\nis then defined on $\\Pi_m$ using Krein's formula \\cite{Akh}.\n\nReturning to the inverse problem, Shondin \\cite{Shond1} considered\nthe inverse scattering problem in $d=3$ for $S$-wave scattering data\nof form\n\\begin{equation}\n\\cot\\delta_0(k) = k^{-1}r(k^2), \\label{eq:GPIlo}\n\\end{equation}\nwhere $r(z)$ is a rational function with real coefficients. Shondin\nrefers to this class of data as the `$R$ class': it corresponds\nexactly with the class of rational $S$-wave $S$-matrices\nsatisfying the usual analytic continuation property $S(k)=S^*(-k)$\n\\cite{Newt}. In particular, the $R$ class contains truncated low\nenergy expansions of form\n\\begin{equation}\n\\cot\\delta_0(k) = -\\frac{1}{kL} + \\frac{1}{2}kr_0 + k^3r_1+\\cdots\n+k^{2n+1}r_n, \\label{eq:low}\n\\end{equation}\nfor low energy parameters $L,r_0,r_1,\\ldots,r_n$ and any $n\\ge 0$,\nand thus furnishes approximations to the low energy $S$-wave\nbehaviour of Schr\\\"{o}dinger operators with short range potentials to\narbitrary order. Shondin's method starts by writing down a candidate\nresolvent on a (positive definite) extension of the usual Hilbert\nspace. (In this respect it resembles the auxiliary space method).\nVarious free functions in this resolvent are then fixed by requiring\nthat the candidate be the resolvent of a self-adjoint operator.\nHowever, this method is limited to those $r(z)$ with negative\nimaginary part in the upper half plane, which is a somewhat\nrestrictive sub-class: for example, scattering data of\nform~(\\ref{eq:low}) is possible only with $r_1,\\ldots ,r_n=0$ and\n$r_0<0$. As we will see later, in the context of our method, more\ngeneral scattering data corresponds to GPI models defined on\nPontryagin spaces. Thus, in order to apply Shondin's method to such\ndata, one would have to guess not only the appropriate extension to\nthe Hilbert space, but also its inner product, thereby rendering it\nmuch less practical as a construction.\n\nIn contrast, the method proposed here allows one to treat the full\n$R$ class. It proceeds from the simple observation that, for\npoint-like interactions, the scattering data $\\delta_0(k)$\ncompletely specify the $S$-wave continuum eigenfunctions as $u_k(r)\n= (2\/\\pi)^{1\/2}\\sin (kr+\\delta_0(k))$, when scattering normalisation\nis imposed. This is quite different from the usual situation in\ninverse scattering theory, where the scattering data specifies (in\nthe first instance) only the asymptotic form of the $u_k(r)$. As a\nresult, the inverse scattering problem for GPI models can be solved\nwithout recourse to the usual Gel'fand-Levitan machinery.\n\nThe $u_k(r)$ define an integral transform ${\\cal T}$ between\n${\\cal H}_r=L^2((0,\\infty),dr)$ and ${\\cal H}_k=L^2((0,\\infty),dk)$, (i.e., the\nradial position and momentum Hilbert spaces) by\n\\begin{equation}\n({\\cal T} \\psi)(k) =\n\\left(\\frac{2}{\\pi}\\right)^{1\/2}\\int_0^\\infty \\sin\n(kr+\\delta_0(k))\\psi(r) dr,\n\\end{equation}\nWe adopt ${\\cal T}$ as a candidate for an eigenfunction transform\nassociated with our desired GPI Hamiltonian $h_{\\rm gpi}$. Of course, ${\\cal T}$\ncannot be the full eigenfunction transform unless it is unitary, in\nwhich case we can define $h_{\\rm gpi}={\\cal T}^* k^2{\\cal T}$. However, when ${\\cal T}$ is\nnon-unitary, we may `dilate' it to a unitary operator between\nenlarged inner product spaces, using the theory of unitary dilations\nas follows. Firstly, we quantify the departure of ${\\cal T}$ from\nunitarity by the operators  $M_1=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-{\\cal T}\\Tt^*$ and\n$M_2=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-{\\cal T}^*{\\cal T}$, and the closures ${\\cal M}_1$ and ${\\cal M}_2$ of their\nrespective ranges. For scattering data in the $R$ class, we will show\nthat the rank and signature of $M_1$ and $M_2$ are finite and given\nin terms of indices reminiscent of Levinson's theorem \\cite{Newt}.\n\nNext, we define indefinite inner products on the ${\\cal M}_i$, together\nwith an operator\n$\\hat{{\\cal T}}:{\\cal H}_r\\oplus{\\cal M}_1\\rightarrow{\\cal H}_k\\oplus{\\cal M}_2$ which is\nunitary with respect to the relevant inner products and satisfies\n$P_{{\\cal H}_k}\\hat{{\\cal T}}|_{{\\cal H}_r}={\\cal T}$, where $P_{{\\cal H}_k}$ is the\northoprojector onto ${\\cal H}_k$. $\\hat{{\\cal T}}$ is said to be a unitary\ndilation\\footnote{See Section~2 for a note on the nomenclature.}  of\n${\\cal T}$. We emphasise that the construction of $\\hat{{\\cal T}}$ and the\nenlarged inner product spaces requires no information beyond that\nencoded in ${\\cal T}$, and is unique up to a unitary equivalence and\nfurther dilation. To complete the construction, we define the GPI\nHamiltonian $h_{\\rm gpi}$ by\n\\begin{equation}\nh_{\\rm gpi} = \\hat{{\\cal T}}^\\dagger\n\\left(\\begin{array}{cc} k^2 & 0 \\\\ 0 & \\Lambda \\end{array}\n\\right)\n\\hat{{\\cal T}},\n\\end{equation}\nwhere the dagger denotes the Pontryagin space adjoint, and we have\nused an obvious block matrix notation.  We will show that $\\Lambda$\nis completely determined by imposing a physical locality requirement:\nthat the `interaction' be localised at the origin. Mathematically,\nthis is expressed by requiring the Hamiltonian agrees with the free\nHamiltonian away from the interaction centre, i.e., $h_{\\rm gpi}(\\psi,0)^T=\n(-\\psi^{\\prime\\prime},0)^T$ if $\\psi\\in C_0^\\infty(0,\\infty)$.\n{}Subject to this locality condition, we have thus constructed both\n$h_{\\rm gpi}$ and its spectral representation. The non-uniqueness in our\nconstruction leads to a family of unitarily equivalent GPI\nHamiltonians with the same spectral and scattering properties.\n\nOur plan is as follows. In Section~2, we briefly describe some\nfeatures of analysis in indefinite inner product spaces, and\nalso describe the construction of unitary dilations, essentially\nfollowing Davis \\cite{Davis}. In addition, we sketch our construction\nin a more abstract setting. Next, in Section~3, we explicitly\nconstruct the operators $M_1$ and $M_2$ (which are of finite rank)\nfor a generic subclass of the $R$ class -- those whose scattering\namplitudes exhibit only simple poles on the physical sheet -- and\ncompute their rank and signature. In Section~4, we construct $h_{\\rm gpi}$ as\ndescribed above. Subject to the locality condition, we show that the\neigenvalues of $h_{\\rm gpi}$ occur at precisely those energies for which the\nscattering amplitude derived from~(\\ref{eq:GPIlo}) exhibits poles on\nthe physical sheet, as is the case for ordinary scattering from `nice'\npotentials. We construct the corresponding eigenfunctions of $h_{\\rm gpi}$,\nand isolate the physical Hilbert space. We also determine the domain\nand resolvent of $h_{\\rm gpi}$, and explicitly construct its M{\\o}ller wave\noperators in a two-space setting, verifying that they exhibit the\nrequired scattering theory.\n\nIn Section 5, we illustrate our procedure by constructing GPI models\nwith scattering behaviour $\\cot\\delta_0(k)=-1\/(kL)+kM$, representing\nthe effective range approximation of low energy scattering theory\n\\cite{Newt}. In the case $M<0$, ${\\cal H}_r$ and ${\\cal H}_k$ are extended to\nlarger Hilbert spaces, and we recover the models of `type $B_2$'\npreviously constructed by Shondin \\cite{Shond1}. These models also\narise as a special case of the auxiliary space construction in\n\\cite{Pav1}. The case $M>0$, for which Pontryagin spaces are\nrequired, appears to be new. Our methods allow the entire  class of\nGPI Hamiltonians to be constructed, along with their spectral\nrepresentations. A particularly interesting subclass of the models\nconstructed corresponds to the case $L=\\infty$, with scattering theory\n$\\cot\\delta_0(k)=kM$. Such models reproduce the leading order\nbehaviour of non-point interactions exhibiting a zero energy\nresonance. We refer to these models as {\\em resonance point\ninteractions} (RPI).\n\nWe also discuss how these GPI models may be used as models for\nSchr\\\"{o}dinger operators with spherically symmetric potentials of\ncompact support. To do this, we employ a general methodology for\ndiscussing the `large scale effects of small objects' developed by\nKay and the author \\cite{KF}. In particular, we develop {\\em fitting\nformulae} (analogous to those given in \\cite{KF}) for matching a\ngiven potential $V(r)$ to the `best fit' GPI model. Finally, in\nSection 6, we conclude by discussing various extensions to our\nmethod.\n\nThe motivation for the present work arose in a consideration of the\nscattering of charged particles off magnetic flux tubes of small\nradius \\cite{FK}, in which it was found that the scattering lengths\nfor spin-$\\frac{1}{2}$ particles generically take the values $0$ or\n$\\infty$ in certain angular momentum sectors. In consequence, the\nanalogue of PI models representing dynamics in\nthe background of an infinitesimally thin wire of flux fails to\ndescribe the leading order scattering theory in these sectors, and\nshould be replaced by models analogous to the RPI models mentioned\nabove. The special nature of this system can be attributed to the\nfact that it is an example of supersymmetric quantum mechanics.\nElsewhere \\cite{F}, we will construct the appropriate class of RPI\nfor this system.\n\n\n\\sect{Preliminaries}\n\\subsection{Unitary Dilations}\n\nWe begin by describing the unitary dilation theory required in the\nsequel. Let ${\\cal H}_1,\\ldots,{\\cal H}_4$ be Hilbert spaces and\n$T\\in{\\cal L}({\\cal H}_1,{\\cal H}_2)$. Then\n$\\hat{T}\\in{\\cal L}({\\cal H}_1\\oplus{\\cal H}_3,{\\cal H}_2\\oplus{\\cal H}_4)$ is called a\n{\\em dilation} of $T$ if\n$T= P_{{\\cal H}_2}\\hat{T}|_{{\\cal H}_1}$\nwhere $P_{{\\cal H}_2}$ is the orthogonal projector onto ${\\cal H}_2$. In\nblock matrix form, $\\hat{T}$ takes form\n\\begin{equation}\n\\hat{T} = \\left(\\begin{array}{cc} T & P \\\\ Q & R \\end{array}\\right).\n\\label{eq:dilfm}\n\\end{equation}\nOur nomenclature follows that of Halmos \\cite{Halmos}. Elsewhere\n(e.g., in the work of Davis \\cite{Davis}), the term `dilation' (or\n`dilatation') often means that $\\hat{T}^n$ is a dilation of $T^n$\nand $(\\hat{T}^*)^n$ is a dilation of $(T^*)^n$ for each $n=1,2,\\ldots$\n(in addition, ${\\cal H}_1={\\cal H}_2$, and ${\\cal H}_3={\\cal H}_4$). We refer to such\noperators as {\\em power dilations}: in the block\nform~(\\ref{eq:dilfm}), this requires $PR^nQ=0$ for each\n$n=0,1,2,\\ldots$.\n\nAccording to a result of Sz.-Nagy \\cite{Nagy}, any contraction $T$\nfrom one Hilbert space to another (i.e., a bounded operator\nsatisfying $\\|T\\|\\le 1$) has a unitary dilation between larger\nHilbert spaces. Subsequently, Davis \\cite{Davis} extended this\nresult to arbitrary closed densely defined operators at the\ncost of introducing indefinite inner product spaces. (It is clear\nthat if $\\|T\\|>1$, no Hilbert space unitary dilation is possible.)\nIn fact, Davis' construction yields a unitary {\\em power} dilation\nof the original operator.  This has no physical relevance in our\nconstruction, and so we use a more economical `cut-down' version of\nDavis' result, described below. First, we briefly review the salient\nfeatures of analysis in indefinite inner product spaces. Full\ntreatments can be found in the monographs of Bogn\\'ar \\cite{Bognar}\nand Azizov and Iokhvidov~\\cite{Azizov}.\n\nWe employ a particular class of indefinite inner product\nspaces known as {\\em $J$-spaces}. Let ${\\cal H}$ be a Hilbert space with\n(positive definite) inner product $\\inner{\\cdot}{\\cdot}$,\nequipped with a unitary involution, $J$. We define a non-degenerate\nindefinite inner product $[\\cdot,\\cdot]$ on ${\\cal H}$ by\n\\begin{equation}\n[x,y]=\\inner{x}{Jy},\n\\end{equation}\nwhich we call the {\\em $J$-inner product}. ${\\cal H}$ equipped with the\n$J$-inner product is called a $J$-space. ${\\cal H}$ admits decomposition\n${\\cal H}={\\cal H}_+\\oplus{\\cal H}_-={\\cal H}_+[+]{\\cal H}_-$ into the eigenspaces ${\\cal H}_\\pm$\nof $J$ with eigenvalue $\\pm 1$, where $[+]$ denotes the orthogonal\ndirect sum in the $J$-inner product. If at least one of the\n${\\cal H}_\\pm$ is finite dimensional, then ${\\cal H}$ is a {\\em Pontryagin\nspace} with respect to $[\\cdot,\\cdot]$ .\n\nThe topology of a $J$-space is determined by the Hilbert space norm;\nhowever, operator adjoints and the notion of unitarity are defined\nrelative to the $J$-inner product. Thus if ${\\cal H}_i$ ($i=1,2$) are\n$J_i$-spaces, and $T\\in{\\cal L}({\\cal H}_1,{\\cal H}_2)$, the {\\em\n$(J_1,J_2)$-adjoint} $T^\\dagger$ of $T$ is defined in terms of the\nHilbert space adjoint $T^*$ by\n\\begin{equation}\nT^\\dagger = J_1T^*J_2.\n\\end{equation}\nEquivalently, $[T^\\dagger x,y]_{{\\cal H}_1}=[x,Ty]_{{\\cal H}_2}$ for all\n$x\\in{\\cal H}_2$, $y\\in{\\cal H}_1$. If $[Ux,Uy]_{{\\cal H}_2}=[x,y]_{{\\cal H}_1}$ for all\n$x,y\\in {\\cal D}\\subset{\\cal H}_1$, $U$ is said to be {\\em\n$(J_1,J_2)$-isometric}; if in addition $U$ is a linear isomorphism of\n${\\cal H}_1$ and ${\\cal H}_2$, and ${\\cal D}={\\cal H}_1$, $U$ is said to be {\\em\n$(J_1,J_2)$-unitary}. Equivalently, $UU^\\dagger = \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal H}_1}$ and\n$U^\\dagger U = \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal H}_2}$. If ${\\cal H}_1={\\cal H}_2$ with $J_1=J_2=J$, terms\nsuch as $(J_1,J_2)$-isometric are abbreviated to $J$-isometric etc.\n\nReturning to the construction of unitary dilations, let $T$ be any\nbounded operator $T\\in{\\cal L}({\\cal H}_1,{\\cal H}_2)$, and define operators\n$M_1 = \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-TT^*$ and $M_2=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-T^*T$. It is trivial to show that the\nrespective closures ${\\cal M}_i=\\overline{{\\rm Ran}\\, M_i}$ of their ranges are\n${\\rm sgn}\\, (M_i)$-spaces, and hence that ${\\cal K}_i={\\cal H}_i\\oplus{\\cal M}_i$ are\n$J_i$-spaces, where $J_i=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal H}_i}\\oplus{\\rm sgn}\\,(M_i)$. We now define\na dilation $\\hat{T}$ of $T$ by\n\\begin{equation}\n\\hat{T} = \\left(\\begin{array}{cc} T & -{\\rm sgn}\\,(M_1)|M_1|^{1\/2} \\\\\n                                  |M_2|^{1\/2} & T^*|_{{\\cal M}_1}\n                \\end{array} \\right),\n\\end{equation}\nwhich has $(J_1,J_2)$-adjoint $\\hat{T}^\\dagger$ equal to\n\\begin{equation}\n\\hat{T}^\\dagger = J_1\\hat{T}^*J_2=\n \\left(\\begin{array}{cc}\n                            T^* & {\\rm sgn}\\, (M_2) |M_2|^{1\/2} \\\\\n                            - |M_1|^{1\/2} & T|_{{\\cal M}_2}\n                        \\end{array} \\right).\n\\end{equation}\nHere, we have used the intertwining relations $Tf(T^*T)=f(TT^*)T$ and\n$T^*f(TT^*)=f(T^*T)T^*$, which hold for any continuous Borel function\n$f$. It is now easy to show that $\\hat{T}^\\dagger\\hat{T}=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal K}_1}$\nand $\\hat{T}\\hat{T}^\\dagger=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal K}_2}$, thus verifying that\n$\\hat{T}$ is a $(J_1,J_2)$-unitary dilation of $T$. In our\napplication, $M_1$ and $M_2$ are finite rank, and so the $J$-spaces\nconstructed above are Pontryagin spaces.\n\nWe briefly consider the uniqueness of the unitary dilations\nconstructed above. Suppose ${\\cal N}_i$ are $J_i$\nspaces ($i=1,2$) and that $\\tilde{T}:{\\cal H}_1\\oplus{\\cal N}_1 \\rightarrow\n{\\cal H}_2\\oplus{\\cal N}_2$ is a unitary dilation of $T$ with matrix\nform~(\\ref{eq:dilfm}). Then, provided that the $M_i$ are finite\nrank, one may show that\n\\begin{equation}\nP_{{\\cal H}_2\\oplus {\\cal Q}}\n\\tilde{T}|_{{\\cal H}_1\\oplus {\\cal P}}=\n\\left(\\begin{array}{cc} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1} & 0 \\\\ 0 & U_2 \\end{array}\\right)\n\\hat{T}\n\\left(\\begin{array}{cc} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1} & 0 \\\\ 0 & U_1^\\dagger\n\\end{array}\\right),\n\\end{equation}\nwhere ${\\cal P}=P^\\dagger\\overline{{\\rm Ran}\\, M_1}$, ${\\cal Q}=Q\\overline{{\\rm Ran}\\,\nM_1}$, and $U_1$ and $U_2$ are unitaries (with respect to the\n$J$-inner products) from ${\\cal M}_1$ and ${\\cal M}_2$ to ${\\cal P}$ and ${\\cal Q}$\nrespectively. In addition, $P_{{\\cal H}_2\\oplus {\\cal Q}}$ is an orthogonal\nprojection onto ${\\cal H}_2\\oplus{\\cal Q}$ in ${\\cal H}_2\\oplus{\\cal N}_2$.\n\nThus $\\hat{T}$ is unique up to further dilation and unitary\nequivalence of the above form. If the $M_i$ are not of finite rank,\nthis statement also holds if the $M_i$ are strictly positive. More\ngenerally, it is not clear whether ${\\cal Q}$ is necessarily\northocomplemented, and therefore whether $P_{{\\cal H}_2\\oplus {\\cal Q}}$ exists.\n\n\\subsection{Abstract Setting}\n\\label{sect:abs}\n\nIn this section, we sketch our construction in a general setting,\nwhich makes clear how it may be extended. In particular, we show how\nthe domain and action of the Hamiltonian is determined.\n\nLet ${\\cal H}_i$ ($i=1,2$) be Hilbert spaces and let $A$ be a densely\ndefined symmetric operator with domain ${\\cal D}\\subset {\\cal H}_1$. Suppose\nthat $A$ possesses two self-adjoint extensions $A_\\pm$ such that\n\\begin{equation}\nA_\\pm = {\\cal T}_\\pm^* \\tilde{A}{\\cal T}_\\pm\n\\end{equation}\nwhere $\\tilde{A}$ is a self-adjoint operator on ${\\cal H}_2$ with\n$(\\tilde{A}+\\omega)^{-1}$ bounded for some $\\omega\\in{\\rm I\\! R}$, and\n${\\cal T}_\\pm$ are unitary operators ${\\cal T}_\\pm :{\\cal H}_1\\rightarrow{\\cal H}_2$. Let\n$a_+$ and $a_-$ be bounded operators on ${\\cal H}_2$ which commute with\n$\\tilde{A}$ and define\n\\begin{equation}\n{\\cal T} = a_+{\\cal T}_+ + a_-{\\cal T}_-.\n\\end{equation}\nIn our application, $a_\\pm$ are determined by the scattering\ndata. We define $M_1$ and $M_2$ as above, for simplicity\nassuming that they are finite rank (as they are in our application).\nThe unitary dilation $\\hat{{\\cal T}}$ derived above is then used to\ndefine a self-adjoint operator $B$ on the Pontryagin space $\\Pi_1\n={\\cal H}_1\\oplus{\\cal M}_1$ by\n\\begin{equation}\nB = \\hat{{\\cal T}}^\\dagger\n\\left(\\begin{array}{cc} \\tilde{A} & 0 \\\\ 0 & \\Lambda \\end{array}\n\\right)\n\\hat{{\\cal T}},\n\\end{equation}\nwhere $\\Lambda$ is a self-adjoint operator on ${\\cal M}_2$ (with respect\nto its inner product). Thus\n\\begin{equation}\nB\\left(\\begin{array}{c} \\varphi\\\\ \\Phi\\end{array}\\right)=\n\\left(\\begin{array}{c}\n{\\cal T}^*\\tilde{A}({\\cal T}\\varphi-\\Theta)\n+{\\rm sgn}\\, M_2|M_2|^{1\/2}\\Lambda(|M_2|^{1\/2}\\varphi+{\\cal T}^*|_{{\\cal M}_1}\\Phi) \\\\\n-|M_1|^{1\/2}\\tilde{A}({\\cal T}\\varphi-\\Theta)\n+{\\cal T}|_{{\\cal M}_2}\\Lambda (|M_2|^{1\/2}\\varphi+{\\cal T}^*|_{{\\cal M}_1}\\Phi)\n\\end{array}\\right),\n\\label{eq:actB}\n\\end{equation}\nwhere $\\Theta={\\rm sgn}\\, M_1 |M_1|^{1\/2}\\Phi$ (considered as an element of\n${\\cal H}_2$), and $B$ has domain\n\\begin{equation}\nD(B) = \\{ (\\varphi,\\Phi)^T\\mid {\\cal T}\\varphi-\\Theta\n\\in D(\\tilde{A})\\}. \\label{eq:domB}\n\\end{equation}\n{}To gain a more explicit description of $D(B)$, we impose the\nrequirement that $B$ be a self-adjoint extension of the\n{\\em non-densely defined} operator $A\\oplus 0$ on ${\\cal D}\\oplus\n0\\subset\\Pi_1$, i.e., $B(\\varphi,0)^T=(A\\varphi,0)^T$ for all\n$\\varphi\\in{\\cal D}$. Later this will carry the physical interpretation of\na locality condition. It is easy to show that this requirement is\nsatisfied if and only if ${\\cal M}_2$ is invariant under $A^*$ and\n\\begin{equation}\n\\Lambda=(|M_2|^{-1\/2}A^*|_{{\\cal M}_2}|M_2|^{1\/2})^*.\n\\end{equation}\n\nAs a consequence of locality, we note that if  $(\\varphi,\\Phi)^T\\in\nD(B)$ with $B (\\varphi,\\Phi)^T=(\\tilde{\\varphi},\\tilde{\\Phi})^T$,\nthen  $\\varphi\\in D(A^*)$, and $\\tilde{\\varphi}=A^*\\varphi$. For take\nany $\\psi\\in {\\cal D}$. Then\n\\begin{equation}\n\\inner{\\tilde{\\varphi}}{\\psi}_{{\\cal H}_1} =\n\\left[\n\\left(\\begin{array}{c}\\tilde{\\varphi}\\\\\n\\tilde{\\Phi}\\end{array}\\right), \\left(\\begin{array}{c}\\psi\\\\\n0\\end{array}\\right)\\right]_{\\Pi_1} =\n\\left[\n\\left(\\begin{array}{c}\\varphi\\\\ \\Phi\\end{array}\\right),\nB\\left(\\begin{array}{c}\\psi\\\\ 0\\end{array}\\right)\n\\right]_{\\Pi_1}= \\inner{\\varphi}{A\\psi}_{{\\cal H}_1}.\n\\end{equation}\nWe may therefore re-write~(\\ref{eq:domB}) as\n\\begin{equation}\nD(B)=\\left\\{ \\left(\\begin{array}{c} \\varphi \\\\ \\Phi \\end{array}\n\\right) \\mid \\varphi\\in D(A^*), \\quad \\Theta_1 \\in D(\\tilde{A})\n\\right\\},\n\\end{equation}\nwhere $\\Theta_1 =a_+{\\cal T}_+\\chi_+ + a_-{\\cal T}_-\\chi_-+\\Theta$ and\n$\\chi_\\pm=(A_\\pm+\\omega)^{-1}(A^*+\\omega)\\varphi-\\varphi$. The\nadvantage of this expression is that $\\chi_\\pm$ can be shown to be the\nunique element of $\\ker (A^*+\\omega)$ such that $\\varphi+\\chi_\\pm\\in\nD(A_\\pm)$. In our application, $\\chi_\\pm$ may be expressed in terms\nof the value of $\\varphi$ and its first derivative at the origin.\n\n{}To determine the action of $B$ more explicitly, we use the fact that\nthe upper component of the right-hand side of~(\\ref{eq:actB}) is\nequal to $A^*\\varphi$ in order to compute $\\tilde{\\Theta}={\\rm sgn}\\,\nM_1|M_1|^{1\/2} \\tilde{\\Phi}$. We obtain\n\\begin{equation}\n\\tilde{\\Theta}=-M_1\\tilde{A}({\\cal T}\\varphi-\\Theta)+\n{\\cal T}(A^*\\varphi-{\\cal T}^*\\tilde{A}({\\cal T}\\varphi-\\Theta))=\n{\\cal T} A^*\\varphi-\\tilde{A}({\\cal T}\\varphi-\\Theta)\n\\end{equation}\nUsing the fact that $\\Theta_1\\in D(\\tilde{A})$, this becomes\n\\begin{equation}\n\\tilde{\\Theta}=\\tilde{A}\\Theta_1 +\\omega(\\Theta_1-\\Theta) +\n{\\cal T}(A^*+\\omega)\\varphi-(\\tilde{A}+\\omega)\n({\\cal T}\\varphi+\\Theta_1-\\Theta).\n\\end{equation}\nThe last two terms cancel by definition of $\\chi_\\pm$ and we\nconclude that\n\\begin{equation}\nB\\left(\\begin{array}{c}\\varphi\\\\ \\Phi\\end{array}\\right)=\n\\left(\\begin{array}{c} A^*\\varphi \\\\\n({\\rm sgn}\\, M_1 |M_1|^{1\/2})^{-1}\\tilde{\\Theta}\n\\end{array}\\right)\n\\end{equation}\nwhere $\\tilde{\\Theta}=\\tilde{A}\\Theta_1 + \\omega(a_+{\\cal T}_+\\chi_+ +\na_-{\\cal T}_-\\chi_-)$.\n\n\\sect{Determination of $M_1$ and $M_2$}\n\nIn this section, we determine the operators $M_1=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-{\\cal T}\\Tt^*$ and\n$M_2=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-{\\cal T}^*{\\cal T}$, where ${\\cal T}$ is an integral transformation\narising from the scattering data in the Shondin $R$ class\n\\cite{Shond1} given by\n\\begin{equation}\n\\cot\\delta_0(k) = k^{-1}\\frac{p(k^2)}{q(k^2)},\\qquad\n\\delta_\\ell(k)\\equiv 0\\quad {\\rm for}~\\ell\\ge 1,\n\\label{eq:GPIlow}\n\\end{equation}\nwhere $p(z)$ and $q(z)$ are coprime polynomials in ${\\rm I\\! R}[z]$, the\nring of polynomials with real coefficients. In particular, we will\nshow how the rank and signature of the $M_i$ are determined by two\n`Levinson indices' defined below. We emphasise that our methods are\nvery different to those of Shondin.\n\nThe scattering amplitude corresponding to $\\delta_0(k)$ is\n\\begin{equation}\nf_0(k) = \\frac{1}{k}e^{i\\delta_0(k)}\\sin\\delta_0(k)=\n\\frac{q(k^2)}{p(k^2) - ikq(k^2)}.\n\\end{equation}\nDefining the polynomial $W(z)$ by\n\\begin{equation}\nW(z) =\\left\\{\\begin{array}{cl} p(-z^2)-zq(-z^2) & p(0)\\not=0 \\\\\n                               p(-z^2)\/z-q(-z^2) & p(0)=0,\n\\end{array}\\right. \\label{eq:W}\n\\end{equation}\nwe note that $f_0(k)$ exhibits poles where $W(ik)=0$. The set\n$\\Omega$ of zeros of $W(z)$ in the left-hand half-plane ${\\rm Re}\\,\nz<0$ corresponds to poles of $f_0(k)$ such that $k^2$ lies on the\nphysical sheet. We refer to the situation where these poles\n(and hence the corresponding zeros of $W(z)$) are simple as the\n{\\em generic case}. In Theorem~\\ref{Thm:local}, we will show\nthat the discrete spectrum of the GPI Hamiltonian is precisely\n$\\{E=-\\omega^2\\mid\\omega\\in\\Omega\\}$ under the requirement of\nlocality.\\footnote{These eigenvalues can be complex: we will return\nto this point in section~5.3.}\n\nThe qualitative features of the scattering data~(\\ref{eq:GPIlow})\nare described by the degrees of $p$ and $q$, two indices $I_L^\\pm$\ndefined below, and the asymptotic behaviour of $\\cot\\delta_0(k)$\ngiven by\n\\begin{equation}\n\\sigma_0 = {\\rm sgn}\\,\\lim_{k\\rightarrow 0^+}\n\\cot\\delta_0(k)\\qquad {\\rm and}\\qquad \\sigma_\\infty\n={\\rm sgn}\\,\\lim_{k\\rightarrow\\infty}\\cot\\delta_0(k),\n\\end{equation}\nwhere the limits are allowed to be $\\pm\\infty$. The indices\n$I_L^\\pm$ are defined by\n\\begin{equation}\nI_L^+= \\frac{\\delta_0(0)-\\delta_0(\\infty)}{\\pi}\\qquad{\\rm and}\n\\qquad\nI_L^-=\\frac{\\zeta(0)-\\zeta(\\infty)}{\\pi},\n\\end{equation}\nwhere the auxiliary scattering data $\\zeta(k)$ is defined\nas a continuous function on ${\\rm I\\! R}^+$ by\n\\begin{equation}\n\\cot\\zeta(k) = -k^{-1}\\frac{p(-k^2)}{q(-k^2)}. \\label{eq:zeta}\n\\end{equation}\nWe refer to $I_L^\\pm$ as the Levinson indices (although Levinson's\ntheorem \\cite{Newt} will not hold in its usual form).\n\nWe now define the integral transform\n${\\cal T}=\\cos\\delta_0(k){\\cal S}+\\sin\\delta_0(k){\\cal C}$, which is suggested by\nthe\nna\\\"{\\i}ve generalised eigenfunctions $u_k(r) =\n(2\/\\pi)^{1\/2}\\sin (kr+\\delta_0(k))$.\nHere, ${\\cal S}$ and ${\\cal C}$ are the sine and cosine transforms,\ndefined by\n\\begin{eqnarray}\n({\\cal S} \\psi)(k) =\n\\sqrt{\\frac{2}{\\pi}}\\int_0^\\infty dr\\, \\psi(r)\\sin kr  & {\\rm and}\n& ({\\cal C} \\psi)(k) =\n\\sqrt{\\frac{2}{\\pi}}\\int_0^\\infty dr\\, \\psi(r)\\cos kr\n\\end{eqnarray}\n(the integrals are intended as limits in $L^2$-norm).\nBoth are unitary maps from ${\\cal H}_r$ to ${\\cal H}_k$;\ntheir inverses have the same form, with $r$ and $k$ exchanged.\nThus ${\\cal T}$ is given explicitly by\n\\begin{equation}\n{\\cal T} = \\frac{p(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}{\\cal S} +\n\\frac{kq(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}} {\\cal C}.\n\\end{equation}\nBecause ${\\cal S}$ and ${\\cal C}$ furnish the spectral representations of\n$-d^2\/dr^2$ on $L^2({\\rm I\\! R}^+)$ with Dirichlet and Neumann boundary\nconditions respectively at the origin, we are in the general situation\nof Section~2.2.\n\nWe now restrict to the generic case and explicitly construct the\n$M_i$ and compute their rank and signature. $M_2$ is given by the\nfollowing proposition, whose proof is given later in this section.\n\\begin{Prop}\n\\label{Prop:M2} In the generic case,\n\\begin{equation}\nM_2 = \\sum_{\\omega\\in\\Omega} \\alpha_\\omega\n\\ket{\\xi_\\omega}\\bra{\\xi_{\\overline{\\omega}}}, \\label{eq:M2}\n\\end{equation}\nwhere $\\xi_\\omega(r) =e^{\\omega r}$, and $\\alpha_\\omega$ is the\nresidue\n\\begin{equation}\n\\alpha_\\omega= {\\rm Res}_\\omega\n2zf_0(-iz). \\label{eq:aw}\n\\end{equation}\nIn addition, ${\\rm Ran}\\,\nM_2={\\rm span}\\,\\{\\xi_\\omega\\mid\\omega\\in\\Omega\\}$,\nand\n\\begin{eqnarray}\n{\\rm rank}\\, M_2 &=& \\frac{1}{2}\\deg W + I_L^+  \\label{eq:M2rnk} \\\\\n{\\rm sig}\\, M_2 &=& \\frac{1}{2}\\left(\\sigma_0^2-\\sigma_\\infty^2\\right)\n-I_L^- . \\label{eq:M2sig}\n\\end{eqnarray}\n\\end{Prop}\n\nNext, define ${\\cal M}_1$ to be the space of all\n$L^2$-vectors of form $Q(k^2)k(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}$,\nsuch that $Q(z)\\in \\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}[z]$ is a polynomial with complex\ncoefficients. Thus\n\\begin{equation}\n{\\cal M}_1 =\n(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}k\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}_{\\mho-1}[k^2]\n\\end{equation}\nwhere $\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}_r[z]$ is the $r+1$-dimensional complex vector space of\npolynomials with complex coefficients and degree at most $r$,\nand $\\mho=\\dim {\\cal M}_1$ is given by\n\\begin{equation}\n\\mho=\\frac{1}{2}\\deg W+\\frac{1}{2}(\\sigma_\\infty^2-\\sigma_0^2) =\n\\max\\{\\deg p ,\\deg q\\}.\n\\end{equation}\n$M_1$ is described by\n\\begin{Prop} \\label{Prop:M1}\nIn the generic case, $M_1$ vanishes on\n${{\\cal M}_1}^\\perp$, and its action on ${\\cal M}_1$ is given by\n$M_1 Q(k^2)k(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}=\\tilde{Q}(k^2)k\n(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}$, where\n\\begin{equation}\n\\tilde{Q}(k^2)=\nQ(k^2)+\\sum_{\\omega\\in\\Omega}\n\\frac{Q(-\\omega^2)\\alpha_\\omega}{q(-\\omega^2)}\n\\frac{p(k^2)-\\omega q(k^2)}{\\omega^2+k^2}.\n\\end{equation}\nMoreover, ${\\rm Ran}\\, M_1={\\cal M}_1$ and\n\\begin{eqnarray}\n{\\rm rank}\\, M_1 & = &\\frac{1}{2}\\deg W +\n\\frac{1}{2}(\\sigma_\\infty^2-\\sigma_0^2) \\label{eq:M1rnk} \\\\\n{\\rm sig}\\, M_1 &=& -(I_L^+ +I_L^-). \\label{eq:M1sig}\n\\end{eqnarray}\n\\end{Prop}\n\nAs an example, let us consider the sub-class of the $R$ class\nconsidered by Shondin \\cite{Shond1}; namely, the case where\n$r(z)=p(z)\/q(z)$ has negative imaginary part in the upper half-plane.\nIn this case, it is easy to show that there can be no solutions to\n$r(-z^2)=z$ and hence to $W(z)=0$ in the left-hand half-plane,\nexcept on the real axis. Moreover, one can show that the residues\n$\\alpha_\\omega$ at these zeros are necessarily positive, so $M_2$ is\na positive operator as a result of~(\\ref{eq:M2}). Accordingly, ${\\cal T}$\nis contractive, and our method yields a unitary dilation defined on\nHilbert spaces. This explains why Shondin was able to construct\nthese GPI models on enlarged {\\em Hilbert} spaces.\n\nWe now prove the above propositions.\n\n{\\noindent\\em Proof of Proposition~\\ref{Prop:M2}:}\n$M_2$ may be written in two equivalent forms:\n\\begin{eqnarray}\nM_2 &=& {\\cal S}^{-1}\\sin^2\\delta_0(k){\\cal S}\n -{\\cal C}^{-1}\\sin^2\\delta_0(k){\\cal C}   \\nonumber \\\\\n& &-{\\cal C}^{-1}\\sin\\delta_0(k)\\cos\\delta_0(k){\\cal S} -\n{\\cal S}^{-1}\\sin\\delta_0(k)\\cos\\delta_0(k){\\cal C} \\label{eq:ker1}\\\\\n&=& {\\cal C}^{-1}\\cos^2\\delta_0(k){\\cal C}  -{\\cal S}^{-1}\\cos^2\\delta_0(k){\\cal S}\n\\nonumber \\\\\n& &-{\\cal C}^{-1}\\sin\\delta_0(k)\\cos\\delta_0(k){\\cal S} -\n{\\cal S}^{-1}\\sin\\delta_0(k)\\cos\\delta_0(k){\\cal C} . \\label{eq:ker2}\n\\end{eqnarray}\n{}To convert this into an integral kernel we use the following Lemma,\nwhich may be proved by standard means (cf. Theorem IX.29 in\n\\cite{RSii}). Here, $v(x)$ and $w(x)$ stand for either $\\sin x$ or\n$\\cos x$, and ${\\cal V}$ and ${\\cal W}$ are the corresponding integral\ntransforms from ${\\cal H}_r$ to ${\\cal H}_k$.\n\\begin{Lem} \\label{Lem:ker}\nLet $g(k)\\in L^2({\\rm I\\! R}^+)\\cap L^\\infty({\\rm I\\! R}^+)$ and define\n$G={\\cal V}^{-1}g(k){\\cal W}$. Then $G$ has integral kernel\n\\begin{equation}\nG(r,r^\\prime) = \\frac{2}{\\pi} \\int_0^\\infty v(kr)w(kr^\\prime)g(k)dk,\n\\end{equation}\n(where the integral is a limit in $L^2$-norm).\n\\end{Lem}\n\nIn the case $\\deg p>\\deg q$, $\\sin^2\\delta_0(k)$ and\n$\\sin\\delta_0(k)\\cos\\delta_0(k)$ are $L^2\\cap L^\\infty$ and so,\napplying Lemma~\\ref{Lem:ker} to~(\\ref{eq:ker1}) and combining\nterms, $M_2$ has integral kernel\n\\begin{equation}\nM_2(r,r^\\prime) =\n\\frac{i}{\\pi}\\int_{-\\infty}^\\infty e^{i\\delta_0(k)}\n\\sin\\delta_0(k) e^{ik(r+r^\\prime)}dk\n=\\frac{1}{\\pi}\\int_{-\\infty}^\\infty\n\\frac{ikq(k^2)e^{ik(r+r^\\prime)}}{p(k^2)-ikq(k^2)} dk.\n\\end{equation}\nMaking the substitution $z=ik$ and closing the contour in the\nleft-hand half-plane, the integrand has a simple pole at each\n$\\omega\\in\\Omega$ and~(\\ref{eq:M2}) follows. If $\\deg q\\ge\\deg p$, we\nargue similarly using~(\\ref{eq:ker2}) to obtain the same result as\nbefore.\n\nBy linear independence of the $\\xi_\\omega$ and non-vanishing of the\n$\\alpha_\\omega$, it follows that ${\\rm Ran}\\,\nM_2={\\cal M}_2={\\rm span}\\,\\{\\xi_\\omega\\mid\\omega\\in\\Omega\\}$, so ${\\rm rank}\\,\nM_2=|\\Omega|$, the cardinality of $\\Omega$.\nUsing residue calculus, one may show that\n\\begin{equation}\n|\\Omega| = \\frac{1}{2}\\deg W +\n\\frac{1}{2\\pi}\\int_{-\\infty}^\\infty \\frac{W^\\prime(ik)}{W(ik)} dk.\n\\end{equation}\nBy rewriting the second term as an integral over $(0,\\infty)$, a\nsmall amount of algebra shows that the integrand is\n$-\\pi^{-1}\\delta^\\prime_0(k)$. Thus~(\\ref{eq:M2rnk}) is established.\n\n{}To compute ${\\rm sig}\\, M_2$, we define the hermitian\nform $m_2(\\varphi,\\psi): {\\cal M}_2\\times{\\cal M}_2\\rightarrow \\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$ by\n$m_2(\\varphi,\\psi)=\\inner{\\varphi}{M_2\\psi}$. Labelling the\nelements of $\\Omega$ as $\\omega_1,\\ldots,\\omega_{|\\Omega|}$, and\nwriting $\\psi=\\sum_i c_i\\xi_{\\omega_i}$, we have\n\\begin{equation}\nm_2(\\psi,\\psi) = \\sum_{i,j,k} \\overline{c_i}\n\\inner{\\xi_{\\omega_i}}{\\xi_{\\omega_j}}\\alpha_j\n\\inner{\\xi_{\\overline{\\omega_j}}}{\\xi_{\\omega_k}}c_k\n= c^\\dagger\\Xi^\\dagger A\\Xi c,\n\\end{equation}\nwhere $A$ and $\\Xi$ are hermitian. $\\Xi$ has components\n$\\Xi_{ij}=\\inner{\\xi_{\\omega_i}}{\\xi_{\\omega_j}}$,\nand is non-singular by\nlinear independence of the $\\xi_\\omega$. By Sylvester's Law of\nInertia \\cite{Cohn}, the signature of $M_2$ equals that of $A$,\nwhich has components\n\\begin{equation}\nA_{ij} = \\left\\{ \\begin{array}{cl}\n\\alpha_{\\omega_i} & \\omega_i=\\overline{\\omega_j} \\\\\n0 & {\\rm otherwise}. \\end{array}\\right.\n\\end{equation}\n$A$ has eigenvalues $\\{\\alpha_\\omega\\mid \\omega\\in{\\rm I\\! R}\\}\\cup \\{\\pm\n|\\alpha_\\omega|\\mid\\omega\\not\\in{\\rm I\\! R}\\}$.\nLabelling the $\\omega_i$ so that $\\omega_1,\\ldots,\\omega_r$ are\nthe real elements of $\\Omega$, we therefore have\n${\\rm sig}\\, M_2={\\rm sig}\\,{\\rm diag}\\, (\\alpha_{\\omega_1}\\ldots,\\alpha_{\\omega_r})$.\n(We have used the fact that\n$\\alpha_{\\overline{\\omega}}=\\overline{\\alpha_\\omega}$, and\nin particular that $\\omega_r\\in{\\rm I\\! R}$ implies\n$\\alpha_r\\in{\\rm I\\! R}$.) Defining $\\zeta(k)$ by~(\\ref{eq:zeta}), it is easy\nto show that $\\cot\\zeta(-\\omega)=1$ for $\\omega\\in\\Omega$, and that\n\\begin{equation}\n\\alpha_\\omega =2\\lim_{z\\rightarrow -\\omega}\n\\frac{z+\\omega}{1-\\cot\\zeta(z)} = \\frac{1}{\\zeta^\\prime(-\\omega)}.\n\\end{equation}\nThus ${\\rm sig}\\,{\\rm diag}\\,(\\alpha_1,\\ldots,\\alpha_r)$ is equal to the number of\ntimes that $\\zeta(k)\\equiv\\pi\/4 \\pmod\\pi$ as $k$\ntraverses ${\\rm I\\! R}^+$, counted according to the sign of\n$\\zeta^\\prime(k)$ at such points. This is related to the Levinson\nindex $I_L^-$ by~(\\ref{eq:M2sig}). $\\vrule height 1.5ex width 1.2ex depth -.1ex $\n\n{\\noindent\\em Proof of Proposition~\\ref{Prop:M1}:} We compute\n\\begin{eqnarray}\nM_1 &=& -\\frac{p(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}{\\cal S}{\\cal C}^{-1}\n\\frac{kq(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}  \\nonumber \\\\\n&& -\\frac{kq(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}{\\cal C}{\\cal S}^{-1}\n\\frac{p(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}},\n\\end{eqnarray}\nwhich vanishes identically on the closure of\n${\\cal D}=(p(k^2)^2+k^2q(k^2))^{1\/2}{\\cal S} C_0^\\infty(0,\\infty)$ as a result\nof elementary properties of the sine and cosine transforms.\nFurthermore, $\\overline{{\\cal D}}^\\perp$ is precisely the space ${\\cal M}_1$\ndefined above, because $\\psi\\perp{\\cal D}$ if and only if\n$(p(k^2)^2+k^2q(k^2))^{1\/2}\\psi$ is the sine transform of a\ndistribution supported at the origin and therefore an odd polynomial\n(cf. Theorem~V.11 in \\cite{RSi}). Hence $M_1$ vanishes on\n${\\cal M}_1^\\perp$ and ${\\rm Ran}\\, M_1\\subset {\\cal M}_1$.\n\nNext, we compute the action of $M_1$ on ${\\cal M}_1$. By contour\nintegration,\n\\begin{equation}\n{\\cal T}^*\\frac{kQ(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}\n=-\\left(\\frac{\\pi}{2}\\right)^{1\/2}\\sum_{\\omega\\in\\Omega}\n\\frac{Q(-\\omega^2)\\alpha_\\omega}{q(-\\omega^2)}\\xi_\\omega(r),\n\\label{eq:Tstr}\n\\end{equation}\nfor polynomials $Q(z)$ such that the operand is in $L^2$. Moreover,\nit is easy to show that\n\\begin{equation}\nT\\xi_{\\omega} = \\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{k}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}\n\\frac{p(k^2)-\\omega q(k^2)}{\\omega^2+k^2},\n\\label{eq:Txi}\n\\end{equation}\nfrom which the action of $M_1$ can be read off as required.\n\n{}To compute the rank and signature of $M_1$, we use the fact that\n\\begin{equation}\n{\\rm rank}\\, M_1 - {\\rm rank}\\, M_2 = {\\rm sig}\\, M_1 -{\\rm sig}\\, M_2 =\n\\dim\\ker {\\cal T}^*-\\dim\\ker{\\cal T} ,\n\\end{equation}\nwhich follows from the intertwining relations $M_1{\\cal T}={\\cal T} M_2$ and\n$M_2{\\cal T}^*={\\cal T}^*M_1$. It therefore remains to determine the dimensions\nof the relevant kernels. Firstly, note that $\\ker {\\cal T}^*\\subset{\\cal M}_1$\nand that (from~(\\ref{eq:Tstr}))\n$\\psi=Q(k^2)k(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}\\in\\ker {\\cal T}^*$ if and only\nif $\\psi\\in{\\cal M}_1$ and $Q(-\\omega^2)=0$ for each $\\omega\\in\\Omega$.\nThus $\\prod_{\\omega\\in\\Omega} (z+\\omega^2)$ divides $Q(z)$ and so\n\\begin{equation}\n\\dim\\ker{\\cal T}^* = \\min\\{\\mho-|\\Omega|,0\\}.\n\\end{equation}\n\nNow consider $\\ker{\\cal T}$. We note that~(\\ref{eq:Txi}) may be rewritten\n\\begin{equation}\nq(-\\omega_i^2){\\cal T}\\xi_{\\omega_i}=\\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{k}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}\n\\frac{p(k^2)q(-\\omega_i^2)-p(-\\omega_i^2)q(k^2)}{k^2+\\omega_i^2},\n\\end{equation}\nand apply the following abstract algebraic result:\n\\begin{Lem}\n\\label{Lem:FW}\nLet $Q,R\\in \\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}[z]$ be coprime with $\\max\\{\\deg Q,\\deg R\\}=k\\ge 0$,\nand let $\\lambda_1,\\ldots,\\lambda_m$ be distinct elements of $\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$.\nThen the polynomials $P_1(z),\\ldots P_m(z)$, defined by\n\\begin{equation}\n(z-\\lambda_i) P_i(z) = R(\\lambda_i)Q(z)-Q(\\lambda_i)R(z)\n\\end{equation}\nspan a $\\min\\{k,m\\}$-dimensional subspace of $\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}_{k-1}[z]$.\n\\end{Lem}\n{\\em Proof:} Let  $n=\\min\\{k,m\\}$. Then it is enough to show that\n$P_1,\\ldots,P_n$ are linearly independent. Assuming that $\\deg Q=k$,\nwe note that  $P_i(z)=R(z) \\tilde{Q}_i(z)-Q(z)\\tilde{R}_i(z)$, where\n$\\tilde{Q}_i(z)=(Q(z)-Q(\\lambda_i))\/(z-\\lambda_i)$ and\n$\\tilde{R}_i(z)=(R(z)-R(\\lambda_i))\/(z-\\lambda_i)$. Suppose the $P_i$\nare linearly dependent. Then $R(z) S(z) = Q(z) T(z)$ where $S(z)\n=\\sum_i \\alpha_i \\tilde{Q}_i(z)$ and $T(z) =\\sum_i\\alpha_i\n\\tilde{R}_i(z)$, for some $0\\not= (\\alpha_1,\\ldots,\\alpha_n)^T\\in\n\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}^n$. Because $Q$ and $R$ are coprime, this implies that $S$ and\n$T$ vanish identically. But one may easily show that the\n$\\tilde{Q}_i$ are linearly independent, by explicitly considering\ntheir coefficients. We therefore obtain a contradiction. $\\vrule height 1.5ex width 1.2ex depth -.1ex $\n\nIn our application, $m=|\\Omega|$ with $\\lambda_i=-\\omega_i^2$ for\neach $i=1,\\ldots,m$ and $k=\\max\\{\\deg p,\\deg q\\}=\\mho$. Thus\n$\\dim{\\cal T}{\\rm Ran}\\, M_2=\\min\\{|\\Omega|,\\mho\\}$ and so\n\\begin{equation}\n\\dim\\ker{\\cal T}= \\min\\{|\\Omega|-\\mho,0\\}.\n\\end{equation}\nIt follows that ${\\rm rank}\\, M_1-{\\rm rank}\\, M_2={\\rm sig}\\, M_1-{\\rm sig}\\,\nM_2=\\mho-|\\Omega|$, from which~(\\ref{eq:M1rnk}) and~(\\ref{eq:M1sig})\nfollow. $\\vrule height 1.5ex width 1.2ex depth -.1ex $\n\n\\sect{The GPI Hamiltonian}\n\\label{sect:GHm}\n\\subsection{Locality and Spectral Properties}\n\nThe results of the previous two sections allow the construction of a\nunitary dilation $\\hat{{\\cal T}}$ of the integral transform ${\\cal T}$. Here,\nwe employ $\\hat{{\\cal T}}$ to define a GPI Hamiltonian consistent with\nscattering theory~(\\ref{eq:GPIlow}). We denote\n$\\Pi_r={\\cal H}_r\\oplus{\\cal M}_1$ and $\\Pi_k={\\cal H}_k\\oplus{\\cal M}_2$ with $J$-inner\nproducts specified by $J_r=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal H}_r}\\oplus{\\rm sgn}\\, (M_1)$, and\n$J_k=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal H}_k}\\oplus{\\rm sgn}\\, (M_2)$. In terms of our general\ndiscussion in Section~2.2, we set $A=-d^2\/dr^2$ on domain\n$C_0^\\infty(0,\\infty)$, and define ${\\cal T}_+={\\cal S}$, ${\\cal T}_-={\\cal C}$, setting\n$a_+$ and $a_-$ to be multiplication by $\\cos\\delta_0(k)$ and\n$\\sin\\delta_0(k)$ respectively. Thus\n$A_+={\\cal S}^*k^2{\\cal S}$, the self-adjoint extension of $A$ with Dirichlet\nboundary conditions at the origin, whilst $A_-={\\cal C}^*k^2{\\cal C}$ is the\nextension with Neumann boundary conditions at the origin. The\noperators $A_\\pm+1$ both have bounded inverse.\n\nThe $S$-wave GPI Hamiltonian is defined by\n\\begin{equation}\nh_{\\rm gpi} = \\hat{{\\cal T}}^\\dagger\\left(\n\\begin{array}{cc} k^2 & 0 \\\\ 0 & \\Lambda \\end{array}\\right)\n\\hat{{\\cal T}}, \\label{eq:GPIham}\n\\end{equation}\nwhere $\\Lambda$ is a ${\\rm sgn}\\, (M_2)$-self-adjoint operator\n$\\Lambda^\\dagger=\\Lambda$ on ${\\cal M}_2$. To fix $\\Lambda$, we require\nthat $h_{\\rm gpi}(\\psi,0)^T=(-\\psi^{\\prime\\prime},0)^T$ for\nall $\\psi\\in C_0^\\infty(0,\\infty)$ as a locality requirement.\nFor general $\\psi\\in{\\cal M}_2$, we have\n\\begin{equation}\nA^*\\psi = -\\sum_{\\omega\\in\\Omega}\\alpha_\\omega \\omega^2\n\\ket{\\xi_\\omega}\\inner{\\xi_{\\overline{\\omega}}}{M_2^{-1}\\psi},\n\\end{equation}\nso ${\\cal M}_2$ is invariant under $A^*$ and it follows immediately from\nSection~2.2 that\n\\begin{Thm} \\label{Thm:local}\nIn the generic case, the unique choice of $\\Lambda$ consistent with\nlocality is\n\\begin{equation}\n\\Lambda= -\\left({\\rm sgn}\\,(M_2) |M_2|^{1\/2}\\right)^{-1}\n\\sum_{\\omega\\in\\Omega} \\alpha_\\omega \\omega^2\\ket{\\xi_\\omega}\n\\bra{\\xi_{\\overline{\\omega}}} |M_2|^{-1\/2}.\n\\end{equation}\n\\end{Thm}\n\nWe proceed to determine the eigenvectors and eigenvalues of\n$\\Lambda$. First note that\n$\\inner{\\xi_{\\overline{\\omega_j}}}{M_2^{-1}\\xi_{\\omega_i}}=\n\\alpha_{\\omega_{i}}^{-1}\\delta_{ij}$, which follows from the identity\n$\\xi_{\\omega_i}=\\sum\\alpha_\\omega\\ket{\\xi_\\omega}\n\\inner{\\xi_{\\overline{\\omega}}}{M_2^{-1}\\xi_{\\omega_i}}$. It is then\na matter of computation to see that $\\varphi_i=\\left({\\rm sgn}\\,(M_2)\n|M_2|^{1\/2}\\right)^{-1}\\xi_{\\omega_i}$ is an eigenvector of $\\Lambda$\nwith eigenvalue $-\\omega_i^2$ for each $i=1,\\ldots,|\\Omega|$. Because\n$\\Lambda$ has rank $|\\Omega|$, this exhausts the discrete spectrum of\n$h_{\\rm gpi}$. The following is then immediate.\n\\begin{Thm}\nIn the generic case, and  with\n$\\Lambda$ is defined as above, $h_{\\rm gpi}$ has the following spectral\nproperties: $\\sigma(h_{\\rm gpi})=\\sigma_{\\rm ac}(h_{\\rm gpi}) \\cup\\sigma_{\\rm\npp}(h_{\\rm gpi})$ where $\\sigma_{\\rm ac}(h_{\\rm gpi})={\\rm I\\! R}^+$ and $\\sigma_{\\rm\npp}(h_{\\rm gpi})$ consists of the $|\\Omega|$ eigenvalues $-\\omega_i^2$, whose\ncorresponding eigenvectors are\n\\begin{equation}\n\\psi_i = \\hat{{\\cal T}}^\\dagger\\varphi_i=\n\\left(\\begin{array}{c} \\xi_{\\omega_i} \\\\\n{\\cal T} \\left({\\rm sgn}\\,(M_2) |M_2|^{1\/2}\\right)^{-1}\\xi_{\\omega_i}\n\\end{array}\\right).\n\\end{equation}\nThe absolutely continuous subspace is the Hilbert space\n$\\hat{{\\cal T}}^\\dagger{\\cal H}_k$.\n\\end{Thm}\n\nThis bears out our earlier statement that the poles of the\nscattering amplitude on the physical sheet correspond to the discrete\nenergy spectrum, if locality is imposed.\n\nThe physical Hilbert space is required to be a positive definite\ninvariant subspace of $\\Pi_r$ relative to $h_{\\rm gpi}$.\\footnote{An\ninvariant subspace ${\\cal L}$ of a $J$-space ${\\cal K}$ relative to a linear\noperator $A$ on ${\\cal K}$ is a subspace of ${\\cal K}$ such that\n$\\overline{D(A)\\cap{\\cal L}}={\\cal L}$ and ${\\rm Ran}\\, A|_{{\\cal L}}\\subset {\\cal L}$, where\nthe closure is taken in the norm topology of ${\\cal K}$.} In $\\Pi_k$, we\nhave the $[\\cdot,\\cdot]_{\\Pi_k}$-orthogonal decomposition\n$\\Pi_k={\\cal H}_k [+] {\\cal M}_2$, where ${\\cal M}_2$ is spanned by the eigenvectors\n$\\varphi_i$ of $\\Lambda$. We compute\n\\begin{equation}\n[\\varphi_i,\\varphi_j]_{{\\cal M}_2}\n=\\inner{\\xi_{\\omega_i}}{M_2^{-1}\\xi_{\\omega_j}} \\nonumber \\\\\n= \\left\\{\\begin{array}{cl} 0 & \\omega_i\\not=\\overline{\\omega_j} \\\\\n\\alpha_{\\omega_j}^{-1} & \\omega_i=\\overline{\\omega_j}.\n\\end{array}\\right.\n\\end{equation}\nHence $\\Pi_k$ is decomposable as\n$\\Pi_k = {\\cal H}_k [+] E_+ [+] E_- [+] H$\nwhere $E_+$ is spanned by the $\\varphi_i$ with\n$[\\varphi_i,\\varphi_i]_{{\\cal M}_2}>0$ ($\\alpha_{\\omega_i}>0$),\n$E_-$ is spanned\nby those with $[\\varphi_i,\\varphi_i]_{{\\cal M}_2}<0$\n($\\alpha_{\\omega_i}<0$), and\n$H$ is the {\\em hyperbolic invariant subspace} spanned by those\n$\\varphi_i$ with $\\omega_i\\not\\in{\\rm I\\! R}$. Moreover, this is a\ndecomposition into invariant subspaces, because $D(k^2)$ is dense in\n${\\cal H}_k$. The physical Hilbert space ${\\cal H}_{\\rm phys}$ is therefore\ndefined by\n\\begin{equation}\n{\\cal H}_{\\rm phys} = \\hat{{\\cal T}}^\\dagger ({\\cal H}_k[+]E_+).\n\\end{equation}\n\nWe briefly discuss the uniqueness of the GPI Hamiltonian constructed\nin this way. As noted in Section~2.1, $\\hat{{\\cal T}}$ is unique up to\nfurther unitary dilation and unitary equivalence because the $M_i$\nare of finite rank. Further dilation merely corresponds to the\n(trivial) freedom to form the direct sum of $h_{\\rm gpi}$ with the\nHamiltonian of an arbitrary independent system. On the other hand,\nreplacing $\\hat{{\\cal T}}$ by $(\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}\\oplus U_2)\\hat{{\\cal T}}(\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}\\oplus U_1)$\nwhere $U_i$ is a ${\\rm sgn}\\, M_i$-unitary operator on ${\\cal M}_i$ for $i=1,2$,\nit is easy to show that the local GPI Hamiltonian $h_{\\rm gpi}^\\prime$\nobtained is given by\n\\begin{equation}\nh_{\\rm gpi}^\\prime=\n\\left(\\begin{array}{cc} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1} & 0\\\\0 & U_1\\end{array}\\right)^\\dagger\nh_{\\rm gpi}\\left(\\begin{array}{cc} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1} & 0\\\\0 & U_1\\end{array}\\right).\n\\end{equation}\nWe have therefore constructed a family of unitarily equivalent GPI\nHamiltonians on $\\Pi_r$ corresponding to the same scattering data. It\nis clearly sufficient to study $h_{\\rm gpi}$ alone in order to determine the\ndomain and scattering properties of $h_{\\rm gpi}^\\prime$.\n\n\\subsection{Domain and Resolvent}\n\nWe now determine the domain and explicit action of the operator\n$h_{\\rm gpi}$ under the locality assumption. Our result is the following:\n\\begin{Thm} \\label{Thm:dom}\nLet $\\Theta_0\n=(2\/\\pi)^{1\/2}k^{2\\mho-1}(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}$. Then in\nthe generic case,\n\\begin{eqnarray}\nD(h_{\\rm gpi})&=&\\left\\{\n\\left( \\begin{array}{c} \\varphi \\\\ \\Phi\\end{array}\\right)\n\\mid \\varphi,\\varphi^\\prime\\in\nAC_{\\rm loc}(0,\\infty),~\\varphi,\\varphi^{\\prime\\prime}\\in L^2;\\quad\n\\Phi\\in{\\cal M}_1, \\right.\\nonumber \\\\\n&&\\qquad\\qquad\n\\left.\\begin{array}{c} \\ \\\\ \\ \\end{array}\n\\Theta -\\lambda[\\varphi]\\Theta_0 \\in\nD(k^2)\\cap {\\cal M}_1\\right\\} ,\n\\end{eqnarray}\nwhere $\\Theta={\\rm sgn}\\, M_1|M_1|^{1\/2}\\Phi$ and\n\\begin{equation}\n\\lambda[\\varphi] = \\left\\{\\begin{array}{cl}\nP\\varphi(0) & \\deg p>\\deg q \\\\\nP\\varphi(0)-Q\\varphi^\\prime(0) & \\deg p=\\deg q \\\\\n-Q\\varphi^\\prime(0) & \\deg p< \\deg q, \\end{array}\\right.\n\\end{equation}\nand $P$ and $Q$ are the leading coefficients of $p(z)$ and $q(z)$\nrespectively. (In the case $M_1=0$,\n$D(h_{\\rm gpi})=\\{\\varphi\\mid\\varphi,\\varphi^\\prime\\in\nAC_{\\rm loc}(0,\\infty),~\\varphi,\\varphi^{\\prime\\prime}\\in\nL^2;~\\lambda[\\varphi]=0\\}$.) Moreover,\n\\begin{equation}\nh_{\\rm gpi}\\left(\\begin{array}{c} \\varphi \\\\ \\Phi\\end{array}\\right)=\n\\left(\\begin{array}{c} -\\varphi^{\\prime\\prime}\\\\ \\tilde{\\Phi}\n\\end{array}\\right) ,\n\\end{equation}\nwhere $\\tilde{\\Phi}$ is given in terms of $\\tilde{\\Theta}={\\rm sgn}\\, M_1\n|M_1|^{1\/2}\\tilde{\\Phi}$ by\n\\begin{equation}\n\\tilde{\\Theta}= k^2(\\Theta-\\lambda[\\varphi]\\Theta_0) +\n\\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{k(\\lambda[\\varphi] k^{2\\mho} -\\varphi(0)p(k^2)\n+\\varphi^\\prime(0)q(k^2))}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}} .\n\\end{equation}\n\\end{Thm}\n{\\em Proof:} The result is a direct application of the discussion in\nSection~2.2. The key point is that, for each $\\varphi\\in\nD(-d^2\/dr^2|_{C_0^\\infty(0,\\infty)}^*)$, the vectors $\\chi_+$ and\n$\\chi_-$ are given by\n\\begin{equation}\n\\chi_+ = -\\varphi(0)e^{-r}\\qquad {\\rm and} \\qquad\n\\chi_-=\\varphi^\\prime(0)e^{-r},\n\\end{equation}\nwhich follows because $\\chi_+$ ($\\chi_-$) is the unique element of\n$\\ker (-d^2\/dr^2|_{C_0^\\infty(0,\\infty)}^*+1)$ such that\n$\\varphi+\\chi_+$ ($\\varphi+\\chi_-$) is in the domain of the Laplacian\nwith Dirichlet (Neumann) boundary conditions at the origin. $\\vrule height 1.5ex width 1.2ex depth -.1ex $\n\nThe resolvent of $h_{\\rm gpi}$ may be written in the form of Krein's formula\nas\n\\begin{equation}\n(h_{\\rm gpi}-z)^{-1} =\n\\left(\\begin{array}{cc} R_0(z) & 0 \\\\ 0 & R_1(z)\\end{array}\\right)\n+\\frac{q(z)}{p(z)+(-z)^{1\/2}q(z)}F(z)F(\\overline{z})^\\dagger.\n\\label{eq:reso}\n\\end{equation}\nHere, $R_0(z)={\\cal S}^{-1}(k^2-z)^{-1}{\\cal S}$ is the free resolvent\nand the defect element $F(z)\\in\\Pi_r$ is given by\n\\begin{equation}\nF(z)=\\left( \\begin{array}{c} e^{-(-z)^{1\/2}r} \\\\\n({\\rm sgn}\\, M_1|M_1|^{1\/2})^{-1}\\Psi(z) \\end{array}\\right),\n\\end{equation}\nwhere $\\Psi(z)\\in{\\cal M}_1$ is\n\\begin{equation}\n\\Psi(z) = \\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{k(p(k^2)q(z)-p(z)q(k^2))}{(k^2-z)(p(k^2)^2+k^2q(k^2)^2)^{1\/2}},\n\\end{equation}\nand the operator $R_1(z)$ is defined on ${\\cal M}_1$ by\n\\begin{equation}\nR_1(z)\\Phi=({\\rm sgn}\\, M_1|M_1|^{1\/2})^{-1}\\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{k(Q(k^2)-Q(z)q(k^2)\/q(z))}{(k^2-z)(p(k^2)^2+k^2q(k^2)^2)^{1\/2}},\n\\end{equation}\nwhere $Q(z)$ is defined in terms of $\\Phi$ by\n\\begin{equation}\n\\Theta={\\rm sgn}\\, M_1|M_1|^{1\/2}\\Phi=\\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{kQ(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}.\n\\end{equation}\nThe above expression for $R(z)$ may be verified directly using\nTheorem~\\ref{Thm:dom}, and the fact that\n\\begin{equation}\n[({\\rm sgn}\\, M_1|M_1|^{1\/2})^{-1}\\Psi(\\overline{z}),\\Phi]_{{\\cal M}_1}=\n-\\frac{Q(z)}{q(z)},\n\\label{eq:clm}\n\\end{equation}\nwhich is required when one takes inner products with\n$F(\\overline{z})$. Using this result, it follows that~(\\ref{eq:reso})\nholds for elements of form $(0,\\Phi)^T$ with $Q(z)=0$; direct\ncomputation establishes it for $Q(z)\\equiv 1$ and also for  vectors\nof form $(\\varphi,0)^T$ with $\\varphi\\in{\\cal H}_r$.\\footnote{Here, it is\nuseful to employ the decomposition ${\\cal H}_r= \\overline{{\\rm Ran}\\,\n(-d^2\/dr^2-z)|_{C_0^\\infty(0,\\infty)}}\\oplus \\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}\ne^{-(-\\overline{z})^{1\/2}r}$.} Thus~(\\ref{eq:reso}) holds on the whole\nof $\\Pi_r$. It remains to establish equation~(\\ref{eq:clm}).\nMultiplying through by $q(z)$, the LHS of~(\\ref{eq:clm}) is equal to\n\\begin{equation}\n\\inner{q(\\overline{z})\\Psi(\\overline{z})}{M_1^{-1}\\Theta}=\n\\inner{{\\cal T}^*q(\\overline{z})\n\\Psi(\\overline{z})}{M_2^{-1}{\\cal T}^*\\Theta}+\\inner{q(\\overline{z})\n\\Psi(\\overline{z})}{\\Theta}. \\end{equation}\nUsing the identity\n$\\inner{\\xi_{\\overline{\\omega_j}}}{M_2^{-1}\\xi_{\\omega_i}}=\n\\alpha_{\\omega_{i}}^{-1}\\delta_{ij}$ and the results of Section~3,\nthe first term is\n\\begin{equation}\n\\inner{{\\cal T}^*q(\\overline{z})\n\\Psi(\\overline{z})}{M_2^{-1}{\\cal T}^*\\Theta}=\n\\sum_{\\omega\\in\\Omega}\\frac{p(z)-\\omega\nq(z)}{q(-\\omega^2)(\\omega^2+z)}Q(-\\omega^2)\\alpha_\\omega.\n\\end{equation}\nThe required result then follows from the calculation\n\\begin{eqnarray}\n\\inner{q(\\overline{z})\n\\Psi(\\overline{z})}{\\Theta} &=&\n\\frac{1}{\\pi}\\int_{-\\infty}^\\infty dk\\frac{ikQ(k^2)\n(p(z)-ikq(z))}{(k^2-z)(p(k^2)-ikq(k^2))} \\nonumber \\\\\n&=& -Q(z)-\\sum_{\\omega\\in\\Omega}\\frac{p(z)-\\omega\nq(z)}{q(-\\omega^2)(\\omega^2+z)}Q(-\\omega^2)\\alpha_\\omega.\n\\end{eqnarray}\n\n\n\\subsection{Scattering Theory}\n\\label{sect:GPIsc}\n\nIn this section, we construct M{\\o}ller wave operators for $h_{\\rm gpi}$\nrelative to the free Hamiltonian $h_0={\\cal S}^{-1} k^2{\\cal S}$ on ${\\cal H}_r$ in\norder to check that $h_{\\rm gpi}$ actually exhibits the required scattering\nbehaviour. Because scattering is a function of the continuous\nspectrum only, our results in this section are actually independent\nof the precise form of $\\Lambda$, and therefore of the locality\nrequirement.\n\nWe work in the $S$-wave, and employ a two space setting: let $B$\nbe self-adjoint on ${\\cal H}_1$, $A$ be self-adjoint on ${\\cal H}_2$ and ${\\cal J}$\nbe a bounded operator from ${\\cal H}_1$ to ${\\cal H}_2$. Then the M{\\o}ller\noperators $\\Omega^\\pm(A,B;{\\cal J})$ are defined by\n\\begin{equation}\n\\Omega^\\pm(A,B;{\\cal J}) = \\lim_{t\\rightarrow\\mp\\infty}\ne^{iAt}{\\cal J} e^{-iBt}P_{\\rm ac}(B),\n\\end{equation}\nand are said to be complete if the closure of ${\\rm\nRan}\\Omega^\\pm(A,B;{\\cal J})$ is equal to ${\\rm Ran} P_{\\rm ac}(A)$.\n\nIn the following,\n${\\cal J}_r$ and ${\\cal J}_k$ are the natural embeddings of ${\\cal H}_r$ and\n${\\cal H}_k$ into $\\Pi_r$ and $\\Pi_k$ respectively.\n\n\\begin{Thm} Let ${\\cal J}:{\\cal H}_r\\rightarrow\\Pi_r$ be given by\n${\\cal J}=\\hat{{\\cal T}}^\\dagger {\\cal J}_k{\\cal T}$. Then\n$\\Omega^\\pm(h_{\\rm gpi},h_0;{\\cal J})$ exist, are complete, and given by\n\\begin{equation}\n\\Omega^\\pm(h_{\\rm gpi},h_0;{\\cal J}) = \\hat{{\\cal T}}^\\dagger {\\cal J}_k\ne^{\\pm i\\delta_0(k)}{\\cal S},\n\\label{eq:Mops}\n\\end{equation}\nwhere $\\delta_0(k)$ is given by~(\\ref{eq:GPIlow}).\n \\end{Thm}\n{\\em Proof:} Writing $U_t$ for multiplication by $e^{-ik^2t}$ on\n${\\cal H}_k$, we have\n\\begin{eqnarray}\ne^{ih_{\\rm gpi} t} {\\cal J} e^{-ih_0t}P_{\\rm ac}(h_0) &=& \\hat{{\\cal T}}^\\dagger\n\\left(\\begin{array}{cc} U_{-t} & 0 \\\\ 0 & \\exp i\\Lambda t\n\\end{array}\\right) \\hat{{\\cal T}} {\\cal J}  {\\cal S}^{-1} U_t {\\cal S}\n\\nonumber \\\\\n&=& \\hat{{\\cal T}}^\\dagger\n{\\cal J}_k U_{-t} {\\cal T}  {\\cal S}^{-1} U_t {\\cal S}.\n\\end{eqnarray}\nNow, for any $u(k)\\in C_0^\\infty(0,\\infty)$,\n\\begin{eqnarray}\n\\| U_{-t}{\\cal T}{\\cal S}^{-1}U_{t}u(k)-e^{\\pm i\\delta_0(k)} u(k)\\|^2 & = &\n\\|\\sin\\delta_0(k){\\cal C} ({\\cal C}^{-1}\\pm i{\\cal S}^{-1})U_t u(k)\\|^2\n\\nonumber\\\\ & \\le & \\frac{2}{\\pi}\\int_0^\\infty dr\n\\left|\\int_0^\\infty dk e^{i(\\pm kr-k^2t)} u(k)\\right|^2,\n\\end{eqnarray}\nwhich vanishes as $t\\rightarrow\\mp\\infty$ by (non)-stationary phase\narguments (see the Corollary to Theorem XI.14 in \\cite{RSiii}). Thus\n$U_{-t}{\\cal T}{\\cal S}^{-1}U_t \\rightarrow e^{\\pm i\\delta_0(k)}$ strongly as\n$t\\rightarrow\\mp\\infty$. The existence and form of the M{\\o}ller\noperators are then immediate. One easily checks that they are unitary\nmaps from ${\\cal H}_r$ to $P_{\\rm ac}(h_{\\rm gpi})=\\hat{{\\cal T}}^\\dagger{\\cal J}_k{\\cal H}_k$,\nto establish completeness. $\\vrule height 1.5ex width 1.2ex depth -.1ex $\n\nWe conclude that our construction does indeed yield the required\nscattering theory, and also that -- as a by-product of the\nconstruction -- complete M{\\o}ller operators may easily and\nexplicitly be determined.\n\n\\sect{Examples}\n\nAs an application, we construct the class of GPI models with\nscattering data\n\\begin{equation}\n\\cot\\delta_0(k) = -\\frac{1}{kL}+kM, \\label{eq:ERlow}\n\\end{equation}\nwhere $L$ is the scattering length, and $M$ is twice the effective\nrange. These models therefore represent the effective range\napproximation to the behaviour of a non-point interaction in the\n$S$-wave. This class of models has been partially studied by\nShondin~\\cite{Shond1}, who considered the case $M<0$ (`models of type\n$B_2$') and also appears as a special case of the models considered\nby Pavlov in \\cite{Pav1}.  (We also note that van Diejen and Tip\n\\cite{Diejen} have constructed models of type $\\cot\\delta_0(k) = (ak\n+ bk^3+ck^5)^{-1}$ using the distributional method.) The case $M>0$\ndoes not appear to have been treated before. Our construction\nprovides a unified construction for all models in the above class,\nand also provides the spectral representation such models as a\nby-product of the construction (although we will not state this\nexplicitly).\n\nThe above class of GPI models contains two interesting\nsub-families: the ordinary point interactions ($M=0$) and also the\nresonance point interactions arising formally by setting $L=\\infty$,\ni.e., $\\cot\\delta_0(k)=kM$ with $M\\in{\\rm I\\! R}\\cup\\{\\infty\\}$. Such models\nare required in situations where the scattering length is\ngenerically forced to be infinite, for example in certain systems of\nsupersymmetric quantum mechanics.\n\nWe begin by briefly treating the point interactions, both for\ncompleteness and also to demonstrate how this class arises in our\nformalism. We then turn to the general case, obtaining RPI models\nin the limit $L\\rightarrow -\\infty$.\n\n\\subsection{Point Interactions}\n\nThe required integral transform is\n\\begin{equation}\n{\\cal T} = (1+(kL)^2)^{-1\/2}{\\cal S} - kL(1+(kL)^2)^{-1\/2}{\\cal C}.\n\\end{equation}\nIn the cases $L=0,\\infty$, ${\\cal T}$ reduces to ${\\cal S}$ and ${\\cal C}$\nrespectively, and the Hamiltonian is given immediately by\n${\\cal T}^*k^2{\\cal T}$. We exclude these cases from the rest of our discussion.\n\nWe therefore apply the construction of Section 3, with $p(z)\\equiv\n-L^{-1}$ and $q(z)\\equiv 1$. We find that $\\mho=0$, so $M_1=0$ (i.e.,\n${\\cal T}\\Tt^*=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}$). Straightforward application of\nProposition~\\ref{Prop:M2} yields\n\\begin{equation}\nM_2=\\left\\{\\begin{array}{cl}\n\\ket{\\chi_L}\\bra{\\chi_L} & L>0 \\\\ 0 & L<0, \\end{array}\\right.\n\\end{equation}\nwhere $\\chi_L(r)=(2\/L)^{1\/2}e^{-r\/L}$ is normalised to unity.\nHence if $L<0$, ${\\cal T}$ is unitary and the Hamiltonian is\n$h_L = {\\cal T}^* k^2 {\\cal T}$,\nwith purely absolutely continuous spectrum ${\\rm I\\! R}^+$. In the case\n$L>0$, the momentum Hilbert space is extended to ${\\cal H}_k\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$,\nrepresenting a single bound state, and the unitary dilation\n$\\hat{{\\cal T}}:{\\cal H}_r\\rightarrow{\\cal H}_k\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$ takes form\n\\begin{equation}\n\\hat{{\\cal T}} =\n\\left(\\begin{array}{c} {\\cal T} \\\\ \\bra{\\chi_L} \\end{array}\\right);\n\\qquad \\hat{{\\cal T}}^*=\n\\left(\\begin{array}{cc} T^* & \\ket{\\chi_L}\\end{array}\\right).\n\\end{equation}\n(${\\cal H}_k\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$ has the obvious inner product.)\nThe Hamiltonian is\n\\begin{equation}\nh_L = \\hat{{\\cal T}}^{-1}\\left(\\begin{array}{cc} k^2 & 0 \\\\ 0 & \\lambda\n\\end{array}\\right) \\hat{{\\cal T}} = {\\cal T}^* k^2{\\cal T} +\n\\lambda\\ket{\\chi_L}\\bra{\\chi_L},\n\\end{equation}\nand the locality requirement fixes $\\lambda= -L^{-2}$, which is, of\ncourse, the usual value. Finally, the domain of $h_L$ is given by\nTheorem~\\ref{Thm:dom} as the space of $\\varphi$ with\n$\\varphi,\\varphi^\\prime\\in AC_{\\rm loc}(0,\\infty)$,\n$\\varphi^{\\prime\\prime}\\in L^2$ and satisfying\nthe well known boundary condition\n\\begin{equation}\n\\varphi(0)+L\\varphi^\\prime(0) = 0.\n\\end{equation}\n{}To summarise, all the well known properties of point interactions\nmay be derived within our formalism.\n\n\\subsection{Effective Range Approximation}\n\nIn this section, we maintain $M\\not=0$, $L\\not=0$, setting\n$p(z)=-L^{-1}+zM$ and $q(z)\\equiv 1$. We will not explicitly\nconstruct the dilation (although this follows immediately from our\ndiscussion), but will use the results of Section~4 to read off the\ndomain and action of the GPI Hamiltonian $h_{L,M}$.\n\nUsing the results of Section~3, we find\n\\begin{equation}\n\\mho=1;\\qquad |\\Omega|=\n\\left\\{\n\\begin{array}{cl} 1+\\frac{1}{2}({\\rm sgn}\\, M+{\\rm sgn}\\, L) & L\\not=\\infty \\\\\n\\frac{1}{2}(1+{\\rm sgn}\\, M) & L=\\infty. \\end{array}\n\\right.\n\\end{equation}\nWriting $W(z)=-M(z-\\omega_1)(z-\\omega_2)$, $\\Omega$ is\nthe subset of $\\{\\omega_1,\\omega_2\\}$ lying in the left-hand\nhalf-plane, and we have $\\omega_1+\\omega_2=-M^{-1}$,\n$\\omega_1\\omega_2=(ML)^{-1}$. The residues $\\alpha_\\omega$ are\n\\begin{equation}\n\\alpha_{\\omega_1}=-\\frac{2\\omega_1}{M(\\omega_1-\\omega_2)},\\qquad\n\\alpha_{\\omega_2}=\\frac{2\\omega_2}{M(\\omega_1-\\omega_2)}.\n\\end{equation}\nIn addition, the space ${\\cal M}_1={\\rm Ran}\\, M_1$ is equal to $\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}\\ket{\\eta}$,\nwhere\n\\begin{equation}\n\\eta(k) = {\\cal N} \\frac{k}{(k^2+(k^2M-L^{-1})^2)^{1\/2}},\n\\end{equation}\nand the normalisation constant is\n\\begin{equation}\n{\\cal N}=\\left\\{\\begin{array}{cl}\n(2|M|\/\\pi)^{1\/2} & ML>0 \\\\\n(2|M|\/\\pi)^{1\/2}(1-4ML^{-1})^{1\/4} & ML<0. \\end{array}\n\\right.\n\\end{equation}\nUsing Proposition~\\ref{Prop:M1}, we obtain\n\\begin{equation}\nM_1 = \\lambda\\ket{\\eta}\\bra{\\eta} ;\n\\qquad \\lambda = \\left\\{\\begin{array}{cl}\n+1 & M<0, L<0 \\\\\n-{\\rm sgn}\\, M (1-4ML^{-1})^{-1\/2} & ML<0 \\\\\n-1 & M>0, L>0.\n\\end{array}\\right.\n\\end{equation}\n\nAccordingly, the extended position inner product space is\n$\\Pi_r={\\cal H}_r\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$ with $J$-inner product specified by\n$J=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}\\oplus (-{\\rm sgn}\\, M)$. The scalar component is the coefficient of\n$\\ket{\\eta}$ in ${\\cal M}_1$. For all generic cases (i.e., all cases\nother than $L=4M>0$) Theorem~\\ref{Thm:dom} entails that the domain\nof $h_{L,M}$ is\n\\begin{equation}\nD(h_{L,M}) = \\left\\{\\left(\\begin{array}{c} \\varphi \\\\ \\Phi\n\\end{array}\\right) \\mid \\varphi,\\varphi^\\prime\\in AC_{\\rm\nloc}(0,\\infty),~\\varphi,\\varphi^{\\prime\\prime}\\in L^2;\\quad \\Phi =\n-|M|^{1\/2}\\varphi(0)  \\right\\},\n\\label{eq:DhLM}\n\\end{equation}\nand that the action is\n\\begin{equation}\nh_{L,M} \\left(\\begin{array}{c} \\varphi \\\\ -|M|^{1\/2}\\varphi(0)\n\\end{array}\\right) =\n\\left(\\begin{array}{c}\n-\\varphi^{\\prime\\prime}\n\\\\ -{\\rm sgn}\\, M |M|^{-1\/2}(\\varphi^\\prime(0)+L^{-1}\\varphi(0))\n\\end{array}\\right).\n\\end{equation}\nMoreover, one may show that these equations also hold in the\nnon-generic case $L=4M>0$.\n\nIt is worth noting how this domain and action correspond to the\nscattering data~(\\ref{eq:ERlow}). Solving the equation\n$h_{L,M}(\\varphi,\\Phi)^T=k^2(\\varphi,\\Phi)^T$ for the generalised\neigenfunctions of $h_{L,M}$, we find\n$\\varphi(r)\\propto\\sin(kr+d(k))$ for some $d(k)$, and also obtain the\nrelation\n\\begin{equation}\n-{\\rm sgn}\\, M |M|^{-1\/2}(\\varphi^\\prime(0)+L^{-1}\\varphi(0))=\n-k^2|M|^{1\/2}\\varphi(0),\n\\end{equation}\nwhich entails that $k\\cot d(k) = \\varphi^\\prime(0)\/\\varphi(0) =\n-L^{-1}+k^2M$. Thus $d(k)$ is precisely the scattering data\n$\\delta_0(k)$.\n\nThe RPI models, which have scattering data $\\cot\\delta_0(k) =kM$ are\nobtained in the same way. The space ${\\cal M}_1$ is spanned by\n$\\psi_M(k) =(2|M|\/\\pi)^{1\/2}(1+(kM)^2)^{-1\/2}$, and the operator\n$M_1$ is found to be $M_1=-({\\rm sgn}\\, M)\\ket{\\psi_M}\\bra{\\psi_M}$. Thus\nthe inner product space is $\\Pi_r={\\cal H}_r\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$ with\n$J=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}\\oplus(-{\\rm sgn}\\, M)$. They have the domain~(\\ref{eq:DhLM}) and\naction\n\\begin{equation}\nh_{{\\rm rpi},M} \\left(\\begin{array}{c} \\varphi \\\\ -|M|^{1\/2}\\varphi(0)\n\\end{array}\\right) =\n\\left(\\begin{array}{c}\n-\\varphi^{\\prime\\prime}\n\\\\ -{\\rm sgn}\\, M |M|^{-1\/2}\\varphi^\\prime(0)\n\\end{array}\\right).\n\\end{equation}\n\nLet us consider the physical Hilbert space for these models. From\nSection~4, this is constructed by projecting out the hyperbolic\ninvariant subspace, and also those eigenfunctions with negative norm\nsquared (if present). The bound states of $h_{L,M}$ are clearly\nvectors of form $(\\xi_\\omega,|M|^{1\/2})^T$ with norm squared equal to\n$-(2{\\rm Re}\\,\\omega)^{-1}-M$, where $\\omega$ is a root of\n$\\omega^2+M^{-1}\\omega+(ML)^{-1}=0$. There are four cases to consider:\n\n{\\noindent \\em Case (i): $M<0$}. $\\Pi_r$ is positive definite so no\nprojection is required.\n\n{\\noindent \\em Case (ii): $M>0$, $L<0$}. There is a unique bound\nstate with\n\\begin{equation}\n\\omega=\\frac{1+(1-4M\/L)^{1\/2}}{-2M}\n\\label{eq:ome}\n\\end{equation}\nand negative norm squared. Projecting this state out, we obtain\n\\begin{equation}\n{\\cal H}_{\\rm phys} =\n\\left\\{\\left(\\begin{array}{c} \\varphi \\\\\nM^{-1\/2}\\inner{\\xi_\\omega}{\\varphi}\\end{array}\\right)\\mid\n\\varphi\\in{\\cal H}_r \\right\\}.\n\\label{eq:HP}\n\\end{equation}\n\n{\\noindent \\em Case (iii): $M>0$, $0<L<4M$}. There are two bound\nstates with complex conjugate eigenvalues. Accordingly, their\neigenfunctions span a hyperbolic invariant subspace. Projecting\nthis subspace out, we find\n\\begin{equation}\n{\\cal H}_{\\rm phys} =\n\\left\\{\\left(\\begin{array}{c} \\varphi \\\\\nM^{-1\/2}\\inner{\n\\frac{1}{2}(\\xi_\\omega+\\xi_{\\overline{\\omega}})}{\\varphi}\n\\end{array}\\right)\\mid\\varphi\\in{\\cal H}_r~{\\rm s.t.}~\n\\inner{(\\xi_\\omega-\\xi_{\\overline{\\omega}})}{\\varphi}=0 \\right\\},\n\\end{equation}\nwhere $\\omega$ is given by equation~(\\ref{eq:ome}).\n\n{\\noindent \\em Case (iv): $M>0$, $L>4M$}. There are two bound\nstates with real eigenvalues. However, only the state specified\nby~(\\ref{eq:ome}) has negative norm. Projecting this out, we arrive\nat the same expression for ${\\cal H}_{\\rm phys}$ as in case (ii).\n\nRPI models are covered by Case (i) for $M<0$, and have ${\\cal H}_{\\rm\nphys}$ given by~(\\ref{eq:HP}) for $M>0$, with $\\omega=-1\/M$.\n\nThe GPI Hamiltonian acts on ${\\cal H}_{\\rm phys}$ by restriction. For\nexample, in case (ii) above, we have\n\\begin{eqnarray}\nD(h_{L,M}|_{{\\cal H}_{\\rm phys}}) &=& \\left\\{\n\\left(\\begin{array}{c} \\varphi \\\\\nM^{-1\/2}\\inner{\\xi_\\omega}{\\varphi}\\end{array}\\right)\\mid\n\\varphi,\\varphi^\\prime\\in AC_{\\rm loc}(0,\\infty),~\n\\varphi,\\varphi^{\\prime\\prime}\\in L^2;\\right. \\nonumber \\\\\n&&\\qquad\\qquad\\qquad\\qquad\\quad\n\\left.\\begin{array}{c} \\ \\\\ \\ \\end{array}\nM\\varphi(0)=-\\inner{\\xi_\\omega}{\\varphi} \\right\\}\n\\end{eqnarray}\non which $h_{L,M}|_{{\\cal H}_{\\rm phys}}$ acts as before. The restricted\noperator has the same continuum spectrum as $h_{L,M}$, but has no\nbound states in this case. Moreover, the property of locality is\npartially lost: it is clear that vectors of form $(\\varphi,0)^T$\nwith $\\varphi\\in C_0^\\infty(0,\\infty)$ are in ${\\cal H}_{\\rm phys}$ only if\n$\\varphi\\perp\\xi_\\omega$. However, for elements of this form in\n${\\cal H}_{\\rm phys}$, it remains the case that $h_{L,M}|_{{\\cal H}_{\\rm\nphys}}(\\varphi,0)^T= (-\\varphi^{\\prime\\prime},0)^T$. Thus the\nproperties of locality and `positivity' are not entirely compatible.\n\n\n\\subsection{Physical Interpretation}\n\nIn this section, we discuss how the effective range models\nconstructed above may be used to model Schr\\\"{o}dinger operators\n$H=-\\triangle+V$, where $V$ is smooth, spherically symmetric and\ncompactly supported within radius $a$ of the origin. Our methodology\nextends that described in~\\cite{KF}, in which the scattering length\napproximation is discussed.\n\nGiven a smooth spherically symmetric potential $V(r)$ supported\nwithin radius $a$ of the origin, we may find the `best fit' GPI model\n$h_{L,M}$ as follows. Let $u_0$ be the $S$-wave zero energy\neigenfunction, i.e., the solution to $-u_0^{\\prime\\prime}+Vu_0=0$\nwith regular boundary conditions at the origin.  Then the arguments\nof Section 11.2 of \\cite{Newt} give the low energy parameters $L$ and\n$M$ as\n\\begin{equation}\nL= a - \\left.\\frac{u_0}{u_0^\\prime}\\right|_{r=a};\n\\label{eq:L}\n\\end{equation}\nand\n\\begin{equation}\nM = a\\left\\{ 1-\\frac{a}{L}\n+\\frac{1}{3}\\left(\\frac{a}{L}\\right)^{2}-\n\\left( 1-\\frac{a}{L}\\right)^2\n\\frac{\\int_0^a |u_0(r)|^2 dr}{a|u_0(a)|^2} \\right\\}.\n\\label{eq:M}\n\\end{equation}\nThus the scattering behaviour is $\\cot\\delta_0(k)=\n-(kL)^{-1}+kM+O(k^3)$ and the best fit GPI model in our class is\n$h_{L,M}$. We refer to equations~(\\ref{eq:L}) and~(\\ref{eq:M}) as\n{\\em fitting formulae}; equation~(\\ref{eq:L}) is the fitting formula\nemployed in \\cite{KF}. The range of energies for which the\napproximation is valid can be determined by a `believability'\nanalysis analogous to that described in \\cite{KF}. We will not do\nthis here.\n\nNote that $M$ obeys the bound\n\\begin{equation}\n-\\infty \\le M < a\\left\\{ 1-\\frac{a}{L}\n+\\frac{1}{3}\\left(\\frac{a}{L}\\right)^{2}\\right\\}.\n\\end{equation}\nMoreover, this bound is best possible: for any\n$L\\in{\\rm I\\! R}\\cup\\{\\infty\\}$ and any $M$ in the above range, one can\nclearly find a smooth function $u_0(r)$ satisfying regular boundary\nconditions at the origin, $u_0\\propto(1-r\/L)$ for $r>a$ and such\nthat~(\\ref{eq:M}) holds. Then the potential defined by\n$V(r)=u_0^{\\prime\\prime}(r)\/u_0(r)$ has $S$-wave scattering behaviour\napproximated to second order by $h_{L,M}$. The contribution to the\ntotal scattering cross section from the effective range term\ngenerally outweighs that from higher angular momenta, so the $S$-wave\nGPI model provides a second order approximation to the full\nscattering behaviour.\n\nFinally, we discuss the interpretation of the discrete spectrum of\n$h_{L,M}$. We have constructed $h_{L,M}$ so that its\nscattering behaviour matches that of a given Schr\\\"{o}dinger\noperator at low energies, $E$. For larger $|E|$, the approximation\nbreaks down -- in the language of \\cite{KF} we say that it is no\nlonger `believable'. Thus, deeply bound states are unlikely to be\nbelievable. In particular, for $0<L<4M$, $h_{L,M}$ exhibits a complex\nconjugate pair of eigenvalues, which can never be\nbelievable.\\footnote{These are not eigenvalues of $h_{L,M}$\nrestricted to the physical Hilbert space. However, they persist as\npoles in the scattering amplitude and our remarks still apply:\n$h_{L,M}$ does not give a reliable approximation to the scattering\ntheory at those scales.} Such phenomena are artifacts of the\nidealisation process, due to the truncation of the low energy\nexpansion. The issue of believability is discussed in \\cite{KF};\nsimilar comments are made in \\cite{Diejen}.\n\n\\sect{Conclusion}\n\nWe begin by discussing various generalisations of our method. There\nare many situations in which the analysis of Section~2.2 may be\napplied. In two dimensions, for example, one can consider radial GPI\nHamiltonians which agree with\n\\begin{equation}\nh = -\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr} + \\frac{\\nu^2}{r^2}\n\\end{equation}\naway from the origin, as models for an infinitesimal `dot' of\nmagnetic flux $\\nu$, with $|\\nu|<1$. In this case, one must employ\nHankel transforms rather than sine and cosine transforms. In \\cite{F}\nwe will implement this programme to construct a models of RPI type\nfor the Dirac equation in the presence of an infinitesimal tube of\nflux. These models provide the leading order approximation to the\nscattering data.\n\nOur method could also be applied to $S$-wave GPI models with a\nCoulombic tail. In this case, the appropriate integral transforms\nwould be based upon Whittaker functions and the scattering data\nwould be specified in terms of Coulomb-modified partial wave shifts.\nIn this case, the dimension of ${\\cal M}_2$ would be countably\ninfinite, due to the countable discrete spectrum of\nsuch models. However, one would expect ${\\cal M}_1$ to remain finite\ndimensional for simple models.\n\nSecondly, it is of interest to generalise the unitary dilation method\nto sectors of higher angular momentum with $\\ell\\ge 1$ (and the\ncorresponding analogues for magnetic flux dots -- i.e., $|\\nu|\\ge 1$\n-- and Coulombic GPI for $\\ell\\ge 1$). This is more problematic,\nbecause the radial Hamiltonian $-d^2\/dr^2+\\ell(\\ell+1)\/r^2$ is\nessentially self-adjoint on $C_0^\\infty(0,\\infty)$ and so the method\nof Section~2.2 does not apply. Here, it might be possible to  obtain\na suitable integral transform by analysing the distributional\nconstruction. We hope to return to this elsewhere.\n\nFinally, we consider applications to the definition of arrays of\npoint scatterers. Here, the most likely use of our methods is to\ngenerate the `monomer' by inverse scattering. By passing to the\nresolvent written in the form of Krein's formula, one can isolate\nthe appropriate `defect element' and proceed to form the array by\nmethods discussed in \\cite{Diejen}, which generalise the procedure\nfor arrays of PI developed in \\cite{Gmann}.\n\n{}To summarise, we have introduced an inverse scattering construction\nfor GPI models using the theory of unitary dilations, and developed\nthe method in detail for the class of single centre $S$-wave GPI\nmodels with rational $S$-matrices. A physical locality requirement\ncompletes the specification of the Hamiltonian, whose scattering,\nspectral and domain properties are explicitly determined from our\nresults.\n\n{\\em Acknowledgements:} The notions of fitting formulae and the\ngeneral methodology of Section~5.3 are due in origin to Bernard Kay\n\\cite{KF}; I thank him and also Graham Allan and Clive Wells for\nuseful conversations. In addition, I thank Churchill College,\nCambridge, for financial support under a Gateway Studentship.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSingle-particle ultrashort pulse X-ray scattering experiments are a promising new approach to the structure determination of biomolecules~\\cite{hajdu_single-molecule_2000, huldt_diffraction_2003, gaffney_imaging_2007, miao_beyond_2015}. \nWhereas most alternative approaches focus on nano-crystals~\\cite{chapman_femtosecond_2006, chapman_femtosecond_2011, boutet_high-resolution_2012, fromme_femtosecond_2011, kirian_femtosecond_2010, schlichting_serial_2015, roedig_high-speed_2017, barends_serial_2022}, X-ray scattering experiments on single molecules can in principle also provide structural information on molecular ensembles and, hence, biomolecular conformational and functional motions. \nIn such 'hit and destroy' experiments, a stream of single molecules is exposed to a beam of high intensity femtosecond X-ray free electron laser (XFEL) pulses (Figure~\\ref{fig: experiment}a), and for each hit the positions of the scattered photons (red dots) on the detector are recorded as a scattering image~\\cite{neutze_potential_2000}. \nImportantly, the ultra-short pulses serve to outrun the subsequent destruction of the particles due to radiation damage, but also\nimply that only very few photons are being recorded for each molecule~\\cite{gaffney_imaging_2007}.\n\nThe feasibility of the method has already been demonstrated by a number of experiments~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015, hosseinizadeh_high-resolution_2014}, but so far only structures of relatively large specimen have been successfully determined, for instance of entire mimivirus particles~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015}  ($450\\,\\text{nm}$ in diameter) and coliphage viruses~\\cite{hosseinizadeh_high-resolution_2014} ($20\\,\\text{nm}$ in diameter). \nWhereas for large specimens many photons are scattered per image, for example $10^7$ for the mimivirus~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015}, for typical proteins only 10-50 coherently scattered photons per image are expected~\\cite{yoon_comprehensive_2016, hantke_condor_2016}, which further complicates structure determination particularly for small molecules. \nSuch images can be obtained at with an intensity of $10^{12}$ photons per pulse at $5\\,\\mathrm{keV}$ and a $1\\,\\mathrm{\\text{\\textmu} m}$ beam diameter~\\cite{von_ardenne_structure_2018}, for example from the XFELs at DESY or SLAC.\n\nMost importantly, the orientation of the molecules at the time of scattering is typically unknown, which poses an additional and substantial refinement challenge.\nAlso, so far it is impossible to systematically include shot noise, incoherently scattered photons, background scattering, or detector noise in the structure refinement.\nThese issues are particularly challenging for the structure refinement of small specimen at high resolution, for example proteins or protein complexes at near atomic resolution. \nA number of methods have been proposed to address these issues, such as orientation determination methods~\\cite{shneerson_crystallography_2008,loh_reconstruction_2009, walczak_bayesian_2014,kassemeyer_optimal_2013,elser_three-dimensional_2011, tegze_atomic_2012, donatelli_reconstruction_2017, flamant_expansion-maximization-compression_2016, ayyer_dragonfly_2016} and manifold embedding algorithms~\\cite{fung_structure_2009,moths_bayesian_2011,schwander_symmetries_2012,giannakis_symmetries_2012}, which, however, typically require 100 to 1000 photons per scattering image.\nAs an alternative, correlation based approaches~\\cite{saldin_structure_2009, saldin_beyond_2010, saldin_new_2011, saldin_structure_2010,kurta_correlations_2017} require, in principle, as little as three photons per image~\\cite{von_ardenne_structure_2018}. \n\nFurther, many biomolecules show structural heterogeneity and conformational dynamics between different distinct structures, which, when resolved, would provide a direct view on biomolecular function. Hence, and similar to the current main challenge in cryogenic electron microscopy~\\cite{guaita_recent_2022}, not one but many structures need to be extracted from the scattering images. Accordingly, in addition to the orientation, also the current conformer for each scattering image is unknown. \nWhereas both orientation determination methods and manifold-based methods have been applied to determine multiple conformational structures~\\cite{zhuang_unsupervised_2022,hosseinizadeh_conformational_2017-1}, the required large number of photons per image precludes their application to single biomolecules.\n\nTo overcome the above issues, we developed and assessed a rigorous Bayesian method for structure determination from single-molecule scattering images. \nWe will demonstrate that this method can not only determine a single structure at high resolution but also determine structural ensembles.\nUnexpectedly, much fewer images are required for refining an ensemble of $n$ conformers consisting of $m$ atoms each than for refining a single structure consisting of $m\\times n$ atoms, which may bring applications to non-synthetic experimental data into reach.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[height = 0.3\\textwidth]{setup.pdf}\n    \\caption{Single molecule scattering experiment. \\textbf{a} A stream of single molecules is hit by femtosecond X-ray pulses, and the scattered photons are recorded as images (image reproduced from von Ardenne et al.~\\cite{von_ardenne_structure_2018}). \\textbf{b} The scattered photons (red dots) are distributed on the Ewald sphere according to the 3D-intensity function $I$ (blue).}\n    \\label{fig: experiment}\n\\end{figure}\n\n\n\\newpage\n\n\\section{Results}\n\n\\paragraph{Summary of the approach.}\n\nFor each scattering image, the positions of the recorded photons specify vectors $\\vec k_1, \\dots, \\vec k_l$ on the Ewald sphere in Fourier space (Figure~\\ref{fig: experiment}b). The probability of observing a photon at a particular position on the detector is proportional to the 3D intensity function $I(\\vec k) \\propto |\\mathcal F\\{\\vec R\\rho_i\\}(\\vec k)|^2$ at the corresponding position $\\vec k$ on the Ewald sphere, which in turn is given by the Fourier transform of the electron density $\\rho_i$ of conformer $i$. Here, $\\vec R$ is the unknown orientation of the molecule for this particular image.\n\nIt follows that the probability of observing an image with photon positions $\\vec k_1, \\dots, \\vec k_l$ is obtained by averaging over both the conformational ensemble $\\boldsymbol\\uprho = \\{\\rho_1, \\dots, \\rho_n\\}$ with weights $\\vec w = \\{w_1, \\dots, w_n\\}$ and all orientations $\\vec R$. Because each scattering image is an independent event, the total probability of the set of all images $\\mathcal I$ reads\n\\begin{equation}\n    P(\\mathcal I \\,|\\, \\boldsymbol\\uprho, \\vec w) \\propto \\prod_{(\\vec k_1, \\dots, \\vec k_l) \\in \\mathcal I} \\sum_{i=1}^n w_i\\int_{\\mathrm{SO}(3)}P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\vec R\\rho_i)\\, \\mathrm d \\vec R\\,.\n    \\label{eq: likelihood}\n\\end{equation}\nThis probability serves to determine either a single structure or a structural ensemble by sampling from the Bayesian posterior probability $P(\\boldsymbol\\uprho, \\vec w \\,|\\, \\mathcal I) \\propto P(\\mathcal I \\,|\\, \\boldsymbol\\uprho, \\vec w)P(\\boldsymbol\\uprho, \\vec w)$ using a Markov chain Monte Carlo approach. For the prior $P(\\boldsymbol\\uprho, \\vec w)$ the orientations are assumed to be uniformly distributed. To minimize the number of required degrees of freedom, and as a means of regularization, we chose a physically motivated representation of each $\\rho_i$ in terms of a sum of Gaussian functions, which also completes the definition of the prior.\n\n\nFor typical cases, the number of required degrees of freedom remains large and poses a formidable sampling challenge. \nTo address this issue, we adopted a hierarchical simulated annealing approach. \nStarting at very low resolution, the macromolecular structures were sampled in multiple hierarchical stages of increasing resolution. To increase the sampling efficiency, in each of these stages, for each Markov step the previous ensemble of structures of maximal posterior probability was used as a proposal density.\nTo this end, the scattering images that would have been observed for a smoothed low resolution copy of the original molecule were obtained from the original images by rejection sampling using the convolution theorem (see the Methods section). \n\nFurther, we adapted the Bayesian formalism such that it only uses those images which contain new information, that is, that contain photons for which the magnitude of $\\vec k$ is larger than the threshold of the resolution from the previous stage.\nWith increasing resolution, the fraction of such useful images becomes very small, thus markedly enhancing computational efficiency up to two orders of magnitude. The approach is described in detail in the Supplementary Information.\n\n\\paragraph{Sample test refinements.}\nBecause our Bayesian approach uses all available information, it we expect it to require fewer scattering images to achieve a certain resolution than, for example, correlation based methods.\nTo assess this aspect, we first tested our method on the single structure level, using the same 46-residue protein crambin~\\cite{jelsch_accurate_2000} as in the study of von Ardenne et al.~\\cite{von_ardenne_structure_2018}.\nA total of $10^8$ noise-free synthetic images were generated, containing a realistic average of $15$ photons each. \nFrom these, the structure was solved in five hierarchical stages (Fig.~\\ref{fig: crambin}a), increasing the number of degrees of freedom by a factor of two in each stage. \nFor the final stage, a representation consisting of $184$ Gaussian functions was used, which is four times the number of residues. For more details see Supplementary Note~\\ref{sup: parameters}. \nIndeed, a similar Fourier shell correlation resolution~\\cite{van_heel_fourier_2005} of $4.2\\,${\\AA} (Fig.~\\ref{fig: crambin}b) was obtained as for the previous correlation based method, using only half of the total number of scattered photons.\n\nNext, to demonstrate that our method can resolve ensembles of multiple conformers, we used three molecular dynamics trajectories of alanine dipeptide~\\cite{scherer_pyemma_2015} of length $250\\,\\text{ns}$ each to generate $10^6$ scattering images, using a randomly chosen snapshot for each image. \nAs before, an average of $15$ photons per image were generated. \nThen, a weighted ensemble of eight conformers was determined from these images (Fig.~\\ref{fig: dipeptide}), each described by a sum of $10$ Gaussian functions. \nTo obtain sufficient statistics, a total of $10$ independent simulated annealing runs were carried out, using the same images.\n\nTo assess the quality of the obtained ensemble, for each of the eight structures the resolution with respect to its nearest neighbor in the input trajectories was calculated using Fourier shell correlations~\\cite{van_heel_fourier_2005} (Fig.~\\ref{fig: dipeptide}b), resulting in a weighted average resolution of $1.6\\,\\text{{\\AA}}$. \nThis result shows that the obtained eight structures are indeed close to the reference ensemble. \nTo also assess the accuracy of the entire ensemble, for each time step in the input trajectories, the resolution with respect to its nearest neighbor among all the determined structures was calculated (Fig.~\\ref{fig: dipeptide}d). \nAs a main result we found that $90\\%$ of the input trajectories are within $2.1\\,${\\AA} Fourier shell correlation resolution of the determined structures, and that all of the trajectory frames are within $2.5\\,${\\AA} resolution of the determined structures. \nFigure~\\ref{fig: dipeptide}e compares the $10$ obtained ensembles with the reference ensemble using a Ramachandran plot~\\cite{ramachandran_stereochemistry_1963} showing the distribution of the torsion angles $\\phi$ and $\\psi$. \nFor each of the determined structures its nearest neighbor in the input trajectories was used to compute these angles. \nAs can be seen, the reference density is well represented by the determined structures. \n\nNext, we asked if our method is also capable of extracting structural ensembles for the larger mini-protein chignolin~\\cite{honda_crystal_2008}, comprising $10$ residues. \nTo that end, $50$ molecular dynamics trajectories of length $10\\,\\text{\\textmu s}$ were used to generate $1.2\\cdot10^7$ images with, on average, $15$ photons each.\nAs a further challenge, this ensemble also contained unfolded structures. \nFrom the obtained images, we determined multiple stages of weighted structural ensembles of increasing resolution and increasing number of conformers (Fig.~\\ref{fig: 5awl}a). \nAs above, resolutions were computed using Fourier shell correlations (Fig.~\\ref{fig: 5awl}b,c), finding a weighted average resolution of $5\\,${\\AA} for the folded conformers, and $9-10\\,${\\AA} for the unfolded conformers.\nInterestingly, in the final stage one of the six determined weights is nearly zero, suggesting that five conformers suffice for the used number of images at this resolution level. \nIt is also remarkable that the $9\\%$ fraction of unfolded states in the reference structure ensemble was correctly identified.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{crambin.pdf}\n    \\caption{Structure determination of Crambin. \\textbf{a} Hierarchical stages of retrieved electron densities. \\textbf{b} Fourier shell correlation between the retrieved densities and the reference density. \\textbf{c} Retrieved electron density. \\textbf{d} Reference electron density.}\n    \\label{fig: crambin}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{dipeptide_weighted_v2.pdf}\n    \\caption{Structural ensemble determination of the alanine dipeptide. \\textbf{a} Reconstructed conformers (green), the corresponding weights, and the nearest neighbors in the input trajectories (blue) with the corresponding resolutions. \\textbf{b} Fourier shell correlations used to compute these resolutions. \\textbf{c} Weighted resolution distribution for $10$ independent runs from the the same data. \\textbf{d} Resolution distribution over the time steps of the input trajectories relative to their nearest neighbors among the determined structures from all $10$ runs. \\textbf{e} Ramachandran plot for the input trajectories (shown as a density) and the determined structures from all $10$ runs (points, the color indicates the different runs). }\n    \\label{fig: dipeptide}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{5awl_tree.pdf}\n    \\caption{Structural ensemble determination for chignolin. \\textbf{a} Hierarchical stages of retrieved structures (green) and their nearest neighbors (blue) in the input trajectories with the corresponding resolutions. \\textbf{b} Fourier shell correlations of the reconstructed structures relative to their nearest neighbors. \\textbf{c} Resolution distribution over the time steps of the input trajectories relative to their nearest neighbors among the determined structures. }\n    \\label{fig: 5awl}\n\\end{figure}\n\n\n\n\n\\paragraph{Scaling.}\nIn the sample applications described above we observed that resolving $n$ conformational structures consisting of $m$ residues each required much fewer scattering images and photons than resolving a single $n\\times m$ residue structure of the same total size and complexity --- even in cases where the members of the former ensemble are very different from each other.\nTo investigate this counterintuitive observation in more detail, small `structures' consisting of randomly placed Gaussian functions were used.  \nFor each combination of parameters, eight independent structure determination runs were performed, and for each run the achieved resolution was determined. \nThe structure weights $w_i$ where chosen to be uniform and kept fixed during the simulated annealing runs.\n\nFigure~\\ref{fig: scaling}c and~\\ref{fig: scaling}f show for each combination of parameters the smallest number of images for which all of the replicas achieved a resolution better than a given threshold. \nAs can be seen in Fig.~\\ref{fig: scaling}c, for the structure ensemble of $n$ conformations with $m$ residues each, the required number of images is approximately proportional to $n^2$, that is, the square of the number of conformations. \nWhereas somewhat unexpected, this finding is in line with a theoretical argument showing that the information content of a single image is in this case proportional to $1\/n^2$ (Supplementary Note \\ref{sup: information}). \nIn contrast, the number of images required to resolve a single structure of $n\\times m$ residues grows even much faster, resembling a power law $m^c$ with an exponent $c \\approx 5$ (Fig.~\\ref{fig: scaling}f). \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{scaling.pdf}\n    \\caption{Dependence of the resolution on the number of images, the number of conformations, and the size of the structure. \\textbf{a} Resolution as a function of the number of images for various numbers of conformations $n$. \\textbf{b} Resolution as a function of the number of conformations for various numbers of images. \\textbf{c} Required number of images to achieve various resolutions as a function of the number of conformations. For comparison, a quadratic relationship is shown (dashed line). \\textbf{d} Resolution as a function of the number of images for various structure sizes (parameterized by the number of Gaussians $m$). \\textbf{e} Resolution as a function of the number of Gaussians for various numbers of images. \\textbf{f} Required number of images to achieve various resolutions as a function of the number of Gaussians. For comparison, a power law $m^5$ is shown (dashed line). }\n    \\label{fig: scaling}\n   \n\\end{figure}\n\n\n\\clearpage\n\\section{Discussion}\nHere we have developed a rigorous Bayesian method for determining biomolecular structures from single molecule X-ray scattering images in the extreme few photon Poisson regime. \nUsing synthetic scattering images generated from simulated X-ray scattering experiments, we have demonstrated that both single structures as well as structural ensembles of small biomolecules can be resolved to near atomic resolution. \n\nOur results for the globular protein crambin show that a similar resolution of $4.2\\,${\\AA} is obtained compared to previous correlation based methods~\\cite{von_ardenne_structure_2018} that require similarly small numbers of photons per scattering image. Because such correlation based methods disregard higher correlations, whereas the full information content of each image is used in our Bayesian approach, we speculated that the latter should require correspondingly fewer images. \nThis was indeed observed for the above protein, for which the number of images required to obtain a resolution of approximately $4.2\\,${\\AA} was reduced from roughly $2\\cdot 10^8$ to $1\\cdot 10^8$. \n\nOur method should be applicable also to much larger molecules, the main bottleneck currently being computational cost. \nHowever, we expect that the factor by which our method requires fewer photons than correlation based methods will increase with the size of the molecule, due to the larger number of photons expected per image.\nFurther, we think that the computational limitations can be overcome by improved optimization or sampling methods or by utilizing prior structural information, either from structure databases, from AlphaFold~\\cite{jumper_highly_2021}, or guided by molecular dynamics force fields.\n\nFor alanine dipeptide the full conformational ensemble generated by an atomistic simulation was extracted at atomistic resolution of on average $1.8\\,${\\AA} from simulated scattering experiments, in which not only the current orientation of the biomolecule but also its current conformer was unknown.\nFor the 10 amino acid protein chignolin~\\cite{honda_crystal_2008} both the folded and unfolded ensembles were resolved, albeit so far at lower resolution. \nNotably, also the weights corresponding to the folded and unfolded conformers were recovered accurately. \nFurthermore, using weighted ensembles allows the number of conformers to be determined dynamically, as some of the determined weights may be zero.\n\nUnexpectedly few images were required to resolve structural ensembles.\nBecause an ensemble of $n$ conformers consisting of $m$ atoms each has the same number of degrees of freedom as a single structure of $n\\times m$ atoms, a similar number of images should be required. \nHowever, closer analysis suggests that roughly $O(m^5)$ images are required to a resolve a single structure with $m$ atoms. One might therefore expect that $O(n^5m^5)$ images are required for an ensemble of $n$ such structures. \nHowever, our analysis suggests that only $O(n^2m^5)$ images are required for this, which is consistent with a theoretical argument that the expected information content of a single scattering image is in this case proportional to $1\/n^2$.\nThis suggests that determining more complex conformational ensembles may be within reach, and that it may be possible to collect the required amount data from experiments. \nOur Bayesian analysis of structural heterogeneity is similar in spirit to approaches that were successfully applied in cryo-electron microscopy~\\cite{cossio_bayesian_2013,grant_cistem_2018,punjani_cryosparc_2017,lyumkis_likelihood-based_2013,scheres_chapter_2016,tang_eman2_2007,zivanov_new_2018,kinman_uncovering_2022}, which, from a mathematical standpoint, shares some similarities with single molecule X-ray scattering, albeit at a much lower noise level. \n\nFrom a more general perspective, our Bayesian approach represents a systematic and rigorous approach to include shot noise in the extreme Poisson regime characteristic for single molecule X-ray scattering experiments. \nIn contrast to other proposed methods, this Bayesian framework will also allow to include other sources of noise and uncertainty in a conceptually straightforward manner, such as incoherently scattered photons, background scattering, detector noise, or scattering by disordered water at the biomolecular surface. \nPrerequisite is the development and calibration of forward noise models such as those documented in, e.g., Ref.~\\cite{yoon_comprehensive_2016}. \n\n\n\\section{Methods}\n\n\\paragraph{Structure and structure ensemble representation.}\n\\label{sec: representation}\nElectron density functions of the reference structures or conformers were described by a sum of $m$ of Gaussian functions with atomic positions $\\vec y_i$, heights $h_i$ and standard deviations $\\sigma_i$,\n\\begin{equation}\\label{eq: representation}\n    \\rho(\\vec r) = \\sum_{i=1}^m \\frac{h_i}{\\left(\\sigma_i\\sqrt{2\\pi}\\right)^3} \\exp \\left(\\frac1{2\\sigma_i^2}\\lVert \\vec r - \\vec y_i\\rVert^2\\right)\\,.\n\\end{equation}\nElectron density functions of the determined structures were described similarly, with one common height $h = h_i$ and one common standard deviation $\\sigma=\\sigma_i$, which is treated as an unknown and determined together with the positions $\\vec y_i$.\nStructural ensembles were represented by a weighted sum of conformers $\\boldsymbol\\uprho = \\{\\rho_1, \\dots, \\rho_n\\}$ with weights $\\vec w = \\{w_1, \\dots, w_n\\}$.\n\n\n\\paragraph{Synthetic data generation.}\n\\label{sec: datagen}\n\nFor each of the synthetic scattering images, the photon positions on the detector $D$ were drawn from a probability distribution proportional to the intensity function $I(\\vec k) = |F\\{\\rho\\}(\\vec k)|^2$ restricted to the appropriate Ewald sphere.\nSpecifically, generation of each image involved the following steps:\n\\begin{enumerate}\n    \\item A conformation of the molecule is selected randomly from the reference ensemble (for example, consisting of molecular dynamics trajectories),\n    \\item a random orientation $\\vec R$ of the molecule is drawn uniformly from the rotation group $\\mathrm{SO}(3)$,\n    \\item the number of scattered photons is drawn from a Poisson distribution with mean $N\\int_D I(\\vec R\\vec k) \\,\\mathrm d \\vec k$, where $N$ is the incoming beam intensity,\n    \\item the position of each scattered photon is drawn from the probability distribution proportional to $(I\\circ \\vec R)|_D$.\n\\end{enumerate}\nThe last two steps were implemented using rejection sampling. To this end, a von Mises-Fisher distribution $p$ on $D$ was chosen with high enough standard deviation that $I(\\vec k) \\leq p(\\vec k)$ everywhere. Then, for each photon, its position $\\vec k$ was drawn from $p$ and it was accepted with probability $I(R\\vec k) \/ p(R\\vec k)$. The beam intensity $N$ was chosen together with a normalization of $\\rho$ such that this procedure accurately produces a Poisson distribution of the desired expected number of photons per image.\n\n\n\\paragraph{Computation of likelihoods.}\n\\label{sec: distribution}\nThe probability density of observing an image defined by photon positions $\\vec k_1, \\dots, \\vec k_l$ given an electron density function $\\rho$ with is corresponding intensity function $I(\\vec k)$ was computed by averaging over all possible orientations $\\vec R\\in \\mathrm{SO}(3)$ of the molecule, \n\\begin{equation}\n\\label{eq: probability density}\n    \\begin{aligned}\n        P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\rho\n        &= \\frac{N^l}{l!}\\int_{\\mathrm{SO}(3)} \\exp\\left(-N\\!\\int_D I(\\vec R\\vec k) \\,\\mathrm d \\vec k\\right) \\left(\\prod_{i=1}^l I(\\vec R\\vec k_i)\\right) \\,\\mathrm d \\vec R, \\\\\n       \n    \\end{aligned}\n\\end{equation}\nwhere for each orientation, the probability is a product of the Poisson distribution for the number of photons $l$ in the image and a factor depending on the photon positions. \nThese integrals were approximated by averaging over a discrete set of typically $r\\approx 10^3$ to $r\\approx 10^5$ rotations $\\vec R_i$ with weights $s_i$,\n\\begin{equation}\n    P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\rho) \\approx \\frac{N^l}{l!}\\sum_{i=1}^r s_i\\exp\\left(-N\\!\\int_D I(\\vec R_i\\vec k) \\,\\mathrm d \\vec k\\right) \\prod_{j=1}^l I(\\vec R_i\\vec k_j).\n\\end{equation}\nThe rotations $\\vec R_i$ and their weights $s_i$ are constructed by combining a Lebedev quadrature rule on $S^2$ with a uniform quadrature rule on $S^1$ via the Hopf map~\\cite{noauthor_sphere_lebedev_rule_nodate,graf_sampling_2009} (Supplementary Note~\\ref{sup: computation}).\n\n\\paragraph{Simulated annealing and hierarchical sampling.}\n\\label{sec: annealing}\nA Monte Carlo simulated annealing approach with the energy function $-\\log P$ was used to sample from or maximize the Bayesian posterior probability, as described in detail in the Supplement.\nTo enhance convergence, Bayesian sampling and maximization were performed in multiple hierarchical resolution stages. \nStarting from a low resolution representation of $\\rho$ with correspondingly few degrees of freedom, the number of Gaussian functions was doubled in each stage and the reduced resolution structure determined by the previous stage was used as a proposal density (see Supplement). \nTo calculate likelihoods for the reduced resolution structures, lower resolution scattering images were generated from the original images by rejection sampling, that is, by removing each photon in the original images with probability $1 - \\exp(-\\sigma^2k^2\/2)$.\nBy construction, this rejection scheme samples from a Fourier transformed density $I_\\rho \\cdot \\exp(-\\sigma^2|\\vec k|^2\/2)$ which, by the convolution theorem, corresponds to a smoothed real space density $\\tilde\\rho = \\rho*\\mathcal N(\\sigma)$ obtained as the convolution of $\\rho$ with a Gaussian kernel with width (resolution) $\\sigma$. \nComputational efficiency was further increased substantially by selecting only those original images for the likelihood computations that actually contain useful information at the respective resolution. As described in the Supplement, the Bayesian formalism allows for removing this selection bias.\n\n\n\\paragraph{Structure alignment and resolution estimate.}\nBecause the orientations of the obtained structures are irrelevant, these were rotationally aligned to each other\nby minimizing the cost function\n\\begin{equation}\n    d(\\vec S) = \\frac1n \\sum_{i=1}^n \\min_{j=1}^m\\, \\lVert\\vec y_i - \\vec S \\vec y_j'\\rVert + \\frac1m \\sum_{j=1}^m \\min_{i=1}^n\\, \\lVert\\vec y_i - \\vec S \\vec y_j'\\rVert.\n\\end{equation}\nHere, the positions $\\vec y_1, \\dots, \\vec y_n$ and $\\vec y_1', \\dots, \\vec y_m'$ define two structures per equation \\eqref{eq: representation} and $\\vec S$ is a rotation matrix $\\vec S \\in \\mathrm{O}(3)$. Both rotations and reflections were included, as X-ray scattering images do not distinguish between mirror images. \n\nThe resolution of the aligned structures was estimated using Fourier shell correlations~\\cite{van_heel_fourier_2005},\n\\begin{equation}\n    \\mathrm{FSC}(k) = \\frac{\\int_{\\lVert\\vec k\\rVert = k} \\hat\\rho_1(\\vec k)^*\\hat\\rho_2(\\vec k)\\,\\mathrm d\\vec k}{\\sqrt{\\int_{\\lVert\\vec k\\rVert = k} |\\hat\\rho_1(\\vec k)|^2\\,\\mathrm d\\vec k}\\sqrt{\\int_{\\lVert\\vec k\\rVert = k} |\\hat\\rho_2(\\vec k)|^2\\,\\mathrm d\\vec k}}\\,,\n\\end{equation}\nwhere $\\rho_1$ are $\\rho_2$ the structures to be compared and $\\hat\\rho$ denotes the Fourier transform of $\\rho$. Accordingly, the achieved resolution was determined as $2\\pi\/k_\\mathrm{fsc}$, where $k_\\mathrm{fsc}$ is the threshold at which the Fourier shell correlation drops below $1\/2$, providing a conservative estimate~\\cite{van_heel_fourier_2005}.\n\n\\paragraph{Molecular dynamics simulations.}\nAll atomistic simulation trajectories were generated using the GROMACS 2018 software package~\\cite{abraham_gromacs_2015} with the Charmm36mm force field~\\cite{huang_charmm36m_2017} and the OPC water model~\\cite{izadi_building_2014}. \nFor chignolin, the starting structure was taken from the Protein Data Bank~\\cite{berman_protein_2000}, entry 5AWL~\\cite{honda_crystal_2008}.\nAll hydrogen atoms were described by virtual sites~\\cite{feenstra_improving_1999}.\nEach protein was placed within a triclinic water box, such that the smallest distance between protein surface and box boundary was larger than $1.5\\,$nm. \nSodium and chloride ions were added to neutralize the system, corresponding to a physiological concentration of $150\\,$mmol\/l. \nEnergy minimization was performed using steepest descent for $5\\cdot 10^4$ steps.\nEach system was subsequently equilibrated for $0.5\\,$ns in the $NVT$ ensemble, and subsequently for $1.0\\,$ns in the $NPT$ ensemble at $1\\,$atm pressure and temperature $300\\,$K using an integration time step of $2\\,$fs. \nThe velocity rescaling thermostat~\\cite{bussi_canonical_2007} and Parrinello-Rahman pressure coupling~\\cite{parrinello_polymorphic_1981} were used with coupling coefficients of $\\tau=0.1\\,$ps and $\\tau=1\\,$ps, respectively.\nAll bond lengths of the solute were constrained using the LINCS algorithm~\\cite{hess_lincs_1997} with an expansion order of 6, and the geometry of the water molecules was constrained using the SETTLE algorithm~\\cite{miyamoto_settle_1992}. \nElectrostatic interactions were calculated using PME~\\cite{darden_particle_1993}, with a real space cutoff of $10\\,${\\AA} and a Fourier spacing of $1.2\\,${\\AA}.\nFor all production runs, a $4\\,$fs integration was used, and the atom coordinates were saved every $100\\,$ps, such that $10^5$ snapshots were available for each trajectory.\nThe trajectories for alanine dipeptide were taken from mdshare \\cite{noauthor_mdshare_nodate-1}. The structure for crambin was taken from PDB entry 1EJG~\\cite{jelsch_accurate_2000}.\n\n\\section{Acknowledgments}\nThis work was financially supported by the Federal Ministry of Education and Research through the joint research project 05K20EGA Fluctuation XFEL, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - CRC 1456\/1 - 432680300. The MD-trajectories were kindly provided by Nicolai Kozlowski. \n\n\n\\printbibliography\n\n\n\n\\clearpage\n\\begin{appendix}\n\n\\section{Introduction}\nSingle-particle ultrashort pulse X-ray scattering experiments are a promising new approach to the structure determination of biomolecules~\\cite{hajdu_single-molecule_2000, huldt_diffraction_2003, gaffney_imaging_2007, miao_beyond_2015}. \nWhereas most alternative approaches focus on nano-crystals~\\cite{chapman_femtosecond_2006, chapman_femtosecond_2011, boutet_high-resolution_2012, fromme_femtosecond_2011, kirian_femtosecond_2010, schlichting_serial_2015, roedig_high-speed_2017, barends_serial_2022}, X-ray scattering experiments on single molecules can in principle also provide structural information on molecular ensembles and, hence, biomolecular conformational and functional motions. \nIn such 'hit and destroy' experiments, a stream of single molecules is exposed to a beam of high intensity femtosecond X-ray free electron laser (XFEL) pulses (Figure~\\ref{fig: experiment}a), and for each hit the positions of the scattered photons (red dots) on the detector are recorded as a scattering image~\\cite{neutze_potential_2000}. \nImportantly, the ultra-short pulses serve to outrun the subsequent destruction of the particles due to radiation damage, but also\nimply that only very few photons are being recorded for each molecule~\\cite{gaffney_imaging_2007}.\n\nThe feasibility of the method has already been demonstrated by a number of experiments~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015, hosseinizadeh_high-resolution_2014}, but so far only structures of relatively large specimen have been successfully determined, for instance of entire mimivirus particles~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015}  ($450\\,\\text{nm}$ in diameter) and coliphage viruses~\\cite{hosseinizadeh_high-resolution_2014} ($20\\,\\text{nm}$ in diameter). \nWhereas for large specimens many photons are scattered per image, for example $10^7$ for the mimivirus~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015}, for typical proteins only 10-50 coherently scattered photons per image are expected~\\cite{yoon_comprehensive_2016, hantke_condor_2016}, which further complicates structure determination particularly for small molecules. \nSuch images can be obtained at with an intensity of $10^{12}$ photons per pulse at $5\\,\\mathrm{keV}$ and a $1\\,\\mathrm{\\text{\\textmu} m}$ beam diameter~\\cite{von_ardenne_structure_2018}, for example from the XFELs at DESY or SLAC.\n\nMost importantly, the orientation of the molecules at the time of scattering is typically unknown, which poses an additional and substantial refinement challenge.\nAlso, so far it is impossible to systematically include shot noise, incoherently scattered photons, background scattering, or detector noise in the structure refinement.\nThese issues are particularly challenging for the structure refinement of small specimen at high resolution, for example proteins or protein complexes at near atomic resolution. \nA number of methods have been proposed to address these issues, such as orientation determination methods~\\cite{shneerson_crystallography_2008,loh_reconstruction_2009, walczak_bayesian_2014,kassemeyer_optimal_2013,elser_three-dimensional_2011, tegze_atomic_2012, donatelli_reconstruction_2017, flamant_expansion-maximization-compression_2016, ayyer_dragonfly_2016} and manifold embedding algorithms~\\cite{fung_structure_2009,moths_bayesian_2011,schwander_symmetries_2012,giannakis_symmetries_2012}, which, however, typically require 100 to 1000 photons per scattering image.\nAs an alternative, correlation based approaches~\\cite{saldin_structure_2009, saldin_beyond_2010, saldin_new_2011, saldin_structure_2010,kurta_correlations_2017} require, in principle, as little as three photons per image~\\cite{von_ardenne_structure_2018}. \n\nFurther, many biomolecules show structural heterogeneity and conformational dynamics between different distinct structures, which, when resolved, would provide a direct view on biomolecular function. Hence, and similar to the current main challenge in cryogenic electron microscopy~\\cite{guaita_recent_2022}, not one but many structures need to be extracted from the scattering images. Accordingly, in addition to the orientation, also the current conformer for each scattering image is unknown. \nWhereas both orientation determination methods and manifold-based methods have been applied to determine multiple conformational structures~\\cite{zhuang_unsupervised_2022,hosseinizadeh_conformational_2017-1}, the required large number of photons per image precludes their application to single biomolecules.\n\nTo overcome the above issues, we developed and assessed a rigorous Bayesian method for structure determination from single-molecule scattering images. \nWe will demonstrate that this method can not only determine a single structure at high resolution but also determine structural ensembles.\nUnexpectedly, much fewer images are required for refining an ensemble of $n$ conformers consisting of $m$ atoms each than for refining a single structure consisting of $m\\times n$ atoms, which may bring applications to non-synthetic experimental data into reach.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[height = 0.3\\textwidth]{setup.pdf}\n    \\caption{Single molecule scattering experiment. \\textbf{a} A stream of single molecules is hit by femtosecond X-ray pulses, and the scattered photons are recorded as images (image reproduced from von Ardenne et al.~\\cite{von_ardenne_structure_2018}). \\textbf{b} The scattered photons (red dots) are distributed on the Ewald sphere according to the 3D-intensity function $I$ (blue).}\n    \\label{fig: experiment}\n\\end{figure}\n\n\n\\newpage\n\n\\section{Results}\n\n\\paragraph{Summary of the approach.}\n\nFor each scattering image, the positions of the recorded photons specify vectors $\\vec k_1, \\dots, \\vec k_l$ on the Ewald sphere in Fourier space (Figure~\\ref{fig: experiment}b). The probability of observing a photon at a particular position on the detector is proportional to the 3D intensity function $I(\\vec k) \\propto |\\mathcal F\\{\\vec R\\rho_i\\}(\\vec k)|^2$ at the corresponding position $\\vec k$ on the Ewald sphere, which in turn is given by the Fourier transform of the electron density $\\rho_i$ of conformer $i$. Here, $\\vec R$ is the unknown orientation of the molecule for this particular image.\n\nIt follows that the probability of observing an image with photon positions $\\vec k_1, \\dots, \\vec k_l$ is obtained by averaging over both the conformational ensemble $\\boldsymbol\\uprho = \\{\\rho_1, \\dots, \\rho_n\\}$ with weights $\\vec w = \\{w_1, \\dots, w_n\\}$ and all orientations $\\vec R$. Because each scattering image is an independent event, the total probability of the set of all images $\\mathcal I$ reads\n\\begin{equation}\n    P(\\mathcal I \\,|\\, \\boldsymbol\\uprho, \\vec w) \\propto \\prod_{(\\vec k_1, \\dots, \\vec k_l) \\in \\mathcal I} \\sum_{i=1}^n w_i\\int_{\\mathrm{SO}(3)}P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\vec R\\rho_i)\\, \\mathrm d \\vec R\\,.\n    \\label{eq: likelihood}\n\\end{equation}\nThis probability serves to determine either a single structure or a structural ensemble by sampling from the Bayesian posterior probability $P(\\boldsymbol\\uprho, \\vec w \\,|\\, \\mathcal I) \\propto P(\\mathcal I \\,|\\, \\boldsymbol\\uprho, \\vec w)P(\\boldsymbol\\uprho, \\vec w)$ using a Markov chain Monte Carlo approach. For the prior $P(\\boldsymbol\\uprho, \\vec w)$ the orientations are assumed to be uniformly distributed. To minimize the number of required degrees of freedom, and as a means of regularization, we chose a physically motivated representation of each $\\rho_i$ in terms of a sum of Gaussian functions, which also completes the definition of the prior.\n\n\nFor typical cases, the number of required degrees of freedom remains large and poses a formidable sampling challenge. \nTo address this issue, we adopted a hierarchical simulated annealing approach. \nStarting at very low resolution, the macromolecular structures were sampled in multiple hierarchical stages of increasing resolution. To increase the sampling efficiency, in each of these stages, for each Markov step the previous ensemble of structures of maximal posterior probability was used as a proposal density.\nTo this end, the scattering images that would have been observed for a smoothed low resolution copy of the original molecule were obtained from the original images by rejection sampling using the convolution theorem (see the Methods section). \n\nFurther, we adapted the Bayesian formalism such that it only uses those images which contain new information, that is, that contain photons for which the magnitude of $\\vec k$ is larger than the threshold of the resolution from the previous stage.\nWith increasing resolution, the fraction of such useful images becomes very small, thus markedly enhancing computational efficiency up to two orders of magnitude. The approach is described in detail in the Supplementary Information.\n\n\\paragraph{Sample test refinements.}\nBecause our Bayesian approach uses all available information, it we expect it to require fewer scattering images to achieve a certain resolution than, for example, correlation based methods.\nTo assess this aspect, we first tested our method on the single structure level, using the same 46-residue protein crambin~\\cite{jelsch_accurate_2000} as in the study of von Ardenne et al.~\\cite{von_ardenne_structure_2018}.\nA total of $10^8$ noise-free synthetic images were generated, containing a realistic average of $15$ photons each. \nFrom these, the structure was solved in five hierarchical stages (Fig.~\\ref{fig: crambin}a), increasing the number of degrees of freedom by a factor of two in each stage. \nFor the final stage, a representation consisting of $184$ Gaussian functions was used, which is four times the number of residues. For more details see Supplementary Note~\\ref{sup: parameters}. \nIndeed, a similar Fourier shell correlation resolution~\\cite{van_heel_fourier_2005} of $4.2\\,${\\AA} (Fig.~\\ref{fig: crambin}b) was obtained as for the previous correlation based method, using only half of the total number of scattered photons.\n\nNext, to demonstrate that our method can resolve ensembles of multiple conformers, we used three molecular dynamics trajectories of alanine dipeptide~\\cite{scherer_pyemma_2015} of length $250\\,\\text{ns}$ each to generate $10^6$ scattering images, using a randomly chosen snapshot for each image. \nAs before, an average of $15$ photons per image were generated. \nThen, a weighted ensemble of eight conformers was determined from these images (Fig.~\\ref{fig: dipeptide}), each described by a sum of $10$ Gaussian functions. \nTo obtain sufficient statistics, a total of $10$ independent simulated annealing runs were carried out, using the same images.\n\nTo assess the quality of the obtained ensemble, for each of the eight structures the resolution with respect to its nearest neighbor in the input trajectories was calculated using Fourier shell correlations~\\cite{van_heel_fourier_2005} (Fig.~\\ref{fig: dipeptide}b), resulting in a weighted average resolution of $1.6\\,\\text{{\\AA}}$. \nThis result shows that the obtained eight structures are indeed close to the reference ensemble. \nTo also assess the accuracy of the entire ensemble, for each time step in the input trajectories, the resolution with respect to its nearest neighbor among all the determined structures was calculated (Fig.~\\ref{fig: dipeptide}d). \nAs a main result we found that $90\\%$ of the input trajectories are within $2.1\\,${\\AA} Fourier shell correlation resolution of the determined structures, and that all of the trajectory frames are within $2.5\\,${\\AA} resolution of the determined structures. \nFigure~\\ref{fig: dipeptide}e compares the $10$ obtained ensembles with the reference ensemble using a Ramachandran plot~\\cite{ramachandran_stereochemistry_1963} showing the distribution of the torsion angles $\\phi$ and $\\psi$. \nFor each of the determined structures its nearest neighbor in the input trajectories was used to compute these angles. \nAs can be seen, the reference density is well represented by the determined structures. \n\nNext, we asked if our method is also capable of extracting structural ensembles for the larger mini-protein chignolin~\\cite{honda_crystal_2008}, comprising $10$ residues. \nTo that end, $50$ molecular dynamics trajectories of length $10\\,\\text{\\textmu s}$ were used to generate $1.2\\cdot10^7$ images with, on average, $15$ photons each.\nAs a further challenge, this ensemble also contained unfolded structures. \nFrom the obtained images, we determined multiple stages of weighted structural ensembles of increasing resolution and increasing number of conformers (Fig.~\\ref{fig: 5awl}a). \nAs above, resolutions were computed using Fourier shell correlations (Fig.~\\ref{fig: 5awl}b,c), finding a weighted average resolution of $5\\,${\\AA} for the folded conformers, and $9-10\\,${\\AA} for the unfolded conformers.\nInterestingly, in the final stage one of the six determined weights is nearly zero, suggesting that five conformers suffice for the used number of images at this resolution level. \nIt is also remarkable that the $9\\%$ fraction of unfolded states in the reference structure ensemble was correctly identified.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{crambin.pdf}\n    \\caption{Structure determination of Crambin. \\textbf{a} Hierarchical stages of retrieved electron densities. \\textbf{b} Fourier shell correlation between the retrieved densities and the reference density. \\textbf{c} Retrieved electron density. \\textbf{d} Reference electron density.}\n    \\label{fig: crambin}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{dipeptide_weighted_v2.pdf}\n    \\caption{Structural ensemble determination of the alanine dipeptide. \\textbf{a} Reconstructed conformers (green), the corresponding weights, and the nearest neighbors in the input trajectories (blue) with the corresponding resolutions. \\textbf{b} Fourier shell correlations used to compute these resolutions. \\textbf{c} Weighted resolution distribution for $10$ independent runs from the the same data. \\textbf{d} Resolution distribution over the time steps of the input trajectories relative to their nearest neighbors among the determined structures from all $10$ runs. \\textbf{e} Ramachandran plot for the input trajectories (shown as a density) and the determined structures from all $10$ runs (points, the color indicates the different runs). }\n    \\label{fig: dipeptide}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{5awl_tree.pdf}\n    \\caption{Structural ensemble determination for chignolin. \\textbf{a} Hierarchical stages of retrieved structures (green) and their nearest neighbors (blue) in the input trajectories with the corresponding resolutions. \\textbf{b} Fourier shell correlations of the reconstructed structures relative to their nearest neighbors. \\textbf{c} Resolution distribution over the time steps of the input trajectories relative to their nearest neighbors among the determined structures. }\n    \\label{fig: 5awl}\n\\end{figure}\n\n\n\n\n\\paragraph{Scaling.}\nIn the sample applications described above we observed that resolving $n$ conformational structures consisting of $m$ residues each required much fewer scattering images and photons than resolving a single $n\\times m$ residue structure of the same total size and complexity --- even in cases where the members of the former ensemble are very different from each other.\nTo investigate this counterintuitive observation in more detail, small `structures' consisting of randomly placed Gaussian functions were used.  \nFor each combination of parameters, eight independent structure determination runs were performed, and for each run the achieved resolution was determined. \nThe structure weights $w_i$ where chosen to be uniform and kept fixed during the simulated annealing runs.\n\nFigure~\\ref{fig: scaling}c and~\\ref{fig: scaling}f show for each combination of parameters the smallest number of images for which all of the replicas achieved a resolution better than a given threshold. \nAs can be seen in Fig.~\\ref{fig: scaling}c, for the structure ensemble of $n$ conformations with $m$ residues each, the required number of images is approximately proportional to $n^2$, that is, the square of the number of conformations. \nWhereas somewhat unexpected, this finding is in line with a theoretical argument showing that the information content of a single image is in this case proportional to $1\/n^2$ (Supplementary Note \\ref{sup: information}). \nIn contrast, the number of images required to resolve a single structure of $n\\times m$ residues grows even much faster, resembling a power law $m^c$ with an exponent $c \\approx 5$ (Fig.~\\ref{fig: scaling}f). \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{scaling.pdf}\n    \\caption{Dependence of the resolution on the number of images, the number of conformations, and the size of the structure. \\textbf{a} Resolution as a function of the number of images for various numbers of conformations $n$. \\textbf{b} Resolution as a function of the number of conformations for various numbers of images. \\textbf{c} Required number of images to achieve various resolutions as a function of the number of conformations. For comparison, a quadratic relationship is shown (dashed line). \\textbf{d} Resolution as a function of the number of images for various structure sizes (parameterized by the number of Gaussians $m$). \\textbf{e} Resolution as a function of the number of Gaussians for various numbers of images. \\textbf{f} Required number of images to achieve various resolutions as a function of the number of Gaussians. For comparison, a power law $m^5$ is shown (dashed line). }\n    \\label{fig: scaling}\n   \n\\end{figure}\n\n\n\\clearpage\n\\section{Discussion}\nHere we have developed a rigorous Bayesian method for determining biomolecular structures from single molecule X-ray scattering images in the extreme few photon Poisson regime. \nUsing synthetic scattering images generated from simulated X-ray scattering experiments, we have demonstrated that both single structures as well as structural ensembles of small biomolecules can be resolved to near atomic resolution. \n\nOur results for the globular protein crambin show that a similar resolution of $4.2\\,${\\AA} is obtained compared to previous correlation based methods~\\cite{von_ardenne_structure_2018} that require similarly small numbers of photons per scattering image. Because such correlation based methods disregard higher correlations, whereas the full information content of each image is used in our Bayesian approach, we speculated that the latter should require correspondingly fewer images. \nThis was indeed observed for the above protein, for which the number of images required to obtain a resolution of approximately $4.2\\,${\\AA} was reduced from roughly $2\\cdot 10^8$ to $1\\cdot 10^8$. \n\nOur method should be applicable also to much larger molecules, the main bottleneck currently being computational cost. \nHowever, we expect that the factor by which our method requires fewer photons than correlation based methods will increase with the size of the molecule, due to the larger number of photons expected per image.\nFurther, we think that the computational limitations can be overcome by improved optimization or sampling methods or by utilizing prior structural information, either from structure databases, from AlphaFold~\\cite{jumper_highly_2021}, or guided by molecular dynamics force fields.\n\nFor alanine dipeptide the full conformational ensemble generated by an atomistic simulation was extracted at atomistic resolution of on average $1.8\\,${\\AA} from simulated scattering experiments, in which not only the current orientation of the biomolecule but also its current conformer was unknown.\nFor the 10 amino acid protein chignolin~\\cite{honda_crystal_2008} both the folded and unfolded ensembles were resolved, albeit so far at lower resolution. \nNotably, also the weights corresponding to the folded and unfolded conformers were recovered accurately. \nFurthermore, using weighted ensembles allows the number of conformers to be determined dynamically, as some of the determined weights may be zero.\n\nUnexpectedly few images were required to resolve structural ensembles.\nBecause an ensemble of $n$ conformers consisting of $m$ atoms each has the same number of degrees of freedom as a single structure of $n\\times m$ atoms, a similar number of images should be required. \nHowever, closer analysis suggests that roughly $O(m^5)$ images are required to a resolve a single structure with $m$ atoms. One might therefore expect that $O(n^5m^5)$ images are required for an ensemble of $n$ such structures. \nHowever, our analysis suggests that only $O(n^2m^5)$ images are required for this, which is consistent with a theoretical argument that the expected information content of a single scattering image is in this case proportional to $1\/n^2$.\nThis suggests that determining more complex conformational ensembles may be within reach, and that it may be possible to collect the required amount data from experiments. \nOur Bayesian analysis of structural heterogeneity is similar in spirit to approaches that were successfully applied in cryo-electron microscopy~\\cite{cossio_bayesian_2013,grant_cistem_2018,punjani_cryosparc_2017,lyumkis_likelihood-based_2013,scheres_chapter_2016,tang_eman2_2007,zivanov_new_2018,kinman_uncovering_2022}, which, from a mathematical standpoint, shares some similarities with single molecule X-ray scattering, albeit at a much lower noise level. \n\nFrom a more general perspective, our Bayesian approach represents a systematic and rigorous approach to include shot noise in the extreme Poisson regime characteristic for single molecule X-ray scattering experiments. \nIn contrast to other proposed methods, this Bayesian framework will also allow to include other sources of noise and uncertainty in a conceptually straightforward manner, such as incoherently scattered photons, background scattering, detector noise, or scattering by disordered water at the biomolecular surface. \nPrerequisite is the development and calibration of forward noise models such as those documented in, e.g., Ref.~\\cite{yoon_comprehensive_2016}. \n\n\n\\section{Methods}\n\n\\paragraph{Structure and structure ensemble representation.}\n\\label{sec: representation}\nElectron density functions of the reference structures or conformers were described by a sum of $m$ of Gaussian functions with atomic positions $\\vec y_i$, heights $h_i$ and standard deviations $\\sigma_i$,\n\\begin{equation}\\label{eq: representation}\n    \\rho(\\vec r) = \\sum_{i=1}^m \\frac{h_i}{\\left(\\sigma_i\\sqrt{2\\pi}\\right)^3} \\exp \\left(\\frac1{2\\sigma_i^2}\\lVert \\vec r - \\vec y_i\\rVert^2\\right)\\,.\n\\end{equation}\nElectron density functions of the determined structures were described similarly, with one common height $h = h_i$ and one common standard deviation $\\sigma=\\sigma_i$, which is treated as an unknown and determined together with the positions $\\vec y_i$.\nStructural ensembles were represented by a weighted sum of conformers $\\boldsymbol\\uprho = \\{\\rho_1, \\dots, \\rho_n\\}$ with weights $\\vec w = \\{w_1, \\dots, w_n\\}$.\n\n\n\\paragraph{Synthetic data generation.}\n\\label{sec: datagen}\n\nFor each of the synthetic scattering images, the photon positions on the detector $D$ were drawn from a probability distribution proportional to the intensity function $I(\\vec k) = |F\\{\\rho\\}(\\vec k)|^2$ restricted to the appropriate Ewald sphere.\nSpecifically, generation of each image involved the following steps:\n\\begin{enumerate}\n    \\item A conformation of the molecule is selected randomly from the reference ensemble (for example, consisting of molecular dynamics trajectories),\n    \\item a random orientation $\\vec R$ of the molecule is drawn uniformly from the rotation group $\\mathrm{SO}(3)$,\n    \\item the number of scattered photons is drawn from a Poisson distribution with mean $N\\int_D I(\\vec R\\vec k) \\,\\mathrm d \\vec k$, where $N$ is the incoming beam intensity,\n    \\item the position of each scattered photon is drawn from the probability distribution proportional to $(I\\circ \\vec R)|_D$.\n\\end{enumerate}\nThe last two steps were implemented using rejection sampling. To this end, a von Mises-Fisher distribution $p$ on $D$ was chosen with high enough standard deviation that $I(\\vec k) \\leq p(\\vec k)$ everywhere. Then, for each photon, its position $\\vec k$ was drawn from $p$ and it was accepted with probability $I(R\\vec k) \/ p(R\\vec k)$. The beam intensity $N$ was chosen together with a normalization of $\\rho$ such that this procedure accurately produces a Poisson distribution of the desired expected number of photons per image.\n\n\n\\paragraph{Computation of likelihoods.}\n\\label{sec: distribution}\nThe probability density of observing an image defined by photon positions $\\vec k_1, \\dots, \\vec k_l$ given an electron density function $\\rho$ with is corresponding intensity function $I(\\vec k)$ was computed by averaging over all possible orientations $\\vec R\\in \\mathrm{SO}(3)$ of the molecule, \n\\begin{equation}\n\\label{eq: probability density}\n    \\begin{aligned}\n        P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\rho\n        &= \\frac{N^l}{l!}\\int_{\\mathrm{SO}(3)} \\exp\\left(-N\\!\\int_D I(\\vec R\\vec k) \\,\\mathrm d \\vec k\\right) \\left(\\prod_{i=1}^l I(\\vec R\\vec k_i)\\right) \\,\\mathrm d \\vec R, \\\\\n       \n    \\end{aligned}\n\\end{equation}\nwhere for each orientation, the probability is a product of the Poisson distribution for the number of photons $l$ in the image and a factor depending on the photon positions. \nThese integrals were approximated by averaging over a discrete set of typically $r\\approx 10^3$ to $r\\approx 10^5$ rotations $\\vec R_i$ with weights $s_i$,\n\\begin{equation}\n    P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\rho) \\approx \\frac{N^l}{l!}\\sum_{i=1}^r s_i\\exp\\left(-N\\!\\int_D I(\\vec R_i\\vec k) \\,\\mathrm d \\vec k\\right) \\prod_{j=1}^l I(\\vec R_i\\vec k_j).\n\\end{equation}\nThe rotations $\\vec R_i$ and their weights $s_i$ are constructed by combining a Lebedev quadrature rule on $S^2$ with a uniform quadrature rule on $S^1$ via the Hopf map~\\cite{noauthor_sphere_lebedev_rule_nodate,graf_sampling_2009} (Supplementary Note~\\ref{sup: computation}).\n\n\\paragraph{Simulated annealing and hierarchical sampling.}\n\\label{sec: annealing}\nA Monte Carlo simulated annealing approach with the energy function $-\\log P$ was used to sample from or maximize the Bayesian posterior probability, as described in detail in the Supplement.\nTo enhance convergence, Bayesian sampling and maximization were performed in multiple hierarchical resolution stages. \nStarting from a low resolution representation of $\\rho$ with correspondingly few degrees of freedom, the number of Gaussian functions was doubled in each stage and the reduced resolution structure determined by the previous stage was used as a proposal density (see Supplement). \nTo calculate likelihoods for the reduced resolution structures, lower resolution scattering images were generated from the original images by rejection sampling, that is, by removing each photon in the original images with probability $1 - \\exp(-\\sigma^2k^2\/2)$.\nBy construction, this rejection scheme samples from a Fourier transformed density $I_\\rho \\cdot \\exp(-\\sigma^2|\\vec k|^2\/2)$ which, by the convolution theorem, corresponds to a smoothed real space density $\\tilde\\rho = \\rho*\\mathcal N(\\sigma)$ obtained as the convolution of $\\rho$ with a Gaussian kernel with width (resolution) $\\sigma$. \nComputational efficiency was further increased substantially by selecting only those original images for the likelihood computations that actually contain useful information at the respective resolution. As described in the Supplement, the Bayesian formalism allows for removing this selection bias.\n\n\n\\paragraph{Structure alignment and resolution estimate.}\nBecause the orientations of the obtained structures are irrelevant, these were rotationally aligned to each other\nby minimizing the cost function\n\\begin{equation}\n    d(\\vec S) = \\frac1n \\sum_{i=1}^n \\min_{j=1}^m\\, \\lVert\\vec y_i - \\vec S \\vec y_j'\\rVert + \\frac1m \\sum_{j=1}^m \\min_{i=1}^n\\, \\lVert\\vec y_i - \\vec S \\vec y_j'\\rVert.\n\\end{equation}\nHere, the positions $\\vec y_1, \\dots, \\vec y_n$ and $\\vec y_1', \\dots, \\vec y_m'$ define two structures per equation \\eqref{eq: representation} and $\\vec S$ is a rotation matrix $\\vec S \\in \\mathrm{O}(3)$. Both rotations and reflections were included, as X-ray scattering images do not distinguish between mirror images. \n\nThe resolution of the aligned structures was estimated using Fourier shell correlations~\\cite{van_heel_fourier_2005},\n\\begin{equation}\n    \\mathrm{FSC}(k) = \\frac{\\int_{\\lVert\\vec k\\rVert = k} \\hat\\rho_1(\\vec k)^*\\hat\\rho_2(\\vec k)\\,\\mathrm d\\vec k}{\\sqrt{\\int_{\\lVert\\vec k\\rVert = k} |\\hat\\rho_1(\\vec k)|^2\\,\\mathrm d\\vec k}\\sqrt{\\int_{\\lVert\\vec k\\rVert = k} |\\hat\\rho_2(\\vec k)|^2\\,\\mathrm d\\vec k}}\\,,\n\\end{equation}\nwhere $\\rho_1$ are $\\rho_2$ the structures to be compared and $\\hat\\rho$ denotes the Fourier transform of $\\rho$. Accordingly, the achieved resolution was determined as $2\\pi\/k_\\mathrm{fsc}$, where $k_\\mathrm{fsc}$ is the threshold at which the Fourier shell correlation drops below $1\/2$, providing a conservative estimate~\\cite{van_heel_fourier_2005}.\n\n\\paragraph{Molecular dynamics simulations.}\nAll atomistic simulation trajectories were generated using the GROMACS 2018 software package~\\cite{abraham_gromacs_2015} with the Charmm36mm force field~\\cite{huang_charmm36m_2017} and the OPC water model~\\cite{izadi_building_2014}. \nFor chignolin, the starting structure was taken from the Protein Data Bank~\\cite{berman_protein_2000}, entry 5AWL~\\cite{honda_crystal_2008}.\nAll hydrogen atoms were described by virtual sites~\\cite{feenstra_improving_1999}.\nEach protein was placed within a triclinic water box, such that the smallest distance between protein surface and box boundary was larger than $1.5\\,$nm. \nSodium and chloride ions were added to neutralize the system, corresponding to a physiological concentration of $150\\,$mmol\/l. \nEnergy minimization was performed using steepest descent for $5\\cdot 10^4$ steps.\nEach system was subsequently equilibrated for $0.5\\,$ns in the $NVT$ ensemble, and subsequently for $1.0\\,$ns in the $NPT$ ensemble at $1\\,$atm pressure and temperature $300\\,$K using an integration time step of $2\\,$fs. \nThe velocity rescaling thermostat~\\cite{bussi_canonical_2007} and Parrinello-Rahman pressure coupling~\\cite{parrinello_polymorphic_1981} were used with coupling coefficients of $\\tau=0.1\\,$ps and $\\tau=1\\,$ps, respectively.\nAll bond lengths of the solute were constrained using the LINCS algorithm~\\cite{hess_lincs_1997} with an expansion order of 6, and the geometry of the water molecules was constrained using the SETTLE algorithm~\\cite{miyamoto_settle_1992}. \nElectrostatic interactions were calculated using PME~\\cite{darden_particle_1993}, with a real space cutoff of $10\\,${\\AA} and a Fourier spacing of $1.2\\,${\\AA}.\nFor all production runs, a $4\\,$fs integration was used, and the atom coordinates were saved every $100\\,$ps, such that $10^5$ snapshots were available for each trajectory.\nThe trajectories for alanine dipeptide were taken from mdshare \\cite{noauthor_mdshare_nodate-1}. The structure for crambin was taken from PDB entry 1EJG~\\cite{jelsch_accurate_2000}.\n\n\\section{Acknowledgments}\nThis work was financially supported by the Federal Ministry of Education and Research through the joint research project 05K20EGA Fluctuation XFEL, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - CRC 1456\/1 - 432680300. The MD-trajectories were kindly provided by Nicolai Kozlowski. \n\n\n\\printbibliography\n\n\n\n\\clearpage\n\\begin{appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{section: Introduction}\n\nBy $\\mathcal H$ we will always denote an undirected graph without multiple edges (and by abuse of notation also denote its set of vertices). In this paper we focus on $\\mathcal H$ which is four-cycle free, that is, it is finite, it has no self-loops and the four-cycle, denoted by $C_4$ is not a subgraph of $\\mathcal H$. Fix $d\\geq 2$. The basic object of study is $X_\\mathcal H$, the space of graph homomorphisms from $\\mathbb{Z}^d$ to $\\mathcal H$. Here by $\\mathbb{Z}^d$ we will mean both the group and its standard Cayley graph.\n\nSuch a space of configurations $X_\\mathcal H$, referred to as a hom-shift, can be obtained by forbidding certain patterns on edges of $\\mathcal H^{\\mathbb{Z}^d}$. If $\\mathcal H$ is a finite graph then $X_\\mathcal H$ is a nearest neighbour shift of finite type. In addition it is also `isotropic' and `symmetric', that is, given vertices $a, b \\in \\mathcal H$ if $a$ is not allowed to sit next to $b$ in $X_\\mathcal H$ for some coordinate direction, then $a$ is not allowed to sit next to $b$ in all coordinate directions. Most of the concepts related to shift spaces are introduced in Section \\ref{section: Hom-shifts}.\n\nRelated to a shift space $X$ is its topological entropy denoted by $h_{top}(X)$ which measures the growth rate of the number of patterns allowed in $X$ with the size of the underlying shape (usually rectangular). For a given shift space, its computation is a very difficult task (look for instance in \\cite{pavlovhardsquare2012} and the references within). We will focus on a different aspect: as in \\cite{covensmital}, a shift space $X$ is called entropy minimal if for all shift spaces $Y\\subsetneq X$, $h_{top}(Y)<h_{top}(X)$. Thus if a shift space has zero entropy and is entropy minimal then it is a topologically minimal system.\n\nFor $d=1$, it is well known that all irreducible nearest neighbour shifts of finite type are entropy minimal \\cite{LM}. However not much is known about it in higher dimensions: Shift spaces with a strong mixing property called uniform filling are entropy minimal \\cite{Schraudner2010minimal}, however nearest neighbour shifts of finite type with weaker mixing properties like block-gluing may not be entropy minimal \\cite{boyle2010multidimensional}. If $\\mathcal H$ is a four-cycle free graph then $X_\\mathcal H$ is not even block-gluing (we do not prove this but is implied by our results). There has been some recent work \\cite{lightwoodschraudnerentropy} which describes some conditions which are equivalent to entropy minimality for shifts of finite type. Our first main result is Theorem \\ref{theorem:four cycle free entropy minimal} which states that $X_\\mathcal H$ is entropy minimal for all connected four-cycle free graphs $\\mathcal H$. Our approach to this seemingly combinatorial question will be via thermodynamic formalism, using measures of maximal entropy, more generally `adapted' Markov random fields. They are defined and described in Section \\ref{section:thermodynamic formalism}.\n\nOur second main result is with regard to the pivot property (Section \\ref{section: the pivot property}). A shift space $X$ is said to have the pivot property if for all distinct configurations $x,y\\in X$ which differ at finitely many sites, there exists a sequence $x=x^1, x^2, \\ldots, x^n=y\\in X$ such that successive configurations $x^i, x^{i+1}$ differ exactly at a single site. We will prove that for four-cycle free graphs $\\mathcal H$, $X_{\\mathcal H}$ has the pivot property (Theorem \\ref{theorem: pivot property for four cycle free}). Many properties similar to the pivot property have appeared in the literature (often by the name local-move connectedness) \\cite{brightwell2000gibbs,dominolocal2010,ordentlich2014data,Shiffieldribbon2002}. For instance consider the following problem: Let $G$ be a finite undirected graph without multiple edges and self-loops. Given two graph homomorphisms $x,y$ from $G$ to $H$, can we find graph homomorphisms $x=x^1, x^2, \\ldots, x^n=y$ from $G$ to $H$ such that successive configurations $x^i, x^{i+1}$ differ exactly at a single site? Such a problem is called a homomorphism reconfiguration problem. If $\\mathcal H$ is four-cycle free it was recently shown in \\cite{Marcinfourcyclefree2014} that the homomorphism reconfiguration problem is solvable in time polynomial in the size of the graph $\\mathcal H$. Our interest in such problems comes from the connections between the pivot property and the study of Markov and Gibbs cocycles \\cite{chandgotia2013Markov}.\n\nA critical part of our proofs for both Theorem \\ref{theorem:four cycle free entropy minimal} and \\ref{theorem: pivot property for four cycle free} depends on the identification of the associated height functions (defined in Section \\ref{Section:heights}). Some of these ideas come from \\cite{chandgotia2013Markov}: Let $C_n$ denote the cycle with vertices $0, 1, \\ldots, n-1$. If $n \\neq 1, 2, 4$ then it was proven that $X_{C_n}$ has the pivot property. Further, Lemma 6.7 in \\cite{chandgotia2013Markov} implied that it is entropy minimal as well. We give a brief description of the latter: Given a configuration $x\\in X_{C_n}$ it was proved that there exists a corresponding height function $h_x \\in X_{\\mathbb{Z}}$ such that $h_x\\mod n= x$. (for $n=3$ also look at \\cite{schmidt_cohomology_SFT_1995}) Further given any ergodic measure $\\mu$ (assuming `adaptedness' cf. Section \\ref{section:thermodynamic formalism}) on $X_{C_n}$ it was shown that a well-defined notion of slope (bounded by $-1$ and $1$) exists which measures the average rate of the increase of the height for every direction. If the slope is maximal in any direction it was proven that $\\mu$ is frozen and otherwise it was shown that it is fully supported; frozen meaning that if $x, y\\in supp(\\mu)$ differ only at finitely many sites then $x=y$. From standard results in thermodynamic formalism it follows that $X_{C_n}$ is entropy minimal.\n\nThese ideas do not immediately translate to four-cycle free graphs. For this we will develop a different notion of `heights': Fix a connected four-cycle free graph $\\mathcal H$. For all $u \\in \\mathcal H$ let $D_\\mathcal H(u)$ denote the ball (for the graph distance) of radius $1$ around $u$. As in algebraic topology, $C$ is a covering space of $\\mathcal H$ if there is a graph homomorphism (called the covering map) $f: C\\longrightarrow \\mathcal H$ such that for all $u \\in \\mathcal H$, $f^{-1}(D_{\\mathcal H}(u))$ is a disjoint union of a constant number of subgraphs of $C$ isomorphic to $D_{\\mathcal H}(u)$ via $f$. Given a graph $\\mathcal H$, its universal cover (denoted by $E_\\mathcal H$) is the unique covering space of $\\mathcal H$ which is a tree. (Section \\ref{section:universal covers} and \\cite{Angluin80}) Denote by $\\pi:E_\\mathcal H\\longrightarrow \\mathcal H$ the corresponding covering map. $\\pi$ induces a map from $X_{E_\\mathcal H}$ to $X_\\mathcal H$ (also denoted by $\\pi$). It is not difficult to see that $\\pi(X_{E_\\mathcal H})\\subset X_\\mathcal H$; we will prove that the map is surjective and that the preimage of a configuration in $X_\\mathcal H$ is unique once we fix the `lift' in $X_{E_\\mathcal H}$ at any vertex of $\\mathbb{Z}^d$. Thus for every $x\\in X_\\mathcal H$ we can associate $\\tilde x \\in X_{E_\\mathcal H}$ such that $\\pi(\\tilde x)= x$ and a `height' function $h_x: \\mathbb{Z}^d\\times \\mathbb{Z}^d \\longrightarrow \\mathbb{Z}$ such that $h_x(\\vec i , \\vec j)$ is the graph distance between $\\tilde x_{\\vec{i}}$ and $\\tilde x_{\\vec{j}}$. From here on the steps for the proofs of Theorems \\ref{theorem:four cycle free entropy minimal} and \\ref{theorem: pivot property for four cycle free} are similar to the steps for the corresponding proofs in \\cite{chandgotia2013Markov}, but the proofs for the individual steps are quite different because unlike in \\cite{chandgotia2013Markov} the height functions are not additive but subadditive, meaning\n$$h_x({\\vec{i}},{\\vec{j}})\\leq h_x({\\vec{i}}, \\vec k)+h_x(\\vec k, {\\vec{j}}).$$\n\nTo streamline the proofs, we use graph folding \\cite{nowwinkler}; much of this is discussed in Section \\ref{section:Folding, Entropy Minimality and the Pivot Property}.\n\n\n\n\n\\section{Shifts of Finite Type and Hom-Shifts} \\label{section: Hom-shifts}\nIn this paper $\\mathcal H$ will always denote an undirected graph without multiple edges and single isolated vertices. For such a graph we will denote the adjacency relation by $\\sim_{\\mathcal H}$ and the set of vertices of $\\mathcal H$ by $\\mathcal H$ (abusing notation). We identify $\\mathbb{Z}^d$ with the set of vertices of the Cayley graph with respect to the standard generators $\\vec e_1,\\vec e_2, \\ldots, \\vec e_d$, that is, $\\vec{i}\\sim_{\\mathbb{Z}^d}\\vec{j}$ if and only if $\\|\\vec{i}-\\vec{j}\\|_1=1$ where $\\|\\cdot\\|_1$ is the $l^1$ norm. We drop the subscript in $\\sim_{\\mathcal H}$ when $\\mathcal H=\\mathbb{Z}^d$. Let $D_n$ and $B_n$ denote the $\\mathbb{Z}^d$-balls of radius $n$ around $\\vec{0}$ in the $l^1$ and the $l^\\infty$ norm respectively. The graph $C_n$ will denote the $n$-cycle where the set of vertices is $\\{0, 1, 2, \\ldots, n-1\\}$ and $i\\sim_{C_n} j$ if and only if $i \\equiv j \\pm 1\\!\\!\\mod n$. The graph $K_n$ will denote the complete graph with $n$ vertices where the set of vertices is $\\{1, 2, \\ldots, n\\}$ and $i \\sim_{K_n} j$ if and only if $i \\neq j$.\n\nLet ${\\mathcal A}$ be a finite \\emph{alphabet} (with the discrete topology) and ${\\mathcal A}^{\\mathbb{Z}^d}$ be given the product topology, making it compact. We will refer to elements $x\\in {\\mathcal A}^{\\mathbb{Z}^d}$ as \\emph{configurations}, denoting them by $(x_{\\vec{i}})_{\\vec{i}\\in \\mathbb{Z}^d}$ where $x_{\\vec{i}}$ is the value of $x$ at $\\vec{i}$. For all $\\vec{i}\\in \\mathbb{Z}^d$ the map $\\sigma^{\\vec{i}}:{\\mathcal A}^{\\mathbb{Z}^d}\\longrightarrow {\\mathcal A}^{\\mathbb{Z}^d}$ given by\n$$(\\sigma^{\\vec{i}}(x))_{\\vec{j}}:=x_{\\vec{i}+\\vec{j}}$$\nis called the \\emph{shift map} and defines a $\\mathbb{Z}^d$-action on ${\\mathcal A}^{\\mathbb{Z}^d}$. Closed subsets of ${\\mathcal A}^{\\mathbb{Z}^d}$ which are invariant under the shift maps are called \\emph{shift spaces}. A \\emph{sliding block code} from a shift space $X$ to a shift space $Y$ is a continuous map $f: X\\longrightarrow Y$ which commutes with the shifts, that is, $\\sigma^{{\\vec{i}}}\\circ f= f\\circ \\sigma^{\\vec{i}}$ for all ${\\vec{i}}\\in \\mathbb{Z}^d$. A surjective sliding block code is called a \\emph{factor map} and a bijective sliding block code is called a \\emph{conjugacy}. We note that a conjugacy defines an equivalence relation; in fact, it has a continuous inverse since it is a continuous bijection between compact sets.\n\nThere is an alternate description of shift spaces using forbidden patterns: A \\emph{pattern} is an element of ${\\mathcal A}^F$ for some finite set $F\\subset \\mathbb{Z}^d$. Given a pattern $a\\in {\\mathcal A}^F$, we will often denote both the pattern and its corresponding cylinder set by $[a]_F$ or by $[x]_F$ when $x|_F=a$. For any set $F\\subset \\mathbb{Z}^d$ and $v\\in {\\mathcal A}$, $x|_F=v$ will mean that $x_{\\vec{i}}=v$ for all ${\\vec{i}}\\in F$. For a set of (forbidden) patterns ${\\mathcal F}$ define\n$$X_{\\mathcal F}:=\\{x\\in {\\mathcal A}^{\\mathbb{Z}^d}\\:\\Big{|}\\: \\sigma^{\\vec{i}}(x)|_F\\notin {\\mathcal F} \\text{ for all }F\\subset \\mathbb{Z}^d\\text{ and }{\\vec{i}} \\in \\mathbb{Z}^d\\}.$$\n\nIt can be proved that a subset $X\\subset {\\mathcal A}^{\\mathbb{Z}^d}$ is a shift space if and only if there exists a set of forbidden patterns ${\\mathcal F}$ such that $X_{\\mathcal F}= X$. Note that given two distinct sets of patterns ${\\mathcal F}_1, {\\mathcal F}_2$ it is possible that $X_{{\\mathcal F}_1}=X_{{\\mathcal F}_2}$. A \\emph{shift of finite type} is a shift space $X$ such that $X=X_{{\\mathcal F}}$ for some finite set ${\\mathcal F}$. A \\emph{nearest neighbour shift of finite type} is a shift of finite type $X$ such that $X=X_{{\\mathcal F}}$ for some set ${\\mathcal F}$ consisting of patterns on single edges and vertices of $\\mathbb{Z}^d$. It follows from a simple recoding argument that any shift of finite type is conjugate to a nearest neighbour shift of finite type \\cite{schimdt_fund_cocycle_98}. In this paper, we will focus on a special class of nearest neighbour shifts of finite type where the forbidden patterns are the same in every `direction':\n\n\nGiven a graph $\\mathcal H$ let\n$$X_{\\mathcal H}:=\\{x\\in\\mathcal H^{\\mathbb{Z}^d}\\:|\\: x_{\\vec {i}}\\sim_\\mathcal H x_{\\vec{j}}\\text{ for all }\\vec{i}\\sim \\vec{j}\\}.$$\nSuch spaces will be called \\emph{hom-shifts}. Note that $X_\\mathcal H$ is the space of graph homomorphisms from $\\mathbb{Z}^d$ to $\\mathcal H$. If $\\mathcal H$ is finite and\n$${\\mathcal F}_\\mathcal H:=\\{[v,w]_{\\vec{0},\\vec{e}_j}\\:|\\:v\\nsim_\\mathcal H w, 1\\leq j\\leq d\\}$$\nthen $X_{\\mathcal H}= X_{{\\mathcal F}_{\\mathcal H}}$. These are exactly the nearest neighbour shifts of finite type with symmetric and isotropic constraints. For example if the graph $\\mathcal H$ is given by Figure \\ref{Figure: Hard Square} then $X_\\mathcal H$ is the hard square shift, that is, configurations with alphabet $\\{0,1\\}$ such that adjacent symbols cannot both be $1$. $X_{K_n}$ is the space of $n$-colourings of the graph, that is, configurations with alphabet $\\{1,2, \\ldots, n\\}$ where all adjacent colours are distinct. We note that the properties, symmetry and isotropy, are not invariant under conjugacy.\n\n${\\mathcal F}$ will always denote a set of patterns and $\\mathcal H$ will always denote a graph, there will not be any ambiguity in the notations $X_{\\mathcal F}, X_\\mathcal H$.\n\nA graph $\\mathcal H$ is called \\emph{four-cycle free} if it is finite, it has no self-loops and $C_4$ is not a subgraph of $\\mathcal H$. For instance $K_4$ is not a four-cycle free graph.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[angle=0,\nwidth=.1\\textwidth]{hardsquare.pdf}\\caption{Graph for the Hard Square Shift}\n\\label{Figure: Hard Square}\n\\end{figure}\n\nThe set of \\emph{globally allowed patterns} of a shift space $X$ on a set $A\\subset \\mathbb{Z}^d$ is\n$$\\mathcal L_A(X):=\\{a\\in {\\mathcal A}^{A}\\:\\big{|}\\:\\text{ there exists }x\\in X\\text{ such that }x|_A=a\\}.$$\nIts \\emph{language} is the set of all finite patterns appearing in $X$ that is\n$$\\mathcal L(X):=\\bigcup_{A\\subset \\mathbb{Z}^d \\text{ finite}}\\mathcal L_A(X).$$\nWe comment that this is different from the set of locally allowed patterns: Let $X$ be a shift space with a forbidden list ${\\mathcal F}$. Given a finite set $A$, a pattern $a\\in {\\mathcal A}^A$ is said to be \\emph{locally allowed} if no pattern from ${\\mathcal F}$ appears in $a$. In general it is undecidable for shifts of finite type whether a locally allowed pattern belongs to $\\mathcal L(X)$ \\cite{Robinson1971}; however it is decidable when $X$ is a hom-shift where it is sufficient to check whether the pattern extends to a locally allowed pattern on $B_n$ for some $n$.\n\nThe topological entropy of the shift space $X$ is the log growth rate of the number of allowed patterns in $X$, that is,\n$$h_{top}(X):=\\lim_{n\\longrightarrow \\infty}\\frac{\\log|\\mathcal L_{B_n}(X)|}{|B_n|}.$$\nThe existence of the limit follows from subadditivity arguments via the well-known multivariate version of Fekete's Lemma \\cite{silviofekete}. Moreover the topological entropy is an invariant under conjugacy (for $d=1$ look at Proposition 4.1.9 in \\cite{LM}, the proof extends to higher dimensions). We remark that the computation of this invariant for shifts of finite type in $d>1$ is a hard problem and very little is known \\cite{pavlovhardsquare2012}, however there are algorithms to compute approximating upper and lower bounds of the topological entropy of the hom-shifts \\cite{symmtricfriedlan1997,louidor2010improved}. Further if $\\mathcal H$ is a finite connected graph with at least two edges, then $h_{top}(X_\\mathcal H)>0$:\n\n\\begin{prop}\\label{proposition: hom-space positive entropy}\nLet $\\mathcal H$ be a finite graph with distinct vertices $a, b$ and $c$ such that $a\\sim_\\mathcal H b$ and $b\\sim_\\mathcal H c$. Then $h_{top}(X_{\\mathcal H})\\geq\\frac{\\log{2}}{2}$.\n\\end{prop}\n\\begin{proof}\n\nIt is sufficient to see this for a graph $\\mathcal H$ with exactly three vertices $a$, $b$ and $c$ such that $a\\sim_\\mathcal H b$ and $b\\sim_\\mathcal H c$. For such a graph any configuration in $X_\\mathcal H$ is composed of $b$ on one partite class of $\\mathbb{Z}^d$ and a free choice between $a$ and $c$ for vertices on the other partite class. Then\n$$|\\mathcal L_{B_n}(X_\\mathcal H)|=2^{{\\lfloor\\frac{(2n+1)^d}{2}\\rfloor}}+2^{{\\lceil\\frac{(2n+1)^d}{2}\\rceil}}$$\nproving that $h_{top}(X_{\\mathcal H})=\\frac{\\log{2}}{2}$.\n\\end{proof}\n\nA shift space $X$ is called \\emph{entropy minimal} if for all shift spaces $Y\\subsetneq X$, $h_{top}(X)>h_{top}(Y)$. In other words, a shift space $X$ is entropy minimal if forbidding any word causes a drop in entropy. From \\cite{quastrow2000} we know that every shift space contains an entropy minimal shift space with the same entropy and also a characterisation of same entropy factor maps on entropy minimal shifts of finite type.\n\nOne of the main results of this paper is the following:\n\\begin{thm}\\label{theorem:four cycle free entropy minimal}\nLet $\\mathcal H$ be a connected four-cycle free graph. Then $X_\\mathcal H$ is entropy minimal.\n\\end{thm}\nFor $d=1$ all irreducible shifts of finite type are entropy minimal \\cite{LM}. A necessary condition for the entropy minimality of $X_\\mathcal H$ is that $\\mathcal H$ has to be connected.\n\\begin{prop}\\label{proposition:entropy requires connectivity}\nSuppose $\\mathcal H$ is a finite graph with connected components $\\mathcal H_1, \\mathcal H_2, \\ldots \\mathcal H_r$. Then $h_{top}(X_\\mathcal H)=\\max_{1\\leq i \\leq r}h_{top}(X_{\\mathcal H_i})$.\n\\end{prop}\nThis follows from the observation that\n$$\\max_{1\\leq i \\leq r}|\\mathcal L_{B_n}({X_{\\mathcal H_i}})|\\leq |\\mathcal L_{B_n}({X_\\mathcal H})|= \\sum_{i=1}^r|\\mathcal L_{B_n}({X_{\\mathcal H_i}})|\\leq r\\max_{1\\leq i \\leq r}|\\mathcal L_{B_n}({X_{\\mathcal H_i}})|.$$\n\n\n\n\n\n\\section{Thermodynamic Formalism}\\label{section:thermodynamic formalism}\nHere we give a brief introduction of thermodynamic formalism. For more details one can refer to \\cite{Rue,walters-book}.\n\nBy $\\mu$ we will always mean a shift-invariant Borel probability measure on a shift space $X$. The \\emph{support} of $\\mu$ denoted by $supp(\\mu)$ is the intersection of all closed sets $Y \\subset X$ for which $\\mu(Y)= 1$. Note that $supp(\\mu)$ is a shift space as well.\nThe \\emph{measure theoretic entropy} is\n\\begin{equation*}\nh_\\mu:=\\lim_{i \\rightarrow \\infty}\\frac{1}{|D_i|}H^{D_i}_{\\mu},\n\\end{equation*}\n\\noindent where $H^{D_i}_{\\mu}$ is the Shannon-entropy of $\\mu$ with respect to the partition of $X$ generated by the cylinder sets on $D_i$, the definition of which is given by:\n\\begin{equation*}\nH^{D_i}_{\\mu}:=\\sum_{a\\in \\mathcal L_{D_i}(X)}-\\mu([a]_{D_i})\\log{\\mu([a]_{D_i})},\n\\end{equation*} with the understanding that $0\\log 0=0$.\n\nA shift-invariant probability measure $\\mu$ is a \\emph{measure of maximal entropy} of $X$ if the maximum of $\\nu \\mapsto h_\\nu$ over all shift-invariant probability measures on $X$ is obtained at $\\mu$. The existence of measures of maximal entropy follows from upper-semi-continuity of the function $\\nu \\mapsto h_\\nu$ with respect to the weak-$*$ topology.\n\n\nFurther the well-known \\emph{variational principle} for topological entropy of $\\mathbb{Z}^d$-actions asserts that if $\\mu$ is a measure of maximal entropy $h_{top}(X)=h_\\mu$ whenever $X$ is a $\\mathbb{Z}^d$-shift space.\n\nThe following is a well-known characterisation of entropy minimality (it is used for instance in the proof of Theorem 4.1 in \\cite{meestersteif2001}):\n\\begin{prop}\n\\label{proposition:entropyviamme}\nA shift space $X$ is entropy minimal if and only if every measure of maximal entropy for $X$ is fully supported.\n\\end{prop}\nWe understand this by the following: Suppose $X$ is entropy minimal and $\\mu$ is a measure of maximal entropy for $X$. Then by the variational principle for $X$ and $supp(\\mu)$ we get\n$$h_{top}(X)=h_\\mu\\leq h_{top}(supp(\\mu))\\leq h_{top}(X)$$\nproving that $supp(\\mu)=X$. To prove the converse, suppose for contradiction that $X$ is not entropy minimal and consider $Y\\subsetneq X$ such that $h_{top}(X)= h_{top}(Y)$. Then by the variational principle there exists a measure $\\mu$ on $Y$ such that $h_\\mu= h_{top}(X)$. Thus $\\mu$ is a measure of maximal entropy for $X$ which is not fully supported.\n\nFurther is known if $X$ is a nearest neighbour shift of finite type; this brings us to Markov random fields which we introduce next.\n\nGiven a set $A\\subset \\mathbb{Z}^d$ we denote the \\emph{$r$-boundary} of $A$ by $\\partial_r A$, that is,\n$$\\partial_r A=\\{w\\in \\mathbb{Z}^d\\setminus A\\:\\Big \\vert\\: \\|w-v\\|_1\\leq r \\text{ for some }v\\in A\\}.$$\nThe \\emph{1-boundary} will be referred to as the \\emph{boundary} and denoted by $\\partial A$.\nA \\emph{Markov random field} on $\\mathcal{A}^{\\mathbb{Z}^d}$ is a Borel probability measure $\\mu$ with the property that\nfor all finite $A, B \\subset \\mathbb{Z}^d$ such that $\\partial A \\subset B \\subset A^{c}$ and $a \\in {\\mathcal A}^A, b \\in {\\mathcal A}^B$ satisfying $\\mu([b]_B)>0$\n\\begin{equation*}\n\\mu([a]_A\\;\\Big\\vert\\;[b]_B)= \\mu([a]_A\\;\\Big\\vert\\;[b]_{ \\partial A}).\n\\end{equation*}\n\nIn general Markov random fields are defined over graphs much more general than $\\mathbb{Z}^d$, however we restrict to the $\\mathbb{Z}^d$ setting in this paper.\n\nA \\emph{uniform Markov random field} is a Markov random field $\\mu$ such that further\n\n\\begin{equation*}\n\\mu([a]_A\\;\\Big\\vert\\;[b]_{ \\partial A})=\\frac{1}{n_{A,b|_{\\partial A}}}\n\\end{equation*}\nwhere $n_{A,b|_{\\partial A}}=|\\{a\\in {\\mathcal A}^A\\:|\\: \\mu([a]_A\\cap [b]_{\\partial A})>0\\}|$.\n\nFollowing \\cite{petersen_schmidt1997, schmidt_invaraint_cocycles_1997}, we denote by $\\Delta_X$ the \\emph{homoclinic equivalence relation} of a shift space $X$, which is given by\n\\begin{equation*}\n\\Delta_X := \\{(x,y)\\in X\\times X\\;|\\; x_{\\vec i}=y_{\\vec i} \\text{ for all but finitely many } \\vec i\\in \\mathbb{Z}^d\\}.\n\\end{equation*}\n\nWe say that a measure $\\mu$ is \\emph{adapted} with respect to a shift space $X$ if\n$supp(\\mu)\\subset X$ and\n\\begin{equation*}\nx\\in supp(\\mu) \\Longrightarrow \\{y\\in X\\:|\\: (x,y)\\in \\Delta_X\\}\\subset supp(\\mu).\n\\end{equation*}\n\nTo illustrate this definition, let $X\\subset \\{0,1\\}^{\\mathbb{Z}}$ consist of configurations in $X$ in which at most a single $1$ appears. $X$ is uniquely ergodic; the delta-measure $\\delta_{0^{\\infty}}$ is the only shift-invariant measure on $X$. But\n$$supp(\\delta_{0^\\infty})= \\{0^\\infty\\}\\subsetneq \\{y\\in X\\;|\\; 0^\\infty_i= y_i \\text{ for all but finitely many } i\\in \\mathbb{Z}\\}=X,$$\nproving that it is not adapted. On the other hand, since the homoclinic relation of ${\\mathcal A}^{\\mathbb{Z}^d}$ is minimal, meaning that for all $x\\in {\\mathcal A}^{\\mathbb{Z}^d}$\n$$\\overline{\\{y\\in {\\mathcal A}^{\\mathbb{Z}^d}\\;|\\; y_{\\vec i}=x_{\\vec i} \\text{ for all but finitely many } {\\vec{i}}\\in \\mathbb{Z}^d\\}}= {\\mathcal A}^{\\mathbb{Z}^d},$$\nit follows that a probability measure on ${\\mathcal A}^{\\mathbb{Z}^d}$ is adapted if and only if it is fully supported.\n\n\n\nThe relationship between measures of maximal entropy and Markov random fields is established by the following theorem. This is a special case of the Lanford-Ruelle theorem \\cite{lanfruell,Rue}.\n\n\\begin{thm} All measures of maximal entropy on a nearest neighbour shift of finite type $X$ are shift-invariant uniform Markov random fields $\\mu$ adapted to $X$.\\label{thm:equiGibbs}\n\\end{thm}\n\nThe converse is also true under further mixing assumptions on the shift space $X$ (called the D-condition). The full strength of these statements is obtained by looking at \\emph{equilibrium states} instead of measures of maximal entropy. The measures obtained there are not uniform Markov random fields, rather Markov random fields where the conditional probabilities are weighted via an \\emph{interaction} giving rise to \\emph{Gibbs states}. Uniform Markov random fields are Gibbs states with interaction zero.\n\nWe will often restrict our proofs to the ergodic case. We can do so via the following standard facts implied by Theorem $14.15$ in \\cite{Georgii} and Theorem 4.3.7 in \\cite{kellerequ1998}:\n\n\\begin{thm}\\label{theorem: ergodic decomposition of markov random fields}\nLet $\\mu$ be a shift-invariant uniform Markov random field adapted to a shift space $X$. Let its ergodic decomposition be given by a measurable map $x\\longrightarrow \\mu_x$ on $X$, that is, $\\mu= \\int_X \\mu_x d\\mu$. Then $\\mu$-almost everywhere the measures $\\mu_x$ are shift-invariant uniform Markov random fields adapted to $X$ such that $supp(\\mu_x)\\subset supp(\\mu)$. Moreover $\\int h_{\\mu_x} d\\mu(x)= h_\\mu$.\n\\end{thm}\n\n\nWe will prove the following:\n\\begin{thm}\\label{theorem: MRF fully supported }\nLet $\\mathcal H$ be a connected four-cycle free graph. Then every ergodic probability measure adapted to $X_\\mathcal H$ with positive entropy is fully supported.\n\\end{thm}\n\nThis implies Theorem \\ref{theorem:four cycle free entropy minimal} by the following: The Lanford-Ruelle theorem implies that every measure of maximal entropy on $X_\\mathcal H$ is a uniform shift-invariant Markov random field adapted to $X_\\mathcal H$. By Proposition \\ref{proposition: hom-space positive entropy} and the variational principle we know that these measures have positive entropy. By Theorems \\ref{theorem: ergodic decomposition of markov random fields} and \\ref{theorem: MRF fully supported } they are fully supported. Finally by Proposition \\ref{proposition:entropyviamme}, $X_\\mathcal H$ is entropy minimal.\n\nAlternatively, the conclusion of Theorem \\ref{theorem: MRF fully supported } can be obtained via some strong mixing conditions on the shift space; we will describe one such assumption. A shift space $X$ is called \\emph{strongly irreducible} if there exists $g>0$ such that for all $x, y \\in X$ and $A, B\\subset \\mathbb{Z}^d$ satisfying $\\min_{\\vec i \\in A, \\vec j \\in B}\\|\\vec i - \\vec j\\|_1\\geq g$, there exists $z\\in X$ such that $z|_{A}= x|_A$ and $z|_B= y|_B$. For such a space, the homoclinic relation is minimal implying the conclusion of Theorem \\ref{theorem: MRF fully supported } and further, that every probability measure adapted to $X$ is fully supported. Note that this does not prove that $X$ is entropy minimal unless we assume that $X$ is a nearest neighbour shift of finite type. Such an argument is used in the proof of Lemma 4.1 in \\cite{meestersteif2001} which implies that every strongly irreducible shift of finite type is entropy minimal. A more combinatorial approach was used in \\cite{Schraudner2010minimal} to show that general shift spaces with a weaker mixing property called uniform filling are entropy minimal.\n\n\n\\section{The Pivot Property}\\label{section: the pivot property}\nA \\emph{pivot} in a shift space $X$ is a pair of configurations $(x,y)\\in X$ such that $x$ and $y$ differ exactly at a single site. A subshift $X$ is said to have \\emph{the pivot property} if for all distinct $(x,y)\\in \\Delta_X$ there exists a finite sequence of configurations $x^{(1)}=x, x^{(2)},\\ldots, x^{(k)}=y \\in X$ such that each $(x^{(i)}, x^{(i+1)})$ is a pivot. In this case we say $x^{(1)}=x, x^{(2)},\\ldots, x^{(k)}=y$ is a \\emph{chain of pivots} from $x$ to $y$.\nHere are some examples of subshifts which have the pivot property:\n\n\\begin{enumerate}\n\\item Any subshift with a trivial homoclinic relation, that is, the homoclinic classes are singletons.\n\\item Any subshift with a safe symbol\\footnote{A shift space $X\\subset {\\mathcal A}^{\\mathbb{Z}^d}$ has a \\emph{safe symbol} $\\star$ if for all $x\\in X$ and $A\\subset \\mathbb{Z}^d$ the configuration $z\\in {\\mathcal A}^{\\mathbb{Z}^d}$ given by\n\\begin{equation*}\nz_{\\vec i}:=\\begin{cases}\nx_{\\vec i} &\\text{ if } \\vec i \\in A\\\\\n\\star &\\text{ if } \\vec i \\in A^c\n\\end{cases}\u2022\n\\end{equation*}\nis also an element of $X$.}.\n\\item The hom-shifts $X_{C_r}$. This was proved for $r\\neq 4$ in \\cite{chandgotia2013Markov}, the result for $r=4$ is a special case of Proposition \\ref{proposition: frozenfoldpivot}.\n\\item $r$-colorings of $\\mathbb{Z}^d$ with $r\\geq 2d+2$. (It is well-known, look for instance in Subsection 3.2 of \\cite{chandgotia2013Markov})\n\\item\\label{item: pivot property list number 5} $X_\\mathcal H$ when $\\mathcal H$ is dismantlable. \\cite{brightwell2000gibbs}\n\\end{enumerate}\nWe generalise the class of examples given by (\\ref{item: pivot property list number 5}) in Proposition \\ref{proposition: frozenfoldpivot}. It is not true that all hom-shifts have the pivot property.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[angle=0,\nwidth=.1\\textwidth]{fivecolouring.pdf}\\caption{Frozen Pattern}\n\\label{Figure: Five colour}\n\\end{figure}\nThe following was observed by Brian Marcus: Recall that $K_n$ denotes the complete graph with $n$ vertices. $X_{K_4}, X_{K_5}$ do not possess the pivot property if the dimension is two. For instance consider a configuration in $X_{K_5}$ which is obtained by tiling the plane with the pattern given in Figure \\ref{Figure: Five colour}. It is clear that the symbols in the box can be interchanged but no individual symbol can be changed. Therefore $X_{K_5}$ does not have the pivot property. However both $X_{K_4}$ and $X_{K_5}$ satisfy a more general property as discussed in Subsection \\ref{subsection: Hom-shifts and the pivot property}.\n\nThe following theorem is another main result in this paper.\n\\begin{thm}\\label{theorem: pivot property for four cycle free}\nFor all four-cycle free graphs $\\mathcal H$, $X_\\mathcal H$ has the pivot property.\n\\end{thm}\nIt is sufficient to prove this theorem for four-cycle free graphs $\\mathcal H$ which are connected because of the following proposition:\n\\begin{prop}\\label{proposition: pivot for disconnected}\nLet $X_1, X_2, \\ldots, X_n$ be shift spaces on disjoint alphabets such that each of them has the pivot property. Then $\\cup_{i=1}^n X_i$ also has the pivot property.\n\\end{prop}\nThis is true since $(x, y)\\in \\Delta_{\\cup_{i=1}^n X_i}$ implies $(x, y)\\in \\Delta_{X_i}$ for some $1\\leq i \\leq n$.\n\n\\section{Folding, Entropy Minimality and the Pivot Property}\\label{section:Folding, Entropy Minimality and the Pivot Property} Given a graph $\\mathcal H$ we say that a vertex $v$ \\emph{folds} into a vertex $w$ if and only if $u \\sim_\\mathcal H v$ implies $u \\sim_\\mathcal H w$. In this case the graph $\\mathcal H\\setminus \\{v\\}$ is called a \\emph{fold} of $\\mathcal H$. The folding gives rise to a `retract' from $\\mathcal H$ to $\\mathcal H\\setminus\\{v\\}$, namely the graph homomorphism from $\\mathcal H$ to $\\mathcal H\\setminus \\{v\\}$ which is the identity on $\\mathcal H\\setminus \\{v\\}$ and sends $v$ to $w$. This was introduced in \\cite{nowwinkler} to help characterise cop-win graphs and used in \\cite{brightwell2000gibbs} to establish many properties which are preserved under `folding' and `unfolding'. Given a finite tree $\\mathcal H$ with more than two vertices note that a leaf vertex (vertex of degree $1$) can always be folded to some other vertex of the tree. Thus starting with $\\mathcal H$, there exists a sequence of folds resulting in a single edge. In fact using a similar argument we can prove the following proposition.\n\n\\begin{prop}\\label{proposition:folding trees into other trees}\nLet $\\mathcal H\\subset \\mathcal H^\\prime$ be trees. Then there is a graph homomorphism $f: \\mathcal H^\\prime \\longrightarrow \\mathcal H$ such that $f|_{\\mathcal H}$ is the identity map.\n\\end{prop}\n\nTo show this, first note that if $\\mathcal H\\subsetneq\\mathcal H^\\prime$ then there is a leaf vertex in $\\mathcal H^\\prime$ which is not in $\\mathcal H$. This leaf vertex can be folded into some other vertex in $\\mathcal H^\\prime$. Thus by induction on $|\\mathcal H^\\prime \\setminus \\mathcal H|$ we can prove that there is a sequence of folds from $\\mathcal H^\\prime$ to $\\mathcal H$. Corresponding to this sequence of folds we obtain a graph homomorphism from $\\mathcal H^\\prime$ to $\\mathcal H$ which is the identity on $\\mathcal H$.\n\nHere we consider a related notion for shift spaces. Given a nearest neighbour shift of finite type $X\\subset {\\mathcal A}^{\\mathbb{Z}^d}$, \\emph{the neighbourhood} of a symbol $v\\in {\\mathcal A}$ is given by\n$$N_X(v):=\\{a \\in {\\mathcal A}^{\\partial \\vec 0}\\:|\\: [v]_{\\vec 0}\\cap [a]_{\\partial \\vec 0}\\in \\mathcal L_{D_1}(X)\\},$$\nthat is the collection of all patterns which can `surround' $v$ in $X$. We will say that $v$ \\emph {config-folds} into $w$ in $X$ if $N_X(v)\\subset N_X(w)$. In such a case we say that $X$ \\emph{config-folds} to $X\\cap({\\mathcal A}\\setminus \\{v\\})^{\\mathbb{Z}^d}$. Note that $X\\cap({\\mathcal A}\\setminus \\{v\\})^{\\mathbb{Z}^d}$ is obtained by forbidding $v$ from $X$ and hence it is also a nearest neighbour shift of finite type. Also if $X=X_\\mathcal H$ for some graph $\\mathcal H$ then $v$ config-folds into $w$ in $X_\\mathcal H$ if and only if $v$ folds into $w$ in $\\mathcal H$. Thus if $\\mathcal H$ is a tree then there is a sequence of folds starting at $X_\\mathcal H$ resulting in the two checkerboard configurations with two symbols (the vertices of the edge which $\\mathcal H$ folds into). This property is weaker than the notion of folding introduced in \\cite{chandgotiahammcliff2014}.\n\nThe main thrust of this property in our context is: if $v$ config-folds into $w$ in $X$ then given any $x\\in X$, every appearance of $v$ in $x$ can be replaced by $w$ to obtain another configuration in $X$. This replacement defines a factor (surjective, continuous and shift-invariant) map $f: X\\longrightarrow X\\cap({\\mathcal A}\\setminus \\{v\\})^{\\mathbb{Z}^d}$ given by\n\\begin{equation*}\n(f(x))_{\\vec i}:=\\begin{cases}\nx_{\\vec i}&\\text{ if } x_{\\vec i}\\neq v\\\\\nw&\\text{ if } x_{\\vec i}= v.\n\\end{cases}\u2022\n\\end{equation*}\nNote that the map $f$ defines a `retract' from $X$ to $X\\cap({\\mathcal A}\\setminus \\{v\\})^{\\mathbb{Z}^d}$. Frequently we will config-fold more than one symbol at once (especially in Section \\ref{section: Proof of the main theorems}):\n\nDistinct symbols $v_1, v_2, \\ldots, v_n$ \\emph{config-fold disjointly} into $w_1, w_2, \\ldots, w_n$ in $X$ if $v_i$ config-folds into $w_i$ and $v_i\\neq w_j$ for all $1\\leq i, j \\leq n$. In this case the symbols $v_1, v_2, \\ldots, v_n$ can be replaced by $w_1, w_2, \\ldots, w_n$ simultaneously for all $x \\in X$. Suppose $v_1, v_2,\\ldots v_n$ is a maximal set of symbols which can be config-folded disjointly in $X$. Then $X\\cap({\\mathcal A}\\setminus \\{v_1, v_2, \\ldots, v_n\\})^{\\mathbb{Z}^d}$ is called a \\emph{full config-fold} of $X$.\n\nFor example consider a tree $\\mathcal H:=(\\mathcal V,\\mathcal E)$ where $\\mathcal V:=\\{v_1, v_2, v_3, \\ldots, v_{n+1}\\}$ and $\\mathcal E:=\\{(v_i, v_{n+1})\\:|\\: 1\\leq i \\leq n\\}$. For all $1\\leq i \\leq n$, $\\mathcal V\\setminus \\{v_i, v_{n+1}\\}$ is a maximal set of symbols which config-folds disjointly in $X_\\mathcal H$ resulting in the checkerboard patterns with the symbols $v_i$ and $v_{n+1}$ for all $1\\leq i \\leq n$. Thus the full config-fold of a shift space is not necessarily unique. However it is unique up to conjugacy:\n\n\\begin{prop}\\label{Proposition: Uniqueness of full config-fold}\nThe full config-fold of a nearest neighbour shift of finite type is unique up to conjugacy via a change of the alphabet.\n\\end{prop}\nThe ideas for the following proof come essentially from the proof of Theorem 4.4 in \\cite{brightwell2000gibbs} and discussions with Prof. Brian Marcus.\n\\begin{proof}\nLet $X\\subset {\\mathcal A}^{\\mathbb{Z}^d}$ be a nearest neighbour shift of finite type and\n$$M:=\\{v\\in {\\mathcal A} \\:|\\: \\text{ for all }w\\in {\\mathcal A},\\  v \\text{ config-folds into }w \\Longrightarrow w \\text{ config-folds into }v\\}.$$\nThere is a natural equivalence relation $\\equiv$ on $M$ given by $v\\equiv w$ if $v$ and $w$ config-fold into each other. Let $A_1, A_2, A_3, \\ldots, A_r\\subset M$ be the corresponding partition. Clearly for all distinct $v, w\\in M$, $v$ can be config-folded into $w$ if and only if $v, w\\in A_i$ for some $i$. It follows that $A\\subset A_i$ can be config-folded disjointly if and only if $\\emptyset\\neq A\\neq A_i$.\n\nLet $v\\in {\\mathcal A}\\setminus M$. We will prove that $v$ config-folds to a symbol in $M$. By the definition of $M$ there exists $v_1\\in {\\mathcal A}$ such that $N_X(v)\\subsetneq N_X(v_1)$. If $v_1\\in M$ then we are done, otherwise choose $v_2\\in {\\mathcal A}$ such that $N_X(v_1)\\subsetneq N_X(v_2)$. Continuing this process recursively we can find a sequence $v= v_0, v_1, v_2, \\ldots, v_n$ such that $N_X(v_{i-1})\\subsetneq N_X(v_i)$ for all $1\\leq i \\leq n$ and $v_n\\in M$. Thus $v$ config-folds into $v_n$, a symbol in $M$. Further if $v$ config-folds to a symbol in $A_i$ it can config-fold to all the symbols in $A_i$. Therefore $B$ is a maximal subset of symbols in ${\\mathcal A}$ which can be config-folded disjointly if and only if $B=\\cup_{i=1}^rB_i\\cup ({\\mathcal A}\\setminus M)$ where $B_i\\subset A_i$ and $|A_i\\setminus B_i|=1$. Let $B'\\subset {\\mathcal A}$ be another such maximal subset, ${\\mathcal A}\\setminus B:=\\{b_1, b_2, \\ldots, b_r\\}$ and ${\\mathcal A}\\setminus B':=\\{b'_1, b'_2, \\ldots, b'_r\\}$ where $b_i, b'_i\\in A_i$. Then the map\n$$f: X\\cap ({\\mathcal A}\\setminus B)^{\\mathbb{Z}^d} \\longrightarrow X\\cap ({\\mathcal A}\\setminus B')^{\\mathbb{Z}^d} \\text{ given by } f(x) := y\\text{ where } y_{\\vec i}=b'_j \\text{ whenever }x_{\\vec i}=b_j$$\nis the required change of alphabet between the two full config-folds of $X$. \\end{proof}\n\nLet $X\\cap({\\mathcal A}\\setminus \\{v_1, v_2, \\ldots, v_n\\})^{\\mathbb{Z}^d}$ be a \\emph{full config-fold} of $X$ where $v_i$ config-folds into $w_i$ for all $1\\leq i\\leq n$. Consider $f_X: {\\mathcal A} \\longrightarrow{\\mathcal A}\\setminus \\{v_1, v_2, \\ldots, v_n\\}$ given by\n\\begin{equation*}\nf_X(v):=\\begin{cases}\nv&\\text{ if } v\\neq v_j \\text{ for all }1\\leq j\\leq n\\\\\nw_j&\\text{ if } v= v_j\\text{ for some }1\\leq j \\leq n.\n\\end{cases}\u2022\n\\end{equation*}\n\\noindent This defines a factor map $f_X: X\\longrightarrow X\\cap({\\mathcal A}\\setminus \\{v_1, v_2, \\ldots, v_n\\})^{\\mathbb{Z}^d}$ given by $(f_X(x))_{{\\vec{i}}}:= f_X(x_{\\vec{i}})$ for all ${\\vec{i}} \\in \\mathbb{Z}^d$. $f_X$ denotes both the factor map and the map on the alphabet; it should be clear from the context which function is being used.\n\n\n\nIn many cases we will fix a configuration on a set $A\\subset \\mathbb{Z}^d$ and apply a config-fold on the rest. Hence we define the map $f_{X,A}: X\\longrightarrow X$ given by\n\\begin{equation*}\n(f_{X,A}(x))_{\\vec i}:=\\begin{cases}\nx_{\\vec i}&\\text{ if } \\vec i \\in A\\\\\nf_X(x_{\\vec i})&\\text{ otherwise.}\n\\end{cases}\u2022\n\\end{equation*}\u2022\n\nThe map $f_{X,A}$ can be extended beyond $X$:\n\n\\begin{prop}\\label{prop: folding_fixing_a_set}\nLet $X\\subset Y$ be nearest neighbour shifts of finite type, $Z$ be a full config-fold of $X$ and $y\\in Y$ such that for some $A\\subset \\mathbb{Z}^d$, $y|_{A^c\\cup\\partial (A^c)}\\in \\mathcal L_{A^c\\cup\\partial (A^c)}(X)$. Then the configuration $z$ given by\n\\begin{equation*}\nz_{\\vec i}:=\\begin{cases}\ny_{\\vec i}&\\text{ if } \\vec i \\in A\\\\\nf_X(y_{\\vec i})&\\text{ otherwise}\n\\end{cases}\u2022\n\\end{equation*}\nis an element of $Y$. Moreover $z|_{A^c}\\in \\mathcal L_{A^c}(Z)$.\n\\end{prop}\nAbusing the notation, in such cases we shall denote the configuration $z$ by $f_{X, A}(y)$.\n\nIf $A^c$ is finite, then $f_{X,A}$ changes only finitely many coordinates. These changes can be applied one by one, that is, there is a chain of pivots in $Y$ from $y$ to $f_{X,A}(y)$.\n\nA nearest neighbour shift of finite type which cannot be config-folded is called a \\emph{stiff shift}. We know from Theorem 4.4 in \\cite{brightwell2000gibbs} that all the stiff graphs obtained by a sequence of folds of a given graph are isomorphic. By Proposition \\ref{Proposition: Uniqueness of full config-fold} the corresponding result for nearest neighbour shifts of finite type immediately follows:\n\\begin{prop}\\label{proposition:uniqueness of stiff shifts}\nThe stiff shift obtained by a sequence of config-folds starting with a nearest neighbour shift of finite type is unique up to conjugacy via a change of the alphabet.\n\\end{prop}\n\nStarting with a nearest neighbour shift of finite type $X$ the \\emph{fold-radius} of $X$ is the smallest number of full config-folds required to obtain a stiff shift. If $\\mathcal H$ is a tree then the fold-radius of $X_\\mathcal H$ is equal to\n$$\\left\\lfloor\\frac{diameter(\\mathcal H)}{2}\\right\\rfloor.$$\nThus for every nearest neighbour shift of finite type $X$ there is a sequence of full config-folds (not necessarily unique) which starts at $X$ and ends at a stiff shift of finite type. Let the fold-radius of $X$ be $r$ and $X= X_0, X_1, X_2, \\ldots, X_r$ be a sequence of full config-folds where $X_r$ is stiff. This generates a sequence of maps $f_{X_i}:X_{i}\\longrightarrow X_{i+1}$ for all $0\\leq i \\leq r-1$. In many cases we will fix a pattern on $D_n$ or $D_n^c$ and apply these maps on the rest of the configuration. Consider the maps $I_{X,n}:X\\longrightarrow X$ and $O_{X,n}:X\\longrightarrow X$ (for $n>r$) given by\n\\begin{equation*}\nI_{X,n}(x):=f_{X_{r-1},D_{n+r-1} }\\left(f_{X_{r-2}, D_{n+r-2}}\\left(\\ldots\\left(f_{X_{0}, D_n}(x)\\right)\\ldots\\right)\\right)\\text{(Inward Fixing Map)}\n\\end{equation*}\u2022\nand\n\\begin{eqnarray*}\nO_{X,n}(x):=f_{X_{r-1}, D_{n-r+1}^c }\\left(f_{X_{r-2}, D_{n-r+2}^c}\\left(\\ldots\\left(f_{X_{0}, D_n^c}(x)\\right)\\ldots\\right)\\right)\\text{(Outward Fixing Map)}.\n\\end{eqnarray*}\u2022\nSimilarly we consider maps which do not fix anything, $F_X: X\\longrightarrow X_r$ given by\n\\begin{eqnarray*}\nF_X(x):= f_{X_{r-1}}\\left(f_{X_{r-2}}\\left(\\ldots\\left(f_{X_{0}}(x)\\right)\\ldots\\right)\\right).\n\\end{eqnarray*}\nNote that $D_k\\cup \\partial D_k= D_{k+1}$ and $D_k^c\\cup \\partial (D_k^c)=D_{k-1}^c$. This along with repeated application of Proposition \\ref{prop: folding_fixing_a_set} implies that the image of $I_{X,n}$ and $O_{X,n}$ lie in $X$. This also implies the following proposition:\n\n\\begin{prop}[The Onion Peeling Proposition]\\label{prop: folding_ to _ stiffness_fixing_a_set}\nLet $X\\subset Y$ be nearest neighbour shifts of finite type, $r$ be the fold-radius of $X$, $Z$ be a stiff shift obtained by a sequence of config-folds starting with $X$ and $y^1, y^2\\in Y$ such that $y^1|_{D_{n-1}^c}\\in \\mathcal L_{D_{n-1}^c}(X)$ and $y^2|_{D_{n+1}}\\in \\mathcal L_{D_{n+1}}(X)$. Let $z^1, z^2\\in Y$ be given by\n\\begin{eqnarray*}\nz^1&:=&f_{X_{r-1},D_{n+r-1} }\\left(f_{X_{r-2}, D_{n+r-2}}\\left(\\ldots\\left(f_{X_{0}, D_n}(y^1)\\right)\\ldots\\right)\\right)\\\\\nz^2&:=&f_{X_{r-1}, D_{n-r+1}^c }\\left(f_{X_{r-2},D_{n-r+2}^c}\\left(\\ldots\\left(f_{X_{0}, D_n^c}(y^2)\\right)\\ldots\\right)\\right)\\text{ for }n>r.\n\\end{eqnarray*}\u2022\nThe patterns $z^1|_{D_{n+r-1}^c}\\in \\mathcal L_{D_{n+r-1}^c}(Z)$ and $z^2|_{D_{n-r+1}}\\in \\mathcal L_{D_{n-r+1}}(Z)$. If $y^1, y^2\\in X$ then in addition\n\\begin{eqnarray*}\nz^1|_{D_{n+r-1}^c}&=&F_X(y^1)|_{D_{n+r-1}^c}\\text{ and}\\\\\nz^2|_{D_{n-r+1}}&=&F_X(y^2)|_{D_{n-r+1}}.\n\\end{eqnarray*}\u2022\n\\end{prop}\n\nAbusing the notation, in such cases we shall denote the configurations $z^1$ and $z^2$ by $I_{X, n}(y^1)$ and $O_{X,n}(y^2)$ respectively. Note that $I_{X, n}(y^1)|_{D_n}= y^1|_{D_n}$ and $O_{X,n}(y^2)|_{D_n^c}= y^2|_{D_n^c}$. Also, $O_{X,n}$ is a composition of maps of the form $f_{X,A}$ where $A^c$ is finite; there is a chain of pivots in $Y$ from $y$ to $O_{X,n}(y)$.\nThere are two kinds of stiff shifts which will be of interest to us: A configuration $x\\in {\\mathcal A}^{\\mathbb{Z}^d}$ is called \\emph{periodic} if there exists $n \\in \\mathbb N$ such that $\\sigma^{n \\vec e_{i}}(x)=x$ for all $1\\leq i \\leq d$. A configuration $x\\in X$ is called \\emph{frozen} if its homoclinic class is a singleton. This notion coincides with the notion of frozen coloring in \\cite{brightwell2000gibbs}. A subshift $X$ will be called \\emph{frozen} if it consists of frozen configurations, equivalently $\\Delta_X$ is the diagonal. A measure on $X$ will be called \\emph{frozen} if its support is frozen. Note that any shift space consisting just of periodic configurations is frozen. All frozen nearest neighbour shifts of finite type are stiff: Suppose $X$ is a nearest neighbour shift of finite type which is not stiff. Then there is a symbol $v$ which can be config-folded to a symbol $w$. This means that any appearance of $v$ in a configuration $x\\in X$ can be replaced by $w$. Hence the homoclinic class of $x$ is not a singleton. Therefore $X$ is not frozen.\n\n\n\\begin{prop}\\label{proposition: periodicfoldentropy} Let $X$ be a nearest neighbour shift of finite type such that a sequence of config-folds starting from $X$ results in the orbit of a periodic configuration. Then every shift-invariant probability measure adapted to $X$ is fully supported.\n\\end{prop}\n\\begin{prop}\\label{proposition: frozenfoldpivot}\nLet $X$ be a nearest neighbour shift of finite type such that a sequence of config-folds starting from $X$ results in a frozen shift. Then $X$ has the pivot property.\n\\end{prop}\n\n\\noindent\\textbf{Examples:}\n\\begin{enumerate}\n\\item\n$X:=\\{0\\}^{\\mathbb{Z}^d}\\cup \\{1\\}^{\\mathbb{Z}^d}$ is a frozen shift space but not the orbit of a periodic configuration. Clearly the delta measure $\\delta_{\\{0\\}^{\\mathbb{Z}^d}}$ is a shift-invariant probability measure adapted to $X$ but not fully supported. A more non-trivial example of a nearest neighbour shift of finite type which is frozen but not the orbit of a periodic configuration is the set of the Robinson tilings $Y$ \\cite{Robinson1971}. There are configurations in $Y$ which have the so-called ``fault lines''; they can occur at most once in a given configuration. Consequently for all shift-invariant probability measures on $Y$, the probability of seeing a fault line is zero. Thus no shift-invariant probability measure (and hence no adapted shift-invariant probability measure) on $Y$ is fully supported.\n\n\\item\\label{Example: Safe Symbol}\nLet $X$ be a shift space with a safe symbol $\\star$. Then any symbol in $X$ can be config-folded into the safe symbol. By config-folding the symbols one by one, we obtain a fixed point $\\{\\star\\}^{\\mathbb{Z}^d}$. Thus any nearest neighbour shift of finite type with a safe symbol satisfies the hypothesis of both the propositions.\n\n\\item \\label{Example: Folds to an edge}Suppose $\\mathcal H$ is a graph which folds into a single edge (denoted by $Edge$) or a single vertex $v$ with a loop. Then the shift space $X_\\mathcal H$ can be\nconfig-folded to $X_{Edge}$ (which consists of two periodic configurations) or the fixed point $\\{v\\}^{\\mathbb{Z}^d}$ respectively. In the latter case, the graph $\\mathcal H$ is called \\emph{dismantlable} \\cite{nowwinkler}. Note that finite non-trivial trees and the graph $C_4$ fold into an edge. For dismantlable graphs $\\mathcal H$, Theorem 4.1 in \\cite{brightwell2000gibbs} implies the conclusions of Propositions \\ref{proposition: periodicfoldentropy} and \\ref{proposition: frozenfoldpivot} for $X_\\mathcal H$ as well.\n\\end{enumerate}\u2022\n\n\\begin{proof}[Proof of Proposition \\ref{proposition: periodicfoldentropy}] Let $\\mu$ be a shift-invariant probability measure adapted to $X$. To prove that $supp(\\mu)= X$ it is sufficient to prove that for all $n\\in \\mathbb N$ and $x\\in X$ that $\\mu([x]_{D_n})>0$. Let $X_0=X$, $X_1$, $X_2$$,\\ldots,$ $X_r$ be a sequence of full config-folds where $X_r:=\\{ \\sigma^{\\vec i_1}(p), \\sigma^{\\vec i_2}(p),\\ldots, \\sigma^{\\vec i_{k-1}}(p) \\}$ is the orbit of a periodic point. For any two configurations $z,w\\in X$ there exists $\\vec i\\in \\mathbb{Z}^d$ such that $F_X(z)= F_X(\\sigma^{\\vec i}(w)).$\nSince $\\mu$ is shift-invariant we can choose $y \\in supp (\\mu)$ such that $F_X(x)= F_X(y).$\nConsider the configurations $I_{X,n}(x)$ and $O_{X,n+2r-1}(y)$. By Proposition \\ref{prop: folding_ to _ stiffness_fixing_a_set} they satisfy the equations\n\\begin{eqnarray*}\nI_{X,n}(x)|_{D_{n+r-1}^c}&=&F_X(x)|_{D_{n+r-1}^c}\\text{ and }\\\\\nO_{X,n+2r-1}(y)|_{D_{n+r}}&=&F_X(y)|_{D_{n+r}}.\n\\end{eqnarray*}\n\\noindent Then $I_{X,n}(x)|_{\\partial D_{n+r-1}}= O_{X,n+2r-1}(y)|_{\\partial D_{n+r-1}}$. Since $X$ is a nearest neighbour shift of finite type, the configuration $z$ given by\n\\begin{eqnarray*}\nz|_{D_{n+r}}&:=&I_{X,n}(x)|_{D_{n+r}}\\\\\nz|_{D_{n+r-1}^c}&:=&O_{X,n+2r-1}(y)|_{D_{n+r-1}^c}\n\\end{eqnarray*}\n\\noindent is an element of $X$. Moreover\n\\begin{eqnarray*}\nz|_{D_{n}}&=&I_{X,n}(x)|_{D_{n}}=x|_{D_n}\\\\\nz|_{D_{n+2r-1}^c}&=&O_{X,n+2r-1}(y)|_{D_{n+2r-1}^c}=y|_{D^c_{n+2r-1}}.\n\\end{eqnarray*}\u2022\nThus $(y, z)\\in \\Delta_X$. Since $\\mu$ is adapted we get that $z\\in supp(\\mu)$. Finally\n$$\\mu([x]_{D_n})=\\mu([z]_{D_n})>0.$$\n\\end{proof}\n\nNote that all the maps being discussed here, $f_X$, $f_{X,A}$, $F_X$, $I_{X,n}$ and $O_{X,n}$ are (not necessarily shift-invariant) single block maps, that is, maps $f$ where $\\left(f(x)\\right)_{\\vec i}$ depends only on $x_{\\vec i}$. Thus if $f$ is one such map and $x|_A= y|_A$ for some set $A\\subset \\mathbb{Z}^d$ then $f(x)|_A=f(y)|_A$; they map homoclinic pairs to homoclinic pairs.\n\n\\begin{proof}[Proof of Proposition \\ref{proposition: frozenfoldpivot}] Let $X_0=X$, $X_1$, $X_2$$,\\ldots,$ $X_r$ be a sequence of full config-folds where $X_r$ is frozen. Let $(x, y) \\in \\Delta_X$. Since $X_r$ is frozen, $F_{X}(x)= F_X(y)$. Suppose $x|_{D_n^c}= y|_{D_n^c}$ for some $n\\in \\mathbb N$. Then $O_{X,n+r-1}(x)|_{D_n^c}=O_{X,n+r-1}(y)|_{D_n^c}$. Also by Proposition \\ref{prop: folding_ to _ stiffness_fixing_a_set},\n$$O_{X, n+r-1}(x)|_{D_n}=F_X(x)|_{D_n}=F_X(y)|_{D_n}= O_{X, n+r-1}(y)|_{D_n}.$$\nThis proves that $O_{X,n+r-1}(x)=O_{X,n+r-1}(y)$. In fact it completes the proof since for all $z\\in X$ there exists a chain of pivots in $X$ from $z$ to $O_{X,n+r-1}(z)$.\n\\end{proof}\n\n\n\n\\section{Universal Covers}\\label{section:universal covers}\nMost cases will not be as simple as in the proof of Propositions \\ref{proposition: periodicfoldentropy} and \\ref{proposition: frozenfoldpivot}. We wish to prove the conclusions of these propositions for hom-shifts $X_\\mathcal H$ when $\\mathcal H$ is a connected four-cycle free graph. Many ideas carry over from the proofs of these results because of the relationship of such graphs with their universal covers; we describe this relationship next. The results in this section are not original; look for instance in \\cite{Stallingsgraph1983}. We mention them for completeness.\n\nLet $\\mathcal H$ be a finite connected graph with no self-loops. We denote by $d_\\mathcal H$ the ordinary graph distance on $\\mathcal H$ and by $D_\\mathcal H(u)$, the \\emph{ball of radius 1} around $u$. A graph homomorphism $\\pi:\\mathcal C\\longrightarrow \\mathcal H$ is called a \\emph{covering map} if for some $n \\in \\mathbb N \\cup \\{\\infty\\}$ and all $u \\in \\mathcal H$, there exist disjoint sets $\\{C_i\\}_{i=1}^n\\subset \\mathcal C$ such that $\\pi^{-1}\\left(D_\\mathcal H(u)\\right)= \\cup_{i=1}^n C_i $ and $\\pi|_{C_i}: C_i\\longrightarrow D_\\mathcal H(u)$ is an isomorphism of the induced subgraphs for $1\\leq i\\leq n$. A \\emph{covering space} of a graph $\\mathcal H$ is a graph $\\mathcal C$ such that there exists a covering map $\\pi: \\mathcal C\\longrightarrow \\mathcal H$.\n\nA \\emph{universal covering space} of $\\mathcal H$ is a covering space of $\\mathcal H$ which is a tree. Unique up to graph isomorphism \\cite{Stallingsgraph1983}, these covers can be described in multiple ways. Their standard construction uses non-backtracking walks \\cite{Angluin80}: A \\emph{walk} on $\\mathcal H$ is a sequence of vertices $(v_1, v_2, \\ldots, v_n)$ such that $v_i\\sim_\\mathcal H v_{i+1}$ for all $1\\leq i \\leq n-1$. The \\emph{length} of a walk $p=(v_1, v_2, \\ldots, v_n)$ is $|p|=n-1$, the number of edges traversed on that walk. It is called \\emph{non-backtracking} if $v_{i-1}\\neq v_{i+1}$ for all $2\\leq i \\leq n-1$, that is, successive steps do not traverse the same edge. Choose a vertex $u \\in \\mathcal H$. The vertex set of the universal cover is the set of all non-backtracking walks on $\\mathcal H$ starting from $u$; there is an edge between two such walks if one extends the other by a single step. The choice of the starting vertex $u$ is arbitrary; choosing a different vertex gives rise to an isomorphic graph. We denote the universal cover by $E_\\mathcal H$. The covering map $\\pi: E_\\mathcal H\\longrightarrow \\mathcal H$ maps a walk to its terminal vertex. Usually, we will denote by $\\tilde u, \\tilde v$ and $\\tilde w$ vertices of $E_\\mathcal H$ such that $\\pi(\\tilde u)= u$, $\\pi(\\tilde v)= v$ and $\\pi(\\tilde w)= w$.\n\nThis construction shows that the universal cover of a graph is finite if and only if it is a finite tree. To see this if the graph has a cycle then the finite segments of the walk looping around the cycle give us infinitely many vertices for the universal cover. If the graph is a finite tree, then all walks must terminate at the leaves and their length is bounded by the diameter of the tree. In fact, the universal cover of a tree is itself while the universal cover of a cycle (for instance $C_4$) is $\\mathbb{Z}$ obtained by finite segments of the walks $(1, 2, 3, 0, 1, 2, 3, 0, \\ldots )$ and $(1, 0, 3, 2, 1, 0, 3, 2, \\ldots )$.\n\nFollowing the ideas of homotopies in algebraic topology, there is a natural operation on the set of walks: two walks can be joined together if one begins where the other one ends. More formally, given two walks $p=(v_1, v_2, \\ldots, v_n)$ and $q=(w_1, w_2, \\ldots, w_m)$ where $v_n=w_1$, consider $p\\star q=(v_1, v_2, \\ldots, v_n, w_2, w_3, \\ldots, w_m)$. However even when $p$ and $q$ are non-backtracking $p\\star q$ need not be non-backtracking. So we consider the walk $[p\\star q]$ instead which erases the backtracking segments of $p \\star q$, that is, if for some $i>1$, $v_{n-i+1}\\neq w_{i}$ and $v_{n-j+1}=w_j$ for all $1\\leq j \\leq i-1$ then\n$$[p\\star q]:=(v_1, v_2, \\ldots, v_{n-i+1}, w_{i-1}, w_{i}, \\ldots, w_m).$$\n\nThis operation of erasing the backtracking segments is called \\emph{reduction}, look for instance in \\cite{Stallingsgraph1983}.\nThe following proposition is well-known (Section 4 of \\cite{Stallingsgraph1983}) and shall be useful in our context as well:\n\\begin{prop}\\label{proposition:isomorphism_of_universal_covering_space}\nLet $\\mathcal H$ be a finite connected graph without any self-loops. Then for all $\\tilde{v}, \\tilde w \\in E_\\mathcal H$ satisfying $\\pi(\\tilde v)= \\pi(\\tilde w)$ there exists a graph isomorphism $\\phi: E_\\mathcal H\\longrightarrow E_\\mathcal H$ such that $\\phi(\\tilde v)= \\tilde w$ and $\\pi \\circ \\phi = \\pi$.\n\\end{prop}\n\nTo see how to construct this isomorphism, consider as an example $ (u)$, the empty walk on $\\mathcal H$ and $(v_1, v_2, \\ldots, v_n)$, some non-backtracking walk such that $v_1=v_n=u$. Then the map $\\phi: E_\\mathcal H\\longrightarrow E_\\mathcal H$ given by\n$$\\phi(\\tilde w):= [(v_1, v_2, \\ldots, v_n) \\star \\tilde w].$$\nis a graph isomorphism which maps $(u)$ to $(v_1, v_2, \\ldots, v_n)$; its inverse is $\\psi: E_\\mathcal H\\longrightarrow E_\\mathcal H$ given by\n$$\\psi(\\tilde w):= [(v_n, v_{n-1}, \\ldots, v_1)\\star \\tilde w].$$\n\nThe maps $\\phi, \\pi$ described above give rise to natural maps, also denoted by $\\phi$ and $\\pi$ where $$\\phi:X_{E_\\mathcal H}\\longrightarrow X_{E_\\mathcal H}$$\nis given by $\\phi(\\tilde x)_{\\vec{i}} := \\phi(\\tilde x_{\\vec{i}})$ and\n$$\\pi: X_{E_\\mathcal H} \\longrightarrow X_{\\mathcal H}$$\nis given by $\\pi(\\tilde x)_{\\vec{i}}:=\\pi(\\tilde x_{\\vec{i}})$ for all ${\\vec{i}} \\in \\mathbb{Z}^d$ respectively. A \\emph{lift} of a configuration $x\\in X_\\mathcal H$ is a configuration $\\tilde{x}\\in X_{E_\\mathcal H}$ such that $\\pi \\circ \\tilde{x}= x$.\n\nNow we shall analyse some consequences of this formalism in our context. More general statements (where $\\mathbb{Z}^d$ is replaced by a different graph) are true (under a different hypothesis on $\\mathcal H$), but we restrict to the four-cycle free condition. We noticed in Section \\ref{section:Folding, Entropy Minimality and the Pivot Property} that if $\\mathcal H$ is a tree then $X_{\\mathcal H}$ satisfies the conclusions of Theorems \\ref{theorem: MRF fully supported } and \\ref{theorem: pivot property for four cycle free}. Now we will draw a connection between the four-cycle free condition on $\\mathcal H$ and the formalism in Section \\ref{section:Folding, Entropy Minimality and the Pivot Property}.\n\n\\begin{prop}[Existence of Lifts]\\label{proposition:covering_space_lifting}\nLet $\\mathcal H$ be a connected four-cycle free graph. For all $x\\in X_\\mathcal H$ there exists $\\tilde{x}\\in X_{E_\\mathcal H}$ such that $\\pi(\\tilde{x})=x$. Moreover the lift $\\tilde{x}$ is unique up to a choice of $\\tilde{x}_{\\vec 0}$.\n\\end{prop}\n\n\\begin{proof}\nWe will begin by constructing a sequence of graph homomorphisms $\\tilde{x}^n:D_n \\longrightarrow E_\\mathcal H$ such that $\\pi \\circ\\tilde{x}^n =x|_{D_n}$ and $\\tilde{x}^m|_{D_n}= \\tilde{x}^n$ for all $m>n$. Then by taking the limit of these graph homomorphisms we obtain a graph homomorphism $\\tilde{x}\\in X_{E_\\mathcal H}$ such that $\\pi \\circ\\tilde{x}=x$. It will follow that given $\\tilde{x}^0$ the sequence $\\tilde{x}^n$ is completely determined proving that the lifting is unique up to a choice of $\\tilde{x}_{\\vec {0}}$.\n\nThe recursion is the following: Let $\\tilde{x}^n: D_n \\longrightarrow E_\\mathcal H$ be a given graph homomorphism for some $n\\in \\mathbb N\\cup \\{0\\}$ such that $\\pi \\circ\\tilde{x}^n=x|_{D_n}$. For any ${\\vec{ i}}\\in D_{n+1}\\setminus D_n$, choose a vertex ${\\vec{j}}\\in D_n$ such that $\\vec{j}\\sim \\vec{i}$. Then $\\pi(\\tilde{x}^n_{\\vec{j}})=x_{\\vec{j}}\\sim x_{\\vec{i}}$. Since $\\pi$ defines a local isomorphism between $E_\\mathcal H$ and $\\mathcal H$, there exists a unique vertex $\\tilde v_{\\vec{i}}\\sim \\tilde{x}^n_{\\vec{j}} \\in E_\\mathcal H$ such that $\\pi(\\tilde v_{\\vec{i}})= x_{\\vec{i}}$. Define $\\tilde{x}^{n+1}: D_{n+1}\\longrightarrow E_\\mathcal H$ by\n\\begin{equation*}\n\\tilde{x}^{n+1}_{\\vec{i}}:=\\begin{cases}\\tilde{x}^{n}_{\\vec{i}} &\\text{if } \\vec{i}\\in D_n\\\\\\tilde v_{\\vec{i}} & \\text{if } \\vec{i}\\in D_{n+1}\\setminus D_n.\\end{cases}\n\\end{equation*}\u2022\nThen clearly $\\pi \\circ \\tilde{x}^{n+1}= x|_{D_{n+1}}$ and $\\tilde{x}^{n+1}|_{D_n}=\\tilde{x}^n$. Note that the extension $\\tilde{x}^{n+1}$ is uniquely defined given $\\tilde{x}^n$.\nWe need to prove that this defines a valid graph homomorphism from $D_{n+1}$ to $E_\\mathcal H$. Let $\\vec{i}\\in D_{n+1}\\setminus D_n$ and $\\vec{j}\\in D_n$ be chosen as described above. Consider if possible any $\\vec{j}^\\prime\\neq \\vec{j} \\in D_n$ such that $\\vec{j}^\\prime \\sim \\vec{i}$. To prove that $\\tilde{x}^{n+1}$ is a graph homomorphism we need to verify that $\\tilde{x}^{n+1}_{\\vec{j}^\\prime}\\sim \\tilde{x}^{n+1}_{\\vec{i}}$.\n\nConsider $\\vec{i}^\\prime\\in D_n$ such that $\\vec{i}^\\prime\\sim \\vec{j}$ and $\\vec{j}^\\prime$. Then $\\vec{i}^\\prime, \\vec{j}, \\vec{i}$ and $\\vec{j}^\\prime$ form a four-cycle. Since $\\mathcal H$ is four-cycle free either $x_{\\vec{i}^\\prime}=x_{\\vec{i}}$ or $x_{\\vec{j}^\\prime}= x_{\\vec{j}}$.\n\nSuppose $x_{\\vec{i}^\\prime}=x_{\\vec{i}}$; the other case is similar. Since $\\pi$ is a local isomorphism and $\\tilde{x}^{n+1}_{\\vec{i}},\\tilde{x}^{n+1}_{\\vec{i^\\prime}} \\sim \\tilde{x}^{n+1}_{\\vec{j}}$, we get that $\\tilde{x}^{n+1}_{\\vec{i}}=\\tilde{x}^{n+1}_{\\vec{i}^\\prime}$. But ${\\vec{i}}', {\\vec{j}}' \\in D_n$ and $\\tilde x^{n+1}|_{D_n}= \\tilde x^{n}$ is a graph homomorphism; therefore $\\tilde{x}^{n+1}_{\\vec{i}}=\\tilde{x}^{n+1}_{\\vec{i}^\\prime}\\sim \\tilde{x}^{n+1}_{\\vec{j}^\\prime}$.\n\\end{proof}\n\n\\begin{corollary}\\label{corollary:covering_space_lifting_homoclinic}\nLet $\\mathcal H$ be a connected four-cycle free graph and $x, y\\in X_\\mathcal H$. Consider some lifts $\\tilde{x}, \\tilde{y} \\in X_{E_\\mathcal H}$ such that $\\pi(\\tilde{x})= x $ and $\\pi(\\tilde{y})=y$. If for some $\\vec{i}_0 \\in \\mathbb{Z}^d$, $\\tilde{x}_{\\vec{i}_0}= \\tilde{y}_{\\vec{i}_0}$ then $\\tilde{x}= \\tilde{y}$ on the connected subset of\n$$\\{\\vec{j} \\in \\mathbb{Z}^d\\:|\\: x_{\\vec{j}}= y_{\\vec{j}}\\}$$ which contains $\\vec{i}_0$.\n\\end{corollary}\n\n\\begin{proof}\nLet $D$ be the connected component of $\\{\\vec{i} \\in \\mathbb{Z}^d \\:|\\: x_{\\vec{i}} = y_{\\vec{i}}\\}$ and $\\tilde D$ be the connected component of $\\{\\vec{i} \\in \\mathbb{Z}^d \\:|\\: \\tilde x_{\\vec{i}} = \\tilde y_{\\vec{i}}\\}$ which contain $\\vec{i}_0$.\n\nClearly $\\tilde D \\subset D$. Suppose $\\tilde D\\neq D$. Since both $D$ and $\\tilde D$ are non-empty, connected sets there exist $\\vec{i} \\in D \\setminus \\tilde D$ and $\\vec{j} \\in \\tilde D$ such that $\\vec{i} \\sim \\vec{j}$. Then $x_{\\vec{i}}= y_{\\vec{i}}$, $x_{\\vec{j}}= y_{\\vec{j}}$ and $\\tilde x_{\\vec{j}}= \\tilde y_{\\vec{j}}$. Since $\\pi $ is a local isomorphism, the lift must satisfy $\\tilde x_{\\vec{i}} = \\tilde y_{\\vec{i}}$ implying $\\vec{i} \\in \\tilde D$. This proves that $D= \\tilde D$.\n\\end{proof}\nThe following corollary says that any two lifts of the same graph homomorphism are `identical'.\n\\begin{corollary}\\label{corollary:lift_are_isomorphic}\nLet $\\mathcal H$ be a connected four-cycle free graph. Then for all $\\tilde x^1,\\tilde x^2 \\in X_{E_\\mathcal H}$ satisfying $\\pi(\\tilde x^1) = \\pi(\\tilde x^2)= x$ there exists an isomorphism $\\phi: E_\\mathcal H \\longrightarrow E_\\mathcal H$ such that $\\phi\\circ \\tilde x^1= \\tilde x^2$.\n\\end{corollary}\n\\begin{proof} By Proposition \\ref{proposition:isomorphism_of_universal_covering_space} there exists an isomorphism\n$\\phi: E_\\mathcal H\\longrightarrow E_\\mathcal H$ such that $\\phi(\\tilde x^1_{\\vec{0}})= \\tilde x^2_{\\vec{0}}$ and $\\pi \\circ \\phi = \\pi$. Then $(\\phi\\circ\\tilde x^1)_{\\vec{0}} = \\tilde x^2_{\\vec{0}}$ and $\\pi (\\phi\\circ\\tilde x^1)= (\\pi \\circ \\phi)(\\tilde x^1)= \\pi (\\tilde x^1)= x$. By Proposition \\ref{proposition:covering_space_lifting} $\\phi\\circ\\tilde x^1= \\tilde x^2$.\n\\end{proof}\n\nIt is worth noting at this point the relationship of the universal cover described here with the universal cover in algebraic topology. Undirected graphs can be identified with $1$ dimensional CW-complexes where the set of vertices correspond to the $0$-cells, the edges to the $1$-cells of the complex and the attaching map sends the end-points of the edges to their respective vertices. With this correspondence in mind the (topological) universal covering space coincides with the (combinatorial) universal covering space described above; indeed a $1$ dimensional CW-complex is simply connected if and only if it does not have any loops, that is, the corresponding graph does not have any cycles; it is a tree. The uniqueness, existence and many such facts about the universal covering space follow from purely topological arguments; for instance look in Chapter $13$ in \\cite{MunkresTopology75} or Chapters $5$ and $6$ in \\cite{Masseyanintroduction1977}.\n\n\n\n\\section{Height Functions and Sub-Cocycles}\\label{Section:heights}\nExistence of lifts as described in the previous section enables us to measure the `rigidity' of configurations. In this section we define height functions and subsequently the slope of configurations, where steepness corresponds to this `rigidity'. The general method of using height\nfunctions is usually attributed to J.H.Conway \\cite{ThurstontilinggroupAMM}.\n\nFix a connected four-cycle free graph $\\mathcal H$. Given $x\\in X_\\mathcal H$ we can define the corresponding \\emph{height function} $h_x:\\mathbb{Z}^d\\times \\mathbb{Z}^d\\longrightarrow \\mathbb{Z}$ given by $h_x({\\vec{i}},{\\vec{j}}):=d_{E_\\mathcal H}(\\tilde{x}_{\\vec{i}},\\tilde{x}_{\\vec{j}} )$ where $\\tilde{x}$ is a lift of $x$. It follows from Corollary \\ref{corollary:lift_are_isomorphic} that $h_x$ is independent of the lift $\\tilde{x}$.\n\n\nGiven a finite subset $A\\subset \\mathbb{Z}^d$ and $x\\in X_\\mathcal H$ we define the \\emph{range of $x$ on A} as\n\\begin{equation*}\nRange_A(x):=\\max_{{\\vec{j}}_1, {\\vec{j}}_2\\in A} h_x({\\vec{j}}_1, {\\vec{j}}_2).\n\\end{equation*}\nFor all $x\\in X_\\mathcal H$\n\\begin{equation*}\nRange_A(x)\\leq Diameter(A)\n\\end{equation*}\nand more specifically\n\\begin{equation} \\label{equation:diameter_bounds_height}\nRange_{D_n}(x)\\leq 2n\n\\end{equation}\nfor all $n \\in \\mathbb N$. Since $\\tilde x\\in X_{E_\\mathcal H}$ is a map between bipartite graphs it preserves the parity of the distance function, that is, if $\\vec i, \\vec j \\in \\mathbb{Z}^d$ and $x\\in X_\\mathcal H$ then the parity of $\\|\\vec i - \\vec j\\|_1$ is the same as that of $h_x(\\vec i, \\vec j)$. As a consequence it follows that $Range_{\\partial D_n}(x)$ is even for all $x\\in X_{\\mathcal H}$ and $n \\in \\mathbb N$. We note that\n$$Range_{A}(x)= Diameter(Image(\\tilde x|_{A})).$$\n\nThe height function $h_x$ is subadditive, that is,\n$$h_x(\\vec i,\\vec j)\\leq h_x(\\vec i, \\vec k)+ h_x(\\vec k, \\vec j)$$\nfor all $x\\in X_\\mathcal H$ and $\\vec i ,\\vec j$ and $\\vec k \\in \\mathbb{Z}^d$. This is in contrast with the usual height function (as in \\cite{chandgotia2013Markov} and \\cite{peled2010high}) where there is an equality instead of the inequality. This raises some technical difficulties which are partly handled by the subadditive ergodic theorem.\n\nThe following terminology is not completely standard: Given a shift space $X$ a \\emph{sub-cocycle} is a measurable map $c: X\\times \\mathbb{Z}^d \\longrightarrow \\mathbb N\\cup \\{0\\}$ such that for all $\\vec i, \\vec j \\in \\mathbb{Z}^d$\n$$c(x, \\vec i +\\vec j)\\leq c(x, \\vec i)+ c(\\sigma^{\\vec i}(x), \\vec j).$$\nSub-cocycles arise in a variety of situations; look for instance in \\cite{Hammersleyfirst1965}. We are interested in the case $c(x, \\vec i)= h_x(\\vec 0, \\vec i)$ for all $x\\in X_\\mathcal H$ and $\\vec i \\in \\mathbb{Z}^d$. The measure of `rigidity' lies in the asymptotics of this sub-cocycle, the existence of which is provided by the subadditive ergodic theorem. Given a set $X$ if $f: X\\longrightarrow \\mathbb R$ is a function then let $f^+:=\\max(0,f)$.\n\n\\begin{thm}[Subadditive Ergodic Theorem]\\label{theorem:Subadditive_ergodic_theorem}\\cite{walters-book}\nLet $(X, \\mathcal B, \\mu)$ be a probability space and let $T: X\\longrightarrow X$ be measure preserving. Let $\\{f_n\\}_{n=1}^\\infty$ be a sequence of measurable functions $f_n: X\\longrightarrow \\mathbb R\\cup \\{-\\infty\\}$ satisfying the conditions:\n\\begin{enumerate}[(a)]\n\\item\n$f_1^+ \\in L^1(\\mu)$\n\\item\nfor each $m$, $n \\geq 1$, $f_{n+m }\\leq f_n + f_m \\circ T^n$ $\\mu$-almost everywhere.\n\\end{enumerate}\u2022\nThen there exists a measurable function $f: X\\longrightarrow \\mathbb R\\cup \\{-\\infty \\}$ such that $f^+\\in L^1(\\mu)$, $f\\circ T=f$, $\\lim_{n\\rightarrow \\infty} \\frac{1}{n}f_n =f$, $\\mu$-almost everywhere\nand\n$$\\lim_{n \\longrightarrow \\infty}\\frac{1}{n}\\int f_n d\\mu = \\inf_{n}\\frac{1}{n}\\int f_n d \\mu= \\int f d\\mu.$$\n\\end{thm}\n\nGiven a direction $\\vec{ i} =(i_1, i_2, \\ldots, i_d)\\in \\mathbb R^d$ let $\\lfloor\\vec{ i}\\rfloor=(\\lfloor i_1\\rfloor, \\lfloor i_2\\rfloor, \\ldots, \\lfloor i_d\\rfloor)$. We define for all $x \\in X_\\mathcal H$ the \\emph{slope of $x$ in the direction $\\vec{ i}$} as\n$$sl_{\\vec {i}}(x):= \\lim_{n \\longrightarrow \\infty}\\frac{1}{n} h_x(\\vec 0, \\lfloor n \\vec{ i}\\rfloor)$$\nwhenever it exists.\n\nIf $\\vec i\\in \\mathbb{Z}^d$ we note that the sequence of functions $f_n: X_\\mathcal H\\longrightarrow \\mathbb N\\cup \\{\\vec 0\\}$ given by\n$$f_n(x)=h_x(\\vec 0, n\\vec i)$$\nsatisfies the hypothesis of this theorem for any shift-invariant probability measure on $X_\\mathcal H$: $|f_1|\\leq \\|\\vec i\\|_1$ and the subadditivity condition in the theorem is just a restatement of the sub-cocycle condition described above, that is, if $T= \\sigma^{\\vec i}$ then\n$$f_{n+m }(x)= h_x(\\vec 0, (n+m)\\vec i)\\leq h_x(\\vec 0, n \\vec i)+ h_{\\sigma^{n \\vec i}x}(\\vec 0,m \\vec i ) =f_n(x) + f_m(T^n(x)).$$\nThe asymptotics of the height functions (or more generally the sub-cocycles) are a consequence of the subadditive ergodic theorem as we will describe next. In the following by an ergodic measure on $X_\\mathcal H$, we mean a probability measure on $X_\\mathcal H$ which is ergodic with respect to the $\\mathbb{Z}^d$-shift action on $X_\\mathcal H$.\n\n\n\n\\begin{prop}[Existence of Slopes]\\label{prop:existence_of_slopes}\nLet $\\mathcal H$ be a connected four-cycle free graph and $\\mu$ be an ergodic measure on $X_\\mathcal H$. Then for all $\\vec{ i}\\in \\mathbb{Z}^d$\n$$sl_{\\vec {i}}(x)=\\lim_{n \\longrightarrow \\infty}\\frac{1}{n} h_x({\\vec{0}}, n \\vec{ i})$$\nexists almost everywhere and is independent of $x$. Moreover if $\\vec {i}= (i_1, i_2\\ldots, i_d)$ then\n$$sl_{\\vec {i}}(x)\\leq \\sum_{k=1}^d |i_k| sl_{\\vec {e}_k}(x).$$\n\\end{prop}\n\\begin{proof}\nFix a direction $\\vec{ i}\\in \\mathbb{Z}^d$. Consider the sequence of functions $\\{f_n\\}_{n=1}^\\infty$ and the map $T: X_\\mathcal H\\longrightarrow X_\\mathcal H$ as described above. By the subadditive ergodic theorem there exists a function $f: X_\\mathcal H\\longrightarrow \\mathbb R\\cup \\{-\\infty\\}$ such that\n$$\\lim_{n \\rightarrow \\infty}\\frac{1}{n}f_n=f\\ almost\\ everywhere.$$\nNote that $f= sl_{\\vec{i}}$. Since for all $x\\in X_\\mathcal H$ and $n \\in\\mathbb N$, $0\\leq f_n\\leq n\\|{\\vec{i}}\\|_1$, $0\\leq f(x)\\leq \\|\\vec i\\|_1$ whenever it exists. Fix any $\\vec{j}\\in \\mathbb{Z}^d$. Then\n\\begin{eqnarray*}\nf_n(\\sigma^{\\vec{j}}(x))&=& h_{\\sigma^{\\vec{j}}(x)}({\\vec{0}}, n \\vec{i})= h_x(\\vec{j}, n \\vec{ i}+\\vec{ j})\n\\end{eqnarray*}\nand hence\n\\begin{eqnarray*}\n-h_x(\\vec{j}, {\\vec{0}}) + h_x({\\vec{0}}, n \\vec{i})- h_x(n \\vec{i}, n \\vec{i}+\\vec{j})&\\leq& f_n(\\sigma^{\\vec{j}}(x))\\\\\n&\\leq& h_x(\\vec{j},{\\vec{0}}) + h_x({\\vec{0}}, n \\vec{i})+ h_x(n \\vec{i}, n \\vec{i}+\\vec{j})\n\\end{eqnarray*}\nimplying\n\\begin{eqnarray*}\n-2\\|\\vec{j}\\|_1+ f_n(x)\\leq & f_n(\\sigma^{\\vec{j}}(x))& \\leq 2\\|\\vec{j}\\|_1+ f_n(x)\n\\end{eqnarray*}\u2022\nimplying\n$$f(x)=\\lim_{n \\longrightarrow \\infty} \\frac{1}{n} f_n(x)= \\lim_{n \\longrightarrow \\infty}\\frac{1}{n} f_n(\\sigma^{\\vec{j}} x)= f(\\sigma^{\\vec{j}}(x))$$\nalmost everywhere. Since $\\mu$ is ergodic $sl_{{\\vec{i}}}= f$ is constant almost everywhere. Let $\\vec{i}^{(k)} = (i_1, i_2, \\ldots, i_k, 0, \\ldots, 0)\\in \\mathbb{Z}^d$. By the subadditive ergodic theorem\n\\begin{eqnarray*}\nsl_{\\vec{i}}(x)= \\int sl_{\\vec{i}}(x) d\\mu&=& \\lim_{n \\longrightarrow \\infty}\\frac{1}{n}\\int h_x({\\vec{0}}, n \\vec{i}) d\\mu\\\\\n&\\leq&\\sum_{k=1}^d \\lim_{n \\longrightarrow\\infty}\\frac{1}{n} \\int h_{\\sigma^{n \\vec{i}^{(k-1)}}(x)}({\\vec{0}}, ni_{k}\\vec{e}_k ) d \\mu\\\\\n&=&\\sum_{k=1}^d \\lim_{n \\longrightarrow\\infty}\\frac{1}{n} \\int h_x({\\vec{0}}, ni_{k}\\vec{e}_k ) d \\mu\\\\\n&\\leq&\\sum_{k=1}^d|i_k|\\lim_{n \\longrightarrow\\infty}\\frac{1}{n} \\int h_x({\\vec{0}}, n\\vec{e}_k ) d \\mu\\\\\n&=&\\sum_{k=1}^d |i_k| sl_{\\vec {e}_k}(x).\n\\end{eqnarray*}\u2022\t\nalmost everywhere.\n\\end{proof}\n\\begin{corollary}\\label{corollary: existence_of _slopes_in_reality}\nLet $\\mathcal H$ be a connected four-cycle free graph. Suppose $\\mu$ is an ergodic measure on $X_\\mathcal H$. Then for all $\\vec{i}\\in \\mathbb R^d$\n$$sl_{\\vec{i}}(x)=\\lim_{n \\longrightarrow \\infty}\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{i}\\rfloor)$$\nexists almost everywhere and is independent of $x$. Moreover if $\\vec{i}= (i_1, i_2,\\ldots, i_d)$ then\n$$sl_{\\vec{i}}(x)\\leq \\sum_{k=1}^d |i_k| sl_{\\vec{e}_k}(x).$$\n\\end{corollary}\n\\begin{proof}\nLet $\\vec{i}\\in \\mathbb Q^d$ and $N\\in \\mathbb N$ such that $N \\vec{i} \\in \\mathbb{Z}^d$. For all $n \\in \\mathbb N$ there exists $k \\in \\mathbb N\\cup\\{0\\}$ and $0\\leq m\\leq N-1$ such that $n = kN+m$. Then\nfor all $x\\in X_\\mathcal H$\n$$h_x({\\vec{0}}, k N\\vec{i})- N\\|\\vec{i}\\|_1\\leq h_x({\\vec{0}}, \\lfloor n\\vec{i} \\rfloor)\\leq h_x({\\vec{0}}, k N\\vec{i})+ N\\|\\vec{i}\\|_1$$\nproving\n$$sl_{\\vec{i}}(x)=\\lim_{n\\longrightarrow\\infty}\\frac{1}{n}h_x({\\vec{0}}, \\lfloor n\\vec{ i} \\rfloor) = \\frac{1}{N}\\lim_{k \\longrightarrow \\infty} \\frac{1}{k}h_x({\\vec{0}}, k N\\vec{i}) = \\frac{1}{N}sl_{N\\vec{i}}(x)$$\nalmost everywhere. Since $sl_{N\\vec{i}}$ is constant almost everywhere, we have that $sl_{\\vec{i}}$ is constant almost everywhere as well; denote the constant by $c_{\\vec{i}}$ . Also\n$$sl_{\\vec{i}}(x)\\leq \\frac{1}{N}\\sum _{l=1}^d|N i_l|sl_{\\vec{e}_l}(x)=\\sum _{l=1}^d|i_l|sl_{\\vec{e}_l}(x).$$\n\nLet $X\\subset X_\\mathcal H$ be the set of configurations $x$ such that\n$$\\lim_{n \\longrightarrow \\infty}\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{i}\\rfloor)=c_{\\vec{i}}$$\nfor all ${\\vec{i}} \\in \\mathbb Q^d$. We have proved that $\\mu(X)=1$.\n\nFix $x\\in X$.\nLet $\\vec i, \\vec{j}\\in \\mathbb R^d$ such that $\\|\\vec{i}- \\vec{j}\\|_1<\\epsilon$. Then\n$$\\left|\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{i}\\rfloor)-\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{j}\\rfloor)\\right|\\leq\\frac{1}{n}\\|\\lfloor n \\vec{i}\\rfloor-\\lfloor n \\vec{j}\\rfloor\\|_1\\leq\\epsilon+\\frac{2d}{n}.$$\nThus we can approximate $\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{i}\\rfloor)$ for ${\\vec{i}} \\in \\mathbb R^d$ by $\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{j}\\rfloor)$ for ${\\vec{j}} \\in \\mathbb Q^d$ to prove that $\\lim_{n \\longrightarrow \\infty}\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{i}\\rfloor)$ exists for all $\\vec i \\in \\mathbb R^d$, is independent of $x\\in X$ and satisfies\n$$sl_{\\vec{i}} (x)\\leq \\sum _{k=1}^d|i_k|sl_{\\vec{e}_k}(x).$$\n\\end{proof}\n\nThe existence of slopes can be generalised from height functions to continuous sub-cocycles; the same proofs work:\n\\begin{prop}Let $c:X\\times \\mathbb{Z}^d \\longrightarrow \\mathbb R$ be a continuous sub-cocycle and $\\mu$ be an ergodic measure on $X$. Then for all $\\vec{i}\\in \\mathbb R^d$\n$$sl^c_{\\vec{i}}(x):=\\lim_{n \\longrightarrow \\infty}\\frac{1}{n} c(x, \\lfloor n \\vec{i}\\rfloor)$$\nexists almost everywhere and is independent of $x$. Moreover if $\\vec{i}= (i_1, i_2\\ldots, i_d)$ then\n$$sl^c_{\\vec{i}}(x)\\leq \\sum_{k=1}^d |i_k| sl^c_{\\vec{e}_k}(x).$$\n\\end{prop}\n\nLet $C_X$ be the space of continuous sub-cocycles on a shift space $X$. $C_X$ has a natural vector space structure: given $c_1, c_2\\in C_X$, $(c_1 +\\alpha c_2)$ is also a continuous sub-cocycle on $X$ for all $\\alpha\\in \\mathbb R$ where addition and scalar multiplication is point-wise. The following is not hard to prove and follows directly from definition.\n\\begin{prop}\\label{proposition: sub-cocycles under conjugacy}\nLet $X, Y$ be conjugate shift spaces. Then every conjugacy $f: X \\longrightarrow Y$ induces a vector-space isomorphism $ f^\\star: C_Y\\longrightarrow C_X$ given by\n$$f^\\star(c)(x, \\vec {i}):= c(f(x), \\vec{i})$$\nfor all $c\\in C_Y$, $x\\in X$ and $\\vec i \\in \\mathbb{Z}^d$. Moreover $sl^c_{\\vec i}(y)=sl^{f^\\star(c)}_{\\vec i}(f^{-1}(y))$ for all $y\\in Y$ and $\\vec i \\in \\mathbb R^d$ for which the slope $sl^c_{\\vec i}(y)$ exists.\n\\end{prop}\n\\section{Proofs of the Main Theorems} \\label{section: Proof of the main theorems}\n\\begin{proof}[Proof of Theorem \\ref{theorem: MRF fully supported }] If $\\mathcal H$ is a single edge, then $X_\\mathcal H$ is the orbit of a periodic configuration; the result follows immediately. Suppose this is not the case. The proof follows loosely the proof of Proposition \\ref{proposition: periodicfoldentropy} and morally the ideas from \\cite{lightwoodschraudnerentropy}: We prove existence of two kind of configurations in $X_\\mathcal H$, ones which are `poor' (Lemma \\ref{lemma:slope 1 is frozen}), in the sense that they are frozen and others which are `universal' (Lemma \\ref{lemma:patching_various_parts}), for which the homoclinic class is dense.\n\nIdeas for the following proof were inspired by discussions with Anthony Quas. A similar result in a special case is contained in Lemma 6.7 of \\cite{chandgotia2013Markov}.\n\n\\begin{lemma}\\label{lemma:slope 1 is frozen} Let $\\mathcal H$ be a connected four-cycle free graph and $\\mu$ be an ergodic probability measure on $X_\\mathcal H$ such that $sl_{\\vec e_k}(x)=1$ almost everywhere for some $1\\leq k \\leq d$. Then $\\mu$ is frozen and $h_\\mu=0$.\n\\end{lemma}\n\\begin{proof}\nWithout loss of generality assume that $sl_{\\vec e_1}(x)=1$ almost everywhere. By the subadditivity of the height function for all $k, n \\in \\mathbb N$ and $x\\in X_\\mathcal H$ we know that\n$$\\frac{1}{kn}h_x(\\vec 0, kn\\vec{e}_1) \\leq \\frac{1}{kn}\\sum_{m=0}^{n-1}h_x(km\\vec{e}_1, k(m+1)\\vec{e}_1)=\\frac{1}{n}\\sum_{m=0}^{n-1}\\frac{1}{k}h_{\\sigma^{km \\vec e_1} (x)}(\\vec 0, k\\vec{e}_1) \\leq 1.$$\nSince $sl_{\\vec{e}_1}(x)= 1$ almost everywhere, we get that\n$$\\lim_{n\\longrightarrow \\infty} \\frac{1}{n}\\sum_{m=0}^{n-1}\\frac{1}{k}h_{\\sigma^{km \\vec e_1} (x)}(\\vec 0, k\\vec{e}_1)=1$$\nalmost everywhere. By the ergodic theorem\n$$\\int \\frac{1}{k}h_{x}(\\vec 0, k\\vec{e}_1) d \\mu= 1.$$\nTherefore $h_{x}(\\vec 0, k\\vec{e}_1)=k$ almost everywhere which implies that\n\\begin{equation}\nh_x(\\vec i, \\vec i+k\\vec{e}_1)=k \\label{eq:slopeoneheightconstantrise}\n\\end{equation}\u2022\nfor all $\\vec i \\in \\mathbb{Z}^d$ and $k \\in \\mathbb N$ almost everywhere. Let $X\\subset supp(\\mu)$ denote the set of such configurations.\n\nFor some $n \\in \\mathbb N$ consider two patterns $a,b \\in \\mathcal L_{B_n\\cup \\partial_2 B_n}(supp(\\mu))$ such that $a|_{\\partial_2 B_n}= b|_{\\partial_2 B_n}$. We will prove that then $a|_{B_n}= b|_{B_n}$. This will prove that $\\mu$ is frozen, and $|\\mathcal L_{B_n}(supp(\\mu))|\\leq|\\mathcal L_{\\partial_2 B_n}(supp(\\mu))|\\leq |{\\mathcal A}|^{|\\partial_2 B_n|}$ implying that $h_{top}(supp(\\mu))=0$. By the variational principle this implies that $h_\\mu=0$.\n\nConsider $x, y \\in X$ such that $x|_{B_n\\cup \\partial_2 B_n}= a$ and $y|_{B_n\\cup \\partial_2 B_n}= b$. Noting that $\\partial_2 B_n$ is connected, by Corollary \\ref{corollary:covering_space_lifting_homoclinic} we can choose lifts $\\tilde x, \\tilde y\\in X_{E_\\mathcal H}$ such that $\\tilde x|_{\\partial_2 B_n}= \\tilde y|_{\\partial_2 B_n}$. Consider any $\\vec i \\in B_n$ and choose $k\\in - \\mathbb N$ such that $\\vec i + k \\vec e_1, \\vec i + (2n+2+k)\\vec e_1 \\in \\partial B_n$. Then by Equation \\ref{eq:slopeoneheightconstantrise} $d_{E_\\mathcal H}(\\tilde x_{\\vec i + k \\vec e_1}, \\tilde x_{\\vec i + (2n+2+k) \\vec e_1})= 2n+2$. But\n$$(\\tilde x_{\\vec i + k \\vec e_1},\\tilde x_{\\vec i + (k+1)\\vec e_1}, \\ldots,\\tilde x_{\\vec i + (2n+2+k) \\vec e_1} )\\text{ and }$$\n$$(\\tilde y_{\\vec i + k \\vec e_1},\\tilde y_{\\vec i + (k+1)\\vec e_1}, \\ldots,\\tilde y_{\\vec i + (2n+2+k) \\vec e_1} )$$\nare walks of length $2n+2$ from $\\tilde x_{\\vec i + k \\vec e_1}$ to $\\tilde x_{\\vec i + (2n+2+k) \\vec e_1}$. Since $E_\\mathcal H$ is a tree and the walks are of minimal length, they must be the same. Thus $\\tilde x|_{B_n}=\\tilde y|_{B_n}$. Taking the image under the map $\\pi$ we derive that\n$$a|_{B_n}=x|_{B_n}=y|_{B_n}= b|_{B_n}.$$\n\\end{proof}\n\nThis partially justifies the claim that steep slopes lead to greater `rigidity'. We are left to analyse the case where the slope is submaximal in every direction. As in the proof of Proposition 7.1 in \\cite{chandgotia2013Markov} we will now prove a certain mixing result for the shift space $X_\\mathcal H$.\n\n\\begin{lemma}\\label{lemma:patching_various_parts} Let $\\mathcal H$ be a connected four-cycle free graph and $|\\mathcal H|= r$. Consider any $x\\in X_\\mathcal H$ and some $y \\in X_\\mathcal H$ satisfying $Range_{\\partial D_{(d+1)n+3r+k}}(y)\\leq 2k$ for some $n \\in \\mathbb N$. Then\n\\begin{enumerate}\n\\item\\label{case:not_bipartite}\nIf either $\\mathcal H$ is not bipartite or $x_{\\vec 0}, y_{\\vec 0}$ are in the same partite class of $\\mathcal H$ then there exists $z\\in X_\\mathcal H$ such that\n$$z_{\\vec i}=\n\\begin{cases}\nx_{\\vec i}& \\ if \\ \\vec i \\in D_n\\\\\ny_{\\vec i} &\\ if \\ \\vec i \\in D_{(d+1)n+3r+k}^c.\n\\end{cases}$$\n\\item \\label{case:bipartite}\nIf $\\mathcal H$ is bipartite and $x_{\\vec 0}, y_{\\vec 0}$ are in different partite classes of $\\mathcal H$ then there exists $z\\in X_\\mathcal H$ such that\n$$z_{\\vec i}=\n\\begin{cases}\nx_{\\vec i+\\vec e_1} &if \\ \\vec i \\in D_n\\\\\ny_{\\vec i} & if \\ \\vec i \\in D_{(d+1)n+3r+k}^c.\n\\end{cases}$$\n\\end{enumerate}\u2022\n\\end{lemma}\nThe separation $dn+3r+k$ between the induced patterns of $x$ and $y$ is not optimal, but sufficient for our purposes.\n\\begin{proof} We will construct the configuration $z$ only in the case when $\\mathcal H$ is not bipartite. The construction in the other cases is similar; the differences will be pointed out in the course of the proof.\n\\begin{enumerate}\n\\item\n\\textbf{Boundary patterns with non-maximal range to monochromatic patterns inside.}\nLet $\\tilde y$ be a lift of $y$ and $\\mathcal T^\\prime$ be the image of $\\tilde y|_{ D_{(d+1)n+3r+k+1}}$. Let $\\mathcal T$ be a minimal subtree of $E_\\mathcal H$ such that\n$$Image(\\tilde y|_{\\partial D_{(d+1)n+3r+k}})\\subset \\mathcal T\\subset \\mathcal T^\\prime.$$\nSince $Range_{\\partial D_{(d+1)n+3r+k}}(y)\\leq 2k$, $diameter(\\mathcal T)\\leq 2k$. By Proposition \\ref{proposition:folding trees into other trees} there exists a graph homomorphism $f:\\mathcal T^\\prime \\longrightarrow \\mathcal T$ such that $f|_\\mathcal T$ is the identity. Consider the configuration $\\tilde y^1$ given by\n$$\\tilde y^1_{\\vec i}= \\begin{cases}\nf(\\tilde y_{\\vec i}) &\\text{ if }\\vec i \\in D_{(d+1)n+3r+k+1}\\\\\n\\tilde y_{\\vec i} &\\text{ otherwise.}\n\\end{cases}\u2022$$\nThe pattern\n$$\\tilde y^1|_{D_{(d+1)n+3r+k+1}}\\in \\mathcal L_{D_{(d+1)n+3r+k+1}}(X_{\\mathcal T})\\subset \\mathcal L_{D_{(d+1)n+3r+k+1}}(X_{E_\\mathcal H}).$$\nMoreover since $f|_\\mathcal T$ is the identity map,\n$$\\tilde y^1|_{D_{(d+1)n+3r+k}^c}=\\tilde y|_{D_{(d+1)n+3r+k}^c}\\in \\mathcal L_{D_{(d+1)n+3r+k}^c}(X_{E_\\mathcal H}).$$\nSince $X_{E_\\mathcal H}$ is given by nearest neighbour constraints $\\tilde y^1\\in X_{E_\\mathcal H}$.\n\nRecall that the fold-radius of a nearest neighbour shift of finite type (in our case $X_\\mathcal T$) is the total number of full config-folds required to obtain a stiff shift. Since $diameter(\\mathcal T)\\leq 2k$ the fold-radius of $X_{\\mathcal T}\\leq k$. Let a stiff shift obtained by a sequence of config-folds starting at $X_{\\mathcal T}$ be denoted by $Z$. Since $\\mathcal T$ folds into a graph consisting of a single edge, $Z$ consists of two checkerboard patterns in the vertices of an edge in $\\mathcal T$, say $\\tilde v_1$ and $\\tilde v_2$. Corresponding to such a sequence of full config-folds, we had defined in Section \\ref{section:Folding, Entropy Minimality and the Pivot Property} the outward fixing map $O_{X_\\mathcal T, (d+1)n+3r+k}$. By Proposition \\ref{prop: folding_ to _ stiffness_fixing_a_set} the configuration $O_{X_\\mathcal T,(d+1)n+3r+k}(\\tilde y^1)\\in X_{E_{\\mathcal H}}$ satisfies\n\\begin{eqnarray*}\nO_{X_\\mathcal T,(d+1)n+3r+k}(\\tilde y^1)|_{D_{(d+1)n+3r+1}}\\in \\mathcal L_{D_{(d+1)n+3r+1}}(Z) \\\\\nO_{X_\\mathcal T,(d+1)n+3r+k}(\\tilde y^1)|_{D_{(d+1)n+3r+k}^c}=\\tilde y^1|_{D_{(d+1)n+3r+k}^c}=\\tilde y|_{D_{(d+1)n+3r+k}^c}.\n\\end{eqnarray*}\n\\noindent Note that the pattern $O_{X_\\mathcal T,(d+1)n+3r+k}(\\tilde y^1)|_{\\partial D_{(d+1)n+3r}}$ uses a single symbol, say $\\tilde v_1$. Let $\\pi (\\tilde v_1)= v_1$. Then the configuration $y^\\prime= \\pi(O_{X_\\mathcal T,(d+1)n+3r+k}(\\tilde y^1))\\in X_\\mathcal H$ satisfies\n\\begin{eqnarray*}\ny^\\prime|_{\\partial D_{(d+1)n+3r}} &=& v_1\\\\\ny^\\prime|_{D_{(d+1)n+3r+k}^c}&=&y|_{D_{(d+1)n+3r+k}^c}.\n\\end{eqnarray*}\u2022\n\\item\n\\textbf{Constant extension of an admissible pattern.}\nConsider some lift $\\tilde x$ of $x$. We begin by extending $\\tilde x|_{B_n}$ to a periodic configuration $\\tilde x^1\\in X_{E_\\mathcal H}$. Consider the map $f: [-n, 3n]\\longrightarrow [-n, n]$ given by\n\\begin{equation*}\nf(k)=\\begin{cases}\nk &\\text{ if } k \\in [-n,n]\\\\\n2n-k &\\text{ if }k \\in [n,3n].\n\\end{cases}\u2022\n\\end{equation*}\n\\noindent Then we can construct the pattern $\\tilde a\\in \\mathcal L_{[-n, 3n]^d}(X_{E_\\mathcal H})$ given by\n$$\\tilde a_{i_1, i_2, \\ldots i_d}= \\tilde x_{f(i_1), f(i_2), \\ldots, f(i_d)}.$$\nGiven $k, l \\in [-n, 3n]$ if $|k-l|=1$ then $|f(k)-f(l)|=1$. Thus $\\tilde a$ is a locally allowed pattern in $X_{E_\\mathcal H}$. Moreover since $f(-n)= f(3n)$ the pattern $\\tilde a$ is `periodic', meaning,\n$$\\tilde a_{i_1, i_2,\\ldots, i_{k-1}, -n, i_{k+1}, \\ldots, i_d }= \\tilde a_{i_1, i_2, \\ldots, i_{k-1}, 3n, i_{k+1}, \\ldots, i_d }$$\nfor all $i_1, i_2, \\ldots, i_d \\in [-n,3n]$. Also $\\tilde a|_{B_n}=\\tilde x|_{B_n}$. Then the configuration $\\tilde x^1$ obtained by tiling $\\mathbb{Z}^d$ with $\\tilde a|_{[-n,3n-1]^d}$, that is,\n$$\\tilde x^1_{\\vec i}= \\tilde a_{(i_1\\!\\!\\!\\mod 4n,\\ i_2\\!\\!\\!\\mod 4n,\\ \\ldots,\\ i_d\\!\\!\\!\\mod 4n)-(n, n, \\ldots, n)}\\text{ for all }\\vec i \\in \\mathbb{Z}^d$$\nis an element of $X_{E_\\mathcal H}$. Moreover $\\tilde x^1|_{B_n}= \\tilde a|_{B_n}= \\tilde x|_{B_n}$ and $Image(\\tilde x^1)= Image(\\tilde x|_{B_n})$. Since $diameter(B_n)=2dn$, $diameter(Image(\\tilde x^1))\\leq 2dn$. Let $\\tilde \\mathcal T= Image(\\tilde x^1)$. Then the fold-radius of $X_{\\tilde \\mathcal T}$ is less than or equal to $dn$. Let a stiff shift obtained by a sequence of config-folds starting at $X_{\\tilde \\mathcal T}$ be denoted by $Z'$. Since $\\tilde \\mathcal T$ folds into a graph consisting of a single edge, $Z^\\prime$ consists of two checkerboard patterns in the vertices of an edge in $\\tilde T$, say $\\tilde w_1$ and $\\tilde w_2$. Then by Proposition \\ref{prop: folding_ to _ stiffness_fixing_a_set}\n\\begin{eqnarray*}\nI_{X_{\\tilde\\mathcal T},n}(\\tilde x^1)|_{D_n}= \\tilde x^1|_{D_n}= \\tilde x|_{D_n}\\\\\nI_{X_{\\tilde\\mathcal T},n}(\\tilde x^1)|_{D_{(d+1)n-1}^c} \\in \\mathcal L_{D_{(d+1)n-1}^c}(Z^\\prime).\n\\end{eqnarray*}\n\\noindent We note that $I_{X_{\\tilde\\mathcal T},n}(\\tilde x^1)|_{\\partial D_{(d+1)n-1}}$ consists of a single symbol, say $\\tilde w_1$. Let $\\pi(\\tilde w_1)= w_1$. Then the configuration $x^\\prime=\\pi(I_{X_{\\tilde\\mathcal T},n}(\\tilde x^1)) \\in X_{\\mathcal H}$ satisfies\n\\begin{eqnarray*}\nx^\\prime|_{D_n}= x|_{D_n}\\text{ and}\\\\\nx^\\prime|_{\\partial D_{(d+1)n-1}}=w_1.\n\\end{eqnarray*}\u2022\n\\item \\textbf{Patching of an arbitrary pattern inside a configuration with non-maximal range.}\nWe will first prove that there exists a walk on $\\mathcal H$ from $w_1$ to $v_1$, $((w_1= u_1), u_2, \\ldots, (u_{3r+2}= v_1))$.\nSince the graph is not bipartite, it has a cycle $p_1$ such that $|p_1|\\leq r-1$ and is odd. Let $v^\\prime$ be a vertex in $p_1$. Then there exist walks $p_2$ and $p_3$ from $w_1$ to $v^\\prime$ and from $v^\\prime$ to $v_1$ respectively such that $|p_2|, |p_3|\\leq r-1$. Consider any vertex $w^\\prime\\sim_\\mathcal H v_1$. If\n$3r+1-|p_2|- |p_3|$ is even then the walk\n$$p_2\\star p_3 (\\star (v_1, w^\\prime, v_1))^{\\frac{3r+1-|p_2|- |p_3|}{2}}$$\nand if not, then the walk\n$$p_2\\star p_1 \\star p_3 (\\star (v_1, w^\\prime, v_1))^{\\frac{3r+1-|p_1|-|p_2|- |p_3|}{2}}$$\nis a walk of length $3r+1$ in $\\mathcal H$ from $w_1$ to $v_1$. This is the only place where we use the fact that $\\mathcal H$ is not bipartite. If it were bipartite, then we would require that $x^\\prime_{\\vec 0}$ and $y^\\prime_{\\vec 0}$ have to be in the same partite class to construct such a walk.\n\nGiven such a walk the configuration $z$ given by\n\\begin{eqnarray*}\nz|_{D_{(d+1)n}}&=& x^\\prime|_{D_{(d+1)n}}\\\\\nz|_{D^c_{(d+1)n +3r}}&= &y^\\prime|_{D^c_{(d+1)n +3r}}\\\\\nz|_{\\partial D_{(d+1)n+i-2}}&=& u_i\\text{ for all } 1\\leq i \\leq 3r+2\n\\end{eqnarray*}\n\\noindent is an element of $X_\\mathcal H$ for which $z|_{D_n}=x^\\prime|_{D_n}=x|_{D_n}$ and $z|_{D_{(d+1)n+3r+k}^c}=y^\\prime|_{D_{(d+1)n+3r+k}^c}=y|_{D_{(d+1)n+3r+k}^c}.$\n\\end{enumerate}\u2022\n\n\\end{proof}\n\nWe now return to the proof of Theorem \\ref{theorem: MRF fully supported }. Let $\\mu$ be an ergodic probability measure adapted to $X_\\mathcal H$ with positive entropy.\n\nSuppose $sl_{\\vec e_i}(x)= \\theta_i$ almost everywhere. By Lemma \\ref{lemma:slope 1 is frozen}, $\\theta_i<1$ for all $1\\leq i \\leq d$. Let $\\theta= \\max_i \\theta_i$ and $0<\\epsilon<\\frac{1}{4}\\left(1- \\theta\\right)$. Denote by $S^{d-1}$, the sphere of radius $1$ in $\\mathbb R^d$ for the $l^1$ norm. By Corollary \\ref{corollary: existence_of _slopes_in_reality} for all $\\vec{v}\\in S^{d-1}$\n$$\\lim_{n\\longrightarrow \\infty }\\frac{1}{n}h_x({\\vec{0}}, \\lfloor n \\vec{v}\\rfloor) \\leq \\theta$$\nalmost everywhere. Since $S^{d-1}$ is compact in $\\mathbb R^d$ we can choose a finite set $\\{\\vec{v}_1, \\vec{v}_2, \\ldots, \\vec{v}_t\\} \\subset S^{d-1}$ such that for all $\\vec{v}\\in S^{d-1}$ there exists some $1\\leq i\\leq t$ satisfying $\\|\\vec{v}_i -\\vec v\\|_1<\\epsilon$. By Egoroff's theorem \\cite{Follandreal1999} given $\\epsilon$ as above there exists $N_0\\in \\mathbb N$ such that for all $n\\geq N_0$ and $1\\leq i \\leq t$\n\\begin{equation}\n\\mu(\\{x\\in X_\\mathcal H\\:|\\:h_x({\\vec{0}}, \\lfloor n \\vec{v}_i\\rfloor)\\leq n\\theta + n\\epsilon\\ for\\ all\\ 1\\leq i\\leq t\\}) >1-\\epsilon.\\label{equation:uniform_continuity_of_heights}\n\\end{equation}\u2022\nLet $\\vec{v} \\in \\partial D_{n-1}$ and $1\\leq i_0\\leq t$ such that\n$\\|\\frac{1}{n}\\vec{v}-\\vec{v}_{i_0}\\|_1<\\epsilon$. If for some $x\\in X_\\mathcal H$ and $n \\in \\mathbb N$\n$$h_x({\\vec{0}}, \\lfloor n \\vec{v}_{i_0}\\rfloor)\\leq n\\theta + n\\epsilon$$\nthen\n$$h_x({\\vec{0}}, \\lfloor \\vec{v}\\rfloor)\\leq h_x({\\vec{0}}, \\lfloor n \\vec{v}_{i_0}\\rfloor) +\\lceil n \\epsilon \\rceil \\leq n\\theta + 2n\\epsilon+1.$$\nBy Inequality \\ref{equation:uniform_continuity_of_heights} we get\n$$\\mu\\left(\\{x\\in X_\\mathcal H\\:|\\: h_x\\left({\\vec{0}}, \\lfloor \\vec{v}\\rfloor\\right)\\leq n\\theta + 2n\\epsilon+1\\ for\\ all\\ \\vec{v}\\in \\partial D_{n-1}\\}\\right) >1-\\epsilon$$\nfor all $n\\geq N_0$. Therefore for all $n\\geq N_0$ there exists $x^{(n)}\\in supp(\\mu) $ such that\n$$Range_{\\partial D_{n-1}}\\left(x^{(n)}\\right)\\leq 2n\\theta + 4 n \\epsilon +2< 2n(1- \\epsilon)+2.$$\nLet $x \\in X_\\mathcal H$ and $n_0\\in \\mathbb N$. It is sufficient to prove that $\\mu([x]_{D_{n_0-1}})>0$. Suppose $r:=|\\mathcal H|$. Choose $k \\in \\mathbb N$ such that\n\\begin{eqnarray*}\nn_0(d+1)+3r+k+1&\\geq&N_0\\\\\n2\\left(n_0(d+1)+3r+k+1\\right)(1-\\epsilon)+2&\\leq&2k.\n\\end{eqnarray*}\u2022\nThen by Lemma \\ref{lemma:patching_various_parts} there exists $z\\in X_\\mathcal H$ such that either\n\\begin{equation*}\nz_{\\vec{j}}=\n\\begin{cases}\nx_{\\vec{j}} \\quad\\quad\\quad\\quad \\quad\\quad\\: &if \\ {\\vec{j}} \\in D_{n_0}\\\\\nx^{\\left(n_0(d+1)+3r+k+1\\right)}_{\\vec{j}} \\ &if \\ {\\vec{j}} \\in D_{n_0(d+1)+3r+k}^c\n\\end{cases}\n\\end{equation*}\u2022\nor\n\\begin{equation*}\nz_{\\vec{j}}=\n\\begin{cases}\nx_{{\\vec{j}} +\\vec e_1}\\quad\\quad\\quad\\quad \\quad\\quad \\:& if \\ {\\vec{j}} \\in D_{n_0}\\\\\nx^{\\left(n_0(d+1)+3r+k+1\\right)}_{\\vec{j}} \\ &if \\ {\\vec{j}} \\in D_{n_0(d+1)+3r+k}^c.\n\\end{cases}\n\\end{equation*}\u2022\nIn either case $(z, x^{\\left(n_0(d+1)+3r+k+1\\right)})\\in \\Delta_{X_\\mathcal H}$. Since $\\mu$ is adapted to $X_\\mathcal H$, $z\\in supp(\\mu)$. In the first case we get that $\\mu([x]_{D_{n_0-1}})=\\mu([z]_{D_{n_0-1}})>0$. In the second case we get that $$\\mu([x]_{D_{n_0-1}})=\\mu(\\sigma^{\\vec e_1}([x]_{D_{n_0-1}}))=\\mu([z]_{D_{(n_0-1)}-\\vec e_1})>0.$$\nThis completes the proof.\n\n\\end{proof}\n\nEvery shift space conjugate to an entropy minimal shift space is entropy minimal. However a shift space $X$ which is conjugate to $X_\\mathcal H$ for $\\mathcal H$ which is connected and four-cycle free need not even be a hom-shift. By following the proof carefully it is possible to extract a condition for entropy minimality which is conjugacy-invariant:\n\n\\begin{thm}\\label{theorem:conjugacy_invariant_entropy minimality condition}\nLet $X$ be a shift of finite type and $c$ a continuous sub-cocycle on $X$ with the property that $c(\\cdot, {\\vec{i}})\\leq \\|{\\vec{i}}\\|_1$ for all ${\\vec{i}} \\in \\mathbb{Z}^d$. Suppose every ergodic probability measure $\\mu$ adapted to $X$ satisfies:\n\\begin{enumerate}\n\\item\nIf $sl^c_{\\vec e_i}(x)=1$ almost everywhere for some $1\\leq i \\leq d$ then $h_\\mu< h_{top}(X)$.\n\\item\nIf $sl^c_{\\vec e_i}(x)<1$ almost everywhere for all $1\\leq i \\leq d$ then $supp(\\mu)=X$.\n\\end{enumerate}\nthen $X$ is entropy minimal.\n\\end{thm}\n\nHere is a sketch: By Proposition \\ref{proposition:entropyviamme} and Theorems \\ref{thm:equiGibbs}, \\ref{theorem: ergodic decomposition of markov random fields} it is sufficient to prove that every ergodic measure of maximal entropy is fully supported. If $X$ is a shift of finite type satisfying the hypothesis of Theorem \\ref{theorem:conjugacy_invariant_entropy minimality condition} then it is entropy minimal because every ergodic measure of maximal entropy of $X$ is an ergodic probability measure adapted to $X$; its entropy is either smaller than $h_{top}(X)$ or it is fully supported. To see why the condition is conjugacy invariant suppose that $f:X\\longrightarrow Y$ is a conjugacy and $c\\in C_Y$ satisfies the hypothesis of the theorem. Then by Proposition \\ref{proposition: sub-cocycles under conjugacy} it follows that ${f^\\star}(c)\\in C_X$ satisfies the hypothesis as well.\n\n\\begin{proof}[Proof of Theorem \\ref{theorem: pivot property for four cycle free}] By Proposition \\ref{proposition: pivot for disconnected} we can assume that $\\mathcal H$ is connected. Consider some $(x, y)\\in \\Delta_{X_\\mathcal H}$. By Corollary \\ref{corollary:covering_space_lifting_homoclinic} there exist $(\\tilde x, \\tilde y)\\in \\Delta_{X_{E_\\mathcal H}}$ such that $\\pi(\\tilde x)= x$ and $\\pi(\\tilde y)=y$. It is sufficient to prove that there is a chain of pivots from $\\tilde x$ to $\\tilde y$. We will proceed by induction on $\\sum_{{\\vec{i}} \\in \\mathbb{Z}^d} d_{E_\\mathcal H}(\\tilde x_{{\\vec{i}}}, \\tilde y_{\\vec{i}})$. The induction hypothesis (on $M$) is : If $\\sum_{{\\vec{i}} \\in \\mathbb{Z}^d} d_{E_\\mathcal H}(\\tilde x_{\\vec{i}}, \\tilde y_{\\vec{i}})= 2M$ then there exists a chain of pivots from $\\tilde x$ to $\\tilde y$.\n\nWe note that $d_{E_\\mathcal H}(\\tilde x_{\\vec{i}}, \\tilde y_{\\vec{i}})$ is even for all ${\\vec{i}} \\in \\mathbb{Z}^d$ since there exists ${\\vec{i}}^\\prime\\in \\mathbb{Z}^d$ such that $\\tilde x_{{\\vec{i}}^\\prime}= \\tilde y_{{\\vec{i}}^{\\prime}}$ and hence $\\tilde x_{\\vec{i}}$ and $\\tilde y_{\\vec{i}}$ are in the same partite class of $E_\\mathcal H$ for all ${\\vec{i}} \\in \\mathbb{Z}^d$.\n\nThe base case $(M=1)$ occurs exactly when $\\tilde x$ and $\\tilde y$ differ at a single site; there is nothing to prove in this case. Assume the hypothesis for some $M\\in \\mathbb N$.\n\nConsider $(\\tilde x, \\tilde y)\\in \\Delta_{X_{E_\\mathcal H}}$ such that\n$$\\sum_{{\\vec{i}} \\in \\mathbb{Z}^d} d_{E_\\mathcal H}(\\tilde x_{\\vec{i}}, \\tilde y_{\\vec{i}})=2M+2.$$\nLet\n$$B=\\{{\\vec{j}} \\in \\mathbb{Z}^d\\:|\\: \\tilde x_{\\vec{j}} \\neq \\tilde y_{\\vec{j}}\\}$$\nand a vertex $\\tilde v\\in E_\\mathcal H$. Without loss of generality we can assume that\n\\begin{equation}\n\\max_{{\\vec{i}} \\in B} d_{E_\\mathcal H}(\\tilde v, \\tilde x_{\\vec{i}})\\geq \\max_{{\\vec{i}} \\in B} d_{E_\\mathcal H}(\\tilde v, \\tilde y_{\\vec{i}}).\\label{equation:assumption_for_pivot}\n\\end{equation}\u2022\nConsider some ${\\vec{i}}_0 \\in B$ such that\n$$d_{E_\\mathcal H}(\\tilde v, \\tilde x_{{\\vec{i}}_0})= \\max_{{\\vec{i}} \\in B}d_{E_\\mathcal H}(\\tilde v, \\tilde x_{{\\vec{i}}}).$$\nConsider the shortest walks $(\\tilde v= \\tilde v_1, \\tilde v_2, \\ldots, \\tilde v_n=\\tilde x_{{\\vec{i}}_0})$ from $\\tilde v$ to $\\tilde x_{{\\vec{i}}_0}$ and $(\\tilde v= \\tilde v^\\prime_1, \\tilde v^\\prime_2, \\ldots, \\tilde v^\\prime_{n^\\prime}=\\tilde y_{{\\vec{i}}_0})$ from $\\tilde v$ to $\\tilde y_{{\\vec{i}}_0}$. By Assumption \\ref{equation:assumption_for_pivot}, $n^\\prime\\leq n$. Since these are the shortest walks on a tree, if $\\tilde v^\\prime_k=\\tilde v_{k^\\prime}$ for some $1\\leq k\\leq n^\\prime$ and $1\\leq k^{\\prime} \\leq n$ then $k =k^{\\prime}$ and $\\tilde v_l = \\tilde v_l^\\prime$ for $1\\leq l \\leq k$. Let\n$$k_0= \\max\\{1\\leq k\\leq n^\\prime\\:|\\: \\tilde v_k^\\prime= \\tilde v_k\\}.$$\nThen the shortest walk from $\\tilde x_{{\\vec{i}}_0}$ to $\\tilde y_{{\\vec{i}}_0}$ is given by $\\tilde x_{{\\vec{i}}_0}=\\tilde v_n, \\tilde v_{n-1}, \\tilde v_{n-2}, \\ldots, \\tilde v_{k_0}, \\tilde v^\\prime_{k_0+1}, \\ldots, \\tilde v^\\prime_{n^\\prime}= \\tilde y_{{\\vec{i}}_0}$.\n\nWe will prove for all $\\vec i \\sim \\vec i_{0}$, $\\tilde x_{\\vec i}= \\tilde v_{n-1}$. This is sufficient to complete the proof since then the configuration\n$$\\tilde x^{(1)}_{{\\vec{j}}}=\n\\begin{cases}\n\\tilde x_{\\vec{j}} \\quad\\ \\:if\\ {\\vec{j}} \\neq {\\vec{i}}_0\\\\\n\\tilde v_{n-2}\\ \\ if\\ {\\vec{j}} = {\\vec{i}}_0,\n\\end{cases}$$\n\\noindent is an element of $X_{E_\\mathcal H}$, $(\\tilde x,\\tilde x^{(1)})$ is a pivot and\n$$n+n^\\prime -2 k_0 -2=d_{E_\\mathcal H}{(\\tilde x^{(1)}_{{\\vec{i}}_0}, \\tilde y_{{\\vec{i}}_0})}< d_{E_\\mathcal H}(\\tilde x_{{\\vec{i}}_0}, \\tilde y_{{\\vec{i}}_0})=n+n^\\prime -2 k_0$$\n\\noindent giving us a pair $(\\tilde x^{(1)}, \\tilde y)$ such that\n$$\\sum_{{\\vec{i}} \\in \\mathbb{Z}^d} d_{E_\\mathcal H}(\\tilde x^{(1)}_{\\vec{i}}, \\tilde y_{\\vec{i}})=\\sum_{{\\vec{i}} \\in \\mathbb{Z}^d} d_{E_\\mathcal H}(\\tilde x_{\\vec{i}}, \\tilde y_{\\vec{i}})-2= 2M$$\n\nto which the induction hypothesis applies. There are two possible cases:\n\\begin{enumerate}\n\\item\n${\\vec{i}} \\in B$: Then $d_{E_\\mathcal H}(\\tilde v, \\tilde x_{\\vec i})=d_{E_\\mathcal H}(\\tilde v, \\tilde x_{\\vec i_0})-1$ and $\\tilde x_{\\vec{i}}\\sim_{E_\\mathcal H} \\tilde x_{\\vec i_0}$. Since $E_\\mathcal H$ is a tree, $\\tilde x_{\\vec{i}}= \\tilde v_{n-1}$.\n\n\n\\item ${\\vec{i}} \\notin B$: Then $\\tilde x_{\\vec{i}}= \\tilde y_{\\vec{i}}$ and we get that $d_{E_\\mathcal H}(\\tilde x_{{\\vec{i}}_0}, \\tilde y_{{\\vec{i}}_0})=2$. Since $\\tilde x_{\\vec{i}}\\sim_{E_\\mathcal H} \\tilde x_{{\\vec{i}}_0}$, the shortest walk joining $\\tilde v$ and $\\tilde x_{{\\vec{i}}}$ must either be $\\tilde v= \\tilde v_1, \\tilde v_2, \\ldots, \\tilde v_{n-1}= \\tilde x_{{\\vec{i}}}$ or $\\tilde v= \\tilde v_1, \\tilde v_2,\\ldots,\\tilde v_{n}= \\tilde x_{{\\vec{i}}_0}, \\tilde v_{n+1}= \\tilde x_{{\\vec{i}}}$. We want to prove that the former is true. Suppose not.\n\nSince $\\tilde y_{{\\vec{i}}_0}\\sim_{E_\\mathcal H} \\tilde x_{{\\vec{i}}}$ and ${\\vec{i}}_0\\in B$, the shortest walk from $\\tilde v$ to $\\tilde y_{{\\vec{i}}_0}$ is $\\tilde v= \\tilde v_1, \\tilde v_2,\\ldots,\\tilde v_{n}= \\tilde x_{{\\vec{i}}_0}, \\tilde v_{n+1}= \\tilde x_{{\\vec{i}}}, \\tilde v_{n+2}=\\tilde y_{{\\vec{i}}_0} $. This contradicts Assumption \\ref{equation:assumption_for_pivot} and completes the proof.\n\n\\end{enumerate}\n\n\\end{proof}\n\n\n\n\\section{Further Directions}\n\\subsection{Getting Rid of the Four-Cycle Free Condition}\n\nIn the context of the results in this paper, the four-cycle free condition seems a priori artificial; we feel that in many cases it is a mere artifact of the proof. To the author, getting rid of this condition is an important and interesting topic for future research. Here we will illustrate what goes wrong when we try to apply our proofs for the simplest possible example with four-cycles, that is, $C_4$.\n\n\nWe have shown (Example \\ref{Example: Folds to an edge}) that $X_{C_4}$ satisfies the hypothesis of Propositions \\ref{proposition: periodicfoldentropy} and \\ref{proposition: frozenfoldpivot} and thus it also satisfies the conclusions of Theorems \\ref{theorem: MRF fully supported } and \\ref{theorem: pivot property for four cycle free}. The proofs of Theorems \\ref{theorem: MRF fully supported } and \\ref{theorem: pivot property for four cycle free} however rely critically on the existence of lifts to the universal cover, that is, Proposition \\ref{proposition:covering_space_lifting}. However the conclusion of this proposition does not hold for $X_{C_4}$: The universal cover of $C_4$ is $\\mathbb{Z}$ and the corresponding covering map $\\pi: \\mathbb{Z} \\longrightarrow C_4$ is given by $\\pi(i)= i \\mod 4$. By the second remark following Theorem 4.1 in \\cite{chandgotia2013Markov} it follows that the induced map $\\pi: X_{\\mathbb{Z}}\\longrightarrow X_{C_4}$ is not surjective disproving the conclusion of Proposition \\ref{proposition:covering_space_lifting} for $X_{C_4}$.\n\n\\subsection{Identification of Hom-Shifts}\n\\noindent\\textbf{Question 1:} Given a shift space $X$, are there some nice decidable conditions which imply that $X$ is conjugate to a hom-shift?\n\nBeing conjugate to a hom-shift lays many restrictions on the shift space, for instance on its periodic configurations. Consider a conjugacy $f:X\\longrightarrow X_\\mathcal H$ where $\\mathcal H$ is a finite undirected graph. Let $Z\\subset X_\\mathcal H$ be the set of configurations invariant under $\\{\\sigma^{2\\vec e_i}\\}_{i=1}^d$. Then there is a bijection between $Z$ and $\\mathcal L_A(X_\\mathcal H)$ where $A$ is the rectangular shape\n$$A:=\\{\\sum_{i=1}^{d}\\delta_i \\vec e_i\\:|\\: \\delta_i\\in \\{0,1\\}\\}$$\nbecause every pattern in $\\mathcal L_A(X_\\mathcal H)$ extends to a unique configuration in $Z$. More generally given a graph $\\mathcal H$ it is not hard to compute the number of periodic configurations for a specific finite-index subgroup of $\\mathbb{Z}^d$. Moreover periodic points are dense in these shift spaces and there are algorithms to compute approximating upper and lower bounds of their entropy \\cite{symmtricfriedlan1997,louidor2010improved}. Hence the same then has to hold for the shift space $X$ as well. We are not familiar with nice decidable conditions which imply that a shift space is conjugate to a hom-shift.\n\n\\subsection{Hom-Shifts and Strong Irreducibility}\\label{subsection: homSI}\n\n\\noindent\\textbf{Question 2:} Which hom-shifts are strongly irreducible?\n\nWe know two such conditions:\n\\begin{enumerate}\n\\item\\cite{brightwell2000gibbs}\nIf $\\mathcal H$ is a finite graph which folds into $\\mathcal H^\\prime$ then $X_\\mathcal H$ is strongly irreducible if and only if $X_{\\mathcal H'}$ is strongly irreducible. This reduces the problem to graphs $\\mathcal H$ which are stiff. For instance if $\\mathcal H$ is dismantlable, then $X_\\mathcal H$ is strongly irreducible.\n\\item\\cite{Raimundo2014}\n$X_\\mathcal H$ is single site fillable. A shift space $X_{\\mathcal F}\\subset {\\mathcal A}^{\\mathbb{Z}^d}$ is said to be \\emph{single site fillable} if for all patterns $a\\in {\\mathcal A}^{\\partial\\{\\vec 0\\}}$ there exists a locally allowed pattern in $X_{\\mathcal F}$, $b\\in {\\mathcal A}^{D_1}$ such that\n$b|_{\\partial\\{\\vec 0\\}}=a$. In case $X_{\\mathcal F}= X_\\mathcal H$ for some graph $\\mathcal H$ then it is single site fillable if and only if given vertices $v_1, v_2, \\ldots, v_{2d}\\in \\mathcal H$ there exists a vertex $v\\in \\mathcal H$ adjacent to all of them.\n\\end{enumerate}\nIt follows that $X_{K_5}$ is single site fillable and hence strongly irreducible for $d=2$. In fact strong irreducibility has been proved in \\cite {Raimundo2014} for shifts of finite type with a weaker mixing condition called TSSM. This does not cover all possible examples. For instance it was proved in \\cite{Raimundo2014} that $X_{K_4}$ is strongly irreducible for $d=2$ even though it is not TSSM and $K_4$ is stiff. We do not know if it is possible to verify whether a given hom-shift is TSSM.\n\n\\subsection{Hom-Shifts and Entropy Minimality}\\textbf{Question 3:} Given a finite connected graph $\\mathcal H$ when is $X_\\mathcal H$ entropy minimal?\n\n\n\nWe have provided some examples in the paper:\n\\begin{enumerate}\n\\item\n$\\mathcal H$ can be folded to a single vertex with a loop or a single edge. (Proposition \\ref{proposition: periodicfoldentropy})\n\\item $\\mathcal H$ is four-cycle free. (Theorem \\ref{theorem:four cycle free entropy minimal})\n\\end{enumerate}\u2022\nAgain this does not provide the full picture. For instance $X_{K_4}$ is strongly irreducible when $d=2$ and hence entropy minimal even though $K_4$ is stiff and not four-cycle free.\nA possible approach might be via identifying the right sub-cocycle and Theorem \\ref{theorem:conjugacy_invariant_entropy minimality condition}.\n\n\\noindent\\textbf{Conjecture:} Let $d=2$ and $\\mathcal H$ be a finite connected graph. Then $X_\\mathcal H$ is entropy minimal.\n\n\n\\subsection{Hom-Shifts and the Pivot Property}\\label{subsection: Hom-shifts and the pivot property}\nWe have given a list of examples of graphs $\\mathcal H$ for which the shift space $X_\\mathcal H$ has the pivot property in Section \\ref{section: the pivot property}. In this paper we have provided two further sets of examples:\n\\begin{enumerate}\n\\item\n$\\mathcal H$ can be folded to a single vertex with a loop or a single edge. (Proposition \\ref{proposition: frozenfoldpivot})\n\\item\n$\\mathcal H$ is four-cycle free. (Theorem \\ref{theorem: pivot property for four cycle free})\n\\end{enumerate}\n\nWe saw in Section \\ref{section: the pivot property} that $X_{K_4}, X_{K_5}$ do not have the pivot property when $d=2$. However they do satisfy a weaker property which we will describe next.\n\nA shift space $X$ is said to have the \\emph{generalised pivot property} if there is an $r\\in \\mathbb N$ such that for all $(x,y)\\in \\Delta_X$ there exists a chain $x^1=x, x^2, x^3, \\ldots, y=x^n\\in X$ such that $x^i$ and $x^{i+1}$ differ at most on some translate of $D_r$.\n\nIt can be shown that any nearest neighbour shift of finite type $X\\subset {\\mathcal A}^\\mathbb{Z}$ has the generalised pivot property. In higher dimensions this is not true without any hypothesis; look for instance in Section 9 in \\cite{chandgotia2013Markov}. It is not hard to prove that any single site fillable nearest neighbour shift of finite type has the generalised pivot property. This can be generalised further: in \\cite{Raimundo2014} it is proven that every shift space satisfying TSSM has the generalised pivot property.\n\n\\noindent\\textbf{Question 4:} For which graphs $\\mathcal H$ does $X_\\mathcal H$ satisfy the pivot property? What about the generalised pivot property?\n\n\\section{Acknowledgments}\nI would like to thank my advisor, Prof. Brian Marcus for dedicated reading of a million versions of this paper, numerous suggestions, insightful discussions and many other things. The line of thought in this paper was begot in discussions with Prof. Tom Meyerovitch, his suggestions and remarks have been very valuable to me. I will also like to thank Prof. Ronnie Pavlov, Prof. Sam Lightwood, Prof. Michael Schraudner, Prof. Anthony Quas, Prof. Klaus Schmidt, Prof. Mahan Mj, Prof. Peter Winkler and Raimundo Brice\\~no for giving a patient ear to my ideas and many useful suggestions. Lastly, I will like to thank Prof. Jishnu Biswas; he had introduced me to universal covers, more generally to the wonderful world of algebraic topology. This research was partly funded by the Four-Year Fellowship at the University of British Columbia. Lastly I would like to thank the anonymous referee for giving many helpful comments and corrections largely improving the quality of the paper.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introduction}\nNeural network based approaches have become popular frameworks in many machine learning research fields, showing its advantages over traditional methods. In NLP tasks, two types of neural networks are widely used: Recurrent Neural Network (RNN) and Convolutional Neural Network (CNN). \n\nRNNs are powerful models in various NLP tasks, such as machine translation \\citep{cho-etal-2014}, sentiment classification \\citep{wang-and-tian-2016,liu-emnlp-2016,wang-etal-2016,zhang-etal-2016,liang-etal-2016}, reading comprehension \\citep{kadlec-etal-2016,dhingra-etal-2016,sordoni-etal-2016,cui-etal-2016,cui-etal-2017-aoa,yang-etal-2016}, etc. \nThe recurrent neural networks can flexibly model different lengths of sequences into a fixed representation. \nThere are two main implementations of RNN: Long Short-Term Memory (LSTM) \\citep{hochreiter-1997} and Gated Recurrent Unit (GRU) \\citep{cho-etal-2014}, which solve the gradient vanishing problems in vanilla RNNs. \n\nCompared to RNN, the CNN model also shows competitive performances in some tasks, such as text classification \\citep{kim-2014}, etc.\nHowever, different from RNN, CNN sets a pre-defined convolutional kernel to ``summarize'' a fixed window of adjacent elements into blended representations, showing its ability of modeling local context.\n\nAs both global and local information is important in most of NLP tasks \\citep{luong-etal-2015}, in this paper, we propose a novel recurrent unit, called Contextual Recurrent Unit (CRU). The proposed CRU model adopts advantages of RNN and CNN, where CNN is good at modeling local context, and RNN is superior in capturing long-term dependencies. We propose three variants of our CRU model: {\\em shallow fusion}, {\\em deep fusion} and {\\em deep-enhanced fusion}. \n\nTo verify the effectiveness of our CRU model, we utilize it into two different NLP tasks: sentiment classification and reading comprehension, where the former is sentence-level modeling, and the latter is document-level modeling. \nIn the sentiment classification task, we build a standard neural network and replace the recurrent unit by our CRU model.\nTo further demonstrate the effectiveness of our model, we also tested our CRU in reading comprehension tasks with a strengthened baseline system originated from Attention-over-Attention Reader (AoA Reader) \\citep{cui-etal-2017-aoa}.\nExperimental results on public datasets show that our CRU model could substantially outperform various systems by a large margin, and set up new state-of-the-art performances on related datasets.\nThe main contributions of our work are listed as follows.\n\\begin{itemize}[leftmargin=*]\n\t\\item We propose a novel neural recurrent unit called Contextual Recurrent Unit (CRU), which effectively incorporate the advantage of CNN and RNN. Different from previous works, our CRU model shows its excellent flexibility as GRU and provides better performance.\n\t\\item The CRU model is applied to both sentence-level and document-level modeling tasks and gives state-of-the-art performances.\n\t\\item The CRU could also give substantial improvements in cloze-style reading comprehension task when the baseline system is strengthened by incorporating additional features which will enrich the representations of unknown words and make the texts more readable to the machine.\n\\end{itemize}\n\n\\section{Related Works}\\label{related-work}\n\nGated recurrent unit (GRU) has been proposed in the scenario of neural machine translations \\citep{cho-etal-2014}. It has been shown that the GRU has comparable performance in some tasks compared to the LSTM. Another advantage of GRU is that it has a simpler neural architecture than LSTM, showing a much efficient computation.\n\nHowever, convolutional neural network (CNN) is not as popular as RNNs in NLP tasks, as the texts are formed temporally. But in some studies, CNN shows competitive performance to the RNN models, such as text classification \\citep{kim-2014}.\n\nVarious efforts have been made on combining CNN and RNN.\n\\citet{wang-etal-2016} proposed an architecture that combines CNN and GRU model with pre-trained word embeddings by word2vec. \n\\citet{liang-etal-2016} proposed to combine asymmetric convolution neural network with the bidirectional LSTM network. \n\\citet{zhang-etal-2016} presented Dependency Sensitive CNN, which hierarchically construct text by using LSTMs and extracting features with convolution operations subsequently. \n\\citet{cai-etal-2016} propose to make use of dependency relations information in the shortest dependency path (SDP) by combining CNN and two-channel LSTM units. \n\\citet{kim-etal-2016} build a neural network for dialogue topic tracking where the CNN used to account for semantics at individual utterance and RNN for modeling conversational contexts along multiple turns in history.\n\n\nThe difference between our CRU model and previous works can be concluded as follows.\n\\begin{itemize}[leftmargin=*]\n  \\item Our CRU model could adaptively control the amount of information that flows into different gates, which was not studied in previous works.\n  \\item Also, the CRU does not introduce a pooling operation, as opposed to other works, such as CNN-GRU \\citep{wang-etal-2016}. Our motivation is to provide flexibility as the original GRU, while the pooling operation breaks this law (the output length is changed), and it is unable to do exact word-level attention over the output. However, in our CRU model, the output length is the same as the input's and can be easily applied to various tasks where the GRU used to. \n  \\item We also observed that by only using CNN to conclude contextual information is not strong enough. So we incorporate the original word embeddings to form a \"word + context\" representation for enhancement.\n\\end{itemize}\n\n\n\\section{Our approach}\\label{cru}\nIn this section, we will give a detailed introduction to our CRU model.\nFirstly, we will give a brief introduction to GRU \\citep{cho-etal-2014} as preliminaries, and then three variants of our CRU model will be illustrated.\n\n\\subsection{Gated Recurrent Unit}\nGated Recurrent Unit (GRU) is a type of recurrent unit that models sequential data \\citep{cho-etal-2014}, which is similar to LSTM but is much simpler and computationally effective than the latter one. We will briefly introduce the formulation of GRU.\nGiven a sequence $x = \\{x_1, x_2, ..., x_n\\}$, GRU will process the data in the following ways. For simplicity, the bias term is omitted in the following equations.\n\\begin{gather}\nz_t = \\sigma(W_z x_t+U_z h_{t-1}) \\\\\nr_t = \\sigma(W_r x_t+U_r h_{t-1}) \\\\\n\\widetilde{h_t} = \\tanh(W x_t+U [r_t \\odot h_{t-1}]) \\\\\nh_t = z_t  h_{t-1} + (1-z_t) \\widetilde{h_t}\n\\end{gather}\n\nwhere $z_t$ is the update gate, $r_t$ is the reset gate, and non-linear function $\\sigma$ is often chosen as $sigmoid$ function.\nIn many NLP tasks, we often use a bi-directional GRU, which takes both forward and backward information into account.\n\n\n\\subsection{Contextual Recurrent Unit}\nBy only modeling word-level representation may have drawbacks in representing the word that has different meanings when the context varies. \nHere is an example that shows this problem.\n\n\\begin{quote}\n\\begin{scriptsize}\\begin{verbatim}\nThere are many fan mails in the mailbox. \nThere are many fan makers in the factory.\n\\end{verbatim}\\end{scriptsize}\n\\end{quote}\n\nAs we can see that, though two sentences share the same beginning before the word {\\em fan}, the meanings of the word {\\em fan} itself are totally different when we meet the following word {\\em mails} and {\\em makers}. The first {\\em fan} means ``a person that has strong interests in a person or thing\", and the second one means ``a machine with rotating blades for ventilation\".\nHowever, the embedding of word {\\em fan} does not discriminate according to the context. \nAlso, as two sentences have the same beginning, when we apply a recurrent operation (such as GRU) till the word {\\em fan}, the output of GRU does not change, though they have entirely different meanings when we see the following words.\n\nTo enrich the word representation with local contextual information and diminishing the word ambiguities, we propose a model as an extension to the GRU, called Contextual Recurrent Unit (CRU).\nIn this model, we take full advantage of the convolutional neural network and recurrent neural network, where the former is good at modeling local information, and the latter is capable of capturing long-term dependencies.\nMoreover, in the experiment part, we will also show that our bidirectional CRU could also significantly outperform the bidirectional GRU model.\n\nIn this paper, we propose three different types of CRU models: {\\em shallow fusion}, {\\em deep fusion} and {\\em deep-enhanced fusion},  from the most fundamental one to the most expressive one. \nWe will describe these models in detail in the following sections.\n\n\\subsubsection{Shallow Fusion}\nThe most simple one is to directly apply a CNN layer after the embedding layer to obtain blended contextual representations. Then a GRU layer is applied afterward. We call this model as {\\em shallow fusion}, because the CNN and RNN are applied linearly without changing inner architectures of both. \n\nFormally, when given a sequential data $x = \\{x_1, x_2, ..., x_n\\}$, a shallow fusion of CRU can be illustrated as follows.\n\\begin{gather} \ne_t = W_e \\cdot x_t ~~;~~ c_t = \\phi(\\widetilde{e_t}) \\\\\nh_t = GRU(h_{t-1}, c_t) \n\\end{gather}\n\nWe first transform word $x_t$ into word embeddings through an embedding matrix $W_e$.\nThen a convolutional operation $\\phi$ is applied to the context of $e_t$, denoted as $\\widetilde{e_t}$, to obtain contextual representations.\nFinally, the contextual representation $c_t$ is fed into GRU units.\n\nFollowing \\cite{kim-2014}, we apply embedding-wise convolution operation, which is commonly used in natural language processing tasks.\nLet $e_{i:j} \\in\\mathbb{R}^{\\mathcal \\\\j*d}$ denote the concatenation of $j-i+1$ consecutive $d$-dimensional word embeddings.\n\\begin{equation} e_{i:j} = concat[e_i, e_{i+1}, ..., e_j] \\end{equation}\n\nThe embedding-wise convolution is to apply a convolution filter {\\bf w} $\\in\\mathbb{R}^{\\mathcal \\\\k*d}$  to a window of $k$ word embeddings to generate a new feature, i.e., summarizing a local context of $k$ words. \nThis can be formulated as\n\\begin{equation} c_i = f({\\bf w} \\cdot e_{i:i+k-1} + b) \\end{equation}\nwhere $f$ is a non-linear function and $b$ is the bias.\n\nBy applying the convolutional filter to all possible windows in the sentence, a feature map $c$ will be generated.\nIn this paper, we apply a {\\em same-length} convolution (length of the sentence does not change), i.e. $c \\in\\mathbb{R}^{\\mathcal \\\\n*1}$.\nThen we apply $d$ filters with the same window size to obtain multiple feature maps.\nSo the final output of CNN has the shape of $C \\in\\mathbb{R}^{\\mathcal \\\\n*d}$, which is exactly the same size as $n$ word embeddings, which enables us to do exact word-level attention in various tasks.\n\n\\subsubsection{Deep Fusion}\nThe contextual information that flows into the update gate and reset gate of GRU is identical in shallow fusion.\nIn order to let the model adaptively control the amount of information that flows into these gates, we can embed CNN into GRU in a deep manner. We can rewrite the Equation 1 to 3 of GRU as follows.\n\\begin{gather}\nz_t = \\sigma(\\phi_z(\\widetilde{e_t})) + U_z h_{t-1}) \\\\\nr_t = \\sigma(\\phi_r(\\widetilde{e_t})) + U_r h_{t-1}) \\\\\n\\widetilde{h_t} = \\tanh(\\phi(\\widetilde{e_t}))+U [r_t \\odot h_{t-1}]) \n\\end{gather}\n\nwhere $\\phi_z, \\phi_r, \\phi$ are three different CNN layers, i.e., the weights are not shared.\nWhen the weights share across these CNNs, the deep fusion will be degraded to shallow fusion.\n\n\\subsubsection{Deep-Enhanced Fusion}\nIn shallow fusion and deep fusion, we used the convolutional operation to summarize the context.\nHowever, one drawback of them is that the original word embedding might be blurred by blending the words around it, i.e., applying the convolutional operation on its context. \n\nFor better modeling the original word and its context, we enhanced the deep fusion model with original word embedding information, with an intuition of ``enriching word representation with contextual information while preserving its basic meaning''.\nFigure \\ref{deep-e-example} illustrates our motivations.\n\n\\begin{figure}[tp]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{example2.pdf}\n  \\caption{\\label{deep-e-example} An intuitive illustration of variants of the CRU model. The gray scale represents the amount of information. a) original sentence; b) original representation of word ``shortcut''; c) applying convolutional filter (length=3); d) adding original word embedding; }\n\\end{figure}\n\nFormally, the Equation 9 to 11 can be further rewritten into\n\\begin{gather}\nz_t = \\sigma(W_z(\\phi_z(\\widetilde{e_t}) + e_t) + U_z h_{t-1} \\\\\nr_t = \\sigma(W_r(\\phi_r(\\widetilde{e_t}) + e_t) + U_r h_{t-1}) \\\\\n\\widetilde{h_t} = \\tanh(W(\\phi(\\widetilde{e_t}) + e_t)+U [r_t \\odot h_{t-1}])\n\\end{gather}\n\nwhere we add original word embedding $e_t$ after the CNN operation, to ``enhance'' the original word information while not losing the contextual information that has learned from CNNs.\n\n\n\\section{Applications}\\label{application}\nThe proposed CRU model is a general neural recurrent unit, so we could apply it to various NLP tasks.\nAs we wonder whether the CRU model could give improvements in both sentence-level modeling and document-level modeling tasks, in this paper, we applied the CRU model to two NLP tasks: sentiment classification and cloze-style reading comprehension.\nIn the sentiment classification task, we build a simple neural model and applied our CRU.\nIn the cloze-style reading comprehension task, we first present some modifications to a recent reading comprehension model, called AoA Reader \\cite{cui-etal-2017-aoa}, and then replace the GRU part by our CRU model to see if our model could give substantial improvements over strong baselines.\n\n\\subsection{Sentiment Classification}\\label{sentiment-classification}\nIn the sentiment classification task, we aim to classify movie reviews, where one movie review will be classified into the positive\/negative or subjective\/objective category.\nA general neural network architecture for this task is depicted in Figure \\ref{sc-arch}.\n\n\\begin{figure}[tp]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{sc-arch2.pdf}\n  \\caption{\\label{sc-arch} A general neural network architecture of sentiment classification task.}\n\\end{figure}\n\nFirst, the movie review is transformed into word embeddings.\nAnd then, a sequence modeling module is applied, in which we can adopt LSTM, GRU, or our CRU, to capture the inner relations of the text.\nIn this paper, we adopt bidirectional recurrent units for modeling sentences, and then the final hidden outputs are concatenated.\nAfter that, a fully connected layer will be added after sequence modeling.\nFinally, the binary decision is made through a single $sigmoid$ unit.\n\nAs shown, we employed a straightforward neural architecture to this task, as we purely want to compare our CRU model against other sequential models.\nThe detailed experimental result of sentiment classification will be given in the next section.\n\n\n\\subsection{Reading Comprehension}\\label{rc-task}\nBesides the sentiment classification task, we also tried our CRU model in cloze-style reading comprehension, which is a much complicated task. \nIn this paper, we strengthened the recent AoA Reader \\cite{cui-etal-2017-aoa} and applied our CRU model to see if we could obtain substantial improvements when the baseline is strengthened.\n\n\\subsubsection{Task Description}\nThe cloze-style reading comprehension is a fundamental task that explores relations between the document and the query.\nFormally, a general cloze-style query can be illustrated as a triple $\\langle {\\mathcal D}, {\\mathcal Q}, {\\mathcal A} \\rangle$, where $\\mathcal D$ is the document, $\\mathcal Q$ is the query and the answer $\\mathcal A$. \nNote that the answer is a {\\em single} word in the document, which requires us to exploit the relationship between the document and query.\n\n\n\\subsubsection{Modified AoA Reader}\nIn this section, we briefly introduce the original AoA Reader \\cite{cui-etal-2017-aoa}, and illustrate our modifications.\nWhen a cloze-style training triple $\\langle \\mathcal D, \\mathcal Q, \\mathcal A \\rangle$ is given, the Modified AoA Reader will be constructed in the following steps.\nFirst, the document and query will be transformed into continuous representations with the embedding layer and recurrent layer.\nThe recurrent layer can be the simple RNN, GRU, LSTM, or our CRU model.\n\nTo further strengthen the representation power, we show a simple modification in the embedding layer, where we found strong empirical results in performance.\nThe main idea is to utilize additional sparse features of the word and add (concatenate) these features to the word embeddings to enrich the word representations. The additional features have shown effective in various models \\cite{dhingra-etal-2016,pengli-etal-2016,yang-etal-2016}.\nIn this paper, we adopt two additional features in document word embeddings (no features applied to the query side).\n\n\\noindent{{$\\bullet$~~ \\bf Document word frequency}}: Calculate each document word frequency. This helps the model to pay more attention to the important (more mentioned) part of the document.\n\\begin{equation} freq(d) = \\frac{word\\_count(d)}{length(\\mathcal D)}, d\\in \\mathcal D \\end{equation}\n\n\\noindent{{$\\bullet$~~ \\bf Count of query word}}: Count the number of each document word appeared in the query. For example, if a document word appears three times in the query, then the feature value will be 3. We empirically find that instead of using binary features (appear=1, otherwise=0) \\cite{pengli-etal-2016}, indicating the count of the word provides more information, suggesting that the more a word occurs in the query, the less possible the answer it will be.\nWe replace the Equation 16 with the following formulation (query side is not changed), \n\\begin{equation}  \\small e(x) = concat[W_e \\cdot x, freq(x), CoQ(x)] , x\\in \\mathcal D \\end{equation}\nwhere $freq(x)$ and $CoQ(x)$ are the features that introduced above. \n\\begin{gather}\n{\\small \\overrightarrow{h_s(x)}} =  {\\small \\overrightarrow{RNN}(e(x)) ; \\overleftarrow{h_s(x)} = \\overleftarrow{RNN}(e(x))} \\\\\nh_s(x) = [\\overrightarrow{h_s(x)}; \\overleftarrow{h_s(x)}]\n\\end{gather}\n\nOther parts of the model remain the same as the original AoA Reader. For simplicity, we will omit this part, and the detailed illustrations can be found in \\citet{cui-etal-2017-aoa}.\n\n\\section{Experiments: Sentiment Classification}\\label{experiments-sc}\n\n\\subsection{Experimental Setups}\nIn the sentiment classification task, we tried our model on the following public datasets.\n\\begin{itemize}[leftmargin=*]\n  \\item {\\bf MR}\\footnote{\\url{http:\/\/www.cs.cornell.edu\/People\/pabo\/movie-review-data\/}} Movie reviews with one sentence each. Each review is classified into positive or negative \\cite{pang-and-lee-2005}.\n  \\item {\\bf IMDB}\\footnote{\\url{http:\/\/ai.stanford.edu\/~amaas\/data\/sentiment\/}} Movie reviews from IMDB website, where each movie review is labeled with binary classes, either positive or negative \\cite{maas-etal-2011}. Note that each movie review may contain several sentences.\n  \\item {\\bf SUBJ}$^1$ Movie review labeled with subjective or objective \\cite{pang-and-lee-2004}. \n\\end{itemize}\n\nThe statistics and hyper-parameter settings of these datasets are listed in Table \\ref{imdb-stats}.\n \n        \\begin{table}[htp]\n        \\begin{center}\n       \n        \\begin{tabular}{lccc}\n        \\toprule\n        & \\bf MR & \\bf IMDB & \\bf SUBJ \\\\\n        \\midrule\n        Train \\# & 10,662 & 25,000 & 10,000 \\\\\n       \n        Test \\# & 10-CV & 25,000 & 10-CV \\\\\n        \\midrule\n        Embed. size & 200 & 256 & 200 \\\\\n        Hidden size & 200 & 256 & 200 \\\\\n        Dropout & 0.3 & 0.3 & 0.4 \\\\\n        Pre-train Embed. & GloVe & - & GloVe \\\\\n        Initial LR & 0.0005 & 0.001 & 0.0005 \\\\\n        Vocab truncation & - & 50,000 & - \\\\\n        \\bottomrule\n        \\end{tabular}\n        \\end{center}\n        \\caption{\\label{imdb-stats} Statistics and hyper-parameter settings of MR, IMDB and SUBJ datasets. 10-CV represents 10-fold cross validation.}\n        \\end{table} \n\n        \nAs these datasets are quite small and overfit easily, we employed $l_2$-regularization of 0.0001 to the embedding layer in all datasets. \nAlso, we applied dropout \\cite{srivastava-etal-2014} to the output of the embedding layer and fully connected layer.\nThe fully connected layer has a dimension of 1024.\nIn the MR and SUBJ, the embedding layer is initialized with 200-dimensional GloVe embeddings (trained on 840B token) \\cite{pennington-etal-2014} and fine-tuned during the training process.\nIn the IMDB condition, the vocabulary is truncated by descending word frequency order.\nWe adopt batched training strategy of 32 samples with ADAM optimizer \\cite{kingma2014adam}, and clipped gradient to 5 \\cite{pascanu-etal-2013}.\nUnless indicated, the convolutional filter length is set to 3, and ReLU for the non-linear function of CNN in all experiments.\nWe use 10-fold cross-validation (CV) in the dataset that has no train\/valid\/test division.     \n\n\\subsection{Results}\\label{result-sc}\n\nThe experimental results are shown in Table \\ref{exp-class}.\nAs we mentioned before, all RNNs in these models are {\\bf bi-directional}, because we wonder if our bi-CRU could still give substantial improvements over bi-GRU which could capture both history and future information.\nAs we can see that, all variants of our CRU model could give substantial improvements over the traditional GRU model, where a maximum gain of 2.7\\%, 1.0\\%, and 1.9\\% can be observed in three datasets, respectively.\nWe also found that though we adopt a straightforward classification model, our CRU model could outperform the state-of-the-art systems by 0.6\\%, 0.7\\%, and 0.8\\% gains respectively, which demonstrate its effectiveness. \nBy employing more sophisticated architecture or introducing task-specific features, we think there is still much room for further improvements, which is beyond the scope of this paper.\n\n\n       \\begin{table}[t]\n        \\begin{center}\n        \\small\n        \\begin{tabular}{lccc}\n        \\toprule\n        \\bf System & \\bf MR & \\bf IMDB & \\bf SUBJ \\\\\n        \\midrule\n       \n        Multi-channel CNN & 81.1 & - & 93.2 \\\\\n       \n        HRL  & - & 90.9 & - \\\\\n        Multi-task arc-II  & - & {\\em 91.2} & {\\em 95.0} \\\\\n        CNN-GRU-wordvec  & 82.3 & - & - \\\\\n        DSCNN-Pretrain  & 82.2 & 90.7 & 93.9 \\\\\n        LR-Bi-LSTM & 82.1 & - & - \\\\\n        AC-BLSTM & 83.1& - & 94.2 \\\\\n        G-AC-BLSTM & {\\em 83.7} & -& 94.3 \\\\\n        \\midrule\\midrule\n        GRU & 81.0 & 90.9 & 93.9 \\\\\n        CRU (shallow fusion) & 82.1 & 91.3 & 95.0 \\\\\n        CRU (deep fusion) & 82.7 & 91.5 & 95.2 \\\\\n        CRU (deep-enhanced, filter=3) & {\\bf 83.7} & {\\bf 91.9} & {\\bf 95.8} \\\\\n        CRU (deep-enhanced, filter=5) & 83.2 & 91.7 & 95.2 \\\\\n        \\bottomrule\n        \\end{tabular}\n        \\end{center}\n        \\caption{\\label{exp-class} Results on MR, IMDB and SUBJ sentiment classification task. Best previous results are marked in italics, and overall best results are mark in bold face. {\\small {\\bf Multi-channel CNN} \\cite{kim-2014}: A CNN architecture with static and non-static word embeddings. {\\bf HRL} \\cite{wang-and-tian-2016}: A hybrid residual LSTM architecture. {\\bf Multi-task arc-II} \\cite{liu-emnlp-2016}: A deep architectures with shared local-global hybrid memory for multi-task learning. {\\bf CNN-GRU-word2vec} \\cite{wang-etal-2016}: An architecture that combines CNN and GRU model with pre-trained word embeddings by {\\em word2vec}. {\\bf DSCNN-Pretrain} \\cite{zhang-etal-2016}: Dependency sensitive convolutional neural networks with pretrained sequence autoencoders. {\\bf AC-BLSTM} \\cite{liang-etal-2016}: Asymmetric convolutional bidirectional LSTM networks. } }\n        \\end{table}\n        \nWhen comparing three variants of the CRU model, as we expected, the CRU with {\\em deep-enhanced fusion} performs best among them. This demonstrates that by incorporating contextual representations with original word embedding could enhance the representation power.\nAlso, we noticed that when we tried a larger window size of the convolutional filter, i.e., 5 in this experiment, does not give a rise in the performance. \nWe plot the trends of MR test set accuracy with the increasing convolutional filter length, as shown in Figure \\ref{mr-length}.\n\n    \\begin{figure}[t]\n      \\centering\n      \\includegraphics[width=0.45\\textwidth]{mr-length.pdf}\n      \\caption{\\label{mr-length} Trends of MR test set accuracy with the increasing convolutional filter length. }\n    \\end{figure}\n    \nAs we can see that, using a smaller convolutional filter does not provide much contextual information, thus giving a lower accuracy.\nOn the contrary, the larger filters generally outperform the lower ones, but not always.\nOne possible reason for this is that when the filter becomes larger, the amortized contextual information is less than a smaller filter, and make it harder for the model to learn the contextual information. However, we think the proper size of the convolutional filter may vary task by task. Some tasks that require long-span contextual information may benefit from a larger filter.\n    \nWe also compared our CRU model with related works that combine CNN and RNN \\cite{wang-etal-2016,zhang-etal-2016,liang-etal-2016}. \nFrom the results, we can see that our CRU model significantly outperforms previous works, which demonstrates that by employing {\\em deep fusion} and enhancing the contextual representations with original embeddings could substantially improve the power of word representations.\n    \nOn another aspect, we plot the trends of IMDB test set accuracy during the training process, as depicted in Figure \\ref{imdb-train}.\nAs we can see that, after iterating six epochs of training data, all variants of CRU models show faster convergence speed and smaller performance fluctuation than the traditional GRU model, which demonstrates that the proposed CRU model has better training stability.\n\n   \\begin{figure}[tb]\n      \\centering\n      \\includegraphics[width=0.45\\textwidth]{imdb-4.pdf}\n      \\caption{\\label{imdb-train} Trends of IMDB test set accuracy with the training time growing.}\n    \\end{figure}\n\n\\section{Experiments: Reading Comprehension}\\label{experiments-rc}        \n    \n    \\begin{table*}[ht]\n    \\begin{center}\n   \n    \\begin{tabular}{p{9cm} ccccc}\n    \\toprule\n    & \\multicolumn{2}{p{2cm}}{\\centering \\bf CBT NE} & \\multicolumn{2}{p{2cm}}{\\centering \\bf CBT CN} \\\\\n    & \\bf Valid & \\bf Test & \\bf Valid & \\bf Test\\\\\n    \\midrule\n    Human \\cite{hill-etal-2015} & - & {\\em 81.6} & - & {\\em 81.6} \\\\\n    MemNN \\cite{hill-etal-2015} & 70.4 & 66.6 & 64.2 & 63.0 \\\\ \n    AS Reader \\cite{kadlec-etal-2016} & 73.8 & 68.6 & 68.8 & 63.4 \\\\\n    GA Reader \\cite{dhingra-etal-2016} & 74.9 & 69.0 & 69.0 & 63.9 \\\\\n   \n    Iterative Attention \\cite{sordoni-etal-2016} & 75.2 & 68.6 & 72.1 & 69.2 \\\\\n    AoA Reader \\cite{cui-etal-2017-aoa} & 77.8 & 72.0 & 72.2 & 69.4 \\\\\n    NSE Adp. Com. \\cite{munkhdalai2016reasoning} & 78.2 & 73.2 & 74.2 & 71.4 \\\\\n    GA Reader + Fine-gating \\cite{yang-etal-2016} & 79.1 & {\\em 75.0} & 75.3 & 72.0  \\\\\n    AoA Reader + Re-ranking \\cite{cui-etal-2017-aoa} & {\\em 79.6} & 74.0 & {\\em 75.7} & {\\em 73.1} \\\\\n    \\midrule\n    M-AoA Reader (GRU) & 78.0 & 73.8 & 72.8 & 69.8 \\\\\n    M-AoA Reader (CRU) & 79.5 & 75.4 & 74.4 & 71.3 \\\\\n    M-AoA Reader (CRU) + Re-ranking & {\\bf 80.6} & {\\bf 76.1} & {\\bf 76.6} & {\\bf 74.5} \\\\\n    \\midrule\\midrule\n    AS Reader (Ensemble) & 74.5 & 70.6 & 71.1 & 68.9 \\\\\n    KnReader (Ensemble) & 78.0 & 73.3 & 72.2 & 70.6 \\\\\n    Iterative Attention (Ensemble) & 76.9 & 72.0 & 74.1 & 71.0 \\\\\n    AoA Reader (Ensemble) & 78.9 & 74.5 & 74.7 & 70.8 \\\\\n    AoA Reader (Ensemble + Re-ranking) & {\\em 80.3} & {\\em 75.7} & {\\em 77.0} & {\\em 74.1} \\\\\n    \\midrule\n    M-AoA Reader (CRU) (Ensemble) & 80.0 & 77.1 & 77.0 & 73.5 \\\\   \n    M-AoA Reader (CRU) (Ensemble + Re-ranking) & {\\bf 81.8} & {\\bf 77.5} & {\\bf 79.0} & {\\bf 76.8} \\\\        \n    \\bottomrule\n    \\end{tabular}\n    \\end{center}\n    \\caption{\\label{public-result} Results on the CBT NE and CN cloze-style reading comprehension datasets.\n     }\n    \\end{table*}\n\n\\subsection{Experimental Setups}   \nWe also tested our CRU model in the cloze-style reading comprehension task.\nWe carried out experiments on the public datasets: CBT NE\/CN \\cite{hill-etal-2015}.\nThe CRU model used in these experiments is the {\\em deep-enhanced} type with the convolutional filter length of 3.\nIn the re-ranking step, we also utilized three features: Global LM, Local LM, Word-class LM, as proposed by \\citet{cui-etal-2017-aoa}, and all LMs are 8-gram trained by SRILM toolkit \\cite{stolcke-2002}.\nFor other settings, such as hyperparameters, initializations, etc., we closely follow the experimental setups as \\citet{cui-etal-2017-aoa} to make the experiments more comparable.\n\n\n\\subsection{Results}\nThe overall experimental results are given in Table \\ref{public-result}. \nAs we can see that our proposed models can substantially outperform various state-of-the-art systems by a large margin.\n\n\\begin{itemize}[leftmargin=*]\n  \\item Overall, our final model (M-AoA Reader + CRU + Re-ranking) could give significant improvements over the previous state-of-the-art systems by 2.1\\% and 1.4\\% in test sets, while re-ranking and ensemble bring further improvements.\n  \\item When comparing M-AoA Reader to the original AoA Reader, 1.8\\% and 0.4\\% improvements can be observed, suggesting that by incorporating additional features into embedding can enrich the power of word representation. Incorporating more additional features in the word embeddings would have another boost in the results, but we leave this in future work.\n  \\item Replacing GRU with our CRU could significantly improve the performance, where 1.6\\% and 1.5\\% gains can be obtained when compared to M-AoA Reader. This demonstrates that incorporating contextual information when modeling the sentence could enrich the representations. Also, when modeling an unknown word, except for its randomly initialized word embedding, the contextual information could give a possible guess of the unknown word, making the text more readable to the neural networks.\n\\item The re-ranking strategy is an effective approach in this task. We observed that the gains in the common noun category are significantly greater than the named entity. One possible reason is that the language model is much beneficial to CN than NE, because it is much more likely to meet a new named entity that is not covered in the training data than the common noun.\n\\end{itemize}\n\n\\section{Qualitative Analysis}\\label{qualitative-analysis}        \nIn this section, we will give a qualitative analysis on our proposed CRU model in the sentiment classification task.\nWe focus on two categories of the movie reviews, which is quite harder for the model to judge the correct sentiment. The first one is the movie review that contains negation terms, such as ``not''. The second type is the one contains sentiment transition, such as ``clever but not compelling''. We manually select 50 samples of each category in the MR dataset, forming a total of 100 samples to see if our CRU model is superior in handling these movie reviews. The results are shown in Table \\ref{quality-result-table}. As we can see that, our CRU model is better at both categories of movie review classification, demonstrating its effectiveness.\n\n        \\begin{table}[h]\n        \\begin{center}\n       \n        \\begin{tabular}{lcc}\n        \\toprule\n        & \\bf GRU & \\bf CRU \\\\\n        \\midrule\n        Negation Term (50) & 37  & 42 \\\\\n        Sentiment Transition (50) & 34  & 40 \\\\\n        \\midrule\n        Total (100) & 71  & 82 \\\\\n        \\bottomrule\n        \\end{tabular}\n        \\end{center}\n        \\caption{\\label{quality-result-table} Number of correctly classified samples.}\n        \\end{table}\n\nAmong these samples, we select an intuitive example that the CRU successfully captures the true meaning of the sentence and gives the correct sentiment label. We segment a full movie review into three sentences, which is shown in Table \\ref{quality-table}.\n\n        \\begin{table}[htbp]\n        \\begin{center}\n        \\small\n        \\begin{tabular}{lcc}\n        \\toprule\n        \\bf Sentence & \\bf GRU & \\bf CRU \\\\\n        \\midrule\n        I like that Smith & POS & POS \\\\\n        \\midrule\n        I like that Smith, \\\\ he's {\\em not making fun of} these people, & POS & POS \\\\\n        \\midrule\n        I like that Smith, \\\\ he's {\\em not making fun of} these people, \\\\he's {\\em not laughing at} them. & {\\em NEG} & POS \\\\\n        \\bottomrule\n        \\end{tabular}\n        \\end{center}\n        \\caption{\\label{quality-table} Predictions of each level of the sentence.  }\n        \\end{table}\n\nRegarding the first and second sentence, both models give correct sentiment prediction. While introducing the third sentence, the GRU baseline model failed to recognize this review as a positive sentiment because there are many negation terms in the sentence. However, our CRU model could capture the local context {\\em during} the recurrent modeling the sentence, and the phrases such as ``not making fun'' and ``not laughing at'' could be correctly noted as positive sentiment which will correct the sentiment category of the full review, suggesting that our model is superior at modeling local context and gives much accurate meaning.\n\n\\section{Conclusion}\\label{conclusion}\nIn this paper, we proposed an effective recurrent model for modeling sequences, called Contextual Recurrent Units (CRU). \nWe inject the CNN into GRU, which aims to better model the local context information via CNN before recurrently modeling the sequence. \nWe have tested our CRU model on the cloze-style reading comprehension task and sentiment classification task.\nExperimental results show that our model could give substantial improvements over various state-of-the-art systems and set up new records on the respective public datasets. In the future, we plan to investigate convolutional filters that have dynamic lengths to adaptively capture the possible spans of its context.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzexbt b/data_all_eng_slimpj/shuffled/split2/finalzzexbt
new file mode 100644
index 0000000000000000000000000000000000000000..a1d700f86905213887836959c7d00050cbad5eef
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzexbt
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\\noindent The identification of time-varying systems plays a key role in different applications, such as adaptive and model predictive control, where a good real-time tracking of the system to be controlled is necessary. In addition, the detection of changes or drifts in plant parameters is crucial in terms of process monitoring and fault detection. Online System Identification (SysId) and the estimation of time-varying systems are typically strictly connected problems: one would like to exploit the new data that become available in order to track possible changes in the system dynamics.\\\\\nRecursive Prediction Error Method (RPEM), a variant of the classical PEM \\cite{Ljung:99,RecursiveBook}, represents nowadays a well-established technique, through which the current estimate can be efficiently updated, as soon as new data are provided. RPEM are parametric approaches, relying on Recursive Least-Squares (RLS) routines, which compute the parameter estimate by minimizing a functional of the prediction errors \\cite{Ljung:99}. An extension of these approaches to the identification of time-varying systems involves the adoption of a forgetting factor, through which old data become less relevant in the estimation criterion. Convergence and stability properties of Forgetting Factor RPEM  have been well-studied within the SysId community \\cite{bittanti1990convergence,guo1993performance}. \\\\\nAlternative approaches model the coefficients trajectories as stochastic processes \\cite{Chow84}, thus exploiting Kalman filtering \\cite{guo1990estimating} or Bayesian inference \\cite{Sarris73} for parameter estimation. Combinations of bases sequences (e.g. wavelet basis \\cite{tsatsanis1993time}) have also been considered to model the parameters time evolution.\n\\\\The above-mentioned parametric procedures share the criticality of the model selection complexity: this step is especially crucial when the model complexity has to be modified in response to changes in the true system dynamics.\nIn addition, classical complexity selection rules (e.g. cross-validation or information criteria) may not be applicable in online settings, due to the excessive computational effort they require. \nModel complexity issues have been partially addressed in the SysId community through the recent introduction of non-parametric methods, relying on Gaussian processes and Bayesian inference \\cite{GP-AC-GdN:11,SurveyKBsysid}. In this framework model complexity is tuned in a continuous manner by estimating the hyper-parameters which describe the prior distribution chosen by the user \\cite{PCAuto2015}. This property makes these new  techniques appealing for the online identification of time-varying systems: indeed, model complexity can be continuously adapted whenever new data become available.\\\\\nIn a previous work \\cite{RPPCECC2016} we started exploring this research direction by adapting the newly introduced Bayesian procedures to an online identification setting. The methodologies proposed in \\cite{RPPCECC2016} are extended in this new paper by dealing with time-varying systems. Two approaches, relying on the use of a forgetting factor, are proposed; in particular, following the approach in \\cite{PA2014}, we investigate the online estimation of the forgetting factor by treating it as a hyper-parameter of the Bayesian inference procedure. These techniques are experimentally compared with the classical parametric counterparts: the results appear favourable and promising  for the methods we propose.\n\\\\\nThe paper is organized as follows. Sec.~\\ref{sec:problem_formulation} presents the online identification framework and the challenges we will try to address. Sec.~\\ref{sec:pem} provides a brief review of parametric real-time identification techniques, while Sec.~\\ref{sec:bayes} illustrates the Bayesian approach to linear SysId, both in the batch and online scenarios. In particular, Sec.~\\ref{sec:time_var} focuses on the estimation of time-varying systems. Experimental results are reported in Sec.~\\ref{sec:experiment}, while conclusions are drawn in Sec.~\\ref{sec:conclusion}.\n\n\n\\section{Problem Formulation}\\label{sec:problem_formulation}\n\\noindent Consider a dynamical system described through an output-error model, i.e.:\n\\begin{equation} \\label{equ:sys}\ny(t) = \\left[h \\ast u\\right](t) + e(t), \\quad y(t), \\ u(t) \\in\\mathbb{R}\n\\end{equation}\nwhere $h(t)$ denotes the model impulse response and $e(t)$ is assumed to be a zero-mean Gaussian noise with variance $\\sigma^2$.\\\\\nSysId techniques aim at estimating the impulse response $h$ of the system, once a set $\\mathcal{D}:=\\left\\{y(t),u(t)\\right\\}_{t=1}^N$ of measurements of its input and output signals is provided.\\\\\nIn this work we consider an online setting, in which a new set of input-output measurements becomes available every $T$ time steps.\nSpecifically, let us define the variable $i:=k\/T$ by assuming w.l.o.g. that $k$ is a multiple of $T$, and the $i^{th}-$dataset as $\\mathcal{D}_i =\\left\\{u(t),y(t)\\right\\}_{t=(i-1)T +1}^{iT}$.\n\\\\We suppose that at time $k$ an impulse response estimate $\\hat{h}^{(i)}$ has been computed using the data coming from a collection of previous datasets $\\bigcup_{l=1}^{i} \\mathcal{D}_l = \\left\\{u(t),y(t)\\right\\}_{t=1}^{i T}$; at time $k+T$ new data $\\mathcal{D}_{i+1}$ become available and we would like to update the previous estimate $\\hat{h}^{(i)}$ by exploiting them. In addition we assume that the underlying system undergoes certain variations that we would like to track: this situation could often arise in practice, due to e.g. variations of the internal temperature, of the masses (e.g. after grasping an object).\\\\\nFurthermore, online applications typically require that the new estimate is available before the new dataset $\\mathcal{D}_{i+2}$ is provided, thus limiting the computational complexity and the memory storage of the adopted estimation methods.\\\\\nIn this paper, the recently proposed Bayesian approach to SysId \\cite{SurveyKBsysid} is adapted in order to cope with the outlined online setting. Its performances are compared with the ones achieved using classical parametric approaches.\n\n\\begin{rem}\nWe stress that in the remainder of the paper we will use the indexes $k$ and $iT$ interchangeably.\n\\end{rem}\n\n\\section{Parametric Approach}\\label{sec:pem}\n\\noindent Standard parametric approaches to SysId rely on the a-priori choice of a model class $\\mathcal{M}$ (e.g. ARX, ARMAX, OE, etc.), which is completely characterized by a parameter $\\theta \\in \\mathbb{R}^m$.\n\n\\subsection{Batch Approach}\n\\noindent In the batch setting, when a dataset $\\mathcal{D}=\t\\left\\{y(t),u(t)\\right\\}_{t=1}^N$ is provided, the identification procedure reduces to estimate $\\theta$ by minimizing the sum of squared prediction errors:\n\\begin{equation}\n\\hat{\\theta} = \\arg\\min_{\\theta\\in \\mathbb{R}^m} V_N (\\theta,\\mathcal{D}) = \\arg\\min_{\\theta\\in \\mathbb{R}^m}  \\frac{1}{2}\\sum_{t=1}^N \\left(y(t)-\\hat{y}(t\\vert \\theta)\\right)^2\n\\end{equation}\nwhere $\\hat{y}(t\\vert \\theta)$ denotes the one-step ahead predictor \\cite{Ljung:99}.\n\n\\subsection{Online Approach}\n\\noindent The extension of these procedures to an online setting relies on RLS (or pseudo LS) methods.\\\\\nFor ease of notation, let us assume $T=1$ in this section. Suppose that at time $k+1$ a new input-output data pair $\\mathcal{D}_{i+1}$ is provided; then $\\hat{\\theta}^{(i)}$ is updated as:\n\\begin{equation}\\label{equ:param_update}\n\\hat{\\theta}^{(i+1)} = \\hat{\\theta}^{(i)} + \\mu^{(i+1)} Q^{(i+1)^{-1}} \\nabla_{\\theta} V_{k+1}(\\hat{\\theta}^{(i)}, \\textstyle{\\bigcup_{l=1}^{i+1}}\\mathcal{D}_{l})\n\\end{equation}\nwhere $\\nabla_\\theta V_{k+1}(\\hat{\\theta}^{(i)}, \\bigcup_{l=1}^{i+1}\\ \\mathcal{D}_{l})$ denotes the gradient of the loss function computed in the previous estimate and in the new data; $\\mu^{(i+1)}\\in \\mathbb{R}$ and $Q^{(i+1)}\\in\\mathbb{R}^{m\\times m}$ are appropriate scalings which assume different shapes according to the specific algorithm which is adopted (see \\cite{RecursiveBook} and \\cite{Ljung:99}, Ch. 11, for further details). Notice that \\eqref{equ:param_update} is simply a scaled gradient step w.r.t. the loss function $V_{k+1}(\\theta,\\bigcup_{l=1}^{i+1}\\mathcal{D}_{l})$.\n\n\\subsection{Dealing with time-varying systems}\n\\noindent In order to cope with time-varying systems, a possible strategy involves the inclusion of a \\textit{forgetting factor} $\\bar{\\gamma}$ in the loss function $V_{k}(\\theta,\\mathcal{D})$:\n\\begin{equation} \\label{equ:loss_ff}\nV_{k}^\\gamma (\\theta,\\mathcal{D}) =  \\frac{1}{2} \\sum_{t=1}^k \\bar{\\gamma}^{k-t} \\left(y(t)-\\hat{y}(t\\vert \\theta)\\right)^2, \\qquad \\bar{\\gamma} \\in (0,1]\n\\end{equation}\nIn this way old measurements become less relevant for the computation of the estimate. A recursive update of the estimate $\\hat{\\theta}^{(i)}$ (as the one in \\eqref{equ:param_update}) can be derived (\\cite{Ljung:99}, Ch. 11).\n\\\\\nAs an alternative, a sliding window approach can be adopted: at each time step only the last $N_w$ data are used for computing the current estimate (with $N_w$ being the window length). However, since this approach does not admit an update rule as the one in \\eqref{equ:param_update}, the computational complexity of the new estimate will depend on the window length.\n\\\\\nA crucial role in the application of parametric SysId techniques is played by the model order selection step: once a model class $\\mathcal{M}$ is fixed, its complexity has to be chosen using the available data. This is typically accomplished by estimating models with different complexities and by applying tools such as cross-validation or information criteria to select the most appropriate one. However, the estimation of multiple models may be computationally expensive, making this procedure not suited for the online identification of time-varying systems. Indeed, in this framework, it should ideally be applied every time new data become available.\n\\\\\nThe recently proposed approach to SysId, relying on regularization\/Bayesian techniques, overpasses the above-described issue by jointly performing estimation and order selection.\nNext section will illustrate how the batch regularization\/Bayesian method can be tailored to the online identification of time-varying systems.\n\n\n\n\\section{Regularization\/Bayesian Approach} \\label{sec:bayes}\n\n\\subsection{Batch Approach}\n\\label{subsec:batch_approach}\n\\noindent We discuss how the regularization\/Bayesian technique works in the standard batch setting, i.e. when data $\\mathcal{D}=\\left\\{y(t),u(t)\\right\\}_{t=1}^N$ are given. For future use, let us define the vector $Y_N =\\left[y(1)\\ ...\\ y(N)\\right]^\\top\\in\\mathbb{R}^N$.\n\\\\\nAccording to the Bayesian estimation, the impulse response $h$ is considered as a realization of a stochastic process with a prior distribution $p_\\eta(h)$, depending on some parameters $\\eta\\in \\Omega$. The prior $p_\\eta(h)$ is designed in order to account for some desired properties of the estimated impulse response, such as smoothness and stability \\cite{GP-AC-GdN:11,SurveyKBsysid}. In the Bayesian framework, the parameters $\\eta$ are known as hyper-parameters and they need to be estimated from the data, e.g. by optimizing the so-called marginal likelihood (i.e. the likelihood once the latent variable $h$ has been integrated out) \\cite{PCAuto2015}:\n\\begin{equation}\n\\hat{\\eta} =\\arg\\max_{\\eta\\in\\Omega} p_\\eta(Y_N) = \\arg\\max_{\\eta\\in\\Omega}  \\int p(Y_N\\vert h)p_\\eta(h)dh\n\\end{equation}\nOnce the hyper-parameters $\\eta$ have been estimated, the minimum variance estimate of $h$ needs to be computed; it coincides with the posterior mean given the observed data:\n\\begin{equation}\\label{equ:post_est}\n\\hat{h} := \\mathbb{E}_{\\hat{\\eta}} \\left[h \\vert Y_N \\right] =\\int h \\frac{p(Y_N\\vert h)p_{\\hat{\\eta}}(h)}{p_{\\hat{\\eta}}(Y_N)} dh\n\\end{equation}\nIn the SysId context, $h$ is typically modelled as a zero-mean Gaussian process (independent of the noise $e(t)$) with covariance $\\mathbb{E}\\left[h(t),h(s)\\right]=\\bar{K}_\\eta(t,s)$ (aka kernel in the Machine Learning literature) \\cite{GP-AC-GdN:11,ChenOL12}. Thanks to this asssumption, the marginal likelihood $p_\\eta(Y_N)$ is Gaussian and the estimate \\eqref{equ:post_est} is available in closed form.\n\\\\\nFurthermore, for simplicity the IIR model in \\eqref{equ:sys} can be accurately approximated by a FIR model of order $n$, whenever $n$ is chosen large enough to catch the relevant components of the system dynamics. By collecting in $\\mathbf{h}:=\\left[h(1)\\ \\cdots\\ h(n)\\right]^\\top\\in\\mathbb{R}^n$ the first $n$ impulse response coefficients, the following Gaussian prior can be defined:\n\\begin{align}\np_\\eta(\\mathbf{h})&\\sim \\mathcal{N}(0,K_\\eta), \\qquad \\eta \\in \\Omega \\subset \\mathbb{R}^d,\\ \\ K_\\eta \\in \\mathbb{R}^{n\\times n}\n\\end{align}\nThe hyper-parameters $\\eta$ can then be estimated by solving\n\\begin{align}\n\\hat{\\eta} &= \\arg\\min_{\\eta\\in\\Omega}\\ - \\ln p_\\eta(Y_N)= \\arg\\min_{\\eta\\in\\Omega}\\ f_N(\\eta)\\label{equ:ml_max}\\\\\nf_N(\\eta) &= Y_N^\\top \\Sigma(\\eta)^{-1} Y_N + \\ln \\det \\Sigma(\\eta)\\label{equ:ml}\\\\\n\\Sigma(\\eta)&= \\Phi_N K_\\eta \\Phi_N^\\top + \\sigma^2 I_N\n\\end{align}\nwhere $\\Phi_N\\in\\mathbb{R}^{N\\times n}$:\n\\begin{equation}\\label{equ:phi}\n\\Phi_N := \\begin{bmatrix}\nu(0) & u(-1) & \\cdots & u(-n+1)\\\\\n\\vdots & \\ddots & \\ddots &\\vdots\\\\\nu(N) & u(N-1) & \\cdots & u(N-n+1)\n\\end{bmatrix} \n\\end{equation}\nIn the batch setting we are considering  the quantities $u(-n+1), ...,u(0)$ can be either estimated or set to zero. Here, we follow the latter option. \nOnce $\\hat{\\eta}$ has been computed, the corresponding minimum variance estimate is given by\n\\resizebox{1.02\\linewidth}{!}{\n  \\begin{minipage}{\\linewidth}\n\\begin{align}\n\\widehat{\\mathbf{h}}  :&=\\mathbb{E}_{\\hat{\\eta}}\\left[\\mathbf{h} \\vert Y_N\\right] = \\arg\\min_{\\mathbf{h}^\\in\\mathbb{R}^n} \\left(Y_N-\\Phi_N\\mathbf{h}\\right)^\\top \\left(Y_N-\\Phi_N\\mathbf{h}\\right) + \\sigma^2 \\mathbf{h}^\\top K_{\\hat{\\eta}}^{-1}\\mathbf{h}\\nonumber \\\\\n&=  (\\Phi_N^\\top \\Phi_N + \\sigma^2 K_{\\hat{\\eta}}^{-1})^{-1}\\Phi_N^\\top Y_N  \\label{equ:h_hat}\n\\end{align}\n\\end{minipage}\n}\n\n\\vspace{1mm}\n\n\\begin{rem}\nThe estimate $\\widehat{\\mathbf{h}}$ in \\eqref{equ:h_hat} can be computed once a noise variance estimate $\\hat{\\sigma}^2$ is available. For this purpose, $\\sigma^2$ can be treated as a hyper-parameter and estimated by solving \\eqref{equ:ml_max} or it can be computed from a LS estimate of $\\mathbf{h}$. In this work the latter option is adopted.\n\\end{rem}\n\n\n\n\\subsection{Online Approach} \\label{subsec:bayes_onine}\n\\noindent We now adapt the batch technique described in Sec.~\\ref{subsec:batch_approach} to the online setting outlined in Sec.~\\ref{sec:problem_formulation}. At time $k+T$, when data $\\mathcal{D}_{i+1}=\\left\\{u(t),y(t)\\right\\}_{t=iT+1}^{(i+1)T}$ are provided, the current impulse response estimate $\\widehat{\\mathbf{h}}^{(i)}$ is updated through formula \\eqref{equ:h_hat}, once the data matrices are enlarged with the new data and a new hyper-parameter estimate $\\hat{\\eta}^{(i+1)}$ is computed. The data matrices are updated through the following recursions\n\\begin{align}\nR^{(i+1)} &:= \\Phi_{(i+1)T}^\\top \\Phi_{(i+1)T} = R^{(i)} + \\left(\\Phi_{iT+1}^{(i+1)T}\\right)^\\top \\Phi_{iT+1}^{(i+1)T}\\label{equ:r}\\\\\n\\widetilde{Y}^{(i+1)} &:= \\Phi_{(i+1)T}^\\top Y_{(i+1)T} =\\widetilde{Y}^{(i)} + \\left(\\Phi_{iT+1}^{(i+1)T}\\right)^\\top Y_{iT+1}^{(i+1)T}\\label{equ:y_tilde}\\\\\n\\xbar{Y}^{(i+1)} &:= Y_{(i+1)T}^\\top Y_{(i+1)T}= \\xbar{Y}^{(i)}  + \\left(Y_{iT+1}^{(i+1)T}\\right)^\\top Y_{iT+1}^{(i+1)T}\\label{equ:y_bar}\n\\end{align}\nwhere $Y_{(i+1)T}=\\left[y(1)\\cdots y(iT+T)\\right]^\\top\\in\\mathbb{R}^{(i+1)T}$, $Y_{iT+1}^{(i+1)T} = \\left[y(iT+1) \\cdots y(iT+T)\\right]$; $\\Phi_i$ is defined as in \\eqref{equ:phi} with $N$ replaced by $(i+1)T$, while $\\Phi_{iT+1}^{(i+1)T}$ has the same structure of matrix \\eqref{equ:phi} but it contains the data from $iT-n+1$ to $(i+1)T$.\nThe computational cost of \\eqref{equ:r}-\\eqref{equ:y_bar} is, $O(n^2T)$, $O(nT)$ and $O(T^2)$, respectively.\\\\\nThe minimization of $f_{(i+1)T}(\\eta)$ in \\eqref{equ:ml}, needed to determine $\\hat{\\eta}^{(i+1)}$, is typically performed through iterative routines, such as 1st or 2nd order optimization algorithms \\cite{BonettiniCPSIAM2014} or the Expectation-Maximization (EM) algorithm \\cite{bottegal2016robust,BOTTEGAL2015466}. Since these methods may require a large number of iterations before reaching convergence, they may be unsuited for online applications. We should recall that, when adopted for marginal likelihood optimization, each iteration of these algorithms has a computational complexity of $O(n^3)$, due to the objective function evaluation. Specifically, $f_{(i+1)T}(\\eta)$ can be robustly evaluated as \\cite{chen2013implementation}\n\\begin{align}\nf_{(i+1)T}(\\eta) = &((i+1)T-n)\\ln \\sigma^2 + 2\\ln\\vert S\\vert \\nonumber\\\\\n&+\\sigma^{-2}(\\ \\xbar{Y}^{(i+1)}- \\| S^{-1}L^\\top \\widetilde{Y}^{(i+1)}\\|_2^2\\ ) \\label{equ:eff_ml}\n\\end{align}\nwhere $L$ and $S$ are Cholesky factors: $K_\\eta =: LL^\\top$ and $\\sigma^2 I_n + L^\\top R^{(i+1)} L =: SS^\\top$ (whose computation is $O(n^3)$).\\\\ \nTo tackle the real-time constraints, the approach proposed in \\cite{RPPCECC2016} is adopted: $\\hat{\\eta}^{(i+1)}$ is computed by running just one iteration of a Scaled Gradient Projection (SGP) algorithm (a 1st order optimization method) applied to solve problem \\eqref{equ:ml_max} \\cite{BonettiniCPSIAM2014}. Algorithm \\ref{alg:1step_grad} summarizes its implementation. Notice that it is initialized with the previous estimate $\\hat{\\eta}^{(i)}$ (obtained using the data $\\bigcup_{l=1}^{i} \\mathcal{D}_l$) which is likely to be close to a local optimum of the objective function $f_{iT}(\\eta)\\equiv f_{k}(\\eta)$. If the number of new data $T << k$, it is reasonable to suppose that $\\arg\\min_{\\eta\\in\\Omega} f_{iT} (\\eta) \\approx \\arg\\min_{\\eta\\in \\Omega} f_{(i+1)T} (\\eta)$. \nTherefore, by just performing one SGP iteration, $\\hat{\\eta}^{(i+1)}$ will be sufficiently close to a local optimum of $f_{(i+1)T}(\\eta)$.\n\n\n\\begin{algorithm}\n\\caption{1-step Scaled Gradient Projection (SGP)}\\label{alg:1step_grad}\n\\begin{algorithmic}[1]\n\\Statex{\\textbf{Inputs:}} previous estimates $\\{ \\hat{\\eta}^{(i)}, \\hat{\\eta}^{(i-1)}\\}$, $\\nabla f_{iT}(\\hat{\\eta}^{(i-1)})$, $R^{(i+1)}$, $\\widetilde{Y}^{(i+1)}$, $\\xbar{Y}^{(i+1)}$, $\\hat{\\sigma}^{(i+1)^2}$\n\\Statex Initialize: $c=10^{-4},\\ \\delta=0.4$\n\\State Compute $\\nabla f_{(i+1)T}(\\hat{\\eta}^{(i)})$\n\\State $r^{(i-1)} \\gets \\hat{\\eta}^{(i)} - \\hat{\\eta}^{(i-1)}$ \\label{alg_step:rf}\n\\State $w^{(i-1)} \\gets \\nabla f_{(i+1)T}(\\hat{\\eta}^{(i)}) - \\nabla f_{iT}(\\hat{\\eta}^{(i-1)})$ \\label{alg_step:w}\n\\State Approximate the inverse Hessian of $f_{(i+1)T}(\\hat{\\eta}^{(i)})$ as $B^{(i)}=\\alpha^{(i)}D^{(i)}$ (using the procedure outlined in \\cite{BonettiniCPSIAM2014})\\label{alg_step:inverse_H}\n\\State Project onto the feasible set:\n\\Statex $z\\gets \\Pi_{\\Omega,D^{(i)}} (\\ \\hat{\\eta}^{(i)} -B^{(i)}\\nabla f_{(i+1)T}(\\hat{\\eta}^{(i)})\\ )$\\label{alg_step:proj}\n\\State $\\Delta\\hat{\\eta}^{(i)} \\gets z - \\hat{\\eta}^{(i)}$\n\\State $\\nu \\gets 1$\n\\If{$f_{(i+1)T}(\\hat{\\eta}^{(i)}+\\nu \\Delta \\hat{\\eta}^{(i)}) \\leq f_{(i+1)T}(\\hat{\\eta}^{(i)})+ c \\nu \\nabla f_{(i+1)T}(\\hat{\\eta}^{(i)})^\\top \\Delta\\hat{\\eta}^{(i)}$}\n\\State Go to step 12\n\\Else\n\\State $\\nu \\gets \\delta \\nu$\n\\EndIf\n\\State $\\hat{\\eta}^{(i+1)} \\gets \\hat{\\eta}^{(i)} + \\nu \\Delta \\hat{\\eta}^{(i)}$\n\\Statex \\textbf{Output:} $\\hat{\\eta}^{(i+1)}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\noindent The key step in Algorithm \\ref{alg:1step_grad}  is \\ref{alg_step:inverse_H}, where the inverse Hessian is approximated as the product between the positive scalar $\\alpha^{(i)}\\in\\mathbb{R}_+$ and the diagonal matrix $D^{(i)}\\in\\mathbb{R}^{d\\times d}$.\n$\\alpha^{(i)}$ is chosen by alternating the so-called Barzilai-Borwein (BB) rules \\cite{barzilai1988two}:\n\\begin{equation}\n\\alpha_1^{(i)} := \\frac{r^{(i-1)^\\top}r^{(i-1)}}{r^{(i-1)^\\top}w^{(i-1)}}, \\qquad \n\\alpha_2^{(i)} :=  \\frac{r^{(i-1)^\\top}w^{(i-1)}}{w^{(i-1)^\\top}w^{(i-1)}}\\label{equ:bb}\n\\end{equation}\nwith $r^{(i-1)}$ and $w^{(i-1)}$ specified at steps \\ref{alg_step:rf} and~\\ref{alg_step:w} of Algorithm \\ref{alg:1step_grad}. The definition of $D^{(i)}$ depends on the constraints set and on the objective function. The authors in \\cite{BonettiniCPSIAM2014} exploit the following decomposition of $\\nabla_\\eta f_{(i+1)T}(\\eta)$ (defined in \\eqref{equ:ml}):\n\\begin{equation}\\label{equ:grad_decomp}\n\\nabla_\\eta f_{(i+1)T}(\\eta) = V(\\eta) - U(\\eta), \\quad V(\\eta)>0, \\  U(\\eta)\\geq 0\n\\end{equation}\nto specify $D^{(i)}$.\n\\noindent We refer the interested reader to \\cite{BonettiniCPSIAM2014} for further details\n\n\\noindent The projection operator adopted at step \\ref{alg_step:proj} of Algorithm \\ref{alg:1step_grad} is\n\\begin{equation}\n\\Pi_{\\Omega,D^{(i)}} (z) = \\textstyle{\\arg\\min_{x\\in\\Omega}} (x-z)^\\top D^{(i)^{-1}}(x-z)\n\\end{equation}\n\n\n\\begin{rem}\nBesides SGP, in \\cite{RPPCECC2016} other inverse Hessian approximations are investigated (e.g. the BFGS formula). In this work we only consider the SGP approximation, since it appears preferable to the others, according to the experiments we performed (both in the time-invariant and -variant domain). \\cite{RPPCECC2016} also considers the EM algorithm as an alternative to 1st order optimization methods to solve problem \\eqref{equ:ml_max}. Even if the results reported for EM in \\cite{RPPCECC2016} are comparable to the ones achieved through SGP, the latter approach appears superior to EM in the time-varying setting we are considering.\n\\end{rem}\n\n\\subsection{Dealing with time-varying systems}\\label{sec:time_var}\n\\noindent In this section we deal with the identification of time-varying systems: specifically, estimators have to be equipped with tools through which past data become less relevant for the current estimation. In the following we propose two routines which combine the ``online Bayesian estimation'' above sketched with the ability to ``forget'' past data.\n\n\\subsubsection{Fixed Forgetting Factor}\nFollowing a classical practice in parametric SysId (see Sec.~\\ref{sec:pem}), we introduce a forgetting factor $\\bar{\\gamma} \\in (0,1]$ into the data we are provided in order to base the estimation mainly on the more recent data. Specifically, we assume that the first $k$ data are generated according to the following linear model:\n\\begin{equation}\\label{equ:ff_model}\n\\bar{G}_k Y_k = \\bar{G}_k \\Phi_k \\mathbf{h} + E, \\ E= \\left[e(1)...e(k)\\right]^\\top \\sim \\mathcal{N}(0,\\sigma^2 I_k)\n\\end{equation}\nwhere $\\bar{G}_k \\bar{G}_k =: \\bar{\\Gamma}_k$  and $\\bar{\\Gamma}_k := diag\\left(\\bar{\\gamma}^{k-1}, \\bar{\\gamma}^{k-2}, ..., \\bar{\\gamma}^0\\right)$. Therefore, when adopting the regularized regression criterion \\eqref{equ:h_hat}, the estimate at time $k$ is computed as:\n\\begin{align}\n\\widehat{\\mathbf{h}}_{\\bar{\\gamma}} &:= \\arg\\min_{\\mathbf{h}\\in\\mathbb{R}^n}\\sum_{t=1}^k \\bar{\\gamma}^{k-t} \\left(y(t) - \\Phi_t^t \\mathbf{h}\\right)^2 + \\sigma^2 \\mathbf{h}^\\top K_{\\hat{\\eta}}^{-1} \\mathbf{h} \\label{equ:regul_probl_ff}\\\\\n&= \\arg\\min_{\\mathbf{h}\\in\\mathbb{R}^n}\\left(Y_k - \\Phi_k \\mathbf{h}\\right)^\\top \\bar{\\Gamma}_k \\left(Y_k - \\Phi_k \\mathbf{h}\\right) + \\sigma^2 \\mathbf{h}^\\top K_{\\hat{\\eta}}^{-1} \\mathbf{h}\\nonumber\\\\\n&= (\\Phi_k^\\top \\bar{\\Gamma}_k \\Phi_k + \\sigma^2 K_{\\hat{\\eta}}^{-1})^{-1} \\Phi_k^\\top \\bar{\\Gamma}_k Y_k \\label{equ:h_hat_ff}\n\\end{align}\nCorrespondingly, the hyper-parameters are estimated solving:\n\\begin{align}\n\\hat{\\eta} &= \\arg\\min_{\\eta\\in\\Omega}\\ Y_k^\\top \\bar{G}_k \\Sigma_{\\bar{\\gamma}}(\\eta)^{-1} \\bar{G}_k Y_k + \\ln \\det \\Sigma_{\\bar{\\gamma}}(\\eta)\\label{equ:eta_hat_ff}\\\\\n\\Sigma_{\\bar{\\gamma}}(\\eta)&= \\bar{G}_k \\Phi_k K_\\eta \\Phi_k^\\top \\bar{G}_k + \\sigma^2 I_k\n\\end{align}\nAlgorithm \\ref{alg:on_line_ff} illustrates the online implementation of the identification procedure based on equations \\eqref{equ:h_hat_ff} and \\eqref{equ:eta_hat_ff}. In particular, it assumes that at time $k$ the estimates $\\widehat{\\mathbf{h}}^{(i)}$ and $\\hat{\\eta}^{(i)}$ are available and they have been computed by solving, respectively, \\eqref{equ:regul_probl_ff} and \\eqref{equ:eta_hat_ff}; these estimates are then online updated after the new data $\\mathcal{D}_{i+1}$ are provided. Once $\\bar{\\gamma}$ is chosen by the user, it is inserted in the data matrices $R_{\\bar{\\gamma}}^{(i+1)}:=\\Phi_{(i+1)T}^\\top \\bar{\\Gamma}_{(i+1)T}\\Phi_{(i+1)T},\\  \\widetilde{Y}_{\\bar{\\gamma}^{(i+1)}}:=\\Phi_{(i+1)T}^\\top \\bar{\\Gamma}_{(i+1)T}Y_{(i+1)T}, \\\n\\xbar{Y}_{\\bar{\\gamma}^{(i+1)}}:=Y_{(i+1)T}^\\top \\bar{\\Gamma}_{(i+1)T}Y_{(i+1)T}$, updated at steps \\ref{alg2_step:r}-\\ref{alg2_step:yb} of the algorithm.\n\n\\begin{algorithm}\n\\caption{Online Bayesian SysId: Fixed Forgetting Factor}\\label{alg:on_line_ff}\n\\begin{algorithmic}[1]\n\\Statex{\\textbf{Inputs:}} forgetting factor $\\bar{\\gamma}$, previous estimates $\\{ \\hat{\\eta}^{(i)}, \\hat{\\eta}^{(i-1)}\\}$, previous data matrices $\\{R_{\\bar{\\gamma}}^{(i)},\\widetilde{Y}_{\\bar{\\gamma}}^{(i)},\\xbar{Y}_{\\bar{\\gamma}}^{(i)}\\}$, new data $\\mathcal{D}_{i+1}=\\left\\{u(t),y(t)\\right\\}_{t=iT+1}^{(i+1)T}$\n\\State $R_{\\bar{\\gamma}}^{(i+1)} \\gets \\bar{\\gamma}^T R_{\\bar{\\gamma}}^{(i)} + \\left(\\Phi_{iT+1}^{(i+1)T}\\right)^\\top \\bar{\\Gamma}_T\\ \\Phi_{iT+1}^{(i+1)T} $ \\label{alg2_step:r}\n\\State $\\widetilde{Y}_{\\bar{\\gamma}}^{(i+1)} \\gets \\gamma^T\\widetilde{Y}_\\gamma^{(i)} + \\left(\\Phi_{iT+1}^{(i+1)T}\\right)^\\top \\bar{\\Gamma}_T\\ Y_{iT+1}^{(i+1)T}$ \\label{alg2_step:yt}\n\\State $\\xbar{Y}_{\\bar{\\gamma}}^{(i+1)} \\gets \\bar{\\gamma}^T \\xbar{Y}_{\\bar{\\gamma}}^{(i)}  + \\left(Y_{iT+1}^{(i+1)T}\\right)^\\top \\bar{\\Gamma}_T\\ Y_{iT+1}^{(i+1)T}$ \\label{alg2_step:yb}\n\\State $\\widehat{\\mathbf{h}}_{LS}^{(i+1)} \\gets R_{\\bar{\\gamma}}^{(i+1)^{-1}} \\widetilde{Y}_{\\bar{\\gamma}}^{(i+1)}$ \\label{alg2_step:ls}\n\\State {\\footnotesize$\\hat{\\sigma}^{(i+1)^2} \\gets \\frac{1}{ (i+1)T - n} \\left(\\bar{Y}_{\\bar{\\gamma}}^{(i+1)}-2\\widetilde{Y}_{\\bar{\\gamma}}^{(i+1)^\\top}\\widehat{\\mathbf{h}}_{LS}^{(i+1)} + \\widehat{\\mathbf{h}}_{LS}^{(i+1)^\\top}R_{\\bar{\\gamma}}^{(i+1)}\\widehat{\\mathbf{h}}_{LS}^{(i+1)} \\right)$}\n\\State $\\hat{\\eta}^{(i+1)}\\gets \\arg\\min_{\\eta\\in\\Omega}\\ f_{(i+1)T}(\\eta)$ (use Algorithm \\ref{alg:1step_grad})\n\\label{alg2_step:ml}\n\\State $\\widehat{\\mathbf{h}}^{(i+1)} \\gets \\left(R_{\\bar{\\gamma}}^{(i+1)} +\\hat{\\sigma}_{\\bar{\\gamma}}^{(i+1)^2} K_{\\hat{\\eta}^{(i+1)}}^{-1}\\right)^{-1}\\widetilde{Y}_{\\bar{\\gamma}}^{(i+1)}$\\label{alg_step:h}\n\\Statex{\\textbf{Output:}} $\\widehat{\\mathbf{h}}^{(i+1)}$, $\\hat{\\eta}^{(i+1)}$\n\\end{algorithmic}\n\\end{algorithm}\n \n\n\n\\subsubsection{Treating the Forgetting Factor as a Hyper-parameter}\nThe Bayesian framework provides the user with the possibility to treat the forgetting factor as a hyper-parameter and to estimate it by solving:\n\\begin{align}\n\\hat{\\eta}, \\hat{\\gamma} &= {\\textstyle\\arg\\min_{\\eta\\in\\Omega, \\gamma\\in(0,1]}}\\ f_k(\\eta,\\gamma)\\label{equ:eta_hat_ff_hyper} \\\\\nf_k(\\eta, \\gamma)&= Y_k^\\top G_k \\Sigma(\\eta,\\gamma)^{-1} G_k Y_k + \\ln \\det \\Sigma(\\eta,\\gamma)\\label{equ:ml_ff_hyper} \\\\\n\\Sigma(\\eta,\\gamma)&= G_k \\Phi_k K_\\eta \\Phi_k^\\top G_k + \\sigma^2 I_k\n\\end{align}\nwhere $G_k G_k =: \\Gamma_k$  and $\\Gamma_k := diag\\left(\\gamma^{k-1}, \\gamma^{k-2}, ..., \\gamma^0\\right)$.\n\\begin{rem} Notice that the model \\eqref{equ:ff_model} is equivalent to\n$$\nY_k =  \\Phi_k \\mathbf{h} + E_{\\bar{\\gamma}}, \\quad E_{\\bar{\\gamma}} = \\left[e_{\\bar{\\gamma}}(1),...,e_{\\bar{\\gamma}}(k)\\right]^\\top \\sim \\mathcal{N}(0,\\sigma^2 \\bar{\\Gamma}_k^{-1})\n$$\nTherefore, treating the forgetting factor as a hyper-parameter is equivalent to modeling the noise with a non-constant variance and to give to the diagonal entries of the covariance matrix an exponential decaying structure.\n\\end{rem}\n\nThe online implementation of this approach is detailed in Algorithm \\ref{alg:on_line_ff_hyper}, where\n\\begin{equation}\nR_{\\hat{\\pmb{\\gamma}}}^{(i)} := \\hat{\\gamma}^{(i)} R_{\\hat{\\pmb{\\gamma}}}^{(i-1)} + \\left(\\Phi_{(i-1)T+1}^{iT}\\right)^\\top \\widehat{\\Gamma}_T^{(i)} \\Phi_{(i-1)T+1}^{iT} \n\\end{equation}\nwith $\\widehat{\\Gamma}_T^{(i)} = diag((\\hat{\\gamma}^{(i)})^{T-1},.. ,(\\hat{\\gamma}^{(i)})^{0})$. $\\widetilde{Y}_{\\hat{\\pmb{\\gamma}}}^{(i)}$ and $\\xbar{Y}_{\\hat{\\pmb{\\gamma}}}^{(i)}$ are analogously defined.\\\\\nWe should stress that the objective function in \\eqref{equ:ml_ff_hyper} does not admit the decomposition \\eqref{equ:grad_decomp}; we have\n\\begin{equation*}\n\\frac{\\partial f_k(\\eta,\\gamma)}{\\partial \\gamma} = V(\\eta,\\gamma) + U(\\eta,\\gamma), \\quad V(\\eta,\\gamma) >0, \\ U(\\eta,\\gamma) \\geq 0 \n\\end{equation*}\nThus, when $\\gamma$ is treated as an hyper-parameter, Algorithm \\ref{alg:1step_grad} is run setting $D^{(i)}=I_d$ at step \\ref{alg_step:inverse_H}; $\\alpha^{(i)}$ is still determined alternating the BB rules \\eqref{equ:bb}.\n\n\n\n\\begin{algorithm}\n\\caption{Online Bayesian SysId: Forgetting Factor as a hyper-parameter}\\label{alg:on_line_ff_hyper}\n\\begin{algorithmic}[1]\n\\Statex{\\textbf{Inputs:}} previous estimates $\\{ \\hat{\\eta}^{(i)}, \\hat{\\eta}^{(i-1)}, \\hat{\\gamma}^{(i)}, \\hat{\\gamma}^{(i-1)}\\}$, previous data matrices $\\{R_{\\hat{\\pmb{\\gamma}}}^{(i)},\\widetilde{Y}_{\\hat{\\pmb{\\gamma}}}^{(i)},\\xbar{Y}_{\\hat{\\pmb{\\gamma}}}^{(i)}\\}$, new data $\\mathcal{D}_{i+1}=\\left\\{u(t),y(t)\\right\\}_{t=iT+1}^{(i+1)T}$\n\\State $R_{\\gamma}^{(i+1)} \\gets \\gamma^T R_{\\hat{\\pmb{\\gamma}}}^{(i)} + \\left(\\Phi_{iT+1}^{(i+1)T}\\right)^\\top \\Gamma_T \\ \\Phi_{iT+1}^{(i+1)T} $ \\label{alg3_step:r}\n\\State $\\widetilde{Y}^{(i+1)}_{\\gamma} \\gets \\gamma^T \\widetilde{Y}_{\\hat{\\pmb{\\gamma}}}^{(i)} + \\left(\\Phi_{iT+1}^{(i+1)T}\\right)^\\top \\Gamma_T \\ Y_{iT+1}^{(i+1)T}$ \\label{alg3_step:yt}\n\\State $\\xbar{Y}^{(i+1)}_\\gamma \\gets \\gamma^T \\xbar{Y}_{\\hat{\\pmb{\\gamma}}}^{(i)}  + \\left(Y_{iT+1}^{(i+1)T}\\right)^\\top \\Gamma_T \\ Y_{iT+1}^{(i+1)T}$ \\label{alg3_step:yb}\n\\State $\\widehat{\\mathbf{h}}_{LS}^{(i+1)} \\gets (R_{\\hat{\\pmb{\\gamma}}}^{(i)})^{-1} \\widetilde{Y}_{\\hat{\\pmb{\\gamma}}}^{(i)}$ \\label{alg3_step:ls}\n\\State {\\footnotesize$\\hat{\\sigma}^{2^{(i+1)}} \\gets \\frac{1}{ (i+1)T - n} \\left(\\xbar{Y}_{\\hat{\\pmb{\\gamma}}}^{(i)}-\n2(\\widetilde{Y}_{\\hat{\\pmb{\\gamma}}}^{(i)})^\\top\\ \\widehat{\\mathbf{h}}_{LS}^{(i+1)} + (\\widehat{\\mathbf{h}}_{LS}^{(i+1)})^\\top R_{\\hat{\\pmb{\\gamma}}}^{(i)} \\widehat{\\mathbf{h}}_{LS}^{(i+1)} \\right)$}\n\\State $\\hat{\\eta}^{(i+1)}, \\hat{\\gamma}^{(i+1)} \\gets \\arg\\min_{\\eta\\in\\Omega, \\gamma\\in(0,1]}\\ f_{(i+1)T}(\\eta,\\gamma)$ \\Statex (use Algorithm \\ref{alg:1step_grad})\n\\label{alg3_step:ml}\n\\State $\\widehat{\\mathbf{h}}^{(i+1)} \\gets \\left(R_{\\hat{\\pmb{\\gamma}}}^{(i+1)} +\\hat{\\sigma}^{2^{(i+1)}}(\\hat{\\gamma}^{(i+1)})\\ K_{\\hat{\\eta}^{(i+1)}}^{-1}\\right)^{-1}\n\\widetilde{Y}_{\\hat{\\pmb{\\gamma}}}^{(i+1)}$\\label{alg_step:h}\n\\Statex{\\textbf{Output:}} $\\widehat{\\mathbf{h}}^{(i+1)}$, $\\hat{\\eta}^{(i+1)}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n \n\n\n\n\n\t\n\n\n\n\\section{Experimental Results}\\label{sec:experiment}\n\\noindent In this section we test the online algorithms for parametric and Bayesian SysId described in Sec.~\\ref{sec:pem} and \\ref{sec:bayes}. Their performance are compared through a Monte-Carlo study over 200 time-varying systems.\n\n\\subsection{Data}\n\\noindent 200 datasets consisting of 3000 input-output measurement pairs are generated. Each of them is created as follows: the first 1000 data are produced by a system contained in the data-bank D4 (used in \\cite{TC-MA-LL-AC-GP:14}), while the remaining 2000 data are generated by perturbing the D4-system with two additional poles and zeros. These  are chosen such that the order of the D4-system changes, thus creating a switch on the data generating system at time $k=1001$. \n\\\\\nThe data-bank D4 consists of 30th order random SISO dicrete-time systems having all the poles inside a circle of radius 0.95. These systems are simulated with a  unit variance  band-limited Gaussian signal with normalized band $[0,0.8]$. A zero mean white Gaussian noise,  with  variance adjusted so that the Signal to Noise Ration (SNR) is always equal to 1, is then added to the output data.\n\n\n\\subsection{Estimators}\n\\noindent The parametric estimators are computed with the \\verb!roe! Matlab routine, using the BIC criterion for the model complexity selection. In the following this estimator will be denoted as \\textit{PEM BIC}. Furthermore, as a benchmark we introduce the parametric oracle estimator, called \\textit{PEM OR}, which selects the model complexity by choosing the model that gives the best fit to the impulse response of the true system. The order selection is performed every time a new dataset becomes available: multiple models with orders ranging from 1 to 20 are estimated and the order selection is performed according to the two criteria above-described.\n\\\\\nFor what regards the methods relying on Bayesian inference we adopt a zero-mean Gaussian prior with a covariance matrix (kernel) given by the so-called TC-kernel:\n\\begin{equation}\nK_\\eta^{TC}(k,j) = \\lambda \\min(\\beta^{k},\\beta^{j}), \\qquad \\eta = [ \\lambda,\\ \\beta]\n\\end{equation}\nwith $\\Omega = \\{(\\lambda,\\beta): \\lambda \\geq 0, 0 \\leq \\beta \\leq 1\\}$ \\cite{ChenOL12}. The length $n$ of the estimated impulse responses is set to 100. In the following, we will use the acronym \\textit{TC} to denote these methods. Furthermore, the notation \\textit{OPT} will refer to the standard Bayesian procedure, in which the SGP algorithm adopted to optimize the marginal likelihood $f_k(\\eta)$ is run until the relative change in $f_k(\\eta)$ is less than $10^{-9}$.\nFrom here on, the online counterpart (illustrated in Sec.~\\ref{sec:bayes}) will be referred to as the \\textit{1-step ML}.\nWe will also use the acronyms \\textit{TC FF} when a fixed forgetting factor is adopted, \\textit{TC est FF} when the forgetting factor is estimated as a hyper-parameter.\n\\\\\nFor each Monte Carlo run, the identification algorithms are initialized using the first batch of data $\\mathcal{D}_{init}=\\left\\{u(t),y(t)\\right\\}_{t=1}^{300}$.\nAfter this initial step, the estimators are updated every $T=10$ time steps, when new data $\\mathcal{D}_{i+1} = \\left\\{u(t),y(t)\\right\\}_{t=iT}^{(i+1)T}$ are provided. The forgetting factor in the \\textit{TC FF} and \\textit{PEM} methods is set to 0.998, while its estimation in \\textit{TC est FF} method is initialized with 0.995.\n\n\\subsection{Performance}\n\\noindent The purpose of the experiments is twofold. First, we will compare the two routines we have proposed in Sec.~\\ref{sec:time_var} to explicitly deal with time-varying systems. Second, we will compare the parametric and the Bayesian identification approaches while dealing with time-varying systems.\\\\\nAs a first comparison, we evaluate the adherence of the estimated impulse response $\\widehat{\\mathbf{h}}$ to the true one $\\mathbf{h}$, measured~as\n\\begin{equation}\n\\label{eq:fit_imp_resp}\n\\mathcal{F}(\\widehat{\\mathbf{h}})= 100 \\cdot \\Big(1- \\frac{\\Vert \\mathbf{h} - \\widehat{\\mathbf{h}} \\Vert_2}{\\Vert \\mathbf{h} \\Vert_2}\\Big)\n\\end{equation}\nFigure \\ref{fig:fit} reports the values of $\\mathcal{F}(\\widehat{\\mathbf{h}})$ at four time instants.\n\n\n\n\n\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[width=.9\\columnwidth]{fitBoxplot_timeVard4_sgp_1i_TC_Nsys200_NestSysOrig1000_NestPerturb3000_SNR1_Nred300_Nstep10_FF998_FFpem998_FINAL-crop.pdf}\n\\caption{Fit $\\mathcal{F}(\\widehat{\\mathbf{h}})$ achieved at four time instants $k$ (corresponding to the number of data available for the estimation).}\n\\label{fig:fit}\n\\end{figure}\n\n\n\\noindent \nIt is interesting to note that immediately before the change in the data generating system ($k = 1000$) the \\textit{TC} methods slightly outperform the ideal parametric estimator \\textit{PEM OR}.\nAfter the switch (occurring at $k=1001$), among regularization\/Bayesian routines \\textit{TC est FF} recovers the fit performance a bit faster than \\textit{TC FF}; even at regime it outperforms the latter because it can choose forgetting factor values that retain a larger amount of data.\n\\\\\nWe also observe how the \\textit{1-step ML} procedures and the corresponding \\textit{OPT} routines provide analogous performance at each time step $k$, validating the method we propose for online estimation and confirming the results in \\cite{RPPCECC2016}.\n\\\\\nThe unrealistic \\textit{PEM OR} represents the reference on the achievable performance of the PEM estimators; it outperforms \\textit{TC} methods in the transient after the switch, while it has comparable performance at regime. Instead, the recursive \\textit{PEM BIC} estimator performs very poorly.\n\n\n\n\\begin{table}\n\\centering\n\\begin{tabularx}{\\columnwidth}{|X|X|X|X|X|X|X|}\n\\hline \n & \\multicolumn{3}{c|}{TC}  & \\multicolumn{2}{c|}{PEM}  \n\\\\ \n & OPT FF & FF & est FF & OR & BIC \\\\ \n\\hline \nmean & 6.70  &       0.44  &       5.45  &   18.44  &      18.44  \\\\\\hline\nstd & 1.28  &       0.03  &       0.67  &     0.69  &       0.69  \\\\\n\\hline \n\\end{tabularx} \n\\caption{Computational cumulative time after data $\\mathcal{D} =\\left\\{u(t),y(t)\\right\\}_{t=1}^{3000}$ are used: mean and std over 200 datasets.}\\label{tab:cumulative_time_TC_PEM}\n\\vspace{-4mm}\n\\end{table}\n\n\n\n\\noindent As a second comparison, Table \\ref{tab:cumulative_time_TC_PEM} reports the computational cumulative time of the proposed algorithms in terms of mean and standard deviation after the estimators are fed with all the data $\\mathcal{D} =\\left\\{u(t),y(t)\\right\\}_{t=1}^{3000}$. \nThe \\textit{1-step ML} methods are one order of magnitude faster than the corresponding \\textit{OPT} ones. The \\textit{TC est FF} estimator is  slower than \\textit{TC FF}: this should be a consequence of having set $D^{(i)}=I_d$ in Algorithm \\ref{alg:1step_grad}. On the other hand the RPEM estimators are three times slower than the \\textit{OPT} ones, thus appearing not particularly appealing for online applications. \n\\section{Conclusion and Future Work}\\label{sec:conclusion}\n\\noindent We have adopted recently developed SysId techniques relying on the Gaussian processes and Bayesian inference to the identification of time-varying systems. Specifically, we have focused on an online setting by assuming that new data become available at predefined time instants. To tackle the real-time constraints we have modified the standard Bayesian procedure: hyper-parameters are estimated by performing only one gradient step in the corresponding marginal likelihood optimization problem.\nIn order to cope with the time-varying nature of the systems to be identified, we propose two approaches, based on the use of a forgetting factor. One of them treats the forgetting factor as a constant, while the other estimates it as a hyper-parameter of the Bayesian inference procedure.\\\\\nWe believe that the preliminary investigation performed in this work may pave the way for further research in this topic. A future research direction could consider the recursive update of the Bayesian estimate, resembling the one which is available for parametric techniques.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nMultiple stellar populations are routinely \nfound in old Galactic and intermediate-age Magellanic Clouds star clusters\n(Piotto 2008 and references therein). \nWhether they are a signature of the cluster formation process or a result of the star formation\nhistory and related stellar\nevolution effects, is still matter of lively discussion (Renzini 2008, Bekki et al. 2008, Decressin et al. 2007).\nThe prototype of globular hosting multiple populations has for long time been \n$\\omega$~ Cen (Villanova et al. 2007), although the current understanding is that it is possibly the remnant\nof a dwarf galaxy (Carraro \\& Lia 2000, Tsuchiya et al. 2004, Romano et al. 2007).\\\\\nMost chemical studies of the stellar population in $\\omega$~ Cen are\nrestricted within 20 arcmin of the cluster radius center (see Norris \\& Da Costa 1995, Villanova et al. 2007),\n where, probably, the diverse stellar components are better mixed and\nless subjected to external perturbations, like the Galactic tidal stress, than the outer\nregions. Assessing whether there are population inhomogeneities in $\\omega$ Cen or gradients\nin metal abundance is a crucial step to progress in our understanding of this fascinating stellar\nsystem.\\\\\nIn Scarpa et al. (2003, 2007) we presented the results of a spectroscopic campaign to\nstudy the stellar radial velocity dispersion profile at $\\sim$ 25 arcmin, half way to  \nthe tidal radius ($\\sim$ 57 arcmin,  Harris 1996), in an attempt to find a new way to verify the \nMOND (Modified Newtonian Dynamics, Milgrom 1983) theory of gravitation.\\\\\nIn this paper we make use of a subsample of those spectra (the ones taken for\nRGB stars) and extract estimates of metal abundances for some of the most\ninteresting elements.\nThe aim is to study the chemical trends of the stellar populations in the cluster periphery,\nto try to learn whether the cluster outskirts contain, both qualitatively\nand quantitatively, the same population mix and to infer from this additional information\non the cluster formation and evolution.\\\\\nThe layout of the paper is as follows. In Sect.~2 we describe observations and data reduction,\nwhile Sect.~3 is dedicated to the derivation of metal abundances.\nThese latter are then discussed in detail in Sect.~4. Sect.~5 is devoted to the comparison\nof the metal abundance trends in the inner and outer regions of $\\omega$~ Cen, and, finally,\nSect.~6 summarizes the findings of this study.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figure1.eps}\n\\caption{The CMD of $\\omega$~ Cen in the optical (left panel) and infrared.\nSolid symbols of different colors indicate stars belonging to the MPP (red),\nIMP (blue) and MRP(green). See text for more details.}\n\\label{f1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figure2.eps}\n\\caption{Distribution of iron abundance for the program stars. A bimodal\nGaussian fit is used to derive the mean iron abundance of the MPP and IMP. \nMean iron abundances of the three peaks are indicated. See text for more details.}\n\\label{f2}\n\\end{figure}\n\n\\begin{table} [!hbp]\n\\caption{Measured Solar abundances ($\\rm\n  {log\\epsilon(X)=log(N_{X}\/N_{H})+12)}$.} \n\\label{t1}\n\\centering\n\\begin{tabular}{lc}\n\\hline\n\\hline\nElement &  log$\\epsilon$(X)\\\\\n\\hline\nNaI & 6.37 \\\\\nMgI & 7.54 \\\\\nSiI & 7.61 \\\\\nCaI & 6.39 \\\\\nTiI & 4.94 \\\\\nTiI & 4.96 \\\\\nCrI & 5.63 \\\\\nFeI & 7.50 \\\\\nNiI & 6.28 \\\\\nZnI & 4.61 \\\\\nYII & 2.25 \\\\\nBaI & 2.31 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\noindent\n\n\n\\section{Observations and Data reduction}\nOur data-set consists of UVES spectra collected in\nAugust 2001, for a project devoted to \nmeasuring radial velocities and establishing membership in the outskirts of the \ncluster. Data were obtained with UVES\/VLT@UT2\n(Pasquini et al.\\ 2002) with a typical seeing of 0.8-1.2 arcsec. \nWe observed isolated stars from the lower red giant branch (RGB) up to the tip\nof the RGB of $\\omega$ Cen, in the magnitude\nrange $11.5<{\\rm V}<16.0$.\n\nWe used the UVES spectrograph in the RED 580 setting. The spectra have a spectral coverage of\n$\\sim$2000 \\AA \\ with the central wavelength at 5800 \\AA. The typical\nsignal to noise ratio is ${\\rm S\/N\\sim 20-30}$.\nFor additional details, the reader is referred to Scarpa et al. (2003).\n\n\nData were reduced using UVES pipelines (Ballester et al.\\ 2000),\nincluding bias subtraction, flat-field correction, wavelength\ncalibration, sky subtraction and spectral rectification. Stars\nwere selected from photographic BV observations (van Leeuwen et al. 2000)\ncoupled with infrared JHK 2MASS photometry (Skrutskie\net al.\\ 2006). Targets are located at a radial distance between 20 and 30\narcmin. The whole sample of stars contain both RGB and horizontal branch (HB) stars.\nIn this paper we focus our attention only on RGB objects, for the sake of comparison\nwith previous studies.\n\n\n\\subsection{Radial velocities and membership}\nIn the present work, radial velocities were used as the membership criterion\nsince the cluster stars all have similar motions with respect to the observer.\nThe radial velocities of the\nstars were measured using the IRAF FXCOR task, which cross-correlates\nthe object spectrum with a template.  As a template, we used a\nsynthetic spectrum obtained through the spectral synthesis code\nSPECTRUM (see {\\sf http:\/\/www.phys.appstate.edu\/spectrum\/spectrum.html} for\nmore details), using a Kurucz model atmosphere with roughly the mean\natmospheric parameters of our stars $\\rm {T_{eff}=4900}$ K, $\\rm {log(g)=2.0}$,\n$\\rm {v_{t}=1.3}$ km\/s, $\\rm {[Fe\/H]=-1.40}$. Each radial velocity was\ncorrected to the heliocentric system. We calculated a first approximation\nmean velocity and the r.m.s ($\\sigma$) of the velocity distribution.\nStars showing $\\rm {v_{r}}$ out of more than\n$3\\sigma$ from the mean value were considered probable field objects and\nrejected, leaving us with 48 UVES spectra of probable members, whose position\nin the CMD are shown in Fig.~\\ref{f1}. Radial velocities\nfor member stars are reported in Tab.~\\ref{t2}\n\n\\section{Abundance analysis}\n\\subsection{Continuum determination}\n\nThe chemical abundances for all elements\nwere obtained from the equivalent widths (EWs) of the spectral lines (see next\nSection for the description of the line-list we used).  \nAn accurate measurement of EWs first requires a good determination\nof the continuum level. Our relatively metal-poor stars\nallowed us to proceed in the following way. First, \nfor each line, we selected a region of 20 \\AA \\ centered on the line itself\n(this value is a good compromise between having enough points, i. e. a good statistic, and \navoiding a too large region where the spectrum might not be flat).\nThen we built the histogram of the distribution of the flux where the peak is a\nrough estimation of the continuum. We refined this determination by fitting a\nparabolic curve to the peak and using the vertex as our continuum estimation. \nFinally, we revised the continuum determination by eye and corrected by hand\nif an obvious discrepancy with the spectrum was  found.\nThen, using the continuum value previously obtained, we fit a Gaussian curve\nto each spectral line and obtained the EW from integration.\nWe rejected lines if they were affected by bad continuum determinations, by non-Gaussian\nshape, if their central wavelength did not agree with that expected from \nour line-list, or if the lines were too broad or too narrow with respect to the\nmean FWHM.\nWe verified that the Gaussian shape is a good approximation for our spectral\nlines, so no Lorenzian correction has been applied.\n\n\\begin{table*}\n\\caption{Stellar parameters. Coordinates are for J2000.0 equinox}\n\\label{t2}\n\\centering\n\\begin{tabular}{cccccccccccc}\n\\hline\n\\hline\nID & $\\alpha$ & $\\delta$ & B & V & J$_{\\rm 2MASS}$ & H$_{\\rm 2MASS}$ & K$_{\\rm 2MASS}$ & T$_{\\rm eff}$& log(g) & v$_{\\rm t}$ & RV$_{\\rm H}$\\\\\n\\hline\n& deg & deg. &  & & & & & $^{0}K$ & & km\/sec & km\/sec \\\\\n\\hline                      \n00006 & 201.27504 & -47.15599 & 16.327 & 15.531 & 13.865 & 13.386 & 13.364 & 5277 & 2.75 & 1.23 & 222.99\\\\\n08004 & 201.07113 & -47.22082 & 15.393 & 14.508 & 12.687 & 12.110 & 12.007 & 4900 & 2.17 & 1.38 & 241.70\\\\\n10006 & 201.16314 & -47.23357 & 14.510 & 13.710 & 11.887 & 11.413 & 11.300 & 5080 & 1.93 & 1.44 & 237.18\\\\\n10009 & 201.24457 & -47.23406 & 13.807 & 12.573 & 10.331 &  9.664 &  9.520 & 4432 & 1.14 & 1.64 & 227.57\\\\\n10010 & 201.33458 & -47.23334 & 14.982 & 13.941 & 11.963 & 11.394 & 11.249 & 4758 & 1.88 & 1.45 & 220.18\\\\\n13006 & 201.13373 & -47.25880 & 16.442 & 15.665 & 14.112 & 13.615 & 13.504 & 5251 & 2.79 & 1.22 & 231.78\\\\\n14002 & 201.16243 & -47.26471 & 15.696 & 14.853 & 13.110 & 12.634 & 12.552 & 5151 & 2.42 & 1.31 & 224.00\\\\\n22007 & 201.08521 & -47.32639 & 14.799 & 13.635 & 11.843 & 11.221 & 11.077 & 4750 & 1.75 & 1.49 & 227.25\\\\\n25004 & 201.18696 & -47.34607 & 15.048 & 14.064 & 12.393 & 11.852 & 11.762 & 5034 & 2.06 & 1.41 & 230.68\\\\\n27008 & 201.16507 & -47.36326 & 15.242 & 14.220 & 12.519 & 12.046 & 11.911 & 5095 & 2.15 & 1.38 & 237.50\\\\\n28009 & 201.13729 & -47.36499 & 15.687 & 14.779 & 13.133 & 12.664 & 12.549 & 5186 & 2.41 & 1.32 & 236.00\\\\\n33006 & 201.12822 & -47.40730 & 13.062 & 11.403 &  8.924 &  8.064 &  7.929 & 4051 & 0.39 & 1.83 & 226.34\\\\  \n34008 & 201.19496 & -47.41343 & 13.803 & 12.629 & 10.510 &  9.897 &  9.749 & 4570 & 1.25 & 1.61 & 232.16\\\\\n38006 & 201.11643 & -47.44354 & 16.289 & 15.436 & 13.822 & 13.304 & 13.263 & 5202 & 2.68 & 1.25 & 222.58\\\\\n39013 & 201.16078 & -47.45089 & 13.950 & 12.755 & 10.690 & 10.097 &  9.935 & 4610 & 1.32 & 1.59 & 231.47\\\\\n42012 & 201.17440 & -47.47487 & 14.468 & 13.379 & 11.299 & 10.705 & 10.613 & 4673 & 1.61 & 1.52 & 232.58\\\\\n43002 & 201.14213 & -47.47916 & 15.313 & 14.365 & 12.597 & 12.113 & 11.956 & 5021 & 2.17 & 1.38 & 229.17\\\\\n45011 & 201.10941 & -47.49389 & 16.208 & 15.346 & 13.630 & 13.146 & 13.146 & 5229 & 2.66 & 1.25 & 224.37\\\\\n45014 & 201.15625 & -47.50013 & 15.894 & 15.066 & 13.316 & 12.803 & 12.720 & 5073 & 2.47 & 1.30 & 249.24\\\\\n46003 & 201.12943 & -47.50252 & 15.640 & 14.788 & 13.073 & 12.578 & 12.455 & 5091 & 2.37 & 1.33 & 242.00\\\\\n48009 & 201.12036 & -47.51844 & 16.504 & 15.616 & 14.125 & 13.602 & 13.537 & 5279 & 2.79 & 1.22 & 222.94\\\\\n49008 & 201.16235 & -47.52717 & 15.657 & 14.687 & 12.799 & 12.256 & 12.210 & 4925 & 2.26 & 1.36 & 238.57\\\\\n51005 & 201.09190 & -47.53945 & 16.140 & 15.292 & 13.551 & 13.005 & 12.913 & 5028 & 2.55 & 1.28 & 221.27\\\\\n57006 & 201.18559 & -47.58523 & 15.906 & 15.046 & 13.320 & 12.797 & 12.757 & 5096 & 2.48 & 1.30 & 234.33\\\\\n61009 & 201.16032 & -47.61620 & 14.488 & 13.496 & 11.533 & 10.947 & 10.890 & 4784 & 1.71 & 1.50 & 239.65\\\\\n76015 & 201.33839 & -47.73435 & 15.602 & 14.604 & 12.839 & 12.355 & 12.231 & 5026 & 2.27 & 1.35 & 241.73\\\\\n77010 & 201.23548 & -47.74124 & 14.992 & 13.641 & 11.886 & 11.269 & 11.133 & 4746 & 1.75 & 1.49 & 238.49\\\\\n78008 & 201.21908 & -47.74676 & 16.088 & 15.001 & 13.484 & 12.990 & 12.909 & 5221 & 2.52 & 1.29 & 223.14\\\\\n80017 & 201.40179 & -47.75878 & 15.250 & 14.294 & 12.481 & 11.989 & 11.896 & 5026 & 2.15 & 1.38 & 231.59\\\\\n82012 & 201.44193 & -47.77921 & 16.094 & 15.298 & 13.558 & 13.059 & 12.947 & 5099 & 2.58 & 1.27 & 232.28\\\\\n85007 & 201.19307 & -47.80062 &   -    &   -    & 14.020 & 13.489 & 13.419 & 4983 & 2.20 & 1.37 & 250.48\\\\\n85014 & 201.37723 & -47.80134 & 15.400 & 14.347 & 12.560 & 11.982 & 11.923 & 4899 & 2.11 & 1.39 & 236.78\\\\\n85019 & 201.53965 & -47.80194 & 15.727 & 14.803 & 12.939 & 12.428 & 12.308 & 4938 & 2.31 & 1.34 & 243.81\\\\\n86007 & 201.22490 & -47.80442 &   -    &   -    & 13.024 & 12.487 & 12.387 & 4914 & 1.88 & 1.45 & 238.70\\\\\n86010 & 201.31217 & -47.80789 & 15.594 & 14.557 & 12.926 & 12.437 & 12.329 & 5115 & 2.29 & 1.35 & 238.05\\\\\n86017 & 201.56208 & -47.80760 & 16.289 & 15.452 & 13.737 & 13.319 & 13.232 & 5290 & 2.73 & 1.24 & 231.74\\\\\n87009 & 201.61710 & -47.81630 & 16.081 & 15.199 & 13.392 & 12.885 & 12.850 & 5082 & 2.54 & 1.29 & 247.67\\\\\n88023 & 201.58521 & -47.82029 & 16.415 & 15.542 & 13.774 & 13.268 & 13.154 & 5050 & 2.66 & 1.25 & 232.87\\\\\n89009 & 201.57067 & -47.83291 & 13.776 & 12.650 & 10.497 &  9.890 &  9.753 & 4568 & 1.25 & 1.61 & 242.07\\\\\n89014 & 201.66544 & -47.83110 & 14.611 & 13.607 & 11.639 & 11.055 & 10.967 & 4774 & 1.75 & 1.49 & 231.57\\\\\n90008 & 201.22516 & -47.83980 &   -    &   -    & 13.209 & 12.703 & 12.591 & 5010 & 1.95 & 1.43 & 240.42\\\\\n90019 & 201.62529 & -47.83825 & 14.462 & 13.509 & 11.537 & 11.018 & 10.911 & 4860 & 1.75 & 1.48 & 232.73\\\\\n90020 & 201.64363 & -47.83814 & 16.305 & 15.563 & 13.858 & 13.395 & 13.292 & 5219 & 2.74 & 1.23 & 240.39\\\\\n93016 & 201.65058 & -47.86211 & 15.342 & 14.479 & 12.620 & 12.107 & 12.031 & 5015 & 2.22 & 1.37 & 230.82\\\\\n94011 & 201.30980 & -47.86480 & 15.217 & 14.151 & 12.462 & 11.911 & 11.842 & 4989 & 2.07 & 1.40 & 241.74\\\\\n95015 & 201.54907 & -47.87303 & 16.122 & 15.264 & 13.475 & 12.977 & 12.884 & 5076 & 2.56 & 1.28 & 239.10\\\\\n96011 & 201.52316 & -47.88203 & 13.954 & 12.975 & 11.027 & 10.514 & 10.416 & 4894 & 1.56 & 1.53 & 229.55\\\\\n98012 & 201.35549 & -47.89600 & 14.561 & 13.623 & 12.034 & 11.552 & 11.471 & 5210 & 1.96 & 1.43 & 229.93\\\\\n\\hline     \n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\caption{Stellar abundances}\n\\label{t3}\n\\centering\n\\begin{tabular}{lccccccccccccc}\n\\hline\n\\hline\nID & FeI & ${\\rm [FeI\/H]}$ & NaI & MgI & SiI & CaI & TiI & TiII & CrI & NiI & ZnI & YII & BaII\\\\\n\\hline                      \n00006 & 6.15 & -1.35 & 4.91 & 6.27 & 6.63 & 5.25 & 3.94 & 3.89 & 4.11 & 4.61 & 3.35 & 1.06 & 1.27\\\\\n08004 & 6.23 & -1.27 & 5.74 & 6.31 & 6.97 & 5.53 & 4.12 & 4.30 & 4.38 & 5.01 & 3.61 & 1.75 & 1.93\\\\\n10006 & 5.80 & -1.70 &  -   &  -   &  -   & 5.03 & 3.77 & 3.64 & 3.76 &  -   & 3.09 &  -   & 1.57\\\\\n10009 & 6.18 & -1.32 & 5.61 & 6.38 &  -   & 5.27 & 3.93 & 4.03 & 4.24 & 5.04 & 3.04 & 1.26 & 1.69\\\\\n10010 & 6.45 & -1.05 & 5.38 & 6.71 &  -   & 5.65 & 3.91 & 4.02 & 4.65 & 5.07 & 3.42 & 1.86 & 2.08\\\\\n13006 & 5.93 & -1.57 &  -   & 6.31 &  -   & 5.06 & 3.88 & 3.61 & 4.04 &  -   & 2.95 &  -   & 0.37\\\\\n14002 & 6.02 & -1.48 &  -   &  -   &  -   & 5.25 & 3.90 & 3.72 & 4.00 & 5.19 & 3.28 & 0.61 & 1.15\\\\\n22007 & 6.43 & -1.07 & 5.80 & 6.88 & 6.85 & 5.61 & 4.07 & 4.17 & 4.51 & 5.09 & 3.54 & 1.80 & 2.00\\\\\n25004 & 6.14 & -1.36 &  -   &  -   &  -   & 5.44 & 4.09 & 3.81 & 3.95 &  -   &  -   &  -   & 1.08\\\\\n27008 & 6.52 & -0.98 &  -   &  -   & 7.29 & 5.60 & 4.30 & 4.24 & 4.91 &  -   &  -   & 1.86 & 2.71\\\\\n28009 & 6.29 & -1.21 &  -   &  -   &  -   & 5.53 & 4.20 & 4.29 &  -   & 4.97 &  -   &  -   & 0.65\\\\\n33006 & 6.07 & -1.43 &  -   & 6.38 &  -   & 5.30 & 4.10 & 4.27 & 4.32 & 4.73 &  -   &  -   & 1.43\\\\ \n34008 & 6.11 & -1.39 & 5.52 & 6.24 &  -   & 5.29 & 3.87 & 3.99 & 4.00 & 4.83 &  -   & 1.27 & 1.54\\\\\n38006 & 5.97 & -1.53 &  -   & 6.19 &  -   & 5.15 & 3.79 & 3.88 & 4.16 & 4.99 & 3.19 & 0.65 & 0.61\\\\\n39013 & 6.01 & -1.49 &  -   & 6.51 & 6.85 & 5.30 & 3.77 & 3.71 & 4.13 & 4.80 &  -   & 1.34 & 1.17\\\\\n42012 & 6.10 & -1.40 & 5.05 & 6.62 & 6.66 & 5.42 & 3.93 & 4.08 & 4.22 & 4.85 & 3.37 & 1.62 & 1.61\\\\\n43002 & 5.94 & -1.56 & 5.10 & 6.09 &  -   & 5.24 & 3.96 & 3.81 & 4.29 &  -   &  -   & 1.52 & 1.15\\\\\n45011 & 6.16 & -1.34 & 5.30 & 6.63 & 6.66 & 5.46 & 4.24 & 4.14 & 4.28 & 4.94 &  -   & 0.98 & 1.64\\\\\n45014 & 5.76 & -1.74 &  -   & 6.25 &  -   & 5.00 & 3.64 & 3.65 & 3.83 &  -   &  -   &  -   & 0.10\\\\\n46003 & 5.81 & -1.69 &  -   & 6.04 &  -   & 5.10 & 3.63 & 3.63 & 4.16 & 4.75 & 3.03 & 0.41 & 0.36\\\\\n48009 & 6.24 & -1.26 & 5.30 & 6.83 &  -   & 5.61 &  -   &  -   & 4.58 & 5.18 & 3.68 & 1.10 & 2.06\\\\\n49008 & 6.09 & -1.41 &  -   & 6.43 &  -   & 5.57 & 4.19 & 4.19 & 4.36 & 5.04 & 4.18 & 2.34 & 1.79\\\\\n51005 & 6.08 & -1.42 &  -   & 6.31 &  -   & 5.56 & 4.01 & 4.34 & 4.55 & 4.97 & 3.59 & 1.28 & 1.20\\\\\n57006 & 5.80 & -1.70 & 5.61 &  -   &  -   & 5.10 & 3.97 & 4.11 & 3.84 &  -   & 2.96 & 0.68 & 0.47\\\\\n61009 & 5.76 & -1.74 &  -   & 6.12 & 6.48 & 5.13 & 3.74 & 3.80 & 4.13 & 5.82 &  -   & 0.38 & 0.50\\\\\n76015 & 5.90 & -1.60 &  -   &  -   &  -   & 5.21 & 3.85 & 3.88 & 4.06 &  -   & 3.40 & 1.12 & 1.70\\\\\n77010 & 6.56 & -0.94 & 5.61 & 7.16 & 7.21 & 5.82 & 4.10 & 4.17 & 4.73 & 5.29 & 3.46 & 1.84 & 1.81\\\\\n78008 & 6.14 & -1.36 &  -   &  -   &  -   & 5.15 & 4.10 & 3.92 & 4.32 & 5.06 &  -   & 0.72 & 0.96\\\\\n80017 & 6.04 & -1.46 &  -   &  -   &  -   & 5.23 & 3.64 & 3.79 & 3.87 &  -   &  -   &  -   & 0.55\\\\\n82012 & 5.86 & -1.64 & 5.63 &  -   &  -   & 5.08 & 3.82 & 3.76 & 4.16 &  -   & 3.72 & 1.06 & 1.29\\\\\n85007 & 5.52 & -1.98 &  -   & 6.29 &  -   & 5.14 & 3.67 & 3.30 & 3.83 & 4.74 &  -   & 0.61 & 0.89\\\\\n85014 & 6.03 & -1.47 &  -   &  -   &  -   & 5.50 & 4.24 & 4.05 & 4.42 &  -   &  -   & 1.86 & 1.62\\\\\n85019 & 5.88 & -1.62 &  -   & 6.59 &  -   & 5.29 & 3.96 & 3.99 & 4.19 &  -   & 3.56 & 1.47 & 1.68\\\\\n86007 & 5.84 & -1.66 & 5.72 & 6.24 & 6.87 & 5.50 & 4.05 & 3.86 & 4.43 & 4.80 & 3.67 & 1.24 & 1.65\\\\\n86010 & 5.87 & -1.63 &  -   &  -   &  -   & 5.17 &  -   &  -   &  -   &  -   &  -   &  -   & 0.71\\\\\n86017 & 6.15 & -1.35 & 5.55 &  -   & 6.84 & 5.54 & 4.19 & 4.21 & 4.17 & 5.07 & 3.33 & 1.22 & 2.30\\\\\n87009 & 6.12 & -1.38 &  -   & 6.59 &  -   & 5.62 & 4.41 & 4.11 & 4.83 & 4.90 & 3.41 & 1.98 & 1.84\\\\\n88023 & 6.10 & -1.40 & 5.08 &  -   &  -   & 5.52 & 4.26 & 4.16 & 4.52 & 4.65 &  -   & 1.80 & 2.20\\\\\n89009 & 5.74 & -1.76 &  -   & 6.28 &  -   & 5.05 & 3.55 & 3.76 & 4.09 & 4.55 &  -   & 0.19 & 0.41\\\\\n89014 & 6.14 & -1.36 & 5.84 &  -   & 6.66 & 5.53 & 4.08 & 4.16 & 4.45 & 4.92 &  -   & 1.18 & 1.64\\\\\n90008 & 5.65 & -1.85 & 5.32 & 6.27 &  -   & 5.35 & 3.82 & 3.29 & 4.20 &  -   & 3.53 & 0.73 & 1.00\\\\\n90019 & 5.83 & -1.67 &  -   &  -   &  -   & 5.16 & 4.10 & 3.70 & 4.14 &  -   &  -   & 0.54 & 0.47\\\\\n90020 & 5.89 & -1.61 &  -   &  -   &  -   & 5.09 & 3.70 & 3.72 &  -   &  -   &  -   & 0.57 & 0.60\\\\\n93016 & 6.20 & -1.30 &  -   & 6.69 &  -   & 5.37 & 4.07 & 3.94 & 4.59 &  -   &  -   & 1.85 & 1.49\\\\\n94011 & 5.93 & -1.57 &  -   &  -   &  -   & 5.17 & 4.02 & 4.00 & 4.12 &  -   &  -   & 0.43 & 0.72\\\\\n95015 & 6.24 & -1.26 &  -   & 6.72 &  -   & 5.32 & 4.31 & 4.28 & 4.71 & 5.12 & 3.74 & 1.33 & 1.48\\\\\n96011 & 5.96 & -1.54 & 5.68 &  -   &  -   & 5.25 & 4.20 & 4.05 & 4.33 & 4.59 &  -   & 1.15 & 1.70\\\\\n98012 & 5.85 & -1.65 &  -   &  -   &  -   & 4.98 &  -   &  -   &  -   &  -   &  -   &  -   & 0.68\\\\\n\\hline \nObs. lines & 30 &    & 2    &  1   &  2   &  10  &  10  &  5   &   5  &   5  &  1   &   4  &   2\\\\      \n\\hline    \n\\end{tabular}\n\\end{table*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=10cm]{figure3.eps}\n\\caption{Trend of Na and $\\alpha-$element abundance ratios as a function of\n  [Fe\/H]. Mean values (continuous lines) are provided for those elements which do not show a \n  sizable scattering. See also Table~3.}\n\\label{f3}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=10cm]{figure4.eps}\n\\caption{Trend of abundance ratios for Iron peak elements (Ni and Cr),\nZn, Y and Ba for the outer region stars.\nMean values (continuous lines) are provided when there is no sizable scatter }\n\\label{f4}\n\\end{figure*}\n\n\\subsection{The linelist}\n\nThe line-lists for the chemical analysis were obtained from the VALD\ndatabase (Kupka et al.\\ 1999) and calibrated using the Solar-inverse technique.\nFor this purpose we used the high resolution, high S\/N Solar spectrum\nobtained at NOAO ($National~Optical~Astronomy~Observatory$, Kurucz et\nal.\\ 1984). The EWs for the reference Solar spectrum were obtained in the same\nway as the observed spectra, with the exception of the strongest lines, where\na Voigt profile integration was used. Lines affected by blends were rejected\nfrom the final line-list.\nMetal abundances were determined by the Local Thermodynamic Equilibrium (LTE)\nprogram MOOG (freely distributed by C. Sneden, University of Texas at Austin),\ncoupled with a solar model atmosphere interpolated from the Kurucz (1992) grids\nusing the canonical atmospheric parameters for the Sun: $\\rm {T_{eff}=5777}$ K,\n$\\rm {log(g)}=4.44$, $\\rm {v_{t}=0.80}$ km\/s and $\\rm {[Fe\/H]=0.00}$.\nIn the calibration procedure, we adjusted the value of the line strength\nlog(gf) of each spectral line in order to report the abundances obtained from\nall the lines of the same element to the mean value.\nThe chemical abundances obtained for the Sun and used in the paper as\nreference are reported in Tab.~\\ref{t1}.\n\n\\subsection{Atmospheric parameters}\n\nEstimates of the atmospheric parameters were derived from \nBVJHK photometry. Effective temperatures (T$_{\\rm eff}$) for each \nstar were derived from the T$_{\\rm eff}$-color relations\n(Alonso et al.\\ 1999, Di Benedetto 1998, and Ramirez \\& M\\'elendez 2005).  \nColors were de-reddened using a reddening given by\nSchlegel et al. (1998). A value E(B-V) = 0.134 mag. was  adopted.\n\nSurface gravities log(g) were obtained from the canonical equation:\n\n\\begin{center}\n${\\rm log(g\/g_{\\odot}) = log(M\/M_{\\odot}) + 4\\cdot\n  log(T_{eff}\/T_{\\odot}) - log(L\/L_{\\odot}) }$\n\\end{center}\n\nFor M\/M$_{\\odot}$ we adopted 0.8 M$_{\\odot}$, which is the\ntypical mass of RGB stars in globular clusters.\nThe luminosity ${\\rm L\/L_{\\odot}}$ was  obtained from the absolute\nmagnitude M$_{\\rm V}$, assuming an absolute distance modulus \nof (m-M)$_{\\rm 0}$=13.75 (Harris 1996), which gives an apparent distance\nmodulus of (m-M)$_{\\rm V}$=14.17 using the adopted reddening. \nThe bolometric correction ($\\rm {BC}$) was  derived by adopting the relation\nBC-T$_{\\rm eff}$ from Alonso et al.\\ (1999). \\\\\nFinally, microturbolence velocity ($\\rm {v_{t}}$) was  obtained from the\nrelation (Marino et al. 2008):\n\n\\begin{center}\n${\\rm v_{t}\\ (km\/s) = -0.254\\cdot log(g)+1.930}$\n\\end{center}\n\nwhich was obtained specifically for old RGB stars, as it is our present sample.\nAdopted atmospheric parameters for each star are reported in Tab.~\\ref{t2}\nin column 9,10,11.\nIn this Table column 1 gives the ID of the star, columns 2 and 3 the\ncoordinates, column 4,5,6,7,8 the B,V,J,H,K magnitudes, column 12 the\nheliocentric radial velocity.\n\n\n\\subsection{Chemical abundances}\n\nThe Local Thermodynamic Equilibrium (LTE) program MOOG (freely\ndistributed by C. Sneden, University of Texas at Austin) has been used to\ndetermine the abundances from EWs, coupled with model atmosphere\ninterpolated from the Kurucz (1992) for the parameters obtained as described\nin the previous Section.\nThe wide spectral range of the UVES data allowed us to derive the\nchemical abundances of several elements. Chemical abundances\nfor single stars we obtained are listed in Tab.~\\ref{t3}.\nThe last line of this table gives the mean number of lines we were able to \nmeasured for each elements.\nTi is the only element for which we could measure neutral and first ionization\nlines. The difference of mean abundances obtained from the two stages is:\n\\begin{center}\n${\\rm \\Delta(TiI-TiII)=0.03\\pm0.01}$\n\\end{center}\nThis difference is small and compatible with zero within 3 $\\sigma$. This confirms\nthat gravities obtained by the canonical equation are not affected by appreciable\nsystematic errors.\n\n\\subsection{Internal errors associated with the chemical abundances}\n\nThe measured abundances of every element vary from\nstar to star as a consequence of both measurement errors and\nintrinsic star to star abundance variations.\nIn this section our goal is to search for evidence of intrinsic\nabundance dispersion in each element by comparing the observed\ndispersion $\\sigma_{\\rm {obs}}$ and that produced by internal errors\n($\\Delta_{\\rm {tot}}$). Clearly, this requires an accurate analysis of\nall the internal sources of measurement errors.\nWe remark here that we are interested in star-to-star intrinsic\nabundance variation, i.e. we want to measure the internal intrinsic\nabundance spread of our sample of stars. For this reason, we\nare not interested in external sources of error which are systematic\nand do not affect relative abundances.\\\\\nIt must be noted that two main sources of errors contribute\nto $\\sigma_{\\rm {tot}}$. They are:\n\\begin{itemize}\n\\item the errors $\\sigma_{\\rm {EW}}$ due to the uncertainties in the\n  EWs measures;\n\\item the uncertainty $\\sigma_{\\rm {atm}}$ introduced by errors in the \n  atmospheric parameters adopted to compute the chemical abundances.\n\\end{itemize}\n\n$\\sigma_{\\rm {EW}}$ is given by MOOG for each element and each star.\nIn Tab.~\\ref{t4} we reported in the second column the average $\\sigma_{\\rm {EW}}$\nfor each element. For Mg and Zn we were able to measure only one line.\nFor this reason their $\\sigma_{\\rm {EW}}$ has been obtained as the mean of\n$\\sigma_{\\rm {EW}}$ of Na and Si multiplied by $\\sqrt 2$. \nNa and Si lines were selected because their strength was similar to that of Mg\nand Zn features. This guarantees that $\\sigma_{\\rm {EW}}$, due to the photon\nnoise, is the same for each single line.\n\nErrors in temperature are easy to determine because, for each star, it is\nthe r.m.s. of the temperatures obtained from the single colours. The mean\nerror $\\Delta$T$_{\\rm eff}$ turned out to be 50 K.\nUncertainty on gravity has been obtained by the canonical formula using the\npropagation of errors. The variables used in this formula that are affected\nby random errors are T$_{\\rm eff}$ and the V magnitude. For temperature we\nused the error previously obtained, while for V we assumed a error of 0.1 mag,\nwhich is the typical random error for photographic magnitudes. Other error\nsources (distance modulus, reddening, bolometric correction) affect\ngravity in a systematic way, so are not important to our analysis.\nMean error in gravity turned out to be 0.06 dex. This implies, in turn,  a mean error\nin microturbolence of 0.02 km\/s.\n \nOnce the internal errors associated with the atmospheric parameters were\ncalculated, we re-derived the abundances of two reference stars (\\#00006 and\n\\#42012) which roughly cover the whole temperature range of our sample \nby using the following combination of atmospheric parameters:\n\\begin{itemize}\n\\item ($\\rm {T_{eff}} \\pm \\Delta (\\rm {T_{eff}})$, $\\rm {log(g)}$,  $\\rm {v_{t}}$)\n\\item ($\\rm {T_{eff}} $, $\\rm {log(g)} \\pm \\Delta (\\rm {log(g)})$,  $\\rm {v_{t}}$)\n\\item ($\\rm {T_{eff}} $, $\\rm {log(g)}$,  $\\rm {v_{t}} \\pm \\Delta (\\rm {v_{t}})$)\n\\end{itemize}\nwhere $\\rm {T_{eff}}$, $\\rm {log(g)}$,  $\\rm {v_{t}}$ are the measures\ndetermined in Section 3.2 .\n\nThe difference of abundance between values obtained with the original\nand those ones obtained with the modified values gives\nthe errors in the chemical abundances due to uncertainties in\neach atmospheric parameter. They are listed in Tab.~\\ref{t4} (columns 3, 4\nand 5) and are the average values obtained from the two stars. \nBecause the parameters were not obtained indipendently we cannot\nestimate of the total error associated with the abundance\nmeasures by simply taking the squadratic sum of all the single errors.\nInstead we calculated the upper limits for the total error as:\n\\begin{center}\n$\\rm {\\Delta_{tot}=\\sigma_{EW}+\\Sigma(\\sigma_{atm})}$\n\\end{center}\nlisted in column 6 of Tab.~\\ref{t4}.\nColumn 7 of Tab.~\\ref{t4} is the observed dispersion.\nComparing $\\sigma_{\\rm obs}$ with $\\Delta_{\\rm tot}$ (wich is an upper limit) we can see\nthat for many elements (Mg, Si, Ca, Ti, Cr, Ni) we do not find any evidence of inhomogeneity\namong the whole Fe range. Some others (Na, Zn, Y, Ba) instead show\nan intrinsic dispersion. This is confirmed also by Figs. \\ref{f3} and \\ref{f4}\n(see next Section).\nFinally we just mention here the problem of the differential reddening. \nSome authors (Calamida et al. 2005) claim that is is of the order of 0.03 mag,\nwhile some others (McDonald et al. 2009) suggest a value lower than 0.02 dex.\nHowever all those results concern the internal part, while no information is\navailable for the region explored in this paper. \nWe can only say that an error on the reddening of 0.03 dex would alter\nthe temperature of 90 degrees.\n\n\\begin{table*}\n\\caption{Internal errors associated with the chemical abundances\ndue to errors in the EW measurement (column 2) and in the atmospheric\nparameters (column 3,4,5) for the studied elements.\n6$^{th}$ column gives the total internal error, while the last\ncolumn gives the observed dispersion of the abundances. See text for\nmore details.\n}\n\n\\label{t4}\n\\centering\n\\begin{tabular}{lcccccc}\n\\hline\n\\hline\nEl. & $\\sigma_{\\rm EW}$ & $\\Delta$T$_{\\rm eff}$ & $\\Delta$log(g) & $\\Delta$v$_{\\rm t}$ & $\\Delta_{\\rm tot}$ & $\\sigma_{\\rm obs}$\\\\\n\\hline                      \n${\\rm [FeI\/H]}$    & 0.05 & 0.05 & 0.01 & 0.02 & 0.13 &  -  \\\\\n${\\rm [NaI\/FeI]}$  & 0.12 & 0.02 & 0.01 & 0.02 & 0.17 & 0.34\\\\\n${\\rm [MgI\/FeI]}$  & 0.18 & 0.02 & 0.01 & 0.02 & 0.23 & 0.18\\\\\n${\\rm [SiI\/FeI]}$  & 0.15 & 0.03 & 0.01 & 0.02 & 0.21 & 0.12\\\\\n${\\rm [CaI\/FeI]}$  & 0.09 & 0.01 & 0.00 & 0.01 & 0.11 & 0.11\\\\\n${\\rm [TiI\/FeI]}$  & 0.14 & 0.04 & 0.01 & 0.01 & 0.20 & 0.19\\\\\n${\\rm [TiII\/FeI]}$ & 0.13 & 0.04 & 0.03 & 0.01 & 0.21 & 0.17\\\\\n${\\rm [CrI\/FeI]}$  & 0.12 & 0.03 & 0.01 & 0.01 & 0.17 & 0.17\\\\\n${\\rm [NiI\/FeI]}$  & 0.13 & 0.01 & 0.01 & 0.01 & 0.16 & 0.14\\\\\n${\\rm [ZnI\/FeI]}$  & 0.19 & 0.04 & 0.03 & 0.02 & 0.28 & 0.32\\\\\n${\\rm [YII\/FeI]}$  & 0.13 & 0.03 & 0.03 & 0.01 & 0.20 & 0.42\\\\\n${\\rm [BaII\/FeI]}$ & 0.14 & 0.02 & 0.03 & 0.00 & 0.19 & 0.50\\\\\n\\hline     \n\\end{tabular}\n\\end{table*}\n\n\n\n\\section{Results of abundance analysis}\nThe results of the abundance analysis are shown in Fig.~\\ref{f2} for [Fe\/H],\nand in Figs.~\\ref{f3} and \\ref{f4} for all the abundance ratios we could derive.\nA Gaussian fit was used to derive the mean metallicity of the three peaks in\nFig.~\\ref{f2}. We found the following values: -1.64 (metal poor\npopulation, {\\it MPP}), -1.37 (intermediate metallicity population, {\\it IMP}), and -1.02\n(metal rich population, {\\it MRP}). Stars belonging to each of the three populations are\nidentified with different colors in Fig.~\\ref{f1}.The population mix is in the proportion \n({\\it MPP:IMP:MRP}) = (21:23:4).\\\\\n\n\\noindent\nThe abundance ratio trends versus [Fe\/H] \nare shown in the various panels in Figs.~\\ref{f3} and \\ref{f4} for all the elements we could measure.\nWhen the abundance ratio scatter is low (lower than 0.2 dex which, according to\nthe previous Section, implies a homogeneous abundance) we also show the mean value\nof the data as a continuous line, to make the comparison with literature easier.\nWhat we find in the outer region of $\\omega$~ Cen is in basic agreement\nwith previous investigations. Comparing our trends with -e.g.- Norris \\& Da Costa (1995)\nvalues( see next Section for a more general comparison with the literature), \nwe find that all the abundance ratios we could measure are in very good\nagreement with that study, except for [Ti\/Fe], \nwhich is slightly larger in our stars, and [Ca\/Fe], \nwhich is slightly smaller in our study. However, within the measurement errors we do not\nfind any significant deviation.\\\\ \nThe $\\alpha-$elements (Mg, Ti, Si and Ca, see Fig.~\\ref{f3}) \nare systematically overabundant with respect to the Sun, \nwhile iron peak elements (Ni and Cr, see Fig.~\\ref{f4}) are basically solar.\nSimilarly, overabundant in average with respect to the Sun are Y, Ba and Zn (see Fig.~\\ref{f4}). \nY abundance ratio show some trend with [Fe\/H], but of the same sign and \ncomparable magnitude to Norris \\& Da Costa (1995).\n\nFinally, we looked for possible correlations between abundance ratios, and compare\nthe outcome from the different populations of our sample. This was possible only for [Y\/Fe] and [Zn\/Fe]\nversus [Ba\/Fe], and it is plotted in Fig.~\\ref{f5}. For {\\it MPP} (filled circles) a trend\nappears both for Zn and Y as a function of Ba (see also value of the slope ($a$) in Fig.~\\ref{f5}), with\nBa-poor stars being also Zn and Y poor. Y-Ba correlation can be easily\nexplained because both are neutron-capture elements.\\\\\nAs for {\\it IMP},  a marginal trend in the Y vs. Ba relation is present,\nwhile no trend appears in the Zn vs. Ba. No trends at all were detected for\nMRP, mostly because our sample of {\\it MRP} stars is too small for any significant\nconclusion. We underline the fact that this different behaviour of {\\it MPP} and {\\it IMP}\nwith respect to their Zn-Y-Ba correlations points to a different chemical\nenrichment history of the two populations.\n\n\\section{Outer versus inner regions}\n\nA promising  application of our data is the comparison of the population mix in the cluster outskirts\nwith the one routinely found in more central regions of the cluster (Norris \\& Da Costa 1995; Smith et al. 2000;\nVillanova et al 2007; Johnson et al. 2009; Wylie-de Boer et al. 2009).\\\\\n\nTo this aim, we firstly compute the fraction of stars\nin the various metallicity  ([Fe\/H]) populations, and compare it with the inner\nregions trends from Villanova et al. (2007), for the sake of homogeneity,\nto statistically test the significance of their similarity or difference. \nWe are aware that this is not much more than a mere exercise.\nFirstly, while our program stars are mostly in the RGB phase, in Villanova et al (2007)\nsample only SGB stars are present. This implies that we are comparing stars in\nslightly different evolutionary phases.\nSecond, and more important, the statistics is  probably too poor. \nIn fact, we report in Table~5 (column 2 and 3) the number of stars\nin the different metallicity bin derived from a Gaussian fit to our and Villanova et al. (2007)\ndata. They  have large errors. We see that within these errors the population mix is basically the\nsame in the inner and outer regions. Therefore, with so few stars we cannot\ndetect easily differences between the inner and outer regions.\nTo check for that, we make use of the Kolmogorov-Smirnov statistics,\nand compare the metallicity distributions of the inner and outer samples, to see\nwhether they come from the same parental distribution. We found that the probability\nthat the two distributions derive from the same underlying distribution is 77$\\%$.\nThis is quite a small number, and simply means that with these samples we cannot\neither disprove or confirm the null hypothesis (say that the two populations have\nsame parental distribution).\nBesides, our sample and that of Villanova et al (2007) do not have stars\nbelonging to the most metal-rich population of Omega centauri (at\n[Fe\/H]$\\sim$-0.6), which therefore we cannot comment on.\\\\\n\n\\noindent\nThat clarified, we then compare in Fig.~6 and Fig.~7 the trend of the various elements we could\nmeasure (see Table~4) in the cluster outskirts with the trends found in the central regions by other\nstudies. In details, in all Fig.~6 panels we indicate with filled circles the data\npresented in this study and  with open circles data from Villanova et al. (2007). Crosses indicate\nWylie-de Boer et al. (2009), stars Norris \\& Da Costa (1995), empty squares \nSmith et al. (2000) and, finally, empty pentagons Johnson et al. (2009).\nWe separate in Fig.~6 elements which do no show significant scatter (see Table~4) from\nelements which do  show a sizeable scatter (see Fig.~7).\nBa abundances from Villanova et al. (2007) were corrected of $\\sim$-0.3 dex,\nto take into account the hyperfine structure that seriously affects the Ba\nline at 4554 \\AA.\n\n\n\n\\begin{table}\n\\centering\n\\label{t5}\n\\begin{tabular}{ccc}\n\\hline\nPopulation & Inner & Outer \\\\\n & $\\%$ & $\\%$ \\\\\n\\hline\nMPP & 46$\\pm$10 & 45$\\pm$10 \\\\\nIMP & 36$\\pm$10 & 47$\\pm$10 \\\\\nMRP & 18$\\pm$10 &  8$\\pm$10 \\\\\n\\hline\n\\end{tabular}\n \\caption{Percentages of different metallicity populations in the inner and outer regions\nof $\\omega$~ Cen.}\n \\end{table}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figure5.eps}\n\\caption{\nAbundance ratios of [Y\/Fe] and [Zn\/Fe] vs. [Ba\/Fe] for our sample.\nFilled circles, open circles, and crosses are MPP, IMP, MRP stars\nrespectively. A straight line has been fitted to MPP stars. The value of the slope\n({\\it a}) is given. In both cases {\\it a} is out of more than 3$\\sigma$ with\nrespect the null trend value, implying that trends for MPP are real.\n}\n\\label{f5}\n\\end{figure}\n\n\n\n\\noindent\nLooking at Fig.~6, we immediately recognize two important facts.\\\\\nFirst, all the studies we culled from the literature for Omega Cen central regions\nshow the same trends.\\\\\nSecond, and more important for the purpose of this paper,\nwe do not see any significant difference bewteen the outer (filled circles)\nand inner  (all the other symbols) regions of the cluster. Only Ti seems to be\nslightly over-abundant in the outer regions with respect to the more central ones.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figure6.eps}\n\\caption{Comparison of abundance ratios in the inner and\nouter stars (filled circles). Symbols are as follows. Empty circles (Villanova et al. 2007),  crosses \n(Wylie-de Boer et al. 2009), stars (Norris \\& Da Costa 1995), empty squares \n(Smith et al. 2000) empty pentagons (Johnson et al. 2009)}\n\\label{f6}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figure7.eps}\n\\caption{Comparison of abundance ratios in the inner\nand outer stars (filled circles). Symbols are as in Fig.~6}\n\\label{f7}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figure8.eps}\n\\caption{Comparison of our Y and Ba abundance ratios (our sample, filled\n  circles) with the literature (inner sample).Symbols are as in Fig.~6}\n\\label{f8}\n\\end{figure}\n\n\\noindent\nAs for the more scattered elements (see Fig.~7) we notice that Na shows  the opposite trend\nin the outer regions with respect to the inner ones, but this is possibly related\nto a bias induced by the signal to noise of our data which does not allow us to detect\nNa-poor stars in the metal poor population.\nOn the other hand, Y and Ba do not show any spatial difference. \\\\\n\n\\noindent\nInterestingly enough, at low metallicity Ba shows quite a significant scattered\ndistribution, expecially for stars more metal-poor than -1.2 dex, covering a\nrange of 1.5 dex. This behaviour is shared with Y and Na, althought at a lower level.\nFurthermore, looking carefully at Fig. 4, it is possible to see a hint of\nbimodality for the Ba content of the stars having [Fe\/H]$<$-1.5 dex\n(i.e. belonging to the MMP), with the presence of a group of objects having\n[Ba\/Fe]$\\sim$1.0 dex, and another having [Ba\/Fe]$\\sim$-0.2 dex.\nThe same trend is visible in all the literature data.\\\\\nWe remind the reader that such a bimodal distribution is similar to that found by \nJohnson et al. (2009, thier Fig.~8) for Al.\\\\\n\n\\noindent\nFinally, we compare our Y vs. Ba trend with literature in Fig. 8. Also in this\ncase the agreement is very good and no radial trend is found.\\\\\n\n\\noindent\nThe stars studied by Wylie-de Boer et\nal. (2009) deserve special attention. They belong to the Kapteyn Group, \nbut their kinematics and chemistry suggest a likely association with $\\omega$ Cen.\nThese stars, in spite of being part of a moving group, exhibit \nquite a large iron abundance spread (see Fig. 6 and 7), totally compatible with the one of\n$\\omega$ Cen. Also their Na and Ba abundance (see Fig. 7) are comparable with\nthose of the cluster. We suggest that the comparison with our data reinforces\nthe idea that the Kapteyn Group is likely formed by stars stripped away from $\\omega$ Cen.\n\n\\section{Conclusions}\nIn this study, we analized a sample of 48 RGB stars located half-way to the tidal\nradius of $\\omega$ Cen, well beyond any previous study devoted to the detailed chemical composition of the\ndifferent cluster sub-populations.\\\\\nWe compared the abundance trends in the cluster outer regions with literature studies which focus\non the inner regions of  $\\omega$ Cen.\\\\\n\n\\noindent\nThe results of this study can be summarized as follows:\n\n\\begin{description}\n\\item $\\bullet$ we could not highlight any difference between the outer and inner regions\nas far as [Fe\/H]is concerned: the same mix of different iron abundance population is present\nin both locations;\n\\item $\\bullet$ most elements appear in the same proportion both in the inner and in the outer\nzone, irrespective of the particular investigation one takes into account for the comparison;\n\\item $\\bullet$ [Ba\/Fe] shows an indication of a bimodal distribution at low metallicity at any location\nin the cluster, which deserves further investigation;\n\\item $\\bullet$ no indications emerge of gradients in the radial abundance trend of the elements we could\nmeasure.\n\\end{description}\n\n\\noindent\nOur results clearly depend on a small data-set, and more extended studies are encouraged to confirm\nor deny our findings.\n\n\n\\section*{Acknowledgements}\nSandro Villanova acknowledges ESO for financial support during several visits to the Vitacura\nScience office. The authors express their gratitude to Kenneth Janes for reading carefully\nthe manuscript.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTransition Metal Dichalcogenides (TMD) have the generic formula MX$_2$, consisting of a transition metal M (Nb, Ta, Ti, Mo, W...) and a chalcogen X (S, Se, Te). They are layered materials with strong in-plane bonds and weak Van der Waals inter-layers interactions providing an important two dimensional (2D) character. The individual layers consist of a triangular lattice of transition metal atoms surrounded by chalcogens, and come in two forms named 1T and 1H. In 1T layers the transition metal atoms are surrounded by six chalcogens in octahedral (O$_h$) coordination, whereas in 1H layers the six chalcogens are in trigonal prismatic (D$_{3h}$) coordination. \nThese two base layers have a wide range of possible stacking arrangements, called polytypes \\cite{Wilson1975AdvPhysCDWTMDReview} (e.g. see Fig.~\\ref{fig:2H3R}), which differ by the translation, rotation and ordering of the two base layers 1H and 1T. \nTMD polytypes are usually classified using Ramsdell's notation\\cite{ramsdell1947studies}, which specifies the number of layers in the unit cell followed by a letter to indicate the lattice type and, when necessary, an additional alphanumeric character to distinguish between stacking sequences. Thus, a 1T polytype has 1 layer in a trigonal unit cell while a 2H polytype has 2 layers in a primitive hexagonal unit cell.\nThis distinction is especially important for TMD as polytypes of the same TMD compound can have dramatically different electronic properties spanning from semiconducting to metallic or superconducting\\cite{Voiry2015TMD_polytypesynthesis_review}. \n\nTMD recently attracted renewed interest because their quasi 2D nature is similar to graphene and the tunability of their electronic properties is promising for novel electronic devices\\cite{Wang2012TMDreviewnature,Vogel2015MRSBull}. In the case of metallic TMD, the 2D character and strong electron-phonon coupling makes them prone to electronic orderings such as Mott insulator or charge density waves (CDW) and superconductivity\\cite{Wilson1975AdvPhysCDWTMDReview}. This multiplicity of possible ground states hold a great technological potential. For instance, a new \\emph{orbitronics} concept has been proposed in TMD such as 1T-TaS$_2$, whereby switching between the orbital configurations and melting the CDW phase using ultrashort laser pulses would yield a complete and reversible semiconductor-to-metal transition\\cite{Ritschel2015Orbitronics}. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{Figure1_1T_2H_3R90dpi.png}%\n\\caption{\\label{fig:2H3R} \\textbf{Cristallographic structures of the three known polytypes of NbS$_2$}: 1T reported only in thin film\\cite{Carmalt2004_1st_1TNbS2synthesis} or monolayer form\\cite{Chen20151TNbS2monolayerforH2}, and 2H and 3R found in bulk crystals form\\cite{Fisher1980NbS2difficultsynthesis}. In the 1T polytype, the transition metal atom is in octahedral coordination and layers are stacked without rotation or in-plane translation. The 2H and 3R polytypes are composed of 1H single layers with the metal atom in trigonal prismatic coordination, but they differ by their stacking: rotation and no in-plane translation for 2H, in-plane translation and no rotation for 3R. The unit cell is indicated by solid red lines.}\n\\end{figure}\n\nCDW are periodic modulations of the electronic density accompanied by a periodic distortion of the crystal lattice. CDW are usually caused either by a nesting vector of the Fermi surface inducing a peak in electronic susceptibility\\cite{gruner}, or by a strong k-dependent electron-phonon coupling\\cite{weber}. TMD are the first 2D compounds where CDW were observed\\cite{Wilson1974PRL}, and TMD appear even more prone to CDW in single layer than in bulk form. For instance, in 2H-NbSe$_2$ the CDW transition temperature increases from 33\\,K in bulk to 145\\,K in monolayer form\\cite{Xi2015highTcdwNbSe2,CalandraPRB2009monolayerNbSe2}.\n\nHowever, among the metallic TMD, NbS$_2$ stands out as none of its polytypes have been reported to have a CDW. \n{In bulk form, only the 2H and 3R polytypes (trigonal prismatic coordination of Nb atoms) have been grown and CDWs have not been reported in either polytytpe\\cite{FriendYoffe1987}}.\nThe trigonal prismatic coordination was found to be thermodynamically stable in bulk by DFT calculations\\cite{LIU2014472}. The 1T polytype (octahedral coordination) has also been reported but only in single layer\\cite{Chen20151TNbS2monolayerforH2} or thin film form\\cite{Carmalt2004_1st_1TNbS2synthesis}, both also without CDW.\n\nIn the 2H polytype of NbS$_2$ that we study here, we previously showed that anharmonic effects prevent the formation of a CDW despite strong phonon modes softening\\cite{Leroux2012anharmonicNbS2}. Thus, 2H-NbS$_2$ is just on the verge of a CDW and DFT calculations also hint at the proximity of density waves instabilities\\cite{Leroux2012anharmonicNbS2, Guller2015DFTsdwNbS2, HeilPRL2017}.\nThe soft phonon modes do contribute to another electronic ordering: the metal-to-superconductor transition below $T_\\mathrm{c}= 6$\\,K\\cite{Wilson1975AdvPhysCDWTMDReview}, in which they are the dominant contributor to anisotropic two-gap superconductivity\\cite{Guillamon2008STMVortexcore,KACMARCIK2010S719,pribulova2010two,Diener2011TDONbS2,Leroux2012Hc1NbS2,HeilPRL2017}.\nYet, no other electronic phase has ever been found experimentally in 2H-NbS$_2$ using either: very pure crystals ($RRR=105$)\\cite{Naito1982ResistivityNbS2}, low temperature (100\\,mK)\\cite{Guillamon2008STMVortexcore}, or high pressure\\cite{JonesMorosinPRB1972pressureNbS2}. This is in contrast with the isoelectronic TMD 2H-NbSe$_2$ and 2H-TaS\/Se$_2$ which all have CDW\\cite{Wilson1975AdvPhysCDWTMDReview,Naito1982ResistivityNbS2}.\n \nHere we find that there are faint traces of CDW in 2H-NbS$_2$ using {diffuse x-ray scattering}. This CDW wavevectors are the same as that of the commensurate CDW in 1T-TaS$_2$ and 1T-TaSe$_2$. Such 1T-like CDW has not been reported before for the NbS$_2$ compound. We suggest two mechanisms to explain both the symmetry and very small amplitude of the CDW we observe. Rotational stacking faults between 2H domains could be locally like a 1T-layer, or a very dilute amount of Nb in the van der Waals interlayer space could also present an octahedral coordination.\n\n\n\n\\section{Materials and methods}\n\nSingle crystals of 2H-NbS$_2$ were synthesized from an appropriate mixture of the elements that was sealed in an evacuated quartz tube. A large excess of sulfur was added to the mixture (20 \\%) to act as a transporting agent and favor the formation of the 2H polytype. The tube was heated to 950$^\\circ$C for 240\\,h, slowly cooled down to 750$^\\circ$C and subsequently quenched to room temperature. This synthesis yielded a powder containing single crystals with lateral sizes exceeding 200\\,$\\mu$m as shown in Ref.\\cite{Diener2011TDONbS2}. Powder x-ray diffraction on several batches showed a predominance by volume of 99\\% of 2H polytype (P6$_3\/mmc$) versus 1\\% of 3R (R$3m$), and the polytype of each single crystal used was checked individually using x-ray diffraction. We find lattice parameters $a = b = 3.33\\,$\\AA~and $c = 11.95\\,$\\AA. Superconducting properties and phonon spectrum of samples from this batch were published elsewhere\\cite{Guillamon2008STMVortexcore,Kacmarcik2010CpNbS2,Diener2011TDONbS2,Leroux2012Hc1NbS2,Leroux2012anharmonicNbS2} and are in agreement with the literature\\cite{Onabe1978ResistivityNbS2,Naito1982ResistivityNbS2}. Typical superconducting transition temperature is $T_\\mathrm{c} = 6.05 \\pm 0.4$\\,K, as determined by AC specific heat\\cite{Kacmarcik2010CpNbS2}. \n\n{Diffuse x-ray scattering} imaging was performed at beamline ID29 at the ESRF at a wavelength of 0.6966 \\AA\\ (17.798 keV) and using a PILATUS 6M detector 200\\,mm away from the sample. 3600 pictures were acquired in three dimensions with 0.1$^\\circ$ oscillations and 0.25\\,s of exposure. Reconstruction of the $(h,k,0)$ plane was performed using CrysAlis software. Final reconstructions were made with locally developed software and Laue symmetry applied to remove the gaps between the detector elements. Inelastic x-ray scattering was performed at beamline ID28 at the ESRF using the Si (9,9,9) monochromator reflection giving an energy resolution of 2.6\\,meV and a photon energy of 17.794 keV. Measurements were performed at 300 and 77\\,K using a nitrogen cryostream cooler.\n\n\\section{Results}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{Figure2_RT_77K.jpg}%\n\\caption{\\label{fig:RT_77K}\\textbf{{diffuse x-ray scattering} of 2H-NbS$_2$.} $(h,k,0)$ plane at 300\\,K \\textbf{(left)} and 77\\,K \\textbf{(right)} showing the hexagonal Brillouin zone, {the reciprocal space base vectors $\\vec{a}^*$ and $\\vec{b}^*$, the high symmetry points $\\Gamma$, M and K, and the (1,0,0) and (2,0,0) Bragg peaks.} Elongated diffuse scattering is visible between Bragg peaks at 300\\,K, and increases in intensity at 77\\,K. It is caused by soft phonon modes and is not visible between each pair of Bragg peaks because of the longitudinal polarization of the soft phonon modes\\cite{Leroux2012anharmonicNbS2}.\nAt 77\\,K, three types of satellite peaks appear: a ring of twelve sharp peaks around each Bragg peak, a peak at the M point and 4 peaks around the M point. One example from each of these three sets of satellite peaks are indicated by white circles on the right panel}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{Graph1.pdf}%\n\\caption{\\label{fig:IXS}\\textbf{Typical IXS spectra to determine the elastic or inelastic nature of the satellite peaks.} IXS energy spectra at 77\\,K around the M point ($\\vec{q} = (0.5,0,0)$) show that the elastic peak at zero energy transfer (static order) is the dominant contribution to the peak observed at the M point in diffuse scattering (energy integrated intensity). But the amplitude of this elastic peak is still comparable to that of a soft phonon. An elastic peak corresponds to a static diffracting object in real space.}\n\\end{figure}\n\nFig.~\\ref{fig:RT_77K} shows the diffuse scattering in the $(h,k,0)$ plane reconstructed from diffuse scattering data, at 300 and 77\\,K. The Bragg peaks amplitude is saturated on these images. The rocking curve of the (1,1,0) spot of a crystal from the same batch has a Full-Width at Half Maximum (FWHM) of 0.12$^\\circ$ at room temperature, implying a Bragg peak FWHM of at most 0.0068\\AA$^{-1}$ or 0.0036 $a^*$, i.e. an in-plane coherence length of at least 276 unit cells. \n\nAt 300\\,K, some diffuse scattering can be seen spanning the length between the different $\\Gamma$M direction around each Bragg peak. This elongated diffuse scattering becomes salient at 77\\,K.\nIt is caused by the broad softening of phonon modes around 1\/3 of $a^*$ (2\/3 of $\\Gamma$M)\\cite{Leroux2012anharmonicNbS2}. \n\nAt 77\\,K, Fig.~\\ref{fig:RT_77K} also shows three types of satellite peaks: a peak at the M point, four peaks around the M point, and a ring of twelve peaks around each Bragg peak.\nInelastic x-ray scattering results, presented in Fig.~\\ref{fig:IXS}, show that these peaks are all of an elastic nature, i.e. reflections of a static order, but with an amplitude similar to that of the soft phonon modes. {Comparison to the (1,1,0) Bragg peak, shows that these peaks are 5 orders of magnitude less intense}. Such a low intensity indicates that these peaks correspond either to very small atomic displacements or to displacements taking place in a very small fraction of the crystal.\n\n\\subsection{Satellite peak at the M point}\n\n\\textit{Ab initio} calculations\\cite{Leroux2012anharmonicNbS2} find that the satellite peak at the M point corresponds to a maximum in electronic susceptibility. Considering its phonon-like amplitude, the peak at the M point could correspond to Friedel's oscillations around impurities. However the peak FWHM is 0.036\\,$a^*$, as shown in the lower panel of Fig.~\\ref{fig:satpeakwidth}, which corresponds to a coherence length of about 30 unit cells. It therefore seems equally likely that the peak at the M point could correspond to an extremely faint CDW with a periodicity of $2\\,a$ induced by the maximum in electronic susceptibility. Such faint $2\\,a$ super-lattice spots have also been reported in 2H-NbSe$_2$\\cite{CHEN1984}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{Figure3_1st2nd3rdorder_reduced.pdf}%\n\\caption{\\label{fig:qvector}\\textbf{Two interwoven commensurate superlattices} Diffuse scattering of 2H-NbS$_2$ in the $(h,k,0)$ plane at 77\\,K.\nThe ring of twelve satellite peaks around each Bragg peak, and the four peaks around the M point can be indexed with two wavevectors. \n\\textbf{(Upper left panel)} Subset of peaks indexed by $\\vec{q_1} = \\frac{3}{13}\\,\\vec{a}^* + \\frac{1}{13}\\,\\vec{b}^*$, with $1^{\\mathrm{st}}$, $2^{\\mathrm{nd}}$ and $3^{\\mathrm{rd}}$ order reflections.\n\\textbf{(Upper right panel)} Both subsets of peaks indexed by $\\vec{q_1}$ in shades of red, and its mirror image $\\vec{q_2} = \\frac{4}{13}\\,\\vec{a}^* - \\frac{1}{13}\\,\\vec{b}^*$ in shades of blue.\n\\textbf{(Lower panel)} $\\vec{q_1}$, and $\\vec{q_2}$ are commensurate with the crystal lattice via $13\\,\\vec{q_1} = 3\\,\\vec{a}^* + \\vec{b}^*$. The wavevectors length is $||\\vec{q_{1,2}}|| =\\frac{1}{\\sqrt{13}}||\\vec{a}^*||$ so that each defines a $\\sqrt{13}\\,a\\times\\sqrt{13}\\,a$ superlattice in real space. This is also geometrically equivalent to $3\\vec{q_1}-\\vec{q_1}'=\\vec{a}^*$, where $\\vec{q_1}'$ is $\\vec{q_1}$ rotated by $+120^\\circ$, which clearly appears in the upper panels.\n{Note that the upper panels show the same region as in Fig.~\\ref{fig:RT_77K}, and that the lower panel is an extended view centered on this same region.}\n}\n\\end{figure}\n\n\\subsection{Other satellite peaks}\n\nThe ring of twelve satellite peaks around each Bragg peak, and the four peaks around the M point can all be indexed with only two wavevectors. The left panel of Fig.~\\ref{fig:qvector} shows the wavevector: $\\vec{q_1} = \\frac{3}{13}\\,\\vec{a}^* + \\frac{1}{13}\\,\\vec{b}^* \\approx 0.231\\,\\vec{a}^* + 0.077\\,\\vec{b}^* $ and its $1^{\\mathrm{st}}$, $2^{\\mathrm{nd}}$ and $3^{\\mathrm{rd}}$ order reflections. This wavevector corresponds to a deviation angle of $\\arctan\\left(\\frac{\\sqrt{3}}{7}\\right)\\approx13.9^{\\circ}$ from $\\vec{a}^*$. The right panel of Fig.~\\ref{fig:qvector} shows both $\\vec{q_1}$ in shades of red, and $\\vec{q_2} = \\frac{4}{13}\\,\\vec{a}^* - \\frac{1}{13}\\,\\vec{b}^*  \\approx 0.308\\,\\vec{a}^* - 0.077\\,\\vec{b}^* $ in shades of blue.\n\nThese two wavevectors are mirror image of each other, with length $||\\vec{q_{1,2}}|| =\\frac{1}{\\sqrt{13}}||\\vec{a}^*||$. They thus correspond to two commensurate $\\sqrt{13}\\,a\\times\\sqrt{13}\\,a$ superlattices in real space.\nNote that the commensurate relation $13\\,\\vec{q_1} = 3\\,\\vec{a}^* + \\vec{b}^*$ is geometrically equivalent to $3\\vec{q_1}-\\vec{q_1}'=\\vec{a}^*$, where $\\vec{q_1}'$ is $\\vec{q_1}$ rotated by $+120^\\circ$. This clearly appears in the upper panels of Fig.~\\ref{fig:qvector} where the $3^{\\mathrm{rd}}$ order reflections from one Bragg peak coincide with the $1^{\\mathrm{st}}$ order reflections from another. \n\nThe presence of high order reflections evidences the long range coherence associated to these peaks or the non-sinusoidal character of the atomic displacements. \nThe long range coherence is also evidenced by the small width of the peaks as shown in the upper panel of Fig.~\\ref{fig:satpeakwidth}. The FWHM along $a^*$ of the satellite peak at (1.231, 0.077, 0) is 0.012 $a^*$ corresponding to a coherence length of $\\approx 83$ unit cells. {These sharp peaks along a* correspond to rods of scattering along c* with FWHM of $\\approx 0.5$\\,c* i.e. 2 unit cells along the c-axis.}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{Satpeakwidth3.pdf}%\n\\caption{\\label{fig:satpeakwidth}\\textbf{Width of the satellite peaks} \\textbf{(Upper panel)} cross-sections of the satellite peak at $(1,0,0)+\\vec{q_1}$, which is part of the ring of 12 peaks around each Bragg peak. \\textbf{(Lower panel)} cross-sections of a satellite peak at the M point. Note that the difference between the cross-section along and perpendicularly to $a^*$ are due to the background of diffuse scattering (caused by soft phonons) which disappears rapidly perpendicularly to $a^*$.}\n\\end{figure}\n\nThe ring of twelve satellite peaks also has an intensity that follows the same extinction pattern as the elongated diffuse scattering, suggesting that it corresponds to a static longitudinal modulation. Indeed, the very specific angles at which the diffuse scattering is extinguished show that the underlying soft phonons (which cause the diffuse scattering) are polarized longitudinally with in-plane niobium displacements\\cite{Leroux2012anharmonicNbS2}.\nIn more details, the scattered intensity depends on phonons polarization via the dynamical structure factor $G(Q,m)$~\\cite{Burkel_IXS}\n\\begin{equation}\nG(Q,m)=\\left|\\sum_{j}^{\\text{unit cell}} f_j(\\vec{Q}).\\mathrm{e}^{-W_j}\\left[\\vec{Q}.\\vec{\\epsilon_j}(\\vec{Q},m) \\right] \\sqrt{M_j} e^{i \\vec{Q}.\\vec{r_j}}\\right|^2\n\\label{eqn:polarisation_phonon}\n\\end{equation}\nwhere $f_j(\\vec{Q})$ is the atomic form factor of atom $j$ at $\\vec{r_j}$ with mass $M_j$; $\\vec{\\epsilon_j}(\\vec{Q},m)$ is the unit displacement vector of atom $j$ in the $m$ phonon branch for a phonon wavevector $\\vec{Q}$; and $\\mathrm{e}^{-W_j}$ is the Debye-Waller factor of atom $j$.\nBecause $\\vec{Q}.\\vec{\\epsilon_j}(\\vec{Q},m)$ is zero for a phonon polarization perpendicular to $\\vec{Q}$ (see Fig.11 in Ref.~\\citenum{Burkel_IXS}), these extinctions indicate that the soft phonons are longitudinally polarized.\n{We cannot distinguish the respective contribution of sulfur and niobium atoms to the longitudinal soft phonon modes in our data. However, the total scattered intensity is dominated by the contribution from niobium atoms as the mass and atomic form factor of niobium are larger than that of sulfur. This suggests that the displacements of the niobium atoms involved in the soft phonons would also be mostly longitudinal, i.e. in the ab plane.}\n\nThe commensurate wavevectors $\\vec{q_1}$ and $\\vec{q_2}$ are the same as those of the low temperature commensurate CDW in 1T-TaS$_2$\\cite{ScrubyPhilMag1975} (semiconducting 1T$_3$ phase) and 1T-TaSe$_2$\\cite{McMillanPRB1975,Wilson1974PRL}.\nIt is worth noting that this CDW is dominated by in-plane longitudinal displacements of Ta atoms in 1T-TaS$_2$\\cite{ScrubyPhilMag1975}. Also, only one set of 6 peaks around each Bragg peaks is observed in the Ta based TMD. {But here we observe that both sets of 6 peaks are equivalently present, evidencing two sets of triple-q CDW, most likely from twinning in the crystal.}\n\nWe therefore conclude that the ring of peaks we observe in 2H-NbS$_2$ is the trace of a faint longitudinal periodic lattice distortion, appearing between 77 and 300\\,K, and corresponding to two commensurate $\\sqrt{13}\\,a\\times\\sqrt{13}\\,a$ CDW identical to that found in 1T Ta based TMD. \nIn addition, as the commensurate CDW becomes incommensurate above 473\\,K in 1T-TaSe$_2$ and 190\\,K in 1T-TaS$_2$, this suggests the possibility of an incommensurate CDW in our crystal as well. As we observed no incommensurate peaks at 300\\,K, this incommensurate CDW would have to occur in a temperature range between 77 and 300\\,K.\n\nInterestingly, in 1T-TaSe$_2$, the thrice degenerate wavevector of the high temperature incommensurate CDW becomes commensurate with the lattice by a rotation of 13.9$^{\\circ}$ because it is not close enough to $1\/3\\,a^*$. {Indeed, according to Landau theory of CDW in TMD\\cite{McMillanPRB1975}, this commensuration by rotation is a feature of the 1T polytype, whereas in the 2H polytype the CDW locks in with 1\/3 of $a^*$\\cite{McMillanPRB1975} (2\/3 of $\\Gamma$M). While we cannot preclude that the CDW we observe is a bulk phenomenon native to the 2H polytype, this would be the first $\\sqrt{13}\\,a\\times\\sqrt{13}\\,a$ in a 2H TMD to our knowledge. In addition, the very short coherence length of the CDW peaks along c* supports the picture of a CDW occurring almost independently in each layer of the crystal. Therefore, we also consider the possibility that this CDW originates from a local 1T-like environment in a 2H crystal and we now discuss the possible origins of such environment.}\n\n\\section{Discussion}\nIn TMD, the dominance of trigonal prismatic (1H) or octahedral (1T) coordination can be classified by transition metal atoms. There are three typical cases. In the first case, as with titanium (Ti), the coordination is generally octahedral so that the dominant polytype is 1T. In the second case, such as niobium (Nb), the coordination is usually trigonal prismatic so that 2H or 3R polytypes are favored. In the third case, as with tantalum (Ta), both coordinations have similar energies, in which case various polytypes can be synthesized: 1T, 2H and mixed stackings of 1T and 1H layers such as 4Hb.\\cite{FriendYoffe1987}\n\nIn NbS$_2$, the 3R polytype is the thermodynamically stable phase at room temperature\\cite{Fisher1980NbS2difficultsynthesis}. The 2H polytype can also be synthesized at room temperature by quenching from $\\approx1000$\\,K. As for the 1T polytype of NbS$_2$, it has never been synthesized in bulk crystal form, but it can be stabilized by strain in thin film\\cite{Carmalt2004_1st_1TNbS2synthesis} or monolayer\\cite{Chen20151TNbS2monolayerforH2} forms.\n\nLooking at the 2H structure in Fig.~\\ref{fig:2H3R}, we emphasize that it has two possible rotational positions for each 1H layer, separated by a 60$^\\circ$ rotation around the c-axis. This rotational position alternates between each 1H layers, so that, once an origin is given, the rotational positions of all 1H layers are fixed in an ideal crystal. In real crystals of the 2H structure, especially if synthesized by quenching, this opens up the possibility of rotational domains, where each domain has a different origin of the rotational positions. \n\nMost interestingly, at the junction of two rotational domains, there should be two 1H layers in the same rotational position stacked one onto the other (i.e. a locally 1H polytype), where the sulfur atoms are facing each other. This locally 1H polytype seems a priori unstable because of the geometrical repulsion between sulfur atoms in adjacent layers. In fact, such stacking of sulfur atoms does not occur in either of the three known polytypes 1T, 2H, or 3R and there are no known purely 1H polytype of NbS$_2$. Energetically, it seems much more likely that one of the sulfur atoms layer will move such that the sulfur atoms of one layer face the center of a triangle of sulfur atoms in the other layer. This reduces the geometrical repulsion, which brings the layers closer together and increases the orbital overlaps and van der Waals interactions. There are, however, several ways to displace the sulfur atoms layer.\n\nOne way, which does not involve changing the coordination of the Nb atoms, is for one of the 1H layer to slide by $(\\frac{1}{3}, \\frac{2}{3}, 0)$ or $(\\frac{2}{3}, \\frac{1}{3}, 0)$, yielding a locally 3R structure (which is non-centrosymmetric, hence the two possible sliding vectors). Such 3R-like stacking faults have actually been studied before in 2H-NbS$_2$. The study\\cite{Katzke2002} concluded to the presence of 15\\% of 3R-like stacking faults in powder samples of 2H-NbS$_2$ {(i.e. any two adjacent layers have a 15\\% chance of having a faulty stacking)}.\nWe performed a similar analysis in our sample and found the presence of 18\\% of 3R-like stacking faults.\\cite{lerouxHALThesis}\n\nA second way the sulfur atoms layer can move to reduce geometrical repulsion at the junction between domains, is a rotation by 60$^\\circ$ around the c-axis. This changes the coordination of the Nb atom from trigonal prismatic to octahedral, and yields a single purely 1T layer. To some extent, this is similar to thin films and monolayer of 1T-NbS$_2$, where 1T layers are stabilized by strain at interfaces. This single 1T layer is only three-fold symmetric and can occur in two types which are mirror image of each other (or, equivalently, rotated by 60$^\\circ$). The junctions between domains would yield both types equiprobably. This would naturally explain the presence of both wavevectors $\\vec{q_1}$ and $\\vec{q_2}$ yielding two $\\sqrt{13}\\,a\\times\\sqrt{13}\\,a$ superlattices, instead of only one in pure 1T-TaS$_2$ and 1T-TaSe$_2$.\n\nTo our knowledge, this type of 1T-like stacking faults has not been studied before. In fact, considering that it involves a change of coordination of the Nb atom, a 1T-like stacking fault seems more energetic than the 3R-like stacking fault considered above. We can therefore expect that the 1T-like stacking fault occurs less frequently than the 3R-like ones. Yet, if the 1T-like CDW we observed in x-ray occurs only on such rare 1T-like stacking fault, it would explain why the CDW x-ray peaks are so faint.\n\nFinally, another explanation for the presence of local 1T-like environment could be based on the presence of small clusters of extra Nb atoms intercalated in the van der Waals gap between layers. Indeed, Meerschaut and Deudon\\cite{Meershault01} have reported that the 3R-NbS$_2$ phase is favored by an overdoping of Nb. This extra Nb is placed in the van der Waals gap between two layer of Nb in a trigonal prismatic coordination. Locally the Nb atom is surrounded by 6 chalcogen atoms in a octahedral coordination. Because of the Nb-Nb repulsion, this extra Nb atom is slightly shifted from the center of the octahedron \\cite{Meershault01}. Thus, in our NbS$_2$ crystal, a local 1T-like environment could be associated to a small amount of extra Nb with a local octahedral coordination lying in the 3R-like stacking fault.\n\n\\section{Conclusions}\nUsing {diffuse x-ray scattering} in 2H-NbS$_2$, we observed very weak superlattice peaks corresponding to two longitudinal commensurate $\\sqrt{13}\\,a\\times\\sqrt{13}\\,a$ periodic lattice distortion, identical to that associated with the CDW of 1T-TaSe$_2$ and 1T-TaS$_2$. Around each Bragg peaks in the $(h,k,0)$ plane, we found a series of 12 satellite peaks at $\\pm$ 13.9$^{\\circ}$ from $\\vec{a}^*$ and $\\vec{b}^*$, commensurate with the lattice through $3\\vec{q_1}-\\vec{q_1}'=\\vec{a}^*$ or, equivalently, $13\\,\\vec{q_1} = 3\\,\\vec{a}^* + \\vec{b}^*$. Inelastic x-ray scattering (IXS) measurements confirmed the predominantly elastic nature of these satellite peaks, but the amplitude of these peaks is almost as faint as that of soft phonons.\nTo our knowledge, no CDW has been reported in any polytypes of NbS$_2$.\nWe suggest that rotational disorder in the stacking of 1H layers, induces 3R-like stacking fault and, less frequently, single 1T layers at the interface between 2H rotational domains. Such rare and dilute 1T layers might be the support of this faint 1T-like CDW. A very dilute amount of Nb in the van der Waals interlayer space of 3R-like stacking fault could also present a 1T-like octahedral coordination.\n\\section{Acknowledgments}\nM.L. acknowledges S. Eley for fruitful discussions.\nThis work was supported by the Neel Institute CNRS - University of Grenoble. We acknowledge the European Synchrotron Radiation Facility for provision of synchrotron radiation facilities. The experiments were performed on beamline ID28 and ID29.\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introducci\u00f3n}\nEn rob\u00f3tica, se conoce como exploraci\u00f3n al proceso mediante el cual se busca aumentar la informaci\u00f3n respecto del entorno del robot para construir un modelo del ambiente que lo rodea.\nSe aplican los algoritmos de exploraci\u00f3n cuando se necesita conocer el estado de una locaci\u00f3n a la que no pueden acceder personas o hay un riesgo latente o manifiesto en su ingreso. \nPor ejemplo en estructuras colapsadas o con posibilidad de colapso en busca de sobrevivientes.\nEn estas circunstancias los robots m\u00e1s utilizados son de dimensiones reducidas por su capacidad de atravesar aberturas peque\u00f1as.\nDebido a estos limitantes geom\u00e9tricos, la capacidad de c\u00f3mputo de estos robots tambi\u00e9n se ve limitada.\n\nLo modelos generados por estos robots son \u00fatiles para poder navegar en estos ambientes, entendi\u00e9ndose por navegar a la tarea que permite llevarlo de un punto a otro de manera segura evadiendo obst\u00e1culos.\nLos modelos pueden ser m\u00e9tricos, donde se trata de representar o reproducir la geometr\u00eda del entorno; o topol\u00f3gicos, donde se describe la relaci\u00f3n espacial entre distintos ambientes.\n\n\nRespecto de los m\u00e9tricos, durante a\u00f1os una de las formas m\u00e1s populares de representar el entorno fue mediante grilla de ocupaci\u00f3n \\cite{moravec_1985}.\nEste m\u00e9todo consiste en representar el ambiente bidimensional con un plano dividido en celdas formando una grilla.\nLas celdas de la grilla son de tama\u00f1o regular y cada una contiene un valor que representa la probabilidad de estar ocupada basada en modelos probabil\u00edsticos de los sensores involucrados.\nDerivaciones de este trabajo usando tres dimensiones con cubos llamados \\textit{voxels} se presentaron en trabajos como \\cite{moravec1996} y \\cite{dryanovski2010} mostraron ser ineficientes respecto del uso de memoria para almacenar el mapa debido al gran tama\u00f1o de los mismos.\n\n\\begin{figure}[tp!]\n\\centering\n  \\includegraphics[width=0.45\\textwidth]{images\/jetson_nano.png}\n  \\caption{Montaje utilizado para las pruebas en un escenario real}\n  \\label{fig:pruebas_real}\n\\end{figure}\n\n\\IEEEpubidadjcol\n\nDe forma m\u00e1s eficiente, el \\textit{framework} Octomap \\cite{hornung2013} usa una estructura en forma de \u00e1rbol con ocho nodos, donde cada nodo comienza con un voxel que es dividido sucesivamente en otros ocho hasta alcanzar la resoluci\u00f3n deseada.\nEsta estructura es llamada \\textit{octrees} y fue propuesta por primera vez en \\cite{doctor1981}.\nEl valor contenido en el voxel puede ser binario, un valor de probabilidad que puede estar basado en distintos criterios o una variante con una cota aplicada a una densidad de probabilidad.\nSi bien el Octomap hace mejor uso de la memoria, el acceso a cada elemento tiene mayor costo computacional que la grilla de ocupaci\u00f3n.\n\nPara mejorar el acceso a cada elemento, en \\cite{museth2013vdb} se propone una variante de \u00e1rboles B+, usados generalmente en sistemas de archivos, llamada VDB por las siglas de \\textit{Volume Dynamic B+tree}.\nCon esta topolog\u00eda se puede modelar un espacio de \u00edndices tridimensional, \\textit{virtualmente infinito} que permite el acceso r\u00e1pido a informaci\u00f3n dispersa.\nAdicionalmente, la implementaci\u00f3n de VDB no impone restricciones de topolog\u00eda sobre los datos de entrada y admite patrones de acceso, inserci\u00f3n y eliminado aleatorio r\u00e1pidos, en promedio $\\mathcal{O}(1)$.\nLa implementaci\u00f3n de esta estructura de datos es conocida como OpenVDB (OVDB) y se presenta en \\cite{museth2019}.\n\nExplotando las caracter\u00edsticas de OVDB, en \\cite{stvl2020steve} se presenta una biblioteca llamada STVL por las siglas de \\textit{Spatio-Temporal Voxel Layer} donde se implementan una serie de \\textit{buffers} que almacenan nubes de puntos provenientes de c\u00e1maras de profundidad u otras fuentes capaces de generar este tipo de datos.\nEstas nubes de puntos se codifican usando OpenVDB logrando formar mapas tridimensionales de manera eficiente.\nDebido a la resoluci\u00f3n y cantidad de sensores, las nube suelen estar compuestas de millones de puntos por lo que es necesario realizar una \\textit{compresi\u00f3n} diezm\u00e1ndola con alg\u00fan criterio.\nEn \\cite{stvl2020steve} esto es realizado mediante un \\textit{filtro de voxelizado}, disponible en la biblioteca \\textit{Point Could Library} \\cite{Rusu_ICRA2011_PCL} el cual corre en CPU.\n\nEn este trabajo se presenta una implementaci\u00f3n de STVL capaz de ser ejecutada en la GPU de la plataforma de desarrollo Jetson Nano de Nvidia, utilizando el arreglo de la Fig.~\\ref{fig:pruebas_real}.\nEsta plataforma fue elegida debido a su reducido tama\u00f1o y gran eficiencia energ\u00e9tica, lo que posibilita su uso en peque\u00f1os robots ya sean voladores o terrestres.\nAdicionalmente, esta plataforma tiene la CPU y la GPU en el mismo chip, por los que comparten f\u00edsicamente la memoria.\nEsto evita que se hagan copias redundantes de bloques de memoria, por ejemplo una imagen, entre el CPU y la GPU.\nEste enfoque es conocido por NVIDIA como \\textit{Zero-Copy}.\nEspec\u00edficamente se presenta una variante de \\cite{bkedkowski2013general} implementada en CUDA para realizar la compresi\u00f3n de los puntos.\nEsta compresi\u00f3n o filtrado se realiza aprovechando el acceso \\textit{Zero-Copy} disponible en la plataforma Jetson Nano.\nNo hay, seg\u00fan el conocimiento de los autores, otra implementaci\u00f3n que pueda ser ejecutada en esta plataforma.\n\nEl trabajo se organiza de la siguiente manera: En la Secci\u00f3n~II se describen las herramientas de \\textit{software} y las plataformas de \\textit{hardware} utilizadas en el trabajo, as\u00ed como la propuesta de modificaci\u00f3n al algoritmo para poder ser utilizado en la plataforma Jetson Nano. Tambi\u00e9n se describen los escenario donde fue probada la implementaci\u00f3n. En la Secci\u00f3n~III se muestran los resultados obtenidos en las distintas plataformas y escenarios, con tres tama\u00f1os de voxel. Las conclusiones del trabajo y futuras l\u00edneas de  investigaci\u00f3n son enumeradas en la Secci\u00f3n~IV.\n\n\\section{Materiales y m\u00e9todos}\n\nPara evaluar el algoritmo propuesto se hicieron pruebas en un ambiente simulado y en un escenario real.\nAmbos escenarios de similares caracter\u00edsticas, est\u00e1ticos y estructurados.\nAdem\u00e1s se compar\u00f3 el tiempo de c\u00f3mputo de la biblioteca original y la propuesta usando distintas plataformas.\n\n\\subsection{Herramientas de software utilizadas}\nLa implementaci\u00f3n se realiz\u00f3 sobre el \\textit{framework} ROS (siglas en ingl\u00e9s de Sistema Operativo para Robots) \\cite{quigley2009ROS}, usando la versi\u00f3n de nombre clave Melodic Morenia.\nEsto permite una r\u00e1pida integraci\u00f3n del m\u00e9todo a evaluar con algoritmos existentes y ampliamente usados en rob\u00f3tica.\nLa mejora propuesta es una variante de m\u00e9todo de filtrado de puntos de PCL~\\cite{pcl} utilizado por la biblioteca STVL disponible en \\cite{stvl}.\nPara la etapa de simulaci\u00f3n, la misma se realiz\u00f3 con el entorno Gazebo \\cite{koenig2004design} versi\u00f3n 9 la cual puede descargarse de \\cite{gazebo} e integrarse a ROS.\n\n\\subsection{Plataformas de hardware utilizadas}\nLa plataforma elegida para realizar las pruebas fue la Jetson Nano de NVIDIA.\nLa misma es una placa de peque\u00f1as dimensiones, tan solo 70mm de largo y 45mm de ancho, lo que posibilita su utilizaci\u00f3n en robots de tama\u00f1o reducido.\nCuenta con una GPU de arquitectura Maxwell de 128 CUDA cores con los cuales puede correr hasta 2048 hilos y como procesador principal tiene un ARM Quad-core Cortex-A57.\nEsta placa cuenta con una memoria de 4GB 64-bit LPDDR4 con un ancho de banda m\u00e1ximo te\u00f3rico 25.6GB\/s.\nCon todo este \\textit{hardware} puede alcanzar 472GFLOPS de rendimiento de c\u00f3mputo en FP16, con 5-10W de consumo de energ\u00eda.\nComo se mencion\u00f3 antes la CPU y la GPU se encuentran en el mismo encapsulado y comparten f\u00edsicamente la misma memoria del sistema, admitiendo una forma limitada de \\textit{memoria unificada}.\nDicha arquitectura permite comunicaciones CPU-GPU, que no son posibles en GPU discretas.\nDe esta forma, se evita la copia de cada punto y solamente es necesario pasar la direcci\u00f3n de memoria del mismo a las funciones de CUDA lo que se traduce en una reducci\u00f3n de tiempo y espacio utilizado.\nEs importante destacar que esta implementaci\u00f3n es en si misma una mejora sobre el algoritmo original presentado por \\cite{bkedkowski2013general}, ya que originalmente estaba pensado para arquitecturas de GPU NVIDIA Fermi y Kepler las cuales son generaciones anteriores y no disponen de memoria unificada.\n\nEl algoritmo tambi\u00e9n fue evaluado en una PC de escritorio con procesador Ryzen 7 1700 y una GPU NVIDIA GTX 1660 Super.\nLa misma cuenta con 6GB de memoria GDDR6 de video, con un ancho de banda te\u00f3rico m\u00e1ximo indicado por el fabricante de hasta 336GB\/s.\nEsta placa pertenece a la familia de micro arquitectura Turing la cual dispone de la tecnolog\u00eda de memoria unificada.\nComo contrapartida, la memoria est\u00e1 separada del procesador de la CPU por lo cual la informaci\u00f3n tiene que ser copiada a trav\u00e9s del bus PCI Express.\n\nPara generar la nube de puntos en el escenario real se utiliz\u00f3 la c\u00e1mara RGB-D Intel RealSense Depth Camera D435.\nEsta c\u00e1mara est\u00e1 formada por un par est\u00e9reo capaz de determinar la distancia al sensor de los puntos dentro de su campo de visi\u00f3n.\nTambi\u00e9n dispone de un proyector infrarrojo, para mejorar la imagen 3D en paredes que no tengan caracter\u00edsticas sobresalientes.\n\n\\subsection{Algoritmo propuesto}\nEn el algoritmo presentado en \\cite{stvl2020steve}, las nubes de puntos provenientes de las c\u00e1maras RGB-D, son ingresadas en \\textit{buffers} de observaciones, los cuales almacenan la posici\u00f3n relativa de la medici\u00f3n con respecto a un sistema de coordenadas globales.\nLuego, estas mediciones pueden ser utilizadas con dos finalidades: operaciones de marcado en la cual se designa la celda como ocupada, u operaciones de liberaci\u00f3n donde se marca la celda como vac\u00eda.\n\nDebido a la implementaci\u00f3n paralela del programa, existen \\textit{buffers} extra destinado a cada una de las operaciones.\nLas nubes de puntos recibidas, al estar en el sistema de referencia de la c\u00e1mara, tienen que ser transformadas al sistema de coordenadas de globales.\nEsta transformaci\u00f3n de un sistema de referencias a otro es realizado mediante la biblioteca \\texttt{tf2} \\cite{foote2013}, la cual permite realizar las transformaciones entre los m\u00faltiples sistemas de referencia presentes en el robot, a lo largo del tiempo.\n\nMediante el an\u00e1lisis de tiempo empleado en cada una de las operaciones del programa mencionadas anteriormente, se determinaron las partes del algoritmo que presentaban mayores demoras.\nComo se puede ver en la Fig.\\ref{fig:tiempo_algoritmos}, las operaciones m\u00e1s costosas desde el punto de vista computacional son la de filtrado de la nube de puntos, llamada \\texttt{filter} y la operaci\u00f3n de marcado\/liberaci\u00f3n, llamada \\texttt{ClearFrustums}.\nEsta \u00faltima comprende el acceso a las celdas del mapa global, utilizando los m\u00e9todos provistos por la biblioteca OpenVDB.\n\n\n\nEstas nubes de puntos pueden llegar a ser muy densas, del orden de millones de puntos, y al ser utilizadas en el marcado de celdas para el mapa global, es necesario realizar una compresi\u00f3n de las mismas.\nEn \\cite{stvl2020steve} esto es realizado mediante un filtro de \\textit{voxelizado}, disponible en la biblioteca de PCL.\nEste filtro recibe como entrada nube de puntos, la cual tiene que estar en el sistema de referencia global, para poder limitar la altura de la nube de puntos.\nEste l\u00edmite en el eje $\\mathbf{z}$, permite al \\texttt{navigation\\_stack} de ROS \\cite{tbd} proyectar la nube de puntos sobre el plano horizontal y generar un mapa de ocupaci\u00f3n, utilizado en la navegaci\u00f3n del robot.\nEl filtro disponible en la biblioteca PCL es calculado en la CPU, y seg\u00fan el conocimiento de los autores, no hay otra implementaci\u00f3n que pueda ser ejecutada en esta plataforma.\n\nPor otro lado, los algoritmos implementados en CUDA para el filtrado de nubes de puntos normalmente se implementan en GPUs de escritorio con gran capacidad de memoria.\nEsto los convierten en poco atractivos para utilizar en plataformas embebidas debido a su limitada capacidad de memoria, generalmente compartida con la CPU.\n\n\\begin{figure}[bt]\n\\centerline{\\includegraphics[width=0.5\\textwidth]{images\/tiempos_funciones_pcl.png}}\n\\caption{Tiempos de procesamiento de las partes de mayor carga computacional del algoritmo.}\n\\label{fig:tiempo_algoritmos}\n\\end{figure}\n\n\nSe implement\u00f3 una versi\u00f3n modificada de \\textit{Cubic Subspaces - Neighboring Buckets} \\cite{bkedkowski2013general}, implementada en CUDA y utilizando el proceso \\textit{Zero-Copy} disponible en plataformas como la Jetson Nano.\nLa idea principal es usar la GPU para descomponer el espacio 3D en una cuadr\u00edcula regular de $2^n \\times 2^n\\times 2^n$ cubos $(n = 4,5,6,7,8,9)$. Por lo tanto, para cada punto, se consideran solo 27 cubos $(3^3)$ para encontrar los vecinos m\u00e1s cercanos.\nPara calcular la distancia entre dos puntos $p1={x_1,y_1,z_1}$ y $p2={x_2,y_2,z_2}$ se utiliza la distancia eucl\u00eddea definida como: %\n{\n\\setlength{\\abovedisplayskip}{1pt}\n\\setlength{\\belowdisplayskip}{15pt}\n\\begin{equation}\nd(p1,p2) = \\Big[ (x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2\\Big]^{\\dfrac{1}{2}}\n\\end{equation}%\n}\ncon $x$, $y$, $z$ y $d(p1,p2)$ pertenecientes al espacio $\\mathbb{R}^3$.\nCada punto del espacio tridimensional $XYZ$ es normalizado, de tal forma que $\\left\\{x,y,z\\in \\mathbb{R}: -1 \\leq x,y,z  \\leq 1\\right\\}$.\nLuego, son clasificados mediante un \u00e1rbol de decisi\u00f3n para determinar a que subdivisi\u00f3n del espacio $2^n \\times 2^n\\times 2^n$ pertenecen.\n\nLa cantidad de puntos que pertenecen a cada subdivisi\u00f3n, es determinada mediante una tabla (\\texttt{tabla\\_subdiv}) ordenada de pares ``clave-valor\"\\,utilizando el conocido algoritmo ``Radix Sort\".\nEs importante notar que esta tabla es almacenada en la memoria global de la GPU, por lo tanto, todos los hilos de CUDA pueden acceder a los datos.\nEl par ``clave-valor\", junto con la informaci\u00f3n de la cantidad de puntos en cada subdivisi\u00f3n, son accesibles mediante la memoria de la GPU, y se utilizar\u00e1 para buscar el vecino m\u00e1s cercano en el algoritmo.\n\n\\begin{figure*}[ht]\n  \\includegraphics[width=\\textwidth,height=4cm]{images\/superposition.png}\n  \\caption{Secuencia de im\u00e1genes del ambiente utilizado para las pruebas. Izquierda: Nube de puntos generada por la c\u00e1mara RGB-D, Centro: Grilla de ocupaci\u00f3n 3D generada por el algoritmo STVL, Derecha: Nube de puntos y mapa superpuestos. }\n  \\label{fig:grilla_3d_generada}\n\\end{figure*}\n\n\\begin{algorithm}[ht]\n\\caption{Filtrado de puntos 3D}\n\\label{alg:filtrado}\n\\begin{algorithmic}[1]\n\\renewcommand{\\algorithmicrequire}{\\textbf{Entrada:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Salida:}}\n\\renewcommand{\\algorithmicfor}{\\textbf{para}}\n\\renewcommand{\\algorithmicdo}{\\textbf{hacer}}\n\\renewcommand{\\algorithmicend}{\\textbf{fin}}\n\\renewcommand{\\algorithmicwhile}{\\textbf{mientras}}\n\\renewcommand{\\algorithmicif}{\\textbf{si}}\n\\renewcommand{\\algorithmicthen}{\\textbf{entonces}}\n\n\\REQUIRE Puntero a la nube de puntos de la c\u00e1mara\n\\ENSURE  Puntero a la de puntos filtrada\n\n\\STATE copiar el puntero del host al device\n\\STATE llamado a la funci\u00f3n de CUDA\n\\FOR {todos los puntos $m^i_{xyz}$ en paralelo}\n\\STATE buscar $subdiv_{m^i}$\n\\STATE actualizar \\texttt{tabla\\_subdiv}\n\\ENDFOR\n\\STATE en paralelo ordenar \\texttt{tabla\\_subdiv} $\\{$radix sort$\\}$\n\n\n\\WHILE {la cantidad de puntos marcados $>$ 1000 $\\{$un kernel CUDA por cada punto $m_{xyz}$ $\\}$}\n\\FOR {todos los puntos $m^i_{xyz}$ en paralelo}\n\\STATE buscar $subdiv$\n\\FOR {todas las $subdiv$ vecinas}\n\\STATE contar la cantidad de vecinos de $\\{$teniendo en cuenta los puntos marcados para borrar$\\}$\n\\ENDFOR $\\{$un kernel de CUDA por un punto$\\}$\n\\STATE marcar $m^i_{xyz}$ para borrar si \\texttt{cont} $>$ \\texttt{umbral}\n\\ENDFOR $\\{$un kernel de CUDA para todos los puntos$\\}$\n\\IF {cantidad de puntos marcada $>$ 1000 }\n\\STATE aleatoriamente elegir 1000 puntos marcados para eliminarlos definitivamente\n\\ENDIF\n\\ENDWHILE\n\n\n\\STATE sincronizar el llamado a los kernels\n\\STATE eliminar los puntos marcados\n\\STATE copiar el puntero del device al host\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\nEl objetivo del filtrado es eliminar puntos, para reducir la densidad de la nube de puntos y al mismo tiempo eliminar el ruido de la medici\u00f3n proveniente de la c\u00e1mara de profundidad.\nUna vez calculados los vecinos m\u00e1s cercanos por el m\u00e9todo anteriormente descripto, las subdivisiones del espacio son iterativamente reducidas hasta obtener la densidad de puntos deseada.\nEste proceso puede ser descripto mediante el Algoritmo \\ref{alg:filtrado}. \n\nLa nube de puntos obtenida del filtro, es una versi\u00f3n comprimida de la nube de entrada, y la densidad de la misma depende del tama\u00f1o de bloque elegido para el filtro.\nEsta dimensi\u00f3n puede ser configurada al inicio del algoritmo, y tambi\u00e9n es utilizada por la estructura de OpenVDB para almacenar los puntos en el mapa global.\n \nDebido a las restricciones impuestas por la evasi\u00f3n de obst\u00e1culos, es una prioridad garantizar c\u00e1lculos completos de la nube de puntos en el menor tiempo posible.\n\nPara asegurar la utilizaci\u00f3n completa de la GPU, el programa determina la cantidad m\u00e1xima de procesadores CUDA disponibles, para distribuir cada \\texttt{subdiv} del espacio.\nEsto se puede ver en el punto n\u00famero 11 del Algoritmo \\ref{alg:filtrado}.\n\n\nSe realizaron experimentos en ambientes simulados y reales para validar el m\u00e9todo de filtrado.\nA continuaci\u00f3n se presenta una breve descripci\u00f3n del sistema utilizado.\n\n\n\\begin{figure}[b]\n\\centering\n  \\includegraphics[width=0.45\\textwidth]{images\/gazebo_stvl.png}\n  \\caption{Robot en el entorno de simulaci\u00f3n. Izquierda: Mapa 3D obtenido por el algoritmo. Derecha: Pasillo de oficina con obst\u00e1culos en sus laterales}\n  \\label{fig:prueba_simulada}\n\\end{figure}\n\n\\subsection{Escenario simulado}\nPara evaluar el algoritmo propuesto, en una primera instancia se utiliz\u00f3 el simulador Gazebo \\cite{koenig2004design}, en su versi\u00f3n 9.\nSe opt\u00f3 por por este simulador ya que es compatible con el protocolo de mensajes de ROS Melodic utilizada en este trabajo, y cuenta con soporte para los sensores requeridos por el algoritmo.\n\nEl robot es del estilo tracci\u00f3n diferencial, con una c\u00e1mara RGB-D montada en su parte superior.\nEl mismo cuenta con odometr\u00eda, con lo cual el mapa generado puede ser extendido m\u00e1s all\u00e1 del alcance de la c\u00e1mara de profundidad.\nEn la Fig.\\ref{fig:prueba_simulada}, se puede observar el mapa generado por el algoritmo.\nEl mismo intenta recrear un ambiente de oficinas, en el cual se encuentran dispuestos mobiliario, en ambos lados.\nEs importante destacar que al ser un sensor simulado, no presenta los artefactos normalmente encontrados en c\u00e1maras RGB-D.\n\n\\subsection{Escenario real}\nPara un an\u00e1lisis cualitativo se realizaron pruebas en un escenario real en donde se tomaron im\u00e1genes de un pasillo con distintos obst\u00e1culos y puertas.\nSe utiliz\u00f3 la configuraci\u00f3n mostrada en la Fig.~\\ref{fig:pruebas_real}.\nLa misma est\u00e1 compuesta por la c\u00e1mara RGB-D con el eje \u00f3ptico alineado con el sentido de circulaci\u00f3n del pasillo.\nLa c\u00e1mara fue montada sobre un perfil de aluminio.\nDebajo de la misma, se coloc\u00f3 la plataforma de desarrollo Jetson Nano.\n\n\\section{Resultados}\n\nComo se explic\u00f3 anteriormente, el mayor c\u00f3mputo del algoritmo est\u00e1 concentrado en dos partes principales, el filtrado y el acceso a la nube de puntos global mediante las funciones provistas por OVDB.\nLuego de la implementaci\u00f3n del algoritmo, en la Fig.~\\ref{fig:pcl_gpu} se muestra el an\u00e1lisis de cada funci\u00f3n comparando la versi\u00f3n original con la versi\u00f3n en GPU ejecut\u00e1ndose en la Jetson Nano.\n\n\\begin{figure}[b]\n\\centerline{\\includegraphics[width=0.5\\textwidth]{images\/tiempos_funciones_pcl_gpu.png}}\n\\caption{Comparaci\u00f3n de las funciones de mayor carga computacional, en el algoritmo con el filtrado en CUDA y en PCL. Se puede apreciar como las funciones las funciones \\texttt{clear\\_frustum} y \\texttt{mark} que est\u00e1n implementadas en CPU, ahora se ejecutan en un menor tiempo, ya que la CPU dispone de m\u00e1s recursos.}\n\\label{fig:pcl_gpu}\n\\end{figure}\n\nEn la Fig.~\\ref{fig:tiempo_filtrado}, se pueden observar los tiempos empleados por el filtrado de la nube de puntos.\nComo se puede ver, la versi\u00f3n utilizando CUDA es m\u00e1s r\u00e1pida que la versi\u00f3n original utilizando el filtrado con la biblioteca PCL.\nEsto era de esperar, ya que el filtrado puede ser descompuesto de forma tal, que se expone el paralelismo del algoritmo.\nEstas tareas masivamente paralelas, hacen que las placas GPU saquen grandes ventajas frente a su versi\u00f3n serializada.\n\n\\begin{figure}[ht!]\n\\centerline{\\includegraphics[width=0.5\\textwidth]{images\/tiempos_filtrado.png}}\n\\caption{Comparaci\u00f3n entre la versi\u00f3n original del algoritmo de filtrado mediante PCL y la versi\u00f3n propuesta en CUDA, para diferentes resoluciones del mapa global}\n\\label{fig:tiempo_filtrado}\n\\end{figure}\n\nPor otro lado, llevar el procesamiento de la nube de puntos a la GPU, implica que la CPU ahora dispone de m\u00e1s recursos, dejando lugar para ejecutar otros algoritmos en forma simultanea.\nEsto se puede ver en la Fig.~\\ref{fig:tiempo_operador_ovdb} como una reducci\u00f3n en el tiempo de procesamiento de la nube de puntos por parte del algoritmo de OVDB, el cual se ejecuta en la CPU.\nPara el mismo tama\u00f1o de grilla elegido, al algoritmo se procesa en menor tiempo en la versi\u00f3n con el filtro paralelizado en CUDA.\n\nEsto tambi\u00e9n se puede ver en la Fig.~\\ref{fig:pcl_gpu}, en la cual las funciones \\texttt{clear\\_frustum} y \\texttt{mark} que est\u00e1n implementadas en CPU, ahora se ejecutan en un menor tiempo, pese a que no se realizaron mejoras en esas funciones.\nEs importante destacar que la mejora se realiz\u00f3 sobre el filtrado de la nube de puntos; y el algoritmo de inserci\u00f3n y eliminaci\u00f3n de puntos, mediantes las funciones de OVDB en el mapa global no presenta modificaciones.\n\n\n\\begin{figure}[b]\n\\centerline{\\includegraphics[width=0.5\\textwidth]{images\/tiempos_operador_ovdb.png}}\n\\caption{Comparaci\u00f3n del tiempo para las operaciones en la CPU por parte de la biblioteca OpenVDB en el mapa global, para diferentes tama\u00f1os de grilla.}\n\\label{fig:tiempo_operador_ovdb}\n\\end{figure}\n\nEn la Fig.~\\ref{fig:grilla_3d_generada} podemos observar el mapa de ocupaci\u00f3n 3D generado por el algoritmo.\nTambi\u00e9n se puede apreciar en la parte inferior, la ausencia de voxels (puntos en el mapa 3D), debido a que es una condici\u00f3n requerida por el \\texttt{navigation\\_stack} para realizar la proyecci\u00f3n sobre el plano y generar el mapa de ocupaci\u00f3n.\nEste mapa generado, puede ser utilizado por algoritmos de planificaci\u00f3n, para desplazarse en el ambiente evitando obst\u00e1culos, y de forma paralela para generar un mapa con la informaci\u00f3n de odometr\u00eda.\nEste escenario ser\u00eda el caso de un robot navegando por un pasillo, en el cual se encuentran presentes obst\u00e1culos que obstruyen su camino.\nEn este caso la resoluci\u00f3n elegida fue de 5cm, la cual no solo permiti\u00f3 representar con efectividad los obst\u00e1culos (Cajas de cart\u00f3n en la imagen), si no que tambi\u00e9n permiti\u00f3 capturar detalles del ambiente.\nPor ejemplo, en la esquina superior derecha, podemos observar como el matafuegos es capturado por el mapa 3D.\nEn el caso de la puerta a la izquierda de la escena, la discontinuidad se puede distinguir por un cambio en el color de la grilla 3D.\nEs importante remarcar que, debido a la presencia de ruido en la c\u00e1mara de profundidad, en la tercera imagen de la Fig.~\\ref{fig:grilla_3d_generada}, algunos voxels del mapa 3D son ocluidos por la nube de puntos.\n\nEl algoritmo de filtrado implementado en CUDA no tiene una gran carga aritm\u00e9tica, debido a que la mayor\u00eda de operaciones corresponden a movimientos de memoria.\nComo se mencion\u00f3 anteriormente, la plataforma utilizada dispone de una memoria LPDDR4 con cuatro canales de 16 bits, capaz de alcanzar un ancho de banda m\u00e1ximo te\u00f3rico de 25.6GB\/s, sin embargo en las pruebas realizadas la velocidad no supera los 16GB\/s para copias entre la misma GPU y de 10GB\/s para copias CPU-GPU.\nLas pruebas fueron realizadas con los ejemplos sugeridos por el fabricante \\cite{cudaSpeed}, para evaluar el ancho de banda.\nEs importante destacar que las transferencias CPU-GPU se deben a no utilizar memoria fijada (\\texttt{pinned\\_memory)}), la cual evita que el sistema operativo la mueva o la cambie al disco.\nLa memoria fija proporciona una mayor velocidad de transferencia para la GPU y permite la copia asincr\u00f3nica\n\n\nComo puede observarse en la Fig.~\\ref{fig:mejora_velocidad}, se obtiene una mejora notable en el tiempo de filtrado de la nube de puntos.\nLa velocidad del filtrado se reduce en un orden de magnitud cuando el algoritmo es ejecutado en la computadora de escritorio como se indica en la Tabla \\ref{tab:mejora_velocidad}.\nEsto tambi\u00e9n se debe al cambio de micro arquitectura, ya que en Turing los procesadores se redise\u00f1aron para unificar la memoria compartida, el almacenamiento en cach\u00e9 de texturas y el almacenamiento en cach\u00e9 de carga de memoria, en una sola unidad.\nEsto se traduce en 2 veces m\u00e1s ancho de banda y m\u00e1s de 2 veces m\u00e1s capacidad disponible para la cach\u00e9 L1 \\cite{cudaTuring}, en comparaci\u00f3n a la arquitectura anterior a Turing.\nPor otra parte, la placa de video cuenta con 22 SM (Streaming Multiprocessors) con con 64 CUDA cores cada uno, lo que mejora el desempe\u00f1o del algoritmo, puesto que cada subdivisi\u00f3n del espacio (\\texttt{subdiv}) es procesada en forma independiente del resto. \n\nCabe aclarar que el algoritmo para este \u00faltimo caso, est\u00e1 utilizando memoria unificada, pero no puede utilizarse el concepto de \\textit{Zero-copy} disponible en la plataforma Jetson.\n\n\\begin{table}[htbp]\n\\caption{Datos estad\u00edsticos obtenidos para la funci\u00f3n de filtrado en la plataforma Jetson Nano y en una GPU discreta, para un tama\u00f1o de 0.02m}\n\\label{tab:mejora_velocidad}\n\\centering\n\\begin{tabular}{l|c|c}\nPlataforma & Media [ms]& Desviaci\u00f3n est\u00e1ndar\\\\\n\\hline\npcl-jetson& $46.88$ & $1.60\\cdot 10^{-3}$\\\\\ncuda-jetson& $23.2$ & $1.38\\cdot 10^{-3}$\\\\\ncuda-desktop& $3.9$ & $1.76\\cdot 10^{-3}$\\\\\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=0.5\\textwidth]{images\/compara_jetson_desktop.png}}\n\\caption{Comparaci\u00f3n de tiempo de filtrado para tres configuraciones con una resoluci\u00f3n de 0.02m: a la izquierda, el algoritmo de filtrado original corriendo en la CPU de la Jetson; al centro, el algoritmo propuesto corriendo en la GPU de la Jetson; a la derecha, el algoritmo propuesto corriendo en la GPU de escritorio}\n\\label{fig:mejora_velocidad}\n\\end{figure}\n\n\n\n\\balance\n\n\\section{Conclusiones}\nEn este trabajo se present\u00f3 una mejora sobre sobre el filtrado de una nube de puntos, en el programa STVL.\nSe realizaron pruebas simuladas como en la vida real, y se analizaron las ventajas de procesar la nube de puntos en la GPU embebida en plataformas como las NVIDIA Jetson.\nTambi\u00e9n se realiz\u00f3 una comparaci\u00f3n de velocidad con una placa de video discreta en una PC de escritorio, en la cual se pudo comprobar que el aumento en la velocidad de memoria disminuye el tiempo de procesamiento del algoritmo.\nDurante el desarrollo del algoritmo, se pudo constatar que es muy importante tener la misma arquitectura de GPU que la de la plataforma sobre la cual se est\u00e1 desarrollando.\nUna discordancia en la misma, podr\u00eda provocar errores conceptuales y algor\u00edtmicos, debido a que algunas funciones pueden no estar disponibles.\nPara el caso de la Jetson Nano, la misma cuenta con una versi\u00f3n limitada de las funciones de memoria unificada.\nEs trabajo futuro implementar mejoras sobre el algoritmo presentado en la plataforma Jetson TX2, la cual cuenta con micro arquitectura Pascal y mejor soporte de memoria unificada, con lo cual se espera mejorar a\u00fan m\u00e1s el rendimiento.\n\nTambi\u00e9n se est\u00e1 trabajando en reemplazar el uso de las funciones de OpenVDB (realizadas en CPU), por las de la biblioteca GVDB provistas por NVIDIA \\cite{gvdb}.\nDebido al uso de funciones implementadas a partir de la micro arquitectura Pascal, se planea utilizar la plataforma Jetson TX2.\n\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nIn recent years, reinforcement learning~\\cite{BertsekasTsitsiklis1996,KaelblingLittmanMoore1996,sutton1998reinforcement,Szepesvari2009} has emerged as a leading framework to learn how to act optimally in unknown environments. Policy gradient methods~\\cite{Sutton2000,Kakade2002,KondaTsitsiklis2003,\nSchulman2015,Schulman2017,SilverSchrittwieserSimonyanEtAl2017} have played a prominent role in the success of reinforcement learning. Such methods have two critical components: policy evaluation and policy improvement. In policy evaluation step, the performance of a parameterized policy is evaluated while in the policy improvement step, the policy parameters are updated using stochastic gradient ascent.\n\nPolicy gradient methods may be broadly classified as Monte Carlo \nmethods and temporal difference methods. In Monte Carlo methods, performance of a\npolicy is estimated using the discounted return of a single sample path; in\ntemporal difference methods, the value(-action) function is guessed and this guess is\niteratively improved using temporal differences. Monte Carlo methods are attractive\nbecause they have zero bias, are simple and easy to implement, and work for\nboth discounted and average reward setups as well as for models with\ncontinuous state and action spaces. However, they suffer from various drawbacks.\nFirst, they have a high variance because a single sample path is used to\nestimate performance. Second, they are not asymptotically optimal for\ninfinite horizon models because it is effectively\nassumed that the model is episodic; in infinite horizon models, the\ntrajectory is arbitrarily truncated to treat the model as an episodic model.\nThird, the policy improvement step cannot be carried out in tandem with\npolicy evaluation. One must wait until the end of the episode to estimate the\nperformance and only then can the policy parameters be updated. It is for\nthese reasons that Monte Carlo methods are largely ignored in the literature\non policy gradient methods, which\nalmost exclusively focuses on temporal difference methods such as actor-critic with eligibility traces~\\cite{sutton1998reinforcement}. \n\nIn this paper, we propose a Monte Carlo method---which we call \\emph{Renewal\nMonte Carlo} (RMC)---for infinite horizon Markov decision processes with\ndesignated start state. Like Monte Carlo, RMC has low bias, is simple and easy\nto implement, and works for models with continuous state and action spaces.\nAt the same time, it does not suffer from the drawbacks of typical Monte Carlo methods.\nRMC is a low-variance online algorithm that works for infinite horizon\ndiscounted and average reward setups. One doesn't have to wait until the end\nof the episode to carry out the policy improvement step; it can be carried out\nwhenever the system visits the start state (or a neighborhood of\nit).\n\nAlthough renewal theory is commonly used to estimate performance of stochastic\nsystems in the simulation optimization community~\\cite{Glynn1986,Glynn1990},\nthose methods assume that the probability law of the primitive random\nvariables and its weak derivate are known, which is not the case in\nreinforcement learning. Renewal theory is also commonly used in\nthe engineering literature on queuing theory and systems and control for\nMarkov decision processes (MDPs) with average reward criteria and a known\nsystem model. There is some prior work on using renewal theory for\nreinforcement learning~\\cite{MarbachTsitsiklis2001,MarbachTsitsiklis2003},\nwhere renewal theory based estimators for the average return and differential\nvalue function for average reward MDPs is developed. In RMC, renewal theory is\nused in a different manner for discounted reward MDPs (and the results\ngeneralize to average cost MDPs).\n\n\\section{RMC Algorithm} \\label{sec:rl}\n\nConsider a Markov decision process (MDP) with state $\\State_t \\in \\mathcal{\\State}$\nand action $\\Action_t \\in \\ACTION$. The system starts in an initial state\n$\\state_0 \\in \\mathcal{\\State}$ and at time $t$:\n\\begin{enumerate}\n  \\item there is a controlled transition from $S_t$ to $S_{t+1}$ according to\n    a transition kernel $P(\\Action_t)$;\n  \\item a per-step reward $R_t =  r(\\State_t, \\Action_t, \\State_{t+1})$ is\n    received.\n\\end{enumerate}\nFuture is discounted at a rate $\\discount \\in (0,1)$. \n\nA (time-homogeneous and Markov) policy $\\policy$ maps the current state to a\ndistribution on actions, i.e., $\\Action_t \\sim \\policy(\\State_t)$. We use\n$\\policy(\\action | \\state)$ to denote \n$\\PR(\\Action_t = \\action | \\State_t = \\state)$.\nThe performance of a policy $\\policy$ is given by\n\\begin{equation}\n  J_\\pi = \n  \\EXPA\\biggl[\\sum_{t=0}^{\\infty}\\discount^{t}\\Reward_t\\biggm|\\State_0 =\n  \\state_0\\biggr]. \\label{eq:Vp-defn}\n\\end{equation}\n\nWe are interested in identifying an optimal policy, i.e., a policy that maximizes the performance. When $\\mathcal{\\State}$ and $\\ACTION$ are Borel spaces, we assume that the model satisfies the standard conditions under which time-homogeneous Markov policies are optimal~\\cite{Hernandez-Lerma1996}.  In the\nsequel, we present a sample path based online learning algorithm, which\nwe call Renewal Monte Carlo (RMC), which identifies a locally optimal policy\nwithin the class of parameterized policies.\n\nSuppose policies are parameterized by a closed and convex subset\n$\\polParSpace$ of the Euclidean space. For example, $\\polParSpace$\n  could be the weight vector in a Gibbs soft-max policy, or the weights\n  of a deep neural network, or the thresholds in a control limit policy, and\nso on. Given $\\polPars \\in \\polParSpace$, we use $\\policy_\\polPars$ to denote\nthe policy parameterized by $\\polPars$ and $J_\\polPars$ to denote\n$J_{\\policy_{\\polPars}}$. We assume that for all policies $\\policy_\\polPars$,\n$\\polPars \\in \\polParSpace$, the designated start state $s_0$ is positive\nrecurrent.\n\nThe typical approach for policy gradient based reinforcement learning is to start with an\ninitial guess $\\polPars_0 \\in \\polParSpace$ and iteratively update it using\nstochastic gradient ascent. In particular, let $\\widehat \\GRAD J_{\\polPars_m}$ be an\nunbiased estimator of $\\GRAD_\\polPars J_\\polPars \\big|_{\\polPars =\n\\polPars_m}$, then update \n\\begin{equation} \\label{eq:J-update}\n  \\polPars_{m+1}\n= \\big[ \\polPars_m + \\alpha_m \\widehat \\GRAD J_{\\polPars_m} \\big]_{\\polParSpace}\n\\end{equation} \nwhere $[\\polPars]_{\\polParSpace}$ denotes the projection of\n$\\polPars$ onto $\\polParSpace$ and $\\{\\alpha_m\\}_{m \\ge 1}$ is the sequence of\nlearning rates that satisfies the standard assumptions of \n\\begin{equation}\\label{eq:lr}\n  \\sum_{m=1}^\\infty \\alpha_m = \\infty \n  \\quad\\text{and}\\quad \n  \\sum_{m=1}^\\infty \\alpha_m^2 < \\infty.\n\\end{equation}\nUnder mild\ntechnical conditions~\\cite{Borkar:book}, the above iteration converges to a $\\polPars^*$ that is\nlocally optimal, i.e., $\\GRAD_\\polPars J_\\polPars \\big|_{\\polPars =\n\\polPars^*} = 0$. In RMC, we approximate $\\GRAD_\\polPars J_\\polPars$ by a\nRenewal theory based estimator as explained below.\n\nLet $\\tau^{(n)}$ denote the stopping time when the system returns to the start\nstate $\\state_0$ for the $n$-th time. In particular, let $\\tau^{(0)}=0$  and\nfor $n \\ge 1$  define \n\\[ \\tau^{(n)} = \\inf\\{t > \\tau^{(n-1)}:\\state_t = \\state_0\\}. \\]\nWe call the sequence of $(\\State_t, \\Action_t,\n\\Reward_t)$ from $\\tau^{(n-1)}$ to ${\\tau^{(n)} - 1}$ as the\n\\emph{$n$-th regenerative cycle}. Let $\\mathsf{R}^{(n)}$ and $\\mathsf{T}^{(n)}$ denote the total\ndiscounted reward and total discounted time of the $n$-th regenerative cycle,\ni.e., \n\\begin{align}\\label{eq:Rn_and_Tn}\n  \\mathsf{R}^{(n)} = \\Gamma^{(n)} \n  \\smashoperator[r]{\\sum_{t = \\tau^{(n-1)}}^{\\tau^{(n)} - 1}}\n  \\discount^{t} R_t\n  \\quad\\text{and}\\quad\n  \\mathsf{T}^{(n)} = \\Gamma^{(n)}\n  \\smashoperator[r]{\\sum_{t = \\tau^{(n-1)}}^{\\tau^{(n)}-1}}\n  \\discount^{t},\n\\end{align}\nwhere $\\Gamma^{(n)}=\\discount^{-\\tau^{(n-1)}}$.\nBy the strong Markov property, $\\{\\mathsf{R}^{(n)}\\}_{n \\ge 1}$ and $\\{\\mathsf{T}^{(n)}\\}_{n \\ge 1}$\nare i.i.d.\\@ sequences. Let $\\mathsf{R}_\\polPars$ and $\\mathsf{T}_\\polPars$ denote $\\EXP[\\mathsf{R}^{(n)}]$\nand $\\EXP[\\mathsf{T}^{(n)}]$, respectively. Define\n\\begin{equation}\n  \\widehat \\mathsf{R} = \\frac 1N \\sum_{n=1}^N \\mathsf{R}^{(n)} \n  \\quad \\hbox{and}\\quad\n  \\widehat \\mathsf{T} = \\frac 1N \\sum_{n=1}^N \\mathsf{T}^{(n)} ,\n  \\label{eq:est}\n\\end{equation}\nwhere $N$ is a large number. \nThen, $\\widehat \\mathsf{R}$ and $\\widehat \\mathsf{T}$ are unbiased and asymptotically\nconsistent estimators of $\\mathsf{R}_\\polPars$ and $\\mathsf{T}_\\polPars$.\n\nFrom ideas similar to standard\nRenewal theory \\cite{Feller1966}, we have the following.\n\\begin{proposition}[Renewal Relationship]\\label{prop:renewal-basic1} The performance of policy $\\policy_\\polPars$ is given by:\n\\begin{equation}\\label{eq:renewal-basic1}\n  J_\\polPars = \\frac { \\mathsf{R}_\\polPars } { (1 - \\discount) \\mathsf{T}_\\polPars }.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\n  For ease of notation, define\n  \\[\n    \\overline \\mathsf{T}_\\polPars = \\EXPB\\big[ \\discount^{\\tau^{(n)} - \\tau^{(n-1)}}\n    \\big]\n  \\]\n  Using the formula for geometric series, we get that $\\mathsf{T}_\\polPars = ( 1 -\n  \\overline \\mathsf{T}_\\polPars)\/(1 - \\discount)$. Hence,\n  \\begin{equation} \\label{eq:Tbar1}\n    \\overline \\mathsf{T}_\\polPars = 1 - (1 - \\discount) \\mathsf{T}_\\polPars.\n  \\end{equation}\n\n  Now, consider the performance:\n  \\begin{align}\n    J_\\polPars &= \\EXPB\\bigg[\n      \\sum_{t=0}^{\\tau^{(1)}-1} \\discount^{t} R_t \n      \\notag \\\\\n      & \\hspace{7em}\n    +  \\discount^{\\tau^{(1)}}\n    \\smashoperator{\\sum_{t = \\tau^{(1)}}^\\infty} \\discount^{t-\\tau^{(1)}} R_t \n    \\biggm| \\State_{0} = \\state_0 \\bigg]\n    \\displaybreak[0]\n    \\notag \\\\\n    &\\stackrel{(a)}= \\mathsf{R}_\\polPars + \n    \\EXPB[ \\discount^{\\tau^{(1)})} ]\\, J_\\polPars\n    \\displaybreak[0]\n    \\notag \\\\\n    &= \\mathsf{R}_\\polPars + \\overline \\mathsf{T}_\\polPars J_\\polPars,\n    \\label{eq:R-11}\n  \\end{align}\n  where the second expression in $(a)$ uses the independence of random\n  variables from $(0, \\tau^{(1)}-1)$ to those from $\\tau^{(1)}$ onwards \n  due to the strong Markov property. Substituting~\\eqref{eq:Tbar1}\n  in~\\eqref{eq:R-11} and rearranging terms, we get the result of the\n  proposition.\n\\end{proof}\nDifferentiating both sides of Equation~\\eqref{eq:renewal-basic1} with respect to $\\polPars$, we get that\n\\begin{equation} \\label{eq:H}\n  \\GRAD_\\polPars J_\\polPars = \\frac{H_\\polPars}{\\mathsf{T}_\\polPars^2(1 - \\discount)},\n  \\enskip\\text{where }\n  H_\\polPars =  \\mathsf{T}_\\polPars \\GRAD_\\polPars \\mathsf{R}_\\polPars\n  - \\mathsf{R}_\\polPars \\GRAD_\\polPars \\mathsf{T}_\\polPars. \n\\end{equation}\n\nTherefore, instead of using  stochastic gradient ascent to find the maximum\nof $J_\\polPars$, we can use stochastic approximation to find\nthe root of $H_\\polPars$. In particular, let $\\widehat H_m$ be an unbiased\nestimator of $H_{\\polPars_m}$. We then use the update\n\\begin{equation} \\label{eq:H-update}\n  \\polPars_{m+1} = \\big[ \\polPars_m + \\alpha_m \\widehat H_m \\big]_{\\polParSpace}\n\\end{equation}\nwhere $\\{\\alpha_m\\}_{m \\ge 1}$ satisfies the standard conditions on learning\nrates~\\eqref{eq:lr}.  The above iteration converges to a locally optimal policy.\nSpecifically, we have the following.\n\n\\begin{theorem}\\label{thm:convergence}\n  Let $\\widehat \\mathsf{R}_m$, $\\widehat \\mathsf{T}_m$, $\\widehat \\GRAD \\mathsf{R}_m$ and $\\widehat\n  \\GRAD \\mathsf{T}_m$ be unbiased estimators of $\\mathsf{R}_{\\polPars_m}$, $\\mathsf{T}_{\\polPars_m}$,\n  $\\GRAD_\\polPars \\mathsf{R}_{\\polPars_m}$, and $\\GRAD_\\polPars \\mathsf{R}_{\\polPars_m}$,\n  respectively such that $\\widehat \\mathsf{T}_m \\perp \\widehat \\GRAD \\mathsf{R}_m$ and\n  $\\widehat \\mathsf{R}_m \\perp \\widehat \\GRAD \\mathsf{T}_m$.\\footnote{The notation $X \\perp Y$\n  means that the random variables $X$ and $Y$ are independent.} Then,\n  \\begin{equation}\n    \\widehat H_m = \\widehat \\mathsf{T}_m \\widehat \\GRAD \\mathsf{R}_m - \\widehat \\mathsf{R}_m \n    \\widehat \\GRAD \\mathsf{T}_m\n    \\label{eq:H-est}\n  \\end{equation}\n  is an unbiased estimator of $H_\\polPars$ and \n  the sequence $\\{\\polPars_m\\}_{m \\ge 1}$ generated\n  by~\\eqref{eq:H-update} converges almost surely and \n  \\[\n    \\lim_{m \\to \\infty} \\GRAD_\\polPars J_\\polPars \\big|_{\\polPars_m} = 0.\n  \\]\n\\end{theorem}\n\\begin{proof}\n  The unbiasedness of $\\widehat H_m$ follows immediately from the\n  independence assumption. The convergence of\n  the $\\{\\polPars_m\\}_{m \\ge 1}$ follows from~\\cite[Theorem 2.2]{Borkar:book}\n  and the fact that the model satisfies conditions (A1)--(A4)\n  of~\\cite[pg~10--11]{Borkar:book}.\n\\end{proof}\n\nIn the remainder of this section, we present two methods for estimating the\ngradients of $\\mathsf{R}_\\polPars$ and $\\mathsf{T}_\\polPars$. \nThe first is a likelihood ratio based\ngradient estimator which works when the policy is differentiable with respect\nto the policy parameters. The second is a simultaneous perturbation based\ngradient estimator that uses finite differences, which is useful when the\npolicy is not differentiable with respect to the policy parameters.\n\n\\subsection{Likelihood ratio based gradient based estimator}\\label{sec:likelihood}\n\nOne approach to estimate the performance gradient is to use likelihood radio\nbased estimates~\\cite{Rubinstein1989,Glynn1990,Williams1992}.\nSuppose the policy $\\policy_\\polPars(\\action | \\state)$ is differentiable with respect to\n$\\polPars$. For any time~$t$, define the likelihood function\n\\begin{equation}\\label{eq:score}\n  \\Score_t = \n  \\GRAD_\\polPars \\log[ \\policy_\\polPars(\\Action_t \\mid \\State_t) ],\n\\end{equation}\nand for $\\sigma \\in \\{ \\tau^{(n-1)}, \\dots, \\tau^{(n)} - 1 \\}$, define\n\\begin{equation}\n  \\mathsf{R}^{(n)}_\\sigma =  \\Gamma^{(n)}\\sum_{t=\\sigma}^{\\tau^{(n)}-1}\\discount^tR_t,\\enskip\n  \\mathsf{T}^{(n)}_\\sigma = \\Gamma^{(n)}\\sum_{t=\\sigma}^{\\tau^{(n)}-1}\\discount^t.\\label{eq:R_T_sigma}\n\\end{equation}\nIn this notation $\\mathsf{R}^{(n)} = \\mathsf{R}^{(n)}_{\\tau^{(n-1)}}$ and $\\mathsf{T}^{(n)} = \\mathsf{T}^{(n)}_{\\tau^{(n-1)}}$.\nThen, define the following estimators for $\\GRAD_\\polPars \\mathsf{R}_\\polPars$ and\n$\\GRAD_\\polPars \\mathsf{T}_\\polPars$:\n\\begin{align}\n\\widehat \\GRAD \\mathsf{R} &= \\frac 1N \\sum_{n=1}^N \\sum_{\\sigma=\\tau^{(n-1)}}^{\\tau^{(n)}-1}\\mathsf{R}^{(n)}_\\sigma  \\Score_{\\sigma},\\label{eq:grad_R_new}\\\\\n\\widehat \\GRAD \\mathsf{T} &= \\frac 1N \\sum_{n=1}^N \\sum_{\\sigma=\\tau^{(n-1)}}^{\\tau^{(n)}-1}\\mathsf{T}^{(n)}_\\sigma  \\Score_{\\sigma},\\label{eq:grad_T_new}\n\\end{align}\nwhere $N$ is a large number. \n\n\\begin{proposition} \\label{prop:estimator}\n  $\\widehat \\GRAD \\mathsf{R}$ and $\\widehat \\GRAD \\mathsf{T}$ defined above are unbiased\n  and asymptotically consistent estimators\n  of $\\GRAD_\\polPars \\mathsf{R}_\\polPars$ and $\\GRAD_\\polPars \\mathsf{T}_\\polPars$. \n\\end{proposition}\n\\begin{proof}\nLet $P_\\polPars$ denote the probability induced on the sample paths when the\nsystem is following policy $\\policy_\\polPars$. For $t \\in \\{ \\tau^{(n-1)},\n\\dots, \\tau^{(n)} - 1\\}$, let $D^{(n)}_{t}$ denote the sample path $(\\State_s,\n\\Action_s, \\State_{s+1})_{s=\\tau^{(n-1)}}^{t}$ for\nthe $n$-th regenerative cycle until time $t$. Then,\n\\[\n  \\let\\smashoperator\\relax\n  P_\\polPars(D^{(n)}_t) = \\smashoperator{\\prod_{s= \\tau^{(n-1)}}^{t} }\n  \\policy_\\polPars(A_s | S_s)\n  \\PR(\\State_{s+1} | S_{s}, A_{s})\n\\]\nTherefore,\n\\begin{equation}  \\label{rel:11}\n  \\GRAD_\\polPars \\log P_\\polPars(D^{(n)}_t) =\n  \\smashoperator{\\sum_{s=\\tau^{(n-1)}}^{t}} \\GRAD_\\polPars \\log\n  \\policy_\\polPars(\\Action_s | \\State_s) = \\smashoperator{\\sum_{s=\\tau^{(n-1)}}^{t}} \\Score_s.\n\\end{equation}\n\nNote that $\\mathsf{R}_\\polPars$ can be written as:\n\\[\n  \\mathsf{R}_\\polPars = \\Gamma^{(n)}\\smashoperator[r]{\\sum_{t = \\tau^{(n-1)}}^{\\tau^{(n)} - 1}}\\discount^{t}\\EXPB[ R_t].\n\\]\nUsing the log derivative trick,\\footnote{Log-derivative trick: For any distribution $p(x|\\theta)$ and any function $f$,\n\\[\n  \\GRAD_\\theta \\EXP_{X \\sim p(X|\\theta)} [ f(X) ]\n  = \n  \\EXP_{X \\sim p(X|\\theta)}[ f(X) \\GRAD_\\theta \\log p(X | \\theta)].\n\\]\n} we get\n\\begin{align} \n  \\GRAD_\\polPars \\mathsf{R}_\\polPars &= \n  \\Gamma^{(n)} \n    \\smashoperator[r]{\\sum_{t = \\tau^{(n-1)}}^{\\tau^{(n)} - 1}}\n    \\discount^{t}\\,\n  \\EXPB[  R_t \\GRAD_\\polPars \\log P_\\polPars(D^{(n)}_t) ] \n  \\notag \\\\\n  &\\stackrel{(a)}= \n  \\Gamma^{(n)} \n  \\EXPB\\bigg[\n  \\sum_{t = \\tau^{(n-1)}}^{\\tau^{(n)} - 1}\n  \\bigg[\n\\discount^{t} R_t{\\sum_{\\sigma=\\tau^{(n-1)}}^t}\\Score_\\sigma\\bigg] \\bigg]\n  \\notag \\\\\n    &\\stackrel{(b)}= \\EXPB\\bigg[ \n  \\sum_{\\sigma = \\tau^{(n-1)}}^{\\tau^{(n)} - 1}\\Score_\\sigma\\bigg[\n  \\Gamma^{(n)}\\sum_{t=\\sigma}^{\\tau^{(n)} - 1}\\discount^{t}  R_t \\bigg] \n  \\bigg] \\notag \\\\\n      &\\stackrel{(c)}= \\EXPB\\bigg[ \n  \\sum_{\\sigma = \\tau^{(n-1)}}^{\\tau^{(n)} - 1}\n  \\mathsf{R}^{(n)}_\\sigma \\Score_\\sigma \\bigg]\n  \\label{rel:12}\n\\end{align}\nwhere $(a)$ follows from~\\eqref{rel:11}, $(b)$ follows from changing the order\nof summations, and $(c)$ follows from the definition of\n$\\mathsf{R}^{(n)}_\\sigma$ in~\\eqref{eq:R_T_sigma}. $\\widehat \\GRAD\n\\mathsf{R}$ is an unbiased and asymptotically consistent estimator of the right hand\nside of the first equation in~\\eqref{rel:12}. The result for $\\widehat \\GRAD\n\\mathsf{T}$ follows from a similar argument.\n\\end{proof}\n\n\\begin{algorithm2e}[!tb]\n\\def\\1#1{\\quitvmode\\hbox to 1em{\\hfill$\\mathsurround0pt #1$}}\n\\SetKwInOut{Input}{input}\n\\SetKwInOut{Output}{output}\n\\SetKwInOut{Init}{initialize}\n\\SetKwProg{Fn}{function}{}{}\n\\SetKwFor{ForAll}{forall}{do}{}\n\\SetKwRepeat{Do}{do}{while}\n\\DontPrintSemicolon\n\\Input{Intial policy $\\polPars_0$, discount factor $\\discount$, \n  initial state~$\\state_0$, number of regenerative cycles $N$}\n\n\\For{iteration $m = 0, 1, \\dots$}{\n  \\For{regenerative cycle $n_1=1$ to $N$}{\n  Generate $n_1$-th regenerative cycle using\n  \\rlap{policy~$\\policy_{\\polPars_m}$.}\n\n  Compute $\\mathsf{R}^{(n_1)}$ and $\\mathsf{T}^{(n_1)}$ using~\\eqref{eq:Rn_and_Tn}.\n  }\n  Set $\\widehat \\mathsf{R}_{m} = \\texttt{average}(\\mathsf{R}^{(n_1)}: n_1 \\in \\{1,\\dots,N\\})$.\n\n  Set $\\widehat \\mathsf{T}_{m} = \\texttt{average}(\\mathsf{T}^{(n_1)}: n_1 \\in \\{1,\\dots,N\\})$.\n\n  \\For{regenerative cycle $n_2=1$ to $N$}{\n    Generate $n_2$-th regenerative cycle using\n    \\rlap{policy~$\\policy_{\\polPars_m}$.}\n\n    Compute $\\mathsf{R}_\\sigma^{(n_2)}$, $\\mathsf{T}_\\sigma^{(n_2)}$ and $\\Score_\\sigma$\n    for all $\\sigma$.\n  }\n   Compute $\\widehat \\GRAD \\mathsf{R}_{m}$ and $\\widehat \\GRAD \\mathsf{T}_m$ \n   using \\eqref{eq:grad_R_new} and~\\eqref{eq:grad_T_new}.\n\n  \\vskip 2pt\n   Set $\\widehat H_m = \\widehat \\mathsf{T}_m \\widehat \\GRAD \\mathsf{R}_m - \\widehat \\mathsf{R}_m \\widehat \\GRAD \\mathsf{T}_m$.\n\n  \\vskip 4pt\n  Update $\\polPars_{m+1} = \\big[ \\polPars_m + \\alpha_m \\widehat H_m\n  \\big]_{\\polParSpace}$.\n}\n\\caption{RMC Algorithm with likelihood ratio based gradient estimates.}\n\\label{alg:likelihood}\n\\end{algorithm2e}\n\nTo satisfy the independence condition of Theorem~\\ref{thm:convergence}, we use two independent sample paths: one to estimate $\\widehat \\mathsf{R}$ and\n$\\widehat \\mathsf{T}$ and the other to estimate $\\widehat \\GRAD \\mathsf{R}$ and $\\widehat\n\\GRAD \\mathsf{T}$. The complete algorithm in shown in Algorithm~\\ref{alg:likelihood}.\nAn immediate consequence of Theorem~\\ref{thm:convergence} is the following. \n\\begin{corollary}\\label{cor:pol_grad_conv}\n  The sequence $\\{\\polPars_m\\}_{m \\ge 1}$ generated by\n  Algorithm~\\ref{alg:likelihood} converges to a local optimal.\n\\end{corollary}\n\n\\begin{remark}\\label{rem:1}\nAlgorithm~\\ref{alg:likelihood} is presented in its simplest form. It is possible to use standard variance reduction techniques such as\nsubtracting a baseline~\\cite{Williams1992,Greensmith2004,Peters2006} to reduce variance. \n\\end{remark}\n\n\\begin{remark}\\label{rem:2} \\label{rem:single_run}\nIn Algorithm~\\ref{alg:likelihood}, we use two separate runs to compute $(\\widehat \\mathsf{R}_m, \\widehat \\mathsf{T}_m)$ and $(\\GRAD \\widehat \\mathsf{R}_m, \\GRAD \\widehat \\mathsf{T}_m)$ to ensure that the independence conditions of Proposition~\\ref{prop:estimator} are satisfied. In practice, we found that using a single run to compute both $(\\widehat \\mathsf{R}_m, \\widehat \\mathsf{T}_m)$ and $(\\GRAD \\widehat \\mathsf{R}_m, \\GRAD \\widehat \\mathsf{T}_m)$ has negligible effect on the accuracy of convergence (but speeds up convergence by a factor of two).\n\\end{remark}\n\n\\begin{remark}\\label{rem:3}\nIt has been reported in the literature~\\cite{Thomas2014} that using a biased estimate of the gradient given by:\n\\begin{equation}\n  \\mathsf{R}^{(n)}_\\sigma = \\Gamma^{(n)} \n  \\sum_{t=\\sigma}^{\\tau^{(n)}-1}\\discount^{t-\\sigma} R_t,\n  \\label{eq:R_sigma_biased} \n\\end{equation}\n(and a similar expression for $T^{(n)}_\\sigma$) leads to faster convergence. We call this variant \\textit{RMC with biased gradients} and, in our experiments, found that it does converge faster than RMC.\n\\end{remark}\n\n\n\\subsection{Simultaneous perturbation based gradient estimator}\nAnother approach to estimate performance gradient is to use simultaneous\nperturbation based estimates\\cite{Spall1992,Maryak2008,Katkovnik1972,Bhatnagar:2013}. The general one-sided form of such estimates is\n\\[\n  \\widehat \\GRAD \\mathsf{R}_\\polPars = \\delta (\n  \\widehat \\mathsf{R}_{\\polPars + c \\delta} - \\widehat \\mathsf{R}_{\\polPars} )\/c\n\\]\nwhere $\\delta$ is a random variable with the same dimension as\n$\\polPars$ and $c$ is a small constant. The expression for $\\widehat \\GRAD\n\\mathsf{T}_\\polPars$ is similar. When $\\delta_i \\sim \n\\text{Rademacher}(\\pm 1)$, the above method corresponds\nto simultaneous perturbation stochastic approximation (SPSA)~\\cite{Spall1992,\nMaryak2008}; when $\\delta \\sim \\text{Normal}(0, I)$, the above method\ncorresponds to smoothed function stochastic approximation\n(SFSA)~\\cite{Katkovnik1972,Bhatnagar:2013}.\n\n\\begin{algorithm2e}[!tb]\n\\def\\1#1{\\quitvmode\\hbox to 1em{\\hfill$\\mathsurround0pt #1$}}\n\\SetKwInOut{Input}{input}\n\\SetKwInOut{Output}{output}\n\\SetKwInOut{Init}{initialize}\n\\SetKwProg{Fn}{function}{}{}\n\\SetKwFor{ForAll}{forall}{do}{}\n\\SetKwRepeat{Do}{do}{while}\n\\DontPrintSemicolon\n\\Input{Intial policy $\\polPars_0$, discount factor $\\discount$, initial state~$\\state_0$, number of regenerative cycles $N$, constant $c$,  perturbation distribution\n$\\Delta$}\n\n\\For{iteration $m = 0, 1, \\dots$}{\n  \\For{regenerative cycle $n_1=1$ to $N$}{\n    Generate $n_1$-th regenerative cycle using\n    \\rlap{policy~$\\policy_{\\polPars_m}$.}\n\n    Compute $\\mathsf{R}^{(n_1)}$ and $\\mathsf{T}^{(n_1)}$ using~\\eqref{eq:Rn_and_Tn}.\n  }\n  Set $\\widehat \\mathsf{R}_{m} = \\texttt{average}(\\mathsf{R}^{(n_1)}: n_1 \\in \\{1,\\dots,N\\})$.\n\n  Set $\\widehat \\mathsf{T}_{m} = \\texttt{average}(\\mathsf{T}^{(n_1)}: n_1 \\in \\{1,\\dots,N\\})$.\n\n  Sample $\\delta \\sim \\Delta$.\n\n  Set $\\polPars_m' = \\polPars_m + c \\delta$.\n\n  \\For{regenerative cycle $n_2=1$ to $N$}{\n    Generate $n_2$-th regenerative cycle using\n    \\rlap{policy~$\\policy_{\\polPars_m}$.}\n\n  Compute $\\mathsf{R}^{(n_2)}$ and $\\mathsf{T}^{(n_2)}$ using~\\eqref{eq:Rn_and_Tn}.\n  }\n  Set $\\widehat \\mathsf{R}'_{m} = \\texttt{average}(\\mathsf{R}^{(n_2)}: n_2 \\in \\{1,\\dots,N\\})$.\n\n  Set $\\widehat \\mathsf{T}'_{m} = \\texttt{average}(\\mathsf{T}^{(n_2)}: n_2 \\in \\{1,\\dots,N\\})$.\n\n  \\vskip 2pt\n  Set $\\widehat H_m = \\delta(\\widehat \\mathsf{T}_m \\widehat \\mathsf{R}'_m \n  - \\widehat \\mathsf{R}_m \\widehat \\mathsf{T}'_m)\/c$.\n\n  \\vskip 4pt\n  Update $\\polPars_{m+1} = \\big[ \\polPars_m + \\alpha_m \\widehat H_m\n  \\big]_{\\polParSpace}$.\n}\n\\caption{RMC Algorithm with simultaneous perturbation based gradient estimates.}\n\\label{alg:SPSA}\n\\end{algorithm2e}\n\n\nSubstituting the above estimates in~\\eqref{eq:H-est} and simplifying, we get\n\\[\n  \\widehat H_\\polPars = \\delta ( \\widehat \\mathsf{T}_\\polPars \\widehat \\mathsf{R}_{\\polPars + c\\delta} \n    - \n  \\widehat \\mathsf{R}_\\polPars \\widehat \\mathsf{T}_{\\polPars + c \\delta} )\/c.\n\\]\nThe complete algorithm in shown in Algorithm~\\ref{alg:SPSA}. Since $(\\widehat\n\\mathsf{R}_\\polPars, \\widehat \\mathsf{T}_\\polPars)$ and $(\\widehat \\mathsf{R}_{\\polPars + c \\delta},\n\\widehat \\mathsf{T}_{\\polPars + c \\delta})$ are estimated from separate sample paths,\n$\\widehat H_\\polPars$ defined above is an unbiased estimator of $H_\\polPars$.\nThen, an immediate consequence of Theorem~\\ref{thm:convergence} is the\nfollowing.\n\\begin{corollary}\n  The sequence $\\{\\polPars_m\\}_{m \\ge 1}$ generated by\n  Algorithm~\\ref{alg:SPSA} converges to a local optimal.\n\\end{corollary}\n\n\\section{RMC for Post-Decision State Model} \\label{sec:post_model}\nIn many models, the state dynamics can be split into two parts: a controlled evolution followed by an uncontrolled evolution. For example, many continuous state models have dynamics of the form\n\\[\nS_{t+1} = f(S_t, A_t) + N_t,\n\\]\nwhere $\\{N_t\\}_{t \\ge 0}$ is an independent noise process. For other examples, see the inventory control and event-triggered communication models in Sec~\\ref{sec:num_exp}. Such models can be written in terms of a post-decision state model described below.\n\nConsider a post-decision state MDP with pre-decision state $\\Prestate_t \\in\n\\PRESTATE$, post-decision state $\\Poststate_t \\in \\POSTSTATE$, action\n$\\Action_t \\in \\ACTION$.\nThe system starts at an initial state $\\poststate_0 \\in \\POSTSTATE$ and at \ntime~$t$: \n\\begin{enumerate}\n  \\item there is a controlled transition from $\\Prestate_t$ to $\\Poststate_t$\n    according to a transition kernel $\\PRE P(\\Action_t)$; \n  \\item there is an uncontrolled transition from $\\Poststate_t$ to\n    $\\Prestate_{t+1}$ according to a transition kernel $\\POST P$;\n  \\item a per-step reward $R_t =  r(\\Prestate_t, \\Action_t,\n    \\Poststate_t)$ is received.\n\\end{enumerate}\nFuture is discounted at a rate $\\discount \\in (0,1)$. \n\\begin{remark}\n  When $\\POSTSTATE = \\PRESTATE$ and $\\PRE P$ is identity, then the above model\n  reduces to the standard MDP model, considered in Sec~\\ref{sec:rl}. When\n  $\\POST P$ is a deterministic transition, the model reduces to a standard MDP\n  model with post decision\n  states~\\cite{VanRoyBertsekasLeeEtAl1997,powell2011approximate}. \n\\end{remark}\n\nAs in Sec~\\ref{sec:rl}, we choose a (time-homogeneous and Markov) policy $\\policy$ that maps the current pre-decision state $\\PRESTATE$ to a\ndistribution on actions, i.e., $\\Action_t \\sim \\policy(\\Prestate_t)$. We use\n$\\policy(\\action | \\prestate)$ to denote \n$\\PR(\\Action_t = \\action | \\Prestate_t = \\prestate)$. \n\nThe performance when the system starts in post-decision state $\\poststate_0 \\in\n\\POSTSTATE$ and follows policy $\\policy$ is given by\n\\begin{equation}\n  J_\\policy =  \n  \\EXPA\\biggl[\\sum_{t=0}^{\\infty}\\discount^{t}\\Reward_t\\biggm|\\Poststate_0 =\n  \\poststate_0\\biggr].\n\\end{equation}\nAs before, we are interested in identifying an optimal policy, i.e., a policy that maximizes the performance. When $\\mathcal{\\State}$ and $\\ACTION$ are Borel spaces, we assume that the model satisfies the standard conditions under which time-homogeneous Markov policies are optimal~\\cite{Hernandez-Lerma1996}.\nLet $\\tau^{(n)}$ denote the stopping times such that $\\tau^{(0)} = 0$ and for\n$n \\ge 1$,\n\\[\n  \\tau^{(n)} = \\inf \\{ t > \\tau^{(n-1)} : \\poststate_{t-1} = \\poststate_0 \\}.\n\\]\nThe slightly unusual definition (using $\\poststate_{t-1} = \\poststate_0$\nrather than the more natural $\\poststate_t = \\poststate_0$) is to ensure that\nthe formulas for $\\mathsf{R}^{(n)}$ and $\\mathsf{T}^{(n)}$ used in Sec.~\\ref{sec:rl} remain\nvalid for the post-decision state model as well. Thus, using arguments similar to Sec.~\\ref{sec:rl}, we can show that both variants of RMC\npresented in Sec.~\\ref{sec:rl} converge to a locally optimal parameter\n$\\polPars$ for the post-decision state model as well.\n\n\\section{Approximate RMC}\\label{sec:approx_rl}\n\nIn this section, we present an approximate version of RMC (for the basic\nmodel of Sec.~\\ref{sec:rl}). Suppose that the state and action spaces\n$\\mathcal{\\State}$ and $\\ACTION$ are separable metric spaces (with metrics $d_S$ and\n$d_A$).\n\nGiven an approximation constant $\\rho \\in \\reals_{> 0}$, let $B^\\rho = \\{s \\in \\mathcal{\\State}: d_S(s,s_0) \\le \\rho\\}$ denote\nthe ball of radius $\\rho$ centered around $s_0$. Given a policy $\\policy$, let\n$\\tau^{(n)}$ denote the stopping times for successive visits to $B^\\rho$,\ni.e., $\\tau^{(0)} = 0$ and for $n \\ge 1$, \n\\[\n  \\tau^{(n)} = \\inf \\{ t > \\tau^{(n-1)} : \\state_t \\in B^\\rho \\}.\n\\]\nDefine $\\mathsf{R}^{(n)}$ and $\\mathsf{T}^{(n)}$ as in~\\eqref{eq:Rn_and_Tn} and let\n$\\mathsf{R}^\\rho_\\polPars$ and $\\mathsf{T}^\\rho_\\polPars$ denote the expected values of\n$\\mathsf{R}^{(n)}$ and $\\mathsf{T}^{(n)}$, respectively. Define\n\\[\n  J^\\rho_\\polPars = \\frac{\\mathsf{R}^\\rho_\\polPars}{ (1-\\discount) \\mathsf{T}^\\rho_\\polPars}.\n\\]\n\n\\begin{theorem}\\label{thm:approx_RMC}\n  Given a policy $\\policy_\\polPars$, let $V_\\polPars$ denote the value\n  function and $\\overline \\mathsf{T}^\\rho_\\polPars = \\EXPB[ \\discount^{\\tau^{(1)}} |\n  \\State_0 = \\state_0 ]$ (which is always less than $\\discount$). Suppose the\n  following condition is satisfied:\n  \\begin{enumerate}\n    \\item[\\textup{(C)}] The value function $V_\\polPars$ is locally Lipschitz\n      in $B^\\rho$, i.e., there exists a $L_\\polPars$ such that for any $s, s'\n      \\in B^\\rho$, \n      \\[\n        | V_\\polPars(s) - V_\\polPars(s') | \\le L_\\polPars d_S(s,s').\n      \\]\n  \\end{enumerate}\n  Then\n  \\begin{equation}\\label{eq:approxJ_bound}\n    \\big| J_\\polPars - J^\\rho_\\polPars \\big| \\le \n    \\frac{ L_\\polPars \\overline \\mathsf{T}^\\rho_\\polPars } { (1-\\discount)\n    \\mathsf{T}^\\rho_\\polPars} \\rho \\le \\frac{\\discount}{(1-\\discount)} L_\\polPars \\rho.\n  \\end{equation}\n\\end{theorem}\n\\begin{proof}\n  We follow an argument similar to Proposition~\\ref{prop:renewal-basic1}.\n  \\begin{align}\n    J_\\polPars &= V_\\polPars(s_0) = \n    \\EXPB\\bigg[\n      \\sum_{t=0}^{\\tau^{(1)}-1} \\discount^{t} R_t \n      \\notag \\\\\n      & \\hskip 6em \n    +  \\discount^{\\tau^{(1)}}\n    \\sum_{t = \\tau^{(1)}}^\\infty \\discount^{t-\\tau^{(1)}} R_t \n    \\biggm| \\State_{0} = \\state_{\\tau^{(1)}} \\bigg]\n    \\notag \\\\\n    &\\stackrel{(a)}= \\mathsf{R}^\\rho_\\polPars + \n\\EXPB[ \\discount^{\\tau^{(1)}} | \\State_0 = \\state_0]\\, V_\\polPars(\\state_{\\tau^{(1)}})\n\\label{eq:approx1}\n  \\end{align}\n  where $(a)$ uses the strong Markov property.\n  Since $V_\\polPars$ is locally Lipschitz with constant $L_\\polPars$ and\n  $s_{\\tau^{(1)}} \\in B^\\rho$, we have that\n  \\[\n    |J_\\polPars - V_\\polPars(s_{\\tau^{(1)}}) | = |V_\\polPars(s_0) -\n    V_\\polPars(s_{\\tau^{(1)}}) | \\le L_\\polPars \\rho. \n  \\]\n  Substituting the above in~\\eqref{eq:approx1} gives\n  \\[\n    J_\\polPars \\le \\mathsf{R}^\\rho_\\polPars + \\overline \\mathsf{T}^\\rho_\\polPars (J_\\polPars +\n    L_\\polPars \\rho).\n  \\]\n  Substituting $\\mathsf{T}^\\rho_\\polPars = (1 - \\overline \\mathsf{T}^\\rho_\\polPars)\/(1 -\n  \\discount)$ and rearranging the terms, we get \n  \\[\n    J_\\polPars \\le J^\\rho_\\polPars + \\frac{L_\\polPars \\overline\n    \\mathsf{T}^\\rho_\\polPars}{(1-\\discount) \\mathsf{T}^\\rho_\\polPars } \\rho.\n  \\]\n  The other direction can also be proved using a similar argument. The second\n  inequality in~\\eqref{eq:approxJ_bound} follows from $\\overline \\mathsf{T}^\\rho_\\polPars \\le \\gamma$ and ${\\mathsf{T}^\\rho_\\polPars \\ge 1}$.\n\\end{proof}\n\nTheorem~\\ref{thm:approx_RMC} implies that we can find an approximately optimal policy by identifying policy parameters $\\polPars$ that minimize $J^\\rho_\\polPars$. To do so, we can appropriately modify both variants of\nRMC defined in Sec.~\\ref{sec:rl} to declare a renewal whenever the state lies\nin $B^\\rho$. \n\nFor specific models, it may be possible to verify that the value function is\nlocally Lipschitz (see Sec.~\\ref{sec:inv_ctrl} for an example). However, we\nare not aware of general conditions that guarantee local Lipschitz\ncontinuity of value functions. It is possible to identify sufficient conditions that guarantee global Lipschitz continuity of value functions (see~\\cite[Theorem 4.1]{Hinderer2005},\n\\cite[Lemma 1, Theorem 1]{Rachelson2010}, \\cite[Lemma 1]{Pirotta2015}). We state these conditions below.\n\\begin{proposition}\\label{prop:Lispschitz}\n  Let $V_\\polPars$ denote the value function for any policy\n  $\\policy_{\\polPars}$. Suppose the model satisfies the following conditions:\n  \\begin{enumerate}\n    \\item The transition kernel $P$ is Lipschitz, i.e., there\n      exists a constant $L_P$ such that for all\n      $s,s' \\in \\mathcal{\\State}$ and $a,a' \\in \\ACTION$, \n      \\[\n        \\mathcal K(P(\\cdot | s,a), P(\\cdot | s',a')) \\le \n        L_P\\big[ d_S(s,s') + d_A(a,a') \\big],\n      \\]\n      where $\\mathcal K$ is the Kantorovich metric (also called\n      Kantorovich-Monge-Rubinstein metric or Wasserstein distance) between\n      probability measures.\n\n    \\item The per-step reward $r$ is Lipschitz, i.e., there exists a constant\n      $L_r$ such that for all $s,s',s_+ \\in \\mathcal{\\State}$ and $a,a' \\in \\ACTION$,\n      \\[\n        | r(s,a,s_+) - r(s',a',s_+) | \\le \n          L_r\\big[ d_S(s,s') + d_A(a,a') \\big].\n      \\]\n  \\end{enumerate}\n  In addition, suppose the policy satisfies the following:\n  \\begin{enumerate}\n    \\setcounter{enumi}{2}\n    \\item The policy $\\policy_\\polPars$ is Lipschitz, i.e., there exists a\n      constant $L_{\\policy_\\polPars}$ such that for any $s,s' \\in \\mathcal{\\State}$,\n      \\[\n        \\mathcal K( \\policy_\\polPars(\\cdot | s), \\policy_\\polPars(\\cdot | s')) \n        \\le\n        L_{\\policy_\\polPars}\\, d_S(s,s').\n      \\]\n    \\item $\\discount L_P(1 + L_{\\policy_\\polPars}) < 1$.\n    \\item The value function $V_\\polPars$ exists and is finite. \n  \\end{enumerate}\n  Then, $V_\\polPars$ is Lipschitz. In particular, for any $s, s' \\in\n  \\mathcal{\\State}$,\n  \\[\n    | V_\\polPars(s) - V_\\polPars(s') | \\le L_\\polPars d_S(s,s'),\n  \\]\n  where\n  \\[\n    L_\\polPars = \\frac{L_r (1 + L_{\\policy_\\polPars})}\n    {1 - \\discount L_P(1 + L_{\\policy_\\polPars}) }.\n  \\]\n\\end{proposition}\n\n\\section{Numerical Experiments}\\label{sec:num_exp}\n\nWe conduct three experiments to evaluate the performance of RMC: a randomly\ngenerated MDP, event-triggered communication, and inventory\nmanagement.  \n\n\\subsection{Randomized MDP (GARNET)} \\label{sec:GARNET}\n\nIn this experiment, we study a randomly generated $\\text{GARNET}(100,10,50)$ \nmodel~\\cite{Bhatnagar2009}, which is an MDP with $100$ states, $10$ actions,\nand a branching factor of $50$ (which means that each row of all transition\n  matrices has $50$ non-zero elements, chosen $\\text{Unif}[0,1]$ and\nnormalized to add to~$1$). For each state-action pair, with probability\n$p=0.05$, the reward is chosen $\\text{Unif}[10,100]$, and with probability\n$1-p$, the reward is~$0$. Future is discounted by a factor of\n$\\discount=0.9$. The first state is chosen as start state. The policy is a Gibbs soft-max distribution\nparameterized by $100 \\times 10$ (states $\\times$ actions) parameters, where\neach parameter belongs to the interval $[-30, 30]$. The temperature of the\nGibbs distribution is kept constant and equal to~$1$.\n\nWe compare the performance of RMC, RMC with biased gradient (denoted by\nRMC-B, see Remark~\\ref{rem:2}), and actor critic with eligibility\ntraces for the critic~\\cite{sutton1998reinforcement} (which we refer to as\nSARSA-$\\lambda$ and abbreviate as S-$\\lambda$ in the plots),\nwith $\\lambda \\in \\{0, 0.25, 0.5, 0.75, 1\\}$.\n For both the RMC algorithms, we use the same runs to estimate the gradients\n (see Remark~\\ref{rem:single_run} in Sec.~\\ref{sec:rl}).\n\\def\\PARAMS{For\nall algorithms, the learning rate is chosen using ADAM~\\cite{ADAM} with\ndefault hyper-parameters and the $\\alpha$ parameter of ADAM equal to $0.05$\nfor RMC, RMC-B, and the actor in SARSA-$\\lambda$ and the learning rate is equal to $0.1$ for the\ncritic in SARSA-$\\lambda$. For RMC and RMC-B, the policy parameters are\nupdated after $N=5$ renewals.}\nEach algorithm\\footnote{\\PARAMS} is run $100$ times and the mean and standard deviation of the\nperformance (as estimated by the algorithms themselves) is shown in\nFig.~\\ref{fig:GARNET-RL-train}. The performance of the corresponding policy\nevaluated by Monte-Carlo evaluation over a horizon of $250$~steps and averaged\nover $100$~runs is shown in Fig.~\\ref{fig:GARNET-RL-eval}. The optimal\nperformance computed using value iteration is also shown.\n\nThe results show that SARSA-$\\lambda$ learns faster (this is expected because\nthe critic is keeping track of the entire value function) but has higher\nvariance and gets stuck in a local minima. On the other hand, RMC and RMC-B\nlearn slower but have a low bias and do not get stuck in a local minima. The\nsame qualitative behavior was observed for other randomly generated models. Policy gradient algorithms only guarantee convergence to a local optimum. We are not sure why RMC and SARSA differ in which local minima they\nconverge to. Also, it was observed that RMC-B (which is RMC with biased evaluation of the gradient)\nlearns faster than RMC.\n\n\\begin{figure}[!t!b]\n  \\centering\n    \\begin{subfigure}{1.0\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{light_garnet_rl_train.pdf}\n    \\caption{}\n    \\label{fig:GARNET-RL-train}\n  \\end{subfigure}\n  \\begin{subfigure}{1.0\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{garnet_rl_eval.pdf}\n    \\caption{}\n    \\label{fig:GARNET-RL-eval}\n  \\end{subfigure}\n    \\caption{Performance of different learning algorithms on\n      $\\text{GARNET}(100,10,50)$ with $p=0.05$ and $\\discount=0.9$. \n      (a)~The performance estimated by the algorithms online. (b)~The\n      performance estimated by averaging over $100$ Monte Carlo evaluations\n    for a rollout horizon of $250$. The solid lines show the mean value and the shaded region shows the $\\pm$ one standard deviation region.}\n\\end{figure}\n\n\\subsection{Event-Triggered Communication} \\label{sec:rem_est}\n\n\\begin{figure}[!t!b]\n  \\centering\n  \\renewcommand\\unitlength{cm}\n  \\includegraphics[width=1.0\\linewidth]{rl_re.pdf}\n  \\caption{Policy parameters versus number of samples (sample values averaged over 100 runs) for event-driven\n    communication using RMC for different values of $p_d$. The solid lines\n  show the mean value and the shaded area shows the $\\pm$ one standard\n  deviation region.}\n  \\label{fig:RE}\n\\end{figure}\n\nIn this experiment, we study an event-triggered communication problem that arises in networked control systems~\\cite{LipsaMartins:2011,CSM:thresholds}. A transmitter observes a first-order autoregressive process $\\{X_t\\}_{t \\ge 1}$, i.e., $X_{t+1} =\n\\alpha X_t + W_t$, where $\\alpha, X_t, W_t \\in \\reals$, and $\\{W_t\\}_{t \\ge 1}$ is\nan i.i.d.\\ process. At each time, the transmitter uses an event-triggered\npolicy (explained below) to determine whether to transmit or not (denoted by\n$A_t = 1$ and $A_t = 0$, respectively). Transmission takes place over an\ni.i.d.\\ erasure channel with erasure probability $p_d$. Let $\\Prestate_t$ and\n$\\Poststate_t$ denote the ``error'' between the source realization and it's\nreconstruction at a receiver. It can be shown that $\\Prestate_t$ and\n$\\Poststate_t$ evolve as follows~\\cite{LipsaMartins:2011,CSM:thresholds}: when $A_t = 0$,\n$\\Poststate_t = \\Prestate_t$; when $A_t = 1$, $\\Poststate_t = 0$ if the\ntransmission is successful (w.p. $(1-p_d)$) and $\\Poststate_t = \\Prestate_t$ if\nthe transmission is not successful (w.p. $p_d$); \nand $\\Prestate_{t+1} = \\alpha \\Poststate_t + W_t$. Note that this is a post-decision state model, where the post-decision\nstate resets to zero after every successful transmission.\\footnote{Had we used\nthe standard MDP model instead of the post-decision state model,\nthis restart would not have always resulted in a renewal.}%\n\nThe per-step cost has two components: a communication cost of\n$\\lambda A_t$, where $\\lambda \\in \\reals_{> 0}$ and an estimation error \n$(\\Poststate_t)^2$. The objective is to minimize the expected discounted cost. \n\nAn event-triggered policy is a threshold policy that chooses $A_t = 1$\nwhenever $|\\Prestate_t| \\ge \\polPars$, where $\\polPars$ is a design choice.\nUnder certain conditions, such an event-triggered policy is known to be\noptimal~\\cite{LipsaMartins:2011,CSM:thresholds}. When the system model is known, algorithms to compute the optimal $\\polPars$ are presented\nin~\\cite{XuHes2004,CM:remote-estimation}. In this section, we use RMC to\nidentify the optimal policy when the model parameters are not known. \n\nIn our experiment we consider an event-triggered model with $\\alpha = 1$,\n$\\lambda = 500$, $p_d \\in \\{0, 0.1, 0.2\\}$, $W_t \\sim {\\cal N}(0, 1)$,\n$\\discount = 0.9$, and use simultaneous perturbation variant of RMC\\footnote{An\nevent-triggered policy is a parametric policy but $\\policy_\\polPars(\\action |\n\\prestate)$ is not differentiable in $\\polPars$. Therefore, the likelihood\nratio method cannot be used to estimate performance gradient.} to identify\n$\\polPars$. We run the algorithm 100 times and the result for different\nchoices of $p_d$ are shown in Fig.~\\ref{fig:RE}.\\footnote{We choose the\nlearning rate using ADAM with default hyper-parameters and the $\\alpha$\nparameter of ADAM equal to 0.01. We choose $c = 0.3$, $N=100$ and $\\Delta =\n\\mathcal{N}(0,1)$ in Algorithm~\\ref{alg:SPSA}.} For $p_d = 0$, the optimal\nthreshold computed using~\\cite{CM:remote-estimation} is also shown.\nThe results show that RMC converges relatively quickly and has low bias across multiple runs.\n\n\\subsection{Inventory Control} \\label{sec:inv_ctrl}\n\nIn this experiment, we study an inventory management problem that arises in operations\nresearch~\\cite{Arrow1951,Bellman1955}. Let $S_t \\in \\subset \\reals$ denote\nthe volume of goods stored in a warehouse, $A_t \\in \\reals_{\\ge 0}$ denote the\namount of goods ordered, and $D_t$ denotes the demand. The state evolves\naccording to $S_{t+1} = S_t + A_t - D_{t+1}$. \n\nWe work with the normalized cost function:\n\\[C(s) = a_p s (1-\\discount)\/\\discount + a_h s \\mathds{1}_{\\{ s \\ge 0\\}} - a_b s \\mathds{1}_{\\{s < 0\\}}, \\] \nwhere $a_p$ is the procurement cost, $a_h$ is the holding cost, and $a_b$ is the backlog cost (see~\\cite[Chapter 13]{Whittle1982}\nfor details).\n\nIt is known that there exists a threshold $\\theta$ such that the optimal\npolicy is a base stock policy with threshold $\\theta$ (i.e., whenever the\ncurrent stock level falls below $\\theta$, one orders up to $\\theta$).\nFurthermore, for $s \\le \\theta$, we have that~\\cite[Sec~13.2]{Whittle1982}\n\\begin{equation}\\label{eq:opt-IC}\n  V_\\polPars(s) = C(s) + \\frac{\\discount}{(1-\\discount)} \\EXP[C(\\polPars - D) ].\n\\end{equation}\nSo, for $B^\\rho \\subset (0, \\theta)$, the value function is locally Lipschitz, with\n\\[\n  L_\\polPars = \\left( a_h + \\frac{1 - \\discount} {\\discount} a_p \\right).\n\\]\nSo, we can use\napproximate RMC to learn the optimal policy.\n\nIn our experiments, we consider an inventory management model with $a_h = 1$, $a_b\n= 1$, $a_p = 1.5$, $D_t \\sim \\text{Exp}(\\lambda)$ with $\\lambda = 0.025$, start\nstate $s_0 = 1$, discount factor $\\discount = 0.9$, and use\nsimultaneous perturbation variant of approximate RMC to identify $\\theta$. \nWe\nrun the algorithm $100$ times and the result is shown in\nFig.~\\ref{fig:inv_ctl-RL}.\\footnote{We choose the learning rate using ADAM\nwith default hyper-parameters and the $\\alpha$ parameter of ADAM equal to\n0.25. We choose $c = 3.0$, $N=100$, and $\\Delta = \\mathcal{N}(0,1)$ in\nAlgorithm~\\ref{alg:SPSA} and choose $\\rho = 0.5$ for approximate RMC\\@. We bound the states within $[-100.0, 100.0]$.} The\noptimal threshold and performance computed using~\\cite[Sec 13.2]{Whittle1982}%\n\\footnote{For $\\text{Exp}(\\lambda)$ demand,\n  the optimal threshold is (see~\\cite[Sec 13.2]{Whittle1982})\n  \\[\\polPars^* = \\frac 1\\lambda \n    \\log\\left( \\frac{a_h + a_b}{a_h + a_p(1-\\gamma)\/\\gamma)} \\right).\\]\n}\nis also shown. \nThe result shows that RMC converges to an approximately optimal parameter value with total cost within the bound predicted in Theorem~\\ref{thm:approx_RMC}.\n\n\\begin{figure}[!t!b]\n  \\centering\n    \\begin{subfigure}{1.0\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{AN_1_inv_ctl_threshold_paper.pdf}\n    \\caption{}\n    \\label{fig:inv_ctl-RL_threshold}\n  \\end{subfigure}\n  \\begin{subfigure}{1.0\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{AN_1_inv_ctl_perf_paper.pdf}\n    \\caption{}\n    \\label{fig:inv_ctl-RL_perf}\n  \\end{subfigure}\n    \\caption{(a) Policy parameters and (b) Performance (total cost) versus\n      number of samples (sample values averaged over 100 runs) for inventory\n      control using RMC\\@. The solid lines show the mean value and the shaded area shows the $\\pm$ one standard\ndeviation region. In (b), the performance is computed using~\\eqref{eq:opt-IC} for the policy parameters given in (a). The red rectangular region shows the total cost bound given by Theorem~\\ref{thm:approx_RMC}.}\n\\label{fig:inv_ctl-RL}\n\\end{figure}\n\n\\section{Conclusions}\n\nWe present a renewal theory based reinforcement learning algorithm called\nRenewal Monte Carlo. RMC retains the key advantages of Monte Carlo methods and\nhas low bias, is simple and easy to implement, and works for models with\ncontinuous state and action spaces. In addition, due to the averaging over\nmultiple renewals, RMC has low variance. We generalized the\nRMC algorithm to post-decision state models and also presented a variant that converges faster to an approximately optimal policy, where the renewal state is replaced by a renewal set. The error in using such an approximation is bounded by the size of the renewal set.\n\nIn certain models, one is interested in the peformance at a reference state\nthat is not the start state. In such models, we can start with an arbitrary\npolicy and ignore the trajectory until the reference state is visited for\nthe first time and use RMC from that time onwards (assuming that the reference\nstate is the new start state).\n\nThe results presented in this paper also apply to average reward models where\nthe objective is to maximize\n\\begin{equation}\n  J_\\pi = \n  \\lim_{t_h \\to\n  \\infty}\\frac{1}{t_h}\\EXPA\\biggl[\\sum_{t=0}^{t_h-1}\\Reward_t\\biggm|\\State_0 =\n  \\state_0\\biggr]. \\label{eq:avg_Vp-defn}\n\\end{equation}\nLet the stopping times $\\tau^{(n)}$ be defined as before. Define the total reward\n$\\mathsf{R}^{(n)}$ and duration $\\mathsf{T}^{(n)}$ of the $n$-th regenerative cycle as\n\\[\n  \\mathsf{R}^{(n)} = \n  \\smashoperator[r]{\\sum_{t = \\tau^{(n-1)}}^{\\tau^{(n)} - 1}} R_t\n  \\quad\\text{and}\\quad\n  \\mathsf{T}^{(n)} =  \\tau^{(n)} - \\tau^{(n-1)}.\n\\]\nLet $\\mathsf{R}_\\polPars$ and $\\mathsf{T}_\\polPars$ denote the expected values of $\\mathsf{R}^{(n)}$\nand $\\mathsf{T}^{(n)}$ under policy $\\policy_{\\polPars}$. Then from standard renewal\ntheory we have that the performance $J_\\polPars$ is equal to \n$\\mathsf{R}_\\polPars\/ \\mathsf{T}_\\polPars$ and, therefore $\\GRAD_\\polPars J_\\polPars =\nH_\\polPars\/T^2_\\polPars$, where $H_\\polPars$ is defined as in~\\eqref{eq:H}. We\ncan use both variants of RMC prosented in Sec.~\\ref{sec:rl} to obtain\nestimates of $H_\\polPars$ and use these to update the policy parameters\nusing~\\eqref{eq:H-update}.\n\n\n\\section*{Acknowledgment}\n\nThe authors are grateful to Joelle Pineau for useful feedback and for suggesting the idea of approximate RMC.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfnbq b/data_all_eng_slimpj/shuffled/split2/finalzzfnbq
new file mode 100644
index 0000000000000000000000000000000000000000..88248c7c31d7467cac4498f970cae2c0fa18f7da
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzfnbq
@@ -0,0 +1,5 @@
+{"text":"\\section{CHIRAL SYMMETRY}\n\\label{s1}\n\nThe outer components of nuclear forces are dominated by pion-exchanges\nand involve just a few basic subamplitudes, describing pion interactions\nwith either nucleons or other pions.\nThe simplest process $N \\!\\rar\\! \\pi N$, corresponding to the emission or \nabsorption of a single pion by a nucleon, is rather well understood and \ngives rise to the one-pion exchange  $N\\!N$ potential ($OPEP$).\nThe scattering reaction $\\pi N \\!\\rar\\! \\pi N$ comes next and determines both \nthe very important two-pion exchange term in the $N\\!N$ force \nand the leading three-body interaction, as shown in Fig.\\ref{F1}.\n\n\\vspace{-4mm}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.7\\columnwidth,angle=0]{Robilotta_Osaka-01.eps}\n\\vspace{-3mm}\n\\caption{Free $\\pi N$ amplitude (a)\nand two-pion exchange two-body (b) and three-body (c) potentials.} \n\\label{F1}\n\\end{center}\n\\end{figure}\n\n\\vspace{-5mm}\n\nThe theoretical understanding of the $\\pi N$ amplitude proved\nto be very challenging and a suitable description was only produced \nby means of chiral symmetry.\nThis framework provides a  natural explanation for the observed \nsmallness of $\\pi N$ scattering lengths and plays a fundamental\nrole in Nuclear Physics.\nNowadays, the use of chiral symmetry in low-energy pion-interactions\nis justified by $QCD$.\n\nThe small masses of the quarks $u$ and $d$, treated as perturbations \nin a chiral symmetric lagrangian, give rise to a well defined chiral \nperturbation theory (ChPT).\nHadronic amplitudes are then expanded in terms of a typical \nscale $q$, set by either pion four-momenta or nucleon three-momenta, \nsuch that $q\\ll 1$ GeV.\nThis procedure is rigorous and many results have the status of {\\em theorems}.\nIn general, these theorems are written as power series \nin the scale $q$ and involve both {\\em leading order terms} and \n{\\em chiral corrections}.\nThe former can usually be derived from tree diagrams, whereas the\nlatter require the inclusion of pion loops and are the main object \nof ChPT. \nAt each order, predictions for a given process must be unique and the \ninclusion of corrections cannot change already existing leading terms.\n\nThe relationship between chiral expansions of the $\\pi N$ amplitude and of \ntwo-pion exchange $(TPE)$ nuclear forces is discussed in the sequence.\nFor the $\\pi N$ amplitude, tree diagrams yield ${\\cal{O}}(q, q^2)$ terms \nand corrections up to ${\\cal{O}}(q^4)$ have already been evaluated, by means of both \ncovariant\\cite{BL}(CF) and heavy baryon\\cite{FM}(HBF) formalisms. \nIn the case of the $N\\!N$ potential, the leading term is ${\\cal{O}}(q^0)$ and\ngiven by the $OPEP$.\nThe tree-level $\\pi N$ amplitude yields $TPE$ contributions at ${\\cal{O}}(q^2,q^3)$\nand corrections at ${\\cal{O}}(q^4)$ are available, based on both \nHBF\\cite{HB} and CF\\cite{HR,HRR}.\nTree-level $\\pi N$ results also determine the leading ${\\cal{O}}(q^3)$ three-body\nforce and partial corrections at ${\\cal{O}}(q^4)$ begin to be derived \n\\cite{IR07,E3NP}.\nAs this discussion suggests, ${\\cal{O}}(q^4)$ corrections to both\ntwo- and three-nucleon forces require just the ${\\cal{O}}(q^3)$\n$\\pi N$ amplitude.\n\nThe full empirical content of the $\\pi N$ amplitude cannot be predicted\nby chiral symmetry alone. \nExperimental information at low energies is usually encoded into the \nsubthreshold coefficients introduced by H\\\"ohler and collaborators\\cite{H83}\nwhich can, if needed, be translated into the low-energy contants (LECs)\nof chiral lagrangians. \nTherefore, in order to construct a ${\\cal{O}}(q^3)$ $\\pi N$ amplitude,\none uses chiral symmetry supplemented by \nsubthreshold information, as indicated in Fig. \\ref{F2}.\nThe first two diagrams correspond to the nucleon pole, \nwhereas the other ones represent a smooth background.\nThe third graph reproduces the Weinberg-Tomozawa contact interaction,\nthe fourth one summarizes LEC contributions and the last two describe \nmedium range pion-cloud effects.\n\n\\vspace{-5mm}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.8\\columnwidth,angle=0]{Robilotta_Osaka-02.eps}\n\\vspace{-4mm}\n\\caption{Representation of the $\\pi N$ amplitude at ${\\cal{O}}(q^3)$.} \n\\label{F2}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{TWO-BODY POTENTIAL}\n\\label{s2}\n\nWith the purpose of discussing the problem of predicted \n$\\times$ observed chiral hierarchies, in this section we review \nbriefly results obtained by our goup\\cite{HR,HRR} for the \n$TPE$-$N\\!N$ potential at ${\\cal{O}}(q^4)$.\nThis component is determined by the three families of diagrams\nshown in Fig. \\ref{F3}. \nFamily $I$ begins at ${\\cal{O}}(q^2)$ and implements the minimal realization of \nchiral symmetry\\cite{RR94},\nwhereas family $I\\!I$ depends on $\\pi\\p$ correlations and is ${\\cal{O}}(q^4)$.\nThey involve only the constants $g_A$ and $ f_\\pi$\nand all dependence on the LECs is concentrated in family $I\\!I\\!I$,\nwhich begins at ${\\cal{O}}(q^3)$.\n\n\\vspace{-2mm}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.7\\columnwidth,angle=0]{Robilotta_Osaka-03.eps}\n\\caption{Dynamical structure of the two-pion exchange potential.} \n\\label{F3}\n\\end{center}      \n\\end{figure}\n\n\\vspace{-5mm}\n\nAs far as chiral orders of magnitude are concerned, on finds that \nthe various components of the force begin as follows\\cite{HRR}:\n${\\cal{O}}(q^2)\\rightarrow V_{SS}^+,  V_T^+, V_C^-$ and \n${\\cal{O}}(q^3)\\rightarrow V_C^+, V_{LS}^+, V_{LS}^-, V_{SS}^-, V_T^-$,\nwhere the superscripts $(+)$ and $(-)$ refer to terms proportional \nto either the identity or $\\mbox{\\boldmath $\\tau$}^{(1)}\\!\\cdot\\! \\mbox{\\boldmath $\\tau$}^{(2)}$ in isospin space.\nAn interesting feature of these results is that the role played by \nfamily $I\\!I$ is completely irrelevant.\nOn the other hand, family $I$ dominates almost completely the components\n$V_{LS}^+$, $V_T^+$, $V_{SS}^+$ and $V_C^-$, \nwhereas family $I\\!I\\!I$ does the same for $V_C^+$, $V_T^-$and $V_{SS}^-$.\n\n\\vspace{-4mm}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.49\\columnwidth,angle=0]{Robilotta_Osaka-04a.eps}\n\\includegraphics[width=0.49\\columnwidth,angle=0]{Robilotta_Osaka-04b.eps}\n\\vspace{-3mm}\n\\caption{$OPEP$ and $TPEP$ contributions to \nspin-spin (left) and tensor (right) NUisovector components.} \n\\label{F4}\n\\end{center}      \n\\end{figure}\n\n\\vspace{-5mm}\n\nThe relationship between the $OPEP[={\\cal{O}}(q^0)]$ and $TPEP[={\\cal{O}}(q^3)]$ \ncontributions to the $V_{SS}^-$ and $V_T^-$ profile functions is \nshown in Fig. \\ref{F4}, where it is possible to see that the chiral\nhierarchy is respected.\n\n\nIn Fig. \\ref{F5}, the two central components $V_C^-[={\\cal{O}}(q^2)]$ and \n$V_C^+[={\\cal{O}}(q^3)]$ are displayed side by side and two features \nare to be noted.\nThe first one concerns the favorable comparison with the empirical\nArgonne\\cite{Arg} potentials in both cases.\nThe second one is that $|V_C^+| \\sim 10 \\, |V_C^-|$ in  regions of physical\ninterest, defying strongly the predicted chiral hierarchy.\nThis problem will be further discussed in the sequence.\n\n\\vspace{-3mm}\n\n\\begin{figure}[h]\n\\begin{center}\n\\hspace*{-4mm}\n\\includegraphics[width=0.50\\columnwidth,angle=0]{Robilotta_Osaka-05a.eps}\n\\includegraphics[width=0.50\\columnwidth,angle=0]{Robilotta_Osaka-05b.eps}\n\\vspace{-2mm}\n\\caption{Isospin odd (left) and even (right) central components of the \ntwo-pion exchange potential.} \n\\label{F5}\n\\end{center}      \n\\end{figure}\n\n\\vspace{-4mm}\n\nViolations of the chiral hierarchy are also present in the \n{\\em drift potential}\\cite{Rdrift}, which corresponds to kinematical\ncorrections due to the fact that the two-body center of mass is allowed to \ndrift inside a larger system. \nIn terms of Jacobi coordinates, it is represented by the operator\n\\begin{eqnarray}\nV(r)^\\pm  =  \\left. V(r)^\\pm \\right] _{cm} + V_D^\\pm \\, \\Omega_{D}\n\\;\\;\\;\\;\\;\\;\\;\\; \\leftrightarrow \\;\\;\\;\\;\\;\\;\\;\\;\n\\Omega_D = \\frac{1}{4\\sqrt{3}}\\, (\\mbox{\\boldmath $\\sigma$}^{(1)}\\!-\\! \\mbox{\\boldmath $\\sigma$}^{(2)}) \\!\\cdot\\! \\mbox{\\boldmath $r$} \\!\\times \\! \\;,\n(-i \\mbox{\\boldmath $\\nabla$}^{^{\\!\\!\\!\\!\\!\\!\\!\\!^\\leftrightarrow}}_\\rho)\\;.\n\\nonumber\n\\end{eqnarray}\n\n\nThe profile function $V_D^+$ together with $V_{LS}^+$,  are \ndisplayed in Fig. \\ref{F6}.\nDrift corrections begin at ${\\cal{O}}(q^4)$ and, in principle, should be smaller \nthan the spin-orbit terms, which begin at ${\\cal{O}}(q^3)$.\nHowever, in this channel, the hierarchy is again not respected.\n\n\\vspace{20mm}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.7\\columnwidth,angle=0]{Robilotta_Osaka-06.EPS}\n\\vspace{-35mm}\n\\caption{Isospin even drift (full and dotted lines)\nand spin-orbit (dashed line) potentials.} \n\\label{F6}\n\\end{center}      \n\\end{figure}\n\n\n\n\n\\section{THREE-BODY POTENTIAL}\n\\label{s3}\n\nThe leading term in the three-nucleon potential, known as $TPE$-$3NP$,\nhas long range and corresponds to the process shown in fig.1c,\nin which a pion is emitted by one of the nucleons, scattered by a second one, \nand absorbed by the last nucleon.\nIs this case, the intermediate $\\pi N$ amplitude, which is ${\\cal{O}}(q)$ for \nfree pions, becomes ${\\cal{O}}(q^2)$ and the three-body force begins at \n${\\cal{O}}(q^3)$.\nThe first modern version of this component of the force was produced by \nFujita and Miyazawa\\cite{F-M}, its chiral structure has bee much debated\nsince the seventies\\cite{3NP} and, nowadays, a sort o consensus has been \nreached about its form\\cite{Rtokyo}. \nThe leading $TPE$-$3NP$ has a generic structure given by\n\\begin{eqnarray}\n&& V_L(123) = -\\,\\frac{\\mu}{(4\\pi)^2}  \n\\left\\{  \\delta_{ab} \\left[  a \\, \\mu - b \\, \\mu^3 \\, \\mbox{\\boldmath $\\nabla$}_{12} \\!\\cdot\\! \\mbox{\\boldmath $\\nabla$}_{23} \\right]  \n+ d\\, \\mu^3 \\; i\\,\\epsilon_{bac} \\tau_c^{(2)}\\;\ni \\, \\mbox{\\boldmath $\\sigma$}^{(2)} \\!\\cdot\\! \\mbox{\\boldmath $\\nabla$}_{12} \\times \\mbox{\\boldmath $\\nabla$}_{23}\\right\\} \n\\nonumber\\\\\n&& \\;\\;\\;\\;\\; \\times \n\\left[  (g_A \\, \\mu\/2 \\,f_\\pi)\\;\\tau_a^{(1)} \\;\\mbox{\\boldmath $\\sigma$}^{(1)} \\!\\cdot\\! \\mbox{\\boldmath $\\nabla$}_{12} \\right] \\;\n\\left[  (g_A \\, \\mu\/2 \\,f_\\pi)\\;\\tau_b^{(3)} \\;\\mbox{\\boldmath $\\sigma$}^{(3)} \\!\\cdot\\! \\mbox{\\boldmath $\\nabla$}_{23} \\right] \\;\nY(x_{12}) \\; Y(x_{23}) \\;,\n\\nonumber\n\\end{eqnarray}\n\n\\noindent\nwhere $\\mu$ is the pion mass and $a$, $b$ and $d$ are strength parameters,\ndetermined by either LECs or subhtreshold coefficients. \n\nThe evaluation of ${\\cal{O}}(q^4)$ corrections requires the inclusion of single\nloop effects and is associated with a large number of \ndiagrams, which are being calculated by Epelbaum and \ncollaborators\\cite{E3NP}.\nIn order to produce a feeling for the structure of these corrections,\nwe discuss a particular set of processes belonging to the \n$TPE$-$3NP$ class, considered recently\\cite{IR07}.\nFull results involve expressions which are too long and cumbersome\nto be displayed here.\nHowever, their main qualitative features can be summarized in the\nstructure  \n$V(123)=V_L(123)+ [V_{\\delta L}(123) + \\delta V(123)]$, \nwhere $V_L$ is the leading term shown above and the factors within \nsquare brackets are ChPT corrections.\nThe function $V_{\\delta L}$ can be obtained directly from $V_L$, by replacing \n$(a,b,c) \\rightarrow (\\delta a,\\delta b, \\delta c)$, where \nthe $\\delta s$ indicate changes smaller than $10\\%$.\nThis part of the ChPT correction corresponds just to shifts in the \nparameters of the leading component.\nThe term $\\delta V(123)$, on the other hand, represents effects associated \nwith new mathematical functions involving both non-local operators \nand complicated propagators containg loop integrals,\nin place of the Yukawa functions.\nThe strengths  of these new functions are determined by a new set of \nparameters $e_i$, which are also typically about $10\\%$ of the \nleading ones.\n\nIn summary, ChPT gives rise both to small changes in already existing \ncoefficients and to the appearance of many new mathematical structures.\nThe latter are the most interesting ones, since they may be instrumental\nin explaining effects such as the $A_y$ puzzle.\n\n\n\n\\section{THE CHIRAL PICTURE}\n\\label{s4}\n\nChiral symmetry has already been applied to about 20 components \nof nuclear forces, allowing a comprehensive picture to be assessed.\nAccording to ChPT, the various effects begin to appear at different\norders and the predicted hierarchy is displayed in the table below.\n\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular} {|c|ccc|}\n\\hline\nbeginning\t  \t& TWO-BODY & TWO-BODY\t& THREE-BODY \t   \\\\ \n\t\t\t\t& $OPEP$   & $TPEP$\t\t& $TPEP$\t\t   \\\\ \\hline \n${\\cal{O}}(q^0)$\t\t& $V_T^-, V_{SS}^-$\t&&    \t  \t   \\\\[1mm] \\hline\n${\\cal{O}}(q^2)$\t\t& $V_D^-$ & $V_C^-; V_T^+, V_{SS}^+$ &\t   \\\\[1mm]\\hline\n${\\cal{O}}(q^3)$\t\t&& $V_{LS}^-, V_T^-, V_{SS}^-; V_C^+, V_{LS}^+$ \n& $d; a, b$ \t\\\\[1mm] \\hline\n${\\cal{O}}(q^4)$\t\t&& $V_D^-; V_Q^+, V_D^+$ & $ e_i $ \\\\[1mm] \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nIn Ref.\\refcite{HRR}, the relative importance of $O(q^2)$, $O(q^3)$ \nand $O(q^4)$ terms in each component of the $TPEP$-$N\\!NP$ has been studied.\nIn general, convergence at distances of physical interest is \nsatisfactory, except for $V_C^+$, where the ratio between \n${\\cal{O}}(q^4)$ and ${\\cal{O}}(q^3)$ contributions \nis larger than $0.5$ for distances smaller than $2.5$ fm.\n\nAs far as the relative sizes of the various dynamical effects are\nconcerned, one finds strong violations of the predicted hierarchy\nwhen one compares $V_C^+$ with $V_C^-$ and $V_D^+$ with $V_{LS}^+$,\nas discussed above. \nIt is interesting to note that, in both cases,\nthe unexpected enhancements occur in the isoscalar sector. \nThe numerical explanation for this behavior is that some of the LECs \nused in the calculation are large and\ngenerated dynamically  by delta intermediate states.\nHowever, it is also possible that perturbation theory may not apply \nto isoscalar interactions at intermediate distances.\nThis aspect of the problem is explored in the next section.\n\n\n\n\n\n\\section{SCALAR FORM FACTOR}\n\\label{s5}\n\n\n\nThe structure of $V_C^+$ was  scrutinized in Ref.\\refcite{HRR}\nand  found to be heavily dominated by a term of the form\n\\begin{eqnarray}\nV_C^+(r) \\sim -\\, (4\/f_\\pi^2)\\;\\left[  (c_3 - 2c_1) - c_3 \\; \\mbox{\\boldmath $\\nabla$}^2\/2 \\right]  \\;\n\\tilde{\\sigma}_{N_N}(r)\\;,\n\\nonumber\n\\end{eqnarray}\nwhere the $c_i$ are LECs and $\\tilde{\\sigma}_{N_N}$ is the leading contribution \nfrom the pion cloud to the nucleon scalar form factor.\nThis close relationship between $\\tilde{\\sigma}_{N_N}$ and $V_C^+$ indicates \nthat the study of the former can shed light into the properties of the \nlatter.\n\nThe nucleon scalar form factor is defined as \n\\begin{eqnarray}\n\\langle  N(p') | \\!-\\! {\\cal{L}}_{sb}\\, | N(p) \\rangle  = \\sigma_N(t) \\; \\bar u(p')\\; u(p) \\;,\n\\nonumber\n\\end{eqnarray}\n\n\\noindent\nwhere ${\\cal{L}}_{sb}$ is the symmetry breaking lagrangian. \nIt has already been expanded\\cite{BL} up to ${\\cal{O}}(q^4)$ and receives its leading \n${\\cal{O}}(q^2)$ contribution from a tree diagram associated with the LEC $c_1$.\nCorrections at ${\\cal{O}}(q^3)$ and ${\\cal{O}}(q^4)$ are produced by two triangle diagrams, \ninvolving nucleon and delta intermediate states.\nIn configuration space\\cite{sigma}, the scalar form factor is denoted\nby $\\tilde{\\sigma}$ and one writes \n\\begin{eqnarray}\n\\tilde{\\sigma}_N(\\mbox{\\boldmath $r$}) = - 4\\, c_1\\, \\mu^2\\, \\delta^3(\\mbox{\\boldmath $r$}) + \n\\tilde{\\sigma}_{N_N}(r) + \\tilde{\\sigma}_{N_\\Delta}(r) \\;,\n\\nonumber\n\\end{eqnarray}\n\n\\noindent\nwhere $\\tilde{\\sigma}_{N_N}$ and $\\tilde{\\sigma}_{N_\\Delta}$ are the finite-range \ntriangle contributions. \n\n\\begin{figure}[t]\n\\begin{center}\n\\hspace*{-4mm}\n\\includegraphics[width=0.35\\columnwidth,angle=-90]{Robilotta_Osaka-07a.ps}\n\\includegraphics[width=0.35\\columnwidth,angle=-90]{Robilotta_Osaka-07b.ps}\n\\caption{Ratios $\\tilde{\\sigma}_N(r)\/(\\mu^2 f_\\pi^2)=(1-\\cos\\theta)$ (left)\nand $\\tilde{\\sigma}_{N_\\Delta}(r)\/\\tilde{\\sigma}_{N_N}(r)$ (right) as functions of \nthe distance $r$.}\n\\label{F7}\n\\vspace{-3mm}\n\\end{center}      \n\\end{figure}\n\nThe symmetry breaking lagrangian can be expressed in terms of the chiral angle \n$\\theta$ as ${\\cal{L}}_{sb} = f_\\pi^2 \\, \\mu^2 \\,(\\cos\\theta-1)$.\nThe ratio $\\tilde{\\sigma}_N(r)\/(\\mu^2 f_\\pi^2)=(1-\\cos\\theta)$ describes the \ndensity of the $q\\bar{q}$ condensate around the nucleon and is \ndisplayed in Fig.~\\ref{F7}.\nOne notes that it vanishes at large distances and increases monotonically \nas one approaches the center.\nThis means that the function $\\tilde{\\sigma}_N(r)$ becomes meaningless beyond \na critical radius $R$, corresponding to $\\theta = \\pi\/2$,\nsince the physical interpretation of the quark condensate \nrequires the condition $q\\bar{q}>0$.\nIn Ref. \\refcite{sigma}, the condensate was assumed to no longer exist in \nthe region $r<R$ and the $\\pi N$ sigma-term was evaluated using the expression\n\\begin{eqnarray}\n\\sigma_N = \\frac{4}{3} \\pi R^3 \\;f_\\pi^2 \\mu^2\n+ 4\\pi \\int_{R}^\\infty dr\\;r^2\\;\\tilde{\\sigma}_N(\\mbox{\\boldmath $r$})\\;.\n\\nonumber\n\\end{eqnarray}\n\nThis procedure yields 43 MeV$<\\sigma_N<\\;$49 MeV, depending on the value \nadopted for the $\\pi N\\Delta$ coupling constant, in agreement  \nwith the empirical value 45$\\pm 8$ MeV.\nThis picture of the nucleon scalar form factor is sound \nand can be used to gain insight about $V_C^+$.\n\nInspecting Fig. \\ref{F7} (right), one learns that the hierarchy predicted \nby ChPT is subverted for distances smaller than 1.5 fm, since the \n${\\cal{O}}(q^4)$ delta becomes more important than the ${\\cal{O}}(q^3)$ nucleon.\nOn the other hand, the good prediction obtained for the nucleon $\\sigma$-term \n(and also for the $\\Delta$ $\\sigma$-term\\cite{sigma}) \nindicates that the functions $\\tilde{\\sigma}_{N_N}(r)$ and $\\tilde{\\sigma}_{N_\\Delta}(r)$\ncan be trusted up to the critical radius $R\\sim$~0.6~fm.\nJust outside this radius, the chiral angle is close to $\\pi\/2$,\nindicating that the pion cloud is non-perturbative in that region.\nThis picture is supported by Fig. \\ref{F5} (right) since, \nat least up to 1 fm, the prediction for $V_C^+$ agrees well with \nthe Argonne phenomenological potentials.\nThis leads to our main conclusion, namely that the range of validity of \ncalculations based on nucleon and delta intermediate states is wider that \nthat predicted by ChPT.\n\n\n\\section*{Acknowledgments}\n\nIt was a great pleasure participating in the Chiral 2007 meeting\nand I would like to thank the organizers for the very nice conference,\nfor the warm and friendly hospitality, and for supporting my \nstay in Osaka.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nDark matter makes up $80$\\% of the total mass budget in the Universe,\nbut its exact nature remains doggedly elusive. Without direct\ndetection of a single dark matter particle we have to infer its\nproperties from its macroscopic distribution, and any observable that\nmight hint at its underlying particulate nature is seized upon with\nvigour. A plethora of observations, such as large scale\nstructures, lensing and scaling relations, and early fluctuations on\nthe Cosmic Microwave Background, agree surprisingly well with\npredictions of a rather simple cosmological model with a cosmological\nconstant, denoted as $\\Lambda$, and a collisionless cold dark matter\n(CDM) component~\\citep[e.g.][]{Ade:2013zuv}. This {$\\Lambda$CDM} ~paradigm has\nbecome the standard and most widely accepted model today for the\nobserved Universe.\n\n{$\\Lambda$CDM}\\ makes testable predictions at galactic scales that can be\ncontrasted with observations. Among the most fundamental of them is\nfor the structure of dark matter halos, with a density profile\nparameterised by a Navarro-Frenk-White (NFW)\nprofile~\\citep{Navarro_1997}, which at the inner radii scales as a power\nlaw $r^{-\\alpha}$ with $\\alpha\\approx1$. Observations, however, of the\nrotation curves of spiral galaxies, including dwarf and low surface\nbrightness galaxies, often exhibit an inner circular velocity of stars and\ngas that increases more mildly than expected from a CDM\nhalo~\\citep[e.g.\n][]{Flores:1994gz,Moore:1994yx,Persic:1995ru,Kuzio_2008,Oh_2015},\nindicating an inner density profile shallower than the NFW cusp, i.e.,\n$\\alpha<1$. This discrepancy can be further generalised as the mass\ndeficit problem: the CDM halo contains too much dark matter mass in\nthe inner regions than inferred from\nobservations~\\citep{MBK_2011,Ferrero_2012, Papastergis_2015, Papastergis2016, Schneider_2016}.\n\nA more intriguing observation is that the galactic rotation curves exhibit a large diversity.\nIndividual fits to galaxy rotation curves \nspan a spectrum from \ncores $\\alpha\\approx 0$ \\citep[e.g.][]{Cote2000,deBlok_2008,deNaray:2009xj} to cusps \\citep[e.g.\n][]{vandenbosch2000,Swaters_2003, Spekkens2005}, and in the case\nof cored profiles, the central densities can differ by factor of $10$\nfor galaxies inhabiting similar halos~\\citep{deNaray:2009xj}.\nRecently, \\cite{Oman_2015} quantified this diversity in rotation\ncurves by comparing ${V_{2\\,\\rm kpc}}$ for a fixed ${V_{\\rm max}}$, where ${V_{2\\,\\rm kpc}}$ is the\ncircular velocity at $2~{\\rm kpc}$ and ${V_{\\rm max}}$ is the peak circular\nvelocity. For ${V_{\\rm max}}$ in the range of $50\\textup{--}250\\;{\\rm km\\, s^{-1}}$,\nthe scatter in ${V_{2\\,\\rm kpc}}$ is a factor of $3\\textup{-}4$ and consequently\nthe mechanism invoked to generate cored profiles must also accommodate\nthis large variations in rotation curve shapes.\nNotably mass modelling is subject to significant uncertainties,\nespecially at the scale of low mass dwarfs where pressure support\neffects, triaxiality, inclination, hidden bars and other\nirregularities hamper the utility of\ncircular velocity profiles for mass modelling\n\\citep{Hayashi2004,Rhee2004,Pineda_2016,Read2016},\nmaking it hard to assess the precise spread in the rotation curves. In this paper, we take the result reported in Oman et al. as our reference.\n\nThe distribution of rotation curves is clearly too heterogeneous to be \nindexed with a single parameter. For example, several different galaxy formation\nmodels seem able to reproduce the Tully-Fisher \\citep{TullyFisher} relation within\n{$\\Lambda$CDM}\\ \\citep{McCarthy_2012,Brook_2016,\n  DiCintio_2016, Sales_2016, Ferrero_2016, Santos-Santos2016,\n  Katz:2016hyb}  provided\nthe stellar feedback populates dark matter halos with the right\nstellar mass and size \\citep[though see also][]{Pace:2016oim},\nyet despite this global consensus the circular velocity profiles\npredicted by each model differ in detail, with some cases leading to the\nformation of cores \\citep[e.g.][]{Governato_2010,\n  Pontzen_2012, DiCintio_2014,Read:2015sta, Wetzel_2016} whereas other\nsimulations preserve the inner dark matter cusps\n\\citep[e.g.][]{Sawala:2014baa, Vogelsberger_2014,Schaller2015}. The\ndisagreement may arise from the inclusion of different physical\nprocesses, different numerical implementation or may even vary with\nstar formation histories~\\citep[e.g.][]{Onorbe2015, Tollet2016}.\n\nIn particular, the feedback model applied by \\citet{Oman_2015} appears\nunable to explain the scatter in observations within the $\\Lambda$CDM\nframework. On the other hand, \\citet{Brook2015b} finds good agreement\nbetween simulations and observations when looking at the scatter in\nthe ratio between the circular velocity at $1$ kpc and ${V_{\\rm max}}$.\nSimilarly \\citet{Read2016} has also found consistency between baryon\ninduced cores within $\\Lambda$CDM and the shape of observed rotation\ncurves. It is unclear, however, that this baryonic solution would hold if the\nsize of the cores measured in observations is $\\gtrsim 1$~kpc\n\\citep[see for instance][]{Tollet2016}. In fact, this may represent a\nserious limitation of this solution. For example IC~2574\nhas an inferred cored inner halo which extends for $8 \\; \\rm kpc$,\nwell beyond the radius of the stars, with a stellar half-mass radius\nof $5\\; \\rm kpc$. This type of object is a challenge to the hypothesis\nof feedback generated cores since, by construction, the extent of the\ncores in such scenarios is limited to the region where stars form and deposit their\nenergy~\\citep{DiCintio_2014}.\n\nAn alternative solution is to consider cores formed out of\nself-interacting dark matter (SIDM), which we explore in this paper.\nThe SIDM model retains important features of the CDM model\nincluding the distribution of halo concentrations as a function of mass  \\citep{Rocha:2012jg}. \nSIDM differs, however, in that scattering between dark matter particles leads to heat transfer and the generation of dense cores in the\ninner regions of halos ~\\citep{Spergel:1999mh,Firmani:2000ce}.\nRecent\nhigh-resolution N-body simulations have shown that the\nself-interaction cross-section per unit mass $\\som\\sim1~{\\rm cm^2\\, g^{-1}}$\nis required to have core densities that are preferred by galaxy\nobservations\n\\citep{Vogelsberger_2012,Zavala:2012us,Rocha:2012jg,Elbert:2014bma}.\n\\cite{Kaplinghat_2016} finds $\\som\\approx1\\textup{--}3~{\\rm cm^2\\, g^{-1}}$\nby directly fitting the rotation curves of dwarf galaxies. On the\nother hand, there are various constraints on $\\som$, including merging\nclusters~\\citep{Randall_2008,Kahlhoefer:2013dca,Robertson_2016,Kim:2016ujt},\nshapes of elliptical galaxies and galaxy\nclusters~\\citep{MiraldaEscude_2002,Feng:2009hw,Peter_2013}, core sizes of\nclusters~\\citep{Yoshida_2000,Rocha:2012jg,Kaplinghat_2016,Elbert_2016},\nand survival of dwarf halos from evaporation~\\citep{Gnedin_2001}.\n\nAmong these constraints the strongest limit is $\\som \\lesssim 0.1\\;{\\rm cm^2\\, g^{-1}}$ in\ngalaxy clusters~\\citep{Yoshida_2000,Kaplinghat_2016,Elbert_2016},\nwhere the relative dark matter velocity $v\\sim1500~{\\rm km~s^{-1}}$.\nAs such, the self-interaction cross-section should have a mild velocity\ndependence, which can be naturally realised in a class of hidden\nsector dark matter models with a light force\nmediator~\\citep[e.g.][]{Feng:2009mn,Feng:2009hw,Buckley:2009in,Loeb:2010gj,Tulin:2013teo,Tulin:2012wi,Boddy:2014yra,Boddy:2016bbu}.\nA velocity-dependent SIDM model, based a Yukawa potential, has been\nimplemented in zoom-in N-body\nsimulations~\\citep{Vogelsberger_2012,Vogelsberger:2015gpr}, and \nfurther generalised in the ETHOS (Effective Theory of\nStructure Formation) framework, mapping\nunderlying particle physics parameters to astrophysical\nobservables~\\citep{Vogelsberger:2015gpr,Cyr-Racine:2015ihg}. \n\nThis approach has several distinct features that help explain the\nrotation curve diversity. Firstly, the\nscatter in the halo concentration leads to variations in core density\ndirectly~\\citep{Kaplinghat_2016}. Secondly, the self-interactions\nthermalise the inner halo with its central dark matter density\n\\emph{dependent} upon the baryonic extent. In dark matter dominated\ngalaxies, a dense core forms whose density is determined by the\nself-interaction cross-section and the halo mass. With a cored profile, the features in\nthe baryon distribution are more prominently reflected in the rotation\ncurves. In contrast, in galaxies where the baryon component dominates\nthe potential, the thermalisation process lead to a denser central\ncore with sizes influenced by the baryonic scale\nradius~\\citep{Kaplinghat_2014}. Analytical calculations to address\nthe diversity problem have been carried out by \\citet{Kamada2016} using isothermal\napproximations to the effects of SIDM with a baryonic potential in\norder to construct fits for a diverse range of individual spiral\ngalaxies.\n\nIn this work we explore the combined effects of SIDM and baryonic\npotentials by performing a series of numerical experiments.\nOur focus is on global trends but we include a\npair of individual fits to test our results on the strongest outliers\nfrom observations. We sample a realistic range of concentrations and\nhalo masses taken from cosmological simulations together with observed\ntrends in the mass-size relation of galaxies. Our numerical approach\nand the analytical one presented in~\\citet{Kamada2016} complement each\nother in addressing the diversity problem in the SIDM model. The\nstructure of this paper is as follows. In Section~\\ref{sec:sims} we\nintroduce in detail our simulations. Sections.~\\ref{sec:V2_Vmax_dm} and\n\\ref{sec:V2_Vmax_bar} explore the diversity problem for a large sample\nof halos with ${V_{\\rm max}}=[30$-$250]$~km~s$^{-1}$ with the latter including the effect\nof baryons. In Sec.~\\ref{sec:obs_curve_sims} we\nconfirm that we can find rotation curves that are reasonable matches\nto a pair of extreme (one cusp-like and one core-like) observed\nrotation curves. In Sec.~\\ref{sec:discussion} we summarise and\nconclude.\n\n\n\n\\section{Numerical simulations}\\label{sec:sims}\n\nWe use a combination of N-body simulations and analytical tools to\nstudy the mass profile of galaxies within the SIDM framework including\nthe effects driven by their baryons. The main focus of this work is on\ndisk dominated galaxies, which we model with a disk\ncomponent embedded within a massive and more extended dark matter halo\n(i.e. no bulge). For the N-body simulations we must produce\ndiscretised initial conditions which we describe for the halo in\nSection~\\ref{sec:bar_ic} in addition to the baryonic potential in\n\\ref{sec:diskpot}. We evolve these initial conditions with and without\nthe effect of a baryonic disk and also with and without\nself-interaction terms using the simulation code described in\nSection~\\ref{sec:nbody}.\n\n\n\\subsection{N-body code}\\label{sec:nbody}\nWe use the {\\sc Arepo} code \\citet{Springel_2010} with the modifications of \n\\citet{Vogelsberger_2012} and \\citet{Vogelsberger:2015gpr} to account for dark matter self-interactions.\nThis methodology uses a Monte-Carlo approach\nto model the scattering of dark matter via the probabilistic\nscattering of the macroscopic dark matter particles in the simulation,\nwhere the density distribution due to an individual particle is\nsmoothed over some kernel which extends over the $k$-nearest\nneighbours. Such an explicit method requires a time step limit $\\Delta\nt_{\\rm sidm}$ such that only $\\lesssim 1$ scattering can occur per\nparticle per timestep. Notably in the cusp region of an NFW halo (i.e.\n$\\rho \\sim r^{-1}$ as $r\\to 0$) as one moves towards the centre one\nfinds that $\\Delta t_{\\rm sidm} \\propto \\frac{1}{\\rho \\sigma} \\sim\nr^{\\frac{1}{2}}$ (with $\\sigma$ the velocity dispersion),\n i.e the centre of the halo is so dynamically cold\nand dense that the mean-free path is below the particle separation. To\navoid such small time steps one additionally needs to limit them to a\nsmall fraction of the acceleration time step.\n\n\nWe have modified this code to include a baryonic potential of a\nMiyamoto-Nagai (hereafter MN) disk (Section~\\ref{sec:diskpot}) and\nused a constant self-interaction cross-section for scattering, which\nis a good approximation in the dark matter low velocity limit, e.g.,\n$v\\lesssim 400 \\; {\\rm km\\, s^{-1}}$ as shown in Fig.~1 of \\citet{Kaplinghat_2016}.\n\nIn our calculations we assume a Hubble constant\n$H_0 = 70\\; {\\rm km\\, s^{-1}}\\,{\\rm Mpc}^{-1}$.  Our simulation period is fixed at\n$10$ Gyr, slightly shorter than the Hubble time\n($H_0^{-1} \\approx 13.96$~Gyr) in order to account, in an approximate\nway, for the assembly time of such galactic disks. It should be\nemphasised that isolated simulations with a static disk potential is\nwell-motivated in the SIDM model; for $\\som\\gtrsim1~{\\rm cm^2\/g}$,\ndark matter self-interactions occur multiple times in the inner halo\nover the age of galaxies, which drive thermalisation of the inner halo\nin the presence of the stellar disk, and as such the final inner SIDM\nhalo profile is more robust to changes in the formation history (given\na known baryon distribution) than that of CDM.\n\n\\subsection{Isolated halo initial conditions}\\label{sec:bar_ic}\n\nFor each of our galaxies, we generate a large set of compound galaxies\nconsisting of an exponential disk embedded in a dark matter halo using\nthe code {\\sc MakeDisk} \\citep{Hernquist_1993, Springel_2005}. The\nsystems are set up in near equilibrium by requiring a joint solution\nto the combined phase space of both disk and halo component.  In\nparticular, the density profile of the halo is set up with a\n\\citet{Hernquist_1990} density profile:\n\\begin{equation}\n\\rho_{\\rm dm}(r) = \\frac{M_{\\rm dm} r_{\\rm hq}}{2\\pi r \\left(r+r_{\\rm hq}\\right)^3} \\, ,\n\\end{equation}\nwith $r_{\\rm hq}$ the scale radius of the halo. \\citet{Hernquist_1990} profiles\nshow a $r^{-1}$ slope in the inner density profile and $r^{-4}$ in the\noutskirts, in close agreement with the inner\/outer slopes of $-1$ and\n$-3$, respectively, of the cosmologically inspired NFW\nprofiles \\citep{Navarro_1997}. The Hernquist profile has the extra\nadvantage of having an analytic expression for its distribution\nfunction, which facilitates the process of setting up the initial\nconditions.\n\n\nHernquist halos can be ``scaled'' to match a given NFW\nprofile quite closely and we therefore report our results in terms of\nvirial mass\\footnote{Virial quantities are defined at the virial\n  radius $r_{200}$, where the enclosed density is $200$ times the critical\n  density of the Universe. } $M_{200}$ and concentration parameter $c$\ncorresponding to NFW halos. Finding the equivalent \nhalo (by matching the dark matter mass within $r_{200}$ and the \nasymptotic density profile as $r\\to0$) gives the relation between parameters \n\\begin{eqnarray}\nM_{\\rm dm} &=& M_{200} - {M_{\\rm d}} \\\\\nr_{\\rm hq} &=& \\left(\\frac{ G M_{200}}{100 H_0^2}\\right)^{1\/3} c^{-1} \\sqrt{2 \\ln (1 + c) - \\frac{2c}{1+c}} \\label{eq:rhq}\n\\end{eqnarray}\ndependent upon the baryonic component ${M_{\\rm d}}$ and the redshift\nzero Hubble constant $H_0$.\n\nThe matching of the internal regions of our Hernquist profile to the\ndesired NFW is less accurate as we move close to the scale radius of\nthe NFW $r_{\\rm s}$, with the density of a Hernquist halo falling off more\nsteeply beyond that point. In terms of the velocity dispersion\nprofile, the Hernquist halo has a peak velocity dispersion shell that \nis more compact and this velocity dispersion is lower than the NFW value.\nThis can affect the heat exchange, which stops once the\nisothermal core has been established. As a consequence of the lower\nvelocity dispersion peak, Hernquist profiles can be more prone to the\nso called ``core collapse'' phenomena, or the runaway collapse in the\ninner regions that ultimately reverses the process of mass scattering\naway from the core and condenses instead more mass in the inner\nregions. This process would be exacerbated by the presence of baryons\nand as $\\som$ increases.\n\nWe have extensively tested this scenario and we include some comparisons\nin Appendix~\\ref{sec:NFW_vs_HQ}. We find \nthat it makes a negligible difference in\nmost of our simulations except for extreme cases with concentrated\nhalos, however as we shall see later it is the \\emph{low} concentration\nhalos where most of the interesting discrepancy between CDM and SIDM predictions lie, i.e. \nthese halos are expected to host the low central (dark matter) \ndensity galaxies that are most challenging to CDM. As such our approach of\nusing Hernquist halos is conservative, as the effect will be small and any\ndeviations will cause us to underestimate our conclusions.\n\nFor the simulations showed in Sec.~\\ref{sec:results}, we set up our\nhalos using $n_p=10^5$ particles to model the dark matter\ndistribution, which results in a spread for the mass per particle\n${\\rm m_p}=8.97\\times 10^4$--$2.66\\times 10^7 \\; \\rm M_\\odot$ according to the mass of the simulated halo.\nWe use a gravitational softening length $\\epsilon=50$~pc, \nand we have tested for numerical convergence using a\nhalo with $M_{200}=1.1\\times 10^{11} \\; \\rm M_\\odot$ and increasing\/decreasing the\nparticle number by a factor $10$ with respect to our \nfiducial runs and find agreement within $10\\%$ in the\n(dark matter only) circular velocity profiles (and by proxy mass) for all radii outside $1.0$~kpc. \n\n\\subsection{Disk potential}\\label{sec:diskpot}\nThe baryonic component of our galaxies is modelled by a fixed disk\npotential, which is not only computationally cheaper, but also allows\nus to maintain full control on the exact distribution of the baryons.\nThis is particularly important for dwarf galaxies which undergo hundreds\nof orbits over a Hubble time and are thus particularly sensitive to \n(unphysical) discretisation-induced instabilities. \nWe therefore strip the disk particles created\nin our initial conditions, and use instead an extra component in the\ngravitational force that accounts for the removed disk particles. \nThis approach is not fully self-consistent since it\nignores the back-reaction of the halo onto the baryons, but it provides\na useful tool to explore the evolution of halo potentials in SIDM in a\nset of controlled experiments where the {\\it final} baryonic\ndistribution is known. \n\nWe implemented a static potential in {\\sc Arepo} following a\n\\citet{Miyamoto_Nagai_1975} disk:\n\\begin{equation}\n\\Phi_{\\rm MN}(R,z) = \\frac{-G{M_{\\rm d}}}{\\sqrt{R^2+\\left({R_{\\rm d}}+\\sqrt{{z_{\\rm d}}^2+z^2}\\right)^2}} \\, ,\n\\end{equation}\nfor a disk of mass ${M_{\\rm d}}$ and scale radius ${R_{\\rm d}}$ and we keep the disk relatively\nthick in all cases by setting ${z_{\\rm d}}\/{R_{\\rm d}}=0.3$ (appropriate to model\ndwarf irregular galaxies). Note that the half-mass radius of the flat \n(${z_{\\rm d}}=0$) MN disk is $\\sqrt{3}{R_{\\rm d}}$ which is very close \nto the ratio of half-mass to scale radius of an exponential disk of \n$1.678$ and so unless ones disk is extremely thick, the scale radius \nof an exponential fit will be within a few percent\\footnote{The\n  differences are much more pronounced in the tails of the\n  distribution, indeed one cannot choose to correspond a flat\n  exponential to a MN density profile via for example the\n  mass weighted root-mean square radii since for the latter that is a\n  divergent quantity.}. From here on we will use ${R_{\\rm d}}$ interchangeably\nto refer to either the scale length of our baryonic disks in simulations\n(the MN scale radius) or the exponential disks associated with observed\nlate-type dwarf galaxies.\n\nNotice that the initial conditions are created assuming an exponential\nprofile while the fixed potential uses an MN model for the\ndisk. We have tested that this small inconsistency does not affect our\nresults.  Due to the diffusive nature of SIDM the resulting deviations\ndue to non-equilibria are of the order of a few percent and quickly\ndamped. Alternative approaches include that of \\citet{Elbert_2016}\nwhere the MN disk is `grown' adiabatically (compared to the orbital\ntimescale) from an initially halo-only setup. We have tested the extreme case \nof inserting the fixed potential instantaneously and find that after 10~Gyr \nevolution in SIDM at 2~kpc this makes a relative difference in the ${V_{2\\,\\rm kpc}}$ \nof $-0.2\\;{\\rm km\\, s^{-1}}$ or $0.6\\%$. Our method, being closer to\nequilibrium, is expected to be lower than these bounds. \n\nWe run the simulations for $10$ Gyr and compute the circular velocity\nprofiles. For the CDM cases this is in the most part a straightforward\nnumerical test since the initial conditions are created in equilibrium\nand we do not see significant evolution of the mass distribution with\ntime. For the most extended disks (${R_{\\rm d}}=6\\; \\rm kpc$), however, the\ncentre-of-mass is not well `tied' to the centre of the (shallow)\nbaryonic potential, and the centre-of-mass can `drift' on the order of\na kpc over 10~Gyr (i.e. an average of $0.1\\;{\\rm km\\, s^{-1}}$) from the centre of the\nbaryonic disk potential, making the assessment of the mass\ndistribution within 2~kpc problematic.  This evolution is expected due\nto discretisation of ICs and asymmetry in tree-based gravitational\nmethods, and is relatively hard to suppress even with a tight timestep\ncriteria. This process can of course also occur to the SIDM halos,\nalthough it is somewhat suppressed since their profiles have less\ncompact centres.  Since evolving CDM halos in extreme cases is\nsomewhat tangential to the purpose of this paper, however, we have\ntested the CDM evolution on the more compact (${R_{\\rm d}}=0.5$ and\n$1.5\\; \\rm kpc$) halos, whose ${V_{2\\,\\rm kpc}}$ evolve by an average of $-7.7\\%$\nover $10$~Gyr. For the ${R_{\\rm d}}=6\\; \\rm kpc$ CDM halos we will use the\ninitial conditions, as we believe those to be more accurate.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\n\n\\subsection{Circular velocity profiles in dark matter only halos}\n\\label{sec:V2_Vmax_dm}\n\nTo illustrate the effect of dark matter self-interactions on the density profile of a dwarf halo without the influence of baryons, we use the analytical Jeans method proposed in~\\citep{Kaplinghat_2014, Kaplinghat_2016} and tested against simulations in~\\citep{Rocha:2012jg,Elbert:2014bma}. We have further checked that its accuracy is within $10\\textup{--}20\\%$ compared to the results from our N-body code~\\citep{Vogelsberger_2012}. The details of the formalism can be found in Appendix~\\ref{sec:jeans}.\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{ccccccc}\n\\hline\n$V_{200}$ & $M_{200}$ &  concentration & $r_{\\rm s}$ (kpc)  & $\\rho_{\\rm s}$ ($10^6 \\; \\rm M_\\odot \\; kpc^{-3}$)$^\\star$ & $\\rm {M_{\\rm d}}$ & ${\\rm med.} \\; {R_{\\rm d}}$\\\\\n(${\\rm km\\, s^{-1}}$) & ($\\rm M_\\odot$) & ($-2\\sigma$,median,$+2\\sigma$) & $-2\\sigma$,med.,$+2\\sigma$ & $-2\\sigma$,med.,$+2\\sigma$ & ($\\rm M_\\odot $) & ($\\rm kpc^{\\star \\star}$)\\\\\n\\hline\n$15$ & $1.12\\times 10^{9}$ & $9.99$, $15.60$, $24.38$ & $2.15$, $1.37$, $0.88$ & $6.07$, $18.42$, $57.80$ & $\\dagger$ & $\\dagger$ \\\\\n$30$ & $8.97\\times 10^{9}$ & $8.32$, $12.99$, $20.30$ & $5.15$, $3.30$, $2.11$ & $3.89$, $11.63$, $36.04$ & $  1.71\\times 10^{8}$ & ($1.93$) \\\\\n$50$ & $4.15\\times 10^{10}$ & $7.27$, $11.36$, $17.74$ & $9.83$, $6.29$, $4.03$ & $2.82$, $8.32$, $25.52$ & $  8.84\\times 10^{8}$ & ($2.53$) \\\\\n$70$ & $1.14\\times 10^{11}$ & $6.65$, $10.39$, $16.23$ & $15.04$, $9.62$, $6.16$ & $2.29$, $6.69$, $20.36$ & $  2.79\\times 10^{9}$ & ($3.03$) \\\\\n$100$ & $3.32\\times 10^{11}$ & $6.05$, $9.46$, $14.78$ & $23.60$, $15.11$, $9.67$ & $1.84$, $5.31$, $16.05$ & $  1.09\\times 10^{10}$ & ($3.60$) \\\\\n$150$ & $1.12\\times 10^{12}$ & $5.44$, $8.50$, $13.28$ & $39.41$, $25.22$, $16.14$ & $1.43$, $4.10$, $12.27$ & $  3.72\\times 10^{10}$ & ($4.09$) \\\\\n$200$ & $2.66\\times 10^{12}$ & $5.04$, $7.88$, $12.30$ & $56.69$, $36.28$, $23.22$ & $1.20$, $3.42$, $10.15$ & $  6.22\\times 10^{10}$ & ($4.53$) \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the halos used in our diversity analysis. $V_{200}$ and $M_{200}$, concentrations with $\\pm 2 \\sigma$ relative to the median relation from \\citet{Ludlow_2014}, and in NFW parameters we give the scale radius $r_{\\rm s}$ and density $\\rho_{\\rm s}$ for this range of concentrations. For the simulations where we include a baryonic disk, $\\rm M_d$ refers to the total baryonic disk mass.\\newline\n  ${}^\\star$ NFW density $\\rho_{\\rm s}$ is computed in the dark matter-only halo case. For the simulations with baryonic disks we fix $M_{200}$ but add a baryonic component, and thus $\\rho_{\\rm s}$ is multiplied by the dark fraction $M_{\\rm dm}\/M_{200}$.\\newline\n  ${}^{\\star\\star}$ We quote median baryonic disk radii only for reference, since we attempt to span the distribution with $0.5$-$6\\;\\rm kpc$ as described in the main text.\\newline\n${}^\\dagger$ This halo was simulated in DM-only.}\n\\label{table:bar_param}\n\\end{table*}\n\n\nOur suite of halos aims to span the `dwarf' galaxy distribution with 7 halos of $V_{200}$ from $15$-$200~{\\rm km\\, s^{-1}}$, the upper limit being near Milky Way-like (where velocity dependent corrections may play some role) down to dwarfs so baryon-poor that we expect negligible deviations from the DM-only results. To encompass the diversity in concentrations we sample $\\pm 2\\sigma$ deviations from the cosmological\nmass-concentration relation presented in \\citet{Ludlow_2014}, whose median\nis well approximated by the linear fit\n\\begin{equation}\\label{eq:linear_ludlow}\nc = 10.51 \\times \\left(\\frac{M_{200}}{10^{11} \\; \\rm M_\\odot}\\right)^{-0.088} \\, ,\n\\end{equation}\nand at each halo mass the distribution is approximately log-normal where $1\\sigma$ corresponds to a ratio of $\\approx 1.25$. The corresponding halo mass ($M_{200}$) and NFW $r_{\\rm s}$ and $\\rho_{\\rm s}$ are given for the \n$70\\; {\\rm km\\, s^{-1}}\\, \\rm Mpc^{-1}$ cosmology in Table~\\ref{table:bar_param}.\n\n\n\\begin{figure*}\n\\includegraphics[width=\\columnwidth]{figures\/dens_vs_rad_70kms.pdf}\n\\includegraphics[width=\\columnwidth]{figures\/V2_Vmax_sig2_sig5_sig10_jeans_sigma_regions.pdf}\n\\caption{\\emph{Left panel:} core formation in a dark matter only halo (NFW) with different\n  $\\som$. Dark matter density profile evolution for SIDM over $10$~Gyr\n  in \\emph{blue}, \\emph{magenta} and \\emph{green} for $\\som=2$, $5$\n  and $10\\; {\\rm cm^2\\, g^{-1}}$ respectively, with CDM ($\\som=0\\; {\\rm cm^2\\, g^{-1}}$) in\n  \\emph{black dashed}. \\emph{Vertical dotted lines} indicate $r_1$ (the radius of unit scattering). \\emph{Right panel:} effects of SIDM on the ${V_{2\\,\\rm kpc}}$ vs ${V_{\\rm max}}$\n  relation compared to CDM for dark matter only\n  halos. \\emph{Black line} is the CDM case, \\emph{blue} the\n  $\\som=2\\; {\\rm cm^2\\, g^{-1}}$ where the shaded regions denote $\\pm 1 \\, \\sigma$\n  scatter (due to halo concentrations; we have interpolated the values from the $\\pm 2\\sigma$ runs in Table~\\ref{table:bar_param}). \\emph{Magenta} and\n  \\emph{green} denote $\\som=5\\; {\\rm cm^2\\, g^{-1}}$ and $10\\; {\\rm cm^2\\, g^{-1}}$\n  respectively. \\emph{Green points} indicate observed values collated in \\citet{Oman_2015}~Table~1, from which we also include the mass deficits, $5\\times 10^8$ and $10^9${$\\rm M_\\odot$} and their effect on ${V_{2\\,\\rm kpc}}$ in \n\\emph{blue shaded regions}.}\n\\label{fig:dm_only}\n\\end{figure*}\n\n\nThe left panel\nof Fig.~\\ref{fig:dm_only} shows the density profiles of a halo with\n$V_{200}=70 \\; {\\rm km\\, s^{-1}}$\n($M_{200}=1.1\\times 10^{11} \\; \\rm M_\\odot$) and a concentration\n$c=6.65$ after $10$~Gyr evolution with CDM ($\\som=0\\; {\\rm cm^2\\, g^{-1}}$, black\ndashed line) vs. SIDM with $\\som=2$, $5$ and $10\\; {\\rm cm^2\\, g^{-1}}$ (see solid\nlines according to labels). \nThe radius of unit scattering, $r_1$ (see also Appendix~\\ref{sec:jeans}) outside of which the halo is undisturbed is marked for the three cases. This radius grows with cross-section, and in the interior we see all\nthree cross-sections produce cores for this halo that extend beyond $2$~kpc.\n\n\nFig.~\\ref{fig:dm_only} (right panel) shows the circular velocity of the SIDM halo profiles evaluated at $2~{\\rm kpc}$ as a function of $V_{\\rm max}$. For the comparison, we also plot the corresponding CDM one with the\nmass-concentration relation from~\\citet{Ludlow_2014} and the range spanned by a compilation of the observed rotation curves in galaxies taken from~\\citet{Oman_2015}. It is clear that ${V_{2\\,\\rm kpc}}$ increases steadily with\n$V_{\\rm max}$ in the case with CDM, although the relation stays below the $1:1$ line. In contrast, SIDM predicts a much shallower relation. ${V_{2\\,\\rm kpc}}$ is almost independent of ${V_{\\rm max}}$ in the range explored, and the median ${V_{2\\,\\rm kpc}}$\nvalue at a given ${V_{\\rm max}}$ depends mildly on cross-section.  For low mass halos (low circular velocity),\n${V_{\\rm max}}$ tends to converge to near ${V_{2\\,\\rm kpc}}$, since the objects are \nsmall and the core size becomes much less than $2~{\\rm kpc}$ and ${V_{\\rm max}}$ is measured near $2\\, \\rm kpc$. It is interesting to note this is a common feature in CDM as well as SIDM halos. Shaded regions indicate\n$\\pm 1 \\sigma$ in concentration (we interpolate from the $\\pm 2 \\sigma$).\n\nThe range of ${V_{2\\,\\rm kpc}}$ and ${V_{\\rm max}}$ statistics of the galaxy rotation curve distribution is shown in Fig.~\\ref{fig:dm_only} \\citep[taken from][]{Oman_2015}.\nThe scatter found in the data is significantly\nlarger than derived purely from that of concentration variations (in either CDM or SIDM). \nA large fraction of the observed ${V_{2\\,\\rm kpc}}$ scatter\nbelow the relation expected for CDM, suggesting that such objects have\na lower dark matter density in the inner regions than predicted by the\nmodel. Instead, self-interactions seem to have the opposite\nbehaviour, providing a better  match to these low density objects but\npredicting too low a ${V_{2\\,\\rm kpc}}$ to explain most of the data. These\ncalculations, however, ignore the contribution of the baryons, and as\nwe argue below, the mass and radial extent of the gas and stars in\ngalaxies may significantly change this prediction to bring SIDM in\ncloser agreement with observations. \n\n\n\\begin{figure*}\n  \\includegraphics[width=2\\columnwidth]{figures\/4panel_V200_70_cc-2.pdf}\n  \\caption{\\emph{Left three panels:} Dark matter contribution to the circular velocity profile \n    for a dwarf galaxy ($V_{200}=70\\; {\\rm km\\, s^{-1}}$, $c=-2\\sigma$)\n    with a fixed baryonic disk component with scale radii\n    $0.5$, $1.5$ and $6\\; \\rm kpc$ from left to right respectively,\n    simulated for 10~Gyr with a cross-section $\\som=2 \\;{\\rm cm^2\\, g^{-1}}$.\n    \\emph{Red lines}\n    indicate the baryonic contribution, \\emph{grey} the\n    dark matter (SIDM), and \\emph{thick solid black} the total. For\n    comparison, the \\emph{thick dashed black} lines indicate the CDM\n    total. \\emph{Right panel} combines the three profiles, with the CDM\n    comparisons in \\emph{grey dashed lines}.}\n\\label{fig:vel_vs_rd}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=\\columnwidth]{figures\/V2_Vmax_sig2_cc-2_0_2.pdf}\n\\includegraphics[width=\\columnwidth]{figures\/V2_Vmax_sig2_likelihood_regions_shen_2sigma.pdf}\n\\caption{N-body simulations in the presence of galactic disks. \\emph{Left panel} shows\n  the individual simulations with disk scale lengths ${R_{\\rm d}}=0.5$, $1.5$ and 6~kpc in \\emph{stars}, \\emph{squares} and \\emph{triangles} respectively, where each is repeated for the three different\n  $-2\\sigma$,  median, and $+2\\sigma$ concentrations and SIDM in \\emph{solid magenta} and CDM in \\emph{open blue symbols}. \\emph{Dashed black line} indicates the CDM relation without baryons, and \\emph{solid black} the corresponding SIDM (with $\\pm 1\\sigma$ shading). \n\\emph{Right panel} shows 2-$\\sigma$ likelihood contours about the median ${R_{\\rm d}}$ and concentration (\\emph{solid lines}) for CDM in the presence of a baryonic disk in \\emph{blue} and\n  SIDM in \\emph{magenta}, inferred from the fits in\n  Eqns.~(\\ref{eq:Shen_size})~and~(\\ref{eq:Shen_size_scatter}), (see also main text). \\emph{Green points} indicate observed values collated in Oman 2015 Table~1.}\n\\label{fig:SIDM_bar_distrib}\n\\end{figure*}\n\n\\subsection{Circular velocity profiles in the presence of baryons}\\label{sec:V2_Vmax_bar}\n\nFor collisionless dark matter halos, the presence of baryons may\nchange the shape  of the {\\it total} circular velocity profile by\nadding the contribution from the gas and the stars. This contribution\nmay or may not change the overall profile, depending on the relative\nfraction of baryonic mass and also the spatial distribution of these\nbaryons\\footnote{Notice that baryons could\nadditionally cause the dark halo to contract \\citep{Blumenthal_1986}, but given our\ninitial condition set-up, the halos will be created already in equilibrium\nwith the baryonic potential}. If gas and stars are very centrally\nconcentrated, they will dominate the contribution to the circular\nvelocity modifying the profile to rise more steeply than the original\nNFW halo. On the other hand, if the baryons are fairly extended, their\ncontribution could be sub-dominant to the dark matter, in which case\nthe total velocity profile including baryons does not deviate\nsignificantly from that of the original NFW halo. \n\nFig.~\\ref{fig:vel_vs_rd} (leftmost three panels) illustrates this\npoint using the N-body simulations set up described in\nSec.~\\ref{sec:sims}. For a fixed halo with $V_{200}=70\\; {\\rm km\\, s^{-1}}$, \n${M_{\\rm d}} = 2.8\\times 10^9$~{$\\rm M_\\odot$}\\ (according to abundance matching\nrelations from \\citet{Behroozi_2013}) and a concentration $-2\\sigma$ below the\nmedian.\nWe plot the equivalent  `circular velocity' inferred for a spherically\naveraged mass distribution\n\\begin{equation}\\label{eq:vcirc}\nV_{\\rm cir}^2 = G M(r)\/r = V_{\\rm cir, dm}^2 + V_{\\rm cir, bar}^2\n\\end{equation}\nwhich is an approximation for non-spherical distributions\n(for a pure thin MN disk without a halo these \n deviations are at maximum $\\approx 15\\%$).\nWe assume that\nthe baryons distribute in a MN disk with scale length\nequivalent ${R_{\\rm d}} = 0.5$, $1.5$ and $6\\; \\rm kpc$ (left to right). The\nlong-dashed lines indicate the final profiles including the baryons\nfor CDM, which show different shapes according to ${R_{\\rm d}}$, with the\ndifferences tracing back to the contribution of the baryons in each\ncase.\n\n\nAs expected, a more concentrated baryonic distribution (small ${R_{\\rm d}}$)\nshows a steep raise of the circular velocity curve, whereas larger\n${R_{\\rm d}}$ values grow more gently. The case with ${R_{\\rm d}}=0.5$~kpc results in\na declining circular velocity profile which might not be the typical\ngalaxy included in studies of rotation curves (probably representing a\nbulge dominated object instead). However, we include this case for\nillustration of the possible extremes and how CDM and SIDM would react to\nthe presence of such a compact galaxy. \n\n\n\nThe same exercise of considering a fixed halo and baryonic content\nbut changing the spatial distribution can be done assuming the SIDM\nscenario. The three left panels in Fig.~\\ref{fig:vel_vs_rd} show the\nexpected total profiles assuming a self-interaction cross-section\n$\\som=2 \\; {\\rm cm^2\\, g^{-1}}$ (see solid black lines). The dark matter component still\nshows a cored profile due to the self-interactions (see thin grey\nline) but this is obscured in the \\emph{total} velocity profile. In fact,\nthe inclusion of baryons creates a fair amount of {\\it diversity} in\nthe shapes of the circular velocity profiles for CDM and SIDM (see\nright panel in the same figure). For example, if we measure ${V_{2\\,\\rm kpc}}$ in\nthese halos (vertical red shaded region), CDM covers a range $\\approx\n45-83$ ${\\rm km\\, s^{-1}}$ which is comparable to that spanned by SIDM, ${V_{2\\,\\rm kpc}}\n\\approx 30-75$ ${\\rm km\\, s^{-1}}$. Encouragingly, the effect of baryons in the case\nwith self-interactions creates a wide range of (total) rotation curves but\nalso maintains a lower central density (in the cases where baryons are\nextended), allowing better accommodation of the low ${V_{2\\,\\rm kpc}}$ values\nthat are occur in observations (see right panel\nFig.~\\ref{fig:dm_only}) and for which CDM alone has no explanation.\n\n\n\n\n\nWe generalise this by simulating all halos introduced in\nSec.~\\ref{sec:V2_Vmax_dm} but including a baryonic disk. The mass of the\ndisk is the sum of a stellar and gaseous component, with the first set\nby the abundance matching relation between $M_*$ and $M_{200}$\nintroduced in \\citet{Behroozi_2013} and the gas mass computed from the\n$M_{\\rm gas}$-$M_\\star$ relations of \\citet{Huang_2012}. To account\nfor variations in halo concentration $c$, we sample three different halos\nof median and $\\pm 2\\sigma$ extremes according\nto the mass-concentration relation from\n\\citet{Ludlow_2014}. Table~\\ref{table:bar_param} summarises the main\nproperties of all our runs. Each halo is then set up in equilibrium\nwith their baryonic MN disk of scale lengths\n${R_{\\rm d}}=0.5,\\; 1.5,\\; \\rm and \\; 6.0$~kpc. We fix the scale\nheight of the disks to ${z_{\\rm d}} = 0.3{R_{\\rm d}}$, a slightly hotter\nstructure than for Milky-Way-like galaxies but that provides a reasonable\ndescription of lower mass galaxies, the main target of this study.\n\n\nWe run our simulations for $10$ Gyr assuming $i)$ collisionless cold dark\nmatter (CDM) and $ii)$ a fiducial self-interaction cross-section\n$\\som = 2\\;{\\rm cm^2\\, g^{-1}}$ for the SIDM case.  We investigate the relation\nbetween $V_{\\rm max}$ and ${V_{2\\,\\rm kpc}}$ of our halos in CDM and SIDM in the\nleft panel of Fig.~\\ref{fig:SIDM_bar_distrib}. Symbol shapes (square,\nstar, triangle) indicate the different scale lengths sampled whereas\nopen-blue and filled-magenta differentiate between CDM and SIDM\nruns. A quick inspection suggests that self-interactions are able to\ngenerate a wider range of ${V_{2\\,\\rm kpc}}$ at fixed $V_{\\rm max}$ which is in\nbetter agreement with observed values from the compilation in\n\\citet{Oman_2015} (green dots).  The diversity in both cases, CDM and\nSIDM, arises from different contribution by the baryons.  Since SIDM\ncores imply a lower dark matter density in the inner regions, the\ncontribution of baryons is \\emph{more} important that in the\ncollisionless case, creating a larger variety in ${V_{2\\,\\rm kpc}}$. Notice that\nfor the more compact disks (${R_{\\rm d}}=0.5$ and $1.5$ kpc) the\npredictions from CDM and SIDM are not that different, and in both\ncases ${V_{2\\,\\rm kpc}}$ can be very close to ${V_{\\rm max}}$, which suggests these\nrotation curves are relatively flat. One should also note that\nin a cosmological context the central density would be even further\npromoted by the adiabatic contraction introduced by baryonic collapse\n\\citep[e.g.][]{Elbert_2016}, which occurs even in the models with\nbursty feedback \\citep{Tollet2016}.\nIn the case with non-dominant\nbaryonic components (triangles) the points stay close to the dark\nmatter-only case (thin solid lines), which is significantly lower in\nthe case of self-interactions, as discussed in\nSec.~\\ref{sec:V2_Vmax_dm}. \n\nThe lower envelope of points in either scenario corresponds to the\nmost extended disks populating the $-2\\sigma$ outliers in halo\nconcentration. A few of the observed points in the left panel of\nFig.~\\ref{fig:SIDM_bar_distrib} still remain unexplained\n(${V_{\\rm max}} \\approx 70 \\; {\\rm km\\, s^{-1}}$ and ${V_{2\\,\\rm kpc}} \\approx 20\\; {\\rm km\\, s^{-1}}$) showing\ncentral densities even lower than can be obtained in SIDM with\n$\\som=2\\;{\\rm cm^2\\, g^{-1}}$. This could be suggesting the need for a larger cross\nsection for self-interactions or more extreme outliers on the\nmass-concentration relation. We should also bear in mind the\npossibility of observational errors as discussed in Sec.~\\ref{sec:intro}.\n\nAlthough the exercise of fixing ${R_{\\rm d}}$ is more clear and intuitive,\nin practise we know that galaxy size will scale with stellar mass, and\nthe average behaviour in ${V_{\\rm max}}$-${V_{2\\,\\rm kpc}}$ plane will depend on the\naverage ${R_{\\rm d}}$ of the population at fixed ${V_{\\rm max}}$.  In the right\npanel of Fig.~\\ref{fig:SIDM_bar_distrib} we take this into account as\nfollows.  The first step is to characterise ${R_{\\rm d}}$ as a function of\n${V_{\\rm max}}$. Because of the tight relation between halo mass and stellar\nmass, this is equivalent to finding a relation between ${R_{\\rm d}}$ and\n$M_*$ or, in our case, ${M_{\\rm d}}$. We compute the stellar size of the\ndisk and then estimate the baryonic one by a simple correction by gas\nfraction. In detail, for our stellar radii we use a fit to the\nlate-type SDSS galaxies of \\citet{Shen_2003}:\n\n\\begin{equation}\\label{eq:Shen_size}\n  R_{\\star} = 0.1  \\, \\left( \\frac{M_\\star}{1 \\; M_\\odot } \\right)^{0.14} \\left( 1 + \\frac{M_\\star}{3.98 \\times 10^{10} \\; M_\\odot} \\right)^{0.25} \\; \\rm kpc\n\\end{equation}\n\n\\noindent\nwith scatter\n\\begin{equation}\\label{eq:Shen_size_scatter}\n  \\sigma_{{\\rm ln} R} = 0.34 + \\frac{0.13}{1 + \\left( \\frac{M_\\star}{3.98 \\times 10^{10} \\; M_\\odot} \\right)^2}\n\\end{equation}\n\n\\noindent\nalong with an estimate of the baryonic size as: ${R_{\\rm d}} = (1+f_{\\rm\n  gas}) R_\\star$, where $f_{\\rm bar}$ is estimated from $M_*$ and\nobservations by \\citet{Huang_2012} as explained before. \nThis procedure then provides a median and $\\pm\n2\\sigma$ disk radii for each halo mass. We then take these dispersions\nin ${R_{\\rm d}}$ and interpolate between our simulations to find the\nappropriate ${V_{\\rm max}}$ and ${V_{2\\,\\rm kpc}}$, finally combining this dispersion in\nquadrature with the dispersion due to the $M_{200}$-concentration\nrelation to infer the median and $\\pm2 \\sigma$ likelihood regions as a\nfunction of halo mass.\n\nThese likelihood regions are shown in the right panel of\nFig.~\\ref{fig:SIDM_bar_distrib}. The expected median in each scenario\nis notably shallower than the points in the left panel, which is\nmainly due to the average disk size rising as ${V_{\\rm max}}$\nincreases. Indeed, as we saw in the left panel of the same figure, at\nfixed ${V_{\\rm max}}$ a more extended baryonic disk will have lower\n${V_{2\\,\\rm kpc}}$. For comparison we include the effects of the mass deficits of \n$5\\times 10^8$ and $10^9${$\\rm M_\\odot$} within 2~kpc.\nSeveral observed points still scatter above and below the median\nrelations for both CDM and SIDM hinting at the need of more extreme outliers\n(more than $2 \\sigma$) to explain such objects. \n\nWe have probably overstated the extremity of these outliers since we \nhave at zero-th order ignored correlations between halo and disk properties \nthat would alleviate the tension. \nFor instance, if more compact disks systematically\npopulate more concentrated halos whereas extended disks live in low\nconcentration halos, the shaded areas could be larger. We have also not \nmade an attempt to account for observational biases in these\ncalculations. Regardless of the absolute value, however, we emphasise that at\nfixed assumptions, SIDM seems to always provide a larger\nrange of diversity than CDM (with scatter at least $150\\%$ of the latter).\n\nAn interesting corollary of the analysis above is that, fixing the\nrelation ${V_{\\rm max}}$-${V_{2\\,\\rm kpc}}$, a given halo in CDM and SIDM would be\npredicted to have different contribution\nfrom the baryons at $2$~kpc, since in SIDM the baryonic matter will have to\n``compensate'' for the lower density of the cored dark matter halo. It\nis therefore important to compare the dark matter fractions predicted\nin both models to check that they are consistent with observed\ngalaxies. Fig.~\\ref{fig:DM_2kpc} shows the dark matter\nfractions $f_{\\rm dm}=M_{\\rm dm}\/M_{\\rm tot}$ measured at $2$~kpc for\nour CDM and SIDM halos (open and filled symbols, respectively)\ncompared to a subsample of galaxies taken from the rotation curves\npresented in \\citep{Oh_2015} and \\citet{Kuzio_2008}.  For the former\nwe use their disk-halo decomposition from the stellar, HI and\nkinematic data, and for \\citet{Kuzio_2008} we use their `popsynth'\nmass modelling rather than the extremal fits. In a couple of cases the\nlast-measured point does not reach 2~kpc and for these we used the\ndark matter fraction at the last measured point. \n\nThere is a significant overlap between the collisionless and the \nself-interacting scenarios. Also, spatial resolution coupled to\nuncertainties in the mass-to-light ratios for the observed objects\nmakes $f_{\\rm dm}$ probably not a good enough indicator to distinguish\nbetween these alternative models. Nonetheless, it is reassuring that\nthe same SIDM objects that reproduce better the observations in the\n${V_{\\rm max}}$-${V_{2\\,\\rm kpc}}$ plane, seem to have dark matter fractions that are\nconsistent with current observations.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/Vmax_V2ratio_dm_plus_CDM.pdf\n\\caption{Dark matter contribution to the rotation curve at 2~kpc for SIDM\n  simulations and observed dwarfs (see main text). \\emph{Filled magenta squares}\n  are the SIDM simulations, in comparison to the CDM in \\emph{open blue squares}.\n  \\emph{Greyscale symbols} refer to observed\n  galaxies from \\citep{Oh_2015} (\\emph{dark circles}),\n  \\citet{Kuzio_2008} (\\emph{light circles}), \\citet{deBlok_2001} \n  (\\emph{light squares}) and \\citet{deBlok_2008} (\\emph{dark squares}).}\n\\label{fig:DM_2kpc}\n\\end{figure}\n\n\n\\subsection{Two extreme examples: IC~2574 and UGC~5721}\\label{sec:obs_curve_sims}\n\nIn Sec.~\\ref{sec:V2_Vmax_bar}, we presented a detailed statistical\nanalysis of a sample of objects simulated within the SIDM\nparadigm. This, however, left unexplained one of the main motivations\nof this work, which is the existence of extreme outliers which are not\npossible to accommodate within the standard CDM scenario. We turn then\nour attention to two particular examples highlighted in\n\\citet{Oman_2015}, UGC~5721 and  IC~2574. These two galaxies have\nsimilar circular velocity measured at the outermost point of the\nrotation curve $\\approx 80$ ${\\rm km\\, s^{-1}}$, which probably indicate that they\npopulate dark matter halos with similar mass, but the {\\it shapes} of\nthe inner regions are remarkably different: whereas UGC~5721 is\nconsistent with a `cuspy' NFW profile, IC~2574 has a very extended core.\nThis pair is an interesting case\nof the aforementioned diversity and we can use it as benchmark for our\nSIDM scenario.\n\nIn Figs.~\\ref{fig:UGC5721} and \\ref{fig:ICgal} we show the data together\nwith the best fit CDM (dashed line) and SIDM (solid black) curves. \nFor UGC~5721 (NGC3724) we use the rotation curve data\nfrom \\citet{Swaters_2003} and stellar and gas densities from Andrew\nPace (priv. communication, for comparison one can see\nFig.~1 in \\citealp{Swaters_2010} which has a slightly higher\nmass-to-light ratio).\n                     \n                     \nFor IC~2574 we use the rotation curve, gas and stellar densities from\n\\citet{Oh_2011}. We multiply the stellar velocity contribution by\n$0.5$ (mass-to-light ratio lower by a factor $4$), so the stellar\ncontribution lies between that in \\citet{Sanders_2002} and\n\\citet{Oh_2011}. For both cases our SIDM halo is run using a cross\nsection for self-interactions of $\\som = 3\\,{\\rm cm^2\\, g^{-1}}$\nfollowing \\citet{Kamada2016}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lccc}\n\\hline\n\n\nProperty & Symbol & IC~2574 & UGC~5721 \\\\\n\\hline\nDisk scale length & ${R_{\\rm d}}$ & $3.0$ & $0.50$ kpc \\\\\nDisk scale height & ${z_{\\rm d}}$ & $0.9$ & $0.15$ kpc \\\\\nStellar mass & $M_\\star$  & $3.00 \\times 10^8$ & $4.48\\times 10^8$ {$\\rm M_\\odot$}\\\\\nGas mass$^\\dagger$ & $M_{\\rm gas}$ & $2.72\\times10^9$  & $8.47\\times10^8 $ {$\\rm M_\\odot$}\\\\\nHalo mass & $M_{200}$ & $9.78\\times10^{10}$ & $5.15\\times 10^{10}$ {$\\rm M_\\odot$} \\\\\nConcentration & $c$ & $6.2$ ($-2.3\\sigma$) & $13.6$ ($+0.9\\sigma$) \\\\\nDM cross-section &$ \\som $ & $3$ & $3$ cm$^2$ g$^{-1}$ \\\\\nNFW radius & $r_{\\rm s}$ & $15.4$ & $5.65\\;\\rm kpc$ \\\\\nNFW density & $\\rho_{\\rm s}$ & $1.9\\times10^{-3}$ & $0.013 \\, \\rm M_\\odot\\, pc^{-3}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the SIDM halos and baryonic potentials for the two galaxies in Fig.~\\ref{fig:UGC5721} and \\ref{fig:ICgal}. \\newline \n$\\dagger$ The gaseous component as added in post-processing due to its disturbed nature, see also main text in Sec.~\\ref{sec:obs_curve_sims}. \\newline \n}\n\\label{table:gal_params_fits}\n\\end{table}\n\nOur fitting process relied upon making an initial guess for the \nhalo mass using ${V_{\\rm max}}$ and sampling the concentration with $\\pm 2.5 \\sigma$ of the \nmass-concentration relation in Eqn.~(\\ref{eq:linear_ludlow}). \nThe combined stellar plus gaseous disks in these dwarfs are not\nwell represented by a single MN disk (especially their gas profiles\nwhich exhibit a much more extended, nearly constant surface density), \nhowever since these differences are primarily in the outer regions where we\nexpect the effects of self-interactions to be sub-dominant, we chose\nto approximate only the \\emph{stellar} distribution with an MN disk \nand apply the gas contribution to the rotation curve in a \npost-processing step (i.e. add in quadrature as in Eqn.~\\ref{eq:vcirc}),\nand after several iterations with the disk component we find the \nparameters indicated in Table~\\ref{table:gal_params_fits}.\nAs noted in Appendix~\\ref{sec:NFW_vs_HQ},\nthe initial conditions are constructed with Hernquist profiles \nconverted from the corresponding NFW parameters, \nbut if one is using a real NFW then in the higher\nconcentration case using a lower $r_{\\rm s}$ would give a\nslightly better match.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/UGC5721.pdf}\n\\caption{Observations for UGC~5721 along with SIDM and CDM simulations. \\emph{Red points} indicate the stellar contribution, \\emph{cyan} the gas and \\emph{grey squares with error bars} the observed rotation curve. The SIDM simulation (with parameters from Table~\\ref{table:gal_params_fits}) has the baryonic mass contribution indicated by the \\emph{green line}, which is a composition of the interpolated gas distribution along with an MN fit to the stellar disk. The halo contribution is given in the \\emph{dotted blue line} and the total curve in \\emph{solid black}. The comparison curve for CDM is given in the \\emph{dashed black line}.}\n\\label{fig:UGC5721}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/IC2574.pdf}\n\\caption{Observations for IC~2574 along with SIDM and CDM simulations. \\emph{Red points} indicate the stellar contribution, \\emph{cyan} the gas and \\emph{grey squares with error bars} the observed rotation curve. The SIDM simulation (with parameters from Table~\\ref{table:gal_params_fits}) has the baryonic mass contribution indicated by the \\emph{green line}, which is a composition of the interpolated gas distribution along with an MN fit to the stellar disk. The halo contribution is given in the \\emph{dotted blue line} and the total curve in \\emph{solid black}. The comparison curve for CDM is given in the \\emph{dashed black line.}}\n\\label{fig:ICgal}\n\\end{figure}\n\nFig.~\\ref{fig:UGC5721} shows that the SIDM fit for UGC~5721 including all components has an\noverall ``cuspy'' profile (see black solid line), which is \\emph{steeper} than the circular\nvelocity expected for the corresponding NFW halo without \nself-interactions (dashed line). For this fit we need a concentrated\nhalo at $+1\\sigma$ above the median mass-concentration relation, which\ntogether with the observed compact stellar disk (scale radius of\n$0.5$~kpc) combine to create a steeply raising velocity curve. \nThis is an excellent example of the importance of baryons in the shape\nof the rotation curves, even for dwarf galaxies. \nThe $\\chi^2\/\\rm dof$ of the SIDM fit is $3.04$ ($4.44$ below the initial, \nwhich is of course not the best individual CDM halo fit). \nThe visual impression of a superior fit is likely due to \nThis $\\chi^2$ is largely driven by the $\\lesssim 0.3$~kpc data with\ntight error bars over a non-monotonic feature, contributing to \na visually impression of a superior fit.\nClearly a more accurate fit to such features would require additional parameters (particularly for the stellar components), however this is outside the scope of this paper.\nWithout the presence of baryons, UGC~5721 is a challenge for SIDM\ndue to the development of an extended low density core at the centre. In our simulations, however, the\nSIDM thermal transfer in cooperation with the baryon dominance in the\ncentral $1.5$~kpc causes the halo to shrink even further than the\npure NFW model. \nThe compact stellar component excavates a potential well sufficient that\nthermalisation of the dark component leads to an almost-NFW final\nconfiguration \\citep[see also ][]{Kamada2016}.\n\n\nOn the other hand, Fig.~\\ref{fig:ICgal} presents the rotation curve\nfit for IC~2574 with an extended and slowly rising core in its\ncentre. The gas has a disturbed profile, but the baryons as a whole\ncontribute little to the total density except within the innermost\n$0.5$~kpc, and make a modest contribution at $r> 6$~kpc. For such a\ndark matter dominated galaxy to deviate so far from a cusp-like\nprofile is clearly a challenge for vanilla CDM \\citep{BlaisOuellette_2001, Sanders_2002,\n  Swaters_2003, Swaters_2010}. Notice that any baryonic solution to\nthis problem within CDM -- for instance assuming that supernova\nfeedback can create a core-- would not work for such object, since the\ncore extends well beyond where the stars are (see discussion in\n\\citealt{Oman_2015}). \nThe $\\chi^2\/\\rm dof$ of the SIDM fit is $1.57$, ($1.39$ below the initial\nconditions), which reflects the introduction of the core\nand quantifies the capacity of SIDM as a solution to this conundrum.\n\n\nWhilst SIDM is relatively efficient in producing cores, this process\nbecomes increasingly sedate as the core size approaches the scale\nradius of the halo (in this case $15.4$~kpc) and so we have had to\nchoose a relatively large scale radius (i.e. low concentration)\nhalo. The ${V_{2\\,\\rm kpc}}$ of our CDM halo is around $35\\; {\\rm km\\, s^{-1}}$ whilst in SIDM\nit is around $27\\; {\\rm km\\, s^{-1}}$, bringing it in better agreement with the\nobserved rotation curve ${V_{2\\,\\rm kpc}} \\approx 23 \\; {\\rm km\\, s^{-1}}$. \n\nWe note that for IC~2574 both the low concentration and self-interactions are required to achieve a good fit. As such one way to test our model is to perform a statistical study of extreme outliers like IC~2574. Since we assume the SIDM model inherits the halo mass-concentration relation of the CDM one, such a study also provides a direct test for {$\\Lambda$CDM}\\ on galactic scales.\n\n\nWe have shown that the SIDM model in the presence of baryons can provide good fits to two extreme outliers. We demonstrate that with SIDM the halo concentration, the baryon concentration and the back-reaction of baryons on the density profile play roles in explaining the rotation curves of these two extremes. SIDM halo profiles are flexible enough to accommodate the diverse rotation curves of spiral galaxies. \n\n\n\\section{Summary and Conclusions}\\label{sec:discussion}\n\n\nThe SIDM model modifies the CDM paradigm by inclusion of non-gravitational interactions between dark matter particles in the halo evolution, and these self-interaction effects are most significant in the regions of halos with the high densities and velocity dispersions, i.e. within the scale radius of the halo. In the outer halo and at large scales SIDM is almost collisionless and retains the successes of {$\\Lambda$CDM}.\nIn the inner regions of halos the dark matter is redistributed into a central core, whose density and size depend on the microscopic particle interaction cross-section, the halo mass and even the baryon concentration. The mass profiles inferred from astronomical observations of the rotation curves of spiral galaxies thus offer an avenue to both constrain the self-interaction cross-section of dark matter and falsify or support specific self-interaction models.\n\n\nObservationally, the shapes of the rotation curves of galaxies exhibit a\nwide diversity. We follow \\citet{Oman_2015} by quantifying the range in the rotation curves using the relation\nbetween the total circular velocity measured within $2~{\\rm kpc}$ and at the peak, ${V_{2\\,\\rm kpc}}\\textup{--}{V_{\\rm max}}$. For a given ${V_{\\rm max}}$, the spread in ${V_{2\\,\\rm kpc}}$ can be a factor of four, which is difficult to reconcile in the prevailing {$\\Lambda$CDM}~paradigm~\\citep{Oman_2015}, where dark matter is assumed to be collisionless over the cosmological timescale. This problem stems from the hierarchical structure formation in {$\\Lambda$CDM}\\ which produces a self-similar halo density profile, which is essentially parameterised by one parameter, the halo mass (or concentration). After we determine the halo mass by ${V_{\\rm max}}$, the halo circular velocity is completely fixed at all radii, up to the scatter, including the inner density cusp that is in contrast to the shallow (cored) one preferred by many dwarf galaxies. In addition, the enclosed mass of a CDM halo cusp tends to overwhelm the baryonic contribution and consequently also the scatter in ${V_{2\\,\\rm kpc}}$ caused by the spread in the baryon concentration.\nThis inflexibility in CDM may be alleviated by introducing particular prescriptions of baryonic feedback processes that can dynamically heat the dark matter \\citep[and others]{Navarro_1996,Pontzen_2012}, however SIDM offers an attractive alternative solution.\n\nIn this paper we have used controlled N-body simulations as a numerical experiment to test the SIDM model's ability to address the diversity problem. We use the {\\sc Arepo} code to run isolated simulations of a live halo in the presence of a static baryonic disk in order to assess the gravitational impact of gas and stars on the halo evolution. This approach is suited to the SIDM model as dark matter self-interactions thermalise the inner halo and the final (inner) density profile is largely insensitive to the formation history. We sample a wide range of halo masses spanning about three orders of magnitude and vary the halo concentration within $\\pm2\\sigma$ from the its mean value. For each halo mass, we use abundance matching relations to choose the disk mass and we take three disk scale radii to span the baryon distributions. This sampling of the model parameter domain enables us to make concrete predictions of the rotation curves in the SIDM model.\n\n\nOur main conclusion (see Fig.~\\ref{fig:SIDM_bar_distrib}) is that the SIDM mechanism accommodates galaxies with a wide range of  ${V_{2\\,\\rm kpc}}$ over the domain of ${V_{\\rm max}}$ from $30$--$250\\;{\\rm km\\, s^{-1}}$. The spread in core densities is due in part to the variance in concentration (within $\\pm2\\sigma$ from the mean), however this alone is insufficient to account for the large relative scatter of the observed values. Including self-interactions allows lower ${V_{2\\,\\rm kpc}}$, which requires low central densities in \\emph{both} baryons and dark matter. High values of ${V_{2\\,\\rm kpc}}$ are still achieved with compact baryonic disks where thermalisation into the deep baryonic potentials can induce NFW-like densities. \nAs a result the predicted range of ${V_{2\\,\\rm kpc}}$ for a fixed ${V_{\\rm max}}$ in SIDM is $50\\%$ larger than the CDM one, leading to a better agreement with the observed rotation curve distribution.\n\nSome observed galaxies in the range ${V_{\\rm max}}=60$--$80\\; {\\rm km\\, s^{-1}}$ and ${V_{2\\,\\rm kpc}}=20$--$30\\;{\\rm km\\, s^{-1}}$ are still challenging.\nThese cases might require \nassumptions about the structure of the copula of the combination of \nhalo concentrations and baryonic disk sizes, in particular that very\nextended disks populate the most under-concentrated halos to reproduce the very low ${V_{2\\,\\rm kpc}}$, which is not an \nunrealistic assumption to first order. \nNevertheless, the SIDM prediction is an improvement over CDM in all cases.\n\n\n\nTo further test our results, we have performed simulations to provide individual fits for two of the most extreme\noutliers highlighted in~\\citet{Oman_2015}, UGC~5721 and IC~2574, cusp- and core-\ndominated galaxies respectively, both with similar ${V_{\\rm max}} \\approx\n80 \\; \\rm {\\rm km\\, s^{-1}}$ (see Figs.~\\ref{fig:UGC5721} and~~\\ref{fig:ICgal}). Picking extreme $+0.9\\sigma$ and $-2.3\n\\sigma$ concentrations from the halo concentration-mass\nrelation, together with a compact ${R_{\\rm d}} = 0.5~{\\rm kpc}$ and\nextended ${R_{\\rm d}} = 3.0~{\\rm kpc}$ baryonic distribution respectively, we find good fits in the SIDM paradigm. Interestingly, we find that the SIDM density profile can be as dense as the NFW one in a baryon-concentrated galaxy such as UGC~5721, because SIDM particles follow an isothermal distribution within the deep baryon potential. Our simulation result confirms the theoretical expectation first predicted in~\\cite{Kaplinghat_2014}. \nAt the other extreme, IC~2574 with an inferred central dark matter dominated core is more straightforwardly tractable with a model that allows dark matter thermalisation, and is more challenging to the collisionless CDM model.\n\nThe implication of these elements is that with the inclusion of realistic baryonic components, SIDM models can produce quantitatively superior fits to rotation curves, both at the level of individual rotation curves and over the statistical distribution of a quantitative measure of their shapes. As one would expect, fitting the most extreme outliers requires an interplay between the combinations of halo concentration, the baryon distribution, and the influence of baryons on the SIDM halo profiles. The results from our numerical experiments fit well with the previous theoretical calculations of \\citet{Kamada2016}. \n\nA number of avenues for future research are apparent. On the observational side, more detailed observations for mass modelling would allow tighter constraints on the dark matter distribution and consequently the interaction cross-section. \nOrthogonally, larger sample sizes would allow the ${V_{2\\,\\rm kpc}}$--${V_{\\rm max}}$ distribution to better discriminate between CDM and SIDM. \nOn the computational side, we have ignored the effects of the hierarchical assembly of dark matter halos. Whilst this is not expected to play a large role in the inner halo, it will nevertheless introduce additional scatter as a function of assembly time. Finally, all these will likely show strong correlations with the baryonic statistics, and investigations of galaxy formation in SIDM using full hydrodynamical cosmological simulations with the baryonic feedback processes are already being explored \\citep[e.g.][]{Vogelsberger_2014,Fry_2015}.\n\n\n\\section*{Acknowledgements}\nThe authors would like to thank Andrew Pace for sharing his collated data set for the rotation curves, \nVolker Springel for making {\\sc Arepo} available for this work and\nJames Bullock, Simon White, Oliver Elbert, and Manoj Kaplinghat for useful discussions. This work was supported by the U.S. Department of Energy under Grant  No. de-sc0008541 (HBY). HBY acknowledges support from the Hellman Fellows Fund. MV acknowledges support through an MIT RSC award and the support of the Alfred P. Sloan Foundation. \n\n\n\\bibliographystyle{mnras}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAstrophysical jets emanate from sources spanning a huge range in energy output \nor length scale - among them young stellar objects (YSO), \nstellar mass compact objects as X-ray binaries or $\\mu$-quasars, \nor the powerhouses of some active galactic nuclei (AGN) which host a super-massive\nblack hole.\nIn particular for radio-loud quasars, \nfor which synchrotron emission dominates the radio spectrum,\nrelativistic jets are a generic feature. \nDue to the omnipresent angular momentum conservation, mass accretion to all of\nthese objects features a disk structure around the central mass.\nIt is commonly believed that jets are launched as disk winds, \nwhich are further accelerated and collimated by magnetic forces\n(see \\citet{1982MNRAS.199..883B, 1983ApJ...274..677P, 1986A&A...162...32C, 1997SvPhU..40..659B, \n2003ApJ...596.1240H, 2007prpl.conf..277P}.\nRelativistic jets may gain further energy by interaction with the\nblack hole magnetosphere \n\\citep{1977MNRAS.179..433B, 1997MNRAS.292..887G, 2005MNRAS.359..801K}.\n\nThe magnetohydrodynamic (MHD) self-collimation of non-relativistic jets has been \nproven in general by \ntime-dependent simulations \\citep{1995ApJ...439L..39U,1997ApJ...482..712O} and \nhave been investigated in further detail considering additional physical effects \nas magnetic diffusivity by \n\\cite{2002A&A...395.1045F}, a variation in \\cite{1999MNRAS.309..233O}, \nnon-axisymmetric instabilities in the launching region \\citep{2003ApJ...582..292O},\nor a variation in the mass flow profile or the magnetic field geometries\n\\citep{2006ApJ...651..272F, 2006MNRAS.365.1131P},\nor the influence of a central magnetic field \\citep{2009ApJ...692..346F,2008A&A...477..521M}.\n\nIn the case of relativistic jets the efficiency of MHD self-collimation is under debate. \nThe main reason is the existence of electric fields which are negligible for non-relativistic\nMHD and which are commonly thought to have a net de-collimating effect on the jet.\nEssentially, \\cite{1991ApJ...377..462C} have demonstrated the current carrying \nrelativistic jet can be highly collimated.\nHowever, the actual structure of these jets still remains unclear - mainly due to \nthe need for simplifying assumptions to solve the corresponding set of MHD equations.\n\nSo far, a variety of theoretical models have been developed for the case of \n{\\em self-similar} jets\n\\citep{1992ApJ...394..459L, 1994ApJ...432..508C, 1995ApJ...446...67C,\n      2003ApJ...596.1104V,2006A&A...447..797M},\nalthough it seems clear that relativity does not obey self-similarity.  \nFully 2.5D theoretical solutions for the internal magnetic jet structure could be\nobtained by neglecting matter inertia\n\\citep{1997A&A...319.1025F, 2001A&A...365..631F}.\nThese force-free solutions for the field structure can in principle be coupled\nto the dynamical wind solution along the field lines\n\\citep{1996A&A...313..591F, 2001A&A...369..308F, 2004ApJ...608..378F}.\n\\cite{1997A&A...319.1025F} obtained solutions for the internal jet force balance \nin Kerr metric with an asymptotically cylindrical jet emerging from a disk-like\nstructure around the central rotating black hole. \nThe shape of the collimating jet boundary was obtained as result of the internal \nforce equilibrium, in particular considering the regularity condition along the \njet outer light surface.\n\nTime-dependent simulations of relativistic MHD jet formation have been performed\nconsidering a general relativistic metric, including also the evolution of \nthe underlying accretion disk.\nEarly - seminal - simulations did last for a few inner disk rotations \nonly \\citep{1998ApJ...495L..63K, 1999ApJ...522..727K}, which is\nsufficient time to demonstrate the launching of an outflow, \nbut hardly sufficient in order to investigate the long-term dynamical evolution of\nthe emerging jet.\n\nMore recent simulations were able to follow several 100 disk rotations and show \nthe formation of a so-called funnel flow origination in the shear layer between \nthe horizon and the inner disk radius\n\\citep{2005ApJ...620..878D, 2007MNRAS.375..513M, 2008MNRAS.388..551T, 2009MNRAS.394L.126M}.\nThese simulations indicate a highly time-variable mass ejections of rather low degree \nof collimation. However, the funnel flow achieves Lorentz factors of up to 50.\nGeneral relativistic MHD simulations are also able to determine the interrelation \nbetween jet formation and the Blandford-Znajek mechanism \n\\citep{2005ApJ...630L...5M, 2007MNRAS.377L..49K}.\n\nUltra-relativistic MHD simulations of accelerating and collimating\njets have been presented by \\citet{2007MNRAS.380...51K},\nspanning over a huge range of length scale and providing jets of \nlarge Lorentz factor $\\Gamma \\sim 10$.\nTheir simulations, however, did not start from the very base of the \njet - the accretion disk, but at some fiducial boundary above the equatorial\nplane.\nSince the jet has been launched already with super-escape speed,\ngravity has not been considered. \nThe jet flow has been confined within a rigid wall of predefined shape \nwhich naturally affects the opening angle of the MHD jet nozzle and thus\njet collimation and acceleration.\n\nThe focus of our present paper is \ni) to concentrate on the formation and acceleration of a relativistic MHD jet right from\nthe launching area the accretion disk surface,\nii) to investigate the (self-) collimation of relativistic MHD jets under the influence\nof de-collimating electric forces and an \"open\" boundary condition for the outflow \niii) to consider gravity as en essential gradient to provide a realistic disk boundary condition in\nequilibrium,\niv) to run long-term simulations lasting more than 1000 inner disk rotations until the\njet reaches steady state,\nv) to concentrate on MHD disk jets as disks are the natural origin for the mass load for\nAGN jets.\n\nThe outline of this paper is as follows: In section \\ref{sec:concepts} we discuss the \nconcepts of ideal special relativistic magnetohydrodynamics in the\nperspective of jet formation.  \nSection \\ref{sec:model setup} is devoted to the initial- and boundary conditions of the\nnumerical simulations, whose results are shown in section \\ref{sec:results}. \nWe conclude in section \\ref{sec:conclusions}.\n\n\\section{Concepts of relativistic MHD jets}\\label{sec:concepts}\n \nIt is well known that relativistic jets must be strongly magnetized \n\\citep{1969ApJ...158..727M, 1986A&A...162...32C, 1993ApJ...415..118L}.\nThis simply reflects the fact that the lower the mass flux, the more electro-magnetic energy (Poynting flux) \ncan be transferred into high kinetic energy per unit mass.\nRelativistic MHD is also limited for very strong magnetization as then\nthe MHD assumption can be violated since a sufficiently large amount of electric charges is lacking which are needed to drive the electric current system. Such a situation might arrise in the ultra-relativistic regime of pulsar-winds but is unlikely for the disk-winds investigated here.  \n\nThis paper deals with the time-dependent formation of relativistic MHD jets \nby using the special relativistic MHD module of the PLUTO code provided by \\cite{2007ApJS..170..228M}\nand applying a Newtonian description of gravity. \n\\subsection{Relativistic MHD equations}\n\nThe relativistic MHD module of PLUTO solves the system of special relativistic conservation laws.\nIn a covariant formulation, \nthe equations follow naturally as a set of hyperbolic equations.\nIt is solved for energy and momentum conservation\n\\begin{equation}\n\\partial_{\\alpha} T^{\\alpha \\beta} = 0 \\label{eq:energy}\n\\end{equation}\nof an ideal magnetized fluid \n\\begin{equation}\nT^{\\alpha \\beta} = (\\rho h + b^2) u^\\alpha u^\\beta + \n\\left(p + \\frac{1}{2} b^2\\right) g^{\\alpha \\beta} -b^\\alpha b^\\beta\n\\end{equation}\nwith the specific plasma enthalpy \n\\begin{equation}\nh=\\frac{\\gamma}{\\gamma-1}\\frac{p}{\\rho}+1,\n\\end{equation}\nthe isotropic gas pressure $p$, density $\\rho$ (both in the local rest-frame), four-velocity $(u^\\alpha)=(\\Gamma,\\Gamma \\mathbf{\\beta})^T$, velocity $\\mathbf{\\beta}=\\mathbf{v}\/c$, Lorentz factor $\\Gamma=(1-\\beta^2)^{-0.5}$ and the magnetic field pseudo vector\n\\begin{equation}\nb^\\alpha = -\\frac{1}{2} \\epsilon^{\\alpha \\beta \\gamma \\delta} u_{\\beta} F_{\\gamma \\delta}.\n\\end{equation}\n\nAssuming infinite conductivity, thus vanishing electric fields in the \nrest-frame of the plasma $F^{\\alpha \\beta} u_{\\beta}= 0$, the \nhomogenous Maxwell equation\n\\begin{equation}\n \\partial_{\\alpha}\\ ^{*}\\!F^{\\alpha \\beta} = 0 \\label{eq:hommaxwell}\n\\end{equation}\ncan be written solely in terms of the magnetic four-vector $b^\\alpha$\n\\begin{equation}\n^{*}\\!F^{\\alpha \\beta} \\equiv \\frac{1}{2} \\epsilon^{\\alpha \\beta \\gamma \\delta} F_{\\gamma \\delta} \n                     =   b^\\alpha u^\\beta - b^\\beta u^\\alpha\\label{eq:hommhd}\n\\end{equation}\nand for the field vector components it \nfollows\\footnote{following the convention that Greek indices run \nfrom 0 to 4 whereas Latin indices go from 1 to 3}\n\\begin{eqnarray}\nB^{i} &=& ^{*}\\!F^{i 0} = b^{i} u^0 - b^{0}u^{i}\\\\\nE^{i} &=& \\epsilon^{ijk}b^j u^k \\label{eq:mhd-cond}.\n\\end{eqnarray}\nEquation \\ref{eq:mhd-cond} represents the ideal MHD condition \n$\\mathbf{E} = - \\mathbf{\\beta} \\times \\mathbf{B}$ \nand is the reason why all electric fields can be eliminated from the equations.  \nThe magnetic four-vector turns out as $b^0 = B^{i} u^{i};\\ \\  b^{i} = (B^{i}+b^0 u^{i})\/u^0$.  \nThe conservation of the Faraday tensor given by Eq.\\,\\ref{eq:hommhd} results in the \nnon-relativistic (ideal) induction equation\nand the solenoidal condition $\\mathbf{\\nabla} \\cdot \\mathbf{B} = 0$.  \nMass conservation is guaranteed by the continuity equation \n\\begin{equation}\n\\partial_{\\alpha} (\\rho u^\\alpha) = 0. \\label{eq:continuity}\n\\end{equation}\nWe apply a polytropic equation of state for the gas with the polytropic\nindex $\\gamma =5\/3$.\n\n\\subsection{Gravity in special relativity}\nThe outcome of MHD simulations is mainly determined by its boundary conditions and therefore requires great care in describing the proper physical state of \ninterest.  \nSince in our simulations the jet is considered to be launched as a wind from a \nrotating disk,\nit is essential to take into account a proper disk model as boundary condition.\nFor the disk boundary we choose a (sub-) Keplerian rotation profile and a\nhydrostatic pressure gradient (see section \\ref{sec:injection}).\nThis choice implies a hot, ``puffed up'' corona with a sound speed \n$c_{s}^2\\simeq 2\/3\\ v_{\\phi}^2$.\nThis is the main reason why we have added gravity to our special relativistic\ntreatment.\nThe direct impact of gravity on the jet dynamics is marginal as the outflow\nis accelerated to super escape speed quickly.\n\nA fully self-consistent relativistic treatment of gravity would imply\na general relativistic approach.\nHowever, we are not interested in the region very close to the horizon \nof the central object, but in the long-term dynamics and evolution of a disk wind into\na relativistic jet.\nWe can therefore neglect general relativistic effects in our simulation \ndomain.\n\nWe apply a softened gravitational potential\n\\begin{equation}\n\\phi = -\\frac{GM}{R+r_{S}}\n\\end{equation}\nwith a softening length of $r_{S}=1\/3$ that may be related to the to the Schwarzschild \nradius of a non-rotating black hole\\footnote{The capital $R=\\sqrt{r^2+z^2}$ denotes the spherical radius throughout this work.}.\n\\red\nThe corresponding acceleration reads\n\\begin{equation}\n\\mathbf{a} = -\\nabla\\phi = -\\frac{GM}{(R+r_{S})^2 R}\\mathbf{r}\n\\end{equation}\nand hence instead of solving equation \\ref{eq:energy}, we solve \n\\begin{equation}\n\\partial_{\\alpha}T^{\\alpha \\beta} = f^\\beta\n\\end{equation}\nwith the four force density \n$\n\t(f^\\beta) = \\Gamma \\rho (\\mathbf{a \\cdot v},\\mathbf{a})^T\n$\nas a local source term on the right-hand side.  \nThis is incorporated in PLUTO as a \"body force\" ($\\Gamma \\mathbf{a}$) using the infrastructure of the code.  \n\\black\n\nOmission of softening would lead to numerical errors \n(due to the unresolved steep gradients in the potential close to the origin), \npiling up to produce artificial acceleration along the spine of the jet close to \nthe axis.\nSoftening is clearly a compromise avoiding the singularity (by limiting the \nrequired resolution) on little cost of realism.  \nAnother choice could be the well-known pseudo-potential by \\cite{1980A&A....88...23P}\nwhich has just the negative softening $\\phi_{PW}=-GM\/(R-r_{S})$.  \nFor the cylindrical geometry of our choice, the singularity would become even more problematic,\ncomplicating the setup a great deal.  \n\n\n\\subsection{Relations in axisymmetric MHD}\nThe region of jet formation may be fairly well approximated in {\\em axisymmetry}.\nIn fact, non-axisymmetric distortions may actually hinder the formation of powerful jets as probably \ndemonstrated by the existence of a variety of strongly magnetized, rapidly rotating accretion disk \nsystems which, however, do not exhibit jets (e.g. cataclysmic variables or most pulsars). \n\nUnder the assumed symmetry in a cylindrical coordinate system, \nthe magnetic field vector can be written as\n\\begin{equation}\n\t\\mathbf{B} = \\mathbf{B}_{p} + B_{\\phi} \\mathbf{e_{\\phi}},\n\\end{equation}\nwhere $B_{\\phi}$ can now be an arbitrary function of $r$ and $z$, as the solenoidal condition\ntranslates to $\\mathbf{\\nabla\\cdot B_{p}}=0$. \nThe stream function \n$\\Psi(r,z)= (1\/2\\pi) \\int \\mathbf{dS \\cdot B_{p}} = rA_{\\phi}$ \nmeasures the magnetic flux through the surface area $S$ \nand follows from the toroidal component of the vector potential,\n\\begin{equation}\n\\mathbf{B_{p}}= \\mathbf{\\nabla}\\times \\mathbf{A_{\\phi}} \n              = \\mathbf{\\nabla}\\times \\frac{\\Psi\\ \\mathbf{e_{\\phi}}}{r} \n              = \\frac{1}{r}\\mathbf{\\nabla}\\Psi \\times\\ \\mathbf{e_{\\phi}}\n\\end{equation}\n\nFor the electric field, the ideal MHD condition $\\mathbf{E = B \\times \\beta}$ gives \n\\begin{equation}\n\\mathbf{E} = \\frac{r\\Omega^{F}}{c} B_{p} \\mathbf{n} = \\frac{r}{r_{\\rm L}} B_{p}\\mathbf{n}\\label{eq:electric-field}\n\\end{equation}\nin terms of the so called angular velocity of the field line \n$\\Omega^F = \\left(v_{\\phi} - v_{p}B_{\\phi}\/B_{p}\\right)\/r$, \nor the so-called light cylinder radius \\footnote{This is the radius at which the hypothetical angular \nvelocity of a field line supercedes the speed of light} of a field line $r_{\\rm L} \\equiv c\/\\Omega^F$.  \nThe direction of the electric field is given by \n$\\mathbf{n = B_{p}}\/B_{p}\\times \\mathbf{e_{\\phi}}$ and is perpendicular to the magnetic\nflux surface $\\Psi(r,z)$.  \nThe poloidal Poynting flux $\\mathbf{\\mathcal{S}} = (c\/4\\pi)\\mathbf{E\\times B_{\\phi}}$ \nsimplifies to \n\\begin{equation}\n\t\\mathbf{\\mathcal{S}} = -\\frac{c}{4\\pi} \\frac{r}{r_{\\rm L}} B_{\\phi}\\mathbf{B_{p}} \n                             = -r\\Omega^{F} \\frac{B_{\\phi}\\mathbf{B_{p}}}{4\\pi}.\n\\end{equation}\n\n\\subsubsection{Perpendicular and parallel force-balance}\\label{sec:force-balance}\nThe processes leading to flow collimation can be identified directly \nfrom the (steady-state) trans-field force-balance equation \n\\citep{1991ApJ...377..462C, 1993A&A...270...71A}.\nHere, we adopt the notation of the latter paper when investigating the \ncollimation behavior in the quasi steady-state time domain of our \nsimulations.\nThe curvature \n$\\kappa \\equiv \\mathbf{n}\\left( \\mathbf{B_{p}} \\cdot \\nabla \\right) \\mathbf{B_{p}} \/B_{p}^2$\nof a flux surface $\\Psi(r,z)$ results from the summation of perpendicular forces,\n\\begin{equation}\n\\begin{split}\n\\kappa \\frac{B_{p}^2}{4\\pi}\\left(1-M^2-\\frac{r^2\\Omega^{F^2}}{c^2}\\right) =  \\\\\n+ \\left(1-\\frac{r^2\\Omega^{F^2}}{c^2}\\right)\\nabla_{\\bot}\\frac{B_{p}^2}{8\\pi} \n+ \\nabla_{\\bot}\\frac{B_{\\phi}^2}{8\\pi}+\\nabla_{\\bot}p \\\\\n+ \\left(\\frac{B_{\\phi}^2}{4\\pi r}-\\frac{\\rho h u_{\\phi}^2}{r}\\right)\\nabla_{\\bot}r\n- \\frac{B_{p}^2\\Omega^{F}}{4\\pi c^2}\\nabla_{\\bot}\\left(r^2\\Omega^F\\right) \\\\\n+ \\Gamma \\rho \\nabla_{\\bot}\\phi, \n\\end{split}\n\\label{eq:trans-field}\n\\end{equation}\nwhere we have added the collimating component of the gravitational force. \nFor the ease of use in section \\ref{sec:trans-field}, we label the terms as \n$(F_{\\rm curv}, F_{\\rm pbp}, F_{\\rm pbphi}, F_{\\rm p}, F_{\\rm pinch}, \nF_{\\rm cf}, F_{\\rm el}, F_{\\rm grav})$ \nin the order of their appearance in Eq.\\ref{eq:trans-field}.\nThe (poloidal) Alfv\\'en Mach number $M$ is relativistically defined as \n\\begin{equation}\n\tM^2 = \\frac{4\\pi \\rho h u_{p}^2}{B_{p}^2}.\n\\end{equation}\nThe gradient $\\nabla_{\\perp}\\equiv\\mathbf{n}\\cdot \\nabla$ \nis projected perpendicular to the magnetic flux surfaces $\\Psi$, and thus along the \n(inward pointing) electric field.  \nThe light surface of a magnetosphere is located where \n$r_{\\rm L}(\\Psi) = r_{\\rm L}(r,z) \\equiv c \/ \\Omega^F(\\Psi)$\n\\footnote{Thus, the light surface consists of the points \nof intersection between the field lines with their corresponding light cylinder}, it hence depends on the flux-geometry as well as on the rotation profile $\\Omega^F$.  \nEach flux surface \/ magnetic field line crosses the light surface at most once (see\nalso the discussion in \\citet{1997A&A...323..999F})\nSome field lines $\\Psi(r,z)$ never cross the light surface, indicating an asymptotic radius\n$r_{\\infty}(\\Psi) < r_{\\rm L,\\Psi}$. For these field lines relativistic effects due\nto rotation (electric fields) are less important.\nFor others, the asymptotic radius is $r_{\\infty}(\\Psi) > r_{\\rm L, \\Psi}$.\nThe light surface constitutes a critical point of the stationary axisymmetric wind equation only\nin the mass-less limit in which it is identical to the modified poloidal Alfv\\'en surface \n$M_{A}^2 = 1-(r_{A}\/r_{L})^2$ \\citep{1986A&A...156..137C, 1986A&A...162...32C}. \nHowever, it is essential to note that at the light surface the dynamical behavior of \nthe poloidal magnetic pressure term changes - the force changes sign.\nThis leads to the existence of three dynamically different regimes in the asymptotic\n(collimated) region of a relativistic jet (see Fig.\\,\\ref{fig:lightcylinder}).\nIn region I, for all field lines $r_{\\infty}(\\Psi) < r_{\\rm L, \\Psi}$ corresponding\nto $(1-(r\/r_{\\rm L})^2)>0$, and, thus, a de-collimating magnetic pressure term.\nField lines in region II do cross their light cylinder, and, since $r> r_{\\rm L}$,\nthe magnetic pressure term acts as collimating for $r > r_{\\rm L}$.\nField lines in Region III  never reach their light cylinder,\nand here the magnetic pressure term is de-collimating again.\nThe slope of the outer part of the light surface critically depends on the dynamics\nand the magnetic field structure of the outflow in the very inner part.\n\nSimilarly, the forces due to the electric field $E = r\/r_{\\rm L} B_{\\rm p}$ \n(second last term in\nEq.\\,\\ref{eq:trans-field}) scale with the relative position to the light surface,\nhence they are important in region II of the jet formation region only.\n\nEquation \\ref{eq:trans-field} together with Fig.\\,\\ref{fig:lightcylinder} once\nmore demonstrates the need to resolve the whole acceleration and collimation\nregion of a jet in radial and vertical direction. \nOnly when the light surface is taken into account self-consistently, the proper\nforce-balance is applied along and across the flow. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=6cm]{f1}\n\\caption{The different dynamical regimes in special relativistic\ndisk-winds. Region I and III stay sub-relativistic, while region II\nis relativistic, i.e. electric forces are not negligible\n(see Sect.\\,\\ref{sec:force-balance} for a discussion).\n\\label{fig:lightcylinder}}\n\\end{figure}\n\nSimilarly one can derive the parallel-field force equation, it becomes:\n\\begin{equation}\n\\begin{split}\n\t\\frac{B_{p}^2}{4\\pi}\\nabla_{||}M^2 = \n\t\\kappa_{||} \\frac{B_{p}^2}{4\\pi}\\left(1-M^2\\right) \\\\\n\t-\\nabla_{||}\\left(p+\\frac{B_{p}^2}{8\\pi}+\\frac{B_{\\phi}^2}{8\\pi}\\right) \\\\\n\t- \\left(\\frac{B_{\\phi}^2}{4\\pi r}-\\frac{\\rho h u_{\\phi}^2}{r}\\right)\\nabla_{||}r \\\\\n\t- \\Gamma \\rho \\nabla_{||} \\phi\n\t\\label{eq:parallel-field}\n\\end{split}\n\\end{equation}\nwith the necessary definitions $\\nabla_{||}\\equiv\\mathbf{B_{p}}\/B_{p}\\nabla$ and \n$\\kappa_{||}B_{p}^2\\equiv \n\\mathbf{B_{p}}\/B_{p}\n\\left(\n\\mathbf{B_{p}}\n\\cdot \n\\nabla\n\\right)\n\\mathbf{B_{p}}\n$.\nWe see how the change in the Mach-number is mediated by the interplay of tension-, pressure-,\npinch-, centrifugal- and gravitational- acceleration.   Electric fields (pointing in the\nperpendicular direction) don't contribute and the equation reduces to the Newtonian case:\nIn steady-state, there is no electric acceleration!\n\n\\red\nA number of self-similar approaches to the relativistic\n jet formation have beed published (eg. \\cite{2003ApJ...596.1104V}).\n While the self-similar ansatz is a powerfull and highly\n successfull tool to solve the non-relativistic MHD problem\n (starting with the Blandford-Payne solution), we believe\n that using self-similarity for relativistic MHD jets is\n problematic.\n \nWe note that neither the light surface nor the relativistic Alfv\\'en\nsurface obeys a self-similar structure.\nIt is well known that forcing self-similarity into the relativistic \nMHD equations constrains\nthe rotation law for the magnetosphere $\\Omega^F(r)  \\propto r^{-1}$.\n(see also discussion in \\citet{1992ApJ...394..459L,1993ApJ...415..118L}).\nThis is a major difference to the non-relativistic self-similar\napproach.\nWe further note that also the scaling for the electric\nfield depends on the radial position of the light surface \n(see Eq.~\\ref{eq:electric-field}).\nThis is, however, of uttermost importance for the structure of relativistic\nmagnetospheres as the electric field forces play a leading role in the\ntrans-field force-balance (equation \\ref{eq:trans-field}).\nSimilar arguments hold for the inner light surfaces around Kerr \nblack holes or the geometry of the black hole ergosphere. \n\nWe therefore believe that a steady-state self-similar relativistic MHD \napproach is intrisically inconsistent with the relativistic \ncharacteristic of the flow.\n\\black\n\\subsubsection{field line constants}\\label{sec:flconst}\nStationary axisymmetric MHD flows conserve the following five quantities along\nthe magnetic flux-function $\\Psi$.\nFrom the iso-rotation law together with the ideal MHD condition follows\nthe rest mass energy-flux per magnetic induction, \n\\begin{equation}\nk = k(\\Psi) \\equiv \\frac{\\rho u_{p}}{B_{p}}\\label{eq:kf}\n\\end{equation}\nand the iso-rotation parameter \n\\begin{equation}\n\\Omega^F = \\Omega^F(\\Psi) = \\frac{1}{r}\\left(v_{\\phi}-v_{p}\\frac{B_{\\phi}}{B_{p}}\\right)\\label{eq:omegaf}\n\\end{equation}\n(often interpreted as angular velocity of the field lines).  \nIn absence of shocks the (pseudo-) entropy \n\\begin{equation}\nQ=\\frac{p}{\\rho^\\gamma} = Q(\\Psi) \\label{eq:Qf}\n\\end{equation}\nis conserved as well as the angular momentum flux\n\\begin{equation}\nl=-\\frac{I}{2\\pi k c} + r u_{\\phi} = l(\\Psi)\\label{eq:lf}\n\\end{equation}\nand the flux ratio of total energy to rest-mass energy,\n\\begin{equation}\n\\mu = \\frac{\\mathcal{S+K+M+T+G}}{\\mathcal{M}} = \\mu(\\Psi) \\label{eq:mu}\n\\end{equation}\nwhere we identify the individual terms as\n(purely) kinetic energy flux\n$\\mathcal{K} \\equiv (\\Gamma-1) \\rho\\ u_{p}$,\nrest-mass energy flux\n$\\mathcal{M} \\equiv \\rho\\ u_{p}$,\nthermal energy flux\n$\\mathcal{T} \\equiv \\Gamma \\frac{\\gamma}{\\gamma-1} p\\ u_{p}$,\nand gravitational energy flux\n$\\mathcal{G} \\equiv \\rho\\ \\phi\\ u_{p}$,\nrespectively.  \nThe cold, asymptotic limit of (\\ref{eq:mu}) is particularly of interest, it reads\n\\begin{equation}\n\\mu = \\Gamma(\\sigma+1)\n\\end{equation}\nwhere $\\sigma=\\mathcal{S\/(K+M)}$ is the customarily defined magnetization parameter\n- the ratio of Poynting to kinetic flux. \nThis simple relation provides a theoretical maximum for the Lorentz-factor \n$\\Gamma^{*}=\\mu$, when the entire electromagnetic energy is converted into kinetic \nenergy. \n\nThe essential point in the quest for relativistic jets is to find a highly energetic \ndisk solution with values of $\\mu$ beyond the anticipated Lorentz-factor.  \nPrevious studies obtaining highly relativistic jets by \\cite{2007MNRAS.380...51K} do\nnot start from a realistic disk solution but inject the jet material with \nan artificial rotation profile, \na high injection speed $v_{\\rm p} > 0.5 c $, and \na density low enough to obtain a high energy flux $12\\leq \\mu_{\\rm max}\\leq 18$.   \nIn the present study we are aiming to improve on the problems just mentioned \nby applying a physical boundary condition as a Keplerian disk-corona in equilibrium.\n\n\n\\red\n\\subsection{Accretion disk coronae}\\label{sec:coronae}\nIt is our ambition to connect the wind solutions to the ambiance of a realistic accretion disk.  \nIn our simulations, the flow originates in the high entropy atmosphere called a corona.\nOptically thin coronae are an integral part in models of the X-ray features of AGN\n\\citep[e.g.][]{1993ARA&A..31..717M}\nand $\\mu$-Quasars \\citep[e.g.][]{2002MNRAS.332..856N,2003NewAR..47..491M}.\n\nWhile Compton cooling can provide the observed spectra, the heating mechanism is not easily found.  \nJust as in the case of the sun, the coronal heat cannot directly be transfered from the colder photosphere\/accretion disk (according to the second law of thermodynamics) and the nature of vertical energy transport is an active field of research.  \nExternal irradiation of flared disks by the central object (or central disk) is certainly present in a multitude of objects \\citep[see][for a review]{2008ChJAS...8..302C} but might not be the primary energy source.  \nAmong the most promising mechanisms we should highlight magnetic reconnection heating as proposed by \\cite{1991ApJ...380L..51H}.  \n\n\nBetween the mid-plane and the coronal point of injection in our simulations, ideal MHD can not provide a realistic picture.  \nIn order for an accretion disk to {\\em work}, a torque of viscous or magnetic origin has to be exerted onto the material. \nAdditionally, the flux-freezing constraint of ideal MHD must be relaxed since it would lead to an accumulating magnetic pressure that ultimately stops the accretion process.  \nStudies modelling both the accretion motion and the super Alfv\\'enic jet based on a stationary self-similar approach (e.g. \\cite{1993ApJ...410..218W, 1993A&A...276..625F, 1995ApJ...444..848L}) rely on global-scale magnetic fields and ad-hoc assumptions on the viscosity and (anisotropic) magnetic diffusion.  \nIn order for these models to be stable against strong magnetic compression on the one side and the magneto rotational instability (MRI; see \\cite{1998RvMP...70....1B}) on the other, \\cite{1995A&A...295..807F} require equipartition for the thermal and magnetic pressure.\n\\cite{2000A&A...361.1178C} demonstrated the importance of heating for the mass-loading or the jet. \nTime-dependend numerical simulations of these magnetized accretion ejection structures (MAES) were presented by \n\\cite{2002ApJ...581..988C, 2004ApJ...601...90C, 2007A&A...469..811Z} \nand adopt a fixed in time anomalous resistivity profile in order to connect the two dynamical regimes.  \n\nAlthough the stationarity of the aforementioned simulations is possibly hampered by numerical diffusion and low resolution effects, MAES with global-scale fields are to date the most successful models to create collimated outflows.  \n\nIt is now widely believed that the source of viscosity and resistivity in weakly magnetized disks is the turbulence seeded by the MRI.  \nLocal, stratified shearing box simulations by \\cite{2000ApJ...534..398M} suggest a quenching of the MRI in the strongly magnetized coronal region, as the  magnetic scale height exceeds the thermal one.  This is in contrast to the equipartition fields proposed for the MAES.  \nMagnetic buoyancy of the large-scale fields eventually created by a turbulent dynamo could then provide the coronal heating.  \nTypically, the MRI results in toroidally dominated coronal fields with $B_{\\phi}^2 > 10 B_{p}^2$ and opening the field lines towards a topology favorable for wind acceleration remains a challenge.  In the context of the solar corona this process is discussed by \\cite{2003ApJ...599.1404W}.  \n \nBy restraining parts of the accretion disk structure, \\cite{2003A&A...398..825V} could achieve outflows in time-dependend simulations of disk-corona structures where the magnetic field is sustained by a mean-field dynamo.  \nAnalytic models of this turbulent disk-corona-outflow connection are however in still its infancy (see the discussion by \\cite{2004ApJ...616..669K} and attempts by \\cite{2009ApJ...704L.113B}).\n\n\n\\black\n\\section{Model setup for the MHD simulations}\\label{sec:model setup}\n\\red\nWith the aforementioned considerations we choose the following model for our investigation.  \nA global-scale poloidal field favorable of wind acceleration is adopted.  Whether it is advected by the accretion flow or created by an underlying dynamo is not of our concern.  \nThe jet base resembles a corona in the sense that it is hot (electron  temperature $\\sim10^9 \\rm K$), has no mechanism of cooling, is non-turbulent (no viscosity) and highly ionized (infinite conductivity).  We choose a Keplerian rotation profile for the field lines.  \nThe flow starts with sub-escape velocity and we investigate sub-magnetosonic injection where mass loading is determined by the internal dynamics as well as mass-fluxes imposed by the boundary condition.  \nWe perform axisymmetric special relativistic MHD simulations of jet formation for a set of different magnetic field geometries and field strengths.  In the following we discuss the numerical realization of our problem.\n\\black\n\\subsection{Boundary conditions}\n\nGiven the 2.5 dimensional nature of the problem, three geometrical boundaries have to be \nprescribed. \nThese are the inlet boundary along $z=0$ from which material is injected into the domain \n({\\em inflow})\nand the two outer boundaries at $r=r_{\\rm end}$ and $z=z_{\\rm end}$ where we expect \nmaterial to leave the computational domain ({\\em outflow}). \nThe boundary condition along $r=0$ ($\\rm R_{beg}$) follows from cylindrical symmetry.  \nFigure \\ref{fig:setup} gives an overview of the different regions.  \n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=7cm]{f2}\n\\caption{Sketch of the different regimes of our grid and boundary.  \nIn both directions we set 20 equidistant cells in $[0,1]$.\nThen follows a stretched grid until we add five equidistant cells one unit-radius before the outflow boundary.  For $r\\in[0,1]$, the hydrostatic corona is fixed to minimize the influence of the central region on the disk-wind.  \n\\label{fig:setup}}\n\\end{figure}\n\n\\subsubsection{Injection boundary ($Z_{beg}$)}\\label{sec:injection}\nPursuing the aim to follow the acceleration of an disk wind from as close to the accretion \ndisk as possible, we start with a sub slow-magnetosonic wind.\nWe are hence free to choose four constraining boundary conditions without overdetermining\nthe system (see \\cite{Bogovalov1997} and Appendix \\ref{sec:MHD-inj} for more details).  \n\nOur choice is to fix the toroidal electric field component $E_{\\phi}=0$. \nThis suppresses the evolution of the bounding poloidal magnetic field and is realized\nby requiring $\\mathbf{v_p || B_p}$.  \nThe field line iso-rotation is kept constant in time and follows a Keplerian rotation law\n$\\Omega^F\\propto r^{-1.5}$.\nUnless specified otherwise, the boundary condition starts out initially in a force-free state with zero toroidal field $\\Omega^F = v_{\\phi}(r)\/r$ corresponding to a disk in hydrodynamic equilibrium.  \nThis is an essential ingredient as - within stationary ideal MHD - $\\Omega^{F}$ just \nequals the mid-plane angular velocity of the material $\\omega(r)$. \nRelaxation of the infinite conductivity constraint would, however, lead to an inequality \n$\\Omega^F(r) \\le \\omega(r)$ owing to the diffusion of magnetic field. \nFor these reasons $\\Omega^F$ should closely follow the expected disk rotational profile \nand should be limited by the maximal velocity in the mid-plane, \ntypically at the inner edge of the disk located at $r=1$.  \n\nA radial force-equilibrium along the whole boundary is enforced by balancing the centrifugal and \npressure support against gravity via the sub-Keplerianity of the\nrotation $\\sqrt{\\chi} = v_\\phi(r=1)\/v_K$ \n\\red\nthat also determines the inlet density, \n\\black\nwhere $v_K$ is the circular velocity that alone sustains against gravity at $r=1$.\n\\red\nA more convenient parametrization is in terms of the\n\trelative temperature \t\n\t$\\epsilon \\equiv c_s^2\/v_\\phi^2 = (\\gamma-1)(1-\\chi)\/\\chi$.\n\\black\nWe investigate two cases - a hot corona with $\\epsilon=2\/3$ and a version with $\\epsilon=1\/6$ \n\\red\n($\\chi=0.5$ and $\\chi=0.8$)\n\\black.\n\nIf we interpret $r=1$ as the innermost stable circular orbit (ISCO) around a black hole, $v_{\\rm K}$ is a measure of the black hole spin.  In the case of a Schwarzschild black hole it is $v_{K}\\simeq0.6c$ while we choose the scaling velocity $v_{\\phi}(r=1) = 0.5 c$ for convenience.  \n\nThe inner disk-edge is numerically difficult to model because of the transition to the inflow of the disk-wind and the steep gradients in gravity.  Within $r<1$, the so-called plunge region, a physical solution would allow for (radial and vertical) accretion onto the central object. \nSince the dynamics in this area would then require a general relativistic treatment  which we cannot provide in this context, we simply minimize the dynamical effect of this region by freezing the hydrostatic solution initially present in the domain.  \nOther authors have assumed a thin funnel flow along the axis (e.g. \\cite{1999ApJ...526..631K}) or added an internal sink-cell (like \\cite{2002ApJ...581..988C}) to circumvent this problem \nThe transition between the ``inner corona'' and disk-wind is smoothed via the Fermi step function \n$F(x) = (1+e^{(1-x)\/0.1})^{-1}$.  \nRotational support is thus turned on by the setting \n\\begin{eqnarray}\n\\rho_{\\rm d}(r,z) &=& \\frac{1}{1-\\chi F^2(R)} (R+r_{S})^{1\/(1-\\gamma)}\\label{eq:rhodisk},\\\\\nv_{\\phi}(r) &=& \\sqrt{\\chi} v_{K} F(R) R^{-0.5}.\n\\end{eqnarray}\nDensity given by equation \\ref{eq:rhodisk} and the coronal pressure $p$ constitute the third and fourth fixed in time conditions.  \n\nWe emphasize that is is not possible to specify both injection velocity and density-profile and thus the mass-flux for sub-(magneto)sonic flows, \nas this is determined by the sonic point.  \nTherefore we match the vertical velocity $v_{z}$ to the domain via $\\partial_{z} v_{z} = 0$, \nwhile the radial component follows from the $E_{\\phi}=0$ condition. \nWe limit the injection speed by the local slow magnetosonic speed in the case when \nthe velocity just above the boundary becomes trans-sonic.\nThis provides the fifth constraint needed in that case.  \n\nWith the induction of a toroidal magnetic field component in the jet, the rotational velocity \nneeds to be adjusted in order to satisfy Eq.\\,\\ref{eq:omegaf},\n\\begin{equation}\nv_{\\phi} = r\\Omega^F + \\frac{v_{p}}{B_{p}}B_{\\phi}\n\\end{equation}\nas we apply the condition $\\partial_{z}B_{\\phi} = \\partial_{z} B_{r} = 0$ \n(while $B_{z}$ then follows from $\\mathbf{\\nabla\\cdot B = 0}$).  \n\nBy letting the jet solution alone determine Poynting- and mass- flux, \nwe loose control over the energy flux parameter $\\mu$ and the limiting asymptotic \nLorentz factor $\\Gamma^*$.  \nIt will rather be a consequence of the MHD under the constraints we have given, while we have used our freedom to provide a boundary most closely resembling a realistic hot disk corona.\nA graphical summary of the disk wind boundary conditions is shown in Fig.\\,\\ref{fig:disk}. \n\nIn order to extend the parameter space towards higher $\\mu$, we also investigate\ncases where we have over-determined the boundary conditions by specifying the mass-flux \nthrough\n\\begin{equation}\nv_{z}(r,0) = v_{\\rm inj} v_{\\phi}(r,0)\n\\end{equation}\nsimilar to e.g. \\citet{1997ApJ...482..712O, 2002A&A...395.1045F}.\nAnother parameter run adopts a $1\/r$ profile for the toroidal \nmagnetic field $B_{\\phi}(r) = - \\eta F(r) \/ r $ in order \nto specify the Poynting flux with an additional parameter $\\eta$.  \nThese runs \n\\red\nand the influence of the over-determination \n\\black\nis discussed separately in section \\ref{sec:extended}.  \n \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f3}\n\\caption{\nProfiles of the fixed in time variables for the inlet in hydrodynamic equilibrium.  Here we give constraints on $\\Omega^F,\\rho,p,E_{\\phi}$ ($E_{\\phi}=0$ not shown).  \nParameters are $v_{\\rm K}=0.5,\\epsilon=2\/3$. \nThe thin dotted line is the Fermi step function used to smooth those variables experiencing a sharp transition at the inner disk radius $r=1$.  \n\\label{fig:disk}}\n\\end{figure}\n\n\\subsubsection{Outflow boundaries ($R_{end},Z_{end}$)}\nThe standard outflow boundary conditions for many numerical codes are zero-gradient \nconditions, which are usually sufficient as the plasma velocities are constrained to be outward-pointing.  \n\n\\red\nIn the case of sub fast-magnetosonic outflows, this strategy is unfortunately insufficient as the flow inside of the domain will depend on the flow beyond the boundary via the incoming characteristics.  \nJust as for the inlet boundary, the now missing information has to be supplied by constraints that describe best the physical conditions downstream of the boundary.  \nIn the case of an outflow, the conditions leading to an untampered flow are however impossible to know a priori.  \nA way to circumvent this unphysical feedback is to avoid any causal contact by moving the boundary far away such that the characteristics will not enter the domain of interest within the simulated time.  \n\\black\n\n\\red\nWhen considering a boundary outside of causal contact ``very far away'', we estimate for Alfv\\'en waves to travel over $10^3$ scale radii within the anticipated simulation time.  The computational effort of such huge grids does not allow a large parameter study at the current time and we must leave this option for future endeavours.  \n\\black\n\n\n\nIn the absence of a substantially better solution, zero-gradients are  used for the \nprimitive variables except for magnetic fields for which this simple approach leads \nto artificial electric currents implying an inward-pointed Lorentz-force.  \nEspecially for low plasma-$\\beta$ this may result in a devastating artificial\ncollimation - preventing any steady state to establish and artificially collimating \nthe outflow increasingly thin with time. \n\n\\cite{1999ApJ...516..221U} have performed a systematic study comparing different \napproaches for outflow conditions including a (toroidal) force-free condition \n$\\mathbf{j_{p}||B_{p} = 0}$ and a more sophisticated version including an additional numerical factor that needs to be determined a posteriori.  \nFor the outflow conditions in our simulations we instead recover the magnetic field \ncomponents by imposing constraints on the poloidal \n($j_{r}=-\\partial_{z} B_{\\phi}$, $j_{z}=r^{-1}\\partial_{r}rB_{\\phi}$) and\ntoroidal $(j_{\\phi}=\\partial_{z}B_{r}-\\partial_{r}B_{z})$ electric  currents.  \nFor the toroidal magnetic field (poloidal electric current) we radially extrapolate \nthe expected $1\/r$ law of a marginal $j_{z}$ at the radial end ($\\rm R_{end})$.\nUsing $\\partial_{z}B_{\\phi}=0$ allows to specify $j_{r}$ at the upper end of the domain ($\\rm Z_{end})$.  \nConcerning the poloidal magnetic field components we implement a current-free \nboundary condition by enforcing $j_{\\phi} = 0$.\nThis is a novel approach designed to minimize spurious effects of collimation.\n\\red\nWe convinced oursevels that boundary effects have only a marginal effect on the solution by varying the grid-size and geometry.  \nFor a detailed discussion and comparison of various outflow conditions we refer to appendix \\ref{sec:zerocurrent}.  \n\\black  \n \nWe note that, as a further complication, a fully relativistic version for a force-free\nor force-balance boundary conditions would also need to take into account electric \nforces. \nWe have estimated the impact of such an upgrade and found that due to the geometry\nof our outflow (in particular the location of the light surface)\nit would play a minor role and is thus not worth the effort to implement.\n\n\\subsection{Initial conditions}\nAs initial state we prescribe a force-free coronal magnetic field,\n$F^{\\alpha \\beta}j_{\\beta} = 0$, together with a gas distribution in hydrostatic\nequilibrium.\nBoth is essential in order to avoid artificial relaxation processes\ncaused by a non-equilibrium initial condition.\nWe apply a polytropic equation of state $p=K\\rho^\\gamma$ with a ``classical''\npolytropic index of $\\gamma=5\/3$ since our flows are always cold when compared\nto the rest-mass.  \nTo further strengthen this choice, we performed a comparison simulation with the \n\\cite{1948PhRv...74..328T} equation of state as described by \\cite{Mignone2005} \nwhich produced an identical jet once the hot shock has passed through.  \nThe constant $K$ is determined by the radial force-balance of the inlet.  \n\nFor the initial magnetic field configuration we apply two different geometries.\n\\red\n\tField configuration A\n\\black\nis a potential field of hourglass shape as applied by \n\\citet{1997ApJ...482..712O} and \\citet{2002A&A...395.1045F} with the magnetic field components\n\\begin{eqnarray}\nB_{r} &=& \\frac{1}{r} \n\t\\left[1-\\frac{(z+z_{d})}{\\left(r^2+(z+z_{\\rm d})^2\\right)^{1\/2}}\\right]\\\\\nB_{z} &=& \\frac{1}{\\left(r^2+(z+z_{\\rm d})^2\\right)^{1\/2}},\n\\end{eqnarray}\ncorresponding to a vector potential \n\\begin{equation}\nA_{\\phi} =\\frac{1}{r} \\left[\\sqrt{r^2+(z+z_{d})^2}-(z+z_{\\rm d})\\right]\\label{eq:aphi-op}\n\\end{equation}\nin cylindrical coordinates with $B_{r} = -\\partial_{z} A_{\\phi}$ and\n$B_{z} = r^{-1}\\partial_{r} \\ r A_{\\phi}$.\nThe dimensionless disk thickness $z_{\\rm d}$ with $(z_{\\rm d}+z)>0$ is introduced to\navoid kinks in the field distribution for $z<0$ (the ghost zones) and we choose $z_{d}=1$ for convenience.  \n\nOur other option for the initial magnetic field \n\\red\n(configuration B)\n\\black\nis the ``split monopole'' \n\\citep{1987PASJ...39..821S} with the magnetic field components  \n\\begin{eqnarray}\nB_{r} &=& \\frac{r}{\\left(r^2+(z+z_{d})^2\\right)^{3\/2}}\\\\\nB_{z} &=& \\frac{z+z_{d}}{\\left(r^2+(z+z_{d})^2\\right)^{3\/2}}.\n\\end{eqnarray}\nIn the split-monopole setup \nthe parameter $z_{\\rm d}$ defines the offset of the \nfiducial center of the monopole from the grid origin, and adjusts \nthe initial angle of the field lines with respect to the disk-surface.\nWe either adopt an angle of $\\theta=85^\\circ$ or $\\theta=77^\\circ$ for the field line passing through $r=1$.\nSimilar to Eq. \\ref{eq:aphi-op}, the split monopole field can be described by a vector potential\n\\begin{equation}\nA_{\\phi} = \\frac{1}{r}\n\\left[1 - \\frac{z+z_{\\rm d}} {\\left(r^2+(z+z_{\\rm d})^2\\right)^{1\/2}}\\right].\n\\end{equation}\n\nThe fields are scaled to satisfy the the choice of the plasma-$\\beta$\n\\begin{equation}\n\\beta \\equiv \\left.\\frac{B_{p}^2}{8\\pi p}\\right|_{r=1,z=0}\n\\end{equation} \nat the inner disk radius.  \n\n\\red\n\tIt should be kept in mind that plasma-$\\beta$ largely varies along the disk boundary.  In configuration A, the profile $\\beta(r)$ monotonically decreases until for large radii it is $\\beta(r)\\propto r^{-0.5}$ leading to a magnetically dominated outer corona.  \n\tIn the split-monopole, $\\beta(r)$ decreases first to a minimum value (at $r^*(\\theta=77^{\\circ})\\approx5$ and $r^*(\\theta=85^{\\circ})\\approx15$) and increases for large radii according to $\\beta(r)\\propto r^{1.5}$ leading to thermal dominance.  \n\\black\n\nIn summary, for our injection boundary condition we are left with the following five dynamical parameters,\n\\begin{equation}\n(v_{\\rm K},\\beta,\\epsilon, v_{\\rm inj},\\eta),\n\\end{equation}\nwhere strictly speaking we are only allowed to choose the first three when launching \nsub-slow. \nAn overview of the simulations performed in this parameterization is shown in \nTab.\\,\\ref{tab_all}.  \n\n\\subsection{Numerical grid and physical scaling}\n\nWe use a numerical grid of $512\\times1024$ cells applying cylindrical coordinates.\nOnward from the inner region $(r<1,z<1)$, which is resolved with $20\\times20$ \nequidistant cells, we apply a stretched grid with the element size increasing \nby a factor of $\\lesssim 1.005$. \nThis leads to a domain size of $(r\\times z)=(100\\times200)r_{\\rm i}$ \ncorresponding to $(300\\times600)\\ r_{\\rm s}$ if $r_{\\rm i}=3r_{\\rm s}$ (see sketch in figure \\ref{fig:setup}).  \nStaggered magnetic fields treated via constrained transport \\citep{1999JCoPh.149..270B} are used to ensure $\\mathbf{\\nabla\\cdot B = 0}$.\n\nBecause of the constraints imposed on the cell aspect ratio by the zero-current \nboundary (appendix \\ref{sec:zerocurrent}), \nwe set the last five grid cells to be equally spaced with maximal aspect \nratios $<3\/1$.  \n\n\\red\nThe dimensionless nature of our simulations allows for various astrophysical interpretations.  We provide a physical scaling of simulation variables (marked with a prime) in the following paragraph.  \n\nSince velocities are given in terms of the speed of light ($c'=1$), relativistic simulations are in need of only two additional scales.    \nThe simulation variables are connected to their physical counterparts via\n\\begin{eqnarray}\nv&=&v'c;\\ \\ l=l'l_{0}; \\ \\ t=t't_{0}=t'l_{0}\/v_{0}; \\ \\ \\rho = \\rho ' \\rho_{0} \\\\\np &=& p'p_{0} = p' \\rho_{0}c^2; \\ \\ B = B'B_{0} = B'\\sqrt{4\\pi\\rho_{0}c^2}.\n\\end{eqnarray}\nIf we assume a Schwarzschild black hole as central body, we may set the spatial scale $l_{0}=6r_{g}$, equating the inner disk radius with the ISCO.  Then it becomes\n\\begin{eqnarray}\nv_{0} &=& 3\\times 10^{10} {\\rm cm}\\ {\\rm s}^{-1}\\\\\nl_{0} &=& 9\\times 10^{5}{\\rm cm} \n                       \\left(\\frac{\\rm M_{\\bullet}}{\\rm M_{\\odot}}\\right)\\\\\nt_{0} &=& 3\\times 10^{-5} {\\rm s} \n                       \\left(\\frac{\\rm M_{\\bullet}}{\\rm M_{\\odot}}\\right).\n\\end{eqnarray}\nAssuming a physical outflow mass-loss rate in terms of the Eddington limited accretion rate\n$\\dot{M}=0.01\\dot{M}_{\\rm edd}$ \nwe can provide a scale for the density by comparison to the mass loss rate of the simulation $\\dot{M}'$\n\\begin{equation}\n\\rho_{0} = 6\\times 10^{-7}\\frac{1}{\\dot{M}'}\\left(\\frac{M_{\\bullet}}{M_{\\odot}}\\right)^{-1}\\rm g\\ cm^{-3}\n\\end{equation}\nwhere we applied a radiative efficiency of $\\eta^*=0.1$.  \nThe scaling of pressure and magnetic fields then follows as \n\n\\begin{eqnarray}\np_{0} &=& 5\\times 10^{14}\\frac{1}{\\dot{M}'}\\left(\\frac{M_{\\bullet}}{M_{\\odot}}\\right)^{-1}\\rm g\\ cm^{-1}\\ s^{-2} \\\\\nB_{0} &=& 8\\times 10^7\\left.\\dot{M}'\\right.^{-0.5}\\left(\\frac{M_{\\bullet}}{M_{\\odot}}\\right)^{-0.5} \\rm Gauss . \n\\end{eqnarray}\n\nUnder these considerations, the only remaining scaling parameter is the mass of the compact object $M_{\\bullet}$.  \nNeglecting additional physical processes as radiation pressure or radiative \ncooling leaves us with a scale-free model that can be applied to any disk-wind\nlaunched jet around compact objects.  \nTable \\ref{tab:scaling} provides a fiducial scaling for a microquasar with $M_{\\bullet}=10 M_{\\odot}$ and for an AGN with $M_{\\bullet}=10^8 M_{\\odot}$.  \n\\black\nThe scale-free nature becomes obvious if we recall the rest-frame temperatures\nfor an ideal gas,\n$\n        T = \\frac{p'\\ c^2}{\\rho' k_{\\rm B}}\\langle \\mu \\rangle \\rm m_{\\rm p}\n$\n\\citep{1989cup..book.....A},\nwith the mean molecular weight $\\langle \\mu\\rangle$,\n    the proton mass ${\\rm m_{\\rm p}}$,\nand the Boltzmann constant $k_{\\rm B}$.\nFor ionized hydrogen $\\langle\\mu\\rangle=0.5$ one would find temperatures of\n$T = (p'\/\\rho')\\ 5.45\\times10^{12} \\rm K$,\nwhile for an electron-positron plasma ($\\mu\\simeq1\/2000$) the temperatures are \nlower by three orders of magnitude.\nThese ultra-high scaling temperatures are only reached in the very inner corona between\nthe central object and the inner disk radius. \nAs we do not intend to follow the dynamics here, this does not really pose a problem. \nIn principle, once $p'\/\\rho'\\gtrsim 1$, the\nequation of state transcends towards $\\gamma = 4\/3$ according to a Synge-gas.\nIn the jet, thermal pressure quickly looses importance and the temperatures \nare significantly lower.\n\n\n\n\\begin{deluxetable}{lllllll}\n\\tablewidth{\\columnwidth}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Fiducial scaling\\label{tab:scaling}\n}\n\\tablehead{\n\\multirow{2}*{ $\\frac{M_{\\bullet}}{M_{\\odot}}$  }&\n$l\/l'$ & \n$t\/t'$ &\n$\\rho\/\\rho'$& \n$p\/p'$ &\n$B\/B'$ \\\\\n& \n$\\rm [cm]$ &\n$\\rm [s]$ &\n$\\rm [g\\, cm^{-3}]$ &\n$\\rm [g\\, s^{-2}\\, cm^{-1}]$ &\n[Gauss]\n}\n\\startdata\n$10^8$ & \n$9\\times10^{13}$&\n$3\\times 10^{3}$ &\n$1.8\\times10^{-16}$  &\n$1.5\\times10^{5}$  &\n$1.4\\times 10^3$\\\\\n$10$ &\n$9\\times10^{6}$ &\n$3\\times10^{-4}$&\n$1.8\\times10^{-9}$ &\n$1.5\\times10^{12} $&\n$4.4 \\times 10^{6}$\n\\enddata\n\\tablecomments{Scaling for simulation WA05 ($\\dot{M}'=32.67$) assuming a physical mass loss rate of $\\dot{M}=1\\%\\dot{M}_{edd}$ with an efficiency of $\\eta*=0.1$.  }\n\\end{deluxetable}\n\n\n\\section{Results and discussion}\\label{sec:results}\n\nWe now present the results of our numerical simulations considering the formation\nof relativistic MHD jets from accretion disks.\nEach simulation consumed approximately 48 hours on 16 processors.  \nThe overall goal is to test whether the paradigm of MHD self-collimation\nof non-relativistic jets established from numerical simulations\n\\cite{1995ApJ...439L..39U, 1997ApJ...482..712O, 1999ApJ...526..631K, 2002A&A...395.1045F}\nalso holds in the relativistic case.\n\n\n\\begin{deluxetable*}{ccccccl||crrrrr}\n\\tablecaption{Parameter summary of our disk-wind simulations. \\label{tab_all}\n}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{\\textwidth}\n\\tablehead{\n \\colhead{ID}                      & \n \\colhead{Top}               &\n \\colhead{$\\beta$}                     & \n \\colhead{$\\epsilon$}       & \n \\colhead{$v_{\\rm inj} $}        & \n \\colhead{$\\eta$}         &\n \\colhead{Remarks}      &\n \\colhead{$\\Gamma_{\\rm max}$} &\n \\colhead{$\\mu_{\\rm max}$} &\n  \\colhead{$\\xi$} &\n \\colhead{$v_{p,\\rm max}$} &\n \\colhead{$r_{\\rm jet}$} &\n \\colhead{$\\dot{M}$}\n }\n\\startdata\n\n\n\n\n\n\n\n\nWA01 & A & 0.2 & 2\/3 & var. & var. & - & \n              1.23&         1.33 &        18.84 &       0.54 &        20.26 &        23.77\n\\\\\n\nWA02 & A & 1 & 2\/3 & var. & var. & - & \n       1.27 &         1.37 &        11.47 &       0.58 &        21.86 &        26.62\n\\\\\nWA03 & A & 2 & 2\/3 & var. & var. & - & \n\n1.26&         1.34 &        10.69 &       0.57 & \n       22.21 &        24.15\n\\\\\nWB01 & B & 1 & 2\/3 & var. & var. & $\\theta=77^\\circ$ & \n       1.33&         1.41 &        8.27 &       0.64 &        24.19 &        16.48\n\\\\\nWB02 & B & 1 & 2\/3 & var. & var. & $\\theta=85^\\circ$ & \n1.29&         1.33 &        8.19 &       0.60 & \n       21.27 &        15.66\n\\\\\nWA04 & A & 0.2 & 1\/6 & var. & var. & - & \n       1.27&         1.38 &        13.40 &       0.58 &        21.14 &        36.08\n\\\\\nWA05 & A & 1 & 1\/6 & var. & var. & - &\n1.25&         1.33 &        9.84 &       0.57 &         22.77 &        32.67\n\\\\\nWA06 & A & 2 & 1\/6 & var. & var. & - &\n1.25&         1.32 &        9.34 &       0.57 & \n       22.92 &        29.27\n\n\n\n\n\n\n\n\\enddata\n\\tablecomments{Columns are from left to right: simulation ID; \n initial magnetic field distribution (Top): potential field (A) or \n     split monopole (B);\n plasma-$\\beta$; accretion disk temperature parameter $\\epsilon$; \n     injection speed $v_{\\rm inj}$ as a fraction of $v_{\\phi}(r)$;\n     toroidal disk-field scaling $\\eta$;  specific remarks; to the right we show values of the steady state, the maximum Lorentz factor $\\Gamma_{\\rm max}$ collimation degree $\\xi$, maximal poloidal velocity $v_{p,\\rm max}$, jet radius $r_{\\rm jet}$ and the total mass flux out of the domain $\\dot{M}$. \n}\n\\end{deluxetable*}\n\n\n\n\\subsection{Overall evolution of the outflow}\n\nThe initial evolution of the disk corona is governed by the propagation of \ntoroidal Alfv\\'en waves launched due to the rotation of the field line \nfoot-points.  \nThe initial force-free magnetic field structure is adapted to a new dynamic\nequilibrium according to a rotating wind magnetosphere.\n\nA wind is launched from the disk boundary and is continuously accelerated \ndriving a shock front through the initial hydrostatic corona and sweeping\nthis material out of the computational domain (Fig.\\,\\ref{fig:simul}).\nThe disk wind evolves into a collimated outflow of super-magnetosonic speed.\nAlong the symmetry axis the hydrostatic initial condition is very well \npreserved.\nOnce the bow shock has passed through the domain, the jet mass flux declines\nto a value which is solely governed by the internal outflow dynamics\nand the injection boundary conditions. \nSimilarly, the post-shock magnetic field distribution follows as well\nfrom the internal outflow dynamics and has in principle little in common with \nthe initial setup.  \nCertain combinations of boundary conditions for mass flux and magnetic field\nwill result in a quasi-stationary\\footnote{We denote the dynamical state as\n{\\em quasi} stationary as due to Keplerian disk rotation the disk outflow\na large radii has evolved for a considerably lower number of disk rotations.\nTherefore, a slight change in the dynamical state of the outer outflow\ncan be expected after another few outer, thus 1000s of inner rotations.}\nstate of the outflow evolution \n(see next section).\nFrom this point onwards we can start our investigations of collimation and\nacceleration.\nIn this paper we concentrate on analysis when the flow has reached a\nquasi-steady state.\nWe usually terminate our simulations after 500 inner disk-rotations $P$, \nwhile a quasi-steady state is established over most of the domain after \nabout 200 rotations.\n\nFigure \\ref{fig:simul} shows the time evolution for two exemplary \nsimulations with an initial hourglass-shaped potential field distribution\n(case A) and a split-monopole field distribution (case B),\neach for the parameter choice $(\\beta,v_{K},\\epsilon) = (1,0.5,2\/3)$.\n\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{f4}\n\n\n\n\n\n\\caption{\nFormation of a relativistic MHD jet.\nShown is the Lorentz factor (color gradient) in terms of $log_{10}(\\Gamma-1)$ \nat the time of 25, 250 and 500 (left to right) inner disk rotations. \n{\\em Top:} Field distribution case A, hourglass-shape potential field \n(run WA02).\n{\\em Bottom:} Split monopole initial field distribution case B (run WB01). \nShown are poloidal magnetic field lines (solid white lines);\nthe critical MHD surfaces (solid black lines);\nthe light surface $r\\cdot \\Omega = c$ (solid black line). \nFor time $T\/P=250$ electric current flow lines are added (solid green line).\nTime step $T\/P=500$ also show the initial field lines for comparison \n(dashed white lines).\n\\label{fig:simul}}\n\\end{figure*}\nThe figure shows the \nLorentz factor, the poloidal magnetic field lines, poloidal electric current flow lines and the critical MHD surfaces.\nIn addition the light surface\nis drawn.\n\nPhenomenologically, the solutions form a magnetic nozzle with, depending on the disk flux distribution, considerable difference in the width, but comparable final opening angles of the fast component.  A broader initial field distribution (case B) also results in broader and faster winds where the material originating from the inner disk is more effectively thinned out.  \nIn analogy to hydrodynamic nozzles, the flow reaches the slow-magnetosonic speed directly above the throat.  \nCollimation happens mainly before the fast-magnetosonic surface is reached. Afterwards, the opening angle of a given field line is approximately conserved.  \n\nOf particular interest is the electric current distribution\n(shown for the time step $T\/P = 250$).\nThe electric current distribution \nis a consequence of the dynamical\nevolution of the outflow and therefore a direct outcome of the\ndisk boundary magnetic flux profile and the Keplerian field line rotation.\n\nIn general, the electric current leaves the outer disk to return within the fast component of the outflow.  \nIt is expected to enter the inner disk and then flow radially outwards \nclosing with the outgoing current.\nSuch butterfly-shaped circuits are expected in Keplerian disks while\nthe $j_{r}$ plays a leading role \nin the disk-jet feedback \\citep{1997A&A...319..340F}.\nA positive radial electric current in the disk-corona supports accretion \nby braking the disk material due to its magnetic torque $j_{r}\\times B_{z}$ \nsimilar to a Barlow-wheel\\footnote{\nHowever, this region is not resolved \nwithin our numerical domain, as it is located below our injection boundary\nas part of the underlying non-ideal MHD accretion disk. \nSee \\cite{2002ApJ...581..988C, 2007A&A...469..811Z} for non-relativistic\nsimulations of the disk-jet interaction}.\n\nThe inclination between the poloidal current vector and the magnetic field line\nindicates the direction of (de-) collimating magnetic forces acting on the flow.\nWhen the inclination becomes less than $90^{\\circ}$, the Lorentz force\n$j_{p}\\times B_{\\phi}$ changes from collimation to de-collimation.  \nThis can be clearly seen in the snapshots at $T\/P=250$ of the case A simulation where actual\nfield lines are indicated in white and initial field lines in red.\nIn the actual field distribution, the field lines are somewhat pushed away from\nthe surface $\\mathbf{j_{p} \\perp B_{p}}$ (this is also where magnetic acceleration is most effective).\nFor case B this happens  \nbeyond the light surface.\nAs a result, both electric and magnetic forces deflect the flow towards \nthe disk boundary which leads to a highly unstable layer just above \nthe outer disk.\nWe will provide an in-depth analysis including all forces acting on the flow\nin section \\ref{sec:trans-field}.\n\nThe locations of the characteristic surfaces are signatures for the MHD flow.\nDepending on the initial magnetic flux distribution (case A,B) and the \nmass flux profile (see also \\citet{2006ApJ...651..272F}), \nthis location may vary a great deal. \nIn our case B simulations, we generally observe surfaces which leave the domain\nin radial direction (parallel to the disk surface).\nFor the case A simulations these surfaces tend to \"collimate\" leaving the domain\nin vertical direction. \nThe latter implies a two-layered structure of the jet - a central super-fast magnetosonic jet surrounded by a \nsub-Alfv\\'enic outflow.\nThis is an interesting aspect for observational modeling \nand for stability analysis of sheath-spine jets \n\\citep{2005MNRAS.356..859P, 2007ApJ...662..835M, 2007ApJ...668L..27K, \n       2007ApJ...664...26H, 2009MNRAS.397.1486B}.\nThe broad wind launched from the outer regions of the disk has much lower velocities, decreasing continously with increasing launching-radius. For example, the terminal velocity of the flow originating from $r_{\\rm fp}>32$ of the case A simulations drops below $0.2c$, consistent with the X-ray absorption features observed in a mounting number of AGN\n\\citep{2006AN....327.1012C, 2009A&ARv..17...47T}.  \nIn principle, our dynamical models can provide basic ingredients (e.g. flow geometries and velocity gradients) for the modeling of spectral line profiles of disk winds \\citep{1995MNRAS.273..225K, 2008MNRAS.388..611S}.  \n\n\nFollowing  \\cite{2006ApJ...651..272F}, we may define an average collimation \ndegree $\\xi$ of the outflow measured as the fraction of vertical and radial \nmass flux through equal-area surfaces at a certain height (here at $z=z_{m}$),\n\\begin{equation}\t\n\t\\xi = \\frac{\n\t\\int_{0}^{r_{m}}r \\Gamma v_{z} \\rho |_{z_{m}}\\ dr\n\t}{\n\t\\int_{z_{m}-r_{m}\/2}^{z_{m}}r_{m}\\Gamma v_{r}\\rho |_{r_{m}}\\ dz\n\t}.\n\\end{equation}\nSimilarly, we define a mass flux weighted jet radius,\n\\begin{equation}\n\tr_{jet} = \\frac{\n\t\t\\int_{0}^{r_{m}}r \\Gamma v_{z} \\rho |_{z_{m}} \\ dr\n\t}{\n\t\\int_{0}^{r_{m}} \\Gamma v_{z} \\rho |_{z_{m}}\\ dr\n\t}.\n\\end{equation}\nThe corresponding values for $\\xi$ and $r_{\\rm jet}$ \nderived for $z_{m}=200$ at the upper end of \nthe domain and for time $t\/P=500$ are given in Tab.\\,\\ref{tab_all}\nalong with the maximum Lorentz factor $\\Gamma_{\\rm max}$, the maximum\npoloidal velocity $v_{p,max}$, and the total mass flux $\\dot{M}$.  \nFigure \\ref{fig:diag} shows the time evolution of these quantities in the top panel.  In general, we observe that the collimation degree $\\xi$ is the most sensitive tracer for secular trends among the observables mentioned.  \nIn the lower panel, we show the evolution of jet-power in the individual energy channels leaving the computational domain (radial and vertical).  \n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f5}\n\\caption{\n\\red\nTime evolution of characteristic quantities.  \\textit{Top panel:} After an initial adjustment till $t\/P\\simeq200$, mass flux $\\dot{M}$, jet radius $r_{jet}$, collimation degree $\\xi$, maximal Lorentz-factor $\\Gamma_{\\rm max}$  and poloidal velocity  $v_{p, \\rm max}$ cease to evolve. \n\\textit{Lower panel}: Power in the individual energy chanels out of  the domain. Thermal power ($T$)\npeaks when the shock reaches the upper boundary and is negligible otherwise.\nThe total outgoing power (labeled accordingly) is dominated by rest-mass ($M$) and Poynting flux ($S$). Also shown are the gravitational ($G$) and (purely) kinetic ($K$) contributions (simulation run WA02).  \n\\black\n\\label{fig:diag}}\n\\end{figure}\nAfter the re-configuration of the initial stationary-state to the dynamical solution, the partitioning of energies is completed at around $t\/P=100$.  Thermal energy-flux peaks when the hot bow-shock passes through the upper boundary.  Far away from the central object, gravitational and thermal energy-flux are negligible\nThe integrated energy-flux is dominated by rest-mass, reflecting the fact that only the inner component reaches significant Lorentz-factors.  \nThe balance between Poynting and kinetic flux is of particular interest. Figure \\ref{fig:diag} shows merely the end result of the spatial conversion history with the remaining electromagnetic energy $\\mathcal{S}$ above the purely kinetic part $\\mathcal{K}$. \nMore detailed insight into how this is established is provided in the following section using an individual field line. \n\\subsection{Stationary state analysis}\nSimulations starting from an initial field distribution A evolve into\na quasi-stationary flow solution after about 200 inner disk rotations.\nFigure \\ref{fig:stationary_flow} shows our reference simulation WA04\nat time $t\/P=250$, including an enlarged subgrid of the innermost area\nof the domain.\n\nSteady state solutions are helpful to understand the flow structure\nfor a number of reasons. \nFirstly, by using MHD conservation laws, the conserved quantities \n(see Sect.\\,\\ref{sec:flconst})\nallow to identify the momentum and {\\em energy channels} of the flow during \nacceleration and collimation.\nSecondly, by using the force-balance equations \n\\ref{eq:trans-field},\\ref{eq:parallel-field} we may identify the\nleading forces on the material along the outflow.\nThirdly, the cross-check for conserved quantities provides another test \nfor the quality of our setup and the numerical approach.\nA secondary indicator of stationarity is the alignment of poloidal velocities\nwith the poloidal magnetic field lines, $E_{\\phi} = 0$.\nFigure \\ref{fig:stationary_flow} shows corresponding velocity vectors\nconfirming this picture.\n\nThis is confirmed by checking in detail the complete set of integrals of motion\nof the MHD-flow $k,\\Omega^F,Q,l,\\mu$ as defined in equations $\\ref{eq:kf}-\\ref{eq:mu}$. \nFigure \\ref{fig:const} shows the relative deviation of these quantities\nfrom their average value along a given field line after $t\/P=500$.  \nThe integrals are conserved within $1\\%$-accuracy already right above the \ninjection boundary - clearly demonstrating the quality the choice \nof our numerical setup, in particular the injection boundary conditions\ncarefully constructed from an equilibrium of Keplerian rotation and \ngas pressure.\n\nDue to the differential rotation law, the number of Keplerian rotations \n$t\/P(r)$ scales with radius as $t\/P(r) = (t\/P) r^{-3\/2}$, implying\nthat at the end of our simulations ($t\/P=500$), \nwe have performed roughly one rotation at $r=64$ and half a rotation \nat $r=100$. \nNonetheless the integrals of motion for the field line $r_{\\rm fp}=64$ are \nconserved within $0.1\\%$.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f6}\n\\caption{Field line constants (conserved quantities).\nShown is the deviation from the average value along the field line\nrooted at $r_{\\rm fp}=2$ (simulation run WA02).\n\\label{fig:const}}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8cm]{f7a}\n\\includegraphics[width=8cm]{f7b}\n\\caption{Logarithmic (rest-frame) density of the stationary \nflow (simulation run WA04).\nShown are poloidal magnetic field lines (solid white), \nelectric current flow lines (solid black), \ncharacteristic MHD surfaces (various dot-dashed green), \nsurface of escape velocity (dotted green), \nlight surface (solid green).  \nArrows in the top plot indicate the velocity field.\nThe bottom figure is an enlarged picture of the central region \nindicating the three regimes defined by the light surface.  \n\\label{fig:stationary_flow}}\n\\end{figure}\n\n\\subsubsection{Collimating and accelerating forces}\\label{sec:trans-field}\nIn this section we identify the forces responsible for jet acceleration\nand collimation applying the steady-state parallel and transversal\nforce-equilibrium Eqs.\\,\\ref{eq:trans-field},\\ref{eq:parallel-field}.\n\nFig.\\,\\ref{fig:f-hot} compares these forces for a number of reference \nsimulations (WB01, WA02, WA05)\nalong a field line rooted at $r_{\\rm fp}=2$.\nAs check for consistency, we also show the gradient of \nthe Mach number $a \\equiv B_{p}^2\/4\\pi\\nabla_{||}M^2$ which \njust coincides with the summation \nof the parallel forces, indicating a steady state\n(see yellow solid and black dashed line).\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{f9a}\n\\includegraphics[width=0.49\\textwidth]{f9b}\n\\includegraphics[width=0.49\\textwidth]{f9c}\n\\includegraphics[width=0.49\\textwidth]{f9d}\n\\includegraphics[width=0.49\\textwidth]{f9e}\n\\includegraphics[width=0.49\\textwidth]{f9f}\n\\caption{Accelerating (left column) and collimating (right column) \nforces along the field line rooted at $r_{\\rm fp}=2$ for the three\nreference simulations (from top to bottom):\n{\\it WB01} (split monopole), \n{\\it WA02} (hourglass potential field, hot case), and \n{\\it WA05} (hourglass potential field, cool case).\nShown are the contributions from gravity, gas pressure gradient, \ncentrifugal force, poloidal magnetic field pressure gradient, \npoloidal field tension and pressure\ngradient, toroidal magnetic field pressure gradient and tension, and\nforces due to the electric field.\nVertical lines indicate the critical surfaces - the Alfv\\'en point along\nthis field line, denoted by 'A', the light cylinder radius denoted by 'lc',\nand the fast magnetosonic point denoted by 'F'.\nIn this logarithmic representation a change of sign in the force\ndirection is indicated by the singularities along the graphs.\n\\label{fig:f-hot}}\n\\end{figure*}\n\nIn general, the outflow starts with sonic speed and is first launched by thermal pressure in the hot disk corona, respectively \nthe centrifugal force in the colder version.\nUntil the Alfv\\'en point, the Lorentz-force of the poloidal electric current \n($F_{\\rm pbphi}+F_{\\rm pinch}$) is the main magnetic driver.\nUltimately the poloidal tension ($F_{\\rm curv}$) keeps the\nacceleration up even above the fast surface.\n\nConcerning the transverse force, we reproduce the expected sign-change \nof the curvature (tension) force (first collimating until the Alfv\\'en \nsurface, de-collimating beyond) and the poloidal pressure force \n(de-collimating until the light cylinder, collimating beyond).\nFor the cross-field balance, we observe the following three regimes: \n\nJust on top of the inlet, the main de-collimating forces besides poloidal magnetic pressure are thermal pressure in the hot case (WB02, WA02) and centrifugal support in the colder case (WA05).  Gravity is here the strongest force towards the origin and the situation just reflects the radial force-equilibrium we have applied for the inlet boundary.  This is the hydrodynamic regime.  \n\nAt the Alfv\\'en point, the residual of the pinch- and toroidal pressure-force ($\\mathbf{j_{p}}\\times B_{\\phi}$) is the main collimator, balanced by the centrifugal term.  Thermal pressure quickly looses importance.  \nThis is the magneto-hydrodynamic regime.  \n\nIn the asymptotic region beyond the light-cylinder, de-collimation by electric forces overcomes the centrifugal force and is balanced by the poloidal magnetic pressure that changes its sign at the light-cylinder (best seen in WB01).  \nThis is the relativistic regime.  \n\nTo get a global impression on the relative importance of the individual\nforces we show a radial cut throughout the asymptotic jet in figure\n\\ref{fig:force-balance}.\nThe strongest forces arise across the inner asymptotic light \nsurface which separates field lines of high angular velocity from \nthose in the non-rotating corona along the axis.\nHere the electric de-collimation is essential.\nThe $B_{\\phi}(r)$ profile is curled up from the inner disk radius along \nthe outflow - resulting in a magnetic pressure gradient that works \nin unison with the toroidal field pinch force until at some radius the \ntoroidal field surpasses its maximum and decreases (negative gradient).\n\nThe strong gradients in toroidal field and rotation induce a current sheet\nand give rise to an electric charge.\nThe space charge $\\rho_{\\rm e}=(1\/4)\\pi\\nabla\\cdot \\mathbf{E}$ is positive\nclose to the axis and changes its sign at a critical line as defined\nby \\cite{1969ApJ...157..869G}.\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f10}\n\\caption{\nTrans-field force cut at $z=200$, inner ($lc_{i}$) and outer ($lc_{o}$) light-cylinder.  The differentially rotating field lines are fastest at the inner disk-radius, resulting in the inner light-cylinder - here electric de-collimation is important.  The $B_{\\phi}(r)$ profile is curled up from the inner disk radius onwards and results in a magnetic pressure gradient that works in unison with the pinch force until the toroidal field surpasses its maximum. Within $r<3$, we omit the curves for thermal and poloidal pressure. These terms fluctuate around $\\pm10^{-5}$ while balancing each other.  \n (run WA02)\n\\label{fig:force-balance}}\n\\end{figure}\n\n\n\\subsubsection{Energy conversion}\nIn the simulations where injection is sub-magnetoslow, the energy flux is \nnot a free parameter, but is consistently determined by the simulation of the disk-wind. \nIt is hence of interest how the partitioning and conversion is realized.    \nFrom the values of $\\mu_{\\rm max}$ given in Tab.\\,\\ref{tab_all} it is obvious \nthat our disk-corona supports only mildly relativistic flows \nbelow $\\Gamma=1.5$ (section \\ref{sec:flconst}).  \n\nIn Fig.\\,\\ref{fig:vp} (bottom left panel) we show the efficiency $\\sigma$ of Poynting flux to kinetic \nflux conversion along the field line with $r_{\\rm fp}=2$ in the fast component of the jet.  \n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{f8a}\n\\includegraphics[width=0.49\\textwidth]{f8b}\n\\caption{ \nDynamical quantities as a function of the distance along the \nfield line rooted at $r_{\\rm fp}=2$ for the model WA02. \nVertical lines indicate the slow-magnetosonic, Alfv\\'en, light surface and fast magnetosonic transitions (from left to right).\n\\textit{Left:} \nIsorotation parameter $\\Omega^F$, poloidal and toroidal velocity are shown in the top panel. \nLorentz factor $\\Gamma$, energy conversion efficiency $\\sigma$, \nand normalized total energy flux $\\mu$ in the bottom panel.  The flow is below equipartition already at the base of the jet.  \n\\textit{Right:}\nComplete energy flux ratios.  In the asymptotic region, thermal and gravitational fluxes are obviously negligible. \n\\label{fig:vp}}\n\\end{figure*}\nHere, $\\sigma$ is below equipartition already at the inlet and it further \ndecreases as $\\Gamma$ approaches $\\mu$.  \nIn Fig.~\\ref{fig:vp} (top left) the toroidal velocity shows that the flow decouples \nfrom co-rotation with the magnetic field at the Alfv\\'en point.\nBeyond the Alfv\\'en point, angular momentum is then carried predominantly\nby the magnetic field.  \nThe poloidal velocity increases from low injection value (sonic velocity) to \n$\\sim0.5c$. \nFurther acceleration can not be expected as the bulk of the energy is already \nin kinetic form.  \nThe right panel of Figure \\ref{fig:vp} shows the individual energy channels compared to the\nrest-mass flux for the same field line. \nAt the base of the jet, the strong poloidal electric currents (a strong toroidal field)\ngive rise to an outflow with $\\mathcal{K}<\\mathcal{T}<\\mathcal{-G}<\\mathcal{S}<\\mathcal{M}$, predominantly transporting energy via rest-mass and Poynting-flux.  \n\nThe kinetic energy flux surpasses the thermal flux at the Alfv\\'en point and \nfurther overcomes the gravitational binding energy term shortly thereafter.  This is not surprising, since the escape-surface can be close to the Alfv\\'en surface at least for the inner field lines (see also Fig. \\ref{fig:stationary_flow}).\n\nOnly then, the cold limit $\\mu = \\Gamma(\\sigma+1)$ is applicable - it is certainly \nvalid in the asymptotical outflow where thermal and gravitational energy fluxes \nare negligible.\n\n\n\\subsection{Dependence on the launching environment}\nFor the simulations described up to now, we have performed in addition several \nparameter runs in order to investigate how the resulting jet dynamics\ndepends on the (prescribed) launching conditions - the disk corona\n(see Tab.~\\ref{tab_all}). \nWe now focus on the impact of the plasma $\\beta$ and the disk temperature \nparameter $\\epsilon$.  \n\nIn general, a low $\\beta$ (a stronger magnetic field) we find that the outflow tends\nto collimate more, as indicated by the higher average collimation degree $\\xi$ \nand a lower momentum weighted jet radius $r_{\\rm jet}$.  \nThis in principle decreases the MHD acceleration efficiency which critically \ndepends on the divergence of flux surfaces.  \nIt is straightforward to define the mass flux-weighted (half-) opening angle of \noutflow,\n\\begin{equation}\n\t\\theta_{\\dot{M}} = \\rm atan\\ \\xi^{-1},\n\\end{equation}\nwhich translates to angles of $3^{\\circ}<\\theta_{\\dot{M}}<7^\\circ$ for the \noutflows under consideration.\n\nThe impact of the magnetic field strength on the amount of mass flux is not clearly visible, \nas the two simulations with $\\epsilon=1\/6, 2\/3$ show a different trend. \nAs the $\\epsilon$-parameter is simply a proxy for the disk corona density, \nit will affect the collimation in the following manner: \nA higher inflow density lowers the Alfv\\'en surface towards the disk\nsurface which in turn broadens the current topology \nand therefore widens the flow.  This is also the trend that we observe in the indicators $\\xi$ and $r_{\\rm jet}$.  \nGiven that the injection speed calculated iteratively from the outflow\nsimulation approaches the slow-magnetosonic speed, \nwe expect the mass flux to scale as $\\dot{M}\\propto \\sqrt{p \\rho}$.  \nIn fact, this is approximately realized since we have \n$\\dot{M}(\\epsilon=1\/6)\/\\dot{M}(\\epsilon=2\/3) =\\sqrt{2.5}\\simeq 1.6$.  \n\nThe change of the initial split-monopole inclination $\\theta$ has little effect\non the overall jet collimation angle.\nIn particular, we observe an opposite trend as\nthe wider initial field with $\\theta=77^\\circ$ ends up slightly more\ncollimated than the one with $\\theta=85^\\circ$.  \nClearly a wider initial field leads to a larger jet radius $r_{\\rm jet}$.  \nHere, we like to stress the point that for the final steady state solutions\nin our simulations the initial field structure is important only insofar\nas it also prescribes the poloidal magnetic field profile along the outflow\nlaunching boundary. \nThe field structure is completely changed from the initial steady structure\nto a new dynamic equilibrium.\nThus it makes no sense to compare the collimation of the initial field\nwith the collimation of the outflow field distribution. \n\n\n\\par \nHaving pointed out the crucial role of the \n\\red\nvertical energy flux from the disk surface \n\\black\n$\\mu$ and the closely\nrelated quantity $\\sigma = \\mathcal{S\/(K+M)}$, \nwe now study the two most promising handles in increasing $\\mu$.  \nThat is (i) a decrease in mass flux $\\mathcal{M}$ and \n        (ii) an increase in Poynting-flux $\\mathcal{S}$.  \nWe first focus on (i) and describe (ii) thereafter.  \n\n\\subsubsection{Towards low mass loading}\\label{sec:light-winds}\n\\red\nA way to obtain high-speed jets seems to be a lower mass\nload injected into a similarly-strong magnetic flux.\nAccording to the well-known Michel-scaling \\citep{1969ApJ...158..727M}, \nthe asymptotic outflow velocity depends on the mass flux \n$u_{\\infty}\\propto \\dot{M}^{-1\/3}$.  \nWe investigate this interrelation running another set of simulations \nwhere we change the mass flux by prescribing a low injection velocity\n(instead of lowering the density of the injected gas,\n see Tab.~\\ref{tab:extended1})\n\n\\begin{deluxetable*}{ccccccl||crrrrr}\n\\tablecaption{Effect of mass-loading\n\\label{tab:extended1}}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{\\textwidth}\n\\tablehead{\n \\colhead{ID}                      & \n \\colhead{Top}               &\n \\colhead{$\\beta$}                     & \n \\colhead{$\\epsilon$}       & \n \\colhead{$v_{\\rm inj} $}        & \n \\colhead{$\\eta$}         &\n \\colhead{Remarks}      &\n \\colhead{$\\Gamma_{\\rm max}$} &\n \\colhead{$\\mu_{\\rm max}$} &\n  \\colhead{$\\xi$} &\n \\colhead{$v_{p,\\rm max}$} &\n \\colhead{$r_{\\rm jet}$} &\n \\colhead{$\\dot{M}$}\n }\n\\startdata\nI001 & A & 0.2  & 2\/3  &  0.01  &  var.  & - &  \n1.66&         2.01 &        25.61 &       0.75 & \n       21.11 &        10.63\n\\\\\n\nI01 & A & 0.2  & 2\/3  &  0.1  &  var.  & - &  \n       1.52&         1.76 &        25.58 &       0.71 & \n       20.49 &        13.78\n\\\\\n\nI02 & A & 0.2  & 2\/3  &  0.2  &  var.  & - &  \n1.38&         1.55 &        26.02 &       0.65 & \n       19.34 &        17.53\n\\\\\n\nI04 & A & 0.2  & 2\/3  &  0.4  &  var.  & - &  \n1.31&         1.45 &        25.30 &       0.60 &        17.07 &        25.84\n\\\\\nI08 & A & 0.2  & 2\/3  &  0.8  &  var.  & - &  \n1.16&         1.24 &        40.05 &       0.47 &        14.74 &        51.30\n\\\\\nI12 & A & 0.2  & 2\/3  &  1.2  &  var.  & - &  \n       1.21&         1.21 &        33.77 &       0.50 &        14.10 &        81.88\n      \n\\enddata\n\\tablecomments{As in table \\ref{tab_all}, but for the extended parameter study with a given $v_{\\rm inj}$.}\n\\end{deluxetable*}\n\n\nHowever, we observe that for low injection speeds $v_{z}<0.4\\ v_{\\phi}$ \nthe resulting (numerical) mass flux does not follow the expected linear \nrelation $\\dot{M} \\propto v_{\\rm inj}$.\nInstead, for lower and lower injection speed it approaches some\nseemingly unphysical offset value (Fig.~\\ref{fig:diag-comp}).\nThis value is unphysical in the sense that it departs from the \nprescribed value at the boundary condition \n$d\\dot{M}_{\\rm inj}\/dr = 2\\pi r \\rho_{\\rm inj} v_{\\rm inj}$.\nThis difficulty arises because the injected flow is sub-magnetosonic\nand thus overdetermined by simultaneously assigning a profile in $\\rho$ and $v_{z}$.\n\n\nAlthough this is in principle problematic, it is not necessarely fatal \nfor the investigation of low mass flux outflows.\nThe mass flux is governed by the magneto-slow point and is thus re-arranged\nalong the flow.\nIndeed we find that above the magneto-slow surface the field line constants \nare very well conserved. \nThis indicates that the flow dynamics transcends at the magneto-slow surface \nfrom a seemingly unphysical state into a MHD flow that satisfies the critical \nconditions at the magnetosonic surfaces.\n\nThis technique of self-adjusting the sub-slow mass inflow, however, turned out\nto be limited towards lower mass fluxes.\nThe resulting mass fluxes are unfortunately, still too high and do not allow \nsubstantially higher magnetisation\n(e.g. $\\mu_{\\rm max}=2.01$ for simulation I001 compared to $\\mu_{\\rm max}=1.33$ in run WA01).  \nFigure \\ref{fig:diag-comp} shows the trends concerning collimation $\\xi$, jet radius $r_{\\rm jet}$,\nand integral mass flux $\\dot{M}$ in the asymptotic flow. \n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f11}\n\\caption{Jet collimation and dynamics against injection speed parameter $v_{\\rm inj}$, The horizontal line indicates the mass flux when $v_{\\rm inj}$ is not specified (WA01).  \nFor $v_{\\rm inj}<0.4$, the mass flux approaches an obviously unphysical offset value.\nAt higher $v_{\\rm inj}$, the mass flux follows the expected linear dependence on the injection parameter (thin solid line).  \n\\label{fig:diag-comp}}\n\\end{figure}\nIncreasing the mass flux enhances collimation while $\\mu_{\\rm max}$ decreases accordingly.  \nFor high injection speed, we observe that the wind originating from the very inner disk evolves\ninto a thin ballistic flow layer that has little in common with the jets we are interested in.  \n\\black\n\\subsubsection{Poynting dominated flows}\\label{sec:extended}\nGiven the limitations mentioned above, we prescribe {\\it a priori} \nthe limiting energy flux parameter $\\mu$ by constraining both the mass flux and \nthe Poynting flux along the injection boundary - with the hope of thus providing \na sufficiently energetic disk wind. \n\nTo achieve this, we adopted a fixed-in-time toroidal magnetic field distribution,\n$B_{\\phi}\\propto -\\eta\/r$ which necessarily changes \n$\\Omega^F(r)$\\footnote{\nIn case of a central black hole causality requires that $r\\Omega^F(r)< 0.6$ \nwhich corresponds to the ISCO velocity for the Schwarzschild case with $r_{ISCO}=1$ \nin our scaling.}.\nThe toroidal field distribution following a $1\/r$ profile corresponds to $j_{z}=0$ \nand has a profound physical motivation as it anticipates the radial currents expected\nin the disk-corona.  \n\nTable \\ref{tab:extended2} summarizes the simulation runs performed within this setup. \nThese simulations have a considerably stronger toroidal magnetic field at the injection\npoint.\nWe apply up to $B_{\\phi}\\sim 4 B_{p}$.\n\\begin{deluxetable*}{ccccccl||crrrrr}\n\\tablecaption{Poynting dominated flows\n\\label{tab:extended2}}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{\\textwidth}\n\\tablehead{\n \\colhead{ID}                      & \n \\colhead{Top}               &\n \\colhead{$\\beta$}                     & \n \\colhead{$\\epsilon$}       & \n \\colhead{$v_{\\rm inj} $}        & \n \\colhead{$\\eta$}         &\n \\colhead{Remarks}      &\n \\colhead{$\\Gamma_{\\rm max}$} &\n \\colhead{$\\mu_{\\rm max}$} &\n  \\colhead{$\\xi$} &\n \\colhead{$v_{p,\\rm max}$} &\n \\colhead{$r_{\\rm jet}$} &\n \\colhead{$\\dot{M}$}\n }\n\\startdata\n\nM02 & A & 0.2  & 2\/3  &  0.1  &  2  & - &\n       2.18&         3.29 &        27.55 &       0.86 &        23.71 &        22.48\n\\\\\n\nM04 & A & 0.2  & 2\/3  &  0.1  &  4  & - & \n       3.51&         7.46 &        8.96 &       0.94 &        29.00 &        48.85\n\\\\\n\nM08 & A & 0.2  & 2\/3  &  0.1  &  8  & - &\n       6.11&         25.61 &        6.8377 &       0.97 &        35.89 &        80.40\n\\enddata\n\\tablecomments{As in table \\ref{tab_all}, but for the extended parameter study with a given $v_{\\rm inj}$ and $\\eta$.}\n\\end{deluxetable*}\n\nAn exemplary process enhancing large-scale toroidal fields in the disk corona \ncould be the MRI-driven dynamo under current investigation by many authors \n\\citep[e.g.][]{2000ApJ...534..398M, 2003A&A...398..825V}.\nThe jet eventually evolving from these disks is not propelled by the\n\\citet{1982MNRAS.199..883B} mechanism, but driven by the toroidal magnetic \npressure \\citep{1996LNP...471...74C, 2003ApJ...599.1238D, 2004ApJ...605..307K}. \nThese so called Tower jets have initially been proposed by \\cite{1996MNRAS.279..389L} and directly extract Poynting-flux from the dist rather than first converting rotational energy into the twisted magnetosphere that is present at the Alfv\\'en surface.\\footnote{See \\cite{2007Ap&SS.307...11K} for a review. \nNote also that these jets have successfully been reproduced by \\cite{2005MNRAS.361...97L} in laboratory experiments with purely radial\ncurrent distributions at the base.}\n\nThe simulations described here are of a mixed type, since they combine large-scale open field lines with a toroidal field emerging from a radial current.  \nFor an example simulation of this type, we show the conversion of energy in \nFig.~ \\ref{fig:vp-ext2} similar to Fig.~\\ref{fig:vp} for the low-energy case.  \nBy design, the injected Poynting flux surpasses the rest mass flux with $\\sigma\\simeq5$ \nwithin the fast outflow component.  \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f12}\n\\caption{As in Fig.~\\ref{fig:vp}, however calculated for simulation run M04.  The injected flow is strongly magnetized and remains Poynting-dominated when leaving the computational domain. \n\\label{fig:vp-ext2}}\n\\end{figure}\n\nThe outflow, which is initially Poynting flux dominated, does not reach \nequipartition at the fast magnetosonic surface $r=r_{\\rm F}$, where we merely find $\\Gamma\\sim\\mu^{1\/3}$ following \\cite{1969ApJ...158..727M,1998MNRAS.299..341B}.  \nIn the asymptotic region $r\\gg r_{F}$, the length scales for additional flow acceleration \nand collimation would increase exponentially with $\\Gamma\\propto (\\mu \\ln r)^{1\/3}$ \n(e.g. \\cite{1994PASJ...46..123T}), which is clearly beyond the reach of our numerical \nmethod.  \n\n\nOur simulations indicate that, given sufficient Poynting flux, bulk Lorentz factors derived\nfrom AGN jet observations can be obtained within several hundred Schwarzschild radii, \n$\\Gamma \\simeq 6$ for model M08. \nSince acceleration has proven to be most effective around the Alfv\\'en point \nwhich is expected to be very close to the central object ($r_{\\rm A}<r_{\\rm lc}$), \nwe expect this conclusion to remain valid also for higher $\\mu$ and thus higher \nterminal $\\Gamma$ flows.  \nHowever, we note that when increasing $\\sigma$, the energy conversion efficiency decreases, \na situation commonly denoted as $\\sigma$-problem in pulsar winds \n\\citep{1974MNRAS.167....1R, 1984ApJ...283..694K}.\nSeveral authors have recently addressed this issue with partly controversial results\n\\citep{2009MNRAS.394.1182K, 2009ApJ...699.1789T, 2009ApJ...698.1570L},\nso that after the successful acceleration towards relativistic speeds, Poynting-flux could still remain and one is tempted to ask: Is there a $\\sigma$-problem for AGN-jets? \nThe simulations presented here are not fit to answer this question satisfactory.\nHowever, we certainly know that the mildly relativistic disk winds presented earlier\ndo not suffer from this, as they are launched already in sub-equipartition.  \n\n\\section{Summary}\\label{sec:conclusions}\nWe have presented ideal MHD simulations of the formation of special relativistic disk \nwinds using the PLUTO 3.0 code.  \nOn the technical side, the key points are: \\\\\n\\textit{i)} \nThe inclusion of (Newtonian) gravity allows us to specify an astrophysically sensible boundary\ncondition of a hydrodynamically stable disk corona.\nWe can thus consistently follow the acceleration from initially sub-escape velocity winds.  \\\\\n\\textit{ii)} \nMuch dedication has been put in the development and testing of a novel realization for the\noutflow boundary that enables us to simulate for hundreds of inner disk rotations while \nminimizing spurious collimation due to artificial boundary currents.  \nOur detailed study of jet collimation is possible only through this effort.  \n\nAs a general result we obtain well collimated jets with a mass flux weighted half-opening\nangle of $3 - 7^\\circ$ and mildly relativistic velocities depending on the launching\nconditions for the outflow.\nThe flow collimation happens mainly in the classical (non-relativistic) regime before the \nlight surface.   \nA major result of our simulations is that we - for the first time - self-consistently \ncalculated the shape of that light surface.\nThe light surface determines the \"relativistic\" charater of the flow. Material which\ntraverses the light surface experiences the full relativistic effects.  \n\nWe can identify three dynamically distinct regions in terms of flow collimation. \\\\\n\\textit{i)} \nIn the hydrodynamic regime upstream of the Alfv\\'en surface, \ngravity balances thermal and magnetic pressure, respectively the centrifugal force in the colder case. \\\\\n\\textit{ii)} \nIn the magneto-hydrodynamic regime following the the Alfv\\'en surface downstream,\nthe residuals of magnetic pinch and the toroidal magnetic pressure gradient balances\nthe centrifugal force.  \\\\\n\\textit{iii)} \nIn the relativistic regime located downstrem of the light surface, the poloidal magnetic \npressure gradients now impose a collimating force against electric field de-collimation.\nElectric forces ultimately overcome the classical magneto-centrifugal contribution.  \n\nA steep rotation profile of the field line as given by a Keplerian disk\nresults in a light surface geometry which steepens for large radii.\nDepending on the magnetic field profile, the light surface may even\ncollimate along the flow for large radii.\nIn such a case the relativistic core inside the light surface is naturally confined \nby a non-relativistic wind.\nThe ability of both the relativistic jets and the non-relativistic disk winds to collimate may provide confining agents for an axial ultra-relativistic funnel \nwhich could probably launched by the Blandford-Znajek process.  \n\nThe relatively slow winds found to arrise at large distances of around 100 Schwarzschild radii may be observed as X-ray absorption winds in radio-quiet AGN.  \n\nIn the case of Blandford-Payne disk winds ($B_{\\phi}=0$ initially), the \noutflow is kinetic energy-dominated with the ratio of electromagnetic energy flux \nto kinetic energy flux $\\sigma<1$ already at the jet base and with this ratio \nfurther de-creasing downstream the outflow.  \nThese disk winds start out at sonic speed and reach only mildly relativistic speeds \nup to Lorentz factors $\\Gamma < 1.5$.\nWe have also investigated cases where the jet Poynting flux is increased by\ndirectly injecting an additional toroidal magnetic field from the disk boundary.\nIn this case we achieve terminal Lorentz-factors up to $\\Gamma_{\\rm max}\\simeq 6$\nwhile the super fast-magnetosonic jet remains Poynting-flux dominated.  \n\n\\begin{acknowledgements}\nWe thank Andrea Mignone and the PLUTO team for the possibility to use the\nPLUTO code and for his support with the relativistic module.  \nNumerical simulations were performed on the PIA cluster of the Max Planck Institute for Astronomy (Heidelberg) located at the Rechen-Zentrum in Garching.\nO.P. likes to thank Bhargav Vaidya for his commitment in numerous discussions.  \nWe are pleased to acknowledge clarifying conversations with Max Camenzind.\n\\end{acknowledgements}\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\n\tWe study $\\cat(0)$ groups and the boundaries of the spaces on which they act. A group is $\\cat(0)$ if it acts geometrically (properly discontinuously, cocompactly, and by isometries) on some $\\cat(0)$ space. The motivating question for this article is the following:\n\n\n\n\\begin{question} \\label{question: pconn}\n\tIf $G$ acts geometrically on a 1-ended $\\cat(0)$ space $X$, when is the visual boundary, $\\partial X$, path connected?\n\\end{question}\n\nThere are two main reasons to investigate this property for boundaries of $\\cat(0)$ spaces. The first comes from the theory of hyperbolic groups. All 1-ended hyperbolic groups have locally connected boundaries  \\cite{BestvinaMess,Swarup,BowCutPoints}. Some natural examples of non-hyperbolic $\\cat(0)$ groups, such as $F_2\\times \\mathbb{Z}$, show that this phenomenon does not occur in general. However, a boundary which is connected and locally connected is also globally path connected. In this sense, path connectivity is the `next best thing' one can hope for in the boundary of a $\\cat(0)$ group. At the same time, there are known examples of $\\cat(0)$ groups which act on spaces with non-path connected visual boundaries (see Example \\ref{example: Croke Kleiner} for one). Therefore, we want to find conditions on $G$ (or on $X$) which determine when $\\partial X$ is path connected.\n\nAnother reason to be interested in path connectivity is due to its relationship with semistability. A proper, 1-ended $\\cat(0)$ space $X$ is \\textit{semistable at infinity} if any two geodesic rays are properly homotopic. Being semistable at infinity is a quasi-isometry invarient, so showing a group is semistable at infinity comes down to finding an appropriate space on which this group acts. Geoghegan  conjectured that all $\\cat(0)$ groups are semistable at infinity (in fact, he has conjectured that all finitely presented groups are semistable at infinity). Geoghegan also shows that given a $\\cat(0)$ space $X$ if $\\partial X$ is path connected, then $X$ is semistable at infinity \\cite{GeoghPconnSemistable}. Piecing all of this together, knowing that $G$ acts geometrically on a $\\cat(0)$ space with a path connected visual boundary shows that $G$ is semistable at infinity.\n\nThe majority of this paper is dedicated to answering Question \\ref{question: pconn} for a class of $\\cat(0)$ groups which exhibit more hyperbolicity than others: $\\cat(0)$ groups with isolated flats.  These groups are hyperbolic relative to a collection of flat stabilizers and therefore share many properties with hyperbolic groups. Hruska and Ruane have found necessary and sufficient conditions for a  1-ended $\\cat(0)$ group with isolated flats to have a locally connected boundary \\cite{HR17}. There are many examples which do not have locally connected boundaries, see Section \\ref{section: examples} for a few. We therefore study the general case. In this paper, we prove the following:\n\n\\begin{customthm}{\\ref{thm main}}Let $G$ be a 1-ended $\\cat(0)$ group with isolated flats acting geometrically on $X$. Then $\\partial X$ is path connected.\n\\end{customthm} \n\nAs an immediate corollary (via the results in \\cite{GeoghPconnSemistable}) we have the following:\n\n\\begin{corollary}\n\tLet $G$ be a 1-ended $\\cat(0)$ group with isolated flats. Then $G$ is semistable at infinity.\n\\end{corollary}\nOther authors have some results which overlap with this corollary. For example, Mihalik and Swenson  show that many 1-ended relatively hyperbolic groups are semistable at infinity \\cite{MihalikSwensonSemistable}. However, they do not cover all cases. They assume that their relatively hyperbolic groups have a trivial maximal peripheral splitting and many $\\cat(0)$ groups with isolated flats have non-trivial maximal peripheral splittings. Hruska and Ruane  show that most groups which are hyperbolic relative to polycyclic groups are semistable at infinity \\cite{HruskaRuaneSemistable}. They assume that these groups have no non-central elements of order 2. We require no assumptions on the $\\cat(0)$ groups with isolated flats to conclude it is semistable at infinity.\n\nAlong the way to proving Theorem \\ref{thm main} we introduce a combination theorem for $\\cat(0)$ spaces and their boundaries:\n\n\\begin{customthm}{\\ref{combo}}\n\n\t\tLet $X$ be a $\\cat(0)$ space and $\\mathcal{B}$ a block decomposition. Suppose the following conditions hold:\n\t\n\t\\begin{enumerate}\n\t\t\\item for each $B\\in \\mathcal{B}$, $\\partial B$ is path connected,\n\t\t\\item for each $W\\in \\mathcal{W}$, $\\partial W$ is non-empty, and\n\t\t\\item all irrational rays are lonely.\n\t\\end{enumerate}\n\tThen $\\partial X$ is path connected.\n\\end{customthm}\n\nFor exact definitions of the terms in Theorem \\ref{combo}, see Section \\ref{section block decomp}. The example to keep in mind is when we have a $\\cat(0)$ group $G$ which is an amalgamated product $G=A\\ast_C B$. In many cases of interest, there is a natural `tree-of-spaces' $X$ on which $G$ acts geometrically and then Theorem \\ref{combo} essentially says if (1) $A$ and $B$ act on spaces $X_A$ and $X_B$ with path connected boundaries, (2) $\\partial X$ is connected,  and (3) a technical condition related to how the boundaries of $X_A$ and $X_B$ interact in $X$, then $\\partial X$ is path connected. Amalgamating $\\cat(0)$ groups is the primary method for forming new $\\cat(0)$ groups. In order to fully understand what types of spaces can appear as the boundary of a $\\cat(0)$ group, we need to understand properties of the boundaries of amalgams.\n\n\\begin{Outline}\n\tIn Section \\ref{section: examples}, we describe three key examples that we return to throughout the paper. Sections \\ref{section visual boundary}, \\ref{rel hyp section}, and \\ref{section:isolated flats} provide the necessary background information on the visual boundary, relatively hyperbolic groups, and $\\cat(0)$ groups with isolated flats. We formally define a block decompositions and prove Theorem \\ref{combo} in Section \\ref{section block decomp}. \n\t\n\tThe remaining sections build up to the proof of Theorem \\ref{thm main}. In Section \\ref{section:fixing a space}, for a given 1-ended $\\cat(0)$ group $G$ with isolated flats, we use the group's maximal peripheral splitting (introduced by Bowditch in \\cite{Bow01}), Hruska-Ruane's convex splitting theorem \\cite{HR17}, and Bridson-Haefliger's Equivariant Gluing Theorem \\cite{BH99} to fix a space $X$ on which $G$ acts geometrically. Hruska-Kleiner's work \\cite{HK05} shows that if $G$ is $\\cat(0)$ with isolated flats, then the boundary is well-defined. Therefore fixing a space on which $G$ acts does not affect the boundary. We then focus on a collection of subsets $Y_v\\subset X$ which cover $X$. These subsets are defined at the end of Section \\ref{section:fixing a space}. In Sections \\ref{section:decomp}, \\ref{section:proper and cocompact}, and \\ref{section:local conn at flats} we show that $\\partial Y_v$ is path connected. Applying  Theorem \\ref{combo} finishes the proof of Theorem \\ref{thm main}. \n\t\n\tOf potentially independent interest is the work in Section \\ref{section:proper and cocompact}. Here, we establish a connection between neighborhoods of points in $\\overline{F}$, the closure of a flat, and neighborhoods in  $\\partial Y_v \\setminus F$. This work was done by Haulmark in the locally connected setting \\cite{Ha17} and we expand the results to our setting.\n\t\n\\end{Outline}\n\n\\begin{ack}\n\tI would like to thank Kim Ruane for her help and guidance throughout this project. I would also like to thank Genevieve Walsh, Mike Mihalik, and Rob Kropholler for many interesting discussions and for feedback on my drafts.\n\\end{ack}\n\n\n\n\\section{Key Examples} \\label{section: examples}\n\nIn this section, we outline three key examples that capture the difficulty in this work. Two are surface group amalgams and the third is a right-angled Coxeter group which is an amalgamation of groups which are virtually surface groups. All three examples are $\\cat(0)$ groups with isolated flats and the boundaries of the spaces on which they act are not locally connected.\n\n\n\\begin{figure}[h]\n\t\\includegraphics[width=.4\\textwidth]{s2-glued-torus}\n\t\\centering\n\t\\caption{A torus and a genus 2 surface with the identification $x=y$.}\n\t\\label{figure: s2 torus}\n\\end{figure}\n\n\n\n\n\n\\begin{example} \\label{example: s2 torus}\n\tLet $\\tilde{X_1}$ be the surface amalgam seen in Figure \\ref{figure: s2 torus}, $G_1$ its fundamental group, and $X_1$ its universal cover. This group is $\\cat(0)$ with isolated flats, with the flats coming from the torus. $X_1$ consists of Euclidean and hyperbolic planes fitting together in a tree-like way. The boundary, $\\partial X_1$, is made up of circles coming from these two types of planes along with the boundary points of rays which pass through infinitely many such planes. It is clear that the union of the circles is path connected since adjacent planes share a pair of boundary points. Because of the isolated flats, if two based rays pass through the same infinite collection of planes, then the rays are asymptotic. Applying Theorem \\ref{combo}, we can conclude that $\\partial X_1$ is path connected.\n\\end{example}\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=.4\\textwidth]{punctured-tori.png}\n\t\\caption{Two tori, each with a boundary component, with the identification $[a,b]^2=[c,d]^2$}\n\t\\label{figure:tori with boundary}\n\t\n\\end{figure}\n\n\\begin{example}\\label{example: punctured tori}\n\tLet $\\tilde{X_2}$ be 2 tori with boundary components as identified in Figure \\ref{figure:tori with boundary} and let $G_2$ be the fundamental group. By identifying $([a,b])^2$ with $([c,d])^2$, we create a Klein bottle group in $G$ generated by $[a,b]$ and $[c,d]$. This subgroup (and its conjugates) are the peripheral subgroups for the relatively hyperbolic structure on $G_2$. Since the Klein bottle group is virtually $\\mathbb{Z}^2$, this makes $G_2$ $\\cat(0)$ with isolated flats and with maximal peripheral splitting seen in Figure \\ref{fig: max periph splitting G2}.\n\t\n\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\n\t\\node[below] at (0,0) (A) {$K=\\langle[a,b],[c,d]\\rangle$};\n\t\\node[below] at (-4,0) (B) {$F_2=\\langle a,b\\rangle$};\n\t\\node[below] at (4,0) (C) {$F_2=\\langle c,d \\rangle$};\n\t\\node[above] at (2,0) {$\\mathbb{Z}=\\langle [c,d] \\rangle$};\n\t\\node[above] at (-2,0) {$\\mathbb{Z}=\\langle [a,b]\\rangle $};\n\n\t\n\t\\draw[fill] (0,0) circle [radius=0.05];\n\t\\draw[fill] (4,0) circle [radius=0.05];\n\t\\draw[fill] (-4,0) circle [radius=0.05];\n\t\\draw (-4,0) -- (0,0) -- (4,0);\n\t\\end{tikzpicture}\n\t\\caption{The maximal peripheral splitting of $G_2$}\n\t\\label{fig: max periph splitting G2}\n\\end{figure}\nA $\\cat(0)$ space $X_2$ on which $G_2$ acts geometrically can be constructed using Bridson and Haefliger's Equivariant Gluing Theorem \\cite{BH99}. This construction captures the isolated flats explicitly since $X_2$ consists of Euclidean planes coming from the Klein bottle group and truncated hyperbolic planes coming from the free groups. Once again, these spaces fit together in a tree-like way and the boundary $\\partial X_2$ is made up of boundary points from the truncated hyperbolic planes, the Euclidean planes, and the rays which pass through infinitely many such spaces.\n\nTo see that $\\partial X_2$ is path connected, consider a truncated hyperbolic plane union every flat which shares a boundary point with it. Call this space $Y$. This is equivalent to taking a $F_2$-vertex $v$ of the Bass-Serre tree for the splitting and letting $Y$ be the of the vertex space for $v$ union the vertex spaces for each of the vertices of the Bass-Serre tree adjacent to $v$. It is clear that $X_2$ is the union of all such $Y$. To see that $\\partial Y$ is path connected, notice that it is a Cantor set coming from the truncated hyperbolic planes union a collection of circles coming from the Euclidean planes. These circle `fill in' the gaps between points of the Cantor set, essentially replacing the missing intervals with circles. Therefore $\\partial Y$ is a circle with a collection of extra arcs attached and is path connected. \n\nSince $\\partial Y_v$ is path connected for each $v$ in the Bass-Serre tree and $X$ is the union of all such $Y_v$, then the only points to check are those which come from rays traveling through infinitely many truncated hyperbolic and Euclidean plans. Once again, the isolated flats condition covers this and we can apply Theorem \\ref{combo} to conclude $\\partial X_2$ is path connected.\n\\end{example}\n\n\\begin{figure}[h]\n\t\\centering\n\t\\begin{tikzpicture}\n\t\\newdimen\\r\n\t\\r=1cm\n\t\\draw (0:\\r)--(60:\\r)--(120:\\r)--(180:\\r)--(240:\\r)--(300:\\r)--(360:\\r);\n\t\\draw (330:1.7)--(300:\\r)--(0:0)--(60:\\r)--(30:1.7)--(0:2)--(330:1.7);\n\t\n\t\\draw[fill] (0:\\r) circle [radius=0.05];\n\t\\draw[fill] (60:\\r) circle [radius=0.05];\n\t\\draw[fill] (120:\\r) circle [radius=0.05];\n\t\\draw[fill] (180:\\r) circle [radius=0.05];\n\t\\draw[fill] (240:\\r) circle [radius=0.05];\n\t\\draw[fill] (300:\\r) circle [radius=0.05];\n\t\\draw[fill] (360:\\r) circle [radius=0.05];\n\t\\draw[fill] (0:0) circle [radius=0.05];\n\t\\draw[fill] (330:1.7) circle [radius=0.05];\n\t\\draw[fill] (30:1.7) circle [radius=0.05];\n\t\\draw[fill] (0:2) circle [radius=0.05];\n\t\n\t\n\t\n\t\\end{tikzpicture}\n\t\\caption{A Right-angled Coxeter group consisting of two hexagons sharing a pair of vertices.}\n\t\\label{figure:RACG}\n\\end{figure}\n\n\\begin{example}\\label{example: RACG}\nLet $G_3$ be the Right-angled Coxeter group in Figure \\ref{figure:RACG} and let $X$ be the Davis complex for $G_3$. This group is $\\cat(0)$ with isolated flats, with the flats coming from the square in the graph. Let $H$ be the RACG whose defining graph is a hexagon, then $G_3$ has the splitting seen in Figure \\ref{figure: $G_3$ splitting}.\n\n\\begin{figure}[h]\n\t\\centering\n\t\n\t\\begin{tikzpicture}\n\t\\node[below] at (-2,0) {$H$};\n\t\\node[below] at (0,0) {$(\\mathbb{Z}_2\\ast \\mathbb{Z}_2)^2$};\n\t\\node[below] at (2,0) {$H$};\n\t\\node[above] at (-1,0) {$\\mathbb{Z}_2\\ast \\mathbb{Z}_2$};\n\t\\node[above] at (1,0) {$\\mathbb{Z}_2\\ast \\mathbb{Z}_2$};\n\t\n\t\\draw[fill] (0,0) circle [radius=0.05];\n\t\\draw[fill] (2,0) circle [radius=0.05];\n\t\\draw[fill] (-2,0) circle [radius=0.05];\n\t\n\t\\draw (-2,0)--(0,0)--(2,0);\n\t\\end{tikzpicture}\n\t\n\t\\caption{A splitting of $G_3$.}\n\t\\label{figure: $G_3$ splitting}\n\\end{figure}\n\n$H$ is virtually a surface group, therefore $X$ consists of spaces quasi-isometric to $\\H^2$, which we will call $\\tilde{H}$, and Euclidean planes. Applying the same reasoning as in Example \\ref{example: s2 torus}, we can conclude that $\\partial X$ is path connected. This example is important for two reasons. First, the way we split the group is vital for applying Theorem \\ref{combo}. If we split the group in the `obvious' way:\n\\[H\\ast_{\\mathbb{Z}_2\\ast \\mathbb{Z}_2} H\\]\nthen the virtual $\\mathbb{Z}^2$ is not captured in the splitting (i.e. this is not a peripheral splitting). Because of this, it is no longer true that if two rays go through the same sequence of $\\tilde{H}$s, then the rays are asymptotic. Second, this is the type of group for which Hruska-Ruane's results on semistability do not apply \\cite{HruskaRuaneSemistable}. However, it is known that RACGs  are semistable at infinity \\cite{MihalikArtinCoxeterSemistable}.\n\n\n\n\\end{example}\n\nWe will refer to all three of these examples again in Section \\ref{section:isolated flats} to describe the isolated flats structure on each group, and again in Section \\ref{section block decomp} to describe the block decompositions of each space.\n\t\n\\section{The Visual Boundary} \\label{section visual boundary}\nWe refer the reader to \\cite{BH99} for an introduction to the theory of $\\cat(0)$ spaces and groups. Throughout this section, $X$ is a proper $\\cat(0)$ space. \n\nWe describe the topology on $\\overline{X}=X\\cup \\partial_{x_0} X$ and then restrict this topology to the visual boundary.\n\n\\begin{defn}[$\\overline{X}$ and the cone topology]\n\t\n\tFix a basepoint $x_0\\in X$. Let $\\partial_{x_0} X$ be the set of geodesic rays $c\\colon [0,\\infty)\\to X$ based at $x_0$. Let $\\overline{B}(x_0,r)$ be the closed ball of radius $r$ and $\\pi_r$ be the projection map to this ball. Define sets of the following form:\n\t\n\t\\[ U(c,r,D):= \\{ x\\in \\overline{X} : d(x,x_0)>r, d(\\pi_r(x),c(r))<D\\}\\]\n\twhere $c$ is a geodesic segment or ray based at $x_0$, $r,D>0$. A neighborhood basis for the \\textit{cone topology} on $\\overline{X}$ consists of sets of the form $U(c,r,D)$ along with metric balls in $X$.\n\t\n\tThe cone topology can be restricted to $\\partial_{x_0} X$ by taking sets of the following form to be a neighborhood basis:\n\t\n\t\\[ U(c,r,D)= \\{ c'\\in \\partial_{x_0} X: d(c(r),c'(r))<D\\}\\]\n\twith $r,D>0$.\n\t\t\\end{defn}\n\t\n\tDue to the follow result in \\cite{BH99}, the topology on $\\partial_{x_0} X$ is independent of choice of basepoint. Therefore we can denote the visual boundary as $\\partial X$ without ambiguity. In practice, however, it often useful to fix a basepoint and work with $\\partial_{x_0} X$.\n\t\n\t\\begin{prop}[{\\cite[II.8]{BH99}}]\n\t\tFor any choice of $x_0$ and $x_1$, $\\partial_{x_0} X$ and $\\partial_{x_1} X$ are homeomorphic.\n\t\\end{prop}\n\n\nWe use a metric on $\\partial X$ which was first introduced by Osajda. We use this metric in Section \\ref{section:decomp}.\n\n\\begin{defn}[Metric on $\\partial X$]\n\tFix $x_0\\in X$ and $D>0$. The metric $d_D$ on $\\partial_{x_0} X$ is defined as follows: given two rays $c,c'\\in \\partial_{x_0} X$, $d_D(c,c')=\\dfrac{1}{r}$ where $r\\in [0,\\infty)$ such that $d(c(r),c'(r))=D$. Since $d(c(t),c'(t))$ increases monotonically with $t$, there is a unique $r$ satisfying $d(c(r),c'(r))=D$.\n\t\n\tThis metric is compatible with the cone topology \\cite{OsajdaSwic}. Furthermore, the spaces $(\\partial X, d_D)$ and $(\\partial X, d_{D'})$ are isometric, so the metric is independent of choice of $D$.\n\\end{defn}\n\n\t\n\\section{Relatively Hyperbolic Group and Peripheral Splittings} \\label{rel hyp section}\n\nIn this section, we define relatively hyperbolic groups, the Bowditch boundary, and state a theorem of Tran which relates the Bowditch boundary to the $\\cat(0)$ boundary. We then introduce the maximal peripheral splitting of a relatively hyperbolic group. We will use the definition of relative hyperbolicity from \\cite{Yaman}. This definition uses an action of a group on a compact, metrizable topological space $M$, which will ultimately be the Bowditch boundary.\n\n\\begin{defn}[Convergence action]\nLet $M$ be a compact metrizable space and $G$ a finitely generated group. An action of $G$ on $M$ is a \\textit{convergence action} if it is properly discontinuous on distinct triples of $M$.\n\\end{defn}\n\n\\begin{defn}[Geometrically finite]\nA convergence group action of $G$ on $M$ is \\textit{geometrically finite} if every point of $M$ is either a bounded parabolic point or a conical limit point. \n\\end{defn}\n\n\n\\begin{defn}[Relatively hyperbolic]\n\tLet the action of $G$ on $M$ be a geometrically finite convergence action and let $\\mathbb{P}$ be a collection of subgroups of $G$. Then $G$ is \\textit{hyperbolic relative to $\\mathbb{P}$} if $\\mathbb{P}$ is exactly the collection of maximal parabolic subgroups. \n\\end{defn}\n\n\\begin{defn}[Bowditch boundary]\n\tThe space $M$ is uniquely determined by the pair $(G,\\mathbb{P})$ and is the \\textit{Bowditch boundary}, denoted $\\partial(G,\\mathbb{P})$.\n\\end{defn}\n\nSince we do not deal with the Bowditch boundary directly, we will not define all the terms above. See \\cite{Yaman} for precise definitions. We need the Bowditch boundary in order to apply Theorem \\ref{Tran Theorem} and to use a handful results of Bowditch which are summarized in Theorem \\ref{thm: peripheral splitting}. Both theorems can be found below.\n\nThere are many other ways to define a relatively hyperbolic group, see for example \\cite{FarbBoundedCoset, GrovesManningCusped}. Each of these ultimately has $G$ acting on a $\\delta$-hyperbolic metric space and takes the boundary of this space to be the Bowditch boundary. This is equivalent to our definition of the Bowditch boundary. Furthermore, all of these definitions of relatively hyperbolic are equivalent.\n\n\nSuppose $G$ is a $\\cat(0)$ group acting geometrically on a $\\cat(0)$ space $X$ and $(G,\\mathbb{P})$ is also relatively hyperbolic. We have so far introduced two different boundaries for $G$: the $\\cat(0)$ boundary and the Bowditch boundary. Work of Tran tells us how to relate these two boundaries:\n\n\n\\begin{thm}[\\cite{Tran13}] \\label{Tran Theorem}\n\tLet $(G,\\mathbb{P})$ be a relatively hyperbolic group acting geometrically on a $\\delta$-hyperbolic or $\\cat(0)$ space $X$. The quotient space formed from $\\partial X$ by collapsing the limit sets of each $P\\in \\mathbb{P}$ to a point is $G$-equivariantly homeomorphic to the Bowditch boundary $\\partial (G,\\mathbb{P})$.\n\\end{thm}\n\nWe will now introduce the maximal peripheral splitting of a relatively hyperbolic group. This plays a vital role in establishing the block decomposition of the relatively hyperbolic groups we study.\n\n\\begin{defn}[Peripheral Splitting]\n\tLet $(G,\\mathbb{P})$ be a relatively hyperbolic group. A \\textit{peripheral splitting} of $(G,\\mathbb{P})$ is a splitting of $G$ as a finite bipartite graph of groups $\\mathcal{G}$ whose vertices have two type: peripheral and component. Furthermore, the collection of subgroups of $G$ which are conjugate to a peripheral vertex group is the same as $\\mathbb{P}$, and $\\mathcal{G}$ does not contain a vertex of degree 1 which is contained in the adjacent peripheral vertex group.\n\\end{defn}\n\nA splitting of this form is called \\textit{trivial} if one of the vertex groups is equal to $G$ and \\textit{nontrivial} otherwise. A splitting $\\mathcal{H}$ is a \\textit{refinement} of another splitting $\\mathcal{G}$ if $\\mathcal{G}$ can be obtained from $\\mathcal{H}$ by a sequence of foldings over edges which preserves the peripheral\/component coloring. A peripheral splitting is \\textit{maximal} if it is not a refinement of any other splitting. \n\n\nWe combine a number of results of Bowditch about peripheral splittings into one large theorem:\n\\begin{thm}[Peripheral Splittings Results] \\label{thm: peripheral splitting}\n\tLet $(G,\\mathbb{P})$ be a 1-ended relatively hyperbolic group. Then the following hold:\n\t\n\t\\begin{enumerate}\n\n\n\t\t\\item if each $P\\in \\mathbb{P}$ is finitely presented, 1- or 2-ended, and does not contain an infinite torsion subgroup, then $\\partial (G,\\mathbb{P})$ is connected \\cite[10.1]{Bow12} and locally connected \\cite[1.5]{Bow01},\n\t\t\n\t\t\\item if $G_v$ is a component vertex group of a peripheral splitting, then $G_v$ is hyperbolic relative to $\\mathbb{P}_v:= \\{ P\\cap G_v: P\\in \\mathbb{P}\\}$ \\cite[1.3]{Bow01},\n\t\t\n\n\t\\item if $G_v$ is a component vertex of a maximal peripheral splitting and both $\\mathbb{P}$ and $\\mathbb{P}_v$ consist of finitely generated, 1- or 2-ended groups with no infinite torsion subgroups, the $\\partial (G_v,\\mathbb{P}_v)$ is connected and locally connected \\cite[1.3 and 1.5]{Bow01}.\n\t\t\n\t\n\t\t\n\n\t\t\n\n\t\t\n\t\t\\item if $G_v$ is a component vertex of a maximal peripheral splitting and both $\\mathbb{P}$ and $\\mathbb{P}_v$ consist of finitely generated, 1- or 2-ended groups with no infinite torsion subgroups, then $\\partial (G_v,\\mathbb{P}_v)$ contains no global cut points \\cite[1.4]{Bow01}.\n\t\\end{enumerate}\n\\end{thm}\n\nIn the setting we are dealing with ($\\cat(0)$ groups with isolated flats) each component vertex is virtually $\\mathbb{Z}^n$ for $n\\geq 2$, and so 1-ended with no infinite torsion subgroups. Furthermore, if $G_v$ is a component vertex, $\\mathbb{P}_v$ consists of 1- or 2-ended groups.  We use the first point of this theorem in order to accurately restate a result of Hruska-Ruane in Section \\ref{section:decomp}. The second point allows us to make sense of the last two. The third point is used at the end of Section \\ref{section:decomp} and the final point is used in a proposition stated in Section \\ref{section:local conn at flats}.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Isolated Flats} \\label{section:isolated flats}\n\nHere we define $\\cat(0)$ isolated flats and outline some results about their large scale geometry. \n\n\\begin{defn}[Flat]\n\tA subset $F\\subset X$ is a \\textit{flat} if it is isometric to $\\mathbb{E}^n$ for some $n\\geq 2$.\n\\end{defn}\n\nThere are many equivalent definitions of isolated flats. Hruska and Kleiner prove the equivalence of these definitions in \\cite{HK05}. We will take the following as our definition:\n\n\n\\begin{defn}[$\\cat(0)$ space with isolated flats]\n\tLet $X$ be a $\\cat(0)$ space admitting a geometric group action by $G$. Let $\\mathcal{F}$ be a $G$-invariant collection of flats. Then $X$ has \\textit{isolated flats} if the following two properties hold:\n\t\n\t\\begin{enumerate}\n\t\t\\item Maximality: There is  $D<\\infty$ such that for any flat $F\\subset X$, there is a flat $F'\\in \\mathcal{F}$ with $F\\subset N_D(F')$. \n\t\t\\item Isolated: For any $r>0$, there is a $\\rho=\\rho(r)>0$ such that for any distinct $F,F'\\in \\mathcal{F}$,\t\n\t\t$$\\text{diam}(N_r(F)\\cap N_r(F'))<\\rho$$\t\t\n\t\\end{enumerate}\n\\end{defn}\n\nWe say a group $G$ is \\textit{$\\cat(0)$ with isolated flats} if it acts geometrically on a $\\cat(0)$ space with isolated flats.\n\n\n\\begin{example}~\n\t\n\t\n\t\\begin{enumerate}\n\t\t\\item Return to Example \\ref{example: s2 torus}. The universal cover of the surface group amalgam in Figure \\ref{figure: s2 torus} is a $\\cat(0)$ space with isolated flats. The fundamental group acts geometrically on the universal cover. The flats, coming from the universal cover of the torus, are isolated in the sense that there is a hyperbolic plane separating them.\n\t\t\n\t\t\\item Consider Example \\ref{example: punctured tori}. The universal cover of the two tori with boundary components with identifications given in Figure \\ref{figure:tori with boundary} is a $\\cat(0)$ space with isolated flats. The $\\mathbb{bR}^2$'s coming from the Klein bottle are separated by truncated $\\H^2$'s.\n\t\t\n\t\t\\item From Example \\ref{example: RACG}, the Davis complex of this RACG is $\\cat(0)$ with isolated flats admitting a geometric group action by the RACG. Here, the flat comes from the square in the defining graph. As with the example above, pairs of flats are separated by (something quasi-isometric to) hyperbolic planes.\n\t\t\n\t\t\\item Let $K$ be the figure 8-knot and $M=S^3\\setminus K$ and let $X$ be the universal cover of $M$. Then $X$ is isometric to truncated $\\H^3$ and is a $\\cat(0)$ space with isolated flats, with the lifts of the torus-boundary of the manifold making up the flats.\n\t\\end{enumerate}\n\\end{example}\n\n\n\nOne of the major theorems from Hruska and Kleiner's work is that there are a number of equivalent characterizations of $\\cat(0)$ spaces with isolated flats. \n\n\\begin{thm}[{\\cite[1.2.1]{HK05}}]\n\tLet $X$ be a $\\cat(0)$ space admitting a geometric group action by $G$. Then the following are equivalent\n\t\n\t\\begin{enumerate}\n\t\t\\item $X$ has isolated flats.\n\t\t\\item Each component of the Tits boundary $\\partial_T X$ is either an isolated point or a standard Euclidean sphere.\n\t\t\\item $X$ is relatively hyperbolic with respect to a family of flats $\\mathcal{F}$.\n\t\t\\item $G$ is relatively hyperbolic with respect to a collection of virtually abelian subgroups of rank at least 2.\n\t\\end{enumerate}\n\\end{thm}\n\nOne key fact we use to establish that $\\cat(0)$ groups with isolated flats satisfy condition (3) in Theorem \\ref{combo} is a quasiconvexity result from Hruska-Ruane. Recall the definition of quasiconvex:\n\n\\begin{defn}[Quasiconvex]\n\tLet $X$ be a geodesic metric space. A subset $C$ is \\textit{$\\kappa$-quasiconvex} if for any pair of points $x,y\\in C$, any geodesic between $x$ and $y$ is in $N_\\kappa(C)$. \n\t\n\tWe say a subset is \\textit{quasiconvex} if it is $\\kappa$-quasiconvex for some $\\kappa$. \n\\end{defn}\n\n\\begin{thm} [{\\cite[5.6]{HR17}}] \\label{Quasi-convex Constant}\n\t\n\tLet $X$ be a $\\cat(0)$ space with isolated flats with respect to $\\mathcal{F}$. There is a constant $\\kappa>0$ such that the following holds:\n\t\\begin{enumerate}\n\t\t\\item Given two flats $F_1,F_2\\in \\mathcal{F}$ with $c$ the shortest length geodesic from $F_1$ to $F_2$, then $F_1\\cup F_2\\cup c$ is $\\kappa$-quasiconvex in $X$.\n\t\t\\item Given a point $p$ and a flat $F\\in \\mathcal{F}$ with $c$ the shortest path from $p$ to $F$, then $c\\cup F$ is $\\kappa$-quasiconvex in $X$. In particular, if $c'$ is any geodesic joining a point of $c$ to a point of $F$, then $c'$ intersects the $\\kappa$-neighborhood of the endpoint in the flat of $c$.\n\t\\end{enumerate}\n\\end{thm}\n\n\n\\section{Block Decompositions} \\label{section block decomp}\n\nThis section is devoted to defining a block decomposition of a space, providing some motivating examples, and concludes with the proof of Theorem \\ref{combo}. A block decomposition was first introduced in \\cite{Moo10} as a generalization of the results from \\cite{CK00}. In both papers, the authors focus on spaces which admit geometric group actions by groups of the form $G=A\\ast_C B$. The spaces on which these groups act can be viewed as the union of closed, convex pieces, called blocks, and the intersection of these blocks follows the subgroup intersection described in the Bass-Serre tree for the splitting. While the definition of a block decomposition is independent of a group action, this paper's motivating examples of such a decomposition come from the splitting of a group.\n\nWe will use the following definition from \\cite{Moo10}:\n\n\\begin{defn} \\label{block decomposition}\n\tA \\textit{block decomposition} of a CAT(0) space $X$ is a collection $\\mathcal{B}$ of closed, convex subsets (called blocks) such that the following conditions hold:\n\t\n\t\\begin{enumerate}\n\t\t\\item \t$X = \\bigcup \\limits_{B\\in \\mathcal{B}} B$,\n\t\t\\item each block intersects at least two other blocks,\n\t\t\\item Parity condition: every block has a $(+)$ or $(-)$ parity such that two blocks intersect only if they have opposite parity,\n\t\t\\item $\\epsilon$-condition: there is an $\\epsilon>0$ such that two blocks intersect iff their $\\epsilon$-neighborhoods intersect.\n\t\\end{enumerate}\n\t\n\\end{defn}\n\nWe will mostly use the facts that each $B$ is closed and convex and that $X=\\bigcup\\limits_{B\\in \\mathcal{B}} B$. Conditions (2)-(4) are there to prevent any degenerate cases (e.g. splitting $X$ into only two pieces, decomposing a plane as a union of lines). \n\n\\begin{defn}\n\tA \\textit{wall} is a non-trivial intersection of blocks and we denote the collection of walls $\\mathcal{W}$. \n\\end{defn}\n\nSince each block is convex, note that each wall is convex as well. The nerve, denoted $\\Nerve(\\mathcal{B})$, is a graph which records block intersections. There is a vertex for each block and an edge if two blocks (non-trivially) intersect. We will use the following notation: If $v$  and $w$ are vertices of $\\Nerve(\\mathcal{B})$, then $B_v, B_w\\in \\mathcal{B}$ are the blocks corresponding to $v$ and $w$, respectively. Given an edge $e$ with vertices $v$ and $w$, then $B_v$ and $B_w$ intersect in a wall by definition and we denote this wall $W_e$.\n\nFollowing results from \\cite{CK00} and \\cite{Moo10}, we get that the nerve is always a tree.\n\\begin{lem}\n\t$\\Nerve(\\mathcal{B})$ is a tree.\n\\end{lem}\n\nSince $\\Nerve(\\mathcal{B})$ is a tree, we will denote it $\\mathcal{T}_\\mathcal{B}$ or $\\mathcal{T}$ when $\\mathcal{B}$ is understood. Here are some examples of block decompositions:\n\n\\begin{figure}[h]\n\t\\includegraphics[width=.4\\textwidth]{torus-complex}\n\t\\centering\n\t\\caption{Three tori with the curve $a$ identified with $b$ and the curve $c$ identified with $d$.}\n\t\\label{fig:torus complex}\n\\end{figure}\n\n\n\n\n\\begin{example} \\label{example: Croke Kleiner}\n\t\n\t\n\t\n\tIn \\cite{CK00}, the authors describe what they call a torus complex, see Figure \\ref{fig:torus complex}. Let $G$ be the fundamental group of the torus complex and $X$ the universal cover. The space $X$ is $\\cat(0)$ and admits a block decomposition coming from a the following splitting of $G$:\n\t$$G = \\left( F_2\\times \\mathbb{Z} \\right) \\ast_{\\mathbb{Z}^2} \\left( F_2\\times \\mathbb{Z} \\right)$$\n\tThe fundamental group of two tori with a pair of curves identified is $F_2\\times \\mathbb{Z}$ which acts geometrically on $T_4\\times \\mathbb{bR}$, where $T_4$ is the valence 4 tree. The blocks come from the lifts of the middle torus and either the left or right torus, meaning each $B\\in \\mathcal{B}$ is isometric to $T_4\\times \\mathbb{bR}$. The walls are the lifts of the middle torus and are all isometric to $\\mathbb{bR}^2$. \n\t\n\t Croke and Kleiner \\cite{CK00} use $\\mathcal{B}$ to show that by changing the angle of intersection between the curves $b$ and $c$ on the middle torus, $G$ can act geometrically on spaces $X_1$ and $X_2$ which have non-homeomorphic boundaries. This answered a question of Gromov. Later, in \\cite{CK02}, Croke and Kleiner vastly extend their findings on this specific example to a class of groups they call `admissible.'\n\\end{example}\n\n\\begin{example} We return to Example \\ref{example: s2 torus}. The universal cover of the surface amalgam from this example is $\\cat(0)$ and has a block decomposition coming from the splitting of the fundamental group. Here, if $G_1$ is the fundamental group, then $G_1$ splits as\n\t$$G_1 = \\pi_1(S_2)\\ast_\\mathbb{Z} \\mathbb{Z}^2$$\n\twhere $S_2$ is the surface with genus 2. The blocks of this decomposition come from the lifts of the two surfaces. Therefore there are two types of blocks: those isometric to $\\H^2$ and those isometric to $\\mathbb{bR}^2$. The walls, however, are all isometric to $\\mathbb{bR}$. Unlike the torus complex, this group is $\\cat(0)$ with isolated flats. \n\\end{example}\n\n\\begin{example}\n\tConsider Example \\ref{example: punctured tori} once again. Let $G_2$ be the fundamental group of this surface amalgam and $X_2$ its universal cover. Recall that this group admits the maximal peripheral splitting as seen in Figure \\ref{fig: max periph splitting G2}. $X_2$ admits a block structure coming from this splitting. The vertex groups which are free act geometrically on truncated hyperbolic space and the vertex group which is the Klein bottle group acts geometrically on a Euclidean plane. These truncated hyperbolic planes and Euclidean planes in $X_2$ make up the block structure. That is, each block is isometric to either a Euclidean plane or a truncated hyperbolic plane.\n\\end{example}\n\n\\begin{example}\n\tReturn now to the RACG in Example \\ref{example: RACG}. Let $G_3$ be the fundamental group and $X_3$ the universal cover of its Davis complex. The block structure on $X_3$ once again comes from a splitting of $G_3$ which we saw in Figure \\ref{figure: $G_3$ splitting}. The blocks of $X_3$ come from the subspaces on which the vertex groups act geometrically. The hexagon subgroup $H$ acts geometrically on a space quasi-isometric to $\\H^2$ while $(\\mathbb{Z}_2\\ast \\mathbb{Z}_2)^2$ acts geometrically on $\\mathbb{bR}^2$. These two types of spaces form the blocks of $X_3$. \n\\end{example}\n\n\n\\begin{example}\n\tMore generally, if $X$ is a $\\cat(0)$ space coming from the construction in the Equivariant Gluing Theorem (\\cite{BH99} Theorem II.11.18), then $X$ admits a block decomposition coming from the splitting of the group $G=G_1\\ast_H G_2$. There is a natural map $p\\colon  X\\to \\mathcal{T}$, where $\\mathcal{T}$ is the Bass-Serre tree for the splitting. If $v$ is a vertex of $\\mathcal{T}$ then the blocks are each of the form  $p^{-1}(N_{1\/2}(v))$. We will be using this in Section \\ref{section:fixing a space}.\n\\end{example}\n\nWith these examples in mind, we can say something about the way block boundaries intersect. We say a wall $W_e\\in \\mathcal{W}$ \\textit{separates} blocks $B_v$ and $B_u$ if any path from $B_v$ to $B_u$ must pass through $W_e$. This is equivalent to $e$ being an edge on any path from $v$ to $u$ in $\\mathcal{T}$. \n\n\\begin{lem} \\label{Block int}\n\tFor two blocks $B_u,B_v\\in \\mathcal{B}$ corresponding to vertices $u,v\\in \\mathcal{T}$, then one of the following holds:\n\t\n\t\\begin{enumerate}\n\t\t\\item $\\partial B_u \\cap \\partial B_v = \\emptyset$\n\t\t\\item The vertices $u$ and $v$ are adjacent and $\\partial B_u \\cap \\partial B_v = \\partial W$ where $W=B_u \\cap B_v$.\n\t\t\\item The vertices $u$ and $v$ are not adjacent in $\\mathcal{T}$ but $\\partial B_u \\cap \\partial B_v$ is non-empty. Then $\\partial B_u \\cap \\partial B_v \\subset \\partial W$ where $W$ is \\textit{any} wall between $B_u$ and $B_v$.\n\t\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n\tWe may assume that $\\partial B_u \\cap B_v \\neq \\emptyset$, so we are in either case (2) or case (3). If $u$ and $v$ are adjacent, then case (2) follows immediately. Suppose then that $\\alpha\\in \\partial B_u \\cap \\partial B_v$ and that $u$ and $v$ are not adjacent in $\\mathcal{T}$. Let $W$ be any wall separating $B_u$ and $B_v$. \n\t\n\tPick basepoints $x_0\\in B_u$ and $x_1\\in B_v$ and let $y$ be a point in $W$ along the geodesic segment from $x_0$ to $x_1$. Let $c_0$ and $c_1$ be rays representing $\\alpha$ based at $x_0$ and $x_1$, respectively. For each $n\\in \\mathbb{N}$, let $y_n$ be a point in $W$ along the geodesic segment from $c_0(n)$ to $c_1(n)$. Since $W$ separates the blocks, these $y_n$ exist. Since $c_0$ and $c_1$ are asymptotic, there is a constant $K$ such that $d(c_0(t),c_1(t))\\leq K$ for all $t$. Therefore $y_n$ is $K$-close to both $c_0$ and $c_1$. The limit of the geodesic segments from $y$ to $y_n$, therefore, is asymptotic to $c_0$ and $c_1$, so is a representative of $\\alpha$. Since $W$ is convex, then $\\alpha\\in \\partial W$.\n\\end{proof}\n\n\n\n\\begin{remark}\n\tThe following should be noted from the proof above: it is possible for $\\alpha\\in \\partial B$ but a ray representing $\\alpha$ is disjoint from $B$ entirely. This happens above with the block $B_u$ and the ray $c_1$.\n\\end{remark}\n\nWe use the following terminology and the definition below from \\cite{CK00}. A ray $c$ \\textit{enters} a block $B$ if for some time $t$, $c(t)\\in B$ and $c(t)$ is not in any other block (and therefore is not in a wall). Similarly, a ray $c$ \\textit{exits} a block $B$ if for some $t$, $c(t)\\in B$ and not in any wall and for some $t'>t$ $c(t')\\notin B$. \n\n\\begin{defn}\n\tLet $x_0\\notin W$ for all $W\\in \\mathcal{W}$. The \\textit{itinerary} of a ray $c$ based at $x_0$, denoted $\\Itin_{x_0}(c)$, is the sequence of blocks $c$ enters. If $x_0$ is understood, then the itinerary will be denoted $\\Itin(c)$. Abusing notation, if the basepoint is understood and $c(\\infty)=\\alpha\\in \\partial X$, then $\\Itin(\\alpha)$ is used to denote the itinerary of the ray representing $\\alpha$. \n\t\n\\end{defn}\n\nIn order for this definition to be well defined, we need the following lemma:\n\n\\begin{lem}\n\tIf $\\mathcal{B}$ is a block decomposition for $X$, then there are points of $X$ which are not in any walls. In fact, for each $B\\in \\mathcal{B}$, there are points of $B$ which do not lie in any walls.\n\\end{lem}\n\n\\begin{proof}\n\tLet $B\\in \\mathcal{B}$ and let $W$ be a wall of $B$ such that $B\\cap B_0 = W$. Define $D=\\inf_{W'\\subset B} d(W,W')$.\n\t\n\tIf $D>0$, then all points in $N_D(W)\\setminus W \\cap B$ are points of $B$ which do not lie in any walls. Since $D>0$, then this set is non-empty.\n\t\n\tAssume, for contradiction, that $D=0$. Then pick $\\delta< \\epsilon$ and consider $N_\\delta(W)$. Since $D=0$, there is some wall $W'\\subset B$ which non-trivially intersects $N_\\delta(W)$. Let $B\\cap B'=W'$. Since $\\delta<\\epsilon$, $N_\\delta(W)\\cap W'$ means that in fact $N_\\epsilon(B_0)\\cap W'\\neq 0$. Since $W'\\subset B'$ and applying the $\\epsilon$-condition on the block decomposition, $B_0\\cap B'\\neq \\emptyset$. But then $B, B', B_0$ all pairwise intersect, contradicting the parity condition. Therefore $D>0$.\n\\end{proof}\t\n\n\\begin{remark} It is important to note that it is possible for $\\alpha\\in \\partial B$ but for $B\\notin \\Itin(\\alpha)$. This can happen as in the proof of Lemma \\ref{Block int}, where $c_0$ represent $\\alpha$, $\\alpha\\in \\partial B_v$, but $\\Itin_{x_0}(c_0)=\\Itin_{x_0}(\\alpha)=\\{B_u\\}$. In other words, this can happen when $\\alpha$ lies in the boundary of multiple blocks.\n\\end{remark}\n\n\nThe lemma below follows exactly in the same way as \\cite{CK00} (Lemma 2), which uses the fact that each $B$ is convex and and $B$'s topological frontier is covered by the collection of blocks corresponding to the link in $\\mathcal{T}$ of the vertex corresponding to $B$.\n\n\\begin{lem} [{\\cite[2]{CK00}}] \\label{Itinerary to Nerve} \n\tLet $X$ be a $\\cat(0)$ space with a block decomposition $\\mathcal{B}$ and corresponding nerve $\\mathcal{T}$. The itinerary for any $\\alpha\\in \\partial X$ consists of blocks which correspond to vertices on a geodesic segment or ray in $\\mathcal{T}$.  \n\\end{lem}\n\n\n\n\nThis lemma gives rise to the following definition:\n\n\\begin{defn}\n\t\n\t\n\t\n\tFix a basepoint $x_0$ which is not in any wall and let $c$ be a geodesic ray based at $x_0$ representing the point $\\alpha\\in \\partial X$. The point $\\alpha$ is \\textit{rational} if $\\Itin(c)$ is finite. The point $\\alpha$ is \\textit{irrational} if $\\Itin(c)$ is infinite.\n\\end{defn}\n\nIn order for this definition to be useful, being rational or irrational must be independent of representative for $\\alpha$.\n\n\\begin{lem}\n\tIf $c$ represents $\\alpha$ and $c$ is rational, then all representatives of $\\alpha$ are rational. \n\\end{lem}\n\n\\begin{proof}\n\tLet $\\alpha \\in \\partial X$ and let $c\\colon [0,\\infty)\\to X$ be a ray representing $\\alpha$ based at $x_0$ be a finite itinerary and let $B$ be the final block of this itinerary. So $\\alpha\\in \\partial B$. Let $c'\\colon [0,\\infty)\\to X$ be a ray representing $\\alpha$ based at $x_1$, and suppose, for contradiction, that $\\Itin_{x_1}(c')$ is infinite. By Lemma \\ref{Itinerary to Nerve}, $\\Itin(c')$ represents a geodesic ray in $\\mathcal{T}$. \n\t\n\tFirst, we show that if $d(v_B,v_{B'})\\geq 2n$ for $v_B,v_{B'}\\in \\mathcal{T}$, then $d(B,B')\\geq 2n\\epsilon$ in $X$. Let $B=B_1,B_2,...,B_{2n}=B'$ denote the $2n$-blocks between $B$ and $B'$. To see $d(B,B')\\geq 2\\epsilon$, notice if $d(v_B,v_{B'})=2$, then by the $\\epsilon$-condition, $d(B,B')\\geq 2\\epsilon$ since the $\\epsilon$-neighborhoods of the blocks do not intersect. So let $x\\in B$ and $x'\\in B'$ and let $[x,x']$ be the geodesic segment between them. For each $0\\leq i \\leq n$, there are $z_i\\in B_{2i}$ such that $d(z_i,z_{i+2})\\geq 2\\epsilon$. Therefore $d(x,x')\\geq 2n\\epsilon$. \n\t\n\tNow, for any $K>0$, there exists some $n$ such that $2n\\epsilon>K$. Furthermore, since $\\Itin(c')$ represent a geodesic ray in $\\mathcal{T}$, there is a block $B'$ such that $d(v_B,v_{B'})\\geq 2n$. Thus $c'$ is not in the $K$-neighborhood of $B$ for any $K$. But $c$ and $c'$ are asymptotic, a contradiction. Thus $\\Itin(c')$ must be finite.\n\t\n\\end{proof}\n\n\\begin{corollary}\n\tIf $\\Itin(c)$ is infinite for some ray $c$, then all rays representing $c(\\infty)$ have infinite itineraries.\n\\end{corollary}\n\n\\begin{proof}\n\tSuppose $c'$ represents $c(\\infty)$ and is finite. Then by the argument above, all rays representing $c'(\\infty)$ have finite itineraries, which cannot happen since $\\Itin(c)$ is infinite.\n\\end{proof}\nLet $\\partial_R X$ and $\\partial_I X$ denote the rational and irrational boundary points of $\\partial X$, respectively. Then we have:\n\\begin{enumerate}\n\t\\item $\\partial_R X = \\bigcup\\limits_{B\\in \\mathcal{B}} \\partial B$\n\t\\item $\\partial_I X = \\partial X \\setminus \\partial_R X$\n\\end{enumerate}\n\nOnce a basepoint is fixed we can use Lemma \\ref{Itinerary to Nerve} to create a map \\linebreak $\\eta\\colon  \\partial_I X \\to \\partial \\mathcal{T}$, mapping a point $\\alpha$ to the end of $\\mathcal{T}$ representing its itinerary at that basepoint. This map is surjective but not necessarily injective. The points in $\\eta ^{-1}(\\alpha)$ are the points of the $\\partial X$ with the same itinerary as $\\alpha$. We will care when $\\alpha$ is the only boundary point with its itinerary.\n\n\\begin{defn}\n\tIf $\\alpha$ is irrational and is the only point in $\\partial X$ with its itinerary, then we say $\\alpha$ is \\textit{lonely}. We will denote the set of lonely rays $\\partial_{I_L} X$.\n\\end{defn}\n\n\n\n\\begin{thm}\\label{combo}\n\tLet $X$ be a $\\cat(0)$ space and $\\mathcal{B}$ a block decomposition. Suppose the following conditions holds:\n\t\n\t\\begin{enumerate}\n\t\t\\item for each $B\\in \\mathcal{B}$, $\\partial B$ is path connected,\n\t\t\\item for each $W\\in \\mathcal{W}$, $\\partial W$ is non-empty,\n\t\t\\item all irrational rays are lonely\n\t\\end{enumerate}\n\tthen $\\partial X$ is path connected.\n\\end{thm}\n\n\\begin{remark}\n\tIt should be noted that we only assume $\\partial W$ is non-empty. We do not need that $\\partial W$ is path connected. The walls having non-empty boundary is equivalent to $\\partial X$ being connected.\n\\end{remark}\n\nGoing forward, we shall refer to condition (3) as the \\textit{lonely condition}.\n\nBefore proving Theorem \\ref{combo}, it should be noted that conditions (1) and (2), without (3), are insufficient for guaranteeing $\\partial X$ being path connected. Returning to Example \\ref{example: Croke Kleiner}, the block decomposition satisfies conditions (1) and (2), but $\\partial X$ is not path connected. In \\cite{CMT}, the authors explicitly construct a pair of distinct irrational rays with the same itinerary which are in a different path component from the union of the block boundaries. In that example, there is something akin to the Topologist's Sine Curve in the boundary, which obstructs path connectivity. Theorem \\ref{combo} shows that the lonely condition is sufficient for avoiding this pathology.\n\nWe will prove Theorem \\ref{combo} in two parts. First we will show that $\\partial_R X$ is path connected. We then show if $\\partial_R X$ is path connected, then $\\partial_R X \\cup \\partial_{I_L} X$ is path connected as well. By the lonely condition, $\\partial_{I_L} X = \\partial_I X$, concluding the proof.\n\n\\begin{proposition}\n\tSuppose for each $B\\in \\mathcal{B}$, $\\partial B$ is path connected and for each $W\\in \\mathcal{W}$, $\\partial W$ is non-empty. Then $\\partial_R X$ is path connected.\n\\end{proposition}\n\n\\begin{proof}\n\tFix $x_0\\in X\\setminus \\mathcal{W}$.\tLet $x_0\\in B$ and $B'$ be an adjacent block. We show that $\\partial B \\cup \\partial B'$ is path connected then proceed by induction. Since block boundaries are path connected, it suffices to show that there is a path from a point in $\\partial B$ to a point in $\\partial B'$. Let $\\alpha\\in \\partial B$ and $\\alpha' \\in \\partial B'$. Since $B$ and $B'$ are adjacent, there is a $W\\in \\mathcal{W}$ such that $B\\cap B' = W$. Let $\\beta\\in \\partial W$, which exists since $\\partial W$ is non-empty. Since $\\partial W = \\partial B \\cap \\partial B'$ and each block has path connected boundary, there are paths from $\\alpha$ to $\\beta$ and from $\\beta$ to $\\alpha'$ in $\\partial B \\cup \\partial B'$. The concatenation of these paths is a path from $\\alpha$ to $\\alpha'$.\n\t\n\tLet $\\Itin(\\alpha')= \\{B_1, B_2, ..., B_n\\}$ with $B_1=B$. The union of the boundary of adjacent blocks is path connected, so $\\partial B_{n-1} \\cup \\partial B_n$ is path connected. By the induction hypothesis, $\\partial B_1 \\cup \\partial B_2 \\cup ... \\cup \\partial B_{n-1}$ is path connected, and therefore $\\partial B_1 \\cup \\partial B_2 \\cup ... \\cup \\partial B_n$ is path connected, which concludes the proof. \n\\end{proof}\n\nBefore proving the next proposition, we need the following lemma from \\cite{CK02}:\n\n\\begin{lem} \\label{Croke Kleiner Converging Rays}\n\tLet $X$ be a locally compact CAT(0) space with a closed, convex subset $Y$. Let $x_0\\in X$ and $\\alpha_i\\in \\partial X$ with $\\alpha_i\\to \\alpha$ and the rays $r_i$ based at $x_0$ representing $\\alpha_i$ and $r$ the ray based at $x_0$ representing $\\alpha$. If $r_i$ intersects $Y$ for all $i$, then either $r$ intersects $Y$ or $\\alpha \\in \\partial Y$.\n\\end{lem}\n\n\\begin{proposition} \\label{lonely plus rational is pconn}\n\tIf $\\partial_R X$ is path connected, then $\\partial_R X \\cup \\partial_{I_L} X$ is path connected.\n\\end{proposition}\n\n\\begin{proof}\n\tFix $x_0\\in X\\setminus \\mathcal{W}$ and let $\\alpha \\in \\partial_{I_L} X$ with $\\Itin(\\alpha) = \\{B_1, B_2, ...\\}$. Let $c\\colon [0,\\infty)$ be a path in $\\partial_R X$ such that $c([n-1,n])\\subset B_n$ for all $n\\in \\mathbb{N}$. Our goal is to extend $c$ to $[0,\\infty]$ such that $c(\\infty)=\\alpha$. \n\t\n\tLet $\\{t_n\\}\\subset [0,\\infty)$ be a monotone sequence tending to infinity. \n\t\n\t\n\tSince $\\partial X$ is compact then, by passing to a subsequence if necessary, $\\{c(t_n)\\}$ converges to some $\\alpha'$. It suffices to show that $\\alpha = \\alpha '$, which we will do by showing $\\Itin(\\alpha) = \\Itin(\\alpha')$.\n\t\n\tFor each $n$, $\\Itin(c(t_n))=\\{B_1, B_2, ..., B_{k_n}\\}$ where $k_n = \\lceil t_n \\rceil$. Let $r_n$ be the ray based at $x_0$ representing $c(t_n)$ and let $r$ be the ray based at $x_0$ representing $\\alpha'$. By our construction, $r_n\\to r$.\n\t\n\tFix a block $B_m$ in $\\Itin(\\alpha)$. There is an $N$ such that for all $n>N$, $B_m\\in \\Itin(c(t_n))$ and so $r_n\\cap B_m \\neq \\emptyset$. By Lemma \\ref{Croke Kleiner Converging Rays}, either $r\\cap B_m\\neq \\emptyset$ or $r\\in \\partial B_m$. If the former, then $B_m\\in \\Itin(\\alpha')$. If the latter is true, pick points $x_n(m)\\in r_n \\cap B_m$ such that $d(x_0, x_n(m))\\to \\infty$. Such a sequence exists since $r\\in \\partial B_m$ and $r_n\\to r$. Then, after possibly passing to a subsequence, the geodesic segments from $x_0$ to $x_n(m)$ converge to a ray $\\rho_m$ with $\\rho_m(\\infty) = \\alpha'$. For each $i<m$, $B_i$ separates $B_m$ from $x_0$, so $B_i\\in \\Itin(\\alpha')$.\n\t\n\tBut if this is true for some $m$, then it is true for all $M>m$ and so $\\alpha \\in \\partial B_M$ for all $M>m$. Consider the ray $\\rho_M$ by the same construction as the previous paragraph. Using the same argument, $B_i\\in \\Itin(\\rho_M(\\infty)) = \\Itin(\\alpha')$ for all $i<M$. Since $m<M$, this means $B_m\\in \\Itin(\\alpha')$. Therefore $\\alpha'$ and $\\alpha$ have the same itinerary. Since $\\alpha$ is lonely, $\\alpha = \\alpha'$.\n\\end{proof}\n\n\\begin{remark}\n\tProving path connectivity of the boundary does not require each block to have a path connected boundary. The proposition only assumes that $\\partial_R X$ is path connected. This distinction will come up later when we show all 1-ended $\\cat(0)$ groups with isolated flats have path connected visual boundaries.\n\\end{remark}\n\n\n\\begin{remark}\n\tWhile the lonely condition is sufficient, it is not necessary for path connectivity. This comes up for the following group\n\t\\[G=F_2\\times \\mathbb{Z} = \\mathbb{Z}^2\\ast_\\mathbb{Z} \\mathbb{Z}^2\\]\n\t$G$ acts the universal cover of two tori with a curve on each identified. The block decomposition comes from viewing $G$ as the amalgamated product of two $\\mathbb{Z}^2$s over a $\\mathbb{Z}$. The boundary is path connected since it is a suspension of a Cantor set, but irrational rays are not lonely.\n\\end{remark}\n\n\n\n\n\\section{Fixing a Space}\\label{section:fixing a space}\n\nNormally when dealing with $\\cat(0)$ groups and their boundaries, one needs to be particularly careful about the space they are working with. This is because Croke and Kleiner \\cite{CK00} showed  that, unlike in the hyperbolic setting, $\\cat(0)$ groups do not have a well defined visual boundary. When it comes to $\\cat(0)$ groups with isolated flats, however, work of Hruska and Kleiner \\cite{HK05} shows that the visual boundary of these groups is well defined . In this section, we will fix the space on which a 1-ended $\\cat(0)$ group with isolated flats will act on and define the block decomposition for this space.\n\nThroughout this section, let $G$ be a $\\cat(0)$ group which is hyperbolic relative to $\\mathbb{P}$, the collection of flat-stabilizers so that $(G,\\mathbb{P})$ is the relatively hyperbolic pair. Let $G$ act geometrically on some $\\cat(0)$ space $X$. \n\nWe want more control over the space on which $G$ acts, so we use the peripheral splitting theorem of Bowditch introduced in Section \\ref{rel hyp section}, a convex splitting theorem of Hruska and Ruane \\cite{HR17}, and a combination theorem of Bridson and Haefliger \\cite{BH99} to fix a different space which admits a geometric group action by $G$. We quote the two remaining theorems below. But first, a definition.\n\n\n\\begin{defn}\n\tLet $G$ be a CAT(0) group acting geometrically on a $\\cat(0)$ space $X$. A subgroup $H\\leq G$ is \\textit{convex} if $H$ stabilizes a closed, convex subspace $Y$ of $X$, and $H$ acts cocompactly on $Y$. Therefore $H$ act geometrically on $Y$, a $\\cat(0)$ space.\n\\end{defn}\n\n\\begin{thm}[Convex Splitting Theorem, {\\cite[1.3]{HR17}}]\n\tLet $G$ act geometrically on a $\\cat(0)$ space $X$. Suppose $G$ splits as the fundamental group of a graph of groups $\\mathcal{G}$ such that each edge group of $\\mathcal{G}$ is convex. Then each vertex group is convex as well. In particular, each vertex group is a $\\cat(0)$ group.\n\\end{thm}\n\n\nLet $\\mathcal{G}$ be the maximal peripheral splitting for our group $G$. Since $G$ has isolated flats, all peripheral subgroups in the graph of groups $\\mathcal{G}$ are virtually abelian and therefore so are all edge groups. By the Flat Torus Theorem \\cite{BH99}, all virtually abelian subgroup of a $\\cat(0)$ group are convex and therefore we can apply the Convex Splitting Theorem to the maximal peripheral splitting of $G$.\n\nLet $\\mathcal{C}$ be the collection of component vertices and $\\P$ the set of peripheral vertices of $\\mathcal{G}$. For each $v\\in \\mathcal{C}$ with vertex group $G_v$, let $X_v$ be the convex subspace of $X$ on which $G_v$ acts geometrically. For $w\\in \\P$, the vertex group $G_w$ is virtually abelian and therefore acts geometrically on some Euclidean space $F_w$. \n\nEach edge $e$ of $\\mathcal{G}$ is incident to a peripheral vertex $w$ and a component vertex $v$. Let $F_e$ be the flat subspace of $F_w$ on which $P_e$ acts geometrically. Each edge also comes equipped with a monomorphism $\\phi_e\\colon P_e\\to G_v$. The Flat Torus Theorem gives a $\\phi_e$-equivariant isometric embedding of $F_e$ into $X_v$. \n\nWith all of this set up, we can use the following theorem to build our desired $X$:\n\n\\begin{thm}[Equivariant Gluing Theorem, {\\cite[II.11.18]{BH99}}]\\label{thm:equivariant gluing}\n\tLet $\\Gamma_0$, $\\Gamma_1$, and $H$ be groups acting geometrically on complete CAT(0) spaces $X_0, X_1$, and $Y$, respectively. Suppose for $j=0,1$, there exists monomorphisms $\\phi_j\\colon H \\hookrightarrow X_j$ and $\\phi_j$-equivariant isometric embeddings $f_j\\colon  Y\\to X_j$, then $\\Gamma=\\Gamma_0\\ast_H \\Gamma_1$ acts geometrically on a CAT(0) space $X$.\n\\end{thm}\n\nThe space uses the Bass-Serre tree $\\mathcal{T}$ for the splitting $\\mathcal{G}$ of $G$. For each component vertex $v$ of $\\mathcal{T}$, take a copy of $X_v$ and for each peripheral vertex $w$ of $\\mathcal{T}$, take a copy of $P_w$. We shall refer to these spaces as \\textit{vertex spaces}. Then for each edge $e$ incident to $v$ and $w$, take a copy of $F_e\\times [0,1]$, which we shall refer to as \\textit{edge spaces}. Identify the $F_e\\times \\{0\\}$ part with the image of $F_e$ in $X_v$ and $F_e\\times \\{1\\}$ is identified with the image of $F_e$ in $F_w$. Following the Bass-Serre tree, all the copies of the different vertex and edge spaces are glued up to form $X$. \n\nWith this $X$ fixed, we can also describe the block decomposition of $X$. For each vertex space $X_v$ or $P_w$, take the closed $\\frac{1}{2}$-tubular neighborhood to be a block $B$. Then each block intersects in $F_e\\times \\{\\frac{1}{2}\\}$. This defines a block decomposition of $X$.\n\n\\begin{remark}\n\tThroughout the remaining sections, when referring to a $\\cat(0)$ group with isolated flats acting geometrically on $X$, we shall be referring to the $X$ from this construction. \n\\end{remark}\n\nOur goal for the next few sections is to show that $\\partial X$ is path connected. In Lemma \\ref{IFP blocks are lonely}, we will show that with this block decomposition, all irrational rays are lonely. Therefore we just need to show that $\\partial_R X$ is path connected.\n\nIn general, each component vertex is not 1-ended. When this happens, $\\partial X_v$ is cannot be path connected and therefore showing $\\partial_R X$ is path connected requires more work. Therefore we introduce $Y_v$:\n\n\\begin{defn}\\label{defn Y}\n\tFor each component vertex $v$, let $Star(v)$ be the star of $v$ in $\\mathcal{T}$. There is a continuous map $p\\colon X\\to \\mathcal{T}$ which maps vertex spaces to vertices and edge spaces to edges in the obvious way. Let $Y_v= p ^{-1} ( Star(v))$.\n\t\n\t Put another way, let $\\mathcal{E}_v$ be the collection of edges incident to $v$ and $\\P_v$ be the collection of vertices of $\\mathcal{T}$ adjacent to $v$. Since $\\mathcal{T}$ is the Bass-Serre tree for the maximal peripheral splitting, each $w\\in \\P_v$ is a peripheral vertex. Then $Y_v$ is\n\t\n\t\\[Y_v := X_v \\cup \\left(\\bigcup_{e\\in \\mathcal{E}_v} F_e\\times[0,1]\\right) \\cup \\left(\\bigcup_{w\\in \\P_w} F_w \\right) \\]\n\tPut a third way, $Y_v$ consists of $X_v$ and every flat of $X$ which shares a boundary point with $Y_v$.\n\t\n\tUltimately, we want to show that $\\partial Y_v$ is path connected for each $v\\in \\mathcal{C}$. When dealing with a generic $Y_v$, we will drop the subscript and denote it $Y$.\n\\end{defn}\n\nThe motivating example for needing to this is Example \\ref{example: punctured tori}. The maximal peripheral splitting of this group is in Figure \\ref{fig: max periph splitting G2}. Since the component vertices are free groups, they have Cantor sets for boundaries. Additionally, there is no obvious intermediate peripheral splitting which has component vertices with path connected visual boundaries. Therefore, one needs to look at one of the Cantor sets along with all the circles attached to it. We will show this space is path connected.\n\n\n\n\\section{Decomposition Spaces}\\label{section:decomp}\nIn this section we will repeat some results from \\cite{HR17} about decompositions of spaces and their relationship to CAT(0) groups with isolated flats. In particular, we establish that the map $\\pi\\colon \\partial X \\to \\partial (G,\\mathbb{P})$ from Theorem \\ref{Tran Theorem} can be viewed through the lens of decomposition theory. We then use this to establish a decomposition of $\\partial Y$ and ultimately conclude that boundary points in $Y$ which are not in the boundary of a flat are locally connected. \n\nMuch of the results here follow from elementary decomposition theory, which can be found in \\cite{Dav86}. For a topological space $M$, a \\textit{decomposition}, $\\mathcal{D}$, is a partition of $M$. The decomposition map $\\pi\\colon M\\to M\/\\mathcal{D}$ is the quotient map where each $d\\in \\mathcal{D}$ is collapsed to a point and the resulting space is given the quotient topology. In this way, decompositions of $M$ are equivalent to quotients of $M$. Any decomposition can be made into a quotient map and any quotient map can be made into a decomposition by taking point pre-images as elements of the decomposition.\n\n\n\\begin{defn}[Upper semicontinuous decomposition]\n\tA decomposition $\\mathcal{D}$ of $M$ is \\textit{upper semicontinuous} if each $d\\in \\mathcal{D}$ is compact and for each $d\\in \\mathcal{D}$ and each open $U\\subset M$ containing $d$, there is an open $V\\subset M$ containing $d$ such that every $d\\in \\mathcal{D}$ which intersects $V$ is contained in $U$. A quotient is \\textit{upper semicontinuous} if the associated decomposition is upper semicontinuous.\n\\end{defn}\n\n\\begin{prop}[{\\cite[I.1.1]{Dav86}}]\n\tLet $\\mathcal{D}$ be a decomposition of a space $M$ with each $d\\in \\mathcal{D}$ compact. The following are equivalent:\n\t\\begin{enumerate}\n\t\t\\item $\\mathcal{D}$ is upper semicontinuous.\n\t\t\\item For an open set $U\\subset M$, let $U^*$ be the union of $d\\in \\mathcal{D}$ such that $d\\subset U$. Then $U^*$ is open.\n\t\t\\item The decomposition map $\\pi\\colon  M\\to M\/\\mathcal{D}$ is closed.\n\t\\end{enumerate}\n\t\n\\end{prop}\n\\begin{comment}\n\\begin{proof} \n\t\n\t$(1)\\Rightarrow (2)$: Let $U$ open in $M$. If there are no $d\\in \\mathcal{D}$ such that $d\\subset U$, then $U^*$ is empty and therefore open. For each $d\\subset U$, let $V_d$ be the open subset of $U$ from the definition of upper semicontinuous. Let $V=\\cup_{d\\subset U}V_d$, then $V$ is open and we claim that $V=U^*$. The inclusion $U^*\\subset V$ is obvious. Suppose $x\\in V$, then $x\\in V_d$ for some $d\\subset U$. Since $\\mathcal{D}$ is a decomposition, $x\\in d'$ for some $d'\\in \\mathcal{D}$. Since $d'\\cap V_d\\neq \\emptyset$, then $d'\\subset  V_d \\subset U$ and therefore $d'\\subset U^*$. This shows the other inclusion.\n\t\n\t$(2)\\Rightarrow (3)$: Let $C\\subset M$ be closed and let $U$ be $C$'s complement in $M$. To show $\\pi\\colon M\\to M\/\\mathcal{D}$ is closed, we must show that $\\pi ^{-1} (\\pi(C))$ is closed. Call this set $C'$, then we have \n\t$$C'= \\{x\\in M: \\pi(x)\\in \\pi(C)\\} = \\cup \\{ d\\in \\mathcal{D}: \\pi(d) \\in \\pi(C) \\}$$\n\tLet $U'= M\\setminus C'$. We claim $U^*=U'$. First, $U^*\\subset U'$ since if $d\\subset U$, then $\\pi(d)\\notin \\pi(C)$ and therefore $d\\cap C'=\\emptyset$. For the other inclusion, suppose $x\\notin C'$, then $x\\in d$ for some $d\\in \\mathcal{D}$ such that $\\pi(d)\\notin \\pi(C)$. Since $\\pi(d)\\notin \\pi(C)$, then $d\\cap C = \\emptyset$, so $d\\subset U$ and thus $d\\subset U^*$. This shows the other inclusion.\n\t\n\t$(3)\\Rightarrow (1)$: Let $d\\in \\mathcal{D}$ and $U$ open in $M$ such that $d\\subset U$. Let $C=M\\setminus U$. Since $\\pi$ is a closed map, then $\\pi(C)$ is closed and therefore $\\pi^{-1}(\\pi(C))$ is closed as well. Let $U'=M\\setminus \\pi^{-1}(\\pi(C))$. Then $U'\\subset U$ and if $d\\cap U' \\neq \\emptyset$, then $d\\subset U$. If $d\\subset U$, then $\\pi(d)\\notin \\pi(C)$ and therefore $d\\subset U'$. Suppose $d'\\cap U' \\neq \\emptyset$, then $\\pi(d') \\notin \\pi(C)$, so $d'\\subset U$. \n\\end{proof}\n\\end{comment}\n\\begin{prop}[{\\cite[I.2.1 and I.2.2]{Dav86}}]\n\tIf $\\mathcal{D}$ is an upper semicontinuous decomposition of a Hausdorff space, then $M\/\\mathcal{D}$ is Hausdorff. If $\\mathcal{D}$ is an upper semicontinuous decomposition of a metric space, then $M\/\\mathcal{D}$ is metrizable.\n\\end{prop}\n\nBoundaries of proper CAT(0) spaces are metrizable, and therefore so are their quotients coming from upper semicontinuous decompositions.\n\n\\begin{prop}[{\\cite[I.3.1]{Dav86}}]\n\tIf $\\mathcal{D}$ is an upper semicontinuous decomposition of $M$, then $\\pi\\colon M\\to M\/\\mathcal{D}$ is a proper map.\n\\end{prop}\n\n\\begin{defn}[Monotone decomposition]\n\tA decomposition $\\mathcal{D}$ of $M$ is \\textit{monotone} if each $d\\in \\mathcal{D}$ is compact and connected.\n\\end{defn}\n\n\\begin{prop} [{\\cite[I.4.1]{Dav86}}] \\label{Montone if and only if connected}\n\tLet $\\mathcal{D}$ be an upper semicontinuous decomposition of a space $M$. Then $\\mathcal{D}$ is monotone if and only if $\\pi ^{-1}(C)$ is connected whenever $C$ is a connected subset of $M\/\\mathcal{D}$.\n\\end{prop}\n\\begin{comment}\n\\begin{proof}\n\t($\\Rightarrow$) Let $C$ be a connected subset of $M\/\\mathcal{D}$ and suppose, for contradiction, that $\\pi^{-1}(C)$ is disconnected. Then there are disjoint closed sets $U$ and $V$ such that $\\pi^{-1}(C)\\subset U \\cup V$. Since each $d\\in \\mathcal{D}$ is connected, if $d\\subset \\pi^{-1}(C)$, then $d\\subset U$ or $d\\subset V$. Furthermore, if $x\\in \\pi^{-1}(C)$ and $x\\in d$ for some $d\\in \\mathcal{D}$, then $d\\subset \\pi^{-1}(C)$. But then $\\pi(U)\\cup \\pi(V)$ are disjoint, cover $C$ and are closed since the decomposition induces a closed map. This is a contradiction, so $\\pi^{-1}(C)$ is connected.\n\t\n\t($\\Leftarrow$) Singletons are connected in $M\/\\mathcal{D}$, therefore $\\pi^{-1}(x)$ is connected for each $x\\in M\/\\mathcal{D}$ and $\\pi^{-1}(x)=d$ for some $d\\in \\mathcal{D}$ by definition of the decomposition map. Thus each $d\\in \\mathcal{D}$ is connected so the decomposition is monotone.\n\\end{proof}\n\\end{comment}\n\n\\begin{prop}[{\\cite[8.5]{HR17}}] \\label{Singleton points are locally connected}\n\tLet $\\mathcal{D}$ be an upper semicontinuous monotone decomposition of $M$. Let $\\{x\\}$ be a singleton member of $\\mathcal{D}$. If $M\/\\mathcal{D}$ is locally connected at $\\pi(x)$, then $M$ is locally connected at $x$.\n\\end{prop}\n\\begin{comment}\n\\begin{proof}\n\tLet $U$ be a neighborhood of $x$ in $M$. Then $U^*$ is an open neighborhood of $x$ contained in $U$. Since $M\/\\mathcal{D}$ is locally connected at $x$, then $\\pi(U^*)$ contains a connected neighborhood $V$ containing $\\pi(x)$. By Proposition \\ref{Montone if and only if connected} $\\pi^{-1}(V)$ is connected and $\\pi^{-1}(V)\\subset U^*\\subset U$. \n\\end{proof}\n\\end{comment}\nThe following definition allows us to construct subspace decompositions:\n\n\\begin{defn}[Subspace Decomposition]\n\tLet $\\mathcal{D}$ be a decomposition of a Hausdorff space $M$. Let $W\\subset M$ such that if $d \\cap W$ is non-empty for any $d\\in \\mathcal{D}$, then $d\\subset W$. The \\textit{induced subspace decomposition} of $W$ is the decomposition consisting of all members of $\\mathcal{D}$ which are contained in $W$. If $\\mathcal{D}$ is upper semi-continuous, then the induced subspace decomposition is as well.\n\\end{defn}\nThis definition will become vital for understanding the decomposition of $\\partial Y\\subset \\partial X$.\n\n\\begin{defn}[Null Family]\n\tA collection of subsets $\\mathcal{A}$ of a metric space is a \\textit{null family} if for each $\\varepsilon>0$, only finitely many $A\\in \\mathcal{A}$ have diameter greater than $\\varepsilon$.\n\\end{defn}\n\n\\begin{prop}[Saturation condition, {\\cite[8.9]{HR17}}] \\label{Null Family Saturation}\n\tLet $\\mathcal{A}$ be a null family of compact sets in a metric space $M$. Suppose $q\\in M$ and $q$ is not in any of the members of $\\mathcal{A}$. Then each neighborhood $U$ of $q$ contains a smaller neighborhood $V$ of $q$ such that for each $A\\in \\mathcal{A}$, if $A\\cap V \\neq \\emptyset$, then $A\\subset U$.\n\\end{prop}\n\\begin{comment}\n\\begin{proof}\n\tLet $U$ a neighborhood of $q$ and suppose $B(q,\\varepsilon)\\subset U$. Then choose $\\delta>0$ such that $\\delta<\\varepsilon\/2$ and such that $d(q,A)>\\delta$ for each of the finitely many $A$ with diameter greater than $\\varepsilon\/2$. If $V=B(q,\\delta)$, then the result follows. \n\\end{proof}\n\\end{comment}\n\n\\begin{remark}\\label{remark: null family condition}\n\t\tIn view of how the metric works on $\\partial X$, the notion of a null family can be reformulated topologically as follows: Fix $x_0$ as the basepoint for the cone topology. A collection $\\mathcal{A}$ of subspaces is a null family in $\\partial X$ if there is some $D>0$ such that for each $r<\\infty$, only finitely many members of $\\mathcal{A}$ are not contained in a set of the form $U(\\cdot, r, D)$. Under the metric $d_D$ on $\\partial_{x_0} X$, only those members of $\\mathcal{A}$ which lie in $U(\\cdot, r, D)$ have diameter at most $1\/r$.\n\t\n\tA similar condition can be put on the cone topology of $\\overline{X}$. We do not need to deal with this directly, however. Work of Bestvina says that $\\overline{X}$ is metrizable and this metric is compatible with the cone topology, which is sufficient for our needs \\cite{Bes96}.\n\\end{remark}\n\n\n\\begin{proposition} [{\\cite[8.12]{HR17}}] \\label{Family of Spheres}\n\tLet $X$ be a CAT(0) space with isolated flats with respect to the family $\\mathbb{P}$. Let $\\mathcal{A}$ be the collection of spheres $\\{ \\partial F: F\\in \\mathbb{P} \\}$. Then $\\mathcal{A}$ is a null family of disjoint compact subsets of $\\partial X$. \n\\end{proposition}\n\n\n\n\\begin{prop}[{\\cite[I.2.3]{Dav86}}]\n\tLet $\\mathcal{D}$ be a decomposition of a metric space $M$ such that each $d\\in \\mathcal{D}$ is compact. Let $\\mathcal{A}$ be the collection of $d\\in \\mathcal{D}$ such that $d$ is not a singleton. Then if $\\mathcal{A}$ is a null family, $\\mathcal{D}$ is an upper semicontinuous decomposition. \n\\end{prop}\n\\begin{comment}\n\\begin{proof}\n\tLet $U$ be an open set in $M$ and $d\\subset U$. For each $x\\in d$, let $\\varepsilon_x$ be such that $B(x,\\varepsilon_x)\\subset U$. For each $x$, let $0<\\delta_x<\\varepsilon_x\/2$ such that $d(x,A)>\\delta$ for each $A\\in \\mathcal{A}$ with diameter at least $\\varepsilon_x\/2$. Then let $V=\\cup_{x\\in d} B(x,\\delta_x)$. It is clear that $V\\subset U$ by our choices of $\\delta_x$. Suppose $y\\in d'\\cap V$ for some $d'\\in \\mathcal{D}$. If $d'=\\{y\\}$, then $d'\\subset U$. If not, then $y\\in B(x,\\delta_x)$ for some $x\\in d$. By our choice of $\\delta_x$, it must be that the diameter of $d'$ is smaller than $\\varepsilon_x\/2$, so $d'\\subset B(x,\\varepsilon)\\subset U$. Therefore $\\mathcal{D}$ defines an upper semicontinuous decomposition of $M$.\n\\end{proof}\n\\end{comment}\n\\begin{corollary}[{\\cite[8.13]{HR17}}]\n\tLet $G$ be a CAT(0) group with isolated flats relative to $\\mathbb{P}$ acting geometrically on a CAT(0) space $X$. Let $\\mathcal{A}$ be the collection of spheres which bound maximal dimensional flats in $\\partial X$. Then the quotient map $\\partial X \\to \\partial X\/ \\mathcal{A} \\cong \\partial (G,\\mathbb{P})$ given in Theorem \\ref{Tran Theorem} is upper semicontinuous and monotone.\n\\end{corollary}\n\n\\begin{corollary}[{\\cite[8.14]{HR17}}]   \\label{Non-flat points are l.c.}\n\tLet $G$ be 1-ended and CAT(0) with isolated flats relative to $\\mathbb{P}$. Then $\\partial X$ is locally connected at each $x$ which is not in the boundary of a flat. \n\\end{corollary}\n\n\\begin{proof}\n\tFrom Theorem \\ref{thm: peripheral splitting}, $\\partial(G,\\mathbb{P})$ is locally connected. Since the decomposition is upper semicontinuous and monotone, the singletons in the decomposition are locally connected as well. In this decomposition, the singletons are exactly the points which are not in the boundary of any flat.\n\\end{proof}\n\n\nRecall the definition of $Y$(Definition \\ref{defn Y}) at the end of Section \\ref{section:fixing a space}: Let $p\\colon X\\to \\mathcal{T}$ be the projection map onto the Bass-Serre tree $\\mathcal{T}$. Then for a component vertex $v\\in \\mathcal{T}$, $Y_v=p^{-1}(Star(v))$. Since we will being working with generic $Y_v$, we shall just denote it as $Y$.\n\\begin{thm} \\label{monotone usc decomp of Y}\n\tThe restriction of $\\pi\\colon \\partial X \\to \\partial (G,\\mathbb{P})$ to $\\partial Y$ defines a monotone upper semicontinuous decomposition and the image of $\\partial Y$ is $\\partial (G_v, \\mathbb{P}_v)$, the Bowditch boundary for the corresponding component vertex. Here \\linebreak $\\mathbb{P}_v : = \\{ P\\cap G_v: P\\in \\mathbb{P}\\}$.\n\\end{thm}\n\n\\begin{proof}\n\tFirst, see that restricting $\\pi$ to $\\partial Y$ is a subspace decomposition for if $d\\cap \\partial Y$, then $d\\subset \\partial Y$. This is clear since for each $d\\in \\mathcal{D}$, either $d$ is a singleton or $d$ is the boundary of a maximal dimensional flat and in either of those cases, $d\\cap \\partial Y$ means $d\\subset \\partial Y$. Additionally this subspace decomposition is monotone. Therefore we have an upper semicontinuous decomposition $\\pi\\colon  \\partial Y \\to \\pi(\\partial Y)$. \n\t\n\tNext, consider the relatively hyperbolic pair $(G_v,\\mathbb{P}_v)$. By the arguments above, the map $\\pi'\\colon  \\partial X_v \\to \\partial (G_v, \\mathbb{P}_v)$ is an upper semicontinuous decomposition. Since the peripheral structure of $G_v$ is inherited from $G$, then $\\pi(\\partial X_v) = \\pi'(\\partial X_v)$.\n\t\n\tLastly, we need to show that $\\pi(\\partial Y) = \\pi (X_v)$. To see this, note that if $d\\subset \\partial Y$, then either $d$ is a singleton or $d$ is the boundary of a flat. In the former case, then $d\\in \\partial X_v$. In the latter case, $d\\cap \\partial X_v \\neq \\emptyset$. The map $\\pi$ collapses all points of $d$ to a single point, therefore $\\pi(d) = \\pi (d\\cap \\partial X_v)$, and therefore $\\pi(\\partial Y) = \\partial (G_v,\\mathbb{P}_v)$.\n\t\n\t\n\t\n\\end{proof}\n\n\\begin{corollary}\\label{Y non-flat points lconn}\n\tIf $x\\in \\partial Y$ and $x\\notin \\partial F$ for any flat $F$, then $\\partial Y$ is locally connected at $x$.\n\\end{corollary}\n\n\\begin{proof}\n\tThis follows from the fact that we have a monotone, upper semicontinuous decomposition to a locally connected space (Theorem \\ref{thm: peripheral splitting}) and that $x$ is a singleton in this decomposition.\n\\end{proof}\n\n\n\n\\section{Proper and Cocompact Action} \\label{section:proper and cocompact}\n\nOur goal of this section is to show that the stabilizer of a flat of $X_v$ also acts properly and cocompactly on $\\partial Y$ minus the higher-dimensional flat. Recall $X_v$ is the vertex-space for a component vertex $v$ of the maximal peripheral splitting of $G$. Since we need to distinguish between flats of $X_v$ and flats of $Y$, we need to establish some notation. Throughout this section, fix a flat $F\\subset X_v$ and let $F^*$ be the maximal dimensional flat of $X$ which contains $F$. By our construction of $Y$, $F^*\\subset Y$ but $F^*$ is not necessarily contained in $X_v$. Let $P$ and $P^*$ stabilize $F$ and $F^*$ respectively. Lastly, define $\\Omega:= \\partial Y \\setminus \\partial F^*$.\n\nThe ideas from this section come from \\cite{Ha17}, which proves the results in the case where $\\partial X$ is locally connected. In that setting, $Y$ and $X_v$ are the same and $F=F^*$.\n\nThe stabilizer $P$ acts properly and cocompactly on $F$. With some work, one can show that $P$ also acts properly and cocompactly on $\\Omega$. Furthermore, the fundamental domains for the action of $P$ on both can be picked in a way which allows us to relate the two spaces. The following lemma of Bowditch  establishes the proper and cocompact action of $P$ on $\\Omega$ \\cite{Bow12}:\n\n\\begin{lem}\\label{Bow1}\n\tLet $P$ be a group acting on topological spaces $C$ and $D$. Define the action of $P$ on $C\\times D$ as the diagonal action and let $\\mathbb{R} \\subset C\\times D$. If $\\mathbb{R}$ is $P$-invariant and the projection maps $\\pi_C$ and $\\pi_D$ from $\\mathbb{R}$ to $C$ and $D$ are both proper and surjective, then the following are equivalent:\n\t\\begin{enumerate}[(a)]\n\t\t\\item $P$ acts properly and cocompactly on $C$\n\t\t\\item $P$ acts properly and cocompactly on $D$\n\t\t\\item $P$ acts properly and cocompactly on $\\mathbb{R}$\n\t\\end{enumerate}\n\\end{lem}\n\nDefine $\\bot(F)$ to be the set of geodesic rays orthogonal to $F$. Recall a geodesic ray $r\\colon [0,\\infty) \\to X$ is \\textit{orthogonal} to a convex set $C\\subset X$ if $r(0)\\in C$ and for every $t>0$ and any $y\\in C$, the Alexandrov angle, $\\angle_{r(0)}(r(t),y)$, is greater than or equal to $\\pi\/2$. \n\nWe will apply this lemma where $F$ and $\\Omega$ are $C$ and $D$ respectively.  Let $\\mathbb{R}:= \\{ (x,q): \\text{there exists } q\\in \\bot(F) \\text{ with } d(x,q(0))\\leq A\\}$, where $A$ is the diameter of the fundamental domain for the action of $P$ on $F$. Throughout, we will assume our basepoint for the topology on $\\partial Y$ is in $F$. Since $P$ acts properly and cocompactly on $F$, we need to show the $P$-invariance of $\\mathbb{R}$ and the projection maps to $F$ and $\\Omega$ are both proper and surjective. \n\n\n\\begin{lem}\n\t$\\mathbb{R}$ is $P$-invariant.\n\\end{lem}\n\n\\begin{proof}\n\t$P$ acts by isometries on $Y$, so if $(x,q)\\in \\mathbb{R}$ and $p\\in P$, then \\linebreak $d(p\\cdot x, p\\cdot q(0))\\leq A$. It is clear that $p\\cdot q$ is a ray, but we need to show that it is orthogonal to $F$.\n\t\n\tFor all $t$, $\\pi_F(q(t))=q(0)$, so $q(0)$ is the closest point of $F$ to each point along the ray. Assume, for contradiction, that $\\pi_F( p\\cdot q(t)) = y$ and \\linebreak $y\\neq p\\cdot q(0)$. Then\t\n\t$$d(p\\cdot q(t), y) < d(p\\cdot q(t), p\\cdot q(0))$$\n\tThis is a strict inequality because of the uniquness of projection points. Applying $p^{-1}$ to all the points gives us:\n\t$$d(q(t),p^{-1} \\cdot  y) < d(q(t),q(0))$$\n\tSince $p^{-1}\\in P$ and $P$ fixes $F$, then $p^{-1} y \\in F$. But $q(0)$ is the closest point of $F$ to $q(t)$, so $y=p\\cdot q(0)$ and therefore $p\\cdot q\\in \\bot(F)$.\n\\end{proof}\n\n\\begin{lem}\\label{lemma: orthogonal rays for all bnd points}\n\tLet $\\alpha\\in \\Omega$. If $r$ is a ray representing $\\alpha$, then there is a ray $q\\in \\bot(F)$ asymptotic to $r$.\n\\end{lem}\n\nTo prove this lemma, we make use of the following lemma from \\cite{Ha17}, which follows closely to Lemma I.5.31 in \\cite{BH99}.\n\n\\begin{lem} \\label{isometric embeddings converge}\n\tIf $(Y,\\rho)$ is a separable metric space, $(X,d)$ a proper metric space, $y_0\\in Y$, and $K$ a compact subset of $X$, then any sequence of isometric embeddings, $c_n\\colon Y \\to X$, with $c_n(y_0)\\in K$ has a subsequence which converges point-wise to an isometric embedding $c\\colon Y\\to X$.\n\\end{lem}\n\nThroughout the remaining of the section, let $[x,y]$ denote the geodesic segment connecting points $x,y\\in X$. \n\n\\begin{proof}[Proof of Lemma \\ref{lemma: orthogonal rays for all bnd points}]\n\tFix a basepoint $x_0$, let  $x_n:=r(n)$ and let $y_n:= \\pi_F(x_n)$ for all $n\\in \\mathbb{N}$. First, assume for contradiction that $y_n$ is unbounded. Then $y_n\\to \\eta \\in \\partial F$. By \\cite{HK09}, there is a constant $M>0$ such that for all $n$ $d(x_0,[x_n,y_n])<M$, where $[x_n,y_n]$ is the geodesic segment in $Y$ from $x_n$ to $y_n$. Let $m$ be the point of this geodesic closest to $x_0$. Since the geodesic $[x_n,y_n]$ is orthogonal to $F$ and $x_0,y_n\\in F$, then the largest side of the triangle with vertices $x_0,y_n,m$ is the edge $[x_0,m]$. Therefore $d(x_0,y_n)<M$, contradicting $y_n$ tending to infinity.\n\t\n\tFor each $n\\in \\mathbb{N}$, let $f_n\\colon [0,n]\\to Y$ be the geodesic $[y_n,x_n]$, which is orthogonal to $F$. For each $k$ and all $n\\geq k$, consider the sequence $(f_n|_{[0,k]})$. By Lemma \\ref{isometric embeddings converge}, these converge to a map $q_k\\colon [0,k]\\to Y$. Notice for each $\\ell < k$, $q_\\ell = q_k|_{[0,\\ell]}$, since the tails of the sequences defining $q_\\ell$ and $q_k$ are identical. Therefore, $q\\colon [0,\\infty)\\to Y$ is a geodesic ray which is the limit of the sequence $(q_n)$. \n\t\n\tIt suffices to show that $q\\in \\bot(F)$ and $q$ is asymptotic to $r$. For the former, fix $y\\in F$ and notice that since $[y_n,x_n]$ is a geodesic segment orthogonal to $F$,  $\\angle_{y_n}(y,x_n) \\geq \\pi\/2$ for all $n\\in \\mathbb{N}$. Then, applying Proposition II.3.3(1) in \\cite{BH99}, see that the map $(x,y,p)\\to \\angle_p(x,y)$ is upper semicontinuous. Therefore the limiting segments $q_n$ are orthogonal to $F$ so $q\\in \\bot(F)$. To see that $q$ and $r$ are asymptotic, notice that $[y_n,x_n]\\subset B_M(r)$, and so $q$ is $M$-close to $r$, thus $q$ and $r$ are asymptotic. \n\t\n\t\n\t\n\\end{proof}\n\n\\begin{corollary}\n\tThe projection maps $\\pi_F$ and $\\pi_\\Omega$ are surjective. \n\\end{corollary}\n\n\\begin{proof}\n\tThe lemma above shows that $\\pi_\\Omega$ is surjective. Using the $P$-action and the fact that $A$ is the diameter of the fundamental domain, then $\\pi_F$ is surjective as well.\n\\end{proof}\n\nPart of showing the projection maps are proper requires an understanding of what happens to the boundary points when a sequence of rays in $\\bot(F)$ converge. \n\n\n\\begin{lem}\n\tIf $(r_n)$ is a sequence of rays orthogonal to $F$ which converge to an element $r\\in \\bot(F)$, then $r_n(\\infty)$ converges to $r(\\infty)$ in the cone topology.\n\\end{lem}\n\n\\begin{proof}\n\tFix $x_0\\in F$ a basepoint for the cone topology on $\\partial Y$. For each $r_n$, let $c_n$ be the ray based at $x_0$ asymptotic to $r_n$. Since $r_n\\to r$, then there is a constant $D$ such that $d(r_n(t),c_n(t))\\leq D$ for all $t$. Applying Lemma \\ref{isometric embeddings converge}, the rays $c_n$ converge to a ray $c$ which is asymptotic to $r$. \n\t\n\tFix $\\epsilon>0$ and let $s>0$. $U(c,s,\\epsilon)$ is a basic neighborhood of $c$ in $\\partial Y$. Since $c_n\\to c$ pointwise, then there is an $N\\in \\mathbb{N}$ such that $d(c_m(s),c(s))<\\epsilon$ for all $m>N$ and therefore $c_n(\\infty)\\to c(\\infty)$.\n\\end{proof}\n\n\nWhat remains is to show that the projection maps are both proper. Recall from Theorem \\ref{Quasi-convex Constant} there is a universal constant $\\kappa>0$ which makes certain subsets of a $\\cat(0)$ space with isolated flats $\\kappa$-quasiconvex. Haulmark extends this result to include geodesic rays which are orthogonal to some flat $F$.\n\n\\begin{lem}[{\\cite[3.7]{Ha17}}] \\label{Haulmark F-q quasiconvex}\n\tLet $F\\in \\mathcal{F}$ and $q\\in\\bot(F)$, then there exists a constant $\\kappa$ such that $q\\cup F$ is $\\kappa$-quasiconvex in $X$.\n\\end{lem}\n\n\\begin{remark}\n\tThe constants $\\kappa$ in Theorem \\ref{Quasi-convex Constant} and Lemma \\ref{Haulmark F-q quasiconvex} above are the same $\\kappa$.\n\\end{remark}\n\n\\begin{corollary}\n\tLet $F^*$ a flat of $X$, $F$ a flat subset of $F^*$ and $q\\in \\bot(F)$, then $q\\cup F$ is $\\kappa$-quasiconvex in $Y$.\n\\end{corollary}\n\n\\begin{proof}\n\tIn $Y$, $\\bot(F)=\\bot(F^*)$ since $F$ separates $F^*$ from the rest of $Y$. $Y$ inherits the metric from $X$, so $q\\cup F^*$ is quasiconvex in $Y$. Any geodesic in $Y$ from points of $F$ to points of $q$ are $\\kappa$-close to $F^*$ or $q$. If they are $\\kappa$-close to $F^*$, then they are $\\kappa$-close to $F$ because $F$ separates $F^*$ from $Y$. Lastly, since $Y$ is convex, all geodesics from between points of $q\\cup F^*$ are in $Y$ to begin with. Thus, $q\\cup F$ is $\\kappa$-quasiconvex.\n\\end{proof}\n\nThe two lemmas below relate how close together a ray $q\\in \\bot(F)$ and $c$ can be, where $c$ is the ray based at $x_0$ asymptotic to $r$.\n\n\\begin{lem}\n\tThere is a constant $M=M(\\kappa)>0$ such that for any ray $q\\in \\bot(F)$, $d(q(0),c(t))< M$ where $c$ is the ray asymptotic to $q$ based at $x_0$ and $t=d(x_0,q(0))$.\n\\end{lem}\n\n\\begin{proof}\n\tLet $\\beta\\colon [0,t]\\to F$ be the geodesic from $x_0$ to $q(0)$. Using the quasiconvexity result above, there is a $\\kappa>0$ such that $c$ is contained in the $\\kappa$-neighborhood of $q\\cup F$. Therefore, there are points $s\\in[0,\\infty)$, $x\\in q$, $y\\in F$ such that $d(c(s),x)\\leq \\kappa$ and $d(c(s),y)\\leq \\kappa$. Using the triangle inequality, $d(x,y)\\leq 2\\kappa$. Since $x\\in q$, $q$ is orthogonal $F$ and $y\\in F$, then $d(x,q(0))\\leq 2\\kappa$ as well and so $d(c(s),q(0))\\leq 3\\kappa$. \n\t\n\tThe triangle $\\triangle(x_0,c(s),q(0))$ has edges with side lengths $t$, $s,$ and at most $3\\kappa$, so $t\\in [s-3\\kappa,s+3\\kappa]$. Thus $|t-s|\\leq 3\\kappa$ and so \\[d(c(t),q(0)) \\leq d(c(t),c(s))+ d(c(s),q(0)) \\leq 3\\kappa+3\\kappa.\\] Setting $M=6\\kappa$ is the desired constant.\n\\end{proof}\n\nThe next lemma relates rays in $\\bot(F)$ and their geodesic representatives, based on how close the former ray's starting point is to a ray in $F$.\n\n\\begin{lem} \\label{delta(epsilon,M)}\n\tFix $\\epsilon>0$ and let $M=M(\\kappa)$ from above. Then there is a constant $\\delta=\\delta(\\epsilon,M)$ such that for any $n>0$ and $\\eta \\in \\partial F$ the following holds: if $q\\in \\bot(F)$ and $q(0)\\in U(\\eta, n, \\epsilon)$, then $c(n)\\in U(\\eta,n,\\delta)$, where $c$ is the ray asymptotic to $q$.\n\\end{lem}\n\n\\begin{proof}\n\tLet $\\beta\\colon  [0,t]\\to F$ be the geodesic from $x_0$ to $q(0)$. The lemma above shows that if $t=n$, then $d(c(t),q(0))\\leq M$ so $d(c(t),\\eta(n))\\leq M+\\epsilon$.\n\t\n\tIf $n<t$, notice that $d(c(t),\\eta(n)) \\leq d(c(t),\\beta(t))+ d(\\beta(t),\\eta(n)) \\leq M+\\epsilon$. Consider the triangle $\\triangle(x_0,c(t),\\eta(n))$. This has side lengths $t$, $n$, and at most $M+\\epsilon$, so $t-n\\leq M+\\epsilon$. With the triangle inequality we have:\t\n\t$$d(c(n),\\eta(n)) \\leq d(c(t),c(n)) + d(c(t), \\eta(n)) \\leq M+\\epsilon + M + \\epsilon$$\n\tLastly, if $t<n$, then $d(\\beta(t),\\eta(n))\\leq \\epsilon$ and $d(\\beta(t),c(t)) \\leq M$ so \\linebreak $d(\\eta(n),c(t))\\leq M+\\epsilon$. Consider the triangle $\\triangle(x_0,\\eta(n),\\beta(t)$, which has sidelengths $t$, $n$, and at most $\\epsilon$. Therefore $n-t< \\epsilon$. Applying the triangle inequalty we get:\t\n\t$$d(c(n),\\eta(n)) \\leq d(c(t),c(n))+ d(c(t),\\eta(n)) \\leq \\epsilon + M+\\epsilon$$\n\tTherefore, letting $\\delta = 2(M+\\epsilon)$ covers all three cases.\n\\end{proof}\n\nWith these two lemmas, we can now prove that the projection maps are proper:\n\n\\begin{prop}\n\tThe projection map $\\pi_\\Omega\\colon \\mathbb{R} \\to \\Omega$ is a proper map.\n\\end{prop}\n\n\\begin{proof}\n\tLet $K\\subset \\Omega$ be a compact set and consider \n\t\\[\\pi_\\Omega^{-1}(K) = \\{ (x,q): q\\in \\bot(F),~ q(\\infty)\\in K,~ d(x,q(0))\\leq A\\}\\subset \\mathbb{R}\\]\n\tLet $C$ be the following set\n\t\\[ C := \\{ x\\in F: \\exists q\\in \\bot(F), ~q(\\infty)\\in K,~ d(q(0),x)\\leq A\\} \\]\n\tWe claim $C$ is compact and we will show this by showing that it is closed and bounded. \n\t\n\tFirst, suppose it is unbounded. Then there is a sequence $x_n$ tending to infinity and so there is a sequence of rays $q_n$ with $q_n(0)$ tending to a point $\\eta\\in \\partial F$. Let $c_n$ be the geodesic ray based at $x_0$ asymptotic to $q_n$. Fix $\\epsilon>0$ and consider $U(\\eta,n,\\epsilon)$. Such a neighborhood contains all but finitely many of $q_i(0)$. By Lemma \\ref{delta(epsilon,M)}, $U(\\eta,n,\\delta)$ contains all but finitely many $c_i$ and therefore $c_i(\\infty)\\to \\eta$. But $c_i(\\infty)\\in K$ and $K$ is compact, so $\\eta\\in K$, a contradiction. Therefore $C$ is bounded.\n\t\n\tNext, let $x_i\\to x$ be a sequence of points in $C$. For each $x_i$, there is a a $q_i\\in \\bot(F)$ with $d(q_i(0),x_i)\\leq A$. So for sufficiently large $i$, $q_i(0)\\in \\overline{B_A(x)}$. Applying Lemma \\ref{isometric embeddings converge}, $q_i$ converge to $q\\in \\bot(F)$ pointwise. Therefore $q(0)\\in \\overline{B_A(x)}$. Since $q_i(\\infty)\\in K$ for all $i$ and $K$ is compact, then $q(\\infty)\\in K$ and so $q(0)\\in C$ and so is $x$.\n\t\n\tNow let $(x_n,q_n)$ be a sequence in $\\pi_\\Omega^{-1}(K)$. Since $q_n(0)\\in C$ and $C$ is compact, by Lemma \\ref{isometric embeddings converge}, after possibly passing to a subsequence, $q_n\\to q$ pointwise. Similarly, $x_n\\in C$, so after passing to a subsequence, $x_n$ converge as well and so $(x_n,q_n)$ can be made to converge to $(x,q)$.\n\t\n\tLastly, to see that $q\\in \\bot(F)$, take any $y\\in F$. Let $p_n=q_n(0)$ and $x_n=q_n(1)$. Since each $q_n$ is orthogonal to $F$, $\\angle_{p_n}(x_,y)\\geq \\pi\/2$. The map $(x,y,p)\\to \\angle_p(x,y)$ is upper semicontinuous (see \\cite{BH99} Proposition II.3.3(1))), so $q\\in \\bot(F)$. This concludes the proof.\n\t\n\\end{proof}\n\\begin{prop}\n\tThe projection map $\\pi_F\\colon \\mathbb{R} \\to F$ is a proper map.\n\\end{prop}\n\n\n\\begin{proof}\n\tLet $C\\subset F$ be compact. Then we have\n\t\\[ \\pi_F^{-1} (C) = \\{ (x,q): q\\in \\bot(F), ~x\\in C, ~d(q(0),x)\\leq A\\}\\]\n\tLet $(x_,q_n)$ be a sequence in $\\pi_F ^{-1}(C)$. Since $C$ is compact, so is $\\overline{B_A(C)}$ and $q_i(0)\\in \\overline{B_A(C)}$ for each $i$. By Lemma \\ref{isometric embeddings converge}, after possibly passing to a subsequence, $q_n\\to q$ pointwise. Furthermore, $x_n\\to x$ since $x_n\\in C$. Therefore, $(x_n,q_n)\\to (x,q)$.\n\t\n\t\tTo see that $q\\in \\bot(F)$, take any $y\\in F$. Let $p_n=q_n(0)$ and $x_n=q_n(1)$. Since each $q_n$ is orthogonal to $F$, $\\angle_{p_n}(x_,y)\\geq \\pi\/2$. The map $(x,y,p)\\to \\angle_p(x,y)$ is upper semicontinuous (see \\cite{BH99} Proposition II.3.3(1))), so $q\\in \\bot(F)$ and thus $(x,q)\\in \\pi_F^{-1}(C)$ making this set compact.\n\\end{proof}\nCombining everything we get the following theorem:\n\n\\begin{thm}\n\tLet $P$ be the stabilizer of a flat $F$ of $X_v$. Then $P$ acts properly and cocompactly on $\\Omega:= \\partial Y \\setminus \\partial F^*$, where $F^*$ is the maximal dimensional flat in $Y$ which contains $F$.\n\\end{thm}\n\nMore importantly, the work above gives the following association between compact sets of $F$ and those of $\\Omega$:\n\n\\begin{corollary}\n\t\\label{Compact Set Association}\n\tGiven a compact set $K\\subset \\Omega$, there exists a compact set $C\\subset F \\subset F^*$ associated to $K$  such that for each $\\eta \\in K$, there is a ray $c\\colon [0,\\infty) \\to Y$ such that $c\\in \\bot(F)=\\bot(F^*)$, $c(0)\\in C$ and $c(\\infty)=\\eta$. Furthermore, the set $C$ can be chosen $P$-equivariantly in the sense that if $C$ is associated to $K$, then $pC$ is associated to $pK$.\n\\end{corollary}\n\n\\begin{proof}\n\tThis follows immediately from the proofs of the projection maps being proper. For any $K\\subset \\Omega$, take $C$ to be $\\pi_F(\\pi_\\Omega^{-1}(K))$, which is compact as seen above and this association is $P$-equivariant since $\\mathbb{R}$ is.\n\\end{proof}\n\nThis corollary is vital for relating neighborhoods of points in $F$ to neighborhoods of points in $\\Omega$, which will ultimately allow us to show that $\\partial Y$ is locally connected.\n\n\\section{Local Connectivity in Flats} \\label{section:local conn at flats}\nThe goal of this section to show that $\\partial Y$ is locally connected at each of the remaining points. Once that is done, we can combine everything we have done to prove Theorem \\ref{thm main}. The decomposition theory gives that for each $\\xi\\in \\partial Y$, if $\\xi\\notin \\partial F^*$ for any flat $F^*$ of $Y$, then $\\partial Y$ is locally connected at $\\xi$. Additionally, if $\\beta\\in \\partial F^*$ and $\\beta \\notin \\partial X_v$, then $\\partial Y$ is locally connected at $\\beta$ since $\\partial F^*$ is a sphere. Therefore all that is left to show is that if $\\alpha \\in \\partial X_v \\cap \\partial F^*= \\partial F$, then $\\partial Y$ is locally connected at $\\alpha$.\n\nInstead of showing local connectivity, we show that if $\\alpha\\in \\partial F$, then $\\partial Y$ is weakly locally connected at $\\alpha$. The distinction between locally connected and weakly locally connected is toward the end of the section, Definition \\ref{def:wlc}.\n\nMany of the ideas in this section come from \\cite{HR17}. The authors work in the setting where $G$ is a $\\cat(0)$ group with isolated flats with a trivial maximal peripheral splitting and ultimately conclude that $\\partial G$ is  locally connected at every point. We show here that the ideas work in our setting as well.\n\nThe approach is to use the association established in Corollary \\ref{Compact Set Association} to relate neighborhoods in $\\overline{F}$ to neighborhoods of $\\overline{\\Omega}$. Recall $F$ is the subflat of $F^*$ such that $\\partial F \\subset \\partial X_v$. Before establishing this connection, we need to understand $\\overline{\\Omega}$.\n\n\\begin{lem}\n\t$\\overline{\\Omega}=\\Omega \\cup \\partial F$.\n\\end{lem}\n\n\\begin{proof}\n\tFirst, we will show that $\\Omega \\cup \\partial F$ is closed. We know that $\\partial F^* $ and $\\partial F$ are both closed in $\\partial Y$. Therefore $\\partial F^* \\setminus \\partial F$ is open. We can also see that \n\t$$\\partial Y \\setminus (\\Omega \\cup \\partial F) = \\partial F^* \\setminus \\partial F$$\n\tTherefore $\\Omega \\cup \\partial F$ is closed since its complement in $\\partial Y$ is open and so $\\overline{\\Omega}\\subset \\Omega \\cup \\partial F$.\n\t\n\tNow suppose $\\alpha\\in \\partial F$, then we want to show that $\\alpha\\in \\overline{\\Omega}$. We will do this by creating a sequence of points in $\\Omega$ which limit to $\\alpha$. Fix $x_0\\in F$ and let $c\\colon [0,\\infty)\\to Y$ be the ray representing $\\alpha$ and let $C$ be a fundamental domain for the action of $P$ on $F$. Using the $P$-action, let $p_i\\cdot C$ cover the ray $c$. For each $p_i$, let $q_i\\in \\bot(F)$ such that $q_i(0)\\in p_i\\cdot C$. Let $r_i$ be the geodesic ray based at $x_0$ asymptotic to $q_i$.\n\t\n\tFix $\\epsilon>0$ and notice that since the translates of $C$ cover $c$, then for all but finitely many $i$, $q_i(0)\\in U(c,n,\\epsilon)$. Applying Lemma \\ref{delta(epsilon,M)}, we have  \\[r_i(n)\\in U(c,n,\\epsilon+\\delta)\\]for all but finitely many $i$. This means that $r_i(\\infty)\\in U(c,n,\\epsilon+\\delta)$ for all but finitely $i$ and thus $r_i(\\infty)\\to \\alpha$ as $n\\to \\infty$, as desired.\t\t\n\t\n\t\n\\end{proof}\n\nNext we restate a result of Haulmark's to emphasize the connection between rays orthogonal to the flat $F$ and their corresponding endpoints:\n\n\\begin{lem} [\\cite{Ha17}] \\label{Haulmark Quasi Convex}\n\tLet $\\kappa$ be the CAT(0) with isolated flats constant as in Theorem \\ref{Quasi-convex Constant}. Suppose $c'\\in \\bot(F)$ and $c$ is some ray contained in $F$. If $c'(0)\\in U(c,r,D)$ for some constants $r,D$, then $c'(\\infty)\\in U(c,r,D+\\kappa)$. Conversely, if $c'(\\infty)\\in U(c,r,D)$, then $c'(0)\\in U(c,r,D+\\kappa)$.\n\\end{lem}\n\n\nWe make use of the following result of Bestvina:\n\n\\begin{prop}[\\cite{Bes96}]\n\tLet $H$ be a group acting geometrically on a $\\cat(0)$ space $Z$. Let $C\\subset Z$ be a compact set. The $H$-translates of $C$ form a null family in the compact space $\\overline{Z}$.\n\\end{prop}\n\n\n\\begin{prop}\n\tIf $K\\subset \\Omega$ is compect, then the collection of $P$-translates of $K$ is a null family in $\\overline{\\Omega}$, where $P$ is the stabilizer of the flat $F\\subset X_v$.\n\\end{prop}\n\n\\begin{proof}\t\n\tLet $K\\subset \\Omega$ be compact and let $C\\subset F$ be the compact set associated to $K$ from Corollary \\ref{Compact Set Association}. Fix a positive constant $D>0$ and let $\\kappa$ be the constant from Theorem \\ref{Quasi-convex Constant}. From the Remark \\ref{remark: null family condition}, it suffices to show that for each $r<\\infty$, only finitely many $P$-translates of $K$ are not contain in any set of the form $U(\\cdot, r, D)$. \n\t\n\tUsing the Bestvina result above, $P^*$-translates of $C$ form a null family in $\\overline{F^*}$. Since $P\\subset P^*$ and $\\overline{F}\\subset \\overline{F^*}$, then the $P$-translates of $C$ form a null family in $\\overline{F}$. So only finitely many of the $P$-translates of $C$ are not in set of the form $U(\\cdot, r,D+\\kappa)$. For any $p\\in P$, if $pK$ lies in a set of the form $U(\\cdot, r, D)$, then $pC$ lies in a set of the form $U(\\cdot, r, D+\\kappa)$ by Lemma \\ref{Haulmark Quasi Convex}. Therefore the $P$-translates of $K$ form a null family in $\\overline{\\Omega}$.\n\\end{proof}\n\n\n\nThe association between compact sets of $F$ and those of $\\Omega$ established in Corollary \\ref{Compact Set Association} can be improved even further: the compact sets chosen can be connected fundamental domains for the action of $P$. To see this is connected, first observe the following:\n\n\\begin{prop} \\label{Connected Fundamental Domain}\n\tLet $(G,\\mathbb{P})$ be a CAT(0) group with isolated flats and peripheral structure $\\mathbb{P}$. Let $(G_v,\\mathbb{P}_v)$ be a component vertex of $G$'s maximal peripheral splitting with the peripheral structure coming from $G$. Let \\linebreak $\\rho \\in \\partial(G_v, \\mathbb{P}_v)$ be a parabolic point stabilized by $P$. Then there exists a connected, compact fundamental domain $C$ for the action of $P$ on $\\partial (G_v, \\mathbb{P}_v) \\setminus \\{\\rho\\}$. \n\\end{prop}\n\n\\begin{proof}\n\t\n\tSince $(G_v,\\mathbb{P}_v)$ is a component vertex of a 1-ended CAT(0) group with isolated flats, we know that $\\partial (G_v, \\mathbb{P}_v)$ is connected and locally connected and therefore path connected. Notice that $\\pi(F)=\\pi(F^*)=\\rho$. We also know that since $G_v$ is a component vertex of a maximal peripheral splitting, then $G_v$ does not split further relative to $\\mathbb{P}_v$. Therefore $\\rho$ is not a global cut point of $\\partial (G_v, \\mathbb{P}_v)$. These facts follow from Theorem \\ref{thm: peripheral splitting}.\n\t\n\tSince $\\rho$ is not a global cut point and $\\partial (G_v, \\mathbb{P}_v)$ is path connected, then $\\partial (G_v, \\mathbb{P}_v)\\setminus \\{\\rho\\}$ is path connected as well. We know that $P$ acts cocompactly on $\\partial (G_v, \\mathbb{P}_v)\\setminus \\{\\rho\\}$. Let $C_0$ be a compact fundamental domain for this action. Our goal is to show that $C_0$ is contained in a connected fundamental domain $C$. \n\t\n\tLet $d= d(C_0, \\rho)$ in $\\partial (G_v, \\mathbb{P}_v)$. We can cover $C_0$ with finitely many open connected sets of diameter at most $d\/2$. The union of the closures of these sets form a compact set $C_1\\subset \\partial (G_v, \\mathbb{P}_v)$ containing $C_0$ and having only finitely many connected components. By our choice of $d$, $C_1 \\subset \\partial (G_v, \\mathbb{P}_v) \\setminus \\{\\rho\\}$. We can then join the finitely many connected components of $C_1$ with finitely many paths in $\\partial (G_v, \\mathbb{P}_v) \\setminus \\{\\rho\\}$, giving us a compact connected fundamental domain $C$.\n\t\n\\end{proof}\n\nUsing this, we can find a connected fundamental domain for $P$'s action on $\\Omega$.\n\n\\begin{prop}\\label{Fundamental Domain on Omega}\n\tThere is a compact, connected fundamental domain $K\\subset \\Omega$ for the action of $P$ on $\\Omega$. Furthermore, if $S$ is any finite generating set for $P$, then $K$ can be chosen such that $sK\\cap K \\neq \\emptyset$ for each $s\\in S$. \n\\end{prop}\n\n\\begin{proof}\n\tWe will use the decomposition map $\\pi\\colon  \\partial Y \\to \\partial (G_v,\\mathbb{P}_v)$ and then pull back the compact, connected fundamental domain for $P$'s action on $\\partial(G_v,\\mathbb{P}_v) \\setminus \\{\\rho \\}$ to get a fundamental domain on $\\Omega$.\n\t\n\tLet $C_0$ be a connected fundamental domain for the action of $P$ on \\linebreak $\\partial (G_v, \\mathbb{P}_v) \\setminus \\{\\rho\\}$ as in Proposition \\ref{Connected Fundamental Domain}. Without loss of generality, we may increase the size of $C_0$ such that $C_0\\cap sC_0 \\neq \\emptyset$ for all $s\\in S$. We can then repeat the process from Proposition \\ref{Connected Fundamental Domain} to construct a connected $C$.\n\t\n\tSince the map $\\pi\\colon \\partial Y \\to \\partial (G_v, \\mathbb{P}_v)$ defines an upper semicontinuous monotone decomposition, then we know that $\\pi ^{-1}(C)$ is compact and connected. \tThe map $\\pi$ is $G$-equivariant, and therefore since $C$ is a fundamental domain, $K$ is as well. Since $C$ satisfies the desired intersection property, then $K$ does as well.\n\\end{proof}\n\nUsing the association between subsets of $F$ and those of $\\Omega$, we can establish a way to determine when subsets of $\\Omega$ are connected.\n\n\\begin{prop} \\label{K and C}\n\tThere exists compact, connected fundamental domains $K$ and $C$ for the action of $P$ on $\\Omega$ and $F$, respectively, such that the following property holds: Let $\\P$ be any subset of $P$. Then:\n\t\n\t\\begin{center}\n\t\tIf $\\bigcup\\limits_{p\\in \\P} pC$ is connected in $F$, then $\\bigcup\\limits_{p\\in \\P} pK$ is connected in $\\Omega$.\n\t\\end{center}\n\\end{prop}\n\n\\begin{proof}\n\tChoose a compact, connected fundamental domain $C$ for the action of $P$ on $F$. Let $S$ be the set of $p\\in P$ such that $pC\\cap C\\neq \\emptyset$. Then $S$ is a finite generating set for $P$. Use Proposition \\ref{Fundamental Domain on Omega} to pick $K$ such that $sK\\cap K \\neq \\emptyset$ for all $s\\in S$. This implies the following, which implies the desired conclusion:\t\n\t\\begin{center}\n\t\tIf $C\\cap pC \\neq \\emptyset$, then $K\\cap pK \\neq \\emptyset$\n\t\\end{center}\n\\end{proof}\n\nWe will now fix the compact, connected fundamental domains $K$ and $C$ for $P$'s action on $\\Omega$ and $F$, respectively. Using Corollary \\ref{Compact Set Association}, let $c'\\colon [0,\\infty)\\to Y$ be a geodesic ray which meets $F$ orthogonally. Using the $P$-action, we may assume that $c'(0)\\in C$ and $c'(\\infty)\\in K$. We will call $c'(0)=q_0$ and use $q_0$ as the basepoint for the cone topology.\n\nLet $\\alpha\\in \\partial F$. Our method is two-fold. First,  associate connected neighborhoods of $\\alpha$ in $\\overline{F^*}$ to neighborhoods of $\\alpha$ in $\\overline{\\Omega}$, which by Proposition \\ref{K and C} will be connected. Then, given a neighborhood of $\\alpha$ in $\\overline{\\Omega}$, find a connected neighborhood of $\\alpha$ in $\\overline{F^*}$ to associate inside of the neighborhood in $\\overline{\\Omega}$. \n\nTo make this possible, we need to find suitably nice neighborhoods in $\\overline{F^*}$. \n\n\n\\begin{defn}\n\tLet $Z$ be a $\\cat(0)$ space and $\\xi\\in \\partial Z$. A neighborhood $\\overline{N}$ of $\\xi$ in $\\overline{Z}$ is \\textit{clean} if the following conditions hold:\n\t\n\t\\begin{enumerate}\n\t\t\\item $\\overline{N}$ is connected.\n\t\t\\item $N=\\overline{N}\\cap Z$ is connected, and\n\t\t\\item Every point of $\\Lambda:=\\overline{N}\\cap \\partial Z$ is a limit point of $N$.\n\t\\end{enumerate}\n\\end{defn}\n\nHruska and Ruane prove  (\\cite{HR17}, Proposition 3.3) that for any $\\xi\\in \\partial Z$, $\\xi$ has a local base of clean connected neighborhoods. We show below that we can pick this neighborhood basis to be clean and connected both in $\\overline{F}$ and $\\overline{F^*}$. \n\n\\begin{lem}\n\tThere is a local base of neighborhoods of $\\alpha$ in $\\overline{F^*}$ which are clean and connected in $\\overline{F}$ and in $\\overline{F^*}$. \n\\end{lem}\n\n\\begin{proof}\n\tFix $q_0$ as the basepoint for the cone topology and let $c\\colon [0,\\infty)\\to F^*$ be the ray based at $q_0$ representing $\\alpha$. It is shown in \\cite{HR17} that basic open sets, $U(c,r,D)$, are clean and connected in $\\overline{F^*}$.  \n\t\n\tFirst note that since $c(0)\\in F$, $c(\\infty)\\in \\partial F$ and $F$ is convex in $F^*$, then $U(c,r,D)\\cap \\overline{F}$ is non-empty. Denote $N_F= F\\cap U(c,r,D)$. \n\t\n\tA point $p\\in F$ is in $N_F$ if and only if the geodesic segment from $c(0)$ to $p$ intersects $B_D(c(r))$. Let $\\gamma\\colon [0,t]\\to F$ be the geodesic segment $[c(0),p]$, then we can choose $s$ so that $d(\\gamma(s),c(r))<D$. It follows that the entire geodesic segment $[\\gamma(s),\\gamma(t)]\\subset U(c,r,D)$, so every point of $N_F$ is contained in a connected subset of $N_F$ which intersects the connected ball $B_D(c(r))$ and thus $N_F$ is connected. All of $U(c,r,D)$ is contained in the closure of $N_F\\subset \\overline{F}$, and therefore is connected as well. Since $U(c,r,D)$ is contained in the closure of $N$, the limit set requirement is satisfied as well.\n\t\n\\end{proof}\n\nFor a given clean, connected neighborhood $\\overline{N}=N\\cup \\Lambda$, let \\linebreak $\\P:=\\{p\\in P: pC\\cap N\\neq \\emptyset\\}$. Since $q_0\\in F$, $\\P$ is non-empty and by Proposition \\ref{K and C}, $\\bigcup\\limits_{p\\in \\P} pC$ is connected. \n\n\\begin{defn}\n\tLet $\\alpha\\in \\partial F$ and $\\overline{N}$ be a clean, connected neighborhood with $\\overline{N}\\cap F = N$ and $\\overline{N}\\cap \\partial F = \\Lambda'$. Let $\\P := \\{p\\in P: pC\\cap N\\neq \\emptyset\\}$. Then the \\textit{neighborhood in $\\overline{\\Omega}$ of $\\alpha$ associated to $\\overline{N}$} is $\\overline{Z}= \\left( \\bigcup\\limits_{p\\in \\P} pK\\right) \\cup \\Lambda'$. \n\\end{defn} \n\nWe now need to show three things: $\\overline{Z}$ is a neighborhood of $\\alpha$, it is connected, and when $\\overline{N}$ can be chosen to be `small,' then $\\overline{Z}$ is `small' as well.\n\n\\begin{lem} \\label{Z is a nieghborhood}\n\tThe set $\\overline{Z}$ associated to $\\overline{N}$ is a neighborhood of $\\alpha$ and is connected.\n\\end{lem}\n\n\\begin{proof}\n\tFirst, we show that $\\overline{Z}$ is a neighborhood of $\\alpha$. Let $c\\colon [0,\\infty)\\to F^*$ be the ray based at $q_0$ representing $\\alpha$ and fix some $D>0$. Since $\\overline{N}$ is a neighborhood of $\\alpha$, we can choose $R$ large enough so that $U(c,R,D+\\kappa)\\cap \\overline{F^*}$ is contained in $\\overline{N}$, and therefore $U(c,R,D)\\cap \\overline{F^*}$ is contained in $\\overline{N}$ as well. The $P$-translates of $K$ form a null family, so there is a neighborhood $V$ of $\\alpha$ with $V\\subset U(c,R,D)$ such that if $pK\\cap V \\neq \\emptyset$, then $pK\\subset U(c,R,D)$.\n\t\n\tNow it suffices to show that $V\\subset \\overline{Z}$. For $\\eta\\in V$, either $\\eta\\in \\Omega$ or $\\eta\\in \\partial F$. In the former case, $\\eta\\in pK$ for some $p\\in P$ and so $pK\\subset U(c,R,D)$. In particular, $p\\cdot q_\\infty\\in U(c,R,D)$ and by Lemma \\ref{Haulmark Quasi Convex}, $p\\cdot q_0\\in U(c,R,D+\\kappa)$. Since $q_0\\in C$ and $U(c,R,D+\\kappa)\\cap \\overline{F^*}\\subset \\overline{N}$, then $pC\\cap N \\neq \\emptyset$. Therefore, $\\eta\\in \\overline{Z}$ by the construction.\n\t\n\tIn the latter case, $\\eta\\in \\partial F\\cap V$, so\n\t\n\t$$\\eta \\in U(c,R,D)\\cap \\partial F \\subset U(c,R,D+\\kappa)\\cap \\partial F \\subset \\overline{N}\\cap \\partial F = \\Lambda'\\subset \\overline{Z}$$\n\tTherefore, $V\\subset \\overline{Z}$ and so it is a neighborhood of $\\alpha$ in $\\overline{\\Omega}$.\n\t\n\tNow we must show that $\\overline{Z}$ is connected. By Proposition \\ref{K and C}, $\\overline{Z}\\cap \\Omega$ is connected. Therefore it suffices to show that every point of $\\Lambda'$ is a limit point of $\\overline{Z}\\cap \\Omega$. Let $\\xi\\in \\Lambda'$. Since $\\overline{N}$ is clean, $\\xi$ is the limit of a sequence $x_i\\in N$. Each $x_i\\in p_iC$ for some $p_i\\in P$. Fix $\\epsilon>0$ and note that since $x_i\\to \\xi$, $x_i\\in U(\\xi,R,\\epsilon)$ for all but finitely many $i$. Let $A$ be the diameter of $C$, then $p_i\\cdot q_0\\in U(\\xi,R,\\epsilon+A)$ for all but finitely many $i$, and so by Lemma \\ref{delta(epsilon,M)}, $p_i\\cdot q(R)\\in U(\\xi,R,\\epsilon+A+\\delta)$ for all but finitely many $i$. Therefore, $p_i\\cdot q(\\infty)\\in U(\\xi,R,\\epsilon+A+\\delta)$ and thus $p_i\\cdot q(\\infty)$ converge to $\\xi$. Since $p_i\\cdot C \\cap N \\neq \\emptyset$, then $p_i\\cdot K\\cap \\overline{Z} \\neq \\emptyset$ and so $\\xi$ is the limit of points in $\\overline{Z}\\cap \\Omega$ as desired.\n\\end{proof}\n\n\\begin{lem} \\label{Z small}\n\tThe neighborhood $\\overline{Z}$ associated to $\\overline{N}$ can be made arbitrarily small by choosing $\\overline{N}$ to be a sufficiently small neighborhood of $\\alpha$ in $\\overline{F^*}$\n\\end{lem}\n\n\\begin{proof}\n\tLet $U$ be a neighborhood of $\\alpha$ in $\\overline{\\Omega}$. Since the $P$-translates of $K$ form a null family. There is a neighborhood $V\\subset U$ such that if $pK\\cap V$ is non-empty, then $pK\\subset U$. Since $V$ is a neighborhood of $\\alpha$, there are constants $R>0$ and $D>0$ such that $U(c,R,D+\\kappa)\\subset V$ and $U(c,R,D)\\cap \\partial F\\subset V$. Then, by Lemma \\ref{Haulmark Quasi Convex}, $U(c,R,D)$ is a neighborhood of $\\alpha$ in $\\overline{F^*}$ such that if $p\\cdot q_0\\in U(c,R,D)$, then $p\\cdot q_\\infty \\in U(c,R,D+\\kappa)$ and so $pK\\subset U$. Furthermore, since the $P$-translates of $C$ form a null family, there is a $W\\subset U(c,R,D)$ such that if $pC\\cap W$ is non-empty, then $pC\\subset U(c,R,D)$. Finally, there is a clean, connected neighborhood $\\overline{N}$ within $W$.\n\t\n\tIt remains to show that the associated neighborhood $\\overline{Z}$ is contained in $U$. Let $\\eta\\in \\overline{Z}$. By our choice of $U(c,R,D)$, if $\\eta\\in \\partial F$, then $\\eta \\in V$ and therefore in $U$. If $\\eta \\in \\Omega$, then $\\eta\\in pK$ for some $p$, so $pK\\cap \\overline{Z}$ is non-empty and thus $pC\\cap N$ is non-empty. This means that $pC\\cap W\\neq \\emptyset$ so $pC\\subset U(c,R,D)$ and thus $pK$ intersects $V$ non-trivially. By the choice of $V$, $pK\\subset U$ and so $\\eta\\in U$ as desired. Thus $\\overline{Z}$ is contained in $U$.\n\t\n\t\n\t\n\\end{proof}\n\nRecall the definitions of weakly locally connected and locally connected:\n\n\\begin{defn}[Weakly locally connected]\\label{def:wlc}\n\tLet $M$ be a topological space. $M$ is \\textit{weakly locally connected} at $m\\in M$ if for every open set $V$, there  is a connected (and not necessarily open) set $N\\subset V$ with $m$ in the interior of $N$.\n\t\n\tWe say a space $M$ is weakly locally connected if it is weakly locally connected at every point.\n\\end{defn}\n\n\n\\begin{defn}[Locally connected]\n\tLet $M$ be a topological space. $M$ is \\textit{locally connected} at $m\\in M$ if for every open set $V$ containing $m$, there is a connected open set $U$ with $m\\in U \\subset V$. \n\t\n\tWe say $M$ is locally connected if it is locally connected at every point.\n\\end{defn}\n\nIn Lemma \\ref{Z is a nieghborhood} and Lemma \\ref{Z small}, we do not know if $\\overline{Z}$ is open or not, so we cannot prove local connectivity in $\\partial Y$. Instead, we show the following:\n\\begin{prop}\\label{Y is wlc}\n\tIf $\\alpha\\in \\partial Y \\cap \\partial F$, then $\\partial Y$ is weakly locally connected at $\\alpha$.\n\\end{prop}\n\n\\begin{proof}\n\tLet $U$ be a neighborhood of $\\alpha$ in $\\partial Y$. Since $\\partial Y = \\overline{\\Omega} \\cup \\partial F^*$, then $U=U_1\\cup U_2$, with $U_1$ a neighborhood of $\\alpha$ in $\\overline{\\Omega}$ and $U_2$ a neighborhood of $\\alpha$ in $\\partial F^*$. Combining Lemma \\ref{Z is a nieghborhood} and Lemma \\ref{Z small}, there is a $V_1\\subset U_1$ which is connected and contains $\\alpha$. Since $\\partial F^*$ is a sphere, it is locally connected, so there is some $V_2\\subset U_2$ which is connected and contains $\\alpha$. Lastly, since $V_1\\cap V_2$ is non-empty, then their union is connected as well. Thus $V_1\\cup V_2\\subset U$ is a connected neighborhood of $\\alpha$.\n\\end{proof}\n\nCombining all of this work leads to the following theorem:\n\n\\begin{thm}\\label{Y is pconn}\n\tLet $(G,\\mathbb{P})$ be a $\\cat(0)$ group with isolated flats acting geometrically on $X$, let $G_v$ be a component vertex in its maximal peripheral splitting, acting geometrically on $X_v\\subset X$ and let $Y$ be as in Definition \\ref{defn Y}. Then $\\partial Y$ is path connected. \n\\end{thm}\n\n\\begin{proof}\n\t\n\tIt suffices to show that $\\partial Y$ is both connected and weakly locally connected at every point. By Theorem 27.16 in \\cite{Wil70}, a space which is weakly locally connected at every point is locally connected at every point. Also by Theorem 31.2 in \\cite{Wil70}, a space which is compact, connected, locally connected and metrizable is locally path connected, which implies globally path connected (by \\cite{Wil70}, Theorem 27.5). Since boundaries of proper metric spaces are compact and metrizable, all we need to show is that $\\partial Y$ is connected and weakly locally connected at each point. \n\t\n\tConnectedness follows from Theorem \\ref{monotone usc decomp of Y}. Since the map $\\pi\\colon \\partial Y \\to \\partial (G_v,\\mathbb{P}_v)$ is monotone and $\\partial(G_v,\\mathbb{P}_v)$ is connected, then $\\partial Y$ is connected as well.\n\t\n\tFor weakly locally connected, there are three types of points in $\\partial Y$ to consider: points in $\\partial X_v$ and not a flat, points just in a flat, and points in both $\\partial X_v$ and $\\partial F$ for some flat $F$. By Corollary \\ref{Y non-flat points lconn}, if $x\\in \\partial Y$ and $x\\notin \\partial F$ for any flat $F$, then $\\partial Y$ is locally connected at $x$ and therefore weakly locally connected at $x$. If $\\xi\\in \\partial F^*$ for some flat $F^*$ and $\\xi \\notin \\partial X_v$, then $\\partial Y$ is locally connected at $\\xi$ and so weakly locally connected at $\\xi$. This is because sufficiently small neighborhoods of $\\xi$ only contain points in $\\partial F^*$, which is locally connected. Lastly, if $\\alpha\\in \\partial X_v \\cap \\partial F$ for some flat $F$, then by Proposition \\ref{Y is wlc}, $\\partial Y$ is weakly locally connected at $\\alpha$.\n\\end{proof}\n\nWe are almost ready to combine all this work to conclude that $\\cat(0)$ groups with isolated flats have path connected visual boundaries. But first we need to show that these groups satisfy the lonely condition.\n\n\\begin{lem} \\label{IFP blocks are lonely}\n\tSuppose $G$ is a 1-ended $\\cat(0)$ group with isolated flats acting geometrically on a $\\cat(0)$ space $X$. Let $\\mathcal{B}$ be the block decomposition coming from the maximal peripheral splitting of $\\mathcal{G}$, then all irrational rays are lonely.\n\\end{lem}\n\n\n\\begin{proof}\n\tFor any flat $F$, let $\\gamma_F\\colon [0,b]\\to X$ be the geodesic segment from $x_0$ to the closest point of $F$. Since $F$ is convex, a closest point exists and is unique. By Theorem \\ref{Quasi-convex Constant}, $\\gamma_F\\cup F$ is $\\kappa$-quaisconvex. In particular, every geodesic starting at $x_0$ which intersects $F$ must be the in $\\kappa$-neighborhood of $\\gamma_F(b)$. \n\t\n\tLet $c,c'$ be rays in $X$ based at $x_0$ with the same, infinite itinerary. Since these rays have the same itinerary, they pass through the same sequence of flats, and must pass through infinitely many flats. But then the rays are $2\\kappa$-close infinitely often. By the convexity of the distance metric, the rays must be $2\\kappa$-close always, and therefore $c$ and $c'$ are asymptotic.\n\t\n\\end{proof}\n\n\n\\begin{thm} \\label{thm main} \n\tAll 1-ended $\\cat(0)$ groups with isolated flats have path connected visual boundaries.\n\\end{thm}\n\n\\begin{proof}\n\tLet $(G,\\mathbb{P})$ be a $\\cat(0)$ group with isolated flats. It suffices to show there is a space $X$ on which $G$ acts which has path connected visual boundary. Let $\\mathcal{G}$ be the maximal peripheral splitting for $(G,\\mathbb{P})$, $X$ be the tree-of-spaces on which $G$ acts geometrically, and $\\mathcal{B}$ the block decomposition coming from this peripheral splitting.\n\t\n\tEach peripheral vertex group in the splitting is virtually $\\mathbb{Z}^n$ for some $n$, so has path connected visual boundary. If each component vertex has path connected visual boundary as well, then by Theorem \\ref{combo} $\\partial X$ is path connected. By Lemma \\ref{IFP blocks are lonely}, the irrational rays are all lonely, so we can apply the theorem.\n\t\n\tIf there is a peripheral vertex which does not have path connected visual boundary, then by Theorem \\ref{Y is pconn}, $\\partial_R X$ is path connected. This is true because\n\t$$\\partial_R X = \\bigcup_{v\\in \\mathcal{C}} \\partial Y_v$$\n\twhere $\\mathcal{C}$ is the collection of component vertices in the tree $\\mathcal{T}$ for the splitting $\\mathcal{G}$. Therefore, we can apply Proposition \\ref{lonely plus rational is pconn} to conclude that $\\partial X$ is path connected.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{alpha\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper, we study the stability of the absolutely \ncontinuous spectrum of one-dimensional Stark operators\nunder various classes of perturbations. Stark Schr\\\"odinger operators describe behavior of the charged particle in the constant electric field. The absolutely continuous spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather fast rate (see, e.g. \\cite{AS}). It is therefore interesting to describe the classes of perturbations which preserve the absolutely continuous spectrum of the Stark operators. In the first part\nof this work, we study perturbations of Stark operators by decaying potetnials.\nThis part is inspired by the recent work of Naboko and Pushnitski \\cite{NaPu}.\nThe general picture that we prove is very similar to the \ncase of perturbations of free Schr\\\"odinger operators \\cite{Kis1}. \nIn accordance with physical intuition, however, the absolutely continuous spectrum is stable under stronger perturbations than in the free case. \nIf in the free case the short range potentials preserving purely absolutely continuous spectrum of the free operator are given by condition (on the power scale)\n$|q(x)| \\leq C(1+|x|)^{-1-\\epsilon},$ in the Stark operator case the corresponding condition reads $|q(x)| \\leq C(1+|x|)^{-\\frac{1}{2}-\\epsilon}.$ If $\\epsilon$ is allowed to be zero in the above bounds, imbedded eigenvalues may occur in both cases (see, e.g. \\cite{NaPu}, \\cite{WN}). Moreover, in both cases if we allow potential to decay slower by an arbitrary function growing to infinity, very rich singular spectrum, such as a dense set of  eigenvalues, may occur (see \\cite{Na} for the free case and \\cite{NaPu} for the Stark case for precise formulation and proofs of these results). The first part of this work draws the paprallel further, showing that the absolutely continuous spectrum of Stark operators is preserved under perturbations satisfying  $|q(x)| \\leq C(1+|x|)^{-\\frac{1}{3} -\\epsilon},$ in particular even in the regimes where a dense \nset of eignevalues occurs; hence in such cases these eigenvalues are genuinely imbedded. Similar results for the free case were proven in \\cite{Kis1}, \\cite{Kis2}. Our main strategy of the proof here is similar to that in \\cite{Kis1} and \\cite{Kis2}: we study the asymptotics of the generalized eigenfunctions and then apply Gilbert-Pearson theory \\cite{GiPe} to derive spectral consequences.\n\nWhile the main new tool we introduce in our treatment of Stark operators is the same as in the free case, namely  the a.e. convergence of the Fourier-type integral operators, there are some major differences. First of all, the spectral parameter enters the final equations that we study in a different way and this makes analysis more complicated. Secondly, we employ a different method to analyze the asymptotics. Instead of Harris-Lutz asymptotic method we study appropriate Pr\\\"ufer transform variables, simplifying the overall consideration. \n\n In the second part of the work we \ndiscuss perturbations by potentials having some additional smoothness properties, but without decay. It turns out that for Stark operators the effects of decay or of additional smoothness of potential on the spectral properties are somewhat similar. \nIt was known for a long time that if a potential perturbing Stark operator has two bounded derivatives the spectrum remains purely absolutely continuous (actually, \ncertain growth of derivatives is also allowed, see Section 3 for details or Walter \\cite{Wa} for the original result). We note that the results similar to Walter's on the preservation on absolutely continuous spectrum were also obtained in \\cite{Ben} by applying different type of technique (Mourre method instead of studying asymptotics of solutions).\n  On the other hand, if the perturbing potential is a sequence of derivatives of $\\delta$ functions in integer points on $R$\nwith certain couplings,   the spectrum may turn pure point \\cite{AEL}, \\cite{Ex}.\n In some sense, the $\\delta'$ interaction is the most singular and least\n``differentiable\" among all available natural perturbations of one-dimensional Schr\\\"odinger operators \\cite{KiSi}. Hence we have very different spectral properties on the very opposite sides of the smoothness scale.  This work closes the part of the gap.  We improve the well-known results of Walter\n\\cite{Wa}  concerning the minimal smoothness required for the preservation of the absolutely continuous spectrum and show that in fact  existence and minimal smoothness of the first derivative is sufficient to imply absolute continuity of the spectrum.      \n\n\n\\section{Decaying perturbations}\n\nConsider a self-adjoint operator $H_{q}$ defined by the differential \nexpression \n\\[ H_{q}u= -u''-xu+q(x)u \\]\non the $L^{2}(-\\infty,\\infty).$  Let us introduce some notation. For the function $f \\in L^{2}$ we denote by $\\Phi f$ its Fourier transform:\n\\[ \\Phi f(k)= L^{2}-\\lim_{N\\rightarrow \\infty} \\int\\limits_{-N}^{N} \\exp(ikx)f(x)\\,dx. \\] \nFor locally integrable function $g$ we denote by $M^{+}g$ the function\n\\[ M^{+}g(x) = \\sup_{1>h>0} \\frac{1}{2h} \\int\\limits_{0}^{h} |g(x+t)+g(x-t)|\\,dt. \\]\nWe denote by ${\\cal M}^{+}f$ the set where $M^{+}f$ is finite. By the general results on the maximal functions (see, e.g. \\cite{Rud}), it is easy to conclude that the complement of the set ${\\cal M}^{+}$ has Lebesgue measure zero. \nWe will prove the following theorem: \\\\\n\\bf Theorem 1.1 \\it Suppose that the potential $q(x)$ satisfies $|q(x)| \\leq C(1+x)^{-\\frac{1}{3}-\\epsilon},$ $\\epsilon >0.$\nThen the absolutely continuous spectrum of multiplicity one fills the whole real axis. Imbedded singular spectrum may occur, but only on the complement of the set \n\\[ S= {\\cal M}^{+}\\left(\\Phi \\left(\\exp \\left(ix^{3}-i\\int_{0}^{x^{3}}q(ct^{\\frac{2}{3}})ct^{-\\frac{2}{3}}\\,dt\\right)q(c x^{2})x^{\\frac{1}{6}}\\right)\\right), \\]\nwhere $c= (\\frac{3}{2})^{\\frac{2}{3}}.$\n\n\\rm We will prove Theorem 1.1 by studying the asymptotics of the  solutions of the equation \n\\begin{equation}\n -u'' -xu+q(x)u=\\lambda u .\n\\end{equation}\nWe will then apply the subordinacy theory developed by Gilbert and Pearson \\cite{GiPe} to derive spectral consequences. For future reference, we summarize here one of the results of the subordinacy theory for Schr\\\"odinger operators defined on the whole axis  due to Gilbert \\cite{Gi} that we will use. \n\nLet us call the solution $u_{1}$ of the equation (1) subordinate on the right if for any different solution $u_{2}$ the limit \n\\[ \\lim_{N \\rightarrow +\\infty} \\frac{\\|u_{1}\\|_{L^{2}(0,N)}}{\\|u_{2}\\|_{L^{2}(0,N)}} \\]\nis equal to zero.  Subordinacy on the left is defined similarly. The set of the energies at which there exists a  solution subordinate both on the right and on the left constitutes the essential support of the absolutely continuous part of the spectral measure  of Schr\\\"odinger operator. Moreover, if for every $\\lambda$ from this set there exists a solution subordinate either on the right or on the left, the absolutely continuous spectrum has multiplicity one. \n\nSince in our case the total potential grows to plus infinity  on the negative  semi-axis, it is clear that for every $\\lambda$ there is a solution subordinate on the left and hence we need only to study the behavior of solutions on the positive semi-axis.\n\nIn studying Schr\\\"odinger operators with  sufficiently smooth potentials going to $-\\infty$ as $x \\rightarrow \\infty$ one of the useful tools is Liouville transform (see, e.g.\n\\cite{Har}). Suppose that  the equation is given by \n\\[ -y''+(v(x)+q(x))y= \\lambda y, \\]\nwhere  $v$ is two times differentiable and goes to $-\\infty$ as $x \\rightarrow \\infty$ sufficiently fast (specifically, $\\int_{1}^{\\xi} dx\/|v(x)|^{\\frac{1}{2}}$ diverges as $\\xi \\rightarrow \\infty$) and $q$ is a perturbation. The Lioville transformation is given\nby\n\\[  \\xi(x) = \\int\\limits_{0}^{x} \\sqrt{|v(s)|} \\,ds, \\,\\,\\,\\,\\,\n\\rho(\\xi) = |v(x(\\xi))|^{\\frac{1}{4}}y(x(\\xi)). \\]\nThe function $\\rho$ satisfies the equation\n\\begin{equation}\n-\\rho''+ Q(\\xi) \\rho = \\rho,\n\\end{equation}\nwhere\n\\[ Q(\\xi)= \\frac{5 |v'(x(\\xi))|^{2}}{16 |v(x(\\xi))|^{3}} - \\frac{v''(x(\\xi))}{4|v(x(\\xi))|^{2}}+\\frac{q(x(\\xi))-\\lambda}{|v(x(\\xi))|}. \\]\nIn some cases this new equation does not contain infinite potential and \nis simpler to deal with.\n\nIn our case $v(x)=-x$ and we  bring the equation (1) to a convenient form using the following explicit Liouville transformation (see also \\cite{NaPu}):\n$\\xi = \\frac{2}{3}x^{\\frac{3}{2}},$ $\\omega (\\xi) = x(\\xi)^{\\frac{1}{4}}u(x(\\xi)).$ Then the function $\\omega$ \nsolves the  Schr\\\"odinger equation\n\\begin{equation}\n -\\omega'' + \\left( \\frac{5}{36\\xi^{2}} + \\frac{-\\lambda + q(c\\xi^{\\frac{2}{3}})}{c\\xi^{\\frac{2}{3}}}\\right)\\omega = \\omega \n\\end{equation}\n(we remind that $c=(\\frac{3}{2})^{\\frac{2}{3}}$).\nWe introduce the short-hand notation\n\\[ b(\\xi)= \\frac{5}{36\\xi^{2}}-\\frac{cq(\\xi^{\\frac{2}{3}})}{c\\xi^{\\frac{2}{3}}} \\]\nand $V(\\xi, \\lambda)$ for the total potential in the equation (3).\nLet us further apply Pr\\\"ufer transformation to the equation for $\\omega,$ setting for each $\\lambda$\n\\[ \\omega(\\xi, \\lambda) = R(\\xi, \\lambda) \\sin (\\theta(\\xi, \\lambda)) \\]\n\\[ \\omega'(\\xi, \\lambda) = R(\\xi, \\lambda) \\cos (\\theta(\\xi, \\lambda)). \\]\nThe equations for $R$ and $\\theta$ are as follows: \n\\begin{equation}\n (\\log R(\\xi, \\lambda))' = \\frac{1}{2}V(\\xi, \\lambda)\\sin (2\\theta(\\xi, \\lambda)) \n\\end{equation}\n\\begin{equation}\n\\theta'(\\xi, \\lambda) = 1- \\frac{1}{2}V(\\xi, \\lambda)(1-\\cos (2\\theta(\\xi, \\lambda))). \n\\end{equation}\nOur goal now is to study the asymptotics of  solutions of (3).\nIn particular, we would like to establish convergence  of the integral\n\\begin{equation}\n \\int\\limits_{0}^{N} \\left( \\frac{5}{36\\xi^{2}} + \\frac{-\\lambda + q(c\\xi^{\\frac{2}{3}})}{c\\xi^{\\frac{2}{3}}}\\right)\\sin (2\\theta(\\xi, \\lambda))\\, d \\xi \n\\end{equation}\nas $N \\rightarrow \\infty$ for a.e.$\\lambda.$ This goal is motivated by the following  \\\\\n\\bf Lemma 1.2. \\it Suppose that for some value of $\\lambda$ the integral (6) converges. Then for this value of $\\lambda,$ there is no subordinate solution of the original equation (1). Moreover, if the integral (6) converges for a.e.~$\\lambda,$ the absolutely continuous part of the spectral measure of the operator $H_{q}$ fills the whole real axis.  \\\\\n\\bf Proof. \\rm If for a given value of $\\lambda$ the integral (6) converges,\nit follows from  (4) and (5) that all solutions of the equation (3) are bounded  and moreover there are two linearly independent solutions with the following asymptotics as $\\xi \\rightarrow +\\infty:$\n\\[ \\omega_{1}(\\xi, \\lambda) = \\sin (\\xi + r_{1}(\\xi, \\lambda))(1+o(1)), \\]\n\\[ \\omega_{2}(\\xi, \\lambda)= \\cos(\\xi + r_{2}(\\xi, \\lambda))(1+o(1)), \\]\nwhere $r_{i}'(\\xi, \\lambda) \\leq C(1+ \\xi)^{-\\frac{2}{3}}.$\nGoing back to the original equation (1), we infer that for a.e. $\\lambda$\nthis equation has only solutions $u_{\\alpha}(x, \\lambda)$ (where $\\alpha$ is some paprametrization of solutions, say by boundary condition at zero) with the following asymptotical behavior as $x \\rightarrow \\infty:$\n\\[ u_{\\alpha}( x, \\lambda) = x^{-\\frac{1}{4}}\\sin (\\frac{2}{3}x^{\\frac{3}{2}}+ f_{\\alpha}(x, \\lambda))(1+o(1)), \\]\nwhere $|f_{\\alpha}'(x, \\lambda)|\\leq C(1+x)^{-\\frac{1}{2}}$ uniformly over $\\alpha.$ \nFor any $u_{\\alpha}$ we then find\n\\[ \\int\\limits_{0}^{N} |u_{\\alpha}(x)|^{2}\\,dx = \\int\\limits_{0}^{N}\nx^{-\\frac{1}{2}}(\\sin(\\frac{2}{3}x^{\\frac{3}{2}}+ f_{\\alpha}(x))^{2}\\, dx \\left( 1+o(1)\\right) = \\]\n\\[ = \\left( N^{\\frac{1}{2}} + \\frac{1}{2}\\int\\limits_{0}^{N}x^{-\\frac{1}{2}}\n\\cos (\\frac{4}{3}x^{\\frac{3}{2}}+2 f_{\\alpha}(x, \\lambda))\\, dx \\right)\\left(1+o(1)\\right)= N^{\\frac{1}{2}}\\left( 1+o(1)\\right). \\]\nThe last equality follows from integrating by parts the integral in the previous expression. Hence, we obtain that for a.e.$\\lambda$ there are no subordinate solutions: as $N \\rightarrow \\infty,$ the $L^{2}$ norm grows at the same rate for all solutions. The last claim of the lemma follows  by direct  application of subordinacy theory. $\\Box$ \\\\\n\\it Remark. \\rm 1. Note that in particular we obtained that for a.e. $\\lambda$ all solutions of the equation (1) are bounded (and even power-decaying). However the results deriving absolute continuity  of the spectrum from the boundedness of solutions (see, e.g., \\cite{Sto}, \\cite{Si2}) are not applicable here because the potential goes to $-\\infty$ on $R^{+}.$ Instead, one has to apply directly Gilbert-Pearson theory. \\\\\n2. Note that while for a.e.$\\lambda$ there is no subordinate solution of equation (1) and hence the absolutely continuous spectrum fills the whole axis, there may be a complementary set of measure zero where the singular part of the spectral measure might be supported. As examples show \\cite{NaPu}, the imbedded spectrum may even be dense. The a.e. convergence of Fourier type integral operators  allows to prove a.e. convergence of the integral (6) while permitting for exceptional measure zero set where convergence may fail. \\\\\n\nWe begin analyzing the integral (6) with the following simple \\\\\n\\bf Lemma 1.3. \\it Suppose that the function $h(\\xi)$ satisfies \n\\[ h(\\xi)= \\xi + g(\\xi), \\]\nwhere $|g'(\\xi)|\\leq C\\xi^{-\\frac{2}{3}}.$ \nThen the integrals\n\\[ \\int\\limits_{1}^{N} \\xi^{-\\frac{2}{3}} \\exp (\\pm i h(\\xi))\\, d\\xi \\]\nconverge and , moreover, \n\\[ \\int\\limits_{N}^{\\infty}\\xi^{-\\frac{2}{3}} \\exp (\\pm i h(\\xi))\\, d\\xi= O(N^{-\\frac{1}{3}}). \\]\n\\bf Proof. \\rm \n\\[ \\int\\limits_{1}^{N} \\xi^{-\\frac{2}{3}} \\exp (\\pm i h(\\xi))\\, d\\xi = N^{-\\frac{2}{3}}\\int\\limits_{1}^{N} \\exp (\\pm i h(\\xi))\\,d\\xi +\\]\n\\[ + \\frac{2}{3}\\int\\limits_{1}^{N}\\xi^{-\\frac{5}{3}}\\int\\limits_{1}^{\\xi}\n\\exp (\\pm i h(\\eta))\\, d\\eta d\\xi. \\]\n Pick $\\xi_{0}$ so that \nfor $\\xi>\\xi_{0}$ $|g'(\\xi)|<\\frac{1}{2}$ ( we can do it by the assumption of Lemma). Then \n\\[ \\int\\limits_{\\xi_{0}}^{N} \\exp(\\pm i h(\\xi))\\,d\\xi=\n\\int\\limits_{h(\\xi_{0})}^{h(\\xi)}\\exp(\\pm i h)\\frac{1}{1+g'(\\xi(h))}\\, dh, \\]\nwhere $|g'(\\xi(h)))| \\leq C_{1}h^{-\\frac{2}{3}}.$\nExpanding the fraction in the last integral, it we find that \n\\[ \\int\\limits_{\\xi_{0}}^{N} \\exp(\\pm  i h(\\xi))\\,d\\xi=O(N^{\\frac{1}{3}}). \\]\nHence we see that the integral in question converges.\nIt is also straightforward to establish the last estimate of the lemma. \n$\\Box$\n\nFrom Lemma 1.3 and (5) it follows that the integrals \n\\[ \\int\\limits_{0}^{N} \\xi^{-\\frac{2}{3}} \\exp(\\pm 2i \\theta(\\xi, \\lambda)) \\,d\\xi \\]\nconverge for every value of $\\lambda.$\nTherefore, we see  that to establish convergence  of the integral (6)\nit suffices to study  the integral\n\\begin{equation}\n \\int\\limits_{1}^{N} \\left( \\frac{5}{36\\xi^{2}}-q(c\\xi^{\\frac{2}{3}})(c\\xi)^{-\\frac{2}{3}}\\right)\\sin(2\\theta(\\xi, \\lambda)\\, d\\xi). \n\\end{equation}\n(Of course, for the fast decaying term  the convergence issue is trivial, but  we incorporate it into the expression for future computational convenience.) \n\n We will need two  lemmata on the a.e. convergence of Fourier-type integrals. The first lemma we need is exactly Lemma 1.3 from \\cite{Kis1}.\nWe refer to that paper for a proof. \\\\\n\\bf Lemma 1.4. \\it Consider the function $f \\in L^{2}(R).$ Then for every $\\lambda_{0} \\in {\\cal M}^{+}(\\Phi(f))$ we have\n\\[ \\int\\limits_{-N}^{N} f(x) \\exp(i\\lambda_{0}x)\\,dx = O(\\log N). \\]\n\n\\rm The second lemma is a variation of \nLemma 1.4 from \\cite{Kis1}. We provide here the statement and a sketch of the proof for the sake of completness. \\\\\n\\bf Lemma 1.5. \\it Suppose that a function $f(x)$ satisfies $|f(x)| \\leq \nC(1+|x|)^{-\\alpha},$ where $\\alpha > \\frac{1}{2}.$ Then for a.e. $\\lambda$ the integral \n\\[ \\int\\limits_{0}^{N} \\exp(i\\lambda x)f(x)\\, dx \\]\nconverges as $x \\rightarrow \\infty$ and moreover for every $\\delta >0$ \n\\begin{equation}\n \\int\\limits_{N}^{\\infty} \\exp(i\\lambda x)f(x)\\, dx = o(N^{-\\alpha+\\frac{1}{2}+\\delta}) \n\\end{equation}\nfor a.e. $\\lambda.$ Explicitely, we can say that (8) holds for every $\\lambda \\in {\\cal M}^{+}(\\Phi (f(x)x^{\\alpha-\\frac{1}{2}-\\epsilon}))$ where $\\epsilon<\\delta.$ \\\\\n\\bf Proof. \\rm   Consider an apriori estimate \n\\[ \\int\\limits_{N}^{\\infty} \\exp (i\\lambda x) x^{\\alpha-\\frac{1}{2}-\\epsilon}f(x)x^{-\\alpha+\\frac{1}{2}+\\epsilon}\\,dx= \nN^{-\\alpha+\\frac{1}{2}+\\epsilon}\\int\\limits_{0}^{N}\\exp(i\\lambda x)\n(f(x)x^{\\alpha-\\frac{1}{2}-\\epsilon})\\,dx+ \\]\n\\[+(\\frac{1}{2}-\\alpha+\\epsilon)\\int\\limits_{N}^{\\infty}x^{-\\alpha-\\frac{1}{2}+\\epsilon}\\left(\\int\\limits_{0}^{x}\\exp(i \\lambda y)(f(y)y^{\\alpha-\\frac{1}{2}-\\epsilon}\\,dy \\right)\\,dx. \\]\nBy Lemma 1.4, for every $\\lambda \\in {\\cal M}^{+}(\\Phi (f(x)x^{\\alpha-\\frac{1}{2}-\\epsilon}))$ we have an estimate\n\\[ \\int\\limits_{0}^{N}\\exp(i \\lambda y)f(y)y^{\\alpha-\\frac{1}{2}-\\epsilon}\\,dy =O(\\log N) \\]\n Therefore for such $\\lambda$\nwe obtain \n\\[ \\int\\limits_{N}^{\\infty}\\exp(i \\lambda x)f(x)\\,dx = O(N^{-\\alpha+\\frac{1}{2}+\\epsilon}\\log N) = o(N^{-\\alpha +\\frac{1}{2} +\\delta}) \\]\nsince $\\delta>\\epsilon.$ The lemma is proven. $\\Box$ \\\\\n\\it Remark. \\rm We note that by Zygmund's theorem \\cite{Zyg} the Fourier integral converges a.e. for every $f \\in L^{p},$ $1 \\leq p<2.$ One can avoid using Lemma 1.4 by applying directly this theorem in the proof of Lemma 1.5. However this approach is less direct and also does not provide any explicit information about the convergence set. \\\\\n\n\\noindent \\bf Corollary 1.6. \\it Suppose that the function $f(\\xi)$ satisfies $|f(\\xi)| \\leq C(1+|\\xi|)^{-\\alpha-\\epsilon},$ where $\\alpha \\geq \\frac{5}{6}$ and $\\epsilon$ is an arbitrarily small positive number. Then for a.e.$\\lambda$ the integral \n\\[ \\int\\limits_{0}^{N} f(\\xi) \\exp(i\\lambda \\xi^{\\frac{1}{3}})\\, d\\xi \\]\nconverges as $N \\rightarrow \\infty.$ Moreover, for every $\\lambda \\in {\\cal M}^{+}(\\Phi (f(\\xi^{3})\\xi^{3\\alpha-\\frac{1}{2}}))$ we have\n\\[ \\left| \\int\\limits_{N}^{\\infty} f(\\xi)\\exp(i\\lambda \\xi^{\\frac{1}{3}})\\,d\\xi \\right| \\leq CN^{-\\alpha +\\frac{5}{6}}. \\]\n\\bf Proof. \\rm \nThe change of variables $y=\\xi^{\\frac{1}{3}}$ transforms the integral to the form\n\\[ \\int\\limits_{0}^{N^{\\frac{1}{3}}} f(y^{3})y^{2}\\exp (6i\\lambda y)\\, dy. \\]\nNote that \n\\[ |f(y^{3})y^{2}| \\leq C(1+|y|)^{-3\\alpha-3\\epsilon+2} \\]\nby assumption. Applying Lemma 1.5 (and remembering to take into account  the lower limit) we obtain the statements of the Corollary. $\\Box$\n\nWe prove the final lemma we need for the proof of the theorem. \nLet us introduce some short-hand notation:\n\\[ \\sigma (x, \\lambda)= 2\\theta(x, \\lambda)- 2x + \\int\\limits_{0}^{x}\n\\left( q(\\xi^{\\frac{2}{3}})\\xi^{-\\frac{2}{3}}-\\lambda\\xi^{-\\frac{2}{3}}\\right)\\,d\\xi+\\lambda \\int\\limits_{0}^{x}\\xi^{-\\frac{2}{3}}\\cos 2\\theta(\\xi, \\lambda)\\,d\\xi; \\]\n\\[ \\gamma(x, \\lambda) = 2\\theta(x, \\lambda) - \\sigma(x, \\lambda). \\] \nFrom the equation (5) we find\n\\begin{equation}\n\\gamma'(x, \\lambda)= b(x, \\lambda) \\cos (\\gamma(x, \\lambda)+\\sigma(x, \\lambda)). \n\\end{equation}\n\\bf Lemma 1.7. \\it Suppose that the potential $q(x)$ satisfies $|q(x)| \\leq C(1+|x|)^{-\\frac{1}{3}-\\epsilon}.$ Then the integrals \n\\[ \\int\\limits_{0}^{N} b(\\xi) \\exp (\\pm \\sigma (\\xi, \\lambda))\\,d\\xi \\]\nconverge as $N \\rightarrow \\infty$ for a.e. $\\lambda$ and moreover\nfor a.e. $\\lambda$ \n\\begin{equation}\n \\int\\limits_{N}^{\\infty} b(\\xi) \\exp (\\pm \\sigma (\\xi, \\lambda))\\,d\\xi = O(N^{-\\frac{1}{18}}). \n\\end{equation}\n\\bf Proof. \\rm We will consider the case of plus sign; the other case is analogous. By Lemma 1.3 and the definitions of $b(\\xi)$ and $\\sigma(\\xi, \\lambda),$ it is clear that it suffices to investigate the convergence of the following integral\n\\begin{equation} \\int\\limits_{0}^{N} q(c\\xi^{\\frac{2}{3}})\\xi^{-\\frac{2}{3}} \\exp\\left(\n2i\\xi -i\\int_{0}^{\\xi}q(\\eta^{\\frac{2}{3}})\\eta^{-\\frac{2}{3}}\\,d\\eta+ 6\\lambda i\\xi^{\\frac{1}{3}} +ic(\\lambda)+i\\xi^{-\\frac{1}{3}}h(\\lambda, \\xi)\\right), \n\\end{equation}\nwhere the function $h(\\xi, \\lambda)$ is bounded for every  $\\lambda.$ \nWe used Lemma 1.3 to   replace \n\\[ \\lambda \\int\\limits_{0}^{\\xi} \\eta^{-\\frac{2}{3}}\\cos (2\\theta (\\eta, \\lambda))\\, d\\eta \\]\nby \\[ c(\\lambda) +  \\xi^{-\\frac{1}{3}}h(\\lambda, \\xi) \\]\nin the exponent in (11).\nLet us denote \n\\begin{equation}\n \\tilde{q}(\\xi)=q(c\\xi^{\\frac{2}{3}})\\xi^{-\\frac{2}{3}}\\exp(2i\\xi-i\\int_{0}^{\\xi}q(c\\eta^{\\frac{2}{3}})c\\eta^{-\\frac{2}{3}}\\,d\\eta).\n\\end{equation}\nWe can rewrite the integral (11) as \n\\[ \\int\\limits_{0}^{N} \\tilde{q}(\\xi) \\exp (6i\\lambda \\xi^{\\frac{1}{3}} -\ni\\xi^{-\\frac{1}{3}}h(\\lambda, \\xi))\\, d\\xi \\]\n(henceforth we omit the constant factor $\\exp (ic(\\lambda))$).\nNote that as $\\xi \\rightarrow \\infty,$ we have  \n\\[ \\exp(i \\xi^{-\\frac{1}{3}} h(\\lambda, \\xi))= 1+ \\xi^{-\\frac{1}{3}}\\tilde{h}(\\lambda,\\xi), \\]\nwhere $\\tilde{h}$ is bounded  in $\\xi$ for every $\\lambda.$\nHence we can write the integral (11) as a sum of two integrals\n\\[ \\int\\limits_{0}^{N}\\tilde{q}(\\xi) \\exp (6i\\lambda \\xi^{\\frac{1}{3}})\\,d\\xi + \\int\\limits_{0}^{N}\\tilde{q}(\\xi)\\xi^{-\\frac{1}{3}}\\tilde{h}(\\xi, \\lambda) \\exp (6i\\lambda, \\xi)\\,d\\xi. \\]\nNote that by assumption on the decay of $q$ and (12) we have \n\\begin{equation}\n \\tilde{q}(\\xi) \\leq C\\xi^{-\\frac{8}{9}-\\frac{2}{3}\\epsilon}. \n\\end{equation}\nTherefore, the  second integral converges for every $\\lambda$ because the integrand is absolutely integrable. It is also easy to estimate the rate of convergence \nfor the second integral:\n\\[ \\left|\\int\\limits_{N}^{\\infty}\\tilde{q}(\\xi)\\xi^{-\\frac{1}{3}}\\tilde{h}(\\xi, \\lambda) \\exp (6i\\lambda \\xi)\\,d \\xi \\right| \\leq C(\\lambda)N^{-\\frac{2}{9}} \\]\nfor every $\\lambda$ because of the assumption on decay of $q.$\nTo study the behavior of the first integral, we note that because of the estimate (13) we can apply Corollary 1.6.\nWe obtain that for a.e. $\\lambda$ the first integral converges, and, moreover, for a.e. $\\lambda$ (explicitely, for every $\\lambda \\in {\\cal M}^{+}(\\Phi (\\tilde{q}(\\xi^{3})\\xi^{\\frac{13}{6}})$) the estimate on the rate of convergence is given by\n\\[ \\int\\limits_{N}^{\\infty}\\tilde{q}(\\xi) \\exp (6i\\lambda \\xi^{\\frac{1}{3}})\\,d\\xi \\leq C(\\lambda)N^{-\\frac{1}{18}}. \\]\nCombining the two estimates, we see that the lemma is proven.\n $\\Box$\n\nWe are now in a position to complete the proof of the theorem. The main idea is to take an advantage of certain symmetry of the equations (4), (5) and estimate (6). \\\\\n\\bf Proof of Theorem 1.1. \\rm  As we already remarked, to complete the proof we only have to study the a.e. convergence of the integral (7). Using (9) we rewrite this integral as\n\\[ \\int\\limits_{1}^{\\infty} b(\\xi) \\sin(\\gamma (\\xi, \\lambda)+\\sigma(\\xi, \\lambda))\\,d\\xi. \\]\nWe will integrate by parts two times, differentiating the terms which contain $\\gamma(\\xi, \\lambda).$ We omit the arguments of $\\gamma$ and $\\sigma$ in the following computations.\n\\[ \\int\\limits_{1}^{N} b(\\xi)\\sin (\\gamma +\\sigma)\\,d \\xi  = \\int\\limits_{1}^{N} b(\\xi)(\\sin (\\gamma )\\cos(\\sigma )+ \\sin (\\sigma)\\cos (\\gamma))\\, d\\xi = \\]\n\\[ = -\\left( \\left. \\cos (\\gamma) \\int\\limits_{\\xi}^{\\infty} b(\\eta)\n\\sin(\\sigma)\\, d\\eta + \\sin (\\gamma) \\int\\limits_{\\xi}^{\\infty} b(\\eta) \\cos (\\sigma)\\,d \\eta \\right) \\right|^{N}_{1}+ \\]\n\\[ + \\int\\limits_{1}^{N} b(\\xi)\\cos(\\gamma+\\sigma)\\left(\\cos (\\gamma )\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\cos(\\sigma)\\,d \\eta-\\sin (\\gamma)\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\sin(\\sigma)\\,d \\eta \\right)\\,d\\xi. \\]\nWe remark that by Lemma 1.7 the conditional integrals are well-defined for a.e. $\\lambda.$ Given that, we see that for a.e. $\\lambda$ the off-diagonal terms stay finite. We rearrange the integral term and write it as follows:\n\\[ \\int\\limits_{1}^{N} (\\sin(\\gamma))^{2} \\left( b(\\xi) \\sin (\\sigma)\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\sin(\\sigma)\\,d \\eta \\right)  d \\xi - \\]\n\\[- \\int\\limits_{1}^{N} (\\sin(\\gamma)\\cos (\\gamma)) \\left( b(\\xi) \\cos (\\sigma)\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\sin(\\sigma)\\,d \\eta +  b(\\xi) \\sin (\\sigma)\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\cos(\\sigma)\\,d \\eta \\right) d \\xi  + \\]\n\\[ + \\int\\limits_{1}^{N} \\cos(\\gamma)^{2} \\left( b(\\xi) \\cos (\\sigma)\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\cos(\\sigma)\\,d \\eta  \\right) d \\xi . \\]\nIntegrating by parts (terms in brackets being integrated), we again  obtain off-diagonal terms which are a.e.$\\lambda$ finite and the following integral term:\n\\[ \\int\\limits_{1}^{N} \\sin (2\\gamma) b(\\xi) \\cos (\\gamma+\\sigma) \\left( \\left( \\int\\limits_{\\xi}^{\\infty}b(\\eta)\\sin (\\sigma)\\,d \\eta \\right)^{2} - \\left( \\int\\limits_{\\xi}^{\\infty}b(\\eta)\\cos (\\sigma)\\,d \\eta \\right)^{2}\\right)\\, d \\xi - \\]\n\\[ - \\int\\limits_{1}^{N} \\cos (2\\gamma) b(\\xi) \\cos (\\gamma+\\sigma)  \\left( \\int\\limits_{\\xi}^{\\infty}b(\\eta)\\sin (\\sigma)\\,d \\eta \\right)\\left(\\int\\limits_{\\xi}^{\\infty}b(\\eta)\\cos (\\sigma)\\,d \\eta \\right) \\, d \\xi. \\]\nHence, the integral (7) and therefore (6) is convergent if \n\\[ b(\\xi) \\left( \\int\\limits_{\\xi}^{\\infty} b(\\eta) \\exp (\\pm \\sigma(\\eta, \\lambda))\\, d \\eta \\right)^{2} \\in L^{1}(0, \\infty). \\]\nBecause of the assumption on the decay of $q$ definition of $b(\\xi)$ and Lemma 1.7, we find that for a.e. $\\lambda \\in R$ the expression above is bounded above by \n\\[ C(\\lambda) \\xi^{-\\frac{2}{3}}\\xi^{-\\frac{2}{9}-\\frac{\\epsilon}{2}}\\xi^{-\\frac{1}{9}}\\leq C(\\lambda) \\xi^{-1-\\frac{\\epsilon}{2}} \\]\nand hence is absolutely integrable for a.e. $\\lambda.$  This proves the convergence of the integral (6) for a.e. $\\lambda.$ Since from the proof of Lemma 1.7  follows that the estimate (10) holds for every $\\lambda \\in {\\cal M}^{+}(\\Phi (\\tilde{q}(\\xi^{3}))\\xi^{-\\frac{13}{6}})),$ we also obtain an explicit set to which the singular part of the spectral measure gives zero weight, as stated in the  theorem. $\\Box$\n\n\\section{The case of sufficiently smooth potential}\n\nIn this section, we take up the study of an alternative kind of conditions implying the absolute continuity of the spectrum of Stark operators. \nWe begin with the remark that if the perturbation $q(x)$ is twice differetiable, $|q'(x)| \\leq C(1+|x|)$ and $|q''(x)| \\leq C(1+|x|)^{\\alpha},$ $\\alpha<\\frac{1}{2},$ we can apply the Liouville transformation directly to the \nwhole potential $-x+q(x)$, obtaining the equation (2) with absolutely integrable \npotential $Q.$ Then by Lemma 1.2, for every $\\lambda$ there is no subordinate solution of the original equation and hence the spectrum is purely absolutely continuous. The situation is more subtle if $q$ does not have two derivatives and hence we cannot apply Liouville transformation directly to the whole potential $-x+q(x).$\nOur goal is to show the following improvement of results proven in  \\cite{Wa}, \\cite{Ben}: \\\\\n\\bf Theorem 2.1. \\it Suppose that the perturbation $q(x)$ is bounded and differentiable, and the derivative $q'(x)$ is bounded and H\\\"older continuous with any exponent $\\alpha >0,$ i.e.\n\\[ \\sup_{x \\in R^{+}} \\left(\\sup_{y} \\frac{|q'(x)-q'(y)|}{|x-y|^{\\alpha}}\\right)<C. \\]\nThen the absolutely continuous part of the spectral measure $\\rho_{\\rm{ac}}^{H_{q}}$ associated with the Stark operator $H_{q}$ fills the whole real axis. \\\\\n\\bf Proof. \\rm \nThe strategy of the proof will be the same as for decaying case. We begin by analyzing equations (4), (5):\n\\[ (\\log R)'(\\xi, \\lambda)= \\frac{1}{2} V(\\xi)\\sin (2\\theta (\\xi, \\lambda)) \\]\n\\[ \\theta'(\\xi, \\lambda) = 1 -\\frac{1}{2}V(\\xi)+\\frac{1}{2}V(\\xi)\\cos(2\\theta(\\xi, \\lambda)). \\]\nWe remind that \n\\[ V(\\xi) =  \\frac{5}{36\\xi^{2}}+\\frac{\\lambda - q(c\\xi^{\\frac{2}{3}})}{c\\xi^{\\frac{2}{3}}}  \\]\nTo treat the phase equation, we introduce quantities similar to $\\gamma$ and $\\sigma$ in the previous proof:\n\\[ \\tilde{\\sigma}(\\xi, \\lambda)= 2 \\xi - \\int\\limits_{0}^{\\xi} V(\\eta)\\,d \\eta \\]\n\\[ \\tilde{\\gamma}(\\xi, \\lambda) = 2\\theta (\\xi, \\lambda) - \\tilde{\\sigma}(\\xi, \\lambda). \\]\n The computation completely repeating that in the proof of Theorem 1.1\nallows us to conclude that $R(\\xi, \\lambda)$ is bounded for a.e. $\\lambda$\nif the expression \n\\begin{equation}\n V(\\xi,\\lambda )\\left[ \\int\\limits_{\\xi}^{\\infty} V(\\eta, \\lambda) \\exp \\left( \\pm \\left( 2i\\eta -\\int\\limits_{0}^{\\eta} V(t, \\lambda)\\, dt \\right)\\right)\\, d\\eta \\right]^{2} \n\\end{equation}\nis well-defined and absolutely integrable for a.e. $\\lambda.$\nLet us now consider this control expression more carefully. \nConsider the  integral \n\\[ w_{\\pm}(\\xi, \\lambda)=\\int\\limits_{\\xi}^{\\infty} V(\\eta, \\lambda) \\exp \\left( \\pm \\left( 2i\\eta -\\int\\limits_{0}^{\\eta} V(t, \\lambda)\\, dt \\right)\\right)\\, d\\eta = \\]\n\\[ = -\\int\\limits_{\\xi}^{\\infty} \\frac{q(c\\eta^{\\frac{2}{3}})}{c\\eta^{-\\frac{2}{3}}} \\exp \\left( \\pm \\left( 2i\\eta -\\int\\limits_{0}^{\\eta} V(t, \\lambda)\\, dt \\right)\\right)\\, d\\eta \\,\\,+ \\] \\[ + \\int\\limits_{\\xi}^{\\infty} \\left(\\frac{5}{36\\eta^{2}} +\\frac{\\lambda}{c\\eta^{\\frac{2}{3}}} \\right) \\exp \\left( \\pm \\left( 2i\\eta -\\int\\limits_{0}^{\\eta} V(t, \\lambda)\\, dt \\right)\\right)\\, d\\eta. \\]\nWe see that the  second integral on the right-hand side converges for all $\\lambda$ and is $O(\\xi^{-\\frac{1}{3}})$ by  Lemma 1.3 \nHence it remains to study the first integral. We rewrite this integral as follows:\n\\begin{equation}\n \\int\\limits_{\\xi}^{\\infty} q(c\\eta^{\\frac{2}{3}}) \\exp \\left( \\pm i \\left(  -\\frac{5}{36\\eta}+ \\int\\limits_{0}^{\\eta}q(ct^{\\frac{2}{3}})ct^{-\\frac{2}{3}}\\,dt \\right)\\right) c\\eta^{-\\frac{2}{3}}\\exp \\left( \\pm \\left(\n2i\\eta + 6i\\lambda \\eta^{\\frac{1}{3}} \\right)\\right)\\, d\\eta. \n\\end{equation}\nLet us introduce a short-hand notation $g(\\eta)$ for the function \n\\[ \\exp \\left( \\pm i \\left(  -\\frac{5}{36\\eta}+ \\int\\limits_{0}^{\\eta}q(ct^{\\frac{2}{3}})ct^{-\\frac{2}{3}}\\,dt \\right)\\right).\\]\nNote that $g$ is continously differentiable and $|g(\\eta)|+|g'(\\eta)|$ is bounded.\nIn (15) we differentiate by parts, differentiating the first two terms and integrating next two. Note that \n\\[ \\int\\limits_{\\xi}^{\\infty} \\eta^{-\\frac{2}{3}} \\exp \\left(\\pm \\left(\n2i\\eta +6i\\lambda \\eta^{\\frac{1}{3}} \\right)\\right) \\, d\\eta = \\frac{1}{2}\\xi^{-\\frac{2}{3}}\\exp \\left( \\pm 2i\\xi \\pm 6i\\lambda \\xi^{\\frac{1}{3}} \\right) +\nO(\\xi^{-\\frac{5}{3}}). \\]\nGiven this observation, we obtain that the integral (15) is equal to\n\\[ O(\\xi^{-\\frac{2}{3}}) + \\left( \\frac{c}{6}\\int\\limits_{\\xi}^{\\infty} \n\\left(2\\eta^{-\\frac{1}{3}}q'(c\\eta^{\\frac{2}{3}}) \\pm i q(c\\eta^{\\frac{2}{3}})\n\\eta^{-\\frac{2}{3}} \\mp i \\frac{5}{36c\\eta^{2}} \\right) \\right. \\times \\]\n\\[ \\times \\left. g(\\eta)\\exp(\\pm 2i \\eta \\pm 6i \\lambda \\eta^{\\frac{1}{3}})\n\\eta^{-\\frac{2}{3}}\\left( 1+ O(\\eta^{-1})\\right) \\right) = \\]\n\\[ = O(\\xi^{-\\frac{1}{3}}) + \\frac{c}{6} \\int\\limits_{\\xi}^{\\infty}q'(c\\eta^{\\frac{2}{3}})\\eta^{-1}g(\\eta)\\exp \\left( \\pm 2i\\eta \\pm 6i\\lambda \\eta^{\\frac{1}{3}} \\right)\\, d \\eta. \\]\nBy assumption, $q'$ is is H\\\"older continuous uniformly over $R$ with \nexponent $\\alpha.$ Hence for every $h>0,$\n\\begin{equation}\n q'(c(\\eta + h)^{\\frac{2}{3}}) - q'(c\\eta^{\\frac{2}{3}}) = C( \\eta, h)\\eta^{-\\frac{\\alpha}{3}}h^{\\alpha}, \n\\end{equation}\nwhere $C(\\eta, h)$ is bounded uniformly for all $\\eta,$ $0<h<1.$ In the rest of the proof we will keep notation $C(\\eta, h)$ for different functions satisfying the same property. \nLet us integrate by parts the last integral ``$\\alpha$\" times. That is, let $h>0$ be fixed, such that $\\exp(ih) -1 \\neq 0.$ Then \n\\[ \\int\\limits_{\\xi}^{\\infty} q'(\\eta^{\\frac{2}{3}})g(\\eta)\\eta^{-1} \\exp (\\pm 6 i\\lambda \\eta^{\\frac{1}{3}})\\exp(\\pm 2i\\eta) \\left( \\exp(\\pm 2ih)-1 \\right)\\, d \\eta= \\]\n\\[ O(\\xi^{-1}) + \\int\\limits_{\\xi}^{\\infty} \\left( q'((\\eta+h)^{\\frac{2}{3}})(\\eta+h)^{-1}g(\\eta+h)\\exp( \\pm 6i\\lambda (\\eta+h)^{\\frac{1}{3}})- \\right. \\] \\[- \\left. q'(\\eta^{\\frac{2}{3}})\\eta^{-1}g(\\eta) \\exp(\\pm 6i\\lambda\\eta^{\\frac{1}{3}}) \\right) \\exp( \\pm 2i\\eta)\\, d \\eta. \\]\nNote that \n\\[ \\exp(\\pm 6i\\lambda(\\eta+h)^{\\frac{2}{3}})-\\exp(\\pm 6i\\lambda\\eta^{\\frac{2}{3}}) = \\eta^{-\\frac{2}{3}}C_{\\lambda}(\\eta, h). \\]\nHence we can rewrite the last integral as\n\\[ O(\\xi^{-\\frac{2}{3}}) + \\int\\limits_{\\xi}^{\\infty} \\left((q'((\\eta+h)^{\\frac{2}{3}})(\\eta+h)^{-1}g(\\eta+h)- q'(\\eta^{\\frac{2}{3}})\\eta^{-1}g(\\eta)\\right)\\exp(\\pm 2i\\eta \\pm 6i\\lambda \\eta^{\\frac{1}{3}})\\, d \\eta.  \\]\nSince we have\n\\begin{equation}\n|(\\eta+h)^{-1}g(\\eta+h)-\\eta^{-1}g(\\eta)| \\leq C\\eta^{-\\frac{2}{3}},\n\\end{equation}\na simple computation taking into account (16) and (17) shows that the last expression is equal to \n\\[  O(\\xi ^{-\\frac{2}{3}})+\\int\\limits_{\\xi}^{\\infty} C(\\eta, h)\\eta^{-1-\\frac{\\alpha}{3}}\\exp(\\pm 2i\\eta \\pm 6i\\lambda \\eta^{\\frac{1}{3}})\\, d \\eta.  \\]\n By Corollary 1.6, we obtain that the last expression converges for a.e. $\\lambda$ and is bounded by $C\\xi^{-\\frac{1}{3}(\\frac{1}{2}+\\alpha)+\\epsilon}$ for every $\\epsilon >0.$\nTherefore we have completed an estimation of the integral $w_{\\pm}(\\xi, \\lambda)$ and obtained that for a.e. $\\lambda$\n\\[ w_{\\pm}(\\xi, \\lambda) \\leq C(\\lambda) (\\xi^{-\\frac{1}{3}+\\epsilon}+\n\\xi^{\\frac{1}{3}(\\frac{1}{2}+\\alpha)+\\epsilon}). \\]\nTaking into account the fact that $|V(\\xi)| \\leq C\\xi^{-\\frac{2}{3}},$ this implies that for a.e. $\\lambda$ the control expression (14)\nis bounded by $C(\\lambda)\\xi^{-1-\\frac{2\\alpha}{3}+\\epsilon}$\nand hence is absolutely integrable. This allows us to conclude the boundedness of solutions of the equation (3) for a.e. $\\lambda.$ Applying Lemma 1.2, we find that for a.e. $\\lambda$ there is no subordinate solution of the original equation (1). This completes the proof of Theorem 2.1. $\\Box$ \\\\\n\n\\section*{Acknowledgment} I am very grateful to Professor  for stimulating discussions and to Professor Stolz for turning author's attention to the problem.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgkae b/data_all_eng_slimpj/shuffled/split2/finalzzgkae
new file mode 100644
index 0000000000000000000000000000000000000000..a04b373171d41451bab7a143998ce20529f97e4c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzgkae
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nIn this paper we study a L\\'evy flights dynamics in an external multi-well \npotential in the annealed regime.\nWe are motivated by the problem of random search of the global \nminimum of an unknown function\n$U$ with help of simulated annealing. For simplicity, we consider a one-dimensional case. \nLet $U$ be a multi-well potential satisfying some regularity conditions.  \nClassical continuous time simulated annealing consists in an examination of a time non-homogeneous Smoluchowski\ndiffusion\n\\begin{equation}\n\\label{eq:1}\nd\\hat{Z}(t)=-U'(\\hat{Z}(t))dt+\\hat{\\sigma}(t)dW(t)\n\\end{equation}\nwith a temperature $\\hat{\\sigma}(t)\\to 0$ as $t\\to+\\infty$. For small values of $\\hat{\\sigma}(t)$,\nthe process $\\hat{Z}$ spends most of the time in small neighbourhoods of the \npotential's local\nminima and makes occasional transitions between the adjacent wells. It is possible to choose\nan appropriate cooling schedule $\\hat{\\sigma}(t)$, such that the diffusion \nsettles down near the\nglobal maximum of $U$. Indeed, one should take \n$\\hat{\\sigma}^2(t)\\approx \\frac{\\theta}{\\ln(\\lambda+t)}$, the parameter $\\theta>0$ being a\ncooling rate and $\\lambda>1$ parameterising the initial temperature. Then there is a critical\nvalue $\\hat{\\theta}>0$, such that $\\hat{Z}(t)$ converges in probability to the global\nminimum of $U$ if $\\theta>\\hat{\\theta}$, and the convergence fails if $0<\\theta<\\hat{\\theta}$. \nMoreover, the critical value $\\hat{\\theta}$ is a logarithmic growth rate of the principal\nnon-zero eigenvalue $\\lambda^1(\\sigma)$ of the generator of the time-homogeneous diffusion\n\\begin{equation}\nd\\hat{X}(t)=-U'(\\hat{X}(t))dt+\\sigma dW(t),\n\\end{equation} \ni.e.\\ $|\\lambda^1(\\sigma)|\\propto \\exp(-\\hat{\\theta}\/\\sigma^2)$.\nHeuristic justification for the convergence is as follows. \nThe principal non-zero eigenvalue $\\lambda^1(\\sigma)$\ndetermines the convergence rate of $\\hat{X}$ to its invariant measure \n$\\mu_\\sigma(dx)=c_\\sigma\\exp(-2U(x)\/\\sigma^2)dx$, $c_\\sigma$ being a normalising factor.\nThus, for any continuous positive function $f$ we have an estimate\n\\begin{equation}\n|\\mathbf{E}_x f(X(t)) - \\int f(x)\\mu_\\sigma(dx)|\n\\leq C e^{-|\\lambda^1(\\sigma)|t}.\n\\end{equation} \nThe weak limit of the invariant measures $\\mu_\\sigma(dy)$ as $\\sigma\\to 0$ \nis a Dirac mass at the potential's global minimum. \nFor small values of $\\sigma(t)$, the dynamics of $\\hat{Z}$ reminds of a \ndynamics of $\\hat{X}$. \nThus  $\\hat{Z}(t)$ has enough time to settle down in the deepest potential well\nif $\\sigma(t)$ is such that\n\\begin{equation}\nt|\\lambda^1(\\sigma(t))|\\to \\infty \\quad \n\\Leftrightarrow\\quad \n\\frac{t}{(\\lambda+t)^{\\hat{\\theta}\/\\theta}}\\to \\infty\n\\quad \n\\Leftrightarrow\\quad \n\\theta>\\hat{\\theta},\\quad t\\to+\\infty.\n\\end{equation}\nThe logarithmic decrease rate of $\\sigma$ was obtained in the seminal paper\n\\cite{GemanG-84}.\nFurther mathematical results on classical continuous time simulated \nannealing can be found in \\cite{ChiangHS-87,HwangS-90,HwangS-92,HolleyS-88,HolleyKS-89}. \nWe also refer the reader to the review paper \\cite{Gidas-95} and further references \ntherein.\n\n\n\nOur research is motivated by the paper \\cite{SzuH-87} by Szu and Hartley, where they\nintroduced the \nso-called \\textit{fast simulated annealing}\nwhich allows to perform a non-local search of the deepest well.\nFast simulated annealing process in the sense of \\cite{SzuH-87} is a discrete time \nMarkov chain, where the states are\nobtained from the Euler approximation of \\eqref{eq:1} driven not by Gaussian noise but \n\\textit{Cauchy} noise. \nThe new state is accepted according to the Metropolis algorithm \n(see \\cite{MetropolisRRT-53}) with Boltzmann acceptance probability which equals $1$, \nif the potential value in this state is smaller, i.e.\\ the new position is `lower'\nin the potential landscape. If the new position is `higher', it is accepted with the probability \n$\\sim \\exp(-\\Delta U\/\\sigma)$, where $\\Delta U$ is the difference of the potential values in the \nnew and the old states, and $\\sigma$ is a decreasing `temperature' parameter. The \nadvantage of this method consists in faster transitions between the potential wells due to \nthe \nheavy tails of Cauchy distribution. Moreover, the authors claim that the optimal cooling\nrate is algebraic, $\\sigma(t)\\approx t^{-1}$, which also accelerates convergence.\n\nIn this paper, we consider a continuous-time \nL\\'evy flights counterpart of the diffusion \\eqref{eq:1}. \nOur goal is to study the asymptotic properties of the system in dependence of a\ncooling schedule. We notify the reader, that in regimes where a L\\'evy flights\nprocess converges to some limiting distribution, it does not locate the global minimum\nof $U$, but reveals the spatial structure of the potential. \n\nThis paper contains a heuristic derivation of results, \nwhich are proved rigorously in \\cite{Pavlyukevich-06a}. \nIt can be seen as a sequel of \\cite{ImkellerP-06,ImkellerP-06a,ImkellerP-06b} where a \nsmall-noise dynamics of L\\'evy flights in external potentials were studied. We emphasise \nthat our methods are purely probabilistic.\n\n\n\\section{Object of study and results}\n\n\\subsection{L\\'evy flights}\n\nLet $L=(L(t))_{t\\geq 0}$ be a L\\'evy flights (LF) process of index $\\alpha\\in(0,2)$, \ni.e.\\ a non-Gaussian stable symmetric L\\'evy\nprocess with marginals having the Fourier transform\n\\begin{equation}\n\\label{eq:f}\n\\mathbf{E} e^{i\\omega L(t)}=e^{-c(\\alpha)t|\\omega|^\\alpha},\n\\quad c(\\alpha)=2\\int_0^\\infty\\frac{1-\\cos{y}}{y^{1+\\alpha}}\\,dy=2|\\cos(\\frac{\\pi\\alpha}{2})\\Gamma(-\\alpha)|.\n\\end{equation}\nIn our analysis we shall use the L\\'evy-Khinchin representation of the characteristic function of $L(t)$, namely\n\\begin{equation}\n\\label{eq:lh}\n\\mathbf E e^{i\\omega L(t)}=\n\\exp\\left\\lbrace t\n\\int_{\\mathbb R\\backslash\\{0\\}} \\left[ e^{i\\omega y}-1\n-i\\omega y \\I{|y|\\leq 1}\\right]\n\\frac{dy}{|y|^{1+\\alpha}}\\right\\rbrace,\n\\end{equation}\nwhere $\\I{A}$ denotes the indicator function of a set $A$.\nThe most important ingredient of the representation\n\\eqref{eq:lh} is the so called \\textit{L\\'evy (jump) measure} of the random process\n$L$ given by\n\\begin{equation}\n\\nu(A)=\\int_{A\\backslash\\{0\\}}\\frac{dy}{|y|^{1+\\alpha}}, \\quad\nA\\,\\,\\mbox{ is a Borel set in}\\,\\,\\mathbb{R}.\n\\end{equation}\nNote, that some authors prefer another parametrisation of LFs with the Fourier transform $\\exp(-t|\\omega|^\\alpha)$.\nIn this paper we use the representation \\eqref{eq:f}\ndue to a simpler form of the L\\'evy measure. \n\nThe measure $\\nu$ controls the intensity and sizes of the jumps of\nthe L\\'evy flights process. Let $\\Delta L(t)=L(t)-L(t-)$ be the random jump \nsize of $L$ at a time instance $t$, $t>0$, and the number of jumps belonging to the set $A$ on the time interval $(0,t]$ be denoted \nby $N(t, A)$, i.e.\\\n\\begin{equation}\nN(t,A)=\\sharp\\{s: (s,\\Delta L_s)\\in (0,t]\\times A\\}.\n\\end{equation}\nThen the random variable $N(t, A)$ has a Poisson distribution with mean\n$t\\nu(A)$ (which can possibly be infinite). \nNote, that for any stability index $\\alpha\\in(0,2)$, the L\\'evy measure of \nany neighbourhood of $0$ is infinite, hence LFs make infinitely many very \nsmall jumps on any time interval.\nThe tails of the density $|y|^{-1-\\alpha}dy$   \ndetermine big jumps of LFs. Thus, $\\mathbf{E}|L(t)|^\\delta<\\infty$, $t>0$, \niff $\\int_{|y|\\geq 1}|y|^\\delta\\nu(dy)<\\infty$\niff $\\delta<\\alpha$.\n\n\n\\subsection{External potential}\n\nWe assume that the external potential $U$ is smooth and has $n$ local minima $m_i$ and $n-1$ local maxima $s_i$ enumerated in the increasing order, i.e.\\\n\\begin{equation}\n-\\infty=s_0<m_1<s_1<\\cdots < s_{n-1}<m_n<s_{n}=+\\infty.\n\\end{equation}\nWe assume also that local extrema are non-degenerate, i.e.\\ $U''(m_i)>0$ and $U''(s_i)<0$, and the potential \nincreases fast at infinity, i.e.\\\n$|U'(x)|>|x|^{1+c}$, $|x|\\to\\infty$ for some $c>0$.\n\nUnder the assumptions on $U$, the deterministic dynamical system\n\\begin{equation}\n\\label{eq:x0}\nX^0_x(t)=x-\\int_0^t U'(X^0_x(u))\\, du\n\\end{equation}\nhas $n$ domains of attraction $\\Omega_i=(s_{i-1},s_i)$ with asymptotically stable \nattractors $m_i$. We note that if $x\\in \\Omega_i$ then $X^0_x(t)\\in \\Omega_i$ for all $t\\geq 0$,\ni.e.\\ the deterministic trajectory cannot pass between different domains of attraction.\nDenote $B_i=\\{x:|m_i-x|\\leq \\Delta\\}$ a $\\Delta$-neighbourhood of the attractor $m_i$.\nWe suppose that $\\Delta$ is small enough, so that $B_i\\subset \\Omega_i$, $1\\leq i\\leq n$.\nDue to the rapid increase of $U'$ at infinity, the return of $X^0_x(t)$ from $\\pm\\infty$\nto $B_1$ or $B_n$ occurs in finite time. \n\n\n\\subsection{Small constant temperature}\n\nFirst, we consider L\\'evy flights $L$ in the potential $U$ in the regime of \nsmall constant temperature.\nThe resulting random dynamics is described by the stochastic differential equation\n\\begin{eqnarray}\nX^\\varepsilon_x(t)&=x-\\int_0^t U'(X^\\varepsilon_x(u-))\\, du+\\varepsilon L(t), \\quad x\\in\\mathbb{R},\\,\\,t\\geq 0.\n\\end{eqnarray} \nThe properties of $X^\\varepsilon$ are studied in our previous \npaper \\cite{ImkellerP-06a} in the case of a double-well potential. \nThe general multi-well case is studied in \\cite{ImkellerP-06b}. Below we formulate \nthe results.\n\nFor any $\\Delta>0$ sufficiently small, in the limit $\\varepsilon\\to 0$, the process $X^\\varepsilon$ spends an overwhelming \nproportion of time in the set $\\cup_{i=1}^n B_i$ making occasional abrupt jumps between \ndifferent neighbourhoods $B_i$.\nThus, the knowledge of the transition times and probabilities is essential\nfor understanding the asymptotic properties of $X^\\varepsilon$.\nLet $T^i_x(\\varepsilon)=\\inf\\{t\\geq 0: X^\\varepsilon_x(t)\\in \\cup_{j\\neq i}B_j\\}$. For $x\\in B_i$, the stopping time\n$T^i_x(\\varepsilon)$ denotes the first transition time to a $\\Delta$-neighbourhood of \na minimum of a different well.\nThen we have the following result.\n\\begin{theorem}[constant temperature, transitions]\n\\label{th:T}\nFor $x\\in B_i$, $1\\leq i\\leq n$, the following estimates hold in the limit $\\varepsilon\\to 0$:\n\\begin{align}\n\\label{eq:t1}\n&\\mathbf{P}_x\\left( X^\\varepsilon(T^i(\\varepsilon))\\in B_j\\right) \\to\\frac{q_{ij}}{q_i}, \\quad i\\neq j,\\\\\n\\label{eq:t2}\n&\\varepsilon^\\alpha T^i_x(\\varepsilon)\\stackrel{d}{\\to} \\exp(q_i),\\\\\n\\label{eq:t3}\n&  \\varepsilon^\\alpha \\mathbf{E}_x T^i(\\varepsilon)\\to \\frac{1}{q_i},\n\\end{align}\nwhere\n\\begin{align}\n\\label{eq:q}\nq_{ij}&\n=\\frac{1}{\\alpha}\n\\left|\\frac{1}{|s_{j-1}-m_i|^\\alpha}-\\frac{1}{|s_j-m_i|^\\alpha}\\right|,\\quad i\\neq j,\\\\\nq_i&=\\sum_{j\\neq i}q_{ij}=\\frac{1}{\\alpha}\\left( \\frac{1}{|s_{i-1}-m_i|^\\alpha}\n+\\frac{1}{|s_i-m_i|^\\alpha}\\right),\n\\end{align}\nand ``$\\stackrel{d}{\\to}$'' denotes convergence in distribution.\n\\end{theorem}\n\nAs we see, the transition times between the wells of $X^\\varepsilon$ are asymptotically \nexponentially distributed in the limit of small noise, and hence \\textit{unpredictable}, \ndue to the memoryless property of the exponential law.\nThe transition probabilities between the wells are noise independent and strictly positive.\nThus, $X^\\varepsilon$ reminds of a Markov process on a finite state space. Indeed, the following theorem \nholds. \n\\begin{theorem}[constant temperature, metastability]\n\\label{th:meta}\nIf $x\\in \\Omega_i$,\n$1\\leq i\\leq n$, then for $t>0$\n\\begin{eqnarray}\nX^\\varepsilon_x\\left( \\frac{t}{\\varepsilon^\\alpha}\\right) \\to Y_{m_i}(t),\\quad \\varepsilon\\to 0,\n\\end{eqnarray}\nin the sense of finite-dimensional distributions, where $Y=(Y_y(t))_{t\\geq 0}$ is a Markov process on a state space $\\{m_1,\\dots, m_n\\}$ with the infinitesimal generator $Q=(q_{ij})_{i,j=1}^n$,\n$q_{ij}$ being defined in \\eqref{eq:q}, $q_{ii}=-q_i$.\n\\end{theorem}\n\nSince none of the entries $q_{ij}$ vanishes, the limiting Markov process $Y$ has a unique invariant distribution \n$\\pi=(\\pi_1,\\dots,\\pi_n)^T$, which can be calculated from the matrix equation $Q^T\\pi=0$.\n\n\n\n\n\n\n\n\\subsection{Decreasing temperature}\n\nIn the annealed regime, the dynamics of L\\'evy flights is characterised by \nthe time non-homogeneous equation\n\\begin{eqnarray}\n\\label{eq:Z}\nZ^\\lambda_{s,z}(t)&=z-\\int_s^t U'(Z^\\lambda_{s,z}(u-))\\, du+\\int_s^t \\frac{dL(u)}{(\\lambda+u)^\\theta},\n\\quad z\\in\\mathbb{R},\\,\\,0\\leq s\\leq t,\n\\end{eqnarray} \nwhere a positive parameter $\\theta$ denotes the \\textit{cooling rate}, and $\\lambda>0$ determines the initial\ntemperature, which equals to $(\\lambda+s)^{-\\theta}$.\n\nIt is easily seen from \\eqref{eq:Z}, that the evolution of the process starting\nat time $s\\geq 0$ is the same as that of the process starting at time zero with a different initial temperature, namely\n\\begin{equation}\n\\label{eq:mp}\n(Z^\\lambda_{s,z}(s+t))_{t\\geq 0}\\stackrel{d}{=}(Z^{\\lambda+s}_{0,z}(t))_{t\\geq 0},\n\\end{equation}\nand thus the particular values of $s$ or $\\lambda$ do not influence asymptotic\nproperties of the \nprocess in the limit $t\\to\\infty$. However, since our theory will work\nfor low temperatures, it is often convenient to study\nthe dynamics not for large values of $s$ and $t$ but for large\nvalues of $\\lambda$.\n\nThe goal of this paper is to study the limiting behaviour of $Z^\\lambda_{0,z}(t)$ as $t\\to \\infty$\nin dependence of the cooling rate $\\theta$, `initial temperature' $\\lambda$ and initial\npoint $z$.\n\n\n\nSimilarly to the classical Gaussian case discussed in the introduction,\nthe candidate for the limiting law of $Z^\\lambda_{0,z}(t)$ is the invariant \ndistribution $\\pi$ of the Markov chain $Y$ from Theorem~\\ref{th:meta}.\nFurthermore, we have to distinguish between two different cooling regimes. \n\nAs in the previous section, for $1\\leq i\\leq n$, consider the stopping times\n\\begin{equation}\n\\tau_{s,z}^{i,\\lambda}=\\inf\\{u\\geq s:Z_{s,z}^\\lambda(u)\\in \\cup_{j\\neq i}B_j\\}.\n\\end{equation}\nIf $z\\in B_i$ then $\\tau_{s,z}^{i,\\lambda}$ denotes the \\textit{transition time} from a\n$\\Delta$-neighbourhood of $m_i$ to a $\\Delta$-neighbourhood of some other potential's minimum. \nFor all $j\\neq i$ we also consider the corresponding\n\\textit{transition probabilities} \n$\\mathbf{P}_{s,z}(Z^\\lambda(\\tau^{i,\\lambda})\\in B_j)$. Then the following analogue of Theorem~\\ref{th:T}\nholds.\n\\begin{theorem}[slow cooling, transitions]\n\\label{th:tau}\nLet $\\theta<1\/\\alpha$. For $z\\in B_i$, $1\\leq i\\leq n$, the following estimates hold in the\nlimit of small initial temperature, i.e. when $\\lambda\\to+\\infty$:\n\\begin{align} \n\\label{eq:p}\n&\\mathbf{P}_{0,z}(Z^\\lambda(\\tau^{i,\\lambda})\\in B_j)\n\\to\\frac{q_{ij}}{q_i},\\quad i\\neq j,\\\\\n\\label{eq:tau}\n&\\frac{\\mathbf{E}_{0,z} \\tau^{i,\\lambda}}{\\lambda^{\\alpha\\theta}}\\to \\frac{1}{q_i},\n\\end{align} \n$q_i$ and $q_{ij}$ being defined in \\eqref{eq:q}.\n\\end{theorem}\n\n\n\n\n\n\n\n\\begin{theorem}[slow cooling, convergence]\n\\label{th:slow}\nLet $\\theta<1\/\\alpha$. \nThen for any $\\lambda>0$, $z\\in\\mathbb{R}$, the law of $Z_{0,z}^\\lambda(t)$ converges weakly to\nthe measure $\\pi$, i.e.\\\nfor any continuous and bounded function $f$ we have\n\\begin{equation}\n\\mathbf{E}_{0,z}f(Z^\\lambda(t))\\to \\sum_{j=1}^n f(m_j)\\pi_j, \\quad t\\to\\infty.\n\\end{equation}\n\\end{theorem}\n\nIf the cooling rate $\\theta$ is above the threshold $1\/\\alpha$, the solution $Z^\\lambda$ gets trapped in one of the wells and thus the convergence fails. Consider the first exit time \nfrom the $i$-th well\n\\begin{equation}\n\\sigma^{i,\\lambda}_{s,z}=\\inf\\{t\\geq 0: Z^\\lambda_{s,z}\\notin \\Omega_i \\}.\n\\end{equation}\n\nThen the following trapping result holds. \n\\begin{theorem}[fast cooling, trapping]\n\\label{th:fast}\nLet $\\theta>1\/\\alpha$.  For $z\\in B_i$, $1\\leq i\\leq n$,  \n\\begin{equation}\n\\mathbf{P}_{0,z}(\\sigma^{i,\\lambda}<\\infty)\n=\\mathcal{O}\\left(\\frac{1}{\\lambda^{\\alpha\\theta-1}}\\right),\\quad\\lambda\\to\\infty.\n\\end{equation}\nConsequently, $\\mathbf{E}_{0,z}\\sigma^{i,\\lambda}=\\infty$.\n\\end{theorem}\n\nIn the subsequent section we sketch the proof of Theorems~\\ref{th:tau}--\\ref{th:fast} and discuss the results.\n\n\\section{Predominant behaviour of the annealed process}\n\nOur study of the random process $Z^\\lambda$ is based on probabilistic analysis of its\nsample paths. We use the decomposition\nof the process $L$ into small- and big-jump parts similar to that used in \\cite{ImkellerP-06a}.\nThus we refer the reader to that paper for details, and sketch the idea briefly.\n\n\\subsection{Big and small jumps of a L\\'evy flights process}\n\nWith help of the L\\'evy-Khinchin formula \\eqref{eq:lh}, we decompose the \nprocess $L$ into a sum of two independent L\\'evy\nprocesses with relatively small and big jumps.  For any cooling rate $\\theta>0$, we\nintroduce two new L\\'evy measures by setting\n\\begin{eqnarray}\n\\nu_\\xi^\\lambda(A) &= \\nu\\left(A\\cap \\{x:|x|\\leq \\lambda^{\\theta\/2}\\}\\right),\\\\\n\\nu_\\eta^\\lambda(A)& = \\nu\\left(A\\cap \\{x:|x|> \\lambda^{\\theta\/2}\\}\\right),\n\\end{eqnarray}\nand two L\\'evy processes $\\xi^\\lambda$ and $\\eta^\\lambda$\nwith the corresponding Fourier transforms:\n\\begin{eqnarray}\n\\mathbf E e^{i\\omega \\xi^\\lambda_t}&=\n\\exp\\left\\lbrace\nt\\int_{\\mathbb R\\backslash\\{0\\}} \\left[e^{i\\omega y}-1-i\\omega y \\I{|y|\\leq 1}\\right]\n\\nu_\\xi^\\lambda(dy)  \\right\\rbrace,\\\\\n\\mathbf E e^{i\\omega \\eta^\\lambda_t}&=\n\\exp\\left\\lbrace\nt\\int_{\\mathbb R\\backslash\\{0\\}} \\left[ e^{i\\omega y}-1-i\\omega y \\I{|y|\\leq 1}\\right]\n\\nu_\\eta^\\lambda(dy)\\right\\rbrace.\n\\end{eqnarray}\nIt is clear that, the processes $\\xi^\\lambda$ and $\\eta^\\lambda$ are independent and\n$L=\\xi^\\lambda+\\eta^\\lambda$.\n\nSince $\\nu^\\lambda_\\xi(\\mathbb{R})=\\infty$, the process\n$\\xi^\\lambda$ makes infinitely many jumps on each time\ninterval. Its jumps  are, however, bounded by the threshold\n$\\lambda^{\\theta\/2}$, i.e.\\ $|\\Delta\\xi_t^\\lambda|\\leq \\lambda^{\\theta\/2}$. \nThus $\\xi^\\lambda_t$ has a finite variance, and more generally moments of all\norders.\n\n\nOn the contrary, the L\\'evy measure of the process\n$\\eta^\\lambda$ is finite, and its mass equals\n\\begin{equation}\n\\beta_\\lambda=\\nu_\\eta^\\lambda(\\mathbb{R})\n=\\int_{-\\infty}^{-\\lambda^{\\theta\/2}} \\frac{dy}{|y|^{1+\\alpha}}\n+\\int_{\\lambda^{\\theta\/2}}^\\infty \\frac{dy}{y^{1+\\alpha}}\n=2\\int_{\\lambda^{\\theta\/2}}^\\infty \\frac{dy}{y^{1+\\alpha}}\n=\\frac{2}{\\alpha}\\lambda^{-\\alpha\\theta\/2}.\n\\end{equation}\nHence, $\\eta^\\lambda$ is a compound Poisson process with jumps\nof absolute value larger than $\\lambda^{\\theta\/2}$. Let $\\tau^\\lambda_k$ and\n$W^\\lambda_k$, $k\\geq 0$, be the jump arrival times and jump sizes of $\\eta^\\lambda$ \nunder the\nconvention $\\tau^\\lambda_0=W^\\lambda_0=0$. Then the inter-arrival times\n$T^\\lambda_k=\\tau^\\lambda_k-\\tau^\\lambda_{k-1}$, $k\\geq 1$, are independent and\nexponentially distributed with mean $\\beta_\\lambda^{-1}$. The jump sizes $W^\\lambda_k$\nare also independent random variables with\nthe probability distribution function given by\n\\begin{equation}\n\\label{eq:w}\n\\mathbf{P}(W^\\lambda_k< u)\n=\\frac{\\nu_\\eta^\\lambda(-\\infty,u)}{\\nu_\\eta^\\lambda(\\mathbb{R})}\n=\\frac{1}{\\beta_\\lambda}\\int_{-\\infty}^u \\I{|y|> \\lambda^{\\theta\/2}}\n\\nu^\\lambda_\\eta(dy).\n\\end{equation}\nFinally, we can represent the random perturbation in \\eqref{eq:Z} as a sum of two processes, namely,\n\\begin{equation}\n\\int_0^t\\frac{dL(u)}{(\\lambda+u)^\\theta}\n=\\int_0^t\\frac{d\\xi^\\lambda_u}{(\\lambda+u)^\\theta}\n+\\sum_{k=1}^\\infty \\frac{W^\\lambda_k}{(\\lambda+\\tau^\\lambda_k)^\\theta}\n\\I{t\\geq \\tau_k}.\n\\end{equation}\n\n\n\n\\subsection{Predominant behaviour}\n\nConsider now the process $Z^\\lambda_{0,z}$ given by equation\n\\eqref{eq:Z}. On the inter-arrival intervals $[\\tau^\\lambda_{k-1}, \\tau^\\lambda_k)$,\n$k\\geq 1$, it is driven only by the process\n$\\varphi_t^\\lambda=\\int_0^t (\\lambda+u)^{-\\theta}d\\xi^\\lambda_u$, \nand at the time instants $\\tau^\\lambda_k$ it\nmakes jumps of the size $W^\\lambda_k\/(\\lambda+\\tau_\\lambda^k)^\\theta$. \nRecall that the jumps of\n$\\xi^\\lambda$ are bounded by $\\lambda^{\\theta\/2}$, hence the jumps sizes \nof $\\varphi^\\lambda$\ntend to zero as $\\lambda\\to\\infty$ for all $t\\geq 0$, i.e.\\ \n\\begin{equation}\n|\\Delta \\varphi^\\lambda_t|\\leq \\frac{\\lambda^{\\theta\/2}}{(\\lambda+t)^\\theta}\\leq \\frac{1}{\\lambda^{\\theta\/2}}.\n\\end{equation}\nThe variance of $\\varphi_t^\\lambda$ tends to zero in the\nlimit of large $\\lambda$, and the random trajectory $Z^\\lambda_{0,z}(t)$\ncan be seen as a small random perturbation of the deterministic\ntrajectory $X^0_z(t)$ of the underlying dynamical system on the\nintervals $[\\tau^\\lambda_{k-1},\\tau^\\lambda_k)$. \nConsider a well $\\Omega_i$ with a minimum $m_i$. Let initial points $z$ be\naway from the unstable points $s_{i-1}$ and $s_i$, namely \n$z\\in(s_{i-1}+\\lambda^{-\\gamma}, s_i-\\lambda^{-\\gamma})$ for some positive $\\gamma$.\nThen the deterministic trajectory $X_z^0(t)$ reaches a $\\lambda^{-\\gamma}$-neighbourhood of $m_i$ in at most logarithmic time $\\mathcal{O}(\\ln\\lambda)$.\nSince the periods between the big jumps are essentially longer, i.e.\\\n\\begin{equation}\n\\mathbf{E} T_k^\\lambda=\\mathbf{E}(\\tau_{k}^\\lambda-\\tau_{k-1}^\\lambda)\n= \\frac{\\alpha}{2}\\lambda^{\\alpha\\theta\/2} \\gg \\mathcal{O}(\\ln\\lambda),\n\\end{equation} \nwe can show that with probability close to $1$, the random trajectory $Z^\\lambda$ is located \nin a small neighbourhood of $m_i$ before the big jump. \n\nThus we can summarise the pathwise behaviour of $Z^\\lambda_{0,z}$ for large values of\n$\\lambda$ as follows: \n\\begin{equation}\n\\begin{aligned}\n\\label{eq:t}\n&Z^\\lambda_{0,z}(0)=z\\in (s_{i-1}+\\lambda^{-\\gamma}, s_i-\\lambda^{-\\gamma}),\\\\\n&Z^\\lambda_{0,z}(\\tau^1_\\lambda-)\\approx m_i,\\\\\n&Z^\\lambda_{0,z}(\\tau^1_\\lambda)\\approx\nm_i+\\frac{W_1^\\lambda}{(\\lambda+\\tau_1^\\lambda)^\\theta}\\in (s_{i-1}+\\lambda^{-\\gamma}, s_i-\\lambda^{-\\gamma})\\\\\n&Z^\\lambda_{0,z}(\\tau^2_\\lambda-)\\approx m_i,\\\\\n&\\cdots\\\\\n&Z^\\lambda_{0,z}(\\tau^k_\\lambda-)\\approx m_i,\\\\\n&Z^\\lambda_{0,z}(\\tau^k_\\lambda)\\approx\nm_i+\\frac{W_k^\\lambda}{(\\lambda+\\tau_k^\\lambda)^\\theta}\\in (s_{j-1}+\\lambda^{-\\gamma}, s_j-\\lambda^{-\\gamma}), \n\\,\\,j\\neq i,\\\\\n&Z^\\lambda_{0,z}(\\tau^{k+1}_\\lambda-)\\approx m_j,\\\\\n&\\cdots,\n\\end{aligned}\n\\end{equation}\nwhereas on the intervals $[\\tau^\\lambda_{k-1},\\tau^\\lambda_k)$ the process $Z^\\lambda$ follows the\ndeterministic trajectory $X^0$.\nThus, since we know the initial location of the particle, \nas well as the jump sizes $W_k^\\lambda$ and jump times $\\tau_k^\\lambda$,\nwe can catch the essential features of the random path $Z^\\lambda$.\n\n\nOf course, we have to be carefull when dealing with trajectories which occasionally enter\nthe $\\lambda^{-\\gamma}$-neighbourhoods of the saddle points $s_i$, where the force field\n$U'$ becomes insignificant. In these neighbourhoods, the L\\'evy particle has no strong \ndeterministic drift which brings it to a certain well's minimum. Thus we cannot decide\nwhether $Z^\\lambda$ converges to $m_i$ or to $m_{i+1}$. However, in the limit\n$\\lambda\\to\\infty$, the probability that $Z^\\lambda$ jumps from a \nneighbourhood of $m_i$ to a \n$\\lambda^{-\\gamma}$-neighbourhood of some $s_j$ is negligible. In our further \nexposition, we do not consider the unstable dynamics in these\n$\\lambda^{-\\gamma}$-neighbourhoods \nand assume that \\eqref{eq:t} holds for all $z\\in\\Omega_i$.\nInterested readers can find rigorous arguments in \\cite{Pavlyukevich-06a}.\n\n\n\n\n\n\\section{Transitions between the wells in the slow cooling regime\\label{s:kl}}\n\nIn this section we justify the limits \\eqref{eq:p} and \\eqref{eq:tau} from \nTheorem~\\ref{th:tau}.\n\nFirst, we obtain the mean value of the first exit time $\\sigma^{i,\\lambda}$\nform the well $\\Omega^i$ in the limit $\\lambda\\to\\infty$.\nIndeed, $Z^\\lambda$ can roughly leave $\\Omega_i$ only\nat one of the time instants $\\tau^\\lambda_k$ when \n$m_i+W^\\lambda_k\/(\\lambda+\\tau^\\lambda_k)^\\theta\\notin \\Omega_i$.\nWe can therefore calculate the mean value of $\\sigma^{i,\\lambda}$\nusing the full probability formula:\n\\begin{equation}\n\\label{eq:m}\n\\begin{aligned}\n&\\mathbf{E}_{0,z}\\sigma^{i,\\lambda}\n\\approx \\sum_{k=1}^{\\infty}\n\\mathbf{E}\\left[\\tau^\\lambda_k \\I{ \\sigma^{i,\\lambda}=\\tau^\\lambda_k} \\right]  \\\\\n&\\approx\\sum_{k=1}^{\\infty}\n\\mathbf{E}\\left[\\tau^\\lambda_k\\cdot\n\\I{m_i+ \\frac{W^\\lambda_1}{(\\lambda+\\tau_1^\\lambda)^\\theta}\\in \\Omega_i,\\dots, \nm_i+ \\frac{W^\\lambda_{k-1}}{(\\lambda+\\tau_{k-1}^\\lambda)^\\theta}\\in \\Omega_i,\nm_i+ \\frac{W^\\lambda_k}{(\\lambda+\\tau_k^\\lambda)^\\theta}\\notin \\Omega_i\n}\\right].\n\\end{aligned}\n\\end{equation}\nSince the arrival times $\\tau_1^\\lambda,\\tau_2^\\lambda,\\dots, \\tau_k^\\lambda$, are dependent, no straightforward\ncalculation of the \nexpectations in the latter sum seems possible. However, we can estimate these expectations\nfrom above and below.\nOur argument is based on the inequalities  \n$0<\\tau_1^\\lambda< \\tau_2^\\lambda<\\cdots<\\tau_k^\\lambda$, and the obvious inclusions\n\\begin{equation}\n\\label{eq:inc}\n\\left\\lbrace  m_i+ \\frac{W^\\lambda_{j}}{\\lambda^\\theta}\\in \\Omega_i\\right\\rbrace \n\\subseteq\n\\left\\lbrace  m_i+ \\frac{W^\\lambda_{j}}{(\\lambda+\\tau_{j}^\\lambda)^\\theta}\\in \\Omega_i\\right\\rbrace \n\\subseteq\n\\left\\lbrace  m_i+ \\frac{W^\\lambda_{j}}{(\\lambda+\\tau_{k}^\\lambda)^\\theta}\\in \\Omega_i\\right\\rbrace,\n\\,\\,\n1\\leq j\\leq k-1,\n\\end{equation}\nwhere the probability of these events can be\ncalculated explicitly from \\eqref{eq:w}, to yield the formula\n\\begin{equation}\n\\quad\\mathbf{P}\\left( m_i+\\frac{W^\\lambda_1}{(\\lambda+t)^\\theta} \n\\notin \\Omega_i\\right)\n=\\frac{1}{\\beta_\\lambda}\n\\left( \\int_{-\\infty}^{-|m_i-s_{i-1}|(\\lambda+t)^\\theta} \n+\\int_{(s_i-m_i)(\\lambda+t)^\\theta}^\\infty \\right)     \\frac{dy}{|y|^{1+\\alpha}}\n=\\frac{q_i}{\\beta_\\lambda(\\lambda+t)^{\\alpha\\theta}}.\n\\end{equation}\n\n\n\n\n\n\\subsection{Mean transition time}\n\n\nLet us obtain an estimate from above. Note that for each $k\\geq 1$, the arrival time \n$\\tau^\\lambda_k$ is a sum of $k$ independent\nexponentially distributed random variables $T^\\lambda_j$ and thus has a    $\\mbox{Gamma}(k,\\beta_\\lambda)$ distribution\nwith a probability density\n$\\beta_\\lambda e^{-\\beta_\\lambda t}(\\beta_\\lambda t)^{k-1}\/(k-1)!$, $t\\geq 0$.\nThen, applying the second inclusion in \\eqref{eq:inc} we obtain\n\\begin{equation}\n\\begin{aligned}\n&\\mathbf{E}\\left[\\tau^\\lambda_k\\cdot\n\\I{m_i+ \\frac{W^\\lambda_1}{(\\lambda+\\tau_1^\\lambda)^\\theta}\\in \\Omega_i,\\dots, \nm_i+ \\frac{W^\\lambda_{k-1}}{(\\lambda+\\tau_{k-1}^\\lambda)^\\theta}\\in \\Omega_i,\nm_i+ \\frac{W^\\lambda_k}{(\\lambda+\\tau_k^\\lambda)^\\theta}\\notin \\Omega_i\n}\\right]\\\\\n&\\leq \\mathbf{E}\\left[\\tau^\\lambda_k\\cdot\n\\I{m_i+ \\frac{W^\\lambda_1}{(\\lambda+\\tau_k^\\lambda)^\\theta}\\in \\Omega_i,\\dots, \nm_i+ \\frac{W^\\lambda_{k-1}}{(\\lambda+\\tau_{k}^\\lambda)^\\theta}\\in \\Omega_i,\nm_i+ \\frac{W^\\lambda_k}{(\\lambda+\\tau_k^\\lambda)^\\theta}\\notin \\Omega_i\n}\\right]\\\\\n&=\\int_0^\\infty \\beta_\\lambda t e^{-\\beta_\\lambda t}\\frac{(\\beta_\\lambda t)^{k-1}}{(k-1)!}\n\\mathbf{P}\\left( m_i+ \\frac{W^\\lambda_1}{(\\lambda+t)^\\theta}\n\\in \\Omega_i\\right)^{k-1} \n\\mathbf{P}\\left( m_i+ \\frac{W^\\lambda_1}{(\\lambda+t)^\\theta}\n\\notin \\Omega_i\\right) dt\\\\\n&=\\int_0^\\infty \\beta_\\lambda t e^{-\\beta_\\lambda t}\\frac{(\\beta_\\lambda t)^{k-1}}{(k-1)!}\n\\left[1-\\frac{q_i}{\\beta_\\lambda(\\lambda+t)^{\\alpha\\theta}} \\right]^{k-1} \n\\frac{q_i}{\\beta_\\lambda(\\lambda+t)^{\\alpha\\theta}} dt.\n\\end{aligned}\n\\end{equation}\nSummation over $k$ yields\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{E}_{0,z}\\sigma^{i,\\lambda}\n&\\lesssim\n\\int_0^\\infty \\beta_\\lambda t e^{-\\beta_\\lambda t} \\frac{q_i}{\\beta_\\lambda(\\lambda+t)^{\\alpha\\theta}} \\sum_{k=1}^{\\infty}   \n\\frac{(\\beta_\\lambda t)^{k-1}}{(k-1)!}\n\\left[1-\\frac{q_i}{\\beta_\\lambda(\\lambda+t)^{\\alpha\\theta}} \\right]^{k-1} dt\\\\\n&=\\int_0^\\infty    \\frac{q_i t }{(\\lambda+t)^{\\alpha\\theta}} \n\\exp\\left( -\\frac{q_i t }{(\\lambda+t)^{\\alpha\\theta}} \\right) dt,\n\\end{aligned}\n\\end{equation}\nwhere `$\\lesssim$' denotes an inequality up to negligible error terms.\nSince $\\alpha\\theta<1$, the latter integral converges for all $\\lambda>0$, and it is \npossible to evaluate it in the limit $\\lambda\\to+\\infty$. \nIntroducing a new variable $u=\\frac{\\lambda+t}{\\lambda}$, we transform it\n to a Laplace type integral with big parameter, \nwhich can be evaluated asymptotically (see \\cite[Chapter 3]{Olver-74}), i.e.\\\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{E}_{0,z}\\sigma^{i,\\lambda}\n&\\lesssim \n\\lambda^{2-\\alpha\\theta}\\int_1^\\infty \\frac{q_i(u-1)}{u^{\\alpha\\theta}}\n\\exp\\left( -\\frac{q_i (u-1)\\lambda^{1-\\alpha\\theta} }{u^{\\alpha\\theta}} \\right) du\\\\\n&\\approx \\lambda^{2-\\alpha\\theta}\\int_1^\\infty q_i(u-1)\n\\exp\\left( -q_i (u-1)\\lambda^{1-\\alpha\\theta}  \\right) du\\\\\n&=q_i^{-1}\\lambda^{\\alpha\\theta}.\n\\end{aligned}\n\\end{equation}\nApplying analogously the first inclusion from \\eqref{eq:inc} to the first $k-1$ jumps, \nwe obtain the estimate from below:\n\\begin{equation}\n\\begin{aligned}\n&\\mathbf{E}_{0,z}\\sigma^{i,\\lambda}\n\\gtrsim \\sum_{k=1}^{\\infty}\n\\mathbf{E}\\left[\\tau^\\lambda_k\\cdot\n\\I{m_i+ \\frac{W^\\lambda_1}{\\lambda^\\theta}\\in \\Omega_i,\\dots, \nm_i+ \\frac{W^\\lambda_{k-1}}{\\lambda^\\theta}\\in \\Omega_i,\nm_i+ \\frac{W^\\lambda_k}{(\\lambda+\\tau_k^\\lambda)^\\theta}\\notin \\Omega_i\n}\\right]\\\\\n&\\geq \\sum_{k=1}^{\\infty}\\int_0^\\infty t\\beta_\\lambda e^{-\\beta_\\lambda t}\n\\frac{(\\beta_\\lambda t)^{k-1}}{(k-1)!}\n\\left[1-\\frac{q_i}{\\beta_\\lambda \\lambda^{\\alpha\\theta}} \\right]^{k-1} \n\\frac{q_i}{\\beta_\\lambda(\\lambda+t)^{\\alpha\\theta}} dt\\\\\n&=\\int_0^\\infty \\frac{q_i t }{(\\lambda+t)^{\\alpha\\theta}} \n\\exp\\left( -\\frac{q_i t }{\\lambda^{\\alpha\\theta}} \\right) dt\n= \\lambda^{2-\\alpha\\theta} \\int_1^\\infty \\frac{q_i(u-1)}{u^{\\alpha\\theta}}\n\\exp\\left( -q_i(u-1)\\lambda^{1-\\alpha\\theta} \\right)\\\\\n&\\approx \\lambda^{2-\\alpha\\theta}\\int_1^\\infty q_i(u-1)\n\\exp\\left( -q_i (u-1)\\lambda^{1-\\alpha\\theta}  \\right) du\n=q_i^{-1}\\lambda^{\\alpha\\theta} .\n\\end{aligned}\n\\end{equation}\nSurprisingly, the estimates from below and above coincide, and thus give the asymptotic value\nof the mean life time of the slowly cooled L\\'evy particle in a potential well.\n\nTo obtain the limit \\eqref{eq:tau} for the mean transition time $\\tau^{i,\\lambda}$\nbetween the sets $B_i$ and $\\cup_{j\\neq i}B_j$,\nwe note that\nat the exit time $\\sigma^{i,\\lambda}$ the process $Z^\\lambda$ enters some of the\nwells $\\Omega_j$, $j\\neq i$, with high probability follows the deterministic trajectory, and reaches a $\\Delta$-neighbourhood\nof a well's minimum in a time of the order $\\mathcal{O}(\\ln\\lambda)$, \nwhich is negligible in comparison with\n$\\lambda^{\\alpha\\theta}$. Thus the limit \\eqref{eq:tau} holds.\n\n\\subsection{Transition probability}\n\n\nTo calculate the transition probability between the wells, it suffices to obtain an estimate from\nbelow. Similarly to the estimate of the mean exit time, we have\n\\begin{equation}\n\\begin{aligned}\n&\\mathbf{P}_{0,z}(Z^\\lambda(\\sigma^{i,\\lambda})\\in\\Omega_j)\n\\gtrsim \\sum_{k=1}^{\\infty}\n\\mathbf{P}\\left(m_i+ \\frac{W^\\lambda_1}{\\lambda^\\theta}\\in \\Omega_i,\\dots, \nm_i+ \\frac{W^\\lambda_{k-1}}{\\lambda^\\theta}\\in \\Omega_i,\nm_i+ \\frac{W^\\lambda_k}{(\\lambda+\\tau_k^\\lambda)^\\theta}\\in \\Omega_j\\right)\\\\\n&\\geq \\sum_{k=1}^{\\infty}\\int_0^\\infty \\beta_\\lambda e^{-\\beta_\\lambda t}\n\\frac{(\\beta_\\lambda t)^{k-1}}{(k-1)!}\n\\left[1-\\frac{q_i}{\\beta_\\lambda \\lambda^{\\alpha\\theta}} \\right]^{k-1} \n\\frac{q_{ij}}{\\beta_\\lambda(\\lambda+t)^{\\alpha\\theta}} dt\\\\\n&=\\int_0^\\infty \\frac{q_{ij} }{(\\lambda+t)^{\\alpha\\theta}} \n\\exp\\left( -\\frac{q_i t }{\\lambda^{\\alpha\\theta}} \\right) dt\n= \\lambda^{1-\\alpha\\theta} \\int_1^\\infty \\frac{q_{ij}}{u^{\\alpha\\theta}}\n\\exp\\left( -q_{i}(u-1)\\lambda^{1-\\alpha\\theta} \\right)\\\\\n&\\approx \\lambda^{1-\\alpha\\theta}\\int_1^\\infty q_{ij}\n\\exp\\left( -q_i (u-1)\\lambda^{1-\\alpha\\theta}  \\right) du\n=q_{ij}q_i^{-1}.\n\\end{aligned}\n\\end{equation}\nWith help of the equality $\\sum_{j\\neq i}q_{ij}q_i^{-1}=1$, we conclude that\n$\\mathbf{P}_{0,z}(Z^\\lambda(\\sigma^{i,\\lambda})\\in\\Omega_j)\\to q_{ij}q_i^{-1}$.\nFinally, since after entering $\\Omega_j$, the process $Z^\\lambda$ reaches $B_j$ with high probability,\n$\\mathbf{P}_{0,z}(Z^\\lambda(\\sigma^{i,\\lambda})\\in\\Omega_j)\\approx\n \\mathbf{P}_{0,z}(Z^\\lambda(\\tau^{i,\\lambda})\\in B_j)$, and \\eqref{eq:p} holds.\n\n\\section{Convergence in the slow cooling regime}\n\nFigure~\\ref{f:slow} illustrates the typical behaviour of $Z$ in the slow cooling \nregime, $\\alpha\\theta<1$. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.9\\textwidth]{150-049.eps}\n\\end{center}\n\\caption{A slow cooling of a L\\'evy particle in a potential with \nlocal minima at $-4$, $0$ and $3$.\\label{f:slow}}\n\\end{figure}\nRoughly speaking, we can distinguish two different behaviours: chaotic and regular.\n\n1.\\ In general case, the initial temperature $\\lambda^{-\\theta}$ can be high, so that \nasymptotics of Theorem~\\ref{th:tau} does not hold. \nThus, the transitions of $Z^\\lambda$ are chaotic until some time instance $T$, when \nthe temperature $(\\lambda+T)^{-\\theta}$ becomes low enough, Theorem~\\ref{th:tau} \nstarts working.\nMoreover, choosing the time $T$ sufficiently large, we\nmake the transition probabilities of $Z$ between the neighbourhoods $B_i$ \nclose to $q_{ij}\/q_i$ with any prescribed precision.\nFor brevity, we can also assume that \n$Z^\\lambda_{0,z}(T)\\in B_i$ for some $i$. \n\n2.\\\nDenote $\\tau(k)$, $k\\geq 0$, successive transition times after $T$ between \ndifferent $B_j$, $1\\leq j\\leq n$, with $\\tau(0)=T$ by convention. \nThe mean values $\\mathbf{E}_{0,z}\\tau(k)$ are finite and can be calculated from Theorem~\\ref{th:tau}. \nIndeed, if $\\tau(k-1)=t_{k-1}$ and $Z^\\lambda(\\tau(k-1))=z_{k-1}\\in B_i$, then\nthe conditional expectation of $\\mathbf{E}_{0,z}\\tau(k)$ equals\n\\begin{equation}\n\\begin{aligned}\n&\\mathbf{E}_{0,z}\\left[\\tau(k)|\\tau(k-1)=t_{k-1},Z^\\lambda(\\tau(k-1))=z_{k-1}\\in B_i\\right]\n=t_{k-1}+\\mathbf{E}_{t_{k-1},z_{k-1}}\\tau^{i,\\lambda}\\\\\n&=t_{k-1}+\\mathbf{E}_{0,z_{k-1}}\\tau^{i,\\lambda+t_{k-1}}\n\\approx t_{k-1}+q_i^{-1}(\\lambda+t_{k-1})^{\\alpha\\theta}<\\infty.\n\\end{aligned}\n\\end{equation}\nFrom the time instance $T$ on, $Z^\\lambda$ makes transitions between \nthe wells with probabilities close to $p_{ij}=q_{ij}\/q_i$, $i\\neq j$, where \n$p_{ii}=0$ by convention. The probabilities $p_{ij}$ determine a discrete time Markov chain $V(k)$ on \n$\\{m_1,\\dots, m_n\\}$, such that $\\mathbf{P}(V(k)=m_j|V(k-1)=m_i)=p_{ij}$. \nIt is clear, that\n$V$ has the unique invariant distribution $\\pi$. Moreover, $V(k)$ converges to the\ninvariant distribution geometrically fast, i.e.\\ there is $0<\\rho<1$ such \nthat for all $1\\leq i,j\\leq n$ and $k\\geq 0$\n\\begin{equation}\n\\label{eq:V}\n|\\mathbf{P}_{m_i}(V(k)=m_j)-\\pi_j|=\\mathcal{O}({\\rho^k}).\n\\end{equation}\nWith help of the asymptotic relation \n\\begin{equation}\n\\mathbf{P}(Z^\\lambda(\\tau(k)\\in B_j)|Z^\\lambda(\\tau(k-1)\\in B_i))\\approx p_{ij}=\\mathbf{P}(V(k)=m_j|V(k-1)=m_i).\n\\end{equation}\none can show that the distributions of $Z(\\tau(k))$ and $V(k)$ are also close\nfor $k\\geq 1$, i.e.\\ \n\\begin{equation}\n\\mathbf{P}(Z^\\lambda_{0,z}(\\tau(k))\\in B_j|Z^\\lambda_{0,z}(T)\\in B_i)\\approx\\mathbf{P}_{m_i}(V(k)=m_j).\n\\end{equation}\nHence, with help of \\eqref{eq:V} for any prescribed accuracy level we can find $k_0\\geq 1$ such that\nfor $k\\geq k_0$ we have\n\\begin{equation}\n\\mathbf{P}_{0,z}(Z^\\lambda(\\tau(k))\\in B_j)\\approx\\pi_j\n\\end{equation}\nindependently on the initial point $z$.\n\nFinally, we note that\nafter time $T$, the process $Z^\\lambda$ spends most of the time in the \nneighbourhoods $B_i$, and \n\\begin{equation}\nZ^\\lambda_{0,z}(t)\\approx Z^\\lambda_{0,z}(\\tau(k)),\\quad \\mbox{for }\\, \nt\\in [\\tau(k),\\tau(k+1)). \n\\end{equation}\nand thus if $t\\geq \\tau(k_0)$ then $\\mathbf{P}_{0,z}(Z^\\lambda(t)\\in B_j)\\approx \\pi^0_j$ .\n\nAs we see, if $\\alpha\\theta<1$,\nthe process $Z^\\lambda$ reminds of a peace-wise constant jump process on the state space\n$\\{m_1,\\dots, m_n\\}$. It\n never stops jumping between the wells of $U$, and the \nrandom sequence $Z^\\lambda(\\tau(k))$, $k\\geq k_0$, behave as a stationary \ndiscrete time Markov chain with a distribution $\\pi$.\n\n\n\\section{Trapping in the fast cooling regime}\n\nThe regime of fast cooling $\\theta>1\/\\alpha$ is more simple.\nWe estimate the probability of the exit from a well. \nSince the exit occurs with high\nprobability only at the arrival times of the big jump process $\\eta^\\lambda$, we estimate\n\\begin{equation}\n\\begin{aligned}\n&\\mathbf{P}_{0,z}(\\sigma^{i,\\lambda}<\\infty)\\approx\n\\sum_{k=1}^\\infty\\mathbf{P}_{0,z}(\\sigma^{i,\\lambda}=\\tau_k^\\lambda)\\\\\n&\\lesssim\\sum_{k=1}^\\infty\n\\mathbf{P}\\left(\nm_i+ \\frac{W^\\lambda_1}{(\\lambda+\\tau_1^\\lambda)^\\theta}\\in \\Omega_i,\\dots, \nm_i+ \\frac{W^\\lambda_{k-1}}{(\\lambda+\\tau_{k-1}^\\lambda)^\\theta}\\in \\Omega_i,\nm_i+ \\frac{W^\\lambda_k}{(\\lambda+\\tau_k^\\lambda)^\\theta}\\notin \\Omega_i\n\\right)\\\\\n&\\leq\\sum_{k=1}^\\infty\n\\mathbf{P}\\left(m_i+ \\frac{W^\\lambda_k}{(\\lambda+\\tau_k^\\lambda)^\\theta}\\notin \\Omega_i\n\\right)\n=\\int_0^\\infty \\beta_\\lambda e^{-\\beta_\\lambda t} \\frac{q_i}{\\beta_\\lambda(\\lambda+t)^{\\alpha\\theta}}  \\sum_{k=1}^\\infty\\frac{(\\beta_\\lambda t)^{k-1}}{(k-1)!} dt\\\\\n&=\\int_0^\\infty    \\frac{q_i }{(\\lambda+t)^{\\alpha\\theta}} dt\n=\\frac{q_i}{\\alpha\\theta-1}\\frac{1}{\\lambda^{\\alpha\\theta-1}}\\to 0, \\quad \\lambda\\to\\infty.\n\\end{aligned}\n\\end{equation} \nAs a consequence we have infinite mean exit times $\\mathbf{E}_{0,z}\\sigma^{i,\\lambda}=\\infty$.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.9\\textwidth]{150-067.eps}\n\\end{center}\n\\caption{A fast cooling of a L\\'evy particle in a potential with \nlocal minima at $-4$, $0$ and $3$.\\label{f:fast}}\n\\end{figure}\nIn other words, if $\\theta>1\/\\alpha$, the dynamics of $Z^\\lambda$ has two\nqualitatively different regimes. First, for high temperatures, the\ntransitions between the wells are chaotic. Second, when the temperature is low enough,\nthe particle gets trapped in one of the wells, see Figure~\\ref{f:fast}. \nIn this case, there is no convergence\nto the invariant measure $\\pi$.\n\n\n\\section{Conclusion and discussion}\n\nIn this paper we studied the large time dynamics of a L\\'evy particle \nin a multi-well external potential, with the temperature decreasing with time\nas $1\/t^\\theta$. \n\nWe discovered, \nthat if the cooling is slow, i.e.\\ $\\theta<1\/\\alpha$, then the system reaches a\nquasi-stationary regime where the transition probabilities between \nthe wells converge to certain values which are explicitly\ndetermined in terms of the potential's spatial geometry. \nMoreover, the mean transition times are finite, \nand between the transitions the process lives in a small neighbourhoods of wells' minima.   \nAs opposed to the Gaussian simulated annealing, the L\\'evy flights process does not settle\ndown near the global maximum of $U$. However, our results can be applied \nfor a search for the global minimum of the \npotentials, which possess the so-called\n``large-rims-have-deep-wells'' property, see \\cite{Schoen-97,Locatelli-02}, i.e.\\ when the \nspatially largest well is at the same time the deepest. \nThen, having empirical estimates of the local minima locations $m_i$ and the invariant distribution $\\pi$, we\ncan derive the coordinates of the saddle points $s_i$, and thus reconstruct the sizes of the wells.\n\nOn the other hand, if the cooling is fast, i.e.\\ $\\theta>1\/\\alpha$, the L\\'evy particle\ngets trapped in one of the wells when the temperature decreases below some critical level.\n\nIn this paper we do not answer the most interesting question: \nis it possible to detect a global minimum of $U$ with\nhelp of a non-local search and L\\'evy flights? The answer to this question is affirmative, and the results will be presented \nin our forthcoming paper.  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{References}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\chapter*{Supplementary Material}\n\\section*{Part A}\n\\textit{In this section we show how the Euler-Lagrange equation (8)\ngenerates the Bose condensate excitation modes (10).}\\\\\nIt is convenient to define the following time independent spinors\n\\begin{equation}\n\\tag{A.1}\nu=\\begin{pmatrix}\n\\cos\\frac{\\theta_0}{2}\\\\\n\\sin\\frac{\\theta_0}{2}e^{i\\varphi_0}\n\\end{pmatrix} \\ , \\ \\ \\ \n\\bar{u}= \\begin{pmatrix}\n-\\sin\\frac{\\theta_0}{2}\\\\\n\\cos\\frac{\\theta_0}{2}e^{i\\varphi_0}\n\\end{pmatrix}\\ .\n\\end{equation}\nHence the Bose condensate field (6) is $z_0=A_0 u$.\nSubstitution of ansatz (9) in the Euler-Lagrange equation (8)\nresults in the following algebraic equation for the ``amplitude'' spinors\n$z_{+}$ and $z_{-}$\n\\begin{align*}\n\\tag{A.2}\n(-\\omega^2+k^2)z_{+} + 2\\omega (\\vec{\\sigma}\\cdot\\vec{{\\cal B}})z_{+} + ({\\cal B}^2-m^2)(z_{-}^\\dag u + u^\\dag z_{+})u&=0\\\\\n(-\\omega^2+k^2)z_{-} - 2\\omega (\\vec{\\sigma}\\cdot\\vec{{\\cal B}})z_{-} + ({\\cal B}^2-m^2)(z_{+}^\\dag u + u^\\dag z_{-})u&=0.\n\\end{align*}\nProjecting each equation (A.2), onto both $u$ and $\\bar{u}$, and using the following definitions;\n\\begin{align*}\n\\tag{A.3}\nu^\\dag z_{+} &= a_1            &&a_1 + a_2^*= a_+\\\\\nu^\\dag z_{-} &= a_2              &&a_1 - a_2^*=a_-\\\\\n\\bar{u}^\\dag z_{+} &= b_1        &&b_1+b_2^*\\,=b_+\\\\\n\\bar{u}^\\dag z_{-} &= b_2  &&b_1-b_2^*\\,=b_-\\\\\n\\end{align*}\nwe get the following matrix equation for the amplitudes\n$a_+$, $a_-$, $b_+$, $b_-$,\n \\begin{align*}\n\\tag{A.4}\\begin{pmatrix} \nk^2 - \\omega^2 + 2({\\cal B}^2-m^2) & 2\\omega {\\cal B}\\cos\\theta_0& 0 & -2\\omega {\\cal B}\\sin\\theta_0 \\\\\n2\\omega {\\cal B}\\cos\\theta_0& k^2 - \\omega^2& -2\\omega {\\cal B}\\sin\\theta_0&0\\\\\n0& -2\\omega {\\cal B}\\sin\\theta_0 & k^2 - \\omega^2& -2\\omega {\\cal B}\\cos\\theta_0\\\\\n-2\\omega {\\cal B}\\sin\\theta_0  & 0 &- 2\\omega{\\cal B}\\cos\\theta_0&k^2 - \\omega^2\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\na_+\\\\\na_-\\\\\nb_+\\\\\nb_-\\\\\n\\end{pmatrix}\n=0.\n\\end{align*}\nFrequencies of normal modes (10) immediately follow from this matrix \nequation.\nAmplitude vectors  $( a_+, a_-, b_+, b_-)$,\ncorresponding to the normal modes are of the following form\n\\begin{align*}\n&|1\\rangle \\propto \n\\begin{pmatrix}\n0\\\\\n\\sin\\theta_0\\\\\n1\\\\\n\\cos\\theta_0\\\\\n\\end{pmatrix}; \\ \\ \\ \\\n|2\\rangle \\propto \n\\begin{pmatrix}\nb_2\\\\\n-\\cos\\theta_0\\\\\n0\\\\\n\\sin\\theta_0\\\\\n\\end{pmatrix}; \\ \\ \\ \\\n|3\\rangle \\propto\n\\begin{pmatrix}\n0\\\\\n\\sin\\theta_0\\\\\n-1\\\\\n\\cos\\theta_0\\\\\n\\end{pmatrix}; \\ \\ \\ \\\n|4\\rangle \\propto\n\\begin{pmatrix}\nb_4\\\\\n-\\cos\\theta_0\\\\\n0\\\\\n\\sin\\theta_0\\\\\n\\end{pmatrix}\\\\\n\\tag{A.5}\n&b_2=\\frac{\\sqrt{(3{\\cal B}^2-m^2)^2+4{\\cal B}^2k^2}-(3{\\cal B}^2-m^2)}\n{2{\\cal B}\\omega_2}\\nonumber\\\\\n&b_4=\\frac{-\\sqrt{(3{\\cal B}^2-m^2)^2+4{\\cal B}^2k^2}-(3{\\cal B}^2-m^2)}\n{2{\\cal B}\\omega_4}\\nonumber\n\\end{align*}\nAt small momenta, $k\\to 0$, the coefficients behave as\n$b_2 \\to 0$, $b_4 \\to const \\ne 0$.\nNote that  there is no naive orthogonality of ``eigenvectors'', in particular \n$\\langle 2|4\\rangle \\ne 0$.\nA similar nonorthogonality is typical for classical (non-quantum)\ncoupled oscillators in magnetic field.\nIn spite of the nonorthogonality, after an appropriate canonical transformation\nthe Hamiltonian is transformed to the sum of independent Hamiltonians\nof the normal modes.\n\n\\newpage\n\\section*{Part B}\n\\textit{\nIn this section we calculate spin vibrations (11),(12)\ncorresponding to different excitation modes.}\\\\\nUsing Eqs. (A.1) and (A.3) we find\n\\begin{align*}\n\\tag{B.1}\nz_+&\\propto \\begin{pmatrix}\n[(a_{+}+a_{-})\\cos\\frac{\\theta_0}{2} - (b_{+}+b_{-})\\sin\\frac{\\theta_0}{2}]\\\\\n[(a_{+}+a_{-})\\sin\\frac{\\theta_0}{2} + (b_{+}+b_{-})\\cos\\frac{\\theta_0}{2}]e^{i\\varphi_0}\\\\\n\\end{pmatrix}\\\\\nz_-&\\propto \\begin{pmatrix}\n[(a_{+}-a_{-})\\cos\\frac{\\theta_0}{2} - (b_{+}-b_{-})\\sin\\frac{\\theta_0}{2}]\\\\\n[(a_{+}-a_{-})\\sin\\frac{\\theta_0}{2} + (b_{+}-b_{-})\\cos\\frac{\\theta_0}{2}]e^{i\\varphi_0}\n\\end{pmatrix}\\\\\n\\end{align*}\nFrom here, using\n\\begin{align*}\n\\tag{B.2}\n\\delta\\vec{\\zeta}&=\\delta z^{\\dag}\\vec{\\sigma}z_0 + z_0^{\\dag}\\vec{\\sigma}\\delta z,\n\\end{align*}\ntogether with Eqs.(9) and(A.5) we find explicit formulas for\n the spin vibration vectors \n$\\delta {\\vec \\zeta}$ presented in Eqs. (11) and (12).\n\n\n\n\\end{document}\n\n\n\n\n\n\n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAhlswede et~al.\\ \\cite{ahlswede00}\nproposed the notion of network coding\nthat multicasts data from a single sender to\nmultiple receivers at a rate at which\nthe ordinary store and forward routing cannot\nmulticast the data.\nSuch high rate multicast becomes feasible by allowing\nintermediate nodes to encode and decode the data.\nA sender is usually called a source and a receiver is called\na sink.\nA network coding is said to be linear if\nevery intermediate node outputs a linear combination\nof its inputs \\cite{li03}.\n\nA study of network coding usually assumes that\nan error does not occur in networks.\nRecently, Cai and Yeung  \\cite{yeung06a,yeung06b}\nconsidered errors in network coding,\nand proposed the network error correcting codes\nthat allow sinks to recover the information even\nwhen errors occur on intermediate edges in the network.\nAfter formulating the network error correction,\nthey proposed the lower and upper bounds\non the number of messages in a network $\\alpha$-error correcting\ncode, \\change{and one of their upper bound was a\nnatural generalization of the Singleton bound\nfor the ordinary error-correcting codes.\nRecently,\nZhang \\cite{zhang06} and Yang et~al.~\\cite{yang07b}\nindependently observed that the Singleton bound\ncan be refined.\nWe note that the problem formulation in \\cite{yeung06a,yeung06b}\nwas later independently presented in \\cite{jaggi05}.\n(The proceedings paper of \\cite{yeung06a,yeung06b} appeared\nin 2002.)}\n\nCai and Yeung  mostly considered the case that\nintermediate nodes perform only simple encoding and\ndecoding without delay, such as computing the output of the node\nas a linear combination of its inputs, and the sinks\nperform complex decoding computation.\nThe network error correcting codes\n\\change{can} avoid  introducing decoding computation and delay into\nintermediate nodes, which is the advantage \nover use of ordinary error correcting codes\nbetween nodes.\n\nNote that\na similar type of network\nfailure in a slightly different context  was considered in\n\\cite[Sect.\\  V]{koetter03}\nand\n\\cite[Sect.\\  VI]{sanders05}\nin which\nevery sink is assumed to know the set of failed edges\nand failed edges are assumed to emit zero symbols.\nNetwork error correction does not assume\nthe knowledge of edges causing errors,\nand the problem formulation is different from \\cite{koetter03,sanders05}.\nNote also that Kurihara \\cite{kurihara06} considered the different\nnotion of robustness. In his paper, he considered network coding\nthat allows sinks to recover partial information with edge failures.\n\n\nFor the construction of the network error-correcting\ncodes, Jaggi et~al.~\\cite{jaggi07}\nproposed a randomized construction that uses\ncoding among different time intervals.\nTheir method produces\ncodes attains the Singleton bound \nwith high probability with sufficiently long\nblock length, where the block length\nrefers to the number of time intervals\namong which coding is done.\nIt is desirable to have a network\nerror-correcting code that does not\ncode among different time intervals and\nthus does not introduce delay.\nConcurrently to this paper,\nYang et~al.~\\cite{yang07b}\nproposed an explicit construction algorithm\nthat produces codes attaining the\nrefined Singleton bound.\nThe idea in \\cite{yang07b} is similar to this paper\nin the sense that they also regard errors as information\nfrom the source and add extra components in the global\nencoding vectors corresponding to errors.\n\n\nIn this paper,\nwe give a \\change{deterministic and\ncentralized} algorithm that constructs\na network error-correcting code that \\change{attains}\nthe Singleton bound \\change{of} network error-correcting codes\nobtained in \\cite{yeung06a}.\n\\change{We also give a relationship between\nthe success probability and the field size\nfor successful construction of network error-correcting\ncodes when intermediate nodes choose their encoding\ncoefficients randomly and independently.}\nThe proposed algorithms are based on \\cite{sanders05}.\nOur network error-correcting codes\nmake multicast robust to errors without introducing\ndelay in the transmission, which is very attractive to\ndelay sensitive multicast applications, such as\nmulticast of video or audio.\nOur method is also useful for cryptographic applications,\nbecause it can tolerate modification and deletion of data\nby an adversary.\n\nThis paper is organized as follows.\nSection 2 introduces notations and the model of errors.\nSection 3 proposes an algorithm for constructing\nnetwork error-correcting codes \\change{attaining}\nthe Singleton bound.\n\\change{Section 4 shows how to modify the algorithm\nin Sect.\\  3 to attain the refined Singleton bound,\nthe success probability of the random\nconstruction of network error-correcting codes,\nand the relationship between\nthe robust network coding \\cite{koetter03,sanders05}\nand the network error-correcting\ncodes with known locations of errors \\cite{yang07}.}\nSection 5 gives concluding remarks.\n\n\\section{Preliminary}\n\\subsection{Basic notations}\nWe consider an acyclic directed graph $G=(V,E)$ with possible parallel\nedges of unit capacity.\n$V \\ni s$ denotes the source  and $V \\supset T$ denotes\nthe set of sinks.\nLet $n$ be the smallest min-cut separating $s$ from any $t\\in T$\n\\change{throughout this paper.}\nFor $v \\in V$, $\\Gamma^+(v)$ (resp.\\ $\\Gamma^-(v)$) denotes the \nset of edges leaving (resp.\\ reaching) the node $v$,\nand $\\mathrm{start}(e)$ (resp.\\ $\\mathrm{end}(e)$)\ndenotes the node at which the edge $e$ starts (resp.\\ ends).\n\nWe consider linear coding over a finite field $\\mathbf{F}_q$ with\n$q$ elements.\nThe source  $s$ gets $k$ ($\\leq n$) input symbols from $\\mathbf{F}_q$.\nThe symbol $y(e)\\in \\mathbf{F}_q$ carried by an edge $e$ is a linear combination of\nthe symbols carried by the edges entering $\\mathrm{start}(e)$.\nThe \\emph{local \\change{encoding} vector\n$m_e: \\Gamma^-(\\mathrm{start}(e))\n\\rightarrow\\mathbf{F}_q$}  determines the coefficients of\nthis linear combination, that is,\n\\[\ny(e) = \\sum_{e'\\in \\Gamma^-(\\mathrm{start}(e))} m_e(e')y(e').\n\\]\n\nIn this paper,\na nonsink node performs only the computation of linear combination\nof its inputs, and they do not correct errors.\nAn error is assumed to occur always at an edge.\nWhen an error occurs at an edge $e$,\nthe symbol received by $\\mathrm{end}(e)$ is different from\none sent by $\\mathrm{start}(e)$,\nand $\\mathrm{end}(e)$ computes its outputs as if there was no error\nat $e$. The error value at an edge $e$\nis defined by the received symbol minus\nthe transmitted symbol at $e$.\nNote that we express\na failure of a node $v \\in V$ in a real network as\nerrors on edges in $\\Gamma^+(v)$ in our model.\nThe number of errors is the number of edges at which errors occur.\nA network code is said to correct $\\alpha$ errors\nif every sink can recover the original information\nsent by the source when $\\alpha$ or less\nerrors occur at arbitrary edges.\n\\change{We call the recovery of information by a sink \\emph{decoding}.}\n\nWe represent errors occurred in the whole network\nby a vector $\\vec{e}$ in $\\mathbf{F}_q^{|E|}$,\nwhere $|E|$ denotes the number of elements in $E$.\nFix some total ordering in $E$, and enumeration of\nthe error values gives $\\vec{e}$.\n\nRegarding on the number of messages in a network $\\alpha$-error\ncorrecting code, Cai and Yeung obtained the following result.\n\\begin{proposition}\\label{lemsingleton} \\textnormal{\\cite{yeung06a}}\nThe number $M$ of messages\nin  a network $\\alpha$-error\ncorrecting code, not necessarily linear, is upper bounded by\n\\[\nM \\leq q^{n-2\\alpha}.\n\\]\n\\end{proposition}\n\n\nVery recently, Zhang~\\cite{zhang06} and\nYang et~al.~\\cite{yang07b} observed that the above\nproposition can be refined as follows.\n\n\\begin{proposition}\\label{lemsingleton2} \\textnormal{\\cite{zhang06,yang07b}}\nLet $n_t$ be the min-cut from the source $s$ to a sink $t$.\nIf the sink $t$ can correct any $\\alpha_t$ errors then\nthe number $M$ of messages\nin  the network \ncorrecting code, not necessarily linear, is upper bounded by\n\\[\nM \\leq q^{n_t-2\\alpha_t}.\n\\]\n\\end{proposition}\n\n\n\\subsection{Jaggi et~al.'s algorithm for construction of\nan ordinary network \\change{code}}\nIn this subsection,\nwe review Jaggi et~al.'s algorithm \\cite{sanders05} for construction of\nan ordinary network coding.\nThe proposed algorithm uses a modified version of their algorithm.\n\nSince linear coding is used,\nthe information carried by an edge $e$\nis a linear combination of $k$ information symbols in $\\mathbf{F}_q$.\nWe can characterize the effect of all the local\n\\change{encoding} vectors on an edge $e$ independently of a\nconcrete $k$ information symbols using \\emph{global\n\\change{encoding} vectors $\\vec{b}(e) \\in \\mathbf{F}_q^k$}.\nWhen the information from the source is $\\vec{i} \\in \\mathbf{F}_q^k$,\nthe transmitted symbol on an edge $e$ is equal to the inner product\nof $\\vec{i}$ and $\\vec{b}(e)$.\n\\change{In order to decide the encoding at the source node $s$,\nwe have to introduce an imaginary source $s'$ and\n$k$ edges of unit capacity from $s'$ to $s$.\nWe regard that $s'$ sends $k$ symbols to $s$ over\n$k$ edges.}\n\nWe initially computes an \\change{$s'$-$t$} flow $f^t$\nof magnitude $k$ for each $t\\in T$ and decomposes this flow\ninto $k$ edge disjoint paths from \\change{$s'$} to $t$.\nIf an edge $e$ is on some flow path $W$ from \\change{$s'$} to\n$t$, let $f_\\leftarrow^t(e)$ denote the predecessor edge of\nthe edge $e$ on the path $W$.\nJaggi et~al.'s algorithm steps through \nthe nodes $v\\in V$ in a topological order\ninduced by the directed graph $G$.\nThis ensures that the global \\change{encoding} vectors of all\nedges reaching $v$ are known when the local \\change{encoding} vectors\nof the edges leaving $v$ are determined.\nThe algorithm defines the coefficients of $m_e$\nfor one edge $e\\in\\Gamma^+(v)$ after the other.\nThere might be multiple flow paths to different sinks\nthrough an edge $e$. Let $T(e)$ denote the set of sinks\nusing $e$ in some flow $f^t$ and\nlet $P(e) = \\{f_\\leftarrow^t(e)\n\\mid t\\in T(e)\\}$ denote the set of\npredecessors edges of $e$ in some flow path.\nThe value $0$ is chosen for $m_e(e')$ with edges\n$e' \\notin P(e)$.\n\nWe introduce two algorithmic variables $B_t$ and $C_t$\nthat are updated by Jaggi et~al.'s algorithm.\n$C_t$ contains one edge from each path in $f^t$,\nnamely the edge whose global \\change{encoding}\nvector was defined most recently in the path.\n$B_t = \\{\\vec{b}(e) \\mid e \\in C_t\\}$ is updated\nwhen $C_t$ is updated.\nThe algorithm determines $m_e$ so that\nfor all $t \\in T$,\n$B_t$ \nis linearly independent.\n\nAfter finishing the algorithm,\nevery sink can \\change{decode} the original information because $B_t$ is\nlinearly independent.\n\n\\section{Construction algorithm}\\label{sec:const}\nWe shall propose an algorithm constructing a network $\\alpha$-error\ncorrecting code carrying $k$ information\nsymbols in $\\mathbf{F}_q$ with $n-k\\geq 2\\alpha$,\nwhich is equivalent to the Singleton bound (Proposition \\ref{lemsingleton}).\nThe proposed construction is based on \\cite{sanders05}.\nWe assume that the size of alphabet $\\mathbf{F}_q$ satisfies\n\\begin{equation}\nq > |T| \\cdot {|E| \\choose 2\\alpha}. \\label{qassumption}\n\\end{equation}\n\n\\begin{definition}\\label{def1}\nFor the original information $\\vec{i} \\in \\mathbf{F}_q^k$ and\nthe error $\\vec{e} \\in \\mathbf{F}_q^{|E|}$,\nlet \\change{$\\phi_t(\\vec{i},\\vec{e}) \\in \\mathbf{F}_q^{|\\Gamma^-(t)|}$}\nbe the vector of symbols carried by the input edges to $t$.\n\\end{definition}\n\n\\begin{lemma}\nIf a sink $t$ can \\change{decode} the original information $\\vec{i}$\nwith any $2\\alpha$ or less errors whose locations are known to the sink $t$,\nthen the sink $t$ can \\change{decode} the original information with any\n$\\alpha$ or less errors without the knowledge of the error locations\nunder the assumption that the number of errors is $\\leq \\alpha$.\n\\end{lemma}\nNote that errors with known locations are called erasures in\n\\cite{yang07} and the properties of erasures are also studied\nin \\cite{yang07}.\n\n\\noindent\\emph{Proof.} \nDenote the Hamming weight of a vector $\\vec{x}$ by $w(\\vec{x})$.\nThe assumption of the lemma implies that\nfor any $\\vec{i} \\neq \\vec{j}$ and $\\vec{e}$ with $w(\\vec{e})\n\\leq 2\\alpha$ we have\n\\begin{equation}\n\\phi_t(\\vec{i},\\vec{e})\\neq \n\\phi_t(\\vec{j},\\vec{0}). \\label{eq10}\n\\end{equation}\nEquation (\\ref{eq10}) implies that for any $\\vec{i} \\neq \\vec{j}$ and\n$\\vec{e}_1$, $\\vec{e}_2$ with $w(\\vec{e}_1) \\leq \\alpha$ and\n$w(\\vec{e}_2)\\leq \\alpha$ we have\n\\[\n\\phi_t(\\vec{i},\\vec{e}_1)\\neq \n\\phi_t(\\vec{j},\\vec{e}_2),\n\\]\nwhich guarantees that $t$ can \\change{decode} the original information\nunder the assumption that the number of errors is $\\leq \\alpha$\nby exhaustive search.\n\\qed\n\n\n\\begin{remark}\nThe above lemma does not guarantee the existence of\nan efficient decoding algorithm.\n\\end{remark}\n\n\nFix $F \\subset E$ with $|F| = 2\\alpha$.\nWe shall show how to construct a network error-correcting code that\nallows every sink to \\change{decode} the original information when the errors can\noccur only at $F$.\n\\change{We call $F$ the \\emph{error pattern}.\nThe following description is a condensed version of the\nproposed algorithm, which is equivalent to the full description with $\\mathcal{F} = \\{ F \\}$ in Fig.~\\ref{fig2} on p.~\\pageref{fig2}.}\n\\begin{enumerate}\n\n\\item\\label{step0} Add the imaginary source $s'$ and draw $k$ edges from\n$s'$ to $s$.\n\n\\item\\label{step1} Add an \\change{imaginary} node $v$ at the midpoint of each $e \\in F$ and add\nan edge of unit capacity from \\change{$s'$} to each $v$.\n\\item\\label{step2} For each sink $t$, do the following:\n\\begin{enumerate}\n\\item\\label{step2a} Draw as many edge disjoint paths\nfrom \\change{$s'$} to $t$ passing through the \\change{imaginary} edges\nadded at Step \\ref{step1}\nas possible.\nLet \\change{$m_t^F (\\leq 2\\alpha)$} be the number of paths.\n\\item\\label{step2b} Draw $k$ edge disjoint paths passing\nthrough $s$ that are also edge disjoint from\nthe \\change{$m_t^F$} paths drawn in the previous step.\n\\end{enumerate}\n\\item\\label{finalstep} Execute the algorithm by Jaggi et~al.\\ \nwith $\\sum_{t\\in T}(k+m_t^F)$ edge disjoint paths constructed in Step \\ref{step2}.\n\\end{enumerate}\n\n\n\\begin{figure}[t!]\n\\psset{unit=0.1\\linewidth}\n\\begin{pspicture}(0,0)(10,13)\n\\rput(5,12){\\circlenode[]{sd}{\\large $s'$}}\n\\rput(5,10){\\circlenode[]{s}{\\large $s$}}\n\\rput(2,8){\\circlenode[]{1}{\\large $1$}}\n\\rput(4,8){\\circlenode[]{2}{\\large $2$}}\n\\rput(6,8){\\circlenode[]{3}{\\large $3$}}\n\\rput(8,8){\\circlenode[]{4}{\\large $4$}}\n\n\\rput(3,9.5){\\dianode[]{A}{\\large $A$}}\n\\rput(2.5,6.5){\\dianode[]{B}{\\large $B$}}\n\n\\rput(3,5){\\circlenode[]{5}{\\large $5$}}\n\\rput(5,5){\\circlenode[]{6}{\\large $6$}}\n\\rput(7,5){\\circlenode[]{7}{\\large $7$}}\n\\rput(3,3){\\circlenode[]{8}{\\large $8$}}\n\\rput(5,3){\\circlenode[]{9}{\\large $9$}}\n\\rput(7,3){\\circlenode[]{10}{\\large $10$}}\n\\rput(1,1){\\circlenode[]{11}{\\large $t_1$}}\n\\rput(9,1){\\circlenode[]{12}{\\large $t_2$}}\n\n\\ncline[linestyle=dashed]{->}{sd}{A}\n\\ncline[linestyle=dashed]{->}{sd}{B}\n\n\\ncarc[arcangle=15.000000]{->}{sd}{s}\n\\ncarc[arcangle=-15.000000]{->}{sd}{s}\n\n\n\\ncline{->}{s}{A}\n\\ncline{->}{A}{1}\n\\ncline{->}{s}{2}\n\\ncline{->}{s}{3}\n\\ncline{->}{s}{4}\n\\ncline{->}{1}{11}\n\\ncline{->}{4}{12}\n\\ncline{->}{1}{B}\n\\ncline{->}{B}{5}\n\\ncline{->}{2}{5}\n\\ncline{->}{2}{6}\n\\ncline{->}{3}{6}\n\\ncline{->}{3}{7}\n\\ncline{->}{4}{7}\n\n\n\\ncline{->}{5}{8}\n\\ncline{->}{6}{9}\n\\ncline{->}{7}{10}\n\\ncline{->}{8}{11}\n\\ncline{->}{9}{11}\n\\ncline{->}{10}{11}\n\\ncline{->}{8}{12}\n\\ncline{->}{9}{12}\n\\ncline{->}{10}{12}\n\\end{pspicture}\n\n\\caption{Example of a network with \\change{imaginary} nodes and edges.\nNodes $A$ and $B$ are the \\change{imaginary} nodes added in Step~\\ref{step1}\nand the dashed lines from \\change{$s'$} to $A$ and $B$ represent\nthe \\change{imaginary} edges added in Step~\\ref{step1}. See Example~\\ref{ex1}\nfor explanation.}\\label{fig1}\n\n\\end{figure}\n\n\n\n\\begin{example}\\label{ex1}\nIn Fig.~\\ref{fig1},\nwe give an example of addition of \\change{imaginary} nodes and edges.\nThe network structure in Fig.~\\ref{fig1}\nis taken from \\cite[Fig.~2]{harada05}.\nNodes $A$ and $B$ are the \\change{imaginary} nodes added in Step~\\ref{step1}\nand the dashed lines from \\change{$s'$} to $A$ and $B$ represent\nthe \\change{imaginary} edges added in Step~\\ref{step1}.\n\nThe min-cut from $s$ to every sink is $4$ in the original network.\nThe set $F$ of edges with errors consists of the edge\nfrom $s$ to node $1$ and the edge from node $1$ to node $5$.\n\nWe denote a path\nby enumerating nodes on the path.\nIn Step~\\ref{step2a} for $t_1$\nwe can find two edge disjoint paths,\nnamely $(s', A,1,t_1)$ and $(s',B,5,8,t_1)$.\nOn the other hand, in Step~\\ref{step2a} for $t_2$,\nwe can find only one edge disjoint path,\nnamely $(s',A,1,B,5,8,t_2)$ \\emph{or} $(s',B,5,8,t_2)$.\nTherefore $m_{t_1}^F=2$ while $m_{t_2}^F = 1$.\n\nIn Step~\\ref{step2b} for $t_1$,\nwe find two edge disjoint paths as\n$(s',s,3,6,9,t_1)$ and $(s',s,4,7,10,t_1)$.\nIn Step~\\ref{step2b} for $t_2$,\nwe find \\emph{three} edge disjoint paths as\n$(s',s,2,6,9,t_2)$, $(s',s,3,7,10,t_2)$, and $(s',s,4,t_2)$.\nWe can use arbitrary two paths among the three paths.\nIn either case, we can find $n-m_t^F$ paths in Step~\\ref{step2b}. \\qed\n\\end{example}\n\nIn Step~\\ref{step2b}, we can guarantee the existence of\n$k$ paths as follows:\nSuppose that edges in \\change{$m_t^F$} paths used in Step~\\ref{step2a}\nare removed from the original network $(V,E)$.\nThen the min-cut from $s$ to a sink $t$ in the original network $(V,E)$\nis at least $n - m_t^F$, which is larger than or equal to\n$k$.\n\nIn Step \\ref{finalstep}\nwe use the algorithm by Jaggi et~al.\\  as if\nthe \\change{imaginary} source\n\\change{$s'$} sent information on the $\\alpha$ \\change{imaginary} edges added\nin Step \\ref{step1}.\nWe denote by $B_t^F$ the set $B_t$ of global encoding vectors\nfor $k+m_t^F$ edge disjoint paths.\n$B_t^F$ consists of $k+m_t^F$ vectors of length $k+2\\alpha$.\nWe require that every sink $t$ is able to decode\n$k$ information symbols, while $t$ may be unable to\ndecode $2\\alpha$ error symbols in general\nbecause $m_t^F \\leq 2\\alpha$.\n\n\nThere are always two edges end at the added \\change{imaginary} node $v$\nand one edge starts from $v$ in Step \\ref{step1}. Since $v$ is \\change{imaginary},\nwe cannot choose local \\change{encoding} vectors at $v$.\nTherefore, in Step \\ref{finalstep},\nall components in the local \\change{encoding} vector at $v$ must be selected\nto $1$, which keeps $B_t$ linearly independent.\n\nThe reason is as follows:\nLet $e$ be the edge from \\change{$s'$} to $v$ added in Step~\\ref{step1}.\nThe global \\change{encoding} vector of $e$ is of the form\n\\[\n(0^{j-1}, 1, 0^{n-j}),\n\\]\nthat is, it has only $1$ at the $j$-th component.\nAll other global \\change{encoding} vectors in $B_t^F$ have zero\nat the $j$-th component, since they are not in downstream\nof $e$ when we choose local \\change{encoding} vectors at $v$.\nTherefore, the added \\change{imaginary} node $v$\ndoes not interfere with the execution of\nJaggi et~al.'s algorithm.\n\n\nObserve also that $q > |T|$ guarantees the successful execution of\nthe algorithm as with the original version of Jaggi et~al.'s algorithm.\n\nWe shall show how each sink $t$ can \\change{decode} the\noriginal information sent from the source $s$.\n\\change{After executing Step~\\ref{finalstep} we have decided all the\nlocal \\change{encoding} vectors in the original network $(V,E)$.}\nConsider the three linear spaces defined by\n\\begin{eqnarray*}\nV_1 &=& \\{ \\phi_t(\\vec{i},\\vec{e}) \\mid \\vec{i}\\in \\mathbf{F}_q^k,\n\\vec{e}\\in\\mathbf{F}_q^{|E|} \\},\\\\\nV_2 &=& \\{ \\phi_t(\\vec{i},\\vec{0}) \\mid \\vec{i}\\in \\mathbf{F}_q^k \\},\\\\\nV_3 &=& \\{ \\phi_t(\\vec{0},\\vec{e}) \\mid \\vec{e}\\in\\mathbf{F}_q^{|E|} \\},\n\\end{eqnarray*}\nwhere components in $\\vec{e}$ corresponding to $E \\setminus F$ are zero,\n\nand $\\phi_t$ is as defined in Definition~\\ref{def1}.\nWe consider $V_1$, $V_2$, and $V_3$ in the original network $(V,E)$\nwithout added \\change{imaginary} nodes and edges.\nThen we have\n\\begin{equation}\nV_1 = V_2 + V_3,\\, \\dim V_2 \\leq k. \\label{eq21}\n\\end{equation}\nSince we keep $B_t^F$ linearly independent,\n\\begin{equation}\n\\dim V_1 \\geq k+m_t^F. \\label{eq22}\n\\end{equation}\nSince the maximum number of edge disjoint paths passing through\nthe \\change{imaginary} edges added in Step \\ref{step1}\nis \\change{$m_t^F$}, we have\n\\begin{equation}\n\\dim V_3 \\leq m_t^F. \\label{eq23}\n\\end{equation}\nEquations (\\ref{eq21}--\\ref{eq23}) imply\n\\begin{eqnarray}\n\\dim V_1 &=& k+m_t^F,\\nonumber\\\\\n\\dim V_2 &=& k,\\label{eq33}\\\\\n\\dim V_3 &=& m_t^F,\\nonumber\\\\\n\\dim V_2 \\cap V_3 &=& 0. \\label{eq12}\n\\end{eqnarray}\nThe number of nonzero components in $\\phi_t(\\vec{i},\\vec{e})$ is $k+m_t^F$\nand the number of unknowns in $\\phi_t(\\vec{i},\\vec{e})$ is $k+2\\alpha$,\nwhich can be larger than $k+m_t^F$.\nHowever, by Eq.~(\\ref{eq12}), the sink $t$ can compute\n$\\phi_t(\\vec{i},\\vec{0})$\nfrom $\\phi_t(\\vec{i},\\vec{e})$ as follows:\nWrite $\\phi_t(\\vec{i},\\vec{e})$ as $\\vec{u} + \\vec{v}$\nsuch that $\\vec{u} \\in V_2$ and $\\vec{v} \\in V_3$.\nBy Eq.~(\\ref{eq12}) $\\vec{u}$ and $\\vec{v}$ are uniquely\ndetermined \\cite[p.19, Theorem 4.1]{lang87}.\nWe have $\\vec{u} = \\phi_t(\\vec{i},\\vec{0})$ and\nthe effect of errors is removed.\nThe sink $t$ can also compute the original\ninformation $\\vec{i}$ from $\\phi_t(\\vec{i},\\vec{0})$\nby Eq.~(\\ref{eq33}).\n\n\nWe shall describe how to construct a network error-correcting\ncode that can correct errors in any edge set $F \\subset E$ with\n$|F| = 2\\alpha$.\n\\change{Let $\\mathcal{F} = \\{ F \\subset E \\,:\\, |F| = 2\\alpha \\}$.}\nThe idea in this paragraph is almost the same as the construction\nof the robust network coding in \\cite[Sect.\\  VI]{sanders05}.\n\\change{Recall that}  $B_t^F$ is the set of global \\change{encoding} vectors\non edge disjoint paths to a sink $t$ with an edge set $F$ of errors.\nExecute Jaggi et~al.'s algorithm keeping $B_t^F$ linearly\nindependent for all $t\\in T$ and all $F\\in \\mathcal{F}$. Then every sink $t$ can \\change{decode} the original information\nwith the knowledge of the edge set $F$ on which errors actually occur.\nAs in \\cite[Sect.\\  VI]{sanders05},\n\\[\nq > |T| \\cdot |\\mathcal{F}| = |T| {|E| \\choose 2\\alpha}\n\\]\nguarantees the successful execution of the algorithm.\n\n\nWe present a pseudo programming code\nof  the proposed algorithm in Fig.\\ \\ref{fig2}.\nIn order to present a detailed description,\nwe introduce new notations.\n$G_F=(V_F,E_F)$ denotes the network\nwith added \\change{imaginary} nodes and edges in  Steps~\\ref{step0}\nand \\ref{step1}\nwith the error pattern $F \\subset E$.\nLet $f^{t,F}$ be the flow established in Steps~\\ref{step2a} and\n\\ref{step2b} in $G_F$.\nLet $f_\\leftarrow^{t,F}(e)$ denote the set of predecessor edges of\nthe edge $e$ in a flow path in $f^{t,F}$.\nLet $T^F(e)$ denote the set of sinks\nusing $e$ in some flow $f^{t,F}$ and\nlet $P^F(e) = \\{f_\\leftarrow^t(e)\n\\mid t\\in T(e)\\}$.\n\n\n\\begin{figure}[t!]\n\n\\begin{tabbing}\n(* Initialization *)\\\\\nAdded \\change{imaginary} node \\change{$s'$} and edges $e_1$, \\ldots, $e_{k}$\nfrom \\change{$s'$} to $s$. $O(k)$\\\\\n\\textbf{foreach} error pattern $F \\in \\mathcal{F}$ \\textbf{do}\\\\\n\\hspace*{4ex}\\= Initialize global \\change{encoding} vector\\\\\n\\>$\\vec{b}^F(e_i) = (0^{i-1},1,0^{k+2\\alpha-i})\\in\\mathbf{F}_q^{k+2\\alpha}$.\\`$O((k+2\\alpha)^2)$\\\\\n\\> \\textbf{foreach} edge  $e \\in F$ \\textbf{do}\\\\\n\\>\\hspace*{4ex}\\= Add an \\change{imaginary} node $v$ at the midpoint of $e \\in F$.\\`$O(1)$\\\\\n\\>\\> Divide $e$ into an edge to $v$ and an edge from $v$.\\`$O(1)$\\\\\n\\>\\> Draw an \\change{imaginary} edge from \\change{$s'$} to $v$.\\` (*) $O(1)$\\\\\n\\>\\textbf{endforeach}\\\\\n\\>\\textbf{foreach} sink $t\\in T$ \\textbf{do}\\\\\n\\>\\> Draw as many edge disjoint paths from \\change{$s'$} to $t$ as possible\\\\\n\\>\\>passing through the edge\nadded in (*).\\\\\n\\` $O(2\\alpha(|E|+k+4\\alpha))$\\\\\n\\>\\> Draw $k$ edge disjoint path from \\change{$s'$} to $t$\npassing through\\\\\n\\>\\>$s$ and\nalso disjoint from\npaths made in the previous step.\\\\\n\\` $O(k(|E|+k+4\\alpha))$\\\\\n\\>\\> Initialize the basis $B_t^F = \\{ \\vec{b}^F(e_i) \\mid e_i$ is on a path to $t \\}$.\\\\\n\\` $O((k+2\\alpha)^2)$\\\\\n\\>\\textbf{endforeach}\\\\\n\\textbf{endforeach}\\\\\n(* Main loop *)\\\\\n\\textbf{foreach} edge $e\\in \\bigcup_{F\\in\\mathcal{F}} E_F\n\\setminus \\{e_1, \\ldots, e_k\\}$ in a topological order \\textbf{do}\\\\\n\\>\\textbf{if} $\\mathrm{start}(e) \\in V$ \\textbf{then}\\\\\n\\>\\>Choose a linear combination $\\vec{b}^F(e) = \\sum_{p\\in P^F(e)}m_e(p)\\vec{b}(p)$\\\\\n\\>\\>such that $B_t^F$ remains linearly independent for all $t$\\\\\n\\>\\>and $F$ by the method in \\cite[Sect.\\  III.B]{sanders05}. \\` (**)\\\\\n\\>\\textbf{else}\\\\\n\\>\\>$m_e(p) = 1$ for all $p\\in P^F(e)$ and\n$\\vec{b}^F(e) = \\sum_{p\\in P^F(e)}\\vec{b}(p).$\\\\\n\\` $O(k+2\\alpha)$\\\\\n\\>\\textbf{endif}\\\\\n\\textbf{endforeach}\\\\\n\\textbf{return} $\\{ m_e(\\cdot) \\mid \\mathrm{start}(e) \\in V \\}$.\n\\end{tabbing}\n\\caption{Construction algorithm for a network $\\alpha$-error correcting code.\nThe rightmost $O(\\cdot)$ indicates the time complexity executing\nthe step.}\\label{fig2}\n\n\\end{figure}\n\nWe shall analyze the time complexity of the proposed\nalgorithm in Fig.~\\ref{fig2}.\nAs in \\cite{sanders05} we assume that any arithmetic\nin the finite field is $O(1)$ regardless of the field size.\nFirst we analyze that of the initialization part.\nObserve that $|E_F| = |E| + k+ 2|F| = |E| + k+4\\alpha$ because\neach edge in $F$ adds two edges to $E$ and\nthere are $k$ edges from \\change{$s'$} to $s$.\nThe most time consuming part in the initialization\nis construction of edge disjoint paths,\nwhose overall time complexity\nis $O((|E|+k+4\\alpha)|\\mathcal{F}||T|(k+2\\alpha))$.\n\nNext we analyze the time complexity of the main loop.\nBy \\cite[Proof of Lemma 8]{sanders05},\nthe time complexity of choosing the local \\change{encoding}\nvector $m_e(p)$ in Step (**) is \n$O((|\\mathcal{F}||T|)^2 (k+2\\alpha))$,\nwhich is the most time consuming part in the main loop.\nChoice of $m_e(p)$\nis executed for $|E|$ edges starting from a real node in $V$.\nThus, the time complexity of the main loop is\n$O(|E|(|\\mathcal{F}||T|)^2 (k+2\\alpha))$,\nand the overall time complexity is\n$\nO(\n|\\mathcal{F}||T|(k+2\\alpha)[\n|E|+k+4\\alpha + |\\mathcal{F}||T|])\n$.\nNote that $|\\mathcal{F}| = {|E| \\choose 2\\alpha}$.\n\nA sink \\change{decode}s the information by exhaustive search.\nSpecifically the sink enumerates all the possible information\nand all the possible errors for all $F\\in\\mathcal{F}$,\nthen compares the resulting symbols on incoming edges with\nthe actual received symbols by the sink.\nThe computation of the resulting symbols can be\ndone by a matrix multiplication in $O((k+\\alpha)^2)$ time complexity.\nThe number of possible information is $q^k$ and\nthe number of possible errors is $\\sum_{j=0}^{\\alpha}{|E| \\choose j}(q-1)^j$.\nThus, the time complexity of decoding by a sink is\n$O(q^k \\sum_{j=0}^{\\alpha}{|E| \\choose j}(q-1)^j(k+\\alpha)^2)$.\n\n\\section{Variants of the proposed method and its relation to\nthe robust network coding}\nWe shall introduce two variants of the proposed method in this section.\n\n\\subsection{Attaining the refined Singleton bound}\nNetwork error-correcting codes constructed by the proposed\nmethod attains the Singleton bound (Proposition \\ref{lemsingleton}),\nwhile they do not necessarily attains the refined Singleton bound\n(Proposition \\ref{lemsingleton2}).\nYang et~al.\\  \\cite{yang07b} concurrently proposed a construction algorithm\nthat produces a code attaining the refined Singleton bound.\nIn this subsection we modify the proposed method so that\nit can produce a code attaining the refined Singleton bound.\n\nLet $n_t$ be the min-cut from $s$ to $t$,\nand suppose that the source $s$ emits $k$ symbols within unit\ntime interval.\nA sink $t$ can correct $\\alpha$ errors if $2\\alpha \\leq n_t -k$.\nLet $\\mathcal{F}_t = \\{ F\\subset E \\,:\\, |F| = n_t - k\\}$ and\n$\\mathcal{F} = \\bigcup_{t\\in T}\\mathcal{F}_t$.\nFor fixed $F \\in \\mathcal{F}$ and $t\\in T$,\nwe cannot garuantee that there exists $k$ edge disjoint\npaths in Step~\\ref{step2b}.\nFor such $F$, the sink $t$ cannot \\change{decode} information\nwith errors occered at $F$. We exclude $B_t^F$ with such $(t,F)$\nfrom the\nalgorithm.\nNote that if $|F| \\leq n_t - k$ then there always exist\n$k$ edge disjoint paths in Step~\\ref{step2b}.\n\nIn order to attain the refined Singleton bound\nwe keep the linear independence of\nall bases in $\\{B_t^F \\mid t\\in T$, $F\\in\\mathcal{F}$,\n$|F| \\leq n_t - k\\}$ in Step (**) in Fig.~\\ref{fig2}.\nBy the exactly same argument, we see that the produced code attains\nthe refined Singleton bound.\n\nBy almost the same argument as Sect.\\ \\ref{sec:const},\nwe see that the modified proposed algorithm runs\nin time complexity \n$\nO(\n|\\mathcal{F}||T|(k+2\\alpha_\\mathrm{max})[\n|E|+k+4\\alpha_\\mathrm{max} + |\\mathcal{F}||T|])\n$,\nwhere $\\alpha_\\mathrm{max} = \\lfloor (\\max_{t\\in T} n_t - k)\/2\\rfloor$. The required field size for successful execution of the algorithm\nis $|T| \\cdot |\\mathcal{F}|$, and in this case\n$|\\mathcal{F}|$ depends on the structure of the network $(V,E)$.\n\n\\begin{table*}[t!]\n\n\\caption{Comparison among the proposed methods and \\cite{yang07b,jaggi07}.\nWe assumed that the min-cut is $n$ for all $t\\in T$\nand $k = n-2\\alpha$. $I$ denotes the maximum of in-degrees of nodes.}\\label{tab1}\n\\begin{tabular}{|p{0.1\\textwidth}|p{0.1\\textwidth}|p{0.2\\textwidth}|p{0.25\\textwidth}|p{0.2\\textwidth}|}\\hline\n&delay&required field size for the success probability of code construction to be\n$\\geq 1-\\delta$&\ntime complexity of code construction&time complexity of decoding by sinks\\\\\\hline\\hline\nFigure~\\ref{fig2}&none&$|T| {|E| \\choose 2\\alpha}$&\n $O(\n{|E| \\choose 2\\alpha}|T|(k+2\\alpha)[\n|E|+k+4\\alpha + {|E| \\choose 2\\alpha}|T|])$&$O(q^k \\sum_{j=0}^{\\alpha}{|E| \\choose j}(q-1)^j(k+\\alpha)^2)$\\\\\\hline\nSect.\\ \\ref{sec:random}&none&$|E||T|{|E| \\choose 2\\alpha}\/\\delta$&\n$O(I)$&$O(q^k \\sum_{j=0}^{\\alpha}{|E| \\choose j}(q-1)^j(k+\\alpha)^2)$\\\\\\hline\nPaper \\cite{yang07b}&none&$|T|{n + |E| -2 \\choose 2\\alpha}$\n&\n$O(|E| |T| q^k\\sum_{j=0}^{2\\alpha} {|E| \\choose j}(q-1)^j)$&$O(q^k \\sum_{j=0}^{\\alpha}{|E| \\choose j}(q-1)^j(k+\\alpha)^2)$\\\\\\hline\nPaper \\cite{jaggi07}&large&not estimated&$O(I)$&\n$O((n \\times \\mathrm{delay})^3)$\\\\\\hline\n\\end{tabular}\n\n\\end{table*}\n\n\n\nOn the other hand,\nthe time complexity of constructing\nlocal \\change{encoding} vectors by the method of Yang et~al.~\\cite{yang07b}\nis\n\\[\nO\\left( |E| q^k \\sum_{t\\in T} \\sum_{j=0}^{n_t-k} {|E| \\choose j}(q-1)^j\\right),\n\\]\nand the required field size is\n\\[\n\\sum_{t\\in T} {n_t + |E| -2 \\choose n_t-k}.\n\\]\nThe time complexity of the proposed algorithm\ncan be smaller or larger depending on the network structure and $q$\nthan Yang et~al.~\\cite{yang07b}.\nThe required field size of the proposed algorithm\ncan also be smaller or larger depending on the network structure.\nHowever, for the special case $n_t = n$ for all $t\\in T$,\nthe required field size of the proposed method is\nsmaller than Yang et~al.~\\cite{yang07b}.\n\n\n\\subsection{Completely randomized construction}\\label{sec:random}\nBy using the idea in the previous section,\nwe can estimate the success probability of\nconstructing a network error-correcting code\nby randomly choosing local \\change{encoding} vectors\nas follows.\nThe idea behind its proof is almost the same as\n\\cite[Theorem 12]{sanders05}.\nObserve that the random choice of local \\change{encoding} vectors\ncompletely remove the time complexity\nof selecting \\change{encoding} vectors in the centralized manner\nat the expense of larger required field size $q$.\n\n\\begin{proposition}\nSuppose that the source $s$ transmits $k$ symbols within\nunit time interval, and let $\\mathcal{F} =\\{F\\subset  E \\,:$\n$|F| = 2\\alpha\\}$\nbe the set of edges on which errors can occur.\nSuppose also that local \\change{encoding} vector coefficients\nare generated at random independently and uniformly over\n$\\mathbf{F}_q$.\nWith this network error-correcting code,\nall sinks can correct errors in any edge set $F\\in \\mathcal{F}$\nwith probability at least $1-\\delta$\nif $q \\geq |E||T||\\mathcal{F}|\/\\delta$.\n\\end{proposition}\n\n\\noindent\\emph{Proof.} \nFirst pick independent random local \\change{encoding}\nvectors for all edges in the network simultaneously.\nThen pick an error pattern $F\\in\\mathcal{F}$.\nFor this $F$, execute Steps 1 and 2 in page~\\pageref{step1}\nand compute the global \\change{encoding} vectors $\\vec{b}^F(e)$'s belonging\nto $\\mathbf{F}_q^{k+|F|}$.\nFor each cut in the network,\ntest whether $B_t^F$'s are linearly independent for all $t$.\nThis test fails with probability at most $|T|\/q$\nby the proof of \\cite[Theorem 9]{sanders05}\nprovided that this tests succeed on all the upstream cuts\nand $n\\geq k+2\\alpha$.\n\nIn the proposed algorithm in Fig.~\\ref{fig2},\nwe test linear independence of $B_t^F$'s on\n$|E|$ cuts in Step (**), which is sufficient to\ngaruantee the decodability of the information by every sink.\nBy the same reason, for each sink to be able to correct errors in $F$,\none needs to consider linear independence only on\nat most $|E|$ such cuts with random\nchoice of local \\change{encoding} vectors.\nBy the union bound, the probability that\nthe the independence tests fails for any of $|T|$ sinks\nin any of the $|E|$ cuts in any of the $|\\mathcal{F}|$\nerror patters is at most $\\delta$\nif $q \\geq |E||T||\\mathcal{F}|\/\\delta$.\n\\qed\n\nJaggi et~al.~\\cite{jaggi07} do not provide an estimate\non the relation between the success probability of\ntheir algorithm and the field size $q$.\nTheir method \\cite{jaggi07} uses coding among\ndifferent time intervals and thus introduces\ndelays while our methods do not introduce extra delay.\nIn addition to this,\n$\\alpha$-error correcting codes by constructed\nby the proposed methods allow sinks to correct\nless than $\\alpha$ errors, while the method\nin \\cite{jaggi07} does not.\nThe advantage of the method in \\cite{jaggi07}\nover the proposed methods in this paper\nis that their method allows efficient decoding of\ninformation by every sink, while our proposed\nmethods require exhaustive search of transmitted\ninformation.\n\nWe summarize the comparison among the proposed algorithms and\n\\cite{jaggi07,yang07b} in Table~\\ref{tab1}.\n\n\n\n\\subsection{Relation to the robust network coding}\nWe clarify the difference between the robust network\ncoding in \\cite[Sect.\\  V]{koetter03},\\cite[Sect.\\  VI]{sanders05} and\nthe network error-correcting codes with known locations\nof errors \\cite{yang07}.\nA network error correcting codes that can correct errors\non a known locations $F\\subset E$\nis a robust network coding tolerating \nedge failures on $F$.\nHowever, the converse is not always true.\nConsider the network consists of three nodes $\\{s,t,v\\}$\nwith two directed edges from $s$ to $v$ and\none directed edge from $v$ to $t$.\nThe source is $s$ and the sink is $t$.\nThe intermediate node $v$ sends to $t$ the sum of two inputs\nfrom $s$. This network coding tolerate single edge failure\nbetween $s$ and $v$ but cannot correct single error between\n$s$ and $v$.\n\n\n\\section{Concluding remarks}\nIn this paper,\nwe proposed an algorithm constructing\nnetwork error-correcting\ncodes attaining the Singleton bound, and\nclarified its relation to the robust network\ncoding \\cite[Sect.\\  VI]{sanders05}.\n\nThere are several research problems that have\nnot been addressed in this paper.\nFirstly, the proposed deterministic algorithm requires\ntests of linear independence against\n${|E| \\choose 2\\alpha}$ sets consisting of \\change{$k+m_t^F$} vectors,\nwhich is really time consuming.\nIt is desirable to have a more efficient \n\\change{deterministic} construction algorithm.\n\nSecondly, since there seems no structure in the constructed\ncode, the \\change{decoding} of the original information at a sink $t$\nrequires the exhaustive search by $t$\nfor possible information from the source and possible errors.\nIt is desirable to have a code with structure that allows\nefficient decoding.\n\nFinally, the case $|T|=1$ and $|E|=n$ includes the ordinary\nerror correcting codes as\na special case.\nSubstituting $|T|=1$, $|E|=n$ and $2\\alpha = n-k$ into Eq.~(\\ref{qassumption})\ngives $q > {n \\choose n-k}$,\nwhich can be regarded as a sufficient condition\nfor\nthe existence of the MDS linear code.\nOn the other hand, a well-known sufficient condition for\nthe existence of the MDS linear code is $q > n-2$,\nwhich suggests that Eq.~(\\ref{qassumption})\nis loose and that there is a room for improvement\nin Eq.~(\\ref{qassumption}).\n\n\\section*{Acknowledgment}\nThe author thanks for constructive criticisms by\nreviewers that improved the presentation of the results\nvery much. He also thanks Dr.\\ Masazumi Kurihara\nfor pointing out ambiguity in the earlier manuscript.\nHe would like to thank Prof.~Kaoru Kurosawa\nfor drawing his attention to the network error correction,\nProf.\\ Olav Geil,\nProf.\\ Toshiya Itoh, Prof.\\ Tomohiko Uyematsu,\nMr.\\ Akisato Kimura, and Dr.\\ Shigeaki Kuzuoka \nfor helpful discussions.\nHe also would like to thank Dr.\\ Sidharth Jaggi and Mr.\\ Allen Min Tan\nfor informing the papers \\cite{jaggi07,yang07b}.\nPart of this research was conducted during the author's\nstay in the Department of Mathematical Sciences,\nAalborg University.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n Recent advances in generative models including GANs \\cite{NIPS2014_5ca3e9b1, DBLP:conf\/iclr\/BrockDS19, Karras_2019_CVPR, Karras_2020_CVPR, Karras2021}, variational autoencoders (VAEs) \\cite{kingma2013auto, rezende2014stochastic, vahdat2020nvae}, and autoregressive models \\cite{van2016conditional,chen2018pixelsnail,henighan2020scaling} have realized high-quality image generation with great diversity. Diffusion probabilistic models \\cite{sohl2015deep} are introduced to match data distributions by learning to reverse multi-step noising processes. Ho et al. \\cite{NEURIPS2020_4c5bcfec} demonstrate the capability of DDPMs to produce high-quality results. Following works \\cite{song2020improved, dhariwal2021diffusion, nichol2021improved,kingma2021variational} further optimize the noise addition schedules, network architectures, and optimization targets of DDPMs. Besides, classifier guidance is added to realize DDPM-based conditional image generation \\cite{dhariwal2021diffusion}. DDPMs have shown excellent results competitive with GANs \\cite{Karras_2020_CVPR, DBLP:conf\/iclr\/BrockDS19} on datasets including CIFAR-10 \\cite{krizhevsky2009learning}, LSUN \\cite{yu2015lsun}, and ImageNet \\cite{van2016conditional}. Moreover, DDPMs have also achieved compelling results in generating videos \\cite{ho2022video, harvey2022flexible, yang2022diffusion, zhang2022motiondiffuse}, audios \\cite{kong2020diffwave, austin2021structured}, point clouds \\cite{zhou20213d, luo2021diffusion, lyu2021conditional, liu2022let}, and biological structures \\cite{xu2022geodiff,hoogeboom2022equivariant}.  \n\nModern DDPMs depend on large amounts of data to train the millions of parameters in their networks like other generative models, which tend to overfit seriously and fail to produce high-quality images with considerable diversity when training data is limited. Unfortunately, it is not always possible to obtain abundant data under some circumstances. A series of GAN-based approaches \\cite{wang2018transferring, ada,mo2020freeze, wang2020minegan, ewc, ojha2021few-shot-gan, zhao2022closer} have been proposed to adapt models pre-trained on large-scale source datasets to target datasets using a few available training samples (e.g., 10 images). These approaches utilize knowledge from source models to relieve overfitting and achieve improvement in generation quality and diversity. Nevertheless, the performance of DDPMs trained on limited data and effective DDPM-based few-shot image generation approaches remain to be investigated.\n\n \n\nTherefore, we first evaluate the performance of DDPMs trained on small-scale datasets and show that DDPMs suffer similar overfitting problems to other modern generative models when trained on limited data. Then we fine-tune pre-trained DDPMs on target domains using limited data directly. The fine-tuned DDPMs achieve faster convergence and greater generation diversity compared with DDPMs trained from scratch but still fail to maintain the relative distances between generation samples during domain adaptation. To this end, we introduce a pairwise similarity loss to keep the distributions of relative distances in target models similar to source models for greater generation diversity, as shown in Fig. \\ref{pairwise}. The main contributions of this paper can be concluded as:\n\n\\begin{itemize}\n    \\item We make the first attempt to study when do DDPMs overfit as training data become scarce and propose a Nearest-LPIPS metric to evaluate generation diversity.\n    \\item We propose a DDPM-PA approach based on the pairwise similarity loss designed for DDPMs to preserve the relative distances between samples during domain adaptation and achieve great generation diversity.\n    \\item  We demonstrate the effectiveness of DDPM-PA qualitatively and quantitatively on a series of few-shot image generation tasks and show that DDPM-PA achieves better generation quality and diversity than current state-of-the-art GAN-based approaches.\n    \n\\end{itemize}\n\n\n\\section{Related Work}\n\\subsection{Diffusion Denoising Probabilistic Models}\n\\textbf{DDPMs Formulation} Given training images $x_0$ following the distribution $q(x_0)$, DDPMs define a forward noising (diffusion) process $q$ adding Gaussian noises with variance $\\beta_t \\in (0,1)$ at diffusion step $t$ and produce the noised image sequences: $x_1,x _2,...,x_T$ as follows:\n\\begin{align}\n\\label{eq1}\n    q(x_1,x_2,...,x_T|x_0) &:= \\prod_{t=1}^{T} q(x_t|x_{t-1}), \\\\\n\\label{eq2}\n    q(x_t|x_{t-1}) &:= \\mathcal{N}(x_t;\\sqrt{1-\\beta_t}x_{t-1},\\beta_t \\mathbf{I}).\n\\end{align}\nWe can sample an arbitrary step of the noising process conditioned on $x_0$ as follows:\n\\begin{align}\n\\label{eq3}\n        q(x_t|x_0) &= \\mathcal{N}(x_t;\\sqrt{\\overline{\\alpha}_t}x_0,(1-\\overline{\\alpha}_t)\\mathbf{I}), \\\\\n        x_t &= \\sqrt{\\overline{\\alpha}_t}x_0 + \\sqrt{1-\\overline{\\alpha}_t}\\epsilon, \\label{eq4}\n\\end{align}\nwhere $\\alpha_t:=1-\\beta_t$, $\\overline{\\alpha}_t:=\\prod_{s=0}^{t}\\alpha_s$, and $\\epsilon$ represents Gaussian distributions following $\\mathcal{N}(0,\\mathbf{I})$. Based on the Bayes theorem, we have the posterior $q(x_{t-1}|x_t,x_0)$:\n\\begin{equation}\n    q(x_{t-1}|x_t,x_0)=\\mathcal{N}(x_{t-1};\\hat{\\mu}_t(x_t,x_0),\\hat{\\beta}_t\\mathbf{I}),\n\\end{equation}\nwhere $\\hat{\\mu}_t(x_t,x_0)$ and $\\hat{\\beta}_t$ are defined in terms of $x_0, x_t$ and the variance as follows:\n\\begin{align}\n    \\hat{\\mu}_t(x_t,x_0)&:=\\frac{\\sqrt{\\overline{\\alpha}_{t-1}}\\beta_t}{1-\\overline{\\alpha}_t}x_0 + \\frac{\\sqrt{\\alpha_t}(1-\\overline{\\alpha}_{t-1})}{1-\\overline{\\alpha}_t}x_t, \\label{eq6}\\\\ \n    \\hat{\\beta}_t&:=\\frac{1-\\overline{\\alpha}_{t-1}}{1-\\overline{{\\alpha}}_t}\\beta_t.\n\\end{align}\n\nWith a large enough $T$, the noised image $x_T$ almost follows an isotropic Gaussian distribution. Therefore, we can randomly sample a $x_T$ from $\\mathcal{N}(0,\\mathbf{I})$ and apply the reverse distribution $q(x_{t-1}|x_t)$ to reverse the diffusion process and get the sample following $q(x_0)$. DDPMs employ a UNet-based neural network to approximate the reverse distribution $q(x_{t-1}|x_t)$ as follows:\n\\begin{align}\n\\label{eq8}\n    p_{\\theta}(x_{t-1}|x_t):=\\mathcal{N}(x_{t-1};\\mu_{\\theta}(x_t,t),\\Sigma_{\\theta}(x_t,t)).\n\\end{align}\nNaturally, we can train the network to predict $x_0$ and apply it to Equation \\ref{eq6} to parameterize the reverse distribution mean $\\mu_{\\theta}(x_t,t)$. Besides, we can also derive the prediction of $\\mu_{\\theta}(x_t,t)$ combing Equations \\ref{eq4} and \\ref{eq6} as:\n\\begin{align}\n    \\mu_{\\theta}(x_t,t)=\\frac{1}{\\sqrt{\\alpha_t}}(x_t-\\frac{\\beta_t}{\\sqrt{1-\\overline{\\alpha}_t}}\\epsilon_{\\theta}(x_t,t)),\n\\end{align}\n if we train the network as a function approximator $\\epsilon_{\\theta}(x_t,t)$ to predict the noise $\\epsilon$ in Equation \\ref{eq4}. Ho et al. \\cite{NEURIPS2020_4c5bcfec} demonstrate that predicting $\\epsilon$ performs well and achieves high-quality results using a reweighted loss function:\n\\begin{align}\n\\label{loss_simple}\n    \\mathcal{L}_{simple}=E_{t,x_0,\\epsilon}\\left[||\\epsilon-\\epsilon_{\\theta}(x_t,t)||\\right]^2.\n\\end{align}\nIn Ho et al.'s work \\cite{NEURIPS2020_4c5bcfec}, the variance $\\Sigma_{\\theta}(x_t,t)$ is fixed as a constant $\\sigma_t^2 \\mathbf{I}$, where $\\sigma_t^2=\\beta_t$ and is not learned. The network is only trained to learn the model mean $\\mu_{\\theta}(x_t,t)$ through predicting noises with $\\epsilon_\\theta(x_t,t)$. Following works \\cite{nichol2021improved} propose to add an additional optimization term $L_{vlb}$ to optimize the variational lower bound (VLB) and guide the learning of $\\Sigma_{\\theta}(x_t,t)$ as follows:\n\\begin{align}\n    L_{vlb}:&=L_0+L_1+...+L_{T-1}+L_T, \\\\\n    L_0:&=-log\\ p_{\\theta}(x_0|x_1), \\\\\n    L_{t-1}:&=D_{KL}(q(x_{t-1}|x_t,x_0)\\,||\\,p_{\\theta}(x_{t-1}|x_t)), \\\\\n    L_T:&=D_{KL}(q(x_T|x_0)\\,||\\,p(x_T)),\n\\end{align}\nwhere $D_{KL}$ represents the Kullback-Leibler (KL) divergence used to evaluate the distance between distributions.\n\n\\textbf{Fast Sampling} DDPMs need time-consuming iterative processes to realize sampling following Equation \\ref{eq8}. Recent works including DDIM and gDDIM \\cite{song2020denoising, zhang2022gddim} extend the original DDPMs to non-Markovian cases for fast sampling. DPM-solver \\cite{lu2022dpm,lu2022dpm2} presents a theoretical formulation for the solution of probability flow ordinary differential equations (ODEs) and achieves a fast ODE solver needing only 10 steps for DDPM sampling. Diffusion Exponential Integrator Sampler (DEIS) \\cite{zhang2022fast} makes use of an exponential integrator to approximately calculate the solution of ODEs. Karras et al. \\cite{karras2022elucidating} present a design space to identify changes in both the sampling and training processes and achieve state-of-the-art FID \\cite{heusel2017gans} on CIFAR-10 \\cite{krizhevsky2009learning} under a class-conditional setting.\n\n\\textbf{Applications} DDPMs have already been applied to many aspects of applications such as image super-resolution \\cite{li2022srdiff, saharia2022image, rombach2022high,ho2022cascaded}, image translation \\cite{saharia2022palette, ozbey2022unsupervised}, semantic segmentation \\cite{baranchuk2021label, asiedu2022decoder}, few-shot generation for unseen classes \\cite{giannone2022few,sinha2021d2c}, and natural language processing \\cite{austin2021structured, li2022diffusion, chen2022analog}. Besides, DDPMs are combined with other generative models including GANs \\cite{xiao2021tackling, wang2022diffusion}, VAEs \\cite{vahdat2021score, huang2021variational, luo2022understanding}, and autoregressive models \\cite{rasul2021autoregressive, hoogeboom2021autoregressive}. Different from existing works, this paper focuses on model-level, unconditional, few-shot image generation with DDPM-based approaches.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{scratch2.jpg}\n    \\caption{ For small-scale datasets including Babies, Sunglasses, and LSUN Church containing 10, 100, and 1000 images, \\textbf{Left}: samples picked from the small-scale datasets, \\textbf{Right}: samples produced by DDPMs trained on the small-scale datasets from scratch.}\n    \\label{scratch}\n\\end{figure*}\n\n\\subsection{Few-shot Image Generation}\n    Few-shot image generation aims to achieve high-quality generation with great diversity using only a few available training samples. However, modern generative models easily overfit and suffer severe diversity degradation when trained on limited data (e.g., 10 images). They tend to replicate training samples instead of generating diverse images following similar distributions. GAN-based few-shot image generation approaches mainly follow TGAN \\cite{wang2018transferring} to adapt GANs pre-trained on large source domains, including ImageNet \\cite{van2016conditional}, LSUN \\cite{yu2015lsun}, and FFHQ \\cite{Karras_2020_CVPR}, to target domains with limited data. Augmentation approaches \\cite{tran2021data, zhao2020differentiable, zhao2020image, ada} also help improve generation diversity with diverse augmented training samples. BSA \\cite{noguchi2019image} fixes all the parameters except for the scale and shift parameters in the generator. FreezeD \\cite{mo2020freeze} freezes parameters in high-resolution layers of the discriminator to relieve overfitting. MineGAN \\cite{wang2020minegan} uses additional fully connected networks to modify noise inputs for the generator. EWC \\cite{ewc} makes use of elastic weight consolidation to regularize the generator by making it harder to change the critical weights which have higher Fisher information values. CDC \\cite{ojha2021few-shot-gan} proposes a cross-domain consistency loss and patch-level discrimination to build a correspondence between source and target domains. DCL \\cite{zhao2022closer} utilizes contrastive learning to push away the generated samples from real images and maximize the similarity between corresponding image pairs in the source and target domain. The proposed DDPM-based approach follows similar strategies to adapt models pre-trained on large source domains to target domains. Our experiments show that DDPMs are appropriate for few-shot image generation tasks and can achieve better results than current state-of-the-art GAN-based approaches in quality and diversity. \n\n\n\\section{DDPMs Trained on Small-scale Datasets}\n\\label{section3}\n\nTo evaluate the performance of DDPMs when training data is limited, we train DDPMs on small-scale datasets containing various numbers of images from scratch. We analyze generation diversity qualitatively and quantitatively to study when do DDPMs overfit as training samples decrease. \n\n\\textbf{Basic Setups} We sample 10, 100, and 1000 images from FFHQ-babies (Babies), FFHQ-sunglasses (Sunglasses) \\cite{ojha2021few-shot-gan}, and LSUN Church \\cite{yu2015lsun} respectively as small-scale training datasets. The image resolution of all the datasets is set as $256\\times 256$. We follow the model setups in prior works \\cite{nichol2021improved, dhariwal2021diffusion} used for LSUN $256^2$ \\cite{yu2015lsun}. The max diffusion step $T$ is set as 1000. Our experiments are carried out on $\\times 8$ NVIDIA RTX A6000 GPUs (with 48 GB of memory each). We use a learning rate of 1e-4 and a batch size of 48. We train DDPMs for $40K$ iterations on datasets containing 10 or 100 images and $60K$ iterations on datasets containing 1000 images, respectively.\n\n\\textbf{Qualitative Evaluation} In Fig. \\ref{scratch}, we visualize the generated samples of DDPMs trained from scratch on small-scale datasets and provide some training samples for comparison (more generation and training samples are provided in Appendix \\ref{appendix_scratch}). We observe that DDPMs overfit and tend to replicate training samples when the datasets are limited to 10 or 100 images. Since some training samples are flipped in the training process as a step of data augmentation, we can also find some generated images symmetric to the training samples. While for datasets containing 1000 images, DDPMs can generate samples following similar distributions of training samples instead of replicating them. The overfitting problem is relatively alleviated. As shown in the bottom row of Fig. \\ref{scratch}, all kinds of babies, people wearing sunglasses, and churches different from training samples can be found in the generated images.\n\n\n\\textbf{Quantitative Evaluation} LPIPS \\cite{zhang2018unreasonable} is proposed to evaluate the perceptual distances between images. We propose a Nearest-LPIPS metric based on LPIPS to evaluate the generation diversity of DDPMs trained on small-scale datasets. More specifically, we first generate 1000 images randomly and find the most similar training sample having the lowest LPIPS distance to each generated sample. Nearest-LPIPS is defined as the LPIPS distances between generated samples and the most similar training samples in correspondence, averaged over all the generated samples. If a generative model reproduces the training samples exactly, the Nearest-LPIPS metric will have a score of zero. Larger Nearest-LPIPS values indicate lower replication rates and greater diversity compared with training samples. \n\n\n\\begin{table}[htbp]\n\\centering\n\\begin{tabular}{c|c|c|c}\nNumber of Samples & Babies & Sunglasses & Church \\\\\n\\hline\n$10$ & $0.2875$ & $0.3030$ & $0.3136$ \\\\\n$100$ & $0.3152$ & $0.3310$ & $0.3327$ \\\\\n$1000$  & $\\pmb{0.4658}$ & $\\pmb{0.4819}$ & $\\pmb{0.5707}$ \\\\\n\\hline\n$10$ (+flip) & $0.1206$ & $0.1217$ & $0.0445$\\\\\n$100$ (+flip) & $0.1556$ & $0.1297$ & $0.1177$ \\\\\n$1000$ (+flip) & $\\pmb{0.4611}$ & $\\pmb{0.4726}$ & $\\pmb{0.5625}$ \\\\\n\\end{tabular}\n\\caption{Nearest-LPIPS ($\\uparrow$) results of DDPMs trained from scratch on several small-scale datasets.}\n\\label{nearlpips}\n\\end{table}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{result1.jpg}\n    \\caption{DDPM-based image generation samples on 10-shot FFHQ $\\rightarrow$ Babies, FFHQ $\\rightarrow$ Sketches, and LSUN Church $\\rightarrow$ Haunted houses.}\n    \\label{result1}\n\\end{figure*}\n\nWe provide the Nearest-LPIPS results of DDPMs trained from scratch on small-scale datasets in the top part of Table \\ref{nearlpips}. For datasets containing 10 or 100 images, we have lower Nearest-LPIPS values. While for datasets containing 1000 images, we get measurably improved Nearest-LPIPS values. To avoid the influence of generated images symmetric to training samples, we flip all the training samples as supplements to the original datasets and recalculate the Nearest-LPIPS metric. The results are listed in the bottom part of Table \\ref{nearlpips}. With the addition of flipped training samples, we find apparently lower Nearest-LPIPS values for datasets containing 10 or 100 images. However, we get almost the same Nearest-LPIPS results for DDPMs trained on larger datasets containing 1000 images, indicating that these models can generate diverse samples different from the original or symmetric training samples.\n\n\nBased on the above analysis of DDPMs trained from scratch on limited data, we conclude that it becomes harder for DDPMs to learn the representations of the datasets as training data become scarce. With extremely limited training samples like 10 or 100 images, DDPMs overfit and can only replicate them but fail to match the data distributions or generate diverse results different from the training samples. \n\n\n\n\n\n\n\n\\section{Few-shot Pairwise DDPM Adaptation}\nFor large-scale datasets with great diversity, DDPMs can learn the representations of datasets and generate high-quality and diverse images following similar distributions to training samples. However, when the datasets are limited to 10 or 100 samples, DDPMs trained from scratch also need large amounts of iterations to generate reasonable images but still lack diversity and can only replicate the training samples as illustrated in Sec. \\ref{section3}. Unfortunately, there is no guarantee of obtaining abundant data in many cases, such as artists' paintings and some other corner cases.\n\nTo realize DDPM-based few-shot image generation, we first fine-tune DDPMs pre-trained on large-scale source datasets on target domains using limited data directly. The fine-tuned models only need about $3K$ iterations to converge. As shown in the middle row of Fig. \\ref{result1}, they can produce diverse samples for target domains utilizing only 10 training samples. However, there still exists room for greater generation diversity. For example, similar hairstyles and facial expressions can be found in the samples produced by the fine-tuned DDPM trained on FFHQ $\\rightarrow$ Babies.\n\n\n\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{degrade.jpg}\n    \\caption{Samples synthesized from fixed noise inputs by the fine-tuned DDPM on 10-shot FFHQ $\\rightarrow$ Amedeo's paintings throughout training.}\n    \\label{degrade}\n\\end{figure}\n\n\n\n\nCompared with pre-trained models, the diversity degradation of fine-tuned models mainly comes from the shortened relative distances between generated samples. As shown in Fig. \\ref{degrade}, two samples synthesized from fixed noise inputs by the fine-tuned DDPM become more and more similar (e.g., eyes, ears, and facial expressions) throughout training. Therefore, we design a pairwise similarity loss to preserve the relative distances between generated samples during domain adaptation and propose the DDPM-PA approach based on that to achieve greater generation diversity.\n\n\n\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{pairwise.jpg}\n    \\caption{The proposed DDPM-PA adds a pairwise similarity loss $\\mathcal{L}_{pair}$ to keep the relative distances between samples generated by the target models similar to the source models. The pairwise similarity between predicted images $\\tilde{x}_0^{i}, \\tilde{x}_0^{j}\\ (0\\leq i,j \\leq N)$ is converted into probability distributions for $\\mathcal{L}_{pair}$ computation. Taking the 10-shot FFHQ $\\rightarrow$ Amedeo's paintings as an example, DDPM-PA can generate diverse samples sharing similar styles with training samples utilizing the knowledge provided by the source model pre-trained on FFHQ.}\n    \\label{pairwise}\n\\end{figure*}\n\n\\subsection{Pairwise Similarity Loss}\n\\label{42}\n\nWe illustrate the pairwise similarity loss in Fig. \\ref{pairwise} using 10-shot FFHQ $\\rightarrow$ Amedeo's paintings as an example. The weights in the target model $\\epsilon_{tar}$ are initialized to the source model $\\epsilon_{sou}$ and adapted to target domains. To construct N-way probability distributions for each image, we sample a batch of noised images $\\left\\lbrace x_t^{n} \\right\\rbrace_{n=0}^{N}$ by randomly adding Gaussian noises to training samples $x_0\\sim q(x_0)$ following Equations \\ref{eq1} and \\ref{eq2}. Then the source and target models are applied to predict the fully denoised images $\\left\\lbrace \\tilde{x}_0^{n} \\right\\rbrace_{n=0}^{N}$. We have the prediction of $\\tilde{x}_0$ in terms of $x_t$ and $\\epsilon_{\\theta}(x_t,t)$ as follows:\n\\begin{align}\n\\label{eq11}\n    \\tilde{x}_0 = \\frac{1}{\\sqrt{\\overline{\\alpha}_t}}x_t-\\frac{\\sqrt{1-\\overline{\\alpha}_t}}{\\sqrt{\\overline{\\alpha}_t}}\\epsilon_{\\theta}(x_t,t),\n\\end{align}\nwhich can be derived from Equation \\ref{eq4}. Cosine similarity is employed to measure the relative distances between the predicted samples $\\tilde{x}_0$.  The probability distribution for $\\tilde{x}_0^{i}\\ (0\\leq i \\leq N)$ in the source and target models can be expressed as follows:\n\\begin{align}\n    p_{i}^{sou} = Softmax(\\left\\lbrace sim(\\tilde{x}_{0_{sou}}^{i},\\tilde{x}_{0_{sou}}^{j})\\right\\rbrace_{\\forall i\\neq j}), \\\\\n    p_{i}^{tar} = Softmax(\\left\\lbrace sim(\\tilde{x}_{0_{tar}}^{i},\\tilde{x}_{0_{tar}}^{j})\\right\\rbrace_{\\forall i\\neq j}),\n\\end{align}\nwhere $sim$ denotes cosine similarity. Then we have the pairwise similarity loss $\\mathcal{L}_{pair}$ as follows:\n\\begin{align}\n    \\mathcal{L}_{pair}(\\epsilon_{sou},\\epsilon_{tar}) = \\mathbb{E}_{t,x_0,\\epsilon} \\sum_{i} D_{KL} (p_{i}^{tar}\\,||\\, p_{i}^{sou}),\n\\end{align}\nwhere $D_{KL}$ represents KL-divergence. The pairwise similarity loss guides the target models to preserve the relative distances between samples and keep similar distributions to source models, leading to greater generation diversity. \n\n\n\n\nThe overall loss function of DDPM-PA is the weighted sum of three terms, including $\\mathcal{L}_{simple}$, $\\mathcal{L}_{vlb}$, and $\\mathcal{L}_{pair}$: \n\\begin{align}\n\\label{loss}\n    \\mathcal{L} = \\mathcal{L}_{simple} + \\lambda_1 \\mathcal{L}_{vlb} + \\lambda_2 \\mathcal{L}_{pair}.\n\\end{align}\nWe follow prior works \\cite{nichol2021improved} to set $\\lambda_1$ as 0.001 to avoid $\\mathcal{L}_{vlb}$ from overwhelming $\\mathcal{L}_{simple}$. The proposed pairwise similarity loss $\\mathcal{L}_{pair}$ is added to relieve overfitting and preserve generation diversity during adaptation when training data is limited. We empirically find $\\lambda_2$ ranging between 0.2 and 5.0 to be effective for few-shot adaptation setups. \n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{result2.jpg}\n    \\caption{DDPM-PA image generation samples on 10-shot FFHQ $\\rightarrow$ Raphael's paintings, FFHQ $\\rightarrow$ Sunglasses, and LSUN Church $\\rightarrow$ Landscape drawings.}\n    \\label{result2}\n\\end{figure*}\n\n\\begin{table*}[htbp]\n\\centering\n\\setlength{\\tabcolsep}{2mm}{\n\\begin{tabular}{l|c|c|c|c|c}\n   Approaches & \\makecell[c]{FFHQ $\\rightarrow$ \\\\ Sketches} & \\makecell[c]{FFHQ $\\rightarrow$ \\\\ Babies} & \\makecell[c]{FFHQ $\\rightarrow$ \\\\ Raphael's paintings} & \\makecell[c]{LSUN Church $\\rightarrow$ \\\\ Haunted houses} & \\makecell[c]{LSUN Church $\\rightarrow$ \\\\ Landscape drawings} \n \\\\\n\\hline\nTGAN \\cite{wang2018transferring} & $0.394 \\pm 0.023$ & $0.510 \\pm 0.026$ & $0.533 \\pm 0.023$ & $0.585 \\pm 0.007$ & $0.601 \\pm 0.030$ \\\\\nTGAN+ADA \\cite{ada} & $0.427 \\pm 0.022$ & $0.546 \\pm 0.033$ & $0.546 \\pm 0.037$ &  $0.615 \\pm 0.018$ & $0.643 \\pm 0.060$ \\\\\nFreezeD \\cite{mo2020freeze} & $0.406 \\pm 0.017$ & $0.535 \\pm 0.021$ & $0.537 \\pm 0.026$ & $0.558 \\pm 0.019$ & $0.597 \\pm 0.032$ \\\\\nMineGAN \\cite{wang2020minegan} & $0.407 \\pm 0.020$ & $0.514 \\pm 0.034$ & $0.559 \\pm 0.031$ & $0.586 \\pm 0.041$ & $0.614 \\pm 0.027$   \\\\\nEWC \\cite{ewc} & $0.430 \\pm 0.018$ & $0.560 \\pm 0.019$ & $0.541 \\pm 0.023$ & $0.579 \\pm 0.035$ & $0.596 \\pm 0.052$ \\\\\nCDC \\cite{ojha2021few-shot-gan} & $0.454 \\pm 0.017$ & $0.583 \\pm 0.014$ & $0.564 \\pm 0.010$ &  $0.620 \\pm 0.029$ & $0.674 \\pm 0.024$ \\\\\nDCL \\cite{zhao2022closer} & $0.461 \\pm 0.021$ & $0.579 \\pm 0.018$ & $0.558 \\pm 0.033$ & $0.616 \\pm 0.043$ & $0.626 \\pm 0.021$ \\\\\n\\hline\nFine-tuned DDPMs & $0.473 \\pm 0.022$ & $0.513 \\pm 0.019$ & $0.466 \\pm 0.018$ & $0.590 \\pm 0.045$ &  $0.666 \\pm 0.044$ \\\\\nDDPM-PA (ours) & $\\pmb{0.509 \\pm 0.054}$ & $\\pmb{0.603 \\pm 0.017}$ &  $\\pmb{0.579 \\pm 0.027}$ & $\\pmb{0.627 \\pm 0.050}$ & $\\pmb{0.686 \\pm 0.032 }$  \\\\\n\\end{tabular}}\n\\caption{Intra-LPIPS ($\\uparrow$) results of DDPM-based approaches and GAN-based baselines on 10-shot image generation tasks adapted from the source domain FFHQ and LSUN Church. Standard deviations are computed across 10 clusters (the same number as training samples). The proposed DDPM-PA achieves state-of-the-art performance in generation diversity compared with modern GAN-based approaches.}\n\\label{intralpips}\n\\end{table*}\n\n\n\n\n\n\n\n\\subsection{Quality and Diversity Evaluation}\n\\label{422}\n\n\\textbf{Basic Setups} To demonstrate the effectiveness of DDPM-PA, we evaluate it with a series of few-shot image generation tasks using extremely few training samples (10 images). The performance of DDPM-PA on relatively abundant data is added in Appendix \\ref{appendix_adaptation}. We choose FFHQ \\cite{Karras_2020_CVPR} and LSUN Church \\cite{yu2015lsun} as source datasets and train DDPMs from scratch on these two datasets for $300K$ and $250K$ iterations as source models. As for the target datasets, we employ 10-shot Sketches \\cite{wang2008face}, FFHQ-babies (Babies), FFHQ-sunglasses (Sunglasses) \\cite{Karras_2020_CVPR}, and face paintings by Amedeo Modigliani and Raphael Peale \\cite{yaniv2019face} in correspondence to the source domain FFHQ. Besides, 10-shot Haunted houses and Landscape drawings are used as the target datasets in correspondence to LSUN Church. The model setups are consistent with the experiments on small-scale datasets in Sec. \\ref{section3}. We train DDPM-PA models for $4K$-$5K$ iterations with a batch size of 24 on $\\times 8$ NVIDIA RTX A6000 GPUs.\n\n\n\\textbf{Evaluation Metrics} We follow Ojha et al.'s work \\cite{ojha2021few-shot-gan} to use Intra-LPIPS for the evaluation of generation diversity. To be more specific, we generate 1000 images and assign them to one of the training samples with the lowest LPIPS \\cite{zhang2018unreasonable} distance. Intra-LPIPS is defined as the average pairwise LPIPS distance within members of the same cluster averaged over all the clusters. If a model exactly replicates the training samples, its Intra-LPIPS will have a score of zero. Larger Intra-LPIPS values correspond to greater generation diversity. We fix the noise inputs for DDPM-based approaches and GAN-based baselines, respectively, to synthesize samples for fair comparison of generation diversity in terms of Intra-LPIPS.\n\nFID \\cite{heusel2017gans} is widely used to evaluate the generation quality of generative models by computing the distribution distance between generated images and training datasets. However, FID would become unstable and unreliable when it comes to datasets containing a few samples (e.g., 10-shot datasets used in this paper). Therefore, we provide visualized samples to evaluate the generation quality of DDPM-PA compared with prior works.\n\n\\textbf{Baselines}\nSince few prior works realize few-shot image generation with DDPM-based models, we employ 7 GAN-based baselines sharing similar targets with us to adapt pre-trained models to target domains using only a few available samples for comparison: TGAN \\cite{wang2018transferring}, TGAN+ADA \\cite{ada}, FreezeD \\cite{mo2020freeze}, MineGAN \\cite{wang2020minegan}, EWC \\cite{ewc}, CDC \\cite{ojha2021few-shot-gan}, and DCL \\cite{zhao2022closer}. All the methods are implemented based on the same StyleGAN2 \\cite{Karras_2020_CVPR} codebase. DDPMs directly fine-tuned on limited data are included for comparison as well. The StyleGAN2 models and DDPMs trained on the large source datasets share similar generation quality and diversity (see more details in Appendix \\ref{appendix_source}).\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{sunglasses.jpg}\n    \\caption{10-shot image generation samples on FFHQ $\\rightarrow$ Sunglasses. All the visualized samples of GAN-based approaches are synthesized from fixed noise inputs (rows 1-7). Visualized samples of the fine-tuned DDPM and DDPM-PA are synthesized from fixed noise inputs as well (rows 8-9). Our approach generates high-quality results with fewer blurs and artifacts and achieves considerable generation diversity.}\n    \\label{sunglass}\n\\end{figure*}\n\n\\textbf{Qualitative Evaluation}\nAs shown in the bottom row of Fig. \\ref{result1}, we visualize the samples of DDPM-PA on 10-shot FFHQ $\\rightarrow$ Babies, FFHQ $\\rightarrow$ Sketches, and LSUN Church $\\rightarrow$ Haunted houses. Compared with fine-tuned DDPMs, DDPM-PA produces more diverse samples different from the training samples under the guidance of $\\mathcal{L}_{pair}$. For example, DDPM-PA generates babies having more diverse hairstyles and facial expressions. More visualized samples of other adaptation setups are provided in Fig. \\ref{result2}. We observe that DDPM-PA generates samples containing hats that cannot be found in training samples when adapting FFHQ to Babies and Sunglasses. The adaptation from LSUN Church to Haunted houses and Landscape drawings retains all kinds of building structures. Moreover, DDPM-PA also performs better on learning target distributions. The target models are encouraged to learn common features of limited training samples instead of generating samples similar to one of them. As shown in the samples of FFHQ $\\rightarrow$ Sketches, DDPM-PA generates images having a more similar style to the training samples while maintaining diversity.\n \n\nFig. \\ref{sunglass} shows the results of GAN-based baselines and DDPM-based approaches on 10-shot FFHQ $\\rightarrow$ Sunglasses. For fair comparison, we fix the noise inputs for GAN-based and DDPM-based approaches, respectively. We observe that GAN-based approaches generate samples containing unnatural blurs and artifacts, while DDPM-based approaches achieve more realistic results. DDPM-PA can generate more diverse images different from the training samples than the fine-tuned DDPM, including hairstyles, the color of sunglasses, et al. Samples produced by DDPM-PA models trained for different iterations and more visualized results comparison on 10-shot FFHQ $\\rightarrow$ Babies, FFHQ $\\rightarrow$ Raphael's paintings, and LSUN Church $\\rightarrow$ Landscape drawings can be found in Appendix \\ref{appendix_results}. \n\n\n\\textbf{Quantitative Evaluation}\nWe provide the Intra-LPIPS results of DDPM-PA under a series of 10-shot adaptation setups in Table \\ref{intralpips}. Benefiting from the pairwise similarity loss $\\mathcal{L}_{pair}$, DDPM-PA realizes a superior improvement of Intra-LPIPS compared with fine-tuned DDPMs. Besides, DDPM-PA outperforms state-of-the-art GAN-based approaches on Intra-LPIPS, indicating its strong capability of maintaining generation diversity. Additional Intra-LPIPS results of other adaptation setups are added in Appendix \\ref{appendix_results}. \n\n\n\n\\subsection{Ablation Analysis}\n\\label{43}\nWe evaluate the generation diversity of DDPM-PA with different $\\lambda_2$, the weight of the proposed $\\mathcal{L}_{pair}$ using 10-shot FFHQ $\\rightarrow$ Raphael's paintings as an example. The quantitative results are listed in Table \\ref{ablation}. We add visualized comparison results in Appendix \\ref{appendix_ablation}. DDPM-PA models with larger $\\lambda_2$ values tend to produce more diverse results and achieve greater diversity. However, too large $\\lambda_2$ values may cause $\\mathcal{L}_{pair}$ to overwhelm $\\mathcal{L}_{simple}$ and prevent the adapted DDPMs from learning the distributions of target domains, leading to negative visual effects and degraded diversity.\n\n\\begin{table}[htbp]\n    \\centering\n    \\small\n    \\begin{tabular}{c|c|c|c}\n        $\\lambda_2$ & $0.01$ & $0.04$ & $0.2$  \\\\\n        \\hline\n        Intra-LPIPS & $0.47 \\pm 0.04 $ & $0.50 \\pm 0.04$ & $0.51 \\pm 0.04$ \\\\\n        \\hline\n        $\\lambda_2$ & $1.0$ & $5.0$ & $10.0$ \\\\\n        \\hline\n        Intra-LPIPS & $0.52 \\pm 0.03$ & $\\pmb{0.58 \\pm 0.03}$ &  $0.55 \\pm 0.05$ \\\\\n    \\end{tabular} \n    \\caption{Intra-LPIPS ($\\uparrow$) results of DDPM-PA trained on 10-shot FFHQ $\\rightarrow$ Raphael's paintings with different $\\lambda_2$.}\n    \\label{ablation}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nThis work focuses on DDPM-based few-shot image generation. We first evaluate the performance of DDPMs on small-scale datasets. When trained on a few samples from scratch, DDPMs need large amounts of iterations to generate reasonable results but can only replicate training samples instead of generating diverse images. Next, we fine-tune pre-trained DDPMs on target domains using limited data directly but still fail to achieve results competitive in diversity. Therefore, we present DDPM-PA using a pairwise similarity loss to preserve the relative distances between samples during domain adaptation. DDPM-PA realizes significant improvement in generation diversity and outperforms current state-of-the-art GAN-based approaches. In addition, DDPM-PA avoids generating blurs or artifacts and achieves more realistic results. We consider our work to be an important step towards more data-efficient diffusion models. The limitations are discussed in Appendix \\ref{appendix_limitations}.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Deep Hawkes Process}\n\\label{sec:DHP_DHP}\nIn this section, we propose that the Deep Hawkes Process (DHP) be used\nto concurrently model the order book event timings and associated event\ntypes. By assimilating it into the order arrival process, the market\nmakers have control over the sending of different orders at specific\npoints in time. The basic idea behind our approach is to view the\nconditional intensity of the Hawkes process as a nonlinear deterministic\nfunction of past history, and to use DLSTM to automatically learn a\nhigh-dimensional representation from the data. We believe that\ncapturing the constraints of the Hawkes process and recurrent network\narchitecture enables a replication of the success of the natural language\nprocess \\citep{Du2016, Mei2017} and time series prediction \\citep{Sagheer2019A, Sagheer2019B} in market making. \n\n\\subsection{Deep Long Short-Term Memory}\n\nDespite the success of recurrent marked temporal point processes and the\nNeural Hawkes process across disciplines, the conventional shallow\nrecurrent network architecture and its variant long short-term memory\n(LSTM) might be inadequate when it comes to modelling noisy\nasynchronous order book events. Lately, the DLSTM architecture \\citep{Sagheer2019A, Sagheer2019B, Zhu2016} used in action modelling\nand multivariate time series forecasting validated its precedence over\ntraditional LSTM architecture. The architecture of the DLSTM is same\nas the previously introduced LSTM, apart from the fact that it involves\nmultiple LSTM layers stacked on top of each other. \n\n\\subsection{DLSTM-SDAE}\n\nIn an ideal setting, the performance of the DLSTM model proved to be\nempirically better than existing contemporary statistical and deep modules. However, as the non-linear variables are modelled in such a way as to be scaled up, the overall learning of the models suffers greatly due to the back-propagation algorithm trapped within multiple local minima \\citep{Sagheer2019B}. This may be the biggest hurdle when it comes to modelling limit order book events that comprise complex, asynchronous non-linear multivariate time series. The ultra-noisy order book data also makes processing more challenging. To circumvent the limitations of the conventional stacked LSTM model, we use the stacked denoising auto-encoder (SDAE) together with DLSTM. The SDAE enables the deep neural networks with multiple nonlinear hidden layers to learn complex features from noisy limit order book data \\citep{Li2019, Vincent2010} and resolves the random weight initialization of LSTM's units problem in DLSTM \\citep{Sagheer2019B}. Figure \\ref{fig:DLSTM} shows the proposed DLSTM- based SDAE architecture for DHP.\n\n\\begin{figure}[H]\n\\centering\n\\resizebox{\\textwidth}{!}{\\begin{tikzpicture}[roundnode\/.style={circle,draw,thick, align=center, text width=6mm, minimum size=6mm}]\n\t \n    \t\n\t\n\t\n\t\\node(h3) at (8,8) {$h^{N}_{t+1}$};\n    \\node(h2) at (4,8) {$h^{N}_{t}$};\n\t\\node(h1) at (0,8) {$h^{N}_{t-1}$};\t\n\t\n\t\\node[draw] (LSTM9) at (8,6) {LSTM  N};\n\t\\node[draw] (LSTM8) at (8,4) {LSTM  2};\n\t\\node[draw] (LSTM7) at (8,2) {LSTM  1};\n    \\node[draw] (LSTM6) at (4,6) {LSTM  N};\t\n\t\\node[draw] (LSTM5) at (0,6) {LSTM  N};\n\t\\node[draw] (LSTM4) at (4,4) {LSTM  2};\n\t\\node[draw] (LSTM3) at (0,4) {LSTM  2};\n\t\\node[draw] (LSTM2) at (4,2) {LSTM  1};\n\t\\node[draw] (LSTM1) at (0,2) {LSTM  1};\n\t\n\t\\node[draw] (SDAE1) at (0,0) {SDAE};\n\t\\node[draw] (SDAE2) at (4,0) {SDAE};\n\t\\node[draw] (SDAE3) at (8,0) {SDAE};\n\t\n\t\\node(H19) at (7.5,5) {$h^{2}_{t+1}$};\n\t\\node(H18) at (7.5,3) {$h^{1}_{t+1}$};\n\t\n\t\\node(H17) at (10,6) {$h^{N}_{t+1}$};\n\t\\node(H16) at (10,4) {$h^{2}_{t+1}$};\n\t\\node(H15) at (10,2) {$h^{1}_{t+1}$};\n\t\\node(H14) at (2.0,6.3) {$h^{N}_{t-1}$};\n\t\\node(H13) at (2.0,4.3) {$h^{2}_{t-1}$};\n\t\\node(H12) at (2.0,2.3) {$h^{1}_{t-1}$};\n\t\\node(H11) at (3.5,5) {$h^{2}_{t}$};\n\t\\node(H10) at (3.5,3) {$h^{1}_{t}$};\n\t\\node(H8) at (-0.5,5) {$h^{2}_{t-1}$};\n\t\\node(H7) at (-0.5,3) {$h^{1}_{t-1}$};\n\t\\node(H6) at (6,6.3) {$h^{N}_{t}$};\n\t\\node(H5) at (-2,6) {$h^{N}_{t-2}$};\n\t\\node(H4) at (6,4.3) {$h^{2}_{t}$};\n\t\\node(H3) at (-2,4) {$h^{2}_{t-2}$};\n\t\\node(H2) at (6,2.3) {$h^{1}_{t}$};\n\t\\node(H1) at (-2,2) {$h^{1}_{t-2}$};\t\n\t\n\t\\draw[->] (LSTM9) to (H17);\n\t\\draw[->] (H5) to (LSTM5);\n\t\\draw[->] (LSTM8) to (H16);\n\t\\draw[->] (H3) to (LSTM3);\n\t\\draw[->] (LSTM7) to (H15);\n\t\\draw[->] (H1) to (LSTM1);\n\t\\draw[->] (LSTM5) to (LSTM6);\n\t\\draw[->] (LSTM5) to (LSTM6);\n\t\\draw[->] (LSTM3) to (LSTM4);\n\t\\draw[->] (LSTM1) to (LSTM2);\n\t\\draw[->] (LSTM9) to (h3);\n\t\\draw[->] (LSTM6) to (h2);\n\t\\draw[->] (LSTM5) to (h1);\n\t\\draw[dotted, very thick] (LSTM4) to (LSTM6);\n\t\\draw[dotted, very thick] (LSTM3) to (LSTM5);\n\t\\draw[dotted, very thick] (LSTM8) to (LSTM9);\n\t\n\t\\draw[->] (LSTM7) to (LSTM8);\n\t\\draw[->] (LSTM6) to (LSTM9);\n\t\\draw[->] (LSTM4) to (LSTM8);\n\t\\draw[->] (LSTM2) to (LSTM7);\n\t\\draw[->] (LSTM2) to (LSTM4);\n\t\\draw[->] (LSTM1) to (LSTM3);\n\t\n\t\n\t\\draw[->] (SDAE3) to (LSTM7);\n\t\\draw[->] (SDAE2) to (LSTM2);\n\t\\draw[->] (SDAE1) to (LSTM1);\n\t\n\t\n\t\\node[roundnode](x1) at (0,-2) {$\\bf{x}_{t-1}$};\n\t\\node[roundnode](x2) at (4,-2) {$\\bf{x}_{t}$};\n\t\\node[roundnode](x3) at (8,-2) {$\\bf{x}_{t+1}$};\n\t\n\t\\draw[->] (x1) to (SDAE1);\n\t\\draw[->] (x2) to (SDAE2);\n\t\\draw[->] (x3) to (SDAE3);\n\t\n\\end{tikzpicture}\n}\n\\caption{The DLSTM-SDAE architecture.}\n\\label{fig:DLSTM}\n\\end{figure}\n\nAs shown in Figure \\ref{fig:DLSTM}, the reconstructed order book data is  denoised by SDAEs layers in in DLSTM-SDAE architecture. In this method, at the first layer the input $\\bf{x_t}$ is corrupted into $\\bf{\\tilde{x_t}}$ using stochastic mapping $\\bf{\\tilde{x_t}} \\sim \\mathcal{S_{\\mathcal{D}}}(\\bf{\\tilde{x_t}}\\mid \\bf{x_t} )$. Then, the autoencoder maps  corrupted input $\\bf{\\tilde{x_t}}$ to a hidden representation $\\mathit{h} = f_{\\theta}(\\bf{\\tilde{x_t}})$ with encoder $f_{\\theta}(\\bf{\\tilde{x_t}})= (W \\bf{\\tilde{x_t}} + b)$. Lastly, the decoder $g_{\\theta^{'}}$ reconstruct $\\bf{z}=g_{\\theta^{'}}(\\bf{\\tilde{x_t}})$ from the hidden representation $\\mathit{h}$. The parameters $\\theta$ and $\\theta^{'}$ are trained using stochastic gradient descent to minimize reconstruction error measured in the squared error loss $L_{2}(\\bf{x},\\bf{z})=\\norm{\\bf{x} -\\bf{z} }^{2}$. Once mapping is learned, the high-level hidden state $\\mathit{h}$ is applied for training the next layer. For detail learning procedure in SDAE, please refer to seminal paper on the subject \\citep{Vincent2010}.\n\nAt time $t$, the denoised input $\\bf{x_t}$ from SDAE is then passed to first layer of LSTM together with previous hidden state $h^{1}_{t-1}$. The hidden state at time $t$, $h^{1}_{t}$ is calculated using recursive LSTM procedure discussed in Equation \\ref{eqn:lstm}. Its is then moved to next time step and LSTM layers. In the second layer, the hidden state $h^{1}_{t}$ and the previous $h^{2}_{t-1}$ is used to compute $h^{2}_{t}$ and procedure repeats until last layer is complied.\n\n\\subsection{Model Formulation}\n\\label{ss:MF}\n\nLet $\\{t_n, \\varkappa_n, k_n\\}_{n \\in \\mathbb{N}}$ be a stream of order book event, where $t_n$ are times of occurrence of an event, its component $\\varkappa_n$, and their corresponding mark $\\{k_n\\}_{n \\in \\mathbb{N}}$ in $k_n := \\{1, \\dots , K\\}$. Then, the probability that the next event occurs at time $t_n$ is of type $k_n$ is $\\mathbb{P}\\{(t_n,\\varkappa_n, k_n) \\mid \\mathcal{H}_n, (t_n - t_{n-1})\\} dt$. We are interested in the model to predict next event stream $\\{t_n, \\varkappa_n, k_n\\}$ given a past history of event $k_n$, evaluate its likelihood and simulate the next event stream by learning from the past event stream. Equation \\ref{eqn:hawkes_mod_a} shows the associated intensity function\nof the DHP with relaxed positivity constraints: \n\n\\begin{equation}\n\\lambda_k(t) = f_{k}(\\bf{w}_{k}^{\\top}\\bf{h}(t))\n\\label{eqn:hawkes_mod_a}\n\\end{equation}\nThe hidden state $\\bf{h}(t)$ is updated from the memory sell $\\bf{c}(t)$ as in Equation \\ref{eqn:hawkes_mod_b}.\n \\begin{equation}\n  \\bf{h}(t) = \\bf{o}_{n} \\odot \\phi (\\bf{c}(t)) \\text{ for } t \\in (t_{n-1}, t_{n}]\n  \\label{eqn:hawkes_mod_b}\n  \\end{equation}\n  \nThe life of interval $(t_{n-1},t_n]$ is determined by the next event $k_n$ at $t_n$, DLSTM reads $\\{t_n, \\varkappa_n, k_n\\}$ and updates the current memory cells $\\bf{c}(t)$ to $\\bf{c}_{n+1}$, associated with hidden state $\\bf{h}(t_n)$. The other parameters of the DLSTM are recursively updated according to the Equation \\ref{eqn:dlstm}.\n\n\\begin{eqnarray}\n\\label{eqn:dlstm}\n\\begin{aligned}\n\\!&{\\bf{i}}_{n+1}= \\sigma \\left( {{\\bf{W}}_{xi}}{{\\bf{x}}_{n}} + {{\\bf{W}}_{hi}}{\\bf{h}}(t_n) + {{\\bf{W}}_{ci}}{\\bf{c}} (t_n) + {\\bf{b}}_i \\right), \\\\ \n\\!&{\\bf{f}}_{n+1} = \\sigma \\left( {{\\bf{W}}_{xf}}{{\\bf{x}}_{n}} + {{\\bf{W}}_{hf}}{\\bf{h}} (t_n) + {{\\bf{W}}_{cf}}{\\bf{c}} (t_n) + {\\bf{b}}_f \\right), \\\\\n\\!&{\\bf{\\overline{c}}}_{n+1} = \\phi \\left( {{\\bf{W}}_{xc}}{{\\bf{x}}_{n}}\\! +\\! {{\\bf{W}}_{hc}}{\\bf{h}}(t_n) \\!+\\! {\\bf{b}}_c \\right), \\\\ \n\\!&{\\bf{c}}_{n+1} = {\\bf{f}}_{n+1}\\!\\odot {\\bf{c}}(t_n)\\!+ {\\bf{i}}_{n+1}\\!\\odot {\\bf{\\overline{c}}}_{n+1}, \\\\ \n\\!&{\\bf{\\widehat{c}}}_{n+1} = {\\bf{\\widehat{f}}}_{n+1}\\!\\odot {\\bf{\\widehat{c}}} (t_n)\\!+ {\\bf{\\widehat{i}}}_{n+1}\\!\\odot {\\bf{\\overline{c}}}_{n+1}, \\\\ \n\\!&{\\bf{o}}_{n+1} = \\sigma \\left( {{\\bf{W}}_{xo}}{{\\bf{x}}_{n}} + {{\\bf{W}}_{ho}}{\\bf{h}}(t_n) + {{\\bf{W}}_{co}}{\\bf{c}}(t_n) + {\\bf{b}}_o \\right), \\\\\n\\!&{\\bf{\\Bar{\\Bar{s}}}}_{n+1} = f \\left( {{\\bf{W}}_{xd}}{{\\bf{x}}_{n}} + {{\\bf{W}}_{hd}}{\\bf{h}}(t_n) + {{\\bf{W}}_{cd}}{\\bf{c}} (t_n) + {\\bf{b}}_d \\right),\n\\end{aligned}\n\\end{eqnarray}\n\nwhere $\\bf{x}_n$ is $n^{th}$ input vector represented by one hot encoding of new order book event $k_n$; the activation functions $\\sigma \\left(x \\right)$ \/ $\\phi \\left(x \\right)$ are sigmoid \/  hyperbolic tangent function, respectively; ${\\bf{W}}_{A B}$ (e.g., ${\\bf{W}}_{c i}$) is the weight matrix from the memory cell to input gate vector; ${\\bf{b}}_{B}$ denotes the bias term of $B$ with $B \\in \\{i,f,c,o,d\\}$, $\\bf{\\Bar{\\Bar{s}}}$ is exponential decay parameter and $f(x)=s \\log(1+\\exp(x\/s))$, $s > 0$ is scaled soft plus function. As it can be seen in the Equation \\ref{eqn:dlstm}, the parameters are updated using the hidden state $\\bf{h}(t_n)$ at time $t_n$, succeeding its decay over interval $t_{n} - t_{n-1}$ rather previous hidden state (Equation \\ref{eqn:lstm}). The memory cell ${\\bf{c}}(t)$ on the interval $(t_{n-1}, t_{n}]$ follows power-law distribution decaying from ${\\bf{c}}_{n+1}$ to ${\\bf{\\widehat{c}}}_{n+1}$ and defined as:\n\n\\begin{equation}\n\\label{eqn:c_decay}\n\\bf{c}(t) = {\\bf{\\widehat{c}}}_{n+1} + \\left( {\\bf{c}}_{n+1} - {\\bf{\\widehat{c}}}_{n+1} \\right)  \\left( \\left( t - t_n \\right)^{{- \\bf{\\Bar{\\Bar{p}}}}_{n+1}} \\right) \\text{ for } t \\in (t_n,t_{n+1}]\n\\end{equation}\n\nThe DHP, with novel discrete update of stacked LSTM state, allows the model to capture a delayed response, fits non-interacting event pairs, and copes with partially observed event streams. \\cite{Mei2017} discuss in detail these benefits of the neural version of the model. In order to ensure mathematical tractability, we have illustrated parameter updates for one of the layers of stacked LSTM, but this can be easily extended to deep architecture. For example, the hidden state ${\\bf{h}}^b$ at block $b$ in stacked LSTM is recursively computed from $b = 1\\,\\text{:}\\,N$ and $t = 1\\,\\text{:}\\,T$ using ${\\bf{h}}^{b}_{t}= \\sigma ({{\\bf{W}}_{h^{b-1}h^{b}}}{\\bf{h}}^{b-1}_t + {{\\bf{W}}_{h^{b}h^{b}}}{\\bf{h}}^{b}_{t-1} + {\\bf{b}}_h)$.\n\n\\subsection{Feedback Loop Exploration}\n\nThe empirical results \\citep{Morariupatrichi2018, Gonzalez2017} indicate the existence of feedback loop between the order flow and the shape of the LOB, together with the self- and cross-excitation effects. In order to efficiently capture this feedback effect in high-dimensional parameter space, we infuse the feedback loop exploration process into the DLSTM-SDAE architecture, as discussed in Figure \\ref{fig:DLSTM}. In connection with designing the deep network architecture and appropriate regularisation, we take into consideration that the network can automatically explore distinct feedback loops for different types of events and their codependency. For example, the feedback effect of market buy and sell orders on price, volume and the bid-ask spread of LOB. \n\nConsequently, we design a fully connected deep network in which\neach neuron represents an LOB state or feedback effect of the preceding\nlayer, in order to automatically explore the feedback loop. Furthermore,\nthe neurons in the same layer are partitioned into $\\mathfrak{B}$ blocks to take into account different combinations of feedback loops. The corresponding regularisation is incorporated into the loss function and described by:\n\n\\begin{equation}\n\\label{eqn:loss}\n\\!\\mathfrak{L}\\!=\\!\\min_{{\\bf{W}}_{xB}}{\\mathfrak{L}}\\!+{\\lambda}_1\\!\\!\\sum_{\\substack{B \\in S}}{\\left\\Vert {\\bf{W}}_{xB}\\right\\Vert}_{1}\\!\\!+\\!{\\lambda}_2\\!\\sum_{\\substack{B \\in S}}\\sum_{{\\mathfrak{b}}=1}^{\\mathfrak{B}}  \\left\\Vert {{\\bf{W}}_{xB,{\\mathfrak{b}}}}^T \\right\\Vert _{2,1}\\!,\n\\end{equation}\n\nwhere $\\mathfrak{L}$ is the loss function of the DLSTM, and other two terms are feedback loop regularization applied to each block in the network \\citep{Zhu2016}. ${\\bf{W}}_{xB}\\!\\in {\\mathbb{R}}^{N_{N} \\times K_{J}} $ is weight matrix, with number of neurons $N_{N}$ and inputs dimension $K_{J}$. The $S$ characterizes the set of gates and cell in LSTM neurons for each block in DLSTM. Lastly, the $\\left\\Vert {\\bf{W}}\\right\\Vert_{2,1}\\!\\!=\\!\\!\\sum_i\\!\\!\\sqrt{\\sum_j\\!\\!w_{i,j}^2}$ is a structural $\\ell_{21}$ norm. The loss function (Equation \\ref{eqn:loss}) was solved by using Adaptive Moment Estimation (Adam) \\citep{Kingma2015}. Adam optimization is an augmentation to stochastic gradient descent that is memory efficient, extremely insensitive to hyperparameters, works with sparse gradients, is appropriate for non-stationary objectives, and learns the learning rates itself on a per-parameter basis. It is well suited for highly noisy and\/or sparse-gradient order book data.\n\n\\subsection{Parameters Estimation}\n\nGiven a collection of sequence of order book events $\\mathcal{S}$, $\\{t_n, \\varkappa_n, k_n\\}_{n \\in \\mathbb{N}}$, the the log-likelihood of the model can be expressed as the sum of the log-intensities of lapsed event minus an integral of the aggregate intensities over the whole interval observed till $T$:\n\\begin{equation}\\label{eqn:loglike}\n\\mathcal{L}\\!=\\!\\  \\sum_{n: t_n \\leq T}\\!\\!\\log \\lambda_{k_n}(t_n)\\! - \\int_{t=0}^{T}\\!\\!\\lambda(t) dt,\n\\end{equation} \n    \nThe parameter $(k,t)$ is estimated maximizing $\\mathcal{L}$ using Adam \\citep{Kingma2015} and Monte Carlo methods \\citep{Mei2017}. The frequently used thinning algorithm adapted from multivariate Hawkes process is then used to sample random sequence from the model.\n\\section*{Abstract}\n\nHigh-frequency market making is a liquidity-providing trading strategy that simultaneously generates many bids and asks for a security at\nultra-low latency while maintaining a relatively neutral position. The\nstrategy makes a profit from the bid-ask spread for every buy and sell\ntransaction, against the risk of adverse selection, uncertain execution\nand inventory risk. We design realistic simulations of limit order\nmarkets and develop a high-frequency market making strategy in which\nagents process order book information to post the optimal price, order\ntype and execution time. By introducing the Deep Hawkes process\nto the high-frequency market making strategy, we allow a feedback loop to be created between order arrival and the state of the limit order book,\ntogether with self- and cross-excitation effects. Our high-frequency market making strategy accounts for the cancellation of orders that influence order queue position, profitability, bid-ask spread and the value of the order. The experimental results show that our trading agent outperforms the baseline strategy, which uses a probability density estimate of the fundamental price. We investigate the effect of cancellations on market quality and the agent's profitability. We validate how closely the simulation framework approximates reality by reproducing stylised facts from the empirical analysis of the simulated order book data.\n\n \n\n\\section{Introduction}\nTechnological innovations and regulatory initiatives in the financial\nmarket have led to the traditional exchange floor being displaced by the\nelectronic exchange. The electronic exchange is a fully automated\ntrading system programmed to incisively enforce order precedence,\npricing and the matching of buy and sell orders. Each order's pricing,\nsubmission and execution is performed using sophisticated algorithmic\ntrading strategies, which account for 85\\% of the equity market's trading\nvolume \\citep{Mukerji2019}. High-frequency trading (HFT, or high-frequency trader), a subset of algorithmic trading, is characterised by exceptionally high speeds, minuscule timeframes and complex programs for initiating and liquidating positions \\citep{SEC2014}. The critical discussion on the role of HFT in a fragmented market has been reignited after the Flash Crash of 6 May 2010 \\citep{Kirilenko2017}. This systemic intra-day anomaly only lasted for a couple of minutes, but temporarily wiped away a trillion dollars in market value. The analysis of agents resolved transaction level data in the E-mini by \\cite{Kirilenko2017} also looks at the behaviour of market makers, whose inventory dynamics remain stationary in conditions of fluctuating liquidity. Even though the market design of E-mini has no high-frequency market maker liability, unlike equity markets, this seminal paper\\citep{Kirilenko2017} gave a boost to research aimed at understanding high-frequency market making or other liquidity-providing strategies in an algorithmic trading setting.\n\nMarket making is a liquidity-providing trading strategy that quotes numerous bids and asks for a security in anticipation of making a profit from a bid-ask spread, while maintaining a relatively neutral position \\citep{Chakrqborty2011}. The high-frequency market making strategy can be characterised as subset of HFT that uses latency, at a scale of nanoseconds, to trade in a fragmented market \\citep{Menkveld2013}. The growing literature reports that the market makers provide quality liquidity, improve market quality, contribute to price efficiency and\nhave a positive but moderate welfare effect \\citep{Kirilenko2017, Brogaard2014, Menkveld2013}. However, there is another strand in the literature that argues that the quality of liquidity is deceptive. The orders are characterised as phantom liquidity, which quickly disappears before other market participants can access it. The optimal design of market making strategies is therefore an important question for practical applicability, market design and security exchange regulations.\n\nThe research on market making spans numerous disciplines, including finance \\citep{Ho1981, Glosten1985, Hara1986, Avellaneda2008, Gueant2013, Cartea2014, Ait2017}, agent-based modeling \\citep{Das2005, Preis2006, Wah2017, Chao2018}, and artificial intelligence \\citep{Spooner2018, Ganesh2019, Kumar2020}. Inspired by seminal work of \\cite{Ho1981} and its mathematical formulation \\citep{Avellaneda2008}, the quintessential research in finance considers market making as a stochastic optimal control problem. In a simplistic setting, the market is modelled as a stochastic process, in which market makers try to maximise the expected utility of their profit and loss under inventory constraints \\citep{Gueant2013}. In parallel to inventory-based models, \\cite{Glosten1985} proposed information-based models, in which market makers face adverse selection risk emerging from informed traders. The unrealistic assumptions placed on market models to mathematically extract the market maker's asset pricing forces researchers to look beyond stochastic optimal control approaches.\n\nMarket making has also been extensively investigated in agent-based modelling (ABM, or agent-based model) literature \\citep{Das2005, Wah2017}. The ABMs in market making evolved from zero intelligence to an intelligent variant by incorporating order book microstructure for order placement, execution and pricing policy. For example, \\cite{Toke2011} reinforced the zero-intelligence market maker model with order arrival following mutually exciting Hawkes processes. The Hawkes process has\nbeen exhaustively used in an empirical estimation and calibration of\nmarket microstructure models deemed essential for designing optimal\nmarket-making strategies \\citep{Toke2011, Alan2018, Morariupatrichi2018}. In these models, the arrival rate of orders is not dependent on the state of the limit order book. However, the empirical results suggest the existence of feedback loop between order arrival and the state of the limit order book, together with self- and cross-excitation effects, for which current models fail to account \\citep{Gonzalez2017, Morariupatrichi2018}. In addition, the Hawkes process constrains the parametric specification for conditional intensity, which\nlimits the model's eloquence. To tackle the parametric specification\nproblem, \\cite{Mei2017} proposed the Neural Hawkes process, in which the Hawkes process is generalised by calculating the event intensities from the hidden state of a long short-term memory (LSTM). Despite the success of the Neural Hawkes process in natural language processing \\citep{Mei2017}, the facile LSTM architecture might be inadequate when it comes to modelling noisy, asynchronous order book events.\n\nIn recent years, deep learning has made significant inroads into high-\nfrequency finance. The Convolutional Neural Network (CNN)\narchitecture and its variants were used to model price-formation\nmechanisms using order book events as input \\citep{Sirignano2019, Cont2019, Tashiro2019, Tsantekidis2017}. However, the CNN architectures are not sophisticated enough to capture self- and cross-excitation effects in the limit order book (LOB) \\citep{Zhang2019}. The deep long-short term memory (DLSTM) architecture performs a hierarchical processing of complex order book events, and as such is able to capture the temporal structures of LOB \\citep{Sagheer2019A, Sagheer2019B}. However, training the DLSTM model directly through stochastic gradient descents, initialised with random parameters, may have led to the backpropagation algorithm being trapped within multiple local minima \\citep{Sagheer2019B, Vincent2010}. To circumvent the aforementioned limitations, the literature proposed an unsupervised pre-training of each layer and a stacking of many convolutional layers \\citep{Vincent2010}. We use Stacking Denoising Autoencoders (SDAEs) together with DLSTM to resolve the random weight initialisation problem in base architecture \\citep{Sagheer2019B}. In addition, SDAEs are quite effective at filtering out noisy order-level data at minuscule resolutions.\n\nThe exemplary predictive performance of deep-learning models has\nencouraged researchers to augment order book data with agent-based\nartificial market simulation, for the purpose of investigating algorithmic trading strategies \\citep{Maeda2019}. The success of the model is dependent on the simulation framework of the financial market being close to realism. However, algorithmic trading research is still waiting for market simulators that could be used for developing, training, and testing algorithms in a manner similar to classic Atari 2600 games simulator \\citep{Mnih2015}. In this paper, we develop realistic simulations of the financial market and use them to design a high-frequency market making agent using the Deep Hawkes process (DHP). The DHP models the streams of order book events by constructing a self-exciting multivariate Hawkes process and a limit order state process, which are coupled and interact with each other. Based on a long stream of high-frequency transaction-level order book data for the different events (e.g. buy, sell, cancel, etc), the high-frequency market makers use DHP\nto accurately predict every held-out event.\n\n\\subsection{Contribution}\n\nThis paper is the first to incorporate DHP into the market making\nstrategy, which allows feedback loops between order arrival and the\nstate of the limit order book, together with self- and cross-excitation\neffects. We extend the neurally self-modulating multivariate point\nprocess \\citep{Mei2017} to the deep framework by stacking SDAE with DLSTM, resulting in DLSTM-SDAE. The SDAE resolves the problem associated with weight initialisation, multiple local minima and ultra-noisy order book data that the stacked recurrent network fails to address. Our approach outperforms the NHP in predicting the next order type and its time. The gained predictive power helps agents to outperform the benchmark market making strategy, and uses a probability density estimate of the fundamental price. We outline our contribution below:\n\\begin{enumerate}\n\\item We designed a multi-asset simulation framework that is scalable and\ncan augment markets of substantial size. The framework is built on\nrealistic market architecture, interface kernels, a matching engine and\nthe Financial Information eXchange (FIX) protocol. \n\\item We are first to introduce a feedback loop between order arrival and the state of the order book using DHP in the high-frequency market making setting.\n\\item We investigate the predictive and trading performance of the agents\nwith the benchmark.\n\\item We explore the effect of cancellation on order queue position, agent's profitability, bid-ask spread, and value of order relative to queue position, in order to verify the existing empirical findings \\citep{Moallemi2017, Dahlstrm2018}.\n\\end{enumerate}\n\n\\subsection{Structure}\n\nThe paper is organized as follow. Section \\ref{sec:DHP_B} presents the background to the limit order book, market making, Hawkes process, Neural Hawkes process, and long short-term memory. Section \\ref{sec:DHP_LR} explores the research stream in market making strategies. Section \\ref{sec:DHP_DHP} explains the novel deep Hawkes process. Section \\ref{sec:DHP_MASF} illustrates the multi-agent simulation framework. Section \\ref{sec:DHP_E} elaborates on the experimental configuration. Section \\ref{sec:DHP_R} provides the results of the experiments. Section \\ref{sec:DHP_C} presents our conclusions.\n\\section{Background}\n\\label{sec:DHP_B}\nIn this section, we first introduce the limit order book together with basic definitions. We then briefly present examples of market making\nstrategy, and review essential tools for designing and investigating the\nclassical market making model. We start with the Hawkes process, the\nassumptions of which are violated in the financial markets. To check the\nmissing links of the Hawkes process, we describe the Neural Hawkes\nprocess. Finally, we discuss the LSTM framework, which represents a\ndivergence from selecting a parametric form for the conditional intensity\nin the Hawkes process variants.\n\n\\subsection{Limit Order Books} \n\nThe LOB is a centralised database for outstanding orders submitted by traders to buy or sell a specified number of securities on an exchange.  Figure \\ref{fig:DHP_LOB} illustrates an example of the reconstructed LOB for Apple securities traded on NASDAQ. The smallest increment by which the price of the security can move is called a \\emph{tick}. The highest price at time $t$ for which there is outstanding buy order is called \\emph{bid price}($168.60$), while the lowest sell price is called \\emph{ask price} ($168.50$). The \\emph{bid-ask spread} ($0.10$) at time $t$ is defined as the difference between the ask and bid prices. The \\emph{mid price} (150.05) at time $t$ is the arithmetic average of the bid and the ask. For an in-depth review of definitions, mechanisms and\nnomenclature, please refer to \\cite{Gould2013}.\n\n\\begin{figure}[H]\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{img\/LOBA}}\n\\caption{ Limit order book at NASDAQ for Apple Inc. (AAPL) on March 29th, 2018.}\\label{fig:DHP_LOB}\n\\end{center}\n\\end{figure}\n\nIn an exchange, the traders can primarily submit three different types\nof orders: \\emph{limit orders}, \\emph{market orders}, and \\emph{cancellation orders}. A \\emph{limit order} is an order to buy or sell a particular number of securities at a specified price or better. A \\emph{market order} is an order to immediately buy or sell a certain number of securities at the best available price in the LOB. The arrival of market orders is instantly executed against the best available price in the limit order book. Orders that exceed the size available at the best price automatically spill over to the next best available price. Unlike market orders, the limit orders rest in the order book, pending\nexecution against a market order or being partially or fully \\emph{canceled} by traders. The limit orders posted near to bid and ask are executed instantly, but may be prolonged if market prices diverge from the requested price. Market makers utilise this attribute to design optimal trading strategies \\citep{Spooner2018}.  \n \nIn an exchange, several orders posted by traders can have the same\nprice at a given time $t$. To effectively match the orders within each\ndiscrete price level, the LOBs employ various priority mechanism\nalgorithms. The algorithm most commonly employed by various\nexchanges is \\emph{price-time}. In this case, for buy or sell orders, the matching algorithms give priority to the orders with the highest or lowest price. In the event of ties, preference is given to orders with the earliest submission time relative to other orders \\citep{Gould2013}. Other prominent priority mechanism algorithms include \\emph{pro-rata} and \\emph{price-size}. Under the \\emph{pro-rata} algorithms, the ties at the given price are broken by distributing orders according to the depth in the LOB, while \\emph{price-size} mechanism algorithms break the ties by giving higher priority to larger orders. Our study makes use of \\emph{price-time} priority mechanism algorithms, as these encourage market makers to submit limit orders when designing their trading strategies. \n\n\\subsection{High-Frequency Market Making}\nMarket making is a liquidity-providing trading strategy that quotes\nnumerous bids and asks for a security in anticipation of making a profit\nin the form of bid-ask spread, while maintaining a relatively neutral\nposition \\citep{Chakrqborty2011}. The modern market making strategy or high-frequency market making strategy can also be characterized as subset of HFT, where traders uses \"lightning fast with a latency (inter-message time) upper bound of 1 millisecond, only engages in proprietary trading, generates many trades (it participates in 14.4\\% of all trades, split almost evenly across both markets), and starts and ends most trading days with a zero net position\" \\citep{Menkveld2013}. The foundation of designing optimal market making strategies is dependent on optimising order-handling costs, adversely selected bid or ask quote costs, and non-zero position risk-averse costs\\citep{Menkveld2013}.\n\nIn order to better understand high-frequency market making, let us consider a simple example from LOB illustrated in Figure \\ref{fig:DHP_LOB}. The market making strategy places a buy order at $100.00$ and sell order at $100.10$. The execution of the both orders will give the traders profit of $00.10$, which is also the spread. As millions of market maker's trades are executed each second, they can amass huge profits.   \n\nHowever, high-frequency market making strategies are always exposed to the risk of adverse price movements, uncertain executions and adverse\nselections \\citep{Penalva2015}. To avoid the above risks, the high-frequency market making strategies seek to compete in the market through lower order-processing costs, fast matching engines and low latency. The fast matching engine reduces the adverse selection risk by facilitating a\nmarket-making strategy to immediately update quotes on the arrival\norder book information, while low latency reduces the search for market\nvenues to mitigate a costly non-zero inventory position \\citep{Penalva2015, Menkveld2013}. The array of sophisticated statistical or machine learning models used on the real-time order book data serve to\noptimise the transaction costs, which are directly related to profitability.\n\n\\subsection{Hawkes Process}\nTo set the stage for Hawkes process, we first explicate key concept related to counting process, point process, and conditional intensity function from the classical literature \\citep{Gao2018, Alan2018, Bacry2015, Embrechts2011}.\n\nLet us consider a positive sequence of event arrival time, $\\{t_n\\}_{n \\in \\mathbb{N}}$, such that $\\forall \\ n \\in \\mathbb{N}$, $t_n < t_{n+1}$, defined on probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$ with almost surely finite, right-continuous step function defined for all $t \\in \\mathbb{R}_+$, and complete information filtration $(\\mathcal{F}_{t})_{t\\geq 0}$, such that $\\mathbb{F} = \\mathcal{F}^N(t)$, $t\\geq 0$. Then the counting process $N(t)$ and its analogous point process $L(t)$ is defined as:\n\n\\begin{equation}\nN(t)=\\sum_{n \\in \\mathbb{N}} \\mathbb{I}_{t_n \\leq t} \\quad \\text{and} \\quad L(t)=\\sum_{n \\in \\mathbb{N}} \\ell_{n}\\cdot \\mathbb{I}_{t_n \\leq t}\n\\end{equation}\nwhere $\\mathbb{I}_{\\{.\\}}$ is the indicator function and $\\{ \\ell_{n}: n \\ge 1 \\}$ is a sequence of non-negative random variable having independent and identical distribution.\n\nIn academic literature, counting and point process terminology is often indistinguishable. The reader is expected to infer the nature of the\nprocess depending on the context. For example, point process are often characterized by the distribution function of the\noccurrence of the event conditioned to the past, but problems associated with the conditional arrival distribution mean that this is not very practical. Instead, the conditional intensity function is used. For counting process $N(t)$ with associated history $\\mathcal{H}(t)$ and adapted to a filtration $\\mathbb{F}$, we define the conditional intensity as: \n\n\\begin{equation}\\label{eq:CIF}\n    \\lambda(t|\\mathbb{F}) = \\lim_{h \\rightarrow 0} \\mathbb{E} \\bigg[ \\frac{(N(t+h) - N(t))|\\mathcal{H}(t)}{h} \\bigg| \\mathbb{F} \\bigg]\n\\end{equation}\n\nWe now define Hawkes process characterized by intensity $\\lambda(t)$ with respect to its natural filtration as:\n\n\\begin{equation}\\label{eq:HP}\n    \\lambda(t|\\mathbb{F}) = \\mu(t) + \\int_{-\\infty}^{t} \\phi(t-\\tau) \\ {dN}(\\tau)  \n\\end{equation}\nwhere $\\mu$ is an the baseline intensity and $\\phi$ is a non-negative kernel function such that $||\\phi||_1 = \\int_0^{\\infty}\\phi(\\tau)d\\tau < 1$. Prominent kernel function  used in finance is exponential decay $\\phi(t) = \\alpha e^{-\\beta t}$, where parameter $\\alpha$ represents previous event weight and  $\\beta$ past event duration. \n\nNow, we consider $N (t) = \\{N_t^i\\}_{i=1}^{M}$ as the M-dimensional  counting process, and its analogous point process as $\\{L_t^i\\}_{i=1}^{M}$. Similarly, we define the multidimensional Hawkes process as:\n\\begin{equation}\\label{eq:MHP}\n    \\lambda_{i}(t|\\mathbb{F})=\\mu_i+\\sum_{j=1}^{M}\\int_0^t \\phi_{i,j}(t-\\tau)d N_j(\\tau),\n    \\end{equation}\nwhere, $\\mu = \\{\\mu_i\\}_{i=1}^{M}$ a baseline intensity vector and $\\phi(t) =  \\{\\phi_{i,j}(t)\\}_{i,j=1}^{M}$ a matrix-valued kernel that is component-wise non-negative, causal, and $L^1$-integrable \\citep{Bacry2015}.\n\nWe can also enrich the Equation \\ref{eq:MHP} by associating each event with time of event $t_n$, its component $\\varkappa _n$ and mark $k_n$. For example, modeling the trades performed at event time $t_n$ with different volume $k_n$ and drawdown intensity $\\varkappa_n$ \\citep{Bacry2015}.  The vector intensity function of the multivariate marked Hawkes process will be defined as: \n\n\\begin{equation}\n    \\lambda_{i}(t,n |\\mathbb{F})=\\mu_i+\\sum_{j=1}^{M} \\int_0^t \\phi_{i,j}(t-\\tau)\\psi_{j}(n)d N_j(\\tau \\times \\delta),\n    \\end{equation}\nwhere, $\\mu = \\{\\mu_i\\}_{i=1}^{M}$ a exogenous intensities vector, $\\phi(t) =  \\{\\phi_{i,j}(t)\\}_{i,j=1}^{M}$ a matrix-valued kernel and $\\psi_{j}(n)$ a impact function of marks.\n\nThe properties of the Hawkes process can be characterized thoroughly in an analytical manner due to the linear structure of the stochastic intensity \\citep{Alan2018, Bacry2015}.The linear properties of a Hawkes process enable linear predictions of the models, given their base intensity and the kernels as parameters (the kernels are non-parametrically calibrated from the data). A Hawkes process can be appropriately approximated as auto-regressive process, wiener processes, and clustering representation \\citep{Bacry2015}. Simply put by \\cite{Mei2017}, \" the Hawkes process supposes that past events can temporarily raise the probability of future events, assuming that such excitation is positive, additive over the past events, and exponentially decaying with time\". In a simplified setting, these properties might be useful for modelling constrained processes, but might not be applicable to real world examples. For example, at large scales, the price formation process's microscopic variables, as they relate to order book events, do not diffuse toward Wiener processes. Similarly, the massive cancellation of orders in LOB might inhibit price rather than exciting it.\n\n\\subsection{Neural Hawkes Process} \\label{ss:NHP}\n\nThe Neural Hawkes Process was (NHP) introduced to fill the gaps in the Hawkes process's unrealistic assumptions. Building on the earlier\nformulation, the positivity constraints on baseline intensity vector $\\mu$, kernels $\\phi$, and decay rate $\\delta$ limit the eloquentness of Hawkes process. It fails to capture inhibition and inherent inertia effect, which are characteristics of realistic financial market \\citep{Mei2017}.\n\nLet $\\{t_n, \\varkappa_n, k_n\\}_{n \\in \\mathbb{N}}$ is a event streams, where $t_n$ are times of occurrence of an event, its component $\\varkappa_n$, and their corresponding mark $\\{k_n\\}_{n \\in \\mathbb{N}}$ in $k_n:= \\{1, \\dots ,K\\}$. Then, the probability of incidence of next event at time $t_n$ of type $k_n$ is $\\mathbb{P}\\{(t_n,\\varkappa_n, k_n) \\mid \\mathcal{H}_n, (t_n - t_{n-1})\\} dt$. The associated intensity function conditioned on the past events $\\overline{h}$ for self-exciting multivariate point process or Hawkes process with exponentially decaying kernel function is:\n\\begin{equation}\n    \\lambda_k(t)= \\mu_k + \\sum_{\\overline{h}: t_{\\overline{h}} < T}= \\alpha_{k_{\\overline{h}},k} \\exp (-\\beta_{k_{\\overline{h}},k} (t-t_{\\overline{h}}) ),\n    \\end{equation}\nThe inhibition and inherent inertia effect are introduced in the self-modulating model, where we  relax positivity constraints on parameters $\\alpha$ and $\\mu$. The negative total resultant activation is then passed through non-linear transfer function (e.g. rectified linear unit (ReLU) function, softplus function etc.) such that:\n\n\\begin{equation}\n    \\lambda_k(t) = f_k(\\tilde{\\lambda}_k(t)),\n    \\end{equation}\n\n\\begin{equation} \\label{eq:NHP}\n    \\tilde{\\lambda}_k(t)= \\mu_k + \\sum_{\\overline{h}: t_{\\overline{h}} < t} \\alpha_{k_{\\overline{h}},k} \\exp (-\\beta_{k_{\\overline{h}},k} (t-t_{\\overline{h}}) ).\n    \\end{equation}\nwhere $\\mu_k < 0$, $\\alpha_{j,k} < 0$ allows inertia and inhibition effect respectively \\citep{Mei2017}.\n    \nThe summation in Equations \\ref{eq:NHP} places a restriction on the $\\tilde{\\lambda}_k(t)$, where past events have an independent and additive influence. This deviates from reality, which is characterised by the existence of complex dependence between the intensities in terms of number of order event types and past event timings \\citep{Bacry2015}.  \\cite{Mei2017} proposed the \\emph{Neural Hawkes Process}  to learn and predict complex dependency by replacing the summation with a novel recurrent neural network. In this novel process, the hidden state vector $\\mathbf{h}(t)$ controls the dynamics of time varying event intensities, which in turn depends on a vector $\\mathbf{c}(t)$ of memory cells in a continuous-time long short-term memory \\citep{Hochreiter1997}.\n \n\\subsection{Long Short-Term Memory} \\label{ss:lstm}\n\nThe central idea of LSTM is the use of {\\em memory cell}, which overcomes the problem associated with {\\em vanishing gradient} \\citep{Arras2019}. The memory cell of LSTM is a complex unit, built from alike nodes in a distinct connectivity pattern, with the novel inclusion of multiplicative nodes, represented in figure \\ref{fig:lstm}. A typical LSTM memory cell architecture contains a cell input activation vector ${{x}}_t$, an input gate ${{i}}_t$, a forget gate ${{f}}_t$, a cell ${{c}}_t$, an output gate ${{o}}_t$ and an output response $h_t$. The distinctive feature of the LSTM approach, input gate and forget gate, govern the information flow into and out of the cell according to gate logic. Whereas, the output gate controls the amount of information flow from the cell to the output ${{h}}_t$. A self-connected recurrent edge with fixed unit weight in the memory cell ensures that error can flow across many time steps without vanishing or exploding.\n\n\\begin{figure}[H]\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{img\/lstm}}\n\\caption{ LSTM Memory Cell \\citep{Zhu2016}.}\\label{fig:lstm}\n\\end{center}\n\\end{figure}\n\nAt each time step $t$, the recursive computation in the LSTM model proceeds according to the following equations:\n\n\\newcommand{\\operatorname{tanh}}{\\operatorname{tanh}}\n\\newcommand{\\mathring} %{\\bar}{\\mathring}\n\n\\begin{eqnarray}\n\\label{eqn:lstm}\n\\begin{aligned}\n\\!&{\\bf{i}}_t = \\sigma \\left( {{\\bf{W}}_{xi}}{{\\bf{x}}_{t}} + {{\\bf{W}}_{hi}}{\\bf{h}}_{t-1} + {{\\bf{W}}_{ci}}{\\bf{c}}_{t-1} + {\\bf{b}}_i \\right), \\\\ \n\\!&{\\bf{f}}_t = \\sigma \\left( {{\\bf{W}}_{xf}}{{\\bf{x}}_{t}} + {{\\bf{W}}_{hf}}{\\bf{h}}_{t-1} + {{\\bf{W}}_{cf}}{\\bf{c}}_{t-1} + {\\bf{b}}_f \\right), \\\\\n\\!&{\\bf{\\overline{c}}}_t = \\phi \\left( {{\\bf{W}}_{xc}}{{\\bf{x}}_{t}}\\! +\\! {{\\bf{W}}_{hc}}{\\bf{h}}_{t-1} \\!+\\! {\\bf{b}}_c \\right), \\\\ \n\\!&{\\bf{c}}_t = {\\bf{f}}_t\\!\\odot {\\bf{c}}_{t-1}\\!+ {\\bf{i}}_t\\!\\odot {\\bf{\\overline{c}}}_t, \\\\ \n\\!&{\\bf{o}}_t = \\sigma \\left( {{\\bf{W}}_{xo}}{{\\bf{x}}_{t}} + {{\\bf{W}}_{ho}}{\\bf{h}}_{t-1} + {{\\bf{W}}_{co}}{\\bf{c}}_{t} + {\\bf{b}}_o \\right), \\\\\n\\!&{\\bf{h}}_t = {\\bf{o}}_t \\odot \\phi \\left( {\\bf{c}}_t \\right),\n\\end{aligned}\n\\end{eqnarray}\nwhere ${{x}}_t$ is input vector at time $t$, the activation function $\\sigma \\left(x \\right)$ \/ $\\phi \\left(x \\right)$ is defined as sigmoid $\\sigma\\left(x\\right)=1\/(1+e^{-x})$ \/ $\\phi \\left(x \\right)=\\tanh$, ${\\bf{W}}_{A B}$ is the weight matrix between $A$ and $B$ (e.g., ${\\bf{W}}_{x i}$ is the weight matrix from the inputs ${\\bf{x}}_{t}$ to the input gates ${\\bf{i}}_t$), ${\\bf{b}}_{B}$ denotes the bias term of $B$ with $B \\in \\{i,f,c,o\\}$ and $\\odot$ denotes point-wise multiplication of the two vectors. To ensure comprehensibility with standard literature, we follow\nthe same naming conventions discussed in the paper by \\cite{Zhu2016}.\n\n\\section{Literature Review}\n\\label{sec:DHP_LR}\n\nIn this section, we briefly review three prominent streams of research\ninto how the Hawkes process is used to design market making strategies.\nWe are well aware of blurred and overlapping boundaries across\nresearch streams, but strongly believe in capturing the essence of\nHawkes models in market making.\n\n\\subsection{Stochastic Optimal Control}\n\nThe classical finance literature employs a stochastic optimal control\nframework to determine market maker's optimal quotes for bid and ask\nin the presence of inventory risk, adverse selection, information\nasymmetry and latency \\citep{Ho1981, Sandas2001, Avellaneda2008, Penalva2015}. The market maker aims to maximise the expected utility of the profit and loss contour at closing time. By integrating a utility framework into the microstructure of LOB, \\cite{Avellaneda2008} were able to analytically derive optimal bid and ask quotes. In the continuous-time model, the mid-prices evolve according to Brownian motion, and the order arrival follows a Poisson process. In one particular case, a non-homogeneous Poisson process is reduced to the Hawkes process \\citep{Bacry2015}. The unrealistic assumptions limit the model's ability to capture adverse selection effects, market impact and autocorrelation structures in the security being traded. Building on the seminal work off \\cite{Avellaneda2008}, numerous researchers looked at price impact, adverse selection effects, and latency, together with different objective functions \\citep{Penalva2015}. The purpose of all of the improvements in the model is to provide numerical approximations for the associated\nstochastic differential equations. However, problems related to model\nambiguity have yet to be addressed \\citep{Nystrom2014}. \n\nThe profusion of transaction-level data at a minuscule resolution\nprovides an unprecedented opportunity to apply Hawkes processes to\nthe study of market microstructure with a view to deciphering price\nformation mechanism, liquidity dynamics and volatility \\citep{Morariupatrichi2018}. The use of Hawkes processes to model\nextreme price moves and order flow is well integrated with the high-\nfrequency market making model \\citep{Nystrom2014, Bacry2015}. For example, \\cite{Nystrom2014} proposed a market making\nmodel based on model risk or uncertainty, in which inventory dynamics,\nfill rates and price formation are modelled using two independent\nHawkes processes. While the market making models based on the\nHawkes processes have been moderately successful in integrating a\nrealistic order arrival process, they fail to incorporate the complex\ninteraction between the self- and cross-excitation effects of the\nendogenous state variables that describe the LOB \\citep{Morariupatrichi2018}. In the classical \"buy low and sell high\" market making strategies, \\citep{Cartea2014} used a multivariate Hawkes process to capture the interaction between market orders and the state of LOB. However, the constraints placed on parametric specification for\nconditional intensity by the Hawkes process limit the model's\neloquence.\n \n\\subsection{Agent-Based Models}\n\nMarket making has been also been extensively investigated in ABM \\citep{Das2005, Wah2017}. Using a bottom-up approach, the ABMs try to artificially simulate the systems' aggregate behaviours caused by the actions and interactions between heterogeneous autonomous agents \\citep{Samanidou2007}. The ABMs in market making range from the simple \"zero intelligence\" of mainstream economics to representing in detail the full complexity of the order book and market microstructure. The seminal work of \\cite{Toke2011} assimilated a microscopic, dynamical statistical model for the continuous double auction \\citep{Smith2003} into  Hawkes framework. The reinforced zero-intelligence market making model\npopulated with the liquidity provider and liquidity taker uses mutually\nexciting Hawkes processes for the order arrival process. However, the\nparametric specification of the exponential kernels contradicts the\nempirical results on the existence of a feedback loop between order\narrival and the state of LOB \\citep{Gonzalez2017, Morariupatrichi2018}. The Neural Hawkes process \\cite{Mei2017} applied a Neural\nHawkes process to natural language processing in order to tackle the\nparametric specifications problem, and generalised the Hawkes process\nby calculating the event intensities from the hidden state of an LSTM.\n\nAnother prominent ABM in market making research \\citep{Wah2017}uses empirical game-theory analysis to examine the effect of\nmarket making on market performance in different scenarios. In the\npaper, the authors use multiple background traders (e.g. traditional zero\nintelligence agents) to produce realistic market microstructure. The\nBayesian market makers were then used to investigate welfare effect,\ntrading gains and strategic behaviour. The simplistic adaptive trading\nstrategies adopted by the market makers ought to be sufficient to\ninvestigate market equilibria, but would face serious challenges in\nrelation to developing realistic market making strategies that also take\naccount of market microstructure. The academic literature needs to\nassimilate the success of artificial intelligence in designing trading\nstrategies that interact with close-to-reality market simulations.\n\n\\subsection{Artificial Intelligence}\n\nThe success of deep reinforcement learning in board games \\citep{Silver2017} and video games \\citep{Mnih2015} soon sent ripples through the world of finance. Drawing on the classical mathematical\nsetup by \\cite{Avellaneda2008}, \\cite{Gueant2019} used a model-based deep actor-critic algorithm to find the optimal bid and ask quotes across a high-dimensional space of corporate bonds. \\cite{Spooner2018} rreconstructed a limit order book from historical data and used it to construct a market making agent using temporal-difference reinforcement learning. The next natural progression, a multi-agent simulation of a dealer market, was developed by \\cite{Ganesh2019} to understand the behaviour of market making agents. The success of this model is dependent on a realistic simulation framework of the financial market. However, algorithmic trading research is evolving, and is still at the stage of exploiting market simulators that could be used to develop training and testing algorithms in a similar vein to the classic Atari 2600 games simulator \\citep{Mnih2015}.\n\nDespite the exemplary predictive performance of deep learning\nmodels, market making has not yet been comprehensively addressed. If\nwe return to stochastic optimal control in market making, we notice that\nthe dynamics of order flow are mostly described by variants of Hawkes\nprocesses, while the resultant control algorithms are associated with\ncontinuous semi-martingales, which are hard to solve recursively \\citep{Gueant2019}. In addition, the unavailability of a realistic\nsimulation framework and high-quality order-level data restricts\nresearchers' ability to replicate the successes of deep learning in market\nmaking.\n\nNevertheless, deep learning models (based on CNN architecture)\nhave done moderately well when it comes to modelling price-formation\nmechanisms using order book events as input \\citep{Sirignano2019, Cont2019, Tashiro2019, Tsantekidis2017}. The literature is proof that empirical investigation of order-level data has always augmented the model of choice. In short, the ABM, statistical modelling and artificial intelligence approaches to market making all have desirable attributes that the others lack. For example, the CNN architectures lacks self- and cross-excitation effects while modelling LOB dynamics \\citep{Zhang2019}. The DLSTM architecture performs hierarchical processing of complex order book events, and as such is able to efficiently capture the temporal structures of LOB \\citep{Sagheer2019A, Sagheer2019B}. As market making involves placing optimal bids and asks, the DLSTM architecture, together with the Hawkes process, would efficiently model order arrival\nand price-formation mechanisms by constructing a self-exciting multivariate Hawkes process with limit order state process, which are\ncoupled and interact with each other.\n\\section{Multi-Agent Simulation Framework}\n\\label{sec:DHP_MASF}\nMulti-agent-based modelling is a bottom-up computational modelling\napproach intended to artificially simulate the aggregate behaviours of\nsystems caused by the actions and interactions between heterogeneous\nautonomous agents in an environment or environments \\citep{Samanidou2007}. In this section, we describe the important components of multi-agent-based modelling, including environment (market\nsimulator), agent ecology (trading strategies) and reward (profit and\nloss), to study the behaviour of market making agents whose strategies\nemploy Deep Hawkes processes.\n\n\\subsection{Market Simulator}\n\nThe market simulator is an essential tool for designing, evaluating and backtesting algorithmic trading strategies under various market scenarios. The market is flooded with various proprietary and open-\nsource financial simulation frameworks, but the constraints associated\nwith licences, application interfaces and software design limit their\nusability \\citep{Maeda2019, Izumi2009}. In this paper, we have designed a multi-asset market simulator from scratch, which is scalable to markets of substantial size. The asynchronous event-based interface is built over realistic market architecture, interface kernels, a matching engine and the Financial Information eXchange (FIX) protocol \u2013 an open electronic communications protocol standard used to carry out trades in electronic exchanges. \n\n\\subsubsection{Market Architecture}\n\nThe market architecture consolidates the communication interface,\nmarket and matching engine. Figure \\ref{fig:DHP_MS} outlines key components of market architecture and their interaction from a high-level perspective. The agents connect to the market via a kernel that hosts order management details. This acts as a transmission channel between agents and markets, thereby providing the extreme throughput and lowest\nlatency for order transactions. It also throttles the amount of\ntransactions, as per the market requirement. As such, there is a guarantee of fairness between agents waiting to place orders. The markets\nrepresent an information interchange, in which heterogeneous agents\ncommunicate through kernels for order transactions, processing and\nexecution according to the matching engine, as per the financial\ninstruments. The markets respond to order status by sending an\nexecution report that covers the period from start to market reset event.\nThis provides opportunities for agents to tweak the parameters in their\ntrading strategies after every trading period if required.\n\n\\tikzstyle{AM} = [rectangle, rounded corners, minimum width=2cm, minimum height=1cm,text centered, draw=black]\n\n\\tikzstyle{arrow} = [thick,->,>=stealth]\n\n\\begin{figure}[H]\n\\centering  \n\\resizebox{\\textwidth}{!}{\\begin{tikzpicture}[node distance=2.0cm]\n\\node (A) [AM] {Agents};\n\\node (K) [AM, right of=A, xshift=1.0cm]{Kernel};\n\\draw [arrow] (A) -- node[anchor=south] {} (K);\n\\draw [arrow] (K) -- node[anchor=north] {} (A);\n\\node (M) [AM, right of=K, xshift=1.0cm]{Markets};\n\\draw [arrow] (K) -- node[anchor=south] {} (M);\n\\draw [arrow] (M) -- node[anchor=north] {} (K);\n\\node (E) [AM, right of=M, xshift=1.0cm]{Matching Engines};\n\\draw [arrow] (M) -- node[anchor=south] {} (E);\n\\draw [arrow] (E) -- node[anchor=north] {} (M);\n\\end{tikzpicture}\n}\n\\caption{Market Simulator.}\n\n\\label{fig:DHP_MS}\n\\end{figure}\n\n\\subsubsection{Communication Interface}\nThe communication interface between agents and markets follows the\nFIX standard protocol to increase efficiency, competition and\ninnovation. It is an electronic communications protocol designed for the\nreal-time exchange of information between agents and exchanges,\nincluding agent identifiers, order identifiers, order handling, trade\nnotifications, broadcasts and execution reports. The widespread\nadoption of the FIX protocol across financial markets reduces costs\nassociated with connectivity, regulatory compliance, liquidity searches\nand transactions. Using the FIX protocol, multiple agents can interact\nwith the market via kernels, simultaneously and independently.\n\n\\subsubsection{Matching Engine}\nAt the core of the market simulator are several \\emph{matching engines} for different financial instruments. Each \\emph{matching engines} matches bids and asks to execute trades in specific instruments. The orders are matched using \\emph{price-time} priority mechanisms. In this context, among bids or asks, the matching algorithms give priority to orders with the highest or lowest price. The ties are broken by giving preference to orders with the earliest submission time compared to other orders. Other well-known priority mechanism algorithms are \\emph{pro-rata} and \\emph{price-size}  \\citep{Gould2013}.\n\n\\subsection{Market Ecology}\nThe success of a financial simulation framework depends on the precise representation of market ecology that can adequately mimic the real\nmarket design. In finance literature, the term \"market ecology\" \\citep{Farmer2002} refers to the composition of heterogeneous trading strategies that keep evolving over time in response to pressure from a contrasting market. The correct mapping of financial market ecology requires the availability of agent-resolved order-level data, which enables identification of sources and events in the market. In the absence of agent-resolved data, the trading strategies are classified by theoretical considerations, the results of surveys, direct investigations of the trading profile of classes of investors and proxies for HFT \\citep{Kirilenko2017, Mankad2013}. In this paper, we adapt the market ecology from the different strands of academic literature \\citep{McGroarty2019, Paulin2019, Musciotto2018, Kirilenko2017, Leal2016, Mankad2013, Toth2012}. \n \n\\subsubsection{Deep Hawkes Market Makers}\n\nDespite the existence of sophisticated market making strategies, the classical \"buy low and sell high\" strategy \\citep{Cartea2014} is a preferred strategy to make money in the securities market. The success of making a profit from short-term price predictions on bids and asks hinges on placing orders at precisely the right time. The mathematical modelling of the complicated order arrival process is done using the Poisson process \\citep{Chakraborti2011E}, which assumes that orders arrive\nrandomly. However, this is contrary to empirical findings, which show that order arrival times are strongly connected \\citep{Rambaldi2017}, have self- and cross-excitation effects \\citep{Morariupatrichi2018} and that a feedback loop exists between the order arrival and the state of the LOB \\citep{Gonzalez2017}. In this article, we incorporate the feedback loop between order arrival and the state of the limit order book, together with self- and cross- excitation effects, using DHP, to design a high-frequency market making strategy.\n\nThe order book events in the securities market are stochastically excited or impeded by a pattern in the past event streams. The market makers are interested in learning the distribution and structure of order book events stream to accurately predict the next order (limit orders, market orders, cancellations, etc.) together with an associated labels (price, volume, etc.). Given a stream of order book events $\\{t_n, \\varkappa_n, k_n\\}_{n \\in \\mathbb{N}}$, the market makers calculate the probability that the next event occurs at time $t_n$ is of type $k_n$ and its probability density  conditioned on the history of events $\\mathcal{H}_n$ by:\n\\begin{equation}\n\\lambda(t)dt = \\mathbb{P}\\{(t_n,\\varkappa_n, k_n) \\mid \\mathcal{H}_n, (t_n - t_{n-1})\\}.\n\\end{equation}\n\n\n\\begin{equation}\nf_n(t) = \\lambda(t) \\exp \\left( -\\int_{t_{n-1}}^{t} \\lambda(\\tau) d\\tau \\right).\n\\end{equation}\n\nTo predict the time and the next event having minimum loss without information about the time $t_n$, we choose $ \\hat{t}_{n} = \\int_{t_{n-1}}^{\\infty} t f_n(t) dt$ and $\\hat{k}_{n} = \\argmax_{k} \\int_{t_{n-1}}^{\\infty} \\frac{\\lambda_k(t)}{\\lambda(t)} f_n(t) dt$. The associated intensity function for calculating the next order book events is the same as DHP as described in Equation \\ref{eqn:hawkes_mod_a}. \n\nThe real securities market has numerous distinct order book events, which is difficult to incorporate in our simulation framework. For the\nsake of computational tractability, we decided to only include limit order buy\/sell, market order buy\/sell, and partial\/full cancellations. We also assume that the high-frequency market maker is trading a single security in a market whose price at time $t$ is denoted by $p_t$. Unlike traditional market making strategies \\citep{Spooner2018}, the DHP market makers can trade with rational limit or market order quantity, price surge and bid-ask spread distributions. \n\nThe deep hawkes market maker (DHMM) place orders at a specified depth relative to the mid-price, $\\overline{p_t}$. At each time step $t$, the DHMM agent's pricing mechanism is given by:\n\n\\begin{equation}\np^{a,b}_t = \\overline{p_t} +  \\sum_{i=1}^{\\overline{\\delta}}i\\cdot\\mathbf{J}^{i,u}_t - \\sum_{i=1}^{\\overline{\\delta}}i\\cdot\\mathbf{J}^{i,d}_t \n\\end{equation}\n\nwhere $\\overline{p_t}$ is the mid price at time $t$, $\\mathbf{J}^{i,u}$ is the number of upward jumps with $i$ ticks, and $\\mathbf{J}^{i,d}$ is the number of downward jumps with $i$ ticks between $0$ and $t$, $i = 1,\\dotsc,\\overline{\\delta}$. The intensities of $\\mathbf{J}^{i,u}$ and $\\mathbf{J}^{i,d}$ are $\\lambda_{k,u}(t)$ and $\\lambda_{k,d}(t)$, respectively,\n\n\\begin{equation}\n\\lambda_{k,i}(t) = f_{k,i}(\\bf{w}_{k,i}^{\\top}\\bf{h}(t)), \\; \\;  k=u,d.\n\\label{eqn:hawkes_mod_A}\n\\end{equation}\n\nThe parameters of the Equation \\ref{eqn:hawkes_mod_A} are calculated as discussed under model formulation in Section \\ref{ss:MF}.\n\nMost of the quantitative finance research into the high-frequency market making problem is based on the assumption of constant order size \\citep{Huang2015}. However, the empirical analyses suggests that the order sizes have striking statistical distribution at different timescales \\citep{Lu2018, Rambaldi2017, Mu2009}. The limit order size follows $\\mathfrak{q}$-Gamma distribution \\citep{Mu2009}. The market maker's willingness to sell or buy specified quantities of securities is defined as:    \n\n\\begin{equation}\n q^{a,b}_{lt}= \\left[\\Gamma (\\alpha, \\beta)\\right]^{q_{max}}_{q_{min}} \\cdot \\left(\\frac{I_t \\pm \\bar{I}}{\\bar{I}}\\right),\n \\label{Eq:G}\n\\end{equation}\nwhere $I_t$ is inventory at time $t$, $\\bar{I}$ maximum inventory, and $\\Gamma (\\alpha, \\beta)$ is $\\mathfrak{q}$-Gamma distribution is described as:\n\n\\begin{equation*}\n \\Gamma (\\alpha, \\beta; q)= \\frac{1}{Z} \\left(\\frac{q}{\\alpha}\\right)^{\\beta}\n \\left[1-(1-\\mathfrak{q}){\\frac{q}{\\alpha}}\\right]^{\\frac{1}{1-\\mathfrak{q}}}~,\n \\label{Eq:qGamma}\n\\end{equation*}\n\n\\begin{equation*}\n Z=\\int_{0}^{\\infty}\\left(\\frac{q}{\\alpha}\\right)^{\\beta}\n \\left[1-(1-\\mathfrak{q}){\\frac{q}{\\alpha}}\\right]^{\\frac{1}{1-\\mathfrak{q}}}{\\rm{d}}q.\n \\label{Eq:Z}\n\\end{equation*}\n\nOne striking feature of equity markets is the existence of short- lived\nlimit orders that are modified or cancelled once every 50 milliseconds \\citep{Dahlstrm2018}. The limit order cancellation is an important\ncharacteristic of market making strategies that are related to expected\nprofit, bid-ask spread and order queue position. We model cancellation\nsizes as follows:\n\n\\begin{equation}\n q^{a,b}_{ct}= \\left[P_c(q;Q)\\right]^{q_{max}}_{q_{min}} \\cdot \\left(\\frac{I_t \\pm \\bar{I}}{\\bar{I}}\\right),\n \\label{Eq:TG}\n\\end{equation}\nwhere $P_c(q;Q)$ is truncated geometric distribution \\citep{Lu2018}. The LOB is represented as $[Q_{-i}:i = 1,\\ldots,L]$ and $[Q_{i}:i=1,\\ldots,L]$ with corresponding quantities $q_i$. The truncated geometric distribution is defined as:\n\\begin{equation*}\nP_c(q;Q)= \\mathbb{P}[q|Q] = \\frac{p_{c}^{0}(1-p^{0}_{c})^{q-1}}{1-(1-p^{0}_{c})^Q}\\mathbbm{1}_{\\{q\\leq Q\\}}.\n\\end{equation*}\n\nFinally, the market order follows a mixture of truncated geometric distribution and the dirac delta distribution \\citep{Lu2018}. The market order size that a market maker is willing to buy or sell is described as: \n\n \\begin{equation}\n q^{a,b}_{mt}= \\left[P_m(q;Q)\\right]^{q_{max}}_{q_{min}} \\cdot \\left(\\frac{I_t \\pm \\bar{I}}{\\bar{I}}\\right),\n \\label{Eq:TGD}\n \\end{equation}\n  \n \\begin{align*}\n P_m(q;Q) &= \\theta_0\\frac{p_m^0(1-p_m^0)^{q-1}}{1-(1-p_m^0)^Q}\\mathbbm{1}_{\\{ q\\leq Q\\}} \\\\\n &+ \\sum_{k=1}^{\\lfloor \\frac{Q-1}{5} \\rfloor} \\theta_k\\mathbbm{1}_{\\{q=5k+1\\}}+\\theta_\\infty \\mathbbm{1}_{\\{q=Q, Q \\neq 5n+1\\}}.\n \\end{align*}\n\nThe parameters $\\{p_c^0, p_m^0, \\theta_0, \\theta_k, \\theta_\\infty\\}$ are estimated using a maximum likelihood method. The details of estimation and calibration can be retrieved from the \\cite{Lu2018}. The market orders are used to clear the unexecuted inventory at the end of trading.\n\n\\subsubsection{Probabilistic Market Makers}\nTo ensure fair competition with DHMM and incorporate the existing state-of-the-art, we include a probabilistic estimate-based benchmark strategy \\citep{Das2005} adapted to our simulation framework. In this market making strategy, the agent attempts to track the fundamental price of securities by maintaining a probability density estimate of the fundamental price.\n\nThe probabilistic market makers (PMM) intent to sell or buy $q$ unit of security at time $t$ for price $p^{a,b}_t$ in a market populated with uninformed, informed and noisy informed agents. Let us assume that the fundamental price of the security at time $t$ is $f_t$, $\\xi$ be the fraction of informed agents and the probability of buy or sell orders by the uninformed agents is $\\zeta$. The noisy informed agents assumes that the price of securities follow normal distribution $p_t \\; = \\; f_t + \\mathcal{N}_s(0,\\sigma^{2}_{n})$. Whereas the fundamental price of security evolves according to a jump process. The order book event defines the jump and prices follow normal distribution. The PMM ask and bid prices at time $t$ are then defined as:\n \n\n\\begin{equation} \n\\begin{split}\np^{a,b}_{t} & = \\frac{1}{P_{Buy, Sell}}\\sum_{f_{t}=f_{min}}^{f_{t}=p^{a,b}_{t}} \\left[ \\left( (1 - \\xi) \\zeta + \\mathsf{Pr}(\\mathcal{N}_s(0,\\sigma^{2}_{n})\\gtrless(p^{a,b}_{t} - f_{t})) \\right) f_{t} \\mathsf{Pr}(f = f_t) \\right]  \\\\\n & + \\frac{1}{P_{Buy, Sell}}\\sum_{f_{t}=p_{t}^{a,b}+1}^{f_{t}=f_{max}} \\left[ \\left( (1 - \\xi) \\zeta + \\mathsf{Pr}(\\mathcal{N}_s(0,\\sigma^{2}_{n})\\lessgtr(f_{t} - p^{a,b}_{t})) \\right) f_{t} \\mathsf{Pr}(f = f_t) \\right] \n\\end{split}\n\\label{Eq:PMM_ab}\n\\end{equation}\n\n\\begin{equation*} \n\\begin{split}\nP_{Buy, Sell} & = \\sum_{f_{t}=f_{min}}^{f_{t}=p^{a,b}_{t}} \\left[ (1 - \\xi) \\zeta + \\mathsf{Pr}(\\mathcal{N}_s(0,\\sigma^{2}_{n})\\gtrless(p^{a,b}_{t} - f_{t})) \\right] \\mathsf{Pr}(f = f_t)   \\\\\n & + \\sum_{f_{t}=p_{t}^{a,b}+1}^{f_{t}=f_{max}} \\left[ (1 - \\xi) \\zeta + \\mathsf{Pr}(\\mathcal{N}_s(0,\\sigma^{2}_{n})\\lessgtr(f_{t} - p^{a,b}_{t})) \\right] \\mathsf{Pr}(f = f_t) \n\\end{split}\n\\label{Eq:P_ab}\n\\end{equation*}\n\nwhere $P_{Buy, Sell}$ is a priori probability of a buy or sell order and $\\mathcal{N}_s(0,\\sigma_{n})$ is sample from normal distribution. The bids\/asks equations derivation, its approximate solutions, density estimate update, and algorithm are discussed in the benchmark paper \\citep{Das2005}. We add layers of complexity on the benchmark algorithm by allowing the PMM to sample order or cancellations size from a normal distribution. The order cancellations size at time $t$ determined as follows:\n\n\\begin{equation}\nq_{t;o,c}^{a,b} = \\eta_{o,c}(\\bar{p}_{t}-\\bar{p}_{t-1})+\\mathcal{N}_s(0,\\sigma^{2}_{o,c}), \\; \\; 0 < \\eta_{o,c} < 1\n\\label{Eq:P_oc}\n\\end{equation}\n\\subsubsection{Fundamental Traders}\n\nFundamental traders decide to trade based on the presumption that the\nsecurities prices will eventually return to their basic, intrinsic or\nfundamental value. Therefore, they strive to buy (sell) the security when\nthe price at time t is below (above) its fundamental value. Fundamental\ntraders are predominantly categorised as buyers or sellers, depending on\nthe inventory at the end of a trading day. The accumulation of directional net positions is an important element in identifying buyers or sellers, since the latter acquire sizable net positions by executing numerous small-size orders, while the former only execute a couple of large orders \\citep{Kirilenko2017, Mankad2013}. According to the agent ecology literature, the fundamental traders assume that fundamental value of a security will follow a random walk:\n\n\\begin{equation}\nf_t = f_{t-1}(1+\\delta_f)(1+x_t), \\; \\; \\; \\delta_f > 0; \\;  x_t  \\sim \\mathcal{N}(0, \\sigma^{2}_x)\n\\end{equation}\n\nGiven last mid-price at time $t$, the limit order price by fundamental traders is determined by:\n\n\\begin{equation}\np_t = \\bar{p}_{t-1}(1+\\delta_f)(1+z_t), \\; \\; z_t  \\sim \\mathcal{N}(0, \\sigma^{2}_z)\n\\end{equation}\n\nFinally, the order under fundamental traders strategy are calculated as follows:\n\n\\begin{equation}\nq_{t;f} = \\eta_{f}(f_{t}-\\bar{p}_{t-1})+\\mathcal{N}_s(0,\\sigma^{2}_{f}), \\; \\; 0 < \\eta_{f} < 1\n\\end{equation}\n\nThe decision to buy or sell is governed by following logic:\n\n\\begin{equation}\n    \\mathcal{D}_t= \n\\begin{cases}\n    Buy ,& q_{t;f} \\geq 0\\\\\n    Sell,              & q_{t;f} < 0\n\\end{cases}\n\\end{equation}\n\n\n\\subsubsection{Chartist Traders}\nUnlike fundamental traders, the chartist or technical trader's strategy depends on predicting future price direction based on past price movement. The chartist traders in our simulation framework use a simple trend-following strategy described in \\cite{Leal2016}. The price, order size and trade direction are described below:\n\n\\begin{equation}\np_t = \\bar{p}_{t-1}(1+\\delta_c)(1+z_t), \\; \\; z_t  \\sim \\mathcal{N}(0, \\sigma^{2}_c)\n\\end{equation}\n\n\\begin{equation}\nq_{t;c} = \\eta_{c}(\\bar{p}_{t-1}-\\bar{p}_{t-2})+\\mathcal{N}_s(0,\\sigma^{2}_{c}), \\; \\; 0 < \\eta_{c} < 1\n\\end{equation}\n\n\\begin{equation}\n    \\mathcal{D}_t= \n\\begin{cases}\n    Buy ,& q_{t;c} \\geq 0\\\\\n    Sell,              & q_{t;c} < 0\n\\end{cases}\n\\end{equation}\n\n\\subsubsection{Noise Traders}\nIn the securities market, noise traders make trading decisions based\nsolely on non-information. In the models, they serve as an essential\nproxy for randomness, no trade and no speculation. We incorporate the\nslightly more evolved noise or background traders from a seminal paper by \\cite{Wah2017}. The noise traders ask or bid price is determined by its fundamental private valuation and trading strategy. The fundamental value evolves according to a mean-reverting stochastic process \\citep{Wah2017}.\n\n\\begin{equation}\nf_t = max\\left[0, \\eta_{n}\\bar{f}+\\eta_{n}(1-f_{t-1})+y_t \\right], \\; 0 < \\eta_{n} < 1; \\; y_t  \\sim \\mathcal{N}(0, \\sigma^{2}_n) \n\\end{equation}\n\nThe private valuation for the noise traders at time $t$ is given by:\n\\begin{equation}\np_v = max \\left[ 0, d_f\\right], \\; d_f \\sim \\mathcal{N}(f_t, \\sigma^{2}_v)\n\\end{equation}\n\nThe noise trader calculates its private value and decide to buy or sell $q_{t;n}$ order sampled from a normal distribution,$\\mathcal{N}(0, \\sigma^{2}_n)$, with equal probability of 1\/2. \n\n\\subsection{Reward Design}\nUnlike traditional reward design, in which an agent's performance is assessed at the end of a trading period, we calculate the agent's\ninstantaneous rewards at each timestep $t$. The reward function for the agents ($j$) comprises profit \\& loss (PnL), inventory cost (IC) and transaction cost (TC). The PnL is simple profit or loss made by the agents through buying or selling security at the exchange. Its is defined as:\n\n\\begin{equation}\nPnL_{t;j} = q^{a}_{t;j}\\left(p^{a}_{t;j} - \\bar{p}_{t} \\right)+q^{b}_{t;j}\\left(\\bar{p}_{t} - p^{b}_{t;j} \\right)\n\\end{equation} \n\nAs a agent's inventory is exposed to the volatility of the market price, we incorporate it our reward design using a term associated with inventory cost. Its given by:\n\n\\begin{equation}\nIC_{t;j} = I_{t;j}\\left(\\bar{p}_{t} - \\bar{p}_{t-1} \\right)\n\\end{equation} \n\nFinally, we consolidate a quadratic penalty on the number of shares executed to account for transaction cost. Specifically, the transaction cost for order executed $q_{t}^{e}$ by agent $j$ till time $t$ is:\n\n\\begin{equation}\nTC_{t;j} = \\daleth \\left( q_{t;j}^{e}\\right)^{2}, \\; 0<\\daleth<1\n\\end{equation}\n\nThe reward function is the sum of orders bought or sold plus inventory cost less a transaction cost penalty.\n\n\\begin{equation}\nR_{t;j} = PnL_{t;j}+IC_{t;j}-TC_{t;j}.\n\\end{equation}\n\n\\subsection{Capital Allocation}\n\\label{sec:CA}\n\nThe amount of currency units held by an agent is represented by capital. Prior to securities market opening in simulation framework, every heterogeneous agents endowed with different amount of capital by a power law distribution. The agent's initial capital $c_a$ follows a power law if it is drawn from drawn from a probability\ndistribution $p(c_a) \\propto {c_a}^{- \\alpha_a}$. The $\\alpha_a$ is referred as scaling parameter which ordinarily lies between 2 and 3 \\citep{Clauset2009}.\n\n\\section{Conclusions}\n\\label{sec:DHP_C}\n\nWe have developed a market making strategy that takes account of the\nfeedback loop between the order arrival and the state of the LOB, with\nself- and cross-excitation effects, while placing an order in our realistic simulation framework. The strategy was designed by integrating the self-modulating multivariate Hawkes process with DLSTM-SDAE. The\ndata-driven approach performed adversely in relation to predicting the\nnext order type and its timestamp when fitted to reconstructed order\nbook data at nanosecond resolution. When trained with millisecond\nresolution data, it outperforms NHP in prediction tasks and benchmark\nmarket making strategies in trading performance. We have demonstrated that extending the DHP in a market making setting accomplished better performance when validating empirical claims about the effect of cancellation on the determinants of order size. Our modelling approach is still far from inferring causality, but does pave the way exploring a range of diverse research avenues. The most important and immediate of these are listed below:\n\\begin{enumerate}\n\\item Apply more advanced pre-processing, architecture, and learning\nalgorithms to filter out ultra-noisy order book data at nanosecond\ntimestamps.\n\\item Explore an intensity-free approach for Hawkes processes with learning mechanisms other than the maximum likelihood approach.\n\\item Embedding DHP within the reinforcement learning framework to learn\nthe optimal policy.\n\\item Extend the model to the deep reinforcement learning framework, in\nwhich the agent's trading action and reward from the simulator are\nasynchronous stochastic events characterised by marked multivariate\nHawkes processes. \n\\item Extract the agent's trading algorithm parameters directly from the order book data , rather than from random seeds or empirical literature.\n\\end{enumerate}\n\n\n\n\n\\section{Experiments}\n\\label{sec:DHP_E}\nIn this section, we elaborate on data, its processing, performance\nmetrics, benchmarks, training and the parameter configuration for the\nproposed model.\n\\subsection{Data}\n\nWe use the publicly available historical Nasdaq TotalView-ITCH 5.0 data feed sample \\footnote{\\url{ftp:\/\/emi.nasdaq.com\/ITCH\/Nasdaq_ITCH\/}} to reconstruct limit order book \\citep{Huang2011}. The reconstructed database provides tick-by-tick details of full order book depth by listing every quote and order at each price level of a specific security in Nasdaq, NYSE, and regional-listed securities on Nasdaq. The raw data feed in the binary format has a series of sequenced messages to describe the system, securities, order, and trade events at a resolution of the nanosecond scale. The event stream at nanosecond timestamp guarantees the inclusion of stochastically missing events which might increase the predictive accuracy of the Deep Hawkes model. Although the neural hawkes model \\citep{Mei2017} is expressive enough to take account of missing event stream, it makes sense to access the performance of deep hawkes model at millisecond resolution order book data as compared to nanoseconds. Nasdaq uses multiple messages to indicate the current order, trading, system, and circuit breakers event's status as discussed in technical report \\citep{Nasdaq2020}. For mathematical tractability, we have sampled high frequency data for hundred most liquid securities over eight days from reconstructed orderbook. The extracted sample data consists of approximately a billion transaction records at nanosecond resolutions together with the possible event of limit order buy\/sell, market order buy\/sell, and cancellations partial\/full. The reconstructed limit order book for Apple at 11:21 on March 29th, 2018 is shown in Figure \\ref{fig:DHP_LOB}.  \n\nThe reconstructed orderbook data is divided into training, validation and test set. For a single security at nanosecond and millisecond resolution, the descriptive statistics are given in Table \\ref{tab:DHP_DS}. The validation set is included to optimize the model's hyper-parameters while training, thus having control at over-fitting. To avoid high variance in the data set, we only record the average value over multiple splits denoted by $\\approx$ in the Table \\ref{tab:DHP_DS}.\n \n\\begin{table}[H]\n\\caption{Descriptive statistics of the orderbook data}\\label{tab:DHP_DS}\n \\resizebox{\\textwidth}{!}{ \\begin{tabular}{clrrrrr}\n    \\toprule\n    \\toprule\n    \\multirow{2}{*}{Data} & \n      \\multicolumn{3}{c}{\\# Orderbook Event Token} &\n      \\multicolumn{3}{c}{Stream Length} \\\\\n      \\cmidrule(l){2-4} \\cmidrule(l){5-7}\n    & Train & Val & Test & Min & Mean & Max \\\\\n    \\midrule\n    Nanosecond & $\\approx 9210480$ & 921000 & 2302620 & 26752 & 85874 & 116217  \\\\\n    \\midrule\n    Millisecond & $\\approx 432116$ & 42252 & 108029 & 1167 & 3670 & 5216 \\\\\n    \n   \\bottomrule\n    \\bottomrule\n  \\end{tabular}\n  }\n  \n\\end{table}\n\n\\subsection{Performance Metrics}\n\nDHMM agents use DHP to accurately predict Experiments order book events (buy, sell or cancel) and their timing. The accuracy of DHMM's predictions in terms of events and time is an important determinant of its trading profitability. The performance metrics are vital components for measuring the performance of the trained model's prediction reliability with observed test data. The widely used scale-\ndependent metric\\citep{Hyndman2006}, root mean square error (RMSE) and classification error rate (ER) were used to evaluate the prediction performance of the Neural Hawkes model. Following \\cite{Mei2017}, we predict each prevailed order book event stream $\\{t_n, \\varkappa_n, k_n\\}$ from the past event stream $\\mathcal{H}_n$ and evaluate prediction using RMSE and ER.\n\nThe predominant metric for evaluating the performance of the agents\nis profit and loss at the end of the trading period. However, this approach may be misleading, as agents are tested across heterogeneous securities, with varying pricing and liquidity structures. Alternatively, to efficiently capture spread, we use a normalised PnL (NPnL) with inventory and quadratic transaction costs. The NPnL is calculated every hour by dividing the total reward by the weighted average market spread. To take account of the small inventories maintained by the market maker,, \\cite{Spooner2018} introduced the mean absolute position (MAP)\nmetric. An extreme score under this metric indicates a risky speculative\nstrategy, while a moderate one indicates a strategy based on a stagnant\nmarket. We record the variability for NPnL and MAP using the standard\ndeviation and mean, respectively.\n\n\\subsection{Benchmarks}\nWe aim to evaluate the performance of the DHMM with a modified\nprobabilistic estimate-based benchmark strategy \\citep{Das2005}. The\nmarket making strategy is an extension of the classic information-based\nmodel \\citep{Glosten1985}, in which agents use the probability\nestimates of the fundamental price of securities to set bid and ask prices. The agents can sample limit, market or cancellation orders from the normal distribution or contradictory to a unit market order. We\nimplement the probabilistic estimate-based strategy at the top of our\nsimulation framework in continuous-time simulation rather than a\ndiscrete-time simulation. This provides the perfect test-bed to assess the performance of the simulation framework in extending the discrete-time mechanisms to continuous-time, where heterogeneous agents interact\nasynchronously. \n\nThe DHP extends the seminal Neural Hawkes process to the deep\nlearning framework in a market making setting. We introduce the novel\narchitecture to circumvent complications related to random weight\ninitialisation, training and noisy order-level data \\citep{Sagheer2019B}. Given that the Neural Hawkes process is the kernel of our proposed deep model, we evaluate the performance of market making agents that use the earlier model in their trading strategies. For comparison purposes, we use the same architecture and training mechanism as discussed in the seminal paper \\citep{Mei2017}. The neurally self-modulating multivariate Hawkes process also acts as benchmark model for evaluating DHP's performance on the prediction of order book events and in terms of time on the reconstructed limit order book data.\n\n\\subsection{Training}\n\nThe high-frequency marker making agents uses DHP to learns from reconstructed limit orderbook data to place bids or asks or cancels at suitable time. The learned prediction is then infused into the market making strategy to trade with the simulation framework. The agents learn the system parameters in a two-step process. Firstly, the preprocessed order book stream, n-th event, $k_n$ is embedded into a latent space before passing into SDAE layer together with timing $t_n$. The deep network, consists of a stack of multiple DAEs, generate higher representation of convoluted order book events interaction. The high level denoised representations are then fed into DLSTM to predict the next order's type and the time to evaluate the loss. The DLSTM-SDAE learns the deep representation in two phases: pre-training and fine-tuning. In pre-training, a greedy layer-wise structure is used to train each layer of DAE iteratively, to form a three-layer SDAE. At the end of pre-training, a stack of three LSTMs is produced as an output of SDAE. Secondly, the parameter of DLSTM-SDAE is then fine-tuned to minimise the error in predicting events and time, using using stochastic gradient-descent and Adam optimisation algorithms. The early stopping methods used on the validation set's log-likelihood performance were also used on the held-out validation set to avoid overfitting. We also add isotropic Gaussian noise to augment generalisation in the performance of the events' classification. Table \\ref{tab:DHP_PC} lists the hyper-parameters tuned by validation set performance for the DLSTM-SDAE network architecture. The other non-LSTM parameters includes  $s_n \\in \\mathbb{R}$ and $\\mathbf{W}_n \\in \\mathbb{R}^D$ as discussed in Section \\ref{sec:DHP_DHP}. The market making agent using NHP uses single layer LSTM  and the number of hidden nodes from a small set $\\left(64, 128, 256, 512, 1024 \\right)$ as described in the base paper \\citep{Mei2017}. The hyperparameters are optimized based on the performance of the validation set.\n\nThe high-frequency market making agents are trained in the simulation framework for 1000 trading days. Each trading day starts at 9:30 and lasts until 16:00. Two hundred trading days were used to fine tune the hyper-parameters using random search. We acknowledge the existence of\ndifferences between the real data and simulated data, but firmly believe\nthat they are generated from the same mechanisms \u2013 a claim substantiated by agent-based models that reproduce stylised facts similar to empirical findings. Taking the above into consideration, we train market makings agents using DHP and NHP for 100 trading days five times. The aim of this exercise is to synchronise the agents' learning over different sets of data generated from the same stochastic process. We then test the performance of the agents against the benchmark for 300 trading days. To ensure fair competition with market making agents, we use heterogeneous market ecology consisting of fundamental, chartist and random agents. The important parameters pertaining to the trading agents in the simulation framework are given in Table \\ref{tab:DHP_PC}.  \n\n\\begin{table}[H]\n    \\centering\n    \\caption{Parameters configuration.}\\label{tab:DHP_PC}\n  \\resizebox{\\textwidth}{!}{  \\begin{tabular}{lll}\n        \\toprule\\toprule\n        Description  & Parameter\/Hyperparameter  & Value \\\\\n        \\midrule\n        Total sessions size & $\\mathsf{T}_{total}$ & 1300 days \\\\\n        Training sample size & $\\mathsf{T}_{train}$ &  1000 days \\\\\n        Testing sample size & $\\mathsf{T}_{test}$ & 300 days \\\\\n        \\midrule\n        Total number of traders &$a_t$& $  \\approx 10^4$ \\\\\n        Number of market makers &$a_m$& 3 \\\\\n        Number of fundamental traders  &$a_f$& 3000 \\\\\n        Number of chartist traders  &$a_c$& 6000 \\\\\n        Number of noise traders  &$a_n$& 4000 \\\\\n        \n        \\midrule\n        Initial capital & $\\mathsf{c}_{a}$ & $ \\sim p(2.3) \\; \\times \\; 10^4$ \\\\\n        Min inventory & $\\min \\Inventory$ & $  \\sim - p(2.3) \\; \\times \\; 10^5$  \\\\\n        Max inventory & $\\max \\Inventory$ & $  \\sim p(2.3) \\; \\times \\;10^5$ \\\\\n        \\midrule\n        Scale factor & $s_n $ & 1 \\\\\n        LSTM weights & $\\mathbf{W}_n$ & $\\sim \\mathcal{N}(0,0.01) $\\\\\n        DHMM limit order parameters & $\\Gamma (\\alpha, \\beta)$ & $\\Gamma (0.07, 1.52)$\\\\\n        DHMM limit order cancellations parameters &$P_c(q;Q)$ & 0.60\\\\\n        \n               \n        Fraction of informed agents & $\\xi$ & 0.33\\\\\n        Probability of buy\/sell by uninformed agents &$\\zeta$ & 0.33\\\\\n        PMM order cancellation size parameter & $\\eta_{o,c}$ & 0.04\\\\\n        Fundamental order size parameter & $\\eta_{f}$ & 0.04\\\\\n        Chartist order size parameter & $\\eta_{c}$ & 0.04\\\\\n        Noise price parameter &$\\eta_{u}$& 0.04 \\\\\n        Transaction cost penalty & $\\daleth$& 0.06\\\\\n        \n        \n        \\midrule \n        Number of layers (DAE\/LSTM) & $N_{L}$ & 3\/3\\\\\n        Number of hidden unit per layer & $N_{H}$1024\\\\\n        Learning rate for pretraining & $\\alpha_{LPT}$ &0.05 \\\\\n        Learning rate for fine-tuning &$\\alpha_{LFT}$& 0.10\\\\\n        Number of pretraining epochs & $\\epsilon_{PT}$&100\\\\\n        Additive isotropic Gaussian noise & $\\eta_{noise}$& $\\sim \\mathcal{N}(0,0.50)$ \\\\\n            \n        \\bottomrule\\bottomrule\n        \n    \\end{tabular}}\n\\end{table}\n\n\\section{Results}\n\\label{sec:DHP_R}\nIn this section, we investigate the performance of market making agents\nin predicting types of order book events and their timestamps. Having\nlearned which orders to send and at what time, we evaluate the agent's\ntrading performance in the simulation framework. We tweak order\ncancellations to examine the impact on the agent's profitability and the\nmicrostructure of the order book. We then check the robustness of the\nmodel by performing sensitivity analysis. Finally, we validate our\nsimulation framework by reproducing stylised facts with our simulated\ndata.\n\n\\subsection{Predictive Performance}\nGiven a stream of order book events $\\{t_n, \\varkappa_n, k_n\\}_{n \\in \\mathbb{N}}$, the market makers seek to predict the next event type and its time. We evaluate predictive performance of $\\hat{t}_{n}$ and $\\hat{k}_{n}$ using RMSE and ER, respectively.To avoid getting entangled in the problem of overfitting, we divide the training set into the sub-training and validation sets. We train DHP and NHP models on the sub-training set so as to choose hyperparameters for validation set. Following the training procedure of \\cite{Mei2017}, we generate the predictive performance of the market making agents on reconstructed order book data at nanosecond resolution in Figure \\ref{fig:PTEN}. As is evident from the figure, neither model is invariably better at predicting events or, in particular, time. It seems that the both models do not explicitly address the complex dynamics of asynchronous order book data at nanosecond resolution. The event dynamics at nanosecond timestamps need much more sophisticated models to filter noise and to model event interaction and non-linearity.\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/IET_N}\n  \\caption{Time Prediction Graph}\\label{fig:IETN}\n      \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/RMSE_N}\n  \\caption{Time Prediction Error}\\label{fig:RMSEN}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/ER_N}\n  \\caption{Event Prediction Error}\\label{fig:ERN}\n      \\end{subfigure}\n          \n   \\caption{Performance evaluation of high-frequency market making agents in predicting order book events and time at nanosecond resolution. The standard deviation over 10 experiments using different train-val-test sample is denoted by error bar. }\\label{fig:PTEN}\n   \\end{figure} \n\n\nIn Figure \\ref{fig:PTEM}, we evaluate the predictive performance of the high-frequency market making agents using millisecond data sampled from the reconstructed order book. Compared to the earlier results, the DHP model's performance has drastically increased. In addition, the time prediction is consistently better than with the NHP. The deep model with novel architecture and pre-training module are sophisticated enough to capture excitation and feedback effects in the order book. The results also substantiate the claim of the NHP model regarding stochastically missing data. The order book events at millisecond timestamps theoretically omit the events at the finer time resolution, but they are generated by a different mechanisms. This is the reason why there are completely different results at the two time resolutions. The Deep Hawkes model presented here is expressive enough to learn true predictive distribution with scholastically missing events, but only if they are generated from the same mechanism. By integrating the predictive  capabilities into the market making strategies, the agents trade in a simulation framework populated with heterogeneous trading strategies. In the next section, we explore the agents' trading performance.\n \n \\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/IET_M}\n  \\caption{Time Prediction Graph}\\label{fig:IETM}\n      \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/RMSE_M}\n  \\caption{Time Prediction Error}\\label{fig:RMSEM}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/ER_M}\n  \\caption{Event Prediction Error}\\label{fig:ERM}\n      \\end{subfigure}\n          \n   \\caption{Performance evaluation of high-frequency market making agents in predicting order book events and time at millisecond resolution. The standard deviation over 10 experiments using different train-val-test sample is denoted by error bar. }\\label{fig:PTEM}\n \n\\end{figure} \n\n\\subsection{Trading Performance}\n\nThe trading performance of the high-frequency market making agents is discussed in  Table \\ref{tab:DHP_AP}. Our simulation framework evaluates various models, including the Neural Hawkes model, the benchmark probabilistic estimate model, and our proposed Deep Hawkes model. According to the performance metrics specified in Table \\ref{tab:DHP_AP}, our proposed agent using DHP (DHMM) consistently outperforms the PMM and NHMM, which suggests that the proposed trading agent benefits by learning the robust microstructure of order book data. Further, the DHP lets the agent capture the self- and cross-excitation effects of the limit order book together with a feedback loop, to place the right order at the right time. We discuss the performance of each agent in detail below.\n\nAs shown in Figure \\ref{fig:PTEM}, the DHMM is better at predicting the type of order and its time compared to NHMM. This is an important element\nof the market making strategy. The novel DLSTM-SDAE architecture allows the agents to learn the hidden representation of the noisy order book data, and therefore to place orders that add to its profitability. Furthermore, DHMM exhibits a faster convergence rate compared to\nNHMM, as shown in Figure \\ref{fig:DHP_TD}.\n\nThe baseline strategy used by PMM maintains a probability density\nestimated on the basis of fundamental price. The fundamental price\nevolves according to the jump process, following a normal distribution.\nThis works in the favor of PMM, which makes more profit at the\nbeginning, as verified in Figure \\ref{fig:DHP_TD}.Over time, however, the DHMM and NHMM learn the art of placing the right order with the right\nintensity. Afterwards, the profitability of the PMM fall dramatically.\nThe PMM might perform better if it took a long position over several\ndays, rather than trading intraday. It would be interesting to check the\nperformance of the PMM agents with different probability density\nestimate conditions on the joint distribution of microstructure features.  \n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/RTR}\n  \\caption{Train}\\label{fig:DHP_RTR}\n      \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/RTS}\n  \\caption{Test}\\label{fig:DHP_RTS}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/RDH}\n  \\caption{Random day}\\label{fig:DHP_RDH}\n      \\end{subfigure}\n          \n   \\caption{Trading agents performance with DHMM, NHMM and PMM while training, testing and random day. }\\label{fig:DHP_TD}\n   \\end{figure} \n   \n   \\begin{table}[H]\n   \\centering\n\\caption{Mean and standard deviation on the daily normalised PnL (PnL) and mean absolute positions (MAP) for different market makers  }\\label{tab:DHP_AP}\n \\resizebox{\\textwidth}{!}{  \\begin{tabular}{cllllll}\n    \\toprule\n    \\toprule\n    \\multirow{2}{*}{Agents} & \n      \\multicolumn{2}{c}{NPnL [$10^5$]} &\n      \\multicolumn{2}{c}{MAP[unit]} \\\\\n      \\cmidrule(l){2-3} \\cmidrule(l){4-5}\n    & Mean & Std.Dev. & Mean & Std.Dev. \\\\\n    \\midrule\n    DHMM & 2.1 & $\\pm 18.26$ & 17 & $\\pm 20$  \\\\\n    \\midrule\n    NHMM & 1.1 & $\\pm 4.09$ & 4 & $\\pm 6$ \\\\\n    \\midrule\n    PMM & -1.6  & $\\pm 79.55$ & 41 & $\\pm 74$  \\\\\n   \\bottomrule\n    \\bottomrule\n  \\end{tabular}\n}  \n\\end{table}\n \n\\subsection{Order Cancellation Effect} \n\nMassive numbers of order cancellations in a short period are a\ndistinctive attribute of the equity market. For example, at Nasdaq\nNordic, order cancellations typically account for 40\\% of submitted limit\norders on a particular trading day. Market making strategies using limit\norder cancellations contribute to the market marker's profit, bid-ask\nspread and order queue position \\citep{Dahlstrm2018}.We study the\ndistribution of profit, bid-ask spread and order queue position by\nremoving the cancellations mechanism in the base simulation\nframework. We estimate the intrinsic value of the order relative to the\nqueue position by applying the model developed by \\cite{Moallemi2017}. The agent's order queue position provides an estimate of number\nof orders ahead of the agent's order at a particular price. A position at\nthe front of the queue guarantees prompt execution, higher fill rate, low\nlatency and lower adverse selection cost. We estimate the queue position\nin the order book by reconstructing the limit order book from the\nsimulated data feed.\n\nLets us suppose that the high-frequency market making agent places a limit order at time $t\\!=0$ seeking best ask price $p_a$ which gets filled or canceled at time $\\tau$. Filling the order pays the agent $p_a$ while cancellations pay nothing. We now describe the value of the order perceived by agents relative to the queue position as:\n\n\\begin{equation}\n\\begin{split}\nV_t &= \\mathbb{E}[  (p_a - p_t)\\mathbb{I}_{\\mathsf{FILL}} -  (p - p_t)\\mathbb{I}_{\\mathsf{FILL}}  \\mid \\mathcal{F}_t ] \\\\\n&= \\mathsf{FP}_t (\\mathsf{LSP}_t - \\mathsf{ASC}_t) \\\\\n&= fill \\; probablity \\left( liquidity \\; spread \\; premium - adverse \\; selection \\; cost \\right)\n\\end{split}\n\\label{EQ:COE}\n\\end{equation} \n\nwhere \n\\begin{eqnarray*}\n\\begin{aligned}\n\\!& \\mathsf{FP}_t  \\triangleq  \\mathbb{P} \\left( \\mathsf{FILL} \\mid \\mathcal{F}_t \\right), \\\\\n\\!& \\mathsf{LSP}_t  \\triangleq  \\left( p_a - p_t \\right), \\\\\n\\!& \\mathsf{ASC}_t  \\triangleq  \\mathbb{E} \\left[ (p_{\\tau} - p_t) \\mid \\mathcal{F}_t, \\mathsf{FILL} \\right]\n\\end{aligned}\n\\end{eqnarray*} \n\nTo empirically calibrate the model (Equation \\ref{EQ:COE}), we take the same parameters used by \\cite{Moallemi2017}. These are exponential order size distribution, trade arrival rate (TAR), average trades size (ATS), trade size in the stan-dard lot (TSS), cancellation arrival rate (CAR), average cancellation size (ACS), price jump arrival rate (PJR), average jump size (AJS), market impact (MI) and average queue size (AQS). The trade size is identified as the limit order or market order, contrary to aggressive market orders as described by \\cite{Moallemi2017}. Table \\ref{tab:OCE} specifies the estimated parameters for simulated data with no cancellation mechanisms (Simulated NC), without cancellation mechanisms (Simulated WC) and an average (Simulated AV) over 21 days. The paper itself provides more detail regarding the parameters, calibration and model fitting \\citep{Moallemi2017}.\n\n\\begin{table}[H]\n\\caption{Estimated parameters for simulated orderbook data.}\\label{tab:OCE}\n\\resizebox{\\textwidth}{!}{  \\begin{tabular}{cccccccccc}\n    \\toprule\n    \\toprule\n    \n     Data & TAR (\/min)& ATS (shares) & TSS (shares) & CAR (\/min) & ACS (shares) & PJR (\/min) & AJS (ticks) & MI   & AQS (shares) \\\\ [0.5ex]\n    \\midrule\n    Simulated NC & 2.53 & 3467 &7664 & 92.72  & 5061 & 1.26 & 0.32 & 10.91 & 16416  \\\\\n    Simulated AV & 2.04 & 4037 & 6901 & 82.21 & 4022 & 1.01 & 0.46 & 8.76 & 23554  \\\\\n    \\midrule\n    Simulated WC & 5.26 & 6329 & 8083 & 43.71 & 1107 & 3.75 & 2.06 & 11.92 & 40191\\\\\n     Simulated AV & 4.07 & 5463 & 9147 & 40.31 & 1560 & 3.01 & 2.06 & 13.02 & 46815 \\\\\n   \\bottomrule\n    \\bottomrule\n  \\end{tabular}}\n  \n\\end{table}\n\nTable \\ref{tab:OCE} shows that an absence of cancellation mechanisms at the high-frequency market maker's end leads to a drastic increase in the average queue size. The decrease in the cancellation rate increases the queue size, which affects the high-frequency market maker's profitability, bid-ask spread and market impact. The value of the order as a function of the queue position, bid-ask spread, and the agent's profit efficiently captures the claim illustrated by the data in Figure \\ref{fig:OCE}. The wider bid-ask spread when agent's are unable to cancel the limit order in Figure \\ref{fig:BAD_A} has negative effect on the profitability (Figure \\ref{fig:RDH_A} ) as compared to scenarios with cancellations (Figure \\ref{fig:BAD_B},\\ref{fig:RDH_B} ). As stated in the model in Equation \\ref{EQ:COE}, the value of an order that is not filled is zero. Figure \\ref{fig:VO_A} shows that an increase in queue length decreases the probability of execution, and therefore the value. The value of the order becomes flat, as the queue length is extremely large. Our results are consistent with the findings of \\cite{Dahlstrm2018} when investigating the determinants of order cancellations. It is difficult to infer causal relationships between cancellations and market microstructure variables based on artificially created scenarios, but this approach nonetheless it paves the way for future investigation using order level data.\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/BAD_B}\n  \\caption{BAD WC}\\label{fig:BAD_B}\n      \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/RDH_B}\n  \\caption{DHMMP WC}\\label{fig:RDH_B}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/VO_B}\n  \\caption{VOPQ WC}\\label{fig:VO_B}\n      \\end{subfigure}\n          \n      \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/BAD_A}\n  \\caption{BAD NC}\\label{fig:BAD_A}\n      \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/RDH_A}\n  \\caption{DHMMP NC }\\label{fig:RDH_A}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/VO_A}\n  \\caption{VOPQ NC}\\label{fig:VO_A}\n      \\end{subfigure}\n          \n   \\caption{Effect of limit order cancellations on the market. The top\nrow (marked WC) represents the distribution when the market maker's\nagents can cancel the limit orders. The bottom row (marked NC)\nrepresents a situation with no cancellations. BAD is intraday bid-ask\ndistribution, DHMMP is profit distribution of DHMM over the trading\nday, and VOPQ is the value of the orders relative to queue position. The\naverage queue length on a particular trading day is represented by a\nblack triangle.}\\label{fig:OCE}\n   \\end{figure} \n \n\n\\subsection{Sensitivity Analysis}\nWe perform sensitivity analysis on the proposed model by changing the\nhyperparameters \u2013 specifically, the number of layers, the number of\nhidden units per layer and the noise level. The aim is to confirm the\nrobustness of the model, rather than overfitting parameters. Table \\ref{tab:SA} shows the performance of our proposed model when there is an increase in the number of layers, hidden units and noise level. The model is robust to the noise level at the optimal choice of numbers of layers, parameters and hidden units.  \n\n\\begin{table}[H]\n\\caption{Sensitivity to the number of hidden units and Gaussian noise.\nThe DLSTM-SDAE used in our model has 3 DAE layers and 3 LSTM layers. In performing sensitivity analysis, we fix the 3 LSTM layers and\nchange only the DAE layer. }\\label{tab:SA}\n \\resizebox{\\textwidth}{!}{ \\begin{tabular}{llll}\n    \\toprule\n    \\toprule\n    & \\# Number of layer 1 & \\# Number of layer 2 & \\# Number of layer 3 \\\\\n    \\midrule \n    Number of hidden unit per layer & $\\left(64, 128, 256, 512, 1024 \\right)$ & $\\left(64, 128, 256, 512, 1024 \\right)$ & $\\left(64, 128, 256, 512, 1024 \\right)$ \\\\\n    \n    \\midrule\n    RMSE &  $\\left( 4.6, 4.4, 4.1, 4.1, 4.0 \\right)$ & $\\left( 3.5, 3.0, 3.0, 2.5, 2.5 \\right)$ & $\\left( 2.0, 2.0, 2.0, 1.5, 1.5 \\right)$ \\\\\n    \\midrule\n    ER & $\\left( 55.7, 55.7, 55.7, 50.4, 54.0 \\right)$ & $\\left( 47.2, 44.4, 44.1, 44.1, 44.1 \\right)$ & $\\left( 37.5, 37.0, 35.0, 35.0, 34.0 \\right)$ \\\\\n   \\midrule\n   Gaussian noise & $\\left(0.10, 0.20, 0.30, 0.40, 0.50 \\right)$ & $\\left(0.10, 0.20, 0.30, 0.40, 0.50 \\right)$& $\\left(0.10, 0.20, 0.30, 0.40, 0.50 \\right)$\\\\\n   \\midrule\n    RMSE & $\\left(7.2,5.9, 4.3, 5.4, 6.5 \\right)$& $\\left(4.1,3.9, 3.0, 4.0, 4.6 \\right)$&$\\left(2.2,2.2, 2.0, 2.1, 2.1 \\right)$ \\\\\n    \\midrule\n    ER & $\\left(60.2,55.9, 50.1, 60.0, 63.5 \\right)$ & $\\left(55.6,52.2, 49.1, 55.3, 67.5 \\right)$& $\\left(37.2,37.9, 37.3, 37.1, 37.2 \\right)$ \\\\\n    \n     \\bottomrule\n    \\bottomrule\n  \\end{tabular}}\n  \n\\end{table}\n\n\n\\subsection{Validation}\n\nThe validation of the trading simulation framework is performed by\nmeasuring how successfully the simulation's output exhibits persis- tent\nempirical patterns in the order book data. Such empirical patterns are\ncommon across various markets and instruments, and even timescales\nare often classified as \"stylised facts\" \\citep{Cont2001}. We present a\nnominal set of stylised facts, reproduced from the empirical analysis of\nsimulated order book data, as shown in Figure \\ref{fig:DHP_SF}.\n\nLet $p(t)$ be the price of a security at time $t$. Given a timescale $\\Delta t$, we define log return at $\\Delta t $ as $ r (t, \\Delta t) = \\ln p(t+\\Delta t) - \\ln p(t)$. The cumulative distribution (CDF) of returns is given as $F_{\\Delta t}(x) = \\mathbb{P}[r(t, \\Delta t) \\leq x]$. The derivative of the earlier gives probability density function (PDF) $F^{'}_{\\Delta t}=f_{\\Delta t}$, empirically estimated for normalized simulated return, as illustrated in Figure \\ref{fig:DHP_N}. The cumulative distribution of return follows power law $F_{\\Delta t} \\sim \\left|r\\right|^{-\\alpha }$ with $2<\\alpha<5$. In Figure \\ref{fig:DHP_NCDF},  the positive tail $F^{+}_{\\Delta t}(x) = \\mathbb{P}[r(t,\\Delta t) \\geq x]$ and the negative tail $F^{-}_{\\Delta t}(x) = \\mathbb{P}[r(t,\\Delta t) \\leq x]$ of cumulative distribution, shown as yellow circles and green squares, exhibit power law, as denoted by the red line with $\\alpha=2.8$. In Figure \\ref{fig:DHP_SFAC}, we show the absence of the autocorrelation  of price change, defined as $\\rho (\\tau) = Corr \\left( r (t, \\Delta t ), r (t + \\tau, \\Delta t )  \\right) $.  The autocorrelation function (ACF) drastically decays to zero in few lags. \n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/SF_Normal}\n  \\caption{PDF}\\label{fig:DHP_N}\n      \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/SF_NCDF}\n  \\caption{CDF}\\label{fig:DHP_NCDF}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.32\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{img\/SF_AC}\n  \\caption{ACF}\\label{fig:DHP_SFAC}\n      \\end{subfigure}\n          \n   \\caption{Stylised facts reproduced from simulated order book data. All graphs were generated on the basis of $\\Delta t = 1\\; minute $. }\\label{fig:DHP_SF}\n   \\end{figure} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhoor b/data_all_eng_slimpj/shuffled/split2/finalzzhoor
new file mode 100644
index 0000000000000000000000000000000000000000..a40d29fa45c1e18919bba0390b8f1d7aa7cabddb
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzhoor
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nLet $H$ be a real separable Hilbert space and $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$ be a stochastic basis satisfiying the usual conditions, $L=(L(t))_{t\\geq 0}$ be a square-integrable cylindrical L\\'evy process in a real separable Hilbert space $U$ with respect to the stochastic basis $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$, taking values in a possibly larger Hilbert space $U_1\\supset U$, and $B:U\\to H$ is a bounded linear operator. Consider an $H$-valued stochastic convolution process\n\\begin{equation}\\label{eq:X}\nX(t)=E(t)X_0+\\int_0^tE(t-s)B\\,\\ensuremath{\\mathrm{d}} L(s)\n\\end{equation}\nwhere $(E(t))_{t\\in [0,T]}$ is a family of bounded linear operators on $H$ and $X_0$ is an $\\cF_0$-measurable $H$-valued random variable. Without loss of generality, all Hilbert spaces are assumed to be infinite-dimensional.\nImportant examples of such processes are weak solutions $(X(t))_{t\\geq0}$ of certain stochastic partial differential equations (SPDEs, for short) driven by additive L\\'{e}vy noise; these can be written as abstract It\\^o stochastic differential equations\n\\begin{equation}\\label{eq:mainEq}\n\\ensuremath{\\mathrm{d}} X(t)+AX(t)\\,\\ensuremath{\\mathrm{d}} t=B\\,\\ensuremath{\\mathrm{d}} L(t),\\quad t\\geq0;\\quad X(0)=X_0,\n\\end{equation}\nwhere $-A$ is the generator of a strongly continuous semigroup $(E(t))_{t\\geq 0}$ on $H$. In particular, we consider the stochastic heat equation\n\\begin{equation}\\label{eq:SHE}\n\\ensuremath{\\mathrm{d}} X(t)+\\Lambda X(t)\\ensuremath{\\mathrm{d}} t= \\ensuremath{\\mathrm{d}} L(t),\\quad t\\geq0;\\quad X(0)=X_0,\n\\end{equation}\nand the stochastic wave equation, written as a first order system,\n\\begin{equation}\\label{eq:SWE}\n\\begin{aligned}\n\\ensuremath{\\mathrm{d}} X_1(t) -X_2(t)\\ensuremath{\\mathrm{d}} t&=0, &&t\\geq0;\\quad X_1(0)=X_{0,1},\\\\\n\\ensuremath{\\mathrm{d}} X_2(t)+\\Lambda X_1(t)\\ensuremath{\\mathrm{d}} t&= \\ensuremath{\\mathrm{d}} L(t),&&t\\geq0;\\quad X_2(0)=X_{0,2}.\n\\end{aligned}\n\\end{equation}\nIn both cases $\\Lambda:=-\\Delta=-\\sum_{j=1}^d\\partial^2\/\\partial\\xi_j^2$ is the Laplace operator on $L^2(\\ensuremath{\\mathcal{O}})$ where $\\ensuremath{\\mathcal{O}}\\subset\\bR^d$ is a bounded domain.\nFor the precise abstract setup of these equations we refer to Sections \\ref{sec:Heat} and \\ref{sec:Wave}.\nIn general, however, we do not require that $(E(t))_{t\\geq 0}$ enjoys the semigroup property so that the abstract framework can accommodate Volterra-type evolution equations as well, see Subsection \\ref{subsec:sve} for details.\n\nConsider an\napproximation $\\tilde X=(\\tilde X(t))_{t\\in [0,T]}$ of the process $(X(t))_{t\\in[0,T]}$ given by\n\\begin{equation}\\label{eq:Xtilde}\n\\tilde X(t)=\\tilde E(t)X_0+\\int_0^t\\tilde E(t-s)B\\,\\ensuremath{\\mathrm{d}} L(s),\n\\end{equation}\nwhere $(\\tilde E(t))_{t\\in[0,T]}$ is a family of bounded linear operators on $H$, which is again not necessarily (extendable to) an operator semigroup. For example, the family  $(\\tilde E(t))_{t\\in[0,T]}$ may be a time-interpolated solution operator family of a space-time discretized stochastic evolution problem, when $H$ is an $L^2$-space of some spatial domain $\\ensuremath{\\mathcal{O}}$. We study the so-called weak error\n\\begin{equation}\\label{eq:defWeakError}\ne(T):=\\bE\\big(G(\\tilde X(T))-G(X(T))\\big)\n\\end{equation}\nfor suitable test functions $G:H\\to \\bR$. At the heart of the paper are the error representation formulae for $e(T)$, Theorem \\ref{thm:errorRep} and Corollary \\ref{cor:errorRep}. The proof of Theorem \\ref{thm:errorRep} is\nbased on Kolmogorov's backward equation for the martingale $Y(t)=E(T)X_0+\\int_0^tE(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s)$, $t\\in[0,T]$, which has the important property that $Y(T)=X(T)$. The introduction of such an auxiliary process $Y$ is well-known for equations with Gaussian noise and has been used by many authors in a weak error analysis, see, for example \\cite{debussche_printems,KovLarLin11,KovLarLin13,KovPri14b} to mention just a few (compare also \\cite{CJK14, DPJR12}). However, the extension of those arguments is not straightforward and the resulting error representation formula differs from the one in the Gaussian case in \\cite{KovLarLin13}. One of the difficulties in the general L\\'evy case (in contrast to the Gaussian case) is that there are no readily available, sufficiently general results on  Kolmogorov's backward equation to suit our analysis. We remedy this, at least for $Y$ as above, in Proposition \\ref{prop:backwardKolm}. Another complication arises from the fact that we use tools from the theory of stochastic integration based on two different settings. One, where we integrate operator-valued processes w.r.t.\\ a Hilbert space-valued L\\'evy process, promoted in the monographs \\cite[Chapter 8]{PesZab07}, \\cite{Met82,MetPel80}, and another one where we integrate Hilbert space-valued integrands w.r.t.\\ a Poisson random measure \\cite{MRT13,Pre10}. The problem occurs because our setting for stochastic differential equations is based on the first approach while the proof of the error representation formula is based on an It\\^o formula which appears in \\cite[Theorem 3.6]{MRT13}; the latter form is well suited for our purposes, but it is formulated using the second approach for stochastic integration. Therefore, in the appendix we link the two stochastic integrals so that we can use the results from both theories.\n\nUsing the abstract error representation we study the weak error of  space-time discretizations for parabolic equations, such as the stochastic heat equation and a stochastic Volterra integro-differential equation, and a hyperbolic equation, the stochastic wave equation. As space discretization we employ a standard continuous finite element method. For the stochastic heat equation we use the backward Euler method, for the Volterra integro-differential equation the backward Euler method combined with a convolution quadrature, and for the stochastic wave equation an $I$-stable rational approximation of the exponential function, such as the Crank-Nicolson scheme, as time integrators. For all equations considered here, the Hilbert-Schmidt norm condition $$\\nnrm{\\Lambda^{(\\beta-1\/\\rho)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(L^2(\\ensuremath{\\mathcal{O}}))}<\\infty, \\quad \\beta>0,$$ determines the rate of convergence, where $\\rho=1$ for the heat and wave equations and $\\rho\\in (1,2)$ for the Volterra equation. Here $U=L^2(\\ensuremath{\\mathcal{O}})$ and $Q\\in\\ensuremath{\\mathscr L}(L^2(\\ensuremath{\\mathcal{O}}))$ is the covariance operator of $L$ as introduced in Section~\\ref{sec:DrivingProcess}.\n\nFor the stochastic heat equation, we show in Theorem \\ref{thm:SHE} that for twice continuously differentiable test functions with bounded second derivatives the rate of weak convergence is essentially twice that of strong convergence.  It is at least $O(h^{2\\beta}+(\\ensuremath{\\Delta t})^\\beta)$, $\\beta\\in(0,1]$, modulo a logarithmic term, where $h$ and $\\ensuremath{\\Delta t}$ are the space- and time-discretization parameters, respectively. This extends the corresponding result from \\cite{LinSch13}, where, in contrast to the present paper, the analysis is restricted to so-called impulsive cylindrical processes on $L^2(\\ensuremath{\\mathcal{O}})$ as driving noise. Moreover, there is a serious restriction on the jump size intensity measure in \\cite[Section~6]{LinSch13}\nadmitting only processes of bounded variation (on finite time intervals).\nHere, the only restriction we have on $L$ is that it is square-integrable, non-Gaussian and has mean zero. Furthermore, we also remove the boundedness assumption on the test functions and their first derivative.\n\nIn Subsection \\ref{subsec:sve} we briefly discuss a stochastic Volterra-type integro-differential equation and obtain a weak rate of order at least $O(h^{2\\beta}+(\\ensuremath{\\Delta t})^{\\rho\\beta})$, $\\beta\\in(0,1\/\\rho]$, where $\\rho$ depends on the order of the convolution kernel appearing in the equation.\n\n\nFor the stochastic wave equation the additional technical condition \\eqref{eq:assgSWE} has to be imposed in order to prove that the weak order is twice the strong order. The order of weak convergence is found to be at least $O(h^{\\min(r, 2\\beta\\frac{r}{r+1})}+(\\ensuremath{\\Delta t})^{\\min(1,2\\beta\\frac{p}{p+1})})$, see Theorem~\\ref{thm:SWE}. Here $p$ and $r$ are the classical orders of the time-discretization and of the finite element method. We would like to point out that, while the extra condition \\eqref{eq:assgSWE} on the second derivative on the test function is restrictive, it trivially holds for the important function $g(x)=\\|x\\|_{L^2(\\ensuremath{\\mathcal{O}})}^2$. We also give a sufficient condition \\eqref{eq:weqii} for the above rate of weak convergence to hold which only involves the jump intensity measure of $L$ and $\\Lambda$, with no further restriction on the test functions. Condition  \\eqref{eq:weqii} is sufficient for $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(L^2(\\ensuremath{\\mathcal{O}}))}<\\infty$ and, in special cases, it is even equivalent to it, see Example~\\ref{ex:spec}.\nFurthermore, as far as the authors know, there are no results available in the literature concerning weak approximation of hyperbolic stochastic partial differential equations driven by L\\'evy noise.\n\nLet us remark that weak error estimates for approximations of L\\'{e}vy-driven stochastic ordinary differential equations have been considered by various authors, see, e.g.~\\cite{JKMP05, MikZha11, PlaBru10, ProTal97} and the references therein. There also exists a series of papers on strong error estimates for approximations of SPDEs  driven by L\\'{e}vy processes or Poisson random measures, see, for example \\cite{Bar10, BarLan12a, BarLan12b, DunHauPro12, Hau08, HauMar06, Lan10} and compare also Remarks \\ref{rem:strongErrorSHE} and \\ref{rem:strongErrorSWE} below. However, to the best of our knowledge, the first steps in a \\emph{weak} error analysis for L\\'{e}vy-driven SPDEs have been done only recently in the already mentioned article \\cite{LinSch13}.\n\nThe present paper\nis organized as follows. In Section \\ref{sec:Setting and preliminaries} we describe the abstract framework of the paper, introduce infinite-dimensional L\\'evy processes with several examples and a framework for linear stochastic partial differential equations driven by additive L\\'evy noise. Assumption \\ref{ass:abstractSetting} summarizes the main assumptions for the general setting of the paper. In Section \\ref{sec:An error representation formula} we state and prove two representation formulae, Theorem~\\ref{thm:errorRep} and Corollary~\\ref{cor:errorRep},  for $e(T)$ given by \\eqref{eq:defWeakError}. The main ingredient in their proofs is Proposition~\\ref{prop:backwardKolm} on Kolmogorov's backward equation. We use the representation formula from Corollary \\ref{cor:errorRep} to establish weak convergence rates for a space-time discretization scheme first for parabolic equations in Section \\ref{sec:Heat}, where we consider the stochastic heat equation (Subsection~\\ref{subsec:she}) and, in less detail, a stochastic Volterra-type integro-differential equation (Subsection~\\ref{subsec:sve}). Then, in Section \\ref{sec:Wave}, we analyse the weak error of a numerical scheme for the stochastic wave equation. In the appendix we link stochastic integration with respect to Poisson random measures to integration with respect to infinite-dimensional L\\'evy processes.\n\n\n\n\n\n\n\n\n\\section{Setting and preliminaries}\n\\label{sec:Setting and preliminaries}\n\nHere we describe in detail our abstract setting and collect some background material from infinite-dimensional stochastic analysis.\\\\\n\n\\noindent\n{\\bf General notation.}\nLet $(\\cH,\\langle\\arg,\\arg\\rangle_{\\cH})$ and $(\\cG,\\langle\\arg,\\arg\\rangle_{\\cG})$ be real, separable Hilbert spaces and denote by $\\ensuremath{\\mathscr L}(\\cH,\\cG)$, $\\ensuremath{\\mathscr L_1}(\\cH,\\cG)$ and $\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$ the spaces of linear and bounded operators, nuclear operators and Hilbert-Schmidt operators from $\\cH$ to $\\cG$, respectively.  The corresonding norms are denoted by $\\nnrm{\\arg}{\\ensuremath{\\mathscr L}(\\cH,\\cG)}$, $\\nnrm{\\arg}{\\ensuremath{\\mathscr L_1}(\\cH,\\cG)}$ and $\\nnrm{\\arg}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}$. If $\\cH=\\cG$, we write $\\ensuremath{\\mathscr L}(\\cH)$, $\\ensuremath{\\mathscr L_1}(\\cH)$ and $\\ensuremath{\\mathscr L_2}(\\cH)$ instead of $\\ensuremath{\\mathscr L}(\\cH,\\cH)$, $\\ensuremath{\\mathscr L_1}(\\cH,\\cH)$ and $\\ensuremath{\\mathscr L_2}(\\cH,\\cH)$. Given a measure space $(M,\\cM,\\mu)$ and $1\\leq p <\\infty$, we denote by $L^p(M;\\cH)=L^p(M,\\cM,\\mu;\\cH)$ the space of all $\\cM\/\\cB(\\cH)$-measurable mappings $f:M\\to\\cH$ with finite norm $\\nnrm{f}{L^p(M;\\cH)}=(\\bE\\nnrm{f}{\\cH}^p)^{1\/p}$, where $\\cB(\\cH)$ denotes the Borel $\\sigma$-algebra on the Hilbert space $\\cH$. By $C^n(\\cH,\\bR)$ we denote the space of all $n$-times continuously Fr\\'{e}chet differentiable functions $f:\\cH\\to\\bR,\\,x\\mapsto f(x)$. By $C_{\\text b}^n(\\cH,\\bR)$ we denote the subspace of functions from $C^n(\\cH,\\bR)$ which are bounded together with their derivatives. Identifying $\\cH$ and $\\ensuremath{\\mathscr L}(\\cH,\\bR)$ via the Riesz isomorphism, we consider for fixed $x\\in\\cH$ the first derivative  $f'(x)$ as an element of $\\cH$. Similarly, the second derivative $f''(x)$ is considered as an element of $\\ensuremath{\\mathscr L}(\\cH)$.\nWe also write $f_x$ and $f_{xx}$ instead of $f'$ and $f''$.\n\n\n\\subsection{The driving L\\'{e}vy process $L$}\n\\label{sec:DrivingProcess}\n\nThe process $L=(L(t))_{t\\geq0}$ in Eq.~\\eqref{eq:mainEq} is a  L\\'{e}vy process with values in a real and separable Hilbert space $U_1$, defined on a filtered probability space $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$ satisfying the usual conditions (cf.~\\cite{PesZab07}).\n$L$ is $(\\cF_t)$-adapted and for $t,h\\geq0$ the increment $L(t+h)-L(t)$ is independent of $\\cF_t$. We always consider a c\\`{a}dl\\`{a}g\n(right continuous with left limits) modification of $L$, i.e., a modification such that $L(t)=\\lim_{s\\searrow t}L(s)$ for all $t\\geq0$ and $L(t-):=\\lim_{s\\nearrow t}L(s)$ exists for all $t>0$, where the limits are pathwise limits in $U_1$.\nOur standard reference for Hilbert space-valued L\\'{e}vy processes is \\cite{PesZab07}.\n\nIn order to keep the exposition simple, we assume that $L$ is square-integrable, i.e., $\\bE\\nnrm{L(t)}{U_1}^2<\\infty$, and that the Gaussian part of $L$ vanishes. Moreover, we assume that $L$ has mean zero, i.e., $\\bE L(t)=0$ in $U_1$.\nLet $\\nu$ be the jump intensity measure (L\\'{e}vy measure) of $L$. Note that the jump intensity measure $\\nu$ of a general L\\'{e}vy process in $U_1$ satisfies $\\nu(\\{0\\})=0$ and $\\int_{U_1}\\min(1,\\nnrm{y}{U_1}^2)\\nu(dy)<\\infty$, cf.~\\cite[Section 4]{PesZab07}. Due to our assumptions we have\n\\begin{equation}\\label{eq:assnu}\n\\int_{U_1}\\nnrm{y}{U_1}^2\\nu(\\ensuremath{\\mathrm{d}} y)<\\infty,\n\\end{equation}\nand the characteristic function of $L$ is given by\n\\begin{equation}\\label{eq:charFunctionL}\n\\bE e^{i\\langle x,L(t)\\rangle_{U_1}}=\\exp\\Big\\{-t\\int_{U_1}\\big(1-e^{i\\langle x,y\\rangle_{U_1}}+i\\langle x,y\\rangle_{U_1}\\big)\\nu(\\ensuremath{\\mathrm{d}} y)\\Big\\},\\quad t\\geq0,\\;x\\in U_1.\n\\end{equation}\nConversely, any $U_1$-valued L\\'{e}vy process $L$ satisfying \\eqref{eq:assnu} and \\eqref{eq:charFunctionL} is square-integrable, with mean zero and vanishing Gaussian part.\n\n\n\nLet $Q_1\\in\\ensuremath{\\mathscr L_1}(U_1)$ be the covariance operator of $L$.\nIt is determined by the jump intensity measure $\\nu$ via\n\\begin{equation}\\label{eq:Q1nu}\n\\langle Q_1x,y\\rangle_{U_1}=\\int_{U_1}\\langle x,z\\rangle_{U_1}\\langle y,z\\rangle_{U_1}\\nu(\\ensuremath{\\mathrm{d}} z),\n\\quad x,y\\in U_1,\n\\end{equation}\nsee~\\cite[Theorem 4.47]{PesZab07}.\nFurther, let\n\\begin{equation*}\n(U_0,\\langle\\arg,\\arg\\rangle_{U_0}):=\\big(Q_1^{1\/2}(U_1),\\langle Q_1^{-1\/2}\\arg,Q_1^{-1\/2}\\arg\\rangle_{U_1}\\big)\n\\end{equation*}\nbe the reproducing kernel Hilbert space of $L$, where $Q_1^{-1\/2}$ denotes the pseudo-inverse of $Q_1^{1\/2}$, see~\\cite[Section 7]{PesZab07}. Recall that the operator $B$ in Eq.~\\eqref{eq:mainEq} is defined on the Hilbert space $U$. We assume that\n\\begin{equation}\\label{eq:U0subsetU}\nU_0\\subset U\\subset U_1,\n\\end{equation}\nand that the inclusions \\eqref{eq:U0subsetU} define continuous embeddings. We denote the embedding of $U_0$ into $U$ by $J_0\\in \\ensuremath{\\mathscr L}(U_0,U)$ and set\n\\begin{equation}\\label{eq:defQ}\nQ:=J_0J_0^*\\in\\ensuremath{\\mathscr L}(U).\n\\end{equation}\nThe nonnegative and symmetric  operator $Q$ is the covariance operator of $L$ considered as a cylindrical process in $U$, cf.~Remark~\\ref{rem:Q} below. As a consequence of Douglas' theorem as stated in \\cite[Appendix A.4]{PesZab07}, compare also \\cite[Corollary C.0.6]{PreRoeck07}, the reproducing kernel Hilbert space of $L$ has the alternative representation\n\\begin{equation*}\n(U_0,\\langle\\arg,\\arg\\rangle_{U_0})=\\big(Q^{1\/2}(U),\\langle Q^{-1\/2}\\arg,Q^{-1\/2}\\arg\\rangle_U\\big).\n\\end{equation*}\n\n\\begin{remark}\\label{rem:Q}\nSuppose w.l.o.g.\\ that $U$ is dense in $U_1$, identify $U$ and $U^*$ via the Riesz isomorphism, and consider the Gelfand triple $U_1^*\\subset U^*\\equiv U\\subset U_1$. Then it is not difficult to see that\n\\[\\bE\\langle L(t),x\\rangle\\langle L(t),y\\rangle=t\\langle Qx,y\\rangle_U,\\quad t\\geq0,\\;x,y\\in U_1^*,\\]\nwhere $\\langle\\arg,\\arg\\rangle:U_1\\times U_1^*\\to\\bR$ is the canonical dual pairing; compare \\cite[Proposition 7.7]{PesZab07}.\nThe unique continuous extensions of the linear mappings $U_1^*\\ni x\\mapsto \\langle L(t),x\\rangle\\in L^2(\\bP)$, $t\\geq0$, to the larger space $U^*$ determine a $2$-cylindrical $U$-process in the sense of \\cite{MetPel80}, compare also \\cite{AppRie10}, \\cite{Rie12}, \\cite{Rie14}.\n\\end{remark}\n\n\\begin{remark}\\label{rem:approxL}\nUnlike in the case of a mean-zero (cylindrical) $Q$-Wiener process in $U$, the covariance operators $Q\\in\\ensuremath{\\mathscr L}(U)$ and $Q_1\\in\\ensuremath{\\mathscr L_1}(U_1)$ do \\emph{not} determine the distribution of the L\\'{e}vy process $L$, but the jump intensity measure $\\nu$ does so according to \\eqref{eq:charFunctionL}. Note that the law of a general L\\'{e}vy process is determined by its characteristics (L\\'{e}vy triplet), cf.~\\cite[Definition~4.28]{PesZab07}, and that the characteristics of $L$ are $(-\\int_{\\{\\nnrm{y}{U_1}\\geq1\\}}y\\,\\nu(\\ensuremath{\\mathrm{d}} y),0,\\nu)$. Nevertheless, the operator $Q$ in $\\eqref{eq:defQ}$ will play an important role in our error analysis.\nLet us briefly make the connection of our setting to the construction of a cylindrical $Q$-Wiener process in $U$ as described in \\cite{DPZ92}, \\cite{PreRoeck07}. To this end, let $(f_k)_{k\\in\\bN}$ be an orthonormal basis of $U_1$ consisting of eigenvectors of $Q_1$ with eigenvalues $(\\lambda_k)_{k\\in\\bN}$ and consider the orthonormal basis $(e_k)_{k\\in\\bN}$ of $U_0$ given by $e_k:=\\lambda_k^{1\/2}f_k$. To simplify notation we suppose for the moment that all eigenvalues $\\lambda_k$ of $Q_1$ are strictly positive. Then, compare \\cite[Section~4.8]{PesZab07}, the real-valued L\\'{e}vy processes $L_k=(L_k(t))_{t\\geq0}$, $k\\in\\bN$, given by\n\\[L_k(t):=\\lambda_k^{-1\/2}\\langle L(t),f_k\\rangle_{U_1}\\]\nare uncorrelated, i.e., $\\bE L_k(t)L_j(s)=0$ if $k\\neq j$, they satisfy $\\bE(L_k^2(t))=t$, and we have\n\\begin{equation}\\label{eq:LPexpansion}\nL(t)=\\sum_{k\\in\\bN}L_k(t)e_k.\n\\end{equation}\nThe infinite sum in \\eqref{eq:LPexpansion} converges for all finite $T>0$ in the space $\\cM^2_T(U_1)$ of  c\\`{a}dl\\`{a}g square-integrable $U_1$-valued $(\\cF_t)$-martingales $M=(M(t))_{t\\in[0,T]}$ with norm $\\nnrm{M}{\\cM^2_T(U_1)}=(\\bE\\nnrm{M(T)}{U_1}^2)^{1\/2}$. In contrast to the Gaussian case, where uncorrelated coordinates are always independent, the coordinate processes $L_k$, $k\\in\\bN$, are in general only uncorrelated but \\emph{not} independent.\n\nConversely, suppose that we are given an arbitrary symmetric and nonnegative operator $Q\\in\\ensuremath{\\mathscr L}(U)$, an orthonormal basis $(e_k)_{k\\in\\bN}$ of $U_0=Q^{1\/2}(U)$, and a family $L_k$, $k\\in\\bN$, of real-valued L\\'{e}vy processes on $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$ that satisfy the following conditions:\n\\begin{itemize}\n\\item Each $L_k$ is $(\\mathcal F_t)$-adapted and for $t,h\\geq0$ the increment $L_k(t+h)-L_k(t)$ is independent of $\\cF_t$;\n\\item each $L_k$ is square-integrable with $\\bE L_k(t)=0$ and $\\bE(L_k^2(t))=t$;\n\\item the processes $L_k$, $k\\in\\bN$, are uncorrelated;\n\\item for all $n\\in\\bN$ the $\\bR^n$-valued process $((L_1(t),\\ldots,L_n(t))^\\top)_{t\\geq 0}$ is a L\\'{e}vy process;\n\\item the Gaussian part of each $L_k$ is zero.\n\\end{itemize}\nThen, if $U_1$ is a Hilbert space containing $U$ such that the natural embedding of $U_0=Q^{1\/2}(U)$ into $U_1$ is Hilbert-Schmidt, the infinite sum in \\eqref{eq:LPexpansion} converges in $\\cM^2_T(U_1)$ and defines a L\\'{e}vy process $L$ with reproducing kernel Hilbert space $U_0$ that fits into our setting.\n\\end{remark}\n\nWe end this subsection with some examples of L\\'{e}vy processes $L$. We suppose that all processes are defined relative to the stochastic basis $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$ and that their increments on time intervals $[t,t+h]$ are independent of $\\cF_t$.\n\n\\begin{example}(\\emph{Subordinate cylindrical $\\widetilde Q$-Wiener process})\nLet $W=(W(t))_{t\\geq0}$ be a cylindrical $\\widetilde Q$-Wiener process in $U$ in the sense of \\cite[Section 2.5.1]{PreRoeck07}, where $\\widetilde Q\\in\\ensuremath{\\mathscr L}(U)$ is a given nonnegative and symmetric operator. Assume that $W$ takes values in a possibly larger Hilbert space $U_1\\supset U$ such that the natural embedding of $U$ into $U_1$ is continuous. Let $\\widetilde Q_1\\in\\ensuremath{\\mathscr L_1}(U_1)$ be the covariance operator of $W$ considered as a Wiener process in $U_1$, i.e., $\\bE\\langle W(t),x\\rangle_{U_1}\\langle W(s),y\\rangle_{U_1}=\\min(s,t)\\langle \\widetilde Q_1x,y\\rangle_{U_1}$ for $x,\\,y\\in U_1$, $s,t\\geq0$. Let $Z=(Z(t))_{t\\geq0}$ be a subordinator, i.e., a real-valued increasing L\\'{e}vy process in the sense of \\cite[Definition 21.4]{Sato13}, \\cite{SSV10}. Assume that $W$ and $Z$ are independent, that the drift of $Z$ is zero, and that the jump intensity measure  $\\rho$ of $Z$ satisfies\n\\begin{equation}\\label{eq:measureSubordinator}\n\\int_0^\\infty s\\,\\rho(\\ensuremath{\\mathrm{d}} s)<\\infty.\n\\end{equation}\nThe latter is equivalent to assuming that $Z$ has first moments, $\\bE|Z(t)|<\\infty$. According to \\cite[Remark~21.6]{Sato13}, the Laplace tranform of $Z(t)$ is given by\n\\begin{equation}\\label{eq:LaplaceSubordinator}\n\\bE(e^{-r Z(t)})=\\exp\\big(-t\\int_0^\\infty(1-e^{-rs})\\rho(\\ensuremath{\\mathrm{d}} s)\\big),\\quad r\\geq0.\n\\end{equation}\nIn this situation, subordinate cylindrical Brownian motion\n\\begin{equation*}\\label{eq:subWP}\nL(t):=W(Z(t)),\\quad t\\geq0,\n\\end{equation*}\ndefines a $U_1$-valued L\\'{e}vy process $L=(L(t))_{t\\geq0}$ that fits into the general framework described above. Indeed, $L$ has stationary and independent increments. Moreover, the independence of $W$ and $Z$, the identity $\\bE e^{i\\langle x,W(s)\\rangle_{U_1}}=e^{-s\\frac12 \\langle \\widetilde Q_1x,x\\rangle_{U_1}}$, Eq.~\\eqref{eq:LaplaceSubordinator} and the symmetry of the distribution $\\bP_{W(1)}=N(0,Q_1)$ imply that characteristic function of $L(t)$ is given by\n\\begin{align*}\n\\bE e^{i\\langle x,L(t)\\rangle_{U_1}}\n&=\\int_0^\\infty e^{-s\\frac12 \\langle\\widetilde Q_1x,x\\rangle_{U_1}}\\,\\bP_{Z(t)}(\\ensuremath{\\mathrm{d}} s)\\\\\n&=\\exp\\Big[-t\\int_0^\\infty(1-e^{-\\frac12\\langle\\widetilde Q_1x,x\\rangle_{U_1}s})\\rho(\\ensuremath{\\mathrm{d}} s)\\Big]\\\\\n&=\\exp\\Big[-t\\int_0^\\infty\\int_{U_1}(1-e^{i\\langle x,\\sqrt s y\\rangle_{U_1}}+i\\langle x,\\sqrt s y\\rangle_{U_1})\\bP_{W(1)}(\\ensuremath{\\mathrm{d}} y)\\rho(\\ensuremath{\\mathrm{d}} s)\\Big].\n\\end{align*}\nAs a consequence, \\eqref{eq:charFunctionL} holds with\n\\begin{equation}\\label{eq:subWPnu}\n\\nu=(\\bP_{W(1)}\\otimes\\rho)\\circ\\kappa^{-1},\n\\end{equation}\nwhere $\\kappa:U_1\\times(0,\\infty)\\to U_1$ is defined by $\\kappa(y,s)=\\sqrt s y$; compare \\cite[Lemma 4.8]{Rie14}. (Note that, by the scaling property of $W$, \\eqref{eq:subWPnu} is equivalent to the standard formula $\\nu=\\int_0^\\infty\\bP_{W(s)}\\,\\rho(\\ensuremath{\\mathrm{d}} s)$, where the measure-valued integral is defined in a weak sense, cf.~\\cite[Section~30]{Sato13}). Moreover, \\eqref{eq:assnu} holds due to \\eqref{eq:measureSubordinator} as we have the equality\n$\\int_{U_1}\\nnrm{y}{U_1}^2\\nu(\\ensuremath{\\mathrm{d}} y)=\\int_0^\\infty s\\,\\rho(\\ensuremath{\\mathrm{d}} s)\\,\\bE(\\nnrm{W(1)}{U_1}^2)$ according to \\eqref{eq:subWPnu}.\nIt follows that $L$ is a $U_1$-valued, square-integrable, mean-zero L\\'{e}vy process with vanishing Gaussian part. It is also not difficult to show that the covariance operators $Q_1\\in\\ensuremath{\\mathscr L_1}(U_1)$ and $Q\\in\\ensuremath{\\mathscr L}(U)$ of $L$ in \\eqref{eq:Q1nu} and \\eqref{eq:defQ} are given by\n$Q_1=\\int_0^\\infty s\\,\\rho(\\ensuremath{\\mathrm{d}} s)\\,\\widetilde Q_1$ and $Q=\\int_0^\\infty s\\,\\rho(\\ensuremath{\\mathrm{d}} s)\\,\\widetilde Q$.\nSubordinate cylindrical Wiener processes have been considered, e.g., in \\cite{BrzZab10}.\n\\end{example}\n\n\\begin{example}(\\emph{Independent one-dimensional L\\'{e}vy processes})\\label{ex:1dlevy}\nLet $Q\\in\\ensuremath{\\mathscr L}(U)$ be symmetric, nonnegative and let $(e_k)_{k\\in\\bN}$ be an orthonormal basis of $U_0:=Q^{1\/2}(U)\\subset U$. Let $L_k=(L(t))_{k\\in\\bN}$, $k\\in\\bN$, be independent real-valued square-integrable L\\'{e}vy processes with vanishing Gaussian part and $\\bE L_k(t)=0$, $\\bE(L_k^2(t))=t$. Let $U_1\\supset U$ be another Hilbert space such that the natural embedding of $U_0$ into $U_1$ is a Hilbert-Schmidt operator. Then, the series \\eqref{eq:LPexpansion} converges for all $T\\in(0,\\infty)$ in the space $\\cM^2_T(U_1)$ and defines a L\\'{e}vy process $L=(L(t))_{t\\geq0}$ satisfiying \\eqref{eq:assnu} and \\eqref{eq:charFunctionL} with jump intensity measure\n\\[\\nu=\\sum_{k\\in\\bN}\\nu_k\\circ\\pi_k^{-1},\\]\nwhere $\\nu_k$ is the L\\'{e}vy measure of $L_k$ and $\\pi_k:\\bR\\to U_1$ is defined by $\\pi_k(\\xi):=\\xi e_k$; compare \\cite[Section 4.8.1]{PesZab07}.\n\\end{example}\n\n\\begin{example}(\\emph{Impulsive cylindrical process})\nLet $\\mu$ be a L\\'{e}vy measure on $\\bR$ such that $\\int_\\bR\\sigma^2\\mu(\\ensuremath{\\mathrm{d}}\\sigma)<\\infty$. Let $\\cO\\subset\\bR^d$ be a bounded domain and $Z=(Z(t))_{t\\geq0}$ an impulsive cylindrical process on $U:=L^2(\\cO)=L^2(\\cO,\\cB(\\cO),\\lambda^d)$ with jump size intensity $\\mu$ in the sense of \\cite[Definition 7.23]{PesZab07}. Here, $\\lambda^d$ denotes $d$-dimensional Lebesgue measure.  The process $Z$ is a measure-valued process defined, informally, by $Z(t,\\ensuremath{\\mathrm{d}}\\xi)=\\int_0^t\\int_\\bR\\sigma\\hat\\pi(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}}\\xi,\\ensuremath{\\mathrm{d}} \\sigma)$, where $\\hat\\pi$ is a compensated Poisson random measure on $[0,\\infty)\\times\\cO\\times\\bR$ with reference measure $\\lambda^1\\otimes\\lambda^d\\otimes\\mu$; see \\cite[Section 7.2]{PesZab07} for details. Let $\\widetilde Q\\in\\ensuremath{\\mathscr L}(U)$ be symmetric and nonnegative, $(b_k)_{k\\in\\bN}$ an orthonormal basis of $U$, and $U_1\\supset U$ a Hilbert space such that the natural embedding of $U_0=\\tilde Q^{1\/2}(U)\\subset U$ into $U_1$ is Hilbert-Schmidt. Then the series\n\\begin{equation}\\label{eq:imulsiveProcess}\nL(t):=\\widetilde Q^{\\frac12} Z(t):=\\sum_{k\\in\\bN}\\int_0^t\\int_\\cO\\int_\\bR\\sigma b_k(\\xi)\\hat\\pi(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}}\\xi,\\ensuremath{\\mathrm{d}} \\sigma)\\widetilde Q^{\\frac12}b_k,\\quad t\\geq0,\n\\end{equation}\nconverges for all $T\\in(0,\\infty)$ in $\\cM^2_T(U_1)$ and defines a L\\'{e}vy process that fits into our general framework with $Q=\\int_\\bR\\sigma^2\\mu(\\ensuremath{\\mathrm{d}}\\sigma)\\widetilde Q$ and\n\\[\\nu=(\\lambda^d\\otimes\\mu)\\circ \\phi^{-1},\\]\nwhere $\\phi\\in L^2(\\cO\\times\\bR,\\lambda^d\\otimes\\mu;U_1)$ is defined by $\\phi(\\xi,\\sigma)=\\sum_{n\\in\\bN}\\sigma b_k(\\xi)\\widetilde Q^{\\frac12}b_k$ (convergence in $L^2(\\cO\\times\\bR,\\lambda^d\\otimes\\mu;U_1)$).\nIn \\cite{LinSch13} we considered the weak approximation of the stochastic heat equation driven by an impulsive process of the form \\eqref{eq:imulsiveProcess}. The results in Section~\\ref{subsec:she} of the present article improve the results of \\cite{LinSch13} in several aspects.\n\\end{example}\n\n\\subsection{Linear stochastic evolution equations with additive noise}\n\\label{sec:LSEE}\n\nWe are mainly interested in equations of the type \\eqref{eq:mainEq}, where $A:D(A)\\subset H\\to H$ is an unbounded linear operator such that $-A$ is the generator of a strongly continuous semigroup $(E(t))_{t\\geq0}\\subset\\ensuremath{\\mathscr L}(H)$, $B\\in\\ensuremath{\\mathscr L}(U,H)$, $L=(L(t))_{t\\geq0}$ is a square-integrable L\\'{e}vy process with reproducing kernel Hilbert space $U_0\\subset U$ as described in Subsection~\\ref{sec:DrivingProcess}, and $X_0\\in L^2(\\Omega,\\cF_0,\\bP;H)$. It is well known that if\n\\begin{equation}\\label{eq:assEB1}\n\\int_0^T \\nnrm{E(t)B}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\ensuremath{\\mathrm{d}} t <\\infty\n\\end{equation}\nfor some (and hence for all) $T>0$, then there exists a unique weak solution $X=(X(t))_{t\\geq0}$ to \\eqref{eq:mainEq} which is given by the variation-of-constants formula \\eqref{eq:X}, see, e.g., \\cite[Chapter~9]{PesZab07}.\nSimilarly, if $(\\tilde E(t))_{t\\in[0,T]}\\subset\\ensuremath{\\mathscr L}(H)$\nis given by some approximation scheme such that $t\\mapsto\\tilde E(t)B$ is a measurable mapping from $[0,T]$ to $\\ensuremath{\\mathscr L_2}(U_0,H)$, then the condition\n\\begin{equation}\\label{eq:assEBtilde1}\n\\int_0^T \\nnrm{\\tilde E(t)B}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\ensuremath{\\mathrm{d}} t <\\infty\n\\end{equation}\nensures that the approximation $\\tilde X=(\\tilde X(t))_{t\\in[0,T]}$ of $(X(t))_{t\\in[0,T]}$ in \\eqref{eq:Xtilde} exists as a square-integrable $H$-valued process. We refer to \\cite[Chapter~8]{PesZab07} for details on the construction and properties of the stochastic integral w.r.t.\\ Hilbert space-valued L\\'{e}vy processes.\n\nIt turns out that our general error representation formula for the weak error $e(T)$ in \\eqref{eq:defWeakError} does not require the semigroup property of the strongly continuous family of operators $(E(t))_{t\\geq0}$. This paves the way for analysing a more general class of L\\'{e}vy-driven linear stochastic evolution equations, including for example stochastic Volterra-type equations as considered in \\cite{KovPri14a}, \\cite{KovPri14b} for the Gaussian case, see Subsection \\ref{subsec:sve} for an example.  For such equations, the weak solution still has the form  \\eqref{eq:X} but the solution operator family $(E(t))_{t\\geq0}\\subset\\ensuremath{\\mathscr L}(H)$ is not a semigroup anymore.  Therefore, we weaken our abstract assumptions and summarize them as follows.\n\n\\begin{assumption} \\label{ass:abstractSetting}\nWe will use the following assumptions:\n\\begin{compactenum}[(i)]\n\\item $H$, $U$ and $U_1$ are real and separable Hilbert spaces;\n\\item\n$L=(L(t))_{t\\geq0}$ is a $U_1$-valued L\\'{e}vy process on $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$ with zero mean and finite second moments and reproducing kernel Hilbert space $U_0$ such that $U_0\\subset U\\subset U_1$ as described in Subsection~\\ref{sec:DrivingProcess};\n\\item\n$X_0\\in L^2(\\Omega,\\cF_0,\\bP;H)$;\n\\item\n$B\\in\\ensuremath{\\mathscr L}(U,H)$ and $(E(t))_{t\\in[0,T]}\\subset \\ensuremath{\\mathscr L}(H)$ is a strongly continuous family of linear operators such that \\eqref{eq:assEB1} holds;\n\\item\nfor all $\\epsilon>0$ there exists $\\Phi_\\epsilon\\in\\ensuremath{\\mathscr L_2}(U_0,H)$ and $C_\\epsilon>0$ such that\n\\[\\nnrm{E(t)Bx}{H}\\leq \\nnrm{\\Phi_\\epsilon x}H,\\quad (t,x)\\in[\\epsilon,T]\\times U_0;\\]\n\\item\n$(\\tilde E(t))_{t\\in[0,T]}\\subset\\ensuremath{\\mathscr L}(H)$ is a family of linear operators such that $t\\mapsto\\tilde E(t)B$ is a measurable mapping from $[0,T]$ to $\\ensuremath{\\mathscr L_2}(U_0,H)$ and \\eqref{eq:assEBtilde1} holds;\n\\item\n$X=(X(t))_{t\\in[0,T]}$ and $\\tilde X=(\\tilde X(t))_{t\\in[0,T]}$ are $H$-valued stochastic processes given by \\eqref{eq:X} and \\eqref{eq:Xtilde}.\n\\end{compactenum}\n\\end{assumption}\n\n\\begin{remark}\nIf $(E(t))_{t\\geq0}$ is an operator semigroup, then \\ref{ass:abstractSetting}(v) is a consequence of \\ref{ass:abstractSetting}(iv). Indeed, if \\eqref{eq:assEB1} holds then $E(t_0)B\\in \\ensuremath{\\mathscr L_2}(U_0,H)$ for some $t_0\\in (0,\\epsilon)$ and hence\n\\[\\nnrm{E(t)Bx}{H}\\le \\|E(t_0)Bx\\|_H\\sup_{t\\in [t_0,T]}\\|E(t-t_0)\\|_{\\ensuremath{\\mathscr L}(H)},\\quad(t,x)\\in[\\epsilon,T]\\times U_0.\\]\nHence, one may take $\\Phi_{\\epsilon}:=cE(t_0)B$ where $c:=\\sup_{t\\in [t_0,T]}\\|E(t-t_0)\\|_{\\ensuremath{\\mathscr L}(H)}$.\n\\end{remark}\n\nTo fix notation, let us briefly recall the It\\^{o} isometry for stochastic integrals w.r.t.\\ $L$. It has the same form as the It\\^{o} isometry for stochastic integrals w.r.t.\\ Hilbert space-valued Wiener processes.\nWe set $\\Omega_T:=\\Omega\\times[0,T]$ and $\\bP_T:=\\bP\\otimes\\lambda$, where $\\lambda$ is Lebesgue measure on $[0,T]$. The predictable $\\sigma$-algebra on $\\Omega_T$ w.r.t.\\ $(\\cF_t)_{t\\in[0,T]}$ is denoted by $\\cP_T$. For operator-valued processes $\\Phi=(\\Phi(t))_{t\\in[0,T]}$ in\n\\[L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H)):=L^2(\\Omega_T,\\cP_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H)),\\]\nwe have\n\\begin{equation}\\label{eq:ItoIsom}\n\\bE\\Big(\\gnnrm{\\int_0^t\\Phi(s)\\ensuremath{\\mathrm{d}} L(s)}H^2\\Big)=\\bE\\int_0^t\\nnrm{\\Phi(s)}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\ensuremath{\\mathrm{d}} s,\\quad t\\in[0,T],\n\\end{equation}\nand the integral process $(\\int_0^t\\Phi(s)\\ensuremath{\\mathrm{d}} L(s))_{t\\in[0,T]}$ belongs to the space $\\cM^2_T(H)$ of  c\\`{a}dl\\`{a}g square-integrable $H$-valued $(\\cF_t)$-martingales. The usual norm in $\\cM^2_T(H)$ is defined by $\\nnrm{M}{\\cM^2_T(H)}=(\\bE\\nnrm{M(T)}{H}^2)^{1\/2}$, $M=(M(t))_{t\\in[0,T]}\\in\\cM^2_T(H)$. Note, however, that the integral processes given by the stochastic integrals in \\eqref{eq:X} and \\eqref{eq:Xtilde} are in general \\emph{not} martingales since the (deterministic) operator-valued integrands also depend on $t$.\n\nWe also recall\nthe definition and some properties of Hilbert-Schmidt operators,\ncf.~\\cite[Chapter~6]{Wei80}. Let $\\cH$ and $\\cG$ be real and separable Hilbert spaces. A linear and bounded operator $C\\in\\ensuremath{\\mathscr L}(\\cH,\\cG)$ belongs to the space $\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$ of Hilbert-Schmidt operators if\n\\[\\nnrm{C}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}:=\\Big(\\sum_{k\\in\\bN}\\nnrm{Ch_k}{\\cG}^2\\Big)^{1\/2}<\\infty\\]\nfor some (and hence for every) orthonormal basis $(h_k)_{k\\in\\bN}$ of $\\cH$. If $C\\in\\ensuremath{\\mathscr L}(\\cH,\\cG)$ and $C^*\\in\\ensuremath{\\mathscr L}(\\cG,\\cH)$ is the adjoint operator, then $C\\in\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$ if and only if $C^*\\in\\ensuremath{\\mathscr L_2}(\\cG,\\cH)$ and one has\n\\begin{equation}\\label{eq:propHSadjoint}\n\\nnrm{C}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}=\\nnrm{C^*}{\\ensuremath{\\mathscr L_2}(\\cG,\\cH)}.\n\\end{equation}\nAlso, if $C\\in\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$, $D\\in\\ensuremath{\\mathscr L}(\\cH)$ and $F\\in\\ensuremath{\\mathscr L}(\\cG)$, then obviously $CD\\in\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$, $FC\\in\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$ and\n\\begin{equation}\\label{eq:estHS}\n\\nnrm{CD}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}\\leq\\nnrm{C}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}\\nnrm{D}{\\ensuremath{\\mathscr L}(\\cH)},\\quad\n\\nnrm{FC}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}\\leq\\nnrm{F}{\\ensuremath{\\mathscr L}(\\cG)}\\nnrm{C}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}.\n\\end{equation}\nIn particular, in our setting we have $\\ensuremath{\\mathscr L}(U_1,H)\\subset\\ensuremath{\\mathscr L_2}(U_0,H)$ since\n\\[\\nnrm{C}{\\ensuremath{\\mathscr L_2}(U_0,H)}=\\nnrm{CQ_1^{1\/2}}{\\ensuremath{\\mathscr L_2}(U_1,H)}\\leq\\nnrm{C}{\\ensuremath{\\mathscr L}(U_1,H)}\\nnrm{Q_1^{1\/2}}{\\ensuremath{\\mathscr L_2}(U_1)}\\]\nfor all $C\\in\\ensuremath{\\mathscr L}(U_1,H)$ and $\\nnrm{Q_1^{1\/2}}{\\ensuremath{\\mathscr L_2}(U_1)}=\\text{Tr\\,}Q_1=\\nnrm{Q_1}{\\ensuremath{\\mathscr L_1}{(U_1)}}<\\infty$.\n\n\\section{An error representation formula}\n\\label{sec:An error representation formula}\n\nIn this section, we state and prove a general representation formula for the weak approximation error $e(T)$ in \\eqref{eq:defWeakError} within the abstract setting described above.\n\n\\subsection{Formulation of the result}\n\nFor the formulation and the proof of the error representation formula, we introduce auxiliary drift-free It\\^{o} processes $Y=(Y(t))_{t\\in[0,T]}$ and $\\tilde Y=(\\tilde Y(t))_{t\\in[0,T]}$ such that\n\\begin{equation*}\nX(T)=Y(T),\\quad \\tilde X(T)=\\tilde Y(T).\n\\end{equation*}\nThe processes $Y$ and $\\tilde Y$ are constructed by applying to $X$ and $\\tilde X$ the deterministic operator-valued processes $(E(T-t))_{t\\in[0,T]}$ and $(\\tilde E(T-t))_{t\\in[0,T]}$. That is, we set\n\\begin{equation}\\label{eq:defY}\nY(t):=E(T)X_0+\\int_0^tE(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s), \\quad t\\in[0,T],\n\\end{equation}\nand\n\\begin{equation}\\label{eq:defYtilde}\n\\tilde Y(t):=\\tilde E(T)X_0+\\int_0^t\\tilde E(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s), \\quad t\\in[0,T].\n\\end{equation}\n\nMoreover, we consider the auxiliary problem\n\\[\\ensuremath{\\mathrm{d}} Z(t)=E(T-t)B\\,\\ensuremath{\\mathrm{d}} L(t),\\quad t\\in[\\tau,T];\\qquad Z(\\tau)=\\xi,\\]\nwhere $\\tau\\in[0,T)$ and $\\xi$ is an $H$-valued $\\cF_\\tau$-measurable random variable. Its solution is given by\n\\begin{equation}\\label{eq:defZttauxi}\nZ(t,\\tau,\\xi):=\\xi+\\int_{\\tau}^t E(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s),\\quad t\\in[\\tau,T],\n\\end{equation}\nand we use it to define for $G\\in C^2(H,\\bR)$ with $\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$ a function $u:[0,T]\\times H\\to\\bR$ by\n\\begin{equation}\\label{eq:defuxt}\nu(t,x):=\\bE\\,G(Z(T,t,x)),\\quad (t,x)\\in [0,T]\\times H.\n\\end{equation}\nNote that the boundedness of $G''$ implies quadratic and linear growth of $G$ and $G'$, respectively. That is,\n\\begin{equation}\\label{eq:growthGG'}\n|G(x)|\\leq C(1+\\nnrm{x}H^2),\\quad G'(x)\\leq C(1+\\nnrm{x}H)\n\\end{equation}\nfor all $x\\in H$ and a constant $C\\in(0,\\infty)$ that does not depend on $x$.\nIt is also not difficult to see that $u$\nis twice Fr\\'{e}chet differentiable w.r.t. $x$ and we have\n\\begin{equation}\\label{eq:u_xu_xx}\nu_x(t,x)=\\bE\\,G'(Z(T,t,x)),\\quad u_{xx}(t,x)=\\bE\\,G''(Z(T,t,x)).\n\\end{equation}\nAll expectations appearing in \\eqref{eq:defuxt} and \\eqref{eq:u_xu_xx} make sense due to \\eqref{eq:growthGG'}, the assumption $\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$, \\eqref{eq:assEB1} and It\\^{o}'s isometry \\eqref{eq:ItoIsom}.\n\nBefore stating the representation formula, we show in the following lemma how operators in $\\ensuremath{\\mathscr L_2}(U_0,H)$ can be identified with functions in\n\\begin{equation*}\nL^2(U_1,\\nu;H):=L^2(U_1,\\mathcal B(U_1),\\nu;H)\n\\end{equation*}\nand how processes in $L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$ can be identified with elements in\n\\begin{equation*}\nL^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H):=L^2(\\Omega_T\\times U_1,\\cP_T\\otimes \\cB(U_1),\\bP_T\\otimes\\nu;H).\n\\end{equation*}\nThese identifications will be used implicitly throughout this article, see Remark~\\ref{not:Phix} below. They also lead to a generic identification of integrals w.r.t.\\ (cylindrical) Hilbert space-valued L\\'{e}vy processes of jump type and integrals w.r.t.\\ the associated Poisson random measures, cf.\\ Appendix~\\ref{sec:PRM+SI}.\n\n\n\\begin{lemma}\\label{lem:integrandIsomorphism}\nLet $(f_k)_{k\\in\\bN}\\subset U_0$ be an orthonormal basis of $U_1$ consisting of eigenvectors of the covariance operator $Q_1\\in\\ensuremath{\\mathscr L_1}(U_1)$ of $L$ and let $(\\lambda_k)_{k\\in\\bN}\\subset[0,\\infty)$ be the corresponding sequence of eigenvalues\n\n\\textbf{(i)}\nGiven $\\Phi\\in\\ensuremath{\\mathscr L_2}(U_0,H)$, the series\n\\[\\iota(\\Phi):=\\sum_{k\\in\\bN,\\lambda_k\\neq 0}\\langle\\arg,f_k\\rangle_{U_1}\\Phi f_k\\]\nconverges in $L^2(U_1,\\nu; H)$.\n\nThe linear mapping\n\\[\\iota:\\ensuremath{\\mathscr L_2}(U_0,H)\\to L^2(U_1,\\nu;H),\\;\\Phi\\mapsto \\iota(\\Phi)\\]\nis an isometric embedding.\n\n\\textbf{(ii)}\nGiven $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$, the series\n\\[\\kappa(\\Phi):=\\sum_{k\\in\\bN,\\lambda_k\\neq0}\\langle\\arg,f_k\\rangle_{U_1}\\Phi(\\arg) f_k\\]\nconverges in $L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H)$.\nThe linear mapping\n\\[\\kappa:L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))\\to L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H),\\;\\Phi\\mapsto \\kappa(\\Phi)\\]\nis an isometric embedding.  For $\\bP_T$-almost all $(\\omega,t)\\in\\Omega_T$ we have\n$\\kappa(\\Phi)(\\omega,t,\\arg)=\\iota(\\Phi(\\omega,t))$\nin $L^2(U_1,\\nu;H)$, where $\\iota$ is the embedding from \\textup{(i)}.\n\\end{lemma}\n\n\\begin{proof\n\\textbf{(i)}\nW.l.o.g.\\ all eigenvalues $\\lambda_k$ are strictly positive. Let $(e_k)_{k\\in\\bN}$ be the orthonormal basis of $U_0$ given by $e_k:=\\lambda_k^{1\/2}f_k$. For $m,n\\in\\bN$ with $m\\leq n$ we have\n\\begin{align*}\n\\sgnnrm{\\sum_{k=m}^n\\langle\\arg,f_k\\rangle_{U_1}\\Phi f_k}{L^2(U_1,\\nu;H)}^2\n&=\\int_{U_1}\\sgnnrm{\\sum_{k=m}^n\\langle x,f_k\\rangle_{U_1}\\Phi f_k}{H}^2\\nu(\\ensuremath{\\mathrm{d}} x)\\\\\n&= \\sum_{j,k=m}^n\\lambda_j^{-1\/2}\\lambda_k^{-1\/2}\\int_{U_1}\\langle x,f_j\\rangle_{U_1}\\langle x,f_k\\rangle_{U_1}\\nu(\\ensuremath{\\mathrm{d}} x)\\,\\langle\\Phi e_j,\\Phi e_k\\rangle_H\\\\\n&=\\sum_{k=m}^n\\nnrm{\\Phi e_k}H^2;\n\\end{align*}\nin the last step we used \\eqref{eq:Q1nu}. Since $\\sum_{k\\in\\bN}\\nnrm{\\Phi e_k}H^2=\\nnrm{\\Phi}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2<\\infty$, this shows that the partial sums $\\sum_{k=1}^n\\langle\\arg,f_k\\rangle_{U_1}\\Phi f_k$, $n\\in\\bN$, are a Cauchy sequence in $L^2(U_1,\\nu;H)$ and\n\\[\\sgnnrm{\\sum_{k=1}^\\infty\\langle\\arg,f_k\\rangle_{U_1}\\Phi f_k}{L^2(U_1,\\nu;H)}=\\nnrm{\\Phi}{\\ensuremath{\\mathscr L_2}(U_0,H)}.\\]\n\n\n\\textbf{(ii)} The first two assertions can be shown as in the proof of (i). The last assertion is due the fact that the iterated integral\n\\[\\int_\\Omega\\int_0^T\\int_{U_1}\\nnrm{\\iota(\\Phi(\\omega,t))(x)-\\kappa(\\Phi)(\\omega,t,x)}H^2\\,\\nu(\\ensuremath{\\mathrm{d}} x)\\,\\ensuremath{\\mathrm{d}} t\\,\\bP(\\ensuremath{\\mathrm{d}}\\omega)\\]\nequals zero, which follows from an approximation argument.\n\\end{proof}\n\n\\begin{remark}\\label{not:Phix}\nFrom now on we will\nidentify operators $\\Phi\\in\\ensuremath{\\mathscr L_2}(U_0,H)$ with the corresponding mappings $\\iota(\\Phi)\\in L^2(U_1,\\nu;H)$ and write\n\\[\\Phi x=\\iota(\\Phi)(x),\\quad x\\in U_1.\\]\nAnalogously, we identify processes $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$ with the corresponding mappings $\\kappa(\\Phi)\\in L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H)$ and write\n\\[\\Phi(\\omega,t)x=\\kappa(\\Phi)(\\omega,t,x),\\quad (\\omega,t,x)\\in \\Omega_T\\times U_1.\\]\nFor processes $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$ both identifications\nare compatible $\\bP\\otimes\\lambda$-almost everywhere on $\\Omega_T$ in the sense that we have $\\kappa(\\Phi)(\\omega,t,\\arg)=\\iota(\\Phi(\\omega,t))$\nin $L^2(U_1,\\nu;H)$ for $\\bP\\otimes\\lambda$-almost all $(\\omega,t)\\in\\Omega_T$.\n\\end{remark}\n\nHere is the main result of this section.\n\\begin{theorem}\\label{thm:errorRep}\nLet Assumption~\\ref{ass:abstractSetting} hold and $G\\in C^2(H,\\bR)$ with $\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$.\nThen, for the process $(\\tilde Y(t))_{t\\in[0,T]}$ from \\eqref{eq:defYtilde} and the function $u:[0,T]\\times H\\to\\bR$ from \\eqref{eq:defuxt} it holds that\n\\begin{equation}\\label{eq:errorRep1}\n\\begin{aligned}\n\\bE\\int_0^T\\int_{U_1}&\\Big|u\\big(t,\\tilde Y(t)+\\tilde E(T-t) By\\big)-u\\big(t,\\tilde Y(t)+E(T-t)By\\big)\\\\\n&-\\big\\langle u_x(t,\\tilde Y(t)),\\big(\\tilde E(T-t) B-E(T-t)B\\big)y\\big\\rangle_H\\Big|\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\;<\\;\\infty.\n\\end{aligned}\n\\end{equation}\nThe weak error $e(T)$ in \\eqref{eq:defWeakError} has the representation\n\\begin{equation}\\label{eq:errorRep2}\n\\begin{aligned}\ne(T)&=\\bE\\big\\{u(0,\\tilde E(T)X_0)-u(0,E(T)X_0)\\big\\}\\\\\n&\\quad +\\bE\\int_0^T\\int_{U_1}\\Big\\{u\\big(t,\\tilde Y(t)+\\tilde E(T-t) By\\big)-u\\big(t,\\tilde Y(t)+E(T-t)By\\big)\\\\\n&\\hspace{3.5cm}-\\big\\langle u_x(t,\\tilde Y(t)),\\big(\\tilde E(T-t) B-E(T-t)B\\big)y\\big\\rangle_H\\Big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t.\n\\end{aligned}\n\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\nThe terms $E(T-t)By$ and $\\tilde E(T-t) By$ appearing in \\eqref{eq:errorRep1} and \\eqref{eq:errorRep2} are defined for $\\lambda\\otimes\\nu$-almost all $(t,y)\\in[0,T]\\times U_1$. This follows from \\eqref{eq:assEB1}, \\eqref{eq:assEBtilde1}, Lemma~\\ref{lem:integrandIsomorphism} and Remark~\\ref{not:Phix},\n\\end{remark}\n\nWe will prove Theorem~\\ref{thm:errorRep} in the next subsection. Let us briefly record an alternative representation of $e(T)$ which follows from Taylor's formula.\nIt will be the starting point for our error estimates in Sections \\ref{sec:Heat} and \\ref{sec:Wave}. For $t\\in [0,T]$, $\\theta\\in[0,1]$ and $y\\in U_1$ set\n\\begin{align*}\nF(t)&:=\\tilde E(t) B-E(t)B,\\\\\n\\Psi_1(t,\\theta,y)&:=(1-\\theta)\\Big\\langle u_{xx}\\big(t,\\tilde Y(t)+ E(T-t)By+\\theta F(T-t)y\\big)F(T-t)y\\,,\\,F(T-t)y\\Big\\rangle_H,\\\\\n\\Psi_2(t,\\theta,y)&:=\\Big\\langle u_{xx}\\big(t,\\tilde Y(t)+\\theta E(T-t)By\\big)E(T-t)By\\,,\\,F(T-t)y\\Big\\rangle_H.\n\\end{align*}\n\n\\begin{corollary}\\label{cor:errorRep}\nIn the setting of Theorem~\\ref{thm:errorRep} we have\n\\begin{equation}\\label{eq:errorRep3}\n\\bE\\int_0^T\\int_{U_1}\\int_0^1\\big\\{|\\Psi_1(t,\\theta,y)|+|\\Psi_2(t,\\theta,y)|\\big\\}\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t<\\infty,\n\\end{equation}\nand the following alternative error representation holds:\n\\begin{equation}\\label{eq:errorRep4}\n\\begin{aligned}\ne(T)&=\\bE\\big\\{u(0,\\tilde E(T)X_0)-u(0,E(T)X_0)\\big\\}\\\\\n&\\quad +\\bE\\int_0^T\\int_{U_1}\\int_0^1\\big\\{\\Psi_1(t,\\theta,y)+\\Psi_2(t,\\theta,y)\\big\\}\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t.\n\\end{aligned}\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof\nThe integrand of the iterated integral in \\eqref{eq:errorRep2} can be rewritten as\n\\begin{align*}\nu\\big(t,\\tilde Y(t)+&\\tilde E(T-t) By\\big)-u\\big(t,\\tilde Y(t)+E(T-t)By\\big)-\\big\\langle u_x(t,\\tilde Y(t)),F(T-t)y\\big\\rangle_H\\\\\n&=\\Big\\{u\\big(t,\\tilde Y(t)+\\tilde E(T-t) By\\big)-u\\big(t,\\tilde Y(t)+E(T-t)By\\big)\\\\\n&\\quad-\\big\\langle u_x\\big(t,\\tilde Y(t)+E(T-t)By\\big),F(T-t)y\\big\\rangle_H\\Big\\}\\\\\n&\\quad+\\big\\langle u_x\\big(t,\\tilde Y(t)+E(T-t)By\\big)-u_x(t,\\tilde Y(t)),F(T-t)y\\big\\rangle_H\\\\\n&=\\int_0^1\\big\\{\\Psi_1(t,\\theta,y)+\\Psi_2(t,\\theta,y)\\big\\}\\,\\ensuremath{\\mathrm{d}}\\theta,\n\\end{align*}\nwhere the last step is due to Taylor's formula. By \\eqref{eq:errorRep1} we have\n\\[\\bE\\int_0^T\\int_{U_1}\\Big|\\int_0^1\\big\\{\\Psi_1(t,\\theta,y)+\\Psi_2(t,\\theta,y)\\big\\}\\,\\ensuremath{\\mathrm{d}}\\theta\\Big|\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t<\\infty.\\]\nThe stronger assertion \\eqref{eq:errorRep3} follows from the boundedness of $G'':H\\to\\ensuremath{\\mathscr L}(H)$, Lemma~\\ref{lem:integrandIsomorphism}, \\eqref{eq:assEB1} and \\eqref{eq:assEBtilde1}.\n\\end{proof}\n\n\n\n\\subsection{Proof of the error representation formula}\n\nIn this subsection, we give the proof of Theorem~\\ref{thm:errorRep}.\n\nFor $\\xi\\in L^2(\\Omega,\\cF_t,\\bP;H)$ we have\n\\begin{equation*}\\label{eq:Eu(t,xi)}\n\\bE\\big(G(Z(T,t,\\xi))\\big)\n=\\int_H\\int_HG(x+y)\\,\\bP_{\\int_t^TE(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s)}(\\ensuremath{\\mathrm{d}} y)\\bP_\\xi(\\ensuremath{\\mathrm{d}} x)\n=\\bE\\big(u(t,\\xi)\\big)\n\\end{equation*}\nby \\eqref{eq:defZttauxi}, \\eqref{eq:defuxt}, the independence of $\\int_t^TE(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s)$ and $\\cF_t$, and Fubini's theorem.\nSince $X(T)=Y(T)$ and $\\tilde X(T)=\\tilde Y(T)$ it follows that\n\\begin{equation}\\label{eq:proofErrRep1}\n\\begin{aligned}\ne(T)&=\\bE\\big(G(\\tilde Y(T))-G(Y(T))\\big)\\\\\n&=\\bE\\big(G(Z(T,T,\\tilde Y(T)))-G(Z(T,0,Y(0)))\\big)\\\\\n&=\\bE\\big(u(T,\\tilde Y(T))-u(0,Y(0))\\big)\\\\\n&=\\bE\\big(u(0,\\tilde Y(0))-u(0,Y(0))\\big)+\\bE\\big(u(T,\\tilde Y(T))-u(0,\\tilde Y(0))\\big).\n\\end{aligned}\n\\end{equation}\nBy \\eqref{eq:defY} and \\eqref{eq:defYtilde}, the first term in the last line equals $\\bE(u(0,\\tilde E(T)X_0)-u(0,E(T)X_0))$.\n\nTo handle the second term in the last line of \\eqref{eq:proofErrRep1}, we first assume that $G:H\\to\\bR$ and $G_x:H\\to H$ are bounded, so that $G\\in C^2_{\\operatorname b}(H,\\bR)$. We will remove this restriction later on. We now want to apply It\\^{o}'s formula to the function $(t,x)\\mapsto u(t,x)$ and the martingale $\\tilde Y=(\\tilde Y(t))_{t\\in[0,T]}$. For this we need the following properties of $u$.\n\n\n\\begin{proposition}\\label{prop:backwardKolm}\nLet Assumption~\\ref{ass:abstractSetting} hold and $G\\in C_{\\operatorname b}^2(H,\\bR)$. The function $u:[0,T]\\times H\\to\\bR,\\;(t,x)\\mapsto u(t,x)$ defined in \\eqref{eq:defuxt} and its Fr\\'{e}chet partial derivatives $u_x$, $u_{xx}$ are continuous and bounded on $[0,T]\\times H$. The time derivative $u_t$ of $u$ exists on $[0,T)\\times H$ and is continuous.\nMoreover, for every $\\epsilon>0$ there exists some $C_\\epsilon<\\infty$ such that\n\\begin{equation}\\label{eq:backwardKolm0}\n\\int_{U_1} \\big|u\\big(t,x+E(T-t)By\\big)- u(t,x)-\\big\\langle u_x(t,x),E(T-t)By\\big\\rangle_H\\big|\\,\\nu(\\ensuremath{\\mathrm{d}} y)<C_\\epsilon\n\\end{equation}\nfor all $t\\in[0,T-\\epsilon]$, and $u$ satisfies the backward Kolmogorov equation\n\\begin{equation}\\label{eq:backwardKolm1}\n\\left.\n\\begin{aligned}\nu_t(t,x)&=-\\int_{U_1} \\Big\\{u\\big(t,x+E(T-t)By\\big)-u(t,x)-\\big\\langle u_x(t,x),E(T-t)By\\big\\rangle_H\\Big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y),\\\\\n&\\hspace{2cm}(t,x)\\in [0,T)\\times H,\\\\\nu(T,x)&=G(x),\\quad x\\in H.\n\\end{aligned}\n\\;\\right\\}\n\\end{equation}\n\\end{proposition}\n\n\n\\begin{proof}\n\nWe begin with the continuity and boundedness of $u$, $u_x$ and $u_{xx}$. The boundedness is obvious by the definition \\eqref{eq:defuxt} of $u$ and by \\eqref{eq:u_xu_xx}. Pick $0\\leq s\\leq t\\leq T$, $x,y\\in H$. Using \\eqref{eq:defuxt}, Jensen's inequality, the mean value theorem, \\eqref{eq:defZttauxi} and It\\^{o}'s isometry,  we have\n\\begin{equation*}\n\\begin{aligned}\n|u(t,x)-u(s,y)|^2&\\leq\\bE\\big(|G(Z(T,t,x))-G(Z(T,s,y))|^2\\big)\\\\\n&\\leq \\sup_{x\\in H}\\nnrm{G'(x)}H^2\\,\\bE\\Big(\\gnnrm{x-y-\\int_s^t E(T-r)B\\,\\ensuremath{\\mathrm{d}} L(r)}H^2\\Big)\\\\\n&\\leq 2\\sup_{x\\in H}\\nnrm{G'(x)}H^2\\Big(\\nnrm{x-y}H^2+\\int_s^t\\nnrm{E(T-r)B}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\,\\ensuremath{\\mathrm{d}} r\\Big).\n\\end{aligned}\n\\end{equation*}\nThus, the continuity of $u$ follows from \\eqref{eq:assEB1} and the boundedness of $G'$.\nSince $u_x(t,x)=\\bE\\,G'(Z(T,t,x))$, the continuity of $u_x:[0,T]\\times H\\to H$ follows analogously from the boundedness of $G''$.\nTo show the continuity of $u_{xx}:[0,T]\\times H\\to\\ensuremath{\\mathscr L}(H)$, define $F\\in C_{\\operatorname b}(H\\times H;\\bR)$ by\n\\[F(x,y):=\\nnrm{G''(x)-G''(y)}{\\ensuremath{\\mathscr L}(H)},\\quad x,\\,y\\in H,\\]\nand fix $(t,x)\\in[0,T]\\times H$, $((t_k,x_k))_{k\\in\\bN}\\subset[0,T]\\times H$ with $(t_k,x_k)\\to (t,x)$ as $k\\to\\infty$. Note that $Z(T,t_k,x_k)\\to Z(T,t,x)$ in $L^2(\\Omega;H)$ by It\\^{o}'s isometry. As a consequence, $(Z(T,t,x),Z(T,t_k,x_k))\\to (Z(T,t,x),Z(T,t,x))$ in distribution (on $H\\times H$) and we obtain\n\\begin{align*}\n\\nnrm{u_{xx}(t,x)-u_{xx}(t_k,x_k)}{\\ensuremath{\\mathscr L}(H)}\n&\\leq \\bE\\, F\\big(Z(T,t,x),Z(T,t_k,x_k)\\big)\\\\\n&\\xrightarrow{k\\to\\infty}\\bE\\, F\\big(Z(T,t,x),Z(T,t,x)\\big)=0,\n\\end{align*}\nyielding the continuity of $u_{xx}$.\n\nBy Taylor's formula and Lemma~\\ref{lem:integrandIsomorphism},\n\\begin{equation*}\n\\begin{aligned}\n\\int_{U_1} \\big|u\\big(t,x+E(T-t)By&\\big)- u(t,x)-\\big\\langle u_x(t,x),E(T-t)By\\big\\rangle_H\\big|\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\leq\\;\\frac12\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_{U_1}\\nnrm{E(T-t)By}{H}^2\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&=\\;\\frac12\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\nnrm{E(T-t)B}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2.\n\\end{aligned}\n\\end{equation*}\nUsing Assumption~\\ref{ass:abstractSetting}(v), condition \\eqref{eq:backwardKolm0} follows with $$C_\\epsilon=1\/2\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\nnrm{\\Phi_\\epsilon}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2.$$\n\nIn order to verify the Kolmogorov equation \\eqref{eq:backwardKolm1}, we first note that for fixed $t\\in[0,T]$ the $H$-valued random variables $\\int_0^tE(s)B\\,\\ensuremath{\\mathrm{d}} L(s)$ and $\\int_{T-t}^{T} E(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s)$ have the same distribution, so that\n\\begin{equation}\\label{eq:backwardKolm2}\nv(t,x):=\\bE\\,G\\big(x+\\int_0^{t}E(s)B\\,\\ensuremath{\\mathrm{d}} L(s)\\big)=u(T-t,x),\\quad (t,x)\\in [0,T]\\times H.\n\\end{equation}\nNext, we fix $x\\in H$ and apply It\\^{o}'s formula \\cite[Theorem~3.6]{MRT13} to the function $y\\mapsto G(x+y)$ and the martingale $M=(M(t))_{t\\in[0,T]}:=(\\int_0^t E(s)B\\,\\ensuremath{\\mathrm{d}} L(s))_{t\\in[0,T]}\\in\\cM^2_T(H)$. Note that $M$ fits into the setting of \\cite{MRT13} since it has the representation\n\\[M(t)=\\int_0^t\\int_{U_1} E(s)By\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} y),\\quad t\\in[0,T],\\]\nwhere $q$ is the compensated Poisson random measure on $[0,\\infty)\\times U_1$ associated to $L$; see the appendix for details.\nWe obtain\n\\begin{equation}\\label{eq:backwardKolm3}\n\\begin{aligned}\nG(x+M(t))= G(x)&+\\int_0^t\\int_{U_1}\\big\\{G\\big(x+M(s-)+E(s)By\\big)-G\\big(x+M(s-)\\big)\\big\\}\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} y)\\\\\n&+\\int_0^t\\int_{U_1}\\big\\{G\\big(x+M(s)+E(s)By\\big)-G\\big(x+M(s)\\big)\\\\\n&\\hspace{4.4cm}-\\big\\langle G'\\big(x+M(s)\\big),E(s)By\\big\\rangle_H\\big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} s,\n\\end{aligned}\n\\end{equation}\nwhere the integrand appearing in the integral w.r.t.\\ $q$ belongs to $L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;\\bR)$ as a consequence of Taylor's formula, the boundedness of $G'$, Lemma~\\ref{lem:integrandIsomorphism} and \\eqref{eq:assEB1}. Similarly, the second integral in \\eqref{eq:backwardKolm3} exists for all $\\omega\\in\\Omega$ and belongs to $L^1(\\Omega;\\bR)$\nsince\n\\begin{equation*}\\label{eq:backwardKolm4}\n\\begin{aligned}\n\\int_0^t\\int_{U_1}\\big|G\\big(x+M(s)+E(s)By\\big)-G\\big(&x+M(s)\\big)-\\big\\langle G'\\big(x+M(s)\\big),E(s)By\\big\\rangle_H\\big|\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} s\\\\\n&\\leq\\;\\frac12\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_0^t\\nnrm{E(s)B}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\ensuremath{\\mathrm{d}} s.\n\\end{aligned}\n\\end{equation*}\nTaking expectations in \\eqref{eq:backwardKolm3} and using the martingale property of the integral w.r.t.\\ $q$ yields\n\\begin{equation}\\label{eq:backwardKolm5}\nv(t,x)=G(x)+\\int_0^t\\int_{U_1}\\big\\{v(s,x+E(s)By)-v(s,x)-\\langle v_x(s,x),E(s)By\\rangle_H\\big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} s.\n\\end{equation}\nBy the fundamental theorem of calculus, \\eqref{eq:backwardKolm1} follows from \\eqref{eq:backwardKolm2}, \\eqref{eq:backwardKolm5} if the mapping\n\\begin{equation}\\label{eq:backwardKolm7}\n(0,T]\\ni s\\to \\int_{U_1}\\big\\{v(s,x+E(s)By)-v(s,x)-\\langle v_x(s,x),E(s)By\\rangle_H\\big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\in\\bR.\n\\end{equation}\nis continuous.\n\nNote that we cannot apply directly the continuity theorem for parameter-dependent integrals to show the continuity of the mapping \\eqref{eq:backwardKolm7}. The reason is that the term $E(s)By$ in the integral in \\eqref{eq:backwardKolm7} is defined only in an $L^2([0,T]\\times U_1,\\lambda\\otimes\\nu;H)$-sense, cf.~Lemma~\\ref{lem:integrandIsomorphism} and Remark~\\ref{not:Phix}, so that we have no information about the continuity of $(0,T]\\in s\\mapsto E(s)By\\in H$ for fixed $y\\in U_1$. Therefore, we use an approximation argument: For $s\\in(0,T]$, $x\\in H$, $y\\in U_1$ and $k\\in\\bN$ set\n\\begin{align*}\nf(s,x,y)&:=v(s,x+E(s)By)-v(s,x)-\\langle v_x(s,x),E(s)By\\rangle_H,\\\\\nf_k(s,x,y)&:=f(s,x,\\Pi_ky),\n\\end{align*}\nwhere $\\Pi_k$ is the orthogonal projection of $U_1$ onto $\\text{span}\\{f_1,\\ldots,f_k\\}$, $(f_k)_{k\\in\\bN}\\subset U_0$ being an orthonormal basis of $U_1$ as in Lemma~\\ref{lem:integrandIsomorphism}. For fixed $x\\in H$,  $f(s,x,y)$ is defined in an $L^2([0,T]\\times U_1,\\lambda\\otimes\\nu;\\bR)$-sense whereas $f_k(s,x,y)$ is defined pointwise. The continuity theorem for parameter-dependent integrals and the strong continuity of $(E(t))_{t\\geq 0}$ yield the continuity of in $\\int_{U_1}f_k(s,x,y)\\nu(\\ensuremath{\\mathrm{d}} y)$ in $(s,x)\\in[0,T]\\times H$. Moreover, we have $f_k(s,x,\\cdot)\\stackrel{k\\to\\infty}{\\longrightarrow} f(s,x,\\cdot)$ in $L^1(U_1,\\nu;\\bR)$, uniformly in $(s,x)\\in[\\epsilon,T]\\times H$ for all $\\epsilon>0$. Indeed, setting $\\Pi^ky:=y-\\Pi_ky$ and using Taylor's theorem, Lemma~\\ref{lem:integrandIsomorphism} and Assumption~\\ref{ass:abstractSetting}(v), we obtain\n\\begin{align*}\n&\\int_{U_1}|f(s,x,y)-f_k(s,x,y)|\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\leq\\int_{U_1}\\int_0^1\\big|\\big\\langle v_{xx}\\big(s,x+E(s)B(\\Pi_ky+\\theta\\Pi^ky)\\big) E(s)B\\Pi^ky,E(s)B\\Pi^ky\\big\\rangle_H\\big|(1-\\theta)\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\quad+\\int_{U_1}\\int_0^1\\big|\\big\\langle v_{xx}\\big(s,x+\\theta E(s)B\\Pi_ky\\big)E(s)B\\Pi_ky,E(s)B\\Pi^ky\\big\\rangle_H\\big|\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\leq\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\big(\\nnrm{E(s)B\\Pi^k}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2+\\nnrm{E(s)B\\Pi_k}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\nnrm{E(s)B\\Pi^k}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\big)\\\\\n&\\leq\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\big(\\nnrm{\\Phi_\\epsilon\\Pi^k}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2+\\nnrm{\\Phi_\\epsilon\\Pi_k}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\nnrm{\\Phi_\\epsilon\\Pi^k}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\big)\n\\end{align*}\nfor all $s\\in[\\epsilon,T]$ and some $\\Phi_\\epsilon\\in\\ensuremath{\\mathscr L_2}(U_0,H)$. The expression in the last line tends to zero as $k\\to\\infty$.\nAs a consequence, $\\int_{U_1}f_k(s,x,y)\\nu(\\ensuremath{\\mathrm{d}} y)\\xrightarrow{k\\to\\infty}\\int_{U_1}f(s,x,y)\\nu(\\ensuremath{\\mathrm{d}} y)$ uniformly in $(s,x)\\in[\\epsilon,T]\\times H$. Thus, the continuity of $\\int_{U_1}f_k(s,x,y)\\nu(\\ensuremath{\\mathrm{d}} y)$ in $(s,x)\\in[0,T]\\times H$ implies the continuity of $\\int_{U_1}f(s,x,y)\\nu(\\ensuremath{\\mathrm{d}} y)$ in $(s,x)\\in(0,T]\\times H$. In particular, we obtain the continuity of the mapping \\eqref{eq:backwardKolm7} as well as the continuity of $u_t$ on $[0,T)\\times H$.\n\\end{proof}\n\n\nFor $G\\in C_{\\operatorname b}^2(H,\\bR)$, the regularity assertions in Propostition~\\ref{prop:backwardKolm} allow us to apply It\\^{o}'s formula \\cite[Theorem~3.6]{MRT13} to the function $(t,x)\\mapsto u(t,x)$ and the $H$-valued martingale $\\tilde Y=(\\tilde Y(t))_{t\\in[0,T]}$ defined in \\eqref{eq:defYtilde}. Note that $\\tilde Y$ fits into the setting of \\cite{MRT13} since it has the representation\n\\begin{equation}\\label{eq:proofErrRep1.5}\n\\tilde Y(t)=\\tilde E(T)X_0+\\int_0^t\\int_{U_1}\\tilde E(T-s) B y\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} y),\\quad t\\in[0,T],\n\\end{equation}\nwhere again $q$ is the compensated Poisson random measure on $[0,\\infty)\\times U_1$ associated with $L$ as described in the appendix. Equality \\eqref{eq:proofErrRep1.5} is a consequence of \\eqref{eq:assEBtilde1}, Lemma~\\ref{lem:integrandIsomorphism}, Remark~\\ref{not:Phix} and\nLemma~\\ref{lem:comparisonIntegrals}. For $T'\\in(0,T)$ we obtain\n\\begin{equation}\\label{eq:proofErrRep2}\n\\begin{aligned}\nu(T',\\tilde Y(T'))\\,&=\\,u(0,\\tilde Y(0))+\\int_0^{T'}u_t(t,\\tilde Y(t))\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\quad+\\int_0^{T'}\\int_{U_1}\\big\\{u\\big(t,\\tilde Y(t-)+\\tilde E(T-t) By\\big)-u(t,\\tilde Y(t-))\\big\\}\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} y)\\\\\n&\\quad+\\int_0^{T'}\\int_{U_1}\\big\\{u\\big(t,\\tilde Y(t)+\\tilde E(T-t) By\\big)-u(t,\\tilde Y(t))\\\\\n&\\quad-\\big\\langle u_x(t,\\tilde Y(t)),\\tilde E(T-t) By\\big\\rangle_H\\big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} s.\n\\end{aligned}\n\\end{equation}\nUsing the boundedness of $u$, $u_x$ and $u_{xx}$, \\eqref{eq:backwardKolm1}, \\eqref{eq:assEB1} and applying similar arguments as in the proof of Proposition~\\ref{prop:backwardKolm}, one sees that all terms in \\eqref{eq:proofErrRep2} are well-defined and integrable w.r.t.\\ $\\bP$. Thus, we can take expectations and use the martingale property of the integral w.r.t.\\ $q$ and the backward Kolmogorov equation \\eqref{eq:backwardKolm1} to obtain\n\\begin{equation}\\label{eq:proofErrRep3}\n\\begin{aligned}\n\\bE\\big(&u(T',\\tilde Y(T'))-u(0,\\tilde Y(0))\\big)=\\\\\n&\\bE\\int_0^{T'}\\int_{U_1}\\Big\\{u\\big(t,\\tilde Y(t)+\\tilde E(T-t) By\\big)-u\\big(t,\\tilde Y(t)+E(T-t)By\\big)\\\\\n&\\qquad\\qquad\\qquad\\qquad-\\big\\langle u_x(t,\\tilde Y(t)),\\big(\\tilde E(T-t) B-E(T-t)B\\big)y\\big\\rangle_H\\Big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\n\\end{aligned}\n\\end{equation}\nfor all $T'\\in(0,T)$. Taking the limit $T'\\to T$ on both sides of \\eqref{eq:proofErrRep3}, we can replace $T'$ by $T$. Here we used the stochastic continuity of $\\tilde Y$ and the continuity of $u$ for the limit on the left hand side. For the limit on the right hand side we used \\eqref{eq:errorRep1}, which is again a consequence of Taylor's formula, the boundedness of $G''$, \\eqref{eq:assEB1}, \\eqref{eq:assEBtilde1} and Lemma~\\ref{lem:integrandIsomorphism}, using similar arguments as in the proof of Proposition~\\ref{prop:backwardKolm}. The combination of \\eqref{eq:proofErrRep1} and \\eqref{eq:proofErrRep3} yields the error representation formula \\eqref{eq:errorRep2} for the case $G\\in C_{\\operatorname b}^2(H,\\bR)$.\n\nFinally, we consider the general case of a test function $G\\in C^2(H,\\bR)$ such that $\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$.\nFor $\\epsilon>0$, define $G_\\epsilon\\in C_{\\operatorname b}^2(H,\\bR)$ by\n\\begin{equation*}\nG_\\epsilon(x):=e^{-\\epsilon\\nnrm{x}H^2} G(x),\\quad x\\in H.\n\\end{equation*}\nThe Fr\\'{e}chet derivatives $G_\\epsilon'(x)\\in H$ and $G_\\epsilon''(x)\\in\\ensuremath{\\mathscr L}(H)$ are given by\n\\begin{align*}\nG_\\epsilon'(x)&=e^{-\\epsilon\\nnrm{x}H^2}\\big(G'(x)-2\\epsilon G(x)x\\big),\\\\\nG_{\\epsilon}''(x)&=e^{-\\epsilon\\nnrm{x}H^2}\\big[G''(x)+2\\epsilon\\big(\\langle G'(x),\\arg\\rangle_H x-\\langle x,\\arg\\rangle_H G'(x)\\big)-4\\epsilon^2 G(x)\\langle x,\\arg\\rangle_Hx\\big],\n\\end{align*}\nso that the boundedness of $G_\\epsilon:H\\to\\bR$, $G_\\epsilon':H\\to H$ and $G_\\epsilon'':H\\to\\ensuremath{\\mathscr L}(H)$ follows from \\eqref{eq:growthGG'}. Moreover, note that $G_\\epsilon(x)\\xrightarrow{\\epsilon\\to0}G(x)$ in $\\bR$ and $G_\\epsilon'(x)\\xrightarrow{\\epsilon\\to0}G'(x)$ in $H$ for all $x\\in H$, as well as\n$\n\\sup_{\\epsilon\\in(0,1]}\\sup_{x\\in H}\\nnrm{G_\\epsilon''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty.\n$\nThe latter is a consequence of \\eqref{eq:growthGG'} and the standard estimate $\\sup_{s>0}s^ne^{-s}<n!$, $n\\in\\bN$.  We set\n\\begin{equation}\\label{eq:defuxtEps}\nu_\\epsilon(t,x):=\\bE\\,G_\\epsilon(Z(T,t,x)),\\quad (t,x)\\in [0,T]\\times H.\n\\end{equation}\nBy what has been shown above, the assertion of Theorem~\\ref{thm:errorRep} holds with $u$ replaced by $u_\\epsilon$. By a dominated convergence argument using \\eqref{eq:growthGG'}, we obtain that $u_\\epsilon(t,x)\\xrightarrow{\\epsilon\\to0} u(t,x)$ in $\\bR$ and $(u_\\epsilon)_x(t,x)\\xrightarrow{\\epsilon\\to0} u_x(t,x)$ in $H$ for all $(t,x)\\in [0,T]\\times H$. For all $t\\in[0,T]$ and $y\\in U_1$, we have\n\\begin{equation}\\label{eq:proofErrRep4}\n\\begin{aligned}\n\\Big|u_\\epsilon\\big(t,&\\tilde Y(t)+\\tilde E(T-t) By\\big)-u_\\epsilon\\big(t,\\tilde Y(t)+E(T-t)By\\big)\\\\\n&\\qquad-\\big\\langle (u_\\epsilon)_x(t,\\tilde Y(t)),\\big(\\tilde E(T-t) B-E(T-t)B\\big)y\\big\\rangle_H\\Big|\\\\\n&\\leq \\sup_{\\epsilon\\in(0,1]}\\sup_{x\\in H}\\nnrm{G_\\epsilon''(x)}{\\ensuremath{\\mathscr L}(H)}\\Big(\\nnrm{\\tilde E(T-t) By}H^2+\\nnrm{E(T-t)B}H^2\\big)\n\\end{aligned}\n\\end{equation}\ndue to Taylor's theorem and the fact that $(u_\\epsilon)_{xx}(t,x)=\\bE G_\\epsilon''(Z(T,t,x))$. Note that the term right hand side of \\eqref{eq:proofErrRep4} is integrable w.r.t. $\\ensuremath{\\mathrm{d}} t\\,\\nu(\\ensuremath{\\mathrm{d}} y)$ as a consequence of \\eqref{eq:assEB1}, \\eqref{eq:assEBtilde1} and Lemma~\\ref{lem:integrandIsomorphism}. Thus, considering \\eqref{eq:errorRep1} and \\eqref{eq:errorRep2} with $u$ replaced by $u_\\epsilon$, we can use dominated convergence as $\\epsilon\\to0$  to finish the proof.\n\n\n\\section{Applications to parabolic equations}\n\\label{sec:Heat}\n\n\\subsection{The stochastic heat equation}\\label{subsec:she} Here we give a detailed error analysis of a space-time discretization of the linear stochastic heat equation with additive L\\'{e}vy noise.\n\nLet $\\ensuremath{\\mathcal{O}}\\subset\\bR^d$ be a bounded convex domain. \nLet $\\Lambda:=-\\Delta=-\\sum_{j=1}^d\\partial^2\/\\partial\\xi_j^2$ be the Laplace operator on $L^2(\\ensuremath{\\mathcal{O}})$ with zero-Dirichlet boundary condition, i.e., with domain $D(\\Lambda):=\\{v\\in H^1_0{(\\ensuremath{\\mathcal{O}})}:~\\Lambda u\\in L^2(\\ensuremath{\\mathcal{O}})\\}$, where $\\Lambda u$ is understood in the distributional sense, see \\cite[Example 3.4.7]{ABHN11}. As usual, $H^n(\\ensuremath{\\mathcal{O}})$ denotes the $L^2$-Sobolev space of order $n\\in\\bN_0$ on $\\ensuremath{\\mathcal{O}}$  and $H^1_0(\\ensuremath{\\mathcal{O}})$ is the $H^1(\\ensuremath{\\mathcal{O}})$-closure of the space $C_c^\\infty(\\ensuremath{\\mathcal{O}})$ of compactly supported test functions.\nThen, setting\n\\[H:=U:=L^2(\\ensuremath{\\mathcal{O}}),\\quad (A,D(A)):=(\\Lambda,D(\\Lambda)),\\quad B:=\\id{L^2(\\ensuremath{\\mathcal{O}})},\\]\nthe abstract equation \\eqref{eq:mainEq} becomes the stochastic heat equation \\eqref{eq:SHE}. It is not difficult to see that the condition $\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(H)}<\\infty$ implies \\eqref{eq:assEB1}, where\n\\begin{equation}\\label{eq:E(t)SHE}\n(E(t))_{t\\geq0}:=(e^{-t\\Lambda})_{t\\geq0}\\subset\\ensuremath{\\mathscr L}(H)\n\\end{equation}\nis the semigroup generated by $-A=-\\Lambda$. Hence, there exists a unique weak solution $X=(X(t))_{t\\geq0}$ to Eq.~\\eqref{eq:SHE}, given by the variation-of-constants formula \\eqref{eq:X}. Furthermore, for some $C>0$ independent of $T$,\n\\begin{equation}\\label{eq:hstab}\n\\nnrm{X(T)}{L^2(\\Omega,H)}\\le C (\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}( H)}+\\nnrm{X_0}{L^2(\\Omega,H)}).\n\\end{equation}\nIn the sequel, we use the smoothness spaces $\\dot H^\\alpha$, $\\alpha\\in\\bR$, defined by\n\\begin{align*}\n\\dot H^\\alpha&:=D(\\Lambda^{\\alpha\/2})\\\\\n&:=\\Big\\{v=\\sum_{k=1}^\\infty v_k\\varphi_k:(v_k)_{k\\in\\bN}\\subset\\bR,\\;|v|_{\\alpha}:=\\nnrm{\\Lambda^{\\alpha\/2}v}{L^2(\\ensuremath{\\mathcal{O}})}=\\Big(\\sum_{k=1}^\\infty \\lambda_k^\\alpha v_k^2\\Big)^{1\/2}<\\infty\\Big\\},\n\\end{align*}\nwhere $(\\varphi_k)_{k\\in\\bN}\\subset D(\\Lambda)$ is an orthonormal basis of $L^2(\\ensuremath{\\mathcal{O}})$ consisting of eigenfunctions of $\\Lambda $ and $(\\lambda_k)_{k\\in\\bN}\\subset(0,\\infty)$ is the corresponding sequence of eigenvalues; compare \\cite[Chapters 3 and 19]{Tho06}. They are Hilbert spaces and one has the identities $\\dot H^0=H=L^2(\\ensuremath{\\mathcal{O}})$, $\\dot H^1=H^1_0(\\ensuremath{\\mathcal{O}})$ and $\\dot H^2=D(\\Lambda)=H^2(\\ensuremath{\\mathcal{O}})\\cap H^1_0(\\ensuremath{\\mathcal{O}})$, where the natural norms of the respective spaces are equivalent. The latter equality is a consequence of the elliptic regularity estimate\n\\begin{equation}\\label{eq:ellreg}\n\\|v\\|_{H^2}\\le C\\|v\\|_{\\dot H^2},\\quad v\\in \\dot{H}^2,\n\\end{equation}\nfor bounded convex domains, see \\cite[Corollary 1]{Fromm93}. Without the convexity assumption one can still define the $\\dot H^\\alpha$ spaces as above but one does not obtain a characterization of $D(\\Lambda)=\\dot{H}^2$ in terms of classical Sobolev spaces.\nFor negative $\\alpha$, the elements of $\\dot H^\\alpha$ are formal sums and we identify them with elements of $L^2(\\ensuremath{\\mathcal{O}})$ if $\\sum_{k=1}^\\infty v_k^2<\\infty$, so that $\\dot H^\\alpha$ is the closure of $L^2(\\ensuremath{\\mathcal{O}})$ w.r.t.\\ the $|\\cdot|_\\alpha$-norm.\n\\begin{remark}\\label{rem:interpolation}\nThe spaces $\\dot H^\\alpha$, $\\alpha\\in\\bR$, can be obtained by both real and complex interpolation: For $\\alpha=(1-\\theta)\\alpha_0+\\theta\\alpha_2$, $\\theta\\in(0,1)$, one has $\\dot H^\\alpha=(\\dot H^{\\alpha_0},\\dot H^{\\alpha_1})_{\\theta,2}=[\\dot H^{\\alpha_0},\\dot H^{\\alpha_1}]_\\theta$ with equivalent norms, where $(\\cdot,\\cdot)_{\\theta,2}$ and $[\\cdot,\\cdot]_\\theta$ denotes real interpolation with summability parameter $q=2$ and complex interpolation, respectively. This follows, e.g., from \\cite[Theorem 1.18.5]{Tri78} and the fact that the spaces $\\dot H^\\alpha$, $\\alpha\\in\\bR$, are isometrically isomorphic to weighted $\\ell^2$-spaces. We will frequently use the corresponding interpolation inequalities in this and the next section.\n\\end{remark}\n\nFor the spatial discretization of Eq.~\\eqref{eq:SHE}, we consider a family of finite-dimensional spaces $(S_h)_{h>0}\\subset H^1_0(\\ensuremath{\\mathcal{O}})$.\n Unless otherwise stated, we endow the spaces $S_h$ with the inner product $\\langle\\cdot,\\cdot\\rangle_H$ and the norm $\\nnrm{\\cdot}H$.  By $P_h:H\\to S_h$ and $\\Pi_h:\\dot H^1\\to S_h$ we denote the orthogonal projections with respect to the inner products in $H$ and $\\dot H^1$, respectively. The discrete Laplacian $\\Lambda_h:S_h\\to S_h$ is defined by\n\\begin{equation}\\label{eq:discreteLaplace}\n\\langle\\Lambda_h v,w\\rangle_{L^2(\\ensuremath{\\mathcal{O}})}=\\langle\\nabla v,\\nabla w\\rangle_{L^2(\\ensuremath{\\mathcal{O}};\\bR^2)},\\quad v,w\\in S_h.\n\\end{equation}\nOur assumption on the spatial approximation is formulated via the following estimate on the Ritz projection $\\Pi_h$,\n\\begin{equation}\\label{eq:basicFEMestimate1}\n\\nnrm{\\Pi_hv-v}{L^2(\\ensuremath{\\mathcal{O}})}\\leq C h^\\beta|v|_\\beta,\\quad v\\in\\dot H^\\beta,\\;1\\leq\\beta\\leq2.\n\\end{equation}\nThis holds for example, if $S_h$ is consisting of piecewise linear functions with respect to a family of triangulations of $\\ensuremath{\\mathcal{O}}$. The parameter $h$ corresponds to the maximal mesh size of the triangulation, see, e.g., \\cite[Lemma~1.1]{Tho06} or \\cite[Section~5.4]{LarTho03}.\n\nThe time discretization of Eq.~\\eqref{eq:SHE} on a finite interval $[0,T]$ is done via the implicit Euler scheme with time step $\\ensuremath{\\Delta t}=T\/N$, $N\\in\\bN$, and grid points $t_n=n\\ensuremath{\\Delta t}$, $n=0,\\ldots N$. For $h>0$ and $N\\in\\bN$, the discretization $(X^n_{h,\\ensuremath{\\Delta t}})_{n=1,\\ldots,N}$ of $(X(t))_{t\\in[0,T]}$ in space and time is given as the solution to\n\\begin{equation}\\label{eq:schemeSHE}\nX^n_{h,\\ensuremath{\\Delta t}}-X^{n-1}_{h,\\ensuremath{\\Delta t}}+\\ensuremath{\\Delta t}\\Lambda_hX^n_{h,\\ensuremath{\\Delta t}}=P_h(L(t_n)-L(t_{n-1})),\\quad n=1,\\ldots,N;\\quad X^0_{h,\\ensuremath{\\Delta t}}=P_hX_0.\n\\end{equation}\n\n\\begin{remark}[strong error] \\label{rem:strongErrorSHE}\nIf the covariance operator $Q\\in\\ensuremath{\\mathscr L}(H)$ of $L$ is such that\n\\begin{equation}\\label{eq:SHEassQ}\n\\nnrm{\\Lambda^{\\frac{\\beta-1}2}Q^{\\frac12}}{\\ensuremath{\\mathscr L_2}(H)}<\\infty\n\\end{equation}\nfor some $\\beta\\geq0$, then the solution $X(t)$ takes values in $\\dot H^\\beta$ for all $t>0$. For the Gaussian case, i.e., the case where $L$ in \\eqref{eq:SHE} is a $Q$-Wiener process, it has been shown in \\cite[Theorem~1.2]{Yan05} that, if \\eqref{eq:SHEassQ} holds and $X_0\\in L^2(\\Omega,\\cF_0,\\bP;\\dot H^\\beta)$ for some $\\beta\\in(0,1]$, then the scheme \\eqref{eq:schemeSHE} has strong convergence of order $\\beta$ in space and $\\beta\/2$ in time:\n\\[\\nnrm{X^n_{h,\\ensuremath{\\Delta t}}-X(t_n)}{L^2(\\Omega;H)}\\leq C(h^\\beta+(\\ensuremath{\\Delta t})^{\\frac\\beta2}),\\quad n=0,\\ldots,N.\\]\nUnlike weak error estimates, strong $L^2$-error estimates are the same in the Gaussian case and in our setting, since the only stochastic tool that is needed is It\\^{o}'s isometry~\\eqref{eq:ItoIsom} which looks the same\nif the driving noise is a L\\'{e}vy process which is an $L^2$-martingale.\nThus the strong error result in \\cite[Theorem~1.2]{Yan05} carries over one-to-one to our setting.\n\\end{remark}\n\n\\begin{remark}\nThe $S_h$-valued random variables $P_h(L(t_n)-L(t_{n-1}))$ in \\eqref{eq:schemeSHE} can be defined in two ways. On the one hand, we may set\n\\[P_h(L(t_n)-L(t_{n-1})):=L^2(\\Omega;S_h)\\text{-}\\lim_{K\\to\\infty}\\sum_{k=1}^K(L_k(t_n)-L_{k}(t_{n-1}))P_he_k,\\]\nwith an orthonormal basis $(e_k)_{k\\in\\bN}$ of $U_0$ and real-valued uncorrelated L\\'{e}vy processes $L_k=(L_k(t))_{t\\geq0}$, $k\\in\\bN$, as in Remark~\\ref{rem:approxL}. The limit exists since, by the finite-dimensionality of $S_h$, one has $P_h\\in\\ensuremath{\\mathscr L_2}(H,S_h)=\\ensuremath{\\mathscr L_2}(U,S_h)\\subset\\ensuremath{\\mathscr L_2}(U_0,S_h)$.\nOn the other hand, we can extend the orthogonal projection $P_h:H\\to S_h$ to a generalized $L^2$-projection $P_h:\\dot H^{-1}\\to S_h$ defined by\n\\[\\langle P_hv,w\\rangle_{H}=\\langle v,w\\rangle_{\\dot H^{-1}\\times\\dot H^1},\\quad v\\in \\dot H^{-1},\\;w\\in S_h.\\] Then, the assumption $\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(H)}<\\infty$ implies that we can take $U_1:=D(\\Lambda^{-1\/2})=\\dot H^{-1}$ as the state space of $L$, so that the expression $P_h(L(t_n)-L(t_{n-1}))$ makes sense $\\omega$-wise. Obviously, both definitions are compatible.\nIn practice, one has to find a suitable way to sample (an approximation of) the discretized noise increment $P_h(L(t_n)-L(t_{n-1}))$. We do not treat this problem in the present paper but refer to \\cite{BarLan12a, DunHauPro12} and \\cite[Remark~4]{LinSch13} for related considerations.\n\\end{remark}\n\n  With $R(\\lambda):=1\/(1+\\lambda)$ and $E_{h,\\ensuremath{\\Delta t}}:=R(\\ensuremath{\\Delta t}\\Lambda_h):=(I+\\ensuremath{\\Delta t}\\Lambda_h)^{-1}$ as well as $E_{h,\\ensuremath{\\Delta t}}^n:=R^n(\\ensuremath{\\Delta t}\\Lambda_h):=((I+\\ensuremath{\\Delta t}\\Lambda_h)^{-1})^n$, the scheme \\eqref{eq:schemeSHE} can be rewritten as\n\\begin{equation}\\label{eq:schemeSHE2}\nX^n_{h,\\ensuremath{\\Delta t}}=E_{h,\\ensuremath{\\Delta t}}^nP_hX_0+\\sum_{j=1}^n E^{n-j+1}_{h,\\ensuremath{\\Delta t}}P_h(L(t_j)-L(t_{j-1})),\\quad n=0,\\ldots,N.\n\\end{equation}\nFor $t\\in[0,T]$, let $\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)\\in\\ensuremath{\\mathscr L}(H)$ be defined by\n\\begin{equation}\\label{eq:EtildeSHE}\n\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t):=\\mathds 1_{\\{0\\}}(t)P_h+\\sum_{n=1}^N\\mathds 1_{(t_{n-1},t_n]}(t)E_{h,\\ensuremath{\\Delta t}}^nP_h\n\\end{equation}\nand set\n\\begin{equation}\\label{eq:XtildeSHE}\n\\tilde X(t)=\\tilde X_{h,\\ensuremath{\\Delta t}}(t):=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)X_0+\\int_0^t\\tilde E_{h,\\ensuremath{\\Delta t}}(t-s)\\,\\ensuremath{\\mathrm{d}} L(s).\n\\end{equation}\nThen $X^n_{h,\\ensuremath{\\Delta t}}=\\tilde X_{h,\\ensuremath{\\Delta t}}(t_n)$ $\\bP$-almost surely. This follows from the construction of the stochastic integral, using an approximation argument and It\\^{o}'s isometry \\eqref{eq:ItoIsom}.\n\nThe following deterministic estimates will be used in the proof of our weak error result stated in Theorem~\\ref{thm:SHE} below.\n\\begin{lemma}\nThe operators $E(t)$ and $\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)$ defined in \\eqref{eq:E(t)SHE} and \\eqref{eq:EtildeSHE} satisfy the error estimates\n\\begin{align}\n\\nnrm{\\tilde E(s)-E(s)}{\\ensuremath{\\mathscr L}(H)}&\\leq C(h^{2}+\\ensuremath{\\Delta t})s^{-1},\\label{eq:detEstSHE1}\\\\\n\\nnrm{\\Lambda^\\alpha E(s)}{\\ensuremath{\\mathscr L}(H)}+\\nnrm{\\Lambda^\\alpha \\tilde E(s)}{\\ensuremath{\\mathscr L}(H)}&\\leq C s^{-\\alpha},\\quad 0\\leq\\alpha\\leq1\/2,\\label{eq:detEstSHE2}\n\\end{align}\n$s\\in(0,T]$, where $C>0$ does not depend on $h$, $\\ensuremath{\\Delta t}$ and $s$.\n\\end{lemma}\n\n\\begin{proof}\nEstimate \\eqref{eq:detEstSHE1} follows from\n\\begin{equation}\\label{eq:hest}\n\\nnrm{E_{h,\\ensuremath{\\Delta t}}^nP_h-E(t_n)}{\\ensuremath{\\mathscr L}(H)}\\leq C(h^2+\\ensuremath{\\Delta t})t_n^{-1},\n\\end{equation}\nsee, for example, \\cite[Theorem~7.7]{Tho06}. We note here that while the latter result is proved under the assumption that $\\ensuremath{\\mathcal{O}}$ has smooth boundary, the proof relies on the availability of \\eqref{eq:basicFEMestimate1}, which is our basic assumption, and hence \\eqref{eq:hest} is valid in our setting.\nFor $s\\in(t_{n-1},t_n]$ we have\n\\begin{align*}\n\\nnrm{(E(t_n)-E(s))v}H&=\\nnrm{\\Lambda E(s)(E(t_n-s)-\\id{H})\\Lambda^{-1}v}H\\\\\n&\\leq\\nnrm{\\Lambda E(s)}{\\ensuremath{\\mathscr L}(H)}\\nnrm{(E(t_n-s)-\\id{H})\\Lambda^{-1}v}H\\\\\n&\\leq Cs^{-1}\\ensuremath{\\Delta t}\\nnrm{v}H,\n\\end{align*}\nwhere we used Theorem~6.13(c),(d) on analytic semigroups in \\cite[Chapter 2]{Paz83}.\nEstimate~\\eqref{eq:detEstSHE2} is due to Theorem~6.13(c) in \\cite[Chapter 2]{Paz83}, Lemma~7.3 in  \\cite{Tho06}, interpolation, and the fact that $\\nrm{A^\\alpha v_h}\\leq\\nrm{A^\\alpha_hv_h}$ for $v_h\\in S_h$, $0\\leq\\alpha\\leq 1\/2$. The latter follows from the basic identity $\\nrm{A^{1\/2}v_h}=\\nrm{A_h^{1\/2}v_h}$ and interpolation.\n\\end{proof}\n\n\nHere is our result for the weak error of the discretization of the stochastic heat equation.\n\\begin{theorem}\\label{thm:SHE}\nAssume that $X_0\\in L^2(\\Omega,\\cF_0,\\bP;H)$ and $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(H)}<\\infty$ for some $\\beta\\in(0,1]$. Let $(X(t))_{t\\geq0}$ be the weak solution \\eqref{eq:X} to Eq.~\\eqref{eq:mainEq} and let $(X_{h,\\ensuremath{\\Delta t}}^n)_{n=0,\\ldots,N}$ be defined by the scheme \\eqref{eq:schemeSHE}. Given $g\\in C^2(H,\\bR)$ with $\\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$, there exists a constant $C=C(g,T)>0$\nthat does not depend on $h$ and $\\ensuremath{\\Delta t}$, such that\n\\[\\big|\\bE\\big(g(X^N_{h,\\ensuremath{\\Delta t}})-g(X(T))\\big)\\big|\\leq C(h^{2\\beta}+(\\ensuremath{\\Delta t})^\\beta)|\\log(h^2+\\ensuremath{\\Delta t})|\\]\nfor $h^{2}+ \\ensuremath{\\Delta t}\\leq 1\/e$.\n\\end{theorem}\n\n\\begin{proof}\nWe are in the setting of Section~\\ref{sec:Setting and preliminaries} with $H=U=L^2(\\ensuremath{\\mathcal{O}})$, $B=\\id H$, and $(E(t))_{t\\geq0}$, $(\\tilde E(t))_{t\\in[0,T]}=(\\tilde E_{h,\\ensuremath{\\Delta t}}(t))_{t\\in[0,T]}$, $(\\tilde X(t))_{t\\in[0,T]}=(\\tilde X_{h,\\ensuremath{\\Delta t}}(t))_{t\\in[0,T]}$ being given by \\eqref{eq:E(t)SHE}, \\eqref{eq:EtildeSHE}, \\eqref{eq:XtildeSHE} respectively. In particular, Assumption~\\ref{ass:abstractSetting} is fulfilled. Since $X_{h,\\ensuremath{\\Delta t}}^N=\\tilde X(T)$, we can use Corollary~\\ref{cor:errorRep} with $G:=g$ to estimate the weak error.\nLet $F(t):=\\tilde E(t)-E(t)$ be the deterministic error operator.\n\nWe begin with the first term on the right hand side of \\eqref{eq:errorRep4} in Corollary~\\ref{cor:errorRep}.\nThe stability estimate \\eqref{eq:hstab} and the deterministic estimate \\eqref{eq:detEstSHE1} yield, for $\\max(h^{2},\\ensuremath{\\Delta t})\\leq 1$,\n\\begin{equation}\\label{eq:proofSHE1}\n\\begin{aligned}\n&\\big|\\bE\\big\\{u(0,\\tilde E(T)X_0)-u(0,E(T)X_0)\\big\\}\\big|\\\\\n&= \\big|\\bE\\big\\{u(0,\\tilde Y(0))-u(0,Y(0))\\big\\}\\big|\\\\\n&= \\Big|\\bE\\int_0^1\\big\\langle u_x\\big(0,Y(0)+\\theta(\\tilde Y(0)-Y(0))\\big),\\tilde Y(0)-Y(0)\\big\\rangle_{ H}\\ensuremath{\\mathrm{d}} \\theta\\Big|\\\\\n&= \\Big|\\bE\\int_0^1\\big\\langle\\bE\\big(g'(Z(T,0,x))\\big)\\big|_{x=Y(0)+\\theta(\\tilde Y(0)-Y(0))},\\tilde Y(0)-Y(0)\\big\\rangle_{H}\\ensuremath{\\mathrm{d}}\\theta\\Big|\\\\\n&\\leq \\int_0^1\\big\\|g'\\big(Z\\big(T,0,Y(0)+\\theta(\\tilde Y(0)-Y(0))\\big)\\big)\\big\\|_{L^2(\\Omega, H)}\\ensuremath{\\mathrm{d}}\\theta\\, \\|(\\tilde E(T)-E(T))X_0\\|_{L^2(\\Omega, H)}\\\\\n&\\leq C\\big(1+\\int_0^1\\big\\|Z\\big(T,0,Y(0)+\\theta(\\tilde Y(0)-Y(0))\\big)\\big\\|_{L^2(\\Omega, H)}\\ensuremath{\\mathrm{d}}\\theta\\big)\\,(h^2+\\ensuremath{\\Delta t})\\,T^{-1}\\,\\nnrm{X_0}{L^2(\\Omega; H)}\\\\\n&\\leq C\\big(1+\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}+\\nnrm{X_0}{L^2(\\Omega; H)}\\big)\\,\\nnrm{X_0}{L^2(\\Omega; H)}\\,T^{-1}\\,(h^{2\\beta}+(\\ensuremath{\\Delta t})^\\beta).\n\\end{aligned}\n\\end{equation}\n\nNext, consider the second term on the right hand side of \\eqref{eq:errorRep4}. We estimate the integrals of the functions $\\Psi_1$ and $\\Psi_2$ separately. Using Lemma~\\ref{lem:integrandIsomorphism} and Remark~\\ref{not:Phix}, we obtain\n\\begin{equation}\\label{eq:proofSHE2}\n\\begin{aligned}\n\\big|\\bE\\int_0^T&\\int_{U_1}\\int_0^1\\Psi_1(t,\\theta,y)\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\big|\\\\\n&\\leq \\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_0^T\\int_{U_1}\\nnrm{F(T-t)y}H^2\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\\\\n&=\\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_0^T\\nnrm{F(T-t)}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\leq C \\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}(h^{2\\beta}+(\\ensuremath{\\Delta t})^\\beta).\n\\end{aligned}\n\\end{equation}\nThe last step is due to the fact that, by It\\^{o}'s isometry \\eqref{eq:ItoIsom}, the integral in the penultimate line is the square of the strong error $\\nnrm{X^N_{h,\\ensuremath{\\Delta t}}-X(T)}{L^2(\\Omega;H)}$ for zero initial condition $X_0=0$, which can be estimated as in the Gaussian case \\cite[Theorem~1.2]{Yan05}, compare Remark~\\ref{rem:strongErrorSHE}. Further, by the Cauchy-Schwarz inequality, Lemma~\\ref{lem:integrandIsomorphism}, and the fact that $U_0=Q^{1\/2}(U)$,\n\\begin{equation}\\label{eq:proofSHE3}\n\\begin{aligned}\n\\big|\\bE&\\int_0^T\\int_{U_1}\\int_0^1\\Psi_2(t,\\theta,y)\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\big|\\\\\n&\\leq \\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_0^T\\int_{U_1}\\nnrm{E(T-t)y}H\\nnrm{F(T-t)y}H\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\leq\\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_0^T\\nnrm{E(T-t)}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\nnrm{F(T-t)}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\leq\\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}\\nnrm{\\Lambda^{\\frac{\\beta-1}2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(H)}^2\\int_0^T\\nnrm{E(t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(H)}\\nnrm{F(t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(H)}\\,\\ensuremath{\\mathrm{d}} t\n\\end{aligned}\n\\end{equation}\nBy \\eqref{eq:detEstSHE2} we have\n\\begin{equation}\\label{eq:detEstSHE3}\n\\nnrm{E(t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(H)}=\\nnrm{\\Lambda^{\\frac{1-\\beta}2}E(t)}{\\ensuremath{\\mathscr L}(H)}\\leq Ct^{-\\frac{1-\\beta}2}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:detEstSHE4}\n\\nnrm{\\Lambda^\\alpha F(t)}{\\ensuremath{\\mathscr L}(H)}\\leq Ct^{-\\alpha},\\quad 0\\leq\\alpha\\leq1\/2.\n\\end{equation}\nInterpolation between \\eqref{eq:detEstSHE1} and \\eqref{eq:detEstSHE4} with $\\alpha=1\/2$ gives\n\\begin{equation}\\label{eq:detEstSHE5}\n\\nnrm{\\Lambda^{\\frac{1-\\beta}2}F(t)}{\\ensuremath{\\mathscr L}(H)}\\leq C\\nnrm{F(t)}{\\ensuremath{\\mathscr L}(H)}^\\beta\\nnrm{\\Lambda^{\\frac12}F(t)}{\\ensuremath{\\mathscr L}(H)}^{1-\\beta}\\leq C(h^2+\\ensuremath{\\Delta t})^\\beta t^{-\\frac{1+\\beta}2}.\n\\end{equation}\nNote that $\\nnrm{F(t)\\Lambda^\\alpha}{\\ensuremath{\\mathscr L}(H)}=\\nnrm{\\Lambda^\\alpha F(t)}{\\ensuremath{\\mathscr L}(H)}$ due to the self adjointness of $\\tilde E(t)$, $E(t)$ and $\\Lambda^\\alpha$. Altogether, using \\eqref{eq:detEstSHE3}, \\eqref{eq:detEstSHE4} and \\eqref{eq:detEstSHE5}, the integral in the last line of \\eqref{eq:proofSHE3} can be estimated by\n\\begin{equation}\\label{eq:proofSHE4}\n\\begin{aligned}\n\\int_0^T&\\nnrm{E(t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(H)}\\nnrm{F(t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(H)}\\,\\ensuremath{\\mathrm{d}} t\\\\\n&=\\Big(\\int_0^{h^2+\\ensuremath{\\Delta t}}+\\int_{h^2+\\ensuremath{\\Delta t}}^T\\Big)\\nnrm{\\Lambda^{\\frac{1-\\beta}2}E(t)}{\\ensuremath{\\mathscr L}(H)}\\nnrm{\\Lambda^{\\frac{1-\\beta}2}F(t)}{\\ensuremath{\\mathscr L}(H)}\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\leq C\\int_0^{h^2+\\ensuremath{\\Delta t}}t^{-\\frac{1-\\beta}2}t^{-\\frac{1-\\beta}2}\\,\\ensuremath{\\mathrm{d}} t+C\\int_{h^2+\\ensuremath{\\Delta t}}^Tt^{-\\frac{1-\\beta}2}(h^2+\\ensuremath{\\Delta t})^\\beta t^{-\\frac{1+\\beta}2}\\,\\ensuremath{\\mathrm{d}} t\\\\\n&= C(h^2+\\ensuremath{\\Delta t})^\\beta(1+|\\log(h^2+\\ensuremath{\\Delta t})|)\\\\\n&\\leq C(h^{2\\beta}+(\\ensuremath{\\Delta t})^\\beta)|\\log(h^2+\\ensuremath{\\Delta t})|.\n\\end{aligned}\n\\end{equation}\nfor $h^2+\\ensuremath{\\Delta t}\\leq 1\/e$, where $C>0$ depends on $T$.\nThe combination of \\eqref{eq:proofSHE1}, \\eqref{eq:proofSHE2}, \\eqref{eq:proofSHE3} and \\eqref{eq:proofSHE4} finishes the proof.\n\\end{proof}\n\\subsection{Stochastic Volterra integro-differential equations}\\label{subsec:sve}\nHere we consider a stochastic integro-differential equation of Volterra-type where the deterministic equation exhibits a parabolic character. The error analysis is basically analogous to the heat equation and therefore we skip some computational details.\n\\label{rem:volterra} The weak solution of the Volterra-type stochastic evolution equation, a simple model of viscoelastic materials in the presence of noise,\n$$\n\\ensuremath{\\mathrm{d}} X(t) + \\left ( \\int_0^t \\frac{1}{\\Gamma(\\rho-1)}(t-s)^{\\rho-2} \\Lambda X(s) \\, \\ensuremath{\\mathrm{d}} s \\right ) \\, \\ensuremath{\\mathrm{d}} t = \\ensuremath{\\mathrm{d}} L(t),~t\\in (0,T]; ~ X(0) =X_0\\in H,\n$$\nis also given by \\eqref{eq:X}, where $(E(t))_{t\\ge 0}$ is the solution operator of the linear, homogeneous deterministic problem (see, for example, \\cite{BGK15,CDaPP,Sperlich} for Gaussian and fractional Brownian noise) and $\\rho\\in (1,2)$. Because of the special convolution kernel above, the equation can be also viewed as a fractional-in-time differential equation.\nUsing the same finite element approximation in space as for the heat equation and a convolution quadrature in time we consider the following recurrence (see \\cite{KovPri14a,KovPri14b} for the Gaussian case),\n\\begin{equation} \\label{eq:full_scheme volterra}\nX^n_{h,\\Delta t} - X^{n-1}_{h,\\Delta t} + \\Delta t \\left ( \\sum_{k=1}^{n} \\omega_{n-k}\\,  \\Lambda_{h} X^k_{h,\\Delta t} \\right ) = P_h(L(t_n)-L(t_{n-1})), \\quad n\\geq 1,\n\\end{equation}\nwith $X^0_{h,\\Delta t}=P_hX_0$ and convolution weights $(\\omega_k)_{k\\ge 0}$ chosen according to (see \\cite{lubich88,lubich88II})\n\\begin{equation} \\label{eq:weight}\n\\left ( \\frac{1 - z}{\\Delta t} \\right )^{1-\\rho} = \\sum_{k\\geq 0} \\omega_k z^k, \\quad |z|<1.\n\\end{equation}\nThe solution to \\eqref{eq:full_scheme volterra} can again be written in the form \\eqref{eq:schemeSHE2} with a suitable operator family $(E^n_{h,\\Delta t})_{n\\in \\mathbb{N}}$, (see \\cite{KovPri14a,KovPri14b} for the Gaussian case). Then, define $(\\tilde{E}(t))_{t\\in [0,T]}$ and $(\\tilde{X}(t))_{t\\in [0,T]}$ according to \\eqref{eq:EtildeSHE} and \\eqref{eq:XtildeSHE}, respectively. In contrast to the heat equation where \\eqref{eq:detEstSHE1} and \\eqref{eq:detEstSHE2} hold, one has the deterministic estimates (see \\cite{Lubich_et_al1996} and \\cite[Theorem 3.1]{KovPri14b})\n\\begin{align}\n\\nnrm{\\tilde E(s)-E(s)}{\\ensuremath{\\mathscr L}(H)}&\\leq C(h^{2\/\\rho}+\\ensuremath{\\Delta t})s^{-1},\\label{eq:detEstSVE1}\\\\\n\\nnrm{\\Lambda^\\alpha E(s)}{\\ensuremath{\\mathscr L}(H)}+\\nnrm{\\Lambda^\\alpha \\tilde E(s)}{\\ensuremath{\\mathscr L}(H)}&\\leq C s^{-\\rho\\alpha},\\quad 0\\leq\\alpha\\leq 1\/{(2\\rho)},\\label{eq:detEstSVE2}\n\\end{align}\n$s\\in(0,T]$, where $C>0$ does not depend on $h$, $\\ensuremath{\\Delta t}$ and $s$. Note further, that if $\\nnrm{\\Lambda^{-1\/(2\\rho)}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}( H)}<\\infty$, then using It\\^o's isometry, we get the stability estimate as in the Gaussian case, see \\cite[page 2333]{KovPri14a},\n$$\n\\nnrm{X(T)}{L^2(\\Omega,H)}\\le C (\\nnrm{\\Lambda^{-1\/(2\\rho)}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}( H)}+\\nnrm{X_0}{L^2(\\Omega,H)}),\n$$\nand, in particular, Assumption \\ref{ass:abstractSetting} (iv) holds. Furthermore, for $t\\in[\\epsilon, T]$ and $x\\in H$, we have that\n\\begin{equation*}\n\\|E(t)x\\|_{H}=\\|\\Lambda^{1\/(2\\rho)}E(t)\\Lambda^{-1\/(2\\rho)}x\\|_H\\le \\|\\Lambda^{1\/(2\\rho)}E(t)\\|_{\\ensuremath{\\mathscr L}(H)}\\|\\,\\|\\Lambda^{-1\/(2\\rho)}x\\|_{H}\\le \\|\\Phi_{\\epsilon}x\\|_H.\n\\end{equation*}\nTherefore, Assumption \\ref{ass:abstractSetting} (v) holds with $\\Phi_{\\epsilon}:=\\sup_{t\\in [\\epsilon, T]}\\|\\Lambda^{1\/(2\\rho)}E(t)\\|_{\\mathcal{L}(H)}\\Lambda^{-1\/(2\\rho)}$, where the supremum is finite because of \\eqref{eq:detEstSVE2}.\nHence, via an analogous calculation as for the heat equation above, setting $H=U=L^2(\\ensuremath{\\mathcal{O}})$, $B=\\id H$, assuming $\\|\\Lambda^{\\frac{\\beta-1\/\\rho}{2}}Q^\\frac12\\|^2_{\\ensuremath{\\mathscr L_2}(H)}<\\infty$, $\\beta\\in (0,1\/\\rho)$, $X_0\\in L^2(\\Omega,\\cF_0,\\bP;H)$, and using \\eqref{eq:detEstSVE1} and \\eqref{eq:detEstSVE2}, one arrives at the weak error estimate\n\\[\\big|\\bE\\big(g(X^N_{h,\\ensuremath{\\Delta t}})-g(X(T))\\big)\\big|\\leq C(h^{2\\beta}+(\\ensuremath{\\Delta t})^{\\rho\\beta})|\\log(h^{2\/\\rho}+\\ensuremath{\\Delta t})|, \\]\nfor $h^{2\/\\rho}+ \\ensuremath{\\Delta t}\\leq 1\/e$. This is essentially twice the strong rate where the latter is the same as in the Gaussian case \\cite{KovPri14a}, as the strong error analysis carries over to our setting, cf.~Remark \\ref{rem:strongErrorSHE}.\n\n\n\\section{Application to the wave equation}\n\n\\label{sec:Wave}\n\nHere, we apply the general error representation from Section~\\ref{sec:An error representation formula} to a discretization of the stochastic wave equation~\\eqref{eq:SWE}.\n\nLet $\\ensuremath{\\mathcal{O}}\\subset\\bR^d$ be a convex bounded domain and let the spaces $\\dot H^\\alpha$, $\\alpha\\in\\bR$, be as in Section~\\ref{sec:Heat}. We use the product spaces\n\\begin{equation*}\n\\cH^\\alpha:=\\dot H^\\alpha\\times\\dot H^{\\alpha-1},\\quad\\alpha\\in\\bR,\n\\end{equation*}\nwith inner product $\\langle v,w\\rangle_{\\cH^\\alpha}:=\\langle v_1,w_1\\rangle_{\\alpha}+\\langle v_2,w_2\\rangle_{\\alpha-1}$, $v=(v_1,v_2)^\\top$, $w=(w_1,w_2)^\\top$ and norm $\\nnrm{v}{\\cH^\\alpha}=(|v_1|_{\\alpha}^2+|v_2|_{\\alpha-1}^2)^{1\/2}$, where $\\langle\\cdot,\\cdot\\rangle_\\alpha$ and  $\\langle\\cdot,\\cdot\\rangle_{\\alpha-1}$ are the inner products in $\\dot H^\\alpha$ and $\\dot H^{\\alpha-1}$ corresponding to the norms $|\\cdot|_\\alpha$ and $|\\cdot|_{\\alpha-1}$ introduced in Section~\\ref{sec:Heat}. We set\n\\[ H:=\\cH^0=\\dot H^0\\times\\dot H^{-1}=L^2(\\ensuremath{\\mathcal{O}})\\times H^{-1}(\\ensuremath{\\mathcal{O}}),\\quad  U:=\\dot H^0=L^2(\\ensuremath{\\mathcal{O}})\\] and define operators $A:D(A)\\subset  H\\to H$ and $B\\in\\ensuremath{\\mathscr L}( U, H)$ by setting $D(A):= \\cH^1$ and\n\\begin{equation*}\nA:=\\begin{pmatrix}0 & -I \\\\ \\Lambda & 0\\end{pmatrix},\\quad B:=\\begin{pmatrix}0\\\\I\\end{pmatrix},\n\\end{equation*}\nwhere the Laplace operator\n$\\Lambda$ from Section~\\ref{sec:Heat} is now considered as an operator from $\\dot H^1$ to $\\dot H^{-1}$. It is well-known that $-A$ generates a strongly continuous semigroup $(E(t))_{t\\geq0}\\subset\\ensuremath{\\mathscr L}( H)$ given by\n\\begin{equation}\\label{eq:E(t)SWE}\nE(t)=\\begin{pmatrix}C(t) & \\Lambda^{-1\/2}S(t) \\\\ -\\Lambda^{1\/2}S(t) & C(t) \\end{pmatrix},\n\\end{equation}\nwhere $C(t):=\\cos(t\\Lambda^{1\/2})$ and $S(t):=\\sin(t\\Lambda^{1\/2})$ are the cosine and sine operators; compare \\cite[Example~B.1]{PesZab07}, \\cite[Section~A.5.4]{DPZ92} and \\cite[Section~3.14]{ABHN11}.\n\nWith these definitions the abstract equation \\eqref{eq:mainEq} becomes the stochastic wave equation~\\eqref{eq:SWE} with $ H$-valued solution $(X(t))_{t\\geq0}=((X_1(t),X_2(t))^\\top)_{t\\geq0}$. As in the Gaussian case, cf.~\\cite[Lemma~4.1]{KovLarLin13}, one sees that the condition $\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$ implies $\\eqref{eq:assEB1}$ and hence the existence of a unique weak solution $X=(X(t))_{t\\geq 0}$, given that the initial condition $X_0=(X_{0,1},X_{0,2})^\\top$ is $ H$-valued and $\\cF_0$-measurable. Furthermore,\n\\begin{equation}\\label{eq:wstab}\n\\nnrm{X(T)}{L^2(\\Omega,H)}\\le T\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}+\\nnrm{X_0}{L^2(\\Omega,H)}.\n\\end{equation}\n\nThe discretization of Eq.~\\eqref{eq:SWE} is done via finite elements\nin space and an $I$-stable rational single step scheme of order in time. (By `$I$-stable' we mean what is called `$I$-acceptable' in \\cite{NorWan79}.) We use the finite element setting introduced in Section~\\ref{sec:Heat}, the only difference being that we assume a possibly higher order approximation property of the Ritz projection $\\Pi_h$ of the form\n\\begin{equation}\\label{eq:basicFEMestimate2}\n\\nnrm{\\Pi_hv-v}{L^2(\\ensuremath{\\mathcal{O}})}\\leq C h^\\beta|v|_\\beta,\\quad v\\in\\dot H^\\beta,\\;1\\leq\\beta\\leq r,\n\\end{equation}\nwith $r\\ge 2$. For $r=2$ this requires no further assumption on the domain $\\ensuremath{\\mathcal{O}}$ (other than convexity) and $S_h$ can be chosen to be the space of continuous piecewise linear functions on a triangulation of $\\ensuremath{\\mathcal{O}}$ with maximal mesh-size $h$, as in the case of the heat equation. For $r>2$ this firstly requires extra assumptions on the domain $\\ensuremath{\\mathcal{O}}$, namely small enough interior angles in case of a polygon, or smooth enough boundary in case of a curved boundary. The reason is that to achieve $r>2$ one needs an elliptic regularity estimate $\\|u\\|_{H^r}\\le C\\|\\Lambda u\\|_{H^{r-2}}$ with $r>2$ instead of \\eqref{eq:ellreg} corresponding to $r=2$. Furthermore, if the boundary is curved, then the triangulation is not exact and one has to be more precise while approximating near the boundary and hence one needs special elements. For example, \\eqref{eq:basicFEMestimate2} holds for $r=4$ for bounded convex domains with smooth boundary and $S_h$ consisting of continuous piecewise cubic polynomials using special, so-called isoparametric elements, near the boundary, see \\cite[Chapter 1]{Tho06}.\nAlthough \\eqref{eq:basicFEMestimate2} does not appear explicitly in the present paper, it is the key ingredient in the proof of the deterministic error estimate for the finite element approximation of the wave equation which we will use later on and hence we state it as our abstract assumption on the finite element spaces $S_h$.\nLet the discretization $A_h:S_h\\times S_h\\to S_h\\times S_h$ of the operator $A:D(A)\\subset H\\to H$ be defined by\n\\begin{equation*}\nA_h:=\\begin{pmatrix}0 & -I \\\\ \\Lambda_h & 0\\end{pmatrix},\n\\end{equation*}\nwhere $\\Lambda_h:S_h\\to S_h$ is the discrete Laplacian introduced in \\eqref{eq:discreteLaplace}.\nThen $-A_h$ generates a strongly continuous semigroup $(E_h(t))_{t\\geq0}\\subset \\ensuremath{\\mathscr L}(S_h\\times S_h)$. As in Section~\\ref{sec:Heat}, we consider for $N\\in\\bN$ a uniform grid $t_n=n\\ensuremath{\\Delta t}=n (T\/N)$, $n=0,\\ldots,N$, on a finite time interval $[0,T]$. We approximate the operators $E_h(t_n)\\in\\ensuremath{\\mathscr L}(S_h\\times S_h)$ by\n\\[E_{h,\\ensuremath{\\Delta t}}^n:=(R(\\ensuremath{\\Delta t} A_h))^n,\\]\nwhere $R$ is a rational function that satisfies the approximation and stability properties\n\\begin{equation*}\n\\begin{aligned}\n|R(iy)-e^{-iy}|&\\leq C |y|^{p+1},\\quad |y|\\leq b,\\\\\n|R(iy)|&\\leq 1,\\quad y\\in\\bR,\n\\end{aligned}\n\\end{equation*}\nfor some positive integer $p$ and some $b>0$; see \\cite{BakBra79, BreTho80} for details. For instance, choosing $R(\\lambda)=1\/(1-\\lambda)$ and  $R(\\lambda)=(2-\\lambda)\/(2+\\lambda)$ yields the backward Euler method ($p=1$) and the Crank-Nicolson method ($p=2$), respectively.\n\nThe numerical scheme for the stochastic wave equation \\eqref{eq:mainEq} can now be formulated as follows:\nFor $h>0$ and $N\\in\\bN$, the discretization $(X^n_{h,\\ensuremath{\\Delta t}})_{n=0,\\ldots,N}$ of $(X(t))_{t\\in[0,T]}$ in space and time is given as the solution to\n\\begin{equation}\\label{eq:schemeSWE1}\nX_{h,\\ensuremath{\\Delta t}}^n=E_{h,\\ensuremath{\\Delta t}}\\big(X_{h,\\ensuremath{\\Delta t}}^{n-1}+P_h B(L(t_n)-L(t_{n-1}))\\big),\\quad n=1,\\ldots,N;\\quad X_{h,\\ensuremath{\\Delta t}}^0=P_hX_0.\n\\end{equation}\nBy slight abuse of notation, we denote here and in the sequel by $P_h$ both the generalized $L^2$-projection from $\\dot H^{-1}$ onto $S_h$ defined by $\\langle P_hv,w\\rangle_{L^2(\\ensuremath{\\mathcal{O}})}=\\langle v,w\\rangle_{\\dot H^{-1}\\times\\dot H^1}$, $v\\in \\dot H^{-1}$, $w\\in S_h$, and the corresponding projection from $ H=\\dot H^{0}\\times\\dot H^{-1}$ onto $S_h\\times S_h$ defined by the action of the former projection on the coordinates of elements in $\\dot H^{0}\\times\\dot H^{-1}$. Moreover, $P^1: H\\to\\dot H^0$ is the projection of elements in $H=\\dot H^0\\times\\dot H^{-1}$ on the first coordinate.\n\n\\begin{remark}[strong error]\\label{rem:strongErrorSWE}\nAs observed for the discretization of the heat equation in Remark~\\ref{rem:strongErrorSHE}, strong $L^2$-error estimates for the scheme \\eqref{eq:schemeSWE1} carry over from the Gaussian case in the L\\'{e}vy $L^2$-martingale case since they only use It\\^{o}'s isometry \\eqref{eq:ItoIsom}. Arguing as in the proof of \\cite[Theorem~4.13]{KovLarLin13}, we obtain that, if\n\\begin{equation}\\label{eq:SWEassQ}\n\\nnrm{\\Lambda^{\\frac{\\beta-1}2}Q^{\\frac12}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty\\quad\\text{ and }\\quad X_0\\in L^2(\\Omega,\\cF_0,\\bP; \\cH^{\\beta})\n\\end{equation}\nfor some $\\beta>0$, then the scheme \\eqref{eq:schemeSWE1} approximates the first component $X_1=P^1X$ of the solution $X$ to \\eqref{eq:mainEq} with strong order $\\min(\\beta r\/(r+1),r)$ in space and $\\min(\\beta p\/(p+1),1)$ in time:\n\\[\\nnrm{X^n_{h,\\ensuremath{\\Delta t},1}-X_1(t_n)}{L^2(\\Omega;\\cdot H^0)}\\leq C\\big(h^{\\min(\\beta\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\beta\\frac{p}{p+1},1)}\\big),\\quad n=0,\\ldots,N.\\]\nHere we have set $X^n_{h,\\ensuremath{\\Delta t},1}:=P^1X^n_{h,\\ensuremath{\\Delta t}}$. The condition \\eqref{eq:SWEassQ} implies that the solution $X=(X(t))_{t\\geq0}$ takes values in $ \\cH^\\beta$, cf.~\\cite[Theorem~3.1]{KovLarSae10}.\n\n\\end{remark}\n\nThe solution to the scheme \\eqref{eq:schemeSWE1} is given by\n\\begin{equation*}\\label{eq:schemeSWE2}\nX^n_{h,\\ensuremath{\\Delta t}}=E_{h,\\ensuremath{\\Delta t}}^nP_hX_0+\\sum_{j=1}^n E^{n-j+1}_{h,\\ensuremath{\\Delta t}}P_hB(L(t_n)-L(t_{n-1})),\\quad n=0,\\ldots,N.\n\\end{equation*}\nFor $t\\in[0,T]$, define operators $\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)\\in\\ensuremath{\\mathscr L}( H)$ by\n\\begin{equation}\\label{eq:EtildeSWE}\n\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t):=\\mathds 1_{\\{0\\}}(t)P_h+\\sum_{j=1}^N\\mathds 1_{(t_{n-1},t_n]}(t)E_{h,\\ensuremath{\\Delta t}}^nP_h,\n\\end{equation}\nwhere the projection $P_h$ is understood as a mapping from $ H=\\dot H^{0}\\times\\dot H^{-1}$ to $S_h\\times S_h$. Then, analogously to the corresponding argument in Section~\\ref{sec:Heat}, one sees that the $S_h\\times S_h$-valued process $(\\tilde X(t))_{t\\in[0,T]}=(\\tilde X_{h,\\ensuremath{\\Delta t}}(t))_{t\\in[0,T]}$ defined by\n\\begin{equation}\\label{eq:XtildeSWE}\n\\tilde X(t)=\\tilde X_{h,\\ensuremath{\\Delta t}}(t):=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)X_0+\\int_0^t\\tilde E_{h,\\ensuremath{\\Delta t}}(t-s)B\\,\\ensuremath{\\mathrm{d}} L(s)\n\\end{equation}\nsatisfies $X^n_{h,\\ensuremath{\\Delta t}}=\\tilde X(t_n)$ $\\bP$-almost surely.\n\nThe proof of the deterministic error estimate in the next lemma is postponed to the end of this section.\n\\begin{lemma} \\label{lem:SWE}\nLet $\\alpha\\geq0$. The operators $E(t)$ and $\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)$ defined in \\eqref{eq:E(t)SWE} and \\eqref{eq:EtildeSWE} satisfy the error estimate\n\\begin{equation}\\label{eq:detEstSWE}\n\\begin{gathered}\n    \\sup_{t\\in[0,T]}\\big(\\nnrm{P^1(\\tilde E(t)-E(t))}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\dot H^0)}\n    +\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\\big)\\\\\n    \\leq C\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},1)}\\big),\n\\end{gathered}\n\\end{equation}\nfor $\\ensuremath{\\Delta t}\\leq1$, where $C=C(T)>0$ does not depend on $h$ and $\\ensuremath{\\Delta t}$\n\\end{lemma}\n\nWe are now in the position to prove the following result concerning the weak error of the approximation $X^N_{h,\\ensuremath{\\Delta t},1}:=P^1X^N_{h,\\ensuremath{\\Delta t}}$ of the first component $X_1(T)=P^1X(T)$ of the solution to the stochastic wave eqation~\\eqref{eq:mainEq} at time $T$.\n\n\\begin{theorem}\\label{thm:SWE}\nLet $X_0\\in L^2(\\Omega,\\cF_0,\\bP; \\cH^{2\\beta})$ for some $\\beta>0$ and $g\\in C^2(H,\\bR)$ with $\\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$. Suppose that either of the following conditions holds.\n\\begin{itemize}\n\\item[\\upshape (i)] $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$ and\n\\begin{equation}\\label{eq:assgSWE}\n\\sup_{x\\in\\dot H^0}\\nnrm{\\Lambda^{\\frac\\beta2}g''(x)\\Lambda^{-\\frac\\beta2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}<\\infty.\n\\end{equation}\n\\item[\\upshape (ii)]\n\\begin{equation}\\label{eq:weqii}\n\\lim_{m\\to\\infty}\\int_{U_1}\\|\\Lambda^{-\\frac12}p_m y\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-\\frac{1}{2}}p_m y\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)<\\infty,\n\\end{equation}\nwhere $p_m$ denotes the orthogonal projection from $\\dot H^0$ to $\\operatorname{span}\\{\\varphi_1,\\ldots,\\varphi_m\\}$, $(\\varphi_k)_{k\\in\\mathbb N}$ being an orthonormal basis of $\\dot H^0$ consisting of eigenvectors of $\\Lambda$.\n\\end{itemize}\nThen, there is a unique weak solution $(X(t))_{t\\geq0}$ to Eq.~\\eqref{eq:mainEq} given by \\eqref{eq:X}. \n\nLet $(X_{h,\\ensuremath{\\Delta t}}^n)_{n=0,\\ldots,N}$ be given by the scheme \\eqref{eq:schemeSWE1}. Then, there exists a constant $C=C(g,T)>0$ that does not depend on $h$ and $\\ensuremath{\\Delta t}$, such that for $\\ensuremath{\\Delta t}\\leq 1$\n$$\n    \\big|\\bE\\big(g(X^N_{h,\\ensuremath{\\Delta t},1})-g(X_1(T))\\big)\\big|\\leq C\\big(h^{\\min(2\\beta\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(2\\beta\\frac{p}{p+1},1)}\\big)\n$$\n\\end{theorem}\n\nBefore proving Theorem~\\ref{thm:SWE}, we state two remarks and discuss some examples where the conditions of Theorem \\ref{thm:SWE}, in particular \\eqref{eq:assgSWE} and \\eqref{eq:weqii}, are satisfied.\n\n\\begin{remark}\nAs a consequence of Lemma~\\ref{lem:integrandIsomorphism} and the fact that $\\Lambda^\\alpha p_m\\in\\ensuremath{\\mathscr L_2}(\\dot H^0)$ for all $\\alpha\\in\\bR$ and $m\\in\\bN$, the terms $\\Lambda^{-1\/2}p_m y$ and $\\Lambda^{\\beta-1\/2}p_m y$ in \\eqref{eq:weqii} are defined in an $L^2(U_1,\\nu(\\ensuremath{\\mathrm{d}} y);\\dot H^0)$-sense. The sequence $\\big(\\|\\Lambda^{-1\/2}p_m y\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-1\/2}p_m y\\|_{\\dot{H}^0}\\big)_{m\\in\\bN}$ is monotonically increasing for $\\nu$-almost all $y\\in U_1$, so that the limit in \\eqref{eq:weqii} is in fact a supremum. Moreover, if we explicitly choose $U_1=\\dot H^{\\beta-1}$ as the state space of $L$, then the condition~(ii) is equivalent to assuming that $\\operatorname{supp}\\nu\\subset \\dot H^{2\\beta-1}$ and\n\\[\\int_{\\dot H^{2\\beta-1}}\\|\\Lambda^{-\\frac12}y\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-\\frac{1}{2}}y\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)<\\infty .\\]\nThis choice of $U_1$ is possible w.l.o.g.\\ whenever $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$, since then the natural embedding of $U_0=Q^{1\/2}U=Q^{1\/2}\\dot H^0$ into $\\dot H^{\\beta-1}$ is Hilbert-Schmidt and we can re-expand $L$ in the form \\eqref{eq:LPexpansion} as an $\\dot H^{\\beta-1}$-valued martingale, compare Remark~\\ref{rem:approxL}. However, in the spirit of, e.g., \\cite{AppRie10,Rie12,Rie14}, we prefer a formulation of our results that is independent of the specific choice of the state space $U_1$.\n\\end{remark}\n\n\\begin{remark}\nInstead of the symmetric condition $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$, the sufficient asymmetric condition $\\nnrm{\\Lambda^{\\beta-1\/2}Q\\Lambda^{-1\/2}}{\\ensuremath{\\mathscr L_1}(\\dot{H}^0)}<\\infty$ is imposed in \\cite{KovLarLin13} in the Wiener case in order to double the rate of strong convergence for the wave equation.\nThe asymmetric condition \\eqref{eq:weqii}\nappearing in (ii) above, which is again sufficient for\\linebreak $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$,\nresembles the same situation in the present case.\n\\end{remark}\n\n\n\\begin{example}\nAn important basic example which satisfies \\eqref{eq:assgSWE} is $g(x)=\\|x\\|^2_{\\dot{H}^0}$.\n\\end{example}\n\n\\begin{example}\nAs another example for a test function $g$ satisfying \\eqref{eq:assgSWE} consider\n\\begin{equation*}\ng(x):=f(\\langle\\varphi_1,x\\rangle_{\\dot H^0},\\ldots,\\langle\\varphi_n,x\\rangle_{\\dot H^0}),\\quad x\\in \\dot H^0,\n\\end{equation*}\nwhere $f\\in C^2(\\bR^n,\\bR)$ has bounded second order derivatives and $(\\varphi_k)_{k\\in\\bN}\\subset D(\\Lambda)$ is an orthonormal basis of $\\dot H^0=L^2(\\ensuremath{\\mathcal{O}})$ consisting of eigenfunctions of $\\Lambda $ with corresponding eigenvalues $(\\lambda_k)_{k\\in\\bN}\\subset(0,\\infty)$. Then, for $x,y\\in\\dot H^0$,\n\\[\n\\Lambda^{\\beta\/2}g''(x)\\Lambda^{-\\beta\/2}y=\\sum_{j,k=1}^n\\lambda_j^{-\\beta\/2}\\lambda_k^{\\beta\/2}(\\partial_j\\partial_kf)\\big(\\langle\\varphi_1,x\\rangle_{\\dot H^0},\\ldots,\\langle\\varphi_n,x\\rangle_{\\dot H^0}\\big)\\langle\\varphi_j,y\\rangle_{\\dot H^0}\\varphi_k\n\\]\nand \\eqref{eq:assgSWE} holds. More generally, the condition \\eqref{eq:assgSWE} is satisfied by all $g\\in C^2(\\dot H^0,\\bR)$ of the form $g=\\tilde g\\circ\\Lambda^{-\\beta\/2}$ with $\\tilde g\\in C^2(\\dot H^0,\\bR)$ satisfying $\\sup_{x\\in\\dot H^0}\\nnrm{\\tilde g''(x)}{\\ensuremath{\\mathscr L}(\\dot H^0)}<\\infty$. For such $g$ we have $g''(x)=\\Lambda^{-\\beta\/2}\\tilde g''(\\Lambda^{-\\beta\/2}x)\\Lambda^{-\\beta\/2}$.\n\\end{example}\n\n\\begin{example}\\label{ex:spec}\nConsider the situation of Example \\ref{ex:1dlevy}; that is when\n\\[\\nu=\\sum_{k\\in\\bN}\\nu_k\\circ\\pi_k^{-1},\\]\nwhere $\\nu_k$ is the L\\'{e}vy measure of $L_k$ and $\\pi_k:\\bR\\to U_1$ is defined by $\\pi_k(\\xi):=\\xi e_k$. Let us assume that $e_k=\\sqrt{q_k}\\,\\varphi_k$, where $(q_k)_{k\\in\\bN}$ is a bounded sequence of positive numbers and $(\\varphi_k)_{k\\in \\bN}$ is an orthonormal basis of $\\dot{H}^0$ consisting of eigenfunctions of $\\Lambda$ with corresponding eigenvalues $(\\lambda_k)_{k\\in \\bN}$.\nThen, with $p_m$ as in (ii) in Theorem \\ref{thm:SWE},\n\\begin{align*}\n&\\lim_{m\\to\\infty}\\int_{U_1}\\|\\Lambda^{-\\frac12}p_m y\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-\\frac{1}{2}}p_m y\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&=\\sum_{k\\in \\bN}\\int_{\\bR}\\xi^2q_k\\|\\Lambda^{-\\frac12}\\varphi_k\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-\\frac12}\\varphi_k\\|_{\\dot{H}^0}\\,\\nu_k(\\ensuremath{\\mathrm{d}} \\xi)\\\\\n&=\\sum_{k\\in \\bN}\\int_{\\bR}\\xi^2q_k\\lambda_k^{-\\frac12}\\|\\varphi_k\\|_{\\dot{H}^0}\\lambda_k^{\\beta-\\frac12}\\|\\varphi_k\\|_{\\dot{H}^0}\\,\\nu_k(\\ensuremath{\\mathrm{d}} \\xi)\\\\\n&=\\sum_{k\\in \\bN}\\int_{\\bR}\\xi^2q_k\\|\\Lambda^{\\frac{\\beta-1}2}\\varphi_k\\|_{\\dot{H}^0}\\|\\Lambda^{\\frac{\\beta-1}2}\\varphi_k\\|_{\\dot{H}^0}\\,\\nu_k(\\ensuremath{\\mathrm{d}} \\xi)\\\\\n&=\\lim_{m\\to\\infty}\\int_{U_1}\\|\\Lambda^{\\frac{\\beta-1}2}p_my\\|_{\\dot{H}^0}\\|\\Lambda^{\\frac{\\beta-1}2}p_my\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\n\\,=\\,\\nnrm{\\Lambda^{\\frac{\\beta-1}2}Q^{\\frac12}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2,\n\\end{align*}\nwhere we used Lemma~\\ref{lem:integrandIsomorphism} in the last step.\nThat is, when $\\nu$ is concentrated on the eigenspaces $\\{r\\varphi_k:r\\in\\bR\\}$, $k\\in\\bN$, of $\\Lambda$, then the abstract asymmetric condition \\eqref{eq:weqii} coincides with the familiar symmetric Hilbert-Schmidt condition. The situation is similar in the Wiener case \\cite{KovLarLin13} when $\\Lambda$ and $Q$ commute.\n\\end{example}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:SWE}]\nFirst suppose that (i) holds. We apply Theorem~\\ref{thm:errorRep} and Corollary~\\ref{cor:errorRep} with $G=g\\circ P^1$. Note that $G'(x)=(P^1)^*g'(P^1x)\\in H$ and $$G''(x)=(P^1)^*g''(P^1x)P^1\\in\\ensuremath{\\mathscr L}( H)$$ for all $x\\in H$, where $(P^1)^*\\in\\ensuremath{\\mathscr L}(\\dot H^0, H)$ is the Hilbert space adjoint of $P^1\\in\\ensuremath{\\mathscr L}( H,\\dot H^0)$. Using \\eqref{eq:u_xu_xx}\none obtains\n\\begin{equation}\\label{eq:proofSWE0}\nu_x(t,\\xi)=\\bE\\big((P^1)^*g'(P^1Z(T,t,x))\\big)\\big|_{x=\\xi},\\quad u_{xx}(t,\\xi)=\\bE\\big((P^1)^*g''(P^1Z(T,t,x))P^1\\big)\\big|_{x=\\xi}\n\\end{equation}\nfor all $H$-valued random variables $\\xi$ and $t\\in[0,T]$.\n\n\nWe combine \\eqref{eq:wstab}, \\eqref{eq:proofSWE0} and the deterministic error estimate \\eqref{eq:detEstSWE} with $\\alpha=2\\beta$ in order to estimate the first term on the right hand side of the error representation formula~\\eqref{eq:errorRep4} in Corollary~\\ref{cor:errorRep}. We have, where the first inequality follows similarly as for the stochastic heat equation, that\n\\begin{equation}\\label{eq:proofSWE1}\n\\begin{aligned}\n\\big|\\bE\\big\\{&u(0,\\tilde E(T)X_0)-u(0,E(T)X_0)\\big\\}\\big|\\\\\n&\\leq \\int_0^1\\big\\|g'\\big(P^1Z\\big(T,0,Y(0)+\\theta(\\tilde Y(0)-Y(0))\\big)\\big)\\big\\|_{L^2(\\Omega, \\dot{H}^0)}\\,\\ensuremath{\\mathrm{d}}\\theta\\\\\n&\\qquad \\times \\|P^1(\\tilde E(T)-E(T))X_0\\|_{L^2(\\Omega, \\dot{H}^0)}\\\\\n&\\leq C\\big(1+\\int_0^1\\big\\|Z\\big(T,0,Y(0)+\\theta(\\tilde Y(0)-Y(0))\\big)\\big\\|_{L^2(\\Omega, H)}\\,\\ensuremath{\\mathrm{d}} \\theta\\big)\\\\\n&\\qquad \\times \\nnrm{P^1(\\tilde E(T)-E(T))}{\\ensuremath{\\mathscr L}( \\cH^{2\\beta},\\dot H^0)}\\,\\nnrm{X_0}{L^2(\\Omega; \\cH^{2\\beta})}\\\\\n&\\leq C\\big(1+\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}+\\nnrm{X_0}{L^2(\\Omega; H)}\\big)\\,\\nnrm{X_0}{L^2(\\Omega; \\cH^{2\\beta})}\\\\\n&\\qquad \\times \\big(h^{\\min(2\\beta\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(2\\beta\\frac{p}{p+1},1)}\\big).\n\\end{aligned}\n\\end{equation}\nUsing \\eqref{eq:proofSWE0}, Lemma~\\ref{lem:integrandIsomorphism} and Remark~\\ref{not:Phix}, the integral of the function $\\Psi_1$ in the second term on the right hand side of the formula \\eqref{eq:errorRep4} can be treated as follows:\n\\begin{equation}\\label{eq:proofSWE2}\n\\begin{aligned}\n\\Big|\\bE&\\int_0^T\\int_{ U_1}\\int_0^1\\Psi_1(t,\\theta,y)\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&= \\Big|\\bE\\int_0^T\\int_{ U_1}\\int_0^1(1-\\theta)\\Big\\langle\\bE\\big(g''\\big(P^1Z(T,t,x+E(T-t)By+\\theta F(T-t)y)\\big)\\big)\\big|_{x=\\tilde Y(t)}\\\\\n&\\quad\\times P^1F(T-t)y,\\;P^1F(T-t)y\\Big\\rangle_{\\dot H^0}\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&\\leq \\sup_{x\\in\\dot H^0}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\int_0^T\\nnrm{P^1F(T-t)}{\\ensuremath{\\mathscr L_2}( U_0,\\dot H^0)}^2\\ensuremath{\\mathrm{d}} t\\\\\n&\\leq \\sup_{x\\in\\dot H^0}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(\\dot H^0)} C \\big(h^{\\min(\\beta\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\beta\\frac{p}{p+1},1)}\\big)^2\n\\end{aligned}.\n\\end{equation}\nThe last step in \\eqref{eq:proofSWE2} is due to the fact that, by It\\^{o}'s isometry \\eqref{eq:ItoIsom}, the integral in the penultimate line is the square of the strong error $\\nnrm{X^N_{h,k,1}-X_1(T)}{L^2(\\Omega;\\dot H^0)}$ for zero initial condition $X_0=0$; it can be estimated as in the Gaussian case \\cite[Theorem~4.13]{KovLarLin13}, compare Remark~\\ref{rem:strongErrorSWE}.\n\nConcerning the integral of the function $\\Psi_2$ in the second term on the right hand side of Eq.~\\eqref{eq:errorRep4}, we have by \\eqref{eq:proofSWE0}, Lemma~\\ref{lem:integrandIsomorphism}, \\eqref{eq:estHS} and since $U_0=Q^{1\/2}(U)=Q^{1\/2}(\\dot H^0)$,\n\\begin{equation}\\label{eq:proofSWE3}\n\\begin{aligned}\n\\Big|\\bE&\\int_0^T\\int_{ U_1}\\int_0^1\\Psi_2(t,\\theta,y)\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&= \\Big|\\bE\\int_0^T\\int_{ U_1}\\int_0^1\\Big\\langle\\bE\\big(g''\\big(P^1Z(T,t,x+\\theta E(T-t)By)\\big)\\big)\\big|_{x=\\tilde Y(t)}\\\\\n&\\quad\\times P^1E(T-t)By,P^1F(T-t)y\\Big\\rangle_{\\dot H^0}\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&= \\Big|\\bE\\int_0^T\\int_{ U_1}\\int_0^1\\Big\\langle\\bE\\big(\\Lambda^{\\frac\\beta2}g''\\big(P^1Z(T,t,x+\\theta E(T-t)By)\\big)\\Lambda^{-\\frac\\beta2}\\big)\\big|_{x=\\tilde Y(t)}\\\\\n&\\quad\\times \\Lambda^{\\frac\\beta2}P^1E(T-t)B\\Lambda^{\\frac{1-\\beta}2}\\Lambda^{\\frac{\\beta-1}2}y,\\;\\Lambda^{-\\frac\\beta2}P^1F(T-t)\\Lambda^{\\frac{1-\\beta}2}\\Lambda^{\\frac{\\beta-1}2}y\\Big\\rangle_{\\dot H^0}\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&\\leq \\sup_{x\\in\\dot H^0}\\nnrm{\\Lambda^{\\frac\\beta2}g''(x)\\Lambda^{-\\frac\\beta2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\nnrm{\\Lambda^{\\frac{\\beta-1}2}Q^{\\frac12}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\\\\\n&\\quad\\times \\int_0^T\\nnrm{\\Lambda^{\\frac\\beta2}P^1E(T-t)B\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\nnrm{\\Lambda^{-\\frac\\beta2}P^1F(T-t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\,\\ensuremath{\\mathrm{d}} t.\n\\end{aligned}\n\\end{equation}\nNote that, by the definition of $B = (0,I)^\\top$ and $E(t)$ from \\eqref{eq:E(t)SWE} we have\n\\begin{equation}\\label{eq:etb}\n\\nnrm{\\Lambda^{\\frac\\beta2}P^1E(T-t)B\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\n=\\nnrm{\\Lambda^{\\frac{\\beta-1}2}S(T-t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\n=\\nnrm{S(T-t)}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\leq1;\n\\end{equation}\nit remains to estimate the integral\n\\begin{equation*}\\label{eq:proofSWE4}\n\\begin{aligned}\n\\int_0^T\\nnrm{\\Lambda^{-\\frac\\beta2}P^1F(T-t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\ensuremath{\\mathrm{d}} t\n&=\\int_0^T\\nnrm{P^1F(t)}{\\ensuremath{\\mathscr L}(\\dot H^{\\beta-1},\\dot H^{-\\beta})}\\ensuremath{\\mathrm{d}} t\\\\\n&=\\int_0^T\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{\\beta-1},\\dot H^{-\\beta})}\\ensuremath{\\mathrm{d}} t.\n\\end{aligned}\n\\end{equation*}\nTo this end, it suffices to apply the deterministic error estimate \\eqref{eq:detEstSWE} with $\\alpha=2\\beta$. The combination of \\eqref{eq:proofSWE1}, \\eqref{eq:proofSWE2} and \\eqref{eq:proofSWE3} finishes the proof.\n\nNext, suppose that (ii) holds.\nBy Lemma~\\ref{lem:integrandIsomorphism} we have\n\\begin{comment}\n\\begin{align*}\n&\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n=\\lim_{m\\to\\infty}\\nnrm{p_m\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n=\\lim_{m\\to\\infty}\\nnrm{\\Lambda^{(\\beta-1)\/2}p_mQ^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\\\\\n&=\\lim_{m\\to\\infty}\\int_{U_1}\\|\\Lambda^{(\\beta-1)\/2}p_my\\|_{\\dot{H}^0}^2\\,\\nu(\\ensuremath{\\mathrm{d}} y)\n=\\lim_{m\\to\\infty}\\int_{U_1}\\langle\\Lambda^{-1\/2}p_my,\\Lambda^{\\beta-1\/2}p_my\\rangle_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\leq\\lim_{m\\to\\infty}\\int_{U_1}\\|\\Lambda^{-1\/2}p_my\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-1\/2}p_my\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)<\\infty,\n\\end{align*}\n\\end{comment}\n\\begin{align*}\n\\nnrm{p_m\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n&=\\nnrm{\\Lambda^{(\\beta-1)\/2}p_mQ^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n=\\int_{U_1}\\|\\Lambda^{(\\beta-1)\/2}p_my\\|_{\\dot{H}^0}^2\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&=\\int_{U_1}\\langle\\Lambda^{-1\/2}p_my,\\Lambda^{\\beta-1\/2}p_my\\rangle_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\leq\\int_{U_1}\\|\\Lambda^{-1\/2}p_my\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-1\/2}p_my\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y),\n\\end{align*}\nhence,\n$$\n    \\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n    =\\lim_{m\\to\\infty}\\nnrm{p_m\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n    < \\infty,\n$$\nproving that there is a unique weak solution $(X(t))_{t\\geq0}$ to Eq.~\\eqref{eq:mainEq} given by \\eqref{eq:X}.\nTo estimate the weak error, we apply an approximation procedure and consider for $m\\in\\bN$ the $H$-valued random variables\n$X^{[m]}(T):=E(T)X_0+\\int_0^TE(T-s)Bp_m\\,\\ensuremath{\\mathrm{d}} L(s)$ and $\\tilde X^{[m]}(T)=\\tilde X^{[m]}_{h,\\ensuremath{\\Delta t}}(T):=\\tilde E(T)X_0+\\int_0^T\\tilde E(T-s)Bp_m\\,\\ensuremath{\\mathrm{d}} L(s)$. Using It\\^{o}'s isometry and the fact that $\\nnrm{(I-p_m)\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}\\to 0$ as $m\\to\\infty$, we get that $X^{[m]}(T)\\xrightarrow{m\\to\\infty} X(T)$ and $\\tilde X^{[m]}(T)\\xrightarrow{m\\to\\infty}\\tilde X(T)$ in $L^2(\\Omega;H)$. As a consequence of this and \\eqref{eq:growthGG'}, we obtain $e^{[m]}(T):=\\bE\\big( G(\\tilde X^{[m]}(T))-G(X^{[m]}(T))\\big)\\xrightarrow{m\\to\\infty}  e(T)$. Thus, it suffices to show the desired decay rate for the error $e^{[m]}(T)$ with a constant that does not depend on $m$. To this end, we observe that the estimate \\eqref{eq:proofSWE1} can be used without any change, and that the analogue to the estimate \\eqref{eq:proofSWE2} gives indeed the desired rate if we use that $\\nnrm{p_m\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}\\leq\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$. Finally, the estimate corresponding to \\eqref{eq:proofSWE3} reads\n\n\n\\begin{equation*}\\label{eq:proofSWE3b}\n\\begin{aligned}\n\\Big|&\\bE\\int_0^T\\int_{U_1}\\int_0^1\\Big\\langle\\bE\\big(g''\\big(P^1Z^{[m]}(T,t,x+\\theta E(T-t)Bp_my)\\big)\\big)\\big|_{x=\\tilde Y^{[m]}(t)}\\\\\n&\\quad\\times P^1E(T-t)Bp_my,P^1F(T-t)p_my\\Big\\rangle_{\\dot H^0}\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&= \\Big|\\bE\\int_0^T\\int_{U_1}\\int_0^1\\Big\\langle\\bE\\big(g''\\big(P^1Z^{[m]}(T,t,x+\\theta E(T-t)Bp_my)\\big)\\big)\\big|_{x=\\tilde Y^{[m]}(t)}\\\\\n&\\quad\\times P^1E(T-t)B\\Lambda^{1\/2}\\Lambda^{-1\/2}p_my,P^1F(T-t)\\Lambda^{1\/2-\\beta}\\Lambda^{\\beta-1\/2}p_my\\Big\\rangle_{\\dot H^0}\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&\\leq \\sup_{x\\in\\dot H^0}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\int_0^T\\nnrm{P^1E(T-t)B\\Lambda^{1\/2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\nnrm{P^1F(T-t)\\Lambda^{1\/2-\\beta}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\quad\\times \\int_{U_1}\\|\\Lambda^{-1\/2}p_my\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-1\/2}p_my\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y),\n\\end{aligned}\n\\end{equation*}\nwhere $Z^{[m]}$ and $\\tilde Y^{[m]}$ are defined by replacing $B$ by $Bp_m$ in the definitions of $Z$ and $\\tilde Y$.\nBy \\eqref{eq:detEstSWE} with $\\alpha=2\\beta$\nand the fact that $\\nnrm{B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1}, \\cH^\\alpha)}=1$ we have that\n\\begin{align*}\n&\\sup_{t\\in[0,T]}\\nnrm{P^1F(T-t)\\Lambda^{1\/2-\\beta}}{\\ensuremath{\\mathscr L}(\\dot H^0)}=\\sup_{t\\in[0,T]}\\nnrm{P^1(\\tilde E(t)-E(t))B\\Lambda^{1\/2-\\beta}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\\\\n&\\quad =\\sup_{t\\in[0,T]}\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{2\\beta-1},\\dot H^0)}\\leq C\\big(h^{\\min(2\\beta\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(2\\beta\\frac{p}{p+1},1)}\\big).\n\\end{align*}\nFinally, by \\eqref{eq:weqii} and \\eqref{eq:etb} with $\\beta =0$, the proof is complete.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:SWE}]\nWe use the estimates\n\\begin{equation}\\label{eq:detEstSWE1}\n\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(E^n_{h,\\ensuremath{\\Delta t}}P_h-E(t_n))}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\dot H^0)}\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big)\n\\end{equation}\nand\n\\begin{equation}\\label{eq:detEstSWE2}\n\\nnrm{E(t)-E(s)}{\\ensuremath{\\mathscr L}( \\cH^\\delta, H)}\\leq C|t-s|^\\delta,\\quad t,\\,s\\geq0,\\;\\delta\\in[0,1].\n\\end{equation}\nfrom Corollary~4.11 and Lemma~4.4 in \\cite{KovLarLin13}. Corollary~4.11 in \\cite{KovLarLin13} is based on an error estimate proved in \\cite{BakBra79}.\n\nBecause of the `piecewise' definition of $\\tilde E(t)$ in \\eqref{eq:EtildeSWE}, the combination of \\eqref{eq:detEstSWE1} and \\eqref{eq:detEstSWE2} gives\n\\begin{equation}\\label{eq:detEstSWE3}\n\\begin{aligned}\n&\\sup_{t\\in[0,T]}\\nnrm{P^1(\\tilde E(t)-E(t))}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\dot H^0)}\\\\\n&\\leq \\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\dot H^0)}+\\sup_{n\\in\\{1,\\ldots,N\\}}\\sup_{t\\in(t_{n-1},t_n)}\\nnrm{E(t_n)-E(t)}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\cH)}\\\\\n&\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha,1)}\\big)\\\\\n&=C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},1)}\\big)\n\\end{aligned}\n\\end{equation}\nfor $\\ensuremath{\\Delta t}\\leq 1$.\nIt remains to show that\n\\begin{equation}\\label{eq:detEstSWE4}\n\\sup_{t\\in[0,T]}\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},1)}\\big).\n\\end{equation}\nTo this end, we will prove the estimate\n\\begin{equation}\\label{eq:detEstSWE5}\n\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big).\n\\end{equation}\nThen, \\eqref{eq:detEstSWE4} follows from \\eqref{eq:detEstSWE5} and \\eqref{eq:detEstSWE2} by estimating analogously to \\eqref{eq:detEstSWE3} and using the fact that\n\\begin{align*}\n\\nnrm{P^1(E(t_n)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\n&=\\nnrm{\\Lambda^{-\\frac\\alpha4}P^1(E(t_n)-E(t))B\\Lambda^{\\frac12-\\frac\\alpha4}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\\\\n&=\\nnrm{P^1(E(t_n)-E(t))B\\Lambda^{\\frac{1-\\alpha}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\\\\n&\\leq \\nnrm{P^1(E(t_n)-E(t))}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\dot H^0)}\\nnrm{B\\Lambda^{\\frac{1-\\alpha}2}}{\\ensuremath{\\mathscr L}(\\dot H^0, \\cH^\\alpha)},\n\\end{align*}\nwhere $\\nnrm{B\\Lambda^{\\frac{1-\\alpha}2}}{\\ensuremath{\\mathscr L}(\\dot H^0, \\cH^\\alpha)}=\\nnrm{B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1}, \\cH^\\alpha)}=1$.\n\nIn order to show \\eqref{eq:detEstSWE5}, we distinguish the cases $\\alpha>2$ and $0\\leq\\alpha\\leq2$.\nFor $\\alpha>2$ we have by \\eqref{eq:detEstSWE1}\n\\begin{equation}\\label{eq:proofSWE6}\n\\begin{aligned}\n\\sup_{n\\in\\{0,\\ldots,N\\}}&\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1},\\dot H^0)}\\\\\n&\\leq\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))}{\\ensuremath{\\mathscr L}( \\cH^{\\alpha},\\dot H^0)}\\nnrm{B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1}, \\cH^\\alpha)}\\\\\n&\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big)\n\\end{aligned}\n\\end{equation}\nAs the operator $P^1(\\tilde E(t)-E(t))B\\in\\ensuremath{\\mathscr L}(\\dot H^0)$ is symmetric in $\\dot H^0$ and since $\\dot H^{-\\alpha+1}$ can be identified with the dual space of $\\dot H^{\\alpha-1}$, we have\n\\begin{equation*}\n\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1},\\dot H^0)}=\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^0,\\dot H^{-\\alpha+1})}\n\\end{equation*}\nand therefore also\n\\begin{equation}\\label{eq:proofSWE7}\n\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^0,\\dot H^{-\\alpha+1})}\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big).\n\\end{equation}\nNext, we use the fact that $\\dot H^{(\\alpha\/2)-1}$ and $\\dot H^{-\\alpha\/2}$ can be represented as the real interpolation spaces $(\\dot H^0,\\dot H^{\\alpha-1})_{\\theta,2}$ and $(\\dot H^{-\\alpha+1},\\dot H^0)_{\\theta,2}$, respectively, where $\\theta=((\\alpha\/2)-1)\/(\\alpha-1)\\in(0,1)$, cf.~Remark~\\ref{rem:interpolation}. Thus, interpolation between \\eqref{eq:proofSWE6} and \\eqref{eq:proofSWE7} yields\n\\begin{equation*}\n\\begin{aligned}\n&\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\\\\\n&\\leq\\sup_{n\\in\\{0,\\ldots,N\\}}C(\\alpha)\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^0,\\dot H^{-\\alpha+1})}^{1-\\theta}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1},\\dot H^0)}^\\theta\\\\\n&\\leq C(T,\\alpha)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big),\n\\end{aligned}\n\\end{equation*}\nsee, e.g., Definition 1.2.2\/2 and Theorem 1.3.3(a) in \\cite{Tri78}.\n\nFor $0\\leq\\alpha\\leq2$, we note that\n\\begin{equation*}\n\\begin{aligned}\n\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^0,\\dot H^{-1})}\n&=\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{1},\\dot H^0)}\\\\\n&\\leq\\nnrm{P^1(\\tilde E(t_n)-E(t_n))}{\\ensuremath{\\mathscr L}( \\cH^2,\\dot H^0)}\\nnrm{B}{\\ensuremath{\\mathscr L}(\\dot H^1, \\cH^2)},\\\\\n\\end{aligned}\n\\end{equation*}\nwhere we used again the symmetry of $P^1(\\tilde E(t)-E(t))B\\in\\ensuremath{\\mathscr L}(\\dot H^0)$.\nBy \\eqref{eq:detEstSWE1} we obtain\n\\begin{equation}\\label{eq:proofSWE8}\n\\sup_{n\\in\\{0,\\ldots,N\\}}\n\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^0,\\dot H^{-1})}\n\\leq C(T)\\big(h^{\\min(2\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(2\\frac{p}{p+1},p)}\\big),\n\\end{equation}\nwhich is \\eqref{eq:detEstSWE5} for $\\alpha=0$.\nMoreover, also by \\eqref{eq:detEstSWE1},\n\\begin{equation}\\label{eq:proofSWE9}\n\\begin{aligned}\n\\sup_{n\\in\\{0,\\ldots,N\\}}&\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{-1},\\dot H^{0})}\\\\\n&\\leq \\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))}{\\ensuremath{\\mathscr L}( H,\\dot H^{0})}\\nnrm{B}{\\ensuremath{\\mathscr L}(\\dot H^{-1}, H)}\n\\;\\leq\\; C(T),\n\\end{aligned}\n\\end{equation}\ni.e., we have \\eqref{eq:detEstSWE5} for $\\alpha=2$. Finally, if $\\alpha\\in(0,2)$, interpolation with $\\theta=(\\alpha\/2)-1\\in(0,1)$ between \\eqref{eq:proofSWE8} and \\eqref{eq:proofSWE9} gives\n\\begin{align*}\n    &\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\\\\\n    &\\quad\\leq\\sup_{n\\in\\{0,\\ldots,N\\}}C(\\alpha)\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{0},\\dot H^{-1})}^{1-\\theta}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{-1},\\dot H^{0})}^{\\theta}\\\\\n    &\\quad\\leq C(T,\\alpha)\\big(h^{\\min(2\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(2\\frac{p}{p+1},p)}\\big)^{\\frac\\alpha2}\\\\\n    &\\quad= C(T,\\alpha)\\big(h^{2\\frac{r}{r+1}}+(\\ensuremath{\\Delta t})^{2\\frac{p}{p+1}}\\big)^{\\frac\\alpha2}\\\\\n    &\\quad\\leq C(T,\\alpha)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big).\\qedhere\n\\end{align*}\n\\end{proof}\n\n\n\n\n\\begin{appendix}\n\\section{Poisson random measures and a comparison of\\\\ stochastic integrals}\n\\label{sec:PRM+SI}\n\nOur proof of Theorem~\\ref{thm:errorRep} is based on It\\^{o}'s formula for Banach space-valued jump processes driven by Poisson random measures as presented in \\cite{MRT13}. Alternatively, one could use It\\^{o}'s formula\nas proved in \\cite{GraPel74}, but the formula in \\cite{MRT13} is more convenient in our setting.\nIn this section, we use Lemma~\\ref{lem:integrandIsomorphism} to relate our setting to the setting in \\cite{MRT13}.\n\nIt is well-known that the jumps of a L\\'{e}vy process determine a Poisson random measure on the product space of the underlying time interval and the state space. We refer to \\cite[Section~6]{PesZab07} for a definition and properties of Poisson random measures. For $(\\omega,t)\\in\\Omega\\times(0,\\infty)$ we denote by $\\Delta L(t)(\\omega):=L(t)(\\omega)-\\lim_{s\\nearrow t}L(s)(\\omega)\\in U_1$ the jump of a trajectory of $L$ at time $t$. Setting\n\\begin{equation*}\nN(\\omega):=\\sum_{\\Delta L(t)(\\omega)\\neq0}\\delta_{(t,\\Delta L(t)(\\omega))},\\quad \\omega\\in\\Omega,\n\\end{equation*}\ndefines a Poisson random measure $N$ on $([0,\\infty)\\times U_1,\\cB([0,\\infty))\\otimes\\cB(U_1))$ with intensity measure (or compensator) $\\lambda\\otimes\\nu$, where $\\lambda$ is Lebesgue measure on $[0,\\infty)$ and $\\nu$ is the jump intensity measure of $L$.\nThis follows, e.g., from Theorem~6.5 in \\cite{PesZab07} together with Theorems~4.9, 4.15, 4.23 and Lemma~4.25 therein.\nWe denote the compensated Poisson random measure by\n\\begin{equation}\nq:=N-\\lambda\\otimes \\nu.\n\\end{equation}\n\nLet $V$ be a (real and separable) Hilbert space. The stochastic integral with respect to $q$ of functions in $L^2(\\Omega_T\\times U_1,\\bP_T\\otimes \\nu;V)=L^2(\\Omega_T\\times U_1,\\cP_T\\otimes\\cB(U_1),\\bP_T\\otimes \\nu;V)$  is constructed as a linear isometry\n\\begin{equation*\nL^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;V)\\to \\cM^2_T(V),\\;\nf\\mapsto\\Big(\\int_0^t\\int_{U_1} f(s,x)\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\Big)_{t\\in[0,T]}.\n\\end{equation*}\nIn particular, the $V$-valued integral processes have c\\`{a}dl\\`{a}g modifications; we will always work with such a c\\`{a}dl\\`{a}g modification.\nUsing a standard stopping procedure, the stochastic integral can be extendend to functions $f\\in L^0(\\Omega_T\\times U_1,\\cP_T\\otimes\\cB(U_1),\\bP_T\\otimes \\nu;V)$ such that\n\\[\\bP\\Big(\\int_0^T\\int_{U_1}\\nnrm{f(s,x)}V^2\\,\\nu(\\ensuremath{\\mathrm{d}} x)\\,\\ensuremath{\\mathrm{d}} s<\\infty\\Big)=1.\\]\nWe refer to \\cite{MRT13}, \\cite{Pre10} and the references therein for details on stochastic integration w.r.t.\\ Poisson random measures, compare also \\cite[Section~8.7]{PesZab07}.\n\n\\begin{remark}\nStricty speaking, in \\cite{MRT13} the integrands $f$ do not have to be predictable but only $\\cF_t\\otimes\\cB(U_1)$-adapted and $\\cF\\otimes\\cB([0,T])\\otimes\\cB(U_1)$-measurable. However, it is clear that in the case of predictable, i.e., $\\cP_T\\otimes\\cB(U_1)$-measurable, and square integrable Hilbert space-valued integrands $f$ the stochastic integral in \\cite{MRT13} coincides with the stochastic integral considered in \\cite{PesZab07}, \\cite{Pre10}. See \\cite{RueTap12} for a detailed comparison of the different spaces of integrands.\n\\end{remark}\n\nSince $\\bE\\int_0^T\\int_{U_1}\\nnrm{x}{U_1}^2\\,\\nu(\\ensuremath{\\mathrm{d}} x)\\ensuremath{\\mathrm{d}} t$ is finite for all $T<\\infty$, it follows that the integral process $(\\int_0^t\\int_{U_1} x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\geq 0}$ is uniquely determined (up to indistinguishability) as a $U_1$-valued square-integrable\nc\\`{a}dl\\`{a}g martingale. Taking into account the assumptions on the L\\'{e}vy process $L$, the L\\'{e}vy-Khinchin decomposition \\cite[Theorem~4.23]{PesZab07}, the definition of $q$, and the construction of the stochastic integral w.r.t.\\ $q$, it is not difficult to see that the processes $L$ and $(\\int_0^{t}\\int_{U_1} x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\geq0}$ are indistinguishable, i.e.,\n\\begin{equation}\\label{eq:L=qInt}\n\\bP\\Big(L(t)=\\int_0^t\\int_{U_1} x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\quad\\forall\\; t\\geq0\\Big)=1.\n\\end{equation}\n\nUsing Lemma~\\ref{lem:integrandIsomorphism}, we are now able to identify stochastic integrals w.r.t.\\ $L$ and stochastic integrals w.r.t.\\ the compensated Poisson random measure $q$. Recall from Remark~\\ref{not:Phix} that we identify processes $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$ with the corresponding functions $\\kappa(\\Phi)\\in L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H)$. Thus, for such $\\Phi$ the integral process $(\\int_0^t\\int_{U_1}\\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\in[0,T]}$ is defined.\n\n\\begin{lemma}\\label{lem:comparisonIntegrals}\nGiven $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$, the $H$-valued c\\`{a}dl\\`{a}g integral processes $(\\int_0^{t}\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s))_{t\\in[0,T]}$ and $(\\int_0^{t\n}\\int_{U_1} \\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\in[0,T]}$ are indistinguishable. That is,\n\\[\\bP\\Big(\\int_0^t\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s)=\\int_0^t\\int_{U_1}\\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\quad \\forall\\,t\\in [0,T]\\Big)=1.\\]\n\\end{lemma}\n\n\\begin{proof}\nWe first assume that $\\Phi$ is a simple $\\ensuremath{\\mathscr L}(U_1,H)$-valued process of the form\n\\begin{equation*\n\\Phi(s)=\\sum_{k=0}^{m-1}\\mathds 1_{F_k}\\mathds 1_{(t_k,t_{k+1}]}(s)\\Phi_k,\\quad s\\in[0,T],\n\\end{equation*}\nwith $0\\leq t_0<t_1<\\cdots<t_m\\leq T$, $m\\in\\bN$, $F_k\\in\\cF_{t_k}$ and $\\Phi_k\\in\\ensuremath{\\mathscr L}(U_1,H)$. Recall from Section~\\ref{sec:LSEE} that $\\ensuremath{\\mathscr L}(U_1,H)$ is a subspace of $\\ensuremath{\\mathscr L_2}(U_0,H)$. Using \\eqref{eq:L=qInt} and applying standard arguments for the evaluation of stochastic integrals, we obtain for fixed $t\\in[0,T]$, $\\bP$-almost surely,\n\\begin{align*}\n\\int_0^t\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s)&=\\sum_{k=0}^{m-1}\\mathds 1_{F_k}\\Phi_k(L(t_{k+1}\\wedge t)-L(t_k\\wedge t))\\\\\n&=\\sum_{k=0}^{m-1}\\mathds 1_{F_k}\\Phi_k\\Big(\\int_0^T\\int_{U_1} \\mathds 1_{(t_k\\wedge t, t_{k+1},\\wedge t]}(s) x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\Big)\\\\\n&=\\sum_{k=0}^{m-1}\\int_0^T\\int_{U_1} \\mathds 1_{F_k}\\mathds 1_{(t_k\\wedge t, t_{k+1},\\wedge t]}(s)\\Phi_k x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\\\\n&=\\int_0^t\\int_{U_1}\\Phi(s) x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x).\n\\end{align*}\nSince both processes are right-continuous, we see that the processes $(\\int_0^{t}\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s))_{t\\in[0,T]}$ and $(\\int_0^{t\n}\\int_{U_1} \\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\in[0,T]}$ are indistinguishable.\n\n\nFor general $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$, take a sequence $(\\Phi_n)_{n\\in\\bN}$ of simple $\\ensuremath{\\mathscr L}(U_1,H)$-valued process\nsuch that $\\Phi_n\\to\\Phi$ in $L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$; see, e.g., \\cite[Proposition~2.3.8]{PreRoeck07} for a proof of the existence of such a sequence.\nThen, the\nprocesses $$\\int_0^{\\arg}\\Phi_n(s)\\,\\ensuremath{\\mathrm{d}} L(s)=(\\int_0^{t}\\Phi_n(s)\\,\\ensuremath{\\mathrm{d}} L(s))_{t\\in[0,T]}$$ and $\\int_0^{\\arg}\\int_{U_1}\\Phi_n(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)=(\\int_0^{t\n}\\int_{U_1} \\Phi_n(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\in[0,T]}$ are indistinguishable for all $n\\in\\bN$, and\nwe have the convergence $\\int_0^{\\arg}\\Phi_n(s)\\,\\ensuremath{\\mathrm{d}} L(s)\\to\\int_0^{\\arg}\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s)$ in $\\cM^2_T(H)$ by the construction of the stochastic integral w.r.t.\\ $L$.\n According to Lemma~\\ref{lem:integrandIsomorphism}, the convergence $\\Phi_n\\to\\Phi$ in $L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$ entails the convergengence $\\kappa(\\Phi_n)\\to \\kappa(\\Phi)$ in $L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H)$, so that we also have $$\\int_0^{\\arg}\\int_{U_1}\\Phi_n(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\to\\int_0^{\\arg}\\int_{U_1}\\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)$$ in $\\cM^2_T(H)$. Thus, $\\int_0^{\\arg}\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s)=\\int_0^{\\arg}\\int_{U_1}\\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)$ as an equality in $\\cM^2_T(H)$, which yields the assertion.\n\\end{proof}\n\n\\end{appendix}\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper, we shall introduce a concept of {\\em fibred coarse embedding into Hilbert space}  for metric spaces (Definition 2.1) and prove the following result:\n\\par\n{\\bf Theorem 1.1.} {\\it Let $X$ be a discrete metric space with bounded geometry. If $X$ admits a fibred coarse embedding into Hilbert space, then the maximal coarse\nBaum-Connes conjecture holds for $X$.}\n\\par\nThe (maximal) coarse Baum-Connes conjecture \\cite{HR, Yu95, GWY} is a geometric analogue of the Baum-Connes conjecture \\cite{BC, BCH} and provides an algorithm  for computing the higher\nindices of generalized elliptic operators on non-compact spaces. It states\nthat a certain assembly map $\\mu$ or $\\mu_{\\max}$ from $\\lim_{d\\to \\infty} K_*(P_d(X))$ to $K_*(C^*(X))$ or $K_*(C^*_{\\max}(X))$ is an isomorphism, where $K_*(P_d(X))$ is the\nlocally finite $K$-homology group\nof the Rips complex $P_d(X)$,  and  $K_*(C^*(X))$ and $K_*(C^*_{\\max}(X))$ are respectively the $K$-theory groups of the Roe algebra and the maximal Roe algebra of $X$.\nThe conjecture has many applications in topology and geometry. In particular, it implies the Novikov conjecture on homotopy invariance of higher signatures,\nthe Gromov conjecture on non-existence of positive scalar curvature metrics on uniformly contractible Riemannian manifolds,\nand the zero-in-the-spectrum conjecture stating that the Laplacian operator acting on\nthe space of all $L^2$-forms of a uniformly contractible Riemannian manifold has zero in its spectrum.\n\\par\nRecall that a discrete metric space $X$ is said to have {\\em bounded geometry} if for any $r>0$ there is $N>0$ such that any ball of radius $r$ in $X$ contains at most $N$ elements.\nA map $f: X\\to Y$ from a metric space $X$ to another $Y$ is said to be a {\\em coarse embedding} or {\\em uniform embedding} \\cite{Grom93} if there exist non-decreasing functions\n$\\rho_1$ and $\\rho_2$ from $[0,\\infty)$ to $(-\\infty, \\infty)$ with $\\displaystyle\\lim_{r\\to \\infty}\\rho_i(r)=\\infty$ ($i=1,2$) such that\n$$\\rho_1\\big( d(x,y) \\big) \\leq d\\big( f(x),f(y) \\big)\\leq \\rho_2\\big(d(x,y)\\big)$$\nfor all $x, y\\in X$.  \\quad M. Gromov suggested that coarse embeddability of a discrete group into a Hilbert space, or a certain Banach space, might be relevant to prove the Novikov conjecture\n\\cite{Grom93}. G. Yu subsequently proved the coarse Baum-Connes conjecture for discrete metric spaces with bounded geometry which are coarsely embeddable into Hilbert space \\cite{Yu00}.\nM. Gromov discovered that  a metric space of expander graphs \\cite{Marg, Lub} cannot be coarsely embedded into a Hilbert space \\cite{Grom00}. N. Higson showed that the assembly map $\\mu$ fails to be\nsurjective for certain Margulis-type of expanders \\cite{Hig99}.\nIn \\cite{Laf}, V. Lafforgue showed that there are residually finite groups whose\nassociated expander graphs cannot be coarsely embedded into any uniformly convex Banach space. Meanwhile, G. Gong, Q. Wang and G. Yu introduced the maximal Roe algebra\nof a metric space with bounded geometry  in \\cite{GWY}, and proved  a version of the maximal coarse Novikov conjecture, i.e. the injectivity of the maximal assembly map $\\mu_{\\max}$,\nfor the box space of a class of residually finite groups, including the expander graphs\nconstructed by V. Lafforgue.\nThereafter, the coarse Novikov conjecture, i.e. the injectivity of the assembly map $\\mu$,\n for a large class of expanders is proved in \\cite{CTWY, GTY, OY}.\nMore recently, H. Oyono-Oyono and G. Yu proved the maximal coarse Baum-Connes conjecture for certain metric spaces constructed from spaces with isometric actions by\nresidually finite groups \\cite{OY}, including a certain class of expander graphs. R. Willett and G. Yu proved the maximal coarse Baum-Connes conjecture for spaces of graphs with large\ngirth \\cite{WiY2}, also including a certain class of expander graphs.\n\\par\nThe notion of fibred coarse embedding into Hilbert space, which we are going to introduce,  generalizes Gromov's notion of coarse embedding into Hilbert space to a great extent.\nRoughly speaking, a metric space $X$ admits such an embedding implies that, although the whole space $X$ may not be coarsely embedded into Hilbert space globally, large bounded subsets of  $X$\ncan be coarsely embedded into Hilbert space within a common distortion as long as these large bounded subsets are far away towards infinity.\nThis feature provides the property with much flexibility, allowing many expander graphs studied above to admit such an embedding.\nIn particular, our result Theorem 1.1 includes the major results of \\cite{OY} and \\cite{WiY2} as special cases.\n\\par\nThis paper is organized as follows. In section 2, we introduce the notion of fibred coarse embedding into Hilbert space, and discuss sevaral situations to which this notion applies.\nIn section 3, we recall the definition of maximal Roe algebra and the formulation of the maximal coarse Baum-Connes conjecture. In section 4, we explain the strategy to prove Theorem 1.1.\nThe problem of proving the maximal coarse Baum-Connes conjecture for a metric space $X$ can be reduced to verifying that the evaluation homomorphism from the $K$-theory of a certain localization\nalgebra to the $K$-theory of the maximal Roe algebra at infinity for the coarse disjoint union of a sequence of finite subspaces of $X$ is an isomorphism.\nIn section 5, we define the twisted Roe algebra at infinity and its localization algebra for a sequence of finite metric spaces\nwhich admits a fibred coarse embedding into Hilbert space. In section 6, we discuss various ideals of the twisted algebras and show that the evaluation map for the twisted algebras\nis an isomorphism. In section 7, we prove a geometric analogue of the Bott periodicity in finite dimension by constructing the Bott maps and the Dirac maps as asymptotic morphisms. This is used to show\nthe evaluation map required in section 4 is an isomorphism, which implies Theorem 1.1.\n\\par\nThroughout this paper, we denote by $\\cK(H)$ or $\\cB(H)$ the algebras of all compact operators or all bounded linear operators on a Hilbert space $H$. We denote by\n$\\cK$ the algebra of all compact operators on a fixed separable Hilbert space $H_0$. Denote by $\\mathbb{N}=\\{0, 1, 2, \\cdots\\}$ the non-negative integers. Denote by $\\#A$ the number of elements in a set $A$.\nFor a metric space $X$, a point $x\\in X$ and $r>0$, we denote the ball of  $X$ with center $x$ and radius $r$  by $B(x, r)$,  or $B_X(x, r)$ if it is necessary to indicate the base space $X$.\n\n\\par\n{\\bf Acknowledgement.} The authors are very grateful to Rufus Willett who carefully read an early version of this paper and suggested very helpful comments.\n\n\n\n\\section{Fibred coarse embedding into Hilbert space}\nIn this section, we introduce the concept of {\\em fibred coarse embedding into Hilbert space} for a metric space, generalizing Gromov's notion of\n{\\em coarse embedding into Hilbert space} \\cite{Grom93}. Let $H$ be a separable Hilbert space, as a model space.\n\\par\n{\\bf Definition 2.1.} A metric space $(X, d)$ is said to admit a {\\em fibred coarse embedding into Hilbert space} if there exist\n\\begin{itemize}\n\\item a field of Hilbert spaces $(H_x)_{x\\in X}$ over $X$;\n\\item a section $s: X\\to \\bigsqcup_{x\\in X} H_x$ (i.e. $s(x)\\in H_x$);\n\\item two non-decreasing functions $\\rho_1$ and $\\rho_2$ from $[0, \\infty)$ to $(-\\infty, \\infty)$ with $\\lim_{r\\to \\infty}\\rho_i(r)=\\infty$ ($i=1, 2$)\n\\end{itemize}\nsuch that for any $r>0$ there exists a bounded subset $K\\subset X$ for which there exists a ``trivialization''\n$$t_C: (H_x)_{x\\in C}\\longrightarrow C\\times H$$\nfor each subset $C\\subset X\\backslash K $ of diameter less than $r$, i.e. a map from $(H_x)_{x\\in C}$ to the constant field $C\\times H$ over $C$ such that the restriction  of $t_C$ to\nthe fiber $H_x$ ($x\\in C$) is an affine isometry $t_C(x): H_x\\to H$, satisfying\n\\begin{itemize}\n\\item[(1)] for any $x, y\\in C$, $\\rho_1(d(x, y))\\leq \\|t_C(x)(s(x))-t_C(y)(s(y))\\|\\leq \\rho_2(d(x, y))$;\n\\item[(2)] for any two subsets $C_1, C_2\\subset X\\backslash K$ of diameter less than $r$ with $C_1\\cap C_2\\not=\\emptyset$, there exists an affine isometry $t_{C_1C_2}: H\\to H$ such that\n\t\t$t_{C_1}(x)\\circ t_{C_2}^{-1}(x)=t_{C_1C_2}$ for all $x\\in C_1\\cap C_2$.\n\\end{itemize}\n\\hfill{$\\Box$}\n\\par\nIn the following we discuss several situations to which the above notion applies. Let $(X_n)_\\nn$ be a sequence of bounded metric spaces.\n\\par\nRecall that {\\em a coarse disjoint union of $(X_n)_\\nn$} is the disjoint union $X=\\bigsqcup_\\nn X_n$ equipped with a metric $d$ such that (1) the restriction of $d$ to each $X_n$ is the original metric\nof $X_n$; (2) $d(X_n, X_m)\\to \\infty$ as $n+m\\to \\infty$ and $n\\neq m$. Note that any two metrics satisfying these two conditions are coarsely equivalent. So we may refer to $X$ as {\\em the} coarse disjoint\nunion of  $(X_n)_\\nn$.\n\\par\nFor each $\\nn$, a metric space $\\wt{X_n}$ is called {\\em a Galois covering of $X_n$} if there exists a discrete group $\\Gamma_n$ acting on $\\wt{X_n}$ freely and properly by isometries\nsuch that $X_n=\\wt{X_n}\/\\Gamma_n$. Denote by $\\pi_n: \\wt{X_n}\\to X_n$ the associated covering map.\n\\par\nA sequence of Galois coverings $(\\wt{X_n})_\\nn$ of $(X_n)_\\nn$ is said to be {\\em asymptotically faithful} (cf. \\cite{WiY1}) if for any $r>0$ there exists $N\\in \\mathbb{N}$ such that, for all\n$n\\geq N$, the covering map $\\pi_n: \\wt{X_n}\\to X_n$ is ``$r$-isometric'', i.e. for all subsets $\\wt{C}\\subset \\wt{X_n}$ of diameter less than $r$, the restriction of $\\pi_n$ to $\\wt{C}$ is an isometry\nonto $C=\\pi_n(\\wt{C})\\subset X_n$.\n\\par\nA sequence of Galois coverings $(\\wt{X_n})_\\nn$ of $(X_n)_\\nn$ is said to admit a {\\em uniform equivariant coarse embedding into Hilbert space} if there exist a map $f_n: \\wt{X_n}\\to H$ for each\n$\\nn$, and two non-decreasing functions $\\rho_1$ and $\\rho_2$ from $[0, \\infty)$ to $(-\\infty, \\infty)$ with $\\lim_{r\\to \\infty}\\rho_i(r)=\\infty$ ($i=1, 2$), such that\n(1) $f_n$ is $\\Gamma_n$-equivariant (this implies $H$ has an action of $\\Gamma_n$ by affine isometries) for all $\\nn$; (2) $\\rho_1(d(x, y))\\leq \\|f_n(x)-f_n(y)\\|\\leq \\rho_2(d(x, y))$\nfor all $x, y\\in \\wt{X_n}$, $\\nn$.\n\\par\n{\\bf Theorem 2.2.} {\\it\nLet $X=\\bigsqcup_\\nn X_n$ be a coarse disjoint union of a sequence of bounded metric spaces. If there exists a sequence of asymptotically faithful Galois coverings\n$(\\wt{X_n})_\\nn$ which admits a uniform equivariant coarse embedding into Hilbert space, then $X$ admits a fibred coarse embedding into Hilbert space.\n}\n\\par\n{\\bf Proof.} For each $\\nn$,  the group $\\Gamma_n$ acts on $\\wt{X_n}\\times H$ by\n$$\\gamma \\cdot (x, h)= (\\gamma x, \\gamma h)$$\nfor $x\\in \\wt{X_n}, h\\in H, \\gamma\\in \\Gamma_n$. Denote the orbit of $(x, h)\\in \\wt{X_n}\\times H$ by $[(x, h)]$, i.e.\n$$[(x, h)]=\\{ \\, (\\gamma x, \\gamma h) \\; | \\; \\gamma\\in \\Gamma_n \\, \\}.$$\nFor any point $a\\in X_n=\\wt{X_n}\/\\Gamma_n$, the action\nof a group element $\\gamma\\in \\Gamma_n$ on $\\wt{X_n}$ permutes the points in $\\pi_n^{-1}(a)\\subset \\wt{X_n}$. Define\n$$H_a:=\\Big( \\pi_n^{-1}(a) \\times H \\Big)\\Big\/ \\Gamma_n.$$\nThen\n$$ (H_a)_{a\\in X_n}=\\Big( \\wt{X_n} \\times H \\Big)\\Big\/\\Gamma_n $$\nis a field of Hilbert spaces over $X_n$. Define a section\n$$s: X_n\\longrightarrow \\Big( \\wt{X_n} \\times H \\Big)\\Big\/\\Gamma_n$$\nby the formula\n$$s(a)=[(x, f_n(x))]\\in H_a$$\nfor any $x\\in \\pi_n^{-1}(a)$, $a\\in X_n$. This is well-defined since the map $f_n: \\wt{X_n}\\to H$ is $\\Gamma_n$-equivariant.\n\\par\nSince the covering sequence $(\\wt{X_n})_\\nn$ is asymptotically faithful, for any $r>0$ there exists $N\\in \\mathbb{N}$ such that\nfor any $n\\geq N$ and any $C\\subset X_n$ of diameter less than $r$,\nthe action of an element $\\gamma \\in \\Gamma_n$ on $\\wt{X_n}$ permutes the (disjoint) components of $\\pi_n^{-1}(C)$, and the restriction of $\\pi_n$ to each of these components is an\nisometry onto $C$. For a point $z\\in \\pi_n^{-1}(C)$, denote by ${\\wt{C}}^z$ the component of $\\pi_n^{-1}(C)$ containing $z$. Then each such\ncomponent ${\\wt{C}}^z$ gives rise to a trivialization:\n$$t_{C, z}: (H_a)_{a\\in C}=\\Big( \\pi_n^{-1}(C) \\times H \\Big)\\Big\/ \\Gamma_n  \\longrightarrow C\\times H, $$\nwhere, for any $a\\in C\\subset X_n$, the affine isometry\n$$t_{C, z}(a): \\; \\; H_a=\\Big( \\pi_n^{-1}(a) \\times H \\Big)\\Big\/ \\Gamma_n     \\longrightarrow H$$\nis given by\n$$\\Big( t_{C, z}(a) \\Big) \\Big( [(x, h)] \\Big) =h$$\nfor $[(x, h)]\\in H_a$ represented by $(x, h)\\in \\wt{C}^z\\times H$.\n\\par\nNow, for any $a, b\\in C$, there exist $x, y\\in \\wt{C}^z$ such that $\\pi_n(x)=a$ and $\\pi_n(y)=b$, so that\n\\[\n\\begin{array}{c}\n  \\Big\\|\\Big(t_{C, z}(a)\\Big) \\big( s(a) \\big) - \\Big( t_{C, z}(b) \\Big) \\big( s(b) \\big) \\Big\\| = \\Big\\| f_n(x)-f_n(y) \\Big\\|              \\\\\n\\in  \\big[ \\rho_1(d(x, y)), \\; \\rho_2(d(x, y)) \\big] =  \\big[ \\rho_1(d(a, b)), \\; \\rho_2(d(a, b)) \\big].\n\\end{array}\n\\]\n\\par\nMoreover, for any $C_1, C_2\\subset X_n$ ($n\\geq N$) of diameter less than $r$ with $C_1\\cap C_2\\not=\\emptyset$, there exist $z, \\gamma z\\in \\wt{X_n}$ for some $\\gamma\\in \\Gamma_n$ such that\n$\\pi_n(z)=\\pi_n(\\gamma z)\\in C_1\\cap C_2$. Then,  for all $a\\in C_1\\cap C_2$,\n$$t_{C_1, z}(a)\\circ t_{C_2, \\gamma z}^{-1}(a)=\\gamma: \\; H\\to H$$\nas an affine isometry mapping $h\\in H$ to $\\gamma h\\in H$. The proof is complete.\n\\hfill{$\\Box$}\n\\par\n{\\bf Example 2.3.} If a  metric space $X$ is coarsely embeddable into Hilbert space, then clearly it admits a fibred coarse embedding into Hilbert space. The maximal coarse Baum-Connes conjecture in this\ncase follows from G. Yu's  work on the coarse Baum-Connes conjecture \\cite{Yu00}, together with a recent result by J. \\v{S}pakula and R. Willett that $K_*(C^*_{\\max}(X))$ is isomorphic to $K_*(C^*(X))$ if $X$ is coarsely embeddable into Hilbert space \\cite{SpWi}.  Note also that for a discrete group, fibred coarse embeddability is the same as usual coarse embeddability.\n\\hfill{$\\Box$}\n\\par\n{\\bf Example 2.4.} Let $X$ be a discrete metric space with bounded geometry. Suppose a residually finite group $\\Gamma$ acts on $X$ freely, properly and cocompactly by isometries,\n such that $X$ is $\\Gamma$-equivariantly coarsely embeddable into Hilbert space. Let $(\\Gamma_n)_\\nn$ be a sequence of finite index normal subgroups of $\\Gamma$ such that for all $r>0$ there\nexists $N\\in \\mathbb{N}$ so that if $B(e, r)$ is the ball in $\\Gamma$ of radius $r$ about the identity, then $\\Gamma_n\\cap B(e, r)=\\{e\\}$ for all $n\\geq N$.\nLet $X_n=X\/\\Gamma_n$ be the quotient space. It follows from Theorem 2.2 that the coarse disjoint union $X=\\bigsqcup_\\nn X_n$ admits a fibred coarse embedding into Hilbert space. Indeed, the constant sequence\n$\\wt{X_n}=X$ ($\\nn$)\nserves as the asymptotically faithful Galois coverings.  Theorem 1.1 implies the maximal coarse Baum-Connes conjecture for such X.\nA special case of this result was proved by H. Oyono-Oyono and G. Yu in \\cite{OY}.\n\\par\nFor a residually finite group $\\Gamma$, the space $X(\\Gamma)=\\bigsqcup_{\\nn} \\Gamma\/\\Gamma_n$ is called {\\em the box space of  $\\Gamma$} (\\cite{Roe03}). It turns out that if the box space $X(\\Gamma)$ is coarsely embeddable into Hilbert space, then $\\Gamma$ is a-T-menable\n(\\cite{Roe03}). It is also easy to see that the converse of this implication is not true. However, it is shown in \\cite{CWW12} that $\\Gamma$ is a-T-menable if and only if the box space $X(\\Gamma)$ is fibred coarsely embeddable into Hilbert space.\n\\hfill{$\\Box$}\n\\par\n{\\bf Example 2.5.} Recall that the {\\em girth} of a graph $G$, denoted by $girth(G)$,  is the shortest length of a cycle in $G$. A sequence of finite connected graphs $(G_n)_\\nn$ is\nsaid to have {\\em large girth} if $girth(G_n)\\to \\infty$ as $n\\to \\infty$. Then the coarse disjoint union $X=\\bigsqcup_{\\nn} G_n$ admits a fibred coarse embedding into Hilbert space.\nIndeed, let $\\wt{G_n}$ be the {\\em universal cover of $G_n$}, which is actually a tree. Then the covering sequence $(\\wt{G_n})_\\nn$ satisfies the conditions in Theorem 2.2. The maximal coarse Baum-Connes\nconjecture for such $X$ was proved by R. Willett and G. Yu in \\cite{WiY2}.\n\\hfill{$\\Box$}\n\\par\nNote that Examples 2.4 and 2.5 imply that a large class of expander graphs admit a fibred coarse embedding into Hilbert space.\n\n\n\n\n\\section{The maximal coarse Baum-Connes conjecture}\n\n\nIn this section, we shall collect from \\cite{GWY} results concerning the maximal Roe algebra of a proper metric space with bounded geometry and the formulation of the maximal coarse Baum-Connes conjecture\n(see also \\cite{OY, WiY1, WiY2}).\n\\par\nLet $X$ be a proper metric space (a metric space is called $proper$ if every closed ball is compact). An $X$-module $H_X$ is a separable Hilbert space\nequipped with a  $*$-representation $\\pi$ of $C_0(X)$, the algebra of all continuous functions on $X$ which vanish\nat infinity. An $X$-module is called non-degenerate if the $*$-representation of $C_0(X)$ is non-degenerate.\nAn $X$-module is said to be standard if no nonzero function in $C_0(X)$ acts as a compact operator.\nWhen $H_X$ is an $X$-module,  for each $f\\in C_0(X)$ and $h\\in H_X$, we denote $(\\pi (f))h$ by $fh$.\n\n\\par\n{\\bf Definition 3.1.} (cf. \\cite{Roe93})  Let $X$ be a standard non-degenerate $X$-module.\n\\\\ \\indent (1) The {\\em support} $\\supp(T)$ of a bounded linear operator $T: H_X\\to H_X$ is defined to be the complement of the set of all points $(x, y)\\in X\\times X$ for which\nthere exist $f, g\\in C_0(X)$  such that $gTf=0$ but $f(x)\\not= 0$, $g(y)\\not=0$.\n\\\\ \\indent (2) A bounded operator $T: H_X\\to H_X$  is said to have {\\em finite propagation}  if\n$$\\sup\\{d(x, y): (x, y)\\in \\Supp(T)\\}<\\infty.$$\nThis number is called the {\\em propagation of $T$}.\n\\\\ \\indent (3) A bounded operator $T: H_X\\to H_X$ is said to be {\\em locally compact} if the operators $fT$ and $Tf$ are compact for all $f\\in C_0(X)$.\n\\par\nDenote by $\\mathbb{C}[X, H_X]$, or simply $\\mathbb{C}[X]$, the set of all locally compact, finite propagation operators on a standard non-degenerate $X$-module $H_X$.  It is straightforward to check that $\\mathbb{C}[X]$ is a $*$-algebra which,\nup to non-canonical isomorphisms, does not depend on the choice of standard non-degenerate $X$-module.\n\\par\n{\\bf Definition 3.2.} A {\\em net} of a metric space $X$ is a countable subset $\\Gamma\\subset X$ such that there exist numbers $\\delta>0$, $R>0$ satisfying (1) $d(\\gamma, \\gamma')>\\delta$\nfor all distinct elements $\\gamma, \\gamma'\\in \\Gamma$; (2) for any $x\\in X$ there exists $\\gamma\\in \\Gamma$ such that $d(x, \\gamma)<R$.  \\quad A metric space $X$ is said to have {\\em bounded geometry}\nif $X$ contains a net with bounded geometry.\n\\par\nThe following result is essentially proved in \\cite{GWY}.\n\\par\n{\\bf Lemma 3.3.} {\\it Let $X$ be a proper metric space with bounded geometry, and let $H_X$ be a standard non-degenerate $X$-module. For any $r>0$ there exists a constant $c>0$ such that\nfor any $*$-representation $\\phi$ of $\\cx$ on a Hilbert space $H_\\phi$ and any $T\\in \\cx$ with propagation less than $r$, we have\n$$||\\phi(T)||_{\\cB(H_\\phi)}\\leq c  \\, || T ||_{\\cB(H_X)} \\,. $$\n}\n\\par\nThis allows us to define the maximal Roe algebra of $X$.\n\\par\n{\\bf Definition 3.4.} \\quad (\\cite{GWY}) Let $X$ be a proper metric space with bounded geometry. The {\\em maximal Roe algebra} of $X$, denoted by $\\cmx$, is the completion of\n$\\cx$ with respect to the $C^*$-norm:\n$$\\big\\|T\\big\\|_{\\max} :=\\sup\\Big\\{ \\big\\|\\phi(T)\\big\\|_{\\cB(H_\\phi)} \\; \\Big | \\; \\phi:\\cx\\to \\cB(H_\\phi), \\mbox{ a $*$-representation} \\Big\\}. $$\n\\par\n{\\bf Definition 3.5.}\nLet $H_X=\\ell^2(Z, H_0)$, where $Z\\subset X$ is a countable dense subset of $X$ and $H_0$ is an infinite dimensional separable Hilbert space. A function $f\\in C_0(X)$ acts on\n$\\ell^2(Z)\\otimes H_0$ by pointwise multiplication on the first component: $f(\\xi\\otimes h)=f\\xi\\otimes h$. Define $\\mathbb{C}_f [X]$ to be the $*$-algebra of all bounded functions\n$T: Z\\times Z\\to \\cK:=\\cK(H_0)$, also viewed as $Z\\times Z$-matrices, such that\n\\begin{itemize}\n\\item for any bounded subset $B\\subset X$, the set\n\t\t$$\\{(x, y)\\in B\\times B\\cap Z\\times Z \\; | \\; T(x, y)\\not= 0 \\}$$\n\t\t is finite;\n\\item there exists $L>0$ such that\n\t\t$$\\#\\{y\\in Z \\; | \\; T(x, y)\\not= 0 \\}<L,  \\quad \\#\\{y\\in Z \\; | \\; T(y, x)\\not= 0 \\}<L $$\n\t\tfor all $x\\in Z$;\n\\item there exists $R>0$ such that $T(x, y)=0$ whenever $d(x, y)>R$ for $x, y\\in Z$.\n\\end{itemize}\nNote that in general $\\mathbb{C}_f [X]$ is a dense $*$-subalgebra of $\\mathbb{C}[X, \\ell(Z, H_0)]$ within $C^*_{\\max}(X)$. In the sequel, we will use $\\mathbb{C}_f [X]$ to replace $\\mathbb{C}[X]$ as a generating subalgebra of $C^*_{\\max}(X)$.\n\\par\n{\\bf Remark 3.6.} (cf. \\cite{HRY} and \\cite{GWY}) \\quad (1) $\\cmx$ does not depend on the choice of countable dense subset $Z\\subset X$ up to non-canonical isomorphisms.\nIf $X$ and $Y$ are coarsely equivalent, then $\\cmx$ is isomorphic to $\\cmy$ via a non-canonical isomorphism. (2) The $K$-theory groups of $\\cmx$ do not depend on the choice of $Z$ up to\ncanonical isomorphisms. If $X$ and $Y$  are coarsely equivalent, then $K_*(\\cmx)$ is isomorphic to $K_*(\\cmy)$ via a canonical isomorphism.\n\\par\nWe next define the assembly map $\\mu_{\\max}$ (also referred to as the index map or the Baum-Connes map) for the maximal Roe algebras.\n\\par\nLet $X$ be a proper metric space. Recall that the locally finite $K$-homology groups $K_i(X)=KK_i(C_0(X),\\mathbb{C})$ $(i=0,1)$ are generated by certain cycles\n(abstract elliptic operators) modulo  certain equivalence relations \\cite{Kas75, Kas88}:\n\\\\ \\indent (1) a cycle for $K_0(X)$ is a pair $(H_X, F)$, where $H_X$ is an $X$-module and $F$ is a bounded linear operator acting on $H_X$ such that $F^*F-I$ and $FF^*-I$ are locally compact,\nand $\\phi F-F\\phi$ is compact for all $\\phi\\in C_0(X)$;\n\\\\ \\indent (2) a cycle for $K_1(X)$ is a pair $(H_X, F)$, where $H_X$ is an $X$-module and $F$ is a {\\em self-adjoint} operator acting on $H_X$ such that $F^2-I$ is locally compact,\nand $\\phi F-F\\phi$ is compact for all $\\phi\\in C_0(X)$.\n\\par\nIn both cases, the equivalence relations on cycles are given by homotopy of the operators $F$, unitary equivalence, and direct sum with ``degenerate\" cycles, those cycles for which\n$F\\phi-\\phi F$, $\\phi(F^* F-I)$ and so on, are not merely compact but actually zero \\cite{Kas75, Kas88}.\n\\par\nThe assembly map $\\mu_{\\max}: K_*(X) \\to K_*(\\cmx)$ is defined as follows.\n\\par\n{\\bf Definition 3.7.} Let $(H_X, T)$ represent a cycle in $K_0(X)$. For any $R>0$, one can always take a  locally finite,\nuniformly bounded open cover $\\{U_i\\}_i$ of $X$ such that\n$$diameter(U_i)<R $$\nfor all $i$, and a continuous partition of unity  $\\{\\phi_i\\}_i$ subordinate to the open cover $\\{U_i\\}_i$. Define $F=\\sum_i \\phi_i^{\\frac{1}{2}}T\\phi_i^{\\frac{1}{2}}$, where the sum converges in\nthe strong operator topology. It is not hard to see that $(H_X, T)$ and $(H_X, F)$ are equivalent in $K_0(X)$. Note that now the propagation of $F$ is less than or equal to $R$, so that\n$F^*F-I$ and $FF^*-I$ are in $\\mathbb{C}[X]$. Let (cf. \\cite{Mil})\n\\[W=\n\\left(\n\\begin{array}{cc}\nI & F \\\\\n0 & I\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n-F^* & I\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{cc}\nI & F \\\\\n0 & I\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{cc}\n0 & -I \\\\\nI & 0\n\\end{array}\n\\right)\n\\in \\cB(H_X\\oplus H_X).\n\\]\nThen both $W$ and $W^{-1}$ have finite propagation (at most $3R$), and\n\\[\nW\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\nW^{-1}-\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\n\\in \\mathbb{C}[X]\\otimes \\cM_2(\\mathbb{C}).\n\\]\nWe then define\n\\[\n\\mu \\big( [(H_X, T)] \\big):=\n\\left[\nW\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\nW^{-1}\n\\right]\n-\n\\left[\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\n\\right]\n\\]\nin $K_0(\\mathbb{C}[X])$. Furthermore, $\\mu \\big( [(H_X, T)] \\big)$ defines an element in $K_0(C^*_{\\max}(X))$ by considering $\\mathbb{C}[X]$ as a  $*$-subalgebra of $C^*_{\\max}(X)$. This element is denoted by $\\mu_{\\max} \\big( [(H_X, T)] \\big) \\in K_0(C^*_{\\max}(X))$.\nThus, we obtain the assembly map $\\mu_{\\max}: K_0(X)\\rightarrow K_0(C^*_{\\max}(X))$. Similarly, we can define $\\mu_{\\max}: K_1(X)\\rightarrow K_1(C^*_{\\max}(X))$.\n\\par\n{\\bf Remark 3.8.}\nLet $X$ be a proper metric space with bounded geometry as above, and $Z$ a countable dense subset of $X$ used to define $\\mathbb{C}_f [X]$ as in Definition 3.5. For any natural number $n>0$, let $Z_n$ be a subset of $Z$ such that $d(x, y)>\\frac{1}{2n}$ for distinct $x, y\\in Z_n$, and $d(x, Z_n)\\leq \\frac{1}{n}$ for all $x\\in X$. Without loss of generality, we may assume $Z=\\cup_{n=1}^\\infty Z_n$. Let $\\mathbb{C}[Z_n]$ be the $*$-algebra of all bounded functions $T: Z_n\\times Z_n\\to \\mathcal{K}$ with finite propagation, i.e., there exists $R>0$ such that $T(x, y)=0$ whenever $d(x, y)>R$ for all $x, y\\in Z_n$. Then $\\mathbb{C}[Z_n]\\subset \\mathbb{C}_f [X]$. Moreover, there exists a non-canonical $*$-isomorphism (cf. \\cite{HRY} or 4.4 in \\cite{GWY}) $Ad(U): \\mathbb{C}[X]\\to \\mathbb{C}[Z_n]$ such that, if $T\\in \\mathbb{C}[X]$ has propagation less than $R$, then the propagation of $(Ad(U))(T)$ is less than $R+\\frac{2}{n}$.\n\\par\nFor any $R>0$, let $\\mu \\big( [(H_X, T)] \\big)\\in K_0(\\mathbb{C}[X])$ be as in Definition 3.7. Then\n$$Ad(U)_*\\big(\\mu \\big( [(H_X, T)] \\big) \\big)\\in K_0(\\mathbb{C}[Z_n]),$$\nwhich also defines an element in $K_0(\\mathbb{C}_f[X])$ via the inclusion $\\mathbb{C}[Z_n]\\hookrightarrow \\mathbb{C}_f [X]$.\nNote that the propagation of the element\n\\[\n\\big( Ad(U) \\big)\\left(\nW\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\nW^{-1}-\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\\right)\n\\in \\mathbb{C}_f[X]\\otimes \\cM_2(\\mathbb{C})\n\\]\nis less than $6R+\\frac{2}{n}$. Note also that $Ad(U)_*$ canonically induces the identity on $K_*(C^*_{\\max}(X))$.\n\\par\n{\\bf Definition 3.9.} Let $X$ be a discrete metric space with bounded geometry. For each $d\\geq 0$, the {\\em Rips complex} $P_d(X)$ at scale $d$ is defined to be the simplicial polyhedron\nin which the set of  vertices is $X$, and a finite subset $\\{x_0, x_1, \\cdots, x_q\\}\\subset X$ spans a simplex if and only if $d(x_i,x_j)\\leq d$ for all $0\\leq i, j\\leq q$.\n\\par\nEndow $P_d(X)$ with the {\\em spherical metric}. Recall that on each path connected component of $P_d(X)$, the spherical\nmetric is the maximal metric whose restriction to each simplex\n$\\{\\sum_{i=0}^q t_ix_i| t_i\\geq 0, \\sum_{i=0}^q t_i=1\\}$ is the metric obtained by identifying the simplex with\n$S_{+}^q$ via the map\n$$\\sum_{i=0}^q t_ix_i \\mapsto \\left( \\frac{t_0}{\\sqrt{\\sum_{i=0}^q t_i^2}}, \\frac{t_1}{\\sqrt{\\sum_{i=0}^q t_i^2}}, \\cdots,\n\\frac{t_q}{\\sqrt{\\sum_{i=0}^q t_i^2}} \\right)\n$$\nwhere $S_+^q:=\\{(s_0, s_1, \\cdots, s_q)\\in \\mathbb{R}^{q+1}, s_i\\geq 0, \\sum_{i=0}^q s_i =1\\}$ is endowed with the standard\nRiemannian metric. If $y_0, y_1$ belong to two different connected components $Y_0, Y_1$ of $P_d(X)$, we define\n$$d(y_0, y_1)= \\min \\{d(y_0, x_0)+d_X(x_0, x_1)+d(x_1, y_1)| x_0\\in X\\cap Y_0, x_1\\in X\\cap Y_1\\}.$$\nThe topology induced by the above metric is the same as the weak topology of the simplicial complex:\na subset $S\\subset P_d(X)$ is closed if and only if the intersection of $S$ with each simplex is closed.\n\\par\nNote that for any $d\\geq 0$, $\\pdx$ is coarsely equivalent to $X$ via the inclusion map. If $d<d'$, then $P_d(X)$ is included in $P_{d'}(X)$ as a subcomplex via a simplicial map.\nPassing to inductive limit, we obtain the assembly map\n$$ \\mu_{\\max}: \\lim_{d\\to \\infty} K_*(\\pdx) \\to  \\lim_{d\\to \\infty} K_*(\\cmpdx) \\cong K_*(\\cmx) .$$\n\\par\n{\\bf The maximal coarse Baum-Connes conjecture:}  {\\it\nIf $X$ is a discrete metric space with bounded geometry, then the assembly map $\\mu_{\\max}$ is an isomorphism.\n}\n\\par\n{\\bf Remark 3.10.} The maximal coarse Baum-Connes conjecture provides a method to compute the higher indices of elliptic differential operators\n(Dirac operators) on non-compact complete Riemannian manifolds and has many applications in topology and geometry. In particular, it implies the Novikov conjecture, the Gromov positive scalar curvature conjecture,\netc. (cf. e.g. \\cite{HR, Roe93, Roe96, Yu95, Yu98, Yu00, Yu06, GWY}).\n\n\n\n\n\n\n\n\n\n\\section{Reduction to coarse disjoint union of finite metric spaces}\n\nThe strategy to prove Theorem 1.1 is as follows. Let $X$ be a discrete metric space with bounded geometry which admits a fibred coarse embedding into Hilbert space.\nWe partition $X$ as $X=X^{(0)}\\cup X^{(1)}$ such that each of $X^{(0)}$, $X^{(1)}$ and $X^{(0)}\\cap X^{(1)}$ is a coarse disjoint union of a sequence of finite subspaces\nof $X$, and the excision pair $(X^{(0)}, \\; X^{(1)})$ respects the Mayer-Vietoris exact sequences on both the $K$-homology for these subspaces and the $K$-theory for the\ncorresponding maximal Roe algebras (the method of cutting the whole space into coarse disjoint unions of finite subspaces has its origin in \\cite{WiY-book}). Consequently, it suffices to prove Theorem 1.1 for $X=\\bigsqcup_{n\\in \\mathbb{N}} X_n$ being a coarse disjoint union of a sequence of finite\nmetric spaces. This case in turn can be reduced to verifying the evaluation homomorphism from the $K$-theory groups of certain localization algebras to that of the corresponding\n{\\em maximal Roe algebra at infinity for the sequence $(X_n)_\\nn$} to be an isomorphism, a fact which will be proved in sections 5, 6 and 7. \n\\par\nIn this section,  we shall explain in detail the above process of reduction. We start with the story for sequences of finite metric spaces. Let $(X_n)_\\nn$ be a sequence of finite (discrete) metric spaces with {\\em uniform bounded geometry},\ni.e. for any $r>0$ there exists $N>0$ such that $\\# B(x, r)<N$ for all $x\\in X_n, \\nn$. For each $d\\geq 0$, let $P_d(X_n)$ be the Rips complex of $X_n$ at scale $d$ endowed with\nthe spherical metric. Take a countable dense subset $\\zdn \\subset \\pdxn$ for each $d\\geq 0$ in such a way that $\\zdn\\subseteq Z_{d', n}$ whenever $d<d'$, for all $\\nn$.\n\\par\n{\\bf Definition 4.1.} For each $d\\geq 0$, define $\\cupdxn$ to be the set of all equivalence classes $\\ttt$ of sequences $(T^{(0)}, \\cdots, T^{(n)}, \\cdots)$ described as follows:\n\\begin{itemize}\n\\item[(1)] $T^{(n)}$ is a bounded function from $\\zdn \\times \\zdn$ to $\\cK$ for all $\\nn$;\n\\item[(2)] for any bounded subset $B\\subset \\pdxn$, the set\n\t\t\t$$\\{ (x, y)\\in B\\times B \\cap \\zdn\\times \\zdn \\; | \\; T^{(n)} (x, y)\\not= 0 \\}$$\n\t\t\tis finite;\n\\item[(3)] there exists $L>0$ such that\n\t\t\t$$\\#\\{y\\in \\zdn \\; | \\; T^{(n)}(x, y)\\not= 0 \\}<L,  \\quad  \\#\\{y\\in \\zdn \\; | \\; T^{(n)}(y, x)\\not= 0 \\}<L$$\n\t\t\tfor all $x\\in \\zdn, \\nn$;\n\\item[(4)] there exists $R>0$ such that $T^{(n)}(x, y)=0$ whenever $d(x, y)>R$ for $x, y\\in \\zdn, \\; \\nn$.\n\\end{itemize}\nThe equivalence relation $\\sim$ on these sequences is defined by\n$$(T^{(0)}, \\cdots, T^{(n)}, \\cdots) \\sim (S^{(0)}, \\cdots, S^{(n)}, \\cdots)$$\n if and only if\n$$\\lim_{n\\to \\infty} \\sup_{x, y\\in \\zdn} \\big\\| T^{(n)}(x, y) - S^{(n)}(x, y) \\big\\|_\\cK =0.$$\nViewing $T^{(n)}$ as $\\zdn\\times \\zdn$ matrices, $\\cupdxn$ is then made into a $*$-algebra by using the usual matrix operations.\n\\par\nDefine $\\cumpdxn$ to be the completion of $\\cupdxn$ with respect to the norm\n\\begin{footnotesize}\n$$\\big\\|T\\big\\|_{\\max} :=\\sup\\Big\\{ \\big\\|\\phi(T)\\big\\|_{\\cB(H_\\phi)} \\; \\Big | \\; \\phi:\\cupdxn \\to \\cB(H_\\phi), \\mbox{ a $*$-representation} \\Big\\}. $$\n\\end{footnotesize}\n\\hfill{$\\Box$}\n\\par\nNote that $\\|T\\|_{\\max}$ is well-defined since $(X_n)_\\nn$ have uniform bounded geometry. Moreover, $(\\pdxn)_\\nn$ is uniformly coarsely equivalent to $(X_n)_\\nn$ for\nany $d>0$, so that\n$\\cumpdxn$ is isomorphic to $C^*_{u, \\max, \\infty}((X_n)_\\nn)$ via a non-canonical isomorphism. Recall from Definition 3.7 and Remark 3.8 that the individual assembly maps\n$$\\mu_{\\max}: K_*(\\pdxn) \\to K_*(\\cmpdxn)$$\n($\\nn$) can be defined by $\\mu_{\\max} ([(H_X, T)])=\\mu_{\\max} ([(H_X, F)])$ such that the propagation of $F$ is less than any given small $R>0$ which is independent of $\\nn$, and that $\\|F\\|\\leq 1$. Consequently,\nwe can obtain the following assembly map at infinity:\n$$\\mu_{\\max, \\infty}: \\; \\; \\frac{\\prod_{n=0}^\\infty K_*(\\pdxn)}{\\oplus_{n=0}^\\infty K_*(\\pdxn)} \\longrightarrow  K_*\\Big(  \\cumpdxn  \\Big).$$\n\\par\nThe following notion of localization algebra has its origin in \\cite{Yu97}.\n\\par\n{\\bf Definition 4.2.}  For each $d\\geq 0$, define $\\culpdxn$  to be the $*$-algebra of all bounded and uniformly norm-continuous functions\n$$f: \\; [0,\\infty)\\longrightarrow \\cupdxn$$\nsuch that $f(t)$ is of the form $f(t)=[(f^{(0)}(t), \\cdots, f^{(n)}(t), \\cdots )]$ for all $t\\in [0, \\infty)$, where the family of functions\n$(f^{(n)}(t))_{\\nn, t\\geq 0}$ satisfy the conditions in Definition 4.1 with uniform constants, and there exists a bounded function $R:\\; [0, \\infty)\\to [0,\\infty)$ with\n$\\lim_{t\\to \\infty} R(t)=0$ such that\n$$ \\big( f^{(n)}(t)\\big) (x, y) =0  \\quad \\mbox{whenever} \\quad d(x, y)>R(t)$$\nfor all $x, y\\in \\zdn$, $\\nn$, $t\\in [0, \\infty)$.\n\\par\nDefine $\\culmpdxn$ to be the completion of $\\culpdxn$ with respect to the norm\n$$\\big\\| f \\big\\|_{\\max} :=\\sup_{t\\in [0, \\infty)} \\big\\|f(t)\\big\\|_{\\max}.$$\n\\hfill{$\\Box$}\n\\par\nWe can also define a local assembly map at infinity as in \\cite{Yu97}:\n$$\\mu_{L, \\max,\\infty}: \\; \\; \\frac{\\prod_{n=0}^\\infty K_*(\\pdxn)}{\\oplus_{n=0}^\\infty K_*(\\pdxn)} \\longrightarrow  K_*\\Big(  \\culmpdxn  \\Big).$$\n\\par\n{\\bf Proposition 4.3.}  {\\it\nSuppose $(X_n)_\\nn$ have uniform bounded geometry. Then the local assembly map $\\mu_{L, \\max,\\infty}$ is an isomorphism.\n}\n\\hfill{$\\Box$}\n\\par\nThe proof is similar to the arguments in \\cite{Yu97}. Note that the evaluation homomorphism\n$$e: \\; \\culmpdxn \\longrightarrow \\cumpdxn $$\ndefined by $e(f)=f(0)$ induces the following commutative diagram:\n\n\\[\n\\xymatrix{\n & \\;\\;\\;\\;\\;\\; \\displaystyle    \\lim_{d\\to\\infty}       K_*\\big(   \\culmpdxn  \\big) \\ar[d]^{e_*}            \\\\\n\\displaystyle\\lim_{d\\to\\infty}     \\frac{\\prod_{n=0}^\\infty K_*(\\pdxn)}{\\oplus_{n=0}^\\infty K_*(\\pdxn)}    \\;\\; \\;\\;\\;  \\ar[ur]^{\\mu_{L, \\max, \\infty}}_\\cong  \\ar[r]^{\\mu_{\\max, \\infty}\\quad\\quad\\quad}\n&\n\\;\\;\\;\\;\\;\\;\\;    \\displaystyle   \\lim_{d\\to\\infty}     K_*\\big(   \\cumpdxn  \\big).\n  }\n\\]\n\n\\par\nWe will devote the second half of the paper, section 5, 6 and 7, to prove the following result.\n\\par\n{\\bf Theorem 4.4.} {\\it\nLet $(X_n)_\\nn$ be a sequence of finite metric spaces with uniform bounded geometry. If the coarse disjoint union $X=\\bigsqcup_{n\\in \\mathbb{N}} X_n$ admits a fibred coarse embedding into\nHilbert space, then\n\\begin{small}\n$$e_*:\\; \\lim_{d\\to\\infty}       K_*\\big(   \\culmpdxn  \\big) \\longrightarrow   \\lim_{d\\to\\infty}     K_*\\big(   \\cumpdxn  \\big)$$\n\\end{small}\nis an isomorphism. Consequently, $\\mu_{\\max, \\infty}$ is an isomorphism.\n}\n\\hfill{$\\Box$}\n\\par\nWe next prove Theorem 1.1 for the case $X=\\bigsqcup_{n\\in \\mathbb{N}} X_n$ being a coarse disjoint union of a sequence of finite metric spaces, by using Theorem 4.4.\nTo do so, we need the following lemma (cf. \\cite{GWY, OY, WiY2}).\n\\par\n{\\bf Lemma 4.5.} {\\it\nLet $X=\\bigsqcup_{n\\in \\mathbb{N}} X_n$ be the coarse disjoint union of a sequence of finite metric spaces with uniform bounded geometry. For each $d\\geq 0$, there is a short exact sequence\n$$0\\longrightarrow \\cK\\longrightarrow \\cmpdx \\longrightarrow \\cumpdxn \\longrightarrow 0$$\nsuch that the inclusion $\\cK\\longrightarrow \\cmpdx$ induces an injection on $K$-theory.\n}\n\\par\n{\\bf Proof.} Let $Z_d\\subset P_d(X)$ be a countable dense subset, and let $\\zdn=Z_d\\cap \\pdxn$ for all $d\\geq 0, \\nn$ as being used in Definition 3.1 and Definition 4.1.\nNote that $\\cK\\cong \\cK(\\ell^2(Z_d)\\otimes H_0)$ is an ideal of $\\cmpdx$.  There is a $*$-homomorphism\n$$\\Phi:\\; \\cpdx \\longrightarrow \\cupdxn$$\ndefined by $\\Phi(T)=[(\\Phi^{(0)}(T), \\cdots, \\Phi^{(n)}(T), \\cdots)]$ for $T\\in \\cpdx,$\nwith\n\\[\n\\Phi^{(n)}(T) = \\left\\{\n\\begin{array}{ll}\n0,                      & \\mbox{ if } n<N_R; \\\\\nT|_{\\zdn\\times \\zdn},   & \\mbox{ if } n\\geq N_R,\n\\end{array}\n\\right.\n\\]\nwhere $R=propagation(T)$ and $N_R\\in \\mathbb{N}$ is large enough such that\n$$d\\Big( X_n, \\, \\bigsqcup_{i=0}^{n-1}X_i \\Big)>2R$$\nfor all $n\\geq N_R$. The $*$-homomorphism $\\Phi$ extends to $C^*_{\\max}$-level\n$$\\Phi:\\; \\cmpdx \\longrightarrow \\cumpdxn$$\nsuch that $\\cK$ lives in the kernel of $\\Phi$. The induced $*$-homomorphism on the quotient\n$$\\Phi:\\; \\cmpdx\/\\cK \\longrightarrow \\cumpdxn$$\nhas an inverse $\\Psi$ defined as follows: for any\n$$\\ttt\\in \\cupdxn,$$\nlet $S=\\bigoplus_\\nn T^{(n)}$. Then $S\\in \\cpdx$ and $\\Phi(S+\\cK)=T$. Define\n$$\\Psi(T)=S+\\cK\\in \\cmpdx\/\\cK.$$\nThen $\\Psi$ extends to a $*$-homomorphism on $C^*_{\\max}$-level:\n$$\\Psi:\\; \\cumpdxn \\longrightarrow \\cmpdx\/\\cK$$\nwhich is the inverse of $\\Phi$. This gives the short exact sequence. The $K$-theory statement follows from \\cite{GWY, OY}.\n\\hfill{$\\Box$}\n\n\\par\n{\\bf Proposition 4.6.} {\\it\nLet $X=\\bigsqcup_{n\\in \\mathbb{N}} X_n$  be the coarse disjoint union of a sequence of finite metric spaces with uniform bounded geometry. If\n$X$ admits a fibred coarse embedding into Hilbert space, then the maximal coarse Baum-Connes conjecture holds for $X$. That is,\n$$ \\mu_{\\max}: \\lim_{d\\to \\infty} K_*(\\pdx) \\to  \\lim_{d\\to \\infty} K_*(\\cmpdx) \\cong K_*(\\cmx) $$\nis an isomorphism.\n}\n\\par\n{\\bf Proof.} For any $d\\geq 0$, there exists $N_d\\in \\mathbb{N}$ large enough such that $d(X_n, X_m)>d$ provided $n, m\\geq N_d$.  Let $X_{N_d}=\\bigcup_{n=0}^{N_d-1} X_n$.\nThen we have\n$$K_*(\\pdx)=K_*(P_d(X_{N_d}))\\bigoplus \\prod_{n=N_d}^\\infty K_*(\\pdxn).$$\nBy the definition of assembly maps and Lemma 4.5, we have the following commutative diagram:\n\n\\[\n\\xymatrix{ 0 \\ar[d]& 0 \\ar[d] \\\\\nK_*(P_d(X_{N_d}))\\oplus\\bigoplus_{n=N_d}^\\infty K_*(P_d(X_n)) \\ar[d] \\ar[r]& K_*(\\mathcal{K}) \\ar[d] \\\\\nK_*(P_d(X)) \\ar[d] \\ar[r]^{\\mu_{\\max} \\quad \\quad} & K_*(C_{max}^*(\\pdx)) \\ar[d]  \\\\\n\\frac{\\prod_{n=0}^\\infty K_*(\\pdxn)}{\\oplus_{n=0}^\\infty K_*(\\pdxn)}\n\\ar[r]^{\\mu_{\\max, \\infty}\\quad\\quad} \\ar[d] & K_*\\Big( \\cumpdxn  \\Big) \\ar[d] \\\\ 0 & 0}\n\\]\n\\par\nPassing to inductive limit as $d\\to \\infty$, the top horizontal arrow is an isomorphism for the following reason. An element in the sum, as a finite sequence, is supported on summands below some\nfixed $m$ and, as $d\\to \\infty$, will eventually be absorbed into the first term on a single simplex. Thus, the assertion reduces to the fact that the assembly map is an isomorphism for a bounded metric\nspace. Now by Theorem 4.4 together with the five lemma, we complete the proof.\n\\hfill{$\\Box$}\n\\par\nFinally, we are able to prove Theorem 1.1 by using Proposition 4.6.\n\\par\n{\\bf Proof of Theorem 1.1.} Fix a point $x_0\\in X$. For $n=0, 1, 2, \\cdots$, let\n$$X_n= \\big\\{ x\\in X \\; \\big| \\; n^3-n\\leq d(x, x_0) \\leq (n+1)^3+(n+1) \\big\\}$$\nand denote\n$$X^{(0)}=\\bigcup_{n:\\; even} X_n; \\quad \\quad X^{(1)}=\\bigcup_{n: \\; odd} X_n.$$\nThen $X=X^{(0)}\\cup X^{(1)}$, and each of $X^{(0)}$, $X^{(1)}$ and  $X^{(0)}\\cap X^{(1)}$ is the coarse disjoint union of a sequence of finite subspaces of $X$, which admits a fibred coarse\nembedding into Hilbert space as restricted from $X$.\n\\par\nMoreover, the pair $(X^{(0)}, X^{(1)})$ is ``$\\omega$-excisive'' (cf. \\cite{HRY}) in the sense that, for any $R>0$ there exists $S>0$\nsuch that\n$$\\Pen(X^{(0)}; R)\\cap \\Pen(X^{(1)}; R) \\subseteq \\Pen(X^{(0)}\\cap X^{(1)}; S),$$\nwhere $\\Pen(Y; R)=\\{x\\in X|d(x, Y)\\leq R\\}$ is the $R$-neighborhood of a subspace. Indeed,\nfor any $R>0$, take an integer $n_R>R$ and let $S=(n_R+1)^3$. Suppose $x\\in \\Pen(X^{(0)}; R)\\cap \\Pen(X^{(1)}; R)$. If $d(x, x_0)\\leq S$, then\n$x\\in \\Pen(X^{(0)}\\cap X^{(1)}; S)$ since $x_0\\in X^{(0)}\\cap X^{(1)}$. If $d(x, x_0)> S$, then there exist\n$$\\bar{x}\\in X^{(0)}-B_X(x_0, n_R^3), \\quad\\quad \\bar{y}\\in X^{(1)}-B_X(x_0, n_R^3) $$\nsuch that $d(x, \\bar{x})\\leq R$ and $d(x, \\bar{y})\\leq R$. Hence, $d(\\bar{x}, \\bar{y})\\leq 2R$. We claim that either $\\bar{x}\\in X^{(0)}\\cap X^{(1)}$, or $\\bar{y}\\in X^{(0)}\\cap X^{(1)}$. Otherwise, we would simultaneously have that\n$$\\bar{x}\\in \\bigcup_{n:even,\\, n\\geq n_R} \\big\\{ x\\in X\\, \\big|\\, n^3+n<d(x, x_0)<(n+1)^3-(n+1) \\big\\}$$\nand\n$$\\bar{y}\\in \\bigcup_{n:odd,\\, n\\geq n_R} \\big\\{ x\\in X\\, \\big|\\, n^3+n<d(x, x_0)<(n+1)^3-(n+1) \\big\\}.$$\nIt follows that $d(\\bar{x}, \\bar{y})\\geq 2\\, n_R>2R$, a contradiction. Consequently, either $\\bar{x}$ or $\\bar{y}$ is in $X^{(0)}\\cap X^{(1)}$, so that\n$x\\in \\Pen(X^{(0)}\\cap X^{(1)}; R) \\subset \\Pen(X^{(0)}\\cap X^{(1)}; S)$. Therefore, the pair $(X^{(0)}, X^{(1)})$ is ``$\\omega$-excisive'' .\n\\par\nFor each $d\\geq 0$, there exists $l_d>0$ such that, for any $x\\in X\\backslash B_X(x_0, l_d)$, the ball $B_X(x, d)$ is contained in either $X^{(0)}$ or $X^{(1)}$. Denote $K_d:=B_X(x_0, l_d)$. Then for Rips complexes, we have\n$$P_d(X)=P_d(K_d\\cup X^{(0)}) \\bigcup P_d(K_d\\cup X^{(1)}).$$\nThis is again an ``$\\omega$-excisive'' decomposition,  into subspaces which are coarsely equivalent to $X^{(0)}$ and $X^{(1)}$ respectively.  As a result (cf. \\cite{HRY}), we have the following\ncommutative diagram, in which the vertical arrows are assembly maps $\\mu_{\\max, \\infty}$ connecting two Mayer-Vietoris exact sequences-----the top on $K$-homology, whereas the bottom on $K$-theory:\n\n\\[\n\\xymatrix\n{\n  & AH_0 \\ar[rr] \\ar'[d][dd]     &  &  BH_0\\ar[rr]\\ar'[d][dd]  &  &  K_0\\Big(  \\pdx \\Big)  \\ar[dd]^{\\mu_{\\max}} \\ar[dl]        \\\\\n    K_1\\Big(  \\pdx \\Big) \\ar[ur]\\ar[dd]_{\\mu_{\\max}}    &  &  BH_1 \\ar[ll]\\ar[dd]     &  &  AH_1 \\ar[ll]\\ar[dd]          &    \\\\\n  & A_0 \\ar'[r][rr]              &  &  B_0 \\ar'[r][rr]         &  &  K_0\\Big(  \\cmx  \\Big) \\ar[dl]                       \\\\\n    K_1\\Big(  \\cmx  \\Big) \\ar[ur]                  &  &  B_1\\ar[ll]              &  &  A_1 \\ar[ll]                  &        }\n\\]\nWhere,\n$$AH_0= K_0\\Big(   P_d(K_d\\cup X^{(0)}) \\bigcap P_d(K_d\\cup X^{(1)}) \\Big) , \\quad BH_0= K_0\\Big(   P_d(K_d\\cup X^{(0)})  \\Big) \\bigoplus   K_0\\Big(   P_d(K_d\\cup X^{(1)})  \\Big) , $$\n$$A_0=K_0\\Big(  C^*_{\\max} ( X^{(0)}\\cap X^{1)})   \\Big),  \\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad  B_0=K_0\\Big(  C^*_{\\max} ( X^{(0)}) \\Big) \\bigoplus K_0\\Big(  C^*_{\\max} ( X^{(1)})\\Big) ,$$\nand similarly for $K_1$. Passing to inductive limit as $d\\to \\infty$, clearly, Theorem 1.1 follows from (the proof of) Proposition 4.6 and the five lemma.\n\\hfill{$\\Box$}\n\n\n\n\n\n\\section{The twisted algebras at infinity}\nIn the rest of this paper, we shall prove that the evaluation homomorphism\n\\begin{small}\n$$e_*: \\lim_{d\\to \\infty} K_*\\Big( \\culmpdxn \\Big) \\longrightarrow \\lim_{d\\to \\infty} K_*\\Big( \\cumpdxn \\Big) $$\n\\end{small}\nis an isomorphism for a sequence of finite metric spaces $(X_n)_\\nn$ with uniform bounded geometry such that the coarse disjoint union $X=\\bigsqcup_\\nn X_n$ admits a fibred coarse embedding\ninto Hilbert space. It follows from the discussion of the last section that this suffices to complete the proof of Theorem 1.1.\n\\par\nIn this section, we shall introduce the twisted Roe algebras at infinity  \\\\\n$\\cumpdxna$ and their localization counterpart \\\\\n$\\culmpdxna$ for $(X_n)_\\nn$. Note that here the\ncoefficient algebras $\\cA(V_n)$ are defined on {\\em finite dimensional} affine subspaces $V_n$ of $H$. In section 6, we study various decompositions\nfor these twisted algebras to show that the evaluation map\n\\begin{small}\n$$e_*: \\lim_{d\\to \\infty} K_*\\Big( \\culmpdxna \\Big) \\longrightarrow \\lim_{d\\to \\infty} K_*\\Big( \\cumpdxna \\Big) $$\n\\end{small}\nfor the twisted algebras is an isomorphism. In section 7, we shall define the Bott maps $\\beta$, $\\beta_L$ and the Dirac maps $\\alpha$, $\\alpha_L$ to build the following\ncommutative diagram\n\\[\n\\xymatrix{\nK_*\\Big( \\culmpdxn \\Big) \\ar[r]^{\\quad e_*} \\ar[d]^{(\\beta_L)_*} & K_*\\Big( \\cumpdxn \\Big) \\ar[d]^{\\beta_*} \\\\\nK_*\\Big( \\culmpdxna \\Big)  \\ar@<1ex>[u]^{(\\alpha_L)_*} \\ar[r]^{\\quad e_*}  & K_*\\Big( \\cumpdxna \\Big )\\;. \\ar@<1ex>[u]^{\\alpha_*}\n}\n\\]\nWe prove a geometric analogue of the Bott periodicity in finite dimensions, i.e. $\\alpha_*\\circ \\beta_*=identity$ and $(\\alpha_L)_*\\circ (\\beta_L)_*=identity$. Passing to the inductive limit\nas $d\\to \\infty$, a diagram chasing argument implies that the top evaluation map is an isomorphism,  as desired.\n\\par\n\\subsection*{{\\it \\S 5.1.  Preliminary}}\n Let $H$ be a separable Hilbert space. Denote by $V_a$, $V_b$ etc. the finite dimensional affine subspaces of $H$. Let $V_a^0$ be the linear subspace of $H$ consisting of differences of elements\nof $V_a$. Let $\\Cliff(V_a^0)$ be the complexified Clifford algebra of $V_a^0$ and $\\cC(V_a)$ the graded $C^*$-algebra of continuous functions vanishing at infinity from $V_a$ into $\\Cliff(V_a^0)$.\nLet $\\cS=C_0(\\mathbb{R})$, graded according to even and odd functions. Define the graded tensor product:\n$$\\cA(V_a)=\\cS\\widehat{\\otimes} \\cC(V_a).$$\nIf $V_a\\subseteq V_b$, then we have a decomposition $V_b=V_{ba}^0\\oplus V_a$, where $V_{ba}^0$ is the orthogonal complement of $V_a^0$ in $V_b^0$. For each $v_b\\in V_b$, we have a corresponding\ndecomposition $v_b=v_{ba}+v_a$, where $v_{ba}\\in V_{ba}^0$ and $v_a\\in V_a$. Every function $h$ on $V_a$ can be extended to a function $\\widetilde{h}$ on $V_b$ by the formula $\\widetilde{h}(v_{ba}+v_a)=h(v_a)$.\n\\par\n{\\bf Definition 5.1.} For affine subspaces $V_a\\subseteq V_b$, denote by $C_{ba}: V_b\\to \\Cliff(V_{ba}^0)$ the function $v_b\\mapsto v_{ba}\\in \\Cliff(V_{ba}^0)$, where $v_{ba}$ is considered as an element\nof $\\Cliff(V_{ba}^0)$ via the inclusion $V_{ba}^0\\subseteq \\Cliff(V_b^0)$. Let $X$ be the unbounded multiplier of $\\cS$ given by the function $t\\mapsto t$. Define a $*$-homomorphism\n$\\beta_{V_b, V_a}:\\cA(V_a)\\to \\cA(V_b)$, or simply denoted by $\\beta_{ba}$, by the formula\n$$\\beta_{ba}(g\\widehat\\otimes h)=g(X\\widehat\\otimes 1+ 1\\widehat\\otimes C_{ba})\\, (1\\widehat\\otimes \\wt h) $$\nfor all $g\\in \\cS, h\\in \\cC(V_a)$, where $g(X\\widehat\\otimes 1+1\\widehat\\otimes C_{ba})$ is defined by functional calculus.\n\\par\n{\\bf Definition 5.2.} If  $V_a\\subseteq V_b$, for any subset $O\\subset \\mathbb{R}_+\\times V_a$, define\n$$\\overline{O}^{\\beta_{ba}}=\\Big\\{ \\; \\big(t, \\;\\; v_{ba}+v_a \\big) \\in \\mathbb{R}_+\\times V_b \\; \\Big| \\;\\; \\Big( \\sqrt{t^2+\\|v_{ba}\\|^2}\\; ,\\;  v_a \\Big)\\in O \\; \\Big\\}.$$\n\\par\nFor any finite dimensional affine subspace $V_a$ of $H$, the algebra $C_0(\\mathbb{R}_+\\times V_a)$ is included in $\\cA(V_a)$ as its center. For any function $a\\in \\cA(V_a)$, the support of $a$,\ndenoted by $\\supp(a)$, is the complement of all points $(t, v)\\in \\mathbb{R}_+\\times V_a$ such that there exists $g\\in C_0(\\mathbb{R}_+\\times V_a)$ such that $g(t, v)\\not=0$ but $g\\cdot a=0$. Note\nthat if $V_a\\subseteq V_b$ and $a\\in \\cA(V_a)$, then\n$$\\supp\\big( \\beta_{ba}(a) \\big) = \\overline{\\supp(a)}^{\\beta_{ba}}.$$\n\\par\n{\\bf Definition 5.3.} Let $W_a, W_b, V_a, V_b, V_c$ be finite dimensional affine subspaces of $H$ with  $W_a\\subseteq W_b\\subset V_c$ and $V_a\\subseteq V_b\\subset V_c$.\nLet $t$ be an affine isometry from $W_b$ onto $V_b$ mapping $W_a$ onto $V_a$. Then $t$ canonically\ninduces a $*$-isomorphism from $\\cA(W_a)$ onto $\\cA(V_a)$. Note that $\\cA(V_a)$ is included in $\\cA(V_c)$ via $\\beta_{ca}$. We denote by $t_*$ the composition\n$$t_*: \\;\\; \\cA(W_a)\\stackrel{\\cong}{\\longrightarrow} \\cA(V_a) \\stackrel{\\beta_{ca}}{\\longrightarrow} \\cA(V_c).$$\nThen we have the following commutative diagram which is useful in the definition of product structure for the twisted Roe algebras.\n\\[\n\\xymatrix{\nt_*: & \\cA(W_a) \\ar[r]^{\\cong} \\ar[d]^{\\beta_{ba}} & \\cA(V_a) \\ar[r]^{\\beta_{ca}} \\ar[d]^{\\beta_{ba}}  & \\cA(V_c) \\ar[d]^{=}   \\\\\n& \\cA(W_b) \\ar[r]^{\\cong}                     & \\cA(V_b) \\ar[r]^{\\beta_{ca}} & \\cA(V_c)\n}\n\\]\n\\hfill{$\\Box$}\n\\par\n\\subsection*{\\it \\S 5.2. The twisted Roe algebras at infinity}\nLet $(X_n)_\\nn$ be a sequence of finite metric spaces with uniform bounded geometry such that the coarse disjoint union $X=\\bigsqcup_\\nn X_n$ admits a fibred coarse embedding into Hilbert space.\nTo fix notations, we note that the notion of fibred coarse embedding into Hilbert space for the coarse disjoint union $X$ is equivalent to the following notion for the sequence\n$(X_n)_\\nn$:\n\\par\n{\\bf Definition 5.4.} A sequence of finite metric spaces $(X_n)\\nn$ with uniform bounded geometry is said to admit a fibred coarse embedding into Hilbert space if there exist\n\\begin{itemize}\n\\item a field of Hilbert spaces $(H_x)_{x\\in X_n, \\nn}$;\n\\item a section $s: X_n\\to\\displaystyle\\bigsqcup_{x\\in X_n} H_x$ for all $\\nn$;\n\\item two non-decreasing functions $\\rho_1$ and $\\rho_2$ from $[0, \\infty)$ to $(-\\infty, \\infty)$ with $\\lim_{r\\to \\infty}\\rho_i(r)=\\infty$ ($i=1, 2$);\n\\item a non-decreasing sequence of numbers $0\\leq \\l_0\\leq \\l_1\\leq \\cdots \\leq \\l_n\\leq \\cdots$ with $\\lim_{n\\to \\infty} \\l_n=\\infty$\n\\end{itemize}\nsuch that for each $x\\in X_n, \\nn$ there exists a trivialization\n$$t_x: (H_z)_{z\\in B(x, l_n)}\\longrightarrow B(x, l_n)\\times H$$\nsuch that the restriction  of $t_x$ to the fiber $H_z$ ($z\\in B(x, l_n)$) is an affine isometry $t_x(z): H_z\\to H$, satisfying\n\\begin{itemize}\n\\item[(1)] $\\rho_1(d(z, z'))\\leq \\|t_x(z)(s(z))-t_x(z')(s(z'))\\|\\leq \\rho_2(d(z, z'))$ for any $z, z'\\in B(x, l_n)$, $x\\in X_n, \\nn$;\n\\item[(2)] for any $x, y\\in X_n$ with $B(x, l_n)\\cap B(y, l_n)\\not=\\emptyset$, there exists an affine isometry $t_{xy}: H\\to H$ such that\n\t\t$t_{x}(z)\\circ t_{y}^{-1}(z)=t_{xy}$ for all $z\\in B(x, l_n)\\cap B(y, l_n)$.\n\\end{itemize}\n\\hfill{$\\Box$}\n\\par\nFor each $d\\geq 0$ and $\\nn$, let $\\pdxn$ be the Rips complex of $X_n$ at scale $d$ endowed with the spherical metric. For each $x\\in X_n$, denote by $\\Star(x)$ the open star of $x$ in\nthe second barycentric subdivision of $\\pdxn$. Take a countable dense subset $\\zdn\\subset \\pdxn$ for each $d\\geq 0$ in such a way that\n$$ (1) \\quad \\zdn\\subset \\displaystyle\\bigsqcup_{x\\in X_n} \\Star(x);  \\quad\\quad\\quad (2) \\quad \\zdn\\subseteq Z_{d', n}  \\mbox{ when }  d<d'. $$\n\\par\nFor any $x\\in \\zdn$, there exists a unique point $\\bar{x}\\in X_n$ such that $x\\in \\Star(\\bar{x})$. We define\n$$H_x=H_{\\bar{x}}, \\quad\\quad s(x)=s(\\bar{x})$$\nfor all $x\\in \\zdn\\cap \\Star(\\bar x)$ and let\n$$t_x(z)=t_{\\bar{x}}(\\bar{z}):\\; H_z\\to H$$\nfor all $x, z\\in \\zdn, \\; \\nn$ with $\\bar z\\in B(\\bar x, l_n)$.\n\\par\nFor each $\\nn$, define $V_n$ to be the {\\em finite dimensional} affine subspace of $H$ spanned by $t_x(z)(s(z))$ for all $z\\in B(x, l_n), \\; x\\in X_n$:\n$$V_n :=\\mbox{ affine-span } \\Big\\{ \\; t_x(z)(s(z))\\; \\Big| \\; z\\in B(x, l_n), \\; x\\in X_n \\Big\\} \\,.$$\nFor each $x\\in \\zdn$, $k\\geq 0$, define\n$$W_k(x)=\\mbox{ affine-span } \\Big\\{ \\; t_x(z)(s(z))\\; \\Big| \\; z\\in B_{\\pdxn} (x, k)  \\; \\Big\\} \\subseteq V_n \\,.$$\nNote that for each $k\\geq 0$, there exists $N\\in \\mathbb{N}$ such that  $W_k(x)$ is well-defined for $x\\in \\zdn$ with $n\\geq N$, and is an affine subspace of $V_n$. By Definition 5.1,\nthe inclusion $W_k(x)\\subseteq V_n$ induces the inclusion\n$$\\beta_{V_n, W_k(x)}:\\; \\cA(W_k(x))\\to \\cA(V_n).$$\n\\par\n{\\bf Definition 5.5.} For each $d\\geq 0$, define $\\cupdxna$ to be the set of all equivalence classes $\\ttt$ of sequences $\\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big)$ described as follows:\n\\begin{itemize}\n\\item[(1)] $T^{(n)}$ is a bounded function from $\\zdn \\times \\zdn$ to $\\cA(V_n)\\widehat\\otimes \\cK$ for all $\\nn$;\n\\item[(2)] for any bounded subset $B\\subset \\pdxn$, the set\n\t\t\t$$\\{ (x, y)\\in B\\times B \\cap \\zdn\\times \\zdn \\; | \\; T^{(n)} (x, y)\\not= 0 \\}$$\n\t\tis finite;\n\\item[(3)] there exists $L>0$ such that\n\t\t\t$$\\#\\{y\\in \\zdn \\; | \\; T^{(n)}(x, y)\\not= 0 \\}<L, \\quad \\#\\{y\\in \\zdn \\; | \\; T^{(n)}(y, x)\\not= 0 \\}<L $$\n\t\t\tfor all $x\\in \\zdn,\\; \\nn$;\n\\item[(4)] there exists $R>0$ such that $T^{(n)}(x, y)=0$ whenever $d(x, y)>R$  for $x, y\\in \\zdn,\\; \\nn$; (the least such $R$ is call the propagation of the sequence\n\t\t\t $\\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big)$ .)\n\\item[(5)] there exists $r>0$ such that\n\t\t\t\t\\[\n\t\t\t\t\t\\begin{array}{rcl}\n\t\t\t\t\t\t\\supp(T^{(n)}(x, y) )  & \\subseteq & B_{\\mathbb{R}_+\\times V_n} \\big( t_x(x)(s(x)), \\; r\\big)     \\\\\n\t\t\t\t\t\t\t\t\t\t  & :=   & \\big\\{ (\\tau, v)\\in \\mathbb{R}_+\\times V_n \\; \\big| \\; \\tau^2+\\|v-t_x(x)(s(x))\\|^2<r^2 \\big\\}\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\]\n\t\tfor all $x, y\\in \\zdn, \\; \\nn$.\n\\item[(6)] there exist $k\\geq 0$ and $K>0$ depending only on the sequence $\\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big)$ (not on $n$) such that, for each $x, y\\in \\zdn$, there exists\n\t\t\t$$T_1^{(n)}(x, y)\\in \\cA(W_k(x))\\widehat\\otimes \\cK\\cong \\cS\\widehat\\otimes \\cC(W_k(x))\\widehat\\otimes \\cK$$\n\t\t\tof the form $\\sum_{i=1}^K g_i\\widehat\\otimes h_i\\widehat\\otimes k_i$ where $g_i\\in \\cS$, $h_i\\in \\cC(W_k(x))$,\n\t\t\t$k_i\\in \\cK$ for $1 \\leq i \\leq K$  such that\n\t\t\t\t$$ T^{(n)}(x, y) = \\Big( \\beta_{V_n, W_k(x)} \\widehat\\otimes 1 \\Big) \\big( T^{(n)}_1 (x, y) \\big) \\,.$$\n\\item[(7)] there exists $c>0$ such that if $T_1^{(n)}(x, y)\\in \\cA(W_k(x))\\widehat\\otimes \\cK$  as above, and $w\\in \\mathbb{R}_+\\times W_k(x)$ is of norm one, then\n\t\tthe derivative of $T^{(n)}_1(x, y)$ in the direction $w$, $\\nabla_w\\big( T^{(n)}_1(x, y) \\big)$, exists in $\\cA(W_k(x))\\widehat\\otimes \\cK$ and is of norm at most $c$.\n\\end{itemize}\nThe equivalence relation $\\sim$ on these sequences is defined by\n$$(T^{(0)}, \\cdots, T^{(n)}, \\cdots) \\sim (S^{(0)}, \\cdots, S^{(n)}, \\cdots)$$\nif and only if\n$$\\lim_{n\\to \\infty} \\sup_{x, y\\in \\zdn} \\big\\| T^{(n)}(x, y) - S^{(n)}(x, y) \\big\\|_{\\cA(V_n)\\widehat\\otimes \\cK} =0.$$\nThe product structure for $\\cupdxna$ is defined as follows. For any two elements $\\ttt$ and $S=\\big[ (S^{(0)}, \\cdots, S^{(n)}, \\cdots) \\big]$ in $\\cupdxna$, their product is defined to be\n$$\\big[ \\big( (TS)^{(0)}, \\cdots, (TS)^{(n)}, \\cdots \\big) \\big] \\,,$$\nwhere there exists a sufficiently large $N\\in \\mathbb{N}$ depending on the propagation of the two representative sequences, such that\n$ (TS)^{(n)}=0$ for all $n=0, 1, 2, \\cdots, N-1$ and\n$$(TS)^{(n)}(x, y)=\\sum_{z\\in \\zdn} \\Big( T^{(n)}(x, z) \\Big) \\cdot \\Big( \\big(t_{xz}\\big)_* \\big( S^{(n)}(z, y) \\big) \\Big)$$\nfor all $x, y\\in \\zdn, \\; n\\geq N$.\n\\par\n{\\bf Remark 5.6.} Some explanations to the above formula are given here. Note that there exists $k\\geq 0$ such that\n\t\t\t\t$ S^{(n)}(z, y) = \\Big( \\beta_{V_n, W_k(z)} \\widehat\\otimes 1 \\Big) \\big( S^{(n)}_1 (z, y) \\big)$ for $z, y\\in \\zdn, \\;\\nn$, where\n\t\t\t\t$S_1^{(n)}(z, y)\\in \\cA(W_k(z))\\widehat\\otimes \\cK$.  For the propagation $R$ of the representative sequence $\\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big)$ and the above $k$, there\nexists $N\\in \\mathbb{N}$ large enough such that for any $n\\geq N$, there exists $\\widetilde{l_n}>0$ (since $\\pdxn$ is coarsely equivalent to $X_n$)\nsuch that the trivialization\n$$t_x: \\;\\; \\Big\\{ H_z \\Big\\}_{z\\in B_{\\pdxn}(x, \\widetilde{l_n})} \\longrightarrow  B_{\\pdxn}(x, \\widetilde{l_n})\\times H$$\nmakes sense on $B_{\\pdxn}(x, \\widetilde{l_n})$, and $R+k<\\widetilde{l_n}$. If $d(x, z)\\leq R$ in $\\pdxn$, then the affine isometry $t_{xz}=t_x(w)\\circ t_z^{-1}(w):\\; H\\to H$ for all\n$$w\\in B_{\\pdxn}(x, \\widetilde{l_n})\\cap B_{\\pdxn}(z, \\widetilde{l_n}).$$\nNote that $t_{xz}$ maps the affine subspace\n$$W_k(z)=\\mbox{ affine-span } \\Big\\{ t_z(w)(s(w)) \\; \\Big| \\; w\\in B_{\\pdxn}(z, k) \\cap \\zdn \\Big\\}$$\nonto an affine subspace of $W_{R+k}(x)$ since $B_{\\pdxn}(z, k)\\subseteq B_{\\pdxn}(x, R+k)$. By Definition 5.3, the composition\n$$t_{xz}: W_k(z)\\to t_{xz}\\big( W_k(z) \\big) \\subseteq W_{R+k}(x) \\subseteq V_n$$\ninduces the $*$-homomorphism\n$$\\big(t_{xz}\\big)_*:\\; \\cA( W_k(z) ) \\to \\cA(V_n).$$\nWe define\n$$\\big(t_{xz}\\big)_*\\Big( S^{(n)}(z, y) \\Big):= \\big(t_{xz}\\big)_*\\Big( S_1^{(n)}(z, y) \\Big)$$\nin the above product formula. Note that for $\\nn$ large enough, this definition of $\\big(t_{xz}\\big)_*\\Big( S^{(n)}(z, y) \\Big)$ does not depend on the choice of $k$\n(see Definition 5.3, and \\cite{HG, HKT}).\n\\hfill{$\\Box$}\n\\par\nThe $*$-structure for $\\cupdxna$ is defined by the formula\n$$ \\big[ \\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big) \\big]^*=   \\big[ \\big( (T^*)^{(0)}, \\cdots, (T^*)^{(n)}, \\cdots \\big) \\big] \\,,$$\nwhere\n$$(T^*)^{(n)}(x, y)=\\big( t_{xy} \\big)_* \\Big( \\big( T^{(n)}(y, x)\\big)^*\\Big)$$\nfor all but finitely many $n$, and $0$ otherwise.\n\\par\nNow, $\\cupdxna$ is made into a $*$-algebra by using the additional usual matrix operations. Define $\\cumpdxna$ to be the completion of $\\cupdxna$ with respect to the norm\n\\begin{footnotesize}\n$$\\big\\|T\\big\\|_{\\max} :=\\sup\\Big\\{ \\big\\|\\phi(T)\\big\\|_{\\cB(H_\\phi)} \\; \\Big | \\; \\phi:\\cupdxna \\to \\cB(H_\\phi), \\mbox{ a $*$-representation} \\Big\\}. $$\n\\end{footnotesize}\n\\hfill{$\\Box$}\n\\par\n{\\bf Definition 5.7.}  For each $d\\geq 0$, define $\\culpdxna$  to be the $*$-algebra of all bounded and uniformly norm-continuous functions\n$$f: \\; [0,\\infty)\\longrightarrow \\cupdxna$$\nsuch that $f(t)$ is of the form $f(t)=[(f^{(0)}(t), \\cdots, f^{(n)}(t), \\cdots )]$ for all $t\\in [0, \\infty)$, where the family of functions\n$(f^{(n)}(t))_{\\nn, t\\geq 0}$ satisfy the conditions in Definition 5.5 with uniform constants, and there exists a bounded function $R:\\; [0, \\infty)\\to [0,\\infty)$ with\n$\\lim_{t\\to \\infty} R(t)=0$ such that\n$$ \\big( f^{(n)}(t)\\big) (x, y) =0  \\quad \\mbox{whenever} \\quad d(x, y)>R(t)$$\nfor all $x, y\\in \\zdn$, $\\nn$, $t\\in [0, \\infty)$.\n\\par\nDefine $\\culmpdxna$ to be the completion of\n$$\\culpdxna$$\nwith respect to the norm\n$$\\big\\| f \\big\\|_{\\max} :=\\sup_{t\\in [0, \\infty)} \\big\\|f(t)\\big\\|_{\\max}.$$\n\\hfill{$\\Box$}\n\\par\nThe evaluation homomorphism\n$$e: \\; \\culmpdxna \\longrightarrow \\cumpdxna $$\ndefined by $e(f)=f(0)$ induces the evaluation homomorphism on $K$-theory:\n\\begin{small}\n$$ \\lim_{d\\to\\infty}       K_*\\big(   \\culmpdxna  \\big) \\stackrel{e_*}{\\longrightarrow}   \\lim_{d\\to\\infty}     K_*\\big(   \\cumpdxna  \\big).$$\n\\end{small}\n\n\n\n\\section{Decompositions of twisted algebras}\nIn this section we shall prove the following result.\n\\par\n{\\bf Theorem 6.1.} {\\it\nLet $(X_n)_\\nn$ be a sequence of finite metric spaces with uniform bounded geometry which admit a fibred coarse embedding into Hilbert space. Then\nthe evaluation homomorphism\n\\begin{small}\n$$ \\lim_{d\\to\\infty}       K_*\\big(   \\culmpdxna  \\big) \\stackrel{e_*}{\\longrightarrow}   \\lim_{d\\to\\infty}     K_*\\big(   \\cumpdxna  \\big).$$\n\\end{small}\nis an isomorphism.\n}\n\\par\nThe proof proceeds by decomposing the twisted algebras into various smaller subalgebras and applying a Mayer-Vietoris argument.\n\\subsection*{\\it \\S 6.1. Coherent system of open subsets of $\\mathbb{R}_+\\times V_n$ ($\\nn$)}\nTo begin with, we shall discuss ideals of the twisted algebras supported on certain open subsets.\n\\par\n{\\bf Definition 6.2.} A collection $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ of open subsets of $\\mathbb{R}_+\\times V_n, \\; \\nn,$ is said to be a {\\em coherent system} if for all but finitely many $\\nn$,\nthe following conditions hold:\n\\begin{itemize}\n\\item[(1)] for any subset $C\\subseteq B(x, l_n)\\cap B(y, l_n)$ with $x, y\\in X_n$, we have\n\t\t$$ O_{n, x}\\cap \\big(\\mathbb{R}_+\\times W_C(x) \\big) = t_{xy} \\Big( O_{n, y} \\cap \\big(\\mathbb{R}_+\\times W_C(y) \\big) \\Big),$$\n\t\twhere\n\t\t$$W_C(x)=\\mbox{ affine-span } \\Big\\{ t_x(z)(s(z)) \\; \\Big| \\; z\\in C \\Big\\}=t_{xy}\\big( W_C(y) \\big),$$\n\t\t$$W_C(y)=\\mbox{ affine-span } \\Big\\{ t_y(z)(s(z)) \\; \\Big| \\; z\\in C \\Big\\}=t_{yx}\\big( W_C(x) \\big),$$\n\t\tand $t_{xy}=t_x(z)\\circ t_y^{-1}(z):\\; H\\to H$ for all $z\\in B(x, l_n)\\cap B(y, l_n)$.\n\\item[(2)] for any $C\\subseteq B(x, l_n)$, where $x\\in X_n, \\;\\nn$, and any affine subspace $W$ with $W_C(x)\\subseteq W\\subseteq V_n$, we have\n\t\t$$ \\overline{O_{n, x}\\cap \\big(\\mathbb{R}_+\\times W_C(x) \\big)}^{\\beta_{W, W_C(x)}} \\subseteq  O_{n, x}\\cap \\big(\\mathbb{R}_+\\times W \\big),$$\n\t\twhere recall (Definition 5.3) that for $V_a\\subseteq V_b$ and $O\\subseteq \\mathbb{R}_+\\times V_a$,\n\t\t$$ \\overline{O}^{\\beta_{ba}} = \\big\\{  (t, v_{ba}+v_a)\\in \\mathbb{R}_+\\times V_b \\; \\big| \\; \\big( \\sqrt{t^2+\\|v_{ba}\\|^2} \\; , \\;v_a \\big)\\in O \\big\\}.$$\n\\end{itemize}\n \\hfill{$\\Box$}\n\\par\n{\\bf Definition 6.3.} Let $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ be a coherent system of open subsets of $\\mathbb{R}_+\\times V_n, \\; \\nn$. For each $d\\geq 0$, define\n$$\\cupdxna_O$$\nto be the $*$-subalgebra of $\\cupdxna$ generated by the equivalence classes of those sequences $\\big[ \\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big) \\big]$ such that\n$$\\supp (T^{(n)}(x, y)) \\subseteq O_{n, \\bar{x}}$$\nfor all $x, y\\in \\zdn$ with $x\\in \\Star(\\bar x)$ for all $n\\geq N$ for some $n\\in \\mathbb{N}$ large enough depending on the sequence  $\\big[ \\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big) \\big]$.\n\\par\nDefine $\\cumpdxna_O$ to be the norm closure of  \\\\\n$\\cupdxna_O$ in $\\cumpdxna$.\n \\hfill{$\\Box$}\n\\par\n{\\bf Lemma 6.4.} {\\it\n$\\cumpdxna_O$ is a two sided ideal of    \\\\\n$\\cumpdxna$.\n}\n\\par\n{\\bf Proof.} Suppose\n$$\\big[ \\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big) \\big]\\in \\cupdxna  \\,,$$\n$$\\big[ \\big( S^{(0)}, \\cdots, S^{(n)}, \\cdots \\big) \\big]\\in \\cupdxna_O  \\,,$$\nso that (1) there exists $R>0$ such that the propagation of $T^{(n)}$ and $S^{(n)}$ is at most $R$ for all $\\nn$; (2) $\\supp(S^{(n)}(z, y))\\subseteq O_{n,z}$ for all $z, y\\in \\zdn, \\,\\nn$;\n(3) there exists $k\\geq 0$ such that $S^{(n)}(z, y)=\\big( \\beta_{V_n, W_k(z)}\\widehat\\otimes 1\\big) \\big( S_1^{(n)}(z, y) \\big)$ for some $S_1^{(n)}(z, y)\\in \\cA(W_k(z))\\widehat\\otimes \\cK$\nfor all $z, y \\in \\zdn \\,\\nn$. It follows that\n$$\\supp \\big(S_1^{(n)}(z, y) \\big) \\subseteq O_{n, z}\\cap \\big( \\mathbb{R}_+\\times W_k(z) \\big), $$\nand\n\\[\n\\begin{array}{rcl}\n\\supp\\Big( \\big( t_{xz} \\big)_* \\big( S_1^{(n)}(z, y) \\big) \\Big) & \\subseteq & t_{xz} \\big(  O_{n, z}\\cap (\\mathbb{R}_+\\times W_k(z)) \\big)     \\\\\n  & = & O_{n, x}\\cap \\big( \\mathbb{R}_+\\times W_{B(z, k)}(x) \\big)  \\,,\n\\end{array}\n\\]\nwhere $B(z, k):=B_{\\pdxn}(z, k)$ and\n$$W_k(z)=\\mbox{ affine-span }\\Big\\{ t_z(w)(s(w)) \\; \\Big| \\; w\\in B_{\\pdxn}(z, k) \\cap \\zdn \\Big\\},$$\n$$W_{B(z, k)}(x) = t_{xz}(W_k(z)).$$\nIt follows from the definition of a coherent system that\n\\[\n\\begin{array}{rcl}\n\\supp\\Big( \\big( t_{xz} \\big)_* \\big( S^{(n)}(z, y) \\big) \\Big) & \\subseteq & \\overline{ \\Big(  O_{n, x}\\cap (\\mathbb{R}_+\\times W_{B(z, k)}(x)) \\Big) } ^{\\beta_{V_n, W_{B(z, k)}(x)}}    \\\\\n  & \\subseteq  & O_{n, x} \\,.\n\\end{array}\n\\]\nHence,\n$$\\supp\\Big( T^{(n)}(x, z) \\cdot \\big( t_{xz} \\big)_* \\big( S^{(n)}(z, y) \\big) \\Big) \\subseteq O_{n, x}, $$\nfor all but finitely many $\\nn$. That is,\n$$\\big[ \\big( (TS)^{(0)}, \\cdots, (TS)^{(n)}, \\cdots \\big) \\big]\\in \\cupdxna_O.$$\nThe proof is complete.\n\\hfill{$\\Box$}\n\\par\n{\\bf Definition 6.5.} Let $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ be a coherent system of open subsets of $\\mathbb{R}_+\\times V_n, \\; \\nn$. For each $d\\geq 0$, define\n$$\\culpdxna_O$$\nto be the $*$-subalgebra of $\\culpdxna$ consisting of all functions\n$$f:\\; [0, \\infty) \\longrightarrow \\cupdxna_O.$$\nDefine $\\culmpdxna_O$ to be the completion of    \\\\\n$\\culpdxna_O$ in $\\culmpdxna$.\n\\hfill{$\\Box$}\n\\par\nNote that $\\culmpdxna_O$ is an ideal of    \\\\\n$\\culmpdxna$. We also have an  evaluation homomorphism\n$$e: \\; \\culmpdxna_O\\rightarrow \\cumpdxna_O$$\ngiven by $e(f)=f(0)$.\n\\par\n\n\\subsection*{\\it \\S  6.2. Strong Lipschitz homotopy invariance}\nIn this subsection, we shall investigate ideals of the twisted algebras supported on those coherent systems which are separated by subsets of $X_n$, $\\nn$. These ideals can be decomposed into certain algebras\ndefined on uniformly bounded subsets of $P_d(X_n)$, $\\nn$, whose $K$-theory turns out to be strongly Lipschitz homotopy invariant. As a result, the evaluation homomorphism for these ideals is\nan isomorphism. This  constitutes a major step towards the proof of Theorem 6.1.\n\\par\nLet $\\Gamma_n$ be a subset of $X_n$ for each $\\nn$ and denote $\\Gamma=(\\Gamma_n)_\\nn$. Let $r>0$.\n\\par\n{\\bf Definition 6.6.} A coherent system $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ of open subsets of  $\\mathbb{R}_+\\times V_n, \\; \\nn,$ is said to be {\\em $(\\Gamma, r)$-separate} if\nthere exist open subsets $\\big( O_{n, x, \\gamma} \\big)_{\\gamma\\in \\Gamma_n\\cap B(x, l_n)}$ of $\\mathbb{R}_+\\times V_n$ for all $x\\in X_n,\\, \\nn$, such that\n\\begin{itemize}\n\\item $O_{n, x}=\\displaystyle\\bigcup_{\\gamma\\in \\Gamma_n\\cap B(x, l_n)} O_{n, x, \\gamma} $ ;\n\\item $ O_{n, x, \\gamma} \\cap O_{n, x, \\gamma'} =\\emptyset $ for distinct $\\gamma, \\gamma'\\in \\Gamma_n\\cap B(x, l_n)$ ;\n\\item $ O_{n, x, \\gamma} \\subseteq B_{\\mathbb{R}_+\\times V_n}\\Big( t_x(\\gamma)(s(\\gamma)) \\, ,  \\, r \\Big)$ for each $\\gamma\\in \\Gamma_n\\cap B(x, l_n)$.\n\\end{itemize}\n\\hfill{$\\Box$}\n\\par\n{\\bf Theorem 6.7.} {\\it\nIf a coherent system $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ is  $(\\Gamma, r)$-separate, then the evaluation homomorphism on $K$-theory\n\\[\n\\begin{array}{ll}\ne_*: &   \\displaystyle \\lim_{d\\to \\infty} K_*\\Big(  \\culmpdxna_{O} \\Big)             \\\\\n&   \\longrightarrow  \\displaystyle \\lim_{d\\to \\infty} K_*\\Big(  \\cumpdxna_{O} \\Big)\n\\end{array}\n\\]\nis an isomorphism.\n}\n\\par\nWe need some preparations before we can prove Theorem 6.8. Suppose $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ is  $(\\Gamma, r)$-separate for some  $\\Gamma=(\\Gamma_n)_\\nn$ and $r>0$ as above.\nFor $d\\geq 0$, let $(Y_\\gamma)_{\\gamma\\in \\Gamma_n}$ be a collection of closed subsets of $\\pdxn$ for each $\\nn$ such that\n\\begin{itemize}\n\\item $(Y_\\gamma)_{\\gamma\\in \\Gamma_n, \\,\\nn}$ is uniformly bounded, i.e. there exists $K>0$ such that $diameter(Y_\\gamma)\\leq K$ for all $\\gamma\\in \\Gamma_n, \\nn$.\n\\item $\\gamma\\in Y_\\gamma$ for all $\\gamma\\in \\Gamma_n, \\nn$.\n\\end{itemize}\nIn particular, we are mainly concerned with the following cases:\n\\begin{itemize}\n\\item[(1)] $Y_\\gamma=B(\\gamma, S):=\\big\\{ x\\in \\pdxn \\;\\big| \\; d(x, \\gamma)\\leq S \\big\\}$ for some common $S>0$ for all $\\gamma\\in \\Gamma_n, ,\\nn$.\n\\item[(2)] $Y_\\gamma=\\Delta_\\gamma(S)$, the simplex in $\\pdxn$ with vertices $\\{x\\in X_n | d(x, \\gamma)\\leq S \\}$ for some common $S>0$ for all $\\gamma\\in \\Gamma_n, \\,\\nn$.\n\\item[(3)] $Y_\\gamma=\\{\\gamma\\}$ for all $\\gamma\\in \\Gamma_n, \\,\\nn$.\n\\end{itemize}\n\\par\n{\\bf Definition 6.8.} For an open subset $O\\subseteq \\mathbb{R}_+\\times V_n$, we denote by $\\cA(O)$ the $C^*$-subalgebra of $\\cA(V_n)$ generated by the functions whose supports are\ncontained in $O$. Note that $\\cA(O)$ is an ideal of $\\cA(V_n)$.\n\\par\n{\\bf Definition 6.9.}  Define $A_\\infty[ \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn]$ to be the $*$-subalgebra of\n\\begin{equation}\\label{qqq}\n\\frac\n{\\prod_\\nn \\Big( \\bigoplus_{\\gamma\\in \\Gamma_n}  C^*_{\\max}(Y_\\gamma)\\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma}) \\Big) }\n{\\bigoplus_\\nn \\Big( \\bigoplus_{\\gamma\\in \\Gamma_n}  C^*_{\\max}(Y_\\gamma)\\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma}) \\Big) }\n\\end{equation}\nconsisting of elements of the form $\\big[ \\big( T^{(0)}, \\cdots, T^{(n)},  \\cdots \\big) \\big]$ where\n$$T^{(n)}=\\bigoplus_{\\gamma_\\in \\Gamma_n} T^{(n)}_\\gamma$$\nwith\n$$T^{(n)}_\\gamma\\in \\mathbb{C}[Y_\\gamma]\\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma}) \\subset C^*_{\\max}(Y_\\gamma)\\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma}),$$\nand when viewed as functions\n$$T^{(n)}_\\gamma:\\; (\\zdn\\times \\zdn)\\cap (Y_\\gamma\\times Y_\\gamma) \\rightarrow \\cA(O_{n, \\gamma, \\gamma}),$$\nthe family  $(T^{(n)}_\\gamma)_{\\gamma\\in \\Gamma_n, \\nn}$ satisfy the conditions in Definition 5.5 with uniform constants.\n\\par\nDefine $A^*_\\infty ( \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn )$ to be the completion of\n$$A_\\infty[ \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn]$$\ninside the $C^*$-algebra (1).\n\\hfill{$\\Box$}\n\\par\n{\\bf Definition 6.10.} Define $A_{L, \\infty}[ \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn]$ to be the $*$-algebra of all bounded and uniformly norm-continuous functions\n$$f: \\; [0,\\infty)\\longrightarrow A_\\infty[ \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn]$$\nwhere $f(t)$ is of the form $f(t)=[(f^{(0)}(t), \\cdots, f^{(n)}(t), \\cdots )]$ for all $t\\in [0, \\infty)$ and\n$$f^{(n)}(t)=\\bigoplus_{\\gamma_\\in \\Gamma_n} f^{(n)}_\\gamma (t)$$\nsuch that the family of functions\n$(f_\\gamma^{(n)}(t))_{\\gamma\\in \\Gamma_n, \\nn, t\\geq 0}$ satisfy the conditions in Definition 5.5 with uniform constants, and there exists a bounded function $R:\\; [0, \\infty)\\to [0,\\infty)$ with\n$\\lim_{t\\to \\infty} R(t)=0$ such that\n$$ \\big( f_\\gamma^{(n)}(t)\\big) (x, y) =0  \\quad \\mbox{whenever} \\quad d(x, y)>R(t)$$\nfor all $x, y\\in \\zdn\\cap Y_\\gamma$, $\\gamma\\in \\Gamma_n$, $\\nn$, $t\\in [0, \\infty)$.\n\\par\nDefine $A^*_{L, \\infty} \\big( \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn \\big)$ to be the completion of\n$$A_{L, \\infty}[ \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn]$$\nwith respect to the norm\n$$\\big\\| f \\big\\|_{\\max} :=\\sup_{t\\in [0, \\infty)} \\big\\|f(t)\\big\\|_{\\max}.$$\n\\hfill{$\\Box$}\n\\par\n{\\bf Proposition 6.11.} {\\it\nSuppose a coherent system $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ is  $(\\Gamma, r)$-separate for some  $\\Gamma=(\\Gamma_n)_\\nn$ and $r>0$ as above. Then\n\\begin{itemize}\n\\item  $\\displaystyle  \\cumpdxna_O \\cong \\lim_{S\\to \\infty} A^*_{\\infty}\\big( \\,\\big( B(\\gamma, S);  \\; \\gamma\\in \\Gamma_n  \\big)_\\nn \\big)  \\, ,$\n\\item  $\\displaystyle  \\culmpdxna_O \\cong \\lim_{S\\to \\infty} A^*_{L, \\infty}\\big( \\,\\big( B(\\gamma, S);  \\; \\gamma\\in \\Gamma_n  \\big)_\\nn \\big)  \\, ,$\n\\item  $\\displaystyle \\lim_{d\\to \\infty}\\cumpdxna_O \\cong \\lim_{S\\to \\infty} A^*_{\\infty}\\big( \\,\\big( \\Delta_\\gamma (S);  \\; \\gamma\\in \\Gamma_n  \\big)_\\nn \\big)  \\, ,$\n\\item  $\\displaystyle  \\lim_{d\\to \\infty}\\culmpdxna_O \\cong \\lim_{S\\to \\infty} A^*_{L, \\infty}\\big( \\,\\big( \\Delta_\\gamma (S);  \\; \\gamma\\in \\Gamma_n  \\big)_\\nn \\big)  \\,. $\n\\end{itemize}\n}\n\\par\n{\\bf Proof.} We shall establish an isomorphism for the first item. Take arbitrarily an element\n$$\\ttt\\in \\cupdxna_O,$$\nwhere the functions $T^{(n)}:\\; \\zdn\\times \\zdn\\rightarrow \\cA(V_n)\\widehat\\otimes \\cK$ have supports\n$$\\supp\\big( T^{(n)}(x, y) \\big) \\subseteq O_{n, x} \\subseteq \\bigsqcup_{\\gamma\\in \\Gamma_n\\cap B_{X_n}(x, l_n)} O_{n, x, \\gamma}$$\nfor all $x, y\\in \\zdn, \\nn, $ since the coherent system $O$ is $(\\Gamma, r)$-separate. Then we have a direct sum decomposition\n$$T^{(n)}(x, y)=\\bigoplus_{\\gamma\\in \\Gamma_n\\cap B_{X_n}(x, l_n)}  T_\\gamma^{(n)}(x, y) \\,,$$\nwhere\n$$T_\\gamma^{(n)}(x, y) = T^{(n)}(x, y)\\Big|_{O_{n, x, \\gamma}}  \\in \\cA(O_{n, x, \\gamma})\\widehat\\otimes \\cK $$\nis the restriction on a subset for all  $x, y\\in \\zdn, \\nn, $ and $\\gamma\\in B_{X_n}(x, l_n)\\cap \\Gamma_n$.  On the other hand, it follows from the support condition (5) in Definition 5.5, there\nexists $\\bar r>0$ such that\n$$\\supp\\big( T^{(n)}(x, y) \\big) \\subseteq B_{\\mathbb{R}_+\\times V_n} \\big(  t_x(x)(s(x)) \\,, \\, \\bar{r} \\big) \\,.$$\nHence, $T_\\gamma^{(n)}(x, y) =0$ whenever\n$$d\\Big( t_x(\\gamma)(s(\\gamma)) \\,, \\, t_x(x)(s(x)) \\Big) > \\bar{r}+r$$\nin $V_n$. It follows that there exists $S>0$ (since the section $s$ is locally a coarse embedding) such that $T_\\gamma^{(n)}(x, y) =0$ whenever $d(x, \\gamma)>S$ for all\n$x, y\\in \\zdn, \\gamma\\in \\Gamma_n,  \\nn $. Now define\n\\begin{equation}\\label{rrr}\nS_\\gamma^{(n)}(x, y)=\\big( t_{\\gamma  x} \\big)_*\\big( T_\\gamma^{(n)}(x, y) \\big) \\,.\n\\end{equation}\nSince $\\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots  \\big)$ has finite propagation, there exists $N>0$ large enough such that, if $n\\geq N$, then $S_\\gamma^{(n)}(x, y)$ is well-defined, and\n$$S_\\gamma^{(n)}(x, y)\\in \\cA(O_{n, \\gamma, \\gamma})\\widehat\\otimes \\cK$$\n  for all $x, y\\in \\zdn\\cap B_{\\pdxn}(\\gamma, S)$ with $\\gamma\\in \\Gamma_n\\cap B_{X_n}(x, l_n)$ and $n\\geq N$.\nTherefore, if we write $B(\\gamma, S)=B_{\\pdxn}(\\gamma, S)$ and view a function as a matrix, we have\n\\[\n\\begin{array}{rcl}\nS_\\gamma^{(n)}  &  =  &  \\Big[ S_\\gamma^{(n)}(x, y)  \\Big] _ {(x, y)\\in (\\zdn\\times \\zdn) \\cap (B(\\gamma, S)\\times B(\\gamma, S))}  \\\\\n  & \\in & \\mathbb{C}\\big[ B(\\gamma, S)  \\big]   \\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma})  \\\\\n  & \\subseteq   & C^*_{\\max}\\big( B(\\gamma, S) \\big)  \\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma}) \\,.\n\\end{array}\n\\]\nDefine $S^{(n)}=0$ for all $n\\leq N-1$ and define\n$$S^{(n)}=\\bigoplus_{\\gamma\\in \\Gamma_n} S_\\gamma^{(n)} \\in  \\bigoplus_{\\gamma\\in \\Gamma_n} C^*_{\\max}\\big( B(\\gamma, S) \\big)  \\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma})$$\nfor $n\\geq N$. Then the class $S=[( S^{(0)}, \\cdots, S^{(n)}, \\cdots )]$ is an element of\n$$A_{\\infty}\\big[ \\, \\big( B(\\gamma, S); \\, \\gamma\\in \\Gamma_n \\big)_\\nn \\big] \\, .$$\nNow the correspondence $T\\mapsto S$ extends to a $*$-isomorphism (see also the proof of Lemma 3.9 in \\cite{SpWi} for essentially the same arguments which can be used to show that the norms in these two $C^*$-algebras agree), as desired in the first item. The remainder isomorphisms in this Proposition follow from the first one.\n\\hfill{$\\Box$}\n\\par\nNow we recall the notion of strong Lipschitz homotopy \\cite{Yu97, Yu98, Yu00}.\n\\par\nLet $(Y_\\gamma)_{\\ggnn}$ and $(\\Delta_\\gamma)_{\\ggnn}$ be two families of uniformly bounded closed subsets of $\\pdxn$, $\\nn$,\nfor some $d\\geq 0$, such that $\\gamma\\in Y_\\gamma$, $\\gamma\\in \\Delta_\\gamma$ for all $\\gamma\\in \\Gamma_n, \\nn$.  A map\n$$g: \\; \\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} Y_\\gamma \\longrightarrow \\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} \\Delta_\\gamma $$\nis said to be {\\em Lipschitz} if\n\\\\ \\indent (1) $g(Y_\\gamma)\\subseteq \\Delta_\\gamma$ for each ${\\gamma\\in \\Gamma_n, \\nn}$;\n\\\\ \\indent (2) there exists a constant $c>0$, independent of $\\ggnn$, such that\n$$ d(g(x),g(y))\\leq c \\cdot d(x,y) $$\nfor all $x,y\\in Y_\\gamma$, $\\ggnn$.\n\\par\nLet $g_1,g_2$ be two Lipschitz maps from $\\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} Y_\\gamma$ to $\\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} \\Delta_\\gamma$.\nWe say $g_1$ is {\\it strongly Lipschitz homotopy} equivalent to $g_2$ if there exists a continuous map\n$$ F: \\; [0,1]\\times \\Big( \\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} Y_\\gamma \\Big)\\longrightarrow \\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} \\Delta_\\gamma $$\nsuch that\n\\\\ \\indent (1) $F(0,x)=g_1(x)$, $F(1,x)=g_2(x)$ for all $x\\in\\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} Y_\\gamma$;\n\\\\ \\indent (2) there exists a constant $c>0$ for which $ d(F(t,x),F(t,y))\\leq c \\cdot d(x,y)$ for all $x,y\\in Y_\\gamma$, $\\ggnn$, $t\\in [0,1]$;\n\\\\ \\indent (3) $F$ is equicontinuous in $t$, i.e., for any $\\varepsilon>0$ there exists $\\delta>0$ such that $d(F(t_1,x),F(t_2,x))<\\varepsilon $ for all\n$x\\in\\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} Y_\\gamma$ whenever $|t_1-t_2|<\\delta$.\n\\par\nWe say $(Y_\\gamma)_{\\ggnn}$ is {\\em strongly Lipschitz homotopy} equivalent to $(\\Delta_\\gamma)_{\\ggnn}$ if there exist Lipschitz maps\n$g_1:\\bigsqcup Y_\\gamma\\to \\bigsqcup \\Delta_\\gamma$ and $g_2:\\bigsqcup \\Delta_\\gamma\\to \\bigsqcup Y_\\gamma$ such that $g_1g_2$ and $g_2g_1$ are\nrespectively strongly Lipschitz homotopy equivalent to identity maps.\n\\par\nDefine $A^*_{L, 0, \\infty}\\big( (Y_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)$ to be the $C^*$-subalgebra of\n$$A^*_{L,  \\infty}\\big( (Y_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)$$\nconsisting of those functions $f$ such that $f(0)=0$.\n\\par\n{\\bf Lemma 6.12.} (cf. \\cite{Yu00})  {\\it\nIf $(Y_\\gamma)_{\\ggnn}$ is strongly Lipschitz homotopy equivalent to $(\\Delta_\\gamma)_{\\ggnn}$, then $K_*\\big(A^*_{L, 0, \\infty}\\big( (Y_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)\\big)$ is isomorphic to $K_*\\big(A^*_{L, 0, \\infty}\\big( (\\Delta_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)\\big)$.\n}\n\\hfill{$\\Box$}\n\\par\nLet $e$ be the evaluation homomorphism from $A^*_{L,  \\infty}\\big( (Y_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)$   to $A^*_{\\infty}\\big( (Y_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)$  given by\n$e(f)=f(0)$.\n\\par\n{\\bf Proposition 6.13.} (cf. \\cite{Yu00}) {\\it For any $d\\geq 0$, let $\\Delta_\\gamma$ be a simplex in $\\pdxn$ for all $\\ggnn$ with $\\gamma\\in \\Delta_\\gamma$. Then the evaluation map\n$$ e_*:\\, K_*\\big(A^*_{L, \\infty} \\big( (\\Delta_\\gamma; \\gamma\\in \\Gamma_n)_\\nn \\big)\\big) \\rightarrow  K_*\\big(A^*_{\\infty} \\big( (\\Delta_\\gamma; \\gamma\\in \\Gamma_n)_\\nn \\big)\\big) $$\nis an isomorphism.\n}\n\\par\n{\\bf Proof.} (cf.  \\cite{Yu00}) Note that $(\\Delta_\\gamma)_{\\ggnn}$ is strongly Lipschitz homotopy equivalent to $(\\{\\gamma\\})_{\\ggnn}$. By an argument of Eilenberg swindle,  we have\n$K_*\\big(A^*_{L,0, \\infty}\\big( (\\{\\gamma\\}; \\gamma\\in \\Gamma_n )_\\nn \\big)\\big)=0$. Consequently, Proposition 6.13 follows from Lemma 6.12 and the six-term exact sequence of $C^*$-algebra $K$-theory.\n\\hfill{$\\Box$}\n\\par\nWe are now ready to give a proof to Theorem 6.7.\n\\par\n{\\bf Proof of Theorem 6.7. }  By Proposition 6.11, we have the following commutative diagram\n\\[\n\\xymatrix{\n  \\displaystyle\\lim_{d\\to\\infty}    \\culmpdxna_O         \\ar[d]_{\\cong}          \\ar[r]^{\\quad e}\n            &    \\displaystyle\\lim_{d\\to\\infty}      \\cumpdxna_O          \\ar[d]^{\\cong}                   \\\\\n  \\displaystyle\\lim_{S\\to\\infty}     A^*_{L, \\infty} \\big( (\\Delta_\\gamma(S); \\gamma\\in \\Gamma_n)_\\nn \\big)        \\ar[r]^{\\quad e}\n            &    \\displaystyle\\lim_{s\\to\\infty}     A^*_{\\infty} \\big( (\\Delta_\\gamma(S); \\gamma\\in \\Gamma_n)_\\nn \\big)\n     }\n\\]\nwhich induces the following commutative diagram on $K$-theory\n\\[\n\\xymatrix{\n  \\displaystyle\\lim_{d\\to\\infty}    K_*\\Big(  \\culmpdxna_O  \\Big)       \\ar[d]_{\\cong}          \\ar[r]^{e_*}\n            &    \\displaystyle\\lim_{d\\to\\infty}    K_*\\Big(   \\cumpdxna_O   \\Big)       \\ar[d]^{\\cong}                   \\\\\n  \\displaystyle\\lim_{S\\to\\infty}  K_*\\Big(   A^*_{L, \\infty} \\big( (\\Delta_\\gamma(S); \\gamma\\in \\Gamma_n)_\\nn \\big)   \\Big)     \\ar[r]^{e_*}\n            &    \\displaystyle\\lim_{s\\to\\infty}    K_*\\Big(    A^*_{\\infty} \\big( (\\Delta_\\gamma(S); \\gamma\\in \\Gamma_n)_\\nn \\big)   \\Big)\n     }\n\\]\nNow Theorem 6.7 follows from Proposition 6.13.\n\\hfill{$\\Box$}\n\\subsection*{\\it \\S 6.3. Proof of Theorem 6.1}\nWe are now able to prove Theorem 6.1, the main result of this section, by gluing together the pieces we studied above.\n\\par\n{\\bf Proof of Theorem 6.1.}  For all $r>0$, define\n$$O^{(r)}_{n, x} \\;:=\\bigcup_{\\gamma\\in B_{X_n}(x, l_n)} B_{\\mathbb{R}_+\\times V_n}\\Big( t_x(\\gamma)(s(\\gamma)), \\, r   \\Big) $$\nfor all $x\\in X_n, \\nn$. Then\n$$O^{(r)}:=\\big( O^{(r)}_{n, x} \\big)_{x\\in X_n, \\nn}$$\nis a coherent system of open subsets. For any $d\\geq 0$, if $r<r'$ then $O^{(r)}_{n, x}\\subseteq O^{(r')}_{n, x}$ so that\n$$\\cumpdxna_{O^{(r)}} \\subseteq \\cumpdxna_{O^{(r')}}   \\;,$$\n$$\\culmpdxna_{O^{(r)}} \\subseteq \\culmpdxna_{O^{(r')}} \\;. $$\nBy the definition of these ideals, we have\n$$\\cumpdxna = \\lim_{r\\to \\infty} \\cumpdxna_{O^{(r)}}  \\;, $$\n$$\\culmpdxna= \\lim_{r\\to \\infty} \\culmpdxna_{O^{(r)}} \\;. $$\nConsequently,  it suffices to show that,  for each $r_0> 0$, the evaluation map\n\\[\n\\begin{array}{ll}\ne_*: &   \\displaystyle \\lim_{d\\to \\infty} K_*\\Big( \\displaystyle \\lim_{r<r_0, r\\to r_0} \\culmpdxna_{O^{(r)}} \\Big)             \\\\\n&   \\longrightarrow  \\displaystyle \\lim_{d\\to \\infty} K_*\\Big(   \\displaystyle \\lim_{r<r_0, r\\to r_0}  \\cumpdxna_{O^{(r)}} \\Big)\n\\end{array}\n\\]\nis an isomorphism.\n\\par\nLet $r_0>0$. Since $(X_n)_\\nn$ has uniform bounded geometry, there exists $N>0$ such that $\\# B(x, r_0)<N$ for all $x\\in X_n, \\nn$.\nIt follows that there exists an integer $J_{r_0}>0$ independent of $n$ such that we have the following decompositions for all $\\nn$:\n\\begin{itemize}\n\\item $X_n=\\bigcup_{j=1}^{J_{r_0}} \\Gamma_n^{(j)}$ for $J_{r_0}$ subspaces $\\Gamma_n^{(j)}$ of $X_n$;\n\\item $\\Gamma_n^{(j)}\\cap \\Gamma_n^{(j')}=\\emptyset$ whenever $j\\not= j'$;\n\\item for any $x\\in X_n$ and any distinct $\\gamma, \\gamma'\\in \\Gamma_n^{(j)}\\cap B_{X_n}(x, l_n)$,\n\t\t\t$$d\\Big( t_x(\\gamma)(s(\\gamma)), \\, t_x(\\gamma')(s(\\gamma'))  \\Big) >2r_0$$\n\t\tin $V_n$.\n\\end{itemize}\n\\par\nFor any $0<r<r_0$ and each $j\\in \\{1, 2, \\cdots, J_{r_0}\\}$, let\n$$O^{(r, j)}_{n,x} = \\bigcup_{\\gamma\\in \\Gamma_n^{(j)}\\cap B_{X_n}(x, l_n)}  B_{\\mathbb{R}_+\\times V_n}\\big( t_x(\\gamma)(s(\\gamma)), \\, r   \\big) $$\nfor all $x\\in X_n, \\nn$. Then the system\n$$O^{(r, j)}=\\Big( O^{(r, j)}_{n,x} \\Big)_{x\\in X_n, \\nn}$$\nis coherent for each $j\\in \\{1, 2, \\cdots, J_{r_0}\\}$, and\n$$O^{(r)}=\\bigcup_{j=1}^{J_{r_0}} O^{(r, j)}.$$\nFor each $j\\in \\{1, 2, \\cdots, J_{r_0}\\}$,  let\n$$\\Gamma^{(j)}=\\big( \\Gamma_n^{(j)} \\big)_\\nn.$$\nIt is clear that the system $O^{(r, j)}$ or\n$$O^{(r, j)} \\cap \\big( \\cup_{i=1}^{j-1} O^{(r, i)}  \\big)$$\nis $(\\Gamma^{(j)}, r)$-separate for each $j\\in \\{1, 2, \\cdots, J_{r_0}\\}$.\n\\par\nBy a construction of ``partition of unity'' as in the proof of Lemma 6.3 in \\cite{Yu00}, for each $j\\in \\{1, 2, \\cdots, J_{r_0}\\}$, we have\n$$\\displaystyle \\lim_{r<r_0, r\\to r_0} \\cumpdxna_{O^{(r, j)}} +\\, \\displaystyle \\lim_{r<r_0, r\\to r_0} \\cumpdxna_{\\cup_{i=1}^{j-1} O^{(r, i)}} $$\n$$  =  \\displaystyle \\lim_{r<r_0, r\\to r_0}  \\cumpdxna_{\\cup_{i=1}^{j} O^{(r, i)}} \\;\\; ,$$\n$$ \\displaystyle \\lim_{r<r_0, r\\to r_0} \\cumpdxna_{O^{(r, j)}}\\cap \\, \\displaystyle \\lim_{r<r_0, r\\to r_0} \\cumpdxna_{\\cup_{i=1}^{j-1} O^{(r, i)}} $$\n$$ = \\displaystyle \\lim_{r<r_0, r\\to r_0} \\cumpdxna_{O^{(r, j)} \\cap \\big( \\cup_{i=1}^{j-1} O^{(r, i)}  \\big)} \\;\\; ,$$\n$$\\displaystyle \\lim_{r<r_0, r\\to r_0} \\culmpdxna_{O^{(r, j)}}+\\, \\displaystyle \\lim_{r<r_0, r\\to r_0} \\culmpdxna_{\\cup_{i=1}^{j-1} O^{(r, i)}} $$\n$$ = \\displaystyle \\lim_{r<r_0, r\\to r_0}  \\culmpdxna_{\\cup_{i=1}^{j} O^{(r, i)}} \\;\\; ,$$\n$$\\displaystyle \\lim_{r<r_0, r\\to r_0} \\culmpdxna_{O^{(r, j)}}\\cap \\,\\displaystyle \\lim_{r<r_0, r\\to r_0} \\culmpdxna_{\\cup_{i=1}^{j-1} O^{(r, i)}} $$\n$$  = \\displaystyle \\lim_{r<r_0, r\\to r_0}  \\culmpdxna_{O^{(r, j)} \\cap \\big( \\cup_{i=1}^{j-1} O^{(r, i)}  \\big)} \\;\\; .$$\n\\par\nNow, Theorem 6.1 follows from Theorem 6.7, together with a Mayer-Vietoris sequence argument (cf. Sect. 3 of \\cite{HRY}).\n\\hfill{$\\Box$}\n\n\n\n\\section{Geometric analogue of Bott periodicity}\nIn this section, we shall define maps $\\alpha$, $\\beta$, $\\alpha_L$, $\\beta_L$ to build the following commutative diagram\n\\[\n\\xymatrix{\nK_*\\Big( \\culmpdxn \\Big) \\ar[r]^{\\quad e_*} \\ar[d]^{(\\beta_L)_*} & K_*\\Big( \\cumpdxn \\Big) \\ar[d]^{\\beta_*} \\\\\nK_*\\Big( \\culmpdxna \\Big)  \\ar@<1ex>[u]^{(\\alpha_L)_*} \\ar[r]^{\\quad e_*}  & K_*\\Big( \\cumpdxna \\Big )\\;. \\ar@<1ex>[u]^{\\alpha_*}\n}\n\\]\nand show a geometric analogue of the Bott periodicity.\n\\par\nThe maps $\\alpha$, $\\beta$, $\\alpha_L$, $\\beta_L$ will be constructed as {\\em asymptotic morphisms}. The reader is referred to\n\\cite{GHT, HK97, HK01, HKT, HG, Yu00} for backgrounds on asymptotic morphisms and the sources of most of the ideas behind the current section. In what follows, we shall use the graded formalism from \\cite{HG}.\n\n\\subsection*{\\it \\S 7.1. The Bott maps $\\beta$ and $\\beta_L$}\nIn this subsection, we define the Bott maps  $\\beta$ and $\\beta_L$.\nRecall that $(X_n)_\\nn$ is a sequence of finite metric spaces with uniform bounded geometry which admit a fibred coarse embedding into Hilbert space as in Definition 5.4.\nFor each $d\\geq 0$, $\\zdn$ is a countable dense subset of $\\pdxn$. And\n$$V_n:= \\mbox{ affine-span }\\Big\\{ t_x(z)(s(z)) \\, \\Big| \\, z\\in B(x, l_n), x\\in X_n \\Big\\}$$\nis a finite dimensional affine subspace of $H$ for each $\\nn$.\n\\par\nBy Definition 5.1,  for each $x\\in \\zdn$, the inclusion of the $0$-dimensional affine subspace $\\{t_x(x)(s(x))\\}$ into $V_n$ induces a $*$-homomorphism\n$$\\beta(x): \\, \\cS\\cong \\cA\\big( \\{t_x(x)(s(x))\\} \\big)  \\rightarrow \\cA(V_n)$$\nby the formula\n$$\\Big( \\beta(x) \\Big) (g)= g \\big(   X\\widehat\\otimes 1+ 1\\widehat\\otimes C_{V_n,t_x(x)(s(x))}   \\big)  \\,,$$\nwhere\n$$C_{V_n,t_x(x)(s(x))}:\\, V_n\\rightarrow V_n^0 \\subseteq \\Cliff(V_n^0)$$\nis the function $v\\mapsto v- t_x(x)(s(x))\\in V_n^0 \\subseteq \\Cliff(V_n^0)$ for all $v\\in V_n$.\n\\par\n{\\bf Definition 7.1.} Let $d\\geq 0$. For each $t\\in [1, \\infty)$, define a map\n$$\\beta_t:\\, \\cS\\widehat\\otimes \\cupdxn \\longrightarrow \\cumpdxna$$\nfor $g\\in \\cS$, $\\ttt\\in \\cupdxn$ by the formula\n$$\\beta_t\\big( g\\widehat\\otimes T \\big)=\\Big[ \\, \\Big(      \\big(  \\beta_t\\big( g\\widehat\\otimes T \\big) \\big)^{(0)}, \\cdots, \\big(  \\beta_t\\big( g\\widehat\\otimes T \\big) \\big)^{(n)}, \\cdots \\Big)\\,\\Big] \\,,$$\nwhere\n$$\\big(  \\beta_t\\big( g\\widehat\\otimes T \\big) \\big)^{(n)}(x, y)=\\big( \\beta(x) \\big) (g_t) \\widehat\\otimes T^{(n)}(x, y)$$\nfor $x, y\\in \\zdn$, $\\nn$, and $g_t(r)=g(\\frac{r}{t})$ for all $r\\in \\mathbb{R}$.\n\\hfill{$\\Box$}\n\\par\n{\\bf Definition 7.2.}  Let $d\\geq 0$. For each $t\\in [1, \\infty)$, define a map\n$$(\\beta_L)_t:\\, \\cS\\widehat\\otimes \\culpdxn \\longrightarrow \\culmpdxna$$\nby the formula\n$$\\Big(  (\\beta_L)_t (f) \\Big) (\\tau)= \\beta_t\\big(  f(\\tau)  \\big)$$\nfor $\\tau\\in \\mathbb{R}_+=[0, \\infty)$.\n\\hfill{$\\Box$}\n\\par\n{\\bf Lemma 7.3.} {\\it\nFor each $d\\geq 0$, the maps $(\\beta_t)_{t\\geq 1}$ and $\\big((\\beta_L)_t\\big)_{t\\geq 1}$ extend to asymptotic morphisms\n$$\\beta:\\; \\cS\\widehat\\otimes \\cumpdxn  \\rightarrow \\cumpdxna   \\,,$$\n$$\\beta_L:\\; \\cS\\widehat\\otimes \\culmpdxn  \\rightarrow \\culmpdxna  \\,.$$\n}\n\\par\n{\\bf Proof.} For any $x, y\\in \\zdn$ with $n\\geq N$ for some $N>0$ large enough such that both  $x$ and $y$ are in the regions of trivializations of $t_x$ and $t_y$, define\n$1$-dimensional affine subspaces of $V_n$:\n$$W^x:=W_{\\{x, y\\}}(x):=\\mbox{ affine-span } \\Big\\{ t_x(x)(s(x)),\\;  t_x(y)(s(y)) \\Big\\} \\,,$$\n$$W^y:=W_{\\{x, y\\}}(y):=\\mbox{ affine-span } \\Big\\{ t_y(x)(s(x)), \\; t_y(y)(s(y)) \\Big\\} \\,.$$\nThen, by Definition 5.1 on $\\beta_{ba}$, we have\n$$\\beta(x)=\\beta_{V_n, W^x}\\circ \\beta_{W^x, \\{ t_x(x)(s(x)) \\}} \\,,$$\n$$\\beta(y)=\\beta_{V_n, W^y}\\circ \\beta_{W^y, \\{ t_y(y)(s(y)) \\}} \\,.$$\nMoreover, we have  $W^x=t_{xy}\\big( W^y \\big)$ and\n$$\\beta_{W^x, \\{ t_x(y)(s(y)) \\}}(g)=\\big( t_{xy} \\big)_* \\Big(  \\beta_{W^y, \\{ t_y(y)(s(y)) \\}}(g)  \\Big)$$\nfor all $g\\in \\cS$, where recall that $t_{xy}=t_x\\circ t_y^{-1}: H\\to H$ and\n$$\\big( t_{xy} \\big)_*: \\;  \\cA(W_C(y)) \\rightarrow \\cA(W_C(x))$$\nmapping $g\\widehat\\otimes h$ to $g\\widehat\\otimes (t_{xy})_*(h)$, is defined in Definition 5.3 for all $C\\subseteq B_{X_n}(x, l_n)\\cap B_{X_n}(y, l_n)$.\n\\par\nFor the generators $g(x)=\\frac{1}{x\\pm i}$ of $\\cS=C_0(\\mathbb{R})$, it is standard argument to verify, for $t\\in [1, \\infty)$, that\n\\[\n\\begin{array}{ll}\n  &  \\Big\\| \\beta_{W^x, \\{ t_x(x)(s(x)) \\}}(g_t) - \\big( t_{xy} \\big)_* \\Big(  \\beta_{W^y, \\{ t_y(y)(s(y)) \\}}(g_t)    \\Big)  \\Big\\|              \\\\\n= &  \\big\\| \\beta_{W^x, \\{ t_x(x)(s(x)) \\}}(g_t) -   \\beta_{W^x, \\{ t_x(y)(s(y)) \\}}(g_t)      \\big\\|              \\\\\n\\leq & \\frac{1}{t} \\|  t_x(x)(s(x)) - t_x(y)(s(y)) \\|     \\\\\n\\leq & \\frac{1}{t} \\rho_2(d(x, y)) \\; \\longrightarrow 0 \\;\\; (t\\to \\infty)  \\,.\n\\end{array}\n\\]\nIt follows from an approximation argument, together with \\cite{Yu00}(Lemma 7.6) and \\cite{HG}(Lemma 3.2), that for all $d\\geq 0$, $R>0, r>0, c>0, \\varepsilon>0$, there exists\n$t_0>1$ such that for all $x, y\\in \\zdn, \\nn, $ with $d(x, y)\\leq R$, for all $t\\geq t_0$ and all $g\\in \\cS$ with\n$$\\supp(g)\\subseteq [-r, r], \\quad\\quad\\quad  \\|g'\\|_\\infty\\leq c  \\,,$$\nwe have\n$$\\Big\\| \\big(  \\beta(x) \\big) (g_t) -\\big( t_{xy} \\big)_*\\Big(   \\big(  \\beta(y) \\big) (g_t)  \\Big)   \\Big\\| < \\varepsilon.$$\n($(t_{xy})_*$ is as in Definition 5.5) As a result, the maps $\\beta_t$ ($t\\geq 1$) define a $*$-homomorphism $\\beta$ from $\\cS\\widehat\\otimes \\cupdxn$ to the asymptotic $C^*$-algebra\n$$\\mathfrak{A}\\Big(  \\cumpdxna  \\Big) := \\frac\n{C_b\\Big( \\big[ 1, \\infty \\big), \\; \\cumpdxna   \\Big)}\n{C_0\\Big( \\big[ 1, \\infty \\big), \\; \\cumpdxna   \\Big)}\n$$\nsatisfying $\\|\\beta(g\\widehat\\otimes T)\\|\\leq \\|g\\|\\cdot \\|T\\|$ for all $g\\in \\cS$ and $T\\in \\cupdxn$. Hence, by the universality of the max norm and the max tensor product,\n$\\beta$ extends to a $C^*$-homomorphism from\n$$\\cS \\widehat\\otimes _{\\max} \\cumpdxn $$\nto $\\mathfrak{A}\\big(  \\cumpdxna  \\big)$. Since $\\cS$ is nuclear, we conclude that $\\beta_t$ extends to an asymptotic morphism from $\\cS \\widehat\\otimes \\cumpdxn$ to\n$\\cumpdxna$. The case for $\\beta_L$ is similar.\n\\hfill{$\\Box$}\n\n\n\n\\subsection*{\\it \\S 7.2. The Dirac maps $\\alpha$ and $\\alpha_L$      }\nIn this subsection, we define the Dirac maps $\\alpha$ and $\\alpha_L$. To begin with, we recall the definition of  the Bott-Dirac operator on a finite dimensional\naffine subspace of the Hilbert space $H$.\n\\par\nLet $V$ be a finite dimensional affine subspace of $H$, and let $V^0$ be the corresponding finite dimensional linear space consisting of differences of elements of $V$. Let\n$$L^2(V):=L^2(V, \\Cliff(V^0))$$\nbe the graded infinite dimensional complex Hilbert space of square integrable $\\Cliff(V^0)$-valued functions on $V$, where $V$ is endowed with the Lebesgue measure induced from the inner\nproduct on $V^0\\subseteq H$. The grading of $L^2(V)$ is inherited from the Clifford algebra $\\Cliff(V^0)$, which is also considered as a graded Hilbert space in such a way that\nthe monomials $e_{i_1}\\cdots e_{i_p}$ associated to an orthonormal basis of $V^0$ are orthonormal.\n\\par\nThe {\\em  Dirac operator $D_V$ of $V$}  is the unbounded operator on $L^2(V)$ defined by the formula\n$$\\big( D_V\\xi \\big) (v) =\\sum_{i=1}^n (-1)^{\\mbox{degree} (\\xi)} \\frac{\\partial \\xi}{\\partial x_i} (v) \\cdot e_i$$\nwhere $\\{e_1, \\cdots, e_n\\}$ is an orthonormal basis for $V^0$, $\\xi\\in L^2(V)$, $v\\in V$, and $\\{x_1, \\cdots, x_n\\}$ is the coordinates dual to  $\\{e_1, \\cdots, e_n\\}$.\nThe domain of $D_V$ is the Schwartz subspace of $L^2(V)$. Note that $D_V$ does not depend on the choice of an orthonormal basis for $V^0$ (and does not depend on a basis point in $V$).\n\\par\nThe {\\em Clifford operator $C_{V, v_0}$ of $V$ at $v_0\\in V$} is an unbounded operator on $L^2(V)$ defined by the formula\n$$\\big( C_{V, v_0} \\xi \\big) (v) = (v-v_0) \\cdot \\xi(v)$$\nfor all $v\\in V$, $\\xi\\in L^2(V)$, where the multiplication $\\cdot$ is the Clifford multiplication of $v-v_0\\in V^0\\subset \\Cliff(V^0)$ and $\\xi(v)\\in \\Cliff(V^0)$.\nThe domain of $C_{V,v_0}$ is also the Schwartz subspace of $L^2(V)$. Note that the Clifford operator  $C_{V, v_0}$  depends on  the choice of the base point $v_0$.\nIf $V$ is actually a linear subspace, and $v_0=0\\in V$, then we also denote $C_V:=C_{V, 0}$.\n\\par\nThe {\\em  Bott-Dirac operator of  $V$ at $v_0\\in V$} is\n$$B_{V, v_0} =D_V+C_{V, v_0}$$\nwith domain again the Schwartz subspace of $L^2(V)$. Denote by $\\xi_{V, v_0}$ the unit vector of $L^2(V)=L^2(V, \\Cliff(V^0))$:\n$$\\xi_{V, v_0}(v)= \\pi^{-\\frac{n}{4}}\\cdot e^{-\\frac{\\|v-v_0\\|^2}{2}} \\; ,$$\n where $n=\\dim(V^0)$. Then $\\xi_{V, v_0}$ spans the kernel of $B_{V, v_0}$, which is a $1$-dimensional subspace of $L^2(V)$.\nIf $V$ is a linear subspace, and $v_0=0\\in V$, then we also denote $B_V:=B_{V, 0}$.\n\\par\nLet $V_a, V_b$ be finite dimensional affine subspaces of $H$ such that $V_a\\subseteq V_b$ and $V_b=V_{ba}^0\\oplus V_a$. Then\n$$L^2(V_b)\\cong L^2(V_{ba}^0)\\widehat\\otimes L^2(V_a)$$\nand $L^2(V_a)$ is regarded as a subspace of $L^2(V_b)$ via the isometry from $L^2(V_a)$ to $L^2(V_b)$ given by\n$$\\xi \\mapsto \\xi_{V_{ba}^0, 0}\\widehat\\otimes \\xi.$$\n\\par\nRecall also that an affine isometry $t: V\\to V$ canonically induces a unitary on $L^2(V)$ and moreover, a $*$-isomorphism\n$$t_*: \\cK(L^2(V))\\rightarrow \\cK(L^2(V))$$\nby conjugation with the unitary.\n\\par\n{\\bf We now come back to the case of interest.} Let $(X_n)_\\nn$ be a sequence of finite metric spaces with uniform bounded geometry which admit a fibred coarse embedding\ninto Hilbert space as described in Definition 5.4. Recall in Definition 5.5, for all $\\nn$,\n$$V_n=\\mbox{ affine-span }\\big\\{ t_x(z)(s(z)) \\,|\\, z\\in B_{X_n}(x, l_n), x\\in X_n \\big\\}.$$\nDefine\n$$E_n:= \\mbox{ linear-span } \\Big\\{ V_n,\\, 0 \\Big\\} \\subseteq H  \\,,$$\nwhich is a  finite dimensional {\\em linear} subspace of $H$ containing $V_n$.\nDenote\n$$L^2_n:=L^2(E_n, \\Cliff(E_n))$$\nand denote by $\\cK(L^2_n)$ the graded $C^*$-algebra of all compact operators on the graded Hilbert space $L^2_n$ for all $\\nn$.\n\\par\n{\\bf Definition 7.4.} For all $x, z\\in X_n (\\nn) $ with $B(x, l_n)\\cap B(z, l_n)\\not=\\emptyset$, we define an isomorphism\n$$(t_{xz})_*:\\, \\cK(L^2_n)\\widehat\\otimes \\cK \\rightarrow \\cK(L^2_n)\\widehat\\otimes \\cK$$\nas follows. Denote\n$$W_x:=\\mbox{ affine-span }\\big\\{ t_x(w)(s(w)) \\,\\big| \\, w\\in  B(x, l_n)\\cap B(z, l_n)  \\,  \\}, $$\n$$W_z:=\\mbox{ affine-span }\\big\\{ t_z(w)(s(w)) \\,\\big| \\, w\\in  B(x, l_n)\\cap B(z, l_n)  \\,  \\}. $$\nThen $W_x=t_{xz}(W_z)$. Denote $W_x^\\perp=E_n\\ominus W_x$ and $W_z^\\perp=E_n\\ominus W_z$ the linear orthogonal complements of $W_x, W_y$ in $E_n$. Choose a unitary operator\n$U_{xz}: \\, W_z^\\perp \\rightarrow W_x^\\perp $. Then\n$$U_{xz}\\oplus t_{xz}: \\, E_n=W_z^\\perp\\oplus W_z \\longrightarrow E_n=W_x^\\perp\\oplus W_x$$\nis an affine isometry from $E_n$ onto $E_n$. We define\n$$(t_{xz})_*:=(U_{xz}\\oplus t_{xz})_*\\widehat\\otimes 1: \\; \\cK(L^2_n)\\widehat\\otimes \\cK \\rightarrow \\cK(L^2_n)\\widehat\\otimes \\cK .$$\nMoreover, in Rips complexes for each $d\\geq 0$, if $x, z\\in \\zdn\\subseteq \\pdxn, \\nn$, with $x\\in \\Star(\\bar x)$ and $z\\in \\Star(\\bar z)$, we define $(t_{xz})_*=(t_{\\bar{x}\\bar{z}})_*$.\n\\par\n{\\bf Remark 7.5.} Let  $\\xi_{W_x^\\perp}$ and $\\xi_{W_z^\\perp}$ be the unit vectors in the kernels of the Bott-Dirac operators at the origin of the linear spaces $L^2(W_x^\\perp)$ and $L^2(W_z^\\perp)$.\nThen  $\\xi_{W_x^\\perp} = U_{xz}\\big( \\xi_{W_z^\\perp} \\big)$. This fact implies in the sequel actual applications,  $(t_{xz})_*$ does not depend on the choice of the unitary $U_{xz}$.\n\\par\n{\\bf Definition 7.6.} For each $d\\geq 0$, define $\\cupdxnk$ to be the set of all equivalence classes $\\ttt$ of sequences\n\t$$(T^{(0)}, \\cdots, T^{(n)}, \\cdots)$$\ndescribed as follows:\n\\begin{itemize}\n\\item[(1)] $T^{(n)}$ is a bounded function from $\\zdn \\times \\zdn$ to $\\cK(L^2_n)\\widehat\\otimes\\cK$ for all $\\nn$;\n\\item[(2)] for any bounded subset $B\\subset \\pdxn$, the set\n\t\t\t$$\\{ (x, y)\\in B\\times B \\cap \\zdn\\times \\zdn \\; | \\; T^{(n)} (x, y)\\not= 0 \\}$$\n\t\t\tis finite;\n\\item[(3)] there exists $L>0$ such that\n\t\t\t$$\\#\\{y\\in \\zdn \\; | \\; T^{(n)}(x, y)\\not= 0 \\}<L, \\quad \\#\\{y\\in \\zdn \\; | \\; T^{(n)}(y, x)\\not= 0 \\}<L $$\n\t\t\tfor all $x\\in \\zdn, \\nn$;\n\\item[(4)] there exists $R>0$ such that $T^{(n)}(x, y)=0$ whenever $d(x, y)>R$ for $x, y\\in \\zdn, \\; \\nn$.\n\\end{itemize}\nThe equivalence relation $\\sim$ on these sequences is defined by\n$$(T^{(0)}, \\cdots, T^{(n)}, \\cdots) \\sim (S^{(0)}, \\cdots, S^{(n)}, \\cdots)$$ if and only if\n$$\\lim_{n\\to \\infty} \\sup_{x, y\\in \\zdn} \\big\\| T^{(n)}(x, y) - S^{(n)}(x, y) \\big\\|_\\cK =0.$$\nThe product structure for $\\cupdxnk$ is defined as follows. For any two elements $\\ttt$ and $S=\\big[ (S^{(0)}, \\cdots, S^{(n)}, \\cdots) \\big]$ in $\\cupdxnk$, their product is defined to be\n$$\\big[ \\big( (TS)^{(0)}, \\cdots, (TS)^{(n)}, \\cdots \\big) \\big], $$\nwhere there exists a sufficiently large $N\\in \\mathbb{N}$ depending on the propagation of the two representative sequences,\nsuch that $ (TS)^{(n)}=0$ for all $n=0, 1, 2, \\cdots, N-1$ and\n$$(TS)^{(n)}(x, y)=\\sum_{z\\in \\zdn} \\Big( T^{(n)}(x, z) \\Big) \\cdot \\Big( \\big(t_{xz}\\big)_* \\big( S^{(n)}(z, y) \\big) \\Big)$$\nfor all $x, y\\in \\zdn, \\; n\\geq N$, where $\\big(t_{xz}\\big)_*$ is defined as in Definition 7.4.\n\\par\nThe $*$-structure for $\\cupdxnk$ is defined by the formula\n$$ \\big[ \\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big) \\big]^*=   \\big[ \\big( (T^*)^{(0)}, \\cdots, (T^*)^{(n)}, \\cdots \\big) \\big] \\,,$$\nwhere\n$$(T^*)^{(n)}(x, y)=\\big( t_{xy} \\big)_* \\Big( \\big( T^{(n)}(y, x)\\big)^*\\Big)$$\nfor all but finitely many $n$, and $0$ otherwise.\n\\par\nNow, $\\cupdxnk$ is made into a $*$-algebra by using the additional usual matrix operations. Define $\\cumpdxnk$ to be the completion of $\\cupdxnk$ with respect to the norm\n\\begin{footnotesize}\n$$\\big\\|T\\big\\|_{\\max} :=\\sup\\Big\\{ \\big\\|\\phi(T)\\big\\|_{\\cB(H_\\phi)} \\; \\Big | \\; \\phi:\\cupdxnk \\to \\cB(H_\\phi), \\mbox{ a $*$-representation} \\Big\\}. $$\n\\end{footnotesize}\n\\hfill{$\\Box$}\n\\par\nSimilarly we define the localization algebra $\\culmpdxnk$.\n\\par\n{\\bf Definition 7.7.} Let $d\\geq 0$. For each $t\\in [1, \\infty)$ define a map\n$$\\alpha_t:\\; \\cupdxna\\rightarrow \\cumpdxnk$$\nby the formula\n$$\\alpha_t(T)=\\Big[ \\Big(  \\big(\\alpha_t(T)\\big)^{(0)}, \\cdots,  \\big(\\alpha_t(T)\\big)^{(n)}, \\cdots     \\Big)  \\Big]$$\nfor $\\ttt\\in \\cupdxna$, with\n$$\\big(\\alpha_t(T)\\big)^{(n)}(x, y)=\\Big( \\theta_t^k(x) \\Big) \\big( T_{1}^{(n)}(x, y)\\big)$$\nfor all $x, y\\in \\zdn, \\nn$, where\n\\begin{itemize}\n\\item[(1)] there exists $k\\geq 0$ independent of $\\nn$ as in condition (6) of Definition 5.5 such that\n\t\t\t\t\t\t$$ T^{(n)}(x, y) = \\Big( \\beta_{V_n, W_k(x)} \\widehat\\otimes 1 \\Big) \\big( T^{(n)}_1 (x, y) \\big)$$\n\t\tfor some $T_1^{(n)}(x, y)\\in \\cA(W_k(x))\\widehat\\otimes \\cK$ of the form $\\sum_{i=1}^K g_i\\widehat\\otimes h_i\\widehat\\otimes k_i$ where $g_i\\in \\cS$, $h_i\\in \\cC(W_k(x))$,\n\t\t\t$k_i\\in \\cK$ for $1 \\leq i \\leq K$.\n\\item[(2)] The map\n\t\t$$ \\theta_t^k(x):\\, \\cA(W_k(x))\\widehat\\otimes \\cK \\rightarrow \\cK(L^2_n)\\widehat\\otimes \\cK$$\n\t\tis defined  by the formula\n\t\t$$\\Big( \\theta_t^k(x) \\Big) \\big( g\\widehat\\otimes h \\widehat\\otimes k \\big)\n\t\t\t= g_t\\big( B_{E_n\\ominus W_k(x)} \\widehat\\otimes 1 + 1\\widehat\\otimes D_{W_k(x)}  \\big)\\big( 1\\widehat\\otimes M_{h_t} \\big) \\widehat\\otimes k$$\n\t\tfor all $g\\in \\cS$, $h\\in \\cC(W_k(x))$, $k\\in \\cK$, $t\\geq 1$, $x\\in \\zdn, \\nn$, where\n\t\\begin{itemize}\n\t\\item[$\\bullet$]  $g_t(s)=g(s\/t)$ for $s\\in \\mathbb{R}$.\t\n\t\\item[$\\bullet$] $D_{W_k(x)}$ is the Dirac operator of the affine space\n\t\t\t\t\t$$W_k(x)=\\mbox{ affine-span } \\big\\{ t_x(z)(s(z)) \\, \\big| \\, z\\in B_{\\pdxn}(x, k) \\big\\} \\subset E_n .$$\n\t\\item[$\\bullet$] $B_{E_n\\ominus W_k(x)}$ is the Bott-Dirac operator at the origin $0$ of the linear orthogonal complement of $W_k(x)$ in $E_n$.\n\t\\item[$\\bullet$] for any $h\\in \\cC(W_k(x))$ and $t\\geq 1$, the function $h_t\\in \\cC(W_k(x))$ is defined by\n\t\t\t\t\t$$ h_t(v)= h\\Big( t_x(x)(s(x))+\\frac{1}{t} \\big( v-t_x(x)(s(x))   \\big)  \\Big)$$\n\t\t\t\tfor all $v\\in W_k(x)$.\n\t\\item[$\\bullet$] $M_{h_t}$ is the pointwise multiplication operator on $L^2(W_k(x), \\Cliff(W_k(x)^0))$\n\t\t\t\t\t$$\\Big( M_{h_t} \\xi \\Big) (v) = h_t(v)\\cdot \\xi(v)$$\n\t\t\t\tfor all $\\xi\\in L^2(W_k(x))$ and $v\\in W_k(x)$.\n\t\\item[$\\bullet$] $1\\widehat\\otimes M_{h_t}:\\, L^2_n\\to L^2_n$ is defined according to the tensor decomposition\n\t\t\t\t\t$$L^2_n\\cong L^2(E_n\\ominus W_k(x)) \\widehat\\otimes L^2(W_k(x)).$$\n\t\\end{itemize}\n\\end{itemize}\n\\par\n{\\bf Remark 7.8.} (1) The fact that $\\Big( \\theta_t^k(x) \\Big) \\big( g\\widehat\\otimes h \\widehat\\otimes k \\big) $ is in $\\cK(L^2_n)\\widehat\\otimes \\cK$ follows from the ellipticity of the Dirac operator\nand the Rellich lemma, see (\\cite{HKT} Definition 8). (2) By the proof of Proposition 4.2 in \\cite{HKT}, we know that $\\alpha_t(T)$ does not asymptotically depend on the choice of $k$.\n\\par\n{\\bf Definition 7.9.} For each $d\\geq 0$ and $t\\geq 1$, define\n$$(\\alpha_L)_t:\\; \\culpdxna\\rightarrow \\culmpdxnk$$\nby the formula\n$$\\Big( (\\alpha_L)_t(f) \\Big)(\\tau)=\\alpha_t\\big(f(\\tau) \\big)$$\nfor all $\\tau\\in [0, \\infty)$.\n\\par\n{\\bf Lemma 7.10.} For each $d\\geq 0$, the maps $(\\alpha_t)_{t\\geq 1}$ and $((\\alpha_L)_t)_{t\\geq 1}$ extend to asymptotic morphisms\n$$\\alpha:\\; \\cumpdxna\\rightarrow \\cumpdxnk \\,,$$\n$$\\alpha_L:\\; \\culmpdxna\\rightarrow \\culmpdxnk  \\,.$$\n\\par\n{\\bf Proof.} We first fix some notations. For a given large $\\ell \\geq 0$, and for any subset\n$$C\\subseteq B_{\\pdxn}(x, \\ell)\\cap B_{\\pdxn}(z, \\ell)$$\nwhere $x, z\\in \\zdn$  (for all but finitely many $n$)  with $d(x, z)<\\ell$ , denote\n$$W_C(x)=\\mbox{ affine-span }\\big\\{ t_x(w)(s(w)) \\, \\big| \\, w\\in C \\big\\},$$\n$$W_C(z)=\\mbox{ affine-span }\\big\\{ t_z(w)(s(w)) \\, \\big| \\, w\\in C \\big\\}.$$\nThen $W_C(x)=t_{xz}(W_C(z))$. For any affine subspace $W$ with $W_C(x)\\subseteq W\\subseteq E_n$, we identify $\\cA(W_C(x))$ with a subalgebra of $\\cA(W)$ via the map\n$\\beta_{W, W_C(x)}$ defined in Definition 5.1.\n\\par\n{\\it Step 1.}  For any $K>0$, $r>0$, $c>0$ and $x\\in \\zdn$, denote by\n\t\t$$\\Big[  \\cA(W_C(x))\\widehat\\otimes \\cK \\Big]_{K, r, c}$$\nthe subset of $\\cA(W_C(x))\\widehat\\otimes \\cK$ consisting of those elements of the form $\\sum_{i=1}^K g_i\\widehat\\otimes h_i\\widehat\\otimes k_i$ where $g_i\\in \\cS$, $h_i\\in \\cC(W_C(x))$,\n$k_i\\in \\cK$ such that (1) each $g_i$ is supported in $[-r, r]$; (2) each $g_i$ and $h_i$ is continuously differentiable and their derivatives satisfy $\\|g'_i\\|_\\infty\\leq c$ and\n $\\|\\nabla_v h_i\\|\\leq c$  for all $v\\in W_C(x)$ such that $\\|v-t_x(x)(s(x))\\|\\leq 1$.\n\\par\nIt follows from (\\cite{Yu00} Lemma 7.5) and (\\cite{HKT} Lemma 2.9) that for any $\\varepsilon>0$ there exists $t_0>1$ such that for all $t\\geq t_0$ and all\n$a, b\\in \\Big[  \\cA(W_C(x))\\widehat\\otimes \\cK \\Big]_{K, r, c} $, we have\n$$\\big\\| \\theta_t^\\ell(x)(ab)-\\theta_t^\\ell(x)(a)\\theta_t^\\ell(x)(b) \\big\\|<\\varepsilon, \\quad\\quad   \\big\\| \\theta_t^\\ell(x)(a^*)-\\theta_t^\\ell(x)(a)^* \\big\\|<\\varepsilon. $$\n\\par\n{\\it Step 2.} The affine isometry $t_{xz}: W_C(z)\\to W_C(x)$ induces a diagram\n\\[\n\\xymatrix{\n\\cA\\big(W_C(z)\\big)\\widehat\\otimes \\cK  \\ar[r]^{\\quad \\theta_t^\\ell(z)} \\ar[d]_{(t_{xz})_*}        &    \\cK(L^2_n)\\widehat\\otimes \\cK  \\ar[d]^{(t_{xz})_*}             \\\\\n\\cA\\big(W_C(x)\\big)\\widehat\\otimes \\cK  \\ar[r]^{\\quad \\theta_t^\\ell(x)}   &  \\cK(L^2_n)\\widehat\\otimes \\cK\n}\n\\]\n\\par\nIt follows from the proof of (\\cite{HKT} Proposition 4.2) and (\\cite{Yu00} Lemma 7.2) that the diagram is asymptotically commutative in the sense that, for any\n$R>0$ (relating propagations in $\\cupdxna$) and any large $\\ell$ (greater than $R+k$ where $k$ is as  Definition 5.5 (6)), for any  $K>0$, $r>0$, $c>0$ as above, and for any\n$\\varepsilon>0$, there exists $t_0>1$ such that for all $t\\geq t_0$ and all $b\\in \\Big[  \\cA(W_C(z))\\widehat\\otimes \\cK \\Big]_{K, r, c} $ , we have\n$$\\Big\\|   \\Big(\\theta_t^\\ell(x)\\Big) \\big( (t_{xz})_*(b) \\big) -  (t_{xz})_* \\Big(  \\big(\\theta_t^\\ell(z)\\big)(b)      \\Big) \\Big\\|<\\varepsilon,$$\nwhenever  $x, z\\in \\zdn$ satisfying  $d(x, z)\\leq R$.\n\\par\n{\\it Step 3.} From the above facts it follows that the maps $(\\alpha_t)_{t\\geq 1}$ define a $*$-homomorphism  $\\alpha$ from $\\cupdxna$ to the asymptotic $C^*$-algebra\n$$\\mathfrak{A}\\Big(  \\cumpdxnk  \\Big) := \\frac\n{C_b\\Big( \\big[ 1, \\infty \\big), \\; \\cumpdxnk   \\Big)}\n{C_0\\Big( \\big[ 1, \\infty \\big), \\; \\cumpdxnk   \\Big)}\n$$\nBy the universality of the max norm, $\\alpha$ extends to a $*$-homomorphism from the $C^*$-algebra $\\cumpdxna$ to $\\mathfrak{A}$. The case for $\\alpha_L$ is similar.\n\\hfill{$\\Box$}\n\n\\subsection*{\\it \\S 7.3. A geometric analogue of the Bott periodicity in finite dimension}\nNote that the asymptotic morphisms $\\beta$, $\\beta_L$, $\\alpha$, $\\alpha_L$  induce the following commutative diagram on $K$-theory:\n\\[\n\\xymatrix{\nK_*\\Big( \\culmpdxn \\Big) \\ar[r]^{\\quad e_*} \\ar[d]^{(\\beta_L)_*} & K_*\\Big( \\cumpdxn \\Big) \\ar[d]^{\\beta_*} \\\\\nK_*\\Big( \\culmpdxna \\Big)  \\ar[d]^{(\\alpha_L)_*} \\ar[r]^{\\quad e_*}  & K_*\\Big( \\cumpdxna \\Big )\\;. \\ar[d]^{\\alpha_*}  \\\\\nK_*\\Big( \\culmpdxnk \\Big)  \\ar[r]^{\\quad e_*}  & K_*\\Big( \\cumpdxnk \\Big )\n}\n\\]\n\\par\nIn this subsection, we shall prove the following result, which is a geometric analogue of the Bott periodicity in finite dimension. The proof is adapted from (\\cite{Yu00} Proposition 7.7).\n\\par\n{\\bf Theorem 7.11.} {\\it\nFor each $d\\geq 0$, the compositions $\\alpha_*\\circ \\beta_*$ and $(\\alpha_L)_*\\circ (\\beta_L)_*$ are the identity.\n}\n\\par\n{\\bf Proof.} {\\it Step 1.} Let $\\gamma$ be the asymptotic morphism\n$$\\gamma:\\, \\cS\\widehat\\otimes \\cumpdxn \\rightarrow \\cumpdxnk$$\ndefined by the formula\n$$\\gamma_t(g\\widehat\\otimes T)= \\Big[ \\, \\Big(    \\big(  \\gamma_t(g\\widehat\\otimes T)   \\big)^{(0)}, \\cdots, \\big(  \\gamma_t(g\\widehat\\otimes T)   \\big)^{(n)}, \\cdots     \\Big) \\Big]$$\nfor all $g\\in \\cS$ and $\\ttt\\in \\cupdxn$, where\n$$\\Big(  \\gamma_t(g\\widehat\\otimes T)   \\Big)^{(n)}(x, y) = g_{t^2}\\big(  B_{E_n, t_x(x)(s(x))}  \\big)\\widehat\\otimes T^{(n)}(x, y)$$\nfor all $t\\geq 1$, $x, y\\in \\zdn, \\nn$, and $B_{E_n, t_x(x)(s(x))}$ is the Bott-Dirac operator of $E_n$ at $t_x(x)(s(x))$.\n\\par\nIt follows from (\\cite{HKT} Proposition 2.13 and Appendix) and (\\cite{Yu00} Proposition 7.7) that the composition $\\alpha\\circ\\beta$ is asymptotically equivalent to\n$\\gamma$ (see also \\cite{SpWi} for a remark on compositions of asymptotic morphisms). Hence, $\\alpha_*\\circ\\beta_*=\\gamma_*$.\n\\par\n{\\it Step 2.} Let $\\delta$ be the asymptotic morphism\n$$\\delta:\\, \\cS\\widehat\\otimes \\cumpdxn \\rightarrow \\cumpdxnk$$\ndefined by the formula\n$$\\delta_t(g\\widehat\\otimes T)= \\Big[ \\, \\Big(    \\big(  \\delta_t(g\\widehat\\otimes T)   \\big)^{(0)}, \\cdots, \\big(  \\delta_t(g\\widehat\\otimes T)   \\big)^{(n)}, \\cdots     \\Big) \\Big]$$\nfor all $g\\in \\cS$ and $\\ttt\\in \\cupdxn$, where\n$$\\Big(  \\delta_t(g\\widehat\\otimes T)   \\Big)^{(n)}(x, y) = g_{t^2}\\big(  B_{E_n, 0}  \\big)\\widehat\\otimes T^{(n)}(x, y)$$\nfor all $t\\geq 1$, $x, y\\in \\zdn, \\nn$, and $B_{E_n, 0}$ is the Bott-Dirac operator of $E_n$ at the origin $0\\in E_n$.\n\\par\nFor each $x\\in \\zdn, \\nn$, let $U_x:\\, L^2_n\\to L^2_n$ be the unitary operator defined by\n$$\\Big( U_x\\xi \\Big)(v)=\\xi\\Big(  v-t_x(x)(s(x)) \\Big)$$\nfor all $\\xi\\in L^2_n:=L^2(E_n, \\Cliff(E_n))$ and $v\\in E_n$. Then\n$$U_x^{-1} \\, B_{E_n, t_x(x)(s(x))} \\, U_x \\, = \\,  B_{E_n, 0} \\,\\,.$$\n\\par\nFor each $s\\in [0, 1]$, define an asymptotic morphism\n$$\\Phi^{(s)}:\\, \\cS\\widehat\\otimes \\cumpdxn \\rightarrow \\cumpdxnk  \\widehat\\otimes \\cM_2(\\mathbb{C})          $$\nby the formula\n\\[\n\\Big(  \\Phi^{(s)}_t(g\\widehat\\otimes T)   \\Big)^{(n)}(x, y)   =  \\Big( U_x^{(s)}\\Big)^{-1}\n\t\t\\left[\n\t\t\\begin{array}{cc}\n\t\t\t\\Big(  \\gamma_t(g\\widehat\\otimes T)   \\Big)^{(n)}\\big(x, y\\big)     &      0                 \\\\\n\t\t\t\t0       &       0\n\t\t\\end{array}\n\t\t\\right]\nU^{(s)}_x\n\\]\nfor all $t\\geq 1$, $x, y\\in \\zdn, \\nn$, where\n\\[\nU^{(s)}_x =R(s) \\left[\n\t\t\t\t\t\\begin{array}{cc}\n\t\t\t\t\t\tU_x\\widehat\\otimes 1     &       0          \\\\\n\t\t\t\t\t\t\t\t\t0        &       1\n\t\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t R(s)^{-1},  \\quad\nR(s)= \\left[\n\t\t\t\t\t\\begin{array}{cc}\n\t\t\t\t\t\t\\cos(\\frac{\\pi}{2}s)        &       \\sin(\\frac{\\pi}{2}s)          \\\\\n\t\t\t\t\t     -\\sin(\\frac{\\pi}{2}s)        &       \\cos(\\frac{\\pi}{2}s)\n\t\t\t\t\t\\end{array}\n\t\t\t\\right].\n\\]\nThen $\\Phi^{(s)}$ is a homotopy between the asymptotic morphisms $\\gamma=\\Phi^{(1)}$ and $\\delta=\\Phi^{(0)}$. Hence, $\\gamma_*=\\delta_*$.\n\\par\n{\\it Step 3.} For each $\\nn$, let $p^{(n)}$ be the projection of $E_n$ onto the 1-dimensional kernel of the Bott-Dirac operator $B_{E_n, 0}$ at the origin $0\\in E_n$.\nFor each  $s\\in [0, 1]$, define an asymptotic morphism\n$$\\Psi^{(s)}:\\, \\cS\\widehat\\otimes \\cumpdxn \\rightarrow \\cumpdxnk   $$\nby the formula\n\\[\n\\Big(  \\Psi^{(s)}_t(g\\widehat\\otimes T)   \\Big)^{(n)}(x, y) =\\left\\{\n\\begin{array}{ll}\ng_{t^2}\\big( \\frac{1}{s} B_{E_n, 0}  \\big) \\widehat\\otimes T^{(n)}(x, y),    &     \\mbox{ if } s\\in (0, 1];       \\\\\ng(0)\\cdot p^{(n)} \\widehat\\otimes T^{(n)}(x, y), &   \\mbox{ if } s=0,\n\\end{array}\n\\right.\n\\]\nfor all $t\\geq 1$, $x, y\\in \\zdn, \\nn$.\nThen $\\Psi^{(s)}$ is a homotopy between the asymptotic morphism $\\delta$ and the $*$-homomorphism\n$$\\sigma:\\,  \\cS\\widehat\\otimes \\cumpdxn \\rightarrow \\cumpdxnk $$\ndefined by the formula\n$$\\sigma \\Big( g\\widehat\\otimes \\big[\\big(  T^{(0)}, \\cdots, T^{(n)}, \\cdots    \\big)\\big] \\Big)\n\t=g(0)\\cdot  \\Big[\\Big(  p^{(0)}\\widehat\\otimes T^{(0)}, \\cdots,  p^{(n)}\\widehat\\otimes T^{(n)}, \\cdots    \\Big)\\Big] .$$\nIt is clear that $\\sigma$ induces identity on $K$-theory. Therefore, we conclude that\n$$\\alpha_*\\circ\\beta_*=\\gamma_*=\\delta_*=\\sigma_*=identity$$\non $K_*\\Big(  \\cumpdxn    \\Big)$.\nThe case for $(\\alpha_L)_*\\circ(\\beta_L)_*$ is similar.\n\\hfill{$\\Box$}\n\\par\n\\vskip 5mm\n\\par\n{\\bf Summary of Proof of Theorem 1.1.}  It follows from Theorem 7.11 and  Theorem 6.1, together with an argument of diagram chasing, that\nthe evaluation map\n\\begin{small}\n$$e_*: \\lim_{d\\to \\infty} K_*\\Big( \\culmpdxn \\Big) \\longrightarrow \\lim_{d\\to \\infty} K_*\\Big( \\cumpdxn \\Big) $$\n\\end{small}\nis an isomorphism for a sequence of finite metric spaces $(X_n)_\\nn$ with uniform bounded geometry such that the coarse disjoint union $X=\\bigsqcup_\\nn X_n$ admits a fibred coarse embedding\ninto Hilbert space. That is, Theorem 4.4 holds, which implies Proposition 4.6 and, subsequently, Theorem 1.1.\n\\hfill{$\\Box$}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nM dwarfs are popular targets for exoplanet research. First, radial velocity (RV) variations induced by the planets around M dwarfs are more significant than those around solar-like stars, making it possible to obtain precise mass measurement towards the terrestrial planet end of the mass distribution. Second, their small stellar radii lead to a large planet-to-star radius ratio, which favors transit detections and further photometric follow-up observations. Planets around M dwarfs are also attractive sources for atmospheric characterization through transmission or emission spectroscopy \\citep{Kempton2018,Batalha2018} as they yield a higher signal-to-noise ratio than equivalent systems with other types of hosts (e.g., LHS 3844b, \\citealt{Vanderspek2019,Kreidberg2019,Diamond2020}). Finally, due to the low stellar luminosity (typically $L<0.1\\ L_{\\odot}$), the habitable zone of M dwarfs is closer to the host star when compared with luminous stars (e.g., TOI-700d, \\citealt{Gilbert2020,Rodriguez2020}), which offers particular advantages to look for planets with potential biosignatures. \n\nOver the last two decades, more than a thousand transiting giant planets (defined as $M_{p}>0.3\\ M_{J}$) have been discovered thanks to successful ground-based surveys, including HATNet \\citep{Bakos2004}, SuperWASP \\citep{Pollacco2006}, KELT \\citep{Pepper2007,Pepper2012} and NGTS \\citep{Chazelas2012,Wheatley2018} as well as space transit missions like {\\it CoRoT} \\citep{Baglin2006}, {\\it Kepler} \\citep{Borucki2010} and {\\it K2} \\citep{Howell2014}. However, even though M dwarfs are the most abundant stellar population in our Milky Way \\citep{Henry2006}, only five giant planets have been confirmed to transit them: Kepler-45b \\citep{Johnson2012}, HATS-6b \\citep{Hartman2015}, NGTS-1b \\citep{Bayliss2018}, HATS-71b \\citep{Bakos2020} and TOI-1899b \\citep{Canas2020}. The deficiency of such systems is thought to be caused by the failed growth of a massive core to start runaway accretion before the gaseous protoplanetary disk dissipates due to the low surface density \\citep{Laughlin2004,Ida2005,Kennedy2008,Liu2020}. Indeed, previous statistical studies of the occurrence rates from {\\it Kepler}\\ show that planets with radii between $1R_{\\oplus}$ and $4R_{\\oplus}$ are frequent around low-mass stars \\citep{Dressing2013,Dressing2015,Hardegree-Ullman2019}. Some of these small planets are possibly the bare cores of failed gas giants. Nevertheless, microlensing surveys have found plenty of cold Jupiters ($a\\gtrsim 1$ AU) around M dwarfs (e.g., \\citealt{Zang2018}), which hints that outer giant planets are probably not rare. A handful of such cases have also been reported by long-term RV observations (e.g., GJ 876b, \\citealt{Marcy2001}; GJ 849b, \\citealt{Butler2006}; GJ 179b, \\citealt{Howard2010}; HIP 79431b, \\citealt{Apps2010}). Gravitational instability is speculated to be the alternative formation mechanism responsible for the surprising number of long-period gas giants around M dwarfs \\citep{Boss2000,Morales2019}. But it is still unclear how these short-period ($P\\lesssim 30$ d) transiting gas giants were formed, and whether they have migrated into their current orbits due to the lack of such systems. Therefore, establishing a well-characterized sample of this kind of planet is an important step to study their formation. Further Rossiter-McLaughlin \\citep{Rossiter1924,McLaughlin1924} or Doppler tomography \\citep{Marsh2001} measurements could reveal the obliquity of these systems, providing important clues about the dynamical history of the planets (e.g., \\citealt{Albrecht2012}).\n\nThe Transiting Exoplanet Survey Satellite ({\\it TESS}, \\citealt{Ricker2014,Ricker2015}), which has performed a two-year all-sky survey, offers exciting opportunities to increase the number of transiting giant planets around M dwarfs. Although {\\it TESS}\\ has already identified several such planet candidates, the intrinsic faintness of their hosts ($V\\gtrsim15$ mag) challenges most ground-based optical spectroscopic facilities to further conduct detailed RV follow-up observations. Some efforts have already been made to validate those planets through multi-color transit modeling and phase curve analysis (e.g., TOI-519b, \\citealt{Parviainen2021}). The new-generation near-infrared spectrograph SPIRou on the Canada-France-Hawaii-Telescope (CFHT) opens a new window to characterize planets around faint stars \\citep{Artigau2014,Donati2020}. It was designed to perform high-precision velocimetry and spectropolarimetry studies. Early observations from SPIRou have shown that it can reach $2\\sim10$ m\/s precision for stars with $H<10$ mag \\citep{Moutou2020,Klein2021,Artigau2021}. Although simulations predict that SPIRou could reach $<2$ m\/s RV precision for inactive M dwarfs with $J<10$ mag (see Figure 5 in \\citealt{Cloutier2018}), precision for faint stars has yet to be determined observationally.\n\nHere we report the discovery of a new transiting giant planet around an M-dwarf star {TOI-530}. We present RV measurements from SPIRou that allow us to obtain a precise companion mass and thus confirm its planetary nature. The rest of the paper is organized as follows. We describe all space and ground-based observational data used in this work in Section \\ref{observations}. Section \\ref{stellar_properties} presents the stellar properties. In Section \\ref{analysis}, we show our analysis of the light curves and RV data. We discuss the prospects of future atmospheric characterization of {TOI-530} b and its potential formation channel in Section \\ref{discussion}. We conclude with our findings in Section \\ref{conclusion}.\n\n\n\n\n\n\n\n\\section{Observations}\\label{observations}\n\\subsection{{\\it TESS}\\ photometry}\n{TOI-530}\\ was observed by {\\it TESS}\\ on its Camera 1 with the two-minute cadence mode in Sector 6 during the primary mission and Sector 33 during the extended mission. The current data span from 2018 December 15th to 2021 January 13th, consisting of 14830 and 17547 measurements, respectively. The target will be revisited in Sectors 44-45 between 2021 October 12th and 2021 December 2nd. Figure \\ref{fov} shows the POSS2 and {\\it TESS}\\ images centered on {TOI-530}.\n\nThe photometric data from Sector 6 were initially reduced by the Science Processing Operations Center (SPOC; \\citealt{Jenkins2016}) pipeline, developed based on the {\\it Kepler}\\ mission's science pipeline. The simple aperture photometry (SAP) flux time series was corrected for instrumental and systematic effects, and for crowding and dilution with the Presearch Data Conditioning (PDC; \\citealt{Stumpe2012,Smith2012,Stumpe2014}) module. Transit signals were searched using the Transiting Planet Search \\citep[TPS;][]{Jenkins2002,Jenkins2017} algorithm on 17 February 2019, yielding a strong transit signal at a period of $\\sim$6.39 days and a transit duration of $\\sim$2.5 hours. The transit signature and pixel data passed all the  validation tests \\citep{Twicken2018,Li2019,Guerrero2021}, including locating the source of the transit signature to within 1 - 3 arcsec of the target star, and no further transiting planet signatures were identified in a search of the residual light curve. The vetting results were reviewed by the TESS Science Office (TSO) and issued an alert for {TOI-530} b as a planet candidate on 28 March 2019.\n\nWe downloaded the Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) light curve from the Mikulski Archive for Space Telescopes\\ (MAST\\footnote{\\url{http:\/\/archive.stsci.edu\/tess\/}}) using the \\code{lightkurve} package \\citep{lightkurvecollaboration,lightkurve}. Combining the datasets of two sectors, we conducted an independent transit search by utilizing the Transit Least Squares (TLS; \\citealt{Hippke2019}) algorithm, which is an advanced version of Box Least Square (BLS; \\citealt{Kovacs2002}), after smoothing the full light curve with a median filter. We recovered the 6.387 d transits with a signal detection efficiency (SDE) of $\\sim50$. After subtracting the TLS model from the {\\it TESS}\\ data, we did not find any other significant transit signals existing in the light curve. We detrended the raw {\\it TESS}\\ light curve by fitting a Gaussian Process (GP) model with a Mat\\'{e}rn-3\/2 kernel using the \\code{celerite} package \\citep{Foreman2017}, after masking out all in-transit data. We show the reprocessed light curve in Figure \\ref{transit_detrend}. \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.94\\columnwidth]{fov.pdf}\n\\includegraphics[width=1.06\\columnwidth]{TPF_Gaia_TIC387690507_S33.pdf}\n\\caption{{\\it Left panel}: The POSS2 blue image of {TOI-530}\\ taken in 1996. The center red dot is the target star in this image and the cyan circle shows its current position, which rules out the unassociated distant eclipsing binary scenario. {\\it Right panel}: Target pixel file (TPF) of {TOI-530}\\ in {\\it TESS}\\ Sector 33 (created with \\code{tpfplotter}, \\citealt{Aller2020}). Different sizes of red circles represent different magnitudes in contrast with {TOI-530}\\ ($\\Delta m$). The aperture used to extract the photometry is overplotted with a red-square region.} \n\\label{fov}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{TOI530_total_lc.png}\n\\caption{{\\it Top panels}: The original {\\it TESS}\\ SAP light curves of {TOI-530}\\ from Sector 6 and 33. {\\it Middle panels}: The PDCSAP light curves of {TOI-530}\\ along with the best-fit GP model shown as red solid lines. {\\it Bottom panels}: The detrended PDCSAP light curves. The transits of {TOI-530} b are marked in blue ticks.} \n\\label{transit_detrend}\n\\end{figure*}\n\n\\subsection{Ground-Based photometry}\\label{gbp}\nWe collected a series of ground-based observations of {TOI-530}, as part of the TESS Follow-up Observing Program (TFOP\\footnote{\\url{https:\/\/tess.mit.edu\/followup}}), to (1) confirm the transit signal on target and rule out nearby eclipsing binary scenario; (2) examine the chromaticity; and (3) refine the transit ephemeris and radius measurement. These observations were scheduled with the help of the {\\it TESS}\\ Transit Finder (\\code{TTF}), which is a customized version of the \\code{Tapir} software package \\citep{Jensen2013}. Due to the observational constraints, unfortunately, we only covered the egress of the event. We summarize the details in Table \\ref{po} and describe individual observations below. We show the raw and detrended ground-based light curves in Figure \\ref{ground_transit_detrend} (see Section \\ref{joint_fit_TESS_ground}).\n\n\\subsubsection{El Sauce}\\label{el}\nAn egress was observed on UT 2019 November 21 in the $R_{c}$ band using the Evans telescope (0.36 m) at the El Sauce Observatory, Chile. The STT 1603 camera has a pixel scale of $\\rm 1.47''$ per pixel. We acquired a total of 65 images over 205 minutes. Photometric analysis was carried out using \\code{AstroImageJ} \\citep{Collins2017} with an uncontaminated aperture of $5.88''$. We excluded all nearby stars within $1'$ as the source causing the {\\it TESS}\\ signal with brightness difference down to $ \\Delta T \\sim 4.1$ mag, and confirmed the signal on target.\n\n\\subsubsection{MuSCAT2}\\label{muscat2}\nWe observed an egress of TOI-530b on the night of UT 2020 January 4 with the multicolor imager MuSCAT2 \\citep{2019JATIS...5a5001N} mounted on the 1.52 m Telescopio Carlos S\\'{a}nchez at Teide Observatory, Tenerife, Spain. MuSCAT2 has a field of view of $7.4' \\times 7.4'$ with a pixel scale of $0.44''$ pixel$^{-1}$ and is able to obtain simultaneous photometry in four bands ($g$, $r$, $i$, and $z_s$). The observations were made with the telescope in optimal focus and the exposure times for each band were 45 s for $g$, 30 s for $r$ and $i$, and 20 s for $z_s$ band. The data were calibrated using standard procedures (dark and flat calibration). Aperture photometry and transit light curve fit was performed using MuSCAT2 pipeline (\\citealp{2020A&A...633A..28P}); the pipeline finds the aperture that minimizes the photometric dispersion while fitting a transit model including instrumental systematic effects present in the time series.  \n\n\\subsubsection{MuSCAT}\\label{muscat}\nWe observed an egress of TOI-530b on UT 2020 March 2 in $g$, $r$, and $z_s$ bands, using the multiband imager MuSCAT \\citep{2015JATIS...1d5001N} mounted on the 188~cm telescope of National Astronomical Observatory of Japan at the Okayama Astro-Complex, Japan. MuSCAT has three CCD cameras, each having a pixel scale of $0.361''$ pixel$^{-1}$ and a field of view of $6.1' \\times 6.1$. We acquired 321, 268, and 474 images with exposure times of 30, 30, and 20~s in $g$, $r$, and $z_s$ bands, respectively. The data were dark-subtracted and flat-field corrected in a standard manner. Aperture photometry was then performed on the reduced images using a custom pipeline \\citep{2011PASJ...63..287F}. The radius of the photometric aperture was chosen to be 18 pixels ($6.5''$) for all bands so that the photometric dispersion was minimized.\n\n\\begin{table*}\n    \\centering\n    \\caption{Ground-based photometric follow-up observations for {TOI-530}}\n    \\begin{tabular}{ccccccccc}\n        \\hline\\hline\n        Telescope &Camera &Filter &Pixel Scale & Aperture Size (pixel) &Coverage      &Date &Duration (minutes) &Total exposures  \\\\\\hline\n        El Sauce (0.36 m) &STT 1603 &$R_{c}$ &1.47 &4 &Egress &2019 November 21 &183 &59\\\\\\hline\n        TCS (1.52 m) &MuSCAT2 &$g$ &0.44 &9.8 &Egress &\t2020 January 4 &237 &305\\\\\n        TCS (1.52 m) &MuSCAT2 &$r$ &0.44 &9.8 &Egress &\t2020 January 4 &237 &456\\\\\n        TCS (1.52 m) &MuSCAT2 &$i$ &0.44 &9.8 &Egress &\t2020 January 4 &237 &456\\\\\n        TCS (1.52 m) &MuSCAT2 &$z_{s}$ &0.44 &9.8 &Egress &\t2020 January 4 &237 &238\\\\\\hline\n        NAOJ (1.88 m) &MuSCAT &$g$ &0.36 &18 &Egress &2020 March 2 &177 &321\\\\\n        NAOJ (1.88 m) &MuSCAT &$r$ &0.36 &18 &Egress &2020 March 2 &177 &268\\\\\n        NAOJ (1.88 m) &MuSCAT &$z_{s}$ &0.36 &18 &Egress &2020 March 2 &177 &474\\\\\n         \\hline\n    \\end{tabular}\n    \\label{po}\n\\end{table*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{ground_only.pdf}\n\\caption{Ground-based light curves for all available instrument. The blue dots are the raw data while the black solid line represents the best fit GP+transit model. The black dots are results after subtracting the GP model (i.e., detrended data). We use these detrended light curves in the final joint-fit (see Section \\ref{joint}).} \n\\label{ground_transit_detrend}\n\\end{figure*}\n\n\\subsection{Spectroscopic Observations}\n\\subsubsection{IRTF}\nWe observed TOI-530 on UT 2019 April 23 with the uSpeX spectrograph \\citep{2003PASP..115..362R, 2004SPIE.5492.1498R} on the 3-m NASA Infrared Telescope Facility (IRTF). Our data was collected in the SXD mode using the $0\\farcs3 \\times 15\\arcsec$ slit and covers a wavelength range of $0.7-2.55$ $\\mu$m. The data was reduced using the \\texttt{Spextool} pipeline \\citep{2004PASP..116..362C}. After reducing, we RV-correct our spectrum using \\texttt{tellrv} \\citep{2014AJ....147...20N}, with which we estimate a systemic radial velocity of $-26 \\pm 5$ km\/s. By comparing our spectrum to those provided by the IRTF library \\citep{Rayner2009}, we determine that our spectrum best matches that of a star of spectral type M0.5V. Lastly, we calculate the metallicity of TOI-530 following the relations defined in \\cite{Mann2013} for cool dwarfs with spectral types between K7 and M5. In performing this calculation, we opted to only use the $Ks$-band spectrum, as \\cite{2019AJ....158...87D} found $Ks$-band spectra to produce more reliable metallicities and suffer less telluric contamination than $H$-band spectra. Our analysis yield metallicities of [Fe\/H] = $0.376 \\pm 0.095$ and [M\/H] = $0.218 \\pm 0.092$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{SpeX_spectrum.pdf}\n\\caption{Renormalized SpeX spectrum of {TOI-530}\\ (red line) and the comparison spectrum (blue line) taken from the IRTF library \\citep{Rayner2009}. The strong atomic features are marked based on the results from \\citet{Cushing2005}. The NIR spectrum of {TOI-530}\\ corresponds to a spectral type of M0.5V.} \n\\label{IRTF}\n\\end{figure}\n\n\\subsubsection{CFHT\/SPIRou}\nWe monitored {TOI-530}\\ over 5 epochs between UT 2020 September 26 and UT 2020 October 5 using SPIRou (standing for SpectroPolarim\\`etre InfraROUge), which is a new-generation high resolution ($64,000$) fiber-fed spectrograph with polarimetric and precision velocimetry capacities, installed at CFHT in 2018 \\citep{Artigau2014,Donati2018}. It has a large bandwidth (from 0.95 to 2.5 $\\mu$m) allowing the detection of several stellar lines in a single shot thus enhancing the precision of the measurement of the stellar radial velocity. For each night, we obtained three sequences, with 975s exposure time for each. The spectroscopic data were reduced using the standard data reduction pipeline (APERO, Cook et al. in prep), which performs the data calibration and corrects the telluric and night-sky emission \\citep{Artigau2021}. For the Night-sky emission, it is corrected using a principal component analysis (PCA) model of OH emission constructed from a library of high-SNR sky observations \\citep{Artigau2014PCA}. The telluric absorption is corrected using a PCA-based approach on residuals after fitting for a basic atmospheric transmission model (TAPAS, \\citealt{Bertaux2014}).\n\nWe extracted the RVs of TOI-530 from the telluric-subtracted SPIRou spectra using \\code{wobble} \\citep{Bedell2019}. Briefly, \\code{wobble} constructs a linear model to infer the stellar and time-varying telluric spectra without requiring any prior knowledge on them, while solving for the RV at each epoch. \n\nWe only used the orders 29--37 (around 1490--1800~nm, all in the H band) to extract RVs for the following reasons. We dropped the orders 13--17, 23--28, 39--43, and 47--49 (around 1130--1230~nm, 1330--1490~nm, 1850--2010~nm, 2326--2510~nm, respectively), because the telluric absorption lines are too heavy. These orders with heavy telluric absorption are basically the wavelength regions in between the photometric bands (Y, J, H, K, and at the end of the K band). Furthermore, we dropped the orders 1--12, 18--22, 44--46 (around 965--1140~nm, 1225--1340~nm, and 2132--2290~nm, respectively), because their signal-to-noise ratios (SNRs) are too low (lower than $\\sim$30). The low SNRs caused poor corrections of the telluric lines by the SPIRou pipeline, which manifests as significant residuals of telluric emission\/absorption, as well as some abnormal features caused by improper telluric subtraction. In addition, the authors of \\code{wobble} also cautioned regarding applying the code to spectral data with SNR less than 50 \\citep[][e.g., in their example using the Barnard's star's data]{Bedell2019}. We extracted RVs from the low-SNR orders and saw little RV variations in these orders, which we believe is due to the fact that \\code{wobble} struggles to recover any RV information from these low-SNR spectra with heavy telluric residuals. \n\nTo pre-process the spectra, we first dropped 500 pixels at both edges of each order and masked out the occasional residual emission lines not fully subtracted by the SPIRou pipeline as follows: We calculated the 80th percentile of the flux (denoted as $q$) in any given order, labeled all pixels with flux larger than $3 \\times q$ as the emission-line pixels, and masked out 5 pixels in total centered around them. Then we scaled the blaze function offered by the SPIRou pipeline to the flux level of the observed spectrum and calculated the flux minus the blaze function at each pixel. If the difference is more than half of the maximum of the blaze function of corresponding order, such pixels were considered as emission-line pixels, and 5 pixels around them were masked out. Then, we used the scaled blaze function to continuum-normalize the spectra. \n\nNext, we passed the natural log of the wavelength, the natural log of the flux, the estimated inverse variance of the flux (set as photon counts at each pixel, assuming Poisson noise on the flux), the time of the observations, the BERVs and the airmass values to \\code{wobble}. We let \\code{wobble} only infer the stellar the spectra to extract RVs because the SPIRou pipeline already divided out telluric absorption. When \\code{wobble} infers the stellar spectrum, it needs optimized L1 and L2 regularization parameters for each orders. For simplicity, we set these regularization parameters to the default values in the \\code{wobble} code, which are the same for all orders.\n\nTo validate our work, we divided each order into the left part and the right part so that the total photon counts of each part are equal. Then we used the same method to get RVs from each part respectively. Comparing the RVs from the left and the right parts of each order, we found that the differences are on par with the RV differences between the three observations taken on the same night (i.e., the intra-night RV variation as derived using the full order). The RV signals are basically consistent cross the nine orders we analyzed. This suggests that our results are unlikely to arise from random noise but instead are of real astrophysical origin. However, we found that the differences between the RVs reported from the left or the right parts of each order (typically 10--30 m\/s) are significantly larger than the RV error bars reported by \\code{wobble} (typically 1.6--1.9 m\/s). Therefore, we calculated the standard deviation of the six RVs from the left and the right of each night (two RVs per observation, 3 observations per night) and used them as the more realistic estimates of the uncertainties of the RVs, which are what we present in Table~\\ref{spirourv}.\n\n\\begin{comment}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{SPIRou_RV_three_part.png}\n\\caption{RVs from the left part and the right part compared with the RVs from the whole spectra. To be clear, the BJDs of the left part and the right part are slightly changed. \\todo{Tianjun: This is an important result. Let's make it a two-column plot. The x axis needs some updates (BJD-2457000). Q: there are no RV error bars for each data point? Mention that error bars are too small and unrealistic so did not plot.}} \n\\label{SPIRou_validation}\n\\end{figure*}\n\\end{comment}\n\n\n\\begin{table}\n     \\centering\n     \\caption{SPIRou RV measurements of {TOI-530}. Each observation took an exposure time of 975s. The RV offset here is arbitrary.}\n     \\begin{tabular}{ccc}\n         \\hline\\hline\n         BJD$_\\mathrm{TDB}$       &RV\\ (m~s$^{-1}$) &$\\sigma_{\\rm RV}$\\ (m~s$^{-1}$) \\\\\\hline\n         2459119.08206 \t&29356.43 \t&20.14 \\\\ \n         2459119.09361 \t&29372.09 \t&20.14 \\\\ \n         2459119.10515 \t&29389.86 \t&20.14 \\\\\n         2459120.06581 \t&29486.04 \t&12.55 \\\\\n         2459120.07736 \t&29497.80 \t&12.55 \\\\\n         2459120.08897 \t&29500.96 \t&12.55 \\\\\n         2459123.06563 \t&29342.45 \t&14.13 \\\\\n         2459123.07718 \t&29358.37 \t&14.13 \\\\\n         2459123.08873 \t&29364.25 \t&14.13 \\\\\n         2459127.07895 \t&29461.84 \t&20.86 \\\\\n         2459127.09089 \t&29495.09 \t&20.86 \\\\\n         2459127.10289 \t&29503.15 \t&20.86 \\\\\n         2459128.06393 \t&29353.01 \t&31.13 \\\\\n         2459128.07548 \t&29374.36 \t&31.13 \\\\\n         2459128.08703 \t&29445.57 \t&31.13 \\\\\n          \\hline\n     \\end{tabular}\n    \n    \n    \n     \\label{spirourv}\n\\end{table}\n\n\\subsection{High Angular Resolution Imaging}\nIf an exoplanet host star has a spatially close companion, that companion (bound or line of sight) can create a false-positive transit signal if it is, for example, an eclipsing binary (EB). For small stars and large planets, this is an especially important check to make, due to the paucity of giant planets orbiting M stars. ``Third-light\" flux from the close companion star can lead to an underestimated planetary radius if not accounted for in the transit model \\citep{Ciardi2015} and cause non-detections of small planets residing with the same exoplanetary system \\citep{Lester2021}. Additionally, the discovery of close, bound companion stars, which exist in nearly one-half of FGK type stars \\citep{Matson2018} and less so for M class stars, provides crucial information toward our understanding of exoplanetary formation, dynamics and evolution \\citep{Howell2021}. Thus, to search for close-in bound companions unresolved in TESS or other ground-based follow-up observations, we obtained high-resolution imaging observations of TOI-530.\n\n\\subsubsection{Keck\/NIRC2 Adaptive Optics Imaging}\nWe observed {TOI-530}\\ with infrared high-resolution adaptive optics (AO) imaging at Keck Observatory \\citep{Ciardi2015,Schlieder2021} on UT 2019 April 7. The observations were made with the NIRC2 instrument on Keck-II behind the natural guide star AO system. The standard 3-point dither pattern was used to avoid the left lower quadrant of the detector which is typically noisier than the other three quadrants. The dither pattern step size\nwas 3$''$ and it was repeated twice, with each dither offset from the previous one by 0.5$''$.\n\nThe observations were taken in the broad-band K ($\\lambda_{o}=2.1956$ $\\mu$m; $\\Delta \\lambda=0.336$ $\\mu$m) with an integration time of 4 s per frame for a total on-source integration time of 36 s. The camera was in the narrow-angle mode with a full field of view of 10$''$ and a pixel scale of approximately 0.009942$''$ per pixel. The Keck AO observations show no additional stellar companions were detected to within a resolution $\\sim0.056''$ FWHM. The sensitivities of the final combined AO image were determined by injecting simulated sources azimuthally around the primary target every $20^{\\circ}$ at radial separations of integer multiples of the FWHM of the central source \\citep{Furlan2017}. The brightness of each injected source was scaled until standard aperture photometry detected it with $5\\sigma$ significance. The resulting brightness of the injected sources relative to the target {TOI-530}\\ was regarded as the contrast limits at that injection location. The final $5\\sigma$ limit at each separation was determined from the average of all of the determined limits at that separation while the uncertainty was given by the RMS dispersion of the results for different azimuthal slices at a given radial distance. We show the 2$\\mu$m sensitivity curve in the left panel of Figure \\ref{imaging} along with an inset image zoomed to primary target, which shows no other companion stars. \n\n\\subsubsection{Gemini-North Speckle Imaging}\nTOI-530 was observed on 2020 February 17 UT using the 'Alopeke speckle instrument on the Gemini North 8-m telescope\\footnote {\\url{https:\/\/www.gemini.edu\/sciops\/instruments\/alopeke-zorro\/}}.  'Alopeke provides simultaneous speckle imaging in two bands (562nm and 832 nm) with output data products including a reconstructed image with robust contrast limits on companion detections (e.g., \\citealt{Howell2016}). Ten sets of $1000\\times0.06$ sec exposures were collected and subjected to Fourier analysis in our standard reduction pipeline \\citep{Howell2011}. The right panel of Figure \\ref{imaging} shows our final contrast curves and the 832 nm reconstructed speckle image. We find that TOI-530 is a single star with no companion brighter than 5-6 magnitudes below that of the target star (earlier than $\\sim$ M4.5V) from the diffraction limit (20 mas) out to 1.2$''$. At the distance of TOI-530 (d=148 pc) these angular limits correspond to spatial limits of 3 to 178 au.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{high_resolution_plot.pdf}\n\\caption{{\\it Left panel}: NIRC2 AO image (inset) and $\\rm K_{s}$-band contrast curve for {TOI-530}. The black line is the $5\\sigma$ sensitivity limit. The shaded purple region represents the azimuthal dispersion ($1\\sigma$) of the contrast determinations. {\\it Right panel}: The $5\\sigma$ 'Alopeke speckle imaging contrast curves in both filters as a function of the angular separation out to 1.2 arcsec, the end of speckle coherence. The inset shows the reconstructed 832 nm image with a 1 arcsec scale bar. The star, TOI-530, was found to have no close companions to within the contrast levels achieved. } \n\\label{imaging}\n\\end{figure*}\n\n\n\\section{Stellar Characterization}\\label{stellar_properties}\n\nWe first use 2MASS observed $m_{K}$ and the parallax from {\\it Gaia}\\ EDR3 to calculate the absolute magnitude, of which we obtain $M_{K}= 5.42 \\pm 0.13$ mag. We then estimate the stellar radius following the polynomial relation between $R_{\\ast}$ and $M_{K}$ derived by \\cite{Mann2015}, and we find $R_{\\ast}=0.54\\pm0.02\\ R_{\\odot}$, assuming a typical uncertainty of 3\\% (see Table 1 in \\citealt{Mann2015}). For comparison, we also estimate the stellar radius $R_{\\ast}=0.55\\pm0.03 R_{\\odot}$ based on the angular diameter relation in \\cite{Boyajian2014}, consistent with our previous estimate within $1\\sigma$. \n\nUsing the empirical polynomial relation between bolometric correction ${\\rm BC}_{K}$ and $V-J$ in \\cite{Mann2015}, we find ${\\rm BC}_{K}$ to be $2.60\\pm0.13$ mag. Thus, we derive a bolometric magnitude $M_{\\rm bol}=8.02\\pm 0.13$ mag, leading to a bolometric luminosity of $L_{\\ast}=0.049\\pm0.005\\ L_{\\odot}$. To estimate the stellar effective temperature of {TOI-530}, we first take use of the Stefan-Boltzmann law. Coupled with the aforementioned stellar radius and bolometric luminosity we derived, we get $T_{\\rm eff}=3666\\pm146$ K. As an independent check, we then obtain $T_{\\rm eff}$ following the empirical relation reported by \\cite{Mann2015} and we find $T_{\\rm eff}=3650\\pm100$ K. Both estimations agree well with the result $T_{\\rm eff}=3663\\pm124$ K from \\cite{Pecaut2013}. \n\nFinally, we evaluate that {TOI-530}\\ has a mass of $M_{\\ast}=0.53\\pm0.01\\ M_{\\odot}$ using Equation 2 in \\cite{Mann2019} according to the $M_{\\ast}$-$M_{K}$ relation. This is consistent with the value $M_\\ast = 0.52 \\pm 0.03\\ M_\\odot$ given by the eclipsing-binary based empirical relation of \\citet{Torres2010}.\n\nAs an independent check, we carry out an analysis of the broadband Spectral Energy Distribution (SED) together with the {\\it Gaia\\\/} EDR3 parallax in order to determine an independent, empirical measurement of the stellar radius, following the procedures described in \\citet{Stassun2016}, \\citet{Stassun2017}, and \\citet{Stassun2018}. We pull the $JHK_S$ magnitudes from {\\it 2MASS} \\citep{Cutri2003,skrutskie2006}, the W1--W3 magnitudes from {\\it WISE} \\citep{wright2010},  the $grizy$ magnitudes from Pan-STARRS \\citep{Magnier2013}, and three {\\it Gaia}\\ magnitudes $G, G_{\\rm BP}, G_{\\rm RP}$ \\citep{GaiaEDR3}. Together, the available photometry spans the full stellar SED over the wavelength range 0.4\\,--\\,10~$\\mu$m (see Figure~\\ref{fig:sed}). \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=10 5 20 15,clip]{TOI530_sed.pdf}\n    \\caption{The best SED fit for {TOI-530}. Red symbols represent the observed photometric measurements, where the horizontal bars represent the effective width of the passband. Blue symbols are the model fluxes from the best-fit NextGen atmosphere model (black). \n\\label{fig:sed}}\n\\end{figure}\n\nWe perform a fit using NextGen stellar atmosphere models, with the $T_{\\rm eff}$, $\\log g$, and [Fe\/H] taken from the spectroscopic analysis. The remaining parameter is the extinction ($A_V$), which we limit to the full line-of-sight extinction from the dust maps of \\citet{schlegel1998}. The resulting fit is shown in Figure.~\\ref{fig:sed} with a reduced $\\chi^2$ of 1.6 and best-fit extinction of $A_V = 0.00^{+0.03}_{-0.00}$. Integrating the model SED gives the bolometric flux at Earth of \\mbox{$F_{\\rm bol} =  7.009 \\pm 0.081 \\times 10^{-11}$ erg~s$^{-1}$~cm$^{-2}$}. Taking the $F_{\\rm bol}$ and $T_{\\rm eff}$ together with the {\\it Gaia\\\/} parallax, with no adjustment for systematic parallax offset \\citep[see, e.g.,][]{Stassun2021}, gives the stellar radius as $R_\\ast = 0.547 \\pm 0.030\\ R_\\odot$. \n\nCombining all the results above, we adopt the weighted-mean values of effective temperature $T_{\\rm eff}$, stellar radius $R_{\\ast}$ and stellar mass $M_{\\ast}$ as listed in Table \\ref{starparam}.\n\nTo identify the Galactic population membership of {TOI-530}, we first calculate the three-dimensional space motion with respect to the LSR based on \\cite{Johnson1987}. We adopt the astrometric values ($\\varpi$, $\\mu_{\\alpha}$, $\\mu_{\\delta}$) from {\\it Gaia}\\ EDR3 and the spectroscopically determined systemic RV from the SpeX spectrum, and we find $U_{\\rm LSR}=48.99\\pm4.59$ km s$^{-1}$, $V_{\\rm LSR}=-20.10\\pm1.92$ km s$^{-1}$, $W_{\\rm LSR}=-6.73\\pm0.56$ km s$^{-1}$. Following the procedure described in \\cite{Bensby2003}, we compute the relative probability $P_{\\rm thick}\/P_{\\rm thin}=0.02$ of {TOI-530}\\ to be in the thick and thin disks by taking use of the recent kinematic values from \\cite{Bensby2014}, indicating that {TOI-530}\\ belongs to the thin-disk population. We further integrate the stellar orbit with the ``MWPotential2014'' Galactic potential using \\code{galpy} \\citep{Bovy2015} following \\cite{Gan2020}, and we estimate that the maximal height $Z_{\\rm max}$ of {TOI-530}\\ above the Galactic plane is about $109$ pc, which agrees with our thin-disk conclusion. \n\nWe finally perform a frequency analysis on the {\\it TESS}\\ PDCSAP photometry after masking the known in-transit data using the generalized Lomb-Scargle periodogram \\citep{Zechmeister2009} to look for stellar activity signals. We find a peak at around $9.4$ d in the {\\it TESS}\\ Sector 6 data, which may be attributed to stellar rotation. However, this periodic signal is not significant in the generalized Lomb-Scargle periodogram of the {\\it TESS}\\ photometry taken in the extended mission. We further analyze the ground-based long-term photometry from the Zwicky Transient Facility (ZTF; \\citealt{Masci2019}). ZTF took a total of 273 exposures for {TOI-530}, which spanned 1036 d. We clip outliers above the $3\\sigma$ level and 242 measurements are left. However, we find that the 9.4 d signal does not show up in the corresponding generalized Lomb-Scargle periodogram, either. Additionally, \\citealt{Newton2018} shows a typical rotational period of $\\sim40$ d for a $0.5\\ M_{\\odot}$ star. We thus conclude that the $9.4$ d signal is probably not associated with stellar rotation. Future {\\it TESS}\\ data to be obtained will allow better identification of the correct rotation period of this target. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{TESS_GLS_updated.pdf}\n\\caption{Generalized Lomb-Scargle periodograms of the TESS PDCSAP photometry from two sectors. The theoretical FAP levels of 10, 1, and 0.1 percent are marked as horizontal solid, dashed, and dot\u2013dashed lines. The vertical red lines mark the maximum peaks of the periodograms.} \n\\label{TESS_GLS}\n\\end{figure}\n\n\n\n\\begin{table}\\label{stellarinfor}\n   \n    \\caption{Basic information of {TOI-530}}\n    \\begin{tabular}{lll}\n        \\hline\\hline\n        Parameter       &Value       \\\\\\hline\n        \\it{Main identifiers}                    \\\\\n        \n         TOI                     &$530$         \\\\\n         TIC                     &$387690507$   \\\\\n         {\\it Gaia}\\ ID            &$3353218995355814656$ \\\\\n         \\it{Equatorial Coordinates} \\\\\n         $\\rm R.A.\\ (J2015.5)$                    &06:53:39.08 \\\\\n         $\\rm DEC.\\ (J2015.5)$                    &12:52:53.68    \\\\\n         \\it{Photometric properties}\\\\\n         ${\\it TESS}$\\ (mag)           &$13.5287\\pm0.0076$   &$\\rm TIC\\ V8^{[1]}$     \\\\\n         ${\\it Gaia}$\\ (mag)           &$14.6217\\pm0.0006$   &{\\it Gaia}\\ EDR3$^{[2]}$   \\\\\n         {\\it Gaia}\\ BP\\ (mag)           &$15.814\\pm0.004$   &{\\it Gaia}\\ EDR3   \\\\\n         {\\it Gaia}\\ RP\\ (mag)           &$13.538\\pm0.002$   &{\\it Gaia}\\ EDR3   \\\\\n         $B$\\ (mag)                    &$16.708\\pm0.044$         &APASS \\\\\n         $V$\\ (mag)                    &$15.403\\pm0.136$          &APASS \\\\\n         $J$\\ (mag)                    &$12.112\\pm0.023$   &2MASS \\\\\n         $H$\\ (mag)                    &$11.468\\pm0.030$   &2MASS \\\\\n         $K$\\ (mag)                    &$11.238\\pm0.020$    &2MASS \\\\\n         \\wise1 (mag)                   &$11.124\\pm0.023$   &{\\it WISE} \\\\\n         \\wise2 (mag)                   &$11.087\\pm0.020$   &{\\it WISE} \\\\\n         \\wise3 (mag)                   &$10.907\\pm0.139$   &{\\it WISE} \\\\\n         \\wise4 (mag)                   &$8.735\\pm0.429$   &{\\it WISE} \\\\\n         \\it{Astrometric properties}\\\\\n         $\\varpi$ (mas)              &$6.77\\pm0.02$  &{\\it Gaia}\\ EDR3  \\\\\n         $\\mu_{\\rm \\alpha}\\ ({\\rm mas~yr^{-1}})$     &$13.62\\pm0.03$   &{\\it Gaia}\\ EDR3   \\\\\n         $\\mu_{\\rm \\delta}\\ ({\\rm mas~yr^{-1}})$     &$-62.52\\pm0.02$   &{\\it Gaia}\\ EDR3  \\\\\n         RV\\ (km~s$^{-1}$)                          &$-25.93\\pm2.00$ &This work  \\\\\n         \\it{Derived parameters} \\\\\n         Distance (pc)                &$147.7\\pm0.6$  &This work     \\\\\n         $U_{\\rm LSR}$ (km~s$^{-1}$)       &$48.99\\pm4.59$     &This work\\\\\n         $V_{\\rm LSR}$ (km~s$^{-1}$)       &$-20.10\\pm1.92$     &This work\\\\\n         $W_{\\rm LSR}$ (km~s$^{-1}$)       &$-6.73\\pm0.56$     &This work\\\\\n         $M_{\\ast}\\ (M_{\\odot})$ &$0.53\\pm 0.02$ &This work       \\\\\n         $R_{\\ast}\\ (R_{\\odot})$ &$0.54\\pm 0.03$ &This work       \\\\\n         $\\rho_\\ast\\ ({\\rm g~cm^{-3}})$ &$4.74\\pm 1.11$ &This work \\\\\n         $\\log g_{\\ast}\\ ({\\rm cgs})$       &$4.70\\pm 0.03$  &This work        \\\\\n         $L_{\\ast}\\ (L_{\\odot})$ &$0.049\\pm0.005$  &This work    \\\\\n         $T_{\\rm eff}\\ ({\\rm K})$           &$3659\\pm 120$  &This work       \\\\\n         $\\rm [Fe\/H]$  &$0.376\\pm 0.095$ &This work \\\\\n         $\\rm [M\/H]$  &$0.218\\pm0.092$ &This work\\\\\n        \n         \\hline\\hline \n    \\end{tabular}\n    \\begin{tablenotes}\n    \\item[1]  [1]\\ \\cite{Stassun2017tic,Stassun2019tic} \n    \\item[2]  [2]\\ \\cite{GaiaEDR3}\n    \\end{tablenotes}\n    \\label{starparam}\n\\end{table}\n\n\n\\section{Analysis and results}\\label{analysis}\n\\subsection{Photometric Analysis}\\label{transit}\n\\subsubsection{{\\it TESS}\\ only}\\label{tess_only}\nWe first model the detrended {\\it TESS}\\ only photometry by utilizing the \\code{juliet} package \\citep{juliet}, which employs \\code{batman} to build the transit model \\citep{Kreidberg2015}. Dynamic nested sampling is applied in \\code{juliet} to determine the posterior estimates of system parameters using the publicly available package \\code{dynesty} \\citep{Higson2019,Speagle2019}.\n\nWe set uninformative uniform priors on both the transit epoch ($T_{0}$) and the orbital period ($P_{b}$), centered on the optimized value obtained from the TLS analysis. Following the approach described in \\cite{Espinoza2018}, instead of directly fitting for the radius ratio ($p=R_{p}\/R_{\\ast}$) and the impact parameter ($b=a\/R_{\\ast}\\cos i$), we apply the new parametrizations $r_{1}$ and $r_{2}$ to sample points, for which we impose uniform priors between 0 and 1. This new parametrization allows us to only sample physically meaningful values of a transiting system with $0 < b < 1 + p$, which reduces the computational cost. We adopt a quadratic limb-darkening law for the {\\it TESS}\\ photometry, where we place a uniform prior on both coefficients ($q_{1}$ and $q_{2}$, \\citealt{Kipping2013}). Since photometric-only data weakly constrain the orbital eccentricity, we fix $e$ at zero and include a non-informative log-uniform prior on stellar density. We fit an extra flux jitter term to account for additional systematics. As the {\\it TESS}\\ PDCSAP light curve has already been corrected for the light dilution, we fix the dilution factors $D$ to 1. Table \\ref{tess_only_fit_priors} summarizes the prior settings we adopt as well as the best-fit value of each parameter. We then rerun the photometry-only fit with free $e$ and $w$ to examine potential evidences of eccentricity by comparing the Bayesian model log-evidence ($\\ln Z$) difference between the circular and eccentric orbit models calculated using the \\code{dynesty} package. Generally, we consider a model is strongly favored than another if $\\Delta \\ln Z>5$ \\citep{Trotta2008}. We find that the circular orbit model is slightly preferred with a Bayesian evidence improvement of $\\Delta \\ln Z=\\ln Z_{\\rm Circular}-\\ln Z_{\\rm Keplerian}=2.8$. We thus conclude that there is no evidence of orbital eccentricity in the {\\it TESS}\\ time-series data. We use the posteriors from the circular orbit fit as a prior to detrend all ground-based photometric data (see next Section).\n\n\n\n\n\n\n\\subsubsection{Ground-based photometric data}\\label{joint_fit_TESS_ground}\nSince all of the eight ground light curves only covered partial transits, the way of detrending generally correlates with the final modeling results. Therefore, we decide to independently detrend all ground photometry in a uniform way using Gaussian processes. As there are no obvious quasi-periodic oscillations existing in data from different facilities, we choose the Mat\\'{e}rn-3\/2 kernel, formulated as:\n\\begin{equation}\n    k_{i,j}(\\tau) = \\sigma^{2}\\left(1+\\frac{\\sqrt{3}\\tau}{\\rho}\\right){\\rm exp}\\left(\\frac{\\sqrt{3}\\tau}{\\rho}\\right),\n\\end{equation}\nwhere $\\tau$ is the time-lag, and $\\sigma$ and $\\rho$ are the covariance amplitude and the correlation timescale of the GP, respectively. Taking the posteriors from the previous {\\it TESS}\\ only fit into account, we put a constraint on the priors to optimize the sampling and reduce the computational time cost. We list our priors in Table \\ref{ground_only_fit_priors} and show the raw and detrended ground light curves in Figure \\ref{ground_transit_detrend}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{TOI530_phase_lc.pdf}\n\\caption{{\\it Top panel}: Phase-folded TESS photometry of {TOI-530}. The red solid line represents the median posterior model. {\\it Bottom panel}: The residuals of the TESS data after subtracting the best-fit transit model. } \n\\label{TESS_transit}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{TOI530_phase_followup_lc_residual.pdf}\n\\caption{Left panels show unbinned phase-folded follow-up transit light curves of {TOI-530}. The instrument and observational band information is presented at the top left of each panel. Our best-fit models are shown as red solid lines. The residuals are shown in the right panels.} \n\\label{ground_transit}\n\\end{figure}\n\n\\subsection{RV-only modeling}\\label{rv}\nWe carry out a preliminary RV-only fit using \\code{juliet}, which utilizes the \\code{radvel} package to build the Keplerian model \\citep{Fulton2018}. In order to reduce the potential errors induced by the orbital period and timing, we fix $P_{b}$ and $T_{0,b}$ at the best-fit transit ephemeris derived from the previous {\\it TESS}\\ only fit. Due to the limited number of RV points and our previous insignificant detection of eccentricity (see Section \\ref{tess_only}), we fit a circular orbit model with $e$ fixed at zero. Since our RV observations only have a short time span, we do not take the RV slope and quadratic term ($\\dot{\\gamma}$ and $\\ddot{\\gamma}$) into consideration in the RV modeling, and we simply fix them at 0. Thus the remaining degrees of freedom are the RV semi-amplitude $K_{b}$, the systemic velocity $\\mu$ and the extra jitter term $\\sigma$, which is used to account for the additional white noise. We adopt wide uniform priors on $K_{b}$ and $\\mu$ but a log-uniform prior on $\\sigma$. Our model reveals that the SPIRou RVs have a semi-amplitude of $K=67.2\\pm15.1$ m\/s. Table \\ref{rvonly_priors} provides our prior settings and the median value of the posterior of each parameter along with their $1\\sigma$ confidence interval. \n\nWe then construct a flat RV model to test the robustness of our RV detection above. Compared with the flat model, we find that our circular orbit model has a $\\ln Z$ improvement of $\\Delta \\ln Z=\\ln Z_{\\rm Circular}-\\ln Z_{\\rm Flat}=4.8$, supporting a significant RV detection.\n\n\n\n\\begin{comment}\n\\begin{table}\n    \\centering\n    \\caption{Model comparison of RV only fits with juliet.}\n    \\begin{tabular}{cccc}\n        \\hline\\hline\n        Model   &Type     &$\\ln Z$    &$\\Delta \\ln Z$     \\\\\\hline\n        $\\rm M1$    &Flat            &-87.5 &-12.3\\\\\n       \n       \n        $\\rm M2$    &Circular                     &-82.7  &-7.5\\\\\n        $\\bf {M3}$    &{\\bf Eccentric}          &{\\bf -75.2}   &{\\bf 0}\\\\\n       \n       \n        \\hline\\hline \n    \\end{tabular}\n   \n   \n   \n   \n    \\label{modelcomp}\n\\end{table}\n\\end{comment}\n\n\\subsection{Joint RV and transit analysis}\\label{joint}\n\nIn order to obtain precise transit ephemeris and physical parameters, we finally jointly model the detrended {\\it TESS}\\ photometry and all ground-based re-processed light curves together with the SPIRou RVs. We adopt the identical priors on planetary and {\\it TESS}\\ photometry parameters as in Section \\ref{tess_only}. While for the ground photometric data, we choose the linear law to parameterize the limb-darkening effect and put a Gaussian prior on the theoretical estimate derived from the \\code{LDTK} package with a width of 0.1 \\citep{Husser2013,Parviainen2015}. Similarly, we also fit an extra flux jitter term for each ground instrument to account for additional white noise. As there are less contamination in the ground data, we fix all dilution factors $D$ to 1. For the SPIRou radial velocities, we adopt the same priors as the circular orbit model in Section \\ref{rv}. We find the {TOI-530} b has a mass of $0.4\\pm0.1\\ M_J$ with a radius of $0.83\\pm0.05\\ R_{J}$, which is the typical size of a giant planet without much inflation. We show the phase-folded light curves along with the best-fit models in Figures \\ref{TESS_transit} and \\ref{ground_transit}. Figure \\ref{SPIROU} shows the SPIRou data and the best-fit RV model.  Table \\ref{tranpriors} summarizes the priors we set in the final joint fit as well as the best-fit value of each parameter. We list the final derived physical parameters in Table \\ref{physical_parameter}.\n\nSince there are a total of 5 nearby stars of {TOI-530}\\ with $T{\\rm mag}<15.5$ located within $1'$ and the light from the brightest star among them (Gaia DR2 3353218784898973312, $T{\\rm mag}=11.0$; star 5 in the right panel of Figure \\ref{fov}) is expected to have a significant contribution of the contamination flux in the photometric aperture due to the large {\\it TESS}\\ pixel scale ($21''$\/pixel), we rerun the joint fit to examine whether additional dilution correction is needed. We set a Gaussian prior on the {\\it TESS}\\ dilution factor $D_{\\rm TESS}$, centered at 1 with a $1\\sigma$ width of 0.1, and keep the left prior settings the same as above. We obtain $D_{\\rm TESS}=0.97\\pm0.03$ and a radius ratio of $R_{p}\/R_{\\ast}=0.156\\pm0.001$, consistent with the result without considering light correction.\n\n\\begin{table*}\n   \n    \\caption{Prior settings and the best-fit values along with the 68\\% credibility intervals in the final joint fit for {TOI-530}. $\\mathcal{N}$($\\mu\\ ,\\ \\sigma^{2}$) means a normal prior with mean $\\mu$ and standard deviation $\\sigma$. $\\mathcal{U}$(a\\ , \\ b) stands for a uniform prior ranging from a to b. $\\mathcal{J}$(a\\ , \\ b) stands for a Jeffrey's prior ranging from a to b.}\n    \\begin{tabular}{lccr}\n        \\hline\\hline\n        Parameter       &Prior &Best-fit    &Description\\\\\\hline\n        \\it{Planetary parameters}\\\\\n       \n        $P_{b}$ (days)  &$\\mathcal{U}$ ($6.2$\\ ,\\ $6.6$)  &$6.387597^{+0.000019}_{-0.000018}$\n        &Orbital period of {TOI-530} b.\\\\\n        $T_{0,b}$ (BJD-2457000)  &$\\mathcal{U}$ ($1468$\\ ,\\ $1472$)   &$1470.1998^{+0.0016}_{-0.0017}$\n        &Mid-transit time of {TOI-530} b.\\\\\n        $r_{1,b}$  &$\\mathcal{U}$ (0\\ ,\\ 1)   &$0.553^{+0.056}_{-0.074}$\n        &Parametrisation for {\\it p} and {\\it b}.\\\\\n        $r_{2,b}$  &$\\mathcal{U}$ (0\\ ,\\ 1)   &$0.155^{+0.002}_{-0.002}$\n        &Parametrisation for {\\it p} and {\\it b}.\\\\\n        $e_{b}$                     &0  &Fixed  &Orbital eccentricity of {TOI-530} b.\\\\\n        $\\omega_{b}$ (deg)          &90 &Fixed  &Argument of periapsis of {TOI-530} b.\\\\\n       \n       \n        \\it{{\\it TESS}\\ photometry parameters}\\\\\n        $D_{\\rm TESS}$  &Fixed    &$1$      &{\\it TESS}\\ photometric dilution factor.\\\\\n        $M_{\\rm TESS}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$-0.00002^{+0.00009}_{-0.00009}$      &Mean out-of-transit flux of {\\it TESS}\\ photometry.\\\\\n        $\\sigma_{\\rm TESS}$ (ppm) &$\\mathcal{J}$ ($10^{-6}$\\ ,\\ $10^{6}$)  &$0.02^{+10.40}_{-0.01}$\n          &{\\it TESS}\\ additive photometric jitter term.\\\\\n        $q_{1}$       &$\\mathcal{U}$ (0\\ ,\\ 1)           &$0.16^{+0.14}_{-0.09}$   &Quadratic limb darkening coefficient.\\\\\n        $q_{2}$       &$\\mathcal{U}$ (0\\ ,\\ 1)                 &$0.46^{+0.33}_{-0.30}$  &Quadratic limb darkening coefficient.\\\\\n        \n        \\it{El Sauce photometry parameters}\\\\\n        $D_{\\rm el}$  &Fixed    &$1$      &El Sauce photometric dilution factor.\\\\\n        $M_{\\rm el}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)    &$-0.0004^{+0.0008}_{-0.0008}$\n        &Mean out-of-transit flux of El Sauce photometry.\\\\\n        $\\sigma_{\\rm el}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$17.3^{+483.7}_{-16.6}$\n          &El Sauce additive photometric jitter term.\\\\\n        $q_{\\rm el}$   &$\\mathcal{N}$ ($0.66$\\ ,\\ $0.1^{2}$)                &$0.74^{+0.07}_{-0.08}$     &Linear limb darkening coefficient.\\\\\n        \n        \\it{MUSCAT2 photometry parameters}\\\\\n        $D_{\\rm MUSCAT2,g}$   &Fixed  &$1$\n          &MUSCAT2 $g$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,g}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$-0.0016^{+0.0006}_{-0.0005}$\n          &Mean out-of-transit flux of MUSCAT2 $g$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,g}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$6868.9^{+517.3}_{-515.7}$\n      &MUSCAT2 $g$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,g}$      &$\\mathcal{N}$ ($0.79$\\ ,\\ $0.1^{2}$)            &$0.67^{+0.07}_{-0.07}$        &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,r}$  &Fixed    &$1$\n          &MUSCAT2 $r$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,r}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$0.0001^{+0.0003}_{-0.0003}$\n       &Mean out-of-transit flux of MUSCAT2 $r$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,r}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$4667.1^{+219.3}_{-203.6}$\n       &MUSCAT2 $r$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,r}$     &$\\mathcal{N}$ ($0.73$\\ ,\\ $0.1^{2}$)             &$0.66^{+0.05}_{-0.05}$   &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,i}$  &Fixed     &$1$\n          &MUSCAT2 $i$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,i}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$0.0002^{+0.0003}_{-0.0003}$\n          &Mean out-of-transit flux of MUSCAT2 $i$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,i}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$5026.9^{+207.0}_{-193.0}$\n          &MUSCAT2 $i$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,i}$     &$\\mathcal{N}$ ($0.54$\\ ,\\ $0.1^{2}$)             &$0.61^{+0.05}_{-0.05}$   &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,z}$   &Fixed  &$1$\n          &MUSCAT2 $z$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,z}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)  &$0.0006^{+0.0002}_{-0.0002}$\n          &Mean out-of-transit flux of MUSCAT2 $z$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,z}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$4504.2^{+144.9}_{-140.6}$\n          &MUSCAT2 $z$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,z}$      &$\\mathcal{N}$ ($0.44$\\ ,\\ $0.1^{2}$)             &$0.58^{+0.05}_{-0.05}$  &Linear limb darkening coefficient.\\\\\n        \n        \\it{MUSCAT photometry parameters}\\\\\n        $D_{\\rm MUSCAT,g}$  &Fixed   &$1$\n          &MUSCAT $g$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,g}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$0.0002^{+0.0009}_{-0.0009}$\n          &Mean out-of-transit flux of MUSCAT $g$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,g}$ (ppm)  &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$27.6^{+711.3}_{-26.8}$\n          &MUSCAT $g$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,g}$   &$\\mathcal{N}$ ($0.79$\\ ,\\ $0.1^{2}$)              &$0.77^{+0.08}_{-0.08}$    &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT,r}$  &Fixed   &$1$\n          &MUSCAT $r$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,r}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$0.0001^{+0.0003}_{-0.0003}$\n          &Mean out-of-transit flux of MUSCAT $r$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,r}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)   &$8.96^{+177.7}_{-8.4}$\n          &MUSCAT $r$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,r}$      &$\\mathcal{N}$ ($0.73$\\ ,\\ $0.1^{2}$)          &$0.76^{+0.06}_{-0.05}$     &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT,z}$  &Fixed   &$1$\n          &MUSCAT $z$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,z}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$0.0002^{+0.0002}_{-0.0002}$\n          &Mean out-of-transit flux of MUSCAT $z$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,z}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$11.7^{+192.4}_{-11.1}$\n          &MUSCAT $z$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,z}$       &$\\mathcal{N}$ ($0.44$\\ ,\\ $0.1^{2}$)            &$0.45^{+0.05}_{-0.05}$   &Linear limb darkening coefficient.\\\\\n        \n        \\it{Stellar parameters}\\\\\n        ${\\rho}_{\\ast}$ ($\\rm kg\\ m^{-3}$) &$\\mathcal{J}$ ($100$\\ ,\\ $\\rm 100^{2}$)   &$4278^{+412}_{-395}$\n         &Stellar density.\\\\\n        \\it{RV parameters}\\\\\n        $K_{b}$ ($\\rm m\\ s^{-1}$)  &$\\mathcal{U}$ ($0$\\ ,\\ $200$)      &$66.5^{+14.1}_{-14.0}$\n        &RV semi-amplitude of {TOI-530} b.\\\\\n        $\\rm \\mu_{SPIRou}$ ($\\rm m\\ s^{-1}$) &$\\mathcal{U}$ ($29300$\\ ,\\ $29500$)   &$29402.4^{+11.1}_{-11.5}$\n        &Systemic velocity for SPIRou.\\\\\n        $\\rm \\sigma_{SPIRou}$ ($\\rm m\\ s^{-1}$) &$\\mathcal{J}$ ($0.1$\\ ,\\ $100$)   &$37.3^{+10.8}_{-8.4}$\n        &Extra jitter term for SPIRou.\\\\\n        \\hline\\hline\n    \\label{tranpriors}    \n    \\end{tabular}\n\\end{table*}\n\n\n\\begin{comment}\n\\begin{table*}\n    \\centering\n    \\caption{Prior settings and the best-fit values along with the 68\\% credibility intervals in the final joint fit for {TOI-530}. $\\mathcal{N}$($\\mu\\ ,\\ \\sigma^{2}$) means a normal prior with mean $\\mu$ and standard deviation $\\sigma$. $\\mathcal{U}$(a\\ , \\ b) stands for a uniform prior ranging from a to b. $\\mathcal{J}$(a\\ , \\ b) stands for a Jeffrey's prior ranging from a to b.}\n    \\begin{tabular}{lccr}\n        \\hline\\hline\n        Parameter       &Best-fit Value       &Prior     &Description\\\\\\hline\n        \\it{Planetary parameters}\\\\\n       \n        $P_{b}$ (days)   &$6.387597^{+0.000016}_{-0.000017}$  \n        &$\\mathcal{U}$ ($6.2$\\ ,\\ $6.6$)\n        &Orbital period of {TOI-530} b.\\\\\n        $T_{0,b}$ (BJD-2457000)    &$1470.1999^{+0.0015}_{-0.0016}$ \n        &$\\mathcal{U}$ ($1468$\\ ,\\ $1472$) \n        &Mid-transit time of {TOI-530} b.\\\\\n        $r_{1,b}$    &$0.554^{+0.054}_{-0.045}$ \n        &$\\mathcal{U}$ (0\\ ,\\ 1)\n        &Parametrisation for {\\it p} and {\\it b}.\\\\\n        $r_{2,b}$    &$0.155^{+0.001}_{-0.001}$ \n        &$\\mathcal{U}$ (0\\ ,\\ 1)\n        &Parametrisation for {\\it p} and {\\it b}.\\\\\n       \n       \n        $\\mathcal{S}_{1,b}$ &$-0.36^{+0.09}_{-0.07}$ &$\\mathcal{U}$ ($-1$\\ ,\\ $1$) &Parametrisation for $e$ and $\\omega$ of {TOI-530} b. \\\\\n        $\\mathcal{S}_{2,b}$ &$0.40^{+0.04}_{-0.04}$ &$\\mathcal{U}$ ($-1$\\ ,\\ $1$) &Parametrisation for $e$ and $\\omega$ of {TOI-530} b. \\\\\n        \\it{{\\it TESS}\\ photometry parameters}\\\\\n        $D_{\\rm TESS}$     &$1$ \n        &Fixed      &{\\it TESS}\\ photometric dilution factor.\\\\\n        $M_{\\rm TESS}$    &$-0.00001^{+0.00009}_{-0.00009}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of {\\it TESS}\\ photometry.\\\\\n        $\\sigma_{\\rm TESS}$ (ppm) &$0.02^{+11.88}_{-0.01}$\n        &$\\mathcal{J}$ ($10^{-6}$\\ ,\\ $10^{6}$)      &{\\it TESS}\\ additive photometric jitter term.\\\\\n        $q_{1}$                &$0.16^{+0.14}_{-0.08}$       &$\\mathcal{U}$ (0\\ ,\\ 1)  &Quadratic limb darkening coefficient.\\\\\n        $q_{2}$                &$0.48^{+0.31}_{-0.30}$       &$\\mathcal{U}$ (0\\ ,\\ 1)  &Quadratic limb darkening coefficient.\\\\\n        \n        \\it{El Sauce photometry parameters}\\\\\n        $D_{\\rm el}$     &$1$ \n        &Fixed      &El Sauce photometric dilution factor.\\\\\n        $M_{\\rm el}$    &$-0.0004^{+0.0007}_{-0.0007}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of El Sauce photometry.\\\\\n        $\\sigma_{\\rm el}$ (ppm) &$24.2^{+495.3}_{-23.2}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &El Sauce additive photometric jitter term.\\\\\n        $q_{\\rm el}$                &$0.73^{+0.07}_{-0.07}$       &$\\mathcal{N}$ ($0.66$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        \\it{MUSCAT2 photometry parameters}\\\\\n        $D_{\\rm MUSCAT2,g}$     &$1$ \n        &Fixed      &MUSCAT2 $g$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,g}$    &$-0.0015^{+0.0005}_{-0.0005}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT2 $g$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,g}$ (ppm) &$6929.7^{+509.1}_{-497.7}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT2 $g$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,g}$                &$0.67^{+0.06}_{-0.05}$       &$\\mathcal{N}$ ($0.79$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,r}$     &$1$ \n        &Fixed      &MUSCAT2 $r$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,r}$    &$0.0002^{+0.0003}_{-0.0003}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT2 $r$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,r}$ (ppm) &$4694.5^{+196.2}_{-186.3}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT2 $r$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,r}$                &$0.65^{+0.04}_{-0.05}$       &$\\mathcal{N}$ ($0.73$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,i}$     &$1$ \n        &Fixed      &MUSCAT2 $i$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,i}$    &$0.0002^{+0.0003}_{-0.0003}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT2 $i$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,i}$ (ppm) &$5090.4^{+190.2}_{-184.3}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT2 $i$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,i}$                &$0.60^{+0.05}_{-0.05}$       &$\\mathcal{N}$ ($0.54$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,z}$     &$1$ \n        &Fixed      &MUSCAT2 $z$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,z}$    &$0.0005^{+0.0002}_{-0.0002}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT2 $z$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,z}$ (ppm) &$4478.3^{+136.2}_{-135.4}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT2 $z$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,z}$                &$0.58^{+0.04}_{-0.05}$       &$\\mathcal{N}$ ($0.44$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        \\it{MUSCAT photometry parameters}\\\\\n        $D_{\\rm MUSCAT,g}$     &$1$ \n        &Fixed      &MUSCAT $g$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,g}$    &$0.0003^{+0.0008}_{-0.0008}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT $g$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,g}$ (ppm) &$41.8^{+799.1}_{-40.6}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT $g$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,g}$                &$0.76^{+0.07}_{-0.08}$       &$\\mathcal{N}$ ($0.79$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT,r}$     &$1$ \n        &Fixed      &MUSCAT $r$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,r}$    &$0.0001^{+0.0003}_{-0.0003}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT $r$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,r}$ (ppm) &$12.6^{+198.2}_{-12.0}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT $r$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,r}$                &$0.75^{+0.06}_{-0.05}$       &$\\mathcal{N}$ ($0.73$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT,z}$     &$1$ \n        &Fixed      &MUSCAT $z$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,z}$    &$0.0002^{+0.0002}_{-0.0002}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT $z$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,z}$ (ppm) &$7.9^{+144.5}_{-7.4}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT $z$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,z}$                &$0.45^{+0.05}_{-0.05}$       &$\\mathcal{N}$ ($0.44$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        \\it{Stellar parameters}\\\\\n        ${\\rho}_{\\ast}$ ($\\rm kg\\ m^{-3}$)   &$4506^{+137}_{-144}$\n        &$\\mathcal{J}$ ($100$\\ ,\\ $\\rm 100^{2}$) &Stellar density.\\\\\n        \\it{RV parameters}\\\\\n        $K_{b}$ ($\\rm m\\ s^{-1}$)       &$86.5^{+6.2}_{-6.7}$\n        &$\\mathcal{U}$ ($0$\\ ,\\ $200$)\n        &RV semi-amplitude of {TOI-530} b.\\\\\n        $\\rm \\mu_{SPIRou}$ ($\\rm m\\ s^{-1}$)    &$29395.4^{+5.2}_{-5.8}$ \n        &$\\mathcal{U}$ ($29300$\\ ,\\ $29500$)\n        &Systemic velocity for SPIRou.\\\\\n        $\\rm \\sigma_{SPIRou}$ ($\\rm m\\ s^{-1}$)    &$5.1^{+6.8}_{-3.2}$ \n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $100$)\n        &Extra jitter term for SPIRou.\\\\\n        \\hline\\hline\n    \\label{tranpriors}    \n    \\end{tabular}\n\\end{table*}\n\\end{comment}\n\n\\begin{table}\n\\caption{Derived physical parameters from the final joint fit for {TOI-530}.}\n\n\n \\begin{tabular}{lcr}   \n    \\hline\\hline\n       \n        Parameter        &Best-fit           &Description\\\\\\hline\n        $R_{p}\/R_{\\ast}$  &$0.155^{+0.002}_{-0.002}$ &Planet radius in units of stellar radius.\\\\\n        $R_{p}$ ($R_{J}$)  &$0.83^{+0.06}_{-0.06}$ &Planet radius.\\\\\n        $M_{P}$ ($M_{J}$)  &$0.40^{+0.09}_{-0.10}$ &Planet mass.\\\\\n        $\\rho_{p}$ ($\\rm g\\ cm^{-3}$)  &$0.93^{+0.49}_{-0.35}$ &Planet density.\\\\\n        ${b}$      &$0.33^{+0.08}_{-0.11}$ &Impact parameter.\\\\\n        $a\/R_{\\ast}$     &$20.97^{+0.65}_{-0.67}$ &Semi-major axis in units of stellar radii.\\\\\n        $a$ (AU)       &$0.052^{+0.005}_{-0.004}$ &Semi-major axis.\\\\\n       \n       \n        $i$ (deg)     &$89.1^{+0.3}_{-0.3}$ &Inclination angle.\\\\\n        $T_{\\rm eq}^{[1]}$ (K)     &$565^{+28}_{-31}$ &Equilibrium temperature.\\\\\n       \n        \\hline\\hline \n    \\end{tabular}\n    \\begin{tablenotes}\n   \n   \n   \n    \\item[1]  [1]\\ We assume there is no heat distribution between the dayside and nightside, and that the albedo is zero.\n    \\end{tablenotes}\n    \\label{physical_parameter}\n\\end{table}\n\n\\begin{comment}\n\\begin{table}\n\\caption{Derived physical parameters from the final joint fit for {TOI-530}.}\n\n\n \\begin{tabular}{lcr}   \n    \\hline\\hline\n       \n        Parameter       &Best Value           &Description\\\\\\hline\n        $R_{p}\/R_{\\ast}$ &$0.155^{+0.001}_{-0.001}$ &Planet radius in units of stellar radius.\\\\\n        $R_{p}$ ($R_{J}$) &$0.83^{+0.05}_{-0.05}$ &Planet radius.\\\\\n        $M_{P}$ ($M_{J}$) &$0.49^{+0.04}_{-0.04}$ &Planet mass.\\\\\n        $\\rho_{p}$ ($\\rm g\\ cm^{-3}$) &$1.14^{+0.35}_{-0.26}$  &Planet density.\\\\\n        ${b}$    &$0.33^{+0.08}_{-0.11}$   &Impact parameter.\\\\\n        $a\/R_{\\ast}$    &$24.85^{+1.62}_{-1.63}$  &Semi-major axis in units of stellar radii.\\\\\n        $a$ (AU)      &$0.062^{+0.008}_{-0.007}$  &Semi-major axis.\\\\\n        $e_{b}$                     &$0.29^{+0.04}_{-0.04}$    &Orbital eccentricity of {TOI-530} b.\\\\\n        $\\omega_{b}$ (deg)          &$7.9^{+17.5}_{-6.4}$  &Argument of periapsis of {TOI-530} b.\\\\\n        $i$ (deg)    &$89.3^{+0.2}_{-0.2}$ &Inclination angle.\\\\\n        $T_{\\rm eq}^{[1]}$ (K)    &$519^{+35}_{-33}$ &Equilibrium temperature.\\\\\n       \n        \\hline\\hline \n    \\end{tabular}\n    \\begin{tablenotes}\n   \n   \n   \n    \\item[1]  [1]\\ We assume there is no heat distribution between the dayside and nightside, and that the albedo is zero.\n    \\end{tablenotes}\n    \\label{physical_parameter}\n\\end{table}\n\\end{comment}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{SPIRou_RV_v2.pdf}\n\\caption{{\\it Left panel}: The systemic velocity-subtracted SPIRou RVs of {TOI-530}\\ as a function of time along with the best-fit circular orbit model from the photometry+RV joint analysis shown as a black solid line. The error bars are the quadrature sum of the instrument jitter term and the measurement uncertainties for all RVs. The orange shaded region represents the $1\\sigma$ confidence interval of the model. {\\it Right panel}: The corresponding phased-folded SPIRou RV data. Residuals are plotted below.} \n\\label{SPIROU}\n\\end{figure*}\n\n\\begin{comment}\n\\begin{table*}\n    \\centering\n    \\caption{Prior settings and the best-fit values along with the 68\\% credibility intervals in the RV modeling for {TOI-530}.}\n    \\begin{tabular}{lccr}\n        \\hline\\hline\n        Parameter       &Best-fit Value       &Prior     &Description\\\\\\hline\n        \\it{Planetary parameters}\\\\\n        $P_{b}$ (days)   &$6.3876$\n        &Fixed\n        &Orbital period of {TOI-530} b.\\\\\n        $T_{0,b}$ (BJD)    &$2458470.199$ \n        &Fixed\n        &Mid-transit time of {TOI-530} b.\\\\\n        $\\mathcal{S}_{1,b}$ &$-0.41^{+0.11}_{-0.10}$ &$\\mathcal{U}$ ($-1$\\ ,\\ $1$) &Parametrisation for $e$ and $\\omega$ of {TOI-530} b. \\\\\n        $\\mathcal{S}_{2,b}$ &$0.38^{+0.05}_{-0.07}$ &$\\mathcal{U}$ ($-1$\\ ,\\ $1$) &Parametrisation for $e$ and $\\omega$ of {TOI-530} b. \\\\\n        \n        \\it{RV parameters}\\\\\n        $\\rm \\mu_{SPIRou}$ ($\\rm m\\ s^{-1}$)    &$29395.2^{+5.3}_{-5.4}$ \n        &$\\mathcal{U}$ ($29300$\\ ,\\ $29500$)\n        &Systemic velocity for SPIRou.\\\\\n        $\\rm \\sigma_{SPIRou}$ ($\\rm m\\ s^{-1}$)    &$1.2^{+4.7}_{-0.9}$ \n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $100$)\n        &Extra jitter term for SPIRou.\\\\\n\n        $K_{b}$ ($\\rm m\\ s^{-1}$)       &$86.5^{+7.0}_{-6.4}$\n        &$\\mathcal{U}$ ($0$\\ ,\\ $200$)\n        &RV semi-amplitude of {TOI-530} b.\n        \n       \n       \n       \n       \n       \n       \n        \n        \\\\ \\hline\n        \\it{Derived parameters}\\\\ \n        $e_{b}$                     &\\multicolumn{2}{c}{$0.31^{+0.05}_{-0.05}$}   &Orbital eccentricity of {TOI-530} b.\\\\\n        $\\omega_{b}$ (deg)          &\\multicolumn{2}{c}{$9.3^{+23.2}_{-6.9}$}  &Argument of periapsis of {TOI-530} b.\\\\\n        $M_{P}$ ($M_{J}$) &\\multicolumn{2}{c}{$0.49^{+0.04}_{-0.04}$} &Planet mass.\\\\\n        $\\rho_{p}$ ($\\rm g\\ cm^{-3}$) &\\multicolumn{2}{c}{$1.14^{+0.35}_{-0.26}$} &Planet density.\\\\\n         \\hline\\hline \n    \\end{tabular}\n   \n   \n   \n   \n   \n    \\label{rvpriors}\n\\end{table*}\n\\end{comment}\n\n\n\\section{Discussion}\\label{discussion}\n\n\\subsection{A lack of hot massive giant planets around M dwarfs?}\nFigure \\ref{aplot} shows the planet-to-star mass ratio ($q$) as a function of separation distance ($a$) of all giant planets ($0.3\\ M_{J}<M_{p}<13.6\\ M_{J}$) around M dwarfs detected by different methods. Regarding the microlensing sample, since most lens systems are still blended with their sources, it is hard to determine the spectral type of the host star in the lens system\\footnote{Even if the lens and their sources have separated after sufficient long time due to the proper motion, it is still difficult as the host stars of the lens systems are very faint (typically $V\\sim25$ mag).}. Thus we simply set a host-mass threshold between 0.08 and 0.65 $M_{\\odot}$, and we filter out targets that meet the mass cut. While for the other three, we pick out the sample mainly based on the spectral information. We only consider the mass ratio here because most microlensing light-curve analyses do not provide the masses of the host and the planet, although the planet-to-host mass ratio, $q$, is well determined \\citep{Mao1991,Gould1992}. To measure the mass of microlensing planet, one needs two observables \\citep{Zang2020}, but most microlensing planets do not have them, and thus a Bayesian analysis is needed to estimate the host mass, which has a typical $1\\sigma$ uncertainty of $\\sim 0.3~M_{\\odot}$. Thus, it is challenging to classify microlensing planets according to different types of host stars. However, several microlensing detections with unambiguous mass measurements demonstrate that gaint planets orbiting M dwarfs are common (e.g., \\citealt{Bennett2020}).\n\nFour giant planets identified by direct imaging that are far from their host M dwarfs are located at the high-mass-ratio region. This is likely caused by observational biases as the imaging method has difficulty to detect low mass Jupiters with $M_{p}$ around $1\\ M_{J}$ (all of these four planets have $M_{p}\\gtrsim10\\ M_{J}$). Microlensing, however, is sensitive to all kinds of widely separated planets with masses ranging from  super-Jupiter down to Earth (e.g., \\citealt{Zang2021}). A total of 55 giant planets harboured by M dwarfs have been discovered with projected separation distance $a_{\\bot}\\gtrsim1$ AU\\footnote{The solutions of 12 microlensing systems have the so-called close-wide degeneracy, shown in pairs as translucent blue squares in Figure \\ref{aplot}.}. There is a wide mass ratio distribution of those microlensing systems, most of which have $q\\lesssim10^{-2}$, indicating that cold Jupiters around M dwarfs are possibly common and diverse. \n\nA similar trend can also been seen in the RV-only sample whose separation distances are between 0.1 and 10 AU, although RV can only determine the minimum mass ratio $q_{\\rm min}$ for those non-transiting systems. Currently, there are no RV-only giant planets with $q_{\\rm min}\\geq10^{-2}$ that have been detected around M dwarfs, which is likely due to observational biases as follows. Unlike microlensing, which is not limited by the lens flux, determining the companion mass spectroscopically requires central stars to be relatively bright (typically $V<13$ mag). Thus the RV-only sample may miss giant planets around faint late-type M dwarfs, which have higher $q_{\\rm min}$ compared with equivalent planets around early-type M dwarfs. For massive early-type M dwarfs, however, some of their companions within that mass ratio range should belong to brown dwarfs, which are not included here. Furthermore, no giant planets have been detected within 0.1 AU of their host M dwarfs from RV-only surveys. This phenomenon can be attributed to the RV observational strategy. Most RV surveys focus on bright nearby M dwarfs and the total sample size is small (roughly $\\sim200$). Thus it is reasonable to find none RV-only giant planets in this region given the low occurrence rate of hot Jupiter ($\\sim0.5\\%$, \\citealt{Fressin2013,Petigura2018,Zhou2019}). Therefore, deep transit surveys play a crucial role in detecting such candidates as they are sensitive to planets with small semi-major axis and large mass ratio, which also more likely to have a large planet-to-star radius ratio. \n\nInterestingly, all five known transiting giant planet systems and {TOI-530} b turn out to have small mass ratio $q\\sim10^{-3}$ and there is a possible dearth at the region with $q>2\\times10^{-3}$ and $a<0.1$ AU (see the shaded region in Figure \\ref{aplot}). Part of that may result from the flux-limit problem above (for $q\\geq10^{-2}$). We note that this deficiency feature may reflect a more fundamental link to the planet formation theory. Recent work from \\cite{Liu2019} constructed a pebble-driven planet population synthesis model, and their simulation results suggest that gas giants may mainly form when the central stars are more massive than $0.3\\ M_{\\odot}$ (see Figure 7 in \\citealt{Liu2019}). This is because planets stop increasing their core masses when they reach the pebble isolation mass $M_{\\rm iso}$, which is proportional to the stellar mass as $M_{\\rm iso}\\propto M_{\\ast}^{4\/3}$. Following gas accretion onto planets with small $M_{\\rm iso} $ is limited due to a slow Kelvin\u2013Helmholtz contraction. Thus they would stop before the runaway gas accretion and be left as rock- or ice-dominated planets with tiny atmospheres. If this is the case, we then expect few giant planets with relatively high mass ratio above $10^{-3}$ when their host masses are below $0.3\\ M_{\\odot}$. Note that the known ``brown dwarf desert'' ($35 \\leq M\\sin i \\leq 55\\ M_{J}$ and orbital periods under 100 days), studied by \\cite{Ma2014} using all available data of close brown dwarfs around solar-type stars, is also located in this region with $q\\gtrsim0.05$. This lower limit is estimated based on the lower limit of the brown dwarf desert $35\\ M_{J}$ and the typical mass upper limit of M dwarfs $0.65\\ M_{\\odot}$. However, it is still unclear whether the deficiency between $2\\times10^{-3}$ and $10^{-2}$ in mass ratio is physical. Compared with the known planets (red squares plus {TOI-530} b in Figure \\ref{aplot}), the transit method is, in principle, more sensitive to giant planets with a larger mass ratio (i.e., larger radius ratio) located in this deficiency region. Additionally, if detected by transit survey, planet candidates within this parameter space range should be easily confirmed by the RV method. Due to the lack of transiting giant planets around M dwarfs, we cannot draw any conclusions yet. Hopefully, the {\\it TESS}\\ QLP Faint Star ($10.5<T<13.5$ mag) Search could provide more such systems and check if this depletion feature is real (Kunimoto et al, in prep). \n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{a_q.pdf}\n\\caption{Planet-to-star mass ratio of all giant planets around M dwarfs as a function of semi-major axis. Different colors represent planets detected by different methods. The two horizontal black dashed lines represent the selection threshold of our sample. The upper limit is $q=13.6\\ M_{J}\/0.08\\ M_{\\odot}=0.17$, while the lower limit corresponds to $q=0.3\\ M_{J}\/0.65\\ M_{\\odot}=4.6\\times10^{-4}$. {TOI-530} b is marked as a magenta circle. The grey shaded region represents a possible paucity of hot massive giant planets around M dwarfs.} \n\\label{aplot}\n\\end{figure}\n\n\\subsection{Metallicity Dependence}\\label{metallicity_dependence}\nAlthough the core accretion theory \\citep{Pollack1996} has predicted rare giant planets around M dwarfs due to their low protoplanetary disk mass as $M_{\\rm disk}$ linearly scales with the stellar mass $M_{\\ast}$ \\citep{Andrews2013}, this defect may be compensated if the parent star is metal rich, which could theoretically supply more solid material used for accretion. Alternatively, gravitational instability (GI) could also form giant planets around M dwarfs \\citep{Boss2006}, though simulation work from \\cite{Cai2006} suggested that GI is unlikely the major mechanism that produce most observed planets. Under the GI hypothesis, we expect that there would not exist a strong dependence between giant planet formation and host metallicity and that even stars with relatively low metallicity should harbour gas giants \\citep{Boss2002}.\n\nIn order to investigate the metallicity dependence observationally, we retrieve a list of solar-type stars (simply selected based on $0.90\\ M_{\\odot}<M_{\\ast}<1.06\\ M_{\\odot}$) hosting giant planets from the NASA Exoplanet Archive \\citep{Akeson2013}. We find a total of 88 transiting and 102 RV-only systems. Most transiting giant planets are hot with semi-major axis $a\\lesssim 0.1$ AU, while the majority of RV-only planets are cold with semi-major axes beyond  $0.1$ AU. Figure~\\ref{fehplot} illustrates the metallicity distribution of their hosts, indicating that hot and cold giant planets do not present much difference in [Fe\/H] preference around solar-type stars (see the transparent points in Figure \\ref{fehplot}). The weighted-mean metallicity of both transiting and RV-only giant planet central stars is above the solar value but almost the same: 0.12 and 0.14, respectively. This is consistent with the conclusions from early work suggesting that the frequency of giant planets increases with stellar metallicity \\citep{Santos2004,Fischer2005,Sousa2011}. However, it is not the same for gas giants around M dwarfs. \n\nAmong all giant planets around M dwarfs, we only find 4 transiting and 9 RV-only systems that have metallicity measurements in the literature. For cold (RV-only) giant planets, the metallicities of their M-dwarf hosts are distributed on both sides of the median value 0.14 (the green dashed line in Figure \\ref{fehplot}) seen for the solar-type stars, though the uncertainties are large. Some of them are formed around metal poor M stars (e.g., GJ 832b, \\citealt{Bailey2009}). This is plausibly in agreement with the previous finding that clump formation is fairly insensitive to the metallicity of the parent star \\citep{Boss2002}, implying that part of the formation of cold giant planets around M dwarfs may take place through GI. Indeed, the recently detected planetary system GJ 3512b, whose host star has a solar metallicity $-0.07\\pm0.16$, favours the GI formation scenario \\citep{Morales2019}.\n\nHowever, the host stars of four known transiting giant planets together with {TOI-530}\\ tend to be metal rich, all of which have [Fe\/H] higher than the aforementioned median value 0.12 (the red dashed line in Figure \\ref{fehplot}). It indicates that the formation of hot (transiting) giant planets around M dwarfs may have a strong dependence on the host metallicity as predicted by the core accretion theory. This is consistent with the recent findings from \\cite{Maldonado2020} (see the left panel of their Figure 14), which reveals a correlation between the metallicities of M dwarfs and their probability of hosting giant planets. Compared with hot giant planets around solar-type stars, the formation of hot giant planets around M dwarfs possibly requires higher host metallicity, though the number of detections is probably too small to confirm this claim. To examine the statistical significance of this feature, we carry out a Kolmogorov-Smirnov (K-S) test. We calculate the K-S statistic between the metallicities of stars in the G- and M-type transiting sample, which yields a $p$ value of 0.023. It roughly corresponds to the $2.5\\sigma$ significance level, at which we can reject the null hypothesis that two samples are from the same distribution. We then adopt the bootstrap method to randomly draw distributions from the G-type parent sample and compute the K-S statistic between these random distributions and the M-type transiting sample. We repeat this procedure for 10,000 times and derive the corresponding $p$ value distribution. We find that 88\\% of the total random samples have $p$ values less than 0.05 ($2\\sigma$ level) while only 0.3\\% of them have $p$ values less than 0.003 ($3\\sigma$ level), indicating a marginal correlation. Future detections of more hot giant planets around M dwarfs will reveal whether this metallicity preference is significant or not.\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{feh_M_G.pdf}\n\\caption{Planet mass $M_{p}$ vs. host star metallicity. Different colors represent planets detected by different methods (transit\/RV). The red and blue dashed lines are the weighted-mean metallicity of solar-type stars hosting giant planets detected by transit and RV-only observations. {TOI-530} b is marked as a magenta circle. All hot giant planet M-dwarf hosts tend to have a higher metallicity than G dwarf hosts (see Section \\ref{metallicity_dependence}).} \n\\label{fehplot}\n\\end{figure}\n\n\n\n\n\\begin{comment}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{P-e.pdf}\n\\caption{Orbital eccentricity of all giant planets around M dwarfs as a function of planet orbital period. Different color represent planets detected by different methods. The grey points mark confirmed giant planets around other types of stars with non-zero eccentricity measurements (uncertainty smaller than 30\n\\%). Data are retrieved from NASA Exoplanet Archive \\citep{Akeson2013}. {TOI-530} b is marked as a magenta circle.} \n\\label{peplot}\n\\end{figure}\n\\end{comment}\n\n\n\\subsection{Prospects for future observations}\nDue to the faintness of the host star, high precision radial velocity spectrographs on large telescopes like the red-optical MAROON-X \\citep{Seifahrt2018} or near-infrared instruments like InfrRed Doppler spectrograph (IRD; \\citealt{Kotani2018}) are required to reach sufficient signal-to-noise ratio. The host star is quiet without strong intrinsic stellar activity as there are no significant flux variations in the {\\it TESS}\\ light curve, making it a suitable target for precise RV follow-up observations. Though a potential 9.4 d modulation signal is shown up in the {\\it TESS}\\ light curve, this is probably not linked to the stellar rotation given our ZTF results and previous findings from \\cite{Newton2018} (see Section \\ref{stellar_properties}).\n\nMeasuring the stellar obliquity of M dwarfs hosting hot giant planets could provide clues about their origins and gain insights into their migration history. To probe the potential opportunities to observe the Rossiter-McLaughlin effect \\citep{Rossiter1924,McLaughlin1924} of {TOI-530}\\ and measure the projected\nangle between the planet orbital and stellar equatorial planes, we estimate the RM semi-amplitude of this system as:\n\\begin{equation}\n    A_{\\rm RM} \\simeq \\frac{2}{3}(R_{p}\/R_{\\ast})^{2}\\sqrt{1-b^{2}}\\times v \\sin i,\n\\end{equation}\nwhere $b$ is the impact parameter and $v \\sin i$ is the projected stellar equatorial rotation velocity. Taking the best-fit values from the light curve modeling, assuming a rotational period of 40 d for a $0.5\\ M_{\\odot}$ star (see Figure 4 in \\citealt{Newton2018}) with an upper and lower limit of 100 and 30 d, we find $A_{\\rm RM}\\sim 10^{+4}_{-6}$ m\/s, making the RM signal detectable with near-infrared RV observations.\n\nFinally, we investigate the atmospheric characterization possibility by calculating the Transmission Spectroscopy Metric (TSM; \\citealt{Kempton2018}) of {TOI-530} b. We obtain a TSM of $41^{+20}_{-13}$, which is much smaller than the recommended threshold of 90 for high-quality sub-Jovians ($4.0<R_{p}<10\\ R_{\\oplus}$) targets from \\cite{Kempton2018}. Thus we conclude that {TOI-530} b is not a promising target for future atmospheric composition studies. \n\n\n\\section{Summary and Conclusions}\\label{conclusion}\nIn this paper, we present the discovery and characterization of a transiting giant planet {TOI-530} b around an early, metal-rich M dwarf. Space and ground photometry as well as SPIRou RVs reveal that {TOI-530} b has a radius of $0.83\\pm0.05\\ R_{J}$ and a mass of $0.4\\pm0.1\\ M_{J}$ on a 6.39-d orbit. Although it is challenging to probe the atmospheric properties of {TOI-530} b due to the faintness of the host star, {TOI-530}\\ is still a suitable target to study the stellar obliquity. Furthermore, we report a potential paucity of hot massive giant planets around M dwarfs with separation distance smaller than $0.1$ AU and planet-to-star mass ratio between $2\\times10^{-3}$ and $10^{-2}$. We also identify a possible correlation between hot giant planet formation and the metallicity of its parent M dwarf. However, due to the current small sample of such systems, we could not make any firm conclusions in this context. Future near-infrared spectroscopic surveys such as SPIRou Legacy Survey-Planet Search (SLS-PS; \\citealt{Moutou2017,Fouque2018}) shall remedy this situation.\n\n\\section*{Affiliations}\n$^{1}$Department of Astronomy, Tsinghua University, Beijing 100084, People's Republic of China\\\\\n$^{2}$Department of Physics, Tsinghua University, Beijing 100084, People's Republic of China\\\\\n$^{3}$National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, People's Republic of China\\\\\n$^{4}$Canada-France-Hawaii Telescope, CNRS, Kamuela, HI 96743, USA\\\\\n$^{5}$Univ. de Toulouse, CNRS, IRAP, 14 Avenue Belin, 31400 Toulouse, France\\\\\n$^{6}$Department of Physics and Astronomy, Vanderbilt University, 6301 Stevenson Center Ln., Nashville, TN 37235, USA\\\\\n$^{7}$Department of Physics, Fisk University, 1000 17th Avenue North, Nashville, TN 37208, USA\\\\\n$^{8}$Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA\\\\\n$^{9}$Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan\\\\\n$^{10}$Instituto de Astrof\\'isica de Canarias (IAC), V\\'ia L\\'actea s\/n, E-38205 La Laguna, Tenerife, Spain\\\\\n$^{11}$Dept. Astrof\\'sica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain\\\\\n$^{12}$NASA Exoplanet Science Institute, Caltech\/IPAC, Mail Code 100-22, 1200 E. California Blvd., Pasadena, CA 91125, USA\\\\\n$^{13}$NASA Ames Research Center, Moffett Field, CA 94035, USA\\\\\n$^{14}$Center for Astrophysics ${\\rm \\mid}$ Harvard {\\rm \\&} Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA\\\\\n$^{15}$Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\\\\\n$^{16}$NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA\\\\\n$^{17}$University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA\\\\\n$^{18}$Department of Physics \\& Astronomy, University of Kansas, 1082 Malott,1251 Wescoe Hall Dr., Lawrence, KS 66045, USA\\\\\n$^{19}$Embry-Riddle Aeronautical University, Prescott, AZ\\\\\n$^{20}$El Sauce Observatory, Coquimbo Province, Chile\\\\\n$^{21}$Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA\\\\\n$^{22}$Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA\\\\\n$^{23}$Okayama Observatory, Kyoto University, 3037-5 Honjo, Kamogatacho, Asakuchi, Okayama 719-0232, Japan\\\\\n$^{24}$Department of Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan\\\\\n$^{25}$Department of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\\\\\n$^{26}$Zhejiang Institute of Modern Physics, Department of Physics \\& Zhejiang University-Purple Mountain Observatory Joint Research Center for Astronomy, Zhejiang University, Hangzhou 310027, China\\\\\n$^{27}$Department of Astronomy, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\\\\\n$^{28}$U.S. Naval Observatory, Washington, D.C. 20392, USA\\\\\n$^{29}$Japan Science and Technology Agency, PRESTO, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan\\\\\n$^{30}$Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan\\\\\n$^{31}$Department of Earth, Atmospheric and Planetary Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA\\\\\n$^{32}$Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA\\\\\n$^{33}$Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA\\\\\n$^{34}$Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA\\\\\n$^{\\ast}$51 Pegasi b Fellow\\\\\n\\section*{Acknowledgements}\nWe are grateful to Coel Hellier for the insights regarding the WASP data. We thank Elisabeth Newton, Robert Wells, Hongjing Yang and Weicheng Zang for useful discussions. We also thank Elise Furlan for the contributions to the speckle data and Nadine Manset for scheduling the SPIRou observations.\nThis work is partly supported by the National Science Foundation of China (Grant No. 11390372 and 11761131004 to SM and TG). \nThis research uses data obtained through the Telescope Access Program (TAP), which has been funded by the TAP member institutes.\nThis work is partly supported by JSPS KAKENHI Grant Numbers JP17H04574 , JP18H05439, 20K14521, JST PRESTO Grant Number JPMJPR1775, and the Astrobiology Center of National Institutes of Natural Sciences (NINS) (Grant Number AB031010).\nThis article is based on observations made with the MuSCAT2 instrument, developed by ABC, at Telescopio Carlos S\u00e1nchez operated on the island of Tenerife by the IAC in the Spanish Observatorio del Teide.\nSome of the observations in the paper made use of the High-Resolution Imaging instrument 'Alopeke obtained under LLP GN-2021A-LP-105. 'Alopeke was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. Data were reduced using a software pipeline originally written by Elliott Horch and Mark Everett. 'Alopeke was mounted on the Gemini North telescope of the international Gemini Observatory, a program of NSF's OIR Lab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\u00f3n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\u00eda e Innovaci\u00f3n (Argentina), Minist\u00e9rio da Ci\u00eancia, Tecnologia, Inova\u00e7\u00f5es e Comunica\u00e7\u00f5es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). \nFunding for the TESS mission is provided by NASA's Science Mission directorate. \nWe acknowledge the use of {\\it TESS}\\ public data from pipelines at the {\\it TESS}\\ Science Office and at the {\\it TESS}\\ Science Processing Operations Center. \nResources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products.\nThis research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. \nThis paper includes data collected by the {\\it TESS}\\ mission, which are publicly available from the Mikulski Archive for Space Telescopes\\ (MAST). \nThis work has made use of data from the European Space Agency (ESA) mission\n{\\it Gaia} (\\url{https:\/\/www.cosmos.esa.int\/gaia}), processed by the {\\it Gaia} Data Processing and Analysis Consortium (DPAC,\n\\url{https:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\\it Gaia} Multilateral Agreement.\nThis work made use of \\texttt{tpfplotter} by J. Lillo-Box (publicly available in www.github.com\/jlillo\/tpfplotter), which also made use of the python packages \\texttt{astropy}, \\texttt{lightkurve}, \\texttt{matplotlib} and \\texttt{numpy}.\n\\section*{Data Availability}\nThis paper includes photometric data collected by the {\\it TESS}\\ mission and ground instruments, which are publicly available in ExoFOP, at \\url{https:\/\/exofop.ipac.caltech.edu\/tess\/target.php?id=387690507}. All spectroscopy data underlying this article are listed in the text. All of the high-resolution speckle imaging data is available at the NASA exoplanet Archive with no proprietary period.\n\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Divisors on smooth manifolds}\n\\label{sec:divonmanifolds}\nIn this section we introduce real and complex divisors on smooth manifolds (see also \\cite{CavalcantiKlaasse18,Klaasse17} and \\cite{CavalcantiGualtieri18, VanderLeerDuran16}). These are extensions of the complex geometric notion of divisors to the smooth setting. The theory of divisors permeates much of this paper, as they will be used to keep track of degeneracies of both Lie algebroids and Poisson structures. Let $X$ be a fixed smooth manifold and let $\\mathcal{C}^\\infty_X$ be its sheaf of smooth functions, with $C^\\infty(X) = \\mathcal{C}^\\infty_X(X)$.\n\nIn this section we will often not specify whether we are dealing with real or complex objects, and treat them simultaneously and on equal footing. However, in the remainder of the paper we will mostly consider real divisors. Their complex counterparts will play a larger role in the follow-up paper \\cite{Klaasse18} (and occur prominently in \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18,VanderLeerDuran16}).\n\\begin{defn} A \\emph{divisor} $(U,\\sigma)$ on $X$ is a line bundle $U \\to X$ together with a section $\\sigma \\in \\Gamma(U)$ with nowhere dense closed zero set $Z_\\sigma := \\sigma^{-1}(0) \\subseteq X$. A \\emph{divisor ideal sheaf} is a locally principal ideal sheaf $\\mathcal{I} \\subseteq \\mathcal{C}^\\infty_X$ generated by nowhere-densely vanishing functions. Its associated \\emph{divisor ideal} is the ideal of its global sections, $I := \\mathcal{I}(X) \\subseteq C^\\infty(X)$.\n\\end{defn}\n\\begin{rem} Recall that a nowhere dense subset of a topological space is one whose closure has empty interior. In other words, its intersection with any nonempty open subset is not dense. Often the subset $Z_\\sigma$ will be a (union of) submanifold(s) of positive codimension.\n\\end{rem}\nAny divisor specifies a divisor ideal $I_\\sigma := \\sigma(\\Gamma(U^*)) \\subseteq C^\\infty(X)$ by evaluation, with associated divisor ideal sheaf $\\mathcal{I}_\\sigma$. Letting $\\alpha$ be a local trivialization of $U^*$, we have $\\alpha(\\sigma) = g$ for some local function $g$, and hence locally $I_\\sigma = \\langle \\alpha(\\sigma) \\rangle = \\langle g \\rangle$. Moreover, note that\n\\begin{equation*}\n\tZ(I) := \\{x \\in X \\, | \\, f(x) = 0 \\text{ for all } f \\in \\mathcal{I}_x\\}\n\\end{equation*}\nis the support of the quotient sheaf $\\mathcal{C}^\\infty_X \/ \\mathcal{I}$, with $Z(I_\\sigma) = Z_\\sigma$ (here $\\mathcal{I}_x$ denotes the stalk of $\\mathcal{I}$ at $x \\in X$). Because of this we will say that $(U,\\sigma)$ is \\emph{supported} on $Z_\\sigma$, and sometimes write $Z_I := Z(I)$. Similarly we say that the divisor ideal $I$ is supported on $Z_I \\subseteq X$.\n\nMorphisms of divisors are most conveniently described through their associated divisor ideals. A \\emph{morphism of divisors} $(U,\\sigma) \\to X$ and $(U',\\sigma') \\to X'$ is a map $f\\colon X \\to X'$ satisfying $f^* I_{\\sigma'} = I_\\sigma$, where $f^* I \\subseteq C^\\infty(X)$ for an ideal $I \\subseteq C^\\infty(X')$ denotes the ideal generated by the pullback. Divisors are \\emph{diffeomorphic} if $f$ is a diffeomorphism, while they are \\emph{isomorphic} (written using $=$) if $X = X'$ and the identity map ${\\rm id}_X$ is a morphism of divisors.\n \nA divisor can be reconstructed up to isomorphism by its divisor ideal. This establishes a bijective correspondence between divisor ideals and isomorphism classes of divisors. This allows us to mostly work with divisor ideals, which carry all essential information. However, often there is a canonical representative of its divisor isomorphism class, for example for Lie algebroid morphisms and Poisson structures.\n\\begin{prop}[\\cite{CavalcantiKlaasse18,VanderLeerDuran16}]\\label{prop:locprincideal} Let $I \\subseteq C^\\infty(X)$ be a divisor ideal. Then there exists a divisor $(U_I,\\sigma)$ on $X$ unique up to isomorphism such that $I_\\sigma = I$.\n\\end{prop}\nDivisors over a fixed manifold $X$ admit a \\emph{product} operation, given by the tensor product $(U,\\sigma) \\otimes (U',\\sigma') := (U \\otimes U', \\sigma \\otimes \\sigma')$. The \\emph{trivial divisor} $(\\underline{\\mathbb{R}},\\underline{1})$ is the unit up to isomorphism. This operation produces another divisor, because the union of nowhere dense subsets is nowhere dense. The product of divisors corresponds to the product of divisor ideals, i.e.\\ $I_{\\sigma \\otimes \\sigma'} = I_\\sigma \\cdot I_{\\sigma'}$, and hence descends to isomorphism classes, with unit $C^\\infty(X)$, the trivial divisor ideal. A divisor ideal $I$ is \\emph{divisible} by $I'$ if $I = I' \\cdot I''$ for some divisor ideal $I''$, and similarly at the level of divisors. We will write $I' < I$ if the divisor ideal $I$ is divisible by $I'$.\n\\begin{rem} Divisors ideals on $X$ with this notion of morphism and product form an abelian category $\\mathfrak{Div}(X)$. Accordingly, we can consider (ir)reducible elements. This leads to asking whether one is able to categorize or classify irreducible divisor ideals. This is akin to the classification of singularities, and related to standardness (\\autoref{defn:divprojstandard}).\n\\end{rem}\n\\begin{defn}\\label{defn:smoothdivisor} A divisor ideal $I \\subseteq C^\\infty(X)$ is \\emph{smooth} if $Z_I \\subseteq X$ is a smooth submanifold.\n\\end{defn}\nHere in the notion of submanifold we allow the codimensions of connected components to vary. Any smooth divisor can be uniquely written as the product of divisors on the connected components of its support, thus we can sometimes assume that the support is connected.\n\\begin{prop}\\label{prop:divvanishingideal} Let $(U,\\sigma)$ be a divisor. Then $I_\\sigma \\subseteq I_{Z_\\sigma}$, the vanishing ideal of $Z_\\sigma$. \n\\end{prop}\n\\bp Locally trivialize $U^*$ by $\\alpha^*$ and let $g := \\alpha(\\sigma) \\in \\mathcal{C}^\\infty_X$. Then locally $I_\\sigma = \\sigma(\\Gamma(U^*)) = \\langle g \\rangle$. As $\\sigma(Z_\\sigma) = 0$, we conclude that $g(Z_\\sigma) = \\sigma(Z_\\sigma) = 0$ so that $g \\in I_{Z_\\sigma}$, hence $I_\\sigma \\subseteq I_{Z_\\sigma}$. \n\\end{proof}\nNote that $I_\\sigma = I_{Z_\\sigma}$ is only possible for $Z_\\sigma$ smooth if $Z_\\sigma$ is of codimension one, as can be seen by counting local generators.\nThere is also a relation between morphisms of divisors and their underlying zero sets. We first introduce the following terminology (c.f.\\ \\cite{CavalcantiKlaasse18}).\n\\begin{defn}\\label{defn:pairsandmaps} A \\emph{pair} $(X,Z)$ is a manifold $X$ with a subset $Z \\subseteq X$. A \\emph{map of pairs} $f\\colon (X,Z) \\to (X',Z')$ is a smooth map $f\\colon X \\to X'$ for which $f(Z) \\subseteq Z'$. A \\emph{strong map of pairs} is a map of pairs $f\\colon (X,Z) \\to (X,Z')$ for which $f^{-1}(Z') = Z$. A map of pairs $f\\colon (X,Z) \\to (X',Z')$ where $Z' \\subseteq X'$ is a submanifold is \\emph{transverse} if $f$ is transverse to $Z'$.\n\\end{defn}\n\\begin{prop}\\label{prop:divmorphstrong} Let $f\\colon (X,U,\\sigma) \\to (X',U',\\sigma')$ be a morphism of divisors. Then $f\\colon (X,Z_{\\sigma}) \\to (X',Z_{\\sigma'})$ is a strong map of pairs, equivalently, satisfies the morphism condition $f^* I_{Z_{\\sigma'}} = I_{Z_{\\sigma}}$.\n\\end{prop}\nIn general, for a map $f\\colon X \\to Y$ and $Z \\subseteq Y$ closed, one can only conclude that $f^* I_Z \\subseteq I_{f^{-1}(Z)}$.\n\\bp As $f$ is a morphism of divisors, we have that $U \\cong f^* U'$ and $\\sigma = g f^* \\sigma'$ for a nonvanishing function $g$. As $Z_{\\sigma}$ is by definition the zero set of $\\sigma$, it is immediate that it equals $f^{-1}(Z_{\\sigma'})$.\n\\end{proof}\n\nGiven a smooth map $f\\colon X \\to X'$ and a divisor $(U',\\sigma')$ on $X'$, one can equip $X$ with a divisor by setting $(U,\\sigma) := (f^* U', f^* \\sigma')$, as long as $f^{-1}(Z_{\\sigma'})$ is nowhere dense (e.g.\\ by assuming $f$ is a transverse map of pairs onto $(X,Z_{\\sigma'})$ if $Z_{\\sigma'}$ is smooth). This automatically makes $f$ a morphism of divisors. If $f$ is transverse to $Z_{\\sigma'}$, the divisor-type is preserved under pullback: by this we mean that the type of degeneracy is similar, so that the class of divisor is preserved.\n\nDivisors admit another type of product. Let $(X,U,\\sigma)$ and $(X',U',\\sigma')$ be two divisors and consider the product $X \\times X'$ with projections $p_{X}\\colon X \\times X' \\to X$ and $p_{X'}\\colon X \\times X' \\to X'$.\n\\begin{defn}\\label{defn:divdirectstum} The \\emph{external tensor product} of $(U,\\sigma)$ and $(U',\\sigma')$ is the divisor\n\\begin{equation*}\n\t(p_X^* U \\otimes p_{X'}^* U', p_X^* \\sigma \\otimes p_{X'}^* \\sigma') \\to X \\times X',\n\\end{equation*}\nwith zero set $Z_\\sigma \\times X' \\cup X \\times Z_{\\sigma'}$ and divisor ideal given by the product $p_X^* I_\\sigma \\cdot p_{X'}^* I_{\\sigma'} \\subseteq C^\\infty(X \\times X')$.\n\\end{defn}\n\\subsection{Examples}\n\\label{sec:divexamples}\nDivisors of various flavors exist, and can naturally be combined using the product operation and external tensor product to form more involved examples. Here we give several examples.\n\\begin{exa}[Trivial divisor] Let $U$ be the trivial line bundle with $\\sigma \\in \\Gamma(U)$ nonvanishing. Then $Z_\\sigma$ is empty, and $(U,\\sigma)$ is called the \\emph{trivial divisor} on $X$. In this case, $I_\\sigma = C^\\infty(X)$.\n\\end{exa}\n\\begin{exa}[Log divisors \\cite{CavalcantiKlaasse18}]\\label{exa:log} A \\emph{log divisor} is a divisor for which $\\sigma$ is transverse to the zero section in $\\Gamma(U)$. Its zero set $Z = Z_\\sigma$ is a smooth hypersurface, and the corresponding divisor ideal is the vanishing ideal $I_Z$, locally generated around $Z$ by $\\langle z \\rangle$, with $z$ a local defining function for $Z$. As shown in \\cite{Klaasse17}, this means that $Z$ specifies a unique isomorphism class of log divisors with zero set $Z$. We will henceforth denote log divisors by $Z = (L_Z,s)$.\n\\end{exa}\n\\begin{exa}[Normal-crossing log divisors \\cite{GualtieriLiPelayoRatiu17}]\\label{exa:normalcrossinglog} The above example can be extended to normal-crossing log divisors $\\underline{Z} := \\bigcup_j Z_j$ associated to a collection $\\{Z_j\\}$ of hypersurfaces which intersect transversely. Such a divisor is the product of the individual log divisors for each $Z_j$, hence locally $I_{\\underline{Z}} = \\langle z_1 \\cdot \\ldots \\cdot z_n \\rangle$. These specify hyperbolic singularities on two-fold intersections: there we can readily change coordinates so that $I_{\\underline{Z}} = \\langle z_1 z_2 \\rangle = \\langle z'^2_1 - z'^2_2 \\rangle$. There is also an extension of these ideas to immersed hypersurfaces with transverse self-crossings \\cite{MirandaScott18}.\n\\end{exa}\nThere is another example resembling \\autoref{exa:normalcrossinglog}, namely that of a \\emph{star log divisor} \\cite{Lanius17}, consisting of the product of log divisors associated to a collection $\\{Z_j\\}$ of pairwise transversely intersecting hypersurfaces $Z_j \\subseteq X$.\n\\begin{exa}[Elliptic divisors \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18}]\\label{exa:elliptic} An \\emph{elliptic divisor} is one for which $\\sigma$ vanishes along a smooth codimension-two critical submanifold along which its normal Hessian\n\\begin{equation*}\n\t{\\rm Hess}(\\sigma) \\in \\Gamma(Z_\\sigma; {\\rm Sym}^2 NZ_\\sigma \\otimes U)\n\\end{equation*}\nis definite. We will denote $Z_\\sigma$ by $D$ and the elliptic divisor by $|D| = (R,q)$. As shown in \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18}, the associated divisor ideal $I_{|D|} := I_q$ is locally generated by $\\langle r^2 \\rangle$ in polar coordinates $(r,\\theta)$ normal to $D$. Note that $I_{|D|}$ is not the vanishing ideal $I_D$ of $D$.\n\\end{exa}\n\\begin{rem} Hypersurfaces carry a unique log divisor ideal structure. This does not hold for codimension-two submanifolds and elliptic divisors. A simple example is provided by $X = \\mathbb{R}^2$ with $D = \\{(0,0)\\}$ and coordinates $(x,y)$. Equip $D$ with the elliptic ideals $I = \\langle x^2 + y^2 \\rangle$ and $I' = \\langle x^2 + 2y^2 \\rangle$. As these ideals are distinct, they supply $D$ with two non-isomorphic elliptic divisor structures. However, it is easy to see that these elliptic divisors are diffeomorphic.\n\\end{rem}\nWe construct a more elaborate class of divisors out of log and elliptic divisors as follows.\n\\begin{exa}[Elliptic-log divisors]\\label{exa:ellipticlog} An \\emph{elliptic-log divisor} is a divisor obtained as the product of a log divisor $Z = (L_Z,s)$ and an elliptic divisor $|D| = (R,q)$ such that $D \\subseteq Z$ (we will write $|D| \\subseteq Z$ as a shorthand for this). We denote this product by $W := Z \\otimes |D|$, with divisor ideal $I_W = I_Z \\cdot I_{|D|}$. As such, pointwise around $D$ there exist normal coordinates $(x,y)$ to $D$ with $\\{x = 0\\}$ cutting out $Z$, for which $I_W = \\langle x (x^2 + y^2) \\rangle$, with $r^2 = x^2 + y^2$.\n\\end{exa}\nIt is also possible for a log divisor $Z$ and an elliptic divisor $|D|$ to either not or only partially intersect. In general, one can readily combine divisors with nowhere intersecting supports using the product operation. However, most interesting are cases where the divisors do intersect (yet retain interesting properties), of which \\autoref{exa:ellipticlog} is a specific instance.\n\\begin{exa}[Complex log divisors \\cite{CavalcantiGualtieri18}]\\label{exa:complexlog} A \\emph{complex log divisor} is a complex divisor $(U,\\sigma)$ for which $\\sigma$ is transverse to zero. This is the complex analogue of \\autoref{exa:log}, and locally we have $I_\\sigma = \\langle w \\rangle$ for $w$ a local complex function cutting out $D$. Its zero set $D$ is smooth of codimension two, and by complex conjugation we obtain another complex log divisor $(\\overline{U},\\overline{\\sigma})$ with $I_{\\overline{\\sigma}} = \\langle \\overline{w} \\rangle$. After taking their product we can extract an elliptic divisor $((U \\otimes \\overline{U})_\\mathbb{R},\\sigma \\otimes \\overline{\\sigma})$ with $I_{\\sigma \\otimes \\overline{\\sigma}} = \\langle w \\overline{w} \\rangle = \\langle r^2 \\rangle$. Given an elliptic divisor and a choice of coorientation for $D$, there is a complex log divisor inducing it which is unique up to diffeomorphism.\n\\end{exa}\nThere is a general class of divisors of \\emph{Morse--Bott type}, i.e.\\ those pairs $(U,\\sigma)$ for which $\\sigma$ is a \\emph{Morse--Bott section}. This means that $Z_\\sigma$ consists of nondegenerate critical submanifolds, i.e.\\ $Z_\\sigma$ is a union of submanifolds $S$ of $\\sigma$, where $\\ker {\\rm Hess}(\\sigma)_p = T_p S$ for all $p \\in S$. If $U$ is trivial, then a Morse--Bott section for $U$ is not the same thing as a Morse--Bott function on $X$, as we only demand a condition on its zero set. In this sense, an alternate way of phrasing the definition is that $\\sigma$ must specify the germ of a Morse--Bott function around $Z_\\sigma$.\n\nThe elliptic divisors of \\autoref{exa:elliptic} are of Morse--Bott type, but normal-crossing log divisors (\\autoref{exa:normalcrossinglog}) are not. The analogue of the linearization result for elliptic divisors (see \\cite[Proposition 2.20]{CavalcantiKlaasse18}) is readily seen to hold for all divisors of Morse--Bott type (see \\cite{Klaasse17}), so that locally the associated divisor ideal $I_\\sigma$ will be given by $\\langle Q_g \\rangle$ with $Q_g$ a homogeneous function in normal bundle coordinates. Note that all Morse--Bott type divisors are smooth.\n\\begin{rem} Elliptic divisors arise as the real divisors which underlie complex log divisors (c.f.\\ \\cite{CavalcantiGualtieri18}). There is another as of yet unexplored class of real divisors, namely those underlying \\emph{quaternionic log divisors}, i.e.\\ coming from a quaternionic line bundle with a transversally vanishing section. These are supported on smooth submanifolds of codimension four, and will be explored further in future work.\n\\end{rem}\n\n\\section{Introduction}\n\\label{sec:introduction}\nPoisson geometry can be viewed as the study of singular symplectic foliations. Poisson structures range from the zero Poisson structure, whose symplectic foliation is by points, to nondegenerate Poisson structures having only one symplectic leaf, the entire manifold. Within this wide range of possibilities, one can impose restrictions on the characteristics of the symplectic foliations that may occur, to obtain subclasses of Poisson manifolds.\n\nIn this paper we focus on Poisson structures whose symplectic foliation is relatively mild. These are the Poisson structures of divisor-type: those Poisson structures which are generically regular, and whose degeneracy is governed by a divisor ideal. These divisor ideals are locally principal ideals whose local generators have nowhere dense zero set, which we will elaborate on below. Examples of such Poisson structures are those which are generically nondegenerate, i.e.\\ those whose union of open symplectic leaves is dense.\n\nThe class of Poisson structures of divisor-type coincides with the almost-regular Poisson structures of Androulidakis--Zambon \\cite{AndroulidakisZambon17}. They define almost-regular Poisson structures from the point of view of singular foliations, namely as those Poisson structures $\\pi$ on a manifold $X$ whose module of Hamiltonian vector fields $\\mathcal{F}_\\pi = \\pi^\\sharp(\\Gamma(T^*X)) \\subseteq \\Gamma(TX)$ is projective. By results of \\cite{Debord01}, this class is characterized by having a smooth associated holonomy groupoid.\n\nOur viewpoint -- considering these Poisson structures through their induced divisor -- allows us to  study them using Lie algebroids which are tailor-made to their degeneracy. In this paper we develop a framework for the study of Poisson structures of divisor-type. Below we outline the ideas underlying this framework, and mention our main results.\n\nThe techniques and language developed in this paper are used to various establish results for almost-regular Poisson structures, including the computation of Poisson cohomology \\cite{KlaasseLanius18}, determining the adjoint symplectic groupoids integrating them \\cite{KlaasseLi18}, and developing topological obstructions to their existence \\cite{Klaasse18three}. Moreover, this framework was used in \\cite{CavalcantiKlaasse18} in the development of Gompf--Thurston methods for symplectic Lie algebroids.\n\\subsection*{Poisson structures of divisor-type}\nIn this paper we adapt the language of divisors to the context of smooth manifolds, following their use in \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18}. A divisor consist of a line bundle together with a section whose zero set is nowhere dense. Each such pair $(L,s)$ specifies a divisor ideal $I_s := s(\\Gamma(L^*)) \\subseteq C^\\infty(X)$. Abstractly, divisor ideals are locally principal ideals $I$ for which the support $Z(I) \\subseteq X$ of the quotient sheaf $C^\\infty(X) \/ \\mathcal{I}$ is nowhere dense, where\n\\begin{equation*}\n\tZ(I) = \\{x \\in X \\, | \\, f(x) = 0 \\text{ for all }f \\in \\mathcal{I}_x\\}.\n\\end{equation*}\nThese ideals keep track of degenerative behavior along this support, and are an extension of the concept of a Cartier divisor in algebraic and complex geometry to the smooth setting.\n\nPoisson structures naturally lead to divisors: Given $\\pi \\in {\\rm Poiss}(X)$ such that $\\dim(X) = 2n$, consider its Pfaffian $\\wedge^n \\pi \\in \\Gamma(\\det(TX))$. Note that $\\pi$ is nondegenerate exactly at those points $x \\in X$ where $\\wedge^n \\pi_x \\neq 0$. Thus $\\pi$ is generically nondegenerate if and only if the pair $(\\det(TX),\\wedge^n \\pi)$ is a divisor. In this case we denote this pair by ${\\rm div}(\\pi)$, with associated divisor ideal $I_\\pi \\subseteq C^\\infty(X)$. This ideal captures the behavior of $\\pi$ along $Z(\\pi) = Z(I_\\pi)$, called the degeneracy locus of $\\pi$, consisting of points where the rank of $\\pi$ is not maximal. \n\nIn general, given $\\pi \\in {\\rm Poiss}(X)$, the subset $X_{\\pi,{\\rm reg}} \\subseteq X$ of points where $\\pi$ is regular is open and dense. In contrast, the subset $X_{\\pi, {\\rm max}} \\subseteq X_{\\pi,{\\rm reg}}$ of points of maximal rank is not. Given an integer $m \\geq 0$, a Poisson structure is of $m$-divisor-type if it is generically of rank $2m$, and its failure to be regular is governed by a divisor. More precisely, for its partial Pfaffians we demand that $\\wedge^{m+1} \\pi \\equiv 0$ and there exists a line subbundle $K \\subseteq \\wedge^{2m} TX$ such that $(K, \\wedge^m \\pi)$ is a divisor, which provides a divisor ideal $I_\\pi$. A consequence of this definition is that $X_{\\pi,{\\rm max}} = X_{\\pi,{\\rm reg}}$ is open and dense, so that $\\pi$ is generically regular of rank $2m$. We stress, however, that not every generically regular Poisson structure is of $m$-divisor-type. This definition is closely related to almost-regularity as defined in \\cite{AndroulidakisZambon17}, as we discuss in \\autoref{sec:almostregularpoisson}.\n\nOne class of generically-nondegenerate Poisson structures is that of log-Poisson structures \\cite{GuilleminMirandaPires14,MarcutOsornoTorres14,Cavalcanti17,GualtieriLi14}, (also called log- or $b$-symplectic). Another class is given by elliptic Poisson structures \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18}, which arise (partially) as the Poisson structures underlying stable generalized complex structures. Further examples include $b^k$-Poisson structures \\cite{Scott16,GuilleminMirandaWeitsman17,Lanius16} and scattering Poisson structures \\cite{Lanius16}. Moreover, in this paper we introduce a novel class called elliptic-log Poisson structures (a more in-depth study of them is deferred to \\cite{KlaasseLanius18}). Each of these Poisson structures of divisor-type has a generically-regular analogue. In particular, there are $m$-log-Poisson structures of generic rank $2m$ (called log-f Poisson structures in \\cite{AndroulidakisZambon17}).\n\nBy keeping track of the divisor associated to a Poisson structure $\\pi$ of divisor-type, we can define almost-injective Lie algebroids $\\mathcal{A} \\to X$ whose degeneracy is similar or identical to that of $\\pi$. These Lie algebroids are candidates for a lifting procedure, whereby $\\pi$ is (partially) desingularized to a Lie algebroid Poisson structure. We first describe such Lie algebroids.\n\\subsection*{Lie algebroids of divisor-type}\nDivisor ideals can also be used to capture the degeneracy of Lie algebroids. This leads to Lie algebroids of divisor-type, which are almost-injective Lie algebroids for which the degeneration of the anchor map is controlled in a precise sense:\n\nGiven a Lie algebroid $\\mathcal{A} \\to X$, one can view its anchor as a section $\\rho_{A} \\in \\Gamma(\\mathcal{A}^* \\otimes TX)$. If ${\\rm rank}(\\mathcal{A}) = \\dim(X)$, then the zero locus of $\\det(\\rho_\\mathcal{A})$ consists of those points where $\\rho_{A}$ is not an isomorphism. Thus $\\mathcal{A}$ is almost-injective if and only if the pair $(\\det(\\mathcal{A}^*) \\otimes \\det(TX), \\det(\\rho_{A}))$ is a divisor, which we denote by ${\\rm div}(\\mathcal{A})$. In fact, divisors can be associated to any bundle map between bundles of the same rank, so that ${\\rm div}(\\mathcal{A}) = {\\rm div}(\\rho_{A})$. With this in mind, a Lie algebroid $\\mathcal{A} \\to X$ is of $m$-divisor-type for some $m \\geq 0$ if there exists a Lie subalgebroid $D \\subseteq TX$ of rank $m$ so that $\\rho_{A}(\\mathcal{A}) \\subseteq D$, and for which the induced Lie algebroid morphism $\\varphi_{\\mathcal{A}}\\colon \\mathcal{A} \\to D$ is of divisor-type. This forces that $\\mathcal{A}$ is of rank $m$, and that it is almost-injective.\n\nAbove we saw how a Lie algebroid $\\mathcal{A}$ has an associated divisor ${\\rm div}(\\mathcal{A})$, and consequently leads to a divisor ideal $I_\\mathcal{A} \\subseteq C^\\infty(X)$. There are also reverse procedures, where divisor ideals are used to construct Lie algebroids of divisor-type. The first of these proceeds as follows: Given a divisor ideal $I \\subseteq C^\\infty(X)$, consider the module of derivations preserving the ideal $I$\n\\begin{equation*}\n\t\\Gamma(TX)_I := \\{V \\in \\Gamma(TX) \\, | \\, \\mathcal{L}_V I \\subseteq I\\}.\n\\end{equation*}\nIf this module is projective, by the Serre--Swan theorem there is a unique almost-injective Lie algebroid $TX_I \\to X$ such that $\\Gamma(TX_I) \\cong \\Gamma(TX)_I$. This is called the ideal Lie algebroid associated to $I$, and we call $I$ projective if $\\Gamma(TX)_I$ is projective. The Lie algebroid $TX_I$ is of divisor-type, with divisor ideal $I_{TX_I}$. Here $I$ and $I_{TX_I}$ might differ, but are often related. This construction recovers the (complex) log- and elliptic tangent bundles \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18,GuilleminMirandaPires14,MarcutOsornoTorres14,Melrose93}.\n\nA second method to construct Lie algebroids  from divisors is the following: Let $\\mathcal{A} \\to X$ be a Lie algebroid, and consider a divisor ideal $I \\subseteq C^\\infty(X)$ whose support $Z(I)$ is a smooth submanifold. Given a Lie subalgebroid $\\mathcal{B}$ of $\\mathcal{A}$ which is supported on $Z(I)$, one can perform elementary modification to construct a new Lie algebroid $[\\mathcal{A}\\:(\\mathcal{B},I)]$ which is isomorphic to $\\mathcal{A}$ outside of $Z(I)$. In fact, there is an induced Lie algebroid morphism $\\varphi\\colon [\\mathcal{A}\\:(\\mathcal{B},I)] \\to \\mathcal{A}$ which is of divisor-type with divisor ideal $I_\\varphi = I^k$, where $k$ is the corank of $\\mathcal{B}$ with respect to $\\mathcal{A}$. This procedure extends the so-called modification procedures (or rescaling) performed in the case where $I = I_Z$, the vanishing ideal of a smooth hypersurface $Z \\subseteq X$ (see \\cite{Li13,GualtieriLi14,Lanius16,Melrose93}).\n\\subsection*{Lifting Poisson structures}\nPoisson structures and Lie algebroids of divisor-type are connected through the process of lifting. Given a Lie algebroid $\\mathcal{A} \\to X$, an $\\mathcal{A}$-Poisson structure $\\pi_{\\mathcal{A}} \\in {\\rm Poiss}(\\mathcal{A})$ is a section $\\pi_{A} \\in \\Gamma(\\wedge^2 \\mathcal{A})$ satisfying $[\\pi_{\\mathcal{A}},\\pi_{\\mathcal{A}}]_{\\mathcal{A}} = 0$, where $[\\cdot,\\cdot]_\\mathcal{A}$ is the Schouten bracket on $\\Gamma(\\wedge^\\bullet \\mathcal{A})$ extending the Lie bracket on $\\Gamma(\\mathcal{A})$. Such a structure turns the tuple $(\\mathcal{A},\\mathcal{A}^*,\\pi_{\\mathcal{A}})$ into a triangular Lie bialgebroid (\\cite{KosmannSchwarzbach95,KosmannSchwarzbachLaurentGengoux05,LiuWeinsteinXu97,MackenzieXu94}). Given $\\pi \\in {\\rm Poiss}(X)$, we say that $\\pi$ is of $\\mathcal{A}$-type if it admits an $\\mathcal{A}$-lift, i.e.\\ an $\\mathcal{A}$-Poisson structure $\\pi_{\\mathcal{A}}$ such that $\\rho_{\\mathcal{A}}(\\pi_{\\mathcal{A}}) = \\pi$.\n\nIf both $\\mathcal{A}$ and $\\pi$ are of divisor-type, then so is $\\pi_{\\mathcal{A}}$, and their divisor ideals are related:\n\\begin{equation*}\n\tI_\\pi = I_{\\pi_{\\mathcal{A}}} \\cdot I_\\mathcal{A}.\n\\end{equation*}\nIn other words, part of the degeneracy of $\\pi$ is absorbed into that of the anchor of $\\mathcal{A}$, making its $\\mathcal{A}$-lift $\\pi_{\\mathcal{A}}$ necessarily more regular. Because of the nature of the Lie algebroids considered in this paper, it is possible to use symplectic techniques to study nondegenerate $\\mathcal{A}$-Poisson structures. Indeed, a nondegenerate $\\mathcal{A}$-Poisson structure is dual to an $\\mathcal{A}$-symplectic structure, making $\\mathcal{A}$ into a symplectic Lie algebroid \\cite{NestTsygan01}. For example, Moser methods are available for such symplectic Lie algebroids $\\mathcal{A}$ of divisor-type (see \\cite[Theorem 5.5]{KlaasseLanius18}), which lead to Darboux-type normal form results for $\\pi_{\\mathcal{A}}$, and hence for the underlying $\\pi \\in {\\rm Poiss}(X)$.\n\nGiven a Poisson structure $\\pi$ of divisor-type, we would like to find a Lie algebroid $\\mathcal{A}$ for which $I_\\mathcal{A} = I_\\pi$, so that any $\\mathcal{A}$-lift of $\\pi$ will be nondegenerate. Sometimes such Lie algebroids can be constructed from the divisor ideal $I_\\pi$. This lifting process necessitates the study of Poisson and symplectic structures in a general Lie algebroid $\\mathcal{A}$, which is the aim of this paper.\n\nWhile most of this paper is written in terms of $\\mathcal{A}$-Lie algebroids and $\\mathcal{A}$-Poisson structures, our main interest is to study Poisson structures $\\pi \\in {\\rm Poiss}(X)$ in the standard sense. The $\\mathcal{A}$-notions have to be developed for their own sake, but are mainly used as auxiliary tools through the process of lifting. Note that there can be multiple non-isomorphic Lie algebroids to which one can nondegenerately lift a given Poisson structure (see \\cite{KlaasseLanius18}).\n\\subsection*{Results}\nThe main aim of this paper is to develop an effective framework for the study of Poisson structures of divisor-type. Because of this we introduce and study:\n\\begin{itemize}\n\t\\item Divisor ideals keeping track of degeneracies;\n\t\\item Lie algebroids of divisor-type, with emphasis on:\n\t%\n\t\\begin{itemize}\n\t\t\\item Ideal Lie algebroids of derivations preserving a divisor ideal;\n\t\t\\item Elementary modification of Lie algebroids using divisor ideals;\n\t\\end{itemize}\n\t\\item Poisson structures of divisor-type, leading to:\n\t%\n\t\\begin{itemize}\n\t\t\\item The process of lifting Poisson structures to their $\\mathcal{A}$-Poisson counterparts;\n\t\\end{itemize}\n\t\\item Residue maps to extract information along the degeneraci loci;\n\t\\item The relation between the lifting Lie algebroid and the modular foliation.\n\\end{itemize}\nHere we highlight several aspects of this undertaking that we provide in this paper. \n\nThe first result concerns lifting for Poisson structures of divisor-type to the ideal Lie algebroids they define. This recaptures the lifting results for log- and elliptic Poisson structures of \\cite{GuilleminMirandaPires14} and \\cite{CavalcantiGualtieri18}. The following is \\autoref{thm:ipoissonaitype}, on lifting to the ideal Lie algebroid $TX_I$.\n\\begin{thmalpha}[Lifting]\\label{thm:introlifting} Let $\\pi \\in {\\rm Poiss}(X)$ be of projective $I$-divisor-type with $TX_I^*$ admitting a local basis of closed sections. Then $\\pi$ is of $TX_I$-type, and of nondegenerate $TX_I$-type if and only if $I$ is standard, i.e.\\ the divisor ideal of $TX_I$ satisfies $I_{TX_I} = I$.\n\\end{thmalpha}\nThe next theorem discusses the residue maps of Lie algebroids $\\mathcal{A} \\to X$. Given an invariant submanifold $D \\subseteq X$ on which the module $(\\rho_\\mathcal{A}|_D)(\\Gamma(\\mathcal{A}|_D)) \\cong \\Gamma(\\mathcal{B})$ is projective, there is an induced surjective morphism $\\widetilde{\\rho}_\\mathcal{A}|_D\\colon \\mathcal{A}|_D \\to \\mathcal{B}$ resulting in a Lie algebroid extension sequence\n\\begin{equation*}\n\t0 \\to E \\to \\mathcal{A}|_D \\to \\mathcal{B} \\to 0.\n\\end{equation*}\nThe resulting residue maps are used in computing the Lie algebroid cohomology of $\\mathcal{A}$, and to extract information from $\\mathcal{A}$-symplectic structures. In \\autoref{sec:residues} we conceptually describe the residue maps found in  \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18,GuilleminMirandaPires14,MarcutOsornoTorres14,Scott16}, and are used in \\cite{KlaasseLanius18}. The following is \\autoref{thm:residues}.\n\\begin{thmalpha}[Residue maps]\\label{thm:introresidues} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $D \\subseteq X$ be a projective $\\mathcal{A}$-invariant submanifold with induced Lie algebroid $\\mathcal{B} \\to D$. Then there exists a residue map\n\\begin{equation*}\n\t{\\rm Res}_D\\colon \\Omega^\\bullet(X,\\mathcal{A}) \\to \\Omega^{\\bullet-\\ell}(D,\\mathcal{B}; \\det(E^*)), \\qquad \\text{where } \\ell = {\\rm rank}(E).\n\\end{equation*}\nThis is a cochain morphism for the natural $\\mathcal{B}$-representation on $\\det(E^*)$, hence descends to\n\\begin{equation*}\n\t[{\\rm Res}_D]\\colon H^\\bullet(X,\\mathcal{A}) \\to H^{\\bullet-\\ell}(D,\\mathcal{B}; \\det(E^*)).\n\\end{equation*}\n\\end{thmalpha}\nWe further discuss the relation between the lifting process and the modular foliation \\cite{GualtieriPym13}, which arises as the sheaf of submodules of $\\Gamma(TX)$ spanned by the modular and Hamiltonian vector fields. The symplectic foliation $\\mathcal{F}_\\pi$ and the modular foliation $\\mathcal{F}_{\\pi,{\\rm mod}}$ of a Poisson structure $\\pi \\in {\\rm Poiss}(X)$ interact with any lifting Lie algebroid. The following is \\autoref{thm:modfoliation}.\n\\begin{thmalpha}[Modular foliation]\\label{thm:intromodfoliation} Let $\\pi \\in {\\rm Poiss}(X)$ be of $\\mathcal{A}$-type for a Lie algebroid $\\mathcal{A} \\to X$. Then $\\mathcal{F}_{\\pi} \\subseteq \\mathcal{F}_{\\pi, {\\rm mod}} \\subseteq \\rho_\\mathcal{A}(\\Gamma(\\mathcal{A})) \\subseteq \\Gamma(TX)$, and the orbits of $\\mathcal{A}$ are $\\pi$-Poisson submanifolds.\n\\end{thmalpha}\nThis result is most interesting when $\\mathcal{A}$ is almost-injective, i.e.\\ when $\\rho_\\mathcal{A}$ is injective on sections. In this case \\autoref{thm:intromodfoliation} gives an intuitive picture of the lifting procedure. It also helps in determining prospective candidate Lie algebroids to lift Poisson structures to.\n\\begin{rem} In this paper we do not discuss Dirac geometry nor generalized complex geometry. Their interaction with divisors and the lifting procedure are instead explored in \\cite{Klaasse18}, see also \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18, Klaasse17}. We do not discuss the Lie groupoids associated to the Lie algebroids we consider, which are all integrable as they are almost-injective \\cite{CrainicFernandes03,CrainicFernandes04,Debord01}. See \\cite{GualtieriLi14,KlaasseLi18,Li13} for explicit constructions of integrations in the setting of log- and elliptic Poisson manifolds.\n\\end{rem}\n\\subsection*{Acknowledgements} This work is partially based on the author's Ph.D.\\ thesis \\cite{Klaasse17}, and was supported by ERC grant 646649, and by VIDI grant 639.032.221 from NWO, the Netherlands Organisation for Scientific Research. The author would like to thank Melinda Lanius, Rui Loja Fernandes, Ioan M{\\u{a}}rcu{\\c{t}}, Brent Pym, Marco Zambon, and especially his Ph.D.\\ advisor Gil Cavalcanti for useful discussions. Moreover, the author would like to thank the University of Illinois at Urbana--Champaign for its hospitality in early 2017 during a part of this project.\n\\subsection*{Organization of the paper}\nThis paper is structured as follows. In \\autoref{sec:divonmanifolds} we discuss the theory of divisors on smooth manifolds, emphasizing especially the notion of a divisor ideal. In \\autoref{sec:liealgebroidsdivtype} we then turn to Lie algebroids of divisor-type, and how divisor ideals can be used to construct them. Then, in \\autoref{sec:poissondivtype} we consider Poisson structures of divisor-type in general Lie algebroids, and discuss the process of lifting Poisson structures along Lie algebroid morphisms. In \\autoref{sec:residues} we then discuss residue maps for Lie algebroids and Poisson modules, and how these can be used to study symplectic Lie algebroids. We further study the modular foliation of a Poisson manifold, and how it interacts with the lifting procedure.\n\n\\section{Lie algebroids of divisor-type}\n\\label{sec:liealgebroidsdivtype}\nIn this section we discuss Lie algebroids and their interaction with divisors, starting in \\autoref{sec:liealgebroids} with how Lie algebroids can specify divisors through the degeneracy of their anchor. \\autoref{sec:aliealgebroids} through \\autoref{sec:singularliealgebroids} discuss several further concepts that we will use.\n\nWe describe two processes by which Lie algebroids can be constructed using divisors. The first of these gives rise to ideal Lie algebroids (in \\autoref{sec:idealliealgebroids}), which are those Lie algebroids whose sections are those preserving a given ideal of functions. The second is that of elementary modification (in \\autoref{sec:elementarymodifications}). Here one uses a injective morphism from, or a surjective comorphism to, a subbundle supported on the zero locus of a smooth divisor ideal. \n\\subsection{Lie algebroids}\n\\label{sec:liealgebroids}\nFor general information regarding Lie algebroids, see \\cite{Mackenzie05}. A vector bundle $\\mathcal{A} \\to X$ is \\emph{anchored} if it has a bundle map $\\rho_{A}\\colon \\mathcal{A} \\to TX$ called its \\emph{anchor}. We refer to morphisms of anchored bundles (those intertwining the anchor maps) by \\emph{anchored morphisms}.\n\\begin{defn} \\label{defn:liealgebroid} A \\emph{Lie algebroid} $(\\mathcal{A},[\\cdot,\\cdot]_{\\mathcal{A}},\\rho_\\mathcal{A})$ is an anchored bundle with a Lie bracket $[\\cdot,\\cdot]_{\\mathcal{A}}$ on $\\Gamma(\\mathcal{A})$ such that $[v,f w]_{\\mathcal{A}} = f [v,w]_{\\mathcal{A}} + \\mathcal{L}_{\\rho_{\\mathcal{A}}(v)} f \\cdot w$ for all $v, w \\in \\Gamma(\\mathcal{A})$, $f \\in C^\\infty(X)$.\n\\end{defn}\nThe bracket $[\\cdot,\\cdot]_\\mathcal{A}$ of a Lie algebroid $\\mathcal{A}$ provides the algebra $\\Omega^\\bullet(\\mathcal{A}) = \\Gamma(\\wedge^\\bullet \\mathcal{A}^*)$ with a differential $d_\\mathcal{A}$ using the Koszul formula. This allows for a simple definition of morphisms.\n\\begin{defn} A \\emph{Lie algebroid morphism} is a bundle morphism $(\\varphi,f)\\colon \\mathcal{A} \\to \\mathcal{A}'$ for which the induced map $\\varphi^*\\colon \\Omega^*(\\mathcal{A}') \\to \\Omega^*(\\mathcal{A})$ is a chain map, i.e.\\ satisfies $\\varphi^* \\circ d_{\\mathcal{A}'} = d_\\mathcal{A} \\circ \\varphi^*$.\n\\end{defn}\n\\begin{rem} Given $v \\in \\Gamma(\\mathcal{A})$, its \\emph{$\\mathcal{A}$-Lie derivative} is the operator $\\mathcal{L}_v = \\{\\iota_v, d_\\mathcal{A}\\}$ on $\\Omega^\\bullet(\\mathcal{A})$, where $\\{\\cdot,\\cdot\\}$ is the graded commutator. It satisfies the usual Cartan relations (see e.g.\\ \\cite{Marle08}).\n\\end{rem}\nLie algebroid morphisms are in particular anchored morphisms, i.e.\\ they satisfy the relation $\\rho_{\\mathcal{A}'} \\circ \\varphi = Tf \\circ \\rho_\\mathcal{A}$. If $(\\varphi,f)\\colon \\mathcal{A} \\to \\mathcal{A}'$ is a base-preserving anchored morphism, it is a Lie algebroid morphism if and only if it moreover induces a Lie algebra homomorphism $\\varphi\\colon \\Gamma(\\mathcal{A}) \\to \\Gamma(\\mathcal{A}')$. Note that the anchor map $\\rho_\\mathcal{A}$ is a Lie algebroid morphism from $\\mathcal{A}$ to $TX$. For base-preserving Lie algebroid morphisms, we keep track of where it is an isomorphism.\n\\begin{defn} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}$ be a Lie algebroid morphism. Its \\emph{isomorphism locus} is the open subset $X_\\varphi \\subseteq X$ where $\\varphi$ is an isomorphism.\n\\end{defn}\nThe complement of the isomorphism locus is denoted by $Z_\\varphi := X \\backslash X_\\varphi$, and is called the \\emph{degeneracy locus} of $\\varphi$. For a Lie algebroid $\\mathcal{A}$ we define $X_{\\mathcal{A}} := X_{\\rho_\\mathcal{A}}$ and refer to this as the isomorphism locus of $\\mathcal{A}$, and similarly for $Z_\\mathcal{A}$. Note that whenever $X_\\varphi$ is nonempty, we have ${\\rm rank}(\\mathcal{A}) = {\\rm rank}(\\mathcal{A}')$ because $X_\\varphi$ is open. Between bundles of the same rank, the condition that $X_\\varphi$ is dense is equivalent to injectivity of the induced map on sections $\\varphi\\colon \\Gamma(\\mathcal{A}') \\to \\Gamma(\\mathcal{A})$.\n\\begin{rem} Given two composable base-preserving Lie algebroid morphisms $\\varphi$ and $\\varphi'$, we have $X_{\\varphi \\circ \\varphi'} \\supseteq X_\\varphi \\cap X_{\\varphi'}$. If we further assume that all Lie algebroids involved are of the same rank (e.g.\\ by each isomorphism locus being nonempty), this becomes an equality. In particular, in this case, if the isomorphism locus $X_{\\varphi \\circ \\varphi'}$ is dense, then so are $X_\\varphi$ and $X_{\\varphi'}$.\n\\end{rem}\nGiven a Lie algebroid morphism $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}$, we can take determinants to obtain $\\det(\\varphi)\\colon \\det(\\mathcal{A}') \\to \\det(\\mathcal{A})$. This specifies a section $\\det(\\varphi) \\in \\Gamma(\\det(\\mathcal{A}'^*) \\otimes \\det(\\mathcal{A}))$. It is now readily verified that $\\varphi$ has dense isomorphism locus if and only if this defines a divisor.\n\\begin{defn}\\label{defn:morphismdivisortype} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}$ be a Lie algebroid morphism with dense isomorphism locus. The \\emph{divisor associated to $\\varphi$} is ${\\rm div}(\\varphi) = (\\det(\\mathcal{A}'^*) \\otimes \\det(\\mathcal{A}), \\det(\\varphi))$, with the associated divisor ideal being denoted by $I_\\varphi$. Given a Lie algebroid $\\rho_\\mathcal{A}\\colon \\mathcal{A} \\to TX$ with dense isomorphism locus, we define ${\\rm div}(\\mathcal{A}) := {\\rm div}(\\rho_\\mathcal{A})$ and $I_\\mathcal{A} := I_{\\rho_\\mathcal{A}}$, and call $\\mathcal{A}$ of \\emph{divisor-type}.\n\\end{defn}\nGiven a divisor ideal $I$ we call a base-preserving Lie algebroid morphism $\\varphi$ \\emph{of $I$-divisor-type} if $I_\\varphi = I$, and similarly for Lie algebroids $\\mathcal{A}$. It is immediate that composition of morphisms of divisor-type corresponds to taking products of their divisors (hence their divisors ideals): after noting that $\\det(\\varphi' \\circ \\varphi) = \\det(\\varphi') \\circ \\det(\\varphi)$ we obtain the equality $I_{\\varphi' \\circ \\varphi} = I_{\\varphi'} \\cdot I_\\varphi$.\n\\begin{prop}\\label{prop:divisorcomposition} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ and $(\\varphi',{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}''$ be Lie algebroid morphisms of divisor-type. Then $\\varphi' \\circ \\varphi$ is of divisor-type, and ${\\rm div}(\\varphi' \\circ \\varphi) \\cong {\\rm div}(\\varphi') \\otimes {\\rm div}(\\varphi)$.\n\\end{prop}\nIn the generically-regular setting, we define a Lie algebroid morphism $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}$ to be of \\emph{$m$-divisor-type} for some $m \\geq 0$ if there exists a regular Lie subalgebroid $D_\\mathcal{A} \\subseteq \\mathcal{A}$ of rank $m$ (see \\autoref{sec:liesubalgds}, in particular \\autoref{exa:adistribution}) such that $\\varphi$ factors through the inclusion $\\varphi_{D_\\mathcal{A}}\\colon D_{\\mathcal{A}} \\to \\mathcal{A}$, i.e.\\ $\\varphi = \\varphi_{D_\\mathcal{A}} \\circ \\varphi_{\\mathcal{A}'}$ for some Lie algebroid morphism $\\varphi_{\\mathcal{A}'}\\colon \\mathcal{A}' \\to D_{\\mathcal{A}}$, and $\\varphi_{\\mathcal{A}'}$ is of divisor-type, with divisor given by ${\\rm div}(\\varphi_{\\mathcal{A}'}) = (\\det(\\mathcal{A}'^*) \\otimes \\det(D_{\\mathcal{A}}), \\det(\\varphi_{\\mathcal{A}'}))$. Alternatively, we can demand the existence of a line subbundle $K_\\mathcal{A} \\subseteq \\wedge^m \\mathcal{A}$ such that $\\wedge^m \\varphi\\colon \\det(\\mathcal{A}') \\to K_\\mathcal{A}$ and $(\\det(\\mathcal{A}'^*)\\otimes K_\\mathcal{A}, \\wedge^m \\varphi)$ is a divisor. These formulations are related via $\\det(D_\\mathcal{A}) = K_\\mathcal{A}$. We then define $\\mathcal{A}$ to be of $m$-divisor-type if its anchor $\\rho_{A}\\colon \\mathcal{A} \\to TX$ is.\n\nWe introduce several further adjectives to describe the behavior of the anchor.\n\\begin{defn}\\label{defn:lalgebroidadjective} Let $\\mathcal{A} \\to X$ be a Lie algebroid. Then $\\mathcal{A}$ is \\emph{regular} if $\\rho_\\mathcal{A}$ has constant rank, \\emph{transitive} if $\\rho_\\mathcal{A}$ is surjective, and \\emph{totally intransitive} if $\\rho_\\mathcal{A} \\equiv 0$. Further, $\\mathcal{A}$ is \\emph{almost-injective} \\cite{Debord01} if $\\rho_\\mathcal{A}\\colon \\Gamma(\\mathcal{A}) \\to \\Gamma(TX)$ is injective, and \\emph{projective} if $\\rho_{A}(\\Gamma(\\mathcal{A})) \\subseteq \\Gamma(TX)$ is projective.\n\\end{defn}\nThese properties are related: transitivity implies regularity, which in turn implies projectivity. If ${\\rm rank}(\\mathcal{A}) = \\dim X$, then $\\mathcal{A}$ is almost-injective if and only if it is of divisor-type.\n\\begin{defn}\\label{defn:laisotropy} The \\emph{isotropy} of $\\mathcal{A} \\to X$ at $x \\in X$ is given by the subspace $\\ker \\rho_{\\mathcal{A},x} \\subseteq \\mathcal{A}_x$.\n\\end{defn}\nThis subspace makes sense for any anchored vector bundle, but for Lie algebroids the isotropy is a Lie algebra, inheriting a bracket from $\\mathcal{A}$. The isotropy Lie algebras $\\ker \\rho_{\\mathcal{A},x}$ may vary in dimension as $x \\in X$ varies, so that in general these may not form a subbundle of $\\mathcal{A}$. A Lie algebroid (or anchored) morphism $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$ restricts to a map $\\varphi\\colon \\ker \\rho_{\\mathcal{A},x} \\to \\ker \\rho_{\\mathcal{A}',f(x)}$ between pointwise isotropies for all $x \\in X$.\n\nThe image of any Lie algebroid under the anchor map determines a singular foliation\n\\begin{equation*}\n\t\\mathcal{F}_\\mathcal{A} := \\rho_{\\mathcal{A}}(\\Gamma(\\mathcal{A})) \\subseteq \\Gamma(TX)\n\\end{equation*}\nin the sense of Androulidakis--Skandalis \\cite{AndroulidakisSkandalis09}. Its leaves, i.e.\\ the associated maximal immersed submanifolds $\\mathcal{O}$ satisfying $T_x \\mathcal{O} = {\\rm im}\\, \\rho_{\\mathcal{A},x}$ for all $x \\in \\mathcal{O}$, are called the \\emph{orbits} of $\\mathcal{A}$.\n\nFor regular Lie algebroids (hence for transitive ones), the kernel of $\\rho_\\mathcal{A}$ is of constant dimension, so that it is a subbundle of $\\mathcal{A}$. If $\\mathcal{A}$ is transitive, the isotropy Lie algebras together exhibit $\\mathcal{A}$ as an abelian extension of $TX$. In other words, there is a short exact sequence:\n\\begin{equation*}\n\t0 \\to \\ker \\rho_\\mathcal{A} \\to \\mathcal{A} \\to TX \\to 0.\n\\end{equation*}\nA similar sequence exists for projective Lie algebroids. In this case (as in \\autoref{sec:singularliealgebroids}), there exists an almost-injective Lie algebroid $\\mathcal{B} \\to X$ with $\\Gamma(\\mathcal{B}) \\cong \\rho_{A}(\\Gamma(\\mathcal{A}))$ which is unique up to isomorphism. This comes with an induced Lie algebroid morphism $\\widetilde{\\rho}_\\mathcal{A}\\colon \\mathcal{A} \\to \\mathcal{B}$ induced by the map on sections $\\rho_{A}\\colon \\Gamma(\\mathcal{A}) \\to \\Gamma(\\mathcal{B})$. This results in a short exact sequence of Lie algebroids:\n\\begin{equation*}\n\t0 \\to \\ker \\widetilde{\\rho}_\\mathcal{A} \\to \\mathcal{A} \\to \\mathcal{B} \\to 0.\n\\end{equation*}\nUnlike the transitive case, the Lie algebroid $\\ker \\widetilde{\\rho}_\\mathcal{A}$ need not be abelian. Following \\cite[Definition 1.10]{AndroulidakisZambon17}, we make the following definition in this case.\n\\begin{defn}\\label{defn:germinalisotropy} The \\emph{germinal isotropy} of $\\mathcal{A} \\to X$ at $x \\in X$ is the subspace $\\ker \\widetilde{\\rho}_{\\mathcal{A},x} \\subseteq \\mathcal{A}_x$.\n\\end{defn}\nIn general one has $\\ker \\widetilde{\\rho}_{\\mathcal{A},x} \\subseteq \\ker \\rho_{\\mathcal{A},x}$. This definition still makes sense when the module $\\mathcal{F}_\\mathcal{A} \\subseteq \\Gamma(TX)$ is projective: if it is not, we can still consider the vector spaces $\\mathcal{F}_\\mathcal{A} \/ I_x \\cdot \\mathcal{F}_\\mathcal{A}$. We will return to the germinal isotropy in \\autoref{sec:singularliealgebroids}.\\\\\n\nLet $\\mathcal{A} \\to X$ be a Lie algebroid with dense isomorphism locus $X_\\mathcal{A}$. Then the inclusion $i\\colon X_{\\mathcal{A}} \\hookrightarrow X$ gives a bijection $\\rho_{\\mathcal{A}}^*\\colon \\Omega^k(X_{\\mathcal{A}}) \\to \\Omega^k_{\\mathcal{A}}(X_{\\mathcal{A}})$ for all $k$. Consequently, we can view $\\mathcal{A}$-forms as smooth forms with certain ``singular'' behavior at its degeneracy locus $Z_\\mathcal{A}$. The isomorphism given by the restricted anchor map $\\rho_\\mathcal{A}\\colon \\mathcal{A}|_{X_\\mathcal{A}} \\to TX|_{X_\\mathcal{A}}$ also implies the following.\n\\begin{prop}\\label{prop:denseisolocusstrongmap} Let $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$ be an anchored morphism between Lie algebroids of divisor-type such that $f^{-1}(X_{\\mathcal{A}'}) = X_\\mathcal{A}$. Then $\\varphi \\equiv Tf$ on sections, and $\\varphi$ is a Lie algebroid morphism.\n\\end{prop}\n\\bp When restricted to the isomorphism loci, we have that $\\varphi \\equiv Tf$ as bundle maps after intertwining with the anchors. Because the isomorphism loci are dense, this means that $\\varphi \\equiv Tf$ on sections, which holds globally on $X$. Because $\\varphi$ is a bundle morphism, the map $\\varphi^*$ is an algebra morphism. In the isomorphism loci, $\\varphi$ must equal $Tf$, and $(Tf,f)$ is a Lie algebroid morphism between $TX$ and $TX'$, which means that $f^*$ is a chain map. By the density of $X_{\\mathcal{A}}$ in $X$, we conclude that the map $\\varphi^*$ is a chain map everywhere.\n\\end{proof}\nThus, for Lie algebroids as \\autoref{prop:denseisolocusstrongmap}, to determine whether there is a Lie algebroid morphism $(\\varphi, f)$, one can check whether $f^*$ extends to a map $\\varphi^*$ on forms. This in turn follows if it holds on generators, so it suffices to verify that $f^*$ extends to a map $\\varphi^*\\colon \\Omega^1(\\mathcal{A}') \\to \\Omega^1(\\mathcal{A})$.\n\\subsection{\\texorpdfstring{$\\mathcal{A}$}{A}-Lie algebroids}\n\\label{sec:aliealgebroids}\nWe now introduce what we call $\\mathcal{A}$-versions of Lie algebroids, where $\\mathcal{A} \\to X$ is a fixed Lie algebroid. The main idea is that these objects come equipped with a preferred Lie algebroid morphism onto $\\mathcal{A}$, called its $\\mathcal{A}$-anchor, through which its own anchor factors. Our interest later will be in those examples for which this $\\mathcal{A}$-anchor is of divisor-type.\n\\begin{defn}\\label{defn:aavala} An \\emph{$\\mathcal{A}$-anchored vector bundle} is an anchored vector bundle $E_\\mathcal{A} \\to X$ with an anchored morphism $\\varphi_{E_\\mathcal{A}} \\colon E_\\mathcal{A} \\to \\mathcal{A}$ called the \\emph{$\\mathcal{A}$-anchor} satisfying $\\rho_{E_\\mathcal{A}} = \\rho_\\mathcal{A} \\circ \\varphi_{E_\\mathcal{A}}$. In turn, an \\emph{$\\mathcal{A}$-Lie algebroid} is an $\\mathcal{A}$-anchored vector bundle $E_\\mathcal{A}$ equipped with a compatible Lie algebroid structure for which $\\varphi_{E_\\mathcal{A}}$ is a Lie algebroid morphism.\n\\end{defn}\nThis concept is useful especially when dealing with Lie algebroids whose sections form a submodule of $\\Gamma(\\mathcal{A})$, i.e.\\ those whose $\\mathcal{A}$-anchor $\\varphi_{E_\\mathcal{A}}\\colon \\Gamma(E_{\\mathcal{A}}) \\to \\Gamma(\\mathcal{A})$ is injective on sections. \\autoref{defn:morphismdivisortype} provides any $\\mathcal{A}$-Lie algebroid $\\mathcal{A}'$ with the notion of being of \\emph{$\\mathcal{A}$-divisor-type} by looking at the pair $(\\det(\\mathcal{A}'^*)\\otimes \\det(\\mathcal{A}), \\det(\\varphi_{\\mathcal{A}}))$, with its \\emph{$\\mathcal{A}$-divisor ideal} being $I_{\\varphi_{\\mathcal{A}}}$. Note also the relevance of \\autoref{prop:divisorcomposition} which shows that $I_{\\mathcal{A}'} = I_{\\varphi_{\\mathcal{A}}} \\cdot I_\\mathcal{A}$ when this makes sense.\n\\begin{rem} Any Lie algebroid $\\mathcal{A} \\to X$ is naturally both an $\\mathcal{A}$- and a $TX$-Lie algebroid.\n\\end{rem}\nRecall that a \\emph{comorphism of anchored vector bundles} $(\\varphi;f)\\colon (E,X) \\dashedrightarrow (E',X')$ consists of a base map $f\\colon X \\to X'$ and a vector bundle morphism $\\varphi\\colon f^* E' \\to E$ such that\n\\begin{equation*}\n\t\\rho_E (\\varphi^*(\\sigma)) \\sim_f \\rho_{E'}(\\sigma) \\qquad \\text{for all } \\sigma \\in \\Gamma(E').\n\\end{equation*}\n We will call these \\emph{anchored comorphisms} for short. An anchored comorphism gives a pullback map $\\varphi^*\\colon \\Gamma(E') \\to \\Gamma(E)$ on sections. A \\emph{Lie algebroid comorphism} $(\\varphi;f)\\colon (E,X) \\dashedrightarrow (E',X')$ is an anchored comorphism for which the induced map $\\varphi^*\\colon \\Gamma(E') \\to \\Gamma(E)$ preserves brackets, i.e.\\ satisfies $\\varphi^*[\\sigma,\\tau]_{E'} = [\\varphi^*(\\sigma),\\varphi^*(\\tau)]_{E}$ for all $\\sigma,\\tau \\in \\Gamma(E')$.\n\\begin{rem} Using the fact that Lie algebroid structures on $\\mathcal{A}$ are in one-to-one correspondence with linear Poisson structures on $\\mathcal{A}^*$, a Lie algebroid comorphism is alternatively defined as a bundle morphism between Lie algebroids whose dual morphism is a Poisson map.\n\\end{rem}\nThe $\\mathcal{A}$-version of the notions of morphism and comorphism are rather straightforward.\n\\begin{defn} An \\emph{$\\mathcal{A}$-anchored morphism} $(\\varphi',\\varphi,f)\\colon (E_\\mathcal{A},\\mathcal{A},X) \\to (E_{\\mathcal{A}'}, \\mathcal{A}',X')$ consists of an anchored morphism $(\\varphi',f)\\colon (E_\\mathcal{A},X) \\to (E_{\\mathcal{A}'},X')$ and a compatible Lie algebroid morphism $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$, i.e.\\ such that we have the intertwining relation $\\varphi \\circ \\varphi_{E_\\mathcal{A}} = \\varphi_{E_\\mathcal{A}'} \\circ \\varphi'$.\n\t\nAn \\emph{$\\mathcal{A}$-Lie algebroid morphism} $(\\varphi',\\varphi,f)\\colon (E_\\mathcal{A},\\mathcal{A},X) \\to (E_{\\mathcal{A}'}, \\mathcal{A}',X')$ is an $\\mathcal{A}$-anchored morphism for which $(\\varphi,f)\\colon (E_\\mathcal{A},X) \\to (E_{\\mathcal{A}'}, X')$ is a Lie algebroid morphism.\n\\end{defn}\n\\begin{defn} An \\emph{$\\mathcal{A}$-anchored comorphism} $(\\varphi';\\varphi,f)\\colon (E_\\mathcal{A},\\mathcal{A},X) \\dashedrightarrow (E_{\\mathcal{A}'}, \\mathcal{A}',X')$ consists of an anchored comorphism $(\\varphi';f)\\colon (E_\\mathcal{A},X) \\dashedrightarrow (E_{\\mathcal{A}'},X')$ and a compatible Lie algebroid morphism $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$, i.e.\\ such that $\\varphi_{E_\\mathcal{A}}(\\varphi^*(\\sigma)) \\sim_{\\varphi'} \\varphi_{E_\\mathcal{A}'}(\\sigma)$ for all $\\sigma \\in \\Gamma(E_{\\mathcal{A}'})$.  An \\emph{$\\mathcal{A}$-Lie algebroid comorphism} $(\\varphi';\\varphi,f)\\colon (E_\\mathcal{A},\\mathcal{A},X) \\dashedrightarrow (E_{\\mathcal{A}'}, \\mathcal{A}',X')$ is an $\\mathcal{A}$-anchored comorphism for which $(\\varphi;f)\\colon (E_\\mathcal{A},X) \\to (E_{\\mathcal{A}'}, X')$ is a Lie algebroid comorphism.\n\\end{defn}\nTo summarize, $\\mathcal{A}$-Lie algebroid (co)morphisms give rise to diagrams as follows:\n\\begin{center}\n\t\\begin{tikzpicture}\n\t\\begin{scope}[xshift=-200]\n\t\\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em,text height=1.5ex, text depth=0.25ex]\n\t{\t\\rho_{E_\\mathcal{A}}\\colon &[-3.1em] E_\\mathcal{A} & \\mathcal{A} & TX \\\\ \\rho_{E_{\\mathcal{A}'}}\\colon &[-3.2em] E_{\\mathcal{A}'} & \\mathcal{A}' & TX'\\\\};\n\t\\path[-stealth]\n\t(m-1-2) edge node [above] {$\\varphi_{E_\\mathcal{A}}$} (m-1-3)\n\t(m-1-2) edge node [left] {$\\varphi$} (m-2-2)\n\t(m-2-2) edge node [above] {$\\varphi_{E_{\\mathcal{A}'}}$} (m-2-3)\n\t(m-1-3) edge node [left] {$\\varphi'$} (m-2-3)\n\t(m-1-3) edge node [above] {$\\rho_\\mathcal{A}$} (m-1-4)\n\t(m-2-3) edge node [above] {$\\rho_{\\mathcal{A}'}$} (m-2-4)\n\t(m-1-4) edge node [left] {$T f$} (m-2-4);\n\t\\end{scope}\n\t%\n\t\\begin{scope}\n\t\\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em,text height=1.5ex, text depth=0.25ex]\n\t{\t\\rho_{E_\\mathcal{A}}\\colon &[-3.1em] E_\\mathcal{A} & \\mathcal{A} & TX\\\\ \\rho_{E_{\\mathcal{A}'}}\\colon &[-3.2em] E_{\\mathcal{A}'} & \\mathcal{A}' & TX'\\\\};\n\t\\path[-stealth]\n\t(m-1-2) edge node [above] {$\\varphi_{E_\\mathcal{A}}$} (m-1-3)\n\t(m-2-2) edge node [above] {$\\varphi_{E_{\\mathcal{A}'}}$} (m-2-3)\n\t(m-1-3) edge node [left] {$\\varphi'$} (m-2-3)\n\t(m-1-3) edge node [above] {$\\rho_\\mathcal{A}$} (m-1-4)\n\t(m-2-3) edge node [above] {$\\rho_{\\mathcal{A}'}$} (m-2-4)\n\t(m-1-4) edge node [left] {$T f$} (m-2-4);\n\t\\path[-stealth, dashed]\n\t(m-1-2) edge node [left] {$\\varphi$} (m-2-2);\n\t\\end{scope}\n\t\\end{tikzpicture}\n\\end{center}\n\\subsection{Fiber products}\n\\label{sec:algoperations}\nWe introduce the fiber product of Lie algebroids over the same base. This is a special case of the fiber product construction for Lie algebroids as found in \\cite{HigginsMackenzie90}.\n\\begin{defn}\\label{def:fiberproduct} Consider a Lie algebroid $\\mathcal{A} \\to X$ and $\\mathcal{A}$-Lie algebroids $(\\varphi_{\\mathcal{A}'}, {\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}$ and $(\\varphi_{\\mathcal{A}''}, {\\rm id}_X)\\colon \\mathcal{A}'' \\to \\mathcal{A}$. Assume that $\\varphi_{\\mathcal{A}'}$ and $\\varphi_{\\mathcal{A}''}$ are transverse in $\\mathcal{A}$, i.e.\\\n\\begin{equation*}\n\t\\varphi_{\\mathcal{A}'}(\\mathcal{A}') + \\varphi_{\\mathcal{A}''}(\\mathcal{A}'') = \\mathcal{A}\n\\end{equation*}\nat each point in $X$. The \\emph{fiber product} of $\\mathcal{A}'$ and $\\mathcal{A}''$ over $\\mathcal{A}$ is the Lie algebroid defined as\n\t%\n\t\\begin{equation*}\n\t\\mathcal{A}' \\times_{\\mathcal{A}} \\mathcal{A}'' = \\{(v,w) \\in \\mathcal{A}' \\times \\mathcal{A}'' \\, | \\varphi_{\\mathcal{A}'}(v) = \\varphi_{\\mathcal{A}''}(w) \\}.\n\t\\end{equation*}\n\t%\n\tThe anchor is inherited from either $\\mathcal{A}'$ or $\\mathcal{A}''$ and its bracket is generated by the expression $[v \\oplus v', w \\oplus w']_{\\mathcal{A}' \\times_{\\mathcal{A}} \\mathcal{A}''} = [v,w]_{\\mathcal{A}'} \\oplus [v',w']_{\\mathcal{A}''}$, for $v,w \\in \\Gamma(\\mathcal{A}')$ and $v',w' \\in \\Gamma(\\mathcal{A}'')$\n\\end{defn}\nThe projections onto $\\mathcal{A}'$ and $\\mathcal{A}''$ turn the fiber product $\\mathcal{A}' \\times_{\\mathcal{A}} \\mathcal{A}''$ into both an $\\mathcal{A}'$- and an $\\mathcal{A}''$-Lie algebroid. There is a natural isomorphism $\\mathcal{A}' \\times_{\\mathcal{A}} \\mathcal{A} \\cong \\mathcal{A}'$. If $\\mathcal{A}$ has dense isomorphism locus, recall that $Z_\\mathcal{A} = X \\backslash X_\\mathcal{A}$  is the degeneracy locus of $\\mathcal{A}$, and the support of its divisor.\n\\begin{prop}\\label{prop:fiberprodisolocus} Let $\\mathcal{A}, \\mathcal{A}' \\to X$ be Lie algebroids of divisor-type with disjoint supports. Then $\\rho_\\mathcal{A}$ and $\\rho_{\\mathcal{A}'}$ are transverse, and $\\mathcal{A} \\times_{TX} \\mathcal{A}'$ is of divisor-type with support $Z_\\mathcal{A} \\cup Z_{\\mathcal{A}'}$.\n\\end{prop}\n\\bp As $Z_\\mathcal{A}$ and $Z_{\\mathcal{A}'}$ are disjoint, pointwise either $\\rho_\\mathcal{A}$ or $\\rho_{\\mathcal{A}'}$ is an isomorphism, so that $\\rho_\\mathcal{A}$ and $\\rho_{\\mathcal{A}'}$ are everywhere transverse. The rest of the statement readily follows.\n\\end{proof}\n\\subsection{Lie subalgebroids}\n\\label{sec:liesubalgds}\nIn this section we consider subobjects of Lie algebroids. Let $\\mathcal{A} \\to X$ be a Lie algebroid and $(\\varphi,f)\\colon (\\mathcal{B}, N) \\hookrightarrow (\\mathcal{A},X)$ be a vector subbundle supported on $N \\subseteq X$. It is an \\emph{anchored subbundle} of $\\mathcal{A}$ supported on $N$ if $\\varphi$ is an anchored morphism, in which case $\\rho_{B} = \\rho_{A}|_N$ restricted to $\\mathcal{B}$. It is easy to see that if $(\\rho_\\mathcal{A}|_N)(\\mathcal{B}) \\subseteq TN$ then $\\mathcal{B}$ carries a unique anchored bundle structure for which $\\varphi$ is such an anchored morphism.\n\nNext, consider the submodule $\\Gamma(\\mathcal{A},\\mathcal{B}) := \\{v \\in \\Gamma(\\mathcal{A}) \\, | \\, v|_N \\in \\Gamma(\\mathcal{B})\\}$ of $\\Gamma(\\mathcal{A})$, which comes equipped with a surjective restriction map $\\Gamma(\\mathcal{A},\\mathcal{B}) \\to \\Gamma(\\mathcal{B})$ with kernel $\\Gamma(\\mathcal{A},0_N)$. Note that $\\mathcal{B}$ is a \\emph{Lie subalgebroid} of $\\mathcal{A}$ supported on $N$ if $\\varphi$ is a Lie algebroid morphism, in which case the restriction map $\\Gamma(\\mathcal{A},\\mathcal{B}) \\to \\Gamma(\\mathcal{B})$ is a Lie algebra homomorphism. The following proposition is noted in \\cite[Proposition 2.15]{Meinrenken17} and in part in \\cite[Remark 2.1.21]{Li13}.\n\\begin{prop}[\\cite{Meinrenken17}]\\label{prop:liesubalgeroid} Let $(\\varphi,f)\\colon (\\mathcal{B},N) \\hookrightarrow (\\mathcal{A},X)$ be an anchored subbundle supported on $N$. Then the following are equivalent:\n\t%\n\t\\begin{itemize}\n\t\\item The submodule $\\Gamma(\\mathcal{A},\\mathcal{B}) \\subseteq \\Gamma(\\mathcal{A})$ is involutive;\n\t\\item $\\mathcal{B}$ carries a unique Lie algebroid structure for which $\\varphi$ is a Lie algebroid morphism.\n\t\\end{itemize}\n\\end{prop}\n\\bp Assume that $\\Gamma(\\mathcal{A},\\mathcal{B})$ is involutive. We show that $\\Gamma(\\mathcal{A},0_N)$ is an ideal in $\\Gamma(\\mathcal{A},\\mathcal{B})$, i.e.\\\n\\begin{equation*}\n\t[\\Gamma(\\mathcal{A},0_N), \\Gamma(\\mathcal{A},\\mathcal{B})]_\\mathcal{A} \\subseteq \\Gamma(\\mathcal{A},\\mathcal{B}).\n\\end{equation*}\nLet $v \\in \\Gamma(\\mathcal{A},0_N)$. Then $v = \\sum_i f_i v_i$ for some $v_i \\in \\Gamma(\\mathcal{A})$ and $f_i \\in I_N$. Given $w \\in \\Gamma(\\mathcal{A},\\mathcal{B})$, we have $\\rho_\\mathcal{A}(w)|_N \\in \\Gamma(TN)$. Consequently $\\mathcal{L}_{\\rho_\\mathcal{A}(w)} f_i|_N = 0$ for all $i$, and the Leibniz rule gives that $[v,w]_\\mathcal{A}|_N = 0$, as we have\n\\begin{equation*}\n\t[v, w]_\\mathcal{A} = [\\sum_i f_i v_i, w]_\\mathcal{A} = \\sum_i (f_i [v_i,w]_{\\mathcal{A}} - \\mathcal{L}_{\\rho_\\mathcal{A}(w)} f_i \\cdot v_i).\n\\end{equation*}\nThus $\\Gamma(\\mathcal{B}) \\cong \\Gamma(\\mathcal{A},\\mathcal{B}) \/ \\Gamma(\\mathcal{A},0_N)$ inherits a Lie bracket, making $\\mathcal{B}$ into a Lie algebroid, since the Leibniz rule for $\\rho_{\\mathcal{A}}$ implies the Leibniz rule is satisfied for $\\rho_{\\mathcal{B}}$. The converse is immediate.\n\\end{proof}\n\\begin{rem}\\label{rem:liesubalgd} Sometimes one includes in the definition, apart from the conditions that $(\\rho_{A}|_N)(\\mathcal{B}) \\subseteq TN$ and involutivity of $\\Gamma(\\mathcal{A},\\mathcal{B})$, the condition that $\\Gamma(\\mathcal{A},0_N)$ is an ideal in $\\Gamma(\\mathcal{A},\\mathcal{B})$. \\autoref{prop:liesubalgeroid} shows this latter condition is a consequence of the first two.\n\\end{rem}\nFor later reference we recall the following special examples of Lie subalgebroids.\n\\begin{defn} A Lie subalgebroid $(\\varphi,f)\\colon (\\mathcal{B},X') \\to (\\mathcal{A},X)$ is \\emph{wide} if $X' = X$ and $f = {\\rm id}_X$.\n\\end{defn}\n\\begin{exa}\\label{exa:regfoliation} Let $\\mathcal{F}$ be a regular foliation on $X$ and $D = T \\mathcal{F} \\subseteq TX$ the associated involutive distribution. Then $\\mathcal{A}_\\mathcal{F} := D$ is a Lie subalgebroid of $TX$. Moreover, if $\\mathcal{A} \\to X$ is a regular Lie algebroid with injective anchor, then $\\mathcal{A} = \\mathcal{A}_\\mathcal{F}$ for $\\mathcal{F}$ associated to the distribution $D = \\rho_\\mathcal{A}(\\mathcal{A})$.\n\\end{exa}\n\\begin{exa}\\label{exa:adistribution} Let $\\mathcal{A} \\to X$ be a Lie algebroid with an \\emph{$\\mathcal{A}$-distribution}, i.e.\\ a smooth subbundle $D_\\mathcal{A} \\subseteq \\mathcal{A}$. Its image $D := \\rho_\\mathcal{A}(D_\\mathcal{A}) \\subseteq TX$, need not be a distribution. If $D_\\mathcal{A} \\subseteq \\mathcal{A}$ is $\\mathcal{A}$-involutive (closed under $[\\cdot,\\cdot]_\\mathcal{A}$), then it is an $\\mathcal{A}$-Lie subalgebroid with $\\mathcal{A}$-anchor the natural inclusion.\n\\end{exa}\n\\subsection{$\\mathcal{A}$-invariant submanifolds}\n\\label{sec:degenloci}\nIn this section we discuss $\\mathcal{A}$-invariant submanifolds, i.e.\\ when one can restrict a given Lie algebroid, as a Lie algebroid. We follow in part \\cite{Pym13}.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid. Then a submanifold $N \\subseteq X$ is \\emph{$\\mathcal{A}$-invariant} if $\\rho_\\mathcal{A}$ is tangent to $N$, i.e.\\ $\\rho_\\mathcal{A}|_N$ has image inside $TN$ (instead of merely $TX|_N$).\n\\end{defn}\nFor closed submanifolds $N \\subseteq X$, a vector field $V \\in \\Gamma(TX)$ preserves its vanishing ideal $I_N$ if and only if it is tangent to $N$. Hence a closed submanifold $N$ is $\\mathcal{A}$-invariant if and only if $I_N$ is preserved by $\\mathcal{A}$, i.e.\\ we have $\\mathcal{L}_{\\rho_\\mathcal{A}(v)} I_N \\subseteq I_N$ for all $v \\in \\Gamma(\\mathcal{A})$. Thus $\\mathcal{A}$-invariant submanifolds consist of orbits of $\\mathcal{A}$, i.e.\\ are unions of (opens of) leaves of the singular foliation given by $\\rho_\\mathcal{A}$.\n\\begin{prop}\\label{prop:ainvariantrestr} Let $\\mathcal{A}$ be a Lie algebroid and $N$ be an $\\mathcal{A}$-invariant submanifold of $X$. Then $(\\mathcal{A}|_N, \\rho_\\mathcal{A}|_N) \\to N$ with restriction of $[\\cdot,\\cdot]_\\mathcal{A}$ is a Lie subalgebroid of $\\mathcal{A}$ along $N$.\n\\end{prop}\n\\bp Using \\autoref{prop:liesubalgeroid}, because $\\mathcal{A}|_N \\to TN$ is an anchored bundle due to $\\mathcal{A}$-invariance of $N$, we need only show that $\\Gamma(\\mathcal{A},\\mathcal{A}|_N)$ is involutive. However, this is immediate.\n\\end{proof}\n\\begin{rem}\\label{rem:inverseimagesmooth} The proof of \\autoref{prop:ainvariantrestr} shows\nmore generally, given\na submanifold $\\iota\\colon N \\hookrightarrow X$, that $\\iota^! \\mathcal{A} := \\rho_\\mathcal{A}^{-1}(TN)$ is a Lie subalgebroid of $\\mathcal{A}$ along $N$ if it is a smooth subbundle of $\\mathcal{A}|_N$ (see \\cite[Proposition 2.17]{Meinrenken17}). Note here that $\\iota^! \\mathcal{A} = \\mathcal{A}|_N$ if and only if $N$ is $\\mathcal{A}$-invariant.\n\\end{rem}\nThe set of $\\mathcal{A}$-invariant closed submanifolds of $X$ admits unions and intersections. We say a subset of $X$ is \\emph{$\\mathcal{A}$-invariant} if it is a union of $\\mathcal{A}$-invariant submanifolds.\n\\begin{lem}\\label{lem:ainvariance} Let $N$ and $N'$ be $\\mathcal{A}$-invariant closed submanifolds of $X$. Then their union $N \\cup N'$ and intersection $N \\cap N'$ are $\\mathcal{A}$-invariant closed subsets.\n\\end{lem}\n\\bp The sets $N \\cup N'$ and $N \\cap N'$ have vanishing ideals given by $I_{N \\cup N'} = I_N \\cap I_{N'} = I_N \\cdot I_{N'}$ and $I_{N \\cap N'} = I_N + I_{N'}$. Using the product rule $\\mathcal{L}_{\\rho_\\mathcal{A}(v)}(f g) = (\\mathcal{L}_{\\rho_\\mathcal{A}(v)}f) \\cdot g + f \\cdot (\\mathcal{L}_{\\rho_{\\mathcal{A}}(v)}g)$, we conclude that $I_{N \\cup N'}$ is $\\mathcal{A}$-invariant. The $\\mathcal{A}$-invariance of $I_{N \\cap N'}$ is immediate by additivity. Alternatively, one can note that $\\mathcal{A}$-invariant submanifolds are precisely unions of $\\mathcal{A}$-orbits.\n\\end{proof}\n\\begin{rem}\\label{rem:preservingideals} The proof of \\autoref{lem:ainvariance} shows that more generally, given a collection of vector fields $\\mathcal{F} \\subseteq \\Gamma(TX)$ preserving two ideals $I$, $I'$, i.e.\\ such that $\\mathcal{L}_v I \\subseteq I$ and $\\mathcal{L}_v I' \\subseteq I'$ for all $v \\in \\mathcal{F}$, the collection $\\mathcal{F}$ also preserves $I + I'$ and $I \\cap I'$ (see \\cite[Proposition 2.1.10]{Pym13}). In \\autoref{lem:ainvariance} this is applied to the collection $\\mathcal{F} := \\rho_{A}(\\Gamma(\\mathcal{A}))$ and ideals $I := I_N$, $I' := I_{N'}$.\n\\end{rem}\nWhen considering residue maps associated to $\\mathcal{A}$-invariant submanifolds in \\autoref{sec:residuemaps}, we will need the following terminology, concerning properties of the restricted anchor map.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $N \\subseteq X$ an $\\mathcal{A}$-invariant submanifold. Then $N$ is \\emph{projective} if the restriction $\\mathcal{A}|_N$ is projective, and \\emph{transitive} if $\\mathcal{A}|_N$ is transitive.\n\\end{defn}\nCertainly any transitive $\\mathcal{A}$-invariant submanifold is in particular projective. Any orbit $\\mathcal{O}$ of $\\mathcal{A}$ is a transitive $\\mathcal{A}$-invariant submanifold. Further examples will be given later in \\autoref{sec:residues}.\n\nGiven a projective $\\mathcal{A}$-invariant submanifold $N \\subseteq X$, by the definition of projectivity, the Serre--Swan theorem provides a Lie algebroid $\\mathcal{B} \\to N$ satisfying $\\Gamma(\\mathcal{B}) \\cong (\\rho_A|_N)(\\Gamma(\\mathcal{A}|_N))$. There is an induced Lie algebroid morphism $\\widetilde{\\rho}_\\mathcal{A} \\colon \\mathcal{A}|_N \\to \\mathcal{B}$ obtained from the natural map on sections. This fits in a short exact sequence of Lie algebroids over $N$, namely\n\\begin{equation*}\n\t0 \\to \\ker \\widetilde{\\rho}_\\mathcal{A} \\to \\mathcal{A}|_N \\to \\mathcal{B} \\to 0.\n\\end{equation*}\nAs we did in \\autoref{sec:liealgebroids}, we call $\\ker \\widetilde{\\rho}_\\mathcal{A}$ the (bundle of) \\emph{germinal isotropy} of $\\mathcal{A}|_N$, after \\cite{AndroulidakisZambon17}. \n\\subsection{Degeneracy loci}\nIn this section we define the degeneracy loci of $\\mathcal{A}$ and show that they are $\\mathcal{A}$-invariant subsets. Given an integer $k \\geq 0$, the anchor of $\\mathcal{A}$ defines a map $\\wedge^k \\rho_\\mathcal{A}\\colon \\wedge^k \\mathcal{A} \\to \\wedge^k TX$. This induces a map $\\wedge^k \\rho_\\mathcal{A}\\colon \\Gamma(\\wedge^k \\mathcal{A} \\otimes \\wedge^k T^*X) \\to C^\\infty(X)$, whose image is an ideal $I_{\\mathcal{A},k}$.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid. The $k$th \\emph{degeneracy locus} of $\\mathcal{A}$ is the subspace $X_{\\mathcal{A},k} \\subseteq X$ defined by the ideal $I_{\\mathcal{A},k+1} \\subseteq C^\\infty(X)$, where the rank of $\\rho_\\mathcal{A}$ is $k$ or less.\n\\end{defn}\n\\begin{prop}[{\\cite[Proposition 2.2.9]{Pym13}}]\\label{prop:degenlocus} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $k \\geq 0$. Then $X_{\\mathcal{A},k}$ is an $\\mathcal{A}$-invariant submanifold whenever it is a smooth submanifold.\n\\end{prop}\n\\bp Assume that $X_{\\mathcal{A},k}$ is smooth and denote the map $\\wedge^{k+1} \\rho_\\mathcal{A}$ by $\\varphi_{k}\\colon \\wedge^{k+1} \\mathcal{A} \\to \\wedge^{k+1} TX$. Let $v \\in \\Gamma(\\mathcal{A})$ and $f \\in I_{\\mathcal{A},k}$. Then by definition there exists $\\xi \\in \\Gamma(\\wedge^{k+1} \\mathcal{A})$ and $\\omega \\in \\Omega^{k+1}(X)$ such that $f = \\langle \\varphi_k(\\xi), \\omega \\rangle$. Using the compatibility of the bracket and anchor we compute that\n\\begin{align*}\n\t\\mathcal{L}_{\\rho_\\mathcal{A}(v)} f &= \\mathcal{L}_{\\rho_\\mathcal{A}(v)} \\langle \\varphi_k(\\xi), \\omega \\rangle = \\langle \\mathcal{L}_{\\rho_\\mathcal{A}(v)} \\varphi_k(\\xi), \\omega \\rangle + \\langle \\varphi_k(\\xi), \\mathcal{L}_{\\rho_\\mathcal{A}(v)} \\omega \\rangle\\\\\n\t&= \\langle \\varphi_k(\\mathcal{L}_{v} \\xi), \\omega\\rangle + \\langle \\varphi_k(\\xi), \\mathcal{L}_{\\rho_\\mathcal{A}(v)} \\omega \\rangle \\in I_{\\mathcal{A},k}.\n\\end{align*}\nAs $I_{\\mathcal{A},k}$ is preserved under the action of $\\mathcal{A}$, we conclude that $X_{\\mathcal{A},k}$ is $\\mathcal{A}$-invariant.\n\\end{proof}\n\\begin{rem}\\label{exa:denseisolocus} Let $\\mathcal{A}^n \\to X^n$ be a Lie algebroid of divisor-type. Then $X \\backslash X_\\mathcal{A} = Z_\\mathcal{A} = X_{\\mathcal{A},n-1}$. In fact, we have an equality of ideals $I_\\mathcal{A} = I_{\\mathcal{A},n}$. Both $X_\\mathcal{A}$ and $Z_\\mathcal{A}$ are $\\mathcal{A}$-invariant subsets.\n\\end{rem}\n\\subsection{Singular Lie subalgebroids}\n\\label{sec:singularliealgebroids}\nIn this section we take a slight step into the world of singular foliations. While those underlying the Lie algebroids of interest to us are relatively mild, it is useful to relate our discussion to the language in existing literature, particularly \\cite{AndroulidakisSkandalis09, AndroulidakisZambon13, AndroulidakisZambon16, AndroulidakisZambon17, LaurentGengouxLavauStrobl18, Zambon18}. To extend the results in this paper to more singular Poisson structures, one is forced to work with sheaves or modules which no longer have an underlying bundle. Because we defer this to future work, we will be somewhat brief here.\n\\begin{rem} As a general comment, projective $C^\\infty(X)$-modules are not necessarily locally finitely-generated, i.e.\\ need not necessarily correspond to section modules of vector bundles. However, our use of the term `projective' will presuppose that the module in question is locally finitely-generated. Indeed, for us such modules are often obtained as images of bundle maps.\n\\end{rem}\nLet $X$ be a manifold and $\\mathcal{C}^\\infty_X$ be the sheaf of smooth functions on $X$, with $C^\\infty(X) = \\mathcal{C}^\\infty_X(X)$. Throughout this section, a \\emph{module} will refer to a $C^\\infty(X)$-module without further mention.\n\nLet $E \\to X$ be a vector bundle and $\\mathcal{E}' \\subseteq \\mathcal{E} := \\Gamma(E)$ be a submodule. Given a smooth map $f \\colon X' \\to X$, let $f^* E \\to X'$ be the pullback bundle. The \\emph{pullback module} $f^* \\mathcal{E}'$ of $\\mathcal{E}'$ along $f$ is\n\\begin{equation*}\n\tf^* \\mathcal{E}' := \\langle g f^* \\xi \\, | \\, g \\in C^\\infty(X'), \\xi \\in \\mathcal{E}' \\rangle \\subseteq \\Gamma(X'; f^*E).\n\\end{equation*}\nIf $\\iota_{X'}\\colon X \\hookrightarrow X$ is a submanifold, we call $\\iota_{X'}^* \\mathcal{E}'$ the \\emph{restriction} of $\\mathcal{E}'$ to $X'$.\nA module is \\emph{finitely generated} if there exist global sections $\\xi_i$ of $\\mathcal{E}$ such that $\\mathcal{E} = C^\\infty(X) \\langle \\xi_i \\rangle$, and \\emph{locally finitely generated} if there exists an open cover $\\{U_i\\}$ of $X$ such that each restriction $i_{U_i}^*\\mathcal{E}$ is finitely generated. Call a module $\\mathcal{E}$ of \\emph{vector bundle-type} when $\\mathcal{E} \\cong \\Gamma(E)$ for a vector bundle $E \\to X$.\n\\begin{defn} A module $\\mathcal{E}$ is \\emph{projective} if it has a global basis of generating sections, or, equivalently, if it is a direct summand of a finitely generated free module. \n\\end{defn}\n\\begin{rem} There is a one-to-one correspondence between locally finitely generated projective modules $\\mathcal{E}$, and locally free sheaves of $\\mathcal{C}^\\infty_X$-modules $\\{U \\mapsto \\mathcal{E}_U\\}$. This extends to singular foliations (adding involutivity and possibly dropping projectiveness), giving a one-to-one correspondence between singular foliations and involutive sheaves of locally finitely generated submodules of $\\mathfrak{X}_X$ \\cite{Wang17}. Adding projectivity (or local freeness), we obtain that projective singular foliations are exactly locally free sheaves of involutive submodules of $\\mathfrak{X}_X$.\n\\end{rem}\nThe \\emph{Serre--Swan correspondence} \\cite{Serre55, Swan62} gives an equivalence of categories between finitely generated projective modules $\\mathcal{E}$, and vector bundles $E \\to X$ via the assignment $\\mathcal{E} \\mapsto \\Gamma(E)$. In other words, locally finitely generated projective modules are of vector bundle-type.\n\nFocusing on involutivity, using a given Lie algebroid $\\mathcal{A} \\to X$, results in the following (we give the definition in terms of sheaves, instead of using compactly-supported sections).\n\\begin{defn}[c.f.\\ \\cite{Zambon18}]\\label{defn:singularliealgebroid} A \\emph{singular subbundle} of $\\mathcal{A}$ is a sheaf of locally finitely generated submodules of its sheaf of sections $\\Gamma(\\mathcal{A})$.  A \\emph{singular Lie subalgebroid} of $\\mathcal{A}$ is an involutive singular subbundle. We also use the associated module of global sections interchangeably.\n\\end{defn}\n\\begin{exa}\tA wide Lie subalgebroid $\\mathcal{A}' \\subseteq \\mathcal{A}$ induces a singular Lie subalgebroid $\\Gamma(\\mathcal{A}')$.\n\\end{exa}\n\\begin{exa} \\autoref{prop:liesubalgeroid} shows that given a Lie subalgebroid $\\mathcal{B}$, the module $\\Gamma(\\mathcal{A},\\mathcal{B})$ is a singular Lie subalgebroid of $\\mathcal{A}$. It is not necessarily projective (see also \\autoref{sec:elementarymodifications}).\n\\end{exa}\n\nA projective singular Lie subalgebroid of $\\mathcal{A}$ defines an $\\mathcal{A}$-Lie algebroid $\\mathcal{A}'$ along with a Lie algebroid morphism $\\mathcal{A}' \\to \\mathcal{A}$. Note however that $\\mathcal{A}'$ is generally \\emph{not} a Lie subalgebroid of $\\mathcal{A}$.\n\nSingular Lie subalgebroids of $TX$ are called \\emph{singular foliations} in the sense of Androulidakis--Skandalis \\cite{AndroulidakisSkandalis09}. Moreover, projective singular foliations $\\mathcal{F}$ are also called \\emph{almost-regular} \\cite{Debord01}, and give rise to almost-injective Lie algebroids $\\mathcal{A}_\\mathcal{F}$ with $\\Gamma(\\mathcal{A}_\\mathcal{F}) \\cong \\mathcal{F}$. Note that underlying any singular Lie subalgebroid $\\mathcal{F}_\\mathcal{A}$ of $\\mathcal{A}$ there is a singular foliation $\\mathcal{F} := \\rho_{A}(\\mathcal{F}_\\mathcal{A})$. However, in general there is no relation between projectivity of the modules $\\mathcal{F}_\\mathcal{A} \\subseteq \\Gamma(\\mathcal{A})$ and $\\mathcal{F} \\subseteq \\Gamma(TX)$.\n\\subsubsection{On projectivity}\nEach submodule $\\mathcal{E}'$ of a projective module $\\mathcal{E}$ has an evaluation map ${\\rm ev}\\colon X \\times \\mathcal{E}' \\to E$ defined by ${\\rm ev}(x,\\xi) := \\xi(x)$. Define ${\\rm ev}_x\\colon \\mathcal{E}' \\to E_x$ by ${\\rm ev}_x := {\\rm ev}(x, \\cdot)$ for $x \\in X$. We will denote by $E_x' := {\\rm ev}_x(\\mathcal{E}')$ the image of this map, which is a vector subspace of the fiber $E_x$ of $E$. Given a point $x \\in X$, let $I_x \\subseteq C^\\infty(X)$ be its vanishing ideal. Moreover, given a submodule $\\mathcal{E}'$ of $\\mathcal{E} = \\Gamma(E)$, consider the quotient vector space $\\mathcal{E}'_x := \\mathcal{E}' \/ I_x \\mathcal{E}'$ of $\\mathcal{E}'$ by $I_x$, called the \\emph{fiber} of $\\mathcal{E}'$ at $x$. Note that $\\mathcal{E}'_x = E'_x$ when $\\mathcal{E}'$ is of vector bundle-type. We have:\n\\begin{prop} Let $\\mathcal{E}' \\subseteq \\mathcal{E}$ be a submodule on $X$ where $\\mathcal{E} = \\Gamma(E)$ is of vector bundle-type. Then $x \\mapsto \\dim E'_x$ is lower semi-continuous while $x \\mapsto \\dim \\mathcal{E}'_x$ is upper semi-continuous.\n\\end{prop}\nWhen $\\mathcal{E}$ is locally finitely generated, the vector spaces $\\mathcal{E}_x$ are finite-dimensional for all $x \\in X$ (see \\cite{AndroulidakisSkandalis09}). The evaluation map ${\\rm ev}_x$ vanishes on $I_x \\mathcal{E}'$, so that it descends to a surjective homomorphism $\\widetilde{\\rm ev}_x\\colon \\mathcal{E}'_x \\to E'_x$. When $\\mathcal{E}$ is a Lie module and $\\mathcal{E}' \\subseteq \\mathcal{E}$ an involutive submodule, the kernel of ${\\rm ev}_x$ is a Lie subalgebra of $\\mathcal{E}'$, and $I_x \\mathcal{E}'$ is an ideal in this subalgebra. This results in the conclusion that $\\mathfrak{a}_x := \\ker {\\rm ev}_x \/ I_x \\mathcal{E}' = \\ker \\widetilde{\\rm ev}_x$ is a Lie algebra, called the \\emph{isotropy} of $\\mathcal{E}'$ at $x \\in X$ when $\\mathcal{E} = \\Gamma(TX)$. There is no induced bracket on $\\mathcal{E}'_x$. There is a short exact sequence\n\\begin{equation*}\n0 \\to \\mathfrak{a}_x \\to \\mathcal{E}'_x \\stackrel{\\widetilde{\\rm ev}_x}{\\to} E_x' \\to 0.\n\\end{equation*}\n\\begin{prop}[{\\cite[Lemma 1.3]{AndroulidakisZambon13}}] Let $\\mathcal{E}' \\subseteq \\mathcal{E}$ be a submodule on $X$ where $\\mathcal{E} = \\Gamma(E)$ is of vector bundle-type. Then $\\dim \\mathfrak{g}_x$ is bounded from above by ${\\rm rank}(E)$.\n\\end{prop}\nThe Serre--Swan correspondence implies that a locally finitely-generated module $\\mathcal{F}$ is projective if and only if the dimension of $\\mathcal{F}_x$ is constant for all $x \\in X$. We can pinpoint the failure of projectivity more precisely, following \\cite{AndroulidakisZambon17}. Let $\\mathcal{F}_\\mathcal{A}$ be a singular Lie subalgebroid of $\\mathcal{A}$. Then the module $\\Gamma(\\mathcal{A})$ is projective, and $\\mathcal{F}_\\mathcal{A}$ is a submodule of it. As such the discussion above leads to a short exact sequence for each point $x \\in X$, namely\n\\begin{equation*}\n\t0 \\to \\mathfrak{a}_x \\to \\mathcal{F}_{\\mathcal{A},x} \\to \\mathcal{A}_{\\mathcal{F}_\\mathcal{A},x} \\to 0,\n\\end{equation*}\nwhere $\\mathcal{A}_{\\mathcal{F}_\\mathcal{A},x} = {\\rm ev}_x(\\mathcal{F}_\\mathcal{A}) \\subseteq \\mathcal{A}_x$ is the evaluation, and $\\mathcal{F}_{\\mathcal{A},x} = \\mathcal{F}_\\mathcal{A} \/ I_x \\mathcal{F}_\\mathcal{A}$. Now assume that $\\mathcal{F}_\\mathcal{A}$ is induced by an $\\mathcal{A}$-Lie algebroid. In other words, assume that  we have $\\varphi_{\\mathcal{A}}\\colon \\mathcal{A}' \\to \\mathcal{A}$ with $\\mathcal{F}_\\mathcal{A} = \\varphi_{\\mathcal{A}}(\\Gamma(\\mathcal{A}')) \\subseteq \\Gamma(\\mathcal{A})$. We then obtain from $\\varphi_{\\mathcal{A}}$ another short exact sequence, namely\n\\begin{equation*}\n\t0 \\to \\mathfrak{g}_x \\to \\mathcal{A}'_x \\to \\mathcal{A}_{\\mathcal{F}_\\mathcal{A},x} \\to 0.\n\\end{equation*}\nThe $\\mathcal{A}$-anchor $\\varphi_{\\mathcal{A}}\\colon \\Gamma(\\mathcal{A}') \\to \\Gamma(\\mathcal{A})$ on sections gives another short exact sequence:\n\\begin{equation*}\n\t0 \\to \\mathfrak{h}_x \\to \\mathcal{A}'_x \\to \\mathcal{F}_{\\mathcal{A},x} \\to 0.\n\\end{equation*}\nWe call the kernel $\\mathfrak{h}_x$ the \\emph{germinal $\\mathcal{A}$-isotropy} of $\\mathcal{F}_\\mathcal{A}$ at $x \\in X$. If the module $\\mathcal{F}_\\mathcal{A}$ is projective, so that $\\mathcal{F}_\\mathcal{A} \\cong \\Gamma(\\mathcal{B})$ for $\\mathcal{B} \\to X$, there is an induced Lie algebroid morphism $\\widetilde{\\varphi}_\\mathcal{A}\\colon \\mathcal{A}' \\to \\mathcal{B}$, and then $\\mathfrak{h}_x = \\ker(\\widetilde{\\varphi}_{\\mathcal{A},x}) \\subseteq \\mathcal{A}'_x$ as in \\autoref{defn:germinalisotropy} for $\\mathcal{A} = TX$. The above sequences are related, as discussed in \\cite[Section 1.3]{AndroulidakisZambon17} for the case of $\\mathcal{A} = TX$. Indeed, we have a short exact sequence\n\\begin{equation*}\n\t0 \\to \\mathfrak{h}_x \\to \\mathfrak{g}_x \\to \\mathfrak{a}_x \\to 0.\n\\end{equation*}\nIn conclusion, projectivity of $\\mathcal{F}_\\mathcal{A}$ is equivalent to $\\dim(\\mathfrak{h}_x)$ being constant for all $x \\in X$. This criterion for projectivity also follows from the fact that the above Lie algebroid morphism $\\widetilde{\\varphi}_\\mathcal{A}$ is surjective by construction, so that it induces a Lie algebroid extension sequence\n\\begin{equation*}\n\t0 \\to \\ker(\\widetilde{\\varphi}_\\mathcal{A}) \\to \\mathcal{A}' \\to \\mathcal{B} \\to 0.\n\\end{equation*}\nIn \\cite{LaurentGengouxLavauStrobl18} the authors discuss the notion of a \\emph{projective resolution} of a singular foliation $\\mathcal{F}$, which is an exact sequence of sheaves of $C^\\infty(X)$-modules:\n\\begin{equation*}\n\t\\dots \\to \\Gamma(E_{-2}) \\to \\Gamma(E_{-1}) \\to \\mathcal{F} \\to 0,\n\\end{equation*}\nwhere $E_{-i}$ for $i \\geq 1$ are vector bundles over $X$, with anchor map $E_{-1} \\to TX$ with image $\\mathcal{F}$.\nIt is clear that this definition makes sense for singular Lie subalgebroids as well upon replacing $TX$ by $\\mathcal{A}$ (giving \\emph{projective $\\mathcal{A}$-resolutions}). The \\emph{projective $\\mathcal{A}$-dimension} of such a module (or sheaf of submodules) is then defined as the minimal length of a projective resolution, minus one. Thus projective singular Lie subalgebroids are those with zero projective $\\mathcal{A}$-dimension. In future work we will treat situations of positive projective dimension.\n\\subsection{Ideal Lie algebroids}\n\\label{sec:idealliealgebroids}\nIn this section we discuss what we call ideal Lie algebroids, which are Lie algebroids defined using a given ideal of functions. Of particular importance are those created using divisor ideals. Let $\\mathcal{A} \\to X$ be a Lie algebroid and let $I \\subseteq C^\\infty(X)$ be an ideal. Consider $I \\cdot \\Gamma(\\mathcal{A}) \\subseteq \\Gamma(\\mathcal{A})$ consisting of all finite sums of products of elements of $I$ and $\\Gamma(\\mathcal{A})$, and let $\\Gamma(\\mathcal{A})_I := \\{v \\in \\Gamma(\\mathcal{A}) \\, | \\, \\mathcal{L}_{\\rho_{A}(v)} I \\subseteq I\\}$ be those sections preserving $I$. Both are $C^\\infty(X)$-submodules of $\\Gamma(\\mathcal{A})$. In this section we sometimes shorten $\\mathcal{L}_{\\rho_{\\mathcal{A}}(v)}$ to $\\mathcal{L}_v$.\n\\begin{lem}\\label{lem:liesubalgs} Let $I \\subseteq C^\\infty(X)$ be an ideal. Then $I \\cdot \\Gamma(\\mathcal{A}) \\subseteq \\Gamma(\\mathcal{A})_I \\subseteq \\Gamma(\\mathcal{A})$ are Lie subalgebras.\n\\end{lem}\n\\bp Let $v, w \\in \\Gamma(\\mathcal{A})_I$ and $f \\in I$. Then $\\mathcal{L}_v f = g$ and $\\mathcal{L}_w f = h$ for $g, h \\in I$. Hence\n\\begin{equation*}\n\t\\mathcal{L}_{[v,w]} f = (\\mathcal{L}_v \\mathcal{L}_w - \\mathcal{L}_w \\mathcal{L}_v) f = \\mathcal{L}_v h - \\mathcal{L}_w g.\n\\end{equation*}\nAs $g, h \\in I$ we have $\\mathcal{L}_{[v,w]} f \\in I$ for all $f \\in I$, hence $[v,w] \\in \\Gamma(\\mathcal{A})_I$.\n\nLet $v, w \\in I \\cdot \\Gamma(\\mathcal{A})$ with $v = \\sum_i f_i v_i$ and $w = \\sum_j f'_j w_j$ for some $f_i, f'_j \\in I$ and $v_i, w_j \\in \\Gamma(\\mathcal{A})$. Then we have $[v,w] \\in I \\cdot \\Gamma(\\mathcal{A})$ by the Leibniz rule as each term starts with an element of $I$,\n\\begin{equation*} \n[v,w] = \\textstyle{\\sum}_{i,j} [f_i v_i, f'_j w_j] = \\textstyle{\\sum}_{i,j} \\left( f_i f'_j [v_i, w_j] + f_i (\\mathcal{L}_{v_i} f'_j) \\cdot w_j - f'_j (\\mathcal{L}_{w_j} f_i) \\cdot v_i\\right).\n\\end{equation*}\nLet $v \\in I \\cdot \\Gamma(\\mathcal{A})$. Then $v = \\sum_i f_i v_i$ for $f_i \\in I$ and $v_i \\in \\Gamma(\\mathcal{A})$. For $f \\in I$ we have $\\mathcal{L}_v f = \\sum_i f_i (\\mathcal{L}_{v_i} f)$. As $f_i \\in I$ for all $i$ and $I$ is an ideal, we conclude that $\\mathcal{L}_v f \\in I$ so that $v \\in \\Gamma(\\mathcal{A})_I$ as desired.\n\\end{proof}\nIt is immediate that if $I$ is locally finitely generated, then both $I \\cdot \\Gamma(\\mathcal{A})$ and $\\Gamma(\\mathcal{A})_I$ are as well. By \\autoref{lem:liesubalgs} they are then both singular Lie subalgebroids of $\\mathcal{A}$. Consequently, if they are also projective, they define $\\mathcal{A}$-Lie algebroids by the Serre--Swan correspondence. Due to \\autoref{lem:liesubalgs}, there is then a sequence of Lie algebroid morphisms as follows:\n\\begin{equation*}\n\tI \\cdot \\mathcal{A} \\to \\mathcal{A}_I \\to \\mathcal{A} \\to TX.\n\\end{equation*}\nWe will mostly focus on the projective case, i.e.\\ when $\\Gamma(\\mathcal{A})_I$ and $I\\cdot \\Gamma(\\mathcal{A})$ are projective.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $I \\subseteq C^\\infty(X)$ be an ideal. The primary and secondary \\emph{ideal Lie algebroids} associated to $(\\mathcal{A},I)$ are the Lie algebroids $\\mathcal{A}_I \\to X$ such that $\\Gamma(\\mathcal{A}_I) \\cong \\Gamma(\\mathcal{A})_I$, and $I \\cdot \\mathcal{A} \\to X$ such that $\\Gamma(I \\cdot \\mathcal{A}) \\cong I \\cdot \\Gamma(\\mathcal{A})$.\n\\end{defn}\nBoth ideal Lie algebroids are unique -- provided they exist -- and their $\\mathcal{A}$-anchors are the natural inclusions on sections. However, their $\\mathcal{A}$-anchors are not injective when viewed as bundle maps, unless $I$ is trivial. The isomorphism locus of the $\\mathcal{A}$-anchor of $\\mathcal{A}_I$, and of $I \\cdot \\mathcal{A}$, as well as the morphism between $I\\cdot \\mathcal{A}$ and $\\mathcal{A}_I$, is given by $X \\backslash Z_I$, the complement of the support of $I$. In other words, these three Lie algebroid morphisms are of divisor-type supported on $Z_I$, which in particular means that the ideal Lie algebroids $\\mathcal{A}_I$ and $I \\cdot \\mathcal{A}$ are both of $\\mathcal{A}$-divisor-type.\n\nOur main interest lies in the case where $\\mathcal{A} = TX$ and $I$ is a divisor ideal. Then both ideal Lie algebroids are of divisor-type, with divisors ${\\rm div}(I \\cdot TX)$ and ${\\rm div}(TX_I)$ supported on $Z_I$, and $I_{I \\cdot TX} = I^n$ if $\\dim X = n$ (and similarly using $\\mathcal{A}$). We introduce the following terminology.\n\\begin{defn}\\label{defn:divprojstandard} A divisor ideal $I \\subseteq C^\\infty(X)$ is said to be \\emph{projective} if $\\Gamma(TX)_I$ is projective. A divisor ideal is \\emph{standard} if it is projective and the divisor ideal of $TX_I$ satisfies $I_{TX_I} = I$.\n\\end{defn}\nNot every divisor ideal is projective, nor is every projective divisor ideal standard. Projectivity of $I \\cdot \\Gamma(A)$ is guaranteed for divisor ideals, as is also noted in \\cite[Lemma 1.11]{Nistor15}.\n\\begin{prop}\\label{prop:dividealalgebroid} Given a Lie algebroid $\\mathcal{A}^n \\to X$ and divisor ideal $I$, $I \\cdot \\Gamma(\\mathcal{A})$ is projective.\n\\end{prop}\n\\bp \nThere is a short exact sequence of sheaves $0 \\to I \\cdot \\Gamma(\\mathcal{A}) \\to \\Gamma(\\mathcal{A}) \\to \\Gamma(\\mathcal{A}) \/ I \\cdot \\Gamma(\\mathcal{A}) \\to 0$. As both $\\Gamma(\\mathcal{A})$ and $I$ are locally free, so is $\\Gamma(\\mathcal{A}) \/ I \\cdot \\Gamma(\\mathcal{A}) \\cong (I^{-1})^n$ (the inverse of $I$ as a sheaf). The same then holds for $I \\cdot \\Gamma(\\mathcal{A})$ as any short exact sequence in a projective module splits.\n\\end{proof}\nWhen $I$ is not necessarily standard, we expect the following relation between $I$ and $I_{TX_I}$.\n\\begin{conj} Let $I \\subseteq C^\\infty(X)$ be a projective divisor ideal. Then $I$ is divisible by $I_{TX_I}$.\n\\end{conj}\nNote that in general, given a Lie algebroid morphism $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ and a divisor ideal $I$, there are induced maps $\\varphi\\colon \\Gamma(\\mathcal{A})_I \\to \\Gamma(\\mathcal{A}')_I$ and $\\varphi\\colon I \\cdot \\Gamma(\\mathcal{A}) \\to I \\cdot \\Gamma(\\mathcal{A}')$. In particular this holds upon taking $\\varphi = \\rho_{A}$ and $\\mathcal{A}' = TX$. The following is based on \\cite[Remark 135]{Frejlich11}.\n\\begin{prop}\\label{prop:transLAprojective} Let $\\mathcal{A} \\to X$ be a transitive Lie algebroid and $I$ a projective divisor ideal. Then $\\Gamma(\\mathcal{A})_I$ is a projective module, and the associated bundle $\\mathcal{A}_I$ is a $TX_I$-Lie algebroid. Moreover, $\\mathcal{A}_I$ is isomorphic to the fiber product $\\mathcal{A} \\times_{TX} TX_I$.\n\\end{prop}\n\\bp As $\\mathcal{A}$ is transitive, there is a short exact sequence $0 \\to K \\to \\mathcal{A} \\to TX \\to 0$ where $K = \\ker \\rho_{A}$ is totally intransitive. We can always split this sequence using an Ehresmann connection giving an isomorphism $\\Gamma(\\mathcal{A}) \\cong \\Gamma(TX) \\oplus \\Gamma(K)$. Using this splitting it follows that $\\Gamma(\\mathcal{A})_I \\cong \\Gamma(TX)_I \\oplus \\Gamma(K)$, showing projectivity of $\\Gamma(\\mathcal{A})_I$ and implying that there is an isomorphism $\\mathcal{A}_I \\cong \\mathcal{A} \\times_{TX} TX_I$, from which the fact that $\\mathcal{A}_I$ is a $TX_I$-Lie algebroid follows.\n\\end{proof}\nGiven a surjective Lie algebroid morphism $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ and divisor ideal $I$, if the module $\\Gamma(\\mathcal{A}')_I$ is projective, the proof of \\autoref{prop:transLAprojective} shows that $\\Gamma(\\mathcal{A})_I$ is also projective, and that $\\mathcal{A}_I \\cong \\mathcal{A} \\times_{\\mathcal{A}'} \\mathcal{A}'_I$. Sadly the transitivity of \\autoref{prop:transLAprojective} is never satisfied for Lie algebroids of nontrivial divisor-type, so that this inheritance property does not apply to them.\nWe next study to what extent $\\Gamma(\\mathcal{A})_I$ depends on the ideal $I$. Recall that the \\emph{radical} of $I$ is defined as $\\sqrt{I} := \\{f \\in C^\\infty(X) \\, | \\, f^n \\in I \\text{ for some } n \\in \\mathbb{N}\\}$. Then $I \\subseteq \\sqrt{I}$ and $Z_I = Z_{\\sqrt{I}}$.\n\\begin{prop}\\label{prop:locprincidealmodule} Let $I \\subseteq C^\\infty(X)$ be a locally principal ideal. Then $\\Gamma(\\mathcal{A})_I = \\Gamma(\\mathcal{A})_{\\sqrt{I}}$.\n\\end{prop}\n\\bp This can be verified locally. Let $f \\in \\sqrt{I}$ be such that $\\sqrt{I} = \\langle f \\rangle$, so that $I = \\langle f^n \\rangle$ for some $n \\in \\mathbb{N}$. Take $V \\in \\Gamma(\\mathcal{A})_I$, so that $\\mathcal{L}_V f^n = g \\in I$. Then $g = g' f^n$ for some function $g'$, hence $n f^{n-1} \\mathcal{L}_V f = g' f^n$, so that $\\mathcal{L}_V f = \\frac1n g' f$, whence $\\mathcal{L}_V f \\in \\sqrt{I}$. As $f$ generates $\\sqrt{I}$, we conclude that $V \\in \\Gamma(\\mathcal{A})_{\\sqrt{I}}$, and thus $\\Gamma(\\mathcal{A})_I \\subseteq \\Gamma(\\mathcal{A})_{\\sqrt{I}}$. For the other inclusion, take $W \\in \\Gamma(\\mathcal{A})_{\\sqrt{I}}$ and $h \\in I$. Then $h = h' f^n$ for some function $h'$, hence\n\\begin{equation*}\n\t\\mathcal{L}_W h = \\mathcal{L}_W h' f^n = n h' f^{n-1} \\mathcal{L}_W f + f^n \\mathcal{L}_W h'.\n\\end{equation*}\nAs $\\mathcal{L}_W f \\in \\sqrt{I}$, write $\\mathcal{L}_W f = f' f$ for some function $f'$. Then $\\mathcal{L}_W h = (n h' f' + \\mathcal{L}_W h') f^n \\in I$, so that $W \\in \\Gamma(\\mathcal{A})_I$. Thus $\\Gamma(\\mathcal{A})_{\\sqrt{I}} \\subseteq \\Gamma(\\mathcal{A})_I$, and equality follows.\n\\end{proof}\nAs a consequence of \\autoref{prop:locprincidealmodule}, the primary ideal Lie algebroids $\\mathcal{A}_I$ associated to divisor ideals $I$ are only sensitive to $I$ up to taking powers; this is not true for the secondary ideal Lie algebroids $I \\cdot \\mathcal{A}$.\n\\begin{rem}\\label{rem:nonstandard} There are many divisor ideals which are not standard. For example, using \\autoref{prop:locprincidealmodule} we obtain that $\\Gamma(\\mathcal{A})_{I^k} = \\Gamma(\\mathcal{A})_I$ for all $k > 1$. Consequently, if $I$ is a projective divisor ideal, we have that $TX_{I^k} = TX_I$. Hence, if $I$ is moreover standard, we get that $I_{TX_{I^k}} = I_{TX_I} = I$, which is not equal to $I^k$ unless $I$ is trivial, i.e.\\ $I^k$ is not standard.\n\\end{rem}\n\\begin{rem}\\label{rem:preservingradical} Analogously to \\autoref{rem:preservingideals}, a slight adaptation of the proof of \\autoref{prop:locprincidealmodule} shows that more generally, given a collection of vector fields $\\mathcal{F} \\subseteq \\Gamma(\\mathcal{A})$ preserving an ideal $I$, i.e.\\ such that $\\mathcal{L}_v I \\subseteq I$ for all $v \\in \\mathcal{F}$, the collection $\\mathcal{F}$ also preserves $\\sqrt{I}$. This is a purely algebraic statement and can be found in \\cite[Lemma 3.3.2]{Dixmier96}.\n\\end{rem}\nAs $Z_I$ is the degeneracy locus of both $I\\cdot \\mathcal{A}$ and $\\mathcal{A}_I$, \\autoref{prop:degenlocus} gives the following.\n\\begin{prop}\\label{prop:ziaiinvariant} Let $I \\subseteq C^\\infty(X)$ be a divisor ideal such that $\\mathcal{A}_I$ exists. Then $Z_I$ is $I \\cdot \\mathcal{A}$- and $\\mathcal{A}_I$-invariant if it is smooth, i.e.\\ $I \\cdot \\Gamma(\\mathcal{A})$ and $\\Gamma(\\mathcal{A}_I)$ have anchor images tangent to $Z_I$.\n\\end{prop}\nLet $I$ and $I'$ be two divisor ideals. If $\\mathcal{A}_{I \\cdot I'}$ exists, then so do $\\mathcal{A}_I$ and $\\mathcal{A}_{I'}$, and we see that $\\mathcal{A}_{I \\cdot I'} = \\mathcal{A}_I \\times_{TX} \\mathcal{A}_{I'}$ is their fiber product. Moreover, $\\mathcal{A}_{I \\cdot I'}$ is both an $\\mathcal{A}_I$- and an $\\mathcal{A}_{I'}$-Lie algebroid (again, if it exists). In favorable cases, the $\\mathcal{A}$-divisor of $\\mathcal{A}_{I \\cdot I'}$ is the product of divisors of $\\mathcal{A}_I$ and $\\mathcal{A}_{I'}$. For example this holds if $\\mathcal{A} = TX$ and $I \\cdot I'$ is standard.\n\\begin{rem} While $\\Gamma(\\mathcal{A})_{I \\cdot I'}$ being projective implies both $\\Gamma(\\mathcal{A})_I$ and $\\Gamma(\\mathcal{A})_{I'}$ are, the converse is not immediate if $Z_I$ and $Z_{I'}$ are not disjoint. For example, one can take $\\mathcal{A} = TX$, $I = I_Z$ and $I' = I_{Z'}$ for hypersurfaces $Z,Z' \\subseteq X$ which have nontransverse intersection.\n\\end{rem}\n\\subsubsection{Examples}\n\\label{exa:ideallalgebroids}\nIn this section we discuss some examples of ideal Lie algebroids constructed using the divisor examples of \\autoref{sec:divexamples}. In general it is quite difficult to determine when a divisor ideal is projective, comparable to finding free divisors in algebraic geometry.\n\\begin{exa}[Trivial divisors] Let $I = C^\\infty(X)$ be the trivial divisor ideal. Then\n\\begin{equation*}\n\tI \\cdot \\Gamma(TX) = \\Gamma(TX)_I = \\Gamma(TX),\n\\end{equation*}\nso that the ideal Lie algebroids satisfy $I \\cdot TX = TX_I = TX$. The same holds for any Lie algebroid $\\mathcal{A}$, in that $I \\cdot \\mathcal{A} = \\mathcal{A}_I = \\mathcal{A}$. In particular $I$ is both projective and standard. \n\\end{exa}\n\\begin{exa}[Log divisors]\\label{exa:zvx} Let $Z \\subseteq X$ be a closed submanifold with vanishing ideal $I_Z$. Then $\\Gamma(TX)_{I_Z}$ consists of all vector fields tangent to $Z$. It is projective if and only if $Z$ has codimension one. In this case $I_Z$ is a log divisor ideal and $\\mathcal{A}_Z := TX_{I_Z}$ is the \\emph{log-tangent bundle}\\footnote{It is also called the \\emph{$b$-tangent bundle} ${}^b TX$, introduced by Melrose when $Z = \\partial X$ (see \\cite{Melrose93} and \\cite{NestTsygan96, GuilleminMirandaPires14}).} $TX(-\\log Z)$. Further, $I_Z \\cdot \\Gamma(TX)$ consists of all vector fields which vanish on $Z$. As with $\\Gamma(TX)_{I_Z}$, it is projective if and only if $Z$ has codimension one, in which case $\\mathcal{B}_Z := I_Z \\cdot TX$ is the \\emph{zero tangent bundle} (also denoted by ${}^0 TX$). Log divisors are standard (\\cite[Proposition 3.4.6]{Klaasse17}), as locally we have\n\\begin{equation*}\n\t\\Gamma(\\mathcal{A}_Z) = \\Gamma(TX)_{I_Z} = \\langle z \\partial_z, \\partial_{x_i} \\rangle \\quad \\text{and} \\quad \\Gamma(\\mathcal{B}_Z) = I_Z \\cdot \\Gamma(TX) = \\langle z \\partial_z, z \\partial_{x_i} \\rangle.\n\\end{equation*}\n\\end{exa}\n\\autoref{exa:zvx} extends to normal-crossing and star log divisors $I_{\\underline{Z}}$. In both cases, there is a projective module $\\Gamma(TX)_{I_{\\underline{Z}}}$ defining the log-tangent bundle $TX(-\\log \\underline{Z})$, which locally is generated by $\\langle z_j \\partial_{z_j}, \\partial_{x_i} \\rangle$ when $\\underline{Z} = \\bigcup_j Z_j$ with $Z_j = \\{z_j = 0 \\}$ (see \\cite{GualtieriLiPelayoRatiu17} and \\cite{Lanius17} respectively). In the former case, the log-tangent bundle $TX(-\\log \\underline{Z})$ is the fiber product of the individual log-tangent bundles $TX(-\\log Z_j)$ associated to the log divisors $I_{Z_j}$ (c.f.\\ below \\autoref{prop:ziaiinvariant}).\n\\begin{exa}[Elliptic divisors]\\label{exa:ellvx} Let $|D| = (R,q)$ be an elliptic divisor with elliptic divisor ideal $I_{|D|}$. Then the module $\\Gamma(TX)_{I_{|D|}}$ is projective, defining the \\emph{elliptic tangent bundle} $\\mathcal{A}_{|D|} = TX(-\\log |D|)$ \\cite{CavalcantiGualtieri18}, which is locally given by $\\Gamma(\\mathcal{A}_{|D|}) = \\Gamma(TX)_{I_{|D|}} = \\langle r \\partial_r, \\partial_{\\theta}, \\partial_{x_i} \\rangle$ in normal polar coordinates $(r,\\theta)$ to $D$ in which $I_{|D|} = \\langle r^2 \\rangle$. From the fact that $r \\partial_r \\wedge \\partial_{\\theta} = r^2 \\partial_x \\wedge \\partial_y$ where $r^2 = x^2 + y^2$ we see that elliptic divisors are standard (see \\cite[Proposition 3.4.17]{Klaasse17}).\n\\end{exa}\n\\begin{rem} Morphisms of divisors often give rise to Lie algebroid morphisms between their primary ideal Lie algebroids. Indeed, in \\cite{Klaasse17,CavalcantiKlaasse18} it is shown that a morphism between log divisor ideals leads to an induced Lie algebroid morphism between their resulting log-tangent bundles. Similarly for morphisms between elliptic divisor ideals. Morphisms between elliptic and log divisors are considered in \\cite{CavalcantiKlaasse18} as boundary maps, and lead to Lie algebroid morphism from the elliptic tangent bundle to the log-tangent bundle. Finally, regarding morphisms between log and elliptic divisors one has to be more careful, see \\cite{KlaasseLi18}.\n\\end{rem}\n\\begin{exa}[Complex log divisors]\\label{exa:complexlogtgnt} Let $D = (U,\\sigma)$ be a complex log divisor with ideal $I_\\sigma$. Then $\\Gamma(TX_\\mathbb{C})_{I_D}$ is a projective module of complex vector fields, defining the \\emph{complex log-tangent bundle} $TX_{\\mathbb{C}}(- \\log D)$, locally given by $\\langle w \\partial_w, \\partial_{\\overline{w}}, \\partial_{x_i} \\rangle$ in coordinates where $I_\\sigma = \\langle w \\rangle$ (see \\cite{CavalcantiGualtieri18}). Because $w \\partial_w \\wedge \\partial_{\\overline{w}} = - 2i w \\partial_x \\wedge \\partial_y$, we see that complex log divisors are standard. This complex Lie algebroid is also denoted by $\\mathcal{A}_D \\to X$, and we will not use it in this paper.\n\\end{exa}\nWe discuss one further example, obtained from an elliptic-log divisor. Let $W = Z \\otimes |D|$ be an elliptic-log divisor with divisor ideal $I_W$. Then the primary ideal Lie algebroid $\\mathcal{A}_W := \\mathcal{A}_{I_W}$, the \\emph{elliptic-log tangent bundle} $TX(-\\log W)$, exists and is both an $\\mathcal{A}_Z$ and an $\\mathcal{A}_{|D|}$-Lie algebroid, as it is the fiber product of the two. The following was pointed out to us by Gil Cavalcanti.\n\\begin{prop}\\label{prop:elllogalgebroid} Let $(X,Z)$ be a log pair and $|D| \\subseteq Z$ an elliptic divisor. Then the module $\\Gamma(TX)_{I_W}$ is projective, so that $\\mathcal{A}_{W}$ is a Lie algebroid. Its divisor ideal is given by $I_{\\mathcal{A}_W} = I_W$.\n\\end{prop}\n\\bp By definition $\\Gamma(TX)_{I_W}$ consists of vector fields preserving $I_W = I_Z \\cdot I_{|D|}$. It is immediate (see \\cite{Klaasse17}) that all vector fields tangent to $D$ belong to $\\Gamma(TX)_{I_W}$. Hence the associated sheaf is locally free if around points in $D$ it contains only two independent vector fields normal to $D$, as $D$ has codimension two. Away from $D$ we have $I_W = I_Z$, so that there $\\Gamma(TX)_{I_W} = \\Gamma(TX)_{I_Z}$ which we know to be locally free. Let $(x,y)$ be normal coordinates around $p \\in D$ such that $I_W = \\langle x (x^2 + y^2) \\rangle$. Set $w(x,y) := x (x^2 + y^2)$ and take $V = a \\partial_x + b \\partial_y$. Note that $V$ belongs to $\\Gamma(TX)_{I_W}$ if and only if the function $f = \\mathcal{L}_V (\\log w)$ is smooth. We compute that\n\\begin{equation*}\nf = \\mathcal{L}_V(\\log w) = \\mathcal{L}_V(\\log x) + \\mathcal{L}_V(\\log(x^2 + y^2)) = a x^{-1} + (2a x + 2 b y)(x^2+y^2)^{-1},\n\\end{equation*}\nso by multiplying by $x^2 + y^2$ and rearranging we obtain $(x^2 + y^2) f - 2a x - 2b y = a x^{-1} (x^2 + y^2)$. The left-hand side is smooth, hence so must be the right-hand side, i.e.\\ $a$ must be divisible by $x$. Hence $a = x \\alpha$ for some smooth $\\alpha$. This implies that $f = \\alpha + (2 x^2 \\alpha + 2 b y)(x^2 + y^2)^{-1}$, so that $\\lambda_1 := (x^2 \\alpha + b y)(x^2 + y^2)^{-1}$ must be smooth. Rewriting this we obtain that\n\\begin{equation*}\n\tb = \\lambda_1 y + (\\lambda_1 - \\alpha)x^2 y^{-1},\n\\end{equation*}\nand as $b$ is smooth, $y$ divides $\\lambda_1 - \\alpha$. Set $\\lambda_2 := (\\lambda_1 - \\alpha) y^{-1}$. Given smooth functions $\\lambda_1$ and $\\lambda_2$ we can now express $a$ and $b$ as $a = x (\\lambda_1 - y \\lambda_2)$ and $b = \\lambda_1 y + \\lambda_2 x^2$. This implies that $V$ is given by $V = \\lambda_1 (x \\partial_x + y \\partial_y) + \\lambda_2 (-x y \\partial_x + x^2 \\partial_y)$. This shows that $V$ must lie in the two-dimensional span $\\langle x \\partial_x + y \\partial_y, x(y \\partial_x - x \\partial_y) \\rangle$. To finish the proof of local freeness, we check that these generators preserve $I_W$. This follows, as we readily compute:\n\\begin{itemize}\n\t\\item \\makebox[3cm][l]{$\\mathcal{L}_{x \\partial_x + y \\partial_y}(\\log w)$} $= x \\partial_x \\log x + (x \\partial_x + y \\partial_y) \\log(x^2 + y^2) = 1 + 2 = 3$,\n\t\\item \\makebox[3cm][l]{$\\mathcal{L}_{x(y \\partial_x - x \\partial_y)}(\\log w)$} $= x y \\partial_x \\log x + x y \\partial_x \\log(x^2 + y^2) - x^2 \\partial_y \\log(x^2 + y^2)$\n\t\\item[] \\makebox[3cm][l]{} $= y + (2 x^2 y - 2 x^2 y)(x^2 + y^2)^{-1} = y$.\n\\end{itemize}\nWe conclude that $\\mathcal{A}_W$ exists and that locally we have (with slight abuse of notation)\n\\begin{equation*}\n\\Gamma(\\mathcal{A}_W) = \\Gamma(TX)_{I_W} = \\langle x \\partial_x + y \\partial_y, x(y \\partial_x - x \\partial_y) \\rangle \\oplus \\Gamma(TD).\n\\end{equation*}\nThe final statement of this proposition follows from the realization that\n\\begin{equation*}\n(x \\partial_x + y \\partial_y) \\wedge x(y \\partial_x - x \\partial_y) = x (x^2 + y^2) \\partial_x \\wedge \\partial_y = w \\partial_x \\wedge \\partial_y. \\qedhere\n\\end{equation*}\n\\end{proof}\nWe can alternatively write the generators of $\\mathcal{A}_W$ using a polar coordinate system $(r,\\theta)$ normal to $D$ supplied by $I_{W} = \\langle r^3 \\cos \\theta \\rangle$. In such a coordinate system, we have that\n\\begin{equation*}\n\\Gamma(\\mathcal{A}_W) = \\langle r \\partial_r, r \\cos \\theta \\partial_\\theta \\rangle + \\Gamma(TD).\n\\end{equation*}\nIn particular, \\autoref{prop:elllogalgebroid} says that $I_W$ is standard. The elliptic-log tangent bundle is further explored in \\autoref{exa:ellipticlogresidue} and \\cite{KlaasseLanius18}, the latter computing its Lie algebroid cohomology.\n\nA slightly different example is provided by the $b^k$-tangent bundles \\cite{Scott16}. The idea here is that instead of considering derivations preserving a certain divisor ideal $I$ (which would give its primary ideal Lie algebroid), we consider those which send $I$ to $I^k$ in a certain sense.\n\nLet $(X,Z)$ be a log pair with $\\iota_Z\\colon Z \\hookrightarrow X$ the inclusion, $k \\geq 1$ be a given integer. The \\emph{sheaf of $k$-jets} at $Z$ is $\\mathcal{J}_Z^k := \\iota_Z^{-1}(C^\\infty(X) \/ I_Z^{k+1})$. A \\emph{$k$-jet} at $Z$ is a global section of $\\mathcal{J}_Z^k$. Denote the set of $k$-jets at $Z$ by $J_Z^k$.\nGiven a function $f$ defined in a neighbourhood of $Z$, let $[f]^k_Z \\in J_Z^k$ denote the $k$-jet at $Z$ represented by $f$. For a given $k$-jet at $Z$, $j \\in J_Z^k$, write $f \\in j$ if $[f]_Z^k = j$. \n\\begin{exa}[$b^k$-tangent bundles, \\cite{Scott16}]\\label{exa:bktangentbundle} Let $j_{k-1} \\in J_Z^{k-1}$ be a choice of $(k-1)$-jet and define the sheaf of vector fields $\\Gamma(TX)_{I^k_Z, j_{k-1}} := \\{v \\in \\Gamma(TX) \\, | \\, \\mathcal{L}_v f \\in I_Z^k \\text{ for all } f \\in j_{k-1}\\}$. One verifies that this is a locally free sheaf of involutive submodules of $\\Gamma(TX)$, and that given $v \\in \\mathcal{V}_X(I_Z)$ and $f \\in C^\\infty(X)$, the jet $[\\mathcal{L}_v f]_Z^{k-1}$ depends only on $[f]_Z^{k-1}$. This defines the \\emph{$b^k$-tangent bundle} $\\mathcal{A}_Z^k \\to X$ by $\\Gamma(\\mathcal{A}_Z^k) \\cong \\Gamma(TX)_{I^k_Z, j_{k-1}}$, which is locally given by $\\Gamma(\\mathcal{A}_Z^k) = \\langle z^k \\partial_z, \\partial_{x_i} \\rangle$ for $z \\in j$.\n\\end{exa}\nNote that when $k = 1$ the jet data is vacuous, so that $\\mathcal{A}_Z^1 = \\mathcal{A}_Z$, the log-tangent bundle. Further discussion of this process can be found in \\cite{Scott16,Klaasse17}. The same construction described above can also be performed with smooth divisor ideals other than $I_Z$, where one uses a choice of $k$-jet at $Z_I$, i.e.\\ a global section of $\\mathcal{J}^k_I := \\iota_{Z_I}^{-1}(C^\\infty(X) \/ I^{k+1})$, for $\\iota_{Z_I}\\colon Z_I \\hookrightarrow X$ the inclusion. The sheaf of vector fields then becomes $\\Gamma(TX)_{I^k, j_{k-1}} := \\{v \\in \\Gamma(TX) \\, | \\, \\mathcal{L}_v f \\in I^k \\text{ for all } f \\in j_{k-1}\\}$.\n\\begin{rem} Let $I \\subseteq C^\\infty(X)$ be a projective smooth divisor ideal. Then its associated primary ideal Lie algebroid $TX_I$ can be restricted to $Z_I$, because it is the degeneracy locus of $TX_I$. Assuming that $Z_I$ is a transitive invariant submanifold for $TX_I$, the restriction $TX_I|_{Z_I}$ is then a Lie subalgebroid of ${\\rm At}(NZ_I) \\to Z_I$, the Atiyah algebroid of the normal bundle. The latter is a transitive Lie algebroid with rank equal to ${\\rm rank}({\\rm At}(NZ_I)) = \\dim(Z_I) + {\\rm codim}(Z_I)^2$. If $Z_I$ is not transitive, then instead there is only an almost-injective Lie algebroid morphism from $TX_I|_{Z_I}$ to ${\\rm At}(NZ_I)$. As ${\\rm rank}(TX_I|_{Z_I}) = {\\rm rank}(TX_I) = \\dim(X)$, this means that in order for $I$ to be projective, the vector fields normal to $Z_I$ which preserve $I$ must cut down the isotropy of ${\\rm At}(NZ_I)$ in dimension from ${\\rm codim}(Z_I)^2$ to ${\\rm codim}(Z_I)$. Related to this is the discussion in \\autoref{sec:divexamples} on Morse--Bott divisor ideals being locally homogeneous around $Z_I$.\n\t\n\tFor example, if $I = I_Z$ is a log divisor ideal, then the above implies that $TX_{I_Z}|_Z \\cong {\\rm At}(NZ)$. For elliptic divisor ideals, instead $TX_{I_{|D|}}|_D$ is a corank-two Lie subalgebroid of ${\\rm At}(ND)$.\n\\end{rem}\n\\subsection{Elementary modifications}\n\\label{sec:elementarymodifications}\nIn this section we discuss a process called \\emph{elementary modification} \\cite{GualtieriLi14, Li13} (partially also \\emph{rescaling} \\cite{Lanius16, Melrose93}), by which, given a Lie algebroid, new Lie algebroids are constructed by specifying a projective singular Lie subalgebroid using extra data. In loc.\\ cit.\\ this is done using Lie algebroids supported on hypersurfaces, but we consider a generalization of this procedure by using Lie algebroids supported on the support of a smooth divisor ideal. When using a log divisor ideal (see \\autoref{exa:log}), one recovers the procedures of loc.\\ cit. There are two procedures which are in a sense dual to another, referred to as lower and upper elementary modification respectively. As noted in \\cite{Li13}, these processes are known in algebraic geometry under the name of Hecke modifications. Recall from \\autoref{defn:smoothdivisor} that a divisor ideal $I \\subseteq C^\\infty(X)$ is smooth if its support $Z_I \\subseteq X$ is a smooth submanifold.\n\\subsubsection{Upper elementary modification}\nWe first describe the process of upper modification (see \\cite{Li13} for the case when $I = I_Z$), which uses surjective comorphisms. Let $I$ be a smooth divisor ideal with ideal sheaf $\\mathcal{I}$, and let $\\mathcal{A} \\to X$ and $\\mathcal{B} \\to Z_I$ be vector bundles. Assume the existence of a surjective bundle comorphism $(\\varphi;i_{Z_I})\\colon (\\mathcal{B},Z_I) \\dashedrightarrow (\\mathcal{A},X)$, i.e.\\ the map $\\varphi^*\\colon i_{Z_I}^* \\mathcal{A} \\to \\mathcal{B}$ is surjective, and has kernel $\\mathcal{K} := \\ker \\varphi \\to Z_I$. From this data, we define a $C^\\infty(X)$-module\n\\begin{equation*}\n\\Gamma(\\mathcal{A};(\\mathcal{B},I)) := \\{v \\in \\Gamma(\\mathcal{A}) \\otimes \\mathcal{I}^{-1} \\, | \\, \\mathcal{I} \\cdot v \\in \\Gamma(\\mathcal{A},\\mathcal{K})\\},\n\\end{equation*}\nwhere it is immediate that $\\Gamma(\\mathcal{A}) \\subseteq \\Gamma(\\mathcal{A};(\\mathcal{B},I))$. Here $\\mathcal{I}^{-1}$ is the inverse of $\\mathcal{I}$ as a sheaf, and we make use of the canonical isomorphism $\\mathcal{I} \\otimes \\mathcal{I}^{-1} \\cong \\mathcal{C}^\\infty_X$. Alternatively, we can choose local generators for the sheaf $\\mathcal{I}$ and its inverse and check that the definition is independent of such a choice, or first realize $\\mathcal{I}$ by a divisor using \\autoref{prop:locprincideal}. It is immediate that $\\Gamma(\\mathcal{A};(\\mathcal{B},I))$ is locally finitely generated because $I$ and $\\Gamma(\\mathcal{A})$ are. The main point of this construction is:\n\\begin{lem}\\label{lem:uppermodprojective} Under the above assumptions, the module $\\Gamma(\\mathcal{A};(\\mathcal{B},I))$ is projective.\n\\end{lem}\n\\bp Trivialize $\\mathcal{B}^m \\to Z_I$ around a point in $Z_I$ with basis of sections $(w_1,\\dots,w_m)$, and let $f \\in \\mathcal{I}$ be a local generator. Choose a complementary basis of sections $(v_{m+1},\\dots,v_n)$ for $\\mathcal{K}$, so that $(w_1,\\dots,w_m,v_{m+1},\\dots,v_n)$ is a local basis of $\\Gamma(\\mathcal{A}^n)$. Then $(w_1,\\dots,w_m, f^{-1} v_{m+1},\\dots, f^{-1} v_n)$ is a local basis of $\\Gamma(\\mathcal{A};(\\mathcal{B},I))$, showing projectivity, as $\\Gamma(\\mathcal{A};(\\mathcal{B},I)) \\cong \\Gamma(\\mathcal{A})$ away from $Z_I$.\n\\end{proof}\nBy the Serre--Swan correspondence there exists a vector bundle\n\\begin{equation*}\n\t\\{\\mathcal{A}\\:(\\mathcal{B},I)\\} \\to X \\quad \\text{defined by} \\quad \\Gamma(\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}) \\cong \\Gamma(\\mathcal{A};(\\mathcal{B},I)).\n\\end{equation*}\nNext, assume that $\\mathcal{A} \\to X$ and $\\mathcal{B} \\to Z_I$ are anchored and $\\varphi$ is an anchored comorphism, i.e.\\ $i_{Z_I}^* (\\mathcal{L}_{\\rho_{A}(v)} f) = \\mathcal{L}_{\\rho_\\mathcal{B}(\\varphi(v))}(i_{Z_I}^*(f))$ for all $v \\in \\Gamma(\\mathcal{A})$ and $f \\in C^\\infty(X)$. Alternatively, the anchor maps should satisfy $\\rho_{\\mathcal{B}}(\\varphi^*(v)) \\sim_\\varphi \\rho_{\\mathcal{A}}(v)$ for all $v \\in \\Gamma(\\mathcal{A})$.\n\\begin{lem}[c.f.\\ \\cite{Li13}]\\label{lem:uppermodanchored} Under the above assumptions, the bundle $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ is anchored.\n\\end{lem}\n\\bp The anchored comorphism condition implies $\\mathcal{K} \\subseteq \\ker(\\rho_{A}|_{Z_I})$, so that for $v \\in \\Gamma(\\mathcal{A},K)$ we have $\\rho_{A}(v) \\in \\Gamma(TX,TZ_I)$, which shows that $\\rho_{A}$ lifts to an anchor for $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$.\n\\end{proof}\nThere is an induced (anchored) morphism $\\mathcal{A} \\to \\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ with isomorphism locus $Z_I$. Finally, assume that $\\mathcal{A} \\to X$ and $\\mathcal{B} \\to Z_I$ are Lie algebroids and $\\varphi$ is a surjective Lie algebroid comorphism, i.e.\\ $\\varphi$ further satisfies $\\varphi^*([v,w]) = [\\varphi^*(v),\\varphi^*(w)]$ for all $v,w \\in \\Gamma(\\mathcal{A})$.\n\\begin{lem}\\label{lem:uppermodinvolutive} Under the above assumptions, the module $\\Gamma(\\mathcal{A};(\\mathcal{B},I))$ has an induced bracket.\n\\end{lem}\n\\bp Because $\\varphi^*$ preserves brackets, there is an induced Lie bracket on $\\Gamma(\\mathcal{A};(\\mathcal{B},I))$ inherited from the module $\\Gamma(\\mathcal{A})$ through the natural map $\\Gamma(\\mathcal{A}) \\to \\Gamma(\\mathcal{A};(\\mathcal{B},I))$.\n\\end{proof}\n\\autoref{lem:uppermodprojective}, \\autoref{lem:uppermodanchored} and \\autoref{lem:uppermodinvolutive} combine to the following definition (c.f.\\ \\cite{Li13}).\n\\begin{defn}\\label{defn:uppermod} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I \\subseteq C^\\infty(X)$ a smooth divisor ideal and $\\mathcal{B} \\to Z_I$ a Lie algebroid with a surjective Lie algebroid comorphism onto $\\mathcal{A}$. The \\emph{$(\\mathcal{B},I)$-upper modification} of $\\mathcal{A}$ is the \n\n\tLie algebroid $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ for which $\\Gamma(\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}) \\cong \\Gamma(\\mathcal{A};(\\mathcal{B},I))$.\n\\end{defn}\nIn this case there is a Lie algebroid morphism $\\mathcal{A} \\to \\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ making $\\mathcal{A}$ into an $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$-Lie algebroid, and the $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$-divisor ideal of $\\mathcal{A}$ is given by $I^k$, where $k = \\dim(\\mathcal{B})$.\n\\begin{rem} From the divisor ideal of the induced morphism $\\mathcal{A} \\to \\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ we see that it is only possible for $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ to be anchored if the divisor of $\\mathcal{A}$ is divisible by $I^k$. This places an often nontrivial restriction on when upper modifications can be Lie algebroids.\n\\end{rem}\nThere is a commutativity property for upper modifications, which is readily verified.\n\\begin{prop} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I,I' \\subseteq C^\\infty(X)$ smooth divisor ideals and $(\\mathcal{B},Z_I)$, $(\\mathcal{B}',Z_{I'})$ Lie algebroids equipped with surjective Lie algebroid comorphisms onto $\\mathcal{A}$. Then upper modification at $Z_I$ and $Z_{I'}$ commute, i.e. we have isomorphisms\n\t%\n\t\\begin{equation*}\n\t\\{\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}\\:(\\mathcal{B}',I')\\} \\cong \\{\\{\\mathcal{A}\\:(\\mathcal{B}',I')\\}\\:(\\mathcal{B},I)\\}.\n\t\\end{equation*}\n\t%\n\\end{prop}\n\\subsubsection{Lower modification}\nWe turn to the dual procedure to upper modification, namely lower modification (or rescaling). This uses injective morphisms instead of surjective comorphisms.\n \nLet $I$ be a smooth divisor ideal and $\\mathcal{B} \\to Z_I$ be a vector subbundle of a vector bundle $\\mathcal{A} \\to X$. To define an appropriate $C^\\infty(X)$-module in this setting, we remark that injective morphisms dualize to surjective comorphisms. That is, the bundle inclusion $(\\varphi,i_{Z_I})\\colon (\\mathcal{B},Z_I) \\to (\\mathcal{A},X)$ dualizes to the surjection $(\\varphi;i_{Z_I})\\colon (\\mathcal{B}^*,Z_I) \\dashrightarrow (\\mathcal{A}^*,X)$. We can thus perform upper modification to obtain the vector bundle\n\\begin{equation*}\n\t\\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\} \\to X \\quad \\text{with} \\quad \\Gamma(\\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\}) \\cong \\Gamma(\\mathcal{A}^*\\:(\\mathcal{B}^*,I)).\n\\end{equation*}\nThis comes equipped with a natural bundle morphism $\\mathcal{A}^* \\to \\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\}$. Dualizing again, using $(\\mathcal{A}^*)^* \\cong \\mathcal{A}$ we obtain a comorphism $\\mathcal{A} \\dashrightarrow \\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\}^*$. Reinterpreted, this says that there is a bundle morphism $\\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\}^* \\to \\mathcal{A}$. This motivates the following definitions:\n\\begin{equation*}\n\t[\\mathcal{A}\\:(\\mathcal{B},I)] := \\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\}^* \\quad \\text{and} \\quad \\Gamma(\\mathcal{A},(\\mathcal{B},I)) := \\Gamma([\\mathcal{A}\\:(\\mathcal{B},I)]).\n\\end{equation*}\n\\begin{rem} We had to proceed by dualizing twice because of the use of a smooth divisor ideal $I$. When $I = I_Z$, one can merely define $\\Gamma(\\mathcal{A},(\\mathcal{B},I_Z)) := \\Gamma(\\mathcal{A},\\mathcal{B})$ as in \\cite{Melrose93,Lanius16,GualtieriLi14}. Note here that the `dual definition' $\\{v \\in \\Gamma(\\mathcal{A}) \\otimes \\mathcal{I} \\, | \\, \\mathcal{I}^{-1} \\cdot v \\in \\Gamma(\\mathcal{A},\\mathcal{B})\\}$ does not make sense.\n\\end{rem}\nNow that we have the module of interest, it can be described locally, as in \\autoref{lem:uppermodprojective}. Trivialize $\\mathcal{B}^m \\to Z_I$ around a point in $Z_I$ with basis of sections $(w_1, \\dots w_m)$, and let $f \\in \\mathcal{I}$ be a local generator. Extend this basis to a local basis $(w_1,\\dots,w_m,v_{m+1},\\dots,v_n)$ of $\\Gamma(\\mathcal{A}^n)$. Then $(w_1,\\dots,w_m, f v_{m+1}, \\dots, f v_n)$ is a local basis for $\\Gamma(\\mathcal{A},(\\mathcal{B},I))$. This also shows projectivity.\n\nWhen $\\mathcal{A} \\to X$ is an anchored bundle and $\\mathcal{B} \\to Z_I$ an anchored subbundle of $\\mathcal{A}$, the bundle $[\\mathcal{A}\\:(\\mathcal{B},I)]$ becomes anchored by inheriting an anchor from $\\rho_{A}\\colon \\mathcal{A} \\to TX$. Note that there is an induced (anchored) morphism $[\\mathcal{A}\\:(\\mathcal{B},I)] \\to \\mathcal{A}$ with isomorphism locus $Z_I$. Note that $[\\mathcal{A}\\:(\\mathcal{B},I)]$ is \\emph{not} an anchored subbundle of $\\mathcal{A}$ (except in the trivial case when $I = C^\\infty(X)$).\n\nNext assume that $\\mathcal{A} \\to X$ is a Lie algebroid and $\\mathcal{B} \\to Z_I$ is a Lie subalgebroid.\n\\begin{lem}\\label{lem:lowermodinvolutive} Under the above assumptions, the submodule $\\Gamma(\\mathcal{A},(\\mathcal{B},I)) \\subseteq \\Gamma(\\mathcal{A})$ is involutive.\n\\end{lem}\n\\bp The submodule $\\Gamma(\\mathcal{A},\\mathcal{B}) \\subseteq \\Gamma(\\mathcal{A})$ is involutive by \\autoref{prop:liesubalgeroid}. To extend this property to $\\Gamma(\\mathcal{A},(\\mathcal{B},I))$ we can make use of the local description above. Let $f \\in \\mathcal{I}$ be a local generator and consider two sections $f s, f s' \\in \\Gamma(\\mathcal{A},(\\mathcal{B},I))$ with $s \\in \\Gamma(\\mathcal{A})$ but not in $\\Gamma(\\mathcal{A},\\mathcal{B})$ (here $g \\in \\mathcal{I}$ can be written as $g = g' f$, but then $g'$ can be absorbed into $s$). Then we have:\n\\begin{equation*}\n\t[f s, f s']_\\mathcal{A} = f^2 [s,s']_\\mathcal{A} + f \\left((\\mathcal{L}_{\\rho_{A}(s)} f) s' - (\\mathcal{L}_{\\rho_{A}(s')} f) s\\right).\n\\end{equation*}\nWe see from this that $[fs,f s']_\\mathcal{A} \\in \\mathcal{I} \\otimes \\Gamma(\\mathcal{A})$, as in \\autoref{lem:liesubalgs}. The other cases are similar.\n\\end{proof}\nThus, making use of \\autoref{lem:lowermodinvolutive} we see that $\\Gamma(\\mathcal{A},(\\mathcal{B},I))$ is a projective singular Lie subalgebroid of $\\mathcal{A}$ in the sense of \\autoref{defn:singularliealgebroid}. This leads to the following.\n\\begin{defn}[{\\cite{GualtieriLi14, Lanius16, Melrose93}}]\\label{defn:lowermod} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I \\subseteq C^\\infty(X)$ a smooth divisor ideal and $\\mathcal{B} \\to Z_I$ a Lie subalgebroid of $\\mathcal{A}$. Then the \\emph{$(\\mathcal{B},I)$-lower modification} of $\\mathcal{A}$ is the $\\mathcal{A}$-Lie algebroid $[\\mathcal{A}\\:(\\mathcal{B},I)] \\to \\mathcal{A}$ for which $\\Gamma([\\mathcal{A}\\:(\\mathcal{B},I)]) \\cong \\Gamma(\\mathcal{A},(\\mathcal{B},I))$.\n\\end{defn}\nIt is immediate that the $\\mathcal{A}$-divisor ideal of $[\\mathcal{A}\\:(\\mathcal{B},I)]$ is given by $I^k$, where $k = {\\rm codim}(\\mathcal{B})$.\n\\begin{rem} When $I = I_Z$, the above procedure is also called \\emph{rescaling} \\cite{Lanius16, Melrose93}; we then denote $[\\mathcal{A}\\:(\\mathcal{B},I_Z)]$ by $[\\mathcal{A}\\:\\mathcal{B}]$ or ${}^{\\mathcal{B}} \\mathcal{A}$ and refer to it as the \\emph{$(\\mathcal{B},Z)$-rescaling} of $\\mathcal{A}$.\n\\end{rem}\n\\begin{rem}\\label{rem:bbprimerescaling} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I$ a smooth divisor ideal and $\\mathcal{B}, \\mathcal{B}' \\to Z_I$ two Lie subalgebroids of $\\mathcal{A}$ such that $\\mathcal{B} \\subseteq \\mathcal{B}'$. Then $[\\mathcal{A}\\:(\\mathcal{B},I)]$ is a $[\\mathcal{A}\\:(\\mathcal{B}',I)]$-Lie algebroid.\n\\end{rem}\nOne can always modify using $0_{Z_I} \\subseteq \\mathcal{A}|_{Z_I}$, the trivial Lie subalgebroid, with\n\\begin{equation*}\n\t[\\mathcal{A}\\:(0_{Z_I},I)] = I \\cdot \\mathcal{A},\n\\end{equation*}\nwhich is the secondary ideal Lie algebroid with section module $I \\cdot \\Gamma(\\mathcal{A})$. Thus lower modification by $0_{Z_I}$ is the harshest, while lower modification using $\\mathcal{B} = \\rho_\\mathcal{A}^{-1}(T(Z_I))$ (if it exists, c.f.\\ \\autoref{rem:inverseimagesmooth}) is the mildest. Note that $0_{Z_I}$-modification is not idempotent.\nThe process of lower modification is commutative when performed relative to disjoint submanifolds.\n\\begin{prop}\\label{prop:lowermodifycommute} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I,I' \\subseteq C^\\infty(X)$ smooth divisor ideals and $(\\mathcal{B}, Z_I)$, $(\\mathcal{B}',Z_{I'})$ Lie subalgebroids of $\\mathcal{A}$ such that $Z_I \\cap Z_{I'} = \\emptyset$. Then lower modification at $Z_I$ and $Z_{I'}$ commute and coincides with taking the fiber product, i.e. we have isomorphisms\n\\begin{equation*}\n\t[[\\mathcal{A}\\:(\\mathcal{B},I)]:(\\mathcal{B}',I')] \\cong [\\mathcal{A}\\:(\\mathcal{B},I)] \\oplus_{\\mathcal{A}} [\\mathcal{A}\\:(\\mathcal{B}',I')] \\cong [[\\mathcal{A}\\:(\\mathcal{B}',I')]:(\\mathcal{B},I)].\n\\end{equation*}\n\\end{prop}\nThe proof of the above is immediate upon noticing we can view $(\\mathcal{B}',Z_{I'})$ as a Lie subalgebroid of $[\\mathcal{A}\\:(\\mathcal{B},I)]$ because the latter is isomorphic to $\\mathcal{A}$ outside $Z_I$, and $Z_I \\cap Z_{I'} = \\emptyset$.\n\\begin{rem} Elementary modification can also be performed in an even slightly more general setting. Namely, instead of a smooth divisor ideal one can use a $C^\\infty(X)$-module $\\mathcal{M}$ that is locally free of rank one, and a suitable vector bundle supported on its smooth support.\n\\end{rem}\nUpper modification is a left inverse to lower modification, and vice versa regarding a right inverse when this makes sense. Often the morphism used is the anchor map on sections.\n\\begin{prop} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I \\subseteq C^\\infty(X)$ a smooth divisor ideal and $\\mathcal{B} \\to Z_I$ a Lie subalgebroid of $\\mathcal{A}$. Then $\\{[\\mathcal{A}\\:(\\mathcal{B},I)]\\:(\\mathcal{B},I)\\} \\cong \\mathcal{A}$, and similarly when there is a surjective Lie algebroid comorphism from $\\mathcal{B}$ onto $\\mathcal{A}$ we have $[\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}\\:(\\mathcal{B},I)] \\cong \\mathcal{A}$.\n\\end{prop}\n\\subsubsection{Examples}\n\\label{sec:examplesmodification}\nWe finish the discussion of modifications by considering some examples. See also \\cite{Melrose95} for further examples when $I = I_Z$, which are of interest in geometric analysis.\n\\begin{exa} Let $(X,Z)$ be a log pair and consider $\\mathcal{A} = TX$ and $\\mathcal{B} = 0_Z$. Then $[\\mathcal{A}\\:\\mathcal{B}] = \\mathcal{B}_Z$, the zero tangent bundle associated to $Z$ (see \\autoref{exa:zvx}).\n\\end{exa}\n\\begin{exa} Let $(X,Z)$ be a log pair and consider $\\mathcal{A} = TX$ and $\\mathcal{B} = TZ$. Then $[\\mathcal{A}\\:\\mathcal{B}] = \\mathcal{A}_Z$, the log-tangent bundle associated to $Z$ (see \\autoref{exa:zvx}).\n\\end{exa}\nThe above example can be extended to cases where $\\mathcal{B} = TF$ for $F$ an involutive distribution on $Z$, for example the kernel of a closed one-form on $Z$. The resulting Lie algebroid $TX(-\\log(Z,F)) := [TX\\:TF]$ occurs in the context of log-Poisson structures (c.f.\\ \\cite{KlaasseLanius18,GualtieriLiPelayoRatiu17}).\n\\begin{exa} Let $(X,Z)$ be a log pair and consider $\\mathcal{A} = \\mathcal{A}_Z$ and $\\mathcal{B} = TZ$ with surjective bundle morphism $\\rho_\\mathcal{A}|_Z\\colon \\mathcal{A}|_Z \\to \\mathcal{B}$. Then $\\rho_\\mathcal{A}$ is a Lie algebroid comorphism and $\\{\\mathcal{A}_Z\\:\\mathcal{B}\\} = TX$.\n\\end{exa}\n\\begin{exa} Let $(X,Z)$ be a log pair and consider $\\mathcal{A} = \\mathcal{A}_Z$ and $\\mathcal{B} = 0_Z$. Then $[\\mathcal{A}\\:\\mathcal{B}] = \\mathcal{C}_Z$, the \\emph{scattering tangent bundle} of \\cite{Lanius16,Melrose95}.\n\\end{exa}\nFurther examples obtained using elliptic divisor ideals can be found in \\cite{KlaasseLanius18}.\n\n\n\\section{Poisson structures of divisor-type}\n\\label{sec:poissondivtype}\nIn this section we discuss the notion of an $\\mathcal{A}$-Poisson structure (c.f.\\ \\cite{CannasDaSilvaWeinstein99}), their interaction with divisors, and the process of lifting using base-preserving Lie algebroid morphisms. Most $\\mathcal{A}$-Poisson structures we will encounter will be generically nondegenerate, or generically regular in a precise sense. At first reading, the reader can let $\\mathcal{A} = TX$ in most of what follows.\n\\begin{defn} An \\emph{$\\mathcal{A}$-Poisson structure} is an $\\mathcal{A}$-bivector $\\pi_\\mathcal{A}\\in \\Gamma(\\wedge^2\\mathcal{A})$ with $[\\pi_\\mathcal{A},\\pi_\\mathcal{A}]_{\\mathcal{A}} = 0$.\n\\end{defn}\nWe denote the space of such $\\mathcal{A}$-bivectors by ${\\rm Poiss}(\\mathcal{A})$, and set ${\\rm Poiss}(X) := {\\rm Poiss}(TX)$. Of particular interest to us are those Poisson bivectors that are \\emph{generically nondegenerate}, i.e.\\ for which $\\pi^\\sharp_{\\mathcal{A}}\\colon \\mathcal{A}^* \\to \\mathcal{A}$ is nondegenerate on an open dense subset of $X$. As for Poisson structures, an $\\mathcal{A}$-Poisson structure induces an $\\mathcal{A}$-Lie algebroid structure on the dual bundle $\\mathcal{A}^* \\to X$.\n\\begin{defn}\\label{defn:apoissonalgebroid} Let $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$ be given. The \\emph{$\\mathcal{A}$-Poisson algebroid} $\\mathcal{A}^*_{\\pi_\\mathcal{A}}$ consists of the bundle $\\mathcal{A}^*$ with $\\mathcal{A}$-anchor map $\\pi_\\mathcal{A}^\\sharp\\colon \\mathcal{A}^* \\to \\mathcal{A}$, whose Lie bracket on $\\Gamma(\\mathcal{A}^*_{\\pi_{\\mathcal{A}}})$ is given by\n\t%\n\t\\begin{equation*}\n\t[v,w]_{\\mathcal{A}^*_{\\pi_\\mathcal{A}}} = \\mathcal{L}_{\\pi_\\mathcal{A}^\\sharp v} w - \\mathcal{L}_{\\pi_\\mathcal{A}^\\sharp w} v - d_{\\mathcal{A}} \\pi_{\\mathcal{A}}(v,w), \\qquad v,w \\in \\Gamma(\\mathcal{A}^*_{\\pi_{\\mathcal{A}}}).\n\t\\end{equation*}\n\\end{defn}\nPoisson structures can be related to the divisors of \\autoref{sec:divonmanifolds} as follows. Denote the space of $\\mathcal{A}$-bivectors by $\\mathfrak{X}^2(\\mathcal{A}) = \\Gamma(\\wedge^2 \\mathcal{A})$ (\\emph{not} $\\Gamma(\\wedge^2 T\\mathcal{A})$), so that we have $\\mathfrak{X}^2(X) = \\mathfrak{X}^2(TX)$, although no confusion should arise. The \\emph{Pfaffian} of $\\pi_\\mathcal{A}\\in \\mathfrak{X}^2(\\mathcal{A}^{2n})$ is the section $\\wedge^n \\pi_\\mathcal{A} \\in \\Gamma(\\det(\\mathcal{A}))$. An $\\mathcal{A}$-Poisson structure $\\pi_{\\mathcal{A}}$ is called \\emph{nondegenerate} if $\\pi_{\\mathcal{A}}^\\sharp\\colon \\mathcal{A}^* \\to \\mathcal{A}$ is an isomorphism, whose existence forces the rank of $\\mathcal{A}$ to be even. Alternatively, one can demand the Pfaffian $\\wedge^n \\pi_\\mathcal{A}$ to be nowhere vanishing, and similarly for $\\wedge^m \\pi_{\\mathcal{A}}$ regarding $2m$-regularity.\n\\begin{defn}\\label{defn:adivisortype} An $\\mathcal{A}$-Poisson structure $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$ is of \\emph{$m$-divisor-type} for some $m \\geq 0$ if $\\wedge^{m+1} \\pi_{\\mathcal{A}} \\equiv 0$ and there exists a line subbundle $K \\subseteq \\wedge^{2m} \\mathcal{A}$ such that $(K, \\wedge^{m} \\pi_\\mathcal{A})$ is a divisor. Denote the divisor of $\\pi_\\mathcal{A}$ by ${\\rm div}(\\pi_\\mathcal{A})$ and the associated divisor ideal by $I_{\\pi_\\mathcal{A}} \\subseteq C^\\infty(X)$.\n\\end{defn}\nNote that if $\\pi_\\mathcal{A}$ is of $m$-divisor-type, it generically has rank $2m$, and if $\\pi_{\\mathcal{A}}$ specifies the trivial divisor, then $\\pi_{\\mathcal{A}}$ is $2m$-regular. In the above definition, when $2m = {\\rm rank}(\\mathcal{A})$, we will simply say that $\\pi_\\mathcal{A}$ is of \\emph{divisor-type}, and then the line bundle $K = \\det(\\mathcal{A})$ can be used.\n\\begin{rem}\\label{rem:apoissontriangbialgebroid} An $\\mathcal{A}$-Poisson structure defines a triangular Lie bialgebroid $(\\mathcal{A}, \\mathcal{A}^*_{\\pi_\\mathcal{A}})$ \\cite{KosmannSchwarzbach95,KosmannSchwarzbachLaurentGengoux05,LiuWeinsteinXu97,MackenzieXu94}: in general, a pair of Lie algebroids $(\\mathcal{A},\\mathcal{A}^*)$ is a Lie bialgebroid if for all $v, w \\in \\Gamma(\\mathcal{A})$\n\t%\n\t\\begin{equation*}\n\td_{\\mathcal{A}^*} [v,w]_\\mathcal{A} = \\mathcal{L}_{v} d_{\\mathcal{A}^*} w - \\mathcal{L}_w d_{\\mathcal{A}^*} v.\n\t\\end{equation*}\n\t%\n\tAny Lie bialgebroid $(\\mathcal{A},\\mathcal{A}^*)$ induces a Poisson structure $\\pi \\in {\\rm Poiss}(X)$ with $\\pi^\\sharp = \\rho_{\\mathcal{A}} \\circ \\rho_{\\mathcal{A}^*}^*$. Not every Lie bialgebroid is triangular (yet $\\pi = \\rho_{\\mathcal{A}}(\\pi_{\\mathcal{A}})$ if it is), and the maps $\\pi_{\\mathcal{A}}^\\sharp \\colon \\mathcal{A}^* \\to \\mathcal{A}$ and $\\rho_{\\mathcal{A}^*}^* \\colon T^*X \\to \\mathcal{A}^*$ are Lie algebroid morphisms. These types of structures are also considered in \\cite{Lanius16,Lanius18}, for Lie algebroids whose isomorphism locus is the complement of a hypersurface.\n\\end{rem}\n\\begin{rem} Given $\\pi_{\\mathcal{A}} \\in {\\rm Poiss}(\\mathcal{A})$ we obtain a (singular) $\\mathcal{A}$-distribution $D_{\\pi_\\mathcal{A}} = \\pi_\\mathcal{A}^\\sharp(\\mathcal{A}^*)$. Regularity of $\\pi_\\mathcal{A}$ is the same as regularity of $D_{\\pi_\\mathcal{A}}$, and then $\\det(D_{\\pi_\\mathcal{A}}) = K$, with $K \\subseteq \\wedge^{2m} \\mathcal{A}$.\n\\end{rem}\n\\begin{rem} If $\\pi_\\mathcal{A}$ is of $m$-divisor-type, it is generically regular of rank $2m$. This is not an if and only if unless $2m = \\dim(X)$: being of $m$-divisor-type for $2m < \\dim(X)$ further demands that the way in which $\\pi_{\\mathcal{A}}$ degenerates is controlled in a precise sense. See also \\autoref{sec:almostregularpoisson}.\n\\end{rem}\nGiven a divisor ideal $I \\subseteq C^\\infty(X)$, we say $\\pi_\\mathcal{A}$ is of \\emph{($m$-)$I$-divisor-type} if $I_{\\pi_\\mathcal{A}} = I$, noting that we must specify both the divisor ideal $I$, and the generic rank $2m$. Given a divisor ideal $I$, we denote by $\\mathfrak{X}^2_I(\\mathcal{A}) \\subseteq \\mathfrak{X}^2(\\mathcal{A})$ the space of divisor $\\mathcal{A}$-bivectors $\\pi_\\mathcal{A}$ such that $I_{\\pi_\\mathcal{A}} = I$. We denote by ${\\rm Poiss}_I(\\mathcal{A})$ the space of $\\mathcal{A}$-Poisson structures of $I$-divisor-type, and by ${\\rm Poiss}_{I,m}(\\mathcal{A})$ the space of $\\mathcal{A}$-Poisson structures of $m$-$I$-divisor-type.\n\\begin{rem} One of our main interests lies in nondegenerate $\\mathcal{A}$-Poisson structures, as we then have access to symplectic techniques (see \\autoref{sec:symplecticliealgebroids}). Indeed, the goal of the lifting process described in Section \\ref{sec:liftingpoisson} is to find a Lie algebroid $\\mathcal{A} \\to X$ for which a given $\\pi \\in {\\rm Poiss}(X)$ can be viewed as a nondegenerate $\\mathcal{A}$-Poisson structure.\n\\end{rem}\nA function $f \\in C^\\infty(X)$ is said to be \\emph{$\\pi_{\\mathcal{A}}$-Casimir} if $\\pi_\\mathcal{A}^\\sharp(d_\\mathcal{A} f) = 0$ (alternatively, $[f,\\pi_{\\mathcal{A}}]_\\mathcal{A} = 0$ using the $\\mathcal{A}$-Schouten bracket). Given an $\\mathcal{A}$-Poisson structure $\\pi_\\mathcal{A}$ of $m$-divisor-type and a $\\pi_\\mathcal{A}$-Casimir function $f$, we can form a new $\\mathcal{A}$-Poisson structure $\\pi'_\\mathcal{A} = f \\cdot \\pi_\\mathcal{A}$. If the zero set of $f$ is nowhere dense, it specifies a divisor $(\\underline{\\mathbb{R}},f)$, so as $\\wedge^{m} \\pi'_\\mathcal{A} = f^m \\cdot \\wedge^m \\pi_\\mathcal{A}$ we conclude that $\\pi'_\\mathcal{A}$ is again of $m$-divisor-type. Moreover, $I_{\\pi'_\\mathcal{A}} =  \\langle f^m \\rangle \\cdot I_{\\pi_\\mathcal{A}}$. See also \\cite[Lemma 3.3]{AndroulidakisZambon17} and \\autoref{sec:almostregularpoisson}.\n\\begin{rem} When $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$ has maximal rank equal to $2$ it follows that $f \\cdot \\pi_\\mathcal{A}$ is $\\mathcal{A}$-Poisson for \\emph{any} function $f \\in C^\\infty(X)$ (as is known for $\\mathcal{A} = TX$): we have $[f\\cdot \\pi_\\mathcal{A}, f \\cdot \\pi_\\mathcal{A}]_\\mathcal{A} = f^2 [\\pi_\\mathcal{A}, \\pi_\\mathcal{A}]_\\mathcal{A} + 2 f \\pi_\\mathcal{A} \\wedge \\pi_\\mathcal{A}^\\sharp(d_\\mathcal{A} f)$, where the latter term vanishes as $\\pi_\\mathcal{A} \\wedge v = 0$ for all $v \\in {\\rm im}(\\pi^\\sharp_\\mathcal{A})$. This can be used to construct examples of $\\mathcal{A}$-Poisson structures of higher divisor-type.\n\\end{rem}\nGiven an $\\mathcal{A}$-Poisson structure, we obtain its \\emph{$\\mathcal{A}$-Hamiltonian vector fields} as the image of the map $f \\mapsto \\pi_\\mathcal{A}^\\sharp(d_\\mathcal{A} f) =: v_{\\mathcal{A}, f} \\in \\mathfrak{X}(\\mathcal{A})$. Similarly we have the \\emph{$\\mathcal{A}$-Poisson vector fields}, i.e.\\ those $v_\\mathcal{A} \\in \\mathfrak{X}(\\mathcal{A})$ such that $\\mathcal{L}_{v_\\mathcal{A}} \\pi_{\\mathcal{A}} = 0$, using the $\\mathcal{A}$-Lie derivative. Define the spaces ${\\rm Ham}(\\pi_\\mathcal{A}) \\subseteq {\\rm Poiss}(\\pi_\\mathcal{A}) \\subseteq \\mathfrak{X}(\\mathcal{A})$ as for Poisson structures. Given $f \\in C^\\infty(X)$ we have $\\pi_{\\mathcal{A}}^\\sharp(d_{\\mathcal{A}} f) = \\pi_{\\mathcal{A}}^\\sharp(\\rho_{\\mathcal{A}}^*(df))$, so that by setting $\\pi := \\rho_\\mathcal{A}(\\pi_\\mathcal{A})$, we see from the relation $\\pi^\\sharp = \\rho_{\\mathcal{A}} \\circ \\pi_{\\mathcal{A}}^\\sharp \\circ \\rho_{\\mathcal{A}}^*$ that $\\rho_\\mathcal{A}(v_{\\mathcal{A},f}) = V_f$. Thus the $\\mathcal{A}$-Hamiltonian vector fields for $\\pi_\\mathcal{A}$ surject onto the Hamiltonian vector fields for $\\pi$.\n\\begin{rem}\\label{rem:productapoisson} Let $(X,\\mathcal{A},\\pi_\\mathcal{A})$ and $(X',\\mathcal{A}',\\pi_{\\mathcal{A}'})$ be two $\\mathcal{A}$-Poisson manifolds. Then their product $(X \\times X', \\mathcal{A} \\oplus \\mathcal{A}', \\pi_{\\mathcal{A}} + \\pi_{\\mathcal{A}'})$ is also $\\mathcal{A}$-Poisson, where $\\mathcal{A} \\oplus \\mathcal{A}' \\to X \\times X'$ is the direct product of Lie algebroids (see \\cite{HigginsMackenzie90}), and both projections are $\\mathcal{A}$-Poisson maps. Thus the class of $\\mathcal{A}$-Poisson structures of divisor-type is closed under products, using the external tensor product of divisors (see \\autoref{defn:divdirectstum}). This is noted in \\cite{AndroulidakisZambon17} for $\\mathcal{A} = TX$ using \\autoref{cor:divtypealmostreg}.\n\\end{rem}\n\\subsection{Degeneraci loci}\nWe discuss degeneraci loci of $\\mathcal{A}$-Poisson structures $\\pi_\\mathcal{A}$, as in \\cite{Pym13}. Given $\\pi_{\\mathcal{A}} \\in \\mathfrak{X}^2(\\mathcal{A}^n)$ with bundle map $\\pi_\\mathcal{A}^\\sharp\\colon \\mathcal{A}^* \\to \\mathcal{A}$, we have both $\\det(\\pi_\\mathcal{A}^\\sharp) \\in \\Gamma(\\det(\\mathcal{A})^2)$ and $\\wedge^n \\pi_\\mathcal{A} \\in \\Gamma(\\det(\\mathcal{A}))$, which are related by $\\det(\\pi_\\mathcal{A}^\\sharp) = \\wedge^n \\pi_\\mathcal{A} \\otimes \\wedge^n \\pi_\\mathcal{A}$. Considering the definition for Lie algebroids, we could define the degeneracy loci of $\\pi_\\mathcal{A}$ using the map $\\pi_\\mathcal{A}^\\sharp$, i.e.\\ as the degeneracy locus of its $\\mathcal{A}$-Poisson algebroid $\\mathcal{A}^*_{\\pi_{\\mathcal{A}}}$. However, $\\pi^\\sharp_\\mathcal{A}$ is skew-symmetric, so its vanishing degree is too high. Instead we will use the Pfaffians $\\wedge^k\\pi_\\mathcal{A} \\in \\Gamma(\\wedge^{2k} \\mathcal{A})$ for $k \\geq 0$. These give rise to maps $\\wedge^k \\pi_\\mathcal{A}\\colon \\Gamma(\\wedge^{2k} \\mathcal{A}^*) \\to C^\\infty(X)$ on sections, and thus define ideals $I_{\\pi_\\mathcal{A},2k}$.\n\\begin{defn} Let $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$. The \\emph{$2k$th degeneracy locus} of $\\pi_\\mathcal{A}$ for $k \\geq 0$ is the subspace $X_{\\pi_\\mathcal{A},2k} \\subseteq X$ of points where ${\\rm rank}(\\pi_{\\mathcal{A}}) \\leq 2k$, defined by the ideal $I_{\\pi_\\mathcal{A},2k+2} \\subseteq C^\\infty(X)$.\n\\end{defn}\nThus the degeneraci loci of $\\mathcal{A}_{\\pi_\\mathcal{A}}^*$ and $\\pi_\\mathcal{A}$ agree as subspaces of $X$, but their ideals do not. Recall that a submanifold $N$ is \\emph{$\\pi$-Poisson} if $\\pi^\\sharp(T_x^* X) \\subseteq T_x N$ for all $x \\in N$. When $N$ is closed this is equivalent to its vanishing ideal $I_N$ being a \\emph{Poisson ideal}, i.e.\\ $\\{I_N,C^\\infty(X)\\}_\\pi \\subseteq I_N$, or to $N$ being $T^*_\\pi X$-invariant. Moreover, note that $I$ being a $\\pi$-Poisson ideal is equivalent to $\\pi^\\sharp(d I) = 0$. When $N$ is $\\pi$-Poisson, it carries a uniquely induced Poisson structure $\\pi_N$ for which the inclusion is a Poisson map. The following is then a consequence of \\autoref{prop:degenlocus}.\n\\begin{prop}\\label{prop:pidegenpipoisson} Each degeneracy locus $X_{\\pi,2k}$ is a $\\pi$-Poisson submanifold if it is smooth.\n\\end{prop}\nFor $\\mathcal{A}$-Poisson structures the notion of a \\emph{$\\pi_{\\mathcal{A}}$-Poisson submanifold} is more involved: it instead is a Lie subalgebroid $(\\varphi,i)\\colon (\\mathcal{B},N) \\hookrightarrow (\\mathcal{A},X)$ with $\\pi_\\mathcal{B} \\in {\\rm Poiss}(\\mathcal{B})$ with $\\varphi_*(\\pi_\\mathcal{B}) = \\pi_\\mathcal{A}$.\nA point $x \\in X$ is $\\pi_{\\mathcal{A}}$-\\emph{regular} if the rank of $\\pi_\\mathcal{A}$ is constant in some open neighbourhood of $x$, and singular otherwise. We denote by $X_{\\pi_\\mathcal{A}, {\\rm reg}} \\subseteq X$ the space of $\\pi_\\mathcal{A}$-regular points, also called the \\emph{regular locus} of $\\pi_\\mathcal{A}$. This is an open dense subspace of $X$, with its complement $X_{\\pi_\\mathcal{A}, {\\rm sing}}$, the \\emph{singular locus} of $\\pi_\\mathcal{A}$, being closed and nowhere dense. Denote by $X_{\\pi_\\mathcal{A}, {\\rm max}} \\subseteq X$ the open subspace where $\\pi_\\mathcal{A}$ has maximal rank. It immediately follows that $X \\backslash X_{\\pi_\\mathcal{A}, {\\rm max}} = X_{\\pi_\\mathcal{A}, 2k-2} = (\\wedge^k \\pi_\\mathcal{A})^{-1}(0)$, where $2k$ is the maximal rank of $\\pi_\\mathcal{A}$. In general we have $X_{\\pi_\\mathcal{A}, {\\rm max}} \\subsetneq X_{\\pi_\\mathcal{A}, {\\rm reg}}$.\n\\subsection{Relation to almost-regularity}\n\\label{sec:almostregularpoisson}\nA Poisson structure $\\pi \\in {\\rm Poiss}(X)$ specifies a singular Lie subalgebroid $\\mathcal{F}_\\pi := \\pi^\\sharp(\\Gamma(T_\\pi^*X))$ of $TX$ consisting of $\\pi$-Hamiltonian vector fields. Following Androulidakis--Zambon \\cite{AndroulidakisZambon17}, we call the Poisson structure $\\pi$ \\emph{almost-regular} if $\\mathcal{F}_\\pi$ is projective. There is a nice criterion for almost-regularity in terms of the $\\pi$-symplectic leaves.\n\\begin{prop}[{\\cite[Theorem 2.8]{AndroulidakisZambon17}}]\\label{prop:almostregulardistr} Let $\\pi \\in {\\rm Poiss}(X)$. Then $\\pi$ is almost-regular if and only if $X_{\\pi, {\\rm max}} \\subseteq X$ is dense and there exists a distribution $D_\\pi \\subseteq TX$ such that $D_{\\pi,x} = T_x L$ for all $x \\in X_{\\pi, {\\rm max}}$, where $L$ is the $\\pi$-symplectic leaf through $x$.\n\\end{prop}\nHence almost-regularity of $\\pi$ depends only on the partition of $X$ into immersed leaves.\n\\begin{rem} As is stressed in \\cite{AndroulidakisZambon17}, in general projectivity of a singular Lie subalgebroid $\\rho_{A}(\\Gamma(\\mathcal{A})) \\subseteq \\Gamma(TX)$ cannot be tested from just the orbits of $\\mathcal{A}$. However, this is possible for those singular Lie subalgebroids arising from Poisson structures due to skew-symmetry of the bivector $\\pi$, which implies that the anchor $\\pi^\\sharp$ of $T^*_\\pi X$ is skew, so that $\\ker(\\pi^\\sharp_x) = {\\rm im}(\\pi^\\sharp_x)^\\circ$.\n\\end{rem}\nBy continuity and density of $X_{\\pi, {\\rm max}}$, the distribution $D_\\pi$ in the above proposition is unique and involutive, and determines a $2k$-dimensional regular foliation by $\\pi$-Poisson submanifolds. Phrased differently, $D_\\pi$ specifies a $2k$-regular Lie algebroid whose orbits are $\\pi$-Poisson submanifolds. Note that $X_{\\pi, {\\rm max}} = X_{\\pi, {\\rm reg}}$ for almost-regular Poisson structures. There is another characterization of almost-regularity, phrased in terms of $\\pi$ itself. Let ${\\rm Poiss}_{2k}(X) \\subseteq {\\rm Poiss}(X)$ be the space of Poisson structures of maximal rank $2k$, i.e.\\ for which $X_{\\pi,{\\rm max}} = X \\backslash X_{2k-2}$.\n\\begin{prop}[{\\cite[Proposition 2.11]{AndroulidakisZambon17}}]\\label{prop:almostregularline} Let $\\pi \\in {\\rm Poiss}_{2k}(X)$. Then $\\pi$ is almost-regular if and only if $X_{\\pi, {\\rm max}}$ is dense and there exists a line bundle $K \\subseteq \\wedge^{2k} TX$ such that $\\wedge^k \\pi \\in \\Gamma(K)$.\n\\end{prop}\nThe above distribution $D_\\pi$ and line bundle $K$ are related via $\\det(D_\\pi) = \\wedge^k D_\\pi = K$. Note that because for $\\pi$ as in \\autoref{prop:almostregularline} we have $X_{\\pi,{\\rm max}} = X_{\\pi,{\\rm reg}} = X \\backslash X_{2k-2} = (\\wedge^{k} \\pi)^{-1}(0)$, so that density of this subspace amounts to saying that the pair $(K,\\wedge^k \\pi)$ is a divisor on $X$.\n\\begin{rem} In work of Lanius \\cite{Lanius16two,Lanius16,Lanius17}, specifically for computing Poisson cohomology, the \\emph{rigged algebroid} $\\mathcal{R}_\\pi$ of a Poisson structure $\\pi$ plays a large role. It is defined as the almost-injective Lie algebroid inducing $\\mathcal{F}_\\pi = \\pi^\\sharp(\\Gamma(T^*X))$. In the joint paper \\cite{KlaasseLanius18} we systematically develop its use in computing the Poisson cohomology of almost-regular Poisson structures.\n\\end{rem}\nTogether with \\autoref{defn:adivisortype} we obtain the following dictionary between concepts.\n\\begin{cor}\\label{cor:divtypealmostreg} Let $\\pi \\in {\\rm Poiss}(X)$ be given. Then the following are equivalent:\n\t%\n\t\\begin{itemize}\n\t\\item $\\pi$ is of $m$-divisor-type for some $m \\geq 0$;\n\t\\item $\\pi$ is almost-regular;\n\t\\item The rigged algebroid $\\mathcal{R}_\\pi$ exists.\n\t\\end{itemize}\n\\end{cor}\nBy defining almost-regularity for $\\mathcal{A}$-Poisson structures as projectivity of $\\pi^\\sharp_{\\mathcal{A}}(\\Gamma(\\mathcal{A}^*))$, and its \\emph{$\\mathcal{A}$-rigged algebroid} $\\mathcal{R}_{\\pi_{\\mathcal{A}}}$ via $\\Gamma(\\mathcal{R}_{\\pi_{\\mathcal{A}}}) \\cong \\pi^\\sharp_{\\mathcal{A}}(\\Gamma(\\mathcal{A}^*))$ \\cite{KlaasseLanius18}, \\autoref{cor:divtypealmostreg} is seen to hold for $\\mathcal{A}$-Poisson structures for any Lie algebroid $\\mathcal{A}$, after adapting the proof of \\cite[Proposition 2.11]{AndroulidakisZambon17}.\n\\subsection{Lifting Poisson structures}\n\\label{sec:liftingpoisson}\nWe next introduce the process of lifting $\\mathcal{A}$-Poisson structures through base-preserving Lie algebroid morphisms (see also the discussion in \\cite{CavalcantiKlaasse18}).\n\\begin{defn}\\label{defn:poissonalift} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ be a Lie algebroid morphism. An $\\mathcal{A}'$-bivector $\\pi_{\\mathcal{A}'} \\in \\Gamma(\\wedge^2 \\mathcal{A}')$ is \\emph{$\\mathcal{A}$-liftable} if there exists an $\\mathcal{A}$-bivector $\\pi_{\\mathcal{A}} \\in \\Gamma(\\wedge^2 \\mathcal{A})$ such that $\\varphi(\\pi_{\\mathcal{A}}) = \\pi_{\\mathcal{A}'}$.\n\\end{defn}\nIn the above situation we again call $\\pi_{\\mathcal{A}}$ the \\emph{$\\mathcal{A}$-lift} of $\\pi_{\\mathcal{A}'}$, and say that $\\pi_{\\mathcal{A}'}$ is of \\emph{$\\mathcal{A}$-type}. We will denote the space of $\\mathcal{A}$-liftable bivectors by $\\mathfrak{X}_\\mathcal{A}^2(X) := \\rho_\\mathcal{A}(\\mathfrak{X}^2(\\mathcal{A})) \\subset \\mathfrak{X}^2(X)$, and the space of $\\mathcal{A}$-liftable Poisson structures by ${\\rm Poiss}_{\\mathcal{A}}(X) \\subseteq {\\rm Poiss}(X)$.\n\\begin{rem} In \\cite[Definition 2.16]{Lanius16} this notion is considered using slightly different terminology: relating it to ours, there $\\pi \\in {\\rm Poiss}(X)$ is called $\\mathcal{A}$-Poisson if it is of $\\mathcal{A}$-type. It seems preferable to describe Poisson structures by their divisor (c.f.\\ \\autoref{defn:adivisortype}) instead of to which Lie algebroid they can be (nondegenerately) lifted, as the latter need not be unique.\n\\end{rem}\nThe $\\mathcal{A}$-lift of a $\\mathcal{A}'$-Poisson structure of $\\mathcal{A}$-type need not necessarily be $\\mathcal{A}$-Poisson (consider $\\mathcal{A}$ with trivial anchor and $\\pi \\equiv 0$). However, this is true when $\\varphi$ is of divisor-type, as will be the case for us (moreover, the $\\mathcal{A}$-lift will then be unique). More generally this holds for almost-injective $\\varphi$. We have $\\pi_{\\mathcal{A}'}^\\sharp = \\varphi \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\varphi^*$ as maps, summarized in the following diagram.\n\\begin{center}\n\t\\begin{tikzpicture}\n\t%\n\t\\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em,text height=1.5ex, text depth=0.25ex]\n\t{\t\\mathcal{A}^* & \\mathcal{A} \\\\ \\mathcal{A}'{}^* & \\mathcal{A}' \\\\};\n\t\\path[-stealth]\n\t(m-1-1) edge node [above] {$\\pi_\\mathcal{A}^\\sharp$} (m-1-2)\n\t(m-2-1) edge node [left] {$\\varphi^*$} (m-1-1)\n\t(m-2-1) edge node [above] {$\\pi_{\\mathcal{A}'}^\\sharp$} (m-2-2)\n\t(m-1-2) edge node [right] {$\\varphi$} (m-2-2);\n\t\\end{tikzpicture}\n\\end{center}\n\\begin{defn}\\label{defn:apoissonmap} An \\emph{$\\mathcal{A}$-Poisson map} $(\\varphi,f)\\colon (\\mathcal{A},X,\\pi_\\mathcal{A}) \\to (\\mathcal{B},X',\\pi_\\mathcal{B})$ is a Lie algebroid morphism between Lie algebroids with $\\mathcal{A}$-Poisson structures such that $\\varphi(\\pi_\\mathcal{A}) = \\pi_\\mathcal{B}$.\n\\end{defn}\nAs for Poisson maps, it is not true that any $\\mathcal{A}$-Poisson structure can be pushed forward to an $\\mathcal{A}'$-Poisson structure along a given Lie algebroid morphism $(\\varphi,f)\\colon \\mathcal{A} \\to \\mathcal{A}'$. However, this is always possible along base-preserving morphisms. In particular, given an $\\mathcal{A}$-Poisson structure, the bivector $\\pi := \\rho_{A}(\\pi_{\\mathcal{A}})$ is always Poisson. The lifting condition of an $\\mathcal{A}'$-Poisson structure to an $\\mathcal{A}$-Poisson structure is exactly that $(\\varphi,{\\rm id}_X)\\colon (\\mathcal{A},\\pi_\\mathcal{A}) \\to (\\mathcal{A}',\\pi_{\\mathcal{A}'})$ is an $\\mathcal{A}$-Poisson map. If $\\pi_{\\mathcal{A}'}$ has a nondegenerate $\\mathcal{A}$-Poisson lift $\\pi_\\mathcal{A}$, we say $\\pi_{\\mathcal{A}'}$ is of \\emph{nondegenerate $\\mathcal{A}$-type}, and similarly for other types (e.g.\\ for regular ones). We often omit writing the Lie algebroid morphism $\\varphi$ (as typically it is induced by an inclusion of modules), and always use the anchor map $\\rho_{\\mathcal{A}}$ when considering $\\mathcal{A}$-lifts of bivectors on $TX$. When $\\pi_{\\mathcal{A}}$ nondegenerately lifts $\\pi_{\\mathcal{A}'}$, we will call its inverse $\\omega_{\\mathcal{A}} := \\pi_\\mathcal{A}^{-1} \\in \\Omega^2(\\mathcal{A})$ the \\emph{dual $\\mathcal{A}$-form} to $\\pi_{\\mathcal{A}'}$. This is an $\\mathcal{A}$-symplectic structure in the sense of \\autoref{sec:symplecticliealgebroids}, making $\\mathcal{A}$ into a symplectic Lie algebroid \\cite{NestTsygan01}.\n\nThus the process of lifting constitutes the existence of a base-preserving $\\mathcal{A}$-Poisson map. In terms of $\\mathcal{A}$-Poisson algebroids, we obtain the following analogue of the well-known fact that a Poisson map induces a Lie algebroid comorphism between the respective Poisson algebroids.\n\\begin{prop}\\label{prop:apoissoncomorphism} Let $(\\varphi,f)\\colon (\\mathcal{A},X,\\pi_\\mathcal{A}) \\to (\\mathcal{B},X',\\pi_\\mathcal{B})$ be an $\\mathcal{A}$-Poisson map. Then there is an induced $\\mathcal{A}$-Lie algebroid comorphism $(\\varphi;\\varphi,f)\\colon (\\mathcal{A}^*_{\\pi_\\mathcal{A}},\\mathcal{A},X) \\dashedrightarrow (\\mathcal{B}^*_{\\pi_\\mathcal{B}},\\mathcal{B},X')$.\n\\end{prop}\nThe lifting process interacts with Poisson structures of divisor-type as follows.\n\\begin{prop}\\label{prop:apoissondivtypelifts} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ be a Lie algebroid morphism of divisor-type, and let $\\pi_{\\mathcal{A}'}$ be an $\\mathcal{A}'$-Poisson structure of $m$-divisor-type for some $m \\geq 0$ that is also of $\\mathcal{A}$-type. Then its unique $\\mathcal{A}$-lift $\\pi_\\mathcal{A}$ is of $m$-divisor-type, and ${\\rm div}(\\pi_{\\mathcal{A}'}) = {\\rm div}(\\pi_\\mathcal{A}) \\cdot {\\rm div}(\\varphi)$ so that $I_{\\pi_{\\mathcal{A}'}} = I_{\\pi_{\\mathcal{A}}} \\cdot I_\\varphi$.\n\\end{prop}\n\\bp Let $\\pi_\\mathcal{A}$ be an $\\mathcal{A}$-lift of $\\pi_{\\mathcal{A}'}$, so that $(\\wedge^2 \\varphi)(\\pi_\\mathcal{A}) = \\pi_{\\mathcal{A}'}$. We have ${\\rm div}(\\pi_{\\mathcal{A}'}) = (K, \\wedge^m \\pi_{\\mathcal{A}'})$ for some line bundle $K \\subseteq \\wedge^{2m} \\mathcal{A}'$. As $\\wedge^{2m} \\varphi\\colon \\wedge^{2m} \\mathcal{A} \\to \\wedge^{2m} \\mathcal{A}'$, we find a line bundle $K_\\mathcal{A} \\subseteq \\wedge^{2m} \\mathcal{A}$ for which $\\wedge^{2m}\\varphi\\colon K_\\mathcal{A} \\to K$ and $\\wedge^m \\pi_\\mathcal{A} \\in \\Gamma(K_\\mathcal{A})$. If $\\pi_\\mathcal{A}$ vanishes then so must $\\pi_{\\mathcal{A}'}$, so that $(K_\\mathcal{A}, \\wedge^m \\pi_\\mathcal{A})$ is a divisor, ${\\rm div}(\\pi_{\\mathcal{A}})$, as a subset of a nowhere dense set is nowhere dense. The statement that ${\\rm div}(\\pi_{\\mathcal{A}'}) = {\\rm div}(\\pi_\\mathcal{A}) \\cdot {\\rm div}(\\varphi)$ then follows from the relation $\\pi_{\\mathcal{A}'}^\\sharp = \\varphi \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\varphi^*$.\n\\end{proof}\nThis proposition makes precise that the process of lifting makes $\\mathcal{A}$-Poisson structures less degenerate, as their degeneracies are absorbed in that of the Lie algebroid morphism (hence, in the Lie algebroid it is lifted to). Moreover, it says that $\\pi_{\\mathcal{A}'}$ is of $m$-regular $\\mathcal{A}$-type if and only if ${\\rm div}(\\pi_{\\mathcal{A}'}) = {\\rm div}(\\varphi)$, once $\\pi_{\\mathcal{A}'}$ is of $\\mathcal{A}$-type (this includes being of nondegenerate $\\mathcal{A}$-type. There is a similar statement for not necessarily base-preserving $\\mathcal{A}$-Poisson maps.\n\\begin{lem} Let $(\\varphi,f)\\colon (\\mathcal{A}^{2n},X,\\pi_{\\mathcal{A}}) \\to (\\mathcal{B}^{2n},X',\\pi_{\\mathcal{B}})$ be an $\\mathcal{A}$-Poisson map between Lie algebroids with $\\mathcal{A}$-Poisson structures of divisor-type. Then $I_{\\pi_{\\mathcal{A}}}$ divides $f^*I_{\\pi_{\\mathcal{B}}}$.\n\\end{lem}\n\\bp As ${\\rm rank}(\\mathcal{A}) = {\\rm rank}(\\mathcal{B})$, the fact that $\\varphi(\\pi) = \\sigma$ implies that $\\det(\\varphi)(\\pi_{\\mathcal{A}}^n) = \\pi_{\\mathcal{B}}^n$. Take local volume forms $\\mu_\\mathcal{A}, \\mu_\\mathcal{B}$ for $\\mathcal{A}$ and $\\mathcal{B}$. For certain functions $g \\in C^\\infty(X)$, $h \\in C^\\infty(X')$:\n\\begin{equation*}\n\t\\langle\\pi_\\mathcal{A}^n,\\mu_\\mathcal{A}\\rangle = g \\text{, so } I_{\\pi_{\\mathcal{A}}} = \\langle g \\rangle, \\qquad \\text{and} \\qquad \\langle\\pi_{\\mathcal{B}}^n,\\mu_\\mathcal{B}\\rangle = h \\text{, so } I_{\\pi_{\\mathcal{B}}} = \\langle h \\rangle.\n\\end{equation*}\nNote that $\\varphi^* \\mu_\\mathcal{B} = w \\mu_\\mathcal{A}$ for some nonnegative function $w \\in C^\\infty(X)$. Thus $I_{\\pi_{\\mathcal{A}}} \\subseteq f^*I_{\\pi_{\\mathcal{B}}}$, as\n\\begin{equation*}\nf^*(h) = f^*\\langle\\pi_{\\mathcal{B}},\\mu_\\mathcal{B}\\rangle = f^*\\langle\\det(\\varphi)(\\pi_{\\mathcal{A}}^n),\\mu_\\mathcal{B}\\rangle = \\langle\\pi_\\mathcal{A}^n, \\varphi^*\\mu_\\mathcal{B}\\rangle = \\langle\\pi_\\mathcal{A}^n, w \\mu_\\mathcal{A}\\rangle = w \\langle \\pi_\\mathcal{A}^n, \\mu_\\mathcal{A}\\rangle = w g.\\qedhere\n\\end{equation*}\n\\end{proof}\nThe above lemma is of particular interest in the standard case when $\\mathcal{A} = TX$ and $\\mathcal{B} = TX'$.\nFor clarity, in the specific case that $\\mathcal{A}' = TX$, \\autoref{prop:apoissondivtypelifts} says that a Poisson structure $\\pi \\in {\\rm Poiss}(X)$ of divisor-type $I$ can only admit nondegenerate $\\mathcal{A}$-lifts to Lie algebroids satisfying $I_\\mathcal{A} = I$.\nWe can iterate the lifting procedure as follows, whose proof is immediate.\n\\begin{prop}\\label{prop:liftiterate} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ and $(\\varphi',{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}''$ be Lie algebroid morphisms, and $\\pi_{\\mathcal{A}''} \\in \\rm{Poiss}(\\mathcal{A}'')$. Then if $\\pi_{\\mathcal{A}''}$ is of $\\mathcal{A}$-type, it is of $\\mathcal{A}'$-type. If $\\pi_\\mathcal{A}$ is an $\\mathcal{A}$-Poisson lift of $\\pi_{\\mathcal{A}''}$, then $\\pi_{\\mathcal{A}'} := \\varphi(\\pi_\\mathcal{A})$ is an $\\mathcal{A}'$-Poisson lift of $\\pi_{\\mathcal{A}''}$ which itself is of $\\mathcal{A}$-type.\n\\end{prop}\nConsequently, an $\\mathcal{A}'$-Poisson structure $\\pi_{\\mathcal{A}'}$ being of $\\mathcal{A}$-type is the same thing as its underlying Poisson structure $\\pi := \\rho_{\\mathcal{A}'}(\\pi_{\\mathcal{A}'})$ being of $\\mathcal{A}$-type. Let $\\pi_{\\mathcal{A}'}$ be a nondegenerate $\\mathcal{A}'$-Poisson structure. Then any $\\mathcal{A}$-Poisson map $(\\varphi,f)\\colon (\\mathcal{A},\\pi_\\mathcal{A}) \\to (\\mathcal{A}',\\pi_{\\mathcal{A}'})$ has to be a Lie algebroid submersion, i.e.\\ the morphism $\\varphi$ must be fiberwise surjective. This implies the following.\n\\begin{prop}\\label{prop:liftingnondegiso} Let $\\pi \\in {\\rm Poiss}(X)$ be of nondegenerate $\\mathcal{A}' $-type and let  $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ be a morphism between Lie algebroids of the same rank. Then the following are equivalent:\n\t%\n\t\\begin{itemize}\n\t\\item $\\varphi$ is an isomorphism;\n\t\\item $\\pi$ is of $\\mathcal{A}$-type;\n\t\\item $\\pi$ is of nondegenerate $\\mathcal{A}$-type.\n\t\\end{itemize}\n\\end{prop}\nAs a consequence, once one has lifted $\\pi$ to being nondegenerate, one cannot (meaningfully) lift further. This makes sense, as the lifting process is meant to desingularize the Poisson structure, and nondegenerate ones are maximally nonsingular. This is consistent with \\autoref{prop:apoissondivtypelifts}, as isomorphisms specify trivial divisors. Moreover, we see that $\\pi_{\\mathcal{A}'}$ is of nondegenerate $\\mathcal{A}$-type if and only if the underlying Poisson structure $\\pi = \\rho_{\\mathcal{A}'}(\\pi_{\\mathcal{A}'})$ is.\n\\begin{rem}\\label{rem:multiplelifts} A Poisson structure can lift nondegenerately to multiple non-isomorphic Lie algebroids of (necessarily) the same divisor-type. The orbits of these Lie algebroids will specify the same partition of $X$ into leaves. For some consequences of this phenomenon, see \\cite{KlaasseLanius18}.\n\\end{rem}\n\\begin{exa}\\label{exa:mregularapoisson} Let $\\pi_\\mathcal{A}$ be an $m$-regular $\\mathcal{A}$-Poisson structure. Then its image $D_{\\pi_\\mathcal{A}} = \\pi_\\mathcal{A}^\\sharp(\\mathcal{A}^*)$ is a regular $\\mathcal{A}$-distribution. By \\autoref{exa:adistribution}, $D_{\\pi_{\\mathcal{A}}}$ is a Lie subalgebroid of $\\mathcal{A}$ and thus carries an $\\mathcal{A}$-Lie algebroid structure $\\varphi\\colon D_{\\pi_\\mathcal{A}} \\to \\mathcal{A}$. It follows that $\\pi_{\\mathcal{A}}$ is of nondegenerate $D_{\\pi_\\mathcal{A}}$-type.\n\\end{exa}\nLet us give another consequence of lifting. Let $\\mathcal{A} \\to X$ be a Lie algebroid and recall that $\\mathcal{F}_\\mathcal{A} = \\rho_{\\mathcal{A}}(\\Gamma(\\mathcal{A}))$ is its induced singular foliation whose leaves are the orbits of $\\mathcal{A}$.\n\\begin{prop}\\label{prop:atypepoissonsubmfd} Let $\\pi \\in {\\rm Poiss}_\\mathcal{A}(X)$. Then the orbits of $\\mathcal{A}$ are $\\pi$-Poisson submanifolds.\n\\end{prop}\n\\bp The lifting condition $\\rho_{\\mathcal{A}}(\\pi_{\\mathcal{A}}) = \\pi$ equivalently reads $\\pi^\\sharp = \\rho_{\\mathcal{A}} \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\rho_{\\mathcal{A}}^*$. As a submanifold $N$ is $\\pi$-Poisson if $\\pi^\\sharp(T^*_x X) \\subseteq T_x N$ for all $x \\in N$, we note that this condition pointwise becomes $\\rho_{\\mathcal{A},x}(\\mathcal{B}_x) \\subseteq T_x N$ for the subspace $\\mathcal{B}_x := \\pi_{\\mathcal{A},x}^\\sharp(\\rho_{\\mathcal{A},x}^*(T^*_x X)) \\subseteq \\mathcal{A}_x$. If $N$ is an orbit of $\\mathcal{A}$, then by definition $T_x N = {\\rm im}(\\rho_{\\mathcal{A},x}) \\subseteq T_x X$, so that this condition is certainly satisfied.\n\\end{proof}\nThis gives an intuitive description of the lifting process when $\\mathcal{A}$ is almost-injective. Namely, a choice of lifting algebroid $\\mathcal{A}$ amounts to a ``grouping'' of the $\\pi$-symplectic leaves into orbits of $\\mathcal{A}$. Unless $\\mathcal{A}$ is regular (with injective anchor), there may be multiple non-isomorphic Lie algebroids with the same orbits (cf.\\ \\autoref{rem:multiplelifts}). There is a partial converse to \\autoref{prop:atypepoissonsubmfd} in the regular injective case, in which case $\\mathcal{A}$ is just a regular foliation $T\\mathcal{F} \\subseteq TX$.\n\\begin{prop}\\label{prop:regularpipoissonlift} Let $\\pi \\in {\\rm Poiss}(X)$ and $\\mathcal{F}$ be a regular foliation by $\\pi$-Poisson submanifolds, with $D = T\\mathcal{F}$ the associated regular Lie algebroid. Then $\\pi$ is of $D$-type.\n\\end{prop}\n\\bp Let $L$ be a leaf of $\\mathcal{F}$. As it is $\\pi$-Poisson, we have $\\pi_x \\in \\wedge^2 T_x L = \\wedge^2 D_x$ for all $x \\in L$. By regularity there exists $\\pi_D \\in {\\rm Poiss}(D)$ agreeing with $\\pi$ on each leaf, which is a $D$-lift of $\\pi$.\n\\end{proof}\nThis result recovers the fact that a regular Poisson structure is equivalently described as a nondegenerate Poisson bivector for the Lie algebroid of tangencies to its symplectic leaves. The consequences of \\autoref{prop:regularpipoissonlift} for almost-regular Poisson structures (i.e., those of $m$-divisor-type), in light of \\autoref{prop:almostregulardistr}, are explored in \\autoref{sec:liftingalmostreg}.\n\\begin{rem} \\autoref{prop:regularpipoissonlift} is false when $\\mathcal{F}$ is no longer regular, as the lifting property cannot be tested purely by looking at the underlying singular foliation of the Lie algebroid. For example, on $\\mathbb{R}^2$ the Lie algebroids $\\mathcal{A}_Z^k = \\langle x^k \\partial_x, \\partial_y \\rangle$ associated to the jets of $x \\in C^\\infty(\\mathbb{R}^2)$ for $Z = \\{x = 0\\}$ all have the same orbits, but $\\pi = x \\partial_x \\wedge \\partial_y$ only admits $\\mathcal{A}_Z^k$-lifts for $k = 0$ or $1$. Essentially what is used in the regular case is that a decomposition into leaves determines a unique module of vector fields if the decomposition is regular (see the discussion in \\cite{AndroulidakisZambon16}).\n\\end{rem}\n\\subsection{Lifting Poisson structures of divisor-type}\n\\label{sec:liftingdivtype}\nIn this section we discuss the process of lifting Poisson structures of divisor-type. Let $I \\subseteq C^\\infty(X)$ be a divisor ideal, and assume that the associated ideal Lie algebroid $TX_I \\to X$ exists. In other words, assume that the divisor ideal $I$ is projective. Recall that given a Lie algebroid $\\mathcal{A}$ we denote the spaces of $\\mathcal{A}$-liftable and $I$-divisor-bivectors by $\\mathfrak{X}^2_{\\mathcal{A}}(X)$ and $\\mathfrak{X}^2_{I}(X)$ respectively. Similarly, we have their Poisson counterparts ${\\rm Poiss}_{\\mathcal{A}}(X) \\subseteq \\mathfrak{X}^2_{\\mathcal{A}}(X)$ and ${\\rm Poiss}_{I}(X) \\subseteq \\mathfrak{X}^2_I(X)$.\n\nWe wish to study the relation between the spaces ${\\rm Poiss}_{TX_I}(X)$ and ${\\rm Poiss}_{I}(X)$. Note that for $\\pi \\in {\\rm Poiss}_I(X)$ to be liftable to $\\mathcal{A}$, by \\autoref{prop:apoissondivtypelifts} $I$ must be $I$ divisible by $I_\\mathcal{A}$. Consequently whether $I$ is standard (\\autoref{defn:divprojstandard}) is relevant, i.e.\\ whether $I_{TX_I} = I$.\n\nDivisor bivectors are generally not liftable to their ideal Lie algebroid, i.e.\\ $\\mathfrak{X}^2_I(X) \\not\\subseteq \\mathfrak{X}^2_{TX_I}(X)$. The other inclusion is also false, $\\mathfrak{X}^2_{TX_I}(X) \\not\\subseteq \\mathfrak{X}^2_I(X)$, as the following examples show.\n\\begin{exa}\\label{exa:logbivector} The bivector $\\pi = x \\partial_x \\wedge \\partial_y + \\partial_z \\wedge \\partial_w + \\partial_x \\wedge \\partial_w$ on $\\mathbb{R}^4$ with coordinates $(x,y,z,w)$ satisfies $\\wedge^2 \\pi = x \\partial_x \\wedge \\partial_y \\wedge \\partial_z \\wedge \\partial_w$, which vanishes transversally on $Z_\\pi = \\{x = 0\\}$. Hence $\\pi$ is of log divisor-type, but does not lift to $\\mathcal{A}_{Z_\\pi} = \\langle x \\partial_x, \\partial_y, \\partial_z, \\partial_w \\rangle$ due to the presence of $\\partial_x \\wedge \\partial_w$, showing that $\\mathfrak{X}^2_I(X) \\not\\subseteq \\mathfrak{X}^2_{TX_I}(X)$ in this case. Note that $\\pi$ is not Poisson, as $[\\pi,\\pi] = -\\partial_x \\wedge \\partial_y \\wedge \\partial_w \\neq 0$.\n\\end{exa}\n\\begin{exa} The Poisson bivector $\\pi = x^2 \\partial_x \\wedge \\partial_y$ on $\\mathbb{R}^2$ with coordinates $(x,y)$ lifts to $\\mathcal{A}_Z$, where $Z = \\{x = 0\\}$, with $\\mathcal{A}_Z$-lift $\\pi_{\\mathcal{A}_Z} = x (x \\partial_x) \\wedge \\partial_y$, but it does not lift nondegenerately, nor does $\\pi$ specify a log divisor structure on $Z$. Thus we see that $\\mathfrak{X}^2_{TX_I}(X) \\not\\subseteq \\mathfrak{X}^2_I(X)$ in this case.\n\\end{exa}\nThe Poisson condition is important for liftability as the above examples show. To study this in more detail, we first discuss which vector fields are liftable to $TX_I$.\n\\begin{prop}\\label{prop:liftifpreserve} Let $\\pi \\in \\mathfrak{X}^2_I(X)$. A vector field $V \\in \\mathfrak{X}(X)$ lies in $\\mathfrak{X}_{TX_I}(X)$ if $\\mathcal{L}_V (\\wedge^n \\pi) = 0$.\n\\end{prop}\n\\bp Denote $\\wedge^n \\pi$ by $\\pi^n$. By definition of $\\Gamma(TX_I) = \\Gamma(TX)_I$, we must show that $\\mathcal{L}_V I \\subset I$, which is a local statement. Let ${\\rm vol}_X$ be a local volume form and define $f := \\pi^n({\\rm vol}_X)$ so that $I = \\langle f \\rangle$. Then $\\mathcal{L}_V f = \\mathcal{L}_V \\pi^n({\\rm vol}_X) = (\\mathcal{L}_V \\pi^n)({\\rm vol}_X) + \\pi^n \\mathcal{L}_V \\rm{vol}_X$. Note that $\\mathcal{L}_V {\\rm vol}_X = g {\\rm vol}_X$ for some local function $g$. From this we conclude that\n\\begin{equation*}\n\t\\mathcal{L}_V f= \\mathcal{L}_V \\pi^n ({\\rm vol}_X) + \\pi^n(g {\\rm vol}_X) = \\mathcal{L}_V \\pi^n ({\\rm vol}_X) + g \\pi^n({\\rm vol}_X) = \\mathcal{L}_V \\pi^n ({\\rm vol}_X) + g f.\n\\end{equation*}\nAs $g f \\in I$, we see that $V$ lifts if $\\mathcal{L}_V \\pi^n = 0$.\n\\end{proof}\nThe assumption in \\autoref{prop:liftifpreserve} seems strong, yet allows for the following conclusion.\n\\begin{cor}\\label{cor:poissonhamiltonian} Let $\\pi \\in {\\rm Poiss}_I(X)$. Then ${\\rm Ham}_\\pi(X) \\subseteq {\\rm Poiss}_{\\pi}(X) \\subseteq \\mathfrak{X}_{TX_I}(X)$.\n\\end{cor}\n\\bp That ${\\rm Ham}_\\pi(X) \\subseteq {\\rm Poiss}_{\\pi}(X)$ is clear. Given $V \\in {\\rm Poiss}_\\pi(X)$ we have $\\mathcal{L}_{V} \\pi = 0$. But then we compute that $\\mathcal{L}_{V} \\pi^n = n (\\mathcal{L}_V \\pi) \\wedge \\pi^{n-1} = 0$, so that $V \\in \\mathfrak{X}_{TX_I}(X)$ by \\autoref{prop:liftifpreserve}.\n\\end{proof}\n\\begin{rem} For a general bivector $\\pi \\in \\mathfrak{X}^2(X)$, the definition of its `Hamiltonian vector fields' $V_f = \\pi^\\sharp(df)$ for $f \\in C^\\infty(X)$ still makes sense. However, the fact that all such vector fields preserve $\\pi$, i.e.\\ $\\mathcal{L}_{V_f} \\pi = 0$, is equivalent to $\\pi$ being Poisson.\n\\end{rem}\nDue to \\autoref{cor:poissonhamiltonian}, we see that for $\\pi \\in {\\rm Poiss}_I(X)$ there exists a map $\\widetilde{\\pi}^\\sharp\\colon T^*X \\to TX_I$ fitting in the following diagram, which is not yet the existence of an $TX_I$-lift.\n\\begin{center}\n\t\\begin{tikzpicture}\n\t%\n\t\\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em,text height=1.5ex, text depth=0.25ex]\n\t{\tTX^*_I & TX_I \\\\ T^*X & TX \\\\};\n\t\\path[-stealth]\n\t(m-2-1) edge node [left] {$\\rho_{TX_I}^*$} (m-1-1)\n\t(m-2-1) edge node [above] {$\\pi^\\sharp$} (m-2-2)\n\t(m-2-1) edge node [above] {$\\widetilde{\\pi}^\\sharp$} (m-1-2)\n\t(m-1-2) edge node [right] {$\\rho_{TX_I}$} (m-2-2);\n\t\\draw[dotted, ->] (m-1-1) to (m-1-2);\n\t\\end{tikzpicture}\n\\end{center}\nOne can obtain an actual $TX_I$-lift under the assumption that $\\Gamma(TX_I^*)$ admits local bases of closed sections. Namely, we can now prove \\autoref{thm:introlifting} from the introduction.\n\\begin{thm}\\label{thm:ipoissonaitype} Let $I \\subseteq C^\\infty(X)$ be a projective divisor ideal with $TX^*_I$ admitting local bases of closed sections. Then ${\\rm Poiss}_I(X) \\subseteq {\\rm Poiss}_{TX_I}(X)$, i.e.\\ those of $I$-divisor-type are of $TX_I$-type.\n\\end{thm}\n\\bp We obtain from the discussion above a map $\\widetilde{\\pi}^\\sharp\\colon T^*X \\to TX_I$. We can dualize the bundle morphism $\\widetilde{\\pi}^\\sharp$ to a map $(\\widetilde{\\pi}^\\sharp)^*\\colon TX_I^* \\to TX$. To test whether this lifts to a map $\\pi_{TX_I}^\\sharp\\colon TX_I^* \\to TX_I$, we proceed as in \\autoref{cor:poissonhamiltonian} by checking whether $(\\widetilde{\\pi}^\\sharp)^*$ maps to $\\Gamma(TX)_I \\cong \\Gamma(TX_I)$ on sections. Assuming that $\\Gamma(TX_I^*)$ admits local bases of closed sections, we can by continuity answer this in the isomorphism locus $X \\backslash Z_I$, where such local sections are exact under the isomorphism with $TX$, and where $(\\widetilde{\\pi}^\\sharp)^* = (\\pi^\\sharp)^*$ using the isomorphism given by $\\rho_{TX_I}$. Here the lifting property follows as in \\autoref{cor:poissonhamiltonian}, using that $(\\pi^\\sharp)^* = - \\pi^\\sharp$ by skew-symmetry. The lifting property then holds in the entirety of $X$ by density of the isomorphism locus.\n\\end{proof}\n\\begin{rem} By inspecting the proof of \\autoref{thm:ipoissonaitype}, we readily see that it still holds for Poisson structures of $m$-$I$-divisor-type, that is, any almost-regular Poisson structure.\n\\end{rem}\nThe converse to the above theorem is false unless $\\pi \\in {\\rm Poiss}(X)$ is of nondegenerate $TX_I$-type and $I$ is standard (see the discussion below \\autoref{prop:apoissondivtypelifts}). Further, we know that if $I$ is standard and $\\pi \\in {\\rm Poiss}_I(X)$ is of $TX_I$-type, then $\\pi$ is in fact of nondegenerate $TX_I$-type.\n\\begin{rem} Following up on \\autoref{rem:nonstandard}, let $I \\subseteq C^\\infty(X)$ be a nontrivial projective divisor ideal and $\\pi \\in {\\rm Poiss}_{I^k}(X)$ for some $k > 1$. While $\\pi$ can be of $TX_I$-type using \\autoref{thm:ipoissonaitype}, it can never be of nondegenerate $TX_I$-type by comparing the divisor ideals of $\\pi$ and $TX_I$.\n\\end{rem}\nNote that if $\\pi$ is of $TX_I$-type, the degeneracy locus $Z_I$ must be a $\\pi$-Poisson subset, i.e.\\ $I_{Z_I}$ is a $\\pi$-Poisson ideal. Similarly we have the following (recall that $I \\subseteq I_{Z_I}$ by \\autoref{prop:divvanishingideal}).\n\\begin{prop}\\label{prop:itypeipoissonideal} Let $I$ be a divisor ideal and $\\pi \\in {\\rm Poiss}_I(X)$. Then $I$ is a $\\pi$-Poisson ideal.\n\\end{prop}\n\\bp Let ${\\rm vol}_X$ be a local volume form, and $g \\in C^\\infty(X)$. Then $I = \\langle \\pi^n({\\rm vol}_X) \\rangle = \\langle f \\rangle$, and\n\\begin{equation*}\n\t\\{g, f\\}_\\pi = \\{g, \\pi^n({\\rm vol}_X)\\}_\\pi = \\pi(dg, d(\\pi^n{\\rm vol}_X)) = \\pm \\mathcal{L}_{V_g} (d \\pi^n({\\rm vol}_X)) = \\pi^n \\mathcal{L}_{V_g} {\\rm vol}_X,\n\\end{equation*}\nusing that as $V_g$ is Hamiltonian, it preserves $\\pi^n$. We have $\\mathcal{L}_{V_g} {\\rm vol}_X = h {\\rm vol}_X$ for some function $h \\in C^\\infty(X)$, so that $\\pi^n \\mathcal{L}_{V_g} {\\rm vol}_X = h f \\in I$. This implies that $\\{g,f\\}_\\pi \\in I$ as desired.\n\\end{proof}\n\\subsection{Examples}\nIn this section we discuss several interesting examples of Poisson structures of divisor-type, and of the lifting procedure to their associated ideal Lie algebroids. We will only do so here for Poisson structures in the usual sense, as this is the main motivation for the development of the theory. However, it is quite possible, and certainly interesting, to consider $\\mathcal{A}$-Poisson structures of divisor-type for given, well-understood, Lie algebroids $\\mathcal{A} \\to X$, in cases where it is difficult to intrinsically capture when a Poisson structure is of $\\mathcal{A}$-type. Recall the examples of divisors and ideal Lie algebroids given in \\autoref{sec:divexamples} and \\autoref{exa:ideallalgebroids}.\n\nThroughout, this section let $X$ be a given $2n$-dimensional manifold.\n\\begin{exa}[Nondegenerate] Let $\\pi \\in {\\rm Poiss}(X)$ be a nondegenerate Poisson structure. Then $\\pi$ is of $C^\\infty(X)$-divisor-type, i.e.\\ $I_\\pi$ is the trivial divisor ideal. By \\autoref{prop:liftingnondegiso}, $\\pi$ is only of $\\mathcal{A}$-type for the Lie algebroid $\\mathcal{A} = TX$, and certainly lifts to it nondegenerately.\n\\end{exa}\n\\begin{exa}[Log-Poisson, \\cite{GuilleminMirandaPires14}]\\label{exa:logpoissonlift} Let $\\pi \\in {\\rm Poiss}(X)$ be a \\emph{log-Poisson structure}, i.e.\\ such that it is of log divisor-type. Then $I_\\pi = I_Z$ is a log divisor ideal. Regarding liftability of $\\pi$, recall from \\autoref{exa:zvx} that the log-tangent bundle $\\mathcal{A}_Z$ is the (primary) ideal Lie algebroid of $I_Z$. By \\autoref{thm:ipoissonaitype} we see that $\\pi$ is of $\\mathcal{A}_Z$-type. As $I_Z$ is standard, it is in fact of nondegenerate $\\mathcal{A}_Z$-type, so that it is dual to a \\emph{log-symplectic structure} \\cite{GuilleminMirandaPires14,GualtieriLi14,Cavalcanti17,MarcutOsornoTorres14}.\n\\end{exa}\nWe can determine the lifting property to log-tangent bundles $\\mathcal{A}_Z \\to X$ more generally.\n\\begin{prop}\\label{prop:azliftpoisson} Let $\\pi \\in {\\rm Poiss}(X)$ be given and let $Z \\subseteq X$ be a hypersurface. Then the following are equivalent:\n\\begin{itemize}\n\t\\item $Z$ is a $\\pi$-Poisson submanifold;\n\t\\item $I_Z$ is a $\\pi$-Poisson ideal;\n\t\\item $\\pi$ of $\\mathcal{A}_Z$-type.\n\\end{itemize}\n\\end{prop}\n\\bp This is a rephrasing of \\cite[Proposition 4.4.1]{Pym13}. The equivalence of the first two points is standard, as $Z$ is a closed submanifold. If $\\pi$ of $\\mathcal{A}_Z$-type then $Z$ is $\\pi$-Poisson, because $Z$ is the degeneracy locus of $\\mathcal{A}_Z$ (c.f.\\ \\autoref{prop:pidegenpipoisson}), or because it is an orbit of $\\mathcal{A}_Z$ (c.f.\\ \\autoref{prop:atypepoissonsubmfd}). To see that if $Z$ is $\\pi$-Poisson, then $\\pi$ is of $\\mathcal{A}_Z$-type, we turn to the proof of \\autoref{thm:ipoissonaitype}: there we showed first that each $\\pi$-Hamiltonian vector field is liftable to $\\mathcal{A}_Z$, before dualizing to obtain liftability of $\\pi$. However, $Z$ being $\\pi$-Poisson is equivalent to all $\\pi$-Hamiltonian vector fields being tangent to $Z$ (or preserving the vanishing ideal $I_Z$).\n\\end{proof}\nFrom \\autoref{prop:azliftpoisson} one can wonder when a Poisson structure $\\pi$ is of nondegenerate $\\mathcal{A}_Z$-type. However, this is already answered by \\autoref{prop:apoissondivtypelifts}: for this $\\pi$ must be of $I_Z$-divisor-type (because $I_{\\mathcal{A}_Z} = I_Z$), which is also sufficient by, for example, \\autoref{prop:itypeipoissonideal}. Hence, $\\pi$ is of nondegenerate $\\mathcal{A}_Z$-type if and only if it of $I_Z$-divisor-type, i.e.\\ is log-Poisson.\n\\begin{rem} Because $I_Z$ is a locally principal (divisor) ideal, \\autoref{prop:azliftpoisson} says that Poisson structures of $I$-divisor-type are liftable to $\\mathcal{A}_Z = TX_{I_Z}$ if $I_Z$ divides $I$. The converse statement is the content of \\autoref{prop:apoissondivtypelifts}. It would be interesting to determine whether there are other Lie algebroids of divisor-type for which a similar statement is true, or to show that this only happens in this case. We seem to crucially use that $I_Z$ is not only a divisor ideal, but in fact the vanishing ideal of its support. This only happens in the log setting.\n\\end{rem}\nWe can determine the lifting property for log-bivectors following \\cite[Lemma 138]{Frejlich11}.\n\\begin{prop}[\\cite{Frejlich11}]\\label{prop:liftazbivectors} Let $\\pi \\in \\mathfrak{X}^2_{I_Z}(X)$ be a bivector of log-divisor-type. Then $\\pi$ is of $\\mathcal{A}_Z$-type if and only if $\\pi^\\sharp(T^*_x X) \\subseteq T_x Z$ for some $x \\in Z_i$ for each connected component $Z_i \\subseteq Z$. \n\\end{prop}\n\\bp Given $x \\in Z$ with local generator $z$ of $I_Z$ we have $\\pi = \\partial_z \\wedge \\pi^\\sharp(dz) + \\nu$ with $\\nu^\\sharp(dz) \\equiv 0$. Then $\\wedge^n \\pi \\sim \\partial_z \\wedge \\pi^\\sharp(dz) \\wedge \\nu^{n-1}$. This decomposition also shows that for all $x \\in Z$, the condition $\\pi^\\sharp(T^*_x X) \\subseteq T_x Z$ is equivalent to the condition $\\pi^\\sharp(d_x I_Z) = 0$. If $\\pi^\\sharp(d_x I_Z) = 0$, then $\\pi^\\sharp(dz)|_Z = 0$, so that in turn $\\pi^\\sharp(dz) = x V$ for some $V \\in \\Gamma(TX)$. Thus $\\pi^\\sharp(d\\log z) = V$ which shows that $\\pi$ is of $\\mathcal{A}_Z$-type (pointwise, that $\\pi_x \\in \\wedge^2 \\mathcal{A}_{Z,x}$). The converse statement is immediate, because $Z$ is an $\\mathcal{A}_Z$-invariant submanifold. Next, the space $\\mathcal{Z}(\\pi) = \\{x \\in Z \\, : \\, \\pi^\\sharp(T^*_x x) \\subseteq T_x Z\\} \\subseteq Z$ is both open and closed, where for openness we use that if $x \\in \\mathcal{Z}(\\pi)$, then $\\pi^\\sharp(d_x z) = 0$, whence $\\nu_x^{n-1} \\neq 0$ by transversality. But then there exists some open $U$ around $x$ on which $\\nu^{n-1}$ is nonvanishing. As $\\wedge^n \\pi$ vanishes on $U \\cap Z$, this implies that $\\pi^\\sharp(dz)$ divides $\\nu^{n-1}$ on $U \\cap Z$. To not contradict transversality of $\\wedge^n \\pi$, we must have that $\\pi^\\sharp(dz) = 0$ on $U \\cap Z$. This in turn implies that $U \\cap Z \\subseteq \\mathcal{Z}(\\pi)$, which shows openness. The rest follows immediately.\n\\end{proof}\nEssentially, the above proposition uses the fact that transversality is an open condition.\n\\begin{rem}\\label{rem:examplelogbivectorlift} Comparing this to \\autoref{exa:logbivector}, we see that there $\\pi^\\sharp(dw) = \\partial_z + \\partial_x \\not\\in TZ$, and indeed the bivector $\\pi$ of that example does not lift to $\\mathcal{A}_Z$, as said by \\autoref{prop:liftazbivectors}.\n\\end{rem}\n\\begin{exa}[Normal-crossing log-Poisson] Let $\\pi\\in {\\rm Poiss}(X)$ be a \\emph{normal-crossing log-Poisson structure}, i.e.\\ such that it is of normal-crossing log divisor-type, with divisor ideal $I_{\\underline{Z}}$. The above discussion (that is, \\autoref{prop:azliftpoisson}) also holds for normal-crossing log-Poisson structures: a Poisson structure is of $\\mathcal{A}_{\\underline{Z}}$-type if and only if $I_{\\underline{Z}}$ is a $\\pi$-Poisson ideal, which is true in this case by \\autoref{prop:itypeipoissonideal}. Being of nondegenerate $\\mathcal{A}_{\\underline{Z}}$-type is again equivalent to being of $I_{\\underline{Z}}$-divisor-type. See also \\cite[Proposition 4.4.1; Proposition 4.4.2]{Pym13}, and \\cite{Klaasse17}.\n\\end{exa}\n\\begin{exa}[$b^k$-Poisson, \\cite{Scott16}]\\label{exa:bkpoissonlift} Another interesting class is given by the \\emph{$b^k$-Poisson structures} $\\pi_k \\in {\\rm Poiss}(X)$, which are of $I^k_Z$-divisor-type for $k \\geq 0$, where $I_Z$ is a log divisor ideal. This does not characterize $b^k$-Poisson structures. To do so, demand that $\\{\\cdot,\\cdot\\}_{\\pi_k}$ satisfies\n\\begin{equation*}\n\t\\{\\cdot,\\cdot\\}_{\\pi_k}\\colon I_Z \\times C^\\infty(X) \\to I_Z^k.\n\\end{equation*}\nIf this holds, then $\\wedge^n \\pi_k$ provides a local generator of $I_Z^k$, and its $(k-1)$-jet can be used to define the $b^k$-tangent bundle $\\mathcal{A}_Z^k$ as in \\autoref{exa:bktangentbundle}. It then follows that $\\pi_k$ is of nondegenerate $\\mathcal{A}_Z^k$-type, or of nondegenerate \\emph{$b^k$-type}. Consequently they are dual to $b^k$-symplectic structures. Note here that $b^1$-Poisson structures are just the log-Poisson structures of \\autoref{exa:logpoissonlift}.\n\\end{exa}\n\\begin{rem} Our notion of nondegenerate $b^k$-type is almost the same as that of $b^k$-type in \\cite{Scott16}. There is one difference, namely that there is a class of (non-canonically) isomorphic Lie algebroids, namely $\\mathcal{A}_Z^k$ with respect to different jet data, and Scott's $b^k$-type demands liftability to any one of these Lie algebroids. We instead extract the jet data from $\\pi_k$ itself.\n\\end{rem}\n\\begin{exa}[Scattering Poisson, \\cite{Lanius16}]\\label{exa:scatteringpoissonlift} Let $(X,Z)$ be a log pair. Then $\\pi \\in {\\rm Poiss}(X)$ is a \\emph{scattering Poisson structure} if its associated Poisson bracket satisfies\n\t%\n\t\\begin{equation*}\n\t\\{\\cdot,\\cdot\\}_\\pi\\colon C^\\infty(X) \\times C^\\infty(X) \\to I_Z, \\qquad \\text{and} \n\t\\qquad \\{\\cdot,\\cdot\\}_\\pi\\colon I_Z \\times C^\\infty(X) \\to I_Z^2.\n\t\\end{equation*}\n\t%\n\tThen $\\pi$ is of nondegenerate $\\mathcal{C}_Z$-type, and also of $\\mathcal{A}_Z$-type (because $\\mathcal{C}_Z$ is an $\\mathcal{A}_Z$-Lie algebroid).\n\\end{exa}\n\\begin{rem} A Poisson structure $\\pi \\in {\\rm Poiss}(X)$ is of $\\mathcal{C}_Z$-type if $Z$ is $\\pi$-Poisson, and the first jet of its $\\mathcal{A}_Z$-lift $\\pi_{\\mathcal{A}_Z}$ vanishes at $Z$ (using \\autoref{prop:azliftpoisson}). Moreover, $\\pi$ is of nondegenerate $\\mathcal{C}_Z$-type if and only if $Z = X_{\\pi_{\\mathcal{A}_Z}, 2n-2}$ as $\\pi_{\\mathcal{A}_Z}$-Poisson divisors (c.f.\\ \\cite[Lemma 5.2]{Lanius16}), which provides an alternative definition of $\\pi$ being scattering Poisson for the log pair $(X,Z)$.\n\\end{rem}\n\\begin{exa}[Elliptic Poisson, \\cite{CavalcantiGualtieri18}]\\label{exa:ellpoissonlift} Let $\\pi \\in {\\rm Poiss}(X)$ be an \\emph{elliptic Poisson structure}, i.e.\\ such that it is of elliptic divisor-type. Then $I_\\pi = I_{|D|}$ for an elliptic pair $(X,|D|)$. The (primary) ideal Lie algebroid of $I_{|D|}$ is the elliptic tangent bundle $\\mathcal{A}_{|D|}$ (see \\autoref{exa:ellvx}). By \\autoref{thm:ipoissonaitype}, the bivector $\\pi$ is of nondegenerate $\\mathcal{A}_{|D|}$-type because $I_{|D|}$ is standard. This recovers \\cite[Lemma 3.4]{CavalcantiGualtieri18}. More discussion of these structures can be found in \\cite{KlaasseLanius18, KlaasseLi18}, determining their Darboux models, Poisson cohomology and adjoint symplectic groupoids.\n\\end{exa}\n\\begin{exa}[Elliptic-log Poisson]\\label{exa:elllogpoissonlift} Let $\\pi \\in {\\rm Poiss}(X)$ be an \\emph{elliptic-log Poisson structure}, i.e.\\ such that it is of elliptic-log divisor-type. Then $I_\\pi = I_W$ for an elliptic-log ideal $I_W = I_Z \\cdot I_{|D|}$. The degeneracy locus of $\\pi$ is then given by $Z$, with $D$ being a $\\pi$-symplectic leaf. Consequently, by \\autoref{prop:azliftpoisson} we know that $\\pi$ is of $\\mathcal{A}_Z$-type. Naturally one wonders whether $\\pi$ is in fact of $\\mathcal{A}_W$-type (noting that $\\mathcal{A}_W = TX_{I_W}$ is an $\\mathcal{A}_Z$-Lie algebroid, c.f.\\ \\autoref{exa:ideallalgebroids}. However, this cannot be immediately answered by \\autoref{thm:ipoissonaitype} because $\\mathcal{A}_W^*$ does not admit local closed generators. However, the first part of the lifting procedure (the map $\\widetilde{\\pi}^\\sharp$) does not use this, so that certainly $\\pi$ lifts to a map $\\widetilde{\\pi}^\\sharp\\colon T^*X \\to \\mathcal{A}_W$. As $I_W$ is standard, we further know that if $\\pi$ is of $\\mathcal{A}_W$-type, then it is of nondegenerate $\\mathcal{A}_W$-type. These types of Poisson structures are studied in more detail in \\cite{KlaasseLanius18}, where in particular their Poisson cohomology is determined.\n\\end{exa}\nIt is very interesting to extend the above list of examples using a new class of divisor ideals.\n\\subsection{Lifting Poisson structures of $m$-divisor-type}\n\\label{sec:liftingalmostreg}\nWe next turn to lifting Poisson structures of $m$-$I$-divisor-type. To this end, let $\\pi \\in {\\rm Poiss}(X)$ be of $m$-divisor-type. Then $\\pi$ is almost-regular by \\autoref{cor:divtypealmostreg}, so that by \\autoref{prop:almostregulardistr} it has a unique associated involutive regular distribution $D_\\pi \\subseteq TX$ of $\\pi$-Poisson submanifolds. We can readily view this distribution as an injective Lie algebroid $\\mathcal{D}_\\pi$, and can wonder whether $\\pi$ can be lifted to it.\n\\begin{prop}\\label{prop:mdivtypelift} Let $\\pi \\in {\\rm Poiss}(X)$ be of $m$-divisor-type with its associated injective Lie algebroid $\\mathcal{D}_\\pi \\to X$. Then $\\pi$ is of $I_\\pi$-$\\mathcal{D}_\\pi$-type, where $I_\\pi$ is the divisor ideal of $\\pi$.\n\\end{prop}\n\\bp As discussed in \\autoref{sec:almostregularpoisson}, the regular distribution $D_\\pi$ obtained from $\\pi$ consists of $\\pi$-Poisson submanifolds. \\autoref{prop:regularpipoissonlift} thus immediately implies that $\\pi$ is of $\\mathcal{D}_\\pi$-type. If we consider the definition of $\\pi$, we note that its associated divisor $(K, \\wedge^m \\pi)$ for $K \\subseteq \\wedge^{2m} TX$ is such that $\\det D_\\pi = K$. From this we see that its $\\mathcal{D}_\\pi$-lift is of $I_\\pi$-divisor-type as desired.\n\\end{proof}\nThis result is noticed without the above precise language in \\cite[Section 4.2]{AndroulidakisZambon17}. Note that the $\\mathcal{D}_\\pi$-lift $\\pi_{\\mathcal{D}} \\in {\\rm Poiss}(\\mathcal{D}_\\pi)$ is automatically generically-nondegenerate by \\autoref{prop:apoissondivtypelifts}.\n\\begin{rem} \\autoref{prop:mdivtypelift} allows for an alternative definition of almost-regularity (or of being of $m$-divisor-type). Namely, an almost-regular Poisson structure is uniquely determined by a regular involutive distribution $D$, and a generically-nondegenerate $\\mathcal{D}$-Poisson structure (with $\\mathcal{D}$ the injective Lie algebroid of $D$). Indeed, if $\\pi_{\\mathcal{D}} \\in {\\rm Poiss}(\\mathcal{D})$ is generically-nondegenerate, then $\\pi := \\rho_D(\\pi_{\\mathcal{D}})$ is almost-regular. \\autoref{prop:mdivtypelift} shows the converse.\n\\end{rem}\nThe above remark shows a useful viewpoint on almost-regular Poisson structures, or those of $m$-divisor-type. Namely, they are just Poisson structures of divisor-type, except using the injective Lie algebroid $\\mathcal{D}$ instead of $TX$. We further remark here that a Poisson structure being almost-regular is different from saying that it is generically regular ($X_{\\pi,{\\rm reg}} = X_{\\pi,{\\rm max}}$), as its degeneracy must further be governed by a divisor (instead of more involved degeneration).\n\nGiven an almost-regular Poisson structure $\\pi$ with its $\\mathcal{D}$-lift $\\pi_{D}$, in light of \\autoref{prop:mdivtypelift} and the discussion in \\autoref{sec:liftingdivtype}, one can try to lift the $\\mathcal{D}$-Poisson structure $\\pi_{\\mathcal{D}}$ further, to the ideal Lie algebroid $\\mathcal{D}_{I_\\pi} \\to \\mathcal{D} \\to TX$. We next consider this for log divisors.\n\\subsubsection{The case of $m$-log-Poisson structures}\nLet $\\pi \\in {\\rm Poiss}(X)$ be an \\emph{$m$-log-Poisson structure}, i.e.\\ such that it is of $m$-log-divisor-type. Such structures are called \\emph{log-f Poisson structure} in \\cite{AndroulidakisZambon17}, but we favor our name due to its consistency with other types of almost-regular Poisson structures. Our discussion here is partially parallel to \\cite[Section 3.4]{AndroulidakisZambon17}. By definition $I_\\pi$ is a log divisor ideal, and the degeneracy locus $Z_\\pi$ of $\\pi$ is a hypersurface. For ease of notation we set $Z := Z_\\pi$ so that $I_\\pi = I_Z$, and moreover let $\\mathcal{D} := \\mathcal{D}_\\pi$ be the associated injective Lie algebroid.\n\nBy \\autoref{prop:mdivtypelift} we know that $\\pi$ is of log-$\\mathcal{D}$-type. A natural question to ask is whether each leaf of $D$ is log-Poisson. However, this need not necessarily be the case.\n\\begin{prop}[{\\cite[Lemma 3.7]{AndroulidakisZambon17}}]\\label{prop:mlogpoissonleaflogpoisson} Let $(X,Z,\\mathcal{D},\\pi)$ be an $m$-log-Poisson manifold. Then a leaf $P$ of $D$ is log-Poisson if and only if for all $x \\in P \\cap Z$, we have that $D_x + T_x Z = T_x X$.\n\\end{prop}\nTo find a Lie algebroid to which $\\pi$ lifts nondegenerately, we can try to form the primary ideal Lie algebroid $\\mathcal{D}_{I_Z}$, which would be the $(TZ \\cap TD,Z)$-rescaling of $\\mathcal{D}$. However, it is not clear a priori that the involutive submodule $\\Gamma(\\mathcal{D})_{I_Z} \\subseteq \\Gamma(\\mathcal{D})$ is projective. For this, we have:\n\\begin{prop}\\label{prop:injectivelogalgebroid} Let $(X,Z)$ be a log pair and $\\mathcal{B} \\to X$ be an injective Lie algebroid. Then the $C^\\infty(X)$-module $\\Gamma(\\mathcal{B})_{I_Z}$ is projective if $Z$ is transverse to the orbits of $\\mathcal{B}$.\n\\end{prop}\n\\bp The orbits of $\\mathcal{B}$ are the leaves of the associated distribution $T\\mathcal{B} := \\rho_{\\mathcal{B}}(\\mathcal{B}) \\subseteq TX$. If $Z$ is transverse to this regular foliation, we can find local bases of generating vector fields $\\partial_{x_i}$ for $\\mathcal{B}$ and $\\partial_{y_j}$ for $TZ$ so that $\\langle \\partial_{x_1}, \\partial_{y_j} \\rangle = \\Gamma(TX)$. Then if locally $I_Z = \\langle z \\rangle$, we have that $\\mathcal{B}_{I_Z} = \\langle z \\partial_{z}, \\partial_{x_i} \\, : \\, i \\geq 2 \\rangle$, with $x_1 = z$, which is thus projective (note that $\\Gamma(TX)_{I_Z} = \\langle z \\partial_{z}, \\partial_{y_j} \\rangle$).\n\\end{proof}\n\\begin{rem}\\label{rem:dtangentZ} When $Z$ is everywhere tangent to the orbits of $\\mathcal{B}$, the module $\\Gamma(\\mathcal{B})_{I_Z}$ is also projective, but just equals $\\Gamma(\\mathcal{B})$. Thus in this case we have that $\\mathcal{B}_{I_Z} = \\mathcal{B}$, i.e.\\ $\\mathcal{B}$ is unchanged.\n\\end{rem}\nWe can apply \\autoref{prop:injectivelogalgebroid} to the case where $\\mathcal{B} = \\mathcal{D}$. This shows that if $Z$ is transverse to the orbits of $\\mathcal{D}$, then $\\mathcal{D}_{I_Z}$ is an almost-injective Lie algebroid for which the natural map $\\mathcal{D}_{I_Z} \\to \\mathcal{D}$ is of $I_Z$-divisor-type. Moreover it follows from the discussion in \\autoref{sec:liftingdivtype} that the $m$-log-Poisson structure $\\pi$ is not only of $\\mathcal{D}$-type, but also of $\\mathcal{D}_{I_Z}$-type. We conclude that when $Z$ is transverse to the orbits of $\\mathcal{D}$, then $\\pi$ is of nondegenerate $\\mathcal{D}_{I_Z}$-type by \\autoref{prop:apoissondivtypelifts}.\n\nAn alternate approach is to consider $\\pi$ and whether it lifts to the usual ideal Lie algebroid $TX_{I_Z} = \\mathcal{A}_Z$, the log-tangent bundle. As $Z$ is by definition the degeneracy locus of $\\pi$, it is a $\\pi$-Poisson submanifold by \\autoref{prop:pidegenpipoisson}. From this we conclude immediately using \\autoref{prop:azliftpoisson} that $\\pi$ is of $\\mathcal{A}_Z$-type with lift $\\pi_{\\mathcal{A}_Z}$. In fact, it follows that $\\pi_{\\mathcal{A}_Z}$ is $2m$-regular when $Z$ is transverse to the orbits of $\\mathcal{D}$. Similarly, when $Z$ is everywhere tangent to the orbits of $\\mathcal{D}$, then $\\pi_{\\mathcal{A}_Z}$ is of $m$-log-divisor-type. By construction $\\mathcal{D}_{I_Z}$ is a $\\mathcal{D}$-Lie algebroid, but from the proof of \\autoref{prop:injectivelogalgebroid} we see that $\\mathcal{D}_{I_Z}$ is also an $\\mathcal{A}_Z$-Lie algebroid (also \\autoref{rem:dtangentZ}). This summarizes into the following diagram.\n\\begin{center}\n\t\\begin{tikzpicture}\n\t%\n\t\\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em,text height=1.5ex, text depth=0.25ex]\n\t{\t(\\mathcal{D}_{I_Z}, \\pi_{\\mathcal{D}_{I_Z}}) & (\\mathcal{D}, \\pi_{\\mathcal{D}}) \\\\ (\\mathcal{A}_Z, \\pi_{\\mathcal{A}_Z}) & (TX, \\pi) \\\\};\n\t\\path[-stealth]\n\t(m-2-1) edge node [above] {$\\rho_{\\mathcal{A}_Z}$} (m-2-2)\n\t(m-1-1) edge node [above] {$\\rho_{\\mathcal{A}_Z}$} (m-1-2);\n\t\\draw[right hook-latex]\n\t(m-1-1) edge (m-2-1)\n\t(m-1-2) edge (m-2-2);\n\t\\end{tikzpicture}\n\\end{center}\nThus, an $m$-log-Poisson structure $\\pi \\in {\\rm Poiss}(X)$ with divisor ideal $I_Z$ and distribution $D$:\n\\begin{itemize}\n\t\\item Is always of log-$\\mathcal{D}$-type;\n\t\\item Is always of $\\mathcal{A}_Z$-type, with $2m$-regular lift when $Z$ is transverse to the $\\mathcal{D}$ orbits;\n\t\\item Is of nondegenerate $\\mathcal{D}_{I_Z}$-type when $Z$ is transverse to the $\\mathcal{D}$-orbits.\n\\end{itemize}\nAt this point it is clear that it is important to understand the relation between transversality of $Z$ and the orbits of $\\mathcal{D}$, and Poisson geometric properties of $\\pi$. The answer to this is provided by the behavior of the modular foliation $\\mathcal{F}_{\\pi,{\\rm mod}}$ of $\\pi$, which in general satisfies $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_{\\pi,{\\rm mod}}$ (discussed in \\autoref{sec:modularfoliation}, see \\autoref{exa:modfolmlog} in particular). Following \\cite{AndroulidakisZambon17}, we denote by\n\\begin{equation*}\n\tZ_{\\rm sing} = \\{x \\in Z \\, | \\, D_x \\subseteq T_x Z\\}\n\\end{equation*}\nthe points on $Z$ where $Z$ does not intersect the orbits of $\\mathcal{D}$ transversely. Due to \\autoref{prop:mlogpoissonleaflogpoisson}, these are the points on $Z$ whose associated leaf $P$ of $D$ is not log-Poisson. The two extreme cases are where $Z_{\\rm sing} = \\emptyset$, so that $Z$ is everywhere transverse to the orbits of $\\mathcal{D}$, and the case $Z_{\\rm sing} = Z$, where $\\mathcal{D}$ is tangent to $Z$. The general setting is more complicated.\n\nWe next follow the above discussion for the three examples contained in \\cite[Example 3.8]{AndroulidakisZambon17}.\n\\begin{exa}\\label{rem:mlogpoissonlocal} Consider the following three $1$-log-Poisson structures on $(X =\\mathbb{R}^3,(x,y,z))$:\n\\begin{itemize}\n\t\\item $\\pi_1 = x \\partial_x \\wedge \\partial_y$;\n\t\\item $\\pi_2 = z \\partial_x \\wedge \\partial_y$;\n\t\\item $\\pi_3 = (z - x^2) \\partial_x \\wedge \\partial_y$.\n\\end{itemize}\nWe then have that $\\Gamma(TX) = \\langle \\partial_x, \\partial_y, \\partial_z \\rangle$ and moreover $\\mathcal{D} := \\mathcal{D}_1 = \\mathcal{D}_2 = \\mathcal{D}_3 = \\langle \\partial_x, \\partial_y \\rangle$. To proceed we list some of the relevant objects for each of these (where $I_i := I_{\\pi_i}$ for $i = 1,2,3$):\n\\begin{itemize}\n\t\\item $I_1 = \\langle x \\rangle$, $\\Gamma(TZ_1) = \\langle \\partial_y, \\partial_z \\rangle$, $Z_{1,{\\rm sing}} = \\emptyset$, $\\Gamma(\\mathcal{A}_{Z_1}) = \\langle x \\partial_x, \\partial_y, \\partial_z \\rangle$, $\\mathcal{D}_{I_1} = \\langle x \\partial_x, \\partial_y \\rangle$;\n\t\\item $I_2 = \\langle z \\rangle$, $\\Gamma(TZ_2) = \\langle \\partial_x, \\partial_y \\rangle$, $Z_{2,{\\rm sing}} = Z_2$, $\\Gamma(\\mathcal{A}_{Z_2})  = \\langle \\partial_x, \\partial_y, z \\partial_z \\rangle$, $\\mathcal{D}_{I_2} = \\langle \\partial_x, \\partial_y \\rangle = \\mathcal{D}$.\n\t\\item $I_3 = \\langle z - x^2 \\rangle$, $\\Gamma(TZ_3) = \\langle \\partial_x + 2x \\partial_z, \\partial_y \\rangle$, $Z_{3,{\\rm sing}} = \\{x_1 = 0\\} \\cap Z$.\n\t\\end{itemize}\nFor $\\pi_1$ and $\\pi_2$, $Z$ is either everywhere transverse, or everywhere tangent to the orbits of $\\mathcal{D}_i$. The case of $\\pi_3$ is more involved, as is the behavior of the modules $\\Gamma(TX)_{I_{Z_3}}$ and $\\Gamma(\\mathcal{D})_{I_{Z_3}}$.\n\nThis shows that $\\pi_i$ lifts to be of log-type to $\\mathcal{D}$ for $i = 1,2,3$, as in \\autoref{prop:mdivtypelift}. Moreover, the $\\mathcal{A}_{Z_i}$-lifts of the Poisson structures $\\pi_i$ for $i = 1,2$ are given by\n\\begin{equation*}\n\t\\pi_{\\mathcal{A}_{Z_1}} = (x \\partial_x) \\wedge \\partial_y, \\quad \\pi_{\\mathcal{A}_{Z_2}} = z \\partial_x \\wedge \\partial_y.\n\\end{equation*}\nThus $\\pi_{\\mathcal{A}_{Z_1}}$ is $2$-regular, $\\pi_{\\mathcal{A}_{Z_2}}$ is of $1$-log-divisor-type, and $\\pi_1$ is of nondegenerate $\\mathcal{D}_{I_1}$-type.\n\\end{exa}\n\\subsection{Symplectic Lie algebroids}\n\\label{sec:symplecticliealgebroids}\nIn this section we briefly discuss symplectic Lie algebroids \\cite{NestTsygan01}, the associated study of $\\mathcal{A}$-symplectic geometry, and its relation with $\\mathcal{A}$-Poisson geometry.\n\\begin{defn}[\\cite{NestTsygan01}] A \\emph{symplectic Lie algebroid} is a Lie algebroid $\\mathcal{A} \\to X$ equipped with an \\emph{$\\mathcal{A}$-symplectic structure}, i.e.\\ a closed nondegenerate $\\mathcal{A}$-two-form ${\\omega_{\\mathcal{A}} \\in \\Omega^2_{\\mathcal{A}}(X)}$.\n\\end{defn}\nIn other words, the $\\mathcal{A}$-two-form $\\omega_\\mathcal{A}$ satisfies $d_\\mathcal{A} \\omega_\\mathcal{A} = 0$ and $\\omega_\\mathcal{A}^n \\neq 0$, where ${\\rm rank}(\\mathcal{A}) = 2n$. The nondegeneracy condition for $\\omega_\\mathcal{A}$ is equivalent to the map $\\omega_\\mathcal{A}^\\flat\\colon \\mathcal{A}^* \\to \\mathcal{A}$, given by $v \\mapsto \\iota_v \\omega_\\mathcal{A}$ for $v \\in \\mathcal{A}$, being an isomorphism. The space of $\\mathcal{A}$-symplectic forms is denoted by ${\\rm Symp}(\\mathcal{A})$, and we say that $X$ is \\emph{$\\mathcal{A}$-symplectic} if it is equipped with a symplectic Lie algebroid $(\\mathcal{A},\\omega_{\\mathcal{A}}) \\to X$.\n\nThe nondegeneracy condition for $\\omega_{\\mathcal{A}}$ implies that the rank of $\\mathcal{A}$ must be even (but note that $\\dim(X)$ need not necessarily be even). Any symplectic Lie algebroid $(\\mathcal{A},\\omega_\\mathcal{A})$ defines an $\\mathcal{A}$-cohomology class $[\\omega_\\mathcal{A}] \\in H^2(\\mathcal{A})$, is naturally oriented by $\\omega_\\mathcal{A}^n \\in \\Gamma(\\det(\\mathcal{A}^*))$, and always admits an \\emph{$\\mathcal{A}$-almost-complex structure}, i.e.\\ a complex structure $J_\\mathcal{A} \\in {\\rm End}(\\mathcal{A})$ for $\\mathcal{A}$ (see \\cite{Klaasse18three}). Lie algebroids $\\mathcal{A}$ of rank two admit $\\mathcal{A}$-symplectic structures if and only if they are orientable, i.e.\\ if $w_1(\\mathcal{A}) = 0$.\nThere is a standard notion of morphism between $\\mathcal{A}$-symplectic manifolds.\n\\begin{defn} Let $(\\varphi,f)\\colon (X, \\mathcal{A}, \\omega_\\mathcal{A}) \\to (X', \\mathcal{A}', \\omega_{\\mathcal{A}'})$ be a Lie algebroid morphism between symplectic Lie algebroids. Then the map $(\\varphi,f)$ is\n\\begin{itemize}\n\t\\item \\emph{$\\mathcal{A}$-symplectic} if $\\varphi^* \\omega_{\\mathcal{A}'} = \\omega_\\mathcal{A}$;\n\t\\item an \\emph{$\\mathcal{A}$-symplectomorphism} if it $\\mathcal{A}$-symplectic, and $\\varphi$ is a fiberwise isomorphism.\n\\end{itemize}\n\\end{defn}\nThe $\\mathcal{A}$-symplectic condition $\\varphi^* \\omega_{\\mathcal{A}'} = \\omega_\\mathcal{A}$ is equivalent to demanding $\\omega_\\mathcal{A}^\\flat = \\varphi \\circ \\omega_\\mathcal{B}^\\flat \\circ \\varphi^*$ as maps. As both $\\omega_\\mathcal{A}$ and $\\omega_{\\mathcal{A}'}$ are nondegenerate, any $\\mathcal{A}$-symplectic map $(\\varphi,f)$ must in particular be fiberwise injective. Consequently, if ${\\rm rank}(\\mathcal{A}) = {\\rm rank}(\\mathcal{A}')$, any $\\mathcal{A}$-symplectic map is necessarily an $\\mathcal{A}$-symplectomorphism, but the base map $f\\colon X \\to X'$ is not necessarily a diffeomorphism. Our main interest however will be in Lie algebroid isomorphisms which are $\\mathcal{A}$-symplectic.\n\\begin{rem} If $\\mathcal{A} \\to X$ is a Lie algebroid of divisor-type, one can perform any symplectic operation to a given $\\mathcal{A}$-symplectic structure in the isomorphism locus $X_\\mathcal{A}$. These include for example the symplectic fiber sum \\cite{Gompf95} or symplectic blow-up \\cite{McDuffSalamon17} procedures.\n\\end{rem}\nLet $(\\mathcal{A},\\omega_{\\mathcal{A}}) \\to X$ be a symplectic Lie algebroid. Then the map $\\omega_{\\mathcal{A}}^\\flat\\colon \\mathcal{A} \\to \\mathcal{A}^*$ can be inverted to a map $(\\omega_{\\mathcal{A}}^\\flat)^{-1}\\colon \\mathcal{A}^* \\to \\mathcal{A}$, which we will denote by $\\pi_{\\mathcal{A}}^\\flat$. This construction defines an $\\mathcal{A}$-Poisson structure $\\pi_{\\mathcal{A}} \\in {\\rm Poiss}(\\mathcal{A})$, which we will also denote by $\\pi_{\\mathcal{A}} = \\omega_{\\mathcal{A}}^{-1}$. Indeed, the conditions $d\\omega_\\mathcal{A} = 0$ and $[\\pi_{\\mathcal{A}},\\pi_{\\mathcal{A}}]_\\mathcal{A} = 0$ are equivalent. As discussed below \\autoref{defn:apoissonmap}, the two-form $\\omega_{\\mathcal{A}}$ is called the \\emph{dual $\\mathcal{A}$-form} to the underlying Poisson structure $\\pi := \\rho_{A}(\\pi_{\\mathcal{A}})$.\n\nThe following is the Moser theorem for Lie algebroids $\\mathcal{A}$ of smooth divisor-type \\cite{KlaasseLanius18}. It covers several results in the literature (\\cite{GuilleminMirandaPires14,Lanius16,MarcutOsornoTorres14,MirandaPlanas18,Moser65,NestTsygan96,Radko02,Scott16}), and is similar to \\cite{MirandaScott18}.\n\\begin{thm}[\\cite{KlaasseLanius18}]\\label{thm:amoser} Let $\\mathcal{A} \\to X$ be a Lie algebroid of smooth divisor-type, and let $k \\in \\{1, 2, \\dim X\\}$. Let $\\omega, \\omega' \\in \\Omega^k(\\mathcal{A})$ be $d_\\mathcal{A}$-closed $\\mathcal{A}$-$k$-forms that are nondegenerate on $Z_\\mathcal{A}$. Assume that either\n\t%\n\t\\begin{itemize}\n\t\\item $[\\omega] = [\\omega'] \\in H^k(\\mathcal{A})$, or\n\t\\item $\\widetilde{\\omega} - \\omega = \\rho_{A}^* \\tau'$ for $\\tau' \\in \\Omega^k(X)$ satisfying $\\tau'|_{Z_\\mathcal{A}} = 0$.\n\t\\end{itemize}\n\t%\n\tThen there exists a Lie algebroid isomorphism $(\\varphi,f)\\colon (\\mathcal{A}|_{U},\\omega) \\to (\\mathcal{A}|_{U'},\\omega')$ on neighbourhoods $U$ and $U'$ of $Z_\\mathcal{A}$ for which $\\varphi^* \\omega' = \\omega$, which in the second case can be chosen such that $f|_{Z_\\mathcal{A}} = {\\rm id}$.\n\\end{thm}\n\\begin{rem} As a consequence of \\autoref{thm:amoser}, to establish an $\\mathcal{A}$-Darboux theorem providing a pointwise normal form for $\\mathcal{A}$-symplectic structures, one need only establish what an $\\mathcal{A}$-symplectic structure must look like locally at a point in $Z_\\mathcal{A}$.\n\\end{rem}\nThis result implies that $\\mathcal{A}$-Nambu structures of top degree (nonvanishing sections of $\\det(\\mathcal{A})$), and hence $\\mathcal{A}$-symplectic structures on surfaces, specifying the same $\\mathcal{A}$-orientation are classified by their $\\mathcal{A}$-cohomology class. Namely, letting $n = \\dim X$, we note that $\\det(\\mathcal{A}^*)$ is a line bundle. Consequently, given cohomologous forms $\\omega,\\omega' \\in \\Omega^n(\\mathcal{A}) = \\Gamma(\\det(\\mathcal{A}^*))$ we have $\\omega = f \\omega'$ for some nonvanishing function $f \\in C^\\infty(X)$. This function must be strictly positive as $\\omega$ and $\\omega'$ give rise to the same $\\mathcal{A}$-orientation, so that $(1-t)\\omega + t \\omega' = ((1-t) + t f) \\omega$ is nondegenerate for all $t \\in [0,1]$. With more knowledge about the Lie algebroid $\\mathcal{A}$, the assumption on induced $\\mathcal{A}$-orientations can sometimes be dropped. This discussion summarizes as follows.\n\\begin{prop} Let $\\mathcal{A} \\to X$ be a Lie algebroid of divisor-type for which $Z_\\mathcal{A}$ is smooth. An $\\mathcal{A}$-Nambu structure $\\Pi$ inducing a given $\\mathcal{A}$-orientation is classified up to $\\mathcal{A}$-orientation-preserving isomorphism by its $\\mathcal{A}$-cohomology class.\n\\end{prop}\nSee also \\cite{MartinezTorres04} and \\cite{MirandaPlanas18} for the cases of log- and $b^k$-Nambu structures of top degree. Let us reiterate this classifies such $\\mathcal{A}$-symplectic structures on surfaces using $H^2(\\mathcal{A})$. Below we give some examples of local Darboux theorems for Poisson structures of divisor-type.\n\\begin{exa}[Nondegenerate] Let $\\pi \\in {\\rm Poiss}(X^{2n})$ be nondegenerate. Then $\\pi$ is dual to a symplectic structure. In other words, it is of nondegenerate $TX$-type. The classical Darboux theorem (e.g.\\ proven using Moser methods) states that around any point in $X$ there are coordinates in which $\\pi = \\omega_0^{-1}$, with $\\omega_0 = \\sum_i dx_i \\wedge dy_i$ the standard symplectic structure in $\\mathbb{R}^{2n}$.\n\\end{exa}\n\\begin{exa}[Log-Poisson] Let $\\pi \\in {\\rm Poiss}(X)$ be log-Poisson, i.e.\\ of log divisor-type with divisor ideal $I_Z$. Then $\\pi$ is of nondegenerate $\\mathcal{A}_Z$-type by \\autoref{exa:logpoissonlift}. Either directly by applying Weinstein's splitting theorem, or by Moser methods for $\\mathcal{A}_Z$ (see \\cite{GuilleminMirandaPires14}), around points in $Z$ there are coordinates $(z,x_i)$ with $I_Z = \\langle z \\rangle$ in which $\\pi = z \\partial_z \\wedge \\partial_{x_2} + \\omega_0^{-1}$.\n\\end{exa}\nThe normal-crossing log case is more involved, as there can be local cohomology of the Lie algebroid $\\mathcal{A}_{\\underline{Z}}$ in contractible neighbourhoods. See the discussions found in \\cite{Lanius16two,Lanius17,MirandaScott18,Radko02two}.\n\\begin{exa}[$b^k$-Poisson]\\label{exa:bkpoissondarboux} Let $\\pi \\in {\\rm Poiss}(X)$ be $b^k$-Poisson as in \\autoref{exa:bkpoissonlift}. Then $\\pi$ is of nondegenerate $\\mathcal{A}_Z^k$-type for the associated $b^k$-tangent bundle. By Moser methods for $\\mathcal{A}_Z^k$ (see \\cite{GuilleminMirandaWeitsman17,Scott16}), around points in $Z$ there are coordinates $(z,x_i)$ with $z \\in j_{k-1}$ such that\n\\begin{equation*}\n\t\\pi = z^k \\partial_z \\wedge \\partial_{x_1} + \\omega_0^{-1}.\n\\end{equation*}\n\\end{exa}\n\\begin{exa}[Scattering Poisson] Let $\\pi \\in {\\rm Poiss}(X)$ be scattering Poisson, so that $\\pi$ is of nondegenerate $\\mathcal{C}_Z$-type (as in \\autoref{exa:scatteringpoissonlift}). By applying Moser methods for $\\mathcal{C}_Z$ (see \\cite{Lanius16}), around points in $Z$ there are coordinates $(z,x_i)$ with $I_Z = \\langle z \\rangle$ in which (for $\\alpha_0$ the standard contact structure, so that $d\\alpha_0 = \\omega_0$)\n\\begin{equation*}\n\t\\pi = z^3 \\partial_z \\wedge \\partial_{x_1} + z^2 \\partial_{z} \\wedge \\alpha_0^{-1} + z^2 \\omega_0^{-1}.\n\\end{equation*}\n\\end{exa}\n\\begin{exa}[Elliptic Poisson]\\label{exa:ellpoissondarboux} Let $\\pi \\in {\\rm Poiss}(X)$ be elliptic Poisson, i.e.\\ of elliptic divisor-type with divisor ideal $I_{|D|}$. Then $\\pi$ is of nondegenerate $\\mathcal{A}_{|D|}$-type by \\autoref{exa:ellpoissonlift}. The local (and global) behavior of $\\pi$ depends on the value(s) of its elliptic residue ${\\rm Res}_q(\\pi)$. This is a residue map associated to the elliptic tangent bundle in \\cite{CavalcantiGualtieri18}, which we recall in \\autoref{exa:elltangentresidues}. There are two qualitatively different local Darboux models for $\\pi$, depending on the vanishing of ${\\rm Res}_q(\\pi)$ (see \\cite{KlaasseLanius18}). Around points in $D$ there are coordinates $(r,\\theta,x_i)$ with $I_{|D|} = \\langle r^2 \\rangle$ with:\n\\begin{equation*}\n\t\\pi = \\begin{cases} r \\partial_r \\wedge \\partial_{x_1} + \\partial_{\\theta} \\wedge \\partial_{x_2} + \\omega_0^{-1} \\qquad &\\text{if } {\\rm Res}_q(\\pi) = 0,\\\\ \\lambda r \\partial_r \\wedge \\partial_\\theta + \\omega_0^{-1} \\qquad &\\text{if } {\\rm Res}_q(\\pi) = \\lambda \\neq 0.\n\\end{cases}\n\\end{equation*}\nThis recovers the classification results of such Poisson structures on surfaces which can be found in \\cite{Radko02two} (where ${\\rm Res}_q(\\pi) \\neq 0$ is forced by nondegeneracy). See also \\cite{KlaasseLi18}.\n\\end{exa}\nThe local behavior of elliptic-log Poisson structures (\\autoref{exa:elllogpoissonlift}) is described in \\cite{KlaasseLanius18}.\n\\section{Residue maps and the modular foliation}\n\\label{sec:residues}\nIn this section we discuss two distinct but related concepts. The first of these is the residue maps one can associate to a Lie algebroid $\\mathcal{A}$ if one is given an appropriate $\\mathcal{A}$-invariant submanifold (\\autoref{defn:ainvariantresidues}). After this we will discuss Lie algebroid modules and consequently the Lie algebroid cohomologies they give rise to. We then discuss when the residue maps, which are initially defined on the levels of Lie algebroid forms, descend to the level of cohomology. After this we proceed to discuss the Lie algebroid modules in the presence of Poisson structures, and in particular consider the modular foliation. We finish by relating the residue maps to symplectic Lie algebroids and the modular foliation in concrete examples.\n\\subsection{Residue maps}\n\\label{sec:residuemaps}\nIn this section we discuss residues of Lie algebroid forms. These are a way to extract relevant information of $\\mathcal{A}$-forms over $\\mathcal{A}$-invariant submanifolds where the restriction of $\\mathcal{A}$ is projective. We further discuss how such residues interact with Lie algebroids morphisms. Residues are used when dealing with geometric structures on $\\mathcal{A}$, in order to extract information along the submanifold (see e.g.\\ \\cite{CavalcantiGualtieri18, CavalcantiKlaasse18}). In \\cite{CavalcantiKlaasse18} the following is discussed for transitive $\\mathcal{A}$-invariant submanifolds, but here we expose a generalization allowing for projective ones, recovering the transitive case when we can take $\\mathcal{B} = TD$ below. As a special case it further covers the case when the restriction of $\\mathcal{A}$ is regular.\n\nLet $\\mathcal{A} \\to X$ be a Lie algebroid and let $D \\subseteq X$ be a projective $\\mathcal{A}$-invariant submanifold (as in \\autoref{defn:lalgebroidadjective}), with $\\mathcal{B} \\to D$ and $(\\rho_A|_D)(\\Gamma(\\mathcal{A}|_D)) \\cong \\Gamma(\\mathcal{B})$. We obtain a short exact sequence\n\\begin{equation*}\n0 \\to \\ker \\widetilde{\\rho}_\\mathcal{A}|_D \\to \\mathcal{A}|_D \\to \\mathcal{B} \\to 0,\n\\end{equation*}\nwhere the $\\widetilde{\\rho}_\\mathcal{A}|_D\\colon \\mathcal{A}|_D \\to \\mathcal{B}$ is the induced bundle map coming from the map on sections. Note that in the transitive case we have that $\\widetilde{\\rho}_{\\mathcal{A}}|_D = \\rho_{A}|_D$.\nWe see that $\\mathcal{A}|_D$ is an extension of $\\mathcal{B}$ by $\\ker \\widetilde{\\rho}_\\mathcal{A}|_D$, the latter being the germinal isotropy of \\autoref{sec:singularliealgebroids}. The case in which we are mainly interested in is when $D = X \\backslash X_\\mathcal{A} = Z_\\mathcal{A}$, the complement of the isomorphism locus of $\\mathcal{A}$ (although in general the restriction need not be transitive nor projective). This is an $\\mathcal{A}$-invariant submanifold of $X$ if it is smooth by \\autoref{prop:degenlocus}, as $X \\backslash X_\\mathcal{A} = X_{\\mathcal{A},n-1}$ where $\\dim X = {\\rm rank}(\\mathcal{A}) = n$ (see \\autoref{exa:denseisolocus}). Note further that $X$ itself is always $\\mathcal{A}$-invariant, as is sometimes useful (say, for regular Lie algebroids). Dualizing the above sequence, we obtain\n\\begin{equation*}\n0 \\to \\mathcal{B}^* \\to \\mathcal{A}^*|_D \\to (\\ker \\widetilde{\\rho}_\\mathcal{A}|_D)^* \\to 0.\t\n\\end{equation*}\nWe are now in the following more general situation: given a short exact sequence $\\mathcal{S}\\colon 0 \\to E \\to W \\to V \\to 0$ of vector spaces, there is an associated dual sequence $\\mathcal{S}^*\\colon 0 \\to V^* \\to W^* \\to E^* \\to 0$. For a given $k \\in \\mathbb{N}$, by taking $k$th exterior powers we obtain a filtration of spaces $\\mathcal{F}^i := \\{\\rho \\in \\wedge^k W^* \\, | \\, \\iota_x \\rho = 0$ for all $x \\in \\wedge^i E \\}$, for $i = 0,\\dots,k+1$. These spaces satisfy $\\mathcal{F}^0 = 0$, $\\mathcal{F}^1 = \\wedge^k V^*$, $\\mathcal{F}^i \\subset \\mathcal{F}^{i+1}$, and $\\mathcal{F}^{i+1} \/ \\mathcal{F}^i \\cong \\wedge^{k-i} V^* \\otimes \\wedge^i E^*$. Setting $\\ell := \\dim E$, we have $\\mathcal{F}^{\\ell + 1} = \\wedge^k W^*$. The \\emph{residue} of an element $\\rho \\in \\wedge^k W^*$ is defined as its equivalence class ${\\rm Res}(\\rho) = [\\rho] \\in \\mathcal{F}^{\\ell + 1} \/ \\mathcal{F}^{\\ell} \\cong \\wedge^{k-\\ell} V^* \\otimes \\wedge^\\ell E^*$. Upon a choice of trivialization of $\\wedge^\\ell E^*$, i.e.\\ a choice of volume element for $E$, one can view the residue ${\\rm Res}(\\rho)$ as an element of $\\wedge^{k-\\ell} V^*$.\n\nWe now define the top residue of a Lie algebroid along a projective $\\mathcal{A}$-invariant submanifold, by performing the above operation to a smoothly varying section of $\\wedge^\\bullet \\mathcal{A}^*$ restricted to $D$.\n\\begin{defn}\\label{defn:ainvariantresidues} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $D \\subseteq X$ a projective $\\mathcal{A}$-invariant submanifold with image $\\mathcal{B}$ and germinal isotropy $E := \\ker \\widetilde{\\rho}_\\mathcal{A}|_D$. The \\emph{residue map} of $\\mathcal{A}$ along $D$ is the map ${\\rm Res}_D\\colon \\Omega^\\bullet(\\mathcal{A}) \\to \\Omega^{\\bullet-\\ell}(\\mathcal{B}; \\wedge^\\ell E^*)$, where $\\ell = \\dim \\ker \\rho_\\mathcal{A}|_D = {\\rm rank}(\\mathcal{A}) - {\\rm rank}(\\mathcal{B})$.\n\\end{defn}\nThere are also lower residues ${\\rm Res}_{-m}\\colon \\wedge^k W^* \\to \\mathcal{F}^{\\ell+1} \/ \\mathcal{F}^{\\ell-m}$ for $m > 0$. These are always defined but have a better description for forms $\\rho \\in \\wedge^k W^*$ whose higher residues vanish, so that ${\\rm Res}_{-m}(\\rho) \\in \\mathcal{F}^{\\ell-m +1} \/ \\mathcal{F}^{\\ell -m} \\cong \\wedge^{k-\\ell+m} V^* \\otimes \\wedge^{\\ell-m} E^*$. This leads to the following.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $D \\subseteq X$ a projective $\\mathcal{A}$-invariant submanifold with image $\\mathcal{B}$ and germinal isotropy $E := \\ker \\widetilde{\\rho}_\\mathcal{A}|_D$. The \\emph{lower residue maps} for $m \\geq 0$ of $\\mathcal{A}$ along $D$ are ${\\rm Res}_{D,-m}\\colon \\Omega^\\bullet_{1-m}(\\mathcal{A}) \\to \\Omega^{\\bullet - \\ell + m}(\\mathcal{B}; \\wedge^{\\ell - m} E^*)$, where $\\Omega^\\bullet_{1-m}(\\mathcal{A})$ is inductively the space of $\\mathcal{A}$-forms all of whose $(1-m)$th or higher residues along $D$ vanish.\n\\end{defn}\n\\begin{rem} In case $D = Z_\\mathcal{A}$ for a Lie algebroid $\\mathcal{A}$ with dense isomorphism locus, one should think of these residues as extracting the coefficients in front of the singular parts of an $\\mathcal{A}$-form along $Z_\\mathcal{A}$, as then $(\\ker \\widetilde{\\rho}_\\mathcal{A})^*|_D$ consists of ``singular'' generators.\n\\end{rem}\n\\begin{rem} Residue maps arise out of any extension of Lie algebroids. This extension need not be induced by the anchor map of the Lie algebroid, even though this is typical.\n\\end{rem}\nGiven a map of short exact sequences $\\Psi\\colon \\widetilde{\\mathcal{S}} \\to \\mathcal{S}$ with dual map $\\Psi^*\\colon \\mathcal{S}^* \\to \\widetilde{\\mathcal{S}}^*$, there is a corresponding map of filtrations $\\Psi^*\\colon \\mathcal{F}^i \\to \\widetilde{\\mathcal{F}}^i$. Setting $\\widetilde{\\ell} := \\dim \\widetilde{E}$, we have the following.\n\\begin{lem}\\label{lem:residues} In the above setting, assume that $\\widetilde{\\ell} > \\ell$. Then $\\widetilde{\\rm Res}(\\Psi^* \\rho) = 0$ for all $\\rho \\in \\wedge^k W^*$.\n\\end{lem}\n\\bp We have $\\rho \\in \\mathcal{F}^{\\ell +1}$ so that $\\Psi^* \\rho \\in \\widetilde{\\mathcal{F}}^{\\ell + 1}$. As $\\widetilde{\\ell} > \\ell$, we have $\\widetilde{\\mathcal{F}}^{\\ell+1} \\subset \\widetilde{\\mathcal{F}}^{\\widetilde{\\ell}} \\subset \\widetilde{\\mathcal{F}}^{\\widetilde{\\ell}+1}$, so that $\\widetilde{\\rm Res}(\\Psi^* \\rho) = [\\Psi^* \\rho] \\in \\widetilde{\\mathcal{F}}^{\\widetilde{\\ell}+1} \/ \\widetilde{\\mathcal{F}}^{\\widetilde{\\ell}}$ vanishes by degree reasons as desired.\n\\end{proof}\nAssuming $\\widetilde{\\ell} > \\ell$, we see from \\autoref{lem:residues} that all lower residues automatically vanish by degree reasons until considering $\\Psi^* \\rho \\in \\widetilde{\\mathcal{F}}^{\\ell +1}$. Hence the first possibly nonzero residue is $\\widetilde{\\rm Res}_{\\ell-\\widetilde{\\ell}}(\\Psi^* \\rho) = [\\Psi^* \\rho] \\in \\widetilde{\\mathcal{F}}^{\\ell+1} \/ \\widetilde{\\mathcal{F}}^{\\ell} \\cong \\wedge^{k-\\ell} \\widetilde{V}^* \\otimes \\wedge^\\ell \\widetilde{E}^*$. As in \\autoref{lem:residues} we obtain the following.\n\\begin{lem}\\label{lem:residuecommute} In the above setting, assuming $\\widetilde{\\ell} \\geq \\ell$, we have $\\Psi^* \\circ {\\rm Res} = \\widetilde{\\rm Res}_{\\ell - \\widetilde{\\ell}} \\circ \\Psi^*$.\n\\end{lem}\nOut of the above discussion the following result on residue maps is immediate.\n\\begin{prop}\\label{prop:ainvsubmfdresidue} Let $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$ be a Lie algebroid morphism and $D \\subseteq X^n$, $D' \\subseteq X'$ transitive $\\mathcal{A}$- respectively $\\mathcal{A}'$-invariant submanifolds such that $f\\colon (X,D) \\to (X',D')$ is a strong map of pairs. Define $\\ell = {\\rm rank}(\\mathcal{A}) - \\dim D$ and $\\ell' = {\\rm rank}(\\mathcal{A}') - \\dim D'$, and assume that $\\ell' \\geq \\ell$. Then ${\\rm Res}_{\\ell - \\ell'} \\circ \\varphi^* = f^* \\circ {\\rm Res}$. Moreover, ${\\rm Res}_{-m} \\circ \\varphi^* = 0$ for $m > \\ell' - \\ell$.\n\\end{prop}\n\\bp As $f$ is a strong map of pairs, we have $T f\\colon TD \\to TD'$. Moreover, as noted below \\autoref{defn:laisotropy}, $\\varphi$ restricts to $\\varphi\\colon \\ker \\rho_\\mathcal{A}|_D \\to \\ker \\rho_{\\mathcal{A}'}|_{D'}$. Consequently, $\\varphi$ induces a map of the relevant short exact sequences defining the residues. The result follows from \\autoref{lem:residuecommute}.\n\\end{proof}\nFor Lie algebroids of divisor-type this specifies to the following.\n\\begin{cor}\\label{cor:residuemapsdenseiso} Let $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$ be a Lie algebroid morphism between Lie algebroids of divisor-type with transitive degeneracy loci. Assume that $f\\colon (X,Z_\\mathcal{A}) \\to (X',Z_{\\mathcal{A}'})$ is a strong map of pairs and that ${\\rm codim}\\, D' \\geq {\\rm codim}\\, D$. Then ${\\rm Res}_{\\ell-\\ell'} \\circ \\varphi^* = f^* \\circ {\\rm Res}$ and ${\\rm Res}_{-m} \\circ \\varphi^* = 0$ for $m > \\ell' - \\ell$, where $\\ell = {\\rm codim}\\, D$ and $\\ell' = {\\rm codim}\\, D'$.\n\\end{cor}\n\\bp This follows from \\autoref{prop:ainvsubmfdresidue}, where we use that degeneracy loci are $\\mathcal{A}$-invariant due to \\autoref{prop:degenlocus}, and that ${\\rm rank}(\\mathcal{A}) = \\dim X$ because $X_\\mathcal{A}$ is nonempty.\n\\end{proof}\n\\begin{rem} When we speak of projective or transitive degeneracy loci, it is implied that we assume that each degeneracy locus has components being smooth submanifolds.\n\\end{rem}\n\\subsection{Lie algebroid modules and cohomology}\n\\label{sec:aconnareps}\nWe next discuss Lie algebroid connections and Lie algebroid modules, which are vector bundles equipped with a flat Lie algebroid connection. We further introduce the associated Lie algebroid cohomologies and characteristic classes, including the modular class. For more information, see e.g.\\ \\cite[Chapter 7]{Mackenzie05} and also \\cite{EvensLuWeinstein99, KosmannSchwarzbachLaurentGengouxWeinstein08,GualtieriPym13}. Let $\\mathcal{A} \\to X$ be a Lie algebroid and $E \\to X$ a vector bundle.\n\\begin{defn} An \\emph{$\\mathcal{A}$-connection} on $E$ is a bilinear map $\\nabla\\colon \\Gamma(\\mathcal{A}) \\times \\Gamma(E) \\to \\Gamma(E)$, $(v,\\sigma) \\mapsto \\nabla_v \\sigma$, such that $\\nabla_{f v} \\sigma = f \\nabla_v \\sigma$ and $\\nabla_v(f\\sigma) = f \\nabla_v \\sigma + (\\rho_\\mathcal{A}(v) f) \\cdot \\sigma$ for all $v \\in \\Gamma(\\mathcal{A})$, $\\sigma \\in \\Gamma(E)$ and $f \\in C^\\infty(X)$. We will also consider $\\nabla$ as a map $\\nabla\\colon \\Gamma(E) \\to \\Omega^1(\\mathcal{A};E)$ via $(\\nabla s)(v) := \\nabla_v s$.\n\\end{defn}\nAny $\\mathcal{A}$-connection $\\nabla$ on $E$ has a \\emph{curvature tensor} $F_\\nabla \\in \\Gamma(\\wedge^2 \\mathcal{A}^* \\otimes {\\rm End}(E))$, given by $F_\\nabla(v,w) := \\nabla_v \\circ \\nabla_w - \\nabla_w \\circ \\nabla_v - \\nabla_{[v,w]_\\mathcal{A}}$ for $v, w \\in \\Gamma(\\mathcal{A})$. An $\\mathcal{A}$-connection $\\nabla$ is \\emph{flat} if $F_\\nabla \\equiv 0$.\n\\begin{defn}\\label{defn:amodule} An \\emph{$\\mathcal{A}$-module} is a vector bundle $E$ equipped with a flat $\\mathcal{A}$-connection $\\nabla$.\n\\end{defn}\nGiven an $\\mathcal{A}$-module $(E,\\nabla)$, there is a differential $d_{\\mathcal{A},\\nabla}$ on $\\Omega^\\bullet(\\mathcal{A};E)$, the space of $\\mathcal{A}$-forms with values in $E$. Recalling that $\\Omega^\\bullet(\\mathcal{A};E) = \\Gamma(\\wedge^\\bullet \\mathcal{A}^*) \\otimes \\Gamma(E)$, we set $d_{\\mathcal{A},\\nabla}(\\eta \\otimes s) := d_{\\mathcal{A}} \\eta \\otimes s + (-1)^{|\\eta|} \\eta \\otimes \\nabla s$, with $|\\eta|$ the degree of $\\eta$. This differential satisfies $d_{\\mathcal{A},\\nabla}(\\eta \\wedge \\xi \\otimes s) = d_{\\mathcal{A}} \\eta \\wedge \\xi \\otimes s + (-1)^{|\\eta|} \\eta \\wedge dt_{\\mathcal{A},\\nabla} (\\xi \\otimes s)$, and $d_{\\mathcal{A},\\nabla}$ squares to zero if and only if $\\nabla$ is flat. In fact, there is a bijective correspondence between such operators on $\\Omega^\\bullet(\\mathcal{A};E)$, and flat $\\mathcal{A}$-connections on $E$.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $(E,\\nabla)$ an $\\mathcal{A}$-module. The \\emph{$\\mathcal{A}$-cohomology with values in $E$} is given by $H^k(\\mathcal{A}; E) := H^k(\\Omega^\\bullet(\\mathcal{A};E), d_{\\mathcal{A},\\nabla})$ for $k \\in \\mathbb{N} \\cup \\{0\\}$.\n\\end{defn}\nNote that $\\mathcal{A}$-modules are also called \\emph{$\\mathcal{A}$-representations}, and that they are Lie algebra representations when $X = \\{{\\rm pt}\\}$. Denote the space of all $\\mathcal{A}$-representations by $\\mathcal{A}$-${\\rm Rep}(X)$.\n\\begin{exa} Let $\\underline{\\mathbb{R}} \\to X$ be the trivial line bundle, carrying a trivial $\\mathcal{A}$-representation structure given by $\\nabla_v f := \\mathcal{L}_{\\rho_{\\mathcal{A}}(v)} f$ for $v \\in \\Gamma(\\mathcal{A})$ and $f \\in \\Gamma(\\underline{\\mathbb{R}}) = C^\\infty(X)$. We then have that $H^\\bullet(\\mathcal{A};\\underline{\\mathbb{R}}) =: H^\\bullet(\\mathcal{A})$, obtaining what is usually called the \\emph{Lie algebroid cohomology} of $\\mathcal{A}$.\n\\end{exa}\nLie algebroid cohomology is generally hard to compute and need not be finite-dimensional. However, the cohomology of most of the Lie algebroids of divisor-type can be determined (see \\cite{KlaasseLanius18} for an overview). Any Lie algebroid $\\mathcal{A} \\to X$ always has a canonical $\\mathcal{A}$-module (see \\cite{EvensLuWeinstein99}).\n\\begin{defn}\\label{defn:canamodule} Let $\\mathcal{A} \\to X$ be a Lie algebroid and set $Q_\\mathcal{A} := \\det(\\mathcal{A}) \\otimes \\det(T^*X)$. Then $Q_\\mathcal{A}$ is  the \\emph{canonical $\\mathcal{A}$-module}, using the $\\mathcal{A}$-connection on $Q_\\mathcal{A}$ that is defined for $v \\in \\Gamma(\\mathcal{A})$, $V \\in \\Gamma(\\det(\\mathcal{A}))$ and $\\mu \\in \\Gamma(\\det(T^*X))$ by the formula $\\nabla_v(V \\otimes \\mu) := \\mathcal{L}_{v} V \\otimes \\mu + V \\otimes \\mathcal{L}_{\\rho_\\mathcal{A}(v)} \\mu$.\n\\end{defn}\nOne readily checks this formula indeed defines a flat $\\mathcal{A}$-connection on the line bundle $Q_\\mathcal{A}$. Several operations can be performed on the class of Lie algebroid representations, such as pullbacks along Lie algebroid morphisms, duals, tensor products, and exterior products.\n\\begin{rem} For Lie algebroids $\\mathcal{A}^n \\to X$ of divisor-type we have $Q_\\mathcal{A}^* \\cong {\\rm div}(\\mathcal{A})$. There should be a relation between this and the Evens--Lu--Weinstein pairing $H^k(\\mathcal{A}) \\otimes H^{n-k}(\\mathcal{A};Q_\\mathcal{A}) \\to \\mathbb{R}$ (see \\cite{EvensLuWeinstein99}), especially when $\\mathcal{A} = TX_I$ for a standard projective divisor ideal $I \\subseteq C^\\infty(X)$.\n\\end{rem}\nAny $\\mathcal{A}$-module $(L,\\nabla)$ of rank $1$ induces a cohomology class in $H^1(\\mathcal{A})$. Given a nonvanishing section $s \\in \\Gamma(L^2)$ and $v \\in \\Gamma(\\mathcal{A})$ (noting that $L^2 = L \\otimes L$ is always trivial) we can write $\\langle v, d_{\\mathcal{A},\\nabla} s\\rangle = \\theta_s(v) s$ for some element $\\theta_s \\in \\Gamma(\\mathcal{A}^*)$. As $d_{\\mathcal{A},\\nabla}$ squares to zero, the section $\\theta_s$ is $d_\\mathcal{A}$-closed, and one verifies that the cohomology class $\\theta_L := \\frac12 [\\theta_s] \\in H^1(\\mathcal{A})$ is independent of $s$. This is the \\emph{characteristic class} of $(L,\\nabla)$. Characteristic classes are natural with respect to tensor products and pullbacks of modules, i.e.\\ they satisfy $\\theta_{L \\otimes L'} = \\theta_L + \\theta_{L'}$ and $\\varphi^* \\theta_L = \\theta_{\\varphi^*L} \\in H^1(\\mathcal{A}')$ given $\\mathcal{A}$-modules $L$ and $L'$ of rank $1$ and a Lie algebroid morphism $\\varphi\\colon \\mathcal{A}' \\to \\mathcal{A}$.\n\nThe characteristic class of the trivial $\\mathcal{A}$-module is zero, and the characteristic class of $Q_\\mathcal{A}$ is called the \\emph{modular class} ${\\rm mod}(\\mathcal{A})$ of $\\mathcal{A}$. One says that $\\mathcal{A}$ is \\emph{unimodular} if ${\\rm mod}(\\mathcal{A}) = 0$. Similarly an $\\mathcal{A}$-module $(L,\\nabla)$ of rank $1$ is \\emph{umimodular} if its characteristic class vanishes. In this case it admits a global nonvanishing section which is $d_{\\mathcal{A},\\nabla}$-closed.\n\\subsubsection{Representations up to homotopy}\nIn studying the residue map of \\autoref{sec:residuemaps} we will have use for a powerful extension of the notion of an $\\mathcal{A}$-representation, introduced in \\cite{AriasAbadCrainic12}. To start, recall that a \\emph{$\\mathbb{Z}$-graded vector bundle} $\\mathbb{E} \\to X$ is a direct sum $\\mathbb{E} = \\bigoplus_{i \\in \\mathbb{Z}} E_i$ of vector bundles, where $E_i$ has \\emph{degree} $i$. Further, ${\\rm End}_k(\\mathbb{E}) \\to X$ is the associated bundle of endomorphisms of $\\mathbb{E}$ of degree $k$. Let $\\mathcal{A} \\to X$ be a Lie algebroid. Then $\\Omega^\\bullet(\\mathcal{A};\\mathbb{E}) := \\Gamma(\\wedge^\\bullet \\mathcal{A}^* \\otimes \\mathbb{E})$ becomes a right graded $\\Omega^\\bullet(\\mathcal{A})$-module in the standard way. With this we have:\n\\begin{defn}[\\cite{AriasAbadCrainic12}] A $\\mathbb{Z}$-graded vector bundle $\\mathbb{E} \\to X$ is an \\emph{$\\mathcal{A}$-representation up to homotopy} if $\\Omega^\\bullet(\\mathcal{A};\\mathbb{E})$ carries a differential $D$ such that it is a right graded $(\\Omega^\\bullet(\\mathcal{A}),d_\\mathcal{A})$-module.\n\\end{defn}\nAs for $\\mathcal{A}$-representations, these lead to cohomology theories $H^\\bullet(\\mathcal{A};\\mathbb{E})$. Moreover, there is a similar bijective correspondence between such differentials $D$, and $\\mathcal{A}$-connections on $\\mathbb{E}$ coming with maps $\\omega_i\\colon \\Gamma(\\wedge^i \\mathcal{A}) \\to {\\rm End}_{1-i}(\\mathbb{E})$ for $i \\geq 2$ satisfying certain coherence conditions.\n\nDenote by $\\mathcal{A}$-${\\rm RepHtpy}$ the space of $\\mathcal{A}$-representations up to homotopy, with $\\mathcal{A}$-${\\rm RepHtpy}_{k,k'}$ for $k,k' \\in \\mathbb{Z}$ the subclasses of those for which the associated $\\mathbb{Z}$-graded vector bundle $\\mathbb{E}$ satisfies $E_i = \\{0\\}$ for all $i$ not satisfying $k \\leq i < k'$. Finally, set $\\mathcal{A}$-${\\rm RepHtpy}_{k,k} := \\mathcal{A}$-${\\rm RepHtpy}_{k,k}$ for those concentrated in degree $k$. Note then that there is a natural inclusion $\\mathcal{A}$-${\\rm Rep} \\hookrightarrow \\mathcal{A}$-${\\rm RepHtpy}_0$ of $E \\mapsto \\mathbb{E}$ with $\\mathbb{E}_0 = E$. The class of $\\mathcal{A}$-representations up to homotopy admits duals, tensor products, and exterior products, similar to what is true for $\\mathcal{A}$-representations.\n\nWe now explain that $\\mathcal{A}$-representations up to homotopy give rise to $\\mathcal{A}$-representations. Given a $\\mathbb{Z}$-graded vector bundle $\\mathbb{E} \\to X$, let $\\mathbb{E}_{\\rm red}$ be the resulting $\\mathbb{Z}_2$-graded vector bundle obtained from the map $\\mathbb{Z} \\to \\mathbb{Z}_2$ of reduction modulo two. Given such a $\\mathbb{Z}_2$-graded vector bundle $\\mathbb{F} \\to X$ with summands $\\mathbb{F} = F_0 \\oplus F_1$, we can consider its \\emph{Berizinian}\n\\begin{equation*}\n\t{\\rm Ber}(F) := \\det(F_0) \\otimes \\det(F_1^*) \\to X.\n\\end{equation*}\nWith this, given a $\\mathbb{Z}$-graded vector bundle $\\mathbb{E} \\to X$, define ${\\rm Ber}(\\mathbb{E}) := {\\rm Ber}(\\mathbb{E}_{\\rm red}) \\to X$.\n\\begin{prop}[\\cite{Mehta15}]\\label{prop:berrephtpy} Let $\\mathbb{E} \\in \\mathcal{A}$-${\\rm RepHtpy}$. Then assigning $\\mathbb{E} \\mapsto {\\rm Ber}(\\mathbb{E})$ defines a map\n\\begin{equation*}\n\t{\\rm Ber}\\colon \\mathcal{A}\\text{-}{\\rm RepHtpy} \\to \\mathcal{A}\\text{-}{\\rm Rep}.\n\\end{equation*}\n\\end{prop}\n\\begin{rem}\\label{rem:adjointrep} We noted in \\autoref{defn:canamodule} that there is a canonical modular $\\mathcal{A}$-module $Q_\\mathcal{A} = \\det(\\mathcal{A}) \\otimes \\det(T^*X)$. We can understand this better using the above framework. Namely, every Lie algebroid $\\mathcal{A} \\to X$ has a canonical $\\mathcal{A}$-representation up to homotopy, namely its \\emph{adjoint representation (up to homotopy)} ${\\rm Adj}(\\mathcal{A})$, given by the anchor sequence $\\mathcal{A} \\to TX$ with its natural differential $D_\\nabla$ \\cite{AriasAbadCrainic12}. Note now that ${\\rm Ber}({\\rm Adj}(\\mathcal{A})) = Q_\\mathcal{A}$, and one readily checks that the resulting induced $\\mathcal{A}$-module structure on $Q_\\mathcal{A}$ is the one of \\autoref{defn:canamodule} (see \\cite{Mehta15}).\n\\end{rem}\n\\begin{rem} The cohomology groups $H^\\bullet(\\mathcal{A}; {\\rm Adj}(\\mathcal{A}))$ of the adjoint representation of \\autoref{rem:adjointrep} form the \\emph{deformation cohomology} $H^\\bullet_{\\rm def}(\\mathcal{A})$ of the Lie algebroid $\\mathcal{A} \\to X$ \\cite{CrainicMoerdijk08}. This important fact motivated the very definition of an $\\mathcal{A}$-representation up to homotopy (see \\cite{AriasAbadCrainic12}).\n\\end{rem}\nThere is a class of representations up to homotopy called \\emph{Serre representations} that is of main importance to us (\\cite[Example 4.15]{AriasAbadCrainic12}). Consider an extension of Lie algebroids over a manifold $X$, i.e.\\ a short exact sequence of Lie algebroid morphisms as follows:\n\\begin{equation*}\n\t0 \\to E \\stackrel{i}{\\to} \\mathcal{A} \\stackrel{p}{\\to} \\mathcal{A}' \\to 0.\n\\end{equation*}\nLet $\\sigma\\colon \\Gamma(\\mathcal{A}') \\to \\Gamma(\\mathcal{A})$ be an Ehresmann connection splitting the above sequence, i.e.\\ such that $p \\circ \\sigma = {\\rm id}_{\\mathcal{A}'}$. This gives a $\\sigma$-dependent identification $\\Gamma(\\mathcal{A}) \\cong \\Gamma(\\mathcal{A}') \\oplus \\Gamma(E)$. Consider the $\\mathbb{Z}$-graded complex $(\\Gamma(\\wedge^\\bullet E^*), d_E)$. Given $v \\in \\Gamma(\\mathcal{A}')$ there is a degree-zero operator $\\nabla_v^\\sigma := {\\rm ad}^*_{\\sigma(v)}\\colon \\Gamma(\\wedge^\\bullet E^*) \\to \\Gamma(\\wedge^\\bullet E^*)$ using the bracket $[\\cdot,\\cdot]_\\mathcal{A}$, which is thus an $\\mathcal{A}'$-connection $\\nabla^\\sigma$ on $\\Gamma(\\wedge^\\bullet E^*)$. Further, the \\emph{curvature} $R_\\sigma \\in \\Gamma(\\wedge^2 \\mathcal{A}'^* \\otimes E)$ of $\\sigma$ is given for $v,v' \\in \\Gamma(\\mathcal{A}')$ by\n\\begin{equation*}\n\tR^\\sigma(v,v') := [\\sigma(v),\\sigma(v')]_\\mathcal{A} - \\sigma([v,v']_{\\mathcal{A}'}).\n\\end{equation*}\nThis naturally turns into a map $i(R^\\sigma)\\colon \\Gamma(\\wedge^2 \\mathcal{A}') \\to {\\rm End}_{-1}(\\wedge^\\bullet E^*)$ after using the contraction $i\\colon E \\to {\\rm End}_{-1}(\\wedge^\\bullet E^*)$. Now define the operator $D := d_E + \\nabla^\\sigma + i(R^\\sigma)$ on $\\mathbb{E} := \\wedge^\\bullet E^*$.\n\\begin{prop}[{\\cite[Example 4.15]{AriasAbadCrainic12}}]\\label{prop:rephtpyextension} The triple $(\\mathbb{E},\\sigma, D)$ defines the structure of an $\\mathcal{A}'$-representation up to homotopy on the exterior algebra $\\mathbb{E} = \\wedge^\\bullet E^*$.\n\\end{prop}\n\\begin{rem} When $E$ is abelian, i.e.\\ has trivial bracket, the above structure in fact defines an $\\mathcal{A}$-module structure on $E$. More generally this is true for the center and abelianization $Z(E)$ and $E\/[E,E]$ of the bundle Lie algebroid $E$ (see \\cite[Proposition 3.3.20]{Mackenzie05}).\n\\end{rem}\n\\subsubsection{Residue maps and representations}\nWe now discuss when the residue maps of \\autoref{sec:residuemaps} induce maps in cohomology. Let $\\mathcal{A} \\to X$ be a Lie algebroid and let $D \\subseteq X$ a projective $\\mathcal{A}$-invariant submanifold, so that we obtain an extension of Lie algebroids\n\\begin{equation*}\n\t0 \\to \\ker\\widetilde{\\rho}_{\\mathcal{A}}|_D \\to \\mathcal{A}|_D \\to \\mathcal{B} \\to 0.\n\\end{equation*}\nBy applying \\autoref{prop:rephtpyextension} we obtain the structure of a $\\mathcal{B}$-representation up to homotopy on $E := \\ker \\widetilde{\\rho}_\\mathcal{A}|_D \\to D$. Using \\autoref{prop:berrephtpy} we obtain a $\\mathcal{B}$-module structure on $\\det(E) \\to D$, and consequently also on its dual $\\det(E^*)$. We can now prove the following (\\autoref{thm:introresidues}).\n\\begin{thm}\\label{thm:residues} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $(\\mathcal{B},D)$ a projective $\\mathcal{A}$-invariant submanifold with $\\ell$-dimensional germinal isotropy $E \\to D$. Then ${\\rm Res}_D\\colon \\Omega^\\bullet(X,\\mathcal{A}) \\to \\Omega^{\\bullet-\\ell}(D, \\mathcal{B}; \\det(E^*))$ is a cochain morphism, hence induces a map $[{\\rm Res}_D]\\colon H^\\bullet(X,\\mathcal{A}) \\to H^{\\bullet-\\ell}(D,\\mathcal{B};\\det(E^*))$.\n\\end{thm}\n\\bp The $\\mathcal{B}$-representation on $\\det(E)$ is induced from the $\\mathcal{B}$-connection $\\nabla^\\sigma = [\\sigma(\\cdot),\\cdot]_\\mathcal{A}$. The condition that ${\\rm Res}_D$ is a cochain morphism is equivalent to the fact that $\\det(E^*) \\subseteq \\wedge^\\bullet \\mathcal{A}^*|_D$ has closed local generating sections. This is implied by \\autoref{prop:berrephtpy}, giving the result.\n\\end{proof}\n\\begin{rem} The above theorem is not true in general for the lower residue maps discussed in \\autoref{sec:residuemaps}. Counterexamples arise from the local generators of $\\wedge^{\\ell-k} E^*$ not being closed. However, by analyzing the proof of \\autoref{thm:residues} we see that it is true for ${\\rm Res}_{D,-k}$ whenever the representation up to homotopy $\\wedge^{\\ell-k} E$ is a $\\mathcal{B}$-module. This property is inherited (i.e.\\ if it holds for some $k$, then also for all $k' \\leq k$), and the above shows that this is true for $k = 0$.\n\\end{rem}\nTo better understand the Lie algebroid cohomology acting as the target for the residue map, it is clear that it is important to determine when, or whether, the $\\mathcal{B}$-representation on $\\det(E)$ is unimodular, so that one can suitably trivialize $\\det(E)$ as a representation.\n\\begin{rem}\\label{rem:residuesymplecticliealgebroid} Let $(\\mathcal{A},\\omega_{\\mathcal{A}}) \\to X$ be a symplectic Lie algebroid and let $D \\subseteq X$ be a transitive $\\mathcal{A}$-invariant submanifold, possibly the degeneracy locus of $\\mathcal{A}$ (see \\autoref{prop:degenlocus}). The residue map gives a form ${\\rm Res}_D(\\omega_\\mathcal{A}) \\in \\Omega^{2-\\ell}(D; \\wedge^\\ell E^*)$, where $E = \\ker \\rho_{\\mathcal{A}}|_D$ and $\\ell = \\dim(E)$. Note however that it can only be nonzero for $\\ell \\leq 2$. One further has the residue ${\\rm Res}_D(\\omega_{\\mathcal{A}}^n) \\in \\Omega^{2n-\\ell}(D;\\wedge^\\ell E^*)$, if ${\\rm rank}(\\mathcal{A}) = 2n$. These describe important features of the induced geometry on $D$, and the latter has a Poisson-geometric interpretation which we discuss in \\autoref{sec:residuesympla}.\n\\end{rem}\n\\begin{rem} While \\autoref{thm:residues} shows the residue map descends to cohomology, it often does not respect the ring structure induced by the wedge product. See also \\autoref{sec:residuesympla}.\n\\end{rem}\n\\subsubsection{Examples of residue maps}\n\\label{sec:residueexamples}\nIn this section we discuss several examples of residue maps, for some of the Lie algebroids of divisor-type that we have discussed in this paper.\n\n\\begin{exa}[$\\mathcal{A}_Z^k$]\\label{exa:bkersiduemaps} For the $b^k$-bundles $\\mathcal{A}_Z^k \\to X$ of \\autoref{exa:bktangentbundle}, the hypersurface $Z \\subseteq X$ is a transitive invariant submanifold. The resulting extension sequence of Lie algebroids\n\\begin{equation*}\n\t0 \\to \\mathbb{L}_{Z,k} \\to \\mathcal{A}_Z^k|_Z \\to TZ \\to 0,\n\\end{equation*}\nis such that $\\mathbb{L}_{Z,k} \\to Z$ is trivial, and has a canonical trivializing section (it is also called the \\emph{$b^k$-normal bundle}, see \\cite[Proposition 4]{GuilleminMirandaPires14} and \\cite[Proposition 3.2]{Scott16}). As such the residue map is a cochain morphism, given by ${\\rm Res}_{Z,k}\\colon \\Omega^\\bullet(\\mathcal{A}_Z^k) \\to \\Omega^{\\bullet-1}(Z)$, and locally $\\frac{dx}{x^k} \\wedge \\alpha + \\beta \\mapsto \\iota_Z^*(\\alpha)$, where $x \\in j_{k-1}$ is a local $(k-1)$-jet generator, $\\alpha,\\beta \\in \\Omega^\\bullet(X)$ and $\\iota_Z\\colon Z \\hookrightarrow X$ is the inclusion. These residue maps along with the dual $\\mathcal{A}_Z^{k-1}$-anchor $\\varphi_{A_Z^k}^*$ fit in the short exact sequence\n\\begin{equation*}\n\t0 \\to \\Omega^\\bullet(\\mathcal{A}_Z^{k-1}) \\to \\Omega^\\bullet(\\mathcal{A}_Z) \\to \\Omega^{\\bullet-1}(Z) \\to 0,\n\\end{equation*}\nwhich is called the \\emph{residue sequence} for $\\mathcal{A}_Z^k$. These sequences split and imply the cohomological consequences $H^\\bullet(\\mathcal{A}_Z^k) \\cong H^\\bullet(\\mathcal{A}_Z^{k-1}) \\oplus H^{\\bullet-1}(Z)$ (c.f.\\ \\cite{GuilleminMirandaPires14,MarcutOsornoTorres14,Melrose93,Scott16}; see \\cite{KlaasseLanius18} for more).\n\\end{exa}\n There are no lower residue maps for $\\mathcal{A}_Z^k \\to X$, as $\\mathbb{L}_{Z,k}$ is one-dimensional. While \\autoref{thm:residues} asserts the existence of the above residue map as stated, it is special in this situation that $\\mathbb{L}_{Z,k}$ is not only (canonically) trivial, but is trivial as a $TZ$-representation, so that the image of the residue map is smooth forms on $Z$, i.e.\\ $\\Omega^{\\bullet-1}(Z)$, instead of $\\Omega^{\\bullet-1}(Z;\\mathbb{L}_{Z,k}^*)$.\n\\begin{exa}[$\\mathcal{A}_{|D|}$]\\label{exa:elltangentresidues} For the elliptic tangent bundle $\\mathcal{A}_{|D|} \\to X$ of \\autoref{exa:ellvx}, the degeneracy locus $D \\subseteq X$ is a transitive invariant submanifold, hence leads to an extension (\\cite{CavalcantiGualtieri18})\n\\begin{equation*}\n\t0 \\to \\underline{\\mathbb{R}} \\oplus \\mathfrak{k} \\to \\mathcal{A}_{|D|}|_D \\to TD \\to 0.\n\\end{equation*}\nNote that $\\ker(\\rho_{\\mathcal{A}_{|D|}}) \\cong \\underline{\\mathbb{R}} \\oplus \\mathfrak{k}$, so that $\\det(\\ker(\\rho_{\\mathcal{A}_{|D|}})) \\cong \\mathfrak{k}$. This results in an \\emph{elliptic residue}\n\\begin{equation*}\t\n\t{\\rm Res}_q\\colon \\Omega^\\bullet(\\mathcal{A}_{|D|}) \\to \\Omega^{\\bullet-2}(D;\\mathfrak{k}^*), \\qquad d\\log r \\wedge d\\theta \\wedge \\alpha + d\\log r \\wedge \\beta + d\\theta \\wedge \\gamma + \\eta \\mapsto \\iota_D^*(\\alpha),\n\\end{equation*}\nfor $\\alpha,\\beta,\\gamma,\\eta \\in \\Omega^\\bullet(X)$ and $\\iota_D\\colon D \\hookrightarrow X$ the inclusion. Letting $\\Omega^\\bullet_0(\\mathcal{A}_{|D|}) = \\ker({\\rm Res}_q)$ we have\n\\begin{equation*}\n\t{\\rm Res}_r\\colon \\Omega^\\bullet_0(\\mathcal{A}_{|D|}) \\to \\Omega^{\\bullet-1}(D), \\qquad d\\log r \\wedge \\beta + d\\theta \\wedge \\gamma + \\eta \\mapsto \\iota_D^*(\\beta),\n\\end{equation*}\nwhich is called the \\emph{radial residue}. A coorientation for $D$ trivializes $\\mathfrak{k}$ and gives further\n\\begin{equation*}\n\t{\\rm Res}_\\theta\\colon \\Omega^\\bullet_0(\\mathcal{A}_{|D|}) \\to \\Omega^{\\bullet-1}(D), \\qquad d\\log r \\wedge \\beta + d\\theta \\wedge \\gamma + \\eta \\mapsto \\iota_D^*(\\gamma),\n\\end{equation*}\nwhich is the \\emph{$\\theta$-residue}. In general ${\\rm Res}_r\\colon \\Omega^\\bullet(\\mathcal{A}_{|D|}) \\to \\Omega^{\\bullet-1}(D,{\\rm At}(S^1 ND))$ and ${\\rm Res}_q = \\iota_{\\partial_{\\theta}} \\circ {\\rm Res}_r$. This is because $\\mathcal{A}_{|D|}|_D$ is an extension by $\\underline{\\mathbb{R}}$ of the Atiyah algebroid ${\\rm At}(S^1 ND) \\to D$, i.e.\\\n\\begin{equation*}\n\t0 \\to \\underline{\\mathbb{R}} \\to \\mathcal{A}_{|D|}|_D \\to {\\rm At}(S^1 ND) \\to 0,\n\\end{equation*}\nso that by the general recipe for residue maps we obtain the map ${\\rm Res}_r$ above, as ${\\rm det}(\\underline{\\mathbb{R}}^*)$ is the trivial representation. Finally ${\\rm At}(S^1 ND)$ has $D$ as transitive invariant submanifold, with\n\\begin{equation*}\n\t0 \\to \\mathfrak{k} \\to {\\rm At}(S^1 ND) \\to TD \\to 0,\n\\end{equation*}\nwhose residue map ${\\rm Res}_{\\rm At}\\colon \\Omega^\\bullet(D, {\\rm At}(S^1 ND)) \\to \\Omega^{\\bullet-1}(D; \\mathfrak{k}^*)$ combines as ${\\rm Res}_q = {\\rm Res}_{\\rm At} \\circ {\\rm Res}_r$, and ${\\rm Res}_{\\rm At} = \\iota_{\\partial_{\\theta}}$ in local coordinates.\nAs in \\autoref{exa:bkersiduemaps} there are cohomological consequences one can draw from this (see \\cite{CavalcantiGualtieri18,CavalcantiKlaasse18}), for example $H^\\bullet(\\mathcal{A}_{|D|}) \\cong H^\\bullet(X\\backslash D) \\oplus H^{\\bullet-1}(S^1 ND)$.\n\\end{exa}\nThe previous example is special as the isotropy bundle is two-dimensional, but splits as a sum of two line bundles. As such we have ${\\rm Res}_{D,0} = {\\rm Res}_q$ and ${\\rm Res}_{D,-1} =  {\\rm Res_r} + {\\rm Res}_\\theta$ here. There are also residue maps for complex Lie algebroids, for example the complex log-tangent bundle $\\mathcal{A}_D$ of \\autoref{exa:complexlogtgnt} with its complex log residue ${\\rm Res}_D\\colon \\Omega^\\bullet(\\mathcal{A}_D) \\to \\Omega^{\\bullet-1}(D;\\mathbb{C})$, which is related to the \\emph{complex residue} ${\\rm Res}_\\mathbb{C} := {\\rm Res}_{D,-1}$ of the elliptic tangent bundle $\\mathcal{A}_{|D|}$ (see \\cite{CavalcantiGualtieri18}).\n\\begin{exa}[$\\mathcal{A}_{\\underline{Z}}$]  Let $\\underline{Z} = \\cup_{j \\in I} Z_j$ be a normal-crossing log divisor with associated log-tangent bundle $\\mathcal{A}_{\\underline{Z}}$ as below \\autoref{exa:zvx}. Set $Z_\\tau = \\cap_{i \\in \\tau} Z_i$ for $\\tau \\subseteq I$ and let $\\underline{Z}_k = \\cup_{|\\tau| = k} Z_\\tau$, so that $\\underline{Z}_k$ consists of all $k$-fold intersections of hypersurfaces. By transversality, these are all smooth submanifolds, with induced normal-crossing log divisors. We have for $\\mathcal{A}_{\\underline{Z}_k} \\to \\underline{Z}_{k}$ that $\\underline{Z}_{k+1}$ is $\\mathcal{A}_{\\underline{Z}_k}$-invariant, with its associated Lie algebroid extension sequence over $\\underline{Z}_{k+1}$ being\n\\begin{equation*}\n\t0 \\to \\mathbb{L}_{\\underline{Z},k} \\to \\mathcal{A}_{\\underline{Z}_k}|_{\\underline{Z}_{k+1}} \\to \\mathcal{A}_{\\underline{Z}_{k+1}} \\to 0.\n\\end{equation*}\nAgain $\\mathbb{L}_{\\underline{Z},k} \\to \\underline{Z}_{k+1}$ is canonically trivial, leading to ${\\rm Res}_{Z,k}\\colon \\Omega^\\bullet(\\underline{Z}_k, \\mathcal{A}_{\\underline{Z}_k}) \\to \\Omega^{\\bullet-1}(\\underline{Z}_{k+1},\\mathcal{A}_{\\underline{Z}_{k+1}})$. This leads to a sequence of composable residue maps, and splitting residue sequences\n\\begin{equation*}\n\t0 \\to \\Omega^\\bullet(\\underline{Z}_k) \\to \\Omega^\\bullet(\\underline{Z}_k, \\mathcal{A}_{\\underline{Z}_k}) \\to \\Omega^{\\bullet-1}(\\mathcal{A}_{\\underline{Z}_{k+1}}) \\to 0,\n\\end{equation*}\nwhich leads to the cohomological consequence $H^\\bullet(\\underline{Z}_k,\\mathcal{A}_{\\underline{Z}_k}) \\cong H^\\bullet(\\underline{Z}_k) \\oplus H^{\\bullet-1}(\\underline{Z}_{k+1}, \\mathcal{A}_{\\underline{Z}_{k+1}})$. These are straightforward analogues of the case of the log-tangent bundle $\\mathcal{A}_Z$ of \\autoref{exa:bkersiduemaps}. By applying induction to $k$ and noting that $X = Z_0$, this gives (c.f.\\ \\cite[Appendix A.24]{GualtieriLiPelayoRatiu17})\n\\begin{equation*}\n\tH^\\bullet(\\mathcal{A}_{\\underline{Z}}) \\cong H^\\bullet(X) \\oplus \\bigoplus_{k \\geq 1} H^{\\bullet-k}(\\underline{Z}_k).\n\\end{equation*}\n\\end{exa}\nThe above can also be done in similar fashion for self-crossing log divisors (see \\cite{MirandaScott18}) instead mapping onto its $k$-strata, but this requires more notation, so that we will not do so here. The other Lie algebroids found in this paper, such as the scattering tangent bundle $\\mathcal{C}_Z \\to X$, also admit residue maps, but these are less useful, because $Z$ is $\\mathcal{C}_Z$-invariant with $\\mathcal{B} = 0_{TZ}$.\n\\begin{exa}[$\\mathcal{A}_{Z,F}$] Consider the Lie algebroid $\\mathcal{A}_{Z,F} := [TX\\:TF] \\to X$ given a codimension-one involutive distribution $TF \\subseteq TZ$ on a hypersurface $Z \\subseteq X$ (see \\autoref{sec:examplesmodification}). Then $\\mathcal{A}_{Z,F}$ has $Z$ as a projective invariant submanifold with $\\mathcal{B} = TF$ (i.e.\\ $\\mathcal{A}_{Z,F}|_Z$ is regular):\n\\begin{equation*}\n\t0 \\to \\ker(\\rho_{\\mathcal{A}_{Z,F}}|_Z) \\to \\mathcal{A}_{Z,F}|_Z \\to TF \\to 0.\n\\end{equation*}\nLocally we have $\\ker(\\rho_{\\mathcal{A}_{Z,F}}|_Z) = \\langle x \\partial_x, x \\partial_y\\rangle$ for $x \\in I_Z$ a local generator, and $\\partial_y \\subseteq \\Gamma(TZ)$ normal to $TF$. The dual generator $dx\/x$ is closed, but the generator $dy\/x$ is not. Thus, while the top residue map is a cochain morphism by \\autoref{thm:residues}, the lower residue map is not. Note that $\\mathcal{A}_{Z,F}$ is an $\\mathcal{A}_Z$-Lie algebroid, and that its $\\mathcal{A}_Z$-anchor is a quasi-isomorphism (c.f.\\ \\cite{GualtieriLiPelayoRatiu17,KlaasseLanius18}).\n\\end{exa}\nSimilar ideas apply to invariantly capture what is done in \\cite{Lanius16}, but we will not do so here.\n\\begin{exa}[$\\mathcal{A}_W$]\\label{exa:ellipticlogresidue} Let $W = Z \\otimes |D|$ be an elliptic-log divisor with associated Lie algebroid $\\mathcal{A}_W \\to X$ as in \\autoref{prop:elllogalgebroid}. The degeneracy locus $Z \\subseteq X$ is $\\mathcal{A}_W$-invariant, and $D \\subseteq Z$ is a hypersurface with log-tangent bundle $\\mathcal{A}_{Z,D} \\to Z$. There is an extension sequence\n\\begin{equation*}\n\t0 \\to E \\to \\mathcal{A}_W|_Z \\to \\mathcal{A}_{Z,D} \\to 0,\n\\end{equation*}\nwith $E$ the rank-$1$ germinal isotropy bundle. This follows from the local description of $\\Gamma(\\mathcal{A}_W) = \\langle x \\partial_x + y \\partial_y, x(y \\partial_x - x \\partial_y) \\rangle \\oplus \\Gamma(TD)$. This results in a residue map\n\\begin{equation*}\n\t{\\rm Res}_Z\\colon \\Omega^\\bullet(\\mathcal{A}_W) \\to \\Omega^{\\bullet-1}(Z,\\mathcal{A}_{Z,D};E^*).\n\\end{equation*}\nThe submanifold $D\\subseteq X$ is also $\\mathcal{A}_W$-invariant, with a transitive extension sequence\n\\begin{equation*}\n\t0 \\to \\ker(\\rho_{\\mathcal{A}_W}|_D) \\to \\mathcal{A}_W|_D \\to TD \\to 0.\n\\end{equation*}\nThis gives another residue map onto $D$, namely\n\\begin{equation*}\n\t{\\rm Res}_D\\colon \\Omega^\\bullet(\\mathcal{A}_W) \\to \\Omega^{\\bullet-2}(D;\\det(F^*)),\n\\end{equation*}\nwith isotropy plane bundle $F = \\ker(\\rho_{\\mathcal{A}_W}|_D) \\to D$. These residue maps are related, because for the log-tangent bundle $\\mathcal{A}_{Z,D} \\to Z$, the submanifold $D \\subseteq Z$ is invariant with sequence\n\\begin{equation*}\n\t0 \\to \\widetilde{E} \\to \\mathcal{A}_{Z,D}|_D \\to TD \\to 0,\n\\end{equation*}\nwhich in turn results in the logarithmic residue map\n\\begin{equation*}\n\t{\\rm Res}_{Z,D}\\colon \\Omega^\\bullet(Z,\\mathcal{A}_{Z,D}) \\to \\Omega^{\\bullet-1}(D;\\widetilde{E}^*).\n\\end{equation*}\nWe have $\\det(F^*) \\cong E^* \\otimes \\widetilde{E}^*$ and through this we have that ${\\rm Res}_D = {\\rm Res}_{Z,D} \\circ {\\rm Res}_Z$. Note that the line bundle $\\widetilde{E} = \\mathbb{L}_D \\to D$ is canonically trivial as in \\autoref{exa:bkersiduemaps}. The cohomological consequences of the existence of these residue maps is discussed in \\cite{KlaasseLanius18}.\n\\end{exa}\nFurther examples of residue maps for other Lie algebroids can be found in \\cite{KlaasseLanius18}.\n\\subsection{Poisson modules and modular residue maps}\n\\label{sec:poissonmodulesmodfoliation}\nIn this section we discuss modules associated to $\\mathcal{A}$-Poisson structures. For simplicity we will mainly focus on the case where $\\mathcal{A} = TX$ \\cite{GualtieriLi14,GualtieriPym13,Polishchuk97}. In essence, given a Lie algebroid $\\mathcal{A} \\to X$, $\\mathcal{A}$-Poisson modules are Lie algebroid modules for the $\\mathcal{A}$-Poisson algebroid $\\mathcal{A}^*_{\\pi_\\mathcal{A}} \\to X$ of an $\\mathcal{A}$-Poisson structure $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$. Unravelling \\autoref{defn:amodule} gives the following definition of an $\\mathcal{A}$-Poisson module.\n\\begin{defn} Let $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$. A \\emph{$\\pi_\\mathcal{A}$-module} is a bundle $E \\to X$ equipped with a flat $\\mathcal{A}$-Poisson connection, i.e.\\ a linear morphism $\\nabla\\colon \\Gamma(E) \\to \\Gamma(\\mathcal{A} \\otimes E)$ satisfying the Leibniz rule\n\t%\n\t\\begin{equation*}\n\t\\nabla(f s) = - \\pi_{\\mathcal{A}}^\\sharp(d_\\mathcal{A} f) \\otimes s + f \\nabla s,\n\t\\end{equation*}\n\tfor all $f \\in C^\\infty(X)$ and $s \\in \\Gamma(E)$, and which for all $\\alpha,\\beta \\in \\Gamma(\\mathcal{A}^*_{\\pi_{A}})$ satisfies (with $\\nabla_\\alpha s = (\\nabla s)(\\alpha)$)\n\t%\n\t\\begin{equation*}\n\t\t\\nabla_{[\\alpha,\\beta]_{\\pi_\\mathcal{A}}} = \\nabla_\\alpha \\circ \\nabla_\\beta - \\nabla_\\beta \\circ \\nabla_\\alpha.\n\t\\end{equation*}\n\t%\n\\end{defn}\nWhen $\\mathcal{A} = TX$, the bracket on $T^*_\\pi X$ is described directly by the relation $[df,dg]_\\pi = d\\{f,g\\}_\\pi$. As such, in this case flatness is given by $\\nabla_{d\\{f,g\\}_\\pi} = \\nabla_{df} \\circ \\nabla_{dg} - \\nabla_{dg} \\circ \\nabla_{df}$ as operators on $\\Gamma(E)$.\n\\begin{exa}\\label{exa:modularreppoisson} Given $\\pi \\in {\\rm Poiss}(X)$, the Lie algebroid $T^*_\\pi X$ has a representation on $\\det(T^*X)$ which is given, for $\\alpha \\in \\Gamma(T^*_\\pi X)$ and $\\mu \\in \\Gamma(\\det(T^*X))$, by\n\\begin{equation*}\n\t\\nabla_\\alpha \\mu = \\mathcal{L}_{\\pi^\\sharp(\\alpha)} \\mu + (\\pi,d\\alpha) \\mu = [\\alpha,\\mu]_\\pi - (\\pi,d\\alpha)\\mu = \\alpha \\wedge d(\\iota_\\pi \\mu).\n\\end{equation*}\nNote that this is essentially the modular representation $Q_{T^*_\\pi X} \\cong \\det(T^*_\\pi X)^2$ of $T^*_\\pi X$.\n\\end{exa}\nThe \\emph{modular class} ${\\rm Mod}(\\pi)$ of $\\pi \\in {\\rm Poiss}(X)$ is given by $2\\, {\\rm Mod}(\\pi) = {\\rm Mod}(T^*_\\pi X) = \\theta_{Q_{T^*_\\pi X}}$.\nGiven a $\\pi_{\\mathcal{A}}$-line bundle $(L,\\nabla)$, a local trivialization $s \\in \\Gamma(L)$ specifies a unique $\\mathcal{A}$-Poisson section $v \\in {\\rm Poiss}(\\pi_{\\mathcal{A}}) \\subseteq \\Gamma(\\mathcal{A})$ by $\\nabla s = v \\otimes s$. In fact, $v$ is $\\mathcal{A}$-Poisson if and only if $\\nabla$ is flat.\n\\begin{rem} For $\\pi \\in {\\rm Poiss}(X)$, in case $L = \\det(T^*X)$ with local covolume form $\\mu$, the above section $v = V_\\mu \\in \\Gamma(TX)$ is called the \\emph{$\\pi$-modular vector field} associated to $\\mu$. For general Lie algebroids $\\mathcal{A}$, the above section then gives the \\emph{$\\pi_{A}$-modular section} $V_{\\mu_\\mathcal{A}}$ associated to $\\mu_\\mathcal{A}$.\n\\end{rem}\n\\begin{exa} Given $\\pi \\in {\\rm Poiss}(X^n)$ locally expressed as $\\pi = \\sum_{ij} \\pi^{ij} \\partial_{x_i} \\wedge \\partial_{x_j}$, the modular vector field associated to $\\mu = \\partial_{x_1} \\wedge \\dots \\wedge \\partial_{x_n}$ is given by $V_\\mu = \\sum_i \\partial_k(\\pi^{ik}) \\partial_{x_i}$.\n\\end{exa}\nWe defined $\\mathcal{A}$-Poisson submanifolds as those equipped with Lie subalgebroids $(Y,\\mathcal{A}',\\pi_{\\mathcal{A}'})$ for which the inclusion is an $\\mathcal{A}$-Poisson map. We now specify to the case $\\mathcal{A} = TX$ to discuss restriction of Poisson modules. Recall that $Y \\subseteq (X,\\pi)$ is $\\pi$-Poisson if $I_Y$ is a $\\pi$-Poisson ideal.\n\\begin{defn}[\\cite{GualtieriPym13,Polishchuk97}] Let $\\pi \\in {\\rm Poiss}(\\mathcal{A})$ be given. A submanifold $Y \\subseteq X$ is a \\emph{strong $\\pi$-Poisson submanifold} if $\\mathcal{L}_V I_Y \\subseteq I_Y$ for all $V \\in {\\rm Poiss}(\\pi)$.\n\\end{defn}\nAs every $\\pi$-Hamiltonian vector field is $\\pi$-Poisson, it is clear that strong $\\pi$-Poisson submanifolds are also $\\pi$-Poisson submanifolds. In general, a \\emph{strong $\\pi$-Poisson ideal} is an ideal $I \\subseteq C^\\infty(X)$ that is preserved by all $\\pi$-Poisson vector fields. \\autoref{rem:preservingideals} shows that strong $\\pi$-Poisson ideals are closed under sums and intersections, and \\autoref{rem:preservingradical} shows the same for taking radicals. As such, we see that (strong) $\\pi$-Poisson submanifolds are closed under unions and intersections. There moreover is a strengthening of \\autoref{prop:pidegenpipoisson} (c.f.\\ \\cite{GualtieriPym13,Polishchuk97}). \n\\begin{prop}[\\cite{Polishchuk97}]\\label{prop:pidegpistrongpoisson} Let $\\pi \\in {\\rm Poiss}(X)$. Then the degeneracy ideals $I_{\\pi,2k}$ are strong $\\pi$-Poisson ideals, hence the degeneracy loci $X_{\\pi,2k}$ are strong $\\pi$-Poisson submanifolds if smooth.\n\\end{prop}\n\\bp Unravelling definitions, for $k \\geq 0$, the ideal $I_{2k,\\pi}$ is defined as the image $\\pi^k(\\Gamma(\\wedge^{2k} T^*X))$. Let $V \\in {\\rm Poiss}(\\pi)$ and $\\alpha \\in \\Gamma(\\wedge^{2k} T^*X)$. Then we readily compute that\n\\begin{equation*}\n\t\\mathcal{L}_V(\\pi^k(\\alpha)) = (\\mathcal{L}_V \\pi^k)(\\alpha) + \\pi^k(\\mathcal{L}_V \\alpha) = \\pi^k(\\mathcal{L}_V \\alpha) \\in I_{2k,\\pi}.\\qedhere\n\\end{equation*}\n\\end{proof}\nEssentially, the above follows because the degeneracy loci are defined directly in terms of the (rank of the) Poisson bivector itself: all $\\pi$-Poisson vector fields preserve all powers of $\\pi$. Turning this on its head, the above proposition says that given $\\pi \\in {\\rm Poiss}(X)$, all $\\pi$-Poisson vector fields must be tangent to all the degeneracy loci of $\\pi$.\n\nConsider a submanifold $Y \\subseteq X$ and the sequence of its conormal bundle:\n\\begin{equation*}\n\t0 \\to N^*Y \\to T^*X|_Y \\to T^*Y \\to 0.\n\\end{equation*}\nWhen $Y$ is a $\\pi$-Poisson submanifold, this becomes an extension sequence of Lie algebroids, using the Poisson algebroids $T^*X_\\pi$ and $T^*Y_{\\pi_Y}$, where $\\pi_Y \\in {\\rm Poiss}(Y)$ is the induced Poisson structure on $Y$. One naturally wonders about restriction of $\\pi$-Poisson line bundles. We have:\n\\begin{prop}[{\\cite[Lem 4.11]{GualtieriPym13}}]\\label{prop:degenlocusstrongpoisson} Let $\\pi \\in {\\rm Poiss}(X)$ and let $(L,\\nabla)$ be a $\\pi$-Poisson line bundle, and let $Y \\subseteq X$ be a strong $\\pi$-Poisson submanifold with induced Poisson structure $\\pi_Y$. Then the restriction $(L|_Y, \\nabla|_Y)$ is a $\\pi_Y$-Poisson module.\n\\end{prop}\n\\bp Let $s \\in \\Gamma(L)$ be a local trivialization. Then $\\nabla s = v \\otimes s$ for a unique $\\pi$-Poisson vector field $v \\in {\\rm Poiss}(\\pi)$. To be able to restrict $\\nabla s$ to $Y$, the image $[v \\otimes s] \\in \\Gamma(Y,{\\rm Hom}(N^*Y \\otimes L|_Y))$ is zero. As $Y$ is a strong $\\pi$-Poisson submanifold we have $\\mathcal{L}_v I_Y \\subseteq I_Y$. Thus $v$ is tangent to $Y$, so that $[v \\otimes s] \\equiv 0$. We conclude that $v \\otimes s|_Y \\in \\Gamma(Y; T^*Y \\otimes L|_Y)$ as required.\n\\end{proof}\nAs explained in \\cite{GualtieriPym13}, a $\\pi$-Poisson line bundle $(L,\\nabla)$ has several \\emph{residue maps}. Given $k \\geq 0$, with $Y := X_{\\pi,2k}$, these are sections ${\\rm Res}_k(\\nabla) \\in \\Gamma(Y,\\wedge^{2k+1} TY)$. They are locally given using a trivialization $s \\in \\Gamma(L)$, so that $\\nabla s = v \\otimes s$ for $v$ a local $\\pi$-Poisson vector field, by the expression\n\\begin{equation*}\n\t{\\rm Res}_k(\\nabla) = v \\wedge \\pi^k_Y.\n\\end{equation*}\nInvariantly, they are obtained by the composition\n\\begin{equation*}\n\t(\\cdot \\wedge \\sigma^k) \\circ \\nabla\\colon \\Gamma(L) \\to \\Gamma(TX \\otimes L) \\to \\Gamma(\\wedge^{2k+1}TX \\otimes L),\n\\end{equation*}\nwhich is then restricted to $Y$ using \\autoref{prop:degenlocusstrongpoisson} to give a morphism\n\\begin{equation*}\n\t\\Gamma(Y, L|_Y) \\to \\Gamma(Y, \\wedge^{2k+1}TY \\otimes L|_Y),\n\\end{equation*}\nfrom which the multiderivation ${\\rm Res}_k(\\nabla) \\in \\Gamma(Y, \\wedge^{2k+1} TY)$ is extracted. During this process it is used that $\\pi^{k+1}|_Y \\equiv 0$ because $Y$ is the $2k$th-degeneracy locus of $\\pi$. Applying the above to $L = \\det(T^*X)$ we obtain the \\emph{$k$th modular residue} ${\\rm Res}_{k,{\\rm mod}}(\\pi)$.\n\nIn \\autoref{sec:residuesympla} we briefly discuss these residue maps and their relation to those in \\autoref{sec:residuemaps} in the context of Poisson structures of nondegenerate $\\mathcal{A}$-type for a Lie algebroid $\\mathcal{A} \\to X$.\n\\subsection{The modular foliation}\n\\label{sec:modularfoliation}\nIn this section we discuss the modular foliation of a Poisson manifold, which was introduced in \\cite{GualtieriPym13}. Our main goal here is to prove \\autoref{thm:modfoliation} which discusses how the modular foliation interacts with the lifting procedure of \\autoref{sec:liftingpoisson}.\n\nLet $\\pi \\in {\\rm Poiss}(X)$ be given and let $(L,\\nabla) \\to X$ be a $\\pi$-Poisson line bundle. Then ${\\rm At}(L) \\to X$, its \\emph{Atiyah algebroid}, is a transitive Lie algebroid. There is a natural section $\\sigma_\\nabla \\in \\Gamma(\\wedge^2 {\\rm At}(L))$, which is in fact an ${\\rm At}(L)$-Poisson structure (\\cite[Corollary 5.3]{Polishchuk97}). It can be viewed as the natural Poisson structure on ${\\rm Tot}(L)$ induced by $\\pi$ and $\\nabla$. In our language, $\\sigma_\\nabla$ is a Poisson lift of the underlying Poisson structure $\\pi$. Due to this, $({\\rm At}(L), {\\rm At}^*(L), \\sigma_\\nabla)$ is a triangular Lie bialgebroid (c.f.\\ \\autoref{rem:apoissontriangbialgebroid}). In \\autoref{sec:poissondivtype} we studied cases where these were of divisor-type, but here instead the Lie algebroid ${\\rm At}(L)$ is transitive. We have the associated module $\\mathcal{F}_{\\sigma_\\nabla} = \\sigma_\\nabla^\\sharp(\\Gamma({\\rm At}(L)^*) \\subseteq \\Gamma({\\rm At}(L))$, which can be pushed down using $\\rho\\colon {\\rm At}(L) \\to TX$.\n\\begin{defn}[\\cite{GualtieriPym13}] Let $\\pi \\in {\\rm Poiss}(X)$ and $(L,\\nabla) \\to X$ a $\\pi$-Poisson line bundle. Then $\\mathcal{F}_{\\pi,(L,\\nabla)} := \\rho(\\mathcal{F}_{\\sigma_\\nabla}) \\subseteq \\Gamma(TX)$ is the singular foliation associated to $(L,\\nabla)$.\n\\end{defn}\nIt is immediate that the module $\\mathcal{F}_{\\pi,(L,\\nabla)}$ is involutive as $\\rho \\circ \\sigma_\\nabla^\\sharp$ is a Lie algebroid morphism.\n\\begin{defn}[\\cite{GualtieriPym13}] Let $\\pi \\in {\\rm Poiss}(X)$. Then the \\emph{modular foliation} $\\mathcal{F}_{\\pi,{\\rm mod}}$ is the singular foliation associated to the modular representation $L = \\det(T^*X)$ of $\\pi$ (see \\autoref{exa:modularreppoisson}).\n\\end{defn}\nLet us describe the modular foliation locally, as in \\cite{PymSchedler18}. Let $\\pi \\in {\\rm Poiss}(X)$ and let $\\mu$ be a local volume form. Then the associated modular vector field $V_{\\pi,\\mu}$ of $\\pi$ is characterized by\n\\begin{equation*}\n\t\\nabla(\\mu) = V_{\\pi,\\mu} \\otimes \\mu, \\qquad \\text{or} \\qquad \\mathcal{L}_{\\pi^\\sharp(df)} \\mu = (\\mathcal{L}_{V_{\\pi,\\mu}} f) \\cdot \\mu.\n\\end{equation*}\nMoreover, it is immediate that $V_{\\pi,\\mu} \\in {\\rm Poiss}(\\pi)$. Consequently, $V_{\\pi,\\mu}$ is always tangent to the degeneracy loci of $\\pi$ by \\autoref{prop:pidegenpipoisson}. The same computation showing that the characteristic class $\\theta_L$ is independent of local generators shows that if we change volume element to $\\mu' = f \\mu$ for some nonvanishing function $f$, we obtain that $V_{\\pi,\\mu'} = V_{\\pi,\\mu} - V_{\\log f}$, i.e.\\ the associated modular vector fields differ by a Hamiltonian vector field. As a result we have\n\\begin{equation*}\n\t\\mathcal{F}_{\\pi,{\\rm mod}} = \\mathcal{F}_\\pi \\cup \\{V_{\\pi,\\mu} \\, | \\, \\mu \\in {\\rm Vol}_{\\rm loc}(X)\\}.\n\\end{equation*}\nThe involutivity of $\\mathcal{F}_{\\pi,{\\rm mod}}$ locally becomes the computation that, given $f \\in C^\\infty(X)$, we have:\n\\begin{equation*}\n\t[V_{\\pi,\\mu}, V_f] = \\mathcal{L}_{V_{\\pi,\\mu}}(\\pi^\\sharp(df)) = \\pi^\\sharp(d \\mathcal{L}_{V_{\\pi,\\mu}} f) = V_{\\mathcal{L}_{V_{\\pi,\\mu}} f}.\n\\end{equation*}\nWe see that $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_{\\pi,{\\rm mod}}$. The leaves of the former are the $\\pi$-symplectic leaves, which are thus necessarily even-dimensional. The leaves of the modular folation come in two flavors:\n\\begin{itemize}\n\t\\item The even-dimensional leaves, where $V_{\\pi,{\\rm mod}}$ is tangent to the symplectic leaves;\n\t\\item The odd-dimensional leaves, where $V_{\\pi,{\\rm mod}}$ is transverse to the symplectic leaves.\n\\end{itemize}\nLet us record the consequence of \\autoref{prop:pidegpistrongpoisson} regarding strong Poisson submanifolds.\n\\begin{prop}\\label{prop:modvfielddegenloci} Let $\\pi \\in {\\rm Poiss}(X)$. Then any local modular vector field for $\\pi$ is tangent to all strong Poisson submanifolds of $\\pi$, hence in particular to all of its degeneracy loci.\n\\end{prop}\nThe following (combined with \\autoref{prop:atypepoissonsubmfd} showing $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_\\mathcal{A}$) is \\autoref{thm:intromodfoliation}.\n\\begin{thm}\\label{thm:modfoliation} Let $\\pi \\in {\\rm Poiss}(X)$ be of $\\mathcal{A}$-type. Then $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_{\\pi,{\\rm mod}} \\subseteq \\mathcal{F}_\\mathcal{A}$.\n\\end{thm}\n\\bp As noted, \\autoref{prop:atypepoissonsubmfd} shows that $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_\\mathcal{A}$, while we always have $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_{\\pi,{\\rm mod}}$. We are left with showing that all local modular vector fields are contained in $\\mathcal{F}_\\mathcal{A}$. Let $\\pi_{A} \\in {\\rm Poiss}(\\mathcal{A})$ be an $\\mathcal{A}$-lift of $\\pi$. Given local trivializing sections $\\nu \\in \\Gamma(\\det(\\mathcal{A}^*))$ and $\\mu \\in \\Gamma(\\det(T^*X))$, note that $\\nu \\otimes \\mu \\in  \\Gamma(\\det(\\mathcal{A}^*) \\otimes \\det(T^*X))$ can serve as a nonvanishing section for the modular representation $Q_{\\mathcal{A}^*_{\\pi_\\mathcal{A}}}$ of the Poisson algebroid $\\mathcal{A}^*_{\\pi_\\mathcal{A}}$. These now give rise to the modular section $v_{\\nu \\otimes \\mu} \\in \\Gamma(\\mathcal{A})$, but also to the $\\pi_\\mathcal{A}$-modular section $V_{\\pi_\\mathcal{A},\\nu} \\in \\Gamma(\\mathcal{A})$ and the $\\pi$-modular vector field $V_{\\pi,\\mu} \\in \\Gamma(TX)$. By \\cite[Equation (64)]{KosmannSchwarzbach00}, these are related by the equality\n\\begin{equation*}\n\t\\rho_\\mathcal{A}(v_{\\nu \\otimes \\mu} - V_{\\pi_\\mathcal{A},\\nu}) = V_{\\pi,\\mu}.\n\\end{equation*}\nThis shows that $V_{\\pi,\\mu} \\in \\mathcal{F}_\\mathcal{A}$, from which $\\mathcal{F}_{\\pi,{\\rm mod}} \\subseteq \\mathcal{F}_\\mathcal{A}$ follows.\n\\end{proof}\nIn \\autoref{cor:poissonhamiltonian} we showed that if $\\pi$ is of $I$-divisor-type, then any $\\pi$-Poisson vector field admits a $TX_I$-lift. This shows that $\\mathcal{F}_{\\pi,{\\rm mod}} \\subseteq \\Gamma(TX)_I = \\rho_{TX_I}(\\Gamma(TX_I))$ because any $\\pi$-modular vector field is $\\pi$-Poisson. Hence, it is not always necessary that $\\pi$ is of $\\mathcal{A}$-type.\n\\begin{rem} It is an interesting question to ask when the modular foliation $\\mathcal{F}_{\\pi,{\\rm mod}}$ is projective. Note that $\\mathcal{F}_\\pi$ is projective if and only if $\\pi$ is of $m$-divisor-type, i.e.\\ is almost-regular (\\autoref{cor:divtypealmostreg}). How is projectivity of $\\mathcal{F}_{\\pi,{\\rm mod}}$ related to projectivity of $\\mathcal{F}_\\pi$?\n\\end{rem}\n\\subsubsection{Relations between modular classes}\nThere are extensions of the above concepts to $\\mathcal{A}$-Poisson structures, using ideas contained in \\cite{CaseiroFernandes13,KosmannSchwarzbachLaurentGengouxWeinstein08,KosmannSchwarzbach00, KosmannSchwarzbach08,KosmannSchwarzbachLaurentGengoux05,KosmannSchwarzbachWeinstein05}. Recall from \\autoref{sec:aconnareps} that any Lie algebroid $\\mathcal{A} \\to X$ has a modular class ${\\rm Mod}(\\mathcal{A})$. As an $\\mathcal{A}$-Poisson structure turns $\\mathcal{A}^*$ into a Lie algebroid (\\autoref{defn:apoissonalgebroid}), there are various modular classes around. In particular, the modular class ${\\rm Mod}(\\pi_{\\mathcal{A}})$ of $\\pi_{\\mathcal{A}}$ is related to ${\\rm Mod}(\\pi)$ through the modular class of $\\mathcal{A}$: an $\\mathcal{A}$-Poisson structure $\\pi_{\\mathcal{A}}$ comes equipped with:\n\\begin{itemize}\n\t\\item A Lie algebroid morphism $\\pi_{\\mathcal{A}}^\\sharp\\colon \\mathcal{A}^*_{\\pi_{\\mathcal{A}}} \\to \\mathcal{A}$;\n\t\\item A Lie algebroid morphism $\\rho_{A}\\colon \\mathcal{A} \\to TX$;\n\t\\item A Lie algebroid morphism $\\pi^\\sharp\\colon T^*X_\\pi \\to TX$;\n\t\\item A Lie algebroid comorphism $\\rho_{A}^*\\colon \\mathcal{A}^*_{\\pi_{\\mathcal{A}}} \\dashrightarrow T^*X_\\pi$.\n\\end{itemize}\nThese are of course related by the lifting relation $\\pi^\\sharp = \\rho_{A} \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\rho_{A}^*$. As $\\rho_{A}^*$ is base-preserving, it can also be considered as a Lie algebroid morphism from $T^*X_\\pi$ to $\\mathcal{A}^*_{\\pi_\\mathcal{A}}$. Each of these four Lie algebroids has a modular class, and each morphism has a relative modular class \\cite{KosmannSchwarzbachWeinstein05}. These are related to (or define) the modular classes ${\\rm Mod}(\\pi_\\mathcal{A})$ and ${\\rm Mod}(\\pi)$ as follows:\n\\begin{itemize}\n\t\\item $2\\, {\\rm Mod}(\\pi_{\\mathcal{A}}) = {\\rm Mod}^{\\pi_{\\mathcal{A}}^\\sharp}(\\mathcal{A}^*_{\\pi_\\mathcal{A}},\\mathcal{A}) = {\\rm Mod}(\\mathcal{A}^*_{\\pi_\\mathcal{A}}) - (\\pi_\\mathcal{A}^\\sharp)^*({\\rm Mod}(\\mathcal{A}))$;\n\t\\item $2\\, {\\rm Mod}(\\pi) = {\\rm Mod}^{\\pi^\\sharp}(T^*X_\\pi, TX) = {\\rm Mod}(T^*X_\\pi) - (\\pi^\\sharp)^*({\\rm Mod}(TX)) = {\\rm Mod}(T^*X_\\pi)$;\n\t\\item ${\\rm Mod}^{\\rho_{A}}(\\mathcal{A},TX) = {\\rm Mod}(\\mathcal{A}) - \\rho_{A}^*({\\rm Mod}(TX)) = {\\rm Mod}(\\mathcal{A})$, because ${\\rm Mod}(TX) = 0$;\n\t\\item ${\\rm Mod}^{\\rho_{A}^*}(T^*X_\\pi, \\mathcal{A}^*_{\\pi_\\mathcal{A}}) = {\\rm Mod}(T^*X_\\pi) - \\rho_{A}({\\rm Mod}(\\mathcal{A}^*_{\\pi_{\\mathcal{A}}}))$, as $(\\rho_\\mathcal{A}^*)^* = \\rho_\\mathcal{A}$.\n\\end{itemize}\nThe equality $\\pi^\\sharp = \\rho_{A} \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\rho_{A}^*$ then implies the following relations:\n\t%\n\t\\begin{align*}\n\t\t2\\, {\\rm Mod}(\\pi) &= {\\rm Mod}^{\\pi^\\sharp}(T^*X_\\pi, TX) = {\\rm Mod}^{\\rho_{A} \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\rho_{A}^*}(T^*X_\\pi, TX)\\\\\n\t\t&= {\\rm Mod}^{\\rho_{A}^*}(T^*X_\\pi, \\mathcal{A}^*_{\\pi_\\mathcal{A}}) + \\rho_{A}({\\rm Mod}^{\\rho_\\mathcal{A} \\circ \\pi_\\mathcal{A}^\\sharp}(\\mathcal{A}^*_{\\pi_\\mathcal{A}}, TX))\\\\\n\t\t&= {\\rm Mod}^{\\rho_{A}^*}(T^*X_\\pi, \\mathcal{A}^*_{\\pi_\\mathcal{A}}) + \\rho_{A}({\\rm Mod}^{\\pi_\\mathcal{A}^\\sharp}(\\mathcal{A}_{\\pi_\\mathcal{A}}^*,\\mathcal{A}) + (\\pi_{A}^\\sharp)^*({\\rm Mod}^{\\rho_\\mathcal{A}}(\\mathcal{A},TX)))\\\\\n\t\t&= {\\rm Mod}^{\\rho_{A}^*}(T^*X_\\pi, \\mathcal{A}^*_{\\pi_\\mathcal{A}}) + \\rho_{A}(2\\,{\\rm Mod}(\\pi_\\mathcal{A}) + (\\pi_{A}^\\sharp)^*({\\rm Mod}(\\mathcal{A}))).\n\t\\end{align*}\nThis type of description can possibly be used in the discussion of liftability of unimodular Poisson structures to unimodular Lie algebroids.\n\\subsubsection{Examples of modular foliations}\nWe finish by describing several examples of modular foliations for Poisson structures of divisor-type, starting locally in dimension two.\n\\begin{exa} As in \\cite[Example 3.1]{PymSchedler18}, given $\\pi = f \\partial_x \\wedge \\partial_y$ on $(\\mathbb{R}^2,(x,y))$, its modular vector field associated to $\\mu = \\partial_x \\wedge \\partial_y$ is given by $V_{\\pi,\\mu} = (\\partial_x f) \\partial_y - (\\partial_y f) \\partial_x$. The symplectic foliation is given by the subset where $f$ is nonvanishing, and the points where $f$ vanishes. The modular vector field is tangent to the zero set of $f$, and vanishes exactly at its critical points.\n\\end{exa}\n\\begin{exa}[Nondegenerate] Let $\\pi \\in {\\rm Poiss}(X)$ be nondegenerate. Then $X$ is a symplectic leaf (or: its connected components). This shows that $\\mathcal{F}_\\pi = \\Gamma(TX)$, as is also immediate as $\\pi^\\sharp\\colon T^*X \\to TX$ is an isomorphism. The only degeneracy locus of $\\pi$ is $X$ itself, and any modular vector field must be tangent to it by \\autoref{prop:modvfielddegenloci}. Thus $\\mathcal{F}_{\\pi,{\\rm mod}} = \\mathcal{F}_{\\pi} = \\Gamma(TX)$. In fact, nondegenerate Poisson manifolds are unimodular, so that their modular class vanishes and there exists a global volume form (the Liouville form) which is preserved by Hamiltonian flows. For this volume form the modular vector field vanishes, which again shows that $\\mathcal{F}_\\pi = \\mathcal{F}_{\\pi, {\\rm mod}}$.\n\\end{exa}\n\\begin{exa}[Log-Poisson] Let $\\pi \\in {\\rm Poiss}(X)$ be a log-Poisson structure. Then $X$ and $Z$ are the only degeneraci loci of $\\pi$. From this it is immediate that any local modular vector field must be tangent to $Z$. However, in fact any local modular vector field is transverse to the symplectic leaves of the induced Poisson structure $\\pi_Z$ \\cite{GualtieriLi14,GuilleminMirandaPires14}. As a consequence we have that $\\mathcal{F}_{\\pi_Z,{\\rm mod}} = \\Gamma(TZ)$, while $\\mathcal{F}_{\\pi,{\\rm mod}} = \\Gamma(TX,TZ) = \\Gamma(\\mathcal{A}_Z)$. By \\autoref{exa:logpoissonlift} we know that $\\pi$ is of $\\mathcal{A}_Z$-type, so that \\autoref{thm:modfoliation} tells us that $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_{\\pi,{\\rm mod}} \\subseteq \\mathcal{F}_{\\mathcal{A}_Z} = \\Gamma(\\mathcal{A}_Z)$. We see that the second inclusion is an equality, and that $\\mathcal{F}_\\pi$ and $\\mathcal{F}_{\\pi, {\\rm mod}}$ are both projective. The symplectic foliation of $\\pi_Z$ is regular of corank-1, with tangencies $TF \\subseteq TZ$, so that $\\mathcal{F}_\\pi = \\Gamma(TX,TF)$.\n\\end{exa}\n\\begin{exa}[$b^k$-Poisson] Let $\\pi \\in {\\rm Poiss}(X)$ be a $b^k$-Poisson structure. Then the behavior of $\\pi$ is similar for all $k \\geq 1$, and in particular to log-Poisson structures (where $k = 1$). Again the only degeneraci loci of $\\pi$ are $X$ and $Z$, but if $k > 1$, then any modular vector field is tangent to the symplectic leaves of $\\pi_Z$. If locally $\\pi = z^k \\partial_z \\wedge \\partial_{x_1} + \\omega_0^{-1}$ as in \\autoref{exa:bkpoissondarboux}, then $V_{\\pi,\\mu} = k z^{k-1} \\partial_{x_1}$ with respect to $\\mu = dz \\wedge dx_i$. In terms of this local description, we then have\n\\begin{equation*}\n\t\\mathcal{F}_\\pi = \\langle z^k \\partial_z, z^k \\partial_{x_1} \\rangle \\oplus \\Gamma(TF), \\quad \\mathcal{F}_{\\pi,{\\rm mod}} = \\langle z^k \\partial_z, z^{k-1} \\partial_{x_1} \\rangle \\oplus \\Gamma(TF), \\quad \\mathcal{F}_{\\mathcal{A}_Z^k} = \\langle z^k \\partial_z, \\partial_{x_1} \\rangle \\oplus \\Gamma(TF),\n\\end{equation*}\n where $TF \\subseteq TX$ are the tangencies to the corank-one symplectic foliation of $\\pi_Z \\in {\\rm Poiss}(X)$.\n\\end{exa}\nThe modular foliations of elliptic and elliptic-log Poisson manifolds are described in \\cite{KlaasseLanius18,KlaasseLi18}.\n\\begin{exa}[$m$-Log-Poisson]\\label{exa:modfolmlog} Let $\\pi \\in {\\rm Poiss}(X)$ be $m$-log-Poisson with associated regular distribution $D$. Then the degeneracy loci of $\\pi$ are $X$ and $Z$, so that any local modular vector field must be tangent to $Z$. However, as $\\pi$ is of $\\mathcal{D}$-type by \\autoref{prop:mdivtypelift}, it must further be tangent to $D$ due to \\autoref{thm:modfoliation}. This suggests that the modular vector fields can detect how the orbits of $\\mathcal{D}$ meet $Z$. In the local examples of \\autoref{rem:mlogpoissonlocal} on $(\\mathbb{R}^3,(x,y,z))$, with respect to the volume form $\\mu = dx \\wedge dy \\wedge dz$, the modular vector fields of $\\pi_1$, $\\pi_2$ and $\\pi_3$ are:\n\\begin{equation*}\n\tV_{\\pi_1,\\mu} = \\partial_y, \\quad V_{\\pi_2,\\mu} = 0, \\quad V_{\\pi_3,\\mu} = - 2x \\partial_y.\n\\end{equation*}\nFrom this we see that the associated modules $\\mathcal{F}_{\\pi_i}$ and $\\mathcal{F}_{\\pi_i, {\\rm mod}}$ are given by\n\\begin{align*}\n\t&\\mathcal{F}_{\\pi_1} = \\langle x \\partial_x, x \\partial_y \\rangle, \\quad \\mathcal{F}_{\\pi_2} = \\langle z \\partial_x, z \\partial_y \\rangle, \\quad \\mathcal{F}_{\\pi_3} = \\langle (z-x^2) \\partial_x, (z-x^2)\\partial_y \\rangle,\\\\\n\t&\\mathcal{F}_{\\pi_1,{\\rm mod}} = \\langle x \\partial_x, \\partial_y \\rangle, \\quad \\mathcal{F}_{\\pi_2,{\\rm mod}} = \\langle z \\partial_x, z \\partial_y \\rangle \\quad \\mathcal{F}_{\\pi_3,{\\rm mod}} = \\langle (z-x^2) \\partial_x, z \\partial_y, x \\partial_y. \\rangle\n\\end{align*}\n\\end{exa}\nIn general, the modular foliation of an almost-regular Poisson manifold behaves as follows: in \\autoref{sec:liftingalmostreg} we saw that if $\\pi \\in {\\rm Poiss}(X)$ is of $m$-$I$-divisor-type with associated involutive distribution $D = D_\\pi$, then $\\pi$ is of both $D$- and $TX_I$-type (assuming projectivity of $I$). Then, \\autoref{thm:modfoliation} implies that $\\mathcal{F}_{\\pi, {\\rm mod}} \\subseteq \\Gamma(\\mathcal{D}_\\pi)$ and $\\mathcal{F}_{\\pi, {\\rm mod}} \\subseteq \\Gamma(TX_I) = \\Gamma(TX)_I$. In particular any modular vector field must be tangent to $D$ and to the subset $Z_I \\subseteq X$ (if it is smooth).\n\\subsection{Residues for symplectic Lie algebroids}\n\\label{sec:residuesympla}\nIn this section we discuss some of the consequences of the fact that symplectic Lie algebroids $\\mathcal{A}$ have residue maps, which can be applied to an $\\mathcal{A}$-symplectic form $\\omega_{\\mathcal{A}}$ or its powers. We recall again that residue maps, while often chain maps, typically do not respect the ring structure coming from the wedge product. We will discuss this process in more detail in \\cite{Klaasse18} in the context of Dirac geometry, yet show here what happens in the case of $b^k$-Poisson or elliptic Poisson structures.\n\nWe will need the following definitions (\\cite[Definitions 3.2.20 and 3.2.21]{OsornoTorres15}).\n\\begin{defn} Let $M^m$ be a manifold equipped with a $(m-2n = 2\\ell)$-regular foliation $\\mathcal{F}$.\n\t%\n\t\\begin{itemize}\n\t\\item an \\emph{$n$-cosymplectic tuple} on $M$ is a tuple $(\\alpha_1, \\dots, \\alpha_n, \\beta)$ of forms, where $\\beta \\in \\Omega^2_{\\rm cl}(M)$ has constant rank $2\\ell$, $\\alpha_i \\in \\Omega^1_{\\rm cl}(M)$ for all $i$, and $\\alpha_1 \\wedge \\dots \\wedge \\alpha_n \\wedge \\beta^{\\ell} \\neq 0$;\n\t\\item an \\emph{$n$-Poisson tuple} on $M$ is a tuple $(V_1, \\dots V_n, \\pi)$, where $\\pi \\in {\\rm Poiss}(M)$ is regular of rank $2\\ell$, $V_i \\in \\Gamma(TM)$ are pairwise commuting $\\pi$-Poisson vector fields, and $V_1 \\wedge \\dots V_n \\wedge \\pi^\\ell \\neq 0$.\n\t\\end{itemize}\n\t%\n\tWe say the tuple is \\emph{adapted to $\\mathcal{F}$} if $T\\mathcal{F}$ is spanned by (the kernel of) the $v_i$ (respectively $\\alpha_i$).\n\\end{defn}\nWhen $n=1$ the above is simply referred to as a \\emph{cosymplectic} or \\emph{corank-$1$ Poisson tuple}. The following correspondence is in full analogy with the case $n=1$ contained in \\cite{GuilleminMirandaPires11}.\n\\begin{lem}[{\\cite[Lemma 3.2.29]{OsornoTorres15}}]\\label{lem:oneonecorrespondence} Let $(M,\\mathcal{F})$ be a manifold with a corank-$2n$ foliation. There is a one-to-one correspondence between $\\mathcal{F}$-adapted $n$-cosymplectic and $n$-Poisson tuples.\n\\end{lem}\nWe wish to reinterpet these structures from a Dirac geometric point of view. Recall that a \\emph{Dirac structure} $E \\subseteq \\mathbb{T}M$ is an involutive Lagrangian subbundle of the standard Courant algebroid structure on $\\mathbb{T}M = TM \\oplus T^*M$. A Dirac structure can equivalently be described by a \\emph{pure spinor line}, as follows (see \\cite{AlekseevBursztynMeinrenken09,Gualtieri11}). Sections $v = V + \\xi \\in \\Gamma(\\mathbb{T}M)$ act on differential forms via Clifford multiplication, given by $v \\cdot \\rho = \\iota_V \\rho + \\xi \\wedge \\rho$ for $\\rho \\in \\Omega^\\bullet(M)$. With this in mind, a pure spinor line is a line bundle $K_E \\subseteq \\wedge^\\bullet T^*M$ that is pointwise generated by a pure spinor $\\rho = e^B \\wedge \\Omega$ with $B$ a two-form and $\\Omega$ a decomposable $k$-form, such that for any nonzero local section $\\rho \\in \\Gamma(K_E)$ we have $d \\rho = v \\cdot \\rho$ for some $v \\in \\Gamma(\\mathbb{T}M)$. The correspondence  is then given by the fact that $E = {\\rm Ann}(K_E)$ is the annihilator of $K_E$ under the Clifford action.\n\\begin{prop} Let $M^m$ be a manifold. Then an $n$-cosymplectic tuple $(\\alpha_1,\\dots,\\alpha_n,\\beta)$ defines a global pure spinor $\\rho := e^\\beta \\wedge (\\alpha_1 \\wedge \\dots \\wedge \\alpha_n)$ for which $\\rho_m \\neq 0 \\in \\Gamma(\\wedge^m T^*M)$. \n\\end{prop}\nWe call the associated pure spinor line $K_\\rho = \\langle \\rho \\rangle$ an \\emph{$n$-cosymplectic structure} on $M$.\n\\bp This is almost immediate. Note that $\\rho_m = \\alpha_1 \\wedge \\dots \\wedge \\alpha_n \\wedge \\beta^\\ell$, which is nonvanishing. Letting $\\Omega := \\alpha_1 \\wedge \\dots \\wedge \\alpha_n$, note that because $\\beta$ and each $\\alpha_i$ is closed we have\n\\begin{equation*}\n\td \\rho = d(e^\\beta \\wedge \\Omega) = d\\beta \\wedge \\rho + e^\\beta \\wedge d\\Omega = 0 = 0 \\cdot \\rho.\n\\end{equation*}\nThe Dirac structure associated to $K_\\rho = \\langle \\rho \\rangle$ is given by $e^\\beta( T\\mathcal{F}) = \\{v + \\beta^\\flat(v) \\, | \\, v \\in \\Gamma(T\\mathcal{F})\\}$.\n\\end{proof}\nFrom the Dirac perspective, the intrinsic object is the pure spinor line $K_\\rho$, instead of $\\rho$ itself. However, $n$-cosymplectic structures have a global generating pure spinor.  In general, this still means that the one-forms $\\alpha_i$ need not exist separately, but only their associated volume form $\\alpha_1 \\wedge \\dots \\wedge \\alpha_n \\in \\Gamma(\\det(T\\mathcal{F})^*)$ (making $\\mathcal{F}$ coorientable). Their separate existence distinguishes $n$-cosymplectic structures from other symplectic foliations (such foliations are called \\emph{unimodular}). Similarly, the two-form $\\beta$ should be viewed as an extension to $M$ of a symplectic form on $\\mathcal{F}$. However, $n$-cosymplectic structures are such that the symplectic form on $\\mathcal{F}$ admits a closed global extension (such symplectic foliations are called \\emph{strong}). Thus, an $n$-cosymplectic structure determines a coorientable and unimodular corank-$2n$ strong symplectic foliation, whose associated pure spinor line has a global generating pure spinor.\n\nWe now discuss how these structures arise in the context of log-Poisson structures, and of elliptic Poisson structures with zero elliptic residue. To start, recall that log-Poisson structures admit nondegenerate lifts to the log-tangent bundles that they define (\\autoref{exa:logpoissonlift}). In other words, $\\pi \\in {\\rm Poiss}(X)$ of $I_Z$-divisor-type has an associated dual $\\mathcal{A}_Z$-symplectic structure $\\omega$.\n\\begin{prop}\\label{prop:logcosymplectic} Let $(X^{2n},Z,\\pi,\\omega)$ be a log-Poisson manifold. Then\n\\begin{equation*}\n\t\\rho := {\\rm Res}_Z(e^\\omega) = {\\rm Res}_Z\\left(\\sum_{k=1}^n \\frac{1}{k!}  \\omega^k\\right) \\in \\Omega^{\\bullet}_{\\rm cl}(Z)\n\\end{equation*}\nforms a $1$-cosymplectic global pure spinor on $Z$, hence defines a $1$-cosymplectic structure. In particular, $Z$ inherits a unimodular corank-$1$ symplectic foliation.\n\\end{prop}\nWe are asserting here that the residue map should be applied to the pure spinor $e^\\omega$ associated to the $\\mathcal{A}_Z$-Dirac structure defined by $\\omega$ (see \\cite{Klaasse18} for more discussion on this process).\n\\bp Because ${\\rm Res}_Z$ is a cochain morphism (due to \\autoref{thm:residues}), we have\n\\begin{equation*}\n\td\\rho = d \\, {\\rm Res}_Z\\left(\\sum_{k=1}^n \\frac{1}{k!} \\omega^k\\right) = {\\rm Res}_Z\\left(\\sum_{k=1}^n \\frac{1}{k!} d (\\omega^k)\\right) = 0.\n\\end{equation*}\n Let $z$ be a local defining function for $\\iota_Z\\colon Z \\hookrightarrow X$, so that locally $\\omega = d\\log z \\wedge \\widetilde{\\alpha} + \\widetilde{\\beta}$. Then \n\\begin{equation*}\n\t\\omega^k = k (d\\log z \\wedge \\widetilde{\\alpha} + \\widetilde{\\beta}) \\wedge \\widetilde{\\beta}^{k-1}, \\qquad 1 \\leq k \\leq n.\n\\end{equation*}\nThus ${\\rm Res}_Z(\\omega^k) = k \\alpha \\wedge \\beta^{k-1} \\neq 0$ for $\\alpha := {\\rm Res}_Z(\\omega) = \\iota_Z^*(\\widetilde{\\alpha})$ and $\\beta := \\iota_Z^*(\\widetilde{\\beta})$ (see \\autoref{exa:bkersiduemaps}). Both $\\rho$ and $\\alpha$ are invariantly defined closed forms on $Z$, and $\\rho$ is a global pure spinor. We see that $\\rho$ is locally given by\n\\begin{equation*}\n\t\\rho = \\alpha \\wedge e^\\beta.\n\\end{equation*}\nMoreover, $\\rho_{2n-1} = \\frac{1}{(n-1)!} \\alpha \\wedge \\beta^{n-1}$, which is nonzero due to nondegeneracy of $\\omega$ (i.e.\\ $\\omega^n \\neq 0$).\n\\end{proof}\n\\begin{rem} \\autoref{prop:logcosymplectic} is the intrinsic reformulation of \\cite[Proposition 10]{GuilleminMirandaPires14}, that log-Poisson structures carry an equivalence class of cosymplectic structures $(\\alpha,\\beta)$, where $(\\alpha,\\beta) \\sim (\\alpha',\\beta')$ if $\\alpha = \\alpha'$ and $\\beta' = \\beta + \\alpha \\wedge df$ for $f \\in C^\\infty(Z)$. There they note that the one-form $\\alpha$ is determined invariantly, and here this is immediate because it arises from the logarithmic residue map ${\\rm Res}_Z$. Any choice of (local) defining function for $Z$ picks out a representative for $\\beta$, but the associated pure spinor $\\rho$ is the invariant object, globally on $Z$. This is again because it is obtained by (repeated) application of the logarithmic residue.\n\\end{rem}\nBy repeating the proof of \\autoref{prop:logcosymplectic} essentially verbatim, we obtain the following invariant statement about the induced geometry on the degeneracy loci of $b^k$-Poisson structures, which lead to symplectic Lie algebroids by \\autoref{exa:bkpoissonlift} (compare this with \\cite{Scott16,GuilleminMirandaWeitsman17}).\n\\begin{prop}\\label{prop:bkpoissoncosymplectic} Let $(X,Z,\\pi)$ be a $b^k$-Poisson manifold. Then $Z$ carries a $1$-cosymplectic global pure spinor, so that it inherits a $1$-cosymplectic structure.\n\\end{prop}\nThere is a similar result for elliptic Poisson manifolds, as follows. Due to \\autoref{exa:ellpoissonlift}, an elliptic Poisson structure $\\pi \\in {\\rm Poiss}(X)$ (with divisor ideal $I_{|D|}$) is of nondegenerate $\\mathcal{A}_{|D|}$-type, with dual $\\mathcal{A}_{|D|}$-symplectic structure $\\omega$. We will focus on the case where ${\\rm Res}_q(\\omega) = 0$, which are those with zero elliptic residue, or \\emph{zero elliptic Poisson structures}. We will moreover require that $D$ is cooriented (which is immediate if $\\pi$ stems from a stable generalized complex structure \\cite{CavalcantiGualtieri18}). As mentioned in \\autoref{exa:elltangentresidues}, this trivializes the isotropy line $\\mathfrak{k} \\to D$, so that the residue one-forms ${\\rm Res}_r(\\omega) \\in \\Omega^1(D)$ and ${\\rm Res}_\\theta(\\omega) \\in \\Omega^1(D)$ are both closed.\n\\begin{prop}\\label{prop:elliptic2cosymplectic} Let $(X^{2n},|D|,\\pi,\\omega)$ be a cooriented zero elliptic Poisson manifold. Then\n\\begin{equation*}\n\t\\rho := {\\rm Res}_q(e^\\omega) = -{\\rm Res}_r(\\omega) \\wedge {\\rm Res}_{\\theta}(\\omega) + {\\rm Res}_q \\left(\\sum_{k=3}^{n} \\frac1{k!} \\omega^k \\right) \\in \\Omega^\\bullet_{\\rm cl}(D)\n\\end{equation*}\nforms a $2$-cosymplectic global pure spinor on $D$, hence defines a $2$-cosymplectic structure. In particular, $D$ inherits a unimodular corank-$2$ symplectic foliation.\n\\end{prop}\nThere are extensions of this statement when $D$ is not cooriented (see \\cite{KlaasseLanius18,KlaasseLi18}). Moreover, this result is known in the context of stable generalized complex geometry due to \\cite{BaileyCavalcantiGualtieri17,CavalcantiGualtieri18}. There are three residue maps, namely ${\\rm Res}_r$, ${\\rm Res}_\\theta$ and ${\\rm Res}_q$, so that the pure spinor above differs slightly from \\autoref{prop:logcosymplectic}. We have that ${\\rm Res}_r(\\omega) \\wedge {\\rm Res}_\\theta(\\omega) = -{\\rm Res}_q(\\omega^2)$. As for \\autoref{prop:logcosymplectic} we apply ${\\rm Res}_q$ to the pure spinor $e^\\omega$ defining the $\\mathcal{A}_{|D|}$-symplectic structure.\n\\bp Let $(r,\\theta)$ be local normal polar coordinates to $\\iota_D\\colon D \\hookrightarrow X$ in which $I_{|D|} = \\langle r^2 \\rangle$. Then locally $\\omega = d\\log r \\wedge \\widetilde{\\alpha}_1 + d\\theta \\wedge \\widetilde{\\alpha}_2 + \\widetilde{\\beta}$. There is no term involving $d\\log r \\wedge d\\theta$ because ${\\rm Res}_q(\\omega) = 0$. We have $\\alpha_1 := {\\rm Res}_r(\\omega) = \\iota_D^*(\\widetilde{\\alpha}_1)$ and $\\alpha_2 := {\\rm Res}_\\theta(\\omega) = \\iota_D^*(\\widetilde{\\alpha}_2)$. These are both closed because the residue maps ${\\rm Res}_r$ and ${\\rm Res}_\\theta$ are cochain morphisms. We have for $2 \\leq k \\leq n$ that\n\\begin{equation*}\n\t\\omega^k = (k d\\log r \\wedge \\widetilde{\\alpha}_1 \\wedge \\beta + k d\\theta \\wedge \\widetilde{\\alpha}_2 \\wedge \\beta + k(k-1) d \\log r \\wedge \\widetilde{\\alpha}_1 \\wedge d\\theta \\wedge \\widetilde{\\alpha}_2 + \\widetilde{\\beta}^2) \\wedge \\widetilde{\\beta}^{k-2}.\n\\end{equation*}\nNote here that $n \\geq 2$ is forced by nondegeneracy and the condition that ${\\rm Res}_q(\\omega) = 0$. Thus\n\\begin{equation*}\n\t{\\rm Res}_q(\\omega^k) = -k (k-1) \\alpha_1 \\wedge \\alpha_2 \\wedge \\beta^{k-2}, \\qquad 2 \\leq k \\leq n,\n\\end{equation*}\nwhere $\\beta := \\iota_D^*(\\widetilde{\\beta})$. In particular this shows that ${\\rm Res}_q(\\omega^2) = -{\\rm Res}_r(\\omega) \\wedge {\\rm Res}_\\theta(\\omega)$. Further, $\\rho$ is closed because ${\\rm Res}_q$ is a cochain morphism. Hence $\\rho$, $\\alpha_1$ and $\\alpha_2$ are invariantly defined forms on $D$, and $\\rho$ is a $2$-cosymplectic global pure spinor. We see that $\\rho$ is locally given by\n\\begin{equation*}\n\t\\rho = -\\alpha_1 \\wedge \\alpha_2 \\wedge e^\\beta.\n\\end{equation*}\nMoreover, we have that $\\rho_{2n-2} = -\\frac{1}{(n-2)!} \\alpha_1 \\wedge \\alpha_2 \\wedge \\beta^{n-2} \\neq 0$, due to nondegeneracy of $\\omega$.\n\\end{proof}\nThere is also a result for elliptic Poisson structures $\\pi$ whose elliptic residue is nonzero, in other words, for \\emph{nonzero elliptic Poisson structures} (for which $D$ is cooriented). Namely, in this case $D$ is a symplectic leaf of $\\pi$. See \\cite{KlaasseLanius18,KlaasseLi18} for more information.\n\nNote that in the presence of $n$ rational one-forms $\\alpha_i \\in \\Omega^1(M)$ whose cohomology classes are linearly independent, Tischler's theorem \\cite{Tischler70} implies that $M$ naturally fibers over $T^n$. An $n$-cosymplectic tuple whose one-forms satisfy this condition is called \\emph{proper}. Any $n$-cosymplectic tuple on a compact manifold can be perturbed slightly to be proper. Call a Poisson structure \\emph{proper} if its induced geometric structure on its degeneracy locus is proper. We obtain the following result for cooriented zero elliptic Poisson manifolds. This is the direct Poisson geometric analogue results in \\cite{BaileyCavalcantiGualtieri17} and a preliminary version of \\cite{CavalcantiGualtieri18} in the setting of generalized complex geometry.\n\\begin{prop} Let $(X,|D|,\\pi)$ be a cooriented zero elliptic Poisson manifold. Then $D$ carries a fibration over $T^2$, which can be made into a symplectic fibration after slight perturbation. Moreover, $\\pi$ can be perturbed through zero elliptic Poisson structures to be proper.\n\\end{prop}\n\\begin{rem} A similar result holds for log-Poisson manifolds (or $b^k$-Poisson) (\\cite{GuilleminMirandaPires14,MarcutOsornoTorres14,Cavalcanti17}): by \\autoref{prop:logcosymplectic} the degeneracy locus $Z$ carries a $1$-cosymplectic structure, hence fibers over $S^1$. This can be perturbed to be proper, making $Z$ a symplectic mapping torus. In \\cite{Cavalcanti17,MarcutOsornoTorres14} it is shown that $\\pi$ can be perturbed through log-Poisson structures so that $Z$ is proper. The argument used there proves essentially verbatim that the same holds for $b^k$-Poisson structure.\n\\end{rem}\n\\subsubsection{Relation with the Poisson residue}\nThere is a relation between the Lie algebroid residue maps that were used above, and the modular Poisson residues of \\autoref{sec:poissonmodulesmodfoliation} due to \\cite{GualtieriLi14,GualtieriPym13}.\n\nLet $\\mathcal{A} \\to X^{2n}$ be a Lie algebroid of smooth divisor-type and let $\\pi \\in {\\rm Poiss}(X)$ be of nondegenerate $\\mathcal{A}$-type with dual $\\mathcal{A}$-symplectic structure $\\omega$. This forces $I_\\pi = I_\\mathcal{A}$ by \\autoref{prop:apoissondivtypelifts}, and it ensures that $Z_\\mathcal{A} = Z_{I_\\mathcal{A}}$ is both the degeneraci locus of $\\mathcal{A}$, and the $(2n-2)$nd degeneracy locus of $\\pi$. Assuming that the restriction of $\\mathcal{A}$ to $Z_\\mathcal{A}$ is transitive with unimodular isotropy, and that $Z_\\mathcal{A}$ is of codimension-one, the residues ${\\rm Res}_{Z_\\mathcal{A}}(\\omega^n) \\in \\Omega^{2n-1}(Z_\\mathcal{A})$ and ${\\rm Res}_{n-1,{\\rm mod}}(\\pi) \\in \\mathfrak{X}^{2n-1}(Z_\\mathcal{A})$ are dual to each other in some sense. In particular, this happens for $b^k$-Poisson structures and their associated $\\mathcal{A}_Z^k$-symplectic structure. From \\autoref{prop:bkpoissoncosymplectic} we see that ${\\rm Res}_Z(\\omega^n) = \\alpha \\wedge \\beta^{n-1}$ if locally $\\omega = dz\/z^k \\wedge \\widetilde{\\alpha} + \\widetilde{\\beta}$ for $z \\in j_{k-1}$. On the other hand, we have that ${\\rm Res}_{n-1,{\\rm mod}}(\\pi) = v \\wedge \\pi_Z^{n-1}$ for $v$ a local modular vector field. Note that $\\beta$ is not globally defined, and nor is $v$. However, their pure spinors make sense, and define the same Dirac structure (the forms we used above as spinors are also called contravariant spinors, with multivector fields being covariant spinors, see \\cite{AlekseevBursztynMeinrenken09,Klaasse18}). See also the discussions in \\cite[Proposition 1.8]{GualtieriLi14} and \\cite[Remark 7.3]{GualtieriPym13}. In general the relation between the Poisson modular residue and the residue maps of the associated symplectic Lie algebroid can be more involved. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nIn the Minimal Supersymmetric Standard Model (MSSM)~\\cite{R1}, if the mixing\nbetween third generation squarks is large, stops\/sbottoms can be rather\nlight~\\cite{qmix} and at the same time, their coupling to Higgs bosons can \nbecome substantial.  This might have a rather large impact on the phenomenology\nof the MSSM Higgs bosons~$^{\\rm 3-7}$. More precisely, concentrating on \nscalar top quarks, their mass eigenvalues are given by\n\\begin{equation}\nm_{\\tilde{t}_{1,2}}^2 = m_t^2 + \\frac{1}{2} \\left[ m_{\\tilde Q_L}^2 + m_{\\tilde\nt_R}^2 +\\cdots \\mp \\sqrt{ (m_{\\tilde Q_L}^2 - m_{\\tilde t_R}^2 +\\cdots)^2 + \n4m_t^2 \\tilde{A}_t^2 } \\right]\\;, \n\\end{equation}\nwhere $m_{\\tilde Q_L}$, $m_{\\tilde t_R}$ are the soft-SUSY breaking scalar \nmasses and the dots stand for the $D$--terms $\\propto M^2_Z \\cos 2\\beta$.  In the \ndecoupling limit (the lightest $h$ boson is Standard Model like and the other \nbosons $A,H$ and $H^\\pm$, are very heavy), the expressions of the coupling $h\n\\tilde{t_1} \\tilde{t_1}$ simply reads ($\\theta_t$ is the mixing angle and $s_W\n\\equiv \\sin\\theta_W$)\n\\begin{equation}\ng_{h \\tilde{t}_1 \\tilde{t}_1 } = \\cos 2\\beta \\left[ \\frac{1}{2} \\cos^2 \\theta_t\n- \\frac{2}{3} s^2_W \\cos 2 \\theta_t \\right] + \\frac{m_t^2}{M_Z^2} + \\frac{1}{2}\n\\sin 2\\theta_t \\frac{m_t \\tilde{A}_t } {M_Z^2}\\;.  \\label{ghtt}\n\\end{equation}\nLarge values of $\\tilde{A}_t\\equiv A_t -\\mu\/ \\tan\\beta$ lead to \n$g_{h \\tilde{t}_1 \\tilde{t}_1} \\sim \\tilde{A}_t$  \nand to an almost maximal $\\tilde{t}$ \nmixing angle, $|\\sin 2 \\theta_t| \\simeq 1$, in particular\nif $m_{\\tilde Q_L} \\simeq m_{\\tilde t_R}$.  \nThe measurement of this important coupling would open a\nwindow to probe directly some of the soft--SUSY breaking terms of the\npotential. To measure Higgs--squarks couplings directly, one needs to consider\nthe three--body associated production of Higgs bosons with scalar quark\npairs~$^{\\rm 4-7}$. This is the supersymmetric analog to the processes\nof Higgs boson radiation from top quark lines~\\cite{ppttH,eetth} which allows \nto probe the $t\\bar{t}$--Higgs Yukawa coupling directly. [At the LHC, the \nprocess $pp \\to t\\bar{t}+$Higgs~\\cite{ppttH} although not competitive with the \ngluon fusion mechanism~\\cite{R7}, can provide a complementary signal since \nbackgrounds are smaller.] \n\nHere, we report on the production of a light Higgs boson $h$ in association\nwith top squarks at future $e^+e^-$ linear machines~\\cite{Xtth}, including the\n$\\gamma \\gamma$ option, both in the unconstrained MSSM and minimal SUGRA cases.\nFor simplicity, we work in the approximation of being close to the decoupling \nlimit, which  implies that we do not consider other Higgs production processes \nwith e.g. the heavy $H$ or $A$ bosons produced in association \nwith top squarks. For large masses, these processes will be suppressed by \nphase--space well before the decoupling regime is reached.\n\\section{Associated production at e$^+$e$^-$ colliders} \nAt future linear $e^+e^-$ colliders, the final state $\\tilde{t}_1 \\tilde{t}_1 h$\nmay be generated in three ways: $(i)$ two--body production of a mixed pair\nof top squarks and the decay of the heaviest stop to the lightest one and a \nHiggs boson, $(ii)$ the continuum production in $e^+e^-$ annihilation $e^+e^- \\rightarrow\n\\tilde{t}_1 \\tilde{t}_1h$ and $(iii)$  the continuum production in \n$\\gamma \\gamma$ collisions $\\gamma \\gamma \\rightarrow \\tilde{t}_1 \\tilde{t}_1h$.\n\n\\vspace*{-4mm}\n\n\\subsection{Two--body production and decay} \nThe total cross section~\\cite{R8} for the process $(i)$ should, in principle, \nbe large enough for the final state to be copiously produced. However, $\\sigma( \ne^+e^- \\rightarrow \\tilde{t}_1 \\tilde{t}_2$) involves the $Z\\tilde{t}_1 \\tilde{t}_2$ \ncoupling, proportional to $\\sin 2 \\theta_t$, while the $\\vert g_{h \\tilde{t}_1 \n\\tilde{t}_2}\\vert $ coupling  in most of the parameter space is proportional \nto $\\cos 2\\theta_t$, such that the cross section times branching ratio will be \nvery small in the no mixing [$\\theta_t \\sim 0$] and maximal mixing $[|\\theta_t| \n\\sim \\pi\/4]$ cases. \n[In addition, the decay width $\\tilde{t}_2 \\rightarrow h\\tilde{t}_1$ is \nin general much smaller~\\cite{R8} than the $\\tilde{t}_2$ decay widths into \nchargino and neutralinos]. Nevertheless, there are regions of the MSSM \nparameter space where\nthe combination $\\sin 2 \\theta_t \\times \\cos 2 \\theta_t$ can be large; this\noccurs typically for a not too small $m_{\\tilde t_L}$--$m_{\\tilde t_R}$ \nsplitting and moderate $\\tilde A_t$ values.\\cite{Xtth} In this case, which is \noften realized in the mSUGRA scenario, this mechanism is visible for the \nexpected high luminosities\\cite{TESLA} $\\int {\\cal L}{\\rm d}t \\sim 500$ fb$^{-1}$.  \n\nThis is illustrated in Fig.~1, where the cross section $e^+e^- \\rightarrow \\tilde{t}_1 \n\\tilde{t}_2$ times the branching ratio BR($\\tilde{t}_2 \\rightarrow \\tilde{t}_1 h)$ is \nshown as a function of the $\\tilde{t}_1$ mass at a c.m. energy of $\\sqrt{s}\n=800$ GeV. We have chosen a mSUGRA scenario with $\\tan\\beta=30$, $m_{1\/2} =$ \n100 GeV, $A_0 = -600$ GeV and sign$(\\mu)= +$. (The dotted lines show the \ncontribution of the  non--resonant process, discussed below, for the same \ninput choice). The cross section can reach the level of 1 fb for relatively\nsmall $m_{\\tilde{t}_1}$ values, leading to thousand events in a few years, for \n$\\int {\\cal L}{\\rm d}t \\sim 500$ fb$^{-1}$. \n\\begin{figure}[htb]\n\\vspace{-.5cm}\n\\begin{center}\n\\mbox{\n\\psfig{figure=ee_sug_1.eps,width=10cm}}\n\\end{center}\n\\vspace{-9.cm}\n\\caption[]{The production cross section $\\sigma (e^+e^- \\rightarrow \\tilde{t}_1 \n\\tilde{t}_1 h$) [in fb] as a function of $m_{\\tilde{t}_1}$ in the mSUGRA \ncase; $\\tan\\beta=$ 30, $m_{1\/2} =$ 100 GeV and $A_0 =-600$ GeV.}\n\\end{figure}\n\\subsection{Production in the continuum in e$^+$e$^-$ collisions}\nThe cross section for the process $e^+e^- \\rightarrow \\tilde{q}_i \\tilde{q}_i \\Phi$ with \n$\\Phi$ the CP--even Higgs boson $h$ or $H$, and $\\tilde{q}_i$ any of the two \nsquarks has been calculated in \\cite{Xtth}.\nWe show in Fig.~2 the rates for the $\\tilde{t}_1 \\tilde{t}_1 h$ final state  as\na function of the $\\tilde{t}_1$ mass in the unconstrained MSSM, at\n$\\sqrt{s}=800$ GeV.  For not too large $\\tilde{t}_1$ masses and large values of\nthe parameter $\\tilde A_t$, the production cross sections can exceed  \n$1$ fb, to be compared to the\nSM--like process $e^+e^- \\rightarrow t \\bar{t}h$ \\cite{eetth} of the order of 2 fb for\n$M_h \\sim 130$ GeV.  This provides more than one thousand events in a few\nyears, with a luminosity  $\\int {\\cal L}{\\rm d}t \\sim 500$ fb$^{-1}$, which should\nbe sufficient to isolate the final state and measure $g_{\\tilde{t}_1\n\\tilde{t}_1 h}$  with some accuracy.  \n\\begin{figure}[htbp]\n\\vspace{-5mm}\n\\begin{center}\n\\mbox{\n\\psfig{figure=ee_mt1.eps,width=10cm}}\n\\end{center}\n\\vspace*{-9cm}\n\\caption[]{The cross section $\\sigma (e^+e^- \\rightarrow \\tilde{t}_1 \\tilde{t}_1 h$) \n[in fb] as a function of the $\\tilde{t}_1$ mass and the choices $\\tan\\beta=30$, \n$A_t=$ 0 (0.5) TeV; $\\tan\\beta=3$, $A_t=$ 1.2 TeV (and $\\mu =-$600 GeV).} \n\\end{figure}\n\nNote however that $\\tilde A_t$ cannot be arbitrarily large without conflicting\npresent constraints: more precisely, the absence of charge and color breaking\nminima (CCB) \\cite{CCB} can put rather stringent bounds on $\\tilde{A}_t$, and\nlarge $\\tilde{A}_t$ values also generate potentially large contributions to\nelectroweak high--precision observables, in particular to the $\\rho$ parameter\n\\cite{drho}, severely constrained  by LEP1 data.\\cite{LEPrho} \nThose constraints, as well as the present \nexperimental lower bounds on the top squark and Higgs \nboson masses, were systematically taken into account in our analysis.\nWe observed\nthat in the unconstrained MSSM case, the continuum production cross section in\n$e^+e^-$ annihilation $\\sigma(e^+e^- \\rightarrow \\tilde{t}_1 \\tilde{t}_1 h$) is often larger\nthan the resonant cross section for the production of $\\tilde{t}_1 \\tilde{t}_2$\nand the subsequent 2--body decay $\\tilde{t}_2 \\rightarrow \\tilde{t}_1 h$, but this is\nnot generic. Indeed, in a situation where both a non-negligible $m_{\\tilde\nt_L}$--$m_{\\tilde t_R}$ splitting and a moderate $\\tilde A_t$ occurs, provided\nthere is sufficient phase space allowed, the production via a resonant $\\tilde\nt_2$ becomes competitive and even dominant, as illustrated in Fig. 1.\n\n\\vspace*{-4mm}\n\n\\subsection{Production in the continuum in $\\gamma \\gamma$ collisions} \nFuture high--energy $e^+e^-$ linear colliders can be turned into high--energy \n$\\gamma \\gamma$ colliders, with the high energy photons coming from Compton\nback--scattering of laser beams.\\cite{laser} The c.m. energy of the $\\gamma \n\\gamma$ collider is expected to be as much as $\\sim 80\\%$ of the one of the \noriginal $e^+e^-$ machine. However, the total luminosity is expected to be \nsomewhat smaller than the one of the $e^+e^-$ mode.\n\\begin{figure}[htb]\n\\begin{center}\n\\vspace{-.5cm}\n\\mbox{\n\\psfig{figure=gaga_mt1.eps,width=10cm}}\n\\end{center}\n\\vspace{-9cm}\n\\caption[]{$\\sigma (\\gamma\\gamma \\rightarrow \\tilde{t}_1 \n\\tilde{t}_1 h$) [in fb] at $\\sqrt{s_{\\gamma \\gamma}} =600$ GeV as a function \nof $m_{\\tilde{t}_1}$. The other parameters have the same values as in Fig.~2.}\n\\end{figure}\nThe total cross section for the subprocess $\\gamma \\gamma \\rightarrow \\tilde{t}_1\n\\tilde{t}_1 h$, calculated in \\cite{Xtth}, is shown in Fig.~3 at a \ntwo--photon c.m. energy $\\sqrt{s}_{\\gamma \\gamma}  \n\\raisebox{-0.13cm}{~\\shortstack{$<$ \\\\[-0.07cm] $\\sim$}}~ 0.8 \\sqrt{s}_{ee} =600$ GeV and as a function of the\n$\\tilde{t}_1$ mass, without convolution with the photon spectrum and with the\nsame inputs and assumptions as in Fig.~2 to compare with the $e^+e^-$ mode. \nBecause the c.m. energy of the $\\gamma \\gamma$ collider is only $\\sim 80\\%$ of\nthe one of the original $e^+e^-$ machine, the process is of course less\nphase--space favored than in the $e^+e^-$ mode.  Nevertheless, the cross section\nfor the $\\tilde{t}_1 \\tilde{t}_1h$ final state is of the same order as in the\n$e^+e^-$ mode for c.m. energies not too close to the kinematical threshold, and\nthe process might be useful to obtain complementary information since it does\nnot involve the $Z$--boson and $\\tilde{t}_2$ exchanges.  If the luminosities of\nthe $\\gamma \\gamma$ and $e^+e^-$ colliders are comparable, a large number of\nevents might be collected for small stop masses and large \n$\\tilde A_t$ values.  \n\n\\vspace*{-4mm}\n\n\\subsection{Decay modes and signal}\nTop squarks in the mass range discussed above will mainly decay~\\cite{R8} into \na $c$ quark + neutralino, $\\tilde{t}_1 \\to c\\chi_1^0$, or a $b$ quark + \nchargino,\n$\\tilde{t}_1 \\rightarrow b\\chi^+$.  In this latter case the lightest chargino,\n$\\chi_1^+$  will decay into the LSP and a real or virtual $W$ boson, leading to\nthe same topology as in the case of the top quark decay, but with a large\namount of missing energy due to the undetected LSP\n(three and four--body decays~\\cite{fourbody} of the stop are also\npossible with the same topology). At $e^+e^-$ colliders one can\nuse the dominant decay mode of the lightest Higgs boson, $h \\rightarrow b\\bar{b}$. The\nfinal state topology will then consist of $4b$ quarks, two of them peaking at\nan invariant mass $M_h$, two real (or virtual) $W$'s and missing energy. With\nefficient micro--vertex detectors, this final state should be rather easy to \ndetect.  \n\n\\vspace*{-4mm}\n\n\\section{Conclusions} \nAt $e^+e^-$ colliders with c.m. energies $\\sqrt{s} \\raisebox{-0.13cm}{~\\shortstack{$>$ \\\\[-0.07cm] $\\sim$}}~ 500$ GeV and with very\nhigh luminosities $\\int {\\cal L}{\\rm d}t \\sim 500$ fb$^{-1}$, the process $e^+e^- \\rightarrow\n\\tilde{t}_1 \\tilde{t}_1h$ can lead to several hundreds of events, since the\ncross sections can exceed the level of a 1 fb for not too heavy top squarks and\nlarge trilinear coupling, $\\tilde A_t \\raisebox{-0.13cm}{~\\shortstack{$>$ \\\\[-0.07cm] $\\sim$}}~ 1$ TeV.  In the case where the top\nsquark decays into a $b$ quark and a real\/virtual chargino, the final state\ntopology with $4b$ quarks, missing energy and additional jets or leptons will\nbe rather spectacular and should be easy to be seen experimentally, thanks to\nthe clean environment of these colliders.  In the $\\gamma \\gamma$ option of the\n$e^+e^-$ collider, the cross sections are similar as previously far from the\nparticle thresholds, but are suppressed for larger masses because of the\nreduced c.m. energies; for $\\gamma \\gamma$ luminosities of the same order as\nthe original $e^+e^-$ luminosities, the $\\tilde{t}_1 \\tilde{t}_1h$ final state\nshould also be observable in this mode, at least in some areas of the MSSM \nparameter space.  \n\nThe production cross section of the $\\tilde{t}_1 \\tilde{t}_1 h$ final state \nis directly proportional to the square of the $\\tilde{t}_1 \\tilde{t}_1 h$\ncouplings, therefore studying this process will allow to measure this important\ncoupling and to probe directly some of the soft--SUSY breaking parameters.  \n\n\\vspace*{-4mm}\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhycl b/data_all_eng_slimpj/shuffled/split2/finalzzhycl
new file mode 100644
index 0000000000000000000000000000000000000000..4a733a5eaaaf04db6684b1d6610f0da33f1ea073
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzhycl
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nLet $X \\subset \\PP^N$ be an $n$-dimensional projective variety over an algebraically closed field $\\Bbbk$ of arbitrary characteristic.\nThe \\emph{Gauss map} $\\gamma$ of $X$\nis defined as a rational map\n\\[\n\\gamma: X \\dashrightarrow \\mathbb{G}(n, \\PP^N),\n\\]\nwhich sends each smooth point $x \\in X$ to the embedded tangent space $\\mathbb{T}_xX$ of $X$ at $x$ in $\\PP^N$.\nThe Gauss map is a classical subject and has been studied by many authors.\nFor example, it is well known\nthat a general fiber of the Gauss map $\\gamma$\nis (an open subset of) a linear subvariety of $\\PP^N$ in characteristic zero\n(P.~Griffiths and J.~Harris \\cite[(2.10)]{GH},\nF.~L.~Zak \\cite[I, 2.3.\\ Theorem (c)]{Zak}).\nThe linearity of general fibers of $\\gamma$ also holds\nin arbitrary characteristic\nif $\\gamma$ is {separable} \\cite[Theorem 1.1]{expshr}.\nWe denote by $\\delta_{\\gamma}(X)$ the dimension of a general fiber of $\\gamma$,\nand call it the \\emph{Gauss defect} of $X$ (see \\cite[2.3.4]{FP}).\nThe Gauss map $\\gamma$ is said to be \\emph{degenerate} if $\\delta_{\\gamma}(X) > 0$.\n\n\\vspace{2mm}\nIn this paper, we investigate the Gauss map of \\emph{toric} $X \\subset \\PP^N$;\nmore precisely, we consider a (not necessarily normal)\ntoric variety $X \\subset \\PP^N$ such that the action of the torus on $X$ extends to the whole space $\\PP^N$.\nIt is known that such $X$ is projectively equivalent to a projective toric variety $X_A$ \nassociated to a finite subset $A$ of a free abelian group $M$ (see \\cite[Ch.~5, Proposition 1.5]{GKZ}).\nThe construction of $X_A$ is as follows.\n\n\nLet $M$ be a free abelian group of rank $n$\nand let $\\Bbbk [M] = \\bigoplus_{u \\in M} \\Bbbk z^u$ be the group ring of $M$ over $\\Bbbk$.\nWe denote by $T_M$ the algebraic torus $\\Spec \\Bbbk[M]$.\nFor a finite subset $A= \\{u_0, \\ldots, u_N\\} \\subset M$,\nwe define the toric variety $X_A$ to be the closure of the image of the morphism\n\\begin{equation}\\label{eq:defphiA}\n  \\varphi_A : T_M  \\rightarrow \\mathbb{P}^N \\ : \\ t \\mapsto [z^{u_0} (t) : \\cdots: z^{u_N}(t) ].\n\\end{equation}\nWe set $  \\langle  A - A  \\rangle \\subset M $ (resp.\\ $ \\langle  A - A  \\rangle_{\\Bbbk} \\subset M_{\\Bbbk}:= M \\otimes_{\\mathbb {Z}} \\Bbbk $)\nto be\nthe subgroup of $M$ (reap.\\ the $\\Bbbk$-vector subspace of $ M_{\\Bbbk}$) generated by $A -A : = \\{ u-u' \\, | \\, u,u' \\in A \\} $.\nThe algebraic torus $T_{ \\langle  A - A  \\rangle}$ acts on $X_A$,\nand $T_{ \\langle  A - A  \\rangle}$ is contained in $X_A$ as an open dense orbit.\nIn this paper, for short, we say a projective variety $X \\subset \\PP^N$ is \\emph{projectively embedded toric}\nif $X$ is projectively equivalent to the projective toric variety $X_A$ associated to some finite subset $A$ of a free abelian group $M$ of finite rank.\n\n\n\n\n\n\n\n\n\n\n\nWe denote by $\\Aff(A)$ (resp.\\ $\\Aff_{\\Bbbk}(A)$)\nthe affine sublattice of $M$ (resp.\\ the $\\Bbbk$-affine subspace of $M_{\\Bbbk}$)\nspanned by $A$.\nIn other words,\n$\\Aff(A)$ (resp.\\ $\\Aff_{\\Bbbk}(A)$) is\nthe set of linear combinations $\\sum_i a_i u_i \\in M$ with $a_i \\in \\mathbb{Z}$ (resp.\\ $a_i \\in \\Bbbk$),\n$\\sum_i a_i = 1$, and $u_i \\in A$. We say that $A$ \\emph{spans} the affine lattice $M$\n(resp.\\ the $\\Bbbk$-affine space $M_{\\Bbbk}$)\nif $\\Aff(A) = M$ (resp.\\ $\\Aff_{\\Bbbk}(A) = M_{\\Bbbk}$).\n\n\\vspace{1em}\n\nProjective geometry of toric $X_A$ has been investigated in view of the projective dual\nin many papers (\\cite{CD}, \n\\cite{DDP}, \\cite{DN}, \\cite{DR}, \\cite{GKZ}, \\cite{MT}, etc.).\nOn the other hand, the Gauss map of $X_A$ has not been well studied yet.\nIn the following result, we describe the structure of Gauss maps of toric varieties\nin terms of combinatorics.\n\n\n\\begin{thm}\\label{thm_structure}\n  Let $\\Bbbk$ be an algebraically closed field of arbitrary characteristic, and\n  let $M$ be a free abelian group of rank $n$.\n  For a finite subset $A= \\{u_0, \\ldots, u_N\\}  \\subset M$ which spans the affine lattice $M$,\n  set\n  \\begin{gather*}\n    B := \\{ u_{i_0} + u_{i_1} + \\cdots + u_{i_n} \\in M  \\, |  \\,  u_{i_0}, u_{i_1} , \\ldots , u_{i_n}  \\text{ span the $\\Bbbk$-affine space } M_{\\Bbbk} \\}\n  \\end{gather*}\n  and let\n  $\\pi: M \\rightarrow M':= M \/ (\\langle B-B \\rangle_{\\mathbb {R}} \\cap M)$ be the natural projection.\n  Let $\\gamma: X_A \\dashrightarrow \\mathbb {G}(n, \\mathbb{P}^N)$ be the Gauss map of the toric variety $X_A \\subset \\PP^N$.\n  Then the following hold.\n  \\begin{enumerate}  \\item\\label{thm:item-image}\n    The closure $\\overline{\\gamma(X_A)}$ of the image of $\\gamma$,\n    which is embedded in a projective space by the Pl\\\"ucker embedding of $ \\mathbb{G} (n, \\mathbb{P}^{N})$,\n    is projectively equivalent to the toric variety $ X_{B} $.\n  \\item\\label{thm:item-map}\n    The restriction of $\\gamma : X_A \\dashrightarrow \\overline{\\gamma (X_A)} \\cong X_{B}$ on $T_M \\subset X_A$\n    is the morphism\n    \\[\n    T_M = \\Spec \\Bbbk[M] \\twoheadrightarrow T_{ \\langle  B-B  \\rangle} = \\Spec \\Bbbk[ \\langle  B-B  \\rangle] \\subset X_{B}\n    \\]\n    induced by the inclusion $ \\langle  B-B \\rangle \\subset M$.\n  \\item\\label{thm:item-fiber}\n    Let $F \\subset T_M$ be an irreducible component of any fiber of $\\gamma|_{T_M}$ with the reduced structure.\n    Let $T_{M'} \\hookrightarrow T_M$ be the subtorus induced by $\\pi$.\n    Then $F$ is a translation of $T_{M'}$ by an element of $T_M$,\n    and \n    the closure $\\overline{F} \\subset X_A$\n    is projectively equivalent to the toric variety $X_{\\pi(A)}$.\n\n  \\end{enumerate}\n  In particular,\n  we have $ \\delta_{\\gamma}(X_{A}) = \\rk M' = n -\\rank  \\langle  B-B  \\rangle$. \n\\end{thm}\n\n\n\n\n\n\n\n\n\nNote that there is no loss of generality in assuming that $A $ spans the affine lattice $M$ in the above theorem (see \\autoref{thm:span-A-M}).\nWe also study when the Gauss map of toric $X_A$ is degenerate (i.e., $\\rank  \\langle  B-B \\rangle < n$),\nand give a developability criterion for covering families of $X_A$\n(see \\autoref{sec:criterion-degeneracy}). \n\n\\vspace{2mm}\n\n\nIn the following example, we illustrate the description in \\autoref{thm_structure}.\n\n\n\\begin{ex}\\label{ex_intro}\n  Let $M=\\mathbb {Z}^2$ and\n  \\[\n  A = \\left\\{\n    \\begin{bmatrix}\n      0 \\\\ 0\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      0 \\\\ 1\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      1 \\\\ -1\n    \\end{bmatrix}\n    ,\\begin{bmatrix}\n      -1 \\\\ -1\n    \\end{bmatrix}\n  \\right\\} \\subset \\mathbb {Z}^2,\n  \\]\n  where we write elements in $\\mathbb {Z}^2$ by column vectors.\n  We consider the Gauss map $\\gamma$ of the toric surface $X_A \\subset \\mathbb{P}^3$.\n  When $\\chara \\Bbbk \\neq 2$,\n  \\[\n  B = \\left\\{\n    \\begin{bmatrix}\n      0 \\\\ -1\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      0 \\\\ -2\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      -1 \\\\ 0\n    \\end{bmatrix}\n    ,\\begin{bmatrix}\n      1 \\\\ 0\n    \\end{bmatrix}\n  \\right\\} \\subset \\mathbb {Z}^2.\n  \\]\n  Hence $ \\langle B-B \\rangle =M =\\mathbb {Z}^2$, and $\\gamma$ is birational due to \\ref{thm:item-map} in \\autoref{thm_structure}.\n  On the other hand, when $\\chara \\Bbbk =2$,\n  \\begin{align*}\n    B &= \\left\\{\n      \\begin{bmatrix}\n        -1 \\\\ 0\n      \\end{bmatrix}\n      ,\\begin{bmatrix}\n        1 \\\\ 0\n      \\end{bmatrix}\n    \\right\\},\\\n    \\lin{B-B} = \\left\\langle\n      \\begin{bmatrix}\n        2\n        \\\\\n        0\n      \\end{bmatrix}\n    \\right\\rangle,\\\n    \\lin{B-B}_{\\mathbb{R}} \\cap M\n    = \\left\\langle\n      \\begin{bmatrix}\n        1 \\\\ 0\n      \\end{bmatrix}\n    \\right\\rangle\n    \\subset \\mathbb {Z}^2,\n  \\end{align*}\n  and $\\pi(A) = \\{0, 1,-1\\} \\subset \\mathbb {Z}^2 \/ (\\langle B-B \\rangle_{\\mathbb {R}} \\cap M) = \\mathbb {Z}^1$.\n  Thus \\ref{thm:item-image} implies that $\\overline{\\gamma(X_A)} \\cong X_{B} =\\mathbb{P}^1$,\n  and \\ref{thm:item-map} implies that\n  $\\gamma|_{T_M}: T_M= (\\KK^{\\times})^2 \\twoheadrightarrow T_{\\lin{B-B}}=\\KK^{\\times}$ is given by $(z_1,z_2) \\mapsto z_1^2$.\n  From \\ref{thm:item-fiber}, \n  a general fiber of $\\gamma$ with the reduced structure is projectively equivalent to the smooth conic $X_{\\pi(A)}$.\n  \\begin{figure}[htbp]\n    \\[\n    \\begin{xy}\n      (10,0)=\"A\",(0,10)=\"B\",\n      (-10,-10)=\"C\",\n      (9.9,0.1)=\"E\",(-5,0.1)=\"F\",\n      (9.8,-0.1)=\"I\",(-5,-0.1)=\"J\",\n      (-20,0)=\"1\",(20,0)=\"2\",\n      (0,-17)=\"3\",(0,18)=\"4\",\n      (50,-17)=\"7\",(50,18)=\"8\",\n      (0,10)*{\\bullet},(0,0)*{\\bullet},\n      (-10,-10)*{\\bullet},(10,-10)*{\\bullet},\n      (-10,0)*{\\times},(10,0)*{\\times},\n      (50,10)*{\\sqbullet},(50,0)*{\\sqbullet},\n      (50,-10)*{\\sqbullet},\n      (53,10)*{1},(53,0)*{0},(55,-10)*{-1},\n      (16,18)*{\\bullet\\;A}, \n      (16,13)*{\\times\\;B},\n      (65,16)*{\\sqbullet\\;\\pi(A)},\n      (35,3)*{\\pi}\n      \\ar \"1\";\"2\"\n      \\ar \"3\";\"4\"\n      \\ar \"7\";\"8\"\n      \\ar (27,0);(43,0)\n    \\end{xy}\n    \\]\n    \\caption{$\\chara \\Bbbk =2$.}\n    \\label{figure1}\n  \\end{figure}\n\\end{ex}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn characteristic zero, it is well known that, if a projective variety\nis the join of some varieties,\nthen its Gauss map is degenerate due to Terracini's lemma (see \\cite[2.2.5]{FP}, \\cite[Ch.\\,II, 1.10.\\,Proposition]{Zak}).\nFor toric varieties in characteristic zero, this is the only case when the Gauss map is degenerate;\nmore precisely, we have:\n\n\n\n\n\n\\begin{cor}\\label{cor_join}\n  Let $X \\subset \\mathbb{P}^N$ be a projectively embedded toric variety in $\\chara\\Bbbk = 0$.\n  Then there exist disjoint torus invariant closed subvarieties $ X_0, \\ldots,X_{\\delta_{\\gamma}(X)} \\subset X$\n  such that $X$ is the join of  $ X_0, \\ldots,X_{\\delta_{\\gamma}(X)}$.\n\\end{cor}\n\nIf the Gauss map of $X_A \\subset \\PP^N$ is separable,\n$A$ is written as a \\emph{Cayley sum} of certain finite subsets $A^0, \\dots, A^{\\delta_{\\gamma}(X)}$ in any characteristic\n(see \\autoref{thm:subvar-in-join} for details).\nHowever, the statement of \\autoref{cor_join} does \\emph{not} hold in general in positive characteristic,\neven if the Gauss map is separable\n(see \\autoref{thm:sep-gamma-X-not-join}).\n\n\\vspace{1em}\n\nNext, let us focus on inseparable Gauss maps.\nA.~H.~Wallace \\cite[\\textsection 7]{Wallace} showed that\nthe Gauss map $\\gamma$ of a projective variety can be \\emph{inseparable}\nin positive characteristic.\nIn this case,\nit is possible that\na general fiber of $\\gamma$ is \\emph{not} a linear subvariety of $\\PP^N$;\nthe fiber can be a union of points\n(H.~Kaji \\cite[Example 4.1]{Kaji1986} \\cite{Kaji1989}, J.~Rathmann \\cite[Example~2.13]{Rathmann}, A.~Noma \\cite{Noma2001}),\nand can be a non-linear variety\n(S.~Fukasawa \\cite[\\textsection{}7]{Fukasawa2005}).\nIn fact, Fukasawa \\cite{Fukasawa2006}\nshowed that \\emph{any} projective variety appears as a general fiber of the Gauss map of some projective variety.\n\n\nAs we will see in \\autoref{cor_rk&deg},\n\\autoref{thm_structure} provides\nseveral computations on the Gauss map $\\gamma$ of toric varieties\n(e.g., the rank, separable degree, inseparable degree). \nWe also obtain the toric version of Fukasawa's result \\cite{Fukasawa2006} as follows:\n\n\n\n\n\n\n\n\n\n\n\n\\begin{thm}[Special case of \\autoref{cor_im&fiber}]\\label{thm_Fukasawa1}\n\n  Assume $\\chara \\Bbbk >0$.\n  Let $Y \\subset \\mathbb{P}^{N'} $ and $Z \\subset \\mathbb{P}^{N''}$ be projectively embedded toric varieties.\n  If $n:=\\dim(Y)+\\dim(Z)$ is greater than or equal to $N'$, then\n  there exists an $n$-dimensional projectively embedded toric variety\n  $X \\subset \\mathbb{P}^{n+N''}$\n  satisfying the following conditions:\n  \\begin{enumerate}[\\normalfont (i)]\n  \\item\n    (The closure of) a general fiber of the Gauss map $\\gamma$ of $X$ with the reduced structure is projectively equivalent to $Y$.\n  \\item \n    (The closure of) the image of $\\gamma$     is projectively equivalent to $Z$.\n  \\end{enumerate}\n\\end{thm}\n\n\n\n\n\n\n\n\nBy \\autoref{thm_Fukasawa1}, any projectively embedded toric variety appears\nas a general fiber and the image of the Gauss map of\na certain projectively embedded \\emph{toric} variety; moreover\nwe can also control the \\emph{rank} of $\\gamma$,\nand the \\emph{number} of the irreducible components of a general fiber of $\\gamma$\n(see \\autoref{sec:posit-char-case}, for details).\n\n\n\\vspace{2mm}\nThis paper is organized as follows.\nIn \\autoref{sec:prel-toric-vari}, we recall some basic properties of toric varieties.\nIn \\autoref{sec:structure-gauss-maps}, we describe the structure of the Gauss maps\nof toric varieties in a combinatorial way, and prove \\autoref{thm_structure}.\nIn \\autoref{sec:degen-gauss-maps}, we investigate when the Gauss maps are degenerate,\nand give a developability criterion.\nAs a result, we show \\autoref{cor_join}.\nIn \\autoref{sec:posit-char-case}, we present two constructions of projectively embedded toric varieties, yielding \\autoref{thm_Fukasawa1}.\n\n\n\\subsection*{Acknowledgments}\n\nThe authors would like to express their gratitude to Professors\nSatoru Fukasawa and Hajime Kaji for their valuable comments and advice.\nIn particular, \\autoref{thm:a-vari-kaji} is due to Professor Kaji.\nThe first author was partially supported by JSPS KAKENHI Grant Number 25800030.\nThe second author was partially supported by JSPS KAKENHI Grant Number 25887010.\n\n\n\n\n\n\\section{Preliminary on toric varieties}\n\\label{sec:prel-toric-vari}\n\n\n\n\n\n\n\nTwo projective varieties $X_1 \\subset \\mathbb{P}^{N_1}$ and $X_2 \\subset \\mathbb{P}^{N_2}$ are said to be\n\\emph{projectively equivalent} if\nthere exist linear embeddings $\\jmath_i: \\mathbb{P}^{N_i} \\hookrightarrow \\PP^N$\n(i.e., $\\jmath_i^*(\\mathscr{O}_{\\PP^N}(1)) = \\mathscr{O}_{\\mathbb{P}^{N_i}}(1)$)\nsuch that $\\jmath_1(X_1) = \\jmath_2(X_2)$.\n(Indeed, we can take $N = \\max  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{N_1,N_2}$.)\n\n\n\\vspace{2mm}\nThe following two lemmas about toric varieties are well known,\nbut we prove them for the convenience of the reader.\n\n\\begin{lem}\\label{lem_lattice_hom}\n  Let $\\pi : M \\rightarrow M'$ be a surjective homomorphism between free abelian groups of finite ranks.\n  Let $\\iota : T_{M'} \\hookrightarrow T_M$ be the embedding induced by $\\pi$.\n  For a finite set $A \\subset M$ with $\\Aff(A)=M$\n  the closure of $\\iota(T_{M'})$ in $X_A$ is projectively equivalent to $X_{\\pi(A)}$.\n  The translations\n  $  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{t \\cdot \\overline{\\iota(T_{M'})}}_{t \\in T_M}$\n  of the closure $\\overline{\\iota(T_{M'})}$\n  under the action of $T_M$ on $X_A$ give a covering family of $X_A$, and\n  each translation is also projectively equivalent to $X_{\\pi(A)}$.\n\\end{lem}\n\n\n\\begin{proof}\n  Let $A=\\{ u_0,\\ldots, u_N\\}$ and $\\pi(A)= \\{u'_0, \\ldots, u'_{N'}\\}$  for $N=\\# A-1$ and $N' = \\# \\pi(A) -1$.\n  We define a linear embedding $\\jmath : \\mathbb{P}^{N'} \\rightarrow \\mathbb{P}^N$ by\n  \\[\n  \\jmath ( [X'_0 : \\cdots: X'_{N'}]) = [X_0 : \\cdots : X_N],\n  \\]\n  where for each $i$, we set $X_i := X'_j$ for $j$ such that $\\pi(u_i)=u'_j$. \n  Then we have the following commutative diagram\n  \\[\n  \\xymatrix{\n    T_{M'} \\ar@{^{(}->}[d]^{\\iota} \\ar[r]^{\\varphi_{\\pi(A)}} \\ar@{}[dr]|\\circlearrowleft &  \\mathbb{P}^{N'} \\ar@{^{(}->}[d]^{\\jmath} \\\\\n    T_M  \\ar[r]^{\\varphi_{A}} & \\mathbb{P}^N.\\\\\n  }\n  \\]\n  Hence\n  $\\overline{\\iota(T_{M'})}$ is projectively equivalent to $X_{\\pi(A)}$.\n  Since the action of $T_M$ on $X_A$ extends to $\\PP^N$ (see \\cite[Ch.~5, Proposition~1.5]{GKZ}), translations of $\\overline{\\iota(T_{M'})}$ are also\n  projectively equivalent to $X_{\\pi(A)}$.\n  Since $\\iota(T_{M'})$ is non-empty and contained in $T_M$,\n  the translations give a covering family of $X_A$.\n\\end{proof}\n\n\n\nLet $f: X \\dashrightarrow Y$ be a rational map between varieties. For a smooth point $x \\in X$,\nwe denote by $d_xf: t_xX \\rightarrow t_{f(x)}Y$\nthe tangent map\nbetween Zariski tangent spaces at $x$ and $f(x)$.\nThe \\emph{rank} of $f$, denoted by $\\rk (f)$, is defined to be the rank of the $\\Bbbk$-linear map\n$d_xf$ for general $x \\in X$.\nRecall that $f$ is said to be \\emph{separable} if \nthe field extension $K(X)\/K(f(X))$ is separable;\nthis condition is equivalent to $\\rk(f) = \\dim(f(X))$.\n\n\n\n\n\\begin{lem}\\label{lem_rk&deg}\n  Let $M$ be a free abelian group of finite rank.\n  Let $M'' $ be a subgroup of $M$ and $g : T_M \\twoheadrightarrow T_{M''} $ be the morphism induced by the inclusion $\\beta : M'' \\hookrightarrow M$.\n  \\begin{enumerate}[\\normalfont \\quad (a)]\n  \\item\\label{item:lem_rk&deg:1}\n    The inclusions $M'' \\subset M''_{\\mathbb {R}} \\cap M \\subset M$ induce\n    a decomposition of $g$\n    \\[\n    T_M \\stackrel{g_1}{\\rightarrow} T_{M''_{\\mathbb {R}} \\cap M} \\stackrel{g_2}{\\rightarrow} T_{M''},\n    \\]\n    where\n    $g_1$ is a morphism with reduced and irreducible fibers, and $g_2$ is a finite morphism.\n\n  \\item\\label{item:lem_rk&deg:2} The rank of $g$ is equal to the rank of the $\\Bbbk$-linear map $\\beta_{\\Bbbk} : M''_{\\Bbbk} \\rightarrow M_{\\Bbbk} $ obtained by tensoring $\\Bbbk$ with $\\beta : M'' \\hookrightarrow M$.\n    In particular, $g$ is separable if and only if\n    $\\rk (\\beta_{\\Bbbk}) = \\rank(M'')$.\n  \\item\\label{item:lem_rk&deg:3} Assume $p = \\chara \\Bbbk > 0$.\n    Let $a$ be the index $ [M''_{\\mathbb {R}} \\cap M : M'']$ and write $a = p^s b$ with integers $s \\geq 0, b \\geq 1 $ such that $ p \\nmid b$.\n    Then the degree, separable degree, inseparable degree of the finite morphism $g_2 $ are $a, b , p^s$ respectively.\n  \\end{enumerate}\n\\end{lem}\n\n\n\n\n\\begin{proof}\n  Set $n= \\rank M, k= \\rank M''$.\n  By the elementary divisors theorem (see \\cite[III, Theorem~7.8]{Lang}, for example),\n  there exists a basis $e_1,\\ldots,e_n $ of $M$ such that\n  \\[\n  M'' = \\mathbb {Z} a_1 e_1 \\oplus \\cdots \\oplus  \\mathbb {Z} a_k e_k \\subset   \\mathbb {Z} e_1 \\oplus \\cdots \\oplus  \\mathbb {Z} e_n =M\n  \\]\n  for some positive integers $a_i$. Set $e''_i := a_i e_i \\in M''$.\n  By the bases $e_1,\\ldots,e_n $ of $M$ and $e''_1,\\ldots, e''_k $ of $M''$,\n  we identify $M $ and $ M'' $ with $\\mathbb {Z}^n$ and $ \\mathbb {Z}^k$ respectively.\n  Then $g : T_M = (\\KK^{\\times})^n  \\rightarrow T_{M''} = (\\KK^{\\times})^k $ is described as\n  \\begin{align}\\label{eq_explicit_g}\n    (\\KK^{\\times})^n \\rightarrow (\\KK^{\\times})^k \\quad : \\quad (z_1, \\ldots,z_n) \\mapsto (z_1^{a_1}, \\ldots,z_k^{a_k}).\n  \\end{align}\n  \\vspace{.5mm}\n\n  \\noindent\n  \\ref{item:lem_rk&deg:1} \n  By \\ref{eq_explicit_g},\n  $g$ is decomposed as\n  \\[\n  (\\KK^{\\times})^n \\rightarrow (\\KK^{\\times})^k  \\rightarrow  (\\KK^{\\times})^k \\quad  : \\quad (z_1, \\ldots,z_n) \\mapsto (z_1, \\ldots,z_k) \\mapsto (z_1^{a_1}, \\ldots,z_k^{a_k}).\n  \\]\n  Since $M''_{\\mathbb {R}} \\cap M = \\mathbb {Z} e_1 \\oplus \\cdots \\oplus  \\mathbb {Z} e_k  \\subset M$,\n  the assertion of \\ref{item:lem_rk&deg:1} \n  follows.\n\n  \\vspace{1ex}\n  \\noindent\n  \\ref{item:lem_rk&deg:3} \n  Write $a_i = p^{s_i} b_i$ by integers $s_i \\geq 0, b_i \\geq 1 $ such that $ p \\nmid b_i$.\n  Since $g_2$ is the morphism\n  \\[\n  (\\KK^{\\times})^k  \\rightarrow  (\\KK^{\\times})^k \\quad  : \\quad  (z_1, \\ldots,z_k) \\mapsto (z_1^{a_1}, \\ldots,z_k^{a_k}) = (z_1^{p^{s_1} b_1}, \\ldots,z_k^{p^{s_k} b_k}),\n  \\]\n  the degree, separable degree, inseparable degree of $g_2 $ are\n  $\\prod_{i=1}^k a_i =a, \\prod_{i=1}^k b_i =b, \\prod_{i=1}^k p^{s_i} = p^s$\n  respectively.\n\n\n\n  \\vspace{1ex}\n  \\noindent\n  \\ref{item:lem_rk&deg:2} \n  If $\\chara \\Bbbk=0$, this statement is clear from \\ref{eq_explicit_g}.\n  Assume $p= \\chara \\Bbbk >0$ and use the notation $a_i, s_i, b_i$ as above.\n  Then the rank of $g$ is equal to\n  $\\#  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}*{1 \\leq i \\leq k}{s_i = 0}$.\n  On the other hand,\n  $\\beta_{\\Bbbk} : M''_{\\Bbbk} = \\Bbbk^{k} \\rightarrow M_{\\Bbbk} = \\Bbbk^n$ is the $\\Bbbk$-linear map defined by $e''_i \\mapsto a_i e_i = p^{s_i} b_i e_i  \\in M_{\\Bbbk}$ for $ 1 \\leq i \\leq k$.\n  Hence the rank of $\\beta_{\\Bbbk}$ is also equal to $\\#  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}*{1 \\leq i \\leq k}{s_i = 0}$.\n  Thus we have $\\rk (g) =\\rk (\\beta_{\\Bbbk})$.\n  The last statement follows from $\\dim g(T_M) =\\dim T_{M''} =\\rk (M'')$.\n\\end{proof}\n\n\n\n\n\n\n\\section{Structure of Gauss maps}\n\\label{sec:structure-gauss-maps}\n\nIn this section,\nwe prove Theorem \\ref{thm_structure}\nand describe several invariants (e.g., the rank) of Gauss maps of toric varieties\nby combinatorial data.\n\n\nIn order to investigate a toric variety $X_A$ for $A \\subset M$,\nwe may assume that $A$ spans the affine lattice $M$ due to the following remark.\n\n\\begin{rem}\\label{thm:span-A-M}\n  For a finite subset $A \\subset M$,\n  let $\\theta : \\mathbb {Z}^m \\rightarrow  \\Aff (A) $ be an affine isomorphism for $ m =\\rank \\langle A -A \\rangle $.\n  Then $ \\theta^{-1}(A)$ spans $\\mathbb {Z}^m$ as an affine lattice and\n  $X_{\\theta^{-1}(A)}$ is naturally identified with $X_A$\n  by \\cite[Chapter 5, Proposition 1.2]{GKZ}.\n  Hence any projectively embedded toric variety $X$ is projectively equivalent\n  to $X_A$ for some $A \\subset M$ with $\\Aff (A)=M$.\n\\end{rem}\n\nLet $A =   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_0, u_1, \\dots, u_N} \\subset M:=\\mathbb{Z}^n$ be a finite subset\nwhich spans the affine lattice $M$.\nWe denote each $u_i$ by a column vector as\n\\[\nu_i =\n\\begin{bmatrix}\n  u_{i,1} \\\\ \\vdots \\\\ u_{i,n}\n\\end{bmatrix}.\n\\]\nThen the morphism $\\varphi_A $, defined by \\ref{eq:defphiA} in \\autoref{sec:introduction},\nis described as\n\\[\n\\varphi_A: (\\KK^{\\times})^n \\rightarrow \\PP^N \\quad :  \\quad z = (z_1, \\dots, z_n) \\mapsto [z^{u_0} : z^{u_1} : \\dots : z^{u_N}],\n\\]\nwhere\n$z^{u_i} := z_1^{u_{i,1}} z_2^{u_{i,2}}\\cdots z_n^{u_{i,n}}$.\nBy the assumption that $A$ spans $\\mathbb{Z}^n$ as an affine lattice,\n$\\phi_A$ is an isomorphism onto an open subset of $X_A$.\n\n\n\nLet us study the Gauss map\n$\\gamma: X_A \\dashrightarrow \\mathbb{G}(n, \\PP^N)$ of $X_A \\subset \\PP^N$.\n\n\n\\begin{lem}\\label{thm:matrix-Gamma}\n  Let $A, \\phi_A$ be as above, and\n  let $x \\in (\\KK^{\\times})^n$. Then\n  $\\gamma(\\varphi_A(x)) \\in \\mathbb{G}(n, \\PP^N)$\n  is expressed by the $\\Bbbk$-valued $(n+1) \\times (N+1)$ matrix $\\Gamma(x)$;\n  more precisely, $\\gamma(\\varphi_A(x))$\n  corresponds to the $n$-plain (i.e., $n$-dimensional linear subvariety of $\\PP^N$)\n  spanned by the $n+1$ points which are given as the row vectors of $\\Gamma(x)$, where\n  \\begin{align*}\n    \\Gamma &: =\n    \\begin{bmatrix}\n      z^{u_0} & z^{u_1} & \\cdots & z^{u_N}\n      \\\\\n      u_{0,1} \\cdot z^{u_0} & u_{1,1} \\cdot z^{u_1} & \\cdots & u_{N,1} \\cdot z^{u_N}\n      \\\\\n      \\vdots & \\vdots && \\vdots\n      \\\\\n      u_{0,n} \\cdot z^{u_0} & u_{1,n} \\cdot z^{u_1} & \\cdots & u_{N,n} \\cdot z^{u_N}\n    \\end{bmatrix}\n    \\\\\n    & \\ =\n\\Bigg[     \\,\n    z^{u_0} \\cdot \n    \\begin{bmatrix}\n      1\n      \\\\\n      u_0\n    \\end{bmatrix}\n    \\ \n    z^{u_1} \\cdot \n    \\begin{bmatrix}\n      1\n      \\\\\n      u_1\n    \\end{bmatrix}\n    \\ \n    \\cdots\n    \\ \n    z^{u_N} \\cdot \n    \\begin{bmatrix}\n      1\n      \\\\\n      u_N\n    \\end{bmatrix}\n    \\,\n  \\Bigg]\n\\end{align*}\n\\end{lem}\n\\begin{proof}\n  Let $L_x \\subset \\PP^N$ be the $n$-plane spanned by\n  the $n+1$ points which are given as the row vectors of\n  \\begin{equation}\\label{eq:mat-TTxX-0}\n    \\begin{bmatrix}\n      z^{u_0} & \\cdots & z^{u_N}\n      \\\\\n      \\tdiff{(z^{u_0})}{z_1} & \\cdots & \\tdiff{(z^{u_N})}{z_1}\n      \\\\\n      \\vdots && \\vdots\n      \\\\\n      \\tdiff{(z^{u_0})}{z_n} & \\cdots & \\tdiff{(z^{u_N})}{z_n}\n    \\end{bmatrix}(x).\n  \\end{equation}\n  Then $L_x$ coincides with the embedded tangent space $\\mathbb{T}_xX$,\n  because of the equality $t_{x'}\\hat L_x = t_{x'}\\hat X$ of Zariski tangent spaces in $t_{x'}\\mathbb {A}^{N+1}$ with $x' \\in \\hat x \\setminus   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{0}$,\n  where $\\hat S \\subset \\mathbb {A}^{N+1}$ means the affine cone of $S \\subset \\PP^N$.\n  On the other hand, \\ref{eq:mat-TTxX-0} is calculated as\n  \\[\n  \\begin{bmatrix}\n    z^{u_0} & \\cdots & z^{u_N}\n    \\\\\n    u_{0,1} \\cdot z^{u_0}\/z_1 & \\cdots & u_{N,1} \\cdot z^{u_N}\/z_1\n    \\\\\n    \\vdots && \\vdots\n    \\\\\n    u_{0,n} \\cdot z^{u_0}\/z_n  & \\cdots & u_{N,n} \\cdot z^{u_N}\/z_n\n  \\end{bmatrix}.\n  \\]\n  Since each row vector means the homogeneous coordinates of a point of $\\PP^N$,\n  by multiplying $z_i$ with the $(i+1)$-th row vector for $1 \\leq i \\leq n$,\n  we have the matrix $\\Gamma$ and the assertion.\n\\end{proof}\n\n\nWe interpret \\autoref{thm:matrix-Gamma} by the Pl\\\"ucker embedding.\nWe regard $\\PP^N$ as ${\\PP_{\\!\\! *}}(V) = (V\\setminus \\{0\\}) \/ \\KK^{\\times}$, the projectivization of $V:=\\Bbbk^{N+1}$.\nLet $\\mathbb{G}(n, \\PP^N) \\hookrightarrow {\\PP_{\\!\\! *}} (\\bigwedge^{n+1} V)$ be the Pl\\\"ucker embedding\nand $[p_{i_0,\\ldots,i_n}]_{(i_0 , i_1 , \\ldots , i_n) \\in I}$ be the Pl\\\"ucker coordinates on ${\\PP_{\\!\\! *}} (\\bigwedge^{n+1} V)$,\nwhere\n\\[\nI :=   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{  ( i_0, i_1, \\dots, i_n ) \\in \\mathbb {N}^{n+1} \\, | \\, 0 \\leq i_0 < i_1 < \\cdots < i_n \\leq N }.\n\\]\n\\begin{lem}\\label{lem_plucker}\n  The composite morphism  $(\\KK^{\\times})^n \\stackrel{\\gamma\\circ \\varphi_A}{\\longrightarrow} \\mathbb{G}(n, \\PP^N) \\hookrightarrow {\\PP_{\\!\\! *}} (\\bigwedge^{n+1} V)$\n  maps $ z = (z_1, \\dots, z_n) \\in (\\KK^{\\times})^n $ to    \\[\n  \\left[\n    \\mu_{i_0, i_1, \\dots, i_n}\n    \\cdot z^{u_{i_0} + u_{i_1} + \\cdots + u_{i_n}}\n  \\right]_{( i_0 ,i_1 , \\ldots , i_n) \\in I} \\in {\\PP_{\\!\\! *}} \\Big( \\bigwedge^{n+1} V \\Big),\n  \\]\n  where\n  \\begin{align*}\n    \\mu_{i_0, i_1, \\dots, i_n} &:=\n    \\det \\begin{bmatrix}\n      1 & 1 & \\cdots & 1\n      \\\\\n      u_{i_0} & u_{i_1} & \\cdots & u_{i_n}\n    \\end{bmatrix}\n    \\in \\Bbbk.\n  \\end{align*}\n\\end{lem}\n\\begin{proof}\n  This directly follows from \\autoref{thm:matrix-Gamma} and the definition of the Pl\\\"ucker embedding.\n\\end{proof}\n\n\n\n\n\n\nSet\n\\[\nJ := \\{  ( i_0, i_1, \\dots, i_n ) \\in I \\, | \\,  \\mu_{i_0, i_1, \\dots, i_n} \\neq 0 \\}.\n\\]\nBy definition, \n$\\mu_{i_0, i_1, \\dots, i_n} \\neq 0$\nin $\\Bbbk$ if and only if $  u_{i_0} , u_{i_1} , \\ldots, u_{i_n}$ span $M_{\\Bbbk}$ as an affine space. Hence\nthe finite set $B$ in \\autoref{thm_structure} is described as\n\\[\nB = \\{ u_{i_0} + u_{i_1} + \\dots + u_{i_n} \\in M \\, | \\, (i_0 , i_1 , \\ldots , i_n) \\in J \\}.\n\\]\nWrite $B = \\{ b_0,b_1,\\cdots,b_{\\# B-1} \\}$ for mutually distinct $b_j \\in M$.\nWe define a linear embedding\n\\begin{equation}\\label{eq:iota-d-hookrar}\n  \\begin{aligned}\n    \\mathbb{P}^{\\# B -1} &\\hookrightarrow {\\PP_{\\!\\! *}} \\Big(\\bigwedge^{n+1} V\\Big)\n    \\\\\n    [Y_0 : Y_1 : \\cdots : Y_{\\# B -1}] &\\mapsto [p_{i_0,i_1,\\ldots,i_n}]_{(i_0 , i_1 , \\ldots , i_n) \\in I}\n  \\end{aligned}\n\\end{equation}\nas follows. \nWhen $(i_0, i_1, \\dots, i_n) \\in J$,\nthere exists a unique \n$0 \\leq j \\leq \\# B -1$ such that $b_j = u_{i_0} + u_{i_1} + \\dots + u_{i_n}$.\nFor this $j$, we set\n$p_{i_0,i_1,\\ldots,i_n} = \\mu_{i_0, i_1, \\dots, i_n} \\cdot Y_j$.\nWhen $(i_0, i_1, \\dots, i_n) \\not\\in J$,\nwe set\n$p_{i_0,i_1,\\ldots,i_n} = 0$.\n\n\nBy \\autoref{lem_plucker} and the definition of the embedding~\\ref{eq:iota-d-hookrar},\nwe have the following commutative diagram:\n\\begin{equation}\\label{eq:pf-thm1}\n  \\begin{split}\n    \\xymatrix{\n      (\\KK^{\\times})^n  \\ar[rrd]^{\\varphi_{B}} \\ar@{^{(}->}[r]^(0.6){\\varphi_A} & X_A \\ar@{-->}[r]^(0.3){\\gamma} & {\\PP_{\\!\\! *}} (\\bigwedge^{n+1} V) \\\\\n      & & \\mathbb{P}^{\\# B -1} \\ar@{^(->}[u]{}.\n    }\n  \\end{split}\n\\end{equation}\n\n\\begin{proof}[Proof of \\autoref{thm_structure}]\n  By taking a basis of $M$,\n  we may assume that $M = \\mathbb {Z}^n$\n  and use the notation as above.\n  From the above diagram~\\ref{eq:pf-thm1}, the embedding $\\mathbb{P}^{\\# B -1} \\hookrightarrow {\\PP_{\\!\\! *}} (\\bigwedge^{n+1} V)$\n  gives an isomorphism between\n  \\[\n  \\overline{\\gamma(X_A)} = \\overline{\\gamma \\circ \\varphi_A(  (\\KK^{\\times})^n)}\n  \\textgene{and} X_{B}=\\overline{\\varphi_{B}((\\KK^{\\times})^n )}.\n  \\]\n  Hence \\ref{thm:item-image} in \\autoref{thm_structure} holds.\n\n\n\n  The toric variety $X_{B}=\\overline{\\varphi_{B}((\\KK^{\\times})^n )}$ contains $T_{\\langle B-B \\rangle}$ as an open dense orbit\n  and\n  the restriction $\\varphi_B |_{T_M}$ is nothing but\n  $T_M= (\\KK^{\\times})^n \\twoheadrightarrow T_{\\langle B-B \\rangle}$ induced by the inclusion $\\langle B-B \\rangle \\hookrightarrow M$. Hence \\ref{thm:item-map} in \\autoref{thm_structure} holds by the diagram \\ref{eq:pf-thm1}.\n\n  To show \\ref{thm:item-fiber}, we use the following claim.\n\n  \\begin{claim}\\label{thm:gammaTM-stein-fact}\n    The morphism $ \\gamma|_{T_M} =\\varphi_B$ is decomposed as\n    \\begin{align*}    T_M \\stackrel{g_1}{\\longrightarrow} T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M} \\stackrel{g_2}{\\longrightarrow} T_{\\langle B-B \\rangle}\n    \\end{align*}\n    by $ \\langle B-B \\rangle \\subset \\langle B-B \\rangle_{\\mathbb {R}} \\cap M \\subset M$,\n    where $g_1$ is a morphism with reduced and irreducible fibers, and $g_2$ is a finite morphism.\n  \\end{claim}\n\n  \\begin{proof}[Proof of \\autoref{thm:gammaTM-stein-fact}]\n    By applying \\ref{item:lem_rk&deg:1} in \\autoref{lem_rk&deg} to $ \\langle B-B \\rangle \\subset M$,\n    we have the assertion.\n  \\end{proof}\n\n\n\n  The short exact sequence\n  \\[\n  0 \\rightarrow  \\langle B-B \\rangle_{\\mathbb {R}} \\cap M \\rightarrow M \\stackrel{\\pi}{\\rightarrow} M \/(\\langle B-B \\rangle_{\\mathbb {R}} \\cap M) \\rightarrow 0\n  \\]\n  induces a short exact sequence of algebraic tori\n  \\begin{align}\\label{eq_alg_gp}\n    1 \\rightarrow T_{M \/(\\langle B-B \\rangle_{\\mathbb {R}} \\cap M)} \\rightarrow T_M \\stackrel{g_1}{\\rightarrow} T_{ \\langle B-B \\rangle_{\\mathbb {R}} \\cap M} \\rightarrow 1.\n  \\end{align}\n  Hence $g_1^{-1}(1_{T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M}}) = T_{M \/(\\langle B-B \\rangle_{\\mathbb {R}} \\cap M)}$ holds for\n  the identity element $1_{T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M}} $ of the torus $T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M}$.\n  Applying \\autoref{lem_lattice_hom} to the surjection $\\pi:  M \\rightarrow M \/(\\langle B-B \\rangle_{\\mathbb {R}} \\cap M) $,\n  it holds that the closure\n  \\[\n  \\overline{g_1^{-1}(1_{T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M}}) }  \\subset X_A\n  \\]\n  is projectively equivalent to $ X_{\\pi(A)}$.\n\n  Let $F $ be an irreducible component of any fiber of $\\gamma |_{T_M}$ with the reduced structure.\n  From \\autoref{thm:gammaTM-stein-fact},\n  $F$ is a fiber of $ g_1$.\n  By \\ref{eq_alg_gp},\n  $F$ is the translation of $g_1^{-1}(1_{T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M}}) = T_{M \/(\\langle B-B \\rangle_{\\mathbb {R}} \\cap M)} $\n  by an element of $ T_M$.\n  Hence the closure $ \\overline{F}$ is projectively equivalent to $ X_{\\pi(A)}$\n  by \\autoref{lem_lattice_hom}.\n\\end{proof}\n\n\n\n\n\\begin{cor}\\label{cor_rk&deg}\n  Let $A, M, \\gamma, B$ be as in \\autoref{thm_structure}.\n  Let $\\gamma |_{T_M} = g_2 \\circ g_1$ be the decomposition of $\\gamma |_{T_M}$ in \\autoref{thm:gammaTM-stein-fact}.\n  Then the following hold.\n\n\n  \\begin{enumerate}\n  \\item\\label{item:rkdeg-rk} The rank of $\\gamma$ is equal to $\\dim (\\Aff_{\\Bbbk} (B))$.\n    In particular, $\\gamma $ is separable if and only if $ \\dim (\\Aff_{\\Bbbk} (B)) = \\rank (\\Aff (B))$.\n\n  \\item\\label{item:rkdeg-deg} Assume $p = \\chara\\Bbbk > 0$. Then we have\n    \\[\n    \\deg(g_2) = [\\lin{B-B}_{\\mathbb{R}} \\cap M : \\lin{B-B}].\n    \\]\n    For the maximum integer $s \\geq 0$ such that $p^s \\mid \\deg(g_2)$,\n    the separable degree and the inseparable degree of $g_2 $\n    are $\\deg(g_2)\/p^s $ and $ p^s$ respectively.\n    Hence the number of the irreducible components of a general fiber of $\\gamma$\n    is equal to $\\deg(g_2)\/p^s$, which is coprime to $p$.\n\n  \\end{enumerate}\n\\end{cor}\n\nWe note that $\\dim (\\Aff_{\\Bbbk}(B) )= \\dim \\lin{B-B}_{\\Bbbk}$  holds since $ \\Aff_{\\Bbbk}(B)$ is a parallel translation of $\\lin{B-B}_{\\Bbbk} $ in $M_{\\Bbbk}$.\n\n\\begin{proof}[Proof of \\autoref{cor_rk&deg}]\n  We apply \\autoref{lem_rk&deg} to $ M'' =\\langle B-B \\rangle \\subset M$.\n  In this case,\n  $g$ in \\autoref{lem_rk&deg} is $\\gamma |_{T_M}$.\n  Since the image of $M''_{\\Bbbk} \\rightarrow  M_{\\Bbbk}$ is nothing but $\\langle B-B \\rangle_{\\Bbbk} \\subset M_{\\Bbbk}$,\n  it holds that $ \\rk (\\gamma) = \\dim \\langle B-B \\rangle_{\\Bbbk} =  \\dim \\Aff_{\\Bbbk} (B)$ by \\ref{item:lem_rk&deg:2} in \\autoref{lem_rk&deg}. This implies \\ref{item:rkdeg-rk}.\n  On the other hand,\n  \\ref{item:rkdeg-deg} follows from \\ref{item:lem_rk&deg:3} in \\autoref{lem_rk&deg} directly.\n\\end{proof}\n\nIn the following examples,\nwe denote the separable degree and the inseparable degree of a finite morphism $f$ by $\\deg_s(f)$ and $\\deg_i(f)$ respectively.\n\n\\begin{ex}\\label{ex_intro:rkdeg:2}\n  Let $A, B$ be as in \\autoref{ex_intro} and\n  assume $\\chara\\Bbbk = 2$.\n  Then $\\rk (\\gamma) = \\dim \\lin{B-B}_{\\Bbbk} = 0$.\n  From\n  \\ref{item:rkdeg-deg} of \\autoref{cor_rk&deg}, we can calculate $\\deg(g_2) = \\deg_i(g_2) = 2$ and $\\deg_s(g_2) = 1$.\n\\end{ex}\n\n\\begin{ex}\\label{thm:kaji's-ex}\n  Kaji's example \\cite[Example 4.1]{Kaji1986} can be interpreted as follows.\n\n  Let $A =   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{0, 1, cp^m, cp^m+1} \\subset M = \\mathbb{Z}^1$,\n  where $c, m$ are positive integers and $p = \\chara\\Bbbk > 0$.\n  Assume that $c$ and $p$ are relatively prime.\n  Then\n  \\[\n  B =   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{1, cp^m+1, 2cp^m+1},\\\n  \\lin{B-B} = \\lin{cp^m},\\\n  \\lin{B-B}_{\\mathbb{R}} \\cap M\n  = M.\n  \\]\n  Therefore $\\deg(\\gamma) = cp^m$, $\\deg_i(\\gamma) = p^m$, $\\deg_s(\\gamma) = c$.\n  In particular, a general fiber of $\\gamma$ with the reduced structure is equal to a set of $c$ points.\n  \\begin{figure}[htbp]\n    \\[\n    \\begin{xy}\n      (-15,0)=\"1\",(110,0)=\"2\",\n      (0,0)*{\\bullet},\n      (12,0)*{\\bullet},\n      (40,0)*{\\bullet},\n      (52,0)*{\\bullet},\n      (12,0)*{\\mbox{\\Large$\\times$}},\n      (52,0)*{\\mbox{\\Large$\\times$}},\n      (92,0)*{\\mbox{\\Large$\\times$}},\n      (0,5)*{0},\n      (12,5)*{1},\n      (40,5)*{cp^m},\n      (55,5)*{cp^m+1},\n      (92,5)*{2cp^m+1},\n      (66,13)*{\\bullet\\;A},\n      (79,13)*{\\mbox{\\Large$\\times$}\\,B},\n      \\ar \"1\";\"2\"\n    \\end{xy}\n    \\]\n    \\caption{Kaji's example.}\n  \\end{figure}\n\n\\end{ex}\n\n\nAs in \\ref{item:rkdeg-deg}  of \\autoref{cor_rk&deg},\nthe number of the irreducible components of a general fiber of $\\gamma$ is coprime to $p = \\chara\\Bbbk$\nin the toric case.\nHowever, the number can be a multiple of $p$ in the non-toric case.\nThe authors learned the following example from H.~Kaji\nin a personal communication.\n\n\\begin{rem}[{A variant of \\cite[Example 4.1]{Kaji1986}}]\n  \\label{thm:a-vari-kaji}\n  We take $f: \\mathbb{P}^1 \\dashrightarrow \\mathbb{P}^1$ to be a separable rational map whose degree is a multiple of $p$,\n  and locally parameterize $g$ by\n  $t \\mapsto [1 : f_1(t)]$.\n  We set $\\phi: \\mathbb{P}^1 \\dashrightarrow \\mathbb{P}^3$ to be \n  the rational map\n  which is locally parameterized by\n  \\[\n  t \\mapsto [1: t: f_1^p: t \\cdot f_1^p]\n  \\]\n  and set $X \\subset \\mathbb{P}^3$ to be the projective curve $\\overline{\\im(\\phi)}$.\n  Then a general fiber of $\\gamma: X \\dashrightarrow \\mathbb{G}(1, \\mathbb{P}^3)$ with the reduced structure\n  is equal to a set of $\\deg(f)$ points.\n  In order to show this, we may check that\n  $\\gamma$ is locally parameterized by\n  $t \\mapsto f_1^p(t)$, whose separable degree is equal to $\\deg(f)$.\n  We leave the details to the reader.\n\n  \n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe following are examples of toric varieties $X_A \\subset \\mathbb{P}^N$ with codimension $1$ or $2$ such that the Gauss maps are birational.\nLater, these examples will be used\nin the proof of \\autoref{cor_rk&fiber}.\n\n\\begin{ex}\\label{prep-cor_rk&fiber}\n  Assume $p = \\chara \\Bbbk >0$.\n  Let $e_1, \\dots, e_n$ be the standard basis of $M:=\\mathbb {Z}^n$, and set\n  \\[\n  A = \\{ 0, e_1,\\ldots,e_n, a_1 e_1 + \\ldots + a_n e_n \\}  \\subset \\mathbb {Z}^n\n  \\]\n  for $a_1 , \\ldots, a_n \\in \\mathbb {Z}$ such that\n  \\begin{itemize}\n  \\item[(i)] $a_1, \\ldots, a_n \\not \\equiv 0 \\mod p$, and \n  \\item[(ii)] $a_1 + \\cdots + a_n  \\not \\equiv 1 \\mod p$.\n  \\end{itemize}\n  Then the Gauss map of the toric hypersurface $X_A \\subset \\mathbb{P}^{n+1}$ is birational\n \n  as follows.\n\n  By condition (i),\n  $A \\setminus \\{ e_i\\}$ spans $M_{\\Bbbk}=\\Bbbk^n$ as an affine space for any $1 \\leq i \\leq n$.\n  Hence it holds that\n  \\begin{equation}\\label{eq:e1+..:1}\n    e_1 + \\cdots + e_n + a_1 e_1 + \\ldots + a_n e_n  - e_i \\in B.\n  \\end{equation}\n  By condition (ii),\n  $A \\setminus \\{ 0\\}$ spans $M_{\\mathbb {R}}=\\Bbbk^n$  as an affine space.\n  Hence \n  \\begin{equation}\\label{eq:e1+..:2}\n    e_1 + \\cdots + e_n + a_1 e_1 + \\ldots + a_n e_n \\in B.\n  \\end{equation}\n  Considering the difference between \\ref{eq:e1+..:1} and \\ref{eq:e1+..:2},\n  we have $e_i \\in B-B$ for any $i$. Therefore  $\\langle B-B \\rangle = M$.\n  By \\ref{thm:item-map} in \\autoref{thm_structure},\n  the Gauss map of $X_A$ is birational.\n\n\n  For example,\n  the conditions (i) and (ii) are satisfied for $a_1= \\cdots = a_n=1$ (resp.\\ $a_1= \\cdots = a_n= -1$)\n  when $n \\not \\equiv 1 \\mod p$ (resp.\\ $n \\not \\equiv  -1 \\mod p$).\n  Hence there exists a toric hypersurface in $\\mathbb{P}^{n+1}$ whose Gauss map is birational\n  if $\\chara \\Bbbk \\neq 2$ or $n$ is even.\n\\end{ex}\n\n\nAs we will see later in \\autoref{rem_rk_in_char2},\nthe Gauss map of any (not necessarily toric) hypersurface in $\\mathbb{P}^{n+1}$ cannot be birational if $\\chara \\Bbbk =2$ and $n$ is odd.\n\n\n\n\n\n\\begin{ex}\\label{ex_char2}\n  Assume $n \\geq 2$.\n  Let\n  \\[\n  A =\\{0, e_1, \\ldots, e_n, e_1 +e_2, e_2+e_3+ \\cdots + e_n \\} \\subset M:=\\mathbb {Z}^n.\n  \\]\n \n \n  For $S := \\{ 0, e_1, \\ldots, e_n, e_1+e_2\\} \\subset A$, each of\n  \\[\n  S\\, \\setminus \\{e_1 + e_2\\},\\ S\\, \\setminus \\{e_1\\},\\ S\\, \\setminus \\{e_2\\}\n  \\]\n  spans the affine space $M_{\\Bbbk}$.\n \n  Hence\n  it holds that $e_1,e_2 \\in  B-B $.\n  In addition,\n  $\\{0, e_1, \\ldots, e_n, e_2+e_3+ \\cdots + e_n \\} \\setminus \\{e_i\\}$ spans the affine space $M_{\\Bbbk}$ for $2 \\leq i \\leq n$.\n  Thus we have $e_i -e_2 \\in  B-B $ for $2 \\leq i \\leq n$.\n  Hence $\\langle B-B \\rangle = M$ and the Gauss map of $X_A \\subset \\mathbb{P}^{n+2}$ is birational. \n\\end{ex}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Degenerate Gauss maps}\n\\label{sec:degen-gauss-maps}\n\n\n\n\n\\subsection{Developability criterion}\n\\label{sec:criterion-degeneracy}\nBy Theorem \\ref{thm_structure},\nthe Gauss map of any given toric variety and\nits general fibers can be explicitly determined\nfrom the computation of $B$ and $\\pi(A)$ as in \\autoref{ex_intro}.\nHowever,\nit is not so clear for what kind of $A$ the Gauss map $\\gamma$ of $X_A \\subset \\PP^N$ is degenerate,\ni.e.,\n$\\rank  \\langle  B-B \\rangle < n$.\nThe following result tells us\na condition that $\\gamma$ is degenerate.\n\n\\begin{prop}\\label{main_thm}\n  Let $M,A,\\pi$ be as in Theorem \\ref{thm_structure}.\n  Let $\\tilde\\pi : M \\rightarrow \\tilde{M}'$ be a surjective homomorphism of free abelian groups.\n  Then $\\tilde\\pi$ \n  factors through $\\pi: M \\rightarrow M'$,\n  i.e., $\\langle B-B \\rangle \\subset \\ker \\tilde \\pi$ \n  if and only if\n  \\[\n  \\sum_{j=0}^{\\tilde{N}'} \\dim \\Aff_{\\Bbbk}(A_j) =n- \\tilde{N}',\n  \\]\n  where $\\tilde{N}' := \\# \\tilde\\pi(A) -1$,\n  $\\tilde\\pi(A ) = \\{ \\tilde u'_0,\\ldots, \\tilde u'_{\\tilde{N}'}\\}$, and $A_j= \\tilde\\pi^{-1}(\\tilde u'_j) \\cap A$.\n\\end{prop}\n\nFor a surjective homomorphism $\\tilde\\pi: M \\rightarrow \\tilde M'$\nof free abelian groups,\nthe toric variety\n$X_A$ is covered by the translations of\n$\\overline{T_{\\tilde M'}} = X_{\\tilde\\pi(A)}$ by elements of $T_M$\ndue to \\autoref{lem_lattice_hom}.\nThe homomorphism $\\tilde\\pi$ factors through $\\pi$\nif and only if\n$\\overline{T_{\\tilde M'}}$ (or equivalently,\nthe translation of $\\overline{T_{\\tilde M'}}$ by any element of $T_M$)\nis  contracted to one point by $\\gamma$.\n\n\n\n\nIn general, a covering family $  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{F_{\\alpha}}$ of a projective variety $X \\subset \\PP^N$ by subvarieties $F_{\\alpha} \\subset X$ is said to be \\emph{developable} if\n$F_{\\alpha}$ is contracted to one point by the Gauss map of $X$\n(i.e., $\\mathbb{T}_xX$ is constant on general $x \\in F_{\\alpha}$) for general $\\alpha$.\n\\autoref{main_thm} is regarded as\na toric version of the developability criterion\n(cf.\\ \\cite[2.2.4]{FP}, \\cite{Fukasawa2005}, \\cite[\\textsection{}4]{expshr}; see also \\autoref{sec:separable-gauss-maps} for the separable case).\n\n\n\nBefore the proof,\nwe illustrate \\autoref{main_thm} by an example.\n\n\n\n\\begin{ex}\\label{ex_intro2}\n  Let $A \\subset \\mathbb {Z}^2$ be as in \\autoref{ex_intro}.\n  For the projection $\\tilde \\pi : \\mathbb {Z}^2 \\rightarrow  \\mathbb {Z}^1$ to the second factor,\n  $\\tilde \\pi(A) = \\{0,1, -1\\}$. Then\n  $A_0 =\\tilde \\pi^{-1}(0) \\cap A,\\, A_1 =\\tilde \\pi^{-1}(1) \\cap A,\\, A_2 =\\tilde \\pi^{-1}(-1) \\cap A$\n  are given by\n  \\[\n  A_0 = \\left\\{\n    \\begin{bmatrix}\n      0 \\\\ 0\n    \\end{bmatrix}\n  \\right\\},\\,\n  A_1 = \\left\\{\n    \\begin{bmatrix}\n      0 \\\\ 1\n    \\end{bmatrix}\n  \\right\\},\\,\n  A_2 = \\left\\{\n    \\begin{bmatrix}\n      1 \\\\ -1\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      -1 \\\\ -1\n    \\end{bmatrix}\n  \\right\\}.\n  \\]\n  Thus\n  \\[\n  \\sum_{j=0}^{\\tilde{N}'} \\dim \\Aff_{\\Bbbk}(A_j) = \\left\\{\n    \\begin{array}{cl} \n      1  & \\text{when }   \\chara \\Bbbk \\neq2,\\\\ 0 & \\text{when } \\chara \\Bbbk=2.\\\\\n    \\end{array} \\right.\n  \\]\n  On the other hand,\n  $n-\\tilde{N}'=2-2=0$ holds.\n  Hence the equality in \\autoref{main_thm} holds if and only if $\\chara \\Bbbk =2$.\n  Note that, in this example, the above\n  $\\tilde\\pi$ can be identified with the natural projection\n  $\\pi: M \\rightarrow M'$ in \\autoref{thm_structure} when $\\chara \\Bbbk=2$.\n\\end{ex}\n\n\nTo prove \\autoref{main_thm}, we need the following lemma.\n\n\n\n\\begin{lem}\\label{thm:AffAj-eq}\n  In the setting of \\autoref{main_thm},\n  the homomorphism $\\tilde \\pi$ factors through $\\pi$ if and only if\n  \\[\n  \\Aff_{\\Bbbk}(A_j) = \\Aff_{\\Bbbk}(  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} \\cap A_j)\n  \\quad (\\RNj)\n  \\]\n  for any $u_{i_0}, \\dots, u_{i_n} \\in A$ which span the affine space $M_{\\Bbbk}$.\n\n\\end{lem}\n\n\\begin{proof}\n  First, we show the ``only if'' part.\n  Assume that $\\tilde \\pi$ factors through $\\pi$.\n  The inclusion ``$\\supset$'' always holds.\n  We show ``$\\subset$''.\n  Let $u \\in A_j$.\n  Since $u_{i_0}, \\dots, u_{i_n}$ span the affine space $M_{\\Bbbk}$,\n  we can write $u = \\sum_{k=0}^n c_{i_k} u_{i_k}$\n  with $\\sum_{k=0}^n c_{i_k} = 1$ and $c_{i_k} \\in \\Bbbk$.\n  For any $k$ with $c_{i_{k}} \\neq 0$, we have $u_{i_{k}} \\in A_j$ as follows.\n\n  If $c_{i_{k}} \\neq 0$, we find that $\\{u\\} \\cup   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_{k'}}}_{0 \\leq k' \\leq n, k' \\neq k}$ span\n  the affine space $M_{\\Bbbk}$. Thus $u + \\sum_{0 \\leq k' \\leq n, k' \\neq k} u_{i_{k'}} \\in B$, \n  and then $u - u_{i_{k}} \\in B-B$.\n  Since $\\tilde \\pi$ factors through $\\pi$, we have $ \\tilde\\pi(u_{i_{k}}) = \\tilde\\pi(u) = \\tilde{u}'_j$ by $u \\in A_j = \\tilde{\\pi}^{-1}(\\tilde{u}'_j) \\cap A$;\n  hence $u_{i_{k}} \\in A_j$.\n  \n  Thus we have $u = \\sum_{u_{i_k} \\in A_j} c_{i_k} u_{i_k}$ with $\\sum_{u_{i_k} \\in A_j} c_{i_k} =1 $,\n  i.e.,\n  $u $ is contained in $ \\Aff_{\\Bbbk}(  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} \\cap A_j)$.\n  This implies the assertion.\n\n  \\vspace{2mm}\n  Next, we show the ``if'' part.\n  For any $b \\in B$,\n  we can write $b= u_{i_0} + \\cdots + u_{i_n}$ with $u_{i_0} , \\ldots , u_{i_n} \\in A$ which span the affine space $M_{\\Bbbk}$.\n  Since the $n$-dimensional affine space $M_{\\Bbbk} $ is spanned by $n+1$ elements $u_{i_0} , \\ldots , u_{i_n} \\in A$,\n  we have\n  \\[\n  \\# (  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, \\dots, u_{i_n} }\\cap A_j) = \\dim  \\Aff_{\\Bbbk}(  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} \\cap A_j) +1.\n  \\] \n  Hence $ \\# (  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, \\dots, u_{i_n} }\\cap A_j)  = \\dim \\Aff_{\\Bbbk}(A_j) + 1 $ holds for each $0 \\leq j \\leq \\tilde{N}'$ by assumption.\n  In particular,\n  it holds that\n  \\[\n  \\tilde \\pi(b)= \\tilde \\pi (u_{i_0} + \\cdots + u_{i_n}) = \\sum_{0 \\leq j \\leq \\tilde{N}'} (\\dim \\Aff_{\\Bbbk}(A_j) + 1) \\cdot u'_j ,\n  \\]\n  which does not depend on $b \\in B$.\n  Thus we have $B-B \\subset \\ker \\tilde \\pi$,\n  i.e.,\n  $\\tilde \\pi $ factors through $\\pi$.\n\\end{proof}\n\n\\begin{rem}\\label{thm:B-B-wr-Aj-Aj}\n  Assume that $\\tilde\\pi$ factors through $\\pi$.\n  From the ``if'' part in the above proof,\n  $\\# (  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, \\dots, u_{i_n} }\\cap A_j)  $ does not depend on $u_{i_0}, \\dots, u_{i_n} \\in A$ which span the affine space $M_{\\Bbbk}$.\n  Thus each element of $B -B$ is written as a linear combination\n  of elements of $\\bigcup_{\\RNj} (A_j-A_j)$.\n\\end{rem}\n\n\n\n\n\n\n\n\n\\begin{proof}[Proof of \\autoref{main_thm}]\n  Let us take $u_{i_0}, \\dots, u_{i_n} \\in A$ which span the affine space $M_{\\Bbbk}$.\n  Then the latter condition of \\autoref{thm:AffAj-eq} holds\n  if and only if\n  \\begin{equation}\\label{eq:sub-AffAj}\n    \\dim \\Aff_{\\Bbbk}(A_j) = \\#(  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} \\cap A_j) - 1\n  \\end{equation}\n  for any $\\RNj$ (we note that ``$\\geq$'' always holds).\n  On the other hand, since $A = A_0 \\sqcup A_1 \\sqcup \\dots \\sqcup A_{\\tilde{N}'}$, we have\n  \\[\n  \\sum_{\\RNj} (\\#(  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} \\cap A_j) -1)\n  = \\#   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} - (\\tilde{N}' + 1) = n-\\tilde{N}'.\n  \\]\n  Hence $\\sum_{\\RNj[\\tilde{N}']} \\dim \\Aff_{\\Bbbk}(A_j) =n- \\tilde{N}'$ holds\n  if and only if the equality~\\ref{eq:sub-AffAj} holds for any $\\RNj$.\n  Therefore this proposition follows from \\autoref{thm:AffAj-eq}.\n\\end{proof}\n\n\\begin{cor}\\label{cor_fiber_separable}\n  Let $A, \\pi: M \\rightarrow M'$ be as in \\autoref{thm_structure}.\n  Then it holds that $\\rk (\\gamma) \\leq n-(\\#\\pi(A)-1)$. Moreover, if $\\gamma$ is separable,\n  then we have $\\# \\pi(A) = \\rank M' +1$, which means $ X_{\\pi(A)}$ is a linear projective space of dimension $\\rank M'$.\n\\end{cor}\n\n\\begin{proof}\n  We apply \\autoref{main_thm} to the homomorphism $\\pi$. Then\n  it holds that $\\sum_{j} \\dim \\lin{A_j-A_j}_{\\Bbbk} =n- (\\#\\pi(A)-1)$.\n  From \\autoref{cor_rk&deg}, we have $\\rk (\\gamma) = \\dim \\Aff_{\\Bbbk} (B) = \\dim \\lin{B-B}_{\\Bbbk}$.\n  By \\autoref{thm:B-B-wr-Aj-Aj},\n  $\\lin{B-B}$ is contained in the space $\\lin{  \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{A_j-A_j}_j}$.\n  Thus $\\rk (\\gamma) \\leq n- (\\#\\pi(A)-1)$ holds.\n\n  If $\\gamma$ is separable, $\\rk (\\gamma) = \\rank \\langle B-B \\rangle = n - \\rank (M')$ holds by \\autoref{cor_rk&deg}.\n  Hence $\\rank(M') \\geq \\#\\pi(A)-1$ holds by the above inequality.\n  The converse inequality ``$\\leq$'' always holds since $\\pi(A)$ spans the affine lattice $M'$.\n  Since $\\pi(A)$ spans the affine lattice $ M'$,\n  the equality $\\# \\pi(A) = \\rank M' +1$ means $ X_{\\pi(A)}$ is a linear projective space of dimension $\\rank M'$.\n\\end{proof}\n\n\n\\begin{rem}\n  The equality ``$\\rk (\\gamma) = n-(\\#\\pi(A)-1)$'' does \\emph{not} hold in general.\n  For example, set $A=\\{0,1,p\\}  \\subset M = \\mathbb {Z}^1$ with $p=\\chara \\Bbbk >0$.\n  Then we have\n  \\[\n  B=\\{1, p+1\\}, \\quad \\lin{B-B} = \\lin{p} , \\quad  \\lin{B-B}_{\\mathbb{R}} \\cap M =M.\n  \\]\n  Thus $\\pi : M=\\mathbb {Z}^1 \\rightarrow M\/(\\lin{B-B}_{\\mathbb{R}} \\cap M) =\\{0 \\}$ is the zero map.\n  Here we have $\\rk (\\gamma) = \\dim \\lin{B-B}_{\\Bbbk} = 0$\n  and \n  $n-(\\#\\pi(A) - 1) = 1-(1-1) = 1$.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\\subsection{Separable Gauss maps, Cayley sums, and joins}\n\\label{sec:separable-gauss-maps}\n\nIn this subsection,\nwe study the case when the Gauss map is separable,\nand prove Corollary \\ref{cor_join} in the characteristic zero case.\n\n\n\n\n\n\\begin{defn}\\label{def_cayley}\n  Let $l \\leq n$ be non-negative integers. Let $e_1,\\ldots,e_l$ be the standard basis of $ \\mathbb {Z}^l$.\n  For finite sets $A^0,\\ldots,A^l \\subset \\mathbb {Z}^{n-l}$,\n  the Cayley sum $A^0 * \\cdots * A^l  $ of $A^0,\\ldots,A^l$ is defined to be\n  \\[\n  A^0 * \\cdots * A^l  := (A^0 \\times \\{0\\}) \\cup (A^1 \\times \\{e_1 \\}) \\cup \\cdots (A^l \\times \\{e_l\\})  \\subset \\mathbb {Z}^{n-l} \\times \\mathbb {Z}^l.\n  \\]\n\\end{defn}\n\nLet $A$ be the Cayley sum of $A^0,\\ldots,A^l \\subset \\mathbb {Z}^{n-l}$,\nand assume that $A$ spans the affine lattice $ \\mathbb {Z}^{n-l} \\times \\mathbb {Z}^l$.\nFor the projection $\\tilde{\\pi} : \\mathbb {Z}^{n-l} \\times \\mathbb {Z}^l \\rightarrow \\mathbb {Z}^l $\nto the second factor, $X_{\\tilde{\\pi}(A)}$ is an $l$-plane\nsince $\\tilde\\pi(A) =   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{0, e_1, \\dots, e_l}$.\nBy \\autoref{lem_lattice_hom}, $X_A$ is covered by $l$-planes,\nwhich are translations of $X_{\\tilde{\\pi}(A)} = \\overline{T_{\\mathbb {Z}^l}}$.\nFor this $\\tilde\\pi$,\n$\\tilde N'$ in \\autoref{main_thm} is equal to $l$.\nThus,\nthe subtorus $ T_{\\mathbb {Z}^l} \\subset T_{\\mathbb {Z}^{n-l} \\times \\mathbb {Z}^l}$ is contracted to one point by the Gauss map of $X_A$\nif and only if\n\\begin{align}\\label{eq_condition_for_sum}\n  \\sum_{j=0}^{l} \\dim \\Aff_{\\Bbbk}(A^j) =n- l.\n\\end{align}\nIn other words,\n\\ref{eq_condition_for_sum} is the condition for the developability\nof the covering family obtained by translations of $\\overline{T_{\\mathbb {Z}^l}}$.\nIn fact,\nany toric variety with separable Gauss map is described by a Cayley sum with the condition~\\ref{eq_condition_for_sum}, as follows.\n\n\n\n\n\n\n\n\n\n\\begin{thm}\\label{thm:subvar-in-join}\n  Let $A, M$ be as in \\autoref{thm_structure}.\n  Assume that the Gauss map $\\gamma$ of $X_A \\subset \\PP^N$ is separable, and set $l = \\delta_{\\gamma}(X)$.\n  Then there exist finite subsets $A^0, \\dots, A^{l} \\subset \\mathbb{Z}^{n-l}$\n  with $\\sum_{j=0}^{l} \\dim \\Aff_{\\Bbbk}(A^j) =n- l$\n  such that $A$ is identified with the Cayley sum of $A^0, \\dots, A^{l} \\subset \\mathbb{Z}^{n-l}$ under some affine isomorphism $M \\simeq \\mathbb{Z}^{n-l} \\times \\mathbb{Z}^{l}$.\n\\end{thm}\n\n\n\n\n\n\\begin{proof}\n  Let $A =   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_0, u_1, \\dots, u_N}$ and $\\pi : M \\rightarrow M'$ be\n  as in \\autoref{thm_structure},\n  where we may assume $u_0=0 \\in M$.\n  From \\autoref{thm_structure} and \\autoref{cor_fiber_separable}, it follows that\n  $\\rank M' = l$ and $\\# \\pi (A) = l+1$ since $\\gamma$ is separable by assumption.\n  Set $\\pi (A)=\\{u'_0, \\ldots,u'_l \\} $.\n  Without loss of generality,\n  we may assume that $u'_0 =\\pi(u_0)=0 \\in M'$.\n  Then $ u'_1, \\ldots,u'_l $ form a basis of $M'$\n  since $\\pi(A) $ spans the affine lattice $M' \\simeq \\mathbb {Z}^l$.\n  Set $A_j = \\pi^{-1}(u'_j) \\cap A$.\n\n\n\n  Fix a splitting $s : M' \\rightarrow M$ of the short exact sequence $0 \\rightarrow \\ker \\pi \\rightarrow M \\stackrel{\\pi}{\\rightarrow} M' \\rightarrow 0$.\n  Then the induced isomorphism\n  \\[\n  M \\stackrel{\\sim}{\\rightarrow} \\ker \\pi \\times M' \\quad : \\quad u \\mapsto (u - s ( \\pi(u)), \\pi(u))\n  \\]\n  gives an identification of $A \\subset M$ with\n  \\begin{equation}\\label{eq:expressA-Cayley}\n    \\bigcup_{j=0}^l \\left( A^j  \\times  \\{u'_j \\}  \\right) \\subset \\ker \\pi \\times M' ,\n  \\end{equation}\n  where $A^j := A_j -s(u'_j) \\subset \\ker \\pi$ is the parallel translation of $A_j$ by $s(u'_j)$.\n  Since\n \n  $ u'_1, \\ldots,u'_l $ form a basis of $M'$, $u_0' = 0$,\n  and $\\ker \\pi \\simeq \\mathbb{Z}^{n-l}$,\n  this theorem follows.\n\\end{proof}\n\n\nIn order to prove \\autoref{cor_join}, we consider a relation between\nCayley sums and joins.\nFor projective varieties $X_1, \\dots, X_m \\subset \\PP^N$,\nwe define the \\emph{join} of $X_1, \\dots, X_m$ to be the closure of\n$\\bigcup_{x_1 \\in X_1, \\dots, x_m \\in X_m} \\Lambda_{x_1, \\dots, x_m} \\subset \\PP^N$,\nwhere $\\Lambda_{x_1, \\dots, x_m}$ is the linear variety spanned by the points $x_1, \\dots, x_m$.\n\n\n\\begin{lem}\\label{lem_torus_inv_sub}\n  Let $A \\subset \\mathbb{Z}^{n-l} \\times \\mathbb{Z}^l$ be the Cayley sum of $A^0, \\dots, A^l \\subset \\mathbb{Z}^{n-l}$ with $\\Aff(A) = \\mathbb{Z}^{n-l} \\times \\mathbb{Z}^l$.\n  Then the following hold.\n  \\begin{enumerate}[\\quad \\normalfont (a)]\n  \\item \\label{lem_torus_inv_sub:1}\n    $X_{A^0}, \\ldots,X_{A^l}$ are embedded into $X_A$ as torus invariant subvarieties,\n    and they are mutually disjoint.\n\n  \\item \\label{lem_torus_inv_sub:2}\n    $X_{A}$ is contained in the join of $X_{A^0}, \\ldots,X_{A^l} \\subset \\PP^N$,\n    and the codimension of $X_{A}$ in the join is\n    $l - n + \\sum_{j=0}^l \\rank \\Aff(A^j)$.\n\n  \\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n  \\begin{inparaenum}[(a)]\n  \\item \n    Write $A^j = \\{u^j_{0}, \\ldots,u^j_{N_j}\\} \\subset \\mathbb {Z}^{n-l}$ for $ N_j= \\# A^j -1$.\n    Set $N=\\# A -1 = \\sum_{j=0}^l (N_j+1) -1$\n    and let $\\{ X^j_{i}\\}_{0 \\leq i \\leq N_j, 0 \\leq j \\leq l}$ be the homogeneous coordinates on $ \\mathbb{P}^{N}$.\n    By the definition of the Cayley sum $A$,\n    it holds that\n    \\begin{equation}\\label{eq:phiAyz}\n      \\begin{aligned}\n        \\varphi_{A }(z, w) &= [w_j z^{u_i^{j}}]_{0 \\leq i \\leq N_j, 0 \\leq j \\leq l} \\\\\n        &= [ w_0z^{u^0_{0}} : \\cdots : w_0 z^{u^0_{N_0}} :\n        w_1 z^{u^1_{0}} : \\cdots : w_1 z^{u^1_{N_1}} :\n        \\cdots: w_l z^{u^l_{0}} : \\cdots : w_l z^{u^l_{N_l}}]\n      \\end{aligned}\n    \\end{equation}\n    in $\\mathbb{P}^N$\n    for $(z, w)=(z_1,\\ldots,z_{n-l}, w_1,\\ldots,w_l)\n    \\in (\\KK^{\\times})^{n-l} \\times (\\KK^{\\times})^{l} = T_{\\mathbb {Z}^{n-l} \\times \\mathbb {Z}^{l}}$,\n    where we set $w_0 = 1$.\n    For fixed $0 \\leq j \\leq l$ and $z \\in (\\KK^{\\times})^{n-l} = T_{\\mathbb {Z}^{n-l}}$,\n    $\\phi_A(z, w)$ converges to\n    \\begin{equation}\\label{eq:phiAjz}\n      [0:\\cdots : 0 : z^{u^j_{0}} : \\cdots : z^{u^j_{N_j}} : 0:\\cdots : 0] \\in \\mathbb{P}^N\n    \\end{equation}\n    when $w_k\/w_j \\rightarrow 0$ for $0 \\leq k \\neq j \\leq l$.\n    Thus, the point \\ref{eq:phiAjz} is\n    contained in the closure $\\overline{\\varphi_{A }(T_{\\mathbb {Z}^l \\times \\mathbb {Z}^{n-l}} )}=X_A$.\n    In other words, $\\varphi_{A^j}(z) = [z^{u^j_{0}} : \\cdots : z^{u^j_{N_j}} ] \\in \\mathbb{P}^{N^j} $ is contained in $X_A$ for any $z \\in (\\KK^{\\times})^{n-l} = T_{\\mathbb {Z}^{n-l}}$\n    by embedding $\\mathbb{P}^{N_j}$ into $\\mathbb{P}^N$ as\n    \\begin{equation}\\label{eq:PNj=Xji}\n      \\mathbb{P}^{N_j}  = (X^{j'}_{i}=0)_{j' \\neq j, 0 \\leq i \\leq N_{j'}} \\subset  \\mathbb{P}^{N} .\n    \\end{equation}\n    Since $X_{A^j}$ is the closure of $\\varphi_{A^j}(T_{\\mathbb {Z}^{n-l}})$,\n    $X_{A^j} \\subset \\mathbb{P}^{N_j}$ is contained in $X_A$.\n\n\n    The action of $T_{\\mathbb {Z}^{n-l} \\times \\mathbb {Z}^{l}}  $ on $X_A$ is described as\n    \\[\n    (z,w) \\cdot [X^j_i]_{0 \\leq i \\leq N_j, 0 \\leq j \\leq l} = [ w_j z^{u^j_i} X^j_i]_{0 \\leq i \\leq N_j, 0 \\leq j \\leq l}\n    \\]\n    for $ (z,w ) = (z_1,\\ldots,z_{n-l}, w_1,\\ldots,w_l) \\in  T_{\\mathbb {Z}^{n-l} \\times \\mathbb {Z}^{l}} $ and $ [X^j_i]_{0 \\leq i \\leq N_j, 0 \\leq j \\leq l} \\in X_A$,\n    where $w_0 = 1$ as before.\n    Therefore $X_{A^j} \\subset X_A$ is a torus invariant subvariety.\n    Since $\\mathbb{P}^{N_0}, \\ldots, \\mathbb{P}^{N_l} \\subset \\mathbb{P}^N$ are mutually disjoint by \\ref{eq:PNj=Xji},\n    so are $X_{A^0}, \\ldots, X_{A^l} \\subset X_A$.\n\n\n    \\vspace{1ex}\n    \\noindent\n  \\item\n    For $(z,w ) \\in  T_{\\mathbb {Z}^{n-l} \\times \\mathbb {Z}^{l}}$,\n    the image $\\varphi_{A}(z,w) \\in X_A \\subset \\PP^N$ is described by \\ref{eq:phiAyz}, and\n    $\\varphi_{A^j}(z) \\in X_{A^j} \\subset \\PP^N$ is described by \\ref{eq:phiAjz}\n    for each $0 \\leq j \\leq l$.\n    Hence $\\varphi_{A}(z,w)  $ is contained in\n    the $l$-plane spanned by $\\varphi_{A^0}(z), \\varphi_{A^1}(z), \\cdots, \\varphi_{A^l}(z) $.\n    Thus $X_{A}$ is contained in the join of $X_{A^0}, \\ldots,X_{A^l}$.\n    From \\ref{eq:PNj=Xji},\n    the dimension of the join of $X_{A^0}, \\ldots,X_{A^l}$ is\n    $l+ \\sum_{j=0}^l \\dim X_{A^j}$,\n    which is equal to $l + \\sum_{j=0}^l \\rank \\Aff(A^j)$.\n    Since $\\dim(X_A) = n$, the assertion about the codimension follows.\n  \\end{inparaenum}\n\\end{proof}\n\n\n\n\n\\begin{ex}\n  Let $A \\subset \\mathbb {Z}^2 \\times \\mathbb {Z}^1$ be the Cayley sum of\n  \\[\n  A^0 = \\Set{\\begin{bmatrix}\n      0 \\\\ 0\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      1 \\\\ 0\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      2 \\\\ 0\n    \\end{bmatrix}\n  }, \\\n  A^1 = \\Set{\\begin{bmatrix}\n      0 \\\\ 0\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      0 \\\\ 1\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      0 \\\\ 2\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      0 \\\\ 3\n    \\end{bmatrix}\n  }\n  \\subset \\mathbb {Z}^2.\n  \\] \n  Then it holds that\n  \\[\n  l - n + \\sum_{j=0}^l \\rank \\Aff(A^j) = 1 - 3 + (1+1) =0.\n  \\]  \n  Hence $X_A$ is the join of $X_{A^0}$ and $X_{A^1}$.\n  In fact,\n  \\[\n  X_A = \\overline{ \\big\\{ [1: x: x^2 : w : w y : w y^2 : w y^3]  \\, | \\,  (x,y,w) \\in (\\KK^{\\times})^3= T_{\\mathbb {Z}^2 \\times \\mathbb {Z}^1} \\big\\} } \\subset \\mathbb{P}^6,\n  \\]\n  and the conic $X_{A^0} \\subset \\mathbb{P}^2$ and the twisted cubic $X_{A^1} \\subset \\mathbb{P}^3$ are embedded into $X_A$ as\n  \\begin{align*}\n    X_{A^0} &=\\overline{ \\big\\{ [1: x: x^2 : 0 : 0 : 0 : 0] \\, | \\, x \\in \\KK^{\\times} \\big\\} } \\hspace{1mm} \\subset X_A, \\\\\n    X_{A^1} &=\\overline{ \\big\\{  [0:0:0 : 1 : y : y^2 : y^3] \\, | \\, y \\in \\KK^{\\times} \\big\\} }  \\subset X_A.\n  \\end{align*}\n\\end{ex}\n\n\\begin{rem}\\label{thm:subvar-in-join:rem}\n  In \\autoref{thm:subvar-in-join},\n  the codimension of $X_A$ in the join of $X_{A^0}, \\dots, X_{A^l}$\n  is\n  $\\sum_{j=0}^l (\\rank \\Aff(A^j) - \\dim \\Aff_{\\Bbbk}(A^j))$\n  by \\autoref{lem_torus_inv_sub}.\n\\end{rem}\n\n\n\n\n\\vspace{1mm}\nNow we can prove \\autoref{cor_join} immediately.\n\n\\begin{proof}[Proof of Corollary \\ref{cor_join}]\n  We may assume that\n  $X=X_A$ for some finite set $A \\subset M$ with $\\Aff (A)=M$.\n  Then the assertion follows from\n  \\autoref{thm:subvar-in-join} and \\autoref{thm:subvar-in-join:rem} since\n  the equality $\\rank \\Aff(A^j) = \\dim \\Aff_{\\Bbbk}(A^j)$ holds in $\\chara\\Bbbk = 0$.\n\\end{proof}\n\n\\begin{cor}\n  Assume $\\chara\\Bbbk=0$.\n  If a toric variety $X_A \\subset \\PP^N$ is the join of some projective varieties,\n  then\n  $X_A$ is the join of some toric varieties $X_{A^0}, X_{A^1}, \\dots, X_{A^{l}}$ for some $l >0$.\n\\end{cor}\n\\begin{proof}\n  Since $X_A$ is the join in $\\chara\\Bbbk=0$,\n  the Gauss defect $\\delta_{\\gamma}(X_A)$ is positive (due to Terracini's lemma). Hence this corollary follows from \\autoref{cor_join}\n  for $l = \\delta_{\\gamma}(X)$.\n\\end{proof}\n\n\n\n\n\n\nThe assumption $\\chara\\Bbbk = 0$ is crucial in the above proof of \\autoref{cor_join}.\nIn positive characteristic,\neven if the Gauss map $\\gamma$ of toric $X_A$ is separable (equivalently, \na general fiber of $\\gamma$ is scheme-theoretically an open subset of a linear variety of $\\PP^N$),\nit is possible that $\\gamma$ is degenerate but $X_A$ is \\emph{not} the join of any varieties, as follows.\n\n\\begin{ex}\\label{thm:sep-gamma-X-not-join}\n  Let $p = \\chara\\Bbbk \\geq 3$.\n  Set\n  \\[\n  A^0=\\{0,1,-1\\}, A^1 =\\{ 0,p\\} \\subset \\mathbb {Z}^1\n  \\]\n  and let $A \\subset \\mathbb {Z}^1 \\times \\mathbb {Z}^1$ be the Cayley sum of $A^0,A^1$.\n  Then\n  \\[\n  \\lin{B-B} = \\mathbb {Z}^1 \\times \\{0\\} \\subset \\mathbb {Z}^1 \\times \\mathbb {Z}^1.\n  \\]\n  Hence $\\pi: M \\rightarrow M\/ (\\lin{B-B}_{\\mathbb{R}} \\cap M)$ coincides with\n  the projection $\\mathbb {Z}^1 \\times \\mathbb {Z}^1 \\rightarrow \\mathbb{Z}^1$ to the second factor.\n  In this setting, the following hold.\n\n  \\begin{enumerate}\n  \\item The Gauss map $\\gamma$ of the surface $X_A \\subset \\mathbb{P}^4$ is separable.\n    A general fiber of $\\gamma$ is a line; in particular,\n    $\\gamma$ is degenerate.\n\n  \\item The conic $X_{A^0}$ and the line $X_{A^1}$ are embedded into $X_A$.\n\n  \\item $X_A$ is of codimension one in the join of $X_{A^0}$ and $X_{A^1}$.\n    On the other hand, $X_A$ itself is not the join of any varieties.\n  \\end{enumerate}\n\n  The reason is as follows.\n  \\begin{inparaenum}\n  \\item The separability of $\\gamma$ follows from \\autoref{cor_rk&deg}.\n    A general fiber of $\\gamma$ is projectively equivalent to $X_{\\pi(A)}$, which is a line.\n  \\item The embedding of $X_{A^j}$ is given as in \n    \\autoref{lem_torus_inv_sub}.\n  \\item It follows from \\autoref{thm:subvar-in-join} that\n    $X_A$ is contained in the join of $X_{A^0}$ and $X_{A^1}$.\n    Since the join is of dimension $3$,\n    the codimension of $X_A$ in the join is equal to $1$. By \\autoref{thm:subvar-in-join:rem}, the codimension is also calculated from\n    \\begin{gather*}\n      \\rk \\Aff (A^0) - \\dim \\Aff_{\\Bbbk} (A^0) = 1-1 = 0,\n      \\\\\n      \\rk \\Aff (A^1) - \\dim \\Aff_{\\Bbbk} (A^1) = 1-0 = 1.\n    \\end{gather*}\n    On the other hand, $X_A$ is not the join of any varieties;\n    this is because,\n    a projective surface $X \\subset \\PP^N$ is the join of some varieties\n    if and only if $X$ is the cone of a curve with a vertex.\n  \\end{inparaenum}\n\n\n \\begin{figure}[htbp]\n    \\[\n    \\begin{xy}\n      (10,0)=\"A\",(0,10)=\"B\",\n      (-10,-10)=\"C\",\n      (9.9,0.1)=\"E\",(-5,0.1)=\"F\",\n      (9.8,-0.1)=\"I\",(-5,-0.1)=\"J\",\n      (-20,0)=\"1\",(65,0)=\"2\",\n      (0,-10)=\"3\",(0,18)=\"4\",\n      (95,-10)=\"7\",(95,18)=\"8\",\n      (10,-4)*{1},\n      (-10,-4)*{-1},\n      (45,0)*{\\mbox{\\scriptsize$\\mid$}},\n      (45,-4)*{p},\n      (0.1,10)*{\\mbox{\\Large$\\times$}},\n      (-10,10)*{\\mbox{\\Large$\\times$}},\n      (10,10)*{\\mbox{\\Large$\\times$}},\n      (35,10)*{\\mbox{\\Large$\\times$}},\n      (45,10)*{\\mbox{\\Large$\\times$}},\n      (55,10)*{\\mbox{\\Large$\\times$}},\n      (45,10)*{\\mbox{\\large$\\circ$}},\n      (0,10)*{\\mbox{\\large$\\circ$}},\n      (0,0)*{\\bullet},\n      (-10,0)*{\\bullet},(10,0)*{\\bullet},\n      (95,10)*{\\sqbullet},(95,0)*{\\sqbullet},\n      (98,10)*{1},(98,0)*{0},\n      (45,-14)*{\n        \\bullet\\;A^0 \\times   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{0}\\quad \n        \\mbox{\\large$\\circ$}\\;A^1 \\times   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{1}\\quad \n        \\mbox{\\Large$\\times$}\\;B\n      }, \n      (110,-14)*{\\sqbullet\\;\\pi(A)},\n      (80,3)*{\\pi}\n      \\ar \"1\";\"2\"\n      \\ar \"3\";\"4\"\n      \\ar \"7\";\"8\"\n      \\ar (72,0);(88,0)\n    \\end{xy}\n    \\]\n    \\caption{Cayley sum: $A = A^0 * A^1$ \\ ($p \\geq 3$).}\n    \\label{figure3}\n  \\end{figure}\n\n\n\n\n\n\n\n\\end{ex}\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{Constructions in positive characteristic}\n\\label{sec:posit-char-case}\n\n\nThis section presents two constructions of projectively embedded toric varieties\nin positive characteristic.\nWe consider whether it is possible to find a toric variety\nwhose Gauss map $\\gamma$ has given data about\n\\begin{enumerate}\n\\item[{(F)}]   each irreducible component of a general fiber of $\\gamma$;\n\\item[{(I)}]   the image of $\\gamma$;\n\\item[{(c)}]\n  the number of the irreducible components of a general fiber of $\\gamma$;\n\\item[{(r)}]\n  the rank of $\\gamma$.\n\\end{enumerate}\nThe statement of \\autoref{thm_Fukasawa1}\nmeans that, a projectively embedded toric variety $X$ is constructed for given (F) and (I).\nIn fact,\nin the construction of \\autoref{cor_im&fiber}, we can control (F), (I), and (c),\nbut not (r) (indeed, $\\rk (\\gamma) = 0$ for $X$ in our proof).\nOn the other hand, in the construction of \\autoref{cor_rk&fiber},\nwe can control (F), (r), and (c).\nHereafter we assume that $p= \\chara \\Bbbk$ is positive.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{thm}\\label{cor_im&fiber}\n  Assume $p = \\chara \\Bbbk >0$.\n  Let $A'$ and $A''$ be finite subsets of free abelian groups $M'$ and $M''$\n  respectively\n  such that $\\Aff (A') = M'$ and $ \\Aff (A'')=M''$.\n  Let $c > 0$ be an integer coprime to $p$.\n  Assume $n:= \\rk(M') + \\rk(M'') \\geq \\#A'-1$ and $\\rk(M'') \\geq 1$.\n  Then there exists a finite subset $A \\subset M := \\mathbb {Z}^n$ with $\\# A = n + \\# A''$ and $\\Aff (A)= M$\n  such that the Gauss map $\\gamma$ of $X_A \\subset \\mathbb{P}^{\\#A-1}$ satisfies the following conditions:\n  \\begin{enumerate}[\\normalfont(i)]\n  \\item (The closure of) each irreducible component of a general fiber of $\\gamma$\n    is projectively equivalent to $X_{A'}$.\n  \\item (The closure of) the image $\\gamma (X_A)$ is projectively equivalent to $X_{A''}$.\n  \\item The number of the irreducible components of a general fiber of $\\gamma$ is equal to $c$.\n  \\end{enumerate}\n\\end{thm}\n\n\\newcommand{N'}\\newcommand{\\sAddm}{N''}{N'}\\newcommand{\\sAddm}{N''}\n\n\\begin{proof}We set $N'}\\newcommand{\\sAddm}{N'' = \\# A'-1$\n  and $A'=\\{ u'_0,\\ldots, u'_{N'}\\newcommand{\\sAddm}{N''}\\}$,\n  and let $e_1, \\ldots, e_{n}$\n  be the standard basis of $M = \\mathbb {Z}^{n}$.\n  Without loss of generality,\n  we may assume that $u'_0=0$.\n  We define a group homomorphism $\\pi$ by\n  \\begin{align*}\n    \\pi :  M \\rightarrow M' \\quad: \\quad   e_i \\mapsto \\left\\{ \n      \\begin{array}{cl} \n        u'_i & \\text{for } 1 \\leq i \\leq N'}\\newcommand{\\sAddm}{N'' ,\\\\ \n        0 & \\text{for }  N'}\\newcommand{\\sAddm}{N'' +1 \\leq i \\leq n .\\\\\n      \\end{array} \\right.\n  \\end{align*}\n  We note that $n \\geq N'}\\newcommand{\\sAddm}{N''$ holds by assumption.\n  Since $\\Aff (A') = M'$ and $u'_0=0 \\in M'$,\n  $\\pi$ is surjective.\n  Hence $\\ker \\pi$ is a free abelian group whose rank is $\\rk (M'')$.\n  Since $\\rk(M'') \\geq 1$, we can take and\n  fix an injective group homomorphism\n  \\[\n  M'' \\hookrightarrow \\ker \\pi\n  \\]\n  whose cokernel is isomorphic to $\\mathbb {Z} \/ \\lin{c}$. \n  Let $A''=\\{ f_0,\\ldots, f_{\\sAddm}\\} \\subset M'' \\subset \\ker \\pi$ for $\\sAddm = \\# A'' -1$.\n  Without loss of generality,\n  we may assume that $f_0=0 \\in M''$.\n  Set\n  \\[\n  A =\\{ e_1,\\ldots, e_{n}, p f_0, \\ldots, p f_{\\sAddm} \\}  \\subset M.\n  \\]\n  Since $e_1, \\ldots, e_{n} $ is a basis of $M$ and $ p f_0 =0 \\in M$,\n  $A$ spans the affine lattice $M$.\n\n  Let $B$ be as in the statement of Theorem \\ref{thm_structure} for the above $A$.\n  Choose $n+1$ elements $u_{i_0}, u_{i_1} , \\ldots , u_{i_{n}} \\in A$ which span the affine space $M_{\\Bbbk}$.\n  Since $p f_s=0 $ in $M_{\\Bbbk} $ for $0 \\leq s \\leq \\sAddm$,\n  at most one element of $\\{p f_0, p f_1,\\ldots, p f_{\\sAddm}  \\}$ is contained in $\\{u_{i_0}, u_{i_1} , \\ldots , u_{i_{n}}\\}$.\n  Hence $\\{u_{i_0}, u_{i_1} , \\ldots , u_{i_{n}}\\} =\\{ p f_s, e_1,\\ldots, e_{n} \\}$ holds for some $0 \\leq s \\leq \\sAddm $.\n  Thus we have\n  \\[\n  B = \\{ p f_0 + e_1+ \\cdots +e_{n} ,\\ldots, p f_{\\sAddm} +e_1+ \\cdots +e_{n}  \\},\n  \\]\n  that is, $B $ is the parallel translation of $p \\cdot A'' $ by $e_1+ \\cdots +e_{n}$.\n  Hence we have $X_B = X_{p \\cdot A''} = X_{A''}$.\n  Since the closure of the image $\\gamma(X_A)$ is projectively equivalent to $X_B$ by \\ref{thm:item-image} in Theorem \\ref{thm_structure},\n  the condition~(ii) in this theorem holds.\n\n\n\n  Since $A''= \\{f_0, \\ldots, f_{\\sAddm} \\}$ spans the affine lattice $M''$,\n  it holds that\n  \\[\n  \\langle B -B \\rangle = p \\cdot \\langle A''-A'' \\rangle =p \\cdot M'' \\subset M'' \\subset \\ker \\pi.\n  \\]\n  Therefore $\\langle B-B \\rangle_{\\mathbb {R}} \\cap M= \\ker \\pi$\n  and the natural projection $M \\rightarrow M \/ (\\langle B-B \\rangle_{\\mathbb {R}} \\cap M)$ coincides with $\\pi : M \\rightarrow M'$.\n  Since $\\pi(A) =A'$ by the definition of $\\pi$ and $A$,\n  the condition~(i) in this theorem follows from \\ref{thm:item-fiber} in Theorem \\ref{thm_structure}.\n\n  Since $ \\ker \\pi \/ M'' \\simeq \\mathbb {Z} \/ \\lin{c}$,\n  the order of the finite group\n  \\[\n  (\\langle B-B \\rangle_{\\mathbb {R}} \\cap \\mathbb {Z}^{n}) \/ \\langle B-B \\rangle = \\ker \\pi \/ (p \\cdot M'')\n  \\]\n  is $p^{\\rk(M'')} c$.\n  Hence (iii) in this theorem follows from (2) in \\autoref{cor_rk&deg}.\n\\end{proof}\n\nNote that, in the above construction, $\\rk (\\gamma) = \\dim\\lin{B-B}_{\\Bbbk} = 0$.\n\n\n\\begin{ex}\n\n\n\n\n\n\n\n  In this example,\n  we illustrate \\autoref{cor_im&fiber} for \n  \\[\n  A' = \\Set{\n    \\begin{bmatrix}\n      0 \\\\ 0\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      1 \\\\ 0\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      0 \\\\ 1\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      1 \\\\ 1\n    \\end{bmatrix}\n  } \\subset \\mathbb{Z}^2,\\\n  A'' =   \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{0, 1, 2, 3} \\subset \\mathbb{Z}^1 \n  \\]\n  and an integer $c >0$ coprime to $p = \\chara \\Bbbk$.\n  In this case, $X_{A'} = \\mathbb{P}^1 \\times \\mathbb{P}^1 \\subset \\mathbb{P}^3$ is a smooth quadric surface\n  and $X_{A''} \\subset \\mathbb{P}^3$ is a twisted cubic curve. \n  Since $n= 2 +1 \\geq \\# A' -1 =4-1$,\n  we can apply \\autoref{cor_im&fiber}.\n\n  \n  \n  \n  We use the notations in the proof of \\autoref{cor_im&fiber}.\n  Since $\\pi : M = \\mathbb {Z}^3 \\rightarrow M' = \\mathbb {Z}^2$ is defined by\n  \\[\n  \\pi(e_1)=\n  \\begin{bmatrix}\n    1 \\\\ 0\n  \\end{bmatrix}, \\quad\n  \\pi(e_2)=\n  \\begin{bmatrix}\n    0 \\\\ 1\n  \\end{bmatrix}, \\quad\n  \\pi(e_3)=\n  \\begin{bmatrix}\n    1 \\\\ 1\n  \\end{bmatrix}\n  \\]\n  for the standard basis $e_1,e_2,e_3$ of $\\mathbb {Z}^3$,\n  $\\ker \\pi $ is generated by $e_1 +e_2 -e_3$.\n  Hence an injection $M'' =\\mathbb {Z}^1 \\hookrightarrow \\ker \\pi$ with cokernel $\\mathbb {Z} \/ \\lin{c}$ is given by mapping $1 \\in \\mathbb {Z}^1$ to $ c (e_1 +e_2 -e_3)$.\n  Thus\n  $A$ in the proof of \\autoref{cor_im&fiber} is \n  \\[\n  A = \\Set{\n    \\begin{bmatrix}\n      1 \\\\ 0 \\\\ 0\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      0 \\\\ 1 \\\\ 0\n    \\end{bmatrix},\n    \\begin{bmatrix}\n      0 \\\\ 0 \\\\ 1\n    \\end{bmatrix},\n    p \\cdot 0 ,\n    p \\cdot  f,\n    p \\cdot  2  f,\n    p \\cdot 3  f\n  }\n  \\text{ for }\n  f= \\begin{bmatrix}\n    c \\\\ c \\\\ -c\n  \\end{bmatrix}.\n  \\]\n\n  \\vspace{2mm}\n  We can see directly that (i) - (iii) in \\autoref{cor_im&fiber} hold for this $A$ as follows:\n  In this case,\n  $X_A$ is the image of \n  $\\phi_A: (\\KK^{\\times})^3 \\hookrightarrow \\mathbb{P}^{6}$ defined by \n  \\begin{align}\\label{eq_varphi}\n    (x,y,z) \\mapsto [x : y : z : 1 : (xyz^{-1})^{pc} : (xyz^{-1})^{2pc} : (xyz^{-1})^{3pc}].\n  \\end{align}\n  We embed $\\mathbb{P}^3 $ into $ \\mathbb{G} (3,\\mathbb{P}^6)$ by mapping $[X:Y:Z:W] \\in \\mathbb{P}^3$ to the $3$-plane in $\\mathbb{P}^6$ spanned by the $4$ points which are given as the row vectors of\n  \\begin{align*}\n    \\begin{bmatrix}\n      0 & 0 & 0 & X & Y & Z & W \n      \\\\\n      1 & 0 & 0 & 0 & 0 & 0 & 0 \n      \\\\\n      0 & 1 & 0 & 0 & 0 & 0 & 0 \n      \\\\\n      0 & 0 & 1 & 0 & 0 & 0 & 0 \n    \\end{bmatrix}.\n  \\end{align*}\n  Then the image of $\\varphi_A(x,y,z)$  by the Gauss map $\\gamma$ of $X_A$ is\n  \\begin{align}\\label{eq_image}\n    [1 : (xyz^{-1})^{pc} : (xyz^{-1})^{2pc} : (xyz^{-1})^{3pc}] \\in \\mathbb{P}^3 \\subset \\mathbb{G}(3, \\mathbb{P}^6)\n  \\end{align}\n  from \\autoref{thm:matrix-Gamma}.\n  Hence the closure $\\overline{\\gamma (X_A)}$ is the twisted cubic curve $X_{A''} \\subset \\mathbb{P}^3$.\n  Thus (ii) holds.\n\n  From \\ref{eq_varphi} and \\ref{eq_image},\n  the fiber of $\\gamma |_{T_M}$ over $[1:1:1:1] \\in \\overline{\\gamma(X_A)}= X_{A''} \\subset \\mathbb{P}^3$ is\n  \\[\n  \\left\\{ [x : y : z : 1 : 1 : 1 : 1] \\in X_A \\subset \\mathbb{P}^6 \\, | \\, (x,y,z ) \\in (\\KK^{\\times})^3, (xyz^{-1})^{pc} =1 \\right\\}.\n  \\]\n  As a set,\n  this is the disjoint union of\n  \\begin{align}\\label{eq_fib_comp}\n    \\left\\{ [s : t : \\zeta^k st : 1 : 1 : 1 : 1] \\in X_A \\subset \\mathbb{P}^6 \\, | \\, (s,t ) \\in (\\KK^{\\times})^2 \\right\\}\n  \\end{align}\n  for $0 \\leq k \\leq c-1$, where $\\zeta \\in \\KK^{\\times}$ is a primitive $c$-th root of unity.\n  Since the closure of each \\ref{eq_fib_comp} is projectively equivalent to $X_{A'}= \\mathbb{P}^1 \\times \\mathbb{P}^1 \\subset \\mathbb{P}^3$,\n  (i) and (iii) are satisfied.\n\\end{ex}\n\n\n\n\n\n\n\n\n\n\n\n\n\nNext, we consider how to construct $X$ for a given integer $r > 0$ such that the rank of the Gauss map of $X$\nis equal to $r$.\nFrom the following remark, we need to assume $r \\neq 1$ if $\\chara\\Bbbk = 2$.\n\n\\begin{rem}\\label{rem_rk_in_char2}\n  In characteristic $2$,  it is known that\n  the rank of the Gauss map of any projective variety $X \\subset \\PP^N$ \\emph{cannot} be equal to $1$.\n  In addition, if $X$ is a hypersurface, then the rank of the Gauss map is \\emph{even}.\n  The reason is as follows.\n  \n  Let $X \\subset \\PP^N$ be a projective variety in $\\chara\\Bbbk = 2$,\n  and let $x \\in X$ be a general point.\n  As in \\cite[\\textsection{}2]{fukaji2010},\n  choosing homogeneous coordinates on $\\PP^N$, we may assume that\n  $X$ is locally parameterized at $x = [1:0:\\dots:0]$\n  by $[1:z_1: \\dots: z_n: f_{n+1}: \\dots: f_{N}]$,\n  where $z_1, \\dots, z_n$ form a regular system of parameters of $\\mathscr{O}_{X,x}$, and\n  $f_{n+1}, \\dots, f_N \\in \\mathscr{O}_{X, x}$.\n  Then\n  $\\rk d_{x}\\gamma$\n  is equal to the rank of the $n \\times (n(N-n))$ matrix\n  \\begin{equation}\\label{eq:Hessians}\n    \\begin{bmatrix}\n      H(f_{n+1}) & H(f_{n+2}) & \\cdots & H(f_{N})\n    \\end{bmatrix},\n  \\end{equation}\n  where\n  $H(f) :=\n  [\\partial^2 f \/ \\partial z_i \\partial z_j]_{1 \\leq i,j \\leq n}\n  $\n  is the Hessian matrix of a function $f$.\n  Assume that $\\rk (\\gamma)$ ($= \\rk d_x \\gamma$) is nonzero.\n  Then the matrix \\ref{eq:Hessians} is nonzero; in particular,\n  one of the Hessian matrix $H(f_{k})$ is nonzero.\n  Since $\\chara\\Bbbk = 2$, we have $\\partial^2 f_k \/ \\partial z_i\\partial z_i = 0$,\n  i.e., the diagonal entries of $H(f_{k})$ are zero. Hence \n  some $\\partial^2 f_k \/ \\partial z_i\\partial z_j$ with $i \\neq j$ must be nonzero.\n  Therefore $H(f_{k})$ has $2\\times 2$ submatrix\n  \\[\n  \\begin{bmatrix}\n    \\partial^2 f_k \/ \\partial z_i\\partial z_i & \\partial^2 f_k \/ \\partial z_j\\partial z_i\n    \\\\\n    \\partial^2 f_k \/ \\partial z_i \\partial z_j & \\partial^2 f_k \/ \\partial z_j\\partial z_j\n  \\end{bmatrix}\n  =\n  \\begin{bmatrix}\n    0 & \\partial^2 f_k \/ \\partial z_i\\partial z_j\n    \\\\\n    \\partial^2 f_k \/ \\partial z_i \\partial z_j & 0\n  \\end{bmatrix},\n  \\]\n  whose determinant is nonzero. This implies that $\\rk (\\gamma) \\geq 2$.\n\n\n  Now assume that $X \\subset \\PP^N$ is a hypersurface.\n  Then $X$ is locally parametrized by $[1:z_1:\\dots:z_n:f_{n+1}]$,\n  and hence $\\rk d_x \\gamma = \\rk H(f_{n+1})$.\n  Since $\\chara\\Bbbk = 2$, the symmetric matrix $H(f_{n+1})$ is skew-symmetric,\n  whose rank is even\n  (for example, see \\cite[\\textsection{}5, n$^\\text{o}$1, Corollaire 3]{bourbaki}).\n\n\\end{rem}\n\n\n\\begin{thm}\\label{cor_rk&fiber}\n  Assume $p = \\chara \\Bbbk > 0$.\n  Let $A'$ be a finite subset of a free abelian group $M'$ with $\\Aff (A') = M'$.\n  Let $ r,c > 0$ be positive integers such that  $(p,r) \\neq(2,1)$ and $c$ is coprime to $p$.\n  Assume that positive integers $n, N$ satisfy\n  \\[\n  n \\geq \\max \\{  (\\# A' -1) +r ,  \\rk (M') + r +1\\}\n  \\]\n  and\n  \\begin{align}\\label{cond_N}\n    N \\geq  \\left\\{ \n      \\begin{array}{cl} \n        2n-\\rk(M')-r+1 & \\text{if } p \\geq 3, \\text{ or } \\ p=2, r : \\text{even} ,\\\\ \n        2n-\\rk(M')-r+2 & \\text{if } p=2, r : \\text{odd} .\\\\ \n      \\end{array} \\right.\n  \\end{align}\n  Then there exists a finite subset $A \\subset M:= \\mathbb {Z}^{n}$\n  with $\\Aff (A)= M$ and $\\# A =N+1$\n  such that the Gauss map $\\gamma$ of $X_A \\subset \\PP^N$ satisfies the following conditions:\n  \\begin{enumerate}[\\normalfont(i)]\n  \\item (The closure of) each irreducible component of a general fiber of $\\gamma$\n    is projectively equivalent to $X_{A'}$.\n  \\item The rank of $\\gamma$ is equal to $r$.\n  \\item The number of the irreducible components of a general fiber of $\\gamma$ is equal to $c$.\n  \\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\n  We set $n' = \\rk(M')$ and  $N'}\\newcommand{\\sAddm}{N'' = \\# A'-1$,\n  and $A'=\\{ u'_0,\\ldots, u'_{N'}\\newcommand{\\sAddm}{N''}\\}$.\n  Let $e_1, \\ldots, e_{n}$ be the standard basis of $M=\\mathbb {Z}^{n}$.\n  Without loss of generality,\n  we may assume that $u'_0=0$.\n  As in the proof of \\autoref{cor_im&fiber},\n  we define a surjective group homomorphism $\\pi : \\mathbb {Z}^n \\rightarrow M' $\n  by $\\pi(e_i) = u'_i$ for $1 \\leq i \\leq N'}\\newcommand{\\sAddm}{N''$ and $\\pi(e_i)=0$ for $N'}\\newcommand{\\sAddm}{N''+1 \\leq i \\leq n$.\n  Since $\\ker \\pi \\simeq \\mathbb {Z}^{n-n'}$ and $e_{N'}\\newcommand{\\sAddm}{N''+1} , \\ldots, e_{N' + r} \\in \\ker \\pi$\n  (note that $N'}\\newcommand{\\sAddm}{N'' +r = (\\# A' -1) +r \\leq n$ holds by assumption),\n  there exist\n  \\[\n  f_1, \\ldots, f_{n-n'-r} \\in \\ker \\pi\n  \\]\n  such that $e_{N'+1} , \\ldots, e_{N'+r}, f_1, \\ldots, f_{n-n'-r} $\n  form a basis of $\\ker \\pi$.\n  By assumption,\n  $n-n' -r = n - \\rk(M') -r \\geq 1$ holds.\n\n\n\n\n\n\n  \\vspace{1mm}\n  First,\n  we consider the case\n  when $N$ is equal to the right hand side of \\ref{cond_N}.\n  Set\n  \\[\n  A = C \\cup D \\subset M,\n  \\]\n  where\n  \\begin{align*}\n    C &= \\{ e_1,\\ldots,e_n, 0, c pf_1, p f_2, \\ldots, p f_{n-n'-r}\\},\n    \\\\\n    D &= \\left\\{ \n      \\begin{array}{cl} \n        \\{e_{N'}\\newcommand{\\sAddm}{N''+1} + \\cdots + e_{N'}\\newcommand{\\sAddm}{N''+r} \\} & \\text{for } r \\not \\equiv 1 \\text{ mod } p ,\\\\ \n        \\{- e_{N'}\\newcommand{\\sAddm}{N''+1} - \\cdots - e_{N'}\\newcommand{\\sAddm}{N''+r} \\} & \\text{for } r \\equiv 1 , r \\not \\equiv - 1 \\text{ mod } p ,\\\\ \n        \\{e_{N'}\\newcommand{\\sAddm}{N''+1} + e_{N'}\\newcommand{\\sAddm}{N''+2}, e_{N'}\\newcommand{\\sAddm}{N''+2} + \\cdots + e_{N'}\\newcommand{\\sAddm}{N''+r} \\} & \\text{for }  p=2, r : \\text{odd}, r \\geq 3.\n      \\end{array} \\right.\n  \\end{align*}\n  By a similar argument in the proof of \\autoref{cor_im&fiber} and by Examples \\ref{prep-cor_rk&fiber}, \\ref{ex_char2},\n  we have\n  \\begin{align}\\label{eq_B-B}\n    \\langle B-B \\rangle = \\bigoplus_{i=N'}\\newcommand{\\sAddm}{N''+1}^{N'}\\newcommand{\\sAddm}{N''+r}  \\mathbb {Z} e_{i} \\oplus \\mathbb {Z} cp f_1 \\oplus \\bigoplus_{j=2}^{n-n'-r}  \\mathbb {Z} p f_j  .\n  \\end{align}\n  Hence \n  $\\langle B-B \\rangle_{\\mathbb {R}} \\cap M= \\ker \\pi$.\n  Since $A' = \\pi (A)$,\n  (i) and (iii) in this corollary follows as in \\autoref{cor_im&fiber}.\n\n  Since $e_{N'+1} , \\ldots, e_{N'+r}, f_1, \\ldots, f_{n-n'-r} $ form a basis of $\\ker \\pi$,\n  we have $\\langle B-B \\rangle_{\\Bbbk} =\\bigoplus_{i=N'}\\newcommand{\\sAddm}{N''+1}^{N'}\\newcommand{\\sAddm}{N''+r}  \\Bbbk e_{j} $.\n  Thus the rank of $\\gamma $ is $\\dim \\langle B-B \\rangle_{\\Bbbk} = r$ by \\autoref{cor_rk&deg}.\n\n  \\vspace{3mm}\n  Next, we consider any integer $N$ satisfying the inequality~\\ref{cond_N}.\n  We take\n  a finite subset $E$ of the right hand side of \\ref{eq_B-B} such that\n  $N = \\# (C \\cup D \\cup E) -1$\n  for the above $C$ and $D$.\n  Set\n  $A := C \\cup D \\cup E \\subset M$.\n  Since\n  $E$ is contained in the right hand side of \\ref{eq_B-B}, the subgroup\n  $\\langle B-B \\rangle $ for this $A$ is the same as \\ref{eq_B-B}.\n  Hence the assertion follows. \\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Muon Ionization Cooling Experiment~\\cite{MICE} is under construction at the UK's Rutherford Appleton Laboratory (RAL) in order to demonstrate for the first time the feasibility and efficacy of ionization cooling of muons~\\cite{cooling1,cooling2}. The MICE apparatus (Fig.~\\ref{fig:MICE}) comprises one cooling lattice cell based on a design from Neutrino Factory Feasibility Study II (FS-II)~\\cite{FSII} sandwiched by the input and output spectrometers and particle identification and timing detectors that will be used to  characterize the ionization-cooling process experimentally. Since an affordable cooling section cools by only $\\sim$\\,10\\% (see Fig.~\\ref{fig:MICE}), too small an effect to measure reliably using standard beam instrumentation,  MICE employs a low-intensity muon beam~\\cite{beam} and measures each muon individually. Once completed, it will thereby demonstrate that the ionization-cooling process is  understood in detail in both its physics and engineering aspects, and that it works as simulated. In order to afford a thorough validation of the codes used to design ionization-cooling channels, MICE will be operated in various modes and optics configurations. Full results are expected by about 2020, with analyses of some configurations available up to five years earlier. Early results will include important validations of the models used in ionization-cooling simulation codes, as well as the first experimental test of muon transverse--longitudinal emittance exchange (needed for six-dimensional cooling, e.g., for a muon collider) in a wedge absorber.\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=.62\\linewidth, trim=450 0 300 0 mm,clip]{MICE-3D-cutaway-new}\\hfill\\includegraphics[width=.37\\linewidth]{MICE-cool-new}}\n\\vspace{-.1in}\n\\caption{Left: cutaway  rendering of  MICE (``Step VI\") apparatus: two precision scintillating-fiber solenoidal spectrometers surround a cooling   cell based on the FS-II ``SFOFO\" design~\\cite{FSII} (particle-ID detectors not shown). Right: normalized-emittance change  vs input emittance in the nominal MICE optics configuration, showing 2.5$\\pi$\\,mm$\\cdot$rad equilibrium  transverse emittance.\\\\[-.15in]\n}\\label{fig:MICE}\n\\end{figure}\n\nIonization cooling is inherently a transverse effect. A simple and intuitive way to see this is to consider that ionization energy loss exerts a braking effect on each muon, reducing both the transverse and longitudinal momentum components, while reacceleration in RF cavities restores only the longitudinal momentum component. With repeated braking and reacceleration the divergence of the beam is progressively reduced. Alternating-gradient focusing brings a concomitant reduction of the beam's cross-sectional area. The process can beneficially continue until an equilibrium is reached between the cooling effect of ionization and the heating effect of multiple scattering, i.e., until \n\\begin{eqnarray} \n\\frac{d\\epsilon_n}{ds}\\approx\n-\\frac{1}{\\beta^2} \\left\\langle\\!\\frac{dE_{\\mu}}{ds}\\!\\!\\right\\rangle\\frac{\\epsilon_n}{E_{\\mu}}\n +\n\\frac{1}{\\beta^3} \\frac{\\beta_\\perp\n(0.014\\,{\\rm GeV})^2}{2E_{\\mu}m_{\\mu}L_R}\\to 0 \\,,\n\\label{eq:cool} \n\\end{eqnarray}  \n $d\\epsilon_n\/ds$ being the rate of normalized-emittance change within the absorber; $\\beta c$,  $E_\\mu$,  and $m_\\mu$\nthe muon velocity, energy, and mass; $\\beta_\\perp$ the lattice betatron function at the absorber; and $L_R$ the  radiation length of the absorber material~\\cite{cooling2}. Ionization cooling works optimally for momenta near the ionization minimum, hence MICE is designed for the momentum range 140 to 240\\,MeV\/$c$.\n\n\n\\section{MICE Steps}\n\nThe MICE precision goal, 0.1\\% emittance resolution, allows even the small emittance decrements (increments) when operating just above (below) the cooling cell's equilibrium emittance to be well measured. This  places a premium on careful spectrometer characterization and calibration. This goal and the anticipated funding schedule led to a planned MICE buildup in a series of six ``Steps,\" but  exigencies of complex apparatus construction have resulted in the abbreviated sequence of Fig.~\\ref{fig:Steps}. Step I has already taken place, and the results presented~\\cite{beam,PID}, with the exception of the  Oct.\\ 2013 EMR (``electron--muon ranger\" total-absorption calorimeter) run. \n\\begin{figure}\n\\begin{minipage}[b]{20pc}\n\\includegraphics[width=1.13\\linewidth\n]{mice-steps-I}\\\\\n\\includegraphics[width=1.03\\linewidth,\n]{mice-steps-IV-VI}\n\\caption{The Steps of MICE}\\label{fig:Steps}\n\\end{minipage}\\hfill\n\\begin{minipage}[b]{16pc}\n\\centerline{\\includegraphics[width=\\linewidth]{SS}}\n\\caption{MICE spectrometer solenoids at vendor}\\label{fig:SS}\n\\end{minipage}\n\n\\end{figure}\n\n\\section{Step IV}\n\nMICE Step IV will represent a first study of muon ionization cooling, with results expected starting in 2015. \nIn addition to the Step I components (pion-production target inserted periodically into the ISIS synchrotron beam, pion-to-muon beamline,  particle-ID and muon-timing detectors), Step IV requires the two trackers with their spectrometer solenoids (SS) and one absorber--focus-coil (AFC) module. The trackers are in hand and have been tested with cosmic rays~\\cite{tracker}. Each SS comprises five superconducting coils: three to provide a 4\\,T field uniform to better than 1\\% over the 1-m-long, 40-cm-diameter tracking volume, and two to match the beam into or out of the  cooling cell. The  spectrometer solenoids (Fig.~\\ref{fig:SS}) have been built by Wang NMR in Livermore, CA, USA, and the focus coils by Tesla Engineering Ltd., UK. As of this writing, after some retrofitting, the first SS has been successfully trained, field-mapped, and delivered to RAL, and training of the second has commenced. The first  AFC has been trained to just above the baseline current in ``flip\" mode (the two coils powered in opposite polarities) and to the full design current in ``solenoid\" mode (same polarities). The training of a second AFC is commencing and the performance of the first will be assessed in the light of that experience.\n\nStep IV will include tests of LH$_2$, LiH, and plastic or other low-$Z$ absorbers. A demonstration of emittance exchange is also planned. Muon colliders require muon cooling in all six phase-space dimensions, and rely on emittance exchange in order to couple the transverse  ionization cooling effect into the longitudinal phase plane. This is accomplished via a suitable correlation between muon momentum and path length in absorbers, and thus requires lattices with dispersion. While there is naturally some dispersion in the MICE beamline, since in MICE each muon is measured individually,  the desired range of dispersion will be created via off-line selection of muons.\n\n\\section{Toward Step VI}\n\nMICE Step IV will allow the detailed characterization of ionization energy loss, multiple Coulomb scattering, and their effect on the ionization cooling process, and will thus validate the models used in muon ionization-cooling simulations. However, it constitutes a ``non-sustainable\" cooling configuration, since the energy lost in the absorber is not replenished.  The first engineering demonstration of  ``sustainable\" cooling will thus be  Step VI (depicted in Fig.~\\ref{fig:MICE}), with a possible ``stopover\" in Step V, depending on component availability. By allowing a thorough exploration of the optics of ionization-cooling lattices, including those with periodic field flips  and those with a modulated solenoid field, Step VI will furnish a complete validation of the simulations.\n\nStep VI requires (besides the components of Step IV) two additional AFC modules and two RF-cavity--coupling-coil (RFCC) modules, each containing four 201\\,MHz RF cavities and one large-diameter coupling-coil (CC) solenoid. All eight RF cavities have been fabricated and the first (Fig.~\\ref{fig:201MHz}) is being readied for testing in the Fermilab MuCool Test Area (MTA). At the FS-II design gradient these cavities each require 4 MW of input power, provided through two coaxial couplers. Prototype tests  at the MTA revealed RF-coupler breakdown as an issue. Couplers of a revised design are now in fabrication and will soon be tested in the MTA. \n\nA full test requires in addition a coupling coil, so that the behavior of the cavity in a multi-tesla solenoid field can be explored. The first CC cold mass (Fig.~\\ref{fig:CC}), built for Harbin Institute of Technology by Qi Huan Co., Beijing, China, and planned for the MTA rather than MICE, is now under test at Fermilab. Once it is trained, three more (including one spare) will be fabricated. Fabrication of the needed cryostats and vacuum vessels \n{(at LBNL, based on initial designs developed by SINAP)}, and of the cavity tuners, is also in progress, as is assembly of the RF control and power distribution systems by LBNL and Daresbury Lab.\n\n\\begin{figure}\n\\begin{minipage}[b]{17pc}\n\\centerline{\\includegraphics[width=\\linewidth,trim=40 0 40 0 mm,clip]{201MHz}}\n\\caption{First MICE RF cavity being fitted with tuners at Fermilab.}\\label{fig:201MHz}\n\\end{minipage}\\hfill\n\\begin{minipage}[b]{17pc}\n\\centerline{\\includegraphics[width=\\linewidth,trim=0 0 0 10 0mm,clip]{CC}}\\vspace{-.1in}\n\\caption{First CC cold mass fitted out at LBNL prior to shipment to Fermilab.}\\label{fig:CC}\n\\end{minipage}\n\\end{figure}\n\nThe FS-II design~\\cite{FSII} calls for 32\\,MW of RF power per cell in order to allow off-crest operation. The MICE Step VI goal is on-crest acceleration, with power provided by four 2\\,MW supplies  from CERN and LBNL refurbished by Daresbury Lab. The first supply has recently reached full power, with a single-supply test  at MICE scheduled for later this year. \n\n\\section{Conclusions}\nMICE is progressing towards the first experimental study of muon ionization cooling. Step IV is planned for 2015 and the concluding Step VI  of MICE by the end of this decade. Once ionization cooling is established, construction of a neutrino factory with cooling could commence. \n\n\\section*{Acknowledgments}\nThe author thanks his MICE collaborators \nfor many stimulating interchanges on these topics. Work supported by the U.S.\\ Dept.\\ of Energy (through the Muon Accelerator Program) and National Science Foundation.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\nNumerous geophysical and astrophysical flows present a two-layer configuration, with a turbulent convective layer standing above or below a stably stratified one. Examples include planetary atmospheres, stars interiors, and possibly the outermost layer of the Earth liquid core \\cite{hirose_composition_2013}. The dynamics of coupled stratified and convective layers are quite complex. Due to the convective motions, internal gravity waves (IGWs) are generated at the interface between the two layers, and propagate in the stratified one. IGWs transport energy and momentum \\cite{rogers_internal_2012,bretherton_momentum_1969} from where they are generated to where they are damped. Thanks to their transport properties and non-linear interactions, IGWs are able to generate and sustain large-scale horizontal flows \\cite{plumb_interaction_1977,rogers_internal_2012}. Examples of such  large-scale flows driven by IGWs are the oscillations of equatorial zonal winds observed in some planets' atmosphere \\cite{fouchet_equatorial_2008,leovy_quasiquadrennial_1991}, including the Earth where it is called the Quasi-Biennial Oscillation (QBO) \\cite{baldwin_quasi-biennial_2001}.\\\\\n\nThe IGWs generation by turbulent dynamics has been studied in various experiments. The generation by a single buoyant plume was experimentally studied by Ansong \\& Sutherland \\cite{ansong_internal_2010}. The penetration of the plume within the stratified layer and the spectral characteristics of the generated IGWs were studied. They found that the peak frequency of the generated IGWs lies in a range close to $0.7N$, where $N$ is the Brunt-V\\\"ais\\\"al\\\"a (or buoyancy) frequency, and that the radial wavenumber is set by the plume cap and not by the width of the plume at the interface.\n\nDeardoff et al. \\cite{deardorff_laboratory_1969} and later Michaelian et al. \\cite{michaelian_coupling_2002} studied the effect of penetrative convection in a stratified layer in a transient, Rayleigh-B\\'enard type experiment. Stratification was initially set up thermally from the top to the bottom of the tank. Then, the fluid was suddenly warmed up at the bottom, triggering Rayleigh-B\\'enard convection. IGWs were measured transiently \\cite{michaelian_coupling_2002} while the stratified (resp. convective) layer was decreasing (resp. increasing) in size. Eventually, there was no more stratified layer to sustain the propagation of IGWs. \n\nTownsend \\cite{townsend_natural_1964} introduced an original set-up to study the quasi-steady generation of IGWs by Rayleigh-B\\'enard convection. Using the fact that water maximum density is around $4^{\\circ}$C, a two-layer system is spontaneously generated by cooling the bottom of a tank at $0^{\\circ}$C and heating its top above $4^{\\circ}$C. The density gradient is unstable at temperature below 4 $^{\\circ}$C and stable at temperature above. This creates a self-organising system, with a turbulent convective layer adjacent to a stratified layer. With dye visualisation and temperature measurements, he observed IGWs propagating close to the interface between the two layers. The $4^\\circ$C convection was also studied experimentally by Le Gal \\cite{legal_penetrative_1997} in a laminar flow, at low Rayleigh number. Convection displayed an hexagonal pattern and viscous entrainment of the fluid above the convective cells was observed. Perrard et al. \\cite{perrard_experimental_2013} and Le Bars et al. \\cite{bars_experimental_2015} re-investigated this setup in a quasi two-dimensional tank using Particle Image Velocimetry (PIV) and  temperature measurements to obtain detailed data of IGWs generated by the convection. They observed a wide spectrum of waves generated at different frequencies. Favoured frequencies were related to the differential attenuation length of the waves depending on frequency, in good agreement with linear theory. No large-scale flow in the stratified layer was observed in this 2D geometry. Numerical simulations of the same configuration were performed by Lecoanet et al. \\cite{lecoanet_numerical_2015}. They demonstrated that IGWs are mainly generated by the Reynolds stresses in the bulk of the convection below the interface, rather than by the displacement of the interface between the two layers. Numerical studies by Couston et al. \\cite{couston_dynamics_2017,couston_order_2018,couston_energy_2018} extended these results by considering a generic non-linear equation of state (piecewise linear around an inversion temperature, with adjustable slopes), both in 2D and 3D horizontally periodic geometries. Various flow regimes and the energy transport by IGWs were quantitatively studied. Interestingly, long simulations -- accessible in 2D studies only -- showed, for low Prandtl numbers ($Pr < 1$), a large-scale horizontal flow with reversals in the stratified layer, similar to the QBO phenomenon introduced above \\cite{couston_order_2018}. \n\\\\\n\nSeveral experiments took interest in the generation and reversal of a large-scale horizontal mean flow in a stratified domain, driven by IGWs. The well-known experiment designed by Plumb and McEwan \\cite{plumb_instability_1978}, later reproduced and extended by Semin et al. \\cite{semin_generation_2016,semin_nonlinear_2018}, is capable of driving a QBO from the mechanical forcing of a standing wave pattern (\\textit{i.e.} two waves with the same frequency and opposite phase speed) in a salty water stratification. With this system, Plumb and McEwan managed to observe few oscillations of the driven large-scale flow before the stratification was destroyed by mixing. The experiment gave results in good agreement with the theory \\cite{richard_s._lindzen_theory_1968,lindzen_updated_1972,plumb_interaction_1977}, notably with reversals starting away from the forcing and propagating towards it.\nSemin et al. \\cite{semin_generation_2016,semin_nonlinear_2018} improved the system by constantly injecting fluid to rebuild the stratification, while removing mixed fluid close to the forcing. This method allowed to run the experiment longer and to study the nature of the bifurcation in the Plumb model which can be either supercritical or subcritical, depending on the dominant dissipative process. In those experimental realisations of the QBO mechanism, the wave forcing remains monochromatic, as opposed to the natural mechanism where it is due to chaotic tropical storms \\cite{baldwin_quasi-biennial_2001}. The forcing is driven by interface displacements, as opposed to the observations of \\cite{lecoanet_numerical_2015}. Besides, only the stratified layer is modelled. It thus remains a challenge to observe experimentally a large-scale, reversing flow from a turbulent source and a wide range of naturally excited IGWs.\n\\\\\n\nIn the present study, we extend the work of Townsend \\cite{townsend_natural_1964,perrard_experimental_2013,bars_experimental_2015} in a cylindrical, 3D geometry reminiscent of Plumb and McEwan's set-up \\cite{plumb_instability_1978,semin_generation_2016,semin_nonlinear_2018}. Our purpose is threefold: to characterise the generation of IGWs in such a self-organising two-layer system, to quantify the coupling between the layers, and to investigate the possible generation of large-scale horizontal flows. Our experiments are complemented by direct numerical simulations of the same configuration. The experimental setup and numerical model are presented in section \\ref{sec:methods}, results are analysed in section \\ref{sec:results}, and conclusions and future works are discussed in section \\ref{sec:discussion}. \n\n\n\n\\section{\\label{sec:methods}Methods}\n\\subsection{Experimental set-up}\nThe set-up consists in a cubic tank whose lateral boundaries are made of 2 cm thick acrylic walls. The bottom boundary is a chrome plated copper plate in which refrigerated fluid is forced to circulate. The top boundary is a commercial, transparent electric heater. The tank inner dimensions are $32 \\times32$ cm for the horizontal section and $H=20$ cm in height. Preliminary experiments were conducted in this cubic geometry. Eventually, a cylinder of outer diameter $D = 29$ cm and thickness $e = 0.4 $ cm was added inside the cubic tank, to reproduce the axisymmetric geometry of \\cite{plumb_instability_1978,semin_generation_2016,semin_nonlinear_2018}, which seems prone to the development of large-scale flows. We are interested in the flow within the cylinder: the fluid in the gap between the cylinder and the cubic tank follows a similar dynamics and thus imposes to the working fluid a (close to) no-flux lateral boundary condition.\n\nThe temperature of the bottom boundary is controlled by water circulating in the copper plate. Water is cooled down by a thermal bath with a fixed temperature set at $-1.25^{\\circ}$C. Due to thermal losses in the pipes alimenting the copper plate, bottom tank temperature is $0.2 \\pm 0.05^\\circ$C. Plate thickness and circulation circuit were optimised so as to ensure a uniform temperature over the whole plate. At the top boundary, the heater is set at a temperature of $35^{\\circ}$C. Its temperature control is custom made: a PT100 probe measures the heater temperature in real time, driving through a feed-back loop the input power injected in the heater. This is a very simple and inexpensive system to impose a temperature while having a transparent top boundary, allowing visualisation and velocity measurements by PIV. Nonetheless, it is necessary to point out that the temperature over the heater area is not perfectly homogeneous. Temperature is maximal at the centre where $T \\sim 38^{\\circ}$C, while the edges are indeed at the requested $T = 35\\pm 0.1^\\circ$C. This inhomogeneity of the top temperature by $\\delta T = 3^{\\circ}$C induces slow convective motions below the heater, in a $\\sim 2$ cm high layer. By performing an experiment where the whole fluid is stably stratified with an overall temperature gradient similar to the one in the stratified layer studied here, but above the density reversal at $4^{\\circ}$C (i.e. bottom boundary set at 10$^{\\circ}$C and top boundary at 70$^{\\circ}$C), we have checked that those top convective motions have no significant impact on the dynamics of the two-layer system. It is also important to say that despite the thick acrylic wall and the intermediate fluid layer between the cylinder and the tank, the working region is not fully thermally insulated on the sides. Nevertheless, our fully stratified, test experiment has shown no motion within the fluid driven by these lateral losses. \n\\\\\n\nThe equation of state of water is non-linear with a  maximum density close to 4$^{\\circ}$C (International Equation of State of Seawater, 1980):\n\\begin{equation}\n    \\begin{split} \n    \\rho(T)= 999.842594+6.793952.10^{-2}T-9.095290.10^{-3}T^2+1.001685.10^{-4}T^3 \\\\\n    -1.120083.10^{-6}T^4+6.536332.10^{-9}T^5.\\\\\n    \\end{split}\n    \\label{eq:eos_eau}\n\\end{equation}\nThus, due to the imposed temperature at top and bottom boundaries, the bottom part of the tank, between 0$^{\\circ}$C and 4$^{\\circ}$C, is convectively unstable (see figure \\ref{fig:setup}). Cold plumes detach from the bottom plate and rise in the tank due to buoyancy. Reciprocally, ``hot'' fluid sinks from the 4$^{\\circ}$C  interface due to gravity. While convective motion takes place in the lower layer, the upper part of the tank, between 4$^{\\circ}$C and 35$^{\\circ}$C, is stably stratified, with an assumed linear temperature profile at equilibrium. The temperature is indeed linear for an ideal case without thermal losses, assuming that the stratified layer has a bottom boundary at fixed temperature $4^\\circ$C and top boundary at $35^\\circ$C (\\textit{i.e.} constant diffusive flux through the whole layer). However, the density profile is not linear, due to the non-linear equation of state of water. Stratification is characterised by the Brunt-V\\\"ais\\\"al\\\"a frequency $N^* = \\frac{1}{2 \\pi} \\sqrt{-\\frac{g}{\\rho_0} \\frac{\\partial \\rho}{\\partial z}}$. Because of the non-linear equation of state, $N^*$ is not constant with depth, as shown in figure \\ref{fig:setup}. For simplicity, we also use below the global buoyancy frequency defined as $N = \\frac{1}{2 \\pi} \\sqrt{-\\frac{g}{\\rho_0} \\frac{\\Delta \\rho}{\\Delta z}}$, where ${\\Delta \\rho}$ is the global density contrast within the stratified layer of depth ${\\Delta z}$.\n\\\\\n\n\\begin{figure}[h]\n\n  \\includegraphics[scale=.3]{schema_cuve_profilN_NB_v3.png}\n  \\caption{2D sketch of the tank. A cylinder (light grey shaded area) is placed in a larger cubic tank. The system is cooled down at the bottom at 0$^{\\circ}$C and heated up at the top at 35$^{\\circ}$C. The bottom half is convective with an almost constant density, apart from the bottom boundary layer. The upper half is stably stratified and waves are generated due to the fluid motions in the convective layer. The graph on the right shows the theoretical profile for the buoyancy frequency $N^*$. It is computed considering a linear temperature profile and the equation of state of water (\\ref{eq:eos_eau}). The dashed line is the global buoyancy frequency $N$ calculated on the stratified layer. The various length scales are the cylinder diameter $D$, the vertical extent of the tank $H$ and the vertical extent of the convective layer $h$. $\\delta$ is the minimal width between the outer square tank and the inner cylinder. The $x$, $y$ and $z$-velocity components are noted $u$, $v$ and $w$ respectively.}\n  \\label{fig:setup}\n\\end{figure}\n\nBefore starting the experiment, the bath and the heater are allowed to reach their assigned temperature. Then, the upper half of the tank is filled with water stratified in temperature from 4$^{\\circ}$C to 35$^{\\circ}$C, using the double bucket technique \\cite{oster_density_1965}. The bottom half is filled with water with a temperature close to 4$^{\\circ}$C. This filling process is used to avoid tremendously long transient before reaching steady state by thermal diffusion. Typically, we fill the tank at the end of a day and start the measurements the next day in order to let the system reach its equilibrium state over night. Each experiment then lasts four days, with no change in the location of the interface. Note that this steady interface position is the result of the heat budget equilibrium between the convective heat flux extracted from the bottom plate, the diffusive heat flux through the stratified layer, and the lateral heat losses.\n\nTo perform PIV measurements, particles are  chosen as small as possible and with a density as close as possible to water density in order to avoid their sedimentation in the stratified layer over the long duration of the experiment. We use fluorescent orange polyethylene micro-spheres whose size ranges from $10 \\, \\mathrm{\\mu m}$ to $20 \\, \\mathrm{\\mu m}$ and density is $1.00 \\pm 0.01$~g\/cc. The fluorescent property allows us, with a high pass filter on the camera, to remove any laser reflection, significantly enhancing the images quality. The tank is illuminated with a green laser $532$~nm. Power is set at $1$~W. \nWe perform side view PIV to measure convection and IGWs spectral characteristics, and top view PIV to observe the large scale flow and its fluctuations over time. The camera used for the side view PIV is a HiSense Zyla $2560 \\times 2160$ pixels recorded on 12 bits. Acquisition rate is $2$~Hz with $100$~ms exposure time. Typical acquisition time for spectral characteristics is $50$~min.\nFor the top view, we use a Point grey camera $1920\\times1080$ pixels on 8~bits. Exposure time is $1$~s, acquisition rate $0.1$~Hz and acquisition time $8$~hours. Captured movies are processed either by the DantecDynamics software DynamicStudio for the side view or by DPIVSoft \\cite{meunier2003analysis} for the top view. Both are resolved into $32\\times32$ pixels boxes with 50\\% overlapping. \n\nSide view PIVs are performed in the middle of the tank at $y=16$~cm in a laser sheet crossing the cylinder along its diameter. This is the case for all figures shown in the $(x,z)$ plane and thus not mentioned in the results section. The vertical fields (see an example in figure \\ref{fig:LSCexpe}) do not show the whole $(x,z)$ plane (where the origin $(0,0)$ is located in the bottom left corner of the cubic tank): it was indeed chosen to zoom in, in order to have the best resolution for the very weak motions in the stratified layer. The interface between the layers is localised between $11 \\mathrm{~cm} \\leqslant z \\leqslant 12$~cm. Typical Rayleigh number for the convection based on this depth is $Ra = 7 \\times 10^6$, and the global Brunt-V\\\"ais\\\"al\\\"a frequency is $N = 1.35 \\times 10^{-1}$~Hz.\n\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[scale = .45]{champ_int_730_colored.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = .45]{LSC_0510_1800_colored.pdf}\n    \\caption{(Left) Instantaneous velocity field. An ascending plume is visible at $x=150$ mm, and transported by the large-scale circulation. (Right) Large-scale circulation in the convective layer obtained by time-averaging velocities over a 50 minutes signal. The large-scale circulation is a counter clockwise cell. No inversion of the circulation has been seen in our experiment. The velocities under $z=10$ mm are noisy and thus not shown here. Maximum instantaneous velocities are $2.5$ times bigger than the maximum averaged velocities. The left edge of the cylinder is located at $x=19$~mm and the centre of the cylinder is at $x=160$~mm. Approximately $40$~mm of the right side of the cylinder is not captured with our PIV measurments.}\n    \\label{fig:LSCexpe}\n\\end{figure*}\n\n\nTo observe the large-scale flow, horizontal views at different heights are performed. A linear translation axis platform from Igus, driven by homemade software, is used to automate the process. With two mirrors, one fixed at the exit of the laser head and the other fixed on the translating platform, it is possible to sweep along one direction with the laser sheet. We typically make measurements during 15~s in each of 11 successive planes separated by 0.5~cm. The full scan duration is about 3~min, and is repeated during at least 8~hours.\n\\\\\n\nThe cylindrical geometry described here differs from the cylindrical shell geometry in \\cite{plumb_instability_1978,semin_generation_2016, semin_nonlinear_2018}. We first tried to work in that annular geometry by adding a second, smaller cylinder in our cubic tank. Three different gap sizes were tested but none showed any interesting dynamics. Indeed, the convection was too confined in the shell to provide an efficient chaotic excitation, and IGWs did not propagate well within the stratification, attenuated quite fast by wall friction. During these tests, we observed the most interesting dynamics within the innermost cylinder, so we decided to use that geometry. The point of the cylindrical shell geometry is that it is a close analogue to the equatorial band of the stratosphere where QBO takes place. By working in a cylinder, the geometry analogy is lost but the physics of the problem remains the same, still a priori allowing for large-scale, reversing, axisymmetric horizontal flows.\n\n\\clearpage\n\\subsection{Numerical model}\\label{sec:methods_num}\n\nTo complement the experiments, we also performed Direct Numerical Simulations (DNS) of the same configuration. We solve the Navier Stokes equations using a non-Oberbeck Boussinesq model. The density variations are consider small compared to the reference density $\\frac{\\Delta \\rho }{\\rho_o} \\ll 1$. Therefore, density fluctuations only appear in the buoyancy force. However, temperature variations affect the value of the thermal expansion coefficient to account for the non-linear equation of state of water. Variations of the thermal diffusivity $\\kappa$ and kinematic viscosity $\\nu$ are neglected. Governing equations are given in dimensionless form by:\n\\begin{align}\n\\label{eq:momentum}\n\\frac{\\partial\\bm{u}}{\\partial t}+\\bm{u}\\cdot\\nabla\\bm{u} = & -\\nabla P+Pr\\nabla^2\\bm{u}-Pr Ra \\ \\theta^2 \\bm{e}_z \\\\\n\\label{eq:heat}\n\\frac{\\partial \\theta}{\\partial t}+\\bm{u}\\cdot\\nabla\\theta = & \\ \\nabla^2 \\theta \\\\\n\\label{eq:mass}\n\\nabla\\cdot\\bm{u} = & \\ 0\n\\end{align}\nwhere we used the depth $H$ of the container as a unit of length, the thermal diffusive timescale $H^2\/\\kappa$ as a unit of time and the difference between the bottom and the inversion temperature $T_0 - T_i$ as a temperature scale. These equations are characterised by the Prandtl number $Pr=\\nu\/\\kappa$, the global Rayleigh number $Ra=\\alpha g (T_0-T_i)^2 H^3\/(\\nu\\kappa)$ and the imposed dimensionless top temperature $\\theta_{top}=(T_{top}-T_i)\/(T_0-T_i)$.\nNote that the quadratic temperature term in the momentum equation is a direct consequence of the nonlinear equation of state of water given by equation~\\eqref{eq:eos_eau}, which is approximed in our model by the quadratic equation $\\rho(T) \\approx \\rho_0 (1-\\alpha(T-T_i)^2)$.\nThe coefficient $\\alpha$ is not the usual thermal expansion coefficient but has a unit of $(\\degree \\mathrm{C})^{-2}$ and is given by $\\alpha\\approx8.1\\times10^{-6}(\\degree \\mathrm{C})^{-2}$ (see also \\cite{lecoanet_numerical_2015}).\n\nWe consider a cylindrical fluid cavity of diameter $D=3H\/2$ as in the experiments.\nBoth horizontal plates are assumed to be no-slip and with fixed temperature.\nThe side wall is assumed to be no-slip and perfectly insulating.\nThis is of course not the case in the experiment, for which lateral heat losses are inevitable and top temperature is not exactly constant, but the objective is to check whether the conclusions drawn from the experimental results are robust and do not depend on these effects.\nSince the experiment runs with water, we use $Pr=7$.\nThe Rayleigh number of the experiment is $Ra=7 \\times 10^7$ while its dimensionless top temperature is $\\theta_{top}=-7.75$. If we were to run the simulation with these parameters, the interface will be located very close to the top boundary. It is not the case in the experiment because of the lateral heat losses, which tend to reduce the effective Rayleigh number. For that reason, and instead of taking into account these losses as in \\cite{lecoanet_numerical_2015}, we kept the insulating lateral boundaries and use the slightly adjusted parameters $Ra=10^7$ and $\\theta_{top} = -11$ instead, which leads to an interface located approximately at $z\\approx120$~mm, as in the experiment. The Rayleigh number could not be lowered under $10^7$ in order to keep the convective flow turbulent enough, thus we had to increase the top temperature to have the interface located at $z\\approx120$~mm.\n\nWe perform DNS of equations~\\eqref{eq:momentum}-\\eqref{eq:mass} using the spectral element solver Nek5000 \\citep{Nek5000}.\nThe global geometry is decomposed into hexahedral elements, with vertical refinement close to the horizontal boundaries and around the mid-plane where the inversion isotherm is expected to be located.\nVelocity, buoyancy and pressure variables are represented as tensor product Lagrange polynomials of order $N$ and $N-2$ based on Gauss or Gauss-Lobatto quadrature points.\nThe total number of grid points is given by $\\mathcal{E}N^3$ where $\\mathcal{E}$ is the number\nof elements.\nFor all the results discussed in this paper, the number of elements is $\\mathcal{E}=8960$ and we use a polynomial order of $N=11$. Numerical convergence was checked by increasing the polynomial order $N$.\nTime integration is performed with a third-order mixed implicit-explicit scheme.\nThe simulations are initialised with a small amplitude buoyancy perturbation and a temperature profile varying linearly between the top and bottom boundaries.\nSimulations are run until a quasi-steady state is reached, after which data is accumulated to compute statistics.\n\n\\section{\\label{sec:results}Results}\n\\subsection{\\label{sec:results_exp}Experiments}\n\\subsubsection{\\label{sec:results_conv}Convection}\nPIV side view is used to quantify horizontal and vertical velocities in the convection zone. Examples of vertical velocities measured at one point in a given location are shown in figure \\ref{fig:panache}, for both ascending cold and descending hot structures. Measurements are consistent with the numerical simulations \\cite{lecoanet_numerical_2015,couston_dynamics_2017} showing intense, localised, cold rising plumes and more diffusive descending plumes. Moreover, these structures are advected by a large-scale circulation encompassing the whole convective layer, as shown in figure \\ref{fig:LSCexpe}.\n\n\\begin{figure}[h]\n    \\centering\n    {\\label{fig:panacheup}\\includegraphics[scale=0.43]{sonde_panache_up.pdf}}\n    \\hspace{.2pt}\n    {\\label{fig:panachedown}\\includegraphics[scale=0.43]{sonde_panache_down.pdf}}\n    \\caption{Time evolution of the vertical velocity $w$ within: (Left) upward plumes at $x = 200$ mm, $z = 45$ mm, (Right) downward structures at $x= 100$ mm, $z = 95$ mm. }\n    \\label{fig:panache}\n\\end{figure}\n\nSpectral analysis is performed to extract power spectral density (PSD) from the velocity signals. Figure \\ref{fig:spectreconv} shows the PSD of the convection vertical velocity $w$. For the two panels, the spectrum is flat with a lot of energy for low frequencies, then the energy drops above some cut-off frequency. Left panel of figure \\ref{fig:spectreconv} shows the vertical velocity PSD at a single point close to the top of the convective layer. A small peak can be seen close to $f = 10^{-2}$~Hz. This is the quasi-periodic signal of the plumes dropping from the top thermal boundary layer. The theoretical characteristic time of convection can be computed from \\cite{gortler_convection_1966}:\n\\begin{equation}\n    \\tau = \\frac{h^2}{\\pi \\kappa}\\left(\\frac{\\Delta T}{\\Delta T_{local}} \\times \\frac{Ra_c}{Ra}\\right)^{2\/3}\n\\end{equation}\n\n\\begin{figure*}[b]\n    \\centering\n    \\includegraphics[scale = .43]{spectre_1_point_cd67_58_vf.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = .43]{spectre_W_50toend_10to57_vf.pdf}\n    \\caption{PSD for the vertical velocity fluctuations. (Left): PSD computed at a single point $x=100$~mm, $z=95$~mm (signal shown in figure \\ref{fig:panache} right). The plume forcing frequency can be seen around $f = 10^{-2}$~Hz (red dashed line). (Right): PSD spatially averaged over the whole convective cell in the measured $(x,z)$ plane (all points above $z=10$~mm and below $z=110$~mm). }\n    \\label{fig:spectreconv}\n\\end{figure*}\n\nwith $h$ the height of the convective layer, $\\kappa$ the thermal diffusivity, $\\Delta T$ the temperature difference between the top and bottom of the convective domain, $\\Delta T_{local}$ the temperature difference between the top and bottom of the thermal boundary layer, and $Ra_c$ the critical Rayleigh number. The critical Rayleigh number in the presence of free and solid interfaces and for fixed temperature is $Ra_c = 1100.65$. For our experiment, the characteristic time is $\\tau = 96$~s, thus characteristic frequency is $1\/ \\tau \\sim 10^{-2}$~Hz, which is close to the observed peak in the left panel of figure \\ref{fig:spectreconv}. At frequencies lower than this characteristic frequency, the spectrum is flat. This is explained by the combined effect of the randomness of the plumes (see figure \\ref{fig:panache}) and of the slow fluctuations of the large-scale circulation. Right panel of figure \\ref{fig:spectreconv} shows the PSD of vertical velocities, averaged over the whole convective cell in the $(x,z)$ plane. It shows a similar trend, with a lower cut-off frequency compared to right panel spectrum. Actually, the plumes signal is more localised and less intense on average than the large-scale circulation signal, which hence dominates the space-averaged PSD.\n\nThe probability density function (PDF) of the vertical velocities in the whole convective layer $\\mathrm{P}(w)$ is computed and shown in figure \\ref{fig:pdf}. It is normalised such that $\\int \\mathrm{P}(w)\\mathrm{d}w = 1$. The PDF describes important features of the convection: it is skewed towards positive values, with positive velocities reaching higher magnitude than negative velocities, \\textit{i.e.} the ascending plumes are stronger than the descending structures. However, the central part of the PDF is close to gaussian profile. The distribution obtained here is in good agreement with the probability density function computed in an idealised 2D numerical model by Couston et al. \\cite{couston_dynamics_2017}. Note that this asymmetry is specific to our model, for which the usual upside-down symmetry in Boussinesq Rayleigh-B\\'enard convection is broken.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale = .6]{pdf_conv.pdf}\n    \\caption{Probability density function of the vertical velocities in the convective layer. All PIV points under $z=110$~mm have been used to compute the PDF.}\n    \\label{fig:pdf}\n\\end{figure}\n\n\\subsubsection{\\label{sec:results:buffer}Buffer layer}\nAn intermediate layer (we name it the buffer layer in the following) is present between the convective layer and the stratified layer. It was first reported in the quasi-2D 4$^{\\circ}$C convection experiment by Perrard et al.  \\cite{perrard_experimental_2013}. Their temperature measurements showed that the buffer layer corresponds to the area where the temperature goes from 4$^{\\circ}$C to 8$^{\\circ}$C. This actually corresponds to the overshooting region for rising cold plumes (note that this type of convection is called \"penetrative convection\" because of this effect). Indeed, since the density of water is close to quadratic around $4^{\\circ}$C, densities at e.g. $0^{\\circ}$C and $8^{\\circ}$C are the same, and the 8$^{\\circ}$C isotherm is the theoretical maximum height reachable by an ascending cold plume at $0^{\\circ}$C in the absence of thermal diffusion. Simultaneously, the overall density profile between $4^{\\circ}$C and $8^{\\circ}$C is stable, as in the stratified layer above $8^{\\circ}$C. The buffer layer is thus a very specific location supporting simultaneously convective overshooting motions and IGWs, as observed with PIV measurements \\cite{perrard_experimental_2013}. \n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale = .6]{buffer_U-xavg.pdf}\n    \\caption{Time evolution of the horizontal average of the horizontal velocity, noted $u_X$. Red (resp. blue) regions correspond to mean flow going towards the right (resp. left).}\n    \\label{fig:bufferlayer}\n\\end{figure}\n\nTo complete the description of this buffer layer now using velocity measurements, we plot in figure \\ref{fig:bufferlayer} the spatio-temporal graph of the horizontal average of the horizontal velocity $u$. The graph exhibits a strong shear around $z=120$ mm. Since the fluid is going in opposite directions above and below $z=120$ mm with a sharp interface, viscous entrainment by the convective layer is excluded. A special kind of thermal coupling might explain the observed dynamics, as sketched in figure \\ref{fig:schema_couplage}. Indeed, when a cold ascending plume from the convection zone reaches the interface and overshoots in the buffer region, its associated velocity perturbation dissipates more rapidly than its temperature perturbation. Due to gravity, the distorted part of the buffer region wants to sink back to its initial state (pictured by the green arrows), while the fluid above the buffer layer moves towards the impact point of the plume to take the place of the falling water (pictured by the red arrows). The buffer layer then needs some compensating fluid from the convective layer. This mechanism works when the velocity perturbation of the plume at the interface dissipates more rapidly than the thermal perturbation, hence for a Prandtl number $Pr \\geq 1$. One might expect the shearing zone to decrease in size and amplitude when thermal diffusion increases (i.e. when the Prandtl number decreases), since the overshooting rising cold plume will then equilibrate thermally during its ascent more rapidly. This may explain why no interfacial shear was reported in the systematic numerical study of \\cite{couston_dynamics_2017,couston_order_2018} where $Pr \\leq 1$. Global temperature field measurements (using e.g. Temperature Dependent, Laser Induced Fluorescence) are now required to confirm or infirm the proposed model, but those are beyond the scope of the present paper. Note that by extension, we call \"buffer layer\" in the following the region including the $T=4^\\circ$C to $T=8^\\circ$C overshooting region and the shear region. In the experiment, the shear region extends from $z=120$~mm to $z \\approx135$~mm.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[scale = .75]{schema_couplage_final2.pdf}\n    \\caption{Sketch of the thermal coupling between the convective and buffer layers. On the left, a cold plume moves upwards towards the interface between the two layers. On the right, isotherms are deflected due to the impact.}\n    \\label{fig:schema_couplage}\n\\end{figure}\n\n\\clearpage\n\\subsubsection{\\label{sec:results_waves}Internal gravity waves}\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale = .83]{visu_onde_colormap_col_02to03_axisequal_schema.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = .53]{panache_v1.pdf\n    \\caption{Velocity fields showing IGWs propagating. (Top) Velocities in $(x,z)$ plane. The signal is frequency-filtered to enhance the visualisation of oscillatory motions: only frequencies between $0.02$ and $0.03$ Hz are shown, propagating at an angle of roughly $75^\\circ$ with the vertical. The angle of propagation is the angle between the constant phase line and the vertical. (Bottom) Velocities in $(x,y)$ plane located at $z \\approx 125$~mm. In the $(x,y)$ plane, IGWs take the form of oscillating rings. Note that this figure is from a previous experiment without any internal cylinder and is therefore only displayed here as an illustration of the IGWs seen from above.}\n    \\label{fig:ondeN}\n\\end{figure}\n\n The convective motions induce a Reynolds stress at and below the interface which generates IGWs propagating in the stratified area \\cite{lecoanet_numerical_2015}. An example is shown in figure \\ref{fig:ondeN}. The vector field has been frequency filtered in the band $[0.02-0.03]$ Hz to isolate a single propagating wave train. We can measure an angle close to $\\theta \\simeq 75^{\\circ}$ between contant phase lines and the vertical. This observation is in good agreement with the inviscid dispersion relation $\\omega = \\pm N \\mathrm{cos}(\\theta)$, which relates the frequency and the propagation angle of IGWs. Indeed, at $z=120$~mm, $N \\sim 0.1$~Hz, thus $\\theta = \\mathrm{cos^{-1}}\\left(\\frac{\\omega}{N}\\right) = 78.5^\\circ$. The motion within the stratified area is a superimposition of many such IGWs oscillating at different frequencies.\n\nTo further investigate the waves signal, waves spectra are plotted in figure \\ref{fig:spectro}, showing the power spectral density of oscillatory motions within the stratified layer at every height, averaged horizontally for each height. The grey line is the theoretically computed buoyancy frequency profile. Figure \\ref{fig:spectro} shows that energy is present in a wide frequency band, from the lowest measured frequencies to the buoyancy frequency $N$. Low frequency motions $f < 4\\times10^{-3}$~Hz are very intense and propagate high in the stratified layer. Motions with frequency ranging from $4 \\times 10^{-3}$~Hz to $N$ are less intense, but still propagate into the stratified layer. Motions propagating at frequencies higher than the buoyancy frequency $N$ are greatly attenuated after a centimetre as IGWs of frequency larger than $N$ are evanescent. The weak signal at low frequencies above $z=180$ mm comes from the convective motions due to the non-homogeneous heating at the top. These motions are confined at the very top of the experimental container. \n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[scale=0.55]{propagation_nonorm_120_vff.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale =.55]{plot_attenuation_norm_v2.pdf}\n    \\caption{(Left) Power spectral density of the absolute velocity $\\sqrt{u^2+w^2}$ above the convective layer. The grey curve shows the theoretical buoyancy frequency profile, assuming a linear profile for the temperature, from $4^\\circ$C to $35^\\circ$C. (Right) Two selected profiles (taken at frequencies shown by dashed lines on the left graph) of the re-scaled PSDs by the PSD at the top of the convective layer, \\textit{i.e.} $z=120$~mm (noted $\\mathrm{PSD_o}$). The PIV measurements are performed for 50 minutes and results (using the pwelch Matlab function) are horizontally averaged at each height to obtain the averaged power spectral density.}\n    \\label{fig:spectro}\n\\end{figure*}\n\nRight panel of figure \\ref{fig:spectro} shows two vertical profiles of the PSD re-scaled by the PSD at the top of the convective layer ($z=120$~mm), taken at two different frequencies. The energy decrease is quite similar between $z=120$~mm and $z=140$~mm for both frequencies. However, for $z>140$~mm, the energy for the higher frequency decreases slower than the energy for the lower frequency. This dependence of the attenuation length regarding the frequency of the signal is a characteristic of IGWs. Indeed, the dispersion relation of IGWs relates the frequency and the wave vector direction. Moreover, energy propagates perpendicularly to the wave vector for IGWs (i.e. group and phase velocities are perpendicular). The closer to $N$ the wave frequency $\\omega$ is, the more horizontal the phase propagates, hence the more vertical energy propagates. High frequency waves are thus capable of transporting energy to high altitudes before being damped. On the opposite, waves with low frequency compared to $N$ propagate energy almost horizontally, and are thus attenuated before reaching high altitudes. At frequencies $f<4\\times 10^{-3}$~Hz, a lot of energy is seen and the attenuation length does not depend on the frequency. There is no reason why IGWs should disappear below a certain frequency, but we would expect to see the attenuation length to keep decreasing with decreasing frequency. We thus deduce that IGWs at frequencies $f \\leqslant 4\\times 10^{-3}$~Hz are hidden in the energy spectrum by some very energetic large-scale slowly-varying flow, which we will describe below.\n\n\nMore than one order of magnitude separates the buoyancy frequency and the fastest large-scale flow fluctuations. The large-scale flow penetrates deep into the stratified layer. It globally decreases in amplitude with height, but with some local increases at $z\\sim 125$ mm (i.e. close to the interface between convective and buffer layers) and $z \\sim 145$ mm.\nThe IGWs signal can be seen between $f = 4 \\times 10^{-3}$ Hz and the buoyancy frequency. A peak that reaches the top of the stratified layer is seen around $f= 1.2 \\times 10^{-2}$ Hz, \\textit{i.e.} the same frequency as the convective forcing discussed in section \\ref{sec:results_conv}. It corresponds to the strong excitation provided by the cold rising and hot sinking turbulent plumes.\nHowever, top panel of figure \\ref{fig:spectro} also shows a sudden drop of the energy at frequencies $f>1.2 \\times 10^{-2}$~Hz. Indeed, wave attenuation is strong at these frequencies, even if they are close to (but below) the buoyancy frequency $N$. Actually, energy dissipation also depends on the norm of the wave vector squared. There is no reason that all excited waves have the same wave vector norm; one could even expect that fastest waves are excited by fastest, hence smallest convective patterns, and are thus also at smallest scale: they then dissipate more rapidly.\n\n\n\\subsubsection{\\label{sec:results_lsf}Large-scale flow in the stratified layer}\nFigure \\ref{fig:spectro} shows an important amount of energy at low frequencies which has been interpreted as the signature of a large-scale slowly-varying flow in the stratified layer. We will now investigate the nature of these fluctuations to see if they relate to reversals similar to the QBO.\n\nFigure \\ref{fig:meanflow} shows horizontal vector fields at the same depth at different times. In figure \\ref{fig:meanflow}(a), the flow goes counter-clockwise inside the cylinder. Figure \\ref{fig:meanflow}(b) shows that two contra-rotating vortices with a smaller amplitude typical velocities have appeared. Figure \\ref{fig:meanflow}(c) shows a mostly clockwise rotating flow, where one of the preceding eddy pairs has nearly disappeared. The large-scale flow thus evolves drastically over time. A criterion is computed to extract a typical mean velocity from those fields that accounts for the ``direction'' of the large-scale flow: as illustrated in figure \\ref{fig:critere}, we compute a mean azimuthal velocity, taken along a ring centred in the cylinder. Other criteria to extract a representative value for the large scale flow direction have also been tested, including: the mean vorticity over the cylinder area, the average of the azimuthal velocity over several rings with different radii, and the azimuthal velocity averaged over thick rings. They all give similar results for the large-scale flow measurement. \n\\\\\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[scale = .8]{evolution_mf_vff.pdf}\n    \\caption{Horizontal velocity vector fields in the stratified layer at different times. The laser sheet is located at $z=150$~mm. The large-scale flow reverses from (a) to (c). Time between (a) and (c) is approximately half an hour. Maximum velocities are $0.1$ mm\/s.}\n    \\label{fig:meanflow}\n\\end{figure*}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale = .45]{critere_mf.pdf}\n    \\caption{Criterion used to extract a significant value for the large scale flow and its direction: the azimuthal velocity is averaged over the ring shown in red.}\n    \\label{fig:critere}\n\\end{figure}\n\nIn order to investigate the vertical phase propagation of the reversals, and thus, to compare the reversal dynamics observed to a QBO-like phenomenon, the setup has been equipped with a linear translating platform that allows us to perform horizontal laser sheet sweeping along the vertical. Horizontal velocities are measured in an horizontal plane, every $5$~mm from the top of the convective layer $z=110$~mm to the middle of the stratified layer $z=160$~mm. Any trace of downward phase propagation of the reversals, as observed on the QBO on Earth \\cite{baldwin_quasi-biennial_2001} and on the historical Plumb experiment \\cite{plumb_interaction_1977, semin_nonlinear_2018}, would be a significant evidence for QBO-like phenomenon in the experiment. \nIndeed, the phase propagation of the reversals due to IGWs non-linear interactions is theorised as follows: an IGW propagating in a stratified layer with an horizontal phase velocity in the same direction as the existing base flow propagates upward until reaching a critical height $z_c$, where it deposits all its energy locally. At $z=z_c$, the flow accelerates. Thus the critical height where the flow is intense enough to damp the wave is lowered. As time goes on, this critical height moves towards the location where the waves are emitted. Here, the waves are emitted at the bottom of the stratified layer. We would expect a downward phase propagation if the reversals are driven by IGWs non-linear interactions. \n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.7]{QBOfinal_avecpentediffusionv2.pdf}\n    \\caption{Reversals of the large-scale flow. $z=115 - 120$ mm is the convective \/ buffer layers interface. Ascending plumes often perturb the buffer layer flow. The velocity was measured at 11 heights, marked by each tick in the vertical axis of the figure, and interpolated in between. The slope of the black dot lines represent the viscous coupling phase velocity.}  \\label{fig:QBO}\n\\end{figure*}\n\n\nWe performed long time experiments (around 8 hours). Typical results extracted from the criterion described above are shown in figure \\ref{fig:QBO}. Blue patches (resp. red patches) represent large scale flow going counter clockwise (resp. clockwise). The present measurements mainly confirm the interpretation of figure \\ref{fig:spectro} for the lowest frequencies: the large-scale flow is horizontal and extends over the whole depth of the stratified layer with an amplitude attenuated with height, and exhibits slow reversals. Additionally, some intense events at $z = 110$ mm are directly related to penetrative plumes from the convection. Reversals times range from $400$s to $1800$s. However, no downward phase propagation of the reversals is observed. On the contrary, the reversals seem to occur along the whole stratification height at the same time, or even with a rapid upward phase propagation. Since the phase propagation is not towards the location where the waves are emitted, the reversals are unlikely driven by the non-linear interactions of IGWs. However, as seen in section \\ref{sec:results_waves}, IGWs propagate in the stratified layer and carry energy. Therefore, they give energy to the large-scale flow through non-linear interactions. Yet, the process is not dominant in the reversals dynamics.\n\\\\\nSince, the reversals observed in figure \\ref{fig:QBO} do not have a downward phase propagation, we look for other mechanisms than the QBO mechanism to explain the reversals. Two other mechanisms can be investigated. The first one relies on a specific convective dynamics within the overall stratified layer driven by horizontal gradients related to imperfect top and side boundary conditions. The second mechanism relies on viscous coupling with the underlying convective and buffer layers. \n\nOur fully stratified reference experiment described in section \\ref{sec:methods} precludes the first scenario. Indeed, setting the bottom boundary at 10$^{\\circ}$C and the top boundary at 70$^{\\circ}$C, no motion is observed for the bottom $3\/4$ of the tank. In this test-experiment, the top $1\/4$ of the tank is animated by convective motions due to the non-homogeneous top heat source (in the standard $4^\\circ$C experiment, where $T_{top} = 35\\degree$C, only $\\sim 2$ cm are affected by the convection at the top of the tank, because the non-homogeneity of the heat source is less important for lower temperature, thus the horizontal convection is weaker). However, these are inefficient to generate waves below and to drive any large-scale flow observable away from the top region.\n\n\n\\begin{figure}[h]\n\\includegraphics[scale = 0.55]{balayage_corr_vf.pdf}\n  \\caption{Velocity vector fields in the horizontal plane. Different columns represent different sweeping cycles $t^*$ (one sweeping cycle corresponds to the 11 steps needed to go from the lowest position $z=110$~mm to the highest position $z=160$~mm). Different rows represent different heights within the same sweeping cycle: first row is the top of the convection $z=110$~mm, second row is in the buffer layer $z=120$~mm and third row is in the stratified layer $z=145$~mm. Convective plumes are easily noticeable on the first row fields.}\n  \\label{fig:champ_correle}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\includegraphics[scale = 0.55]{balayage_NOcorr_vf.pdf}\n  \\caption{Same as figure \\ref{fig:champ_correle} but for different sweeping cycles. Note that in this set of velocity fields, the buffer and stratified layers are less correlated than they are in figure \\ref{fig:champ_correle}.}\n  \\label{fig:champ_pascorrele}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[scale=0.6]{correProdscalaire_BW_vf.pdf}\n  \n  \\caption{Velocity correlation between the three layers. Dashed black lines are the $-0.5$ and $0.5$ values. Correlation coefficient are  window-averaged over $10$~min to smooth the curve.}\n  \\label{fig:correlation}\n\\end{figure}\n\nThis leaves viscous entrainment as a possible driving mechanism. The dotted lines on figure \\ref{fig:QBO} show a theoretical viscous time, computed from the time for viscous entrainment to drive 20\\% of the horizontal velocity at $z = 115$ mm to $z=160$ mm, starting from a base state flow at rest. The 20\\% value corresponds to the measured value of the large scale flow at $z=160$ mm compared to the value at $z = 115$ mm (noted $u_b$). The theoretical viscous time entrainment is given by $t = \\frac{z^2}{4\\,\\nu\\, \\mathrm{erf}^{-2}\\left(\\left(\\frac{u}{u_b}-1\\right)\\right)}$. The reversals occur in a time scale comparable with this theoretical viscous time. The similarity between the slope of the dashed lines and the slope of the upward phases suggests that reversals are driven viscously.\n\nHowever, the existence of the buffer layer and its associated intense shear, with opposite horizontal velocities with the convective layer below (see figure \\ref{fig:bufferlayer}) precludes direct viscous coupling between the convective and stratified layers. Besides, no reversal has been observed in the convective region. We thus propose a thermal coupling between the convective and buffer layers as seen in section \\ref{sec:results:buffer}, associated to a viscous coupling between the buffer and stratified ones. To further quantify this possibility, figures \\ref{fig:champ_correle} and \\ref{fig:champ_pascorrele} show horizontal velocity fields at different heights and at different times. For each of the columns shown, first row is the mean flow in the convective layer at depth $z=110$~mm, second row is the mean flow in the buffer layer at depth $z=120$~mm and third row is the mean flow in the stratified layer at depth $z=145$~mm. The correlation coefficients through time between (i) the convective and buffer layers, (ii) the buffer and stratified layers, and (iii) the convective and stratified layers have been computed. It consists in a scalar product of the velocity vector for each position at two different heights rescaled by the product of the norm of the velocity vector at the two heights, \\textit{i.e.}: \n\n\\begin{equation}\nR_{ij} = R(x_i,y_j) = \\frac{u(x_i,y_j,z_1) \\times u(x_i,y_j,z_2) + v(x_i,y_j,z_1) \\times v(x_i,y_j,z_2)}{\\left(u(x_i,y_j,z_1)^2+v(x_i,y_j,z_1)^2\\right)^{1\/2} \\times \\left(u(x_i,y_j,z_2)^2 + v(x_i,y_j,z_2)^2 \\right)^{1\/2}}\n\\end{equation}\nThis gives a correlation coefficient $R_{ij}$ for each PIV position in the horizontal plane. The global correlation coefficient $R$ is computed by spatially averaging the local correlation coefficients. \n\nResults are shown in figure \\ref{fig:correlation}. The convective and buffer layers are negatively correlated: the correlation coefficient is most of the time close to $R=-0.5$. This can also be seen at all times in figures \\ref{fig:champ_correle} and \\ref{fig:champ_pascorrele}, where horizontal velocities in the convective and buffer layers have opposite direction. A diverging flow coming from an impinging plume in the convective zone corresponds to a converging flow in the buffer layer towards the impact zone, hence confirming the thermal coupling mechanism described in section \\ref{sec:results:buffer}. This converging flow may lead either to a clockwise or anticlockwise azimuthal mean flow, depending on the details of the chaotic excitation from the convective plumes. The correlation coefficient between the convective and stratified layers can be positive or negative, and is anyway most of the time less than 0.2, in absolute value. The correlation coefficient between the buffer and stratified layers shows a lot of temporal variations. However, it remains always positive. At a given time, the large-scale flow in the stratified layer may switch between a regime strongly dominated by the buffer layer (see also figure \\ref{fig:champ_correle}), and a second regime where the flow in the stratified layer is quite different from the flow in the buffer layer (see also figure \\ref{fig:champ_pascorrele}). \n\nWe thus conclude that the stratified layer is globally viscously driven by the buffer layer. However, the stratified layer exhibits additional complexities. These might be due to IGWs interacting with the large-scale flow. The results from Couston et al. \\cite{couston_order_2018} show that the lower the Prandtl number, the more regular the QBO. In the experiment, the Prandtl number is close to $Pr = 7$: the typical associated QBO-type flow is irregular, with low amplitude. We thus propose that large-scale flow driven by IGWs non-linear interaction superimposes on the viscously driven flow, but remains secondary. We do not know at this point how to disentangle those two potential contributions from the available data. \n\n\n\\clearpage\n\\subsection{\\label{sec:results_num} Numerical simulations}\nThe experimental results are not fully sufficient to explain, with complete certainty, the origin of the buffer layer and of the large-scale flow observed in the stratified layer. In addition, the effects of the lateral heat losses and top temperature heterogeneity are difficult to distinguish. To answer these questions, 3D DNS of a configuration similar to our experiments are performed, reproducing the 4$^{\\circ}$C convection but with idealised boundary conditions (i.e. no flux on the sides, and fixed temperature at the top and bottom). As mentioned in section \\ref{sec:methods_num}, the Rayleigh number $Ra$ and $T_{top}$ are tuned so that the interface depth in the experiment and the numerical simulation are similar. We have $Ra=10^7$ and $T_{top}=48^\\circ$C. All the numerical simulations are run dimensionless, but results are shown in dimensional values. The length scale is $H = 200$~mm, the vertical extent of the whole domain (hence diameter is $D=300$~mm), the timescale is the thermal diffusive time $\\tau = \\frac{H^2}{\\kappa} = \\frac{0.2^2}{1.5 \\, 10^{-7}} = 2.67 \\times 10^{5}$~s, and the temperature is given by the dimensionless temperature $\\theta = \\frac{T - T_i}{T_0 - T_i}$, where $T, T_i, T_0$ are respectively the dimensional temperature, the inversion temperature of the equation of state (i.e. $4^\\circ$C), and the bottom temperature (i.e. $0^\\circ$C). Results for sections \\ref{sec:results_num_conv} - \\ref{sec:results_num_igw} are computed from a $(x,z)$ vertical plane located along a cylinder diameter.\n\n\\subsubsection{\\label{sec:results_num_conv}Large-scale circulation in the convection zone and buffer layer}\nFigure \\ref{fig:LSC_num} shows that a large-scale circulation takes place in the convective layer. It consists of a cell filling the whole convective layer, and exhibits no reversal over the whole course of the simulation. The fluid rotates counter clockwise in the vertical plane. This is qualitatively consistent with the mean flow observed in the experiment and shown in the right panel of figure \\ref{fig:LSCexpe}. As in the experiments, a counter current exists on the top of the convective layer at $z= 120$~mm, creating a strong shear and demonstrating the existence of a buffer layer in the numerical simulation as well. \n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale = .47]{quiver_inst_color.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = .47]{quiver_moyen_color.pdf}\n    \\caption{(Left) Instantaneous velocity field. An ascending plume is visible at $x=230$ mm. (Right) Large-scale circulation in the convective layer obtained by time-averaging velocities over a 50 minutes recording. The large-scale circulation is a counter clockwise cell. Maximum instantaneous velocities are $3$ times bigger than the maximum averaged velocities.}\n    \\label{fig:LSC_num}\n\\end{figure}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[scale = 0.5]{Spatio_temp_Ux_v3_dim.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = 0.5]{profil_T_all.pdf}\n    \\caption{(Left) Horizontal average of the horizontal velocity $u$ over a vertical cross-section in the middle of the tank. The buffer layer can be seen above $z=120$~mm. A stationary large-scale circulation is present in the convective layer, even if it appears quite perturbed at the end of the signal. (Right) Temperature profiles along the $z$-axis.}\n    \\label{fig:num_buffer}\n\\end{figure*}\n\nThe space-time diagram of the mean horizontal flow shown in figure \\ref{fig:num_buffer} confirms it. Observing the buffer layer in the absence of side thermal losses and top temperature heterogeneity is an additional argument accounting for the fact that it is not an artefact driven by imperfect experimental conditions.\nWe also observe that the flow within the convection stays positive through time at the bottom and negative at the top. This is evidence of the steady large scale circulation taking place in the convective layer. Some events appear at $t > 1.42 \\times 10^{5}$~s and are interpreted as quasi-reversal of the large-scale circulation. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=.5]{quiverIsotherme.pdf}\n    \\caption{Velocity field and temperature isotherms at the end of an upward plume impact on the interface.}\n    \\label{fig:num_quiverisoT}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=.6]{corre_WetU.pdf}\n    \\caption{(Left) Spatio-temporal diagram of the horizontal velocity $u$ at $z=128$~mm. (Right) Spatio-temporal diagram of the vertical velocity $w$ at $z=108$~mm. The event at $t \\approx 1.052 \\times 10^5$~s is shown in figure \\ref{fig:num_quiverisoT}.}\n    \\label{fig:num_correWU}\n\\end{figure}\n\nThe temperature profile along the $z$ axis is also plotted on the right panel of figure \\ref{fig:num_buffer}. The figure shows a temporal and horizontal average of the temperature field (black thick curve), two temporal averages at two different positions $x=14$~mm (left side of the tank - dashed grey) and $x=277$~mm (right side - dotted grey) and an instantaneous profile at $x=145$~mm (middle of the tank - thick grey with crosses). The thermal boundary layer can be seen, between $z=0$~mm and $z=10$~mm. Then, between $z=10$~mm and $z=100$~mm lies a layer of constant temperature $T \\sim 2.8^\\circ$C. Between $100 \\mathrm{~mm} \\leqslant z \\leqslant 115 \\mathrm{~mm}$, the temperature profile evolves from constant to linear for $z > 115$ ~mm. The $T=4^\\circ$C (respectively $T=8^\\circ$C) isotherm is located at $z = 110$~mm (resp. $z = 120$~mm). Note that the temporal average of the temperature profiles are different on the left and right sides of the tank. Indeed, the constant temperature height goes to $z=90$~mm for the left side whereas it goes to $z=115$~mm for the right side. This suggests that the convective \/ buffer layer interface does not lies at one height over the whole tank but is a function of time and space. This is very likely due to the large-scale circulation. Thus, the thermal coupling described in \\ref{sec:results:buffer} will likely occur at different heights, depending on time and horizontal position.\n\n\nThe thermal coupling as schematised in figure \\ref{fig:schema_couplage} can be found in the numerical simulation. This is represented in figure \\ref{fig:num_quiverisoT}. An upward plume impacting the convective \/ buffer layer interface is seen. The isotherms ranging from $T=4^\\circ$C to $T=11^\\circ$C are deflected upward, due to the plume bringing cold fluid upward. On the contrary, the isotherms $T = 12 -14^\\circ$C are deflected downward by the converging flow. Isotherms at $T \\geqslant 15^\\circ$C remain horizontal. After the impact on the interface, the plume is deflected outwards. One could expect the fluid above the impact to be viscously entrained by this outward deflection. However, as observed in figure \\ref{fig:num_quiverisoT} for the simulation and figures \\ref{fig:champ_correle}-\\ref{fig:champ_pascorrele} for the experiment, the fluid above the interface is going towards the plume, \\textit{i.e.} in the opposite direction of the fluid below, hence explaining the observed shear (see figures \\ref{fig:num_quiverisoT} and \\ref{fig:schema_couplage}). The time evolution of these dynamics is shown in figure \\ref{fig:num_correWU}.\n\n\nFigure \\ref{fig:num_correWU} shows the time evolution of the horizontal velocity $u$ in the shear layer at $z=128$~mm and the time evolution of the vertical velocity $w$ in the convective layer at $z=108$~mm. Comparing the two panels of figure \\ref{fig:num_correWU} shows that upward plumes are concomitant with converging horizontal velocities towards the plume impact. Indeed, the spatio-temporal diagram of $w$ exhibits local strong upward plumes. These plumes, as suggested by the dashed black lines, are correlated in time and space with converging horizontal velocities. For instance, an upward plume is seen at $x\\approx220$~mm and $t\\approx1.043 \\times 10^5$~s. At the same horizontal position and time, the positive horizontal velocity becomes stronger and the negative horizontal velocity patch increases in size to reach $x\\approx220$~mm. The converging horizontal velocities event occurs a short time after the impact of the plumes. Thus, it can be concluded that the plume induces the converging flow, as suggested by our explanation in section \\ref{sec:results:buffer}.\n\n\\clearpage\n\n\\subsubsection{\\label{sec:results_num_igw}Internal gravity waves}\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale = 0.55]{spectro_vfinale.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = .55]{plot_attenuation_norm_num_v2.pdf}\n    \\caption{(Left) Power spectral density of the absolute velocity $\\sqrt{u^2+w^2}$ in the buffer and stratified layers. The grey curve shows the buoyancy frequency profile computed from the spatial and temporal average of the temperature field. (Right) Two selected profiles (taken at frequencies shown by dashed lines on the left graph) of the re-scaled PSDs by the PSD at the top of the convective layer, \\textit{i.e.} $z=118$~mm.}\n    \\label{fig:num_spectro}\n\\end{figure}\n\nPSDs are computed in the stratified and buffer layers and are plotted in figure \\ref{fig:num_spectro}. As for the experiment (figure \\ref{fig:spectro}), numerical results show oscillatory motions at different frequencies attenuated with height. Experimental results (figure \\ref{fig:spectro}) and numerical results (figure \\ref{fig:num_spectro}) show strongly similar dynamics: most of the energy is present at low frequencies ($f<3 \\times 10^{-3}$~Hz). The motion with frequencies ranging from $3 \\times 10^{-3}$~Hz to $N$ are less intense, and almost no energy is seen at frequencies $f>N$. \n\nRight panel of figure \\ref{fig:num_spectro} shows two selected vertical profiles (shown by the white dashed line on the left panel figure) of the the PSD re-scaled by the PSD at $z=118$~mm. The energy for the higher frequency ($f=1.4\\times10^{-2}$~Hz) decreases slower than the energy for the lower frequency ($f=5.0\\times10^{-3}$~Hz). This is, as experimental results, in agreement with the dispersion relation of IGWs.\nThe overall behaviour of waves spectra is similar in experiment and numerical simulation, with an attenuation length independent of the frequency in the low-frequency signal thus confirming a viscous coupling origin of the large-scale flow, and increasing when the frequency goes towards $N$ in the wave domain.\n\n\\subsubsection{Large-scale flow within the stratified layer}\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[scale = 0.6]{Spatiotemp_Vtheta_cyl_v2_square_dim.pdf}\n    \\hspace{.1cm}\n      \\includegraphics[scale = 0.6]{Spatiotemp_Vtheta_cyl_v2_zoom_dim.pdf}\n    \\caption{Spatio-temporal diagrams of the azimuthal averaging of the azimuthal velocity inside a virtual cylinder of radius $r = 140~$mm. The bottom figure is a zoom on the stratified zone, delimited in the top figure by the black square. The slope of the black lines show the theoretical viscous diffusive time.}\n    \\label{fig:num_vthetacyl}\n\\end{figure*}\nSimilarly to what has been done for the experimental data, figure \\ref{fig:num_vthetacyl} shows the mean azimuthal velocity over the whole height of a virtual cylinder of radius $r = 140$ mm. We observe reversals within the convective layer ($z<120$ mm), which are not systematically correlated with the signal in the stratified layer. The mean velocity in the stratified layer also exhibits reversals. They are characterised by an upward phase propagation from the buffer zone at $z=120$ mm, as shown in the zoom (bottom panel of figure \\ref{fig:num_vthetacyl}). The phase velocity seen in figure \\ref{fig:num_vthetacyl} is in good agreement with the theoretical time for viscous propagation $t = \\frac{z^2}{4\\,\\nu\\, \\mathrm{erf}^{-2}\\left(\\left(\\frac{u}{u_b}-1\\right)\\right)}$. This corroborates the fact that the reversals observed within the stratified layer are viscously driven from the dynamics occurring in the buffer layer, as it has been seen for the experiment. Reversals time ranges from $300$~s to $1500$~s. Those reversals times are similar to the experimental ones, though slightly shorter (numerical reversals are $\\sim20$\\% faster than experimental reversals). \n\n\\clearpage\n\\section{Conclusion}\\label{sec:discussion}\nThe 4$^{\\circ}$C convection experiment, originally performed by Townsend \\cite{townsend_natural_1964}, has been re-investigated using long-term PIV measurements in a vertical cross-section, and in several horizontal cross-sections within the stratified layer. This last type of measurements has allowed to investigate for the first time the long-term horizontal mean flow in the stratified layer. Experiments have been complemented by direct numerical simulations. The first result of this paper is the confirmation, in 3D and with ideal boundary conditions, of the presence of a buffer layer, including an overshooting region as first observed by Perrard \\cite{perrard_experimental_2013}, and a shear region. We have argued that the buffer layer is driven by thermal coupling with the convection, due to the non-linear equation of state of water, and that this mechanism is a priori related to a Prandtl number larger than one.  The second result is that the buffer layer viscously drives slow reversals of the horizontal large-scale flow within the stratified layer. \n\n\nAdditionally, IGWs at different frequencies propagate in the stratified layer. They likely interact with the horizontal large-scale flow, and probably also produce a reversing flow, which superimposes to the viscously driven one. From Couston et al. \\cite{couston_order_2018}, we know that the Prandtl number has a strong influence on this QBO-like mechanism: the lower the Prandtl number, the stronger the amplitude of the QBO. In water, $Pr \\sim 7$, and the expected amplitude of the large-scale QBO flow is weak, hence dominated by the viscous driving. Further experimental studies at lower Prandtl number should allow deciphering the two contributions. One could for instance suggest using gas as a working fluid; however, the absence of density reversal around a given temperature will necessitate to consider either transient experiments like \\cite{deardorff_laboratory_1969, michaelian_coupling_2002}, or two-gas experiments which might then be prone to double diffusive instabilities. Experimentally, the question also remains to understand why the only successful QBO experiment has been performed in salty water, hence with a Schmidt number (equivalent to Pr) of 700.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=.6]{U_avgX.pdf}\n    \\caption{Horizontal average of the horizontal velocity $u$ over a vertical cross section in the middle of the numerical domain for $Pr = 0.1$.}\n    \\label{fig:num_pr01}\n\\end{figure}\n\nIn the meantime, it is straightforward to change the Prandtl number in the numerical simulation of our set-up. We have thus run a second simulation with the same Rayleigh number $Ra = 10^7$ and top temperature $\\theta_{top} = 11$ but with $Pr = 0.1$. In this simulation, as shown in figure \\ref{fig:num_pr01}, no buffer layer is observed, but strong signatures of a QBO like mechanism are visible, marked by downward phase propagation of the reversals of the large-scale flow. This configuration thus deserves a more systematic study in the future.\n\n\\section*{Acknowledgements} \nThe authors acknowledge funding by the European Research Council under the European Union's Horizon 2020 research and innovation program through Grant No. 681835-FLUDYCO-ERC-2015-CoG. This work was granted access to the HPC resources of Aix-Marseille Universit\\'e financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program \"Investissements d'Avenir\" supervised by the Agence Nationale de la Recherche. Computations were also conducted with the support of the HPC resources of GENCI-IDRIS (Grant No.A0060407543).\n\n\n\\section{\\label{sec:intro}Introduction}\nNumerous geophysical and astrophysical flows present a two-layer configuration, with a turbulent convective layer standing above or below a stably stratified one. Examples include planetary atmospheres, stars interiors, and possibly the outermost layer of the Earth liquid core \\cite{hirose_composition_2013}. The dynamics of coupled stratified and convective layers are quite complex. Due to the convective motions, internal gravity waves (IGWs) are generated at the interface between the two layers, and propagate in the stratified one. IGWs transport energy and momentum \\cite{rogers_internal_2012,bretherton_momentum_1969} from where they are generated to where they are damped. Thanks to their transport properties and non-linear interactions, IGWs are able to generate and sustain large-scale horizontal flows \\cite{plumb_interaction_1977,rogers_internal_2012}. Examples of such  large-scale flows driven by IGWs are the oscillations of equatorial zonal winds observed in some planets' atmosphere \\cite{fouchet_equatorial_2008,leovy_quasiquadrennial_1991}, including the Earth where it is called the Quasi-Biennial Oscillation (QBO) \\cite{baldwin_quasi-biennial_2001}.\\\\\n\nThe IGWs generation by turbulent dynamics has been studied in various experiments. The generation by a single buoyant plume was experimentally studied by Ansong \\& Sutherland \\cite{ansong_internal_2010}. The penetration of the plume within the stratified layer and the spectral characteristics of the generated IGWs were studied. They found that the peak frequency of the generated IGWs lies in a range close to $0.7N$, where $N$ is the Brunt-V\\\"ais\\\"al\\\"a (or buoyancy) frequency, and that the radial wavenumber is set by the plume cap and not by the width of the plume at the interface.\n\nDeardoff et al. \\cite{deardorff_laboratory_1969} and later Michaelian et al. \\cite{michaelian_coupling_2002} studied the effect of penetrative convection in a stratified layer in a transient, Rayleigh-B\\'enard type experiment. Stratification was initially set up thermally from the top to the bottom of the tank. Then, the fluid was suddenly warmed up at the bottom, triggering Rayleigh-B\\'enard convection. IGWs were measured transiently \\cite{michaelian_coupling_2002} while the stratified (resp. convective) layer was decreasing (resp. increasing) in size. Eventually, there was no more stratified layer to sustain the propagation of IGWs. \n\nTownsend \\cite{townsend_natural_1964} introduced an original set-up to study the quasi-steady generation of IGWs by Rayleigh-B\\'enard convection. Using the fact that water maximum density is around $4^{\\circ}$C, a two-layer system is spontaneously generated by cooling the bottom of a tank at $0^{\\circ}$C and heating its top above $4^{\\circ}$C. The density gradient is unstable at temperature below 4 $^{\\circ}$C and stable at temperature above. This creates a self-organising system, with a turbulent convective layer adjacent to a stratified layer. With dye visualisation and temperature measurements, he observed IGWs propagating close to the interface between the two layers. The $4^\\circ$C convection was also studied experimentally by Le Gal \\cite{legal_penetrative_1997} in a laminar flow, at low Rayleigh number. Convection displayed an hexagonal pattern and viscous entrainment of the fluid above the convective cells was observed. Perrard et al. \\cite{perrard_experimental_2013} and Le Bars et al. \\cite{bars_experimental_2015} re-investigated this setup in a quasi two-dimensional tank using Particle Image Velocimetry (PIV) and  temperature measurements to obtain detailed data of IGWs generated by the convection. They observed a wide spectrum of waves generated at different frequencies. Favoured frequencies were related to the differential attenuation length of the waves depending on frequency, in good agreement with linear theory. No large-scale flow in the stratified layer was observed in this 2D geometry. Numerical simulations of the same configuration were performed by Lecoanet et al. \\cite{lecoanet_numerical_2015}. They demonstrated that IGWs are mainly generated by the Reynolds stresses in the bulk of the convection below the interface, rather than by the displacement of the interface between the two layers. Numerical studies by Couston et al. \\cite{couston_dynamics_2017,couston_order_2018,couston_energy_2018} extended these results by considering a generic non-linear equation of state (piecewise linear around an inversion temperature, with adjustable slopes), both in 2D and 3D horizontally periodic geometries. Various flow regimes and the energy transport by IGWs were quantitatively studied. Interestingly, long simulations -- accessible in 2D studies only -- showed, for low Prandtl numbers ($Pr < 1$), a large-scale horizontal flow with reversals in the stratified layer, similar to the QBO phenomenon introduced above \\cite{couston_order_2018}. \n\\\\\n\nSeveral experiments took interest in the generation and reversal of a large-scale horizontal mean flow in a stratified domain, driven by IGWs. The well-known experiment designed by Plumb and McEwan \\cite{plumb_instability_1978}, later reproduced and extended by Semin et al. \\cite{semin_generation_2016,semin_nonlinear_2018}, is capable of driving a QBO from the mechanical forcing of a standing wave pattern (\\textit{i.e.} two waves with the same frequency and opposite phase speed) in a salty water stratification. With this system, Plumb and McEwan managed to observe few oscillations of the driven large-scale flow before the stratification was destroyed by mixing. The experiment gave results in good agreement with the theory \\cite{richard_s._lindzen_theory_1968,lindzen_updated_1972,plumb_interaction_1977}, notably with reversals starting away from the forcing and propagating towards it.\nSemin et al. \\cite{semin_generation_2016,semin_nonlinear_2018} improved the system by constantly injecting fluid to rebuild the stratification, while removing mixed fluid close to the forcing. This method allowed to run the experiment longer and to study the nature of the bifurcation in the Plumb model which can be either supercritical or subcritical, depending on the dominant dissipative process. In those experimental realisations of the QBO mechanism, the wave forcing remains monochromatic, as opposed to the natural mechanism where it is due to chaotic tropical storms \\cite{baldwin_quasi-biennial_2001}. The forcing is driven by interface displacements, as opposed to the observations of \\cite{lecoanet_numerical_2015}. Besides, only the stratified layer is modelled. It thus remains a challenge to observe experimentally a large-scale, reversing flow from a turbulent source and a wide range of naturally excited IGWs.\n\\\\\n\nIn the present study, we extend the work of Townsend \\cite{townsend_natural_1964,perrard_experimental_2013,bars_experimental_2015} in a cylindrical, 3D geometry reminiscent of Plumb and McEwan's set-up \\cite{plumb_instability_1978,semin_generation_2016,semin_nonlinear_2018}. Our purpose is threefold: to characterise the generation of IGWs in such a self-organising two-layer system, to quantify the coupling between the layers, and to investigate the possible generation of large-scale horizontal flows. Our experiments are complemented by direct numerical simulations of the same configuration. The experimental setup and numerical model are presented in section \\ref{sec:methods}, results are analysed in section \\ref{sec:results}, and conclusions and future works are discussed in section \\ref{sec:discussion}. \n\n\n\n\\section{\\label{sec:methods}Methods}\n\\subsection{Experimental set-up}\nThe set-up consists in a cubic tank whose lateral boundaries are made of 2 cm thick acrylic walls. The bottom boundary is a chrome plated copper plate in which refrigerated fluid is forced to circulate. The top boundary is a commercial, transparent electric heater. The tank inner dimensions are $32 \\times32$ cm for the horizontal section and $H=20$ cm in height. Preliminary experiments were conducted in this cubic geometry. Eventually, a cylinder of outer diameter $D = 29$ cm and thickness $e = 0.4 $ cm was added inside the cubic tank, to reproduce the axisymmetric geometry of \\cite{plumb_instability_1978,semin_generation_2016,semin_nonlinear_2018}, which seems prone to the development of large-scale flows. We are interested in the flow within the cylinder: the fluid in the gap between the cylinder and the cubic tank follows a similar dynamics and thus imposes to the working fluid a (close to) no-flux lateral boundary condition.\n\nThe temperature of the bottom boundary is controlled by water circulating in the copper plate. Water is cooled down by a thermal bath with a fixed temperature set at $-1.25^{\\circ}$C. Due to thermal losses in the pipes alimenting the copper plate, bottom tank temperature is $0.2 \\pm 0.05^\\circ$C. Plate thickness and circulation circuit were optimised so as to ensure a uniform temperature over the whole plate. At the top boundary, the heater is set at a temperature of $35^{\\circ}$C. Its temperature control is custom made: a PT100 probe measures the heater temperature in real time, driving through a feed-back loop the input power injected in the heater. This is a very simple and inexpensive system to impose a temperature while having a transparent top boundary, allowing visualisation and velocity measurements by PIV. Nonetheless, it is necessary to point out that the temperature over the heater area is not perfectly homogeneous. Temperature is maximal at the centre where $T \\sim 38^{\\circ}$C, while the edges are indeed at the requested $T = 35\\pm 0.1^\\circ$C. This inhomogeneity of the top temperature by $\\delta T = 3^{\\circ}$C induces slow convective motions below the heater, in a $\\sim 2$ cm high layer. By performing an experiment where the whole fluid is stably stratified with an overall temperature gradient similar to the one in the stratified layer studied here, but above the density reversal at $4^{\\circ}$C (i.e. bottom boundary set at 10$^{\\circ}$C and top boundary at 70$^{\\circ}$C), we have checked that those top convective motions have no significant impact on the dynamics of the two-layer system. It is also important to say that despite the thick acrylic wall and the intermediate fluid layer between the cylinder and the tank, the working region is not fully thermally insulated on the sides. Nevertheless, our fully stratified, test experiment has shown no motion within the fluid driven by these lateral losses. \n\\\\\n\nThe equation of state of water is non-linear with a  maximum density close to 4$^{\\circ}$C (International Equation of State of Seawater, 1980):\n\\begin{equation}\n    \\begin{split} \n    \\rho(T)= 999.842594+6.793952.10^{-2}T-9.095290.10^{-3}T^2+1.001685.10^{-4}T^3 \\\\\n    -1.120083.10^{-6}T^4+6.536332.10^{-9}T^5.\\\\\n    \\end{split}\n    \\label{eq:eos_eau}\n\\end{equation}\nThus, due to the imposed temperature at top and bottom boundaries, the bottom part of the tank, between 0$^{\\circ}$C and 4$^{\\circ}$C, is convectively unstable (see figure \\ref{fig:setup}). Cold plumes detach from the bottom plate and rise in the tank due to buoyancy. Reciprocally, ``hot'' fluid sinks from the 4$^{\\circ}$C  interface due to gravity. While convective motion takes place in the lower layer, the upper part of the tank, between 4$^{\\circ}$C and 35$^{\\circ}$C, is stably stratified, with an assumed linear temperature profile at equilibrium. The temperature is indeed linear for an ideal case without thermal losses, assuming that the stratified layer has a bottom boundary at fixed temperature $4^\\circ$C and top boundary at $35^\\circ$C (\\textit{i.e.} constant diffusive flux through the whole layer). However, the density profile is not linear, due to the non-linear equation of state of water. Stratification is characterised by the Brunt-V\\\"ais\\\"al\\\"a frequency $N^* = \\frac{1}{2 \\pi} \\sqrt{-\\frac{g}{\\rho_0} \\frac{\\partial \\rho}{\\partial z}}$. Because of the non-linear equation of state, $N^*$ is not constant with depth, as shown in figure \\ref{fig:setup}. For simplicity, we also use below the global buoyancy frequency defined as $N = \\frac{1}{2 \\pi} \\sqrt{-\\frac{g}{\\rho_0} \\frac{\\Delta \\rho}{\\Delta z}}$, where ${\\Delta \\rho}$ is the global density contrast within the stratified layer of depth ${\\Delta z}$.\n\\\\\n\n\\begin{figure}[h]\n\n  \\includegraphics[scale=.3]{schema_cuve_profilN_NB_v3.png}\n  \\caption{2D sketch of the tank. A cylinder (light grey shaded area) is placed in a larger cubic tank. The system is cooled down at the bottom at 0$^{\\circ}$C and heated up at the top at 35$^{\\circ}$C. The bottom half is convective with an almost constant density, apart from the bottom boundary layer. The upper half is stably stratified and waves are generated due to the fluid motions in the convective layer. The graph on the right shows the theoretical profile for the buoyancy frequency $N^*$. It is computed considering a linear temperature profile and the equation of state of water (\\ref{eq:eos_eau}). The dashed line is the global buoyancy frequency $N$ calculated on the stratified layer. The various length scales are the cylinder diameter $D$, the vertical extent of the tank $H$ and the vertical extent of the convective layer $h$. $\\delta$ is the minimal width between the outer square tank and the inner cylinder. The $x$, $y$ and $z$-velocity components are noted $u$, $v$ and $w$ respectively.}\n  \\label{fig:setup}\n\\end{figure}\n\nBefore starting the experiment, the bath and the heater are allowed to reach their assigned temperature. Then, the upper half of the tank is filled with water stratified in temperature from 4$^{\\circ}$C to 35$^{\\circ}$C, using the double bucket technique \\cite{oster_density_1965}. The bottom half is filled with water with a temperature close to 4$^{\\circ}$C. This filling process is used to avoid tremendously long transient before reaching steady state by thermal diffusion. Typically, we fill the tank at the end of a day and start the measurements the next day in order to let the system reach its equilibrium state over night. Each experiment then lasts four days, with no change in the location of the interface. Note that this steady interface position is the result of the heat budget equilibrium between the convective heat flux extracted from the bottom plate, the diffusive heat flux through the stratified layer, and the lateral heat losses.\n\nTo perform PIV measurements, particles are  chosen as small as possible and with a density as close as possible to water density in order to avoid their sedimentation in the stratified layer over the long duration of the experiment. We use fluorescent orange polyethylene micro-spheres whose size ranges from $10 \\, \\mathrm{\\mu m}$ to $20 \\, \\mathrm{\\mu m}$ and density is $1.00 \\pm 0.01$~g\/cc. The fluorescent property allows us, with a high pass filter on the camera, to remove any laser reflection, significantly enhancing the images quality. The tank is illuminated with a green laser $532$~nm. Power is set at $1$~W. \nWe perform side view PIV to measure convection and IGWs spectral characteristics, and top view PIV to observe the large scale flow and its fluctuations over time. The camera used for the side view PIV is a HiSense Zyla $2560 \\times 2160$ pixels recorded on 12 bits. Acquisition rate is $2$~Hz with $100$~ms exposure time. Typical acquisition time for spectral characteristics is $50$~min.\nFor the top view, we use a Point grey camera $1920\\times1080$ pixels on 8~bits. Exposure time is $1$~s, acquisition rate $0.1$~Hz and acquisition time $8$~hours. Captured movies are processed either by the DantecDynamics software DynamicStudio for the side view or by DPIVSoft \\cite{meunier2003analysis} for the top view. Both are resolved into $32\\times32$ pixels boxes with 50\\% overlapping. \n\nSide view PIVs are performed in the middle of the tank at $y=16$~cm in a laser sheet crossing the cylinder along its diameter. This is the case for all figures shown in the $(x,z)$ plane and thus not mentioned in the results section. The vertical fields (see an example in figure \\ref{fig:LSCexpe}) do not show the whole $(x,z)$ plane (where the origin $(0,0)$ is located in the bottom left corner of the cubic tank): it was indeed chosen to zoom in, in order to have the best resolution for the very weak motions in the stratified layer. The interface between the layers is localised between $11 \\mathrm{~cm} \\leqslant z \\leqslant 12$~cm. Typical Rayleigh number for the convection based on this depth is $Ra = 7 \\times 10^6$, and the global Brunt-V\\\"ais\\\"al\\\"a frequency is $N = 1.35 \\times 10^{-1}$~Hz.\n\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[scale = .45]{champ_int_730_colored.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = .45]{LSC_0510_1800_colored.pdf}\n    \\caption{(Left) Instantaneous velocity field. An ascending plume is visible at $x=150$ mm, and transported by the large-scale circulation. (Right) Large-scale circulation in the convective layer obtained by time-averaging velocities over a 50 minutes signal. The large-scale circulation is a counter clockwise cell. No inversion of the circulation has been seen in our experiment. The velocities under $z=10$ mm are noisy and thus not shown here. Maximum instantaneous velocities are $2.5$ times bigger than the maximum averaged velocities. The left edge of the cylinder is located at $x=19$~mm and the centre of the cylinder is at $x=160$~mm. Approximately $40$~mm of the right side of the cylinder is not captured with our PIV measurments.}\n    \\label{fig:LSCexpe}\n\\end{figure*}\n\n\nTo observe the large-scale flow, horizontal views at different heights are performed. A linear translation axis platform from Igus, driven by homemade software, is used to automate the process. With two mirrors, one fixed at the exit of the laser head and the other fixed on the translating platform, it is possible to sweep along one direction with the laser sheet. We typically make measurements during 15~s in each of 11 successive planes separated by 0.5~cm. The full scan duration is about 3~min, and is repeated during at least 8~hours.\n\\\\\n\nThe cylindrical geometry described here differs from the cylindrical shell geometry in \\cite{plumb_instability_1978,semin_generation_2016, semin_nonlinear_2018}. We first tried to work in that annular geometry by adding a second, smaller cylinder in our cubic tank. Three different gap sizes were tested but none showed any interesting dynamics. Indeed, the convection was too confined in the shell to provide an efficient chaotic excitation, and IGWs did not propagate well within the stratification, attenuated quite fast by wall friction. During these tests, we observed the most interesting dynamics within the innermost cylinder, so we decided to use that geometry. The point of the cylindrical shell geometry is that it is a close analogue to the equatorial band of the stratosphere where QBO takes place. By working in a cylinder, the geometry analogy is lost but the physics of the problem remains the same, still a priori allowing for large-scale, reversing, axisymmetric horizontal flows.\n\n\\clearpage\n\\subsection{Numerical model}\\label{sec:methods_num}\n\nTo complement the experiments, we also performed Direct Numerical Simulations (DNS) of the same configuration. We solve the Navier Stokes equations using a non-Oberbeck Boussinesq model. The density variations are consider small compared to the reference density $\\frac{\\Delta \\rho }{\\rho_o} \\ll 1$. Therefore, density fluctuations only appear in the buoyancy force. However, temperature variations affect the value of the thermal expansion coefficient to account for the non-linear equation of state of water. Variations of the thermal diffusivity $\\kappa$ and kinematic viscosity $\\nu$ are neglected. Governing equations are given in dimensionless form by:\n\\begin{align}\n\\label{eq:momentum}\n\\frac{\\partial\\bm{u}}{\\partial t}+\\bm{u}\\cdot\\nabla\\bm{u} = & -\\nabla P+Pr\\nabla^2\\bm{u}-Pr Ra \\ \\theta^2 \\bm{e}_z \\\\\n\\label{eq:heat}\n\\frac{\\partial \\theta}{\\partial t}+\\bm{u}\\cdot\\nabla\\theta = & \\ \\nabla^2 \\theta \\\\\n\\label{eq:mass}\n\\nabla\\cdot\\bm{u} = & \\ 0\n\\end{align}\nwhere we used the depth $H$ of the container as a unit of length, the thermal diffusive timescale $H^2\/\\kappa$ as a unit of time and the difference between the bottom and the inversion temperature $T_0 - T_i$ as a temperature scale. These equations are characterised by the Prandtl number $Pr=\\nu\/\\kappa$, the global Rayleigh number $Ra=\\alpha g (T_0-T_i)^2 H^3\/(\\nu\\kappa)$ and the imposed dimensionless top temperature $\\theta_{top}=(T_{top}-T_i)\/(T_0-T_i)$.\nNote that the quadratic temperature term in the momentum equation is a direct consequence of the nonlinear equation of state of water given by equation~\\eqref{eq:eos_eau}, which is approximed in our model by the quadratic equation $\\rho(T) \\approx \\rho_0 (1-\\alpha(T-T_i)^2)$.\nThe coefficient $\\alpha$ is not the usual thermal expansion coefficient but has a unit of $(\\degree \\mathrm{C})^{-2}$ and is given by $\\alpha\\approx8.1\\times10^{-6}(\\degree \\mathrm{C})^{-2}$ (see also \\cite{lecoanet_numerical_2015}).\n\nWe consider a cylindrical fluid cavity of diameter $D=3H\/2$ as in the experiments.\nBoth horizontal plates are assumed to be no-slip and with fixed temperature.\nThe side wall is assumed to be no-slip and perfectly insulating.\nThis is of course not the case in the experiment, for which lateral heat losses are inevitable and top temperature is not exactly constant, but the objective is to check whether the conclusions drawn from the experimental results are robust and do not depend on these effects.\nSince the experiment runs with water, we use $Pr=7$.\nThe Rayleigh number of the experiment is $Ra=7 \\times 10^7$ while its dimensionless top temperature is $\\theta_{top}=-7.75$. If we were to run the simulation with these parameters, the interface will be located very close to the top boundary. It is not the case in the experiment because of the lateral heat losses, which tend to reduce the effective Rayleigh number. For that reason, and instead of taking into account these losses as in \\cite{lecoanet_numerical_2015}, we kept the insulating lateral boundaries and use the slightly adjusted parameters $Ra=10^7$ and $\\theta_{top} = -11$ instead, which leads to an interface located approximately at $z\\approx120$~mm, as in the experiment. The Rayleigh number could not be lowered under $10^7$ in order to keep the convective flow turbulent enough, thus we had to increase the top temperature to have the interface located at $z\\approx120$~mm.\n\nWe perform DNS of equations~\\eqref{eq:momentum}-\\eqref{eq:mass} using the spectral element solver Nek5000 \\citep{Nek5000}.\nThe global geometry is decomposed into hexahedral elements, with vertical refinement close to the horizontal boundaries and around the mid-plane where the inversion isotherm is expected to be located.\nVelocity, buoyancy and pressure variables are represented as tensor product Lagrange polynomials of order $N$ and $N-2$ based on Gauss or Gauss-Lobatto quadrature points.\nThe total number of grid points is given by $\\mathcal{E}N^3$ where $\\mathcal{E}$ is the number\nof elements.\nFor all the results discussed in this paper, the number of elements is $\\mathcal{E}=8960$ and we use a polynomial order of $N=11$. Numerical convergence was checked by increasing the polynomial order $N$.\nTime integration is performed with a third-order mixed implicit-explicit scheme.\nThe simulations are initialised with a small amplitude buoyancy perturbation and a temperature profile varying linearly between the top and bottom boundaries.\nSimulations are run until a quasi-steady state is reached, after which data is accumulated to compute statistics.\n\n\\section{\\label{sec:results}Results}\n\\subsection{\\label{sec:results_exp}Experiments}\n\\subsubsection{\\label{sec:results_conv}Convection}\nPIV side view is used to quantify horizontal and vertical velocities in the convection zone. Examples of vertical velocities measured at one point in a given location are shown in figure \\ref{fig:panache}, for both ascending cold and descending hot structures. Measurements are consistent with the numerical simulations \\cite{lecoanet_numerical_2015,couston_dynamics_2017} showing intense, localised, cold rising plumes and more diffusive descending plumes. Moreover, these structures are advected by a large-scale circulation encompassing the whole convective layer, as shown in figure \\ref{fig:LSCexpe}.\n\n\\begin{figure}[h]\n    \\centering\n    {\\label{fig:panacheup}\\includegraphics[scale=0.43]{sonde_panache_up.pdf}}\n    \\hspace{.2pt}\n    {\\label{fig:panachedown}\\includegraphics[scale=0.43]{sonde_panache_down.pdf}}\n    \\caption{Time evolution of the vertical velocity $w$ within: (Left) upward plumes at $x = 200$ mm, $z = 45$ mm, (Right) downward structures at $x= 100$ mm, $z = 95$ mm. }\n    \\label{fig:panache}\n\\end{figure}\n\nSpectral analysis is performed to extract power spectral density (PSD) from the velocity signals. Figure \\ref{fig:spectreconv} shows the PSD of the convection vertical velocity $w$. For the two panels, the spectrum is flat with a lot of energy for low frequencies, then the energy drops above some cut-off frequency. Left panel of figure \\ref{fig:spectreconv} shows the vertical velocity PSD at a single point close to the top of the convective layer. A small peak can be seen close to $f = 10^{-2}$~Hz. This is the quasi-periodic signal of the plumes dropping from the top thermal boundary layer. The theoretical characteristic time of convection can be computed from \\cite{gortler_convection_1966}:\n\\begin{equation}\n    \\tau = \\frac{h^2}{\\pi \\kappa}\\left(\\frac{\\Delta T}{\\Delta T_{local}} \\times \\frac{Ra_c}{Ra}\\right)^{2\/3}\n\\end{equation}\n\n\\begin{figure*}[b]\n    \\centering\n    \\includegraphics[scale = .43]{spectre_1_point_cd67_58_vf.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = .43]{spectre_W_50toend_10to57_vf.pdf}\n    \\caption{PSD for the vertical velocity fluctuations. (Left): PSD computed at a single point $x=100$~mm, $z=95$~mm (signal shown in figure \\ref{fig:panache} right). The plume forcing frequency can be seen around $f = 10^{-2}$~Hz (red dashed line). (Right): PSD spatially averaged over the whole convective cell in the measured $(x,z)$ plane (all points above $z=10$~mm and below $z=110$~mm). }\n    \\label{fig:spectreconv}\n\\end{figure*}\n\nwith $h$ the height of the convective layer, $\\kappa$ the thermal diffusivity, $\\Delta T$ the temperature difference between the top and bottom of the convective domain, $\\Delta T_{local}$ the temperature difference between the top and bottom of the thermal boundary layer, and $Ra_c$ the critical Rayleigh number. The critical Rayleigh number in the presence of free and solid interfaces and for fixed temperature is $Ra_c = 1100.65$. For our experiment, the characteristic time is $\\tau = 96$~s, thus characteristic frequency is $1\/ \\tau \\sim 10^{-2}$~Hz, which is close to the observed peak in the left panel of figure \\ref{fig:spectreconv}. At frequencies lower than this characteristic frequency, the spectrum is flat. This is explained by the combined effect of the randomness of the plumes (see figure \\ref{fig:panache}) and of the slow fluctuations of the large-scale circulation. Right panel of figure \\ref{fig:spectreconv} shows the PSD of vertical velocities, averaged over the whole convective cell in the $(x,z)$ plane. It shows a similar trend, with a lower cut-off frequency compared to right panel spectrum. Actually, the plumes signal is more localised and less intense on average than the large-scale circulation signal, which hence dominates the space-averaged PSD.\n\nThe probability density function (PDF) of the vertical velocities in the whole convective layer $\\mathrm{P}(w)$ is computed and shown in figure \\ref{fig:pdf}. It is normalised such that $\\int \\mathrm{P}(w)\\mathrm{d}w = 1$. The PDF describes important features of the convection: it is skewed towards positive values, with positive velocities reaching higher magnitude than negative velocities, \\textit{i.e.} the ascending plumes are stronger than the descending structures. However, the central part of the PDF is close to gaussian profile. The distribution obtained here is in good agreement with the probability density function computed in an idealised 2D numerical model by Couston et al. \\cite{couston_dynamics_2017}. Note that this asymmetry is specific to our model, for which the usual upside-down symmetry in Boussinesq Rayleigh-B\\'enard convection is broken.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale = .6]{pdf_conv.pdf}\n    \\caption{Probability density function of the vertical velocities in the convective layer. All PIV points under $z=110$~mm have been used to compute the PDF.}\n    \\label{fig:pdf}\n\\end{figure}\n\n\\subsubsection{\\label{sec:results:buffer}Buffer layer}\nAn intermediate layer (we name it the buffer layer in the following) is present between the convective layer and the stratified layer. It was first reported in the quasi-2D 4$^{\\circ}$C convection experiment by Perrard et al.  \\cite{perrard_experimental_2013}. Their temperature measurements showed that the buffer layer corresponds to the area where the temperature goes from 4$^{\\circ}$C to 8$^{\\circ}$C. This actually corresponds to the overshooting region for rising cold plumes (note that this type of convection is called \"penetrative convection\" because of this effect). Indeed, since the density of water is close to quadratic around $4^{\\circ}$C, densities at e.g. $0^{\\circ}$C and $8^{\\circ}$C are the same, and the 8$^{\\circ}$C isotherm is the theoretical maximum height reachable by an ascending cold plume at $0^{\\circ}$C in the absence of thermal diffusion. Simultaneously, the overall density profile between $4^{\\circ}$C and $8^{\\circ}$C is stable, as in the stratified layer above $8^{\\circ}$C. The buffer layer is thus a very specific location supporting simultaneously convective overshooting motions and IGWs, as observed with PIV measurements \\cite{perrard_experimental_2013}. \n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale = .6]{buffer_U-xavg.pdf}\n    \\caption{Time evolution of the horizontal average of the horizontal velocity, noted $u_X$. Red (resp. blue) regions correspond to mean flow going towards the right (resp. left).}\n    \\label{fig:bufferlayer}\n\\end{figure}\n\nTo complete the description of this buffer layer now using velocity measurements, we plot in figure \\ref{fig:bufferlayer} the spatio-temporal graph of the horizontal average of the horizontal velocity $u$. The graph exhibits a strong shear around $z=120$ mm. Since the fluid is going in opposite directions above and below $z=120$ mm with a sharp interface, viscous entrainment by the convective layer is excluded. A special kind of thermal coupling might explain the observed dynamics, as sketched in figure \\ref{fig:schema_couplage}. Indeed, when a cold ascending plume from the convection zone reaches the interface and overshoots in the buffer region, its associated velocity perturbation dissipates more rapidly than its temperature perturbation. Due to gravity, the distorted part of the buffer region wants to sink back to its initial state (pictured by the green arrows), while the fluid above the buffer layer moves towards the impact point of the plume to take the place of the falling water (pictured by the red arrows). The buffer layer then needs some compensating fluid from the convective layer. This mechanism works when the velocity perturbation of the plume at the interface dissipates more rapidly than the thermal perturbation, hence for a Prandtl number $Pr \\geq 1$. One might expect the shearing zone to decrease in size and amplitude when thermal diffusion increases (i.e. when the Prandtl number decreases), since the overshooting rising cold plume will then equilibrate thermally during its ascent more rapidly. This may explain why no interfacial shear was reported in the systematic numerical study of \\cite{couston_dynamics_2017,couston_order_2018} where $Pr \\leq 1$. Global temperature field measurements (using e.g. Temperature Dependent, Laser Induced Fluorescence) are now required to confirm or infirm the proposed model, but those are beyond the scope of the present paper. Note that by extension, we call \"buffer layer\" in the following the region including the $T=4^\\circ$C to $T=8^\\circ$C overshooting region and the shear region. In the experiment, the shear region extends from $z=120$~mm to $z \\approx135$~mm.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[scale = .75]{schema_couplage_final2.pdf}\n    \\caption{Sketch of the thermal coupling between the convective and buffer layers. On the left, a cold plume moves upwards towards the interface between the two layers. On the right, isotherms are deflected due to the impact.}\n    \\label{fig:schema_couplage}\n\\end{figure}\n\n\\clearpage\n\\subsubsection{\\label{sec:results_waves}Internal gravity waves}\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale = .83]{visu_onde_colormap_col_02to03_axisequal_schema.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = .53]{panache_v1.pdf\n    \\caption{Velocity fields showing IGWs propagating. (Top) Velocities in $(x,z)$ plane. The signal is frequency-filtered to enhance the visualisation of oscillatory motions: only frequencies between $0.02$ and $0.03$ Hz are shown, propagating at an angle of roughly $75^\\circ$ with the vertical. The angle of propagation is the angle between the constant phase line and the vertical. (Bottom) Velocities in $(x,y)$ plane located at $z \\approx 125$~mm. In the $(x,y)$ plane, IGWs take the form of oscillating rings. Note that this figure is from a previous experiment without any internal cylinder and is therefore only displayed here as an illustration of the IGWs seen from above.}\n    \\label{fig:ondeN}\n\\end{figure}\n\n The convective motions induce a Reynolds stress at and below the interface which generates IGWs propagating in the stratified area \\cite{lecoanet_numerical_2015}. An example is shown in figure \\ref{fig:ondeN}. The vector field has been frequency filtered in the band $[0.02-0.03]$ Hz to isolate a single propagating wave train. We can measure an angle close to $\\theta \\simeq 75^{\\circ}$ between contant phase lines and the vertical. This observation is in good agreement with the inviscid dispersion relation $\\omega = \\pm N \\mathrm{cos}(\\theta)$, which relates the frequency and the propagation angle of IGWs. Indeed, at $z=120$~mm, $N \\sim 0.1$~Hz, thus $\\theta = \\mathrm{cos^{-1}}\\left(\\frac{\\omega}{N}\\right) = 78.5^\\circ$. The motion within the stratified area is a superimposition of many such IGWs oscillating at different frequencies.\n\nTo further investigate the waves signal, waves spectra are plotted in figure \\ref{fig:spectro}, showing the power spectral density of oscillatory motions within the stratified layer at every height, averaged horizontally for each height. The grey line is the theoretically computed buoyancy frequency profile. Figure \\ref{fig:spectro} shows that energy is present in a wide frequency band, from the lowest measured frequencies to the buoyancy frequency $N$. Low frequency motions $f < 4\\times10^{-3}$~Hz are very intense and propagate high in the stratified layer. Motions with frequency ranging from $4 \\times 10^{-3}$~Hz to $N$ are less intense, but still propagate into the stratified layer. Motions propagating at frequencies higher than the buoyancy frequency $N$ are greatly attenuated after a centimetre as IGWs of frequency larger than $N$ are evanescent. The weak signal at low frequencies above $z=180$ mm comes from the convective motions due to the non-homogeneous heating at the top. These motions are confined at the very top of the experimental container. \n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[scale=0.55]{propagation_nonorm_120_vff.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale =.55]{plot_attenuation_norm_v2.pdf}\n    \\caption{(Left) Power spectral density of the absolute velocity $\\sqrt{u^2+w^2}$ above the convective layer. The grey curve shows the theoretical buoyancy frequency profile, assuming a linear profile for the temperature, from $4^\\circ$C to $35^\\circ$C. (Right) Two selected profiles (taken at frequencies shown by dashed lines on the left graph) of the re-scaled PSDs by the PSD at the top of the convective layer, \\textit{i.e.} $z=120$~mm (noted $\\mathrm{PSD_o}$). The PIV measurements are performed for 50 minutes and results (using the pwelch Matlab function) are horizontally averaged at each height to obtain the averaged power spectral density.}\n    \\label{fig:spectro}\n\\end{figure*}\n\nRight panel of figure \\ref{fig:spectro} shows two vertical profiles of the PSD re-scaled by the PSD at the top of the convective layer ($z=120$~mm), taken at two different frequencies. The energy decrease is quite similar between $z=120$~mm and $z=140$~mm for both frequencies. However, for $z>140$~mm, the energy for the higher frequency decreases slower than the energy for the lower frequency. This dependence of the attenuation length regarding the frequency of the signal is a characteristic of IGWs. Indeed, the dispersion relation of IGWs relates the frequency and the wave vector direction. Moreover, energy propagates perpendicularly to the wave vector for IGWs (i.e. group and phase velocities are perpendicular). The closer to $N$ the wave frequency $\\omega$ is, the more horizontal the phase propagates, hence the more vertical energy propagates. High frequency waves are thus capable of transporting energy to high altitudes before being damped. On the opposite, waves with low frequency compared to $N$ propagate energy almost horizontally, and are thus attenuated before reaching high altitudes. At frequencies $f<4\\times 10^{-3}$~Hz, a lot of energy is seen and the attenuation length does not depend on the frequency. There is no reason why IGWs should disappear below a certain frequency, but we would expect to see the attenuation length to keep decreasing with decreasing frequency. We thus deduce that IGWs at frequencies $f \\leqslant 4\\times 10^{-3}$~Hz are hidden in the energy spectrum by some very energetic large-scale slowly-varying flow, which we will describe below.\n\n\nMore than one order of magnitude separates the buoyancy frequency and the fastest large-scale flow fluctuations. The large-scale flow penetrates deep into the stratified layer. It globally decreases in amplitude with height, but with some local increases at $z\\sim 125$ mm (i.e. close to the interface between convective and buffer layers) and $z \\sim 145$ mm.\nThe IGWs signal can be seen between $f = 4 \\times 10^{-3}$ Hz and the buoyancy frequency. A peak that reaches the top of the stratified layer is seen around $f= 1.2 \\times 10^{-2}$ Hz, \\textit{i.e.} the same frequency as the convective forcing discussed in section \\ref{sec:results_conv}. It corresponds to the strong excitation provided by the cold rising and hot sinking turbulent plumes.\nHowever, top panel of figure \\ref{fig:spectro} also shows a sudden drop of the energy at frequencies $f>1.2 \\times 10^{-2}$~Hz. Indeed, wave attenuation is strong at these frequencies, even if they are close to (but below) the buoyancy frequency $N$. Actually, energy dissipation also depends on the norm of the wave vector squared. There is no reason that all excited waves have the same wave vector norm; one could even expect that fastest waves are excited by fastest, hence smallest convective patterns, and are thus also at smallest scale: they then dissipate more rapidly.\n\n\n\\subsubsection{\\label{sec:results_lsf}Large-scale flow in the stratified layer}\nFigure \\ref{fig:spectro} shows an important amount of energy at low frequencies which has been interpreted as the signature of a large-scale slowly-varying flow in the stratified layer. We will now investigate the nature of these fluctuations to see if they relate to reversals similar to the QBO.\n\nFigure \\ref{fig:meanflow} shows horizontal vector fields at the same depth at different times. In figure \\ref{fig:meanflow}(a), the flow goes counter-clockwise inside the cylinder. Figure \\ref{fig:meanflow}(b) shows that two contra-rotating vortices with a smaller amplitude typical velocities have appeared. Figure \\ref{fig:meanflow}(c) shows a mostly clockwise rotating flow, where one of the preceding eddy pairs has nearly disappeared. The large-scale flow thus evolves drastically over time. A criterion is computed to extract a typical mean velocity from those fields that accounts for the ``direction'' of the large-scale flow: as illustrated in figure \\ref{fig:critere}, we compute a mean azimuthal velocity, taken along a ring centred in the cylinder. Other criteria to extract a representative value for the large scale flow direction have also been tested, including: the mean vorticity over the cylinder area, the average of the azimuthal velocity over several rings with different radii, and the azimuthal velocity averaged over thick rings. They all give similar results for the large-scale flow measurement. \n\\\\\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[scale = .8]{evolution_mf_vff.pdf}\n    \\caption{Horizontal velocity vector fields in the stratified layer at different times. The laser sheet is located at $z=150$~mm. The large-scale flow reverses from (a) to (c). Time between (a) and (c) is approximately half an hour. Maximum velocities are $0.1$ mm\/s.}\n    \\label{fig:meanflow}\n\\end{figure*}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale = .45]{critere_mf.pdf}\n    \\caption{Criterion used to extract a significant value for the large scale flow and its direction: the azimuthal velocity is averaged over the ring shown in red.}\n    \\label{fig:critere}\n\\end{figure}\n\nIn order to investigate the vertical phase propagation of the reversals, and thus, to compare the reversal dynamics observed to a QBO-like phenomenon, the setup has been equipped with a linear translating platform that allows us to perform horizontal laser sheet sweeping along the vertical. Horizontal velocities are measured in an horizontal plane, every $5$~mm from the top of the convective layer $z=110$~mm to the middle of the stratified layer $z=160$~mm. Any trace of downward phase propagation of the reversals, as observed on the QBO on Earth \\cite{baldwin_quasi-biennial_2001} and on the historical Plumb experiment \\cite{plumb_interaction_1977, semin_nonlinear_2018}, would be a significant evidence for QBO-like phenomenon in the experiment. \nIndeed, the phase propagation of the reversals due to IGWs non-linear interactions is theorised as follows: an IGW propagating in a stratified layer with an horizontal phase velocity in the same direction as the existing base flow propagates upward until reaching a critical height $z_c$, where it deposits all its energy locally. At $z=z_c$, the flow accelerates. Thus the critical height where the flow is intense enough to damp the wave is lowered. As time goes on, this critical height moves towards the location where the waves are emitted. Here, the waves are emitted at the bottom of the stratified layer. We would expect a downward phase propagation if the reversals are driven by IGWs non-linear interactions. \n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.7]{QBOfinal_avecpentediffusionv2.pdf}\n    \\caption{Reversals of the large-scale flow. $z=115 - 120$ mm is the convective \/ buffer layers interface. Ascending plumes often perturb the buffer layer flow. The velocity was measured at 11 heights, marked by each tick in the vertical axis of the figure, and interpolated in between. The slope of the black dot lines represent the viscous coupling phase velocity.}  \\label{fig:QBO}\n\\end{figure*}\n\n\nWe performed long time experiments (around 8 hours). Typical results extracted from the criterion described above are shown in figure \\ref{fig:QBO}. Blue patches (resp. red patches) represent large scale flow going counter clockwise (resp. clockwise). The present measurements mainly confirm the interpretation of figure \\ref{fig:spectro} for the lowest frequencies: the large-scale flow is horizontal and extends over the whole depth of the stratified layer with an amplitude attenuated with height, and exhibits slow reversals. Additionally, some intense events at $z = 110$ mm are directly related to penetrative plumes from the convection. Reversals times range from $400$s to $1800$s. However, no downward phase propagation of the reversals is observed. On the contrary, the reversals seem to occur along the whole stratification height at the same time, or even with a rapid upward phase propagation. Since the phase propagation is not towards the location where the waves are emitted, the reversals are unlikely driven by the non-linear interactions of IGWs. However, as seen in section \\ref{sec:results_waves}, IGWs propagate in the stratified layer and carry energy. Therefore, they give energy to the large-scale flow through non-linear interactions. Yet, the process is not dominant in the reversals dynamics.\n\\\\\nSince, the reversals observed in figure \\ref{fig:QBO} do not have a downward phase propagation, we look for other mechanisms than the QBO mechanism to explain the reversals. Two other mechanisms can be investigated. The first one relies on a specific convective dynamics within the overall stratified layer driven by horizontal gradients related to imperfect top and side boundary conditions. The second mechanism relies on viscous coupling with the underlying convective and buffer layers. \n\nOur fully stratified reference experiment described in section \\ref{sec:methods} precludes the first scenario. Indeed, setting the bottom boundary at 10$^{\\circ}$C and the top boundary at 70$^{\\circ}$C, no motion is observed for the bottom $3\/4$ of the tank. In this test-experiment, the top $1\/4$ of the tank is animated by convective motions due to the non-homogeneous top heat source (in the standard $4^\\circ$C experiment, where $T_{top} = 35\\degree$C, only $\\sim 2$ cm are affected by the convection at the top of the tank, because the non-homogeneity of the heat source is less important for lower temperature, thus the horizontal convection is weaker). However, these are inefficient to generate waves below and to drive any large-scale flow observable away from the top region.\n\n\n\\begin{figure}[h]\n\\includegraphics[scale = 0.55]{balayage_corr_vf.pdf}\n  \\caption{Velocity vector fields in the horizontal plane. Different columns represent different sweeping cycles $t^*$ (one sweeping cycle corresponds to the 11 steps needed to go from the lowest position $z=110$~mm to the highest position $z=160$~mm). Different rows represent different heights within the same sweeping cycle: first row is the top of the convection $z=110$~mm, second row is in the buffer layer $z=120$~mm and third row is in the stratified layer $z=145$~mm. Convective plumes are easily noticeable on the first row fields.}\n  \\label{fig:champ_correle}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\includegraphics[scale = 0.55]{balayage_NOcorr_vf.pdf}\n  \\caption{Same as figure \\ref{fig:champ_correle} but for different sweeping cycles. Note that in this set of velocity fields, the buffer and stratified layers are less correlated than they are in figure \\ref{fig:champ_correle}.}\n  \\label{fig:champ_pascorrele}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[scale=0.6]{correProdscalaire_BW_vf.pdf}\n  \n  \\caption{Velocity correlation between the three layers. Dashed black lines are the $-0.5$ and $0.5$ values. Correlation coefficient are  window-averaged over $10$~min to smooth the curve.}\n  \\label{fig:correlation}\n\\end{figure}\n\nThis leaves viscous entrainment as a possible driving mechanism. The dotted lines on figure \\ref{fig:QBO} show a theoretical viscous time, computed from the time for viscous entrainment to drive 20\\% of the horizontal velocity at $z = 115$ mm to $z=160$ mm, starting from a base state flow at rest. The 20\\% value corresponds to the measured value of the large scale flow at $z=160$ mm compared to the value at $z = 115$ mm (noted $u_b$). The theoretical viscous time entrainment is given by $t = \\frac{z^2}{4\\,\\nu\\, \\mathrm{erf}^{-2}\\left(\\left(\\frac{u}{u_b}-1\\right)\\right)}$. The reversals occur in a time scale comparable with this theoretical viscous time. The similarity between the slope of the dashed lines and the slope of the upward phases suggests that reversals are driven viscously.\n\nHowever, the existence of the buffer layer and its associated intense shear, with opposite horizontal velocities with the convective layer below (see figure \\ref{fig:bufferlayer}) precludes direct viscous coupling between the convective and stratified layers. Besides, no reversal has been observed in the convective region. We thus propose a thermal coupling between the convective and buffer layers as seen in section \\ref{sec:results:buffer}, associated to a viscous coupling between the buffer and stratified ones. To further quantify this possibility, figures \\ref{fig:champ_correle} and \\ref{fig:champ_pascorrele} show horizontal velocity fields at different heights and at different times. For each of the columns shown, first row is the mean flow in the convective layer at depth $z=110$~mm, second row is the mean flow in the buffer layer at depth $z=120$~mm and third row is the mean flow in the stratified layer at depth $z=145$~mm. The correlation coefficients through time between (i) the convective and buffer layers, (ii) the buffer and stratified layers, and (iii) the convective and stratified layers have been computed. It consists in a scalar product of the velocity vector for each position at two different heights rescaled by the product of the norm of the velocity vector at the two heights, \\textit{i.e.}: \n\n\\begin{equation}\nR_{ij} = R(x_i,y_j) = \\frac{u(x_i,y_j,z_1) \\times u(x_i,y_j,z_2) + v(x_i,y_j,z_1) \\times v(x_i,y_j,z_2)}{\\left(u(x_i,y_j,z_1)^2+v(x_i,y_j,z_1)^2\\right)^{1\/2} \\times \\left(u(x_i,y_j,z_2)^2 + v(x_i,y_j,z_2)^2 \\right)^{1\/2}}\n\\end{equation}\nThis gives a correlation coefficient $R_{ij}$ for each PIV position in the horizontal plane. The global correlation coefficient $R$ is computed by spatially averaging the local correlation coefficients. \n\nResults are shown in figure \\ref{fig:correlation}. The convective and buffer layers are negatively correlated: the correlation coefficient is most of the time close to $R=-0.5$. This can also be seen at all times in figures \\ref{fig:champ_correle} and \\ref{fig:champ_pascorrele}, where horizontal velocities in the convective and buffer layers have opposite direction. A diverging flow coming from an impinging plume in the convective zone corresponds to a converging flow in the buffer layer towards the impact zone, hence confirming the thermal coupling mechanism described in section \\ref{sec:results:buffer}. This converging flow may lead either to a clockwise or anticlockwise azimuthal mean flow, depending on the details of the chaotic excitation from the convective plumes. The correlation coefficient between the convective and stratified layers can be positive or negative, and is anyway most of the time less than 0.2, in absolute value. The correlation coefficient between the buffer and stratified layers shows a lot of temporal variations. However, it remains always positive. At a given time, the large-scale flow in the stratified layer may switch between a regime strongly dominated by the buffer layer (see also figure \\ref{fig:champ_correle}), and a second regime where the flow in the stratified layer is quite different from the flow in the buffer layer (see also figure \\ref{fig:champ_pascorrele}). \n\nWe thus conclude that the stratified layer is globally viscously driven by the buffer layer. However, the stratified layer exhibits additional complexities. These might be due to IGWs interacting with the large-scale flow. The results from Couston et al. \\cite{couston_order_2018} show that the lower the Prandtl number, the more regular the QBO. In the experiment, the Prandtl number is close to $Pr = 7$: the typical associated QBO-type flow is irregular, with low amplitude. We thus propose that large-scale flow driven by IGWs non-linear interaction superimposes on the viscously driven flow, but remains secondary. We do not know at this point how to disentangle those two potential contributions from the available data. \n\n\n\\clearpage\n\\subsection{\\label{sec:results_num} Numerical simulations}\nThe experimental results are not fully sufficient to explain, with complete certainty, the origin of the buffer layer and of the large-scale flow observed in the stratified layer. In addition, the effects of the lateral heat losses and top temperature heterogeneity are difficult to distinguish. To answer these questions, 3D DNS of a configuration similar to our experiments are performed, reproducing the 4$^{\\circ}$C convection but with idealised boundary conditions (i.e. no flux on the sides, and fixed temperature at the top and bottom). As mentioned in section \\ref{sec:methods_num}, the Rayleigh number $Ra$ and $T_{top}$ are tuned so that the interface depth in the experiment and the numerical simulation are similar. We have $Ra=10^7$ and $T_{top}=48^\\circ$C. All the numerical simulations are run dimensionless, but results are shown in dimensional values. The length scale is $H = 200$~mm, the vertical extent of the whole domain (hence diameter is $D=300$~mm), the timescale is the thermal diffusive time $\\tau = \\frac{H^2}{\\kappa} = \\frac{0.2^2}{1.5 \\, 10^{-7}} = 2.67 \\times 10^{5}$~s, and the temperature is given by the dimensionless temperature $\\theta = \\frac{T - T_i}{T_0 - T_i}$, where $T, T_i, T_0$ are respectively the dimensional temperature, the inversion temperature of the equation of state (i.e. $4^\\circ$C), and the bottom temperature (i.e. $0^\\circ$C). Results for sections \\ref{sec:results_num_conv} - \\ref{sec:results_num_igw} are computed from a $(x,z)$ vertical plane located along a cylinder diameter.\n\n\\subsubsection{\\label{sec:results_num_conv}Large-scale circulation in the convection zone and buffer layer}\nFigure \\ref{fig:LSC_num} shows that a large-scale circulation takes place in the convective layer. It consists of a cell filling the whole convective layer, and exhibits no reversal over the whole course of the simulation. The fluid rotates counter clockwise in the vertical plane. This is qualitatively consistent with the mean flow observed in the experiment and shown in the right panel of figure \\ref{fig:LSCexpe}. As in the experiments, a counter current exists on the top of the convective layer at $z= 120$~mm, creating a strong shear and demonstrating the existence of a buffer layer in the numerical simulation as well. \n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale = .47]{quiver_inst_color.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = .47]{quiver_moyen_color.pdf}\n    \\caption{(Left) Instantaneous velocity field. An ascending plume is visible at $x=230$ mm. (Right) Large-scale circulation in the convective layer obtained by time-averaging velocities over a 50 minutes recording. The large-scale circulation is a counter clockwise cell. Maximum instantaneous velocities are $3$ times bigger than the maximum averaged velocities.}\n    \\label{fig:LSC_num}\n\\end{figure}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[scale = 0.5]{Spatio_temp_Ux_v3_dim.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = 0.5]{profil_T_all.pdf}\n    \\caption{(Left) Horizontal average of the horizontal velocity $u$ over a vertical cross-section in the middle of the tank. The buffer layer can be seen above $z=120$~mm. A stationary large-scale circulation is present in the convective layer, even if it appears quite perturbed at the end of the signal. (Right) Temperature profiles along the $z$-axis.}\n    \\label{fig:num_buffer}\n\\end{figure*}\n\nThe space-time diagram of the mean horizontal flow shown in figure \\ref{fig:num_buffer} confirms it. Observing the buffer layer in the absence of side thermal losses and top temperature heterogeneity is an additional argument accounting for the fact that it is not an artefact driven by imperfect experimental conditions.\nWe also observe that the flow within the convection stays positive through time at the bottom and negative at the top. This is evidence of the steady large scale circulation taking place in the convective layer. Some events appear at $t > 1.42 \\times 10^{5}$~s and are interpreted as quasi-reversal of the large-scale circulation. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=.5]{quiverIsotherme.pdf}\n    \\caption{Velocity field and temperature isotherms at the end of an upward plume impact on the interface.}\n    \\label{fig:num_quiverisoT}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=.6]{corre_WetU.pdf}\n    \\caption{(Left) Spatio-temporal diagram of the horizontal velocity $u$ at $z=128$~mm. (Right) Spatio-temporal diagram of the vertical velocity $w$ at $z=108$~mm. The event at $t \\approx 1.052 \\times 10^5$~s is shown in figure \\ref{fig:num_quiverisoT}.}\n    \\label{fig:num_correWU}\n\\end{figure}\n\nThe temperature profile along the $z$ axis is also plotted on the right panel of figure \\ref{fig:num_buffer}. The figure shows a temporal and horizontal average of the temperature field (black thick curve), two temporal averages at two different positions $x=14$~mm (left side of the tank - dashed grey) and $x=277$~mm (right side - dotted grey) and an instantaneous profile at $x=145$~mm (middle of the tank - thick grey with crosses). The thermal boundary layer can be seen, between $z=0$~mm and $z=10$~mm. Then, between $z=10$~mm and $z=100$~mm lies a layer of constant temperature $T \\sim 2.8^\\circ$C. Between $100 \\mathrm{~mm} \\leqslant z \\leqslant 115 \\mathrm{~mm}$, the temperature profile evolves from constant to linear for $z > 115$ ~mm. The $T=4^\\circ$C (respectively $T=8^\\circ$C) isotherm is located at $z = 110$~mm (resp. $z = 120$~mm). Note that the temporal average of the temperature profiles are different on the left and right sides of the tank. Indeed, the constant temperature height goes to $z=90$~mm for the left side whereas it goes to $z=115$~mm for the right side. This suggests that the convective \/ buffer layer interface does not lies at one height over the whole tank but is a function of time and space. This is very likely due to the large-scale circulation. Thus, the thermal coupling described in \\ref{sec:results:buffer} will likely occur at different heights, depending on time and horizontal position.\n\n\nThe thermal coupling as schematised in figure \\ref{fig:schema_couplage} can be found in the numerical simulation. This is represented in figure \\ref{fig:num_quiverisoT}. An upward plume impacting the convective \/ buffer layer interface is seen. The isotherms ranging from $T=4^\\circ$C to $T=11^\\circ$C are deflected upward, due to the plume bringing cold fluid upward. On the contrary, the isotherms $T = 12 -14^\\circ$C are deflected downward by the converging flow. Isotherms at $T \\geqslant 15^\\circ$C remain horizontal. After the impact on the interface, the plume is deflected outwards. One could expect the fluid above the impact to be viscously entrained by this outward deflection. However, as observed in figure \\ref{fig:num_quiverisoT} for the simulation and figures \\ref{fig:champ_correle}-\\ref{fig:champ_pascorrele} for the experiment, the fluid above the interface is going towards the plume, \\textit{i.e.} in the opposite direction of the fluid below, hence explaining the observed shear (see figures \\ref{fig:num_quiverisoT} and \\ref{fig:schema_couplage}). The time evolution of these dynamics is shown in figure \\ref{fig:num_correWU}.\n\n\nFigure \\ref{fig:num_correWU} shows the time evolution of the horizontal velocity $u$ in the shear layer at $z=128$~mm and the time evolution of the vertical velocity $w$ in the convective layer at $z=108$~mm. Comparing the two panels of figure \\ref{fig:num_correWU} shows that upward plumes are concomitant with converging horizontal velocities towards the plume impact. Indeed, the spatio-temporal diagram of $w$ exhibits local strong upward plumes. These plumes, as suggested by the dashed black lines, are correlated in time and space with converging horizontal velocities. For instance, an upward plume is seen at $x\\approx220$~mm and $t\\approx1.043 \\times 10^5$~s. At the same horizontal position and time, the positive horizontal velocity becomes stronger and the negative horizontal velocity patch increases in size to reach $x\\approx220$~mm. The converging horizontal velocities event occurs a short time after the impact of the plumes. Thus, it can be concluded that the plume induces the converging flow, as suggested by our explanation in section \\ref{sec:results:buffer}.\n\n\\clearpage\n\n\\subsubsection{\\label{sec:results_num_igw}Internal gravity waves}\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale = 0.55]{spectro_vfinale.pdf}\n    \\hspace{.1cm}\n    \\includegraphics[scale = .55]{plot_attenuation_norm_num_v2.pdf}\n    \\caption{(Left) Power spectral density of the absolute velocity $\\sqrt{u^2+w^2}$ in the buffer and stratified layers. The grey curve shows the buoyancy frequency profile computed from the spatial and temporal average of the temperature field. (Right) Two selected profiles (taken at frequencies shown by dashed lines on the left graph) of the re-scaled PSDs by the PSD at the top of the convective layer, \\textit{i.e.} $z=118$~mm.}\n    \\label{fig:num_spectro}\n\\end{figure}\n\nPSDs are computed in the stratified and buffer layers and are plotted in figure \\ref{fig:num_spectro}. As for the experiment (figure \\ref{fig:spectro}), numerical results show oscillatory motions at different frequencies attenuated with height. Experimental results (figure \\ref{fig:spectro}) and numerical results (figure \\ref{fig:num_spectro}) show strongly similar dynamics: most of the energy is present at low frequencies ($f<3 \\times 10^{-3}$~Hz). The motion with frequencies ranging from $3 \\times 10^{-3}$~Hz to $N$ are less intense, and almost no energy is seen at frequencies $f>N$. \n\nRight panel of figure \\ref{fig:num_spectro} shows two selected vertical profiles (shown by the white dashed line on the left panel figure) of the the PSD re-scaled by the PSD at $z=118$~mm. The energy for the higher frequency ($f=1.4\\times10^{-2}$~Hz) decreases slower than the energy for the lower frequency ($f=5.0\\times10^{-3}$~Hz). This is, as experimental results, in agreement with the dispersion relation of IGWs.\nThe overall behaviour of waves spectra is similar in experiment and numerical simulation, with an attenuation length independent of the frequency in the low-frequency signal thus confirming a viscous coupling origin of the large-scale flow, and increasing when the frequency goes towards $N$ in the wave domain.\n\n\\subsubsection{Large-scale flow within the stratified layer}\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[scale = 0.6]{Spatiotemp_Vtheta_cyl_v2_square_dim.pdf}\n    \\hspace{.1cm}\n      \\includegraphics[scale = 0.6]{Spatiotemp_Vtheta_cyl_v2_zoom_dim.pdf}\n    \\caption{Spatio-temporal diagrams of the azimuthal averaging of the azimuthal velocity inside a virtual cylinder of radius $r = 140~$mm. The bottom figure is a zoom on the stratified zone, delimited in the top figure by the black square. The slope of the black lines show the theoretical viscous diffusive time.}\n    \\label{fig:num_vthetacyl}\n\\end{figure*}\nSimilarly to what has been done for the experimental data, figure \\ref{fig:num_vthetacyl} shows the mean azimuthal velocity over the whole height of a virtual cylinder of radius $r = 140$ mm. We observe reversals within the convective layer ($z<120$ mm), which are not systematically correlated with the signal in the stratified layer. The mean velocity in the stratified layer also exhibits reversals. They are characterised by an upward phase propagation from the buffer zone at $z=120$ mm, as shown in the zoom (bottom panel of figure \\ref{fig:num_vthetacyl}). The phase velocity seen in figure \\ref{fig:num_vthetacyl} is in good agreement with the theoretical time for viscous propagation $t = \\frac{z^2}{4\\,\\nu\\, \\mathrm{erf}^{-2}\\left(\\left(\\frac{u}{u_b}-1\\right)\\right)}$. This corroborates the fact that the reversals observed within the stratified layer are viscously driven from the dynamics occurring in the buffer layer, as it has been seen for the experiment. Reversals time ranges from $300$~s to $1500$~s. Those reversals times are similar to the experimental ones, though slightly shorter (numerical reversals are $\\sim20$\\% faster than experimental reversals). \n\n\\clearpage\n\\section{Conclusion}\\label{sec:discussion}\nThe 4$^{\\circ}$C convection experiment, originally performed by Townsend \\cite{townsend_natural_1964}, has been re-investigated using long-term PIV measurements in a vertical cross-section, and in several horizontal cross-sections within the stratified layer. This last type of measurements has allowed to investigate for the first time the long-term horizontal mean flow in the stratified layer. Experiments have been complemented by direct numerical simulations. The first result of this paper is the confirmation, in 3D and with ideal boundary conditions, of the presence of a buffer layer, including an overshooting region as first observed by Perrard \\cite{perrard_experimental_2013}, and a shear region. We have argued that the buffer layer is driven by thermal coupling with the convection, due to the non-linear equation of state of water, and that this mechanism is a priori related to a Prandtl number larger than one.  The second result is that the buffer layer viscously drives slow reversals of the horizontal large-scale flow within the stratified layer. \n\n\nAdditionally, IGWs at different frequencies propagate in the stratified layer. They likely interact with the horizontal large-scale flow, and probably also produce a reversing flow, which superimposes to the viscously driven one. From Couston et al. \\cite{couston_order_2018}, we know that the Prandtl number has a strong influence on this QBO-like mechanism: the lower the Prandtl number, the stronger the amplitude of the QBO. In water, $Pr \\sim 7$, and the expected amplitude of the large-scale QBO flow is weak, hence dominated by the viscous driving. Further experimental studies at lower Prandtl number should allow deciphering the two contributions. One could for instance suggest using gas as a working fluid; however, the absence of density reversal around a given temperature will necessitate to consider either transient experiments like \\cite{deardorff_laboratory_1969, michaelian_coupling_2002}, or two-gas experiments which might then be prone to double diffusive instabilities. Experimentally, the question also remains to understand why the only successful QBO experiment has been performed in salty water, hence with a Schmidt number (equivalent to Pr) of 700.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=.6]{U_avgX.pdf}\n    \\caption{Horizontal average of the horizontal velocity $u$ over a vertical cross section in the middle of the numerical domain for $Pr = 0.1$.}\n    \\label{fig:num_pr01}\n\\end{figure}\n\nIn the meantime, it is straightforward to change the Prandtl number in the numerical simulation of our set-up. We have thus run a second simulation with the same Rayleigh number $Ra = 10^7$ and top temperature $\\theta_{top} = 11$ but with $Pr = 0.1$. In this simulation, as shown in figure \\ref{fig:num_pr01}, no buffer layer is observed, but strong signatures of a QBO like mechanism are visible, marked by downward phase propagation of the reversals of the large-scale flow. This configuration thus deserves a more systematic study in the future.\n\n\\section*{Acknowledgements} \nThe authors acknowledge funding by the European Research Council under the European Union's Horizon 2020 research and innovation program through Grant No. 681835-FLUDYCO-ERC-2015-CoG. This work was granted access to the HPC resources of Aix-Marseille Universit\\'e financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program \"Investissements d'Avenir\" supervised by the Agence Nationale de la Recherche. Computations were also conducted with the support of the HPC resources of GENCI-IDRIS (Grant No.A0060407543).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe study of universal scaling behaviors associated with the nonequilibrium critical phenomena is an attractive and fascinating field of statistical physics that has attracted a considerable attention in recent years  \\cite{kardar1998nonequilibrium,fisher1998collective,amaral1995scaling,kawamura2012statistical}. \n Indeed, it is expected that a wide range of models at the critical point could well be characterized by the same universal parameters, as is well known from equilibrium critical phenomena \\cite{odor2004universality}. \nIs it possible to derive these parameters to determine the universality of critical phase transitions in out-of-equilibrium models? Recent studies have focused on the dynamical characteristics of a vast range of problems, including fracture propagation in solids \\cite{gao1989first,schmittbuhl1995interfacial,tanguy1998individual,alava2006statistical,kawamura2012statistical}, charge density waves in anisotropic\nconductors \\cite{fisher1985sliding,gruner1988dynamics}, vortices in type-II superconductors \\cite{blatter1994vortices}, domain walls\nin ferromagnetic \\cite{lemerle1998domain} or ferroelectric \\cite{tybell2002domain} systems, \nthe contact line of a fluid drop on a disordered substrate \\cite{cieplak1988dynamical,moulinet2004width,alava2004imbibition},\nthe deformation of crystals \\cite{alava2006statistical}, crackling noise in a wide range of physical systems from magnetic materials to paper crumpling\n \\cite{sethna2001crackling,bonamy2008crackling}, friction and lubrication \\cite{cule1996tribology,kawamura2012statistical,vanossi2011modeling}, \nthe motion of geological faults \\cite{fisher1997statistics,kawamura2012statistical}, tumour growth \\cite{bru2004pinning}, and many others. This diverse set of processes may be described as an extended elastic manifold driven over quenched disorder, which has a complicated dynamics that includes non-equilibrium phase transitions.\n\nThe competition between the deformation induced by quenched disorder (induced by the presence of impurities in the host environment) and the elastic material's response to an applied driving force is the key factor determining their dynamical behaviour in all of these complex non-linear systems.\n\nThe \"depinning transition\" phenomenon is a significant result of this competition \\cite{surface2}. \nAt zero temperature and in the absence of an external driving force $F$, \nthe system is disordered but it does not move and remains \\textit{pinned}\nby the quench disorder. When the external force is increased from zero, \nthe elastic object \\textit{unpins} and reaches a finite steady-state velocity \\cite{fisher1998collective}.\nThis describes the critical phase transition of the elastic interface \nat the critical force $F = F_c$, where the driving force $F$ plays the\nrole of the control parameter and the mean velocity $v$ is the\norder parameter \\cite{kardar1998nonequilibrium}. Note that the critical value of the external force $F_c$ is not\nuniversal and its value depends on the details of the model. \nThe steady-state average velocity follows a power-law characteristic as $v\\sim (F-F_c)^\\theta$ while approaching the critical point from above, where $theta$ is a universal parameter. Other measures, such as the local width, the correlation functions, the correlation length, and the structure factor, may be used to extract the exponents associated to the criticality of the elastic interface. These techniques have been extensively used to investigate the self-affine surface structure's scaling properties \\cite{surface2,mckane2013scale,krug1997origins}.\n\nConsider a single-valued function $u(\\textbf{x},t)$ that describes an elastic interface.  \nThe global surface width $W= \\sqrt{\\langle (u(\\textbf{x},t) - \\langle u \\rangle_{\\textbf{x}})^2 \\rangle_{\\textbf{x}}}$, is the simplest quantity used to characterize the scaling characteristics of elastic interfaces near the critical point,  which is defined as the standard deviation around the mean position. \nFor a finite system of size $L$, the roughening of $u$ from a \nflat initial condition, scales as:  \n\\begin{eqnarray}\nW(L,t) \\sim t^\\beta f(L\/t^{1\/\\nu}),\n\\end{eqnarray}\nwhere the exponents $\\beta$  and $\\nu$ are called the growth and the dynamical exponent. The scaling \nfunction $f(x)$ is such that $f(x)\\sim \\textrm{const}$ for $x\\gg 1$, and $f(x) \\sim x^{\\zeta_{g}}$ \nfor $x\\ll 1$ (the exponent $\\zeta_g$ is known as the global roughness exponent). Finite size effects, as expected, occur when $t^{\\times}_W \\sim L^\\nu$. The self-affine\nscaling relates now $\\zeta_g$, $\\beta$, and the dynamical exponent $\\nu$ through $\\nu = \\zeta_g\/\\beta$ \\cite{surface2}.\n\nThe average velocity, which corresponds to the order parameter of the pinning-depinning transition of a driven interface, may be assumed to be a homogeneous function of time $t$ and $|F-F_c|$, similar to critical phenomena as:\n\\begin{eqnarray}\nv(t,F) \\sim t^{-\\sigma} g(|F-F_c|t^{\\sigma\/\\theta}),\n\\end{eqnarray}\nwhere $\\sigma$ is a universal scaling exponent. For $F > F_c$ there is a crossover time-scale, $t^{\\times}_v\\sim |F-F_c|^{\\theta\/\\sigma}$ between two regimes: $g(x) \\rightarrow \\textrm{const}$ for $t\\ll t^{\\times}_v$ and $g(x)\\sim x^\\theta$ for $t \\gg t^{\\times}_v$.  \n\nThe equilibrium configuration of an elastic rough interface in the critical point is expected to be self-affine, and the tow-point correlation function is supposed to obey the scaling form\n\\begin{eqnarray}\nC(r)=\\langle [ u(\\textbf{x})- u(\\textbf{x}^\\prime) ] ^2\\rangle \\sim |\\textbf{x}-\\textbf{x}^\\prime |^{2\\zeta_l},\n\\end{eqnarray}\nwhere $\\zeta_l$ is the local roughness exponent. \n\nVarious experimental, analytical, and numerical works have been proposed to compute the critical exponents $theta$, $beta$, $zeta_g$, $zeta_l$, and $sigma$ characterizing the ``pinning-depinning'' phase transition, in a similar fashion to the equilibrium critical phenomena.\n\nThe purpose of this research is to describe and investigate the statics and dynamics of a generalized model for the investigation of a range of other reported phenomena in which the pinning-deppinig phase transition may occur.\nThe paper is organized as follows. Section \\ref{Definition sec} introduces the model. \nSection\n\\ref{Numerical method sec} describes the numerical formalism. \nIn Sec. \\ref{Numerical results sec}  we discuss our findings. In the final section, we summarize the obtained results and our conclusions.\n\n\n\n\\section{Definition of the model}\\label{Definition sec}\nDespite the significant variations in theoretical models, many of the computations were performed using the linear assumption of the elasticity $u(\\textbf{x},t)$. The following equation can be used to explain the motion of an interface in an isotropic disordered material at this level of precision \\cite{fisher1998collective}:\n\\begin{eqnarray}\n\\label{fisher_formula}\n\\frac{\\partial u(\\textbf{x},t)}{\\partial t} = F  + f_p(\\mathbf{x},u(\\mathbf{x},t)) -\\mathcal{K}\\left[ u(\\mathbf{x},t) \\right]~,\n\\end{eqnarray} \nwhere $F$ is a uniform external force which is also the control parameter and $f_p$ representing\nthe ``non-thermal'' quenched random forces due to the randomness and impurities of the heterogeneous medium. \nThe quenched random noise $f_p(\\mathbf{x},u(\\mathbf{x},t))$, can be taken to have zero mean satisfying the relation \n$\\langle f_p(\\mathbf{x},u) f_p(\\mathbf{x}^\\prime,u^\\prime) \\rangle = 2D \\delta (\\mathbf{x}-\\mathbf{x}^\\prime) \\mathcal{R} (u-u^\\prime)$, where $\\mathcal{R}(u-u^\\prime)$ assumed to decay rapidly for large values of its argument. The final term $\\mathcal{K}\\left[ u(\\mathbf{x},t) \\right]$ in Eq. (\\ref{fisher_formula}) describes the elastic forces between different parts. It has the form\n\\begin{eqnarray}\n\\mathcal{K}\\left[ u(\\mathbf{x},t) \\right] = \\int d^D\\textbf{x}^\\prime \\int dt^\\prime \\mathcal{J}(\\textbf{x}-\\textbf{x}^\\prime,t-t^\\prime)\\nonumber \\\\\n\\times \\left[ u(\\textbf{x}^\\prime,t^\\prime) - u(\\textbf{x},t) \\right],\n\\end{eqnarray}\nwhere $D$ is the space dimension and $\\mathcal{J}(\\textbf{x}-\\textbf{x}^\\prime,t-t^\\prime)$  is defined as the propagation kernel to transmit the stress on the interface from its elasticity. Moreover, systems\nwith short range elasticity of the interface are\ncharacterized by $\\mathcal{J}(\\textbf{x},t)\\propto \\delta (t) \\nabla^2 \\delta (\\textbf{x})$ \\cite{fisher1998collective}.\n\nTheoretical studies on quenched disordered systems, such as a contact line of a liquid meniscus on a disordered substrate \\cite{rosso2002roughness,moulinet2004width}, \ncrack propagation \\cite{rosso2002roughness,laurson2010avalanches} \nand solid friction \\cite{moretti2004depinning}, have shown that it is possible to express\nthe kernel $\\mathcal{K}[u]$ in a long-range form \n\\begin{eqnarray}\n\\label{long range kernel}\n\\mathcal{K}\\left[ u(\\mathbf{x},t) \\right]\\propto \\int d^D\\textbf{x}^\\prime \\frac{u(\\textbf{x},t)-u(\\textbf{x}^\\prime,t)}{|\\textbf{x}-\\textbf{x}^\\prime|^{D+z}},\n\\end{eqnarray}\nwhere the exponent $z$ is a variable that depends on the model chosen to represent the elastic interface \\cite{tanguy1998individual}. The most important aspect of the singular integration Eq. (\\ref{long range kernel}) is that it may be used to rewrite the elastic force $\\mathcal{K}[u]$ as \n\\begin{eqnarray}\\label{fractional fisher}\n\\mathcal{K}\\left[ u(\\mathbf{x},t) \\right] = \\left( - \\Laplace \\right) ^{z\/2} \nu(\\mathbf{x},t),\n\\end{eqnarray}\nwhere $\\left( - \\Laplace \\right) ^{z\/2}$ is the fractional Laplacian \ndefined by its Fourier transform \n$\\widehat{\\left( - \\Laplace \\right)} ^{z\/2} \\Phi (\\textbf{k}) = \n|\\textbf{k}|^z \\widehat{\\Phi }(\\textbf{k})$ \\cite{fractiobaloperators}.\nAccording to Eqs. (\\ref{fisher_formula}) and (\\ref{fractional fisher}),\none can rewrite the Eq. (\\ref{fisher_formula}) as follows:\n\\begin{eqnarray}\n\\label{generalized_fisher_formula}\n\\frac{\\partial u(\\mathbf{x},t)}{\\partial t} =F +f_p(\\mathbf{x},u(\\mathbf{x},t))-\n\\left( - \\Laplace \\right) ^{z\/2} u(\\mathbf{x},t) \n~.\n\\end{eqnarray}\n\n\nIt is indeed worth mentioning that the dynamics given by Eq. (\\ref{generalized_fisher_formula} are essentially generalizations of the quenched Edwards-Wilkinson (qEW) and quenched Mullins-Herring (qMH) equations, which are the simplest and most often used equations to explain the interface pinning-depinning transition in quenched random media, with $z=2 \\textrm{ and } 4$, respectively.\n\nMany researches have been carried on the qEW and qMH equations, as well as the related models. Early studies investigated numerically the crucial characteristics of the qEW equation \\cite{leschhorn1993interface},\nand it has been the subject of many\ntheoretical and numerical studies in recent years \n\\cite{ramasco2000generic,rosso2001origin,lacombe2001force,rosso2003depinning,kolton2006short,kolton2009universal}.\nRecently, a novel and very efficient approach investigated the qEW equation's depinning threshold and critical exponents \\cite{ferrero2013numerical,ferrero2013nonsteady}.\nThe scaling properties of the qMH equation at the critical point of the pinning-depinning transition have been quantitatively explored \\cite{lee2000growth,lee2006depinning,boltz2014depinning}.\nIt is worth noting that, for the \nso-called space-fractional quenched equation Eq. (\\ref{generalized_fisher_formula}), the scaling hypothesis has been  established in Ref. \\cite{xia2012depinning} \n(The fractional power $z$ is expected to be in the range $1.5 \\leqslant z\\leqslant 2$). The Grunwald-Letnikov form of a fractional derivative has been used to discretize the space-fractional quenched equation, which is essentially an integro-differential equation, as noted in Ref.\\cite{xia2012depinning}.\n\nDespite the success of\nEqs. (\\ref{fisher_formula}) and (\\ref{generalized_fisher_formula}) \nin describing the dynamics of elastic interfaces driven through a disordered medium, this toy model had one weakness: hydrodynamic interactions were not included. This is the case, for instance, of polymers \\cite{haidara2008competitiv,d2010single},\nmembranes \\cite{nissen2001interface,verma2014rough}, the dynamics of colloid suspensions, macromolecular solutions and multicomponent systems \\cite{clague1996hindered,miguel2003deblocking,cui2004anomalous,Sbragaglia2014,stannard2011dewetting}. \nBecause of the long-range hydrodynamic interaction, the dynamical behavior of these systems is correlated via flows.\n\n\nThe generalized elastic model (GEM), proposed in Ref. \\cite{Taloniprl}, is a suitable linear model that may capture the essence of criticality and phase transition (see \\cite{Talonipre,Taloniepl,Taloniperturb,Talonirev} for more details). In this case, we used this model in the presence of a quenched disorder. The quenched form of the generalized elastic model (qGEM) is represented by the stochastic linear integrodifferential equation shown below\n\\begin{eqnarray}\n\\label{qGEM}\n\\frac{\\partial u(\\mathbf{x},t)}{\\partial t} =F +\n\\int d^d x^\\prime \\Lambda(\\vert \\mathbf{x}-\\mathbf{x}^\\prime\\vert)\\frac{\\partial^z}\n{\\partial |\\mathbf{x}^\\prime | ^z}u(\\mathbf{x}^\\prime,t)\\nonumber \\\\\n+f_p(\\mathbf{x},u(\\mathbf{x},t))\n~,\n\\end{eqnarray}\nwhere the dynamical variables of the system $u(\\mathbf{x},t)$ describes \nan elastic interface driven through a disordered media. \n$F$ is the driving force on the interface and\n$f_p$ represents the quenched pinning forces\nwhich its distribution can be chosen Gaussian with the\nfirst two moments, $\\langle f_p (\\mathbf{x},u)\\rangle = 0$ and\n$\\langle f_p(\\mathbf{x},u) f_p(\\mathbf{x}^\\prime,u^\\prime)\\rangle \\propto \\delta (\\mathbf{x}-\\mathbf{x}^\\prime)\\delta(u-u^\\prime) $.\nThe hydrodynamic interaction term $\\Lambda(\\vert \\mathbf{x}-\n\\mathbf{x}^\\prime\\vert)$, corresponds to the non-local coupling of \ndifferent sites $\\mathbf{x}$ and $\\mathbf{x}'$.\nHere, $\\partial^z\/\\partial|\\mathbf{x}|^z$ is the multidimensional\nRiesz-Feller fractional derivative operator, which is  \ndefined via its\nFourier transform \n$\\mathcal{F}\\left\\{\\frac{\\partial^z}{\\partial|\\mathbf{x}|^z}\\Phi (\\textbf{x})\\right\\}\n\\equiv-|\\mathbf{k}|^z  \\Phi (\\textbf{k})$, \nimmediately implies that the Riesz-Feller fractional derivative has \nthe same meaning as the \nfractional Laplacian operator $\\partial^z\/\\partial|\\mathbf{x}|^z:=-\\left( - \n\\Laplace \\right) ^{z\/2}$ \\cite{fractiobaloperators}.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=7cm,clip]{vel_z_1_alpha_05.eps}\n\\includegraphics[width=7cm,clip]{vel_z_3_alpha_05.eps}\n\\includegraphics[width=7cm,clip]{vel_z_4_alpha_05.eps}\n\\end{center}\n\\caption{(Color online)The numerical evaluation for the average velocity $v(t,F) = \\frac{d}{dt} \\langle \\int  u(x,t)dx \\rangle$ for the generalized elastic model with quenched disorder which corresponds to the order parameter of the pinning-depinning transition of the visco-elastic interface driven through a disordered media. The behavior of the order parameter is strongly influenced by the hydrodynamic interaction parameter $\\alpha$ and the fractional power $z$. Top: The order parameter $v(t)$ as a function of time for three different values of the external force $F$. It goes to zero, when $t\\to \\infty$. The saturation values of the order parameter $v(F)$ is shown in the inset. Note that there is no phase transition for the values $\\alpha = 0.5$ and $z=1.0$ in Eq. (\\ref{qGEM}). Middle: The order parameter as a function of time and force for the so called $1$st order phase transition for the values $\\alpha = 0.5$ and $z=3.0$ Bottom: The same analysis for  the values $\\alpha = 0.5$ and $z=4.0$  shows an ordinary pinning-depinnig phase transition. Solid red line corresponds to the scaling relation $v(t) \\sim t^{-\\sigma}$ for the critical point $F=F_c$. \n}\n\\label{Figure:1}\n\\end{figure} \n\n\nAt this point a specification\nof the hydrodynamic interaction kernel $\\Lambda (\\vec{r})$ is called for. \nFor no fluid-mediated interactions, one may suppose that the friction kernel is local, $\\Lambda (\\vec{r})=\\delta (|\\vec{r}|)$ (examples). \nFor systems having non-local interactions, such as membranes, polymers, or viscoelastic surfaces, where the hydrodynamic interactions take on a long-range power-law form, a different scenario \n\\begin{eqnarray}\\label{non local hydrodynamics kernel}\n\\Lambda (\\vec{r}) \\sim \\frac{1}{|\\vec{r}|^{\\alpha}},\n\\end{eqnarray}\noccurs (where $\\frac{D-1}{2}<\\alpha<D$) \\cite{Taloniprl}. \nIt should be emphasized that the $D$-dimensional Fourier transform of the \nhydrodynamic friction kernel Eq. (\\ref{non local hydrodynamics kernel}) \nis given by $\\Lambda (\\textbf{q}) = A |\\textbf{q}|^{\\alpha -D}$ \n($A=\\textrm{const}$). It's clear that the local hydrodynamic interaction corresponds to the act of taking $\\alpha = D$ and $A=1$  \n($\\Lambda (\\textbf{q}) = 1$) \\cite{Talonipre}. \n\nThe next section will present a detailed description of the discretization approach used to numerically explore the generalized elastic model in the presence of quenched disorder Eq. (\\ref{qGEM}) for various values of the fractional order $z$ and the non-local hydrodynamic interaction strength $\\alpha$.\n\n\\section{Numerical algorithm}\\label{Numerical method sec}\nWe consider here the qGEM (\\ref{qGEM}) in one spatial dimension $D=1$.  \nThe interface position $u(x_i,t_n)$ is specified on a lattice of size $L$, where $x_i= i\\Delta x$ and $t_n = n\\Delta t$ are defined with $i=0,\\dots, L$ and $u_i^n$ is kept as a continuous variable.\n\nTo solve Eq (\\ref{qGEM}) in discretized time and space, use the finite difference approximation to estimate the time derivative (forward Euler method): \n\\begin{eqnarray}\\label{time finite difference}\n\\frac{\\partial u(x_i,t_n)}{\\partial t} = \\frac{u(x_i,t_{n+1})-u(x_i,t_n)}{\\Delta t}~.\n\\end{eqnarray}\nThe discrete space Riesz-Feller fractional operator $\\partial^z \/\\partial|x|^z$ in the Eq. (\\ref{qGEM})\ncan be approximated using the matrix transform method proposed \nby Ili\\'c \\textit{et al} \\cite{ilic2005numerical,ilic2006numerical}. \nMoreover, there are many other different numerical methods \nhave been proposed to simulate such fractional operators \\cite{Yang:thesis}.\nLet us first consider the common notation for the Riesz-Feller derivative in terms of the \nLaplacian $\\partial^z \/\\partial \\vert {x} \\vert^z:=-(-\\Laplace)^{z\/2}$ \n\\cite{saichev1997fractional}. \nThe matrix transform algorithm is based on the following definition: \nFirst consider the usual finite\ndifference scheme for Laplacian in one dimension\n\\begin{eqnarray}\\label{fourier laplace}\n\\Laplace\\phi(x) = \\frac{1}{(\\Delta x)^2} \\lbrace \\phi(x-\\Delta x)-2\\phi(x)+\\phi(x+\\Delta x) \\rbrace,\n\\end{eqnarray} \nwhere $\\lbrace \\phi(x) \\rbrace$ is the complete set of orthogonal functions. \nUsing the Fourier transform \n$\\phi(x) = \\frac{1}{2\\pi}\\int \\widehat{\\phi} (q)e^{-iqx}dq$, \nthe discretized Laplacian Eq. (\\ref{fourier laplace}) in \nthe Fourier representation can be rewritten as,\n\\begin{eqnarray}\n\\widehat{(\\Laplace)}\\phi(q) = -(2-2\\cos (q\\Delta x))\\phi(q),\n\\end{eqnarray}\nwhere $\\Delta x$ corresponds to the lattice constant. \n\nOne might start with the Fourier representation of the discretized Laplacian to approximate the Fourier representation of the discretized fractional Laplacian $(-\\Laplace)$  as: $\\lambda(q) = 2(1-\\cos(q))$, and raise it to appropriate power, \n $(2(1-\\cos(q)))^{z\/2}$. This technique has been invented by Ili\\'c \\textit{et al} \n (for more details see Refs. \\cite{ilic2005numerical,ilic2006numerical,Yang:thesis}).   \n\nThe matrix transform approach proposes that one can obtain the elements of the matrix representation of the Laplacian $\\mathbb{A}_{l,m}=-\\int_0 ^{2\\pi}\\frac{dq}{2\\pi} \n(2-2\\cos (qa)) e^{iq(l-m)}$ where $\\mathbb{A}\\equiv\\text{tridiag}(1,-2,1)$.\nThe elements of the matrix $\\mathbb{K}$, representing the discretized fractional Laplacian $-(-\\Laplace)^{z\/2}$ are then \n\\begin{eqnarray} \\label{HOLR} \n\\mathbb{K}_{l,m}&= -\\int_0^{2\\pi} \\frac{dq}{2\\pi} e^{iq(l-m)} \\left[ 2(1-\\cos(q))\\right]^{\\frac{z}{2}}\\nonumber \\\\\n&=\\frac{\\Gamma(-\\frac{z}{2}+n)\\Gamma(z +1)}{\\pi \\Gamma(1+\\frac{z}{2}+n)} \\sin(\\frac{z}{2}\\pi),\n\\end{eqnarray}\nwhere $n=\\vert l-m\\vert$, and fractional order $z\\geq 1$. In the special case $z=2$ the $\\mathbb{K}$ matrix is equal to the matrix $\\mathbb{A}$ of a simple Laplacian. On the other hand, if $\\alpha\/2$ is an integer, then \n$\\mathbb{K}(n) = (-1)^{\\alpha-n+1}C_{\\alpha,\\frac{\\alpha}{2}+n}$ for $n\\leq\\alpha\/2$ and $\\mathbb{K}(n)=0$ for $n>\\alpha\/2$, where $C_{\\alpha,\\frac{\\alpha}{2}+n}$ are binomial coefficients \\cite{zoia2007fractional}. \n \nBringing together Eqs. (\\ref{time finite difference}) and (\\ref{HOLR})\nand substitution into the Eq. (\\ref{qGEM}) leads to the discrete version of \nthe qGEM.\nWe employ the finite difference method to investigate the numerical \ndiscretization of Eq.(\\ref{qGEM}), in the form\n\\begin{eqnarray}\\label{discretqGEM}\nu_i^{n+1} =u_i^n &+ \\Delta t \\lbrace   \\frac{1}{(\\Delta x)^z} \\sum_{j= 0}^{L}\\sum_{k= 0}^{L} \\Lambda (\\vert i-j\\vert)\\mathbb{K}_{j,k}u_k^n \\nonumber \\\\\n& +F +  f_p(x_i,u_i^n)\\rbrace\n ~, \n\\end{eqnarray}\nwhere $u_i^n$ approximates the interface profile $u(x_i , t_n )$ at the $i$th lattice point and the $n$th time step. The lattice constant $\\Delta x$ has been set equal \nto one and the grid steps  $\\Delta t$ in time was chosen small enough to \navoid numerical instabilities. \n\nIn order to numerically generate quenched random field \n $f_p(x_i,u_i^n)$, without loss of generality we assumed the continuous stochastic variables $u(x_i , t_n )$ are discretized into a finite numbers of integer values $[u_i^n\/\\epsilon]$ where $\\epsilon \\ll 1$ is an arbitrary small parameter and $[\\dots]$ represents the bracket notation for the integer part of a given continuous variable.\nThen the quenched random field $f_p$ is defined on a square array where each cell $[i,h]$ ($1\\leqslant i \\leqslant L$ and $h =[u_i^n\/\\epsilon ]$)  is assigned an identically distributed random variables $\\eta(i,h)$ with normal Gaussian\ndistribution with zero mean and unit variance. The random disorder $f_p(x_i,u_i^n)$ is obtained by the linear interpolation of the random\nforce between two random variables $\\eta(i,h)$ and $\\eta(i,h+1)$ where $h = [u_i^n\/\\epsilon]$.\n\nThe numerical investigation of the scaling characteristics and critical exponents of the quenched generalized elastic model for different values of the fractional order $z$ and the non-local hydrodynamic interaction power $\\alpha$ is presented in detail in the next section.\n\n\n\n\\section{Numerical results}\\label{Numerical results sec}\nTo determine the time evolution of the interface specified by $u(x,t)$\nand to obtain the critical properties of the qGEM, \nthe simulation is started with initial condition $u(x,0) = 0$ , and boundary condition $u(x,t) = u(x+L,t)$.\nWe simulated this model on a lattice of size $L = 256$. In addition, we carefully\nchoose the  time increment $\\Delta t$ small enough to\nensure the stability of the numerical algorithm.\n\nIn order to determine the criticality of the qGEM (\\ref{qGEM}) and (\\ref{discretqGEM}) for various parameter\nvalues of the fractional order $z$ and the hydrodynamic interaction parameter $\\alpha$, we first compute the average velocity\n$v(t,F) = \\frac{d}{dt} \\langle \\int  u(x,t)dx \\rangle$ as a\nfunction of time for various values of the external homogeneous force $F$. \n\n\nSurprisingly,\nour simulations indicate that, the qGEM model in the limit $t\\rightarrow \\infty$ exhibits three quite different behaviours depending on the values of $z$ and $\\alpha$. \nWhen hydrodynamic interactions is strongly long-range $\\alpha \\ll 1$ and the fractional power $z\\ll 4$,  there exists\nno phase transition between a pinned phase and a moving phase. In this regime $\\lim _{t\\rightarrow\\infty} v(t,F)=0$ for an arbitrary external driving force $F$. Such a behaviour is shown on the top panel of Fig. (\\ref{Figure:1}) for $\\alpha =0.5$ and $z = 1.0$. \nIn the opposite limit when the parameters $\\alpha \\leq 1 $ and $z \\gg 1 $ the velocity of the interface remains zero (\\textit{pinned} phase) up to a critical force $F_c$ and above $F_c$  \nthe velocity $v(t)$ decreases\nas a power-law at the beginning and then becomes constant at all later time \\textit{i.e.} $\\lim _{t\\rightarrow\\infty} \\frac{d}{dt} v(t,F)=0$ (\\textit{moving} phase). \nAs indicated in the bottom panel Fig. (\\ref{Figure:1}), $v(F)$ is a continuous function of $F$. Thus the transition,\nlooks similar to the continuous phase transition in the context of the critical phenomena. \nAnother surprising features of the qGEM model is the anomalous pinning-depinning transition for some specific values of the parameters $\\alpha$ and $z$ in the ($\\alpha$,$z$) plane. In the anomalous regime, an elastic interface which exhibits non-trivial phase transition behavior is pinned when $F<F_c$. But for $F > F_c$ we observe a jump in the average velocity as a function of $F$ (see Fig. (\\ref{Figure:1})), may lead to a first order\nphase transition in which the order parameter of the system changes discontinualy from zero to a finite value. Note that above $F_c$ the average velocity varies with time $\\lim _{t\\rightarrow\\infty} \\frac{d}{dt} v(t,F)\\neq 0$, which is noticeably\n different from a standard pinning-depinning phase transition appears in\n the elastic interface models. In Fig. (\\ref{Figure:2}) one may see a so-called phase diagram calculated\nfor the generalized elastic model with quenched disorder. \n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8cm,clip]{phase_diagram_1.eps}\n\\end{center}\n\\caption{(Color online)The phase diagram of the generalized elastic model with \nquenched disorder. There are three different regimes, depending on the values of\nthe parameters $z$ and $\\alpha$ in Eq. (\\ref{qGEM}):\nThe first regime is when $z\\ll 4$ and $\\alpha<1$ where there is no phase\ntransition between pinned and moving phases. The second regime is when  \n$z < 4$ and $\\alpha \\gg 0$ where the order parameter of the system $v(F)$ \nas a function of the control parameter $F$ changes\ncontinuously from zero to the non-zero values ($2nd$ order phase transition).\nIn the third regime the mean velocity $v(F)$ \nas a function of $F$ changes\ndiscontinuously from zero to the non-zero values ($1st$ order phase transition).    \n}\n\\label{Figure:2}\n\\end{figure} \n \n  \nWe here focus on one aspect of the\nproblem namely the scaling\nbehavior with characteristic critical exponents of the qGEM close to the depinning critical point (This part is incomplete and we will put our results here very soon). \n\n\n\\section{Conclusions}\\label{Conclusion sec}\nIn this paper we have studied the the depinning transition of the elastic interface with non-local hydrodynamic interactions. As we mentioned earlier this model is called generalized elastic model in the presence of quenched disorder. We numerically studied different aspects of this model for different values of the fractional order $z$ and the non-local hydrodynamic interaction power $\\alpha$.\nWe found that the behaviour of order parameter $v(F)$ as function of the external force $F$  highly depends on the values of $z$ and $\\alpha$. There are three distinct phases in the phase space $(z-\\alpha)$. In the small values of $z$ and $alpha$ the order parameter vanishes and in the thermodynamic limit the steady-state order parameter approaches zero.\nIn opposite limit, where $\\alpha \\sim 1$ and $z>>1$, the model exhibits second-order phase transition and the order parameter $v(F)$ continuously changes from zero to none-zero values. And finally there is an additional phase with the order parameter changes discontinuously changes from zero to non-zero values, which is characterized by a first-order phase transition. We have analysed in detail the steady state of the model\nin the second-order phase transition regime. Our model displays naturally scaling features near critical point $F_c$. We measured different scaling exponents as functions of $z$ and $\\alpha$. Our results are in a good agreements with the well-known models.  \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nThe Sun is powered by nuclear fusion reactions occurring in its core~\\cite{Bethe:1939bt,Bahcall:2004pz,Bahcall:2000nu,Antonelli:2012qu,Davis:2016hil}. This involves hydrogen and helium predominantly, but also heavier elements such as lithium, carbon, nitrogen, oxygen, beryllium and boron. Each heavier element is produced through nuclear fusion from the lighter ones in a chain-reaction, with some steps in the chain releasing neutrinos of a characteristic energy spectrum. There are two main chains which convert hydrogen to helium in stars: the proton-proton chain and the carbon-nitrogen-oxygen (CNO) cycle~\\cite{Bethe:1939bt}, both of which lead to neutrino production.\nFor the proton-proton chain there are five neutrino components (pp, $^7$Be, $^8$B, pep and hep) each with a different spectrum, while for the CNO cycle there are three ($^{13}$N, $^{15}$O and $^{17}$F), and we refer to the sum of the latter as the CNO neutrinos.\n\nTheoretical predictions for the fluxes of these components depend on solar models, which themselves depend on various inputs such as the abundance of heavier ``metal'' elements (specifically all elements heavier than $^4$He, for example $^{12}$C, $^{13}$N and $^{15}$O) i.e. the metallicity. Solar models based on abundances that were inferred from earlier observations and modelling of the photosphere were also in excellent agreement  with helioseismological observations (e.g. GS98 \\cite{Basu1997,Basu2004}). However, advances in photosphere and line-formation modelling led to a downward revision of most of the abundances of elements heavier than helium, leading to the more recent ``low-metallicitly'' models~\\cite{Asplund:2009fu}, now incompatible with helioseismological data~\\cite{2009ASPC..416..193B,Vinyoles:2017bqj}. This disagreement is known as the solar metallicity, or solar abundance problem \\cite{Bahcall:2004yr, Bahcall06, Yang07, Basu08, Serenelli:2009yc}, and its solution will require additional, independent data (it may also be a sign of new physics, see e.g. ref.~\\cite{Frandsen:2010yj,Vincent:2014jia}). In particular, the CNO neutrino flux is very sensitive to the metallicity of the solar core, as can be seen in Table~\\ref{cno_flux_table}, and hence a precise measurement of this flux would help in improving solar models, by providing another metallicity measurement against which they can be tested~\\cite{Song:2017kvf,Bergstrom:2016cbh}. \n\n\nWhile the components of the proton-proton chain have all been measured (pp~\\cite{Bellini:2014uqa}, pep~\\cite{Collaboration:2011nga}, $^7$Be~\\cite{Bellini:2013lnn}, $^8$B~\\cite{Agostini:2017cav,Aharmim:2011vm,Abe:2016nxk}, hep at $1\\sigma$~\\cite{Bergstrom:2016cbh,Aharmim:2006wq}), a measurement of the CNO neutrino flux has not yet been achieved~\\cite{Bergstrom:2016cbh}. This is primarily because CNO neutrinos have neither a high energy, like $^8$B neutrinos, nor a huge flux, like pp neutrinos, but instead form a sub-dominant component of the total solar neutrino spectrum at energies below approximately $1.5$~MeV.\nFurthermore, due to effects from the finite energy-resolution and the adopted detection reaction of experiments, the observed spectrum in a detector resulting from CNO neutrinos is expected to resemble strongly that from pep neutrinos, leading to systematic uncertainties in determining the CNO flux even for a background-free experiment.\n\nMany current and future experiments will attempt to measure the CNO flux by looking for neutrinos scattering on electrons, such as Borexino~\\cite{Bellini:2013lnn}, SNO+~\\cite{Andringa:2015tza}, the Jinping Neutrino Experiment~\\cite{JinpingNeutrinoExperimentgroup:2016nol} and liquid argon experiments such as DarkSide and Argo~\\cite{Aalseth:2017fik,Franco:2015pha}. It should also be possible to look for CNO neutrinos through scattering with nuclei, in a similar manner to direct searches for dark matter. However this would likely require new technologies to achieve the required low energy threshold~\\cite{Strigari:2016ztv}, for example refs.~\\cite{Strauss:2017cam,Angloher:2017sxg,Maris:2017xvi,Petricca:2017zdp,Agnese:2017jvy,Schutz:2016tid,Budnik:2017sbu,Aguilar-Arevalo:2016ndq,Arnaud:2017bjh,Agnese:2016cpb,Yang:2017yaw,Arnaud:2017usi}.\n\nIn this work we determine the accuracy to which these future experiments can measure the CNO neutrino flux, and whether this is enough to distinguish between the two solar metallicity scenarios.\nFor experiments looking for electron-recoils from solar neutrinos (e.g. Borexino, Argo and SNO+), some individual projections have been carried out by the experimental collaborations themselves for their respective experiments~\\cite{Bellini:2013lnn,Andringa:2015tza,Franco:2015pha}. \nThe purpose of this work is not to refute these estimates, but rather to gather the predictions in the same place and to compare as closely as possible the discovery possibilities from a theory perspective. \n\nOur main aim is to obtain a time-scale for when the CNO flux will be measured. Furthermore, as we will see, it is difficult but not impossible for nuclear recoil experiments to see this flux.  By establishing a time-frame we hope to provide crucial input for experimentalists who plan to run a low-threshold nuclear-recoil experiment, which may also be designed to search for light dark matter.\n\n\n\n\n\n\n\\begin{table}[bt]\n\\centering\n \\renewcommand{\\arraystretch}{1.4}\n\\begin{tabular}{ p{3cm} || c | c | c  }\n\\multirow{2}{3cm}{\\textbf{Solar Metallicity}} & \\multicolumn{3}{c }{\\textbf{CNO Neutrino Flux [cm$^{-2}$ s$^{-1}$] }}   \\\\  \n\\cline{2-4}\n& $^{13}$N [$10^8$] &  $^{15}$O [$10^8$] &  $^{17}$F [$10^6$]  \\\\  \\hline\nHigh & 2.78 $\\pm$ 0.42  & 2.05 $\\pm$ 0.35  & 5.92 $\\pm$ 1.06  \\\\ \\hline\nLow & 2.04 $\\pm$ 0.29  & 1.44 $\\pm$ 0.23 & 3.26 $\\pm$ 0.59  \\\\\n\\hline\n\\end{tabular}\n\\caption{Predicted fluxes of each component of the total neutrino emission from the CNO cycle, used in this work, for the high and low solar metallicity models (GS98 and AGSS09met respectively)~\\cite{Vinyoles:2017bqj,Basu1997,Basu2004}.}\n\\label{cno_flux_table}\n\\end{table}\n\n\n\n\\section{Electron-recoil experiments searching for CNO neutrinos}\nExperiments searching for electronic recoils induced by interactions with neutrinos will be sensitive to CNO neutrinos, provided their low-energy threshold is below approximately $1.5$~MeV. Here we consider only elastic scattering i.e. $\\nu_e + e^- \\rightarrow \\nu_e + e^-$, and not inelastic scattering as will be seen for example in the DUNE experiment~\\cite{Acciarri:2015uup}.\n\nFor these experiments there exist various backgrounds whose spectra are similar to that expected from CNO neutrino-induced electronic recoils, and which are often specific to the detector or target. In some cases, the rates of these backgrounds can be determined externally, for example through measurements of the decay rate of daughter nuclei in a radioactive decay chain. It is vital that not only should these background rates be kept as low as possible, to reduce statistical uncertainties, but also ideally that they are known \\emph{a priori} to keep systematic uncertainties small.\n\n\nThe aim of our analysis is to quantify the precision with which the CNO flux can be measured for up-coming experimental runs, given the uncertainties on the other solar neutrino fluxes and backgrounds in the energy region of interest. Of particular importance are the systematic uncertainties on determining the CNO neutrino flux, which arise through degeneracies between the CNO flux and both the other neutrino fluxes and the detector-specific background rates. Motivated by this fact, we perform a Markov Chain Monte Carlo (MCMC) parameter scan over all of the neutrino fluxes and background rates, by comparing the spectra of these sources to simulated data with a Poisson likelihood.\nEach relevant background rate and neutrino flux has one parameter which determines its total energy-integrated rate, and the spectra are kept fixed up to this normalisation i.e. we fit spectra, but vary only the total rates for backgrounds or fluxes for solar neutrinos.\n\n\n\n\nThe MCMC analysis requires that each parameter has with it an associated prior distribution, which reflects the amount of knowledge we have about this parameter before the analysis is performed. For each solar neutrino flux we assume that we have no knowledge and therefore assume a uniform (or flat) prior. In practice all uniform priors are constant between zero and a value much greater than the fiducial value used to generate the simulated data. We will also use a Gaussian prior for some backgrounds when their rates are known to a given precision.\n\nAfter the analysis is complete the MCMC returns sampled values of each of the parameters, whose histogram is the posterior distribution, which tells us which values of the various parameters fit best to the simulated data. The posterior has a number of dimensions equal to the number of free parameters, and so in order to find the precision with which the CNO flux can be measured we marginalise the posterior over all other parameters.\n\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{hist_fiducial_Borexino.pdf}\n\\caption{Differential electron recoil event rate from CNO neutrinos (solid black) compared with various backgrounds  for Borexino~\\cite{Agostini:2017aaa,Collaboration:2011nga,Bellini:2013lnn}. }\n\\label{fig:spec_borexino}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_borexino_ny_10_0_lab_gran_sassofixed_C11_flag_1_0_v3.pdf} \\\\\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_borexino_ny_10_0_lab_gran_sassovery_fixed_C11_flag_1_0_v3.pdf}\n\\caption{Two-dimensional marginalised posterior distributions for Borexino running for 10~years with the pessimistic prior set, assuming $10\\%$ uncertainty on the $^{210}$Bi rate (top) or the optimistic prior set assuming $1\\%$ uncertainty on the $^{210}$Bi rate  (bottom).The dashed black lines show the fiducial values used to generate simulated data. In each panel the CNO flux is compared to different background rates and other solar neutrino fluxes, where the red  contour bounds $68\\%$ of the posterior distribution, while yellow bounds $95\\%$ and blue $99\\%$.}\n\\label{fig:borexino_mcmc}\n\\end{figure*}\n\nIn the rest of this section we detail, for each experiment, the detector-specific backgrounds and their priors, and the results of our MCMC analysis on the precision with which the CNO neutrino flux can be measured. We consider Borexino, SNO+ and dual-phase liquid argon experiments such as DarkSide or Argo. We do not perform a dedicated analysis for the Jinping Neutrino Experiment. However its projected sensitivity should be similar to that of SNO+, corrected for the different target mass~\\cite{JinpingNeutrinoExperimentgroup:2016nol}.\n\n\n\\subsection{Borexino}\n\nBorexino is a liquid scintillator experiment situated in the Gran Sasso laboratory, with a target mass of 278 tons and a fiducial mass in the most recent run of 71.3 tons~\\cite{Bellini:2013lnn,Agostini:2017ixy}. Neutrinos are detected by observing scintillation light from the electrons, which have been given MeV-levels of energy though elastic scattering.\nIn their previous Phase-I run, the Borexino collaboration did not observe CNO neutrinos and so set a limit on the CNO interaction rate of $R_{\\mathrm{CNO}} < 7.9~[100 \\, \\mathrm{ton} \\, \\mathrm{day}]^{-1}$ at $95\\%$ confidence~\\cite{Bellini:2013lnn}, with a similar limit set in Phase-II~\\cite{Agostini:2017ixy}. Here we consider a future run of Borexino, taking into account various improvements as detailed below.\nAs the only running experiment considered in this work, it will have the most realistic projections for measuring the CNO flux, since for example its backgrounds and energy-resolution are well-understood already.\n\n\nAs shown in figure~\\ref{fig:spec_borexino}, apart from the other solar neutrino fluxes ($^7$Be and pep) the largest backgrounds in the region of interest for a CNO neutrino search by Borexino arise from $^{210}$Bi beta-decays (originating from the slow decay of $^{210}$Pb) and $^{11}$C decay to positrons~\\cite{Agostini:2017ixy,Agostini:2017aaa,Collaboration:2011nga}. The former is particularly troublesome as its spectrum follows closely that expected from the CNO neutrinos, and so any measurement of the CNO neutrino rate will be partially degenerate with the rate of $^{210}$Bi beta-decay electrons. \nFor Borexino, it has been suggested that the magnitude of this $^{210}$Bi background can be inferred to an accuracy of $10\\%$ (i.e. around $6.4$ counts per ton per year) or better through measurements of the alpha-decay of its daughter nucleus $^{210}$Po~\\cite{Villante:2011zh}. Recent upgrades to the Borexino experiment, to establish secular equilibrium in the $^{210}$Pb decay chain, mean that such a measurement is now possible~\\cite{calaprice_talk_borexino}.\nThe $^{11}$C positron background arises from cosmogenic muons interacting with $^{12}$C nuclei. Its contamination in the region of interest for a CNO search is reduced by an order of magnitude using ``three-fold'' coincidence cuts on the cosmogenic muons and neutrons which are commonly produced with the $^{11}$C nuclei~\\cite{Agostini:2017ixy,Agostini:2017aaa,Collaboration:2011nga}. \n\n\n\n\\begin{table}[tb]\n\\centering\n \\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{ p{2.5cm} || c | c | c | c }\n\\multirow{3}{3cm}{\\textbf{Background}} & \\multicolumn{2}{c |}{\\textbf{Optimistic}} & \\multicolumn{2}{c}{\\textbf{Pessimistic}}  \\\\  \n\\cline{2-5}\n& Value & Error & Value & Error  \\\\ \n& [(ton yr)$^{-1}$] & [1$\\sigma$] & [(ton yr)$^{-1}$] & [1$\\sigma$] \\\\ \\hline\n$^{210}$Bi& 64 & $1\\%$ & 64 & $10\\%$  \\\\\n\\hline\n$^{210}$Po& 950 & Free & 950 & Free \\\\ \\hline\n$^{11}$C& 9.49 & Free & 9.49  & Free \\\\ \n\\hline\n\\end{tabular}\n\\caption{Fiducial values and relative uncertainties for the backgrounds in Borexino based on Phase-II data~\\cite{Agostini:2017ixy}. We provide either the one-sigma error on the \nGaussian prior, with central value equal to the fiducial value, or allow ``Free'' errors, meaning the prior distribution is uniform and unconstrained.}\n\\label{borexino_priors}\n\\end{table}\n\nIn order to understand the projected sensitivity of Borexino to CNO neutrinos in the future, and the effect of this technique to measure the $^{210}$Bi background, we perform two MCMC runs each with a different prior distribution for the $^{210}$Bi rate. In the first case, we assume that the method proposed in ref.~\\cite{Villante:2011zh} allows the $^{210}$Bi background rate to be measured to a precision of $10\\%$ of the total rate (i.e. around $6.4$ counts per ton per year), and so the $^{210}$Bi rate is given a Gaussian prior with this value as its one-sigma standard deviation. While in the second case we assume an optimistic scenario in which the $^{210}$Bi rate will be measured to $1\\%$ accuracy (i.e. around $0.64$ counts per ton per year).  Our different priors and background rates are summarized in Table~\\ref{borexino_priors}, based on the Phase-II\\footnote{We use Phase-II as this is the most recent published Borexino data, though note that the backgrounds in the next phase of Borexino will likely be smaller due to the slow decay of $^{210}$Pb and its daughters.} run in ref.~\\cite{Agostini:2017ixy}, which we also use to obtain the spectra of the background components. \n\n\n\n\n\n\n\nFigure~\\ref{fig:borexino_mcmc} shows a comparison of the posterior distributions from our MCMC projection for Borexino with either the optimistic or pessimistic prior sets, assuming $10$~years of data-taking. Each panel is a two-dimensional projection of the total posterior distribution i.e. the posterior distribution summed over all free parameters except those on the $x$ and $y$ axes. This allows us to see degeneracies between the CNO flux and the different backgrounds and other solar neutrino spectra. For example, the left-most panel on each plot shows that the CNO flux measurement is strongly degenerate with the $^{210}$Bi background, as expected since they have similar spectra. The precision on the CNO flux measurement is then obtained by marginalising over the other parameters, for example by summer over the $x$-axis in any of the two-dimensional panels, and so projecting onto the $y$-axis.\n\nFor the pessimistic priors there is a wide range of pairs of values for the $^{210}$Bi rate and CNO flux which provide a good fit to the data, which means that we have only weak constraints on the CNO flux, since it can be compensated by changing the $^{210}$Bi rate. However, with the optimistic prior set values of the $^{210}$Bi rate close to its fiducial value are strongly preferred, since we know those values further away can not be physical as we assume the $^{210}$Bi rate has been measured to $1\\%$ accuracy. Hence the degeneracy is broken and the allowed range of CNO flux values is much smaller.  Importantly for Borexino this degeneracy is very strong, which is why the effect of the optimistic priors is so clear, indicating that the ability of Borexino to measure the CNO flux will be extremely sensitive to the precision with which the $^{210}$Bi rate is measured, as well as to the total rate of this background.\n\nSimilarly, in the right-most panel we show the degeneracy between the CNO and pep neutrinos fluxes, resulting from their similar elastic-scattering spectra. This significantly degrades the ability of Borexino, and indeed all neutrino detectors, to measure the CNO flux. For example, if we knew the pep neutrino flux precisely then the CNO flux could be constrained to be within the region where the vertical dashed line in the right-most panel intersects with the shaded regions, projected onto the $y$-axis, but unfortunately disentangling these two neutrinos fluxes is not possible. \n\n\\subsection{SNO+}\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{hist_fiducial_SNO.pdf}\n\\caption{The spectrum of electronic recoil events from CNO neutrinos (solid black) compared with various backgrounds expected for SNO+~\\cite{Andringa:2015tza}. }\n\\label{fig:spec_snoplus}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_sno_plus_ny_0_5_lab_sno_lab_priors_fixed_realistic.pdf} \\\\\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_sno_plus_ny_3_0_lab_sno_lab_priors_fixed_realistic.pdf}\n\\caption{Marginalised posteriors for SNO+ with 6 months (top) or 3 years (bottom) of data and pessimistic priors. A diagonal shaped distribution means the parameters have some degeneracy between them. The dashed black lines show the fiducial values used to generate simulated data. In each panel the CNO flux is compared to different background rates and other solar neutrino fluxes, where the red  contour bounds $68\\%$ of the posterior distribution, while yellow bounds $95\\%$ and blue $99\\%$. Note that all total rates are calculated above an energy of $0.3$~MeV.}\n\\label{fig:sno_plus}\n\\end{figure*}\n\nSNO+ is an upcoming liquid scintillator experiment situated at SNOLAB, with a 780 ton target mass~\\cite{Andringa:2015tza}. \nThe expected neutrino signals and backgrounds are shown in figure~\\ref{fig:spec_snoplus}, where following ref.~\\cite{Andringa:2015tza} the background rates are based on those already achieved in Borexino. Due to its location within a deeper site compared with Borexino, its cosmogenic backgrounds such as $^{11}$C are smaller. The primary purpose of the SNO+ experiment is to measure neutrino-less double-beta decay using $^{130}$Te loaded into the liquid scintillator~\\cite{Andringa:2015tza}. Due to this, the SNO+ detector may only be sensitive to CNO neutrinos for a fraction of its total running time, since this $^{130}$Te beta-decay leads to a large background in the CNO neutrino energy range. Hence in this work we consider only SNO+ in its pure scintillator mode i.e. without $^{130}$Te doping.\n\nAs with Borexino, and shown in figure~\\ref{fig:spec_snoplus}, the SNO+ detector will have a difficult background from $^{210}$Bi, although it should have better energy-resolution making this background easier to separate from the CNO neutrino spectrum~\\cite{Andringa:2015tza}. It is also expected to have backgrounds originating from the thorium and uranium decay chains and $^{85}$Kr, $^{210}$Po and $^{11}$C. As detailed in ref.~\\cite{Andringa:2015tza}, each of these backgrounds can be measured to some extent using coincidence decays and tagging of daughter nuclei. For $^{210}$Po, $\\alpha$-tagging will be used to reduce this background by $95\\%$. Hence we adopt Gaussian priors on these background rates which reflect the expected precision to which these backgrounds will be determined, as shown in table~\\ref{sno_plus_priors}. We perform two runs, an optimistic and pessimistic scenario, where in the former case the  $^{210}$Bi background can be measured to an accuracy of $1\\%$, while in the latter case it will be measured to $10\\%$ precision, both employing the method proposed in ref.~\\cite{Villante:2011zh}.\nIt remains to be seen\nwhether the SNO+ background projections\nare actually achievable in practice.\n\n\\begin{table}[t]\n\\centering\n \\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{ p{3cm} || c | c | c | c }\n\\multirow{3}{3cm}{\\textbf{Background}} & \\multicolumn{2}{c |}{\\textbf{Optimistic}} & \\multicolumn{2}{c}{\\textbf{Pessimistic}}  \\\\  \n\\cline{2-5}\n& Value & Error & Value & Error  \\\\ \n& [(ton yr)$^{-1}$] &[1$\\sigma$]  & [(ton yr)$^{-1}$] & [1$\\sigma$] \\\\ \\hline\n$^{210}$Bi& 45.4 & $1\\%$ & 45.4 & $10\\%$  \\\\\n\\hline\n$^{210}$Po& 1530 & $20\\%$ & 1530 & $20\\%$ \\\\ \\hline\n$^{11}$C& 1.74 & Free & 1.74  & Free \\\\ \\hline\n$^{85}$Kr& 96.4 & $50\\%$ & 96.4  & $50\\%$ \\\\ \\hline\nU chain& 74.4 & $7\\%$ & 74.4  & $7\\%$ \\\\ \\hline\nTh chain& 11.1 & $25\\%$ & 11.1  & $25\\%$ \\\\ \n\\hline\n\\end{tabular}\n\\caption{Fiducial values and relative uncertainties for the backgrounds in SNO+ above a threshold energy of $0.3$~MeV (without $^{130}$Te doping)~\\cite{Andringa:2015tza}. A percentage uncertainty means the one-sigma error on the \nGaussian prior, with central value equal to the fiducial value. Columns with ``Free'' errors mean the prior distribution is uniform\ni.e. unconstrained.}\n\\label{sno_plus_priors}\n\\end{table}\n\n\n\n\nFigure~\\ref{fig:sno_plus} shows the results of our MCMC analysis for SNO+, assuming a $50\\%$ fiducial mass cut. There is a significant systematic uncertainty on the CNO flux arising from the $^{210}$Bi background, as can be seen by the degeneracy between these two parameters in the left-most panel. The uncertainty on the CNO flux alone is then obtained by marginalising over the $^{210}$Bi rate, and all of the other parameters. We have made the assumption that the $^{210}$Bi background rate can be measured to an accuracy of $10\\%$ i.e. the pessimistic priors, which makes the CNO-$^{210}$Bi degeneracy clear. With six months of data, the degeneracy is such that a zero CNO flux can not be excluded with high significance. \n\n\nThe improvement on the CNO flux measurement going from 6 months to 3 years of SNO+ data is large. The reason for this is subtle, as more data also helps relieve the partial degeneracy between both $^{210}$Bi and pep and CNO, especially when the tails of their spectra can be measured precisely. This can be seen in the left-most panel of each plot, where the best-fit region between CNO and $^{210}$Bi rotates towards the horizontal, meaning the degeneracy between these two parameters has been reduced. This is because the increased statistics means that the spectra of the CNO and $^{210}$Bi components, as shown in figure~\\ref{fig:spec_snoplus}, can be more easily distinguished. Hence an increased amount of data has not only reduced statistical uncertainties, but also systematics .\nThere is little degeneracy between the CNO flux and the $^{11}$C or U-chain backgrounds, since their spectra are not expected to be similar to that from CNO. though there is a small amount of degeneracy for the 3~year run between the Th-chain and CNO.\n\n\n\n\\subsection{Liquid argon TPCs}\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{hist_fiducial_argon.pdf}\n\\caption{The spectrum of electronic recoil events from CNO neutrinos (solid black) compared with various backgrounds expected for a liquid argon experiment operating in the Gran Sasso lab~\\cite{Franco:2015pha}. The $^{222}$Rn contamination is assumed to be $10$~$\\mu$Bq per $100$~ton. }\n\\label{fig:spec_argon_gs}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_argon_ny_50_0_lab_gran_sasso_priors_free.pdf} \\\\\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_argon_ny_50_0_lab_gran_sasso_priors_fixed_radon.pdf} \\\\\n\\caption{Contours showing the degeneracy between the CNO flux and various backgrounds and other solar neutrino fluxes for a liquid argon electronic-recoil experiment running in the Gran Sasso lab, with an exposure of 1000 ton-years with an unknown radon background (top) or a radon background known to a precision of $10\\%$ at one-sigma (bottom). The dashed black lines show the fiducial values used to generate simulated data. In each panel the red  contour bounds $68\\%$ of the posterior distribution, while yellow bounds $95\\%$ and blue $99\\%$.}\n\\label{fig:argon_GS_1000tonyr}\n\\end{figure*}\n\n\n\nIn addition to the currently-running DEAP-3600~\\cite{Amaudruz:2017ekt} there exist several upcoming or proposed experiments, such as DarkSide-20k (with a fiducial mass of 20~tons) and Argo (with a fiducial mass of 100~tons)~\\cite{Aalseth:2017fik,Franco:2015pha}, based on a dual-phase argon\\footnote{We do not consider liquid xenon experiments for an electronic recoil CNO neutrino search due to their large backgrounds, especially from $^{136}$Xe double-beta decay~\\cite{Baudis:2013qla}.} time-projection chamber (TPC) set-up which will look for both dark matter, primarily through nuclear-recoils, and neutrinos~\\cite{Aalseth:2017fik}. Here we take a more general view and will not look at a specific experiment in detail, with the motivation of determining the best experimental set-up to maximize sensitivity to CNO neutrinos for an argon experiment. Additionally, although we know that DarkSide-20k will be housed in the Gran Sasso lab~\\cite{Aalseth:2017fik}, we do not know if this will be the case for Argo, and so we will consider also the case where Argo is housed in SNOLAB or Jinping~\\cite{JinpingNeutrinoExperimentgroup:2016nol}. \n\nThe analysis of ref.~\\cite{Franco:2015pha} identified two major sources of backgrounds relevant to a search for CNO neutrinos using electronic recoils in a liquid argon experiment: cosmogenic backgrounds and $^{222}$Rn  from the detector environment. The background from $^{222}$Rn comes from $\\beta$-decay electrons emitted by $^{214}$Pb and $^{214}$Bi, two of the daughters of $^{222}$Rn $\\alpha$-decay. In this work we use the background spectra calculated in ref.~\\cite{Franco:2015pha}, which we show in figure~\\ref{fig:spec_argon_gs}. \nAs can be seen from\nthe figure, the excellent energy resolution\nmakes the pep spectrum easy to distinguish\nfrom CNO.\n\nThe size of the background from the $^{222}$Rn decay chain depends on the specific detector set-up of the argon experiment, and needs to be smaller than $\\sim 100$~$\\mu$Bq per $100$~ton to make a measurement of the CNO flux achievable~\\cite{Franco:2015pha}. The DEAP-3600 experiment had a $^{222}$Rn contamination level of $(1.8 \\pm 0.2) \\cdot 10^4$~$\\mu$Bq per $100$~ton in its most recent run~\\cite{Amaudruz:2017ekt}, however a larger experiment such as DarkSide-20k or Argo should have a smaller contamination by volume, as the $^{222}$Rn enters through the walls of the argon tank and scales less strongly with increasing experiment size than the total target mass. Throughout this work we assume a $^{222}$Rn contamination of $10$~$\\mu$Bq per $100$~ton.\nThe magnitude of the radon background can be measured through observations of the delayed coincidence between $^{214}$Bi $\\beta$-decay and $^{214}$Po $\\alpha$-decay, another daughter of $^{222}$Rn $\\alpha$-decay~\\cite{Franco:2015pha,Amaudruz:2017ekt}. Hence in order to understand the effect of this measurement on the CNO sensitivity, we perform analyses with the $^{222}$Rn freely-varying and with it fixed \\emph{a priori} to a given precision. All of the fiducial values and priors we use in our analysis are given in table~\\ref{argon_priors}. \n\n\n\n\n\n\\begin{table}[b]\n\\centering\n \\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{ p{3.5cm} || c | c | c | c }\n\\multirow{3}{3.5cm}{\\textbf{Background}} & \\multicolumn{2}{c |}{\\textbf{Optimistic}} & \\multicolumn{2}{c}{\\textbf{Pessimistic}}  \\\\  \n\\cline{2-5}\n& Value & Error & Value & Error  \\\\ \n& [(ton yr)$^{-1}$] &  & [(ton yr)$^{-1}$] &  \\\\ \\hline\n$^{222}$Rn daughters& 0.84 & $10\\%$ & 0.84 & Free  \\\\\n\\hline\n$^{32}$P& 0.78  & Free & 0.78 & Free \\\\ \\hline\nOther cosmogenics& 5.4 & Free & 5.4  & Free \\\\ \n\\hline\n\\end{tabular}\n\\caption{Fiducial values and uncertainties for the backgrounds in a liquid argon experiment, based in Gran Sasso~\\cite{Franco:2015pha}, where ``Other cosmogenics\" refers to all cosmogenic backgrounds except for $^{32}$P. A percentage uncertainty means the one-sigma error on the \nGaussian prior, with central value equal to the fiducial value. Columns with ``Free'' errors mean the prior distribution is uniform\ni.e. unconstrained.}\n\\label{argon_priors}\n\\end{table}\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_argon_ny_50_0_lab_sno_lab_priors_free.pdf} \\\\\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_argon_ny_50_0_lab_sno_lab_priors_fixed_radon.pdf} \n\\caption{Contours showing the degeneracy between the CNO flux and various backgrounds and other solar neutrino fluxes for a liquid argon electronic-recoil experiment running in SNOLAB or Jinping, with an exposure of 1000 ton-years with an unknown radon background (top) or a radon background known to a precision of $10\\%$ at one-sigma (bottom). The dashed black lines show the fiducial values used to generate simulated data. In each panel the red  contour bounds $68\\%$ of the posterior distribution, while yellow bounds $95\\%$ and blue $99\\%$.}\n\\label{fig:argon_SNOlab_1000tonyr}\n\\end{figure*}\n\n\nThe result of our MCMC scan for a liquid argon experiment based in Gran Sasso lab with a $1000$~ton-year exposure is shown in figure~\\ref{fig:argon_GS_1000tonyr}. It is clear that the CNO flux is degenerate with measurements of the radon background and the background from $^{32}$P decays, as expected from their similar spectra at lower energies seen in figure~\\ref{fig:spec_argon_gs}. Indeed, as can be seen in the second panel from the left in each plot, knowledge of the radon background vastly improves the precision to which the CNO flux can be measured, by eliminating regions of parameter space where a larger radon background can compensate for a smaller CNO flux, and vice versa. For the pessimistic case, the $99\\%$ region extends to values where the CNO flux is zero, and the radon background is around $5$ counts per day per ton, and so when marginalising over the radon background SNO+ can not exclude the possibility of a zero CNO flux at $99\\%$ confidence. By contrast, when we fix the radon background for the optimistic case, a zero CNO flux will be strongly disfavoured by the experiment. Hence in order to measure the CNO neutrino flux at high-precision with an argon-based experiment, it is crucial that the radon background is kept as small as possible and is measured.\n\n\n\n\n\nIn the case where the liquid argon experiment, such as Argo, is based in SNOLAB or Jinping, we assume that the $^{32}$P and ``Other cosmogenics'' background rates in Table~\\ref{argon_priors} are 100 times lower owing to the smaller atmospheric muon flux~\\cite{JinpingNeutrinoExperimentgroup:2016nol}. As shown in figure~\\ref{fig:argon_SNOlab_1000tonyr}, the improvement gained by this reduction in the cosmogenic background is larger than one might expect, and is due to two effects: the first is the reduction in statistical uncertainty from the lower total background. The second is a partial breaking of the degeneracy between the CNO and $^{222}$Rn spectra, as the radon background spectrum has a predominant tail which will be difficult to measure above the cosmogenic background in Gran Sasso (see figure~\\ref{fig:spec_argon_gs}), but is easily visible for Argo, or a similar experiment, based in SNOLAB or Jinping. This also means that an external measurement of the $^{222}$Rn background through coincidence decays is less important for an experiment based in SNOLAB or Jinping, since its rate can be determined well from the beta-spectrum alone. Indeed as can be seen in the second-from-left panel in figure~\\ref{fig:argon_SNOlab_1000tonyr} there is still degeneracy between the CNO flux and the $^{222}$Rn rate in the pessimistic case, but it no longer extends to values of zero for the CNO flux. This is due to the fact that a zero CNO flux  would need to be compensated by a larger radon background which would also introduce too many events at higher energies above the cut-off of the CNO spectrum, while in the Gran Sasso case these extra events would not be easily visible above the large cosmogenic background. \n\n\n\n\\subsection{Comparison of potential CNO flux measurements}\n\n\n\n\nFigure~\\ref{fig:ER_comparison} shows the expected precision with which our various projected experimental runs can measure the CNO neutrino flux at $3 \\sigma$ confidence. For each run, the error bars bracket the region where the value of the CNO neutrino flux provides a fit to the simulated data which deviates by $3 \\sigma$ or less from the best-fit value, marginalising over the other parameters in the MCMC fit. We have chosen $3 \\sigma$ since it is generally considered to be a sufficient level of confidence for discovery in astrophysics. \n\nA liquid argon electronic recoil experiment based in Gran Sasso lab will obtain the required precision with a $1000$~ton~year exposure and a radon background which has been measured \\emph{a priori} to a precision of $10\\%$ at $1 \\sigma$ (or better). This exposure could be obtained by the proposed Argo experiment in just over $3$~years if it has a mass of $300$~tons, but will likely take up to $10$~years since the plan is to use $100$~tons as the fiducial mass~\\cite{Davini:2016vpd,Aalseth:2017fik,Franco:2015pha}. With such an exposure Argo in Gran Sasso will be able to distinguish between the low and high metallicity scenarios for the CNO flux at $3 \\sigma$ confidence, though as expected from figure~\\ref{fig:argon_GS_1000tonyr} this is not the case if the size of the radon background is unknown or poorly measured. Unfortunately for a lower exposure the precision degrades significantly, and so there is no prospect for measuring the CNO flux with DarkSide-20k. \nIn all such instances the argon experiments\ngain a significant advantage from their excellent energy resolution.\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.98\\textwidth]{CNO_categories_all_v5_metallicity_low_nsigma_3_v2_best_fit_and_err.pdf}\n\\caption{Comparison of experiments searching for CNO neutrinos using electronic recoils, with either exposure or running-time labelled. The error-bars show the projected $3 \\sigma$ precision with which each experimental run will be able to measure the flux of CNO neutrinos.\nThe optimistic and pessimistic scenarios differ by how accurately key backgrounds will be measured, and are detailed in Table~\\ref{argon_priors} for liquid argon experiments (e.g. DarkSide-20k or Argo)~\\cite{Aalseth:2017fik,Franco:2015pha}, Table~\\ref{sno_plus_priors} for SNO+~\\cite{Andringa:2015tza} and Table~\\ref{borexino_priors} for Borexino~\\cite{Bellini:2013lnn,Agostini:2017ixy}. The horizontal lines labelled ``High\" and ``Low\" refer to the high and low metallicity scenarios respectively~\\cite{Vinyoles:2017bqj,Basu1997,Basu2004}. Each column and colour represents a different experiment or lab combination.}\n\\label{fig:ER_comparison}\n\\end{figure*}\n\nFor Argo, or a similar experiment, based in SNOLAB or Jinping the prospects are even better. This is mainly because the $^{222}$Rn background is much easier to distinguish from the CNO spectrum, compared with Gran Sasso where the larger cosmogenic background makes it difficult to precisely measure the radon from its beta-spectrum alone. In this case a precise determination of the CNO flux can be made in approximately half the time, with a $500$~ton~year exposure, even if the radon background is not measured using observations of coincidence decays. This would take Argo (in SNOLAB or Jinping) $5$~years with a $100$~ton fiducial mass. In both cases this requires a radon background at the level of $10 \\mu$Bq per 100 tons contamination.\n\n\n\n\n\nSNO+ in its pure liquid-scintillator mode (i.e. without $^{130}$Te added) should achieve a good measurement of the CNO flux with three years of running, and with five years of data will be able to distinguish the two scenarios for the solar metallicity using CNO neutrinos at $3 \\sigma$, provided that the $^{210}$Bi background can be measured to an accuracy of $1\\%$. As can be seen in figure~\\ref{fig:ER_comparison}, without this constraint on the $^{210}$Bi background rate, the measurement of the CNO flux will be much less precise due to the large degeneracy between the CNO flux and the $^{210}$Bi rate. Hence an accurate determination of the $^{210}$Bi rate in SNO+ is crucial for a CNO neutrino search.\nA SNO+ run of six months should be able to detect CNO neutrinos with 99\\% confidence, but will lead to only a modest constraint on the absolute flux, which is unlikely to be better than the limit already set by Borexino~\\cite{Bellini:2013lnn,Agostini:2017ixy}. Hence SNO+ would need to run for between $3$ and $5$ years to be confident of measuring the CNO flux to enough precision to solve the solar metallicity problem, provided that its background levels are as low as those already measured in Borexino.\n\nAfter ten years of running, Borexino could measure the CNO flux with enough precision to separate the two solar models, provided that the $^{210}$Bi background is measured to a precision of $1\\%$ i.e. the optimistic prior scenario.\nHowever, this projection is extremely sensitive to the level of precision to which the backgrounds, especially $^{210}$Bi can be measured to i.e. for Borexino the difference in CNO flux precision between the optimistic and pessimistic cases is particularly large.\nIf the future run of Borexino has larger background rates than we have assumed or if these are not measured to the precision assumed in our optimistic prior case, then Borexino is unlikely to measure the CNO flux even after ten years, as is clear from the large error bars for the pessimistic case. \n\n\nNote also that we have only fit energy spectra in our analysis of each experiment, while the experimental collaborations will have access to additional information. Hence our projections should be considered as conservative estimates. \nFor example, the Borexino collaboration will have access to more data such as the spatial position of each interaction, information on coincidences between different detected events and the pulse-shape of each event, which may improve their sensitivity to CNO neutrinos~\\cite{Agostini:2017ixy,Agostini:2017aaa,Collaboration:2011nga}. The SNO+ and Argo groups would likely have access to similar information, though in all cases the improvement is not likely to be large, considering that we have made efforts to implement the effects of these cuts where possible.\n\nWhich of these experimental runs gets to the CNO flux measurement first, especially with enough precision to solve the solar metallicity problem, depends on both background control and on the amount of running time dedicated to a CNO search. If the SNO+ collaboration commit most of their experimental run-time to a search for neutrino-less double-beta decay~\\cite{Andringa:2015tza} then it is possible (though perhaps unlikely) they will be overtaken by Argo, or potentially the Jinping Neutrino Experiment~\\cite{JinpingNeutrinoExperimentgroup:2016nol}, despite the fact that SNO+ is in a more advanced stage of development. As an estimate, if SNO+ finishes its neutrino-less double-beta decay search in 2022, then a CNO flux measurement may be possible by around 2025 to 2027, but possibly later. This relies crucially on the estimates of the $^{210}$Bi rate in ref.~\\cite{Andringa:2015tza} being correct. The Borexino experiment has the advantage that it is already running, and may be the first to exclude a zero CNO flux, provided its backgrounds are kept under control.  \nImportantly, this also means that its background rates will be the most realistic of\nthe experiments considered in this work, in\ncontrast to the projections of SNO+ and\nArgo, and so for a fair comparison this fact\nmust be taken into account.\n\n\nAs a final point, we note that it is possible that new technologies may allow the CNO flux to be measured by electron-recoil experiments sooner, in particular the development of experiments which can detect both scintillation and Cherenkov light, such as THEIA~\\cite{Gann:2015fba,Alonso:2014fwf,Caravaca:2016fjg}. This would mean that the direction of the recoiling electrons could be measured in addition to their energies, which would break the degeneracy between solar neutrinos and background such as $^{210}$Bi.\n\n\\section{Nuclear-recoil experiments searching for CNO neutrinos}\nIn the previous section we considered experiments looking for CNO neutrinos through their scattering with electrons, now we turn to nuclear-recoil searches. For the case of CNO neutrinos scattering with nuclei the energy range of interest is below a  few hundred eV, with the exact value depending on the particular target nucleus~\\cite{Strigari:2016ztv} i.e. targets with heavier nuclei require lower thresholds to observe CNO neutrinos, as shown in figure~\\ref{fig:CNO_NR}. Such low-energy recoils are difficult to observe, limiting the sensitivity of these searches. However nuclear-recoil searches have the advantage of lower backgrounds, and a larger cross section of interaction between neutrinos and nuclei (compared with electrons), arising from coherent enhancement by approximately the square of the number of neutrons in the target nucleus. This also means that target exposures (effective mass times experiment live time) can in principle be much lower. Hence, in contrast to the previous section, our focus here will not be on background control, but instead we are predominantly concerned with finding the required combination of target nucleus, low-energy threshold and exposure in order to measure the CNO neutrino flux with precision.\n\n\\begin{figure}[bt]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{CNO_NR.pdf}\n\\caption{The expected spectrum of nuclear-recoils from elastic scattering by CNO neutrinos in the high-metallicity case, for various different target nuclei.}\n\\label{fig:CNO_NR}\n\\end{figure}\n\nCurrent nuclear-recoil experiments focus on searching for dark matter, but they also work well as neutrino detectors~\\cite{Cerdeno:2016sfi,Lang:2016zhv,Strigari:2016ztv,Billard:2013qya,Billard:2014yka}. \nThe energy thresholds of the most successful dark matter direct detection experiments are currently too high for a CNO neutrino search, for example the liquid-xenon-based LZ and XENON1T experiments have thresholds around a keV, meaning they will not see any CNO neutrinos~\\cite{Akerib:2016vxi,Aprile:2017iyp,Mount:2017qzi}. However there has been much recent progress on the development of experiments with lower thresholds. For example, the CRESST-III experiment which looks for small temperature changes of a cryogenically-cooled CaWO$_4$ crystal, had a low-energy threshold conservatively set\nat 100 eV for their most recent analysis with $2.39$~kg-days of data~\\cite{Petricca:2017zdp}. In addition, the same collaboration has developed the $\\nu$-cleus detector with a $20$~eV threshold using a $0.5$g Al$_2$O$_3$ crystal target~\\cite{Angloher:2017sxg}. There has also been progress using germanium-based experiments such as SuperCDMS, which has achieved energy thresholds as small as 75eV and 56eV in their most recent runs with an exposure around 70kg-days~\\cite{Agnese:2017jvy}. This has been possible thanks to a special (high-voltage) operation mode and new efforts in understanding and reducing the overall background rate.\n\nThe next stage in this experiment, SuperCDMS in SNOLAB, could go as low as a 40eV threshold with an exposure in high-voltage detectors around $44$kg-yr for germanium and $10$~kg-yr for silicon~\\cite{Agnese:2016cpb}. In addition, the next phase of the EDELWEISS experiment, which also uses a germanium target, could have a mass as large as 100kg~\\cite{Arnaud:2017usi}.  Finally, the NEWS-G experiment, based on the technology of gaseous spherical detectors, had a 720eV threshold in their recent run with  a 9.7kg-days exposure using neon and CH$_4$ targets~\\cite{Arnaud:2017bjh}. In this kind of detector, the low energy threshold is limited only by the mean ionization energy of the gas mixture, which, depending on the specific target, can be as low as a few eV \\cite{Bougamont:2011zz,Savvidis:2016wei}.\n\nIn the rest of this section we will focus on making projections for future versions of such searches, since at present their low target masses, much smaller than for the electron-recoil experiments, will not provide enough data for a precise measurement of the CNO flux. We will determine exactly what exposures will be needed, for a given low-energy threshold and target nucleus, for a positive detection of the CNO flux.\n\n\n\n\\subsection{Statistical procedure}\n\n\n\n\nIn order to do so, we test the ability of an experiment with a given target nucleus, recoil detection threshold and exposure to discriminate the CNO flux from the pp, pep, $^8$B and $^7$Be neutrino fluxes originating in the pp chain. Thus, we\n\\begin{enumerate}\n\\item randomly generate events based on the total expected spectrum, from both the proton-proton-chain and CNO. For each pseudoexperiment, the normalizations of the proton-proton-chain fluxes are randomly obtained, within measured uncertainties \\cite{Bergstrom:2016cbh}\\footnote{11\\%, 8.5\\%,15\\% and 3\\% for pp, $^8$B, pep and $^7$Be, respectively.}. The CNO flux is fixed to the sum of the $^{13}$N and $^{15}$O fluxes. \n\\item We construct an extended unbinned likelihood: \n\\begin{equation}\nL = e^{-N_{\\mathrm{exp}}}\\prod_{i}^{N_{\\mathrm{obs}}}\\left[\\sum_c \\phi_{0,c}\\frac{dR}{dE_R}(E_i)\\right] P(\\phi_{0,c}),\n\\label{eq:LLNR}\n\\end{equation}\nwhere $\\frac{dR}{dE_R}$ is the expected differential neutrino-nucleus scattering rate, $N_{\\mathrm{obs}}$ is the number of ``observed'' events in the sample, and $P(\\phi_{0,c})$ is a Gaussian prior on the neutrino fluxes, with mean and width based on the currently measured fluxes. The index c runs over the components $c = \\{$pp, pep, $^8$B, $^7$Be, ($^{13}$N + $^{15}$O)  $\\}$. The total solar fluxes $\\phi_{0,c}$ are normalized such that the total number of expected events is:\n\\begin{equation}\nN_{\\mathrm{exp}} = \\sum_c N_{\\mathrm{exp},c}= \\sum_c \\phi_{0,c'}\\int_{E_{th}}^{E_{\\mathrm{max}}}\\frac{dR}{dE_R}dE_R\n\\end{equation}\nWe do not consider background events, which implies a certain level of discrimination between electronic and nuclear recoils.\n\nThis is because solar neutrinos from other populations, especially $pp$, will induce electron recoils in the energy range relevant for a nuclear-recoil search for CNO neutrinos.  Despite $pp$ neutrinos being much more abundant, coherent scattering with nuclei benefits from a larger cross section than for electronic scattering, including a factor of the nucleon number squared.  \nWith this consideration, the rate expected from $pp$ neutrino-induced electron scattering is smaller than the one for CNO neutrino nuclear recoils, by approximately a factor 10 in xenon for example, in the region-of-interest~\\cite{Cerdeno:2016sfi,Akerib:2018lyp}. \nThus, although a certain level of discrimination between electron and nuclear recoils would be desirable, it is only really important for the lightest nuclei, where one would preferably have around $90\\%$ electron-recoil rejection or better.\n\n\\item The likelihood in equation \\eqref{eq:LLNR} is then maximised setting $\\phi_{0,\\mathrm{CNO}} = 0$, then again for  allowing the total CNO flux to vary freely. The quantity $ts = 2(\\log(L_{\\mathrm{CNO}}) - \\log(L_{\\mathrm{CNO}=0}))$ is constructed. We count an experiment as successful when $ts > 3.84$, corresponding to a 95\\% CL detection ($ts$ is indeed distributed as a chi-squared with one degree of freedom). \n\\item Steps 1-3 are repeated 5000 times for each point in exposure-threshold parameter space. \n\\end{enumerate}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{cleanedNR}\n\\caption{Required exposure versus threshold to discover the CNO neutrino flux via coherent neutrino-nucleus interaction at 95\\% CL, in the high metallicity scenario. For each target element all parameters above the lines will lead to a CNO detection. In the low-metallicity case, we are unable to significantly distinguish the CNO flux from the other solar neutrino components.}\n\\label{fig:NR_results_high}\n\\end{figure}\n\n\\subsection{Results}\n\nFig.~\\ref{fig:NR_results_high} shows the result of this procedure, for nuclear recoil experiments using helium, oxygen, neon, silicon, germanium and xenon in the high-metallicity scenario. The contours represent the line above which 90\\% of the pseudoexperiments are able to see the CNO flux at 95\\% CL or more. \nAs expected, lighter targets are able to measure the CNO flux with a higher threshold, as larger momenta can be transferred from the neutrinos. The drawback is a suppression in event rate due to the smaller $N^2$ enhancement, where $N$ is the number of neutrons. \n\n\n\n\n\n\n\n\nFigure~\\ref{fig:NR_results_high} allows us to assess the suitability of the different experimental techniques to look for CNO neutrinos. For example, despite their large projected exposures, liquid xenon detectors would require a threshold energy smaller than 5eV, which is three orders of magnitude smaller than their current values. It is not clear if this is physically possible, see e.g. ref. \\cite{Baldini:2004td}.\nConversely, the threshold needed for gaseous spherical detectors with light targets seems within the reach of future experiments, however, the required exposures are still very far from those projected in future detectors, and it is not clear whether discrimination from electron recoils is possible. The situation is similar for experiments with oxygen-based crystal targets, such as CRESST and $\\nu$-cleus~\\cite{Angloher:2017sxg}.\n\n\nOf all the nuclear-recoil technologies, low-temperature solid state detectors appear to be best suited for the observation of CNO neutrinos. Indeed, the exposure needed for germanium is just a factor of three larger than what SuperCDMS SNOLAB will achieve, although the threshold would need to be reduced to $\\sim 10$eV. Likewise, silicon would require a threshold of approximately 30eV, smaller than the projected 70eV of SuperCDMS SNOLAB, and the exposure needed is approximately ten times larger.\nIn principle, these thresholds could be reduced since the required energy to produce a single electron-hole ($e-h$) pair in germanium and silicon is of the order of 3eV and 3.6eV, respectively. Although this is not within the reach of current technology, future improvements could make this possible. \n\n\nFor detection of CNO neutrinos in the low-metallicity scenario, the prospects of direct detection experiments are less promising. The $^{13}$N and $^{15}$O fluxes decrease substantially and the resulting spectrum blends in to that of the pep neutrinos. As a result, much larger exposures are needed, as well as much lower thresholds. As an example, a silicon experiment would require a threshold of approx 10~eV and a minimum exposure of $0.5$~ton-yr to start observing these events, whereas a germanium-based experiment would require a threshold close to the minimum energy to excite an $e-h$ pair.\n\n\n\n\\section{Conclusion}\nThere is no consistent model of the Sun which explains both helioseismological data and measurements of the solar metallicity using spectroscopic data, leading to the so-called ``solar metallicity problem''~\\cite{Asplund:2009fu,2009ASPC..416..193B}.\nThe aim of this paper has been to work out when experiments will measure the CNO neutrino flux with enough precision to help solve this problem, and the challenges this involves. We have studied both nuclear recoil experiments, which need to be sensitive to very low-energy recoils but have small backgrounds, and electronic recoil experiments, which generally have larger target masses but also higher background rates. We determine whether such nuclear-recoil technologies, which are important to the search for light dark matter, can catch up with the electronic-recoil experiments.\n\nFor experiments searching for CNO neutrinos using electron-recoils, we have made projections with an MCMC analysis, to compare how precisely they will be able to measure the CNO flux, given various assumptions about the backgrounds and how well their rates will be measured. Figure~\\ref{fig:borexino_mcmc} shows our results for Borexino, figure~\\ref{fig:sno_plus} for SNO+ and figures~\\ref{fig:argon_GS_1000tonyr} and \\ref{fig:argon_SNOlab_1000tonyr} for a liquid argon experiment, such as Argo~\\cite{Aalseth:2017fik,Franco:2015pha}, in Gran Sasso and SNOLAB respectively. Each two-dimensional panel shows the range of parameters which provide a good fit to simulated data, highlighting the degeneracies between parameters which lead to systematic uncertainties. For Borexino and SNO+ the main degeneracy is between the CNO flux and the $^{210}$Bi background, while for argon experiments the CNO flux is degenerate with the background from the decay of the $^{222}$Rn daughters. Any technology which could break this degeneracy would significantly improve the accuracy of a CNO flux measurement (see e.g. refs.~\\cite{Gann:2015fba,Alonso:2014fwf,Caravaca:2016fjg}). In all cases the CNO flux is strongly degenerate with the pep neutrino flux.\n\nSince we have performed our own analysis, we are able to compare our projected sensitivities for each experiment. Our comparison is shown in figure~\\ref{fig:ER_comparison}. For each experimental run we have chosen an optimistic and pessimistic prior set, with the former imposing strong constraints on the sizes of key backgrounds and thereby reducing systematic uncertainties, while in the latter case we use only weak constraints. The comparison between these different assumptions makes it clear that controlling key systematics is just as vital as obtaining more data.\n\n\n\nFor experiments looking for CNO neutrinos through their scattering with nuclei we  focused on future searches, and determined the required low-energy threshold and exposure for different target nuclei, in order to measure the CNO neutrino flux precisely. Our results for nuclear-recoils are shown in figure~\\ref{fig:NR_results_high}.\n\n\n\n\nOur analysis highlights several ways in which the CNO flux may be measured to the precision needed to separate the two solar metallicity scenarios, which we list below:\n\\begin{itemize}\n\\item If Argo is built in Gran Sasso lab with a $100$~ton fiducial mass then it will measure the CNO flux to the required precision after ten years, provided that the $^{222}$Rn-chain background is measured to a accuracy of $10\\%$ or better.\n\\item Building Argo in SNOLAB or Jinping, where the cosmogenic backgrounds will be smaller, should lead to an accurate CNO flux measurement in only $5$~years, assuming a $100$~ton fiducial mass. In all cases the argon experiments rely on their extremely good projected energy resolution.\n\\item SNO+ will measure the CNO flux in $5$~years provided that the $^{210}$Bi rate is known to at least $1\\%$ accuracy. However this is only if it is kept running in its liquid scintillator mode, and not while doped with $^{130}$Te for the neutrino-less double-beta decay search. It is likely the best candidate to see CNO neutrinos, with a potential for detection between 2025 to 2027 if its double-beta search ends in 2022, but only if the $^{210}$Bi rate meets projections to be at least as low as that already achieved in Borexino~\\cite{Andringa:2015tza}.\n\\item Borexino will set strong limits on the CNO neutrino flux, and could obtain a good measurement after ten years, provided it can measure the $^{210}$Bi rate to an accuracy of better than $0.5$~counts per day. It is the most sensitive of the experiments to systematics from backgrounds, and so an accurate measurement of $^{210}$Bi is particularly important. Crucially though, it is the only experiment we have considered which is currently running, and so has the most realistic background assumptions.\n\\item Nuclear recoils of CNO neutrinos could also be within the reach of dark matter experiments, although these would require very low-thresholds and large exposures. Low-temperature solid state detectors seem the best alternative to confirm or rule-out the high-metallicity scenario, but the technology will have to improve to be able to lower the experimental threshold down to 10eV for germanium or 30eV for silicon. Probing the low-metallicity scenario is much more challenging, as lower thresholds and larger exposures are needed. Such an experiment would also revolutionize constraints on light dark matter by many orders of magnitude~\\cite{Agnese:2017jvy}.\n\\end{itemize}\n\nIt is clear is that, despite the challenges, neutrino experiments will be able to contribute significantly to the solar metallicity problem in the near future. A full solution will likely also need us to understand why helioseismology disagrees with the standard solar model. Which experiment gets there first will depend on the amount of time dedicated to a CNO neutrino search and crucially, how well the backgrounds can be controlled for electron-recoil searches, and how low the energy threshold will be for nuclear-recoil searches.\n\n\\acknowledgments\nWe thank Davide Franco, Gabriel Orebi Gann, Gilles Gervais, El\\'ias L\\'opez-Asamar, Valentina Lozza, Ryan Martin and Oleg Smirnov.\nDGC acknowledges support from STFC.\nJD and MF are funded by the European Research Council through the project DARKHORIZONS under the European Union's Horizon 2020 program (ERC Grant Agreement no.648680).  The work of MF was also supported partly by the STFC Grant  ST\/L000326\/1 and this project has benefited from a Durham IPPP Fellowship.   \n\n\\bibliographystyle{JHEP}\n\n\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzifra b/data_all_eng_slimpj/shuffled/split2/finalzzifra
new file mode 100644
index 0000000000000000000000000000000000000000..2eae508bd2ed7b59ed5d8177ccf49a57ba78ce36
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzifra
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\n\n\n\\section{Introduction}\n\n\nIron--chromium alloys are not only scientifically interesting due to their peculiar material properties, but also play an important technological role as the base component for stainless steels \\cite{Vit2011.chapter,Wen2011.book}. \nFrom a basic-science perspective,\nFe--Cr alloys at varying relative compositions and structures exhibit intricate phenomena such as giant magnetoresistance~\\cite{Gru1986.PRL57.2442}, a spin-ice phase~\\cite{Bur1983.JPhysF13.451}, and the ``\\SI{475}{\\celsius} embrittlement'' effect~\\cite{Sah2009.MaterSciEngA508.1}. \nFrom an application point of view,\nprecision design of steels---beyond the traditional method of empirical trial and error---benefits from an atomic-level understanding of the ins and outs of Fe--Cr alloys.\nOf particular importance is to explore the mechanisms by which the Cr atoms render the open surfaces of Cr-containing steels \nresistant to corrosion \nat and above Cr concentrations of 9--10\\%~\\cite{Lev1989.Wear131.39}.\n\n\n\nThe static properties of the iron--chromium system have been extensively studied at the atomic level. For example, surface~\\cite{Rop2011.JPCM23.265004} and interface~\\cite{Lu2011.PhysStatusSolidiB248.2087} energies and surface segregation energies of chromium~\\cite{Lev2012.PRB85.064111} have been calculated for Fe--Cr alloys using \\emph{ab initio} methods.\nImportantly, such calculations have shown that the aforementioned critical Cr concentration for the onset of the corrosion resistance in stainless steels closely coincides with an onset of anomalous surface segregation of Cr in Fe--Cr alloys~\\cite{Rop2007.PRB76.220401, Lev2013.PRB87.075409}. This observation suggests that the segregation of Cr toward the surface plays a crucial role in facilitating the formation of the protective, self-healing layer of iron and chromium oxides that is known to be the reason for the corrosion resistivity of stainless steels~\\cite{Ole1980.MaterSciEng42.161,Lin1992.SurfSci277.43,Gon2007.InorgMater43.515,Don2009.CorrosSci51.827,Rop2021.SciRep11.6046}. \n\n\nThe current \\emph{ab initio} modeling methods give reasonably accurate predictions of the static properties of Fe--Cr alloys. The drawback of these methods is that they are computationally heavy, \nto the extent that the length scales in the modeling can be no larger than a few nanometers and that the costs of studying time-dependent phenomena are still largely prohibitive.\nIn an attempt to overcome these limitations, semi-empirical potential models have been developed for the Fe--Cr system.\nThe most recent of these \nare the concentration-dependent embedded-atom model~(CDEAM)~\\cite{Car2005.PRL95.075702,*Stu2009.ModelSimulMaterSc17.075005}, the two-band embedded-atom model~(2BEAM)~\\cite{Bon2011.PhilosMag91.1724}, and \nthe Tersoff potential~\\cite{Hen2013.JPCM25.445401}. \nThese potentials have been fitted to various basic material properties such as cohesion energies, lattice constants, and elastic properties, and, in general, they describe the point-defect energetics and the solubility of chromium in iron at low concentrations fairly well.\nAs such, the semi-empirical Fe--Cr potential models are well suited for modeling bulk Fe--Cr alloys and have been employed, in conjunction with Monte Carlo methods, to study \ndiffusion coefficients and precipitation kinetics~\\cite{Sen2014.ActaMater73.97}, vacancy migration near grain boundaries~\\cite{Cas2014.ComputMaterSci84.217}, and equilibrium configurations of phase-separated Fe--Cr alloys~\\cite{Zhu2011.JNuclMater417.1082} (using an earlier 2BEAM parametrization~\\cite{Ols2005.PRB72.214119}).\n\n\n\nThe aforementioned semi-empirical potential models~\\cite{Bon2011.PhilosMag91.1724,Car2005.PRL95.075702,Stu2009.ModelSimulMaterSc17.075005,Hen2013.JPCM25.445401} turn out to be, however, much less successful in predicting\nproperties of Fe--Cr surfaces~\\cite{Kur2015.PRB92.214113}. \nTheir disagreement with \n\\emph{ab initio} calculations is succinctly demonstrated by inspecting the segregation of a Cr atom from the bulk to the surface of a bcc Fe crystal: all three semi-empirical models predict the Cr atom to segregate to the second layer of Fe atoms, whereas according to \\emph{ab initio} calculations the first layer (i.e., the surface itself) should be preferred~\\cite{Kur2015.PRB92.214113}. In addition, all three models fail to reproduce the \\emph{ab initio} results of Fe--Cr interface energies~\\cite{Lu2011.PhysStatusSolidiB248.2087}. As a consequence of these shortcomings, no Monte Carlo simulations of atom kinetics \nnear Fe--Cr surfaces\nhave been performed, and the current atomic-level knowledge of surface physics in Fe--Cr alloys is far from satisfactory, \ndespite the practical importance of the topic in relation to the\ncorrosion of steel surfaces.\n\n\n\n\nIn this paper, we develop a new semi-empirical potential model that is suitable for investigating surface physics in Fe--Cr alloys at the atomic level,\nyet offers performance on par with the existing models in describing the bulk alloy.\nWe choose to work in the Tersoff formalism and use previously developed potentials for the homonuclear Fe--Fe~\\cite{Mul2007.JPCM19.326220,Bjo2007.NIMB259.853} and Cr--Cr~\\cite{Hen2013.JPCM25.445401} interactions while constructing a new one for the heteronuclear Fe--Cr interaction.\nThis choice \nof homonuclear potentials \nmakes the new potential model directly compatible with the previously developed Fe--C~\\cite{Hen2009.PRB79.144107,Hen2013.JPCM25.445401} and Cr--C~\\cite{Hen2013.JPCM25.445401} models~\\footnote{The Fe--C and Cr--C models both use the same C--C potential from Refs.~\\cite{Bre1990.PRB42.9458,Jus2005.JApplPhys98.123520}, and are therefore mutually compatible.}, so that they can all be combined into an Fe--Cr--C potential model for stainless steel.  \n\n\n\n\nThe remainder of this paper is organized as follows. In Sec.~\\ref{sec:methods}, we \nintroduce \nthe Tersoff potential formalism, describe the fitting procedure by which we develop the new Fe--Cr potential, and \noutline \nthe DFT methods \nthat we use to generate the target data for the fitting procedure.\nSection~\\ref{sec:results} is devoted to presenting the new potential and benchmarking it against the pre-existing potential models and DFT calculations, \nwith both bulk and surface properties considered. \n\\softpekcom{Removed a false sentence about the contents. The results mentioned could possibly be added into the paper, though.}\nFinally, we summarize our main results and discuss their implications in Sec.~\\ref{sec:discussion}.\n\n\\section{Theory and methods}\\label{sec:methods}\n\n\\subsection{Potential formalism}\\label{subsec:potential_formalism}\n\nThe reactive Tersoff formalism~\\cite{Ter1988.PRB37.6991,Bre1990.PRB42.9458,Alb2002.PRB65.195124} adopted in this work originates from Pauling's concept of bond order~\\cite{Pau1960.book,Abe1985.PRB31.6184};\nit can also be formally linked~\\cite{Alb2002.PRB65.195124} to\nboth the tight-binding scheme~\\cite{Cle93.PRB48.22} and the embedded-atom method~\\cite{Daw1984.PRB29.6443,Bre1989.PRL63.1022}. Since the same formalism has already been described extensively elsewhere~\\cite{Alb2002.PRB65.195124,Alb2002.PRB66.035205,Nor2003.JPCM15.5649,Jus2005.JApplPhys98.123520}, we will give here only a brief overview.\nIn the molecular dynamics code {\\footnotesize{LAMMPS}}~\\cite{Pli1995.JCompPhys117.1}, this formalism is available as the potential style \\texttt{tersoff\/zbl}.\nAlthough the only atomic types considered in this work are the elements Fe and Cr, we present the potential formalism below for a general system with an arbitrary number of atomic types.\n\nLet each atom in the system (regardless of its type) be assigned a unique ordinal number, which we will denote using the roman indices $\\inda,\\indb,\\indc\\in\\mathbb{N}$. Furthermore, let $\\left(\\elemindex{\\inda}\\right)_{\\inda}$ be a finite sequence such that $\\elemindex{\\inda}$ gives the type of the $\\inda$th atom, with $\\inda \\in \\left\\{1,\\dots,N\\right\\}$ and $N$ denoting the total number of atoms in the system~\\footnote{In our case, $\\elemindex{\\inda}\\in \\left\\{ \\mathrm{Fe},\\mathrm{Cr} \\right\\}\\ \\forall \\inda\\left\\{1,\\dots,N\\right\\}$ and $N=N_\\mathrm{Fe}+ N_\\mathrm{Cr}$.}. \nIn the Tersoff formalism, the total potential energy $E_\\mathrm{tot}$ of the system can then be written as a sum of individual bond energies:\n\\begin{equation}\\label{eq:total_potential}\nE_\\mathrm{tot}=\\sum_{\\inda=1}^N \\sum_{\\indb = \\inda+1}^N \\left\\{ \\VZBL{\\elemindex{\\inda}}{\\elemindex{\\indb}} \\bigl(r_{\\inda\\indb}\\bigr)\\left[1-F_{\\elemindex{\\inda}\\elemindex{\\indb}} \\bigl(r_{\\inda\\indb}\\bigr)\\right]+\\VABOP{\\elemindex{\\inda}}{\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr)F_{\\elemindex{\\inda}\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr)\\right\\},\n\\end{equation}\nwhere $r_{\\inda\\indb}$ is the distance between atoms $\\inda$ and $\\indb$, $\\VZBL{\\elemindex{\\inda}}{\\elemindex{\\indb}}$ is the universal Ziegler--Biersack--Littmark (ZBL) potential~\\cite{Zie1985_book,*Zie1985.incollection} for elements $\\elemindex{\\inda}$ and $\\elemindex{\\indb}$, \\(\\VABOP{\\elemindex{\\inda}}{\\elemindex{\\indb}}\\) is the pure Tersoff potential for these elements,\nand\n\\begin{equation}\\label{eq:fermi}\nF_{\\elemindex{\\inda}\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr)=\\left\\{1+\\exp\\left[-b_{\\mathrm{F},\\elemindex{\\inda}\\elemindex{\\indb}}\\left(r_{\\inda\\indb}-r_{\\mathrm{F},\\elemindex{\\inda}\\elemindex{\\indb}}\\right)\\right]\\right\\}^{-1}\n\\end{equation}\nis a Fermi function used to join the short-range ZBL and longer-range Tersoff parts smoothly together.  The values of the parameters $b_\\mathrm{F}$ and $r_\\mathrm{F}$ are chosen manually such that the potential is essentially the unmodified Tersoff potential at and past equilibrium bonding distances and that a smooth transition to the ZBL potential at short separations is obtained for all realistic coordination numbers. \\softpekcom{Refer to other papers utilizing this same interpolation approach?}\n\nIncorporation of the ZBL potential [as done in Eq.~\\eqref{eq:total_potential}] is needed to make the potential formalism suitable for modeling nonequilibrium phenomena such as melting or high-energy particle irradiation processes, which typically involve repulsive short-distance interactions originating mainly from the screened Coulomb repulsion between the positively charged nuclei. The ZBL potential is written as\n\\begin{equation}\\label{eq:ZBL}\n\\VZBL{\\elemindex{\\inda}}{\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr)=\\frac{e^2}{4\\pi \\varepsilon_0}\\frac{Z_{\\elemindex{\\inda}}Z_{\\elemindex{\\indb}}}{r_{\\inda\\indb}} \\phi\\bigl(r_{\\inda\\indb}\/a_{\\elemindex{\\inda}\\elemindex{\\indb}} \\bigr),\n\\end{equation}\nwhere $Z_\\elemindex{\\inda}$ is the atomic number of element $\\elemindex{\\inda}$,\n\\begin{equation}\na_{\\elemindex{\\inda}\\elemindex{\\indb}}= \\frac{\\SI{0.8854}{\\bohr}}{Z_{\\elemindex{\\inda}}^{0.23}+Z_{\\elemindex{\\indb}}^{0.23}}\n\\end{equation}\nwith $\\si{\\bohr}$ denoting the Bohr radius, and $\\phi$ is the universal screening function\n\\begin{equation}\n\\begin{split}\n\\phi\\left(x\\right)&=0.1818\\,e^{-3.2\\, x}+ 0.5099\\, e^{-0.9423\\,x} \\\\ & \\quad + 0.2802\\,  e^{-0.4028\\,x} + 0.02817\\,e^{-0.2016\\,x}.\n\\end{split}\n\\end{equation}\nThis screening function has been fitted to the interaction energy between ions, and its accuracy is $\\sim\\mkern-6mu 10\\%$~\\cite{Zie1985_book,*Zie1985.incollection}.\n\nThe Tersoff part $\\VABOP{}{}$ is what chiefly determines the equilibrium properties of the system. It is written as\n\\begin{equation}\\label{eq:ABOP}\n\\VABOP{\\elemindex{\\inda}}{\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr) = f^\\mathrm{c}_{\\elemindex{\\inda}\\elemindex{\\indb}} \\bigl(r_{\\inda\\indb}\\bigr)\\left[V_{\\elemindex{\\inda}\\elemindex{\\indb}}^\\mathrm{R}\\bigl(r_{\\inda\\indb}\\bigr) - \\frac{b_{\\elemindex{\\inda}\\elemindex{\\indb}}+b_{\\elemindex{\\indb}\\elemindex{\\inda}}}{2} V_{\\elemindex{\\inda}\\elemindex{\\indb}}^\\mathrm{A}\\bigl(r_{\\inda\\indb}\\bigr) \\right],\n\\end{equation} \nwhere $f^\\mathrm{c}$ is a cutoff function for the pair interaction, $V^\\mathrm{R}$ is a repulsive and $V^\\mathrm{A}$ an attractive pair potential, and $b$ is a bond-order term that describes three-body interactions and angularity. The pair potentials are of the Morse-like form\n\\begin{subequations}\\label{eq:pair_potentials}\n\\begin{align}\n\\label{eq:pair_potentialsR}\nV^\\mathrm{R}_{\\elemindex{\\inda}\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr) &=\\frac{D_{0,\\elemindex{\\inda}\\elemindex{\\indb}}}{S_{\\elemindex{\\inda}\\elemindex{\\indb}}-1} \\exp\\left[- \\sqrt{2S_{\\elemindex{\\inda}\\elemindex{\\indb}}}\\beta_{\\elemindex{\\inda}\\elemindex{\\indb}}\\left(r_{\\inda\\indb}-r_{0,\\elemindex{\\inda}\\elemindex{\\indb}}\\right)\\right], \\\\ \n\\label{eq:pair_potentialsA}\nV^\\mathrm{A}_{\\elemindex{\\inda}\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr) &=\\frac{S_{\\elemindex{\\inda}\\elemindex{\\indb}}D_{0,\\elemindex{\\inda}\\elemindex{\\indb}}}{S_{\\elemindex{\\inda}\\elemindex{\\indb}}-1} \\exp\\left[-\\frac{\\sqrt{2}\\beta_{\\elemindex{\\inda}\\elemindex{\\indb}}}{\\sqrt{S_{\\elemindex{\\inda}\\elemindex{\\indb}}}}\\left(r_{\\inda\\indb}-r_{0,\\elemindex{\\inda}\\elemindex{\\indb}}\\right)\\right],\n\\end{align}\n\\end{subequations}\nwhere $D_0$ and $r_0$ are the bond energy and length of the dimer molecule, respectively, and $S>1$ is \na dimensionless parameter that adjusts the relative strengths of the repulsive and attractive terms. \nThe parameter $\\beta$ is related to the ground-state vibrational frequency $\\omega$ and the reduced mass $\\mu$ of the dimer according to \n\\begin{equation} \n\\beta_{\\elemindex{\\inda}\\elemindex{\\indb}}=\\frac{\\sqrt{2\\mu_{\\elemindex{\\inda}\\elemindex{\\indb}}}\\pi\\omega_{\\elemindex{\\inda}\\elemindex{\\indb}}}{\\sqrt{D_{0,\\elemindex{\\inda}\\elemindex{\\indb}}}}.\n\\end{equation}\nThe bond-order term is given by \n\\begin{equation}\nb_{\\elemindex{\\inda}\\elemindex{\\indb}} = \\frac{1}{\\sqrt{1+\\chi_{\\elemindex{\\inda}\\elemindex{\\indb}}}},\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq:chi}\n\\chi_{\\elemindex{\\inda}\\elemindex{\\indb}}=\\sum_{\\indc\\left(\\neq \\inda,\\indb\\right)} f^\\mathrm{c}_{\\elemindex{\\inda}\\elemindex{\\indc}}\\bigl(r_{\\inda\\indc}\\bigr) g_{\\elemindex{\\inda}\\elemindex{\\indc}}\\bigl(\\theta_{\\inda\\indb\\indc}\\bigr)\\exp\\bigl[\\alpha_{\\elemindex{\\inda}\\elemindex{\\indb}\\elemindex{\\indc}}\\bigl(r_{\\inda\\indb}-r_{\\inda\\indc} \\bigr)\\bigr].\n\\end{equation} \nIn Eq.~\\eqref{eq:chi}, $\\theta_{\\inda\\indb\\indc}$ is the angle between the vectors $\\mathbf{r}_{\\inda\\indb}= \\mathbf{r}_{\\indb} - \\mathbf{r}_{\\inda}$ and $\\mathbf{r}_{\\inda\\indc}$, and $g$ is the angular function \n\\begin{equation}\\label{eq:g}\ng_{\\elemindex{\\inda}\\elemindex{\\indc}}\\bigl(\\theta_{\\inda\\indb\\indc}\\bigr)=\\gamma_{\\elemindex{\\inda}\\elemindex{\\indc}}\\left[1+\\frac{c_{\\elemindex{\\inda}\\elemindex{\\indc}}^2}{d_{\\elemindex{\\inda}\\elemindex{\\indc}}^2}-\\frac{c_{\\elemindex{\\inda}\\elemindex{\\indc}}^2}{d_{\\elemindex{\\inda}\\elemindex{\\indc}}^2+\\left(h_{\\elemindex{\\inda}\\elemindex{\\indc}}+\\cos\\theta_{\\inda\\indb\\indc}\\right)^2}\\right],\n\\end{equation}\nwhere $\\gamma$, $c$, $d$, and $h$ are adjustable parameters. \n\nThe cutoff function $f^\\mathrm{c}$ appearing in Eqs.~\\eqref{eq:ABOP} and~\\eqref{eq:chi} is continuously differentiable and is defined piecewise as\n\\begin{equation}\\label{eq:cutoff}\nf^\\mathrm{c}_{\\elemindex{\\inda}\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr) =\n\\begin{cases}\n1,&  r_{\\inda\\indb} < R_{\\elemindex{\\inda}\\elemindex{\\indb}}-D_{\\elemindex{\\inda}\\elemindex{\\indb}}, \\\\\n\\frac{1}{2}-\\frac{1}{2}\\sin\\frac{\\pi\\left(r_{\\inda\\indb}-R_{\\elemindex{\\inda}\\elemindex{\\indb}}\\right)}{2D_{\\elemindex{\\inda}\\elemindex{\\indb}}},& |r_{\\inda\\indb}-R_{\\elemindex{\\inda}\\elemindex{\\indb}}|\\leq D_{\\elemindex{\\inda}\\elemindex{\\indb}}, \\\\\n0,& r_{\\inda\\indb} > R_{\\elemindex{\\inda}\\elemindex{\\indb}}+D_{\\elemindex{\\inda}\\elemindex{\\indb}},\n\\end{cases}\n\\end{equation} \nwhere $R$ and $D$ determine, respectively, the center and width of the cutoff interval. Typically, $R$ is chosen to lie midway between the second- and third-nearest neighbors in the relevant equilibrium crystal.\n\n\\subsection{Fitting procedure}\\label{subsec:fitting}\n\nIn order to devise a well-performing Fe--Cr potential in the Tersoff formalism, we use the following approach: \nThe parameters for the Cr--Cr interaction are all taken completely unchanged from Ref.~\\cite{Hen2013.JPCM25.445401}, while the parameters for the Fe--Fe interaction are taken unchanged from Refs.~\\cite{Mul2007.JPCM19.326220,Bjo2007.NIMB259.853} with the exception that the value of the cutoff distance $R_{\\mathrm{Fe}\\,\\mathrm{Fe}}$ is increased to \\SI{3.5}{\\angstrom} from the original \\SI{3.15}{\\angstrom} to avoid an unphysical increase in Young's modulus of elasticity at elevated temperatures~\\cite{[{This same cutoff adjustment has been previously made by, e.g., }]Kuo2016.CMS111.525}. \nFurthermore, the two parameters $b_{\\mathrm{F},\\,\\mathrm{Fe}\\,\\mathrm{Cr}}$ and $r_{\\mathrm{F},\\,\\mathrm{Fe}\\,\\mathrm{Cr}}$ appearing in the Fermi function [Eq.~\\eqref{eq:fermi}] of the Fe--Cr potential are chosen to be the same as in Ref.~\\cite{Hen2013.JPCM25.445401}. \nAfter making these initial choices, we are left with \na total of 17 Fe--Cr potential parameters \nthat we then determine by numerical optimization.\n\n\nWe implement the numerical optimization in {\\footnotesize MATLAB}~\\cite{matlab}. To this end, the fitting of the 17 non-predetermined potential parameters is formulated as the nonlinear constrained least-squares minimization problem\n\\begin{equation}\\label{eq:minimization_problem}\n\\min_{\\xi_i\\in[0,1]}T(\\xi_1,\\dots,\\xi_{17}),\n\\end{equation}\nwhere $\\xi_i\\in[0,1]$  is a normalized optimization variable that maps to the closed interval between the minimum and maximum values we allow for the $i$th potential parameter. \nThe minimum (maximum) allowed values are chosen small (large) enough to have no significant impact on the optimal solution.\nThe target function $T$ is the weighted square sum of the differences between the target values $t_j$ and the potential model's predictions $p_j$,\n\\begin{equation}\\label{eq:target_function}\nT(\\xi_1,\\dots,\\xi_{17})=\\sum_{j=1}^{N_{\\mathrm{data}}}w_j \\left[ t_j - p_j\\left(\\xi_1,\\dots,\\xi_{17}\\right)\\right]^2,\n\\end{equation}\nwhere $w_j \\geq 0$ and $N_{\\mathrm{data}}$ is the number of evaluated quantities in the fitting database. The constrained minimization problem \\eqref{eq:minimization_problem} is solved using the trust-region-reflective algorithm~\\cite{Mor1983.SJSSC3.553,Byr1988.MaPr40.247,Bra1999.SJSC21.1} implemented in the {\\footnotesize MATLAB} function \\texttt{lsqnonlin}. The potential predictions $p_j$ are evaluated by calling the \\texttt{minimize} command in {\\footnotesize LAMMPS}~\\cite{Pli1995.JCompPhys117.1}.\nThe target values in our fitting database are determined beforehand by DFT calculations, with the database consisting of the following quantities ($N_{\\mathrm{data}}=40$):\n\\begin{itemize}\n    \\item FeCr dimer bond length $r_0$ (target value \\SI{2.1428}{\\angstrom});\n   \n    \\item formation energies of nine Cr point defects in bcc Fe (see Table~\\ref{table:defect_energies} for the specific defects and the target formation energies);\n    \\item mixing energy of Fe\\textsubscript{52}Cr\\textsubscript{2} (7 different configurations);\n    \\item mixing energy of Fe\\textsubscript{51}Cr\\textsubscript{3} (16 different configurations);\n    \\item mixing energy of Fe\\textsubscript{40}Cr\\textsubscript{14} (SQS cell);\n    \\item mixing energy of Fe\\textsubscript{27}Cr\\textsubscript{27} (SQS cell);\n    \\item mixing energy of Fe\\textsubscript{14}Cr\\textsubscript{40} (SQS cell);\n    \\item Cr segregation energies to the first four layers of a (100) surface of bcc Fe.\n\\end{itemize}\nHere the alloy mixing energies are calculated for a periodically repeating 54-atom bcc \ncell; the subscripts indicate the numbers of Fe and Cr atoms per cell. \nFor computational efficiency, the mixing energies corresponding to the combinatorially challenging intermediate Cr concentrations from 25.9\\% to 74.1\\% are estimated by means of special quasirandom structures (SQSs)~\\cite{Zun1990.PRL65.353}. For a given Cr concentration $c_{\\mathrm{Cr}} \\coloneqq N_\\mathrm{Cr}\/\\left(N_\\mathrm{Fe}+N_\\mathrm{Cr}\\right)$, the SQS cell is a single, computationally constructed 54-atom cell whose lattice sites are occupied by Fe and Cr atoms in such a way that the resulting periodic lattice mimics as closely as possible the first few, physically most relevant radial correlation functions of a perfectly random lattice with the same $c_{\\mathrm{Cr}}$. We generate the SQS cells using the \\texttt{mcsqs} code~\\cite{Wal2013.Calphad42.13} of the Alloy Theoretic Automated Toolkit~\\cite{Wal2002.Calphad26.539}.\n\n\n\n\n\\subsection{DFT calculations}\\label{subsec:dft_methods}\n\nBefore solving the optimization problem~\\eqref{eq:minimization_problem}, we carry out \\emph{ab initio} DFT calculations to determine the target values $t_j$ in Eq.~\\eqref{eq:target_function}.\nAll our DFT calculations are performed using the GPAW code~\\cite{Mor2005.PRB71.035109,gpaw2} (version 1.1.0) and the Atomic Simulation Environment (ASE)~\\cite{ase} (version 3.11). Valence--core interactions are modeled with the projector augmented-wave method (GPAW\/PAW version 0.8). \nThe generalized-gradient approximation is used in the form of the Perdew--Burke--Ernzerhof exchange\u2013correlation functional~\\cite{pbe}.\n\nAll the bulk DFT calculations are carried out using a $3\\times 3\\times 3$ cubic simulation cell of 54 atoms. \nWave functions are represented on a real-space grid of $48\\times 48 \\times 48$ points, and Brillouin-zone integrations are performed on \na Monkhorst--Pack grid~\\cite{Mon1976.PRB13.5188} of $4\\times 4 \\times 4$ $k$ points.\nAll atomic coordinates are relaxed using the Broyden--Fletcher--Goldfarb--Shanno algorithm \nuntil the Hellmann--Feynman forces are less than \\SI[per-mode=symbol]{0.05}{\\electronvolt\\per\\angstrom}. \n\nThe DFT calculations involving an Fe(100) surface are performed for a $2\\times 2\\times 5$ slab geometry with a distance of \\SI{24}{\\angstrom} between the two surfaces. \nA real-space grid of $48\\times 48 \\times 288$ points and a Monkhorst--Pack grid of $4 \\times 4 \\times 1$ $k$ points are used. \nThe two centermost atomic layers of the slab are fixed to their bulk positions, while all other atoms are relaxed using the {\\footnotesize{FIRE}} algorithm~\\cite{Bit2006.PRL97.170201} with a force tolerance of \\SI[per-mode=symbol]{0.05}{\\electronvolt\\per\\angstrom}. \n\n\n\n\n\n\n\\section{Results}\\label{sec:results}\n\n\\subsection{New Fe--Cr potential}\n\n\\softpekcom{If needed, discuss the values of the cutoff parameters here.}\n\nThe numerically optimized parameter values for the new Fe--Cr Tersoff potential are shown in Tables~\\ref{table:abop_parameters} and~\\ref{table:alpha_parameters}. The corresponding potential input file for the LAMMPS potential style \\texttt{tersoff\/zbl} is available at [URL].\n\nIt is worth noting from Table~\\ref{table:alpha_parameters} that the values of the six three-body parameters $\\alpha$ pertaining to the heteronuclear Fe--Cr part have apparently not been numerically optimized in the old Tersoff parametrization of Ref.~\\cite{Hen2013.JPCM25.445401}, on the grounds of them all having the same exact value of 1. \nThis should be contrasted with \nthe new parametrization, where all six heteronuclear $\\alpha$ parameters have been fully incorporated into the optimization. \nThe resulting increase in the dimensionality of the optimization space may in part explain why we have succeeded in significantly improving the Tersoff potential's agreement with the \\emph{ab initio} target data \nfor both bulk and surface properties, as demonstrated below.\n\n\n\n\n\n\n\n\\begin{table}[htb]\n\\caption{\\label{table:abop_parameters}Parameters for the new Tersoff potential of the Fe--Cr system. \nThe Fe--Fe part is the same as originally introduced in Ref.~\\cite{Mul2007.JPCM19.326220} and subsequently augmented with the additional parameters $b_\\mathrm{F}$ and $r_\\mathrm{F}$ [Eq.~\\eqref{eq:fermi}] in Ref.~\\cite{Bjo2007.NIMB259.853}, except for a slightly larger cutoff $R$ that we adopt to avoid an unphysical increase in Young's modulus of elasticity at elevated temperatures; \nthe Cr--Cr part is from Ref.~\\cite{Hen2013.JPCM25.445401}; the Fe--Cr part given in the last column is derived in this work. \nNote that the three-body parameter $\\alpha$ has only one value associated with each of the two homonuclear potentials ($\\alpha_{\\mathrm{Fe}\\Fe\\mathrm{Fe}}$ and $\\alpha_{\\mathrm{Cr}\\Cr\\mathrm{Cr}}$, as shown here) but assumes six distinct values for the heteronuclear Fe--Cr part (as listed separately in Table~\\ref{table:alpha_parameters}). \nAll the two-body Fe--Cr parameters  \nare symmetric with respect to interchange of the atomic types (i.e., $\\gamma_{\\mathrm{Fe}\\,\\mathrm{Cr}}=\\gamma_{\\mathrm{Cr}\\,\\mathrm{Fe}}$, and similarly for the others).\n}\n\\begin{ruledtabular}\n\\begin{tabular}{llddd}\n& & \\multicolumn{3}{c}{Interaction}\\\\\n\\cline{3-5}\n\\multicolumn{2}{c}{Parameter}&\\multicolumn{1}{c}{Fe--Fe} &\\multicolumn{1}{c}{Cr--Cr}&\\multicolumn{1}{c}{Fe--Cr}\\\\\n\\hline\n$D_0$ &(eV) & 1.5 & 4.0422 & 1.2277 \\\\\n$r_0$ &(\\AA) & 2.29 & 2.1301 & 2.2320 \\\\ \n$\\beta$ &(\\AA$^{-1}$)& 1.4 & 1.6215 &0.8957\\\\\n$S$ & & 2.0693 & 3.3679 & 3.1743\\\\  \n$\\gamma$ & & 0.01158 &  0.1577 & 0.0996 \\\\ \n$c$ & & 1.2899 & 1.0329 & 0.0794\\\\ \n$d$ & &  0.3413 & 0.1381 & 5.9464 \\\\  \n$h$ & &  -0.26 & -0.2857 &0.2952 \\\\\n$R$ &(\\AA) & 3.5 & 3.2 & 3.15 \\\\\n$D$ &(\\AA) & 0.2 & 0.2  & 0.15\\\\\n$\\alpha$ & & 0 & 1.3966 & - \\\\\n$b_\\mathrm{F}$ &($\\mathrm{\\AA}^{-1}$) & 2.9 & 12.0 & 10.0 \\\\\n$r_\\mathrm{F}$ &(\\AA) & 0.95 & 1.7 & 1.0\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table}[htb]\n\\caption{\\label{table:alpha_parameters} Three-body coefficients $\\alpha$ pertaining to the Fe--Cr part of the potential for both the old and the new Tersoff parametrizations. The rest of the parameter values for the new Tersoff potential are given in Table~\\ref{table:abop_parameters}, \nwhile the old Tersoff potential is presented in its entirety in Ref.~\\cite{Hen2013.JPCM25.445401}.}\n\\begin{ruledtabular}\n\\begin{tabular}{ldd}\n& \\multicolumn{2}{c}{Fe--Cr parametrization}\\\\\n\\cline{2-3}\n\\multicolumn{1}{c}{Parameter}&\\multicolumn{1}{c}{Old Tersoff} &\\multicolumn{1}{c}{New Tersoff}\\\\\n\\hline\n$\\alpha_{\\mathrm{Fe}\\,\\mathrm{Fe}\\,\\mathrm{Cr}}$ & 1.0 & -0.356010 \\\\\n$\\alpha_{\\mathrm{Fe}\\,\\mathrm{Cr}\\,\\mathrm{Fe}}$ & 1.0 & -3.282748 \\\\\n$\\alpha_{\\mathrm{Cr}\\,\\mathrm{Fe}\\,\\mathrm{Fe}}$ & 1.0 & -1.181222 \\\\\n$\\alpha_{\\mathrm{Fe}\\,\\mathrm{Cr}\\,\\mathrm{Cr}}$ & 1.0 & -1.340769 \\\\\n$\\alpha_{\\mathrm{Cr}\\,\\mathrm{Fe}\\,\\mathrm{Cr}}$ & 1.0 & -0.233643 \\\\\n$\\alpha_{\\mathrm{Cr}\\,\\mathrm{Cr}\\,\\mathrm{Fe}}$ & 1.0 & -1.455751 \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\subsection{Comparison of the potential models in regard to bulk Fe--Cr}\\label{subsec:bulk_results}\n\n\\softpekcom{Although it might be beneficial in some sense to start with the surface properties, I think it is conceptually more straightforward to start with the bulk properties first. If you think otherwise, I'm open to changing this. One might also question if the bulk\/surface division is the best way to go for us. As for now, there are quite a lot more bulk than surface results---regardless of our project aims.}\n\n\n\\softpekcom{Start the bulk results with introducing Fig.~\\ref{fig:mixing_energy} and defining the mixing energy and formation energy.}\n\nLet us start with the properties of bulk Fe--Cr alloys and compare the predictions of the new Tersoff potential to those of our DFT calculations, the CDEAM potential, the 2BEAM potential, and the old Tersoff potential. \nIn particular, Figs.~\\ref{fig:mixing_energy} and~\\ref{fig:sro_700K} show, respectively, the mixing energy and the Cowley short-range order parameter of the bcc Fe--Cr alloy as a function of the chromium concentration, and Table~\\ref{table:defect_energies} lists the formation energies of various isolated Cr defects in bcc iron.\nIn Fig.~\\ref{fig:mixing_energy}, the alloy mixing energy is determined as the formation energy per atom averaged over different random-alloy configurations \nat a given chromium concentration $c_{\\mathrm{Cr}} \\coloneqq N_{\\mathrm{Cr}}\/\\left(N_{\\mathrm{Fe}}+N_{\\mathrm{Cr}}\\right)$. \nThe formation energy of a given structure, in turn, is defined in this work as\n\\begin{equation}\\label{eq:formation_energy}\nE_{\\mathrm{f}} = E_{\\mathrm{tot}}\\left(\\mathrm{Fe}_{1-c_{\\mathrm{Cr}}}\\mathrm{Cr}_{c_{\\mathrm{Cr}}}\\right) - N_{\\mathrm{Fe}}E_{\\mathrm{coh}} \\left(\\mathrm{Fe}\\right) - N_{\\mathrm{Cr}}E_{\\mathrm{coh}}\\left(\\mathrm{Cr}\\right),\n\\end{equation}\nwhere $E_{\\mathrm{tot}}$ is the total potential energy of the computational cell, $N_\\elemindex{}$ is the number of atoms of element $\\elemindex{}$ in the cell, and $E_{\\mathrm{coh}}\\left(\\elemindex{}\\right)$ is the cohesive energy of element $\\elemindex{}$. \nThe cohesive energy $E_{\\mathrm{coh}}\\left(\\elemindex{}\\right)$ is determined as the total potential energy of a cell containing only atoms of element $\\elemindex{}$ divided by the number of atoms in that cell.\nFor the Fe--Fe and Cr--Cr potentials in Table~\\ref{table:abop_parameters}, \n$E_{\\mathrm{coh}}\\left(\\mathrm{Fe}\\right)=\\SI{-4.179}{\\electronvolt}$ (bcc lattice constant $a_\\mathrm{Fe}=\\SI{2.889}{\\angstrom}$) and $E_{\\mathrm{coh}}\\left(\\mathrm{Cr}\\right)=\\SI{-4.099}{\\electronvolt}$ ($a_\\mathrm{Cr}=\\SI{2.872}{\\angstrom}$). \n\n\\begin{figure}[htb]\n\\includegraphics[width=0.99\\columnwidth,keepaspectratio]{mixing_energy_full_scale_panel_a_20200818_cropped}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.99\\columnwidth,keepaspectratio]{mixing_energy_small_scale_panel_b_20200818_cropped}\n\\caption{\\label{fig:mixing_energy}(a)~Mixing energy of the bcc Fe--Cr alloy as a function of the chromium concentration as given by our DFT calculations and the four Fe--Cr potential models. (b)~Close-up of the region below 15\\% Cr concentration. The calculations are performed using a 54-atom computational cell. A total of 131 different 54-atom configurations are used for the whole Cr concentration range from 0 to 1, with the error bars corresponding to the standard error of the mean at the given Cr concentration. The lines drawn through the data points are guides to the eye.\n\\softpekcom{I should say something about the configurations used either here or in the body text.} \n}\n\\end{figure}\n\nAs reported by Olsson, Abrikosov, Vitos, and Wallenius~\\cite{Ols2003.JNuclMater321.84}, \\emph{ab initio} calculations predict a negative (positive) mixing energy of Fe--Cr alloys for chromium concentrations below (above) $\\sim 6\\%$. As can be seen from Fig.~\\ref{fig:mixing_energy}(b), our DFT calculations corroborate this result. Although the zero-crossing behavior is qualitatively reproduced by all four potential models under consideration, \nthere are quantitative differences: the new Tersoff potential yields the best fit to the \\emph{ab initio} mixing energies for Cr concentrations $<6$\\%, closely followed by the CDEAM potential, while the 2BEAM and old Tersoff potentials predict the zero crossing to lie at a higher Cr concentration [Fig.~\\ref{fig:mixing_energy}(b)]. For chromium-rich alloys, which are included in Fig.~\\ref{fig:mixing_energy}(a), the new Tersoff potential matches the DFT values significantly better than the other three potentials.\nFor example, at 50\\% Cr concentration, where the alloy mixing energy has been calculated with the SQS cell~\\cite{Zun1990.PRL65.353}, the mixing energy given by the new Tersoff potential is only 0.2\\% larger than the DFT value of \\SI{4.663}{\\electronvolt}, whereas the CDEAM, 2BEAM and old Tersoff potentials differ, respectively, by 15\\%, $-9.7$\\%, and $-65$\\% from the DFT result.\n\\softpekcom{Let me know if you think I should invert the order of the panels in Fig.~\\ref{fig:mixing_energy}. \\akcom{Ok for me.}}\n\n\\softpekcom{As the second set of bulk data, present the point defect energies of Table \\ref{table:defect_energies}. Check also if increasing the computational cell size changes the formation energies appreciably, i.e., whether there is interactions between the defects.}\n\n\n\\begin{table*}[tb]\n\\caption{\\label{table:defect_energies}Energies of Cr point defects in bcc Fe, as per our DFT calculations and the CDEAM, 2BEAM, the old Tersoff, and the new Tersoff potentials. For consistency with the DFT treatment, all the formation energies are calculated using a 54-atom cell. The last line is for the Fe substitutional defect in bcc Cr.}\n\\begin{ruledtabular}\n\\begin{tabular}{lddddd}\n& \\multicolumn{5}{c}{Formation energy (eV)} \\\\\n\\cline{2-6}\n\\multicolumn{1}{c}{Defect} & \\multicolumn{1}{c}{DFT (GPAW)}&  \\multicolumn{1}{c}{CDEAM} & \\multicolumn{1}{c}{2BEAM} & \\multicolumn{1}{c}{Old Tersoff} & \\multicolumn{1}{c}{New Tersoff} \\\\\n\\hline\n$\\langle 100 \\rangle$ Fe--Cr & 5.327 & 3.56 & 3.66 &4.96 &5.15  \\\\\n$\\langle 110 \\rangle$ Fe--Cr & 4.090 & 3.20 & 3.18 & 4.06 &4.22 \\\\\n$\\langle 111 \\rangle$ Fe--Cr & 4.596 & 3.19 & 3.54 & 4.66 &4.88 \\\\\nOctahedral Cr & 5.339 & 3.56 & 3.33 & 6.01 & 5.30 \\\\\nTetrahderal Cr & 4.611 & 3.50 & 3.32 & 4.44 & 4.83\\\\\nSubstitutional Cr & -0.1444 & -0.112 & -0.233 & -0.0629  & -0.136\\\\\n$\\langle 100 \\rangle$ Cr--Cr & 5.43 &4.49  & 3.98 & 4.88 & 5.45 \\\\\n$\\langle 110 \\rangle$ Cr--Cr & 4.35 & 3.81 & 3.26 & 3.80 & 4.09\\\\\n$\\langle 111 \\rangle$ Cr--Cr & 4.699 & 4.63 & 3.51 & 4.58 & 5.00 \\\\\n\\hline\nSubstitutional Fe in Cr & 0.4825 & 0.498 & 0.357 & 0.235  & 0.511\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\nThe formation energies of selected Cr point defects in bcc Fe are presented in Table~\\ref{table:defect_energies}. In general, all four potential models are in fairly good agreement with the target DFT values, with percentage errors typically $<10$\\%. If we take the nine Cr point defects listed in Table~\\ref{table:defect_energies} and compute the symmetric mean absolute percentage error (SMAPE) of the predictions of each potential model~\\cite{[{The SMAPE is defined here as \n$100\\%\\sum_{j=1}^{N_E} \\abs{t_j-p_j}\/\\bigl(N_E \\abs{t_j}+N_E \\abs{p_j}\\bigr)$, where the sum is over the $N_E=9$ formation energies and $t_j$ and $p_j$ are their target values and potential-model predictions, respectively. See, e.g., }]Arm1985.book,*Flo1986.Omega14.93},\nwe obtain the following ranking (from best to worst): new Tersoff (SMAPE of 2.1\\%), old Tersoff (7.2\\%), CDEAM (13\\%), and 2BEAM (17\\%). On average, the new Tersoff potential is therefore in a better agreement with our \\emph{ab initio} point-defect energies than the other three potential models. We note in particular that the new Tersoff potential provides the best match for the negative DFT value of the Cr substitutional defect.\n\n\\softpekcom{Next, define the SRO parameter and present and analyze the SRO data (at three different temperatures?). In particular, compare the various model predictions with the experimental data at \\SI{700}{\\kelvin} (Fig.~\\ref{fig:sro_700K}).} \n\n\\begin{figure}[b]\n\\includegraphics[width=0.99\\columnwidth,keepaspectratio]{sro_beta_at_700K_vs_small_cr_concentration_20200818_cropped}\\\\\n\\vspace{4mm}\n\\includegraphics[width=0.99\\columnwidth,keepaspectratio]{sro_beta_at_700K_vs_cr_concentration_20200818_cropped}\n\\caption{\\label{fig:sro_700K}Short-range order parameter $\\beta$ [Eq.~\\eqref{eq:sro_beta}] at \\SI{700}{\\kelvin} in a bcc Fe--Cr alloy as a function of the chromium concentration $c_{\\mathrm{Cr}}$ for (a)~$0.02 \\leq c_{\\mathrm{Cr}} \\leq 0.16$ and (b)~$0.02 \\leq c_{\\mathrm{Cr}} \\leq 0.9$. The lines are guides to the eye. The experimental data points at $c_{\\mathrm{Cr}}=0.05$, $0.1$, and $0.15$ are from the diffuse--neutron-scattering measurements of Mirebeau, Hennion, and Parette~\\cite{Mir1984.PRL53.687}; fits A and B correspond to least-squares fits of the nuclear cross section with three and four order parameters, respectively.\n\\softpekcom{Could present these at other temperatures as well, but there is no experimental data to compare against.}\n} \n\\end{figure}\n\nBesides comparing formation energies, we may examine the degree of short-range ordering in a thermally equilibrated Fe--Cr alloy. To this end, the Cowley short-range order parameters $\\alpha$ can be defined as~\\cite{Cow1950.PR77.669}\n\\begin{equation}\n\\alpha_{\\mathrm{Cr}}^{(k)} = 1 - \n\\frac{Z^{(k)}_{\\mathrm{Fe}}}{\\left(Z^{(k)}_{\\mathrm{Fe}}+Z^{(k)}_{\\mathrm{Cr}}\\right)\\left(1-c_{\\mathrm{Cr}}\\right)},\n\\end{equation}\nwhere $Z^{(k)}_{\\mathrm{Fe}}$ and $Z^{(k)}_{\\mathrm{Cr}}$ are the average numbers of Fe and Cr atoms in the $k$th neighbor shell.\nWe further define the linear combination\n\\begin{equation}\\label{eq:sro_beta}\n\\beta=\\frac{8 \\alpha_{\\mathrm{Cr}}^{(1)} + 6\\alpha_{\\mathrm{Cr}}^{(2)}}{14}\n\\end{equation}\nrelevant to bcc lattices. If $\\beta < 0$, a Cr atom prefers to have Fe atoms as its nearest neighbors; if $\\beta > 0$, Cr prefers Cr neighbors; $\\beta=0$ corresponds to a random alloy. The configurations used to determine $\\beta$ are computed with a Monte Carlo method where possible moves \nconsist of atom displacements and exchanges of types (Fe or Cr) of pairs of atoms.\nThe displacements are performed with short sequences of MD simulations in the canonical ensemble, \nbecause this \nhas been found to be more efficient in moving the atoms than the conventional Metropolis algorithm~\\cite{Kur2015.PRB92.214113}. It should be noted that these Monte Carlo--MD calculations are pure equilibrium simulations with no kinetics involved.\n\nFigure~\\ref{fig:sro_700K} shows the order parameter $\\beta$ at \\SI{700}{\\kelvin} as a function of the Cr concentration \nfor the four potential models, along with three experimental data points determined by \ndiffuse--neutron-scattering measurements~\\cite{Mir1984.PRL53.687}.\nWhile none of the four potential models yields a particularly good fit to the experimental data, the CDEAM and 2BEAM potentials at least qualitatively reproduce the observed change of sign of $\\beta$ from negative to positive around $c_{\\mathrm{Cr}} = 0.1$  [Fig.~\\ref{fig:sro_700K}(a)].\nFor both Tersoff potentials, $\\beta$ remains negative up to high Cr concentrations, eventually turning positive at $c_{\\mathrm{Cr}}\\approx 0.37$ for the new Tersoff potential and at $c_{\\mathrm{Cr}}\\approx 0.76$ for the old Tersoff potential [Fig.~\\ref{fig:sro_700K}(b)].\nWe thus conclude that the new Tersoff potential, although largely failing to match the experimental data, still performs noticeably better than the old Tersoff potential in describing the short-range ordering in Fe--Cr alloys.\n\n\\softpekcom{As the final set of bulk results, present migration energy barriers in bulk bcc Fe (Fig.~\\ref{fig:energy_barriers} and Table~\\ref{table:migration_barriers}).}\n\nWe have also computed the migration energy barries related to the diffusion of a vacancy--Cr-substitutional pair in bulk bcc Fe using the nudged elastic band (NEB) method~\\cite{Mil1995.SurfaceScience324.305,Jon1998.chapter} implemented in LAMMPS. We consider both a process where a Cr subsitutional moves to a vacancy in the nearest-neighbor site and a process where the vacancy migrates from the nearest-neighbor to the second-nearest-neighbor site of the Cr atom (see Fig.~\\ref{fig:energy_barriers}). The obtained barrier energies for the four potential models are listed in Table~\\ref{table:migration_barriers} along with DFT results by Messina, Nastar, Garnier, Domain, and Olsson~\\cite{Mes2014.PRB90.104203}.\nBy far the best agreement with the DFT values is given by the 2BEAM potential (with the largest relative difference being $<6\\%$), followed by the CDEAM potential, which overestimates the Cr--vacancy migration energy by 53\\% but is otherwise close to the target values. The new Tersoff potential is in fairly good agreement with DFT for the Cr--vacancy migration energy $E_2^\\mathrm{mig}$ but noticeably overestimates both barries $E_{12}^\\mathrm{mig}$ and $E_{21}^\\mathrm{mig}$. The old Tersoff potential is also off by significant margins (26\\% for $E_2^\\mathrm{mig}$, $-32\\%$ for $E_{12}^\\mathrm{mig}$, and $16\\%$ for $E_{21}^\\mathrm{mig}$). \n\n\\begin{figure}[htb]\n\\includegraphics[width=0.65\\columnwidth,keepaspectratio]{migration_barriers_schematic}\n\\caption{\\label{fig:energy_barriers}Nomenclature used for the migration energy barriers. The Cr atom is shown in blue and the Fe atoms in red. $E_2^\\mathrm{mig}$ denotes the migration energy barrier for the solute--vacancy jump, and $E_{ij}^\\mathrm{mig}$ is for an iron atom moving from site $j$ to a vacant site $i$. Table~\\ref{table:migration_barriers} lists the values of these energies barriers within the different models under consideration. \n}\n\\end{figure}\n\n\\begin{table}[tbh]\n\\caption{\\label{table:migration_barriers}Migration energy barriers for Cr--vacancy diffusion and Fe self-diffusion in the neighboring site (see Fig.~\\ref{fig:energy_barriers}).}\n\\begin{ruledtabular}\n\\begin{tabular}{lddd}\nMethod & E_2^\\mathrm{mig} &  E_{12}^\\mathrm{mig}   & E_{21}^\\mathrm{mig}  \\\\\n\\cline{1-4}\nDFT~\\cite{Mes2014.PRB90.104203} & 0.575(45) & 0.69 & 0.64(1) \\\\\nCDEAM & 0.8772 & 0.6467 & 0.6160 \\\\\n2BEAM & 0.5725 & 0.6511 & 0.6260 \\\\\nOld Tersoff & 0.7267 & 0.4666 & 0.7413 \\\\\nNew Tersoff & 0.632 & 1.09 & 0.941 \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\subsection{Comparison of the potential models for surface-related properties}\\label{subsec:surface_results}\n\nWe now move to scenarios involving surfaces \nof Fe--Cr alloys and investigate whether we have achieved our goal of improving \nupon the performance of the existing potential models in \npredicting properties of such surfaces. \n\nLet us first consider the segregation of Cr atoms to the (100) surface of bcc Fe. The segregation energy of Cr from a given reference region A to another region B is defined as \nthe net change in energy when a Cr atom is transferred from A to B and an Fe atom from B to A,\n\\begin{equation}\nE^{\\mathrm{Cr}}_{\\mathrm{segr},\\mathrm{A}\\to\\mathrm{B}}=E_{\\mathrm{tot}}(\\textrm{Fe at A, Cr at B})-E_{\\mathrm{tot}}(\\textrm{Fe at B, Cr at A}),\n\\end{equation}\nwhere $E_{\\mathrm{tot}}$ is the total energy of the computational cell.  In our case, B will be one of the top surface layers and A will be a bulk site. The energies are calculated using a slab geometry with periodic boundary conditions in $y$ and $z$ directions and the slab thickness $L_x$ large enough that further increases in $L_x$ cause no changes in the segregation energies. The reference bulk position A is taken from the middle of the slab. For nonzero background Cr concentration, the segregation energy values are averaged over 10\\,000 random alloy configurations.\n\n\\begin{figure}[htb]\n\\includegraphics[width=0.891\\columnwidth,keepaspectratio]{segregation_energy_layer1_vs_cr_concentration_20200818_cropped}\n\\includegraphics[width=0.891\\columnwidth,keepaspectratio]{segregation_energy_layer2_vs_cr_concentration_20200818_cropped}\n\\includegraphics[width=0.891\\columnwidth,keepaspectratio]{segregation_energy_difference_vs_cr_concentration_20200818_cropped}\n\\caption{\\label{fig:segregation_energy}Segregation energy of a Cr atom in a bcc Fe--Cr alloy from the bulk to (a) the surface layer and (b) the first subsurface layer as a function of the chromium concentration $c_{\\mathrm{Cr}}$. (c) The difference between the surface- and first-subsurface-layer segregation energies. The asterisk is the GPAW \\emph{ab initio} result at $c_{\\mathrm{Cr}}=0$ and the squares are the \\emph{ab initio} results from Ref.~\\cite{Kur2015.PRB92.214113} obtained using the EMTO method. The lines are guides to the eye.}\n\\end{figure}\n\nThe Cr segregation energies to the topmost surface layer ($\\eseg{1}$) and the second topmost layer ($\\eseg{2}$) are shown in Figs.~\\ref{fig:segregation_energy}(a) and ~\\ref{fig:segregation_energy}(b), respectively, as a function of the Cr concentration $c_{\\mathrm{Cr}}$ for both the DFT and the four potential models. Due to computational limitations, the GPAW DFT calculations are limited to zero background Cr concentration. The DFT results for $c_{\\mathrm{Cr}} > 0$ are obtained \nwith a basis set of exact muffin-tin orbitals (EMTO)~\\cite{And1994.incollection,And2000.PRB62.R16219,Vit2007.book} in combination with the coherent potential approximation~\\cite{Sov1967.PR156.809,Vit2001.PRL87.156401}, \nwhich circumvents the need to average over a large number of alloy configurations as done for the potential models. \nNote that only the zero-concentration segregation energies of the new Tersoff potential have been explicitly fitted (with the target values being the GPAW ones); the data for $c_{\\mathrm{Cr}} > 0$ should be regarded as predictions of the model.\n\nThe old Tersoff potential from Ref.~\\cite{Hen2013.JPCM25.445401} is observed to be far away from the \\emph{ab initio} segregation energies for both layers.\nThe CDEAM and 2BEAM potentials give a fairly good fit to the EMTO results for $\\eseg{1}$ but are far off from both the GPAW and EMTO results for $\\eseg{2}$. \nImportantly, as can be observed from Fig.~\\ref{fig:segregation_energy}(c), all three pre-existing models (CDEAM, 2BEAM, and old Tersoff) yield $\\eseg{2}-\\eseg{1} < 0$ at all Cr concentrations shown and thus predict segregation of Cr to the second atomic layer instead of the topmost one. This is in stark contrast to the \\emph{ab initio} results, for which  $\\eseg{2}-\\eseg{1}$ is positive at all investigated concentrations. Fortunately, this shortcoming is fixed by our new Tersoff potential, for which $\\eseg{2}-\\eseg{1} > 0$ in the entire range $0 \\leq c_{\\mathrm{Cr}} \\leq 0.48$ and the fit to the GPAW value of  $\\eseg{2}-\\eseg{1}=\\SI{0.3696}{\\electronvolt}$ at $c_{\\mathrm{Cr}}=0$ is excellent, the relative error being $2.5\\%$.\n\nWe have also investigated the migration of Cr in the vicinity of an Fe(100) surface. \nIn Table~\\ref{table:surface_migration_barriers}, we list the migration energy barriers for the process where a surface Cr atom moves to a vacancy at a nearest-neighbor site in the second layer. Although we do not have a DFT value for this migration energy to compare against, it is worthwhile to note that the 2BEAM potential and the new Tersoff potential both yield \nan energy barrier close to zero \nwhereas the value given by the old Tersoff potential is much higher than the others.\n\n\\begin{table}[tbh]\n\\caption{\\label{table:surface_migration_barriers}Migration energy barriers for a surface Cr atom moving to a nearest-neighbor second-layer vacancy.}\n\\begin{ruledtabular}\n\\begin{tabular}{ld}\nPotential & \\multicolumn{1}{c}{Energy barrier (eV)}  \\\\\n\\cline{1-2}\nCDEAM & 0.2624 \\\\\n2BEAM & 0.0035 \\\\\nOld Tersoff & 1.1866 \\\\\nNew Tersoff & 0.0023 \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\softpekcom{Here we can also present the migration energy barriers as a function of the layer index (cf. Daniel's presentation). No DFT data to compare against, though.}\n\n\n\\section{Discussion}\\label{sec:discussion}\n\nIn summary, we have presented a new interatomic Fe--Cr potential that is suitable for molecular dynamics simulations of surface physics in Fe--Cr alloys. The potential was formulated in the Tersoff formalism, allowing it to be combined with previously developed Fe--C and Cr--C potentials~\\cite{Hen2013.JPCM25.445401} into a model of the stainless-steel system Fe--Cr--C. The new potential parameters were optimized by fitting to a structural database consisting of \\emph{ab initio} results not only for bulk alloys but also for the segregation of Cr atoms \nto the (100) surface of an Fe--Cr alloy. \nWe compared the performance of the new potential to that of three pre-existing Fe--Cr potentials with regard to the fitting database as well as to bulk- and surface-related quantities not involved in the fitting.\n\nFor the bulk properties, the new Tersoff potential was found to perform at the same overall level as the pre-existing EAM potentials considered, namely, the CDEAM and the 2BEAM potentials. On the one hand, the new Tersoff potential provided the closest match of all the tested potentials to our DFT-calculated mixing energies and point-defect energies. On the other hand, the 2BEAM potential was the best performer when it came to the short-range ordering in random Fe--Cr alloys \n(qualitatively reproducing the experimentally observed sign change of the order parameter at $c_{\\mathrm{Cr}} \\approx 0.1$)\nand to the vacancy--Cr diffusion in bcc Fe \n(yielding energy barriers within 6\\% of the DFT values). \nFor the tested bulk data, the new Tersoff potential performed significantly better than the old Tersoff potential from Ref.~\\cite{Hen2013.JPCM25.445401}, even though only the latter was fitted exclusively to bulk quantities.\n\nThe main objective we set at the beginning was\nfor the new potential model\nto outperform \nthe existing models \nby correctly predicting the surface-segregation behavior of Cr in Fe--Cr alloys.\n\nWe achieved this goal in the sense that the new Tersoff potential yields the same ordering of Cr segregation energies as the DFT calculations, namely, $\\eseg{1} < \\eseg{2}$. This is in contrast to the other potential models, for which $\\eseg{1} > \\eseg{2}$ \nand which thus predict Cr to segregate primarily to the second layer instead of the first.\nThe agreement between the new Tersoff potential and the DFT calculations is far from perfect, however. For example, the new Tersoff potential predicts \na sign change in the first-layer segregation energy\nat a significantly smaller Cr concentration ($\\sim 2.5\\%$) than do the DFT-based EMTO calculations ($\\sim 8\\%$)~\\cite{Kur2015.PRB92.214113,Rop2007.PRB76.220401}.\n\nIn light of the above, we conclude that the new Tersoff potential appears to be the best potential for scenarios where both surface and bulk properties of Fe--Cr alloys are of importance. In bulk systems without boundaries, however, the 2BEAM potential is perhaps the most optimal choice, especially if we take into account its lower computational cost compared to the Tersoff potentials. The old Tersoff potential performs worse than the new one in almost all of our tests, and hence \nit seems difficult to justify its future use.\n\n\\begin{acknowledgments}\nThis work was supported by the Academy of Finland (Grant No.~308632). The computational resources granted by the CSC -- IT Center for Science, Finland, and by the Finnish Grid and Cloud Infrastructure project (FGCI; urn:nbn:fi:research-infras-2016072533) are gratefully acknowledged, as are the facilities provided by the Turku University Centre for Materials and Surfaces (MatSurf).\n\\end{acknowledgments}\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sintro}\n\n\nProtoplanetary disks originate from dense cloud material consisting of\nsub-$\\mu$m sized, almost completely amorphous interstellar medium\n(ISM) dust grains \\citep{BE00,LD01,KE04,HG10}. The dust and gas in\nthese disks form the basic matter from which planets may form. At the\nsame time, mineralogical studies of primitive solar system bodies\nsuggest that a considerable fraction of the silicate grains in these\nobjects are of crystalline nature (\\citealt{WO07,PB10}, and references\ntherein). It is then naturally implied that the crystallinity fraction\nincreases, through thermal and chemical modification of these solids\nduring the general planet formation process, commonly referred to in\nthe literature as ``disk evolution''.\n\nAs time passes, the small dust responsible for the infrared (IR)\nexcess observed around young stars is subjected to different processes\nthat affect, and will eventually determine how this progression will\nend. Planets and planetary systems have been observed around hundreds\nof stars other than the Sun, showing that this result is rather common\n\\citep{US07}. IR observations have revealed a great number of {\\it\n  debris disks}, composed of large planetesimal rocks and smaller\nbodies, around a variety of stars spanning a large range in spectral\ntypes and ages \\citep{RI05,BR06,SU06,GA07,CA09}. A few debris disks\nare known to harbor planets (e.g., $\\beta$ Pictoris and Fomalhaut,\n\\citealt{LG10,KA08}), although it is still unclear whether this is\noften true \\citep{KO09}. The majority of main-sequence stars show no\nsigns of planets or debris within the current observational\nlimitations, however, indicating that the disks around such stars at\nthe time of their formation have dissipated completely, leaving no\ndust behind to tell the story. Which processes are important and\ndeterminant for the aftermath of disk evolution are still under\ndebate, and this topic is the subject of many theoretical and\nobservational studies over the last decade, stimulated in large by\nrecent IR and (sub-)millimeter facilities.\n\nSpecifically on the subject of the mineralogical composition, spectra\nfrom the ground and the {\\it Infrared Space Observatory} gave the\nfirst clues of a potential link between crystalline material in\nprotoplanetary disks and comets. A great similarity was noted between\nthe spectra of the disk around the Herbig star HD 100546 and that of\ncomet Hale-Bopp \\citep{CR97,MA98}. More recently, the InfraRed\nSpectrograph (IRS, 5 -- 38 $\\mu$m, \\citealt{HO04}) on-board the {\\it\n  Spitzer Space Telescope} allowed an unprecedented combination of\nhigh sensitivity and the ability to observe large numbers of disks,\ndown to the brown dwarf limit. The shape of the silicate features\nprobed by the IRS spectra at 10 and 20 $\\mu$m is affected by the\ncomposition, size and structure of its emitting dust. Amorphous\nsilicates show broad smooth mid-IR features, while the opacities of\ncrystalline grains show sharp features due to their large-scale\nlattice arrangement, such that even small fractions of crystalline\ngrains produce additional structure in the silicate features\n\\citep{MI05,BO08,JU09,OF10}. Because most protoplanetary disks are\noptically thick at optical and IR wavelengths, the silicate features\nobserved in the mid-IR are generally emitted by dust in the optically\nthin disk surface only. To probe the disk midplane, observations at\nlonger wavelengths are necessary. Additionally, the emission at 10 and\n20 $\\mu$m has been shown to arise from different grain populations,\nprobing different radii \\citep{KE06,OF09,OF10}. While the 10 $\\mu$m\nfeature probes a warmer dust population, at $\\leq$ 1 AU for T Tauri\nstars, the dust emitting at 20 $\\mu$m is colder, further out and\ndeeper into the disk \\citep{KE07}.\n\nTwo methods have been proposed to explain the formation of crystal\ngrains: thermal annealing of amorphous grains or vaporization followed\nby gas-phase condensation. Both methods require high temperatures\n(above $\\sim$1000 K, \\citealt{FA00,GA04}) which is inconsistent with\nouter disk temperatures. However, crystalline grains have been\nobserved in outer, as well as in inner disks \\citep{VB04}. Large-scale\nradial mixing has been invoked to explain the presence of crystals at\nlow temperatures in the outer disk \\citep{BM00,GA04,CI09}. A third\nproposed formation mechanism for crystal formation is that shock waves\ncould locally heat amorphous silicates and crystallize them\n\\citep{DC02,HD02}.\n\nFrom protoplanetary disks to comets, several authors have attempted to\ninfer the dust composition from IRS spectra and laboratory data on\namorphous and crystalline silicate dust, using a variety of analysis\ntechniques. Whether for individual objects\n\\citep{FO04,ME07,PI08,BY08}, for mixed disk samples\n\\citep{BO01,AP05,VB05,BO08,OF09,OF10,JU10}, or systematic studies of\nthe disk population of a given star-forming region\n\\citep{SI09,WA09,ST09}, it has been shown that a significant mass\nfraction of the dust in those disks must be in crystalline\nform. However, the many studies dealing with the mineralogical\ncomposition of dust to date focus on a specific region or object,\nfailing to investigate the hypothesis that the crystallinity fraction\nis a measure of the evolutionary stage of a region. That is, no study\nin the literature has yet investigated an increase of crystallinity\nfraction with cluster age.\n\nMineralogical studies of Solar System bodies show a range of\ncrystallinity fractions. Evidence from primitive chondrites shows that\nthe abundance of crystalline silicate material varies from nearly\nnothing up to 20 -- 30~\\% (e.g. Acfer 094 and ALH77307, \\citealt{PB10}\nand references therein). Oort cloud comets, with long periods and\nlarge distances from the Sun, have inferred crystallinity fractions up\nto 60 -- 80 \\% (e.g. Hale-Bopp, \\citealt{WO99,WO07}). Jupiter-family,\nor short period comets, have lower fractions, up to $\\sim$35 \\%\n(e.g. 9P\/Tempel 1, \\citealt{HA07}; 81P\/Wild 2, \\citealt{ZO06}). This\ndiscrepancy in fractions points to the existence of a radial\ndependence in crystallinity fraction in the protoplanetary disk around\nthe young Sun \\citep{HA05}. It is important to note that those values\nare model dependent, and the use of large amorphous grains (10 -- 100\n$\\mu$m) can lead to systematically lower crystalline fractions\n\\citep{HA02}. This is evident for Hale-Bopp, where \\citet{MI05a} find\na much lower fraction ($\\sim$7.5 \\%) than other authors, using a\ndistribution of amorphous grain sizes up to 100 $\\mu$m. What is clear\nis that even within the discrepancies, the crystallinity fractions\nderived for Solar System bodies are appreciably higher than those\nderived for the ISM dust ($< 2 \\%$, \\citealt{KE04}). Recent {\\it\n  Spitzer} data indicate further similarities between crystalline\nsilicate features seen in comets or asteroids with those seen in some\ndebris disks around solar mass stars \\citep{BE06,LI07,LI08}. One\nproposed explanation is that the observed spectral features in the\ndisk result from the catastrophic break-up of a single large body (a\n`super comet') which creates the small dust particles needed for\ndetection. At the even earlier protoplanetary disk stage, there is\nlimited observational evidence for radial gradients in crystallinity\nfrom mid-infrared interferometry data, with higher crystallinity\nfractions found closer to the young stars \\citep{VB04,SC08}. All of\nthis suggests that the crystallization occurs early in the disk\nevolution and is then incorporated into larger solid bodies.\n\nBesides dust composition, the evolution of grain sizes is an essential\nindicator of disk evolution. The initially sub-$\\mu$m size ISM grains\nmust grow astounding 14--15 orders of magnitude in diameter if they\nare to form planets. If grains were to grow orderly and steadily,\ntheoretical calculations predict disks to have fully dissipated their\nsmall grains within $\\sim$10$^5$ years \\citep{WE80,DD05}. The fact\nthat many disks a few Myr old are observed to have small grains\n\\citep{HE08} poses a serious problem for the paradigm that grain\ngrowth is a steady, monotonic process in disk evolution and planet\nformation. Additionally, small dust has been observed in the surface\nlayers of disks in clusters of different ages and environments for\nhundreds of systems. The implications, as discussed most recently by\n\\citet{OL10} and \\citet{OF10}, is that small grains must be\nreplenished by fragmentation of bigger grains, and that an equilibrium\nbetween grain growth and fragmentation is established. \\citet{OL10}\nhave shown that this equilibrium is maintained over a few million\nyears, as long as the disks are optically thick, and is independent of\nthe population or environment studied.\n\nIn this paper we present a comprehensive study of the mineralogical\ncomposition of disks around stars in young star-forming regions (where\nmost stars are still surrounded by optically thick disks) and older\nclusters (where the majority of disks has already dissipated).\nCorrelating the results on mean size and composition of dust grains\nper region, obtained in a homogeneous way using the same methodology,\nwith the properties of small bodies in our own Solar System can put\nconstraints on some of the processes responsible for disk evolution\nand planet formation. The Serpens Molecular Cloud, whose complete\nflux-limited YSO population has been observed by the IRS instrument\n\\citep{OL10}, is used as a prototype of a young star-forming region,\ntogether with Taurus, the best studied region to date. The sources\nthat have retained their protoplanetary disks in the $\\eta$\nChamaeleontis and Upper Scorpius clusters are used to probe the\nmineralogy in the older bin of disk evolution.\n\nSection~\\ref{sdata} describes the YSO samples in the 4 regions\nmentioned. The {\\it Spitzer} IRS observations and reduction are\nexplained. The spectral decomposition method B2C \\citep{OF10} is\nbriefly introduced in \\S~\\ref{s_spdecomp}, and its results for\nindividual and mean cluster grain sizes and composition are shown in\n\\S~\\ref{sres}. In \\S~\\ref{sdis} the results are discussed in the\ncontext of time evolution. There we demonstrate that no evolution is\nseen in either mean grain sizes or crystallinity fractions as clusters\nevolve from $\\sim$1 to 8 Myr. The implications for disk formation and\ndissipation, and planet formation are discussed. In \\S~\\ref{scon} we\npresent our conclusions.\n\n\n\\section{Spitzer IRS data}\n\\label{sdata}\n\n\nThe four regions presented here were chosen due to the availability of\ncomplete sets of IRS spectra of their IR-excess sources, while\nspanning a wide range of stellar characteristics, environment, mean\nages and disk fractions (the disk fraction of Serpens is still\nunknown, see Table \\ref{t_overview}).\n\nThe IRS spectra of a complete flux-limited sample of young stellar\nobjects (YSO) in the Serpens Molecular Cloud have been presented by\n\\citet{OL10}, based on program ID \\#30223 (PI: Pontoppidan). As\ndetailed there, the spectra were extracted from the basic calibration\ndata (BCD) using the reduction pipeline from the Spitzer Legacy\nProgram ``From Molecular Cores to Planet-Forming Disks'' (c2d,\n\\citealt{LH06}). A similarly large YSO sample in the Taurus\nstar-forming region has been presented by \\citet{FU06}. IRS spectra of\nall 18 members of the $\\eta$ Chamaeleontis cluster were first shown by\n\\citet{SI09}, while the spectra of 26 out of the 35 IR-excess sources\nin the Upper Scorpius OB association were shown by \\citet{DC09} (the\nremaining 9 objects were not known at the time the observations were\nproposed). For the latter 3 regions, the post-BCD data were downloaded\nfrom the SSC pipeline (version S18.4) and then extracted with the\nSpitzer IRS Custom Extraction software (SPICE, version 2.3) using the\nbatch generic template for point sources. As a test, the IRS spectra\nof the YSOs in Serpens were also reduced using SPICE to ensure that\nboth pipelines produce nearly identical results. On visual inspection,\nno discrepancies were found between the results from the two\npipelines, all objects showed the exact same features in both\nspectra. The similarity in outputs is such that the effects on the\nspectral decomposition results are within the cited error bars.\n\nSince the spectral decomposition method applied here aims to reproduce\nthe silicate emission from dust particles in circumstellar disks, the\nsample has been limited to spectra that show clear silicate features.\nThe few sources with PAH emission have been excluded from the\nsample. PAH sources amount to less than 8\\% in low-mass star-forming\nregions \\citep{GE06,OL10}. Furthermore, spectra with very low\nsignal-to-noise ratios (S\/N) are excluded from the analysed sample in\norder to guarantee the quality of the results. In addition, for\nobjects \\#114 and 137 in Serpens, and 04370+2559 and V955Tau in Taurus\nthe warm component fit contributes to most of the spectrum, leaving\nvery low fluxes to be fitted by the cold component. This produces\nlarge uncertainties in the cold component fit, and they are therefore\nnot further used in the analysis. The low S\/N objects rejected amount\nto less than 10~\\% of each of the Serpens and Taurus samples, so the\nstatistical results derived here should not be affected by this\nremoval. The final sample of 139 sources analysed is composed of 60\nobjects in Serpens, 66 in Taurus, 9 objects in Upper Scorpius, and 4\nin $\\eta$ Chamaeleontis. The statistical uncertainties of the spectra\nwere estimated as explained in \\citet{OF09}.\n\nThe great majority of the objects studied here are low-mass stars\n(spectral types K and M, see Table \\ref{t_comp}). The study of\nmineralogical evolution across stellar mass is not the focus of this\npaper. Such a study would require a separate paper, in which the same\ntechniques are used for low- and intermediate-mass stars. Thus, the\nstatistical results derived in the following sections concern T Tauri\nstars, and not necessarily apply to intermediate-mass Herbig Ae\/Be\nstars.\n\n\n\\section{Spectral Decomposition and the B2C method}\n\\label{s_spdecomp}\n\n\nIn order to reproduce the observed IRS spectra of these circumstellar\ndisks the B2C decomposition method, explained in detail and tested\nextensively in \\citet{OF10}, is applied. Two dust grain populations,\nor components, at different temperatures (warm and cold) are used in\nthe method, in addition to a continuum emission. The warm component\nreproduces the 10 $\\mu$m feature, while the cold component reproduces\nthe non-negligible residuals at longer wavelengths, over the full\nspectral range (see Figure \\ref{f_fit}). Each component, warm and\ncold, is the combination of five different dust species and three\ngrain sizes for amorphous silicates or two grain sizes for crystalline\nsilicates.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{f1.ps}\n\\end{center}\n\\caption{\\label{f_fit} Example of the B2C modeling for object \\#15 in\n  Serpens. The black line is the estimated continuum for this\n  source. The red line is the fit to the warm component and the blue\n  line is the fit to the cold component. The green line is the final\n  fit to the entire spectrum. The original spectrum is shown in black\n  with its uncertainties in light grey.}\n\\end{figure}\n\nThe three amorphous species are silicates of olivine stoichiometry\n(MgFeSiO$_4$), silicates of pyroxene stoichiometry (MgFeSiO$_6$), and\nsilica (SiO$_2$). The two crystalline species are both Mg-rich end\nmembers of the pyroxene and olivine groups, enstatite (MgSiO$_3$) and\nforsterite (Mg$_2$SiO$_4$). As further explained in \\citet{OF10}, the\ntheoretical opacities of the amorphous species are computed assuming\nhomogeneous spheres (Mie theory), while those for the crystalline\nspecies use the distribution of hollow spheres (DHS, \\citealt{MI05})\ntheory so that irregularly shaped particles can be simulated.\n\nIn addition, the three grain sizes used are 0.1, 1.5 and 6.0 $\\mu$m,\nrepresenting well the spectroscopic behaviour of very small,\nintermediate-sized and large grains. For the crystalline species,\nhowever, the code is limited to only 2 grain sizes (0.1 and 1.5\n$\\mu$m). This restriction is imposed because large crystalline grains\nare highly degenerate with large amorphous grains (as can be seen in\nFigure 1 of \\citealt{OF10}), and because the production of large 6.0\n$\\mu$m pure crystals is not expected via thermal annealing\n\\citep{GA04}.\n\nThe B2C method itself consists of three steps. First, the continuum is\nestimated and subtracted from the observed spectrum. The adopted\ncontinuum is built by using a power-law plus a black-body at\ntemperature $T_{\\rm cont}$. The power-law represents the mid-IR tail\nof emission from the star and inner disk rim. The black-body is\ndesigned to contribute at longer wavelengths, and is therefore\nconstrained to be less than 150 K. Each dust component is then fitted\nseparately to the continuum-subtracted spectrum.\n\nThe second step is to fit the warm component to reproduce the 10\n$\\mu$m silicate feature between $\\sim$7.5 and 13.5 $\\mu$m. This is\ndone by summing up the 13 mass absorption coefficients ($N_{\\rm\n  species}$ = 5, $N_{\\rm sizes}$ = 3 or 2, for amorphous and\ncrystalline species, respectively), multiplied by a black-body\n$B_{\\nu} (T_{\\rm w})$ at a given warm temperature $T_{\\rm w}$.\n\nThe third step is to fit the residuals, mostly at longer wavelengths,\nover the entire spectral range (5 -- 35 $\\mu$m). This is done in a\nsimilar manner, for a given cold temperature $T_{\\rm c}$. The final\nfit is a sum of the three fits described, as can be seen in Figure\n\\ref{f_fit}. The entire fitting process is based on a Bayesian\nanalysis, combined with a Monte Carlo Markov chain, in order to\nrandomly explore the space of free parameters. The resulting mean\nmass-average grain size is the sum of all sizes fitted, each size\nbeing weighted by their corresponding masses, as:\n\n\\begin{tiny}\n\\begin{eqnarray}\n  \\langle a_{\\rm warm\/cold} \\rangle = \n  \\left(\\sum_{j=1}^{\\mathrm{N_{\\rm sizes}}}  a_{j} \n    \\sum_{i=1}^{\\mathrm{N_{\\rm species}}}  \n    M_{{\\rm w\/c},i}^{j} \\right)\n  \\times \\left( \\sum_{j=1}^{\\mathrm{N_{\\rm sizes}}} \n    \\sum_{i=1}^{\\mathrm{N_{\\rm species}}}  \n    M_{{\\rm w\/c},i}^{j} \\right)^{-1}\n  \\label{eq:meana}\n\\end{eqnarray}\n\\end{tiny}\nwhere $a_1 = 0.1\\,\\mu$m (small grains), $a_2 = 1.5\\,\\mu$m\n(intermediate-sized grains) and $a_3 = 6\\,\\mu$m (large grains).\nFurther details and tests of the B2C procedure can be found in\n\\citet{OF10}. That paper also demonstrates that the procedure is\nrobust for statistical samples, and that the relative comparisons\nbetween samples, which are the focus of this paper, should not suffer\nfrom the assumptions that enter in the procedure. The robustness of\nthe procedure is evaluated by fitting synthetic spectra, and is\ndiscussed in detail in their Appendix A. The influence of the\ncontinuum estimate is also discussed, especially for the cold\ncomponent for both grain sizes and crystallinity fractions, and it is\nshown that prescriptions that do not use large 6 $\\mu$m grains (which\nare, to some degree, degenerate with the continuum) give fits that are\nnot so good.\n\nFor the amorphous grains, the B2C procedure uses the Mie scattering\ntheory to compute mass absorption coefficients. However, \\citet{MI07}\nfound that they could best reproduce the extinction profile toward the\ngalactic center using the DHS scattering theory, with a maximum\nfilling factor of 0.7. The most striking difference between Mie and\nDHS mass absorption coefficients is seen for the O--Si--O bending mode\naround 20 $\\mu$m. Here we investigate the influence of the use of DHS\ninstead of Mie for amorphous grain with an olivine or pyroxene\nstoichiometry. We conducted tests on a sub-sample of 30 objects (15 in\nSerpens and 15 in Taurus). The conclusion of such tests is that it has\na small influence on the quantities we discuss in this study. For the\nwarm component of the 30 objects, we find a change in the mean\ncrystallinity fraction of -1.6\\% (the mean crystallinity for this\nsub-sample using DHS is 9.3\\% versus 10.9\\% using Mie), which is in\nthe range of uncertainties claimed in this study. We also computed the\nmean slope of grain size distributions to gauge the effect of using\nDHS on grain sizes. On average, the grain size distribution indices\nare steeper by $\\sim$0.2 (with a mean slope of -3.01 for this\nsub-sample using DHS versus -2.80 using Mie). Therefore, our main\nconclusions are preserved for the warm component. Concerning the cold\ncomponent, the inferred crystallinity fraction using DHS is 22.5\\%\nversus 15.1\\% with Mie, a mean increase of 7.4\\%. For the mean slope\nof grain size distributions, a negligible decrease is found (-3.07\nusing DHS versus -3.01 for Mie). Again, the differences found are\nwithin our significant errors for the cold component and do not change\nany of our conclusions.\n\nIt is important to note that the S\/N generally degrades at longer\nwavelengths when compared to shorter wavelengths. The lower S\/N\nreflect on the cold component fits and will most likely result in\nlarger uncertainties. We evaluate that the fits to the cold component\nare reliable and add important information on the dust mineralogy\n(albeit with larger uncertainties) and thus those results are included\nin the following discussion.\n\n\n\\section{Results}\n\\label{sres}\n\n\nThe IRS spectra of the 139 YSOs with IR excess discussed in\n\\S~\\ref{sdata} were fitted with the B2C spectral decomposition\nprocedure. The relative abundances derived for all objects are shown\nin Appendix \\ref{sabun}. The S\/N drops considerably for the long\nwavelength module of some of the objects studied (including all\nobjects in Upper Scorpius and $\\eta$ Chamaeleontis). For this reason,\nthe cold component could not be satisfactorily fitted and no results\nfor this component are presented for these sources (see Appendix\n\\ref{sabun}).\n\nDue to the large number of objects, these results allow statistical\nstudies on both the mineralogy and size distribution of the grains\nthat compose the optically thin surface layers of disks in each\ncluster studied. The mean abundances of each species per region are\npresented in Table \\ref{t_mean_comp}, where it can be seen that the\nmajority of the dust studied is of amorphous form. In Table\n\\ref{t_mean} the mean mass-average grain sizes and crystallinity\nfractions per region are shown. Mean sizes are in the range 1 -- 3\n$\\mu$m, without significant difference between regions. These results\nare discussed in detail in the following sections.\n\nIt is important to note that the comparison of results derived here\nfor the different regions is valid because the same method, with exact\nsame species, is used for all sources. The comparison of samples\nanalyzed in distinct ways can lead to differences in results that do\nnot correspond to real differences in composition. Nevertheless, in\n\\S~\\ref{scomp} the results presented here are compared to literature\nresults for the same objects, when available, with generally good\nagreement.\n\n\n\\subsection{Grain Sizes}\n\\label{s_gsize}\n\n\nThe mean mass-averaged grain sizes for the warm ($\\langle a_{\\rm warm}\n\\rangle$) and cold ($\\langle a_{\\rm cold} \\rangle$) components are\nshown in Figure \\ref{f_size}, for Serpens and Taurus (for the objects\nin Upper Sco and $\\eta$ Cha no results for the cold component are not\navailable, see Appendix \\ref{sabun}). It is seen that the two clouds\noverlap greatly, and that the grain sizes derived from the different\ntemperature components do not seem to correlate. To quantify this\ncorrelation, a Kendall $\\tau$ correlation coefficient can be computed\ntogether with its associated probability $P$ (between 0 and 1). $\\tau$\n= 1(-1) defines a perfect correlation (anti-correlation), and $\\tau$ =\n0 means that the datasets are completely independent. A small $P$, on\nthe other hand, testifies to how tight the correlation is. For the\nwarm and cold mean mass-averaged grain sizes for both clouds, $\\tau$\nis found to be 0.14, with $P$ = 0.07. This lack of correlation\nindicates that different processes are likely responsible for\nregulating the size distribution at different radii \\citep{OF10}.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{f2.ps}\n\\end{center}\n\\caption{\\label{f_size} Mass-averaged mean grain sizes for the warm\n  ($\\langle a_{\\rm warm} \\rangle$) and cold ($\\langle a_{\\rm cold}\n  \\rangle$) components. Black dots are the objects in Serpens, and red\n  squares are the objects in Taurus.}\n\\end{figure}\n\nAlthough the average grain size in the warm component is bigger than\nthat in the cold component within a given star-forming region, as\nshown in Table \\ref{t_mean}, this difference is mostly not\nsignificant. However, Figures \\ref{f_size} and \\ref{f_sizedist}\nclearly show a difference between the range of grain sizes spanned in\nboth components, with $\\langle a_{\\rm cold} \\rangle$ never reaching\nnear the biggest grain size modeled (6.0 $\\mu$m) for any object. A\npossible explanation for larger grains at smaller radii, suggested by\n\\citet{ST09}, is that grains coagulate faster in the inner disk where\ndynamical timescales are shorter.  However as discussed by\n\\citet{OL10} and in \\S~\\ref{sover}, the mean dust size at the disk\nsurface is not regulated by grain growth alone, but also by\nfragmentation and vertical mix. This means that faster coagulation at\nsmaller radii cannot be uniquely responsible for bigger grains in the\ninner disk. Future modeling should try to understand this difference\nin mean grain sizes observed.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{f3.ps}\n\\end{center}\n\\caption{\\label{f_sizedist} Distribution of mass-averaged mean grain\n  sizes for the warm ($\\langle a_{\\rm warm} \\rangle$, left panel) and\n  cold ($\\langle a_{\\rm cold} \\rangle$, right panel) components. Due\n  to the low number statistics, the objects in Upper Sco and $\\eta$\n  Cha have been merged together as an older cluster.}\n\\end{figure}\n\nFurthermore, Serpens and Taurus occupy an indistinguishable locus in\nFigure \\ref{f_size}, explicitly seen in Figure \\ref{f_sizedist}. A two\nsample Kolmogorov-Smirnov test (KS-test) was performed and the results\nshow that the null hypothesis that the two distributions come from the\nsame parent population cannot be rejected to any significance\n(14\\%). The older regions, although lacking statistical significance,\nshow a distribution of mass-average grain sizes in the same range\nprobed by the young star-forming regions (Figure\n\\ref{f_sizedist}). This supports the evidence that the size\ndistribution of the dust in the surface layers of disks is\nstatistically the same independent of the population studied\n\\citep{OL10}. \n\nThe results here confirm those from \\citet{OF10} that the mean\ndifferential grain size distributions slope for the three grain sizes\nconsidered are shallower than the reference MRN differential size\ndistribution ($\\alpha$ = -3.5). The mean grain size distributions\nslopes ($\\alpha$) for each region can be found in Table \\ref{t_mean}.\n\n\n\\subsection{Disk Geometry}\n\\label{s_dgeo}\n\n\nThe amount of IR-excess in a disk is directly related to its geometry\n\\citep{KE87,GM01,DU01}. Specifically using the IRS spectra, disk\ngeometry can be inferred from the flux ratio between 30 and 13 $\\mu$m\n($F_{30}\/F_{13}$, \\citealt{JB07,OL10,ME10}). A flared geometry ($1.5\n\\lesssim F_{30}\/F_{13} \\lesssim 5$), with considerable IR excess and\nsmall dust, allows the uppermost dust layers to intercept stellar\nlight at both the inner and outer disk. For flat disks ($F_{30}\/F_{13}\n\\lesssim 1.5$) with little IR excess, only the inner disk can easily\nintercept the stellar radiation as the outer disk is\nshadowed. Moreover, cold or transitional disks are interesting objects\nthat present inner dust gaps or holes, producing a region with little\nor no near-IR excess ($5 \\lesssim F_{30}\/F_{13} \\lesssim 15$). It is\ninteresting to explore the effect of disk geometry on both the mean\nmass-average grain sizes and crystallinity fractions of the disks\nstudied.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{f4.ps}\n\\end{center}\n\\caption{\\label{f_f30} Top: Flaring index $F_{30}\/F_{13}$, used as a\n  proxy for disk geometry, versus warm (left) and cold (right)\n  mass-averaged mean grain sizes. Bottom: $F_{30}\/F_{13}$ versus warm\n  (left) and cold (right) crystallinity fractions. The YSOs in Serpens\n  (black dots), Taurus (red squares), Upper Sco (blue stars), and\n  $\\eta$ Cha (green triangles) are compared. }\n\\end{figure}\n\nFigure \\ref{f_f30} shows $F_{30}\/F_{13}$ as a proxy for disk geometry\ncompared with the mean mass-averaged grain sizes and crystallinity\nfractions for both components and all regions studied here. No\npreferential grain size (correlation coefficient $\\tau$ = -0.14, $P$ =\n0.02, and $\\tau$ = 0.07, $P$ = 0.33 for warm and cold components,\nrespectively) nor crystallinity fraction ($\\tau$ = 0.09, $P$ = 0.10\nfor the warm, and $\\tau$ = -0.19, $P$ = 0.01 for the cold component)\nis apparent for any given disk geometry. Similar scatter plots result\nfor the mean mass-average grains sizes for only amorphous ($\\tau$ =\n-0.12, $P$ = 0.08 for the warm, and $\\tau$ = 0.13, $P$ = 0.11 for the\ncold component), or only crystalline grains ($\\tau$ = 0.08, $P$ = 0.17\nfor the warm, and $\\tau$ = -0.13, $P$ = 0.10 for the cold\ncomponent). Furthermore, no clear separation is seen between the\ndifferent regions studied. The statistically relevant samples in\nSerpens and Taurus define a locus where the majority of the objects is\nlocated in each plot, which is followed by the lower number statistics\nfor older regions. Figure \\ref{f_f30} therefore shows not only that\ngrain size and crystallinity fraction are not a function of disk\ngeometry, but also that younger and older regions show similar\ndistributions of those two parameters.\n\n\n\\subsection{Crystallinity Fraction}\n\\label{s_cryst}\n\n\nThe crystallinity fractions derived from the warm and cold components\n($C_{\\rm Warm}$ and $C_{\\rm Cold}$, respectively) for Serpens and\nTaurus are show in Figure \\ref{f_cryst}. No strong trend of warm and\ncold crystallinity fractions increasing together is seen ($\\tau$ =\n0.10, with $P$ = 0.10 for the entire sample). This fact implies that,\nif an unique process is responsible for the crystallization of dust at\nall radii, this process is not occurring at the same rate in the\ninnermost regions as further out in the disk. This is opposite to the\nconclusion of \\citet{WA09}, who derive a correlation between inner and\nouter disk crystallinity from the simultaneous presence of the 11.3\nand 33 $\\mu$m features. The opacities of the crystalline species are\nmore complex than those two features alone, making the analysis here\nmore complete than that of \\citet{WA09}. Our finding that the fraction\nof crystalline material in disk surfaces varies with radius can\nconstrain some of the mechanisms for formation and distribution of\ncrystals.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{f5.ps}\n\\end{center}\n\\caption{\\label{f_cryst} Crystalline fraction of the warm and cold\n  components in Serpens (black dots) and Taurus (red squares).}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{f6.ps}\n\\end{center}\n\\caption{\\label{f_warmdist} Distribution of crystalline fractions for\n  Serpens (top), Taurus (middle), and Upper Sco and $\\eta$ Cha\n  combined (bottom). Similar distributions and the same range of\n  fractions are seen for all clusters.}\n\\end{figure}\n\nA wider spread in crystallinity fraction is observed for the cold\ncomponent than for the warm component (Figure \\ref{f_warmdist}), which\nis reflected in the mean crystallinity fractions for each sample\n(Table \\ref{t_mean}). This discrepancy could be real, or an artifact\ndue to the signal-to-noise ratio (S\/N) being frequently lower at\nlonger wavelengths (cold component) than that at shorter wavelengths\n(warm component), introducing a larger scatter. The difference in\nSerpens is more significant ($\\langle C_{\\rm warm} \\rangle \\simeq$\n11.0\\% and $\\langle C_{\\rm cold} \\rangle \\simeq$ 17.5\\%). The left\npanel of Figure \\ref{f_cum} shows the cumulative fractions as\nfunctions of crystallinity fractions. Despite small differences\nbetween the warm (red line) and cold (blue line) components, the two\ncumulative fractions have similar behavior. If this difference is\ntrue, there is a small fraction of T Tauri disks with a higher cold\n(outer) than warm (inner) crystallinity fraction. This finding\ncontrasts with that derived by \\citet{VB04} for the disks around 3\nHerbig stars. Their spatially resolved observations infer higher\ncrystallinity fractions in the inner than in the outer disks, albeit\nbased on only 10 $\\mu$m data. A larger sample of objects with good S\/N\nincluding both 10 and 20 $\\mu$m data is needed to better constrain\nthis point. In addition, Figure \\ref{f_warmdist} shows that younger\nand older clusters have similar distributions of crystallinity\nfractions.\n\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{f7.ps}\n\\end{center}\n\\caption{\\label{f_cum} Left: Cumulative fractions of the crystallinity\n  fractions, for Serpens (dashed line) and Taurus (dotted\n  line). Right: Cumulative fraction of the ration between the\n  forsterite and enstatite fractions, for Serpens (dashed line) and\n  Taurus (dotted line). The warm component is shown in red while the\n  cold component is blue.}\n\\end{figure}\n\nAs discussed by many authors, both the grain size and the degree of\ncrystallinity affect the silicate features, therefore it is\ninteresting to search for trends between these two parameters. In\nFigure \\ref{f_warm}, the mass-average grain sizes are compared to the\ncrystallinity fraction for both warm (left panel) and cold (right\npanel) components. No obvious trends are seen in either component,\nneither any separation between regions. This result supports the\ndiscussion of \\citet{OF10} that whatever processes govern the mean\ngrain size and the crystallinity in disks, they are independent from\neach other.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{f8.ps}\n\\end{center}\n\\caption{\\label{f_warm} Mass-averaged mean grain sizes versus the\n  crystalline fraction for Serpens (black dots), Taurus (red squares),\n  Upper Sco (blue stars), and $\\eta$ Cha (green triangles).}\n\\end{figure}\n\n\n\\subsubsection{Enstatite vs. Forsterite}\n\\label{scomp2}\n\n\nThe disk models of \\citet{GA04} consider chemical equilibrium of a\nmixture of solid and gas at high temperatures, allowing radial mixing\nof material. These models predict a predominance of forsterite in the\ninnermost regions of the disk, while enstatite dominates at lower\ntemperatures (being converted from forsterite). From the observational\npoint of view, data on disks around T Tauri \\citep{BO08} and Herbig\nAe\/Be stars \\citep{JU10} have shown the opposite trend: enstatite is\nmore concentrated in the inner disk, while forsterite dominates the\ncolder, outer disk region. \\citet{BO08} interpret this result as a\nradial dependence of the species formation mechanisms, or a\nnon-equilibrium of the conditions under which the species formed,\ncontrary to the models assumptions.\n\nFor the regions presented in this study, it can be seen in Table\n\\ref{t_mean_comp} for mean cluster values and in Table \\ref{t_comp}\nfor individual objects that the results derived from this study\ngenerally follow those of \\citet{BO08}, with more enstatite in the\nwarm component and, to a lesser extent, more forsterite in the cold\ncomponent. The right panel of Figure \\ref{f_cum} illustrates this for\nthe cumulative fraction of the forsterite over enstatite ratios for\nindividuals disks. However, this trend is not very significant given\nthe uncertainties.\n\n\n\\subsection{The Silicate Strength-Shape Relation}\n\\label{ssil}\n\n\nA correlation between the shape and the strength of the 10 $\\mu$m\nsilicate feature from disks has been discussed extensively in the\nliterature \\citep{VB03,KE06,OF09,PA09,OL10,OF10}. Synthetic 10 $\\mu$m\nfeatures generated for different grain sizes and compositions have\nbeen shown to fit well with observations, yielding grain size as the\nimportant parameter responsible for such a relationship. The degree of\ncrystallinity of the dust also plays a role on the shape of this\nfeature. However, as clearly shown for EX Lup \\citep{AB09}, and\nsupported by models \\citep{MI08,OF09}, an increase in crystallinity\nfraction does not change the strength of the feature, even though its\nshape does change. Crystallinity is then understood as responsible for\nthe scatter in the strength-shape relationship, and not the\nrelationship itself. As a result, the strength and shape of the 10\n$\\mu$m silicate feature yield the typical size of the grains in the\nupper layers of the disk at a few AU from the star \\citep{KE07}. The\ntop panel of Figure \\ref{f_s10} shows the results for Serpens, Taurus,\nUpper Sco and $\\eta$ Cha. The bottom panel presents the median values\nper region, indicating the 15 -- 85 percentile ranges of the\ndistributions. Overlaid are the models of \\citet{OF09} for different\ngrain sizes (0.1 -- 6.0 $\\mu$m) generated for amorphous silicates of\nolivine and pyroxene stoichiometry, and a 50:50 mixture. The\ndifference in mean ages does not correspond to a significant\ndifference in mean grain sizes between the different regions.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{f9.ps}\n\\end{center}\n\\caption{\\label{f_s10} Top: The ratio of normalized fluxes at 11.3 to\n  9.8 $\\mu$m ($S_{11.3}\/S_{9.8}$) is plotted against the peak at 10\n  $\\mu$m ($S^{10\\mu{\\rm m}}_{{\\rm peak}}$) for Serpens (black dots),\n  Taurus (red squares), Upper Scorpius (blue stars), $\\eta$\n  Chamaeleontis (green triangles). Bottom: Squares show the median\n  values and crosses indicate the 15 -- 85 percentile ranges of the\n  distributions (top panel). Colored curves are derived from\n  theoretical opacities for different mixtures by \\citet{OF09}. The\n  open circles correspond to different grain sizes, from left to right\n  6.25, 5.2, 4.3, 3.25, 2.7, 2.0, 1.5, 1.25, 1.0 and 0.1 $\\mu$m. }\n\\end{figure}\n\nWith the mean grain sizes derived from the spectral decomposition, it\nis possible to further explore the validity of using the strength of\nthe 10 $\\mu$m silicate feature to trace the sizes of grains in the\nsurfaces of disks. The left panel of Figure \\ref{f_s10_2} shows the\ncorrelation between $\\langle a_{\\rm warm} \\rangle$ and $S^{10\\mu{\\rm\n    m}}_{{\\rm peak}}$ for all 4 samples. The Kendall $\\tau$\ncoefficient of -0.29, $P$ = 0.01 supports the effectiveness of\n$S^{10\\mu{\\rm m}}_{{\\rm peak}}$ as a proxy for grain sizes, with\nsmaller values of $S^{10\\mu{\\rm m}}_{{\\rm peak}}$ implying larger\ngrain sizes.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{f10.ps}\n\\end{center}\n\\caption{\\label{f_s10_2} Left panel: Strength of the 10 $\\mu$m silicate\n  feature ($S^{10\\mu{\\rm m}}_{{\\rm peak}}$) versus the mass-averaged\n  mean grain size for the warm component. Right panel: Strength of the\n  10 $\\mu$m silicate feature versus crystalline fraction for the warm\n  component. The best fit relationships are shown for reference. }\n\\end{figure}\n\nOn the other hand, it is also possible to test how the degree of\ncrystallinity can influence the strength of the 10 $\\mu$m silicate\nfeature. The lack of correlation between $C_{\\rm warm}$ and\n$S^{10\\mu{\\rm m}}_{{\\rm peak}}$ ($\\tau$ = -0.07, $P$ = 0.14), shown in\nthe right panel of Figure \\ref{f_s10_2} for all samples, supports that\nthe degree of crystallinity is not the dominant parameter setting the\nstrength of the 10 $\\mu$m silicate feature. These results argue\nagainst the results of \\citet{ST09} that find a high crystallinity\nfraction and small grains fitting low strengths of the 10 $\\mu$m\nsilicate feature. Although it may be possible to fit a few spectra\nwith a certain prescription, a good model should be able to explain\nthe robust relationship between the strength and shape of the 10\n$\\mu$m silicate feature observed for large numbers of disks. Despite\nthe many processes able to change the shape or the strength of this\nfeature, only grain size has so far demonstrated capability to explain\nthe observed trend. Our conclusion is that $S^{10\\mu{\\rm m}}_{{\\rm\n    peak}}$ and dust sizes are appropriately correlated.\n\n\n\\subsection{Comparison with other studies}\n\\label{scomp}\n\n\nDust composition results are available in the literature for the disks\nin Taurus and $\\eta$ Cha (see Table \\ref{t_lit} for an overview).\n\\citet{SI09} present their analysis in $\\eta$ Cha considering the same\n5 dust species and three grain sizes (enstatite in their model is the\nonly species for which only the 2 smaller grain sizes are considered),\nbut for a distribution of temperatures derived using the Two Layer\nTemperature Distribution (TLTD, \\citealt{JU09}) decomposition\nprocedure. For the same 4 objects, their mean amorphous fraction is\n80.1 $\\pm$ 9.3 \\%. This result is consistent with the 82.8 $\\pm$ 12.9\n\\% mean amorphous fraction found here. The mean crystalline fractions\nderived are 18.4 $\\pm$ 10.7 \\% with TLTD and 17.1 $\\pm$ 12.8 \\%\nderived here.\n\n\\citet{ST09} present their decomposition procedure for 65 YSOs in\nTaurus. This method also takes into consideration a warm and a cold\ntemperature, and makes use of two amorphous species (olivine and\npyroxene) with two grain sizes (small and large), and 3 crystalline\nspecies (enstatite, forsterite and crystalline silica) of a single\nsize. Their mean warm amorphous fraction is 82.9 $\\pm$ 19.3 \\% and\nwarm crystalline fraction is $17.1 \\pm 19.3$ \\%, while here the\nderived fractions are 89.0 $\\pm$ 6.6 \\% and 10.9 $\\pm$ 6.6 \\% for the\nwarm amorphous and crystalline fractions, respectively. For the cold\ncomponent, \\citet{ST09} derive a mean cold amorphous fraction of 77.3\n$\\pm$ 19.9 \\% and cold crystalline fraction of 22.6 $\\pm$ 19.9 \\%,\nwhile here the values are 85.9 $\\pm$ 10.6 \\% and 13.9 $\\pm$ 10.5 \\%,\nrespectively. The consistently lower amorphous (higher crystalline)\nfractions found by \\citet{ST09} could be a result of their choice to\nuse silica in crystalline rather than amorphous form (as used\nhere).\n\n\n\\section{Discussion}\n\\label{sdis}\n\n\\subsection{Dust Characteristics}\n\\label{sover} \n\nSection \\ref{sres} has shown that the disk populations in the four\nregions presented here, young and older, have very similar\ndistributions in the two main dust parameters: grain size and\ncomposition. The large number of objects in the two young regions\nstudied occupy a region in parameter space of either grain size or\ncrystallinity fraction that is also populated by the small number of\nolder disks. The grain sizes derived for the cold component never\nreach the biggest grain size modeled (6 $\\mu$m), different from the\nwarm component results that span the entire range in sizes. The\ncrystallinity fraction does not seem to be correlated with mean grain\nsize, warm or cold. Whatever processes are responsible for the\ncrystallization of the initially amorphous grains, they should not\nonly be independent from the processes that govern the grain size\ndistribution, but they should also be able to work on bigger amorphous\ngrains. Alternatively, the crystalline lattice should be able to keep\nitself regular during the coagulation of small crystalline dust to\ncreate big crystalline grains.\nThe correlation between the strength of the 10 $\\mu$m feature and the\nmean grain size in disk surfaces, combined with the lack of\ncorrelation between crystallinity fraction and $S^{10\\mu{\\rm m}}_{{\\rm\n    peak}}$, supports the wide usage of $S^{10\\mu{\\rm m}}_{{\\rm\n    peak}}$ as a proxy for dust size in literature\n\\citep{VB03,KE06,PA09}.\n\n\\citet{BO08} found a strong correlation between disk geometry and the\nstrength of the 10 $\\mu$m silicate feature for a very small sample of\nT Tauri stars (7 disks), which points to flatter disks having\nshallower 10 $\\mu$m features (i.e., big grains in the disk\nsurface). Using results from similar decomposition procedures,\n\\citet{OF10} and \\citet{JU10} confirm this trend for larger samples of\nT Tauri (58 disks) and Herbig Ae\/Be stars (45 disks),\nrespectively. Those trends are much weaker than that found by\n\\citet{BO08}, showing a larger spread. For the current even larger\nsample (139 disks), no significant trend is seen, indicating that the\nearlier small sample trends may have been affected by a few\noutliers. This result is similar to that found by \\citet{OL10} for a\nlarge YSO sample ($\\sim$~200 objects) using the strength of the 10\n$\\mu$m silicate feature as a proxy for grain size (Figure 14 in that\npaper). As discussed by Oliveira et al., the sedimentation models of\n\\citet{DD08} expect a strong correlation of larger grains in flatter\ndisks that is not seen. This means that sedimentation alone cannot be\nresponsible for the distribution of mean grain sizes in the upper\nlayers of protoplanetary disks around T Tauri stars. Furthermore, the\nlack of correlation between crystallinity fraction and disk geometry\nis not in support of the results of \\citet{WA09} and \\citet{ST09}, who\nfind a link between increasing crystallinity fraction and dust\nsedimentation.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{f11.ps}\n\\end{center}\n\\caption{\\label{f_grain} Left: Mean mass-average grain sizes vs. disk\n  fraction. Serpens is not included because its disk fraction is not\n  yet known. Filled circles represent mean warm grain sizes, and open\n  triangles represent mean cold grain sizes. Error bars for the mean\n  mass-average grain sizes are estimated using a Monte Carlo approach,\n  sampling the errors of the individual objects. Right: Mean grain\n  sizes vs. mean cluster age. Filled circles represent results for the\n  warm component, while open triangles represent the cold\n  component. The black points are YSOs in Serpens, red in Taurus, blue\n  in Upper Sco and green in $\\eta$ Cha. }\n\\end{figure}\n\nAs discussed in \\citet{OL10} for Serpens and Taurus, and confirmed by\nthe addition of considerably older samples, there is no clear\ndifference in the mean grain sizes in the disk surfaces with mean\ncluster age, which can be seen in Figure \\ref{f_grain}. This evidence\nsupports the discussion in that paper that the dust population\nobserved in the disk surface cannot be a result of a progressive,\nmonotonic change of state from small amorphous grains, to large, more\ncrystalline grains, or `grain growth and processing'. The fact that\nthe distribution of grain sizes in the upper layers of disks does not\nchange with cluster age implies that an equilibrium of the processes\nof dust growth and fragmentation must exist, which also supports the\nexistence of small grains in disks that are millions of years old whereas\ndust growth is a rapid process \\citep{WE80,DD05}. That small dust is\nstill seen in disks in older regions like Upper Sco and $\\eta$ Cha\nargues that this equilibrium of processes is maintained for millions\nof years, as long as the disks are optically thick, but independent of\nthem having a flared or flatter geometry.\n\n\n\\subsection{Evolution of Crystallinity with Time?}\n\\label{sevol}\n\n\nLiterature studies of disk fractions of different YSO clusters with\ndifferent mean ages show a trend of decreasing disk fraction,\ni.e. disks dissipating with time, over some few millions of years\n\\citep{HA01,HE08}. This decrease is clearly confirmed by the lower\nfraction of disks still present in the older regions studied here\n(Upper Sco and $\\eta$ Cha). According to current planet formation\ntheories, if giant planets are to be formed from gas rich disks, the\noptically thin, gas-poor disks in those older regions should already\nharbor (proto-)planets. Considering the evidence from small bodies in\nour own Solar System that suggest considerably higher crystallinity\nfractions than ISM dust (see \\citealt{WO07} and \\citealt{PB10} for\nreviews of latest results), a crystallinity increase must occur.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{f12.ps}\n\\end{center}\n\\caption{\\label{f_comp2} Left: Crystallinity fraction vs. disk\n  fraction. Serpens is not included because its disk fraction is not\n  yet known. Filled circles represent mean warm crystallinity, and\n  open triangles represent mean cold crystallinity. Uncertainty for\n  crystallinity fractions are estimated using a Monte Carlo approach,\n  sampling the errors of the individual objects. Right: Crystallinity\n  fraction vs. mean cluster age. Filled circles represent mean warm\n  crystallinity, and open triangles represent mean cold\n  crystallinity. The black points are YSOs in Serpens, red in Taurus,\n  blue in Upper Sco and green in $\\eta$ Cha. }\n\\end{figure}\n\nIn Figure \\ref{f_comp2}, the mean crystallinity fraction per region is\nplotted against two evolutionary parameters: disk fraction (left) and\nmean age (right). Within the spread in individual fractions it is seen\nthat, just as for grain sizes, there is no strong evidence of an\nincrease of crystallinity fraction with either evolutionary parameter.\nThis implies that there is no evolution in grain sizes or\ncrystallinity fraction for the dust in the surface of disks over\ncluster ages in the range 1 -- 8 Myr, as probed by the observations\npresented here. Essentially, there is no change in these two\nparameters until the disks disperse. Starting from the assumption that\ninitially the dust in protoplanetary disks is of ISM origin\n(sub-$\\mu$m in size and almost completely amorphous), it appears that\na modest level of crystallinity is established in the disk surface\nearly in the evolution ($\\leq$ 1 Myr) and then reaches some sort of\nsteady state, irrespective of what is taking place in the disk\nmidplane. Thus, the dust in the upper layers of disks does not seem to\nbe a good tracer of the evolution that is taking place in the disk\ninterior, where dust is growing further for the formation of\nplanetesimals and planets, at many times higher crystallinity\nfractions, to be consistent with evidence from Solar system bodies.\n\nIf this is the case, within 1 Myr this surface dust must be\ncrystallized to the observed fraction ($\\sim$10 -- 20~\\%). This result\nputs constraints on the formation of circumstellar disks. One\npossibility is that this crystallization of the dust in disks mostly\noccurs during the embedded phase. In this early stage of star\nformation, where large quantities of material are still accreting\ntowards the protostar, a fraction of the infalling material comes very\nclose to the protostar and is heated to temperatures $>$800 K before\nit moves outwards in the disk. Alternatively, accretion shocks or\nepisodic heating events could be responsible for thermally annealing\nthe dust in the disk surface.\n\nThe 2-D models of \\citet{VD10} treat the radial evolution of crystals\nin time. According to these models, 100~\\% of the dust in the inner\ndisk ($\\leq$ 1 AU) is crystallized within 1 Myr. With time, the inner\ndisk crystalline fraction drops as the disk spreads, and crystalline\nmaterial is transported to outer parts of the disk. These models can\nhelp explain the rapid crystallization required to account for our\nresults. However, the models do show a decrease in inner disk ($\\leq$\n1 AU) crystallinity fraction with time, which is not supported by our\nresults. Since these models do not discriminate on vertical structure,\nbut rather present crystallinity fractions that are integrated over\nall heights at a given radius, this decrease in crystallinity fraction\nis not necessarily connected to the surface of the disk. Thus the\ndecrease in crystallinity fraction with time found in the models of\n\\citet{VD10} could be explained as a decrease in crystallinity\nfraction just in the disk midplane where the bulk of the mass resides,\nbut not in the surface layers, as our data indicate. That would imply\nthat radial mixing of these crystals is more efficient than vertical\nmixing, which is responsible for the crystallinity fraction decrease\nin the disk midplane.\n\nAccording to the models of \\citet{CI07} for outward transport of high\ntemperature materials, variations in radial transport dynamics with\nheight produce vertical gradients in the crystalline fractions, such\nthat the upper layers of the disk will have lower crystallinity\nfractions than the midplane population. If that is the case, the\nobservations discussed here, which probe the disk surface only, lead\nto lower limits on the real crystalline fraction of disk midplanes. In\nthis scenario, planets (and comets) forming in the disk midplane would\nhave higher crystalline abundances than those derived here for the\ndisk surfaces, which are compatible with what has been observed in our\nSolar System. However, this model does not make predictions for the\ntime evolution of the systems. Combining the vertical and radial\nmixing processes with evolutionary models such as those of\n\\citet{VD10} are needed to investigate whether older and younger disks\ncould still show the same distribution of crystallinity fractions in\nthe upper layers of disks, as observed here.\n\n\n\\section{Conclusions}\n\\label{scon}\n\n\nThis paper presents the spectral decomposition of Spitzer\/IRS spectra\nusing the B2C decomposition model of \\citet{OF10}. Mineralogical\ncompositions and size distributions of dust grains in the surface\nlayers of protoplanetary disks are derived for 139 YSOs belonging to\nfour young star clusters using the same method.\n\nSerpens and Taurus are used as prototypes of young regions, where most\nstars are still surrounded by disks, while Upper Sco and $\\eta$ Cha\nrepresent the older bin of disk evolution, where a large fraction of\nthe disks have already dissipated but some massive protoplanetary\ndisks are left. The large number of objects analyzed allows\nstatistical results that point to the main processes that affect the\ngrain size distribution and composition of dust in protoplanetary\ndisks. Furthermore, the usage of the same analysis method for regions\nof different mean ages allow a study of evolution of the dust\nparameters with time.\n\nOur large sample does not show a preferential grain size or\ncrystallinity fraction with disk geometry, contrary to earlier\nanalyses based on smaller samples. Also, younger and older regions\nhave very similar distributions. The difference between mean\nmass-averaged grain sizes for the warm and cold components of a given\nstar-forming region is small, however a considerable difference is\nseen between the ranges of grain sizes spanned in both components. The\ncold mass-averaged grain sizes never reach the biggest size modelled\n(6 $\\mu$m) while the warm mass-averaged grain sizes span the entire\nrange of sizes modeled. The crystallinity fractions derived for inner\n(warm) and outer (cold) disks are typically 10 -- 20\\%{}, and not\ncorrelated. The cold crystallinity fraction shows a larger spread than\nthe warm. No strong difference is seen between the overall mean warm\nand cold crystallinity fraction. Within the crystalline dust\npopulation, more enstatite is found in the warm component and more\nforsterite in the cold component. The differences are not very\nsignificant, however.\n\nThe results of the spectral decomposition support the usage of the\nstrength of the 10 $\\mu$m silicate feature ($S^{10\\mu{\\rm m}}_{{\\rm\n    peak}}$) as a proxy of the mean grain size of dust in the disk\nsurface. This is supported by the correlation between $S^{10\\mu{\\rm\n    m}}_{{\\rm peak}}$ and mean grain size and lack of correlation with\nthe mean crystallinity.\n\nMean cluster ages and disk fractions are used as indicators of the\nevolutionary stage of the different populations. Our results show that\nthe different regions have similar distributions of mean grain sizes\nand crystallinity fractions regardless of the spread in mean ages of 1\n-- 8 Myr. Thus, despite the fact that the majority of disks dissipate\nwithin a few Myr, the surface dust properties do not depend on age for\nthose disks that have not yet dissipated in the 1 -- 8 Myr range. This\npoints to a rapid change in the composition and crystallinity of the\ndust in the early stages ($\\leq$ 1 Myr) that is maintained essentially\nuntil the disks dissipate.\n\n\n\\acknowledgements Astrochemistry at Leiden is supported by a Spinoza\ngrant from the Netherlands Organization for Scientific Research (NWO)\nand by the Netherlands Research School for Astronomy (NOVA)\ngrants. This work is based on observations made with the Spitzer Space\nTelescope, which is operated by the Jet Propulsion Laboratory,\nCalifornia Institute of Technology under a contract with NASA. Support\nfor this work was provided by NASA through an award issued by\nJPL\/Caltech.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nEver since L. Gross proved that a logarithmic Sobolev inequality is equivalent to hypercontractivity of the associated semigroup (see \\cite{G}), these inequalities have been the subject of much research and interest.  They have proved extremely useful as a tool in the control of the rate of convergence to equilibrium of spin systems, and were extensively studied (see for example \\cite{B-H1},\\cite{G-Z}, \\cite{Led}, \\cite{O-R}, \\cite{S-Z1},\\cite{Yo2}, \\cite{Ze}).  Up until recently, however, most of the attention has been focused on the case of elliptic generators, for which there are some very powerful methods for proving such inequalities.  Our aim here is to show that a certain class of infinite dimensional measures corresponding to non-elliptic H\\\"ormander type generators satisfy logarithmic Sobolev inequalities.\n\nOne method that exists for proving coercive inequalities such as the logarithmic Sobolev inequality and the spectral gap inequality, as well as gradient bounds (which are closely related) involves showing that the so-called $CD(\\rho, \\infty)$ condition holds (see \\cite{A-B-C-F-G-M-R-S}, \\cite{Bakry}).  Indeed, let $L$ be the generator of a Markov semigroup $P_t$, and define the operators\n\n\\begin{align*}\n\\Gamma(f,f) &= \\frac{1}{2}\\left(L(f^2) - 2fLf\\right)\\\\\n\\Gamma_2(f,f) &= \\frac{1}{2}\\left[L\\Gamma(f,f) - 2\\Gamma(f, Lf)\\right].\n\\end{align*}\nWe say that the $CD(\\rho, \\infty)$ holds when there exists $\\rho\\in\\mathbb{R}$ such that\n\\[\n\\Gamma_2(f,f) \\geq \\rho \\Gamma(f,f).\n\\]\nWhen $L$ is elliptic such a condition holds in many situations.  In the case when $M$ is a complete connected Riemannian manifold, and $\\nabla$ and $\\Delta$ are the standard Riemannian gradient and Laplace-Beltrami operators, taking $L=\\Delta$, the condition reads\n\\[\n|\\nabla\\nabla f|^2 + \\mathrm{Ric}(\\nabla f, \\nabla f) \\geq \\rho|\\nabla f|^2.\n\\]\nThis holds for some $\\rho\\in\\mathbb{R}$ when $M$ is compact, or for $\\rho =0$ when $M=\\mathbb{R}^n$ with the usual metric, since $\\mathrm{Ricci}=0$.\n\nHowever, in this paper we will consider non-elliptic H\\\"ormander generators. For such generators these methods do not work, since the $CD(\\rho, \\infty)$ condition does not hold.  Indeed, the Ricci tensor of our generators can be thought of as being $-\\infty$ almost everywhere.  \n\nWe consider an $N$-dimensional lattice and impose interactions between points in the lattice described by an unbounded quadratic potential.  In the standard case where the underlying space is Euclidean, the $CD(\\rho, \\infty)$ condition allows us to prove that the finite dimensional measures on the lattice, which depend on the boundary conditions, satisfy logarithmic Sobolev inequalities uniformly on the boundary conditions.  It is then possible to pass to the infinite dimensional measure.  We aim for a comparable result in a more complicated sub-Riemannian setting, using different methods.\n\nIn \\cite{L-Z} a similar situation is studied, in that the authors consider a system of H\\\"ormander generators in infinite dimensions and prove logarithmic Sobolev inequalities as well as some ergodicity results.  The main difference between the present set up and their situation is that we consider a non-compact underlying space, namely the Heisenberg group, in which the techniques of \\cite{L-Z} cannot be applied.\n\n\n\n\n\\section{Logarithmic Sobolev inequalities on the Heisenberg group}\n\\label{Logarithmic Sobolev inequalities on the Heisenberg group}\nWe consider the Heisenberg group, $\\mathbb{H}$, which can be described as $\\mathbb{R}^3$ with the following group operation:\n$$x\\cdot\\tilde{x} = (x_1, x_2, x_3)\\cdot(\\tilde{x}_1, \\tilde{x}_2, \\tilde{x}_3) = (x_1 + \\tilde{x}_1, x_2 + \\tilde{x}_2, x_3 + \\tilde{x}_3 + \\frac{1}{2}(x_1\\tilde{x}_2 - x_2\\tilde{x}_1)).$$\n$\\mathbb{H}$ is a Lie group, and its Lie algebra $\\mathfrak{h}$ can be identified with the space of left invariant vector fields on $\\mathbb{H}$ in the standard way.  By direct computation we see that this space is spanned by \n\\begin{eqnarray}\nX_1 &=& \\partial_{x_1} - \\frac{1}{2}x_2\\partial_{x_3} \\nonumber \\\\\nX_2 &=& \\partial_{x_2} + \\frac{1}{2}x_1\\partial_{x_3} \\nonumber \\\\\nX_3 &=& \\partial_{x_3} = [X_1, X_2]. \\nonumber\n\\end{eqnarray}\nFrom this it is clear that $X_1, X_2$ satisfy the H\\\"ormander condition (i.e. $X_1, X_2$ and their commutator $[X_1, X_2]$ span the tangent space at every point of $\\mathbb{H}$).  It is also easy to check that the left invariant Haar measure (which is also the right invariant measure since the group is nilpotent) is the Lebesgue measure $dx$ on $\\mathbb{R}^3$. \n\n$\\mathbb{H}$ is naturally equipped with a 1-parameter family of automorphisms $\\{\\delta_\\lambda\\}_{\\lambda>0}$ defined by\n\\[\n\\delta_\\lambda(x) := \\left(\\lambda x_1, \\lambda x_2, \\lambda^2 x_3\\right).\n\\]\n$\\{\\delta_\\lambda\\}_{\\lambda>0}$ is called a family of \\textit{dilations}.  Thus $\\mathbb{H}$ is an example of a homogeneous Carnot group (see \\cite{B-L-U} for an extensive study of such groups).\n \nOn $C^\\infty_0(\\mathbb{H})$, define the \\textit{sub-gradient} to be the operator given by\n$$\\nabla := (X_1,X_2)$$ \nand the \\textit{sub-Laplacian} to be the second order operator given by \n$$\\Delta := X_1^2 + X_2^2.$$\n$\\nabla$ can be treated as a closed operator from $L^2(\\mathbb{H}, dx)$ to $L^2(\\mathbb{H}; \\mathbb{R}^2, dx)$.  Similarly, since $\\Delta$ is densely defined and symmetric in $L^2(\\mathbb{H}, dx)$, we may treat $\\Delta$ as a closed self-adjoint operator on $L^2(\\mathbb{H}, dx)$ by taking the Friedrich extension.  \n\nWe introduce the logarithmic Sobolev inequality on $\\mathbb{H}$ in the following way.\n\n\\begin{definition}\nLet $q\\in(1,2]$, and let $\\mu$ be a probability measure on $\\mathbb{H}$.  $\\mu$ is said to satisfy a \\textit{$q$-logarithmic Sobolev inequality} $(LS_q)$ on $\\mathbb{H}$ if there exists a constant $c>0$ such that for all smooth functions $f: \\mathbb{H}\\to \\mathbb{R}$\n\\begin{equation}\n\\label{LSq}\n\\mu\\left(|f|^q\\log\\frac{|f|^q}{\\mu|f|^q}\\right) \\leq c\\mu\\left(|\\nabla f|^q\\right)\n\\end{equation}\nwhere $\\nabla$ is the sub-gradient on $\\mathbb{H}$.\n\\end{definition}\n\n\\begin{rem}\nThe $(LS_q)$ was introduced in \\cite{B-L} and further studied in \\cite{B-Z}, as a variation of the more standard $(LS_2)$ inequality.  Here it is noted that for $q<2$, the $(LS_q)$ inequality serves as a certain sharpening of $(LS_2)$, at least when the underlying space is finite dimensional.\n\\end{rem}\n\n\\begin{rem}\n\\label{tensorisation}\nWe recall four important standard properties of $(LS_q)$ inequalities that will be used below (see \\cite{B-Z} and \\cite{G-Z}):\n\\begin{itemize}\n\\item[\\rm{(i)}] {\\rm$(LS_q)$ is stable under tensorisation:} Suppose $\\mu_1$ and $\\mu_2$ satisfy $(LS_q)$ inequalities with constants $c_1$ and $c_2$ respectively.  Then $\\mu_1\\otimes\\mu_2$ satisfies an $(LS_q)$ inequality with constant $\\max\\{c_1, c_2\\}$.\n\\item[\\rm{(ii)}] {\\rm$(LS_q)$ is stable under bounded perturbations:} Suppose $d\\mu = \\frac{e^{-U}}{Z}dx$ satisfies an $(LS_q)$, and that $W$ is bounded.  Then $\\tilde\\mu(dx) = \\frac{e^{-U - W}}{\\tilde Z}dx$ satisfies an $(LS_q)$ inequality.\n\\item[\\rm{(iii)}] {\\rm$(LS_q) \\Rightarrow (SG_q)$:} Suppose $\\mu$ satisfies an $(LS_q)$ inequality with constant $c$.  Then $\\mu$ satisfies a $q$-spectral gap inequality (we say $\\mu$ satisfies an $(SG_q)$ inequality) with constant $\\frac{4c}{\\log2}$ i.e. \n\\[\n\\mu\\left|f - \\mu f\\right|^q \\leq \\frac{4c}{\\log2} \\mu\\left(|\\nabla f|^q\\right)\n\\]\nfor all smooth $f$.\n\\item[\\rm{(iv)}] When the underlying space is finite dimensional, $(LS_q) \\Rightarrow (LS_{q'})$ and $(SG_q) \\Rightarrow (SG_{q'})$ for $q<q'$.\n\\end{itemize}\n\\end{rem} \n\nThe operator $\\Delta$ is non-elliptic, but by H\\\"ormander's theorem it is hypoelliptic, so that the associated heat semigroup $P_t$ has a smooth convolution kernel with respect to the Haar measure.  Note that we can calculate the $\\Gamma$ and $\\Gamma_2$ functions for this generator explicitly, and we can easily see that there does not exist a constant $\\rho\\in\\mathbb{R}$ such that $\\Gamma_2\\geq\\rho\\Gamma$.  However, despite this, in \\cite{B-B-B-C} and \\cite{Li} it was recently shown that there exists a constant $C$ such that\n\\[\n|\\nabla P_t f|(x) \\leq CP_t(|\\nabla f|)(x), \\qquad \\forall x\\in \\mathbb{H},\n\\]\nwhich is a surprising result.  It follows directly from this gradient bound that the heat kernel measure on $\\mathbb{H}$ satisfies an $(LS_2)$ inequality on $\\mathbb{H}$.\n\n\\begin{rem}\nSince we have the sub-gradient on the right-hand side, \\eqref{LSq} is a logarithmic Sobolev inequality corresponding to a H\\\"ormander type generator.  Indeed,  if $\\mu(dx) = \\frac{e^{-U}}{Z}dx$ then it is clear that $\\mathcal{L} = \\Delta - \\nabla U.\\nabla$ is a Dirichlet operator satisfying\n\\[\n\\mu\\left(f\\mathcal{L}f\\right) = - \\mu\\left(|\\nabla f|^2\\right),\n\\]\nwhere $\\Delta$ is the sub-Laplacian, and $\\nabla$ the sub-gradient.\n\\end{rem}\n\nIndependently, and by very different methods, in \\cite{H-Z} the authors were able to show that a related class of measures on $\\mathbb{H}$ satisfy $(LS_q)$ inequalities (see Theorem \\ref{hz} below).  To describe these we first need to introduce the natural distance function on $\\mathbb{H}$, which is the so-called \\textit{Carnot-Carath\\'eodory distance}.  This distance is more natural than the usual Euclidean one, since it takes into account the extra structure that the Heisenberg group posseses.  \n\nWe define the Carnot-Carath\\'eodory distance between two points in $\\mathbb{H}$ by considering only \\textit{admissible} curves between them.  A Lipschitz curve $\\gamma:[0,1]\\to\\mathbb{H}$ is said to be \\textit{admissible} if $\\gamma'(s) = a_1(s)X_1(\\gamma(s)) + a_2(s)X_2(\\gamma(s))$ almost everywhere with measurable coefficients $a_1, a_2$ i.e. if $\\gamma'(s) \\in sp\\{X_1(\\gamma(s)),X_2(\\gamma(s))\\}$ a.e.  Then the length of $\\gamma$ is given by\n$$l(\\gamma) = \\int_0^1\\left(a_1^2(s) + a_2^2(s)\\right)^{1\/2}ds$$\nand we define the Carnot-Carath\\'eodory distance between two points $x,y\\in\\mathbb{H}$ to be \n$$d(x,y) := \\inf\\{l(\\gamma):\\gamma\\ \\textrm{is an admissible path joining}\\ x\\ \\textrm{to}\\ y\\}.$$\nWrite $d(x)=d(x,e)$, where $e$ is the identity.\n\n\n\\begin{rem}\nThis distance function is well defined as a result of Chow's theorem, which states that every two points in $\\mathbb{H}$ can be joined by an admissible curve (see for example \\cite{B-L-U},\\cite{Grom}).\n\\end{rem}\n \nIt is well know that $d$ is a homogeneous norm on $\\mathbb{H}$ i.e. $d(\\delta_\\lambda(x)) = \\lambda d(x)$ for all $\\lambda>0$, where $\\delta_\\lambda$ is the dilation as defined above (see for example \\cite{B-L-U}).\n \nGeodesics are smooth, and are helices in $\\mathbb{R}^3$.  They have an explicit parameterisation.  For details see \\cite{B-B-B-C}, \\cite{Beals}, \\cite{Bel}, \\cite{Monti}.\nWe also have that $x=(x_1, x_2, x_3)\\mapsto d(x)$ is smooth for $(x_1, x_2) \\neq 0$, but not at points $(0,0, x_3)$, so that the unit ball has singularities on the $x_3$-axis (one can think of it as being 'apple' shaped).\n\nIn our analysis, we will frequently use the following two results.  The first is the well-known fact that the Carnot-Carath\\'eodory distance satisfies the eikonal equation (see for example \\cite{Monti}):\n\n\\begin{proposition}\n\\label{eik}\nLet $\\nabla$ be the sub-gradient on $\\mathbb{H}$.  Then $|\\nabla d(x)|=1$ for all $x=(x_1, x_2, x_3)\\in\\mathbb{H}$ such that $(x_1, x_2)\\neq0$.\n\\end{proposition}\n\nWe must be careful in dealing with the notion of $\\Delta d$, since it will have singularities on the $x_3$-axis.  However, the following (proved in \\cite{H-Z}) provides some control of these singularities.\n\n\\begin{proposition}\n\\label{sub-laplacian lem}\nLet $\\Delta$ be the sub-Laplacian on $\\mathbb{H}$. There exists a constant $K$ such that $\\Delta d \\leq \\frac{K}{d}$ in the sense of distributions.\n\\end{proposition}\n\n\\begin{proof}\nFor the sake of completeness, we recall part of the proof given in \\cite{H-Z}.  It suffices to show that $\\Delta d \\leq K$ on $\\{d(x) =1\\}$.  Indeed, using dilations and homogeneity, we have that \n\\[\n\\Delta d(x)=\\lambda\\Delta d(\\delta_\\lambda(x))\n\\]\nfor all $x\\neq0, \\lambda>0$, so that for any $x\\in\\mathbb{H}\\backslash\\{0\\}$\n\\begin{equation}\n\\label{laplacian estimate}\n\\Delta d(x) \\leq \\frac{1}{d(x)}\\sup_{\\{d(y) = 1\\}}\\Delta d(y).\n\\end{equation}\nSince everything is smooth away from the $x_3$ axis, in order to prove that  $\\Delta d \\leq K$ on $\\{d(x) =1\\}$, it suffices to look at what happens in a small neighbourhood of $(0,0, z)$, where $z$ is such that $d\\left((0,0,z)\\right)=1$.\nTo do this, let\n\\[\nA_\\eta := \\left\\{(r,s) \\in \\mathbb{R}^2: s>0, r>-\\eta s\\right\\},\n\\]\n and for $x = (x_1, x_2, x_3)\\in\\mathbb{H}$ write $\\|x\\| := \\left(x_1^2 + x_2^2\\right)^{1\/2}$.  Then it is shown that there exists $\\eta>0$ and a smooth function $\\psi(r,s)$ defined on $A_\\eta$ such that for $x=(x_1, x_2, x_3)\\in \\mathbb{H}$,\n\\[\nd(x) = \\psi(\\|x\\|, |x_3|),\n\\]\nand moreover that $\\partial_r\\psi<0$ when $r=0$.  One can then compute that\n\\begin{equation}\n\\label{sub-laplacian}\n\\Delta d(x) = \\frac{1}{\\|x\\|}\\partial_r\\psi(\\|x\\|, |x_3|) + \\partial_r^2\\varphi(\\|x\\|, |x_3|) + \\frac{\\|x\\|^2}{4}\\partial_{s}\\varphi(\\|x\\|, |x_3|).\n\\end{equation}\nFrom \\eqref{sub-laplacian} it follows that $\\Delta d$ is bounded from above in a small neighbourhood of $(0,0,z)$, since although the first term is unbounded, it is negative.\n\\end{proof}\n\nThe following result is also found in \\cite{H-Z}.\n\n\\begin{theorem}\n\\label{hz}\nLet $\\mu_p$ be the probability measure on $\\mathbb{H}$ given by \n\\[\n\\mu_p (dx) = \\frac{e^{-\\beta d^p(x)}}{\\int_\\mathbb{H} e^{-\\beta d^p(x)}dx}dx\n\\]\nwhere $p\\geq2$, $\\beta >0$, $dx$ is the Lebesgue measure on $\\mathbb{R}^3$ and $d(x)$ is the Carnot-Carath\\'edory distance.  Then $\\mu_p$ satisfies an $(LS_q)$ inequality, where $\\frac{1}{p}+\\frac{1}{q} = 1$.\n\\end{theorem}\n\nIn the remainder of this paper we use the methods contained in \\cite{H-Z}, together with an iterative procedure based on ideas contained in \\cite{G-Z}, \\cite{Led}, \\cite{Ze2} and \\cite{Ze} to prove an $(LS_q)$ inequality for a class of infinite dimensional measures on $(\\mathbb{H})^{\\mathbb{Z}^N}$. \n\n\n\n\n\n\n\n\n\n\n\\section{Infinite dimensional setting and main result}\n\\label{Infinite dimensional setting}\n\\paragraph{\\textit{The Lattice:}}\nLet $\\mathbb{Z}^N$ be the $N$-dimensional square lattice, for some fixed $N\\in\\mathbb{N}$.  We equip $\\mathbb{Z}^N$ with the $l^1$ lattice metric $dist(\\cdot,\\cdot)$, defined by\n\\[\ndist(i,j) := \\sum_{l=1}^N|i_l - j_l|\n\\]\nfor $i=(i_1,\\dots, i_N), j=(j_1, \\dots, j_N) \\in \\mathbb{Z}^N$.  For $i,j\\in\\mathbb{Z}^N$ we will also write \n\\[\ni\\sim j \\qquad \\Leftrightarrow \\qquad dist(i,j) = 1\n\\]\ni.e. $i\\sim j$ when $i$ and $j$ are nearest neighbours in the lattice.\n\n\nFor $\\Lambda \\subset \\mathbb{Z}^N$, we will write $|\\Lambda|$ for the cardinality of $\\Lambda$, and $\\Lambda \\subset \\subset \\mathbb{Z}^N$ when $|\\Lambda|<\\infty$.\n  \n\\paragraph{\\textit{The Configuration Space:}}\nLet $\\Omega = (\\mathbb{H})^{\\mathbb{Z}^N}$ be the \\textit{configuration space}.  We introduce the following notation.  Given $\\Lambda \\subset \\mathbb{Z}^N$ and $\\omega = (\\omega_i)_{i\\in\\mathbb{Z}^N}\\in\\Omega$, let $\\omega_\\Lambda := (\\omega_i)_{i\\in\\Lambda} \\in\\mathbb{H}^\\Lambda$ (so that $\\omega\\mapsto \\omega_\\Lambda$ is the natural projection of $\\Omega$ onto $\\mathbb{H}^\\Lambda$).  \n\n\nLet $f\\colon \\Omega \\to \\mathbb{R}$.  Then for $i\\in\\mathbb{Z}^N$ and $\\omega \\in \\Omega$ define $f_i(\\cdot|\\omega)\\colon\\mathbb{H} \\to \\mathbb{R}$ by\n$$f_i(x|\\omega) := f(x\\bullet_i\\omega)$$\nwhere the configuration $x\\bullet_i\\omega\\in \\Omega$ is defined by declaring its $i$th coordinate to be equal to $x\\in\\mathbb{H}$ and all the other coordinates coinciding with those of $\\omega\\in\\Omega$.  Let $C^{(n)}(\\Omega)$, $n\\in\\mathbb{N}$ denote the set of all functions $f$ for which we have $f_i(\\cdot|\\omega) \\in C^{(n)}(\\mathbb{H})$ for all $i\\in\\mathbb{Z}^d$ .  For $i\\in\\mathbb{Z}^N, k\\in\\{1,2\\}$ and $f\\in C^{(1)}(\\Omega)$, define\n$$X_{i,k}f(\\omega) := X_kf_i(x|\\omega)\\vert_{x = \\omega_i},$$\nwhere $X_1, X_2$ are the left invariant vector fields on $\\mathbb{H}$ defined in section \\ref{Logarithmic Sobolev inequalities on the Heisenberg group}.\n\n\nDefine similarly $\\nabla_if(\\omega) := \\nabla f_i(x|\\omega)\\vert_{x = \\omega_i}$ and $\\Delta_if(\\omega) := \\Delta f_i(x|\\omega)\\vert_{x = \\omega_i}$ for suitable $f$, where $\\nabla$ and $\\Delta$ are the sub-gradient and the sub-Laplacian on $\\mathbb{H}$ respectively.  For $\\Lambda \\subset \\mathbb{Z}^N$, set $\\nabla_\\Lambda f = (\\nabla_if)_{i\\in\\Lambda}$ and\n$$|\\nabla_\\Lambda f|^q := \\sum_{i\\in\\Lambda}|\\nabla_if|^q.$$\nWe will write $\\nabla_{\\mathbb{Z^d}} = \\nabla$, since it will not cause any confusion.\n\nFinally, a function $f$ on $\\Omega$ is said to be \\textit{localised} in a set $\\Lambda\\subset\\mathbb{Z}^N$ if $f$ is only a function of those coordinates in $\\Lambda$.\n\n\\paragraph{\\textit{Local Specification and Gibbs Measure:}}\nLet $\\Phi = (\\phi_{\\{i,j\\}})_{\\{i,j\\}\\subset\\mathbb{Z}^N, i\\sim j}$ be a family of $C^2$ functions such that $\\phi_{\\{i,j\\}}$ is localised in $\\{i,j\\}$.  Assume that there exists an $M\\in(0,\\infty)$ such that $\\|\\phi_{\\{i,j\\}}\\|_\\infty \\leq M$ and $\\|\\nabla_i\\nabla_j\\phi_{\\{i,j\\}}\\|_\\infty\\leq M$ for all $i,j\\in\\mathbb{Z}^N$ such that $i\\sim j$.  We say $\\Phi$ is a bounded potential of range 1.  For $\\omega\\in\\Omega$, define\n$$H_\\Lambda^\\omega(x_\\Lambda) = \\sum_{\\substack{\\{i,j\\}\\cap\\Lambda\\neq\\emptyset \\\\ i\\sim j}}\\phi_{\\{i,j\\}}(x_i, x_j),$$\nfor $x_\\Lambda = (x_i)_{i\\in\\Lambda} \\in \\mathbb{H}^\\Lambda$, where the summation is taken over couples of nearest neighbours $i\\sim j$ in the lattice with at least one point in $\\Lambda$, and where $x_i = \\omega_i$ for $i\\not\\in \\Lambda$.\n\nNow let $(\\mathbb{E}^{\\omega}_\\Lambda)_{\\Lambda\\subset\\subset\\mathbb{Z}^N,\\omega\\in\\Omega}$ be the local specification defined by\n\\begin{equation}\n\\label{local spec}\n\\mathbb{E}^{\\omega}_\\Lambda(dx_\\Lambda)=\\frac{e^{-U^{\\omega}_\\Lambda(x_\\Lambda)}}{\\int e^{-U^{\\omega}_\\Lambda(x_\\Lambda)}dx_\\Lambda} dx_{\\Lambda}\\equiv\\frac{e^{-U^{\\omega}_\\Lambda(x_\\Lambda)}}{Z^\\omega_\\Lambda} dx_{\\Lambda}\n\\end{equation}\nwhere $dx_\\Lambda$ is the Lebesgue product measure on $\\mathbb{H}^\\Lambda$ and \n\\begin{equation}\n\\label{hamiltonian}\nU^{\\omega}_\\Lambda (x_\\Lambda) = \\alpha\\sum_{i\\in\\Lambda}d^p(x_i) + \\varepsilon\\sum_{\\substack{ \\{i,j\\}\\cap\\Lambda \\neq \\emptyset \\\\ i\\sim j}}(d(x_i) + \\rho d(x_j))^{2} + \\theta H_\\Lambda^\\omega(x_\\Lambda),\n\\end{equation}\nfor $\\alpha>0$, $\\varepsilon, \\rho, \\theta \\in\\mathbb{R}$, and $p\\geq2$, where as above  $x_i = \\omega_i$ for $i\\not\\in \\Lambda$.\n\n\\begin{rem}\n\\label{indices}\nIn the case when $p=2$, we must have that $\\varepsilon> -\\frac{\\alpha}{2N}$ to ensure that $\\int e^{-U_\\Lambda}dx_\\Lambda <\\infty$.\n\\end{rem}\n\nWe define an infinite volume \\textit{Gibbs measure} $\\nu$ on $\\Omega$ to be a solution of the (DLR) equation:\n\\[\n\\nu \\mathbb{E}^{\\cdot}_\\Lambda f = \\nu f\n\\]\nfor all bounded measurable functions $f$ on $\\Omega$.  $\\nu$ is a measure on $\\Omega$ which has $\\mathbb{E}^{\\omega}_\\Lambda$ as its finite volume conditional measures.\n\n~\n\nThe main result of this paper is the following:\n\n\\begin{theorem}\n\\label{main}\nLet $\\nu$ be a Gibbs measure corresponding to the local specification defined by \\eqref{local spec} and \\eqref{hamiltonian}.  Let $q$ be dual to $p$ i.e. $\\frac{1}{p}+\\frac{1}{q} =1$ and suppose $\\varepsilon\\rho>0$, with $\\varepsilon>-\\frac{\\alpha}{2N}$ if $p=2$. Then there exists $\\varepsilon_0, \\theta_0>0$ such that for $|\\varepsilon|<\\varepsilon_0$ and $|\\theta|<\\theta_0$, $\\nu$ is unique and satisfies an $(LS_q)$ inequality i.e. there exists a constant $C$ such that\n\\[\n\\nu\\left(|f|^q\\log\\frac{|f|^q}{\\nu |f|^q}\\right) \\leq C\\nu\\left(\\sum_i|\\nabla_if|^q\\right)\n\\]\nfor all $f$ for which the right-hand side is well defined.\n\\end{theorem}\n\nWe briefly mention some consequences of this result.  The first follows directly from Remark \\ref{tensorisation} part (iii).\n\n\\begin{corollary}\nLet $\\nu$ be as in Theorem \\ref{main}.  Then $\\nu$ satisfies the $q$-spectral gap inequality.  Indeed\n\\[\n\\nu\\left|f-\\nu f\\right|^q \\leq \\frac{4C}{\\log2}\\nu\\left(\\sum_i|\\nabla_if|^q\\right)\n\\]\nwhere $C$ is as in Theorem \\ref{main}.\n\\end{corollary}\n\nThe proofs of the next two can be found in \\cite{B-Z}.\n\n\\begin{corollary}\nLet $\\nu$ be as in Theorem \\ref{main} and suppose $f:\\Omega \\to \\mathbb{R}$ is such that $\\||\\nabla f|^q\\|_\\infty  < 1$. Then\n\\[\n\\nu\\left( e^{\\lambda f}\\right) \\leq \\exp\\left\\{\\lambda \\nu(f) +   \\frac{C}{q^q(q-1)}\\lambda^q\\right\\}\n\\]\nfor all $\\lambda>0$ where $C$ is as in Theorem \\ref{main}.  Moreover, by applying Chebyshev's inequality, and optimising over $\\lambda$, we arrive at the following `decay of tails' estimate\n\\[\n\\nu\\left\\{\\left|f - \\int fd\\nu\\right| \\geq h\\right\\} \\leq 2\\exp\\left\\{-\\frac{(q-1)^p}{C^{p-1}}h^p\\right\\}\n\\]\nfor all $h>0$, where $\\frac{1}{p} + \\frac{1}{q} =1$.\n\\end{corollary}\n\n\\begin{corollary}\nSuppose that our configuration space is actually finite dimensional, so that we replace $\\mathbb{Z}^N$ by some finite graph $G$, and $\\Omega = (\\mathbb{H})^G$.  Then Theorem \\ref{main} still holds, and implies that if $\\mathcal{L}$ is a Dirichlet operator satisfying\n\\[\n\\nu\\left(f\\mathcal{L} f\\right) = -\\nu\\left(|\\nabla f|^2\\right),\n\\]\nthen the associated semigroup $P_t = e^{t\\mathcal {L}}$ is ultracontractive.\n\\end{corollary}\n\n\\begin{rem}\nIn the above we are only considering interactions of range $1$, but we can easily extend our results to deal with the case where the interaction is of finite range $R$.\n\\end{rem}\n\n\\section{Results for the single site measure}\n\\label{results for single site measure}\nThe aim of this section is to show that the single site measures\n\\[\n\\mathbb{E}^{\\omega}_{\\{i\\}}(dx_i) =: \\mathbb{E}^{\\omega}_i(dx_i) = \\frac{e^{-U^{\\omega}_i(x_i)}}{Z^{\\omega}_i}dx_i, \\qquad i\\in\\mathbb{Z}^N,\n\\]\nsatisfy an $(LS_q)$ inequality uniformly on the boundary conditions $\\omega\\in\\Omega$.  We will often drop the $\\omega$ in the notation for convenience.  The work is strongly motivated by the methods of Hebisch and Zegarlinski described in \\cite{H-Z}.\n\\begin{theorem}\n\\label{uniform LSq}\nLet $\\frac{1}{q} +\\frac{1}{p}=1$, and $\\varepsilon\\rho>0$ with $\\varepsilon>-\\frac{\\alpha}{2N}$ if $p=2$.  Then there exists a constant $c$, independent of the boundary conditions $\\omega \\in \\Omega$ such that\n\\[\\mathbb{E}^{\\omega}_i\\left(|f|^q\\log\\frac{|f|^q}{\\mathbb{E}_i^\\omega|f|^q}\\right) \\leq c \\mathbb{E}^{\\omega}_i(|\\nabla_if|^q)\\]\nfor all smooth $f:\\Omega \\to \\mathbb{R}$.\n\\end{theorem}\n\nIt will be convenient to work with alternative measures to the ones defined above.  Indeed, if we can prove uniform $(LS_q)$ inequalities for the single site measures when $\\theta=0$ (so that we no longer have the the bounded interaction term in \\eqref{hamiltonian}), then by Remark \\ref{tensorisation} (ii), which states that $(LS_q)$ inequalities are stable under bounded perturbations, Theorem \\ref{uniform LSq} will hold.  Moreover, it is clear that\n\\[\n\\frac{e^{-\\alpha d^p(x_i) - \\varepsilon\\sum_{j:j\\sim i}(d(x_i) + \\rho d(\\omega_j))^{2}}}{\\int e^{-\\alpha d^p(x_i) - \\varepsilon\\sum_{j:j\\sim i}(d(x_i) + \\rho d(\\omega_j))^{2}}dx_i} = \\frac{e^{-\\tilde{U}^\\omega_i}}{\\int e^{-\\tilde{U}^\\omega_i}dx_i},\n\\]\nwhere \n$$\n\\tilde{U}_i(x_i) = \\alpha d^p(x_i) + 2N\\varepsilon d^{2}(x_i) + 2\\varepsilon\\rho d(x_i)\\sum_{j:j\\sim i}d(\\omega_j).\n$$\nIt is therefore sufficient to work with the measures defined by $\\tilde{\\mathbb{E}}_i^\\omega(dx_i) = (\\tilde{Z}_i^\\omega)^{-1}e^{-\\tilde{U}_i^\\omega}$, instead of $\\mathbb{E}_i^\\omega$.\n\nThe proof of the theorem will be in three steps.  We first prove the following inequality, designated a `$U$-bound' in \\cite{H-Z}.\n\n\n\\begin{lemma}\n\\label{lem q U-Bound}\nLet $\\frac{1}{p}+\\frac{1}{q} =1$ and suppose $\\varepsilon\\rho>0$, with $\\varepsilon>-\\frac{\\alpha}{2N}$ if $p=2$.  Then the following inequality holds:\n\\[\n\\tilde{\\mathbb{E}}_i^\\omega\\left(|f|^q\\left(d^{p}  + d \\sum_{j:j\\sim i}d(\\omega_j)\\right)\\right) \\leq A\\tilde{\\mathbb{E}}_i^\\omega|\\nabla_if|^q + B\\tilde\\mathbb{E}_i^\\omega |f|^q,\n\\]\nfor all smooth $f:\\Omega\\to\\mathbb{R}$, and some constants $A, B\\in(0,\\infty)$ independent of $\\omega$. \n\\end{lemma}\n\n\n\n\n\\begin{proof}\nWithout loss of generality assume $f\\geq0$.  By the Liebniz rule, we have\n\\begin{equation}\n\\label{leibniz}\n(\\nabla_i f)e^{-\\tilde{U}_i} = \\nabla_i(fe^{-\\tilde{U}_i}) + f\\nabla_i \\tilde{U}_ie^{-\\tilde{U}_i}. \n\\end{equation}\n\nTaking the inner product of both sides of this equation with $d(x_i)\\nabla_id(x_i)$ and integrating yields\n\\begin{align*}\n\\int_{\\mathbb{H}}fd\\nabla_id.\\nabla_i\\tilde{U}_ie^{-\\tilde{U}_i}dx_i &\\leq \\int_{\\mathbb{H}}d|\\nabla_id||\\nabla_i f|e^{-\\tilde{U}_i}dx_i - \\int_{\\mathbb{H}}d\\nabla_id.\\nabla_i\\left(fe^{-\\tilde{U}_i}\\right)dx_i \\\\\n&= \\int_{\\mathbb{H}}d|\\nabla_i f|e^{-\\tilde{U}_i}dx_i + \\int_{\\mathbb{H}}f\\nabla_i\\cdot(d\\nabla_id)e^{-\\tilde{U}_i}dx_i,\n\\end{align*}\nwhere we have used integration by parts and Proposition \\ref{eik}.  Now by Proposition \\ref{sub-laplacian lem}, we have\n\\[\n\\nabla_i\\cdot(d\\nabla_id) = |\\nabla_id|^2 + d\\Delta_id \\leq 1+K\n\\]\nin terms of distributions.  Therefore we have\n\\begin{align*}\n\\int_{\\mathbb{H}}fd\\nabla_id.\\nabla_i\\tilde{U}_ie^{-\\tilde{U}_i}dx_i &\\leq  \\int_{\\mathbb{H}}d|\\nabla_i f|e^{-\\tilde{U}_i}dx_i + (1+K)\\int_{\\mathbb{H}}fe^{-\\tilde{U}_i}dx_i.\n\\end{align*}\n\nReplacing $f$ by $f^q$ in this inequality, and using Young's inequality, we arrive at\n\\begin{align}\n\\label{U-bound one}\n\\int_{\\mathbb{H}}f^qd\\nabla_id.\\nabla_i\\tilde{U}_ie^{-\\tilde{U}_i}dx_i &\\leq \\frac{1}{\\tau}\\int_{\\mathbb{H}}|\\nabla_i f|^qe^{-\\tilde{U}_i}dx_i + \\frac{q}{p}\\tau^{p-1}\\int_{\\mathbb{H}}f^qd^pe^{-\\tilde{U}_i}dx_i \\nonumber \\\\\n& \\qquad+ (1+K)\\int_{\\mathbb{H}}f^qe^{-\\tilde{U}_i}dx_i,\n\\end{align}\nfor all $\\tau>0$.\n\nWe now calculate that\n\\begin{align*}\n\\nabla_i d(x_i).\\nabla_i\\tilde{U}_i(x_i) &= p\\alpha d^{p-1}(x_i) + 4N\\varepsilon d(x_i) + 2\\varepsilon\\rho \\sum_{j:j\\sim i}d(\\omega_j),\n\\end{align*}\nalmost everywhere, again using Proposition \\ref{eik}.\n\nFor $\\varepsilon\\rho>0$, we therefore have that there exist constants $a_1, b_1 \\in (0, \\infty)$ such that\n\\begin{equation}\n\\label{lower estimate1}\nd(x_i)\\nabla_i d(x_i).\\nabla_i\\tilde{U}_i(x_i) \\geq a_1\\left(d^{p}(x_i)  + d(x_i) \\sum_{j:j\\sim i}d(\\omega_j)\\right) - b_1.\n\\end{equation}\nThis is clear if $\\varepsilon>0$.  If $\\varepsilon<0$ and $p>2$ then we use the fact that for any $\\delta\\in(0,1)$ there exists a constant $C(\\delta)$ such that $d \\leq \\delta d^{p} + C(\\delta)$.  If $p=2$, recall from Remark \\ref{indices} that we must assume $\\varepsilon>-\\frac{\\alpha}{2N}$, and the assertion follows.\n\n\nUsing the estimate \\eqref{lower estimate1} in \\eqref{U-bound one} and taking $\\tau$ small enough, we see that there exist constants $A, B\\in(0, \\infty)$ independent of $\\omega$  such that\n\\begin{align*}\n&\\int f^q\\left(d^{p}(x_i)  +  d(x_i) \\sum_{j:j\\sim i}d(\\omega_j)\\right)e^{-\\tilde{U}_i}dx_i \\\\\n& \\qquad \\leq A\\int |\\nabla_if|^qe^{-\\tilde{U}_i}dx_i + B\\int f^qe^{-\\tilde{U}_i}dx_i,\n\\end{align*}\nwhich proves the lemma.\n\\end{proof}\n\n\nThe second step is to use this to prove that $\\tilde\\mathbb{E}_{i}^{\\omega}$ satisfies a $q$-spectral gap inequality uniformly on the boundary conditions $\\omega$.\n\n\n\\begin{lemma}\n\\label{qSG}\nLet $\\frac{1}{p} +\\frac{1}{q}=1$ and suppose $\\varepsilon\\rho>0$, with $\\varepsilon>-\\frac{\\alpha}{2N}$ if $p=2$.  Then $\\tilde{\\mathbb{E}}_i^\\omega$ satisfies the q-spectral gap inequality uniformly on the boundary conditions i.e. there exists a constant $c_0\\in(0,\\infty)$ independent of $\\omega$ such that\n\\[\\tilde\\mathbb{E}^{\\omega}_i|f-\\tilde\\mathbb{E}^{\\omega}_if|^q \\leq c_0 \\tilde\\mathbb{E}^{\\omega}_i|\\nabla_i f|^q\\]\nfor all smooth $f:\\Omega\\to\\mathbb{R}$.\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nFirst note that\n\\begin{equation}\n\\label{note}\n\\tilde{\\mathbb{E}}_i|f-\\tilde{\\mathbb{E}}_if|^q \\leq 2^{q} \\tilde{\\mathbb{E}}_i|f-m|^q\n\\end{equation}\nfor any $m\\in\\mathbb{R}$.\n\nLet $\\mathcal{W}_i^\\omega(x_i) := d^{p}(x_i)  +  d(x_i) \\sum_{j:j\\sim i}d(\\omega_j)$.   Then, for any $L\\in(0,\\infty)$, we have\n\\begin{align*}\n\\tilde{\\mathbb{E}}_i|f-m|^q & = \\tilde{\\mathbb{E}}_i|f - m|^q\\mathbf{1}_{\\{\\mathcal{W}_i \\leq L\\}} + \\tilde{\\mathbb{E}}_i|f - m|^q\\mathbf{1}_{\\{\\mathcal{W}_i \\geq L\\}}\n\\end{align*}\nwhere $\\mathbf{1}_{\\{\\mathcal{W}_i \\leq L\\}}$ is the indicator function of the set $A^\\omega(L) := \\{x_i \\in \\mathbb{H}: \\mathcal{W}^\\omega_i(x_i) \\leq L\\}$.  Let\n\\[\nI_1:= \\tilde{\\mathbb{E}}_i|f - m|^q\\mathbf{1}_{\\{\\mathcal{W}_i  \\leq L\\}}, \\qquad I_2 :=\\tilde{\\mathbb{E}}_i|f - m|^q\\mathbf{1}_{\\{\\mathcal{W}_i\\geq L\\}}.\n\\]\nWe estimate each of these terms separately.  We can treat $f$ as a function of $x_i$ only by fixing all the others.  Take \n\\[\nm = m(f) := \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)} f(x_i)dx_i\n\\]\nwhere $|A^\\omega(L)| = \\int_{A^\\omega(L)}dx_i$ is the Lebesgue measure of $A^\\omega(L)$.  Then we have that\n\\begin{align}\n\\label{I_1}\nI_1 & = \\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^q\\frac{e^{-\\tilde U^\\omega_i(x_i)}}{\\tilde Z_i}dx_i\\nonumber\\\\\n&\\leq \\frac{e^{2N|\\varepsilon| L^\\frac{2}{p}}}{\\tilde Z_i} \\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^qdx_i\n\\end{align}\nsince on $A^\\omega(L)$ we have that $\\tilde U^\\omega_i \\geq - 2N|\\varepsilon|L^{\\frac{2}{p}}$.  Now, using the invariance of the Lebesgue measure with respect to the group translation\n\\begin{align}\n\\label{interp1}\n&\\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^qdx_i\\nonumber\\\\\n&\\qquad \\leq \\frac{1}{|A^\\omega(L)|^q} \\int_{A^\\omega(L)}\\left(\\int_\\mathbb{H}|f(x_i) - f(x_iy_i)|\\mathbf{1}_{A^\\omega(L)}(x_iy_i)dy_i\\right)^qdx_i\\nonumber\\\\\n&\\qquad \\leq \\frac{1}{|A^\\omega(L)|}\\int_\\mathbb{H}\\int_\\mathbb{H}|f(x_i) - f(x_iy_i)|^q \\mathbf{1}_{A^\\omega(L)}(x_iy_i)\\mathbf{1}_{A^\\omega(L)}(x_i)dy_idx_i\n\\end{align}\nusing H\\\"older's inequality.  Let $\\gamma : [0, t] \\to \\mathbb{H}$ be a geodesic in $\\mathbb{H}$ from $0$ to $y_i$ such that $|\\dot\\gamma(s)|\\leq1$.  Then\n\\begin{align*}\n|f(x_i) - f(x_iy_i)|^q &= \\left| \\int_0^t \\frac{d}{ds}f(x_i\\gamma(s))ds\\right|^q \\\\\n&=  \\left| \\int_0^t \\nabla_i f(x_i\\gamma(s)).\\dot\\gamma(s)ds\\right|^q\\\\\n&\\leq t^\\frac{q}{p}\\int_0^t|\\nabla_i f(x_i \\gamma(s))|^qds\n\\end{align*}\nagain by H\\\"older's inequality, where $\\frac{1}{p} +\\frac{1}{q}=1$.  Here $t = d(y_i)$.  Using this estimate in \\eqref{interp1} we see that\n\\begin{align}\n\\label{interp2}\n&\\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^qdx_i\\nonumber\\\\\n&\\qquad \\leq \\frac{1}{|A^\\omega(L)|}\\int_\\mathbb{H}\\int_\\mathbb{H} d^\\frac{q}{p}(y_i)\\int_0^t|\\nabla_i f(x_i \\gamma(s))|^qds \\mathbf{1}_{A^\\omega(L)}(x_iy_i)\\mathbf{1}_{A^\\omega(L)}(x_i)dy_idx_i.\n\\end{align}\nNote that when $x_iy_i\\in A^\\omega(L)$ and $x_i\\in A^\\omega(L)$ then we have $d(x_iy_i)\\leq L^{\\frac{1}{p}}$ and $d(x_i)\\leq L^\\frac{1}{p}$, so that\n\\[\nd(y_i) = d(x_i^{-1}x_iy_i) \\leq d(x_i) + d(x_iy_i) \\leq 2L^\\frac{1}{p}.\n\\]\nTherefore, continuing \\eqref{interp2}, \n\\begin{align}\n\\label{interp3}\n&\\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^qdx_i\\nonumber\\\\\n&\\qquad \\leq\\frac{(2L^\\frac{1}{p})^\\frac{q}{p}}{|A^\\omega(L)|}\\int_\\mathbb{H}\\int_\\mathbb{H}\\int_0^t|\\nabla_i f(x_i \\gamma(s))|^q \\mathbf{1}_{A^\\omega(L)}(x_iy_i)\\mathbf{1}_{A^\\omega(L)}(x_i)dsdy_idx_i.\n\\end{align}\n\nNext we note that for $x_iy_i\\in A^\\omega(L)$ and $x_i\\in A^\\omega(L)$ we have\n\\[\nd^p(y_i) + d(y_i) \\sum_{j:j\\sim i}d(\\omega_j) \\leq 2^pL\n\\]\nand \n\\begin{align*}\nd^p(x_i\\gamma(s)) + d(x_i\\gamma(s)) \\sum_{j:j\\sim i}d(\\omega_j) &\\leq 2^{p-1}\\left(d^p(x_i) + d(x_i) \\sum_{j:j\\sim i}d(\\omega_j)\\right) \\\\\n&\\quad + 2^{p-1}\\left(d^p(\\gamma(s)) + d(\\gamma(s)) \\sum_{j:j\\sim i}d(\\omega_j)\\right)\\\\\n&\\leq 2^{p-1}L + 2^{p-1}\\left(d^p(y_i) + d(y_i) \\sum_{j:j\\sim i}d(\\omega_j)\\right)\\\\\n&\\leq 2^{p-1}L(1+2^p) =: R.\n\\end{align*}\nThus we can continue \\eqref{interp3} by writing\n\\begin{align}\n\\label{interp4}\n&\\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^qdx_i\\nonumber\\\\\n&\\quad \\leq\\frac{(2L^\\frac{1}{p})^\\frac{q}{p}}{|A^\\omega(L)|}\\int_\\mathbb{H}\\int_\\mathbb{H}\\int_0^t|\\nabla_i f(x_i \\gamma(s))|^qds \\mathbf{1}_{A^\\omega(R)}(x_i\\gamma(s))\\mathbf{1}_{A^\\omega(2^pL)}(y_i)dy_idx_i\\nonumber\\\\\n& \\quad \\leq\\frac{(2L^\\frac{1}{p})^\\frac{q}{p}}{|A^\\omega(L)|}\\int_\\mathbb{H} d(y_i) \\left(\\int_\\mathbb{H}|\\nabla_i f(x_i)|^q\\mathbf{1}_{A^\\omega(R)}(x_i)dx_i\\right)\\mathbf{1}_{A^\\omega(2^pL)}(y_i)dy_i\\nonumber\\\\\n& \\quad \\leq\\frac{(2L^\\frac{1}{p})^{\\frac{q}{p}+1}}{|A^\\omega(L)|}\\int_\\mathbb{H} \\left(\\int_\\mathbb{H}|\\nabla_i f(x_i)|^q\\mathbf{1}_{A^\\omega(R)}(x_i)dx_i\\right)\\mathbf{1}_{A^\\omega(2^pL)}(y_i)dy_i\\nonumber\\\\\n& \\quad = 2^qL^\\frac{q}{p}\\frac{|A^\\omega(2^pL)|}{|A^\\omega(L)|}\\int_{A^\\omega(R)}|\\nabla_i f(x_i)|^qdx_i\\nonumber\\\\\n& \\quad \\leq  2^qL^\\frac{q}{p}e^R\\frac{|A^\\omega(2^pL)|}{|A^\\omega(L)|}\\int |\\nabla_i f(x_i)|^qe^{-\\tilde U_i^\\omega}dx_i\n\\end{align}\nwhere in the last line we have used the fact that on the set $A^\\omega(R)$ we have $e^{-\\tilde U_i^\\omega} \\geq e^{-R}$.  We finally note that $|A^\\omega(2^pL)|\/|A^\\omega(L)|$ can be bounded above by a constant $C_1$ independent of $\\omega$.  This is because $|A^\\omega(2^pL)|\/|A^\\omega(L)|\\geq 1$ and\n\\[\n\\frac{|A^\\omega(2^pL)|}{|A^\\omega(L)|} \\to 1 \\quad {\\rm as} \\quad \\sum_{j:j\\sim i}d(\\omega_j) \\to \\infty.\n\\]\nThen, using \\eqref{interp4} in \\eqref{I_1} yeilds\n\\[\nI_1 \\leq C_2 \\tilde\\mathbb{E}_i|\\nabla_i f|^q\n\\]\nwhere $C_2 = 2^qL^\\frac{q}{p}e^{2N|\\varepsilon|L^\\frac{2}{p} + R}C_1$ is independent of $\\omega$.\n\nFor the second term, we have that\n\\begin{align*}\nI_2 &\\leq \\frac{1}{L}\\tilde\\mathbb{E}_i\\left(|f -m|^q\\mathcal{W}_i\\right) \\\\\n& \\leq \\frac{A}{L}\\tilde\\mathbb{E}_i\\left|\\nabla f \\right|^q + \\frac{B}{L}\\tilde\\mathbb{E}_i\\left|f-m\\right|^q\n\\end{align*}\nwhere we have used Lemma \\ref{lem q U-Bound}.  Putting the estimates for $I_1$ and $I_2$ together, we see that\n\\begin{align*}\n\\tilde\\mathbb{E}_i\\left|f-m\\right|^q & \\leq  \\left(C_2 + \\frac{A}{L}\\right) \\tilde\\mathbb{E}_i|\\nabla_i f|^q + \\frac{B}{L}\\tilde\\mathbb{E}_i\\left|f-m\\right|^q \\\\\n\\Rightarrow \\tilde\\mathbb{E}_i\\left|f-m\\right|^q & \\leq \\frac{C_2 + \\frac{A}{L}}{1-  \\frac{B}{L}}\\tilde\\mathbb{E}_i|\\nabla_i f|^q\n\\end{align*}\nfor $L>B$, where all constants are independent of $\\omega$.  We can finally use this in \\eqref{note} to get the result. \n\n\\end{proof}\n\nWe can now prove Theorem \\ref{uniform LSq}:\n\\begin{proof}[of Theorem  \\ref{uniform LSq}]\n\nOur starting point is the classical Sobolev inequality on the Heisenberg group for the Lebesgue measure (\\cite{Var}):  there exists a $t>0$ such that\n\\begin{equation}\n\\label{CS}\n\\left(\\int|f|^{1+t}dx_i\\right)^{\\frac{1}{1+t}} \\leq a \\int|\\nabla_if|dx_i + b\\int|f|dx_i,\n\\end{equation}\nfor some constants $a,b\\in(0,\\infty)$. Without loss of generality, we may assume that $f\\geq0$.  Suppose also, to begin with, that $\\tilde\\mathbb{E}_i(f)=1$.  Now, if we set\n\\[\ng\\equiv \\frac{fe^{-\\tilde{U}_i}}{\\tilde{Z}_i}\n\\]\nthen\n\\begin{equation}\n\\label{LS1}\n\\tilde\\mathbb{E}_i(f\\log f) = \\int_\\mathbb{H} g\\log g dx_i + \\tilde\\mathbb{E}_i(f\\tilde{U}_i) + \\log \\tilde{Z}_i.\n\\end{equation}\nNow by Jensen's inequality\n\\begin{align*}\n\\int g\\log gdx_i & = \\frac{1}{t}\\int g\\log g^tdx_i  \\\\\n& = \\frac{1}{t}\\int g\\log \\left(d^{1+t}g^t\\right)dx_i - \\frac{1+t}{t}\\int g\\log ddx_i \\\\\n&\\leq \\frac{1+t}{t}\\log\\left(\\int (dg)^{1+t}dx_i\\right)^{\\frac{1}{1+t}} +\\frac{1+t}{t}\\int g ddx_i \\\\\n& \\leq\\frac{1+t}{t}\\left(\\int (dg)^{1+t}dx_i\\right)^{\\frac{1}{1+t}} +\\frac{1+t}{t}\\tilde\\mathbb{E}_i(f d) \\\\\n&\\leq \\frac{a(1 + t)}{t}\\int |\\nabla_i (dg)|dx_i + \\frac{1 +t}{t}(b+1)\\tilde\\mathbb{E}_i(f d)\\\\\n&\\leq  \\frac{a(1 + t)}{t}\\int d|\\nabla_i g|dx_i + \\frac{1 +t}{t}(b+1)\\tilde\\mathbb{E}_i(f d) + \\frac{a(1 + t)}{t},\n\\end{align*}\nwhere we have used the classical Sobolev inequality \\eqref{CS}, the fact that we have assumed $\\tilde\\mathbb{E}_i(f)=1$, and the elementary inequality $\\log x\\leq x$.  Hence by \\eqref{LS1}\n\\begin{align}\n\\label{LS2}\n\\tilde\\mathbb{E}_i(f\\log f) &\\leq \\frac{a(1 + t)}{t}\\int d\\left|\\nabla_i \\left(\\frac{fe^{-\\tilde{U}_i}}{\\tilde{Z}_i}\\right)\\right|dx_i + \\tilde\\mathbb{E}_i(f\\tilde{U}_i) + \\frac{1 +t}{t}(b+1)\\tilde\\mathbb{E}_i(f d)\\nonumber\\\\\n&\\quad +  \\frac{a(1 + t)}{t} + \\log \\tilde{Z}_i \\nonumber\\\\\n& \\leq \\frac{a(1 + t)}{t}\\tilde\\mathbb{E}_i(d |\\nabla_i f|) + \\frac{a(1 + t)}{t}\\tilde\\mathbb{E}_i(f d|\\nabla_ i \\tilde U_i|) + \\tilde\\mathbb{E}_i(f\\tilde{U}_i)  \\nonumber\\\\\n& \\qquad + \\frac{1 +t}{t}(b+1)\\tilde\\mathbb{E}_i(fd^p) + 1 +  \\frac{a(1 + t)}{t} + \\log \\tilde{Z}_i.\n\\end{align}\n\nNow, since $\\varepsilon\\rho>0$ we have that $\\tilde Z_i^\\omega \\leq C_3$ for some constant $C_3\\in(0, \\infty)$ independent of $\\omega$.\n\nMoreover, we can directly calculate that\n\\begin{align}\n\\label{est1}\nd(x_i)|\\nabla_i \\tilde{U}^\\omega_i|(x_i) &\\leq \\alpha pd^p(x_i) + 4N|\\varepsilon|d^2(x_i) + 2\\varepsilon\\rho d(x_i)\\sum_{j:j\\sim i}d(\\omega_j)\\nonumber\\\\\n&\\leq  \\left(\\alpha p + 4N|\\varepsilon|\\right)d^p(x_i) + 2\\varepsilon\\rho d(x_i)\\sum_{j:j\\sim i}d(\\omega_j) + 4N|\\varepsilon|\\nonumber\\\\\n& \\leq a_3\\mathcal{W}_i^\\omega(x_i) + b_3\n\\end{align}\nalmost everywhere, where $\\mathcal{W}_i^\\omega(x_i) = d^{p}(x_i) + d(x_i)\\sum_{j:j\\sim i}d(\\omega_j)$ as in Lemma \\ref{qSG}, and $a_3 = \\max\\{\\alpha p + 4N|\\varepsilon|, 2\\varepsilon\\rho\\}$ and $b_3=4N|\\varepsilon|$ are constants independent of $\\omega$.\nSimilarly there exist constants $a_4, b_4\\in[0, \\infty)$ independent of $\\omega$ such that\n\\begin{equation}\n\\label{est2}\n\\tilde{U}^\\omega_i(x_i) \\leq a_4\\mathcal{W}_i^\\omega(x_i) + b_4.\n\\end{equation}\n\nWe can substitute estimates \\eqref{est1} and \\eqref{est2} into \\eqref{LS2}.  This yields\n\\begin{align}\n\\label{LS2.5}\n\\tilde\\mathbb{E}_i(f\\log f) &\\leq c_1\\tilde\\mathbb{E}_i(d |\\nabla_i f|) + c_2\\tilde\\mathbb{E}_i(f \\mathcal{W}_i) + c_3\n\\end{align}\nwhere \n\\begin{align*}\nc_1= &\\frac{a(1 + t)}{t}, \\qquad c_2 =  \\frac{aa_3(1 + t)}{t} + a_4 + \\frac{1+t}{t}(b+1)\\\\\n&c_3 =  1 +  \\frac{a(1 + t)}{t} + \\log C_3 + b_3 + b_4\n\\end{align*}\nare all independent of $\\omega$.\n\n\nNow, by fiirst replacing $f$ by $\\frac{f}{\\tilde\\mathbb{E}_i f}$ and then $f$ by $f^q$ in \\eqref{LS2.5}, after an application of Young's inequality we see that\n\\begin{align}\n\\label{LS3}\n\\tilde\\mathbb{E}_i\\left(f^q\\log \\frac{f^q}{\\tilde\\mathbb{E}_i f^q}\\right) & \\leq \\frac{c_1}{q}\\tilde\\mathbb{E}_i(|\\nabla_i f|^q) + \\left(c_2 + \\frac{c_1}{p}\\right)\\tilde\\mathbb{E}_i\\left(f^q\\mathcal{W}_i\\right) + c_3\\tilde\\mathbb{E}_i(f^q).\n\\end{align}\n\nWe recognise that the second term in \\eqref{LS3} can be bounded using Lemma \\ref{lem q U-Bound}.  Indeed, using this estimate\n\n\\begin{equation}\n\\label{GLS}\n\\tilde\\mathbb{E}_i\\left(f^q\\log \\frac{f^q}{\\tilde\\mathbb{E}_i f^q}\\right)  \\leq \\tilde c_1\\tilde\\mathbb{E}_i(|\\nabla_i f|^q) + \\tilde c_2\\tilde\\mathbb{E}_i(f^q),\n\\end{equation}\nwhere $\\tilde c_1 = \\frac{c_1}{q} + A\\left(c_2 + \\frac{c_1}{p}\\right)$ and $\\tilde c_2 = c_3 + B\\left(c_2 + \\frac{c_1}{p}\\right)$. Thus $\\tilde\\mathbb{E}^\\omega_i$ satisfies the generalised $(LS_q)$ inequality uniformly on the boundary conditions.\n\nFinally, we have the $q$-Rothaus inequality (see \\cite{B-Z}, \\cite{Ro}), which states that\n\\[\n\\tilde\\mathbb{E}_i\\left(f^q\\log\\frac{f^q}{\\tilde\\mathbb{E}_if^q}\\right) \\leq \\tilde\\mathbb{E}_i\\left(\\left|f-\\tilde\\mathbb{E}_if\\right|^q\\log\\frac{\\left|f-\\tilde\\mathbb{E}_if\\right|^q}{\\tilde\\mathbb{E}_i\\left|f-\\tilde\\mathbb{E}_if\\right|^q}\\right) + 2^{q+1}\\tilde\\mathbb{E}_i\\left|f-\\tilde\\mathbb{E}_if\\right|^q.\n\\]\nUsing this together with \\eqref{GLS} and the $q$-spectral gap inequality proved in Lemma \\ref{qSG} we thus arrive at a constant $c$ independent of $\\omega$ such that\n\\[\n\\tilde\\mathbb{E}_i\\left(f^q\\log\\frac{f^q}{\\tilde\\mathbb{E}_if^q}\\right) \\leq c\\tilde\\mathbb{E}_i(|\\nabla_i f|^q),\n\\]\nwhich proves Theorem \\ref{uniform LSq}.\n\\end{proof}\n\n\\section{The logarithmic Sobolev inequality for Gibbs measures}\n \n  \nIn this section we show how to pass from the uniform $(LS_q)$ inequality for the single site measures $\\mathbb{E}^{\\omega}_i$, to the $(LS_q)$ inequality for the corresponding Gibbs measure $\\nu$ on the entire configuration space $\\Omega = (\\mathbb{H})^{\\mathbb{Z}^N}$.  In the more standard Euclidean model, this problem has been extensively studied in the case $q=2$ , for example in \\cite{B-H1}, \\cite{G-Z}, \\cite{Led}, \\cite{Marton} and more recently in \\cite{O-R}, as well as in many of the afore mentioned papers.  The case $q<2$ was looked at in \\cite{B-Z}.  The following argument is strongly related to these methods, though it is based on the work contained in \\cite{Ze2} and \\cite{Ze}.\n\nWe work in greater generality than is required for Theorem \\ref{main}, though the results of section \\ref{results for single site measure} show that in the specific case where the local specification is defined by \\eqref{local spec} and \\eqref{hamiltonian}, the hypotheses \\textbf{(H0)} and \\textbf{(H1)} below are satisfied.  Then Theorem \\ref{main} follows as an immediate corollary of Theorem \\ref{thm1}.\n\nConsider a local specification $\\{\\mathbb{E}^{\\omega}_\\Lambda\\}_{\\Lambda\\subset\\subset\\mathbb{Z}^N, \\omega\\in\\Omega}$ defined by \n\\begin{equation}\n\\label{gen local spec}\n\\mathbb{E}^{\\omega}_\\Lambda(dx_{\\Lambda})=\\frac{e^{-\\sum_{i\\in\\Lambda}\\phi(x_i)-\\sum_{\\{i,j\\}\\cap\\Lambda\\neq\\emptyset, i\\sim j}J_{ij}V(x_i,x_j)}dx_\\Lambda} {Z^{\\omega}_\\Lambda}\n\\end{equation}\nwhere $Z^{\\omega}_\\Lambda$ is the normalisation factor and the summation is taken over couples of nearest neighbours $i\\sim j$ in the lattice with at least one point in $\\Lambda$ and where $x_i = \\omega_i$ for $i\\not\\in \\Lambda$, as before.  We suppose that $|J_{ij}| \\in [0, J_0]$ for some $J_0>0$.\n\n~\n\nWe will work with the following hypotheses:\n \n \\begin{itemize}\n \\item[\\textbf{(H0):}] The one dimensional single site measures $\\mathbb{E}^{\\omega}_i$ satisfy $(LS_q)$ with a constant $c$ which is independent of the boundary conditions $\\omega$.  \n \n~\n \n\\item[\\textbf{(H1):}] The interaction $V$ is such that\n$$\n\\left\\Vert \\nabla_i \\nabla_j V(x_i,x_j) \\right\\Vert_{\\infty}<\\infty.\n$$\n\\end{itemize}\n \n\\begin{rem}\nIn the situation where {\\rm \\textbf{(H1)}} is not satisfied, i.e. when the interaction potential grows faster than quadratically, a number of results have been obtained in \\cite{Ioannis1} and \\cite{Ioannis2} under some additional assumptions.\n\\end{rem}\n \n\n \n \n \n \n \n \n \n\\begin{theorem}\n\\label{thm1}\nSuppose the local specification $\\{\\mathbb{E}^{\\omega}_\\Lambda\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^N,\\omega \\in\n\\Omega}$ defined by \\eqref{gen local spec} satisifies {\\rm\\textbf{(H0)}} and {\\rm \\textbf{(H1)}}.  Then, for sufficiently small $J_0$, the corresponding infinite dimensional Gibbs measure $\\nu$ is unique and satisfies the $(LS_q)$ inequality \n$$\\nu \\left(|f|^q \\log\\frac{|f|^q}{\\nu |f|^q}\\right)\\leq C \\nu (\\left| \\nabla f\\right|^q)$$                               \nfor some positive constant $C$. \n\\end{theorem}\n\nFor notational sake we only prove this result for the case $N=2$, but our methods are easily generalised.  Before proving Theorem \\ref{thm1} we will present some useful lemmata.  \n \n\\subsection{Lemmata:}\n\nDefine the following sets\n\\begin{align*}\n&\\Gamma_0=(0,0)\\cup\\{j \\in \\mathbb{Z}^2 : dist(j,(0,0))=2m \\text{\\; for some \\;}m\\in\\mathbb{N}\\}, \\\\  \n&\\Gamma_1=\\mathbb{Z}^2\\smallsetminus\\Gamma_0 .\n\\end{align*}\nwhere $dist(\\cdot, \\cdot)$ is as in section \\ref{Infinite dimensional setting}.  Note that $dist(i,j)>1$ for all $i,j \\in\\Gamma_k,k=0,1$ and $\\Gamma_0\\cap\\Gamma_1=\\emptyset$.  Moreover $\\mathbb{Z}^2=\\Gamma_0\\cup\\Gamma_1$.  As above, for the sake of notation, we will write $\\mathbb{E}_{\\Gamma_k}=\\mathbb{E}_{\\Gamma_k}^{\\omega}$ for $k=0,1$. We will also define \n$$ \\mathcal{P}:=\\mathbb{E}_{\\Gamma_1}\\mathbb{E}_{\\Gamma_{0}}.$$\n \n\\begin{lemma} \n\\label{lem1}\nIf the local specification $\\{\\mathbb{E}^{\\omega}_\\Lambda\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^N,\\omega \\in\n\\Omega}$ satisfies {\\rm \\textbf{(H0)}} and {\\rm \\textbf{(H1)}}, then, for sufficiently small $J_0$, there exist constants $\\tilde D>0$ and $\\tilde\\eta\\in(0,1)$ such that\n\\begin{equation}\n\\label{first lem}\n\\nu\\left\\vert \\nabla_{\\Gamma_k}(\\mathbb{E}_{\\Gamma_{l}}f)\n\\right\\vert^q \\leq \\tilde D\\nu\\left\\vert \\nabla_{\\Gamma_k} f\n\\right\\vert^q+\\tilde\\eta\\nu\\left\\vert \\nabla_{\\Gamma_l} f\n\\right\\vert^q\\end{equation}\nfor $k,l\\in\\{0,1\\}$ such that $k\\neq l$.\n\\end{lemma}\n\n\n\n\n \\begin{proof}   \n \nFor convenience, suppose $k=1$ and $l=0$. The case $k=0,l=1$ follows similarly. We can write \n\\begin{align}\n\\label{lem1-1}\n\\mathcal{I} &:=\\nu\\left\\vert \\nabla_{\\Gamma_1 }\n(\\mathbb{E}_{\\Gamma_0}f) \\right\\vert ^q = \\nu\\sum_{i\\epsilon \\Gamma_1}\\left\\vert \\nabla_{i }\n(\\mathbb{E}_{\\Gamma_0}f) \\right\\vert ^q \\nonumber \\\\\n& \\leq \\nu\\sum_{i\\epsilon \\Gamma_1}\n\\left\\vert \\nabla_{i }\n(\\mathbb{E}_{\\{\\sim i\\}}f) \\right\\vert ^q \\nonumber \\\\\n&\\leq2^{q-1}\\nu\\sum_{i\\epsilon \\Gamma_1}\n\\left\\vert \\mathbb{E}_{\\{\\sim i\\}}\\nabla_{i } f\n\\right\\vert\n^q+\n2^{q-1}J_{0}^{q}\\nu\\sum_{i\\epsilon \\Gamma_1}\n\\left\\vert\\mathbb{E}_{\\{\\sim i\\}}(f\\mathcal{U}_i) \\right\\vert ^q\n\\end{align}\nwhere above we have denoted $\\{ \\sim i\\}=\\{j:j \\sim i\\}$, $W_i=\\sum_{j\\in\\{\\sim i\\}}\\nabla_iV(x_{i}, x_{j})$ and \n$$\\mathcal{U}_i=W_i-\\mathbb{E}_{\\{ \\sim i\\}}W_i.$$\nThen\n\\begin{align}\n\\label{lem1-2}\n\\mathcal{I} &\\leq2^{q-1}\\nu\\sum_{i\\epsilon \\Gamma_1}\n\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert \\nabla_{i } f \\right\\vert^q + 2^{q-1}J_{0}^q\\nu\\sum_{i\\epsilon\\Gamma_1}\\left\\vert\\mathbb{E}_{\\{\\sim i\\}}(f-\\mathbb{E}_{\\{\\sim i\\}}f)\\mathcal{U}_i \\right\\vert ^q \\nonumber \\\\ \n&\\leq 2^{q-1}\\nu\\sum_{i\\epsilon \\Gamma_1}\n\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert \\nabla_{i } f\n\\right\\vert^q+2^{q-1}J_{0}^q\\nu\\left(\\sum_{i\\epsilon \\Gamma_1}\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q\\left(\\mathbb{E}_{\\{\\sim i\\}}\\left|\\mathcal{U}_i\\right|^p\\right)^{q\/p}\\right)\n\\end{align}\nusing H\\\"older's inequality and the fact that $\\mathbb{E}_{\\{\\sim i\\}}\\mathcal{U}_i =0$.  Since interactions occur only between nearest neighbours in the lattice, we have that no interactions occur between points of the set $\\{\\sim i\\}$.  Hence the measure $\\mathbb{E}_{\\{\\sim i\\}}^{\\omega}$ is the product measure of the single site measures i.e.  $\\mathbb{E}_{\\{\\sim i\\}}^{\\omega}=\\otimes_{j \\in\\{\\sim i\\}}\\mathbb{E}^{\\omega}_j$.\nMoreover, by \\textbf{(H0)}, all measures $\\mathbb{E}^{\\omega}_j, j \\in\\{\\sim i\\}$ satisfy the $(LS_q)$ inequality with a constant $c$ uniformly on the\nboundary conditions.  Therefore, since the $(LS_q)$ inequality is stable under tensorisation (see Remark \\ref{tensorisation} (i)), we have that the product measure $\\mathbb{E}_{\\{\\sim i\\}}^{\\omega}$ also satisfies the $(LS_q)$ inequality with the same constant $c$.  By Remark \\ref{tensorisation} (iii), it follows that $\\mathbb{E}_{\\{\\sim i\\}}^{\\omega}$ also satisfies the $q$-spectral gap inequality with constant $c_0 = \\frac{4c}{\\log2}$.\n\nHence we have\n\\begin{equation}\n\\label{lem1-3}\n\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q\\leq c_0\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert\n\\nabla_{\\{\\sim i\\}} f\\right\\vert^q.\n\\end{equation}\nMoreover, by Remark \\ref{tensorisation} (iv), since $q<p$ and $\\mathbb{E}_{\\{\\sim i\\}}$ is a measure on a finite dimensional space, we have there exists a constant $\\tilde c_0$ such that\n\\begin{align}\n\\label{lem1-4}\n\\mathbb{E}_{\\{\\sim i\\}}\\left|\\mathcal{U}_i\\right|^p=& \\mathbb{E}_{\\{\\sim i\\}}\\left|W_i-\\mathbb{E}_{\\{\\sim\ni\\}}W_i\\right|^p\\leq \\tilde{c}_0\\sum_{_{ j \\in\\{\\sim i\\}}}\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert \\nabla_{j} W_i\\right\\vert\n^p \\nonumber\\\\\n&\\leq \\tilde{c}_0\\sum_{_{ j \\in\\{\\sim i\\}}}\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert \\nabla_{j} \\nabla_iV(x_{i}, x_{j})\\right\\vert\n^p\\leq 4\\tilde{c}_0M^p\\end{align}\nwhere $M=\\|\\nabla_i\\nabla_j V(x_i, \\omega_j)\\|_\\infty < \\infty$ by \\textbf{(H1)}. \n\nIf we combine \\eqref{lem1-2}, \\eqref{lem1-3} and \\eqref{lem1-4} we obtain\n \\begin{align*}\n\\nu\\left\\vert \\nabla_{\\Gamma_1 }\n(\\mathbb{E}_{\\Gamma_0}f) \\right\\vert ^q &\\leq 2^{q-1}\\nu\\left(\\sum_{i\\epsilon \\Gamma_1}\n\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert \\nabla_{i } f\n\\right\\vert^q\\right)  \\\\\n&\\qquad + 2^{q-1}c\\left(4\\tilde{c}_0\\right)^{q\/p}M^qJ^q_0\\nu\\left(\\sum_{i\\epsilon \\Gamma_1}\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert\n\\nabla_{\\{\\sim i\\}} f\\right\\vert^q\\right)\\\\\n&\\leq  2^{q-1}\\nu\\left(\\sum_{i\\epsilon \\Gamma_1}\n\\left\\vert \\nabla_{i } f\n\\right\\vert^q\\right)  \\\\\n&\\qquad + 2^{q+1}c\\left(4\\tilde{c}_0\\right)^{q\/p}M^qJ^q_0\\nu\\left(\\sum_{i\\epsilon \\Gamma_0}\\left\\vert\n\\nabla_{i} f\\right\\vert^q\\right).\n\\end{align*}\nTherefore, choosing $J_0$ sufficiently small so that  $ 2^{q+1}c\\left(4\\tilde{c}_0\\right)^{q\/p}M^qJ^q_0<1$, we see that\n$$\\nu\n\\left\\vert \\nabla_{\\Gamma_1 }\n(\\mathbb{E}^{\\Gamma_0}f) \\right\\vert ^q\\leq \\tilde D\\nu\\left\\vert \\nabla_{ \\Gamma_1 } f\n\\right\\vert^q+\\tilde\\eta\\nu\\left\\vert\\nabla_{ \\Gamma_0} f\\right\\vert^q$$\n with $\\tilde{D} =2^{q-1}$ and $\\tilde\\eta= 2^{q+1}c\\left(4\\tilde{c}_0\\right)^{q\/p}M^qJ^q_0<1$, as required.\n\n \\end{proof}\n \n \\begin{lemma}\n \\label{lem2}\nSuppose the local specification  $\\{\\mathbb{E}_{\\Lambda}^{\\omega}\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^N, \\omega \\in\n\\Omega}$ satisfies {\\rm \\textbf{(H0)}} and {\\rm \\textbf{(H1)}}, and let $W_i=\\sum_{j\\in\\{\\sim i\\}}\\nabla_iV(x_{i}, x_{j})$ be as in the previous lemma.  Then there exists a constant $\\kappa$, independent of the boundary conditions, such that\n\\begin{align*}\n\\mathbb{E}_{\\{\\sim i\\}}&\\left(|f|^q;  W_{i}\\right) \\leq \\left(\\mathbb{E}_{\\{\\sim i\\}}f^q\\right)^{\\frac{1}{p}}\\left( \\kappa\\mathbb{E}_{\\{\\sim i \\}}\\left|\\nabla_{\\{ \\sim i\\}} f\\right|^q\\right)^\\frac{1}{q},\n\\end{align*}\nwhere $\\frac{1}{p}+\\frac{1}{q}=1$ and $\\mathbb{E}_{\\{\\sim i \\}}(g;h):= \\mathbb{E}_{\\{\\sim i \\}}(gh) - \\mathbb{E}_{\\{\\sim i \\}}(g)\\mathbb{E}_{\\{\\sim i \\}}(h)$ for any functions $g,h$.\n\\end{lemma}\n \\begin{proof} \n \n Without loss of generality, we may suppose that $f\\geq0$.  Let $\\hat\\mathbb{E}_{\\{\\sim i\\}}$ be an isomorphic copy of $\\mathbb{E}_{\\{\\sim i\\}}$. Then we have \n\\begin{align*}\n\\mathbb{E}_{\\{\\sim i\\}}(f^q;W_{i}) &=\\frac{1}{2}\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(\\left(f^q- \\hat f^{q}\\right)(W_{i}-\\hat W_i  )\\right) \\\\\n& = \\frac{1}{2}\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left[\\left(\\int_0^1\\frac{d}{ds}F_s^qds\\right)\\left(W_i - \\hat W_i\\right)\\right]\n\\end{align*}\nwhere $F_s = sf + (1-s)\\hat f$ for $s\\in[0,1]$.  Then\n\\begin{align}\n\\label{covlem1}\n\\mathbb{E}_{\\{\\sim i\\}}(f^q;W_{i}) &= \\frac{q}{2} \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left[\\left(\\int_0^1F_s^{q-1}ds\\right)\\left(f-\\hat f\\right)\\left(W_i - \\hat W_i\\right)\\right] \\nonumber\\\\\n&\\leq \\frac{q}{2}\\left\\{ \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(\\int_0^1F_s^{q-1}ds\\right)^p\\right\\}^\\frac{1}{p} \\nonumber\\\\\n& \\quad \\times \\left\\{ \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\hat f\\right|^q\\left|W_i - \\hat W_i\\right|^q\\right\\}^{\\frac{1}{q}}.\n\\end{align}\n\nNow by Jensen's inequality and convexity of the function $y\\mapsto y^q$ we have\n\\begin{align}\n\\label{covlem1a}\n\\left\\{ \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(\\int_0^1F_s^{q-1}ds\\right)^p\\right\\}^\\frac{1}{p} &\\leq \\left\\{ \\int_0^1\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}F_s^{q}ds\\right\\}^\\frac{1}{p}\\nonumber\\\\\n&\\leq  \\left\\{ \\int_0^1\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(sf^{q} + (1-s)\\hat f^q\\right)ds\\right\\}^\\frac{1}{p}\\nonumber\\\\\n&= \\left(\\mathbb{E}_{\\{\\sim i\\}} f^q\\right)^{\\frac{1}{p}}.\n\\end{align}\n\nMoreover,\n\\begin{align}\n\\label{covlem2}\n&\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\hat f\\right|^q\\left|W_i - \\hat W_i\\right|^q \\nonumber \\\\\n& \\qquad \\leq  2^q\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q\\left|W_i - \\hat W_i\\right|^q.\n \\end{align}\n\nWe have the following relative entropy inequality (see eg \\cite{A-B-C-F-G-M-R-S}, \\cite{D-S}): if $\\mu$ is a probability measure then\n\\[\n\\mu(uv) \\leq \\frac{1}{\\tau}\\mu(u)\\log\\mu(e^{\\tau v}) + \\frac{1}{\\tau}\\mu\\left(u\\log\\frac{u}{\\mu (u)}\\right), \\qquad \\forall \\tau>0.\n\\]\nApplying this to the right hand side of \\eqref{covlem2} with $\\mu = \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}$ we see that $\\forall \\tau>0$\n\\begin{align}\n\\label{covlem3}\n&\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\hat f\\right|^q\\left|W_i - \\hat W_i\\right|^q \\nonumber\\\\\n&\\qquad \\qquad \\leq \\frac{2^q}{\\tau}\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q\\log \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(e^{\\tau|W_i - \\hat W_i|^q}\\right) \\nonumber \\\\\n& \\quad\\quad\\quad + \\frac{2^q}{\\tau} \\mathbb{E}_{\\{\\sim i\\}}\\left(\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q\\log\\frac{\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q}{\\mathbb{E}_{\\{\\sim i\\}} \\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q}\\right).\n\\end{align}\n\nNow, by the Herbst argument, see for example \\cite{Helffer} or \\cite{Ledconc} , and using both \\textbf{(H0)} and \\textbf{(H1)}, we have that for some $\\tau>0$ there exists a constant $\\Theta>0$ independent of $\\omega$ such that\n\\[\n\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(e^{\\tau|W_i - \\hat W_i|^q}\\right) \\leq \\Theta.\n\\]\nWe can also use \\textbf{(H0)} to bound the second term of \\eqref{covlem3}.  This gives\n\\begin{align}\n\\label{covlem4}\n\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\hat f\\right|^q\\left|W_i - \\hat W_i\\right|^q \\nonumber &\\leq \\frac{2^q\\log\\Theta}{\\tau}\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q \\nonumber \\\\\n& \\quad\\quad + \\frac{2^qc}{\\tau}\\mathbb{E}_{\\{\\sim i\\}}\\left|\\nabla_{\\{\\sim i\\}}f\\right|^q\\nonumber\\\\\n&\\leq \\frac{2^q}{\\tau}\\left(c_0\\log\\Theta + c\\right) \\mathbb{E}_{\\{\\sim i\\}}\\left|\\nabla_{\\{\\sim i\\}}f\\right|^q\n\\end{align}\nwhere $c_0=\\frac{4c}{\\log2}$ as above, by Remark \\ref{tensorisation} (iii).\n\nPutting estimates \\eqref{covlem1a} and \\eqref{covlem4} into \\eqref{covlem1} we see that \n\\begin{align*}\n\\mathbb{E}_{\\{\\sim i\\}}(f^q;W_{i}) &\\leq \\left(\\mathbb{E}_{\\{\\sim i\\}} f^q\\right)^{\\frac{1}{p}}\\left(\\frac{q^q}{\\tau}(c_0\\log\\Theta +c)\\mathbb{E}_{\\{\\sim i\\}}\\left|\\nabla_{\\{\\sim i\\}}f\\right|^q\\right)^{\\frac{1}{q}}\n\\end{align*}\nwhich gives the desired result.\n\\end{proof}\n \n \n \n \n\\begin{lemma}\n\\label{lem3}\nSuppose the local specification  $\\{\\mathbb{E}_{\\Lambda}^{\\omega}\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^N,\\omega \\in\n\\Omega}$ satisfies {\\rm \\textbf{(H0)}} and {\\rm \\textbf{(H1)}}.  Then, for sufficiently small $J_0$, there exist constants $D >0$ and $\\eta\\in(0,1)$ such that\n\\begin{equation}\n\\label{lem3ineq}\n\\nu\\left\\vert \\nabla_{\\Gamma_k}(\\mathbb{E}_{\\Gamma_{l}}|f|^q)^\\frac{1}{q}\n\\right\\vert^q \\leq D\\nu\\left\\vert \\nabla_{\\Gamma_k} f\n\\right\\vert^q+\\eta\\nu\\left\\vert \\nabla_{\\Gamma_l} f\n\\right\\vert^q\\end{equation} \nfor $k,l\\in\\{0,1\\}, k\\neq l$. \n\\end{lemma}\n\n\\begin{proof} \nAgain we may suppose $f\\geq0$.  For $k=1,l=0$ (the other case is similar),  we can write \n\\begin{align}\n\\label{lem3-1}\n\\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_0}  f^q)^{\\frac{1}{q}} \\right\\vert\n^q &\\leq \\nu\\sum_{i\\epsilon\\Gamma_1}\\left\\vert \\nabla_i (\\mathbb{E}_{\\{\\sim i\\}}f^q)^{\\frac{1}{q}}\\right\\vert ^q \\nonumber \\\\ \n&= \\nu\\sum_{i\\epsilon\\Gamma_1} \\frac{1}{q^q}(\\mathbb{E}_{\\{\\sim i\\}}f^q)^{-\\frac{q}{p}}  \\left\\vert\\nabla_i( \\mathbb{E}_{\\{\\sim i\\}}  f^q)\\right\\vert ^q.\n\\end{align}  \n\n\nWe will compute the terms in the sum on the right hand side of \\eqref{lem3-1}.\nFor $i \\in \\Gamma_1$, we have\n\\begin{align*}\n\\nabla_i (\\mathbb{E}_{\\{\\sim i\\}} f^q) &=  q(\\mathbb{E}_{\\{\\sim i\\}} f^{q-1} \\nabla_i f )-\\sum_{ j\\in\\{\\sim i\\}}J_{i,j}\\mathbb{E}_{\\{\\sim i\\}}\\left(f^q; \\nabla_iV(x_i, x_j)\\right) \\\\\n\\Rightarrow \\left|\\nabla_i (\\mathbb{E}_{\\{\\sim i\\}} f^q)\\right| \n&\\leq q\\left( \\mathbb{E}_{\\{\\sim i\\}} f^q\\right)^{1\/p} \\left( \\mathbb{E}_{\\{\\sim i\\}}|\\nabla_if|^q\\right)^{1\/q} + J_0\\left|\\mathbb{E}_{\\{\\sim i\\}}\\left(f^q; W_i\\right)\\right|,\n\\end{align*}\nso that\n\\begin{align*}\n \\left|\\nabla_i (\\mathbb{E}_{\\{\\sim i\\}} f^q)\\right|^q &\\leq 2^{q-1}q^q\\left( \\mathbb{E}_{\\{\\sim i\\}} f^q\\right)^{\\frac{q}{p}} \\left( \\mathbb{E}_{\\{\\sim i\\}}|\\nabla_if|^q\\right)\\\\\n&\\qquad + 2^{q-1}J^q_0\\left|\\mathbb{E}_{\\{\\sim i\\}}\\left(f^q; W_i\\right)\\right|^q,\n\\end{align*}\nwhere $W_i = \\sum_{j\\in\\{\\sim i\\}} \\nabla_iV(x_i, x_j)$ as above.  \n\n\nWe can use Lemma \\ref{lem2} to bound the correlation in the second term.  Indeed, this gives\n\\begin{align*}\n\\left|\\nabla_i (\\mathbb{E}_{\\{\\sim i\\}} f^q)\\right|^q &\\leq  \\left(\\mathbb{E}_{\\{\\sim i\\}} f^q\\right)^{\\frac{q}{p}} \\left(2^{q-1}q^q\\mathbb{E}_{\\{\\sim i\\}}|\\nabla_if|^q + 2^{q-1}\\kappa J^q_0\\mathbb{E}_{\\{\\sim i\\}}\\left|\\nabla_{\\{\\sim i\\}} f\\right|^q\\right).\n\\end{align*}\n\nUsing this in \\eqref{lem3-1} yields\n\\begin{align*}\n\\nu \\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_0}  f^q)^{\\frac{1}{q}} \\right\\vert\n^q &\\leq \\nu\\sum_{i\\in\\Gamma_1}\\left(2^{q-1}\\mathbb{E}_{\\{\\sim i\\}}|\\nabla_if|^q + \\frac{2^{q-1}}{q^q}\\kappa J^q_0\\mathbb{E}_{\\{\\sim i\\}}\\left|\\nabla_{\\{\\sim i\\}} f\\right|^q\\right) \\\\\n& = 2^{q-1} \\nu\\left|\\nabla_{\\Gamma_1} f\\right|^q + \\frac{2^{q-1}}{q^q}\\kappa J_0^q\\nu\\sum_{i\\in\\Gamma_1}\\left|\\nabla_{\\{\\sim i\\}} f\\right|^q \\\\\n& = 2^{q-1} \\nu\\left|\\nabla_{\\Gamma_1} f\\right|^q + \\frac{2^{q+1}}{q^q}\\kappa J_0^q\\nu\\left|\\nabla_{\\Gamma_0} f\\right|^q.\n\\end{align*}\nFinally, taking $J^q_0<\\frac{q^q}{2^{q+1}\\kappa}$ we see that\n\\[\n\\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_0}  f^q)^{\\frac{1}{q}} \\right\\vert\n^q \\leq D\\nu \\left|\\nabla_{\\Gamma_1} f\\right|^q + \\eta\\nu\\left|\\nabla_{\\Gamma_0} f\\right|^q,\n\\]\nwhere $D=2^{q-1}$ and $\\eta= \\frac{2^{q+1}}{q^q}\\kappa J_0^q <1$, as required.\n\\end{proof}    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \\begin{lemma}\n \\label{lem4}\n Suppose the local specification  $\\{\\mathbb{E}_{\\Lambda}^{\\omega}\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^N,\\omega \\in\\Omega}$ satisfies {\\rm \\textbf{(H0)}} and {\\rm \\textbf{(H1)}}.  Then $\\mathcal{P}^nf$ converges\n $\\nu$-almost everywhere to $\\nu f$,  where we recall that $\\mathcal{P}=\\mathbb{E}_{\\Gamma_1}\\mathbb{E}_{\\Gamma_{0}}$.  In particular, $\\nu$ is unique.\n \\end{lemma}\n \n \n \\begin{proof}\n \n\\noindent  \nWe will follow \\cite{G-Z}. We have \n \\begin{align}\\nonumber\\nu\\left|f- \\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}} f\\right|^q&\\leq 2^{q-1}\\nu\\mathbb{E}_{\\Gamma_{0}}\\left|f- \\mathbb{E}_{\\Gamma_{0}} f\\right|^q+2^{q-1}\\nu\\mathbb{E}_{\\Gamma_{1}}\\left|\\mathbb{E}_{\\Gamma_{0}}f- \\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}} f\\right|^q\\\\ &\n \\leq2^{q-1}c_0\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q+2^{q-1}c_0\\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_{0}}f\n)\\right\\vert^q,\n\\end{align}           \nsince by \\textbf{(H0)} and Remark \\ref{tensorisation} both the measures $\\mathbb{E}_{\\Gamma_{0}}$ and $\\mathbb{E}_{\\Gamma_{1}}$ satisfy the $(SG_q)$ inequality with constant $c_0 = \\frac{4c}{\\log2}$ independant of the boundary conditions. If we use Lemma \\ref{lem1} we get \n\\[\n\\nu\\left|f- \\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}} f\\right|^q \\leq 2^{q-1}c_0\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q+2^{q-1}c_0(\\tilde D\\nu\\left\\vert \\nabla_{\\Gamma_1} f\n\\right\\vert^q+\\tilde\\eta\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q)\n\\]\nFrom the last inequality we obtain that for any   $n\\in \\mathbb{N}$,  \n\\begin{align*}\n\\nu\\left|\\mathcal{P}^nf- \\mathcal{P}^{n+1} f\\right|^q &\\leq 2^{q-1}c_0\\nu\\left\\vert \\nabla_{\\Gamma_0} \\mathcal{P}^nf\n\\right\\vert^q+2^{q-1}c_0\\tilde\\eta\\nu\\left\\vert \\nabla_{\\Gamma_0} \\mathcal{P}^nf\n\\right\\vert^q \\\\\n&= 2^{q-1}c_0(1+\\tilde\\eta)\\nu\\left\\vert \\nabla_{\\Gamma_0} \\mathcal{P}^nf\\right\\vert^2,\n\\end{align*}\nusing the fact that $\\mathcal{P}^n$ does not depend on coordinates in $\\Gamma_1$ by definition, so that $\\nabla_{\\Gamma_1}\\mathcal{P}^n = 0$.  By repeated applications of Lemma \\ref{lem1} we see that,\n\n\\begin{align*}\n\\nu\\left|\\mathcal{P}^nf- \\mathcal{P}^{n+1} f\\right|^q &\\leq 2^{q-1}c_0(1+\\tilde\\eta)\\tilde\\eta^{2n-1}\\nu\\left|\\nabla_{\\Gamma_1}\\mathbb{E}_{\\Gamma_0}f\\right|^q \\\\\n&\\leq 2^{q-1}c_0(1+\\tilde\\eta)\\tilde\\eta^{2n-1}\\left(\\tilde D\\nu\\left|\\nabla_{\\Gamma_1}f\\right|^q + \\tilde\\eta\\nu\\left|\\nabla_{\\Gamma_0}f\\right|^q\\right).\n\\end{align*}\n\nSince $\\tilde\\eta<1$, this clearly tends to zero as $n\\to\\infty$, so that the sequence $\\{\\mathcal{P}^n\\}$ is Cauchy in $L^q(\\nu)$.  Moreover, by the Borel-Cantelli lemma, the sequence\n $$\\{ \\mathcal{P}^n f-\\nu \\mathcal{P}^n f\\}_{n \\in \\mathbb{N}}$$  \n converges $\\nu-a.s$. The limit of $\\mathcal{P}^n f-\\nu \\mathcal{P}^n f=\\mathcal{P}^n f-\\nu  f$    is therefore constant and hence identical to zero. \n \\end{proof}      \n \n \n \n \n \n \n \n \n \n \\subsection{Proof of Theorem \\ref{thm1} }\n \n \n \n \n \n \n \n\\begin{proof}\nRecall that we want to extend the $(LS_q)$ inequality from the single-site measures $\\mathbb{E}_{i}^{\\omega}$ to the Gibbs measure corresponding to the local specification $\\{\\mathbb{E}_{\\Lambda}^{\\omega}\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^2,\\omega \\in\n\\Omega}$ on the entire lattice (since we are taking $N=2$ for convenience).  As mentioned, to do so, we will follow the iterative method developed by B. Zegarlinski in  \\cite{Ze2} and \\cite{Ze}. \n\nAgain without loss of generality, suppose $f\\geq0$.  We can write \n \\begin{align}\n \\label{thm1-1}\n \\nonumber \\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right)=&\\nu\\mathbb{E}_{\\Gamma_0} \\left(f^q \\log\\frac{f^q}{\\mathbb{E}_{\\Gamma_0} f^q}\\right)+\\nu\\mathbb{E}_{\\Gamma_{1}} \\left(\\mathbb{E}_{\\Gamma_0}f^q \\log\\frac{\\mathbb{E}_{\\Gamma_0}f^q}{\\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_0} f^q}\\right) \\\\ \n&\\qquad + \\nu \\left(\\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}}f^q \\log\\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}}f^q\\right)- \\nu\\left(f^q \\log \\nu f^q\\right).\n\\end{align}\nAs mentioned above, by \\textbf{(H0)} and since the measures $\\mathbb{E}_{\\Gamma_0}$\nand $\\mathbb{E}_{\\Gamma_1}$ are in fact product measures, we know that they both satisfy $(LS_q)$ with constant $c$ independent of the boundary conditions.  Using this fact in \\eqref{thm1-1} yields\n\n\\begin{align}\n\\label{thm1-2}\n\\nonumber  \\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right) \\leq c\\nu(\\mathbb{E}_{\\Gamma_0}&\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q)+c\\nu \\mathbb{E}_{\\Gamma_1}\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_0} f^q)^{\\frac{1}{q}}\n\\right\\vert^q\\\\\n& \\qquad + \\nu \\left(\\mathcal{P}f^q \\log\\mathcal{P}f^q\\right)-\\nu \\left(f^q \\log \\nu f^q\\right).\n\\end{align}\nFor the third term of \\eqref{thm1-2} we can similarly write \n\\begin{align}\n\\nonumber\\nu \\left(\\mathcal{P}f^q \\log \\mathcal{P} f^q\\right)&= \\nu \\mathbb{E}_{\\Gamma_{0}}\\left(\\mathcal{P}f^q \\log \\frac{\\mathcal{P} f^q}{\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P} f^q}\\right)+\\nu \\mathbb{E}_{\\Gamma_{1}}\\left(\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P}f^q \\log \\frac{\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P} f^q}{\\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P} f^q}\\right)\\\\\n& \\qquad + \\nonumber\n\\nu\\left(\\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P}f^q \\log \\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P}f^q \\right).\n\\end{align}\nIf we use again the $(LS_q)$ inequality  for the measures $\\mathbb{E}_{\\Gamma_{k}},k=0,1$\nwe get\n \\begin{equation}\n \\label{thm1-3}\n \\nu\\left(\\mathcal{P}f^q \\log \\mathcal{P} f^q\\right)\\leq c \\nu\\left\\vert \\nabla_{\\Gamma_0}(\\mathcal{P}f^q)^\\frac{1}{q}\n\\right\\vert^q+c \\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_{0}} \\mathcal{P}f^q)^\\frac{1}{q}\n\\right\\vert^q+\\nu \\left(\\mathcal{P}^2f^q \\log \\mathcal{P}^2f^q\\right).\n\\end{equation}\nWorking similarly for the last term $\\nu \\left(\\mathcal{P}^2f^q \\log \\mathcal{P}^2f^q\\right)$  of \\eqref{thm1-3} and inductively for any term $\\nu (\\mathcal{P}^kf^q \\log \\mathcal{P}^kf^q)$, then after $n$ steps \\eqref{thm1-2} and \\eqref{thm1-3} will give\n\\begin{align}\n\\label{thm1-4}\n\\nonumber\\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right) &\\leq c \\sum_{k=0}^{n-1} \\nu \\left\\vert \\nabla_{\\Gamma_0}(\\mathcal{P}^kf^q)^\\frac{1}{q}\\right\\vert^q+c \\sum_{k=0}^{n-1} \\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_{0}} \\mathcal{P}^kf^q)^\\frac{1}{q}\\right\\vert^q \\\\\n& \\qquad + \\nu \\left(\\mathcal{P}^n f^q \\log\\mathcal{P}^n f^q\\right)-\\nu \\left(f^q \\log \\nu f^q\\right).\n \\end{align}\n \nIn order to deal with the first and second term on the right-hand side of \\eqref{thm1-4} we will use Lemma \\ref{lem3}.    If we apply inductively relationship \\eqref{lem3ineq}, for any $k\\in\\mathbb{N}$ we obtain\n\\begin{equation}\n\\label{thm1-5}\n\\nu\\left\\vert \\nabla_{\\Gamma_0}(\\mathcal{P}^kf^q)^\\frac{1}{q}\n\\right\\vert^q \\leq \\eta^{2k-1}D\\nu\\left\\vert \\nabla_{\\Gamma_1} f\n\\right\\vert^q+\\eta^{2k}\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q\n\\end{equation}\nand\n\\begin{equation}\n\\label{thm1-6}\n\\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P}^kf^q)^\\frac{1}{q}\n\\right\\vert^q \\leq \\eta^{2k}D\\nu\\left\\vert \\nabla_{\\Gamma_1} f\n\\right\\vert^q+\\eta^{2k+1}\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q.\n\\end{equation}\n \n \n \n \n \nUsing \\eqref{thm1-5} and \\eqref{thm1-6} in \\eqref{thm1-4} we see that\n\\begin{align}\n\\label{thm1-7}\n\\nonumber\\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right)& \\leq cD\\left(\\eta^{-1}  + 1\\right)\\left(\\sum_{k=0}^{n-1}\\eta^{2k}\\right)\\nu\\left\\vert \\nabla_{\\Gamma_1} f\\right\\vert^q \\\\\n& \\qquad + c\\left(1+\\eta\\right)\\left(\\sum_{k=0}^{n-1}\\eta^{2k}\\right)\\nu\\left\\vert \\nabla_{\\Gamma_0} f\\right\\vert^q \\nonumber \\\\ \n& \\qquad + \\nu \\left(\\mathcal{P}^n f^q \\log \\mathcal{P}^n f^q\\right)-\\nu (f^q \\log \\nu f^q).\n\\end{align}\n \n \n \n \n \n \nBy Lemma \\ref{lem4} we have that $\\lim_{n\\to \\infty}\\mathcal{P}^nf^q=\\nu f^q$,  $\\nu-a.s$.   Therefore, taking the limit as $n\\to\\infty$ in \\eqref{thm1-7} yields\n\\[\n\\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right) \\leq cD\\left(\\frac{1}{\\eta}+1\\right)K\\nu\\left\\vert \\nabla_{\\Gamma_1} f\n\\right\\vert^q+c(1 + \\eta)K\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q\n\\]\n where $K=\\sum_{k=0}^{\\infty}\\eta^{2k} = \\frac{1}{1-\\eta^2}$ for  $\\eta<1$.  Hence\n \\[\n\\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right) \\leq C\\nu|\\nabla f|^q\n\\]\nfor $C = \\max\\left\\{ cD\\left(\\frac{1}{\\eta}+1\\right)K, c(1 + \\eta)K\\right\\}$, as required.\n\\end{proof}  \n\n\n\n\n\\bibliographystyle{siam}     \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nAround half of the abundances of the heaviest isotopes in the periodic table, including gold and europium, are produced through the rapid neutron-capture process \\citep[$r$-process,][]{Burbidge57,Cameron57}. Since the first discussion of the $r$-process in the 1950s, there has been debate over which astrophysical sites produce $r$-process material. \nRecently, the detection of an optical transient associated with the neutron star merger GW170817 \\citep{LIGOGW170817a,coulter17} provided strong evidence for $r$-process production in neutron star mergers \\citep[e.g.,][]{Drout17, Pian17}. Neutron star mergers thus appear to be a source of $r$-process elements, but it is unclear if they are the dominant source in the early universe. One concern stems from observations of $r$-process abundances in metal-poor ([Fe\/H] $< -2.5$) stars in the Galactic halo.\n\nIt is unclear whether the delay time to form and coalesce a binary neutron star system is too long to provide $r$-process material to near-pristine gas before the formation of metal-poor stars \\citep[e.g.,][]{Argast04,Skuladottir19,Cescutti15,Wehmeyer15,Haynes19,Kobayashi20}. Possible solutions include processes like inhomogeneous metal mixing or inefficient star formation mitigating the delay time \\citep[e.g.,][]{Ishimaru15,Shen15,vandeVoort15,RamirezRuiz15,Ji16b,Dvorkin20} or common envelope producing a large number of rapidly merging neutron star binaries \\citep[e.g.,][]{Beniamini16,Safarzadeh19,Zevin19,Andrews20}, but concerns have not been eradicated.\n\nNatal kicks received from the supernova explosions that give birth to neutron stars may have also made it unlikely for small, early galaxies to retain neutron star binaries \\citep{Bramante16,Beniamini16,Bonetti19}. For example, the highly $r$-process-enriched metal-poor stars in the ultra-faint dwarf galaxy Reticulum II could potentially be explained by a neutron star merger \\citep{Ji16b}, but the natal kick would have to have been very small ($v < v_{esc} \\sim 10-20 $ km s\\textsuperscript{-1}) and\/or the merger time extremely short to avoid kicking the binary out of the tiny galaxy \\citep{Tarumi20,Safarzadeh19b,Bramante16}. This is in contrast to larger estimates of $20-140$ km s$^{-1}$ based on the offset distribution of short-duration $\\gamma$-ray bursts from their host galaxies \\citep{Fong2013}, and $5-5450$ km s$^{-1}$ from galactic double neutron star systems \\citep{Wong2010}.\n\nIn light of these concerns --- coupled with the inference that the ejecta from GW170817 was dominated by an accretion disk wind, rather than dynamical tidal tails \\citep[e.g.][]{siegel2019b} --- \\citet{siegel19} revived the idea that collapsars (the supernova- and $\\gamma$-ray-burst-triggering collapse of rapidly rotating massive stars) may be an important source of $r$-process material (see also, e.g., \\citealt{MacFadyen99,McLaughlin2005}). In particular, the accretion disks formed in collapsars can have similar conditions to the $r$-process producing disk of GW170817. \\citet{siegel19} found that for accretion rates $\\gtrsim$ 10$^{-3}$ M$_\\odot$ s$^{-1}$, these disks produce neutron-rich outflows that synthesize heavy $r$-process nuclei. They also found that collapsars can yield sufficient $r$-process material to explain over 80\\% of the $r$-process content of the Universe. \nAlthough the electron fraction in collapsar disk winds is still debated \\citep{Surman06,Miller19}, this is currently one of the most promising ways for core-collapse supernovae (CCSN) to make $r$-process elements (other than magnetorotationally driven CCSN, e.g., \\citealt{Nishimura15}, but see \\citealt{Mosta18}).\n\n\nThe scatter in the abundances of metal-poor stars is a useful probe of different $r$-process origins.\nIn this paper, we investigate collapsars as a source of the $r$-process in the early universe by investigating whether they can self-consistently reproduce the scatter of europium (Eu, Z = 63) in the most metal-poor stars. Our results also hold for any prompt $r$-process site with a power law distribution of effective $r$-process yields. As our representation was directly inspired by collapsar properties for the purposes of studying $r$-process collapsars, though, we call the $r$-process site in our model ``collapsars\" and discuss alternative interpretations in more detail in Section \\ref{subsec:NSMvscollapsar}.\n\nOur model assumes the $r$-process material in metal-poor stars was formed exclusively in collapsars with stochastic $r$-process yields. Previous stochastic models primarily assume the $r$-process is produced in fixed amounts, but comes from multiple different sources and\/or mixes into different environments \\citep[e.g.,][]{Tsujimoto+Shigeyama14,Cescutti15,Wehmeyer15,Shen15}. \nIn contrast, our model assumes the $r$-process source has an intrinsically stochastic production: each collapsar synthesizes a different amount of $r$-process material.\n\nIn Section \\ref{sec:model}, we outline our stochastic collapsar enrichment model in which we assume each collapsar contributes an $r$-process yield that is independently drawn from a power law distribution, inspired by models of collapsar jet fits to $\\gamma$-ray burst data. Our model is constrained using stellar abundance data described in Section \\ref{sec:data}, and the parameter constraints are described in Section \\ref{sec:results}. The implications of these results are discussed in Section \\ref{sec:disc}, where we put our results in context with collapsar jet property distributions, different types of core-collapse supernovae, and different estimates for the amounts of $r$-process material which may be produced by collapsars. Our conclusions are summarized in Section \\ref{sec:conc}.\n\n\n\\section{Collapsar $r$-Process Yield Model} \\label{sec:model}\n\n\\begin{figure*}[t!]\n\\gridline{\n\t\\fig{schematic.pdf}{0.45\\textwidth}{(a) Schematic of our model. Core-collapse supernovae explode into the total gas, yielding iron (Fe), and some fraction of these are collapsars which also yield stochastic amounts of europium (Eu). This produces stars with different [Eu\/Fe] and [Fe\/H] abundances.}\n\t\\fig{stellar_iqr.pdf}{0.4\\textwidth}{(b) The $r$-process abundance scatter in the RPA data. The data is shown as dots (Eu detections) and triangles (Eu upper limits). The grey boxes show the estimated $\\text{IQR}_{\\text{Eu}}$ (the abundance scatter) in several metallicity bins, which decreases with increasing metallicity.}}\n\\caption{Schematic of our theoretical model and scatter plot of the stellar data our models attempts to reproduce. Our model attempts to reproduce the observed Eu scatter at low metallicity by assuming all Eu is produced by collapsars, which are a fraction of all core-collapse supernovae.\n\\label{fig:stellarscatter}}\n\\end{figure*}\n\n\nThe purpose of our model is to determine the distribution of $r$-process abundances (as measured by [Eu\/Fe]) in a fixed metallicity bin (as measured by [Fe\/H]). A schematic of the model can be seen in Figure \\ref{fig:stellarscatter}a. The novel feature of our model is that we explicitly study whether variable $r$-process yields from a single class of r-process events could produce observed abundance scatter. This is in contrast to previous models \\citep[e.g.,][]{Cescutti15,Ojima18,Shen15,vandeVoort15} which generally had a fixed yield per event and produced scatter through different $r$-process sites and\/or different galactic environments.\n\n\\subsection{Defining ``Collapsar''} \\label{subsec:definecollap}\n\nThe term ``collapsar'' typically refers to the collapse of a massive, rapidly-rotating star in which accretion onto a central black hole can produce a beamed jet, commonly evoked as the progenitors of long-duration $\\gamma$-ray bursts (LGRBs). In the model described below, we more broadly use the term to encompass a population of core-collapse supernovae that produce heavy r-process material with a power-law distribution of yields. \nThis definition is motivated by the traditional collapsar picture in which rapid accretion onto a compact object launches a collimated outflow wherein both the duration and luminosities of LGRBs are well-described by power laws \\citep{petropoulou17,sobacchi17}. Connecting jet properties to $r$-process production is inspired by the possible connection between the accretion phase during which $r$-process material is produced (due to a sufficiently high accretion rate that neutronizes the disk) and the phase during which the collapsar jet is launched. In particular, \\citet{siegel19} finds that the production of heavy r-process material requires $\\dot{M} \\gtrsim$ 10$^{-3}$ M$_\\odot$ s$^{-1}$, closely matched to the accretion rates required for jet production \\citep{MacFadyen99}. The results described below also hold for any prompt $r$-process site (e.g., occurring roughly concurrently with CCSN; this could potentially include fast-merging neutron stars) that lead to a power law distribution of heavy $r$-process material. We discuss other interpretations in Section \\ref{subsec:NSMvscollapsar}.\n\n\nIn addition, our model does not require that a jet successfully breaks out of the progenitor star. While the most extreme $r$-process producing events require large fallback accretion disks and are likely associated with LGRBs, our model also includes events that eject smaller amounts of $r$-process material. Such systems may produce weaker outflows\/jets and be observed as low-luminosity GRBs \\citep[ll-GRBs; e.g.,][]{Bromberg11,petropoulou17}, relativistic supernovae \\citep[e.g.,][]{Soderberg2010,Margutti2014}, or broad-lined Type Ic supernovae \\citep[Type Ic-BL; e.g.,][]{Milisavljevic2015,Modjaz2016}.\n\n\n\\subsection{Basic Physical Set-up and Model Parameters}\n\nWe analytically model the abundance distribution as arising from a burst of core-collapse supernova enrichment, some fraction of which are collapsars that produce non-zero $r$-process yields.\nEach core-collapse supernova produces an iron yield and each collapsar also produces an $r$-process yield that is independently and identically drawn from a power law distribution. We assume the typical star formed in these galaxies forms after the metal yields from all of the supernovae fall into and mix within the hydrogen gas of the system. During this process, some fraction of the metals are permanently lost from the galaxy due to gas outflows.\n\nThis model has five free parameters: \n\\begin{enumerate}\n\\item $N_{\\text{SN}}$: the number of core-collapse supernovae enriching the gas.\n\\item $\\avg{N_r}$: the average number of collapsars (i.e. supernovae that produce non-zero amounts of $r$-process material).\n\\item $M_{r,\\text{min}}$: the minimum mass of $r$-process material that can be produced by a collapsar. \n\\item $\\alpha$: the power law exponent for our power law distribution of $r$-process yield produced per collapsar.\n\\item $y_{\\rm{Fe,eff}}$: the effective iron yield per supernovae per unit gas mass. $y_{\\rm{Fe,eff}} = y_{\\text{Fe}}f_{\\text{retained}}\/M_{\\text{gas}}$, where $y_{\\text{Fe}}$ is the iron yield per supernova, $f_{\\text{retained}}$ is the fraction of iron retained in the galaxy and not carried out of the system by gas outflows, and $M_{gas}$ is the total gas mass in the system.\n\\end{enumerate}\n\nThese parameters combine to yield the mean gas metallicity, the the fraction of supernovae that are collapsars ($f_r$), and the average yield per collapsar ($\\avg{M_r}$), as described below.\n\n\\subsection{Gas Enrichment}\n\nWe determine the distribution of europium abundances produced by this model at a given metallicity. The mean metallicity of our stars is found from $M_{\\text{Fe}}\/M_{\\text{H}} = N_{\\text{SN}} y_{\\rm{Fe,eff}}$. The mass of iron and hydrogen is converted to [Fe\/H] using a mean molecular weight of $\\mu_{\\text{Fe}}=56$ for Fe, $\\log \\epsilon_\\odot(\\text{Fe}) = 7.50$, and $\\log \\epsilon_\\odot(\\text{H}) = 12.00$ \\citep{Asplund09}, where we use the stellar spectroscopist notation $\\mbox{[X\/Y]} \\equiv \\log N_{\\text{X}} \/ N_{\\text{Y}} - \\log \\left(N_{\\text{X}}\/N_{\\text{Y}}\\right)_\\odot = \\log(\\frac{M_{\\text{X}}\\mu_{\\text{Y}}}{M_{\\text{Y}}\\mu_{\\text{X}}}) - (\\log\\epsilon_\\odot(\\text{X}) - \\log\\epsilon_\\odot(\\text{Y}))$. \n\nIn order to determine the distribution of europium values, we enrich this gas with $N_r$ collapsars, which we draw stochastically from a Poisson distribution with mean $\\avg{N_r} = f_r N_{\\text{SN}}$.\nEach collapsar contributes an $r$-process yield $M_r$ that is independently drawn from a power law distribution.\n\n\\begin{equation}\n  p(M_r) \\propto \\left(\\frac{M_r}{M_{r,\\text{min}}}\\right)^{-\\alpha} \\quad M_r \\geq M_{r,\\text{min}}  \\label{eq:Mr}\n\\end{equation}\n\nThe $r$-process yield for a single explosion, $M_r$, can be converted to $M_{\\text{Eu}}$ by using the solar $r$-process mass fraction of europium compared to all nuclei with mass number A $> 70$. The mass fraction, $X_{\\text{Eu}}$, is approximately $10^{-3}$ ($1.75 \\times 10^{-3}$, \\citealt{Arnould07}; $9.77 \\times 10^{-4}$, \\citealt{Sneden08}). When converting total europium mass to [Eu\/Fe], we use a mean molecular weight of $\\mu_{\\text{Fe}}=152$ for Eu and $\\log \\epsilon_\\odot(\\text{Eu}) = 0.52$. We also assume that europium and iron have the same retention fraction, $f_{\\text{retained}}$, meaning the same fraction of both is lost from the galaxy.\n\nNote that if $\\alpha \\leq 2$, then the average yield produced per collapsar diverges and our model would also require an upper cutoff to the amount of r-process material that can be produced by a single collapsar, $M_{r,\\text{max}}$. However, when we compare to observed data in Section \\ref{sec:results}, it will turn out our results imply $\\alpha > 2$, in which case the average yield per collapsar is: \n\n\\begin{equation}\n\\avg{M_r} = M_{r,\\text{min}} \\frac{\\alpha-1}{\\alpha-2}\n\\end{equation}\n\nWhile in principle $y_{\\text{Fe}}$ is also stochastic, for simplicity we hold it constant. This is fine as long as $f_r$ is small, since variations in the Fe yield will average out.\n\nOperationally, we create a model [Eu\/Fe] distribution by considering several thousand instances of supernova enrichment. Each instance is a single data point in our modeled cumulative distribution function. For each instance, we draw an $N_r$ value and then draw $M_r$ for each of the $N_r$ collapsars. The total europium and iron masses retained in the galaxy in each instance are transformed into a [Eu\/Fe] measurement. We also add a 0.1 dex Gaussian uncertainty to mimic observational errors.\n\n\\subsection{Constraining Model Parameters: Literature Estimates for Effective Iron Yields} \\label{subsec:res_yfM}\n\nThe effective iron yield of core-collapse supernova per unit gas mass cannot be directly constrained from a sample of stellar abundance data. We constrain its value by combining estimates for each component parameter (recall $y_{\\rm{Fe,eff}} = y_{\\text{Fe}}f_{\\text{retained}}\/M_{\\text{gas}}$) from the literature. \n\nThe fraction of retained metals is set to $f_{\\text{retained}} = 10^{-2 \\pm 0.5}$,  assuming that metal-poor stars form early in small galaxies. Observationally, individual faint galaxies have $f_{\\text{retained}}$ in this range: the Milky Way's moderately faint dSphs (e.g., Ursa Minor) have kept less than 1\\% of their metals \\citep{Kirby11outflow}; while the faint but still star-forming galaxy Leo P has kept about 5\\% of its metals \\citep{McQuinn15}.\nTheoretically, retaining about 1\\% of metals in small galaxies reproduces the slope and normalization of the mass-metallicity relation \\citep[e.g.,][]{Dekel03,Robertson05}. The retention fraction is also borne out in hydrodynamic galaxy simulations \\citep[e.g.,][]{Emerick18}. \n\n$M_{\\text{gas}}$ is set by models of how supernovae dilute metals into a mixing mass of gas. For small, early galaxies that form metal-poor stars, the mixing mass is $M_{\\text{gas}} \\sim 10^6 \\, M_\\odot$ \\citep{2015MNRAS.454..659J}. The strict lower limit on this mass is the mass contained in a single final supernovae remnant, a minimum of around $\\sim10^{4.5}~M_\\odot$ \\citep[e.g., ][]{Magg20,Macias18}, with a range of average mixing masses for metal-poor stars of $10^{5}$ to $10^{8} M_\\odot$ of gas.\nFor systems with higher $M_{\\text{gas}}$, more metals are retained, resulting in a higher retention fraction $f_{\\text{retained}}$ (and vice versa).\n\nTo estimate an average iron yield from CCSN, we calculate a weighted average between observations of H-rich CCSN and H-poor CCSN. A detailed discussion can be found in Appendix~\\ref{subsec:yFe}, but find that the average yield is $y_{\\text{Fe}} \\approx 0.1 M_\\odot$, with the uncertainty in $f_{\\text{retained}}$ and $M_{\\text{gas}}$ far outweighing that of $y_{\\text{Fe}}$.\n\nAltogether, $y_{\\rm{Fe,eff}}$ has a wide range of possible values ($10^{-10} - 10^{-7}$), but our fiducial choice is $y_{\\rm{Fe,eff}} = 10^{-9}$. This choice is validated by an independent estimation of the frequency of $r$-process events in ultra-faint dwarf galaxies in Section \\ref{subsec:res_NSNfr}. We note, however, that there is tension between the values expected for $y_{\\rm{Fe,eff}}$ in very low mass galaxies based on the theoretical breakdown described in this section and comparisons to several external constraints. For example, the number of supernovae predicted in an ultra-faint dwarf galaxy using the Salpeter initial mass function suggests an effective iron yield closer to $\\sim10^{-7.5}$, and a simulation of extremely metal-poor ([Fe\/H] $=-3.42$) stars forming after a single supernova gives an estimated effective iron yield as high as $\\sim10^{-6.5}$ \\citep{Chiaki19}. This is not fully unexpected as $y_{\\rm{Fe,eff}}$ differs in different galaxies and the lowest mass galaxies will have the highest effective yields, but we note that this parameter remains uncertain and may trend higher than its fiducial value.\n\n\\subsection{Constraining Model Parameters: Fitting Stellar Abundance Data} \\label{subsec:model_stellar}\n\nAfter fixing the effective iron yield, stellar abundances are used to constrain the other model parameters. The stellar abundance data provides us effectively four observable quantities of interest:\n\n\\begin{enumerate}\n\\item the mean metallicity of the stars, $\\avg{\\text{[Fe\/H]}}$, \n\\item the mean $r$-process abundance, $\\avg{\\text{[Eu\/Fe]}}$, \n\\item the estimated fraction of stars that formed from gas not enriched by an $r$-process event, $f_0$, \n\\item the observed scatter in $r$-process abundance between stars, $\\text{IQR}_{\\text{Eu}}$. \n\\end{enumerate}\nRather than quantifying scatter with standard deviation, $\\sigma(\\text{Eu})$, we use the more robust interquartile range, a measure of statistical dispersion equal to the difference between the 75th and 25th percentiles, denoted $\\text{IQR}_{\\text{Eu}}$. Model parameters are then determined as follows: \n\n$\\avg{N_r}$ and $\\alpha$: The average number of collapsars, $\\avg{N_r}$, and the exponent of the $r$-process yield power law distribution, $\\alpha$, are determined by comparing the observed $f_0$ and $r$-process scatter, $\\text{IQR}_{\\text{Eu}}$, to those predicted by our models with varying $\\avg{N_r}$ and $\\alpha$. In a small metallicity bin, the shape of the [Eu\/Fe] distribution function is dependent on only these two parameters. The other potentially relevant parameters contribute only to shifting the distribution to higher or lower [Eu\/Fe]. By focusing on only the shape of the distribution, we can avoid making assumptions about any additional parameters when determining $\\avg{N_r}$ and $\\alpha$.\n\n$N_{SN}$ and $f_r$: The number of SN enriching the gas, $N_{SN}$, is determined from the mean metallicity of the stars and the effective iron yield using $\\avg{M_{\\text{Fe}}\/M_{\\text{H}}} = N_{SN} \\times y_{\\rm{Fe,eff}}$. The fraction of supernova that are collapsars, $f_r=\\avg{N_r}\/N_{SN}$, is then found by combining $N_{SN}$ and the average number of collapsars, $\\avg{N_r}$, from above.\n\n$M_{r,\\text{min}}$ and $\\avg{M_r}$: We first determine the average $r$-process yield produced per collapsar, $\\avg{M_r}$, using the relationship: $\\avg{M_r} f_{\\text{retained}} X_{\\text{Eu}} \\approx \\avg{M_{\\text{Eu}}} \/ \\avg{N_r}$. In this equation, $\\avg{M_{\\text{Eu}}}$ is found by considering the mean $r$-process abundance $\\avg{\\text{[Eu\/Fe]}}$ and mean metallicity $\\avg{\\text{[Fe\/H]}}$ in combination with $M_{\\text{H}} \\approx M_{\\text{gas}}$. The minimum $r$-process yield is then $M_{r,\\text{min}} = \\avg{M_r} \\frac{\\alpha-2}{\\alpha-1}$.\n\nWe do not attempt to model higher moments of the [Eu\/Fe] distribution beyond the mean and scatter because we expect selection effects in the data to dominate. We also do not attempt to model the shape of the distribution tails for both observational and theoretical reasons. Observationally, the low end of the [Eu\/Fe] distribution cannot be well known without a robust selection function. Theoretically, the low and high ends of our distribution are not robust due to our assumption that model stars form after all of the supernova yields have fallen into and mixed with the hydrogen gas. This is because our assumption precludes outlier stars that, for example, could have more or less europium due to inhomogeneous mixing.\n\nFuture work will address these concerns through a more detailed treatment of enrichment that incorporates our variable-yield work into a more complete picture that includes scatter due to differences in galaxy formation (e.g., different environments or metal mixing). This will allow for a better determination of whether our assumption of a power law is an appropriate shape for the $r$-process yield distribution and improved constraints on $\\alpha$ and $N_r$.\n\n\\section{Stellar Abundance Samples} \\label{sec:data}\n\n\\subsection{Sample Selection}\n\nWe use a stellar abundance sample from the $R$-process Alliance (RPA), a collection of detailed abundances of 601 halo stars \\citep{Hansen18,Sakari18,rpa3,RPA4}. The RPA stars are bright (V $<$ 13.5), metal-poor ([Fe\/H] $\\lesssim -2$) red giant stars in the Milky Way stellar halo. They were observed with a focus on obtaining a statistically complete sample of europium abundances. To verify the RPA data, we also consider a sample of 228 metal-poor red giant halo stars from \\citet{Roederer14b} \\citepalias[henceforth][]{Roederer14b}. Both of these samples report europium measurements or upper limits for every star.\n\nThe \\citetalias{Roederer14b} sample has [Eu\/Fe] abundances that are $0.22$ dex lower and [Fe\/H] abundances that are $0.19$ dex lower from other samples due to using a much cooler effective temperature scale and isochrone-based surface gravities \\citep{2014MNRAS.445.2946R}. We thus shift the reported measurements up by these amounts when plotting in Figure \\ref{fig:starCDF} and reporting values in Table \\ref{tab:CDF}.\n\nWe restrict most of our analysis to very metal-poor ([Fe\/H] $< -2.5$) stars, and the highest metallicity we consider is [Fe\/H] $< -1.75$ (when analyzing the evolution of the Eu scatter with increasing metallicity in Section \\ref{subsec:aN}). \nWe only consider stars with barium-to-europium abundance ratios that could be produced by the $r$-process ($-0.9 \\lesssim $ [Ba\/Eu] $ \\lesssim -0.4$). [Ba\/Eu] higher than $\\sim -0.4$ indicates contamination from the $s$-process, another nucleosynthetic process which forms europium. The solar $r$-process barium-to-europium ratio is [Ba\/Eu] $ \\approx -0.8$ \\citep{Sneden08}, and stars with much lower [Ba\/Eu] cannot be explained by the $r$-process pattern. We note that small variations in these purity cuts do not significantly change our results.\n\nTaking into account these restrictions (with [Fe\/H] $<-2.5$), the RPA sample includes 83 stars with Eu measurements and an additional 11 stars with Eu upper limits. The \\citetalias{Roederer14b} sample includes 36 stars with Eu measurements and 4 with Eu upper limits. The RPA sample (up to [Fe\/H] $<-1.75$) and its $\\text{IQR}_{\\text{Eu}}$ in different metallicity bins can be seen in Figure \\ref{fig:stellarscatter}b.\n\n\\subsection{Construction of Statistical Distributions}\n\n\n\\begin{figure*}[!htb]\n\\plotone{rpa_roederer_kme.pdf}\n\\caption{Cumulative distribution functions for the RPA and \\citetalias{Roederer14b} samples. Both CDFs are determined using the Kaplan-Meier estimator, which takes into account detections and upper limits to estimate the true distribution. The shaded regions show 95\\% confidence on the CDF estimate. Grey lines outline the interquartile range (25\\%-75\\%) for the RPA Kaplan-Meier CDF. The CDFs have been extended to the \\textit{y}-axis to show the estimated fraction of stars in each sample that have no $r$-process elements.\n\\label{fig:starCDF}}\n\\end{figure*}\n\n\\begin{table*}[!htb]\n\\centering\n\\caption{Interquartile ranges and fraction of stars formed from gas with no $r$-process enrichment for different [Eu\/Fe] CDFs from observational stellar samples with [Fe\/H] $< -2.5$. The distributions can be seen in Figure \\ref{fig:starCDF}. $\\text{IQR}_{\\text{Eu}}$ uncertainties are due to both KME confidence levels and uncertain observations. The $f_0$ values are upper limits as the distribution could continue to lower [Eu\/Fe] with lower $f_0$.\n} \\label{tab:CDF}\n\\begin{tabular}{c|c|c|c|c}\n\\tablewidth{0pt}\n\\hline\n\\hline\nStellar Abundance Sample & $\\text{IQR}_{\\text{Eu}}$ & $f_0$ & $\\avg{\\text{[Eu\/Fe]}}$ & $\\avg{\\text{[Fe\/H]}}$ \\\\\n\\hline\nRPA & $0.50^{+0.15}_{-0.10}$ & $0.04^{+0.10}_{-0.04}$ & $0.3^{+0.1}_{-0.1}$ & $-2.7^{+0.1}_{-0.1}$ \\\\\nR14 & $0.38^{+0.51}_{-0.14}$ & $0.04^{+0.11}_{-0.04}$ & $0.2^{+0.2}_{-0.2}$ & $-2.9^{+0.1}_{-0.1}$ \\\\\n\\hline\n\\end{tabular} \n\\end{table*}\n\n\nTo combine the mixture of measurements and upper limits into a statistical distribution of europium for each sample, we employ survival statistics, a branch of statistics that deals with censored datasets, e.g., upper limits. The most general single variate survival statistic is the Kaplan-Meier estimator (KME), which provides a non-parametric maximum likelihood estimate of a distribution from observed data. The Kaplan-Meier estimator and survival statistics have been used extensively in astronomical literature \\citep[e.g.][]{upperlimits1,upperlimits2,wardle86,Simcoe04}. We use the \\texttt{KaplanMeierFitter} from the survival analysis python package \\texttt{lifelines} \\citep{lifelines}. For this estimate to be valid, two assumptions about the distribution of upper limits must hold. First, the upper limits should be independent of each other, which is true here as the stars are independent. Second, the upper limits should be random -- i.e., the probability that a measurement will be censored should not correlate with the measurement value itself. This assumption may not hold because lower [Eu\/Fe] values are more likely to be censored.\nIdeally, we would fully forward model and censor our theoretical results, but that requires many additional assumptions including a completeness function (probability of measuring any value given [Eu\/Fe]), an error function (the value we measure for [Eu\/Fe] given its true value), and an upper limit function (the probability of setting a [Eu\/Fe] upper limit at a specific value given its true value). Fully forward modeling the observational sample is beyond the scope of this paper. We thus use the Kaplan-Meier estimate while keeping in mind that this may not be a perfect estimate.\n\nFigure \\ref{fig:starCDF} shows the [Eu\/Fe] cumulative distribution functions for the RPA and R14 samples. The interquartile range, $\\text{IQR}_{\\text{Eu}}$, differs slightly for the different samples but is consistent within the uncertainty. The mean [Eu\/Fe] and [Fe\/H] also differ slightly. The zero-limit $f_0$, the estimated fraction of stars that formed from gas that was not enriched by an $r$-process event, is the same in both samples. In our model, $f_0$ is the fraction of stars with no europium enrichment ([Eu\/Fe] $= -\\infty$), but we cannot identify if real stars have no $r$-process enrichment (and stars could receive trace amounts of europium enrichment through other processes despite the [Ba\/Eu] cuts we applied to purify our sample). We thus estimate $f_0$ in the data by taking the lowest CDF value from the observed distribution as estimated by survival statistics. This assumes that the CDF immediately plateaus at lower [Eu\/Fe] instead of continuing to decrease. Because the distribution could continue to decrease with lower [Eu\/Fe], the observed $f_0$ values are upper limits. Realistically, the real distribution certainly does not fully plateau even if our $f_0$ estimate is correct because of the possible other trace sources of europium, but for the purposes of this analysis and because we cannot estimate the CDF to extremely low [Eu\/Fe] regardless, we ignore those minor effects. These values are shown in Table \\ref{tab:CDF}.\n\n\\section{Results} \\label{sec:results}\n\nWe use the stellar abundance data to constrain the model parameters. The results are summarized in Table \\ref{tab:results}.\n\n\\begin{table*}[!htb]\n\\centering\n\\caption{Model parameters determined from observations. The wide ranges of $N_{\\text{SN}}$, $M_{r,\\text{min}}$, and $y_{\\rm{Fe,eff}}$ encompass broad uncertainty in the fraction of metals retained in each galaxy and each galaxy's gas mass. To be thorough we include these full ranges.\nWe also validate our fiducial values for $y_{\\rm{Fe,eff}}$, $f_r$, and $\\avg{N_r}$ (which also validates $N_{\\text{SN}}$, $M_{r,\\text{min}}$, and $\\avg{M_r}$). For $\\alpha$, the full range of values produce similar distribution shapes. Derived parameter values are shown below the double line.} \\label{tab:results}\n\\begin{tabular}{c|c|c|c}\n\\tablewidth{0pt}\n\\hline\n\\hline\n  & Description & Range & Fiducial Value \\\\\n\\hline\n$N_{\\text{SN}}$ & \nNumber of core-collapse supernovae & $30-30000$ & $3000$ \\\\\n\\hline\n$\\avg{N_r}$ & Average number of $r$-process collapsars & $2 - 4$ & 3 \\\\\n\\hline\n$M_{r,\\text{min}}$ & Minimum $r$-process yield produced per collapsar (see Eq. \\ref{eq:Mr}) & $3\\times10^{-4} - 3\\times10^{-1}$ & $3\\times10^{-2}$ \\\\\n\\hline\n$\\alpha$ & Power law exponent of $M_r$ distribution (see Eq. \\ref{eq:Mr}) & $2.2 - 6$ & $2.8$ \\\\\n\\hline\n$y_{\\rm{Fe,eff}}$\\footnote{determined from literature values} & Effective supernovae iron yield into the total gas mass, $y_{\\text{Fe}}f_{\\text{retained}}\/M_{\\text{gas}}$ & $10^{-10} - 10^{-7}$ & $10^{-9}$ \\\\\n\\hline\n\\hline\n$f_r$ & Fraction of supernovae that are collapsars, $\\avg{N_r}\/N_{\\text{SN}}$ & $10^{-4} - 10^{-1}$ & $10^{-3}$ \\\\\n\\hline\n$\\avg{M_r}$ & Average $r$-process yield produced per collapsar & $7\\times10^{-4} - 7\\times10^{-1}$ & $7\\times10^{-2}$ \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{$\\avg{N_r}$ and $\\alpha$} \\label{subsec:aN}\n\nWe use the model described in Section \\ref{sec:model} to calculate theoretical cumulative distribution functions (CDF) of stellar [Eu\/Fe] abundances. CDFs resulting from different representative choices of $\\avg{N_r}$ and $\\alpha$ can be seen in Figure \\ref{fig:modelCDFs}. Each model CDF has an arbitrary offset that shifts the CDF left or right for plotting purposes. Recall that $\\avg{N_r}$ and $\\alpha$ can be constrained using only the shape of the distribution (i.e., the $\\text{IQR}_{\\text{Eu}}$ and $f_0$). A higher $\\avg{N_r}$ causes both a lower $f_0$ since fewer stars will form from un-enriched gas and a narrower distribution due to the central limit theorem. A higher $\\alpha$  also narrows the distribution by increasing the rarity of high $M_r$. When constraining these parameters with the $\\text{IQR}_{\\text{Eu}}$, a higher $\\avg{N_r}$ thus corresponds to a lower $\\alpha$ and vice versa.\n\n\\begin{figure*}[!htb]\n\\centering\n\\includegraphics[width=0.875\\linewidth]{model_CDF.pdf}\n\\caption{Stellar [Eu\/Fe] abundance cumulative distribution functions (colored lines) for models with different $\\avg{N_r}$ and $\\alpha$ values. The observed [Eu\/Fe] CDF for the RPA sample is shown in black with grey uncertainty. Our fiducial model, $\\avg{N_r}=3$ and $\\alpha=2.8$, is shown in solid blue in both plots. When either $\\avg{N_r}$ or $\\alpha$ is not specified, the fiducial value is used. The model CDFs have an arbitrary offset to shift the distribution left or right for plotting purposes, so only the shape (i.e., the IQR and zero-fraction) is relevant.\n\\label{fig:modelCDFs}}\n\\end{figure*}\n\n\\begin{figure*}[!htb]\n\\gridline{\n\t\\fig{model_f0s_labels.pdf}{0.4\\textwidth}{(a) Values of the zero-fraction, $f_0$, for different models. The heat map colors are normalized to $f_0 = 0.04^{+0.10}_{-0.04}$, the $f_0$ of the RPA stellar abundance sample (see Table \\ref{tab:CDF}). Red is higher than observed $f_0$, blue is lower. $\\avg{N_r}$ alone affects the $f_0$ value.}\n\t\\fig{model_IQRs_labels.pdf}{0.4\\textwidth}{(b) Values of the IQR for different models. The heat map colors are normalized to $\\text{IQR}_{\\text{Eu}} = 0.50^{+0.15}_{-0.10}$, the IQR of the RPA stellar abundance sample (see Table \\ref{tab:CDF}). Red is higher than observed IQR, blue is lower. For $\\avg{N_r}=1$, we determined the IQR by assuming symmetry and doubling the 50\\%-75\\% range (because the distribution is always above 25\\%).}}\n\\caption{Heat maps showing how the cumulative distribution IQR and zero-fractions of our model vary with $\\avg{N_r}$ and $\\alpha$. Black boxes outline the parameter combinations that can explain the stellar data within observational uncertainty (using the RPA sample; see Table \\ref{tab:CDF}). Note: The plotted $\\alpha$ values increase by 0.2 until $\\alpha = 3.0$, at which point they increase by 0.5 due to increasingly slower variation in $\\text{IQR}_{\\text{Eu}}$. \\label{fig:modelHeatmaps}\n}\n\\end{figure*}\n\n$\\avg{N_r}$, the average number of $r$-process collapsars enriching our stellar population, is constrained by the estimated fraction of stars which formed from gas not enriched by an $r$-process event, $f_0$. For $f_0 = 0.04$, $\\avg{N_r} = 3$ is the best fit value. Figure \\ref{fig:modelHeatmaps}a shows how the value of $f_0$ changes with $\\avg{N_r}$, independently of $\\alpha$. In this figure, the black boxes outline the parameter values which explain the observed $f_0$ or $\\text{IQR}_{\\text{Eu}}$.\n$\\avg{N_r} = 2$ to 4 can also explain observations. Note that the observed $f_0$ is an upper limit as the distribution could smoothly continue to lower [Eu\/Fe] with a lower $f_0$. The constraint on $\\avg{N_r}$ from $f_0$ is thus a lower bound.\n\n\\begin{figure*}[!htb]\n\\gridline{\\fig{scatter_evolution_Nr_3_lines.pdf}{0.45\\textwidth}{(a) The evolution of the Eu scatter with increasing metallicity can be well explained by our fiducial model choices of $\\avg{N_r}=3$ at $\\avg{\\text{[Fe\/H]}} = -2.7$ (the mean metallicity of our RPA sample) and $\\alpha = 2.8$.}\n\t\\fig{scatter_evolution_Nr_30_lines.pdf}{0.45\\textwidth}{(b) If $\\avg{N_r}$ is increased to $\\avg{N_r} = 30$ at $\\avg{\\text{[Fe\/H]}} = -2.7$, no single choice of $\\alpha$ well explains the evolution of the Eu scatter. Higher $\\avg{N_r}$ choices result in poorer matches to observations.}}\n\\caption{The decrease in the [Eu\/Fe] scatter with higher metallicity seen in the data (hollow circles and squares) is reproduced by our model (colored dots). Each [Fe\/H] bin of 0.3 dex corresponds to approximately a factor of 2 increase in supernovae, hence why we double the number of $r$-process collapsars in each bin. Reproducing the evolution in the scatter at higher metallicity as well as low metallicity increases confidence in our fiducial model choices of $\\avg{N_r}$ and $\\alpha$. \\label{fig:scatterevol}\n}\n\\end{figure*}\n\nTo validate our fiducial value of $\\avg{N_r}=3$, we also reproduce the evolution in europium scatter with increasing metallicity. This gives an upper bound to the constraint. We examine the RPA stellar abundance sample in several metallicity bins (up to [Fe\/H]  $= -1.65$; see Figure \\ref{fig:scatterevol}). We compare model scatter to the scatter of the RPA distribution as determined by both the Kaplan-Meier estimator (which takes into account europium detections and upper limits) and as determined by only europium detections. The \\citetalias{Roederer14b} sample is excluded from this plot because it has too few stars in each bin to determine distributions.\n\nAs metallicity increases, $\\avg{N_r}$ should increase linearly, but the scatter should decrease with $\\sqrt{\\avg{N_r}}$. Reproducing the $\\text{IQR}_{\\text{Eu}}$ in several metallicity bins thus suggests our model uses the correct $\\avg{N_r}$. When binning on metallicity, our model with $\\avg{N_r}=3$ at $\\avg{[\\text{Fe\/H}]} =-2.7$ well reproduces the observed decrease in scatter. If we increase $\\avg{N_r}$ by a factor of 10 or more, our model no longer well reproduces the observed decrease in scatter unless $\\alpha$ is allowed to vary with metallicity. Considering all uncertainty from the Kaplan Meier Estimator, the upper bound is $\\avg{N_r} \\approx 50$, but the model favors a much lower $\\avg{N_r}$. This suggests our fiducial $\\avg{N_r}=3$ is roughly correct despite being a lower bound.\n\nTo be thorough, we also explore the extreme case where \\textit{all} core-collapse supernovae result in $r$-process collapsars -- i.e., all core-collapse supernovae form an accretion disk that is able to synthesize a non-zero amount of $r$-process material ($\\avg{N_r} = N_{\\text{SN}}$ and $f_r=1$). We consider the case where $\\avg{N_r}=3000$, where 3000 is our fiducial value of $N_{\\text{SN}}$ (see Section \\ref{subsec:res_NSNfr}). In this extreme case, the vast majority of collapsars would produce extremely small amounts of $r$-process material. As seen in Figure~\\ref{fig:CDFfr1}, this extreme case can explain the observed $\\text{IQR}_{\\text{Eu}}$ with a scatter of $\\text{IQR}_{\\text{Eu}}=0.45$ for $\\alpha = 1.8$. The upper bound set on $\\avg{N_r}$ by the evolution of scatter with metallicity (Figure \\ref{fig:scatterevol}) disfavors this model, however. The situation where all core-collapse supernovae produce $r$-process is only favored if $N_{\\text{SN}}$ is below 30, lower than even our most extreme $N_{\\text{SN}}$ value.\nThis extreme model also does not reproduce the observed distribution at low [Eu\/Fe] as well as the fiducial model, though we note that the tails of the observed distribution are less trustworthy than the $\\text{IQR}_{\\text{Eu}}$. We thus keep $\\avg{N_r}=3$ for our results.\n\n$\\alpha$, the exponent of the $r$-process yield power law distribution, is constrained by the $\\text{IQR}_{\\text{Eu}}$ value of the distribution, which varies with both $\\alpha$ and $\\avg{N_r}$ as shown in Figure \\ref{fig:modelHeatmaps}b. For $\\avg{N_r} = 3$ and $\\text{IQR}_{\\text{Eu}} = 0.50^{+0.15}_{-0.10}$, the constrained value is $\\alpha = 2.8^{+4.2}_{-0.6}$.\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{CDF_fr1.pdf}\n\\caption{Distribution for an extreme model where all core-collapse supernovae produce some $r$-process material ($f_r=1$). This extreme model can explain the observed $\\text{IQR}_{\\text{Eu}}$. It cannot well explain the evolution of scatter with increasing metallicity, however (e.g., Figure \\ref{fig:scatterevol}). It also cannot well explain the low [Eu\/Fe] tail. For this model, the minimum amount of $r$-process material per collapsar is extremely small, $M_{r,\\text{min}} \\approx 10^{-6} M_\\odot$. \\label{fig:CDFfr1}}\n\\end{figure}\n\n\\subsection{$N_{\\text{SN}}$ and $f_r$} \\label{subsec:res_NSNfr}\n\nThe number of supernovae, $N_{\\text{SN}}$, is linearly related to $y_{\\rm{Fe,eff}}$. To explain the mean metallicity $\\avg{\\text{[Fe\/H]}}=-2.7 \\pm 0.1$ (using the method described in Section \\ref{subsec:model_stellar}), on the extreme ends of $y_{\\rm{Fe,eff}}$ values we need anywhere from 30 to 30,000 supernovae; lower $y_{\\rm{Fe,eff}}$ corresponds to higher $N_{\\text{SN}}$ as more supernovae are needed to explain the mean metallicity. For our fiducial value of $y_{\\rm{Fe,eff}} = 10^{-9}$, $N_{\\text{SN}} \\approx 3000$.\n\nWith values for $\\avg{N_r}$ (from Section \\ref{subsec:aN}) and $N_{\\text{SN}}$, we can determine the fraction of supernovae that result in $r$-process material producing collapsars, $f_r = \\avg{N_r}\/N_{\\text{SN}}$. Considering the extremes of the possible values of $y_{\\rm{Fe,eff}}$, $f_r \\approx 0.0001$ to $0.1$. For our fiducial values ($\\avg{N_r} = 3$ and $y_{\\rm{Fe,eff}} = 10^{-9}$), $f_r \\approx 0.001$.\n\nTo validate our fiducial choice of $y_{\\rm{Fe,eff}}$, we also estimate $f_r$ using observations of ultra-faint dwarf galaxies around the Milky Way. There are now high-resolution spectroscopic abundances for stars in 19 surviving ultra-faint dwarfs.\nOf these, three of the dwarfs (Grus~II, Reticulum~II, and Tucana~III) exhibit $r$-process enrichment \\citep{Hansen20_Grus,Ji16b,Hansen17}. Since these are extremely small systems, we assume each of these three dwarfs experienced one $r$-process event (as in \\citealt{Ji16b,Brauer19}), and then estimate the total number of supernovae that contributed to all of their stellar populations to estimate $f_r$. We combined literature values of their absolute magnitudes $M_V$ \\citep{Munoz18,Torrealba18,Drlica15,Bechtol15,Mutlu18} with a Salpeter individual mass function that predicts $0.02 L_0$ supernovae where $L_0$ is the present-day luminosity in $L_\\odot$ \\citep{Ji16b}. The ultra-faint dwarfs cumulatively experienced about $1800$ supernovae. The fraction of supernovae that result in $r$-process material producing collapsars is thus $f_r \\sim 3\/1800 = 0.002$. This validates our fiducial model values of $f_r \\approx 0.001$ and $y_{\\rm{Fe,eff}} \\approx 10^{-9}$. We note again, however, that there is tension between our fiducial estimate of $y_{\\rm{Fe,eff}}$ and several external constraints in very low mass galaxies, as discussed in Section \\ref{subsec:res_yfM}, so we continue to report the full uncertainty in these parameters.\n\n\\subsection{$M_{r,\\text{min}}$}\n\nThe minimum $r$-process yield produced per collapsar, $M_{r,\\text{min}}$ (see Eq. \\ref{eq:Mr}), depends on $\\alpha$ and varies linearly with $y_{\\rm{Fe,eff}}$. To transform between total $r$-process yield (nuclei with A $\\geq 70$) and europium yield, we use the solar $r$-process europium mass fraction $X_{\\text{Eu}} \\approx 10^{-3}$. To explain the observed mean europium-iron abundance ratio $\\avg{\\text{[Eu\/Fe]}}$, on the extreme ends of $y_{\\rm{Fe,eff}}$ values, we find that $M_{r,\\text{min}} \\approx 0.0003 - 0.3 M_\\odot$; lower $y_{\\rm{Fe,eff}}$ corresponds to higher $M_{r,\\text{min}}$ both because a lower $f_{\\text{retained}}$ causes less europium to be retained in the galaxy and because higher $M_{\\text{gas}}$ requires a higher mass of iron and europium to explain the mean [Fe\/H] and [Eu\/Fe] abundances. For our fiducial values of $y_{\\rm{Fe,eff}} = 10^{-9}$, $\\avg{N_r} = 3$, and $\\alpha = 2.8$, we find $M_{r,\\text{min}} \\approx 0.03 M_\\odot$, or a mean $r$-process yield per collapsar of $\\avg{M_r} \\approx 0.07 M_\\odot$.\n\nIn the extreme case where all core-collapse supernovae produce a nonzero amount of $r$-process material (Figure \\ref{fig:CDFfr1}), the minimum amount of $r$-process material per collapsar would be extremely small, $M_{r,\\text{min}}\\approx 10^{-6} M_\\odot$ for $\\avg{N_r} \\approx 3000$. This situation is disfavored because it does not reproduce the observed decrease of Eu scatter with increasing metallicity unless $y_{\\rm{Fe,eff}}$ is much higher than our fiducial value.\n\n\\section{Discussion} \\label{sec:disc}\n\nUsing stellar abundance data to constrain parameters in our stochastic collapsar chemical enrichment model produces a self-consistent physical picture, which was not guaranteed a priori. We now discuss this in more detail and place our results in context with other potentially physically relevant values. We also discuss the limitations of this model in Section \\ref{subsec:caveats}.\n\n\\begin{figure*}[t!]\n\\plotone{constraints_new.pdf}\n\\caption{Constraints on $\\avg{M_r}$, $f_r$, and $\\alpha$ from our model (blue) in context of potentially physically relevant values (black dotted lines). For descriptions of the reference values, see Section \\ref{sec:disc}. Our fiducial model values are plotted as a blue dot, while the dark blue shaded region represents an order of magnitude uncertainty around our fiducial values and the light blue shaded region represents the full uncertainty.  \\label{fig:constraints}}\n\\end{figure*}\n\n\n\\subsection{Implications of $f_r$: The Fraction of CCSN that Produce Collapsars} \\label{subsec:disc_fr}\n\nWhat fraction of core-collapse supernovae (CCSN) produce collapsars? Recall that our definition of collapsar is motivated by physical picture in which a rapid fallback accretion onto a black hole simultaneously produces heavy r-process material via accretion disk winds and launches a collimated outflow, but does \\emph{not} require that a jet successfully break out of the progenitor star. The power law distribution yields adopted in Section~\\ref{sec:model} naturally includes less extreme explosions that produce smaller amounts of $r$-process material via disk outflows.\n\nTo reproduce the observed scatter in [Eu\/Fe] abundances at low metallicity with a collapsar-like model, we require that the fraction of CCSN producing $r$-process material is between 10$^{-4}$ and 10$^{-1}$ with a fiducial value of 10$^{-3}$. We now compare these values to the observed rates for various classes of transients that have previously been proposed to be powered by collapsars or jet-driven explosions.\n\nWe begin with long-duration $\\gamma$-ray bursts (LGRBs). Current measurements of the local (z=0) rate for LGRBs beamed towards Earth from the \\emph{Swift} satellite range from $1.3^{+0.6}_{-0.7}$ Gpc$^{-3}$ yr$^{-1}$ for L$>$10$^{50}$ ergs \\citep{Wanderman2010} to $0.42^{+0.9}_{-0.4}$ Gpc$^{-3}$ yr$^{-1}$ accounting for complex \\emph{Swift} trigger criteria \\citep{Lien2014}. Correcting these for a canonical beaming factor of 50 \\citep{Guetta2005}, results in inferred intrinsic rates of $65^{+0.30}_{-0.35}$ Gpc$^{-3}$ yr$^{-1}$ and $21^{+4.5}_{-2}$ Gpc$^{-3}$ yr$^{-1}$, respectively. Comparing these to the local CCSN rate from the Lick Observatory SN Search (LOSS) of 0.705 ($\\pm$0.089) $\\times$10$^{-4}$ Mpc$^{-3}$ yr$^{-1}$ \\citep{Li2011} yields R$_{\\rm{LGRB}}$\/R$_{\\rm{CCSN}}$ values of (2$-$9)$\\times$10$^{-4}$. This range of values is only slightly lower than our fiducial value of $f_r$. LGRBs could thus be linked to $r$-process production. Our uncertainty on $f_r$ errs toward a higher value for the $r$-process fraction, though, so $f_r$ could very well be larger than $f_{\\text{LGRB}}$. In that case, we would require that massive stars beyond those that launch successful GRBs form accretion disks with physical conditions capable of producing heavy $r$-process material.\n\nIn particular, Type Ic-BL supernovae are a class of hydrogen-poor SN that display high ejecta velocities (hence ``broad-lined'') and kinetic energies ($\\sim$10$^{52}$ ergs) for which central engines are commonly evoked. While the nature of the central engine is still debated \\citep{Thompson2004,MacFadyen99,Barnes2018}, the fact that all SN observed in association with LGRBs have been Type Ic-BL supernovae has lead to the hypothesis that all events of this class are powered by jets. Differences in the detailed manifestation of these explosions (LGRBs, low-luminosity GRBs, relativistic SN, or ``ordinary'' Ic-BL SN) would then be driven by a distribution of engine timescales or progenitor radii \\citep[e.g.][]{Lazzati2012,Margutti2014}. We therefore compare our constraints on $f_r$ to the rates of Type Ic-BL SN to investigate if they are consistent with all Type Ic-BL SN harboring collapsar engines.\n\nBased on the full LOSS sample, \\citet{2017PASP..129e4201S} find that Type Ic-BL SN account for a fraction of (1.1 $\\pm$ 0.8) $\\times 10^{-2}$ of CCSN.  However, the LOSS sample was a targeted survey, biased towards high metallicity galaxies and it is well established that Type Ic-BL SN show a preference for low metallicity environments \\citep[e.g.][]{Modjaz2020}. It is therefore possible that Type Ic-BL SN represent a higher fraction of all CCSN at low metallicity, which is what our parameter $f_r$ actually constrains. Unfortunately, to date, there has been no untargeted, volume-limited study that examines the fraction of CCSN that are Type Ic-BL at low-metallicity. \\citet{Graur2017b} and \\citet{Arcavi2010} examine relative rates of different core collapse SN subtype in ``high'' and ``low'' mass galaxies for the LOSS and early PTF samples, respectively. \\citet{Graur2017b} find no significant difference in the Type Ic-BL fraction (1-2\\%), while \\citet{Arcavi2010} find that Type Ic-BL may make up a significantly higher fraction of all SN ($\\sim 10-13\\%$) in low luminosity galaxies. We caution, however, that both samples contain only 2-3 Type Ic-BL events and are therefore dominated by low number statistics. More recently, \\citet{2020arXiv200805988S} investigate the host galaxies of the full sample of 888 SN identified by PTF, including 36 Type Ic-BL. They find that Type Ic-BL production is significantly stifled above a galaxy mass of $\\log{M\/M_\\odot} = 10$, with Type Ic-BL comprising $\\gtrsim 5$\\% of their observed CCSN sample below this threshold compared to $\\lesssim 2$\\% above.\n\nR$_{\\rm{Ic-BL}}$\/R$_{\\rm{CCSN}}$ values of 0.01--0.1 fall within the range of $f_r$ found by our model (see Figure~\\ref{fig:constraints}). However, the latter is at the extreme high end, implying that while our model is consistent with all Type Ic-BL SNe producing europium, it favors a scenario in which $\\lesssim$10\\% do. We note that this would not preclude the possibility that all Type Ic-BL SNe harbor jets, but rather require that some lack the accretion disk properties necessary for the production of \\emph{heavy} r-process material. This could imply that a subset of Type Ic-BL SNe (a) harbor accreting black holes, but do not reach sufficiently high accretion rates ($> 10^{-3}$ M$_\\odot$ s$^{-1}$) to proceed past $^{56}$Ni-rich outflows \\citep{siegel19}, or (b) harbor magnetar central engines for which neutrino irradiation can limit neucleosynthesis from disk ejecta to the light r-process \\citep[e.g.][]{Margalit17,Radice18}.\n\nFor comparison, we also calculate a rough effective rate of neutron star mergers per CCSN. The cosmic NSM rate from the second LIGO-Virgo gravitational wave transient catalog is $320^{+490}_{-240}$ Gpc$^{-3}$ yr$^{-1}$ \\citep{2020arXiv201014533T}. When comparing this to the LOSS rate of galactic CCSN, the estimated NSM fraction is $4.5^{+7.0}_{-3.4} \\times 10^{-3}$. This is higher than our fiducial value of $f_r$, but within model uncertainties. The LIGO rate of NSMs could thus potentially account for the rate of $r$-process events required by our model to explain metal-poor star abundances, though it is not favored by our fiducial results. We also note that the rate of NSMs in the early universe likely differs from the rate found by LIGO, and that NSMs would need to be fast-merging to be described by our model.\n\n\n\\subsection{Implications of $\\avg{M_r}$: The Amount of $r$-Process Yield Produced per Collapsar}\n\nOur determination of the minimum and average amounts of $r$-process yield produced per collapsar ($M_{r,\\text{min}} \\approx 0.03 M_\\odot$ and $\\avg{M_r} \\approx 0.07 M_\\odot$, respectively) is based entirely on our analysis of RPA stellar abundance data, independent of any previous estimates in literature of the amount of $r$-process material that might be produced by such events. To place our results in context, we compare them to several reference estimates of $r$-process yields from single events (see Figure \\ref{fig:constraints}). Note again that we define $r$-process yield as the yield of nuclei with mass number A $\\geq 70$.\n\n\\citet{siegel19} demonstrated that accretion disk outflows in collapsars could produce significant amounts of $r$-process material. For different presupernova models, they found the amount of europium varied from $6.0 \\times 10^{-6} \\, M_\\odot$ to $5.8 \\times 10^{-4} \\, M_\\odot$, or $M_r = 0.006$ to $0.579 \\, M_\\odot$ (for our definition $M_r$). Their fiducial model corresponds to $M_r = 0.27  M_\\odot$. Their fiducial yield is about four times larger than our fiducial average yield, but our $M_{r,\\text{min}}$ and $\\avg{M_r}$ values fall within their range of yields.\nIn Figure \\ref{fig:constraints}, the shaded region correspond to the spread of $r$-process yields found by the \\citet{siegel19} simulations.\n\nFurthermore, if we assume that the isotropic energy of a $\\gamma$-ray burst roughly traces the amount of $r$-process yield, we can compare the energies of LGRBs to that of GW170817 to estimate the $M_r$ from collapsars in which the associated jet successfully breaks out of the progenitor star. This assumption predicates on the ideas that (1) the same physical processes act in both short and long GRBs and (2) the accretion phase during which europium is produced roughly coincides with the phase during which the GRB occurs in the source frame, matching assumptions of \\citet{siegel19}. \\citet{cote18} infer that $\\sim 3-15 \\times 10^{-6} M_\\odot$ of europium was ejected from the post-merger accretion disk of GW170817. This translates to $\\sim 0.01 M_\\odot$ of heavy $r$-process material for a europium mass fraction of $X_{\\text{Eu}}$~$=$~$10^{-3}$.\nThe istropic $\\gamma$-ray energy of GW170817 was $E_{\\gamma,iso,GW170817} = 2.1^{+6.4}_{-1.5} \\times 10^{52}$ ergs \\citep{GW170817Eiso}, and from a sample of 468 LGRBs, the mean istropic energy of LGRBs is $E_{\\gamma,iso,LGRBs} \\approx 2.6^{+2.7}_{-0.5} \\times 10^{53}$ ergs \\citep{LGRB_stats}. With these values:\n$$M_{r,collapsar}\\sim M_{r,GW170817}\\frac{E_{\\gamma,iso,LGRBs}}{E_{\\gamma,iso,GW170817}} \\sim 0.1 M_{\\odot}$$ \n(see also \\citealt{Siegel20}). This aligns with our fiducial value of $\\avg{M_r}$. In particular, our fiducial results lie near the intersection of the $r$-process yield expected per LGRB and the fraction of LGRBs per CCSN (see Figure \\ref{fig:constraints}). This supports the possibility that LGRBs are linked to $r$-process production.\n\nFor the final reference mass, we compare to the amount of $r$-process yield that was produced in the $r$-process event that enriched the ultra-faint dwarf galaxy Reticulum II \\citep{Ji16b}. This galaxy preserves $r$-process enrichment from a single prolific event in the early universe. To explain the europium abundances of its stars, it likely experienced an event with a europium yield of $10^{-4.3}$ to $10^{-4.6} M_\\odot$ \\citep{Ji16b}. With $X_{\\text{Eu}}=10^{-3}$, this corresponds to $M_r \\sim 0.04 M_\\odot$. Our $\\avg{M_r}$ value is only slightly higher than this mass. This yield is also consistent with that expected for neutron star mergers \\cite[e.g., the yield estimated from GW170817,][]{siegel2019b,cote18}.\n\n\\subsection{Implications of $\\alpha$: Learning About Collapsar Properties from $r$-Process Abundance Scatter} \\label{subsec:disc_a}\n\nUnfortunately, the current precision on the shape of the [Eu\/Fe] distribution does not provide tight constraints on $\\alpha$, the exponent of our $r$-process yield power law distribution. For $\\avg{N_r} = 3$, any $\\alpha = 2.2-6.0$ can  explain the observed scatter. Our fiducial value of $\\alpha = 2.8$ best fits the data, but the full range of possible values produces similar distribution widths (see Figure \\ref{fig:modelHeatmaps}b).\n\nThe $\\alpha$ constraints from metal-poor stars can be compared to power law distributions of long $\\gamma$-ray burst (LGRB) engine duration, engine luminosity, and isotropic energy. Figure \\ref{fig:constraints} shows our constraints on $\\alpha$ in context with the exponents from these distributions.\n\n\\citet{petropoulou17} modeled the central engines which power LGRBs, determining power law distributions for both the engine luminosities and engine activity times:  $p(L_{engine})\\propto L^{-\\alpha_L}$ and $p(t_{engine})\\propto t^{-\\alpha_t}$. By assuming that more powerful engines can more quickly break out of the collapsing star to produce $\\gamma$-ray signals (with a breakout time that scales with jet luminosity as $L^{-\\chi}$), they show that the shape of the $\\gamma$-ray duration distribution can be uniquely determined by the observed GRB luminosity function. In particular, they determine the power law indexes of the $L_{engine}$ and $t_{engine}$ distributions by connecting them with the observed distributions of luminosities and durations of LGRBs. For $\\chi=1\/3$, \\citet{petropoulou17} find $\\alpha_L=2.4$ and $\\alpha_t=3.5$, while for $\\chi=1\/2$, they constrain $\\alpha_L=2.4$ and $\\alpha_t=4.6$. In addition, by assuming a single breakout time, \\citet{sobacchi17} find a power law distribution for $t_{engine}$ consistent with $\\alpha_t\\sim4$. \n\nFurthermore, we can determine the isotropic energy distribution of LGRBs since $E \\propto L \\times t$. Because both $L_{engine}$ and $t_{engine}$ draw from power law distributions, the distribution of their product follows the distribution of the variable with a smaller $\\alpha$, in this case $\\alpha_L = 2.4$.\n\nOur $\\alpha$ constraint overlaps with all of these values, with the fiducial value falling closer to $L_{engine}$ or $E_{iso}$. Any of these properties could therefore potentially trace the $r$-process yield.\nFor a better constraint on $\\alpha$, we need a significantly lower uncertainty on the observed $\\text{IQR}_{\\text{Eu}}$. Figure \\ref{fig:IQRuncertainty} shows how tightly $\\text{IQR}_{\\text{Eu}}$ must be measured for the stellar samples to improve the $\\alpha$ constraint. This plot was constructed assuming the $\\text{IQR}_{\\text{Eu}}$ is centered on $\\text{IQR}_{\\text{Eu}}=0.50$, as found for the RPA sample. To differentiate between the distributions for $t_{engine}$ and $L_{engine}$ or $E_{iso}$, the $\\text{IQR}_{\\text{Eu}}$ must be measured with uncertainty $<0.05$ dex. This abundance precision is better than what current measurements can achieve in metal-poor stars, though it may become achievable in the future as stellar spectroscopy methods improve.\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{IQR_uncertainty.pdf}\n\\caption{To improve our constraint on $\\alpha$, we must improve our measurement of the stellar $\\text{IQR}_{\\text{Eu}}$ for metal-poor stars. Here we show how the $\\alpha$ constraint is improved for several different $\\text{IQR}_{\\text{Eu}}$ uncertainties. \\label{fig:IQRuncertainty}}\n\\end{figure}\n\nThe $\\text{IQR}_{\\text{Eu}}$ is a robust but very inefficient estimator of the distribution shape. Alternatively, we could use the full distribution shape. This requires a reliable selection function, but would likely not demand 0.05 dex precision.\n\n\\subsection{Neutron Star Mergers vs. Collapsars} \\label{subsec:NSMvscollapsar}\n\nHere we focused on collapsars, demonstrating that a $r$-process production site with a power law distribution inspired by LGRB jet properties can self-consistently reproduce the abundances and scatter observed in metal-poor stars. These results would also apply to another prompt $r$-process site ejecting a Solar $r$-process abundance pattern that scales with a power law, however.\n\nThe possibility that collapsars produce $r$-process material is debated. For example, earlier semi-analytic work on collapsar disk winds by \\citet{Surman06} found that collapsar outflows are too neutron-poor to produce heavy $r$-process isotopes. A recent study by \\citet{Miller19} that investigated the \\citet{siegel19} results with more detailed modeling of neutrino transport also found that collapsar outflows are incapable of producing third peak $r$-process material. Furthermore, \\citet{Macias19} found that any $r$-process site that also produces large amounts of iron is disfavored by observations of metal-poor stars. Collapsars that do not produce large amounts of iron (e.g., LGRBs without an associated supernovae; \\citealt{2006Natur.444.1047F}) would avoid the dilution problems discussed by Macias \\& Ramirez-Ruiz, but the topic is unsettled.\n\nNeutron star mergers are a demonstrated source of $r$-process thanks to GW170817 and, in principle, their europium yields can vary as well with different neutron star binary masses, mass ratios \\citep{Korobkin12,Bauswein13,Hotokezaka13,dietrich15,Sekiguchi16}, and eccentricities \\citep{Chaurasia18,Papenfort18}.\nFurthermore, the $M_{r,\\text{min}}$ and $\\avg{M_r}$ values in our model are roughly consistent with the $r$-process yield estimated for GW170817 \\citep{siegel2019b,cote18}. Because of this, variable-yield neutron star mergers could also potentially explain the $r$-process scatter in metal-poor stars via a similar model to that presented in this paper. More work is needed to determine a reasonable distribution of $r$-process effective yields from neutron star mergers, combining input distributions of binary neutron star properties and yields (e.g., those from the numerical simulations cited above) and kick velocities \\cite[e.g.,][]{Tarumi20, Safarzadeh19b, Bonetti19}.\n\n\\subsection{Limitations of Initial Model} \n\\label{subsec:caveats}\n\nOur initial model is purposefully simple in order to act as a focused exploration of variable-yield collapsars. In particular, the model assumes all abundance scatter is due to variable stellar populations. This assumption allows us to expressly investigate variable yields as a source of scatter, but it does not consider possible effects due to differences in galaxy formation. Real dwarf galaxies have differences in their hierarchical assembly, small amounts of cross-pollution, and experience inhomogeneous mixing \\cite[e.g.,][]{Venn04,Ji15,Griffen18}. Abundance scatter is likely affected by these complexities. Inhomogeneous enrichment has been included in some previous models \\cite[e.g.,][]{Cescutti15,Wehmeyer15}, but it is not a solved problem. In this very metal-poor regime, theoretical work has not yet given a simple way to model the amount of scatter from galaxy formation effects.\n\nThis model also assumes that each star probes an independent gas reservoir. For every star in our model, we assume that it originates from a different dwarf galaxy in which a number of SN exploded over some time, the metals fell back down into the galaxy and fully mixed, and then our model star formed from the mixed gas. This approximates the average star that formed in a given gas reservoir. Real stellar samples likely contain stars that originated together, though. Observational work that studies the accretion origin of stars through, for example, analysis of stellar streams and kinematic clustering will inform the quality of this assumption in the future.\n\nTo transcend the limitations of this initial model, future models will consider scatter due to differences in galaxy formation and include a more detailed treatment of chemical enrichment and star formation. We are currently developing high-resolution hydrodynamic simulations of dwarf galaxy evolution that will study these effects and further explore the origins of $r$-process material.\n\n\\section{Conclusions} \\label{sec:conc}\n\nWe have produced a self-consistent model in which collapsars synthesize all of the $r$-process material in the early universe. By assuming the $r$-process material in metal-poor ([Fe\/H] $< -2.5$) stars was formed exclusively in collapsars with stochastic yields, we can reproduce the observed distribution of europium abundances with parameter values that are consistent with other independently determined reference values. This was not guaranteed a priori.\n\nThis is not evidence that collapsars dominantly produce $r$-process material in the early universe, however. Neutron star mergers with variable effective europium yields may also be able to explain the $r$-process scatter. More work is needed on the effective europium yields of neutron star mergers. In particular, the retention fraction is important for the collapsar model, but it becomes even more important for neutron star mergers with different natal kick velocities and coalescence times.\n\nAbundance scatter of metal-poor stars is an important window into the different mechanisms producing $r$-process elements. Individual mechanisms can produce scatter without the need for multiple sources. In this paper, we assume a power law distribution of collapsar $r$-process yields. The range of constrained values for the exponent $\\alpha$ is comparable to those of the distributions for long $\\gamma$-ray burst isotropic energies, engine luminosities, and engine times (Section \\ref{subsec:disc_a}). Improved constraints on $\\alpha$ could allow us to investigate which, if any, of these collapsar properties trace $r$-process yield. \n\nLastly, in our model, the fraction of core-collapse supernovae that result in $r$-process collapsars, $f_r$, is comparable to the fraction of core-collapse supernovae that result in long $\\gamma$-ray bursts. This could indicate a link between LGRBs and $r$-process. The uncertainty in our model errs to higher $f_r$, though, and if $f_r$ is higher then we would require a significant number of $r$-process collapsars that do not produce long $\\gamma$-ray bursts. Our model also favors a scenario in which $\\lesssim 10$\\% of Type Ic-BL supernovae produce europium. This does not preclude all Ic-BL SNe from harboring choked jets, but would imply that some Ic-BL SNe lack the accretion disk properties to synthesize heavy $r$-process isotopes.\n\n       \n\\acknowledgments\nWe thank Daniel Siegel for providing nucleosynthesis products from his group's collapsar simulations and for helpful discussions.\nWe also thank Paz Beniamini Tony Piro, \nEnrico Ramirez-Ruiz, Phillip Macias, and Anne Kolborg for helpful discussions.\nKB acknowledges support from the United States Department of Energy grant DE-SC0019323.\nAPJ acknowledges support by NASA through Hubble Fellowship grant HST-HF2-51393.001, awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.\nAPJ also acknowledges a Carnegie Fellowship and the Thacher Research Award in Astronomy.\nMRD acknowledges support from the NSERC through grant RGPIN-2019-06186, the Canada Research Chairs Program, the Canadian Institute for Advanced Research (CIFAR), and the Dunlap Institute at the University of Toronto.\nAF acknowledges support from NSF grant AST-1716251, the Silverman (1968) Family Career Development Professorship and thanks the Wissenschaftskolleg zu Berlin for their wonderful Fellow's program and generous hospitality.\n\n\n\\section{} \n\n\\textit{Research Notes of the \\href{https:\/\/aas.org}{American Astronomical Society}}\n(\\href{http:\/\/rnaas.aas.org}{RNAAS}) is a publication in the AAS portfolio\n(alongside ApJ, AJ, ApJ Supplements, and ApJ Letters) through which authors can \npromptly and briefly share materials of interest with the astronomical community\nin a form that will be searchable via ADS and permanently archived.\n\nThe astronomical community has long faced a challenge in disseminating\ninformation that may not meet the criteria for a traditional journal article.\nThere have generally been few options available for sharing works in progress,\ncomments and clarifications, null results, and timely reports of observations\n(such as the spectrum of a supernova), as well as results that wouldn't\ntraditionally merit a full paper (such as the discovery of a single exoplanet\nor contributions to the monitoring of variable sources). \n\nLaunched in 2017, RNAAS was developed as a supported and long-term\ncommunication channel for results such as these that would otherwise be\ndifficult to broadly disseminate to the professional community and persistently\narchive for future reference.\n\nSubmissions to RNAAS should be brief communications - 1,000 words or fewer\n\\footnote{An easy way to count the number of words in a Research Note is to use\nthe \\texttt{texcount} utility installed with most \\latex\\ installations. The\ncall  \\texttt{texcount -incbib -v3 rnaas.tex}) gives 57 words in the front\nmatter and 493 words in the text\/references\/captions of this template. Another\noption is by copying the words into MS\/Word, and using ``Word Count'' under the\nTool tab.}, and no more than a single figure (e.g. Figure \\ref{fig:1}) or table\n(but not both) - and should be written in a style similar to that of a\ntraditional journal article, including references, where appropriate, but not\nincluding an abstract.\n\nUnlike the other journals in the AAS portfolio, RNAAS publications are not\npeer reviewed; they are, however, reviewed by an editor for appropriateness\nand format before publication. If accepted, RNAAS submissions are typically\npublished within 72 hours of manuscript receipt. Each RNAAS article is\nissued a DOI and indexed by ADS \\citep{2000A&AS..143...41K} to create a\nlong-term, citable record of work.\n\nArticles can be submitted in \\latex\\ (preferably with the new \"RNAAS\"\nstyle option in AASTeX v6.2), MS\/Word, or via the direct submission in the\n\\href{http:\/\/www.authorea.com}{Authorea} or\n\\href{http:\/\/www.overleaf.com}{Overleaf} online collaborative editors.\n\nAuthors are expected to follow the AAS's ethics \\citep{2006ApJ...652..847K},\nincluding guidance on plagiarism \\citep{2012AAS...21920404V}.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.85,angle=0]{aas.pdf}\n\\caption{Top page of the AAS Journals' website, \\url{http:\/\/journals.aas.org},\non October 15, 2017.  Each RNAAS manuscript is only allowed one figure or\ntable (but not both). Including the\n\\href{http:\/\/journals.aas.org\/\/authors\/data.html\\#DbF}{data behind the figure}\nin a Note is encouraged, and the data will be provided as a link in the\npublished Note.\\label{fig:1}}\n\\end{center}\n\\end{figure}\n\n\n\\acknowledgments\n\nAcknowledge people, facilities, and software here but remember that this counts\nagainst your 1000 word limit.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Intro}\n\nNew physics searches are often accomplished by looking for excess in cross-section pertaining to specific decay channels of new physics particles. Constraining the measured cross-section to specific decay channels can often reduce the background significantly enough to identify the excess with high confidence level. Thus, it is usually cleaner to search for new physics particles in the leptonic or photonic channels as they offer very small background at the Large Hadron Collider (LHC). However, the cleaner channels often have extremely small cross-section that it makes the searches for new physics difficult or requires a large amount of data to get a significant sample of the signal events. Additionally, leptonic searches can be difficult if the physics or decay channels of the new particle are not known. Further difficulties may arise if there is a missing transverse energy in a leptonic channel. Similar difficulties also arise for the Standard Model (SM) heavy particles, namely, $W,~Z$ bosons, Higgs boson \nand top quarks, except that the physics of these particles is well known. These heavy particles dominantly decay to light quarks and gluons (collectively partons) which then shower and fragment into jets of hadrons. Since jets are invariably produced at the LHC due to hard scattering of the constituent partons, heavy SM particles are often faked by light quark\/gluon radiations and radiations coming from other sources, such as initial state radiation, underlying event\\cite{Barnafoldi:2011ad, Cacciari:2009dp} and pile-up \\cite{Krohn:2013lba}.\nAt present collider energies, these heavy SM particles are often produced with a large Lorentz boost factor and their hadronic decay products are realised as a single collimated 'fat jet'~\\cite{Adams:2015hiv, Altheimer:2013yza, Abdesselam:2010pt, Altheimer:2012mn}. In such cases, the mass of the heavy particle should be reflected in the mass of the fat jet. One may expect to disentangle the signal from the large background of light parton jets, using the jet mass cut\\footnote{In this work, we mainly focus on gluon jets as the background since their cross-section is much larger.}, but as it happens to be the case, gluon jet cross-section has a significant long tail even after accounting for underlying event \\cite{Barnafoldi:2011ad, Cacciari:2009dp} and pile-up \\cite{Krohn:2013lba}, which can be dealt with effectively using the jet grooming techniques, such as soft drop~\\cite{Larkoski:2014wba, Marzani:2017kqd, Dreyer:2018tjj}, pruning \\cite{Ellis:2009su, Ellis:2009me}, trimming \\cite{Krohn:2009th} and mass drop\/filtering \\cite{Dasgupta:2013ihk, Butterworth:2008iy, Marzani:2017mva}.\nFurther discussion on underlying event and pile-up is beyond the scope of this work. As far as this paper is concerned, we will assume that we start with a well-groomed jet. Since a simple mass cut is not effective in identifying the particle originating the jet inspite of grooming, thus a great deal of effort has been made in jet identification by constructing observables that can reject the parton jets and reduce the background~\\cite{Altheimer:2013yza, Abdesselam:2010pt, Altheimer:2012mn, Adams:2015hiv, Larkoski:2017jix, Marzani:2019hun}. The emphasis of such observables is to improve the signal rate to mistag rate ratio. This will also be the spirit of our work. The SM heavy particles provide a perfect playground for improving on such jet identification strategies as their decay channels are well understood. The purpose of our work is to improve upon these strategies by studying a new non-linear jet observable {\\em zest} dependent only on the transverse momentum distribution of particles in the jet, similar to transverse-zeal introduced in the context of jet quenching studies \\cite{Gavai:2015pka}.\n\n\nThe remainder of the paper is organized as follows. In Sec.~\\ref{Zest}, we define zest and discuss its properties. In Sec.~\\ref{sec:gluzest}, we outline the features exhibited by zest distribution of gluon-initiated jets that make it suitable for vetoing the gluon jets. In Sec.~\\ref{Simulation}, we discuss the details of our simulations and provide the principal results of our analysis in Sec.~\\ref{zestFilter}. In Sec.~\\ref{bibFilter}, we introduce another substructure observable {\\it bib} which also depends only on the transverse momentum but linearly and is largely uncorrelated to zest. We contrast the two observables and perform a bi-variate analysis to improve the discrimination ability in Sec.~\\ref{bivariate}. In Sec.~\\ref{pzest}, we  generalize zest by introducing a real parameter $p$ that can be tuned to modify the contribution of the soft particles in the jet. The parameter $p$ can be optimized to provide improved discrimination for the heavy SM particle jets from the gluon background, as discussed in Sec.~\\ref{Z} and Sec.~\\ref{top}. We show that for the top quark-initiated jets, the discrimination provided by generalized zest is in close comparison to recent machine learning (ML) approaches. We propose that studying such infrared and collinear (IRC) unsafe observables may help to uncover the hidden physics behind ML approaches. Since, zest is collinear unsafe and is computable for hadronic final states only, we study its dependence on different hadronization models in Sec.~\\ref{hadModel}. We conclude in Sec.~\\ref{Conclusion}.\n\n\n\\section{Zest of a Jet}\n\\label{Zest}\n\nGiven a well-groomed jet composed of hadrons, reconstructed through a suitable jet algorithm like the anti-k$_t$ clustering algorithm~\\cite{Cacciari:2008gp}, {\\em zest} of the jet is defined as\n\n\\begin{eqnarray}\n\\label{eq:zest}\n\\zeta = \\frac{-1}{\\log \\big (\\sum_{i \\in  \\rm{Jet}} e^{-P_T\/\\vert  {\\bf p}_{T i} \\vert}\\big )} \\, ,\n\\end{eqnarray}\n\nwhere $ P_{T} = \\sum_{i \\in \\rm{Jet}} \\vert \\textbf{p}_{T i} \\vert $ and $\\textbf{p}_{T i}$ is the transverse momentum of the $i^{th}$ particle in the jet with respect to the jet axis.  \nNote that zest is composed purely out of the transverse momenta of the final state particles. For the extreme case of a jet composed of a single energetic particle, it is straightforward to see that $\\zeta$ reduces to $1$. Similarly, for two leading particles in the jet carrying equal fractions of energy, i.e. $P_{T} = 2\\, p_{T}$, we get\n\\begin{equation}\n\\label{eq:zest2pt}\n\\zeta = \\frac{-1}{\\log(2)-2} \\approx 0.765\\, ,\n\\end{equation}  \nThis value remains roughly the same even if the two particles do not carry equal fractions of energy, i.e. for a generic break up of $P_{T} = x\\, p_{T} + (1-x)\\, p_{T}$ with $0<x<1$.\nThis is confirmed also by looking at FIG.~\\ref{fig:2ptzest} where we have plotted the normalized distribution of $10^5$ randomly generated pairs of particles with total $P_{T}=1$. From the plot, we see that the two particle distribution has a sharp peak at about $\\sim 0.77$. A similar analysis done for only three particles in a jet gives a slightly broader multiplicity peak at $\\zeta \\sim 0.53$. Therefore, we see that the zest distribution captures multiplicity peaks whenever there are a few energetic particles in the jet. These multiplicity peaks shift towards smaller and smaller value as the number of particles contributing to the jet increases. In the special case of a jet containing $n$ particles each carrying equal $\\vert  {\\bf p}_{T i} \\vert$, $\\zeta$ reduces to $\\frac{1}{n - \\log n}$. Thus we expect zest to be strongly correlated to {\\em multiplicity of a jet}. Later we will generalize zest such that multiplicity will emerge as a limiting case.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.65]{Zest2pt}\n\\caption{Zest distribution for two leading particles in a jet with a fixed $P_{T}$. We see a dominant two-particle peak at a zest value of 0.76.}\n\\label{fig:2ptzest}\n\\end{figure} \n\nZest has certain interesting properties which makes it a useful observable to study jet substructure : (1) It is invariant under boosts made along the jet axis as it is composed entirely of transverse momentum components of the jet constituents, (2) It is sensitive to transverse momentum distribution of energetic particles while it is largely insensitive to the soft particles in the jet since their contribution to the observable is exponentially suppressed, and (3) Zest is also mostly insensitive to the global color flow of partons. In other words, the zest distribution of the gluon-initiated jets (or any other colored particle) is stable against the change of color flow direction of the colored particles forming the jet. A colored parton that evolves into a color neutral hadron jet recoils against another colored parton. The color neutral final state is obtained due to the exchange of several soft partons. Changing the direction of the recoiling colored partner effects only the exchange of a few soft particles between the partners. Since zest is stable against a few changes in the soft sector, it also becomes stable to the color recoil.\n\nWhile zest is a non-linear and collinear unsafe jet observable due to which it cannot be calculated in perturbation theory, it can be computed using Monte Carlo based event generators~\\cite{Sjostrand:2007gs, Bellm:2015jjp}. Although collinear unsafe, zest may provide a new perspective into the jet substructure that may not be accessible through IRC safe jet observables. In particular, the jet observables based on machine learning techniques, while not calculable through perturbation theory, provide the best discrimination ability~\\cite{Kasieczka:2019dbj,Amram:2020ykb,Bradshaw:2019ipy,Dolen:2016kst,Chen:2019uar,Moreno:2019bmu,CMS:2019gpd}. The physical features extracted by such observables that enable the discrimination remains unknown. We anticipate that studying zest and other IRC unsafe observables may shed some light on the black box offered by neural-network-based observables.  It goes without saying that zest is hadronization model dependent, however we expect that its ability to discriminate between the originating particles will be independent of hadronization models used in a Monte Carlo simulator, as is demonstrated later in Sec.~\\ref{hadModel}.\n\n\\section{Zest distribution of gluon-initiated jets}\n\\label{sec:gluzest}\n\nAs gluon jets, on an average, consist of a larger number of particles, the zest distribution of gluon-initiated jets is expected to peak at smaller values of the observable.  \n\\begin{figure}[t]\n \\centering\n  \\includegraphics[width=0.45\\textwidth]{Gluon_Zest_v3}\n \\caption{Zest distribution of gluon-initiated jets with a jet mass around $W$ boson mass, $Z$ boson mass and top quark mass.}\n \\label{zplot}\n \\end{figure}\nWe validate this in FIG.~\\ref{zplot} by plotting zest distribution for jets initiated by offshell gluons of masses around $W,~Z$ boson and top quark masses. \nWe see that the zest distribution for gluon jets peaks at small $\\zeta \\sim$ 0.1, as expected. Interestingly, we also see that these zest distributions are nearly independent of the jet mass. To investigate this, we have shown the ratio of the mean of $n$ maximum $p_{T i }$ to $P_T$ for $n=1,~2,~5$ and $10$ for various jet masses in TABLE~\\ref{tab:veto}. We find that these ratios are nearly independent of jet mass, which explains why the zest distribution for gluons is insensitive to the jet mass.\n\n\n\n\\begin{table*}\n\\begin{tabular}{|m{2cm}||*{7}{c|}}\\hline\n\\backslashbox[1.5cm]{$m_{J}$}{$n$}\n&\\makebox[5em]{1}&\\makebox[3em]{2}&\\makebox[3em]{5}\n&\\makebox[3em]{10}\\\\\\hline\\hline\n\\center gluons around W mass &0.121 $\\pm$ 0.049 &0.105 $\\pm$ 0.035 &0.081 $\\pm$ 0.018 &0.0617 $\\pm$ 0.0091\\\\\\hline\n\\center gluons around Z mass &0.120 $\\pm$ 0.048 &0.104 $\\pm$ 0.035 &0.079 $\\pm$ 0.018 &0.0603 $\\pm$ 0.0094\\\\\\hline\n\\center gluons around top mass &0.116 $\\pm$ 0.051 &0.099 $\\pm$ 0.037 &0.074 $\\pm$ 0.019 &0.0552 $\\pm$ 0.0102\\\\\\hline\n\\end{tabular}\n\\\\~\\\\\n\\caption{The entries in the table give the ratio of mean of $n$ maximum $\\vert p_{T i}\\vert$ to $P_{T}$ of the jet, that is $\\frac{\\langle max_{n}\\vert p_{T i}\\vert \\rangle}{P_{T}}$. The first set of numbers specify the mean value and the second is the standard deviation calculated over a set of 10000 jets with the specified jet mass.}\n\\label{tab:veto}\n\\end{table*}\n\n\\begin{figure}[t]\n \\centering\n  \\includegraphics[width=0.44\\textwidth]{SoftvsHard}\n  \\includegraphics[width=0.43\\textwidth]{ColorFlow}\n \\caption{(a) Zest distribution of gluon-initiated jets with a hard gluon of energy about 300 GeV when the recoiling gluon is soft (blue) and hard (pink). (b) Zest distribution of gluon-initiated jets of energy about 300 GeV when the soft recoiler is taken at $30^{\\circ}$ (green), $60^{\\circ}$ (red), $90^{\\circ}$ (light blue), $180^{\\circ}$ (black) vs. a hard recoiler at $180^{\\circ}$ (purple). } \n \\label{colorflow}\n \\end{figure}\n \n \\begin{figure}[t]\n \\centering\n  \\includegraphics[width=0.44\\textwidth]{Gluon_Zest_incl_excl}\n \\caption{Zest distribution of gluon-initiated jets around top quark mass by including or excluding a few soft particles $< 3$ GeV to the jet.}\n \\label{softphysics}\n \\end{figure}\n\n\nWe also verify color flow independence of zest, discussed in the previous section, by looking at the zest-distribution of gluon jets recoiling against complementary color gluon at various angles. The result is shown in FIG.~\\ref{colorflow}(a) and \\ref{colorflow}(b). In FIG.~\\ref{colorflow} we have considered jets of jet energy about 300 GeV, for two different cases: (i) a hard gluon of energy 300 GeV recoiling a soft gluon of energy 1 MeV at $180^\\circ$, and (ii) a hard gluon of energy 300 GeV recoiling a hard gluon of energy 200 GeV at $180^\\circ$. From the plot in FIG.~\\ref{colorflow}(a), we note that the choice of the recoiling partner energy does not strongly effect the zest distribution. Furthermore, from  FIG.~\\ref{colorflow} (b), we observe the color stability of gluon-initiated jets against the direction of its recoiling color neutralising partner. Since, the energy as well as the direction of the recoiling colored particle does not largely effect the analysis, for the rest of the paper, we will take the recoiling gluon as soft and back-to-back to the primary hard gluon.\n\nAs zest is largely insensitive to soft particles in the jet, its distribution curve is not effected by including or excluding a few soft particles in the jet. We verify this in FIG.~\\ref{softphysics} where we have plotted the zest distribution curves for gluon-initiated jets of mass around the top quark mass with following three variations: (i) including all the particles in the jet, (ii) excluding all the soft particles less than $3$ GeV from the jet (iii) including few extra randomly generated soft particles less than $3$ GeV in the jet. We see that the zest distribution shifts only very slightly. \n\n\nThe stability of the gluon zest distribution against jet mass variation, color flow direction, and inclusion\/exclusion of few soft particles makes it a potentially useful observable for vetoing gluon jets at the LHC.\nFor instance, a zest cut at about $\\zeta \\sim 0.3$ is expected to largely cut out the gluon background to heavy SM particles and hopefully also for heavier new physics resonances.\n\n\n\\section{Simulation Details}\n\\label{Simulation}\n\nThe details of our Monte Carlo simulations are presented below.\n\n\\begin{enumerate}\n\n\\item All jets are simulated using {\\sc Pythia} 8 \\cite{Sjostrand:2007gs} by inserting $W$ and $Z$ bosons with energy 500 GeV along the central axis (y-axis) and allowing them to decay to hadronic modes only.\\footnote{For the study presented here, we have turned off all the strong, weak and electromagnetic decays of the primary hadrons formed from showering and subsequent hadronization. \nWe do not study detector analysis in this paper.\n} \n\n\\item We collect all the particles in the forward hemisphere and call it a jet. Apart from hemisphere jets, sometimes we also construct anti-k$_t$ jets with different jet radii, R = \\{0.5, 0.7, 0.9, 1.2\\}, using the {\\sc Fastjet} package~\\cite{Cacciari:2011ma} in {\\sc Pythia}.\n\n\\item For simulating a color-singlet top quark jet, we perform an $e^+\\,e^-$ annihilation and allow the intermediate $Z$ boson to decay to a top-antitop pair at 500 GeV each, travelling back-to-back. We further constrain the top decay channels to hadrons only. \nJets are constructed by dividing the event into two hemispheres using the thrust axis.\n\n\\item To shower a color singlet gluon-initiated jet, we insert an energetic gluon of energy 500 GeV and offshellness same as the heavy particle mass in the event record along the y-axis, while a soft onshell gluon of energy 1 MeV with opposite color is inserted in the opposite direction to the +y-axis. The direction of the color conserving gluon partner of the hard gluon does not effect the analysis, as zest is stable to global color flow of the partons (see FIG.~\\ref{colorflow}). For convenience, we choose the direction of the recoiling gluon opposite to that of hard gluon.\n\n\\item The energetic gluon jet is identified by constructing the thrust axis and then taking all particles in the forward hemisphere.\n\n\\item The resulting gluon jets are accepted if their masses are within 10 GeV of the corresponding heavy particle mass. Although the energy of the jet does not effect zest, we observe that the selected gluons jets have energy ranging between 485 GeV to 500 GeV. \n\n\n\n \\begin{figure*}\n\\centering\n  \\includegraphics[width=0.62\\textwidth]{zest1}\n  \\caption{Zest distribution for $W,~Z$ boson and top quark initiated jets along with gluon jets of corresponding masses.}\n \\label{zestplot}\n \\end{figure*}\n\n\\item  In order\nto test the sensitivity of our observable to non-perturbative physics, we also generate jets in two distinct ways: (a) by varying the default parameters of the hadronization model implemented in {\\sc Pythia}, thereby producing a new set of final state events, and (b) by utilizing a different model for non-perturbative hadronization effects by using the {\\sc Herwig}~\\cite{Bellm:2015jjp} event generator. We study this non-perturbative physics dependence for $Z$ boson-initiated jet along with corresponding gluon background in Sec.~\\ref{hadModel}.\n\n\\item The {\\sc Herwig} event is sampled using the process $e^+\\, e^- \\to {\\rm Z} \\to {\\rm jets}$ at a center of mass energy equal to the mass of the $Z$ boson, i.e., $E_{\\rm cm} = m_Z$. We then boost all the final state particles along a direction, which we take as the +y-axis. The resulting boosted jet has a mass about the $Z$ boson and energy around 500 GeV.\n\n\\item In this work, we will not study Higgs-initiated jets because a Higgs boson dominantly decays to a $b\\,\\bar{b}$ pair which receives a large background from $Z\\rightarrow b\\,\\bar{b}$ and $g\\rightarrow b\\,\\bar{b}$, in which case the analysis proposed here does not offer a substantially good discrimination ability.\n\\end{enumerate}\n\n\\section{Zest as Filter}\n\\label{zestFilter}\n\n \nAs noted earlier, the zest distribution of gluon-initiated jets peaks at small values of the observable in comparison to their heavy particle counterparts. This accounts for an appreciable distinguishing criterion offered by zest in characterization of heavy particle jets from the background gluon jets.\nWe analyze this discriminating ability provided by zest by looking at the $\\zeta$-distribution curves for $W,~Z$ boson and top quark-initiated jets along with the corresponding gluon-initiated jets with a mass about the heavy particle masses. This comparison is shown in FIG.~\\ref{zestplot}. From the plot, we see that the zest distribution of the gluon-initiated jets peaks at relatively smaller values of $\\zeta$ compared to their heavy particle counterparts. In addition, the zest distribution of gluon-initiated jets is found to be narrow, independent of jet energy and has a little overlap with a similar distribution of most other heavy particle jets. Moreover, we find that for the top quark-initiated jet, the zest distribution peaks at a slightly higher value than $W$ or $Z$ boson-initiated jets.\n\nThe tagging efficiency of a zest-based binary classifier can be further evaluated by comparing the performance curves of these jets. In FIG.~\\ref{ROC}, we present the relative operating characteristic (ROC) curves for the zest-based filter. We note that zest provides good signal statistics along with a high background rejection rate. For example, we note that for all the heavy particle jets considered, nearly 80 to 90\\% of the signal stays in the accepted sample after applying a zest cut to remove 90\\% of the gluons.\n\n\\begin{figure}\n\\centering\n \\includegraphics[scale=0.35]{zestROC1}\n  \\caption{ROC curves for $W,~Z$ boson and top quark-initiated jets using {\\it zest} as the binary classifier.}\n \\label{ROC}\n\\end{figure}\n\n \\begin{figure}\n\\centering\n  \\includegraphics[width=0.47\\textwidth]{ZJets_VaryingR}\n  \\caption{Zest distribution for $Z$ boson-initiated jets with different jet radii. Here jets are constructed using the anti-kT clustering algorithm for different $R$ values.}\n \\label{ZJetsR}\n \\end{figure}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{bibplot}\n\\caption{$bib$-distribution of heavy particle initiated jets and gluon jets forming the background.}\n\\label{bibplot}\n\\end{figure*}\n\n\n\n\nIn experiments, jets are often constructed by a suitable jet algorithm employing a cut on the jet cone radius $R$. At large jet radius, the wide angle soft particles contribute to the jet, while as the jet radius decreases, the observable receives contribution from collinear particles in the jet and a few soft particles. Since zest is not significantly effected by the inclusion or exclusion of a few soft particles, we anticipate the zest distribution to be stable against change in the jet cone radius. This is confirmed by FIG.~\\ref{ZJetsR}, where $\\zeta$-distribution curves with different jet radii, namely R = \\{0.5, 0.7, 1.0, 1.2\\}, are presented for $Z$-boson initiated jets. From the plot, we also note that the two particle peak at $\\zeta \\approx 0.77$ gets enhanced when the jet radius is sufficiently small, aligned with our expectation.\n\\section{Boost-invariant Broadening as Filter}\n\\label{bibFilter}\n\nTo contrast zest with an observable linearly dependent on transverse momentum $\\vert {\\bf{p}}_{T}\\vert$ of the particles, we introduce a simple observable \\textit{boost-invariant broadening} or $bib$ (for brevity) which is defined by the equation,\n\\begin{equation}\nbib = \\frac{1}{m_{J}} \\sum_{i\\epsilon Jet} \\vert {\\bf p}_{T i} \\vert = \\frac{P_T}{m_J}\\,.\n\\label{bib}\n\\end{equation}\nHere as well, all transverse momenta are measured with respect to the jet axis and $P_T$ is the same quantity as defined in the context of zest. It is interesting to note that $bib$ is similar to the event shape observable \\textit{broadening}~\\cite{Dokshitzer:1998kz}, however the use of $ m_{J}$ instead of $E_{J}$ renders it invariant under boosts made along the jet axis. Zest and $bib$ have boost-invariance along the jet axis and jet energy independence in common with each other. Although there can be other observables which can be chosen for a bi-variate analysis, $bib$ provides a simple contrast to zest : (i) $bib$ is a linear function of the transverse momentum of the particles while zest is not, (ii) $bib$ is calculable using standard perturbative techniques\\footnote{As $bib$ is defined by the ratio of two IRC safe observables, it is essentially Sudakov safe~\\cite{Larkoski:2013paa, Larkoski:2015lea} which implies that even though the observable does not has a valid fixed order expansion in $\\alpha_s$, it can still be calculated using all-order resummation. This ensures that the singular region of the phase space for the observable is exponentially suppressed.}, while zest is collinear unsafe and relies on a suitable Monte Carlo generator that encodes a definite model to capture the hadronization effects for its computation, and (iii) $bib$ is sensitive to the global color flow of partons while zest is insensitive to it. The results presented in this section and for the  bi-variate analysis in the next section, are strictly for the case in which the soft gluon is back-to-back with the primary hard gluon.   \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47\\textwidth]{bibROC1}\n\\caption{ROC curves for heavy particle jets with $bib$ as the classifier.}\n\\label{bibROC}\n\\end{figure}\n\nThe physics of the primary splitting of heavy particles leading to partonic decays is different from that of branching of quarks and gluons. Parton splittings dictated by Quantum Chromodynamics (QCD) are dominated by soft and collinear radiation having small transverse momenta, thus we expect gluon initiated jets to have a peak at a relatively smaller value in $bib$-distribution compared to jets initiated by heavy particles at the same jet mass.\nThe comparison of $bib$-distributions for heavy particles and corresponding gluon jets is shown in FIG.~\\ref{bibplot}.\nWe also present the ROC curves for $bib$ when used as a binary classifier in FIG.~\\ref{bibROC}. From the plot, we find that $bib$ acts as an appreciable classifier for $W,~Z$ boson-initiated jets but does not offer good discrimination for heavier top quark-initiated jets. This is in contrast with the zest ROC curves of FIG.~\\ref{ROC}. Overall, zest provided better classifier performance compared to $bib$. Nonetheless, $bib$ also contrasts zest in the stability of gluon distributions with respect to the jet mass. We have also verified that the two observables do not have a significant linear correlation. Thus, we expect that the two observables in conjunction can be used to improve the signal to background ratio. We present this bi-variate analysis in the next section.\n\n\n\n\\section{Bi-variate Analysis}\n\\label{bivariate}\n\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.48\\textwidth]{scatterPlot}\\\\\n  \\includegraphics[width=0.35\\textwidth]{scatterplot1}\n \\caption{(a)-(c) Scatter plots in $bib$-zest plane for heavy particle and corresponding gluon jets.}\n \\label{scatter}\n\\end{figure}\n\n\nDiscrimination for the originating particle can be further improved with a multivariate analysis. Recently, multivariate approaches are becoming extremely popular as they are expected to provide a more detailed characterization of the QCD radiation pattern within a jet which can be exploited to further enhance the new physics searches at the LHC~\\cite{Larkoski:2017jix, Marzani:2019hun}. For the bi-variate analysis proposed here, we generate scatter plots in the \\textit{bib}-zest plane for gluon initiated jets and heavy particle jets, as shown in FIG.~\\ref{scatter} (a)-(c).  \nWe note that, in general, gluon jets occupy the extreme left corner of the $bib$-zest plane, while heavy particle jets are mostly concentrated towards the upper right region in the plot, thus providing statistical discrimination between the two types of jets.\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.48\\textwidth]{ExclInclW.pdf}\n \n \\caption{Scatter plot in $bib$-zest plane for $W$-boson jet and the corresponding gluon jet, with the 90\\% Gaussian contour and inclusion and exclusion-zone cuts specified on it.}\n \\label{Wscatter}\n\\end{figure}\n\n\n\\begin{figure*}\n\\centering\n \\includegraphics[width=0.7\\textwidth]{fig1b}\n \\caption{ROC curves for heavy particle jets with zest, inclusion and exclusion$-$filter based binary classifiers.}\n \\label{allRoC}\n\\end{figure*}\n\nThe zest-$bib$ distributions shown in FIG.~\\ref{scatter} are reasonably well-approximated by an asymmetric two-dimensional Gaussian distribution. We fit both the heavy particle and gluon jet distributions with an asymmetric Gaussian distribution. On the fitted Gaussian distributions, we draw contours of different heights. An example for the 90\\% contour, illustrated with a dashed line, is shown in FIG.~\\ref{Wscatter} for the $W$-boson initiated jets along with the corresponding gluon background. By 90\\% contour, we refer to the closed curve that includes 90\\% of the jets inside it. The tangents are drawn parallel to $bib$-axis and zest-axis on the gluon distribution contour to intercept with the plot frame at the bottom and to the left, respectively. The solid line represents an open curve made by combining the tangents and right-side of the contour meeting the tangents. The background region is defined as the area to the left of the solid curve. Similar exercise is also done on the heavy particle initiated jet distribution. The signal region is defined as the region to the right of the solid curve on the heavy particle distribution. We define two new cuts (1) the  `inclusion-zone' cut and (2) the `exclusion-zone' cut. The `inclusion-zone' cut is the solid curve on the signal while the `exclusion-zone' cut is the solid curve on the background. We note that for the case of a $W$ boson-initiated jet with 90\\% of the signal as the desired value, an exclusion-zone cut provides better statistics over an inclusion-zone cut.\n\n\n\n\n\nThe relative performance of these bi-variate cuts in comparison to zest alone is studied through ROC curves in FIG.~\\ref{allRoC}. We present three types of ROC curves for each type of heavy particle jet (a) zest based cuts only (represented by dashed lines), (b) cuts based on inclusion-zone obtained by including the percentage of events contained within the primary contour (represented by dotted lines), and (c) cuts based on exclusion-zone obtained by excluding the percentage of events contained within the primary contour (represented by solid lines). From the plot, we note that for a high signal rate, exclusion-zone statistics provides slightly better discrimination than only zest-based cuts for the $W,~Z$ bosons, however inclusion\/exclusion-zone provide no significant improvement in comparison to a zest only filter for the top quark-initiated jet.\nOn the other hand, inclusion-zone statistics is more efficient when high gluon rejection is required and small signal rate is acceptable.\n\n\n\\section{Generalized Zest ($p$-zest)}\n\\label{pzest}\nAs pointed out earlier in Sec.~\\ref{Zest}, in the special case of $n$-equal transverse momenta particles in the jet, zest is a function of {\\it particle multiplicity} in a jet. In this section, we aim to generalize the observable such that the particle multiplicity emerges as a limiting case of generalized zest. We define the generalization of zest (or, `$p$-zest' for brevity) as follows:\n\n\\begin{equation}\n\\zeta_p = -\\frac{1}{\\log\\Big(\\sum_i\\, e^{-P_T^{(p)}\/\\vert {\\bf p}_{T i}^{p}\\vert}\\Big)}\\, .\n\\end{equation}\nHere the sum is over all the final state particles in the jet, as earlier, and $P_{T}^{(p)}$ is defined as\n\\begin{equation}\nP_T^{(p)} = \\sum_i \\vert {\\bf p}_{T\\,i}^{\\,p} \\vert \\, ,\n\\end{equation}\nwhere $p > 0$. The ability to tune the parameter $p$ may also provide a new way of looking at the jet substructure by allowing us to vary the net contribution of the soft sector.\n\nFor $p > 1$, the contribution of the soft particles to the determination of $P_T^{(p)}$ will reduce. \nThus, mostly collinear or energetic particles will contribute to $p$-zest.\n\n \\begin{figure}\n \\centering\n  \\includegraphics[width=0.475\\textwidth]{zvsmult_Mass_1}\n  \\includegraphics[width=0.48\\textwidth]{zvsmult_Mass_2}\n \\caption{(a)Zest distribution of gluon jets at various heavy particle mass windows, and (b) $1\/n$ distribution of gluon jets at various heavy particle mass windows.}\n \\label{zvsmult_Mass}\n \\end{figure}  \n \n\nIn the extreme limit of $p \\to \\infty$, only the leading parton contributes and $\\zeta_{\\infty}$ approaches 1. Therefore, this limit offers no useful jet discrimination ability.   \nIn contrast, in the limit $p \\to 0$, all particles contribute equally giving\n\\begin{equation}\n\\zeta_0 = \\frac{1}{n-\\log n} \\, ,\n\\label{eq:mult}\n\\end{equation}\nwhere $n$ is the number of particles in the jet. For a large enough $n$ such that $n \\gg \\log n$, the above expression reduces simply to the inverse of particle multiplicity.  \nThus, we note that closer the value of $p$ is to 0, the stronger is the correlation of $p$-zest and multiplicity. As $p$ is increased, the correlation becomes smaller and vanishes for the extreme limit of $p \\to \\infty$.\nThough the observables zest (with $p=1$) and multiplicity are correlated,  \nzest exhibits interesting properties such as stability against change in the jet mass and stability against change in the jet radius for gluon-initiated jets, which are absent in the $1\/n$ distribution. This is confirmed by the plots in FIG.~\\ref{zvsmult_Mass} and FIG.~\\ref{zvsmult_R}. In FIG.~\\ref{zvsmult_Mass}(a) and~\\ref{zvsmult_Mass}(b), the zest and the $1\/n$ distribution curves for gluon-initiated jets with masses about heavy particle mass, are presented.   \nFrom the plots, we see that the $1\/n$ distribution shifts to the left as the mass of the jet is increased while the zest distribution remains more or less unchanged. \\footnote{Note that this is not to say that $1\/n$ does not offer gluon jet vetoing ability, instead it may very well provide the ability to veto gluon jets like zest. From the trend observed in FIG.~\\ref{zvsmult_Mass}(b), we expect the heavier jets to shift further to the left and \nhence a cut at around 0.04 may cut out the gluon jets largely. However, as we will show in Sec.~\\ref{top}, for the heavy top quark-initiated jets, zest offers an appreciable improvement over multiplicity.}\n \\begin{figure}[t]\n \\centering\n  \\includegraphics[width=0.48\\textwidth]{zvsmult_R_1}\n  \\includegraphics[width=0.47\\textwidth]{zvsmult_R_2}\n \\caption{(a)Zest distribution of $Z$ boson-initiated jets for different $R$ values, and (b) $1\/n$ distribution of $Z$ boson-initiated jets for varying jet radii.}\n \\label{zvsmult_R}\n \\end{figure}  \nSimilarly, the stability of the zest distribution curves to varying the jet radius is shown in FIG.~\\ref{zvsmult_R}(a) for a $Z$ boson-initiated jet of jet energy 300 GeV while the corresponding $1\/n$ curves are shown in FIG.~\\ref{zvsmult_R}(b)\\footnote{Here we have used jets of initial energy 300 GeV to demonstrate this difference. This is because a $Z$ boson at 500 GeV is extremely collimated and $R = 0.5$ is sufficient to capture all the radiations originating from this extremely boosted $Z$ boson.}. From the plots, we clearly see that the zest distribution curves are mostly stable to the change in jet radius while for the $1\/n$ distribution, the curve for $R=0.5$ appears to be shifted away from other $R$ values. This is expected as zest is mostly uneffected by the inclusion or exclusion of a few soft particles while the multiplicity curves are somewhat sensitive to it.\n\nNoting these differences, we investigate the discrimination ability offered by $p$-zest over particle multiplicity in a jet and zest ($p$=1). It will be interesting to find $p$ that provides the best discrimination. For this,  we consider two type of heavy particle jets : (a) a $Z$ boson-initiated jet, and (b) a top quark-initiated jet. \nThe detailed analysis of these results is presented in the following subsections.\n\n\n\\subsection{$Z$ boson-initiated jet}\n\\label{Z}\nFor the $Z$ boson-initiated jet, we generate the ROC curves for $p \\!=\\!\\{0.3, 0.5, 1, 1.5, 2, 3\\}$ and particle multiplicity, as shown in FIG.~\\ref{ZFilter}. The solid cyan line with open squares on it illustrates the multiplicity distribution while the other solid lines with the corresponding legends represent the $p$-zest distributions. From the plots, we find that multiplicity provides a better discrimination in comparison to zest ($p$ = 1) while $p$-zest with $p = 0.3-0.5$ provides the optimal discrimination.\nThis is also confirmed by looking at FIG.~\\ref{ZFilter1} where with 80\\% of signal efficiency as the accepted value, we plot the curve for the background jet rejection offered by $p$-zest with varying values of parameter $p$. From the plot, we find that $p=0.3-0.5$ provides the best discriminating ability for a $Z$-boson initiated jet, although it is not significantly better than just multiplicity.\n\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.5\\textwidth]{ZFilters}\\\\\n\\caption{ROC curves for $Z$ boson-initiated jets with various choices of $p$ compared against the inverse of multiplicity as a filter.}\n \\label{ZFilter}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47\\textwidth]{ZDiscriminate}\n\\caption{Discrimination power offered by $p$-zest at fixed signal efficiency of 80\\% for $Z$-boson initiated jets.}\n \\label{ZFilter1}\n\\end{figure}\n\n\\subsection{top quark-initiated jet}\n\\label{top}\n\nThe ROC curves for top-initiated jets with $p$-zest and $1\/n$ as the observables are presented in FIG.~\\ref{TopFilter}. Interestingly, here we find that $p$-zest for $p= 0.3 - 1.0$ performs significantly better than the $1\/n$ distribution. The same is confirmed further by the background rejection efficiency curve with respect to $p$ at fixed signal efficiency of 80\\%, as shown in FIG.~\\ref{TopDiscriminate}.\n\n\nFrom this analysis, we find that $p$-zest allows us to find the $p$ value that optimizes the discrimination. We note that a $p$ value of about $0.3$ to $0.5$ is optimal for the heavy particles studied here. \n\n\nIt is interesting to note that the discrimination ability offered by our preliminary $p$-zest study for the top quark-initiated jets is comparable with that of a class of ML-based top taggers ~\\cite{Kasieczka:2019dbj}. Although our study does not include proton-proton collisions and detector effects considered in~\\cite{Kasieczka:2019dbj}, we are not claiming a strict comparison of our results with ML-based techniques. While these ML algorithms offer the highest discrimination ability for heavy particle jets, the particular features that offer this improvement remain largely unknown. $p$-Zest may offer some insight into the physics of ML-based taggers. Therefore, we propose that studying IRC unsafe observables based on physical principles may provide a window into the physics hidden in ML based techniques.\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.5\\textwidth]{TopFilters}\n\\caption{ROC curves for top quark-initiated jets with various choices of $p$ compared against the inverse of multiplicity as a filter.}\n \\label{TopFilter}\n\\end{figure} \n\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.47\\textwidth]{TopDiscriminate}\n\\caption{Discrimination power offered by $p$-zest at fixed signal efficiency of 80\\% for top quark initiated jets.}\n \\label{TopDiscriminate}\n\\end{figure}\n\n\\section{Hadronization Model dependence}\n\\label{hadModel}\n\nAs pointed out earlier, $p$-zest is collinear unsafe and cannot be calculated by standard perturbative techniques, therefore we rely on Monte Carlo event generators that incorporate a suitable hadronization model for its computation. In this section, we study the effect of varying hadronization models on zest. This will allow us to understand how the observed value of zest is sensitive to the modeling of non-perturbative effects. This check is important as the observable can only be computed for hadronic final states.\n\n\\begin{figure*}\n\\centering\n \\includegraphics[width=0.7\\textwidth]{hadronPlot}\\\\\n \\includegraphics[width=0.7\\textwidth]{hadron}\n\n \\caption{(a) Zest distribution curves for $Z$ boson-initiated jets with various choices of modelling the non-perturbative physics, and (b) ROC curves for $Z$ boson-initiated jets by varying the hadronization parameters in {\\sc Pythia}.}\n \\label{hadronization}\n\\end{figure*}\nTo study the hadronization model dependence of zest, we vary the non-perturbative physics modelling by: a) changing the hadronization parameters in {\\sc Pythia}, and b) changing the hadronization model by using the {\\sc Herwig} event generator to simulate the event. This will modify the spectrum of the final state primary hadrons, therefore allowing us to study the hadronization model dependence for zest.\n\nIn FIG.~\\ref{hadronization} (a), we present the zest-distribution curves for a $Z$ boson-initiated jet with different clustering mechanisms for implementing the long distance physics. The {\\sc Pythia} 8 event generator that we use for simulating the events, incorporates the Lund string model of hadronization \\cite{Andersson:1983ia} to cluster final state partons into hadrons. This model is built upon a ``string\" analogy, i.e. as the separation between the two partons increases, the potential energy stored in the string rises linearly. At some point, the potential energy stored is so large that the string breaks forming a $q\\bar{q}$ pair leading to individual hadrons, with the `fragmentation function' given by,\n\\begin{equation}\n  f(z) \\propto z^{-1} (1-z)^{a} \\exp(-b\\, m_{\\perp}^{2}\/z)\\, ,\n  \\label{lund}\n\\end{equation}\nwhere $z$ is the energy fraction carried by the hadron and $m_{\\perp}$ is its transverse mass while $a, b$ are some adjustable parameters.   \nChanging $a$ and\/or $b$ in the above equation, changes the way fragmentation happens, hence modifying the final state spectrum.\\footnote{The default values for the parameters $a$ and $b$ used in {\\sc Pythia} 8 are 0.3 and 0.8 respectively. Here\nwe will vary these values to their extremum limits by setting $a$ to 1.8 and $b$ to 2.} The zest-distribution shifts as shown in FIG.~\\ref{hadronization}(a). Moreover, we observe a similar shift for the gluon zest distribution (background) and the performance curves change only very slightly, as shown in FIG.~\\ref{hadronization}(b). For {\\sc Herwig}, we cannot simulate single offshell gluons, thus we do not have corresponding curve for the gluon zest and ROC. However, from the heavy particle distribution for the {\\sc Herwig} simulated event shown by the solid brown curve (with filled triangles) in FIG.~\\ref{hadronization}(a), we see that the zest distribution lies well within the extreme bounds of hadronization parameters used in {\\sc Pythia}, shown by the solid magenta (filled squares) and green (filled diamonds) curves in FIG.~\\ref{hadronization}(a). \nThus, from our analysis we conclude that although zest is not calculable perturbatively, but zest based discrimination is stable against different hadronization models.\n\nSince zest is collinear unsafe, its discrimination ability may depend upon the resolution of the detector. We have also verified that while the zest curves shift if we coarse-grain the angular resolution between the particles for both the gluon-initiated and heavy-particle-initiated jets, they shift systematically such that the discrimination ability of the observable remains unaffected.\n\n\n\\section{Conclusion}\n\\label{Conclusion}\n\nWe have presented a new jet substructure observable, {\\em zest}, and discussed its potential for discriminating standard model heavy particles forming jets from the QCD background of gluon-initiated jets. We have shown that the zest distribution of gluon-initiated jets is stable against the change in jet mass, change of global color flow of the partons and inclusion or exclusion of a few soft particles into\/from the jet. These properties make it a suitable observable for vetoing the large gluon background at the colliders. Though zest is a non-linear and collinear unsafe observable, we have demonstrated that zest-based discrimination is largely insensitive to different hadronization models. We have shown that zest can be generalized through a real parameter, $p$, which can be optimized to further enhance the discrimination ability. A $p$ value between 0.3 to 0.5 is found to provide the optimal discrimination ability for all the heavy particle jets. We have also shown that for a top quark-initiated jet, $p$-zest provides a significant improvement over particle multiplicity in a jet. The discrimination provided by $p$-zest (with $p=0.3-0.5$) for the top quark-initiated jets approaches ML-based results, and we propose that looking at other IRC unsafe jet observables may help to uncover the physics hidden through such ML-based techniques.\n\n\\begin{acknowledgments}\nWe thank Tuhin Roy for discussions on some of the aspects of this paper and comments on the manuscript.\n\\end{acknowledgments}\n\n\n\\bibliographystyle{unsrt}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzziheu b/data_all_eng_slimpj/shuffled/split2/finalzziheu
new file mode 100644
index 0000000000000000000000000000000000000000..7e89bcb7f62cab25149a8db97f86023399e181ed
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzziheu
@@ -0,0 +1,5 @@
+{"text":"\\section{Lie-Rinehart algebras} \\label{sec1}\n\nThroughout this paper $k$ will denote a field\nof characteristic $0$.\nIn this section we will briefly recall the definition\nof Lie-Rinehart algebra and mention some basic examples.\nLie-Rinehart algebras were introduced and studied\nby Herz \\cite{Her}, Palais \\cite{Pal} and Rinehart \\cite{Rin}.\nThey form the algebraic counterpart of the more geometric\nnotion of Lie algebroid, which has become better known.\nIt was Huebschmann \\cite{Hue} who gave Lie-Rinehart algebras their name,\nand also emphasized their advantages over Lie algebroids\nin contexts where singularities arise.\nSee also \\cite{Hue2} for an excellent survey.\n\nLet $R$ be a unital {\\em commutative} algebra over $k$.\nA {\\em Lie-Rinehart algebra over} $R$ is a Lie algebra\n$L$ over $k$, equipped with a structure of a unital left $R$-module\nand a homomorphism of Lie algebras $\\rho\\!:L\\to\\mathord{\\mathrm{Der}}_{k}(R)$\ninto the Lie algebra of derivations on $R$,\nwhich is a map of left $R$-modules\nsatisfying the\nfollowing Leibniz rule\n$$ [X,rY]=r[X,Y]+\\rho(X)(r)Y $$\nfor any $X,Y\\in L$ and $r\\in R$. We shall write $\\rho(X)(r)=X(r)$.\n\nNote that in \\cite{Hue} and \\cite{Rin},\na Lie-Rinehart algebra over a fixed $R$ is\nreferred to as $(k,R)$-Lie algebra.\n\n\n\\begin{ex} \\label{ex1.1} \\rm\n(1)\nIf $R=k$, then $\\mathord{\\mathrm{Der}}_{k}(R)=\\{0\\}$, and a Lie-Rinehart\nalgebra over $R$ is simply a Lie algebra over $k$.\n\n(2)\nFor arbitrary $R$, the Lie algebra $\\mathord{\\mathrm{Der}}_{k}(R)$ is\nitself a Lie-Rinehart algebra over $R$ if one takes $\\rho$\nto be the identity map.\n\n(3)\nLet $k=\\mathbb{R}$. If $\\pi\\!:E\\to M$ is a vector bundle\nover a smooth manifold $M$, then the structure\nof a Lie-Rinehart algebra on the $\\mathord{\\mathit{C}^{\\infty}}(M)$-module\n$\\Gamma(E)$ of sections of $E$ is the same as the structure\nof a Lie algebroid on $E$.\n\n(4)\nLet $L$ be a Lie-Rinehart algebra over $R$ and\n$\\tau\\!:R\\to S$ a homomorphism of unital commutative\n$k$-algebras. Then we can form a Lie-Rinehart\nalgebra $\\tau_{!}(L)$ over $S$ as the kernel\nof the map $\\varphi$,\n$$\n\\xymatrix{\n0 \\ar[r] & \\tau_{!}(L) \\ar[r] &\n(S\\mathbin{\\otimes}_{R}L)\\oplus \\mathord{\\mathrm{Der}}_{k}(S) \\ar[r]^-{\\varphi} &\n\\mathord{\\mathrm{Hom}}_{k}(R,S)\\;,\n}\n$$\ndefined by\n$$ \\varphi(s\\mathbin{\\otimes} X,D)(r)=s\\tau(X(r))-D(\\tau(r))\\;.$$\nThe bracket on $\\tau_{!}(L)$ is given by\n$$ [(s\\mathbin{\\otimes} X,D),(t\\mathbin{\\otimes} Y,E)]=\n   (st\\mathbin{\\otimes} [X,Y] + D(t)\\mathbin{\\otimes} Y - E(s)\\mathbin{\\otimes} X , [D,E] )\\;,$$\nwhile the representation of $\\tau_{!}(L)$ on $S$\nis given by the projection.\nIn the special case where $\\tau$ is a localization\n$R\\to R_{\\mathfrak{p}}$ of $R$ at a prime ideal $\\mathfrak{p}$ of $R$,\none can check that $\\tau_{!}(L)$ is isomorphic\nto the localization $R_{\\mathfrak{p}}\\mathbin{\\otimes}_{R}L=L_{\\mathfrak{p}}$, with the bracket\n$$ [s^{-1}X,t^{-1}Y] =\n   (st)^{-1}[X,Y] - s^{-1}t^{-2}X(t)Y + t^{-1}s^{-2}Y(s)X $$\nand representation $\\rho_{\\mathfrak{p}}\\!:L_{\\mathfrak{p}}\\to\\mathord{\\mathrm{Der}}_{k}(R_{\\mathfrak{p}})$ induced\nby $\\rho\\!:L\\to\\mathord{\\mathrm{Der}}_{k}(R)$ and the canonical map\n$R_{\\mathfrak{p}}\\mathbin{\\otimes}_{R}\\mathord{\\mathrm{Der}}_{k}(R)\\to\\mathord{\\mathrm{Der}}_{k}(R_{\\mathfrak{p}})$.\nThis agrees with the definition given in \\cite{Rin}.\n\\end{ex}\n\nThe Lie-Rinehart algebras over $R$ form a category\n$$ \\mathsf{LieRinAlg}_{R} \\;,$$\nwhere a morphism $\\phi\\!:L\\to L'$ is a homomorphism\nof Lie algebras over $k$ as well as a map\nof $R$-modules which intertwines the representations,\n$\\rho'\\mathbin{{\\scriptstyle \\circ }} \\phi=\\rho$.\nThe operation $\\tau_{!}$, induced by\na homomorphism $\\tau\\!:R\\to S$\nof unital commutative $k$-algebras,\nis a functor $\\mathsf{LieRinAlg}_{R}\\to\\mathsf{LieRinAlg}_{S}$.\nMoreover, for another homomorphism\n$\\sigma\\!:S\\to T$ of unital commutative\n$k$-algebras there is a canonical map\n$\\sigma_{!}(\\tau_{!}(L))\\to (\\sigma\\mathbin{{\\scriptstyle \\circ }}\\tau)_{!}(L)$.\nUsing these canonical maps, the\ncategories $\\mathsf{LieRinAlg}_{R}$,\nfor varying $k$-algebras $R$, can be\nassembled into one big (fibered) category\n$$ \\mathsf{LieRinAlg}\\;,$$\nin which a map $(R,L)\\to (S,K)$\nis a pair $(\\tau,\\phi)$, consisting of a homomorphism\nof unital $k$-algebras $\\tau\\!:R\\to S$ and a homomorphism\n$\\phi\\!:\\tau_{!}(L)\\to K$ of Lie-Rinehart algebras over $S$.\n\n\n\\section{The universal enveloping algebra} \\label{sec2}\n\nLet $L$ be a Lie-Rinehart algebra over $R$.\nThe left $R$-module $R\\oplus L$ has a natural\nLie algebra structure, given by\n$$[(r,X), (s,Y)]=(X(s)-Y(r),[X,Y]) $$\nfor any $r,s\\in R$ and $X,Y\\in L$.\nLet $U(R\\oplus L)$ be its universal enveloping algebra over $k$,\nobtained as the quotient of the tensor algebra\nof $R\\oplus L$ over $k$ with respect\nto the usual ideal.\nWrite $i\\!:R\\oplus L\\to U(R\\oplus L)$ for the canonical inclusion\nand $\\bar{U}(R\\oplus L)$ for\nthe subalgebra of $U(R\\oplus L)$ generated by\n$i(R\\oplus L)$.\nThe {\\em universal enveloping algebra} of the Lie-Rinehart algebra\n$L$ over $R$ (see \\cite{Rin})\nis the quotient algebra\n$$ \\mathord{\\mathscr{U}}(R,L)=\\bar{U}(R\\oplus L)\/I $$\nover $k$,\nwhere $I$ is the two-sided\nideal in $\\bar{U}(R\\oplus L)$\ngenerated by the elements\n$i(s,0)\\cdot i(r,X)-i(sr,sX)$,\nfor all $r,s\\in R$ and $X\\in L$. The natural map\n$\\iota_{R}\\!:R\\to\\mathord{\\mathscr{U}}(R,L)$, $r\\mapsto i(r,0)+I$,\nis a homomorphism of unital $k$-algebras, while\n$\\iota_{L}\\!:L\\to\\mathord{\\mathscr{U}}(R,L)$,\n$X\\mapsto i(0,X)+I$, is a homomorphism of Lie algebras.\nFurthermore, we have $\\iota_{R}(r)\\iota_{L}(X)=\\iota_{L}(rX)$\nand $[\\iota_{L}(X),\\iota_{R}(r)]=\\iota_{R}(X(r))$.\n\nThe universal enveloping algebra $\\mathord{\\mathscr{U}}(R,L)$ is characterized by the\nfollowing universal property:\nif $A$ is any unital $k$-algebra,\n$\\kappa_{R}\\!:R\\to A$ a homomorphism of unital $k$-algebras\nand $\\kappa_{L}\\!:L\\to A$ a homomorphism of Lie algebras\nsuch that $\\kappa_{R}(r)\\kappa_{L}(X)=\\kappa_{L}(rX)$\nand $[\\kappa_{L}(X),\\kappa_{R}(r)]=\\kappa_{R}(X(r))$\nfor any $r\\in R$ and $X\\in L$, then\nthere exists a unique homomorphism of\nunital algebras $f\\!:\\mathord{\\mathscr{U}}(R,L)\\to A$ such that\n$f\\mathbin{{\\scriptstyle \\circ }} \\iota_{R}=\\kappa_{R}$ and\n$f\\mathbin{{\\scriptstyle \\circ }} \\iota_{L}=\\kappa_{L}$.\n\nIn particular, the universal property of $\\mathord{\\mathscr{U}}(R,L)$\nimplies that there exists a unique\nrepresentation\n$$ \\varrho\\!:\\mathord{\\mathscr{U}}(R,L)\\to \\mathord{\\mathrm{End}}_{k}(R) $$\nsuch that $\\varrho\\mathbin{{\\scriptstyle \\circ }}\\iota_{L}=\\rho$ and\n$\\varrho\\mathbin{{\\scriptstyle \\circ }}\\iota_{R}$ is the canonical\nrepresentation given by the\nmultiplication in $R$. Since\nthe canonical\nrepresentation of $R$ is faithful, we see that\nthe map $\\iota_{R}$ is injective.\nWe shall therefore identify $\\iota_{R}(R)$ with $R$,\n$\\iota_{R}(r)=r$.\nWe shall often denote\n$\\iota_{L}(X)$ by $\\iota(X)$ or simply by $X$.\nIn this notation, the algebra $\\mathord{\\mathscr{U}}(R,L)$\nis generated by elements $X\\in L$ and $r\\in R$,\nwhile $r\\cdot X=rX$ and\n$[X,r]=X\\cdot r - r\\cdot X=X(r)$ in $\\mathord{\\mathscr{U}}(R,L)$.\nAs a $k$-linear space, $\\mathord{\\mathscr{U}}(R,L)$ is generated by $R$ and\nthe powers $\\iota(L)^{n}$, $n=1,2,\\ldots\\,$.\nThe algebra $\\mathord{\\mathscr{U}}(R,L)$ also has a natural filtration\n$$ R=\\mathord{\\mathscr{U}}_{(0)}(R,L)\\subset\\mathord{\\mathscr{U}}_{(1)}(R,L)\\subset\\mathord{\\mathscr{U}}_{(2)}(R,L)\\subset\\cdots\\;,$$\nwhere $\\mathord{\\mathscr{U}}_{(n)}(R,L)$ is spanned by $R$ and the\npowers $\\iota(L)^{m}$, for $m=1,2,\\ldots,n$.\nWe define the associated graded algebra as\n$$ \\mathord{\\mathrm{gr}}(\\mathord{\\mathscr{U}}(R,L))=\\bigoplus_{n=0}^{\\infty} \\mathord{\\mathscr{U}}_{(n)}(R,L)\/\\mathord{\\mathscr{U}}_{(n-1)}(R,L)\\;,$$\nwhere we take $\\mathord{\\mathscr{U}}_{(-1)}(R,L)=\\{0\\}$.\nIt is a commutative unital algebra over $R$.\n\n\\begin{ex} \\rm \\label{ex2.1}\n(1) Let $V$ be a left $R$-module. With zero bracket and representation,\n$V$ is a Lie-Rinehart algebra. The corresponding universal enveloping algebra\n$\\mathord{\\mathscr{U}}(R,V)$ is in this case the symmetric algebra $S_{R}(V)$\n(see the appendix below).\n\n(2) For any Lie algebra over $k$, the universal enveloping algebra\n$\\mathord{\\mathscr{U}}(k,L)$ is the classical\nuniversal enveloping algebra $U(L)$ of $L$.\n\n(3) If $G$ is a Lie groupoid over a smooth manifold $M$\nand $\\mathfrak{g}$ its Lie algebroid, then the algebra of right\ninvariant tangential differential operators on $G$ is\na concrete model for the universal enveloping algebra $\\mathord{\\mathscr{U}}(\\mathord{\\mathit{C}^{\\infty}}(M),\\Gamma(\\mathfrak{g}))$\n(see \\cite{NWX}).\n\\end{ex}\n\nAs for the classical universal enveloping algebra of a Lie algebra,\nthere is a Poincar\\'{e}-Birkhoff-Witt theorem for the\nuniversal enveloping algebra of a Lie-Rine\\-hart algebra \\cite{Rin}:\nif the Lie-Rinehart algebra $L$ is projective as a left $R$-module,\nthen the\nnatural map $\\theta\\!:S_{R}(L)\\to\\mathord{\\mathrm{gr}}(\\mathord{\\mathscr{U}}(R,L))$ is an isomorphism of\nalgebras. In particular, this implies that\n$\\iota_{L}\\!:L\\to\\mathord{\\mathscr{U}}(R,L)$ is in this case injective.\n\n\n\n\\section{Rinehart bialgebras} \\label{sec3}\n\n\nThe universal enveloping algebra of a Lie algebra\nis a Hopf algebra, as is the group ring of a discrete group.\nIn this section we will identify the algebraic structure\ncommon to the universal enveloping algebra of a Lie-Rinehart algebra\nand the convolution algebra of an \\'{e}tale groupoid. This\nstructure has occurred in the literature under various names,\nsee \\cite{Kap,Lu,Mal,Mrcun,Mrcun2,Tak,Xu}. We suggest the name\n{\\em Rinehart bialgebra}.\n\nLet $R$ be a unital commutative algebra over $k$ as before.\nAll modules considered will be unital left $R$-modules, and\n$\\mathbin{\\otimes}_{R}$ will always denote the tensor product of left\n$R$-modules.\nSuppose that $A$ is a unital $k$-algebra\nwhich {\\em extends} $R$, i.e.\\ such that\n$R$ is a unital subalgebra of $A$.\nIn particular, $A$ is an $R$-$R$-bimodule.\nThe left $R$-module $A\\mathbin{\\otimes}_{R}A$\n(tensor product of $A$, viewed as a left $R$-module, with itself)\nis also a right $R$-module in two ways.\nObserve, that $A\\mathbin{\\otimes}_{R}A$ is not necessarily an algebra in a\nnatural way unless $R$ lies in the centre of $A$.\nFollowing \\cite{Kap},\nwe denote by $A\\bar{\\mathbin{\\otimes}}_{R}A$ the submodule of\n$A\\mathbin{\\otimes}_{R}A$ given by the kernel of the map $\\vartheta$,\n$$\n\\xymatrix{\n0 \\ar[r] & A\\bar{\\mathbin{\\otimes}}_{R}A \\ar[r] &\nA\\mathbin{\\otimes}_{R}A \\ar[r]^-{\\vartheta} & \\mathord{\\mathrm{Hom}}_{k}(R,A\\mathbin{\\otimes}_{R}A)\\;,\n}\n$$\ndefined by $\\vartheta(a\\mathbin{\\otimes} b)(r)=ar\\mathbin{\\otimes} b - a\\mathbin{\\otimes} br$.\nThe $R$-module $A\\bar{\\mathbin{\\otimes}}_{R}A$ has a natural structure of\na $k$-algebra.\nIf $R$ is in the centre of $A$, then\n$A\\bar{\\mathbin{\\otimes}}_{R}A=A\\mathbin{\\otimes}_{R}A$.\n\n\\begin{dfn} \\rm \\label{dfn3.1}\nA {\\em Rinehart bialgebra over} $R$\nis a unital $k$-algebra $A$ which extends $R$,\nwith a {\\em compatible} structure of a cocommutative\ncoalgebra in the category of left $R$-modules.\n\nIf we denote the comultiplication\nby $\\mathord{\\Delta}\\!:A\\to A\\mathbin{\\otimes}_{R} A$ and the counit\nby $\\mathord{\\epsilon}\\!:A\\to R$, the compatibility conditions\nare\n\\begin{enumerate}\n\\item [(i)] $\\mathord{\\Delta}(A)\\subset A\\bar{\\mathbin{\\otimes}}_{R}A$,\n\\item [(ii)] $\\mathord{\\epsilon}(1)=1$,\n\\item [(iii)] $\\mathord{\\Delta}(1)=1\\mathbin{\\otimes} 1$,\n\\item [(iv)] $\\mathord{\\epsilon}(ab)=\\mathord{\\epsilon}(a\\mathord{\\epsilon}(b))$, and\n\\item [(v)] $\\mathord{\\Delta}(ab)=\\mathord{\\Delta}(a)\\mathord{\\Delta}(b)$\n\\end{enumerate}\nfor any $a,b\\in A$.\n\\end{dfn}\n\nObserve that the condition (v) makes sense because\nof (i) and the fact that $A\\bar{\\mathbin{\\otimes}}_{R}A$ is a $k$-algebra.\nNote, however, that (iv) does not express that $\\mathord{\\epsilon}$ is\nan algebra map.\n\nThe Rinehart bialgebras over $R$ form a category\n$$ \\mathsf{RinBiAlg}_{R}\\;, $$\nin which a morphism $f\\!:A\\to B$ is a map\nwhich is at the same time a homomorphism of $k$-algebras\nwith unit and a homomorphism of coalgebras in the category\nof left $R$-modules.\n\n\\begin{ex} \\label{ex3.2} \\rm\n(1)\nIf a unital $k$-algebra $A$ extends $R$ such that $R$ lies in the centre\nof $A$, a Rinehart bialgebra structure on $A$\nis the same as an ordinary $R$-bialgebra structure on $A$.\n\n(2)\nLet $\\mathord{\\mathscr{U}}(R,L)$ be the universal enveloping algebra of\na Lie-Rinehart algebra $L$ over $R$ (Section \\ref{sec2}).\nThe universal property implies that there exists a unique\nhomomorphism of algebras\n$$ \\mathord{\\Delta}\\!:\\mathord{\\mathscr{U}}(R,L)\\to \\mathord{\\mathscr{U}}(R,L)\\bar{\\mathbin{\\otimes}}_{R}\\mathord{\\mathscr{U}}(R,L)\n   \\subset \\mathord{\\mathscr{U}}(R,L)\\mathbin{\\otimes}_{R}\\mathord{\\mathscr{U}}(R,L) $$\nsuch that $\\mathord{\\Delta}(r)=1\\mathbin{\\otimes} r=r\\mathbin{\\otimes} 1$ and\n$\\mathord{\\Delta}(X)=1\\mathbin{\\otimes} X + X\\mathbin{\\otimes} 1$ for any $r\\in R$ and $X\\in L$.\nWith the counit $\\mathord{\\epsilon}\\!:\\mathord{\\mathscr{U}}(R,L)\\to R$ given by\n$$ \\mathord{\\epsilon}(u)=\\varrho(u)(1) \\;,$$\none can check that $\\mathord{\\mathscr{U}}(R,L)$ is a Rinehart bialgebra over $R$.\nFurthermore, a morphism of Lie-Rinehart algebras\n$\\phi\\!:L\\to L'$ induces, by the universal property,\na morphism of Rinehart bialgebras\n$\\mathord{\\mathscr{U}}(R,\\phi)\\!:\\mathord{\\mathscr{U}}(R,L)\\to\\mathord{\\mathscr{U}}(R,L')$, and this gives a functor\n$$ \\mathord{\\mathscr{U}}\\!:\\mathsf{LieRinAlg}_{R}\\to\\mathsf{RinBiAlg}_{R} \\;.$$\n\n(3)\nThe convolution algebra of\nsmooth functions with compact support on\nan \\'{e}tale Lie groupoid $G$\nover a compact manifold $M$\nis a Rinehart bialgebra over $\\mathord{\\mathit{C}^{\\infty}}(M)$\n(see \\cite{Mrcun,Mrcun2}).\n\\end{ex}\n\n\nLet $A$ be a Rinehart bialgebra over $R$. Then $A$ splits\nas an $R$-module as\n$$ A=R\\oplus\\bar{A}\\;,$$\nwhere $\\bar{A}=\\ker\\mathord{\\epsilon}$. The $R$-submodule $\\bar{A}$\nis a subalgebra of $A$ and carries a\ncocommutative coassociative coproduct\n$\\bar{\\mathord{\\Delta}}\\!:\\bar{A}\\to \\bar{A}\\mathbin{\\otimes}_{R}\\bar{A}$, defined by\n$$ \\bar{\\mathord{\\Delta}}(a)=\\mathord{\\Delta}(a)- a\\mathbin{\\otimes} 1 - 1\\mathbin{\\otimes} a\\;.$$\nThe bialgebra $A$ can be reconstructed from $\\bar{A}$, $\\bar{\\mathord{\\Delta}}$\nand the multiplication on $\\bar{A}$.\n\nThe $R$-module $\\bar{A}$ has a filtration\n$$ \\{0\\}=\\bar{A}_{0} \\subset \\bar{A}_{1} \\subset \\bar{A}_{2} \\subset \\cdots $$\nwith $\\bar{A}_{n}=\\ker\\bar{\\mathord{\\Delta}}^{(n)}$, where\n$\\bar{\\mathord{\\Delta}}^{(n)}$ denotes the iterated coproduct\n$\\bar{A}\\to \\bar{A}\\mathbin{\\otimes}_{R}\\cdots\\mathbin{\\otimes}_{R}\\bar{A}$ ($n+1$ copies).\nWe refer to this filtration as the {\\em primitive filtration}\nof $\\bar{A}$, and also write $A_{n}=R\\oplus\\bar{A}_{n}$ ($n\\geq 0$).\nThe\nsubmodule $\\bar{A}_{1}$ is called the submodule of {\\em primitive}\nelements, and also denoted by $\\mathord{\\mathscr{P}}(A)$.\nWe observe that $\\mathord{\\mathscr{P}}(A)$ is a Lie-Rinehart algebra over $R$. Its\nLie bracket is given by the commutator in $A$, and its representation\n$\\rho\\!:\\mathord{\\mathscr{P}}(A)\\to \\mathord{\\mathrm{Der}}_{k}(R)$ is given by\n$\\rho(a)(r)=\\mathord{\\epsilon}(a r)$.\nIndeed, note that\n\\begin{equation*}\n\\begin{split}\na r      & = (\\mathord{\\epsilon}\\mathbin{\\otimes} 1)(\\mathord{\\Delta}(a  r)) \\\\\n         & = (\\mathord{\\epsilon}\\mathbin{\\otimes} 1)(\\mathord{\\Delta}(a)  \\mathord{\\Delta}(r)) \\\\\n         & = (\\mathord{\\epsilon}\\mathbin{\\otimes} 1)((a\\mathbin{\\otimes} 1 + 1\\mathbin{\\otimes} a) (r\\mathbin{\\otimes} 1)) \\\\\n         & = (\\mathord{\\epsilon}\\mathbin{\\otimes} 1)(a r\\mathbin{\\otimes} 1 + r\\mathbin{\\otimes} a) \\\\\n         & = \\mathord{\\epsilon}(a  r)+r a\\;,\n\\end{split}\n\\end{equation*}\nand from this it follows easily that $\\rho$\nis an $R$-linear homomorphism of Lie algebras.\n\nWe call $A$ (or $\\bar{A}$) {\\em cocomplete} if\n$A=\\bigcup_{n=0}^{\\infty}A_{n}$\n(or $\\bar{A}=\\bigcup_{n=0}^{\\infty}\\bar{A}_{n}$).\nThe Rinehart algebra $A$ is {\\em graded projective}\nif each of the subquotients\n$A_{n+1}\/A_{n}=\\bar{A}_{n+1}\/\\bar{A}_{n}$\nis a projective $R$-module.\nWe will use these notions in\nseveral theorems stated below.\n\n\\begin{ex} \\label{ex3.3} \\rm\nIt follows from the Poincar\\'{e}-Birkhoff-Witt theorem\nthat the primitive filtration of the\nuniversal enveloping algebra $\\mathord{\\mathscr{U}}(R,L)$, associated to\na Lie-Rinehart algebra $L$ over $R$,\ncoincide with its natural filtration\nif $L$ is projective as a left $R$-module.\nFurthermore,\nthe universal enveloping algebra $\\mathord{\\mathscr{U}}(R,L)$ is in this case\ncocomplete and graded projective.\n\\end{ex}\n\n\\begin{rem} \\rm \\label{rem3.4}\nLet $A$ be a cocomplete graded projective\nRinehart bialgebra over $R$. In particular, this implies that\n$A$ and all $A_{n}$ are projective $R$-modules.\nWe write $\\mathord{\\mathrm{gr}}(\\bar{A})=\\bigoplus_{n=1}^{\\infty}\\mathord{\\mathrm{gr}}_{n}(\\bar{A})$,\nwhere $\\mathord{\\mathrm{gr}}_{n}(\\bar{A})=\\bar{A}_{n}\/\\bar{A}_{n-1}$.\nThere is a cocommutative coassociative comultiplication\n$\\bar{\\mathord{\\Delta}}^{\\mathrm{gr}}$ on $\\mathord{\\mathrm{gr}}(\\bar{A})$,\ninduced by $\\bar{\\mathord{\\Delta}}$, such that\n$$ \\bar{\\mathord{\\Delta}}^{\\mathrm{gr}}(\\mathord{\\mathrm{gr}}_{n}(\\bar{A}))\\subset\n   \\bigoplus_{p+q=n}\\mathord{\\mathrm{gr}}_{p}(\\bar{A})\\mathbin{\\otimes} \\mathord{\\mathrm{gr}}_{q}(\\bar{A}) $$\nand $\\ker(\\bar{\\mathord{\\Delta}}^{\\mathrm{gr}})=\\mathord{\\mathrm{gr}}_{1}(\\bar{A})$.\nFurthermore, the non-counital coalgebra\n$\\mathord{\\mathrm{gr}}(\\bar{A})$ is cocomplete, i.e.\\\n$\\bigcup_{n=1}^{\\infty}\\ker((\\bar{\\mathord{\\Delta}}^{\\mathrm{gr}})^{(n)})=\\mathord{\\mathrm{gr}}(\\bar{A})$\n(see the appendix).\nNote that\nany morphism $f\\!:A\\to B$ of cocomplete graded projective\nRinehart bialgebras over $R$ induces a\nmorphism of non-counital coalgebras\n$\\mathord{\\mathrm{gr}}(\\bar{f})\\!:\\mathord{\\mathrm{gr}}(\\bar{A})\\to\\mathord{\\mathrm{gr}}(\\bar{B})$.\n\\end{rem}\n\n\n\n\\section{A Cartier-Milnor-Moore theorem} \\label{sec4}\n\n\nWe have already seen that the universal enveloping algebra construction\ndefines a functor\n$$ \\mathord{\\mathscr{U}}\\!:\\mathsf{LieRinAlg}_{R}\\to\\mathsf{RinBiAlg}_{R}\\;.$$\nIn the other direction, there is a functor\n$$ \\mathord{\\mathscr{P}}\\!:\\mathsf{RinBiAlg}_{R}\\to\\mathsf{LieRinAlg}_{R}\\;,$$\nwhich assigns to a Rinehart bialgebra $A$\nits Lie-Rinehart algebra $\\mathord{\\mathscr{P}}(A)$ of\nprimitive elements.\n\n\\begin{theo} \\label{theo4.1}\nThe functor $\\mathord{\\mathscr{U}}$ is left adjoint to $\\mathord{\\mathscr{P}}$.\nFurthermore, the functors $\\mathord{\\mathscr{U}}$ and $\\mathord{\\mathscr{P}}$ restrict\nto an equivalence between the full subcategory\nof Lie-Rinehart algebras over $R$\nwhich are projective as left $R$-modules\nand that\nof cocomplete graded projective Rinehart bialgebras\nover $R$.\n\\end{theo}\n\nThe second part of the theorem in particular implies\nthe following property of the counit of the adjunction,\nwhich is an analogue of the Cartier-Milnor-Moore\ntheorem for Hopf algebras:\n\n\\begin{cor} \\label{cor4.2}\nLet $A$ be a Rinehart bialgebra over $R$.\nIf $A$ is cocomplete and graded projective, then\nthere is a canonical isomorphism of\nRinehart bialgebras\n$\\mathord{\\mathscr{U}}(R,\\mathord{\\mathscr{P}}(A))\\to A$.\n\\end{cor}\n\n\\begin{proof}[Proof of Theorem \\ref{theo4.1}]\nFor a Lie-Rinehart algebra $L$ over $R$, the\ncanonical map $L\\to\\mathord{\\mathscr{U}}(R,L)$ clearly lands\nin the submodule of primitive elements, and this\ndefines the unit of the adjunction,\n$$ \\alpha_{L}\\!:L\\to\\mathord{\\mathscr{P}}(\\mathord{\\mathscr{U}}(R,L))\\;.$$\nFor a Rinehart bialgebra $A$ over $R$, the inclusion\n$\\mathord{\\mathscr{P}}(A)\\to A$ induces, by the universal property\nof the universal enveloping algebra, a canonical algebra map\n$$ \\beta_{A}\\!:\\mathord{\\mathscr{U}}(R,\\mathord{\\mathscr{P}}(A))\\to A\\;,$$\nwhich in fact is clearly a map of Rinehart\nbialgebras. This defines the counit of the adjunction.\n\nThe first part of the theorem states that these two maps\nsatisfy the triangular identities\n$$ \\mathord{\\mathscr{P}}(\\beta_{A})\\mathbin{{\\scriptstyle \\circ }}\\alpha_{\\mathord{\\mathscr{P}}(A)}=\\mathord{\\mathrm{id}}_{\\mathord{\\mathscr{P}}(A)} $$\nand\n$$ \\beta_{\\mathord{\\mathscr{U}}(R,L)}\\mathbin{{\\scriptstyle \\circ }}\\mathord{\\mathscr{U}}(R,\\alpha_{L})=\\mathord{\\mathrm{id}}_{\\mathord{\\mathscr{U}}(R,L)}\\;. $$\nThese both hold by the (uniqueness part of the) universal\nproperty of the universal enveloping algebra.\n\nIf $L$ is projective as a left $R$-module,\nthen the Poincar\\'{e}-Birkhoff-Witt theorem\nimplies that $\\mathord{\\mathscr{U}}(R,L)$ is graded projective and cocomplete\n(because the natural and primitive filtrations coincide),\nand that $\\alpha_{L}$ is an isomorphism.\nFor a graded projective cocomplete Rinehart bialgebra\n$A$ over $R$, the $R$-module $\\mathord{\\mathscr{P}}(A)$ is obviously\nprojective, and it remains to show that in this case the map\n$\\beta=\\beta_{A}$ is an isomorphism.\n\nIt suffices to prove that the map of reduced\nnon-counital coalgebras\n$$ \\bar{\\beta}\\!:\\bar{\\mathord{\\mathscr{U}}}(R,\\mathord{\\mathscr{P}}(A))\\to\\bar{A} $$\nis an isomorphism.\nFor this, in turn, it is enough to show that the induced map\nof non-counital coalgebras\n$$ \\mathord{\\mathrm{gr}}(\\bar{\\beta})\\!:\\mathord{\\mathrm{gr}}(\\bar{\\mathord{\\mathscr{U}}}(R,\\mathord{\\mathscr{P}}(A)))\\to\\mathord{\\mathrm{gr}}(\\bar{A}) $$\nis an isomorphism.\nThe projection\n$\\mathord{\\mathrm{gr}}(\\bar{A})\\to \\mathord{\\mathrm{gr}}_{1}(\\bar{A})=\\mathord{\\mathscr{P}}(A)$ induces a map\nof non-counital coalgebras $\\gamma\\!:\\mathord{\\mathrm{gr}}(\\bar{A})\\to \\bar{S}_{R}(\\mathord{\\mathscr{P}}(A))$, by\nthe universal property of $\\bar{S}_{R}(\\mathord{\\mathscr{P}}(A))$ (Proposition \\ref{propA.2}).\nNow consider the diagram\n$$\n\\xymatrix{\n\\mathord{\\mathrm{gr}}(\\bar{\\mathord{\\mathscr{U}}}(R,\\mathord{\\mathscr{P}}(A))) \\ar[r]^-{\\mathord{\\mathrm{gr}}(\\bar{\\beta})} \\ar[dr]_-{\\bar{\\theta}^{-1}} &\n\\mathord{\\mathrm{gr}}(\\bar{A}) \\ar[d]^{\\gamma} \\\\\n& \\bar{S}_{R}(\\mathord{\\mathscr{P}}(A))\n}\n$$\nwhere $\\bar{\\theta}^{-1}$ is the inverse of the Poincar\\'{e}-Birkhoff-Witt\nisomorphism $\\theta$ (Section \\ref{sec2}) restricted to\n$\\bar{S}_{R}(\\mathord{\\mathscr{P}}(A))$.\nAll maps in the diagram\nare maps of non-counital coalgebras.\nThus, to see that the diagram commutes, it suffices\n(again by the universal property of $\\bar{S}_{R}(\\mathord{\\mathscr{P}}(A))$ stated\nin Proposition \\ref{propA.2}) that\n$$ \\mathord{\\mathrm{pr}}_{1}\\mathbin{{\\scriptstyle \\circ }}\\gamma\\mathbin{{\\scriptstyle \\circ }}\\mathord{\\mathrm{gr}}(\\bar{\\beta})=\\mathord{\\mathrm{pr}}_{1}\\mathbin{{\\scriptstyle \\circ }}\\bar{\\theta}^{-1} $$\nfor the projection $\\mathord{\\mathrm{pr}}_{1}\\!:\\bar{S}_{R}(\\mathord{\\mathscr{P}}(A))\\to\\mathord{\\mathscr{P}}(A)$,\nwhich is clear from the explicit definitions.\nTo finish the proof,\nrecall from Remark \\ref{rem3.4}\nthat $\\mathord{\\mathrm{gr}}(\\bar{A})$ is cocomplete and $\\ker(\\bar{\\mathord{\\Delta}}^{\\mathrm{gr}})=\\mathord{\\mathscr{P}}(A)$,\nso that\nby Lemma \\ref{lemA.1} the map\n$\\gamma$ is injective, while by the commutativity\nof the diagram it is also surjective. Thus $\\gamma$ is\nan isomorphism, and hence so is $\\mathord{\\mathrm{gr}}(\\bar{\\beta})$.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nBlack holes provide a powerful laboratory for testing our ideas about space and time at both the theoretical as well as the observational frontier. A striking theoretical prediction based on quantum field theory in curved spacetime is that black holes are not entirely black: their event horizon emits black body radiation with a temperature inversely proportional to the mass of the black hole \\cite{Hawking:1975vcx,haw1,haw2,haw3}. Owed to this so-called Hawking effect, the black hole loses mass and eventually evaporates completely within a finite time-span. The robustness of this scenario has been corroborated in a number of different ways comprising perturbative computations in a fixed background spacetime \\cite{Hawking:1975vcx}, the detector approach \\cite{detector0,detector00,detector,detector1}, as well as by analogy to the Unruh effect \\cite{Unruh:1976db}.\n\nThe Hawking effect gives rise to a series of theoretical puzzles though. Firstly, one encounters the black hole information paradox reviewed in \\cite{chen}. The picture above suggests that the physical information about how the black hole was formed could permanently disappear, allowing many physical states to evolve into the same state. Basically, there are three viewpoints on this problem \\cite{stev1,dan}: 1) information is indeed lost after the black holes has evaporated completely, 2) evaporation stops and information is preserved inside a stable remnant, and 3) information may be returned outside via Hawking radiation. In particular, with regard to option 2) it is important to understand the final configuration emanating from the black hole evaporation process. \n\nThe second puzzle comes from the increase of the horizon temperature as the black hole becomes lighter and lighter. For instance, a black hole with a mass of the order of the Planck mass would have a temperature $T_h \\approx 10^{31}$K. Theoretically, it is then predicted that the final stage of the black hole evaporation process generates a so-called thunderbolt singularity. From the observational perspective, this feature leads to the prediction that a black hole reaching a mass range between $10^9 - 10^{13}$g should create powerful short-lived gamma-ray bursts with energy of a few hundred MeV \\cite{haw}. So far, all attempts to detect such high energy bursts have failed and resulted only in upper bounds on black hole evaporation rate in the vicinity of Earth \\cite{gamma,gamma1}.\n\nThirdly, a cold phase in a black hole's life gives an interesting perspective on dark matter \\cite{mc,rov,dm}. If the evaporation process is halted as some mass (potentially set by the Plack-mass) one may end up with a stable configuration which, except for its gravitational interaction, has no or extremely small interaction with ordinary matter and hence fits perfectly into the definition of a Weakly Interaction Massive Particle  (WIMP). This has led many authors to claim that in fact such tiny black holes could explain the mystery of dark matter in our universe, see, e.g., \\cite{mc,rov,dm} for selected references. In reference \\cite{revi} it is shown that no major constrain can be cast upon the properties of Planck-size remnants if they play the role of dark matter at a cosmological scale; nonetheless, the way these remnants can be produced and their stability could be potential weak spots of such scenarios \\cite{rov}. \n\nThese puzzles, clearly ask for a better understanding of the black hole evaporation process  beyond the quantum field theory in curved spacetime analysis. It is conceivable, that the ultimate answer lies in the realm of a theory of quantum gravity. Since there are currently many different routes at various stages of development, we take a different angle on the problem: quite strikingly many quantum gravity programs, including string theory\n\\cite{Becker:2006dvp,Zwiebach:2004tj,Palti:2019pca},\n loop quantum gravity and spin foams \\cite{Ashtekar:2021kfp,Perez:2012wv,Rovelli:2014ssa,Ashtekar:2021kfp},\n\n asymptotically safe gravity \\cite{Niedermaier:2006wt,Reuter:2012id,Percacci:2017fkn,Eichhorn:2018yfc,Reuter:2019byg,Pereira:2019dbn,Reichert:2020mja,Pawlowski:2020qer},\nCausal Dynamical Triangulations \\cite{Ambjorn:2012jv,Loll:2019rdj},\nand Ho\\v{r}ava-Lifshitz gravity \\cite{Hor,Wang:2017brl,Barvinsky:2021ubv} predict a dynamical dimensional reduction of the theories momentum space \\cite{Carlip:2017eud,Carlip:2019onx} (also see \\cite{tHooft:1993dmi} for an early account of this idea). Specifically, the spectral dimension $D_s$, probing the effective dimension experienced by a random walk, drops from $D_s = 4$ at macroscopic scales to $D_s \\approx 2$ at short distances. For instance, the analysis  of geometries obtained from the Causal Dynamical Triangulations program reported a scale-dependent spectral dimension \\cite{Ambjorn:2005db},\n\\begin{equation}\nD_s(T) = a - \\frac{b}{c+T} \\, , \n\\ee\nwhere $a=4.02$, $b=119$, and $c =54$. A similar analysis within Euclidean Dynamical Triangulations \\cite{Laiho:2017htj} obtained $D_s(T) = 3.94 \\pm 0.16$ and $D_s(T) = 1.44 \\pm 0.19$ for the large and small distance values when extrapolating to the continuum and infinite volume limits. Generalizing \\cite{Lauscher:2005qz}, the scale-dependent spectral dimension along a fixed asymptotically safe renormalization group trajectory has been computed in \\cite{Reuter:2011ah}, leading to a three plateau structure with $D_s(T) =4$, $D_s(T) = 4\/3$, and $D_s(T)=2$ at large, intermediate, and short distances, respectively (also see \\cite{Reuter:2012xf} for a review).\n\n\n\nIn \\cite{Barcaroli:2015xda} the scale-dependence of the spectral dimension has been linked to the dimensionality of the theories momentum space: a drop of the spectral dimension indicates that there are less degrees of freedom at high-energy as compared to the expectation based on the dimension of spacetime observed at macroscopic scales. Thus the dynamical dimensional reduction provides a powerful mechanism for eliminating divergences occurring at high energies. Within the realm of multi-scale models \\cite{cal,Calcagni:2013vsa,Calcagni:2016azd,Calcagni:2015xcf}, the phenomenological consequences of this mechanism have been explored, e.g., in the context of quantum field theory \\cite{cal-standard}, cosmology \\cite{cal-cosmo,cal-cosmo2}, and also for the Unruh effect \\cite{Alkofer:2016utc}, see \\cite{Calcagni:2021ipd} for an up-to-date review.\\footnote{Along different lines, fractal aspects of black holes have been considered within the ``un-gravity program'' \\cite{grav1,spec1}.}\n\nIn \\cite{carlip} the authors suggested an intricate connection between dynamical dimensional reduction and the formation of cold remnants formed at the end of the black hole evaporation process based on a two-dimensional dilaton-gravity model. The goal of our work is to complement this analysis by implementing the effect of a drop in the spectral dimension in the thermodynamic properties of a four-dimensional Schwarzschild black hole. As our main result, we demonstrate that the mechanism of dynamical dimensional reduction removes the thunderbolt singularity appearing in the last stages of the black hole evaporation process. It does not lead to the formation of long-lived black hole remnants though. The latter requires additional ingredients, with a change in the topology of the black hole solution being the most probable one.\n\nThe rest of our work is organized as follows. Sect.\\ \\ref{sect.2} and Sect.\\ \\ref{sect.3} provide a brief introduction to black hole thermodynamics and the concept of generalized dimensions, respectively. Our analysis is presented in Sect.\\ \\ref{sect.4} and we conclude with a brief discussion and outlook in Sect.\\ \\ref{sect.5}.   \n\n\\section{Black hole thermodynamics in a nutshell} \n\\label{sect.2}\nWe start by reviewing the basics of black hole thermodynamics, referring to \\cite{Davies,Wald:1995yp,Raine:2005bs} for more detailed, pedagogical accounts. For simplicity, we consider spherically symmetric black holes described by the Schwarzschild solution. In natural units where $G=c=\\hbar=k_b=1$, the resulting line-element is\n\\begin{equation}\\label{eq:sssol}\nds^2 = (1 - \\frac{2M}{r}) dt^2 - (1 - \\frac{2M}{r})^{-1} dr^2 - r^2 d\\Omega^2  \\, . \n\\ee \nHere $d\\Omega^2 = d\\theta^2 + \\sin^2\\theta d\\phi^2$ is the line-element on the unit two-sphere and $M$ is the mass of the black hole. The geometry \\eqref{eq:sssol} possesses an event horizon at\n\\begin{equation}\nr_h = 2M \\, . \n\\ee\nBased on \\eqref{eq:sssol} one readily deduces that the area of this horizon is\n\\begin{equation}\\label{eq:horizonarea}\nA_h = 4 \\pi r_h^2 = 16 \\pi M^2 \\, . \n\\ee\nAny object or photon crossing this horizon inevitably has to move inward, eventually ending at the curvature singularity at $r=0$. Classically, signals emitted at $r \\le r_h$ can not reach an observer stationed at $r > r_h$. Hence the terminology ``black hole''.\n\nThe analysis within the framework of quantum field theory in curved spacetime \\cite{Hawking:1975vcx} shows, however, that the event horizon emits black body radiation (Hawking radiation) with a temperature proportional to the surface gravity at the  horizon. For the Schwarzschild black hole \\eqref{eq:sssol}, this results in\n\\begin{equation}\\label{eq:Hawkingtemp}\nT_h = \\frac{1}{8\\pi M} \\, . \n\\ee\nThe resulting luminosity $L$ is then given by\n\\begin{equation}\\label{eq:Lgeneral}\nL = A_h \\, I \\, , \n\\ee\nwhere $I$ is the integrated black body spectrum. For bosonic fields as the scalar field considered in this work\n\\begin{equation}\\label{eq:bbfactor}\nI = \\int \\frac{d^3p}{(2\\pi)^3} \\frac{E}{e^{E\/T_h} - 1} \\, . \n\\ee\nFor a massless photon $E = |\\vec{p}| = \\omega$ and one recovers the standard result \\cite{Raine:2005bs}:\\footnote{In general, the power contained in the Hawking radiation associated with a massless scalar field has the form $P = \\sum_l \\int_0^\\infty d\\omega P_l(\\omega)$ with the $l$th multipole contributing with $P_l(\\omega) = \\frac{A_h}{8\\pi^2} T_l(\\omega) \\omega^3 (e^{\\omega\/T_h}-1)^{-1}$. Our analysis focuses on the $l=0$ sector and neglects the gray-body corrections $T_l(\\omega)$. Since the latter encode the transmission probability of Hawking radiation reaching future infinity without being backscattered by the gravitational barrier surrounding the black hole, one expects that these will lead to an additional suppression of the massive contributions as compared to the massless ones. Based on eq.\\ \\eqref{eq:Lspec} one then expects that the inclusion of the $T_l(\\omega)$ will further inhibit the formation of remnants while leaving the leading order analysis unaffected.}\n\\begin{equation} \\label{Lumin}\n\t\\begin{split}\n\tL_{\\rm massless} \\,= & \\, \\frac{8 M^2}{\\pi} \\int_{0}^{\\infty} d\\omega \\, \\frac{\\omega^3}{e^{8\\, \\pi\\, M\\, \\omega}-1} \\\\\n\t= & \\, \\frac{1}{7680 \\pi M^2} \\, . \n\t\\end{split} \n\\end{equation}\nHere we have performed the integral over frequencies and expressed the result in terms of the black hole mass $M$ by substituting \\eqref{eq:Hawkingtemp}. $L_{\\rm massless}$ as a function of $M$ is then illustrated as the blue straight line in Fig.\\ \\ref{Hlumin}. Eq.\\ \\eqref{Lumin} exhibits the curious feature that black holes become more and more luminous the lighter they become. In particular $L$ diverges as $M \\rightarrow 0$. The presence of this so-called thunderbolt singularity suggests that the semi-classical analysis breaks down when describing the final stage of black hole evaporation \\cite{Hawking:1992ti,Piran:1993tq,Ashtekar:2010hx,Lowe:1993zw}.\n\\begin{figure}[h] \n\t\\includegraphics[scale=0.60]{lumin1} \\\\\n\t\\caption{\\label{Hlumin} Luminosity of a Schwarzschild black hole as a function of the mass $M$. The case of a massless and a massive scalar field with $m^2 = 1$ are illustrated by  the straight blue and orange lines, respectively.}\n\\end{figure} \n\nThe life-time of a black hole with initial mass $M_0$ can then be obtained by integrating the mass-loss formula\n\\begin{equation}\\label{eq:massloss}\n\\frac{d M}{dt} = - L \\, . \n\\ee\nSubsituting \\eqref{Lumin} gives the black hole evaporation time\n\\begin{equation}\nt_{\\rm evap} = 2560 \\, \\pi \\, M_0^3 \\, . \n\\ee\nThus the emission of Hawking radiation renders the life-time of the black hole finite.\n\nThe luminosity formula \\eqref{eq:Lgeneral} is readily generalized to the case of a scalar field with mass $m$. In this case the energy appearing in \\eqref{eq:bbfactor} is replaced by the relativistic dispersion relation $E^2 = \\vec{p}^2 + m^2$. In general, $I$ does not admit an simple analytic expression. Nevertheless, it is instructive to study the following limits: if $m\/T_h \\ll 1$ the particle is relativistic and one essentially recovers the massless case. For temperatures $m\/T_h \\gg 1$ the particle is non-relativistic and the Bose-Einstein distribution in \\eqref{eq:bbfactor} may be approximated by the Boltzmann distribution. This shows that the contribution of a massive mode is actually exponentially suppressed at temperatures $T_h \\lesssim m$. The full expression for $L_{\\rm massive}$ as a function of $M$ is obtained by numerical integration. The result is the orange line shown in Fig.\\ \\ref{Hlumin}.\n  \n\\section{The Spectral Dimension in a nutshell} \n\\label{sect.3}\nAn intuitive picture about quantum gravity is that spacetime at short distances will develop non-manifold like features. A first step towards characterizing the resulting structures is to generalize the notion of ``dimension'' borrowing concepts from fractal geometry \\cite{book-diffusion}. In this way one naturally distinguishes between the Hausdorff dimension (based on covering a set of points with balls of decreasing radius), the spectral dimension (measuring the dimension ``felt'' by a diffusing particle), or the walk dimension (related to the expectation value of the distance traveled by a random walk as function of the diffusion time) of an Euclidean space. While all of these dimensions agree when working on manifolds, they characterize distinct properties of fractal spaces. In the present work, the key role is played by the spectral dimension $D_s$ which may be interpreted as the dimension of the theories momentum space \\cite{Barcaroli:2015xda}. A decrease of the spectral dimension at high energy may then be interpreted as ``the theory possessing less degrees of freedom than its analogue defined on a background manifold''. \n\nFormally, the spectral dimension $d_s$ and its scale-dependent generalization $D_s(T)$ is introduced by studying the diffusion of a test particle on a $d$-dimensional Euclidean spacetime with metric $g_{\\mu\\nu}$ with respect to the fiducial diffusion time $T$. Denoting the Laplacian constructed from $g_{\\mu\\nu}$ by $\\Delta \\equiv - g^{\\mu\\nu}D_\\mu D_\\nu$, and introducing $F(\\Delta) \\equiv G(\\Delta)^{-1}$ with $G(\\Delta)$, being the position-space representation of the particles propagator, the motion of the test particle is captured by the generalized heat equation\n\\begin{equation}\\label{diff}\n\t\\frac{\\partial}{\\partial T} K_{g}(\\xi,\\xi_0;T)= -F(\\Delta) K_{g}(\\xi,\\xi_0;T)\n\\end{equation} \nsubject to the boundary condition\n\\begin{equation}\nK_{g}(\\xi,\\xi_0;T)|_{T = 0} = \\delta^d(\\xi - \\xi_0) \\, . \n\\ee\nHere $K_{g}(\\xi,\\xi_0;\\sigma)$ is the heat-kernel associated with $F(\\Delta)$. It describes the probability of the particle defusing from the initial point $\\xi_0$ to $\\xi$ during the time-interval $T$. In particular, one recovers the standard heat-equation for $F(\\Delta) = \\Delta$. The return probability $P_g(T)$ is then defined by the particle returning to its initial point after time $T$\n\\begin{equation}\\label{eq:return}\n P_g(T) \\equiv V^{-1} \\int d^d\\xi \\sqrt{g} \\, K_{g}(\\xi,\\xi;T) \\, . \n\\ee\nHere $V \\equiv \\int d^d\\xi \\sqrt{g}$ is the volume of the space. Based on \\eqref{eq:return}\nthe spectral dimension $d_s$ is then defined as\n\\begin{equation}\\label{def:specdim}\nd_s \\equiv -2 \\lim_{T \\rightarrow 0} \\, \\frac{d \\ln P_g(T)}{d \\ln T} \\, . \n\\ee\nFor the standard heat-equation on a smooth manifold $d_s = d$ agrees with the topological dimension of the manifold. At this stage, it is convenient to generalize \\eqref{def:specdim}, allowing for a scale-dependent spectral dimension\n\\begin{equation}\\label{def:specdimT}\n\tD_s(T) \\equiv -2\\frac{d \\ln P_g(T)}{d \\ln T} \\, . \n\\end{equation}\n$D_s(T)$ takes into account the possibility that long random walks my experience a different spectral dimension than the infinitesimal ones entering in the definition \\eqref{def:specdim}.\n\nOn a flat Euclidean space $\\mathbb{R}^d$ with metric $\\delta_{\\mu\\nu}$ the generalized heat-equation \\eqref{diff} can be solved using Fourier-techniques\n\\begin{equation}\\label{eq:diffkernel}\n\tK_{\\delta}(\\xi,\\xi_0;T)=\\int \\frac{d^dp}{(2\\pi)^d}e^{ip(\\xi-\\xi_0)} e^{-T F(p^2)} \\, . \n\\end{equation}\nThe resulting return probability is\n\\begin{equation}\\label{eq:retprob}\n\tP_\\delta(T)=\\int \\frac{d^dp}{(2\\pi)^d} e^{-T F(p^2)} \\, . \n\\end{equation}\nFor $F(p^2) = p^2$, the return probability evaluates to\n\\begin{equation}\nP_\\delta(T)= (4 \\pi T)^{-d\/2} \\, . \n\\ee\nSubstituting this result into \\eqref{def:specdimT} shows that $D_s(T) = d$ is independent of $T$ and agrees with the topological dimension $d$. \n\nThe computation is readily generalized to the case where the function $F(p^2)$ has a fixed scaling behavior $F(p^2) = p^{2+\\delta}$.\\footnote{Generically, any function $F(p^2)$ for which the integral \\eqref{eq:diffkernel} is not the Fourier-transform of a Gaussian will result in diffusion kernels $K_g(\\xi,\\xi_0;T)$ which are not positive semi-definite. The occurrence of negative probabilities can be cured by going to fractional calculus \\cite{cal}. Since this is not relevant in the present analysis, we do not dwell on this technical feature at this point.}\n Rewriting the integral in \\eqref{eq:retprob} in terms of the dimensionless variable $x = p^{2+\\delta} T$, one readily finds that \\cite{Reuter:2011ah}\n\\begin{equation}\\label{eq:dsT}\n\tD_s(T)=\\frac{2d}{2+\\delta} \\, , \n\\end{equation}\nwhich is again independent of the diffusion time $T$.\n\nBased on \\eqref{eq:dsT} it is then straightforward to construct a simple multi-scale model which interpolates between $D_s(T) = 2$ at microscopic and $D_s(T) = 4$ at macroscopic scales \\cite{Alkofer:2016utc}. Starting from the momentum-space propagator\n\\begin{equation}\\label{mod}\n\tG(p^2)=\\frac{1}{p^2}-\\frac{1}{p^2+m^2} \\, , \n\\end{equation}\none obtains $F(p^2) = \\frac{1}{m^2} \\, p^2 (p^2 + m^2)$. Thus $F(p^2)$ interpolates between $F(p^2) \\propto p^2$ for $p^2 \\ll m^2$ and $F(p^2) \\propto p^4$ for $p^2 \\gg m^2$. Evaluating \\eqref{eq:dsT} in these scaling regimes suggests that one recovers the desired behavior of the spectral dimension at microscopic and macroscopic scales. The integrals determining $P_\\delta(T)$ can be performed analytically and can be expressed in terms of error-functions. The resulting spectral dimension is shown in Fig.\\ \\ref{crossover}. This confirms that the model indeed interpolates between $D_s = 4$ for $T\/m \\gg 1$ and $D_s = 2$ for $T\/m \\ll 1$ \n\\begin{figure}[h] \n\t\\includegraphics[scale=0.60]{crossover2.pdf}\n\t\\caption{\\label{crossover} Illustration of the scale-dependent spectral dimension $D_s(T)$ obtained from the two-scale model \\eqref{mod} with $m^2 =1$. $D_s(T)$ interpolates smoothly between $D_s = 4$ for $T\/m \\gg 1$ and $D_s = 2$ for $T\/m \\ll 1$.}\n\\end{figure}\nEq.\\ \\eqref{mod} then defines a generic toy model realizing the dynamical dimensional reduction encountered in the full-fledged quantum gravity analysis. The latter may also fix the cross-over scale $m$ based on microscopic considerations.\n\n\\section{Black hole thermodynamics including a non-trivial spectral dimension} \n\\label{sect.4}\nAt this stage we are in a position to combine our discussions on the thermodynamical properties of black holes and dynamical dimensional reduction. Throughout this section we will assume that the radiation emitted by the event horizon remains thermal also for very light black holes, see \\cite{thermality,thermality1,thermality2} for a detailed analysis supporting this assumption. The goal of this section is then to go beyond the semi-classical analysis utilizing the concepts of generalized dimensions and dynamical dimensional reduction.\n\\subsection{Single-scale analysis}\n\\label{sect.4.1}\nFrom the perspective of generalized dimensions the luminosity formula \\eqref{eq:Lgeneral} contains two distinguished elements. Firstly, the black body factor $I$ contains an integral over the theories momentum space. This suggests that the dimension appearing in this term is the spectral dimension\n\\begin{equation}\\label{eq:Ispec}\nI_{d_s} = \\int \\frac{d^{d_s-1}p}{(2\\pi)^{d_s-1}} \\frac{E}{e^{E\/T_h} - 1} \\, . \n\\ee\nSecondly, the horizon area is related to position space properties. This suggests that this term is sensitive to the Hausdorff dimension of the (quantum) spacetime. Owed to the lack of a concrete model which would allow to describe such an effect for the event horizon, we refrain from including such an effect. If a concrete model is available, this feature may be included rather straightforwardly in the present setting by considering an effective dimension build from a linear combination of the spectral and Hausdorff dimension.\n\nThe generalization \\eqref{eq:Ispec} then allows to determine the threshold on the spectral dimension $d_s$ required for creating a long-lived black hole remnant from dynamical dimensional reduction in the momentum space. For this purpose we consider \\eqref{eq:Ispec} for a massless scalar field in a scaling regime where $d_s$ is constant but not necessary identical to the topological dimension $d$ of the spacetime. Convergence of the integral requires $d_s > 1$. Assuming convergence, $I_{d_s} = \\tilde{c} M^{-d_s}$ where $\\tilde{c}$ is a numerical constant. In combination with the classical horizon area \\eqref{eq:horizonarea} the luminosity obtained from the spectral dimension is\n\\begin{equation}\nL_{d_s} = 16 \\pi \\tilde{c} \\, M^{2-d_s} \\, . \n\\ee\nThe mass-loss formula \\eqref{eq:massloss} then shows that the generation of a remnant for which $t_{\\rm evap}$ is infinite requires $d_s-2 \\le -1$ or, equivalently, $d_s \\le 1$. This is, however, in conflict with requiring convergence of $I_{d_s}$. Thus just modifying the spectral dimension for the fields constituting the Hawking radiation is not sufficient to create a long-lived remnant. In particular, the black hole evaporation does not stop if $d_s$ drops below three. \n\\subsection{Dynamical dimensional reduction}\n\\label{sect.4.2}\nNotably, the luminosity \\eqref{eq:Lgeneral} \\emph{is linear} in the two-point correlation function of the corresponding fields. The Euclidean multi-scale model following from \\eqref{mod} then suggests the following generalization to the black hole context. First, the Euclidean flat-space propagators are analytically continued to Lorentzian signature using a standard Wick rotation. Subsequently, the principle of covariance is used to promote the derivatives appearing in the position space representation to covariant derivatives. In this way one naturally arrives at the conclusion that the luminosity $L_{D_s}$ of a black hole, in a situation where the scalar field modeling the Hawking radiation exhibits dynamical dimensional reduction, is given by the sum of a massless and massive contribution weighted by a relative minus sign:\n\\begin{equation}\\label{eq:Lspec}\nL_{D_s}(M;m) = L_{\\rm massless}(M) - L_{\\rm massive}(M;m) \\, . \n\\ee\n\\begin{figure}[h] \n\t\\includegraphics[scale=0.60]{lumin2}\n\t\\caption{\\label{luminds} Luminosity $L_{D_s}$ of a Schwarzschild black hole with mass $M$ arising from the two-scale model \\eqref{mod} with $m^2 = 1$ (blue line). The inclusion of the massive mode triggering the dynamical dimensional reduction renders the luminosity finite as $M \\rightarrow 0$. The massless case is added as the dashed line for reference.}\n\\end{figure} \nFrom the general analysis in Sect.\\ \\ref{sect.2} we then conclude that the contribution of the second term is exponentially suppresses for $M \\gg m$. As a result, $L_{D_s}(M;m)$ agrees with the semi-classical analysis in this regime. Conversely, for $M \\lesssim m$ both terms contribute with equal magnitude. As a result $L_{D_s}$ remains finite as $M \\rightarrow 0$. The crossover between these two regimes together with the removal of the thunderbolt singularity is illustrated in Fig.\\ \\ref{luminds}, where $L_{D_s}$ has been evaluated numerically for $m^2=1$.\n\nThe taming of the luminosity for light black holes arises from the interplay of the massless and massive degree of freedom which provides a Pauli-Villars-type regularization for $L_{\\rm massless}$. We stress that, from a quantum gravity perspective, this feature does not result from introducing an additional ghost field, but is a direct result of the reduced number of degrees of freedom exhibited by the theory at scales $|\\vec{p}| \\gtrsim m$.  \n\nThe result shown in Fig.\\ \\ref{luminds} together with a detailed numerical analysis reveals that $L_{D_s}(M;m)$ can be very well approximated by a simple interpolating function\n\\begin{equation}\\label{eq:Lspecana}\nL_{D_s}(M;m) = \\frac{c}{b \\, m^{-2}  + M^2}\n\\ee\nwhere\n\\begin{equation}\nc = \\frac{1}{7680 \\pi} \\, , \\qquad b \\approx 0.0125 \\, . \n\\ee\nThe value for $b$ has been obtained from fitting the ansatz \\eqref{eq:Lspecana} to $L_{D_s}(M,m)$, obtained via numerical integration, at values $M \\ll 1$.  \n\nThe analytic formula \\eqref{eq:Lspecana} again allows to compute the Hawking evaporation time of the black hole analytically. Integrating eq.\\ \\eqref{eq:massloss} yields\n\\begin{equation}\\label{eq:tevapspec}\nt_{\\rm evap} = 2560 \\pi M_0^3 + \\frac{b}{c} \\frac{M_0}{m^2} \\, . \n\\ee\nThus the dynamical dimensional reduction leads to an increase of the black hole lifetime. Since the scale $m$ where the dynamical dimensional reduction sets in is expected to be the Planck scale, this is a rather tiny effect though. Evaluating the term linear in $M_0$ for $m = m_{\\rm Planck} = 2.18 \\times 10^{-5}$ g and $M_0 = 10^9$ g yields that the change in the lifetime of the black hole is given by $\\Delta t_{\\rm evap} = 7.46 \\times 10^{-28}$ s. Thus the structure of \\eqref{eq:tevapspec} indicates that a luminosity which is constant as $M \\rightarrow 0$ does not lead to a long-lived remnant with a life-time comparable to cosmic time-scales.\n\n\\section{Conclusions and Discussion} \n\\label{sect.5}\nMotivated by the observation that light black holes with a mass given by the Planck mass $M_{\\rm Pl} \\approx 10^{-5}$g may constitute valid dark matter candidates \\cite{mc,rov,dm,mur}, we used black hole thermodynamics to investigate the luminosity and lifetime of spherically symmetric black hole solutions. Our work stepped out of the perturbative framework of quantum field theory in curved spacetime by including the effect of a dynamical dimensional reduction of the theories momentum space. As this is a feature shared by many approaches to quantum gravity \\cite{Carlip:2017eud,Carlip:2019onx}, it is intriguing to investigate whether this mechanism leads to the formation of long-lived black hole remnants. While we showed that the dynamical dimensional reduction mechanism generically removes the divergences in the black hole luminosity encountered in the last stage of the evaporation process, the results do not provide any evidence supporting the formation of long-lived remnants. \n\nAt this point the following remarks on the scope and limitations of our analysis are in order. Our work implemented the mechanism of dynamical dimensional reduction at the level of the degrees of freedom constituting the Hawking radiation. In this course \\emph{we did not modify the topology of the background black hole solution} which is given by the Schwarzschild solution. This entails that our spacetimes exhibit just one horizon, the event horizon and there is no inner horizon. This feature is at variance with many proposals for quantum gravity inspired black hole metrics as, e.g., the Hayward metric \\cite{Hayward}, the renormalization group improved black hole solutions constructed by Bonanno and Reuter \\cite{martin}, or the Planck stars inspired by Loop Quantum Gravity \\cite{loop-rem}. A direct consequence of our topology is that the black holes in our work do not have a critical mass where the two horizons coincide and the surface gravity (and hence the Hawking temperature) is zero. By disentangling the effects of dynamical dimensional reduction and the topology of spacetime, our analysis clearly reveals that it is the latter ingredient which is decisive for forming a light black hole remnant  during the final stages of black hole evaporation. \n\nThis observation is also the key for reconciling our results with the ones reported by Carlip and Grummiller \\cite{carlip}. In this case the effect of dynamical dimensional reduction was essentially incorporated through generalizing the scaling law for the event horizon,\n\\begin{equation}\\label{eq:eh}\nA_h = 4 \\pi r_h^{d_h - 2} \\, , \n\\ee\nand subsequently identifying $d_h = d_s$. Within the single-scale analysis of Sect.\\ \\ref{sect.4.1} this modification with $d_h = 2$ or $d_h = 3$ would not lead to a stop of the Hawking evaporation process. A careful analysis of the dilaton model used in \\cite{carlip} shows, however, that the underlying black hole solutions must come with a second horizon: in this way one can approach a critical configuration if the generalized dimension of the dilaton model $D(X) = 3$. This picture then corroborates our conclusion that it is actually the topology of the (quantum) black hole and not the effect of a dynamical dimensional reduction that is crucial for forming light long-lived remnants.\n\nNaturally, it would be interesting to base our conclusion on a first-principle derivation from a full-fledged theory of quantum gravity. Such a derivation will require detailed knowledge about the momentum-dependence of the theories two-point functions. Notably, the form factor program for Asymptotic Safety \\cite{Knorr:2019atm} has recently made substantial progress along these lines \\cite{Knorr:2018kog,Bosma:2019aiu,Draper:2020bop,Platania:2020knd,Bonanno:2021squ,Knorr:2021niv}. Clearly, it would then be interesting to evaluate the black hole luminosity as an observable sensitive to a non-trivial momentum dependence in the propagators of the fields. We hope to come back to this point in the future.\n\n\n\\begin{acknowledgments}\n\tWe thank C.\\ Laporte for interesting discussions. The work by F.S.\\ is supported by the NWA-grant ``The Dutch Black Hole Consortium''.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe semiconductor quantum dots (QDs) resulting from the quantum\nconfinement of heterostructures exhibit atom-like discrete\nelectron energy levels. High-efficiency single-QD devices show the\nfunctionalities of low electrical and optical power outputs. These\nsingle-QD devices include single electron\ntransistors[\\onlinecite{Guo}-\\onlinecite{Kuba}], single photon\nsources[\\onlinecite{Michler}-\\onlinecite{Chang}], single photon\ndetectors[\\onlinecite{Gustavsson}] and single electron heat\nengines[\\onlinecite{Josefsson}]. Some applications of QD devices\nrequire both high efficiency and significant output power.\nTherefore, one needs QD solids that can retain the size tunable\nproperties of the QDs while exhibiting the band transport\ncharacteristic of bulk semiconductors.[\\onlinecite{Kagan}]\nAlthough much effort has been devoted to producing such QD solids,\nstudies of the thermoelectric properties of such 2D QD arrays have\nbeen lacking.[\\onlinecite{Harman},\\onlinecite{Talgorn}]\n\nDesigning a thermoelectric material with a high figure of merit\n($ZT$) and optimized power output is under\npursuit.[\\onlinecite{Kuo1}-\\onlinecite{Kuo2}] The dimensionless\nfigure of merit $ZT=S^2G_eT\/\\kappa$ depend on the Seeback\ncoefficient ($S$), electrical conductance ($G_e$) and thermal\nconductance ($\\kappa$) of the material. Although 1D QD arrays have\nvery high $ZT$ values, there exist many limitations in the\nimplementations of thermoelectric devices.[\\onlinecite{Kagan}] 2D\nand 3D QD arrays are required for realistic applications. The\n$\\kappa$ of a 2D QD array is smaller than that of bulk\nmaterial.[\\onlinecite{ChenG}] This low dimensional system has the\npotential to realize high $ZT$ values.[\\onlinecite{Chen}]\nTherefore, it is desirable to investigate the power factor\n($PF=S^2G_e$) of 2D QD arrays, which directly affects the\nelectrical power output. The enhancement of $G_e$ calls for a\nlarge number of electronic states (band-like). However a large $S$\nvalue occurs in dilute electronic states (atom-like). Therefore,\nenhancing one of these physical quantities will unavoidably\nsuppress the other. This study theoretically investigated the\nthermoelectric properties of a finite 2D QD array coupled to\nelectrodes, as shown in Fig. 1. The electrons of the QD array are\ninjected from the electrodes.[\\onlinecite{Mahan}] We demonstrated\nthat $G_e$ in band-like situation and $S$ in atom-like situation\ncan happen simultaneously when the miniband center of a 2D QD\narray remains a certain distance from the Fermi level of the\nelectrodes. These results will improve the thermoelectric\nperformance of 2D materials such as $SnSe$ and\n$MoS_2$.[\\onlinecite{Zhao}-\\onlinecite{Fan}]\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[trim=2.5cm 0cm 2.5cm 0cm,clip,angle=-90,scale=0.3]{Fig1}\n\\caption{(a) Schematic diagram of a two dimensional (2D) quantum\ndot (QD) array coupled to electrodes. $\\Gamma_{L}$ ($\\Gamma_R$)\ndenotes the tunneling rate of the electrons between the left\n(right) electrode and the leftmost (rightmost) QDs. Energy diagram\nof a 2D QD array coupled to electrodes with different equilibrium\ntemperatures ($T_L$ and $T_R$). }\n\\end{figure}\n\n\n\n\\section{Formalism}\n\nTo model the thermoelectric properties of a 2D QD array connected\nto the electrodes, the Hamiltonian of the system shown in Fig. 1\nis given by $H=H_0+H_{QD}$,[\\onlinecite{Haug}] where\n\\begin{small}\n\\begin{eqnarray}\nH_0& = &\\sum_{k,\\sigma} \\epsilon_k\na^{\\dagger}_{k,\\sigma}a_{k,\\sigma}+ \\sum_{k,\\sigma} \\epsilon_k\nb^{\\dagger}_{k,\\sigma}b_{k,\\sigma}\\\\ \\nonumber\n&+&\\sum_{\\ell}^{N_y}\\sum_{k,\\sigma}\nV^L_{k,\\ell,j}d^{\\dagger}_{\\ell,j,\\sigma}a_{k,\\sigma}\n+\\sum_{\\ell}^{N_y}\\sum_{k,\\sigma}V^R_{k,\\ell,j}d^{\\dagger}_{\\ell,j,\\sigma}b_{k,\\sigma}+H.c.\n\\end{eqnarray}\n\\end{small}\nThe first two terms of Eq.~(1) describe the free electron gas in\nthe left and right electrodes. $a^{\\dagger}_{k,\\sigma}$\n($b^{\\dagger}_{k,\\sigma}$) creates  an electron of momentum $k$\nand spin $\\sigma$ with energy $\\epsilon_k$ in the left (right)\nelectrode. $V^L_{k,\\ell,j}$ ($V^R_{k,\\ell,j}$) describes the\ncoupling between the left (right) lead with its adjacent QD in the\n$\\ell$th row, which counts from 1 to $N_y$.\n\\begin{small}\n\\begin{eqnarray}\nH_{QD}&= &\\sum_{\\ell,j,\\sigma} E_{\\ell,j}\nd^{\\dagger}_{\\ell,j,\\sigma}d_{\\ell,j,\\sigma}\\\\ \\nonumber&+&\n\\sum_{\\sigma}\\sum_{\\ell 1,\\ell 2}^{N_y}\\sum_{j1,j2}^{N_x} t_{\\ell\n1,\\ell 2, j1, j2} d^{\\dagger}_{\\ell 1,j1,\\sigma} d_{\\ell\n2,j2,\\sigma}+H.c,\n\\end{eqnarray}\n\\end{small}\n\\begin{equation}\nt_{\\ell 1,\\ell 2,j1, j2}= \\{ \\begin{array}{ll} -t_{y} &\nif~j1=j2,  |\\ell 1-\\ell 2|=1\\\\\n-t_{x} & if~\\ell 1=\\ell 2, |j1-j2|=1\n\\end{array},\n\\end{equation}\nwhere { $E_{\\ell,j}$} is the energy level of QD  in the\n${\\ell}$-th row and $j$-th column. The spin-independent $t_{\\ell\n1, \\ell 2, j1, j2}$ describes the electron hopping strength, which\nis limited to the nearest neighboring sites. $d^{\\dagger}_{\\ell\n1,j1,\\sigma} (d_{\\ell 2,j2,\\sigma})$ creates (destroys) one\nelectron in the QD at the $\\ell$th row and $j$th column. If the\nwave functions of the electrons in each QD are localized, the\nelectron Coulomb interactions are strong. Their effects on\nelectron transport are significant in the scenario of weak hopping\nstrengths.[\\onlinecite{Kuo3}] On the other hand, the wave\nfunctions of the electrons are delocalized in the scenario of\nstrong hopping strengths to form minibands; hence their weak\nelectron Coulomb interactions can be ignored.\n\nTo study the transport properties of a 2D QD array junction\nconnected to electrodes, it is convenient to use the\nKeldysh-Green's function\ntechnique[\\onlinecite{Haug},\\onlinecite{Meir}]. Electron and heat\ncurrents leaving electrodes can be expressed as\n\\begin{eqnarray}\nJ &=&\\frac{2e}{h}\\int {d\\varepsilon}~\nT_{LR}(\\varepsilon)[f_L(\\varepsilon)-f_R(\\varepsilon)],\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n& &Q_{e,L(R)}\\\\ &=&\\frac{\\pm 2}{h}\\int {d\\varepsilon}~\nT_{LR}(\\varepsilon)(\\varepsilon-\\mu_{L(R)})[f_L(\\varepsilon)-f_R(\\varepsilon)]\\nonumber\n\\end{eqnarray}\nwhere\n$f_{\\alpha}(\\varepsilon)=1\/\\{\\exp[(\\varepsilon-\\mu_{\\alpha})\/k_BT_{\\alpha}]+1\\}$\ndenotes the Fermi distribution function for the $\\alpha$-th\nelectrode, where $\\mu_\\alpha$  and $T_{\\alpha}$ are the chemical\npotential and the temperature of the $\\alpha$ electrode. $e$, $h$,\nand $k_B$ denote the electron charge, the Planck's constant, and\nthe Boltzmann constant, in that order. $T_{LR}(\\varepsilon)$\ndenotes the transmission coefficient of a 2D QD array connected to\nelectrodes, which can be solved by the formula $\nT_{LR}(\\varepsilon)=4Tr[\\hat{\\Gamma}_{L}\\hat{G}^{r}_{D,A}(\\varepsilon)\\hat{\\Gamma}_{R}\\hat{G}^{a}_{D,A}(\\varepsilon)]$,\nwhere the matrix of tunneling rates ($\\hat{\\Gamma}_L$ and\n$\\hat{\\Gamma}_R$) and Green's functions\n($\\hat{G}^{r}_{D,A}(\\varepsilon)$ and\n$\\hat{G}^{a}_{D,A}(\\varepsilon)$) can be constructed by\ncoding.[\\onlinecite{Kuo4}]\n\nThe electrical conductance ($G_e$), Seebeck coefficient ($S$) and\nelectron thermal conductance ($\\kappa_e$) can be evaluated by\nusing Eqs. (4) and (5) with a small applied bias $\\Delta\nV=(\\mu_L-\\mu_R)\/e$ and cross-junction temperature difference\n$\\Delta T=T_L-T_R$. We obtain these thermoelectric coefficients\n$G_e=e^2{\\cal L}_{0}$, $S=-{\\cal L}_{1}\/(eT{\\cal L}_{0})$ and\n$\\kappa_e=\\frac{1}{T}({\\cal L}_2-{\\cal L}^2_1\/{\\cal L}_0)$. ${\\cal\nL}_n$ is given by\n\\begin{equation}\n{\\cal L}_n=\\frac{2}{h}\\int d\\varepsilon~\nT_{LR}(\\varepsilon)(\\varepsilon-E_F)^n\\frac{\\partial\nf(\\varepsilon)}{\\partial E_F},\n\\end{equation}\nwhere $f(\\varepsilon)=1\/(exp^{(\\varepsilon-E_F)\/k_BT}+1)$ is the\nFermi distribution function of electrodes at equilibrium\ntemperature $T$.\n\n\\section{ Results and discussion}\nAccording to Eq. (6), transmission coefficient plays a significant\nrole for electron transport between the electrodes. To illustrate\nthe electronic states of finite 2D QD array, we have calculated\nand shown in Fig. 2 transmission coefficient $T_{LR}(\\varepsilon)$\nas a function of $\\varepsilon$ for different tunneling rates\n($\\Gamma_{L(R),\\ell,j}(\\varepsilon)=2\\pi\\sum_{k}\n|V^{L(R)}_{k,\\ell,j}|^2\n\\delta(\\varepsilon-\\varepsilon_k)=\\Gamma_t$). A square lattice\nwith homogenous electron hopping strengths $t_x=t_y=t_c=6\\Gamma_0$\nand site-independent QD energy level $E_{\\ell,j}=E_0=E_F$ has been\nconsidered in the calculation of $T_{LR}(\\varepsilon)$. All\nphysical parameters are in units of $\\Gamma_0$. In Fig. 2(a),\n$T_{LR}(\\varepsilon)$ reveals the tunneling probability of the\nelectrons of the electrodes through the electronic states of 2D QD\narray, those energy is described by\n$\\varepsilon=E_0-2t_c(cos(\\frac{n_x\\pi}{N_x+1})+\ncos(\\frac{n_y\\pi}{N_y+1}))$, where $n_x=1,2,..N_x$ and\n$n_y=1,2,..N_y$.  Because the QD array is connected to the\nelectrodes, these electronic states have inhomogeneous broadening.\nThey are also restricted within the range between $-4t_c$ and\n$4t_c$. We can tune the distribution of electronic states by\nchanging $N$, $t_c$ and $\\Gamma_t$.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[angle=-90,scale=0.3]{Fig2}\n\\caption{Transmission coefficient $T_{LR}(\\varepsilon)$ as a\nfunction of $\\varepsilon$ for different $N$ values  at\n$t_x=t_y=t_c=6\\Gamma_0$ and $E_0=E_F$. Diagrams (a) and  (b)\nconsider tunneling rates $\\Gamma_L=\\Gamma_R=\\Gamma_t=1\\Gamma_0$\nand $\\Gamma_t=6\\Gamma_0$, respectively. To prevent the curves from\noverlapping each other, we shifted the curve of N = 8.}\n\\end{figure}\n\nFrom Eqs. (4) and (5), the maximum electron current and heat\ncurrent occur at $T_{LR}(\\varepsilon)$ with the maximum area. The\nauthors of Ref. [\\onlinecite{Kuo4}] proved two results: the\nmaximum area of $T_{LR}(\\varepsilon)$ can be reached at the\ncondition of $\\Gamma_t=t_c$ and the maximum area increases with\nincreasing $N$, as seen in Fig. 2(b). Note that the 2D\ntight-binding electronic states show the Van Hove singularity in\nthe density of states (DOS) as $N \\rightarrow \\infty$ (DOS\ndiverges at $E_0$). At zero temperature, the electrical\nconductance is given by the transmission coefficient\n$G_e=\\frac{2e^2}{h}T_{LR}(E_F)$. We now clarify how the electronic\nstates influence the thermoelectric coefficients of a finite 2D QD\narray.\n\nFig. 3 shows the calculated $G_e$, $\\kappa_e$ and Lorenz number\n($L_0=\\kappa_e\/(TG_e)$ at functions of the QD energy level for\nvarious values of $\\Gamma_t$ at $k_BT=1\\Gamma_0$, $t_c=6\\Gamma_0$\nand $N=8$. As a result of temperature effect ($\\frac{\\partial\nf(\\varepsilon)}{\\partial E_F}$), the electronic states shown in\nFig. 2(a) can not be resolved in Fig. 3(a). It is not easy to\njustify a finite 2D QD array in the band-like or molecule-like\nsituation from $G_e$ at finite temperature, especially at high\ntemperatures. The curves of $\\kappa_e$ in Fig. 3(b) are similar to\nthose of $G_e$. According to the Wiedemann-Franz law,\n$L_0\/(k^2_B\/e^2)=\\frac{\\pi^2}{3}$ is a temperature-independent\nquantity. In Fig. 3(c), the $L_0$ curve corresponding to\n$\\Gamma_t=6\\Gamma_0$ is approximately $\\pi^2\/3$. For comparison,\nwe also add the calculated $G_e$, $\\kappa_e$ and $L_0$ for 1D QD\narray with $N_x=100$ and $t_c=\\Gamma_t=12\\Gamma_0$ (the band width\nof $48\\Gamma_0$ in this 1D miniband). As seen in Fig. 3(c) , 1D QD\narray yields a Lorenz number\n$L_0=\\frac{k^2_B}{e^2}\\frac{\\pi^2}{3}$ between $-10\\Gamma_0 \\le\n\\Delta \\le 10\\Gamma_0$. This can be regarded as a manifested\nband-like transport.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[angle=-90,scale=0.3]{Fig3}\n\\caption{(a) Electrical conductance $G_e$, (b) electron heat\nconductance $\\kappa_e$ and (c) Lorenz number\n($L_0=\\kappa_e\/(TG_e)$) as functions of $\\Delta=E_0-E_F$ for\nvarious $\\Gamma_t$ values  at $k_BT=1\\Gamma_0$, $t_c=6\\Gamma_0$\nand $N=8$. The red curves correspond to a 1D QD array with\n$N_x=100$, $t_c=\\Gamma_t=12\\Gamma_0$ and $k_BT=1\\Gamma_0$.}\n\\end{figure}\n\nFurthermore, we have calculated $G_e$, $\\kappa_e$ and $L_0$ for 2D\nQD array with $N=8$ as functions of temperature for different\n$\\Gamma_t$ values at $E_0=E_F$ and $t_c=6\\Gamma_0$ in Fig. 4. The\nred curves are the results of the 1D QD array corresponding to\nthose of Fig. 3. One-dimensional QD arrays have a\ntemperature-independent $G_e$ and a linear temperature-dependent\n$\\kappa_e$. As a consequence, $L_0=\\kappa_e\/(TG_e)$ leads to a\ntemperature-independent behavior. According to the results of\nFigs. 3 and 4, the thermoelectric properties of 2D QD arrays with\n$t_c=6\\Gamma_0$, $\\Gamma_t=6\\Gamma_0$ and $N=8$ are very similar\nto those of 1D QD arrays with minibands. We deduce that finite 2D\nQD arrays have band-like characteristics when\n$t_c=\\Gamma_t=6\\Gamma_0$ and $N=8$.\n\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[angle=-90,scale=0.3]{Fig4}\n\\caption{(a) Electrical conductance, (b) electron heat conductance\nand (c) Lorenz number  as functions of temperature for various\n$\\Gamma_t$ values at $E_0=E_F$, $t_c=6\\Gamma_0$ and $N=8$. The red\ncurves correspond to those of Fig. 3 with $E_0=E_F$.}\n\\end{figure}\n\n\nBecause many thermoelectric devices operate at high temperatures,\nit is important to examine the power factor of 2D QD arrays in\nthis regime. In Figs. 2-4 we have focused on the electron\ntransport in resonant tunneling procedure (RTP) in which the\nSeebeck coefficient is very small. To obtain a large $PF$ value at\nhigh temperature, we considered the electron transport in the\nthermionic-assisted tunneling procedure (TATP) where the band\ncenter ($E_0$) is far away from the Fermi level $E_F$ in the\nelectrodes. In Fig. 5, we have calculated $G_e$, $S$, $PF$ and\n$L_0$ as functions of temperature for various $t_c$ values at\n$E_0-E_F=30\\Gamma_0$. Because the maximum $T_{LR}(\\varepsilon)$\narea occurs at $\\Gamma_t=t_c$, we adopted this condition for all\nthe subsequent steps. As seen in Fig. 5(a), $G_e$ is vanishingly\nsmall at low temperature due to the electronic states of 2D QD\narrays being kept a certain distance from the $E_F$. The\nenhancement of $G_e$ with increasing temperature is a typical\ncharacteristic arising from the TATP. To understand the\ntemperature behavior of $G_e$ at $\\Gamma_t=t_c=1\\Gamma_0$, we have\nthe expression of $G_{e,atom}=\\frac{e^2}{h} \\frac{\\pi \\Gamma_t}\n{2k_BT cosh^2((E_0-E_F)\/(2k_BT))}$ when the transmission\ncoefficient is approximated as\n$T_{LR}(\\varepsilon)=4\\Gamma^2_t\/((\\varepsilon-E_0)^2+(2\\Gamma_t)^2)$\nin Eq. (6). In addition, $S_{atom}=-\\Delta\/T=-(E_0-E_F)\/T$, which\nexplains the behavior of $S$ at $t_c=1\\Gamma_0$ and $k_BT \\ge\n2\\Gamma_0$ in Fig. 5(b). Although $G_e$ is highly enhanced with\nincreasing $t_c$, $S$ is not so sensitive to $t_c$ for $k_BT \\ge\n10 \\Gamma_0$. This explains why the trend of maximum $PF$ for\n$t_c$ shown in Fig. 5(c) is determined by $G_e$. In Fig. 5(d),\nthree $L_0$ curves violate the Wiedemann-Franz law. Note that\n$\\Gamma_t=t_c=6\\Gamma_0$ provides the band-like characteristic\n(see Fig. 3(c)).\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[angle=-90,scale=0.3]{Fig5}\n\\caption{(a) Electrical conductance, (b) Seeback coefficient, (c)\npower factor ($PF=S^2G_e$) and (d) Lorentz number  as functions of\n$T$ for various $t_c$ values at $E_0-E_F=30\\Gamma_0$ and $N=8$.\nMeanwhile, we have adopted $\\Gamma_t=t_c$.}\n\\end{figure}\n\nFig. 6 shows the calculated $G_e$, $S$ and $PF$ as functions of\n$E_0$ for various $t_c$ values at $k_BT=25\\Gamma_0$ to reveal the\neffect of the band center. As seen in Fig. 6(a), $G_e$ has a\nmaximum value when the band center ($E_0$) is located at $E_F$.\nThe Seeback coefficients in Fig. 6(b) are zero at $E_0=E_F$. It is\nattributed to the symmetrical distribution of the electrons and\nholes on the electronic states of the 2D QD array. Here, the holes\nare defined as the empty states below $E_F$. For\n$k_BT=25\\Gamma_0$, the Seebeck coefficients are well described by\n$S_{atom}=-\\Delta\/T$. We find that the maximum values of $PF$\noccur near $\\Delta=60\\Gamma_0$, as indicated in Fig. 6(c). When\napproaching the atomic limit ($t_c \\rightarrow 0$), one can prove\nthat the optimization of $PF$ is given by $\\Delta\/k_BT=2.4$. The\nresults of Fig. 6(c) imply that 2D QD arrays with minibands\n($t_c=6\\Gamma_0$) preserve the atomic thermoelectric properties\nwhen the band center is far away from the $E_F$ of the electrodes.\nIn Fig. 2(b), $T_{LR}(\\varepsilon)$ depends on $N$. Therefore, we\nadd in Fig. 6 the curves with triangle marks for $N=7$ and\n$t_c=6\\Gamma_0$. From the curves of $N=7$ and $N=8$, we see that\nthe enhancement of $G_e$ resulting from the increase of electronic\nstates does not suppress $S$. It is worthy noting that a single 1D\nQD array does not exist such a behavior. We reinvestigate $PF$ as\nfunctions of $t_c$ for $N=7,8$ at $\\Delta=60\\Gamma_0$ and\n$k_BT=25\\Gamma_0$ in Fig. 6(d). $PF$ is a linear function of $t_c$\nas $t_c\\le 6\\Gamma_0$. Meanwhile, the maximum $PF$ is given by\n$t_c=\\Delta\/4$. The red curves represent the case where $t_y=0$ to\nclarify the geometer effects. When $t_c \\le 6\\Gamma_0$, the\ngeometry effects can be ignored.\n\n\nBecause $T_{LR}(\\varepsilon)$ lacks an analytical form, it is not\neasy to illustrate the complex behavior of $PF$ shown in Fig.\n6(d). If we make the assumption that minibands have homogenous\nelectronic states and consider the square-form\n$T_{LR}(\\varepsilon)$ given by\n\\begin{equation}\nT_{LR}(\\varepsilon)=\\left\\{ \\begin{array}{ll}\nN_y& \\mbox {if $-2t_c\\le \\varepsilon-E_0 \\le 2t_c,$}\\\\\n0 &\\mbox{otherwise}\\end{array} \\right.\n\\end{equation}\nthe analytical forms of $G_e$ and $S$ can be derived as\n\\begin{equation}\nG_e=\\frac{2e^2N_y}{h}~(tanh(y_1)-tanh(y_2))\n\\end{equation}\nand\n\\begin{equation}\nS=\\frac{2ek_BN_y}{h}~\\frac{(S_1(y_1)-S_2(y_2))}{G_e}\n\\end{equation}\nwhere $S_i(y_i)=y_i~tanh(y_i)-log(cosh(y_i))$,\n$y_1=\\frac{\\Delta+2t_c}{2k_BT}$ and\n$y_2=\\frac{\\Delta-2t_c}{2k_BT}$.  In Fig 6(d), the blue curves are\ncalculated by using Eqs. (8) and (9). Because Eq. (7) considers\n$N_y$ 1D QD arrays with homogenous electric states, it is expected\nthat the $PF$ given by Eqs. (8) and (9) is overestimated. However,\nEqs. (8) and (9) provide a clear picture that the enhancement of\n$PF$ follows the enhancement of $G_e$. Meanwhile, $S\\approx\nS_{atom}$ as long as $\\frac{t_c}{k_BT} < 0.25$ and\n$\\frac{\\Delta}{k_BT} \\ge 2.4$. We deduce that $G_e$ in a band-like\ntransport situation and $S$ in an atomic-like situation can\ncoexist for finite 2D QD arrays with $t_c=6\\Gamma_0$ and\n$\\Delta=60\\Gamma_0$ at $k_BT=25\\Gamma_0$. If we set\n$\\Gamma_0=1~meV$, then our analysis in Fig. 6 becomes a very\nuseful guideline for thermoelectric devices operated at room\ntemperature.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[angle=-90,scale=0.3]{Fig6}\n\\caption{(a) Electrical conductance, (b) Seebeck coefficient and\n(c) power factor as functions of $E_0$ for different $t_c$ values\nat $N=8 $ and $k_BT=25\\Gamma_0$. The curves with black triangle\nmarks correspond to the case of $N=7$ and $t_c=6\\Gamma_0$.(d) $PF$\nas functions of $t_c$ for $\\Delta=60\\Gamma_0$ and\n$k_BT=25\\Gamma_0$. The red curves correspond to $t_y=0$. The blue\ncurves are calculated using Eq. (7).}\n\\end{figure}\n\n\n\n\n\\section{Conclusion}\nWe have theoretically investigated the thermoelectric properties\nof 2D QD arrays. In RTP, the Lorenz number with a value near\n$\\pi^2\/3$ and a temperature-independent behavior demonstrates that\n2D QD arrays with $t_c=\\Gamma_t=6\\Gamma_0$ and $N=8$ indeed form\nminibands. When this miniband center is far away from the Fermi\nlevel of the electrodes, TATP dominates the electron transport\nbetween the electrodes and $L_0$ violates the Wiedemann-Franz law.\nIn TATP, $G_e$ is enhanced as the number of electronic states\nincreases, whereas the $S$ values remain in an atom-like\nsituation. This is a remarkable property that would lead to a\nhigh-efficiency thermoelectric devices made of QDs with large\nelectrical power output. This interesting phenomenon exists not\nonly for 2D QD arrays with square-lattices but also\ntriangular-lattices, which will be reported in elsewhere.\n\n\n{\\bf Acknowledgments}\\\\\nThis work was supported under Contract No. MOST 107-2112-M-008\n-023MY2\n\\mbox{}\\\\\nE-mail address: mtkuo@ee.ncu.edu.tw\\\\\n\n\\setcounter{section}{0}\n\n\\renewcommand{\\theequation}{\\mbox{A.\\arabic{equation}}}\n\\setcounter{equation}{0}\n\n\\mbox{}\\\\\n\n{\\bf Data Availability Statements}\\\\\n\nThe data that supports the findings of this study are available\nwithin the article.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn recent years, high-dimensional data have flooded almost all fields of science and technology. These types of data are being generated in massive quantities and share a common high-throughput mechanism. Generally speaking, a prominent number of measurements of features, often in the millions, are gathered for a single subject.\nThe goal of the inherent analysis for these large datasets is to make useful inferences for each subject or a future subject. The tools for solving the \nproblems in this area play particularly important roles, especially in this\n``data science'' era.\n\nOne of the fundamental problems in large-scale inference \\citep{large-infer} is classification. Consider a two-class classification problem, where we have $n$ labeled training samples $(X_i,Y_i)$, $i=1, \\dots, n$. Here, $X_i$'s are $p$-dimensional feature vectors and $Y_i\\in\\{0,1\\}$ are the corresponding class labels. The goal is to estimate the label of a new observation $X$. A significant amount of work has been done in this field: see \\citep{Anderson, friedman1989regularized, lachenbruch1979discriminant}. \n\nIn modern analytical approaches, usually the number of features is huge but only a small portion of them are regarded as relevant to the classification decision, although these are not known in advance. \nIn this sense, the methods will lose power because of the large amount of noise. \nA common way to solve this problem is to perform dimension reduction such that the number of features is greatly reduced, and then use only the resultant features for classification; see \\citep{FJY, PCS, Wasserman}. \n\n\nFisher's Linear Determinant Analysis (LDA) in \\cite{Fisher} utilizes a weighted average of the features of the test sample to make a prediction. In the traditional setting where  $n \\gg p$, the optimal weight vector for LDA where the two classes are assumed to share the same correlation structure $\\Omega$ satisfies\n\\begin{equation} \\label{Fisherw}\nw \\propto \\Omega  (\\mu_1 - \\mu_0)\n\\end{equation}\nwhere the mean vectors $\\mu_0$ and $\\mu_1$, as well as $\\Omega$, can be easily estimated, and therefore Fisher's LDA is approachable. \n\nHowever, for the high-dimensional setting where  $p \\gg n$, LDA faces immediate problems:\n\\begin{itemize}\n\\item It is practically infeasible to estimate $\\Omega$ due to the limited sample size $n$.\n\\item It is difficult to account for the information from different $\\Omega_0$ and $\\Omega_1$. The difference affects two areas: the difference in the covariance matrix term and the estimation of $\\omega$. The classification criteria have to be improved to combine these two parts. \n\\item The analysis for error rate is quite complicated even in the simplest case, where $\\Omega_0= \\Omega_1 = I_p$, as the signals are rare and weak. An extensive discussion may be found in  \\cite{DJ08, Jin-survey1, FJY, Jin-survey2, JKW}.\n\n\\end{itemize}\n\nWith recent developments in the estimation of the precision matrix in high dimensions \\cite{PCS}, the first problem has been alleviated; that is, Partial Correlation Screening (PCS) is computationally efficient in estimating a row of $\\Omega$, and it needs only a few rows of the empirical covariance matrix. PCS has been demonstrated to be capable of executing the estimation of a large precision matrix ($p$ = 10K), and has been applied with evident success in classification. \n\nFor the second problem, one variant of LDA that incorporates the covariance-matrix information is proposed as Quadratic Discriminant Analysis (QDA) in \\cite{RegularizedQDA}; with extensions for the high-dimensional data; see \\cite{aoshima2019high, wu2019quadratic, RFQD}. When the signals are sparse, the original QDA method is modified accordingly with sparsity assumptions of the difference of the precision matrices and $\\mu_1 - \\mu_0$; see \\cite{fan2015innovated, jiang2018direct, li2015sparse}. However, to the best of our knowledge, the theoretical discussions for the error rate for the high-dimensional QDA with rare and weak signals have not been thoroughly discussed.\n\nTherefore, this paper primarily focuses on addressing the second and third problems. Starting from the classical QDA, we propose a new QDA with the feature-selection approach, where we estimate $\\Omega_1$ and $\\Omega_2$ with PCS and $\\mu$ with sample mean for the quadratic term (second-order information), and estimate the linear term (first-order information) with $d^\\top X$, where $d$ is found by thresholding $\\Omega_2 \\mu_2 - \\Omega_1 \\mu_1$. \nNumerical analysis of a real data set suggests that our new approach should incorporate the second-order information, as it will then perform better than the optimal high-dimensional LDA method. \n\nFor the third problem, we propose the rare and weak model on both the mean vector and the precision matrix, tying their rareness and weakness to the parameter $p$. \nBased on the model, we carefully quantify the possibility and impossibility regions for the two-class classification by simultaneously exploiting both the difference in $\\mu$ and the difference in $\\Omega$. In addition, we carefully analyze how the errors in QDA may affect the classification results, which has not been done previously in the rare and weak signal model.\n\n\nWe calibrate the effect of quadratic terms on classification, in terms of  \n\\begin{itemize}\n\\item the possibility and impossibility region under the ideal case, where both $\\mu_k$ and $\\Sigma_k$ ($k=0,1$) are known;\n\\item the possibility and impossibility region for the newly proposed QDA with feature selection, under other alternative cases where $\\mu_k$ and $\\Sigma_k$ ($k=0,1$) are only partially known;\n\\item the successful classification region for the newly proposed QDA with feature selection when both $\\mu_k$ and $\\Omega_k$ are unknown. \n\\end{itemize}\nTo highlight our findings, we show that the QDA classification rule achieves a misclassification rate of 0 asymptotically ($p \\rightarrow \\infty$) in some parameter regions, and for other regions has asymptotic constant error in the ideal case (Theorems \\ref{thm:equal}-\\ref{thm:idealunequal}). The results for other alternative cases are subsequently analyzed in Theorems \\ref{thm:unknownmu}-\\ref{thm:unknownomega}. \n\n\n\\subsection{Quadratic Discriminant Analysis} \nConsider the two-class classification problem mentioned above, where we observe $n$ training samples $(X_i,Y_i)$, $i=1, \\dots, n$. \nGiven $Y_i=k$, we assume the feature vector $X_i \\in \\mathcal{R}^p$ follows a multivariate normal distribution with mean $\\mu_k$ and covariance matrix $\\Omega_k^{-1}$. \nLet $X$ denote an independent test sample from the same population; then, \n\\begin{equation} \\label{T0}\nX|Y \\sim (1 - Y) N(\\mu_0, \\Omega_0^{-1})+ Y N(\\mu_1,\\Omega_1^{-1}).\n\\end{equation}\nWe would like to classify $X$ as being from either $Y=0$ or $Y=1$. \n\nFor two-population classification problems, the QDA method is commonly used to exploit both the mean and covariance information; see \\cite{mclachlan2004}. Similarly, when $\\mu_k$ and $\\Omega_k$ are all known, $k = 0, 1$, we derive the likelihood function of the two populations, and the ratio gives the criteria for the current problem. \nThe classification rule is that\n\\begin{equation} \\label{CR2}\n\\begin{array}{l}\n\\displaystyle \\hat{Y} = I \\left\\{X^\\top\\left (\\Omega_0 -\\Omega_1\\right )X-2\\bigl(\\mu_0^\\top\\Omega_0 -\\mu_1^\\top\\Omega_1 \\bigr)X\\right. \\\\\n\\displaystyle \\left. \\ \\ \\ + \\bigl(\\mu_0^\\top\\Omega_0\\mu_0- \\mu_1^\\top\\Omega_1\\mu_1+\\ln|\\Omega_1|- \\ln|\\Omega_0|\\bigr) \\ > \\ 0\\right\\} ,\n\\end{array}\n\\end{equation} \nwhere $I(A)$ is the indicator function of event $A$. If $P(Y_i = 1) = q\\neq 0.5$, an additional term $2\\ln \\frac{q}{1-q}$ on the right-hand side of the inequality will improve the accuracy. However, as this does not have a significant effect on possibility and impossibility regions, our analysis applied to this case (i.e. $q=0.5$) will suffice. \n\nWhen $\\mu_k$ and $\\Omega_k$ are unknown in reality, we have to use the estimated parameters instead. When $p \\ll n$, the sample mean and covariance are accurate. However, in the high-dimensional setting with $p \\gg n$ and the signals being rare and weak, the estimation becomes difficult.\n\n\n\n\\subsection{Rare and Weak Signal Model}\nIn the high-dimensional setting, the number of features $p \\rightarrow \\infty$ and the signals are rare and weak. We need a model to catch the rareness and weakness in both the mean part and the covariance part. Since both parts include the rareness and weakness, the general model is complicated. Here, we consider them separately and make some simplifications. \n\nIn \\cite{FJY}, the case that $\\mu_0 \\neq \\mu_1$ but $\\Omega_0 = \\Omega_1$ has been studied. We start with the model from there, and assume $\\mu_0=-\\mu$ and $\\mu_1=\\mu$. If not, we could simply do a location shifting of the distance $\\frac{1}{2}|\\mu_1-\\mu_0|$ to the left or the right. \nWe model the contrast mean vector $\\mu$ as\n\\begin{equation}\\label{Param1}\n\\mu_i \\ \\overset{i.i.d.}\\sim \\ (1-\\epsilon)\\mathcal M_0+\\epsilon \\mathcal M_\\tau, \\ \\ \\ \\ i=1,\\dots,p,\n\\end{equation}\nwhere $\\mathcal M_0$ and $\\mathcal M_\\tau$ are the point mass at 0 and $\\tau>0$ respectively. Here, we assume the signals have the same signs and strengths. This is largely for simplicity, and, as discussed in \\citep{FJY} and our remarks following Theorem \\ref{thm:equal}, $\\mathcal M_\\tau$ can be replaced by a random variable with support $[-\\tau,\\tau]$; all the results discussed in this paper will still hold. \nThe signals are rare and weak in the sense that, when the sample size $n$ approaches infinity, we assume that the strength and density of the signals converges to 0, i.e., \n\\begin{equation}\n\\epsilon \\ \\rightarrow \\ 0, \\ \\ \\ \\ \\tau \\ \\rightarrow \\ 0. \n\\end{equation}\n\nTo examine the effect of the quadratic term, we also model the difference between $\\Omega_0$ and $\\Omega_1$. Here we use the precision matrix instead of the correlation matrix because the estimations of the two matrices are totally different in the high-dimensional setting, and the former one has a direct effect on the classification results.\nWe are interested in the difference between $\\Omega_0$ and $\\Omega_1$, and so we assume $\\Omega_0$ is known and focus on the parameterization of $\\Omega_1 - \\Omega_0$. \nWithout loss of generality, we assume $\\Omega_0=I$. Hence, we write $\\Omega = \\Omega_1$ and $\\Sigma = \\Sigma_1$ in the following context for notational simplicity. \n\nWe introduce a characterization of rareness and weakness for $\\Omega$. Let $W(\\nu)$ denote a $p$-dimensional Wigner matrix with parameter $\\nu$. Specifically, let $W(\\nu)$ be symmetric with 0's on the diagonals and $w_{ij}$ on the off-diagonals $w_{ij}$, following the distribution \n\\begin{equation}\\label{Omega0}\nw_{ji} = w_{ij} \\ \\overset{i.i.d.}\\sim \\ (1-\\nu)\\mathcal M_0+\\frac{\\nu}{2} \\mathcal M_1 + \\frac{\\nu}{2} \\mathcal M_{-1}, \\ \\ \\ \\ 1 \\leq i < j \\leq p,\n\\end{equation}\nwhere $\\mathcal M_1$ and $\\mathcal M_{-1}$ are the point masses at 1 and $-1$, respectively.\nThen, we model $\\Omega$ as \n\\begin{equation}\\label{Omega1}\n\\Omega \\ = \\ \\Sigma^{-1} \\ = \\ cI+\\eta W\n\\end{equation}\nfor some constant $c=c(n, p)$ close to 1. Here, the parameters $1 - c$ and $\\eta$ represent the signal weakness, and $\\nu$ represents the signal rareness. We assume $\\nu \\rightarrow 0$, $\\eta \\rightarrow 0$ and $c \\rightarrow 1$ when the sample size $n$ goes to infinity. \n\nNow, with a realization of $W$, the model under consideration is the mixture model\n\\begin{equation}\\label{M12}\nX_i|(W,Y_i) \\ \\sim \\ (1-Y_i) N(-\\mu,I) + Y_i N \\left(\\mu,[cI+\\eta W]^{-1}\\right), \\ \\ i=1,\\dots,n.\n\\end{equation}\nWe are interested in the possibility and impossibility regions under the current model and successful and unsuccessful classification regions of QDA classifier. \n\nWe tie all the parameters to $p$ by some constant parameters, as follows, and explore how these parameters influence the classification results. For the mean vector, the parameters $\\epsilon$ and $\\tau$ depict the signal sparsity and weakness, respectively. We define them as\n\\begin{equation}\\label{Param2}\n\\epsilon \\ = \\ p^{-\\zeta}, \\ \\ \\ \\ \\tau \\ = \\ g_p p^{-\\theta}, \\ \\ \\ \\ 0<\\zeta, \\theta <1,\n\\end{equation}\nwhere $g_p$ can be $1$ or a $\\ln p$ term, which will be discussed in Section \\ref{sec:main}.  \nFor the precision matrix $\\Omega$, we take the following parameterization in a similar manner to that of the mean part: \n\\begin{equation}\\label{Param}\n\\eta=f_p p^{-\\alpha}, \\ \\ \\ \\nu=p^{-\\beta}, \\ \\ \\ \\xi=1-c=p^{-\\gamma}, \\ \\ \\ \\ \\ 0<\\alpha, \\gamma<1, 0< \\beta <2,\n\\end{equation}\nwhere $f_p$ can be $1$ or a $\\ln p$ term. Details are provided in  Section \\ref{sec:main}. \nHere, $\\alpha$, $\\beta$, $\\gamma$, $\\zeta$, and $\\theta$ are all constants. It is of interest to explore the performance diagram of the QDA classifier on the space formed by these parameters. \n\nTo guarantee that $\\Omega$ is a precision matrix, $\\eta W$ must be weak enough for $\\Omega = cI + \\eta W$ to be positive definite. According to Lemma \\ref{lemma1} in Section \\ref{sec:proof}, this requirement is satisfied with high probability under the condition\n\\begin{equation}\\label{con1}\n\\beta > 1 - 2\\alpha. \n\\end{equation}\nHence, we discuss only the classification possibility and impossibility regions under this condition. \n\nFinally, we tie the sample size $n$ to $p$ by\n\\begin{equation}\\label{Param3}\nn = p^{\\delta}, \\qquad 0 < \\delta < 1. \n\\end{equation}\nWhen $p \\rightarrow \\infty$, $n$ goes to infinity at a much slower rate. \n\n\n\n\n\n\\subsection{QDA with Feature Selection}\nFor the high-dimensional setting, we have to incorporate feature selection in the proposed QDA classifier. Consider a new  observation $X$ from the mixture population (\\ref{M12}); i.e., $X \\sim N(-\\mu, I)$ when $Y = 0$ and $X \\sim N(\\mu, \\Omega^{-1})$ when $Y = 1$. Under the current model, the classification rule (\\ref{CR2}) is reduced to \n\\begin{equation}\\label{eqn:qda}\n\\hat{Y}(X; \\mu, W)= I \\{X^\\top(I - \\Omega) X + 2\\mu^\\top (I+\\Omega) X + \\mu^\\top(I - \\Omega) \\mu + \\ln|\\Omega| > 0\\}.\n\\end{equation}\nIn this rule, the unknown $\\mu$ and $\\Omega$ need to be estimated. We propose the following algorithm for QDA classification with feature selection:\n\\begin{itemize}\n\\item[] {\\bf Algorithm 1}.\n\\item[0.] Find the corresponding precision matrix $\\hat{\\Omega}$ with a precision-matrix-estimation method. Calculate the average vector of the two classes as $\\hat{\\mu}_0 = \\frac{1}{n_0} \\sum_{i: Y_i = 0} X_i$ and $\\hat{\\mu}_1 = \\frac{1}{n_1} \\sum_{i: Y_i = 1} X_i$, where $n_1 = \\sum_{i =1}^n Y_i$ and $n_0 = n - n_1$. \n\\item[1.] Let $d_0=\\hat{\\mu}_0$, $d_1= \\hat{\\Omega} \\hat{\\mu}_1$ and $d = d_1 - d_0=(d(1),\\dots, d(p))^\\top$. \n\\item[2.] With a threshold $t$, let $d^{(t)}$ denote the indicator vector of feature selection, i.e. for $j=1,\\dots,p$,\n\\begin{equation}\nd^{(t)}(j)= \\left\\{\\begin{array}{ll}\n1, & |d(j)| > t, \\\\\n0, & |d(j)| < t. \n\\end{array}\n\\right.\n\\end{equation*}\nLet $\\hat \\mu_d^{(t)}$ be the hard-thresholded mean difference vector $d$, i.e. $\\hat \\mu_d^{(t)}=d\\circ d^{(t)}$.\n\\item[3.] Let $C = \\mu^\\top(I - \\Omega) \\mu + \\ln |\\Omega|$. When $C$ is unknown, we use grid search to figure out $C$ with best performance.\n\\item[4.] (Prediction step) For any fresh data vector $X$, calculate the QDA score\n$$ \nQ=X^\\top(I - \\hat{\\Omega}) X + 2( \\hat\\mu_d^{(t)})^\\top X + C\n$$\nand estimate the label of $X$ as $\\hat{Y}$ with $\\hat Y = I\\{Q > 0\\}$. \n\\end{itemize}\n\nIn this procedure, Step 0 can adopt any estimation method for the high-dimensional sparse precision matrix, such as the one in \\cite{PCS}. The theoretical limit of the proposed method depends on the estimation method applied. \n\nIn Steps 1\u20132, we propose a feature-selection component for $d =  \\hat{\\Omega}\\hat{\\mu}_1 - \\hat{\\mu}_0$. This step is quite standard in high-dimensional data analysis, and yet is different from the high-dimensional LDA approach in two aspects. First, we consider $\\hat{\\Omega}\\hat{\\mu}_1 - \\hat{\\mu}_0$ instead of $\\hat{\\mu}_1 - \\hat{\\mu}_0$ directly, since the latter provides less information. Second, we apply the hard-thresholding method instead of the clipping method, which also provides the estimation of quantity that helps when we calculate the term $X'\\Omega X$. \n\n\\subsection{A Real Data Example}\\label{sec:quickdata}\nWe use a quick example to demonstrate how this works on the real data. We consider the rats dataset with summaries given in Table \\ref{table:data}. The original rats dataset was collected in a study of gene expressions of live rats in response to different drugs and a toxicant; we use the cleaned version by \\cite{liver}. This dataset consists of $181$ samples measured on the same set of $8491$ genes, with $61$ samples labeled by \\cite{liver} as toxicants and the other $120$ as other drugs.  \n\n\\begin{table}[ht]\n\\caption{A gene-expression microarray rats dataset.}\n\\begin{tabular}{|l|l||c|c|}\n\\hline\nData Name & Source  &    $n$ ($\\#$ of subjects)  & $p$ ($\\#$ of genes)   \\\\\n\\hline\nRats & Yousefi et al. (2010)    &  181 &  8491  \\\\\n\\hline\n\\end{tabular}\n\\label{table:data}\n\\end{table}\n\n\nThis dataset has been carefully studied in \\cite{PCS}, with the performance of the two-class classification compared among a sequence of popular classifiers, including SVM, Random Forest, and HCT-PCS. The HCT-PCS, which achieves optimal classification when it adapts  LDA \\citep{DJ08, FJY} in the rare and weak signal setting, was shown to have very promising classification results with this data. \n\nThat said, in HCT-PCS, all samples of the two classes are assumed to share the same precision matrix, leaving room for improvement. We now focus on how to use the difference in precision matrix between the two classes to improve the classification of this data and compare the results here with those from the LDA setting. Here, we leave out all the implementation details, which will be introduced in Section \\ref{sec:rats}, and only highlight our findings for this rats data:\n\\begin{itemize}\n\\item QDA further outperforms LDA in terms of both best error rate and average errors, and produces better results than those other methods in \\cite{PCS}, including SVM and Random Forest, suggesting that QDA gives a better separation by taking into account the second-order difference between the two classes.\n\n\\end{itemize}\n\nAs illustrated in Figure \\ref{figure:errorrats-k30-intro} (below, left), we can see that the performances (in terms of test error) of LDA are all above those of QDA at every data splitting of 15 splittings for the rats data, given that their precision matrices were estimated by PCS separately. In fact, this finding holds for both the best error rate and average error of the two classifiers. Figure \\ref{figure:errorrats-k30-intro} (below, right) demonstrates the surface of the test error between LDA and QDA, by varying the estimated covariance matrices. This Zoom-in plot shows the error rates of the two classifiers in great detail, and supports our finding that the QDA does bring necessary improvement over LDA when the precision matrices are appropriately estimated. \n\\begin{figure}[htb!]\n    \\centering\n    \\subfigure[]{\\includegraphics[scale=0.35]{rats2.pdf}} \\quad\n\t   \\subfigure[]{\\includegraphics[scale=0.4]{plot1_LQDA2_k30_new}}\n\t   \\caption{Comparison of testing errors for the rats data: (a)  error rates (y-axis) of LDA and QDA for 15 data splittings at a certain sparsity of the precision matrix; (b) Zoom-in errors for the rats data for varying choices of parameters in estimating the precision matrices for one splitting of 15 splittings.}\n    \\label{figure:errorrats-k30-intro}\n\\end{figure}\n\n\n\\subsection{Main results} This paper considered several scenarios, from the simplest case, in which both the mean vector and precision matrix are known, to the most general case, in which both are unknown. The results for the unknown precision matrix (Theorem \\ref{thm:unknownomega}) make use of current advances in the area of precision-matrix estimation. To avoid discussing these advances (which are beyond the scope of this paper) and focus on the main results of our research, we now examine the case in which the mean vector is unknown while the precision matrix is known (i.e., Theorem \\ref{thm:unknownmu}). \n\nWhen $\\gamma <1\/2$, the signal in variances is so large that even for a uninformative mean vector the QDA classifier can achieve successful classification (Theorem \\ref{thm:equal}). Thus we consider the non-trivial case that $\\gamma\\geq 1\/2$ (i.e., (\\ref{eqn:gamma})). On the other hand, since the mean vector is unknown, its estimation accuracy depends on the sample size $n$. Let \n\\begin{equation}\\label{kappa}\n\\kappa=\\max\\{\\kappa_1, \\kappa_2\\},\n\\end{equation}\nwhere \n\\begin{equation}\\label{kappa12}\n\\kappa_1=2-2\\alpha-\\beta \\ \\ {\\rm and } \\ \\  \\kappa_2=1-2\\theta-\\zeta.\n\\end{equation}\nOur results show that $\\kappa$ is a key quantity in the phase transition whether the mean vector is known or unknown. Here, $\\kappa_1$ and $\\kappa_2$ are the synchronized indexes of signal weakness and sparsity in the mean difference and the covariance matrix difference, respectively. In detail, $p^{\\kappa_1}\\sim p^2\\eta^2\\nu$, which is the order of the squared $L_2$-norm of the eigenvalues of $\\eta W$, while $p^{\\kappa_2}\\sim p\\tau^2\\epsilon$, which is the order of the squared $L_2$-norm of the mean vector $\\mu$. Under model (\\ref{M12}) with balanced sample ($q=1\/2$) and the parameterization (\\ref{Param1}),  (\\ref{Omega1})--(\\ref{Param3}), and (\\ref{eqn:gamma}), we examine two cases:\n\\begin{itemize}\n\\item[(i)] When $\\theta < \\delta\/2$, the signals in the mean difference of the two populations are so strong that the QDA classifier achieves the same classification results for the unknown mean vector as for the known mean vector (Theorem \\ref{thm:idealunequal}). More specifically, the QDA misclassification rate converges to 0 when $\\kappa > 0$, converges to positive values (between 0 and $1\/2$) when $\\kappa<0$, and converges to either 0 or positive values on the boundary $\\kappa=0$ depending on whether a $\\ln p$ term is introduced in the signal strength. Thus the boundary $\\kappa=0$ partitions the phase space into successful and unsuccessful classification regions of QDA classifier, which are further proved the respective possibility and impossibility regions. In another word, the QDA classifier achieves the largest successful classification region. \n\n\\item[(ii)] When $\\theta \\geq \\delta\/2$, the signals in the mean difference of the two populations are weak, and so the unknown mean vector reduces the QDA classification successful region for the case of the known mean vector (Theorem \\ref{thm:idealunequal}). More specifically, the QDA misclassification rate converges to 0 when $\\kappa > (1-\\delta)\/2$, converges to $1\/2$ when $\\kappa<(1-\\delta)\/2$, and converges to either to 0 or positive values on the boundary $\\kappa=(1-\\delta)\/2$ depending on whether a $\\ln p$ term is introduced in the signal strength.\n\n\\end{itemize}\nIn summary, when the sample size is large enough relative to signal strength in the mean difference ($\\theta < \\delta\/2$), the QDA classification result when the mean vector is unknown is the same as when it is known (`oracle' property) and QDA achieves the possibility region; otherwise, the QDA successful classification region will be reduced roughly from $\\kappa>0$ to $\\kappa>(1-\\delta)\/2$.  \n\nTo visualize the relationships among these parameters and the possibility\/impossibility and QDA classification successful\/unsuccessful regions, some of the results in Theorem  \\ref{thm:unknownmu} are given in Figures \\ref{figure: phase} and \\ref{figure: parameter}.  Figure \\ref{figure: phase} depicts those regions in terms of $\\kappa_1$ and $\\kappa_2$ values, with subfigure (a) for $\\theta<\\delta\/2$ and (b) for $\\theta\\geq \\delta\/2$. Note that $\\kappa_2>1-\\delta-\\zeta>-\\delta$ when $\\theta<\\delta\/2$, $\\kappa_2\\leq 1-\\delta-\\zeta<1-\\delta$ when $\\theta\\geq \\delta\/2$, and $\\kappa_1<1$ by (\\ref{con1}). In Figure \\ref{figure: phase}, we observe that, when the two cases $\\theta<\\delta\/2$ and $\\theta\\geq \\delta\/2$ are compared, although the QDA successful region is reduced from $\\kappa>0$ to $\\kappa>(1-\\delta)\/2$, the region is not simply shrunk in size. Instead, some of the successful region is actually removed while another area is added to it, when we switch from the case of $\\theta<\\delta\/2$ to the case of $\\theta\\geq \\delta\/2$. The added area in the case of $\\theta\\geq \\delta\/2$ is the green area over $\\kappa_2\\in [-2, -\\delta]$. This added area is simply due to the fact that $\\theta\\geq \\delta\/2$ imposes only an upper bound $1-\\delta$ on $\\kappa_2$ and lower bound is still $-2$, while $\\theta<\\delta\/2$ imposes a lower bound $-\\delta$ on $\\kappa_2$. \n\n\\begin{figure}[htb!]\n    \\centering\n    \\subfigure[Strong signal region ($\\theta<\\delta\/2$)]{\\includegraphics[scale=0.35]{DeltaLarge.pdf}} \\hspace{0.2cm}\n    \\subfigure[Weak signal region ($\\theta\\geq \\delta\/2$)]{\\includegraphics[scale=0.35]{DeltaSmall.pdf}} \n      \\caption{The possibility\/impossibility regions and QDA successful\/unsuccessful classification regions derived in Theorem \\ref{thm:unknownmu} and defined in terms of $\\kappa_1=2-2\\alpha-\\beta$ and $\\kappa_2=1-2\\theta-\\zeta$ for fixed $\\delta$ and for the two cases: (a) strong signal region ($\\theta<\\delta\/2$) and (b) weak signal region ($\\theta\\geq \\delta\/2$).}\n    \\label{figure: phase}\n\\end{figure}\n\nTo provide a sense of the relationship between the two parameters $\\alpha$ and $\\beta$ from the covariance matrix and, separately, the relationship between the two parameters $\\theta$ and $\\zeta$ from the mean vector, for fixed $\\delta$ Figure \\ref{figure: parameter} displays the QDA successful and unsuccessful classification regions of one set of parameters when the other set of parameters is fixed. Subfigures (a) and (b) are for fixed $\\theta$ and $\\zeta$ (and thus $\\kappa_2$) values, while (c) and (d) are for fixed $\\alpha$ and $\\beta$ (and thus $\\kappa_1$) values. Note that we consider only the cases in which $\\kappa_2\\leq 0$ in (a) and $\\kappa_2\\leq (1-\\delta)\/2$ in (b), since otherwise the QDA successful region of $\\alpha$ and $\\beta$ will be the collection of all possible values. Similarly, we do not consider the case $\\kappa_1>(1-\\delta)\/2$, which would make the QDA successful region of $\\theta$ and $\\zeta$ be the whole square. Note that (a) and (b) are under the constraint (\\ref{con1}). From (a) and (b) we can see that, when $\\theta$ increases from less than $\\delta\/2$ to greater than $\\delta\/2$, the QDA successful region of $\\alpha$ and $\\beta$ decreases. From (c) and (d) we observe that, when $\\kappa_1$ decreases (i.e., $2\\alpha+\\beta$ increases) from a positive value to a negative value, the QDA successful region of $\\theta$ and $\\zeta$ decreases.\n\n\\begin{figure}[htb!]\n    \\centering\n    \\subfigure[$\\theta<\\delta\/2$, $\\kappa_2\\leq 0$]{\\includegraphics[scale=0.4]{CovDeltaLarge.pdf}}\n               \\hspace{0.2cm}\n               \\subfigure[$\\theta\\geq \\delta\/2$, $\\kappa_2\\leq (1-\\delta)\/2$]{\\includegraphics[scale=0.4]{CovDeltaSmall.pdf}} \n    \\subfigure[$0<\\kappa_1\\leq (1-\\delta)\/2$]{\\includegraphics[scale=0.4]{MeanKappa1Large.pdf}} \n                \\hspace{0.7cm}\n                \\subfigure[$\\kappa_1\\leq 0$]{\\includegraphics[scale=0.4]{MeanKappa1Small.pdf}} \n      \\caption{The possibility\/impossibility regions and QDA successful\/unsuccessful classification regions derived in Theorem \\ref{thm:unknownmu} when $\\delta$ and part of the rest parameters are fixed: (a) $\\delta$, $\\theta$ and $\\zeta$ are fixed, $\\theta<\\delta\/2$ and $1-2\\theta-\\zeta\\leq 0$; (b) $\\delta$, $\\theta$ and $\\zeta$ are fixed, $\\theta\\geq \\delta\/2$ and $1-2\\theta-\\zeta \\leq (1-\\delta)\/2$; (c) $\\delta$, $\\alpha$ and $\\beta$ are fixed, and $0<2-2\\alpha-\\beta \\leq (1-\\delta)\/2$; (d) $\\delta$, $\\alpha$ and $\\beta$ are fixed, and $2-2\\alpha-\\beta\\leq 0$.}\n    \\label{figure: parameter}\n\\end{figure}\n\nThe methods employed and results obtained in this work are unique compared with other literatures on QDA methods for high dimensional data with sparse signals. In \\cite{li2015sparse}, sparsity assumptions are made on the covariance matrices $\\Sigma_0$, $\\Sigma_1$ and $\\Sigma_0 - \\Sigma_1$, instead of on the precision matrices in our work. In \\cite{wu2019quadratic} and \\cite{fan2015innovated}, sparsity assumptions are made on the precision matrices. However, \\cite{wu2019quadratic} requires a stronger signal that $\\max\\{\\kappa_1, \\kappa_2\\} \\geq 1$ while we only need $\\max\\{\\kappa_1, \\kappa_2\\} > 0$. In \\cite{fan2015innovated}, sparsity condition is expressed as that the number of rows in $\\Omega_i$ that have non-zero off-diagonal entries are at the order of $o(\\min(n, p))$. In our work, we allow each entry to be independently symmetrically distributed, which is more general.\n\n\n\n\n\\subsection{Content and Notations}\nThe main results for the phase diagram of the QDA method and the one with feature selection are discussed in Section \\ref{sec:main}. In Section \\ref{sec:proof}, we show the proofs of the main theorems. Next, numerical results of the proposed methods and algorithms on real data are presented in Section \\ref{sec:rats}. In Section \\ref{sec:dis}, some concluding remarks and potential directions of future work are discussed. The details of the proofs are provided in the appendices. \n\nHere we list the notations used throughout the paper. Let the eigenvalues of $W$ be denoted by $\\lambda_1 \\geq \\lambda_2 \\geq \\cdots \\geq \\lambda_p$. For a matrix $M$, we use $\\|M\\|$ and $\\|M\\|_F$ to denote its spectral normal and Frobenius norm, respectively, and $Tr(M)$ to denote its trace which equals the summation of the eigenvalues of $M$. We use $diag(c_1,\\dots, c_p)$ to denote a diagonal matrix with diagonal elements $c_1,\\dots, c_p$, and use $I(A)$ to denote an indicator function over event $A$. For two vectors or matrices $a$ and $b$ of same dimension, $a\\circ b$ denotes the Hadamard (entrywise) product.   \n\n\n\\section{Phase transition for QDA}\\label{sec:main}\nThe theoretical analysis consists of three parts. First, in Section \\ref{sec:ideal}, we discover the possibility and impossibility regions in the ideal case where all the parameters $\\mu$ and $\\Omega$ are known, and show that QDA achieves the largest successful classification region. Second, in Section \\ref{sec:mean}, we further characterize the QDA successful and unsuccessful classification regions when $\\mu$ have to be estimated but $\\Omega$ are known.  Last, in Section \\ref{sec:unknownpre}, we analyze the QDA successful region when both $\\mu$ and $\\Omega$ are unknown. \n\n\n\\subsection{Ideal case}\\label{sec:ideal}\nIn the ideal case, we assume that $\\mu$ and $\\Omega$ are known. \nWe begin with the position in which $\\mu = 0$ to examine the effect of the quadratic terms on classification. In the general case, it means that the mean vectors from both classes are the same. Hence, the rare and weak signal model (\\ref{M12}) is reduced to the mixture model \n\\begin{equation}\\label{M1}\nX_i|(W,Y_i) \\ \\sim \\ (1-Y_i) N(0,I) + Y_i N \\left(0,[cI+\\eta W]^{-1}\\right).\n\\end{equation}\n\nFor any classifier $T$, we denote the estimated class label of a fresh data $X$ as $\\hat Y^T(X)$. The misclassification rate of $T$ is defined as \n\\begin{equation}\\label{eqn:error}\nMR(T) = \\frac{1}{2}P_{\\epsilon, \\tau, \\eta, \\nu, \\xi}(\\hat Y^T(X) = 0|Y = 1) + \\frac{1}{2}P_{\\epsilon, \\tau, \\eta, \\nu, \\xi}(\\hat Y^T(X)= 1|Y = 0).\n\\end{equation}\nUnder the current model, we identify the regions where the QDA method succeeds and fails. The results are given in the following theorem. \n\nFor notation simplicity, we use $C_p$ to denote any constant term satisfying $\\underset {p\\to\\infty}{\\lim\\inf} \\; C_p>0$ and use $L_p$ to denote any $\\ln p$ term such that $\\underset {p\\to\\infty}{\\lim\\inf} \\; \\frac{L_p}{(\\ln p)^r}>0$ for some constant $r>0$.\n\n\n\n\\begin{theorem}\\label{thm:equal}\nConsider model (\\ref{M1}) with the parameterizations (\\ref{Omega1})--(\\ref{con1}).\n\\begin{itemize}\n\\item[(i)] If one of the following conditions is satisfied,\n\\begin{itemize}\n\\item [(1)] $\\gamma < 1\/2$, $f_p=C_p$;\n\\item [(2)] $\\gamma \\geq 1\/2$, $\\beta<2-2\\alpha$, $f_p=C_p$;\n\\item [(3)] $\\gamma \\geq 1\/2$, $\\beta=2-2\\alpha$, $f_p=L_p$;\n\\end{itemize}\nthe QDA classification rule (\\ref{eqn:qda}) under model (\\ref{M1}) has a misclassification rate (MR) that converges to 0 as $p\\to\\infty$.\n\n\\item[(ii)] If $\\gamma> 1\/2$ and $\\beta>2-2\\alpha$, then for any classifier $L$, the misclassification rate (\\ref{eqn:error}) of $L$ converges to $1\/2$ when $p\\to\\infty$.\n\\end{itemize}\n\\end{theorem}\n\n\nTheorem \\ref{thm:equal} depicts an exact phase diagram of the QDA method when $f_p$ is a $\\ln p$ term. When $\\gamma <1\/2$ or $\\beta \\leq 2 - 2\\alpha$, the QDA method achieves a misclassification rate of 0 asymptotically. Otherwise ($\\gamma>1\/2$ and $\\beta>2-2\\alpha$), all classifiers, including the QDA method, will fail. Note that, when $\\gamma < 1\/2$, the deviation in diagonal elements of the precision matrices is large enough for successful classification. \nHence, the interesting case is when\n\\begin{equation}\\label{eqn:gamma}\n\\gamma > 1\/2,\n\\end{equation}\nand the following analysis is predicated under this assumption. On the boundary $\\gamma=1\/2$, the analysis of the possibility and impossibility regions is much more complicated and one needs to zoom in and may introduce more parameters. Given that we already have six parameters, we would put aside this special case and only consider the case of $\\gamma>1\/2$.\n\n\n\nTheorem \\ref{thm:equal} tells us that under (\\ref{eqn:gamma}), the QDA classifier succeeds in the region $\\beta\\leq 2-2\\alpha$ while all classifiers fail in the region $\\beta>2-2\\alpha$. This indicates that the possibility and impossibility regions are $\\beta\\leq 2-2\\alpha$ and $\\beta> 2-2\\alpha$ respectively and QDA succeeds in the whole possibility region and thus is optimal in this sense. \n\n\nIn the proof (see Appendix A in Supplementary Material), it may be observed that the result in Theorem \\ref{thm:equal} depends on $\\xi^2 = (c - 1)^2$ instead of $\\xi$ itself. Thus, if we set\n\\[\n\\Omega \\ = \\  I +diag(\\pm \\xi)+\\eta W, \n\\]\nTheorem \\ref{thm:equal} still holds. This makes more sense since now we allow either a $\\xi$ or a $-\\xi$ deviation from $1$ for any of the $p$ diagonal elements. Even more generally, $\\xi$ can be replaced by a random variable with support $[-\\xi,\\xi]$. For example, $\\Omega =I+diag(U_1,\\dots,U_p)+\\eta W$ with $U_i$ being i.i.d. uniform random variables over $[-\\xi,\\xi]$.\n\nWe next consider the case where the means are unequal but still known, i.e., $\\mu \\neq 0$. In such a case, the mixture model is back to (\\ref{M12}). The regions of possibility and impossibility under (\\ref{eqn:gamma}) and the optimality of QDA are characterized in the following theorem.\n\n\n\\begin{theorem}\\label{thm:idealunequal}\nConsider model (\\ref{M12}) with the parameterizations (\\ref{Param1}), (\\ref{Omega1})--(\\ref{con1}), and (\\ref{eqn:gamma}). \n\\begin{itemize}\n\\item[(i)] If one of the following conditions is satisfied, \n\\begin{itemize}\n\\item [(1)]  $\\beta<2-2\\alpha$, $f_p=C_p$;\n\\item [(2)]  $\\beta=2-2\\alpha$, $f_p=L_p$; \n\\item [(3)] $0<\\zeta<1-2\\theta$, $g_p=C_p$;\n\\item [(4)] $\\zeta=1-2\\theta$, $g_p=L_p$;\n\\end{itemize}\nthe QDA classification rule (\\ref{eqn:qda}) has a misclassification rate that converges to 0 as $p\\to\\infty$. \n\\item[(ii)] If $\\beta>2-2\\alpha$ and  $\\zeta>1-2\\theta$, then, for any classifier $L$, the misclassification rate (\\ref{eqn:error}) of $L$ converges to $1\/2$ when $p\\to\\infty$. \n\\end{itemize}\n\\end{theorem}\n\n\n{\\bf Remark 1}. Theorem \\ref{thm:idealunequal}  demonstrates that under the assumptions stated therein,  the possibility and impossibility regions are $\\{\\beta\\leq 2-2\\alpha\\}\\cup \\{\\zeta\\leq 1-2\\theta\\}$ and $\\{\\beta>2-2\\alpha,\\zeta>1-2\\theta\\}$ respectively. More importantly, Theorem \\ref{thm:idealunequal} shows that the QDA classifier achieves the possibility region and thus is optimal this sense. \n \n\n{\\bf Remark 2}. Note that conditions (3) and (4) in Theorem \\ref{thm:idealunequal} on the parameters modelling the difference in mean vectors are weaker than the phase diagram of the LDA in \\cite{FJY}. This makes sense, as here we assume $\\mu$ is known when applying the QDA classification rule (\\ref{eqn:qda}), whereas \\cite{FJY} considered trained LDA classifiers with $\\mu$ unknown. \n\n{\\bf Remark 3}. From the results in Theorem \\ref{thm:idealunequal}, we observe that the possibility region for the mean part, $0<\\zeta<1-2\\theta$, and that for the precision matrix part, $1-2\\alpha<\\beta<2-2\\alpha$, are independent. This again could be explained by the fact that the classification rule (\\ref{eqn:qda}) assumes known $\\mu$ and $\\Omega$. When we consider data-trained classifiers, the two possibility regions will interact and there will be a phase diagram over the two regions showing a balance between the two sets of sparsity\/weakness indices. \n\n\n{\\bf Remark 4}. In model  (\\ref{M12}), we can also introduce the sparsity and weakness in the diagonals of the precision matrix $\\Omega$. However, the model thus raised will have six sparsity and weakness indices in total: 2 for means, 2 for diagonals, and 2 for off-diagonals of the precision matrix. We can readily obtain the possibility region for this model, but it will be difficult to visualize and is thus omitted here.  \n\n\n\n\n\\subsection{Unknown Mean Vector}\\label{sec:mean}\n\nIn this section, we discuss the case in which $\\mu$ is unknown but $\\Omega$ is known. Again, without loss of generality, we assume $\\Omega_0 = I$, and consider the model (\\ref{M12}). \nFurthermore, since the estimation accuracy is closely related to the sample size $n$, we employ the relationship $n = p^{\\delta}$ in (\\ref{Param3}).\nHence, the sample size is always smaller than $p$ by an order. \n\nWhen $\\Omega$ is known but $\\mu$ is unknown, we modify Algorithm 1 to adopt the current setting, as follows:\n\\begin{itemize}\n\\item[] {\\bf Algorithm 2}.\n\\item[0.] Calculate the average vector of the two classes as $\\hat{\\mu}_0 = \\frac{1}{n_0} \\sum_{i: Y_i = 0} X_i$ and $\\hat{\\mu}_1 = \\frac{1}{n_1} \\sum_{i: Y_i = 1} X_i$, where $n_1 = \\sum_{i =1}^n Y_i$ and $n_0 = n - n_1$. \n\\item[1.] Let $d_0=\\hat{\\mu}_0$, $d_1= \\Omega \\hat{\\mu}_1$ and $d=d_1-d_0=(d(1),\\dots, d(p))^\\top$.\n\\item[2.] Define $t = \\frac{2\\ln p}{\\sqrt{n}} \\cdot 1\\{\\max_{1\\leq j \\leq p} |d_j| > 2\\ln p\/\\sqrt{n}\\}$ and let $d^{(t)}$ denote the indicator vector of feature selection, i.e. for $j=1,\\dots,p$,\n\\begin{equation}\nd^{(t)}(j)= \\left\\{\\begin{array}{ll}\n1, & |d(j)| > t, \\\\\n0, & |d(j)| < t. \n\\end{array}\n\\right.\n\\end{equation*}\nLet $\\hat{\\mu}^{(t)}$ and $\\hat\\mu_d^{(t)}$ be the hard-thresholded vectors $\\hat\\mu_0$ and $d$ respectively with feature selection, i.e.  \n\\begin{equation*}\n\\hat{\\mu}^{(t)}=\\hat\\mu_0\\circ d^{(t)}, \\ \\ \\ \\hat\\mu_d^{(t)}=d\\circ d^{(t)}.\n\\end{equation*}\n\\item[3.] Define $C = (\\hat{\\mu}^{(t)})^\\top (I - \\Omega) \\hat{\\mu}^{(t)} +\\ln |\\Omega|$.\n\\item[4.] (Prediction step) For any fresh data vector $X$, calculate the QDA score\n$$ \nQ=X^\\top(I -\\Omega) X + 2(\\hat\\mu_d^{(t)})^\\top X + C\n$$\nand estimate the label of $X$ as $\\hat{Y}$ with $\\hat{Y} = I\\{Q > 0\\}$. \n\\end{itemize}\n\n\n\\begin{theorem}\\label{thm:unknownmu}\nUnder model (\\ref{M12}) and the parameterization (\\ref{Param1}),  (\\ref{Omega1})--(\\ref{Param3}), and (\\ref{eqn:gamma}), \n\\begin{itemize}\n\\item [(i)] When $\\theta \\geq \\delta\/2$, i.e., the signals are weak, \n\\begin{itemize}\n\\item[(1)] If $\\max\\{2-2\\alpha-\\beta, 1-2\\theta-\\zeta\\} > (1-\\delta)\/2$, the QDA classification rule (\\ref{eqn:qda}) with feature selection in Algorithm 2 has a misclassification rate that converges to 0 as $p\\to\\infty$. \n\n\\item[(2)] If $\\max\\{2-2\\alpha-\\beta, 1-2\\theta-\\zeta\\}< (1-\\delta)\/2$, then $MR(QDA) \\geq \\Phi(-1\/2)\/16$ when $p \\rightarrow \\infty$.\n\n\\item[(3)] If $\\max\\{2-2\\alpha-\\beta, 1-2\\theta-\\zeta\\}=(1-\\delta)\/2$, the misclassification rate depends on $f_p$ or $g_p$ in the following ways:\n\\begin{itemize}\n\\item [(A)] If $2 - 2\\alpha - \\beta = (1 - \\delta)\/2$ and $f_p=L_p$, then $MR(QDA) \\rightarrow 0$;\n\\item [(B)] If $1-2\\theta-\\zeta = (1 - \\delta)\/2$ and $g_p=L_p$, then $MR(QDA) \\rightarrow 0$;\n\\item [(C)] If $2 - 2\\alpha - \\beta = (1 - \\delta)\/2$ and $f_p = C_p$, then $MR(QDA) \\geq c$ for a constant $c > 0$; \n\\item [(D)] If $1-2\\theta-\\zeta = (1 - \\delta)\/2$ and $g_p=C_p$, then $MR(QDA) \\geq c$ for a constant $c > 0$.\n\\end{itemize} \n\\end{itemize}\n\n\\item[(ii)] When $\\theta < \\delta\/2$, i.e., the signals are strong, the results in Theorem \\ref{thm:idealunequal} hold.\n\n\\end{itemize}\n\\end{theorem}\n\n\nWhen the estimated mean is introduced, the parameterization of the sample size $n$ also has an effect on the QDA successful classification region. Comparing the results in Theorem \\ref{thm:idealunequal} and Theorem \\ref{thm:unknownmu}, there are two differences. First, the signal strength parameter $\\theta$ becomes an important factor in characterizing the region. When $\\theta > \\delta\/2 $, the signals are weak and impossible to identify, so $\\beta$ or $\\zeta$ must be small enough (so the signals are dense) for the classification to be successful. On the other hand, when $\\theta < \\delta\/2$, the signals are so strong that can afford to be relatively sparse.\nSecond, the QDA successful classification region is reduced by $\\frac{1 - \\delta}{2}$ for both parameters $\\beta$ and $\\zeta$ when signals are weak, unlike when the signals are strong. The reduction of the successful classification region is a result of the estimation error for the mean vectors. \n\n\n\n\n\n\n\\subsection{Unknown mean and unknown covariance}\\label{sec:unknownpre}\nIn this section, we discuss the case in which $\\mu_k$ are unknown, $\\Sigma_0 = I$, and $\\Sigma_1$ is also unknown. The estimation of the precision matrix in the high-dimensional case is still restricted to the sparse case for now. Hence, here we still assume $\\Sigma = I$ and $\\Omega - I = (c - 1)I + \\eta W$, so that both of them are sparse. Under model (\\ref{M12}), we modify Algorithm 1 to adopt the current setting, as follows: \n\\begin{itemize}\n\\item[] {\\bf Algorithm 3}.\n\\item[0.] Calculate the estimated mean vector for the two classes as $\\hat{\\mu}_0 = \\frac{1}{n_0} \\sum_{i=1}^{n_0} X_i$ and $\\hat{\\mu}_1 = \\frac{1}{n_1} \\sum_{i=n_0+1}^{n} X_i$, where $n_1 = \\sum_{i =1}^n Y_i$ and $n_0 = n - n_1$. Estimate the precision matrix $\\Omega$ with some method and let $\\hat{\\Omega}$ denote the estimation.\n\\item[1.] Let $d_0=\\hat{\\mu}_0$, $d_1= \\hat{\\Omega}\\hat{\\mu}_1$ and $d=d_1-d_0=(d(1),\\dots,d(p))^\\top$.\n\\item[2.] Define $t = \\frac{2\\ln p}{\\sqrt{n}} \\cdot 1\\{\\max_{1\\leq j \\leq p} |d(j)| > 2\\ln p\/\\sqrt{n}\\}$ and let $d^{(t)}$ denote the indicator vector of feature selection, i.e. for $j=1,\\dots,p$,\n\\begin{equation}\nd^{(t)}(j)= \\left\\{\\begin{array}{ll}\n1, & |d(j)| > t, \\\\\n0, & |d(j)| < t. \n\\end{array}\n\\right.\n\\end{equation*}\nLet $\\hat{\\mu}^{(t)}$ and $\\hat\\mu_d^{(t)}$ be the hard-thresholded vectors $\\hat\\mu_0$ and $d$ respectively with feature selection, i.e.  \n\\begin{equation*}\n\\hat{\\mu}^{(t)}=\\hat\\mu_0\\circ d^{(t)}, \\ \\ \\ \\hat\\mu_d^{(t)}=d\\circ d^{(t)}.\n\\end{equation*}\n\\item[3.] Define $C = (\\hat{\\mu}^{(t)})^\\top (I - \\hat{\\Omega})\\hat{\\mu}^{(t)} +\\frac{1}{n_1}Tr(\\hat{\\Omega} - I) + \\ln |\\hat{\\Omega}|$.\n\\item[4.] (Prediction step) For any fresh data vector $X$, calculate the QDA score\n$$ \nQ=X^\\top(I -\\hat{\\Omega}) X + 2(\\hat{\\mu}_d^{(t)})^\\top X + C\n$$\nand estimate the label of $X$ as $\\hat Y$ with $\\hat Y=I\\{Q>0\\}$. \n\\end{itemize}\n\nIn Step 0, the estimation of the sparse precision matrix can be performed via any suitable approach. This has been discussed in numerous publications in the literature, such as \\cite{CLIME, FJY, glasso, PCS}. Here, our goal is to develop the QDA approach with feature-selection step, instead of designing a new precision-matrix-estimation approach. In the following theorem, we consider a general precision-matrix-estimation approach, and let $\\Delta_{\\hat{\\Omega}}$ denote the estimation error. We give the region that $MR(QDA) \\rightarrow 0$ based on $\\Delta_{\\hat{\\Omega}}$. \n\n\n\n\\begin{theorem}\\label{thm:unknownomega}\nConsider model (\\ref{M12}) and the parameterization (\\ref{Param1}),  (\\ref{Omega1})--(\\ref{con1}), (\\ref{eqn:gamma}), and (\\ref{Param3}). Assume $1 < \\beta < 2$. For the employed precision-matrix-estimation approach, let $\\Delta_{\\hat\\Omega} = \\|\\Omega - \\hat{\\Omega}\\|$ be the spectral norm of the error. \nSuppose $\\Delta_{\\hat\\Omega} \\rightarrow 0$ when $p \\rightarrow \\infty$, and it satisfies that \n\\[\n\\Delta_{\\hat{\\Omega}} (p\\eta  + p \\tau^2 \\epsilon + \\sqrt{p}\\ln p)+ p \\Delta_{\\hat{\\Omega}}^2 +p\/n \\ll p\\xi^2 + \\eta^2 p^2 \\nu + \\tau^2 p \\epsilon. \n\\]\nThen,  the QDA classification rule (\\ref{eqn:qda}) with feature selection in Algorithm 3 has a misclassification rate that converges to 0 as $p\\to\\infty$. \n\\end{theorem}\n\nThe proof of Theorem \\ref{thm:unknownomega} can be found in Appendix \\ref{sec:proof4}. Here, we develop the general rule for the QDA classification with feature selection method. For the weak signal case that $\\theta > \\delta\/2$, the condition can be relaxed by replacing $p\/n$ to be $p\\xi\/n$. However, the term $p\/n$ is not the dominating term when we apply the PCS method and the CLIME method in Algorithm 3. Therefore, we didn't differentiate the two cases. \n\nCompared to Theorems \\ref{thm:idealunequal} and \\ref{thm:unknownmu}, a big difference here is that the condition is an inequality that containing both the precision matrix parameters and the mean vector parameters. In the following two corollaries, we can see that the condition $\\zeta < \\alpha + \\delta\/2 - 2\\theta$ indicates an intervention between the precision matrix weakness parameter $\\alpha$, the mean vector sparsity and weakness parameters $\\zeta$ and $\\theta$, and the sample size parameter $\\theta$. Under this situation, the dominating error term comes from $X(\\hat{\\Omega} - I)X$, which contains both $\\Omega$ and $\\mu$ (in $X$). \n \nWe apply the PCS approach in \\cite{PCS} and the CLIME approach in \\cite{CLIME} to be the precision-matrix-estimation approach. The results can be found in the following corollaries.  \n\\begin{cor}\\label{cor:PCS}\nUnder the conditions of Theorem \\ref{thm:unknownomega} and that $\\alpha < \\delta\/2$, and PCS is employed for precision-matrix estimation.  Consider the conditions that \n\\begin{itemize}\n\\item [(i)] $\\beta < 1  - \\alpha + \\delta\/2$ and $f_p=C_p$;\n\\item [(ii)]  or, $\\zeta < \\alpha + \\delta\/2 - 2\\theta$ and $g_p=C_p$.\n\\end{itemize} \nIf one of the above conditions is satisfied, then the QDA classification rule (\\ref{eqn:qda}) with feature selection in Algorithm 3 has a misclassification rate that converges to 0 as $p\\to\\infty$. \n\\end{cor}\n\n\n\\begin{cor}\\label{cor:CLIME}\nUnder the conditions of Theorem \\ref{thm:unknownomega}, and that CLIME is employed for precision-matrix estimation.  Assume $\\alpha < \\delta\/2$, and consider the conditions that \n\\begin{itemize}\n\\item [(i)] $\\beta < 1  - \\alpha + \\delta\/2$ and $f_p=C_p$;\n\\item [(ii)]  $\\zeta < \\alpha + \\delta\/2 - 2\\theta$ and $g_p=C_p$;\n\\end{itemize} \nIf one of the above conditions is satisfied, then the QDA classification rule (\\ref{eqn:qda}) with feature selection in Algorithm 3 has a misclassification rate that converges to 0 as $p\\to\\infty$. \n\\end{cor}\n\n\n\n\n\n\n\\section{Proofs}\\label{sec:proof}\nIn this section, we present the proof of Theorems \\ref{thm:idealunequal}--\\ref{thm:unknownmu}. Theorem \\ref{thm:equal} is a special case of Theorem \\ref{thm:idealunequal}, and the proof can be found in the Appendix. \n\n\nWe first write out the misclassification rate. Given $\\mu$ and $W$ (and thus $\\Omega$), the two types of misclassification rates are defined as \n\\begin{equation}\\label{eqn:p}\np_{0,\\mu,W} = P_{Y = 0}(Q>0|\\mu, W), \np_{1, \\mu, W} = P_{Y = 1}(Q < 0|\\mu, W).\n\\end{equation}\nThen, the population misclassification rate ($MR$) is \n\\begin{equation}\\label{eqn:mr}\nMR \\ = \\ \\frac{1}{2}E_{\\mu, W}[p_{0,\\mu,W}]+ \\frac{1}{2} E_{\\mu,W}[p_{1,\\mu,W}].\n\\end{equation}\nWe wish to find a boundary that divides the parameter space into two regions. In one region, $MR$ converges to 0. In the other region, $MR$ is always larger than some positive constant. \n\n\nThis section is structured as follows. In Section \\ref{sec:lemmas}, we present some mathematical results as the preparations, and the proof can be found in the Appendix. Based on these preparations, Theorem \\ref{thm:idealunequal} is then proved in Section \\ref{sec:thmideal}. In Section \\ref{sec:thmmuposs}, we proceed to discuss the proof of Theorem \\ref{thm:unknownmu}, where $\\mu$ is unknown and we have to analyze the difference term $Q(X, \\hat{\\mu}, W) - Q(X, \\mu, W)$. \n\n\n\n\n\\subsection{Preparation}\\label{sec:lemmas}\nTo prove the theorems, we need the following propositions and lemmas for $\\mu$ and $W$. The propositions can be easily proved, and so we have omitted their proofs. The proofs of the lemmas can be found in the Appendix. \n\nFor a fixed $W$, let the eigenvalue decomposition of $W$ be \n \\[\n W=U\\Lambda U^\\top, \n\\] \n where $U^\\top U=U U^\\top=I$ and $\\Lambda=diag(\\lambda_1, \\lambda_2, \\dots, \\lambda_p)$. Here,  $\\lambda_i = \\lambda_i(W)$ are the eigenvalues of $W$ such that $\\lambda_1 \\geq \\lambda_2 \\geq \\cdots \\geq \\lambda_p$. Let the spectral norm of $W$ be denoted by $\\|W\\|$. \n\nSince $\\Omega = cI + \\eta W$, $\\Omega=U(cI+\\eta\\Lambda)U^\\top$. Therefore, $\\Omega - I= \\xi I + \\eta W$, where the eigenvalues are defined as \n\\[\nm_i = \\xi+\\eta\\lambda_i, \\qquad 1 \\leq i \\leq p. \n\\]\nLet $\\tilde{\\mu}=U^\\top \\mu=(\\tilde{\\mu}_1, \\dots, \\tilde{\\mu}_p)^\\top$ and $\\tilde{X} = U^\\top X=(\\tilde{x}_1 \\dots, \\tilde{x}_p)^\\top$. Note that here we use the subscripts to denote the vector elements. \n\n\n\n\\begin{prop}\\label{prop1}\n$\\sum_{i=1}^p \\lambda_i = Tr(W) = 0$. \n\\end{prop}\n\\begin{prop}\\label{prop2}\n$\\sum_{i=1}^p \\lambda_i^2 = Tr(W^TW) \\sim 2Binomial(p(p-1)\/2, \\nu)$. \n\\end{prop}\n\n\n\\begin{lemma}\\label{lemma1}\nUnder models (\\ref{Omega1}) and (\\ref{Param}), when $p\\to\\infty$, with probability $1-o(1)$, \n\\begin{equation*}\n\\|W\\| \\leq b(p, \\beta) = \\left\\{\\begin{array}{ll}\n3\\sqrt{p\\nu} = 3p^{(1-\\beta)\/2}, & 0<\\beta < 1,\\\\\n2\\sqrt{\\frac{\\ln p}{\\ln\\ln p}}, & \\beta = 1,\\\\\n\\frac{2}{\\beta - 1}, & 1 < \\beta \\leq 2.\n\\end{array}\n\\right.\n\\end{equation*}\nLet $R_W$ be a function of $\\lambda_i$'s defined by $R_W=\\frac{\\frac{1}{p}\\sum_{i=1}^p |\\lambda_i |^3}{\\left(\\frac{1}{p}\\sum_{i=1}^p \\lambda_i^2\\right)^{3\/2}}$; \nthen, with probability $1 - o(1)$, \n\\begin{equation*}\nR_W \\leq r(p, \\beta) = \\left\\{\\begin{array}{ll}\n216, & 0<\\beta < 1,\\\\\n4\\sqrt{\\frac{\\ln p}{\\ln\\ln p}}, & \\beta=1, \\\\\n\\frac{4}{\\beta-1}p^{(\\beta-1)\/2}, & 1 < \\beta \\leq 2.\n\\end{array}\n\\right.\n\\end{equation*}\n\\end{lemma}\n\nNow, we consider $\\Omega = cI + \\eta W$. \nIn view of the result of $\\|W\\|$ and the fact that $m_i = \\xi + \\eta \\lambda_i$ under model (\\ref{Omega1}), when the condition (\\ref{con1}) holds, with probability $1 - o(1)$,  \n\\begin{equation}\\label{eqn:smallm}\n\\max_{1 \\leq i \\leq p} |m_i|  = \\|\\xi I + \\eta W\\| = \\xi + \\eta \\|W\\| \\rightarrow 0. \n\\end{equation}\nAccording to basic matrix theory, \n\\begin{equation}\\label{eqn:summ}\n\\sum_{i=1}^p m_i^2 = Tr((\\Omega - I)^{\\top}(\\Omega - I)) = p\\xi^2+\\eta^2 \\overset p{\\underset{i=1}\\sum} \\lambda_i^2, \n\\end{equation}\nwhere $\\sum_{i = 1}^p \\lambda_i^2 \\sim 2 Binomial(p(p-1)\/2, \\nu)$, according to Proposition \\ref{prop2}. \n\nFinally, we consider the signals in the mean part. For $\\mu$, we have \n\\begin{equation}\\label{eqn:normmu}\n\\mu^\\top \\mu = \\overset p{\\underset{i=1}\\sum} \\mu_i^2 \\ = \\ \\tau^2 \\mathcal B_p, \\quad \\mbox{where }\\mathcal B_p \\sim Binomial(p,\\epsilon).\n\\end{equation}\n\nAs we frequently use these terms in the following analysis, we provide a summary of the magnitude of these terms in the following lemma.  \n\\begin{lemma}\\label{lemmaeb}\nConsider model (\\ref{M12}) with the parameterizations (\\ref{Param1}), (\\ref{Omega1})--(\\ref{con1}), and (\\ref{eqn:gamma}). With probability $1 + o(1)$, we have\n\\begin{equation}\\begin{array}{lll}\n \\overset p{\\underset{i=1}\\sum}\\lambda_i^2 =  \\eta^2 p^2 \\nu(1 + o(1)), \\\\\n \\overset p{\\underset{i=1}\\sum} m_i^2 = p\\xi^2+ \\eta^2 p^2 \\nu(1 + o(1)),\\quad \\max |m_i| = o(1), \\\\\n \\overset p{\\underset{i=1}\\sum} \\mu_i^2 \\ = \\ p\\tau^2  \\epsilon(1 + o(1)). \n \\end{array}\n\\end{equation}\n\\end{lemma}\nIn addition, there is a lemma about $\\tilde{\\mu} = U^{\\top} \\mu$, which is proved in Appendix \\ref{sec:lemma2}. \n\\begin{lemma}\\label{lemma2} \nAs $p\\to\\infty$, under models (\\ref{Param1}), (\\ref{Param2}), (\\ref{Param}), and (\\ref{con1}), \n\\begin{equation*}\nR_{\\mu} := \\frac{1}{\\sqrt p} \\cdot \\frac{\\frac{1}{p}\\sum_{i=1}^p |\\tilde{\\mu}_i|^3}{\\left(\\frac{1}{p}\\sum_{i=1}^p |\\tilde{\\mu}_i|^2\\right)^{3\/2}} \\ \\overset{\\mathcal P}\\longrightarrow \\ 0.\n\\end{equation*}\n\\end{lemma}\n\nFinally, we state a lemma about $\\Phi(x_p)$, where $\\Phi(\\cdot)$ is the cumulative density function (c.d.f.) of the standard normal distribution. \n\\begin{lemma}\\label{lemmaphi}\nConsider a random variable $x_p$ and $E[\\Phi(x_p)]$:\n\\begin{itemize}\n\\item If $x_p \\rightarrow -\\infty$ in probability as $p \\rightarrow \\infty$, i.e., $P(x_p > -M) \\rightarrow 0$ for any constant $M > 0$ when $p \\rightarrow \\infty$, then $E[\\Phi(x_p)] \\rightarrow 0$ when $p \\rightarrow \\infty$.\n\\item If $x_p \\rightarrow \\infty$ in probability as $p \\rightarrow \\infty$, i.e., $P(x_p < M) \\rightarrow 0$ for any constant $M > 0$ when $p \\rightarrow \\infty$, then $E[\\Phi(x_p)] \\rightarrow 1$ when $p \\rightarrow \\infty$. \n\\end{itemize}\n\\end{lemma}\n\n\\subsection{Proof of Theorem \\ref{thm:idealunequal}}\\label{sec:thmideal}\nThere are two parts of the proof. For the region of possibility, we want to show that $MR(QDA)$ converges to 0 in this region. For the region of impossibility, we want to show any classifier $L$ has $MR(L)$ that achieves $1\/2$ if the parameters fall in this region. \n\nWe first work on the region of possibility. For this region, we want to find out $MR(QDA)$. Recall that QDA estimates the label by \n\\[\nQ = Q(X, \\mu, W) = - X^\\top(\\Omega - I) X + 2 \\mu^{\\top} (\\Omega + I) X - \\mu^\\top (\\Omega - I)\\mu  + \\ln|\\Omega|. \n\\]\nTherefore, we should analyze $Q$.\n\nAccording to (\\ref{eqn:mr}), we first find the misclassification rate when $\\mu$ and $W$ are given, and then take the expectation. It indicates only $X$ is the random variable, and the random part is \n\\begin{equation}\\label{eqn:S}\nS = - X^\\top(\\Omega - I) X + 2 \\mu^{\\top} (\\Omega + I) X. \n\\end{equation}\n$S$ can be viewed as a summation of squared normal random variables. Actually, we have the lemma about the asymptotic distribution of $S$. \n\\begin{lemma}\\label{lemma:S}\nGiven $Y = i$, $i = 0, 1$, let $\\mu_S = E[S|Y = i]$ and $\\sigma^2_S = \\mathrm{Var}(S|Y = i)$. Define $Z = \\frac{S - \\mu_S}{\\sigma_S}$, then \n\\[\n\\sup_{x}|F_{Z|Y = i}(x) - \\Phi(x)| \\leq C(\\frac{1 + R_W}{\\sqrt{p}} + R_{\\mu}), \n\\]\nwhere $F_{Z|Y = i}(x) = P(Z \\leq x|Y = i)$ and $\\Phi(\\cdot)$ is the c.d.f. of the standard normal distribution. \n\\end{lemma}\nHere, $\\mu_S$ may differ when $Y = 0$ and $Y = 1$. Since $Y$ is given here and afterwards, we use $\\mu_S$ to denote this conditional expectation without confusion. For $\\sigma^2_S$, the situation is the same. \n\nAccording to Lemma \\ref{lemma1} and Lemma \\ref{lemma2}, $C(\\frac{1 + R_W}{\\sqrt{p}} + R_{\\mu}) = o(1)$ with probability $1 - o(1)$. Therefore, we only need to consider the event that $C(\\frac{1 + R_W}{\\sqrt{p}} + R_{\\mu}) = o(1)$, which means $Z|Y = i$ converges to the standard normal distribution in distribution. \nTherefore, we have that \n\\begin{equation}\\label{p0thm2}\n\\begin{array}{lll}\np_{0, \\mu, W} & = & P_{Y = 0}(Q > 0 |\\mu, W)\\\\\n& = & P_{Y = 0}(S - \\mu^\\top (\\Omega - I)\\mu  + \\ln|\\Omega|  > 0 |\\mu, W)\\\\\n& = & P_{Y = 0}(\\sigma_S Z + \\mu_S - \\mu^\\top (\\Omega - I)\\mu  + \\ln|\\Omega|  > 0 |\\mu, W)\\\\\n& = & 1 - \\Phi(\\frac{\\mu^\\top (\\Omega - I)\\mu  - \\ln|\\Omega| - \\mu_S}{\\sigma_S}) + o(1)\\\\\n& = & \\Phi(-\\frac{\\mu^\\top (\\Omega - I)\\mu  - \\ln|\\Omega| - \\mu_S}{\\sigma_S}) + o(1).\n\\end{array}\n\\end{equation}\nSimilarly, for $p_{1,\\mu, W}$, we have that \n\\begin{equation}\\label{p1thm2}\np_{1, \\mu, W} = P_{Y = 1}(Q < 0 |\\mu, W)= \\Phi(\\frac{\\mu^\\top (\\Omega - I)\\mu  - \\ln|\\Omega| - \\mu_S}{\\sigma_S}) + o(1).\n\\end{equation}\nHence, we need to derive the performance of $p_{0,\\mu,W}$ and $p_{1, \\mu, W}$. To derive them, we should check the term that \n\\begin{equation}\\label{eqn:T}\nT = \\frac{\\mu^\\top (\\Omega - I)\\mu  - \\ln|\\Omega| - \\mu_S}{\\sigma_S}.\n\\end{equation}\n\nWe figure out $T$ for the two cases. \n\\begin{itemize}\n\\item Case 0: $Y = 0$. We first figure out $\\mu_S$ and $\\sigma_S$. According to Property 1 in Appendix \\ref{sec:gchi}, \n\\begin{equation}\\label{zmeanvar1}\n\\mu_S = Tr(I - \\Omega) - \\mu^\\top(3\\Omega + I) \\mu, \\, \\sigma^2_S = 2Tr((\\Omega - I)^2) + 16 \\mu^\\top \\Omega^2 \\mu.\n\\end{equation}\nIntroduce (\\ref{zmeanvar1}) into the formula of $T$ in (\\ref{eqn:T}), then \n\\[\nT_0 := T|Y = 0  = \\frac{4\\mu^\\top \\Omega \\mu  - \\ln|\\Omega| + Tr(\\Omega - I) }{\\sqrt{2Tr((\\Omega - I)^2) + 16 \\mu^\\top \\Omega^2 \\mu}}.\n\\]\nRecall that $\\Omega = U((1+\\xi)I + \\eta \\Lambda) U^\\top$ with eigenvalues $1 + m_i = 1 + \\xi + \\eta \\lambda_i$, and $\\tilde{\\mu} = U^\\top \\mu$. Introduce the terms into the above equation and there is \n\\begin{equation}\\label{c0form}\nT_0 = \\frac{-\\overset p{\\underset {i=1}\\sum}\\ln\\bigl[\\frac{1+m_i}{e^{m_i+4(1+m_i)\\tilde{\\mu}_i^2}}\\bigr]}{\\sqrt{2\\sum_{i=1}^p \\left[m_i^2 +8(1+m_i)^2\\tilde{\\mu}_i^2 \\right] }}.\n\\end{equation}\nFrom Lemma \\ref{lemmaeb}, $\\max_{1 \\leq i \\leq p}|m_i| = o(1)$ with high probability. Hence, \n\\begin{equation}\\label{conbound}\n\\ln\\biggl[\\frac{1+m_i}{e^{m_i+4(1+m_i)\\tilde{\\mu}_i^2}}\\biggr] = -\\frac{1}{2}m_i^2(1 + o(1)) - 4(1+m_i)\\tilde{\\mu}_i^2.\n\\end{equation}\nPlugging (\\ref{conbound}) into (\\ref{c0form}), we have \n\\begin{equation}\\label{p1constant}\nT_0 =  \\frac{1}{2\\sqrt 2} \\sqrt{\\sum\\nolimits_{i = 1}^p (m_i^2+8\\tilde{\\mu}_i^2)}(1+o(1)).\n\\end{equation}\n\n\n\n\\item Case 1: $Y = 1$. Similarly, we first derive the mean and variance of $S$, then figure out $T|Y = 1$. According to Property 2 in Appendix \\ref{sec:gchi},\n\\begin{equation}\\label{zmeanvar2}\n\\mu_S = Tr(\\Omega^{-1} - I) + \\mu^\\top(\\Omega + 3I) \\mu, \\, \\sigma^2_S = 2Tr((\\Omega^{-1} - I)^2) + 16 \\mu^\\top \\Omega^{-1} \\mu.\n\\end{equation}\nIntroduce (\\ref{zmeanvar2}) into the formula of $T$ in (\\ref{eqn:T}), then \n\\[\nT_1 := T|Y = 1  = \\frac{-4\\mu^\\top  \\mu  - \\ln|\\Omega| + Tr(I - \\Omega^{-1}) }{\\sqrt{2Tr((\\Omega^{-1} - I)^2) + 16 \\mu^\\top \\Omega^{-1} \\mu}}.\n\\]\nIntroducing in the terms $m_i$'s and $\\tilde{\\mu}_i$'s and combining the results with Lemma \\ref{lemmaeb}, similarly we have \n\\begin{equation}\\label{p2cons}\nT_{1} = \\frac{-\\overset p{\\underset {i=1}\\sum}\\ln\\left[\\frac{1+m_i}{e^{m_i\/(1+m_i)-4\\tilde{\\mu}_{i}^2}}\\right ] }{\\left[2\\overset p{\\underset{i=1}\\sum} \\left[\\frac{m_i^2}{(1+m_i)^2} + \\frac{8}{1+m_i}\\tilde{\\mu}_{i}^2 \\right] \\right]^{1\/2}} = -\\frac{1}{2\\sqrt{2}}  \\sqrt{\\sum\\nolimits_{i = 1}^p (m_i^2+8\\tilde{\\mu}_i^2)}(1+o(1)).\n\\end{equation}\n\\end{itemize}\nIntroducing (\\ref{p1constant}) and (\\ref{p2cons}) into (\\ref{p0thm2}) and (\\ref{p1thm2}) with (\\ref{eqn:T}), there is \n\\begin{equation}\\label{eqn:thm2p01}\n\\begin{array}{lll}\np_{0,\\mu,W} = \\Phi(-T_0) + o(1) & = & \\Phi(-\\frac{1}{2\\sqrt 2} \\sqrt{\\sum\\nolimits_{i = 1}^p (m_i^2+8\\tilde{\\mu}_i^2)}(1+o(1))) + o(1), \\\\\np_{1,\\mu,W} = \\Phi(T_1) + o(1) & = & \\Phi(-\\frac{1}{2\\sqrt 2} \\sqrt{\\sum\\nolimits_{i = 1}^p (m_i^2+8\\tilde{\\mu}_i^2)}(1+o(1))) + o(1).\\\\\n\\end{array}\n\\end{equation}\nTherefore, for the misclassification rate, there is \n\\begin{equation}\\label{eqn:mr2}\nMR = \\frac{1}{2} E[p_{0,\\mu,W}] + \\frac{1}{2} E[p_{1,\\mu,W} ] = E[\\Phi(-\\frac{1}{2\\sqrt 2} \\sqrt{\\sum\\nolimits_{i = 1}^p (m_i^2+8\\tilde{\\mu}_i^2)}(1+o(1)))] + o(1). \n\\end{equation}\n\nAccording to Lemma \\ref{lemmaeb}, with probability $1 - o(1)$, $\\sum_{i=1}^p m_i^2 = p\\xi^2 + \\eta^2p^2\\nu(1 + o(1))$ and $\\sum_{i=1}^p \\tilde{\\mu}_i^2 = \\sum_{i=1}^p {\\mu}_i^2 = p\\tau^2\\epsilon(1 + o(1))$. Therefore, with probability $1 - o(1)$, \n\\[\n-\\frac{1}{2\\sqrt 2} \\sqrt{\\sum\\nolimits_{i = 1}^p (m_i^2+8\\tilde{\\mu}_i^2)}(1+o(1)) =  \\bigl[-\\frac{\\sqrt{p\\xi^2+\\eta^2 p^2 \\nu +8\\tau^2p \\epsilon}}{2\\sqrt 2} \\bigr](1+o(1)).\n \\]\n\nNow we consider the term \n\\[\np\\xi^2+\\eta^2 p^2 \\nu +8\\tau^2p = p^{1 - 2\\gamma} + f_p p^{2-2\\alpha - \\beta} + g_p p^{1 - \\zeta - 2\\theta}. \n\\]\nRecall that $\\gamma > 1\/2$ so that $p^{1 - 2\\gamma} \\rightarrow 0$. Now recall the region of possibility: \n\\begin{itemize}\n\\item [(1)]  $\\beta<2-2\\alpha$, $f_p=C_p$;\n\\item [(2)]  $\\beta=2-2\\alpha$, $f_p=L_p$;\n\\item [(3)] $0<\\zeta<1-2\\theta$, $g_p=C_p$;\n\\item [(4)] $\\zeta=1-2\\theta$, $g_p=L_p$.\n\\end{itemize}\nWhen one of the above conditions hold, obviously $p\\xi^2+\\eta^2 p^2 \\nu +8\\tau^2p \\rightarrow \\infty$ when $p \\rightarrow \\infty$. According to (\\ref{eqn:mr2}) and Lemma \\ref{lemmaphi}, $MR \\rightarrow 0$. \n\n\\vspace{1em}\nNext we consider the impossibility. To show that the classification error goes to $1\/2$ for any classifier $L$ in this region, we have to introduce a new lemma. Let $f$ be the density function of $X \\sim N(-\\mu, I)$ and $g$ be the density function of $X \\sim N(\\mu, \\Omega^{-1})$. The Hellinger affinity between $f$ and $g$ is defined as $H(f, g) = \\int \\sqrt{f(x) g(x)} dx$. We have the following lemma. \n\\begin{lemma}\\label{lemma:H} \nFor any classifier $L = L(X|\\mu, \\Omega)$, \n\\[\n|MR(L) - 1\/2 | \\leq C(1 - H(f, g))^{1\/2}. \n\\]\n\\end{lemma}\n\nThis lemma is well known, and so we omit the proof. According to this lemma, to prove the impossibility, we only need to prove that $H(f, g) = 1 + o(1)$. \n\nNext, we check $H(f, g)$. According to the definition of multivariate normal distribution,\n\\begin{equation*}\n\\begin{array}{lll}\nH(f, g) & = & \\int \\frac{|\\Omega|^{1\/4}}{(\\sqrt{2\\pi})^p} \\exp\\{-\\frac{1}{4}\\bigl[(x + \\mu)^\\top (x + \\mu) + (x - \\mu)^\\top \\Omega (x - \\mu)\\bigr]\\}dx\\\\\n& = & \\frac{|\\Omega|^{1\/4}}{(\\sqrt{2\\pi})^p} \\exp\\{-\\frac{1}{4}\\mu^{\\top} (I + \\Omega - (I - \\Omega) (I + \\Omega)^{-1} (I - \\Omega))\\} \\\\\n&& \\times \\int \\exp\\{-\\frac{1}{2}\\bigl[(x - (I + \\Omega)^{-1}(I - \\Omega) \\mu)^\\top \\frac{I + \\Omega}{2} (x - (I + \\Omega)^{-1}(I - \\Omega) \\mu) \\bigr]\\} dx\\\\\n& = & \\frac{|\\Omega|^{1\/4}}{|(\\Omega + I)\/2|^{1\/2}} \\exp\\{-\\frac{1}{4}\\mu^{\\top} (I + \\Omega - (I - \\Omega) (I + \\Omega)^{-1} (I - \\Omega))\\} \\\\\n& = & \\exp\\{-\\frac{1}{4} \\bigl[ \\sum_{i = 1}^p \\frac{4(1 + m_i)}{2 + m_i} \\tilde{\\mu}_i^2 + \\sum_{i = 1}^p \\ln \\frac{(1 + m_i\/2)^2}{1 + m_i} \\bigr]\\}. \n\\end{array}\n\\end{equation*}\nTo show that $H(f, g) = 1 + o(1)$, we only need to prove $R = \\sum_{i = 1}^p \\frac{4(1 + m_i)}{2 + m_i} \\tilde{\\mu}_i^2 + \\sum_{i = 1}^p \\frac{(1 + m_i\/2)^2}{1 + m_i} = o(1)$. Now, we have\n\\begin{equation*}\n\\begin{array}{lll}\nR & = & 2\\sum_{i = 1}^p \\tilde{\\mu}_i^2(1 + o(1)) + \\sum_{i = 1}^p \\ln (1 + \\frac{m_i^2}{4(1 + m_i)})\\\\\n& =  & 2 \\|\\mu\\|^2(1+o(1)) + \\sum_{i = 1}^p \\frac{m_i^2}{4(1 + m_i)} (1+o(1))\\\\\n& = & 2 \\|\\mu\\|^2(1+o(1)) + \\frac{1}{4} \\|\\Omega - I\\|^2_F. \n\\end{array}\n\\end{equation*}\nTherefore, when $\\|\\mu\\|^2 = o(1)$ and $\\|\\Omega - I\\|^2_F = o(1)$, the misclassification error from any classifier will be close to 1\/2. \n\nAccording to Lemma \\ref{lemmaeb}, with probability $1 + o(1)$, $\\|\\mu\\|^2 = p\\tau^2\\epsilon (1 + o(1))$ and $\\|\\Omega - I\\|^2_F = \\sum_{i = 1}^p m_i^2 = p\\xi^2 + \\eta^2p^2 \\nu (1 + o(1))$. In the region of impossibility, $p\\tau^2\\epsilon + p\\xi^2 + \\eta^2p^2 \\nu \\rightarrow 0$, and so $H(f, g) = 1 + o(1)$. As a result, $MR(L) \\rightarrow 1\/2$ for any classifier $L$. \n\nCombining the results of the regions of possibility and impossibility, Theorem \\ref{thm:idealunequal} is proved. {\\unskip\\nobreak\\hfil\\penalty50\\hskip2em\\vadjust{\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{thm:unknownmu}: the weak signal region}\\label{sec:thmmuposs}\n\nIn this section, we focus on the algorithm for QDA with feature selection, when $\\mu$ is unknown. To estimate $\\mu$, we use $\\mu^* = -\\hat{\\mu}_0$ for the quadratic part and $d$ for the linear part:\n\\begin{equation}\\label{eqn:d}\nd \\ = \\ \\Omega \\hat{\\mu}_1 - \\hat{\\mu}_0 \\ \\sim \\ N\\bigl((I + \\Omega)\\mu, \\ \\frac{1}{n_0}I+\\frac{1}{n_1}\\Omega\\bigr).\n\\end{equation} \nFor a threshold $t$, we let $\\hat{d}_{j} = d_j*I(|d_j| \\geq t)$. When $\\max_{1 \\leq j \\leq p} |d_i| \\leq 2{\\ln p}\/\\sqrt{n}$, we take $t = 0$; otherwise we take $t = 2\\sqrt{\\ln p}\/\\sqrt{n}$. \nTherefore, our discussion includes the case in which $t = 0$ and the case in which $t \\neq 0$. \n\nIn Appendix \\ref{sec:fs}, it is shown that $t = 0$ happens with probability $1 - o(1)$ in the weak signal region, and $t \\neq 0$ happens with probability $1 - o(1)$ in the strong signal region. For the latter case, the signals can be exactly recovered with probability $1 - o(1)$. \n\n\n\n\n\n\\subsubsection{The region of successful classification}\\label{sec:weaksig}\nConsider the event $\\{t = 0\\}$. It happens with probability $1 - o(1)$, so we focus on this event only. \n\nWhen $t = 0$, the estimated label is $I\\{Q > 0\\}$ where \n\\[\nQ = Q(X, \\hat{\\mu}, W) = X^\\top(I -{\\Omega}) X + 2d^\\top X + \n \\hat{\\mu}_0^\\top (I - {\\Omega}) \\hat{\\mu}_0 + \\ln |{\\Omega}|.\n\\]\nAccording to the definition of $Z$, $\\mu_S$ and $\\sigma_S$ in Lemma \\ref{lemma:S}, we rewrite it as \n\\begin{equation}\\label{eqn:deltaq}\nQ(X, \\hat{\\mu}, W)  =\n  \\sigma_S Z + \\mu_S - \\mu^\\top (\\Omega - I)\\mu  + \\ln|\\Omega| + \\Delta Q, \n\\end{equation}\nwhere \n\\begin{equation}\\label{Decomp1}\n\\Delta Q  =   2(d^\\top -\\mu^\\top(I+\\Omega))X+\\left[\\hat{\\mu}_0^\\top (I - {\\Omega}) \\hat{\\mu}_0 -\\mu^\\top(I-\\Omega)\\mu\\right]. \n\\end{equation} \n\nAccording to Lemma \\ref{lemma:S}, with probability $1 - o(1)$, $Z$ uniformly converges to the standard normal distribution . Therefore, with the same definition of $T_i$ in (\\ref{eqn:T}), (\\ref{c0form}), and (\\ref{p2cons}), we have \n\\[\np_{i, \\mu, W} = \\Phi((-1)^i*(-T_i + \\frac{\\Delta Q}{\\sigma_S})) + o(1), \\quad i = 0, 1.\n\\]\nNote that the QDA successful region in Theorem \\ref{thm:unknownmu} is a subset of that in Theorem \\ref{thm:idealunequal}. According to Section \\ref{sec:thmideal}, $\\sigma_S = \\sqrt{p\\xi^2 + \\eta^2 p^2 \\nu + 8 \\tau^2 p \\epsilon}$, and $-T_0= T_1 = -C\\sqrt{p\\xi^2 + \\eta^2 p^2 \\nu + 8 \\tau^2 p \\epsilon}$. Further, later we will show that \n\\begin{equation}\\label{eqn:deltaq} \n\\Delta Q  = o({p\\xi^2 + \\eta^2 p^2 \\nu + 8 \\tau^2 p \\epsilon}). \n\\end{equation}\nTherefore, noting that $p\\xi^2 + \\eta^2 p^2 \\nu + 8 \\tau^2 p \\epsilon \\rightarrow \\infty$ in the successful region, we have that \n\\[\np_{i, \\mu, W} = \\Phi( -C\\sqrt{p\\xi^2 + \\eta^2 p^2 \\nu + 8 \\tau^2 p \\epsilon}(1 + o(1))) \\rightarrow 0, \\qquad i = 0, 1.\n\\]\nThe misclassification rate now is that \n\\begin{equation}\\label{eqn:mr3}\nMR(QDA) = \\frac{1}{2} E_{\\mu, W} p_{0, \\mu, W} + \\frac{1}{2} E_{\\mu, W} p_{1, \\mu, W} \\rightarrow 0.\n\\end{equation}\n\nNow, we only need to show (\\ref{eqn:deltaq}) holds in the successful region. We first introduce a lemma about $\\Delta Q$, where the proof can be found in Appendix. \n\\begin{lemma}\\label{lemma:DeltaQ}\nUnder the model assumptions and the definition of $\\Delta Q$ in (\\ref{Decomp1}), there is\n\\begin{equation}\\label{eqn:lemmaDeltaQ}\n \\Delta Q   \\lesssim  \\sqrt{p^2 \\eta^2 \\nu \\ln p}\/n + \\sqrt{p\/n \\ln\\ln p}.\n\\end{equation}\n\\end{lemma}\n\nWe compare (\\ref{eqn:lemmaDeltaQ}) with (\\ref{eqn:deltaq}) in the  successful region in Theorem \\ref{thm:unknownmu}, that \n\\begin{equation}\\label{eqn:weakposs}\n\\left\\{\\begin{array}{ll}\n\\frac{1-\\delta}{2} < \\max \\{2 - 2\\alpha - \\beta, 1 - 2\\theta - \\zeta\\}, \\\\\n\\frac{1-\\delta}{2} = 2 - 2\\alpha - \\beta > 1 - 2\\theta - \\zeta, f_p = L_p, \\\\\n\\frac{1-\\delta}{2} = 1 - 2\\theta - \\zeta > 2 - 2\\alpha - \\beta, g_p = L_p. \\\\\n\\end{array}\n\\right.\n\\end{equation}\nNote that $p\\xi^2 + p^2 \\eta^2 \\nu + 8 p\\tau^2 \\epsilon = p^{1 - 2\\gamma} + f_p^2 p^{2 - 2\\alpha - \\beta} + g_p^2 p^{1 - 2\\theta - \\zeta} \\rightarrow \\infty$ in this region. \n\nThe first term $\\sqrt{p^2 \\eta^2 \\nu \\ln p}\/n \\ll \\sqrt{p^2 \\eta^2 \\nu}$. When $p^2 \\eta^2 \\nu \\rightarrow 0$, this term converges to 0, which is at a lower order of  $p\\xi^2 + p^2 \\eta^2 \\nu + 8 p\\tau^2 \\epsilon$. When $p^2 \\eta^2 \\nu \\rightarrow \\infty$, then $\\sqrt{p^2 \\eta^2 \\nu \\ln p}\/n \\ll p^2 \\eta^2 \\nu$. Hence, it is always at $o(p\\xi^2 + p^2 \\eta^2 \\nu + 8 p\\tau^2 \\epsilon )$ in this region. \nThe second term $\\sqrt{p\/n \\ln \\ln p} = p^{\\frac{1-\\delta}{2}} \\sqrt{\\ln \\ln p} \\ll L_p p^{\\max \\{2 - 2\\alpha - \\beta, 1 - 2\\theta - \\zeta\\}}$ according to the definition of $L_p$. \nCombining the analysis of these two terms, $\\Delta Q = o(p\\xi^2 + p^2 \\eta^2 \\nu + 8 p\\tau^2 \\epsilon)$. Hence, (\\ref{eqn:deltaq}) is proved.\n\nCombining the results, in the successful classification regions described in part (i) of Theorem \\ref{thm:unknownmu}, $MR(QDA) \\rightarrow 0$. \n\n\n\n\n\n\\subsubsection{The region of unsuccessful classification} \nAgain, we consider the event $\\{t = 0\\}$ only. \nAccording to the definition that $d = \\Omega \\hat{\\mu}_1 - \\hat{\\mu}_0$, there is \n\\begin{equation}\nQ  =  X^\\top (I - \\Omega) X + 2 X^{\\top} (\\Omega \\hat{\\mu}_1 - \\hat{\\mu}_0) + \\hat{\\mu}_0^\\top (I - \\Omega) \\hat{\\mu}_0 + \\ln |\\Omega| \n\\end{equation}\nWe discuss the case when $X$ is given first, and then add in the randomness of $X$. When $X$ is given, we have the following lemma, which is proved in Appendix. \n\\begin{lemma}\\label{lemma:weakfail}\nWith probability at least $\\Phi(-1\/2)\/4$, there is \n\\begin{eqnarray}\\label{eqn:qimposs}\nQ \\geq S + \\sqrt{X^\\top \\Omega X \/n} + \\mu^{\\top}(I - \\Omega)\\mu+ \\ln|\\Omega|+\\frac{1}{n_0} Tr(I - \\Omega), \n\\end{eqnarray}\nwhere $S$ is defined in (\\ref{eqn:S}).\n\\end{lemma}\n\nNow we consider the randomness of $X$. We focus on the case that $X \\sim N(-\\mu, I)$, and the case that $X \\sim N(\\mu, \\Omega^{-1})$ is very similar. \nConsider the case that $X \\sim N(\\mu, I)$. According to Lemma \\ref{lemma:S}, $\\frac{S - E[S]}{\\sigma_S}$ converges to the standard normal distribution uniformly. Therefore, $P(S \\geq E[S]) = 1\/2 + o(1)$. According to (\\ref{zmeanvar1}), $E[S] = Tr(I - \\Omega) - \\mu^\\top(3\\Omega + I) \\mu$. Hence, we have that \n\\begin{equation}\\label{eqn:x1}\nP(S + \\mu^\\top (I - \\Omega)\\mu \\geq -4\\mu^\\top \\Omega \\mu+ Tr(I - \\Omega)) = 1\/2 + o(1). \n\\end{equation}\nThe other term including $X$ is that $\\sqrt{X^\\top \\Omega X\/n}$. According to the properties of non-central chi-square distribution, $E[X^\\top \\Omega X\/n] = \\mu^\\top \\Omega \\mu\/n + Tr(\\Omega)\/n$ and $\\mathrm{Var}(X^\\top \\Omega X\/n) = 4\\mu^\\top \\Omega^2 \\mu\/n^2 + 2Tr(\\Omega^2)\/n^2 = p\/n^2 (1 + o(1))$. \nBy Markov inequality, \n\\begin{equation}\\label{eqn:x2}\nP(X^\\top \\Omega X\/n \\geq \\mu^\\top \\Omega \\mu\/n + Tr(\\Omega)\/n - 2\\sqrt{p\/n^2}) \\geq 3\/4. \n\\end{equation}\n\nCombining (\\ref{eqn:x1}) and (\\ref{eqn:x2}), the probability that both inequality holds is no smaller than $1\/4$. Introducing them into (\\ref{eqn:qimposs}), with probability at least $c\/4$, there is \n\\begin{eqnarray}\\label{eqn:qresult}\nQ & \\geq & -4\\mu^\\top \\Omega \\mu+ Tr(I - \\Omega)+ \\sqrt{\\mu^\\top \\Omega \\mu\/n + Tr(\\Omega)\/n - 2\\sqrt{p\/n^2}}\\\\\n&&  + \\ln|\\Omega|+\\frac{1}{n_0} Tr(I - \\Omega)\\nonumber\\\\\n& \\gtrsim & -4 g_p^2 \\tau^2 p \\epsilon + \\sqrt{p\/n} - f_p^2 \\eta^2p^2\\nu.\n\\end{eqnarray}\n\nNow we consider the region defined in Theorem \\ref{thm:unknownmu}. \n\\begin{itemize}\n\\item $\\max\\{2-2\\alpha-\\beta, 1-2\\theta-\\zeta\\}< (1-\\delta)\/2$;\n\\item or, $2 - 2\\alpha - \\beta = (1 - \\delta)\/2$ and $f_p = C_p$; \n\\item or, $1-2\\theta-\\zeta = (1 - \\delta)\/2$ and $g_p=C_p$.\n\\end{itemize} \nIn the first region, there is $\\tau^2 p \\epsilon  + \\eta^2p^2\\nu \\ll \\sqrt{p\/n}$, then the misclassification rate when $X \\sim N(-\\mu, I)$ is $P(Q > 0) \\geq c\/4$ according to (\\ref{eqn:qresult}). \nIn the second and third region, the coefficient of $\\sqrt{p\/n}$ can be adjusted with a smaller probability. Therefore, the misclassification rate is still no smaller than a constant. \nWhen $X \\sim N(\\mu, \\Omega^{-1})$, the same result can be got. \nHence, in this region, $MR(QDA) \\geq c$ where $c > 0$ is a constant. {\\unskip\\nobreak\\hfil\\penalty50\\hskip2em\\vadjust{\n\n\n\n\\subsection{Proof of Theorem \\ref{thm:unknownmu}: the strong signal region}\\label{sec:thm3strong} \nWhen the signals are strong, with the threshold $t = \\sqrt{2\\log p\/n}$, all the signals $\\mu_i \\neq 0$ are exactly recovered with probability $1 + o(1)$. Hence, we only consider the event that $\\{t = \\sqrt{2\\log p\/n}\\}$ and all the signals are exactly recovered.  \n\nFor the case that $t \\neq 0$, $\\hat{\\mu}^{(t)} = \\hat{\\mu}_0 \\circ d^{(t)}$ and $\\hat{\\mu}_d^{(t)} = d \\circ d^{(t)}$. \nFor simplicity, in this section, we use $\\hat{\\mu}_0$ and $d$ to denote $\\hat{\\mu}_0 \\circ d^{(t)}$ and $\\hat{\\mu}_d^{(t)}$, respectively. Note that since all the signals are exactly recovered, $\\hat{\\mu}_0$ has zeros on the non-signal entries and non-zeros on the signals, and the same for $d$. \n\n\nLet $k=\\|\\mu\\|_0$ denote the number of non-zeros in $\\mu$. \nWithout loss of generality, we permute $\\mu$ such that the first $k$ entries are the non-zeros and the rest are the zeros. We also permute $W$, $\\Omega$, and $X$ accordingly, and rewrite $W$ and $\\Omega$ as $2\\times 2 $ block matrices $W=\\left(^{W_{11} \\ W_{12}}_{W_{21} \\ W_{22}}\\right)$ and $\\Omega=\\left(^{\\Omega_{11} \\ \\Omega_{12}}_{\\Omega_{21} \\ \\Omega_{22}}\\right)$ respectively, where $W_{11}$ and $\\Omega_{11}$ are $k\\times k$ sub-matrices of $W$ and $\\Omega$,  respectively. Let $X^{(k)}$, $d^{(k)}$, $\\mu^{(k)}$, and $\\hat{\\mu}_0^{(k)}$ denote, respectively, $X$, $d$, $\\mu$, and $\\hat{\\mu}_0$ restricted on the first $k$ entries, and let $X^{(p-k)}$ denote $X$ restricted on the last $(p-k)$ entries. Then $\\mu^{(k)}$ is a length $k$ vector with all elements as $\\tau$. \n\n\n\nSimilar as the derivation in Section \\ref{sec:weaksig}, we have \n\\begin{equation}\\label{eqn:deltaqstrong}\nQ(X, \\hat{\\mu}, W)  =\n  \\sigma_S Z + \\mu_S - \\mu^\\top (\\Omega - I)\\mu  + \\ln|\\Omega| + \\Delta Q, \n\\end{equation}\nwhere \n\\begin{equation}\\label{Decomp2}\n \\Delta Q =  \\displaystyle 2(d - (I + \\Omega)\\mu)^\\top X+\\left[\\hat{\\mu}_0^\\top(I-\\Omega)\\hat{\\mu}_0-\\mu^\\top(I-\\Omega)\\mu\\right] .\n\\end{equation}\nWe want to show that in the region of success, \n\\begin{equation}\\label{eqn:deltaqmag} \n\\Delta Q  = o({p\\xi^2 + \\eta^2 p^2 \\nu + 8 \\tau^2 p \\epsilon}). \n\\end{equation}\nTherefore, noting that $p\\xi^2 + \\eta^2 p^2 \\nu + 8 \\tau^2 p \\epsilon \\rightarrow \\infty$ in the region of possibility, we have $MR(QDA) \\rightarrow 0$. \n\nTo prove (\\ref{eqn:deltaqmag}), we introduce a lemma about $\\Delta Q$ for the strong signal case, where the proof can be found in Appendix. \n\\begin{lemma}\\label{lemma:StrongDeltaQ}\nUnder the model assumptions and the definition of $\\Delta Q$ in (\\ref{Decomp2}), there is\n\\begin{equation}\\label{eqn:lemmaDeltaQ2}\n \\Delta Q   \\leq   c p\\epsilon \\xi\/n + O(p \\epsilon\\sqrt{\\eta^2 \\nu}\/n) + \\sqrt{p\\epsilon} \\tau \\ln p (1 + o(1)).\n\\end{equation}\n\\end{lemma}\nConsider the three terms in (\\ref{eqn:lemmaDeltaQ2}). The first term is $c n^{-1}p\\epsilon \\xi \\ll \\sqrt{p}\\epsilon\/n \\ll \\tau^2 \\sqrt{p}\\epsilon$, since $\\xi \\ll p^{-1\/2}$ and $1\/\\sqrt{n} \\ll \\tau $. The second term is $n^{-1}p \\epsilon\\sqrt{\\eta^2 \\nu} = \\frac{\\epsilon}{n}\\sqrt{\\eta^2 p^2 \\nu}$. Hence, it is at a smaller order if $\\eta^2 p^2 \\nu \\rightarrow \\infty$. If $\\eta^2 p^2 \\nu \\rightarrow 0$, $p\\xi^2 + p\\tau^2 \\epsilon \\rightarrow \\infty$ in the region of possibility, so that $\\frac{\\epsilon}{n}\\sqrt{\\eta^2 p^2 \\nu}$ is still at a smaller order. For the last term $\\sqrt{p\\epsilon}\\tau \\ln p = \\sqrt{p \\tau^2\\epsilon }\\ln p$, the analysis is the same. \n\nTherefore, in the region of possibility identified by part (ii) of Theorem \\ref{thm:unknownmu}, (\\ref{eqn:deltaqmag}) always holds, and hence $MR$ converges to 0. {\\unskip\\nobreak\\hfil\\penalty50\\hskip2em\\vadjust{\n\n\\subsubsection{The region of Impossibility}\nWhen the mean vector $\\mu$ is unknown, the region of impossibility is no smaller than the case that the mean vector $\\mu$ is already known (\\cite{LeCam}). For the latter, the region of impossibility is depicted in Theorem \\ref{thm:idealunequal}. \n\nWhen the signals are strong, the region of impossibility is the same with that in Theorem \\ref{thm:idealunequal}. Hence, when the mean vector is unknown, any classifier $L$ has $MR(L) \\rightarrow 1\/2$ in this region.  {\\unskip\\nobreak\\hfil\\penalty50\\hskip2em\\vadjust{\n\n\n\n\\section{Real Data Analysis}\\label{sec:rats}\nIn this paper, we consider the rats dataset present in \\cite{liver}. As we introduced in Section \\ref{sec:quickdata}, this data set record the gene expressions of live rats in response to different drugs and toxicant. There are 181 samples and 8491 genes, where 61 samples are labeled as toxicant and the other 120 are labeled as other drugs. \nWe compare QDA with LDA, where the latter one is shown to enjoy the best performance compared to classifiers such as SVM, RandomForest, GLasso and FoBa. \nThe QDA with feature selection for the real data is discussed in Section \\ref{sec:real-proc} and the implementation details and results are in Section \\ref{sec:dataresults}. \n\n\n\\subsection{Procedure for the real data}\\label{sec:real-proc}\nHere, we present a procedure for the classification based on QDA for the real data. For the real data, we have to estimate $\\Omega_0$, $\\Omega_1$, $\\mu_0$ and $\\mu_1$ separately. Further, we need to eliminate the effect of the feature variances. Hence, there is an additional scaling step in the following algorithm. \n\n\\begin{itemize}\n\\item[0.] \nFind the corresponding precision matrix ($\\hat{\\Omega}_0$, $\\hat{\\Omega}$). Find the sample mean vectors $\\hat{\\mu}_0$ and $\\hat{\\mu}_1$ for groups 0 and 1. \n\\item[1.] Estimate the difference $\\Omega_0-\\Omega_1$ by $\\Omega_ {\\rm diff}=\\hat\\Omega_0-\\hat\\Omega_1$. Set the diagonals of $\\Omega_{\\rm diff}$ to be zeros.  \n\\item[2.] Let $d_0= \\hat{\\Omega}_0\\hat{\\mu}_0$ and $d_1= \\hat{\\Omega}\\hat{\\mu}_1$. Scale them by standard deviation and sample size, i.e., \n\\[\nd_0 = \\hat{\\Omega}_0\\hat{\\mu}_0\/s_0, \\quad d_1 = \\hat{\\Omega}\\hat{\\mu}_1\/s_1,\n\\]\nwhere $s_i$ are the standard deviation vector of the train data from class $i$, $i=0, 1$, and the devision means element-wise devision. \n\n\\item[3.] Let $d = d_1 - d_0$. For a threshold $t$, define the vector $d^{(t)}$ where $d^{(t)}_j = d_j*I(|d_j| \\geq t)$. \n\n\\item[4.] For any fresh data vector $X$, compute the scaled test vector \n\\[\nx_j = [X_j - \\bar{\\mu}_j]\/s_j,\n\\]\nwhere $\\bar{\\mu}=\\frac{\\hat{\\mu}_1+\\hat{\\mu}_0}{2}$ and $s$ is the standard error of the pooled data\n$$\ns_j   =    \\sqrt{[(n_0-1) ((s_0)_j)^2+(n_1-1)((s_1)_j)^2] \/ (n_0+n_1-2)}.\n$$\n\\item[5.] Calculate the Z-score\n$$ \nZ=x^\\top \\hat{\\Omega}_{\\mbox{{\\footnotesize diff}}} x+2(d^{(t)})^\\top x+C\n$$\nand classify $X$ to be in class  0 if $Z < 0$ (or in class 1 if $Z > 0$). \n\\end{itemize}\n\nThere are two tuning parameters in the algorithm, $t$ in Step 3 and $C$ in Step 5. In the implementations, we use a grid search to find the optimal values. Details in Section \\ref{sec:dataresults}. \n\nIn Step 1, we set all the diagonals of $\\Omega_{\\rm diff}$ to be 0. The reason is that the sample size is limited. The training data has 90 samples for class 0 and  46 samples for class 1 only, so the elementary-wise error for $\\hat{\\Omega}_0$ and $\\hat{\\Omega}$ are $\\sim 1\/\\sqrt{50}$. It will cause a comparatively large error on the diagonals of $\\hat{\\Omega}_0 - \\hat{\\Omega}$, and hence the classification criteria. So for this data set, we set all the diagonals of $\\Omega_{\\rm diff}$ to be 0. It is not necessary for large data sets. \n\n\\subsection{Implementation and Results}\\label{sec:dataresults}\nFollowing the setup of the data analysis in \\cite{PCS}, we apply 4-fold data splitting to the sample. \nFor each class, we randomly draw one fourth of the samples, and then combine them to be the test data while using the leftover to be the training data. We do the splitting for 15 times independently and record the error with QDA and LDA for each splitting. \nThe data (sample indices for the 15 splittings is available at \\url{https:\/\/zhigang-yao.github.io\/research.html}. \n\nIn the real data analysis section, we focus on comparing QDA and LDA. The LDA is implemented within the setting of QDA, where in Step (3)  of the algorithm in Section \\ref{sec:real-proc} we use clipping thresholding instead of hard thresholding, and in Step (5) we set  $\\hat{\\Omega}_{\\mbox{{\\footnotesize diff}}}=\\mathbf{0}$ for LDA. \nThe clipping threshold is employed since it gives much more satisfactory results than hard thresholding for LDA; details in \\cite{PCS}. For QDA, the two ways give similar results. \nSince the calculation of $\\hat{d}$ involves the calculation of $\\hat{\\Omega}_0$ and $\\hat{\\Omega}$ and the thresholding, LDA algorithm has exactly the same tuning parameters with QDA. \nThe procedure of determining these tuning parameters are the same for both algorithms, so that the results are comparable.\n\n\nThere are two set of parameters in the algorithm. One is in the estimation of PCS, and the other are $C$ and $t$ in the algorithm. For PCS, there are four tuning parameters $(q_1, q_2, \\delta, L)$. Here we use the same set of tuning parameters for the estimation of both $\\hat{\\Omega}_0$ and $\\hat{\\Omega}$, since the two classes are from the same data set and the performance of PCS is not sensitive to the choice of these parameters (\\cite{PCS}). \nFollowing the setting in \\cite{PCS}, we set $(\\delta, L)=(.1, 30)$, and also tried $(\\delta, L)=(.1, 50)$. The selection of $(q_1, q_2)$ is done with $C$ and $t$ by grid search. \n\nIn Step (3) and Step (5) of the algorithm, there are two tuning parameters $t$ and $C$. We set the ranges $[t_{\\mbox{{\\footnotesize min}}}, t_{\\mbox{{\\footnotesize max}}}]=[0, \\mbox{max}_{1\\leq j \\leq p} |d_j|]$ with an increment of .1 and $[C_{\\mbox{{\\footnotesize min}}}, C_{\\mbox{{\\footnotesize max}}}]=[-50, 50]$ with an increment of 1. For $(q_1, q_2)$, we consider $.1 \\leq q_k \\leq 1$, with an increment of .1, $k = 0, 1$. \nThe smallest error is obtained over a grid search of $t$, $C$, and $(q_1, q_2)$. \nThis step is the same for both QDA and LDA to be fair. We compare the smallest error that LDA and QDA can achieve. \n\n\n\n\n\\begin{figure}[htb!]\n    \\centering\n   \\subfigure[ $(C_{\\mbox{{\\footnotesize min}}}, C_{\\mbox{{\\footnotesize max}}})=(-50, 50)$ ]{ \\includegraphics[width= 2.37 in]{rats2.pdf}} \\quad\n\t\t\\subfigure[$(C_{\\mbox{{\\footnotesize min}}}, C_{\\mbox{{\\footnotesize max}}})=(-100, 100)$]{\\includegraphics[width= 2.37 in]{rats1.pdf}}\n      \\caption{Comparison of testing error rate (y-axis) of LDA and QDA for the rats data with $(\\delta_1, L_1)=(\\delta_2, L_2)=(.1, 30)$ and 15 data splittings.}\n    \\label{figure:errorrats-k30}\n\\end{figure}\n\n\\begin{figure}[htb!]\n    \\centering\n    \\subfigure[]{ \\includegraphics[width= 2.37in]{plot1_LQDA2_k30_new}} \\quad\n\t\t\\subfigure[]{\\includegraphics[width= 2.37in]{plot1_LQDA3_k30_new}} \n\t\t\\subfigure[]{\\includegraphics[width= 2.37in]{plot1_LQDA4_k50_new}} \\quad\n\t\t\\subfigure[]{\\includegraphics[width= 2.37in]{plot1_LQDA5_k50_new}}\n\t\t\t \\caption{Zoom-in errors for the rats data for varying choices of $(q_1, q_2) \\in (.1,1) \\times (.1, 1)$ for one splitting of 15 splittings in Figure 4(a)-(b) and Figure 6(a)-(b).}\n    \\label{figure:errorrats-zoomin}\n\\end{figure}\n\nBoth the LDA test error (the best error) and the QDA test error (the best error) over all 15 data splittings are reported in Figure \\ref{figure:errorrats-k30}.\nIn the left panel of Figure \\ref{figure:errorrats-k30}, we can see that the error rates of LDA are all above QDA at every data splitting.  To better show the difference between them, we also plot the testing error rate in the right panel of Figure \\ref{figure:errorrats-k30} for a wider grid-search range that $[C_{\\mbox{{\\footnotesize min}}}, C_{\\mbox{{\\footnotesize max}}}]=[-100, 100]$. \nWhen $L$ changes from 30 to 50, the results are summarized in Figure \\ref{figure:errorrats-k50}, which is similar. This comparison clearly demonstrates the expected superiority of QDA over LDA. \nThe results suggest that, for rats data, QDA  outperforms LDA in terms of both best error rate and average error; with the results in \\cite{PCS} for other methods, where the authors have shown that HCT-based LDA significantly outperforms all other HCT-based methods as well as SVM and RF, our findings also suggest that the QDA gives a better separation than the LDA  by taking into account the second order difference between the two classes.\n\n\\begin{figure}[htb!]\n    \\centering\n    \\subfigure[$(C_{\\mbox{{\\footnotesize min}}}, C_{\\mbox{{\\footnotesize max}}})=(-50, 50)$]{\\includegraphics[width= 2.4 in]{rats2-k50.pdf}}\\quad\n\t\t\\subfigure[$(C_{\\mbox{{\\footnotesize min}}}, C_{\\mbox{{\\footnotesize max}}})=(-100, 100)$]{\\includegraphics[width= 2.4 in]{rats1-k50.pdf}}\n      \\caption{Comparison of testing error rate (y-axis) of LDA and QDA for the rats data with $(\\delta_1, L_1)=(\\delta_2, L_2)=(.1, 50)$ and 15 data splittings.}\n    \\label{figure:errorrats-k50}\n\\end{figure}\n\n\n\n\n\\section{Discussion}\\label{sec:dis}\nThis paper focuses on the classification problem associated with the use of QDA and feature selection for data of rare and weak signals. We derived the successful and unsuccessful classification regions, by using first the case of a known mean vector and covariance matrix, then the case of an unknown mean vector but known covariance matrix, and finally the case in which both mean vector and covariance matrix were unknown. We also proved that these regions were actually the possibility and impossibility regions under the same modelling, which indicates that QDA achieves the optimal classification results in this manner. In addition, we developed computing and classification algorithms that incorporated feature selection for rare and weak data. With these algorithms, our real data analysis showed that QDA had much improved performance over LDA. \n\nOur theoretical results showed that the two sets of signal weakness and sparsity parameters, one set from the mean vector and the other set from the covariance matrix, influence the possibility\/impossibility regions or QDA successful\/unsuccessful regions almost independently (except for a $\\max$ operator over the two sets of parameters) when the covariance matrix is known. When both the mean vector and covariance matrix are unknown, the two sets of parameters interact with each other as indicated in Theorem \\ref{thm:unknownomega}. For the latter case, the analysis of the misclassification rate is very complicated and we only obtained partial results for this most general case; further study is therefore warranted. Also, for the precision matrix $\\Omega$ given in (\\ref{Omega1}), we can introduce sparsity and weakness in the diagonal elements of $I-\\Omega$, the difference in precision matrices, instead of using a constant $\\xi=1-c$ for all diagonal elements.  \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nEmergent scale invariance is a key to many of our current scientific and engineering challenges, including cell membranes~\\cite{machta2012critical}, turbulence~\\cite{canet2016fully}, fracture and plasticity~\\cite{shekhawat2013damage, chen2013scaling}, and also the more traditional continuous thermodynamic phase transitions. \nThe current formulation of the field has an elegant framework which can explain observables that scale as power laws times homogeneous functions. However, the literature on corrections to this result, including logarithms and exponentially diverging quantities, is much more scattered and does not have a similarly systematic framework. \n\nThe renormalization group (RG) is our tool for understanding emergent\nscale invariance. At root, despite challenges of implementation, the renormalization group (RG)\ncoarse grains and rescales the system to generate ordinary differential equations (ODEs) for model parameters as a\nfunction of the observed log length scale $\\ell$. A fixed point of these\nflows represents a system which looks the same at different length scales;\nsystems near criticality flow near to this fixed point. In cases where the\nflow can be linearized around the fixed point, the RG\nimplies that observables near criticality are given by a power law times\na universal function of an invariant combinations of variables; {\\em e.g.}\nthe Ising model has magnetization $m \\sim t^{\\beta} \\mathcal{M}(L t^\\nu)$ where $L$ is the system size and $t = (T - T_c)\/T_c$ is the deviation of the temperature $T$ from the critical temperature $T_c$.\n\nSurprisingly often, this scenario of universal critical exponents\nand scaling functions is violated; free energies and correlation lengths\nscale with logarithms or exponentials, and the proper form of the \nuniversal scaling functions is often unknown. \nSpecifically,\ndeviations arise in the Ising model in $d = 1$~\\cite{ising1925beitrag}, 2~\\cite{salas2002exact}, \\& 4~\\cite{larkin1995phase},\nthe tricritical Ising model in $d = 3$~\\cite{Wegner73}, the $d=2$ XY\nmodel~\\cite{KosterlitzT73}, the surface critical behavior of\npolymers~\\cite{Diehl87, Eisenriegler88}, van der Waals interactions in 3-d spherical model~\\cite{Dantchev06}, finite size scaling of the random field\nIsing model (RFIM) in $d = 6$~\\cite{Ahrens10}, thermodynamic Casimir effects in\nslabs with free surfaces~\\cite{Diehl12,Diehl14}, the $d = 2$,\n4-state Potts model~\\cite{Salas97,Shchur09,Berche13}, percolation and the 6-d Potts model \\cite{PhysRevE.68.036129}, and many other systems. \nEach such system has hitherto been treated as a special case. \n\nHere we use the fact that the predictions of the RG can be written down as a set of differential equations in the abstract space of Hamiltonians. This allows us to apply a branch of dynamical systems theory, normal form theory~\\cite{murdock2006normal, PNFT1} to provide a unified description applicable to all of these systems. We arrange these systems into {\\em universality\nfamilies} of theories, each defined by its normal form. Each family has \n{\\em universal terms} (linear and nonlinear), whose values \ndetermine a system's universality\nclass within the family. Finally, each family's normal form predicts the\nnatural {\\em invariant scaling\ncombinations} governing universal scaling functions.\n\n\nThe perspective we present here is transformative: unifying, simplifying,\nand systematizing a previously technical subject and promising new developments\nin the field. Our best analogy is to the introduction of homotopy theory in the\n1970's~\\cite{toulouse1976principles,rogula1976large,Merm79,GoldbartK19} \nas a systematic method that unified the treatment of some of the many defect\nstructures studied in materials and field theories. Just as there have\nbeen several previous works that correctly identified the universal effects\nof nonlinear terms for phase transitions where an analytic RG \napproach is available~\\cite{Wegner72,Meinke05,sonoda,magradze,pelissetto2013renormalization, barma1984corrections, barma1985two, hasenbusch2008universal, Salas97}, the \nBurger's vector, winding number, and wrapping number of dislocations in \ncrystals, disclinations in liquid crystals, and Skyrmions in nuclei were \nunderstood individually before the mathematics of homotopy theory was seen\nas the natural mathematical framework. Just as homotopy theory facilitated the\nstudy of defects in more complex systems (metallic glasses, cosmic strings,\nquasicrystals), so our normal\nform methods are allowing the correct identification\nand characterization of the singularity in systems in experimental and \nnumerical explorations where analytic RG calculations do not yet\nexist~\\cite{Lorien17}. Finally, homotopy theory quickly uncovered\nthe fascinating entanglement and transformation properties of non-abelian\ndefects~\\cite{Merm79}, with early speculative applications in glass\nphysics~\\cite{Nelson83} and eventually \ninspiring the closely related nonabelian braiding being developed for \ntopological quantum computing. Similarly, we demonstrate here that our methods\nallow, for what appears to be the first time, the use of the correct,\nremarkably rich, invariant scaling variables in the universal scaling\nfunctions for systems where universal nonlinear RG terms are needed, and\nwe have discussed elsewhere~\\cite{raju2018reexamining} how our methods can\nbe powerful tools for systematically incorporating corrections to\nscaling near critical points even when universal nonlinear terms are not\nneeded. For example, in the future the normal-form change of variables \nwe introduce here could become an inner expansion matched to \nseries and virial expansions at extremes of the phase diagram; this would\nallow rapid and accurate convergent characterizations of materials systems\nclose to and far from criticality.\n\nOur machinery provides a\nstraightforward method to determine the complete form of the critical\nsingularity in these challenging cases. Our initial results are complex and\ninteresting; they pose challenges which we propose to address in future\nwork. The coordinate transformation to the normal form embodies analytic corrections to scaling, which allow us to address experimental\nsystems as they vary farther from the critical point. Finally, bifurcation theory\nis designed to analyze low-dimensional dynamical systems without detailed understanding of the underlying equations;\nour methods should improve scaling collapses in critical phenomena\nlike 2-d jamming~\\cite{goodrich2014jamming}\nwhere there is numerical evidence for logarithms but no RG framework is available.\n\nWe begin by distinguishing our work from previous literature connecting the RG to normal form theory. The previous approach~\\cite{deville2008analysis, ei2000renormalization, ziane2000certain} compared the application of RG-like methods and normal form theory to solving nonlinear differential equations using perturbation theory. The connection we are making is different. We are applying normal form theory to the RG flow equations. Hence, our approach is to apply normal form theory to make predictions about the general structure of the flows given the topology (nature and number of fixed points), rather than to apply it to the model that produces these flows. \n\nWe give an introduction to normal form theory in Section~\\ref{sec:normalform}. We give a survey of the previous literature on nonlinear scaling in the RG in section~\\ref{sec:earlierwork}. We show how the application of normal form theory allows us to define universality families of fixed points in Section~\\ref{sec:universality}. We present several worked out examples starting with the 4-d Ising model in Section~\\ref{sec:4dising} and the Random Field Ising model in Section~\\ref{sec:randomfieldising}. We then work out the application of normal form theory to the Ising model in dimensions $1$, $2$ and $3$ in Sections~\\ref{sec:3dising}--~\\ref{sec:2dising}.\n\n\n\\section{Normal Form Theory}\\label{sec:normalform}\n\nNormal form theory~\\cite{PNFT1} is a technique to reduce differential equations to a `normal form' by change of coordinates, often the simplest possible form. This is achieved by making near-identity coordinate transformations to get rid of as many terms as possible from the equation. It was developed initially by Poincar\\'{e} to integrate nonlinear systems~\\cite{poincare, chenciner2015poincare}. The physical behavior should be invariant under analytic changes of coordinates, and the length (or time) parameter should stay the same, \nwhich the mathematical literature addresses by perturbative polynomial changes\nof coordinates (attempting removal of $n$th order nonlinearities in the flow\nby using $n$th order or lower terms in the change of variables). To any \nfinite order\nthis gives an analytic change of coordinates, but it is not in general\nguaranteed to converge to an analytic transformation; we will thus\ncall it a polynomial change of coordinates. \n\nWe give a brief introduction to normal form theory here for completeness. A more detailed treatment can be found in Ref.~\\cite{PNFT1}. Typically one starts with a set of differential equations of the form \n\\begin{equation}\n \\frac{d \\bm{\\theta}}{d \\ell} = \\bm{g}(\\bm{\\theta}, \\epsilon) ,\n \\end{equation}\n where $\\epsilon$ is some parameter, $\\bm{\\theta} = \\{\\theta_i\\}$ is the vector of state variables and the vector field $\\bm{g}$ defines the flow. In the context of statistical mechanics and renormalization group flows, $\\theta_i$'s are parameters or coupling constants that enter into the free energy and $\\epsilon$ is the difference in dimension from the lower or upper critical dimensions. Let us first work with the case where $\\epsilon$ does not enter into the equations. The first step is to find the fixed point of the equation and use translations to set the fixed point of each $\\theta_i$ is at 0. The next step is to linearize about the fixed point and reduce the linear part to the simplest possible form. In general, this is the Jordan canonical form but it is often just the eigenbasis. Then, the equation is \n \\begin{equation}\n \\frac{d \\bm{\\theta}}{d \\ell} = J \\bm{\\theta} + \\bm{f}(\\bm{\\theta}) ,\n\\end{equation}\nwhere $J$ is the linearized matrix of the flow and the remaining terms are in the vector field $\\bm{f}(\\bm{\\theta}) \\sim \\mathcal{O}(\\theta^2)$. Terms of order $k$ are defined to be made up of homogeneous polynomials of order $k$. So for $\\bm{\\theta} = (\\theta_1, \\theta_2, \\theta_3)$, $\\theta_1^2 \\theta_2 \\theta_3 \\sim \\mathcal{O}(\\theta^4)$. We will denote terms of order $k$ by a lower index. So \n\\begin{equation}\n \\bm{f}(\\bm{\\theta}) = \\sum_{k \\geq 2} \\bm{f}_k (\\bm{\\theta}) .\n\\end{equation}\nNote the index is giving the order of the polynomial and not enumerating the components of the vector field. Let the lowest non-zero term be at some order $k \\geq 2$ (usually 2). Then we can write\n\\begin{equation}\n \\frac{d \\bm{\\theta}}{d \\ell} = J \\bm{\\theta} + \\bm{f}_k (\\bm{\\theta}) + \\mathcal{O}(\\theta^{k+1}) .\n \\end{equation}\nThe idea is to try and remove higher order terms by making coordinate changes. To remove the term $\\bm{f}_k$, we try to do a coordinate change of order $k$,  \n\\begin{equation}\n\\label{changecoord}\n \\bm{\\theta} = \\bm{\\tilde \\theta} + \\bm{h}_k (\\bm{\\tilde \\theta}) ,\n\\end{equation}\nwhere $\\bm{h}_k (\\bm(\\tilde \\theta))$ is a polynomial in $\\bm{\\tilde \\theta}$. This construction is similar  to nonlinear scaling fields~\\cite{cardy1996scaling, Wegner72} which try to linearize the RG flow equations with a subtle difference that we will remark on later. The higher order terms which we can remove by coordinate changes correspond to analytic corrections to scaling. Then, to find the equations in the new variables.\n\\begin{equation}\n \\frac{d \\bm{\\theta}}{d \\ell} = \\frac{d \\bm{\\tilde \\theta}}{d \\ell} + (\\mathcal{D} \\bm{h}_k) \\frac{d \\bm{\\tilde \\theta}}{d\\ell} .\n\\end{equation}\n$\\mathcal{D} \\bm{h}_k$ is the matrix of partial derivatives of the vector field $\\bm{h_k}$ with respect to the parameters $\\bm{\\theta}$. Now, substituting this into the equation\n\\begin{equation}\n (1 + \\mathcal{D} \\bm{h}_k) \\frac{d \\bm{\\tilde \\theta}}{d \\ell} = J (\\bm{\\tilde\n    \\theta} + \\bm{h}_k) + \\bm{f}(\\bm{\\tilde \\theta} + \\bm{h}(\\bm{\\tilde\n    \\theta})) + \\mathcal{O}({\\tilde \\theta}^{k+1}), \n \\end{equation}\nwhich upon simplification gives\n \\begin{equation}\n\n\n \\frac{d \\bm{\\tilde \\theta}}{d \\ell} = J \\bm{\\tilde \\theta} - (\\mathcal{D} \\bm{h}_k) J \\bm{\\tilde \\theta} + (\\mathcal{D} J \\bm{\\tilde \\theta}) \\bm{h}_k + \\bm{f}_k (\\bm{\\tilde \\theta})+ \\mathcal{O}({\\tilde \\theta}^{k+1}) .\n\\end{equation}\nFor the last line, notice that the matrix $J$ is the same as $\\mathcal{D} J \\bm{\\tilde \\theta}$ (i.e. the same as the matrix of partial derivatives with respect to parameters $\\bm{\\tilde \\theta}$ of the vector $J \\bm{\\tilde \\theta}$). \nTwo of the terms can be written as the Lie bracket (a commutator for vector fields) defined as $[\\bm{h}_k, J \\bm{\\tilde \\theta}] = - ( (\\mathcal{D} \\bm{h}_k) J \\bm{\\tilde \\theta} - (\\mathcal{D} J \\bm{\\tilde \\theta}) \\bm{h}_k)$  to give the final equation \n\\begin{equation}\n \\frac{d \\bm{\\tilde \\theta}}{d \\ell} = J \\bm{\\tilde \\theta} + [\\bm{h}_k, J \\bm{\\tilde \\theta}] + \\bm{f}_k + \\mathcal{O}({\\tilde \\theta}^{k+1}) .\n\\end{equation}\nSo, if we want to remove the term $\\bm{f}_k$, we need to solve the equation  $[\\bm{h}_k, J \\bm{\\tilde \\theta}] =  -\\bm{f}_k$ for $\\bm{h}_k$. It's important to note that whether this equation can be solved or not depends only on the linear part of the equation given by the matrix J. That is, within the space of transformations that we are considering, the linear part of the equation completely determines how much the equation can be simplified and how many terms can be removed. This is not true if there are zero eigenvalues and one then has to consider a broader space of transformations which we will consider later.\n\nTo see when the equation can be solved, we first note that the space of homogeneous polynomials is a vector space with a basis constructed in the obvious way $\\theta_1^{\\alpha_1}...\\theta_n^{\\alpha_n}$. Any term at order $k$ can be written as a sum of such terms for which $\\sum_i \\alpha_i = k$. The Lie bracket can be thought of as a linear operator on this space. To find the set of possible solutions is to find the range of this linear operator. Let us take the case where the linear part is diagonalizable and so just consists of the eigenvalues $\\lambda_i$. Let us say for simplicity that the $j$ component of the vector $\\bm{f_k}$ $(f_k)^j = c {\\tilde \\theta}_1^{\\alpha_1}...{\\tilde \\theta}_n^{\\alpha_n}$ for some set of \\{$\\alpha_i$\\}. Then, the $j$th component of the matrix equation reduces to\n\n\n\\begin{equation}\n \\lambda_j (h_k)^j - \\left(\\sum_i \\lambda_i \\alpha_i\\right) (h_k)^j = c \\theta_1^{\\alpha_1}...\\theta_n^{\\alpha_n} .\n\\end{equation}\nThis can be solved easily by choosing $(h_k)^j = a {\\tilde \\theta}_1^{\\alpha_1}...{\\tilde \\theta}_n^{\\alpha_n}$ and\n\\begin{equation}\n\\label{normalformequation}\n a = \\frac{c}{\\lambda_j - \\sum_i \\lambda_i \\alpha_i} .\n\\end{equation}\nWhen all nonlinear terms can be removed by such a coordinate transformation, then the usual case of power law scaling is obtained. The fixed point, in this case, is called hyperbolic. Alternatively, if we have a term with  $\\lambda_j = \\sum_i \\lambda_i \\alpha_i$ (a linear combination of other eigenvalues $\\lambda_i$ with positive integer coefficients $\\alpha_i$), this term is called a resonance and cannot be removed from the equation for $d \\theta_j\/d \\ell$. This contributes to the singularity at the fixed point which is no longer given by power law combinations. \n\n\\subsection{Bifurcations}\n\nNotice a special case of these equations when for some $k$, a particular $\\lambda_k = 0$. In this case, it is possible to get an infinite number of resonances because the equation $\\lambda_i = \\lambda_i + \\alpha_k \\lambda_k$ is also true for all $\\alpha_k$ and $\\lambda_i$. This case, when one of the eigenvalues goes to 0 depending on some parameter $\\epsilon$ is called a \\textit{bifurcation}. If all linear eigenvalues $\\lambda_i$ of the flows are distinct and\nnon-zero, which terms can be removed using polynomial coordinate changes\ndepends only on these $\\lambda_i$. As we saw, this approach can be formulated\nelegantly as a linear algebra problem of the Lie bracket on the space of\nhomogeneous polynomials.  For more\ngeneral cases---including bifurcations---`hypernormal form'~\\cite{murdock2004hypernormal, yu2007simplest,\nyu2002computation} theory develops a systematic but somewhat more\nbrute-force machinery to identify which terms can and cannot be removed\nperturbatively by polynomial changes of coordinates. Classic bifurcations include the pitchfork bifurcation, the transcritical bifurcation, the saddle node and the Hopf bifurcation~\\cite{GuckenheimerH13}. \n\nConfusingly, bifurcation theory separately has its own `normal form' of bifurcations. These normal forms are derived in a very different way using the implicit function theorem. The basic idea is to ask for the smallest number of terms in the equation which will preserve the qualitative behavior of the fixed points (e.g. exchange of stability of fixed points), and then map any other equation on to this simple equation using the implicit function theorem. This mapping allows for too broad a class of transformations to be useful for our purposes. An important feature of the flows that we want to preserve is their \\textit{analyticity}, we therefore only consider polynomial changes of coordinates.\n\nAn explicit example is given by the 4-d Ising model. It is known that the magnetization $M \\sim t^{1\/2} (\\log t)^{1\/3}$ with $\\log \\log$ corrections. The quartic coupling $u$ and the temperature $t$ have flow equations which traditional bifurcation theory would simplify to\n\\begin{align}\n \\frac{d u}{d \\ell} &= - \\bar{B} u^2 , \\\\\n \\frac{d t}{d \\ell} &= 2 t .\n\\end{align}\nCalculating the magnetization with this set of flow equations leads to the wrong power of logarithmic corrections. By allowing too broad a class of coordinate transformations, bifurcation theory hides the true singularity in the non-analytic coordinate change. We will show that normal form theory instead predicts  \n\\begin{align}\n \\frac{d u}{d \\ell} &= -\\bar{B} u^2 + \\bar{D} u^3 , \\\\\n \\frac{d t}{d \\ell} &= 2 t - \\bar{A} t u ,\n\\end{align}\nwhich does predict the correct behavior. We will present the explicit solution of this equation in Section~\\ref{sec:4dising}. Here, we just note that the traditional $\\log$ and $\\log \\log$ terms follow from the solution's  asymptotic behavior.  To get these equations, we will remove higher order terms in $u$ by using a coordinate change that is lower in order (broadening the formalism we considered in Section~\\ref{sec:normalform}). Using lower order terms to remove higher order terms is part of hypernormal form theory. For our purposes, the distinction is somewhat artificial and  here we simply use normal form theory to denote any procedure that uses only polynomial changes of coordinates to change terms in flow equations. \n\n\nIn Sections~\\ref{sec:4dising} and~\\ref{sec:randomfieldising}, we will explicitly work out the case of a single variable undergoing a bifurcation for the 4d Ising model and the 2d Random Field Ising model and show how there are only a finite number of terms which cannot be changed or removed. It is worth mentioning here that there can be cases in which two variables simultaneously have 0 eigenvalues. The XY model~\\cite{kosterlitz1974critical} offers an example where this happens. The dATG transition in 6 dimensions has two variables that simultaneously go through a transcritical bifurcation~\\cite{charbonneau2017nontrivial, Yaida18}. Polynomial changes of coordinates in both variables can be used here too, but because there are generically more terms at higher order than at lower order (there are many more ways to combine two variables into a sixth order polynomial than there are to combine them into a third order polynomial), we usually do not have enough freedom to remove all terms. Therefore, simultaneous bifurcations in more than one variable often have an infinite number of terms in their flow equations that cannot be removed.\n\n\n\n\n\n\\section{Earlier work} \\label{sec:earlierwork}\n\nThe approach we take is inspired by Wegner's early work~\\cite{Wegner72,\nWegner73}, subsequent developments by Aharony and\nFisher~\\cite{aharony1980university, aharony1983nonlinear}, and by studies of Barma and Fisher on\nlogarithmic corrections to scaling~\\cite{barma1984corrections,\nbarma1985two}. The approach of Salas and Sokal on the 2-d Potts model~\\cite{Salas97}, and  of Hasenbusch~\\cite{hasenbusch2008universal} et al. on the 2d Edwards-Anderson model \nis similar in spirit to ours.\n\nWegner~\\cite{Wegner72} first constructed nonlinear scaling fields which transform linearly under an arbitrary renormalization group. His construction is very similar to the coordinate changes we considered above for normal form theory. The one difference is that Wegner explicitly allows the new coordinates to depend on the coarse graining length $\\ell$. We will not allow this explicit dependence on $\\ell$ in our change of coordinates, as it doesn't seem to offer any advantage over regular normal form theory.\n\n\n\n\n\nEventually, the goal of using normal form theory to understand the differential equations that describe RG flow is to simplify and systematize scaling collapses. This requires a systematic way of dealing with corrections to scaling beyond the usual power laws. There are three different types of corrections to scaling that have appeared in the literature. These include logarithmic, singular and analytic corrections to scaling. Logarithms in the scaling behavior typically occur at an upper critical dimension or in the presence of a resonance. Wegner and Riedel~\\cite{Wegner73} considered the case of a zero eigenvalues which occurs at the upper critical dimension of Ising and tri-critical Ising models. They derived the form of the scaling in terms of logarithmic corrections to scaling. However, they used perturbation theory to ignore higher order terms in the flow equations rather than only keeping those terms which cannot be removed by an application of normal form theory. Here, we will solve the full flow equations and see that the logarithmic corrections to scaling are better incorporated as part of the true singularity using normal form theory.\n\nAnalytic corrections to scaling were explored by Aharony and Fisher~\\cite{aharony1983nonlinear} who gave a physical interpretation of the nonlinear scaling fields (see below Eq.(~\\ref{changecoord})) in terms of analytic corrections to scaling in the Ising model. Analytic corrections to scaling capture the difference between the physical variable $T$ and $H$ (that your thermometer or gaussmeter measures) and the symbols $\\tilde{t}$ and $\\tilde{h}$ in the theory of the magnet. The liquid gas transition is in the Ising universality class but a theory of the liquid gas transition has to include analytic corrections to scaling to match with the universal predictions of the Ising model. Moreover, such corrections are also needed to explain the non-universal behavior away from the fixed point. Analytic corrections to scaling will correspond to terms in the differential equations that can be removed by coordinate changes.\n\nThe singular corrections to scaling are also incorporated as part of the true singularity with the addition of irrelevant variables.  Finally, the ability to change the renormalization scheme leads to what are called redundant variables. In related work~\\cite{raju2018reexamining}, we argue that these variables can be seen as a gauge choice which contributes to the corrections of scaling. In forthcoming work~\\cite{Clement18}, we will explore the consequence of this distinction between gauge corrections and genuine singular corrections to scaling further.\n\n\n\n\n\n\n\n\nFinally Salas and Sokal, in the context of the 2-d Potts model, derive the normal form of the flow equations for a transcritical bifurcation. Similarly, Hasenbusch et al. derive the normal form for the 2-d Edwards-Anderson model which is also a transcritical bifurcation. Both of these do not solve the full flow equations but end up approximating the solution by logarithms. In the context of QCD, Sonoda derived the solution for the flow of a coupling which undergoes a transcritical bifurcation. \n\nDespite similar inclinations, none of these works make the complete\nconnection to normal form theory. One advantage of our approach is precisely that it brings together this disparate literature into a unified framework.  The\nanalysis presented here is general and applicable to all kinds of\nsituations, ranging from old problems like the nonequilibrium random field Ising\nmodel (NERFIM)~\\cite{PerkovicDS95}, to newer research problems like jamming~\\cite{goodrich2014jamming}. \n\n\n\\section{Universality Families} \\label{sec:universality}\n\n\\begin{table*}\n\\begin{center}\n\\resizebox{\\linewidth}{!}{\n  \\begin{tabular}{ | l | c | c| c| }\n    \\hline\n    Universality family & Systems & Normal form & Invariant scaling combinations \\\\ \\hline\n     \\makecell{\\raisebox{-0.4\\height}{\\includegraphics[width=.06\\textwidth]{hyperbolicbifn2.png}}}  \n     \n    & \\makecell{\\textbf{3-d Ising Model $(t)$} \\\\ 3-d RFIM $(w)$} & $dt\/dl = (1\/\\nu) t$ &\n\n\n\t\t $L t^{\\nu}$ \\\\ \\hline\n    \\makecell{\\raisebox{-0.4\\height}{\\includegraphics[width=.06\\textwidth]{pitchforkbifn2.png}}}\n    & \\makecell{\\textbf{2-d RFIM $(w)$} \\\\ 6-d Potts model $(q)$} & $dw\/dl =  w^3 + \\textcolor{blue}{B w^5}$ & \n\n\t\t$L e^{1\/(2 w^2)} \n\t\t(\\textcolor{darkGreen}{1\/w^2} \\textcolor{blue}{+ B})^{\\textcolor{blue}{-B\/2}}$\\\\ \\hline\n     \\makecell{\\raisebox{-0.4\\height}{\\includegraphics[width=.09\\textwidth]{transcriticalbifn2.png}}}\n    & \\makecell{ \\textbf{4-d Ising model $(u,t)$} \\\\ 2-d NERFIM $(-w,S)$ \\\\ 1-d Ising model $(-t,h)$ }  & \\makecell{$du\/dl = -u^2 + \\textcolor{darkGreen}{D u^3}$ \\\\ $dt\/dl = 2 t \\textcolor{darkGreen}{- A t u}$}  & \n\t\\makecell{$L e^{1\/u - D} \\textcolor{blue}{(1\/(D u) - 1)^{D}} = L y^{D}$  \\\\\n\t $t L^2\\textcolor{blue}{(W(y L^{1\/D})\/(1\/(D u) - 1))^{-A} }$}\n       \\\\ \\hline\n      \\makecell{Resonance} & \\textbf{2-d Ising model} \n    & \\makecell{$df\/dl = 2 f \\textcolor{darkGreen}{- t^2} \\textcolor{darkGreen}{- L^{-2}}$ \\\\ $dt\/dl = t\\textcolor{darkGreen}{+AL^{-1}}$} & \n\t$tL \\textcolor{darkGreen}{+ A\\log L}$ \\\\ \\hline\n    \\makecell{\\raisebox{-0.4\\height}{\\includegraphics[width=.1\\textwidth]{xybifn.png}}}\n    & \\textbf{2-d XY model} &\n    \\makecell{$dx\/dl=-y^2(1\\textcolor{darkGreen}{+xf(x^2)})$ \\\\\n  $dy\/dl = -xy$} &\n   \\makecell{$y^2 -2\\int_0^x s\/(1\\textcolor{darkGreen}{+sf(s^2)})\\,ds$\\\\ \n\t$=y^2-x^2\\textcolor{darkGreen}{-(2f(0)\/3)x^3+(f(0)^2\/2)x^4+\\mathcal\n  O(x^5)} $}\\\\ \\hline\n \n  \\end{tabular}}\n  \\end{center}\n  \\caption{Normal forms and universal invariant scaling combinations for \ntraditional and intrinsically nonlinear renormalization-group critical points.\nThe universal scaling of most critical points are power-law combinations\nof the control variables, derived from the linearized normal-form equations\nof hyperbolic RG fixed points. Many systems have well-studied\nlogarithmic corrections, exponentially diverging correlations, or other\nsingularities that we attribute to intrinsic nonlinearities in the \nRG flow equations. \nIn blue are new universal terms predicted by our analysis of the\ncorresponding dynamical system normal forms, which appear not to have\nbeen hitherto discussed in the literature.\nIn green are terms we explain which have been previously observed using other\nmethods~\\cite{Wegner72,Meinke05,sonoda,magradze,pelissetto2013renormalization}. The normal form equations are shown for the system in bold. Other systems in the same universality family have the same equations associated with different variables (shown in parenthesis). The invariant scaling combination for the transcritical family requires the Lambert $W$ function defined by the equation $W(x) \\exp(W(x)) = x$. Many of the results quoted in the table were obtained in disparate literatures (QCD, glasses, critical phenomena etc.) but are united in this common framework. Other families are possible, the flow equations for the replica symmetry breaking transition in disordered media have a simultaneous transcritical bifurcation and possibly also have a Hopf bifurcation~\\cite{Yaida18}} .\n\\label{systemstable}\n\\end{table*}  \n\n\n\nTraditionally, the RG contains the concept of a universality class. The universality class is essentially determined by the critical exponents which explain the scaling behavior of a model, i.e. by linearized RG eigenvalues. Normal form theory suggests another possible classification. Each fixed point can be classified by the bifurcation or resonance that it is at. The simplest case, which is also the traditional one, is the hyperbolic universality family. In the hyperbolic case, it is possible to remove all nonlinear terms in the flow equations by changes of coordinates. Hence, the RG can be written as a linear flow to all orders in perturbation theory. Different values for the linear eigenvalues correspond to different universality classes. While traditionally this is a statement about the linearization of the RG, here it is a statement about the only terms in the flow equations that are \\textit{universal} in the sense that they can not be removed by a coordinate change. \n\nThe need for this generalization becomes clear when we examine cases which are not traditional. In  Table~\\ref{systemstable} we present common universality families \nand well-studied statistical mechanics systems governed by each.\nThe pitchfork bifurcation shows up\nin the 2-d Random Field Ising model; it has a cubic term in the equations\nfor $w$, the ratio of the disorder to the coupling~\\cite{Bray85}. We\nhave derived that the correct equations require an additional $w^5$\nterm~\\cite{Lorien17}, which was not included in previous work. The 2-d\nIsing model has a well known logarithmic correction to the specific\nheat, which Wegner associated with a $t^2$ resonance term in the flow\nequation~\\cite{Wegner72}. The 1-d and 4-d Ising models have transcritical\nbifurcations. The 1-d Ising case is somewhat special and we will cover it later in Section~\\ref{sec:1dising}. These cover all the important bifurcations with one variable~\\footnote{We have not studied any example of a saddle node bifurcation which would require a transition from a critical point to no critical point.}.\n\n\nWhen more than one variable is undergoing a bifurcation, or if more than one variable has an inherently nonlinear flow, the analysis becomes considerably more complicated. This is evidenced in the the 2-d XY model at the Kosterlitz--Thouless (KT) transition~\\cite{kosterlitz1977d}.  It has been shown that the simplest normal form\nof its flow equations (in the inverse-temperature-like variable\n$x\\sim1\/T-1\/T_c$ and the fugacity $y$) has an infinite number of\nuniversal terms, which can be rearranged into an analytic\nfunction $f$~\\cite{pelissetto2013renormalization} (Table~\\ref{systemstable}).\nWe conjecture that the very similar transition observed in randomly grown\nnetworks~\\cite{callaway2001randomly,dorogovtsev2001anomalous} is not\nin the KT-universality class, but rather is in the same universality\nfamily. It is not to be expected that a percolation transition for\ninfinite-dimensional networks should flow to the same fixed point\nas a 2-d magnetic system, but it is entirely plausible that they share\nthe same normal form with a different universal function $f$.\n\nDifferent universality classes within the same universality family, such as those of the 4-d Ising model and the 2-d NERFIM have different power laws and scaling functions. However, as shown in Table~\\ref{systemstable}, because they both have a transcritical bifurcation the two classes have the same complicated invariant scaling combinations~\\footnote{A correlation length $y^{-D}$ from Table~\\ref{systemstable} defined in terms of the marginal variable in both cases diverges exponentially; in terms of the temperature the correlation length is a power law}. This hidden connection is made apparent in the shared normal form, where the quartic coupling and temperature ($u,T$) in the first class are associated with the (negative of) disorder strength and avalanche size ($-w,S$) in the second~\\footnote{The minus sign on $w$ and $t$ for the 1-d Ising\nand the NERFIM is because $w$ and $t$ are \nmarginally relevant whereas $u$ is marginally irrelevant for 4-d Ising.}.\n\n\n\nIndeed, the normal form not only unites these universality classes, but allows a more precise handling of their singularity. It is usually stated that the magnetization $M \\sim t^{1\/2} (\\log\nt)^{1\/3}$, the specific heat $C \\sim (\\log t)^{1\/3}$ and the\nsusceptibility $\\chi \\sim (\\log t)^{1\/3}\/t$ with $\\log \\log$ corrections~\\cite{Wegner73}. We show in the supplementary material that\nthe true singularity of the magnetization\nat the critical point is $M \\sim t^{1\/2} W(x t^{-27\/25})^{1\/3}$,\nwhere $W$ is the Lambert-W function defined by $W(z) e^{W(z)} = z$, and\n$x(u)$ is a complicated but explicit function of the irrelevant variable $u$.\n(The traditional log\nand log-log terms follow from the asymptotic behaviors of $W(x)$ at large\nand small $x$. The universal power $27\/25$ becomes manifest in the \ncomplete singularity, but is disguised into a constant factor up to\nleading logs.) We now show how to apply normal form theory to specific examples. \n\n\n\n\\section{Application to specific systems}\nIn the sections below, we derive in detail the scaling form for the entries shown in Table~\\ref{systemstable}. Our archetypal example is the 4-d Ising model. For this, we derive the scaling forms and use them to perform scaling collapses of numerical simulations. We then discuss the scaling of the Random Field Ising model, the XY model and the Ising model in dimensions $1$, $3$ and $2$. \n\n\\subsection{4-d Ising} \\label{sec:4dising}\n\nThe study of critical points using the renormalization group was turned into a dynamical system problem by Wilson~\\cite{wilson1974renormalization}. These RG calculations are done by first expressing the Ising model as a field theory with a quartic potential $u \\phi^4$. Then by coarse-graining in momentum space and rescaling, one obtains the flow equations\n\\begin{align}\n {d t}\/{d \\ell} =& 2 t -  \\bar{A} t u + \\bar{C} t u^2 + \\bar{E} t u^3 \\nonumber \\\\ &+ \\bar{G} t u^4 + \\bar{I} t u^5 + \\bar{K} t u^6 ... , \\\\\n {d u}\/{d \\ell} =& \\epsilon u - \\bar{B} u^2 + \\bar{D} u^3 + \\bar{F} u^4 \\nonumber \\\\ &+ \\bar{H} u^5 + \\bar{J} u^6 + \\bar{L} u^6  ... , \\\\\n {d f}\/{d \\ell} =& (4 - \\epsilon) f + ...,\n \\label{wilsoneqs}\n\\end{align}\nwhere $t$ is the temperature, $f$ is the free energy and $u$ is the leading irrelevant variable. This is the highest order to which the flow equations are known as of now. The coefficients take the values, $\\bar{A} = 1$, $\\bar{B} = 3$, $\\bar{C} = 5\/6$, $\\bar{D} = 17\/3$, $\\bar{E} = -7\/2$, $\\bar{F} \\approx 32.54$, $\\bar{G} \\approx 19.96$, $\\bar{H} \\approx -271.6$, $\\bar{I} \\approx -150.8$, $\\bar{J} \\approx 2849$, $\\bar{K} \\approx 1355$, $\\bar{L} \\approx -34776$~\\cite{kompaniets2016renormalization, chetyrkin1983five}. The flow equation for $u$ in this case takes the form of a transcritical bifurcation with parameter $\\epsilon = 4 - d$ tuning the exchange of stability between the Gaussian ($u = 0$) and Wilson-Fisher fixed point ($u \\neq 0$). \n\nConsider these equations for $\\epsilon = 0$ which is the point at which it undergoes a transcritical bifurcation. To derive the normal form, one considers a change of variables of the form \n\\begin{align}\n t &= \\tilde t + a_1 \\tilde t \\tilde u + a_2 \\tilde t \\tilde u^2 + ... , \\\\\n u &= \\tilde u + b_1 \\tilde u^2 + b_2 \\tilde u^3 + b_3 u^4 + ...\n\\end{align}\nThis gives the equations up to order $u^4$, \n\\begin{align}\n {d \\tilde t}\/{d \\ell} =& 2 \\tilde t -  \\bar{A} \\tilde t \\tilde u + (-\\bar{A} b_1 + a_1 \\bar{B} + \\bar{C}) t u^2 + ... , \\\\\n {d \\tilde u}\/{d \\ell} =& - \\bar{B} \\tilde{u}^2 + \\bar{D} \\tilde{u}^3 \\nonumber \\\\ &+ (-b_1^2 \\bar{B} + b_2 \\bar{B} + b_1 \\bar{D} + \\bar{E}) \\tilde{u}^4 + ...\n\\end{align}\nNote that any term of the form $u^m t$ in the equations for $d t\/d \\ell$ and any term of the form $u^m$ in the equations for $d u\/d \\ell$ is a resonance. Hence, the coefficients $\\bar{A}$, $\\bar{B}$ and $\\bar{D}$ remain unchanged with this change of variables. However, the coefficients $\\bar{C}$ and $\\bar{E}$ are changed (though the change is independent of $a_2$ and $b_3$ because they are resonances) and in particular, can be set to 0 by an appropriate choice of coefficients. \n\nThis creates a general procedure for reducing this flow to its simplest possible form. First, all terms that are not resonances are removed in the usual way by solving Eq.(~\\ref{normalformequation}). Then, we perturbatively remove most of the resonances using the following procedure. First consider the $u$ flow. Suppose the lowest order term in the flow after the $u^3$ term is $u^n$, i.e.\n\n\\begin{equation}\n \\frac{d u}{d \\ell} = - \\bar{B} u^2 + \\bar{D} u^3 + \\bar{N_n} u^n + \\mathcal{O}(u^{n+1})\n\\end{equation}\nwith $n > 3$. Consider a change of variables of the form $u = \\tilde{u} + b_{n-2} \\tilde{u}^{n-1}$. Then\n\\begin{widetext}\n\\begin{align}\n (1 + (n-1) b_{n-2} \\tilde{u}^{n-2}) \\frac{d \\tilde u}{d \\ell} &= - \\bar{B} (\\tilde{u} + b_{n-2} \\tilde{u}^{n-1})^2 + \\bar{D} ((\\tilde{u} + b_{n-2} \\tilde{u}^{n-1})^3 \\nonumber \\\\ &+ \\bar{N_n} (\\tilde{u} + b_{n-2} \\tilde{u}^{n-1})^n + \\mathcal{O}(\\tilde{u}^{n+1}) , \\\\\n \\frac{d \\tilde u}{d \\ell} &= \\frac{- \\bar{B} \\tilde{u}^2 + \\bar{D} \\tilde{u}^3 + \\bar{N_n} \\tilde{u}^n - 2 \\bar{B} b_{n-2} \\tilde{u}^n}{(1 + (n-1) b_{n-2} \\tilde{u}^{n-2}) } + \\mathcal{O}(\\tilde{u}^{n+1}) , \\\\\n &= - \\bar{B} \\tilde{u}^2 + \\bar{D} \\tilde{u}^3 + (\\bar{N_n} - 2 \\bar{B} b_{n-2} + (n-1) b_{n-2} \\bar{B}) \\tilde{u}^n + \\mathcal{O}(\\tilde{u}^{n+1}) .\n\\end{align}\n\\end{widetext}\nEvidently, the coefficient of the $\\tilde{u}^n$ term can be set to 0 with an appropriate choice  $b_{n-2} = N_n\/(\\bar{B} (3 - n))$.\n\nSo all terms of the form $u^n$ with $n > 3$ can be removed by a change of coordinates. Incidentally, this derivation also shows why it is not possible to remove the $u^3$ term. Now consider the $t$ equation with\n\\begin{equation}\n \\frac{dt}{d \\ell} = 2 t - \\bar{A} t \\tilde u + M_n t \\tilde u^{n-1} + \\mathcal{O} (t \\tilde u^{n}) .\n\\end{equation}\nWe consider a change of coordinates\n\\begin{equation}\n t = \\tilde t + a_{n-2} \\tilde t \\tilde{u}^{n-2} .\n\\end{equation}\nIt is then straightforward to show\n\\begin{equation}\n \\frac{d \\tilde t}{d \\ell} = 2 \\tilde t - \\bar{A} \\tilde t \\tilde u + (M_n + \\bar{B} (n-2) a_{n-2} + a_{n-2} \\bar{A}) \\tilde t \\tilde u^{n-1} + \\mathcal{O} (t \\tilde u^{n}) .\n\\end{equation}\nSo setting $a_{n-2} = - M_n\/(\\bar{B}(n-2) + A)$ sets the coefficient of the $t u^{n-1}$ term with $n > 2$ to 0. \n\nAny term which is not of this form can be removed in the usual way by solving Eq.~\\ref{normalformequation}. Finally, we have another degree of freedom that we have used. We can rescale $u$ and $t$ to set some of the nonlinear coefficients to 1. This reflects the fact that the original coefficients $\\bar{A}$, $\\bar{D}$ depend on an arbitrary scale of $u$ and $t$ that we have chosen. By choosing the scale so $\\bar{B} = 1$ and the coefficient of the resonance is -1, defines $D = \\bar{D}\/\\bar{B}^2$ and $A = \\bar{A}\/\\bar{B}$. Hence, by considering all such polynomial change of coordinates, we can reduce this set of equations to their normal form \n\n\n\n\n\n\n\n\\begin{align}\n {d \\tilde t}\/{d \\ell} &= 2 \\tilde{t} - A \\tilde{u} \\tilde{t} , \\label{isinghnf} \\\\ \n {d \\tilde u}\/{d \\ell} &= - \\tilde{u}^2 + D \\tilde{u}^3, \\label{isinghnf2} \\\\\n {d \\tilde f}\/{d \\ell} &= 4 \\tilde f - {\\tilde t}^2 \\label{isinghnf3} .\n\\end{align} \n\n\nThe resultant equations then have 2 parameters $A$ and $D$ which\nare \\textit{universal}, in a way that is similar to the eigenvalues of the RG flows as in Table~\\ref{systemstable}. The normal form variables $\\tilde t$, $\\tilde u$, $\\tilde f$ are equal to the physical variables $t$, $u$ and $f$ to linear order (up to a rescaling). Corrections to these are analytic corrections to scaling. Hence, we will henceforth simply refer to the normal form variables as $t$, $u$ and $f$. It is important to note that we are making a particular choice for the analytic corrections to scaling by setting them equal to zero. It is possible to make a different choice for the higher order coefficients. In particular, the equation for $d u\/d \\ell$ goes to $\\infty$ at finite $\\ell$ if $u$ starts at a large enough value. Hence, it may be more useful to make a different choice for the higher order coefficients. All of these choices will agree close to the critical point but will have different behavior away form the critical point. Later, we will consider a different choice for the higher order terms.\n\nThe 4-d Ising model has both a bifurcation and a resonance. The $u^2$, $u^3$ and $A u t$ terms come from the bifurcation and cannot be removed\nby an analytic change of coordinates. The $t^2$ term is a consequence of\nan integer resonance between the temperature and free energy eigenvalue,\n$\\lambda_t = 1\/\\nu = 2, \\lambda_f = d = 4$. \n\n\n\\begin{figure}\n \\includegraphics[width=.85\\linewidth]{magcollapsen2.pdf}\n\\caption{Scaling collapses for the magnetization and susceptibility using the scaling form given by the normal form Eqs.(~\\ref{isinghnf}~--~\\ref{isinghnf3}). Simulations are done on a 4-d lattice using a Wolff algorithm for lattice sizes ranging from $L = 4$ to $L = 32$. Here $M_\\mathrm{scaling}$ $=$ $L ((W(y L^{1\/D})+1))^{1\/4}$ and $t_{\\mathrm{scaling}}$ $=$ $L^{-2} (W(y L^{1\/D})\/(1\/D u_0 - 1))^{1\/3} ((W(y L^{1\/D})+1))^{-1\/2}$. We find $u_0 = 0.4\\pm0.1$ for the 4-d nearest-neighbor hypercubic-lattice. An estimate of the error is given by estimating $u_0$ with a different choice of normal form which gives $u_0 = 0.5$.  }\n\\label{collapses1}\n \\end{figure}\n\n\\begin{figure}\n \\includegraphics[width=.85\\linewidth]{susccollapsen2.pdf}\n \\caption{Scaling collapse for the susceptibility using the scaling form given by the normal form Eqs.(~\\ref{isinghnf}~--~\\ref{isinghnf3}). Simulations are done on a 4-d lattice using a Wolff algorithm for lattice sizes ranging from $L = 4$ to $L = 32$. Here $\\chi_\\mathrm{scaling}$ $=$ $L^2 ((W(y L^{1\/D}) + 1))^{1\/2}$ and $t_{\\mathrm{scaling}}$ $=$ $L^{-2} (W(y L^{1\/D})\/(1\/D u_0 - 1))^{1\/3} ((W(y L^{1\/D})+1))^{-1\/2}$. We find $u_0 = 0.4$ for the 4-d nearest-neighbor hypercubic-lattice.}\n\\label{collapses2}\n \\end{figure}\n\n\n\n\n\nBefore examining the full solution\nof Eqs.~(\\ref{isinghnf}~--~\\ref{isinghnf3}), we will first study the effect\nof each part of the RG flows. First, considering only the linear terms and\ncoarse-graining until $t(\\ell^*) = 1$, the free energy is given by \n$f \\sim t^2$. This is the mean-field result and also the traditional scaling\nform that RG results take in the absence of nonlinear terms in the flow\nequations. Second, we include the resonance between the temperature and\nfree energy eigenvalue, which leads to an irremovable $t^2$ term in\nthe flow equation for the free energy. This term cannot be removed by analytic\ncoordinate changes, and yields a $\\log$ correction to the specific heat.\nThird, the irrelevant variable $u$ undergoes a transcritical\nbifurcation. Results in the hyper-normal form theory literature, as well\nas some articles in the high-energy theory\nliterature~\\cite{sonoda,magradze} recognize that the simplest form that \nthe equation can be brought into is Eq.~\\ref{isinghnf2}.  The solutions\nof Eqs.(~\\ref{isinghnf}~--~\\ref{isinghnf2}) are $u(\\ell) = 1 \/(D (1 + W(y e^{\\ell\/D})))$ and\n$t(\\ell) = t_0 e^{2 \\ell} (W(y e^{ \\ell\/D})\/(1\/(D u_0) - 1))^{-A}$  where $y[u_0]$ is again\na messy but explicit function: $y = (1\/(D u_0)-1) \\exp(1\/(D u_0)-1)$. We show how to derive this in the supplementary material. \nThe traditional log and log-log corrections are\nderived by expanding the $W$ function for large $\\ell$.\n\n\nLet us use this to derive the finite-size scaling form of the free energy. Early finite-size scaling work~\\cite{aktekin2001finite, lai1990finite, montvay1987numerical} attempted scaling collapses with logs; recent work does not attempt collapses at all~\\cite{lundow2009critical}. Finite-size scaling requires an equation for the magnetic field, $h$, given by $dh\/d\\ell = 3 h$. Explicit calculations show that the coefficient of the $h u$ term is zero (see below). The free energy is then a function of three scaling variables, $u(\\ell)$, $t(\\ell)$ and $h(\\ell)$. It is given by \n\\begin{align}\n    f(t_0, u_0) &= e^{-4 \\ell} f(t(\\ell), u(\\ell), h(\\ell))\\nonumber\\\\\n    &-  W(y e^{\\ell\/D})^{-A} \\left( \\frac{W(y e^{\\ell\/D})^{-A}}{1 - A}-\\frac{1}{A}\\right).\n\\end{align}\n\n To get a finite size scaling form, we coarse grain until $\\ell = \\log L$, the system size. Note that $u(L)$ cannot just be ignored because it is a dangerous irrelevant variable. However we can account for it by taking the combination $t(L)\/(u(L))^{1\/2}$ and $h(L)\/(u(L))^{1\/4}$ as our scaling variables~\\cite{binder1985finite}. The scaling form of the free energy then depends on $u_0$ which we do not have a way to change or set in the simulation. Instead, we treat $u_0$ as a fit parameter in the scaling form of the susceptibility: \n\n\\begin{equation}\n \\chi = L^2 \\left(W(y L^{\\frac{1}{D}}) + 1\\right)^{\\frac{1}{2}} \\Phi \\left( t_0 L^2 \\left(\\frac{W(y L^{1\/D})}{1\/(D u_0) - 1}\\right)^{-A}\\right).\n\\end{equation}\n\n\\noindent At the critical point $t = 0$, the function $\\Phi$ must be analytic for\nfinite $L$ (since non-analyticity requires an infinite system size).\n$\\Phi(0)$ is therefore a constant independent of $L$ and $u_0$ at $t =\n0$. Using this, $u_0$ may be estimated from $\\chi$ at different values\nof $L$ by fitting to its predicted dependence \n$\\chi \\propto L^2 (W(y[u_0] L^{1\/D}) + 1)^{1\/2}$ where \n$y[u_0]$ is defined above.\n\n\nFigures~\\ref{collapses1}--\\ref{collapses2} shows the scaling collapse of the magnetization\nand susceptibility. The magnetization is collapsed using the best-fit value of $u_0=0.4$. Though our collapses are not significantly\nbetter than the traditional logarithmic forms, the\ncorrect form of the singularity will be more apparent at larger values\nof $u_0$. This is because the $\\log \\log$ term which is the second term in the asymptotic expansion of the $W$ function is very small compared to the $\\log$ except at large $u_0$ and small $L$. Changing the value of $u_0$ will require a model different from the nearest neighbor square lattice Ising model.\n\nSo far, we have been considering the effects of changing coordinates in the control variables on the predictions of the theory. Wegner~\\cite{wegner1974some} had also considered changing coordinates in the degrees of freedom of the theory. These changes lead to `redundant' variables, the corrections from which can be removed by coordinate changes. We discuss them in separate work. Here we merely note that they can be used to explain some features of the scaling, like the fact that the coefficient of the $h u$ term is zero.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Choice of normal form} \n\nThere are certain choices we have made in our application of normal form theory. One is to keep the flow parameter $\\ell$ unchanged. Some of the dynamical systems literature considers changing $\\ell$ to depend on other parameters. This would be unusual since the coarse graining length would depend on the physical parameters but does not seem to be disallowed.\nWe show in the supplementary material that this does not change the predictions for the 4-d Ising model.\n\nNormal form theory makes a particular choice for what to do with the coefficients that can be changed by coordinate changes: it sets them equal to zero. In general, however, it is not clear that the best choice to make is to set them equal to zero. Consider the equation\n\\begin{equation}\n \\frac{d u}{d \\ell} = - u^2 + D u^3 \n \\label{ueqchange}\n\\end{equation}\nwhich, as we saw, has the solution $u(\\ell) = 1 \/(D (1 + W(y e^{\\ell\/D})))$. Here, $y = (1\/(D u_0)-1) \\exp(1\/(D u_0)-1)$. Note that $u_0 > 0$ as a requirement for the stability of the free energy. If $u_0 < 1\/D$, then $y > 0$, and if $u_0 \\leq 1\/D$, $y \\leq 0$. Hence, the domain of attraction of the fixed point at $u_0 = 0$ has a length $1\/D$. If we have a system where $u_0 > 1\/D$, then this will lead to $u(\\ell) \\rightarrow \\infty$ in a finite coarse graining length. This is reflected in the branch cut of the $W$ function at $-1\/e$. In the context of high energy physics, some have tried to find deep meaning in this pole~\\cite{magradze}. \n\nHowever, for scaling purposes, we generally prefer a choice of coordinates for which there is no such unphysical behavior. One natural choice is to instead use the equations \n\\begin{equation}\n \\frac{d u}{d \\ell} = - \\frac{u^2}{1 + D u} . \n \\label{newnormalform}\n\\end{equation}\nFor small $u$, this has the same behavior as Eq.~\\ref{ueqchange}. However, the behavior at large $u$ is now well behaved. The solution of this equation is \n\\begin{equation}\n u(\\ell) = \\frac{1}{D W(e^{\\ell\/D} 1\/(D u_0) e^{1\/(D u_0)})} ,\n\\end{equation}\nwhich is in fact somewhat simpler that the solution to Eq.(~\\ref{ueqchange}). Scaling collapses with this choice of normal form for the susceptibility are shown in the case of the Ising model in Figure~\\ref{collapses2ndmethod}. Better numerics are needed to tell if this choice of normal form is really useful. It turns out that this form has been implicitly used before in the Random Field Ising model as we show explicitly in the next section.\n\n\\begin{figure}\n \\includegraphics[width=.85\\linewidth]{suscepcollapse2ndmethod.pdf}\n \\caption{Scaling collapse for the susceptibility using the scaling form given by a different choice of normal form derived from Eq.(~\\ref{newnormalform}). Simulations are done on a 4-d lattice using a Wolff algorithm for lattice sizes ranging from $L = 4$ to $L = 32$. Here $\\chi_\\mathrm{scaling}$ $=$ $L^2 ((W(y L^{1\/D}))^{1\/2}$ and $t_{\\mathrm{scaling}}$ $=$ $L^{-2} (W(y L^{1\/D}))^{-1\/6}\/(1\/D u_0))^{1\/3} \\exp(1\/(3 W(y L^{1\/D}))) $ with $y = 1\/(D u_0) e^{1\/(D u_0)}$. We find $u_0 = 0.5$ for the 4-d nearest-neighbor hypercubic-lattice using this method.}\n\\label{collapses2ndmethod}\n \\end{figure}\n\n\n\\subsection{Random Field Ising model} \\label{sec:randomfieldising}\n\nFinding critical exponents for the random field Ising model has been a longstanding challenge in physics. Some initial results used supersymmetry to prove an equivalence of the Random Field Ising model in dimensions $d+2$ with the Ising model in dimensions $d$~\\cite{parisi1979random, de2006random}. It was later shown that the lower critical dimension of the Random Field Ising model is not $3$ (as would be expected from such a correspondence) but rather $2$~\\cite{imbrie1984lower}. The upper critical dimension is 6. Here, we will look at the scaling behavior of the Random Field Ising model at its \\textit{lower} critical dimension, $d = 2$. \n\nConsider a spin system with a random field.  \n\\begin{equation}\n \\mathcal{H} = -\\sum_{\\langle i j \\rangle} J s_i s_j + \\sum_i h_i s_i ,\n\\end{equation}\nwhere, $J$ is the nearest neighbor coupling and $h_i$ is a random field chosen from a Gaussian distribution with width $r$. A phenomenological theory for the RG was formulated by Bray and Moore~\\cite{Bray85}. It turns out to be useful to define a quantity $w = r\/J$. Then, using heuristic arguments on the stability of domain walls, they derive\n\\begin{equation}\n \\frac{d w}{d \\ell} = -\\epsilon\/2 w + A w^3 ,\n \\label{rfimwrongeq}\n\\end{equation}\nwith $\\epsilon = d-2$, and $d$ is the dimension. Note that the flow equations have a symmetry under $w \\rightarrow -w$ because the physics is invariant under $r \\rightarrow -r$ about the critical point at $r = 0$. This is an example of a pitchfork bifurcation. Bray and Moore argue for this scaling form by looking at the scaling of $r$ and $J$ separately. The scaling of $J$ is given by looking at the energy of a domain wall of size $b^d$. The energy of the domain wall is proportional to $b^{d-1}$. By considering the cost of roughening the domain wall because of the presence of random fields, which goes as $r^2$, they are able to derive the next term in the equation for $J$ which is now\n\\begin{equation}\n \\frac{d J}{d \\ell} = (d - 1) J + D w^2 J + \\mathcal{O}(w^4) .\n \\label{nnequation}\n\\end{equation}\nFor the random field $r$, the energy of a region of size $b^d$ is proportional to $b^{d\/2}$. Any corrections requires forming a domain of `wrong spins' which, being akin to a barrier crossing problem, is exponentially suppressed. Hence the equation for $r$ is given by \n\\begin{equation}\n\\frac{d r}{d \\ell} = \\frac{d}{2} r \n\\label{rnequation}\n\\end{equation}\nwith exponentially small corrections. These two equations together can be used to derive Eq.(~\\ref{rfimwrongeq}). Bray and Moore conjecture that Eq.(~\\ref{rnequation}( holds exactly to all orders in $w$ (up to exponential corrections). However, it is possible for Eq.~(\\ref{nnequation}) to have higher order terms in $w$ and thus Eq.(~\\ref{rfimwrongeq}) is only correct to order $w^5$. Integrating Eq.(~\\ref{rfimwrongeq}), we get $\\ell \\sim -1\/(2 A w^2) + 1\/(2 A w_0^2)$. This implies that the correlation length is \n\\begin{equation}\n \\xi \\sim e^{1\/(2 A w_0^2)} .\n\\end{equation}\nFor finite size systems, the system size $L \\sim \\exp(1\/(2 A w_0^2)$. Meinke and Middleton~\\cite{Meinke05} showed that their finite size data was much better fit by a function of the form $w_0^{-2 y} \\exp(C\/w_0^2)$ where $C$ is a constant they fit to ($C=\\Delta_0$ in their notation) and $y$ = 1.07. We will show that this prediction is consistent with the results of normal form theory. \n\nAs we have already argued, there is no reason Eq.(~\\ref{rfimwrongeq}) is true to all orders in $w$. Indeed the, normal form prediction for the flow equations can be derived in a straightforward way. Consider adding a term $A_n w^n$ to Eq.(~\\ref{rfimwrongeq}) at $\\epsilon = 0$. This is a resonance and cannot be removed usually under normal form theory.  Suppose we make a change of coordinates $w = \\tilde{w} + a_n \\tilde{w}^{n -2}$. Then, to order $\\mathcal{O}(\\tilde{w}^n)$, we get\n\\begin{equation}\n \\frac{d \\tilde w}{d \\ell} = A \\tilde{w}^3 + (3 A a_n - A a_n (n -2) + A_n) \\tilde{w}^n + \\mathcal{O}(\\tilde{w}^{n+1}) .\n\\end{equation}\n\nWe can set the coefficient of $\\tilde{w}^n = 0$ if we use $a_n = A_n\/((n-5) A)$. This procedure fails for $n = 5$ but works for all $n > 5$. \\footnote{We note that we are assuming here that the coordinate transformations respect the symmetry of the problem $w \\rightarrow -w$. Otherwise, it is possible to remove the $\\tilde{w}^5$ term at the cost of introducing a $\\tilde{w}^4$ term.} Hence, the normal form of the equilibrium RFIM is given by \n\\begin{equation}\n \\frac{d \\tilde w}{d \\ell} = \\tilde{w}^3 - D \\tilde{w}^5 .\n\\end{equation}\nAs before, we have used the freedom to rescale $w$ to set the coefficient of the $\\tilde{w}^3$ term to 1. \n\nThe solution of this equation gives us an expression for the correlation length\n\\begin{equation}\n \\xi \\sim (1\/w^2 - D)^{D\/2} e^{1\/w^2} .\n\\end{equation}\nThis scaling form could explain the data in Meinke and Middleton with $D$ as a fit parameter. Notice that for this to work, $D$ must be positive. However, this solution has the strange property that the correlation length goes to $0$ for $w^2 = 1\/D$. If $w^2 > 1\/D$, $w^2(l)$ decreases till it reaches $1\/D$. If $w^2 < 1\/D$, it increases till it reaches $1\/D$. As in the 4-d Ising model, it may be more useful to consider instead the flow equation\n\\begin{equation}\n\\frac{d \\tilde w}{d \\ell} = \\frac{\\tilde{w}^3}{1 + D \\tilde{w}^2} .\n\\end{equation}\nThis gives the scaling form \n\\begin{equation}\n \\xi \\sim e^{1\/(2 w^2)} (w^2)^{-D\/2} .\n\\end{equation}\nThis is exactly consistent with the scaling form Meinke and Middleton use to collapse their data. Their data would predict the universal value for $D = 2.14$~\\footnote{Note that they also have a fit parameter which sets the scale of the exponential. However, this parameter is not universal since it depends on the scale of $w$ unlike $D$}. Any system in the same universality class should see a value of $D$ consistent with this value. However, different values of $D$ would correspond to different universality families within the same class. We now turn to discussing the XY model before returning to the Ising model in dimensions 3, 2, and 1.\n\n\\subsection{2-D XY Model}\n\nThe 2-d XY model is a remarkable system for several reasons. It was the site\nof recently celebrated insight into the connection between ground-state\ntopology and phase transitions \\cite{kosterlitz2017nobel}. Thermodynamic\nquantities have essential singularities at its phase transition, not ordinary\npower laws, and their derivatives remain continuous to arbitrary order, making\nits phase transition infinite order \\cite{berezinskii1970destroying,\nkosterlitz1973ordering, kosterlitz1974critical}.  This is related the fact\nthat its RG flow equations are inherently nonlinear: they have no relevant and\ntwo marginal state variables and the procedure laid out by\n\\eqref{normalformequation} for removing higher order terms from the flow\nequations contributes nothing to their simplification.\n\nThe XY model is usually posed as ferromagnetically interacting planar spins.\nIts partition function is exactly equivalent to the product of a trivial\nGaussian model---corresponding to spin wave degrees of freedom---with a\nneutral Coulomb gas---corresponding to the interaction of spin vortices\n\\cite{knops1977exact}. The latter component contains the interesting critical\nbehavior, which is characterized by these vortices going through an unbinding\ntransition. The flow equations for a Coulomb gas in dimension $d$ are given by\n\\begin{align}\n  dK\/dl&=-K(\\tfrac14Ky^2+d - 2)+\\cdots\\\\\n  dy\/dl&=-y(K-d)+\\cdots,\n\\end{align}\nwhere $K\\sim T^{-1}$ and $y$ is the fugacity of the vortices\n\\cite{kosterlitz1977d}, which for an XY model is a function of temperature and\ncannot be tuned independently but is a free parameter in other equivalent\nmodels, e.g., the Coulomb gas itself. For $d>2$ there is no phase transition\nin this system, and for $d<2$ a nontrivial unstable fixed point appears and\nthere is a phase transition in the hyperbolic universality family. It is worth\nnoting that these flow equations do not describe the XY model for any\ndimension besides $d=2$; 2 is the \\emph{upper} critical dimension of the\nCoulomb gas and these flow equations, while it is the \\emph{lower} critical\ndimension for the XY model. At $d=2$ the flow equations undergo a novel\nbifurcation: there appears a line of stable fixed points at $y=0$ for all\n$K>2$, terminating at $K=2$. This termination is the\nBerezinskii--Kosterlitz--Thouless (BKT) critical point. The flow equation near this point with $x=K-2$ is\n\\begin{align}\n  \\label{eq:kt-truncated:1}\n  dx\/dl&=-y^2+\\cdots\\\\\n  \\label{eq:kt-truncated:2}\n  dy\/dl&=-xy+\\cdots.\n\\end{align}\nThese flow equations are zero to linear order and have zero Jacobian at the fixed point.\n\nIn principle arbitrary higher-order terms in these equations exist, but there\nare several constraints on their form. There is a symmetry $y\\to-y$ in the\npartition function arising from the neutrality condition---$y$ enters the\npartition function in factors of $y^{-\\sum_rn_r^2}$ for $\\sum_rn_r=0$---which\nimplies that $dx\/dl$ be even in $y$ and $dy\/dl$ be odd. In addition, when the\nfugacity is zero the model is trivial and $x$ cannot flow, meaning that\n$dx\/dl$ must only have terms proportional to $y$.  Having applied these\nconstraints, the simplest normal form has been proven by induction in\npolynomial order (Appendix A of \\cite{pelissetto2013renormalization}) to take\nthe form\n\\begin{align}\n  \\label{eq:kt-normalform:1}\n  d\\tilde x\/dl&=-\\tilde y^2-b_0\\tilde x\\tilde y^2-b_1\\tilde x^3\\tilde y^2+\\cdots\\\\\n  \\label{eq:kt-normalform:2}\n       &=-\\tilde y^2\\big(1+\\tilde xf(\\tilde x^2)\\big)\\\\\n  \\label{eq:kt-normalform:3}\n  d\\tilde y\/dl&=-\\tilde x\\tilde y.\n\\end{align}\nFor the BKT point in the sine--Gordon model, which is thought to display to\nthe same universality as the XY model, it is known that $b_0=3\/2$\n\\cite{balog2000intrinsic, pelissetto2013renormalization}. An infinite number\nof coefficients remain, represented here in the form of the Taylor\ncoefficients of an analytic function $f(x^2)$. These numbers are universal in the\nsense that there is no redefinition of $\\tilde x$ and $\\tilde y$ such that the\nflow equations take on the form above and contain different coefficient\nvalues.  Unlike those in the previous sections, this bifurcation does not have\na named classification as far as the authors know. \n\nA constant of the RG flow can be found by integrating these forms. First,\ndividing the equations \\eqref{eq:kt-normalform:3} by\n\\eqref{eq:kt-normalform:2} (and dropping the tildes), we find\n\\begin{equation}\n  \\frac{dy}{dl}\\bigg\/\\frac{dx}{dl}=\\frac x{y(1+xf(x^2))},\n\\end{equation}\nwhich separates into\n\\begin{equation}\n  y\\frac{dy}{dl}=\\frac x{1+xf(x^2)}\\frac{dx}{dl}.\n\\end{equation}\nIntegrating both sides and choosing $l_0$ such that $x(l_0)=0$, we find\n\\begin{align}\n  \\frac12&\\big(y(l)^2-y(l_0)^2\\big)=\\int_{y(l_0)}^{y(l)}y\\,dy=\\int_{l_0}^ly\\frac{dy}{dl}dl\\\\\n  &=\\int_{l_0}^l\\frac x{1+xf(x^2)}\\frac{dx}{dl}dl\n  =\\int_0^{x(l)}\\frac x{1+xf(x^2)}dx.\n\\end{align}\nIt follows that \n\\begin{align}\n  y(l_0)^2&=y(l)^2-2\\int_0^{x(l)}\\frac x{1+xf(x^2)}dx\\\\\n          &=y(l)^2-x(l)^2+\\frac23b_0x(l)^3-\\frac12b_0x(l)^4\\\\\n          &\\hspace{4em}+\\frac25(b_0^3+b_1)x(l)^5+O(x(l)^6)\n\\end{align}\nis a constant of the flow. The expansion of the integral can be taken to\narbitrary order with ordinary computer algebra software. The finite-size\nbehavior of the flow is rather complicated and doesn't yield closed form\nresults, details can be found in \\cite{pelissetto2013renormalization}.\n\nThe XY model and other infinite-order transitions are usually characterized by\nthe anomalous exponent $\\sigma$ parametrizing the essential singularity in the\ncorrelation length,\n\\begin{equation}\n  \\xi\\sim e^{at^{-\\sigma}},\n  \\label{eq:kt-corr}\n\\end{equation}\nwhich for the BKT transition is $\\sigma=1\/2$ \\cite{kosterlitz1974critical}.\nConformal field theory predicts the presence of infinitely many models with\nthis anomalous exponent \\cite{ginsparg1988applied}.\nThe value of $\\sigma$ been shown to be fixed by the quadratic-order truncation\nof the system's flow equation, independent of any higher-order terms\n\\cite{itoi1999renormalization}. There are six possible quadratic-order terms in flow equations with two variables. Of these, two can be removed by linear transformations of the two variables. Two more can be set to 1 by rescaling the variables. Hence, there are two parameters at quadratic order which determine the universality family that the system belongs to, and infinite number of subsequent terms which determine the universality class. Giving a full classification of possibilities is beyond the scope of this paper but we give some examples below.\n\n\nFor instance, when the requirement of symmetry under $y\\to-y$ is lifted, the\nflow equations can no longer be brought to the form Eqs.~\\eqref{eq:kt-normalform:2}\nand \\eqref{eq:kt-normalform:3}; though the simplest form that results isn't\nyet known, it is certainly different from the symmetric case, a fact that can\nbe found by simply trying to eliminate the nonsymmetric cubic terms. In such a\ncase the codimension of the bifurcation would likely be different, corresponding\nto the fact that no \\emph{a priori} reason exists for the vanishing of\nthe term linear in $y$ in $dx\/dl$. Such linear terms would change the universality family. Among the infinite collection of BKT-like\nconformal theories---including the many physical models identified as having a\nBKT-like transition because their behavior resembles Eq.~\\eqref{eq:kt-corr}, like\npercolation in grown networks \\cite{callaway2001randomly,\ndorogovtsev2001anomalous}. These may already be examples of models with\n$\\sigma=1\/2$ but belonging to another universality class. It could\nalso be the case that all BKT-like transitions are in fact members of the same\nuniversality family and class.\n\n\nOther universality classes and families definitely do exist, characterized by novel values\nfor $\\sigma$. The level-1 $\\mathrm{SU}(N)$ Wess--Zumino--Witten model has been\nfound to be characterized by $\\sigma=N\/(N+2)$ \\cite{itoi1997extended}.\nDislocated-mediated melting alone has produced a melange of anomalous\nexponents, with $\\sigma=1\/2$, $\\sigma=2\/5$, and $\\sigma=0.369\\,63\\ldots$\ndepending on precise specification of the model and the lattice geometry\n\\cite{nelson1979dislocation, young1979melting}. Topological transitions in\nsystems whose vortices are non-Abelian produce several series of $\\sigma$\nvalues dependent on particular symmetry \\cite{bulgadaev1999berezinskii}. Each\nvalue of $\\sigma$ indicates either a different universality family or merely a different class within the same family depending on how it affects the terms at quadratic order. A classification of possible\nbifurcations and corresponding simplest normal forms is in order for flow\nequations whose leading order is quadratic, and whose expansions are\nconstrained or not by various symmetries. This would be the first step in developing techniques for\ndistinguishing between universality classes and families of this type using\nexperimental or simulation data. \n\n\n\\subsection{3-d Ising model} \\label{sec:3dising}\n\nThere is a sense in which the Ising model is simplest in 3 dimensions because it is part of the hyperbolic universality family. It is also the first natural application of the $\\epsilon$ expansion. The transcritical bifurcation at 4 dimensions leads to an exchange of stabilities of the Gaussian fixed point and the Wilson-Fisher fixed point at a non-zero value of $u = u^*$. About this Wilson-Fisher fixed point, the flow equations of the 3-d Ising model are in the hyperbolic universality class with linear coefficients which define the Ising universality class. \n\n\\begin{figure}\n \\includegraphics[width=.85\\linewidth]{fixedpointsdimension.png}\n\\caption{Fixed points as a function of dimension in the Ising model. There is a transcritical bifurcation in both 4 and 1 dimensions, leading to $W$ functions and exponential correlation lengths respectively. The fixed point in 3 d is hyperbolic and the flow can be linearized. The fixed point in 2 d has a resonance which leads to a logarithmic specific heat. The challenge is to find a scaling form which interpolates between dimensions giving the correct behavior in all of these dimensions.}\n \\end{figure}\n\nHowever, another approach is to consider the scaling form as a function of the dimension $\\epsilon$ in a way that is well defined even at $\\epsilon = 0$. Doing this, naturally requires us to keep nonlinear terms in the equation because we already know that the 4-d Ising model has nonlinear terms in its flow equations.\n\nWe want to write the flow equations about the 3-d fixed point but keep the nonlinear terms required for the scaling form to have the correct limiting behavior in 2-d and in 4-d. We can write the normal form of the flow equations as \n \\begin{align}\n {d \\tilde t}\/{d \\ell} &= \\lambda_t \\tilde{t} - A \\tilde{u} \\tilde{t} ,  \\\\ \n {d \\tilde u}\/{d \\ell} &= \\lambda_u \\tilde{u}- \\tilde{u}^2 + D \\tilde{u}^3,  \\\\\n {d \\tilde f}\/{d \\ell} &= d \\tilde f - {\\tilde t}^2  , \\\\\n {d \\tilde h}\/{d \\ell} &= \\lambda_h h .\n\\end{align}\n\nWe have included the nonlinear terms in $u$ required for the correct scaling behavior and the resonance between the temperature and the free energy. As usual, we switch notation to $t$, $h$ and $u$ with the understanding that they are different from the normal form variables by analytic corrections. Let us look at the scaling variable formed with $t$ and $u$ which can be obtained by solving\n\n\\begin{equation}\n \\frac{d \\tilde t}{d \\tilde u} = (\\lambda_t \\tilde{t} - A \\tilde{u} \\tilde{t})\/(\\lambda_u \\tilde{u} - \\tilde{u}^2 + D \\tilde{u}^3) .\n \\label{scalingequation3d}\n\\end{equation}\nThe solution of this equation gives the scaling variable\n\\begin{equation}\n t (u)^{-\\frac{\\lambda_t}{D u_1 u_2}} (u - u_1)^{-\\frac{(\\lambda_t - A u_1)}{D u_1 (u_1 - u_2)}} (u - u_2)^{\\frac{\\lambda_u - A u_2}{D (u_2 - u_1) u_2}} = \\textrm{const} \n \\label{scaling3d}\n\\end{equation}\nwhere $u_1$ and $u_2$ are the two non-zero roots of the denominator on the r.h.s of Eq.\\ref{scalingequation3d} which to first order in $\\lambda_u$ are given by $u_1 = \\lambda_u$ and $u_2 = 1\/D - \\lambda_u$. The form of the scaling variable is interesting, it is essentially given by a product of the linearized scaling variables at the three fixed points that the equation has. Taking the limit $\\epsilon \\rightarrow 0$, we get \n\\begin{align}\n t e^{-2\/u} u^{2 D - A} (1 - D u)^{A-2 D} = \\textrm{const}\n\\end{align}\nwhich is the right scaling variable in $4$-d. We have not yet been able to obtain an analytical form for the scaling variable involving $t$ and $h$. This is because the equation for $u(l)$ does not seem to have a closed-form solution here (unlike the 4-d case). Nevertheless, we are motivated by an attempt to create scaling variables which interpolate between different dimensions and have the correct scaling behavior in many dimensions going down from $4$ to $1$. Once the full scaling variables are written down, a first test would be to see if these scaling variables do better collapsing the numerical data in 3-d. \n\n\n\\subsection{1-d Ising model} \\label{sec:1dising}\n\nThe 1-d Ising model is somewhat different because it is the lower critical dimension and does not have a phase transition. The 1-d Ising model has an exact solution which can be obtained by using transfer matrices. The partition function can be written as the trace of a transfer matrix $\\mathcal{T}^N$ where $N$ is the number of spins in the system. The matrix $\\mathcal{T}_{i j} = e^{-\\beta \\mathcal{H}(s_i s_j)}$. Coarse graining here can be done by a well-defined procedure, the coarse grained transfer matrix is defined as $\\tilde{\\mathcal{T}} = \\mathcal{T}^b$ where $b$ is the coarse graining length scale. Defining $\\ell = \\log b$ and expanding for $b$ close to 1, we can get flow equations for the temperature $T$\n\\begin{equation}\n \\frac{d T}{d \\ell} = - \\frac{T^2}{2} \\sinh \\left(\\frac{2}{T} \\right) \\log \\left (\\tanh \\left(\\frac{1}{T}\\right) \\right) .\n \\label{1dfullflow}\n\\end{equation}\nThis is different from the flow equations we have considered so far because of the presence of non-analytic terms in the flow. The non-analytic term which multiplies the $T^2$ term is $= -1$ at $T = 0$. So, this equation corresponds to a transcritical bifurcation\n\\begin{equation}\n \\frac{d T}{d \\ell} =  \\frac{T^2}{2} + ...\n\\end{equation}\nwhere the additional terms are non-analytic at $T = 0$. This can be used to derive a correlation length $\\chi \\sim \\exp(2\/T)$.  To interpret the flow further, consider the change of coordinates $\\kappa = \\exp(-2\/T)$. In these variables, the flow is \n\\begin{equation}\n \\frac{d \\kappa}{d \\ell} = (\\kappa^2 - 1) \\log \\left( \\frac{1-\\kappa}{\\kappa + 1} \\right) .\n\\end{equation}\nEvidently, the flow is analytic in this variable. Solving the full flow Eq.(~\\ref{1dfullflow}) gives $\\chi \\sim -1\/(\\log \\tanh(1\/T))$.\n\nFor non-zero $\\epsilon$, this argument is usually extended in what is called a Migdal-Kadanoff procedure for doing RG~\\cite{kadanoff1976notes, chaikin1995principles}. The flow equations are identical except for the presence of a $- \\epsilon T$ term which serves as the bifurcation parameter. The $1+\\epsilon$ expansion can be summed completely because the flow equation is known to all orders. It does not yield very accurate critical exponents though it gives the exact value of the critical temperature in 2-d (because it respects duality symmetry). Several people have improved the expansion~\\cite{martinelli1981systematical, bruce1981droplet}. \n\nThe presence of non-analytic terms in the flow equations complicates the application of normal form theory. We will come back to it when discussing Legendre transform of flow equations.\n\n\n\\subsection{2-d Ising model} \\label{sec:2dising}\nThe 2-d Ising model is a particularly nice example because it has an exact solution in the absence of a magnetic field. All predictions then can be compared to the exact solution. Surprisingly, despite the known exact solution, the scaling behavior of the 2-d Ising model is still not completely understood. A full discussion of the 2-d Ising model will be given in separate work~\\cite{Clement18}. Here, we give a brief summary of the issues involved.\n\nThe only variable required to describe the 2-d Ising model in the absence of a field is the temperature $t$. The linear eigenvalues of the free energy and the temperature are $2$ and $1$ respectively. The normal form of the flow equations can be written as \n\\begin{align}\n \\frac{d \\tilde f}{d \\ell} &= 2  \\tilde f - \\tilde t^2 , \\\\\n \\frac{d \\tilde t}{d \\ell} &=  \\tilde t .\n\\end{align}\nWe have used the fact that the only term which cannot be removed by traditional normal form analysis is the resonance $t^2$. In fact, it cannot be removed by any analytic change of variables. We have also used the freedom to rescale $t$ to set the coefficient of the resonance equal to -1~\\footnote{The sign is set to match the exact solution of the square lattice nearest neighbor Ising model}. The solution to this can be written as $\\tilde t = \\tilde t_0 e^{\\ell}$ and the free energy \n\n\\begin{equation}\n\\label{2disingeq}\n \\tilde f(\\tilde t_0, \\ell) = e^{-2 \\ell} \\tilde f(\\tilde t_0 e^{\\ell}) - \\tilde t_0^2 \\ell .\n\\end{equation}\n\nCoarse graining until $\\tilde t(\\ell) = 1$ or $l = -\\log(\\tilde t_0)$, we get\n\\begin{equation}\n \\tilde f(\\tilde t_0) = \\tilde t_0^2 \\tilde f(1) + \\tilde t_0^2 \\log \\tilde t_0 .\n\\end{equation}\n\nNow, the normal form variable $\\tilde t_0$ is some analytic function of the physical variable $t_0$. It is linear to first order in $t_0$. Hence, we can write it as $\\tilde t_0 = t_0 (1 + c(t_0))$ where $c$ is some analytic function. Then, we can expand \n\\begin{widetext}\n\\begin{align}\n \\tilde f(t_0) &= t_0^2 (1 + c(t_0))^2 f(1) +  t_0^2 (1 + c(t_0))^2 \\log t_0 (1 + c(t_0)) , \\\\\n &= t_0^2 (1 + c(t_0))^2 f(1) + t_0^2 (1 + c(t_0))^2 \\log (1 + c(t_0)) +  t_0^2  (1 + c(t_0))^2 \\log t_0 , \\\\\n &= a(t_0) + b(t_0) \\log t_0 \n\\end{align}\n\\end{widetext}\nwhere both $a(t_0)$ and $b(t_0)$ are some analytic functions of $t_0$. Meanwhile any change of coordinates which adds an analytic function of $t_0$ to $\\tilde f$ can be absorbed in the definition of $a(t_0)$. Hence, we can write the final most general form of the free energy of the 2-d Ising model as $f = a(t_0) + b(t_0) \\log t_0$. Indeed, the exact solution of the 2-d Ising model can be written in this form~\\cite{caselle2002irrelevant}. \n\nWhile the basic solution of the 2-d Ising model is simple, some challenges still remain. The scaling form in the presence of other variables (like the magnetic field and other irrelevant variables) which has so far only been conjectured~\\cite{aharony1983nonlinear, caselle2002irrelevant}  naturally follows from an application of normal form theory. It is given simply by including other variables in the argument of the free energy in Eq.(~\\ref{2disingeq}) before coarse graining till $t(\\ell) = 1$. Irrelevant variables are the source of singular corrections to scaling. An interesting unresolved issue is the presence of higher powers of logarithms in the susceptibility which are not found in the free energy~\\cite{orrick2001susceptibility, chan2011ising}. This is usually attributed to the presence of irrelevant variables. Here it is possible to show that the irrelevant variables which are derived from conformal field theory~\\cite{caselle2002irrelevant} would in fact lead to higher powers of logarithms in the free energy which are not observed. Hence, they cannot explain the higher powers of logarithms in the susceptibility. It is possible that there are other irrelevant variables in the 2-d square lattice nearest neighbor Ising model with a field which are not predicted by conformal field theory but can capture the higher powers of logarithms in the susceptibility, as they turn on with a field.\n\nThe logarithm due to the resonance in the 2-d Ising model is most apparent in the specific heat. It is easy to derive the flow equations for the inverse specific heat which have the form\n\\begin{equation}\n \\frac{d C^{-1}}{d \\ell} = 2 C^{-2} ,\n\\end{equation}\nand has a transcritical bifurcation in two dimensions. This raises a question, is it legitimate to talk about a bifurcation in two dimensions for the Ising model if it happens in the space of results rather than the space of control variables? Intriguingly, though perhaps unrelated, a bifurcation has been observed in 2 dimensions using methods of conformal bootstrap~\\cite{golden2015no, el2014conformal}. In thermodynamics, a natural framework which interchanges between results and control parameters is given by Legendre transforms. However, the flow equations for the Legendre transformed coordinates generically have non-analyticities in them. We suspect that the variable $t$ (and $h$ etc.) is uniquely specified as the correct variable for RG.  It is possible that it is more natural to consider removing degrees of freedom in the canonical ensemble ($t$ and $f$), then in a microcanonical one ($E$ and $S$)~\\footnote{In fact, there is an interesting connection here with information geometry. It is much more natural to talk about the Fisher Information metric in the canonical ensemble. To be able to talk about the uncertainty in a thermodynamic quantity, one has to be able to exchange that quantity with the environment. Otherwise, calculating the Fisher Information Metric can give ill-defined answers. This is further motivation to consider that a particular thermodynamic ensemble may be more suitable for some purposes.}. A fuller discussion will be given in forthcoming work~\\cite{Clement18}.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nWe have shown how normal form theory leads to systematic procedure for handling the singularity in RG flows. The concept of universality families broadens the notion of a universality class and we have elucidated it with several different examples.  We have focused on getting a precise handle on the singularity at the critical point. However, normal form theory also gives an elegant way to fit corrections to scaling. Interestingly, even the scaling of the 2-d Ising model which has an exact solution has some unresolved mysteries which we are exploring.  It is possible that interpolating between dimensions in a way that\ncaptures the correct singularities can improve scaling collapses\nin all dimensions. Finally, we are exploring the application of our methods\nto systems like jamming in 2-d~\\cite{goodrich2014jamming}, where logarithmic\ncorrections are observed but no renormalization-group theory is available.  In general, we expect this fruitful confluence of dynamical systems theory and\nthe renormalization group will not only clarify and illuminate previously\nknown technical calculations, but will also facilitate quantitative analysis\nof experimental and theoretical systems farther from their critical points\nand before the underlying field theory is well understood.\n\n\\section{Acknowledgements}\n\nWe thank Tom Lubensky, Andrea Liu, John Guckenheimer, Randall Kamien and Cameron Duncan for useful conversations. AR, CBC, LXH, JPK, DBL and JPS were supported by the National Science Foundation through Grant No. NSF DMR-1719490. LXH was supported by a fellowship from Cornell University. DZR was supported by the Bethe\/KIC Fellowship and the National Science Foundation through Grant No. NSF DMR-1308089.\n\n\n\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzjuog b/data_all_eng_slimpj/shuffled/split2/finalzzjuog
new file mode 100644
index 0000000000000000000000000000000000000000..f8fc292cee6bb491d2a1810095ff684382d61bc8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzjuog
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\\label{sec:intro}\nWith the proliferation of big data matrices, dimension reduction has become an important tool in many data analysis applications. There are several methods of dimension reduction for a given problem; however, the approximated data often consists of derived features that are either no longer interpretable or difficult to interpret in the original context.  For example, while the singular value decomposition (SVD) provides an optimal approximation and compression of data, it may be difficult for domain experts to directly draw conclusions or interpret the singular vectors. In some applications, it is necessary to find a dimension reduction method that preserves the original properties (such as sparsity, nonnegativity, being integer-valued) of the data and ensures interpretability. In an attempt to solve this difficulty, one possibility for a low-rank representation of a given data matrix is to use a subset of the original columns and rows of the matrix itself: a CUR decomposition; see, e.g., Mahoney and Drineas \\cite{Drineas}. The selected subsets of rows and columns capture the most relevant information of the original matrix.\n\nA CUR decomposition of rank $k$ of a (square or rectangular) $m \\times n$ matrix $A$ is of the form\n\\begin{equation}\n\\label{cur}\nA \\ \\approx \\ CMR \\ := \\ A P \\, \\cdot \\, M \\, \\cdot \\, S^TA.\n\\end{equation}\n Here, $P$ is an $n \\times k$ (where $k < \\min(m,n)$) {\\em index selection matrix} with some columns of the identity matrix $I_n$ that selects certain columns of $A$.\nSimilarly, $S$ is an $m \\times k$ matrix with columns of $I_m$\nthat selects certain rows of $A$; so $C$ is $m \\times k$ and $R$ is $k \\times n$. We construct the $k \\times k$ matrix $M$ in such a way that the decomposition has some desirable approximation properties that we will discuss in \\cref{sec:GCUR}. (In line with \\cite{Str19}, we will use the letter $M$ rather than $U$.) It is also possible to select a different number of columns and rows; here, $M$ will not be of dimension $k\\times k$. It is worth noting that given $k$, this decomposition is not unique; there are several ways to obtain this form of approximation to $A$ with different techniques of choosing the representative columns and rows. Many algorithms for this decomposition using the truncated SVD (TSVD) as a basis have been proposed \\cite{Boutsidis, Zhang, Mahoney, Drineas}. In \\cite{Sorensen}, Sorensen and Embree present a CUR decomposition inspired by a discrete empirical interpolation method (DEIM) on a TSVD. Let the rank-$k$ TSVD be\n\\begin{equation}\n\\label{svd}\nA \\approx W_k \\Psi_k Z_k^T,\n\\end{equation}\nwhere the columns of $W_k$ and $Z_k$ are orthonormal,\nwhile $\\Psi_k$ is diagonal with nonnegative elements. Throughout the paper we assume a unique truncated singular value decomposition, i.e., the $k$th singular value is not equal to the $(k+1)$st singular value.\n\nThere is extensive work on CUR-type decompositions in both numerical linear algebra and theoretical computer science. In this paper, we develop a generalized CUR decomposition (GCUR) for matrix pair, for any two matrices $A$ and $B$ with the same number of columns: $A$ is $m\\times n$, $B$ is $d\\times n$ and of full rank. The intuition behind this generalized CUR decomposition is that we can view it as a {\\em CUR decomposition of $A$ relative to $B$}. As we will see in \\cref{pp3}, when $B$ is square and nonsingular, the GCUR decomposition has a close connection with the CUR of $AB^{-1}$. The GCUR is also applicable to nonsquare matrices $B$; see the examples in \\cref{sec:EXP}. We show in \\cref{pp3} that if $B$ is nonsquare but of a full rank, we still have a close connection between the CUR decomposition of $AB^+$ (where $B^+$ denotes the pseudoinverse of $B$) and the GCUR decomposition. Another intuition for this GCUR decomposition comes from a footnote remark by Mahoney and Drineas \\cite[p.~700]{Mahoney}; for data sets where a low-dimensional subspace obtained by the SVD cannot capture class or subgroup separation, an SVD-based CUR decomposition performs equally poorly. This is evident in \\cref{exp:3}.\n\nInspired by the work of Sorensen and Embree \\cite{Sorensen}, we present a generalized CUR decomposition using the discrete empirical interpolation method.\nThe DEIM algorithm for interpolation indices, presented in \\cite{Chaturantabut}, is a discrete variant of empirical interpolation proposed in \\cite{Barrault} as a method for model order reduction for nonlinear dynamical systems. In \\cite{Sorensen}, the authors used DEIM as an index selection technique for constructing the $C$ and $R$ factors of a CUR decomposition. The DEIM algorithm independently selects the column and row indices based on the right and left singular vectors of a data matrix $A$, respectively. Our new GCUR method uses the matrices obtained from the GSVD instead. Besides using DEIM on the GSVD for index selection, we can also use other CUR-type index selection strategies for the GCUR. The proposed method can be used in situations where a low-rank matrix is perturbed with noise, where the covariance of the noise is not a multiple of the identity matrix. It may also be appropriate for applications where one is interested in extracting the most discriminative information from a data set of interest relative to another data set. We will see examples of these in \\cref{sec:EXP}. \n\n\\begin{example}\nThe following simple example shows that using the matrices obtained from the GSVD instead of the SVD can lead to more accurate results when approximating data with colored noise. As in \\cite[p.~55]{hansen} and \\cite{park1994}, we use the term ``colored noise'' for noise of which the covariance matrix is not a multiple of the identity.\n\nWe are dealing with a full-rank matrix $A_E$ representing low-rank data. We want to try to recover an original low-rank matrix perturbed by colored noise. Our test matrix $A_E$ is a rank-2 matrix $A$ of size $3\\times3$ perturbed by additive colored noise $E$ with a given desired covariance structure. We take  \n\n\\[A= {\\footnotesize \\begin{bmatrix*}[r]\n      1  &    0  &  1\\\\\n     0   &  2    & 2\\\\\n     1   & 1 &  2\n \\end{bmatrix*}}, \n \\quad  E^T\\!E= {\\footnotesize \\begin{bmatrix*}[l]\n      1.0 & 0.8 & 0.3\\\\\n    0.8 &  1.0 & 0.8\\\\\n    0.3 & 0.8  & 1.0\n \\end{bmatrix*}}.\\]\nWe generate the colored noise as an additive white Gaussian noise multiplied by the Cholesky factor $(R)$ of the desired covariance matrix. The matrix $A_E$ is, as a result, a sum of a rank-2 matrix and a correlated Gaussian noise matrix. We compute the SVD of both $A$ and $A_E$. The $k$ dominant left singular vectors of $A$ are denoted by $W_k$ while those of $A_E$ are $\\widetilde{W}_k$. We also compute the GSVD of $(A_E, R)$ and denote the $k$ dominant left generalized singular vectors by $U_k$. Since we are interested in recovering $A$, we examine the angle between the leading $k$-dimensional exact left singular subspace \\text{Range}$(W_k)$ and its approximations \\text{Range}$(\\widetilde{W}_k)$ and \\text{Range}$(U_k)$. We generate 1000 different test cases and take the average of the subspace angles.\n\n\\Cref{tab:0} shows the results for $k=2$ and three different noise levels. We observe that the approximations obtained using the GSVD in terms of subspace angles are more accurate than those from the SVD; about 40\\% gain in accuracy. This illustrates the potential advantage of using generalized singular vectors in the presence of colored noise.\n\\begin{table}[htb!]\n\\centering\n{\\caption{The average angle between the leading two-dimensional exact singular subspace \\text{Range}$(W_2)$ (which is the range of $A$) and its approximations \\text{Range}$(\\widetilde{W}_2)$ and \\text{Range}$(U_2)$, for different values of the noise level $\\varepsilon$. The subspaces \\text{Range}$(U_2)$ and \\text{Range}$(\\widetilde{W}_2)$ are from the SVD of $A_E$ and the GSVD of $(A_E, R)$, respectively.}\\label{tab:0}\n{\\footnotesize\n\\begin{tabular}{clc} \\hline \\rule{0pt}{3mm}%\n$\\varepsilon$ & Method & Subspace angle    \\\\ \\hline \\rule{0pt}{3.5mm}%\n$5\\cdot 10^{-2}$ & SVD  &$1.7\\cdot 10^{-2}$   \\\\\n& GSVD & $1.2\\cdot 10^{-2}$ \\\\[0.5mm] \\hline \\rule{0pt}{3.5mm}%\n$5\\cdot 10^{-3}$ & SVD &$1.7\\cdot 10^{-3}$   \\\\\n& GSVD & $1.1\\cdot 10^{-3}$ \\\\[0.5mm] \\hline \\rule{0pt}{3.5mm}%\n$5\\cdot 10^{-4}$ & SVD & $1.7\\cdot 10^{-4}$  \\\\\n& GSVD & $1.1\\cdot 10^{-4}$ \\\\[0.5mm]\\hline %\n\\end{tabular}}}\n\\end{table}\n\\end{example}\n\nInspired by this example, we expect that the GCUR compared to the CUR may produce better approximation results in the presence of non-white noise, as it is based on the GSVD instead of the SVD. We show in \\cref{sec:EXP} that the GSVD and the GCUR may provide equally good approximation results even when we use an inexact Cholesky factor. \n\nThroughout the paper, we denote the 2-norm by $\\norm{\\cdot}$ and the inf-norm by $\\norm{\\cdot}_\\infty$. We use MATLAB notation to index vectors and matrices; thus, $A(:,p)$ denotes the $k$ columns of $A$ whose corresponding indices are in vector $p \\in \\mathbb N^k$.\n\n\\textbf{Outline.}\nWe give a brief introduction to the generalized singular value decomposition in \\cref{sec:GSVD}. We also discuss the truncated GSVD and its approximation error bounds. We summarize the DEIM technique we use for index selection in \\cref{sec:DEIM}. \\Cref{sec:GCUR} introduces the new generalized CUR decomposition with an analysis of its error bounds. In \\cref{alg:GCUR-DEIM}, we present a DEIM-type GCUR decomposition algorithm. Results of numerical experiments are presented in \\cref{sec:EXP}, followed by conclusions in \\cref{sec:Con}. \n\n\\section{ Generalized singular value decomposition}\\label{sec:GSVD}\nThe GSVD appears throughout this paper since it is a key building block of the proposed algorithm. This section gives a brief overview of this decomposition. The original proof of the existence of the GSVD has first been introduced by Van Loan in \\cite{Van}. Paige and Saunders \\cite{Paige} later presented a more general formulation without any restrictions on the dimensions except for both matrices to have the same number of columns. Other formulations and contributions to the GSVD have been proposed in \\cite{stewart1982,sun,van1985}. For our applications in this paper, let $A\\in \\mathbb R^{m\\times n}$ and $B\\in \\mathbb R^{d\\times n}$ with both $m\\ge n$ and $d\\ge n$. Following the formulation of the GSVD proposed by Van Loan \\cite{Van}: there exist matrices $U \\in{ \\mathbb R ^{m \\times m}}$, $V \\in{ \\mathbb R^{d \\times d}}$ with orthonormal columns and a nonsingular $X \\in {\\mathbb R ^{n \\times n}}$ such that\n\\begin{equation}\n\\begin{aligned}\n\\label{gsvd}\nU^T\\!AX &= \\Gamma = \\text{diag}(\\gamma_1,\\dots,\\gamma_n),  \\qquad &\\gamma_i\\in [0,1],\\\\\nV^T\\!BX &=\\Sigma = \\text{diag}(\\sigma_1,\\dots,\\sigma_n), \\qquad &\\sigma_i\\in [0,1],\n\\end{aligned}\n\\end{equation}\nwhere $\\gamma_i^{2}+\\sigma_i^{2} = 1 $. Although traditionally the ratios $\\gamma_i\/\\sigma_i$ are in a nondecreasing order, for our purpose we will instead maintain a nonincreasing order. The matrices $U$ and $V$ contain the left generalized singular vectors of $A$ and $B$, respectively; and similarly, $X$ contains the right generalized singular vectors and is identical for both decompositions. While the SVD provides two sets of linearly independent basis vectors, the GSVD of $(A, B)$ gives three new sets of linearly independent basis vectors (the columns of $U, V,$ and $X$) so that the two matrices $A$ and $B$ are diagonal when transformed to these new bases.\nWe note that only the reduced GSVD is needed, so that we can assume that $U \\in \\mathbb R^{m \\times n}$, $V \\in \\mathbb R^{d \\times n}$, and $\\Gamma$ and $\\Sigma$ are $n \\times n$.\n\nOur analysis is based on the following formulation of the GSVD presented in \\cite{van1985}. Let $Y := X^{-T}$ in the GSVD of \\eqref{gsvd}, then $A = U \\Gamma Y^T$ and $B = V \\Sigma Y^T$. Let us characterize matrix $Y$. (In fact, Matlab's {\\tt gsvd} routine renders $Y$ instead of $X$.) Since\n\\begin{equation}\\label{eq:mgsvd}\n  A=U \\, \\Gamma \\, Y^T, \\qquad B=V \\, \\Sigma \\, Y^T  \n\\end{equation}\nthis implies that we have the following congruence transformation\n\\[A^T\\!A=Y(\\Gamma^T\\Gamma)Y^T, \\qquad B^T\\!B=Y(\\Sigma^T\\Sigma)Y^T.\\]\nFrom the above, it follows that $A^T\\!A$ has the same inertia as $\\Gamma^T\\Gamma$ and the same holds for $B^T\\!B$ and $\\Sigma^T\\Sigma$ (here this mainly gives information on the number of zero eigenvalues).\nWe also see that, provided $A$ and $B$ are of full-rank, these similarity transformations hold:\n\\begin{equation}\n\\label{eqn:matrix_Y}\n\\begin{aligned}\n (B^T\\!B)(A^T\\!A)^{-1} &=Y(\\Sigma^T\\Sigma)(\\Gamma^T\\Gamma)^{-1}Y^{-1} = Y\\,\\text{diag}(\\sigma_i^2\/\\gamma_i^2)\\, Y^{-1},\\\\\n (A^T\\!A)(B^T\\!B)^{-1} &=Y(\\Gamma^T\\Gamma)(\\Sigma^T\\Sigma)^{-1}Y^{-1}=Y\\,\\text{diag}(\\gamma_i^2\/\\sigma_i^2)\\, Y^{-1}.\n\\end{aligned}\n\\end{equation}\nThe columns of the matrix $Y$ are therefore the eigenvectors for both $(A^T\\!A)(B^T\\!B)^{-1}$ and its inverse $(B^T\\!B)(A^T\\!A)^{-1}$. The GSVD avoids the explicit formation of the cross-product matrices $A^T\\!A$ and $B^T\\!B$ (see also \\cref{exp:3}).\n\n\\textbf{Truncated GSVD.}\nIn some practical applications it could be of interest to approximate both matrices $(A, B)$ by other matrices $(A_k, B_k)$, said truncated, of a specific rank $k$. For use in \\cref{sec:GCUR}, we define the truncated GSVD (TGSVD) for $(A,B)$ as (cf.~\\cite[(2.34)]{hansen})\n\\begin{equation}\n    \\label{eq:tgsvd}\n    A_k := U_k \\Gamma_k Y_k^T, \\qquad  B_k := V_k \\Sigma_k Y_k^T,\n\\end{equation}\nwhere $k < n$. If we partition\n\\begin{equation}\\label{eq:pgsvd}\nU = [U_k \\ \\widehat{U}], \\ V = [V_k \\ \\widehat{V}], \\ Y = [Y_k \\ \\widehat{Y}], \\ \\Gamma = \\text{diag} (\\Gamma_k, \\widehat{\\Gamma}), \\ \\Sigma = \\text{diag} (\\Sigma_k, \\widehat{\\Sigma})\n\\end{equation}\nthen it follows that $A - A_k = \\widehat{U} \\, \\widehat{\\Gamma} \\, \\widehat{Y}^T.$\nThe following proposition is useful for understanding the error bounds for the GCUR. In line with \\cite[p.~495]{pchansen},\nlet $\\psi_i(A)$ and $\\psi_i(Y)$ be the singular values of matrix $A$ and $Y$, respectively (cf.~also \\eqref{svd}). The first and second statements of the following proposition are from \\cite{pchansen}; while the third statement may not be present in the literature yet, it is straightforward.\n\\begin{proposition}\\label{pp1}\nLet $A = U\\, \\Gamma \\, X^{-1} = U \\, \\Gamma \\, Y^T$ as in \\eqref{gsvd}, with $Y = X^{-T}$, then for $i=1,\\dots,n$ (see, e.g., \\cite[pp.~495--496]{pchansen})\n\\[\\gamma_i\\cdot\\psi_{\\min}(Y)\\leq \\psi_i(A)=\\psi_i(U \\, \\Gamma \\, Y^T) \\leq \\psi_i(\\Gamma) ~ \\norm{Y}=\\gamma_i\\cdot\\norm{Y}\\]\nso\n\\[ \\frac{\\psi_i(A)}{\\|Y\\|} \\le\n\\gamma_i= \\psi_i(\\Gamma) =\\psi_i(U^T\\!A Y^{-T}) \\leq \\psi_i(A)~\\norm{Y^{-1}}.\\]\nMoreover,\n\\[ {\\gamma_{k+1}}\\cdot\\psi_{\\min}(\\widehat{Y}) \\leq \\norm {A - A_k} \\leq  {\\gamma_{k+1}}\\cdot \\norm{\\widehat{Y}}.  \\]\n\\end{proposition}\n\\begin{proof}\nThis follows from \\eqref{eq:mgsvd} and the well-known property that, for the product of two matrices we have $\\psi_i(A) \\, \\psi_{\\min}{(B)} \\le \\psi_i(AB) \\leq \\psi_i(A) \\, \\norm{B}$ (see, e.g., \\cite[p.~89]{Householder}).\n\\end{proof}\n\nThe results above are relevant tools for the analysis and understanding of generalized CUR and its error bounds which we will introduce in \\cref{sec:GCUR}. \n\\section{Discrete empirical interpolation method}\\label{sec:DEIM} We now summarize the tool from existing literature \\cite{Sorensen, Chaturantabut} that we use to select columns and or rows from matrices. Besides the GSVD, the DEIM algorithm plays an important role in the proposed method. The DEIM procedure works on the columns of a specified basis vectors sequentially. The basis vectors must be linearly independent. Assuming we have a full-rank basis matrix $U \\in \\mathbb R^{m\\times k}$ with $k\\le m$, to select $k$ rows from $U$, the DEIM procedure constructs an index vector $s\\in \\mathbb N^k$ such that it has non-repeating values in $\\{1,\\dots,m\\}$. Defining the selection matrix $S$ as an $m\\times k$ identity matrix indexed by $s$, i.e., $S=I(:,s)$ and $\\mathbf x(s) = S^T\\mathbf x$ (cf.~\\cite{Sorensen}), we have an {\\em interpolatory projector} defined through the DEIM procedure as\n\\[\\mathbb{S}=U(S^TU)^{-1}S^T.\\]\nWe can show that $S^TU$ is nonsingular (see \\cite[Lemma~3.2]{Sorensen}). The term ``interpolatory projector\" stems from the fact that for any $\\mathbf x\\in \\mathbb R^m$ we have\n\\[(\\mathbb{S}\\mathbf x)(s)=S^T\\mathbb{S}\\mathbf x=S^TU(S^TU)^{-1}S^T\\mathbf x=S^T\\mathbf x=\\mathbf x(s),\\]\nimplying the projected vector $\\mathbb{S}\\mathbf x$ matches $\\mathbf x$ in the $s$ entries \\cite{Sorensen}. \n\nTo select the indices contained in $s$, the columns of $U$ are considered successively. The first interpolation index corresponds to the index of the entry with the largest magnitude in the first basis vector. The rest of the interpolation indices are selected by removing the direction of the interpolatory projection in the previous bases from the subsequent one and finding the index of the entry with the largest magnitude in the residual vector. The index selection using DEIM is limited by the rank of the basis matrix, i.e., the number of indices selected can be no more than the number of vectors available.\n\nTo form $s$, let $\\mathbf u_j$ denote the $j$th column of $U$ and $U_j$ be the matrix of the first $j$ columns of $U$. Similarly, let $s_j$ contain the first $j$ entries of $s$, and let $S_j=I(:,s_j)$. More precisely, we define $s_1$ such that\n$|\\mathbf u_1(s_1)|=\\norm{\\mathbf u_1}_\\infty$ and the $j$th interpolatory projector $\\mathbb{S}_j$ as \n\\[\\mathbb{S}_j=U_j(S_j^TU_j)^{-1}S^T_j.\\]\nTo select $s_j$, remove the $u_{j-1}$ component from $u_j$ by projecting $u_j$ onto indices \\{1, \\dots, $s_{j-1}$\\}, thus \n\\[\\mathbf r_j=\\mathbf u_i-\\mathbb{S}_{j-1}\\mathbf u_i,\\]\nthen take the index of the entry with the largest magnitude in the residual, i.e., $s_j$ such that  \n\\[|\\mathbf r_j(s_j)|=\\norm{\\mathbf r_j}_\\infty.\\]\nIf there are multiple options, we take the smallest index. In a nutshell, we find the indices via a non-orthogonal Gram--Schmidt-like process (oblique projections) on the $\\mathbf u$-vectors. Since the input vectors are linearly independent, the residual vector $\\mathbf r$ is guaranteed to be nonzero. This DEIM algorithm forces the selection matrix $S$ to find $k$ linearly independent rows of $U$ such that the local growth of $\\norm{(S^TU)^{-1}}$ is kept modest via a greedy search \\cite[p.~2748]{Chaturantabut} as implemented in \\cref{alg:CUR-DEIM}.\n\n\\begin{tcbverbatimwrite}{tmp_\\jobname_alg.tex}\n\\begin{algorithm}\n\\caption{DEIM index selection \\cite{Sorensen}}\n\\label{alg:CUR-DEIM}\n {\\bf Input:} $U \\in \\mathbb R^{m \\times k}$, with $k\\le m$ (linearly independent columns)\\\\\n {\\bf Output:} Indices $s \\in \\mathbb N^k$ with distinct entries in $\\{1,\\dots,m\\}$ \n\\begin{algorithmic}[1]\n\t\\STATE{$\\mathbf u = U(:,1)$}\n\t\\STATE{ $s_1 = \\argmax_{1\\le i\\le m}~|(\\mathbf u)_i|$}\n\t\\FOR{ $j = 2, \\dots, k$ }\n\t\\STATE{$\\mathbf u = U(:,j)$ }\n\t\\STATE{$\\mathbf c=U(s,1:j-1)^{-1}\\mathbf u(s)$}\n\t\\STATE{$\\mathbf r=\\mathbf u-U(:,1:j-1)\\,\\mathbf c$}\n\t\\STATE{$s_j$ = $\\argmax_{1\\le i\\le m}~|(\\mathbf r)_i|$}\n\t\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\\end{tcbverbatimwrite}\n\n\\input{tmp_\\jobname_alg.tex}\n\nAlthough the DEIM index selection procedure is basis-dependent, if the interpolation indices are determined, the DEIM interpolatory projector is independent of the choice of basis spanning the space \\text{Range}$(U)$.\n\\begin{proposition}\\label{pp2}(\\cite[Def.~3.1, (3.6)]{Chaturantabut} ). \nLet $Q$ be an orthonormal basis of \\text{Range}$(U)$ where $Q_i = [\\mathbf q_1,\\dots,\\mathbf q_i]$ for $1 \\leq i \\leq k$, then\n\\[U(S^T U)^{-1}S^T = Q(S^T Q)^{-1}S^T.\\]\n\\end{proposition}\nThis proposition allows us to take advantage of the special properties of an orthonormal matrix in cases where our input basis matrix is not (see \\cref{pp4}).\n\\section{Generalized CUR decomposition and its approximation properties}\\label{sec:GCUR}\nIn this section we describe the proposed generalized CUR decomposition and provide a theoretical analysis of its error bounds.\n\\subsection{Generalized CUR decomposition}\nWe now introduce a new generalized CUR decomposition of matrix pairs $(A, B)$, where $A$ is $m \\times n$ $(m\\ge n)$ and $B$ is $d \\times n$ $(d\\ge n)$, and $B$ is of full rank. This GCUR is inspired by the truncated generalized singular value decomposition for matrix pairs, as reviewed in \\cref{sec:GSVD}. We now define a generalized CUR decomposition (cf.~\\eqref{cur}).\n\\begin{definition} \\label{Dfn4}\nLet $A$ be $m \\times n$ and $B$ be $d \\times n$ and of full rank, with $m\\ge n$ and $d\\ge n$.\nA generalized CUR decomposition of $(A, B)$ of rank $k$ is a matrix approximation of $A$\nand $B$ expressed as\n\\begin{equation}\n  \\label{eq:gcur}\n  \\begin{aligned}\n A_k := C_A\\, M_A\\, R_A = AP \\, M_A \\, S_A^TA ~ ,  \\\\\n B_k := C_B\\, M_B\\, R_B = BP \\, M_B \\, S_B^TB. \n\\end{aligned}\n\\end{equation}\nHere $S_A \\in \\mathbb R^{m \\times k}$, $S_B \\in \\mathbb R^{d \\times k}$, and $P \\in \\mathbb R^{n \\times k}$ are index selection matrices $(k < n)$. \n\\end{definition}\nIt is key that {\\em the same} columns of $A$ and $B$ are selected;\nthis gives a coupling between the decomposition of $A$ and $B$.\n\nThe matrices $C_A, C_B$ and $R_A, R_B$ are subsets of the columns and rows, respectively, of the original matrices. In the rest of the paper, we will focus on the matrix $A$;  we can perform a similar analysis for the matrix $B$. As in \\cref{sec:DEIM}, we again have the vectors $s_A, p$ as the indices of the selected rows and columns such that $C_A = AP$ and $R_A = S_A^TA$, where $S_A=I(:,s_A)$ and $P=I(:,p)$. The choice of $p$ and $s_A$ is based on the transformation matrices from the rank-$k$ truncated GSVD.\n\nGiven $P$ and $S_A$, the middle matrix $M_A$ can be constructed in different ways to satisfy certain desirable approximation properties. In \\cite{Sorensen}, the authors show how setting $M=A(s,p)^{-1}$ leads to a CUR decomposition corresponding to the $p$ columns and $s$ rows of $A$. Instead, following these authors \\cite{Sorensen} and others \\cite{Mahoney,Stewart}, we choose to construct the middle matrix $M_A$ as $(C_A^T C_A)^{-1}C_A^T A R_A^T(R_A R_A^T )^{-1}$. This possibility, as shown by Stewart \\cite{Stewart}, minimizes $\\norm{A-CMR}$ for a given $p$ and $s$. Computing the middle matrix as such yields a decomposition that can be viewed as first projecting the columns of $A$ onto $\\text{Range}(C)$ and then projecting the result onto the row space of $R$, both steps being optimal for the 2-norm error. \n\nThe following proposition establishes a connection between the GCUR of $(A, B)$ and the CUR of $AB^{-1}$ and $AB^+$ for a square and nonsingular $B$ and a nonsquare but full-rank $B$, respectively.\n\\begin{proposition}\\label{pp3} If $B$ is a square and nonsingular matrix, then the selected row and column indices from the CUR decomposition of $AB^{-1}$ are the same as index vectors $s_A$ and $s_B$ obtained from the GCUR decomposition of $(A, B)$, respectively. Moreover in the special case where $B$ is the identity, the GCUR decomposition of $A$ coincides with the CUR decomposition of $A$, in that the factors $C$ and $R$ of $A$ for both methods are the same: the first line of \\eqref{eq:gcur} is equal to \\eqref{cur}.  In addition, if $B$ is nonsquare but of a full rank, we have this same connection between the indices from the CUR decomposition of $AB^+$ and the index vectors $s_A$ and $s_B$ obtained from the GCUR decomposition of $(A, B)$.\n\\end{proposition}\n\\begin{proof} We start with the GSVD \\eqref{eq:mgsvd} . If $B$ is square and nonsingular, then the SVD of $G=AB^{-1}$ can be expressed in terms of the GSVD of $(A,B)$, and is equal to $G = U (\\Gamma\\Sigma^{-1}) V^T$ \\cite{matrixcomp}.\nTherefore, the row index selection matrix for $G$ is $S_A$, and the column index selection matrix is $S_B$ since they are determined using $U$ and $V$, respectively. \n\nIf $B$ is the identity, then the second statement follows from $G = A$. From the second line of \\eqref{eq:mgsvd} we have that $Y=V\\Sigma^{-1}$. This implies that the index of the largest entries in the columns of $V$ would be the same as that of $Y$. As previously shown the left and right singular vectors of $A$ are equivalent to the $U$ and $V$ matrices from the GSVD. Hence the selection matrix $P$ in \\eqref{eq:gcur} obtained by performing DEIM on $Y$ would be the same as the selection matrix $P$ in \\eqref{cur} obtained by applying DEIM to the right singular vectors of $A$.\n\nIf $B$ is nonsquare but of full rank $n$, then we still have a similar connection between the GSVD of $(A,B)$ and the SVD of $AB^+$ because of the following. Since the factors in the reduced GSVD $B = V\\Sigma Y^T$ are of full rank, we have $B^+ = Y^{-T} \\Sigma^{-1} V^T$. This means that $AB^+ = U \\Gamma \\Sigma^{-1} V^T$, so the index vectors $s_A$ and $s_B$ from GCUR of $(A,B)$ are equivalent to the selected column and row indices from CUR of $AB^+$, respectively.\n\\end{proof}\n\n\n\nAlthough we can obtain indices of a CUR decomposition of $AB^{-1}$ using the GCUR of $(A, B)$, the converse does not hold. We emphasize that we need the GSVD for the GCUR decomposition and cannot use the SVD of $AB^{-1}$ or $AB^+$ instead, since the GCUR decomposition requires the $Y$ matrix from \\eqref{eq:tgsvd} to find the column indices. While we used the generalized singular vectors here, in principle one could use other vectors, e.g., an approximation to the generalized singular vectors.\n\nTo build the decomposition, it is relevant to know the dominant rows and columns of $A$ and $B$ in their rank-$k$ approximations. Given that $A_k$ and $B_k$ are rank-$k$ approximations of $A$ and $B$, respectively, how should the columns and rows be selected? \\Cref{alg:GCUR-DEIM} is a summary of the procedure\\footnote{Note that the backslash operator used in \\cref{alg:GCUR-DEIM} is a Matlab type notation for solving linear systems and least-squares problems.}.\nWe emphasize that we can parallelize the work in \\cref{line3,line4,line5,line6,line7,line8} since it consists of three independent runs of DEIM. Note that if we are only interested in approximating the matrix $A$ from the pair $(A, B)$, we can omit \\cref{line5,line8} as well as the second part of \\cref{line9}; thus saving computational cost. \n\nIn some applications, one might be interested in a generalized interpolative decomposition, of which the column and row versions are of the form \n\\begin{equation}\\label{eq:ID}\nA \\approx C_A\\widetilde M_A, \\ B \\approx C_B\\widetilde M_B\n\\qquad \\text{or} \\qquad\nA\\approx \\widehat M_A R_A, \\ B\\approx \\widehat M_B R_B.\n\\end{equation}\nHere $\\widetilde M_A=C_A^+\\!A$ is $k \\times n$ and $\\widehat M_A=AR^+_A$ is $m \\times k$; similar remarks hold for $\\widetilde M_B$ and $\\widehat M_B$. As noted in \\cite{Sorensen}, since the DEIM index selection algorithm identifies the row and column indices independently, this form of decomposition is relatively straightforward.\n\nIn terms of computational complexity, the dense GSVD method requires $\\mathcal O((m+d)n^2)$ work and the three runs of DEIM together require $\\mathcal O((m+n+d)k^2)$ work, so the overall complexity of the algorithm is dominated by the construction of the GSVD. (This might suggest iterative GSVD approaches; see \\cref{sec:Con}.) \n\n\\begin{tcbverbatimwrite}{tmp_\\jobname_alg2a.tex}\n\\begin{algorithm}\n\\caption{DEIM type GCUR decomposition}\n\\label{alg:GCUR-DEIM}\n {\\bf Input:} $A \\in \\mathbb R^{m \\times n}$, $B \\in \\mathbb R^{d \\times n}$, desired rank $k$ \\\\\n{\\bf Output:} rank-$k$ generalized CUR decomposition \\\\\n$A_k = A(:,p) \\, \\cdot \\, M_A \\, \\cdot \\, A(s_A,:)$, \\quad\n$B_k = B(:,p) \\, \\cdot \\, M_B \\, \\cdot \\, B(s_B,:)$\n\\begin{algorithmic}[1]\n\\setcounter{ALC@unique}{0}\n\t\\STATE{$[U, V, Y] = {\\sf gsvd}(A,B)$ \\hfill (according to nonincreasing generalized singular values)}\n\t\\FOR{ $j = 1, \\dots, k$ }\n\t\\STATE\\label{line3}{ $p(j) = \\argmax_{1\\le i\\le n}~|(Y(\\,:,j))_i|$ \\quad \\hfill (Iteratively pick indices)}\n\t\\STATE\\label{line4}{$s_A(j) = \\argmax_{1\\le i\\le m}~|(U(\\,:,j))_i|$}\n\t\\STATE\\label{line5}{$s_B(j) = \\argmax_{1\\le i\\le d}~|(V(\\,:,j))_i|$}\\\\\n\t \\quad \\hfill (Update new columns) \\\\\n\t\\STATE\\label{line6}{$Y(\\,:,~j+1) = Y(\\,:,~j+1)-Y(:,~1:j)\\cdot (Y(p,1:j)\\ \\backslash  \\ Y(p,~j+1)$)}\n\t\\STATE\\label{line7}{$U(\\,:,~j+1) = U(\\,:,~j+1)-U(:,~1:j)\\cdot (U(s_A,~1:j)\\ \\backslash  \\ U(s_A,~j+1)$)}\n\t\\STATE\\label{line8}{$V(\\,:,~j+1) = V(\\,:,~j+1)-V(:,~1:j)\\cdot (V(s_B,~1:j)\\ \\backslash \\ V(s_B,~j+1)$)} \n\t\\ENDFOR\n\t\\STATE\\label{line9}{ $M_A = A(\\,:,p) \\ \\backslash \\ (A \\ \/ \\ A(s_A,:\\,))$, \\quad $M_B = B(\\,:,p) \\ \\backslash \\ (B \\ \/ \\ B(s_B,:\\,))$}\n\\end{algorithmic}\n\\end{algorithm}\n\\end{tcbverbatimwrite}\n\n\\input{tmp_\\jobname_alg2a.tex}\n\nThe pseudocode in \\cref{alg:GCUR-DEIM} assumes the matrices from the GSVD (i.e., $U$, $V$, and $Y$) corresponds to a nonincreasing order of the generalized singular values. \n\nIn generalizing the DEIM-inspired CUR decomposition, we also look for a generalization of the related theoretical results. While the results presented in \\cite{Sorensen} express the error bounds in terms of the optimal rank-$k$ approximation, for our generalized CUR factorization, the most relevant quantity is the rank-$k$ GSVD approximation. In the following subsection, we present the theoretical results for bounding the GCUR approximation error. \n\n\\subsection{Error Bounds in terms of the SVD approximation} The error bounds for any rank-$k$ matrix approximation are usually expressed in terms of the rank-$k$ SVD approximation error. We will show this in the following proposition and also discuss its limitations. \nFor this proposition, we introduce the following notation.\nLet $A = W \\Psi Z^T = W_k \\Psi_k Z_k^T + W_{\\perp} \\Psi_{\\perp} Z_{\\perp}^T$ be the SVD of $A$ (see \\eqref{svd}), where $Z_k$ contains the largest $k$ right singular vectors.\nLet $Q_k$ be an $n \\times k$ matrix with orthonormal columns. It turns out in both \\cite{Sorensen} and this section that $\\norm{A(I-Q_kQ_k^T)}$ is a central quantity in the analysis. In the DEIM-induced CUR decomposition work \\cite{Sorensen}, we take the right singular vectors to be $Q_k$, but here we study this quantity for general $Q_k$. In our context, we are particularly interested in $Q_k$ as the orthogonal basis of the matrix $Y_k$ in \\eqref{eq:tgsvd}.\nDenote $\\mathcal Q_k = \\text{span}(Q_k)$ and $\\mathcal Z_k = \\text{span}(Z_k)$. Recall that $\\psi_i(A)$ are the singular values of $A$.\n\n\\begin{proposition}\nLet $Q_k$ be an $n \\times k$ matrix with orthonormal columns, and let $Z_k$ contain the largest $k$ right singular vectors of $A$. Then\n\\[\n\\psi_{k+1}^2(A) \\le \\|A(I-Q_kQ_k^T)\\|^2 \\le\n\\psi_{k+1}^2(A) + \\|A\\|^2 \\cdot \\sin^2(\\mathcal Z_k, \\mathcal Q_k).\n\\]\nMore precisely, we have\n\\[\n\\|A(I-Q_kQ_k^T)\\|^2 \\le \\psi_{k+1}^2(A) + \n\\sum_{j=1}^k \\psi_j(A)^2 \\cdot \\sin^2(\\mathbf z_j, \\mathcal Q_k).\n\\]\n\\end{proposition}\n\\begin{proof}\nThe lower bound follows from the SVD; the optimal $\\mathcal Q_k$ is $\\mathcal Z_k$.\nWe can derive the upper bounds from\n\\begin{align*}\n\\|A(I-Q_kQ_k^T)\\|^2 & = \\|W_k \\Psi_k Z_k^T (I-Q_kQ_k^T)\\|^2\n+ \\|W_{\\perp} \\Psi_{\\perp} Z_{\\perp}^T (I-Q_kQ_k^T)\\|^2 \\\\\n& \\le \\|A\\|^2 \\cdot \\sin^2(\\mathcal Z_k, \\mathcal Q_k)\n+ \\psi_{k+1}^2 \\cdot \\sin^2(\\mathcal Z_{\\perp}, \\mathcal Q_k) \\\\\n& \\le \\|A\\|^2 \\cdot \\sin^2(\\mathcal Z_k, \\mathcal Q_k) + \\psi_{k+1}^2.\n\\end{align*}\nFurthermore, more specifically,\n\\[\n\\|A(I-Q_kQ_k^T)\\|^2 = \\sum_{j=1}^k \\psi_j^2(A) \\ |\\mathbf z_j^T (I-Q_kQ_k^T)|^2\n+ \\|W_{\\perp} \\Psi_{\\perp} Z_{\\perp}^T (I-Q_kQ_k^T)\\|^2.\\]\n\\end{proof}\n\nThe significance of this result is that $\\|A(I-Q_kQ_k^T)\\|$ may be close to $\\psi_k(A)$ when $\\mathcal Q_k$ captures the largest singular vectors of $A$ well. For instance, in the standard CUR, $Q_k$ is equivalent to $Z_k$ so the quantity $\\sin^2(\\mathcal Z_k, \\mathcal Q_k)$ equals 0. Suppose the matrix $B$ from \\eqref{eq:mgsvd} is close to the identity or is a scaled identity, we expect that $\\sin^2(\\mathcal Z_k, \\mathcal Q_k)$ will be approximately zero. However, this sine will generally not be small, as we illustrate by the following example.\n\n\\begin{example} {\\rm\nLet $A = \\text{diag}(1,2,3)$, and $B = \\text{diag}(1,20,300)$.\nDenote by $\\mathbf e_j$ the $j$th standard basis vector.\nThen clearly $Z_1 = \\mathbf z_1 = \\mathbf e_3$, while the largest right generalized singular vector $\\mathbf q_1$ is equal to the largest right singular vector of $AB^{-1} = \\text{diag}(1, 0.1, 0.01)$, and hence $Q_1 = \\mathbf q_1 = \\mathbf e_1$.\nThis implies that $\\sin(\\mathcal Z_1, \\mathcal Q_1) = \\sin(\\mathbf z_1, \\mathbf q_1)$ is large.}\n\\end{example}\n\n\\subsection{Error Bounds in terms of the GSVD approximation}\nWith the above results in mind, instead of using the rank-$k$ SVD approximation error, we will derive error bounds for $\\norm{A-CMR}$ (see \\eqref{eq:gcur}) in terms of the error bounds of a rank-$k$ GSVD approximation of $A$ (see \\cref{pp1}).\nThe matrices $C$ and $R$ are of full-rank $k$ determined by the row and column index selection matrices $S$ and $P$, respectively and $M = C^+\\!AR^+$.  From \\cref{alg:GCUR-DEIM}, we know that $S$ and $P$ are derived using the $k$ columns of the matrices $U$ and $Y$, respectively, corresponding to the largest generalized singular value (see \\eqref{eq:tgsvd}).\n\nWe use the interpolatory projector given in \\cref{pp2}. Therefore instead of $Y$ (see \\eqref{eq:mgsvd}), we use its orthonormal basis $Q$, to exploit the properties of an orthogonal matrix.\n\nWe will now analyze the approximation error between $A$ and its interpolatory projection $A \\mathbb{P}$. The proof of the error bounds for the proposed method closely follows the one presented in \\cite{Sorensen}.\nThe second inequality of the first statement of \\cref{pp4} is in \\cite[Lemma~4.1]{Sorensen}. The first inequality of the first statement is new but completely analogous. In the second statement, we use the GSVD.\nFor the analysis, we need the following QR-decomposition of $Y$ (see \\eqref{eq:pgsvd}):\n\\begin{equation} \\label{eq:QR}\n[Y_k \\ \\ \\widehat Y] = Y = QT = [Q_k \\ \\ \\widehat Q]\n\\begin{bmatrix} T_k & T_{12} \\\\ 0 & T_{22} \\end{bmatrix} = [Q_k T_k \\ \\ Q \\widehat T],\n\\end{equation}\nwhere we have defined\n\\begin{equation} \\label{eq:T_hat}\n\\widehat T := \\begin{bmatrix} T_{12} \\\\ T_{22} \\end{bmatrix}.\n\\end{equation}\nThis implies that\n\\[\nA = A_k + \\widehat U \\, \\widehat \\Gamma \\, \\widehat Y^T = U_k \\Gamma_k Y_k^T + \\widehat U \\, \\widehat \\Gamma \\, \\widehat Y^T\n= U_k \\Gamma_k T_k Q_k^T + \\widehat U \\, \\widehat \\Gamma \\, \\widehat T Q^T.\n\\]\n\n\\begin{proposition} \\label{pp4} (Generalization of \\cite[Lemma~4.1]{Sorensen})\nGiven $A \\in \\mathbb R^{m\\times n}$ and $Q_k \\in \\mathbb R^{n\\times k}$ with orthonormal columns, let $P \\in \\mathbb R^{n\\times k}$ be a selection matrix and $Q_k^TP$ be nonsingular. Let $\\mathbb{P} = P(Q_k^TP)^{-1}Q_k^T$, then\n\\[\\psi_{\\min}(A(I - Q_kQ_k^T))~\\norm{(Q_k^TP)^{-1}}\\le \\norm{A-A\\mathbb{P}} \\leq \\norm{A(I - Q_kQ_k^T)}~ \\norm{(Q_k^TP)^{-1}}.\\]\nIn particular, if $Q_k$ is an orthonormal basis for $Y_k$, the first $k$ columns of $Y$, then\n\\[\n\\gamma_{k+1}\\cdot \\psi_{\\min}(T_{22}) \\cdot \\norm{(Q_k^TP)^{-1}}\\le \\norm{A-A\\mathbb{P}} \\le \\gamma_{k+1} \\cdot \\norm{T_{22}} \\cdot \\norm{(Q_k^TP)^{-1}}.\n\\]\n\\end{proposition}\n\\begin{proof} We have that $Q_k^T\\mathbb{P}=Q_k^TP(Q_k^TP)^{-1}Q_k^T=Q_k^T$ implies $Q_k^T(I-\\mathbb{P}) = 0$.\nTherefore,\n\\[\\norm{A-A\\mathbb{P}}=\\norm{A(I-\\mathbb{P})} = \\norm{A(I - Q_kQ_k^T)(I-\\mathbb{P})}\\leq \\norm{A(I - Q_kQ_k^T)}~\\norm{I-\\mathbb{P}}\\]\nand also\n\\[\\norm{A(I - Q_kQ_k^T)(I-\\mathbb{P})} \\ge \\psi_{\\min}(A(I - Q_kQ_k^T))~\\norm{I-\\mathbb{P}} \\ .\\]\nNote that, if $\\mathbb{P} \\ne 0$ and $\\mathbb{P} \\ne I$ (see, e.g., \\cite{Daniel}) then\n\\[\\norm{I-\\mathbb{P}} =\\norm{\\mathbb{P}} =\\norm{(Q_k^TP)^{-1}}.\\]\nWith $A = U \\, \\Gamma \\,Y^T$, $A_k = U_k\\Gamma_kY_k^T$, $Y=QT$ and $Y_k=Q_kT_k$ we have\n\\begin{align*}\nA\\ Q_k\\, Q_k^T &= \\big[U_k \\ \\ \\widehat U \\big] \\begin{bmatrix} \\Gamma_k & 0\\\\ 0 & \\widehat \\Gamma \\end{bmatrix} \\begin{bmatrix} T_k^T & 0 \\\\[0.5mm] T_{12}^T & T_{22}^T \\end{bmatrix} \\begin{bmatrix} I_k \\\\ 0 \\end{bmatrix} Q_k^T \\\\\n&= U_k\\Gamma_kT_k^TQ_k^T + \\widehat U \\, \\widehat \\Gamma \\, T_{12}^TQ_k^T\n\\end{align*}\nand hence\n\\begin{align*}\nA\\,(I-Q_kQ_k^T) & = (A-A_k)-\\widehat U \\, \\widehat \\Gamma \\, T_{12}^TQ_k^T \\\\\n& = \\widehat U \\, \\widehat \\Gamma \\, \\widehat T^T Q^T - \\widehat U \\, \\widehat \\Gamma \\, T_{12}^TQ_k^T = \\widehat U \\, \\widehat \\Gamma \\, T_{22}^T \\widehat Q^T.\n\\end{align*}\nThis implies\n\\[\\norm{A\\,(I - Q_kQ_k^T)} \\le \\gamma_{k+1} \\cdot \\norm{T_{22}}\\]\nand\n\\[\\norm{A\\,(I - Q_kQ_k^T)} \\ge \\gamma_{k+1} \\cdot \\psi_{\\min}(T_{22}). \\]\n\\end{proof}\n\nLet us now consider the operation on the left-hand side of $A$. Given the set of interpolation indices $\\{s_1, \\dots, s_k\\}$ determined from $U_k$, $S = [\\mathbf e_{s_1},\\dots,\\mathbf e_{s_k}]$  and for a nonsingular $S^TU_k$, we have the DEIM interpolatory projector $\\mathbb{S}=U_k(S^TU_k)^{-1}S^T$. Since $U_k$ consists of the dominant $k$ left generalized singular vectors of $A$ and has orthonormal columns, it is not necessary to perform a QR-decomposition as we did in \\cref{pp4}.\n\nThe following proposition is analogous to \\cref{pp4}. The results are similar to those in \\cite[p.~A1461]{Sorensen} except that here, we use the approximation error of the GSVD instead of the SVD.\n\\begin{proposition}\n\\label{pp4b} Given $U_k \\in \\mathbb R^{m\\times k}$ with orthonormal columns, let $S \\in \\mathbb R^{m\\times k}$ be a selection matrix and  $S^TU_k$ be nonsingular. Furthermore, let $\\mathbb{S} = U_k(S^TU_k)^{-1}S^T$, then, with $\\widehat T$ as in \\eqref{eq:T_hat},\n\\[\\gamma_{k+1}\\cdot\\psi_{\\min}(\\widehat{T})\\cdot\\norm{(S^T\\!U_k)^{-1}}\\le\\norm{A-\\mathbb{S}A}\\leq \\gamma_{k+1}\\cdot\\norm{\\widehat{T}} \\cdot \\norm{(S^T\\!U_k)^{-1}}. \\]\n\\end{proposition}\n\\begin{proof}\nWe have\n\\[\\norm{A-\\mathbb{S}A} =\\norm{(I-\\mathbb{S})A}=\\norm{(I-\\mathbb{S})(I-U_kU_k^T)A}. \\]\nSimilar to before, if $\\mathbb{S} \\ne 0$ and $\\mathbb{S} \\ne I$ then \n\\[\\norm{I-\\mathbb{S}} =\\norm{\\mathbb{S}} =\\norm{(S^T\\!U_k)^{-1}}.\\] \nSince $(I-U_k U_k^T) A = A-U_k \\Gamma_k Y_k^T = \\widehat U \\, \\widehat \\Gamma \\, \\widehat Y^T = \\widehat U \\, \\widehat \\Gamma \\, \\widehat T^T\\!Q^T$ we get\n\\[\\norm{(I-U_kU_k^T)A}=\\norm{A-A_k} \\leq \\gamma_{k+1}\\cdot\\norm{\\widehat{T}},\\]\nand $\\norm{(I-U_kU_k^T)A} \\ge \\gamma_{k+1} \\cdot \\psi_{\\min}(\\widehat{T})$, from which the result follows. \n\\end{proof}\n\nWe will now use \\cref{pp4,pp4b}  to find a bound for the approximation error of the GCUR of $A$ relative to $B$. As in \\cite{Sorensen} we first show in the following proposition that the error bounds of the interpolatory projection of $A$ onto the chosen rows and columns apply equally to the orthogonal projections of $A$ onto the same row and column spaces. \n\\begin{proposition}\\label{pp5}(Generalization and slight adaptation of \\cite[Lemma~4.2]{Sorensen}) Given the selection matrices $P$, $S$, let $C=AP$ and $R=S^T\\!A$. Suppose that $C \\in \\mathbb R^{m \\times k}$ and $R \\in \\mathbb R^ {k \\times n}$ are full rank matrices with $k< \\min(m,n)$, and that $Q_k^TP$ and $S^TU_k$ are nonsingular. With $\\widehat T$ and $T_{22}$ as in \\eqref{eq:QR}--\\eqref{eq:T_hat}, we have the bound for the orthogonal projections of $A$ onto the column and row spaces: \n\\begin{align*}\n\\norm{(I-CC^+)A} & \\le \\gamma_{k+1} \\cdot \\norm{T_{22}} \\cdot \\norm{(Q_k^T\\!P)^{-1}}, \\\\\n\\norm{A(I-R^+\\!R)} & \\le \\gamma_{k+1} \\cdot \\norm{\\widehat{T}} \\cdot \\norm{(S^T\\!U_k)^{-1}}.\n\\end{align*}\n\\end{proposition}\n\\begin{proof} This proof is a minor modification of that of \\cite[Lemma~4.2]{Sorensen}, we closely follow their proof technique.  With $C=AP$ of full rank, we have $C^+=(P^T\\!A^T\\!AP)^{-1}(AP)^T$. With this, the orthogonal projection of $A$ onto $C$ can be stated as\n\\[CC^{+\\!}A=(AP(P^T\\!A^T\\!AP)^{-1}P^T\\!A^T)A.\\]\nLet $\\Pi_P =P(P^T\\!A^T\\!AP)^{-1}P^T\\!A^T\\!A$, note that $\\Pi_P P = P$ since $\\Pi_P$ is an oblique projector on Range$(P)$. We can rewrite $CC^+\\!A$ as $CC^+\\!A=A\\Pi_P$.\nHence the error in the orthogonal projection of $A$  will be\n$(I-CC^+)A=A(I-\\Pi_P)$.\nSince $\\Pi_P  \\mathbb{P}= \\mathbb{P}$, we have\n\\[A(I-\\Pi_P)=A(I-\\Pi_P)(I-\\mathbb{P})=(I-CC^+)A(I-\\mathbb{P}),\\]\ntherefore\n\\begin{align*}\n\\norm{(I-CC^+)A} &=\\norm{A(I-\\Pi_P)}=\\norm{(I-CC^+)A(I-\\mathbb{P})} \\\\\n&\\leq \\norm{(I-CC^+)}~\\norm{A(I-\\mathbb{P})}.\n\\end{align*}\nWith $C$ being nonsquare, $\\norm{I-CC^+} =1$ (see, e.g., \\cite{Daniel})\nand $\\norm{A(I-\\mathbb{P})}\\leq \\gamma_{k+1}\\cdot \\norm{T_{22}} \\cdot \\norm{(Q_k^T\\!P)^{-1}}$ from \\cref{pp4}, we have\n\\[\\norm{(I-CC^+)A}\\leq \\gamma_{k+1}\\cdot \\norm{T_{22}} \\cdot \\norm{(Q_k^T\\!P)^{-1}}.\\]\nIn a similar vein, with $R=S^T\\!A$ and $R^+=R^T(RR^T)^{-1}$ we have $R^+=A^TS(S^T\\!AA^T\\!S)^{-1}$ and the error in the orthogonal projection of $A$ is $A(I-R^+\\!R) =(I-\\Pi_S)A$, where $\\Pi_S = AA^T\\!S(S^T\\!AA^T\\!S)^{-1}S^T$, so that\n\\[(I-\\Pi_S)A=(I-\\mathbb{S})(I-\\Pi_S)A=(I-\\mathbb{S})A(I-R^+\\!R)\\]\nand\n\\[\n\\norm{A(I-R^+\\!R)}\\leq \\norm{(I-\\mathbb{S})A}~\\norm{(I-R^+\\!R)} \\leq \\gamma_{k+1} \\cdot \\norm{\\widehat{T}} \\cdot \\norm{(S^T\\!U_k)^{-1}}. \n\\]\n\\end{proof}\n\nThis result helps to prove an error bound for the GCUR approximation error. For the following theorem, we again closely follow the approach of \\cite{Sorensen} which also follows a procedure in \\cite{Mahoney}.\nAs stated in \\cref{Dfn4} the middle matrix can be computed as $M=(C^T\\!C)^{-1}C^T\\!A R^T(R R^T )^{-1}=C^+\\!AR^+$.\n\\begin{theorem} \\label{theorem1} (Generalization of \\cite[Thm.~4.1]{Sorensen}) Given $A \\in \\mathbb R^{m\\times n}$ and $Y_k$, $U_k$ from \\eqref{eq:tgsvd}, let $P$ and $S$ be  selection matrices so that $C=AP$ and $R=S^T\\!A$ are of full rank. Let $Q_k \\in \\mathbb R^{n\\times k}$ be the $Q$-factor of $Y_k$, and $\\widehat{T}$ and $T_{22}$ as in \\eqref{eq:QR}--\\eqref{eq:T_hat}. Assuming $Q_k^T\\!P$ and $S^T\\!U_k$ are nonsingular, then with the error constants\n\\[\\eta_p := \\norm{(Q_k^T\\!P)^{-1}}, \\qquad \\eta_s := \\norm{(S^T\\!U_k)^{-1}},\\]\nwe have\n\\[\\norm{A-CMR} \\leq \\gamma_{k+1} \\cdot (\\eta_p\\cdot\\norm{T_{22}} +\\eta_s\\cdot\\norm{\\widehat{T}}) \\le \\gamma_{k+1} \\cdot (\\eta_p + \\eta_s) \\cdot \\norm{\\widehat{T}}.\\]\n\\end{theorem}\n\\begin{proof}\nBy the definition of $M$, we have\n\\[A-CMR=A-CC^+\\!AR^+\\!R=(I-CC^+)A+CC^+\\!A(I-R^+\\!R),\\]\nusing the triangle inequality, it follows that\n\\[\\norm{A-CMR}=\\norm{A-CC^+\\!AR^+\\!R}\\leq \\norm{(I-CC^+)A}+\\norm{CC^+}~\\norm{A(I-R^+\\!R)}\\]\nand the fact that $CC^+$ is an orthogonal projection with $\\norm{CC^+}=1$,\n\\[\\norm{A-CMR} \\le \\gamma_{k+1} \\cdot \\norm{T_{22}} \\cdot \\norm{(Q_k^T\\!P)^{-1}}+\\norm{(S^T\\!U_k)^{-1}}~\\norm{\\widehat{T}} \\cdot \\gamma_{k+1}.\\]\n\\end{proof}\n\nThe last line of \\cref{theorem1} can be related to the results in \\cite[Thm~4.1]{Sorensen}; both theorems have the factors $\\eta_p$ and $\\eta_s$. In \\cite{Sorensen} the error of the CUR approximation of $A$ is within a factor of $\\eta_p + \\eta_s$ of the best rank-$k$ approximation (SVD). Here we have the bounds in terms of $\\gamma_{k+1}$ from the GSVD \\eqref{gsvd} and the additional factors $\\norm{\\widehat{T}}$ and $\\norm{T_{22}}$. The results presented in this subsection suggest that a good index selection procedures that minimize $\\norm{(Q_k^T\\!P)^{-1}}$ and $\\norm{(S^T\\!U_k)^{-1}}$ are theoretically desirable. We note that where these results have been presented for matrix $A$ in \\eqref{eq:gcur}, similar results can be obtained for $B$. \n\n\\section{Numerical experiments}\\label{sec:EXP}\nWe now present the results of a few numerical experiments to illustrate the performance of GCUR for low-rank matrix approximation. For the first two experiments, we consider a case where a data matrix $A$ is corrupted by a random additive noise $E$ and the covariance of this noise (the expectation of $E^T\\!E$) is not a multiple of the identity matrix. We are therefore interested in a method that can take the actual noise into account. Traditionally, a pre-whitening matrix $R^{-1}$ (where $R$ is the Cholesky factor of the noise's covariance matrix) may be applied to the perturbed matrix \\cite{hansen}, so that one can use SVD-based methods on the transformed matrix. With a GSVD formulation, the pre-whitening operation becomes an integral part of the algorithm \\cite{hansen2007subspace}; we do not need to explicitly compute $R^{-1}$ and transform the perturbed matrix. We show in the experiments that using SVD-based methods without pre-whitening the perturbed data yields less accurate approximation results of the original matrix. \n\nFor the last two experiments, we consider a setting where there are two data sets collected under different conditions e.g., treatment and control experiment where the former has distinct variation caused by the treatment or signal-free and signal recordings where the former contains only noise. We are interested in exploring and identifying patterns and discriminative features that are specific to one data set.  \n\n\\begin{experiment}\\label{exp:1} {\\rm\nThis experiment is an adaptation of experiments in \\cite[p.~66: Sect.~3.4.4]{hansen} and \\cite[Ex.~6.1]{Sorensen}. We construct matrix $A$ to be of a known modest rank. We then perturb this matrix with a noise matrix $E \\in \\mathbb R^{m \\times n}$ whose entries are correlated. Given $A_E= A+E$, we evaluate and compare the GCUR and the CUR decomposition on $A_E$  in terms of recovering the original matrix $A$. Specifically, the performance of each decomposition is assessed based on the 2-norm of the relative matrix approximation error i.e., $\\norm{A-\\widetilde {A} }\/\\norm{A}$, where $\\widetilde {A}$ is the approximated low-rank matrix. \nWe present the numerical results for four noise levels; thus $E=\\varepsilon\\, \\frac{\\norm{F}}{\\norm{A}} F$ where $\\varepsilon$ is the parameter for the noise level and $F$ is a randomly generated correlated noise. We first generate a sparse, nonnegative rank-50 matrix $A \\in \\mathbb R^{m \\times n}$, with $m=100000$ and $n=300$, of the form \n\\[A=\\sum_{j=1}^{10}\\frac{2}{j}\\, \\mathbf x_j\\ \\mathbf y_j^T + \\sum_{j=11}^{50}\\frac{1}{j}\\, \\mathbf x_j\\ \\mathbf y_j^T,\\]\nwhere $\\mathbf x_j \\in \\mathbb R^{m}$ and $\\mathbf y_j \\in \\mathbb R^{n}$ are sparse vectors with random nonnegative entries (i.e., $\\mathbf x_j={\\sf sprand}(m,1,0.025)$ and $\\mathbf y_j={\\sf sprand}(n,1,0.025)$, just as in \\cite{Sorensen}. Unlike \\cite{Sorensen} we then perturb the matrix with a correlated Gaussian noise $E$ whose entries have zero mean and a Toeplitz covariance structure (in MATLAB $\\text{desired-cov}(F)={\\sf toeplitz}(0.99^0$, $\\dots$, $0.99^{n-1})$, $R={\\sf chol}(\\text{desired-cov}(F))$, and $F= {\\sf randn}(m,n)\\cdot R)$ and $\\varepsilon \\in \\{0.05$, $0.1$, $0.15$, $0.2\\}$. We compute the SVD of $A_E$  and the GSVD of $(A_E,R)$ to get the input matrices for the CUR and the GCUR decomposition respectively. \\Cref{fig:a,fig:b,fig:c,fig:d} compare the relative errors of the proposed DEIM-GCUR (see \\cref{alg:GCUR-DEIM}) and the DEIM-CUR (see \\cref{alg:CUR-DEIM}) for reconstructing the low-rank matrix $A$ for different noise levels. \nWe observe that for higher noise levels the GCUR technique gives a more accurate low-rank approximation of the original matrix $A$. The DEIM-GCUR scheme seems to perform distinctly well for higher noise levels and moderate values of $k$. As indicated in \\cref{sec:GCUR}, the GCUR method is slightly more expensive since it requires the computation of the TGSVD instead of the TSVD. We observe that, as $k$ approaches rank$(A)$, the relative error of the TGSVD continues to decrease; this is not true for the GCUR. We may attribute this phenomenon to the fact that the relative error is saturated by the noise considering we pick actual columns and rows of the noisy data. Since $\\varepsilon$ indicates the relative noise level, it is, therefore, natural that for increasing $k$, the quality of the TSVD approximation rapidly approaches $\\varepsilon$. For this experiment, we assume that an estimate of the noise covariance matrix is known and therefore we have the exact Cholesky factor; we stress that this may not always be the case. \n\n We now show an example where we use an inexact Cholesky factor $\\widehat R$. We derive $\\widehat R$ by multiplying all off-diagonal elements of the exact Cholesky factor $R$ by factors which are uniformly random from the interval $[0.9, 1.1]$. Here, the experiment setup is the same as described above with the difference that we compute the GSVD of $(A_E,\\widehat R)$ instead. In \\cref{fig:2a,fig:2b}, we observe that the GCUR and the GSVD still deliver good approximation results even for an inexact Cholesky factor $\\widehat R$ which may imply that we do not necessarily need the exact noise covariance. \n\n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\subfloat[$\\varepsilon=0.2$.]{\\label{fig:a}{\\includegraphics[width=0.48\\textwidth]{Plots\/GCUR_plots\/0.2.png}}}\\hspace{0.1pt}\n\t\\subfloat[$\\varepsilon=0.15$.]{\\label{fig:b}{\\includegraphics[width=0.48\\textwidth]{Plots\/GCUR_plots\/0.15.png} }}\n\t\n\t\\subfloat[$\\varepsilon=0.1$.]{\\label{fig:c}{\\includegraphics[width=0.48\\textwidth,]{Plots\/GCUR_plots\/0.1.png} }}\\hfill\n\t\\subfloat[$\\varepsilon=0.05$.]{\\label{fig:d}{\\includegraphics[width=0.48\\textwidth]{Plots\/GCUR_plots\/0.05.png} }}\n\t\\caption{Accuracy of the DEIM-GCUR approximations compared with the standard DEIM-CUR approximations in recovering a sparse, nonnegative matrix $A$ perturbed with correlated Gaussian noise (\\cref{exp:1}) using exact Cholesky factor of the noise covariance. The relative errors $\\norm{A-\\widetilde {A}_k }\/\\norm{A}$ (on the vertical axis) as a function of rank $k$ (on the horizontal axis) for $\\varepsilon=0.2$, $0.15$, $0.1$, $0.05$, respectively.\\label{fig:1}}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\subfloat[$\\varepsilon=0.15$.]{\\label{fig:2a}{\\includegraphics[width=0.48\\textwidth]{Plots\/GCUR_plots\/0.15_inexact.png}}}\\hspace{0.1pt}\n\t\\subfloat[$\\varepsilon=0.1$.]{\\label{fig:2b}{\\includegraphics[width=0.48\\textwidth]{Plots\/GCUR_plots\/0.1_inexact.png} }}\n\t\n\t\\caption{Accuracy of the DEIM-GCUR approximations compared with the standard DEIM-CUR approximations in recovering a sparse, nonnegative matrix $A$ perturbed with correlated Gaussian noise (\\cref{exp:1}) using an inexact Cholesky factor of the noise covariance. The relative errors $\\norm{A-\\widetilde {A}_k }\/\\norm{A}$ (on the vertical axis) as a function of rank $k$ (on the horizontal axis) for $\\varepsilon=0.15$, $0.1$, respectively.\\label{fig:2}}\n\\end{figure}\n}\n\\end{experiment}\n\n\n\\begin{experiment}\\label{exp:2}{\\rm For this experiment, we maintain all properties of matrix $A_E$ mentioned in the preceding experiment except for the column size that we reduce to 10000 (i.e., $A_E \\in \\mathbb R^{10000 \\times 300}$) and instead of a sparse nonnegative matrix $A$, we generate a dense random matrix $A$. As in \\cite[Ex.~6.2]{Sorensen}, we also modify $A$ so that there is a significant drop in the 10th and 11th singular values. The matrix $A$ is now of the form  \n\\[A=\\sum_{j=1}^{10}\\frac{1000}{j}\\, \\mathbf x_j\\ \\mathbf y_j^T + \\sum_{j=11}^{50}\\frac{1}{j}\\, \\mathbf x_j\\ \\mathbf y_j^T,\\]\nFor each fixed $\\varepsilon$ and $k$, we repeat the process 100 times and then compute the average relative error. The results in \\cref{tab:1} show that the advantage of the GCUR over the CUR still remains even when singular values of the original matrix $A$ decrease more sharply. We observe that the difference in the relative error of the GCUR and the CUR is quite significant when the rank of the recovered matrix $\\widetilde A$ is lower than that of $A$ (i.e., $k \\ll 50$).\n\n\\begin{table}[ht!]\n\n\\centering\n{\\caption{Comparison of the qualities $\\norm{A-\\widetilde {A}_k }\/\\norm{A}$ of the TSVD, TGSVD, CUR, and GCUR approximations as a function of index $k$ and noise level $\\varepsilon$ in \\cref{exp:2}. The relative errors are the averages of 100 test cases.}\\label{tab:1}\n{\\footnotesize\n\\begin{tabular}{clcccc} \\hline \\rule{0pt}{3mm}\n$k$ & Method $\\backslash \\ \\varepsilon$ & $0.05$ & $0.1$ & $0.15$ & $0.2$ \\\\ \\hline \\rule{0pt}{3.5mm}\n10 & TSVD & $0.008$ &   $0.045$    & $0.150$   & $0.200$\\\\\n& TGSVD & $0.002$ &    $0.003$ &    $0.005$ & $0.007$\\\\[0.5mm]\n& CUR & $0.052$   & $0.118$ & $0.141$ &  $0.186$\\\\\n& GCUR & $0.053$ &   $0.088$ &   $0.112$ &   $0.134$\\\\[0.5mm] \\hline \\rule{0pt}{3.5mm}\n15 & TSVD & $0.050$ &   $0.100$    & $0.150$   & $0.200$\\\\\n& TGSVD & $0.009$ &    $0.017$ &    $0.026$ & $0.035$\\\\[0.5mm]\n& CUR & $0.049$   & $0.097$ & $0.146$ &  $0.196$\\\\\n& GCUR & $0.046$ &   $0.091$ &   $0.138$ &   $0.185$\\\\[0.5mm] \\hline \\rule{0pt}{3.5mm}\n20 & TSVD & $0.050$ &  $0.100$  &  $0.150$   &  $0.200$\\\\\n& TGSVD & $0.011$  &  $0.023$  &  $0.034$  &  $0.015$\\\\[0.5mm]\n& CUR & $0.050$  &  $0.099$  &  $0.149$  &  $0.199$\\\\\n& GCUR & $0.049$   &  $0.097$  &  $0.146$  & $ 0.198$ \\\\[0.5mm] \\hline\n\\rule{0pt}{3.5mm}\n30 & TSVD & $0.050$ &   $0.100$    & $0.150$  &  $0.200$\\\\\n& TGSVD & $0.016$ &    $0.031$ &    $0.047$ & $0.063$\\\\[0.5mm]\n& CUR & $0.050$   & $0.100$ & $0.150$ &  $ 0.199$\\\\\n& GCUR & $0.050$ &   $0.099$ &   $0.149$ &   $0.199$\\\\ \\hline\n\\end{tabular}}}\n\\end{table}\n\n\nThe higher the noise level, the more advantageous the GCUR scheme may be over the CUR one. Especially for moderate values of $k$ such as $k=10$, the GCUR approximations are of better quality than those based on the CUR. For higher values of $k$ such as $k=30$, the approximation quality of the CUR and GCUR method become comparable since they both start to pick up the noise in the data columns. In this case, the GCUR does not improve on the CUR. Since it is a discrete method, picking indices for columns instead of generalized singular vectors, we see that the GCUR method yields worse results than the TGSVD approach.\n}\n\\end{experiment}\n\n\\begin{experiment}\\label{exp:3}{\\rm Our next experiment is adapted from \\cite{Abid}. We create synthetic data sets which give an intuition for settings where the GSVD and the GCUR may resolve the problem of subgroups. Consider a data set of interest (target data), $A$, containing 400 data points in a 30-dimensional feature space. This data set has four subgroups ({\\color{aqua} blue}, {\\color{yellow} yellow}, {\\color{orange} orange}, and {\\color{darkmagenta} purple}), each of 100 data points. The first 10 columns for all 400 data points are randomly sampled from a normal distribution with a mean of 0 and a variance of 100. The next 10 columns of two of the subgroups ({\\color{aqua} blue} and {\\color{orange} orange}) are randomly sampled from a normal distribution with a mean of 0 and a unit variance  while the other two subgroups ({\\color{yellow} yellow} and {\\color{darkmagenta} purple}) are randomly sampled from a normal distribution with a mean of 6 and a unit variance. The last 10 columns of subgroups {\\color{aqua} blue} and {\\color{yellow} yellow} are sampled from a normal distribution with a mean of 0 and a unit variance and those of {\\color{darkmagenta} purple} and {\\color{orange} orange} are sampled from  a normal distribution with a mean of 3 and a unit variance.\n\nOne of the goals of the SVD (or the related concept principal component analysis) in dimension reduction is to find a low-dimensional rotated approximation of a data matrix while maximizing the variances.\nWe are interested in reducing the dimension of $A$. If we project the data onto the two leading right singular vectors, we are unable to identify the subgroups because the variation along the first 10 columns is significantly larger than in any other direction, so some combinations of those columns are selected by the SVD. \n\nSuppose we have another data set $B$ (a background data set), whose first 10 columns are sampled from a normal distribution with a mean of 0 and a variance of 100, the next 10 columns are sampled from a normal distribution with a mean of 0 and a variance of 9 and the last 10 columns are sampled from a normal distribution with a mean of 0 and a unit variance. The choice of the background data set is key in this context. Generally, the background data set should have the structure we would like to suppress in the target data, which usually corresponds to the direction with high variance but not of interest for the data analysis \\cite{Abid}. With the new data, one way to extract discriminative features for clustering the subgroups in $A$ is to maximize the variance of $A$ while minimizing that of $B$, which leads to a trace ratio maximization problem \\cite{chen}\n\\[\\widehat U := \\argmax_{U \\in \\mathbb R^{n \\times k}, \\ U^TU=I_k}~ \\text{Tr}~\\big[(U^T\\!B^T\\!B\\,U)^{-1}(U^T\\!A^T\\!A\\,U)\\big],\\]\nwhere $n=30$ and $k=5$ or $k=10$.  \nBy doing this, the first dimensions are less likely to be selected because they also have a high variance in data set $B$. Instead, the middle and last dimensions of $A$ are likely to be selected as they have the dimensions with the lowest variance in $B$, thereby allowing us to separate all four subgroups. The solution $\\widehat U \\in \\mathbb R^{n \\times k}$ to the above problem is given by the $k$ (right) eigenvectors of $(B^T\\!B)^{-1}(A^T\\!A)$ corresponding to the $k$ largest eigenvalues (cf., \\cite[pp.~448--449]{fukunaga2013}); this corresponds to the (``largest'') right generalized singular vectors of $(A,B)$ (the transpose of \\eqref{eqn:matrix_Y}). As seen in \\cref{fig:3}, projecting $A$ onto the leading two right generalized singular vectors produces a much more clearer subgroup separation (top-right figure) than projecting onto the leading two right singular vectors (top-left figure). Therefore, we can expect that a CUR decomposition based on the SVD will also perform not very well with the subgroup separation. In the bottom figures is a visualization of the data using the first two important columns selected using the DEIM-CUR (left figure) and the DEIM-GCUR (right figure). To a large extent, the GCUR is able to differentiate the subgroups while the CUR fails to do so. We investigate this further by comparing the performance of subset selection via DEIM-CUR on $A$ (\\cref{alg:CUR-DEIM}) and DEIM-GCUR on $(A, B)$ (\\cref{alg:GCUR-DEIM}) in identifying the subgroup or class representatives of $A$; we select a subset of the columns of $A$ (5 and 10) and compare the classification results of each method. We center the data sets by subtracting the mean of each column from all the entries in that column. Given the class labels of the subgroups, we perform a ten-fold cross-validation (i.e., split the data points into 10 groups and for each unique group take the group as test data and the rest as training \\cite[p.~181]{james2013introduction}) and apply two classifiers on the reduced data set: ECOC (Error Correcting Output Coding) \\cite{dietterich1994} and {\\em classification tree} \\cite{banfield2006} using the functions {\\sf fitcecoc} and {\\sf fitctree} with default parameters as implemented in MATLAB. It is evident from \\cref{tab:2} that the TGSVD and the GCUR achieve the least classification error rate, e.g., for reducing the dimension from 30 to 10; 0\\% and 6.3\\% respectively, using the ECOC classifier and 0\\% and 9.5\\%  respectively, using the tree classifier. The standard DEIM-CUR method achieves the worst classification error rate.\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=\\textwidth]{Plots\/GCUR_plots\/svd_gsvd_2.png}\\\\\n    \\includegraphics[width=\\textwidth]{Plots\/GCUR_plots\/cur_gcur.png}\n    \\caption{(Top-left) We project the synthetic data containing four subgroups onto the first two dominant right singular vectors. The lower-dimensional representation using the SVD does not effectively separate the subgroups. In the bottom-left figure, we visualize the data using the first two columns selected by DEIM-CUR. (Top-right) We illustrate the advantage of using GSVD by projecting the data onto the first two dominant right generalized singular vectors corresponding to the two largest generalized singular values. In the bottom-right figure, we visualize the data using the first two columns selected by DEIM-GCUR. The lower-dimensional representation of the data using the GSVD-based methods clearly separates the four clusters while the SVD-based methods fail to do so.}\n    \\label{fig:3}\n\\end{figure}\n\n\\begin{table}[ht!]\n\\centering\n{\\caption{$k$-Fold loss is the average classification loss overall 10-folds using SVD, GSVD, CUR, and GCUR as dimension reduction in \\cref{exp:3}. The second and third columns give information on the number of columns selected from the data set using the CUR and GCUR plus the number of singular and generalized singular vectors considered for the ECOC classifier. Likewise, the fifth and sixth columns for the tree classifier.}\\label{tab:2}\n{\\footnotesize\n\\begin{tabular}{lcclcc} \\hline \\rule{0pt}{3mm}%\nMethod & \\multicolumn{2}{c}{$k$-Fold Loss} & Method & \\multicolumn{2}{c}{$k$-Fold Loss}\\\\\n & 5 & 10 & & 5 &10\\\\ \\hline \\rule{0pt}{3.5mm}%\nTSVD+ECOC &$0.638$ &$0.490$ & TSVD+Tree  & $0.693$ & $0.555$  \\\\\nTGSVD+ECOC & $0\\phantom{.000}$& $0\\phantom{.000}$ & TGSVD+Tree  & $0\\phantom{.000}$ & $0\\phantom{.000}$ \\\\[0.5mm]\nCUR+ECOC & $0.793$ &$0.485$  & CUR+Tree  & $0.793$  & $0.540$  \\\\\n GCUR+ECOC & $0.055$&$0.063$  & GCUR+Tree & $0.075$  &$0.095$   \\\\[0.5mm] \\hline\n\\end{tabular}}}\n\n\\end{table}\n}\n\\end{experiment}\n\\begin{experiment}\\label{exp:4}{\\rm We will now investigate the performance of the GCUR compared to the CUR on a higher-dimensional public data sets. The data sets consists of single-cell RNA expression levels of bone marrow mononuclear cells (BMMCs) from an acute myeloid leukemia (AML) patient and two healthy individuals. We have data on the BMMCs before stem-cell transplant and the BMMCs after stem-cell transplant. We preprocess the data sets as described by the authors in \\cite{boileau2020} \\footnote{\\url{https:\/\/github.com\/PhilBoileau\/EHDBDscPCA\/blob\/master\/analyses\/}} keeping the 1000 most variable genes measured across all 16856 cells (patient-035: 4501 cells and two healthy individuals; one of 1985 cells and the other of 2472 cells). The data from the two healthy patients are combined to create a background data matrix of dimension $4457 \\times 1000$ and we use patient-035 data set as the target data matrix of dimension $4501 \\times 1000$ . Both data matrices are sparse: the  patient-035 data matrix has 1,628,174 nonzeros; i.e., about 36\\% of all entries are nonzero and the background data matrix has 1,496,229 nonzeros; i.e., about 34\\% of all entries are nonzero. We are interested in exploring the differences in the AML patient's BMMC cells  pre- and post-transplant.\nWe perform SVD, GSVD, CUR and GCUR on the target data (AML patient-035) to see if we can capture the biologically meaningful information relating to the treatment status. For the GSVD and the GCUR procedure the background data is taken into account. As evident in \\cref{fig:4}, the GSVD and the GCUR produce almost linearly separable clusters which corresponds to pre- and post-treatment cells. These methods evidently capture the biologically meaningful information relating to the treatment and are more effective at separating the pre- and post-transplant cell samples compared to the other two. For the SVD and the CUR, we observe that both cell types follow a similar distribution in the space spanned by the first three dominant right singular vectors and the first three important gene columns, respectively. Both methods fail to separate the pre- and post-transplant cells.  \n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=\\textwidth]{Plots\/GCUR_plots\/cell_035.png}\n    \\caption{Acute myeloid leukemia patient-035 scRNA-seq data.(Top-left) A 3-D projection of the patient's BMMCs on the first three dominant right singular vectors. In the bottom-left figure, we visualize the data using the first three genes selected by DEIM-CUR. The lower-dimensional representation using the SVD-based methods does not effectively give a discernible cluster of the pre- and post-transplant cells. (Top-right) We illustrate the advantage of using GSVD by projecting the patient's BMMCs onto the first three dominant right generalized singular vectors corresponding to the three largest generalized singular values. In the bottom-right figure, we visualize the data using the first three genes selected by DEIM-GCUR. The lower-dimensional representation using the GSVD-based methods produce a linearly separable clusters.}\n    \\label{fig:4}\n\\end{figure}\n}\n\\end{experiment}\n\\section{Conclusions}\\label{sec:Con} In this paper we propose a new method, the DEIM-induced GCUR (generalized CUR) factorization with pseudocode in \\cref{alg:GCUR-DEIM}. It is an extension of the DEIM-CUR decomposition for matrix pairs. When $B$ is square and nonsingular, there are close connections between the GCUR of $(A,B)$ and the DEIM-induced CUR of $AB^{-1}$. When $B$ is the identity, the GCUR decomposition of $A$ coincides with the DEIM-induced CUR decomposition of $A$. There exist a similar connection between the CUR of $AB^+$ and the GCUR of $(A, B)$ for a nonsquare but full-rank matrix $B$. \n\nThe DEIM index selection procedure independently selects the row and column indices defining the GCUR, which facilitates the parallelization of the decomposition. We note that we can restrict the generalized CUR decomposition discussed in this paper to a generalized interpolative decomposition (see \\eqref{eq:ID}). Although we used the generalized singular vectors here, in principle one could use other vectors, e.g., an approximation to the generalized singular vectors. Instead of the DEIM procedure, we can also use other index selection procedures of the  CUR-type factorization for the GCUR.  In our experiments, we choose the same number of columns and rows for the approximation of the original matrix. However, we do not need to choose the same number of columns and rows. We have extended the existing theory concerning the DEIM-CUR approximation error to this DEIM generalized CUR factorization; we derived the bounds of a rank-$k$ GCUR approximation of $A$ in terms of a rank-$k$ GSVD approximation of $A$.\n\nWhile a CUR decomposition acts on one data set, a GCUR factorization decomposes two data sets together. An implication of this is that we can use it in selecting discriminative features of one data set relative to another. For subgroup discovery and subset selection in a classification problem where two data sets are available, the new method can perform better than the standard DEIM-CUR decomposition as shown in the numerical experiments. The GCUR algorithm can also be useful in applications where a data matrix suffers from non-white (colored) noise. The GCUR algorithm can provide more accurate approximation results compared to the DEIM-CUR algorithm when recovering an original matrix with low rank from data with colored noise. For the recovery of data perturbed with colored noise, we need the Cholesky factor of an estimate of the noise covariance. However, as shown in the experiments, even for an inexact Cholesky factor the GCUR may still give good approximation results. We note that, while the GSVD always provides a more accurate result than the SVD regardless of the noise level, the GCUR decomposition  is particularly attractive for higher noise levels and moderate values of the rank of the recovered matrix compared to the CUR factorization. In other situations, both methods may provide comparable results. In addition, the GCUR decomposition is a discrete method, so choosing indices for columns and rows instead of the generalized singular vectors leads to worse results than the GSVD approach. \n\nComputationally, the DEIM-GCUR algorithm requires the input of the GSVD, which is of the same complexity but more expensive than the SVD required for DEIM-CUR. For the case where we are only interested in approximating the matrix $A$ from the pair $(A, B)$, we can omit some of the lines in \\cref{alg:GCUR-DEIM}; thus saving computational cost. \nIn the case that the matrices $A$ and $B$ are so large, a full GSVD may not be affordable; in this case, we can consider iterative methods (see, e.g., \\cite{Zwaan, hochstenbach, Zha}).\n\nWhile in this work we used the GCUR method in applications such as extracting information from one data set relative to another, we expect that its promise may be more general.\n\n\\bibliographystyle{siam}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Gamma rays from DM spikes around IMBHs}\n\\label{normalization}\nWe give simple numerical estimates that illustrate how the IMBH-spike scenario can readily account for the \\textit{Fermi} ``excess\" for very small annihilation cross-sections. Following  \\citet{agrawal17}, we  estimate the DM parameters that can reproduce the flux of one point source from the analysis of \\citet{lee16}. About $10^{3}$ such point sources are then needed to contribute of the order of the residual $\\gamma$-ray emission.\nThe integrated photon flux for a spike around an individual IMBH---referred to as a mini-spike in the following---between $E_{\\gamma,\\mathrm{min}} = 1.893\\, \\mathrm{GeV}$ and $E_{\\gamma,\\mathrm{max}} = 11.943\\, \\mathrm{GeV}$ \\citep{lee16} is given as usual by \n\\begin{equation}\n\\Phi_{\\mathrm{sp}} = \\dfrac{\\left\\langle \\sigma v \\right\\rangle}{2 m_{\\mathrm{DM}}^{2} d^{2}} \\left( \\int_{E_{\\gamma,\\mathrm{min}}}^{E_{\\gamma,\\mathrm{max}}} \\! \\dfrac{\\mathrm{d}N}{\\mathrm{d}E_{\\gamma}} \\, \\mathrm{d}E_{\\gamma}\\right) \\left( \\int_{0}^{R_{\\mathrm{sp}}} \\! r^{2} \\rho^{2}(r) \\, \\mathrm{d}r \\right),\n\\end{equation}\nwhere $\\left\\langle \\sigma v \\right\\rangle$ is the velocity-averaged annihilation cross-section, $m_{\\mathrm{DM}}$ the mass of the DM candidate, $\\mathrm{d}N\/\\mathrm{d}E_{\\gamma}$ the $\\gamma$-ray spectrum per annihilation---taken from \\citet{cirelli2011}, and $d \\approx 8.32\\ \\rm kpc$ \\citep{gillessen2017}, the distance between Earth and the IMBH. Our benchmark scenario is a DM candidate with $m_{\\mathrm{DM}} = 30\\ \\rm GeV$ annihilating into $b\\bar{b}$, compatible with the spectral properties of the GC residual $\\gamma$-ray emission. The DM profile in the mini-spike is defined as follows:\\footnote{We use a simplified expression based on \\citet{gondolo99}. The inner boundary of the spike is related to capture of DM particles by the BH and is given by $2R_{\\mathrm{S}}$ for a non-rotating BH \\citep{sadeghian13}. The precise value of this inner radius has no impact on our calculations. The DM profile outside the spike is not relevant since the dominant part of the annihilation flux comes from the inner region of the spike.} \n\\begin{equation}\n\\rho(r) = \n\\begin{cases}\n0 & r \\leqslant 2 R_{\\mathrm{S}} \\\\\n\\rho_{\\mathrm{sat}} & 2 R_{\\mathrm{S}} < r \\leqslant R_{\\mathrm{sat}} \\\\\n\\rho_{0} \\left( \\dfrac{r}{R_{\\mathrm{sp}}} \\right)^{-\\gamma_{\\mathrm{sp}}} & R_{\\mathrm{sat}} < r \\leqslant R_{\\mathrm{sp}}\n\\end{cases},\n\\end{equation}\nwhere the saturation density is given by $\\rho_{\\mathrm{sat}} = m_{\\mathrm{DM}}\/(\\left\\langle \\sigma v \\right\\rangle t_{\\mathrm{BH}})$ with $t_{\\mathrm{BH}}$ the  BH age, and $R_{\\mathrm{sat}} = R_{\\mathrm{sp}} (\\rho_{\\mathrm{sat}}\/\\rho_{0})^{-1\/\\gamma_{\\mathrm{sp}}}$ by continuity. The radial extension of the spike $R_{\\mathrm{sp}}$ is of the order of the BH influence radius, $G M_{\\mathrm{BH}}\/\\sigma_{*}^{2}$ \\citep{peebles72}. The extended $M_{\\mathrm{BH}}$-$\\sigma_{*}$ relation for IMBHs \\citep{tremaine2002} gives an estimated value of $\\sigma_{*} \\approx 10\\ \\rm km\\ s^{-1}$, and $R_{\\mathrm{sp}} \\approx 0.043\\, \\rm pc$. Then, $\\rho_{0} \\approx (3-\\gamma_{\\mathrm{sp}}) M_{\\mathrm{sp}}\/(4 \\pi R_{\\mathrm{sp}}^{3})$ by the requiring the mass inside the spike $M_{\\mathrm{sp}}$ be of the order of the BH mass, with $M_{\\mathrm{sp}} \\approx M_{\\mathrm{BH}} \\approx 10^{2}$--$10^{3}\\  M_{\\odot}$. For $\\gamma_{\\mathrm{sp}} > 3\/2$, the integrated flux for a single mini-spike reads\\footnote{For $\\gamma_{\\mathrm{sp}} = 3\/2$, the integrated flux for a single mini-spike is given by\n\\begin{align}\n\\Phi_{\\mathrm{sp}} \\approx \\ & 7 \\times 10^{-12} \\, \\mathrm{cm^{-2} \\, s^{-1}} \\ln \\left( 6 \\times 10^{6} \\left( \\dfrac{M_{\\mathrm{sp}}}{10^{3}\\, \\mathrm{M_{\\odot}}} \\right)^{-1}  \\left( \\dfrac{R_{\\mathrm{sp}}}{0.043\\, \\mathrm{pc}} \\right)^{3} \\right. \\nonumber \\\\\n& \\times \\left. \\left( \\dfrac{m_{\\mathrm{DM}}}{30\\, \\mathrm{GeV}} \\right) \\left( \\dfrac{\\left\\langle \\sigma v \\right\\rangle}{3 \\times 10^{-31} \\, \\mathrm{cm^{3}\\, s^{-1}}} \\right)^{-1} \\left( \\dfrac{t_{\\mathrm{BH}}}{10^{10}\\, \\mathrm{yr}} \\right)^{-1}  \\right) \\nonumber \\\\\n& \\times \\left( \\dfrac{d}{8.32\\, \\mathrm{kpc}} \\right)^{-2} \\left( \\dfrac{M_{\\mathrm{sp}}}{10^{3}\\, \\mathrm{M_{\\odot}}} \\right)^{2} \\left( \\dfrac{R_{\\mathrm{sp}}}{0.043\\, \\mathrm{pc}} \\right)^{-3} \\nonumber \\\\\n& \\times \\left( \\dfrac{m_{\\mathrm{DM}}}{30\\, \\mathrm{GeV}} \\right)^{-2} \\left( \\dfrac{\\left\\langle \\sigma v \\right\\rangle}{3 \\times 10^{-31} \\, \\mathrm{cm^{3}\\, s^{-1}}} \\right) \\left( \\dfrac{N_{\\gamma}^{(\\mathrm{tot})}}{5.7} \\right). \\nonumber\n\\end{align}}\n\\begin{align}\n\\label{spike_flux}\n\\Phi_{\\mathrm{sp}} =\\ & \\dfrac{\\gamma_{\\mathrm{sp}}}{3(2 \\gamma_{\\mathrm{sp}}-3)} \\left( \\dfrac{3-\\gamma_{\\mathrm{sp}}}{4\\pi} \\right)^{\\frac{3}{\\gamma_{\\mathrm{sp}}}} \\dfrac{1}{d^{2}} M_{\\mathrm{sp}}^{\\frac{3}{\\gamma_{\\mathrm{sp}}}} \\nonumber \\\\\n& \\times R_{\\mathrm{sp}}^{3\\left( 1-\\frac{3}{\\gamma_{\\mathrm{sp}}}\\right) } t_{\\mathrm{BH}}^{\\frac{3}{\\gamma_{\\mathrm{sp}}}-2} m_{\\mathrm{DM}}^{-\\frac{3}{\\gamma_{\\mathrm{sp}}}} \\left\\langle \\sigma v \\right\\rangle^{\\frac{3}{\\gamma_{\\mathrm{sp}}}-1} N_{\\gamma}^{(\\mathrm{tot})},\n\\end{align}\nwhere \n$N_{\\gamma}^{(\\mathrm{tot})} = \\int_{E_{\\gamma,\\mathrm{min}}}^{E_{\\gamma,\\mathrm{max}}} \\! (\\mathrm{d}N\/\\mathrm{d}E_{\\gamma}) \\, \\mathrm{d}E_{\\gamma}$. More specifically, for $\\gamma_{\\mathrm{sp}} = 9\/4$, the flux from a mini-spike is\n\\begin{align}\n\\Phi_{\\mathrm{sp}} \\approx & \\ 1 \\times 10^{-10}\\, \\mathrm{cm^{-2}\\,s^{-1}} \\left( \\dfrac{d}{8.32\\, \\mathrm{kpc}} \\right)^{-2} \\left( \\dfrac{M_{\\mathrm{sp}}}{10^{3}\\, \\mathrm{M_{\\odot}}}\\right) ^{4\/3} \\nonumber \\\\\n& \\times \\left( \\dfrac{R_{\\mathrm{sp}}}{0.043\\, \\mathrm{pc}} \\right)^{-1} \\left( \\dfrac{m_{\\mathrm{DM}}}{30\\, \\mathrm{GeV}} \\right)^{-4\/3} \\nonumber \\\\\n& \\times \\left( \\dfrac{\\left\\langle \\sigma v \\right\\rangle}{2 \\times 10^{-40} \\, \\mathrm{cm^{3}\\, s^{-1}}} \\right)^{1\/3} \\left( \\dfrac{t_{\\mathrm{BH}}}{10^{10}\\, \\mathrm{yr}} \\right)^{-2\/3} \\left( \\dfrac{N_{\\gamma}^{(\\mathrm{tot})}}{5.7} \\right).\n\\end{align}\nLet us assume that the entire point-source contribution to the \\textit{Fermi} excess fluctuations is given by the IMBH spikes. The resulting limits on the annihilation cross-section are as follows. The upper limit on $\\left\\langle \\sigma v \\right\\rangle$ is extremely small ($\\sim 10^{-40}\\, \\mathrm{cm^{3}\\, s^{-1}}$) for a steep mini-spike with $\\gamma_{\\mathrm{sp}} = 9\/4$ and $M_{\\mathrm{BH}} = 10^{3}\\, M_{\\odot}$. This is related to the very weak dependence of $\\Phi_{\\mathrm{sp}}$ on the cross-section. For a relaxed spike with $\\gamma_{\\mathrm{sp}} = 3\/2$, the upper limit on the cross-section is of the order of $10^{-31} \\, \\mathrm{cm^{3}\\, s^{-1}}$ for a population of $10^{3}\\, M_{\\odot}$ IMBHs. For $M_{\\mathrm{BH}} = 10^{2}\\, M_{\\odot}$, the best-fit cross-sections become $2 \\times 10^{-36}\\, \\mathrm{cm^{3}\\, s^{-1}}$ for $\\gamma_{\\mathrm{sp}} = 9\/4$ and $3 \\times 10^{-29}\\, \\mathrm{cm^{3}\\, s^{-1}}$ for $\\gamma_{\\mathrm{sp}} = 3\/2$.\n\n\n\n\\section{Global signal and spatial morphology}\nWe now consider a distribution of IMBHs that collect in the inner galaxy. Most of them are failed mergers, as mentioned above, with a total mass amounting to of the order of the mass of the central SMBH, $4 \\times 10^6\\, M_\\odot$. First, we note that to compute the radial profile of $\\gamma$-rays, we need to convolve the radial distribution of the IMBHs with the radial dependence of the mini-spike flux.\n\nThe DM interpretation of the \\textit{Fermi} excess works for the morphology because it naturally gives the $\\gamma$-ray profile as roughly the square of the NFW profile, or $r^{-2}$. The present model needs to address this point. However, the case for the DM interpretation may not be that strong. Firstly, the MWG may have a DM core \\citep{Portail17}. Second, the \\textit{Fermi} excess can be fit, according to a reanalysis, by a stellar mass (bulge)-related profile \\citep{bartels17}.  \n\nThe point sources (IMBHs) have a $r^{-3\/2}$ density profile toward the GC. This  is a dynamically relaxed profile that follows the Bahcall--Wolf solution for a stellar cusp \\citep{BW76}. This might not match the observed profile if the mini-spike masses and luminosities are independent of radius. In fact,\nthere will be mass segregation, the more massive IMBHs falling in closer to the GC but stalling at\/near the final parsec. IMBHs are point masses, and too dense to be tidally disrupted. Let us estimate the radial dependence of the mini-spike luminosity. The radial flux profile for a set of mini-spikes around BHs is $\\Phi_{r} \\propto \\Phi_{\\mathrm{sp}} r^{-3\/2}$. From Eq.~(\\ref{spike_flux}), the flux for an individual mini-spike is $\\Phi_{\\rm sp}  \\propto M_{\\mathrm{BH}}^{3-6\/\\gamma_{\\rm sp}}$, where $\\gamma_{\\mathrm{sp}}= (9-2\\gamma)\/(4-\\gamma)$. Here, we assume $M_{\\mathrm{sp}} = M_{\\mathrm{BH}}$  and $R_{\\mathrm{sp}} =GM_{\\mathrm{BH}}\/\\sigma_{*}^2$, with $\\sigma_{*} \\propto M_{\\mathrm{BH}}^{1\/4}$ \\citep{tremaine2002} so that $R_{\\mathrm{sp}} \\propto M_{\\mathrm{BH}}^{1\/2}$. Hence, $\\Phi_{\\mathrm{sp}}  \\propto M_{\\mathrm{BH}}^{(3\/2)(1-1\/\\gamma_{\\mathrm{sp}})}$. In addition, $M_{\\mathrm{BH}}$ increases as $r$ decreases because of mass segregation by settling. The two-body relaxation time-scale is $\\propto t_{\\mathrm{r}} \\propto 1\/M_{\\mathrm{BH}}$, as is the dynamical friction time that is $\\sim M_{\\mathrm{SMBH}}\/M_{\\mathrm{BH}}$ orbital times. One needs a simple diffusion model to go further, but a rough guess using adiabatic invariants might be $r v M_{\\mathrm{BH}} = \\mathrm{const}$ (conservation of angular momentum), so that $M_{\\mathrm{BH}} \\propto r^{-1\/2}$. Hence, the radial flux profile is \n\\begin{equation}\n\\Phi_{r} \\propto r^{-\\frac{9}{4} \\left( 1 - \\frac{1}{3\\gamma_{\\mathrm{sp}}} \\right) }.\n\\end{equation}\nGenerally, $\\gamma_{\\mathrm{sp}}= (9-2\\gamma)\/(4-\\gamma)$, so that for a core, $\\gamma=0$ and $\\gamma_{\\mathrm{sp}}=9\/4$, while for $\\gamma = 1$, $\\gamma_{\\mathrm{sp}}=7\/3$ and for $\\gamma=3\/2$, $\\gamma_{\\mathrm{sp}}=12\/5$. Hence, for adiabatic mini-spikes, the radial profile is $\\Phi_{r}^{(\\gamma=0)} \\propto r^{-23\/12}$, $\\Phi_{r}^{(\\gamma=1)} \\propto r^{-27\/14}$ and $\\Phi_{r}^{(\\gamma=3\/2)} \\propto r^{-31\/16}$. Therefore, $\\Phi_{r} \\propto r^{-2}$ for IMBH mini-spikes, always approximating the observed $\\gamma$-ray profile independently of the DM halo profile.\n\n\n\n\\begin{figure}[h!]\n\\centering \n\\includegraphics[width=\\linewidth]{GCE_profile.pdf} \n\\caption{\\label{profile}Angular profiles for the total $\\gamma$-ray emission at 2 GeV with bright point sources masked (black solid line; \\citealp{ackermann17}) and for various components of the $\\gamma$-ray emission. Green squares: MSP-like component extracted from the data \\citep{ackermann17}. Red dashed line: GeV excess in the sample model from \\citet{ackermann17} corresponding to a generalized NFW profile template with slope $\\gamma = 1.25$. Magenta dotted-dashed line: GeV excess in the sample model but for a regular NFW profile ($\\gamma = 1$). Yellow line: prediction for MSPs in the bulge of the Milky Way from disrupted globular clusters \\citep{brandt15}. Our IMBH--mini-spike model is depicted by the blue shaded area for benchmark slopes discussed in the text. Here, we are mostly interested in illustrating the spatial morphology of the signal in our model, so we arbitrarily rescaled the angular IMBH--mini-spike profile at the level of the first MSP-like point.}\n\\end{figure}\n\n\n\n\n\nThese predictions are illustrated in Fig.~\\ref{profile}, which shows the angular profile of the total GC $\\gamma$-ray data at 2 GeV with bright point sources masked \\citep{ackermann17}, along with the profiles of various components of the $\\gamma$-ray emission. Our model typically gives an angular profile that is consistent with expectations from bulge sources like MSPs. \n\nThe situation is complicated by the fact that the DM spikes may be heated---for relaxed mini-spikes $\\Phi_{r} \\propto r^{-1.75}$---and partially stripped as the IMBHs fall into the GC region, although tidal disruption of PBH clusters and dynamical friction may in turn steepen the IMBH profile, as discussed in \\citet{fragione17} for MSPs. Regardless, it seems plausible, pending detailed simulations, that our model gives a good approximation to the \\textit{Fermi} $\\gamma$-ray excess profile.\n\n\n\\section{Discussion}\nOne  attractive model  for the LIGO events argues that hard massive BH binaries form in dense stellar clusters. This scenario has one advantage over rivals: it was proposed before the aLIGO detection \\citep{bae14} to give acceptable rates and masses. Protoglobular clusters are likely pregalactic sites\n and are dispersed as substructure disrupts when the  bulge formed. Stellar cluster-enhanced  formation of massive BH binaries quantitatively accounts for the observed LIGO rates, when integrated  out to several hundred Mpc \\citep{park17}.  Such massive BHs may have formed prolifically at high redshift, when there was most likely a top-heavy initial mass function, providing  a possible pathway to forming IMBHs. In the PBH case, one appeals to BH binary formation by early capture in the first bound DM substructures  at the onset of matter domination \\citep{sasaki16}. Some subsets of these (one needs of the order of 10\\%) might have merged to form IMBHs.\n\nWe expect that massive binaries should be enhanced in number near the GC where the most massive protoglobulars dispersed to form the central NSC. These would generate MSPs as well as BH binaries. Hence, these two populations should track each other. Neither would have a significant disk component. Another consequence would be an enhanced rate of BH mergers in galactic nuclei that might be detectable by LIGO \\citep{nishikawa2017}. These LIGO events  occur  within the IMBH sphere of influence. This could lead to enhanced drag and affect the gravitational-wave signal phase evolution. This could potentially be seen as a cumulative phase shift by LISA over many cycles \\citep{yue2017}.\n\nWe showed that mini-spikes around a population of  hundreds or thousands  IMBHs can significantly contribute to the GC emission and can readily account for both the normalization and spatial morphology of the $\\gamma$-ray excess for very small annihilation cross-sections. The expected morphology of the predicted excess  does not necessarily follow the standard DM halo profile, for instance, it can effectively trace the Galactic bulge due to mass segregation and the dependence of mini-spike luminosities on BH mass. This circumvents the issue raised by the observation of an excess of $\\gamma$-rays in control regions in the disk where no significant contribution from DM is expected \\citep{ackermann17}. IMBHs would appear naturally in central regions due to three-body encounters and ejections. This distinctive morphology also allows the model to evade the constraints of \\citet{clark16} that ruled out a DM interpretation of the excess in terms of ultra-compact mini-halos. We note that the constraints of \\citet{clark16} do not account for more recent studies of the GC emission that revealed a more complex spatial morphology \\citep{ackermann17,bartels17}. Finally, we expect the central massive BHs seen in nearby galactic centers, if indeed formed in the early universe, to have DM spikes, and hence to be \\textit{Fermi} $\\gamma$-ray sources. 47 Tuc  is a possible example \\citep{abdo09}, although one cannot easily distinguish a possible $\\gamma$-ray point source from the expected  population of MSPs. Future observations may help us elucidate this point.\n\n  \n  \n\\section*{Acknowledgments}\nT.L. receives financial support from CNRS-IN2P3. T.L. also acknowledges support from the European Union's Horizon 2020 research and innovation program under the Marie Sk\\l{}odowska-Curie grant agreement Nos. 690575 and 674896; beside recurrent institutional funding by CNRS-IN2P3 and the University of Montpellier.  The work of J.S. has been supported in part by European Research Council (ERC) Project No. 267117 (DARK) hosted by Universit\\'{e} Pierre \\& Marie Curie -- Paris VI, Sorbonne Universit\\'{e}s.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Appendix}\n\n\\subsection{Experiment settings}\n\\section{Approaches}\n\\label{sec:approaches}\n\nIn this setting, we hope to investigate the ability of {\\texttt{BERT}}~ model to predict events in a sentence without training. \nTo this end, we attempt to convert event detection in the following 3 well-known problems: entailment prediction, question answering, and masked token prediction. \n\\begin{enumerate}\n    \\item \\textbf{Textual entailment (TE).}\n        In standard textual entailment prediction setting, as depicted in ~\\cite{mnli}, the model is given a pair of sentences, called ``premise'' and ``hypothesis'', and the model is tasked to predict whether the hypothesis is entailed by, contradicts to, or is irrelevant to the premise. \n        We convert event detection to this format by treating each sentence as the premise, and for each event, a statement that this event has occurred as the hypothesis. \n        Ideally, the model would classify that the sentence entails the event statement if the event indeed occurred in the context of the sentence. \n        \n        We assume that the model predicts that the input sentence pair belongs to one of the three categories: \\emph{contradictory}, \\emph{entailment}, or \\emph{neutral}. \n        Given a query-text pair constructed from Section~\\ref{sec:preprocess:query_event} and ~\\ref{sec:preprocess:query_argument}, we predict that the corresponding event or argument occurs if and only if the textual entailment model predicts ``entailment''. \n    \\item \\textbf{Question answering (QA).}\n        The input in this setting includes a pair of sentences, ``question'' and ``text'', as described in ~\\cite{rajpurkar2016squad}. The model is supposed to detect a span of words in the text that best answers the question, if there is any. \n        We have two type of questions: the ``Wh''-type and the Yes-No type, as described in Section~\\ref{sec:method:arg_query}. We refer them as SQ-QA and PQ-QA, respectively. \n        For the first type, we apply the standard span detection approach. We treat the second type as a natural 0-1 classification problem, classifying the pair of sentences as either ``Yes'' or ``No''.\n    \\item \\textbf{Masked token prediction. }\n        This approach formulates event detection as a kind of ``fill in the blank''problem. \n        {\\texttt{BERT}}~ is essentially a \\emph{masked language model}, and it is supposed to predict masked tokens given the context. \n        We combine sentences with event queries, but instead of filling in actual event types, we place the special \\texttt{[MASK]} token in place of event types and feed it to {\\texttt{BERT}}~. We then use {\\texttt{BERT}}~'s pretrained weights to predict the most probable tokens on placeholders. \n        We propose this method mostly as a probing technique to understand what information about events {\\texttt{BERT}}~ is able to summarize from texts and justify the performance of other models. \n\\end{enumerate}\n\nThroughout this paper, we used pretrained DistilBert model on relevant data and tasks for finetuning and evaluation. \nSpecifically, for TE, we use DistilBert trained on MultiNLI~\\cite{mnli} data~\\footnote{https:\/\/huggingface.co\/ishan\/distilbert-base-uncased-mnli}. For both types of QA, we use DistilBert trained on SQuAD~\\cite{rajpurkar2016squad} data~\\footnote{https:\/\/huggingface.co\/distilbert-base-cased-distilled-squad}. \n\n\\section{Conclusion}\nWe propose a reading comprehension framework for event extraction tasks. \nWe design a textual entailment based method for event detection and question answering for argument detection. \nExperiment findings suggest our framework can effectively distill knowledge from {\\texttt{BERT}}~ as well as guide the model with semantic information, achieving state-of-the-art results on few-shot and supervised settings.\n\nOur experiemnts also suggest several promising reseach directions. We summarize them here. \nIt is clear that while the current framework with mostly template-based queries can achieve superior performance already, \nthe queries are not ideal since they do not relate to actual sentence context and the casual relation between distinctive events and arguments. \nOne promising direction is generating queries that are 1) more flexible and authentic, 2) relevant to input sentence's context, and 3) reveals causal relations between events and arguments. We believe this would enable a more label efficient and robust zero-shot and few-shot learning framework.\n\nIn our current framework and, indeed, all existing reading comprehension frameworks to our best knowledge, \none must construct multiple queries for one input instance for complete classification results. \nThis could be burdensome when there's a large number of queries to be constructed, usually a result of large number of label types. \nOne problem is how to do so efficiently by enabling information sharing between different queries. \n\nExperiments show that the current {\\texttt{BERT}}~ model cannot learn efficiently from long event descriptions. A siginificant advancement in language modeling would be enabling it for zero- or few-shot learning with only a few descriptions and annotation guides. \n\nWe manually select reading comprehension tasks for two event extraction tasks. \nA key ingredient to a more general reading comprehension solution to NLP tasks is a principle to measure the transferibility between tasks. \n\n\n\\subsection{Argument Detection Experiments}\n\\label{sec:arg}\n\\subsubsection{Probing without Supervision}\n\\label{sec:arg:zero_shot}\n\nLike in event detection experiments, we first probe pretrained {\\texttt{BERT}}~ for QA tasks on SQuAD~\\cite{rajpurkar2016squad} and see if they can predict arguments without training~\\footnote{We used deepset\/bert-large-uncased-whole-word-masking-squad2. }.\nWe experiment on the following query types:\n\\begin{enumerate}\n    \\item QA-Template. This constructs questions from templates in Section~\\ref{sec:method:arg}\n    \\label{qa_exp_temp}\n    \\item QA-Guide. This constructs questions from descriptions in the annotation guide~\\cite{ace05guide}.\n    \\label{qa_exp_guide}\n    \\item QA-Trigger. This constructs questions from a template similar to the one in Section~\\ref{sec:method:arg}, but the question is asked about the trigger, like: ``What is the \\texttt{[ARGUMENT]} in \\texttt{[TRIGGER]}?''.\n    \\label{qa_exp_trigger}\n    \\item QA-Trigger-Plus. This is based on a similar template which includes both trigger word and event name. For example: \n    ``What is the \\texttt{[ARGUMENT]} in event \\texttt{[EVENT]} triggered by `\\texttt{[TRIGGER]}'?''\n    \\label{qa_exp_trigger_plus}\n\\end{enumerate}\n\\ref{qa_exp_trigger} and ~\\ref{qa_exp_trigger_plus} is the method used by ~\\cite{du2020event}. Since we are doing event detection without triggers, it is necessary to include \\ref{qa_exp_trigger} \nas well to demonstrate the effect of removing trigger words on argument detection.\n\nTable~\\ref{tab:zero_shot_arg_results} shows results of models without training. Unlike in event detection, no threshold probability was manually chosen. We see that all four query templates perform similarly, and are all much better than random baseline. \nAlso, we observe that questions with trigger words perform slightly better than those without. However, this is based on ground-truth annotation trigger words. \n\nIn Table~\\ref{tab:exp:arg_examples}, we show some examples of zero-shot predictions by generated queries.\nAdditionally, we manually wrote customized queries for each sentence based on the principle that the query should specify as much information as possible about the event, including other relevant arguments, like ``London's financial world'', and ''by the end of the year'', that are present in the input sentence. \n\nFrom Table~\\ref{tab:exp:arg_examples}, we can observe that the custom query is much more semantically and contextually related to the input sentence. Compared with template-based queries which might ask ``Who started a new position'', or ``What is the artifact in Transfer-Ownership'', the written query is much more context specific. \nIt provides stronger guidance for {\\texttt{BERT}}~ model to extract the actual answer. \nUnfortunately, such queries are specific to sentence contexts and requires human writing, and it is hard to scale-up to the entire dataset at this time. \n\n\\begin{table}[tbp]\n    \\centering\n    \\begin{tabular}{c|ccccc}\n        & Precision~\\(\\uparrow\\) & Recall~\\(\\uparrow\\) & F1~\\(\\uparrow\\) & p-value~\\(\\downarrow\\) \\\\ \\hline\n        Random & 10.64 & 15.53 & 12.63  &  0.56 \\\\\n        QA-Temp & 31.83 & 22.96 & 26.67 & 0.00 \\\\ \n        QA-Guide & \\textbf{31.89} & 23.08 & 26.78 & 0.00 \\\\ \n        QA-Trig & 28.03 & \\textbf{26.26} & 27.12 & 0.00\\\\ \n        \\rowcolor{LightCyan} QA-Trig-Plus & 30.05 & 24.80 & \\textbf{27.17} & 0.00\n    \\end{tabular}\n    \\caption{Zero-shot learning for argument detection. We report Precision, Recall, F1, and \\(p\\)-values. The arguments are predicted based on ground-truth event labels and trigger word annotations. }\n    \\label{tab:zero_shot_arg_results}\n\\end{table}\n\n\\begin{table*}[tbp]\n    \\small\n    \\centering\n    \\begin{tabular}{p{0.43\\textwidth}|c|c|c|c}\n        Sentence & Event, Argument, \\& Answer & Query Type & Prediction & Result \\\\ \\hline\n        \\multirow{4}{0.43\\textwidth}{\\textbf{What is the organization he said that is going to start?} ``Prostitution is completely discriminalised in Sydney and we are going to build a monster,'' he said.}  & \\multirow{4}*{\\shortstack{Start-Organization, \\\\ Organization, \\\\ ``a monster''}}  & QA-Temp & ``a monster'' & Correct \\\\ \n        & & QA-Guide & None & Wrong \\\\\n        & & QA-Trigger & ``a monster'' & Correct \\\\ \n        & & Custom & ``Universal \\(\\ldots\\) assets'' & Partial correct \\\\\\hline\n\n        \\multirow{4}{0.43\\textwidth}{\\textbf{Who started a new job in London's financial world?} Former senior banker Callum McCarthy begins what is one of the most important jobs in London 's financial world in September , when incumbent Howard Davies steps down.  }  & \\multirow{4}*{\\shortstack{Start-Position, \\\\ Person, \\\\ ```Former \\(\\ldots\\) McCarthy''}}  & QA-Temp & ``Former \\(\\ldots\\) McCarthy'' & Correct \\\\ \n        & & QA-Guide & ``Former \\(\\ldots\\) McCarthy'' & Correct \\\\\n        & & QA-Trigger & None & Wrong \\\\ \n        & & Custom & ``Callum McCarthy'' & Partial correct \\\\\\hline\n\n        \\multirow{4}{0.43\\textwidth}{\\textbf{What are the entertainment assets Vivendi tries to sell by the end of the year?} Vivendi confirmed that it planned to shed its entertainment assets by the end of the year, including its famed Universal movie studio and television assets.}  & \\multirow{4}*{\\shortstack{Transfer-Ownership, \\\\ Artifact, \\\\ ``its famed\\(\\ldots\\) studio''}}  & QA-Temp & None & Wrong \\\\ \n        & & QA-Guide & None & Wrong \\\\\n        & & QA-Trigger & None & Wrong \\\\ \n        & & Custom & ``Universal \\(\\ldots\\) assets'' & Partial correct \\\\\\hline\n\n        \\multirow{4}{0.43\\textwidth}{\\textbf{When did the meeting that Jean-Rene Fourtou participated in take place?} Chief executive Jean - Rene Fourtou told shareholders at the group 's annual general meeting Tuesday that \\hide{the sale of Vivendi Universal Entertainment was a major goal for 2003 , and that} negotiations were already under way.}  & \\multirow{4}*{\\shortstack{Meet, \\\\ Time-Within, \\\\ ``Tuesday''}}  & QA-Temp & None & Wrong \\\\ \n        & & QA-Guide & None & Wrong \\\\\n        & & QA-Trigger & None & Wrong \\\\ \n        & & Custom & ``Tuesday'' & Correct \\\\\\hline\n    \\end{tabular}\n    \\hide{\n    \\begin{tabular}{l|c}\n        Index & Custom Query \\\\ \\hline\n        1 & \\\\\\hline\n        2 & What are the entertainment assets Vivendi are trying to sell by the end of the year? \\\\\\hline\n        3 & When did the meeting that Jean-Rene Fourtou participated in take place \\\\  \n    \\end{tabular}\n    }\n    \\caption{Zero-shot learning for question answering examples. The bold first sentence is the ``Custom'' query that human manually wrote given the context. We list predictions by three generated queries and the custom query. Some non-essential parts are removed from sentences due to space limitations.  }\n    \\label{tab:exp:arg_examples}\n\\end{table*}\n\n\\subsubsection{Few-shot learning}\n\\label{sec:arg:sup}\nTo the best of our knowledge, we are the first to do few-shot argument detection without using external semantic role labeling tools. We show our results in Table~\\ref{tab:few_shot_arg_results}.\n\nIn Table~\\ref{tab:few_shot_arg_results}, we report rgument prediction scores based on both event predictions using TE model from Section~\\ref{sec:event:sup} and on ground-truth event labels. \nErrors in event predictions would propagate in the first scenario. \nWe see that the hand-written query based on guides perform a lot better, especially when \\(K\\) is low, than template-based queries. \n\n\n\\begin{table*}[htbp]\n    \\centering\n    \\begin{tabular}{c|c|ccccccccc}\n    Events & \\(K\\)-shot & \\(K=1\\) & \\(K=3\\) & \\(K=5\\) & \\(K=7\\) & \\(K=9\\) \\\\ \\hline\n     \\multirow{2}*{Predicted}  & QA-Temp & 1.35 \\std{2.34} & 37.27 \\std{2.28} & 44.08 \\std{0.74} & 44.74 \\std{2.87} & 46.59 \\std{0.94} \\\\\n      &  QA-Guide & 19.58 \\std{2.45} & 41.38 \\std{1.61} & 45.41 \\std{1.48} & 44.92 \\std{1.12} & 46.86 \\std{0.74} \\\\ \\hline\n    \\multirow{2}*{Ground Truth} & QA-Temp & 1.62 \\std{2.81} & 45.49 \\std{2.46} & 52.49 \\std{1.32} & 54.44 \\std{3.19} & 56.15 \\std{1.13} \\\\\n       &  QA-Guide & 22.67 \\std{3.03} & 50.97 \\std{2.42} & 55.48 \\std{1.68} & 54.39 \\std{0.85} & 57.66 \\std{1.23} \\\\ \n     \n      \n\\end{tabular}\n    \\caption{Results on few-shot argument detection. The first part predicts and compute metrics based on predicted events with the best text entailment model trained in Section~\\ref{sec:method:event}. The second part is based on ground-truth event labels. }\n    \\label{tab:few_shot_arg_results}\n\\end{table*}\n\n\\subsubsection{Fully Supervised}\nFor fully supervised argument detection, we compare our approach to DMCNN~\\cite{chen2015event} and VERB-QA~\\cite{du2020event} described in Section~\\ref{sec:event:sup}.\n\nVERB-QA~\\cite{du2020event} curates the argument with a trigger-word based QA framework. Specifically, their template is: \n``\n\\texttt{[Wh-word]} is the \\texttt{Argument} in \\texttt{Trigger}?\n''\nwhere ``Wh-word'' means interrogative words such as ``What'', ``Who'' and ``When'', and ``Trigger'' is the detected trigger word. This is essentially equivalent to our QA-Trig.\nIn ~\\ref{tab:arg_sup} we report the experiment results. Our method with \n\n\\begin{table}\n    \\small\n    \\begin{tabular}{l|l|ccc}\n      Events & Method & Precision & Recall & F1\\\\ \\hline\n       \\multirow{4}*{Pred.} &  DMCNN & \\textbf{62.2} & 46.9 & 53.5 \\\\ \n        & VERB-QA & 56.77  & 50.24 & 53.31 \\\\\n       & QA-Temp &  56.24~\\std{1.21} & \\textbf{52.14}~\\std{3.52} & 54.01~\\std{2.13} \\\\ \n        \\rowcolor{LightCyan}& QA-Guide & 57.69~\\std{0.45} & 51.90~\\std{3.39} & \\textbf{54.61}~\\std{1.67} \\\\ \n        \\hline\n        & QA-Temp & 71.06~\\std{1.58} & \\textbf{65.84}~\\std{2.14} & 68.25~\\std{0.98} \\\\ \n        \\rowcolor{LightCyan}\n         \\multirow{1}{*}{G. Truth} & QA-Guide & \\textbf{72.20}~\\std{2.45} & 65.79~\\std{3.09} & \\textbf{68.79}~\\std{0.57}\n    \\end{tabular}\n    \\caption{Fully supervised learning for argument detection. Similar to in few-shot setting, we report our scores on both ground-truth and predicted event labels. }\n    \\label{tab:arg_sup}\n\\end{table}\n\n\n\\subsubsection{Discussions}\n\\label{sec:arg:ablation}\n\n\\paragraph{Does trigger word help?}\nThe potential drawback of event detection without triggers is that it might lose the trigger information that could be important for subsequent argument detection tasks. \nIn Section~\\ref{sec:arg:zero_shot}, we see that questions with trigger words perform slightly better than questions with only event and argument names. \nSince trigger words contain the only clue in the query about the context of the sentence, it is indeed reasonable that questions with triggers should perform better.\n\nHowever, this is based on golden trigger words. In real application, predicting trigger words tend to propagate more errors than predicting sentence-level event labels.\nHence, in supervised setting, we see that methods based on sentence-level event predictions (QA-Temp, QA-Guide) perform slightly better than VERB-QA, which predicts arguments based on predicted trigger words and questions constructed with them. \nWe may conclude that trigger words are non-essential to both event detection and argument detection. \n\\subsection{Event Detection Experiments}\n\\label{sec:event_detection}\n\\hide{For event detection, we use {\\texttt{BERT}}~ and {\\texttt{DistilBERT}}~ models finetuned on MNLI. Since we could not find a pretrained {\\texttt{BERT-Large}}~ on MNLI and finetuning it exceeds overburden our computation resources, we do not experiemnt with {\\texttt{BERT-Large}}~. }\n\n\\begin{table}[tbp]\n    \\small\n    \\centering\n    \\begin{tabular}{c|ccccc}\n        & Precision~\\(\\uparrow\\) & Recall~\\(\\uparrow\\) & F1~\\(\\uparrow\\) & Threshold & p-value~\\(\\downarrow\\) \\\\ \\hline\n        Random & 5.75 & 86.28 & 10.79 & 0.157 & 0.63 \\\\\n        MTP + TE & 12.50 & 29.09 & 17.40 & 0.825 & 0.00 \\\\ \n        MTP + QA & 15.13 & 17.05 & 16.03 & 0.849 & 0.00 \\\\\n        QA & 1.03 & 100 & 2.03 & 0.0 & 0.00 \\\\\n        \\rowcolor{LightCyan}\n        TE & 16.84 & 36.89 & \\textbf{23.12} & 0.385 & 0.00 \\\\\n    \\end{tabular}\n    \\caption{Zero-shot learning for event detection. Precision, Recall, F1-scores are evaluated on test set, with Threshold chosen to maximize F1-score on the development set. Any instance with predicted probability greater than the Threshold is classified as positive. \\(p\\)-value indicates the the level of statistical significance that the model does separate ground-truth events from false events. \\textit{Random} is the F1-score obtained by randomly generating scores from a uniform distribution on \\([0, 1]\\). }\n    \\label{tab:zero_shot_results}\n\\end{table}\n\n\\begin{table*}[tbp]\n\\begin{tabular}{p{0.25\\textwidth}p{0.10\\textwidth}p{0.25\\textwidth}p{0.25\\textwidth}}\n Sentence & Ground-truth & Top candidates from vocab & Top candidates from event names \\\\  \\hline\n    Jay Garner the retired general will go into Iraq soon with his troops soon. & \\multirow{3}{0.10\\textwidth}{Movement: Transport} & \n    just time to never what war has this today now & Attack (0.98) Injure (0.96) Transport (0.92) \\\\ \\cline{1-4} \n     It would not have been necessary to fire those 17 people right away. & Personnel: End-Position & never just to time had christmas not certainly have now & Die (0.97) Injure (0.92) Trial-Hearing (0.91) \\\\ \\cline{1-4}\n     \\hide{If he is willing to file a lawsuit on the basis of New York non-profit organization law, which might or might not be applicable to the USCF,}\n     Why wouldn't he file a lawsuit on the basis of the USCF's violation of its own bylaws, which unquestionably ARE applicable to the USCF? & \\multirow{4}*{Justice: Sue} & just never to has what not have already having had & Trial-Hearing (0.98) Sue (0.98) Injure (0.96) Arrest-Jail (0.90) Transfer-Money (0.85) \\\\ \\hline\n\\end{tabular}\n\\caption{Examples of masked token predictions. We show the original sentence, ground-truth labels, top candidate for the [MASK] placeholder from both the entire vocabulary and only from the set of event names. For each top candidate event name, we also show their predicted probabilities. }\n\\label{table:mtp_examples}\n\\end{table*}\n\n\n\\begin{table*}[tbp]\n    \\centering\n    \\begin{tabular}{l|ccccccccc}\n    \\(K\\)-shot & \\(K=1\\) & \\(K=3\\) & \\(K=5\\) & \\(K=7\\) & \\(K=9\\) \\\\\n    {\\texttt{DistilBERT}}~ & 0.42~\\std{0.29} & 0.42~\\std{0.29} & 0.42~\\std{0.29} & 0.42~\\std{0.29} & 0.42~\\std{0.29} s\\\\\n        TED & \\textbf{43.02} \\std{2.36} & \\textbf{51.60} \\std{2.15} & 51.51 \\std{2.59} & 56.20 (\\(\\pm\\) 3.11)  & 56.83 (\\(\\pm\\) 4.36) \\\\\n        TED+D (1) & 23.10 (\\(\\pm\\) 5.20) & 45.94 (\\(\\pm\\) 4.80)  & 51.92 (\\(\\pm\\) 4.07) & 54.88 (\\(\\pm\\) 2.82) & 57.11 (\\(\\pm\\) 2.90) \\\\\n        TED+D (5) & 17.80 (\\(\\pm\\) 4.29) & 45.76 (\\(\\pm\\) 5.17) & 52.44 (\\(\\pm\\) 2.47) & 54.68 (\\(\\pm\\) 3.55) & 59.62 (\\(\\pm\\) 2.63) \\\\\n       \n       \n       \n        \\hline\n        {\\texttt{BERT}}~ & 0.70~\\std{0.40} & 0.70~\\std{0.40} & 0.42~\\std{0.29} & 0.42~\\std{0.29} & 0.42~\\std{0.29} \\\\ \n        TE & 36.00 \\std{3.56} & 50.68 \\std{1.51} & 53.31 \\std{2.40} & 56.01 (\\(\\pm\\) 2.52)  & 57.46 (\\(\\pm\\) 3.24) \\\\\n        TE+D (1) & 25.19 (\\(\\pm\\) 3.28) & 46.94 (\\(\\pm\\) 5.99)  & 53.96 (\\(\\pm\\) 2.96) & 55.33 (\\(\\pm\\) 2.99) & 59.56 (\\(\\pm\\) 1.91) \\\\\n        TE+D (5) & 26.66 (\\(\\pm\\) 1.74) & 49.48 (\\(\\pm\\) 1.95) & \\textbf{54.47} (\\(\\pm\\) 3.25) & \\textbf{57.00} (\\(\\pm\\) 2.03) & \\textbf{61.75}f (\\(\\pm\\) 2.12) \\\\\n    \\end{tabular}\n    \\caption{\n        Results on few-shot event detection. We show mean scores and standard variances on micro F1-scores. \n        We tried both {\\texttt{DistilBERT}}~ and {\\texttt{BERT}}~ as the backbone model. ``TED'' means pretrained {\\texttt{DistilBERT}}~ on textual entailment, and ``TE'' means pretrained {\\texttt{BERT}}~ on the same task. \n        ``+D'' means trained with event description, and the number in the following parenthesis means the max number of used sentences in the description. \n    }\n    \\label{tab:few_shot_results}\n\\end{table*}\n\n\\begin{table}[tbp]\n    \\small\n    \\centering\n    \\begin{tabular}{l|ccc}\n        Method & Precision & Recall & F1 \\\\  \\hline\n        DMCNN~\\cite{chen2015event} & 75.6 & 63.6 & 69.1 \\\\ \n        Delta~\\cite{lu2019distilling} & 67.30 & 69.62 & 68.44 \\\\ \n        VERB-QA~\\cite{du2020event} & 71.12 & 73.70 & 72.39  \\\\\n      \n        \\hline\n        {\\texttt{BERT}}~~\\cite{devlin2018bert_} & 71.52 \\std{0.19} &  70.48 \\std{1.65} & 70.99 \\std{0.82} \\\\ \n        Delta~\\cite{lu2019distilling} & 70.97 & 70.78 & 70.88 \\\\ \n        DS-DMCNN~\\cite{liu2019event} &  \\textbf{75.7} & 66.0 & 70.5 \\\\ \n        \\rowcolor{LightCyan}\n        TE & 73.28 \\std{2.13} & 76.29 \\std{1.30} & \\textbf{75.43} \\std{1.48} \\\\\n        TE-D (1) & 73.04 \\std{3.21} & 75.82 \\std{1.41} & 74.39 \\std{2.29} \\\\ \n        TE-D (5) & {72.95} \\std{3.03} & \\textbf{78.20} \\std{3.89} & {75.38} \\std{0.53}  \n    \\end{tabular}\n    \\caption{Supervised results. The first group of methods are based on trigger detection, while the second group predicts events without triggers.}\n    \\label{table:supervised_eventsp}\n\\end{table}\n\n\\begin{figure*}[tbp]\n    \\centering\n    \\subfigure[Without description.]{\n    \\label{fig:distil_vs_bert:nodesc}\n    \\includegraphics[width=0.3\\textwidth]{figs\/distil_vs_bert_no_desc.pdf}\n    }\n    \\subfigure[With description, one sentence. ]{\n    \\label{fig:distil_vs_bert:desc1}\n    \\includegraphics[width=0.3\\textwidth]{figs\/distil_vs_bert_desc_1.pdf}\n    }\n    \\subfigure[With full description. ]{\n    \\label{fig:distil_vs_bert:desc5}\n    \\includegraphics[width=0.3\\textwidth]{figs\/distil_vs_bert_desc_5.pdf}\n    }\n    \\caption{{\\texttt{DistilBERT}}~ vs {\\texttt{BERT}}~ on event detection in few-shot settings, with or without descriptions. Green lines with square marks are scores with {\\texttt{BERT}}~ as backbone model, and blue lines with circle marks with are {\\texttt{DistilBERT}}~. Strips represent confidence intervals. }\n    \\label{fig:distil_vs_bert}\n\\end{figure*}\n\n\\subsubsection{Probing without Supervision}\n\\label{sec:event:zero_shot}\nWe first experiment on predicing events without any training, but only exploiting potential knowledge in pretrained models alone. \n\nConcretely, we mostly experiment with {\\texttt{BERT}}~ and the smaller {\\texttt{DistilBERT}}~ models pretrained on textual entailment prediction dataset, MNLI~\\cite{mnli}\\footnote{We used https:\/\/huggingface.co\/textattack\/bert-base-uncased-MNLI and https:\/\/huggingface.co\/ishan\/distilbert-base-uncased-mnli.}\n\nWe report performance on these following methods:\n\\begin{itemize}\n    \\item {Random}. This method simply generates random scores from uniform distribution on \\([0, 1]\\) as predictive probability as a refernce. \n    \\item Textual entailment (TE). This is the our approach introduced in Section~\\ref{sec:method:event} with {\\texttt{BERT}}~.\n    \\item Question answering (QA). We ask models question like ``What is the trigger for \\texttt{[EVENT]}?'' and let the model do span detection of the trigger word, following the same method as in ~\\ref{sec:method:arg}. We convert the prediction to sentence-level detection by predicing an event if any span with scores greater than threshold is predicted for that event. For this particularly model, we use {\\texttt{BERT}}~ pretrained on SQuAD~\\cite{rajpurkar2016squad}. \n    \\item Masked token prediction (MTP). Instead of filling in event labels in query templates, we fill in \\([MASK]\\) special token as a placeholder and let {\\texttt{BERT}}~ model predict what this token might be. \n    Events are classified based on the event label name's average score predicted by {\\texttt{BERT}}~ to replace \\([MASK]\\). \n    For this model, the original pretrained {\\texttt{BERT}}~ is used~\\cite{devlin2018bert_}.\n    The scores here are obtained by sigmoid function on individual logits, instead of softmax across entire vocabulary, to avoid arithmetic underflow. Specifically,\n    \\begin{itemize}\n        \\item \\texttt{MTP+TE} predicts the \\([MASK]\\) token based on textual entailment formulation. \n        \\item \\texttt{MTP+QA} predicts the \\([MASK]\\) token based on a question answering formulation, where we attach a question like ``Did any event about \\texttt{[MASK]} happen?'' before the input sentence. \n    \\end{itemize}\n\\end{itemize}\n\nIn common binary classification settings, one predicts positive when the predicted probability is greater than \\(0.5\\). \nHowever, when the model is not trained the concerned data, although the model may contain certain bias towards the task, it is not necessarily \\emph{calibrated} - i.e. it might predict favorably towards the right answer, but not necessarily to the extent that the predicted probability is always larger than \\(0.5\\). \nHence, we manually choose a threshold \\(p_0\\), \nfrom 0 to 1 with step 0.01 that maximizes F1-score on the development set, and we use this threshold to separate predicted negative and positive class and report the performance on test set using this threshold.\n\nTable~\\ref{tab:zero_shot_results} shows the results for zero-shot event detection. We show the following metrics: Precision, Recall, F1, the optimal threhsold on the development set. \nIn addition, for each of the methods, we performed a statistcal test on whether the distribution of scores on the ground-truth events is significantly different from the distribution of scores on false events. We use the standard Kolgomorov-Smirnov 2-sample test~\\cite{massey1951kolmogorov,hodges1958significance} and report the \\(p\\)-value, the lower the better. \n\nFrom the results, we clearly see that entailment is a more natural formulation for {\\texttt{BERT}}~ model as it contains more prior knowledge than question answering framework. If we use a question answering model to predict trigger span, the best performance is obtained when \\(p_0=0.0\\), effectively providing no useful information at all. \nWe believe that it is because trigger words are not natural ``answers'' to questions, events are, but event names are not necessarily present in the text where events occur. Textual entailment does not rely on this, hence it is more able than question answering to solve event detection.\n\nIn Table~\\ref{table:mtp_examples} we showcase some examples of wrong predictions made by masked token prediction based on textual entailment. \nWe see that when \\({\\texttt{BERT}}~\\) doesn't predict the correct event as the one with the top probability, it still chooses the events that are semantically related to the sentence, although the corresponding event might not have actually happened. \n\n\n\\subsubsection{Few-shot Learning}\n\\label{sec:event:sup}\nFollowing zero-shot learning, we naturally want to test the model's ability to transfer knowledge to event detection given a few examples for each event type. \nWe naturally want to test the model's ability to transfer to event detection task with a few learning examples and if the model can build upon a few samples the link between model's semantic knowledge and the reasoning of events. \n\nTable~\\ref{tab:few_shot_results} shows experiemnt results on few-shot learning. \nWe would like to highlight the following observations:\n\\begin{enumerate}\n    \\item {\\texttt{BERT}}~ and {\\texttt{DistilBERT}}~ on their own is not capable of few-shot learning for our tasks, resulting in converging to trivial solutions. \n    \\item {\\texttt{BERT}}~ model does significantly outperform {\\texttt{DistilBERT}}~. When \\(K=1\\), {\\texttt{DistilBERT}}~ works even better in textual entailment without description. \n    \\item {\\texttt{BERT}}~'s becames advantageous when the number of training data increases.\n    \\item Both {\\texttt{BERT}}~ and {\\texttt{DistilBERT}}~ cannot efficiently utilize description information when \\(K\\) is small, resulting in a lower F1-score. When \\(K\\) increases, the performance gradually increases and matches the model without description. \n\\end{enumerate}\nGenerally, in few-shot settings, using \\({\\texttt{DistilBERT}}~\\) could be a better choice, balancing performance and efficiency. \nAs a comparison, \\cite{peng2016event} achieved 67.6. They prefilled possible argument slots for candidate events using semantic role labeling tools, effectively summarizing event structures, and this piece of information is not readily available to {\\texttt{BERT}}~ model. In the future, it would be an interesting direction to investigate if it is possible to use {\\texttt{BERT}}~ model to more effectively extract event structures. \n\n\n\\hide{\nIn Table~\\ref{tab:few_shot_results}, we show few-shot learning results of different models with or without event description. \nEvent descriptions in the annotation guide~\\cite{ace05guide} can contain as many as 5 sentences, exceeding the text length of the sentence greatly. \nThis makes harnessing the information in description hard when the number of training data is small. \nTherefore, in addition to training with full event descriptions, we truncate event descriptions to contain their the first sentences in hope of making the model easier to learn how to use information from descriptions. \nWe make the following observations. \nFirst, when the training samples are too small (from \\(K=1\\) to \\(K=5\\)), adding event descriptions undermines model performance, especially in by a large margin when \\(K=1\\). \nFurthermore, when training with only the first sentence in description, the performance drop is much lower. \nHowever, when we have more samples per class, the models with event descriptions gradually outperform those without, and models with full descriptions outperform with only one sentence. \nFurthermore, when trained without descriptions, one observes counter-intuitively that the standard variance across multiple experiments increase with the number of samples. \n}\n\n\\subsubsection{Fully Supervised.}\n\\paragraph{Baselines.}\nBecause the commonly used data for event detection, ACE05, is not freely available, there are only so few works that are open source whose code are complete and immediately runnable given data. This creates a problem for us to evaluate traditional methods in terms of sentence-level event detection performance.\nHence, for many of them, we could only report the results from the original paper. \n\nDuring our attempts to reproduce the experiments, event with official code, we still struggle to produce results consist with what was reported in papers. The same phenomenon was observed by ~\\cite{orr2018event} where metrics can drop from 5 to 10 points in rigorously designed testing compared with reported results. \nBased on observations, we suspect such inconsistencies could be a result from lack of adherence to standard practices in terms of evaluation schemes, data processing, split of training\/dev\/test sets, and model selection, which all could influence the variance of validity of reported performance.\n\n\n\\hide{\nIt is worth noting that even when reproducing results for trigger-based detection, as is the setting of most previous works, we struggle to reproduce their reported results consistently using official code because of the inconsistencies and sometimes dubious practices we observed in terms of model selection, prediction and evaluation schemes, data processing, and the split of training\/development\/test sets, which all could influence the variance and validity of reported performance. \nSimilar observation was made by ~\\cite{orr2018event}, where they found that the performance of most previous works drop by a large margin, from 5 to even 10 points, in rigorous evaluation settings. \nWe believe it is indeed necessary in this field to set standards on these matters\nif any further progress in this field is to be made. }\n\nBased on these reasons, we could only vouch for the validity of performances methods reported with standard variances, and the same observations and conclusion extends to argument detection as well. \nThe following baselines are used as comparable sentence-level event detection methods without triggers: \n\\begin{itemize}\n    \\item {\\texttt{BERT}}~~\\cite{devlin2018bert_}. This based on the original pretrained {\\texttt{BERT}}~ model, where event classification is done by learning a linear classifier on top of the pooled sentence embeddings. \n    \\item Delta~\\footnotemark~\\cite{lu2019distilling}. This method is based on an adversarial framework where the model learns both discriminative and generalizable information. Although it is originally a trigger-based event detection method, we evaluate it here in terms of sentence-level scores. \n    \\item DS-DMCNN~\\cite{liu2019event}. This method performs event detection without trigger based on ~\\cite{chen2015event}. \n\\end{itemize}\n\nThe following baselines are trigger-based event detection methods, which we report here as a reference:\n\\begin{itemize}\n    \\item DMCNN~\\cite{chen2015event}. This method proposed a dynamic pooling layer for CNN for event detection. \n    \\item Delta\\footnotemark[\\value{footnote}]~\\cite{lu2019distilling}. This is the same method as mentioned above. \n    \\item VERB-QA~\\cite{du2020event}. This is a QA based event detection framework where ``verb'' is used as a query to hint the model for trigger word detection and classification. \n\\end{itemize}\n\\footnotetext{This method was originally proposed to detect and classify triggers. We report the performance of their method on both trigger-based event detection and sentence-level event detection, while the former is computed by ignoring if the span matches. The performance on this method is reported based on reproduction of their method of the authors' published code. The paper reported about 74.7 F1-score on trigger detection and classification.}\n\nOur methods include TE, TE-D (1) and TE-D (5), which are the same as in the few-shot learning setting with {\\texttt{BERT}}~ model. \nWe observed significant performance drop with {\\texttt{DistilBERT}}~, with only 70.04 averge F1 score. Therefore, we focus on performances on {\\texttt{BERT}}~ model only, which can apparently utilize more data more effectively. \nFrom Table~\\ref{table:supervised_eventsp}, our model achieves the best performance among baselines. \n\n\\hide{\n\\begin{table}\n    \\centering\n    \\begin{tabular}{l|l|ccc}\n        Setting & Method & Precision & Recall & F1 \\\\  \\hline\n        \\multirow{4}{*}{w\/ triggers} & DMCNN~\\cite{chen2015event} & 75.6 & 63.6 & 69.1 \\\\ \n        & Delta~\\cite{lu2019distilling} & 67.30 & 69.62 & 68.44 \\\\ \n        & VERB-QA~\\cite{du2020event} & 71.12 & 73.70 & 72.39  \\\\\n        & {\\texttt{BERT}}~~\\cite{devlin2018bert_} & & & \\\\\n        \\hline\n        \\multirow{6}{*}{w\/o triggers} & {\\texttt{BERT}}~~\\cite{devlin2018bert_} & & & \\\\ \n        & DS-DMCNN~\\cite{liu2019event} &  75.7 & 66.0 & 70.5 \\\\ \n        & Delta~\\cite{lu2019distilling} & 70.97 & 70.78 & 70.88 \\\\ \n        & TE & & & \\\\\n        & TE-D (1) & 73.04 \\std{3.21} & 75.82 \\std{1.41} & 74.39 \\std{2.29} \\\\ \n        & TE-D (5) & 72.95 \\std{3.03} & 78.20 \\std{3.89} & 75.38 \\std{0.53}  \n    \\end{tabular}\n    \\caption{Supervised results.}\n    \\label{tab:my_label}\n\\end{table}\n}\n\n\\subsubsection{Discussions}\n\\label{sec:event:ablation}\n\\paragraph{Effect of query structure. }\nIn query templates we infused two kinds of knowledge: 1) information about the event detection task by providing a statement sentence, and 2) information about labels (event types) by filling in the statement event names. \nA natural question one would ask is if the statement structure is actually helpful or is it just the event names are enough.\n\n\\paragraph{Do event descriptions help?}\nBased on experiments on few-shot (Table~\\ref{tab:few_shot_results}) and fully supervised (Table~\\ref{table:supervised_eventsp}) settings, \nwe see that event descriptions can be a burden to model learning when the number of data is extremely small. While with increased data size and in the fully supervised setting, model with descriptions could have comparable perforance with the model without descriptions, it does not outperform the latter. \n\nHowever, event descriptions do provide valuable information on what are events and is sufficient on its own for a human annotator.  \nThis could suggest that {\\texttt{BERT}}~ model cannot learn from descriptions more than it can learn from the data. This suggests that more work should be done on improving language models in understanding how to ``read'' a ``description'' or ``manual''. \n\n\\paragraph{{\\texttt{BERT}}~ vs. {\\texttt{DistilBERT}}~}\nIn Figure~\\ref{fig:distil_vs_bert} we compare the performance of {\\texttt{BERT}}~ and {\\texttt{DistilBERT}}~. \nIt is clear from the figures that in few-shot settings, there is no statistically significant difference in performance, except for \\(1\\)-shot learning, where {\\texttt{DistilBERT}}~ is better without description and {\\texttt{BERT}}~, however, is better with descriptions. \nIn fully supervised learning, the difference is significant where \\({\\texttt{DistilBERT}}~\\) achieves only 70 F1-score in average and {\\texttt{BERT}}~ achieves 75. \nTherefore, when training labels are extremely scarce, it is sufficient to use \\({\\texttt{DistilBERT}}~\\) to effectively learn from these samples with less memory consumption and computational burden. \nIn fully supervised learning, however, {\\texttt{BERT}}~'s ability to utilize massive data is significantly better than {\\texttt{DistilBERT}}~. \n\n\\section{Experiments}\n\\label{sec:exp}\nWe report in this section the experiemnt results on event detection.\nIn Section~\\ref{sec:data} we briefly introduce the dataset, ACE 2005, that we experiment on. \nFor both of the two tasks, \nfirst, we probe the pretrained language model's ability to infer events without specific training in Section~\\ref{sec:event:zero_shot},~\\ref{sec:arg:zero_shot}.\nThen, in Section~\\ref{sec:event:sup},\\ref{sec:arg:sup}, we report results when the model is finetuned on event detection and argument detection.\nAblation studies were reported separately in Section~\\ref{sec:event:ablation},~\\ref{sec:arg:ablation}.\n\n\\subsection{Data}\n\\label{sec:data}\nWe use ACE05 Multilingual Training Corpus~\\cite{doddington2004automatic} for event detection and argument detection. \nIt contains 33 event types, 28 argument types, and sentences that come with various documents. \n\nACE2005 is observed to have domian shifts between its official training, development, and test set split: the training and dev. set contain informal documents such as web logs, while the test set is a collection of newswire articles~\\cite{doddington2004automatic,orr2018event}. \n\nWe follow the data split used by ~\\cite{chen2015event}. Table~\\ref{tab:data_stat} gives data statistics. \n\n\\begin{table}\n    \\begin{tabular}{l|ccc}\n       Data split & \\# sentences & \\# events & \\# arguments \\\\ \\hline\n       Train &  14626 & 4309 & 7702 \\\\ \n       Dev & 870 & 492 & 923 \\\\ \n       Test & 708 & 422 & 887 \n    \\end{tabular}\n    \\caption{Data statistics for training, development, and test set. We list here number of sentences, events, and arguments. }\n    \\label{tab:data_stat}\n\\end{table}\n\n\\input{exp_event}\n\\input{exp_arg}\n\\section{Few-shot learning}\n\\label{sec:few_shot}\nSince that unlike in many other tasks, pretrained language models does not significantly outperform random guessing as explained in Section~\\ref{sec:zero_shot}, \nwe naturally want to test the model's ability to transfer to event detection task with a few learning examples and if the model can build upon a few samples the link between model's semantic knowledge and the reasoning of events. \n\nConcretely, we experiment on TE and PQ-QA and see if the \n\\newcommand{\\(\\pm\\)}{\\(\\pm\\)}\n\\newcommand{\\std}[1]{\\((\\pm#1)\\)}\n\n\\begin{table*}[htbp]\n    \\centering\n    \\begin{tabular}{c|cccccc}\n    \\(K\\)-shot & \\(K=1\\) & \\(K=3\\) & \\(K=5\\) & \\(K=7\\) & \\(K=9\\) & K=N\\\\  \\hline\n    {\\texttt{BERT}}~ & & & & & \\\\\n        TE & 43.02 \\std{2.36} & 51.60 \\std{2.15} & 51.51 \\std{2.59} & 56.20 (\\(\\pm\\) 3.11)  & 56.83 (\\(\\pm\\) 4.36) & \\\\\n        TE+D (1) & 23.10 (\\(\\pm\\) 5.20) & 45.94 (\\(\\pm\\) 4.80)  & 51.92 (\\(\\pm\\) 4.07) & 54.88 (\\(\\pm\\) 2.82) & 57.11 (\\(\\pm\\) 2.90) & \\\\\n        TE+D (5) & 17.80 (\\(\\pm\\) 4.29) & 45.76 (\\(\\pm\\) 5.17) & 52.44 (\\(\\pm\\) 2.47) & 54.68 (\\(\\pm\\) 3.55) & 59.62 (\\(\\pm\\) 2.63) & \\\\\n        PQ-QA & 40.23 \\std{1.30} & 48.88 \\std{3.27} & 50.05 \\std{1.40} & 54.59 \\std{1.88} & 55.77 \\std{3.83} & \\\\\n        PQ-QA+D (1) & 11.81 \\std{13.45} & 47.86 \\std{2.27} & 49.72 \\std{4.57} & 52.89 \\std{3.36} & 54.40 \\std{4.41} & \\\\\n        PQ-QA+D (5) & 6.34 \\std{13.09} & 42.60 \\std{3.01} & 49.34 \\std{2.86} & 52.48 \\std{3.79} & 54.97 \\std{2.38} &\n    \\end{tabular}\n    \\caption{Results on few-shot learning. Here, we show the average micro F1-scores and standard deviations for each model and different number of training samples per class. TE and PQ-QA stand for textual entailment and polar-question question answering, +D means with event description, the number in the parenthesis means the maximum number of sentences in description. }\n    \\label{tab:few_shot_results}\n\\end{table*}\n\nIn Table~\\ref{tab:few_shot_results}, we show few-shot learning results of different models with or without event description. \nEvent descriptions in the annotation guide~\\cite{ace05guide} can contain as many as 5 sentences, exceeding the text length of the sentence greatly. \nThis makes harnessing the information in description hard when the number of training data is small. \nTherefore, in addition to training with full event descriptions, we truncate event descriptions to contain their the first sentences in hope of making the model easier to learn how to use information from descriptions. \n\nWe make the following observations. \nFirst, when the training samples are too small (from \\(K=1\\) to \\(K=5\\)), adding event descriptions undermines model performance, especially in by a large margin when \\(K=1\\). \nFurthermore, when training with only the first sentence in description, the performance drop is much lower. \nHowever, when we have more samples per class, the models with event descriptions gradually outperform those without, and models with full descriptions outperform with only one sentence. \nFurthermore, when trained without descriptions, one observes counter-intuitively that the standard variance across multiple experiments increase with the number of samples. \n\n\\section{Introduction}\n\nEvent extraction is one of the most important and challenging tasks in information extraction.\nEvent extraction consists of two subtasks: 1) The first task, event detection, is\nto detect if natural language text describes the\noccurrence of certain events. 2)\nThe second task, argument detection, aims to find the attributes and participants, such as ``when'', and ``where'', and  ``who'', to the events.\nTypically, both tasks are formulated as \\textit{supervised sequence labeling} problems:\nevent detection is usually formulated as detecting the trigger words or phrases that best ``indicate'' an event;\nand  argument detection is formulated as identifying entities that serve as arguments, such as ``who'', ``when'', and ``where'' for that event type.\nVarious sequence labeling models~\\cite{yang2019exploring,orr2018event,hong2011using,liao2010using,chen2015event,chen2017automatically,nguyen2015event,nguyen2016joint,wang2019adversarial,sha2019jointly,yang2016hierarchical,mehta2019event,nguyen2018graph,lu2019distilling,du2020event}\nhave been proposed, which can be trained\non an annotated corpus and then used for recognizing event triggers and arguments at test time.\n\nFor example, the following sentence is extracted from ACE05 event extraction benchmark~\\cite{doddington2004automatic}:\n\\begin{quotation}\n   ``Orders went out today to \\underline{deploy} 17,000 US soldiers in the \\textit{Persian Gulf region.} ''\n\\end{quotation}\nIn the above example, an event ``Transport'' happened, \\underline{deploy} is annotated as its trigger word, and \\textit{Persian Gulf region} is the\n\\textit{Destination} argument to the event, and \\textit{17,000 US soliders} would be the \\textit{Artifact} (interpreted as passengers). \n\nHowever, formulating event extraction as supervised sequence labeling tasks have several drawbacks.\nFor event detection,\nthe annotation of event trigger words is of high variance. As the definition of ``trigger words'' is the word that most ``clearly'' expresses the occurrence of the event~\\cite{ace05guide,liu2019event}, it is inherently noisy and time-consuming to label, especially in complex documents~\\cite{liu2019event}.\nThis requires developing a specific set of rules governing ambiguity during annotation. Even so, the model is not necessarily able to recognize, or benefited from, such knowledge.\nThe annotation of arguments suffers from the same problem, as the span can be arbitrary (is it ``U.S. soldiers'' or ``17,000 U.S. soldiers'', or just simply ``soldiers''? All of them are\nvalid answers.)\nSome existing approaches~\\cite{liao2010using,sha2019jointly,duan2017exploiting,yan2019event,liu2018jointly}\nattempt to resolve these issues by introducing more complex structural features (such as dependency parses and document-level information) or use more complex neural sequential\nmodels.\nBut as a result, learning such complex models becomes highly label hungry, even when powerful pretrained language models are used~\\cite{wang2019adversarial,yang2019exploring,du2020event}.\nFurthermore, the learned model can easily overfit and be vulnerable to domain shift.\nAs reported in \\cite{orr2018event}, when rigorously tested with multiple random initializations, many works suffer from a severe performance drop compared with the best performance reported in their papers.\n\nWe propose to formulate event extraction as a machine reading comprehension task.\nThe general format of machine reading comprehension is that given a \\emph{query} and a context sentence, the algorithm finds the answer to the query conditioned on the context sentence.\nSpecifically, we formulate event detection as a textual entailment prediction task. \\hide{, where given each sentence as the context,\nthe model predicts if an event occurs by predicting if a description of the event is entailed by the sentence, without detecting or using the event's trigger word.}\nThe underlying intuition is that, suppose a sentence describes an event, then a statement that this event has happened would be a natural entailment to the first sentence.\nCompared with formulating event extraction as a sequence labeling problem,\nthe benefit of such an entailment formulation is twofold.\nFirst,\nthe model can  distill knowledge from pretrained language models and be appealing for few-shot or even zero-shot settings. Second, it can avoid\nhaving to pinpoint event triggers and be able to flexibly exploit complex event semantics in the sentence.\n\n\n\\begin{figure}[t]\n  \\includegraphics[width=0.9\\linewidth]{figs\/model}\n  \\caption{Illustration of our framework. \n  Given an input sentence, our framework predicts an event by predicting if a statement about the event is a logical entailment to the sentence with a textual entailment prediction module. \n  To extract arguments, we construct natural language questions about the argument and extract the answer with a question answering module from the sentence. \n  }\n  \\label{fig:illustration}\n\\end{figure}\n\nWe formulate argument extraction as a question answering problem. Finding an argument to an event can be naturally posed as finding an answer to a\nquestion like ``Who was the attacker?'' or ``Where is the destination of the transport?''. \nIn principle, argument extraction can be formulated as a textual entailment problem as well, but this requires generating a statement for each event-argument-entity pair. This would require pre-identify entities and the training could be inefficient. \nBy reframing argument extraction as QA, we can again transfer knowledge from pretrained models on how to find answers from text data.\nIn Section~\\ref{sec:method}, we give detailed descriptions on how to pose event extraction tasks as reading comprehension tasks, and how to solve them.\nThe illustration of our model can be found in Figure~\\ref{fig:illustration}.\n\nWe conducted experiments and analyzed model performances in zero-shot, few-shot, and fully-supervised settings in Section~\\ref{sec:exp}. Our major\nfindings from the experiments are: (1) Our proposed approach of formulating event extraction as reading comprehension can\neffectively distill knowledge by probing pre-trained models, and\nachieve strong performance for zero-shot event detection without training data. (2) The  performance of our model increases largely when the model is\nfed a small amount of labeled data. In fully-supervised settings, it achieve state-of-the-art performance. (3) We found that trigger words do not\nimprove performance for either event detection or argument detection, which justifies our design of discarding trigger words for event detection.\n\nThe major contributions of this paper are as follows:\n\\begin{enumerate}\n    \\item We propose a reading comprehension framework for event and argument detection. Comparing with existing works, our framework is more capable in\n      exploiting pre-trained reading comprehension models on tasks and event semantics.\n    \\item We are the first to propose a reading comprehension framework of event detection and argument detection without trigger words.\n    \\item\nBy probing and fine-tuning\n      pretrained reading comprehension models,\n      our approach achieves much stronger performance for\n low-shot event extraction compared with the baselines; and it\n achieves\n state-of-the-art\nperformance for fully-supervised event detection on the ACE 2005 benchmark.\n\\end{enumerate}\n\n\\section{Our Approach}\n\\label{sec:method}\nIn this section, we introduce our framework to solve event extraction tasks by formulating them into reading comprehension problems. We cast\nevent detection as a textual entailment prediction problem based on the intuition that a sentence should entail an event if the latter is described in\nthat sentence. We cast argument detection as a question answering problem since questions can be naturally asked about the specific arguments of\nan event. In the following, we first provide high-level descriptions of such a framework in Section \\ref{sec:method:event} and Section \\ref{sec:method:arg}. We\ndescribe how we generate queries for these two subtasks in Section \\ref{sec:query_generation}, and finally detail our model along with its probing and\ntraining procedures in Section \\ref{sec:method:bert_backbone}.\n\n\n\\subsection{Event Detection as Textual Entailment}\n\\label{sec:method:event}\nTo pose event detection as textual entailment prediction, given sentence \\(X\\) and an event type,\nwe first construct a statement \\(Q\\) that claims the event type has happened. The task of determining whether an event has\nhappened in \\(X\\) is then translated to judging if \\(Q\\) is a natural entailment to \\(X\\). For example:\n\\begin{quotation}\n  (\\(X\\)) David is leaving to become chairman of London School of Economics. (\\(Q\\)) \\textit{Hence, an event about \\texttt{Start of Position} occurred.}\n\\end{quotation}\nIn the above, the first sentence is the original input sentence to event detection, and the second italic sentence is a statement constructed for the queried event type, \\texttt{Start-Position}.\nIf the statement is entailed by the first sentence, we predict that the event has happened.\nWe will describe in detail how to construct such \\(Q\\) statements in Section~\\ref{sec:query_generation}.\n\n\\subsection{Argument Detection as Question Answering}\n\\label{sec:method:arg}\nTo convert argument detection to question answering, we construct a ``Wh''-question \\(Q\\) (a question that starts with an interrogative word, such as ``What'', ``Who'', and 'Where', etc.) about the concerned event and argument type, like the following:\n\\begin{quotation}\n    \\textit{\n      (\\(Q\\))   Where did the \\texttt{Meeting} take place?\n    }\n    (\\(X\\))    But the Saint Petersburg summit ended without any formal declaration on Iraq.\n\\end{quotation}\nWhere the first italic sentence is a question about event \\texttt{Meet} and the queried argument type is ``Place''.\nWe construct questions from fixed templates as well as manually written question forms for each possible combination of events and argument types. See Section~\\ref{sec:query_generation} for details.\n\n\n\\subsection{Query Generation}\n\\label{sec:query_generation}\n\nWe introduced event extraction and machine reading comprehension tasks\nin Section~\\ref{sec:tasks},\nand how can the former be transformed as the latter in Section~\\ref{sec:method:event},~\\ref{sec:method:arg}.  \nIn this section, we describe how to generate queries for the two event extraction tasks, and how are combined with input sentences. \nAfter this, all preparations would be made for the model computation flow. \n\nIt is of crucial importance \nto generate high quality queries, i.e. statements about events for entailment-prediction-based event detection, and questions about events and argument types for question-answering-based argument detection, because the quality of queries determines 1) how the model distills information about tasks and labels from pretrained weights, and 2) how the model connect the semantics of events, arguments, and the sentence context. \n\n\\vpara{Statements for events.}\nWe assume that each event type has a label name. To construct a statement about an event, we simply fill in the label name in the following template:\n\\begin{quotation}\n    Hence, an event about \\texttt{[EVENT]} happened.\n\\end{quotation}\nwhere \\texttt{[EVENT]} is the placeholder for event names.\n\nThis statement is supposed to serve as a guide to distill knowledge from the language model about both the \\emph{task} and the \\emph{label}.\nFor the  \\emph{task}, i.e. event detection,  an ideal model should be informed that it is supposed to recognize ``what has happened''.\nFor the  \\emph{label}, the model should recognize clues on the semantics of the event by its label name.\nIn addition to label name, we expect that an natural langauge description of the event would further help us distill knowledge from the model.\nSo, we append an optional piece of description on the event, acquired from the data's annotation guide~\\cite{ace05guide}, to the above statement. Some examples of event descriptions are given in Table~\\ref{tab:event_desc}.\n\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{c|p{60mm}}\n    Event & Description \\\\ \\hline\n       Be-Born  & \\textit{A \\textbf{Be-Born} Event occurs whenever a PERSON Entity is given birth to.} Please note that we do not include the birth of other things or ideas. \\\\ \\hline\n       Marry & \\textit{\\textbf{Marry} events are official Events, where two people are married under the legal definition.} \\\\ \\hline\n       Divorce & \\textit{\\textbf{A Divorce} event occurs whenever two people are officially divorced under the legal definition of divorce. We do not include separations or church annulments. }\\\\ \\hline\n      Transfer-Money & \\textit{\\textbf{Transfer-Money} events refer to the giving, receiving, borrowing, or lending money when it is not in the context of purchasing something.} The canonical examples are: (1) people giving money to organizations (and getting nothing tangible in return); and (2) organizations lending money to people or other orgs.  \\\\\n    \\end{tabular}\n    \\caption{Example event descriptions.}\n    \\label{tab:event_desc}\n\\end{table}\n\n\\vpara{Questions for events and argument types.}\nSimilar to events, each argument type has a label name as well.\nWe have two options to construct a question for a pair of event and argument.\nFirst and most straightforwardly,\nwe could use a fixed template similar to event statement:\n\\begin{quotation}\n    Who or what participated as role\n    \\texttt{[ARGUMENT]} in the event \\texttt{[EVENT]}?\n\\end{quotation}\nwhere argument names and event names are filled in respective slots.\n\nThis rather inflexible approach does not provide much information on the relation between events and arguments,\nsince we assume the same query structure between all event-argument paris. It is more natural to ask questions differently, specific to the concerned event and argument. For example, ``What is the Person in event Start-Position?'' is a lot less natural than ``Who started a new position?''.\nSo, we manually composed a question for every pair of events and arguments based on descriptions in the annotation guide~\\cite{ace05guide}. We do so by converting the description text to a question with minimal edit while ensuring that the concerned argument type appears in the question. For example, from ``The people who are married.'' we construct ``Who are the married person?''. ``People'' are changed to ``Person'' to match the concerned argument type name. \nExamples\nare given in Table~\\ref{tab:arg_questions}. \n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{c|c|p{50mm}}\n    Event & Arg. Type & Question \\\\ \\hline\n       \\multirow{2}{*}{Marry} & Person & Who are the married person? \\\\\n       & Where & Where does the marriage take place?\\\\ \\hline\n       \\multirow{2}{*}{Attack} & Attacker & Who is the attacker? \\\\\n       & Target & Who is attacked?\n    \\end{tabular}\n    \\caption{Example questions for event-argument pairs. }\n    \\label{tab:arg_questions}\n\\end{table}\n\n\\subsection{Model Details}\n\\label{sec:method:bert_backbone}\nIn the previous section we described how to transform event extraction tasks to reading comprehension tasks.\nIn this section, we detail our model for solving these tasks, as well as how the model works in zero-shot and supervised settings.\n\n\\vpara{{\\texttt{BERT}}~ masked language model. }\nFirst, we give an introduction to the backbone {\\texttt{BERT}}~ model that outputs hidden representations for tokens and sentences as the backbone for textual entailment prediction and question answering.\n\n{\\texttt{BERT}}~~\\cite{devlin2018bert_} model is a masked language model that consists of deep multihead attention layers.\nGiven a pair of input sentences \\(X^{(1)}, X^{(2)}\\), {\\texttt{BERT}}~ first runs WordPiece~\\cite{schuster2012wordpiece} tokenizer to tokenize the both sentences into a sequence of token ids: \\( [ x_{1}^{(i)}, \\ldots, x_{n_i}^{(i)} ], i=1,2 \\). Then, two sentences are concatenated into one sequence:\n\\begin{equation}\n    S = [[\\mathrm{CLS}], x^{(1)}_1,\\ldots, x^{(1)}_{n_1}, [\\mathrm{SEP}], x_1^{(2)}, \\ldots, x^{(2)}_{n_{2}}, [\\mathrm{SEP}]]\n\\end{equation}\nwhere \\([\\mathrm{CLS}]\\) is a special token at the beginning that is supposed to aggregate information in the two sentences, \\([\\mathrm{SEP}]\\) is another special token used to inform separation between sentences. This combined sequence of tokens is the input to {\\texttt{BERT}}~ model.\n\n\\vpara{Textual entailment for event detection.}\nTo perform textual entailment prediction,\nwe attach a linear classifier on top of the sentence embedding.\nTo obtain sentence embedding, we use {\\texttt{BERT}}~'s default pooling method:\n\\begin{equation}\n    \\vec{x} = \\texttt{Pool}({\\texttt{BERT}}~([X, Q]))\n    \\label{eqn:bert_sentence_embed}\n\\end{equation}\nwhere \\(\\texttt{Pool}\\) operator just extracts the output embedding of the \\([\\mathrm{CLS}]\\) token.\n\nA linear layer transforms the sentence embedding to logits:\n\\begin{equation}\n    [l_0, l_1, l_2] = \\vec{x}^T \\mathbf{W}_E\n\\end{equation}\nwhere \\(\\mathbf{W}_E\\in\\mathbb{R}^{d\\times 3}\\), \\(d\\) is the dimension of the hidden spaces, and \\(l_0,l_1,l_2\\) are logits for predicting contraditory, entailment, and neutral, respectively. We consider that the underlying event to happen if and only if it predicts entailment. Hence,\nthe final scores for predicting the described event in the statement is obtained by a softmax function:\n\\begin{equation}\n    [p_0, p_1] = \\texttt{softmax}(l_0+l_2, l_1)\n\\end{equation}\nThe final loss function is the cross entropy function:\n\\begin{equation}\n    \\ell_E = \\texttt{cross\\_entropy}(p_0, p_1; y) = -((1-y)\\log p_0 + y \\log p_1)\n    \\label{eqn:entail_loss}\n\\end{equation}\nwhere \\(y\\) is a binary label that indicates whether the described event happens.\n\n\\vpara{Question answering for argument detection.}\nGiven a question about an event and argument type, we need to predict the probability for each token as the starting or ending of the answer span.\nFirst, we collect the output embeddings from {\\texttt{BERT}}~ for each token in the original sentence \\(X\\):\n\\begin{equation}\n    [\\vec{x}_1,\\ldots,\\vec{x}_n] = {\\texttt{BERT}}~(S)\n\\end{equation}\nTo obtain possible spans, we use two linear classifiers on token embeddings:\n\\begin{equation}\n    \\begin{aligned}\\\n    [\\mathbf{s}^l_1,\\ldots, \\mathbf{s}^l_n] & = \\texttt{softmax}([\\vec{x}_1,\\ldots, \\vec{x}_n]^T \\mathbf{W}_l) \\\\\n    [\\mathbf{s}^r_1,\\ldots, \\mathbf{s}^r_n] & = \\texttt{softmax}([\\vec{x}_1,\\ldots, \\vec{x}_n]^T \\mathbf{W}_r)\n    \\end{aligned}\n\\end{equation}\nwhere \\(\\mathbf{W}_l, \\mathbf{W}_r \\in \\mathbb{R}^{d\\times 1} \\), \\texttt{softmax} normalizes logits across tokens, and \\(s_i^l, s_i^r\\) are scores of the \\(i\\)th token being the start or end of the answer span.\nGiven ground-truth labels of span starts \\(y_i^l\\) and ends \\(y_i^r\\),\nthe model optimizes the cross entropy loss:\n\\begin{equation}\n    \\begin{aligned}\n        \\ell_Q & = \\ell_Q^l + \\ell_Q^r \\\\\n        \\ell_Q^l & = - \\sum_{i=1}^n y_i^l \\log s_i^l \\\\\n        \\ell_Q^r & = - \\sum_{i=1}^n y_i^r \\log s_i^r\n    \\end{aligned}\n    \\label{eqn:qa_loss}\n\\end{equation}\nto make a prediction,\nwe follow ~\\cite{devlin2018bert_} and\npredicts the span as \\(i, j\\) with the highest \\(s_i^l + s_j^r\\) satisfying \\(i<j\\).\nAdditionally, unlike in the QA setting in SQuAD~\\cite{rajpurkar2016squad} and adopted by {\\texttt{BERT}}~~\\cite{devlin2018bert_},\nnot every queried event and argument type pair has an answer.\nHence, we only predict the answer when both \\(s_i^l\\) and \\(s_j^r\\) are greater than a certain threshold, typically \\(0.5\\).\n\\section{Preliminary}\n\n\n\n\\subsection{Tasks Definition}\n\\label{sec:tasks}\n\\vpara{Event extraction.}\nThe event extraction task consists of two subtasks: event detection and argument detection. We describe these tasks as follows.\n\\begin{itemize}\n    \\item \\textit{Event Detection.}\n        Following ~\\cite{liu2019event}, we formulate event detection as a multi-class sentence classification problem.\n        Each input sentence \\(X=[x_1,\\ldots,x_n]\\) is associated with a binary label vector \\(y\\in\\mathbb{R}^m\\) where \\(m\\) is the number of event types. Each component corresponds to an event type and is \\(1\\) if and only if the input sentence describes that event.\n        Our goal is to predict for each event whether it is described in the given sentence.\n    \\item \\textit{Argument detection.}\n        Following most related works, e.g. ~\\cite{chen2015event}, each sentence is annotated \\emph{entities} as candidate arguments. Argument\n        detection aims to determine, given the sentence, a described event and an entity, whether the entity serves as one of the predefined argument\n        roles to that  event.\n\\end{itemize}\n\n\\vpara{Machine Reading Comprehension.}\n\\hide{\nIn this paper, we provide a working definition of machine reading comprehension tasks (MRC) as, given a context sentence \\(X\\), we feed the machine, or\nmodel, a query \\(Q\\), and it is supposed to return an answer to \\(Q\\). \\zc{this sentence is too fragmented.}\n}\nWe here give a working definition of machine reading comprehension (MRC) tasks.\nGiven a context sentence \\(X\\), we feed the machine (model) a query \\(Q\\) in form of natual language sentences. The model is expected to return the answer to \\(Q\\) conditioned on the context \\(X\\).\n\nSpecifically, we consider two MRC tasks: 1) textual entailment prediction and 2) question answering.\n\n\\begin{enumerate}\n    \\item \\textit{Textual Entailment Prediction (TE).}\n    Textual entailment prediction, as the name suggests, aims at classifying if a sentence is the entailment of another.\n    Specifically, given sentence \\(X\\) and \\(Q\\), we query the model if \\(Q\\) is contradictory to, entailed by, or neutral to \\(X\\).\n    Recognizing textual entailment can be formulated as a three-way classification task over sentence pair \\((X, Q)\\).\n    \\item \\textit{Question Answering (QA).} We consider span-selection based question answering. In other words, we assume that the answer to the question can be found as a span of words in a given piece of text.\n    Specifically, given sentence \\(X\\) and question \\(Q\\), the model finds a span in \\(X\\) by predicting the starting and ending token of the span and returns it as the answer.\n\\end{enumerate}\n\\section{Related Work}\n\nIn this section, we review related works from closely relevant fields. First, we review the most relevant reading comprehension framework and applications to NLP tasks (Section~\\ref{sec:related:rc}). \nSecond, we review relevant papers from event extraction,  in both standard supervised setting (Section~\\ref{sec:related:ed}) and few-shot learning setting (Section~\\ref{sec:related:fsl}). \nAt last, since our method implicitly treat {\\texttt{BERT}}~ as a knowledge base, we review relevant papers in Section~\\ref{sec:related:knowledge}. \n\n\\subsection{Reading Comprehension Frameworks}\n\\label{sec:related:rc}\nWe first review methods that reframe various NLP tasks as machine reading comprehension. \nMost trending natural language processing tasks can be formulated as reading comprehension tasks. \\cite{levy2017zero_shot_re} used pre-{\\texttt{BERT}}~ model for question answering for zero-shot relation extraction. \nLater on, more works formulated tasks reading comprehension, such as \\cite{keskar2019qa_sentence} for sentence classification, \\cite{li2019unified_mrc_ner} for NER, \\cite{obamuyide2018zero_shot_re_entail} for relation extraction with entailment prediction, \\cite{wu2020corefqa-query} for coreference resolution, and \\cite{wu2019zero_qa_slot_filling} for slot filling. \\cite{mccann2018natural_qa_multitask} unified 10 tasks into one question answering based multi-task learning framework. \\cite{das2018building_kg} used reading comprehension as a tool to build knowledge graphs. \\cite{gardner2019question_useful} probed whether or when the format of question answering is useful.\n\\cite{chen2019reading_manual_ee} designed a pipeline to do zero-shot event detection by reading information from the annotation guide. \n\nFormulating tasks as machine reading comprehension has one key advantage, that it is able to better utilize knowledge in modern language models, such as {\\texttt{BERT}}~~\\cite{devlin2018bert_}, ideally improving label efficiency. \nFor example, among above mentioned works, \\cite{levy2017zero_shot_re,obamuyide2018zero_shot_re_entail,wu2019zero_qa_slot_filling,chen2019reading_manual_ee} all employed reading comprehension as a tool for zero- or few-shot learning. \n\n\\cite{du2020event} is closley related to our work in that they also developed a reading comprehension framework for event detection. The key difference is that, they used a QA framework for event trigger detection and classification. As shown in Section~\\ref{sec:event_detection}, this isn't necessarily the most natural way to utilize {\\texttt{BERT}}~. Indeed, \\cite{du2020event} found that the most effective question form is a single word ``verb'', essentailly not a question and only hinting the model that verbs are more possible trigger words, which is not the ideal kind of knowledge one expects to extract from {\\texttt{BERT}}~.\n\n\\subsection{Event Detection}\n\\label{sec:related:ed}\nThe early attempts to solve event detection rely on hand-crafted features ~\\cite{li2013joint} and probabilisitc rules ~\\cite{liu2016probabilistic}.\nMore recent works utilize neural networks for learning salient representations for event detection, such as CNN~\\cite{chen2015event,nguyen2015event,nguyen2015event}, RNN~\\cite{nguyen2016joint,duan2017exploiting,hong2018self,sha2019jointly}, and GNN~\\cite{liu2018jointly,nguyen2018graph,yan2019event}. \n\nUnlike other sequence tagging problems, such as NER, event extraction relies more heavily on the structural information in the sentence. Hence, one way to predict events and arguments jointly is by utilizing dependency parse~\\cite{nguyen2018graph,sha2019jointly,liu2018jointly,yan2019event}. \nMany researchers exploit the hierachical information as well~\\cite{liao2010using,yang2018dcfee,duan2017exploiting,yang2016hierarchical,mehta2019event}.\n\nSimilarly to our setting, \\cite{liu2019event} considers event detection a sentence classification task, citing that the subjectivity in tagging event triggers can harm model performance and the knowledge of triggers is ``non-essential to the task''~\\cite{liu2019event}. However, they did not further extend the framework to include argument detection.\nTo the best of our knowledge, we are the first to implement argument detection without relying on trigger information.\n\n\\subsection{Few-shot Learning}\n\\label{sec:related:fsl}\nThere exists two approaches to few-shot event detection.\nThe first is based on defining prototype vectors for events~\\cite{peng2016event,huang2017zero}.\nThey both extract a graphical structure for each candidate trigger words with external resources, including semantic role labeling~\\cite{peng2016event} and abstract meaning representation~\\cite{huang2017zero}. \nSRL and AMR themselves are tantamount to argument detection already, so their works can be viewed as event detectoin based on given arguments. Since we focus on doing both event detection and argument detection from scratch, their methods aren't directly comparable to ours. It would be a promising direction to incorporate their structural knowledge into our framework. \n\nThe second approach that has been applied to few-shot event detection is meta-learning in ~\\cite{deng2020meta}. They have built their own data based on ACE05~\\cite{doddington2004automatic}. Since we do not have access to their customized data or their implementation, we could not compare our method with theirs. \n\n\\subsection{{\\texttt{BERT}}~ as a Knowledge Base}\n\\label{sec:related:knowledge}\nThere have been works that investigate whether Bert is a knowledge base itself.\nThe hypothesis is that Bert, while learning from massive corpora, could memorize factual and commonsense knowledge about the world. ~\\cite{petroni2019language,roberts2020much}.\n\\cite{petroni2019language}, Bert might have inferred right relations, without having the right understanding, but instead based on ``learned associations of objects with subjects from co-occurrence patterns''. Still, the work of \\cite{bouraoui2019inducing} shows that fine-tuned Bert model can consistently outperform simple word-vector-based models in inferring relations. \\cite{roberts2020much} extends to use Bert to answer more complex natural language queries, instead of traditional triplet queries, and shows that Bert outperforms open-domain retrieval baselines by a large margin.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $\\Omega \\subseteq\\mathbb R^N$ be a bounded domain with  $C^2$-boundary $\\partial\\Omega$. In this paper we study the following nonlinear parametric singular Dirichlet problem:\n\\begin{equation}\\tag{$P_{\\lambda}$}\\label{eqp}\n\t\\left\\{\\begin{array}{l}\n\t\t-\\Delta_pu(z)=\\lambda u(z)^{-\\gamma}+f(z,u(z))\\ \\mbox{in}\\ \\Omega,\\\\\n\t\tu|_{\\partial\\Omega}=0,\\ u>0,\\ \\lambda>0,\\ 0<\\gamma<1.\n\t\\end{array}\\right\\}\n\\end{equation}\n\nIn this problem, $\\Delta_p$ denotes the $p$-Laplacian differential operator defined by\n$$\\Delta_pu={\\rm div}\\,(|Du|^{p-2}Du)\\ \\mbox{for all}\\ u\\in W^{1,p}_0(\\Omega),\\ 1<p<\\infty.$$\n\nOn the right-hand side of \\eqref{eqp} (the reaction of the problem), we have a parametric singular term $u\\mapsto \\lambda u^{-\\gamma}$ with $\\lambda>0$ being the parameter and $0<\\gamma<1$. Also, there is a Carath\\'eodory perturbation $f(z,x)$ (that is, for all $x\\in\\mathbb R$ the mapping $z\\mapsto f(z,x)$ is measurable and for almost all $z\\in\\Omega$ the mapping $x\\mapsto f(z,x)$ is continuous). We assume that $f(z,\\cdot)$ exhibits $(p-1)$-linear growth near $+\\infty$.\n\n We are looking for positive solutions of problem \\eqref{eqp}. Our aim is to describe in a precise way the dependence on the parameter $\\lambda>0$ of the set of positive solutions.\n\n \\textcolor{black}{\nWe prove a bifurcation-type property, which is the main result of our paper. Concerning the hypotheses $H(f)$ on the perturbation $f(z,x)$ and the other notation used in the statement of the theorem, we refer to Section~2.\nThe main result of the present paper is stated in the following theorem.}\n\n\\textcolor{black}{\n{\\bf Theorem A.} {\\sl If hypotheses $H(f)$ hold, then there exists $\\lambda^*\\in(0,+\\infty)$ such that\n\t\\begin{itemize}\n\t\t\\item [(a)] for every $\\lambda\\in(0,\\lambda^*)$, problem \\eqref{eqp} has at least two positive solutions\n\t\t\t$$\n\t\t\tu_\\lambda,\\hat{u}_\\lambda\\in {\\rm int}\\,C_+,\\ u_\\lambda\\neq\\hat{u}_\\lambda, \\ u_\\lambda\\leq\\hat{u}_\\lambda;\n\t\t\t$$\n\t\t\\item [(b)] for $\\lambda=\\lambda^*$, problem \\eqref{eqp} has at least one positive solution\n\t\t\t$$\n\t\t\tu^*_\\lambda\\in {\\rm int}\\,C_+;\n\t\t\t$$\n\t\t\\item [(c)] for $\\lambda>\\lambda^*$, problem \\eqref{eqp} has no positive solutions.\n\t\\end{itemize}}\n}\n\nIn the past, singular problems were studied in the context of semilinear equations (that is, $p=2$). We mention the works of Coclite \\& Palmieri \\cite{2}, Ghergu \\& R\\u{a}dulescu \\cite{5}, Hirano, Saccon \\& Shioji \\cite{10}, Lair \\& Shaker \\cite{11}, Sun, Wu \\& Long \\cite{20}. A detailed bibliography and additional topics on the subject, can be found in the book of Ghergu \\& R\\u{a}dulescu \\cite{6}. For nonlinear equations driven by the $p$-Laplacian, we mention the works of Giacomoni, Schindler \\& Taka\\v{c} \\cite{7}, Papageorgiou, R\\u{a}dulescu \\& Repov\\v{s} \\cite{16, 16bis}, Papageorgiou \\& Smyrlis \\cite{17}, Perera \\& Zhang \\cite{18}. Of the aforementioned papers, closest to our work here is that of Papageorgiou \\& Smyrlis \\cite{17}, where the authors also deal with a parametric singular problem and prove a bifurcation-type result. In their problem, the perturbation $f(z,x)$ is ($p-1$)-superlinear in $x\\in\\mathbb R$ near $+\\infty$. So, our present work complements the results of \\cite{17}, by considering equations in which the reaction has the competing effects of a singular term and of a $(p-1)$-linear term.\n\nOur approach uses variational tools together with suitable truncation and comparison techniques.\n\n\\section{Preliminaries and hypotheses}\n\nLet $X$ be a Banach space and $X^*$ its topological dual. By $\\left\\langle \\cdot,\\cdot\\right\\rangle$ we denote the duality brackets of the pair $(X^*,X)$. Given $\\varphi\\in C^1(X,\\mathbb R)$, we say that $\\varphi$ satisfies the ``Cerami condition\" (the ``C-condition\" for short), if the following property holds:\n\\begin{center}\n``Every sequence $\\{u_n\\}_{n\\geq 1}\\subseteq X$ such that\n$$\\{\\varphi(u_n)\\}_{n\\geq 1}\\subseteq\\mathbb R\\ \\mbox{is bounded and}\\\n\\mbox{$(1+||u_n||)\\varphi'(u_n)\\rightarrow 0$ in $X^*$ as $n\\rightarrow\\infty$,}$$\nadmits a strongly convergent subsequence.\"\n\\end{center}\n\nUsing this notion, we can state the ``mountain pass theorem\".\n\\begin{theorem}\\label{th1} {\\bf (Mountain pass theorem)}\n\tAssume that $\\varphi\\in C^1(X,\\mathbb R)$ satisfies the C-condition, $u_0,u_1\\in X$, $||u_1-u_0||>\\rho>0$,\n\t$$\\max\\{\\varphi(u_0),\\varphi(u_1)\\}<\\inf\\{\\varphi(u):||u-u_0||=\\rho\\}=m_{\\rho}$$\n\tand $c=\\inf\\limits_{\\gamma\\in\\Gamma}\\max\\limits_{0\\leq t\\leq 1}\\ \\varphi(\\gamma(t))$ with $\\Gamma=\\{\\gamma\\in C([0,1],X):\\gamma(0)=u_0,\\gamma(1)=u_1\\}$. Then $c\\geq m_{\\rho}$ and $c$ is a critical value of $\\varphi$ (that is, we can find $\\hat{u}\\in X$ such that $\\varphi'(\\hat{u})=0$ and $\\varphi(\\hat{u})=c$).\n\\end{theorem}\n\nThe analysis of problem \\eqref{eqp} will involve the Sobolev space $W^{1,p}_0(\\Omega)$ and the Banach space $$C^1_0(\\overline{\\Omega})=\\{u\\in C^1(\\overline{\\Omega}):u|_{\\partial\\Omega}=0\\}.$$ We denote by $||\\cdot||$ the norm of $W^{1,p}_0(\\Omega)$. On account of the Poincar\\'e inequality, we have\n$$||u||=||Du||_p\\ \\mbox{for all}\\ u\\in W^{1,p}_0(\\Omega).$$\n\nThe space $C^1_0(\\overline{\\Omega})$ is an ordered Banach space with positive (order) cone $$C_+=\\{u\\in C^1_0(\\overline{\\Omega}):u(z)\\geq 0\\ \\mbox{for all}\\ z\\in\\overline{\\Omega}\\}.$$ This cone has a nonempty interior given by\n$${\\rm int}\\, C_+=\\left\\{u\\in C_+:u(z)>0\\ \\mbox{for all}\\ z\\in\\Omega,\\ \\left.\\frac{\\partial u}{\\partial n}\\right|_{\\partial\\Omega}<0\\right\\}.$$\n\nHere, $n(\\cdot)$ denotes the outward unit normal on $\\partial\\Omega$.\n\nLet $h_1,h_2\\in L^{\\infty}(\\Omega)$. We write $h_1\\prec h_2$, if for every compact $K\\subseteq\\Omega$, we can find $c_K>0$ such that $c_K\\leq h_2(z)-h_1(z)$ for almost all $z\\in K$. Note that, if $h_1,h_2\\in C(\\Omega)$ and $h_1(z)<h_2(z)$ for all $z\\in\\Omega$, then $h_1\\prec h_2$.\n\nThe next strong comparison principle can be found in Papageorgiou \\& Smyrlis \\cite[Proposition 4]{17} (see also Giacomoni, Schindler \\& Taka\\v{c} \\cite[Theorem 2.3]{7}).\n\n\\begin{prop}\\label{prop2}\n\tIf $\\hat{\\xi}\\geq 0,h_1,h_2\\in L^{\\infty}(\\Omega)$, $h_1\\prec h_2,u_1\\in C_+$ with $u_1(z)>0$ for all $z\\in\\Omega$, $u_2\\in {\\rm int}\\, C_+$ and\n\t\\begin{eqnarray*}\n\t\t&&-\\Delta_pu_1(z)+\\hat{\\xi}u_1(z)^{p-1}-\\lambda u_1(z)^{-\\gamma}=h_1(z),\\\\\n\t\t&&-\\Delta_pu_2(z)+\\hat{\\xi}u_2(z)^{p-1}-\\lambda u_2(z)^{-\\gamma}=h_2(z)\\ \\mbox{for almost all}\\ z\\in\\Omega,\n\t\\end{eqnarray*}\n\tthen $u_2-u_1\\in {\\rm int}\\, C_+.$\n\\end{prop}\n\nWe denote by $A:W^{1,p}_0(\\Omega)\\rightarrow W^{-1,p'}(\\Omega)=W^{1,p}_0(\\Omega)^*\\left(\\frac{1}{p}+\\frac{1}{p'}=1\\right)$  the nonlinear map defined by\n$$\\left\\langle A(u),h\\right\\rangle=\\int_{\\Omega}|Du|^{p-2}(Du,Dh)_{\\mathbb R^N}dz\\ \\mbox{for all}\\ u,h\\in W^{1,p}_0(\\Omega).$$\n\nThis map has the following properties (see Motreanu, Motreanu \\& Papageorgiou \\cite[p. 40]{15}).\n\\begin{prop}\\label{prop3}\n\tThe map $A:W^{1,p}_0(\\Omega)\\rightarrow W^{-1,p'}(\\Omega)$ is bounded (that is, $A$ maps bounded sets to bounded sets), continuous, strictly monotone and of type $(S)_+$, that is, if $u_n\\stackrel{w}{\\rightarrow}u$ in $W^{1,p}_0(\\Omega)$ and $\\limsup\\limits_{n\\rightarrow\\infty}\\left\\langle A(u_n),u_n-u\\right\\rangle\\leq 0$, then $u_n\\rightarrow u$ in $W^{1,p}_0(\\Omega)$.\n\\end{prop}\n\nConsider the following nonlinear eigenvalue problem\n\\begin{equation}\\label{eq1}\n\t-\\Delta_pu(z)=\\hat{\\lambda}|u(z)|^{p-2}u(z)\\ \\mbox{in}\\ \\Omega,\\ u|_{\\partial\\Omega}=0.\n\\end{equation}\n\nWe say that $\\hat{\\lambda}\\in\\mathbb R$ is an ``eigenvalue\" of ($-\\Delta_p,W^{1,p}_0(\\Omega)$) if problem (\\ref{eq1}) admits a nontrivial solution $\\hat{u}\\in W^{1,p}_0(\\Omega)$, known as an ``eigenfunction\" corresponding to $\\hat{\\lambda}$. The nonlinear regularity theory (see Gasinski \\& Papageorgiou \\cite[pp. 737-738]{3}) implies that $\\hat{u}\\in C^1_0(\\overline{\\Omega})$. There is a smallest eigenvalue $\\hat{\\lambda}_1>0$ with the following properties:\n\\begin{itemize}\n\t\\item $\\hat{\\lambda}_1>0$ is isolated (that is, if $\\hat{\\sigma}(p)$ denotes the spectrum of ($-\\Delta_p,W^{1,p}_0(\\Omega)$) then we can find $\\epsilon>0$ such that $(\\hat{\\lambda}_1,\\hat{\\lambda}_1+\\epsilon)\\cap\\hat{\\sigma}(p)=0$);\n\t\\item $\\hat{\\lambda}_1$ is simple (that is, if $\\hat{u},\\hat{v}\\in C^1_0(\\overline{\\Omega})$ are eigenfunctions corresponding to $\\hat{\\lambda}_1$, then $\\hat{u}=\\xi\\hat{v}$ for some $\\xi\\in \\mathbb R\\backslash\\{0\\}$);\n\t\\begin{equation}\\label{eq2}\n\t\t\\bullet\\hspace{3cm}\\hat{\\lambda}_1=\\inf\\left\\{\\frac{||Du||^p_p}{||u||^p_p}:u\\in W^{1,p}_0(\\Omega),u\\neq 0\\right\\}.\\hspace{4cm}\n\t\\end{equation}\n\\end{itemize}\n\nIt follows from the above properties that the eigenfunctions corresponding to $\\hat{\\lambda}_1$ do not change sign. We denote by $\\hat{u}_1$ the positive, $L^p$-normalized (that is, $||\\hat{u}_1||_p=1$) eigenfunction corresponding to $\\hat{\\lambda}_1>0$. From the nonlinear maximum principle (see, for example, Gasinski \\& Papageorgiou \\cite[p. 738]{3}), we have $\\hat{u}_1\\in {\\rm int}\\, C_+$. Any eigenfunction corresponding to an eigenvalue $\\hat{\\lambda}\\neq\\hat{\\lambda}_1$, is nodal (that is, sign-changing). More details about the spectrum of $(-\\Delta_p,W^{1,p}_0(\\Omega))$ can be found in \\cite{3, 15}.\n\nWe can also consider a weighted version of the eigenvalue problem (\\ref{eq1}). So, let $m\\in L^{\\infty}(\\Omega)$, $m(z)\\geq 0$ for almost all $z\\in\\Omega,\\ m\\neq 0$. We consider the following nonlinear eigenvalue problem:\n\\begin{equation}\\label{eq3}\n\t-\\Delta_pu(z)=\\tilde{\\lambda}m(z)|u(z)|^{p-2}u(z)\\ \\mbox{in}\\ \\Omega,\\ u|_{\\partial\\Omega}=0.\n\\end{equation}\n\nThis problem has the same properties as (\\ref{eq1}). So, there is a smallest eigenvalue $\\tilde{\\lambda}_1(m)>0$ which is isolated, simple and admits the following variational characterization\n$$\\tilde{\\lambda}_1(m)=\\inf\\left\\{\\frac{||Du||^p_p}{\\int_{\\Omega}m(z)|u|^pdz}:u\\in W^{1,p}_0(\\Omega),u\\neq 0\\right\\}.$$\n\nAlso the eigenfunctions corresponding to $\\tilde{\\lambda}_1(m)$ have a fixed sign and we denote by $\\tilde{u}_1(m)$  the positive, $L^p$-normalized eigenfunction. We have $\\tilde{u}_1(m)\\in {\\rm int}\\, C_+$. These properties lead to the following monotonicity property of the map $m\\mapsto\\tilde{\\lambda}_1(m)$.\n\n\\begin{prop}\\label{prop4}\n\tIf $m_1,m_2\\in L^{\\infty}(\\Omega),0\\leq m_1(z)\\leq m_2(z)$ for almost all $z\\in\\Omega$ and both inequalities are strict on the sets of positive measure, then $\\tilde{\\lambda}_1(m_2)<\\tilde{\\lambda}_1(m_1)$.\n\\end{prop}\n\nGiven $x\\in\\mathbb R$, we set $x^{\\pm}=\\max\\{\\pm x, 0\\}$. Then for $u\\in W^{1,p}_0(\\Omega)$, we set $u^{\\pm}(\\cdot)=u(\\cdot)^{\\pm}$. We know that\n$$u^{\\pm}\\in W^{1,p}_0(\\Omega),\\ |u|=u^++u^-,\\ u=u^+-u^-.$$\n\nIf $g:\\Omega\\times\\mathbb R$ is a measurable function (for example, a Carath\\'eodory function) then by $N_g(\\cdot)$ we denote the Nemytski map corresponding to $g(\\cdot,\\cdot)$ defined by\n$$N_g(u)(\\cdot)=g(\\cdot,u(\\cdot))\\ \\mbox{for all}\\ u\\in W^{1,p}_0(\\Omega).$$\n\nGiven $v,u\\in W^{1,p}_0(\\Omega)$ with $v\\leq u$, we define the order interval $[v,u]$ by\n$$[v,u]=\\{y\\in W^{1,p}_0(\\Omega):v(z)\\leq y(z)\\leq u(z)\\ \\mbox{for almost all}\\ z\\in\\Omega\\}.$$\n\nThe hypotheses on the perturbation $f(z,x)$ are the following:\n\n\\smallskip\n$H(f):$ $f:\\Omega\\times\\mathbb R\\leftarrow\\mathbb R$ is a Carath\\'eodory function such that $f(z,0)=0$ for almost all $z\\in\\Omega$ and\n\\begin{itemize}\n\t\\item[(i)] for every $\\rho>0$, there exists $a_{\\rho}\\in L^{\\infty}(\\Omega)$ such that\n\t$$|f(z,x)|\\leq a_{\\rho}(z)\\ \\mbox{for almost all}\\ z\\in\\Omega,\\ \\mbox{and all}\\ 0\\leq x\\leq\\rho;$$\n\t\\item[(ii)] $\\hat{\\lambda}_1<\\eta\\leq\\liminf\\limits_{x\\rightarrow+\\infty}\\frac{f(z,x)}{x^{p-1}}\\leq\\limsup\\limits_{x\\rightarrow+\\infty}\\frac{f(z,x)}{x^{p-1}}\\leq\\hat{\\eta}$ uniformly for almost all $z\\in\\Omega;$\n\t\\item[(iii)] there exists a function $w\\in C^1(\\overline{\\Omega})$ such that\n\t$$w(z)\\geq c_0>0\\ \\mbox{for all}\\ z\\in\\overline{\\Omega},\\ \\Delta_pw\\in L^{\\infty}(\\Omega)\\ \\mbox{with}\\ \\Delta_pw(z)\\leq 0\\ \\mbox{for almost all}\\ z\\in\\Omega,$$\n\tand for every compact $K\\subseteq\\Omega$ we can find $c_K>0$ such that\n\t$$w(z)^{-\\gamma}+f(z,w(z))\\leq-c_K<0\\ \\mbox{for almost all}\\ z\\in K;$$\n\t\\item[(iv)] there exists $\\delta_0\\in(0,c_0)$ such that for every compact $K\\subseteq\\Omega$\n\t$$f(z,x)\\geq\\hat{c}_K>0\\ \\mbox{for almost all}\\ z\\in K,\\ \\mbox{and all}\\ x\\in\\left(0,\\delta_0\\right];$$\n\t\\item[(v)] for every $\\rho>0$, there exists $\\hat{\\xi}_{\\rho}>0$ such that for almost all $z\\in\\Omega$ the function\n\t$$x\\mapsto f(z,x)+\\hat{\\xi}_{\\rho}x^{p-1}$$\n\tis nondecreasing on $[0,\\rho]$.\n\\end{itemize}\n\n\\begin{remark}\n\tSince we are looking for positive solutions and all the above hypotheses concern the positive semiaxis $\\mathbb R_+=\\left[0,+\\infty\\right)$, we may assume without any loss of generality that\n\t\\begin{equation}\\label{eq4}\n\t\tf(z,x)=0\\ \\mbox{for almost all}\\ z\\in\\Omega,\\ \\mbox{and all}\\ x\\leq 0.\n\t\\end{equation}\n\\end{remark}\n\nHypothesis $H(f)(iii)$ implies that asymptotically at $+\\infty$ we have uniform nonresonance with respect to the principal eigenvalue $\\hat{\\lambda}_1>0$ of $(-\\Delta_p,W^{1,p}_0(\\Omega))$. The resonant case was recently examined  for nonparametric singular Dirichlet problems by Papageorgiou, R\\u{a}dulescu \\& Repov\\v{s} \\cite{16}.\n\n\\begin{ex}\nThe following functions satisfy hypotheses $H(f)$. For the sake of simplicity we drop the $z$-dependence:\n$$f(x)=\\left\\{\\begin{array}{ll}\n\tx^{\\tau-1}-3x^{\\vartheta-1}&\\mbox{if}\\ 0\\leq x\\leq 1\\\\\n\t\\eta x^{p-1}-(\\eta+2)x^{q-1}&\\mbox{if}\\ 1<x\n\\end{array}\\right\\}\\ (\\mbox{see (\\ref{eq4})})$$\nwith $1<\\tau<\\vartheta$, $1<q<p$ and $\\eta>\\hat{\\lambda}_1$; and\n\n\\textcolor{black}{\n$$f(x)=\\left\\{\\begin{array}{ll}\n\t2\\sin(2\\pi x)&\\mbox{if}\\ 0\\leq x\\leq 1\\\\\n\t\\eta (x^{p-1}-x^{q-1})&\\mbox{if}\\ 1<x\n\\end{array}\\right.$$\nwith $\\eta>\\hat{\\lambda}_1$, $1<q<p$. }\n\\end{ex}\n\n\\section{A purely singular problem}\n\nIn this section we deal with the following purely singular parametric problem:\n\\begin{equation}\\tag{$Au_{\\lambda}$}\\label{eqa}\n\t\\left\\{\\begin{array}{l}\n\t\t-\\Delta_pu(z)=\\lambda u(z)^{-\\gamma}\\ \\mbox{in}\\ \\Omega\\\\\n\t\tu|_{\\partial\\Omega}=0,\\ u>0,\\ \\lambda>0,\\ 0<\\gamma<1.\n\t\\end{array}\\right\\}\n\\end{equation}\n\nThe next proposition establishes the existence and $\\lambda$-dependence of the positive solutions for problem \\eqref{eqa}.\n\\begin{prop}\\label{prop5}\n\tFor every $\\lambda>0$ problem \\eqref{eqa} admits a unique solution $\\tilde{u}_{\\lambda}\\in {\\rm int}\\, C_+$, the map $\\lambda\\mapsto\\tilde{u}_{\\lambda}$ is nondecreasing from $(0,\\infty)$ into $C^1_0(\\overline{\\Omega})$ (that is, if $0<\\vartheta<\\lambda$, then $\\tilde{u}_{\\vartheta}\\leq\\tilde{u}_{\\lambda}$) and $||\\tilde{u}_{\\lambda}||_{C^1_0(\\overline{\\Omega})}\\rightarrow 0$ as $\\lambda\\rightarrow 0^+$.\n\\end{prop}\n\\begin{proof}\n\tThe existence of a unique solution $\\tilde{u}_{\\lambda}\\in {\\rm int}\\, C_+$ follows from Proposition 5 of Papageorgiou \\& Smyrlis \\cite{17}.\n\t\n\tLet $0<\\vartheta<\\lambda$ and let $\\tilde{u}_{\\vartheta},\\tilde{u}_{\\lambda}\\in {\\rm int}\\, C_+$ be the corresponding unique solutions of problem \\eqref{eqa}. Evidently, $\\tilde{u}^{p'}_{\\vartheta}\\in {\\rm int}\\, C_+\\left(\\frac{1}{p}+\\frac{1}{p'}=1\\right)$ and so by Proposition 2.1 of Marano \\& Papageorgiou \\cite{14}, we can find $c_1>0$ such that\n\t\\begin{eqnarray*}\n\t\t&&\\hat{u}_1\\leq c_1\\tilde{u}^{p'}_{\\vartheta},\\\\\n\t\t&\\Rightarrow&\\hat{u}_1^{1\/p'}\\leq c_1^{1\/p'}\\tilde{u}_{\\vartheta},\\\\\n\t\t&\\Rightarrow&\\tilde{u}^{-\\gamma}_{\\vartheta}\\leq c_2\\hat{u}_1^{-\\gamma\/p'}\\ \\mbox{for some}\\ c_2>0.\n\t\\end{eqnarray*}\n\t\n\t Lemma of Lazer \\& McKenna \\cite[p. 726]{12}, implies that $\\hat{u}_1^{-\\gamma\/p'}\\in L^{p'}(\\Omega)$. Therefore $\\tilde{u}_{\\vartheta}^{-\\gamma}\\in L^{p'}(\\Omega)$. We introduce the Carath\\'eodory function $g_{\\lambda}(z,x)$ defined by\n\t\\begin{equation}\\label{eq5}\n\t\tg_{\\lambda}(z,x)=\\left\\{\\begin{array}{ll}\n\t\t\t\\lambda\\tilde{u}_{\\vartheta}^{-\\gamma}&\\mbox{if}\\ x\\leq\\tilde{u}_{\\vartheta}(z)\\\\\n\t\t\t\\lambda x^{-\\gamma}&\\mbox{if}\\ \\tilde{u}_{\\vartheta}(z)<x.\n\t\t\\end{array}\\right.\n\t\\end{equation}\n\t\n\tWe set $G_{\\lambda}(z,x)=\\int^x_0g_{\\lambda}(z,s)ds$ and consider the functional $\\hat{\\psi}_{\\lambda}:W^{1,p}_0(\\Omega)\\rightarrow\\mathbb R$ defined by\n\t$$\\hat{\\psi}_{\\lambda}(u)=\\frac{1}{p}||Du||^p_p-\\int_{\\Omega}G_{\\lambda}(z,u)dz\\ \\mbox{for all}\\ u\\in W^{1,p}_0(\\Omega).$$\n\t\n\tProposition 3 of Papageorgiou \\& Smyrlis \\cite{17} implies that $\\hat{\\psi}_{\\lambda}\\in C^1(W^{1,p}_0(\\Omega))$. From (\\ref{eq5}) and since $\\tilde{u}_{\\vartheta}^{-\\gamma}\\in L^{p'}(\\Omega)$ it follows that $\\hat{\\psi}_{\\lambda}(\\cdot)$ is coercive. Also, via the Sobolev embedding theorem, we see that $\\hat{\\psi}_{\\lambda}(\\cdot)$ is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find $\\bar{u}_{\\lambda}\\in W^{1,p}_0(\\Omega)$ such that\n\t\t\\begin{eqnarray}\\label{eq6}\n\t\t\t&&\\hat{\\psi}_{\\lambda}(\\bar{u}_{\\lambda})=\\inf\\{\\hat{\\psi}_{\\lambda}(u):u\\in W^{1,p}_0(\\Omega)\\},\\nonumber\\\\\n\t\t\t&\\Rightarrow&\\hat{\\psi}'_{\\lambda}(\\bar{u}_{\\lambda})=0,\\nonumber\\\\\n\t\t\t&\\Rightarrow&\\left\\langle A(\\bar{u}_{\\lambda}),h\\right\\rangle=\\int_{\\Omega}g_{\\lambda}(z,\\bar{u}_{\\lambda})hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega).\n\t\t\\end{eqnarray}\n\t\t\n\t\tIn (\\ref{eq3}) we choose $h=(\\tilde{u}_{\\vartheta}-\\bar{u}_{\\lambda})^+\\in W^{1,p}_0(\\Omega)$. We have\n\t\t\\begin{eqnarray}\\label{eq7}\n\t\t\t\\left\\langle A(\\bar{u}_{\\lambda}),(\\tilde{u}_{\\vartheta}-\\bar{u}_{\\lambda})^+\\right\\rangle&=&\\int_{\\Omega}\\lambda\\tilde{u}_{\\vartheta}^{-\\gamma}(\\tilde{u}_{\\vartheta}-\\bar{u}_{\\lambda})^+dz\\ (\\mbox{see (\\ref{eq5})})\\nonumber\\\\\n\t\t\t&\\geq&\\int_{\\Omega}\\vartheta\\tilde{u}_{\\vartheta}^{-\\gamma}(\\tilde{u}_{\\vartheta}-\\bar{u}_{\\lambda})dz\\ (\\mbox{since}\\ \\vartheta<\\lambda)\\nonumber\\\\\n\t\t\t&=&\\left\\langle A(\\tilde{u}_{\\vartheta}),(\\tilde{u}_{\\vartheta}-\\bar{u}_{\\lambda})^+\\right\\rangle,\\nonumber\\\\\n\t\t\t\\Rightarrow\\tilde{u}_{\\vartheta}\\leq\\bar{u}_{\\lambda}.\n\t\t\\end{eqnarray}\n\t\t\n\t\tFrom (\\ref{eq5}), (\\ref{eq6}), (\\ref{eq7}), we have\n\t\t\\begin{eqnarray*}\n\t\t\t&&-\\Delta_p\\bar{u}_{\\lambda}(z)=\\lambda\\bar{u}_{\\lambda}(z)^{-\\gamma}\\ \\mbox{for almost all}\\ z\\in\\Omega,\\left.\\bar{u}_{\\lambda}\\right|_{\\partial\\Omega}=0,\\\\\n\t\t\t&\\Rightarrow&\\bar{u}_{\\lambda}=\\tilde{u}_{\\lambda},\\\\\n\t\t\t&\\Rightarrow&\\tilde{u}_{\\vartheta}\\leq\\tilde{u}_{\\lambda}\\ (\\mbox{see (\\ref{eq7})}).\n\t\t\\end{eqnarray*}\n\t\t\n\t\tTherefore the map $\\lambda\\mapsto\\tilde{u}_{\\lambda}$ is nondecreasing from $(0,+\\infty)$ into $C^1_0(\\overline{\\Omega})$.\n\t\t\n\t\tWe have\n\t\t$$\\left\\langle A(\\tilde{u}_{\\lambda}),h\\right\\rangle=\\int_{\\Omega}\\lambda\\tilde{u}_{\\lambda}^{-\\gamma}hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega).$$\n\t\t\n\t\tChoosing $h=\\tilde{u}_{\\lambda}\\in W^{1,p}_0(\\Omega)$, we obtain\n\t\t\\begin{eqnarray}\\label{eq8}\n\t\t\t&&||D\\tilde{u}_{\\lambda}||^p_p=\\lambda\\int_{\\Omega}\\tilde{u}_{\\lambda}^{1-\\gamma}dz\\leq \\lambda c_3||\\tilde{u}_{\\lambda}||_p\\ \\mbox{for some}\\ c_3>0\\nonumber\\\\\n\t\t\t&&(\\mbox{see Theorem 13.17 of Hewitt \\& Stromberg \\cite[p. 196]{9}}),\\nonumber\\\\\n\t\t\t&\\Rightarrow&\\{\\tilde{u}_{\\lambda}\\}_{\\lambda\\in\\left(0,1\\right]}\\subseteq W^{1,p}_0(\\Omega)\\ \\mbox{is bounded and }||\\tilde{u}_{\\lambda}||\\rightarrow 0\\ \\mbox{as}\\ \\lambda\\rightarrow 0^+.\n\t\t\\end{eqnarray}\n\t\t\n\t\tAs in the first part of the proof, using Proposition 2.1 of Marano \\& Papageorgiou \\cite{14}, we show that $\\tilde{u}_{\\lambda}^{-\\gamma}\\in L^r(\\Omega)$ for $r>N$. Then Proposition 1.3 of Guedda \\& V\\'eron \\cite{8} implies that\n\t\t\\begin{equation}\\label{eq9}\n\t\t\t\\tilde{u}_{\\lambda}\\in L^{\\infty}(\\Omega)\\ \\mbox{and}\\ ||\\tilde{u}_{\\lambda}||_{\\infty}\\leq c_4\\ \\mbox{for some}\\ c_4>0,\\ \\mbox{and all}\\ 0<\\lambda\\leq 1.\n\t\t\\end{equation}\n\t\t\n\t\tLet $k_{\\lambda}=\\lambda\\tilde{u}^{-\\gamma}_{\\lambda}\\in L^r(\\Omega),\\lambda\\in\\left(0,1\\right]$ and consider the following linear Dirichlet problem\n\t\t\\begin{equation}\\label{eq10}\n\t\t\t-\\Delta v(z)=k_{\\lambda}(z)\\ \\mbox{in}\\ \\Omega,\\ v|_{\\partial\\Omega}=0,\\ 0<\\lambda\\leq 1.\n\t\t\\end{equation}\n\t\t\n\t\tStandard existence and regularity theory (see, for example, Struwe \\cite[p. 218]{19}), implies that problem (\\ref{eq10}) has a unique solution $v_{\\lambda}(\\cdot)$ such that\n\t\t$$v_{\\lambda}\\in W^{2,r}(\\Omega)\\subseteq C^{1,\\alpha}_{0}(\\overline{\\Omega})=C^{1,\\alpha}(\\overline{\\Omega})\\cap C^1_0(\\overline{\\Omega}),\\ ||v_{\\lambda}||_{C^{1,\\alpha}_0(\\overline{\\Omega})}\\leq c_5$$\n\t\tfor some $c_5>0$, all $\\lambda\\in\\left(0,1\\right]$, and with $\\alpha=1-\\frac{N}{r}\\in(0,1)$ (recall that $r>N$). Let $\\beta_{\\lambda}(z)=Dv_{\\lambda}(z)$. Then $\\beta_{\\lambda}\\in C^{0,\\alpha}(\\overline{\\Omega})$ for every $\\lambda\\in\\left(0,1\\right]$. We have\n\t\t$$-{\\rm div}\\,[|D\\tilde{u}_{\\lambda}|^{p-2}D\\tilde{u}_{\\lambda}-\\beta_{\\lambda}]=0\\ \\mbox{in}\\ \\Omega,\\left.\\ \\tilde{u}_{\\lambda}\\right|_{\\partial\\Omega}=0\\ (\\mbox{since}\\ \\tilde{u}_{\\lambda}\\ \\mbox{solves}\\ \\eqref{eqa}).$$\n\t\t\n\t\tThen Theorem 1 of Lieberman \\cite{13} (see also Corollary 1.1 of Guedda \\& V\\'eron \\cite{8}) and (\\ref{eq9}), imply that we can find $s\\in(0,1)$ and $c_6>0$ such that\n\t\t$$\\tilde{u}_{\\lambda}\\in C^{1,s}_0(\\overline{\\Omega})\\cap {\\rm int}\\, C_+,\\ ||\\tilde{u}_{\\lambda}||_{C^{1,s}_0(\\overline{\\Omega})}\\leq c_6\\ \\mbox{for all}\\ \\lambda\\in\\left(0,1\\right].$$\n\t\t\n\t\tFinally, the compact embedding of $C^{1,s}_0(\\overline{\\Omega})$ into $C^1_0(\\overline{\\Omega})$ and (\\ref{eq8}) imply that\n\t\t$$||\\tilde{u}_{\\lambda}||_{C^1_0(\\overline{\\Omega})}\\rightarrow 0\\ \\mbox{as}\\ \\lambda\\rightarrow 0^+.$$\nThis completes the proof.\n\\end{proof}\n\n\n\\section{Bifurcation-type theorem}\n\nLet $$\\mathcal{L}=\\{\\lambda>0:\\ \\mbox{problem \\eqref{eqp} admits a positive solution}\\}$$ \\begin{center}\n$S_{\\lambda}=\\mbox{the set of positive solutions for problem \\eqref{eqp}}$.\n  \\end{center}\n\\begin{prop}\\label{prop6}\nIf hypotheses $H(f)$ hold, then $\\mathcal{L}\\neq\\emptyset$.\n\\end{prop}\n\\begin{proof}\nUsing Proposition \\ref{prop5}, we can find $\\lambda_0\\in\\left(0,1\\right]$ such that\n\\begin{equation}\\label{eq11}\n\\tilde{u}_{\\lambda}(z)\\in\\left(0,\\delta_0\\right]\\ \\mbox{for all}\\ z\\in\\Omega,\\ \\mbox{all}\\ \\lambda\\in\\left(0,\\lambda_0\\right].\n\\end{equation}\n\nHere, $\\delta_0>0$ is as postulated by hypothesis $H(f)(iv)$.\n\nWe fix $\\lambda\\in\\left(0,\\lambda_0\\right]$ and we consider the following truncation of the reaction in problem \\eqref{eqp}:\n\\begin{eqnarray}\\label{eq12}\n\\hat{k}_{\\lambda}(z,x)=\\left\\{\\begin{array}{ll}\n    \\lambda\\hat{u}_{\\lambda}(z)^{-\\gamma}+f(z,\\tilde{u}_{\\lambda}(z))&\\mbox{if}\\ x<\\tilde{u}_{\\lambda}(z)\\\\\n    \\lambda x^{-\\gamma}+f(z,x)&\\mbox{if}\\ \\tilde{u}_{\\lambda}\\leq x\\leq w(z)\\\\\n    \\lambda w(z)^{-\\gamma}+f(z,w(z))&\\mbox{if}\\ w(z)<x\n\\end{array}\n\\right.\n\\end{eqnarray}\n(recall that $\\delta_0<c_0\\leq w(z)$ for all $z\\in\\overline{\\Omega}$). This is a Carath\\'eodory function. We set $\\hat{K}_{\\lambda}(z,x)=\\int^x_0\\hat{k}_{\\lambda}(z,s)ds$ and consider the function $\\hat{\\varphi}_{\\lambda}:W^{1,p}_0(\\Omega)\\rightarrow \\mathbb R$ defined by\n$$\\hat{\\varphi}_{\\lambda}(u)=\\frac{1}{p}||Du||^p_p-\\int_{\\Omega}\\hat{K}_{\\lambda}(z,u)dz\\ \\mbox{for all}\\ u\\in W^{1,p}_0(\\Omega).$$\n\nAs before, we have $\\hat{\\varphi}_{\\lambda}\\in C^1(W^{1,p}_0(\\Omega))$. Also,  it follows from (\\ref{eq12}) that\n$$\\hat{\\varphi}(\\cdot)\\ \\mbox{is coercive}.$$\n\nIn addition, we have that\n$$\\hat{\\varphi}_{\\lambda}(\\cdot)\\ \\mbox{is sequentially lower semicontinuous}.$$\n\nTherefore, we can find $\\hat{u}_{\\lambda}\\in W^{1,p}_0(\\Omega)$ such that\n\\begin{eqnarray}\\label{eq13}\n&&\\hat{\\varphi}_{\\lambda}(\\hat{u}_{\\lambda})=\\inf[\\hat{\\varphi}_{\\lambda}(u):u\\in W^{1,p}_0(\\Omega)],\\nonumber\\\\\n&\\Rightarrow &\\hat{\\varphi}'_{\\lambda}(\\hat{u}_{\\lambda})=0,\\nonumber\\\\\n&\\Rightarrow &\\left\\langle A(\\hat{u}_{\\lambda}),h\\right\\rangle=\\int_{\\Omega}\\hat{k}_{\\lambda}(z,\\hat{u}_{\\lambda})hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega).\n\\end{eqnarray}\n\nIn (\\ref{eq13}) we choose $h=(\\tilde{u}_{\\lambda}-\\hat{u}_{\\lambda})^+\\in W^{1,p}_0(\\Omega)$. Then\n\\begin{eqnarray*}\n\t\\left\\langle A(\\hat{u}_{\\lambda}),(\\tilde{u}_{\\lambda}-\\hat{u}_{\\lambda})^+\\right\\rangle&=&\\int_{\\Omega}[\\lambda\\tilde{u}_{\\lambda}^{-\\gamma}+f(z,\\tilde{u}_{\\lambda})](\\tilde{u}_{\\lambda}-\\hat{u}_{\\lambda})^+dz\\ (\\mbox{see (\\ref{eq12})})\\\\\n\t&\\geq&\\int_{\\Omega}\\lambda\\tilde{u}_{\\lambda}^{-\\gamma}(\\tilde{u}_{\\lambda}-\\hat{u}_{\\lambda})^+dz\\\\\n\t&&(\\mbox{see (\\ref{eq11}) and hypothesis H(f)(iv)})\\\\\n\t&=&\\left\\langle A(\\tilde{u}_{\\lambda}),(\\tilde{u}_{\\lambda}-\\hat{u}_{\\lambda})^+\\right\\rangle\\ (\\mbox{see Proposition \\ref{prop5}}),\\\\\n\t\\Rightarrow\\tilde{u}_{\\lambda}\\leq\\hat{u}_{\\lambda}.&\n\\end{eqnarray*}\n\nNext, we choose   $h=(\\hat{u}_{\\lambda}-w)^+\\in W^{1,p}_0(\\Omega)$ in (\\ref{eq13}). Then\n\\begin{eqnarray*}\n\t\\left\\langle A(\\hat{u}_{\\lambda}),(\\hat{u}_{\\lambda}-w)^+\\right\\rangle&=&\\int_{\\Omega}[\\lambda w^{-\\gamma}+f(z,w)](\\hat{u}_{\\lambda}-w)^+dz\\ (\\mbox{see (\\ref{eq12})})\\\\\n\t&\\leq&\\left\\langle A(w),(\\hat{u}_{\\lambda}-w)^+\\right\\rangle\n\\end{eqnarray*}\n(see hypothesis $H(f)(iii)$ and use the nonlinear Green identity, see \\cite[p. 211]{3})\n$$\\Rightarrow\\tilde{u}_{\\lambda}\\leq w.$$\n\nSo, we have proved that\n\\begin{equation}\\label{eq14}\n\t\\hat{u}_{\\lambda}\\in[\\tilde{u}_{\\lambda},w].\n\\end{equation}\n\nUsing (\\ref{eq14}) and (\\ref{eq12}), equation (\\ref{eq13}) becomes\n\\begin{eqnarray}\\label{eq15}\n\t&&\\left\\langle A(\\hat{u}_{\\lambda}),h\\right\\rangle=\\int_{\\Omega}[\\lambda\\hat{u}_{\\lambda}^{-\\gamma}+f(z,\\hat{u}_{\\lambda})]hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega),\\nonumber\\\\\n\t&\\Rightarrow&-\\Delta_p\\hat{u}_{\\lambda}(z)=\\lambda\\hat{u}_{\\lambda}(z)^{-\\gamma}+f(z,\\hat{u}_{\\lambda}(z))\\ \\mbox{for almost all}\\ z\\in\\Omega,\\left.\\hat{u}_{\\lambda}\\right|_{\\partial\\Omega}=0.\n\\end{eqnarray}\n\nFrom (\\ref{eq14}), (\\ref{eq15}) and Theorem 1 of Lieberman \\cite{13}, we infer that\n\\begin{eqnarray*}\n\t&&\\hat{u}_{\\lambda}\\in[\\tilde{u}_{\\lambda},w]\\cap {\\rm int}\\, C_+,\\\\\n\t&\\Rightarrow&\\lambda\\in\\mathcal{L},\\hat{u}_{\\lambda}\\in S_{\\lambda}.\n\\end{eqnarray*}\nThis completes the proof.\n\\end{proof}\n\nA byproduct of the above proof is the following corollary.\n\\begin{corollary}\\label{cor7}\n\tIf hypotheses $H(f)$ hold, then $S_{\\lambda}\\subseteq {\\rm int}\\, C_+$ for all $\\lambda>0$.\n\\end{corollary}\n\nThe next proposition shows that $\\mathcal{L}$ is an interval.\n\\begin{prop}\\label{prop8}\n\tIf hypotheses $H(f)$ hold, $\\lambda\\in\\mathcal{L}$ and $\\vartheta\\in(0,\\lambda)$, then $\\vartheta\\in\\mathcal{L}.$\n\\end{prop}\n\\begin{proof}\n\tSince $\\lambda\\in\\mathcal{L}$, we can find $u_{\\lambda}\\in S_{\\lambda}\\subseteq {\\rm int}\\, C_+$. Proposition \\ref{prop5} implies that we can find $\\tau\\in[0,\\lambda_0]$ (see (\\ref{eq11})) such that\n\t$$\\tau<\\vartheta\\ \\mbox{and}\\ \\tilde{u}_{\\tau}\\leq u_{\\lambda}.$$\n\t\n\tWe introduce the Carath\\'eodory function $e(z,x)$ defined by\n\t\\begin{equation}\\label{eq16}\n\t\te_{\\vartheta}(z,x)=\\left\\{\\begin{array}{ll}\n\t\t\t\\vartheta\\tilde{u}_{\\tau}(z)^{-\\gamma}+f(z,\\tilde{u}_{\\tau}(z))&\\mbox{if}\\ x<\\tilde{u}_{\\tau}(z)\\\\\n\t\t\t\\vartheta x^{-\\gamma}+f(z,x)&\\mbox{if}\\ \\tilde{u}_{\\tau}(z)\\leq x\\leq u_{\\lambda}(z)\\\\\n\t\t\t\\vartheta u_{\\lambda}(z)^{-\\gamma}+f(z,u_{\\lambda}(z))&\\mbox{if}\\ u_{\\lambda}(z)<x.\n\t\t\\end{array}\\right.\n\t\\end{equation}\n\t\n\tWe set $E_{\\vartheta}(z,x)=\\int^x_0 e_{\\vartheta}(z,s)ds$ and consider the functional $\\hat{\\psi}_{\\vartheta}:W^{1,p}_0(\\Omega)\\rightarrow\\mathbb R$ defined by\n\t$$\\hat{\\psi}_{\\vartheta}(u)=\\frac{1}{p}||Du||^p_p-\\int_{\\Omega}E_{\\vartheta}(z,u)dz\\ \\mbox{for all}\\ u\\in W^{1,p}_0(\\Omega).$$\n\t\n\tWe know that $\\hat{\\psi}_{\\vartheta}\\in C^1(W^{1,p}_0(\\Omega))$. Moreover, $\\hat{\\psi}_{\\vartheta}$ is coercive (see (\\ref{eq16})) and sequentially weakly lower semicontinuous. So, we can find $u_{\\vartheta}\\in W^{1,p}_0(\\Omega)$ such that\n\t\\begin{eqnarray}\\label{eq17}\n\t\t&&\\hat{\\psi}_{\\vartheta}(u_{\\vartheta})=\\inf\\{\\hat{\\psi}_{\\vartheta}(u):u\\in W^{1,p}_0(\\Omega)\\},\\nonumber\\\\\n\t\t&\\Rightarrow&\\hat{\\psi}'_{\\vartheta}(u_{\\vartheta})=0,\\nonumber\\\\\n\t\t&\\Rightarrow&\\left\\langle A(u_{\\vartheta}),h\\right\\rangle=\\int_{\\Omega}e_{\\vartheta}(z,u_{\\vartheta})hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega).\n\t\\end{eqnarray}\n\t\n\tIn (\\ref{eq17})  we first choose $h=(\\tilde{u}_{\\tau}-u_{\\vartheta})^+\\in W^{1,p}_0(\\Omega)$. Then\n\t\\begin{eqnarray*}\n\t\t\\left\\langle A(u_{\\vartheta}),(\\tilde{u}_{\\tau}-u_{\\vartheta})^+\\right\\rangle&=&\\int_{\\Omega}[\\vartheta\\tilde{u}_{\\tau}^{-\\gamma}+f(z,\\tilde{u}_{\\tau})](\\tilde{u}_{\\tau}-u_{\\vartheta})^+dz\\ (\\mbox{see (\\ref{eq16})})\\\\\n\t\t&\\geq&\\int_{\\Omega}\\vartheta\\tilde{u}_{\\tau}^{-\\gamma}(\\tilde{u}_{\\tau}-u_{\\vartheta})^+dz\\\\\n\t\t&&(\\mbox{since}\\ \\tau\\leq\\lambda_0,\\ \\mbox{see (\\ref{eq11}) and hypothesis}\\ H(f)(iv))\\\\\n\t\t&\\geq&\\int_{\\Omega}\\tau\\tilde{u}_{\\tau}^{-\\gamma}(\\tilde{u}_{\\tau}-u_{\\vartheta})^+dz\\ (\\mbox{recall that}\\ \\tau<\\vartheta)\\\\\n\t\t&=&\\left\\langle A(u_{\\tau}),(\\tilde{u}_{\\tau}-u_{\\vartheta})^+\\right\\rangle\\ (\\mbox{see Proposition \\ref{eq5}}),\\\\\n\t\t\\Rightarrow\\tilde{u}_{\\tau}\\leq u_{\\vartheta}.&&\n\t\\end{eqnarray*}\n\t\n\tNext, in (\\ref{eq17}) we choose $h=(u_{\\vartheta}-u_{\\lambda})^+\\in W^{1,p}_0(\\Omega)$. Then\n\t\\begin{eqnarray*}\n\t\t\\left\\langle A(u_{\\vartheta}),(u_{\\vartheta}-u_{\\lambda})^+\\right\\rangle&=&\\int_{\\Omega}[\\vartheta u_{\\lambda}^{-\\gamma}+f(z,u_{\\lambda})](u_{\\vartheta}-u_{\\lambda})^+dz\\ (\\mbox{see (\\ref{eq16})})\\\\\n\t\t&\\leq&\\int_{\\Omega}[\\lambda u_{\\lambda}^{-\\gamma}+f(z,u_{\\lambda})](u_{\\vartheta}-u_{\\lambda})^+dz\\ (\\mbox{since}\\ \\vartheta<\\lambda)\\\\\n\t\t&=&\\left\\langle A(u_{\\lambda}),(u_{\\vartheta}-u_{\\lambda})^+\\right\\rangle\\ (\\mbox{since}\\ u_{\\lambda}\\in S_{\\lambda}),\\\\\n\t\t\\Rightarrow u_{\\vartheta}\\leq u_{\\lambda}.&&\n\t\\end{eqnarray*}\n\t\n\tSo, we have proved that\n\t\\begin{equation}\\label{eq18}\n\t\tu_{\\vartheta}\\in[\\tilde{u}_{\\tau},u_{\\lambda}].\n\t\\end{equation}\n\t\n\tIt follows from (\\ref{eq16}), (\\ref{eq17}) and (\\ref{eq18}) that\n\t$$\\vartheta\\in\\mathcal{L}\\ \\mbox{and}\\ u_{\\vartheta}\\in S_{\\vartheta}\\subseteq {\\rm int}\\, C_+.$$\nThe proof is now complete.\n\\end{proof}\n\nAn interesting byproduct of the above proof is the following result.\n\\begin{corollary}\\label{cor9}\n\tIf hypotheses $H(f)$ hold, $\\lambda\\in\\mathcal{L},u_{\\lambda}\\in S_{\\lambda}\\subseteq {\\rm int}\\, C_+$, and $\\vartheta<\\lambda$, then $\\vartheta\\in\\mathcal{L}$ and we can find $u_{\\vartheta}\\in S_{\\vartheta}\\subseteq {\\rm int}\\, C_+$ such that $u_{\\vartheta}\\leq u_{\\lambda}$.\n\\end{corollary}\n\nIn fact, we can improve the above result as follows.\n\\begin{prop}\\label{prop10}\n\tIf hypotheses $H(f)$ hold, $\\lambda\\in\\mathcal{L},u_{\\lambda}\\in S_{\\lambda}\\subseteq {\\rm int}\\, C_+$, and $\\vartheta<\\lambda$, then $\\vartheta\\in\\mathcal{L}$ and we can find $u_{\\vartheta}\\in S_{\\vartheta}\\subseteq {\\rm int}\\, C_+$ such that $u_{\\lambda}-u_{\\vartheta}\\in {\\rm int}\\, C_+$.\n\\end{prop}\n\\begin{proof}\n\tFrom Corollary \\ref{cor9} we know that $\\vartheta\\in\\mathcal{L}$ and we can find $u_{\\vartheta}\\in S_{\\vartheta}\\subseteq {\\rm int}\\, C_+$ such that\n\t\\begin{equation}\\label{eq19}\n\t\tu_{\\vartheta}\\leq u_{\\lambda}.\n\t\\end{equation}\n\t\n\tLet $\\rho=||u_{\\lambda}||_{\\infty}$ and let $\\hat{\\xi}_{\\rho}>0$ be as postulated by hypothesis $H(f)(v)$. Then\n\t\\begin{eqnarray*}\n\t\t&&-\\Delta_pu_{\\vartheta}+\\hat{\\xi}_pu_{\\vartheta}^{p-1}-\\lambda u_{\\vartheta}^{-\\gamma}\\\\\n\t\t&=&-(\\lambda-\\vartheta)u_{\\vartheta}^{-\\gamma}+f(z,u_{\\vartheta})+\\hat{\\xi}_{\\rho}u_{\\vartheta}^{p-1}\\\\\n\t\t&\\leq&f(z,u_{\\lambda})+\\hat{\\xi}_{\\rho}u_{\\lambda}^{p-1}\\ (\\mbox{recall that}\\ \\vartheta<\\lambda\\ \\mbox{and see (\\ref{eq19}) and hypothesis}\\ H(f)(v))\\\\\n\t\t&=&-\\Delta_pu_{\\lambda}+\\hat{\\xi}_{\\rho}u_{\\lambda}^{p-1}-\\lambda u_{\\lambda}^{-\\gamma}\\ (\\mbox{since}\\ u_{\\lambda}\\in S_{\\lambda}).\n\t\\end{eqnarray*}\n\t\n\tWe set\n\t\\begin{eqnarray*}\n\t\t&&h_1(z)=f(z,u_{\\vartheta}(z))+\\hat{\\xi}_{\\rho}u_{\\vartheta}(z)^{p-1}-(\\lambda-\\vartheta)u_{\\vartheta}(z)^{-\\gamma}\\\\\n\t\t&&h_2(z)=f(z,u_{\\lambda}(z))+\\hat{\\xi}_{\\rho}u_{\\lambda}(z)^{p-1}.\n\t\\end{eqnarray*}\n\t\n\tWe have\n\t$$h_2(z)-h_1(z)\\geq(\\lambda-\\vartheta)u_{\\vartheta}(z)^{-\\gamma}\\geq(\\lambda-\\vartheta)\\rho^{-\\gamma}\\ \\mbox{for almost all}\\ z\\in\\Omega$$\n\t(see (\\ref{eq19}) and hypotheses $H(f)(v)$).\n\t\n\tWe can apply Proposition \\ref{prop2} and conclude that\n\t$$u_{\\lambda}-u_{\\vartheta}\\in {\\rm int}\\, C_+.$$\nThe proof is now complete.\n\\end{proof}\n\nDenote $\\lambda^*=\\sup\\mathcal{L}.$\n\\begin{prop}\\label{prop11}\n\tIf hypotheses $h(f)$ hold, then $\\lambda^*<+\\infty$.\n\\end{prop}\n\\begin{proof}\n\tLet $\\epsilon>0$ be such that $\\hat{\\lambda}_1+\\epsilon<\\eta$ (see hypothesis $H(f)(ii)$). We can find $M>0$ such that\n\t\\begin{equation}\\label{eq20}\n\t\tf(z,x)\\geq[\\hat{\\lambda}_1+\\epsilon]x^{p-1}\\ \\mbox{for almost all}\\ z\\in\\Omega,\\ \\mbox{and all}\\ x\\geq M.\n\t\\end{equation}\n\t\n\tAlso, hypothesis $H(f)(i)$ implies that we can find large enough $\\tilde{\\lambda}>0$ such that\n\t\\begin{equation}\\label{eq21}\n\t\t\\tilde{\\lambda}M^{-\\gamma}+f(z,x)\\geq[\\hat{\\lambda}_1+\\epsilon]M^{p-1}\\ \\mbox{for almost all}\\ z\\in\\Omega,\\ \\mbox{and all}\\ 0\\leq x\\leq M.\n\t\\end{equation}\n\t\n\tIt follows from (\\ref{eq20}) and (\\ref{eq21}) that\n\t\\begin{equation}\\label{eq22}\n\t\t\\tilde{\\lambda}x^{-\\gamma}+f(z,x)\\geq[\\hat{\\lambda}_1+\\epsilon]x^{p-1}\\ \\mbox{for almost all}\\ z\\in\\Omega,\\ \\mbox{and all}\\ x\\geq 0.\n\t\\end{equation}\n\t\n\tLet $\\lambda>\\tilde{\\lambda}$ and suppose that $\\lambda\\in\\mathcal{L}$. Then we can find $u_{\\lambda}\\in S_{\\lambda}\\subseteq {\\rm int}\\, C_+$. We have\n\t\\begin{equation}\\label{eq23}\n\t\t-\\Delta_pu_{\\lambda}=\\lambda u_{\\lambda}^{-\\gamma}+f(z,u_{\\lambda})>\\tilde{\\lambda}u_{\\lambda}^{-\\gamma}+f(z,u_{\\lambda})\\geq[\\hat{\\lambda}_1+\\epsilon]u_{\\lambda}^{p-1}\\ \\mbox{for a.a.}\\ z\\in\\Omega\\ (\\mbox{see (\\ref{eq22})}).\n\t\\end{equation}\n\t\n\tSince $u_{\\lambda}\\in {\\rm int}\\, C_+$, we can find $t\\in(0,1)$ so small that\n\t\\begin{equation}\\label{eq24}\n\t\t\\hat{y}_1=t\\hat{u}_1\\leq u_{\\lambda}\n\t\\end{equation}\n\t(see Proposition 2.1 of Marano \\& Papageorgiou \\cite{14}). We have\n\t\\begin{equation}\\label{eq25}\n\t\t-\\Delta_p\\hat{y}_1=\\hat{\\lambda}_1\\hat{y}_1^{p-1}<[\\hat{\\lambda}_1+\\epsilon]\\hat{y}_1^{p-1}\\ \\mbox{for almost all}\\ z\\in\\Omega.\n\t\\end{equation}\n\t\n\tUsing (\\ref{eq24}), we can define the Carath\\'eodory function $\\beta(z,x)$ as follows\n\t\\begin{eqnarray}\\label{eq26}\n\t\t\\beta(z,x)=\\left\\{\n\t\t  \\begin{array}{lll}\n\t\t\t & [\\hat{\\lambda}_1 + \\epsilon]\\hat{y}_1(z)^{p-1}&  \\mbox{if}\\ x<\\hat{y}_1(z)\\\\\n\t\t\t & [\\hat{\\lambda}_1 + \\epsilon]x^{p-1}           &  \\mbox{if}\\ \\hat{y}_1(z)\\leq x\\leq u_{\\lambda}(z)\\\\\n\t\t\t & [\\hat{\\lambda}_1 + \\epsilon]u_{\\lambda}(z)^{p-1}&  \\mbox{if}\\ u_{\\lambda}(z)<x.\n\t\t  \\end{array}\\right.\n\t\\end{eqnarray}\n\t\n\tWe set $B(z,x)=\\int^x_0\\beta(z,s)ds$ and consider the $C^1$-functional $\\sigma:W^{1,p}_0(\\Omega)\\rightarrow\\mathbb R$ defined by\n\t$$\\sigma(u)=\\frac{1}{p}||Du||^p_p-\\int_{\\Omega}B(z,u)dz\\ \\mbox{for all}\\ u\\in W^{1,p}_0(\\Omega).$$\n\t\n\tFrom (\\ref{eq26}) it is clear that $\\sigma(\\cdot)$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\\bar{u}\\in W^{1,p}_0(\\Omega)$ such that\n\t\\begin{eqnarray}\\label{eq27}\n\t\t&&\\sigma(\\bar{u})=\\inf\\{\\sigma(u):u\\in W^{1,p}_0(\\Omega)\\},\\nonumber\\\\\n\t\t&\\Rightarrow&\\sigma'(\\bar{u})=0,\\nonumber\\\\\n\t\t&\\Rightarrow&\\left\\langle A(\\bar{u}),h\\right\\rangle=\\int_{\\Omega}\\beta(z,\\bar{u})hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega).\n\t\\end{eqnarray}\n\t\n\tIn (\\ref{eq27}) we first choose $h=(\\hat{y}_1-\\bar{u})^+\\in W^{1,p}_0(\\Omega)$. Then\n\t\\begin{eqnarray*}\n\t\t\\left\\langle A(\\bar{u}),(\\hat{y}_1-\\bar{u})^+\\right\\rangle &=&\\int_{\\Omega}[\\hat{\\lambda}_1+\\epsilon]\\hat{y}_1^{p-1}(\\hat{y}_1-\\bar{u})^+dz\\ (\\mbox{see (\\ref{eq26})})\\\\\n\t\t&\\geq&\\left\\langle A(\\hat{y}_1),(\\hat{y}_1-\\hat{u})^+\\right\\rangle\\ (\\mbox{see (\\ref{eq25})}),\\\\\n\t\t\\Rightarrow\\hat{y}_1\\leq\\bar{u}.&\n\t\\end{eqnarray*}\n\t\n\tAlso, in (\\ref{eq27}) we choose $h=(\\bar{u}-u_{\\lambda})^+\\in W^{1,p}_0(\\Omega)$. Then\n\t\\begin{eqnarray*}\n\t\t\\left\\langle A(\\bar{u}),(\\bar{u}-u_{\\lambda})^+\\right\\rangle&=&\\int_{\\Omega}[\\hat{\\lambda}_1+\\epsilon]u_{\\lambda}^{p-1}(\\bar{u}-u_{\\lambda})^+dz\\ (\\mbox{see (\\ref{eq26})})\\\\\n\t\t&\\leq&\\left\\langle A(u_{\\lambda}),(\\bar{u}-u_{\\lambda})^+\\right\\rangle\\ (\\mbox{see (\\ref{eq23})}),\\\\\n\t\t\\Rightarrow\\bar{u}\\leq u_{\\lambda}.&&\n\t\\end{eqnarray*}\n\t\n\tSo, we have proved that\n\t\\begin{equation}\\label{eq28}\n\t\t\\bar{u}\\in[\\hat{y}_1,u_{\\lambda}].\n\t\\end{equation}\n\t\nIt follows\tfrom (\\ref{eq26}), (\\ref{eq27}) and (\\ref{eq28}) that\n\t\\begin{eqnarray*}\n\t\t&&-\\Delta_p\\bar{u}(z)=[\\hat{\\lambda}_1+\\epsilon]\\bar{u}(z)^{p-1}\\ \\mbox{for almost all}\\ z\\in\\Omega,\\ \\bar{u}|_{\\partial\\Omega}=0,\\\\\n\t\t&\\Rightarrow&\\bar{u}\\in C^1_0(\\overline{\\Omega})\\ \\mbox{must be nodal, a contradiction (see (\\ref{eq28}))}.\n\t\\end{eqnarray*}\n\t\n\tTherefore we have $\\lambda^*\\leq\\tilde{\\lambda}<+\\infty$.\n\\end{proof}\n\nNext, we show that the critical parameter $\\lambda^*>0$ is admissible.\n\\begin{prop}\\label{prop12}\n\tIf hypotheses $H(f)$ hold, then $\\lambda^*\\in\\mathcal{L}$.\n\\end{prop}\n\\begin{proof}\n\tLet $\\{\\lambda_n\\}_{n\\geq1}\\subseteq(0,\\lambda^*)$ and assume that $\\lambda_n\\rightarrow(\\lambda^*)^{-}$ as $n\\rightarrow\\infty$. We can find $u_n=u_{\\lambda_n}\\in S_{\\lambda_n}\\subseteq {\\rm int}\\,C_+$ for all $n\\in\\mathbb N$. Then\n\t\\begin{equation}\\label{eq29}\n\t\t\\langle A(u_n),h\\rangle = \\int_\\Omega[\\lambda_n u_n^{-\\gamma}+f(z,u_n)]hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\lambda),\\ \\mbox{all}\\ n\\in\\mathbb N.\n\t\\end{equation}\n\t\n\tSuppose that $||u_n||\\rightarrow\\infty$. We set $y_n=\\frac{u_n}{||u_n||}\\ n\\in\\mathbb N$. Then $||y_n||=1, y_n\\geq0$ for all $n\\in\\mathbb N$. So, we may assume that\n\t\\begin{equation}\\label{eq30}\n\t\ty_n\\xrightarrow{w}y\\ \\mbox{in}\\ W^{1,p}_0(\\Omega)\\ \\mbox{and}\\ y_n\\rightarrow y\\ \\mbox{in}\\ L^p(\\Omega)\\ \\mbox{as}\\ n\\rightarrow\\infty.\n\t\\end{equation}\n\t\n\tFrom (\\ref{eq29}) we have\n\t\\begin{equation}\\label{eq31}\n\t\t\\langle A(y_n),h\\rangle = \\int_\\Omega\\left[\\frac{\\lambda_n}{||u_n||^{p+\\gamma-1}}y^{-\\gamma}_n + \\frac{N_f(u_n)}{||u_n||^{p-1}}\\right]hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega),\\ n\\in\\mathbb N.\n\t\\end{equation}\n\t\n\tHypotheses $H(f)(i),(ii)$ imply that\n\t$$\n\t|f(z,x)|\\leq c_7[1+x^{p-1}]\\ \\mbox{for almost all}\\ z\\in\\Omega,\\ \\mbox{all}\\ x\\geq0,\\ \\mbox{and some}\\ c_7>0.\n\t$$\n\t\n\tThis growth condition implies that\n\t\\begin{equation}\\label{eq32}\n\t\t\\left\\{\\frac{N_f(u_n)}{||u_n||^{p-1}}\\right\\}_{n\\geq1}\\subseteq L^{p'}(\\Omega)\\ \\mbox{is bounded}.\n\t\\end{equation}\n\t\n\tThen (\\ref{eq32}) and hypothesis $H(f)(ii)$ imply that at least for a subsequence, we have\n\t\\begin{eqnarray}\\label{eq33}\n\t\t\\frac{N_f(u_n)}{||u_n||^{p-1}}\\xrightarrow{w}\\eta_0(z)y^{p-1}\\ \\mbox{in}\\ L^{p'}(\\Omega)\\ \\mbox{as}\\ n\\rightarrow\\infty,\\\\\n\t\t\\mbox{with}\\ \\eta\\leq\\eta_0(z)\\leq\\hat{\\eta}\\ \\mbox{for almost all}\\ z\\in\\Omega \\nonumber \\\\\n\t\t\\mbox{(see Aizicovici, Papageorgiou \\& Staicu \\cite{1}, proof of Proposition 16)}. \\nonumber\n\t\\end{eqnarray}\n\t\n\tIn (\\ref{eq31}) we choose $h=y_n-y\\in W^{1,p}_0(\\Omega)$, pass to the limit as $n\\rightarrow\\infty$, and use (\\ref{eq30}) and (\\ref{eq32}). Then\n\t\\begin{eqnarray}\n\t\t& \\lim_{n\\rightarrow\\infty}\\langle A(y_n),y_n-y\\rangle=0, \\nonumber \\\\\n\t\t\\Rightarrow & y_n\\rightarrow y\\ \\mbox{in}\\ W^{1,p}_0(\\Omega)\\ \\mbox{(see Proposition \\ref{prop3}), hence}\\ ||y||=1,\\ y\\geq0. \\label{eq34}\n\t\\end{eqnarray}\n\t\n\tTherefore, if in (\\ref{eq31}) we pass to the limit as $n\\rightarrow\\infty$ and use (\\ref{eq34}) and (\\ref{eq33}), then\n\t\\begin{eqnarray}\n\t\t& \\langle A(y),h\\rangle = \\int_\\Omega\\eta_0(z)y^{p-1}hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega), \\nonumber \\\\\n\t\t\\Rightarrow & -\\Delta_py(z)=\\eta_0(z)y(z)^{p-1}\\ \\mbox{for almost all}\\ z\\in\\Omega, \\ y|_{\\partial\\Omega}=0. \\label{eq35}\n\t\\end{eqnarray}\n\t\n\tSince $\\eta\\leq\\eta_0(z)\\leq\\hat{\\eta}$ for almost all $z\\in\\Omega$ (see (\\ref{eq33})), using Proposition \\ref{prop4}, we have\n\t$$\n\t\\tilde{\\lambda}_1(\\eta_0)\\leq\\tilde{\\lambda}_1(\\eta)<\\tilde{\\lambda}_1(\\hat{\\lambda_1})=1.\n\t$$\n\t\n\tSo, from (\\ref{eq35}) and since $||y||=1$ (see (\\ref{eq34})), it follows that $y$ must be nodal, a contradiction (see (\\ref{eq34})). Therefore\n\t$$\n\t\\{u_n\\}_{n\\geq1}\\subseteq W^{1,p}_0(\\Omega)\\ \\mbox{is bounded}.\n\t$$\n\t\n\tHence, we may assume that\n\t\\begin{equation}\\label{eq36}\n\t\tu_n\\xrightarrow{w}u^*\\ \\mbox{in}\\ W^{1,p}_0(\\Omega)\\ \\mbox{and}\\ u_n\\rightarrow u^*\\ \\mbox{in}\\ L^p(\\Omega)\\ \\mbox{as}\\ n\\rightarrow\\infty.\n\t\\end{equation}\n\t\n\tOn account of Corollary \\ref{cor9}, we may assume that $\\{u_n\\}_{n\\geq1}$ is nondecreasing. Therefore $u^*\\neq0$. Also, we have\n\t\\begin{equation}\\label{eq37}\n\t\t0\\leq(u^*)^{-\\gamma}\\leq u_n^{-\\gamma}\\leq u_1^{-\\gamma}\\in L^{p'}(\\Omega)\\ \\mbox{for all}\\ n\\in\\mathbb N.\n\t\\end{equation}\n\t\n\tFrom (\\ref{eq36}) and by passing to a subsequence if necessary, we can say that\n\t\\begin{equation}\\label{eq38}\n\t\tu_n(z)^{-\\gamma}\\rightarrow u^*(z)^{-\\gamma}\\ \\mbox{for almost all}\\ z\\in\\Omega.\n\t\\end{equation}\n\t\n\tFrom (\\ref{eq37}), (\\ref{eq38}) and Problem 1.19 of Gasinski \\& Papageorgiou \\cite{4}, we have that\n\t\\begin{equation}\\label{eq39}\n\t\tu_n^{-\\gamma}\\xrightarrow{w}(u^*)^{-\\gamma}\\ \\mbox{in}\\ L^{p'}(\\Omega)\\ \\mbox{as}\\ n\\rightarrow\\infty.\n\t\\end{equation}\n\t\n\tIf in (\\ref{eq29}) we choose $h=u_n-u^*\\in W^{1,p}_0(\\Omega)$, pass to the limit as $n\\rightarrow\\infty$ and use (\\ref{eq39}) and the fact that $\\{N_f(u_n)\\}_{n\\geq1}\\subseteq L^{p'}(\\Omega)$ is bounded, then\n\t\\begin{eqnarray}\n\t\t& & \\lim_{n\\rightarrow\\infty}\\langle A(u_n),u_n-u^*\\rangle=0, \\nonumber \\\\\n\t\t& \\Rightarrow & u_n\\rightarrow u^*\\ \\mbox{in}\\ W^{1,p}_0(\\Omega)\\ \\mbox{(see Proposition \\ref{prop3})}. \\label{eq40}\n\t\\end{eqnarray}\n\t\n\tFinally, in (\\ref{eq29}) we pass to the limit as $n\\rightarrow\\infty$ and use (\\ref{eq39}) and (\\ref{eq40}). We obtain\n\t\\begin{eqnarray*}\n\t\t& & \\langle A(u^*),h\\rangle = \\int_\\Omega[\\lambda^*(u^*)^{-\\gamma} + f(z,u^*)]hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega), \\\\\n\t\t& \\Rightarrow & u^*\\in S_{\\lambda^*}\\subseteq {\\rm int}\\,C_+\\ \\mbox{and}\\ \\lambda^*\\in\\mathcal{L}.\n\t\\end{eqnarray*}\nThis completes the proof.\n\\end{proof}\nWe have proved that\n$$\n\\mathcal{L}=(0,\\lambda^*].\n$$\t\n\\begin{prop}\\label{prop13}\n\tIf hypotheses $H(f)$ hold and $\\lambda\\in(0,\\lambda^*)$, then problem  $(P_\\lambda)$ admits at least two positive solutions\n\t$$\n\tu_\\lambda,\\hat{u}_\\lambda\\in {\\rm int}\\,C_+, \\hat{u}_{\\lambda}-u_\\lambda\\in C_+\\backslash\\{0\\}.\n\t$$\n\\end{prop}\n\\begin{proof}\n\tLet $u^*\\in S_{\\lambda^*}\\subseteq {\\rm int}\\,C_+$ (see Proposition 12). Invoking Proposition \\ref{prop10}, we can find $u_\\lambda\\in S_\\lambda\\subseteq {\\rm int}\\,C_+$ such that\n\t\\begin{equation}\\label{eq41}\n\t\tu^*-u_\\lambda\\in {\\rm int}\\,C_+.\n\t\\end{equation}\n\t\n\tWe consider the Carath\\'eodory function $\\tau_\\lambda(z,x)$ defined by\n\t\\begin{equation}\\label{eq42}\n\t\t\\tau_\\lambda(z,x)=\\left\\{\n\t\t\t\\begin{array}{ll}\n\t\t\t\t\\lambda u_\\lambda(z)^{-\\gamma} + f(z,u_\\lambda(z)) & \\mbox{if}\\ x\\leq u_\\lambda(z) \\\\\n\t\t\t\t\\lambda x^{-\\gamma} + f(z,x) & \\mbox{if}\\ u_\\lambda(z)<x.\n\t\t\t\\end{array}\n\t\t\\right.\n\t\\end{equation}\n\t\n\tRecall that $u_\\lambda^{-\\gamma}\\in L^{p'}(\\Omega)$ (see the proof of Proposition \\ref{prop5}). We set $T_\\lambda(z,x)=\\int^x_0\\tau_\\lambda(z,s)ds$ and consider the functional $\\tilde\\varphi_\\lambda:W^{1,p}_0(\\Omega)\\rightarrow\\mathbb R$ defined by\n \t$$\n\t\\tilde{\\varphi}_\\lambda(u)=\\frac{1}{p}||Du||^p_p - \\int_\\Omega T_\\lambda(z,u)dz\\ \\mbox{for all}\\ u\\in W^{1,p}_0(\\Omega).\n\t$$\n\t\n\tWe know that $\\tilde{\\varphi}_\\lambda\\in C^1(W^{1,p}_0(\\Omega))$. Let $K_{\\tilde{\\varphi}_\\lambda}=\\{u\\in W^{1,p}_0(\\Omega):\\tilde{\\varphi}_\\lambda'(u)=0\\}$ (the critical set of $\\tilde{\\varphi}_\\lambda$). Also, for $u\\in W^{1,p}_0(\\Omega)$, we set $$[u)=\\{v\\in W^{1,p}(\\Omega):u(z)\\leq v(z)\\ \\mbox{for almost all}\\ z\\in\\Omega\\}.$$\n\t\\begin{claim}\\label{claim1}\n\t\t$K_{\\tilde{\\varphi_\\lambda}}\\subseteq[u_\\lambda)\\cap {\\rm int}\\,C_+$.\n\t\\end{claim}\n\n\tLet $u\\in K_{\\tilde{\\varphi}_\\lambda}$. We have\n\t\\begin{equation}\\label{eq43}\n\t\t\\langle A(u),h\\rangle = \\int_\\Omega \\tau_\\lambda(z,u)hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega).\n\t\\end{equation}\n\t\n\tWe choose $h=(u_\\lambda-u)^+\\in W^{1,p}_0(\\Omega)$. Then\n\t\\begin{eqnarray}\n\t\t\\langle A(u),(u_\\lambda-u)^+\\rangle & = & \\int_\\lambda [\\lambda u_\\lambda^{-\\gamma} + f(z,u_\\lambda)](u_\\lambda-u)^+dz\\ \\mbox{(see (\\ref{eq42}))} \\nonumber \\\\\n\t\t& = & \\langle A(u_\\lambda), (u_\\lambda-u)^+\\rangle\\ \\mbox{(since $u_\\lambda\\in S_\\lambda$)}, \\nonumber \\\\\n\t\t& \\Rightarrow & u_\\lambda\\leq u. \\label{eq44}\n\t\\end{eqnarray}\n\t\n\tFrom (\\ref{eq42}), (\\ref{eq43}) and (\\ref{eq44}), we obtain\n\t\\begin{eqnarray*}\n\t\t& & \\langle A(u),h\\rangle = \\int_\\Omega [\\lambda u^{-\\gamma} + f(z,u)]hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega), \\\\\n\t\t& \\Rightarrow & u\\in S_\\lambda\\subseteq {\\rm int}\\,C_+\\ \\mbox{and}\\ u_\\lambda\\leq u, \\\\\n\t\t& \\Rightarrow & u\\in[u_\\lambda)\\cap {\\rm int}\\,C_+.\n\t\\end{eqnarray*}\n\tThis proves Claim 1.\n\t\n\tNote that $u_\\lambda\\in K_{\\tilde{\\varphi}_\\lambda}$. We may assume that\n\t\\begin{equation}\\label{eq45}\n\t\tK_{\\tilde{\\varphi}_\\lambda}\\cap[u_\\lambda,u^*]=\\{u_\\lambda\\},\n\t\\end{equation}\n\tor otherwise we already have a second positive smooth solution for problem \\eqref{eqp} (see (\\ref{eq42})) and so we are done.\n\t\n\tWe introduce the following Carath\\'eodory function\n\t\\begin{equation}\\label{eq46}\n\t\t\\hat{\\tau}_\\lambda(z,x)=\\left\\{\n\t\t\\begin{array}{ll}\n\t\t\t\\tau_\\lambda(z,x) & \\mbox{if}\\ x\\leq u^*(z) \\\\\n\t\t\t\\tau_\\lambda(z,u^*(z)) & \\mbox{if}\\ u^*(z)<x.\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{equation}\n\t\n\tWe set $\\hat{T_\\lambda}(z,x)=\\int^x_0\\hat{\\tau}_\\lambda(z,s)ds$ and consider the $C^1$-functional $\\hat{\\varphi}_\\lambda:W^{1,p}(\\Omega)\\rightarrow\\mathbb R$ defined by\n\t$$\n\t\\hat{\\varphi}_\\lambda(u)=\\frac{1}{p}||Du||^p_p - \\int_\\Omega\\hat{T_\\lambda}(z,u)dz\\ \\mbox{for all}\\ u\\in W^{1,p}_0(\\Omega).\n\t$$\n\t\n\tThis functional is coercive (see (\\ref{eq46})) and sequentially weakly lower semicontinuous. Hence we can find $\\tilde{u}_\\lambda\\in W^{1,p}_0(\\Omega)$ such that\n\t\\begin{eqnarray}\n\t\t& & \\hat{\\varphi}_\\lambda(\\tilde{u}_\\lambda) = \\inf\\{\\hat{\\varphi}_\\lambda(u):u\\in W^{1,p}_0(\\Omega)\\}, \\nonumber \\\\\n\t\t& \\Rightarrow & \\hat{\\varphi}'_{\\lambda}(\\tilde{u}_\\lambda)=0, \\nonumber \\\\\n\t\t& \\Rightarrow & \\langle A(\\tilde{u}_\\lambda),h\\rangle = \\int_\\Omega\\hat{\\tau}_\\lambda(z,\\tilde{u}_\\lambda)hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega). \\label{eq47}\n\t\\end{eqnarray}\n\t\n\tIn (\\ref{eq47}) we choose $h=(u_\\lambda-\\tilde{u}_\\lambda)^+\\in W^{1,p}_0(\\Omega)$ and $h=(\\tilde{u}_\\lambda-u^*)^+\\in W^{1,p}_0(\\Omega)$ and obtain that\n\t\\begin{equation}\\label{eq48}\n\t\t\\tilde{u}_\\lambda\\in[u_\\lambda,u^*].\n\t\\end{equation}\n\t\n\tFrom (\\ref{eq46}), (\\ref{eq47}), (\\ref{eq48}) we infer that\n\t\\begin{eqnarray*}\n\t\t& & \\tilde{u}_\\lambda\\in K_{\\tilde{\\varphi}_\\lambda}\\cap[u_\\lambda,u^*], \\\\\n\t\t& \\Rightarrow & \\tilde{u}_\\lambda = u_\\lambda\\ \\mbox{(see (\\ref{eq45}))}.\n\t\\end{eqnarray*}\n\t\n\tFrom (\\ref{eq42}) and (\\ref{eq46}) it is clear that\n\t$$\n\t\\tilde{\\varphi}_\\lambda|_{[0,u^*]}=\\hat{\\varphi}_\\lambda|_{[0,u^*]}.\n\t$$\n\t\n\tAlso, $u_\\lambda$ is a minimizer of $\\hat{\\varphi}_\\lambda$. Since $u^*-u_\\lambda\\in {\\rm int}\\,C_+$ (see (\\ref{eq41})), it follows that\n\t\\begin{eqnarray}\n\t\t& & u_\\lambda\\ \\mbox{is a local}\\ C^1_0(\\overline{\\Omega})-\\mbox{minimizer of}\\ \\tilde{\\varphi}_\\lambda, \\nonumber \\\\\n\t\t& \\Rightarrow & u_\\lambda\\ \\mbox{is a local}\\ W^{1,p}_0(\\Omega)-\\mbox{minimizer of}\\ \\tilde{\\varphi}_\\lambda. \\label{eq49} \\\\\n\t\t&& \\mbox{(see Motreanu, Motreanu \\& Papageorgiou \\cite[Theorem 12.18, p. 409]{15})}. \\nonumber\n\t\\end{eqnarray}\n\t\n\tWe assume that $K_{\\tilde{\\varphi}_\\lambda}$ is finite or otherwise on account of Claim \\ref{claim1}, we already have an infinity of positive smooth solutions for problem \\eqref{eqp} bigger than $u_\\lambda$ and so we are done. Because of (\\ref{eq49}), we can find $\\rho\\in(0,1)$ small such that\n\t\\begin{eqnarray}\n\t\t\\tilde{\\varphi}_\\lambda(u_\\lambda)<\\inf\\{\\tilde{\\varphi}_\\lambda(u):||u-u_\\lambda||=\\rho\\}=\\tilde{m}_\\lambda \\label{eq50} \\\\\n\t\t\\mbox{(see Aizicovici, Papageorgiou \\& Staicu \\cite{1}, proof of Proposition 29)}. \\nonumber\n\t\\end{eqnarray}\n\tHypothesis $H(f)(ii)$ implies that\n\t\\begin{equation}\\label{eq51}\n\t\t\\tilde{\\varphi}_\\lambda(t\\hat{u}_1)\\rightarrow-\\infty\\ \\mbox{as}\\ t\\rightarrow+\\infty.\n\t\\end{equation}\n\t\n\t\\begin{claim}\\label{claim2}\n\t\t$\\tilde{\\varphi}_\\lambda$ satisfies the $C$-condition.\n\t\\end{claim}\n\t\n\tLet $\\{u_n\\}_{n\\geq1}\\subseteq W^{1,p}_0(\\Omega)$ such that $\\{\\tilde{\\varphi}_\\lambda(u_n)\\}_{n\\geq1}\\subseteq\\mathbb R$ is bounded and\n\t$$\n\t(1+||u_n||)\\tilde{\\varphi}_\\lambda'(u_n)\\rightarrow0\\ \\mbox{in}\\ W^{-1,p'}(\\Omega)=W^{1,p}_0(\\Omega)^*\\ \\mbox{as}\\ n\\rightarrow\\infty.\n\t$$\n\t\n\tWe have\n\t\\begin{equation}\\label{eq52}\n\t\t|\\langle A(u_n),h\\rangle - \\int_\\Omega\\tau_\\lambda(z,u_n)hdz| \\leq \\frac{\\varepsilon_n||h||}{1+||u_n||}\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega),\\ \\mbox{with}\\ \\varepsilon_n\\rightarrow0^+.\n\t\\end{equation}\n\t\n\tWe choose $h=-u^-_n\\in W^{1,p}_0(\\Omega)$ in (\\ref{eq52}) and also use (\\ref{eq42}). Then\n\t\\begin{eqnarray}\n\t\t& & ||Du^-_n||^p_p \\leq c_8||u^-_n||\\ \\mbox{for some}\\ c_8>0, \\mbox{and all}\\ n\\in\\mathbb N, \\nonumber \\\\\n\t\t& \\Rightarrow & \\{u^-_n\\}_{n\\geq1}\\subseteq W^{1,p}_0(\\Omega)\\ \\mbox{is bounded}. \\label{eq53}\n\t\\end{eqnarray}\n\t\n\tSuppose that $||u^+_n||\\rightarrow\\infty$ and let $y_n=\\frac{u^+_n}{||u^+_n||}\\ n\\in\\mathbb N$. Then $||y_n||=1,y_n\\geq0$ for all $n\\in\\mathbb N$. So, we may assume that\n\t\\begin{equation}\\label{eq54}\n\t\ty_n\\xrightarrow{w}y\\ \\mbox{in}\\ W^{1,p}_0(\\Omega)\\ \\mbox{and}\\ y_n\\rightarrow y\\ \\mbox{in}\\ L^p(\\Omega),\\ y\\geq0.\n\t\\end{equation}\n\t\n\tFrom (\\ref{eq52}) and (\\ref{eq53}), we have\n\t\\begin{equation}\\label{eq55}\n\t\t|\\langle A(y_n),h\\rangle - \\int_\\Omega\\frac{N_{\\tau_\\lambda}(u_n^+)}{||u_n^+||^{p-1}}hdz| \\leq \\varepsilon_n'||h||\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega),\\ \\mbox{with}\\ \\varepsilon_n'\\rightarrow0.\n\t\\end{equation}\n\tFrom (\\ref{eq42}) and hypothesis $H(f)(ii)$, we have\n\t\\begin{eqnarray}\\label{eq56}\n\t\t\\frac{N_{\\tau_\\lambda}(u_n^+)}{||u_n^+||^{p-1}}\\xrightarrow{w} \\eta_0(z)y^{p-1}\\ \\mbox{in}\\ L^{p'}(\\Omega)\\ \\mbox{as}\\ n\\rightarrow\\infty\\ \\\\\n\t\t\\mbox{with}\\ \\eta\\leq\\eta_0(z)\\leq\\hat{\\eta}\\ \\mbox{for almost all}\\ z\\in\\Omega.\\ \\mbox{(see (\\ref{eq33}))}.\\nonumber\n\t\\end{eqnarray}\n\t\n\tIn (\\ref{eq55}) we choose $h=y_n-y\\in W^{1,p}_0(\\Omega)$ and pass to the limit as $n\\rightarrow\\infty$. Then\n\t\\begin{eqnarray}\n\t\t& & \\lim_{n\\rightarrow\\infty}\\langle A(y_n),y_n-y\\rangle=0, \\nonumber \\\\\n\t\t& \\Rightarrow & y_n\\rightarrow y\\ \\mbox{in}\\ W^{1,p}_0(\\Omega)\\ \\mbox{(see Proposition \\ref{prop3}), hence}\\ ||y||=1, y\\geq0. \\label{eq57}\n\t\\end{eqnarray}\n\t\n\tThen passing to the limit as $n\\rightarrow\\infty$ in (\\ref{eq55}) and using (\\ref{eq56}) and (\\ref{eq57}), we obtain\n\t\\begin{eqnarray}\n\t\t& & \\langle A(y),h\\rangle = \\int_\\Omega\\eta_0(z)y^{p-1}hdz\\ \\mbox{for all}\\ h\\in W^{1,p}_0(\\Omega), \\nonumber \\\\\n\t\t& \\Rightarrow & -\\Delta_py(z)=\\eta_0(z)y(z)^{p-1}\\ \\mbox{for almost all}\\ z\\in\\Omega, y|_{\\partial\\Omega}=0. \\label{eq58}\n\t\\end{eqnarray}\n\t\n\tAs before, using Proposition 4, we have\n\t\\begin{eqnarray*}\n\t\t& & \\tilde{\\lambda}_1(\\eta_0)\\leq\\tilde{\\lambda}_1(\\eta)<\\tilde{\\lambda}_1(\\hat{\\lambda}_1)=1, \\\\\n\t\t& \\Rightarrow & y\\ \\mbox{must be nodal (see (\\ref{eq58}), (\\ref{eq57})), a contradiction (see (\\ref{eq57}))}.\n\t\\end{eqnarray*}\n\t\n\tThis proves that $\\{u^+_n\\}_{n\\geq1}\\subseteq W^{1,p}_0(\\Omega)$ is bounded. Hence\n\t$$\n\t\\{u_n\\}_{n\\geq1}\\subseteq W^{1,p}_0(\\Omega)\\ \\mbox{is bounded (see (\\ref{eq53}))}.\n\t$$\n\t\n\tSo, we may assume that\n\t\\begin{equation}\\label{eq59}\n\t\tu_n\\xrightarrow{w}u\\ \\mbox{in}\\ W^{1,p}_0(\\Omega)\\ \\mbox{and}\\ u_n\\rightarrow u\\ \\mbox{in}\\ L^p(\\Omega)\\ \\mbox{as}\\ n\\rightarrow\\infty.\n\t\\end{equation}\n\t\n\tIn (\\ref{eq52}) we choose $h=u_n-u\\in W^{1,p}_0(\\Omega)$, pass to the limit as $n\\rightarrow\\infty$ and use (\\ref{eq59}). Then\n\t\\begin{eqnarray*}\n\t\t& & \\lim_{n\\rightarrow\\infty}\\langle A(u_n),u_n-u\\rangle = 0, \\\\\n\t\t& \\Rightarrow & u_n\\rightarrow u\\ \\mbox{in}\\ W^{1,p}_0(\\Omega)\\ \\mbox{(see Proposition 3)}.\n\t\\end{eqnarray*}\n\t\n\tThis proves Claim \\ref{claim2}.\n\t\n\tOn account of (\\ref{eq50}), (\\ref{eq51}) and Claim \\ref{claim2} we can apply Theorem \\ref{th1} (the mountain pass theorem) and find $\\hat{u}_\\lambda\\in W^{1,p}_0(\\Omega)$ such that\n\t\\begin{eqnarray*}\n\t\t\\hat{u}_\\lambda\\in K_{\\tilde{\\varphi}_\\lambda}\\subseteq[u_\\lambda)\\cap {\\rm int}\\,C_+\\ \\mbox{(see Claim \\ref{claim1})}, \\\\\n\t\t\\tilde{m_\\lambda}\\leq\\tilde{\\varphi}_\\lambda(\\hat{u}_\\lambda)\\ \\mbox{(see (\\ref{eq50})), hence}\\ \\hat{u}_\\lambda\\neq u_\\lambda.\n\t\\end{eqnarray*}\n\t\n\tTherefore $\\hat{u}_\\lambda\\in {\\rm int}\\,C_+$ is the second positive solution of \\eqref{eqp} and\n\t$$\n\t\\hat{u}_\\lambda - u_\\lambda\\in C_+\\backslash\\{0\\}.\n\t$$\nThe proof is now complete.\n\\end{proof}\t\n\n\\textcolor{black}{Therefore we have also proved Theorem A, which is the main result of this paper.}\n\n\\begin{remark}\n\tAn interesting open problem is whether there is such a bifurcation-type theorem for resonant problems, that is,\n\t$$\n\t\\hat{\\lambda}_1\\leq\\liminf_{x\\rightarrow +\\infty}\\frac{f(z,x)}{x^{p-1}} \\leq\\limsup_{x\\rightarrow +\\infty}\\frac{f(z,x)}{x^{p-1}}\\leq\\hat{\\eta}\\ \\mbox{uniformly for almost all}\\ z\\in\\Omega\n\t$$\n\tor even for the nonuniformly nonresonant problems, that is,\n\t$$\\eta(z)\\leq\\liminf\\limits_{x\\rightarrow+\\infty}\\frac{f(z,x)}{x^{p-1}}\\leq\\limsup\\limits_{x\\rightarrow +\\infty}\\frac{f(z,x)}{x^{p-1}}\\leq\\hat{\\eta}\\ \\mbox{uniformly for almost all}\\ z\\in\\Omega$$\n\twith $\\eta\\in L^\\infty(\\Omega)$ such that\n\t$$\n\t\\hat{\\lambda}_1\\leq\\eta(z)\\ \\mbox{for almost all}\\ z\\in\\Omega,\\ \\eta\\not\\equiv\\hat{\\lambda}_1.\n\t$$\n\\end{remark}\n\nIn both cases it seems to be difficult to show that $\\lambda^*<\\infty$. Additional conditions on $f(z,\\cdot)$ might be needed.\n\n\\medskip\n{\\bf Acknowledgements.} \\textcolor{black}{The authors wish to thank the referee for his\/her remarks and suggestions.}\nThis research was supported by the Slovenian Research Agency grants\nP1-0292, J1-8131, J1-7025, N1-0064, and N1-0083. V.D.~R\\u adulescu acknowledges the support through a grant of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130,\nwithin PNCDI III.\n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nSymmetry breaking and formation of ordered structures~(patterns) in spatially extended dissipative systems driven away from equilibrium via some instability mechanisms are very interesting and important phenomena, occurring widely in physics, chemistry, biology, cosmology, and even economics and sociology,\netc.~\\cite{Nicolis1977,Haken1987,Murray1989,Cross1993,Bowman1998,\nWeidman2000,Cross2009}.  Well-known instability mechanisms for pattern formations include the Rayleigh-B\\'{e}nard instability in thermal fluid convection~\\cite{Benard1901,Drazin1981}, the\nTaylor-Couette instability in rotating fluids~\\cite{Taylor1923}, the\nelectrohydrodynamic instability in nematic liquid crystals~\\cite{Dubois1978},\nand the Faraday instability for parametric waves~\\cite{Faraday1831}; other typical examples are the lasing instability in laser devices~\\cite{Newell1990,Arecchi1999,Lugiato1999}, the\nMullins-Sekerka instability for solidification pattern growth (e.g., snowflakes)~\\cite{Mullins1964},\nand the Turing instability for structures created in chemical reaction and living systems (e.g., animal coats)~\\cite{Turing1952}. For details, see Refs.~\\cite{Nicolis1977,Haken1987,Murray1989,Cross1993,Bowman1998,\nWeidman2000,Cross2009} and references cited therein.\nOne of main characters of these pattern forming systems is that an external drive (stress) must be applied, and the induced instability trigers symmetry-breaking causing the dissipative structures appearing immediately in the linear regime.\n\n\nBesides the linear instability in driven dissipative systems, much attention were also paid to the research of modulational instability (MI) in nonlinear systems~\\cite{Segur2007,Zakharov2009,Zakharov2013,Kibler2015,Biondini2016,\nConforti2016,Mosca2018,Kraych20191}. MI is a typical nonlinear instability discovered firstly in the study of water waves~\\cite{Benjamin1967},\nand may apply to non-dissipative and non-driven systems, where a plane wave of finite amplitude may undergo an instability and lose its energy to  sideband components, resulting in a nonlinear modulation of the plane wave. Usually, MI is considered as wave dynamics problems for  conservative nonlinear systems, where wave envelopes are controlled typically by cubic nonlinear Schr\\\"{o}dinger~(NLS) equation, with local and attractive Kerr nonlinearities. For such systems, the existence of MI is thought to be the major reason for the formation of solitons\n(see Refs.~\\cite{Segur2007,Zakharov2009,Zakharov2013,Kibler2015,Biondini2016,Conforti2016,Mosca2018,Kraych20191,Reece2007,Solli,\nNguyen2017,Kraych20192,Benjamin1967,Kamchatnov1997,Kartashov2011,Kivshar2006,Agrawal2007,Pethick2008} and references cited therein).\n\n\nIn recent years, considerable efforts were made on the study of\nMI in conservative nonlinear systems described by cubic NLS equations with nonlocal Kerr nonlinearity. It was found that MI may occur in such systems even the Kerr nonlinearity is repulsive, or has both repulsive and attractive parts~\\cite{Krolikowski2001,Wyller2002,Krolikowski2004,\nDoktorv2007,Henkel2010,Esbensen2011,Tiofack2015,Maucher2016,\nMaucher2017}, which provides the possibility to realize spontaneous symmetry breaking~\\cite{Malomed2013} and generate spatially extended, ordered structures in nonlocal nonlinear media. Various self-organized patterns were found~\\cite{Tiofack2015,Maucher2016,Maucher2017,Mottl2012,Labeyrie2014,\nZhangYC2018}, especially the ones of atomic density formed in Rydberg-dressed~\\cite{Henkel2010,Cinti2010,Cinti2014,Henkel2012,Hsuch2012,\nHsuch2017,Li2018} and dipolar~\\cite{Saito2009,Lu2015,Kadau2016,Wachtler2016,Xi2018,Zhang2019} Bose-Einstein condensates.\n\n\nOn the other hand, in the past two decades a large amount research works were focused on the investigation of cold Rydberg atomic gases~\\cite{Gallagher2006,Saffman2010} working under condition of electromagnetically induced transparency (EIT). EIT is an important quantum destruction interference effect occurring typically in resonant three-level atomic systems, by which the absorption of a probe laser field can be largely eliminated by a control laser field~\\cite{Fleischhauer2005}. Due to strong, long-range atom-atom interaction~(also called Rydberg-Rydberg interaction),\nsuch systems are desirable nonlinear optical media with strong, nonlocal Kerr nonlinearity if the Rydberg-Rydberg interaction is suitably mapped to photon-photon interaction via EIT~\\cite{Fir2016,Mur2016}. In an interesting work, Sevinli {\\it et al.}~\\cite{Sevincli2011} reported  a self-organized hexagonal optical pattern via a MI of plane-wave probe beam in a cold Rydberg atomic gas with a repulsive Rydberg-Rydberg interaction.\n\n\nIn this work, we propose and analyze a scheme for realizing various self-organized optical structures and their structural phase transition in a cold Rydberg atomic gas via a Rydberg-EIT~\\cite{Mohapatra2007,Pritchard2010}. By exploiting a microwave dressing (i.e., a microwave field couples two electrically excited Rydberg states)~\\cite{Tana2011,Sedlacek2012,Yu2013,Maxwell2013,Petrosyan2014,Pohl2014,Li2014,\nLi2015,Rao2014,Adams2014,Liu2015,Thompson2017,Votg2018,Vogt2019,Jing2020}, we show that the nonlocal Kerr nonlinearity of the Rydberg gas (which has only a repulsive Rydberg-Rydberg interaction in the absence of the microwave field) is significantly modified, and its strength and sign can be tuned actively. Based on such nonlocal Kerr nonlinearity, we demonstrate that a homogeneous (plane wave) state of probe laser field can undergo MI and hence spontaneous symmetry breaking, which may result in the formation of various ordered optical patterns.\n\n\nThrough detailed analytical and numerical analysis, we find that a homogeneous state of the probe field is firstly transited into a hexagonal lattice pattern (which is the only lattice pattern when the microwave dressing is absent). Interestingly, this hexagonal lattice pattern may undergo a structural phase transition and develop into several types of square lattice patterns when the microwave field is applied and its strength is increased. Moreover, as a outcome of the MI the formation of nonlocal spatial optical solitons is also possible by a suitable choice of system parameters. Different from the results reported before, the optical patterns and nonlocal optical solitons found here can be flexibly manipulated via the adjustment of the effective probe-field intensity, nonlocality degree of the Kerr nonlinearity, and the strength of the microwave field. Our study opens a way for actively controlling the self-organization and structural phase transition of optical patterns through microwave-dressing on Rydberg gases, which are not only of fundamental interest, but also useful for potential applications in optical information processing and transmission.\n\n\nThe remainder of the article is arranged as follows. In Sec.~\\ref{sec2}, we describe the physical model, discuss the modification and enhancement of the Kerr nonlinearity contributed by the microwave field, and derive the nonlinear envelope equation of the probe field. In Sec.~\\ref{sec3}, we consider the MI of a plane-wave state, investigate the formation and structural phase transitions of optical patterns controlled by the microwave field, effective probe-field intensity, the nonlocal Kerr nonlinearity and its nonlocality degree. The result on the formation of nonlocal spatial optical solitons is also presented. The last section (Sec.~\\ref{sec4}) gives a summary of our main research results.\n\n\n\\section{Physical model, nonlinear envelope equation, and enhanced Kerr nonlinearity}\\label{sec2}\n\n\\subsection{Physical model}\\label{sec21}\n\nWe consider an ensemble of lifetime-broadened four-level atomic gas with an ladder-type level configuration, shown schematically in Fig.~{\\ref{Fig1}}{\\color{blue}(a)}.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{fig1.eps}\n\\caption{\\footnotesize Schematics of the model.\n(a)~Ladder-type four-level atomic configuration for realizing the microwave-dressed Rydberg-EIT. Here, the weak probe laser field~(blue), strong control laser field~(red), and strong microwave field~(green) with\nhalf Rabi frequencies ${\\Omega}_{p}$, $\\Omega_c$, and $\\Omega_m$ drive the transitions $|1\\rangle\\leftrightarrow|2\\rangle$, $|2\\rangle\\leftrightarrow|3\\rangle$, and $|3\\rangle\\leftrightarrow|4\\rangle$, respectively;\nStates $|1\\rangle$ and $|2\\rangle$ are respectively ground and excited states, both  $|3\\rangle$ and $|4\\rangle$ are highly excited Rydberg states; $\\Delta_2$, $\\Delta_3$, and $\\Delta_4$ are respectively the one-, two-, and three-photon detunings; $\\Gamma_{12}$, $\\Gamma_{23}$, and $\\Gamma_{24}$ are the spontaneous emission decay rates from $|2\\rangle$ to $|1\\rangle$,  $|3\\rangle$ to $|2\\rangle$ and\n$|4\\rangle$ to $|2\\rangle$, respectively.\nTwo Rydberg atoms locating respectively at position ${\\bf r}$ and ${\\bf r}'$ interact through van der Waals potential $\\hbar {\\mathcal V}_{\\rm vdw}^{l}({\\bf r}^{\\prime}-{\\bf r})$ ($l=s,d,e$; see text).\n(b)~Possible experimental geometry, where small solid circles denote atoms and large dashed circles denote Rydberg blockade spheres.\n(c)~Emergence of an optical pattern via modulation instability.\n}\\label{Fig1}\n\\end{figure}\nHere, the weak probe laser field with central angular frequency $\\omega_{p}$, wavevector ${\\bf k}_{p}$, and half Rabi frequency\n$\\Omega_p$ drives the transition from atomic ground state $|1\\rangle$ to  intermediate state $|2\\rangle$, and the strong control laser field with\ncentral angular frequency $\\omega_c$, wavevector ${\\bf k}_c$, and\nhalf Rabi frequency $\\Omega_c$ drives the transition $|2\\rangle$ to the first highly excited Rydberg state $|3\\rangle$. This ladder-type three level EIT is dressed by a microwave field with central angular frequency\n$\\omega_m$, wavevector ${\\bf k}_m$, and half Rabi frequency\n$\\Omega_m$, which couples the transition between the Rydberg state $|3\\rangle$ and another Rydberg state $|4\\rangle$.\nThe total electric fields acting in the atomic system can be written as ${\\bf E(r},t)=\\sum_{j}{\\bf e}_{j}{\\cal E}_{j}e^{i({\\bf k}_{j}\\cdot{\\bf r}-\\omega_{j}t)} +{\\rm  H.c.}$, with ${\\bf e}_{j}$ and ${\\cal E}_{j}$ respectively the polarization unit vector and the envelope of $j$-th laser field ($j=p,c,m$).  $\\Delta_2$, $\\Delta_3$, and $\\Delta_4$ are respectively the one-, two-, and three-photon detunings; $\\Gamma_{12}$, $\\Gamma_{23}$, and $\\Gamma_{24}$  are the spontaneous emission decay rates from $|2\\rangle$ to $|1\\rangle$,  $|3\\rangle$ to $|2\\rangle$, and $|4\\rangle$ to $|2\\rangle$, respectively.\nThe microwave field employed here is to realize a microwave-dressed Rydberg-EIT~\\cite{Tana2011,Sedlacek2012,Yu2013,Maxwell2013,Petrosyan2014,Pohl2014,Li2014,Li2015,\nRao2014,Adams2014,Liu2015,Thompson2017,Votg2018,Vogt2019,Jing2020} and thus to modify the Rydberg-Rydberg interaction,\nwhich, in turn, can manipulate the interaction strength and sign for the photons in the probe field and hence realize self-organized optical structures not discovered before.\n\n\nThe dynamics of the system is controlled by the Hamiltonian $\\hat{H}=\\mathcal{N}_a\\int d^3{\\bf r}\\hat{\\mathcal{H}}({\\bf r},t)$, where $\\hat{\\mathcal{H}}({\\bf r},t)$ is Hamiltonian density and $\\mathcal{N}_a$ is atomic density. Under the electric-dipole and rotating-wave approximations, the Hamiltonian density in interaction picture reads\n\\begin{align}\\label{Hamiltonian}\n\\hat{\\mathcal{H}}\\equiv\\hat{\\mathcal{H}}_1+\\hat{\\mathcal{H}}_{{\\rm vdW}},\n\\end{align} \\noindent\nwhere Hamiltonian $\\hat{\\mathcal{H}}_1$ describes unperturbed atoms as well as the interaction between the atoms and the laser fields,  $\\hat{\\mathcal{H}}_{{\\rm vdW}}$ describes the Rydberg-Rydberg interaction, respectively given by\n{\\small\n\\begin{subequations}\n\\begin{align}\n&\\hat{\\mathcal{H}}_1=-\\hbar\\sum_{\\alpha=2}^4\\Delta_{\\alpha}\\hat{S}_{\\alpha\\alpha} -\\hbar(\\Omega_p \\hat{S}_{12}+\\Omega_c\n\\hat{S}_{23}+\\Omega_m \\hat{S}_{34}+{\\rm H.c.}),\\\\\n&\\hat{\\mathcal{H}}_{{\\rm vdW}}=\\hbar\\mathcal{N}_a\\int d^3{ r}^\\prime\\bigg\\{\\sum_{\\alpha=3,4}\\hat{S}_{\\alpha\\alpha}({\\bf\nr}^{\\prime},t)\\mathcal{V}_{\\alpha\\alpha}^s({\\bf r}^\\prime-{\\bf r})\\hat{S}_{\\alpha\\alpha}({\\bf r},t)\\notag\\\\\n&\\qquad\\,+\\mathcal{V}_{34}^d({\\bf r}^\\prime-{\\bf r})\\left[\\hat{S}_{33}({\\bf\nr^\\prime},t)\\hat{S}_{44}({\\bf r},t)+\\hat{S}_{44}({\\bf r^\\prime},t)\\hat{S}_{33}({\\bf r},t)\\right]\\notag\\\\\n&\\qquad\\,+\\mathcal{V}_{34}^e\n({\\bf r}^\\prime-{\\bf r})\\left[\\hat{S}_{43}({\\bf\nr^\\prime},t)\\hat{S}_{34}({\\bf r},t)+\\hat{S}_{34}({\\bf r^\\prime},t)\\hat{S}_{43}({\\bf r},t)\\right]\\bigg\\}.\n\\end{align}\n\\end{subequations}}\\noindent\nHere $d^3 r'=dx' dy' dz'$;\n$\\hat{S}_{\\alpha\\beta}=|\\beta\\rangle \\langle \\alpha|\\exp\\{i[({\\bf k}_{\\beta}-{\\bf k}_{\\alpha})\\cdot{\\bf r}-(\\omega_{\\beta}-\\omega_{\\alpha}+\\Delta_{\\beta}-\\Delta_{\\alpha})t]\\}$\nis the transition operator satisfying the commutation relation\n$[\\hat{S}_{\\alpha\\beta}({\\bf\nr},t),\\hat{S}_{\\alpha^\\prime\\beta^\\prime}({\\bf r}^\\prime,t)]=\\mathcal{N}_a^{-1}\n\\delta ({\\bf r}-{\\bf r}{^\\prime})(\\delta_{\\alpha\\beta'}\\hat{S}_{\\alpha'\\beta}({\\bf r},t)-\\delta_{\\alpha'\\beta}\\hat{S}_{\\alpha\\beta'}({\\bf r'},t))$;\nthe one-, two-, and three-photon detuings are respectively given by\n$\\Delta_2=\\omega_p-(\\omega_2-\\omega_1)$, $\\Delta_3= \\omega_c+\\omega_p-(\\omega_3-\\omega_1)$, and\n$\\Delta_4=\\omega_c+\\omega_p+\\omega_m-(\\omega_4-\\omega_1)$, with\n$E_{\\alpha}=\\hbar\\omega_{\\alpha}$ the eigenenergy of the atomic state $|\\alpha\\rangle$.\nThe half Rabi frequencies of the probe, control, and microwave fields are, respectively, $\\Omega_p=({\\bf e}_{p}\\cdot{\\bf p}_{21}){\\cal\nE}_{p}\/\\hbar$, $\\Omega_c=({\\bf e}_{c}\\cdot{\\bf p}_{32}){\\cal E}_{c}\/\\hbar$, and $\\Omega_m=({\\bf e}_{m}\\cdot{\\bf p}_{43}){\\cal\nE}_{m}\/\\hbar$, with ${\\bf p}_{\\alpha\\beta}$  the electric dipole matrix element associated with the transition between the states $|\\alpha \\rangle$ and $|\\beta\\rangle$.\n\n\nThe Hamiltonian density $\\hat{\\mathcal{H}}_{{\\rm vdW}}$ is the contribution by the Rydberg-Rydberg interaction, which contains\nfour parts, represented by $\\mathcal{V}_{33}^s$, $\\mathcal{V}_{44}^s$, $\\mathcal{V}_{\\rm 34}^d$, and $\\mathcal{V}_{\\rm 34}^e$,\nrespectively; the term $\\mathcal{V}_{\\rm 33}^s=-C_{33}^{s}\/|{\\bf r}'-{\\bf r}|^6$\\,\\,  ($\\mathcal{V}_{44}^s=-C_{44}^{s}\/|{\\bf r}'-{\\bf r}|^6$)\ndescribes the van der Waals interaction between the two\natoms located respectively at positions ${\\bf r}'$ and ${\\bf r}$\nand excited to the same Rydberg state $|3\\rangle$\\, ($|4\\rangle$); the term $\\mathcal{V}_{\\rm 34}^{d}=-C_{34}^{d}\/|{\\bf r}'-{\\bf r}|^6$\\,\\,\n($\\mathcal{V}_{\\rm 34}^e=-C_{34}^{e}\/|{\\bf r}'-{\\bf r}|^3$)\ndescribes the direct non-resonant van der Waals interaction\n(resonant exchange dipole-dipole interaction)\nbetween the two atoms\nexcited to different Rydberg states (i.e. $|3\\rangle$ and $|4\\rangle$). Here $C_{\\alpha\\beta}^{l}$   ($\\{\\alpha\\beta\\}=\\{33,44,34\\}$; $l=s,d,e$) are dispersion parameters~\\cite{Petrosyan2014,Li2014,Li2015}.\n\n\nThe time evolution of the atoms in the system is governed by the optical\nBloch equation\n\\begin{equation}\\label{Bloch0}\n\\frac{\\partial\\rho}{\\partial t}=-\\frac{i}{\\hbar} \\left[{\\hat{ H}},\\rho\\right]-\\Gamma\\left[\\rho\\right],\n\\end{equation}\nwhere $\\rho ({\\bf r},t)=\\langle \\hat{S}({\\bf r},t)\\rangle$~\\cite{note0} is a $4\\times 4$ density matrix (with density matrix elements $\\rho_{\\alpha\\beta}({\\bf r},t)=\\langle \\hat{S}_{\\alpha\\beta} ({\\bf r},t)\\rangle$; $\\alpha,\\beta=1,2,3,4$)\ndescribing the atomic population and coherence, $\\Gamma$ is a $4\\times 4$ relaxation matrix describing the spontaneous emission\nand dephasing. Explicit expressions of $\\rho_{\\alpha\\beta}({\\bf r},t)$ are  presented in Appendix~\\ref{app1}.\n\n\nThe propagation of the probe field is controlled by Maxwell equation, which under paraxial and slowly-varying envelope approximations is reduced into~\\cite{Mur2016}\n\\begin{align}\\label{Maxwell}\n  i\\left(\\frac{\\partial}{\\partial z}+\\frac{1}{c}\\frac{\\partial}{\\partial\n  t}\\right)\\Omega_p+\\frac{c}{2\\omega_p}\\nabla_{\\perp}^2\\Omega_p\n  +\\kappa_{12}\\rho_{21}=0,\n\\end{align}\\noindent\nwhere $\\nabla_{\\perp}^2=\\partial^2\/\\partial x^2+\\partial^2\/\\partial y^2$ describes diffraction, $\\kappa_{12}=\\N_a\\omega_p|({\\bf e}_p\\cdot{\\bf p}_{12})|^2\/(2\\epsilon_0c\\hbar)$ is a parameter describing\nthe coupling between the atoms and the probe field,\nand $c$ is the light speed in vacuum.\nWithout loss of generality, we assume the probe field propagates along $z$ direction, i.e., ${\\bf k}_p=(0,0,\\omega_p\/c)$; to suppress Doppler effect, the microwave field is along the $z$ direction but the control field is along the negative $z$ direction [i.e., ${\\bf k}_m=(0,0,\\omega_m\/c)$ and ${\\bf k}_c=(0,0,-\\omega_c\/c)$]. A possible experimental arrangement is given in Fig.~\\ref{Fig1}(b).\n\n\nNote that the physical model described above is valid for any microwave-dressed Rydberg atomic gas. But for latter calculations where numerical values of the system are needed, we take cold $^{87}$Rb atomic gas~\\cite{Petrosyan2014} (which has density-density interaction in the absence of the microwave field) as a realistic example. The assigned atomic levels in Fig.~\\ref{Fig1}(a) are $|1\\rangle=|5 S_{1\/2}\\rangle$, $|2\\rangle=|5P_{3\/2}\\rangle$,  $|3\\rangle=|nS_{1 \/ 2}\\rangle$, and $|4\\rangle=|n P_{3 \/ 2}\\rangle$. For example, for principal quantum number $n=60$, the dispersion parameters are\n$C_{33}^{s}=- 2\\pi \\times 140\\,{\\rm GHz}~\\mu {\\rm {m^6}}$~(repulsive  interaction), $C_{44}^{s}=2\\pi \\times 295  \\,{\\rm GHz}~\\mu {\\rm\n{m^6}}$~(attractive interaction), $C_{34}^{d}=-2\\pi \\times 3 \\,{\\rm GHz}~\\mu {\\rm {m^6}}$ (repulsive interaction), and\n$C_{34}^{e}=-2\\pi \\times  3.8 \\,{\\rm GHz}~\\mu {\\rm {m^3}}$~(repulsive  interactions)~\\cite{Petrosyan2014,Rao2014,note1}, respectively.\nTypical system parameters are chosen as follows:\n$\\Delta_2=3.17\\times 10^2$\\,MHz,\n$\\Delta_3=15.3$\\,MHz,\n$\\Delta_4=1.32$\\,MHz;\n$\\Gamma_{12}=2\\pi \\times 6.1$\\,MHz,\n$\\Gamma_3=\\Gamma_{4}=2\\pi \\times 1.67\\times 10^{-2}$\\,MHz;\n$\\Omega_{c}=20\\,{\\rm MHz}$;\n$\\N_a=1.0\\times 10^{11}\\,{\\rm cm^{-3}}$.\n\n\nWe stress that although the Bloch Eq.~(\\ref{Bloch0}) is for the evolution of one-body density-matrix elements $\\rho_{\\alpha\\beta}({\\bf r},t)$, it involves two-body density-matrix elements $\\rho_{\\alpha\\beta,\\mu\\nu}({\\bf r^{\\prime},r},t)= \\langle\n\\hat{S}_{\\alpha\\beta}({\\bf r^{\\prime}},t) \\hat{S}_{\\mu\\nu}({\\bf r},t)\\rangle$ due to the Rydberg-Rydberg interaction; furthermore, the equation of motion for $\\rho_{\\alpha\\beta,\\mu\\nu}({\\bf r^{\\prime},r},t)$ involves three-body density-matrix elements $\\rho_{\\alpha\\beta,\\mu\\nu,\\gamma\\delta}({\\bf r}^{\\prime\\prime},{\\bf r}^{\\prime}, {\\bf r},t)=\\langle\n\\hat{S}_{\\alpha\\beta}({\\bf r}^{\\prime\\prime},t) \\hat{S}_{\\mu\\nu}({\\bf r}',t)\\rangle \\hat{S}_{\\gamma\\delta}({\\bf r},t)\\rangle$, and so on. Thus an effective approach for solving such a hierarchy of infinite equations involving many-atom correlations is needed.\n\n\n\\subsection{Enhanced Kerr nonlinearity by the microwave dressing}\\label{sec22}\n\nWe first consider the modification of Kerr nonlinearity of the system induced by the microwave field based on the physical model described above. For simplicity, we assume that the control and microwave fields are strong enough, so that they are not depleted during the propagation of the probe field. Since the probe field is weak, a perturbation expansion can be applied to solve the Maxwell-Bloch (MB) equations (\\ref{Bloch0}) and (\\ref{Maxwell}) by taking $\\Omega_p$ as a small expansion parameter. Generalizing the approach developed in Refs.~\\cite{Bai2016,Zhang2018,Bai2019,Bai2020}, where MB equations for Rydberg atomic gases without microwave dressing are solved beyond mean-field approximation in a self-consistent and effective way, we can obtain the solutions of the Bloch Eq.~(\\ref{Bloch0}) using the perturbation expansion up to third-order approximation. In particular, the result of the one-body density matrix element $\\rho_{21}$ can be obtained analytically (see the Appendix~\\ref{app1} for detail).\n\n\nWith the expression of $\\rho_{21}$ and the definition of probe-field susceptibility, i.e. $\\chi= \\mathcal{N}_a({\\bf e\\cdot p}_{12})\\rho_{21}\/(\\epsilon_0\\mathcal{E}_p)$, it is easy to obtain the optical susceptibility of the probe field, which reads\n\\begin{equation}\n\\chi=\\chi^{(1)}+\\left(\\chi_{\\rm loc}^{(3)}+\\chi_{\\rm nloc}^{(3)}\\right)|\\mathcal{E}_p|^2,\n\\end{equation}\nwhere $\\chi^{(1)}$ is the linear susceptibility;\n$\\chi_{\\rm loc}^{(3)}$ and $\\chi_{\\rm nloc}^{(3)}$ are local and nonlocal third-order nonlinear (Kerr) susceptibilities, originated respectively from non-zero two-photon detuning (i.e. $\\Delta_3\\neq 0$)~\\cite{Bai2016,Zhang2018,Bai2019,Bai2020,Wang2001,Chen2014}) and from the Rydberg-Rydberg interaction in the system.\nExpressions of $\\chi^{(1)}$, $\\chi_{\\rm loc}^{(3)}$, and $\\chi_{\\rm nloc}^{(3)}$ are given in Appendix~\\ref{app3}; see\nEqs.~(\\ref{chi3s1}), (\\ref{chi3s2}), and (\\ref{chi3n}), respectively.\nUsing the system's parameters given at the final part of the last subsection and taking\n$\\Omega_m=18$\\,MHz, we obtain\n$\\chi^{(3)}_{\\rm loc} \\approx (5.08+0.012i)\\times 10^{-11}\\, {\\rm m^{2}\/V^2}$,\n$\\chi^{(3)}_{\\rm nloc}\\approx (3.05+0.022i)\\times 10^{-8}\\,\n{\\rm m^{2}\/V^2}$.\nWe see the imaginary parts of $\\chi^{(3)}_{\\rm loc}$ and $\\chi^{(3)}_{\\rm nloc}$ are much smaller than their real parts,\nwhich is due to the EIT effect contributed by the control field; moreover, the nonlocal Kerr nonlinearity is three orders of magnitude larger than the local one, which is due to the strong Rydberg-Rydberg interaction together with the microwave dressing.\n\n\nIt is helpful to reveal how the microwave-dressing modify the Kerr effect\nof the system. Fig.~\\ref{Fig2}(a) shows\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\textwidth]{fig2.eps}\n\\caption{\\footnotesize\nKerr nonlinearity enhancement, interaction potential energy, and normalized nonlinear response function $\\Re$ of the probe field in the presence of the microwave dressing.\n(a)~Real part Re($\\chi^{(3)}_{\\rm loc}$)~(solid red line) and  imaginary part Im($\\chi^{(3)}_{\\rm loc})$~(dotted blue line) of the local nonlinear susceptibility $\\chi_{\\rm\nloc}^{(3)}$ as a function of the half Rabi frequency $\\Omega_m$ of the microwave field.\n(b)~The same as (a) but for the nonlocal nonlinear susceptibility $\\chi_{\\rm nloc}^{(3)}$.\n(c)~Potential-energy curves $E_{1}$ (solid blue line), $E_2$ (solid black line), and $E_3$ (solid red line) of two Rydberg atoms\nas functions of the interatomic distance $r= |{\\bf r'-r}|$ for $\\Omega_m=10~$MHz and $\\Delta=13.98$ MHz; $E_{33}$ (dashed blue line), $E_{44}$ (dashed black line), and $E_{34}^{+}$ (dashed red line) are for the case without the microwave field.\n(d)~Normalized response function $\\Re\/\\Re_{\\rm max}$ as a function of the dimensionless coordinate $\\xi= x\/R_0$ ($R_0$ is the typical transverse beam radius of the probe field) with $\\Omega_m=0$~(dotted black line), 5~MHz~(solid red line), and 15~MHz~(solid blue line), respectively.\nInset: the normalized response function $\\tilde{\\Re}\/\\tilde{\\Re}_{\\rm max}$ in momentum space (i.e. the Fourier transformation of\n$\\Re\/\\Re_{\\rm max}$)\nas a function of the dimensionless wavenumber\n$\\beta_1=R_0 k_x$ ~($k_x$ is the dimensional wavenumber  in $x$ direction) with $\\Omega_m=0$, 5, and 15\\,MHz, respectively.\n}\n\\label{Fig2}\n\\end{figure}\nthe real part Re($\\chi^{(3)}_{\\rm loc}$)~(solid red line) and  imaginary part Im($\\chi^{(3)}_{\\rm loc})$~(dotted blue line) of the local nonlinear susceptibility $\\chi_{\\rm loc}^{(3)}$ as a function of the half Rabi frequency $\\Omega_m$ of the microwave field. Shown in Fig.~\\ref{Fig2}(b)\nis same as that in Fig.~\\ref{Fig2}(a) but for the nonlocal nonlinear susceptibility $\\chi_{\\rm nloc}^{(3)}$.\nFrom the figure we see that the nonlinear optical susceptibilities have two evident features:\n(i)~Both the real parts of $\\chi_{\\rm\nloc}^{(3)}$ and $\\chi_{\\rm nloc}^{(3)}$ are much larger than the corresponding imaginary parts, contributed by the EIT effect;\n(ii)~In the value range of $\\Omega_m$ taken here,\n${\\rm Re(\\chi_{loc}^{(3)})}$ is an increasing function; however,\n${\\rm Re(\\chi_{\\rm nloc}^{(3)}}$) increases firstly, then arrives a maximum at some value of $\\Omega_m$, and decreases when $\\Omega_m$ is increased further. At the point of the maximum, where\n$\\Omega_m\\approx 18$ MHz, ${\\rm Re}(\\chi_{\\rm nloc}^{(3)})\n\\approx 3.05\\times 10^{-8}\\,{\\rm m}^2\/{\\rm V}^2$, which is 15 times larger than the case without the microwave field\n[${\\rm Re}(\\chi_{\\rm nloc}^{(3)})\\approx 0.2\\times 10^{-8}~{\\rm m^2\/V^2}$ for $\\Omega_m=0$]. Thus, microwave-dressing can be used to modify the Kerr effect of the system greatly.\n\n\nTo support the above conclusion, a calculation is carried out for the interaction potential  of two Rydberg atoms\nlocated respectively at positions ${\\bf r}$ and\n${\\bf r}'$, which may occupy in the Rydberg states $|3\\rangle$ and $|4\\rangle$. In the absence of the microwave field (i.e., $\\Omega_m=0$), the basis set of the such two-atom system consists of states\n$|33\\rangle= |3_13_2\\rangle$,\n$|44\\rangle= |4_14_2\\rangle$, and\n$|34_\\pm \\rangle= 1\/\\sqrt{2}(|3_14_2\\rangle\\pm|3_24_1\\rangle )$,\nwith the subscript 1 and 2 representing atom 1 and 2, respectively.\nThe (bare state) eigen energies of the system are $E_{33}=-\\hbar C_{33}^s\/{\\bf |r'-r|}^6$, $E_{44}=2\\hbar\\Delta -\\hbar C_{44}^s\/{\\bf |r'-r|}^6$, $E_{34}^{+}=\\hbar \\Delta+ \\hbar C_{34}^e\/|{\\bf r'-r}|^3$, and $E_{34}^{-}=\\hbar \\Delta- \\hbar C_{34}^d\/|{\\bf r'-r}|^6$, with $\\Delta=\\Delta_3-\\Delta_4=13.98$ MHz.\nSince antisymmetric state $|34_{-}\\rangle$ is nearly not coupled to  laser field, one can disregard it if the microwave field is present (i.e., $\\Omega_m\\neq 0$). Then, the Hamiltonian in the two-atom basis set $\\{|33\\rangle$, $|34_{+}\\rangle$, $|44\\rangle\\}$ takes the form\n\\begin{align}\\label{HTA}\n  \\mathcal{H}=\\hbar\\left(\n  \\begin{array}{ccc}\n    -\\frac{C_{33}^s}{|{\\bf r'-r}|^6}&\\sqrt{2}\\Omega_m&0\\\\\n    \\sqrt{2}\\Omega_m&\\Delta+\\frac{C_{34}^e}{|{\\bf r'-r}|^3}&\\sqrt{2}\\Omega_m\\\\\n    0&\\sqrt{2}\\Omega_m&2\\Delta-\\frac{C_{44}^s}{|{\\bf r'-r}|^6}\\\\\n  \\end{array}\n  \\right).\n\\end{align}\nAfter diagonalization, we can obtain the energies $E_{1}$, $E_2$, and $E_3$ of the Hamiltonian (\\ref{HTA}).\nPotential-energy curves of $E_{1}$, $E_2$, and $E_3$ as functions of the interatomic separation $r= |{\\bf r'-r}|$ for $\\Omega_m=10~$MHz are shown in Fig.~\\ref{Fig2}(c). For comparison,\nthe bare potential-energy curves $E_{33},\\, E_{44}$, and $E_{34}^+$ (for $\\Omega_m=0$) are also shown. We see that, compared with the case without the microwave field, the potential-energy curves are modified largely by the introduction of the microwave field, especially for small interatomic separation $r$. The reason is that the microwave dressing brings a coupling between the Rydberg states $|3\\rangle$ and $|4\\rangle$, and thereby a modification of the Rydberg-Rydberg interaction. It is the use of the microwave dressing that brings the significant change and enhancement of the nonlocal Kerr nonlinearity.\n\n\n\\subsection{Nonlinear envelope equation and the property of nonlinear response function}\\label{sec23}\n\n\nWe now derive the envelope equation which controls the dynamics of the probe field. By substituting the solution of $\\rho_{21}$ into\nthe Maxwell Eq.~(\\ref{Maxwell}) and making a local approximation along the $z$ direction on the nonlocal nonlinear response function~(see the Appendix~\\ref{app3}),\nwe obtain the following three-dimensional (3D) nonlocal nonlinear Schr\\\"odinger (NNLS) equation\n{\\small\n\\begin{align}\n& i\\frac{\\partial \\Omega_p }{\\partial z}+\\frac{c}{2\\omega_p}\\nabla_{\\perp}^{2}\\Omega_p + {W_1}|\\Omega_p{|^2}\\Omega_p\\notag\\\\\n& \\qquad+\\int d^2 {r'}\nG({\\bf r_{\\perp}'-r_{\\perp}})|\\Omega_p({\\bf r_{\\perp}'},z)|^2\\Omega_p({\\bf r_{\\perp}},z)=0,\\label{NNLS1}\n\\end{align}}\\noindent\nwith ${\\bf r}_{\\perp}=(x,y)$,  $d^2{r'}=dx' dy'$.\nThe third and forth terms on the left hand side of this equation describe two types of self-phase modulations of the probe field, contributed respectively by the local Kerr nonlinearity (originated from non-zero two-photon detuning (i.e., $\\Delta_3\\neq 0$)~\\cite{Bai2016,Zhang2018,Bai2019,Bai2020,Wang2001,Chen2014}) and the nonlocal Kerr nonlinearity (originated from the Rydberg-Rydberg interaction). In the integral of the forth term of the NNLS equation, $G$ is a reduced nonlinear response function, taking the form\n$G({\\bf r_{\\perp}'-r_{\\perp}})=\\sum_{\\alpha=3,4}G_{\\alpha\\alpha}^s({\\bf r_{\\perp}'-r_{\\perp}})+\\sum_{l={d,e}}G_{34}^l({\\bf r_{\\perp}'-r_{\\perp}})$,\nwith\n$G_{\\alpha\\beta}^l({\\bf r_{\\perp}'-r_{\\perp}})=\\int dz'R_{\\alpha\\beta}^l({\\bf r'-r})$\n($\\{\\alpha\\beta\\}=\\{33,44,34\\}$; $l=s,d,e$). Explicit expressions\nof the matrix elements of nonlinear response function\n$R_{\\alpha\\beta}^{l}({\\bf r}^\\prime-{\\bf r})$ and local nonlinear coefficient $W_1$ are given in the Appendix~\\ref{app3}~[see Eq.~(\\ref{W1}) and Eqs.~(\\ref{eqb5}), respectively]. Due to the microwave dressing,\nthe nonlocal Kerr nonlinearity consists of four parts;  the first~(second) part $G_{33}^s({\\bf r_{\\perp}'-r_{\\perp}})$\n[$G_{44}^s({\\bf r_{\\perp}'-r_{\\perp}})$] is contributed by the interaction of the atoms lying in the same Rydberg state $|3\\rangle$~($|4\\rangle$);\nthe third (forth) part $G_{34}^d({\\bf r_{\\perp}'-r_{\\perp}})$~[$G_{34}^e({\\bf r_{\\perp}'-r_{\\perp}})$] is contributed by the interaction of the atoms lying in the different Rydberg states $|3\\rangle$ and $|4\\rangle$.\n\n\nFor the convenience of later discussions and numerical calculations, we rewrite the 3D NNLS Eq.~(\\ref{NNLS1}) into the non-dimensional form\n\\begin{align}\\label{NNLS2}\n&i\\frac{\\partial u}{\\partial s} + \\tilde{\\nabla}_{\\perp}^2u+\\int d^2\\zeta'\n\\Re(\\vec{\\zeta}'-\\vec{\\zeta})|u(\\vec{\\zeta}',s){|^2}\nu(\\vec{\\zeta},s)=0,\n\\end{align}\nwith $s = z\/(2{L_{\\rm diff}})$, $u=\\Omega_p\/U_0$,\n$\\tilde{\\nabla}_{\\perp}^2=\\partial^2\/\\partial\\xi^2\n+\\partial^2\/\\partial\\eta^2$,\n$\\vec{\\zeta}= (\\xi,\\eta)=(x,y)\/{R_0}$, and\n$d^2 {\\zeta}^{\\prime}= d\\xi^{\\prime}d \\eta^{\\prime}$.\nHere ${L_{\\rm diff}} = {\\omega _p}R_0^2\/c$ is the typical diffraction length, which is 1.61 mm in our system;\n$U_0$ is the typical Rabi frequency of the probe field;\n$R_0$ is the typical beam radius of the probe field;\nthe non-dimensional nonlinear response function is defined by\n$\\Re(\\vec{\\zeta}'-\\vec{\\zeta})=2L_{\\rm diff}U_0^2R_0^2 G[{(\\vec{\\zeta}'-\\vec{\\zeta})}R_0]$. Note that in writing Eq.~(\\ref{NNLS2})\nwe have neglected the term related to $W_1$ because the\nlocal Kerr nonlinearity is much smaller than the nonlocal one~\\cite{Bai2019}.\n\n\nThe property of the nonlocal Kerr nonlinearity of the system is characterized by the nonlinear response function $\\Re(\\vec{\\zeta})$.\nComparing with the case without the microwave field ($\\Omega_m=0$), $\\Re(\\vec{\\zeta})$ is largely modified and can be  manipulated by the use of the microwave field ($\\Omega_m\\neq 0$). To demonstrate this, the normalized response function $\\Re\/\\Re_{\\rm max}$ as a function for $\\xi= x\/R_0$ is shown in Fig.~\\ref{Fig2}(d), where the dotted black line, solid red line, and solid blue line are for $\\Omega_m=0$, $\\Omega_m=5$~MHz, and $\\Omega_m=$15~MHz, respectively.\nWe see that, due to the role played by the microwave field, the shape of\n$\\Re\/\\Re_{\\rm max}$ is changed significantly. Especially, for a larger  microwave field, the $\\Re\/\\Re_{\\rm max}$ curve\nbecomes negative for the small value of $\\xi$ (not shown here), consistent with the result  obtained in Ref.~\\cite{Petrosyan2014}.\nPlotted in the inset of the figure is the normalized response function in momentum space $\\tilde{\\Re}\/\\tilde{\\Re}_{\\rm max}$\n(i.e., the Fourier transformation of $\\Re\/\\Re_{\\rm max}$)\nas a function of the non-dimensional wavenumber\n$\\beta_1=R_0 k_x$ ~($k_x$ is the wavenumber in $x$ direction) with $\\Omega_m=0$, 5, and 15\\,MHz, respectively.\nOne sees that $\\tilde{\\Re}\/\\tilde{\\Re}_{\\rm max}$ has only one change in sign for $\\Omega_m=0$; however, more changes in sign arise when $\\Omega_m$ takes nonzero values. Such behavior of $\\tilde{\\Re}\/\\tilde{\\Re}_{\\rm max}$ is due to the joint action by the nonlocal Kerr nonlinearity and the microwave field, through which the MI of the plane-wave probe field may occur~(see the next section).\n\n\n\nExcept for the significant dependence on the microwave field $\\Omega_m$,\nthe property of the response function depends also on another parameter, i.e., the nonlocality degree of the Kerr nonlinearity, defined by\n\\begin{equation}\\label{NLD}\n\\sigma\\equiv {R_b}\/{R_0},\n\\end{equation}\nwhere $R_b$ is the radius of Rydberg blockade sphere, given by\n$R_b=|C_{33}^{s}\/\\delta_{\\rm EIT}|^{1\/6}$~\\cite{Saffman2010,Fir2016,Mur2016}, with\n$\\delta_{\\rm EIT}$ the width of EIT transparency window. One has\n$\\delta_{\\rm EIT}=|\\Omega_c|^2\/\\gamma_{21}$ for\n$\\Delta_2=0$, and $\\delta_{\\rm EIT}=|\\Omega_c|^2\/\\Delta_2$\nfor $\\Delta_2\\gg\\gamma_{21}$. With the system parameters used here,\nwe have $R_b\\approx 8.34 ~\\mu{\\rm m}$.\nIn the next section, we shall show that the structural phase transitions of the optical patterns of the system depend strongly not only on the microwave field $\\Omega_m$ but also on the nonlocality degree $\\sigma$ of the Kerr nonlinearity.\n\n\n\\section{Modulational instability, emergence of optical patterns and solitons}\\label{sec3}\n\n\\subsection{Modulation instability}\\label{sec31}\nMI is a nonlinear instability of constant-amplitude continuous waves  under long-wavelength perturbations, occurring in a variety of contexts where Kerr nonlinearity is attractive and local~\\cite{Zakharov2009,Biondini2016}; it can also arise in systems with repulsive but nonlocal Kerr nonlinearity when the perturbations have both long~\\cite{Krolikowski2001,Krolikowski2004} and short~\\cite{Maucher2016,Maucher2017,Sevincli2011} wavelengths. To explore the MI in the our system, we consider the MI of the plane-wave solution of the NNLS Eq.~(\\ref{NNLS2}), i.e.,\n\\begin{equation}\\label{PW}\nu_{\\rm pw}(\\vec{\\zeta},s)=A_0\\exp\\left[-is A_0^2\\int \\Re(\\vec{\\zeta})d^2 \\zeta\\right],\n\\end{equation}\nwhere $A_0$ is a real number.\nSince any perturbation can be expanded as a superposition of many Fourier modes, we make the MI analysis of the plane wave by taking only a periodic mode as the perturbation, i.e.,\n\\begin{eqnarray}\\label{PTS}\n\\tilde{u}(\\vec{\\zeta},s)=\n&&\\left[A_0+a_1e^{i\\vec{\\beta}\\cdot \\vec{\\zeta}+\\lambda\ns}+a_2^*e^{-i\\vec{\\beta}\\cdot \\vec{\\zeta}+\\lambda^* s}\\right]\\nonumber\\\\\n&& \\times\n\\exp\\left[-is A_0^2\\int \\Re(\\vec{\\zeta})d^2\\zeta\\right],\n\\end{eqnarray}\nwhere $a_1$ and $a_2$ are small complex amplitudes of the perturbation and $\\vec{\\beta}= (\\beta_1,\\beta_2)$\\, ($\\beta_1\\equiv R_0 k_x$, $\\beta_2\\equiv R_0 k_y$; $k_x$ $k_y$ are wavenumbers in $x$ and $y$ directions, respectively) is non-dimensional 2D wavevector and $\\lambda$ is the growth rate of the perturbation, to be determined yet.\n\n\nSubstituting the perturbation solution (\\ref{PTS}) into Eq.~(\\ref{NNLS2}) and keeping only linear terms of $a_1$ and $a_2$, it is easy to obtain the expression of the growth rate\n\\begin{align}\n\\lambda^2=-{\\beta}^2\\left[\\beta^2-2A_0^2\\,\\tilde{\\Re}(\\vec{\\beta})\\right],\n\\end{align}\nwhere ${\\beta}=\\sqrt{\\beta_1^2+\\beta_2^2}$ and $\\tilde{\\Re}(\\vec{\\beta})$  is the response function in momentum space [i.e., the Fourier transformation of $\\Re(\\vec{\\zeta})$].\n\n\nThe property of the growth rate $\\lambda$ depends on the plane-wave intensity $A_0^2$, the shape of the response function $\\tilde{\\Re}$ where the microwave field $\\Omega_m$ plays an important role.\nShown in Fig.~\\ref{Fig3}(a)\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fig3.eps}\n\\caption{\\footnotesize Modulation instability and its manipulation for the plane-wave probe field in the Rydberg atomic gas with the microwave dressing.\n(a)~$-\\lambda^2$ ($\\lambda$ is the growth rate) as a function of $\\beta=\\sqrt{\\beta_1^2+\\beta_2^2}$  [$\\beta_{j}\\equiv R_0k_j$; $k_j$ is the wavenumber along $j$th direction ($j=x,y$); $R_0$ is the typical transverse beam radius of the probe field], for\nthe microwave field $\\Omega_m=0$~(dotted black line), 5~MHz~(dashed red line), and 15~MHz~( solid blue line), respectively; shadow regions are ones where MI occurs.\n(b)~Real part of the growth rate ${\\rm Re}(\\lambda$) as  a function of $\\beta$ and the effective probe-field intensity $I_{\\rm eff}=\\alpha A_0^2$\\, [with $A_0$ the amplitude of the plane wave and $\\alpha=-\\int{\\Re}(\\vec{\\zeta})d^2\\zeta$\\,], for the microwave field $\\Omega_m=10$~MHz. The colorful region is the one for ${\\rm Re}(\\lambda)>0$, where MI occurs.\n(c)~${\\rm Re}(\\lambda$) as a function of $\\beta$ and $\\Omega_m$ for $I_{\\rm eff}=20$; the colorful region is the one where MI occurs.\n}\\label{Fig3}\n\\end{figure}\nis the curve of $-\\lambda^2$ as a function of the non-dimensional wavenumber $\\beta$ for the microwave field $\\Omega_m=0$~(dotted black line), 5~MHz~(dashed red line), and 15~MHz~( solid blue line), respectively. The shadow regions in the figure are ones for ${\\rm Re}(\\lambda)>0$. That is to say, MI occurs in these shadow regions and hence the plane-wave state of the probe field is unstable. The MI will lead to a symmetry breaking of the system and hence a phase transition to new states. As a result, new optical self-organized structures (or pattern formation) appear in the system (see next section). We note that, different from the cases reported in Refs.~\\cite{Krolikowski2001,Krolikowski2004} but similar to those considered in Refs.~\\cite{Maucher2016,Maucher2017,Sevincli2011},\nthe MI in the present system arises for the perturbation of short wavelengths.\n\n\nTo obtain a further understanding of the MI, Fig.~\\ref{Fig3}(b) shows\nthe real part of the growth rate, ${\\rm Re}(\\lambda$), as  a function of $\\beta$ and the effective probe-field intensity\n\\begin{equation}\\label{I}\nI_{\\rm eff} =\\alpha A_0^2\n\\end{equation}\nfor $\\Omega_m=10$~MHz, where $\\alpha=-\\int{\\Re}(\\vec{\\zeta})d^2\\zeta$ is a parameter characterizing the role by the nonlocal Kerr nonlinearity.\nThe colorful region in the figure is the one where ${\\rm Re}(\\lambda)>0$ and hence MI occurs.  Fig.~\\ref{Fig3}(c) shows ${\\rm Re}(\\lambda$) as a  function of $\\beta$ and $\\Omega_m$ for $I_{\\rm eff}=20$, with the colorful region denoting the one where the MI happens.  From these results we see that the MI depends not only on the effective probe-field intensity\n$I_{\\rm eff}$ but also on the microwave field $\\Omega_m$, which provides\nways to manipulate the MI and thereby the emergence of the optical patterns in the system.\n\n\n\\subsection{Pattern formation controlled by the Kerr nonlinearity and the microwave field}\\label{sec32}\n\n\nWe now turn to consider the outcome of the MI in the system. Note that\nin the absence of the microwave field the system is reduced to a three-level one (i.e. conventional Rydberg-EIT) and the atom-atom interaction Hamiltonian $\\hat{\\cal H}_{\\rm vdw}$ owns only the term\n$\\hbar\\mathcal{N}_a\\int d^3{ r}^\\prime \\hat{S}_{33}({\\bf\nr}^{\\prime},t)\\mathcal{V}_{33}^s({\\bf r}^\\prime-{\\bf r})\\hat{S}_{33}({\\bf r},t)$; however, in the presence of the microwave field, the state $|4\\rangle$ may have a significant population and hence it plays an important role for the dynamics of the probe field. In this case $\\hat{\\cal H}_{\\rm vdw}$ owns four terms,\nwhich may be comparable through the tuning of the system parameters.\nAs a result, the  nonlinear response function $G$ in the envelope Eq.~(\\ref{NNLS1}) contains four terms, i.e., $G=G_{33}^s+G_{44}^s+G_{34}^d+G_{34}^e$; the nonlocal Kerr nonlinearities contributed by $G_{33}^s$, $G_{34}^d$, and $G_{34}^e$ are repulsive, but the one contributed by $G_{44}^s$ is attractive. Therefore, depending on system parameters and based on the competition among these four terms in $G$, the total Kerr nonlinearity of the system may be type of self-defocusing or self-focusing, which means that the system may support very rich nonlinear structures after the occurrence of the MI, including the emergence of various optical patterns and solitons.\nGenerally, when the repulsive part (contributed by $G_{33}^s$, $G_{34}^d$, and $G_{34}^e$)\nplays a dominant role over the attractive part (contributed by $G_{44}^s$), the MI results in the formation of optical patterns; on the contrary, when the attractive part is dominant over the repulsive part, the MI gives rise to the formation of bright solitons.\n\n\nAs a first step, we focus on the case of pattern formation, for which the whole Kerr nonlinearity must be type of self-defocusing. This can be realized by choosing suitable system parameters to make the repulsive part in $G$\\, (i.e., $G_{33}^s$, $G_{34}^d$, and $G_{34}^e$)  is larger than the attractive part (i.e., $G_{44}^s$). In fact, the system parameters given at the final part of Sec.~\\ref{sec21} fulfill such requirement. Except for these parameters, other three parameters, i.e., $I_{\\rm eff}$ (the effective probe-field intensity), $\\sigma$ (the nonlocality degree of the Kerr nonlinearity), and $\\Omega_m$ (the microwave field), play significant roles for determining the types of optical patterns in the system. Based on such consideration and for obtaining the optical patterns, we seek the ground-state solution of the system by a numerical simulation solving Eq.~(\\ref{NNLS2}) via an imaginary evolution and split-step Fourier methods~\\cite{YangJK2010}, for which the total energy of the system\n{\\small\n\\begin{align}\nE=\n&\\int |\\tilde{\\nabla}_{\\perp}u(\\vec{\\zeta},s)|^2 d^2\\zeta\\nonumber\\\\\n&\n+\\frac{1}{2}\\iint \\Re({\\vec{\\zeta}^\\prime-\\vec{\\zeta}})|u(\\vec{\\zeta},s)|^2\n  |u(\\vec{\\zeta}^{\\prime},s)|^2 d^2\\zeta^{\\prime}  d^2 \\zeta\\label{E}\n\\end{align}}\\noindent\nis minimum. The initial condition used in the simulation is the plane wave (\\ref{PW}), perturbed by a random noise.\n\n\nShown in Fig.~\\ref{Fig4}(a)\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\textwidth]{fig4.eps}\n\\caption{\\footnotesize Pattern formation and phase diagram controlled by the effective intensity of the probe field $I_{\\rm eff}=\\alpha A_0^2$\nand the microwave field $\\Omega_m$, for  the nonlocality degree $\\sigma\\equiv R_b\/R_0=1$. Here, $A_0$ is the amplitude of the plane-wave state; $\\alpha= -\\int{\\Re}(\\vec{\\zeta})d^2\\zeta$; $R_b$ and $R_0$ are Rydberg blockade radius and the transverse radius of the probe beam, respectively.\n(a)~Phase diagram of the structural transition of optical patterns, where different regions (phases) are obtained by changing the values of $I_{\\rm eff}$ and $\\Omega_m$.\nRegion $\\circled{1}$: the homogeneous (i.e., the plane wave) state;\nRegion $\\circled{2}$: the hexagonal lattice;\nRegion $\\circled{3}$: the type I square lattice;\nRegion $\\circled{4}$: the type II square lattice.\n(b)~The hexagonal lattice of the normalized probe-field amplitude $|u|$ as a function of $\\xi=x\/R_0$ and $\\eta=y\/R_0$ for $\\Omega_m=10$~MHz and $I_{\\rm eff}=15$, corresponding to the region $\\circled{2}$ in panel (a).\n(c)~The type I square lattice of $|u|$ as a function of $\\xi$ and $\\eta$ for $\\Omega_m=13$~MHz and $I_{\\rm eff}=25$, corresponding to the region $\\circled{3}$ in panel (a).\n(d)~~The type II square lattice of $|u|$ as a function of $\\xi$ and $\\eta$ for $\\Omega_m=15$~MHz and $I_{\\rm eff}=35$, corresponding to the region $\\circled{4}$ in panel (a).\n}\\label{Fig4}\n\\end{figure}\nis the phase diagram describing the phase transition of self-organized optical structures, which are controlled by the effective intensity of the probe field $I_{\\rm eff}=\\alpha A_0^2$ and the microwave field $\\Omega_m$.\nThe dashed lines in the figure are boundaries of different phases.\nWhen obtaining the phase diagram, the nonlocality degree of the Kerr nonlinearity, i.e., $\\sigma= R_b\/R_0$, is fixed to be 1. From the figure, we see that several structural transitions of optical patterns\nemerge when $I_{\\rm eff}$ and $\\Omega_m$ are changed in the following ways:\n(i)~from the homogeneous state $\\circled{1}$ to the hexagonal lattice $\\circled{2}$;\n(ii)~from the hexagonal lattice  $\\circled{2}$ to the type I square lattice $\\circled{3}$;\n(iii)~from the type I square lattice $\\circled{3}$ to the type II square lattice $\\circled{4}$).\nHere $\\circled{1}$, $\\circled{2}$, $\\circled{3}$, and $\\circled{4}$ represent regions of the homogeneous state, hexagonal lattice, type I square lattice, and type II square lattice, respectively.\n\n\nTo be more concrete, we give several examples for illustrating the optical lattice patterns that correspond to the self-organized structures indicated in the different regions of Fig.~\\ref{Fig4}(a).  Fig.~\\ref{Fig4}(b) shows a hexagonal lattice pattern, where the amplitude $|u|$ of the probe field is normalized  as a function of $\\xi=x\/R_0$ and $\\eta=y\/R_0$; it is obtained by taking $\\Omega_m=10$~MHz and $I_{\\rm eff}=15$, located in the region $\\circled{2}$ of Fig.~\\ref{Fig4}(a). Such hexagonal lattice pattern\nwas found by Sevinli {\\it et al.}~\\cite{Sevincli2011} where no microwave dressing is used (i.e., $\\Omega_m=0$); in this case the hexagonal lattice pattern is the only one that can be obtained via the MI of the homogeneous (plane wave) state.\n\n\nPlotted in Fig.~\\ref{Fig4}(c) is the optical pattern by taking $|u|$ as a function of $\\xi$ and $\\eta$, for $\\Omega_m=13$~MHz and $I_{\\rm eff}=25$ [which locates in the region $\\circled{3}$ of Fig.~\\ref{Fig4}(a)]. We see that in this case a new optical structure, called the type I square lattice, emerges. Obviously, such new optical structure, which does not exist if the microwave dressing is absent, arises due to the symmetry breaking induced by the introduction of the microwave field.\n\n\nFig.~\\ref{Fig4}(d) gives the result for the optical pattern with increasing microwave field and the effective intensity of the probe field,\nby taking $\\Omega_m=15$~MHz and $I_{\\rm eff}=35$ [which is in the region\n$\\circled{4}$ of Fig.~\\ref{Fig4}(a)]. One sees that in this situation another type of optical structure, called the type II square lattice, appears. By comparing the type I square lattice pattern of Fig.~\\ref{Fig4}(c), we see that there is an angle difference (around $45^\\circ$) between the type I and type II square lattices; furthermore, there are also differences for the normalized probe-field amplitudes and the lattice constants  between these two types of square lattice patterns (for detail, see Table~\\ref{tab1} below).\n\n\n\n\n\\subsection{Pattern formation controlled by the nonlocality degree of the Kerr nonlinearity and the microwave field}\\label{sec33}\nTo explore the structural phase transition of the optical patterns further, we now fix the effective probe-field intensity\n($I_{\\rm eff}=35$) but take the nonlocality degree of the Kerr nonlinearity $\\sigma$ and the microwave field $\\Omega_m$ as control parameters. Similar to the last subsection, we seek the spatial distribution of the probe field for which the total energy (\\ref{E}) of the system is minimum, through a numerical simulation of Eq.~(\\ref{NNLS2}).\n\n\nShown in Fig.~\\ref{Fig5}(a) is the phase diagram of the structural transition of optical patterns, where different regions (phases) are obtained by changing the values of $\\sigma$ and $\\Omega_m$, separated by\ndashed lines (i.e., boundaries of different phases).\nWe see that several structural transitions [i.e., from the homogeneous state $\\circled{1}$ to the hexagonal lattice $\\circled{4}$, from the hexagonal lattice $\\circled{4}$ to the type I square lattice $\\circled{3}$, and from the type I square lattice $\\circled{3}$ to the type II square lattice $\\circled{2}$\\,] of the optical patterns arise  when $\\sigma$ and $\\Omega_m$ are varied.\n\n\n\\begin{figure}[htpb]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fig5.eps}\n\\caption{\\footnotesize Pattern formation and phase diagram controlled by the nonlocality degree of the Kerr nonlinearity $\\sigma=R_b\/R_0$ and the microwave field $\\Omega_m$, for\nthe effective probe-field intensity $I_{\\rm eff}=35$.\n(a)~Phase diagram of the structural transition of optical patterns, where different regions (phases) are obtained by changing the values of $\\sigma$ and $\\Omega_m$.\nRegion $\\circled{4}$: the pattern with hexagonal lattice structure;\nRegion $\\circled{3}$: the pattern with type I square lattice structure;\nRegion $\\circled{2}$: the pattern with type II square lattice structure;\nRegion $\\circled{1}$: the homogeneous (i.e. the plane wave) state.\n(b)~The hexagonal structure of the normalized probe-field amplitude $|u|$ as a function of $\\xi=x\/R_0$ and $\\eta=y\/R_0$ for $\\Omega_m=10$~MHz and $\\sigma=2$, corresponding to the region $\\circled{4}$ in panel (a).\n(c)~The type I square structure of $|u|$ as a function of $\\xi$ and $\\eta$ for $\\Omega_m=12$~MHz and  $\\sigma=1$, corresponding to the region $\\circled{3}$ in panel (a).\n(d)~~The type II square structure of $|u|$ as a function of $\\xi$ and $\\eta$ for $\\Omega_m=18$~MHz and $\\sigma=1$, corresponding to the region $\\circled{4}$ in panel (a).\n}\\label{Fig5}\n\\end{figure}\n\nWe also give several examples for illustrating the optical patterns corresponding to the self-organized structures indicated in the different regions of Fig.~\\ref{Fig5}(a). Fig.~\\ref{Fig5}(b) shows a hexagonal lattice pattern, obtained by taking the normalized amplitude\nof the probe field $|u|$ as a function of $\\xi=x\/R_0$ and $\\eta=y\/R_0$, for $\\Omega_m=10$~MHz and $\\sigma=2$ [located in the region $\\circled{4}$ of Fig.~\\ref{Fig5}(a)].\n\n\nShown in Fig.~\\ref{Fig5}(c) is the optical pattern for $\\Omega_m=12$~MHz and  $\\sigma=1$ [located in the region $\\circled{3}$ of Fig.~\\ref{Fig5}(a)]; one sees that in this case the lattice pattern is a type I square lattice, which is absent without microwave field. Illustrated in Fig.~\\ref{Fig5}(d) is the optical pattern with  $\\Omega_m=18$~MHz and $\\sigma=1$, which is in the region\n$\\circled{2}$ of Fig.~\\ref{Fig5}(a); in this case\nthe type II square lattice structure appears.\n\n\nTo see clearly the differences between the two types of square lattice patterns, a quantitative comparison is made for the normalized probe-field amplitude $|u|$, microwave field $\\Omega_m$, effective probe-field intensity $I_{\\rm eff}$, and lattice constant $l$ (i.e., the distance between the maximums of two adjacent optical spots) between the type I and type II square lattice patterns obtained in Fig.~\\ref{Fig4} and Fig.~\\ref{Fig5} by taking the nonlocality degree of the Kerr nonlinearity $\\sigma=1$, with the result presented in Table~\\ref{tab1}.\n\\begin{table}[htbp]\n\\centering\\caption{\\footnotesize Differences between the type I and type II square lattice patterns. Values of the normalized probe-field amplitude $|u|$, microwave field $\\Omega_m$, effective probe-field intensity $I_{\\rm eff}$, and lattice constant $l$ for the two types of square lattice patterns.\n}\\label{tab1}\n\\vspace{4mm}\n\\renewcommand\\tabcolsep{9.5pt}\n\\begin{tabular}{ccccc}\n\\Xhline{1pt}\nType\n&$|u|_{\\rm max}$\n& $\\Omega_m$\n& $I_{\\rm eff}$\n& $l$\\\\\n\\Xhline{1pt}\nType I\\,\\,\\,\\, [Fig.~\\ref{Fig4}(c)]\n&28.5\n&13\n&25\n&1.74\\\\\nType II\\,\\,\\,\\,[Fig.~\\ref{Fig4}(d)]\n&54.4\n&15\n&35\n&1.66\\\\\nType I\\,\\,\\,\\, [Fig.~\\ref{Fig5}(c)]\n&37.9\n&12\n&35\n&2.82\\\\\nType II\\,\\,\\,\\,[Fig.~\\ref{Fig5}(d)]\n&69.5\n&18\n&35\n&2.35\\\\\n\\Xhline{1pt}\n\\end{tabular\n\\end{table\nWe see that:\n(i)~the lattice constant $l$ of the type I square lattice pattern is larger than that of the type II one;\n(ii)~comparing Fig.~\\ref{Fig5}(c) with Fig.~\\ref{Fig5}(d) [Fig.~\\ref{Fig4}(c) with Fig.~\\ref{Fig4}(d)], the lattice constant $l$\nis larger for smaller microwave field $\\Omega_m$.\nThe physical reason for such differences is that the nonlocal nonlinear response function $\\Re\/\\Re_{\\rm max}$ has a significant dependence on the microwave field. When the microwave filed $\\Omega_m$ is increased, the shape of $\\Re\/\\Re_{\\rm max}$ is largely modified [i.e., it becomes narrower and steeper; see Fig.~\\ref{Fig2}(d)], which makes the system change into a new state and thereby a new type of square lattice pattern emerges.\n\n\nCombining Fig.~\\ref{Fig4} and Fig.~\\ref{Fig5}, which are the key results of this work, we see that, in the parameter domains considered here, the system supports three types of self-organized optical structures (i.e., the hexagonal lattice, the type I and type II square lattices), and their phase transitions can be controlled by actively manipulating the microwave field ($\\Omega_m$), the effective probe-field intensity ($I_{\\rm eff}$), and the nonlocality degree of the Kerr nonlinearity ($\\sigma$).\nThe basic physical mechanism of the MI and the formation of the optical patterns found here may be understand as follows. When the plane-wave probe field with a finite amplitude is applied to and propagates in the Rydberg atomic gas along the $z$ direction, the nonlocal Kerr nonlinearity coming from the Rydberg-Rydberg interaction brings a phase modulation to the probe field; due to the role played by the diffraction in the transverse (i.e., $x$ and $y$) directions, the phase modulation is converted into  amplitude modulation. Because of the joint effect of the phase and amplitude modulations, in some parameter domains the probe field undergoes MI and re-organizes its spatial distribution and hence the formation of optical patterns occurs.\n\n\nThe emergence of the different self-organized structures (i.e. the hexagonal and square lattice optical patterns) and related phase changes are originated from the spatial symmetry breaking of the system.\nTo understand this and illustrate furthermore the differences between various optical lattice patterns, a detailed theoretical analysis on the ground-state energy of the system for different spatial distributions of the probe-field intensity is given in Appendix~\\ref{appC}.\n\n\\subsection{Formation of nonlocal spatial optical solitons}\\label{sec34}\n\nSpatial optical solitons, i.e., localized nonlinear optical structures resulting from the balance between nonlinearity and diffraction, can form\nthrough the MI of plane waves~\\cite{Kivshar2006,Chen2012}. However,\na necessary condition for the formation of an optical soliton is that the Kerr nonlinearity in the system should be of the type of self-focusing. As indicated in Sec.~\\ref{sec32}, due to the microwave dressing in our system there exist four kinds of nonlocal Kerr nonlinearities, which are described by the four response functions (i.e., $G_{33}^s$, $G_{44}^s$, $G_{34}^d$, $G_{34}^e$), and one of them (i.e., $G_{44}^s$) is attractive. Therefore, it is possible to make the total Kerr nonlinearity of the system to be a self-focused one if suitable system parameters are chosen.\n\n\nIn fact, a self-focused total Kerr nonlinearity can indeed be obtained by choosing the following system parameters, i.e.\n$\\Delta_2=6.28\\times 10^2\\,{\\rm MHz}$,\n$\\Delta_3=6.92\\,{\\rm MHz}$,\n${\\rm \\Delta_4=1\\times 10^4\\,Hz}$,\n ${\\rm \\Gamma_{12}=2\\pi\\times 6}$ MHz,\n$\\Gamma_3=\\Gamma_{4}=2\\pi\\times 1.67\\times 10^{-2}$ MHz, $\\Omega_{c}=90\\,{\\rm MHz}$, and $\\N_a=1.0\\times 10^{11}\\,{\\rm\ncm^{-3}}$.  In this situation, the attractive interaction contributed by $G_{44}^s$ plays a dominant role, and hence the plane-wave state (\\ref{PW}) will undergo a MI and is possible to be squeezed into a soliton by the Kerr nonlinearity.\n\n\nTo confirm the MI, a numerical simulation based on an imaginary-time propagation method is carried out by solving the NNLS equation (\\ref{NNLS2}) with the above parameters. Shown in Fig.~\\ref{Fig6}(b)\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig6.eps}\n\\caption{\\footnotesize Formation and propagation of nonlocal spatial optical solitons in the presence of the microwave dressing.\n(a)~Initial condition ($|u|=1.3~\\text{sech}[(\\xi^2+\\eta^2)^{1\/2}]$ at $s=0$ ($s= z\/2L_{\\rm diff}$; $L_{\\rm diff}$ is diffraction length).\n(b)~Spatial distribution of the soliton when propagating to the position $s=5$, by taking the normalized probe-field amplitude $|u|$ as a function of non-dimensional coordinates $\\xi= x\/R_0$ and $\\eta= y\/R_0$\n($R_0$ is the typical transverse beam radius of the probe field)\nfor the microwave field $\\Omega_m=10$~MHz and the nonlocality degree $\\sigma=1$.\n(c)~Random initial condition $|u|= 1+ 0.05f$ ($f$ is a Gaussian noise)\nand (d)~The soliton (generated by the random initial condition) when  propagating to $s=5$.\n(e), (f), and (g) are two-, three-, and four-soliton solutions when they propagating to $s=5$ for $(\\Omega_m, \\sigma)=(10,1.2)$, $(15,1.4)$, and $(18,1.8)$, respectively.\n}\n\\label{Fig6}\n\\end{figure}\\noindent\nis the spatial distribution of the probe-field envelope when it propagates 10 diffraction length (i.e., $s\\equiv z\/2L_{\\rm diff}=5)$, by taking  $|u|$ (the normalized probe-field amplitude) as a function of non-dimensional coordinates $\\xi= x\/R_0$ and $\\eta=y\/R_0$, for the microwave field $\\Omega_m=10$~MHz and the nonlocality degree of the Kerr nonlinearity $\\sigma=1$. The initial condition used in the simulation is $|u|=1.3~\\text{sech}[(\\xi^2+\\eta^2)^{1\/2}]$\\, [Fig.~\\ref{Fig6}(a)]. We see that a nonlocal spatial optical soliton can indeed form in the system and it is quite stable during propagation.\nNote that solitons can be also generated by using random initial conditions. Fig.~\\ref{Fig6}(d) shows the spatial distribution of a soliton when it is created and propagates 10 diffraction length, for which the initial condition used is of the form $|u|= 1+ 0.05f$, where $f$ is Gaussian noise\\, [Fig.~\\ref{Fig6}(c)].\n\n\nDifferent from the studies considered before, the nonlocal spatial optical solitons found here can be actively manipulated by actively tuning $\\Omega_m$ and $\\sigma$.  Shown in the panels (e), (f), and (g) of Fig.~\\ref{Fig6} are two-soliton, three-soliton, and four-soliton solutions when they propagating to 10 diffraction length (i.e., $s=5$), obtained for $(\\Omega_m, \\sigma)=(10,1.2)$, $(15,1.4)$, and $(18,1.8)$, respectively.\nOther multi-soliton solution solutions of the system may also be obtained.\n\n\n\\section{Summary}\\label{sec4}\n\nIn this work, we have proposed a scheme for the realization of optical pattern formation and spatial solitons via a Rydberg-EIT. Through the use of a microwave dressing, we have shown that the nonlocal Kerr nonlinearity of the system can be manipulated actively and its magnitude can be enhanced significantly. Based on such nonlocal and tunable Kerr nonlinearity, we have demonstrated that a plane-wave probe field can undergo MI and spontaneous symmetry breaking, and thereby various self-organized optical patterns may emerge in the system. In particular, we have found that a hexagonal lattice pattern, which appears after the MI when the repulsive part of the nonlocal nonlinear response function is larger than its attractive part, may develop into several types of square lattice patterns when the microwave field is applied and tuned actively. Furthermore, through the MI the formation of nonlocal spatial optical solitons has also been found when the attractive part of the nonlocal nonlinear response function is dominant over its repulsive part. Different from the results reported before, the optical patterns and nonlocal optical solitons discovered here can be flexibly adjusted and controlled through the change of the effective probe-field intensity, nonlocality degree of the Kerr nonlinearity, and the strength of the microwave field. Our work opens a way for a versatile control of the self-organizations and structural phase transitions of laser light based on microwave-dressed Rydberg gases, which may have potential applications in optical information processing and transmission.\n\n\n\n\\acknowledgments\n\nAuthors thank Y.-C. Zhang and Z. Bai for useful discussions. This work was supported by the National Natural Science Foundation of China under Grant\nNo.~11474099 and No.~11975098.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzkdfw b/data_all_eng_slimpj/shuffled/split2/finalzzkdfw
new file mode 100644
index 0000000000000000000000000000000000000000..3f9c1785d4f1cded56630dda77f1a6777318aeb2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzkdfw
@@ -0,0 +1,5 @@
+{"text":"\\section{A Dark Lens candidate}\n\nIn a campaign to study the optical properties of faint VLA FIRST \n(Faint Images of the Radio Sky at Twenty-Centimeters) sources with the \nHubble Space Telescope, Russell et al. [\\pos{1}] identified an optical arc\n4 arcseconds away from FIRST J121839.7 +295325 (hereafter J1218+2953). \nThere was no optical identification of the radio source itself.\nThe possible relation between the two objects was further investigated by\nRyan et al. [\\pos{2}], who proposed that the arc may be a gravitationally lensed\nimage and the radio source may belong to the lensing object. There are two \nmajor difficulties with this interpretation. First of all the lensing galaxy\nis not seen. Moreover the redshifts of the radio source and the optical arc\nare not known. Ryan et al. [\\pos{2}] estimated a redshift range of\n$0.8<z<1.5$ for the radio source on the basis that the galaxy is not seen \nin the optical ($z>0.8$), but it has a relatively bright radio flux density ($z<1.5$). \nThey carried out photometric redshift measurements of the optical arc with \nthe SAO Multi-Mirror Telescope. A weak spectral break was seen at around\n4300 \\AA, which could be due to the Balmer\/4000 \\AA ~break at $z\\sim0.13$, \nor due to the Lyman-break at $z\\sim2.5$. Probability analysis showed \nthat the Lyman-break is much more likely, which suggests that the arc is \nlocated at a high redhsift. Obviously, if the spectral break is related to \nthe Balmer series then the redshift is much smaller and the two objects are \nunrelated.\n\nRyan et al. [\\pos{2}] carried out gravitational lens modelling and found that\nthe mass distribution must be elliptical to produce such a long arc. However \nthe observed sub-structure within the arc, and especially a bright knot at the\nend, cannot be easily explained by gravitational lensing.\nThey predict a secondary image which appears close to a pointlike source in the\nHST image, but this is likely due to a \"hot pixel\". According to the model, the enclosed \nmass within the Einstein-radius of 1.3 arcseconds is $10^{12\\pm0.5}M_{\\odot}$.\n\nIt is not clear how such a massive object may remain hidden. Even if there is strong\nobscuration by dust, there should be an infrared counterpart detected. IR imaging with \nthe SAO Wide-field camera showed no galaxy with limiting magnitudes of $J=22.0$\\,mag \nand $H=20.7$\\,mag [\\pos{2}]. Ryan et al. conclude that either there is an early-type\ngalaxy with significant amount of dark matter, or this could be a massive system\nwith an AGN that is completely obscured, with dynamic mass-to-light ratio exceeding\n100 $M_{\\odot}L_{\\odot}^{-1}$.\n\n\\section{VLBI imaging of FIRST J1218+2953}\n\n   \\begin{figure*}\n   \\centering\n   \\vspace{20pt}\n   \\includegraphics[bb=33 120 582 714,clip,angle=0,width=7cm]{SHORT.ps}\n   \\includegraphics[bb=36 123 531 641,clip,angle=0,width=7cm]{UVPLOT.ps}\n   \\caption{\\label{short} Left: naturally weighted image of J1218+2953 from 2 hours of short \ne-EVN observations at 1.6 GHz. The first contour is drawn at 0.4 mJy\/beam, the following ones \nare multiples of this by $\\sqrt{2}$. The beam was 30.3 $\\times$ 23.8 mas, oriented at \nPA 3.3 degrees. The peak brightness was about 2.3 mJy\/beam. The total cleaned flux density \nrecovered was less then 15 mJy. Right: $uv$-coverage of the observations. The shortest MERLIN \nbaselines (Jb2-Cm-Kn) are a great value for imaging of sources extending to several mas. \nHowever in this case the short observations were still not enough to recover most of the \nflux density.} \n    \\end{figure*}\n\n\\subsection{Short e-EVN observations}\n\nWe applied for short e-EVN observations in December 2008, to test the AGN hypothesis\nfor J1218+2953. The total flux density of the source at 1.6 GHz is 33 mJy, which \ncan be easily detected with limited EVN resources if it is compact. The observations\ntook place on 23 January 2009 at a data rate of 512 Mbps with 2-bit sampling, dual \npolarization, and lasted for 2 hours. The array consisted of Cambridge and  Knockin \n(MERLIN telescopes, limited to 128 Mbps), Jodrell Bank MkII, Medicina, Onsala, Torun and \nthe Westerbork Synthesis Radio Telescope (WSRT). \nThe target was phase-referenced to the nearby calibrator J1217+3007. \nThe data were pipelined and then imaged in Difmap [\\pos{3}]. The source was detected\nand was resolved to two components, separated by about 500~mas (see Fig. \\ref{short}). \n\nThis result opened up new possibilities for the interpretation of the radio source.\nAlthough it was not predicted from the \"best-fit\" gravitational lens model of Ryan et al.,\none may speculate that these two components are gravitationally lensed images of the\nsame background source that gives rise to the optical arc images, or perhaps a completely\nunrelated background source. Alternatively, we may see a core-jet system or\na medium-size symmetric object (MSO). To distinguish between these scenarios we put in an\nobserving proposal by the 1 February 2009 deadline (just 8 days following the e-EVN\nobservations), for full-track e-EVN observations at 5 and 1.6 GHz.\n\n\\subsection{Follow-up experiments}\n\nJ1218+2953 was observed at 5 GHz on 24-25 March for 8 hours. The array this time\nincluded the 100m Effelsberg telescope as well. Four telescopes (Ef, On, Tr, Wb) sent data \nto the correlator at 1024 Mbps rate, the rest at 512 Mbps or lower. The 1.6 GHz\nobservations were carried out on 21-22 April 2009 at 512 Mbps data rate. In the array\nKnockin was replaced by Darnhall, the 76m Lovell Telescope was used instead of the MkII\nin Jodrell Bank, and Arecibo joined as well (for 2 hours and 20 minutes). These observations \nand the data processing were similar to the short project described above. In addition, we \nreduced the synthesis array data from the WSRT that was obtained during the VLBI run.\n\nThe 5 GHz image resolves the South-East component into an elongated, slightly curved jet-like\nstructure, which does not point towards the North-West component (see Fig.~\\ref{fulltracks}). \nThere is no very compact component that could be firmly identified as a core. The total \ncleaned flux density is only 3 mJy compared to the total WSRT flux density of 9 mJy, indicating \nthat most of the source is resolved out in this image. The 1.6 GHz image reveals an even more \ncomplex, but more continuous structure. The total cleaned flux density was about 20 mJy, close\nto the total WSRT flux density of 27 mJy. \n\n\\section{Interpretation of the results}\n\nThese preliminary e-EVN results show that the radio source near the optical arc has a complex\nstructure. The spectrum of the components is steep; that of the faint component near the phase \ncentre is somewhat flatter. Because of the apparent quasi-continuous structure, the various \ncomponents are likely not gravitationally lensed images of an unrelated background source. \n\nComparing the total cleaned flux density of about 20 mJy to the WSRT flux density of 27 mJy, \nit is evident that most of the flux density is recovered at 1.6 GHz with the EVN. This indicates \nthat the radio source and the optical arc cannot be lensed image pairs of the same background \nobject, because in that case most of the radio flux density would be present near the optical \narc since that image is strongly magnified (if the arc is indeed a lensed image). \n\nFurther, gravitational lensing should be achromatic; thus were the pair of components to the \nSE and NW in the 1.6 GHz image (Fig.~\\ref{fulltracks}, bottom panel) lensed images, the flux \ndensity ratio of the inner:outer components of each should be similar, a condition that is \nclearly violated. \n\nThe most likely scenario is that the images show a single compact steep-spectrum (CSS) source \n(projected linear size < 20 $h^{-1}$ kpc), that might be categorized as a medium-size symmetric \nobject (MSO, projected linear size > 1 $h^{-1}$ kpc) as well [\\pos{4}]. There is thus evidence \nfor AGN activity in the radio source. Note that if the optical arc is lensed by an object related \nto this AGN, then the mass-centre of the lens should be at the position of the AGN, which \nconstrains further modelling of the system. \n\nFinally we note that this research benefited strongly from two aspects of the (e-)EVN: \nthe additional short MERLIN spacings were of great use in recovering flux density on \nseveral-hundred mas scales, and the simultaneous recording of WSRT synthesis array data was \nvery important for the intepretation of our results.   \n\n   \\begin{figure*}\n   \\centering\n   \\vspace{20pt}\n   \\includegraphics[bb=33 138 582 685,clip,angle=0,width=9.5cm]{5GHz_map.ps}\n   \\includegraphics[bb=33 138 582 685,clip,angle=0,width=9.5cm]{1.6GHz_map.ps}\n   \\caption{\\label{fulltracks} Full-track 5 GHz (top) and 1.6 GHz (bottom) e-EVN maps of J1218+2953.\n   The peak brightnesses are 360 $\\mu$Jy\/beam and and 2.7 mJy\/beam, respectively. The lowest contour\n   levels are set at the 3~sigma noise level (49 $\\mu$Jy\/beam and 75 $\\mu$Jy\/beam, respectively)\n   and they increase by a factor of $\\sqrt{2}$. Both maps were naturally weighted; at 1.6 GHz\n   a Gaussian taper was applied additionally, with half amplitude at 10 M$\\lambda$. \n   The beam was 15.4 $\\times$ 8.8 mas, oriented at PA $-$30.5 degrees at 5 GHz, and \n   30.9 $\\times$ 23.3 mas, oriented at PA $-$13.3 degrees at 1.6 GHz.} \n    \\end{figure*}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro} Exponential random graphs are among the most\nwidespread models for real-world networks as they are able to\ncapture a broad variety of common network tendencies by\nrepresenting a complex global structure through a set of tractable\nlocal features. These rather general models are exponential\nfamilies of probability distributions over graphs, in which\ndependence between the random edges is defined through certain\nfinite subgraphs, in imitation of the use of potential energy to\nprovide dependence between particle states in a grand canonical\nensemble of statistical physics. They are particularly useful when\none wants to construct models that resemble observed networks as\nclosely as possible, but without going into details of the\nspecific process underlying network formation. For history and a\nreview of developments, see e.g. Fienberg \\cite{F1} \\cite{F2},\nFrank and Strauss \\cite{FS}, H\\\"{a}ggstr\\\"{o}m and Jonasson\n\\cite{HJ}, Newman \\cite{N}, and Wasserman and Faust \\cite{WF}.\n\nMany of the investigations into exponential random graphs employ\nthe elegant theory of graph limits as developed by Lov\\'{a}sz and\ncoauthors (V.T. S\\'{o}s, B. Szegedy, C. Borgs, J. Chayes, K.\nVesztergombi, ...) \\cite{BCLSV1} \\cite{BCLSV2} \\cite{BCLSV3}\n\\cite{Lov} \\cite{LS}. Building on earlier work of Aldous\n\\cite{Aldous} and Hoover \\cite{Hoover}, the graph limit theory\nconnects sequences of graphs to a unified graphon space equipped\nwith a cut metric. Though the theory itself is tailored to dense\ngraphs, serious attempts have been made at formulating parallel\nresults for sparse graphs \\cite{AL} \\cite{BS} \\cite{BCCZ1}\n\\cite{BCCZ2} \\cite{CD2}. Since networks are often very large in\nsize, a pressing objective in studying exponential models is to\nunderstand their asymptotic tendencies. From the point of view of\nextremal combinatorics and statistical mechanics, emphasis has\nbeen made on the variational principle of the limiting\nnormalization constant, concentration of the limiting probability\ndistribution, phase transitions and asymptotic structures. See\ne.g. Aristoff and Zhu \\cite{AZ}, Chatterjee and Diaconis\n\\cite{CD}, Kenyon et al. \\cite{KRRS}, Kenyon and Yin \\cite{KY},\nLubetzky and Zhao \\cite{LZ1} \\cite{LZ2}, Radin and Sadun\n\\cite{RS1} \\cite{RS2}, and Radin and Yin \\cite{RY}.\n\nDespite their flexibility, conventionally used exponential random\ngraphs admittedly have one shortcoming. They cannot directly model\nweighted networks as the underlying probability space consists of\nsimple graphs only, i.e., edges are either present or absent.\nSince many substantively important networks are weighted, this\nlimitation is especially problematic. The need to extend the\nexisting exponential framework is thus justified, and several\ngeneralizations have been proposed \\cite{K} \\cite{RPW}\n\\cite{WDBCD}. An alternative interpretation for simple graphs is\nsuch that the edge weights are iid and satisfy a Bernoulli\ndistribution. This work will instead assume that the iid edge\nweights follow a generic common distribution and rigorously\nanalyze the associated phase transitions and critical phenomena.\n\nThe rest of this paper is organized as follows. In Section\n\\ref{weight} we provide basics of graph limit theory and introduce\nkey features of edge-weighted exponential random graphs. In\nSection \\ref{statement} we summarize some important general\nresults, including a variational principle for the limiting\nnormalization constant (Theorems \\ref{main1} and \\ref{main2}) and\nan associated concentration of measure (Theorem \\ref{main3})\nindicating that almost all large graphs lie near the maximizing\nset. Theorems \\ref{main4} and \\ref{gen} then give simplified\nversions of these theorems in the ``attractive'' region of the\nparameter space where the parameters $\\beta_2,...,\\beta_k$ are all\nnon-negative. In Section \\ref{app} we specialize to exponential\nmodels where the edge weights are uniformly distributed and show\nin Theorem \\ref{phase} the existence of a first order phase\ntransition curve ending in a second order critical point. Lastly,\nin Section \\ref{discuss} we investigate the asymptotic phase\nstructure of a directed model where a large deviation principle is\nmissing.\n\n\\section{Edge-weighted exponential random graphs}\n\\label{weight} Let $G_n\\in \\mathcal{G}_n$ be an edge-weighted\nundirected labeled graph on $n$ vertices, where the edge weights\n$x_{ij}$ between vertex $i$ and vertex $j$ are iid real random\nvariables having a common distribution $\\mu$. Any such graph\n$G_n$, irrespective of the number of vertices, may be represented\nas an element $h^{G_n}$ of a single abstract space $\\mathcal{W}$\nthat consists of all symmetric measurable functions $h(x,y)$ from\n$[0,1]^2$ into $\\mathbb{R}$ (referred to as ``graph limits'' or\n``graphons''), by setting $h^{G_n}(x,y)$ as the edge weight\nbetween vertices $\\lceil nx \\rceil$ and $\\lceil ny \\rceil$ of\n$G_n$. For a finite simple graph $H$ with vertex set\n$V(H)=[k]=\\{1,...,k\\}$ and edge set $E(H)$ and a simple graph\n$G_n$ on $n$ vertices, there is a notion of density of graph\nhomomorphisms, denoted by $t(H, G_n)$, which indicates the\nprobability that a random vertex map $V(H) \\to V(G_n)$ is\nedge-preserving,\n\\begin{equation}\n\\label{t} t(H, G_n)=\\frac{|\\text{hom}(H,\nG_n)|}{|V(G_n)|^{|V(H)|}}.\n\\end{equation}\nFor a graphon $h\\in \\mathcal{W}$, define the graphon homomorphism\ndensity\n\\begin{equation}\n\\label{tt} t(H, h)=\\int_{[0,1]^k}\\prod_{\\{i,j\\}\\in E(H)}h(x_i,\nx_j)dx_1\\cdots dx_k.\n\\end{equation}\nThen $t(H, G_n)=t(H, h^{G_n})$ by construction, and we take\n(\\ref{tt}) as the definition of graph homomorphism density $t(H,\nG_n)$ for an edge-weighted graph $G_n$. This graphon\ninterpretation enables us to capture the notion of convergence in\nterms of subgraph densities by an explicit ``cut distance'' on\n$\\mathcal{W}$:\n\\begin{equation}\nd_{\\square}(f, h)=\\sup_{S, T \\subseteq [0,1]}\\left|\\int_{S\\times\nT}\\left(f(x, y)-h(x, y)\\right)dx\\,dy\\right|\n\\end{equation}\nfor $f, h \\in \\mathcal{W}$. The common distribution $\\mu$ for the\nedge weights yields probability measure $\\mathbb P_n$ and the associated\nexpectation $\\mathbb E_n$ on $\\mathcal{G}_n$, and further induces\nprobability measure $\\mathbb Q_n$ on the space $\\mathcal{W}$ under the\ngraphon representation.\n\nA non-trivial complication is that the topology induced by the cut\nmetric is well defined only up to measure preserving\ntransformations of $[0,1]$ (and up to sets of Lebesgue measure\nzero), which may be thought of vertex relabeling in the context of\nfinite graphs. To tackle this issue, an equivalence relation\n$\\sim$ is introduced in $\\mathcal{W}$. We say that $f\\sim h$ if\n$f(x, y)=h_{\\sigma}(x, y):=h(\\sigma x, \\sigma y)$ for some measure\npreserving bijection $\\sigma$ of $[0,1]$. Let $\\tilde{h}$\n(referred to as a ``reduced graphon'') denote the equivalence\nclass of $h$ in $(\\mathcal{W}, d_{\\square})$. Since $d_{\\square}$\nis invariant under $\\sigma$, one can then define on the resulting\nquotient space $\\tilde{\\mathcal{W}}$ the natural distance\n$\\delta_{\\square}$ by $\\delta_{\\square}(\\tilde{f},\n\\tilde{h})=\\inf_{\\sigma_1, \\sigma_2}d_{\\square}(f_{\\sigma_1},\nh_{\\sigma_2})$, where the infimum ranges over all measure\npreserving bijections $\\sigma_1$ and $\\sigma_2$, making\n$(\\tilde{\\mathcal{W}}, \\delta_{\\square})$ into a metric space.\nWith some abuse of notation we also refer to $\\delta_{\\square}$ as\nthe ``cut distance''. After identifying graphs that are the same\nafter vertex relabeling, the probability measure $\\mathbb P_n$ yields\nprobability measure $\\tilde{\\mathbb P}_n$ and the associated expectation\n$\\tilde{\\mathbb E}_n$ (which coincides with $\\mathbb E_n$). Correspondingly,\nthe probability measure $\\mathbb Q_n$ induces probability measure\n$\\tilde{\\mathbb Q}_n$ on the space $\\tilde{\\mathcal{W}}$ under the\nmeasure preserving transformations.\n\nBy a $k$-parameter family of exponential random graphs we mean a\nfamily of probability measures $\\mathbb P_n^{\\beta}$ on $\\mathcal{G}_n$\ndefined by, for $G_n\\in\\mathcal{G}_n$,\n\\begin{equation}\n\\label{pmf} \\mathbb P_n^{\\beta}(G_n)=\\exp\\left(n^2\\left(\\beta_1\nt(H_1,G_n)+\\cdots+\n  \\beta_k t(H_k,G_n)-\\psi_n^{\\beta}\\right)\\right)\\mathbb P_n(G_n),\n\\end{equation}\nwhere $\\beta=(\\beta_1,\\dots,\\beta_k)$ are $k$ real parameters,\n$H_1,\\dots,H_k$ are pre-chosen finite simple graphs (and we take\n$H_1$ to be a single edge), $t(H_i, G_n)$ is the density of graph\nhomomorphisms, $\\mathbb P_n$ is the probability measure induced by the\ncommon distribution $\\mu$ for the edge weights, and\n$\\psi_n^{\\beta}$ is the normalization constant,\n\\begin{equation}\n\\label{psi} \\psi_n^{\\beta}=\\frac{1}{n^2}\\log \\mathbb E_n\n\\left(\\exp\\left(n^2 \\left(\\beta_1 t(H_1,G_n)+\\cdots+\\beta_k\nt(H_k,G_n)\\right) \\right)\\right).\n\\end{equation}\nWe say that a phase transition occurs when the limiting\nnormalization constant $\\displaystyle\n\\psi^\\beta_\\infty:=\\lim_{n\\to\n  \\infty}\\psi_n^{\\beta}$ has a singular point, as it is the\ngenerating function for the limiting expectations of other random\nvariables,\n\\begin{equation}\n\\label{E} \\lim_{n\\to \\infty}\\mathbb E_n^\\beta t(H_i, G_n)=\\lim_{n\\to\n\\infty}\\frac{\\partial}{\\partial\n\\beta_i}\\psi_n^\\beta=\\frac{\\partial}{\\partial\n\\beta_i}\\psi_\\infty^\\beta,\n\\end{equation}\n\\begin{equation}\n\\label{Cov} \\lim_{n\\to \\infty}n^2\\left(\\mathbb C\\textrm{ov}_n^\\beta\n\\left(t(H_i, G_n), t(H_j, G_n)\\right)\\right)=\\lim_{n\\to\n\\infty}\\frac{\\partial^2}{\\partial \\beta_i\n\\partial \\beta_j}\\psi_n^\\beta=\\frac{\\partial^2}{\\partial \\beta_i\n\\partial \\beta_j}\\psi_\\infty^\\beta.\n\\end{equation}\nThe exchange of limits in (\\ref{E}) and (\\ref{Cov}) is nontrivial,\nbut may be justified using similar techniques as in Yang and Lee\n\\cite{YL}. Since homomorphism densities $t(H_i, G_n)$ are\npreserved under vertex relabeling, the probability measure\n$\\tilde{\\mathbb P}_n^\\beta$ and the associated expectation\n$\\tilde{\\mathbb E}_n^\\beta$ (which coincides with $\\mathbb E_n^\\beta$) may\nlikewise be defined.\n\n\\begin{definition}\nA phase is a connected region of the parameter space $\\{\\beta\\}$,\nmaximal for the condition that the limiting normalization constant\n$\\psi_\\infty^\\beta$ is analytic. There is a $j$th-order transition\nat a boundary point of a phase if at least one $j$th-order partial\nderivative of $\\psi_\\infty^\\beta$ is discontinuous there, while\nall lower order derivatives are continuous.\n\\end{definition}\n\nMore generally, we may consider exponential models where the terms\nin the exponent defining the probability measure contain functions\non the graph space other than homomorphism densities. Let $T:\n\\tilde{\\mathcal{W}} \\rightarrow \\mathbb{R}$ be a bounded\ncontinuous function. Let the probability measure $\\mathbb P^T_n$ and the\nnormalization constant $\\psi^T_n$ be defined as in (\\ref{pmf}) and\n(\\ref{psi}), that is,\n\\begin{equation}\n\\label{pmf2}\n\\mathbb P^T_n(G_n)=\\exp\\left(n^2(T(\\tilde{h}^{G_n})-\\psi^T_n)\\right)\\mathbb P_n(G_n),\n\\end{equation}\n\\begin{equation}\n\\label{psi2} \\psi^T_n=\\frac{1}{n^2}\\log \\mathbb E_n \\exp\\left(n^2\nT(\\tilde{h}^{G_n}) \\right),\n\\end{equation}\nThen the probability measure $\\tilde{\\mathbb P}_n^T$ and the associated\nexpectation $\\tilde{\\mathbb E}_n^T$ (which coincides with $\\mathbb E_n^T$) may\nbe defined in a similar manner.\n\nWe will assume that the common distribution $\\mu$ on the edge\nweights has \\textit{finite support}, which implies that the\ngraphon space $\\mathcal{W}$ under consideration is a finite subset\nof $\\mathbb{R}$. These $L^\\infty$ graphons generalize graphons\nthat take values in $[0,1]$ only and are better suited for generic\nedge-weighted graphs instead of just simple graphs. The ``finite\nsupport'' assumption also assures that the moment generating\nfunction $M(\\theta)=\\int e^{\\theta x}\\mu(dx)$ is finite for all\n$\\theta$ and the conjugate rate function of Cram\\'{e}r, $I:\n\\mathbb{R}\\rightarrow \\mathbb{R}$, where\n\\begin{equation}\n\\label{I} I(x)=\\sup_{\\theta\\in \\mathbb{R}}\\left(\\theta x-\\log\nM(\\theta)\\right)\n\\end{equation}\nis nicely defined. The domain of the function $I$ can be extended\nto $\\tilde{\\mathcal{W}}$ in the usual manner:\n\\begin{equation}\n\\label{II} I(\\tilde{h})=\\int_{[0,1]^2}I(h(x,y))dxdy,\n\\end{equation}\nwhere $h$ is any representative element of the equivalence class\n$\\tilde{h}$. It was shown in \\cite{CV} that $I$ is well defined on\n$\\tilde{\\mathcal{W}}$ and is lower semi-continuous under the cut\nmetric $\\delta_\\square$. The space $(\\tilde{\\mathcal{W}},\n\\delta_{\\square})$ enjoys many other important properties that are\nessential for the study of exponential random graph models. It is\na compact space and homomorphism densities $t(H, \\cdot)$ are\ncontinuous functions on it. In fact, homomorphism densities\ncharacterize convergence under the cut metric: a sequence of\ngraphs converges if and only if its homomorphism densities\nconverge for all finite simple graphs, and the limiting\nhomomorphism densities then describe the resulting graphon.\n\n\\section{Statement of results}\n\\label{statement} The normalization constant plays a central role\nin statistical mechanics because it encodes essential information\nabout the structure of the probability measure; even the existence\nof its limit bears important consequences. In the case of\nexponential random graphs, the computational tools currently used\nby practitioners to compute this constant become unreliable for\nlarge networks \\cite{BBS} \\cite{SPRH}. This problem however can be\ncircumvented if we know a priori that the limit of the\nnormalization constant exists. One can then choose a ``scaled\ndown'' network model with a smaller number of vertices and use the\nexact value of the normalization constant in the scaled down model\nas an approximation to the normalization constant in the larger\nmodel, and a computer program that can evaluate the exact value of\nthe normalization constant for moderate sized networks would serve\nthe purpose \\cite{Hunter}. The following Theorem \\ref{main1} is a\ngeneralization of the corresponding result (Theorem 3.1) in\nChatterjee and Diaconis \\cite{CD}, where they assumed that the\nedge weights $x_{ij}$ between vertices $i$ and $j$ are iid real\nrandom variables satisfying a special common distribution --\nBernoulli having values $1$ and $0$ each with probability $1\/2$.\nFor the sake of completeness and to motivate further discussions\nin Theorem \\ref{main2}, we present the proof details below.\n\n\\begin{theorem}\n\\label{main1} Let $T: \\tilde{\\mathcal{W}} \\rightarrow \\mathbb{R}$\nbe a bounded continuous function. Let $\\psi_n^T$ and $I$ be\ndefined as before (see (\\ref{psi2}), (\\ref{I}) and (\\ref{II})).\nThen the limiting normalization constant $\\displaystyle\n\\psi^T_\\infty:=\\lim_{n\\rightarrow \\infty}\\psi_n^T$ exists, and is\ngiven by\n\\begin{equation}\n\\label{setmax} \\psi^T_\\infty=\\sup_{\\tilde{h}\\in\n\\tilde{\\mathcal{W}}}\\left(T(\\tilde{h})-\\frac{1}{2}I(\\tilde{h})\\right).\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nFor each Borel set $\\tilde{A} \\subseteq \\tilde{\\mathcal{W}}$ and\neach $n$, define\n\\begin{equation}\n\\label{n} \\tilde{A}^n=\\{\\tilde{h}\\in \\tilde{A}:\n\\tilde{h}=\\tilde{h}^{G_n} \\text{ for some } G_n\\in\n\\mathcal{G}_n\\}.\n\\end{equation}\nFix $\\epsilon>0$. Since $T$ is a bounded function, there is a\nfinite set $R$ such that the intervals $\\{(c, c +\\epsilon): c\\in\nR\\}$ cover the range of $T$. Since $T$ is a continuous function,\nfor each $c\\in R$, the set $\\tilde{U}_{c}$ consisting of reduced\ngraphons $\\tilde{h}$ with $c<T(\\tilde{h})<c+\\epsilon$ is an open\nset, and the set $\\tilde{H}_{c}$ consisting of reduced graphons\n$\\tilde{h}$ with $c\\leq T(\\tilde{h})\\leq c+\\epsilon$ is a closed\nset. Let $\\tilde{U}_{c}^n$ and $\\tilde{H}_{c}^n$ be respectively\ndefined as in (\\ref{n}). The large deviation principle, Theorem\n1.2 of \\cite{CV}, implies that:\n\\begin{equation}\n\\limsup_{n\\rightarrow \\infty}\\frac{\\log\n\\tilde{\\mathbb Q}_n(\\tilde{H}_{c}^n)}{n^2}\\leq -\\inf_{\\tilde{h}\\in\n\\tilde{H}_{c}}\\frac{1}{2}I(\\tilde{h}),\n\\end{equation}\nand that\n\\begin{equation}\n\\liminf_{n\\rightarrow \\infty}\\frac{\\log\n\\tilde{\\mathbb Q}_n(\\tilde{U}_{c}^n)}{n^2}\\geq -\\inf_{\\tilde{h}\\in\n\\tilde{U}_{c}}\\frac{1}{2}I(\\tilde{h}),\n\\end{equation}\nwhere $\\tilde{Q}_n$ denotes the family of probability measures on\n$\\tilde{\\mathcal{W}}$ that is inherited from the corresponding\nprobability measures $\\tilde{P}_n$ on $\\mathcal{G}_n$.\n\nWe first consider $\\limsup \\psi_n^T$.\n\\begin{equation}\n\\exp(n^2\\psi_n^T)\\leq \\sum_{c\\in\nR}\\exp(n^2(c+\\epsilon))\\tilde{\\mathbb Q}_n(\\tilde{H}_{c}^n)\\leq\n|R|\\sup_{c\\in\nR}\\exp(n^2(c+\\epsilon))\\tilde{\\mathbb Q}_n(\\tilde{H}_{c}^n).\n\\end{equation}\nThis shows that\n\\begin{equation}\\label{supce}\n\\limsup_{n\\rightarrow \\infty}\\psi^{T}_{n}\\leq \\sup_{c\\in\nR}\\left(c+\\epsilon-\\inf_{\\tilde{h}\\in\n\\tilde{H}_{c}}\\frac{1}{2}I(\\tilde{h})\\right).\n\\end{equation}\nEach $\\tilde{h}\\in \\tilde{H}_{c}$ satisfies $T(\\tilde{h})\\geq c$.\nConsequently,\n\\begin{equation}\n\\sup_{\\tilde{h}\\in\n\\tilde{H}_{c}}(T(\\tilde{h})-\\frac{1}{2}I(\\tilde{h}))\\geq\n\\sup_{\\tilde{h}\\in\n\\tilde{H}_{c}}(c-\\frac{1}{2}I(\\tilde{h}))=c-\\inf_{\\tilde{h}\\in\n\\tilde{H}_{c}}\\frac{1}{2}I(\\tilde{h}).\n\\end{equation}\nSubstituting this in (\\ref{supce}) gives\n\\begin{eqnarray}\n\\limsup_{n\\rightarrow \\infty}\\psi^{T}_{n}&\\leq&\n\\epsilon+\\sup_{c\\in R}\\sup_{\\tilde{h}\\in\n\\tilde{H}_{c}}(T(\\tilde{h})-\\frac{1}{2}I(\\tilde{h}))\\\\\\notag&=&\\epsilon+\\sup_{\\tilde{h}\\in\n\\tilde{\\mathcal{W}}}(T(\\tilde{h})-\\frac{1}{2}I(\\tilde{h})).\n\\end{eqnarray}\n\nNext we consider $\\liminf \\psi^{T}_{n}$.\n\\begin{equation}\n\\exp(n^2\\psi^{T}_{n})\\geq \\sup_{c\\in R}\n\\exp(n^2c)\\tilde{\\mathbb Q}_n(\\tilde{U}_{c}^n).\n\\end{equation}\nTherefore for each $c\\in R$,\n\\begin{equation}\\label{infc}\n\\liminf_{n \\rightarrow \\infty}\\psi^{T}_{n}\\geq\nc-\\inf_{\\tilde{h}\\in \\tilde{U}_{c}}\\frac{1}{2}I(\\tilde{h}).\n\\end{equation}\nEach $\\tilde{h}\\in \\tilde{U}_{c}$ satisfies\n$T(\\tilde{h})<c+\\epsilon$. Therefore\n\\begin{equation}\n\\sup_{\\tilde{h}\\in\n\\tilde{U}_{c}}(T(\\tilde{h})-\\frac{1}{2}I(\\tilde{h}))\\leq\n\\sup_{\\tilde{h}\\in\n\\tilde{U}_{c}}(c+\\epsilon-\\frac{1}{2}I(\\tilde{h}))=c+\\epsilon-\\inf_{\\tilde{h}\\in\n\\tilde{U}_{c}}\\frac{1}{2}I(\\tilde{h}).\n\\end{equation}\nTogether with (\\ref{infc}), this shows that\n\\begin{eqnarray}\n\\liminf_{n \\rightarrow \\infty}\\psi^{T}_{n}&\\geq&\n-\\epsilon+\\sup_{c\\in R}\\sup_{\\tilde{h}\\in\n\\tilde{U}_{c}}(T(\\tilde{h})-\\frac{1}{2}I(\\tilde{h}))\\\\\\notag&=&-\\epsilon+\\sup_{\\tilde{h}\\in\n\\tilde{\\mathcal{W}}}(T(\\tilde{h})-\\frac{1}{2}I(\\tilde{h})).\n\\end{eqnarray}\n\nSince $\\epsilon$ is arbitrary, this completes the proof.\n\\end{proof}\n\nWhen $T$ is not continuous, we lose control of the openness and\nclosedness of the sets in $\\tilde{\\mathcal{W}}$, and hence can not\nresort to the large deviation principle of \\cite{CV} to find an\nexplicit characterization of the limiting normalization constant\n$\\psi^T_\\infty$, but we can still conclude that this important\nlimit exists under mild conditions.\n\n\\begin{theorem}\n\\label{main2} Let $T: \\tilde{\\mathcal{W}} \\rightarrow \\mathbb{R}$\nbe a bounded function. Let $I$ and $\\psi_n^T$ be defined as before\n(see (\\ref{psi2}), (\\ref{I}) and (\\ref{II})). Then the limiting\nnormalization constant $\\psi^T_\\infty$ exists.\n\\end{theorem}\n\n\\begin{proof}\nSince $T$ is a bounded function, there is a finite set $R$ such\nthat the intervals $\\{(c, c +\\epsilon): c\\in R\\}$ cover the range\nof $T$. Similarly as in the proof of Theorem \\ref{main1}, we have\n\\begin{equation}\n\\exp(n^2\\psi_n^T)\\leq |R|\\sup_{c\\in\nR}\\exp(n^2(c+\\epsilon))\\tilde{\\mathbb Q}_n(\\tilde{U}_{c}^n),\n\\end{equation}\nand\n\\begin{equation}\n\\exp(n^2\\psi^{T}_{n})\\geq \\sup_{c\\in R}\n\\exp(n^2c)\\tilde{\\mathbb Q}_n(\\tilde{U}_{c}^n).\n\\end{equation}\nThis implies that\n\\begin{equation}\n\\limsup_{n\\rightarrow \\infty}\\psi^T_n-\\liminf_{n\\rightarrow\n\\infty}\\psi^T_n\\leq \\frac{\\log |R|}{n^2}+\\epsilon.\n\\end{equation}\nSince $\\epsilon$ is arbitrary, this completes the proof.\n\\end{proof}\n\n\\begin{remark}\nThe boundedness condition on $T$ can be relaxed even further.\n$(\\log |R|)\/(n^2) \\rightarrow 0$ is sufficient.\n\\end{remark}\n\nLet $\\tilde{H}$ be the subset of $\\tilde{\\mathcal{W}}$ where\n$\\psi_\\infty^T$ is maximized. By the compactness of\n$\\tilde{\\mathcal{W}}$, the continuity of $T$ and the lower\nsemi-continuity of $I$, $\\tilde{H}$ is a nonempty compact set. The\nset $\\tilde{H}$ encodes important information about the\nexponential model (\\ref{pmf2}) and helps to predict the behavior\nof a typical random graph sampled from this model. The following\nTheorem \\ref{main3} states that in the large $n$ limit, the\nquotient image $\\tilde{h}^{G_n}$ of a random graph $G_n$ drawn\nfrom (\\ref{pmf2}) must lie close to $\\tilde{H}$ with probability\n$1$. Especially, if $\\tilde{H}$ is a singleton set, the theorem\ngives a law of large numbers for $G_n$; while if $\\tilde{H}$ is\nnot a singleton set, the theorem points to the existence of a\nfirst order phase transition.\n\n\\begin{theorem}\n\\label{main3} Let $\\tilde{H}$ be defined as in the above\nparagraph. Let $\\mathbb P_n^T$ (\\ref{pmf2}) be the probability measure\non $\\mathcal{G}_n$. Then\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty}\\delta_\\square(\\tilde{h}^{G_n},\n\\tilde{H})=0 \\text{ almost surely}.\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nThe proof of this theorem follows a similar line of reasoning as\nin the proof of the corresponding result (Theorem 3.2) in\nChatterjee and Diaconis \\cite{CD}. The key observation is that the\ndistance between the graph and the maximizing set decays\nexponentially as the number of vertices of the graph grows. For\nany $\\eta>0$ there exist $C, \\gamma>0$ such that for all $n$ large\nenough,\n\\begin{equation}\n\\mathbb P_n^T(\\delta_\\square(\\tilde{h}^{G_n}, \\tilde{H})>\\eta)\\leq\nCe^{-n^2\\gamma}.\n\\end{equation}\nThis is stronger than ordinary convergence in probability, which\nonly shows that the probability decays to zero but does not give\nthe speed. Utilizing the fast exponential decay rate, we can then\nresort to probability estimates in the product space. By\nBorel-Cantelli,\n\\begin{equation}\n\\sum_{n=1}^\\infty \\mathbb P_n^T(\\delta_\\square(\\tilde{h}^{G_n},\n\\tilde{H})>\\eta)<\\infty \\text{ implies that }\n\\mathbb P_n^T(\\delta_\\square(\\tilde{h}^{G_n}, \\tilde{H})>\\eta \\text{\ninfinitely often})=0.\n\\end{equation}\nThe conclusion hence follows.\n\\end{proof}\n\nWhen the bounded continuous function $T$ on $\\tilde{\\mathcal{W}}$\nis given by the sum of graph homomorphism densities\n$T=\\sum_{i=1}^k \\beta_i t(H_i, \\cdot)$, the statement of Theorems\n\\ref{main1} and \\ref{main3} may be simplified in the\n``attractive'' region of the parameter space where the parameters\n$\\beta_2,...,\\beta_k$ are all non-negative, as seen in the\nfollowing Theorems \\ref{main4} and \\ref{gen}.\n\n\\begin{theorem}\n\\label{main4} Consider a general $k$-parameter exponential random\ngraph model (\\ref{pmf}). Suppose $\\beta_2,...,\\beta_k$ are\nnon-negative. Then the limiting normalization constant\n$\\displaystyle \\psi_\\infty^\\beta$ exists, and is given by\n\\begin{equation}\n\\label{lmax} \\psi_{\\infty}^{\\beta}=\\sup_{u}\\left(\\beta_1\nu^{e(H_1)}+\\cdots+\\beta_k u^{e(H_k)}-\\frac{1}{2}I(u)\\right)\\\\,\n\\end{equation}\nwhere $e(H_i)$ is the number of edges in $H_i$, $I$ is the\nCram\\'{e}r function (\\ref{I}) (\\ref{II}), and the supremum is\ntaken over all $u$ in the domain of $I$, i.e., where $I<\\infty$.\n\\end{theorem}\n\n\\begin{proof}\nThe proof of this theorem follows a similar line of reasoning as\nin the proof of the corresponding result (Theorem 4.1) in\nChatterjee and Diaconis \\cite{CD}, where the exact form of $I$ for\nthe Bernoulli distribution was given. The crucial step is\nrecognizing that $I$ is convex. So though the exact form of $I$ is\nnot always obtainable for a generic common distribution $\\mu$, the\nlog of the moment generating function $M(\\theta)$ is convex and\n$I$, being the Legendre transform of a convex function, must also\nbe convex.\n\\end{proof}\n\n\\begin{theorem}\n\\label{gen} Let $G_n$ be an exponential random graph drawn from\n(\\ref{pmf}). Suppose $\\beta_2,...,\\beta_k$ are non-negative. Then\n$G_n$ behaves like an Erd\\H{o}s-R\\'{e}nyi graph $G(n, u^*)$ in the\nlarge n limit, where $u^*$ is picked randomly from the set $U$ of\nmaximizers of (\\ref{lmax}).\n\\end{theorem}\n\n\\begin{proof}\nThe assertions in this theorem are direct consequences of Theorems\n\\ref{main3} and \\ref{main4}.\n\\end{proof}\n\n\\section{An application: Uniformly distributed edge weights}\n\\label{app} Consider the set $\\mathcal{G}_n$ of all edge-weighted\nundirected labeled graphs on $n$ vertices, where the edge weights\n$x_{ij}$ between vertex $i$ and vertex $j$ are iid real random\nvariables uniformly distributed on $(0,1)$. The common\ndistribution for the edge weights yields probability measure\n$\\mathbb P_n$ and the associated expectation $\\mathbb E_n$ on $\\mathcal{G}_n$.\nGive the set of such graphs the probability\n\\begin{equation}\n\\mathbb P_n^\\beta(G_n)=\\exp\\left(n^2\\left(\\beta_1 t(H_1, G_n)+\\beta_2\nt(H_2, G_n)-\\psi_n^\\beta\\right)\\right)\\mathbb P_n(G_n),\n\\end{equation}\nwhere $\\beta=(\\beta_1, \\beta_2)$ are $2$ real parameters, $H_1$ is\na single edge, $H_2$ is a finite simple graph with $p\\geq 2$\nedges, and $\\psi_n^\\beta$ is the normalization constant,\n\\begin{equation}\n\\label{dpsi} \\psi_n^\\beta=\\frac{1}{n^2}\\log\n\\mathbb E_n\\left(\\exp\\left(n^2\\left(\\beta_1 t(H_1, G_n)+\\beta_2 t(H_2,\nG_n)\\right)\\right)\\right).\n\\end{equation}\nThe associated expectation $\\mathbb E_n^\\beta$ may be defined\naccordingly, and a phase transition occurs when the limiting\nnormalization constant $\\displaystyle \\psi^\\beta_\\infty=\\lim_{n\\to\n  \\infty}\\psi_n^{\\beta}$ has a singular point.\n\n\\begin{theorem}\n\\label{phase} For any allowed $H_2$, the limiting normalization\nconstant $\\psi_\\infty^\\beta$ is analytic at all $(\\beta_1,\n\\beta_2)$ in the upper half-plane $(\\beta_2\\geq 0)$ except on a\ncertain decreasing curve $\\beta_2=r(\\beta_1)$ which includes the\nendpoint $(\\beta_1^c, \\beta_2^c)$. The derivatives\n$\\frac{\\partial}{\\partial \\beta_1}\\psi_\\infty^\\beta$ and\n$\\frac{\\partial}{\\partial \\beta_2}\\psi_\\infty^\\beta$ have (jump)\ndiscontinuities across the curve, except at the end point where,\nhowever, all the second derivatives $\\frac{\\partial^2}{\\partial\n\\beta_1^2}\\psi_\\infty^\\beta$, $\\frac{\\partial^2}{\\partial \\beta_1\n\\partial \\beta_2}\\psi_\\infty^\\beta$ and $\\frac{\\partial^2}{\\partial\n\\beta_2^2}\\psi_\\infty^\\beta$ diverge.\n\\end{theorem}\n\n\\begin{corollary}\nFor any allowed $H_2$, the parameter space $\\{(\\beta_1, \\beta_2):\n\\beta_2\\geq 0\\}$ consists of a single phase with a first order\nphase transition across the indicated curve $\\beta_2=r(\\beta_1)$\nand a second order phase transition at the critical point\n$(\\beta_1^c, \\beta_2^c)$.\n\\end{corollary}\n\n\\begin{figure}\n\\centering\n\\includegraphics[clip=true, height=3.5in]{curve.pdf}\n\\caption{The V-shaped region (with phase transition curve\n$r(\\beta_1)$ inside) in the $(\\beta_1, \\beta_2)$ plane. Graph\ndrawn for $p=2$.} \\label{Vshape}\n\\end{figure}\n\n\\begin{proof}\nThe moment generating function for the uniform $(0,1)$\ndistribution is given by $M(\\theta)=(e^\\theta-1)\/\\theta$. The\nassociated Cram\\'{e}r function $I$ is finite on $(0,1)$,\n\\begin{equation}\n\\label{Iu} I(u)=\\sup_{\\theta\\in \\mathbb{R}}\\left(\\theta u-\\log\n\\frac{e^\\theta-1}{\\theta}\\right),\n\\end{equation}\nbut does not admit a closed-form expression. As shown in Theorem\n\\ref{main4}, the limiting normalization constant\n$\\psi_\\infty^\\beta$ exists and is given by\n\\begin{equation}\n\\label{smax} \\psi_\\infty^\\beta=\\sup_{0\\leq u \\leq 1}\n\\left(\\beta_1u+\\beta_2u^p-\\frac{1}{2}I(u)\\right).\n\\end{equation}\nA significant part of computing phase boundaries for the\n$2$-parameter exponential model is then a detailed analysis of the\ncalculus problem (\\ref{smax}). However, as straightforward as it\nsounds, since the exact form of $I$ is not obtainable, getting a\nclear picture of the asymptotic phase structure is not that easy\nand various tricks need to be employed.\n\nConsider the maximization problem for $l(u; \\beta_1,\n\\beta_2)=\\beta_1u+\\beta_2u^p-\\frac{1}{2}I(u)$ on the interval $[0,\n1]$, where $-\\infty<\\beta_1<\\infty$ and $0\\leq \\beta_2<\\infty$ are\nparameters. The location of maximizers of $l(u)$ on the interval\n$[0, 1]$ is closely related to properties of its derivatives\n$l'(u)$ and $l''(u)$,\n\\begin{equation}\nl'(u)=\\beta_1+p\\beta_2u^{p-1}-\\frac{1}{2}I'(u),\n\\end{equation}\n\\begin{equation*}\nl''(u)=p(p-1)\\beta_2u^{p-2}-\\frac{1}{2}I''(u).\n\\end{equation*}\n\n\\begin{figure}\n\\centering\n\\includegraphics[clip=true, width=6in, height=2.5in]{below.pdf}\n\\caption{Outside the V-shaped region, $l(u)$ has a unique local\nmaximizer (hence global maximizer) $u^*$. Graph drawn for\n$\\beta_1=-5$, $\\beta_2=3.5$, and $p=2$.} \\label{below}\n\\end{figure}\n\nFollowing the duality principle for the Legendre transform between\n$I(u)$ and $\\log M(\\theta)$ \\cite{ZRM}, we first analyze\nproperties of $l''(u)$ on the interval $[0, 1]$. Recall that\n\\begin{equation}\nI(u)+\\log \\frac{e^\\theta-1}{\\theta}=\\theta u,\n\\end{equation}\nwhere $\\theta$ and $u$ are implicitly related. Taking derivatives,\nwe have\n\\begin{equation}\n\\label{dual} u=\\left.\\left(\\log\n\\frac{e^\\theta-1}{\\theta}\\right)\\right|_\\theta',\n\\end{equation}\n\\begin{equation*}\nI''(u)\\cdot \\left.\\left(\\log\n\\frac{e^\\theta-1}{\\theta}\\right)\\right|_\\theta''=1.\n\\end{equation*}\nConsider the function\n\\begin{equation}\nm(u)=\\frac{I''(u)}{2p(p-1)u^{p-2}}\n\\end{equation}\non $[0, 1]$. By (\\ref{dual}), analyzing properties of $m(u)$\ntranslates to analyzing properties of the function\n\\begin{eqnarray}\nn(\\theta)&=&2p(p-1)\\left.\\left(\\log\n\\frac{e^\\theta-1}{\\theta}\\right)\\right|_\\theta''\\cdot\n\\left(\\left.\\left(\\log\n\\frac{e^\\theta-1}{\\theta}\\right)\\right|_\\theta'\\right)^{p-2}\\\\\\notag\n&=&2p(p-1)\\left(\\frac{e^{2\\theta}-e^\\theta(\\theta^2+2)+1}{(e^\\theta-1)^2\\theta^2}\\right)\\left(\\frac{e^\\theta(\\theta-1)+1}{(e^\\theta-1)\\theta}\\right)^{p-2}\n\\end{eqnarray}\nover $\\mathbb{R}$, where $m(u)n(\\theta)=1$ and $u$ and $\\theta$\nsatisfy the dual relationship. We recognize that\n\\begin{equation}\n\\lim_{\\theta\\rightarrow -\\infty}n(\\theta)=0,\n\\end{equation}\n\\begin{equation*}\n\\lim_{\\theta\\rightarrow\n0}n(\\theta)=\\frac{p(p-1)}{3}\\left(\\frac{1}{2}\\right)^{p-1},\n\\end{equation*}\n\\begin{equation*}\n\\lim_{\\theta\\rightarrow \\infty}n(\\theta)=0,\n\\end{equation*}\nwhich implies that $n(\\theta)$ achieves a finite global maximum.\nWe claim that the global maximum can only be attained at\n$\\theta_0\\geq 0$. This would further imply that there is a finite\nglobal minimum for $m(u)$, and it can only be attained at $u_0\\geq\n\\frac{1}{2}$. First suppose $p=2$. Then $n'(\\theta_0)=0$ easily\nimplies that $\\theta_0=0$. Now suppose $p>2$. Then since\n$n'(\\theta_0)=0$, we have\n\\begin{equation}\n\\label{compare}\n\\left.\\left(\\frac{e^{2\\theta}-e^\\theta(\\theta^2+2)+1}{(e^\\theta-1)^2\\theta^2}\\right)\\right|_{\\theta_0}'\\left.\\left(\\frac{e^\\theta(\\theta-1)+1}{(e^\\theta-1)\\theta}\\right)\\right|_{\\theta_0}\n=-(p-2)\\left.\\left(\\frac{e^{2\\theta}-e^\\theta(\\theta^2+2)+1}{(e^\\theta-1)^2\\theta^2}\\right)^2\\right|_{\\theta_0}<0.\n\\end{equation}\nThis says that $\\theta_0$ is on the portion of $\\mathbb{R}$ where\n$\\left(e^{2\\theta}-e^\\theta(\\theta^2+2)+1\\right)\/\\left((e^\\theta-1)^2\\theta^2\\right)$\nis decreasing, i.e., $\\theta_0>0$. As $\\theta$ goes from $0$ to\n$\\infty$, the left hand side of (\\ref{compare}) first decreases\nfrom $0$ then increases to $0$ and is always negative, whereas the\nright hand side of (\\ref{compare}) monotonically increases from\n$-(p-2)\/144$ to $0$ and approaches $0$ at a rate faster than the\nleft hand side, and so there exists a $\\theta_0$ which makes both\nsides equal. Our numerical computations show that $n'(\\theta_0)=0$\nis uniquely defined, and $n(\\theta)$ increases from $-\\infty$ to\n$\\theta_0$ and decreases from $\\theta_0$ to $\\infty$. See Table\n\\ref{table1} for some of these critical values.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ccccc}\n$p$ & $\\theta_0$ & $n(\\theta_0)$ & $u_0$ & $m(u_0)$ \\\\\n\\hline \\hline \\\\\n$2$ & $0$ & $0.3333$ & $0.5$ & $3$ \\\\\n$3$ & $1.3251$ & $0.5575$ & $0.6073$ & $1.7937$ \\\\\n$5$ & $2.9869$ & $0.8324$ & $0.7183$ & $1.2014$ \\\\\n$10$ & $5.6256$ & $1.0894$ & $0.8259$ & $0.9180$ \\\\ \\\\\n\\end{tabular}\n\\end{center}\n\\caption{Critical values for $m(u)$ and $n(\\theta)$ as a function\nof $p$.} \\label{table1}\n\\end{table}\n\nThe above analysis shows that for $\\beta_2\\leq m(u_0)$,\n$l''(u)\\leq 0$ over the entire interval $[0, 1]$; while for\n$\\beta_2>m(u_0)$, $l''(u)$ takes on both positive and negative\nvalues, and we denote the transition points by $u_1$ and $u_2$\n($u_1<u_0<u_2$). Properties of $l''(u)$ on $[0, 1]$ entails\nproperties of $l'(u)$ over the same interval. For $\\beta_2 \\leq\nm(u_0)$, $l'(u)$ is monotonically decreasing. For\n$\\beta_2>m(u_0)$, $l'(u)$ is decreasing from $0$ to $u_1$,\nincreasing from $u_1$ to $u_2$, and then decreasing again from\n$u_2$ to $1$.\n\nThe analytic properties of $l''(u)$ and $l'(u)$ help us\ninvestigate properties of $l(u)$ on the interval $[0, 1]$. Being\nthe Legendre transform of a smooth function, $I(u)$ is a smooth\nfunction, and grows unbounded when $u$ approaches $0$ or $1$. By\nthe duality principle for the Legendre transform, $I'(u)=\\theta$\nwhere $\\theta$ and $u$ are linked through (\\ref{dual}), so\n$I'(0)=-\\infty$ and $I'(1)=\\infty$. This says that $l(u)$ is a\nsmooth function, $l(0)=l(1)=-\\infty$, $l'(0)=\\infty$ and\n$l'(1)=-\\infty$, so $l(u)$ can not be maximized at $0$ or $1$. For\n$\\beta_2\\leq m(u_0)$, $l'(u)$ crosses the $u$-axis only once,\ngoing from positive to negative. Thus $l(u)$ has a unique local\nmaximizer (hence global maximizer) $u^*$. For $\\beta_2>m(u_0)$,\nthe situation is more complicated. If $l'(u_1)\\geq 0$ (resp.\n$l'(u_2)\\leq 0$), $l(u)$ has a unique local maximizer (hence\nglobal maximizer) at a point $u^*>u_2$ (resp. $u^*<u_1$). See\nFigures \\ref{below} and \\ref{above}. If $l'(u_1)<0<l'(u_2)$, then\n$l(u)$ has two local maximizers $u_1^*$ and $u_2^*$, with\n$u_1^*<u_1<u_0<u_2<u_2^*$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[clip=true, width=6in, height=2.5in]{above.pdf}\n\\caption{Outside the V-shaped region, $l(u)$ has a unique local\nmaximizer (hence global maximizer) $u^*$. Graph drawn for\n$\\beta_1=-2.5$, $\\beta_2=4$, and $p=2$.} \\label{above}\n\\end{figure}\n\nLet\n\\begin{equation}\nf(u)=\\frac{uI''(u)}{2(p-1)}-\\frac{1}{2}I'(u)\n\\end{equation}\nso that $l'(u_1)=\\beta_1+f(u_1)$ and $l'(u_2)=\\beta_1+f(u_2)$.\nAgain by the duality principle for the Legendre transform,\nanalyzing properties of the function $f(u)$ on $[0, 1]$ translates\nto analyzing properties of the function\n\\begin{equation}\ng(\\theta)=\\frac{\\theta\\left(e^\\theta-1\\right)\\left(e^\\theta(\\theta-1)+1\\right)}{2(p-1)\\left(e^{2\\theta}-e^\\theta(\\theta^2+2)+1\\right)}-\\frac{1}{2}\\theta\n\\end{equation}\nover $\\mathbb{R}$, where $f(u)=g(\\theta)$ and $u$ and $\\theta$\nsatisfy the dual relationship. We recognize that\n\\begin{equation}\n\\lim_{\\theta\\rightarrow -\\infty}g(\\theta)=\\infty,\n\\end{equation}\n\\begin{equation*}\n\\lim_{\\theta\\rightarrow 0}g(\\theta)=\\frac{3}{p-1},\n\\end{equation*}\n\\begin{equation*}\n\\lim_{\\theta\\rightarrow \\infty}g(\\theta)=\\infty,\n\\end{equation*}\nwhich implies that $g(\\theta)$ achieves a finite global minimum.\nWe claim that the global minimum can only be attained at the same\nunique $\\theta_0$ that makes $n'(\\theta)=0$. This can be checked\nby identifying the condition for $g'(\\theta)=0$. For $p=2$,\n$g'(\\theta_0)=0$ easily implies that $\\theta_0=0$. For $p>2$,\nsince $g'(\\theta_0)=0$, we have\n\\begin{multline}\n\\left.\\left(\\frac{e^\\theta(\\theta-1)+1}{(e^\\theta-1)\\theta}\\right)\\right|_{\\theta_0}'\\left.\\left(\\frac{e^{2\\theta}-e^\\theta(\\theta^2+2)+1}{(e^\\theta-1)^2\\theta^2}\\right)\\right|_{\\theta_0}-\n\\left.\\left(\\frac{e^{2\\theta}-e^\\theta(\\theta^2+2)+1}{(e^\\theta-1)^2\\theta^2}\\right)\\right|_{\\theta_0}'\\left.\\left(\\frac{e^\\theta(\\theta-1)+1}{(e^\\theta-1)\\theta}\\right)\\right|_{\\theta_0}\n\\\\=(p-1)\\left.\\left(\\frac{e^{2\\theta}-e^\\theta(\\theta^2+2)+1}{(e^\\theta-1)^2\\theta^2}\\right)^2\\right|_{\\theta_0},\n\\end{multline}\nwhich yields the same solution as (\\ref{compare}) after a simple\ntransformation. Thus $g(\\theta)$ decreases from $-\\infty$ to\n$\\theta_0$ and increases from $\\theta_0$ to $\\infty$. This further\nimplies that there is a finite global minimum for $f(u)$ attained\nat $u_0$, and $f(u)$ decreases from $0$ to $u_0$ and increases\nfrom $u_0$ to $1$. Both $\\theta_0$ and $u_0$ coincide with the\ncritical values listed in Table \\ref{table1}. See Table\n\\ref{table2}. We conclude that $l'(u_1)\\geq 0$ for $\\beta_1\\geq\n-f(u_0)$ and $\\beta_2=m(u_0)$. The only possible region in the\n$(\\beta_1, \\beta_2)$ plane where $l'(u_1)<0<l'(u_2)$ is bounded by\n$\\beta_1<-f(u_0):=\\beta_1^c$ and $\\beta_2>m(u_0):=\\beta_2^c$.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ccccc}\n$p$ & $\\theta_0$ & $g(\\theta_0)$ & $u_0$ & $f(u_0)$ \\\\\n\\hline \\hline \\\\\n$2$ & $0$ & $3$ & $0.5$ & $3$ \\\\\n$3$ & $1.3251$ & $1.3222$ & $0.6073$ & $1.3222$ \\\\\n$5$ & $2.9869$ & $0.1059$ & $0.7183$ & $0.1059$ \\\\\n$10$ & $5.6256$ & $-1.1723$ & $0.8259$ & $-1.1723$ \\\\ \\\\\n\\end{tabular}\n\\end{center}\n\\caption{Critical values for $f(u)$ and $g(\\theta)$ as a function\nof $p$.} \\label{table2}\n\\end{table}\n\nWe examine the behavior of $l'(u_1)$ and $l'(u_2)$ more closely\nwhen $\\beta_1$ and $\\beta_2$ are chosen from this region. Recall\nthat $u_1<u_0<u_2$. By monotonicity of $f(u)$ on the intervals\n$(0, u_0)$ and $(u_0, 1)$, there exist continuous functions\n$a(\\beta_1)$ and $b(\\beta_1)$ of $\\beta_1$, such that $l'(u_1)<0$\nfor $u_1>a(\\beta_1)$ and $l'(u_2)>0$ for $u_2>b(\\beta_1)$. As\n$\\beta_1\\rightarrow -\\infty$, $a(\\beta_1)\\rightarrow 0$ and\n$b(\\beta_1)\\rightarrow 1$. $a(\\beta_1)$ is an increasing function\nof $\\beta_1$, whereas $b(\\beta_1)$ is a decreasing function, and\nthey satisfy $f(a(\\beta_1))=f(b(\\beta_1))=-\\beta_1$. The\nrestrictions on $u_1$ and $u_2$ yield restrictions on $\\beta_2$,\nand we have $l'(u_1)<0$ for $\\beta_2<m(a(\\beta_1))$ and\n$l'(u_2)>0$ for $\\beta_2>m(b(\\beta_1))$. As $\\beta_1\\rightarrow\n-\\infty$, $m(a(\\beta_1))\\rightarrow \\infty$ and\n$m(b(\\beta_1))\\rightarrow \\infty$. $m(a(\\beta_1))$ and\n$m(b(\\beta_1))$ are both decreasing functions of $\\beta_1$, and\nthey satisfy $l'(u_1)=0$ when $\\beta_2=m(a(\\beta_1))$ and\n$l'(u_2)=0$ when $\\beta_2=m(b(\\beta_1))$. As $l'(u_2)>l'(u_1)$ for\nevery $(\\beta_1, \\beta_2)$, the curve $m(b(\\beta_1))$ must lie\nbelow the curve $m(a(\\beta_1))$, and together they generate the\nbounding curves of the $V$-shaped region in the $(\\beta_1,\n\\beta_2)$ plane with corner point $(\\beta_1^c, \\beta_2^c)$ where\ntwo local maximizers exist for $l(u)$. See Figures \\ref{Vshape},\n\\ref{lower} and \\ref{upper}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[clip=true, width=6in, height=2.5in]{lower.pdf}\n\\caption{Along the lower bounding curve $m(b(\\beta_1))$ of the\nV-shaped region, $l'(u)$ has two zeros $u_1^*$ and $u_2^*$, but\nonly $u_1^*$ is the global maximizer for $l(u)$. Graph drawn for\n$\\beta_1=-5.7$, $\\beta_2=5$, and $p=2$.} \\label{lower}\n\\end{figure}\n\nFix an arbitrary $\\beta_1<\\beta_1^c$, we examine the effect of\nvarying $\\beta_2$ on the graph of $l'(u)$. It is clear that\n$l'(u)$ shifts upward as $\\beta_2$ increases and downward as\n$\\beta_2$ decreases. As a result, as $\\beta_2$ gets large, the\npositive area bounded by the curve $l'(u)$ increases, whereas the\nnegative area decreases. By the fundamental theorem of calculus,\nthe difference between the positive and negative areas is the\ndifference between $l(u_2^*)$ and $l(u_1^*)$, which goes from\nnegative ($l'(u_2)=0$, $u_1^*$ is the global maximizer) to\npositive ($l'(u_1)=0$, $u_2^*$ is the global maximizer) as\n$\\beta_2$ goes from $m(b(\\beta_1))$ to $m(a(\\beta_1))$. Thus there\nmust be a unique $\\beta_2$: $m(b(\\beta_1))<\\beta_2<m(a(\\beta_1))$\nsuch that $u_1^*$ and $u_2^*$ are both global maximizers, and we\ndenote this $\\beta_2$ by $r(\\beta_1)$. See Figures \\ref{Vshape}\nand \\ref{along}. The parameter values of $(\\beta_1, r(\\beta_1))$\nare exactly the ones for which positive and negative areas bounded\nby $l'(u)$ equal each other. An increase in $\\beta_1$ induces an\nupward shift of $l'(u)$, and may be balanced by a decrease in\n$\\beta_2$. Similarly, a decrease in $\\beta_1$ induces a downward\nshift of $l'(u)$, and may be balanced by an increase in $\\beta_2$.\nThis justifies that $r(\\beta_1)$ is monotonically decreasing in\n$\\beta_1$.\n\nThe rest of the proof follows as in the proof of the corresponding\nresult (Theorem 2.1) in Radin and Yin \\cite{RY}, where some\nprobability estimates were used. A (jump) discontinuity in the\nfirst derivatives of $\\psi_\\infty^\\beta$ across the curve\n$\\beta_2=r(\\beta_1)$ indicates a discontinuity in the expected\nlocal densities (\\ref{E}), while the divergence of the second\nderivatives of $\\psi_\\infty^\\beta$ at the critical point\n$(\\beta_1^c, \\beta_2^c)$ implies that the covariances of the local\ndensities go to zero more slowly than $1\/n^2$ (\\ref{Cov}). We omit\nthe proof details.\n\\end{proof}\n\n\\begin{remark}\nThe maximization problem (\\ref{smax}) is solved at a unique value\n$u^*$ off the phase transition curve $\\beta_2=r(\\beta_1)$, and at\ntwo values $u_1^*$ and $u_2^*$ along the curve. As\n$\\beta_1\\rightarrow -\\infty$ (resp. $\\beta_2\\rightarrow \\infty$),\n$u_1^*\\rightarrow 0$ and $u_2^*\\rightarrow 1$. The jump from\n$u_1^*$ to $u_2^*$ is quite noticeable even for small parameter\nvalues of $\\beta$. For example, taking $p=2$, $\\beta_1=-5$, and\n$\\beta_2=5$, numerical computations yield that $u_1^* \\approx\n0.137$ and $u_2^* \\approx 0.863$ with $l(u_1^*)\\approx l(u_2^*)\n\\approx -1.0854$.\n\\end{remark}\n\n\\begin{figure}\n\\centering\n\\includegraphics[clip=true, width=6in, height=2.5in]{upper.pdf}\n\\caption{Along the upper bounding curve $m(a(\\beta_1))$ of the\nV-shaped region, $l'(u)$ has two zeros $u_1^*$ and $u_2^*$, but\nonly $u_2^*$ is the global maximizer for $l(u)$. Graph drawn for\n$\\beta_1=-4.3$, $\\beta_2=5$, and $p=2$.} \\label{upper}\n\\end{figure}\n\n\\begin{proposition}\nThe Cram\\'{e}r function $I(u)$ (\\ref{Iu}) for the uniform\ndistribution on $(0, 1)$ is symmetric about the line\n$u=\\frac{1}{2}$.\n\\end{proposition}\n\n\\begin{proof}\nRecall that $I(u)$ is the Legendre transform of the log moment\ngenerating function $\\log M(\\theta)$ for the uniform $(0, 1)$\ndistribution. Take $0<u<\\frac{1}{2}$. Though the exact form of $I$\nis not obtainable, following the duality principle, we have\n\\begin{equation}\nI(u)=\\theta_1 u-\\log\\frac{e^{\\theta_1}-1}{\\theta_1},\n\\end{equation}\n\\begin{equation*}\nI(1-u)=\\theta_2 (1-u)-\\log\\frac{e^{\\theta_2}-1}{\\theta_2},\n\\end{equation*}\nwhere $\\theta_1$ and $\\theta_2$ are both finite and uniquely\ndefined, and satisfy $I'(u)=\\theta_1$ and $I'(1-u)=\\theta_2$. We\nshow that $I(u)=I(1-u)$ when $\\theta_2=-\\theta_1$. The calculation\nis straightforward:\n\\begin{equation}\nI(1-u)=-\\theta_1 (1-u)-\\log\\frac{e^{-\\theta_1}-1}{-\\theta_1}\n=\\theta_1 u-\\log\\frac{e^{\\theta_1}-1}{\\theta_1}=I(u).\n\\end{equation}\nThis verifies our claim.\n\\end{proof}\n\n\\begin{corollary}\nTake $H_1$ a single edge and $H_2$ a finite simple graph with\n$p\\geq 2$ edges. The associated phase transition curve\n$\\beta_2=r(\\beta_1)$ lies above the straight line\n$\\beta_2=-\\beta_1$ when $p\\geq 3$, and is exactly portion of the\nstraight line $\\beta_2=-\\beta_1$ ($\\beta_1\\leq -3$) when $p=2$.\n\\end{corollary}\n\n\\begin{proof}\nFrom the proof of Theorem \\ref{phase}, there are two global\nmaximizers $u_1^*$ and $u_2^*$ for $l(u)$ along the phase\ntransition curve $\\beta_2=r(\\beta_1)$, with $u_0\\geq \\frac{1}{2}$\nand $0<u_1^*<u_0<u_2^*<1$. Furthermore, the $y$-coordinate\n$\\beta_2^c$ of the critical point $(\\beta_1^c, \\beta_2^c)$ is\nalways positive. On the straight line $\\beta_1+\\beta_2=0$, we\nrewrite $l(u)=\\beta_1(u-u^p)-\\frac{1}{2}I(u)$. First suppose\n$p=2$. Since $I(u)$ and $u-u^2$ are both symmetric about the line\n$u=\\frac{1}{2}$, two global maximizers $u_1^*$ and $u_2^*$ exist\nfor $l(u)$. Next consider the generic case $p>2$. Analytical\ncalculations give that $u-u^p<(1-u)-(1-u)^p$ for\n$0<u<\\frac{1}{2}$. Since $I(u)$ is symmetric, this says that for\n$\\beta_1<0$ (resp. $\\beta_2>0$), the global maximizer $u^*$ of\n$l(u)$ satisfies $u^*\\leq \\frac{1}{2}$. This implies the desired\nresult.\n\\end{proof}\n\n\\begin{figure}\n\\centering\n\\includegraphics[clip=true, width=6in, height=2.5in]{along.pdf}\n\\caption{Along the phase transition curve $r(\\beta_1)$, $l(u)$ has\ntwo local maximizers $u_1^*$ and $u_2^*$, and both are global\nmaximizers for $l(u)$. Graph drawn for $\\beta_1=-5$, $\\beta_2=5$,\nand $p=2$.} \\label{along}\n\\end{figure}\n\n\\section{Further discussion}\n\\label{discuss} But what if the common distribution $\\mu$ on the\nedge weights does not have finite support? Unfortunately, we do\nnot have a generic large deviation principle in this case. As an\nexample, we will examine the asymptotic phase structure for a\ncompletely solvable exponential model and hope the analysis would\nprovide some inspiration. Let $G_n\\in \\mathcal{G}_n$ be an\nedge-weighted directed labeled graph on $n$ vertices, where the\nedge weights $x_{ij}$ from vertex $i$ to vertex $j$ are iid real\nrandom variables whose common distribution $\\mu$ is standard\nGaussian. As in the undirected case, the common distribution for\nthe edge weights yields probability measure $\\mathbb P_n$ and the\nassociated expectation $\\mathbb E_n$ on $\\mathcal{G}_n$. Give the set of\nsuch graphs the probability\n\\begin{equation}\n\\mathbb P_n^\\beta(G_n)=\\exp\\left(n^2\\left(\\beta_1e(G_n)+\\beta_2s(G_n)-\\psi_n^\\beta\\right)\\right)\\mathbb P_n(G_n),\n\\end{equation}\nwhere $\\beta=(\\beta_1, \\beta_2)$ are $2$ real parameters, $H_1$ is\na directed edge, $H_2$ is a directed $2$-star, $e(G_n)$ and\n$s(G_n)$ are respectively the directed edge and $2$-star\nhomomorphism densities of $G_n$,\n\\begin{equation}\ne(G_n)=\\frac{1}{n^2}\\sum_{1\\leq i,j\\leq n} x_{ij}, \\hspace{0.2cm}\ns(G_n)=\\frac{1}{n^3}\\sum_{1\\leq i,j,k \\leq n} x_{ij}x_{ik},\n\\end{equation}\nand $\\psi_n^\\beta$ is the normalization constant,\n\\begin{equation}\n\\label{dpsi} \\psi_n^\\beta=\\frac{1}{n^2}\\log\n\\mathbb E_n\\left(\\exp\\left(n^2\\left(\\beta_1e(G_n)+\\beta_2s(G_n)\\right)\\right)\\right).\n\\end{equation}\nThe associated expectation $\\mathbb E_n^\\beta$ may be defined\naccordingly, and a phase transition occurs when the limiting\nnormalization constant $\\displaystyle \\psi^\\beta_\\infty=\\lim_{n\\to\n  \\infty}\\psi_n^{\\beta}$ has a singular point.\n\nPlugging the formulas for $e(G_n)$ and $s(G_n)$ into (\\ref{dpsi})\nand using the iid property of the edge weights, we have\n\\begin{eqnarray}\n\\psi_n^\\beta&=&\\frac{1}{n^2}\\log\n\\mathbb E_n\\left(\\exp\\left(\\beta_1\\sum_{i=1}^n\\left(\\sum_{j=1}^n\nx_{ij}\\right)+\\frac{\\beta_2}{n}\\sum_{i=1}^n\\left(\\sum_{j=1}^n\nx_{ij}\\right)^2\\right)\\right)\\\\\n&=&\\frac{1}{n}\\log\\mathbb E \\left(\\exp\\left(\\beta_1\nY+\\frac{\\beta_2}{n}Y^2\\right)\\right),\\notag\n\\end{eqnarray}\nwhere $Y=\\sum_{j=1}^n x_{1j}$ satisfies a Gaussian distribution\nwith mean $0$ and variance $n$, and $\\mathbb E$ is the associated\nexpectation. We compute, for $\\beta_2<\\frac{1}{2}$,\n\\begin{equation}\n\\mathbb E \\left(\\exp\\left(\\beta_1\nY+\\frac{\\beta_2}{n}Y^2\\right)\\right)=\\int_{-\\infty}^\\infty\ne^{\\beta_1y+\\frac{\\beta_2}{n}y^2}\\frac{1}{\\sqrt{2\\pi\nn}}e^{-\\frac{y^2}{2n}}dy\n=\\frac{1}{\\sqrt{1-2\\beta_2}}e^{\\frac{n\\beta_1^2}{2(1-2\\beta_2)}},\n\\end{equation}\nwhich implies that\n\\begin{equation}\n\\psi_\\infty^\\beta=\\frac{\\beta_1^2}{2(1-2\\beta_2)}.\n\\end{equation}\nThis is a smooth function in terms of the parameters $\\beta_1$ and\n$\\beta_2$, and so $\\psi_\\infty^\\beta$ does not admit a phase\ntransition.\n\n\\section*{Acknowledgements}\nThis work originated at the Special Session on Topics in\nProbability at the 2016 AMS Western Spring Sectional Meeting,\norganized by Tom Alberts and Arjun Krishnan. Mei Yin's research\nwas partially supported by NSF grant DMS-1308333. She thanks Sean\nO'Rourke and Lingjiong Zhu for helpful conversations.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Intro}\n\nSequence modelling is an important problem in NLP, as many NLP tasks can be modelled as sequence-to-sequence decoding.\nAmong them are POS tagging, chunking, named entity recognition \\cite{Collobert-2011-NLP-1953048.2078186}, Spoken Language Understanding (SLU) for human-computer interactions \\cite{demori08-SPM}, and also machine translation \\cite{Sutskever-2014-SSL-2969033.2969173,DBLP-journals-corr-BahdanauCB14}.\n\nIn other cases, NLP tasks can be decomposed, at least in principle, in several subtasks, the first of which is a sequence modelling problem.\nFor instance, syntactic parsing can be performed by applying syntactic analysis to POS-tagged sentences \\cite{Collins-1997-TGL}; coreference chain detection \\cite{Soon2001,Ng2002,Grouin.etAL:I2B2:2011} can be decomposed into mention detection and coreferent mention linking; and structured named entity detection \\cite{Grouin.etAL:I2B2:2011,dinarelli2012-eacl,DINARELLI-ROSSET:OCR-NER:LREC2012}, can be done by first detecting simple entity components then combining them to construct complex tree-shaped entities.\n\nMost of these tasks can also be performed by a single model: either as a joint architecture like the joint model for POS tagging and syntactic analysis from \\cite{Rush-2012-IPP-2390948.2391112} or with a fully end-to-end model like the one developed by \\cite{D17-1018} for coreference detection.\nIn any case, these models still include at some point a sequence modelling module that could be improved by studying successful models for the related sequence labelling tasks.\n\nThis is even more true for neural models, since designing a single complex neural architecture for a complex problem may indeed lead to sub-optimal learning.\nFor this reason, it may be more desirable to train a sequence labelling model alone at first and to learn to perform the other steps using the pre-trained parameters of the first step's model, as is done for instance when using pre-trained lexical embeddings in a downstream model \\cite{lample2016neural,Ma-Hovy-ACL-2016}.\nIn that case, care must be taken to avoid too unrelated downstream tasks that could lead to \\emph{Catastrophic forgetting} \\cite{kemker2018measuring}, though some hierarchical multi-task architectures have proven successful \\cite{N18-1172}.\n\nFinally, \\cite{DBLP-journals-corr-VinyalsKKPSH14} has shown that it is possible to model syntactic analysis as a sequence labelling problem by adapting a \\emph{Seq2seq} model. As a consequence, we could actually design a unified multi-task learning neural architecture for a large class of NLP problems, by recasting them as sequence decoding tasks.\n\nRecurrent Neural Networks (RNNs) hold state-of-the-art results in many NLP tasks, and in particular in sequence modelling problems \\cite{lample2016neural,Ma-Hovy-ACL-2016,dinarelli_hal-01553830,D17-1018}.\nGated RNNs such as GRU and LSTM are particularly effective for sequence labelling thanks to an architecture that allows them to use long-range information in their internal representations \\cite{werbos-bptt,Hochreiter-1997-LSTM,Cho-2014-GatedRecurrentUnits}.\n\nIn this paper we focus our work to searching for more effective neural models for sequence labelling tasks such as POS tagging or Spoken Language Understanding (SLU).\nSeveral very effective solutions already exist for these problems, in particular the sequence-to-sequence model \\cite{Sutskever-2014-SSL-2969033.2969173} (\\emph{Seq2seq} henceforth), the \\emph{Transformer} model \\cite{46201}, and the whole family of models using a neural CRF layer on top of one or several LSTM or GRU layers \\cite{Hochreiter-1997-LSTM,Cho-2014-GatedRecurrentUnits,lample2016neural,Ma-Hovy-ACL-2016,Vukotic.etal_2016,LSTM-CNN-NER-2015,huang2015bidirectional}.\n\nWe propose an alternative neural architecture to those mentioned above.\nThis architecture uses GRU recurrent layers as internal memory capable of taking into account arbitrarily long contexts of both input (words and characters), and output (labels).\nOur architecture is a variant of the \\emph{Seq2seq} model where two different decoders are used instead of only one of the original architecture. The first decoder goes backward through the sequence, outputting label predictions, using the hidden states of the encoder and its own previous hidden states and label predictions as input.\nThe second decoder is a more standard forward decoder that uses the hidden states of the encoder, the hidden states and \\emph{future} predictions generated by the backward decoder and its own previous hidden states and predictions to output labels.\nWe name this architecture \\emph{Seq2biseq}, as it generates output sequences from output-wise bidirectional, global decisions.\n\nOur work is inspired by previous work published in \\cite{dinarelli_hal-01553830,Dupont-etAl-LDRNN-CICling2017,2016:arXiv:DinarelliTellier:NewRNN,DinarelliTellier:RNN:CICling2016}, where bidirectional output-wise decisions were taken using a simple recurrent network.\nA similar idea, called \\textit{deliberation network}, has been proposed in \\cite{NIPS2017_6775}, where however two forward decoders were used. In this respect we believe that using a backward decoder for the first pass may encode more different, expressive information for the second, forward pass.\nOur architecture takes global decisions like a \\emph{LSTM+CRF} model \\cite{lample2016neural} thanks to the use of the two decoders. These take global context into account on both sides of a given position of the input sequence.\n\nWe compare our solution with state-of-the-art models for SLU and POS-tagging in particular the models described in \\cite{dinarelli_hal-01553830,Dupont-etAl-LDRNN-CICling2017} and in \\cite{lample2016neural}.\nIn order to make a direct comparison, we evaluate our models on the same tasks: a French SLU task provided with the MEDIA corpus \\cite{Bonneau-Maynard2006-media}, and the well-known task of POS-tagging of the Wall Street Journal portion of the Penn Treebank \\cite{Marcus93buildinga}.\n\nOur results are all reasonably close to the state of the art, and most of them are actually better.\n\nThe paper is organized as follows: in the next section we describe the state-of-the-art of neural models for sequence labelling.\nIn the section~\\ref{sec:GRU-IRNN} we describe the neural model we propose in this paper, while in the section~\\ref{sec:eval} we describe the experiments we performed to evaluate our models.\nWe draw our conclusions in the section~\\ref{sec:Conclusions}\n\n\\section{State of the Art}\n\\label{sec:SOTA}\n\nThe two main neural architectures used for sequence modelling are the \\emph{Seq2seq} model \\cite{Sutskever-2014-SSL-2969033.2969173} and a group of models where a neural CRF output layer is stacked on top of one or several LSTM or GRU layers \\cite{Hochreiter-1997-LSTM,Cho-2014-GatedRecurrentUnits,lample2016neural,Ma-Hovy-ACL-2016,Vukotic.etal_2016,LSTM-CNN-NER-2015,huang2015bidirectional}.\n\nThe \\emph{Seq2seq} model, also known as \\emph{encoder-decoder}, uses a first module to encode the input sequence as a single vector $c$.\nIn the version of this model proposed in \\cite{Sutskever-2014-SSL-2969033.2969173} $c$ is the hidden state of the encoder after seeing the whole input sequence.\nA second module decodes the output sequence using its previous predictions and $c$ as input.\n\nThe subsequent work of \\cite{DBLP-journals-corr-BahdanauCB14} extends this model with an attention mechanism.\nThis mechanism provides the decoder with a dynamic representation of the input that depends on the decoding step, which proved to be more efficient for translating long sentences.\n\nThis mechanism has also been turned out to be effective for other NLP tasks \\cite{D17-1018,DBLP-journals-corr-KimDHR17,simonnet-hal-01433202}.\n\nConcerning models using a neural CRF output layer \\cite{Ma-Hovy-ACL-2016,lample2016neural}, a first version was already described in \\cite{Collobert-2011-NLP-1953048.2078186}.\nThese solutions use one or more recurrent hidden layers to encode input items (words) in context. Earlier simple recurrent layers like \\emph{Elman} and \\emph{Jordan} \\cite{Elman90findingstructure,jordan-serial}, which showed limitations for learning long-range dependencies \\cite{Bengio-1994-RNN-Learning-Difficulty}, have been replaced by more sophisticated layers like LSTM and GRU \\cite{Hochreiter-1997-LSTM,Cho-2014-GatedRecurrentUnits}, which reduced such limitations by using gates.\n\nIn this type of neural models, a first representation of the prediction is computed with a local output layer.\nIn order to compute global predictions with a CRF neural layer, the \\emph{Viterbi} algorithm is applied over the sequence of local predictions \\cite{Collobert-2011-NLP-1953048.2078186,Mesnil-RNN-2015}.\n\nA more recent neural architecture for sequence modelling is the \\emph{Transformer} model \\cite{46201}. This model use an innovative deep non-recurrent neural architecture, relying heavily on attention mechanisms \\cite{DBLP-journals-corr-BahdanauCB14} and skip connections \\cite{Bengio03aneural} to overcome limitations of recurrent networks in propagating the learning signal over long paths. The Transformer model has been designed for computational efficiency reasons, but it captures long-range contexts with multiple attention mechanisms (multi-head attention) applied to the whole input sequence. Skip-connections guarantee that the learning signal is back-propagated effectively to all the network layers.\n\nConcerning previous works on the same tasks used in this work, namely MEDIA \\cite{Bonneau-Maynard2006-media} and the Penn Treebank (WSJ) \\cite{Marcus93buildinga}, several publications have been produced starting from $2007$ (MEDIA) and $2002$ (WSJ) \\cite{raymond07-luna,dinarelli09:Interspeech,Hahn.etAL-SLUJournal-2010,Dinarelli2010.PhDThesis,dinarelli2011:emnlp,Dinarelli.etAl-SLU-RR-2011}, applying several different models like \\emph{SVM} and \\emph{CRF} \\cite{Vapnik98-book,lafferty01-crf}.\nStarting from 2013 several works also focused on neural models. At first simple recurrent networks have been used \\cite{RNNforSLU-Interspeech-2013,RNNforLU-Interspeech-2013,Vukotic.etal_2015}. In the last few years also more sophisticated models have been studied \\cite{SLUwithLSTM-NN-IEEEWshop-2014,Vukotic.etal_2016,dinarelli_hal-01553830}.\n\n\\section{The Seq2biseq Neural Architecture}\n\\label{sec:GRU-IRNN}\n\n\\begin{figure}\n\\center\n\\begin{tikzpicture}\n\t\\def$h_{w_1}$, $h_{w_2}$, $h_{w_3}${$w_1$, $w_2$, $w_3$}\n\t\\def$\\overrightarrow{e_1}$, $\\overrightarrow{e_2}$, $\\overrightarrow{e_3}${$\\overrightarrow{e_1}$, $\\overrightarrow{e_2}$, $\\overrightarrow{e_3}$}\n\t\\def$\\overleftarrow{e_1}$, $\\overleftarrow{e_2}$, $\\overleftarrow{e_3}${$\\overleftarrow{e_1}$, $\\overleftarrow{e_2}$, $\\overleftarrow{e_3}$}\n\t\n\t\\begin{scope}[local bounding box=net]\n\t\\foreach \\w [count=\\wi from 1, remember=\\wi as \\lastwi] in $h_{w_1}$, $h_{w_2}$, $h_{w_3}$ {\n\t\\ifnum\\wi>1\n            \t\\node[right=6.5em of w\\lastwi.center, text height=1.5ex, text depth=0.25ex, anchor=center] (w\\wi) {\\w};\n            \\else\n            \t\\node[text height=1.5ex, text depth=0.25ex] (w\\wi) {\\w};\n            \\fi\n\t}\n\t\\foreach \\w [count=\\wi from 1, remember=\\wi as \\lastwi] in $h_{w_1}$, $h_{w_2}$, $h_{w_3}$ {\n\t\t\\draw[->] (w\\wi) -- +(0, 2em) node[draw, anchor=south, rounded corners=1pt, inner sep=0.3em] (ew\\wi) {encoder};\n\t\t\\ifnum\\wi>1\n\t\t\t\\draw[<->] (ew\\lastwi) -- (ew\\wi);\n\t\t\\fi\n\t}\n\t\n\t\\foreach \\w [count=\\wi from 1, remember=\\wi as \\lastwi] in $h_{w_1}$, $h_{w_2}$, $h_{w_3}$ {\n\t\t\\draw[->] (ew\\wi) -- +(-0.5, 2em) node[draw, anchor=south, rounded corners=1pt, inner sep=0.3em] (bdec\\wi) {$\\overleftarrow{decoder}$};\n\t\t\\ifnum\\wi>1\n\t\t\t\\draw[->] (bdec\\wi) -- (bdec\\lastwi);\n\t\t\\fi\n\t}\n\t\n\t\\foreach \\w [count=\\wi from 1, remember=\\wi as \\lastwi] in $h_{w_1}$, $h_{w_2}$, $h_{w_3}$ {\n\t\t\\draw[->,red] (ew\\wi) to[bend right=30] +(+0.5, 5em) node[draw, anchor=south, rounded corners=1pt, inner sep=0.3em] (fdec\\wi) {$\\overrightarrow{decoder}$};\n\t\t\\draw[->] (bdec\\wi) -- (fdec\\wi);\n\t\t\\ifnum\\wi>1\n\t\t\t\\draw[->] (fdec\\lastwi) -- (fdec\\wi);\n\t\t\\fi\n\t}\n\t\n\t\\foreach \\y [count=\\yi from 1, remember=\\yi as \\lastyi] in $\\overrightarrow{e_1}$, $\\overrightarrow{e_2}$, $\\overrightarrow{e_3}$ {\n            \t\\draw[red,->] (fdec\\yi) -- +(0, 3em) node[text height=1.5ex, text depth=0.25ex, anchor=south] (fy\\yi) {\\y};\n            \t\\ifnum\\yi>1\n            \t\t\\draw[red,->] (fy\\lastyi) to[bend right=30] (fdec\\yi);\n            \t\\fi\n\t}\n\t\n\t\\foreach \\y [count=\\yi from 1, remember=\\yi as \\lastyi] in $\\overleftarrow{e_1}$, $\\overleftarrow{e_2}$, $\\overleftarrow{e_3}$ {\n\t\t\\draw[->] (bdec\\yi) -- +(0, 5.3em) node[text height=1.5ex, text depth=0.25ex, anchor=south] (by\\yi) {\\y};\n\t}\n\t\n\t\\draw[->] (by3) to[bend left=42] (bdec2.east);\n\t\\draw[->] (by2) to[bend left=42] (bdec1.east);\n\n        \\end{scope}\n\n\\end{tikzpicture}\n    \\caption{Overall network structure}\\label{fig:network-structure}\n\\end{figure}\n\nAs an alternative to the \\emph{Seq2seq} and \\emph{LSTM+CRF} neural models for sequence labelling, we propose in this paper a new neural architecture inspired from the original \\emph{Seq2seq} model and from models described in \\cite{dinarelli_hal-01553830,Dupont-etAl-LDRNN-CICling2017}.\nFigure~\\ref{fig:network-structure} shows the overall architecture.\n\nOur architecture is similar to the \\emph{Seq2seq} model in that we use modules to encode a long-range context on the output side similar to the decoder of the \\emph{Seq2seq} architecture.\nThe similarity with respect to models described in \\cite{dinarelli_hal-01553830,Dupont-etAl-LDRNN-CICling2017} is the use of a bidirectional context on the output side in order to take into account previous, but also future predictions for the current model decision. Future predictions are computed by an independent decoder which processes the input sequence backward.\n\nOur architecture extends the \\emph{Seq2seq} original model through the use of an additional backward decoder that allows taking into account both past and future information at decoding time.\nOur architecture also improves the models described in \\cite{dinarelli_hal-01553830,Dupont-etAl-LDRNN-CICling2017} since it uses more sophisticated layers to model long-range contexts on the output side, while previous models used fixed-size windows and simple linear hidden layers.\nThanks to these modifications our model makes predictions informed by a global distributional context, which approximates a global decision function.\nWe also improve the character-level word representations by using a similar solution to the one proposed in \\cite{Ma-Hovy-ACL-2016}.\n\nOur neural architecture is based on the use of GRU recurrent layers at word, character and label levels.\nGRU is an evolution of the LSTM recurrent layer which has often shown better capacities to model contextual information \\cite{Cho-2014-GatedRecurrentUnits,Vukotic.etal_2016}.\n\nIn order to make notation clear, in the following sections, bidirectional GRU hidden layers are noted $\\GRU$, while we use $\\fGRU$ and $\\bGRU$ for a forward and backward hidden layer respectively.\nFor the output of these layers we use respectively $h_{w_i}$, $\\overrightarrow{h_{e_i}}$ and $\\overleftarrow{h_{e_i}}$, with a letter as index to specialize the GRU layer for a specific input (e.g. $w$ for the GRU layer used for words, $e$ for labels, or entities, and so on), and an index $i$ to indicate the index position in the current sequence.\nFor example $\\overleftarrow{h_{e_i}}$ is the backward hidden state, at current position ($i$), of the GRU layer for labels.\nThe models described in this work always use as input words, characters and labels. Their respective embedding matrices are all noted $E_x$, with $x$ denoting the different input unit types (e.g. $E_w$ is the embedding matrix for words), and their dimensions $D_x$.\n\n\\subsection{Character-level Representations}\n\\label{subsec:char_rep}\n\nThe character-level representation of words was computed at first as in \\cite{Ma-Hovy-ACL-2016}, substituting a GRU to the LSTM layer:\nthe characters $c_{w, 1}, \\dots, c_{w, n}$ of a word $w$ are first represented as a sequence $S_c(w)$ of $n$ $D_c$-dimensional embeddings. These are fed to the $\\GRU_c$ layer. The final state $h_c(w)$ is kept as the character level representation of $w$.\n\nWe improved this module so that it generates a character-level representation using all the hidden states generated by $\\GRU_c$:\n\\begin{equation}\\label{eq:charrep}\n    \\begin{aligned}\n        S_c(w) &= (E_{c}(c_{w, 1}), \\dots, E_c(c_{w, n})) \\\\\n        (h_c(c_{w, 1}), \\dots, h_c(c_{w, n})) &= \\GRU_c(S_{c}(w), h_0^c) \\\\\n        h_c(w) &= \\FFNN( Sum( h_c(c_{w, 1}), \\dots, h_c(c_{w, n}) ) )\n    \\end{aligned}\n\\end{equation}\n$\\FFNN$ is again a general, possibly multi-layer Feed-Forward Neural Network with non-linear activation functions. This new architecture was inspired by \\cite{46201}, where $\\FFNN$s were used to extract deeper features at each layer.\n\nPreliminary experiments have shown that this character-level representation is more effective than the one inspired by the work of \\cite{Ma-Hovy-ACL-2016}.\n\n\\subsection{Word-level Representations}\n\\label{subsec:word_rep}\n\nWords are first mapped into embeddings, then the embedding sequence is processed by a $\\GRU_{w}$ bidirectional layer.\nUsing the same notation as for characters, a sequence of words $S = w_1, \\dots, w_N$ is converted into embeddings $E_w(w_i)$ with $1\\leq i \\leq N$.\nWe denote $S_i = w_1, \\dots, w_i$ the sub-sequence of $S$ up to the words $w_i$.\nIn order to augment the word representations with their character-level representations, and to use a single distributed representation, we concatenate the character-level representations $h_{c}(w_i)$ (eq.~\\ref{eq:charrep}) to the word embeddings before feeding the $\\GRU_w$ layer with the whole sequence. Formally:\n\\begin{equation}\\label{eq:lex-rep}\n    \\begin{aligned}\n        S_w &= (E_{w}(w_{1}), \\dots, E_{w}(w_{N})) \\\\\n        S^{lex} &= ([E_{w}(w_{1}), h_{c}(w_1)], \\dots, [E_{w}(w_{N}), h_{c}(w_N)]) \\\\\n        h_{w_i} &= \\GRU_w(S_i^{lex}, h_{w_{i-1}})\n    \\end{aligned}\n\\end{equation}\nWhere we used $S_w$ for the whole sequence of word embeddings generated from the word sequence $S$.\n\nIn the same way, $S^{\\mathrm{lex}}$ is the sequence obtained concatenating word embeddings and character-level representations, which constitute the lexical-level information given as input to the model.\n$[~]$ is the matrix (or vector) concatenation, and we also used the notation $S_i^{\\mathrm{lex}}$ for the sub-sequence of $S^{\\mathrm{lex}}$ up to position $i$.\n\n\\subsection{Label-level Representations}\n\\label{subsec:label_rep}\n\nIn order to obtain label representations encoding long-range contexts, we use a $\\GRU$ hidden layer also on label embeddings.\nWe apply first a backward step on label embeddings in order to compute representations that will be used as future label predictions, or right context, in the following forward step.\nUsing the same notation as used previously, we have:\n\\begin{equation}\\label{eq:label-bw}\n    \\overleftarrow{h_{e_i}} = \\bGRU_e(E_l(e_{i+1}), \\overleftarrow{h_{e_{i+1}}})\n\\end{equation}\nfor $i = N \\dots 1$.\nWe note that here we use the label on the right of the current position, $e_{i+1}$, $e_{i}$ is not known at time step $i$.\n\nThe hidden state $\\overleftarrow{h_{e_{i+1}}}$ is the hidden state computed at previous position in the backward step, thus associated to the label on the right of the current label to be predicted. In other words we interpret $\\overleftarrow{h_{e_{i}}}$ as the right context of the (unknown) label $e_i$, instead of as the in-context representation of $e_i$ itself, and similarly for $\\overleftarrow{h_{e_{i+1}}}$.\nThe right context of $e_i$, $\\overleftarrow{h_{e_i}}$, is used to predict $e_i$ at time step $i$.\n\nIn the same way, we compute the representation of the left context of the label $e_i$ by scanning the input sequence forward, which gives:\n\\begin{equation}\\label{eq:label-fw}\n    \\overrightarrow{h_{e_i}} = \\fGRU_{e}(E_l(e_{i-1}), \\overrightarrow{h_{e_{i-1}}})\n\\end{equation}\nfor $i = 1 \\dots N$.\nThe neural components described so far are already sufficient to build rich architectures.\nHowever, we believe that the information from the lexical context is useful not only to disambiguate the current word in-context, but also to disambiguate the contextual representations used for label prediction.\nIndeed, in sequence labelling labels only provide abstract lexical or semantic information. It thus seems reasonable to think that they are not sufficient to effectively encode features in the label context representations $\\overleftarrow{h_{e_i}}$  and $\\overrightarrow{h_{e_i}}$.\n\nFor this reason, we add to the input of the layers $\\bGRU_{e}$ and $\\fGRU_{e}$ the lexical hidden representation $h_{w_i}$ computed by the $\\GRU_{w}$ layer.\nTaking this into account, the computation of the right context for the current label prediction becomes:\n\\begin{equation}\\label{eq:label-bw-le}\n    \\overleftarrow{h_{e_i}} = \\bGRU_{e}([h_{w_i}, E_l(e_{i+1})], \\overleftarrow{h_{e_{i+1}}})\n\\end{equation}\nThe computation of the left context is done in a similar way.\n\nThis modification makes the layers $\\bGRU_{e}$ and $\\fGRU_{e}$ in our architecture similar to the decoder of a \\emph{Seq2seq} architecture \\cite{Sutskever-2014-SSL-2969033.2969173}.\nThe modules $\\bGRU_{e}$ and $\\fGRU_{e}$ are indeed like two decoders from an architectural point of view, but also they encode the contextual information in the same way using gated recurrent layers.\n\nHowever, the full architecture differs from a traditional \\emph{Seq2seq} model by the use of an additional decoder, capable of modelling the right label context, while the original model used a single decoder, modelling only the left context.\nThe idea of using two decoders is inspired mainly by the evidence that both left and right output-side contexts are equally informative for the current prediction.\n\nAnother difference with respect to the \\emph{Seq2seq} model is that the $\\bGRU_{e}$ and $\\fGRU_{e}$ layers have access to the lexical-level hidden states $h_{w_i}$.\nThis allows these layers to take the current lexical context into account and is thus more adapted to sequence labelling than using the same representation of the input sentence for all the positions, which is the solution of the original \\emph{Seq2seq} model.\n\nAs we mentioned above, the \\emph{Seq2seq} model has been improved with an attention mechanism \\cite{DBLP-journals-corr-BahdanauCB14}, which is another way to provide the model with a lexical representation focusing dynamically on different parts of the input sequence depending on the position $i$.\nThis attention mechanism has also proved to be efficient for sequence labelling, and it might be that our architecture could benefit from it too, but this is out of our scope for this article and we leave it for future work.\\footnote{This is currently in progress}\n\nWe can motivate the use of the lexical information $h_{w_i}$ in the decoders $\\bGRU_{e}$ and $\\fGRU_{e}$ with complex systems theory considerations, as suggested in \\cite{2017-NoRNN}.\n\\cite{Holland-1999-ECO-520475} state that a complex system, either biological or artificial, is not equal to the sum of its components.\nMore precisely, the behaviour of a complex system evolves during its existence and shows the emergence of new functionalities, which can not be explained by simply considering the system's components individually.\n\\cite{RePEc-wop-safiwp-93-11-070} qualitatively characterizes the evolution of a complex system's behaviour with three different types of adaptation, two of which are particularly interesting in the context of this work and can be concisely named \\emph{aggregation} and \\emph{specialization}.\n\nIn the first, several components of the system adapt in order to become a single \\emph{aggregated} component from a functioning point of view.\nIn \\emph{specialization}, several initially identical components of the system adapt to perform different functionalities.\nThese adaptations may take place at different unit levels, a neuron, a simple layer, or a whole module.\n\nThe most evident cases of \\emph{specialization} are the gates of the LSTM or GRU layers \\cite{Cho-2014-GatedRecurrentUnits}, as well as the attention mechanism \\cite{DBLP-journals-corr-BahdanauCB14}.\nIndeed, the $\\mathbf{z}$ and $\\mathbf{r}$ gates of a GRU recurrent layer are defined in the exact same way, with the same number of parameters, and they use exactly the same input information.\n\nHowever, during the evolution of the system \u2014 that is, during the learning phase \u2014 the $\\mathbf{r}$ gate adapts (specialises) to become the reset gate, which allows the network to forget the past information, when it is not relevant for the current prediction step.\nIn the same way, the $\\mathbf{z}$ gate becomes the equivalent of the input gate of a LSTM, which controls the amount of input information that will affect the current prediction.\n\nIn our neural architecture we can observe \\emph{aggregation}: the layers $\\bGRU_{e}$ and $\\fGRU_{e}$ adapt at the whole layer level, they become like gates which filter label-level information that is not useful for the current prediction.\nIn the same way as the input to gates of GRU or LSTM is made of current input and previous hidden state, the input to the $\\bGRU_{e}$ and $\\fGRU_{e}$ layers is made of lexical level and previous label level information, both needed to discriminate the abstract semantic information provided by the labels alone.\nWe will show in the evaluation section the effectiveness provided by this choice.\n\nWhile both of the two decoders used in our models are equivalent to the decoder of the original \\emph{Seq2seq} architecture, we believe it is interesting to analyse the contribution of each piece of information given as input to this component, which we will show in the evaluation section.\n\n\\subsection{Output Layer}\n\\label{subsec:output}\n\nOnce all pieces of information needed to predict the current label are computed, the output of the backward step is computed as follows:\n\\begin{equation}\\label{eq:backward-model}\n    \\begin{aligned}\n        o_{bw} &= W_{bw} [h_{w_{i}}, \\overleftarrow{h_{e_i}}] + b_{bw} \\\\\n        e_{i} &= \\argmax(\\logsoftmax(o_{bw}))\n    \\end{aligned}\n\\end{equation}\nWe start the backward step using a conventional symbol (\\texttt{<EOS>}) as end-of-sentence marker.\nWe repeat the backward step prediction for the whole input sequence. The process is shown in figure~\\ref{pic:back-dec}.\n\n\\begin{figure}\n\\center\n\\begin{tikzpicture}\n\t\\def$h_{w_1}$, $h_{w_2}$, $h_{w_3}${$h_{w_1}$, $h_{w_2}$, $h_{w_3}$}\n\t\\def$\\overrightarrow{e_1}$, $\\overrightarrow{e_2}$, $\\overrightarrow{e_3}${$\\overleftarrow{e_{1}}$, $\\overleftarrow{e_{2}}$, $\\overleftarrow{e_{3}}$}\n\t\n\t\\begin{scope}[local bounding box=net]\n\t\\foreach \\w [count=\\wi from 1, remember=\\wi as \\lastwi] in $h_{w_1}$, $h_{w_2}$, $h_{w_3}$ {\n\t\\ifnum\\wi>1\n            \t\\node[right=6.5em of w\\lastwi.center, text height=1.5ex, text depth=0.25ex, anchor=center] (w\\wi) {\\w};\n            \\else\n            \t\\node[text height=1.5ex, text depth=0.25ex] (w\\wi) {\\w};\n            \\fi\n\t}\n\t\n\t\\foreach \\w [count=\\wi from 1, remember=\\wi as \\lastwi] in $h_{w_1}$, $h_{w_2}$, $h_{w_3}$ {\n\t\t\\draw[->] (w\\wi) -- +(0, 2em) node[draw, anchor=south, rounded corners=1pt, inner sep=0.3em] (gruw\\wi) {$\\overleftarrow{GRU_l}$};\n\t\t\\ifnum\\wi>1\n\t\t\t\\draw[<-] (gruw\\lastwi) -- (gruw\\wi);\n\t\t\\fi\n\t}\n\t\n\t\\foreach \\w [count=\\wi from 1, remember=\\wi as \\lastwi] in $\\overrightarrow{e_1}$, $\\overrightarrow{e_2}$, $\\overrightarrow{e_3}$ {\n\t\t\\draw[->] (gruw\\wi) -- +(0, 2em) node[draw, text width=1.5cm, text centered, anchor=south, rounded corners=1pt, inner sep=0.3em] (lin\\wi) {log-soft linear};\n\t\t\\draw[->] (w\\wi) to[bend left=55] (lin\\wi);\n\t}\n\t\n\t\\foreach \\w [count=\\wi from 1, remember=\\wi as \\lastwi] in $\\overrightarrow{e_1}$, $\\overrightarrow{e_2}$, $\\overrightarrow{e_3}$ {\n\t\t\\draw[->] (lin\\wi) -- +(0, 2.5em) node[text height=1.5ex, text depth=0.25ex, anchor=south] (o\\wi) {\\w};\n\t\t\\ifnum\\wi>1\n\t\t\t\\draw[->,dashed] (o\\wi) -- (gruw\\lastwi);\n\t\t\\fi\n\t}\n\t\n\t\\node[right=6.5em of o3.center, text height=1.5ex, text depth=0.25ex, anchor=center] (eos) {$<EOS>$};\n\t\\draw[->,dashed] (eos) to[bend left] (gruw3);\n\n        \\end{scope}\n\n\\end{tikzpicture}\n\\caption{Structure of the backward decoder}\\label{pic:back-dec}\n\\end{figure}\n\nThis allows to have all the pieces of information needed to predict the current label in the forward step, at character and word level, but also at right and left label context level, with respect to the current position to be labeled:\n\\begin{equation}\\label{eq:bidirectional-model}\n    \\begin{aligned}\n        o_i &= W_o [\\mathbf{\\overrightarrow{h_{e_i}}}, h_{w_i}, \\mathbf{\\overleftarrow{h_{e_i}}}] + b_o \\\\\n        e_i &= \\argmax(\\logsoftmax(o_i))\n    \\end{aligned}\n\\end{equation}\nA high-level schema of the forward pass is shown in figure~\\ref{pic:for-dec}.\n\n\\begin{figure}\n\\center\n\\begin{tikzpicture}\n\t\\def$h_{w_1}$, $h_{w_2}$, $h_{w_3}${$h_{w_1}$, $h_{w_2}$, $h_{w_3}$}\n\t\\def$\\overrightarrow{e_1}$, $\\overrightarrow{e_2}$, $\\overrightarrow{e_3}${$\\overrightarrow{e_1}$, $\\overrightarrow{e_2}$, $\\overrightarrow{e_3}$}\n\t\n\t\\begin{scope}[local bounding box=net]\n\t\\foreach \\w [count=\\wi from 1, remember=\\wi as \\lastwi] in $h_{w_1}$, $h_{w_2}$, $h_{w_3}$ {\n\t\\ifnum\\wi>1\n            \t\\node[right=6.5em of w\\lastwi.center, text height=1.5ex, text depth=0.25ex, anchor=center] (ew\\wi) {\\w};\n            \\else\n            \t\\node[text height=1.5ex, text depth=0.25ex] (ew\\wi) {\\w};\n            \\fi\n\t}\n\t\n\t\\foreach \\w [count=\\wi from 1, remember=\\wi as \\lastwi] in $h_{w_1}$, $h_{w_2}$, $h_{w_3}$ {\n\t\t\\draw[->] (ew\\wi) -- +(-0.5, 2em) node[draw, anchor=south, rounded corners=1pt, inner sep=0.3em] (bdec\\wi) {$\\overleftarrow{decoder}$};\n\t\t\\ifnum\\wi>1\n\t\t\t\\draw[->] (bdec\\wi) -- (bdec\\lastwi);\n\t\t\\fi\n\t}\n\t\n\t\\foreach \\w [count=\\wi from 1, remember=\\wi as \\lastwi] in $h_{w_1}$, $h_{w_2}$, $h_{w_3}$ {\n\t\t\\draw[->,red] (ew\\wi) to[bend right=30] +(+0.5, 5em) node[draw, anchor=south, rounded corners=1pt, inner sep=0.3em] (fdec\\wi) {$\\overrightarrow{decoder}$};\n\t\t\\draw[->] (bdec\\wi) -- (fdec\\wi);\n\t\t\\ifnum\\wi>1\n\t\t\t\\draw[->] (fdec\\lastwi) -- (fdec\\wi);\n\t\t\\fi\n\t}\n\t\n\t\\foreach \\y [count=\\yi from 1, remember=\\yi as \\lastyi] in $\\overrightarrow{e_1}$, $\\overrightarrow{e_2}$, $\\overrightarrow{e_3}$ {\n            \t\\draw[red,->] (fdec\\yi) -- +(0, 3em) node[text height=1.5ex, text depth=0.25ex, anchor=south] (fy\\yi) {\\y};\n            \t\\ifnum\\yi>1\n            \t\t\\draw[red,->] (fy\\lastyi) to[bend right=30] (fdec\\yi);\n            \t\\fi\n\t}\n\n        \\end{scope}\n\n\\end{tikzpicture}\n\\caption{High-level schema of the forward pass}\\label{pic:for-dec}\n\\end{figure}\n\nThe $\\logsoftmax$ function computes log-probabilities and it is thus suited for the loss-function used to learn the model described in the next section.\n\nWe note that the forward decoder is in fact a bidirectional decoder, as it uses both backward and forward hidden states $\\overrightarrow{h_{e_i}}$ and $\\overleftarrow{h_{e_i}}$ for the current prediction.\n\nThe hypothesis motivating the architecture of our neural models is the following: gated hidden layers such as LSTM and GRU can keep relatively long contexts in memory and to extract from them the information that is relevant to the current model prediction.\nThis is supported by the findings in recent works, such as \\cite{P18-2116}, which shows that most of the modelling power of gated RNN comes from their ability to compute at each step a context-dependent weighted sum on their inputs, in a way that is akin to dynamical attention mechanism.\nAs an immediate consequence, we think that using such hidden layers is an effective way to keep in memory a relatively long context on the output item level, that is labels, as well as on the input item level, that is words, characters and possibly other information.\n\nAn alternative, non-recurrent architecture, the Transformer model \\cite{46201} has been proposed with the goal of using attention mechanisms to overcome the learning issues of RNN in contexts where the learning signal has to back-propagate through very long paths.\nHowever, the recent work of \\cite{Dehghani2018UniversalT} shows that integrating a concept of recurrence in Transformers can improve their performances in some contexts.\nThis leads us to believe that recurrence is a fundamental feature for neural architectures for NLP and all of the domains where data are sequential by nature.\n\nAs a side note, the main features of the Transformer model - the multi-head attention mechanism and the skip connections \\cite{46201} - could in principle be integrated into our architecture.\nInvestigations of the costs and benefits of such additions is left for future work.\n\nFinally, while the decision function of our model remains local, its decisions are informed by global information at the word, character and label level thanks to the use of long-range contexts encoded by the GRU layers.\nIn that sense, it can be interpreted as an approximation of a global decision function and provides a viable alternative to the use of a CRF output layer \\cite{lample2016neural,Ma-Hovy-ACL-2016}.\n\n\\subsection{Learning}\n\\label{subsec:learning}\n\nOur models are learned by minimizing the negative log-likelihood $\\mathcal{LL}$ with respect to the data. Formally:\n\\begin{equation}\\label{eq:LL}\n    -\\mathcal{LL}(\\Theta | D) = -\\sum_{d=1}^{|D|} \\sum_{i=1}^{N_d} \\frac{1}{2}(\\text{log-p}(\\overrightarrow{e_i})+\\text{log-p}(\\overleftarrow{e_i}))  + \\frac{\\lambda}{2} \\left | \\Theta \\right |^2 )\n\\end{equation}\n$\\text{log-p}(\\overrightarrow{e_i})$ and $\\text{log-p}(\\overleftarrow{e_i})$ are the log-probabilities over predictions of the forward and backward decoders, respectively, we thus strengthen the global character of our model's predictions.\nThe first sum scans the learning data $D$ of size $|D|$, while the second sum scans each learning sequence $S_d$, of size $N_d$.\n\nGiven the relatively small size of the data we use for the evaluation, and the relatively high complexity of the models proposed in this paper, we add a $L_2$ regularization term to the cost function with a $\\lambda$ coefficient.\nThe cost-function is minimized with the \\emph{Back-propagation Through Time} algorithm (BPTT) \\cite{werbos-bptt}, provided natively by the \\emph{Pytorch} library (see section~\\ref{subsec:settings}).\n\n\\section{Evaluation}\n\\label{sec:eval}\n\n\\subsection{Data}\n\\label{subsec:data}\n\nWe evaluate our models on two tasks, one of Spoken Language Understanding (SLU), and one of POS tagging, namely \\emph{MEDIA} and \\emph{WSJ} respectively.\nThese tasks have been widely used in the literature \\cite{Vukotic.etal_2015,Vukotic.etal_2016,dinarelli_hal-01553830,Ma-Hovy-ACL-2016,2018-ijcai-lstm-ldrnn} and allow thus for a direct comparison of results.\n\n\\textbf{The French MEDIA corpus} \\cite{Bonneau-Maynard2006-media} was created for the evaluation of spoken dialog systems in the domain of hotel information and reservation in France.\nIt is made of $1~250$ human-machine dialogs acquired with a \\textit{Wizard-of-OZ} approach, where \\num{250} users followed \\num{5} different reservation scenarios.\n\nData have been manually transcribed and annotated with domain concepts, following a rich ontology.\nSemantic components can be combined to build relatively complex semantic classes.\\footnote{For example, the label \\texttt{localisation} can be combined with the components \\texttt{ville} (city), \\texttt{distance-relative} (relative-distance), \\texttt{localisation-relative-g\u00e9n\u00e9rale} (general-relative-localisation), \\texttt{rue} (street), etc.}\n\nStatistics on the training, development and test data of the MEDIA corpus are shown in table~\\ref{tab:MEDIAStats}.\nThe MEDIA task can be modelled as a sequence labelling task by segmenting concepts over words with the BIO formalism \\cite{Ramshaw95-BIO}.\nAn example of sentence with its semantic annotation is shown in table~\\ref{tab:ATIS-MEDIA-exemple}.\nFor exhaustive, we also show some word-classes available for this task, allowing models for a better generalization.\nHowever, our model does not use these classes, as explained in section~\\ref{subsec:settings}.\n\n\\textbf{The English corpus Penn Treebank} \\cite{Marcus93buildinga}, and in particular the section of the corpus corresponding to the articles of Wall Street Journal (WSJ), is one of the most known and used corpus for the evaluation of models for sequence labelling.\n\nThe task consists of annotating each word with its Part-of-Speech (POS) tag. We use the most common split of this corpus, where sections from \\num{0} to \\num{18} are used for training ($38~219$ sentences, $912~344$ tokens), sections from \\num{19} to \\num{21} are used for validation ($5~527$ sentences, $131~768$ tokens), and sections from \\num{22} to \\num{24} are used for testing ($5~462$ sentences, $129~654$ tokens).\n\n\\begin{table}[t]\n\t    \\centering\n\t    \\scriptsize\n\t    \\begin{tabular}{|ccc|}\n\t    \\hline\n\t    \\multicolumn{3}{|c|}{MEDIA corpus example} \\\\\n\t    \\textbf{Words} & \\textbf{Classes} & \\textbf{Labels} \\\\\n\t    \\hline\n        Oui & - & Answer-B \\\\\n        l' & - & BDObject-B \\\\\n        hotel & - & BDObject-I \\\\\n        le & - & Object-B \\\\\n        prix & - & Object-I \\\\\n        \u00e0 & - & Comp.-payment-B \\\\\n        moins & relative & Comp.-payment-I \\\\\n        cinquante & tens & Paym.-amount-B \\\\\n        cinq & units & Paym.-amount-I \\\\\n        euros & currency & Paym.-currency-B \\\\\n        \\hline\n\t    \\end{tabular}\n\t    \\caption{An example of sentence with its semantic annotation and word classes, taken from the French corpus MEDIA. The translation of the sentence in English is ``Yes, the hotel with a price less than fifty euros per night''}\n\t    \\label{tab:ATIS-MEDIA-exemple}\n    \\end{table}\n\n\\begin{table}[t]\n\\begin{minipage}{1.0\\linewidth}\n    \\centering\n    \\scriptsize\n    \\begin{tabular}{|l|rr|rr|rr|}\n      \\hline\n      & \\multicolumn{2}{|c|}{Training} & \\multicolumn{2}{|c|}{Validation} & \\multicolumn{2}{|c|}{Test}\\\\\n      \\hline\n      \\# sentences     &\\multicolumn{2}{|c|}{12~908} &\\multicolumn{2}{|c|}{1~259}&\\multicolumn{2}{|c|}{3~005} \\\\\n      \\hline\n      \\hline\n      & \\multicolumn{1}{|c}{Words} & \\multicolumn{1}{c|}{Concepts} &  \\multicolumn{1}{|c}{Words} & \\multicolumn{1}{c|}{Concepts} &\n      \\multicolumn{1}{|c}{Words} & \\multicolumn{1}{c|}{Concepts} \\\\\n      \\hline\n      \\# words          & 94~466 & 43~078 & 10~849 & 4~705 & 25~606 & 11~383 \\\\\n      \\# dict.         &  2~210 &     99 &    838 &    66 &  1~276 &     78 \\\\\n      \\# OOV\\%   & --     & --     &  1,33  & 0,02  &  1,39  &  0,04  \\\\\n      \\hline\n    \\end{tabular}\n    \\caption{Statistics on the French MEDIA corpus}\n  \\label{tab:MEDIAStats}\n  \\end{minipage}\n\\end{table}\n\n\\subsection{Experimental settings}\n\\label{subsec:settings}\n\nIn order to keep our architecture as general as possible, we limit our model inputs to the strict word (and character) information available in the raw text data and ignore the additional features available in the MEDIA dataset.\n\nFor convenience, the hyperparameters of our system have been tuned by simple independent linear searches on the validation data \u2014 rather than a grid search on the full hyperparameters space.\n\nAll of the parameters of neural layers are initialised with the Pytorch 0.4.1 default initializers\\footnote{Uniform random initialization for the GRU layers and \\cite{LeakyReLU-PReLU-2015} initialization for the linear layers.} and trained by SGB with a \\num{0.9} momentum for \\num{40} epochs on MEDIA, and ADAM optimizer for \\num{52} epochs on WSJ, keeping the model that gave the best accuracy on the development data set.\n\nFor training, we start with a learning rate of \\num{0.125} that we decay linearly after each epoch to end up at \\num{0} at the end of the chosen number of training epochs.\nFollowing \\cite{PracticalRecommendations-Bengio-2012}, we also apply a random dropout to the embeddings and the output of the hidden layers that we optimized to a rate of \\num{0.5}, and $L_2$ regularization to all the parameters with an optimal coefficient of \\num{e-4}.\n\nFinally, we have conducted experiments to find the optimal layer sizes, which gave us \\num{200}, \\num{150} and \\num{30} for word, labels and character embeddings respectively, \\num{100} for the $\\GRU_c$ layer and \\num{300} for all the other GRU layers.\nThose values are for the MEDIA task; for WSJ only the word embeddings and hidden layer sizes (respectively \\num{300} and \\num{150}) are different.\n\nIn order to reduce the training time, we use mini-batches of size\\footnote{Using larger batches is faster but degrades the overall accuracy.} \\num{100}.\nIn the current neural network frameworks, all the sequences in a mini-batch must have the same length, which we enforced at first by padding all of the sentences with the conventional symbol \\texttt{<s>} to the length of the longest one.\nHowever this caused two problems: first, there are a few unusually long sentences in the datasets we used, for instance, there is a single sentence of \\num{198} words in MEDIA.\nSecondly, in order to compute automatically the gradients of the parameters, Pytorch keeps in memory the whole graph of operations performed on the input of the model \\cite{paszke2017automatic}, which was far too large for the hardware we used, since for our model, we have to keep track of all the operations at all of the timesteps.\n\nWe found two solutions to these problems.\nThe first was to train on fixed-length, overlapping sub-sequences, or segments\\footnote{Shifting each segment one token ahead with respect to the previous}, truncated from the whole sentences, which did not appear to impair the performances significantly and allowed us to avoid more involved solutions such as back-propagation through time with memorization \\cite{NIPS2016_6221}.\nThe second was to cluster sentences by their length. This makes small clusters for unusually long sentences, which fit thus in memory, and big clusters of average-length sentences, which are further split into sub-clusters to have an optimal balance between the learning signals of different clusters, and alleviate us to find adaptive learning rates for different clusters.\n\nIn the optimization phase, we found out that the first solution works far better for the MEDIA task.\nWe believe that this is due to the noisy nature of the corpus (speech transcription), and to its relatively small size\nUsing fixed-length segments reduces the amount of noise the network must filter, while the fact that segments shift and overlap makes the network more robust, as it can see any token as the beginning of a segment, which in turns helps overcoming scarcity of the dataset.\nThis robustness is not needed when using bigger amount of grammatically well-formed textual data, like the WSJ corpus.\nIndeed the two solutions gave similar results on this corpus, we thus preferred sentence clusters which is a more intuitive solution and may better fit bigger data sets.\n\nAfter performing these optimization on the development set for each task, we kept the best models and evaluated them on the corresponding test sets, which we report and discuss in the next section.\n\nAll of our development and experiments were done on \\emph{2,1 GHz Intel Xeon E5-2620} CPUs and \\emph{GeForce GTX 1080} GPUs.\\footnote{1600 MHz, 2560 cores}.\n\n\\subsection{Results}\n\\label{subsec:results}\n\nResults presented in this section on the MEDIA corpus are means over \\num{10} runs, while results on the WSJ corpus are obtained in a single run, as it seems the most common practice.\\footnote{We can note that results over different runs on the WSJ have a very small variation, less or equal to \\num{0.01} accuracy points}\n\nConcerning the MEDIA task, since the model selection during the training phase is done based on the accuracy on the development data, we show accuracy in addition to F1 measure and Concept Error Rate (CER) as it is common practice in the literature on this task.\nF1 measure is computed with the script made available to the community for the \\emph{CoNLL} evaluation campaign.\\footnote{\\url{https:\/\/github.com\/robertostling\/efselab\/blob\/master\/3rdparty\/conlleval.perl}}. CER is computed by Levenshtein alignment between reference annotation and model hypothesis, with an algorithm much similar to the one implemented in the \\emph{sclite} toolkit.\\footnote{\\url{http:\/\/www1.icsi.berkeley.edu\/Speech\/docs\/sctk-1.2\/sclite.htm}}\n\nSince our model is similar to \\emph{Seq2seq} model, but it uses two decoders, in the remainder of this paper our model will be named \\emph{Seq2Biseq}.\nThe model training is performed using gold labels in the training data, while in test phase the model uses predicted labels to build left and right label-level contexts. This corresponds to the best strategy, according to \\cite{RNNforSLU-Interspeech-2013}.\n\nWe compare our results to those obtained by running the software developed for \\cite{dinarelli_hal-01553830}\\footnote{Available upon request at \\url{http:\/\/www.marcodinarelli.it\/software.php}} and tuning its hyperparameters\\footnote{The optimal settings being more or less those provided in the original article}.\n\n\\begin{table}[t]\n\\centering\n    \\begin{tabular}{|l|r|c|c|}\n\n        \\hline\n        \\textbf{Model}  & \\textbf{Accuracy} & \\textbf{F1 measure} & \\textbf{CER} \\\\ \\hline\n        \\hline\n        \\multicolumn{4}{|c|}{\\textbf{MEDIA DEV}}\\\\ \\hline\n        \\hline\n        Seq2Biseq           \t\t\t\t&\t89.11\t\t&\t85.59\t&\t11.46\t\t\\\\\n        Seq2Biseq$_{le}$      \t\t\t&\t89.42\t&\t86.09\t&\t10.58\t\\\\\n        Seq2Biseq$_{le}$ seg-len 15\t\t&\t\\textbf{89.97}\t&\t\\textbf{86.57}\t&\t\\textbf{10.42}\t\\\\\n        \\emph{fw}-Seq2Biseq$_{le}$ seg-len 15\t&\t89.51\t&\t85.94\t&\t11.40 \\\\\n        \\hline\n\n    \\end{tabular}\n\\caption{Comparison of results on the development data of the MEDIA corpus, with and without the lexical information (Seq2Biseq$_{le}$) as input to the modules $\\bGRU_{e}$ and $\\bGRU_{e}$}\n\\label{tab:lex-label-gru}\n\\end{table}\n\nConcerning our hypothesis about the capability of our models to encode a long-range context, and to filter out useless information with respect to the current labelling decision, we show results of two (sets of) experiments to validate such hypothesis.\n\nIn the first one, we compare the results obtained by models with and without the use of the lexical information as input to the decoders $\\bGRU_{e}$ and $\\bGRU_{e}$ (section~\\ref{subsec:label_rep}).\nThese results are shown in the first two lines of the table~\\ref{tab:lex-label-gru}.\nThe model using the lexical information is indicated with Seq2Biseq$_{le}$ in the table (for \\textbf{l}abels and l\\textbf{e}xical information).\nAs we can see in the table, this model obtains much better results than the one not using the lexical information as input to the label decoders.\nThis confirms that this information helps discriminating the semantic information provided by labels at a given processing step of the input sequence.\n\nIn the second experiment, we test the capability of our models to filter out useless semantic information, that is on the label side, for the current labelling decision.\nIn order to do this, we increase the size of the segments in the learning phase: \\num{15} instead of \\num{10} by default.\nIt is important to note that in the context of a SLU task, where input sequences are transcriptions of human speech, using longer segments is possibly risky, since a longer context may be much more noisy even if it is slightly more informative.\n\nMoreover, the models in the literature applied to the MEDIA task and using a fixed-size window to capture contextual information, never use a window wider than \\num{3} tokens around the current token to be labelled.\nThis confirms the difficulty to extract useful information from a longer context.\nResults of this experiment are shown in the third line of table~\\ref{tab:lex-label-gru}.\nOur hypothesis seems to be also valid in this case, as models using segments of length \\num{15} obtain better results than those using the default size of \\num{10} and this with respect to all the evaluation metrics.\n\nWe note that, while the effectiveness of the decoder's architecture of the \\emph{Seq2seq} model does not need any more to be proved, these results still provide possibly interesting analyses in the particular context of sequence labelling.\\footnote{The \\emph{Seq2seq} model has been designed and mainly used for machine translation}\n\nIn order to show the advantage provided by the use of two decoders instead of only one like in the original \\emph{Seq2seq} model, we show results obtained using only one decoder for the left label-side context in table~\\ref{tab:lex-label-gru}\nThese results are indicated in the table with \\emph{fw-Seq2Biseq$_{le}$ seg-len 15} (this model corresponds basically to the original \\emph{Seq2seq}). This model is exactly equivalent to our best model \\emph{Seq2Biseq$_{le}$ seg-len 15}, the only difference is that it uses only the left label context. As we can see, this model is much less effective than the version using two decoders, which also confirms that the right context on the output side (labels) is very informative.\n\nOur hypothesis concerning the \\emph{aggregation} specialization of our model during the learning phase seems also confirmed (section~\\ref{subsec:label_rep}).\nThe fact that the Seq2Biseq$_{le}$ model obtains better results than the simpler model Seq2Biseq tends to confirm the hypothesis.\n\nIndeed, if the model Seq2Biseq$_{le}$ gave more importance to the lexical information than the semantic information given by labels at the input of the decoders $\\bGRU_{e}$ and $\\fGRU_{e}$, its better results would not have a clear explanation, as both Seq2Biseq$_{le}$ and Seq2Biseq models (table~\\ref{tab:lex-label-gru}) use the lexical information separately (indicated with $h_{w_i}$ in the equation~\\ref{eq:lex-rep}).\n\nSince the information provided by labels alone is already taken into account by the model Seq2Biseq, we can deduct that the Seq2Biseq$_{le}$ model can extract more effective semantic representations, and this even when we provide it with longer contexts (with segments of size $15$).\n\n\\begin{table}[t]\n\\centering\n    \\begin{tabular}{|l|r|c|c|c|}\n\n        \\hline\n        \\textbf{Model}  & \\textbf{Accuracy} & \\textbf{F1 measure} & \\textbf{CER} & \\textbf{p-value}\\\\ \\hline\n        \\hline\n        \\multicolumn{5}{|c|}{\\textbf{MEDIA DEV}}\\\\ \\hline\n        \\hline\n        LD-RNN$_{\\mathrm{deep}}$\t\t\t\t&\t89.26 (0.16)\t&\t85.79 (0.24)\t&\t10.72 (0.14) & -- \\\\ \\hline\n        Seq2Biseq$_{le}$ seg-len 15     \t& 89.97 (0.20)\t& 86.57 (0.22)\t& 10.42 (0.26) & 0.043 \\\\\n        Seq2Biseq$_{\\text{2-opt}}$     \t& \\textbf{90.22} (0.14)\t& \\textbf{86.88} (0.16)\t& \\textbf{9.97} (0.24) & -- \\\\ \\hline\n        \\hline\n        \\multicolumn{5}{|c|}{\\textbf{MEDIA TEST}}\\\\ \\hline\n        \\hline\n        LD-RNN$_{\\mathrm{deep}}$\t\t\t\t&\t89.51 (0.21)\t&\t87.31 (0.19)\t& 10.02 (0.17) & -- \\\\ \\hline\n        Seq2Biseq$_{le}$ seg-len 15     \t& 89.57 (0.12)\t& 87.50 (0.17)\t& 10.26 (0.19) & 0.047 \\\\\n        Seq2Biseq$_{\\text{2-opt}}$     \t& \\textbf{89.79} (0.22)\t& \\textbf{87.69} (0.20)\t& \\textbf{9.93} (0.28) & -- \\\\\n        \\hline\n\n    \\end{tabular}\n    \\caption{Comparison of results obtained on the MEDIA corpus by the system LD-RNN$_{\\mathrm{deep}}$, ran by ourselves for this work, and our model Seq2Biseq$_{le}$, using segments of size $15$ (see section~\\ref{subsec:settings}).}\n\\label{tab:ldrnn-vs-gru-ldrnn-clen15}\n\\end{table}\n\nIn another set of experiments, we compared our model with the one proposed in \\cite{dinarelli_hal-01553830}, from which we inspired our neural architecture.\nWe downloaded the software associated to the paper\\footnote{Described at \\url{http:\/\/www.marcodinarelli.it\/software.php} and available upon request}, and we ran experiments on the MEDIA corpus in the same conditions as our experiments. We used the deep variant of the model described in \\cite{dinarelli_hal-01553830}, LD-RNN$_{\\mathrm{deep}}$, which gives the best results on MEDIA. The results of these experiments are shown in the table~\\ref{tab:ldrnn-vs-gru-ldrnn-clen15}.\nAs we can see in the table, on the development data of the MEDIA task (MEDIA DEV), our model is more effective than the LD-RNN$_{\\mathrm{deep}}$ of \\cite{dinarelli_hal-01553830}, which holds the state-of-the-art on this task.\nThese gains are also present for the test data (MEDIA TEST), even if they are smaller, and the LD-RNN$_{\\mathrm{deep}}$ model is still the more effective in terms of Concept Error Rate (CER).\n\nWe would like to underline that we did not perform an exhaustive optimization of all the hyper-parameters.\\footnote{This because it takes a lot of time, but more importantly because we believe a good model should give good results without too much effort, otherwise a previous model which already proved comparably effective should be preferred}\nAs we can see in table~\\ref{tab:ldrnn-vs-gru-ldrnn-clen15}, results obtained with the model LD-RNN$_{\\mathrm{deep}}$ on the test data are always better than those obtained on the development data. In contrast, our model obtains a worse accuracy, which leads the model selection in the training phase, on the test data. This lack of generalization may indicate a sub-optimal parameter choice or an over-training problem.\n\nIn the table~\\ref{tab:ldrnn-vs-gru-ldrnn-clen15} we also report standard deviations on the \\num{10} experiments (between parentheses), and the results of the significance tests performed on the output of our model and of the model LD-RNN$_{\\mathrm{deep}}$. We used the significance test described in \\cite{Yeh00moreaccurate}, which applies on the output of the two compared systems, and it is suited for the evaluation metrics used most often in NLP.\\footnote{In contrast to several other significance tests, this test doesn't make any assumption on the classes independence, nor on the representative coverage of the sample} We re-implemented the significance test script based on the one described in \\cite{sigf06}.\\footnote{\\url{https:\/\/nlpado.de\/~sebastian\/software\/sigf.shtml}}\nOur model is compared to the LD-RNN$_{\\mathrm{deep}}$ model in terms of F1 measure, which is more constraining than the accuracy and as constraining as the CER.\nThe result of the significance test is given in the column \\emph{p-value} of the table, and it represents the probability that the gain is not significant. Most often the gains are considered significant with a p-value equal or smaller than $0.05$.\n\n\\begin{table}[t]\n\\centering\n    \\begin{tabular}{|l|rcc|}\n\n        \\hline\n        \\textbf{Model}  & \\textbf{Accuracy} & \\textbf{F1 measure} & \\textbf{CER} \\\\ \\hline\n        \\hline\n        \\multicolumn{4}{|c|}{\\textbf{MEDIA TEST}}\\\\ \\hline\n        \\hline\n        BiGRU+CRF \\cite{dinarelli_hal-01553830}\t\t&\t-- \t&\t86.69\t\t&\t10.13\t\\\\\\hline\n        LD-RNN$_{\\mathrm{deep}}$ \\cite{dinarelli_hal-01553830}\t& -- \t\t&\t87.36\t\t&\t\\textbf{9.8}\t\t\\\\\n        LD-RNN$_{\\mathrm{deep}}$\t\t\t\t&\t89.51\t\t&\t87.31\t\t&\t10.02\t\t\\\\\\hline\n        Seq2Biseq$_{le}$ seg-len 15     \t&\t89.57\t&\t87.50\t&\t10.26\t\\\\\n        Seq2Biseq$_{\\text{2-opt}}$     \t& \\textbf{89.79}\t& \\textbf{87.69}\t& 9.93 \\\\ \\hline\n\n    \\end{tabular}\n\\caption{Comparison of results on MEDIA with our best models and the best models in the literature}\n\\label{tab:gru-ldrnn-clen15-vs-SOTA-media}\n\\end{table}\n\nWe ran another set of experiments on the MEDIA task with our best model in order to compare to the best models in the literature on this task, which are those described in \\cite{dinarelli_hal-01553830}.\nIn particular we compared our results to the models using a neural CRF output layer for modelling label sequences and take global decisions.\n\nThe results of these experiments are shown in the table~\\ref{tab:gru-ldrnn-clen15-vs-SOTA-media}.\nIn this table we indicate simply with LD-RNN$_{\\mathrm{deep}}$ the results obtained in our experiments using the software \\emph{LD-RNN}\\footnote{\\url{http:\/\/www.marcodinarelli.it\/software.php}}, while we add the reference \\cite{dinarelli_hal-01553830} after LD-RNN$_{\\mathrm{deep}}$ to indicate that results have been taken directly from the reference.\nAs we can see, the only new outcome in this table with respect to those already shown in previous tables, is the best CER of $9.8$ obtained by the model LD-RNN$_{\\mathrm{deep}}$ published in \\cite{dinarelli_hal-01553830}.\nThese results are obtained however using also the word-classes available with the MEDIA corpus. Our model is still more effective than the others in terms of accuracy and F1 measure, providing thus the new state-of-the-art results on this task.\n\nThe experiments performed on the MEDIA task with different variants of our model allowed us to find the best neural architecture for sequence modelling. In order to have a more general view on the effectiveness of our model on the problem of sequence labelling, we performed some experiments of POS tagging on the WSJ corpus, which is a well-known benchmark for sequence labelling, used since more than $15$ years.\nIn order to show the effectiveness of the model alone, without the impact of any external resources, we performed experiments without using pre-trained embeddings. This is however a quite common practice and can lead to remarkable improvements \\cite{Ma-Hovy-ACL-2016}.\n\nOn this task we compare to the model \\emph{LD-RNN$_{\\mathrm{deep}}$} of \\cite{dinarelli_hal-01553830}, and to the model \\emph{LSTM-CRF} of \\cite{Ma-Hovy-ACL-2016}. To the best of our knowledge the latter is one of the rare work on neural models where results are given also without pre-trained embeddings, allowing a direct comparison. The \\emph{LSTM-CRF} model is moreover one of the best models on the WSJ corpus when using embeddings pre-trained with GloVe \\cite{pennington2014glove}.\n\nThe results of the POS tagging task on the WSJ corpus are shown in the table~\\ref{tab:comp-WSJ}. As we can see our model obtains the best results among those not using any pre-trained embeddings.\nOur results are however worse than those obtained with pre-trained embeddings, which constitute the state-of-the-art on this task.\nIn this respect, we would like to underline that the overall best results are obtained with a neural model described in \\cite{2018-ijcai-lstm-ldrnn}. This model is only slightly better than the \\emph{LSTM-CRF} model, which we outperform when not using pre-trained embeddings. Moreover the model proposed in \\cite{2018-ijcai-lstm-ldrnn} (\\emph{LSTM+LD-RNN} in the table) is very similar to our model.\n\n\\begin{table}[t]\n\\centering\n    \\begin{tabular}{|l|c|c|}\n\n        \\hline\n        \\textbf{Model}  & \\multicolumn{2}{|c|}{\\textbf{Accuracy}} \\\\ \\hline\n        \\hline\n        & WSJ DEV & WSJ TEST \\\\ \\hline\n        \\hline\n        LD-RNN$_{\\mathrm{deep}}$\t\t\t\t\t\t\t& 96.90\t& 96.91 \\\\\n        LSTM+CRF \\cite{Ma-Hovy-ACL-2016}\t\t\t\t& -- \t\t& 97.13 \\\\\n        Seq2Biseq\t\t\t\t\t\t\t\t& 97.13\t& 97.20 \\\\\n        Seq2Biseq$_{\\text{2-opt}}$\t\t\t\t\t\t& \\textbf{97.33}\t& \\textbf{97.35} \\\\\n        \\hline\n        \\hline\n        LSTM+CRF + Glove \\cite{Ma-Hovy-ACL-2016}\t\t& 97.46 \t& 97.55 \\\\\n        LSTM+LD-RNN + Glove \\cite{2018-ijcai-lstm-ldrnn}\t& -- \t\t& 97.59 \\\\ \\hline\n\n    \\end{tabular}\n\\caption{Comparison of our model with the model \\emph{LD-RNN$_{\\mathrm{deep}}$}, and the best models of the literature, on the POS tagging task of the WSJ corpus}\n\\label{tab:comp-WSJ}\n\\end{table}\n\nIn order to compare our model to the model LD-RNN$_{\\mathrm{deep}}$ also in terms of complexity and computation efficiency, we show in the table~\\ref{tab:comp-params-train-time} the number of parameters as well as the training time on the MEDIA and WSJ corpora.\nFor the sake of completeness, we also report the number of parameters of the other models mentioned in this paper. Except for the model \\emph{GRU+CRF} for which we took the number of parameters from the reference \\cite{dinarelli_hal-01553830} (hidden layers of size $200$), all the other numbers are computed based on the same layer sizes.\n\nWe can see in the table~\\ref{tab:comp-params-train-time} that the training time for our model is longer than for the model LD-RNN$_{\\mathrm{deep}}$ on the MEDIA task.\nThis is because our neural architecture is quite more complex, and since the corpus is relatively small, we can not fuly take advantage of GPU parallelism.\n\nThis is confirmed on the WSJ corpus, where the training time of our model is much smaller than the time needed by the LD-RNN$_{\\mathrm{deep}}$ model, despite this corpus is quite bigger than MEDIA.\\footnote{The model LD-RNN$_{\\mathrm{deep}}$ is coded in Octave, and while it can run on GPUs, this framework is not fully optimized to scale on GPUs}\nThe time needed for testing are not reported in the table, we can note that they are negligible for both models, as it never exceeded a few minutes\n\n\\begin{table}[t]\n\\centering\n    \\begin{tabular}{|l|c|c|c|}\n\n        \\hline\n        \\textbf{Model}  & \\textbf{\\# of parameters} & \\multicolumn{2}{|c|}{\\textbf{Training time}} \\\\ \\hline\n        \\hline\n        & MEDIA & MEDIA & WSJ \\\\ \\hline\n        \\hline\n        Seq2Biseq$_{le}$\t\t\t\t\t& 2,139,950 & 3h30' & 16h-17h \\\\\n        LD-RNN$_{\\mathrm{deep}}$\t\t\t\t\t& 2,551,700 & 1h30' & $>$ 6 days \\\\\n        \\hline\n        \\hline\n        GRU+CRF \\cite{dinarelli_hal-01553830}\t& 2,328,360 & -- & -- \\\\\n        \\emph{Seq2seq}\t\t\t\t\t\t& 1,703,450 & -- & -- \\\\\n        \\emph{Seq2seq+Att.}\t\t\t\t\t& 2,244,050 &-- & -- \\\\ \\hline\n\n    \\end{tabular}\n\\caption{Comparison of the neural models proposed or mentioned in this paper, in terms of number of parameters, and of training time for our model and the the model LD-RNN$_{\\mathrm{deep}}$}\n\\label{tab:comp-params-train-time}\n\\end{table}\n\nWhile the results described in this paper can be considered satisfactory, considering the complexity of our neural network with respect to the LD-RNN$_{\\mathrm{deep}}$ model, we were  surprised to find out that the gains were not larger on the MEDIA task.\nAt first we thought that our network suffered from overfitting on such a small task, and given the complexity of our network, nothing could be done to solve this problem beyond reducing the total number of parameters.\nHowever, after a quick analysis of the output of our model on the MEDIA development data, we found clear signs revealing that our model was actually ignoring the learning signal coming from the backward decoder (eq.~\\ref{eq:backward-model}).\n\nSince our neural network was explicitly designed to take both left and right label-side contexts into account, we thought that the problem was coming from the learning phase. In particular we thought that our model was underfitting due to the problem of \\textit{very-long back-propagation paths} described in \\cite{46201}, and which motivated the design of the Transformer model, without recurrent layers and with skip connections to enforce the back-propagation of the learning signal.\nWe adopted a different approach: we applied two different optimizers to the two decoders, one for a negative log-likelihood computed with the output of the backward decoder (only $\\text{log-p}(\\overleftarrow{e_i})$, see eq.~\\ref{eq:backward-model}), and another one for the global negative log-likelihood computed from the output of both forward and backward decoders (see equation~\\ref{eq:LL}).\nWe note that the forward decoder also uses predictions and hidden states of the backward decoder, the second optimizer thus also refines the parameters of the backward decoder with left, forward information.\n\nWe ran new experiments in exactly the same conditions as described before, the only difference being that we used these two optimizers.\nThe final results are reported in table~\\ref{tab:ldrnn-vs-gru-ldrnn-clen15} for MEDIA and in the table~\\ref{tab:comp-WSJ} for the WSJ, where the model learned using two optimizers is indicated with Seq2Biseq$_{\\text{2-opt}}$.\n\nAs we can see in the tables, the results improved on both tasks, on both development and test data, and in terms of all the evaluation metrics.\nTo the best of our knowledge, the results obtained on MEDIA are the best on this task, except for the CER where the model LD-RNN$_{\\mathrm{deep}}$ using class features is still the best (9.8 vs. our 9.93 on the test set). Also, the results obtained on the WSJ corpus are the best obtained without any external resource and without pre-trained embeddings. We leave the integration of pre-trained embeddings as future work.\n\n\\section{Conclusions}\n\\label{sec:Conclusions}\n\nIn this article, we propose a new neural architecture for sequence modelling heavily based on GRU recurrent hidden layers.\nWe use these layers to encode long-range contextual information at several levels: words, characters and labels.\n\nOur main contributions are the use of two different decoders for label prediction, one modelling a backward (future, or right) label context, and one for a forward label context.\nThe combination of the two contexts allow our model to take labelling decisions informed by a global context, approximating a global decision function.\nAnother contribution is the use of two different optimizers to optimize separately the two decoders.\nThis improves even further the results obtained on the two evaluation tasks studied in this work.\n\nThe results obtained are state-of-the-art on the MEDIA task.\nOn the POS tagging task of the WSJ corpus, our results are state-of-the-art if we do not consider the models that use pre-trained word embeddings, and still close to the state-of-the-art if we do so.\n\n\\section{Acknowledgements}\n\nThis work is part of the \"Investissements d'Avenir\" overseen by the French National Research Agency ANR-10-LABX-0083 (Labex EFL), and is also supported by the ANR DEMOCRAT (Describing and Modelling Reference Chains: Tools for Corpus Annotation and Automatic Processing) project ANR-15-CE38-0008.\n\n\\bibliographystyle{splncs2016}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.8\\columnwidth]{images\/new-teaser-v1.PNG}\n    \\caption{Our VRAG encodes video-level embeddings from spatial intra-frame and temporal inter-frame information for CBVR.} \\vspace{-3mm}\n    \\label{fig:teaser}\n\\end{figure}\n\nThe volume of videos on the Internet has grown exponentially with the inception of media-sharing websites such as Facebook, Twitch and Youtube. \nContent-based Video Retrieval (CBVR) is important on these platforms for applications such as video recommendation and video filtering.\n\nIn CBVR, evaluating the video similarity is a key component. There are predominantly two types of approaches for inferring video-to-video similarity: frame-level approaches and video-level approaches. Video-level methods encode videos into fixed size embeddings, and measure video similarity through the similarity of their embeddings. On the other hand, frame-level methods derive video similarity by aggregating pairwise similarities between video frames. These two approaches form a dichotomy. Video-level approaches offer faster evaluation speeds, while frame-level methods provide more accurate video similarities at significant overheads. Although practical implementations of frame-level methods may employ optimizations such as summarizing the video into a shorter sequence of frames~\\cite{vid-summary:1708-09545, vid-summary:1812-01969, vid-summary:zhou2018deep}, video-level methods remain orders of magnitude more efficient than frame-level methods when the number of videos scale to the size of billions. Consequently, video-level methods remain highly relevant for real-world applications despite its inferior performance to frame-level methods.\n\nRecently, several video-level CBVR works~\\cite{lbow, Baraldi2018LAMVLT, kordopatis2017dml, baseline:tmk} propose the same design principle of extracting local frame-level features, and then aggregating these features into fixed-size video embeddings. As a result, most of these works focus on proposing a better frame-level feature extractor and\/or video-level feature pooling methods. For example, DML~\\cite{kordopatis2017dml} and LBoW~\\cite{lbow} extract Maximum Activation of Convolution (MAC) features from each frame and then apply mean pooling and bag-of-words, respectively, to get video-level representations. LAMV~\\cite{Baraldi2018LAMVLT} later propose representing frames at a finer granularity using Region Maximum Activation of Convolutions (R-MAC) and learnable pooling of these frame descriptors in the Fourier Domain~\\cite{baseline:tmk}. While prior works have made strides in frame-level feature extraction and video-level pooling, these approaches encode each frame separately, and do not model the spatial and temporal interactions inherent within videos. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{images\/shot-level-video-retrieval-smaller.PNG}\n    \\caption{Our VRAG in a shot-level video retrieval setting.} \\vspace{-3mm}\n    \\label{fig:shot-retrieval-diagram}\n\\end{figure}\n\nIn view that prior video-level approaches lack modelling of spatio-temporal interactions within videos, we propose VRAG: a region attention graph-based framework for content-based video retrieval. As shown in Figure~\\ref{fig:teaser}, our VRAG models videos at the fine-grained region-level as a graph. Each node of the graph represents a R-MAC feature vector. To model interactions between region nodes, we encode spatial relations through complete subgraphs between regions in the same frame and temporal relations through fully connected edges across adjacent frames. We further augment these spatial and temporal connections through self-attention, which modulates the strength of these associations through the affinities between their region-level content. Consequently, we transform each region into context-aware embeddings by selectively aggregating features from neighboring regions via Graph Attention~\\cite{graph-attention} layers. To generate video-level embeddings, we learn attention weights for each region and selectively aggregate region-level embeddings into a fixed size representation. We model our attention pooling on the intuition that important video regions add significant context to other regions and derive the attention weights from the pairwise affinities between regions across multiple Graph Attention layers. \n\nOur VRAG improves the state-of-the-art for video-level approaches across video retrieval tasks~\\cite{dataset:fivr200k, dataset:evve, dataset:cc-web-video}, and reduces the performance gap between video-level and frame-level retrieval approaches. On Event Video Retrieval~\\cite{dataset:evve}, our VRAG also achieves higher retrieval scores over the state-of-the-art for frame-level retrieval, ViSiL~\\cite{kordopatiszilos2019visil}, while being more than 50$\\times$ faster. We further propose to reduce the gap between video-level and frame-level approaches by segmenting videos into shots and representing videos over multiple shot embeddings using our VRAG, as illustrated in Figure~\\ref{fig:shot-retrieval-diagram}. Our shot-level VRAG evaluates the similarity between videos by aggregating pairwise similarities between their shot embeddings. In our experiments, we show that our shot-level VRAG bridges the gap between video-level and frame-level approaches on most video retrieval tasks~\\cite{dataset:fivr200k, dataset:cc-web-video}\nat faster evaluation speeds than frame-level methods, and higher retrieval precision over video-level approaches. \n\n\\section{Related Work}\n\n\\subsection{Video Retrieval}\n\n\\textbf{Multi-modal video retrieval} performs video retrieval through multi-modal embeddings. Recent methods use pretrained expert networks to extract video representations across different modalities, e.g. optical flow, audio, and introduce novel techniques to fuse these representations into a multi-modal embedding~\\cite{collaborative-experts, moEE, mm-transformer}. \\cite{collaborative-experts} models pairwise interactions between modalities and aggregates across modalities via attention pooling while \\cite{mm-transformer} use Transformers~\\cite{attention-is-all-you-need} to fuse features from different modalities.\n\n\\textbf{Video-to-text retrieval} is a popular instance of cross-modal video retrieval, where relevant videos are retrieved given a query sentence and vice versa. The video-to-text pipeline extracts video and text representations and map related video-text pairs into the same representation~\\cite{collaborative-experts, howto100m}.\n\n\\subsection{Content-based Video Retrieval (CBVR)}\nCBVR methods can be broadly classified into video-level~\\cite{baseline:tmk, Baraldi2018LAMVLT, kordopatis2017dml, lbow} and frame-level~\\cite{temporal_network, baseline:dp, kordopatiszilos2019visil} approaches.\n\n\\textbf{Video-level approaches} encode videos into fixed size embeddings, i.e.  vectors. These methods extract features from video frames and temporally pool frame features into a fixed size representation. The similarity between videos is then determined through the similarities between their embeddings e.g. cosine similarity. Compared to prior video-level~\\cite{kordopatis2017dml, lbow, baseline:tmk, Baraldi2018LAMVLT} methods that encode video frames independently, our VRAG models spatio-temporal relations between\/within frames through Graph Convolution, and adds video-level context into the frame representations.\n\n\\textbf{Frame-level approaches}~\\cite{temporal_network, baseline:dp} represent videos as sequences of video frame features that scale with the video length. To evaluate video similarity, these methods aggregate pairwise similarities between video frames to derive the similarity between videos. Recently, ViSiL~\\cite{kordopatiszilos2019visil} propose using Convolutional Neural Networks (CNN) layers and the (Symmetric) Chamfer Similarity to aggregate pairwise frame similarities, and achieves state-of-the-art frame-level CBVR. However, ViSiL requires $n^2$ forward passes to evaluate pairwise similarities between $n$ videos, while our VRAG is orders of magnitude faster since it uses only $n$ forward passes to generate video embeddings.\n\n\\section{Our Approach}\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.85\\textwidth]{images\/wider-fig.PNG}\n    \\caption{Our Video Region Attention Graph Network (VRAG). (Left) Graph structure. (Right) VRAG Network.}\n    \\label{fig:VRAG-architecture}\n\\end{figure*}\n\nWe present our Video Region Attention Graph (VRAG) network in Figure~\\ref{fig:VRAG-architecture}. \n\n\\textbf{\\label{sect:frame-repr}Input Frame Representation.} Following ViSiL, we represent each video frame using concatenated Region Maximum Activation of Convolution~\\cite{rmac} (R-MAC) features from intermediate convolution layers. We pass each video frame through a pretrained ConvNet, i.e. ResNet50~\\cite{resnet} pre-trained on ImageNet, with $L$ layers to generate intermediate activation maps, $f_\\text{conv}: \\mathcal{I}^{(t)} \\mapsto \\left\\{\\mathcal{M}^{(t)}_1, \\ldots, \\mathcal{M}^{(t)}_L\\right\\}$. Given that each intermediate activation map $\\mathcal{M}_l^{(t)} \\in \\mathbb{R}^{H_l \\times W_l \\times C_l}$ has varying spatial dimensions, we transform the activation maps into the same spatial dimension by defining different R-MAC kernel sizes for each layer. Subsequently, we concatenate the intermediate R-MAC features channel-wise to represent each video frame, i.e. $f_\\text{rmac}: \\left\\{\\mathcal{M}^{(t)}_1, \\ldots, \\mathcal{M}^{(t)}_L\\right\\} \\mapsto \\mathcal{X}^{(t)} \\in \\mathbb{R}^{R \\times C}$, where $R$ refers to the number of R-MAC features in each frame, and \n$C=\\sum_{l=1}^{L}C_l$\nis the dimensions of the concatenated vector. We then encode a video of length $T$ into $N=T\\times R$ R-MAC features, i.e. $\\mathbf{X} = \\left[\\mathcal{X}^{(1)}|\\mathcal{X}^{(2)}| \\ldots| \\mathcal{X}^{(T)}\\right] \\in \\mathbb{R}^{N \\times C}$.\n\n\\textbf{Spatial and Temporal Graph Structure.}\nWe represent each R-MAC feature vector $\\mathbf{x}_i \\in \\mathbf{X}=\\left[\\mathbf{x}_1, \\ldots, \\mathbf{x}_N\\right] \\in \\mathbb{R}^{N \\times C}$ as a node $\\mathcal{V}_i$ in the graph $\\mathcal{G} = \\{\\mathcal{V}, \\mathcal{E}_\\text{spatial}, \\mathcal{E}_\\text{temporal}\\}$ of our VRAG.\nWe then add two sets graph edges: 1) $\\mathcal{E}_\\text{spatial}$ is a set of edges that connects region nodes in the same frame $\\mathcal{I}^{(t)}$ to capture spatial relationships between these nodes with a frame and this includes a self referencing edge. 2) $\\mathcal{E}_\\text{temporal}$ refers to edges that connect the regions between consecutive frames $\\mathcal{I}^{(t-1)}$ and $\\mathcal{I}^{(t+1)}$ to capture temporal relations between video frames. \n\n\\textbf{Learning Region Embeddings.} From the video region graph $\\mathcal{G} = \\{\\mathcal{V}, \\mathcal{E}_\\text{spatial}, \\mathcal{E}_\\text{temporal}\\}$ constructed earlier, we learn region-level embeddings using Graph Convolutional Network~\\cite{gcn, graph-attention} (GCN) layers. To increase the duration of videos that can be processed, we first reduce the dimensions of R-MAC region vectors, i.e. $\\mathbf{X} \\in \\mathbb{R}^{N\\times C}$ using a fully-connected layer with non-linear activation, i.e. $f_{r}: \\mathbf{X} \\mapsto \\mathbf{X}^{(0)} \\in \\mathbb{R}^{N\\times C^\\prime}$.\nWe then pass the region vectors through $K$ Graph Attention~\\cite{graph-attention} layers to generate intermediate region features, i.e. $g^{(k)}: \\mathbf{X}^{(k-1)} \\mapsto \\mathbf{X}^{(k)} \\in \\mathbb{R}^{N \\times C^\\prime}$ for $k \\in \\left[1, \\ldots, K\\right]$. Specifically, we aggregate the features from neighboring regions as follows: \n\n\\paragraph{1)} To enable interactions between neighboring regions with complementary representations, we use the key-query self-attention mechanism~\\cite{attention-is-all-you-need} to describe the similarity between regions. Specifically, we generate key and query embeddings using linear transformations from the input region vectors $\\mathbf{x}_i^{(k-1)}$, i.e.:\n\\begin{equation}\n\\small\n   \n        f_\\text{query}^{(k)} : \\mathbf{x}_i^{(k-1)} \\mapsto \\mathbf{a}_i^{(k)} \\in \\mathbb{R}^{C^\\prime}, \\quad\n        f_\\text{key}^{(k)} : \\mathbf{x}_i^{(k-1)} \\mapsto \\mathbf{b}_i^{(k)} \\in \\mathbb{R}^{C^\\prime}.\n   \n    \\label{eqn:kq-embeddings}\n\\end{equation}\n\n\\vspace{-2mm}\n\\paragraph{2)} The similarity between a region $i$ and its neighboring region $j \\in \\mathcal{N}(i)$ is defined using the dot product of their query and key embeddings, i.e. $s_{ij}^{(k)}= \\mathbf{a}_i^{(k)} \\cdot \\mathbf{b}_{j}^{(k)}$. The output of the Graph Attention layer $g^{(k)}$ is derived as:\n\\begin{equation}\n   \n       w_{ij}^{(k)} = \\frac{e^{s_{ij}^{(k)}}}{\\sum_{n \\in \\mathcal{N}(i)}{e^{s_{in}^{(k)}}}}, \\quad\n       f^{(k)} : \\sum_{j \\in \\mathcal{N}(i)} w_{ij}^{(k)} \\mathbf{x}_{j}^{(k-1)} \\mapsto \\mathbf{x}_i^{(k)},\n   \n\\end{equation}\nwhere $f^{(k)}$ is a linear layer with a non-linear activation.\n\n\\vspace{-2mm}\n\\paragraph{3)} Finally, given that adding graph convolution layers may lead to over-smoothing of region representations~\\cite{gcn:over-smoothing1, chen2019measuring}, we propose concatenating region representations along the depth of our network to preserve discriminative features, i.e. $f_\\text{concat}: \\left\\{\\mathbf{X}, \\mathbf{X}^{(0)}, \\ldots, \\mathbf{X}^{(K)}\\right\\} \\rightarrow \\mathbf{R} \\in \\mathbb{R}^{N \\times \\left(C + (K+1)C^\\prime\\right)}$.\n\n\\textbf{Video Embedding.} We aggregate region embeddings $\\mathbf{R}=[\\mathbf{r}_1, \\ldots, \\mathbf{r}_N] \\in \\mathbb{R}^{N \\times \\left(C + (K+1)C^\\prime\\right)}$ using weighted sum aggregation to generate a pooled region representation:\n    $\\overline{\\mathbf{r}}=\\sum^{N}_{i=1}\\beta_i \\mathbf{r}_i$,\nwhere $\\beta_i$ is the attention weight for region $\\mathbf{r}_i$. The pooled representation $\\overline{\\mathbf{r}}$ is fed into two linear layers to generate the video-level embedding $\\mathbf{v}$, i.e.: $f_\\text{mlp}:\\overline{\\mathbf{r}} \\mapsto \\mathbf{v} \\in \\mathbb{R}^D$. We model our attention weights $\\boldsymbol{\\beta} = [\\beta_1, \\ldots, \\beta_N] \\in \\mathbb{R}^N$ from the intuition that important regions in a video act as anchor regions that add context to other regions, and should have high affinities with other regions in the video. Therefore, we compute the attention weights $\\boldsymbol{\\beta}$ as follows:\n\\vspace{-2mm}\n\\paragraph{1)} We reuse the key and query embeddings of each Graph Attention layer $g^{(k)}$ from Equation~\\ref{eqn:kq-embeddings} to compute $K$ pairwise affinity matrices $\\mathcal{A}^{(k)}$ between regions, i.e.\n\\begin{equation}\n    \\begin{aligned}\n    \\mathcal{A}^{(k)}&=\\mathbf{A}^{(k)} \\times \\left(\\mathbf{B}^{(k)}\\right)^\\top \\in \\mathbb{R}^{N \\times N}, \\\\\n    \\mathbf{A}^{(k)} &= [\\mathbf{a}_i^{(k)}, \\ldots, \\mathbf{a}^{(k)}_N] \\in \\mathbb{R}^{N \\times C^\\prime}, \\\\\n    \\mathbf{B}^{(k)} &= [\\mathbf{b}_i^{(k)}, \\ldots, \\mathbf{b}^{(k)}_N] \\in \\mathbb{R}^{N \\times C^\\prime}. \\\\\n    \\end{aligned}\n\\end{equation}\nThe row $\\mathcal{A}_i^{(k)}$ returns the pairwise affinity between region $i$ and every other region $j \\in [1, \\ldots, N]$ at Graph Attention layer $k$. For each region $i$, we average the affinities in $\\mathcal{A}_i^{(k)}$ and concatenate the averaged affinities over $K$ Graph Attention layers to obtain the affinity vector $\\boldsymbol{\\alpha}_i \\in \\mathbb{R}^{K}$. \n\n\\vspace{-2mm}\n\\paragraph{2)} We feed $\\boldsymbol{\\alpha}_i$ into a single linear layer to derive the unnormalized attention weights i.e. $f_\\text{att}: \\boldsymbol{\\alpha}_i \\mapsto \\tilde{\\beta}_i$ and normalize the region attention weights \nvia softmax, i.e.:\n\\begin{equation}\n   \\beta_i = \\frac{e^{\\tilde\\beta_i}}{\\sum^{N}_{j=1}e^{\\tilde\\beta_j}}.\n\\end{equation}\n\n\\subsection{Training VRAG}\nWe train VRAG using the triplet margin loss~\\cite{triplet-margin-loss}. From a triplet of video-level embeddings $\\left(\\mathbf{v}, \\mathbf{v}^+, \\mathbf{v}^-\\right)$ that correspond to the anchor, positive and negative video respectively, we compute the loss:\n\\begin{equation}\n    \\mathcal{L} = \\left\\lfloor c\\left(\\mathbf{v}, \\mathbf{v}^-\\right) - c\\left(\\mathbf{v}, \\mathbf{v}^{+}\\right) + m\\right\\rfloor_+,\n\\end{equation}\nwhere $c(\\cdot, \\cdot)$ computes the cosine similarity between embeddings, $m$ refers to the margin. During training, we sample triplets using the triplet mining scheme consistent with ~\\cite{kordopatiszilos2019visil}.\n\n\\section{Shot Representations}\n\nIn this section, we introduce an intermediate approach utilizing shot representations for video retrieval. We use the shot-boundary algorithm in Section~\\ref{sect:sba} to divide a video $\\left[\\mathcal{I}^{(1)}, \\ldots, \\mathcal{I}^{(T)}\\right]$ into non-overlapping shots and use VRAG to encode each shot into embeddings. Using shots can reduce noise within VRAG embeddings by removing spurious edges across shot boundaries.\nIn our shot-level approach, we evaluate video similarity by aggregating the pairwise similarities between shot-level embeddings.\n\n\\textbf{\\label{sect:sba}Shot Boundary Algorithm.} In our shot boundary algorithm, we identify the boundaries between video shots by comparing the similarities between consecutive frames. Specifically, we flatten the R-MAC features from each frame $\\mathcal{X}^{(t)} \\in \\mathbb{R}^{R \\times C}$ into the frame representation vector $\\mathcal{F}^{(t)} \\in \\mathbb{R}^{(R\\times C)\\times1}$. The similarities between consecutive frames is then the cosine similarity between their representations vectors $\\mathcal{F}^{(t)}$. We mark the frame $t$ as the start of a new video shot when the cosine similarity between $\\mathcal{F}^{(t)}$ and $\\mathcal{F}^{(t-1)}$ is lower than the minimum cosine similarity threshold $\\tau_s$. Generally, a higher $\\tau_s$ creates more shots and resembles frame-level approaches while a smaller $\\tau_s$ reduces the number of shots and closely approximates video-level methods.\n\n\\textbf{\\label{sect:aggregate-shots}Aggregating Shot Similarities.} Under our shot-level approach, we evaluate the similarity between videos $\\mathbf{X}$ and $\\mathbf{X}^\\prime$ from the pairwise similarities between their shot embeddings, i.e. $\\left[\\mathbf{o}_1, \\ldots, \\mathbf{o}_N\\right]$ and $\\left[\\mathbf{o}_1^\\prime, \\ldots, \\mathbf{o}_M^\\prime\\right]$. We compute the cosine similarity between pairs of shots and build a pairwise shot cosine similarity matrix $\\mathcal{S} \\in [-1, 1]^{N \\times M}$ where $\\mathcal{S}_{ij}$ returns the cosine similarity between shots $\\mathbf{o}_i$ and $\\mathbf{o}_j^\\prime$.\n\nWe aggregate these pairwise similarities into a similarity metric between videos. We use two aggregation schemes proposed in \\cite{kordopatiszilos2019visil}: Chamfer similarity (CS) which takes the maximum similarity along the columns of $\\mathcal{S}$ followed by average pooling of the maximum similarity vector, i.e. \n\\begin{equation}\n    s = \\operatorname{CS}(\\mathbf{X}, \\mathbf{X}^\\prime)=\\frac{1}{N}\\sum_{i=1}^N \\left(\\max_{j \\in 1, \\ldots, M} \\mathcal{S}_{ij}  \\right),\n\\end{equation}\nSymmetric Chamfer Similarity (SCS) which computes the average Chamfer similarity from $\\mathcal{S}$ and $\\mathcal{S^\\top}$, i.e.:\n\\begin{equation}\n    \\begin{aligned}\n    s = \\operatorname{SCS}(\\mathbf{X}, \\mathbf{X}^\\prime) = \\frac{\\operatorname{CS}(\\mathbf{X}, \\mathbf{X}^\\prime) + \\operatorname{CS}(\\mathbf{X}^\\prime, \\mathbf{X})}{2}.\n    \\end{aligned}\n\\end{equation}\n\n\\section{Experiments}\n\nWe evaluate video retrieval performance using mean Average Precision~\\cite{dataset:cc-web-video} (mAP). Additionally, we provide qualitative results in our supplementary material.\n\n\\subsection{Experiment Settings}\n\nWe evaluate VRAG and our shot-level approach over several video retrieval settings: Near-Duplicate Video Retrieval (NDVR); Event Video Retrieval (EVR); and Fine-grained Incident Video Retrieval (FIVR).\n\n\\vspace{0mm}\n\\paragraph{Datasets.} For evaluation, we use CC\\_WEB\\_VIDEO~\\cite{dataset:cc-web-video} for NDVR, EVVE~\\cite{dataset:evve} for EVR, and FIVR200K~\\cite{dataset:fivr200k} and its subset FIVR5K~\\cite{dataset:fivr200k, kordopatiszilos2019visil} for FIVR. During training, we sample triplets of videos from VCDB~\\cite{dataset:vcdb}.\n\n\\noindent \\textbf{CC\\_WEB\\_VIDEO}~\\cite{dataset:cc-web-video} contains 13,129 videos with 24 query videos. In this dataset, near-duplicate videos correspond to videos stored in different formats e.g. .flv; .wmv; and videos with minor content differences e.g. photometric variations like lighting. We evaluate on the original annotations from \\cite{dataset:cc-web-video} and the cleaned annotations from \\cite{kordopatiszilos2019visil}. \n\n\\noindent\\textbf{EVVE}~\\cite{dataset:evve} contains 2,995 videos from Youtube with 620 query videos over 13 events. \n\n\\noindent \\textbf{FIVR200K}~\\cite{dataset:fivr200k} contains 100 query videos and a total of 225,960 videos from 4,687 Wikipedia events. The dataset groups similar videos into four categories: 1) Near-Duplicate (ND) videos which share all scenes with the query video; 2) Duplicate Scene (DS) videos that share at least one scene with the query video; 3) Complementary Scene (CS) videos that share at least segment with the query video, but from a different viewpoint; 4) Incident Scene (IS) videos that capture the same incident as the query video without sharing segments.\nFrom the video categories, the dataset introduces three video retrieval settings: 1) Duplicate Scene Video Retrieval (DSVR) which only considers ND and DS videos as positives; 2) Complementary Scene Video Retrieval (CSVR) that extends from DSVR to include CS videos; and 3) Incident Scene Video Retrieval (ISVR) which marks all four categories of videos as similar videos.\n\n\\vspace{0mm}\n\\noindent \\textbf{FIVR5K}~\\cite{dataset:fivr200k, kordopatiszilos2019visil} is a subset of FIVR200K, with 50 query videos and 5,000 database videos. To create FIVR5K, the authors~\\cite{kordopatiszilos2019visil} select the 50 hardest query videos on the DSVR task from FIVR200K using \\cite{lbow} to measure difficulty.\n\n\\vspace{0mm}\n\\noindent \\textbf{VCDB}~\\cite{dataset:vcdb} consists of a core dataset with 528 videos and a background dataset with 100,000 distractor videos. The core dataset videos has a total of 9,236 pairs of partial copies between the 528 videos in the core dataset.\n\n\\paragraph{Implementation Details.}\nFor each video, we sample frames at one second intervals. We extract R-MAC features from each bottleneck layer of ResNet50~\\cite{resnet}, i.e. $L=4$ and $C=3840$. We set the number of regions per frame $R=9$, intermediate dimensions $C^\\prime=512$, video-level embedding dimensions $D=4096$. We use $K=3$ Graph Attention~\\cite{graph-attention} layers and the ELU~\\cite{nonlinearity:elu} non-linearity.\nWe train VRAG on a single GTX 1080 Ti using a batch size of one over 120 epochs. The maximum duration of each video clip in the training triplet is $W=64$s, and we sample 1,000 triplets from each triplet pool, giving a total of 2,000 training iterations per epoch. We optimize VRAG using the triplet margin loss with $m=0.2$ and the Adam~\\cite{optimizer:adam} optimizer using a fixed learning rate at $3 \\times 10^{-7}$. During inference, VRAG processes up to 2,000 frames on 11GB of GPU memory, which amounts to over 30 minutes of video.\n\n\n\\subsection{Ablation Studies}\n\n\n\n\\paragraph{Our VRAG.}\nAt the region-level, we use attention to aggregate features from neighboring regions. We compare our attention aggregation mechanism with alternatives such as max aggregation and average aggregation, i.e. GCN~\\cite{gcn}. In Table~\\ref{tab:ablation:region-aggregation}, we show that our choice of region-level attention aggregation improves performance across all FIVR5K settings.\n\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|}\n    \\hline\n         Method & Region Agg. & DSVR & CSVR & ISVR \\\\\n         \\hline\\hline\n        \n       \n         \\multirow{3}{*}{VRAG} & Max & 0.520 & 0.537 & 0.508 \\\\\n         \\cline{2-5}\n         &  Average & 0.518 & 0.532 & 0.506 \\\\\n         \\cline{2-5}\n          &  Attention & \\textbf{0.532} & \\textbf{0.548} & \\textbf{0.518} \\\\\n         \\hline\n    \\end{tabular}\n    \\caption{Comparison on FIVR5K over different choices of region-level aggregation. Attention-weighted summation is used to pool region embeddings for all choices of region-level aggregation.}\n    \\label{tab:ablation:region-aggregation}\n\\end{table}\n\nAt the video-level, we pool region embeddings to a fixed size vector $\\bar{\\mathbf{r}}$ using an attention-weighted summation. We compare our attention pooling with standard methods, i.e. max pooling and average pooling. In Table~\\ref{tab:ablation:region-pooling}, we demonstrate that our learnable pooling gives better performance over conventional pooling techniques.  \n\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|}\n    \\hline\n         Method & Region Pooling & DSVR & CSVR & ISVR \\\\\n         \\hline\\hline\n         \\multirow{3}{*}{VRAG} & Max & 0.423 & 0.435 & 0.415 \\\\\n        \\cline{2-5}\n          & Average & 0.505 & 0.513 & 0.499 \\\\\n         \\cline{2-5}\n          &  Attention & \\textbf{0.532} & \\textbf{0.548} & \\textbf{0.518} \\\\\n         \\hline\n    \\end{tabular}\n    \\caption{\\label{tab:ablation:region-pooling} {Comparison on FIVR5K over different video-level pooling techniques.}}\n\\end{table}\nWe also compare our choice of self-attention to an alternative with reduced parameters. In Table~\\ref{tab:ablation:attention}, we observe that using separate key\/query parameters allows for more complex relations between regions and significantly improves retrieval performance in FIVR5K.\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|}\n    \\hline\n         Method & Attention & DSVR & CSVR & ISVR \\\\\n         \\hline\\hline\n         \\multirow{2}{*}{VRAG} & $f_\\text{query}^{(k)} = f_{key}^{(k)}$ & 0.493 & 0.510 & 0.490 \\\\\n         \\cline{2-5}\n          &  $f_\\text{query}^{(k)} \\neq f_{key}^{(k)}$ & \\textbf{0.532} & \\textbf{0.548} & \\textbf{0.518} \\\\\n         \\hline\n    \\end{tabular}\n    \\caption{Comparison on FIVR5K using different implementations of attention.}\n    \\label{tab:ablation:attention}\n\\end{table}\n\nFinally, we propose concatenating the region-level embeddings to reduce over-smoothing effects observed in GCNs~\\cite{gcn:over-smoothing1, chen2019measuring}. We compare against three baselines: 1) Using only the output from the final Graph Attention layer; 2) Concatenating the outputs from all Graph Attention layers; 3) Concatenating the outputs from all Graph Attention layers and $f_r$. In Table~\\ref{tab:ablation:region-embed}, we show that  multi-layer concatenation of region-level embeddings greatly improves retrieval performance. Furthermore, we observe a monotonic increase in performance as we concatenate  region representations across more layers.\n\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|}\n    \\hline\n         Method & Concat. Layers & DSVR & CSVR & ISVR \\\\\n         \\hline\\hline\n         \\multirow{4}{*}{VRAG} & Final Graph Attn. & 0.452 & 0.463 & 0.442 \\\\\n        \\cline{2-5}\n         & All Graph Attn. & 0.481 & 0.497 & 0.477 \\\\\n        \\cline{2-5}\n         & All Graph Attn. + $f_r$ & 0.507 & 0.526 & 0.498 \\\\\n        \\cline{2-5}\n         &  All Layers & \\textbf{0.532} & \\textbf{0.548} & \\textbf{0.518} \\\\\n         \\hline\n    \\end{tabular}\n    \\caption{Comparison on FIVR5K over different levels of depth-wise concatenation.}\n    \\label{tab:ablation:region-embed}\n\\end{table}\n\n\\paragraph{Shot-level Video Retrieval.}\nIn shot-level video retrieval, i.e. VRAG-S, we compare different choices of $\\tau_s$ for our shot boundary algorithm. In Table~\\ref{tab:ablation:shot-level}, we demonstrate that increasing the number of shots i.e $\\tau_s = 0.75$ results in better performance in FIVR5K. Our results are also consistent with the frame-level approach ViSiL~\\cite{kordopatiszilos2019visil}, where aggregating similarities using Chamfer Similarity (CS) has better performance than Symmetric Chamfer Similarity (SCS). We also observe that our shot-level approach bridges the large gap in performance between video level approaches, i.e. VRAG and frame-level approaches, i.e. ViSiL~\\cite{kordopatiszilos2019visil}, in FIVR5K.\n\n\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|}\n    \\hline\n         Method & $\\tau_s$ & Shot Agg. & DSVR & CSVR & ISVR \\\\  \n         \\hline\\hline\n         VRAG & - & - & 0.532 & 0.548 & 0.518 \\\\\n         \\hline \\hline\n        \n       \n        \n        \n         \\multirow{4}{*}{VRAG-S} & \\multirow{2}{*}{0.5} & CS & 0.493 & 0.512 & 0.483 \\\\\n          \\cline{3-6}\n         && SCS & 0.463 & 0.477 & 0.464 \\\\\n          \\cline{2-6}\n        \n       \n        \n        \n         & \\multirow{2}{*}{0.75} & CS & \\textbf{0.709} & \\textbf{0.704} & \\textbf{0.636} \\\\\n         \\cline{3-6}\n        & & SCS & 0.596 & 0.606 & 0.575 \\\\\n         \\hline \\hline \n         \\multirow{2}{*}{ViSiL} & \\multirow{2}{*}{-} & CS & \\textbf{0.880} & \\textbf{0.869} & \\textbf{0.777} \\\\\n         \\cline{3-6}\n         & & SCS & 0.830 & 0.823 & 0.731 \\\\\n         \\hline \\hline\n    \\end{tabular}\n    \\caption{Comparison on FIVR5K over different shot-level hyperparameters.}\n    \\label{tab:ablation:shot-level}\n\\end{table}\n\n\\subsection{Comparison with Baseline Methods\n\nWe compare VRAG and our shot-level approach with existing approaches as baselines across several video retrieval tasks. We evaluate our performance on CC\\_WEB\\_VIDEO~\\cite{dataset:cc-web-video} for NDVR, EVVE~\\cite{dataset:evve} for EVR, and use FIVR200K~\\cite{dataset:fivr200k}, and FIVR5K~\\cite{kordopatiszilos2019visil, dataset:fivr200k}, for FIVR.\n\\vspace{0mm}\n\\paragraph{Video-level Methods.} For LBoW~\\cite{lbow}, we use the results from \\cite{video-verification-fake-news} for CC\\_WEB\\_VIDEO and FIVR200K. We re-implement LBoW, i.e. LBoW$^\\dagger$, with 1000 codebooks built on  VCDB using KMeans++, for EVVE and FIVR5K. For DML~\\cite{kordopatis2017dml}, we obtain results for CC\\_WEB\\_VIDEO and FIVR200K from \\cite{video-verification-fake-news}, and use the released source code\\footnote{https:\/\/github.com\/MKLab-ITI\/ndvr-dml} for EVVE and FIVR5K, i.e. DML*. \nFor TMK~\\cite{baseline:tmk} and LAMV~\\cite{Baraldi2018LAMVLT} on EVVE, we found discrepancies between their evaluation script and the original script from \\cite{dataset:evve}. We provide corrected results, i.e. TMK*, LAMV*, using the original EVVE evaluation script~\\cite{dataset:evve}. LAMV was not available in the released source code\\footnote{\\label{footnote:lamv}https:\/\/github.com\/facebookresearch\/videoalignment}, and we use TMK with frequency normalization (0.534 mAP on EVVE) as a close proxy for LAMV (0.536 mAP on EVVE~\\cite{Baraldi2018LAMVLT}). We also evaluate TMK* and LAMV* on FIVR5K and FIVR200K. For other video-level methods, we report results from \\cite{kordopatiszilos2019visil}.\n\n\\vspace{-3mm}\n\\paragraph{Frame-level Methods.} On EVVE, the authors of ViSiL~\\cite{kordopatiszilos2019visil} were only able to download, process and evaluate on ~80\\% of the dataset. We provide complete results for ViSiL using the released source code\\footnote{https:\/\/github.com\/MKLab-ITI\/visil.git}, i.e. ViSiL$_\\text{sym}$* and ViSiL$_{v}$*. For other frame-level methods, we report the updated results using deep network features in \\cite{kordopatiszilos2019visil}.\n\n\\textbf{Near-duplicate Video Retrieval.}\n\nIn Table~\\ref{tab:results:ccweb}, we compare our approach with other video-level and frame-level approaches on CC\\_WEB\\_VIDEO~\\cite{dataset:cc-web-video}, and demonstrate that our approach achieves state-of-the-art performance for video-level NDVR. Similar to FIVR5K, we observe that our shot-level approach has intermediate performance to frame-level and video-level approaches. \nWe also compare the run times for video-level VRAG and shot-level VRAG with ViSiL$_v$~\\cite{kordopatiszilos2019visil} after extracting R-MAC features from each video frame. Video-level and shot-level VRAG takes 33mins and 52mins, respectively, while ViSiL uses 109mins to infer video similarities. \n\n\\begin{table}[!htbp]\n\\small\n\\setlength\\tabcolsep{3pt}\n    \\begin{tabular}{|l|l|c|c|c|c|}\n        \\hline \\hline\n        Type & Method & cc\\_web & cc\\_web* & cc\\_web$_c$ & cc\\_web$_c$* \\\\\n        \\hline \\hline\n        \\multirow{3}{*}{Video} & \\text{LBoW} & 0.957 & 0.906 & - & - \\\\\n        & \\text{DML} & \\textbf{0.971} & 0.941 & 0.979 & 0.959 \\\\\n        & \\textbf{Ours} & \\textbf{0.971} & \\textbf{0.952} & \\textbf{0.980} & \\textbf{0.967} \\\\\n        \\hline \\hline\n       \n        \\multirow{2}{*}{Shot} & \\textbf{Ours (CS)} & 0.975 & 0.955 & \\textbf{0.987} & \\textbf{0.977} \\\\\n        & \\textbf{Ours (SCS)} & \\textbf{0.976} & \\textbf{0.959} & 0.986 & \\textbf{0.977} \\\\\n        \\hline\\hline\n        \\multirow{6}{*}{Frame} & \\text{CTE} & \\textbf{0.996} & - & - & - \\\\\n        & \\text{DP} & 0.975 & 0.958 & 0.990 & 0.982 \\\\\n        & \\text{TN} & 0.978 & 0.965 & 0.991 & 0.987 \\\\\n        & \\text{ViSiL}$_f$ & 0.984 & 0.969 & 0.993 & 0.987 \\\\\n        & \\text{ViSiL}$_\\text{sym}$ & 0.982 & 0.969 & 0.991 & 0.988 \\\\\n        & \\text{ViSiL}$_v$ & 0.985 & \\textbf{0.971} & \\textbf{0.996} & \\textbf{0.993} \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{Results on four different versions of CC\\_WEB\\_VIDEO. (*) denotes evaluation on the entire data set and the subscript $c$ uses cleaned annotations.} \\vspace{-3mm}\n    \\label{tab:results:ccweb}\n\\end{table}\n\n\\begin{table*}[tb]\n\\setlength\\tabcolsep{5pt}\n\\footnotesize\n\\centering\n\\begin{tabular}{|l||l||c||ccccccccccccc|}\n\\hline\nApproach & Method & mAP & \\multicolumn{13}{c|}{per event class} \\\\\n\\hline \\hline\n\\multirow{9}{*}{Video}\n    & LBoW$^\\dagger$ & 0.469  & 0.323 & 0.373 & 0.062 & 0.392 & 0.306 & 0.232 & 0.205 & 0.127 & 0.060 & 0.376 & 0.233 & 0.769 & 0.713 \\\\    \n    & \\text{DML*} & 0.472  & 0.437 & 0.368 & 0.052 & 0.385 & 0.242 & 0.275 & 0.205 & 0.105 & 0.085 & 0.414 & 0.245 & 0.783 & 0.656 \\\\\n    &\\text{TMK*} & 0.469  & 0.508 & 0.306 & 0.139 & 0.366 & 0.294 & 0.244 & 0.208 & 0.125 & 0.152 & 0.287 & 0.213 & 0.810 & 0.614 \\\\\n    & \\text{LAMV*} & 0.493 & 0.649 & 0.321 & 0.157 & 0.411 & 0.319 & 0.241 & 0.224 & 0.124 & 0.194 & 0.257 & 0.191 & 0.857 & 0.660 \\\\\n    & \\text{LAMV+QE*} & 0.541 & 0.795 & 0.413 & \\textbf{\\underline{0.160}} & \\underline{0.546} & \\underline{0.376} & 0.297 & 0.235 & 0.124 & 0.236 & 0.257 & 0.185 & 0.907 & 0.754 \\\\\n    & \\text{LAMV} & 0.536 & 0.715 & 0.383 & 0.158 & 0.461 & 0.387 & 0.227 & 0.247 & 0.138 & 0.222 & 0.273 & 0.273 & 0.908 & 0.691 \\\\\n    & \\text{LAMV+QE} & 0.587 & 0.837 & 0.500 & 0.126 & \\textbf{0.588} & \\textbf{0.455} & \\textbf{0.343} & \\textbf{0.267} & 0.142 & 0.230 & 0.293 & 0.216 & \\textbf{0.950} & 0.770 \\\\\n    & \\textbf{Ours} & 0.623 & 0.792  & 0.675 & 0.072 & 0.496 & 0.329 & 0.292 & \\underline{0.256} & 0.241 & \\underline{\\textbf{0.497}} & 0.692 & 0.378 & 0.928 & 0.770 \\\\\n    & \\textbf{Ours+QE} & \\textbf{\\underline{0.653}}  & \\textbf{\\underline{0.888}} & \\textbf{\\underline{0.743}} & 0.042 & 0.505 & 0.342 & \\underline{0.304} & 0.247 & \\underline{\\textbf{0.280}} & 0.489 & \\underline{\\textbf{0.782}} & \\underline{\\textbf{0.410}} & \\underline{0.943} & \\underline{\\textbf{0.835}} \\\\\n    \\hline\\hline\n   \n    \\multirow{2}{*}{Shot} & \\textbf{Ours (CS)} & 0.539 & 0.796 & 0.599 & 0.077 & \\textbf{0.515} & 0.203 & \\textbf{0.266} & 0.190 & 0.098 & 0.222 & 0.589 & 0.299 & 0.836 & \\textbf{0.775} \\\\\n    & \\textbf{Ours (SCS)}  & \\textbf{0.606} & \\textbf{0.832} &\\textbf{ 0.722} & \\textbf{0.155} & 0.494 & \\textbf{0.336} & 0.265 & \\textbf{0.236} & \\textbf{0.177} & \\textbf{0.366} & \\textbf{0.620} & \\textbf{0.304} & \\textbf{0.925} & 0.670 \\\\\n    \\hline\\hline\n    \\multirow{3}{*}{Frame} & \\text{ViSiL}$_{f}$ & 0.589 & 0.889 & 0.570 & 0.169 & 0.432 & 0.345 & \\textbf{0.393} & \\textbf{0.297} & 0.181 & 0.479 & 0.564 & 0.369 & 0.885 & 0.799 \\\\\n    & \\text{ViSiL}$_\\text{sym}$* & 0.612 & \\textbf{0.923} & \\textbf{0.724} & \\textbf{0.301} & 0.573 & \\textbf{0.418} & 0.276 & 0.291 & \\textbf{0.200} & \\textbf{0.544} & 0.396 & 0.339 & \\textbf{0.938} & 0.753 \\\\\n    & \\text{ViSiL}$_\\text{v}$* & \\textbf{0.618} & 0.920 & 0.713 & 0.222 & \\textbf{0.589} & 0.350 & 0.345 & 0.276 & 0.169 & 0.444 & \\textbf{0.567} & \\textbf{0.375} & 0.909 &\\textbf{0.842} \\\\\n    \\hline \\hline\n  \\end{tabular}\n  \\caption{Comparison with state-of-the-art EVR approaches on EVVE. We use the same event class ordering as \\cite{dataset:evve}. For video-level approaches, we also underline the result with the highest mAP, excluding the results with discrepancies in the evaluation script, i.e. LAMV and LAMV+QE. We report results obtained from the original EVVE evaluation script.} \\vspace{-2mm}\n  \\label{tab:results:evve}\n\\end{table*}\n\n\\textbf{Fine-grained Incident Video Retrieval.} We evaluate on FIVR5K and FIVR200K in Table \\ref{tab:results:fivr5k} and Table \\ref{tab:results:fivr200k} respectively. Our results include the run times for available frame-level and video-level methods after extracting frame-level features. VRAG achieves state-of-the-art performance over video-level methods on FIVR datasets.\nGenerally, we observe a huge difference in retrieval performance between video-level and frame-level approaches on FIVR. Although our shot-level approach bridges the gap between video-level and frame-level while taking  $1.4\\times$ longer than video-level VRAG, the dramatic disparity between the frame-level and video-level approaches warrants a qualitative inspection of FIVR200K.\n\n\\begin{table}[ht]\n    \\footnotesize\n    \\begin{tabular}{|l|l|c|c|c|c|}\n        \\hline \\hline\n        Type & Method & DSVR & CSVR & ISVR & Time\\\\\n        \\hline \\hline\n        \\multirow{6}{*}{Video} & \\text{LBoW$^\\dagger$} & 0.351 & 0.320 & 0.298 & 4m 9s \\\\\n        & \\text{DML*} & 0.354 & 0.351 & 0.331 & 3m 29s \\\\\n        & \\text{TMK*} & 0.411 & 0.416 & 0.388  & 19m 9s \\\\ \n        & \\text{LAMV*} & 0.498 & 0.488 & 0.426 & 20m 24s \\\\\n        & \\textbf{Ours} & \\textbf{0.532} & \\textbf{0.548} & \\textbf{0.518} & 8m 27s \\\\\n        \\hline \\hline\n       \n       \\multirow{2}{*}{Shot} & \\textbf{Ours (CS)} & \\textbf{0.709} & \\textbf{0.704} & \\textbf{0.636}  &  \\multirow{2}{*}{10m 35s}\\\\\n        & \\textbf{Ours (SCS)} & 0.596 & 0.606 & 0.575 &  \\\\\n        \\hline\\hline\n        \\multirow{3}{*}{Frame} \n        & \\text{ViSiL}$_{f}$ & 0.838 & 0.832 & 0.739 & - \\\\\n        & \\text{ViSiL}$_\\text{sym}$ & 0.830 & 0.823 & 0.731 & 45m 24s \\\\\n        & \\text{ViSiL}$_v$ & \\textbf{0.880} & \\textbf{0.869} & \\textbf{0.777} & 46m 39s  \\\\\n        \\hline\n    \\end{tabular}\n   \n    \\caption{Results on FIVR5K} \\vspace{-3mm}\n    \\label{tab:results:fivr5k}\n\\end{table}\n\\vspace{-3mm}\n\\begin{table}[ht]\n    \\footnotesize\n    \\begin{tabular}{|l|l|c|c|c|c|}\n        \\hline \\hline\n        Type & Method & DSVR & CSVR & ISVR & Time\\\\\n        \\hline \\hline\n        \\multirow{6}{*}{Video} & \\text{LBoW} & 0.378 & 0.361 & 0.297 & 3h 50m \\\\\n        & \\text{DML} & 0.398 & 0.351 & 0.331 & 3h 31m \\\\\n        & \\text{TMK*} & 0.417 & 0.394 & 0.319  & 18h 51m \\\\ \n        & \\text{LAMV*} & \\textbf{0.489} & 0.459 & 0.364 & 20h 30m \\\\\n        & \\textbf{Ours} & 0.484 & \\textbf{0.470} & \\textbf{0.399} & 6h 50m \\\\\n        \\hline \\hline\n       \\multirow{2}{*}{Shot} & \\textbf{Ours (CS)} & \\textbf{0.723} & \\textbf{0.678} & \\textbf{0.554} & \\multirow{2}{*}{9h 34m} \\\\\n        & \\textbf{Ours (SCS)} & 0.536 & 0.504 & 0.422 &  \\\\\n        \\hline\\hline\n        \\multirow{5}{*}{Frame}\n        & \\text{DP} & 0.775 & 0.740 & 0.632 & - \\\\\n        & \\text{TN} & 0.724 & 0.699 & 0.589 & - \\\\\n        & \\text{ViSiL}$_{f}$ & 0.843 & 0.797 & 0.660 & - \\\\\n        & \\text{ViSiL}$_\\text{sym}$ & 0.830 & 0.823 & 0.731 & 63h 31m \\\\\n        & \\text{ViSiL}$_{v}$ & \\textbf{0.880} & \\textbf{0.869} & \\textbf{0.777} & 66h 18m\\\\\n        \\hline \\hline\n    \\end{tabular}\n   \n    \\caption{Results on FIVR200K} \\vspace{-2mm}\n    \\label{tab:results:fivr200k}\n\\end{table}\n\nWe found that most FIVR200K queries comprise of single shots while their Duplicate Scene (DS) videos are sequences of visually diverse shots. Additionally, most shots in the DS videos have low visual and semantic correspondence with the query segment. Consequently, video-level methods are likely to yield noisy representations that are distinct to the query. On the other hand, higher fidelity approaches, i.e. frame-level and shot-level retrieval, that segment the video into multiple representations can preserve the query video segment and obtain more accurate video similarities. We show examples in the supplementary material.\n\n\n\n\n\\subsubsection{Event Video Retrieval}\nIn Table~\\ref{tab:results:evve}, we evaluate our performance on EVVE~\\cite{dataset:evve}. Without test augmentations, i.e. Query Expansion, VRAG outperforms state-of-the-art video-level and frame-level methods. We apply Average Query Expansion~\\cite{query-expansion} on VRAG, i.e. VRAG+QE and it demonstrates further performance gains. Our shot-level approach is competitive with video-level and frame-level approaches and use 75mins of evaluation time. In contrast, video-level VRAG takes approximately 12mins for inference on EVVE, while ViSiL~\\cite{kordopatiszilos2019visil} uses 18hrs. \nWe attribute the difference in efficiency to paradigm differences between video-level and frame-level approaches: Our video-level method requires $620 + 2375 = 2995$ forward passes through VRAG to encode $n$ video embeddings. In contrast, ViSiL directly outputs the similarity between every query-candidate video pair. This requires $620 \\times 2375 \\approx 1.5\\times10^6$ forward passes through ViSiL, which is orders of magnitude larger than the number of videos $n=2995$. \n\n\n\n\n\\section{Conclusion}\n\nIn this work, we introduce Video Region Attention Graph Network (VRAG) that utilizes self-attention during region aggregation and region pooling to generate video-level embeddings for efficient video retrieval. \nSpecifically, we represent videos at a finer granularity through region-level features and encode video spatio-temporal dynamics through region-level relations. \nOur VRAG improves the state-of-the-art performance of video-level methods across multiple video retrieval datasets. We also introduce an intermediate approach to video-level and frame-level video retrieval that utilizes shots for video retrieval. We demonstrate that our shot-level approach bridges the gap in performance between video-level and frame-level approaches, where it obtains higher video retrieval performance at marginal increase in computational costs over video-level approaches.\n\n\n\\section{Qualitative Results}\n\n\\subsection{T-SNE Embeddings}\n\nIn this section, we visualize our video-level representations into T-SNE embeddings for EVVE~\\cite{dataset:evve} in Figure~\\ref{fig:tsne-evve} and FIVR5K~\\cite{dataset:fivr200k, kordopatiszilos2019visil} in Figure~\\ref{fig:tsne-fivr5k}.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=1\\columnwidth]{images\/evve-final.png}\n    \\caption{T-SNE embedding visualizations on EVVE. (Left) DML~\\cite{kordopatis2017dml}. (Right) Our VRAG.}\n    \\label{fig:tsne-evve}\n\\end{figure}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=1\\columnwidth]{images\/fivr5k-final.png}\n    \\caption{T-SNE embedding visualizations on FIVR-5K. (Left) DML~\\cite{kordopatis2017dml}. (Right) Our VRAG.}\n    \\label{fig:tsne-fivr5k}\n\\end{figure}\n\nAcross both datasets, we observe that our VRAG forms more visible and well-separated T-SNE embedding clusters compared to the existing video-level method DML~\\cite{kordopatis2017dml}.\n\n\n\\subsection{Retrieval Results}\n\nAdditionally, we include examples of retrieval results on the EVVE~\\cite{dataset:evve} dataset. In our retrieval examples, we visualize the five closest database videos to the query videos. We demonstrate positive results in Figures~{\\ref{fig:pos:geyser} to \\ref{fig:pos:performance}}. From Figures~\\ref{fig:pos:flood} and \\ref{fig:pos:performance}, we see that our VRAG can retrieve videos of similar events taken from different viewpoints. In Figures~\\ref{fig:pos:geyser} and \\ref{fig:pos:water-park}, VRAG retrieves videos of the same event taken from different points in time.\n\nFigures~\\ref{fig:neg:geyser} to \\ref{fig:neg:performance} demonstrate some failure cases of our method with negatively-related videos marked in red borders. In Figures~{\\ref{fig:neg:geyser}, \\ref{fig:neg:riot} and \\ref{fig:neg:performance}}, we see that our VRAG pulls negative videos that are semantically related to the query video. Specifically, Figure~\\ref{fig:neg:geyser} includes a negative video from a different geyser, Figure~\\ref{fig:neg:riot} has a negative video from another riot, and Figure~\\ref{fig:pos:performance} shows a negative video from a separate live performance. However, our VRAG may confuse videos \nin instances where distractor videos contain similar elements to query videos as positives. For example in Figure~\\ref{fig:neg:protest}, VRAG retrieves a negative video depicting a crowded sports stadium as a close positive to a query video of a protest with multiple scenes of crowds.\n\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[width=1.5\\columnwidth]{images\/qualitative\/positives\/0bqBMJHjqdo_censored.jpg}\n    \\caption{Positive geyser event retrieval example.}\n    \\label{fig:pos:geyser}\n\\end{figure*}\n\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[width=1.5\\columnwidth]{images\/qualitative\/positives\/5aiVFD2jOKE_censored.jpg}\n    \\caption{Positive flooding event retrieval example.}\n    \\label{fig:pos:flood}\n\\end{figure*}\n\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[width=1.5\\columnwidth]{images\/qualitative\/positives\/GOg-JbebB3w.png}\n    \\caption{Positive water park event retrieval example.}\n    \\label{fig:pos:water-park}\n\\end{figure*}\n\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[width=1.5\\columnwidth]{images\/qualitative\/positives\/H3m66ZlyIhs.png}\n    \\caption{Positive live performance retrieval example.}\n    \\label{fig:pos:performance}\n\\end{figure*}\n\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[width=1.5\\columnwidth]{images\/qualitative\/negatives\/0Y5NdarmalI.png}\n    \\caption{Negative geyser event retrieval example. Negative videos have red borders.}\n    \\label{fig:neg:geyser}\n\\end{figure*}\n\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[width=1.5\\columnwidth]{images\/qualitative\/negatives\/2sr4odOy33I_censored.jpg}\n    \\caption{Negative protests event retrieval example.  Negative videos have red borders.}\n    \\label{fig:neg:protest}\n\\end{figure*}\n\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[width=1.5\\columnwidth]{images\/qualitative\/negatives\/BOAU395mfJw_censored.jpg}\n    \\caption{Negative riots event retrieval example. Negative videos have red borders.}\n    \\label{fig:neg:riot}\n\\end{figure*}\n\n\\begin{figure*}[h]\n    \\centering\n    \\includegraphics[width=1.5\\columnwidth]{images\/qualitative\/negatives\/LY9lCuarBOA.png}\n    \\caption{Negative live performance retrieval example. Negative videos have red borders.}\n    \\label{fig:neg:performance}\n\\end{figure*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\setcounter{equation}{0}  Consider the linear parabolic equation\n\\begin{align}\\label{PDE0}\n\\left\\{\n\\begin{array}{ll}\n\\displaystyle\n\\partial_tu-\\sum_{i,j=1}^d\\partial_{i}\\big(a_{ij}(x)\\partial_j\nu\\big)+c(x)u =f-\\sum_{j=1}^d\\partial_jg_j\n&\\mbox{in}~~\\Omega\\times(0,T),\\\\[5pt]\n\\displaystyle \\sum_{i,j=1}^da_{ij}(x)n_i\\partial_j u=\\sum_{j=1}^dg_jn_j\n&\\mbox{on}~~\\partial\\Omega\\times(0,T),\\\\[8pt]\nu(\\cdot,0)=u^0 &\\mbox{in}~~\\Omega,\n\\end{array}\n\\right.\n\\end{align}\nwhere $\\Omega$ is a bounded smooth domain in $\\mathbb{R}^d$ $(d\\geq 2)$, $T$\nis a given positive number, $f$ and ${\\bf g}=(g_1,\\cdots,g_d)$ are\ngiven functions. The Galerkin finite element method (FEM) for the\nabove equation seeks $\\{u_h(t)\\in S_h\\}_{t> 0}$\nsatisfying the parabolic finite element equation:\n\\begin{align}\\label{FEEq0}\n\\left\\{\n\\begin{array}{ll}\n\\displaystyle \\big(\\partial_tu_h,v_h\\big)\n+\\sum_{i,j=1}^d\\big(a_{ij}\\partial_j\nu_h,\\partial_iv_h\\big)+\\big(cu_h,v_h\\big)\n=(f,v_h)+\\sum_{j=1}^d(g_j,\\partial_jv_h), ~~\\forall~v_h\\in\nS_h,\\\\[8pt]\nu_h(0)=u^0_h ,\n\\end{array}\n\\right.\n\\end{align}\nwhere $S_h$, $0<h<h_0$, denotes a finite element subspace of\n$H^1(\\Omega)$ consisting of continuous piecewise polynomials of\ndegree $r\\geq 1$ on certain quasi-uniform triangulations of \n$\\Omega$ which fit the boundary exactly.\nThe coefficients $a_{ij}=a_{ji}$, $i,j=1,\\cdots,d$, and $c$ in the\nabove equations are assumed to satisfy\n\\begin{align}\\label{coeffcond}\n\\Lambda_1|\\xi|^2\\leq\n\\sum_{i,j=1}^da_{ij}(x)\\xi_i\\xi_j\n\\leq \\Lambda_2|\\xi|^2\n\\quad\\mbox{and}\\quad\nc(x)\\geq c_0,\\quad\\mbox{for}~~x\\in\\Omega,\n\\end{align}\nfor some positive constants $\\Lambda_1,\\Lambda_2$ and $c_0$.\n\nDefine the\nelliptic operator $A: H^1(\\Omega)\\rightarrow H^1(\\Omega)'$ and its\nfinite element approximation $A_h: S_h\\rightarrow S_h$\nby\n\\begin{align}\n& (Aw,v):=\\sum_{i,j=1}^d(a_{ij}\\partial_j w,\\partial_iv)+(cw,v),\n~~~~~\\qquad\\qquad \\forall~w,v\\in\nH^1(\\Omega),\\\\\n &(A_hw_h,v_h):=\\sum_{i,j=1}^d(a_{ij}\\partial_j\nw_h,\\partial_{i}v_h)+(cw_h,v_h),\\qquad~ \\forall~w_h,v_h\\in S_h .\n\\end{align}\nFor homogeneous equations, i.e. $f\\equiv g_j\\equiv 0$, the solutions of\n(\\ref{PDE0}) and (\\ref{FEEq0}) can be expressed as $u(t)=E(t)u^0$\nand $u_h(t)=E_h(t)u^0_h$, where $\\{E(t)=e^{-tA}\\}_{t>0}$ and\n$\\{E_h(t)=e^{-tA_h}\\}_{t>0}$ denote the semigroups generated by the\noperators $-A$ and $-A_h$, respectively.\nFrom the theory of parabolic equations, we know that\n$\\{E(t)\\}_{t>0}$ extends to an analytic semigroup on\n$C(\\overline\\Omega)$, satisfying\n\\begin{align}\n&\\|E(t)v\\|_{L^\\infty}\n+t\\|\\partial_tE(t)v\\|_{L^\\infty}\\leq C\\|v\\|_{L^\\infty}\n,\\quad\\forall~v\\in C(\\overline\\Omega) .\n\\end{align}\nIts counterpart for the discrete finite element\noperator is the analyticity of the semigroup $\\{E_h(t)\\}_{t>0}$\non $L^\\infty\\cap S_h$:\n\\begin{align}\n&\\|E_h(t)v_h\\|_{L^\\infty} +t\\|\\partial_tE_h(t)v_h\\|_{L^\\infty}\\leq\nC\\|v_h\\|_{L^\\infty} ,\\quad\\forall~v_h\\in S_h,~ \\forall~t>0.\n\\label{STLEst}\n\\end{align}\nAlong with the approach of analytic semigroup, one may reach more\nprecise analysis of the finite element solution, such as\nmaximum-norm error estimates of semi-discrete Galerkin FEMs\n\\cite{Tho, TW1,Wah}, resolvent estimates of elliptic finite element\noperators \\cite{Bak, BTW,CLT}, error analysis of fully discrete FEMs for\nparabolic equations \\cite{LN, Pal,Tho}, and the discrete maximal\n$L^p$ regularity \\cite{Gei1,Gei2}.\n\nA related topic is the space-time maximum-norm\nstability estimate for inhomogeneous\nequations ($f$ or $g_j$ may not be identically\nzero):\n\\begin{align}\n\\|u_h\\|_{L^\\infty(\\Omega\\times(0,T))}\\leq\nC_T\\|u_h^0\\|_{L^\\infty}+C_Tl_h\\|u\\|_{L^\\infty(\\Omega\\times(0,T))},\n\\quad\\forall~T>0\n. \\label{STLEst2}\n\\end{align}\nUnder certain regularity assumptions on $u$, a straightforward\napplication of the above inequality is the maximum-norm error\nestimate:\n\\begin{align}\n\\| u - u_h \\|_{L^\\infty(\\Omega\\times(0,T))} \\leq C_T\\| u^0 - u_h^0\n\\|_{L^\\infty} +C_Tl_h h^{r+1}  \\| u \\|_{L^\\infty( (0,T);\nW^{r+1,\\infty})}  . \\label{error}\n\\end{align}\n\n\nIn the last several decades, many efforts have been devoted to the\nstability-analyticity estimate (\\ref{STLEst}) and the space-time\nstability estimate (\\ref{STLEst2}). Schatz et. al. \\cite{STW1}\nestablished (\\ref{STLEst}) for $d=2$ and $r=1$, with constant\ncoefficients $a_{ij}$, by using a weighted-norm technique. Later,\nNitsche and Wheeler \\cite{NW} proved (\\ref{STLEst2}) for $d=2,3$ and\n$r\\geq 4$ with constant coefficients. Rannacher \\cite{Ran} proved\n(\\ref{STLEst})-(\\ref{STLEst2}) in convex polygons with\nconstant coefficients, and  Chen \\cite{Chen} improved the results to\n$1\\leq d\\leq 5$. A more precise analysis was given by Schatz et al\n\\cite{STW2}, where they proved that \\refe{STLEst}-\\refe{STLEst2}\nhold with $l_h=1$ for $r\\geq 2$ and $l_h=\\ln(1\/h)$ for $r=1$, and\nthey showed that the logarithmic factor is necessary for $r=1$. In\n\\cite{STW2}, the proof was given under the condition that the parabolic\nGreen's function\nsatisfies \n\\begin{align}\n|\\partial_t^\\alpha\\partial_x^\\beta G(t, x, y)|\\leq\nC(t^{1\/2}+|x-y|)^{-(d+2\\alpha+|\\beta|)} e^{-\\frac{|x-y|^2}{Ct}}\n\\label{g-cond} ,~~\\forall~\\alpha\\geq 0,|\\beta|\\geq 0,\n\\end{align}\nwhich holds when the coefficients $a_{ij}(x)$\nare smooth enough \\cite{EI70}. The stability\nestimate (\\ref{STLEst}) was also studied in \\cite{Bak, TW2}\nfor the Dirichlet boundary condition and in \\cite{Cro} for\na lumped mass method. Moreover, Leykekhman \\cite{Ley}\nshowed the stability estimate (\\ref{STLEst2}) in a more general weighted norm,\nand Hansbo \\cite{Han} investigated the related $L^s\\rightarrow L^r$ stability\nestimate.  Also see \\cite{TW1,Wah} for some\nworks in the one-dimensional space. Clearly, all these results were\nestablished for parabolic equations with the coefficient $a_{ij}(x)$\nbeing smooth enough. Related maximum-norm error estimates of\nGalerkin FEMs in terms of an elliptic projection and the associated elliptic Green's function\ncan be found in \\cite{BSTW,ELWZ,Lin2, LTW, Nit,Ran,Tho, Whe}.\nSome other nonlinear models were analyzed in\n\\cite{Dob2}. Again, these works were\nbased on the assumption that the coefficients $a_{ij}$ are smooth\nenough.\n\n\nIn many physical applications, the coefficients $a_{ij}$ may\nnot be smooth enough. One of examples is the\nincompressible miscible flow in porous media \\cite{Dou,LS1},\nwhere $[a_{ij}]_{i,j=1}^d$ denotes the diffusion-dispersion tensor\nwhich is Lipschitz continuous in many cases. In this case,\nthe solution is in $L^p((0,T);W^{2,q})$ for $1<p,q<\\infty$ \n(see Lemma 2.1 in Section 2).\nAs a first attempt towards this direction, in this paper,\nwe prove the maximum-norm stability estimates\n(\\ref{STLEst})-(\\ref{STLEst2}) for parabolic equations with\n Lipschitz continuous coefficients $a_{ij}\\in\nW^{1,\\infty}(\\Omega)$, and (\\ref{error}) follows immediately. Moreover,\nalong with these maximum-norm estimates we also obtain a\nsemigroup estimate:\n\\begin{align}\\label{smgest}\n\\|\\sup_{t>0}|E_h(t)v_h|\\big\\|_{L^q}\\leq C_q\\|v_h\\|_{L^q},\n~~\\forall~v_h\\in S_h,~~1<q\\leq \\infty .\n\\end{align}\nBased on these results, we establish the $L^p$ error estimate\n\\begin{align}\n&\\|P_hu - u_h \\|_{L^p((0,T);L^q)} \\leq  C_{p,q}\\|P_hu^0 - u_h^0\n\\|_{L^q} +C_{p,q}\\|P_hu-R_hu \\|_{L^p((0,T);L^q)} , \\label{LpqSt1}\n\\end{align}\nand the maximal $L^p$ regularity\n\\begin{align}\n&\\|u_h \\|_{L^p((0,T);W^{1,q})} \\leq C_{p,q}\\|{\\bf\ng}\\|_{L^p((0,T);L^q)} ,\\quad\\mbox{when}~~ u^0_h\\equiv f\\equiv 0 ,\n\\label{LpqSt2}\\\\\n&\\|\\partial_tu_h \\|_{L^p((0,T);L^q)}+\\|A_hu_h \\|_{L^p((0,T);L^q)}\n\\leq C_{p,q}\\|f\\|_{L^p((0,T);L^q)} ,\\quad\\mbox{when}~~ u^0_h\\equiv\n{\\bf g}\\equiv 0, \\label{LpqSt3}\n\\end{align}\nfor all $1<p,q<\\infty$,\nwhere $R_h$ is the Ritz projection operator\nassociated with the elliptic operator $A$, and $P_h$ is the\n$L^2$ projection operator onto the finite element space.\nNote that the inequality (\\ref{LpqSt3}) was studied by\nGeissert \\cite{Gei1,Gei2} by assuming that (\\ref{g-cond})\nholds for $\\alpha=0$, $|\\beta|\\leq 2$ and $0\\leq\n\\alpha\\leq 2$, $|\\beta|= 2$, where a sufficient condition $a_{ij}\\in\nC^{2+\\alpha}(\\overline\\Omega)$ was given.\nThe estimates (\\ref{LpqSt2})-(\\ref{LpqSt3}) resemble the\nmaximal $L^p$ regularity of the continuous parabolic\nproblem. As far as we know, the estimate\n(\\ref{LpqSt1})-(\\ref{LpqSt2}) have not been proved, which imply\noptimal error estimates of the finite element solution and\ncan be regarded as the stability of the parabolic projection\nonto the finite element space.\nThese results are required in \\cite{LS2} to establish\noptimal $L^p((0,T);L^q)$ error estimates\nof FEMs for parabolic equations with time-dependent\nnonsmooth coefficients.\n\nThe rest part of this paper is organized as follows. In Section 2,\nwe introduce some notations and present our\nmain results. In Section 3, we present some new estimates for parabolic\nGreen's functions. Based on these new estimates, we prove a key lemma\nin establishing the the maximum-norm stability estimates.\nIn Section 4, we prove the maximum-norm stability estimates,\n$L^p$ error estimates and maximal $L^p$ regularity for the finite element solution.\n\n\n\n\\section{Notations and main results}\n\\setcounter{equation}{0}\nLet $\\Omega$ be a bounded smooth domain in $\\mathbb{R}^d$ ($d\\geq 2$).\nFor any integer $k\\geq 0$ and $1\\leq p\\leq\\infty$, let\n$W^{k,p}(\\Omega)$ be the usual Sobolev space \\cite{Adams} of\nfunctions defined in $\\Omega$ equipped with the norm\n$$\n\\|f\\|_{W^{k,p}(\\Omega)}=\\left\\{\n\\begin{array}{ll}\n\\biggl(\\sum_{|\\beta|\\leq k}\\int_\\Omega|\\partial^\\beta f|^p{\\rm d}\nx\\biggl)^\\frac{1}{p},\n&1\\leq p<\\infty\n,\\\\[10pt]\n\\displaystyle\n\\sum_{|\\beta|\\leq k}{\\rm ess}\\,\\sup_{x\\in\n\\Omega}\\,|\\partial^\\beta\nf(x)|, & p=\\infty,\n\\end{array}\n\\right.\n$$\nwhere\n$$\\partial^\\beta=\n\\frac{\\partial^{|\\beta|}}{\\partial\nx_1^{\\beta_1}\\cdots\\partial x_d^{\\beta_d}}\n$$\nfor the multi-index $\\beta=(\\beta_1,\\cdots,\\beta_d)$,\n$\\beta_1\\geq\n0$, $\\cdots$, $\\beta_d\\geq 0$, and\n$|\\beta|=\\beta_1+\\cdots+\\beta_d$.\nIn particular, we set $L^p(\\Omega):=W^{0,p}(\\Omega)$ for\n$1\\leq p\\leq\\infty$ and\n$W^{-k,p}(\\Omega):=(W^{k,p'}(\\Omega))'$ for any positive\ninteger $k$, and we set $H^k(\\Omega):=W^{k,2}(\\Omega)$ for any\ninteger $k$.\n\nFor any integer $k\\geq 0$ and $0<\\alpha<1$, let\n$C^{k+\\alpha}(\\Omega)$ denote the usual H\\\"{o}lder\nspace of functions defined in $\\Omega$ equipped\nwith the norm\n$$\n\\|f\\|_{C^{k+\\alpha}(\\Omega)}=\\sum_{|\\beta|\\leq k}\\|D^\\beta\nf\\|_{L^\\infty(\\Omega)}+\\sum_{|\\beta|=\nk}\\sup_{x,y\\in\\Omega}\\frac{|D^\\beta f(x)-D^\\beta\nf(y)|}{|x-y|^\\alpha} .\n$$\n\nFor any Banach space $X$ and a given $T>0$, the Bochner spaces\n\\cite{Yos} $L^p((0,T);X)$ and $W^{1,p}((0,T);X)$ are equipped with\nthe norms\n\\begin{align*}\n&\\|f\\|_{L^p((0,T);X)} =\\left\\{\n\\begin{array}{ll}\n\\displaystyle\\biggl(\\int_0^T\n\\|f(t)\\|_X^pdt\\biggl)^\\frac{1}{p},\n&\n1\\leq p<\\infty\n,\\\\[10pt]\n\\displaystyle{\\rm ess\\,sup}_{t\\in(0,T)}\\|f(t)\\|_X,\n& p=\\infty,\\end{array}\n\\right.\n\\\\[8pt]\n&\\|f\\|_{W^{1,p}((0,T);X)}=\n\\|f\\|_{L^p((0,T);X)}\n+\\|\\partial_tf\\|_{L^p((0,T);X)} ,\n\\end{align*}\nand we set\n$ Q_T:=\\Omega\\times(0,T)$. For\nnonnegative integers $k_1$ and $k_2$, we define\n\\begin{align*}\n&\\|f\\|_{W^{k_1,k_2}_{p,q}( Q_T)}\n:=\\|f\\|_{L^p((0,T);L^{q}(\\Omega))}\n+\\|\\partial_t^{k_1}f\\|_{L^p((0,T);L^{q}(\\Omega))}\n+\\|f\\|_{L^p((0,T);W^{k_2,q}(\\Omega))} ,\n\\end{align*}\nand\n\\begin{align*}\n&\\|f\\|^{(h)}_{W^{k,0}_{p}( Q_T)}\n:=\\|f\\|_{L^p( Q_T)}\n+\\biggl(\\int_0^T\\sum_{|\\alpha|\\leq k}\\sum_{l=1}^L\n\\int_{\\tau^h_l}|\\partial^\\alpha f|^p{\\rm d} x{\\rm d} t \\biggl)^{\\frac{1}{p}} ,\n\\end{align*}\nwhere $\\tau^h_l$, $l=1,\\cdots,L$, denote elements of a quasi-uniform triangulation of $\\Omega$.\n\nFor the simplicity of notations,\nin the following sections, we\nwrite $L^p$, $W^{k,p}$, $C^{k+s}$ and\n$W^{k_1,k_2}_{p,q}$ as the abbreviations of $L^p(\\Omega)$,\n$W^{k,p}(\\Omega)$, $C^{k+s}(\\overline\\Omega)$ and\n$W^{k_1,k_2}_{p,q}( Q_T)$, respectively. We also set\n$L^p( Q_T)=\nL^p((0,T);L^p)$,\n$W^{k_1,k_2}_{p}\n=W^{k_1,k_2}_{p,p}$\nfor nonnegative integer $k_1,k_2$ and\n$1\\leq p\\leq\\infty$, and\n $L^p_h:=L^p\\cap S_h$. For any domain $Q\\subset\n Q_T$, we define $$Q^t:=\\{x\\in\\Omega:~ (x,t)\\in Q\\}$$\nand\n$$\n\\|f\\|_{L^{p,q}(Q)}:=\\biggl(\\int_0^T\\biggl(\\int_{Q^t}|f(x,t)|^q{\\rm d}\nx\\biggl)^{\\frac{p}{q}}{\\rm d} t\\biggl)^{\\frac{1}{p}}\n,\\quad\\mbox{for}~ 1\\leq p,q<\\infty ,\n$$\nand we use the abbreviations\n$$\n(\\phi,\\varphi):=\\int_\\Omega \\phi(x)\\varphi(x){\\rm d} x,\\qquad\n[u,v]:=\\iint_{ Q_T} u(x,t)v(x,t){\\rm d} x{\\rm d} t .\n$$\nWe write $w(t)=w(\\cdot,t)$ as abbreviation for any function $w$ defined on $ Q_T$.\n\n\nMoreover, we set $a(x)=[a_{ij}(x)]_{d\\times d}$ as\na coefficient matrix and define the operators\n\\begin{align*}\n&A:H^1\\rightarrow H^{-1},\\qquad~\\, A_h:S_h\\rightarrow S_h,\\\\\n&R_h:H^1\\rightarrow S_h, \\qquad~~\\, P_h:L^2\\rightarrow S_h, \\\\\n&\\overline\\nabla\\cdot:(H^1)^d\\rightarrow H^{-1},\\quad\n\\overline\\nabla_h\\cdot:(H^1)^d\\rightarrow S_h,\n\\end{align*}\nby\n\\begin{align*}\n&\\big(A\\phi,v\\big)= \\big(a\\nabla \\phi,\\nabla\nv\\big)+\\big(c\\phi,v\\big) \\qquad~~~\n\\mbox{for all~~$\\phi ,v \\in H^1$},\\\\[3pt]\n&\\big(A_h\\phi_h,v\\big)= \\big(a\\nabla \\phi_h,\\nabla\nv\\big)+\\big(c\\phi_h,v\\big) \\quad\\,\n\\mbox{for all~~$\\phi_h\\in S_h$, $v\\in S_h$},\\\\[3pt]\n&\\big(A_hR_hw,v\\big)=\\big(Aw,v\\big) \\qquad\\qquad\\qquad~~ \\mbox{for\nall~~$w \\in H^1$ and $v\\in S_h$} ,\n\\\\[3pt]\n&\\big(P_h\\phi,v\\big)=\\big(\\phi,v\\big) \\qquad\\qquad\\qquad\\qquad~~~\n\\mbox{for all~~$\\phi \\in L^2$\nand $v \\in S_h$} ,\\\\\n&\\big(\\overline\\nabla\\cdot{\\bf w},v\\big)=-\\big({\\bf w},\\nabla v\\big)\n\\qquad\\qquad\\qquad~\\, \\mbox{for all~~${\\bf w} \\in (H^1)^d$ and\n$v \\in H^1$,} \\\\\n&\\big(\\overline\\nabla_h\\cdot{\\bf w},v\\big)=-\\big({\\bf w},\\nabla\nv\\big) \\qquad\\qquad\\qquad \\mbox{for all~~${\\bf w} \\in (H^1)^d$ and $v\n\\in S_h$} .\n\\end{align*}\nClearly,\n$R_h$ is the Ritz projection operator associated to the elliptic\noperator $A$ and $P_h$ is the $L^2$ projection operator onto the finite element space,\nwhich satisfy\n\\begin{align*}\n&\\|u-P_hu\\|_{W^{m,p}} \\leq C\\|u\\|_{W^{m,p}},\\qquad\\quad\\, 1\\leq p\\leq\\infty,\\\\\n&\\|u-R_hu\\|_{W^{m,p}} \\leq Ch^{1-m}\\|u\\|_{W^{1,p}},\\quad 1<p\\leq\\infty,~(m,p)\\neq (0,\\infty) \n\\end{align*}\nwhere $m=0,1$ and $C$ is some positive constant\nindependent of the mesh size $h$. \n\nWith these notations, for $f\\in L^2((0,T);L^2)$, ${\\bf g}\\in L^2((0,T);L^2)^d$\nand $u_0\\in L^2$, the problem\n\\begin{align}\\label{sdfjisloudf9890}\n\\left\\{\n\\begin{array}{ll}\n\\partial_tu+Au=f-\\overline\\nabla\\cdot{\\bf g}\n ,\\\\[3pt]\nu(0)=u_0 ,\n\\end{array}\n\\right.\n\\end{align}\nadmits a unique solution $u\\in L^2((0,T);H^1)\\cap\nH^1((0,T);H^{-1})$, which coincides with the weak solution of the\ninitial-boundary\nvalue problem \\refe{PDE0}.\nThe following lemma gives the maximal $L^p$ regularity of the\ncontinuous parabolic problem \\cite{KW04}.\n\\begin{lemma}\\label{s00}\n{\\bf$\\!\\!\\!$(Maximal $\\bf L^p$ regularity)}~~\\\\\n{\\it Let $u\\in L^2((0,T);H^1)\\cap H^1((0,T);H^{-1})$\n be the solution to the problem\n\\begin{align}\\label{saf9}\n\\left\\{\n\\begin{array}{ll}\n\\partial_tu+Au=f-\\overline\\nabla\\cdot{\\bf g},\\\\[3pt]\nu(0)=0   .\n\\end{array}\n\\right.\n\\end{align}\nThen the following inequalities hold:\n\\begin{align}\n&\\!\\!\\!\\|\\partial_tu\\|_{L^p((0,T);L^q)}+\\|u\\|_{L^p((0,T);W^{2,q})}\n\\leq C_{p,q}\\|f\\|_{L^p((0,T);L^q)}, ~~~\n1<p,q<\\infty,~~\\mbox{if}~~{\\bf g}\\equiv 0,\n\\label{dsk2}\\\\\n&\\!\\!\\!\\|u\\|_{L^p((0,T);W^{1,q})} \\leq C_{p,q}\\|{\\bf\ng}\\|_{L^p((0,T);L^q)},\\qquad\\qquad\\qquad\\qquad\\quad 1<p,q<\\infty\n,~~\\mbox{if}~~f\\equiv 0. \\label{dsk3}\n\\end{align}\n}\n\\end{lemma}\n\nSince $L^q((0,T);W^{2,q})\\cap \nW^{1,q}((0,T);L^{q})\\hookrightarrow\nL^p((0,T);W^{1,p})\\cap W^{1,p}((0,T);W^{-1,p}) $, \nthe inequality \\refe{dsk2} implies that\n\\begin{align}\\label{dsk4}\n&\\|\\partial_tu\\|_{L^p((0,T);W^{-1,p})}\n+\\|u\\|_{L^p((0,T);W^{1,p})}\n\\leq C_{p,q}\\|f\\|_{L^q((0,T);L^q)}, ~~~\n\\mbox{if}~~{\\bf g}\\equiv 0,\n\\end{align} \nfor any $p>2$ and $q=(d+2)p\/(d+2+p)<p$.\n\n\n\n\\subsection{Main results}\nThe main results of this paper are given below and the proofs are presented in\nSection 3--4.\n\n\\begin{theorem}\\label{MainTHM1}\n{\\it If $a_{ij}=a_{ji}\\in W^{1,\\infty}$ and $c\\in L^\\infty$ satisfy the condition {\\rm\n(\\ref{coeffcond})}, then the solutions of\n{\\rm(\\ref{PDE0})}-{\\rm(\\ref{FEEq0})} satisfy the\nmaximum-norm estimates {\\rm(\\ref{STLEst})}-{\\rm(\\ref{STLEst2})} with $l_h=\\ln(2+1\/h)$,\nthe semigroup estimate {\\rm(\\ref{smgest})}, the $L^p$ stability\nestimate {\\rm(\\ref{LpqSt1})} and the maximal $L^p$\nregularity {\\rm(\\ref{LpqSt2})}-{\\rm(\\ref{LpqSt3})}.\n}\n\\end{theorem}\n\n\nIn the proof of our main results, we can assume that the functions $f$ and $g$ are smooth enough and the exact solution $u$ satisfies $u\\in L^p((0,T);W^{2,p})$ and $\\partial_tu\\in L^p((0,T);L^p)$ for arbitrarily large $p$. However, the generic positive constant $C$ in this paper does not depend on the regularity of $f$, $g$ or $u$.\nTherefore, by a passing to a limit, one can see that\n(1.8) defines a parabolic projection for $u^0_h\\in S_h$ and \n$u\\in C(\\overline \\Omega\\times[0,T])$,\n(1.13) holds for ${\\bf g}\\in L^p((0,T);L^q)^d$ and (1.14)\nholds for $f\\in L^p((0,T);L^q)$.\n\nUnlike \\cite{STW2},\nwe are not able to remove the logarithmic factor $l_h$ in \n{\\rm(\\ref{STLEst})}-{\\rm(\\ref{STLEst2})} for finite element spaces \nof polynomial \ndegree $r\\geq 2$, due to the low regularity of the coefficients.\n\n\\subsection{Further notations}\\label{fnot}\nTo prove our main results, we present some further notations,\nwhich were introduced in \\cite{STW2,SW1} and also used in \n\\cite{Ley,Sol}.\n\nFor an element $\\tau_l^h$ and a point $x_0\\in \\tau_l^h$, we\nlet $\\delta_{x_0}$ denote the Dirac Delta function centered at\n$x_0$, i.e. $\\int_\\Omega\\delta_{x_0}(y)\\varphi (y){\\rm d} y=\\varphi(x_0)$ for\nany $\\varphi\\in C(\\overline\\Omega)$, and we denote by\n$\\widetilde\\delta_{x_0}$ a regularized\nDelta function satisfying the following\nconditions:\n\\begin{align}\n&\\mbox{$\\widetilde\\delta_{x_0}$ is\nsupported in $\\tau_l^h$ ,}\\\\\n&\\chi(x_0)=\\int_{\\tau_l^h}\\chi\\,\n\\widetilde\\delta_{x_0}\\,{\\rm d} x,\\quad\n\\mbox{for all}~\\chi\\in S_h ,\n\\label{chidef}\\\\\n&\\|\\widetilde\\delta_{x_0}\\|_{W^{m,p}}\\leq\n Ch^{-m-d(1-1\/p)}~~\\mbox{for}~~1\\leq p\\leq\n \\infty,~m=0,1,2,3 . \\label{Deltadef}\n\\end{align}\nLet $G(t,x,x_0)$ be {\\it Green's function} of the parabolic\nequation, defined by\n\\begin{align}\\label{GFdef}\n&\\partial_tG(t,\\cdot,x_0)+ AG(t,\\cdot,x_0)=0\\quad\n\\mbox{for $t>0$ with $G(0,\\cdot,x_0)=\\delta_{x_0}$}\n.\n\\end{align}\nThe corresponding {\\it regularized Green's function} $\\Gamma(t,x,x_0)$\nis defined by\n\\begin{align}\\label{GMFdef}\n&\\partial_t\\Gamma(\\cdot,\\cdot,x_0)+A\\Gamma(\\cdot,\\cdot,x_0)=0\\quad\n\\mbox{for~ $t>0$~ with~\n$\\Gamma(0,\\cdot,x_0)=\\widetilde\\delta_{x_0}$},\n\\end{align}\nand the {\\it discrete Green's function} $\\Gamma_h(\\cdot,\\cdot,x_0)$\nis defined as the solution of the equation\n\\begin{align}\n&\\partial_t\\Gamma_{h}(\\cdot,\\cdot,x_0)\n+ A_h\\Gamma_{h}(\\cdot,\\cdot,x_0)=0 \\label{GMhFdef}\n\\quad\\mbox{for~ $t>0$~ with~\n$\\Gamma_{h}(0,\\cdot,x_0)=P_h\\delta_{x_0}=P_h\\widetilde\\delta_{x_0}$},\n\\end{align}\nwhere $P_h$ is the $L^2$ projection onto the finite element space.\nNote that\n$\\Gamma(t,x,x_0)$ and $\\Gamma_h(t,x,x_0)$ are symmetric with respect\nto $x$ and $x_0$.\n\nBy the fundamental estimates of parabolic equations \\cite{FS}\nand from Appendix B of \\cite{Gei1}, we know that the\nGreen's function $G$ satisfies\n\\begin{align}\n&|G(t,x,y)|\\leq\nC(t^{1\/2}+|x-y|)^{-d}e^{-\\frac{|x-y|^2}{Ct}},\\label{FEstP}\\\\\n&|\\partial_tG(t,x,y)|\\leq Ct^{-d\/2-1}e^{-\\frac{|x-y|^2}{Ct}} ,\\label{FtEstP}\\\\\n&|\\partial_{tt}G(t,x,y)|\\leq Ct^{-d\/2-2}e^{-\\frac{|x-y|^2}{Ct}} .\\label{FtEstP2}\n\\end{align}\nBy estimating $\\Gamma(t,x,x_0)=\\int_\\Omega\nG(t,x,y)\\widetilde\\delta_{x_0}(y){\\rm d} y$, we see that\n(\\ref{FEstP})-(\\ref{FtEstP2}) also hold when $G$ is replaced by\n$\\Gamma$ and when $\\max(t^{1\/2},|x-y|)\\geq 2h$.\n\nFor any open subset $D\\subset\\Omega$, we set $\\overline\nD^\\partial ={\\rm int}(D)\\cup (\\overline D \\cap\n\\partial\\Omega)$. Let $S_h(D)$ denote the restriction of\nfunctions in $S_h$ to $D$, and let $S_h^0(\\overline\nD^\\partial)$ denote the functions in $S_h$ with the support in\n$\\overline D^\\partial$. For a given subset $D\\subset\\Omega$,\nwe set $D_{d} =\\{x\\in \\Omega : {\\rm dist}(x,D)\\leq d\\}$\nfor $d> 0$.\nWe denote by $I_h : W^{1,1}(\\Omega) \\rightarrow S_h$ the\noperator given in \\cite{STW2} having the following properties:\nif $d \\geq kh$, then\n$$\\| I_hv - v\\|_{W^{s,p}(D_d)}^{(h)} \\leq\nCh^{l-s}\\|v\\|_{W^{l,p}(D_{2d})},\n\\quad~\\mbox{for~ $0\\leq s\\leq l\\leq r $ ~and~ $ 1\\leq p\\leq\n\\infty$,}$$\nand if supp$(v)\\subset \\overline D ^\\partial_d$, then\n$I_hv\\in S^0_h(\\overline D_{2d}^\\partial)$; also, if\n$v|_{D_d}\\in S_h(D_d)$, then $I_hv = v $ on $D$ and the\nbound above may be replaced by\n$Ch^{l-s}\\|v\\|_{W^{l,p}(D_{2d}\\backslash D)}$.\n\nFor any integer $j$, we define $d_j=2^{-j}$.\nLet $J_1=1$ and $J_0=0$, and let $J_*$ be an integer\nsatisfying $2^{-J_*}= C_*h$ with $C_*\\geq 16$ to\nbe determined later, thus $J_*=\\log_2[1\/(C_*h)]\\leq 2\\ln(2+1\/h)$.\nFor the given constant $C_*$, we have $J_1<J_*$ when $h<1\/(4C_*)$,\nand for a given $x_0\\in\\Omega$ and $j\\geq\nJ_1$ we define the subsets $Q_*,Q_j\\subset\\Omega_T$ and\n$\\Omega_*,\\Omega_j\\subset\\Omega$ by\n\\begin{align*}\n&Q_*(x_0)=\\{(x,t)\\in\\Omega_T: \\max (|x-x_0|,t^{1\/2})\\leq\nd_{J_*}\\},\\\\\n&\\Omega_*(x_0)=\\{x\\in \\Omega: |x-x_0|\\leq d_{J_*}\\},\\\\\n&Q_j(x_0)=\\{(x,t)\\in \\Omega_T:d_j\\leq\n\\max (|x-x_0|,t^{1\/2})\\leq2d_j\\},\\\\\n&\\Omega_j(x_0)=\\{x\\in \\Omega: d_j\\leq|x-x_0|\\leq2d_j\\} ,\\\\\n&Q_{J_0}(x_0)=\\Omega_T\\big\\backslash\\big(\n\\cup_{j=J_1}^{J_*}Q_{j}(x_0)\\cup Q_*(x_0)\\big) ,\\\\\n&\\Omega_{J_0}(x_0)=\\Omega\\big\\backslash\\big(\n\\cup_{j=J_1}^{J_*}\\Omega_{j}(x_0)\\cup \\Omega_*(x_0)\\big).\n\\end{align*}\nFor $j<J_0$, we simply define\n$Q_{j}(x_0)=\\Omega_{j}(x_0)=\\emptyset$ and for any integer $j$ we define\n\\begin{align*}\n&\\Omega_j'(x_0)=\\Omega_{j-1}(x_0)\\cup\\Omega_{j}(x_0)\\cup\\Omega_{j+1}(x_0),\n~~~Q_j'(x_0)=Q_{j-1}(x_0)\\cup Q_{j}(x_0)\\cup Q_{j+1}(x_0), \\\\\n&\\Omega_j''(x_0)=\\Omega_{j-2}(x_0)\\cup\\Omega_{j}'(x_0)\\cup\\Omega_{j+2}(x_0),\n~~~\nQ_j''(x_0)=Q_{j-2}(x_0)\\cup Q_{j}'(x_0)\\cup Q_{j+2}(x_0),\\\\\n&\\Omega_j'''(x_0)=\\Omega_{j-2}(x_0)\\cup\\Omega_{j}''(x_0)\\cup\\Omega_{j+2}(x_0),\n~~~\nQ_j'''(x_0)=Q_{j-2}(x_0)\\cup Q_{j}''(x_0)\\cup Q_{j+2}(x_0) .\n\\end{align*}\nThen we have\n\\begin{align*}\n&\\Omega_T=\\bigcup^{J_*}_{j=J_0}Q_j(x_0)\\,\\cup Q_*(x_0)\n\\quad\\mbox{and}\\quad\n\\Omega=\\bigcup^{J_*}_{j=J_0}\\Omega_j(x_0)\\,\\cup \\Omega_*(x_0),\\end{align*}\nWe refer to $Q_*(x_0)$ as the ``innermost\" set.\nWe shall write $\\sum_{*,j}$ when the innermost set is included and\n$\\sum_j$ when it is not. When $x_0$ is fixed and there is no ambiguity, we simply write\n$Q_j=Q_j(x_0)$, $Q_j'=Q_j'(x_0)$, $Q_j''=Q_j''(x_0)$ and\n$\\Omega_j=\\Omega_j(x_0)$, $\\Omega_j'=\\Omega_j'(x_0)$,\n$\\Omega_j''=\\Omega_j''(x_0)$.\n\nWe shall write\n\\begin{align*}\n&\\|v\\|_{k,D}=\\biggl(\\int_D\\sum_{|\\alpha|\\leq\nk}|\\partial^\\alpha v|^2{\\rm d} x\\biggl)^{\\frac{1}{2}},\\qquad\n\\vertiii{v}_{k,Q}=\\biggl(\\int_Q\\sum_{|\\alpha|\\leq\nk}|\\partial^\\alpha v|^2{\\rm d} x{\\rm d} t\\biggl)^{\\frac{1}{2}} ,\n\\end{align*}\nfor any domain $D\\subset\\Omega$ and $Q\\subset\\Omega\\times(0,T)$.\nThe time derivative will always be displayed explicitly.\nWe denote by $C$ a generic positive constant, which\nwill be independent of $h$, $x_0$, and\nthe undetermined constant $C_*$ until\nit is determined at the end of Section \\ref{dka7}.\n\n\n\\section{Estimates of the parabolic Green's function}\n\\setcounter{equation}{0}\nTo prove our main results, we need the following lemma.\nThe proof of the lemma will be given in the next two subsetions.\n\\begin{lemma}\\label{GMhEst}\n{\\it Let $x_0\\in\\Omega$ and $T=1$.\nLet $\\Gamma(t,x,x_0)$ and $\\Gamma_h(t,x,x_0)$ be defined in\n{\\rm(\\ref{GMFdef})-(\\ref{GMhFdef})}, and set\n$F(t,x)=\\Gamma_h(t,x,x_0)-\\Gamma(t,x,x_0)$.\nThen there exists a positive constant $h_0>0$\nsuch that when $h<h_0$ we have\n\\begin{align}\n&\n\\|\\partial_tF\\|_{L^1( Q_T)}+\n\\|t\\partial_{tt}F\\|_{L^1( Q_T)}\n+\nh^{-1}l_h^{-1}\n\\|F\\|_{W^{1,0}_1( Q_T)}\\leq C,\\label{FFEst1}\\\\\n& \\|\\partial_tF\\|_{L^1(\\Omega\\times(0,\\infty))}\n+ \\|t\\partial_{tt}F\\|_{L^1 (\\Omega\\times(0,\\infty))} \\leq C ,\n\\label{FFEst2}\n\\end{align}\nwhere $l_h=\\ln(2+1\/h)$ and the constant $C$ does not depend on $x_0$.\n}\n\\end{lemma}\n\nThe estimates in the lemma were proved in \\cite{STW2} \nfor parabolic equations with smooth coefficients \nfor which the Green function satisfies (\\ref{g-cond}).\nSince $x_0$ is fixed, we simply write\n$G$ and $\\Gamma$ as abbreviations for\nthe functions $G(\\cdot,\\cdot,x_0)$ and\n$\\Gamma(\\cdot,\\cdot,x_0)$, respectively,\nwhen there is no ambiguity.  We shall assume that\n$h< 1\/(4C_*)$, so that $Q_j(x_0)$, $J_0\\leq j\\leq J_*$,\nare well defined as in the last section.\nIn the rest part of this section, we set $T=1$.\n\n\n\\subsection{Estimates of the Green's functions}\n\\label{EGF}\nIn this subsection, we present some new estimates for the Green's\nfunction, the regularized Green's function and the discrete Green's\nfunction, which will be used in the next subsection to prove Lemma \\ref{GMhEst}.\n\n\\begin{lemma}\\label{GFEst1}\n{\\it\nThere exist $p_1>d$ and $\\alpha>0$ such that \nfor any integer $J_0\\leq j\\leq J_*$, we have\n\\begin{align}\n&d_j^{-4}\\vertiii{\\Gamma(\\cdot,\\cdot,x_0)}_{2,Q_j(x_0)}\n+d_j^{-2}\\vertiii{\\partial_{t}\n\\Gamma(\\cdot,\\cdot,x_0)}_{2,Q_j(x_0)} \\nonumber\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n+\\vertiii{\\partial_{tt}\\Gamma(\\cdot,\\cdot,x_0)}_{2,Q_j(x_0)}\n\\leq Cd_j^{-d\/2-5}, \\label{GFest01}\\\\\n&\\|\\partial_t\\partial_{x_i}G(\\cdot,\\cdot,x_0)\n\\|_{L^\\infty(Q_j(x_0))}+\\|\\partial_t\n\\partial_{x_i}\\Gamma(\\cdot,\\cdot,x_0)\\|_{L^\\infty(Q_j(x_0))}\\leq\nCd_j^{-d -3},\\label{GFest02}\\\\\n&\\|\\partial_{x_i x_l}G(\\cdot,\\cdot,x_0) \\|_{L^{\\infty,p_1}(\\cup_{k\\leq\nj}Q_k(x_0))}\\leq Cd_j^{-d-2+d\/p_1} \\label{GFest03}\n,\\\\\n&\\|\\partial_t\\Gamma(\\cdot,\\cdot,x_0)\\|_{L^1(\\Omega\\times(T,\\infty))}\n+\\|t\\partial_{tt}\\Gamma(\\cdot,\\cdot,x_0)\\|_{L^1(\\Omega\\times(T,\\infty))}\\leq\nC ,\\label{GFest0423}\\\\\n&d_j^{-\\alpha}\\|\\partial_{x_i y_l}\nG(\\cdot,\\cdot,x_0)\\|_{L^\\infty(Q_k(x_0))} +\\|\\partial_{x_i\ny_l}G(\\cdot,\\cdot,x_0)\\|_{C^{\\alpha,\\alpha\/2}(\\overline\nQ_k(x_0))}\\leq Cd_j^{-d -2-\\alpha}\\label{GFest05} ,\\\\\n&\\|\\Gamma_h(1,\\cdot,x_0)\\|_{L^2}+\\|\\partial_t\\Gamma_h(1,\\cdot,x_0)\\|_{L^2}\n+\\|\\partial_{tt}\\Gamma_h(1,\\cdot,x_0)\\|_{L^2}\\leq\nC\\|\\Gamma_h(\\cdot,\\cdot,x_0)\\|_{L^2(\\Omega\\times(1\/2,1])} ,\n\\label{dlky60}\n\\end{align}\nfor $i,l=1,2,\\cdots,d$.\n}\n\\end{lemma}\n\\noindent{\\it Proof}~~~\nFor the given $x_0$ and $j$,  we define a coordinate transformation\n$x-x_0=d_j\\widetilde x$ and $t=d_j^2\\widetilde t$, and $\\widetilde\nG(\\widetilde t,\\widetilde x):=G(t,x,x_0)$, $\\widetilde\nG_{y_l} (\\widetilde t,\\widetilde x):=\\partial_{y_l} G(t,x,y)|_{y=x_0}$, $\\widetilde\na(\\widetilde x):=a(x)$, \n$\\widetilde\nc(\\widetilde x):=c(x)$,\n$\\widetilde Q_k=\\{(\\widetilde x,\\widetilde\nt)\\in\\mathbb{R}^{d+1}:(x,t)\\in Q_k\\}$, $\\widetilde Q_k'=\\widetilde\nQ_{k-1}\\cup \\widetilde Q_k\\cup \\widetilde Q_{k+1}$,  \n$\\widetilde \\Omega_k=\\{(\\widetilde x,\\widetilde\nt)\\in\\mathbb{R}^{d+1}:(x,t)\\in \\Omega_k\\}$, $\\widetilde \\Omega_k'=\\widetilde\n\\Omega_{k-1}\\cup \\widetilde \\Omega_k\\cup \\widetilde \\Omega_{k+1}$,\n$\\widetilde \\Omega=\\{\\widetilde x\\in\\mathbb{R}^d: x \\in\n\\Omega\\}$, and $\\widetilde Q_{\\widetilde T}=\\{(\\widetilde\nx,\\widetilde t)\\in\\mathbb{R}^{d+1}: (x,t) \\in  Q_T\\}$.\nThen $\\widetilde G(\\widetilde t,\\widetilde\nx)$ and $\\widetilde G_{y_l} (\\widetilde t,\\widetilde\nx)$ are solutions of the equations\n\\begin{align}\n&\\partial_{\\widetilde t}\\widetilde G-\\nabla_{\\widetilde\nx}\\cdot(\\widetilde a\\nabla_{\\widetilde x}\n\\widetilde G)+\\widetilde c\\widetilde G=0 \\quad\\mbox{in}~~\\widetilde Q_{j}', \\label{GFest06}\\\\\n&\\partial_{\\widetilde t}\\widetilde G_{y_l}-\\nabla_{\\widetilde\nx}\\cdot(\\widetilde a\\nabla_{\\widetilde x}\n\\widetilde G_{y_l})\n+\\widetilde c\\widetilde G_{y_l}=0 \\quad\\mbox{in}~~\\widetilde Q_{j}' .\n\\label{GFest066}\n\\end{align}\n\nBy the estimates of parabolic equations \n(see Lemma A.1 in Appendix), \nwe have\n\\begin{align}\n&|\\!|\\!|\\partial_{\\widetilde t}\\widetilde\nG|\\!|\\!|_{\\widetilde Q_j}+|\\!|\\!|\\widetilde\nG|\\!|\\!|_{2,\\widetilde Q_j}\n+|\\!|\\!|\\partial_{\\widetilde t}\\widetilde\nG|\\!|\\!|_{2,\\widetilde Q_j}\n+|\\!|\\!|\\partial_{\\widetilde t\\widetilde t}\\widetilde\nG|\\!|\\!|_{2,\\widetilde Q_j}\n\\leq  C|\\!|\\!|\\widetilde G|\\!|\\!|_{\\widetilde Q_j'}  , \\label{dkjq}\\\\\n&\\|\\partial_{\\widetilde x_i}\\widetilde\nG\\|_{L^\\infty(\\widetilde Q_j)}\n+\\|\\partial_{\\widetilde x_i}\\widetilde\nG\\|_{C^{\\alpha,\\alpha\/2}(\\overline{\\widetilde Q}_j)}\n+\\|\\partial_{\\widetilde x_i}\\partial_{\\widetilde x_l}\\widetilde\nG\\|_{L^{\\infty,p_1}(\\widetilde Q_j)}\n \\leq C|\\!|\\!|\\widetilde G|\\!|\\!|_{\\widetilde Q_j'}\n\\\\\n&\\|\\partial_{\\widetilde x_i}\\widetilde\nG_{y_l}\\|_{L^\\infty(\\widetilde Q_j)} \n+\\|\\partial_{\\widetilde x_i}\\widetilde\nG_{y_l}\\|_{C^{\\alpha,\\alpha\/2}(\\overline{\\widetilde Q}_j)}\n\\leq C|\\!|\\!|\\widetilde G_{y_l}|\\!|\\!|_{\\widetilde Q_j'}  .\n\\label{dkjq3}\n\\end{align}\n \nTransforming back to the $(x,t)$ coordinates, \\refe{dkjq}-\\refe{dkjq3}\nreduce to\n\\begin{align*}\n&d_j^{-4}\\vertiii{ G}_{2,Q_j}\n+d_j^{-2}\\vertiii{\\partial_{t}\nG}_{2,Q_j}+\\vertiii{\\partial_{tt}G}_{2,Q_j}\\leq\nCd_j^{-6}\\vertiii{ G}_{Q_j'} ,\\\\\n& d_j^{-2}\\|\\partial_{x_i}G\\|_{L^\\infty( Q_j)}\n+\\|\\partial_t\\partial_{x_i}G\\|_{L^\\infty( Q_j)}\\leq Cd_j^{-d\/2\n-4}\\vertiii{ G}_{Q_j'},\\\\\n&\n\\|\\partial_{x_i}\\partial_{x_l}G\\|_{L^{\\infty,p_1}( Q_j)}\\leq Cd_j^{-d\/2\n-3+d\/p_1}\\vertiii{ G }_{Q_j'} ,\\\\\n& d_j^{-\\alpha}\\|\\partial_{x_i}\n\\partial_{y_l}G\\|_{L^\\infty(Q_j)}\n+\\|\\partial_{x_i}\\partial_{y_l}G\n\\|_{C^{\\alpha,\\alpha\/2}(\\overline\nQ_j)} \\leq Cd_j^{-d\/2 -2-\\alpha}\n\\vertiii{ \\partial_{y_l}G }_{Q_j'}\\leq Cd_j^{-1-\\alpha}\n\\|\\partial_{y_l}G \\|_{L^\\infty(Q_j')}.\n\\end{align*}\nFrom the Green function estimate (\\ref{FEstP}), we see that\n$\\vertiii{ G}_{Q_j'}\\leq Cd_j^{-d\/2+1}$ and so\n\\begin{align}\n&d_j^{-4}\\vertiii{ G}_{2,Q_j}\n+d_j^{-2}\\vertiii{\\partial_{t}\nG}_{2,Q_j}+\\vertiii{\\partial_{tt}G}_{2,Q_j}\n\\leq Cd_j^{-d\/2-5}, \\label{GFest07}\\\\\n&d_j^{-2}\\|\\partial_{x_i}G\\|_{L^\\infty(Q_j)}\n+\\|\\partial_t\\partial_{x_i}G\\|_{L^\\infty(Q_j)}\\leq Cd_j^{-d\n-3},\\label{GFes03}\\\\\n&\\|\\partial_{x_i x_l}G\\|_{L^{\\infty,p_1}(Q_j)}\\leq Cd_j^{-d\n-2+d\/p_1} \\label{Gsnn} ,\\\\\n&d_j^{-\\alpha}\\|\\partial_{x_i}\\partial_{y_l}\nG\\|_{L^\\infty(Q_j)}+\\|\\partial_{x_i}\\partial_{y_l}\nG\\|_{C^{\\alpha,\\alpha\/2}(\\overline Q_j)} \\leq Cd_j^{\n-1-\\alpha}\\|\\partial_{y_l}G \\|_{L^\\infty(Q_j')}\n\\leq\nCd_j^{-d-2-\\alpha} , \\label{Gmm2}\n\\end{align}\nwhere we have used (\\ref{GFes03}) in deriving (\\ref{Gmm2}).\nClearly, (\\ref{Gsnn}) further implies that\n\\begin{align}\\label{fd00}\n&\\|\\partial_{x_i x_l}G\\|_{L^{\\infty,p_1}(\\cup_{k\\leq j}Q_k)}\\leq\nCd_j^{-d -2+d\/p_1} .\n\\end{align}\n\n\n\nBy estimating $\\Gamma(t,x)=\\int_\\Omega\nG(t,x,y)\\widetilde\\delta_{x_0}(y){\\rm d} y$, we can see that the\nestimates (\\ref{GFest07})-(\\ref{fd00}) also hold when $G$ is\nreplaced by $\\Gamma$.\n\nFrom the inequalities\n(\\ref{FtEstP})-(\\ref{FtEstP2}) we derive that\n\\begin{align}\\label{dlk80}\n&\\|\\partial_tG(t,\\cdot,x_0)\\|_{L^\\infty}\n+\\|\\partial_t\\Gamma (t,\\cdot,x_0)\\|_{L^\\infty}\n\\nonumber\\\\\n&+\\|t\\partial_{tt}G(t,\\cdot,x_0)\\|_{L^\\infty}\n+\\|t\\partial_{tt}\\Gamma (t,\\cdot,x_0)\\|_{L^\\infty}\\leq\nCt^{-d\/2-1}~~\\mbox{for}~~t\\geq 1\/4,\n\\end{align}\nwhich implies (\\ref{GFest0423}).\n\nFinally, we note that the inequality (\\ref{dlky60}) follows\nfrom basic energy estimates.\n\nThe proof of Lemma \\ref{GFEst1} is complete.\n~\\vrule height8pt width 5pt depth 0pt\\bigskip\n\n\n\n\n\n\\subsection{Proof of Lemma \\ref{GMhEst}}\n\\label{dka7}\n\n\\begin{lemma}\\label{LocEst}\n{\\it\nSuppose that $z(\\cdot,t)\\in H^1$, $z_t(\\cdot,t)\\in L^2$ and $z_h(\\cdot,t)\\in S_h$\nfor each fixed $t\\in[0,T]$, and suppose that $e=z_h-z$ satisfies the equation\n$$\n(e_t,\\chi)+(a\\nabla e,\\nabla \\chi)+(c\ne,\\chi)=0,\\quad\\forall~\\chi\\in S_h,~t>0 ,\n$$\nwith $z(\\cdot,0)=0$ and $z_h(\\cdot,0)=z_{0h}$ on $\\Omega_j'$.\nThen for any $q>0$ there exists a constant $C_q$ such that\n\\begin{align*}\n\\vertiii{e_t}_{Q_j} + d_j^{-1}\\vertiii{e}_{1,Q_j}\n\\leq\nC_q\\big(I_j(z_{0h})+X_j(I_hz-z)\n+H_j(e)+d_j^{-2}\\vertiii{e}_{Q_j'}\\big),\n\\end{align*}\nwhere\n\\begin{align*}\n&I_j(z_{0h})=\\|z_{0h}\\|_{1,\\Omega_j'}\n+d_j^{-1}\\|z_{0h}\\|_{\\Omega_j'},\\\\\n&X_j(I_hz-z)=d_j\\vertiii{\\partial_t(I_hz-z)}_{1,Q_j'}\n+\\vertiii{\\partial_t(I_hz-z)}_{Q_j'}\n+d_j^{-1}\\vertiii{I_hz-z}_{1,Q_j'}+\nd_j^{-2}\\vertiii{I_hz-z}_{Q_j'},\\\\\n&H_j(e)=(h\/d_j)^q\\big(\\vertiii{e_t}_{Q_j'}\n+d_j^{-1}\\vertiii{e}_{1,Q_j'}\\big) .\n\\end{align*}\n}\n\\end{lemma}\n\nThe above lemma was proved in \\cite{STW2} (Section 5 and Section 6) only for\nparabolic equations with smooth coefficients. However, we can see from the proof\nthat the lemma still holds when $a_{ij}\\in W^{1,\\infty}(\\Omega)$ and\n$c\\in L^\\infty(\\Omega)$ satisfy (\\ref{coeffcond}).\nMoreover, for parabolic equations with smooth coefficients,\nLemma \\ref{GMhEst} was proved in \\cite{STW2} by applying Lemma\n\\ref{LocEst} with the additional assumption (\\ref{g-cond}). Here, we\nshall prove Lemma \\ref{GMhEst} directly from Lemma \\ref{GFEst1} and\nLemma \\ref{LocEst}.\n\nFirst, we prove \\refe{FFEst1}. \nLet $\\mu_j=[h\\ln(2+1\/h)]^{-1}+d_j^{-1}$ and define\n\\begin{align}\\label{KdKj0}\nK_j=\n\\vertiii{\\partial_tF}_{Q_j}+\nd_j^2\\vertiii{\\partial_{tt}F}_{Q_j}\n+\n\\mu_j\n\\vertiii{F}_{1,Q_j} ,\n\\end{align}\nand\n\\begin{align}\\label{KdKj}\n{\\cal K}:=\\sum_{j}d_j^{1+d\/2}K_j.\n\\end{align}\nFrom Section 4 of \\cite{STW2} we see that (\\ref{FFEst1}) holds if we can prove that ${\\cal K}\\leq C$ for some positive constant $C$ which is independent of $h$, $J_*$ and $C_*$.\n\nTo prove the boundedness of ${\\cal K}$, we set $e=F$ and $e=F_t$\nin Lemma \\ref{LocEst}, respectively. Since in either case $z(0)=0$ on $\\Omega_j'$,\nwe obtain\n\\begin{align}\nK_j\\leq\nC(\\widehat{I_j}+\\widehat{X_j}+\\widehat{H_j}+\\mu_jd^{-1}_j\\vertiii{\nF }_{Q'_j} ),\n\\end{align}\nwhere, by using the exponential decay of \n$|P_h\\widetilde\\delta_{x_0}(y)|\\leq Ch^{-d}e^{-C|y-x_0|\/h}$ \n\\cite{STW2}, we have\n\\begin{align*}\n&\\widehat{I_j}=d_j^2\\|F_t(0)\\|_{1,\\Omega_j'}\n+d_j\\|F_t(0)\\|_{\\Omega_j'}\n+d_j\\mu_j\\|F(0)\\|_{1,\\Omega_j'}+\\mu_j\\|F(0)\\|_{\\Omega_j'}\n\\leq Ch^{-1-d\/2}e^{-Cd_j\/h},\\\\\n&\\widehat{X_j}=\nd_j^3\\vertiii{(I_h\\Gamma-\\Gamma)_{tt}}_{1,Q_j'}\n+d_j^2\\vertiii{(I_h\\Gamma-\\Gamma)_{tt}}_{Q_j'}\n+\\mu_jd_j^2\\vertiii{(I_h\\Gamma-\\Gamma)_t}_{1,Q_j'}\n+\\mu_jd_j\\vertiii{(I_h\\Gamma-\\Gamma)_t}_{Q_j'}\\\\\n&\\qquad~\n+\\mu_j\\vertiii{I_h\\Gamma-\\Gamma}_{1,Q_j'}\n+\\mu_jd_j^{-1}\\vertiii{I_h\\Gamma-\\Gamma}_{Q_j'}\\\\\n&\\quad~\n\\leq (d_j^3h+d_j^2h^2)\\vertiii{\\partial_{tt}\\Gamma}_{2,Q_j''}\n+(\\mu_jd_j^2h+\\mu_jd_jh^2)\\vertiii{\\partial_{t}\\Gamma}_{2,Q_j''}\n+(\\mu_jh+\\mu_jd_j^{-1}h^2)\\vertiii{\\Gamma}_{2,Q_j''}\\\\\n&\\quad~\n\\leq C hd_j^{-d\/2-2}+C(\n[\\ln(2+1\/h)]^{-1}+h\/d_j)d_j^{-d\/2-1},\\\\\n&\\widehat{H_j}=\\big(h\/d_j\\big)^q \\big(d_j^2\\vertiii{F_{tt}}_{Q_j'}\n+d_j\\vertiii{F_{t}}_{1,Q_j'}\n+\\mu_jd_j\\vertiii{F_t}_{Q_j'}+\\mu_j\\vertiii{F}_{1,Q_j'}\n\\big)\\\\\n&\\quad~\n\\leq \\big(h\/d_j\\big)^q\n\\big(d_j^2\\vertiii{F_{tt}}_{Q_T}+d_j\\vertiii{F_{t}}_{1,Q_T}\n+\\mu_jd_j\\vertiii{F_t}_{Q_T}+\\mu_j\\vertiii{F}_{1,Q_T}\n\\big).\n\\end{align*}\nThe last term $\\widehat{H_j}$ was estimated in \\cite{STW2} via\nenergy estimates, with $\\sum_{j}d_j^{1+d\/2}\\widehat{H_j}\\leq C$.\nTherefore,\n\\begin{align}\\label{dlj6}\n{\\cal K}=\\sum_{j}d_j^{1+d\/2}K_j\\leq\nC+C\\sum_{j}d_j^{d\/2}\\mu_j\\vertiii{F}_{Q_j'} .\n\\end{align}\n\nTo estimate $\\vertiii{F}_{Q_j'}$, we apply a duality argument.\nLet $w$ be the solution of the backward parabolic equation\n$$\n-\\partial_tw+Aw=v\\quad\\mbox{with}~~w(T)=0,\n$$\nwhere $v$ is a function which is supported in $Q_j'$ and\n$\\vertiii{v}_{Q_j'}=1$. Multiplying the above equation by $F$,\nwith integration by parts we get\n\\begin{align}\\label{dka6}\n[F,v]=(F(0),w(0))+[F_t,w]+\\sum_{i,j=1}^d[a_{ij}\\partial_j\nF,\\partial_i w]+[cF,w],\n\\end{align}\nwhere\n\\begin{align*}\n(F(0),w(0))&=(P_h\\widetilde\\delta_{x_0}-\\widetilde\\delta_{x_0},w(0))\\\\\n&=(P_h\\widetilde\\delta_{x_0}-\\widetilde\\delta_{x_0},w(0)-I_hw(0))\\\\\n&=\n(P_h\\widetilde\\delta_{x_0},w(0)-I_hw(0))_{\\Omega_j''}\n+(P_h\\widetilde\\delta_{x_0}-\\widetilde\\delta_{x_0},\nw(0)-I_hw(0))_{(\\Omega_j'')^c}\\\\\n&:=I_1+I_2 .\n\\end{align*}\nSince\n$|P_h\\widetilde\\delta_{x_0}(y)|\\leq Ch^{-d}e^{-C|y-x_0|\/h}$ \n\\cite{STW2}, we derive\nthat\n\\begin{align}\n&|I_1|\\leq\nCh\\|P_h\\widetilde\\delta_{x_0}\\|_{L^2(\\Omega_j'')}\\|w(0)\\|_{H^1(\\Omega)}\\leq\nCd_j^{d\/2}h^{-d+1}e^{-Cd_j\/h}\\vertiii{v}_{Q_j'}\\leq\nCh^{2}d_j^{-d\/2 -1}, \\label{SF2}\\\\\n&|I_2|\\leq C\\|P_h\\widetilde\\delta_{x_0}-\n\\widetilde\\delta_{x_0}\\|_{L^{p_1'}}\n\\|w(0)-I_hw(0)\\|_{L^{p_1}((\\Omega_j'')^c)}\\leq\nCh^{2-d\/p_1}\\|w(0)\\|_{W^{2,p_1}((\\Omega_j'')^c)} .\n\\label{SF22}\n\\end{align}\nWe proceed to estimate $\\|w(0)\\|_{W^{2,p_1}((\\Omega_j'')^c)}$.\nLet $D_j$ be a set containing $(\\Omega_j'')^c$ but its\ndistance to $\\Omega_j'$ is larger than $C^{-1}d_j$. Since\n$$\n\\partial_{x_i}\\partial_{x_j}w(x,0)\n=\\int_0^{T}\\int_{\\Omega}\n\\partial_{x_i}\\partial_{x_j}G(s,x,y)v(y,s){\\rm d} y{\\rm d} s ,\n$$\nby taking the $L^p(D_j)$ norm with respect to $x$ we obtain\n$$\n|x-y|+s^{1\/2}\\geq C_1^{-1}d_j \\quad\\mbox{for $x\\in D_j$ and $(y,s)\\in\nQ_j$}\n$$\nfor some positive constant $C_1$.  Using (\\ref{GFest03}) we\nfurther derive that\n\\begin{align}\n\\|\\partial_{x_i}\\partial_{x_j}w(0)\\|_{L^{p_1}(D_j)}\n&\\leq C\\sup_{y\\in\\Omega}\\|\\partial_{x_i}\\partial_{x_j}\nG(\\cdot,\\cdot,y)\\|_{L^{\\infty,p_1}(\\cup_{k\\leq\nj+\\log_2C_1}Q_k(y))}\\|v\\|_{L^{1}(Q_j')}\\nonumber\\\\\n&\\leq C d_j^{-d -2+d\/p_1}\\|v\\|_{L^{1}(Q_j')}\n \\nonumber\\\\\n&\\leq C d_j^{-d\/2 -1+d\/p_1}\\vertiii{v}_{Q_j'} , \\nonumber\\\\\n&=C d_j^{-d\/2 -1+d\/p_1} . \\label{SF3}\n\\end{align}\nFrom (\\ref{SF2})-(\\ref{SF3}), we see that the \nfirst term on the right-hand side of \\refe{dka6} is bounded by\n\\begin{align}\n|(F(0),w(0))| \\leq  Ch^{2}d_j^{-d\/2 -1}+Ch^{2}d_j^{-d\/2\n-1}(h\/d_j)^{-d\/p_1} \\leq  Ch^{2}d_j^{-d\/2 -1}(h\/d_j)^{-d\/p_1}  ,\n\\label{f69}\n\\end{align}\nand the rest terms are bounded by\n\\begin{align}\\label{sd80}\n&[F_t,w]+\\sum_{i,j=1}^d[a_{ij}\\partial_j F,\\partial_iw]+[cF,w]\\nonumber\\\\\n&=[F_t,w-I_hw]+\\sum_{i,j=1}^d[a_{ij}\\partial_j F,\\partial_i\n(w-I_hw)] +[cF,w-I_hw]\\nonumber\\\\\n&\\leq\n\\sum_{*,i}C(h^2\\vertiii{F_t}_{Q_i}+h\\vertiii{F}_{1,Q_i})\\vertiii{w}_{2,Q_i'}\n.\n\\end{align}\nTo estimate $\\vertiii{w}_{2,Q_i'}$ we consider the expression\n$$\n\\partial_{x_i}\\partial_{x_j}w(x,t)\n=\\int_0^{T}\\int_{\\Omega}\n\\partial_{x_i}\\partial_{x_j}G(s-t,x,y)v(y,s)1_{s>t}\\,{\\rm d} y{\\rm d} s .\n$$\nIf $i\\leq j-2$ (so that $d_i>d_j$), then $w(t)=0$ for\n$t>d_j^2$ (because $v$ is supported in $Q_j$), $|x-y|\\sim d_i$\nand $s-t\\in(0,d_i^2)$ for $t< d_j^2$, $(x,t)\\in Q_i$ and\n$(y,s)\\in Q_j'$.\nwe obtain\n\\begin{align*}\n\\vertiii{\\partial_{x_i}\\partial_{x_j}w}_{Q_i'}\n\\leq\\sup_{y}\\vertiii{\\partial_{x_i}\\partial_{x_j}G(\\cdot,\\cdot,y)}_{Q_i(y)}\n\\|v\\|_{L^1(Q_j)}\\leq\nCd_i^{-d\/2-1}d_j^{d\/2+1}\\vertiii{v}_{Q_j}\\leq\nC(d_j\/d_i)^{d\/2+1} .\n\\end{align*}\nIf $i\\geq j+2$ (so that $d_i\\leq d_j$), then\n$\\max(|s-t|^{1\/2},|x-y|)\\geq d_{j+2}$ for $(x,t)\\in Q_i$, thus for\n$1\/2=1\/\\bar p_1+1\/p_1$ we have\n\\begin{align*}\n\\vertiii{\\partial_{x_i}\\partial_{x_j}w}_{Q_i'}\n&=\\sup_{(y,s)\\in Q_T}\\vertiii{\\partial_{x_i}\\partial_{x_j}G(s-\\cdot,\\cdot,y)\n1_{\\cup_{k\\leq j+2}Q_{k}(y)}}_{Q_i'}\n\\|v\\|_{L^1(Q_j')}\\\\\n&\\leq Cd_i^{ 1+d \/\\bar p_1}\\sup_{y}\\|\\partial_{x_i}\n\\partial_{x_j}G(\\cdot,\\cdot,y)\\|_{L^{\\infty,p_1}(\\cup_{k\\leq j+2}Q_{k}'(y))}\n\\vertiii{v}_{Q_j'}d_j^{d\/2+1}\n\\\\\n&\\leq Cd_i^{1+d \/\\bar p_1}d_j^{-d -2+d\/p_1}d_j^{d\/2+1}\\\\[5pt]\n&= C(d_i\/d_j)^{1+d\/2-d\/p_1} .\n\\end{align*}\nIf $|i-j|\\leq 1$, then by applying the standard energy estimate we get\n$\\vertiii{w}_{2,Q_T}\\leq C\\vertiii{v}_{Q_T}=C$.\nCombining the three cases, we have proved\n$$\\vertiii{w}_{2,Q_i'}\\leq C\\min\\big(d_i\/d_j,d_j\/d_i\\big)^{1+d\/2-d\/p_1}:=C m_{ij}.$$\nSubstituting \\refe{f69}-\\refe{sd80} into (\\ref{dka6}) gives \n\\begin{align}\n\\vertiii{F}_{Q_j'}\\leq Ch^2d_j^{-d\/2-1}\n(h\/d_j)^{-d\/p_1}+C\\sum_{*,i}m_{ij}\n(h^2\\vertiii{F_t}_{Q_i}\n+h\\vertiii{F}_{1,Q_i}) .\n\\end{align}\nBy noting that $p_1>d$, \\refe{dlj6} reduces to\n\\begin{align*}\n{\\cal K}&\\leq C+C\\sum_j (h\/d_j)^{1-d\/p_1}+C\\sum_j d_j^{d\/2}\n\\mu_j \\sum_{*,i}m_{ij}\\big(h^2\\vertiii{F_t}_{Q_i}\n+h\\vertiii{F}_{1,Q_i} \\big)\\\\\n&\\leq\nC+C\\sum_{*,i}\\left(h^2\\vertiii{F_t}_{Q_i}+h\\vertiii{F}_{1,Q_i}\n\\right)\\sum_jd^{d\/2}_j\\mu_jm_{ij}\\\\\n&\\leq\nC+C\\sum_{*,i}\\left(h^2\\vertiii{F_t}_{Q_i}+h\\vertiii{F}_{1,Q_i}\n\\right)\nd^{1+d\/2}_i\\mu_id^{-1}_i\\\\\n&\\leq C+\\left(h\\vertiii{F_t}_{Q_*}\n+\\vertiii{F}_{1,Q_*} \\right)d_{J_*}^{d\/2}\n\\big(1\/\\ln(2+1\/h)+h\/d_{J_*}\\big)\\\\\n&~~~+C\\sum_id^{1+d\/2}_i\n\\left(\\vertiii{F_t}_{Q_i}+\\mu_i\\vertiii{F}_{1,Q_i}\\right\n)\\left(\\frac{h}{d_i}\\right)\\\\\n&\\leq\nC+CC^{d\/2}_*+C\\sum_id^{1+d\/2}_iK_i\\left(\\frac{h}{d_i}\\right)\\\\\n&\\leq C_2+C_2C^{d\/2}_*+C_2C^{-1}_*{\\cal K}\n\\end{align*}\nfor some positive constant $C_2$.  \nBy choosing $C_*=16+2C_2$, the above inequality shows\nthat ${\\cal K}$ is bounded. As we have mentioned,\nthe boundedness of ${\\cal K}$ implies (\\ref{FFEst1}).\n\nNext, we prove \\refe{FFEst2}. From the definition of ${\\cal K}$ in (\\ref{KdKj0})-(\\ref{KdKj}),\nwe further derive that\n\\begin{align*}\n\\vertiii{\\partial_tF}_{L^2(\\Omega\\times(1\/4,1))}+\n \\vertiii{\\partial_{tt}F}_{L^2(\\Omega\\times(1\/4,1))} \\leq C .\n\\end{align*}\nThe above inequality and (\\ref{dlk80}) imply that\n\\begin{align*}\n\\vertiii{\\partial_t\\Gamma_h}_{L^2(\\Omega\\times(1\/4,1))}+\n \\vertiii{\\partial_{tt}\\Gamma_h}_{L^2(\\Omega\\times(1\/4,1))} \\leq C ,\n\\end{align*}\nwhich together with (\\ref{dlky60}) gives\n\\begin{align*}\n\\|\\partial_t\\Gamma_h(1,\\cdot,x_0)\\|_{L^2}+\n\\|\\partial_{tt}\\Gamma_h(1,\\cdot,x_0)\\|_{L^2} \\leq C .\n\\end{align*}\nDifferentiating the equation (\\ref{GMhFdef}) with respect to $t$ and\nmultiplying the result by $\\partial_t\\Gamma_h$, we get\n\\begin{align*}\n&\\frac{{\\rm d}}{{\\rm d} t}\\|\\partial_t\\Gamma_h(t,\\cdot,x_0)\\|_{L^2}^2\n+c_0\\|\\partial_t\\Gamma_h(t,\\cdot,x_0)\\|_{L^2}^2  \\\\\n&\\leq \\frac{{\\rm d}}{{\\rm d} t}\\|\\partial_t\\Gamma_h(t,\\cdot,x_0)\\|_{L^2}^2\n+(A_h\\partial_t\\Gamma_h(t,\\cdot,x_0),\\partial_t\\Gamma_h(t,\\cdot,x_0)) \\\\\n&= 0\n\\end{align*}\nfor $t\\geq 1$, which further gives\n\\begin{align*}\n\\|\\partial_t\\Gamma_h(t,\\cdot,x_0)\\|_{L^2}^2\\leq\ne^{-c_0(t-1)}\\|\\partial_t\\Gamma_h(1,\\cdot,x_0)\\|_{L^2}^2\n\\leq Ce^{-c_0(t-1)} .\n\\end{align*}\nSimilarly, we can prove that\n\\begin{align*}\n\\|\\partial_{tt}\\Gamma_h(t,\\cdot,x_0)\\|_{L^2}^2\n\\leq Ce^{-c_0(t-1)} .\n\\end{align*}\nFrom (\\ref{FFEst1}), (\\ref{GFest0423})\nand the last two inequalities, we derive (\\ref{FFEst2}) for the case\n$h<h_0:=1\/(4C_*)$.\n\n\nThe proof of Lemma \\ref{GMhEst} is completed. ~\\vrule height8pt width 5pt depth 0pt\n\n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{MainTHM1}}\n\\label{pkq}\n\\setcounter{equation}{0}\nIn this section, we assume that $C_*=16+2C_2$ as chosen in the last section.\n\n\\subsection{Proof of (\\ref{STLEst})-(\\ref{STLEst2})}\nSince\n$$(E_h(t)v_h)(x_0) = (F (t), v_h) + (\\Gamma(t), v_h)\n=\\int_0^t(\\partial_tF (s), v_h){\\rm d} s+(F(0), v_h) + (\\Gamma(t), v_h)$$\nwith $\\|F(0)\\|_{L^1}+\\|\\Gamma(t)\\|_{L^1}\\leq C$, it follows that\n(\\ref{STLEst}) is a consequence of (\\ref{FFEst2}) when $h<h_0$.\n\nFrom Page 1360 of \\cite{STW2} we also see that, for $t\\in(0,T)$,\n\\begin{align*}\nu_h(t)(x_0) = (u(t), \\widetilde \\delta_h) + \n\\int_0^t(u(t), \\partial_tF(t-s)) {\\rm d} s\n+\\int_0^t (u(s),AF(t-s)){\\rm d} s ,\n\\end{align*}\nwhere the first two terms on the right-hand side\nare bounded by $C\\|u\\|_{L^\\infty(Q_T)}$ and\nthe third term satisfies that\n\\begin{align*}\n&\\bigg|\\int_0^t (u(s),AF(t-s)){\\rm d} s \\bigg|\\\\\n&\\leq C\\|u\\|_{L^\\infty(Q_T)}\n\\big(h^{-1}\\|F\\|_{W^{1,0}_1( Q_T)}+\\|I_h\\Gamma-\\Gamma\\|_{W^{2,0}_1(Q_T)}^{(h)}\n+h^{-1}\\|I_h\\Gamma-\\Gamma\\|_{W^{1,0}_1(Q_T)}\\big)\\\\\n&\\leq C\\|u\\|_{L^\\infty(Q_T)}\n\\big(l_h+\\|I_h\\Gamma-\\Gamma\\|_{W^{2,0}_1(Q_T)}^{(h)}\n+h^{-1}\\|I_h\\Gamma-\\Gamma\\|_{W^{1,0}_1(Q_T)}\\big) ,\n\\end{align*}\nwhere we have used (\\ref{FFEst1}) in the last inequality. We\nsee that (\\ref{STLEst2}) is a consequence of the following inequality:\n\\begin{align}\\label{dso7}\n\\|I_h\\Gamma-\\Gamma\\|_{W^{2,0}_1(Q_T)}^{(h)}\n+h^{-1}\\|I_h\\Gamma-\\Gamma\\|_{W^{1,0}_1(Q_T)}\n\\leq Cl_h .\n\\end{align}\nTo check the above inequality, we simply note that\n\\begin{align*}\n\\|I_h\\Gamma-\\Gamma\\|_{W^{2,0}_1(Q_*)}^{(h)}\n+h^{-1}\\|\\nabla(I_h\\Gamma-\\Gamma)\\|_{L^1(Q_*)}^{(h)}\n&\\leq\nCC_*^{1+d\/2}h^{1+d\/2}\\vertiii{\\Gamma}_{2,Q_T}\\\\\n&\\leq\nCC_*^{1+d\/2}h^{1+d\/2}\\|\\widetilde\\delta_{x_0}\\|_{H^1(\\Omega)}\\\\\n&\\leq CC_*^{1+d\/2},\n\\end{align*}\nand by Lemma \\ref{GFEst1} we have\n\\begin{align*}\n&\\|I_h\\Gamma-\\Gamma\\|_{W^{2,0}_1(Q_T\\backslash\nQ_*)}^{(h)}+h^{-1}\\|I_h\\Gamma-\\Gamma\\|_{W^{1,0}_1(Q_T\\backslash\nQ_*)} \\\\[8pt]\n&\\leq C\\sum_j d_j^{1+d\/2}\\vertiii{\\Gamma}_{2,Q_j} \\leq C\\sum_j\nd_j^{1+d\/2}d_j^{-1-d\/2} \\\\\n&\\leq CJ_* \\leq Cl_h .\n\\end{align*}\nTherefore, (\\ref{dso7}) is proved for $T=1$, which implies (\\ref{STLEst2}) for $T=1$\nand $h<h_0$.\nThe case $T>1$ follows from the case $T=1$ by iterations:\n\\begin{align*}\n\\|u_h\\|_{L^\\infty(\\Omega\\times(k,k+1])}\\leq\nC\\|u_h\\|_{L^\\infty(\\Omega\\times(k-1,k])}+Cl_h\\|u\\|_{L^\\infty(\\Omega\\times(0,T))},\n\\quad\\forall~k\\geq 1.\n\\end{align*}\n\nWhen $h\\geq h_0$ and $f\\equiv g_j\\equiv 0$, the standard energy\nestimates of (\\ref{FEEq0}) give\n\\begin{align*}\n\\|u_h(t)\\|_{L^2}+t\\|\\partial_tu_h(t)\\|_{L^2}\\leq C\\|u_h(0)\\|_{L^2} .\n\\end{align*}\nBy using an inverse inequality, we further derive that\n\\begin{align*}\n\\|u_h(t)\\|_{L^\\infty}+t\\|\\partial_tu_h(t)\\|_{L^\\infty}\n\\leq Ch_0^{-d\/2}(\\|u_h(t)\\|_{L^2}+t\\|\\partial_tu_h(t)\\|_{L^2})\n\\leq C\\|u_h(0)\\|_{L^2} \\leq C\\|u_h(0)\\|_{L^\\infty},\n\\end{align*}\nwhich implies (\\ref{STLEst}).\n\nWhen $h\\geq h_0$ while $f$ or $g_j$ may not be identically zero, we decompose\nthe solution of (\\ref{FEEq0})  as\n$u_h=\\widetilde u_h+v_h$, where $\\widetilde u_h$ and $v_h$ are solutions of\nthe equations\n\\begin{align}\\label{Ga513}\n&\\left\\{\n\\begin{array}{ll}\n\\partial_t\\widetilde u_h+A_h\\widetilde u_h = f_h-\\overline\\nabla_h\\cdot{\\bf g} ,\n\\\\[3pt]\n\\widetilde u_h(0)=P_hu^0 ,\n\\end{array}\n\\right.\n\\end{align}\nand\n\\begin{align}\\label{Eqv9}\n&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\left\\{\n\\begin{array}{ll}\n\\partial_tv_h+A_hv_h = 0 ,\n\\\\[3pt]\nv_h(0)=u^0_h-P_hu^0,\n\\end{array}\n\\right.\n\\end{align}\nrespectively.  Write the equation (\\ref{PDE0}) as\n\\begin{align}\\label{Ga903}\n&\\left\\{\n\\begin{array}{ll}\n\\partial_tu+Au\n=f -\\overline\\nabla\\cdot {\\bf g} &\\mbox{in}~\\Omega,\n\\\\[3pt]\nu(0)=u^0 &\\mbox{in}~\\Omega ,\n\\end{array}\n\\right.\n\\end{align}\nand let $w_h=\\widetilde u_h-P_hu$. The difference of (\\ref{Ga513}) and\n(\\ref{Ga903}) gives\n\\begin{align*}\n&\\left\\{\n\\begin{array}{ll}\n\\partial_t w_h+A_h w_h =   A_h(R_hu-P_hu) ,\n\\\\[3pt]\nw_h(0)=0 .\n\\end{array}\n\\right.\n\\end{align*}\nMultiplying the above equation by $w_h$, we obtain\n\\begin{align*}\n\\|w_h\\|_{L^\\infty((0,T);L^2)}\n&\\leq C\\|R_hu-P_hu\\|_{L^2((0,T);H^1)}\\\\\n&\\leq Ch_0^{-1}\\|R_hu-P_hu\\|_{L^2((0,T);L^2)}\\\\\n&\\leq C_{T}\\|R_hu-P_hu\\|_{L^\\infty((0,T);L^\\infty)}\\\\\n&\\leq C_{T}\\|u\\|_{L^\\infty( Q_T)} ,\n\\end{align*}\nwhere we have used the inequality \n$\\|R_hu\\|_{L^\\infty}\\leq C_{h_0}\\|u\\|_{L^\\infty}$\nin the last step. By using an inverse inequality we further derive that\n\\begin{align*}\n\\|w_h\\|_{L^\\infty( Q_T)}\\leq\nCh_0^{-d\/2}\\|w_h\\|_{L^\\infty((0,T);L^2)}\\leq C_T\\|u\\|_{L^\\infty( Q_T)}.\n\\end{align*}\n\nApplying (\\ref{STLEst}) to the equation (\\ref{Eqv9}) we obtain\n$$\n\\|v_h\\|_{L^\\infty( Q_T)}\\leq C\\|u_h^0-P_hu^0\\|_{L^\\infty}\n\\leq C\\|u_h^0\\|_{L^\\infty}+C\\|u\\|_{L^\\infty( Q_T)}  .\n$$\nThe last two inequalities imply (\\ref{STLEst2}) for the case $h\\geq h_0$.\n\n\\subsection{Proof of (\\ref{smgest})}\n\n\nWe define the truncated Green function $G_{\\rm tr}^*$ in the\nfollowing way. Let $\\eta$ be a nonnegative smooth function on $\\mathbb{R}$\nsuch that $\\eta(\\rho)=0$ for $|\\rho|\\leq 1\/2$ and $\\eta(\\rho)=1$ for\n$|\\rho|\\geq 1$. If we set $\\chi(t,x,y)=\\eta\\big(|x-y|^4+t^2\\big)$\nand $\\chi_\\epsilon(t,x,y)=\n\\chi(t\/\\epsilon^2,x\/\\epsilon,y\/\\epsilon)$, then $\\chi_\\epsilon$ is a\n$C^\\infty$ function of $x,y$ and $t$. It is easy to see that $\\chi_\\epsilon=0$ when\n$\\max(|x-y|,\\sqrt{t})<\\epsilon\/2$, and $\\chi_\\epsilon=1$ when\n$\\max(|x-y|,\\sqrt{t})>\\epsilon$, and $|\\partial^{\\alpha_1}_t\n\\partial^{\\beta_1}_x\\partial^{\\beta_2}_y\n\\chi_\\epsilon(t,x,y)|\\leq C\\epsilon^{-2\\alpha_1\n-|\\beta_1|-|\\beta_2|}$.\n\nFor  $d_{J_*}=C_*h$,\n$\\chi_{d_*}(\\cdot,\\cdot,y)=0$ in the domain\n$Q_{*\/2}(y):=\\{(x,t)\\in Q_T: \\max(|x-y|,\\sqrt{t})<d_{J_*}\/2\\}$.\nWe define a truncated Green's function by\n\\begin{align}\nG_{\\rm tr}^*(t,x,y)=\nG(t,x,y)\\chi_{d_{J_*}}(t,x,y) .\n\\end{align}\nCheck that $G_{\\rm tr}^{*}(t,x,y)$ is symmetric with respect to $x$\nand $y$, $G_{\\rm tr}^{*}(\\cdot,\\cdot,y)\\equiv 0$ in $Q_{*\/2}(y)$, $0\\leq G^*_{\\rm\ntr}(t,x,y)\\leq G(t,x,y)$ and it obeys (\\ref{FEstP})-(\\ref{FtEstP}) when\n$\\max(|x-y|,\\sqrt{t})>d_{J_*}$.\n\nFor the fixed trianglular element $\\tau_l^h$ and\nthe point $x_0\\in \\tau_l^h$, the function\n$\\widetilde \\delta_{x_0}$ is supported in\n$\\tau_l^h\\subset \\Omega_*(x_0)$ with\n$\\int_\\Omega\\widetilde\\delta_{x_0}(y){\\rm d} y=1$\n(see the notations in Section \\ref{fnot}). Therefore, by using\nLemma \\ref{GFEst1}  we see that\n\\begin{align*}\n&\\iint_{\\Omega_\\infty\\backslash\nQ_*(x_0)}|\\partial_t\\Gamma(\\tau,x,x_0)\n-\\partial_tG(\\tau,x,x_0)|{\\rm d} x{\\rm d}\\tau\\\\\n&= \\iint_{ Q_T\\backslash Q_*(x_0)}\\biggl|\\int_\\Omega\n\\partial_tG(\\tau,x,y)\\widetilde\\delta_{x_0}(y){\\rm d}\ny-\\partial_tG(\\tau,x,x_0)\\biggl|{\\rm d} x{\\rm d}\\tau\\\\\n&~~~\n+\\iint_{\\Omega\\times(T,\\infty)}|\\partial_t\\Gamma(\\tau,x,x_0)\n-\\partial_tG(\\tau,x,x_0)|{\\rm d} x{\\rm d}\\tau\\\\\n&\\leq\nCh\\iint_{\\max(|x-y|,\\tau^{1\/2})>\\frac{1}{2}C_*h}\\sup_{y\\in\\tau^l}\\big|\n\\nabla_y\\partial_tG(\\tau,x,y)\\big|{\\rm d} x{\\rm d}\\tau\n+C   \\\\\n&=Ch\\sum_{j}\\iint_{Q_j'(y)}\\sup_{y\\in\\tau^l}\\big|\n\\nabla_y\\partial_tG(\\tau,x,y)\\big|{\\rm d} x{\\rm d}\\tau\n+C\\\\\n&\\leq C\\sum_{j}\\frac{h}{d_j} +C \\\\\n&\\leq C .\n\\end{align*}\n\nMultiplying (\\ref{GMFdef}) by $\\partial_t\\Gamma$ and integrating the\nresult, we get\n\\begin{align*}\n&\\|\\partial_t\\Gamma(\\cdot,\\cdot,x_0)\\|_{L^2(Q_T)}\\leq\nC\\|\\widetilde \\delta_{x_0}\\|_{H^1(\\Omega)}\\leq Ch^{-d\/2-1} ,\n\\end{align*}\nwhich implies that\n\\begin{align*}\n&\\iint_{Q_*(x_0)}|\\partial_t\\Gamma(\\tau,x,x_0)|{\\rm d} x{\\rm d}\\tau\n\\leq\nd_{J_*}^{d\/2+1}\\|\\partial_t\\Gamma(\\cdot,\\cdot,x_0)\\|_{L^2(Q_T)}\\leq\nC .\n\\end{align*}\nEasy to check that\n\\begin{align}\n|\\partial_tG_{\\rm tr}^*(t,x,y)|\\leq Cd_{J_*}^{-d-2}\n\\quad\\mbox{for}~\\max(|x-y|,t^{1\/2}) \\leq d_{J_*}\n\\end{align}\nand  so\n\\begin{align}\n&\\iint_{Q_*(x_0)} |\\partial_tG_{\\rm tr}^*(t,x,x_0)|{\\rm d}\nx{\\rm d}\\tau\\leq Cd_{J_*}^{-d-2} d_{J_*}^{d+2} \\leq C .\n\\end{align}\nIt follows that\n\\begin{align*}\n&\\iint_{\\Omega_\\infty}|\\partial_t\\Gamma(\\tau,x,x_0)\n-\\partial_tG_{\\rm tr}(\\tau,x,x_0)|{\\rm d} x{\\rm d}\\tau\\\\\n&= \\iint_{\\Omega_\\infty\\backslash\nQ_*}|\\partial_t\\Gamma(\\tau,x,x_0)\n-\\partial_tG(\\tau,x,x_0)|{\\rm d} x{\\rm d}\\tau\n+\\iint_{Q_*}(|\\partial_t\\Gamma(\\tau,x,x_0)|+\n|\\partial_tG_{\\rm tr}(\\tau,x,x_0)|){\\rm d} x{\\rm d}\\tau\\\\\n&\\leq C .\n\\end{align*}\nFrom Lemma \\ref{GMhEst} and the last inequality, we see that\n\\begin{align*}\n&\\iint_{\\Omega_\\infty}|\\partial_t\\Gamma_{h}(\\tau,x,x_0)\n-\\partial_tG_{\\rm tr}(\\tau,x,x_0)|{\\rm d} x{\\rm d}\\tau\\\\\n&\\leq\\iint_{\\Omega_\\infty}|\\partial_t\\Gamma_{h}(\\tau,x,x_0)\n-\\partial_t\\Gamma(\\tau,x,x_0)|{\\rm d} x{\\rm d}\\tau\n+\\iint_{\\Omega_\\infty}|\\partial_t\\Gamma(\\tau,x,x_0)\n-\\partial_tG_{\\rm tr}(\\tau,x,x_0)|{\\rm d} x{\\rm d}\\tau\\\\\n&\\leq C.\n\\end{align*}\n\nSince both $\\Gamma_h(\\tau,x,y)$ and $G^*_{\\rm tr}(\\tau,x,y)$\nare symmetric with respect to $x$ and $y$, from the last\ninequality we see that the kernel\n$K(x,y)=\\int_0^\\infty|\\partial_t\\Gamma_h(\\tau,x,y)\n-\\partial_tG^*_{\\rm tr}(\\tau,x,y)|{\\rm d}\\tau$\nsatisfies\n\\begin{align*}\n&\\sup_{y\\in\\Omega}\\int_\\Omega K(x,y){\\rm d}\nx+\\sup_{x\\in\\Omega}\\int_\\Omega K(x,y){\\rm d} y\\leq C .\n\\end{align*}\nBy Schur's lemma \\cite{Kra}, the operator $M_K$ defined by\n$M_Ku_h(x)=\\int_\\Omega K(x,y)u_h(y){\\rm d} y$ is bounded on\n$L^q(\\Omega)$ for any $1\\leq q\\leq\\infty$, i.e.\n\\begin{align}\\label{Ms1}\n\\|M_Ku_h\\|_{L^q}\\leq C\\|u_h\\|_{L^q} ,\\quad 1\\leq q\\leq \\infty\n.\n\\end{align}\n\nLet\n$E^*_{\\rm tr}(t)u_h(x)=\\int_\\Omega G_{\\rm tr}^*(t,x,y)u_h(y){\\rm d} y$.\nWe have\n\\begin{align*}\n&\\sup_{t>0}|E_h(t)u_h(x)|\\\\\n&\n\\leq\\sup_{t>0}|(E_h(t)-E^*_{\\rm\ntr}(t))u_h(x)|+\\sup_{t>0}|E^*_{\\rm tr}(t)u_h(x)|\\\\\n&\n\\leq\n|(P_h\\delta_{x},u_h)|\n+\\sup_{t>0}\\biggl|\\int_0^t\n\\int_\\Omega(\n\\partial_t\\Gamma_h(\\tau,\\cdot,y)\n-\\partial_tG^*_{\\rm tr}(\\tau,x,y)u_h(y)){\\rm d}\ny{\\rm d}\\tau\\biggl|+\\sup_{t>0}|E^*_{\\rm tr}(t)u_h(x)|\\\\\n&\\leq\n|(P_h\\delta_{x},u_h)|\n+\\int_0^\\infty\n\\int_\\Omega\n|\\partial_t\\Gamma_h(\\tau,x,y)\n-\\partial_tG^*_{\\rm tr}(\\tau,x,y)|u_h(y){\\rm d} y{\\rm d}\n\\tau+\\sup_{t>0}E(t)|u_h|(x),\\\\\n&\n=:|u_h(x)|+M_Ku_h(x)+\\sup_{t>0}E(t)|u_h|(x)\n\\end{align*}\nwhere\n\\begin{align*}\n&\\|M_Ku_h\\|_{L^q}\\leq C\\|u_h\\|_{L^q} , \\qquad\\qquad\\forall~\n1\\leq q\\leq \\infty,~~\\mbox{by (\\ref{Ms1})},\\\\[5pt]\n&\\|\\sup_{t>0}E(t)|u_h|\\|_{L^q}\\leq C_q\\|u_h\\|_{L^q},\n\\quad\\forall~1<q<\\infty ,~~\\mbox{by \\cite{Gra}} ,\\\\\n&\\|\\sup_{t>0}E(t)|u_h|\\|_{L^\\infty}\\leq \\|u_h\\|_{L^\\infty},\n\\quad~ \\mbox{by the maximum principle} .\n\\end{align*}\nThis proves (\\ref{smgest}) for the case $h<h_0$.\n\nOn the other hand, when $h\\geq h_0$,  from (\\ref{STLEst}) we see that\n\\begin{align*}\n\\|\\sup_{t>0}|E_h(t)v_h|\\big\\|_{L^q}\n\\leq C\\sup_{t>0}\\|E_h(t)v_h\\|_{L^\\infty}\n\\leq C\\|v_h\\|_{L^\\infty}\\leq Ch_0^{-d\/q}\\|v_h\\|_{L^q} .\n\\end{align*}\n\nThe proof of (\\ref{smgest}) is completed.\n\n\n\\subsection{Proof of (\\ref{LpqSt1})-(\\ref{LpqSt3})}\n\n\nSince the operator $E_h(t)$ is symmetric, i.e.\n$(E_h(t)u_h,v_h)=(u_h,E_h(t)v_h)$ for any $u_h,v_h\\in S_h$,\nfrom (\\ref{STLEst}) we derive that, by a\nduality argument and by interpolation \\cite{BL},\n\\begin{align}\n&\\|E_h(t)v_h\\|_{L^q} + t\\|\\partial_tE_h(t)v_h\\|_{L^q} \\leq\nC\\|v_h\\|_{L^q}, \\quad\\mbox{for}~~ 1\\leq q\\leq\\infty , \\label{anf3}\n\\end{align}\nwhich means that $\\{E_h(t)\\}_{t>0}$ is an analytic semigroup on $L^q_h$.\n\nFirst, we prove (\\ref{LpqSt3}).\nFor the case $u^0_h\\equiv {\\bf g}\\equiv 0$,\nwe rewrite the equation (\\ref{FEEq0}) as\n\\begin{align}\\label{Ga5}\n&\\left\\{\n\\begin{array}{ll}\n\\partial_tu_h+A_hu_h=f_h  ,\n\\\\[5pt]\nu_h(0)=0 ,\n\\end{array}\n\\right.\n\\end{align}\nwhere $f_h=P_hf$.\nFrom \\cite{Weis1,Weis2}, we know that the maximal $L^p$\nregularity (\\ref{LpqSt3}) holds iff one of the following sets is\n$R$-bounded in ${\\cal L}(L^q_h,L^q_h)$ independent of $h$:\\\\\n(i)~ $\\{\\lambda(\\lambda+A_h)^{-1}:|{\\rm arg}(\\lambda)|<\n\\pi\/2+\\theta\\}$ for some\n$0<\\theta< \\pi\/2 \\,$,\\\\\n(ii)~ $\\{E_h(t),~tA_hE_h(t):t>0\\} \\,$, \\\\\n(iii)~ $\\{E_h(z):|{\\rm arg}(z)|<\\theta\\}$ for some $0<\\theta< \\pi\/2\n\\,$.\n\nMoreover, from Lemma 4.c in \\cite{Weis2} we know that the set in (iii) is $R$-bounded in\n${\\cal L}(L^q_h,L^q_h)$ for some $\\theta=\\theta_{\\kappa_q}>0$ if the\nanalytic semigroup $\\{E_h(z)\\}$ satisfies the maximal estimate:\n\\begin{align*}\n\\biggl\\|\\sup_{t>0}\\biggl|\\frac{1}{t} \\int_0^tE_h(s)u_h{\\rm d}\ns\\biggl|\\biggl\\|_{L^q}\\leq \\kappa_q\\|u_h\\|_{L^q},\\quad\\forall~u_h\\in\nL^q_h(\\Omega) .\n\\end{align*}\nSince the last inequality is a consequence of the maximal semigroup estimate\n(\\ref{smgest}), we thus proved the maximal $L^p$ regularity\n(\\ref{LpqSt3}).\n\nSecondly, we prove \\refe{LpqSt1} and \\refe{LpqSt2}. For the general case $u^0_h\\neq 0$\nor ${\\bf g}\\neq 0$, we let $u_h=\\widetilde u_h+v_h$, where\n$\\widetilde u_h$ and $v_h$ are the solutions of the equations\n\\begin{align}\\label{Ga51}\n&\\left\\{\n\\begin{array}{ll}\n\\partial_t\\widetilde u_h+A_h\\widetilde u_h = f_h-\\overline\\nabla_h\\cdot{\\bf g} ,\n\\\\[3pt]\n\\widetilde u_h(0)=P_hu^0 ,\n\\end{array}\n\\right.\n\\end{align}\nand\n\\begin{align}\n&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\left\\{\n\\begin{array}{ll}\n\\partial_tv_h+A_hv_h = 0 ,\n\\\\[3pt]\nv_h(0)=u^0_h-P_hu^0,\n\\end{array}\n\\right.\n\\end{align}\nrespectively.  Write the equation (\\ref{PDE0}) as\n\\begin{align}\\label{Ga90}\n&\\left\\{\n\\begin{array}{ll}\n\\partial_tu+Au\n=f -\\overline\\nabla\\cdot {\\bf g} &\\mbox{in}~\\Omega,\n\\\\[3pt]\nu(0)=u^0 &\\mbox{in}~\\Omega ,\n\\end{array}\n\\right.\n\\end{align}\nand let $w_h=\\widetilde u_h-P_hu$. The difference of (\\ref{Ga51}) and\n(\\ref{Ga90}) gives\n\\begin{align*}\n&\\left\\{\n\\begin{array}{ll}\n\\partial_t w_h+A_h w_h =   A_h(R_hu-P_hu) ,\n\\\\[3pt]\nw_h(0)=0 .\n\\end{array}\n\\right.\n\\end{align*}\nMultiplying the above equation by $A_h^{-1}$, we get\n\\begin{align*}\n&\\left\\{\n\\begin{array}{ll}\n\\partial_t A_h^{-1}w_h+A_h A_h^{-1}w_h = R_hu-P_hu ,\n\\\\[3pt]\nA_h^{-1}w_h(0)=0 ,\n\\end{array}\n\\right.\n\\end{align*}\nand using (\\ref{LpqSt3}) we derive that\n\\begin{align*}\n\\|w_h\\|_{L^p((0,T);L^q)}\\leq C_{p,q}\\|R_hu-P_hu \\|_{L^p((0,T);L^q)} .\n\\end{align*}\nOn the other hand, it is easy to derive that $\\|v_h(t)\\|_{L^2}\\leq\nCe^{-t\/C}\\|u^0_h-P_hu^0\\|_{L^2}$, which with (\\ref{anf3})\ngives (via interpolation)\n\\begin{align*} \\|v_h(t)\\|_{L^q}\\leq\nCe^{-t\/C_q}\\|u^0_h-P_hu^0\\|_{L^q} ~~~\\mbox{for}~~1< q<\\infty .\n\\end{align*}\nThe last two inequalities imply (\\ref{LpqSt1}).\n\nIf $u^0_h\\equiv u^0\\equiv f\\equiv 0$, then $v_h=0$\nand by using Lemma \\ref{s00} we derive that\n\\begin{align*}\n\\|u_h\\|_{L^p((0,T);W^{1,q})}\n&=\\|\\widetilde u_h\\|_{L^p((0,T);W^{1,q})} \\\\\n&\\leq \\|w_h\\|_{L^p((0,T);W^{1,q})}\n+\\|P_hu\\|_{L^p((0,T);W^{1,q})}\\\\\n&\\leq Ch^{-1}\\|w_h\\|_{L^p((0,T);L^q)}\n+C\\|u\\|_{L^p((0,T);W^{1,q})}\\\\\n&\\leq C_{p,q}h^{-1}\\|R_hu-P_hu \\|_{L^p((0,T);L^q)}\n+C\\|u\\|_{L^p((0,T);W^{1,q})} \\\\\n&\\leq C_{p,q}\\|u\\|_{L^p((0,T);W^{1,q})} \\\\\n&\\leq C_{p,q}\\|{\\bf g}\\|_{L^p((0,T);L^q)} .\n\\end{align*}\nThis proves the inequality (\\ref{LpqSt2}).\n\nThe proof of Theorem \\ref{MainTHM1} is completed. ~\\vrule height8pt width 5pt depth 0pt\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Appendix --- \nInterior estimates of parabolic equations on $\\bf\\widetilde Q_j'$}\n\\renewcommand{\\thelemma}{A.\\arabic{lemma}}\n\\renewcommand{\\theproposition}{A.\\arabic{lemma}}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\\setcounter{lemma}{0} \\setcounter{equation}{0}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzmtnu b/data_all_eng_slimpj/shuffled/split2/finalzzmtnu
new file mode 100644
index 0000000000000000000000000000000000000000..cc0e43093246c10b6b2fb22848fed726810fdf92
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzmtnu
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\\label{s1}\nThe quiet Sun covers most of the solar surface, in particular at activity minimum, but \nalso plays an important role even during the active phase of the solar cycle. The magnetic \nfield in the quiet Sun is composed of the network \\citep{1967SoPh....1..171S}, \ninternetwork \\citep[IN,][]{1971IAUS...43...51L, 1975BAAS....7..346L}, and the ephemeral \nregions \\citep{1973SoPh...32..389H}. For an overview of the small-scale magnetic \nfeatures, see \\citet{1993SSRv...63....1S, 2009SSRv..144..275D, 2014masu.book.....P, \n2014A&ARv..22...78W}.\n\nThe IN features are observed within the supergranular cells and carry hecto-Gauss fields  \n\\citep[][]{1996A&A...310L..33S, 2003A&A...408.1115K, 2008A&A...477..953M} although \nkilo-Gauss fields have also been observed in the IN  \\citep{2010ApJ...723L.164L, \nLagg16}. They evolve as unipolar and bipolar features with typical lifetimes of less than \n10 minutes \\citep{2010SoPh..267...63Z, 2013apj...774..127l,lsa}, i.e., they continuously \nbring new flux to the solar surface, either flux that has been either freshly generated, \nor recycled. They carry fluxes $\\le 10^{18}$ Mx, with the lower limit on the smallest \nflux decreasing with the increasing spatial resolution and polarimetric sensitivity of \n the observing instruments, although the identification technique also plays an important \nrole. \n\nEphemeral regions are bipolar magnetic features appearing within the supergranular cells \ncarrying fluxes $\\approx 10^{19}$ Mx \\citep{2001ApJ...548..497C, 2003ApJ...584.1107H} and \nare much longer-lived compared to the IN features, with lifetimes of 3 -- 4.4 hours \n\\citep{2000RSPTA.358..657T, 2001ApJ...555..448H}. The ephemeral regions also bring new \nmagnetic flux to the solar surface. \n\nThe network is more stable, with typical lifetimes of its structure of a few \nhours \nto a day, although the individual kG magnetic elements within the network live for a much \nshorter time, as the entire flux within the network is exchanged within a period of 8--24 \nhr \\citep{2003ApJ...584.1107H, 2014apj...797...49g}. The flux in the network is fed by \nephemeral regions \\citep{0004-637x-487-1-424, 2001ApJ...555..448H} and IN features \n\\citep[][]{2014apj...797...49g}. The network features are found along the supergranular \nboundaries and carry fields of kG strength with a typical flux of $10^{18}$ Mx \n\\citep{1995soph..160..277w}. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{f1}\n\\caption{Two sample magnetograms recorded at $t$ = 00:47 UT and 00:58 UT on 2009 June 9 \nwith the \\textsc{Sunrise}\/IMaX instrument during its first science flight.}\n\\label{magnetogram}\n\\end{figure*}\n\nThe magnetic flux is produced by a dynamo, the location of which is currently the subject \nof debate, as is whether there is only a single dynamo acting in the Sun \n\\citep[e.g.,][]{2012A&A...547A..93S} or whether there is a small-scale dynamo acting in \naddition to a global dynamo \\citep{1993A&A...274..543P, 1999ApJ...515L..39C, \n2001A&G....42c..18C, 2007A&A...465L..43V, 2008A&A...481L...5S, 2010A&A...513A...1D, \n2013A&A...555A..33B, 2015ApJ...803...42H, 2016ApJ...816...28K}. In addition, it is unclear \nif all the magnetic flux appearing on the Sun is actually new flux produced by a dynamo, \nor possibly recycled flux transported under the surface to a new location, where it \nappears again \\citep[e.g.,][]{2001ASPC..236..363P}. This may be particularly important at \nthe smallest scales. \n\nAn important parameter constraining the production of magnetic flux is the amount of \nmagnetic flux appearing at the solar surface. In particular, the emergence of magnetic \nflux at very small scales in the quiet Sun provides a probe for a possible small-scale \ndynamo acting at or not very far below the solar surface. The deep minimum between solar \ncycles 23 and 24 offered a particularly good chance to study such flux emergence, as the \nlong absence of almost any activity would suggest that most of the emerging flux is newly \nproduced one and is not flux transported from decaying active regions to the quiet Sun \n(although the recycling of some flux from ephemeral regions cannot be ruled out).\n\nThe IN quiet Sun displays by far the largest magnetic flux emergence rate (FER). Already, \n\\citet{1987SoPh..110..101Z} pointed out that two orders of magnitude more flux appears in \nephemeral regions than in active regions, while the FER in the IN is another two orders \nof \nmagnitude larger. This result is supported by more recent studies \n\\citep[e.g.,][]{2002ApJ...565.1323S, 2009SSRv..144..275D, 2009ApJ...698...75P, \n2011SoPh..269...13T}.  Given the huge emergence rate of the magnetic flux in the IN, it \nis of prime importance to measure the amount of flux that is brought to the surface \nby these features.\n \nThe current estimates of the FER in the IN vary over a wide range, which include: \n$10^{24}\\rm{\\,Mx\\, day^{-1}}$ \\citep[][]{1987SoPh..110..101Z}, $3.7\\times \n10^{24}\\rm{\\,Mx\\,day^{-1}}$ \n\\citep[$120\\rm{\\,Mx\\,cm^{-2}\\,day^{-1}}$,][]{2016ApJ...820...35G} and \n$3.8\\times10^{26}\\rm{\\,Mx\\,day^{-1}}$ \\citep[][]{2013SoPh..283..273Z}.\nBy considering all the magnetic features (small-scale features and active regions), \n\\citet{2011SoPh..269...13T} measure a global FER of $3\\times 10^{25}\\rm{\\,Mx\\,day^{-1}}$ \n($450 \\rm{\\,Mx \\,cm^{-2} \\,day^{-1}}$), while \\citet{th-thesis} measures $3.9 \\times \n10^{24} \\rm{\\,Mx \\,day^{-1}}$ ($64\\rm{\\,\\,Mx\\,cm^{-2}\\,day^{-1}}$), whereby almost all of \nthis flux emerged in the form of small IN magnetic features. The FER depends on the \nobservations and the method used to measure it. A detailed comparison of the FERs from \ndifferent works is presented Section~\\ref{s6}.\n\nTo estimate the FER, \\citet{1987SoPh..110..101Z} and \\citet{2011SoPh..269...13T}  \nconsidered features with fluxes $\\ge 10^{16}$ Mx, while \\citet{2013SoPh..283..273Z} and \n\\citet{2016ApJ...820...35G} included features with fluxes as low as $6\\times 10^{15}$ \nMx and $6.5\\times10^{15}$ Mx (M. Go\\v{s}i\\'{c}, priv. comm.), respectively. However, with \nthe launch of the balloon-borne \\textsc{Sunrise}{} observatory in 2009 \\citep{2010ApJ...723L.127S, \n2011SoPh..268....1B, 2011SoPh..268..103B, 2011SoPh..268...35G} carrying the Imaging \nMagnetograph eXperiment \\citep[IMaX,][]{2011SoPh..268...57M}, it has now become possible \nto estimate the FER including the contribution of IN features with fluxes as low as \n$9\\times10^{14}$ Mx \\citep[][hereafter referred to as LSA17]{lsa}. The IMaX instrument \nhas \nprovided unprecedented high-resolution magnetograms of the quiet Sun observed at 5250\\, \n\\AA. The high resolution is the main reason for the lower limiting flux. A detailed \nstatistical analysis of the IN features observed in Stokes $V$ recorded by \\textsc{Sunrise}\/IMaX \nis carried out in LSA17. In the present paper we estimate the FER in the IN region using \nthe same data.\n\nIn Section~\\ref{s2} we briefly describe the employed \\textsc{Sunrise}{} data. The IN features \nbringing flux to the solar surface that are considered in the estimation of FER are \noutlined in Section~\\ref{s3}. The FER from \\textsc{Sunrise}{} are presented, discussed and \ncompared with previously obtained results in Section~\\ref{s4} while our conclusions \nare presented in Section~\\ref{s8}.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{f2}\n\\caption{Panels a--d: Schematic representation of the different paths by \nwhich magnetic flux is brought to the solar surface and of its subsequent evolution.  The \nquantities $f_i$ and $f_m$ are the instantaneous and maximum fluxes of a feature, \nrespectively, where instantaneous means just after it appears. {Panel e}: Typical \nvariation in the flux of a feature, born by  {unipolar and bipolar} appearances \n(top) and born by splitting\/merging (bottom), over time. The flux gained by them after \nbirth is $\\Delta f=f_m - f_i$. The features born by splitting carry fluxes $f_{i1,2}$ at \nbirth and reach $f_{m1,2}$ in the course of their lifetime, gaining flux $\\Delta f_{1,2}$ \nafter birth. $f_{i1}+f_{i2}$ is equal to the flux of the parent feature at the time of its \nsplitting. The feature born by merging carries flux equal to the sum of the fluxes of the \nparent features $f_{i1}+f_{i2}$. The blue line indicates the merging of the two features.}\n\\label{flux}\n\\end{figure*}\n\n\\section{Data}\n\\label{s2}\nThe data used here were obtained during the first science flight of \\textsc{Sunrise}{} described \nby \\citet{2010ApJ...723L.127S}. We consider 42 maps of the line-of-sight (LOS) magnetic \nfield, $B_{\\rm LOS}$, obtained from sets of images in the four Stokes parameters  \nrecorded with the IMaX instrument between 00:36 and 00:59 UT on 2009 June 9 at the solar \ndisk center, with a cadence of 33\\,s,  {a spatial resolution of \n$0.^{\\prime\\prime}15-0.^{\\prime\\prime}18$ (plate scale is $0.^{\\prime\\prime}054$ per \npixel), and an effective field of view (FOV) of $43^{\\prime\\prime} \\times \n43^{\\prime\\prime}$ after phase diversity reconstruction}.  {The data were \nreconstructed with a point spread function determined by in-flight phase diversity \nmeasurements to correct for the low-order aberrations of the telescope \\citep[defocus, \ncoma, astigmatism, etc., see][]{2011SoPh..268...57M}. The instrumental noise of the \nreconstructed data was $3\\times10^{-3}$ in units of continuum intensity. For \nidentification of the features, spectral averaging was done which further reduced the \nnoise to $\\sigma=1.5\\times10^{-3}$. All features with signals above a $2\\sigma$ \nthreshold, which corresponds to 12\\,G,  were used \\citep[][]{2011SoPh..268...57M}.}\n\nTo measure the flux, we use $B_{\\rm LOS}$ determined with the center-of-gravity (COG) \nmethod \n\\citep[][LSA17]{1979A&A....74....1R, 2010A&A...518A...2O}.  {The inclination of the \nIN fields has been under debate, with several studies dedicated to measuring their \nangular distribution. By analysing the data from \\textit{Hinode}, \n\\citet{2007ApJ...670L..61O, 2008ApJ...672.1237L} concluded the IN fields to be \npredominantly horizontal. However, using the same dataset, \\citet{2011ApJ...735...74I} \nfound some of the IN fields to be vertical. \\citet{2014A&A...569A.105J} arrived at similar \nconclusions (vertical inclination) by analysing the magnetic bright points observed from \nthe first flight of \\textsc{Sunrise}{}. Variations in the inclination of the IN fields with \nheliocentric angle ($\\mu$) have been reported by \\citet{2012ApJ...751....2O, \n2013A&A...550A..98B, 2013A&A...555A.132S}. Isotropic and quasi-isotropic distribution of \nthe IN field inclinations is favoured by \\citet{2008A&A...479..229M}, using the Fe\\,{\\sc \ni} $1.56\\,\\mu$m infrared lines, and by \\citet{2009ApJ...701.1032A,2014A&A...572A..98A} \nusing \\textit{Hinode} data. More recently, \\citet{2016A&A...593A..93D} found the \ndistribution of IN field inclination to be quasi-isotropic by applying 2D inversions on \n\\textit{Hinode} data and comparing them with 3D magnetohydrodynamic simulations. For a \ndetailed review on this, see \\citet{2015SSRv..tmp..113B}.}\n\n {In the determination of the FER, we use $B_{\\rm LOS}$ for consistency and for \neasier comparison with earlier studies on the FER. Also, the determination of the exact \namount of flux carried in horizontal field features is non-trivial and requires estimates \nof the vertical thickness of these features and the variation of their field strength with \nheight. In addition, if they are loop-like structures \\citep[as is suggested by \nlocal-dynamo simulations, e.g.,][]{2007A&A...465L..43V}, then there is the danger of \ncounting the flux multiple times if one or both of their footpoints happen to be resolved \nby the \\textsc{Sunrise} data. We avoid this by considering only the vertical component of the \nmagnetic field. It is likely that we miss the flux carried by unresolved magnetic loops by \nconcentrating on Stokes $V$, but this problem is suffered by all previous studies of FER \nand should decrease as the spatial resolution of the observations is increased. With the \n\\textsc{Sunrise}{} I data analyzed here having the highest resolution, we expect them to see more \nof the flux in the footpoints of the very small-scale loops that appear as horizontal \nfields in \\textit{Hinode} and \\textsc{Sunrise}{} data \\citep{2010A&A...513A...1D}.}\n\nThe small-scale features were identified and tracked using the feature tracking code \ndeveloped in LSA17.  {For the sake of completeness we summarize the most \nrelevant results from LSA17 as follows. All the features covering at least 5 pixels \nwere considered with Stokes $V$ being larger than $2\\sigma$ in each pixel. To determine \nthe flux per feature,  the $B_{\\rm LOS}$ averaged over the feature, denoted as \n\\textlangle$B_{\\rm LOS}$\\textrangle{} was used.} \\textlangle$B_{\\rm LOS}$\\textrangle{} \nhad values up to 200\\,G, even when the maximum field strength in the core of the feature \nreached kG values.  {A total of 50,255 features of both polarities were identified. \nThe sizes of the features varied from 5-1,585 pixels, corresponding to an area range of \n$\\approx 8\\times 10^{-3}-2.5$\\,Mm$^2$. The tracked features had lifetimes ranging from \n0.55 to 13.2 minutes.} The smallest detected flux of a feature was $9\\times10^{14}$ Mx \nand \nthe largest $2.5\\times10^{18}$ Mx.\n\n \nAt the time of the flight of \\textsc{Sunrise}{} in 2009 the Sun was extremely quiet, with no signs \nof activity on the solar disk. Two sample magnetograms at  00:47 UT and 00:58 UT are shown \nin Figure~\\ref{magnetogram}. Most of the features in these maps are part of the IN and in \nthis paper, we determine the rate at which they bring flux to the solar surface. \n\n\\section{Processes increasing magnetic flux at the solar surface}\n\\label{s3}\nThe different processes increasing the magnetic flux at the solar surface are \nschematically represented in Figure~\\ref{flux}a -- \\ref{flux}d. In the figure, $f_i$ \nrefers to the flux of the feature at its birth and $f_m$ is the maximum flux that a \nfeature attains over its lifetime. A typical evolution of the flux of a feature born by\n {unipolar or bipolar} appearances, and by splitting\/merging is shown in \nFigure~\\ref{flux}e (top and bottom, respectively). The gain in the flux of a feature \nafter its birth is $f_m-f_i$. Magnetic flux at the surface increases through the \nfollowing processes:\n\n    \\begin{table*}\n          \\centering\n          \\caption{The instantaneous and maximum fluxes of the features, \n  {integrated over all features and time frames}, in different \n processes measured in LSA17}\n            \\begin{tabular}{|c|c|c|c|c|}\n            \\hline\n Process & Instantaneous flux & Maximum flux & Flux gain & Factor of increase\\\\\n & ($f_i$ in Mx) & ($f_m$ in Mx) & ($\\Delta f= f_m-f_i$ in Mx) & ($f_m\/f_i$)\\\\\n \\hline\n   {Unipolar appearance} & $4.69\\times10^{19}$ & $9.69\\times10^{19}$ &  \n$4.99\\times10^{19}$ &2.06\\\\\n  Splitting & $1.76\\times10^{20}$& $2.12\\times10^{20}$&$3.60\\times10^{19}$&1.20\\\\\n  Merging & $1.64\\times 10^{20}$& $1.85\\times 10^{20}$&$2.20\\times10^{19}$&1.31\\\\\n   {Bipolar appearance} & $3.85\\times10^{18}$ & $9.53\\times10^{18}$& \n$5.67\\times10^{18}$&2.47\\\\\n  \\hline\n            \\end{tabular}\n            \\label{table1}\n        \\end{table*}\n\n\n\\begin{enumerate}\n \\item{ {Unipolar appearance}}: The birth of an isolated feature with no spatial \noverlap with  any of the existing features in the current and\/or previous time frame \n(Figure~\\ref{flux}a). \n \\item { {Bipolar appearance}}: Birth of bipolar features, with the two polarities \nclosely spaced, and either appearing simultaneously or separated by a couple of time \nframes ( {referred to as} time symmetric and asymmetric emergence in LSA17; \nsee also Figure~\\ref{flux}b). \n\\item { {Flux gained by features in the course of their \nlifetime}}: The gain in the \nflux of a feature in the course of its lifetime, i.e. the increase in flux between its \nbirth and the time it reaches its maximum flux, before dying in one way or another, either \nby interacting with another feature, or by disappearing.\n\n This gain can take place in features born in different ways, be it by growth, or \nthrough the merging or splitting of pre-existing features (Figures~\\ref{flux}c, \n\\ref{flux}d and \\ref{flux}e).\n\\end{enumerate}\n\n {Note that the bipolar appearance of magnetic flux is often referred to as \n`emergence' in earlier papers including LSA17. However, the term `emergence' in FER \ndescribes the appearance of new flux at the solar surface from all the three processes \ndescribed above. To avoid confusion, we refer to the emergence of  bipolar features as \nbipolar appearance.}  {Of all the newly born features over the entire time series, \n19,056 features were unique (for area ratio 10:1, see Section~\\ref{s4}). Among them 48\\% \n(8728 features) were unipolar and 2\\% (365 features) were part of bipolar appearances. \nFeatures born by splitting constituted 38\\% (6718 features), and 12\\% (2226 features) \nwere born by merging. The remaining 1019 features correspond to those alive in the \nfirst frame. A comparison of the rates of birth and death of the features by various \nprocesses for different area ratio criteria is given in Table~2 of LSA17.}\n\nIn the FER estimations, the flux brought by the features born by \n {unipolar and bipolar} appearances is the maximum flux that they attain \n($f_m$) over their lifetime. In the case of features born by splitting or merging, the \nflux gained after birth is taken as the flux brought by them to the surface. This gain is \nthe difference between their flux at birth $f_i$ and the maximum flux they attain $f_m$, \ni.e. $f_m - f_i$. \n\n\\begin{figure}[h!]\n\\centering\n \\includegraphics[width=0.3\\textwidth]{f3}\n \\caption{Schematic representation of multiple peaks in the flux of a feature occurring \nin the course of its lifetime. In the FER estimations, we consider only the largest gain, \ni.e., flux increase during first peak in this example.}\n \\label{fg}\n\\end{figure}\n\nOur approach is conservative in the sense that if a feature reaches multiple peaks of flux \nin the course of its lifetime, as in the example shown in Figure~\\ref{fg}, then we \nconsider only \nthe largest one (the flux gained during the first peak in Figure~\\ref{fg}), and neglect \nincreases in flux contributing to smaller peaks such as the second and third peaks in \nFigure~\\ref{fg}.  {Multiple peaks in the flux of a feature (shown in \nFigure~\\ref{fg}) are rarely seen, as most features do not live long enough to display \nthem (see LSA17). }\n\n {Changes in the flux of a feature in the course of its lifetime can cause \nit to seemingly appear and disappear with time if its total flux is close to the \nthreshold set in the study (given by a signal level twice the noise in at least five\ncontiguous pixels). If it disappears and reappears again, then it will be counted twice. \nThis introduces uncertainties in the measurement of FER. Uncertainties are discussed in \nSection~\\ref{s7}.}\n\n\\section{Results and Discussions}\n\\label{s4}\n\n\\subsection{Flux emergence rate (FER)}\n\\label{s5}\nIn this work, we consider the results from the area ratio criterion 10:1 of LSA17. In \nthat paper, the authors devise area ratio criteria (10:1, 5:1, 3:1 and 2:1) to avoid that \na feature dies each and every time that a tiny feature breaks off, or merges with it. \nFor example in a splitting event, the largest of the features formed by splitting \nmust have an area less than $n$ times the area of the second largest, under the $n:1$ area \nratio criterion.  {We have verified that the choice of the area ratio criterion does \nnot drastically alter the estimated FER, with variations being less than 10\\% for area \nratios varying between 10:1 and 2:1.}\n\nThe  {instantaneous} and maximum fluxes of the features in different processes are \ngiven in Tables~$1-5$ of LSA17. A summary is repeated in Table~\\ref{table1} for \nconvenience {, where fluxes are given for features born by the four processes listed \nin the first column. The instantaneous flux, in the second column, refers to the flux of \na feature at its birth ($f_i$ in Figure~\\ref{flux}). In the third column is the maximum \nflux of a feature during its lifetime ($f_m$ in Figure~\\ref{flux}). The flux gain in the \nfourth column is the difference of the second and third columns ($\\Delta f=f_m-f_i$). \nIn the fifth column is the factor by which the flux increases from its birth to its peak \n($f_m\/f_i$). The fluxes given in this table are the sum total over all features in the \nentire time series, for each process.} \n\nTo compute the FER, we add the fluxes from the \nvarious processes described in the previous section. For features born by appearance \n {(unipolar and bipolar), we take their maximum flux of the features ($f_m$)} to be \nthe fresh flux emerging at the surface. For the features formed by merging or splitting \nonly the flux increase after the birth  {($\\Delta f$)} is considered. From the  \nfirst two processes alone, the total flux brought to the surface is $1.1\\times10^{20}$ Mx \nover an FOV of $43^{\\prime\\prime} \\times 43^{\\prime\\prime}$ in 22.5 minutes. This gives \nan \nFER of $700 \\rm{\\, \\, Mx\\, cm^{-2}\\,day^{-1}}$. Including the flux gained by split\/merged \nfeatures increases the FER to $1100 \\rm{\\,\\,Mx\\,cm^{-2}\\,day^{-1}}$. Figure~\\ref{flux1} \nshows the contribution from each process to the total FER. The isolated features appearing \non the solar surface contribute the largest, nearly 60\\%. Given that the emerging bipoles \ncontain only 2\\% of the total observed flux (Table~5 in LSA17), they contribute only \nabout \n5.7\\% to the FER. \n\nHowever, the flux brought to the solar surface by features born by splitting or merging, \nafter their birth is $5\\times10^{19}$ Mx, which is quite significant and contributes \n$\\approx35\\%$ to the FER. The contribution to solar surface flux by this process is \ncomparable to the flux brought to the surface by features born by  {unipolar} \nappearance ($9.7\\times10^{19}$ Mx) and nearly an order of magnitude higher than that flux \nfrom features born by  {bipolar appearance} ($9.5\\times10^{18}$ Mx). \n\nOver their lifetimes, the features born by splitting and by merging gain 1.2 times their \ninitial flux (i.e., $f_m = 1.2 \\times f_i$). The fluxes gained by features born by \nappearance relative to their flux at birth is slightly higher ($\\approx2$ times, i.e., \n$f_m = 2\\times f_i$). This is because the initial magnetic flux of the features born by \nappearance is quite low.  The flux at birth of split or merged features is already quite \nhigh because the parent features which undergo splitting or merging are at later stages in \ntheir lives (see Figure~\\ref{flux}e). This is also evident from the fact that the average \ninitial flux per feature of the features born by splitting or merging ($2.9\\times10^{16}$ \nMx and $7.4\\times10^{16}$ Mx, respectively) is an order of magnitude higher than the \naverage initial flux per feature of the appeared  {unipolar or bipolar} features \n($5.4\\times10^{15}$ Mx, see Table~2 of LSA17).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.3]{f4}\n\n \\caption{The percentage of contribution to the flux emergence rate (FER) from different \nprocesses bringing flux to the solar surface. In the case of  {unipolar and bipolar} \nappearances, the maximum flux of the feature is used to determine the FER. For the \nfeatures born by splitting\/merging, the flux gained by them after birth is considered. \nThis gain is the difference $f_m - f_i$ in Figures~\\ref{flux}c, \\ref{flux}d and \n\\ref{flux}e.}\n \\label{flux1}\n\\end{figure}\n\nThe fact that the small-scale magnetic features are the dominant source of fresh flux in \nthe quiet photosphere is discussed in several publications \\citep{2002ApJ...565.1323S, \n2009SSRv..144..275D, 2009ApJ...698...75P, 2011SoPh..269...13T}. Our results extend these \nearlier findings to lower flux per feature values. As shown in Figure~\\ref{hist}, over the \nrange $10^{15}-10^{18}$ Mx, nearly $65\\%$ of the detected features carry a flux $ \\le \n10^{16}$ Mx (left panel). They are also the dominant contributors to the FER (right \npanel). In this figure, only the features that are born by  {unipolar and bipolar} \nappearances are considered. Below $2\\times10^{15}$ Mx, we see a drop as we approach the \nsensitivity limit of the instrument. \n\n\\subsubsection{Flux loss rate}\nFlux is lost from the solar surface by disappearance, cancellation of opposite \npolarity features, and decrease in the flux of the features in the course of their \nevolution (i.e. the opposite process to the ``flux gain'' described earlier in \nSection~\\ref{s3}). As seen from Tables~3 and 4 of LSA17, the increase in flux at the \nsolar surface balances the loss of flux, as it obviously must if the total amount of flux \nis to remain unchanged. To compute the flux loss rate, we take the maximum flux of the \nfeatures that die by cancellation and by disappearance to be the flux lost by them in the \ncourse of their lifetime and during disappearance or cancellation. For the \nfeatures that die by splitting\/merging we take the difference between the maximum flux of \nthe features and the flux at their death as a measure of the flux lost during their \nlifetimes. By repeating the analyses for the 10:1 area ratio criterion, we find that the \nflux is lost from the solar surface at a rate of 1150$\\rm{\\, \\, Mx\\, cm^{-2}\\,day^{-1}}$ \nwhich corresponds within 4.5\\% to the obtained FER. This agreement serves as a \nconsistency check of the FER value that we find.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{f5}\n \\caption{{Left panel:} histogram of the number of features born by \n {unipolar and bipolar} appearances, carrying fluxes in the range $10^{15}-10^{18}$ \nMx. {Right panel:} the flux emergence rate from the features born by \n {unipolar and bipolar} appearances as a function of their flux.}\n \\label{hist}\n\\end{figure*}\n\n\n\\subsection{Uncertainties}\n\\label{s7}\n {Although most of the uncertainties and ambiguities that arise during feature \ntracking have been carefully taken care of, as discussed in LSA17, some additional \nones which can affect the estimated FER are addressed below.}\n\n {In our computation of the FER, the features born before the time series began and \nthe features still alive at the end are not considered. According to LSA17, the first and \nthe last frames of the time series had 1019 and 1277 features, respectively.  To estimate \ntheir contribution, we assume that the features still living at the end have a similar \nlifetime, size, flux distribution and formation mechanism as the total number of features \nstudied. We attribute the appropriate  average flux at birth and the average flux gain \nfor \nfeatures born by splitting, merging, unipolar and bipolar appearance. After including \nthese additional fluxes, we get an FER of $\\approx 1150 \\rm{\\, \\, \nMx\\,cm^{-2}\\,day^{-1}}$, \ncorresponding to a $4-5\\%$ increase. With this method, we are associating the features \nwith flux gain than they might actually contribute (as many of them are likely to reach \ntheir maximum flux only after the end of the time series). This will be balanced out by \nnot considering the features that are already alive at the beginning (also, it is \nimpossible to determine the birth mechanism of these features).}\n\n {Furthermore, in the analysis of LSA17, the features touching the spatial \nboundaries were not counted. An estimate of their contribution, in ways similar to the \nabove, leads to a further increase of the FER by $5-6\\%$. Thus combining the features in \nthe first and last frames and the features touching spatial boundaries together increase \nthe FER by $\\approx 10\\%$. } \n\nMeanwhile, as discussed in Section~\\ref{s3}, in the case of flux gained after birth by \nfeatures born from splitting or merging, we consider only the gain to reach the maximum \nflux in the feature and not the smaller gains required to reach secondary maximum of flux \nin the feature, if any (see Figure~\\ref{fg}).  {These instances are quite rare. \nTo estimate their contribution, we consider all features living for at least four minutes \n(eight time steps) so as to distinguish changes in flux from noise fluctuations. They \nconstitute a small fraction of $\\approx 4\\%$. If all these features are assumed to show \ntwo maxima of equal strength, then they increase the FER by $\\approx \n1.5\\%$. This is a generous estimate and both these assumptions are unlikely to be met. \nHowever this is balanced out by not considering the features that have more than two \nmaxima. Thus the increase in FER is quite minor.} \n\nAdditionally, some of the features identified in a given time frame could disappear, i.e. \ndrop below the noise level, for the next couple of frames, only to reappear after that. \n {This is unlikely to happen due to the thermal or mechanical changes for the  \n\\textsc{Sunrise}{} observatory, flying in a highly stable environment at float altitude, and with \nactive thermal control of critical elements in the IMaX instrument. As mentioned in \nSection~\\ref{s3}, the appearance and disappearance of features could also occur due to \nthe \napplied threshold on the signal levels. In our analyses, the reappeared features are \ntreated as newly appeared. This leads to a higher estimation of the FER. \n\\citet{2016ApJ...820...35G} have estimated that accounting for reappeared features \ndecreases the FER by nearly $10\\%$. If we assume the same amount of decrease in the FER \nfrom the reappeared features in our dataset, then we finally obtain an FER of 1100$ \n\\rm{\\, \n\\, Mx\\, cm^{-2}\\,day^{-1}}$.}\n\n\n\\subsection{Comparison with previous studies}\n\\label{s6}\nBelow, we compare our results from \\textsc{Sunrise}{} data with those from the \\textit{Hinode} \nobservations analysed in three recent publications. Although all these papers use \nobservations from the same instrument, they reach very different estimates of FERs. The \nimportant distinguishing factor between them is the method that is used to identify the \nmagnetic features and to calculate the FER. For comparison we summarize the main \nresult that we have obtained here. We find that in the quiet Sun (composed dominantly of \nthe IN) the FER is $1100 \\rm{\\,Mx\\,cm^{-2}\\,day^{-1}}$. This corresponds \nto $6.6 \\times 10^{25} \\rm{\\,Mx\\,day^{-1}}$ under the assumption that the whole Sun is as \nquiet as the  {very tranquil} \\textsc{Sunrise}\/IMaX FOV. \n\nAccording to \\cite{2016ApJ...820...35G}, the flux appearance or emergence rate in \nthe IN region is $120 \\rm{\\,\\,Mx\\,cm^{-2}\\,day^{-1}}$, which corresponds to $3.7 \n\\times 10^{24} \\rm{\\,\\,Mx\\,day^{-1}}$ over the whole surface and the contribution from \nthe \nIN is assumed to be $\\approx50\\%$. The authors track individual features and measure \ntheir \nfluxes, which is similar to the method used in LSA17. Their estimate is an order of \nmagnitude lower than the FER obtained in the present paper. This difference can be \nexplained by the higher spatial resolution of \\textsc{Sunrise}{} compared to \\textit{Hinode}. The \nisolated magnetic feature with the smallest flux detected in \\textsc{Sunrise}\/IMaX data is \n$9\\times10^{14}$ Mx  {(see LSA17)}, which is nearly an order of magnitude smaller \nthan the limit of $6.5\\times10^{15}$ Mx (M. Go\\v{s}i\\'{c}, priv. comm.), underlying the \nanalysis of \\citet{2016ApJ...820...35G}. Additionally, the IMaX data are recorded with \n33\\,s cadence, while the two data sets analysed by the above authors have cadences of 60 \nand 90\\,s each. A higher cadence helps in better tracking of the evolution of features \nand \ntheir fluxes.  {Also, a significant number of the very short-lived features that we \nfind may have been missed by \\citet{2016ApJ...820...35G}.}\n\n\\citet{2011SoPh..269...13T}, also using \\textit{Hinode} observations, estimate the FER \nby fitting a power law to the distribution of frequency of emergence ($\\rm{Mx^{-1} \n\\,cm^{-2} \\,day^{-1}}$) over a wide range of fluxes ($10^{16}-10^{22}$ Mx, which covers \nboth, small-scale features as well as active regions). It is shown that a single power \nlaw index of -2.7 can fit the entire range. Depending on the different emergence \ndetection methods used and described by these authors, such as Bipole Comparison (BC), \nTracked Bipolar (TB) and Tracked Cluster (TC), the authors find a wide range of FERs from \n32 to $470 \\rm{\\,Mx \\,cm^{-2}\\,day^{-1}}$ which correspond to 2.0 to $28.7 \\times 10^{24} \n\\rm{\\,Mx \\,day^{-1}}$ over the whole solar surface \\citep[Table 2 \nof][]{2011SoPh..269...13T,th-thesis}.  To match their results from \\textit{Hinode} with \nother studies, the authors choose an FER of $450\\rm{\\,\\,Mx\\,cm^{-2}\\,day^{-1}}$, from the \nhigher end of the range (C. Parnell, priv. comm.). This is nearly four times higher than \nthe value quoted in \\cite{2016ApJ...820...35G}, who also used the \\textit{Hinode} \nobservations and a smaller minimum flux per feature, so that they should in principle \nhave caught more emerging features. However \\citet{th-thesis}, using a power law \ndistribution similar to \\citet{2011SoPh..269...13T} and a slightly different index of\n-2.5, estimates an FER of $64 \\rm{\\,Mx \\,cm^{-2}\\,day^{-1}}$.  {A possible reason \nfor this difference, as briefly discussed in both these studies, could be the different \nfeature tracking and identification methods used. In \\citet{2011SoPh..269...13T}, all the \nfeatures identified by BC, TB and TC methods are considered in determining the FER. \nAccording to the authors, the BC method counts the same feature multiple times and \nover-estimates the rate of flux emergence. However in \\citet{th-thesis}, only the \nfeatures \ntracked by TB method are used. The large differences in the FERs from the three detection \nmethods quoted in Table~2 of \\citet{2011SoPh..269...13T}, support this line of \nreasoning. \n The FER in \\citet{th-thesis} is roughly half that found by \\citet{2016ApJ...820...35G} \nand hence is at least in qualitative agreement.} The FER estimated by us is 2.5 times \nhigher than  {the largest value obtained by} \\citet{2011SoPh..269...13T} and 17 \ntimes higher than that of \\citet{th-thesis}.\n\nAnother recent estimate of the FER is by \\citet{2013SoPh..283..273Z}. Using \n\\textit{Hinode} observations, they estimate that the IN fields contribute up to \n$3.8\\times10^{26} \\rm{\\,Mx\\,day^{-1}}$ to the solar surface. This is an order of \nmagnitude higher than the global FER of $3 \\times 10^{25}\\rm{\\,Mx\\,day^{-1}}$ published \nby \\citet{2011SoPh..269...13T} and is two orders of magnitude higher than the \n$3.7\\times10^{24}\\rm{\\,Mx\\,day^{-1}}$ obtained by \\citet{2016ApJ...820...35G}.  In \n\\citet{2013SoPh..283..273Z}, it is assumed that every three minutes, the IN features \nreplenish the flux at the solar surface  with an average flux density of \n12.4\\,G ($\\,\\rm{\\,Mx\\,cm^{-2}})$. Here, three minutes is taken as the average lifetime of \nthe IN features \\citep{2010SoPh..267...63Z}.  {Their FER is nearly six times higher \nthan our estimate, although the lowest flux per feature to which \\textit{Hinode}\/SOT is \nsensitive is significantly larger than for \\textsc{Sunrise}\/IMaX (due to the lower spatial \nresolution of \nthe former). To understand this difference, we applied the method of \n\\citet{2013SoPh..283..273Z} to the \\textsc{Sunrise}\/IMaX observations. From the entire time \nseries, the total sum of the flux in all features with flux $> 9\\times10^{14}$ Mx is \n$1.1\\times10^{21}$\\,Mx over an area of $3.9\\times10^{20}$\\,cm$^{2}$. This gives us an \naverage flux density of 2.8\\,G, which is  4.5 times smaller than 12.4\\,G of \n\\citet{2013SoPh..283..273Z}. If the IN features are assumed to have an average lifetime \nof three minutes, similar to \\citet{2013SoPh..283..273Z}, then FER over the whole solar \nsurface is $8.2\\times10^{25}$\\,Mx\\,day$^{-1}$. This is 1.2 times higher than our original \nestimate from the feature tracking method. If instead, we take the average lifetime of \nthe features in our dataset from first appearance to final disappearance at the surface \nof \n$\\approx1.8$ minutes, we get an FER of $1.38\\times10^{26}$\\,Mx\\,day$^{-1}$, nearly 1.9 \ntimes higher than our original estimate and still 2.8 times smaller than \n\\citet{2013SoPh..283..273Z}. This is longer than 1.1 minute quoted in LSA17, which \nincludes death of a feature by splitting or merging (see LSA17), i.e. processes that do \nnot remove flux from the solar surface.}\n\n {To be sure that the problem does not lie in the COG technique employed here, we \nalso estimated the average flux density by considering the $B_{\\rm LOS}$ from the \nrecently \navailable inversions of the \\textsc{Sunrise}{} data \\citep{fatima}. The $B_{\\rm LOS}$ values \nreturned by the inversions differ from those given by the COG technique by about 5\\% on \naverage (individual pixels show much larger differences, of course), so that this cannot \nexplain the difference  to the value adopted by \\citet{2013SoPh..283..273Z}. If all the \npixels, including noise, are considered then the average flux density is 10.7\\,G. This is \nan absolute upper limit of the average flux density as a large part of it is due to noise \n and it is still lower than the IN signal of  12.4\\,G, estimated by \n\\citet{2013SoPh..283..273Z}.}\n {Thus the high value of FER from \\citet{2013SoPh..283..273Z} is at least partly \ndue to their possibly too high value of average flux density. The observations analysed \nby these authors clearly show network and enhanced network features. If some of these \nare misidentified, then this would result in a higher average flux density. If this is \nindeed the case, then the estimate of the lifetime of 3 minutes may also be too short \n\\citep[the technique of][neglects any possible correlation between magnetic flux and \nlifetime of a feature]{2013SoPh..283..273Z}. Although the amount of flux in IN \nfields is not expected to change significantly with time or place \n\\citep[see][]{2013A&A...555A..33B}, this is not true for the amount of \nflux in the network, which changes significantly. For example another time series taken \nby \\textsc{Sunrise}{} during its first flight, having slightly more network in the FOV, is found \nto \nhave an average $B_{\\rm LOS}$ of around 16\\,G (including noise), which is higher than the \n12.4\\,G used by \\citet{2013SoPh..283..273Z}. However, this is just a qualitative \nassessment and the very large FER found by \\citet{2013SoPh..283..273Z} needs to be probed \nquantitatively in a future study.}\n\n\\section{Conclusions}\n\\label{s8}\nIn this paper, we have estimated the FER in the quiet Sun from the IN features \nusing the observations from \\textsc{Sunrise}\/IMaX recorded during its first science flight in \n2009. We have included the contribution from features with fluxes in the range \n$9\\times10^{14} - 2.5\\times10^{18}$ Mx, whose evolution was followed directly. By \naccounting for the three important processes that bring flux to the solar surface: \n {unipolar and bipolar} appearances, and flux gained after birth by features born by \nsplitting or merging over their lifetime, we estimate an FER of $1100 \n\\rm{\\,Mx\\,cm^{-2}\\,day^{-1}}$. The third process is found to contribute significantly to \nthe FER. The smaller features with fluxes $\\le10^{16}$ Mx bring most of the flux to the \nsurface. Since our studies include fluxes nearly an order of magnitude smaller than the \nsmallest flux measured from the \\textit{Hinode} data, our FER is also an order of \nmagnitude higher when compared to studies using a similar technique \n\\citep[i.e.,][]{2016ApJ...820...35G}.  {We compare also with other estimates of the \nFER in the literature. \\citet{2011SoPh..269...13T} obtained a range of values. Those near \nthe lower end of the range \\citep[also quoted by ][]{th-thesis}, which are possibly the \nmore reliable ones, are roughly consistent with the results of \n\\citet{2016ApJ...820...35G}. The high FER of $3.8\\times10^{26}\\,{\\rm Mx\\, day^{-1}}$ \nfound by \\citet{2013SoPh..283..273Z} is, however, difficult to reconcile with any other \nstudy. It is likely so high partly due to the excessively large $B_{\\rm LOS}$ of \nIN fields of 12.4\\,G used by these authors, which is more than a factor of 4 \ntimes larger than the averaged $B_{\\rm LOS}$ of 2.8 G that we find. Even the absolute \nupper limit of the spatially averaged $B_{\\rm LOS}$ in our data (including noise) is \nbelow the value used by \\citet{2013SoPh..283..273Z}. We therefore \nexpect that they have overestimated the FER.}\n\n {There is clearly a need for further investigation, not only to quantify  the \nreasons for the different results obtained by different techniques. There are also still \nmultiple open questions. What is the cause of the increase and decrease of the flux of a \nfeature during its lifetime? Is this due to interaction with ``hidden'' flux? Is this \nhidden flux not visible because it is weak and thus below the noise threshold, or because \nit is structured at very small scales, i.e. it is below the spatial resolution? How \nstrongly does the ``hidden'' or missed flux change with changing spatial resolution? The \nmost promising approach to answering these and related questions is to study the flux \nevolution in an magnetohydrodynamic simulation that includes a working small-scale \nturbulent dynamo.}\n\n\\begin{acknowledgements}\n {We thank F. Kahil for helping with the comparison of COG and inversion results.} \nH.N.S. and  {L.S.A.} acknowledge the financial support from the Alexander von \nHumboldt foundation. The German contribution to \\textsc{Sunrise}{} and its reflight was funded by \nthe Max Planck Foundation, the Strategic Innovations Fund of the President of the Max \nPlanck Society (MPG), DLR, and private donations by supporting members of the Max Planck \nSociety, which is gratefully acknowledged. The Spanish contribution was funded by the \nMinisterio de Econom\\'i\u00ada y Competitividad under Projects ESP2013-47349-C6 and \nESP2014-56169-C6, partially using European FEDER funds. The HAO contribution was partly \nfunded through NASA grant number NNX13AE95G. This work was partly supported by the BK21 \nplus program through the National Research Foundation (NRF) funded by the Ministry of \nEducation of Korea.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\nWillis materials \\cite{Willis_WM_1981, Milton_NJP_2006, Srivastava_IJSNM_2015, Koo_NatComms_2016, Nassar_JMPS_2017, Sieck_PRB_2017, Muhlestein_NatComms_2017, DiazRubio_PRB_2017, Ponge_EML_2017, Li_NatComms_2018, Quan_PRL_2018} are complex media in which the unusual coupling between stress and particle velocity on one hand and linear momentum and strain on the other hand enable unique properties that only recently began to be explored experimentally. These include unsurpassed control over the propagation of elastic waves \\cite{Milton_NJP_2006}, independent engineering of transmitted and reflected wave fronts \\cite{Koo_NatComms_2016, Muhlestein_NatComms_2017, Li_NatComms_2018}, and broadband sound isolation in thin structures \\cite{Popa_NatComms_2018}.\n\nWe uncover in this article a previously unexplored property of Willis materials, namely the ability to produce linear and broadband non-reciprocal sound transport through remarkably thin metamaterial layers, a feature unachievable through other methods. Acoustic non-reciprocity has recently become an appealing concept because it enables improved sound control, acoustic imaging, and signal processing in acoustic devices \\cite{Haberman_Fleury_review}. The initial work involved non-linear structures in which the behavior of higher harmonics was non-reciprocal while the frequency of the impinging excitation remained reciprocal \\cite{liu2000locally, liang2009pass, liang2010pass, boechler2011bifurcation, Popa_NatComms_2014}. However, truly unidirectional devices require an asymmetric material response at the fundamental frequency in linear media. This has been a more difficult task. Nevertheless, linear non-reciprocal devices based on acoustic circulator-like structures have been demonstrated \\cite{fleury2014circulator} but they are narrow band and bulky. Topological insulators have been shown to break sound propagation reciprocity \\cite{yang2015topological, ni2015topologically} but these techniques provide narrow band solutions which are not applicable to three-dimensional waves. Materials with time modulated properties support non-reciprocal broadband sound, however, the modulation must occur on a significant length scale, therefore these materials necessarily occupy a large volume which becomes an issue especially at low frequencies \\cite{Trainiti_NJP_2016,attarzadeh_AA_2018,Nanda_JASA_2018}.\n\nHere we show that active metamaterials supporting decoupled Willis material parameters overcome the limitations of previous approaches. Specifically, this article brings three major contributions. First, essentially all theoretical analysis on Willis materials assume unbiased passive media in which the two Willis coupling tensors are related (coupled) and satisfy a strict set of constraints \\cite{Quan_PRL_2018}. We demonstrate experimentally that breaking the constraints of passivity leads to highly non-reciprocal and broadband linear media whose deeply subwavelength dimensions are unachievable through other methods. Second, we present a method to extract the effective material properties of a metamaterial in which the Willis coupling terms are allowed to be decoupled. We employ this method to show that our metamaterial is characterized by one very large and one negligible Willis coupling term in a broad band of frequencies, a feature not possible in passive media. Third, we show experimentally for the first time non-reciprocal sound transport in a multi-dimensional space. Interestingly, our metamaterial replicates in acoustics a mechanism for highly non-reciprocal electromagnetic wave transport based on electromagnetic bianisotropy \\cite{Popa_PRB_2012}, which is a solid experimental justification of the term acoustic bianisotropy recently given to Willis materials \\cite{Sieck_PRB_2017}. However, unlike the electromagnetics case, the acoustic metamaterial is very broadband having an operation bandwidth of at least one octave. This suggests that broadband non-reciprocal bianisotropic metamaterials in electromagnetics are within reach.\n\n\\section{Willis Coupling \\& Non-reciprocity}\\label{media}\nThe constitutive relations in Willis acoustic media can be written in the following form using standard index notation \\cite{nemat2011overall}\n\\begin{equation} \\label{eq1}\n\\begin{split}\n    -p &= Bu_{k,k}+S_kv_k, \\\\                               \n    \\mu_{i} &= D_{i}u_{k,k}+ \\rho_{ij}v_j, \\\\\n\\end{split}\n\\end{equation}\nwhere the acoustic pressure $p$ is related to the volumetric strain $u_{k,k}$ via the bulk modulus $B$ and to the particle velocity vector $v_k$ via the Willis coupling vector $S_{k}$. The momentum density vector $\\mu_{i}$ is related to the particle velocity vector $v_j$ via the second order mass density tensor $\\rho_{ij}$ and the volumetric strain $u_{k,k}$ via the Willis coupling  vector $D_{i}$. \n\nThe coupling between adjacent unit cells inside a metamaterial means that the dynamics of the unit cells inside the middle of the metamaterial are slightly different from the dynamics of edge unit cells when the unit cell size is larger than roughly a tenth of the operating wavelength \\cite{Sieck_PRB_2017}. Therefore, the description of the metamaterial acoustic behavior in terms of macroscopic material parameters becomes insufficient. A better description of the metamaterial dynamics is given in terms of microscopic polarizabilities \\cite{Sieck_PRB_2017, Popa_NatComms_2018}. According to this description, acoustic Willis metamaterials are modeled as collections of highly subwavelength sources that sense the monopole (i.e., local pressure field $p_{loc}$) or dipole moments (i.e., local velocity $v_{loc}$) in the surrounding acoustic field and create local monopole (i.e., pressure) or dipole moments (i.e., particle velocity) in response according to the following equations\n\\begin{equation}\n\\begin{aligned}\np_d&=0, &(v_d)_i&=(\\alpha_d)_{ij}(v_{loc})_j,\\\\\np_m&=\\alpha_mp_{loc}, &(v_m)_i&=0,\\\\\np_{dm}&=(\\alpha_{dm})_i Z_0 (v_{loc})_i, &(v_{dm})_i&=0,\\\\\np_{md}&=0, &(v_{md})_i&=(\\alpha_{md})_i Z_0^{-1} p_{loc},\n\\end{aligned}\n\\label{p&v}\n\\end{equation}\nwhere the subscripts $d$ and $m$ refer to the conventional dipole-to-dipole and monopole-to-monopole sources and $dm$ and $md$ refer to the Willis dipole-to-monopole and monopole-to-dipole sources. The local pressure generated by these sources are $p_d$, $p_m$, $p_{dm}$ and $p_{md}$ and the generated local particle velocities are $v_d$, $v_m$, $v_{dm}$ and $v_{md}$. The characteristic acoustic impedance of air is $Z_0$. Finally, the unitless polarizabilities $\\alpha_d$, $\\alpha_m$, $\\alpha_{dm}$ and $\\alpha_{md}$ quantify the linear relationship between the generated pressure and particle velocity and the sensed local fields.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=3.3truein]{figures\/configml.eps}\n  \\caption{Metasurface represented by collocated polarizability sheets having the acoustic polarizabilities $\\alpha_d$, $\\alpha_m$, $\\alpha_{md}$, and $\\alpha_{dm}$. The pressure and velocity fields excited by the incident field $p_i$ contribute to the local pressure $p_{loc}$ and velocity $v_{loc}$ at the position of the sheets.}\n  \\label{microconfig}\n\\end{figure}\n\nPassive materials undergoing no external biasing require correlated Willis coupling polarizabilities $\\alpha_{dm}$ and $\\alpha_{md}$ \\cite{muhlestein2016macro}. We show here that we can break this correlation in active metamaterials. Namely, we generate strong non-reciprocal sound transport in metamaterials in which we control $\\alpha_{dm}$ and $\\alpha_{md}$ independently.\n\nTo simplify the analysis, we consider a one-dimensional scenario in which a plane wave $p_i$ is normally incident on a one-unit-cell-thick metasurface modeled as four co-located polarizability sheets (see Fig. \\ref{microconfig}). In this scenario the vector and tensor quantities in Eqs. (\\ref{p&v}) become scalars. Consequently, the local pressure ($p_{loc}$) and particle velocity ($v_{loc}$) at the location of the polarizability sheets are the superposition of the incident and generated fields. Their expressions are given by the following equations.\n\\begin{equation}\n\\begin{aligned}\np_{loc}&= p_i+\\alpha_{m}p_{loc}+\\alpha_{dm}v_{loc}Z_0,\\\\\nv_{loc}&= s Z_0^{-1} p_i+\\alpha_dv_{loc}+\\alpha_{md} Z_0^{-1} p_{loc},\n\\end{aligned}\n\\label{pi}\n\\end{equation}\nwhere $s$ is a parameter that quantifies the direction of the incident wave, i.e., $s=1$ if the incident wave propagates in $+x$ direction and $s=-1$ if the incident wave propagates in $-x$ direction.\n\nEquations (\\ref{p&v}) and (\\ref{pi}) lead to the following relationships between the local pressure, local particle velocity, and the incident acoustic pressure at the location of the polarizability sheets\n\\begin{equation}\n\\begin{aligned}\np_{loc}&= \\frac{s\\alpha_{dm}+1-\\alpha_{d}}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}p_i,\\\\\nv_{loc}&= \\frac{s(1-\\alpha_{m})+\\alpha_{md}}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}Z_0^{-1}p_i.\n\\end{aligned}\n\\label{local}\n\\end{equation}\n\nAccording to Fig. \\ref{microconfig}, the relationship between the incident pressure $p_i$, reflected pressure $p_r$, and transmitted pressure $p_t$ can be described as follows.\n\\begin{equation}\n\\begin{aligned}\np_t&=p_i+p_m+p_{dm}+s(v_{md}+v_d)Z_0,\\\\\np_r&=p_m+p_{dm}-s(v_{md}+v_d)Z_0.\\\\\n\\end{aligned} \n\\label{Ps}\n\\end{equation}\n\nEquations (\\ref{p&v}), (\\ref{local}), and (\\ref{Ps}) lead to the following scattering parameters calculated in terms of polarizabilities:\n\\begin{equation}\n\\begin{aligned}\nS_{21}&\\equiv\\frac{p_t}{p_i}|_{s=+1}=\\frac{(1+\\alpha_{md})(1+\\alpha_{dm})-\\alpha_m\\alpha_d}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}, \\\\\nS_{11}&\\equiv\\frac{p_r}{p_i}|_{s=+1}=\\frac{\\alpha_m-\\alpha_d+\\alpha_{dm}-\\alpha_{md}}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}, \\\\\nS_{22}&\\equiv\\frac{p_r}{p_i}|_{s=-1}=\\frac{\\alpha_m-\\alpha_d+\\alpha_{md}-\\alpha_{dm}}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}, \\\\\nS_{12}&\\equiv\\frac{p_t}{p_i}|_{s=-1}=\\frac{(1-\\alpha_{md})(1-\\alpha_{dm})-\\alpha_m\\alpha_d}{(1-\\alpha_{d})(1-\\alpha_{m})-\\alpha_{dm} \\alpha_{md}}, \n\\end{aligned}\n\\label{S-p}\n\\end{equation}\nwhere $S_{ij}$ ($i,j=\\overline{1,2}$) are the usual $S$-parameters representing the pressure reflection and transmission coefficients for propagation in the $+x$ and $-x$ directions.\n\nAll passive materials under no external biasing satisfy $\\alpha_{dm}=-\\alpha_{md}$ and the material is reciprocal, i.e., $S_{12}=S_{21}$. However, if $\\alpha_{dm}$ and $\\alpha_{md}$ could be controlled independently, the reciprocity could be broken. For instance, letting $\\alpha_{dm}=0$ and $\\alpha_{md}\\neq 0$ results in very different transmission coefficients $S_{21}$ and $S_{12}$, which means that the polarizability sheets behave as a highly non-reciprocal medium. We demonstrate this point experimentally by constructing a non-reciprocal active acoustic metamaterial in which $\\alpha_{dm}\\approx 0$ and $\\alpha_{md}\\neq 0$. The value of $\\alpha_{md}$ will be chosen to maximize the amplitude of the non-reciprocity factor, defined as $S_{21}\/S_{12}$.\n\n\\section{Active metamaterial design and testing}\\label{method}\nBianisotropic (Willis) metamaterials in which the Willis coupling terms are independently controlled have been shown to provide excellent sound isolation capabilities \\cite{Popa_NatComms_2018}. We reconfigure this metamaterial platform to generate highly non-reciprocal responses. Specifically, the unit cell shown in Fig. \\ref{fi3-1}a consists of a 3.5 cm by 3.5 cm printed circuit board (PCB) designed to implement a monopole-to-dipole source. An omnidirection micro-electro-mechanical (MEMS) transducer senses the local pressure, i.e., the monopole moment in the field, and drives a dipole source composed of two back-to-back flat transducers working $180^\\circ$ out-of-phase via an amplifier circuit of impulse response $g$. The sensing transducer is placed in the plane of symmetry of the dipole in order to ensure that the generated local particle velocity does not contribute to the local pressure $p_{loc}$ sensed by the MEMS transducer. As a result, the unit cell's monopole-to-dipole polarizability is proportional to $g$ and thus it is controlled by the electronics. At the same time the one-directional nature of the amplifier enforces $\\alpha_{dm}=0$.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=3.3truein]{figures\/unitcell_waveguide.eps}\n  \\caption{Metamaterial fabrication and measurements. (a) Unit cell schematic; (b) Constructed 5-unit-cell metasurface; (c) The experimental setup shows the metasurface placed inside a two-dimensional acoustic waveguide. The pressure fields are measured in the highlighted areas.}\n  \\label{fi3-1}\n\\end{figure}\n\nA Willis metasurface consisting of five identical unit cells is constructed to demonstrate its non-reciprocal nature (see Fig. \\ref{fi3-1}b). The thickness of the metasurface with all components installed is approximately 9 mm. The metasurface is tested inside a two-dimensional waveguide of dimensions 1.2 m by 1.2 m. A schematic of the experimental setup is presented in Fig. \\ref{fi3-1}c. An external source speaker is placed at the edge of the waveguide. The metasurface is placed 58 cm away from the speaker in a window cut in the middle of a wall dividing the waveguide into two regions. The metasurface is designed so that its effective polarizabilities $\\alpha_m$ and $\\alpha_d$ match those of the wall. The external speaker produces short Gaussian pulses centered on 3000 Hz and having a width of 5 periods at the center frequency.\n\nWe scan the pressure fields in the vicinity of the metasurface in the region highlighted in Fig. \\ref{fi3-1}c in two scenarios. In the first scenario, the external speaker faces the side of the metasurface that is opaque to sound, which we call the reverse orientation in analogy to the reverse polarization of an electronic diode. In the second scenario, the speaker faces the transparent side (forward orientation). \n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=7truein]{figures\/mega.eps}\n  \\caption{\n    Measured transmitted sound pressure distribution. (a) Set up of the waveguide indicating the region measured for transmitted sound pressures; (b) Transmitted sound pressure distribution at one time point, left: pressure distribution for reverse orientation, right: pressure distribution for forward orientation; (c) Transmitted sound pressure distribution at 3000 Hz, left: pressure distribution for reverse orientation, right: pressure distribution forforward orientation; the regions surrounded by the dotted lines are the areas affected by the metasurface (i.e.,, effective area).\n  }\n    \\label{merit}\n\\end{figure*}\n\\section{Experimental Results}\\label{result}\nThe metasurface non-reciprocal behavior is measured directly by comparing the transmitted fields obtained in these two orientations (see Fig. \\ref{merit}a). Figure \\ref{merit}b shows the spatial distribution of the transmitted acoustic waves for the reverse and forward orientations and measured at time instances that better show the transmitted pulses. The spatial region influenced by the metasurface is marked by the dotted lines obtained by connecting the position of the external speaker and the edges of the metasurface. These measurements confirm qualitatively that the metasurface is transparent in the forward orientation and opaque in the reverse orientation. In contrast, outside the dotted lines the measured pressure is the same for both orientations which confirms the reciprocal behavior of the conventional wall.\n\nTo quantify the performance of the metasurface as a broadband non-reciprocal medium, we compute the spectra of the transmitted waves from the time domain measurements. Figure \\ref{merit}c shows the spectra obtained for the reverse (left) and forward orientations (right) at the incident pulse center frequency of 3000 Hz, which further confirms the strongly asymmetric transmission characteristics. Furthermore, we define the non-reciprocity factor as the ratio between the frequency domain pressure measured in the reverse orientation ($p_t^{rev}$) to the pressure measured in the forward orientation ($p_t^{fwd}$) 4 cm behind the metasurface (point A in Fig. \\ref{merit}c), namely $p_t^{rev}\/p_t^{fwd}$, which is a quantity equal to $S_{21}\/S_{12}$.\n\n\\begin{figure}\n  \\includegraphics[width=3.2truein]{figures\/nonr.eps}\n  \\caption{%\n   Amplitude and phase of the non-reciprocity factor retrieved from acoustic pressure field measurement.\n  }\n  \\label{nonr}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=3.4truein]{figures\/polarizabilities.eps}\n  \\caption{%\n    Acoustic polarizabilities calculated from measurements.\n  }\n  \\label{alpha}\n\\end{figure}\n Figure \\ref{nonr} shows the amplitude and phase of the non-reciprocity factor in the 2000 Hz to 4000 Hz octave excited by the external source. The figure confirms that the transmitted pressure is significantly different in the entire octave (i.e.,, $p_t^{rev}\/p_t^{fwd}\\neq 1$). The measured acoustic pressure amplitudes are significantly different at most frequencies. For instance, the amplitudes are more than 5 dB apart in the entire 2600 Hz to 3800 Hz band. Interestingly, whenever the $p_t^{fwd}$ and $p_t^{rev}$ amplitudes become comparable, their phases are almost $180^\\circ$ out-of-phase. This is explained by a transmitted field dominated by the dipole moment created by the active metasurface in response to the sensed monopole moment. The fields produced in the transmission directions have the same magnitude but different signs for two opposite directions of propagation.\n \nA key advantage of our active Willis material is its ability to control the Willis polarizabilities $\\alpha_{md}$ and $\\alpha_{dm}$ independently, which  is not possible in passive media \\cite{Sieck_PRB_2017, Quan_PRL_2018}. We compute the acoustic polarizabilities by inverting Eqs. (\\ref{S-p}) and obtain\n\n\\begin{equation}\n\\begin{aligned}\n\\alpha_m&=\\frac{S_{12}S_{21}-(1-S_{11})(1-S_{22})}{(1+S_{12})(1 + S_{21}) - S_{11}S_{22}},\\\\\n\\alpha_d&=\\frac{S_{12}S_{21}-(1+S_{11})(1+S_{22})}{(1+S_{12})(1 + S_{21}) - S_{11}S_{22}},\\\\\n\\alpha_{md}&=\\frac{S_{21}-S_{12} + S_{22}-S_{11}}{(1+S_{12})(1 + S_{21}) - S_{11}S_{22}},\\\\\n\\alpha_{dm}&=\\frac{S_{21}-S_{12} + S_{11}-S_{22}}{(1+S_{12})(1 + S_{21}) - S_{11}S_{22}},\n\\end{aligned} \n\\label{alpha-S}\n\\end{equation}\nwhere the numerical values of the $S$-parameters are evaluated from the fields measured behind and in front of the metasurface using a standard approach \\cite{Popa_PRB_2013, Popa_JASA_2016, Muhlestein_NatComms_2017}. Specifically, we measure the reflection ($S_{11}$) and transmission ($S_{21}$) coefficients in the the reverse orientation by exciting the metasurface with a speaker placed on the $x<0$ side of the metasurface (see Fig. \\ref{merit}a). The reflection and transmission coefficients measured in the forward orientation ($S_{22}$ and $S_{12}$) are obtained by placing the excitation speaker on the other side of the metasurface in the $x>0$ region. The speaker is positioned several wavelengths away from the metasurface to assure that the wavefront curvature at the metasurface is relatively small. The small curvature  justifies our one-dimensional analysis as explained in past work describing the process of measuring reflection and transmission coefficients from field measurements \\cite{Popa_JASA_2016}.\n\nThe polarizabilities computed with Eqs. (\\ref{alpha-S}) are shown in Fig. \\ref{alpha}. The results demonstrate our ability to control the Willis polarizabilities independently. Specifically, $\\alpha_{dm}\\approx 0$ in the selected frequency range while $\\alpha_{md}$ is significantly different from $\\alpha_{dm}$, a feature not possible in passive structures. Remarkably, $\\alpha_{md}$ is the second highest polarizabilitiy generated in a very compact metasurface. The largest polarizability is $\\alpha_{d}$ which confirms that the unpowered metasurface behaves like a membrane that produces significant dipole moment but small monopole moment, i.e., $\\alpha_{m}$ is small. The measurements also show that the effective polarizabilities are relatively constant in frequency, which demonstrates the broadband nature of the metamaterial. The measured variation with frequency is caused by numerical artifacts typically appearing in the process of extracting the effective material properties of opaque and acoustically thin structures \\cite{lee2010composite, Zigonenau_JAP_2011}.\n\n\\section{Conclusion}\nTo conclude, we demonstrated experimentally that the independent control of the two acoustic Willis coupling terms enables extremely non-reciprocal linear media composed of highly subwavelength and broadband unit cells, which are features unachievable in passive structures. Interestingly, our approach mirrors a method to obtain large non-reciprocal electromagnetic wave transport in media having asymmetric magneto-electric coupling terms, which is a strong experimental confirmation of the connection between electromagnetic bianisotropy and Willis media. Furthermore, we showed how the four basic acoustic polarizabilities of acoustic media can be retrieved from sound field measurements performed around the metamaterial. The extracted polarizabilities confirm our ability to design broadband and non-resonant Willis materials having strong bianisotropic responses. We believe that the range of effective acoustic properties enabled by active Willis materials will afford an unparalleled level of control over the sound propagation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe goal of quantum computing is to implement a programmable quantum information processor. Such a processor requires access to a universal gate set from which any quantum algorithm can be constructed. Universal gate sets can be formed from single-qubit gates supplemented by a two-qubit entangling gate\\cite{divincenzo1995universalqc}.\nFurthermore, fault-tolerance is necessary in order to perform arbitrarily long and precise computations, which, for the most lenient error correcting surface codes, puts a lower bound of around 0.99 on the required gate fidelities\\cite{raussendorf2007, corcoles2015, obrien2017, fowler2012}.\nExtensible high-fidelity entangling two-qubit gates are thus key elements in any multi-purpose quantum information processor.\n\nSingle-qubit gate operations are routinely performed with fidelities above 0.99\\cite{buluta2011review, gustavsson2013, reagor2018rigetti, rol2017, sheldon2016, chen2016, barends2014cz, wright2019, harty2014, itoh2014, zajac2018}, but pushing two-qubit gate fidelities above 0.99 still proves a daunting task.\nDespite the challenges in realizing a low loss environment while at the same time having high control of two-qubit operations, several two-qubit gates have been reported to do so.\nThe first group to accomplish this was Benhelm \\textit{et al.}, who in 2008 demonstrated a M\u00f8lmer-S\u00f8rensen-type entangling gate\\cite{sorensen1999, sorensen2000} with a fidelity of 0.993 using laser-controlled trapped calcium ions\\cite{benhelm2008}.\nSince then, similar ion trap experiments have realized high-fidelity two-qubit gates\\cite{gaebler2016, ballance2016, erhard2019, harty2016, ballance2015}.\nAnother promising qubit architecture is silicon-based quantum dots\\cite{koiller2001, friesen2003, itoh2014, zajac2018}, where controlled-rotation gates were recently benchmarked with a fidelity of 0.98\\cite{huang2019}.\n\n\nIn superconducting qubits the controlled-phase (\\textsf{CZ}) gate~\\cite{barends2014cz, kelly2014cz, Yu2014, rol2019, kjaergaard2019} and the cross-resonance (\\textsf{CR}) gate\\cite{Sheldon2016crgate} have been shown to exceed a fidelity of 0.99.\nOther two-qubit gates, like the $i\\textsf{SWAP}$ and $\\sqrt{i\\textsf{SWAP}}$ gates\\cite{mckay2016iswap, Dewes2012sqrtiswap, salathe2015iswap, reagor2018rigetti, caldwell2017}, $b\\textsf{SWAP}$ gate\\cite{poletto2012bswap}, the resonator induced phase (\\textsf{RIP}) gate\\cite{paik2016ripgate}, and a parametric \\textsf{CZ} gate\\cite{reagor2018rigetti, caldwell2017}, have been demonstrated with fidelities in the 0.9's.\nThese quantum gates are typically performed with transmons\\cite{transmon_original, you2007pretransmon, Barends2013xmon, Yu2014}, coupled directly to each other or via a separate coupling element, e.g. a transmission line resonator or a tunable coupler.\n\n\n\n\nIn this work, we propose the implementation of controlled two-qubit operations utilizing quantum interference patterns in a network of four qubits.\nAs a specific architecture, where this four-qubit gate can be implemented natively, we consider superconducting transmon qubits placed in a diamond-shaped geometry.\nThe qubits are coupled only through simple capacitive couplings. A similar 2D array of transmons was considered in Refs.~\\cite{lloyd2016, ciani2019}, but with different couplings and purpose.\nThe system comprises a four-qubit quantum gate (`the diamond gate'), where the state of two qubits control a two-qubit gate operation on the remaining two qubits.\nSince the diamond gate natively implements multiple unitaries, it is a useful addition to the gate set used for quantum simulation and quantum compilation.\nDue to its ability to perform (controlled) two-qubit entangling operations, supplementing the diamond gate with single-qubit operations allows for universal quantum computing on the target qubits.\n\nIn Sec.~\\ref{sec:system} we discuss the operation of the diamond gate, and in Sec.~\\ref{sec:extensibility} how it can constitute a building block in an extensible quantum computer.\nIn Sec.~\\ref{sec:simulations} we simulate the transmon implementation of the gate, using parameters from state-of-the-art superconducting qubits, in a Lindblad master equation simulation.\nWe find that the gate generally operates with fidelity around 0.99 in less than 100 ns.\nFinally, in Sec.~\\ref{sec:qutrits} we consider the effects of couplings to higher-energy states in the transmon spectrum, leading to undesired leakage across the control. We show how this behavior can be counteracted by engineering a cross-coupling to cancel the effects.\nThis is a passive scheme, in contrast to the microwave pulse-based scheme recently shown to reduce leakage in the \\textsf{CZ} gate\\cite{rol2019}.\n\nThroughout this paper, we use units where $\\hbar = 1$.\n\n\n\n\n\n\\section{Results}\n\n\\subsection{Four-qubit diamond gate}\n\\label{sec:system}\n\nConsider the four-qubit Hamiltonian being a sum of the non-interacting part\n\\begin{equation}\n   H_0 = -\\frac{1}{2} (\\Omega + \\Delta) (\\sigma_z^\\text{T1} + \\sigma_z^\\text{T2})\n   - \\frac{1}{2} \\Omega (\\sigma_z^\\text{C1} + \\sigma_z^\\text{C2}) \\; ,\n   \\label{eq:H0}\n\\end{equation}\nwhere $\\Omega + \\Delta$ ($\\Omega$) is the fixed frequency of the target (control) qubits, and the interaction terms\n\\begin{equation}\n   H_\\text{int} = J_\\text{C} \\, \\sigma_y^\\text{C1}\\sigma_y^\\text{C2}\n   + J (\\sigma_y^{\\text{T}1} + \\sigma_y^{\\text{T}2}) (\\sigma_y^{\\text{C}1} + \\sigma_y^{\\text{C}2}) \\; .\n   \\label{eq:Hint}\n\\end{equation}\nHere $\\sigma_z^j = \\dyad{0}{0}_j - \\dyad{1}{1}_j$ and $\\sigma_y^j = i\\dyad{1}{0}_j - i\\dyad{1}{0}_j$ are Pauli operators on qubit $j$, and the qubit frequencies are assumed positive such that $\\ket{0}_j$ is the non-interacting qubit ground state. For simplicity we have assumed that the two target (control) qubits are on resonance, and that all the couplings between the target and control qubits have the same strength $J$, although, as we will show later, this contraint is not needed for high performance of the gate. The four-qubit system is sketched in Figure~\\ref{fig:superconducting_circuit}a.\nAs we will discuss in the following, the system implements a four-qubit gate, which we will refer to as `the diamond gate' due to the geometry of the system.\n\nSuperconducting circuits offer a natural platform for implementing this type of Hamiltonian\\cite{krantz2019}. Specifically, by truncating the Hilbert space for each degree of freedom to qubits, the circuit of four capacitively coupled transmon qubits in Figure~\\ref{fig:superconducting_circuit}b implements the Hamiltonian.\nLater, we analyze the model including the second excited state of the transmon qubits.\n\n\\begin{figure*}\n  \\includegraphics[width=\\textwidth]{figure1.pdf}\n  \\caption{\n  \\textbf{(a)} The diamond gate: Four-qubit system consisting of two target qubits (T1 and T2) and control qubits (C1 and C2) coupled through exchange interactions (dashed lines) with the indicated strengths.\n  \\textbf{(b)} Lumped element superconducting circuit diagram of four capacitively coupled transmons, where each colored subcircuit corresponds to the same-colored qubit in \\textbf{(a)}.\n  \\textbf{(c)--(d)} Example transformations implemented by the diamond gate, $U$, of Eq.~\\eqref{eq:U}.}\n  \\label{fig:superconducting_circuit}\n\\end{figure*}\n\nWe now consider the interaction Hamiltonian, $H_\\text{int}$, in the frame rotating with $H_0$ and simplify the expression by assuming $\\lvert 2\\Omega \\rvert \\gg \\lvert J \\rvert$ (rotating wave approximation), which allows us to ignore the most rapidly oscillating terms. The system Hamiltonian is then\n\\begin{equation}\n   H = J_\\text{C} \\, \\sigma_+^\\text{C1}\\sigma_-^\\text{C2}\n   + J \\, e^{i\\Delta t}\n   (\\sigma_+^{\\text{T}1} + \\sigma_+^{\\text{T}2}) (\\sigma_-^{\\text{C}1} + \\sigma_-^{\\text{C}2})\n   + \\text{H.c.} \\; ,\n   \\label{eq:H}\n\\end{equation}\nwith $\\sigma_+^j = \\dyad{1}{0}_j$ and $\\sigma_-^j = \\dyad{0}{1}_j$ on qubit $j$. This Hamiltonian governs the dynamics resulting from the interactions in the model. We show in Appendix~\\ref{sec:unitary_gate_analysis} that the effective unitary time-evolution of $H$ gives rise to a four-qubit gate operating by means of controlled quantum interference (the diamond gate).\nThe analysis is based on a Magnus expansion of $H$ within Floquet theory, which assumes $\\lvert \\Delta \\rvert \\gg \\lvert J \\rvert, \\lvert J_\\text{C} \\rvert$, i.e. a qubit detuning much larger than the coupling strengths.\n\nThe diamond gate is a four-way controlled two-qubit gate operation on the target qubits T1 and T2.\nConsider the following gates in the target qubit computational basis, $\\{ \\ket{00}_\\text{T}, \\ket{01}_\\text{T}, \\ket{10}_\\text{T}, \\ket{11}_\\text{T} \\}$, where the superscripts refer to the control setting (discussed below):\n\\begin{align}\n    \\label{eq:UT00}\n    U^{00}_\\text{T} ={} &\n    \\begin{pmatrix}\n        \\makebox[1.1em]{$1$} & 0 & 0 & 0 \\\\[.2em]\n        0 & 0 & \\makebox[1.1em]{$-1$} & 0 \\\\[.2em]\n        0 & \\makebox[1.1em]{$-1$} & 0 & 0 \\\\[.2em]\n        0 & 0 & 0 & \\makebox[1.1em]{$-1$}\n    \\end{pmatrix}\n    = \\textsf{ZZ} \\cdot \\textsf{CZ} \\cdot \\textsf{SWAP}\n    \\; , \\\\\n    \\label{eq:UT11}\n    U^{11}_\\text{T} ={} &\n    \\begin{pmatrix}\n        \\makebox[1.1em]{$-1$} & 0 & 0 & 0 \\\\[.2em]\n        0 & 0 & \\makebox[1.1em]{$-1$} & 0 \\\\[.2em]\n        0 & \\makebox[1.1em]{$-1$} & 0 & 0 \\\\[.2em]\n        0 & 0 & 0 & \\makebox[1.1em]{$1$}\n    \\end{pmatrix}\n    = - \\textsf{CZ} \\cdot \\textsf{SWAP}\n    \\; , \\\\\n    \\label{eq:UTpl}\n    U^{\\Psi^+}_\\text{T} ={} &\n    \\begin{pmatrix}\n        \\makebox[1.1em]{$-1$} & 0 & 0 & 0 \\\\[.2em]\n        0 & \\makebox[1.1em]{$1$} & 0 & 0 \\\\[.2em]\n        0 & 0 & \\makebox[1.1em]{$1$} & 0 \\\\[.2em]\n        0 & 0 & 0 & \\makebox[1.1em]{$-1$}\n    \\end{pmatrix}\n    e^{-it_g J_\\text{C}}\n    = - \\textsf{ZZ} \\, e^{-it_g J_\\text{C}}\n    \\; , \\\\\n    \\label{eq:UTmi}\n    U^{\\Psi^-}_\\text{T} ={} &\n    \\begin{pmatrix}\n        \\makebox[1.1em]{$1$} & 0 & 0 & 0 \\\\[.2em]\n        0 & \\makebox[1.1em]{$1$} & 0 & 0 \\\\[.2em]\n        0 & 0 & \\makebox[1.1em]{$1$} & 0 \\\\[.2em]\n        0 & 0 & 0 & \\makebox[1.1em]{$1$}\n    \\end{pmatrix}\n    e^{+it_g J_\\text{C}}\n    = \\textsf{II} \\, e^{+it_g J_\\text{C}}\n    \\; .\n\\end{align}\nHere $t_g$ is the gate time given by\n\\begin{equation}\n   t_g = \\frac{\\pi\\lvert \\Delta\\rvert}{4J^2} \\; .\n   \\label{eq:tg}\n\\end{equation}\nEqs.~\\eqref{eq:UT00}--\\eqref{eq:UTmi} show the two-qubit operations in terms of well-known gates from the literature, see e.g. Ref.~\\cite{nielsen_chuang_2010}. Here $\\textsf{ZZ}$ is understood as a $\\textsf{Z}$ gate on each target qubit. Thus we see that $U^{00}_\\text{T}$ and $U^{11}_\\text{T}$ are two different combined swap and phase operations.\nAccess to just one of these entangling gates will facilitate universal quantum computing.\nThe third gate, $U^{\\Psi^-}_\\text{T}$, is a phase operation distinguishing target states with different parity (addition of T1 and T2's bit value modulo 2) by application of a relative sign.\nThe final gate, $U_\\text{T}^{\\Psi^-}$, which just adds a global phase, is the identity gate. We can therefore regard the preceding three gates as actual computational gates, while $U_\\text{T}^{\\Psi^-}$ is the idle position of the device.\n\nThe above two-qubit gates are controlled by the state of the control qubits, which we describe in the following orthonormal basis: $\\{ \\ket{00}_\\text{C}, \\ket{11}_\\text{C}, \\ket{\\Psi^+}_\\text{C}, \\ket{\\Psi^-}_\\text{C} \\}$.\nWe refer to this basis, which mixes computational basis states and the Bell states $\\ket{\\Psi^\\pm}_\\text{C} = (\\ket{01}_\\text{C} \\pm \\ket{10}_\\text{C})\/\\sqrt 2$, as the control basis. The full four-qubit unitary operation of the diamond gate is\n\\begin{equation}\n    \\label{eq:U}\n    \\begin{aligned}\n        U ={} &\\dyad{00}_\\text{C} U^{00}_\\text{T} + \\dyad{11}_\\text{C} U^{11}_\\text{T} \\\\\n              &+ \\dyad{\\Psi^+}_\\text{C} U^{\\Psi^+}_\\text{T} + \\dyad{\\Psi^-}_\\text{C} U^{\\Psi^-}_\\text{T} \\; .\n    \\end{aligned}\n\\end{equation}\nCast this way, it is evident that $U$ describes a four-way controlled operation on the target qubits. If the control qubits are initialized in one of the control basis states, only the corresponding gate among \\eqref{eq:UT00}--\\eqref{eq:UTmi} is performed. The control state is unchanged after the gate operation.\nFigure~\\ref{fig:superconducting_circuit}c--d illustrate the gate operation on the target state $\\ket{01}_\\text{T}$ in the cases where the control is $\\ket{00}_\\text{C}$ and $\\ket{\\Psi^+}_\\text{C}$, respectively. However, these gate diagrams only show the gate operation for these two control states, and in general the diamond gate performs a unitary operation on any initial four-qubit state.\nA more sophisticated decomposition of the full unitary $U$ is given i Figure~\\ref{fig:gate_circuit} in Appendix~\\ref{sec:unitary_gate_analysis}, where we note that the complexity in terms of number of $\\textsf{CNOT}$ gates is 42.\nHave access to four controlled two-qubit operations natively is useful for quantum simulation and may ease quantum gate compilation significantly.\n\n\n\nAs shown in Appendix~\\ref{sec:unitary_gate_analysis}, the unitary time-evolution under the Hamiltonian of Eq.~\\eqref{eq:H} approximately gives rise to $U$. Within the first order Magnus expansion, the approximation is exact when $J_\\text{C} = 0$, however a non-zero coupling between the control qubits is needed in order to initialze the control Bell states.\nSuch a coupling allows the triplet states $\\{ \\ket{00}_\\text{C} , \\ket{11}_\\text{C}, \\ket{\\Psi^+}_\\text{C} \\}$ to mix slightly during the gate operation, in which case the separation of control states in Eq.~\\eqref{eq:U} is no longer exact.\nThis leads to small gate infidelities of the order $(2J\/\\Delta)^2 = \\pi\/(t_g\\Delta)$ when then control qubits are initialized in $\\ket{00}_\\text{C}$ or $\\ket{11}_\\text{C}$, and twice as large when the control is in $\\ket{\\Psi^+}_\\text{C}$.\nFor typical superconducting circuit parameter values, like the ones used in the following section, these infidelities are on the order $10^{-3}$ to $10^{-2}$.\nNotice that the infidelity scales inversely with the gate time, leading to a trade-off between a fast gate and high-fidelity coherent operations.\nSince the singlet state $\\ket{\\Psi^-}_\\text{C}$ does not mix with the triplet states, the idle gate operation is not affected by the coupling $J_\\text{C}$, and the gate fidelity is only limited by other factors, e.g. qubit decoherence.\n\nAs mentioned above, the performance of the gate is increased if $J_\\text{C} = 0$, however a non-zero direct coupling between the control qubits is necessary if we wish to preparate the entangled Bell states.\nIn the following, we will assume a fixed value of $J_\\text{C}$, although ideally a tunable coupler\\cite{yan2018} can be used to turn on the coupling only during control state preparation.\nIf the control qubits are detuned from the target qubits, $\\lvert \\Delta \\rvert \\gg \\lvert J \\rvert$, we can initialize the control state without affecting the target qubits.\nThis detuning can be achieved by flux tunable devices, or by fabricating single-junction qubits with different frequencies.\nThus, ignoring the oscillating terms of Eq.~\\ref{eq:H}, we have effectively decoupled the control and target qubits.\nWe note that the effective Hamiltonian of the control qubits in the rotating frame, $J_\\text{C} (\\sigma_+^\\text{C1}\\sigma_-^\\text{C2} + \\sigma_-^\\text{C1}\\sigma_+^\\text{C2})$, has a zero-energy subspace spanned by $\\ket{00}_\\text{C}$ and $\\ket{11}_\\text{C}$, and eigenstates $\\ket{\\Psi^\\pm}_\\text{C}$ of energy $\\pm J_\\text{C}$.\nAn energy separation of $J_\\text{C}\/2\\pi \\sim \\SI{20}{\\mega\\hertz}$ allows us to initialize the control in $\\ket{\\Psi^\\pm}_\\text{C}$ by driving energy transitions\\cite{poletto2012bswap, Sheldon2016crgate}.\nTo initialize the control in $\\ket{00}_\\text{C}$ or $\\ket{11}_\\text{C}$, we can induce Rabi oscillations between these two states by driving the control qubits similarly to the procedure analyzed in Ref.~\\cite{didier2018}.\n\n\n\\subsection{Extensible quantum computer}\n\\label{sec:extensibility}\n\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{figure2.pdf}\n    \\caption{Proposed architecture for an extensible quantum computer. \\textbf{(a)} Four connected copies of the four-qubit diamond gate device. Detuning the qubits on the plaquettes A from the qubits on the plaquettes B allows each four-qubit device to run the diamond gate independently, while tuning the connecting qubits into resonance allows swap operations between plaquettes A and B. \\textbf{(b)} A sequence of diamond gates $U$ of Eq.~\\eqref{eq:U} in each plaquette and two-qubit swaps between the plaquettes running on the 16-qubit quantum computer.}\n    \\label{fig:extensibility}\n\\end{figure}\n\nThe four-qubit quantum interference device can constitute a building block in an extensible quantum computer by connecting several copies.\nOne possible architecture is illustrated in Figure~\\ref{fig:extensibility}a, where a 16-qubit quantum computer is constructed by connecting four copies of the four-qubit device, for instance through capacitive couplings.\nOn the plaquettes labelled A the control qubits are oriented vertically (1, 2, 13 and 14) and the target qubits horizontally (3, 4, 15 and 16), while the diamond gate devices on the plaquettes B are rotated by ninety degrees, such that control and target qubits from different plaquettes are connected.\nThis design of alternating A and B plaquettes can be extended in a straight-forward manner in one or two dimensions.\n\n\nThe quantum algorithm shown in Figure~\\ref{fig:extensibility}b is a generic algorithm spreading entanglement in the computer. Supplemented with single-qubit rotations, it may serve as a variational quantum eigensolver\\cite{Abhinav2017ibm}.\nThe algorithim can be implemented in the following way. Initially, the plaquette A qubits are far detuned from the plaquette B qubits, allowing each four-qubit diamond gate device to run the unitary gate $U$ of Eq.~\\eqref{eq:U} independently. After the completion of the gates, we can prevent further dynamics within each plaquette by switching the controls to the idle state. Then, by tuning pairs of connected qubits from different plaquettes into resonance, for instance 4 and 5, we can perform swap gates or use a suitable microwave driving to perform other desired two-qubit operations.\nFinally, by tuning the qubits out of resonance, and potentially switching certain controls, we are ready to run the diamond gate again.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Numerical simulations}\n\\label{sec:simulations}\n\nAlthough the analytic results suggest a functioning four-qubit diamond gate, we use numerical simulations to quantify the performance of the gates for state-of-the-art superconducting qubit parameters\\cite{wang2018, kounalakis2018, Wendin2016cqed_review}. Decoherence is included via the Lindblad master equation,\n\\begin{equation}\n  \\dot \\rho = - i [H, \\rho]\n  + \\sum_n \\Big[ C_n \\rho C_n^\\dagger\n  - \\frac{1}{2} (\\rho C_n^\\dagger C_n +  C_n^\\dagger C_n \\rho)\n  \\Big] \\; .\n  \\label{eq:master-equation}\n\\end{equation}\nHere $\\rho$ is the density matrix, $H$ is the Hamiltonian of Eq.~\\eqref{eq:H}, and the sum is taken over the following eight collapse operators, $C_n$: $\\sqrt{\\gamma} \\, \\sigma_z^{i}$ inducing pure dephasing and $\\sqrt{\\gamma} \\, \\sigma_-^{i}$ inducing qubit relaxation (photon loss), with $i$ running over all four qubits, denoting by $\\gamma$ the decoherence rate. We solve the master equation numerically using the Python toolbox QuTiP\\cite{qutip}.\n\n\n\n\n\\begin{table}\n    \\begin{tabular}{lll}\n        \\toprule\n        & Parameter set 1 & Parameter set 2 \\\\\n        \\midrule\n        $J_\\text{C} \/ 2\\pi \\, \\si{\\mega\\hertz}$ & $20$ & $20$ \\\\\n        $J \/ 2\\pi \\, \\si{\\mega\\hertz}$          & $65$ & $45$ \\\\\n        $\\Delta \/ 2\\pi \\, \\si{\\giga\\hertz}$     & $2$  & $0.5$ \\\\\n        $\\gamma \/ \\si{\\mega\\hertz}$             & $0.01$ & $0.01$ \\\\\n        \\midrule\n        Predicted $t_g \/ \\si{\\nano\\second}$     & $59.2$ & $30.9$ \\\\\n        Simulated $t_g \/ \\si{\\nano\\second}$     & $59.3$ & $31.5$ \\\\\n        \\midrule\n        $F_{00}(t_g)$          & $0.9943$ & $0.9662$ \\\\\n        $F_{11}(t_g)$          & $0.9931$ & $0.9668$ \\\\\n        $F_{\\Psi^+}(t_g)$      & $0.9881$ & $0.9348$ \\\\\n        $F_{\\Psi^-}(t_g)$      & $0.9968$ & $0.9983$ \\\\\n        $F(t_g)$               & $0.9923$ & $0.9637$ \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Two sets of model parameters and their corresponding gate times and gate fidelities. The gate fidelities are found at the simulated $t_g$.}\n    \\label{tab:parameters}\n\\end{table}\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{figure3.pdf}\n    \\caption{Fidelities versus time for the individually controlled gates ($F_{00}$, $F_{11}$, $F_{\\Psi^+}$, $F_{\\Psi^-}$) and the total diamond gate ($F$). Insets show zooms around the gate time.\n    The parameters used in \\textbf{(a)} are set 1 from Table~\\ref{tab:parameters}, and in \\textbf{(b)} they are set 2.}\n    \\label{fig:fid_vs_time}\n\\end{figure}\n\n\n\\begin{figure*}[t]\n   \\includegraphics[width=2\\columnwidth]{figure4.pdf}\n   \\caption{Simulations varying the model parameters $J_\\text{C}$, $J$ and $\\Delta$, with qubit decoherence of rate $\\gamma = \\SI{0.01}{\\mega\\hertz}$. While one parameter is varied, the remaining two are fixed at the values marked by the gray vertical lines (parameter set 1 of Table~\\ref{tab:parameters}). \\textbf{(a)--(c)} Gate times, also showing the prediction of Eq.~\\eqref{eq:tg} as the dashed line. \\textbf{(d)--(f)} Gate fidelities, i.e. the fidelities at the simulated gate time.}\n   \\label{fig:simulation_params}\n\\end{figure*}\n\n\n\nAs a quality measure of the gate, we consider the average fidelity\\cite{Nielsen2002avfid} (or simply `fidelity' in the following),\n\\begin{equation}\n    \\label{eq:av_fidelity_def}\n    F(t) \\equiv\n    \\int d\\psi \\matrixel{\\psi}{U_\\text{target}^\\dagger \\mathcal{E}_t(\\dyad{\\psi}) U_\\text{target}}{\\psi} \\; ,\n\\end{equation}\nwhich quantifies how well the quantum map $\\mathcal{E}_t$ approximates the target unitary gate $U_\\text{target}$ over a uniform distribution of input quantum states.\nIf the diamond gate is run with an arbitrary initial state, the integral is taken over all possible four-qubit states, and can be reduced to a sum over a density matrix basis, as shown in Ref.~\\cite{Nielsen2002avfid}.\nPutting $U_\\text{target} = U$ from Eq.~\\eqref{eq:U} and $\\mathcal{E}_t(\\rho(0)) = \\rho(t)$ found from solving Eq.~\\eqref{eq:master-equation}, the computed fidelity quantifies the overall performance of the diamond gate with arbitrary initial states.\nWe denote this fidelity by $F$. Its maximum value (the gate fidelity) defines the gate time, which generally matches the predicted value of Eq.~\\eqref{eq:tg} within a few percent.\nThe sources of gate infidelity are qubit decoherence and state mixing accommodated by a non-zero $J_\\text{C}$.\n\n\n\n\n\n\n\n\n\nIn order to study the performance of the four individual gates of Eqs.~\\eqref{eq:UT00}--\\eqref{eq:UTpl}, we initialize the control qubits in $\\ket{\\phi}_\\text{C} \\in \\{ \\ket{00}_\\text{C}, \\ket{11}_\\text{C}, \\ket{\\Psi^+}_\\text{C}, \\ket{\\Psi^-}_\\text{C} \\}$.\nIn this case the target operation is a single term in Eq.~\\eqref{eq:U}, $U_\\text{target} = \\dyad{\\phi}_\\text{C} U_\\text{T}^\\phi$, and the integral is taken over all states on the form $\\ket{\\phi}_\\text{C}\\ket{\\psi}_\\text{T}$, i.e. only varying the target qubits' state, $\\ket{\\psi}_\\text{T}$.\nThese states span a subspace of the entire four-qubit Hilbert space characterized by the fixed control state, however couplings to other control states leads to leakage out of the subspace, which we take into account with the appropriate modification of the sum formula in Ref.~\\cite{Nielsen2002avfid}. The resulting fidelity is denoted $F_\\phi$, and the value at the gate time is denoted the gate fidelity for the associated gate.\n\n\n\n\nTwo example parameter sets relevant for superconducting qubits are shown in Table~\\ref{tab:parameters}. We use the state-of-the art decoherence rate $\\gamma = \\SI{0.01}{\\mega\\hertz}$, corresponding to a qubit life-time of $\\gamma^{-1} = \\SI{100}{us}$\\cite{wang2018}. Figure~\\ref{fig:fid_vs_time} shows the simulated fidelities as functions of time. As expected, there is a trade-off between a fast gate and high-fidelity operations.\nParameter set 1 operates in $\\SI{59.3}{\\nano\\second}$ with gate fidelities $\\sim 0.99$, which decreases to $\\sim 0.96$ for the very fast $\\SI{31.5}{\\nano\\second}$ gate of parameter set 2.\nThe gate infidelities for each controlled gate follow the expectations discussed in the previous section.\nIn particular, the idle gate fidelity, $F_{\\Psi^-}(t)$, is only limited by qubit decoherence, reducing its value from $1$ to $0.9983$ and $0.9968$, respectively, during the operation time in the two cases.\nFor the remaining three controlled gates, a longer gate time can improve the gate fidelity, with the drawback of increased susceptibility to qubit decoherence.\nUltimately this limits the number of computations the diamond gate device can run successfully.\nFor the purpose of demonstrating the model, we will use parameter set 1 in the following, unless otherwise stated.\n\n\n\n\n\n\nTo probe the sensitivity to the model parameters, we vary each of $\\Delta$, $J$ and $J_\\text{C}$.\nAs is evident from Figure~\\ref{fig:simulation_params}a--c, the simulated gate times follow closely the prediction of Eq.~\\eqref{eq:tg}. Specifically, the gate time is tunable through $\\Delta$ and $J$.\nThe gate fidelities for the individually controlled gates and the total diamond gate are shown in Figure~\\ref{fig:simulation_params}a--f. Except for the phase gate controlled by $\\ket{\\Psi^+}_\\text{C}$, which is affected most strongly by couplings to other control states, the fidelities are above 0.99 over a wide range of parameters.\nDue to the mathematical equivalence between the two swapping gates controlled by $\\ket{00}_\\text{C}$ and $\\ket{11}_\\text{C}$, the gate fidelities for these operations are very similar. We attribute the difference to qubit relaxation, which only affects $\\ket{11}_\\text{C}$ and becomes more pronounced as the gate time increases.\nThe identity gate controlled by $\\ket{\\Psi^-}_\\text{C}$ is only limited by decoherence, and its gate fidelity decreases linearly with the gate time.\n\n\n\\begin{figure}\n   \\includegraphics[width=\\columnwidth]{figure5.pdf}\n   \\caption{Investigating gate stability for the following system infidelities: \\textbf{(a)} Crosstalk coupling between the target qubits. \\textbf{(b)} Random asymmetric noise in the couplings between the target and control qubits. \\textbf{(c)} Control state infidelity. \\textbf{(d)} Qubit decoherence with rate $\\gamma$.}\n   \\label{fig:simulation_noise}\n\\end{figure}\n\n\nWith a superconducting circuit implementation in mind, we consider a variety of system infidelites and their impact on the gate fidelities, see Figure~\\ref{fig:simulation_noise}. Most harmful is a direct capacitive coupling between the target qubits (Figure~\\ref{fig:simulation_noise}a), which allows the target qubits to bypass the control qubits, thereby circumventing the interference condition set by the control qubits.\nThe gate fidelities roughly decrease with the square of the cross-coupling strength $J_\\text{T}$, leading to noticable gate infidelities even for a relatively weak coupling. However, as we will show in the next section, crosstalk should not be suppressed, but rather utilized to combat another effect appearing in superconducting qubits: couplings to higher-energy states in the qubits' spectrum.\n\nFigure~\\ref{fig:simulation_noise}b shows simulation results with random noise on the couplings between the target and control qubits emulating asymmetries present in an actual circuit due to fabrication limits.\nEach data point in the plot corresponds to a simulation with random deviations from the noiseless value, $J$, denoting by $\\delta J$ the maximum deviation over the four couplings.\nThe gate performance is very robust towards this type of noise.\n\nBell state generation, which is required for the control states $\\ket{\\Psi^\\pm}_\\text{C}$, has been shown with a state infidelity of $\\sim 0.005$\\cite{barends2014cz}.\nWe introduce control state infidelity in the following way. For each data point in Figure~\\ref{fig:simulation_noise}c we contruct a random four-by-four Hermitian matrix $M$, from which we construct a unitary matrix $V = e^{i\\epsilon M}$, where $\\epsilon$ is a small real parameter.\nIn the simulations, we apply $V$ to the initial state of the control qubits in order to model imperfect state preparation.\nThe resulting gate fidelity is shown as a function of the maximum infidelity among the four control states.\nThe diamond gate suffers a linear decrease in gate fidelity, but remains high-performing for realistic control state infidelity.\n\nQubit decoherence in the form of relaxation and dephasing is included in the master equation~\\eqref{eq:master-equation} with rate $\\gamma$.\nIn Figure~\\ref{fig:simulation_noise}d we see that the gate fidelity decreases linearly with $\\gamma$.\nEven for qubits with $\\gamma = \\SI{0.05}{\\mega\\hertz}$, corresponding to a lifetime of $\\gamma^{-1} = \\SI{20}{\\micro\\second}$, the gate fidelity is $\\sim 0.98$.\nWe attribute this robustness to the relatively short gate time of $\\SI{59.3}{\\nano\\second}$.\n\n\n\n\n\n\n\n\n\n\\subsection{Higher-energy states}\n\\label{sec:qutrits}\n\nIn the previous section, we treated a model for four coupled qubits. In the superconducting circuit implementation of Figure~\\ref{fig:superconducting_circuit}, these qubits are comprised of the two lowest energy states of the each transmon, $\\ket{0}$ and $\\ket{1}$.\nHowever, in an actual superconducting circuit, the qubits may couple to higher-energy states in the transmon spectrum, which is the spectrum of a slightly anharmonic oscillator\\cite{transmon_original}.\nIn this section, we analyse the effects from including the second excited state, $\\ket{2}$, in the spectrum, thereby turning each qubit into a qutrit.\n\nThe full analysis of the circuit of Figure~\\ref{fig:superconducting_circuit}b is given in Appendix~\\ref{sec:qutrit_analysis}. The resulting four-qutrit Hamiltonian is a sum of the non-interacting part\n\\begin{equation}\n    \\label{eq:qutrit_H0}\n    \\tilde H_0 = -\\frac{1}{2} \\Omega_\\text{T} (\\tilde\\sigma_z^\\text{T1} + \\tilde\\sigma_z^\\text{T2})\n                 -\\frac{1}{2} \\Omega_\\text{C} (\\tilde\\sigma_z^\\text{C1} + \\tilde\\sigma_z^\\text{C2}) \\; ,\n\\end{equation}\nand the interaction terms\n\\begin{equation}\n    \\label{eq:qutrit_Hint}\n    \\tilde H_\\text{int} = J_\\text{T} \\tilde\\sigma_y^\\text{T1} \\tilde\\sigma_y^\\text{T2}\n                        + J_\\text{C} \\tilde\\sigma_y^\\text{C1} \\tilde\\sigma_y^\\text{C2}\n                        + J (\\tilde\\sigma_y^\\text{T1} + \\tilde\\sigma_y^\\text{T2}) (\\tilde\\sigma_y^\\text{C1} + \\tilde\\sigma_y^\\text{C2}) \\; ,\n\\end{equation}\nwhich are analogous to Eqs.~\\eqref{eq:H0}--\\eqref{eq:Hint}. The `Pauli $z$-operator' on qutrit $j$, denoted $\\tilde \\sigma_j$, includes $\\ket{2}_j$ in such a way that it has an energy $\\Omega_j + \\alpha_j$ above $\\ket{1}_j$, with $\\Omega_j$ and $\\alpha_j$ the frequency and anharmonicity, respectively. Typically $\\alpha_j \/ \\Omega_j \\sim -0.05$, yielding a small detuning of the second excited state compared to an equidistant spectrum (i.e. to vanishing anharmonicity). The operator is given as\n\\begin{equation}\n    \\label{eq:qutrit_sigmaz}\n    \\tilde\\sigma_z^j = \\dyad{0}{0}_j - \\dyad{1}{1}_j - \\left( 3 + \\frac{2\\alpha_j}{\\Omega_j} \\right) \\dyad{2}{2}_j \\; ,\n\\end{equation}\nThe `Pauli $y$-operator' on qutrit $j$ is\n\\begin{equation}\n    \\label{eq:qutrit_sigmay}\n    \\tilde\\sigma_y^j = i T_0^j \\dyad{1}{0}_j + i T_2^j \\dyad{2}{1}_j + \\text{H.c.} \\; ,\n\\end{equation}\nwhere $T_0^j \\approx 1$ and $T_2^j \\approx \\sqrt 2$ can be expressed in terms of $\\Omega_j$ and $\\alpha_j$ (see Appendix~\\ref{sec:qutrit_analysis}).\nHence, the coupling between the first and second excited state is as strong as the coupling between the two lowest (qubit) levels.\nDue to the small anharmonicity in transmons, i.e. that the energy separation between the qubit levels almost equals the separation between the first and second excited state, couplings that exchange a single excitation like $\\ket{11} \\rightarrow \\ket{02}$ are not strongly energetically suppressed.\nIn fact, this transition is sometimes used for the \\textsf{CZ} gate\\cite{krantz2019}.\nNotice that this lack of suppression holds for transmons in general, and is not a consequence of the specific model considered here.\n\nThis has two undesired consequences. Firstly, unless $\\abs{J_\\text{C}\/\\alpha_\\text{C}} \\ll 1$, it allows the control state $\\ket{11}_\\text{C}$ to mix with $\\ket{02}_\\text{C}$ and $\\ket{20}_\\text{C}$, leading to a non-conserved control state during the gate operation. This can be resolved by redefining the control state as\n\\begin{equation}\n    \\label{eq:tilde11}\n    \\ket{\\tilde{11}}_\\text{C} = \\cos\\tilde\\theta \\ket{11}_\\text{C} + \\sin\\tilde\\theta \\frac{1}{\\sqrt 2} (\\ket{02}_\\text{C} + \\ket{20}_\\text{C}) \\; ,\n\\end{equation}\nwith the mixing angle $\\tilde\\theta = -\\frac{1}{2} \\arctan (2\\sqrt 2 J_\\text{C} T_1^\\text{C} T_2^\\text{C} \/ \\alpha_\\text{C}) \\sim 0.5$, such that it is an eigenstate of an effective control state Hamiltonian.\nThis introduces a significant component of $(\\ket{02}_\\text{C} + \\ket{20}_\\text{C})\/\\sqrt{2}$, which is avoided if $J_\\text{C} = 0$.\nDetails are found in Appendix~\\ref{sec:qutrit_analysis}.\n\n\\begin{figure}\n   \\includegraphics[width=\\columnwidth]{figure6.pdf}\n   \\caption{Swap rate, found as the inverse of the smallest time $t$ where the swap fidelity (probability) $\\vert \\bra{\\phi}_\\text{C}\\bra{01}_\\text{T} e^{-i (\\tilde H_0 + \\tilde H_\\text{int})t} \\ket{10}_\\text{T} \\ket{\\phi}_\\text{C} \\vert^2$ becomes close to unity, versus crosstalk strength $J_\\text{T}$.\n   Data points are shown with the control state $\\ket{\\phi}_\\text{C}$ set to each of the displayed states.\n   The parameters used in the simulation are\n   $J_\\text{C}\/2\\pi = \\SI{20}{\\mega\\hertz}$, $J\/2\\pi = \\SI{65}{\\mega\\hertz}$, $\\Omega_\\text{C}\/2\\pi = \\SI{7}{\\giga\\hertz}$,\n   $\\Omega_\\text{T}\/2\\pi = \\SI{9}{\\giga\\hertz}$,\n   $\\alpha_\\text{C} = \\SI{-270}{\\mega\\hertz}$ and $\\alpha_\\text{T} = \\SI{-280}{\\mega\\hertz}$.\n   The optimal value of Eq.~\\eqref{eq:JTopt} is marked with a vertical line, $J_\\text{T}^\\text{opt}\/2\\pi = \\SI{-3.66}{\\mega\\hertz}$.}\n   \\label{fig:qutrit_swap_time_vs_JT}\n\\end{figure}\n\n\\begin{figure*}\n   \\includegraphics[width=2\\columnwidth]{figure7.pdf}\n   \\caption{Fidelity for swapping $\\ket{\\psi}_\\text{T} \\leftrightarrow \\ket{\\psi'}_\\text{T}$ for the indicated processes, computed as\n   $\\vert \\bra{\\phi}_\\text{C}\\bra{\\psi'}_\\text{T} e^{-i (\\tilde H_0 + \\tilde H_\\text{int})t} \\ket{\\psi}_\\text{T} \\ket{\\phi}_\\text{C} \\vert^2$, with the control state $\\ket{\\phi}_\\text{C}$ indicated above each column.\n   The parameters used in the simulation are\n   $J_\\text{C}\/2\\pi = \\SI{20}{\\mega\\hertz}$, $J\/2\\pi = \\SI{65}{\\mega\\hertz}$, $\\Omega_\\text{C}\/2\\pi = \\SI{7}{\\giga\\hertz}$,\n   $\\Omega_\\text{T}\/2\\pi = \\SI{9}{\\giga\\hertz}$,\n   $\\alpha_\\text{C} = \\SI{-270}{\\mega\\hertz}$ and $\\alpha_\\text{T} = \\SI{-280}{\\mega\\hertz}$.\n   \\textbf{(a)--(d)} No crosstalk, $J_\\text{T} = 0$. \\textbf{(e)--(h)} Crosstalk is set to its optimal value of Eq.~\\eqref{eq:JTopt}, $J_\\text{T}^\\text{opt}\/2\\pi = \\SI{-3.66}{\\mega\\hertz}$.}\n   \\label{fig:qutrit_swap_fids_vs_time}\n\\end{figure*}\n\nSecondly, excitations to the second excited states allow unwanted processes which bypass the control.\nFor instance, when the diamond gate is desired to be idle, leakage across the control can occur via:\n\\begin{equation}\n    \\label{eq:qutrit_leakage}\n    \\ket{\\Psi^-}_\\text{C} \\ket{10}_\\text{T}\n    \\rightarrow\n    \\frac{1}{\\sqrt 2} ( \\ket{02}_\\text{C} - \\ket{20}_\\text{C} ) \\ket{00}_\\text{T}\n    \\rightarrow\n    \\ket{\\Psi^-}_\\text{C} \\ket{01}_\\text{T} \\; .\n\\end{equation}\nSince this is a second order process in the qutrit model Hamiltonian, it would not pose a threat to the functionality of the diamond gate if it only relied on (generally faster) first order processes. However, the swap operations of Eqs.~\\eqref{eq:UT00}--\\eqref{eq:UT11} are also second order processes, leading to a failure of the idle diamond gate on the same time-scale as the operation of the swap gates. Similarly, the control state $\\ket{\\Psi^+}$ fails to prevent excitation leakage across the control, corrupting the operation of  Eq.~\\eqref{eq:UTpl}.\n\n\n\nHowever, these undesired processes can be mitigated by taking advantage of the effects of crosstalk. The circuit analysis in Appendix~\\ref{sec:qutrit_analysis} reveals a weak unavoidable crosstalk coupling of strength $J_\\text{T}$ in the interaction Hamiltonian~\\eqref{eq:qutrit_Hint}, which by itself has a significant negative impact on the gate fidelities, c.f. Figure~\\ref{fig:simulation_noise}a.\nThis leads directly to leakage across the control through processes of the type\n\\begin{equation}\n    \\label{eq:crosstalk_leakage}\n    \\ket{\\Psi^-}_\\text{C} \\ket{10}_\\text{T}\n    \\rightarrow\n    \\ket{\\Psi^-}_\\text{C} \\ket{01}_\\text{T} \\; .\n\\end{equation}\nThis process has the same unwanted outcome as the one of Eq.~\\ref{eq:qutrit_leakage}. As we show below, we can therefore restore the gate functionality by tuning the value of $J_\\text{T}$ such that these two unwanted leakage processes cancel each other.\nAnalyzing the problem with second order perturbation theory in order to calculate the amplitude of the leaked state (see Appendix~\\ref{sec:qutrit_analysis}), we find destructive interference between these processes when the crosstalk strength takes the optimal value\n\\begin{equation}\n    \\label{eq:JTopt}\n    \\begin{aligned}\n        J_\\text{T}^\\text{opt} ={}\n        &\\frac{(J T_2^\\text{C})^2}{\\Omega_\\text{C} + \\Omega_\\text{T} + \\alpha_\\text{C} + J_\\text{C}(T_1^\\text{C})^2} \\\\\n        &+ \\frac{(J T_2^\\text{C})^2}{\\Omega_\\text{C} - \\Omega_\\text{T} + \\alpha_\\text{C} + J_\\text{C}(T_1^\\text{C})^2} \\; .\n    \\end{aligned}\n\\end{equation}\nThus by tuning the crosstalk strength to $J_\\text{T} = J_\\text{T}^\\text{opt}$, we expect the fidelity for the target qubit swap $\\ket{01}_\\text{T} \\leftrightarrow \\ket{10}_\\text{T}$ to diminish, or equivalently a vanishing swap rate, when the control state is $\\ket{\\Psi^\\pm}_\\text{C}$.\nFigure~\\ref{fig:qutrit_swap_time_vs_JT} shows the swap rate for varying $J_\\text{T}$, with control qubits in each of the four control states.\nWe find two distinct zero-points, one for the data related to the control states $\\ket{00}_\\text{C}$ and $\\ket{\\Psi^\\pm}_\\text{C}$ at the expected value $J_\\text{T}^\\text{opt}$ (vertical line), and one for $\\ket{\\tilde{11}}_\\text{C}$.\nThus, it is possible to prevent the unwanted swap operation for the control states $\\ket{\\Psi^\\pm}_\\text{C}$, but as a consequence also the swap operation controlled by $\\ket{00}_\\text{C}$ is obstructed.\nOn the other hand, the swap operation controlled by $\\ket{\\tilde{11}}_\\text{C}$ is preserved at $J_\\text{T} = J_\\text{T}^\\text{opt}$, although the gate time is prolonged to around $\\SI{220}{\\nano\\second}$.\nRemarkably, for $J_\\text{T}\/2\\pi \\approx \\SI{-2.5}{\\mega\\hertz}$ the situation is reversed. Here, putting the control in $\\ket{\\tilde{11}}_\\text{C}$ prevents swapping, while the three remaining control states permit it.\nAt each zero-point, the gate time (inverse swap rate) for the swapping gate(s) is prolonged compared to the results in the previous section.\nTo reduce the gate time, one should pick parameters such that the zero-points are further apart, or such that the inclination of the graphs are steeper.\n\nFigure~\\ref{fig:qutrit_swap_fids_vs_time} illustrates in more detail the cancellation of unwanted transfer by crosstalk engineering.\nEach subfigure shows the swap fidelity for different initial target qubit states. The control is initialized in the state indicated above each column.\nFigure~\\ref{fig:qutrit_swap_fids_vs_time}a--d (the top row) show simulations for $J_\\text{T} = 0$, while the crosstalk has been put to its optimal value, $J_\\text{T} = J_\\text{T}^\\text{opt}$, in Figure~\\ref{fig:qutrit_swap_fids_vs_time}e--h (the bottom row).\nAs expected from Figure~\\ref{fig:qutrit_swap_time_vs_JT}, the swap $\\ket{01}_\\text{T} \\leftrightarrow \\ket{10}_\\text{T}$ (dark lines) occurs for any control state when there is no crosstalk, but is controlled uniquely by $\\ket{\\tilde{11}}_\\text{C}$ when the crosstalk is at the optimal value.\nIn the cases of $\\ket{00}_\\text{T}$ and $\\ket{11}_\\text{T}$, we wish to maintain a unit fidelity across all control states, i.e. the states should acquire at most a phase.\nTuning the crosstalk to $J_\\text{T}^\\text{opt}$ also improves the gate operation in this regard.\n\nEngineering crosstalk to mitigate unwanted leakage through higher-excited states is killing two birds with one stone: Each process is harmful to the functionality of the diamond gate, but letting them cancel each other preserves the ability to control the swap operation.\nThe price is the loss of swap functionality in the gate controlled by $\\ket{00}_\\text{C}$, and an increased gate time for the model parameters considered here.\nGenerally, the phases applied to each target state will be modified for all four controlled gates, but we do not pursue an analysis here, as other factors specific to the implementation will contribute to this as well.\nRather, our main goal was to demonstrate a passive method for dealing with undesired leakage processes.\n\n\n\n\n\\section{Discussion}\n\nWe have proposed a quantum interference device by coupling four qubits with exchange interactions.\nBy analyzing the unitary dynamics of the system, we have shown that it realizes the diamond gate: a four-way controlled two-qubit gate, with the ability to run two different entangling swap and phase operations, a (parity) phase operation, an idling gate with no dynamics, or an arbitrary superposition of these.\nWe considered an implementation in superconducting qubits using transmon qubits, and found that it generally operated fast and with high fidelity using state-of-the-art model and noise parameters. When taking second excited states into account, we had to prevent leakage across the control by engineering crosstalk, demonstrating a general method to avoid leakage in superconducting qubit systems.\nThe cost of this was a single redefined control state, one swap gate turning into a phase gate, altered phases on the gates, and a slower gate for the considered parameters.\nHowever, we only consider this analysis a starting point for an actual implementation, which might also include active microwave driving to optimize the operations or to prevent certain transitions.\nIt might also be worthwile to consider other types of superconducting qubits with larger anharmonicity, or entirely different platforms such as lattices of ultracold atoms or ions, where qubit encoded in hyperfine states or vibrational modes are far detuned from the rest of the spectrum.\n\nWe illustrated how the four-qubit diamond gate device can constitute an essential building block in an extensible quantum computer, and proposed a simple scheme where quantum algorithms are run on the computer by parallel processing on each four-qubit module interspersed with two-qubit operations spreading entanglement in the system, and single-qubit operations.\nEvidently, this scheme is adaptable to many different algorithms, and future work will investigate which algorithms are suitable to be implemented in the diamond-plaquette device.\n\n\n\n\n\n\n\\begin{acknowledgments}\n  This research was funded in part by the U.S. Army Research Office Grant No. W911NF-17-S-0008. N.J.S.L., L.B.K., and N.T.Z. acknowledge support from the Carlsberg Foundation and The Danish National Research Council under the Sapere Aude program. M.K. acknowledges support from the Carlsberg Foundation. T.W.L. acknowledges support from Microsoft. N.J.S.L. ackowledge discussions with Al\u00e1n Aspuru-Guzik, Daniel Kyungdeock Park, Kasper Sangild, and Stig Elkj\u00e6r Rasmussen. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements of the US Government.\n\\end{acknowledgments}\n\n\\onecolumngrid\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Flux Qubits}\n\nWe describe our design of the SC flux qubit that prepares an SC island with SC\nphase $\\varepsilon=\\varepsilon^{0}$ or $\\varepsilon^{1}$ depending on the flux\nqubit state $\\left\\vert 0\\right\\rangle _{\\mathrm{flux}}$ or $\\left\\vert\n1\\right\\rangle _{\\mathrm{flux}}$. The phase of the SC island can coherently\ncontrol the coupling between the two Majorana fermions at the end of the STIS\nwire. We would like to have a small phase separation $\\Delta\\varepsilon\n\\equiv\\varepsilon^{1}-\\varepsilon^{0}\\ll\\pi\/2$, so that we can easily switch\noff the coupling $H_{12}^{\\mathrm{MF}}$ when we do not want to couple the\nMajorana fermions (MFs).\n\nThe previous design of flux qubit with three JJs \\cite{Mooij99,Orlando99},\nhowever, is not amenable to achieving a small phase separation $\\Delta\n\\varepsilon\\ll\\pi\/2$, because of the following reason. The three JJs have\nJosephson energy $E_{J,1}=E_{J,2}=E_{J}$ and $E_{J,3}=\\alpha E_{J}$\n\\cite{Mooij99,Orlando99}. The phase difference across the first junction is\n$\\theta=\\pm\\cos^{-1}\\frac{1}{2\\alpha}$. By choosing $\\alpha=\\eta+1\/2$ and\n$\\eta\\ll1$, there is a small phase separation $\\Delta\\varepsilon=2\\left\\vert\n\\theta\\right\\vert \\approx4\\eta^{1\/2}$. However, the energy barrier for the\ntunneling is significantly suppressed $\\Delta U=2\\alpha-\\left(  2-\\frac\n{1}{2\\alpha}\\right)  E_{J}\\approx4\\eta^{2}E_{J}\\sim\\theta^{4}E_{J}$. The\naction associated with the tunneling is $S\\approx\\theta\\sqrt{\\Delta U\/E_{C}%\n}\\approx\\theta^{3}\\sqrt{E_{J}\/E_{C}}\\propto\\theta^{3}E_{J}$, where the last\nstep uses the property that $E_{C}\\propto1\/E_{J}$. In order to maintain a\nsimilar tunneling matrix element between the two potential minima, it requires\nthe unfavorable scaling $E_{J}\\propto1\/\\theta^{3}$. For example, to achieve\n$\\theta=0.1$, we have to increase the area of the Josephson junction by\n$\\sim10^{3}$. Practically, it would also be very challenging to have a precise\nvalue of $\\alpha=\\frac{1}{2}+\\frac{\\theta^{2}}{4}=0.5025$, which fine tunes\nthe phase separation $\\Delta\\varepsilon$. In contrast, the four-junction\ndesign of flux qubit has a favorable scaling of $E_{J,4}=\\beta E_{J}%\n\\propto1\/\\theta$ and there is no fine-tuned parameter. This motivates us to\nredesign the flux qubit with more Josephson junctions.\n\n\\begin{figure}[b]\n\\begin{center}\n\\includegraphics[width=6cm]{fig_FluxQubit.pdf}\n\\end{center}\n\\caption[fig:FluxQubit]{The design of flux qubit consists of a loop of four\nJJs in series that encloses an external magnetic flux $f\\Phi_{0}$. Schematic\nillustration in terms of (a) JJs and (b) SC islands. }%\n\\label{fig:FluxQubit}%\n\\end{figure}\n\nAs shown in Fig.~\\ref{fig:FluxQubit}, the flux qubit consists of a loop of\nfour JJs in series that encloses an external magnetic flux $f\\Phi_{0}$\n($f\\approx1\/2$ and $\\Phi_{0}=h\/2e$ is the SC flux quantum). The first two JJs\nhave equal Josephson coupling energy $E_{J,1}=E_{J,2}=E_{J}$; the third JJ has\ncoupling energy $E_{J,3}=\\alpha E_{J}$, with $0.5<\\alpha<1$; the coupling in\nthe fourth JJ is $E_{J,4}=\\beta E_{J}$, with $\\beta\\gg1$. For JJs with the\nsame thickness but different junction area $\\left\\{  A_{j}\\right\\}  $,\n$E_{J,j}\\propto A_{j}$ and $C_{j}\\propto A_{j}$. The charging energies can be\ndefined as $E_{C,1}=E_{C,2}=E_{C}=\\frac{e^{2}}{2C_{1}}$, $E_{C,3}=\\alpha\n^{-1}E_{C}$ and $E_{C,4}=\\beta^{-1}E_{C}$. Notice that $E_{J,j}E_{C,j}%\n=E_{J}E_{C}$ is independent of $j$. The system may have two stable states with\npersistent circulating current of opposite sign.\n\nHere is a summary of the key results:\n\n1.\\qquad For $\\beta\\rightarrow\\infty$, we may neglect the fourth JJ and reduce\nthe system to the well-studied flux qubit with three JJs\n\\cite{Mooij99,Orlando99}. For $\\beta\\gg1$, there is only a small phase\ndifference across the fourth junction, with $\\theta_{4}=\\pm\\theta_{4}^{\\ast\n}=\\mathbf{Z}_{\\mathrm{flux}}\\theta_{4}^{\\ast}$ depending on the sign of the\ncirculating current (i.e., the state of flux qubit, with $\\mathbf{Z}%\n_{\\mathrm{flux}}=\\left(  \\left\\vert 0\\right\\rangle \\left\\langle 0\\right\\vert\n-\\left\\vert 1\\right\\rangle \\left\\langle 1\\right\\vert \\right)  _{\\mathrm{flux}%\n}$). We show that the magnitude of the phase difference can be small\n$\\theta_{4}^{\\ast}\\approx\\frac{\\sqrt{4\\alpha^{2}-1}}{2\\alpha\\beta}\\propto\n\\beta^{-1}$. As shown in Fig.~\\ref{fig:FluxQubit}, two SC islands ($c$ and\n$d$) are connected by this junction, if we fix the SC phase $\\phi_{c}$, then\n$\\phi_{d}=\\varepsilon^{0,1}=\\phi_{c}\\pm\\theta_{4}^{\\ast}$ has phase separation\n$\\Delta\\varepsilon=2\\theta_{4}^{\\ast}\\approx\\frac{\\sqrt{4\\alpha^{2}-1}}%\n{\\alpha\\beta}$. The SC island $d$ can be used to coherently control the\ncoupling between the Majorana fermions of the STIS wire.\n\n2.\\qquad There are quantum fluctuations for the phase of the SC island. The\nmagnitude of quantum fluctuations depends on $\\left\\{  E_{C,j}\\right\\}  $ and\n$\\left\\{  E_{J,j}\\right\\}  $. For $\\beta\\gg1$, the dynamics associated with\n$\\theta_{4}$ can be characterized by a harmonic oscillator (HO) Hamiltonian%\n\\[\nH^{\\mathrm{HO}}=\\frac{p_{\\theta_{4}}^{2}}{2M_{4}}+\\frac{E_{J,4}}{2}\\left(\n\\theta_{4}-\\mathbf{Z}_{\\mathrm{flux}}\\theta_{4}^{\\ast}\\right)  ^{2},\n\\]\nwhere the effective mass is $M_{4}=\\frac{1}{8E_{c_{4}}}$ and the canonical\nmomentum $p_{\\theta_{4}}$ satisfies $\\left[  \\theta_{4},p_{\\theta_{4}}\\right]\n=i$ (with $\\hbar\\equiv1$). We may rewrite $H^{\\mathrm{HO}}=\\left(  a^{\\dag\n}a+1\/2\\right)  \\omega$ and $\\theta_{4}=\\mathbf{Z}_{\\mathrm{flux}}\\theta\n_{4}^{\\ast}+\\zeta\\left(  a^{\\dag}+a\\right)  \/\\sqrt{2}$, where the oscillator\nfrequency is $\\omega=\\sqrt{8E_{J}E_{C}}$ and the magnitude of quantum\nfluctuations is $\\zeta=\\left(  \\frac{8E_{C}}{E_{J}}\\right)  ^{1\/4}\\beta\n^{-1\/2}$. We justify that this simple model agrees very well with the general\nmodel characterizing the quantum fluctuations for the flux qubit with coupled JJs.\n\n3.\\qquad Various parameters characterizing the flux qubit are also calculated,\nincluding the plasma frequencies $\\left\\{  \\omega_{i}\\right\\}  _{i=1,2,3}$,\nbarrier height $\\Delta U$, and the tunneling matrix element $t$. For example,\ngiven parameters $\\alpha=0.8$ and $\\beta=10$, we compute $\\left\\{  \\omega\n_{i}\\right\\}  \\approx\\left(  2.8,2.3,1.8\\right)  \\sqrt{E_{J}E_{C}}\\sim\n\\sqrt{E_{J}E_{C}}$, $\\Delta U\\approx0.26E_{J}$, and $t\\approx1.8\\sqrt\n{E_{J}E_{C}}\\exp\\left[  -0.7\\left(  E_{J}\/E_{C}\\right)  ^{1\/2}\\right]  $. We\nnotice that these parameters for the flux qubit hardly depend on $\\beta$ when\n$\\beta>10$, which verifies the intuition that inserting a large Josephson\njunction to the loop has almost no effect to the properties of the flux qubit.\n\n4.\\qquad We propose two schemes to implement the SC phase-controller, which\ncan fix the phase difference between an SC island and a big SC reservoir.\n\nIn the following, we provide detailed analysis to justify our design of flux\nqubit with four JJs. First, we give the Hamiltonian description for the\nsystem. Then, we calculate the phase separation and quantum fluctuations.\nNext, we numerically obtain various quantities such as plasma frequencies,\nbarrier height, and tunneling matrix element. Our numerical calculation also\nverifies our estimates on phase separation and magnitude of quantum\nfluctuations. After that, we propose two implementations of the SC\nphase-controller. Finally, we derive the energy splitting function $E\\left(\n\\varepsilon\\right)  $ that is highly non-linear in terms of $\\varepsilon$.\n\n\\section{Hamiltonian for Flux Qubit}\n\nThe Hamiltonian for a flux qubit consisting of four JJs in series is%\n\\begin{equation}\nH^{\\mathrm{flux}}=T+U,\n\\end{equation}\nwith the Josephson potential%\n\\begin{equation}\nU=\\sum_{j=1,2,3,4}E_{J,j}\\left(  1-\\cos\\theta_{j}\\right)  ,\n\\end{equation}\nand the capacitive charging energy%\n\\begin{equation}\nT=\\frac{1}{2}\\sum_{j=1,2,3,4}C_{j}V_{j}^{2}.\n\\end{equation}\nHere for the $j$-th Josephson junction, $E_{J,j}$ is the Josephson coupling,\n$\\theta_{j}$ is the gauge-invariant phase difference, $C_{j}$ is the\ncapacitance, and $V_{j}$ is the voltage across the junction. Suppose that all\nthe junctions have the same thickness, we have $E_{J,j}$ $\\propto A_{j}$ and\n$E_{C,j}\\equiv\\frac{e^{2}}{2C_{j}}\\propto A_{j}^{-1}$ where $A_{j}$ is the\narea of the junction and $C_{j}$ is the junction capacitance \\cite{Tinkham96}.\nThe quantity $E_{J,j}E_{C,j}=E_{J}E_{C}$ does not depend on $j$.\n\nThe Josephson potential is constraint by the phase relation\n\\begin{equation}\n\\sum_{j}\\theta_{j}+2f\\pi\\equiv0\\left(  \\operatorname{mod}2\\pi\\right)  ,\n\\label{eq:PhaseConstraint}%\n\\end{equation}\nThis phase relation removes one degree of freedom. We may introduce a 3-vector\n$\\vec{\\theta}=\\left(  \\theta_{1},\\theta_{2},\\theta_{3}\\right)  $ to\ncharacterize the system with four JJs. (Note that we only need to choose three\nindependent phases for 3-vector, e.g., $\\vec{\\theta}=\\left(  \\theta_{1}%\n,\\theta_{2},\\theta_{4}\\right)  $ is also a valid choice.) For $f=1\/2$, the\nJosephson potential is%\n\\begin{align}\nU  &  =-E_{J,1}\\cos\\theta_{1}-E_{J,2}\\cos\\theta_{2}-E_{J,3}\\cos\\theta_{3}\\\\\n&  +E_{J,4}\\cos\\left(  \\theta_{1}+\\theta_{2}+\\theta_{3}\\right)  .\\nonumber\n\\end{align}\nBy appropriately choosing the parameter $\\left\\{  E_{J,j}\\right\\}  $, there\nare only two minima for the Josephson energy at $\\pm\\left\\{  \\theta_{j}^{\\ast\n}\\right\\}  =\\pm\\left\\{  \\theta_{1}^{\\ast},\\theta_{2}^{\\ast},\\theta_{3}^{\\ast\n},\\theta_{4}^{\\ast}\\right\\}  $ (see Fig.~\\ref{fig:Potentialn6}ab). We may\nidentify the two levels of the flux qubit as $\\left\\vert 0\\right\\rangle\n_{\\mathrm{flux}}=\\left\\vert \\left\\{  \\theta_{i}^{\\ast}\\right\\}  \\right\\rangle\n$ and $\\left\\vert 1\\right\\rangle _{\\mathrm{flux}}=\\left\\vert \\left\\{\n-\\theta_{i}^{\\ast}\\right\\}  \\right\\rangle $. Thus, we may denoted the two\npotential minima as $\\left\\{  \\mathbf{Z}_{\\mathrm{flux}}\\theta_{j}^{\\ast\n}\\right\\}  $, with $\\mathbf{Z}_{\\mathrm{flux}}=\\left(  \\left\\vert\n0\\right\\rangle \\left\\langle 0\\right\\vert -\\left\\vert 1\\right\\rangle\n\\left\\langle 1\\right\\vert \\right)  _{\\mathrm{flux}}$.\n\n\\begin{figure*}[ptbh]\n\\begin{center}\n\\includegraphics[width=16cm]{fig_Potential_n6.pdf}\n\\end{center}\n\\caption[fig:Potentialn6]{Contour plots of Josephson energy as a function of\n(a) $\\left\\{  \\theta_{1},\\theta_{2},\\theta_{3}\\right\\}  $ (with $\\theta\n_{4}=\\pi-\\theta_{1}-\\theta_{2}-\\theta_{3}$). (b) $\\left\\{  \\theta_{1}%\n,\\theta_{2},\\theta_{4}\\right\\}  $ (with $\\theta_{3}=\\pi-\\theta_{1}-\\theta\n_{2}-\\theta_{4}$). (c) $\\left\\{  \\theta_{1},\\theta_{4}\\right\\}  $ (with\n$\\theta_{3}=\\pi-\\theta_{1}-\\theta_{2}$ and $\\theta_{4}=0$) (d,e) $\\left\\{\n\\theta_{1},\\theta_{4}\\right\\}  $ (with $\\theta_{1}=\\theta_{2}$ and $\\theta\n_{3}=\\pi-2\\theta_{1}-\\theta_{4}$). (f) Marginal probability distributions of\n$\\theta_{4}$ associated with states $\\left\\vert 0\\right\\rangle _{\\mathrm{flux}%\n}$ (blue solid line) and $\\left\\vert 1\\right\\rangle _{\\mathrm{flux}}$ (red\ndashed line). The parameters are $E_{J}\/E_{C}=80$ and $\\left\\{  E_{J,i}%\n\/E_{J}\\right\\}  _{i=1,2,3,4}=\\left\\{  1,1,\\alpha=0.8,\\beta=10\\right\\}  $.}%\n\\label{fig:Potentialn6}%\n\\end{figure*}\n\nThe capacitive charging energy can be regarded as the kinetic energy\nassociated with the dynamics of $\\vec{\\theta}$. This is because the voltage\nacross the junction is given by the Josephson voltage-phase relation\n$V_{j}=\\left(  \\frac{\\Phi_{0}}{2\\pi}\\right)  \\dot{\\theta}_{j}$\n\\cite{Tinkham96} and the time derivatives $\\left\\{  \\dot{\\theta}_{j}\\right\\}\n$ obey the constraint $\\dot{\\theta}_{1}+\\dot{\\theta}_{2}+\\dot{\\theta}_{3}%\n+\\dot{\\theta}_{4}=0$ (derived from Eq.(\\ref{eq:PhaseConstraint})). Thus, we\nmay write the capacitive charging energy as%\n\\begin{align}\nT  &  =\\frac{1}{2}\\left(  \\frac{\\Phi_{0}}{2\\pi}\\right)  ^{2}\\sum\n_{i,j=1,2,3}C_{ij}\\dot{\\theta}_{i}\\dot{\\theta}_{j}\\\\\n&  =\\frac{1}{2}\\overrightarrow{\\dot{\\theta}}^{T}\\cdot\\mathbf{M}\\cdot\n\\overrightarrow{\\dot{\\theta}}\\\\\n&  =\\frac{1}{2}\\vec{p}^{T}\\cdot\\mathbf{M}^{-1}\\cdot\\vec{p},\n\\end{align}\nwhere the capacitive matrix is%\n\\begin{equation}\nC_{ij}=C_{i}\\delta_{ij}+C_{4},\n\\end{equation}\nthe effective mass tensor is\n\\begin{equation}\n\\mathbf{M}=\\left(  \\frac{\\Phi_{0}}{2\\pi}\\right)  ^{2}\\mathbf{C},\n\\end{equation}\nand the canonical momentum is%\n\\begin{equation}\n\\vec{p}=\\mathbf{M}\\cdot\\overrightarrow{\\dot{\\theta}}.\n\\end{equation}\nTherefore, we have reduced the problem to the canonical model of the quantum\nsystem with Hamiltonian\n\\begin{equation}\nH=\\frac{1}{2}\\vec{p}^{T}\\cdot\\mathbf{M}^{-1}\\cdot\\vec{p}+U\\left(  \\vec{\\theta\n}\\right)  , \\label{eq:HamGeneral}%\n\\end{equation}\nwhere the operators satisfy the commutation relation $\\left[  \\theta_{j}%\n,p_{k}\\right]  =i\\delta_{jk}$.\n\nBased on this model, we obtain the phase separation and the magnitude of\nquantum fluctuations in the next two sections.\n\n\\section{Phase Separation between the Potential Minima}\n\nWe now calculate the potential minimum $\\left\\{  \\theta_{j}^{\\ast}\\right\\}  $\nby introducing a Lagrange variable $\\lambda$ associated with the phase\nrelation (Eq.(\\ref{eq:PhaseConstraint})). We study the function%\n\\begin{equation}\nF=-\\sum_{j=1,2,3,4}E_{J,j}\\cos\\theta_{j}-\\lambda\\left(  \\sum_{j}\\theta_{j}%\n-\\pi\\right)  .\n\\end{equation}\nThe first derivatives all vanish at the extreme point:%\n\\begin{equation}\nE_{J,j}\\sin\\theta_{j}^{\\ast}=\\lambda.\n\\end{equation}\nFrom the phase relation $\\sum_{j}\\theta_{j}^{\\ast}=\\sum_{j}\\sin^{-1}%\n\\frac{\\lambda}{E_{J,j}}=\\pi$, we may solve for $\\lambda$.\n\nFor our system with four JJs, we can also calculate $\\theta_{4}^{\\ast}$ by\nseries expansion with respect to the small parameter $\\beta^{-1}$. For\n$\\beta\\rightarrow\\infty$, we have the zeroth order expansion $\\lambda^{\\left(\n0\\right)  }=\\frac{\\sqrt{4\\alpha^{2}-1}}{2\\alpha}$, $\\theta_{1}^{\\left(\n0\\right)  }=\\theta_{2}^{\\left(  0\\right)  }=\\sin^{-1}\\lambda^{\\left(\n0\\right)  }=\\cos^{-1}\\frac{1}{2\\alpha}$, $\\theta_{3}^{\\left(  0\\right)  }%\n=\\pi-2\\cos^{-1}\\frac{1}{2\\alpha}$ and $\\theta_{4}^{\\left(  0\\right)  }=0$.\nThen, to the first order of $\\beta^{-1}$, we have $\\theta_{4}^{\\left(\n1\\right)  }=\\sin^{-1}\\frac{\\lambda^{\\left(  0\\right)  }}{\\beta}\\approx\n\\frac{\\sqrt{4\\alpha^{2}-1}}{2\\alpha\\beta}$. Therefore,\\ the phase difference\nfor the fourth junction is\n\\begin{equation}\n\\theta_{4}^{\\ast}=\\frac{\\sqrt{4\\alpha^{2}-1}}{2\\alpha\\beta}+\\left(  \\beta\n^{-2}\\right)  .\n\\end{equation}\nAs plotted in Fig.~\\ref{fig:BetaDependence2}b, the numerically obtained\nquantity $\\frac{1}{\\Delta\\varepsilon}=\\frac{1}{2\\theta_{4}^{\\ast}}\\propto\n\\beta$.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8cm]{fig_BetaDependence_v4.pdf}\n\\end{center}\n\\caption[fig:BetaDependence2]{Parameters for the flux qubit as a function of\n$\\beta$. (a,b,c) Phase separation $\\Delta\\varepsilon$ (dark solid line) and\nthe magnitude of quantum fluctuations $\\zeta$ (purple dashed line) from\nnumerical calculation. The theoretical prediction from\nEq.~(\\ref{eq:QuantumFluctuations}) (gray solid line in (c)) agrees very well\nwith the numerical calculation. (d,e,f) The two ground states of the flux\nqubit can be coupled by intra-cell tunneling (red solid lines) and inter-cell\ntunneling (blue dashed lines) \\cite{Mooij99,Orlando99}, characterized by the\naction $S$, potential barrier $\\Delta U$, and tunneling rate $t$. Panels (a)\nand (f) assume $E_{J}\/E_{C}=80$.}%\n\\label{fig:BetaDependence2}%\n\\end{figure}\n\n\\section{Models for Quantum Fluctuations of the SC Phase}\n\nIn this section, we consider two models for quantum fluctuations across the\nJJs. The quantum fluctuations across the JJs are due to the finite mass matrix\nfor the Hamiltonian (Eq.(\\ref{eq:HamGeneral})), which is proportional to the\ncapacitance matrix. In the following, we first provide a simple, intuitive\nmodel to characterize the quantum fluctuations associated with $\\theta_{4}$\nfor $\\beta\\gg1$. Then, we consider the general model of multiple coupled\nharmonic oscillators, and obtain the formula for the magnitude of quantum\nfluctuations associated with a projected degree of freedom. We find very good\nagreement between the two models when $\\beta\\gg1$, which justifies the simple model.\n\n\\subsection{Simple Model -- One Harmonic Oscillator}\n\nGiven very large $\\beta$, we may neglect the higher order couplings to other\nJJs and consider the reduced Hamiltonian for the fourth JJ%\n\\begin{equation}\nH_{J,4}=\\frac{1}{2}C_{4}V_{4}^{2}+E_{J,4}\\left(  1-\\cos\\left(  \\theta\n_{4}-\\mathbf{Z}_{\\mathrm{flux}}\\theta_{4}^{\\ast}\\right)  \\right)  ,\n\\end{equation}\nwith displaced potential minimum at $\\mathbf{Z}_{\\mathrm{flux}}\\theta\n_{4}^{\\ast}$. We then use the harmonic approximation and obtain%\n\\begin{equation}\nH^{\\mathrm{HO}}=\\frac{p_{\\theta_{4}}^{2}}{2M_{4}}+\\frac{E_{J,4}}{2}\\left(\n\\theta_{4}-\\mathbf{Z}_{\\mathrm{flux}}\\theta_{4}^{\\ast}\\right)  ^{2},\n\\label{eq:Osc2}%\n\\end{equation}\nwhere $M_{4}=\\frac{1}{8E_{C,4}}$ and $E_{C,4}=\\frac{e^{2}}{2C_{4}}$. The\noscillator has frequency $\\omega=\\sqrt{8E_{J,4}E_{C,4}}=\\sqrt{8E_{J}E_{C}}$\nand characteristic length\n\\begin{equation}\n\\zeta=\\left(  \\frac{8E_{C,4}}{E_{J,4}}\\right)  ^{1\/4}=\\left(  \\frac{8E_{C}%\n}{E_{J}}\\right)  ^{1\/4}\\beta^{-1\/2}. \\label{eq:QuantumFluctuations}%\n\\end{equation}\nHere $\\zeta$ is also the magnitude of quantum fluctuations.\n\n\\subsection{General Model -- Coupled Harmonic Oscillators}\n\nWe now consider the general model of multiple coupled harmonic oscillators,\nand obtain the formula for the magnitude of quantum fluctuations associated\nwith a projected degree of freedom.\n\nClose to the potential minimum $\\left\\{  \\mathbf{Z}_{\\mathrm{flux}}\\theta\n_{j}^{\\ast}\\right\\}  $, we may expand the potential function to the second\norder of $\\vec{x}\\equiv\\vec{\\theta}-\\mathbf{Z}_{\\mathrm{flux}}\\vec{\\theta\n}^{\\ast}$%\n\\begin{equation}\nU=U_{\\min}+\\frac{1}{2}\\sum_{i,j=1,2,3}K_{ij}x_{i}x_{j}+O\\left(  x^{3}\\right)\n\\end{equation}\nwith%\n\\begin{equation}\nK_{ij}=\\left.  \\frac{d^{2}U}{d\\theta_{i}d\\theta_{j}}\\right\\vert _{\\vec{\\theta\n}=\\vec{\\theta}^{\\ast}}.\n\\end{equation}\nThe effective Hamiltonian around the minimum describes a system of coupled\nHarmonic oscillators:%\n\\begin{align}\nH_{\\mathrm{oscillator}}  &  =\\frac{1}{2}\\sum_{i,j=1,2,3}M_{ij}\\dot{x}_{i}%\n\\dot{x}_{j}+\\frac{1}{2}\\sum_{i,j=1,2,3}K_{ij}x_{i}x_{j}\\\\\n&  =\\frac{1}{2}\\vec{p}^{T}\\cdot\\mathbf{M}^{-1}\\cdot\\vec{p}+\\frac{1}{2}\\vec\n{x}^{T}\\cdot\\mathbf{K}\\cdot\\vec{x}%\n\\end{align}\nwhere we have used the definition $\\vec{p}=\\mathbf{M}\\cdot\\overrightarrow\n{\\dot{\\theta}}=\\mathbf{M}\\cdot\\overrightarrow{\\dot{x}}$.\n\nTo solve for the system of coupled oscillators, we perform the following\ntransformation to make the mass matrix\/tensor isotropic. For real and\nsymmetric mass matrix ($M_{ij}=M_{ji}$), there is an orthogonal transformation\n$V_{1}$ (with $V_{1}^{-1}=V_{1}^{T}$) that diagonalizes the mass matrix%\n\\begin{equation}\nV_{1}\\mathbf{M}V_{1}^{T}=\\mathbf{\\Lambda}%\n\\end{equation}\nwhere $\\Lambda_{ij}=\\lambda_{i}\\delta_{ij}$ is the diagonal matrix. The\neigenvalue $\\lambda_{i}$ is effective mass along the $i$-th principle axis. By\ninverting both sides, we have $V_{1}\\mathbf{M}^{-1}V_{1}^{T}=\\mathbf{\\Lambda\n}^{-1}$. For diagonal matrix, we may also define $\\left(  \\Lambda^{\\pm\n1\/2}\\right)  _{ij}=\\lambda_{i}^{\\pm1\/2}\\delta_{ij}$. Introducing the\ntransformation%\n\\begin{align}\n\\vec{x}^{\\prime}  &  =\\Lambda^{1\/2}V_{1}\\cdot\\vec{x}\\\\\n\\vec{p}^{\\prime}  &  =\\Lambda^{-1\/2}V_{1}\\cdot\\vec{p}%\n\\end{align}\nwe have%\n\\begin{equation}\nH_{\\mathrm{oscillator}}=\\frac{1}{2}\\vec{p}^{\\prime T}\\cdot\\vec{p}+\\frac{1}%\n{2}\\vec{x}^{\\prime T}\\cdot\\mathbf{\\tilde{K}}\\cdot\\vec{x}%\n\\end{equation}\nwhere%\n\\begin{equation}\n\\mathbf{\\tilde{K}=}\\Lambda^{-1\/2}\\left(  V_{1}KV_{1}^{T}\\right)\n\\Lambda^{-1\/2}.\n\\end{equation}\nFinally, we diagonalize the real symmetric matrix $\\mathbf{\\tilde{K}}$ via\northogonal transformation $V_{2}$%\n\\begin{equation}\nV_{2}\\mathbf{\\tilde{K}}V_{2}^{T}=\\mathbf{\\Omega}%\n\\end{equation}\nwhere $\\Omega_{ij}=\\omega_{i}^{2}\\delta_{ij}$ is the diagonal matrix. Overall,\nthe position and momentum transform as\n\\begin{subequations}\n\\label{eq:Transformation}%\n\\begin{align}\n\\vec{y}  &  =V_{2}\\vec{x}^{\\prime}=V_{2}\\Lambda^{1\/2}V_{1}\\cdot\\vec{x}\\\\\n\\vec{q}  &  =V_{2}\\vec{p}^{\\prime}=V_{2}\\Lambda^{-1\/2}V_{1}\\cdot\\vec{p}%\n\\end{align}\nThe eigenvalue $\\omega_{i}^{2}$ is the square of the $i$-th oscillator\nfrequency (also called plasma frequency). Given parameters $\\alpha=0.8$ and\n$\\beta=10$, we calculate the plasma frequencies $\\left\\{  \\omega_{i}\\right\\}\n_{i=1,2,3}=\\left\\{  2.8,2.3,1.8\\right\\}  \\sqrt{E_{J}E_{C}}$. We may also vary\nthe parameter $\\beta$, and observe that the plasma frequencies only depend\nvery weakly for $\\beta\\gg1$.\n\nNote that position and momentum have different transformations $V_{2}%\n\\Lambda^{1\/2}V_{1}$ and $V_{2}\\Lambda^{-1\/2}V_{1}$. Furthermore, these\ntransformations are not orthogonal transformations. However, as long as the\ntransformations preserve the commutation relation $\\left[  \\hat{y}_{j},\\hat\n{q}_{k}\\right]  =\\left[  \\hat{x}_{j},\\hat{p}_{k}\\right]  =i\\delta_{jk}$, we\ncan still perform quantization over the transformed coordinate.\n\nFollowing this procedure, we can numerically compute the magnitude of quantum\nfluctuations of $\\theta_{4}$ as detailed below.\n\n\\subsection{Quantum Fluctuations in SC Phase}\n\nWe now quantize the phase difference across the fourth junction $\\theta_{4}$.\nNear the potential minimum at $\\left\\{  \\theta_{i}^{\\ast}\\right\\}  $, we\nperform the transformation of Eq.\\ (\\ref{eq:Transformation}) along with the\nquantization $\\hat{y}_{i}=\\frac{\\hat{a}_{i}^{\\dag}+\\hat{a}_{i}}{\\sqrt{2}}$ and\n$\\hat{q}_{i}=\\frac{\\hat{a}_{i}^{\\dag}-\\hat{a}_{i}}{\\sqrt{2}i}$, we obtain the\nHamiltonian for three uncoupled harmonic oscillators%\n\\end{subequations}\n\\begin{equation}\n\\tilde{H}_{\\mathrm{oscillator}}=\\sum_{i=1,2,3}\\hbar\\omega_{i}\\left(  \\hat\n{a}_{i}^{\\dag}\\hat{a}_{i}+\\frac{1}{2}\\right)  ,\n\\end{equation}\nwith eigenfrequencies of $\\left\\{  \\omega_{i}\\right\\}  _{i=1,2,3}$. Each\noscillatory mode may induce quantum fluctuations in $\\theta_{4}$, with\ncharacteristic length scale $\\zeta_{i}$. The operator form of $\\theta_{4}$ can\nbe written as%\n\\begin{equation}\n\\theta_{4}=\\mathbf{Z}_{\\mathrm{flux}}\\theta_{4}^{\\ast}+\\sum_{i=1,2,3}\\zeta\n_{i}\\frac{\\hat{a}_{i}^{\\dag}+\\hat{a}_{i}}{\\sqrt{2}}%\n\\end{equation}\n\n\nWe calculate the values of $\\zeta_{i}$ as the following. In the $y$%\n-coordinate, the characteristic displacement vector for the $i$-th mode is\n$\\vec{l}_{i}^{\\left(  y\\right)  }=\\sqrt{\\frac{\\hbar}{\\omega_{i}}}\\vec{e}%\n_{i}^{\\left(  y\\right)  }$, with unit vector $\\vec{e}_{i}$ along the $i$-th\ndirection. We may transform this back to the $x$-coordinate, $\\vec{l}%\n_{i}^{\\left(  x\\right)  }=\\left(  V_{2}\\Lambda^{1\/2}V_{1}\\right)  ^{-1}%\n\\cdot\\sqrt{\\frac{\\hbar}{\\omega_{i}}}\\vec{e}_{i}^{\\left(  y\\right)  }$. Note\nthat $\\vec{l}_{i}^{\\left(  x\\right)  }$ is no longer orthogonal. From $\\vec\n{l}_{i}^{\\left(  x\\right)  }$, we can obtain the characteristic fluctuating\nscale of $\\zeta_{i}=\\left\\vert \\sum_{k=1,2,3}\\left(  \\vec{l}_{i}^{\\left(\nx\\right)  }\\right)  _{k}\\right\\vert $. The magnitude of quantum fluctuations\nof $\\theta_{4}$ can be computed as\n\\begin{equation}\n\\zeta=\\left(  \\sum_{i=1,2,3}\\left\\vert \\zeta_{i}^{2}\\right\\vert \\right)\n^{1\/2} \\label{eq:QuantumFluctuations2}%\n\\end{equation}\nfor independent fluctuations from the three uncoupled harmonic oscillators.\n\nAs plotted in Fig.~\\ref{fig:BetaDependence2}c, the numerically obtained value\nfor $\\zeta$ (using Eq.~(\\ref{eq:QuantumFluctuations2})) agrees very well with\nthe prediction from the simple model (using Eq.~(\\ref{eq:QuantumFluctuations}%\n)), which scales as $\\beta^{-1\/2}$. Therefore, the simple model of\nEq.~(\\ref{eq:QuantumFluctuations}) provides a reliable description to\ncharacterize the dynamics associated with $\\theta_{4}$.\n\n\\section{Tunneling Matrix Element}\n\n\\subsection{WKB method}\n\nWe use the WKB method to estimate the tunneling matrix element\n\\cite{Orlando99}. The action associated with the pathway $\\vec{\\theta}\\left(\nr\\right)  $ from $\\vec{\\theta}\\left(  0\\right)  =\\vec{\\theta}_{a}$ to\n$\\vec{\\theta}\\left(  1\\right)  =\\vec{\\theta}_{b}$ is%\n\\begin{equation}\nS=\\int_{\\vec{\\theta}_{a}}^{\\vec{\\theta}_{b}}\\sqrt{2\\left(  U-E\\right)  }%\n\\sqrt{d\\vec{\\theta}^{T}\\cdot\\mathbf{M}\\cdot d\\vec{\\theta}},\n\\end{equation}\nand the tunneling matrix element can be estimated as\n\\begin{equation}\nt\\approx\\frac{\\hbar\\omega}{2\\pi}~e^{-S\/\\hbar}.\n\\end{equation}\nThe phase space $\\vec{\\theta}$ has period $2\\pi~$for all three directions. We\nmay introduce the unit cell with volume $\\left(  2\\pi\\right)  ^{3}$ and three\nbasis vectors $\\vec{a}_{1}=2\\pi\\left(  1,0,0\\right)  $, $\\vec{a}_{2}%\n=2\\pi\\left(  0,1,0\\right)  $, and $\\vec{a}_{3}=2\\pi\\left(  0,0,1\\right)  $.\n(Note that we may choose the shape of the unit cell for our convenience.)\nRegarding the tunneling pathway, we may choose the initial point $\\vec{\\theta\n}_{a}^{\\ast}=-\\vec{\\theta}^{\\ast}$, while the choice for final point is not\nunique as $\\vec{\\theta}_{b}=\\vec{\\theta}^{\\ast}+\\sum_{i}n_{i}\\vec{a}_{i}$ for\nintegers $\\left\\{  n_{i}\\right\\}  $. However, we may require that the final\npoint, $\\vec{\\theta}_{b}^{\\ast}$, be the one that has the minimum action from\nthe initial point, and we choose the unit cell so that it includes the minimum\naction pathway that connects $\\vec{\\theta}_{a}^{\\ast}$ and $\\vec{\\theta}%\n_{b}^{\\ast}$. After this procedure, the intra-cell tunneling has the minimum\naction, compared to all inter-cell tunneling pathways.\n\n\\subsection{Pathway with minimum action}\n\nWe should use the pathway $\\vec{\\theta}\\left(  r\\right)  $ with extreme action\nfor the WKB method, i.e.,%\n\\[\n\\frac{\\delta S}{\\delta\\vec{\\theta}\\left(  r\\right)  }=0.\n\\]\nWe obtain these extreme pathways using the following approach. First, we\ndiscretize the integration%\n\\begin{equation}\nS=\\sum_{i=1}^{N}\\left\\vert \\vec{x}_{i}-\\vec{x}_{i-1}\\right\\vert ~f\\left(\n\\frac{\\vec{x}_{i}+\\vec{x}_{i-1}}{2}\\right)  ,\n\\end{equation}\nwith $\\vec{x}_{0}=\\vec{\\theta}_{a}$ and $\\vec{x}_{N}=\\vec{\\theta}_{b}$. Then\nwe calculate the derivatives with respect to $\\vec{r}_{i}$%\n\\begin{align}\n\\frac{\\delta S}{\\delta\\vec{r}_{i}} &  =\\frac{\\Delta\\vec{x}_{i}}{\\left\\vert\n\\Delta\\vec{x}_{i}\\right\\vert }f\\left(  \\frac{\\vec{x}_{i}+\\vec{x}_{i-1}}%\n{2}\\right)  -\\frac{\\Delta\\vec{x}_{i+1}}{\\left\\vert \\Delta\\vec{x}%\n_{i+1}\\right\\vert }f\\left(  \\frac{\\vec{x}_{i+1}+\\vec{x}_{i}}{2}\\right)\n\\nonumber\\\\\n&  +\\left\\vert \\Delta\\vec{x}_{i}\\right\\vert \\bigtriangledown f\\left(\n\\frac{\\vec{x}_{i}+\\vec{x}_{i-1}}{2}\\right)  +\\left\\vert \\Delta\\vec{x}%\n_{i+1}\\right\\vert \\bigtriangledown f\\left(  \\frac{\\vec{x}_{i+1}+\\vec{x}_{i}%\n}{2}\\right)  .\n\\end{align}\nFor extreme pathway, these derivatives should vanish. If we start with an\nnon-extreme pathway with non-vanishing derivatives, we can update the pathway\nso that the updated pathway becomes closer to the extreme pathway. By\nrepeating the update procedure many times, we will obtain a pathway that is\nvery close to the extreme pathway. Since we know in advance that we are\nlooking for the pathway that gives the minimum action, we apply the following\nupdate rules for the $k$th update:%\n\\begin{equation}\n\\vec{r}_{i}^{\\left(  k+1\\right)  }=\\vec{r}_{i}^{\\left(  k\\right)  }%\n-\\epsilon\\frac{\\delta S}{\\delta\\vec{r}_{i}}%\n\\end{equation}\nwhere $\\epsilon$ determines the evolution rate and $i=1,2,\\cdots,N-1$. For\nsufficiently small evolution rate%\n\\begin{equation}\nS\\left[  \\left\\{  \\vec{r}_{i}^{\\left(  k+1\\right)  }\\right\\}  \\right]\n-S\\left[  \\left\\{  \\vec{r}_{i}^{\\left(  k\\right)  }\\right\\}  \\right]\n=-\\epsilon\\left(  \\frac{\\delta S}{\\delta\\vec{r}_{i}}\\right)  ^{2}+O\\left(\n\\epsilon^{2}\\right)  \\leq0,\n\\end{equation}\nwhich ensures continuous reduction of the action.\n\n\\subsection{Tunneling rates and $\\beta$ Dependence}\n\nThis algorithm gives us the correct pathway that locally minimize the action\nbetween neighboring potential wells. We find that the pathways are essentially\nthe straight lines connecting the different minima, with no significant\ndifference in terms of the action values. For example, given parameters\n$\\alpha=0.8$ and $\\beta=10$, the intra-cell action is $S_{in}\\approx\n0.7\\hbar\\sqrt{E_{J}\/E_{C}}$ and the smallest inter-cell action is\n$S_{out}\\approx1.4\\hbar\\sqrt{E_{J}\/E_{C}}$. For $E_{J}\/E_{C}\\approx80$, we\nwill have $t_{2}\/t_{1}\\approx\\exp\\left[  \\left(  S_{1}-S_{2}\\right)\n\/\\hbar\\right]  \\sim10^{-3}\\ll1$. The barrier height for intra-cell tunneling\nis $\\Delta U=0.25E_{J}$.\n\nWhen $\\beta\\rightarrow\\infty$, all quantities (potential minimum position,\nplasma frequencies, action for tunneling, energy barrier, and tunneling matrix\nelement) reduce to the case with three JJs. As illustrated in\nFig.~\\ref{fig:BetaDependence2}, the deviation scales as $\\beta^{-1}$. For\n$\\beta\\geq10$, the perturbation from $\\beta$ is very small.\n\nFor practical parameters of mesoscopic aluminum junctions with critical\ncurrent density $500$ A\/cm$^{2}$ \\cite{Mooij99,Orlando99}, a junction with an\narea of $A=0.2\\times0.4$ $\\mu m^{2}$ can achieve $E_{J}\\approx200$ GHz and\n$E_{J}\/E_{C}\\approx80$, which corresponds to the first two junctions\n$E_{J,1}=E_{J,2}=E_{J}$. Since $E_{J,j}\\propto A_{j}$, the fourth junction\n$E_{J,4}=\\beta E_{J}$ should have an area of approximately $1\\times1$ $\\mu\nm^{2}$ to achieve $\\beta\\approx10.$\n\n\\section{Phase-Controllers}\n\nFor the STIS quantum wire, we would like to fix the phase difference between\ntwo disconnected SC islands. For example, we would like have $\\phi_{l}%\n-\\phi_{u}=\\pi\/2$, $\\phi_{r}-\\phi_{u}=\\pi\/2$, $\\phi_{c}-\\phi_{u}=\\phi_{c}+\\pi$,\nas shown in Fig.~\\ref{fig:TopoFluxQubits}ac. The idea is to connect the two SC\nislands via a phase-controller. The phase-controller has two large SC islands\nwith a controllable phase difference $\\gamma$. Using large SC islands for\nphase-controllers reduces quantum fluctuations in the SC phase.\n\nIn this section, we consider two approaches to building a phase-controller\nusing Josephson junctions (JJs). The phase difference $\\gamma$ between the two\nlarge SC islands can be controlled by either the external magnetic flux\n$\\gamma=\\gamma\\left(  \\Phi_{x}\\right)  $ or the electric current\n$\\gamma=\\gamma\\left(  I\\right)  $, as detailed below.\n\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=8.7cm]{fig_PhaseController.pdf}\n\\end{center}\n\\caption[fig:PhaseController]{The phase-controllers can establish desired\nphase difference $\\gamma$ between SC islands $a$ and $b$. For example, (a) the\nflux phase-controller with external magnetic flux $\\Phi_{x}$, and (b) the\ncurrent phase-controller with external electric current $I$.}%\n\\label{fig:PhaseController}%\n\\end{figure}\n\n\\subsection{Flux phase-controller}\n\nThe \\emph{flux phase-controller} uses an rf SQUID loop that is interrupted by\na single Josephson junction (Fig.~\\ref{fig:PhaseController}a). We may change\nthe external magnetic flux enclosed by the loop, to induce the desired phase\ndifference between the two SC\\ islands, $a$ and $b$. The rf SQUID loop has\ninductance $L$, and the JJ has Josephson coupling energy $E_{J}$ and\ncapacitive charging energy $E_{C}$ ($=e^{2}\/2C$). The Hamiltonian for the flux\nphase-controller is%\n\\begin{equation}\nH=U+T,\n\\end{equation}\nwhere the potential energy is%\n\\begin{equation}\nU=\\frac{1}{2L}\\left(  \\Phi_{x}-\\frac{\\Phi_{0}}{2\\pi}\\gamma\\right)  ^{2}%\n+E_{J}\\left(  1-\\cos\\gamma\\right)\n\\end{equation}\nwith $\\Phi_{x}$ for the external flux enclosed by the loop and $\\gamma$ for\nthe gauge invariant phase difference across the junction, and the capacitive\ncharging energy is%\n\\begin{equation}\nT=\\frac{1}{2}CV^{2}%\n\\end{equation}\nwith the voltage $V=\\frac{\\Phi_{0}}{2\\pi}\\frac{d\\gamma}{dt}$ and the effective\nmass $m_{eff}=C\\left(  \\frac{\\Phi_{0}}{2\\pi}\\right)  ^{2}.$\n\nWe find that the potential minimum satisfies the condition%\n\\begin{equation}\n\\Phi_{x}=\\frac{\\Phi_{0}}{2\\pi}\\gamma+E_{J}L\\frac{2\\pi}{\\Phi_{0}}\\sin\\gamma,\n\\end{equation}\nThis expression can be used to determine $\\gamma$ as a function of $\\Phi_{x}$.\nFor $L\\rightarrow0$, we have $\\gamma=-2\\pi\\frac{\\Phi_{x}}{\\Phi_{0}}$. For\nsufficiently small $L$ (satisfying $L<\\frac{\\Phi_{0}^{2}}{4\\pi^{2}E_{J}}$),\nthis relation is still single valued for all $\\gamma$. Therefore, we can\ndeterministically control the gauge invariant phase difference $\\gamma$ by\napplying an appropriate $\\Phi_{x}$.\n\nWe then consider the perturbation around the potential minimum and obtain the\nplasma frequency $\\omega_{p}\\approx\\left(  LC\\right)  ^{-1\/2}$ for $L\\ll\n\\frac{\\Phi_{0}^{2}}{4\\pi^{2}E_{J}}$. The quantum fluctuations of $\\gamma$\naround the potential minimum has characteristic scale\n\\begin{equation}\n\\zeta_{\\mathrm{flux}}=\\sqrt{\\frac{\\hbar}{m_{eff}\\omega_{p}}}\\approx2\\sqrt{\\pi\n}\\left(  \\frac{E_{c}}{E_{L}}\\right)  ^{1\/4}.\n\\end{equation}\n\n\n\\subsection{Current phase-controller}\n\nAlternatively, we may also control the phase differences via the external\ncurrent $I$ through a Josephson junction (Fig.~\\ref{fig:PhaseController}a).\nThe Hamiltonian for such a \\emph{current phase-controller} is%\n\\begin{equation}\nH=U+T,\n\\end{equation}\nwhere the potential energy is%\n\\begin{equation}\nU=-I\\frac{\\Phi_{0}}{2\\pi}\\gamma+E_{J}\\left(  1-\\cos\\gamma\\right)\n\\end{equation}\nwith $\\gamma$ for the gauge invariant phase difference across the junction,\nand the capacitive charging energy is%\n\\begin{equation}\nT=\\frac{1}{2}CV^{2}%\n\\end{equation}\nwith the voltage $V=\\frac{\\Phi_{0}}{2\\pi}\\frac{d\\gamma}{dt}$ and the effective\nmass $m_{eff}=C\\left(  \\frac{\\Phi_{0}}{2\\pi}\\right)  ^{2}$.\n\nAs long as $I<I_{c}\\equiv\\frac{2\\pi}{\\Phi_{0}}E_{J}$, we have the potential\nminimum at\n\\begin{equation}\n\\gamma=\\sin^{-1}I\/I_{c}.\n\\end{equation}\nWe then consider the perturbation around the potential minimum and obtain the\nplasma frequency $\\omega_{p}\\approx\\left(  \\frac{2eI_{c}\\cos\\gamma^{\\ast}%\n}{\\hbar C}\\right)  ^{1\/2}$. The quantum fluctuations of $\\gamma$ around the\npotential minimum has characteristic scale\n\\begin{equation}\n\\zeta_{\\mathrm{current}}=\\sqrt{\\frac{\\hbar}{m_{eff}\\omega_{p}}}=\\left(\n\\frac{8}{\\cos\\gamma^{\\ast}}\\frac{E_{c}}{E_{J}}\\right)  ^{1\/4}.\n\\end{equation}\n\n\n\\subsection{Comparison between flux and current phase-controllers}\n\nBoth flux and current phase-controllers enable us to reliably induce phase\ndifference between two SC islands. We compare the two phase-controllers in the\nfollowing aspects:\n\n(1) Tunable phase rage: A single flux phase-controller can create all desired\nphase differences in the range $\\left[  -\\pi,\\pi\\right]  $. In contrast, a\nsingle current phase-controller can only create phase difference in the range\n$\\left(  -\\pi\/2,\\pi\/2\\right)  $. However, by using two or more current\nphase-controllers in series, it is possible to create arbitrary phase differences.\n\n(2) Quantum fluctuations: Both controllers have similar scaling for the\nquantum fluctuations, which scales as $\\left(  E_{c}\/E_{L,J}\\right)  ^{1\/4}$.\nIn order to minimize the quantum fluctuations, we may use the JJ loops with a\nsmall inductance (i.e., $E_{L}>E_{J}$) for flux phase-controller, while we may\nuse the JJ loops with large Josephson coupling energy $E_{J}$ for current\nphase-controller. Note that for current phase-controller the quantum\nfluctuations becomes unfavorably large when $\\gamma\\approx\\pi\/2$, which can be\novercome by using two or more controllers in series to reduce the fluctuations.\n\n\\subsection{Parameters}\n\nWe may estimate the quantum fluctuations for practical devices. According to\nthe experimental parameters of the large Josephson junctions \\cite{Martinis02}%\n: the charging energy $E_{c}\\equiv e^{2}\/2C=0.15$ mK and the Josephson\ncoupling energy $E_{J}=I_{c}\\Phi_{0}\/2\\pi=500$ K, which gives us\n$\\zeta_{\\mathrm{current}}\\approx(8\\ast\\frac{0.00015}{500})^{1\/4}=0.04$. It is\nalso possible to build a SQUID loop with very small inductive energy\n$E_{L}\\equiv\\Phi_{0}^{2}\/2L=645$ K \\cite{Friedman00}, and we can obtain\n$\\zeta_{\\mathrm{flux}}\\approx2\\sqrt{\\pi}\\ast(\\frac{0.00015}{645})^{1\/4}%\n\\approx0.08$. By further increasing the junction area, we may further decrease\n$E_{c}$ and increase $E_{J}$, which should give us more reduced quantum\nfluctuations from the phase-controller.\n\n\\section{Brief Derivation for Energy Splitting $E\\left(  \\varepsilon\\right)\n$}\n\nWe now briefly derive the energy splitting function $E\\left(  \\varepsilon\n\\right)  $, which is a highly non-linear function of $\\varepsilon$. The\nderivation mostly follows Ref.~\\cite{FuL08}.\n\nWe start with the effective\\ Hamiltonian \\cite{FuL08}%\n\\begin{equation}\nH^{\\mathrm{STIS}}=-iv_{F}\\tau^{x}\\partial_{x}+\\delta_{\\varepsilon}\\tau\n^{z}\\mathrm{,\\ }%\n\\end{equation}\nwith $\\delta_{\\varepsilon}=-\\Delta_{0}\\sin\\varepsilon\/2$ (differed from\n\\cite{FuL08} by a minus sign, due to a slightly different assignment of SC\nphases). The Hamiltonian can be written as%\n\\begin{equation}\nH^{\\mathrm{STIS}}\\left(  k\\right)  =v_{F}k\\tau^{x}+\\delta_{\\varepsilon}%\n\\tau^{z}=\\left(\n\\begin{array}\n[c]{cc}%\n\\delta_{\\varepsilon} & v_{F}k\\\\\nv_{F}k & -\\delta_{\\varepsilon}%\n\\end{array}\n\\right)  ,\n\\end{equation}\nwhere $k$ is the wave vector in the quantum wire. The eigen-energies for\n$H^{\\mathrm{STIS}}\\left(  k\\right)  $ are $E^{\\pm}\\left(  \\delta_{\\varepsilon\n},k\\right)  =\\pm\\sqrt{\\delta_{\\varepsilon}^{2}+v_{F}^{2}k^{2}}$. For a finite\nSTIS quantum wire with length $L$, it can only support a discretized set of\nwave vectors satisfying the boundary condition \\cite{FuL08}%\n\\begin{equation}\n\\tan kL=-\\frac{v_{F}k}{\\delta_{\\varepsilon}}=kL\/\\Lambda_{\\varepsilon},\n\\label{eq:BC}%\n\\end{equation}\nwith a dimensionless parameter\n\\begin{equation}\n\\Lambda_{\\varepsilon}\\equiv-\\delta_{\\varepsilon}L\/v_{F}.\n\\end{equation}\nFor given $\\Lambda_{\\varepsilon}$, we may solve Eq. (\\ref{eq:BC}) and obtain a\nset of solutions $kL=f_{n}\\left(  \\Lambda_{\\varepsilon}\\right)  $ with index\n$n=0,1,2,\\cdots$ for different bands. The function $f_{n}\\left(  y\\right)  $\nis just the inverse function of $y=x\/\\tan\\left(  x\\right)  $ associated with\nthe $n$th invertible domain.\n\nFor the lowest band, we have\n\\begin{equation}\nE\\left(  \\varepsilon\\right)  \\equiv E^{+}\\left(  \\delta_{\\varepsilon}%\n,k_{0}\\right)  =\\Delta E\\sqrt{\\Lambda_{\\varepsilon}^{2}+f_{0}^{2}\\left(\n\\Lambda_{\\varepsilon}\\right)  },\n\\end{equation}\nwith $\\Delta E=v_{F}\/L$. Note that $kL=f_{0}\\left(  \\Lambda_{\\varepsilon\n}\\right)  $ is purely imaginary for $\\Lambda_{\\varepsilon}\\in\\left(\n1,\\infty\\right)  $, and it is real for $\\Lambda_{\\varepsilon}\\in\\left(\n-\\infty,1\\right)  $. Physically, imaginary $kL$ corresponds to localized MFs\nat the ends of the quantum wire, and real $kL$ indicates delocalized MFs.\nThose higher bands (with $n\\geq1$) are associated with the excitation modes of\nthe quantum wire, with excitation energy at least $\\Delta E$ \\cite{FuL08}.\n\nIn summary, we have%\n\\begin{equation}\n\\frac{E\\left(  \\varepsilon\\right)  }{\\Delta E}=\\sqrt{\\Lambda_{\\varepsilon}%\n^{2}+f_{0}^{2}\\left(  \\Lambda_{\\varepsilon}\\right)  },\n\\end{equation}\nwhich is plotted in Fig~\\ref{fig:Potential}a.\n\n\n\\end{document} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\n\nIn this work, we introduce a unified 3D pipeline for OAR localization-segmentation rather than novel localization or segmentation architectures. The proposed localization framework and the  unified two-step approach for multi-organ segmentation work directly in 3D volumetric data to exploit 3D context information. We have validated our method in a public benchmark data set, which shows consistent improvement in Dice score compared to the baseline, especially for the small structures. Future work includes directly predicting the extension of the bounding boxes along the centroid prediction to discard the need for population statistics.\n\n\n\n\n\\section{Experiments and Discussion}\\label{chapter:experiments}\n\nTo validate the performance of the proposed approach, we perform the segmentation of 16 anatomical structures considered as OAR: lungs, kidneys, liver, spleen, pancreas, aorta, urinary bladder, sternum, thyroid, vertebra, right-, and left-psoas and rectus abdominal muscles.\n\n\n\n\\textbf{Data set:} The data set used in the experiments consists of CT scans from the gold corpus, and silver corpus in the VISCERAL dataset \\cite{Jimenez2016}. 74 CT scans from the silver corpus data are used for training and hyper-parameter tuning. The annotations in this set are automatically labeled by fusing the results of multiple algorithms. 23 CT scans from the gold corpus, which contains manually annotated labels, are used for testing. \n\n\n\n\nAll volumes are resampled to $3mm^3$ and $1mm^3$ for the localization and segmentation stages, respectively. We normalized each volume to zero mean and unit variance.\n\n\nTo evaluate the overall performance of the proposed two-stage approach for multi-organ segmentation, we report the average dice score between the ground truth and the predicted segmentation map. In addition, organ-wise dice scores are reported using box plots.\n\n\\textbf{Optimization:} We train the networks using Adam optimizer with a decaying learning rate initialized at $1e^{-3}$ and $1e^{-5}$ for the multi-variate regression model and segmentation model, respectively. We use a Mini-batch size of 1 in all experiments. The training was continued till validation loss converged. All the experiments are conducted on an NVIDIA Titan Xp GPU with 12GB vRAM.\n\n\n\\textbf{Gaussian Variance:} For the generation of ground-truth Gaussian heat maps, several approaches were tested. Experimental results showed that an isotropic variance of \\(150mm^2\\) gives the best performance. However, larger variances resulted in less precise predictions, while smaller variances tended to cause localization failures, where no strong centroid prediction was generated. Organ-size-specific heat-maps had the tendency to suffer from the same pitfalls, being either too large or too small.\n\n\n\\textbf{Threshold for Centroid Prediction:} We found an empirical value of $\\tau = 0.1$ to obtain the organ centroid prediction using the validation set. Furthermore, we found that heat maps' values  $<0.1$ translate to the absence of organ or detection failure.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\textbf{Quantitative Evaluation:} For the quantitative evaluation, we report our results and compare them with two approaches. Here is the definition of the experiments:\n\\begin{itemize}\n\\renewcommand{\\labelitemi}{$\\bullet$}\n  \\item \\textbf{Baseline (3D U-Net)}: segmentation directly in 3D, this is a 3D version of the original U-Net with weighted cross-entropy for class imbalance.\n  \\item \\textbf{2D+3D Two-step}: 2D localization proposed in \\cite{DeVos2017} followed by 3D U-Net segmentation.\n  \\item \\textbf{3D+3D Two-step}: Proposed 3D localization followed by 3D U-Net segmentation.\n\\end{itemize}\n\n\n\nIn \\textbf{Table} \\ref{tab:total_dice} we report the overall Dice score for every method. We can observe that our proposed approach results in an improvement of $1.4\\%$ compared to the baseline. Similarly, the 2D+3D Two-step approach results in a similar performance to our proposed approach. Nevertheless, notice that the 2D localization proposed by DeVos et al. \\cite{DeVos2017} requires the training of three independent 2D networks in axial, coronal, and sagittal views to generate the final bounding box, while our approach only requires one network.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{ll}\n\\textbf{Method}                              & \\textbf{Dice Score} \\\\\n\\noalign{\\hrule height 0.8pt}\nBaseline (3D U-Net)                           & 0.9120 \\(\\pm\\) 0.026                        \\\\ \\hline\n2D+3D Two-step            & \\textbf{0.9274 \\(\\pm\\) 0.016 }                       \\\\ \\hline\n3D+3D Two-step & \\textbf{0.9260 \\(\\pm\\)  0.018}                       \\\\ \\hline\n\\end{tabular}\n\\caption{ \\small{Quantitative results: the table describes the mean and standard deviation of the global Dice scores for all evaluated models.}}\n    \\label{tab:total_dice}\n\\end{center}\n\\end{table}\nIn \\textbf{Fig.} \\ref{fig:dice_scores} we can observe the organ-wise performance across different evaluated methods using box plots. We can notice that Dice scores for big structures, for instance, lungs and liver, do not improve significantly compared to the baseline. In contrast, the proposed two-step approach provides significant improvement for small structures such as the spleen, urinary bladder, aorta, trachea, thyroid gland, first lumbar vertebra, and muscles. We attribute this to the fact that training a network to segment all structures at once creates class imbalance and small structures are underrepresented. The proposed two-step approach resolves this problem, delivering better performance for these structures.\n\n\\textbf{Qualitative Evaluation:} In \\textbf{Fig} \\ref{fig:segimg}  we show a qualitative comparison between the baseline, the two-step approach with 2D localization, and the proposed approach. We highlight ROIs with green boxes to show regions where our approach improves the segmentation and red boxes to indicate regions where baseline produces false positives. We can observe the correlation of Dice score improvement in the quantitative results with the visual assessment, particularly for the small organs. For instance, the highlighted red boxes show examples of false positives for the trachea and lungs. Moreover, observing the last two columns of \\textbf{Fig} \\ref{fig:segimg}, we can visually recognize that the proposed approach results in more smooth and continuous segmentation compared to both the baselines and 2D+3D approach. This is clearly observed on the boundaries of the lungs' segmentations indicated with yellow arrows in \\textbf{Fig.} \\ref{fig:segimg}.\n\n\n\n\n\n\n\n\n\\begin{figure}[h]\n  \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/img_comparison.png}\n   \n    \\caption{\\small{Qualitative results: ROIs in green indicate regions where the proposed approach improves the segmentation. ROIs in red indicate regions where the baseline produces false positives. Yellow arrows point to regions where the proposed approach presents smooth and continuous segmentations.}}\n  \\label{fig:segimg}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\nLocalization and segmentation of OAR are two necessary pre-processing steps for many automatic medical image analysis tasks, such as radiation therapy planning \\cite{lipkova2019personalized,ezhov2020real} and lesion detection \\cite{bilic2019liver}. Expert radiologists perform these steps manually, making the process time-consuming, costly, and dependent on the level of expertise. These problems call for computer-aided systems for automatic analysis. \n\n\\begin{figure}[ht!]\n  \\centering\n    \\includegraphics[width=1\\linewidth]{figures\/method_overview_two_rows.png}\n    \\caption{\\small{Overview of the proposed approach. The input is a 3D CT scan. In the localization step, our proposed multi-variate regression network together with population statistics predicts organs' bounding boxes. In the segmentation step, segmentation is performed on organ-wise ROIs. The output of the network is the merged segmentation map after aggregation.}}\n  \\label{fig:overview}\n\\end{figure}\nIn recent years, deep convolutional neural networks have been widely adopted in medical imaging applications for both localization \\cite{DeVos2017,ren2015faster,xu2019efficient,navarro2020deep,waldmannstetter2020reinforced} and segmentation \\cite{ronneberger2015u,cciccek20163d,Navarro2019,qasim2020red}. These methods focus either on localization or segmentation. While the detection task prioritizes coarse-level organ representation to estimate relative position, the segmentation task requires fine feature representation to determine organ boundaries. We argue that in a holistic pipeline, prior knowledge about organ location obtained from the localization task is useful for the segmentation model to specialize in organ-specific feature learning solely. In this vein, few approaches aim at jointly solving the two tasks \\cite{he2017mask,zhao2019knowledge, dabiri2020deep,hussain2021cascaded}.\nNevertheless, the model complexity of Mask-RCNN \\cite{he2017mask} makes it unsuitable for 3D volumetric analysis. Likewise, methods similar to the proposed by Zhao et al. \\cite{zhao2019knowledge} require a registration step resulting in a time-consuming approach. On the other hand, \\cite{hussain2021cascaded, dabiri2020deep}  use 2D networks for localization using axial, coronal, and sagittal views without exploiting the 3D information. Other approaches like in \\cite{vesal2020fully} segments the organ of interest twice: one in a coarse resolution using the gradcam for localization and later used a crop around the attention map to segment the organ. In \\cite{liu2019automatic} patches generated from superpixels are classified as candidate regions of the organ of interest in the localization step followed by 2.5D segmentation networks. In contrast, our approach uses one network for centroids prediction and one for organs-specific segmentation.\n\nThe aforementioned two-step approaches are either too computational expensive for medical imaging or lack ability to fully exploit 3D information. In this work, we focus on bringing together 3D localization and segmentation pipelines to fully exploit 3D context information, rather than proposing novel localization or segmentation architectures. With this intention, we propose a holistic two-step approach to efficiently exploit the organ-size invariant relative positional information in a 3D context. Firstly, inspired by \\cite{payer2020coarse}, we formulate the localization problem as a 3D multi-variate regression problem to predict the organs' centroids and hence the bounding boxes. Secondly, we leverage 3D segmentation networks to obtain the final segmentation map around the localized region obtained from the first step. Lastly, we aggregate each organ segmentation, resulting in a multi-organ segmentation map.\n\n\n\n\n\n\n\n\n\n\n\n\\section{METHODOLOGY}\n\\label{chapter:methodology}\nOur proposed holistic approach consists of two stages: the localization and the segmentation stage. We obtain the final segmentation map by aggregating the individual segmentation predictions. Fig \\ref{fig:overview}. shows an overview of the proposed approach. In the rest of the section, we describe the details of each component.\n\n\n\\subsection{Localization}\n\n\\subsubsection{\\textbf{Multi-variate Gaussian Regression}}\nWe formulate the organ localization task as a multi-variate Gaussian regression problem for organs' centroids. These predictions in combination with average organs' bounding boxes enables the exploitation of 3D context information. Formally, we define the input 3D medical image as $\\mathbf{X} \\in \\mathbb{R}^{(h \\times w \\times d)}$. For each organ, we define the ground truth as a Gaussian heat map of the form \n$\\mathbf{y_i}= -e^{{\\parallel x- \\mu_{i} \\parallel}^{2} \/\\ 2 \\sigma^2}$, where $\\mu_{i}$ is the centroid of the $i^{th}$ organ and $\\sigma^2$ controls the variance of the Gaussian. The ground truth annotation results in $\\mathbf{Y} \\in \\mathbb{R}^{(h \\times w \\times d \\times n)}$, where $n$ is the number of organs to be localized.\n\nThe proposed network architecture is inspired by the seminal works in Sekuboyina et al. \\cite{Sekuboyina2018,sekuboyina2020labeling,payer2020coarse} for vertebrae localization and segmentation. We created a 3D version of the authors' approach \\cite{Sekuboyina2018,sekuboyina2020labeling} to work directly in 3D volumetric data, instead of 2D projections, depicted in \\textbf{ Fig.} \\ref{fig:architecture}. In contrast to \\cite{payer2020coarse}, we tailored the 3D localization proposed for vertebrae and adapted it to organs. This architecture allows the network to exploit the 3D context information and efficiently use computational resources for different CT organs. The loss function used for optimization is the $\\mathcal{L}_{2}$ loss:\n\\begin{equation}\n\\mathcal{L}_{2} = \\frac{1}{h\\times w\\times d \\times n} \\sum_{} {\\parallel \\mathbf{Y} - \\tilde{\\mathbf{Y}} \\parallel}^2\n\\end{equation}\nwhere $\\mathbf{\\tilde{Y}}$ stands for the predicted Gaussian heat map. \n\n\n\n\\begin{figure}[h!]\n  \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/network.png}\n    \\caption{\\small{Proposed 3D Multi-variate Gaussian regression network. Each output channel corresponds to one organ heat map.}}\n  \\label{fig:architecture}\n\\end{figure}\n\n\\subsubsection{\\textbf{Bounding Box Generation}}\nTo obtain bounding boxes from the predictions of the multi-variate Gaussian regression model $\\mathbf{\\tilde{Y}} \\in \\mathbb{R}^{(h \\times w \\times d \\times n)}$, the following steps are executed:\n\\begin{itemize}\n\\renewcommand{\\labelitemi}{$\\bullet$}\n  \\item Voxels with Gaussian heat map's values above a threshold $\\tau$ form a region potentially containing the organ centroid. This method is preferred over taking the highest heat-map probability as it is sensitive to outliers and false positives.\n  \\item We define the mid-point of this region as the organ centroid prediction.\n  \\item For each organ, an average bounding box size is computed from the training set.\n  \\item Finally, this bounding box is centered around the aforementioned centroid prediction plus $v$ voxels in each direction resulting in the final box. Adding this margin ensures the organ is contained inside the box.\n\\end{itemize}\n\n\n\n\n\n\n\n\\subsection{Organ-wise 3D Segmentation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFrom the predicted organ's bounding boxes, we crop organ-wise ROIs from the CT scans. Then, we use these ROIs to train organ-specific 3D segmentation networks. This model segregation enables us to customize the networks to learn organ-specific feature representations. Furthermore, the localization stage endows the networks to process high-resolution 3D information efficiently. In this work, we adopted 3D U-Nets \\cite{cciccek20163d} to generate the organ segmentation maps. \n\n\n\nWe use a combination of binary cross-entropy and Dice as loss function:\n\\begin{equation}\n\\mathcal{L} = \\underbrace{- \\sum_{x}^{} {g_{}{(x)} \\log{p_{}{(x)}}}}_\\text{Cross-Entropy Loss} - \\underbrace{{\\frac{2 \\sum_{x}^{} {p_{}{(x)}} g_{}{(x)}} {\\sum_{x}^{} {p_{}^{2}{(x)}} + \\sum_{x}^{} {g_{}^{2}{(x)}} } }}_\\text{Dice Loss}\n\\end{equation}\nwhere $p(x)$ is the probability of a pixel in location $x$ to belong to the foreground class and $g(x)$ is the ground truth for the corresponding pixel.\n\n\\begin{figure*}[h]\n  \\centering\n    \\includegraphics[width=0.7\\linewidth]{figures\/dice_scores.png}\n    \\caption{\\small{Organ-wise box plots comparison across all tested methods. A constant improvement of  Dice score is obtained with the proposed approach compared to the baseline.}}\n  \\label{fig:dice_scores}\n\\end{figure*}\nWe aggregate the segmented crops from every organ to generate the final segmentation map, bringing the crops back to the original image space. During this aggregation process, there may exist overlap between organs that are close to each other. Those overlapping voxels, for which segmentation networks disagree, are set to the predictions with the highest probability value.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzoght b/data_all_eng_slimpj/shuffled/split2/finalzzoght
new file mode 100644
index 0000000000000000000000000000000000000000..d2ed777465922eab745426193891f49f04b76818
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzoght
@@ -0,0 +1,5 @@
+{"text":"\n\\section{Introduction}\n\\label{introduction}\n\\textls[-25]{{Since the first direct observation of gravitational waves (GWs), carried out by the interferometers of the LIGO\/VIRGO collaboration \\cite{abbott2016}, gravitational radiation has proved to be a precious messenger of information for astrophysics, allowing not only to confirm the predictions of general relativity \\cite{einstein1916,depa2022} but also to study extraordinary phenomena, such as the merger of neutron stars \\cite{abbott2017}, and to discover several black holes \\cite{abbott2020}. However, gravitational astronomy is still in its infancy and to date, only the tip of the iceberg of the GW spectrum has been scratched. Additionally, low-frequency GWs are not accessible by ground-based interferometers, which are currently only sensitive to high-frequency GWs (in the range 10--$10^{4}$ Hz) emitted in the final stages of the coalescence of compact objects \\cite{maggiore2008a}. To explore the low-frequency side of the GW spectrum (in the range $10^{-5}$--1 Hz), space-based gravitational interferometers, such as Laser Interferometer Space Antenna (LISA) \\cite{lisa2017}, are required and, in addition, for very low-frequency GWs (in the range $10^{-10}$--$10^{-6}$ Hz), pulsar timing arrays (PTAs) must be used, which are currently the only instruments sensitive to them \\cite{moore2015}.}\n\nPTAs are arrays of extremely regular millisecond pulsars (MSPs), constantly monitored by radio telescopes to measure the variations in the times of arrival (ToAs) of the emitted radio pulses, commonly referred to as timing residuals (TRs). Since GWs can influence the radio pulse ToAs, their signature can be found by studying the TRs \\cite{estabrook1975,sazhin1978,detweiler1979}, once cleaned from other effects that can affect them. At present, the main purpose of PTAs is to reveal the gravitational wave background (GWB) due to the superposition of GWs emitted by the supermassive black hole binary systems (SMBHBs) that, according to the current cosmological model, populate the Universe \\cite{maggiore2008b}. The GWB should induce spatially correlated TRs in all the MSPs of PTAs \\cite{mashhoon1982}. Since this correlation should take the form described by the Hellings and Downs function \\cite{hellings1983}, the latter is considered the smoking gun for the GWB \\cite{maiorano2021a}. Although from the data collected by the main PTA collaborations (the European Pulsar Timing Array (EPTA) \\cite{desvignes2016}, the Indian Pulsar Timing Array (InPTA) \\cite{joshi2018}, the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) \\cite{arzoumanian2018}, and the Parkes Pulsar Timing Array (PPTA) \\cite{reardon2016}, which join their efforts as the International Pulsar Timing Array (IPTA) \\cite{verbiest2016}) it has been possible to obtain strong constraints on both the frequency and the amplitude of the GWB, and despite the emergence from them of a common red noise compatible with that expected for the GWB, the smoking gun is still missing, and the first GW detection by PTAs has not yet been claimed. Furthermore, the first detection of continuous GW emitted by a SMBHB seems even more distant, as it may be subject to knowledge of the GWB.\n\n\\textls[-25]{This article is motivated by the exigency to look for additional instruments and tools for detecting very low-frequency GWs \\cite{maiorano2021b}. This necessity is made more urgent by recent research works, which show the limits and criticalities of the investigation principles adopted up to now~\\cite{zic2022}. For this purpose, the potential of the surfing effect for the detection of the continuous GWs emitted by a SMBHB has been analyzed. Although the surfing effect is present in some works on PTAs, the literature to date is lacking in articles dedicated to its discussion in the context of general relativity, and in which a possible concrete application is presented. For this reason, this paper is structured as follows: Section \\ref{sec:1} is dedicated to a detailed description of the surfing effect, Section \\ref{sec:2} describes an application of the surfing effect to the SMBHB candidate \\linebreak PKS 2131--021, in Section \\ref{sec:3} the signatures of the GWs possibly emitted by the SMBHB candidate \\linebreak PKS 2131--021, have been searched in the averaged narrowband and wideband post-fit whitened TRs obtained by NANOGrav collaboration on the MSP J2145--0750, and finally, the main conclusions are presented Section \\ref{sec:4}.}}\n\n\\section{An Overview on the Surfing Effect}\n\\label{sec:1}\nThe surfing effect occurs when the angle $\\theta$ seen by an observer on Earth between the travel direction of the radio pulses emitted by a MSP and the travel direction of the GWs emitted by a single GW source is sufficiently small. In this case, the radio pulses behave as if they were surfing the GWs, producing an increased value of the pulsar TRs. That can be shown by considering the function $R\\left(t,n_i\\right)$ which describes the pulsar TRs induced by the GWs emitted by a single source \\cite{estabrook1975, sazhin1978, detweiler1979}:\n\\begin{equation}\n\\label{residualsfunction}\nR\\left(t,n_i\\right)=\\sum_{A=+,\\times}F^A\\left[r^A_e\\left(t,n_i\\right) -r^A_p\\left(t,n_i\\right)\\right]\n\\end{equation}\n\\textls[-25]{where $t$ is the time, $n_i$ is the versor oriented toward the source of the GWs\\endnote{Sometimes, in the literature, the versor oriented along the travel direction of the GWs $\\Omega_i$ is considered in place of $n_i$. \\linebreak Since the two versors point toward opposite directions, the relation between them is $\\Omega_i=-n_i$.}, $A$ is the GW polarization state index, which is $+$ in the case of the plus-polarization state, and $\\times$ in the case of the cross-polarization state, and $F^A$ is the antenna pattern function, defined as\\endnote{In this paper, the Einstein notation, which indicates the sum over repeated indices, has been adopted.}:\n\\begin{equation}\n\\label{antennafunction}\nF^A=\\frac{1}{2}\\frac{p^ip^je_{ij}^A}{(1-n_ip^i)}\n\\end{equation}\n\nHere, $p^i$ is the versor oriented toward the MSP, and $e_{ij}^A$ is the GW polarization tensor:\n\\begin{equation}\n\\label{polarizationplus}\ne_{ij}^+=\\begin{pmatrix}\n1& 0& 0 \\\\ \n0& -1& 0 \\\\ \n0& 0& 0 \n\\end{pmatrix}\n\\end{equation}\n\\begin{equation}\n\\label{polarizationscross}\ne_{ij}^\\times=\\begin{pmatrix}\n0& 1& 0 \\\\ \n1& 0& 0 \\\\ \n0& 0& 0 \n\\end{pmatrix}\n\\end{equation}\n\n\\textls[-35]{Moreover, $r^A_e\\left(t,n_i\\right)$ and $r^A_p\\left(t,n_i\\right)$ are the Earth and the pulsar terms defined respectively by\\endnote{In this paper, the geometrical units c=G=1 have been adopted.}:}\n\\begin{equation}\n\\label{earthterm}\nr^A_e\\left(t,n_i\\right)=\\int^t_0\\dd{t}h^A_e\\left(\\omega_e t\\right)\n\\end{equation}\n\\begin{equation}\n\\label{pulsarterm}\nr^A_p\\left(t,n_i\\right)=\\int^t_0\\dd{t}h^A_p\\left(\\omega_p t-\\omega_p t_p\\left(1-n_ip^i\\right)\\right)\n\\end{equation}\nwhere $\\omega_e$ and $\\omega_p$ are the GW angular frequency, evaluated at the Earth and MSP positions, respectively, $t_p$ is the time distance from the MSP to the Earth, and $h^A_e\\left(\\omega_e t\\right)$ and $h^A_p\\left(\\omega_p t-\\omega_p t_p\\left(1-n_ip^i\\right)\\right)$ are the perturbations on the flat space-time metric induced by the GWs, evaluated at the Earth and MSP positions, respectively\\endnote{For an exhaustive derivation of Equation \\eqref{residualsfunction} see ref. \\cite{maggiore2008b}.}.\n\nEquation \\eqref{residualsfunction} is particularly useful to describe the pulsar TRs induced by the GWs emitted by a circularized SMBHBs in the continuous emission regime. In the literature, this assumption is often made in order to simplify the description of the principles on which PTAs are based. That is reasonable because, according to the most widely accepted models, SMBHBs form during the collision of galaxies by dynamic friction, which, as it is widely known, tends to circularize their orbits \\cite{bonetti2020}.\n\nIn this case, it is convenient to adopt a coordinate system $Oxyz$ with its origin on the Earth\\endnote{Such a choice would induce spurious TRs, with a period of one year, due to the motion of the Earth relative to the Solar System Barycenter. Although accounting for that is crucial from the experimental point of view and is done during the data processing phase, this issue can be safely ignored in this theoretical context.}, so that:\n\\begin{equation}\n\\label{nversor}\nn_i\\equiv\\left(0,0,1\\right)\n\\end{equation}\n\\begin{equation}\n\\label{pversor}\np^i\\equiv\\left(\\sin{\\theta_p}\\cos{\\phi_p},\\sin{\\theta_p}\\sin{\\phi_p},\\cos{\\theta_p}\\right)\n\\end{equation}\n\\textls[-35]{where $\\theta_p$ is the angle between the versor $p^i$ and the positive $z$-axis, and $\\phi_p$ is the angle between the $x$-axis and the projection on the $xy$-plane of the versor $p^i$ (see Figure \\ref{fig:reference}). With that choice, the product $n_ip^i$, which appears in Equations \\eqref{antennafunction}, \\eqref{earthterm}, and \\eqref{pulsarterm}, is simply $\\cos\\theta_p$.}\n\n\\begin{figure}[H]\n\\hspace{-30pt}\n\\includegraphics[scale=0.65]{reference.png}\n\\caption{Scheme of the adopted $Oxyz$ coordinate system, chosen so that the versor $n_i$ is aligned with the $z$-axis. The black circle pair indicates the SMBHB, pointed by the versor $n_i$, while the white circle indicates the MSP, pointed by the versor $p_i$. In the figure, the distances of the SMBHB and the MSP from the origin $O$, where the observer is placed, are not in scale.}\n\\label{fig:reference}\n\\end{figure}\n\nThe antenna pattern function in Equation \\eqref{antennafunction} can be rewritten in this coordinate system, taking into account the Equations \\eqref{polarizationplus} and \\eqref{polarizationscross}, for both the polarization states:\n\\begin{equation}\n\\label{antennaplus}\nF^+=\\frac{1}{2}\\frac{\\sin^2\\theta_p}{1-\\cos\\theta_p}\\cos(2\\phi_p)\n\\end{equation}\n\\begin{equation}\n\\label{antennacross}\n F^\\times=\\frac{1}{2}\\frac{\\sin^2\\theta_p}{1-\\cos\\theta_p}\\sin(2\\phi_p)\n\\end{equation}\n\nMoreover, the perturbations on the flat space-time metric induced by the GWs evaluated at the Earth and MSP positions are \\cite{perrodin2018}:\n\\begin{equation}\n\\label{hearthterm}\nh^A_e\\left(\\omega_e t\\right)=\\mathcal{F}^A\\mathcal{A}\\sin{\\left(\\omega_e t+\\alpha^A\\right)}\n\\end{equation}\n\\begin{equation}\n\\label{hpulsarterm}\nh^A_p\\left(\\omega_p t-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)\\right)=\\mathcal{F}^A\\mathcal{A}\\sin{\\left(\\omega_p t-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)+\\alpha^A\\right)}\n\\end{equation}\nwhere $\\mathcal{F}^A$ is a factor that accounts for the orbital inclination angle $\\iota$ of the SMBHB \\cite{sesana2010}:\n\\begin{equation}\n\\label{inclinationplus}\n\\mathcal{F}^+=1+\\cos^2{\\iota}\n\\end{equation}\n\\begin{equation}\n\\label{inclinationcross}\n\\mathcal{F}^\\times=-2\\cos{\\iota}\n\\end{equation}\nand $\\mathcal{A}$ is the perturbation amplitude, given by:\n\\begin{equation}\n\\label{amplitude}\n\\mathcal{A}=2\\frac{\\mathcal{M}^{5\/3}}{D}\\left(\\frac{\\omega}{2}\\right)^{2\/3}\n\\end{equation}\nwhere $\\mathcal{M}$ is the chirp mass of the SMBHB, $D$ is the luminosity distance, and $\\omega$ is the angular frequency of GWs, and $\\alpha^A$ is the initial phase, with $\\alpha^+=0$ and $\\alpha^\\times=\\pi\/2$. Note that all the quantities in Equation \\eqref{amplitude} are red-shifted \\cite{perrodin2018}.\n\nThe difference between the Earth term, in Equation \\eqref{earthterm}, and the pulsar term, \\linebreak in Equation \\eqref{pulsarterm}, can be determined for both the polarization states by taking into account Equations \\eqref{hearthterm}--\\eqref{amplitude}, obtaining:\n\\begin{equation}\n\\label{differenceplus}\n\\begin{split}\n&r^+_e\\left(t,n_i\\right) -r^+_p\\left(t,n_i\\right)=-\\frac{\\mathcal{A}}{\\omega_e}\\left(1+\\cos^2{\\iota}\\right)\\left[\\cos{\\left(\\omega_e t\\right)}-1\\right]+\\\\\n&+\\frac{\\mathcal{A}}{\\omega_p}\\left(1+\\cos^2{\\iota}\\right)\\left[\\cos{\\left(\\omega_p t-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)\\right)}-\\cos{\\left(-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)\\right)}\\right]\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\label{differencecross}\n\\begin{split}\n&r^\\times_e\\left(t,n_i\\right) -r^\\times_p\\left(t,n_i\\right)=-\\frac{\\mathcal{A}}{\\omega_e}\\left(-2\\cos{\\iota}\\right)\\left[-\\sin{\\left(\\omega_e t\\right)}\\right]+\\\\\n&+\\frac{\\mathcal{A}}{\\omega_p}\\left(-2\\cos{\\iota}\\right)\\left[-\\sin{\\left(\\omega_p t-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)\\right)}+\\sin{\\left(-\\omega_p t_p\\left(1-\\cos\\theta_p\\right)\\right)}\\right]\n\\end{split}\n\\end{equation}\n\nThen, the full expression of the function $R\\left(t,n_i\\right)$ is found by substituting the Equations \\eqref{antennaplus}, \\eqref{antennacross}, \\eqref{differenceplus} and \\eqref{differencecross} in Equation \\eqref{residualsfunction}. The result obtained above can be simplified by introducing some additional working hypotheses. First of all, the SMBHB can be assumed to be face-on, which means that $\\iota=\\pi\/2$. In this case, as can be seen from Equations \\eqref{inclinationplus} and \\eqref{inclinationcross}, $\\mathcal{F}^+=1$ and $\\mathcal{F}^\\times=0$. Secondly, since only one MSP is under consideration, the angle $\\theta_p$ coincides with the angle $\\theta$, so $\\theta_p=\\theta$, the coordinate system is rotated to have $\\phi_p=0$, and $t_p$ is replaced by $T$ to simplify the notation. In this case, as can be seen by Equations $\\eqref{antennaplus}$ and \\eqref{antennacross}, $F^+=\\sin^2\\theta\\left[2\\left(1-\\cos\\theta\\right)\\right]^{-1}$ and\n$F^\\times=0$. Thirdly, the SMBHB can be assumed to be very far from its coalescence, as most of the SMBHBs observable with PTAs should be. Under this assumption, the variation of the angular frequency of the GWs~\\cite{shapiro1983} during the interval of time $\\Delta t=T\\left(1-\\cos\\theta\\right)$, which denotes the time delay between the Earth epoch and the pulsar epoch \\cite{jenet2004}, can be neglected, so $\\omega_e\\approx\\omega_p$ and the angular frequency of GWs can be denoted by just $\\omega$. Using all these assumptions, one gets:\n\\begin{equation}\n\\label{compactresiduals}\n\\begin{split}\n&R\\left(t,\\theta\\right)=-\\frac{1}{2}\\frac{\\sin^2\\theta}{1-\\cos\\theta}\\frac{\\mathcal{A}}{\\omega}\\left[\\cos{\\left(\\omega t\\right)}-1\\right]+\\\\&-\\frac{1}{2}\\frac{\\sin^2\\theta}{1-\\cos\\theta}\\frac{\\mathcal{A}}{\\omega}\\left[-\\cos{\\left(\\omega t-\\omega T\\left(1-\\cos\\theta\\right)\\right)}+\\cos{\\left(-\\omega T\\left(1-\\cos\\theta\\right)\\right)}\\right]\n\\end{split}\n\\end{equation}\nwhere the dependence on $n_i$ has been replaced by the dependence on $\\theta$. As can be seen from Equation \\eqref{compactresiduals}, the way the function $R\\left(t,\\theta\\right)$ depends on the angle $\\theta$ makes its use for the detection of GWs rather problematic. For example, it is interesting to note that $R\\left(t,\\theta\\right)$ has a discontinuity for $\\theta=0$, which can be removed by placing $R\\left(t,0\\right)\\equiv\\lim_{\\theta\\to 0} R\\left(t,\\theta\\right)=0$. Moreover, for values of $\\omega$ and $T$ of the same order of magnitude of the average angular frequency of the GWs emitted by SMBHB observable by PTAs and of the average time distance of the MSPs included in PTAs, respectively, the function $R\\left(t,\\theta\\right)$ oscillates extremely rapidly with $\\theta$.\n \nSince the function $R(t,\\theta)$ carries all the information about the GWs emitted by the SMBHB, it is crucial to have a clear understanding of its behavior. The easiest way to do this is by considering, instead of $R\\left(t,\\theta\\right)$, the function $S(\\zeta,\\eta)$, which, using the substitutions:\n\\begin{equation}\n\\label{sostitutionzeta}\n\\zeta\\equiv\\chi\\frac{t}{T}\n\\end{equation}\n\\begin{equation}\n\\label{sostitutioneta}\n\\eta\\equiv\\chi\\left(1-\\cos\\theta\\right)\n\\end{equation}\nwhere $\\chi\\equiv\\omega T$, $\\zeta\\in\\left[0,\\infty\\right[$ for $t\\in\\left[0,\\infty\\right[$ and $\\eta\\in\\left[0,2\\chi\\right]$ for $\\theta\\in\\left[0,\\pi\\right]$, can be defined by:\n\\begin{equation}\n\\label{sfunctiondef}\nS\\left(\\zeta,\\eta\\right)\\equiv2\\frac{\\chi}{\\mathcal{A}T}\\left|R\\left(\\zeta,\\eta\\right)\\right|\n\\end{equation}\n\nThe function defined in Equation \\eqref{sfunctiondef} can be written more explicitly as:\n\\begin{equation}\n\\label{sfunction}\nS\\left(\\zeta,\\eta\\right)=\\left(2-\\frac{\\eta}{\\chi}\\right)\\left|\\cos\\zeta-1-\\cos\\left(\\zeta-\\eta\\right)+\\cos(-\\eta)\\right|\n\\end{equation}\n\\textls[-25]{where the first factor in Equation \\eqref{sfunction} has taken out the absolute value since it is always positive in the defined domain. The TRs induced by GWs are enhanced for the values which maximize the function $S\\left(\\zeta,\\eta\\right)$ in Equation \\eqref{sfunction}. The function $S\\left(\\zeta,\\eta\\right)$ is a periodic function characterized by an amplitude that depends only on $\\eta$ through the first linear factor:}\n\\begin{equation}\n\\label{samplitude}\na\\left(\\eta\\right)\\equiv 2-\\frac{\\eta}{\\chi}\n\\end{equation}\nand its shape depends on both the variables $\\zeta$ and $\\eta$ through the second oscillating factor:\n\\begin{equation}\n\\label{soscillating}\nf\\left(\\zeta,\\eta\\right)\\equiv\\left|\\cos\\zeta-1-\\cos\\left(\\zeta-\\eta\\right)+\\cos(-\\eta)\\right|\n\\end{equation}\nso that:\n\\begin{equation}\n\\label{shortsfunction}\nS\\left(\\zeta,\\eta\\right)=a\\left(\\eta\\right)f\\left(\\zeta,\\eta\\right)\n\\end{equation}\n\nEquation \\eqref{shortsfunction} allows to find the global maximum of the function $S\\left(\\zeta,\\eta\\right)$ by studying the maxima of the functions $a\\left(\\eta\\right)$ and $f\\left(\\zeta,\\eta\\right)$. The function $a\\left(\\eta\\right)$ is maximized when: \n\\begin{equation}\n\\label{firstfactor}\na\\left(\\eta\\right)=2\\text{ for }\\eta=0\n\\end{equation}\nthat occurs when:\n\\begin{equation}\n\\label{surfingeffect}\n\\theta=0\n\\end{equation}\n while the function $f\\left(\\zeta,\\eta\\right)$ is maximized when: \n\\begin{equation}\n\\label{secondfactor}\nf\\left(\\zeta,\\eta\\right)=4\\text{ for }\\left(\\zeta,\\eta\\right)=\\left(\\pi+2\\pi n_\\zeta,\\pi+2\\pi n_\\eta\\right)\n\\end{equation}\nthat occurs when:\n\\begin{equation}\n\\label{timeposition}\nt=\\frac{\\pi T}{\\chi}\\left(1+2n_\\zeta\\right)\\equiv t_{n_\\zeta}\n\\end{equation}\n\\begin{equation}\n\\label{angularposition}\n\\theta=\\arccos\\left(1-\\frac{\\pi}{\\chi}\\left(1+2n_\\eta\\right)\\right)\\equiv\\theta_{n_\\eta}\n\\end{equation}\n\nTherefore, the results in Equations \\eqref{surfingeffect} and \\eqref{angularposition} imply that, for each value of $t$, the global maximum of the function $R\\left(\\zeta,\\eta\\right)$ can be found placing\n$n_{\\eta}\\equiv 0$. Then, since for $n_{\\eta}= 0$ results $\\theta_0\\ll 1$, this property is referred to as surfing effect, and $\\theta_0=\\arccos\\left(1-\\pi\/\\chi\\right)$ is the surfing effect angle.\n\n\\section{An Application of the Surfing Effect to the Case of the Supermassive Black Hole Binary Candidate PKS 2131--021}\n\\label{sec:2}\nThe existence of SMBHBs is not only predicted by most of the hierarchical structure formation models \\cite{sesana2013} but also supported by observational evidence. In some rare cases, the inferred orbital parameters suggest that, if the data analysis is correct, the SMBHB should emit GWs observable with PTAs. In these cases, as has already been conducted with the SMBHB candidate in the Seyfert galaxy $3$C $66$B, the data collected by PTAs can be used to rule out the GW emission \\cite{jenet2004,depa2004,arzoumanian2020a}.\n\nAn interesting recent study on the blazar PKS 2131--021 opens the possibility for an application of the surfing effect \\cite{oneill2022}. In fact, according to the data analysis of the blazar PKS 2131--021 radio emission, it can be identified as an SMBHB candidate located at a redshift $z=1.285$ \\cite{wu2007}, with an observed orbital period of $1760.4^{+5.3}_{-5.3}$ d and a non-red-shifted chirp mass $\\lesssim 5.4 \\times 10^9 M_\\odot$ (see ref. \\cite{oneill2022}). Adopting the most recent values for cosmological constants determined by the Planck collaboration \\cite{planck2020}, the luminosity distance of the SMBHB candidate is $D=9.2$ Gpc, while the angular frequency and the amplitude of the expected GW emission are $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ Hz and $\\mathcal{A}\\lesssim 2.4\\times 10^{-15}$, respectively.\n\nAn optimal strategy to confirm these results, by taking advantage of the surfing effect, is to perform a single-pulsar search of continuous GWs focusing on the MSP with the smallest angular distance from the SMBHB candidate PKS 2131--021 among the ones currently included in PTAs. For this reason, the best MSP is the MSP J2145--0750, which lies at the angle $\\theta=0.1156$, corresponding to about $6.6234^{\\circ}$, from the SMBHB candidate PKS 2131--021 \\cite{gaia,deller2019}. The MSP J2145--0750 is a well-studied MSP, lying in a binary system with a white dwarf, at a distance of $0.62^{+0.00}_{-0.02}$ kpc \\cite{deller2019} from the Earth and timed for about $12.5$~years by the NANOGrav collaboration \\cite{alam2021a, alam2021b}. The choice of the MSP J2145--0750 is also convenient because it allows neglecting the GW angular frequency variation between the Earth and the pulsar terms due to the SMBHB PKS 2131--021 orbital evolution. In fact, by considering the equation describing the GW angular frequency variation \\cite{shapiro1983}:\n\\begin{equation}\n\\label{frequencyevolution}\n\\dot{\\omega}=\\frac{12}{5}2^{1\/3}\\mathcal{M}^{5\/3}\\omega^{11\/3}\\left(1+z\\right)\n\\end{equation}\nand substituting in the integral of Equation \\eqref{frequencyevolution} the time delay between the Earth epoch and the pulsar epoch, which is $\\Delta t = 13.5\\pm 0.4$ yr, the GW angular frequency variation between the Earth and the pulsar terms turns out to be $\\Delta\\omega\\lesssim0.042\\times 10^{-8}$ Hz.\n\nWith all these data in hand, Equation \\eqref{angularposition} can be used to determine the function $S\\left(\\zeta,\\eta\\right)$ and then to find out the surfing effect angle (see Figure \\ref{fig:sfunction}).\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.85]{sfunction.png}\\hfill\n\\caption{Plot of the $S\\left(\\zeta,\\eta\\right)$ function relative to the SMBHB candidate PKS 2131--021 and the MSP J2145--0750. The axes on the plane indicate the values assumed by the $\\zeta$ and $\\eta$ variables defined in Equations \\eqref{sostitutionzeta} and \\eqref{sostitutioneta}, respectively, expressed in radians. The vertical axis and the color, described by the color bar, indicate the values assumed by $S\\left(\\zeta,\\eta\\right)$, defined in Equation \\eqref{sfunctiondef}, dimensionless.}\n\\label{fig:sfunction}\n\\end{figure}\nOnce $t$ has been chosen to satisfy Equation \\eqref{timeposition}, by setting, for example, $n_\\zeta\\equiv0$, it is possible to define the function $R\\left(\\theta\\right)\\equiv R\\left(t_0,\\theta\\right)$, useful for verifying whether the angular separation $\\theta$ between the MSP and the SMBHB candidate is such as to maximize the TRs. In this case, $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ Hz and $T=2024^{+16}_{-65}$ yr, therefore $\\chi=5274^{+58}_{- 186}$, and at the angle $\\theta_0=0.0345$, corresponding to about $1.9778^{\\circ}$, one has $R\\left(\\theta_0\\right)\\lesssim 0.116$ $\\upmu$s. Even if the actual angle between the SMBHB candidate PKS 2131--021 and the MSP J2145--0750, which is $\\theta=0.1156$, is larger than $\\theta_0$, it can be found that this is very close to $\\theta_5=0.1145$, which identifies the sixth peak of the function $R\\left(\\theta\\right)$. In fact, for $\\theta=0.1156$ results $R\\left(\\theta\\right)\\lesssim0.104$ $\\upmu$s (see Figures \\ref{fig:seangle} and \\ref{fig:seanglezoom}).\n\\begin{figure}[H]\n\\hspace{-5pt}\n\\includegraphics[scale=0.53]{seangle.png}\\hfill\n\\caption{{Plot} \n of the $R\\left(\\theta\\right)$ function relative to the SMBHB candidate PKS 2131--021 and the MSP J2145--0750. The horizontal axis indicates the values assumed by the $\\theta$ variable, expressed in radians. The vertical axis indicates the values assumed by $R\\left(\\theta\\right)$, expressed in nanoseconds. The orange dot indicates the value assumed by the $R\\left(\\theta\\right)$ function when $\\theta$ coincides with the actual angle between the SMBHB candidate PKS 2131--021 and the MSP J2145--0750, which is $\\theta=0.1156$.}\n\\label{fig:seangle}\n\\end{figure}\n\\begin{figure}[H]\n\\hspace{-5pt}\n\\includegraphics[scale=0.53]{seanglezoom.png}\\hfill\n\\caption{Zoomed plot of the $R\\left(\\theta\\right)$ function relative to the SMBHB candidate PKS 2131--021 and the MSP J2145--0750. The horizontal axis indicates the values assumed by the $\\theta$ variable, expressed in radians. The vertical axis indicates the values assumed by the $R\\left(\\theta\\right)$ function, expressed in nanoseconds. The orange dot indicates the value assumed by $R\\left(\\theta\\right)$ when $\\theta$ coincides with the actual angle between the SMBHB candidate PKS 2131--021 and the MSP J2145--0750, which is $\\theta=0.1156$.}\n\\label{fig:seanglezoom}\n\\end{figure}\nWe emphasize that this is a lucky coincidence and allows one to say that if the chirp mass of the SMBHB candidate PKS 2131--021 coincides with the estimated upper limit, the TRs induced by the GWs emitted by the SMBHB candidate PKS 2131--021 on the MSP J2145--0750 might be observable even at the current PTA sensitivity level due to the surfing effect. In addition, as can be easily deduced from the Figures \\ref{fig:seangle} and \\ref{fig:seanglezoom}, if $\\theta$, whether small or large, is in a node of the function $R\\left(\\theta\\right)$, where it vanishes, in the TRs of the considered MSP there will be no trace of the influence of GWs. Eventually, it is necessary to stress the fact that the larger is $\\theta$, the higher the precision required for $\\chi$. In fact, as it is shown in \\mbox{Figure \\ref{fig:seanglezoomerror}}, for the same value of $\\theta$ the function $R\\left(\\theta\\right)$ can assume sensibly different values within the uncertainty on $\\chi$. However, if $\\theta$ is in the first peak, the role played by the error on $\\chi$ is marginal.\n\\begin{figure}[H]\n\\hspace{-5pt}\n\\includegraphics[scale=0.53]{seanglezoomerror.png}\\hfill\n\\caption{Comparative zoomed plot of the $R\\left(\\theta\\right)$ function relative to the SMBHB candidate PKS 2131--021 and the MSP J2145--0750. The horizontal axis indicates the values assumed by the $\\theta$ variable, expressed in radians. The vertical axis indicates the values assumed by the $R\\left(\\theta\\right)$ function, expressed in nanoseconds. The green and red dotted lines indicate the values assumed by the $R\\left(\\theta\\right)$ function for $\\chi=5274-186$ and $\\chi=5274+58$, respectively. The orange dot indicates the value assumed by $R\\left(\\theta\\right)$ when $\\theta$ coincides with the actual angle between the SMBHB candidate PKS 2131--021 and the MSP J2145--0750, which is $\\theta=0.1156$. The green and red dots indicate the values assumed by $R\\left(\\theta\\right)$ when $\\theta$ coincides with the actual angle between the SMBHB candidate PKS 2131--021 and the MSP J2145--0750, which is $\\theta=0.1156$, for $\\chi=5274-186$ and $\\chi=5274+58$, respectively.}\n\\label{fig:seanglezoomerror}\n\\end{figure}\n\n\\section{A Look to NANOGrav Data}\n\\label{sec:3}\nThe NANOGrav collaboration has recently published the $12.5$ yr Data Set,\nwhich is publicly available (see ref. \\cite{nanoweb}), making it possible to examine the TRs of the MSP J2145--0750. In particular, the datasets useful for searching GWs are the averaged narrowband \\cite{alam2021a} and wideband \\cite{alam2021b} post-fit whitened TRs (see Figures \\ref{fig:narrow1} and \\ref{fig:wide1}) since they are characterized by a low weighted root mean squared value $\\sigma_w$, which has been evaluated by using the following definition:\n\\begin{equation}\n\\label{wrms}\n\\sigma_w = \\sqrt{\\frac{\\sum_{i}^n{\\frac{\\left(x_i-\\Bar{x}\\right)^2}{\\sigma_i^2}}}{\\sum_{i}^n(\\frac{1}{\\sigma_i^2})}}\n\\end{equation}\nwhich is valid for any set of quantities $x_1, x_2, \\ldots, x_n$ characterized by the errors $\\sigma_1, \\sigma_2, \\ldots, \\sigma_n $ and by an average $\\Bar{x}$. In this case, using Equation \\eqref{wrms} it has been found $\\sigma_w=0.333$ $\\upmu$s for narrowband, and $\\sigma_w=0.276$ $\\upmu$s for wideband.\n\\begin{figure}[H]\n\\hspace{-5pt}\n\\includegraphics[scale=0.53]{narrow1.png}\\hfill\n\\caption{Plot of the averaged narrowband post-fit whitened TRs of the MSP J2145--0750, which are from the $12.5$ yr Data Set published by the NANOGrav collaboration. The horizontal axis indicates the time, expressed in years. The vertical axis indicates the averaged narrowband post-fit whitened TRs, expressed in~nanoseconds.}\n\\label{fig:narrow1}\n\\end{figure}\\vspace{-6pt}\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{wide1.png}\\hfill\n\\caption{Plot of the wideband post-fit whitened TRs of the MSP J2145--0750, which are from the $12.5$ yr Data Set published by the NANOGrav collaboration. The horizontal axis indicates the time, expressed in years. The vertical axis indicates the wideband post-fit whitened TRs, expressed in~nanoseconds.}\n\\label{fig:wide1}\n\\end{figure}\nIn Figures \\ref{fig:narrow1} and \\ref{fig:wide1}, the origin of time is at Modified Julian Date $53267$, corresponding to 19 September $2004$, from which the observation started. Equation \\eqref{compactresiduals} implies that the TRs induced by the GWs emitted by a SMBHB are periodic, with an angular frequency that is the same as the angular frequency $\\omega$ of the GWs. This periodic signature can be searched in TRs using periodic analysis algorithms, such as the Lomb-Scargle (LS) algorithm \\cite{lomb1976,scargle1982}.\n\nThe LS periodograms of the TRs plotted in Figures \\ref{fig:narrow1} and \\ref{fig:wide1} have been obtained and plotted in Figures \\ref{fig:narrow2} and \\ref{fig:wide2}, respectively.\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{narrow2.png}\\hfill\n\\caption{Plot\nof the LS periodogram of the averaged narrowband post-fit whitened TRs of the MSP J2145--0750. The black-dotted lines indicate the angular frequencies found by the LS algorithm. The red-dotted line indicates the angular frequency $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ expected for the GWs possibly emitted by the SMBHB candidate PKS 2131--021. The gray shaded area indicates uncertainty on $\\omega$ at the $3\\sigma$ confidence level. The horizontal axis indicates the GW angular frequency, expressed in Hertz. The vertical axis indicates the normalized amplitude, dimensionless.}\n\\label{fig:narrow2}\n\\end{figure}\n\\vspace{-12pt}\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{wide2.png}\\hfill\n\\caption{Plot\nof the LS periodogram of the wideband post-fit whitened TRs of the MSP J2145--0750. The black-dotted lines indicate the angular frequencies found by the LS algorithm. The red-dotted line indicates the angular frequency $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ expected for the GWs possibly emitted by the SMBHB candidate PKS 2131--021. The gray shaded area indicates uncertainty on $\\omega$ at the $3\\sigma$ confidence level. The horizontal axis indicates the GW angular frequency, expressed in Hertz. The vertical axis indicates the normalized amplitude, dimensionless.}\n\\label{fig:wide2}\n\\end{figure}\nIn both cases (see Figures \\ref{fig:narrow2} and \\ref{fig:wide2}), among the periodicities with the highest normalized amplitude, the LS algorithm finds one associated with an angular frequency close to \\linebreak $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ Hz, expected for GWs possibly emitted by the SMBHB candidate PKS 2131--021. Specifically, it finds $\\omega=8.006\\times 10^{-8}$ Hz in the case of the narrowband dataset and $\\omega=8.0139\\times 10^{-8}$ Hz in the case of the wideband dataset. At this point, a shuffling test was conducted to verify that these values are not artifacts and emerge from true periodicities in the datasets. To this aim, several datasets were created by shuffling the starting datasets, and, for each of the shuffled datasets, the LS periodogram was obtained. After doing so, the periodogram associated with the starting dataset was compared with each periodogram associated with the shuffled datasets to establish which is characterized by the largest normalized amplitude at the frequency value under consideration. \\linebreak This procedure was iterated 10,000 times, with the result that the former normalized amplitude was larger with respect to the latter the $\\simeq$87\\% of cases for narrowband and the $\\simeq$93\\% of cases for wideband.\n\nAlthough it can be read as an encouraging result, it is important to stress that the two periodograms have been obtained using the complete datasets, provided by the NANOGrav collaboration. However, from Figures \\ref{fig:narrow1} and \\ref{fig:wide1}, it can be noted that, in the first observation years, the cadence of TRs was sporadic and irregular, and the error bands were significantly large than in the last observation years. Therefore, it may be convenient to reanalyze the datasets after applying some filters. Based on the previous considerations, one possibility is to remove both the TRs falling in the first six observation years and those associated with an uncertainty larger than $3\\sigma_w$ (see Figures \\ref{fig:narrow3} and \\ref{fig:wide3}). Applying this filter also assures that, as shown in ref. \\cite{oneill2022}, the dataset is limited to a time interval from 2010 onwards when the SMBHB candidate should have been in the continuous emission regime.\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{narrow3.png}\\hfill\n\\caption{Plot of the averaged narrowband post-fit whitened TRs of the MSP J2145--0750, filtered by removing the TRs associated with an uncertainty larger than $3\\sigma_w$ and the TRs before the sixth year of observation. The horizontal axis indicates the time, expressed in years. The vertical axis indicates the averaged narrowband post-fit whitened TRs, expressed in nanoseconds.}\n\\label{fig:narrow3}\n\\end{figure}\n\\vspace{-12pt}\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{wide3.png}\\hfill\n\\caption{Plot of the wideband post-fit whitened TRs of the MSP J2145--0750, filtered by removing the TRs associated with an uncertainty larger than $3\\sigma_w$ and the TRs before the sixth year of observation. The horizontal axis indicates the time, expressed in years. The vertical axis indicates the averaged narrowband post-fit whitened TRs, expressed in nanoseconds.}\n\\label{fig:wide3}\n\\end{figure}\nThe LS periodograms of the TRs plotted in Figures \\ref{fig:narrow3} and \\ref{fig:wide3} have been obtained and plotted in Figures \\ref{fig:narrow4} and \\ref{fig:wide4}, respectively.\n\\begin{figure}[H]\n\n\\includegraphics[scale=0.53]{narrow4.png}\\hfill\n\\caption{Plot\nof the LS periodogram of the averaged narrowband post-fit whitened TRs of the MSP J2145--0750, filtered by removing the TRs associated with an uncertainty larger than $3\\sigma_w$ and the TRs before the sixth year of observation. The black-dotted lines indicate the angular frequencies found by the LS algorithm. The red-dotted line indicates the angular frequency $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ expected for the GWs possibly emitted by the SMBHB candidate PKS 2131--021. The gray shaded area indicates uncertainty on $\\omega$ at the $3\\sigma$ confidence level. The horizontal axis indicates the GW angular frequency, expressed in Hertz. The vertical axis indicates the normalized amplitude, dimensionless.}\n\\label{fig:narrow4}\n\\end{figure}\n\\vspace{-12pt}\n\\begin{figure}[H]\n\\includegraphics[scale=0.53]{wide4.png}\\hfill\n\\caption{Plot\nof the LS periodogram of the wideband post-fit whitened TRs of the MSP J2145--0750, filtered by removing the TRs associated with an uncertainty larger than $3\\sigma_w$ and the TRs before the sixth year of observation. The black-dotted lines indicate the angular frequencies found by the LS algorithm. The red-dotted line indicates the angular frequency $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ expected for the GWs possibly emitted by the SMBHB candidate PKS 2131--021. The gray shaded area indicates uncertainty on $\\omega$ at the $3\\sigma$ confidence level. The horizontal axis indicates the GW angular frequency, expressed in Hertz. The vertical axis indicates the normalized amplitude, dimensionless.}\n\\label{fig:wide4}\n\\end{figure}\nIn both cases (see Figures \\ref{fig:narrow4} and \\ref{fig:wide4}), the LS algorithm allows finding the periodicity with the highest normalized amplitude associated with an angular frequency even closer to $\\omega=8.262^{+0.025}_{-0.025}\\times 10^{-8}$ Hz. Specifically, it is found $\\omega=8.147\\times 10^{-8}$ Hz in the case of the narrowband dataset, and $\\omega=8.205\\times 10^{-8}$ Hz in the case of the wideband dataset. Even if these values are so close to the angular frequency of the GW emission expected for the SMBHB candidate PKS 2131--021, this is not enough for claiming its detection. Further, more sophisticated analyses must be conducted in order of ensuring the solidity of this result and the nature of this periodicity.\n\n\\section{Conclusions}\n\\label{sec:4}\nIn this article, a possible investigation method to search for continuous GWs, based on the surfing effect, has been proposed. The surfing effect, described in Section \\ref{sec:1}, is usually ignored in most of the studies on the GWB since it is not relevant when the entire array of MSPs is used for its detection \\cite{hellings1983,lentati2015,arzoumanian2020,goncharov2021}. However, when performing a single-pulsar search of GWs emitted by a circularized SMBHBs in the continuous emission regime, it has to be taken into consideration. In fact, in this case, as shown in Sections \\ref{sec:1} and \\ref{sec:2}, the global maximum of the function $R\\left(t,\\theta\\right)$, which describes the TRs induced by GWs, corresponds, at any time, to an angular separation between the SMBHB and the MSPs almost null. However, it is important to keep in mind that, at any time, the function $R\\left(t,\\theta\\right)$ is characterized by very rapid angular oscillations, the more frequent the greater the $\\chi$ factor, as shown in Figures \\ref{fig:seangle} and \\ref{fig:seanglezoom}. Therefore, the function $R\\left(\\theta\\right)$ represents a sort of map, which can be used to verify if a MSP is more or less suitable for performing a single-pulsar search of continuous GWs, so MSPs that fall on one of the peaks will be preferred, and those that fall into one of the nodes will be discarded.\n\nAs an example, in Sections \\ref{sec:2} and \\ref{sec:3}, the case of the SMBHB candidate PKS 2131--021 was considered. This system, based on the orbital parameters determined by the radio analysis of its emission \\cite{oneill2022}, should be an emitter of GWs satisfying the hypotheses made in Section \\ref{sec:1} for surfing effect. By a lucky coincidence, as shown in \\mbox{Figures \\ref{fig:seangle} and \\ref{fig:seanglezoom}}, among the MSPs currently included in PTAs, the angularly closest MSP to the SMBHB candidate PKS 2131--021, which is the MSP J2145--0750, is such that its angular separation falls into a peak of the function $R\\left(\\theta\\right)$. This makes it worthy of attention since in its TRs a periodic signature induced by the GWs emitted by the SMBHB candidate PKS 2131--021 may be present. For this reason, in Section \\ref{sec:4}, the LS algorithm was used to search for such periodic signature. The periodograms plotted in Figures \\ref{fig:narrow2} and \\ref{fig:wide2} have been obtained from the complete datasets, provided by the NANOGrav collaboration \\linebreak (see Figures \\ref{fig:narrow1} and \\ref{fig:wide1}), while the periodograms plotted in Figures \\ref{fig:narrow4} and \\ref{fig:wide4} have been obtained from the filtered datasets (see Figures \\ref{fig:narrow3} and \\ref{fig:wide3}), discussed in Section \\ref{sec:3}. Interestingly, the LS algorithm finds in each dataset a periodicity associated with an angular frequency very close to the angular frequency of the GW emission expected for the SMBHB candidate PKS 2131--021. Although this result does not constitute proof of the SMBHB candidate PKS 2131--021 GW emission, it is certainly worthy of further investigation. A more detailed and rigorous analysis of the MSP J2145--0750 TRs, using the NANOGrav $12.5$ yr Data Set in conjunction with the IPTA Data Release 2 \\cite{perera2019}, will be proposed in a future paper entirely dedicated to searching for GWs potentially emitted by the SMBHB candidate PKS 2131--021. There, it will also be discussed the use of other MSPs not too angularly far from the SMBHB candidate PKS 2131--021, as well as the possible improvements that will be brought on this kind of search by next-gen PTAs, such as Square-Kilometer Array (SKA) \\cite{johnston2007,kramer2015,weltman2020,padmanabhan2022}.\n\n\n\\vspace{6pt} \n\\authorcontributions{Conceptualization, M.M. and F.D.P.; Formal analysis, M.M. and F.D.P.; Writing---original draft, M.M.; Writing---review \\& editing, M.M. and F.D.P.; Supervision, A.A.N. All authors have read and agreed to the published version of the manuscript.}\n\n\\funding{This research received no external funding.}\n\\dataavailability{The data used in this paper were collected by the NANOGrav collaboration and are publicly available.\n} \n\\acknowledgments{We warmly acknowledge Andrea Possenti (INAF-OAC) for the enlightening discussions and comments on the paper. We acknowledge the support of the Theoretical Astroparticle Physics (TAsP) and Euclid projects of the Istituto Nazionale di Fisica Nucleare (INFN).}\n\\conflictsofinterest{The authors declare no conflict of interest.}\n\\abbreviations{Abbreviations}{\nThe following abbreviations are used in this manuscript:\\\\\n\n\\noindent \n\\begin{tabular}{@{}ll}\nEPTA~~~~~~~~~~ & European Pulsar Timing Array\\\\\nGW & Gravitational Wave\\\\\nGWB & Gravitational Wave Background\\\\\nINAF & Istituto Nazionale di Astrofisica\\\\\nINFN & Istituto Nazionale di Fisica Nucleare\\\\\nInPTA & Indian Pulsar Timing Array\\\\\n\n\\end{tabular}}\n\n\\abbreviations{}{%\n\\noindent\n\\begin{tabular}{@{}ll}\nLIGO & Laser Interferometer Gravitational-Wave Observatory\\\\\nLISA & Laser Interferometer Space Antenna\\\\\nMDPI & Multidisciplinary Digital Publishing Institute\\\\\nMSP & Millisecond Pulsar\\\\\nNANOGrav & North American Nanohertz Observatory for Gravitational Waves\\\\\nOAC & Osservatorio Astronomico di Cagliari\\\\\nPPTA & Parkes Pulsar Timing Array\\\\\nPTA & Pulsar Timing Array\\\\\nSMBHB & Supermassive Black Hole Binary\\\\\nSKA & Square Kilometre Array\\\\\nTAsP & Theoretical Astroparticle Physics\\\\\nToA & Time of Arrival\\\\\nTR & Timing Residual\\\\\n\\end{tabular}}\n\\begin{adjustwidth}{-\\extralength}{0cm}\n\\printendnotes[custom]\n\\reftitle{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Mathematica codes}\n\\label{appendFloquet}\n\\justifying\n\n\n\\section{Electric and magnetic fields}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\mathit{r}=\\sqrt{x^2+y^2+z^2};}\\\\\n\\pmb{\\gamma =\\frac{2\\ell ^2}{\\sqrt{4\\ell ^2t^2+\\left(\\mathit{r}^2-t^2+\\ell ^2\\right)^2}};}\\\\\n\\pmb{\\omega _1=\\gamma \\frac{x}{\\ell };}\\\\\n\\pmb{\\omega _2=\\gamma \\frac{y}{\\ell };}\\\\\n\\pmb{\\omega _3=\\gamma \\frac{z}{\\ell };}\\\\\n\\pmb{\\omega _4=\\gamma \\frac{\\mathit{r}^2-t^2-\\ell ^2}{2\\ell ^2};}\\\\\n\\pmb{J_-[\\text{f$\\_$}] \\text{:=} \\frac{1}{\\sqrt{2}}(\\text{c$\\alpha $} D[f,\\beta ] - \\text{c$\\beta $} D[f,\\alpha ]);}\\\\\n\\pmb{\\text{Il}_-[\\text{f$\\_$}] \\text{:=} \\frac{1}{\\sqrt{2}}(\\text{c$\\alpha $} D[f,\\text{c$\\beta $}] - \\beta  D[f,\\alpha ]);}\\\\\n\\pmb{\\text{expi$\\tau $} = \\frac{(\\ell +I*t)^2+\\mathit{r}^2}{\\sqrt{4\\ell ^2t^2+\\left(\\mathit{r}^2-t^2+\\ell ^2\\right)^2}};}\\\\\n\\pmb{\\text{Sin$\\tau $}[\\text{n$\\_$}]\\text{:=}\\text{ComplexExpand}\\left[\\text{Im}\\left[\\text{expi$\\tau $}^n\\right]\\right]\\text{\/\/}\\text{FullSimplify};}\\\\\n\\pmb{\\text{Cos$\\tau $}[\\text{n$\\_$}]\\text{:=}\\text{ComplexExpand}\\left[\\text{Re}\\left[\\text{expi$\\tau $}^n\\right]\\right]\\text{\/\/}\\text{FullSimplify};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{ClearAll}[Y];}\\\\\n\\pmb{Y[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}0\\text{\/;}(j<0 \\|m>j \\| m<-j \\| n>j \\| n<-j);}\\\\\n\\pmb{Y[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=} }\\\\\n\\pmb{Y[j,m,n]=\\text{Simplify}\\left[\\left(\\sqrt{\\frac{2 \\pi ^2}{2j+1}}\\right)^{-1}\\sqrt{\\frac{2^{2 j - m - n}(j+m)!(j+n)!}{(2j)!(2j)!(j-m)!(j-n)!}}\\text{Nest}\\left[\\text{Il}_-,\\text{Nest}\\left[J_-,\\alpha\n^{2 j},j-n\\right],j-m\\right]\\text{\/.}\\right.}\\\\\n\\pmb{\\left.\\left\\{\\alpha \\to  \\omega _1+I \\omega _2,\\text{c$\\alpha $}\\to  \\omega _1-I \\omega _2,\\beta \\to  \\omega _3+I \\omega _4,\\text{c$\\beta $}\\to\n \\omega _3-I \\omega _4\\right\\}\\right];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{Z_+[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(j-n)(j-n+1)\/2}Y[j,m,n+1];}\\\\\n\\pmb{Z_3[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(j+1)^2-n^2}Y[j,m,n];}\\\\\n\\pmb{Z_-[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}-\\sqrt{(j+n)(j+n+1)\/2}Y[j,m,n-1];}\\\\\n\\pmb{X[1][\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}Z[1][j,m,n]=\\text{Cos$\\tau $}[2j+2]\\left(Z_+[j,m,n]+Z_-[j,m,n]\\right)\/\\sqrt{2};}\\\\\n\\pmb{X[2][\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}Z[2][j,m,n]=\\text{Cos$\\tau $}[2j+2]I\\left(-Z_+[j,m,n]+Z_-[j,m,n]\\right)\/\\sqrt{2};}\\\\\n\\pmb{X[3][\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}Z[3][j,m,n]=\\text{Cos$\\tau $}[2j+2]Z_3[j,m,n];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{HopfRanadaX}[1]=-\\frac{1}{8}\\text{Sin$\\tau $}[2];}\\\\\n\\pmb{\\text{HopfRanadaX}[2]=-\\frac{1}{8}\\text{Cos$\\tau $}[2];}\\\\\n\\pmb{\\text{HopfRanadaX}[3]=0;}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{ClearAll}[e];}\\\\\n\\pmb{e[\\text{a$\\_$}]\\text{:=}}\\\\\n\\pmb{e[a]=}\\\\\n\\pmb{\\text{Simplify}[}\\\\\n\\pmb{\\frac{\\gamma ^2}{\\ell ^3}\\left(t*R[a]d[t]-\\frac{1}{2}\\text{Sum}\\left[\\left(t^2-\\mathit{r}^2+\\ell ^2\\right)\\text{KroneckerDelta}[a,k]d[R[k]],\\{k,1,3\\}\\right]-\\right.}\\\\\n\\pmb{\\text{Sum}[R[a]R[k]d[R[k]],\\{k,1,3\\}]-\\text{Sum}[\\ell *\\text{LeviCivitaTensor}[3][[a,j,k]]R[j]d[R[k]],}\\\\\n\\pmb{\\{k,1,3\\},\\{j,1,3\\}])\\text{\/\/.}\\{R[0]\\to t,R[1]\\to x,R[2]\\to y,R[3]\\to z\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Evec}[\\text{X1$\\_$},\\text{X2$\\_$},\\text{X3$\\_$}]\\text{:=}\\text{Module}[\\{X,\\mathcal{A},A,F\\},}\\\\\n\\pmb{X[1]\\text{:=}\\text{X1};}\\\\\n\\pmb{X[2]\\text{:=}\\text{X2};}\\\\\n\\pmb{X[3]\\text{:=}\\text{X3};}\\\\\n\\pmb{\\mathcal{A}\\text{:=}\\text{FullSimplify}[\\text{Sum}[X[a]e[a],\\{a,1,3\\}]];}\\\\\n\\pmb{A[0]\\text{:=}\\mathcal{A}\\text{\/.}\\{d[t]\\to 1,d[x]\\to 0,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[1]\\text{:=}\\mathcal{A}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 1,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[2]\\text{:=}\\mathcal{A}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 1,d[z]\\to 0\\};}\\\\\n\\pmb{A[3]\\text{:=}\\mathcal{A}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 0,d[z]\\to 1\\};}\\\\\n\\pmb{F[\\text{u$\\_$},\\text{v$\\_$}]\\text{:=}D[A[u],Y[v]]-D[A[v],Y[u]];}\\\\\n\\pmb{\\{\\text{Simplify}[F[0,1]],\\text{Simplify}[F[0,2]],\\text{Simplify}[F[0,3]]\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{ClearAll}[\\text{EMfields}];}\\\\\n\\pmb{\\text{EMfields}[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$},\\text{L$\\_$}]\\text{:=}\\text{EMfields}[j,m,n,L]=}\\\\\n\\pmb{\\text{Module}[\\{R,\\mathcal{A},A,F\\},}\\\\\n\\pmb{R[0]=t;}\\\\\n\\pmb{R[1]=x;}\\\\\n\\pmb{R[2]=y;}\\\\\n\\pmb{R[3]=z;}\\\\\n\\pmb{\\mathcal{A}[\\text{a$\\_$}]\\text{:=}\\text{Simplify}[X[a][j,m,n]*e[a]\\text{\/.}\\{\\ell \\to L\\}];}\\\\\n\\pmb{\\text{FullGauge}=\\text{Sum}[\\mathcal{A}[a],\\{a,1,3\\}];}\\\\\n\\pmb{A[0]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 1,d[x]\\to 0,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[1]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 1,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[2]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 1,d[z]\\to 0\\};}\\\\\n\\pmb{A[3]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 0,d[z]\\to 1\\};}\\\\\n\\pmb{F[\\text{u$\\_$},\\text{v$\\_$}]\\text{:=}D[A[u],R[v]]-D[A[v],R[u]];}\\\\\n\\pmb{\\text{Ef}[1]\\text{:=}F[0,1]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Ef}[2]\\text{:=}F[0,2]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Ef}[3]\\text{:=}F[0,3]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[1]\\text{:=}F[3,2]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[2]\\text{:=}F[1,3]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[3]\\text{:=}F[2,1]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\{\\{\\text{Ef}[1],\\text{Ef}[2],\\text{Ef}[3]\\},\\{\\text{Bf}[1],\\text{Bf}[2],\\text{Bf}[3]\\}\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Ef}[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$},\\ell \\_]\\text{:=}\\text{EMfields}[j,m,n,\\ell ][[1]]\\text{\/\/}\\text{Re}\\text{\/\/}\\text{ComplexExpand}\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$},\\ell \\_]\\text{:=}\\text{EMfields}[j,m,n,\\ell ][[2]]\\text{\/\/}\\text{Re}\\text{\/\/}\\text{ComplexExpand}\\text{\/\/}\\text{Simplify};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{En}[\\text{j$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\frac{1}{2}\\left(\\text{Sum}\\left[\\text{Ef}[j,m,n,1][[a]]^2+\\text{Bf}[j,m,n,1][[a]]^2,\\{a,1,3\\}\\right]\\right)\\text{\/\/}\\text{FullSimplify}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{ClearAll}[\\text{HRfields}];}\\\\\n\\pmb{\\text{HRfields}[\\text{L$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{R,\\mathcal{A},A,F\\},}\\\\\n\\pmb{R[0]=t;}\\\\\n\\pmb{R[1]=x;}\\\\\n\\pmb{R[2]=y;}\\\\\n\\pmb{R[3]=z;}\\\\\n\\pmb{\\mathcal{A}[\\text{a$\\_$}]\\text{:=}\\text{Simplify}[\\text{HopfRanadaX}[a]*e[a]\\text{\/.}\\{\\ell \\to L\\}];}\\\\\n\\pmb{\\text{FullGauge}=\\text{Sum}[\\mathcal{A}[a],\\{a,1,3\\}];}\\\\\n\\pmb{A[0]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 1,d[x]\\to 0,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[1]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 1,d[y]\\to 0,d[z]\\to 0\\};}\\\\\n\\pmb{A[2]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 1,d[z]\\to 0\\};}\\\\\n\\pmb{A[3]\\text{:=}\\text{FullGauge}\\text{\/.}\\{d[t]\\to 0,d[x]\\to 0,d[y]\\to 0,d[z]\\to 1\\};}\\\\\n\\pmb{F[\\text{u$\\_$},\\text{v$\\_$}]\\text{:=}D[A[u],R[v]]-D[A[v],R[u]];}\\\\\n\\pmb{\\text{Ef}[1]\\text{:=}F[0,1]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Ef}[2]\\text{:=}F[0,2]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Ef}[3]\\text{:=}F[0,3]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[1]\\text{:=}F[3,2]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[2]\\text{:=}F[1,3]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{Bf}[3]\\text{:=}F[2,1]\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\{\\{\\text{Ef}[1],\\text{Ef}[2],\\text{Ef}[3]\\},\\{\\text{Bf}[1],\\text{Bf}[2],\\text{Bf}[3]\\}\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{EfHR}[\\ell \\_]\\text{:=}\\text{HRfields}[\\ell ][[1]]\\text{\/\/}\\text{Re}\\text{\/\/}\\text{ComplexExpand}\\text{\/\/}\\text{Simplify};}\\\\\n\\pmb{\\text{BfHR}[\\ell \\_]\\text{:=}\\text{HRfields}[\\ell ][[2]]\\text{\/\/}\\text{Re}\\text{\/\/}\\text{ComplexExpand}\\text{\/\/}\\text{Simplify};}\\)\n\\end{doublespace}\n\n\n\\vspace{12pt}\n\\section{Field lines}\n\\vspace{2pt}\n\nThis code is taken from the following open access platform:\\\\\n\\href{https:\/\/mathematica.stackexchange.com\/questions\/687\/id-like-to-display-field-lines-for-a-point- charge-in-3-dimensions}{https:\/\/mathematica.stackexchange.com\/questions\/687\/ \\\\ id-like-to-display-field-lines-for-a-point- charge-in-3-dimensions}\n\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{fieldSolve}\\text{::}\\text{usage}=}\\\\\n\\pmb{\\text{$\\texttt{\"}$fieldSolve[f,x,x0,$\\backslash $!$\\backslash $($\\backslash $*SubscriptBox[$\\backslash $(t$\\backslash $), $\\backslash $(max$\\backslash\n$)]$\\backslash $)] symbolically takes a vector field }}\\\\\n\\pmb{\\text{f with respect to the vector variable x, and then finds a vector curve r[t] starting at }}\\\\\n\\pmb{\\text{the point x0 satisfying the equation dr\/dt=$\\alpha $ f[r[t]] for t=0...$\\backslash $!$\\backslash $($\\backslash $*SubscriptBox[$\\backslash\n$(t$\\backslash $), }}\\\\\n\\pmb{\\text{$\\backslash $(max$\\backslash $)]$\\backslash $). Here $\\alpha $=1\/$|$f[r[t]]$|$ for normalization. To get verbose output add debug=True\n}}\\\\\n\\pmb{\\text{to the parameter list.$\\texttt{\"}$};}\\\\\n\\pmb{}\\\\\n\\pmb{\\text{fieldSolve}[\\text{field$\\_$},\\text{varlist$\\_$},\\text{xi0$\\_$},\\text{tmax$\\_$},\\text{debug$\\_$}: \\text{False}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{xiVec},\\text{equationSet},t\\},}\\\\\n\\pmb{\\text{If}[\\text{Length}[\\text{varlist}]\\neq \\text{Length}[\\text{xi0}],\\text{Print}[\\text{$\\texttt{\"}$Number of variables must equal number of\ninitial \n}}\\\\\n\\pmb{\\text{  conditions $\\backslash $nUSAGE:$\\backslash $n$\\texttt{\"}$}<>\\text{fieldSolve}\\text{::}\\text{usage}];\\text{Abort}[]];}\\\\\n\\pmb{\\text{xiVec}=\\text{Through}[\\text{varlist}[t]];}\\\\\n\\pmb{\\text{equationSet}=\\text{Join}[\\text{Thread}[\\text{Map}[D[\\#,t]\\&,\\text{xiVec}]==}\\\\\n\\pmb{\\text{Normalize}[\\text{field}\\text{\/.}\\text{Thread}[\\text{varlist}\\to \\text{xiVec}]]],\\text{Thread}[(\\text{xiVec}\\text{\/.}t\\to 0)==\\text{xi0}]];}\\\\\n\\pmb{\\text{If}[\\text{debug},}\\\\\n\\pmb{\\text{Print}[\\text{Row}[\\{\\text{{``}Numerically solving the system of equations$\\backslash $n$\\backslash $n{''}},}\\\\\n\\pmb{\\text{TraditionalForm}[(\\text{Simplify}[\\text{equationSet}]\\text{\/.}t\\to \\text{t})\\text{\/\/}\\text{TableForm}]\\}]]];}\\\\\n\\pmb{\\text{Map}[\\text{Head},\\text{First}[\\text{xiVec}\\text{\/.}\\text{Quiet}[\\text{NDSolve}[\\text{equationSet},\\text{xiVec},\\{t,0,\\text{tmax}\\}]]],2]]}\\\\\n\\pmb{}\\\\\n\\pmb{\\text{fieldLinePlot}[\\text{field$\\_$},\\text{varList$\\_$},\\text{seedList$\\_$},\\text{opts}:\\text{OptionsPattern}[]]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{sols},\\text{localVars},\\text{var},\\text{localField},\\text{plotOptions},\\text{tubeFunction},\\text{tubePlotStyle},}\\\\\n\\pmb{\\text{postProcess}=\\{\\}\\},\\text{plotOptions}=\\text{FilterRules}[\\{\\text{opts}\\},\\text{Options}[\\text{ParametricPlot3D}]];}\\\\\n\\pmb{\\text{tubeFunction}=\\text{OptionValue}[\\text{{``}TubeFunction{''}}];}\\\\\n\\pmb{\\text{If}[\\text{tubeFunction}\\text{=!=}\\text{None},\\text{tubePlotStyle}=\\text{Cases}[\\text{OptionValue}[\\text{PlotStyle}],\\text{Except}[\\text{$\\_$Tube}]];}\\\\\n\\pmb{\\text{plotOptions}=\\text{FilterRules}[\\text{plotOptions},\\text{Except}[\\{\\text{PlotStyle},\\text{ColorFunction},}\\\\\n\\pmb{\\text{ColorFunctionScaling}\\}]];}\\\\\n\\pmb{\\text{postProcess}=\\text{Line}[\\text{x$\\_$}]:\\to \\text{Join}[\\text{tubePlotStyle},\\{\\text{CapForm}[\\text{{``}Butt{''}}],}\\\\\n\\pmb{\\text{Tube}[x,\\text{tubeFunction}\\text{@@@}x]\\}]];}\\\\\n\\pmb{\\text{If}[\\text{Length}[\\text{seedList}[[1,1]]]\\neq \\text{Length}[\\text{varList}],}\\\\\n\\pmb{\\text{Print}[\\text{$\\texttt{\"}$Number of variables must equal number of initial conditions\n}}\\\\\n\\pmb{\\text{   $\\backslash $nUSAGE:$\\backslash $n$\\texttt{\"}$}<>\\text{fieldLinePlot}\\text{::}\\text{usage}];\\text{Abort}[]];}\\\\\n\\pmb{\\text{localVars}=\\text{Array}[\\text{var},\\text{Length}[\\text{varList}]];}\\\\\n\\pmb{\\text{localField}=\\text{ReleaseHold}[\\text{Hold}[\\text{field}]\\text{\/.}\\text{Thread}[\\text{Map}[\\text{HoldPattern},}\\\\\n\\pmb{\\text{Unevaluated}[\\text{varList}]]\\to \\text{localVars}]];}\\\\\n\\pmb{\\text{Show}[}\\\\\n\\pmb{\\text{ParallelTable}[}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\text{Evaluate}[\\text{Through}[\\#[t]]],\\{t,\\#[[1,1,1,1]],\\#[[1,1,1,2]]\\},}\\\\\n\\pmb{\\text{Evaluate}@\\text{Apply}[\\text{Sequence},\\text{plotOptions}]]\\&[}\\\\\n\\pmb{\\text{fieldSolve}[\\text{localField},\\text{localVars},\\text{seedList}[[i,1]],\\text{seedList}[[i,2]]]]\\text{\/.}\\text{postProcess},}\\\\\n\\pmb{\\{i,\\text{Length}[\\text{seedList}]\\}]]];}\\\\\n\\pmb{}\\\\\n\\pmb{\\text{Options}[\\text{fieldLinePlot}]=\\text{Append}[\\text{Options}[\\text{ParametricPlot3D}],\\text{{``}TubeFunction{''}}\\to \\text{None}];}\\\\\n\\pmb{}\\\\\n\\pmb{\\text{SyntaxInformation}[\\text{fieldLinePlot}]=\\{\\text{{``}LocalVariables{''}}\\to \\{\\text{{``}Solve{''}},\\{2,2\\}\\},}\\\\\n\\pmb{\\text{{``}ArgumentsPattern{''}}\\to \\{\\_,\\_,\\_,\\text{OptionsPattern}[]\\}\\};}\\\\\n\\pmb{}\\\\\n\\pmb{\\text{SetAttributes}[\\text{fieldSolve},\\text{HoldAll}];}\\)\n\\end{doublespace}\n\n\\vspace{12pt}\n\\section{Rotation of indices}\n\\vspace{2pt}\n\n\\subsection*{j=0 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rules0}=\\{}\\\\\n\\pmb{Y[\\text{n$\\_$}]:\\to  0\\text{\/;}\\text{Not}@(n\\text{==}0),}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_$} \\text{c$\\_$}):\\to  b \\delta [a]\\cdot c\\text{\/;}\\text{NumericQ}[b], \\text{(* linearity on numbers *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_$} + \\text{c$\\_$}):\\to  \\delta [a]\\cdot b +\\delta [a]\\cdot c,\\text{(* linearity on the sum *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot \\text{b$\\_$}:\\to  0\\text{\/;} \\text{NumericQ}[b],\\text{(* 0 on numbers *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot e[\\text{b$\\_$}]:\\to 2\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,b,c]]e[c],\\{c,3\\}],\\text{(* action on the forms *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_$} \\text{c$\\_$}):\\to  (\\delta [a]\\cdot b) c+b(\\delta [a]\\cdot c),\\text{(* product rule *)}}\\\\\n\\pmb{\\delta [1]\\cdot Y[\\text{n$\\_$}]:\\to  Y[n+1]+Y[n-1],\\text{(*} \\text{action} \\text{on} \\text{harmonics}, \\text{calculated} \\text{by} \\text{hand}\n\\text{*)}}\\\\\n\\pmb{\\delta [2]\\cdot Y[\\text{n$\\_$}]:\\to  I(Y[n-1]-Y[n+1]),}\\\\\n\\pmb{\\delta [3]\\cdot Y[\\text{n$\\_$}]:\\to  -2I n Y[n]}\\\\\n\\pmb{\\};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{Z_+[\\text{n$\\_$}]\\text{:=}\\sqrt{(-n)(-n+1)\/2}Y[n+1]; }\\\\\n\\pmb{Z_3[\\text{n$\\_$}]\\text{:=}\\sqrt{(1)^2-n^2}Y[n];}\\\\\n\\pmb{Z_-[\\text{n$\\_$}]\\text{:=}-\\sqrt{(n)(n+1)\/2}Y[n-1];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{A_-=\\text{Simplify}@\\text{Sum}\\left[Z_-[n]\\Lambda [n],\\{n,-1,1\\}\\right]\\text{\/\/.}\\text{rules0}; \\text{(* Combining solutions *)}}\\\\\n\\pmb{A_+=\\text{Simplify}@\\text{Sum}\\left[Z_+[n]\\Lambda [n],\\{n,-1,1\\}\\right]\\text{\/\/.}\\text{rules0};}\\\\\n\\pmb{A[3]=\\text{Simplify}@\\text{Sum}\\left[Z_3[n]\\Lambda [n],\\{n,-1,1\\}\\right]\\text{\/\/.}\\text{rules0};}\\\\\n\\pmb{A[1]=\\frac{\\left(A_++A_-\\right)}{\\sqrt{2}};}\\\\\n\\pmb{A[2]=\\frac{I*\\left(-A_++A_-\\right)}{\\sqrt{2}};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Acomp}[\\text{a$\\_$},\\text{n$\\_$}]\\text{:=}\\partial _{\\Lambda [n]}A[a]\\text{(*} \\text{Defining} A_a{}^{\\text{mn}} \\text{*)}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{variation}[\\text{arot$\\_$}]\\text{:=}\\text{(*} \\text{Fully} \\text{opening} D_a\\left(\\Lambda _{\\text{mn}}A_a{}^{\\text{mn}}e^a\\right)\n\\text{*)}}\\\\\n\\pmb{\\text{Sum}[\\text{d$\\Lambda $}[\\text{arot},n]\\text{Acomp}[a,n]e[a]+\\Lambda [n](\\delta [\\text{arot}]\\cdot \\text{Acomp}[a,n])e[a]}\\\\\n\\pmb{+\\Lambda [n]\\text{Acomp}[a,n](\\delta [\\text{arot}]\\cdot e[a]),\\{n,-1,1\\},\\{a,3\\}]\\text{\/\/.}\\text{rules0}; }\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{eqnsd$\\Lambda $}[\\text{arot$\\_$},\\text{bform$\\_$}]\\text{:=}\\text{Simplify}@D[D[\\text{variation}[\\text{arot}],e[\\text{bform}],Y[0]]];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{d$\\Lambda $list}=\\text{Flatten}@\\text{Table}[\\text{d$\\Lambda $}[a,n],\\{a,3\\},\\{n,-1,1\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{alleqnsd$\\Lambda $}=\\text{Flatten}@\\text{Table}[\\text{eqnsd$\\Lambda $}[a,b]==0,\\{a,3\\},\\{b,3\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Solve}[\\text{alleqnsd$\\Lambda $},\\text{d$\\Lambda $list}] }\\)\n\\end{doublespace}\n\n\\subsection*{j=1\/2 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rules}=\\{}\\\\\n\\pmb{Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  0\\text{\/;}\\text{Not}@((m\\text{==}0.5\\lor m\\text{==}-0.5)\\land (n\\text{==}0.5\\lor n\\text{==}-0.5)),}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} \\text{c$\\_$}):\\to  b \\delta [a]\\cdot c\\text{\/;}\\text{NumericQ}[b], \\text{(* linearity on numbers\n*)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} + \\text{c$\\_\\_$}):\\to  \\delta [a]\\cdot b +\\delta [a]\\cdot c,\\text{(* linearity on the sum *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot \\text{b$\\_$}:\\to  0\\text{\/;} \\text{NumericQ}[b],\\text{(* 0 on numbers *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot e[\\text{b$\\_$}]:\\to 2\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,b,c]]e[c],\\{c,3\\}],\\text{(* action on the forms *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} \\text{c$\\_\\_$}):\\to  (\\delta [a]\\cdot b) c+b(\\delta [a]\\cdot c),\\text{(* product rule *)}}\\\\\n\\pmb{\\delta [1]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  -I(Y[m+1,n]+Y[m,n+1]+Y[m-1,n]+Y[m,n-1]),}\\\\\n\\pmb{\\text{(*} \\text{action} \\text{on} \\text{harmonics}, \\text{calculated} \\text{by} \\text{hand} \\text{*)}}\\\\\n\\pmb{\\delta [2]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  Y[m-1,n]+Y[m,n-1]-Y[m+1,n]-Y[m,n+1],}\\\\\n\\pmb{\\delta [3]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  -2I(m+n)Y[m,n]}\\\\\n\\pmb{\\};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{Z_+[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(1\/2-n)(1\/2-n+1)\/2}Y[m,n+1]; }\\\\\n\\pmb{Z_3[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(1\/2+1)^2-n^2}Y[m,n];}\\\\\n\\pmb{Z_-[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}-\\sqrt{(1\/2+n)(1\/2+n+1)\/2}Y[m,n-1];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{A_-=\\text{Simplify}@\\text{Sum}\\left[Z_-[m,n]\\Lambda [m,n],\\{m,-1\/2,1\/2\\},\\{n,-3\/2,3\/2\\}\\right]\\text{\/\/.}\\text{rules}; }\\\\\n\\pmb{\\text{(* Combining solutions *)}}\\\\\n\\pmb{A_+=\\text{Simplify}@\\text{Sum}\\left[Z_+[m,n]\\Lambda [m,n],\\{m,-1\/2,1\/2\\},\\{n,-3\/2,3\/2\\}\\right]\\text{\/\/.}\\text{rules};}\\\\\n\\pmb{A[3]=\\text{Simplify}@\\text{Sum}\\left[Z_3[m,n]\\Lambda [m,n],\\{m,-1\/2,1\/2\\},\\{n,-3\/2,3\/2\\}\\right]\\text{\/\/.}\\text{rules};}\\\\\n\\pmb{A[1]=\\frac{\\left(A_++A_-\\right)}{\\sqrt{2}};}\\\\\n\\pmb{A[2]=\\frac{I*\\left(-A_++A_-\\right)}{\\sqrt{2}};}\\\\\n\\pmb{\\text{Acomp}[\\text{a$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\partial _{\\Lambda [m,n]}A[a]\\text{(*} \\text{Defining} A_a{}^{\\text{mn}} \\text{*)}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{variation}[\\text{arot$\\_$}]\\text{:=}\\text{(*} \\text{Fully} \\text{opening} D_a\\left(\\Lambda _{\\text{mn}}A_a{}^{\\text{mn}}e^a\\right)\n\\text{*)}}\\\\\n\\pmb{\\text{Sum}[\\text{d$\\Lambda $}[\\text{arot},m,n]\\text{Acomp}[b,m,n]e[b]+\\Lambda [m,n](\\delta [\\text{arot}]\\cdot \\text{Acomp}[b,m,n])e[b]+}\\\\\n\\pmb{\\Lambda [m,n]\\text{Acomp}[b,m,n](\\delta [\\text{arot}]\\cdot e[b]),\\{m,-1\/2,1\/2\\},}\\\\\n\\pmb{\\text{     }\\{n,-3\/2,3\/2\\},\\{b,3\\}]\\text{\/\/.}\\text{rules}; }\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{eqnsd$\\Lambda $}[\\text{arot$\\_$},\\text{bform$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\text{Simplify}@D[D[\\text{variation}[\\text{arot}],e[\\text{bform}],Y[m,n]]];}\\\\\n\\pmb{\\text{(*} \\text{Taking} \\text{the} \\text{term} \\text{of} D_a \\text{with} Y_{\\text{mn}} \\text{and} e^b \\text{*)}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{d$\\Lambda $list}=\\text{Flatten}@\\text{Table}[\\text{d$\\Lambda $}[a,m,n],\\{a,3\\},\\{m,-1\/2,1\/2\\},\\{n,-3\/2,3\/2\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{alleqnsd$\\Lambda $}=\\text{Flatten}@\\text{Table}[\\text{eqnsd$\\Lambda $}[a,b,m,n]==0,\\{a,3\\},\\{b,3\\},\\{m,-1\/2,1\/2\\},}\\\\\n\\pmb{\\{n,-1\/2,1\/2\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sol}=\\text{Solve}[\\text{alleqnsd$\\Lambda $},\\text{d$\\Lambda $list}][[1]]}\\)\n\\end{doublespace}\n\n\\subsection*{j=1 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rules}=\\{}\\\\\n\\pmb{Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  0\\text{\/;}}\\\\\n\\pmb{\\text{Not}@((m==-1\\lor m==0\\lor m==1)\\land (n==-1\\lor n==0\\lor n==1)),}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} \\text{c$\\_$}):\\to  b \\delta [a]\\cdot c\\text{\/;}\\text{NumericQ}[b], \\text{(* linearity on numbers\n*)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} + \\text{c$\\_\\_$}):\\to  \\delta [a]\\cdot b +\\delta [a]\\cdot c,\\text{(* linearity on the sum *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot \\text{b$\\_$}:\\to  0\\text{\/;} \\text{NumericQ}[b],\\text{(* 0 on numbers *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot e[\\text{b$\\_$}]:\\to 2\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,b,c]]e[c],\\{c,3\\}],\\text{(* action on the forms *)}}\\\\\n\\pmb{\\delta [\\text{a$\\_$}]\\cdot (\\text{b$\\_\\_$} \\text{c$\\_\\_$}):\\to  (\\delta [a]\\cdot b) c+b(\\delta [a]\\cdot c),\\text{(* product rule *)}}\\\\\n\\pmb{\\delta [1]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  -I*\\sqrt{2}(Y[m+1,n]+Y[m,n+1]+Y[m-1,n]+Y[m,n-1]),}\\\\\n\\pmb{\\text{(*} \\text{action} \\text{on} \\text{harmonics}, \\text{calculated} \\text{by} \\text{hand} \\text{*)}}\\\\\n\\pmb{\\delta [2]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  \\sqrt{2}(Y[m-1,n]+Y[m,n-1]-Y[m+1,n]-Y[m,n+1]),}\\\\\n\\pmb{\\delta [3]\\cdot Y[\\text{m$\\_$},\\text{n$\\_$}]:\\to  -2I(m+n)Y[m,n]}\\\\\n\\pmb{\\};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{Z_+[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(1-n)(1-n+1)\/2}Y[m,n+1]; }\\\\\n\\pmb{Z_3[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\sqrt{(1+1)^2-n^2}Y[m,n];}\\\\\n\\pmb{Z_-[\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}-\\sqrt{(1+n)(1+n+1)\/2}Y[m,n-1];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{A_-=\\text{Simplify}@\\text{Sum}\\left[Z_-[m,n]\\Lambda [m,n],\\{m,-1,1\\},\\{n,-2,2\\}\\right]\\text{\/\/.}\\text{rules};}\\\\\n\\pmb{\\text{(* Combining solutions *)}}\\\\\n\\pmb{A_+=\\text{Simplify}@\\text{Sum}\\left[Z_+[m,n]\\Lambda [m,n],\\{m,-1,1\\},\\{n,-2,2\\}\\right]\\text{\/\/.}\\text{rules};}\\\\\n\\pmb{A[3]=\\text{Simplify}@\\text{Sum}\\left[Z_3[m,n]\\Lambda [m,n],\\{m,-1,1\\},\\{n,-2,2\\}\\right]\\text{\/\/.}\\text{rules};}\\\\\n\\pmb{A[1]=\\frac{\\left(A_++A_-\\right)}{\\sqrt{2}};}\\\\\n\\pmb{A[2]=\\frac{I*\\left(-A_++A_-\\right)}{\\sqrt{2}};}\\\\\n\\pmb{\\text{Acomp}[\\text{a$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\partial _{\\Lambda [m,n]}A[a]\\text{(*} \\text{Defining} A_a{}^{\\text{mn}} \\text{*)}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{variation}[\\text{arot$\\_$}]\\text{:=}\\text{(*} \\text{Fully} \\text{opening} D_a\\left(\\Lambda _{\\text{mn}}A_a{}^{\\text{mn}}e^a\\right)\n\\text{*)}}\\\\\n\\pmb{\\text{Sum}[\\text{d$\\Lambda $}[\\text{arot},m,n]\\text{Acomp}[b,m,n]e[b]+\\Lambda [m,n](\\delta [\\text{arot}]\\cdot \\text{Acomp}[b,m,n])e[b]+}\\\\\n\\pmb{\\Lambda [m,n]\\text{Acomp}[b,m,n](\\delta [\\text{arot}]\\cdot e[b]),\\{m,-1,1\\},\\{n,-2,2\\},\\{b,3\\}]\\text{\/\/.}\\text{rules}; }\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{eqnsd$\\Lambda $}[\\text{arot$\\_$},\\text{bform$\\_$},\\text{m$\\_$},\\text{n$\\_$}]\\text{:=}\\text{Simplify}@D[D[\\text{variation}[\\text{arot}],e[\\text{bform}],Y[m,n]]];}\\\\\n\\pmb{\\text{(*} \\text{Taking} \\text{the} \\text{term} \\text{of} D_a \\text{with} Y_{\\text{mn}} \\text{and} e^b \\text{*)}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{d$\\Lambda $list}=\\text{Flatten}@\\text{Table}[\\text{d$\\Lambda $}[a,m,n],\\{a,3\\},\\{m,-1,1\\},\\{n,-2,2\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{alleqnsd$\\Lambda $}=\\text{Flatten}@\\text{Table}[\\text{eqnsd$\\Lambda $}[a,b,m,n]==0,\\{a,3\\},\\{b,3\\},\\{m,-1,1\\},\\{n,-1,1\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sol}=\\text{Solve}[\\text{alleqnsd$\\Lambda $},\\text{d$\\Lambda $list}][[1]]}\\)\n\\end{doublespace}\n\n\n\\vspace{12pt}\n\\section{Trajectories}\n\\vspace{2pt}\n\n\\subsection*{Single particle}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{RelSoln}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa\n\\_]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{r,v,\\Gamma ,\\text{initcond},\\text{eom},\\text{vars}\\},}\\\\\n\\pmb{r[\\text{t$\\_$}]=\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\};}\\\\\n\\pmb{v[\\text{t$\\_$}]=\\{\\text{v1}[t],\\text{v2}[t],\\text{v3}[t]\\};}\\\\\n\\pmb{\\text{initcond}=\\{r[0]==\\text{pos0},v[0]\\text{==}\\text{vel0}\\};}\\\\\n\\pmb{\\Gamma [\\text{t$\\_$}]\\text{:=}\\frac{1}{\\sqrt{1-\\text{v1}[t]^2-\\text{v2}[t]^2-\\text{v3}[t]^2}};}\\\\\n\\pmb{\\text{eom}=\\left\\{\\partial _tr[t]==v[t],\\partial _t(\\Gamma [t]v[t])==\\right.}\\\\\n\\pmb{\\kappa  \\text{Simplify}[((\\text{Ef}[\\text{config}[[1]],\\text{config}[[2]],\\text{config}[[3]],\\ell ]}\\\\\n\\pmb{+v[t]\\times \\text{Bf}[\\text{config}[[1]],\\text{config}[[2]],\\text{config}[[3]],\\ell ])}\\\\\n\\pmb{\\text{\/.}\\{x\\to \\text{r1}[t],y\\to \\text{r2}[t],z\\to \\text{r3}[t]\\})]\\};}\\\\\n\\pmb{ \\text{vars}=\\text{Flatten}@\\{r[t],v[t]\\};}\\\\\n\\pmb{\\text{    }\\text{NDSolve}[\\text{Flatten}@\\text{Join}[\\text{eom},\\{\\text{initcond}\\}],\\text{vars},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{Method}\\text{-$>$}\\{\\text{{``}EquationSimplification{''}}\\text{-$>$}\\text{{``}Residual{''}}\\}]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Soln}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa\n\\_]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{r,v,\\text{initcond},\\text{eom},\\text{vars}\\},}\\\\\n\\pmb{r[\\text{t$\\_$}]=\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\};}\\\\\n\\pmb{v[\\text{t$\\_$}]=\\{\\text{v1}[t],\\text{v2}[t],\\text{v3}[t]\\};}\\\\\n\\pmb{\\text{initcond}=\\{r[0]==\\text{pos0},v[0]\\text{==}\\text{vel0}\\};}\\\\\n\\pmb{\\text{eom}=\\left\\{\\partial _tr[t]==v[t],\\partial _t(v[t])==\\right.}\\\\\n\\pmb{\\kappa  \\text{Simplify}[((\\text{Ef}[\\text{config}[[1]],\\text{config}[[2]],\\text{config}[[3]],\\ell ]+v[t]\\times }\\\\\n\\pmb{\\text{Bf}[\\text{config}[[1]],\\text{config}[[2]],\\text{config}[[3]],\\ell ])\\text{\/.}\\{x\\to \\text{r1}[t],y\\to \\text{r2}[t],z\\to \\text{r3}[t]\\})]\\};}\\\\\n\\pmb{ \\text{vars}=\\text{Flatten}@\\{r[t],v[t]\\};}\\\\\n\\pmb{\\text{    }\\text{NDSolve}[\\text{Flatten}@\\text{Join}[\\text{eom},\\{\\text{initcond}\\}],\\text{vars},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{Method}\\text{-$>$}\\{\\text{{``}EquationSimplification{''}}\\text{-$>$}\\text{{``}Residual{''}}\\}]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{RelPlots}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa\n\\_:1]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\text{pos0},\\text{vel0},\\kappa ]\\},}\\\\\n\\pmb{\\{\\text{Plot}[\\text{r1}[t]\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v1}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube},}\\\\\n\\pmb{\\left.\\left.\\text{Plot}\\left[\\sqrt{\\text{v1}[t]^2+\\text{v2}[t]^2+\\text{v2}[t]^2}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to\n\\text{All}\\right]\\right\\}\\right]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Plots}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa\n\\_:1]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,\\text{pos0},\\text{vel0},\\kappa ]\\},}\\\\\n\\pmb{\\{\\text{Plot}[\\text{r1}[t]\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v1}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\left.\\left.\\text{Plot}\\left[\\sqrt{\\text{v1}[t]^2+\\text{v2}[t]^2+\\text{v2}[t]^2}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to\n\\text{All}\\right]\\right\\}\\right]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{RelSolnHR}[\\text{tf$\\_$},\\ell \\_,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa \\_]\\text{:=}\\text{Module}[\\{r,v,\\Gamma\n,\\text{initcond},\\text{eom},\\text{vars}\\},}\\\\\n\\pmb{r[\\text{t$\\_$}]=\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\};}\\\\\n\\pmb{v[\\text{t$\\_$}]=\\{\\text{v1}[t],\\text{v2}[t],\\text{v3}[t]\\};}\\\\\n\\pmb{\\text{initcond}=\\{r[0]==\\text{pos0},v[0]\\text{==}\\text{vel0}\\};}\\\\\n\\pmb{\\Gamma [\\text{t$\\_$}]\\text{:=}\\frac{1}{\\sqrt{1-\\text{v1}[t]^2-\\text{v2}[t]^2-\\text{v3}[t]^2}};}\\\\\n\\pmb{\\text{eom}=\\left\\{\\partial _tr[t]==v[t],\\partial _t(\\Gamma [t]v[t])==\\kappa  \\text{Simplify}[((\\text{EfHR}[\\ell ]+v[t]\\times \\text{BfHR}[\\ell\n])\\right.}\\\\\n\\pmb{\\text{\/.}\\{x\\to \\text{r1}[t],y\\to \\text{r2}[t],z\\to \\text{r3}[t]\\})]\\};}\\\\\n\\pmb{ \\text{vars}=\\text{Flatten}@\\{r[t],v[t]\\};}\\\\\n\\pmb{\\text{    }\\text{NDSolve}[\\text{Flatten}@\\text{Join}[\\text{eom},\\{\\text{initcond}\\}],\\text{vars},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{Method}\\text{-$>$}\\{\\text{{``}EquationSimplification{''}}\\text{-$>$}\\text{{``}Residual{''}}\\}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{SolnHR}[\\text{tf$\\_$},\\ell \\_,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa \\_]\\text{:=}\\text{Module}[\\{r,v,\\text{initcond},\\text{eom},\\text{vars}\\},}\\\\\n\\pmb{r[\\text{t$\\_$}]=\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\};}\\\\\n\\pmb{v[\\text{t$\\_$}]=\\{\\text{v1}[t],\\text{v2}[t],\\text{v3}[t]\\};}\\\\\n\\pmb{\\text{initcond}=\\{r[0]==\\text{pos0},v[0]\\text{==}\\text{vel0}\\};}\\\\\n\\pmb{\\text{eom}=\\left\\{\\partial _tr[t]==v[t],\\partial _t(v[t])==\\kappa  \\text{Simplify}[((\\text{EfHR}[\\ell ]+v[t]\\times \\text{BfHR}[\\ell ])\\right.}\\\\\n\\pmb{\\text{\/.}\\{x\\to \\text{r1}[t],y\\to \\text{r2}[t],z\\to \\text{r3}[t]\\})]\\};}\\\\\n\\pmb{ \\text{vars}=\\text{Flatten}@\\{r[t],v[t]\\};}\\\\\n\\pmb{\\text{    }\\text{NDSolve}[\\text{Flatten}@\\text{Join}[\\text{eom},\\{\\text{initcond}\\}],\\text{vars},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{Method}\\text{-$>$}\\{\\text{{``}EquationSimplification{''}}\\text{-$>$}\\text{{``}Residual{''}}\\}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{RelPlotsHR}[\\text{tf$\\_$},\\ell \\_:1,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa \\_:1]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,\\text{pos0},\\text{vel0},\\kappa ]\\},}\\\\\n\\pmb{\\{\\text{Plot}[\\text{r1}[t]\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v1}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube},}\\\\\n\\pmb{\\left.\\text{Plot}\\left[\\sqrt{\\text{v1}[t]^2+\\text{v2}[t]^2+\\text{v2}[t]^2}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to\n\\text{All}\\right]\\right\\}}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsHR}[\\text{tf$\\_$},\\ell \\_:1,\\text{pos0$\\_$}?\\text{ListQ},\\text{vel0$\\_$}?\\text{ListQ},\\kappa \\_:1]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\text{pos0},\\text{vel0},\\kappa ]\\},}\\\\\n\\pmb{\\{\\text{Plot}[\\text{r1}[t]\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{r3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v1}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v2}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],\\text{Plot}[\\text{v3}[t]}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}],}\\\\\n\\pmb{\\left.\\text{Plot}\\left[\\sqrt{\\text{v1}[t]^2+\\text{v2}[t]^2+\\text{v2}[t]^2}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotRange}\\to\n\\text{All}\\right]\\right\\}}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\subsection*{Multi-particle (line)}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{GenerateLine}[\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{list}=\\{\\{0,0,0\\}\\},\\text{elem}\\},}\\\\\n\\pmb{\\text{Do}[\\text{elem}=i \\text{distance} \\text{Normalize}[\\text{direc}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{list},\\text{elem}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{list},-\\text{elem}],\\{i,\\text{npts}\\}];}\\\\\n\\pmb{\\text{list}}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nStarting at rest and positioned on the line\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosLine}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},}\\\\\n\\pmb{\\text{PlotRange}\\to \\text{All}]\\text{\/.}\\text{Line}\\text{-$>$}(\\text{Tube}[\\#,.005]\\&)],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ListLinePlot3D}[\\text{listpos},\\text{PlotStyle}\\to \\text{RGBColor}[0,0.4,0.6]]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosLine2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},}\\\\\n\\pmb{\\text{PlotRange}\\to \\text{All}]\\text{\/.}\\text{Line}\\text{-$>$}(\\text{Tube}[\\#,.005]\\&)],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ListLinePlot3D}[\\text{listpos},\\text{PlotStyle}\\to \\text{RGBColor}[0,0.4,0.6]]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosLineHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]}\\\\\n\\pmb{\\text{\/.}\\text{Line}\\text{-$>$}(\\text{Tube}[\\#,.005]\\&)],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ListLinePlot3D}[\\text{listpos},\\text{PlotStyle}\\to \\text{RGBColor}[0,0.4,0.6]]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosLineHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,-\\text{tf},\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]}\\\\\n\\pmb{\\text{\/.}\\text{Line}\\text{-$>$}(\\text{Tube}[\\#,.005]\\&)],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ListLinePlot3D}[\\text{listpos},\\text{PlotStyle}\\to \\text{RGBColor}[0,0.4,0.6]]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nStarting at origin with velocities along the line\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelLine}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},\\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},\\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelLine2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},\\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},\\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelLineHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},\\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},\\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelLineHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direc$\\_$}?\\text{ListQ},\\text{distance$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{GenerateLine}[\\text{direc},\\text{distance},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},\\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},\\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\subsection*{Multi-particle (circle)}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{CirclePoints3D}[\\text{direcN$\\_$},\\text{npts$\\_$}]\\text{:=} \\text{Module}[\\{\\text{list}=\\text{Table}[\\{0,0,0\\},\\text{npts}]\\},}\\\\\n\\pmb{\\text{Which}[\\text{direcN}\\text{==}1,\\text{Do}[\\text{list}[[n,2]]=\\text{CirclePoints}[\\text{npts}][[n,1]];}\\\\\n\\pmb{\\text{list}[[n,3]]=\\text{CirclePoints}[\\text{npts}][[n,2]],\\{n,1,\\text{npts}\\}],}\\\\\n\\pmb{\\text{direcN}\\text{==}2,\\text{Do}[\\text{list}[[n,1]]=\\text{CirclePoints}[\\text{npts}][[n,1]];}\\\\\n\\pmb{\\text{list}[[n,3]]=\\text{CirclePoints}[\\text{npts}][[n,2]],\\{n,1,\\text{npts}\\}],}\\\\\n\\pmb{\\text{direcN}\\text{==}3,\\text{Do}[\\text{list}[[n,2]]=\\text{CirclePoints}[\\text{npts}][[n,2]];}\\\\\n\\pmb{\\text{list}[[n,1]]=\\text{CirclePoints}[\\text{npts}][[n,1]],\\{n,1,\\text{npts}\\}]];}\\\\\n\\pmb{\\text{list}}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{circle3D}[\\text{centre$\\_$}:\\{0,0,0\\},\\text{radius$\\_$}:1,\\text{normal$\\_$}:\\{0,0,1\\},\\text{angle$\\_$}:\\{0,2 \\text{Pi}\\}]\\text{:=}}\\\\\n\\pmb{\\text{Composition}[\\text{Line},\\text{Map}[\\text{RotationTransform}[\\{\\{0,0,1\\},\\text{normal}\\},\\text{centre}],\\#]\\&,}\\\\\n\\pmb{\\text{Map}[\\text{Append}[\\#,\\text{Last}@\\text{centre}]\\&,\\#]\\&,\\text{Append}[\\text{DeleteDuplicates}[\\text{Most}@\\#],\\text{Last}@\\#]\\&,}\\\\\n\\pmb{\\text{Level}[\\#,\\{-2\\}]\\&,\\text{MeshPrimitives}[\\#,1]\\&,\\text{DiscretizeRegion},}\\\\\n\\pmb{\\text{If}}\\\\\n\\pmb{][\\text{First}@\\text{Differences}@\\text{angle}\\text{$>$=}2 \\text{Pi},\\text{Circle}[\\text{Most}@\\text{centre},\\text{radius}],}\\\\\n\\pmb{\\text{Circle}[\\text{Most}@\\text{centre},\\text{radius},\\text{angle}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nStarting at rest and located on the circle\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosCircle}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{CirclePoints3D}[\\text{direcN},\\text{npts}],}\\\\\n\\pmb{\\text{TempSoln},N=\\{\\{1,0,0\\},\\{0,1,0\\},\\{0,0,1\\}\\}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]}\\\\\n\\pmb{\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube}],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],}\\\\\n\\pmb{\\text{Tube}[\\text{circle3D}[\\{0,0,0\\},r,N[[\\text{direcN}]]],0.001]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosCircle2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{CirclePoints3D}[\\text{direcN},\\text{npts}],}\\\\\n\\pmb{\\text{TempSoln},N=\\{\\{1,0,0\\},\\{0,1,0\\},\\{0,0,1\\}\\}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},}\\\\\n\\pmb{\\text{PlotRange}\\to \\text{All}]\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube}],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],}\\\\\n\\pmb{\\text{Tube}[\\text{circle3D}[\\{0,0,0\\},r,N[[\\text{direcN}]]],0.001]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosCircleHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],}\\\\\n\\pmb{\\text{TempSoln},N=\\{\\{1,0,0\\},\\{0,1,0\\},\\{0,0,1\\}\\}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]}\\\\\n\\pmb{\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube}],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],}\\\\\n\\pmb{\\text{Tube}[\\text{circle3D}[\\{0,0,0\\},r,N[[\\text{direcN}]]],0.002]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosCircleHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],}\\\\\n\\pmb{\\text{TempSoln},N=\\{\\{1,0,0\\},\\{0,1,0\\},\\{0,0,1\\}\\}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]}\\\\\n\\pmb{\\text{\/.}\\text{Line}\\text{-$>$}\\text{Tube}],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],}\\\\\n\\pmb{\\text{Tube}[\\text{circle3D}[\\{0,0,0\\},r,N[[\\text{direcN}]]],0.002]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nStarting at origin with velocities radially outward\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelCircle}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelCircle2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelCircleHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelCircleHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{direcN$\\_$},\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{CirclePoints3D}[\\text{direcN},\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/10],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/10],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\subsection*{Multi-particle (sphere)}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{RandomSpherePoints}[\\text{n$\\_$}]\\text{:=}\\text{Module}[\\{\\text{list}=\\{\\}\\},}\\\\\n\\pmb{\\text{Do}[\\text{AppendTo}[\\text{list},\\text{RandomPoint}[\\text{Sphere}[]]],\\{i,1,n\\}];\\text{list}]}\\)\n\\end{doublespace}\n\nStarting at rest and located on the circle\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereSymmetric}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{SpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereSymmetric2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{SpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereRandom}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},}\\\\\n\\pmb{\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereRandom2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],}\\\\\n\\pmb{\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosInhomogeneous}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{range$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\text{RandomReal}[\\{\\text{range}\/10,\\text{range}\\}]}\\\\\n\\pmb{ \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];\\text{AppendTo}[\\text{listplots},}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{npts}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},\\text{range}]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosInhomogeneous2}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{range$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}= \\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{Soln}[\\text{tf},\\text{config},\\ell ,\\text{RandomReal}[\\{\\text{range}\/10,\\text{range}\\}] }\\\\\n\\pmb{\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];\\text{AppendTo}[\\text{listplots},}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{npts}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},\\text{range}]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereSymmetricHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{SpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[1,0,1],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereSymmetricHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{SpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[1,0,1],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereRandomHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[1,0,1],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosSphereRandomHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{r$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,r \\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{Length}@\\text{listpos}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[1,0,1],\\text{Sphere}[\\{0,0,0\\},r]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosInhomogeneousHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{range$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSolnHR}[\\text{tf},\\ell , \\text{RandomReal}[\\{\\text{range}\/10,\\text{range}\\}]}\\\\\n\\pmb{\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];\\text{AppendTo}[\\text{listplots},}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{npts}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},\\text{range}]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsPosInhomogeneousHR2}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{range$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listplots}=\\{\\},\\text{listpos}=\\text{RandomSpherePoints}[\\text{npts}],\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell , \\text{RandomReal}[\\{\\text{range}\/10,\\text{range}\\}]}\\\\\n\\pmb{\\text{listpos}[[i]],\\{0,0,0\\},\\kappa ];\\text{AppendTo}[\\text{listplots},}\\\\\n\\pmb{\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}}\\\\\n\\pmb{\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},\\text{PlotStyle}\\to \\text{Red},\\text{Axes}\\to \\text{True},}\\\\\n\\pmb{\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]],\\{i,\\text{npts}\\}];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{RGBColor}[0,0.4,0.6],\\text{Sphere}[\\{0,0,0\\},\\text{range}]\\}]];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nStarting at origin with velocities radially outward\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelSphereSymmetric}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{SpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelSphereRandom}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{RandomSpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelInhomogeneous}[\\text{tf$\\_$},\\text{config$\\_$}?\\text{ListQ},\\ell \\_:1,\\kappa \\_,\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{RandomSpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{RelSoln}[\\text{tf},\\text{config},\\ell ,\\{0,0,0\\},(i\/100) \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},(i\/100) \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\nHopf--Ran\\~{a}da\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelSphereSymmetricHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{SpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelSphereRandomHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{v$\\_$},\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{RandomSpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},v \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},v \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{PlotsVelInhomogeneousHR}[\\text{tf$\\_$},\\ell \\_:1,\\kappa \\_,\\text{npts$\\_$}]\\text{:=}}\\\\\n\\pmb{\\text{Module}[\\{\\text{listvel}=\\text{RandomSpherePoints}[\\text{npts}],\\text{listplots}=\\{\\},\\text{TempSoln}\\},}\\\\\n\\pmb{\\text{Do}[\\text{TempSoln}=\\text{SolnHR}[\\text{tf},\\ell ,\\{0,0,0\\},(i\/100) \\text{listvel}[[i]],\\kappa ];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{ParametricPlot3D}[\\{\\text{r1}[t],\\text{r2}[t],\\text{r3}[t]\\}\\text{\/.}\\text{TempSoln},\\{t,0,\\text{tf}\\},}\\\\\n\\pmb{\\text{PlotStyle}\\to \\text{Hue}[i\/20],\\text{Axes}\\to \\text{True},\\text{AxesLabel}\\to \\{x,y,z\\},\\text{PlotRange}\\to \\text{All}]];}\\\\\n\\pmb{\\text{AppendTo}[\\text{listplots},\\text{Graphics3D}[\\{\\text{Hue}[i\/20],\\text{Arrowheads}[0.03],}\\\\\n\\pmb{\\text{Arrow}[\\text{Tube}[\\{\\{0,0,0\\},(i\/100) \\text{listvel}[[i]]\\}]]\\}]],\\{i,\\text{Length}@\\text{listvel}\\}];}\\\\\n\\pmb{\\text{Show}[\\text{listplots},\\text{PlotRange}\\text{-$>$}\\text{All}]}\\\\\n\\pmb{]}\\)\n\\end{doublespace}\n\n\n\n\n\\vspace{12pt}\n\\section{Fluctuation matrix}\n\\vspace{2pt}\n\n\\subsection*{j=0 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{U=\\text{Array}\\left[\\Phi _0,3,1\\right];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{V=\\text{Flatten}[\\text{Array}[\\Phi ,\\{3,3\\},\\{1,1\\}]];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rem3}[\\text{m$\\_$}]\\text{:=}\\text{QuotientRemainder}[m-1,3][[2]]+1}\\\\\n\\pmb{\\text{quo3}[\\text{n$\\_$}]\\text{:=}\\text{Quotient}[n-1,3]+1}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{num}[\\text{p$\\_$},\\text{a$\\_$}]\\text{:=}3\\text{QuotientRemainder}[p-1,3][[2]]}\\\\\n\\pmb{+\\text{QuotientRemainder}[a-1,3][[2]]+4}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00} =2(1+\\psi [\\tau ])^2\\text{IdentityMatrix}[3];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $01} = \\text{Table}[0,\\{m,3\\},\\{n,9\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}\\left[\\text{sq$\\Omega $01}[[m,n]]=\\left(-2 \\psi '[\\tau ]\\right)D[\\text{Sum}[\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,m,k]]\\right.}\\\\\n\\pmb{\\Phi [k,a],\\{k,1,3\\}],\\{a,1,3\\}],V[[n]]],\\{m,1,3\\},\\{n,1,9\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $10}=\\text{Transpose}[\\text{sq$\\Omega $01}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11d}=(2\\psi [\\tau ]^2+4\\psi [\\tau ]+6) \\text{IdentityMatrix}[9];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11od}=\\text{Table}[0,\\{m,9\\},\\{n,9\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}\\left[\\text{sq$\\Omega $11od}[[m,n]]=2\\left(\\psi [\\tau ]^2-\\psi [\\tau ]-2\\right)\\right.}\\\\\n\\pmb{D[\\text{Sum}[\\text{LeviCivitaTensor}[3][[k,\\text{quo3}[m],p]]}\\\\\n\\pmb{\\text{LeviCivitaTensor}[3][[k,\\text{rem3}[m],a]]}\\\\\n\\pmb{\\Phi [p,a],\\{k,1,3\\},\\{p,1,3\\},\\{a,1,3\\}],}\\\\\n\\pmb{V[[n]]],\\{m,1,9\\},\\{n,1,9\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11}=\\text{sq$\\Omega $11d}+\\text{sq$\\Omega $11od};}\\)\n\\end{doublespace}\n\n\\subsection*{j=1\/2 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{j=\\frac{1}{2};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{U=\\text{Flatten}\\left[\\text{Array}\\left[\\Phi _0,\\{2,3\\},\\{-1\/2,1\\}\\right]\\right]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{V=\\text{Flatten}[\\text{Array}[\\Phi ,\\{2,3,3\\},\\{-1\/2,1,1\\}]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{preFac}[\\text{exp$\\_$}]\\text{:=}\\exp \\text{\/.}\\{U[[1]]\\to 1,U[[2]]\\to 1,U[[3]]\\to 1,U[[4]]\\to 1,}\\\\\n\\pmb{U[[5]]\\to 1,U[[6]]\\to 1,V[[1]]\\to 1,V[[2]]\\to 1,V[[3]]\\to 1,V[[4]]\\to 1,}\\\\\n\\pmb{V[[5]]\\to 1,V[[6]]\\to 1,V[[7]]\\to 1,V[[8]]\\to 1,V[[9]]\\to 1,V[[10]]\\to 1,}\\\\\n\\pmb{V[[11]]\\to 1,V[[12]]\\to 1,V[[13]]\\to 1,V[[14]]\\to 1,V[[15]]\\to 1,}\\\\\n\\pmb{V[[16]]\\to 1,V[[17]]\\to 1,V[[18]]\\to 1\\}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Pos1}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,1\\right]}\\\\\n\\pmb{\\text{Pos2}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,2\\right]}\\\\\n\\pmb{\\text{Pos3}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,3\\right]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapRp}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j-\\text{Pos1}[\\exp ])(j+\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi _0[\\text{Pos1}[\\exp ]+1,\\text{Pos2}[\\exp ]]}\\\\\n\\pmb{\\text{UmapRm}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j+\\text{Pos1}[\\exp ])(j-\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi _0[\\text{Pos1}[\\exp ]-1,\\text{Pos2}[\\exp ]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapR}[1][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{1}{\\sqrt{2}}(\\text{UmapRm}[\\exp ]+\\text{UmapRp}[\\exp\n]),0\\right]}\\\\\n\\pmb{\\text{UmapR}[2][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{I}{\\sqrt{2}}(\\text{UmapRm}[\\exp ]-\\text{UmapRp}[\\exp ]),0\\right]}\\\\\n\\pmb{\\text{UmapR}[3][\\text{exp$\\_$}]\\text{:=}\\text{If}[\\text{Pos1}[\\exp ]<100,2I \\text{Pos1}[\\exp ]\\exp ,0]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapR}[1][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{UmapR}[2][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{UmapR}[3][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapRp}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j-\\text{Pos1}[\\exp ])(j+\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi [\\text{Pos1}[\\exp ]+1,\\text{Pos2}[\\exp ],\\text{Pos3}[\\exp ]]}\\\\\n\\pmb{\\text{VmapRm}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j+\\text{Pos1}[\\exp ])(j-\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi [\\text{Pos1}[\\exp ]-1,\\text{Pos2}[\\exp ],\\text{Pos3}[\\exp ]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapR}[1][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{1}{\\sqrt{2}}(\\text{VmapRm}[\\exp ]+\\text{VmapRp}[\\exp\n]),0\\right]}\\\\\n\\pmb{\\text{VmapR}[2][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{I}{\\sqrt{2}}(\\text{VmapRm}[\\exp ]-\\text{VmapRp}[\\exp ]),0\\right]}\\\\\n\\pmb{\\text{VmapR}[3][\\text{exp$\\_$}]\\text{:=}\\text{If}[\\text{Pos1}[\\exp ]<100,2I \\text{Pos1}[\\exp ]\\exp ,0]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapR}[1][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{VmapR}[2][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{VmapR}[3][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rem3}[\\text{i$\\_$}]\\text{:=}\\text{QuotientRemainder}[i-1,3][[2]]+1}\\\\\n\\pmb{\\text{quo3}[\\text{i$\\_$}]\\text{:=}\\text{Quotient}[i-1,3]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{j3}[\\text{m$\\_$}]\\text{:=}\\text{Quotient}[m-1,9]-1\/2}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{pNum}[\\text{m$\\_$}]\\text{:=}\\text{QuotientRemainder}[\\text{Quotient}[m-1,3],3][[2]]+1}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{num}[\\text{x$\\_$},\\text{p$\\_$},\\text{a$\\_$}]\\text{:=}3\\text{QuotientRemainder}[p-1,3][[2]]}\\\\\n\\pmb{+\\text{QuotientRemainder}[a-1,3][[2]]+9x+\\frac{23}{2}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00d} = \\left(3+2(1+\\psi [\\tau ])^2\\right)\\text{IdentityMatrix}[6];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00od} =\\text{Table}[0,\\{a,1,6\\},\\{b,1,6\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $00od}[[m,n]]=2(1+\\psi [\\tau ])D[\\text{Sum}[\\text{UmapR}[a][\\text{Sum}[}\\\\\n\\pmb{\\left.\\left.\\text{LeviCivitaTensor}[3][[a,\\text{rem3}[m],k]]\\Phi _0[\\text{quo3}[m]-1\/2,k],\\{k,1,3\\}\\right]\\right],}\\\\\n\\pmb{\\{a,1,3\\}],U[[n]]],\\{m,1,6\\},\\{n,1,6\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00}=\\text{sq$\\Omega $00d}+\\text{sq$\\Omega $00od};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $01} = \\text{Table}[0,\\{i,6\\},\\{j,18\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $01}[[m,n]]=(-2 \\psi '[\\tau ])D[\\text{Sum}[\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,\\text{rem3}[m],k]]}\\\\\n\\pmb{\\Phi [\\text{quo3}[m]-1\/2,k,a],\\{k,1,3\\}],\\{a,1,3\\}],V[[n]]],\\{m,1,6\\},\\{n,1,18\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $10}=\\text{Transpose}[\\text{sq$\\Omega $01}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11d}=\\left(7+2(1+\\psi [\\tau ])^2\\right)\\text{IdentityMatrix}[18];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11od}=\\text{Table}[0,\\{m,18\\},\\{n,18\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $11od}[[m,n]]=D[\\text{Sum}[2(1+\\psi [\\tau ])\\text{VmapR}[b][\\text{Sum}[}\\\\\n\\pmb{\\text{LeviCivitaTensor}[3][[b,\\text{pNum}[m],p]]\\Phi [\\text{j3}[m],p,\\text{rem3}[m]],\\{p,1,3\\}]]}\\\\\n\\pmb{+2\\text{VmapR}[b][\\text{Sum}[\\text{LeviCivitaTensor}[3][[b,\\text{rem3}[m],a]]\\Phi [\\text{j3}[m],}\\\\\n\\pmb{\\text{pNum}[m],a],\\{a,1,3\\}]]+2\\left(\\psi [\\tau ]^2-\\psi [\\tau ]-2\\right)}\\\\\n\\pmb{\\text{Sum}[\\text{LeviCivitaTensor}[3][[b,\\text{pNum}[m],p]]\\text{LeviCivitaTensor}[3][[b,\\text{rem3}[m],a]]}\\\\\n\\pmb{\\Phi [\\text{j3}[m],p,a],\\{a,1,3\\},\\{p,1,3\\}],\\{b,1,3\\}],V[[n]]],}\\\\\n\\pmb{\\{m,1,18\\},\\{n,1,18\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11}=\\text{sq$\\Omega $11d}+\\text{sq$\\Omega $11od};}\\)\n\\end{doublespace}\n\n\\subsection*{j=1 case}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{j=1;}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{U=\\text{Flatten}\\left[\\text{Array}\\left[\\Phi _0,\\{3,3\\},\\{-1,1\\}\\right]\\right]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{V=\\text{Flatten}[\\text{Array}[\\Phi ,\\{3,3,3\\},\\{-1,1,1\\}]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{preFac}[\\text{exp$\\_$}]\\text{:=}\\exp \\text{\/.}\\{U[[1]]\\to 1,U[[2]]\\to 1,U[[3]]\\to 1,U[[4]]\\to 1,}\\\\\n\\pmb{U[[5]]\\to 1,U[[6]]\\to 1,U[[7]]\\to 1,U[[8]]\\to 1,U[[9]]\\to 1,V[[1]]\\to 1,}\\\\\n\\pmb{V[[2]]\\to 1,V[[3]]\\to 1,V[[4]]\\to 1,V[[5]]\\to 1,V[[6]]\\to 1,V[[7]]\\to 1,}\\\\\n\\pmb{V[[8]]\\to 1,V[[9]]\\to 1,V[[10]]\\to 1,V[[11]]\\to 1,V[[12]]\\to 1,V[[13]]\\to 1,}\\\\\n\\pmb{V[[14]]\\to 1,V[[15]]\\to 1,V[[16]]\\to 1,V[[17]]\\to 1,V[[18]]\\to 1,V[[19]]\\to 1,}\\\\\n\\pmb{V[[20]]\\to 1,V[[21]]\\to 1,V[[22]]\\to 1,V[[23]]\\to 1,}\\\\\n\\pmb{V[[24]]\\to 1,V[[25]]\\to 1,V[[26]]\\to 1,V[[27]]\\to 1\\}}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Pos1}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,1\\right]}\\\\\n\\pmb{\\text{Pos2}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,2\\right]}\\\\\n\\pmb{\\text{Pos3}[\\text{exp$\\_$}]\\text{:=}\\text{Extract}\\left[\\text{preFac}[\\exp ]^{-1}\\exp ,3\\right]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapRp}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j-\\text{Pos1}[\\exp ])(j+\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi _0[\\text{Pos1}[\\exp ]+1,\\text{Pos2}[\\exp ]]}\\\\\n\\pmb{\\text{UmapRm}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j+\\text{Pos1}[\\exp ])(j-\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi _0[\\text{Pos1}[\\exp ]-1,\\text{Pos2}[\\exp ]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapR}[1][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{1}{\\sqrt{2}}(\\text{UmapRm}[\\exp ]+\\text{UmapRp}[\\exp\n]),0\\right]}\\\\\n\\pmb{\\text{UmapR}[2][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{I}{\\sqrt{2}}(\\text{UmapRm}[\\exp ]-\\text{UmapRp}[\\exp ]),0\\right]}\\\\\n\\pmb{\\text{UmapR}[3][\\text{exp$\\_$}]\\text{:=}\\text{If}[\\text{Pos1}[\\exp ]<100,2I \\text{Pos1}[\\exp ]\\exp ,0]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{UmapR}[1][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{UmapR}[2][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{UmapR}[3][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapRp}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j-\\text{Pos1}[\\exp ])(j+\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi [\\text{Pos1}[\\exp ]+1,\\text{Pos2}[\\exp ],\\text{Pos3}[\\exp ]]}\\\\\n\\pmb{\\text{VmapRm}[\\text{exp$\\_$}]\\text{:=}-2I\\sqrt{\\frac{(j+\\text{Pos1}[\\exp ])(j-\\text{Pos1}[\\exp ]+1)}{2}}\\text{preFac}[\\exp ]}\\\\\n\\pmb{\\Phi [\\text{Pos1}[\\exp ]-1,\\text{Pos2}[\\exp ],\\text{Pos3}[\\exp ]]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapR}[1][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{1}{\\sqrt{2}}(\\text{VmapRm}[\\exp ]+\\text{VmapRp}[\\exp\n]),0\\right]}\\\\\n\\pmb{\\text{VmapR}[2][\\text{exp$\\_$}]\\text{:=}\\text{If}\\left[\\text{Pos1}[\\exp ]<100,\\frac{I}{\\sqrt{2}}(\\text{VmapRm}[\\exp ]-\\text{VmapRp}[\\exp ]),0\\right]}\\\\\n\\pmb{\\text{VmapR}[3][\\text{exp$\\_$}]\\text{:=}\\text{If}[\\text{Pos1}[\\exp ]<100,2I \\text{Pos1}[\\exp ]\\exp ,0]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{VmapR}[1][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{VmapR}[2][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\\\\n\\pmb{\\text{VmapR}[3][\\text{exp$\\_$}]\\text{:=}0\\text{\/;}\\exp \\text{==}0}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{rem3}[\\text{i$\\_$}]\\text{:=}\\text{QuotientRemainder}[i-1,3][[2]]+1}\\\\\n\\pmb{\\text{quo3}[\\text{i$\\_$}]\\text{:=}\\text{Quotient}[i-1,3]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{j3}[\\text{m$\\_$}]\\text{:=}\\text{Quotient}[m-1,9]-1}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{pNum}[\\text{m$\\_$}]\\text{:=}\\text{QuotientRemainder}[\\text{Quotient}[m-1,3],3][[2]]+1}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00d} = \\left(8+2(1+\\psi [\\tau ])^2\\right)\\text{IdentityMatrix}[9];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00od} =\\text{Table}[0,\\{a,1,9\\},\\{b,1,9\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $00od}[[m,n]]=2(1+\\psi [\\tau ])D[\\text{Sum}[\\text{UmapR}[a][\\text{Sum}[}\\\\\n\\pmb{\\left.\\left.\\text{LeviCivitaTensor}[3][[a,\\text{rem3}[m],k]]\\Phi _0[\\text{quo3}[m]-1,k],\\{k,1,3\\}\\right]\\right],}\\\\\n\\pmb{\\{a,1,3\\}],U[[n]]],\\{m,1,9\\},\\{n,1,9\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00}=\\text{sq$\\Omega $00d}+\\text{sq$\\Omega $00od};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00d} = \\left(8+2(1+\\psi [\\tau ])^2\\right)\\text{IdentityMatrix}[9];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00od} =\\text{Table}[0,\\{a,1,9\\},\\{b,1,9\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $00od}[[m,n]]=2(1+\\psi [\\tau ])D[\\text{Sum}[\\text{UmapR}[a][\\text{Sum}[}\\\\\n\\pmb{\\left.\\left.\\text{LeviCivitaTensor}[3][[a,\\text{rem3}[m],k]]\\Phi _0[\\text{quo3}[m]-1,k],\\{k,1,3\\}\\right]\\right],}\\\\\n\\pmb{\\{a,1,3\\}],U[[n]]],\\{m,1,9\\},\\{n,1,9\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $00}=\\text{sq$\\Omega $00d}+\\text{sq$\\Omega $00od};}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $01} = \\text{Table}[0,\\{i,9\\},\\{j,27\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $01}[[m,n]]=(-2 \\psi '[\\tau ])D[\\text{Sum}[\\text{Sum}[\\text{LeviCivitaTensor}[3][[a,\\text{rem3}[m],k]]}\\\\\n\\pmb{\\Phi [\\text{quo3}[m]-1,k,a],\\{k,1,3\\}],\\{a,1,3\\}],V[[n]]],\\{m,1,9\\},\\{n,1,27\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $10}=\\text{Transpose}[\\text{sq$\\Omega $01}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11d}=\\left(12+2(1+\\psi [\\tau ])^2\\right)\\text{IdentityMatrix}[27];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11od}=\\text{Table}[0,\\{m,27\\},\\{n,27\\}];}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{Do}[\\text{sq$\\Omega $11od}[[m,n]]=D[\\text{Sum}[2(1+\\psi [\\tau ])\\text{VmapR}[b][\\text{Sum}[}\\\\\n\\pmb{\\text{LeviCivitaTensor}[3][[b,\\text{pNum}[m],p]]\\Phi [\\text{j3}[m],p,\\text{rem3}[m]],\\{p,1,3\\}]]}\\\\\n\\pmb{+2\\text{VmapR}[b][\\text{Sum}[\\text{LeviCivitaTensor}[3][[b,\\text{rem3}[m],a]]}\\\\\n\\pmb{\\Phi [\\text{j3}[m],\\text{pNum}[m],a],\\{a,1,3\\}]]}\\\\\n\\pmb{+2\\left(\\psi [\\tau ]^2-\\psi [\\tau ]-2\\right)\\text{Sum}[\\text{LeviCivitaTensor}[3][[b,\\text{pNum}[m],p]]}\\\\\n\\pmb{\\text{LeviCivitaTensor}[3][[b,\\text{rem3}[m],a]]}\\\\\n\\pmb{\\Phi [\\text{j3}[m],p,a],\\{a,1,3\\},\\{p,1,3\\}],\\{b,1,3\\}],V[[n]]],}\\\\\n\\pmb{\\{m,1,27\\},\\{n,1,27\\}]}\\)\n\\end{doublespace}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $11}=\\text{sq$\\Omega $11d}+\\text{sq$\\Omega $11od};}\\)\n\\end{doublespace}\n\n\\subsection*{Matrix eigenvalues}\n\n\\begin{doublespace}\n\\noindent\\(\\pmb{\\text{sq$\\Omega $} =\\{\\{\\text{sq$\\Omega $00},\\text{sq$\\Omega $01}\\},\\{\\text{sq$\\Omega $10},\\text{sq$\\Omega $11}\\}\\}\\text{\/\/}\\text{ArrayFlatten};}\\) \\\\\n\\(\\pmb{\\text{sq$\\Omega $no$\\psi $dot}=\\text{sq$\\Omega $}\\text{\/.}\\psi '[\\tau ]\\to 0;}\\) \\\\\n\\(\\pmb{\\text{sq$\\Omega $no$\\psi $}=\\text{sq$\\Omega $}\\text{\/.}\\{\\psi [\\tau ]\\to 0,\\psi '[\\tau ]\\to 0\\};}\\) \\\\\n\\(\\pmb{\\text{Eigenvalues}[\\text{sq$\\Omega $no$\\psi $}]}\\) \\\\\n\\(\\pmb{\\text{Eigenvalues}[\\text{sq$\\Omega $no$\\psi $dot}]}\\) \\\\\n\\(\\pmb{\\text{Eigenvalues}[\\text{sq$\\Omega $}]}\\)\n\\end{doublespace}\n\n\\chapter{Polynomials in the characteristic equation}\n\\label{appendPolynomials}\n\n\n\\begin{center}\n\\resizebox{\\linewidth}{10cm}{\n\\begin{tabular}{| P{10mm} | P{25mm} | P{45mm} | P{80mm} |} \n\\hline\n$j\\ ,\\ v $ & P($\\lambda$) & Q($\\lambda$)  & R($\\lambda$) \\\\ [0.5ex] \n \\hline\\hline\n$0\\ ,\\ 0$ \\phantom{\\Big|} & N\/A & $(-2 + 6 \\psi^2) - \\lambda$ & $-(2 - 6 \\psi^2) + \\lambda$ \\\\\n \\hline\n$0\\ ,\\ 1$ \\phantom{\\Big|} & 1 & $(2 + 2 \\psi + 4 \\psi^2) - \\lambda$ & $(4 + 12\\psi + 20 \\psi^2 + 20 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) - (4 + 6 \\psi + 6 \\psi^2) \\lambda + \\lambda^2$ \\\\\n \\hline\n$0\\ ,\\ 2$ \\phantom{\\Big|} & N\/A & $(10 + 6 \\psi) - \\lambda$ & N\/A \\\\\n \\hline\n$\\sfrac12\\ ,\\ \\sfrac12$ \\phantom{\\Big|} & $-(1 + 6 \\psi^2) + \\lambda$ & $-(1 + 2 \\psi^2 + 24 \\psi^4) + (2 + 10 \\psi^2) \\lambda - \\lambda^2$ & $-(1 + 4 \\psi^2 + 28 \\psi^4 + 48 \\psi^6 - 8 \\dot{\\psi}^2 - 48 \\psi^2 \\dot{\\psi}^2) + (3 + 16 \\psi^2 + 44 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (3 + 12 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\ \n \\hline\n$\\sfrac12\\ ,\\ \\sfrac32$ \\phantom{\\Big|} & $-(7 + 3 \\psi) + \\lambda$ & $-(49 + 42 \\psi + 32 \\psi^2 + 12 \\psi^3)   + (14 + 6 \\psi + 4 \\psi^2) \\lambda - \\lambda^2$ & $-(343 + 588 \\psi + 574 \\psi^2 + 360 \\psi^3 + 136 \\psi^4 + 24 \\psi^5 - 56 \\dot{\\psi}^2 - 24 \\psi \\dot{\\psi}^2) + (147 + 168 \\psi + 124 \\psi^2 + 48 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (21 + 12 \\psi + 6 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\ \n \\hline\n$\\sfrac12\\ ,\\ \\sfrac52$ \\phantom{\\Big|} & N\/A & $(17 + 8 \\psi) - \\lambda$ & N\/A \\\\\n \\hline\n$1\\ ,\\ 0$ \\phantom{\\Big|} & $1$ & $(2 - 2 \\psi + 4 \\psi^2) - \\lambda$ & $( 4 - 12 \\psi + 20 \\psi^2 - 20 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) - (4 - 6 \\psi + 6 \\psi^2) \\lambda + \\lambda^2$ \\\\\n \\hline\n$1\\ ,\\ 1$ \\phantom{\\Big|} & $(36 + 36 \\psi^2) - (12 + 6\\psi^2) \\lambda + \\lambda^2$ & $(216 + 192 \\psi^2 + 104 \\psi^4) - (108 + 92 \\psi^2 + 24 \\psi^4) \\lambda + (18 + 10 \\psi^2) \\lambda^2 - \\lambda^3$ & $(1296 + 1584 \\psi^2 + 1008 \\psi^4 + 208 \\psi^6 - 288 \\dot{\\psi}^2 - 288 \\psi^2 \\dot{\\psi}^2) - (864 + 960 \\psi^2 + 432 \\psi^4 + 48 \\psi^6 - 96 \\dot{\\psi}^2 - 48 \\psi^2 \\dot{\\psi}^2) \\lambda + (216 + 188 \\psi^2 + 44 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda^2 - (24 + 12\\psi^2) \\lambda^3 + \\lambda^4$ \\\\\n \\hline\n$1\\ ,\\ 2$ \\phantom{\\Big|} & $-(14 + 4 \\psi) + \\lambda$ & $-(196 + 112 \\psi + 60 \\psi^2 + 16 \\psi^3) + (28 + 8 \\psi + 4 \\psi^2) \\lambda - \\lambda^2$ & $-(2744 + 3136 \\psi + 2128 \\psi^2 + 928 \\psi^3 + 248 \\psi^4 + 32 \\psi^5 - 112 \\dot{\\psi}^2 - 32 \\psi \\dot{\\psi}^2) + (588 + 448 \\psi + 236 \\psi^2 + 64 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (42 + 16 \\psi + 6 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\\n \\hline\n$1\\ ,\\ 3$ \\phantom{\\Big|} & N\/A & $(26 + 10 \\psi) - \\lambda$ & N\/A \\\\ \n \\hline\n$\\sfrac32\\ ,\\ \\sfrac12$ \\phantom{\\Big|} & $-(7 - 3 \\psi ) + \\lambda$ & $-(49 - 42 \\psi + 32 \\psi^2 - 12 \\psi^3) + (14 - 6 \\psi+ 4 \\psi^2) \\lambda - \\lambda^2$ & $-(343 - 588 \\psi + 574 \\psi^2 - 360 \\psi^3 + 136 \\psi^4 - 24 \\psi^5 - 56 \\dot{\\psi}^2 + 24 \\psi \\dot{\\psi}^2) + (147 - 168 \\psi + 124 \\psi^2 - 48 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (21 - 12 \\psi + 6 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\ \n \\hline\n$\\sfrac32\\ ,\\ \\sfrac32$ \\phantom{\\Big|} & $(169 + 78 \\psi^2) - (26 + 6 \\psi^2) \\lambda + \\lambda^2$ & $(2197 + 962 \\psi^2 + 216 \\psi^4) - (507 + 204 \\psi^2 + 24 \\psi^4) \\lambda + (39 + 10 \\psi^2) \\lambda^2 - \\lambda^3$ & $(28561 + 16900 \\psi^2 + 4732 \\psi^4 + 432 \\psi^6 - 1352 \\dot{\\psi}^2 - 624 \\psi^2 \\dot{\\psi}^2) + (-8788 - 4628 \\psi^2 - 936 \\psi^4 - 48 \\psi^6 + 208 \\dot{\\psi}^2 + 48 \\psi^2 \\dot{\\psi}^2) \\lambda + (1014 + 412 \\psi^2 + 44 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda^2 - (52 + 12 \\psi^2) \\lambda^3 + \\lambda^4$ \\\\\n \\hline\n$\\sfrac32\\ ,\\ \\sfrac52$ \\phantom{\\Big|} & $-(23 + 5 \\psi) + \\lambda$ & $-(529 + 230 \\psi + 96 \\psi^2 + 20 \\psi^3) + (46 + 10 \\psi + 4 \\psi^2) \\lambda - \\lambda^2$ & $-(12167 + 10580 \\psi + 5566 \\psi^2 + 1880 \\psi^3 + 392 \\psi^4 + 40 \\psi^5 - 184 \\dot{\\psi}^2 + 40 \\psi\\dot{\\psi}^2) + (1587 + 920 \\psi + 380 \\psi^2 + 80 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda + (-69 - 20 \\psi - 6 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\\n \\hline\n$\\sfrac32\\ ,\\ \\sfrac72$ \\phantom{\\Big|} & N\/A & $(37 + 12 \\psi) - \\lambda$ & N\/A \\\\ \n \\hline\n$2\\ ,\\ 0$ \\phantom{\\Big|} & N\/A & $(10- 6 \\psi) - \\lambda$ & N\/A \\\\\n \\hline\n$2\\ ,\\ 1$ \\phantom{\\Big|} & $-(14 - 4 \\psi) + \\lambda$ & $-(196 - 112 \\psi + 60 \\psi^2 - 16 \\psi^3) + (28 - 8 \\psi + 4 \\psi^2) \\lambda - \\lambda^2$ & $-(2744 - 3136 \\psi + 2128 \\psi^2 - 928 \\psi^3 + 248 \\psi^4 - 32 \\psi^5 - 112 \\dot{\\psi}^2 + 32 \\psi \\dot{\\psi}^2) + (588 - 448 \\psi + 236 \\psi^2 - 64 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (42 - 16 \\psi + 6 \\psi^2) \\lambda^2 + \\lambda^3$ \\\\\n \\hline\n$2\\ ,\\ 2$ \\phantom{\\Big|} & $(484 + 132 \\psi^2) - (44 + 6 \\psi^2) \\lambda + \\lambda^2$ & $(10648 + 2816 \\psi^2 + 360 \\psi^4) - (1452 + 348 \\psi^2 + 24 \\psi^4) \\lambda + (66 + 10 \\psi^2) \\lambda^2 - \\lambda^3$ & $(234256 + 83248 \\psi^2 + 13552 \\psi^4 + 720 \\psi^6 - 3872 \\dot{\\psi}^2 - 1056 \\psi^2 \\dot{\\psi}^2) - (42592 + 13376 \\psi^2 + 1584 \\psi^4 + 48 \\psi^6 - 352 \\dot{\\psi}^2 - 48 \\psi^2 \\dot{\\psi}^2) \\lambda + (2904 + 700 \\psi^2 + 44 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda^2 - \n (88 + 12 \\psi^2) \\lambda^3 + \\lambda^4$ \\\\\n \\hline\n$2\\ ,\\ 3$ \\phantom{\\Big|} & $-(34 + 6 \\psi) + \\lambda$ & $-(1156 + 408 \\psi + 140 \\psi^2 + 24 \\psi^3) + (68 + 12 \\psi + 4 \\psi^2) \\lambda - \\lambda^2$ & $-(39304 + 27744 \\psi + 11968 \\psi^2 + 3312 \\psi^3 + 568 \\psi^4 + 48 \\psi^5 - 272 \\dot{\\psi}^2 - 48 \\psi \\dot{\\psi}^2) + (3468 + 1632 \\psi + 556 \\psi^2 + 96 \\psi^3 + 8 \\psi^4 - 8 \\dot{\\psi}^2) \\lambda - (102 + 24 \\psi + 6\\psi^2) \\lambda^2 + \\lambda^3$ \\\\ \n \\hline\n$2\\ ,\\ 4$ \\phantom{\\Big|} & N\/A & $(50 + 14 \\psi) - \\lambda$ & N\/A \\\\ \n \\hline\n\\end{tabular}\n}\n\\end{center}\n\n\\restoregeometry\n\\chapter{Rotation of indices}\n\n\\label{appendRotation}\n\\noindent\nBy construction the gauge potential ${\\cal A}$ is $SO(4)$ invariant, which means it is also invariant under the action of $SO(3)$ generators $\\mathcal{D}_a$ \\eqref{DPdef}. For a complex-valued ${\\cal A} = {\\cal A}_a\\,e^a$ expanded using \\eqref{Asep} (for type I) and \\eqref{XY} at a fixed $j$ this means \n\\begin{equation}\n\\begin{aligned}\n    0 &\\= \\mathcal{D}_a ({\\cal A}) \\\\\n    &\\= \\sum_{m,n,\\tilde{n}} C^{n,\\tilde{n}}_{j,b}\\, \\mathrm{e}^{\\mathrm{i}\\Omega\\tau}\\, \\left( \\mathcal{D}_a(\\Lambda_{m,\\tilde{n}})\\,e^b\\,Y_{j;m,n} + \\Lambda_{m,\\tilde{n}}\\,\\mathcal{D}_a(e^b)\\,Y_{j;m,n} + \\Lambda_{m,\\tilde{n}}\\,e^b\\,\\mathcal{D}_a(Y_{j;m,n})  \\right)\n\\end{aligned}    \n\\end{equation}\nwhere $\\mathcal{D}_a(e^b)$ are determined from \\eqref{LieActionL} and \\eqref{LieActionR} while $\\mathcal{D}_a(Y_{j;m,n})$ are determined from (\\ref{Y-action}-\\ref{JpmAction}). By collecting the coefficients of various linearly independent $e^b$ and $Y_{j;m,n}$ terms in the above expansion for a fixed $\\mathcal{D}_a$ one gets a set of coupled linear equations for $\\mathcal{D}_a(\\Lambda_{m,\\tilde{n}})$, which can be easily solved. The action of the generators $\\mathcal{D}_a$ on $\\Lambda_{m,\\tilde{n}}$ for $j=0,~ 1\/2 ~\\textrm{and}~ 1$ is given in the following table.\n\n\\begin{center}\n\\begin{adjustbox}{width=\\linewidth}\n    \\begin{tabular}{|P{3.25cm}|P{6.3cm}|P{6.25cm}|P{1.8cm}|}\n        \\hline\n      & $\\mathcal{D}_1$ & $\\mathcal{D}_2$ & $\\mathcal{D}_3$ \\\\ [0.5ex] \\hline \\hline\n     $\\left.j=0: \\right.                                               \n     \\begin{aligned}\n        &\\Lambda_{0,-1} \\mapsto \\\\\n        &\\Lambda_{0,0} \\mapsto \\\\\n        &\\Lambda_{0,1} \\mapsto\n     \\end{aligned}$ & \n     $\\begin{aligned}                                              \n         &\\sqrt{2}\\mathrm{i} \\Lambda_{0,0} \\\\\n         &\\sqrt{2}\\mathrm{i} (\\Lambda_{0,1}+\\Lambda_{0,-1}) \\\\\n         &\\sqrt{2}\\mathrm{i} \\Lambda_{0,0}\n     \\end{aligned}$ & \n     $\\begin{aligned}                                              \n         &-\\sqrt{2}  \\Lambda_{0,0} \\\\\n         &\\sqrt{2}  (\\Lambda_{0,-1} - \\Lambda_{0,1}) \\\\\n         &\\sqrt{2}  \\Lambda_{0,0} \n     \\end{aligned}$ &\n     $\\begin{aligned}                                              \n         &-\\sqrt{2}\\mathrm{i}  \\Lambda_{0,-1} \\\\\n         &\\quad 0 \\\\\n         &\\sqrt{2}\\mathrm{i}  \\Lambda_{0,1}\n     \\end{aligned}$ \\\\ \\hline\n     $\\left.\\begin{aligned}\n        &\\quad j=\\sfrac12: \\\\                                               \n        &\\underbrace{\\textrm{Notation}}\\\\\n        &\\pm\\sfrac12\\equiv\\pm \\\\\n        &\\pm\\sfrac32\\equiv\\uparrow\\downarrow\n     \\end{aligned}\\right\\}\n     \\begin{aligned}\n        &\\Lambda_{-,\\downarrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,-} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,+} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,\\uparrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,\\downarrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,-} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,+} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,\\uparrow} \\mapsto\n     \\end{aligned}$ & \n     $\\begin{aligned}                                              \n         &\\mathrm{i}(\\sqrt{3}\\Lambda_{-,-} + \\Lambda_{+,\\downarrow} ) \\\\\n         &\\mathrm{i}(\\sqrt{3}\\Lambda_{-,\\downarrow} + 2\\Lambda_{-,+} + \\Lambda_{+,-} ) \\\\\n         &\\mathrm{i}(2\\Lambda_{-,-} + \\sqrt{3}\\Lambda_{-,\\uparrow} + \\Lambda_{+,+} ) \\\\\n         &\\mathrm{i}(\\sqrt{3}\\Lambda_{-,+} + \\Lambda_{+,\\uparrow} ) \\\\\n         &\\mathrm{i}(\\Lambda_{-,\\downarrow} + \\sqrt{3}\\Lambda_{+,-} ) \\\\\n         &\\mathrm{i}(\\Lambda_{-,-} + \\sqrt{3}\\Lambda_{+,\\downarrow} + 2 \\Lambda_{+,+} ) \\\\\n         &\\mathrm{i}(\\Lambda_{-,+} + 2\\Lambda_{+,-} + \\sqrt{3}\\Lambda_{+,\\uparrow} ) \\\\\n         &\\mathrm{i}(\\Lambda_{-,\\uparrow} + \\sqrt{3}\\Lambda_{+,+} ) \\\\\n     \\end{aligned}$ &\n     $\\begin{aligned}                                              \n         &-\\sqrt{3}\\Lambda_{-,-} - \\Lambda_{+,\\downarrow} \\\\\n         &\\sqrt{3}\\Lambda_{-,\\downarrow} - 2\\Lambda_{-,+} - \\Lambda_{+,-} \\\\\n         &2\\Lambda_{-,-} - \\sqrt{3}\\Lambda_{-,\\uparrow} - \\Lambda_{+,+} \\\\\n         &\\sqrt{3}\\Lambda_{-,+} - \\Lambda_{+,\\uparrow} \\\\\n         &\\Lambda_{-,\\downarrow} - \\sqrt{3}\\Lambda_{+,-} \\\\\n         &\\Lambda_{-,-} + \\sqrt{3}\\Lambda_{+,\\downarrow} -2\\Lambda_{+,+} \\\\\n         &\\Lambda_{-,+} +2\\Lambda_{+,-} -\\sqrt{3}\\Lambda_{+,\\uparrow} \\\\\n         &\\Lambda_{-,\\uparrow} + \\sqrt{3}\\Lambda_{+,+} \\\\\n     \\end{aligned}$ &\n     $\\begin{aligned}                                              \n         &-4\\mathrm{i}\\Lambda_{-,\\downarrow} \\\\[.2em]\n         &-2\\mathrm{i}\\Lambda_{-,-} \\\\[.2em]\n         &0 \\\\[.2em]\n         &2\\mathrm{i}\\Lambda_{-,\\uparrow} \\\\[.2em]\n         &-2\\mathrm{i}\\Lambda_{+,\\downarrow} \\\\[.2em]\n         &0 \\\\[.2em]\n         &2\\mathrm{i}\\Lambda_{+,+} \\\\[.2em]\n         &4\\mathrm{i}\\Lambda_{+,\\uparrow}\n     \\end{aligned}$ \\\\ \\hline\n     $\\left.\\begin{aligned}\n        &\\quad j=1: \\\\                                               \n        &\\underbrace{\\textrm{Notation}}\\\\\n        &\\pm1\\equiv\\pm \\\\\n        &\\pm2\\equiv\\uparrow\\downarrow\n     \\end{aligned}\\right\\}\n     \\begin{aligned}\n        &\\Lambda_{-,\\downarrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,-} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,0} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,+} \\mapsto \\\\[.2em]\n        &\\Lambda_{-,\\uparrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{0,\\downarrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{0,-} \\mapsto \\\\[.2em]\n        &\\Lambda_{0,0} \\mapsto \\\\[.2em]\n        &\\Lambda_{0,+} \\mapsto \\\\[.2em]\n        &\\Lambda_{0,\\uparrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,\\downarrow} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,-} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,0} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,+} \\mapsto \\\\[.2em]\n        &\\Lambda_{+,\\uparrow} \\mapsto\n     \\end{aligned}$ & \n     $\\begin{aligned}                                              \n         &\\mathrm{i}(2\\Lambda_{-,-} + \\sqrt{2}\\Lambda_{0,\\downarrow} ) \\\\\n         &\\mathrm{i}(2\\Lambda_{-,\\downarrow} + \\sqrt{6}\\Lambda_{-,0} + \\sqrt{2}\\Lambda_{0,-} ) \\\\\n         &\\mathrm{i}(\\sqrt{6}\\Lambda_{-,-} + \\sqrt{6}\\Lambda_{-,+} + \\sqrt{2}\\Lambda_{0,0} ) \\\\\n         &\\mathrm{i}(\\sqrt{6}\\Lambda_{-,0} + 2\\Lambda_{-,\\uparrow} + \\sqrt{2}\\Lambda_{0,+} ) \\\\\n         &\\mathrm{i}(2\\Lambda_{-,+} + \\sqrt{2}\\Lambda_{+,\\uparrow} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{-,\\downarrow} + 2\\Lambda_{0,-} + \\sqrt{2}\\Lambda_{+,\\downarrow} ) \\\\\n         &\\mathrm{i}\\sqrt{2}(\\Lambda_{-,-} + \\sqrt{2}\\Lambda_{0,\\downarrow} + \\sqrt{3}\\Lambda_{0,0} + \\Lambda_{+,-} ) \\\\\n         &\\mathrm{i}\\sqrt{2}(\\Lambda_{-,0} + \\sqrt{3}\\Lambda_{0,-} + \\sqrt{3}\\Lambda_{0,+} + \\Lambda_{+,0} ) \\\\\n         &\\mathrm{i}\\sqrt{2}(\\Lambda_{-,+} + \\sqrt{3}\\Lambda_{0,0} + \\sqrt{2}\\Lambda_{0,\\uparrow} + \\Lambda_{+,+} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{-,\\uparrow} + 2\\Lambda_{0,+}  + \\sqrt{2}\\Lambda_{+,\\uparrow} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{0,\\downarrow} + 2\\Lambda_{+,-} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{0,-} + 2\\Lambda_{+,\\downarrow} + \\sqrt{6}\\Lambda_{+,0} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{0,0} + \\sqrt{6}\\Lambda_{+,-} + \\sqrt{6}\\Lambda_{+,+} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{0,+} + \\sqrt{6}\\Lambda_{+,0} + 2\\Lambda_{+,\\uparrow} ) \\\\\n         &\\mathrm{i}(\\sqrt{2}\\Lambda_{0,\\uparrow} + 2\\Lambda_{+,+} ) \\\\        \n     \\end{aligned}$ &\n     $\\begin{aligned}                                              \n         &-2\\Lambda_{-,-} - \\sqrt{2}\\Lambda_{0,\\downarrow} \\\\\n         &2\\Lambda_{-,\\downarrow} - \\sqrt{6}\\Lambda_{-,0} - \\sqrt{2}\\Lambda_{0,-} \\\\\n         &\\sqrt{6}\\Lambda_{-,-} - \\sqrt{6}\\Lambda_{-,+} - \\sqrt{2}\\Lambda_{0,0} \\\\\n         &\\sqrt{6}\\Lambda_{-,0} - 2\\Lambda_{-,\\uparrow} - \\sqrt{2}\\Lambda_{0,+} \\\\\n         &2\\Lambda_{-,+} - \\sqrt{2}\\Lambda_{0,\\uparrow} \\\\\n         &\\sqrt{2}\\Lambda_{-,\\downarrow} - 2\\Lambda_{0,-} - \\sqrt{2}\\Lambda_{+,\\downarrow} \\\\\n         &\\sqrt{2}(\\Lambda_{-,-} + \\sqrt{2}\\Lambda_{0,\\downarrow} - \\sqrt{3}\\Lambda_{0,0} - \\Lambda_{+,-}) \\\\\n         &\\sqrt{2}(\\Lambda_{-,0} + \\sqrt{3}\\Lambda_{0,-} - \\sqrt{3}\\Lambda_{0,+} - \\Lambda_{+,0}) \\\\\n         &\\sqrt{2}(\\Lambda_{-,+} + \\sqrt{3}\\Lambda_{0,0} - \\sqrt{2}\\Lambda_{0,\\uparrow} - \\Lambda_{+,+}) \\\\\n         &\\sqrt{2}\\Lambda_{-,\\uparrow} + 2\\Lambda_{0,+} - \\sqrt{2}\\Lambda_{+,\\uparrow} \\\\\n         &\\sqrt{2}\\Lambda_{0,\\downarrow} + -2\\Lambda_{+,-} \\\\\n         &\\sqrt{2}\\Lambda_{0,-} + 2\\Lambda_{+,\\downarrow} - \\sqrt{6}\\Lambda_{+,0} \\\\\n         &\\sqrt{2}\\Lambda_{0,0} + \\sqrt{6}\\Lambda_{+,-} - \\sqrt{6}\\Lambda_{+,+} \\\\\n         &\\sqrt{2}\\Lambda_{0,+} + \\sqrt{6}\\Lambda_{+,0} - 2\\Lambda_{+,\\uparrow} \\\\\n         &\\sqrt{2}\\Lambda_{0,\\uparrow} + 2\\Lambda_{+,+} \\\\\n     \\end{aligned}$ &\n     $\\begin{aligned}                                              \n         &-6\\mathrm{i}\\Lambda_{-,\\downarrow} \\\\[.2em]\n         &-4\\mathrm{i}\\Lambda_{-,-} \\\\[.2em]\n         &-2\\mathrm{i}\\Lambda_{-,0} \\\\[.2em]\n         &0 \\\\[.2em]\n         &2\\mathrm{i}\\Lambda_{-,\\uparrow} \\\\[.2em]\n         &-4\\mathrm{i}\\Lambda_{0,\\downarrow} \\\\[.2em]\n         &-2\\mathrm{i}\\Lambda_{0,-} \\\\[.2em]\n         &0 \\\\[.2em]\n         &2\\mathrm{i}\\Lambda_{0,+} \\\\[.2em]\n         &4\\mathrm{i}\\Lambda_{0,\\uparrow} \\\\[.2em]\n         &-2\\mathrm{i}\\Lambda_{+,\\downarrow} \\\\[.2em]\n         &0 \\\\[.2em]\n         &2\\mathrm{i}\\Lambda_{+,0} \\\\[.2em]\n         &4\\mathrm{i}\\Lambda_{+,+} \\\\[.2em]\n         &6\\mathrm{i}\\Lambda_{+,\\uparrow}\n     \\end{aligned}$ \\\\ \\hline\n    \\end{tabular}\n\n\n\\end{adjustbox} \n\\end{center}\n\n\\restoregeometry\n\n\\nopagebreak[0]\n\\chapter{Carter-Penrose transformation}\n\\label{appendCPtransf}\n\\noindent\nThe metric on the Minkowski space $\\mathds R^{1,3}$ in polar coordinates is given by\n\\begin{equation}\n    \\mathrm{d} s_{Mink}^2 \\= -\\mathrm{d} t^2 + \\mathrm{d} r^2 + r^2\\,\\mathrm{d}\\Omega_2^2\\ ;\\qquad \\mathrm{d}\\Omega_2^2 \\= \\mathrm{d}\\theta^2 + \\sin^2\\theta\\,\\mathrm{d}\\phi^2 ,\n\\end{equation}\nwhere $-\\infty < t < \\infty,\\ 0 \\leq r < \\infty,\\ 0 \\leq \\theta \\leq \\pi$, and $0 \\leq \\phi \\leq 2\\pi$. We first employ the light-cone coordinates $(u,v)$ to transform the metric in the following way\n\\begin{equation}\\label{transf1}\n\\begin{aligned}\n    &u\\ :=\\ t-r\\ ,\\quad v\\ :=\\ t+r\\ ;\\quad  -\\infty < u \\leq v < \\infty\\  \\\\[2pt]\n    &\\implies \\mathrm{d} s_{Mink}^2 \\= -\\mathrm{d} v\\, \\mathrm{d} u + \\sfrac14 (v-u)^2\\,\\mathrm{d}\\Omega_2^2\\ .\n\\end{aligned}\n\\end{equation}\nIn the second step we compactify the spacetime with the help of the coordinate (U,V) as follows:\n\\begin{equation}\\label{transf2}\n\\begin{aligned}\n   &U\\ :=\\ \\arctan u\\ ,\\quad V\\ :=\\ \\arctan v\\ ;\\quad  -\\sfrac\\pi2 < U \\leq V < \\sfrac\\pi2\\  \\\\[2pt]\n    &\\implies \\mathrm{d} s_{Mink}^2 \\= \\frac{1}{4\\cos^2V\\cos^2U}\\left[-4\\,\\mathrm{d} V\\, \\mathrm{d} U +  \\sin^2(V-U)\\,\\mathrm{d}\\Omega_2^2\\right]\\ .\n\\end{aligned}\n\\end{equation}\nFinally, we rotate the coordinate system back using $(\\tau,\\chi)$ to obtain the desired form of the metric:\n\\begin{equation}\\label{transf3}\n\\begin{aligned}\n   &\\tau\\ :=\\ V + U\\ ,\\quad \\chi\\ :=\\ \\pi + U - V\\ ;\\quad  0 < \\chi \\leq \\pi\\ , |\\tau| < \\chi\\  \\\\[2pt]\n    &\\implies \\mathrm{d} s_{Mink}^2 \\= \\gamma^{-2}\\left[-\\mathrm{d} \\tau^2 + \\mathrm{d}\\chi^2 +  \\sin^2\\chi\\,\\mathrm{d}\\Omega_2^2\\right]\\ ; \\quad \\gamma \\= \\cos\\tau - \\cos\\chi\\ .\n\\end{aligned}\n\\end{equation}\nWe realize that the Minkowski metric is conformally equivalent to a Lorentzian cylinder $\\mathcal{I}\\times S^3\\ ;\\ \\mathcal{I}=(-\\pi,\\pi)$ with a conformal factor that can be recasted in terms of Minkowski coordinates using the above transformations (\\ref{transf1}-\\ref{transf3}):\n\\begin{equation}\\label{metricCyl}\n      \\mathrm{d} s_{cyl}^2\\ :=\\ -\\mathrm{d}\\tau^2 + \\mathrm{d}\\Omega_3^2 \\= \\gamma^2\\, \\mathrm{d} s_{Mink}^2\\ ;\\quad \\gamma = \\frac{2\\ell^2}{\\sqrt{4\\,t^2\\,\\ell^2 + (r^2-t^2+\\ell^2)^2}}\\ ,\n\\end{equation}\nwhere we have made $\\gamma$ dimensionless using the de Sitter radius $\\ell$. The lightcone structure of the spacetime as presented in the Penrose diagram (see Figure \\ref{CPplot}) is a direct consequence of \\eqref{transf3}.  \n\n\\chapter{Conclusion \\& Outlook}\n\\label{Chapter7} \n\nWe have studied stability behaviour of some well known solutions of $SU(2)$ Yang--Mills fields in $4$-dimensional de Sitter space $\\mathrm{d}{S}_4$ under generic gauge perturbation. These solutions could be of relevance to early time cosmology (before the electro-weak symmetry breaks down) in a scenario recently presented by Friedan \\cite{Friedan}. An $SO(4)$ symmetric sector is analytically solvable and reduces to three coupled anharmonic oscillators (for the metric, an $SU(2)$ Yang\u2013Mills field and the Higgs field, the latter being frozen to its vacuum state). We have presented a complete analysis of the linear gauge-field perturbations of the time-dependent Yang\u2013Mills solution, by diagonalizing the fluctuation operator and studying the long-time behavior of the ensuing Hill's equations using the stroboscopic map and Floquet theory. For parametrically large gauge-field energy (as is required in Friedan's setup) the natural frequencies and monodromies become universal, and some unstable perturbation modes survive even in this limit. This provides strong evidence that such oscillating cosmic Yang\u2013Mills fields are unstable against small perturbations, although we have not yet included metric fluctuations here. Their influence will be analyzed in\nfollow-up work.\n\nWe have also analyzed a family of electromagenetic knot configurations recently developed by making use of a conformal equivalence between $\\mathrm{d}{S}_4$, Minkwoski space $\\mathds R^{1,3}$ and a finite Lorentzian $S^3$-cylinder. These solutions are constructed on the cylinder in an $SO(4)$ covariant way and are then pulled back to the Minkowski space using the conformal map (that leaves the Maxwell's theory invariant). These ``basis-knot\" solutions of Maxwell's equation are labelled with $S^3$ harmonics $Y_{j,m,n}$ and give rise to field configurations of knotted field lines when pulled back to the Minkowski space. We have analyzed the symmetry feature of these basis knotted electromagnetic field configurations with the isometry group $SO(1,4)$ of the de Sitter space. We have further studied, numerically, the effect of these basis configurations on the trajectories of multiple identical charged particles with different initial conditions. Various behaviors were obtained, including a separation of trajectories into different \"solid angle regions\" that converge asymptotically into a beam of charged particles along a few particular regions of space, an ultrarelativistic acceleration of particles and coherent twists\/turns of the trajectories before they go off asymptotically. The results contribute to an effort to better understand the interactions between electromagnetic knots and charged particles \\cite{AT10}. This becomes increasingly relevant as laboratory generation of knotted\nfields progresses \\cite{LSMetal18}. Further work in this direction could be to analyze a single Fourier mode of these\nsolutions to understand its experimental realization via monochromatic laser beams.\n\nFurthermore, we have considered complex linear combinations of these basis-knot field configurations (that can model any finite-energy field configuration) and characterized the corresponding moduli space of null fields. We have also computed Noether charges of such a linearly combined configuration with fixed $j$ for the conformal group $SO(2,4)$, which is the largest symmetry group for the Maxwell theory. Here again the ``de Sitter\" method proved to be advantageous in that the expressions of these charge densities simplifies immensely on the cylinder and, thus, can be computed with ease. We found that many of these charges vanish owing to the orthogonality of the harmonics and that the energy and momentum are the only independent charged in many cases. A nice geometric structure of $1$-forms facilitated the computation of spherical components of vector charges as well. We verified our results against the results for some modified Hopf--Ran{\\~a}da field configurations of \\cite{HSS15}. We would like to further check the validity of our results by comparing them with other solutions presented in \\cite{HSS15}.\n\\chapter{Non-Abelian solution: $SU(2)$}\n\\label{Chapter6} \n\\justifying\n\nHere we present cosmic Yang--Mills solution with $SU(2)$ gauge group and study their stability behaviour. The contents of this chapter, in large parts, are taken from the published work (in Nucl. Phys. B) \\cite{KLP21}. All the graphics in this chapter is due to Gabriel Pican\\c{c}o Costa. I have been thoroughly involved at all stages of this work.\n\n\n\n\\vspace{12pt}\n\\section{SO(4)-symmetric cosmic Yang--Mills solution}\n\\vspace{2pt}\nThe Yang--Mills action on this ansatz simplifies to\n\\begin{equation}\\label{YMaction}\n   S \\= \\frac{-1}{4g^2}\\int_{{\\cal I}\\times S^3} \\!\\!\\! \\mathrm{tr}\\ {\\cal F}\\wedge *{\\cal F} \\=\n   \\frac{6\\pi^2}{g^2} \\int_{\\cal I} \\!\\mathrm{d}\\tau\\ \\bigl[ \\sfrac12\\dot\\psi^2 - V(\\psi) \\bigr]\n   \\quad\\textrm{with}\\quad V(\\psi)\\=\\sfrac12(\\psi^2{-}1)^2\\ ,\n\\end{equation}\nwhere $g$ here denotes the gauge coupling.\nDue to the principle of symmetric criticality~\\cite{Palais}, solutions to the mechanical problem \n\\begin{equation} \\label{Newton}\n   \\ddot{\\psi} + V'(\\psi) \\= 0\n\\end{equation}\nwill, via (\\ref{Aansatz}), provide Yang--Mills configurations which extremize the action.\nConservation of energy implies that\n\\begin{equation} \\label{energylaw}\n   \\sfrac12\\dot{\\psi}^2 +V(\\psi) \\= E \\= \\textrm{constant}\\ ,\n\\end{equation}\nand the generic solution in the double-well potential~$V$ is periodic in~$\\tau$ with a period~$T(E)$.\n\nHence, fixing a value for~$E$ and employing time translation invariance to set $\\dot{\\psi}(0)=0$\nuniquely determines the classical solution~$\\psi(\\tau)$ up to half-period shifts.\nIts explicit form is\n\\begin{equation} \\label{backgrounds}\n   \\psi(\\tau) \\= \n \\begin{cases}\n   \\ \\sfrac{k}{\\epsilon}\\,\\mathrm{cn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)  \n   \\qquad\\qquad\\ \\ \\,\\textrm{with}\\quad T=4\\,\\epsilon\\,K(k) \n   & \\textrm{for}\\quad \\sfrac12<E<\\infty \\\\[4pt]\n   \\ 0 \\qquad\\qquad\\qquad\\qquad\\quad\\! \\textrm{with}\\quad T=\\infty \n   & \\textrm{for}\\quad E=\\sfrac12 \\\\[4pt]\n   \\ \\pm\\sqrt{2}\\,\\mathrm{sech}\\bigl(\\sqrt{2}\\,\\tau\\bigr) \n   \\qquad\\ \\textrm{with}\\quad T=\\infty \n   & \\textrm{for}\\quad E=\\sfrac12 \\\\[4pt]\n   \\ \\pm\\sfrac{k}{\\epsilon}\\,\\mathrm{dn}\\bigl(\\sfrac{k\\,\\tau}{\\epsilon},\\sfrac1k\\bigr)  \n   \\qquad\\quad\\ \\textrm{with}\\quad T=2\\,\\sfrac{\\epsilon}{k}\\,K(\\sfrac1k) \\quad\n   & \\textrm{for}\\quad 0<E< \\sfrac12 \\\\[4pt]\n   \\ \\pm 1 \\qquad\\qquad\\qquad\\qquad \\textrm{with}\\quad T=\\pi\n   &  \\textrm{for}\\quad E=0 \n \\end{cases}\\ ,\n\\end{equation}\nwhere cn and dn denote Jacobi elliptic functions, $K$ is the complete elliptic integral of the first kind, and\n\\begin{equation}\n   2\\,\\epsilon^2 \\= 2k^2{-}1 \\= 1\/\\sqrt{2E} \\qquad\\textrm{with}\\quad\n   k=\\sfrac{1}{\\sqrt{2}},1,\\infty \\quad\\Leftrightarrow\\quad E=\\infty,\\sfrac12,0\\ .\n\\end{equation}\nFor $E{\\gg}\\frac12$, we have $k^2{\\to}\\frac12$, and the solution is well approximated by \n$\\frac{2}{\\epsilon}\\cos\\bigl(\\frac{2\\sqrt{\\pi^3}}{\\Gamma(1\/4)^2}\\frac{\\tau}{\\epsilon}\\bigr)$.\nAt the critical value of $E{=}\\sfrac12$ ($k{=}1$), the unstable constant solution coexists \nwith the celebrated bounce solution, and below it the solution bifurcates into oscillations \nin the left or right well of the double-well potential, which halfens the oscillation period. \nThe two constant minima $\\psi=\\pm1$ correspond to the vacua ${\\cal A}=0$ and ${\\cal A}=g^{-1}\\mathrm{d} g$.\nActually, the time translation freedom is broken by the finite range of~${\\cal I}$,\nso that time-shifted solutions differ in their boundary values \nand also in their values for the total energy and action.\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=0.9\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{psi_large.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{psi_medium+.pdf} \\\\[12pt]\n\\includegraphics[width = 0.35\\paperwidth]{psi_medium-.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{psi_small.pdf}\n\\caption{Plots of $\\psi(\\tau)$ over one period, for different values of $k^2$:\n$0.500001$ (top left), $0.9999999$ (top right), $1.0000001$ (bottom left) and $2$ (bottom right).}\n\\end{figure}\n\nThe corresponding color-electric and -magnetic field strengths read\n\\begin{equation}\n   {\\cal E}_a \\= {\\cal F}_{0a} \\= \\sfrac12\\dot\\psi\\,T_a \\quad\\quad\\textrm{and}\\quad\\quad\n   {\\cal B}_a \\= \\sfrac12\\varepsilon_a^{\\ bc}{\\cal F}_{bc} \\= \\sfrac12(\\psi^2{-}1)\\,T_a\\ ,\n\\end{equation}\nwhich yields a finite total energy (on the cylinder) of $6\\pi^2 E\/g^2$ and a finite action~\\cite{ILP17prl,ILP17jhep}\n\\begin{equation}\n   g^2 S[\\psi] \\= 6\\pi^2\\int_{\\cal I} \\!\\mathrm{d}\\tau\\ \\bigl[E-(\\psi^2{-}1)^2\\bigr]\n   \\=6\\pi^2\\int_{\\cal I} \\!\\mathrm{d}\\tau\\ \\bigl[\\dot{\\psi}^2-E\\bigr]\\ \\ge\\ -3\\pi^2|{\\cal I}|\\ .\n\\end{equation}\nThe energy-momentum tensor of our SO(4)-symmetric Yang--Mills solutions is readily found as\n\\begin{equation}\n   T \\= \\frac{3\\,E}{g^2 a^2} \\bigl( \\mathrm{d}\\tau^2 + \\sfrac13\\,\\mathrm{d}\\Omega_3^2 \\bigr)\\ ,\n\\end{equation}\nwhich is traceless as expected.\n\nThe Einstein equations for a closed FLRW universe with cosmological constant~$\\Lambda$\nreduce to two independent relations, which can be taken to be its trace and its time-time component.\nIn conformal time one gets, respectively,\n\\begin{equation} \\label{friedmann}\n   \\left.\\begin{cases}\n   \\quad -R+4\\,\\Lambda \\= 0 \\\\[4pt]\n   \\quad R_{\\tau\\tau}+\\sfrac12R\\,a^2-\\Lambda\\,a^2\\= \\kappa\\,T_{\\tau\\tau}\n   \\end{cases} \\right\\}\n   \\quad\\Leftrightarrow\\quad\n   \\left.\\begin{cases}\n   \\quad \\ddot{a}+W'(a)\\=0 \\\\[4pt]  \\quad \\sfrac12\\dot{a}^2+W(a) \\= \\frac{\\kappa}{2g^2}E \\ =:\\ E'\n   \\end{cases} \\right\\}\n\\end{equation}\nwith a gravitational coupling $\\kappa=8\\pi G$, \na gravitational energy $E'$\nand a cosmological potential \n\\begin{equation} \\label{gravpot}\n   W(a) \\= \\sfrac12 a^2 - \\sfrac{\\Lambda}{6} a^4\\ .\n\\end{equation}\nThe two anharmonic oscillators, with potential~$V$ for the gauge field and potential~$W$ for gravity,\nare coupled only via the balance of their conserved energies,\n\\begin{equation}\n   \\frac1\\kappa \\bigl[ \\sfrac12\\dot{a}^2+W(a)\\bigr] \\= \\frac1{2g^2} \\bigl[ \\sfrac12\\dot{\\psi}^2+V(\\psi)\\bigr]\\ ,\n\\end{equation}\nwhich is nothing but the Wheeler--DeWitt constraint~$H=0$.\n\\begin{figure}[h!]\n\\captionsetup{width=\\linewidth}\n\\centering\n\\includegraphics[width = 0.35\\paperwidth]{Wplot.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{Vplot.pdf} \n\\caption{Plots of the cosmological potential $W(a)$ for $\\lambda{=}1$ and the double-well potential $V$.}\n\\end{figure}\n\nThe Friedmann equation~(\\ref{friedmann}), being a mechanical system with an inverted anharmonic\npotential~(\\ref{gravpot}), is again easily solved analytically,\n\\begin{equation} \\label{universalscale}\n   a(\\tau) \\= \\begin{cases}\n   \\ \\sqrt{\\sfrac{3}{\\Lambda}}\\,\\sfrac{1}{2\\epsilon'}\\,\n   \\sqrt{\\frac{1-\\mathrm{cn}\\bigl(\\sfrac{\\tau}{\\epsilon'},k'\\bigr)}{1+\\mathrm{cn}\\bigl(\\sfrac{\\tau}{\\epsilon'},k'\\bigr)}}\n   \\qquad\\qquad  \\textrm{with}\\quad T'=2\\,\\epsilon'K(k') \n   & \\textrm{for}\\quad \\sfrac{3}{8\\Lambda}<E'<\\infty \\\\[4pt]\n   \\ \\sqrt{\\sfrac{3}{2\\Lambda}}\\,\\tanh\\bigl(\\tau\/\\sqrt{2}\\bigr) \\qquad\\qquad\\quad\\  \\textrm{with}\\quad T'=\\infty \n   & \\textrm{for}\\quad E'=\\sfrac{3}{8\\Lambda} \\\\[4pt]\n   \\ \\sqrt{\\sfrac{3}{\\Lambda}}\\,\\sfrac{1}{2\\epsilon'}\\,\n   \\sqrt{\\frac{1-\\mathrm{dn}\\bigl(\\sfrac{k'\\tau}{\\epsilon'},\\sfrac{1}{k'}\\bigr)}\n   {1+\\mathrm{dn}\\bigl(\\sfrac{k'\\tau}{\\epsilon'},\\sfrac{1}{k'}\\bigr)}}\n   \\qquad\\quad \\textrm{with}\\quad T'=2\\,\\sfrac{\\epsilon'}{k'}K(\\sfrac{1}{k'}) \\quad\n   & \\textrm{for}\\quad 0<E'< \\sfrac{3}{8\\Lambda} \\\\[10pt]\n   \\ 0 \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\ \\; \\textrm{with}\\quad T'=\\pi\n   &  \\textrm{for}\\quad E'=0 \n   \\end{cases}\\ ,\n\\end{equation}\nwhere we abbreviated~\\footnote{\nOur ${k'}^2$ should not be confused with the dual modulus $1{-}k^2$, which is often denoted this way.}\n\\begin{equation}\n   2\\,{\\epsilon'}^2 \\= 2{k'}^2{-}1 \\= 1\/\\sqrt{\\smash{\\sfrac{8\\Lambda}{3}}\\, E'}\n   \\qquad\\textrm{so that}\\quad\n   k'=\\sfrac{1}{\\sqrt{2}},1,\\infty \\quad\\Leftrightarrow\\quad E'=\\infty,\\sfrac{3}{8\\Lambda},0\\ .\n\\end{equation}\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{a_large.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{a_medium+.pdf} \\\\[12pt]\n\\includegraphics[width = 0.35\\paperwidth]{a_medium-.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{a_small.pdf}\n\\caption{Plots of $a(\\tau)$ over one lifetime, for $\\Lambda{=}1$ and different values of ${k'}^2$:\n$0.505$ (top left), $0.9999999$ (top right), $1.0000001$ (bottom left) and $1.1$ (bottom right).}\n\\end{figure}\nFor $E'{\\gg}\\frac{3}{8\\Lambda}$, we have ${k'}^2{\\to}\\frac12$, and the solution is well approximated by\n$\\sqrt{\\frac{3}{\\Lambda}}\\frac{1}{2\\epsilon'}\\tan\\bigl(\\frac{\\sqrt{\\pi^3}}{\\Gamma(1\/4)^2}\\frac{\\tau}{\\epsilon'}\\bigr)$.\nWe only listed solutions with initial value $a(0)=0$ (big bang). \nThere exist also (for $E'<\\sfrac{3}{8\\Lambda}$) bouncing solutions,\nwhere the universe attains a minimal radius \n$a_{\\textrm{min}}=\\bigl[\\frac{3}{2\\Lambda}(1+\\sqrt{1-\\smash{\\sfrac{8\\Lambda}{3}}E'})\\bigr]^{1\/2}$\nbetween infinite extension in the far past ($t{=}{-}\\infty\\leftrightarrow\\tau{=}0$) and the\nfar future ($t{=}{+}\\infty\\leftrightarrow\\tau{=}T'$). For $E'{>}0$ they are obtained by \nsending $\\mathrm{dn}\\to-\\mathrm{dn}$ in~(\\ref{universalscale}) above.\nThe quantity~$T'$ listed there is the (conformal) lifetime of the universe, \nfrom the big bang until either the big rip (for $E'>\\frac{3}{8\\Lambda}$) \nor the big crunch of an oscillating universe (for $E'<\\frac{3}{8\\Lambda}$).\nThe solution relevant to our Einstein--Yang--Mills system is entirely determined by the Newtonian energy~$E$\ncharacterizing the cosmic Yang--Mills field: above the critical value of\n\\begin{equation}\n   E_{\\textrm{crit}} \\= \\frac{2\\,g^2}{\\kappa}\\,\\frac{3}{8\\,\\Lambda}\n\\end{equation}\nthe universe expands forever (until $t_{\\textrm{max}}{=}\\infty$), \nwhile below this value it recollapses (at $t_{\\textrm{max}}{=}\\int_0^{T'}\\!\\mathrm{d}\\tau\\,a(\\tau)$).\nIt demonstrates the necessity of a cosmological constant (whose role may be played\nby the Higgs expectation value) as well as the nonperturbative nature of the cosmic Yang--Mills field,\nwhose contribution to the energy-momentum tensor is of~$O(g^{-2})$.\n\n\\vspace{12pt}\n\\section{Natural perturbation frequencies}\n\\vspace{2pt}\n\\noindent\nOur main task in this paper is an investigation of the stability of the cosmic Yang--Mills solutions\nreviewed in the previous section. For this, we should distinguish between global and local stability.\nThe former is difficult to assess in a nonlinear dynamics but clear from the outset in case of a compact\nphase space. The latter refers to short-time behavior induced by linear perturbations around\nthe reference configuration. We shall look at this firstly, in the present section and the following one. \nHere, we set out to diagonalize the fluctuation operator for our time-dependent Yang--Mills backgrounds\nand find the natural frequencies. \n\nEven though our cosmic gauge-field configurations are SO(4)-invariant, we must allow for all kinds of\nfluctuations on top of it, SO(4)-symmetric perturbations being a very special subclass of them.\nA generic gauge potential ``nearby'' a classical solution~${\\cal A}$ on ${\\cal I}\\times S^3$ can be expanded as\n\\begin{equation}\n   {\\cal A}+\\Phi \\= {\\cal A}(\\tau,g)\\ +\\ \\sum_{p=1}^3 \\Phi_0^p(\\tau,g)\\,T_p\\,\\mathrm{d}\\tau\\\n   +\\ \\sum_{a=1}^3\\sum_{p=1}^3 \\Phi_a^p(\\tau,g)\\,T_p\\,e^a(g)\n\\end{equation}\nwith, using $(\\mu)=(0,a)$,\n\\begin{equation}\n   \\Phi_\\mu^p(\\tau,g) \\= \\sum_{j,m,n} \\Phi_{\\mu|j;m,n}^p(\\tau)\\,Y_{j;m,n}(g)\\ ,\n\\end{equation}\non which we notice the following actions (suppressing the $\\tau$ and $g$ arguments),\n\\begin{equation}\n\\begin{aligned}\n   &(L_a \\Phi_\\mu^p)_{j;m,n} \\= \\Phi_{\\mu|j;m',n}^p \\bigl(L_a)^{m'}_{\\ m}\\ ,\\qquad\n   (S_a \\Phi)_0^p \\= 0\\ ,\\\\\n   &(S_a \\Phi)_b^p \\= -2\\varepsilon_{abc}\\,\\Phi_c^p \\ ,\\qquad\n   (T_a \\Phi)_\\mu^p \\= -2\\varepsilon_{apq}\\,\\Phi_\\mu^q \\ ,\n\\end{aligned}\n\\end{equation}\nwhere the $L_a$ matrix elements are determined from \\eqref{Y-action} and \\eqref{JpmAction},\nand $S_a$ are the components of the spin operator.\nThe (metric and gauge) background-covariant derivative reads\n\\begin{equation}\n   D_\\tau \\Phi \\= \\partial_\\tau \\Phi \\quad\\textrm{and}\\quad\n   D_a \\Phi \\= L_a \\Phi + [ {\\cal A}_a,\\Phi ] \\quad\\textrm{with}\\quad\n   {\\cal A}_a \\= \\sfrac12\\bigl(1+\\psi(\\tau)\\bigr)\\,T_a\\ ,\n\\end{equation}\nwhich is equivalent to\n\\begin{equation}\n   D_a\\Phi_b^p \\= L_a\\Phi_b^p - \\varepsilon_{abc}\\Phi_c^p + [ {\\cal A}_a,\\Phi_b]^p\n   \\quad\\textrm{since}\\quad \n   D_a\\,e^b \\= L_a\\,e^b -  \\varepsilon_{abc}\\,e^c \\= \\varepsilon_{abc}\\,e^c \\ .\n\\end{equation}\n\nThe background ${\\cal A}$ obeys the Coulomb gauge condition,\n\\begin{equation}\n   {\\cal A}_\\tau =0 \\qquad\\quad\\textrm{and}\\quad\\qquad L_a{\\cal A}_a = 0\\ ,\n\\end{equation}\nbut we cannot enforce these equations on the fluctuation~$\\Phi$. \nHowever, we may impose the Lorenz gauge condition,\n\\begin{equation} \\label{gauge}\n   D^\\mu \\Phi^p_\\mu \\= 0 \\qquad\\Rightarrow\\qquad\n   \\partial_\\tau \\Phi_0^p - L_a\\Phi^p_a - \\sfrac12(1{+}\\psi)(T_a\\Phi_a)^p \\= 0\\ ,\n\\end{equation}\nwhich is seen to couple the temporal and spatial components of~$\\Phi$ in general.\nWe then linearize the Yang--Mills equations around ${\\cal A}$ and obtain\n\\begin{equation}\n   D^\\nu D_\\nu \\Phi_\\mu - R_{\\mu\\nu}\\Phi^\\nu + 2[{\\cal F}_{\\mu\\nu},\\Phi^\\nu] \\= 0\n\\end{equation}\nwith the Ricci tensor\n\\begin{equation}\n   R_{\\mu 0} \\= 0 \\qquad\\quad\\textrm{and}\\quad\\qquad R_{ab} \\= 2\\delta_{ab}\\ .\n\\end{equation}\nAfter a careful evaluation, the $\\mu{=}0$ equation yields\n\\begin{equation} \\label{temporal}\n   \\bigl[\\partial_\\tau^2 - L_b L_b + 2(1{+}\\psi)^2\\bigr] \\Phi_0^p \n   - (1{+}\\psi)L_b(T_b\\Phi_0)^p - \\dot{\\psi}(T_b\\Phi_b)^p \\= 0\\ ,\n\\end{equation}\nwhile the $\\mu{=}a$ equations read\n\\begin{equation} \\label{spatial}\n\\begin{aligned}\n   \\bigl[\\partial_\\tau^2 - L_b L_b + 2(1{+}\\psi)^2{+}4\\bigr]\\Phi_a^p \n   &-(1{+}\\psi)L_b(T_b\\Phi)_a^p - L_b(S_b\\Phi)_a^p \\\\\n   &- \\sfrac12(1{+}\\psi)(2{-}\\psi)(S_bT_b\\Phi)_a^p\n   - \\dot{\\psi}(T_a\\Phi_0)^p \\= 0\\ .\n\\end{aligned}\n\\end{equation}\nIt is convenient to package the orbital, spin, isospin, and fluctuation triplets into formal vectors,\n\\begin{equation}\n   \\vec{L}=(L_a)\\ ,\\qquad \\vec{S}=(S_a)\\ ,\\qquad \\vec{T}=(T_a)\\ ,\\qquad \\vec{\\Phi}=(\\Phi_a)\\ ,\n\\end{equation}\nrespectively, but they act in different spaces, hence on different indices,\nsuch that $\\vec{S}^2=\\vec{T}^2=-8$ on~$\\Phi$. \nIn this notation, (\\ref{gauge}), (\\ref{temporal}) and~(\\ref{spatial}) take the compact form\n(suppressing the color index~$p$)\n\\begin{fleqn}[\\parindent]\n\\begin{equation}\n   \\partial_\\tau\\Phi_0 - \\vec{L}{\\cdot}\\vec{\\Phi} - \\sfrac12(1{+}\\psi)\\vec{T}{\\cdot}\\vec{\\Phi} \\= 0\\ ,\n   \\label{gauge2}\n\\end{equation}\n\\begin{equation}\n   \\bigl[\\partial_\\tau^2{-}\\vec{L}^2+2(1{+}\\psi)^2\\bigr] {\\Phi}_0 \n   - (1{+}\\psi)\\vec{L}{\\cdot}\\vec{T}\\,\\Phi_0 - \\dot{\\psi}\\,\\vec{T}{\\cdot}\\vec{\\Phi} \\= 0 \\ ,\n   \\label{mu=0}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n   \\bigl[\\partial_\\tau^2{-}\\vec{L}^2{-}\\sfrac12\\vec{S}^2{+}2(1{+}\\psi)^2\\bigr] {\\Phi}_a\n   &- (1{+}\\psi)\\vec{L}{\\cdot}\\vec{T}\\,\\Phi_a - \\vec{L}{\\cdot}(\\vec{S}\\,\\Phi)_a \\\\\n   &- \\sfrac12(1{+}\\psi)(2{-}\\psi)\\vec{T}{\\cdot}(\\vec{S}\\,\\Phi)_a - \\dot{\\psi}\\,T_a\\Phi_0 \\= 0\\ .\n\\end{split}\n\\label{mu=a}\n\\end{equation}\n\\end{fleqn}\n\n\nA few remarks are in order. \nFirst, except for the last term, (\\ref{mu=0}) is obtained from (\\ref{mu=a}) by setting $\\vec{S}=0$,\nsince $\\Phi_0$ carries no spin index. \nSecond, both equations can be recast as\n\\begin{fleqn}\n\\begin{equation} \\label{2spins}\n   \\bigl[ \\partial_\\tau^2 - \\sfrac{1{-}\\psi}{2}\\vec{L}^2 - \\sfrac{1{+}\\psi}{2}(\\vec{L}{+}\\vec{T})^2 \n   - 2(1{+}\\psi)(1{-}\\psi) \\bigr] \\Phi_0 \\= \\dot{\\psi}\\,\\vec{T}\\cdot\\vec{\\Phi}\\ ,  \n\\end{equation}\n\\begin{equation} \\label{3spins}\n\\begin{split}\n   \\bigl[ \\partial_\\tau^2 - \\sfrac{(1{-}\\psi)(2{+}\\psi)}{4}\\vec{L}^2 - \\sfrac{\\psi(1{+}\\psi)}{4}(\\vec{L}{+}\\vec{T})^2\n   + \\sfrac{\\psi(1{-}\\psi)}{4}(\\vec{L}&{+}\\vec{S})^2 - \\sfrac{(1{+}\\psi)(2{-}\\psi)}{4}(\\vec{L}{+}\\vec{T}{+}\\vec{S})^2 \\\\\n   &- 2(1{+}\\psi)(1{-}\\psi) \\bigr] \\vec{\\Phi} = \\dot{\\psi}\\vec{T}\\Phi_0\\ ,\n\\end{split}\n\\end{equation}\n\\end{fleqn}\nwhich reveals a problem of addition of three spins and a corresponding symmetry under\n\\begin{equation} \\label{symmetry}\n   \\psi\\ \\leftrightarrow\\ -\\psi\\ ,\\qquad\n   \\vec{L}\\ \\leftrightarrow\\ \\vec{L}{+}\\vec{T}{+}\\vec{S} \\quad\\quad\\textrm{and}\\quad\\quad\n   \\vec{L}{+}\\vec{S}\\ \\leftrightarrow\\ \\vec{L}{+}\\vec{T}\\ .\n\\end{equation}\nThird, for constant backgrounds $(\\dot{\\psi}{=}0)$ the temporal fluctuation $\\Phi_0$ decouples \nand may be gauged away. \nStill, the fluctuation operator in~(\\ref{3spins}) is easily diagonalized only when \nthe coefficient of one of the first three spin-squares vanishes, i.e.~for $\\vec{L}{=}0$ ($j{=}0$),\nfor the two vacua $\\psi{=}{\\pm}1$, or for the ``meron'' $\\psi{=}0$. \nThe latter case has been analyzed by Hosotani~\\cite{Hosotani}.\n\nLet us decompose the fluctuation problem (\\ref{gauge2})--(\\ref{mu=a}) into finite-dimensional blocks\naccording to a fixed value of the spin~$j\\in\\sfrac12\\mathds N$,\n\\begin{equation}\n   \\vec{L}^2\\, \\Phi^p_{\\mu|j} \\= -4\\,j(j{+}1)\\,\\Phi^p_{\\mu|j}\n\\end{equation}\nand suppress the $j$ subscript. We employ the following coupling scheme,\\footnote{\nAnother (less convenient) scheme couples $\\vec{L}{+}\\vec{S}$, then $(\\vec{L}{+}\\vec{S}){+}\\vec{T}=:\\vec{V}$.}\n\\begin{equation}\n   \\vec{L}{+}\\vec{T}=:\\vec{U} \\qquad\\textrm{then}\\qquad \\vec{U}{+}\\vec{S}=(\\vec{L}{+}\\vec{T}){+}\\vec{S}=:\\vec{V}\\ .\n\\end{equation}\nClearly, $\\vec{U}$ and $\\vec{V}$ act on $\\vec{\\Phi}$ in $su(2)$ representations \n$j\\otimes 1$ and $j\\otimes 1\\otimes 1$, respectively. On $\\Phi_0$, we must put~$\\vec{S}{=}0$\nand have just $\\vec{V}{=}\\vec{U}$ act in a $j\\otimes 1$ representation.\nCombining the coupled equations (\\ref{mu=0}) and (\\ref{mu=a}) to a single linear system for\n$(\\Phi^p_\\mu)=(\\Phi^p_0,\\Phi^p_a)$, we get a $12(2j{+}1)\\times 12(2j{+}1)$ fluctuation matrix~$\\Omega^2_{(j)}$,\n\\begin{equation} \\label{fluctop}\n   \\bigl[ \\delta_{\\mu\\nu}^{pq}\\,\\partial_\\tau^2\\ +\\ (\\Omega^2_{(j)})_{\\mu\\nu}^{pq} \\bigr]\\,\\Phi_\\nu^q \\= 0\\ .\n\\end{equation}\nActually, there is an additional overall $(2j{+}1)$-fold degeneracy present due to the trivial action of\nthe $su(2)_{\\textrm{R}}$ generators~$R_a$, which plays no role here and will be suppressed.\nRoughly speaking, the $3(2j{+}1)$ modes of $\\Phi_0$ are related to gauge modes,\\footnote{\nStrictly, they are gauge modes only when $\\dot\\psi{=}0$. \nOtherwise, the gauge modes are mixtures with the $\\Phi_a$ modes.}\nand we still must impose the gauge condition~(\\ref{gauge2}), which also has $3(2j{+}1)$ components.\nTherefore, a subspace of dimension $6(2j{+}1)$ inside the space of all fluctuations will \nrepresent the physical gauge-equivalence classes in the end.\n\nOur goal is to diagonalize the fluctuation operator~(\\ref{fluctop}) for a given fixed value of~$j$.\nIt has a block structure,\n\\begin{equation} \\label{block}\n   \\Omega^2_{(j)} \\= \\begin{pmatrix} \\bar{N} & -\\dot{\\psi}\\,T^\\top \\\\[6pt] -\\dot{\\psi}\\,T & N \\end{pmatrix}\\ ,\n\\end{equation}\nwhere $\\bar{N}$ and $N$ are given by the left-hand sides of (\\ref{2spins}) and~(\\ref{3spins}), respectively.\nWe introduce a basis where $\\vec{U}^2$, $\\vec{V}^2$ and $V_3$ are diagonal, i.e.\n\\begin{equation}\n   \\vec{U}^2\\,|uvm\\> \\= -4\\,u(u{+}1)\\,|uvm\\> \\quad\\quad\\textrm{and}\\quad\\quad \n   \\vec{V}^2\\,|uvm\\> \\= -4\\,v(v{+}1)\\,|uvm\\>\\ ,\n\\end{equation}\nwith $m=-v,\\ldots,v$ and denote the irreducible $su(2)_v$ representations with those quantum numbers as $\\rep{v}{u}$.\nOn the $\\Phi_0$ subspace, $u$ is redundant since $u{=}v$ as $\\vec{S}{=}0$.\nWorking out the tensor products, we encounter the values\n\\begin{equation} \\label{uvreps}\n\\begin{aligned}\n   \\rep{v}{u} &\\= \\rep{j{-}2}{j{-}1}\\ ;\\!\\!\\!&\\rep{j{-}1}{j{-}1}&\\ ,\\ \\rep{j{-}1}{j}\\ ;\\ \\rep{j}{j{-}1}\\ ,\n   \\!\\!\\!&\\rep{j}{j}&\\ ,\\ \\rep{j}{j{+}1}\\ ;\\ \\rep{j{+}1}{j}\\ ,\\!\\!\\!&\\rep{j{+}1}{j{+}1}&\\ ;\\ \\rep{j{+}2}{j{+}1} \n   &\\quad\\textrm{on}\\ \\vec{\\Phi}\\ ,& \\\\\n   \\rep{v}{u} &\\= &\\rep{j{-}1}{j{-}1}&\\ ; &\\rep{j}{j}&\\ ; &\\rep{j{+}1}{j{+}1}&\n   &\\quad\\textrm{on} \\ \\Phi_0&\\ ,\n\\end{aligned}\n\\end{equation}\nwith some representations obviously missing for $j{<}2$.\n\nLet us treat the $\\dot{\\psi}\\,T$ term in~(\\ref{block}) as a perturbation and momentarily put it to zero,\nso that $\\Omega^2_{(j)}$ is block-diagonal for the time being.\nThen, it is easy to see from (\\ref{2spins}) and (\\ref{3spins}) that $[\\vec{V},\\bar{N}]=[\\vec{U},\\bar{N}]=0$\nand $[\\vec{V},N]=0$, even though $[\\vec{U},N]\\neq0$ because $(\\vec{L}{+}\\vec{S})^2$ \nis not diagonal in our basis. Therefore, we have a degeneracy in~$m$.\nFurthermore, both $\\bar{N}$ and $N$ decompose into at most three respectively five blocks \nwith fixed values of~$v$ ranging from $j{-}2$ to~$j{+}2$ and separated by semicolons in~(\\ref{uvreps}). \nMoreover, the $\\bar{N}$~blocks are irreducible and trivially also carry a value of~$u{=}v$. \nIn contrast, $N$ is not {\\it simply\\\/} reducible; its $\\vec{V}$ representations have multiplicity one, two or three. \nOnly the $N$~blocks with extremal $v$~values in~(\\ref{uvreps}) are irreducible. \nThe other ones are reducible and contain more than one $\\vec{U}$~representation, \nhence the $u$-spin distinguishes between their (two or three) irreducible $v$~subblocks.\nThe only non-diagonal term in~$N$ is the $(\\vec{L}{+}\\vec{S})^2$ contribution, which couples different copies\nof the same $v$-spin to each other, but of course not to any $u{=}v$ block of~$\\bar{N}$, \nand does not lift the $V_3{=}m$~degeneracy. \nAs a consequence, the unperturbed fluctuation equations \nfor $\\Phi_0{=}\\Phi_{(\\bar{v})}$ and $\\vec{\\Phi}{=}\\Phi_{(v,\\alpha)}$ take the form \n(suppressing the $m$ index)\n\\begin{fleqn}\n\\begin{equation}\n\\begin{split}\n   \\mathbbm{1}_{(\\bar{v})}\\bigl[ \\partial_\\tau^2\\ +\\ \\bar\\omega^2_{(\\bar{v})}\\bigr]\\,\\Phi_{(\\bar{v})} \\= 0 \\qquad\\quad\\textrm{and}\\quad\\qquad\n   \\mathbbm{1}_{(v)}\\bigl[ \\partial_\\tau^2\\ +\\ \\omega^2_{(v,\\alpha)}\\bigr]\\,\\Phi_{(v,\\alpha)} \\= 0 \\\\[4pt]\n   \\quad\\textrm{for}\\quad \\dot{\\psi}=0 \\quad\\textrm{with}\\quad \\bar{v} \\in \\{ j{-}1,\\ j,\\ j{+}1\\} \\quad\\textrm{and}\\quad\n   v \\in \\{ j{-}2,\\ j{-}1,\\ j,\\ j{+}1,\\ j{+}2 \\} \\ ,\n\\end{split}\n\\end{equation}\n\\end{fleqn}\nwhere $\\mathbbm{1}_{(v)}$ denotes a unit matrix of size~$2v{+}1$, and $\\alpha$ counts the multiplicity \nof the $v$-spin representation in~$N$ (between one and three). \nAccording to~(\\ref{2spins}) the unperturbed frequency-squares for $\\bar{N}$ are the eigenvalues\n\\begin{equation}\n   \\bar{\\omega}^2_{(\\bar{v})} \\= \n   2(1{-}\\psi)\\,j(j{+}1) + 2(1{+}\\psi)\\,\\bar{v}(\\bar{v}{+}1) - 2(1{+}\\psi)(1{-}\\psi)\n\\end{equation}\nwith multiplicity  $2\\bar{v}{+}1$, hence we get\n\\begin{equation}\n\\begin{aligned}\n   \\bar{\\omega}^2_{(j-1)} &\\= 2\\,\\psi^2-4j\\,\\psi+2(2j^2{-}1)\\ ,\\\\\n   \\bar{\\omega}^2_{(j)} &\\= 2\\,\\psi^2+2(2j^2{+}2j{-}1)\\,\\\\\n   \\bar{\\omega}^2_{(j+1)} &\\= 2\\,\\psi^2+4(j{+}1)\\,\\psi +2(2j^2{+}4j{+}1)\\ .\n\\end{aligned}\n\\end{equation}\nConsidering $N$ in (\\ref{3spins}), we can read off the eigenvalues at $v=j{\\pm}2$\nbecause in these two extremal cases $(\\vec{L}{+}\\vec{S})^2=\\vec{U}^2$ \nis already diagonal in the $\\bigl\\{ |uvm\\>\\bigr\\}$ basis. For the other $v$-values\nwe must diagonalize a $2{\\times}2$ or $3{\\times}3$ matrix to find\n\\begin{equation}\n\\begin{aligned}\n   \\omega^2_{(j-2)} &\\= \\textrm{root of}\\ Q_{j-2}(\\lambda)\n   \\= -2(2j{-}1)\\,\\psi + 2(2j^2{-}2j{+}1)\\ ,\\\\\n   \\omega^2_{(j-1,\\alpha)} &\\= \\textrm{two roots of}\\ Q_{j-1}(\\lambda)\\ ,\\\\\n   \\omega^2_{(j,\\alpha)} &\\= \\textrm{three roots of}\\ Q_j(\\lambda)\\ ,\\\\\n   \\omega^2_{(j+1,\\alpha)} &\\= \\textrm{two roots of}\\ Q_{j+1}(\\lambda)\\ ,\\\\\n   \\omega^2_{(j+2)} &\\= \\textrm{root of}\\ Q_{j+2}(\\lambda)\n   \\= 2(2j{+}3)\\,\\psi + 2(2j^2{+}6j{+}5)\\ ,\n\\end{aligned}\n\\end{equation}\neach with multiplicity $2v{+}1$, where $Q_v$ denotes a linear, quadratic or cubic polynomial.\\footnote{\nFor $j{<}2$ some obvious modifications occur due to the missing of $v{<}0$ representations.}\n\nLet us now turn on the perturbation $\\dot{\\psi}\\,T$, which couples $N$ with~$\\bar{N}$,\nand consider the characteristic polynomial~${\\cal P}_j(\\lambda)$ of our fluctuation problem,\n\\begin{equation}\n\\begin{aligned}\n   {\\cal P}_j(\\lambda) &\\ :=\\\n   \\det\\,\\Bigl(\\begin{smallmatrix} \\bar{N}{-}\\lambda & -\\dot{\\psi}T^\\top \\\\[4pt] \n   -\\dot{\\psi}T & N{-}\\lambda \\end{smallmatrix} \\Bigr)\n   \\= \\det (N{-}\\lambda) \\cdot \\det\\bigl[ (\\bar{N}{-}\\lambda) - \\dot{\\psi}^2\\,T^\\top (N{-}\\lambda)^{-1} T \\bigr] \\\\[4pt]\n   &\\= \\bigl[{\\textstyle\\prod}_v \\det (N_{(v)}{-}\\lambda)\\bigr] \\cdot \n   \\det\\bigl[ (\\bar{N}{-}\\lambda) - \\dot{\\psi}^2\\, T^\\top \\{{\\textstyle\\bigoplus}_{v} (N_{(v)}{-}\\lambda)^{-1}\\}\\,T\\bigr] \\ ,\n\\end{aligned}\n\\end{equation}\nwhere we made use of\n\\begin{equation}\n\\< u\\,v\\,m|\\,N\\,|u'v'm'> \\= \\bigl(N_{(v)}\\bigr)_{uu'}\\,\\delta_{vv'}\\delta_{mm'}\\ .\n\\end{equation}\nSince $T$ furnishes an $su(2)$ representation (and not an intertwiner) \nit must be represented by square matrices and thus cannot connect different $v$ representations.\nHence the perturbation does not couple different $v$ sectors but only links $N$ and $\\bar{N}$\nin a common $\\bar{v}{=}v$~sector. Therefore, it does not affect the extremal sectors $v=j{\\pm}2$.\nMoreover, switching to a diagonal basis $\\{|\\alpha vm\\>\\}$ for $N$ we can simplify to\n\\begin{equation}\n\\begin{aligned}\n   T^\\top \\{{\\textstyle\\bigoplus}_{v} (N_{(v)}{-}\\lambda)^{-1}\\}\\,T\\bigr]\n   &\\= {\\textstyle\\bigoplus}_{\\bar{v}}\\{T^\\top (N{-}\\lambda)^{-1} T\\}_{(\\bar{v})} \\\\\n   &\\={\\textstyle\\bigoplus}_{\\bar{v}}\\Bigl\\{ {\\textstyle\\sum}_\\alpha \n   (\\omega^2_{(\\bar{v},\\alpha)}{-}\\lambda)^{-1} \\bigl(T^\\top|\\alpha\\>\\!\\<\\alpha|\\,T\\bigr)_{(\\bar{v})}\\Bigr\\}\\ .\n\\end{aligned}\n\\end{equation}\nObserving that \n$\\bigl(T^\\top|\\alpha\\>\\!\\<\\alpha|\\,T\\bigr)_{(\\bar{v})}=\n-t_{\\bar{v},\\alpha}\\bigl(\\vec{T}^2\\bigr)_{(\\bar{v})}=8\\,t_{\\bar{v},\\alpha}\\mathbbm{1}_{(\\bar{v})}$\nwith some coefficient functions $t_{\\bar{v},\\alpha}(\\psi)$, with $\\sum_\\alpha t_{\\bar{v},\\alpha}=1$,\nwe learn that the $V_3$ degeneracy remains intact and arrive at ($\\bar{v}\\in\\{j{-}1,j,j{+}1\\}$)\n\\begin{equation} \\label{charP}\n\\begin{aligned}\n   {\\cal P}_j(\\lambda) &\\= \\bigl[{\\textstyle\\prod}_v Q_v(\\lambda)^{2v+1}\\bigr] \\cdot\n   {\\textstyle\\prod}_{\\bar{v}} \\bigl\\{(\\bar{\\omega}^2_{(\\bar{v})}{-}\\lambda) \n   - 8\\dot{\\psi}^2 {\\textstyle\\sum}_\\alpha t_{\\bar{v},\\alpha} \n   (\\omega^2_{(\\bar{v},\\alpha)}{-}\\lambda)^{-1} \\bigr\\}^{2\\bar{v}+1}\\\\[4pt]\n   &\\= (\\omega^2_{(j-2)}{-}\\lambda)^{2j-3}\\cdot(\\omega^2_{(j+2)}{-}\\lambda)^{2j+5}\\cdot\n   {\\textstyle\\prod}_{\\bar{v}} \\bigl\\{ (\\bar{\\omega}^2_{(\\bar{v})}{-}\\lambda)\\,Q_{\\bar{v}}(\\lambda) - 8\\dot{\\psi}^2 P_{\\bar{v}}(\\lambda) \\bigr\\}^{2\\bar{v}+1} \\\\[4pt]\n   &\\= (\\omega^2_{(j-2)}{-}\\lambda)^{2j-3}\\cdot(\\omega^2_{(j+2)}{-}\\lambda)^{2j+5}\\cdot\n   {\\textstyle\\prod}_{\\bar{v}} R_{\\bar{v}}(\\lambda)^{2\\bar{v}+1}\\ ,\n\\end{aligned}\n\\end{equation}\nwhere $P_{\\bar{v}}=Q_{\\bar{v}}\\sum_\\alpha t_{\\bar{v},\\alpha} (\\omega^2_{(\\bar{v},\\alpha)}{-}\\lambda)^{-1}$\nis a polynomial of degree one less than $Q_{\\bar{v}}$ since all poles cancel, and $R_{\\bar{v}}$ is a\npolynomial of one degree more.\nWe list the polynomials $Q_v$, $P_{\\bar{v}}$ and $R_{\\bar{v}}$ for $j{\\le}2$ in the Appendix.\n\nTo summarize, by a successive basis change ($m'=-j,\\ldots,j$ and $m=-v,\\ldots,v$)\n\\begin{equation}\n   \\bigl\\{ |\\mu\\,p\\,m'\\> \\bigr\\} \\quad\\Rightarrow\\quad \n   \\bigl\\{ |\\bar{v} m\\>, |u v m\\> \\bigr\\} \\quad\\Rightarrow\\quad\n   \\bigl\\{ |\\bar{v} m\\>, |\\alpha v m\\> \\bigr\\} \\quad\\Rightarrow\\quad\n   \\bigl\\{ |\\beta v m\\> \\bigr\\}\n\\end{equation}\nwe have diagonalized~(\\ref{fluctop}) to\n\\begin{equation} \\label{diagonalized}\n   \\bigl[ \\partial_\\tau^2 - \\Omega^2_{(j,v,\\beta)} \\bigr] \\, \\Phi_{(v,\\beta)} \\= 0\n   \\qquad\\textrm{with}\\quad v\\in\\{j{-}2,j{-}1,j,j{+}1,j{+}2\\} \\ ,\n\\end{equation}\nwhere $\\Omega^2_{(j,v,\\beta)}$ are the distinct roots of the characteristic polynomial~${\\cal P}_j$ in~(\\ref{charP}),\nand (for $j{\\ge}2$) the multiplicity $\\beta$ takes $1,3,4,3,1$ values, respectively:\n\\begin{equation}\n   \\Omega^2_{(j,j\\pm2)} = \\omega^2_{(j\\pm2)}\\ ,\\quad\n   \\Omega^2_{(j,j\\pm1,\\beta)} = \\textrm{three roots of}\\ R_{j\\pm1}(\\lambda)\\ ,\\quad\n   \\Omega^2_{(j,j,\\beta)} = \\textrm{four roots of}\\ R_{j}(\\lambda)\\ .\n\\end{equation}\nThe reflection symmetry~(\\ref{symmetry}) implies that \n$\\Omega^2_{(v,j,\\cdot)}(\\psi)=\\Omega^2_{(j,v,\\cdot)}(-\\psi)$.\nFor $j{<}2$, obvious modifications occur due to the absence of some $v$~representations.\n\nWe still have to discuss the gauge condition~(\\ref{gauge2}), which can be cast into the form\n\\begin{equation}\n   0 \\= \\mbox{$\\partial$}_\\tau\\Phi_0 - \\bigl[ \\sfrac12(1{-}\\psi)\\vec{L}+\\sfrac12(1{+}\\psi)\\vec{U}\\bigr]\\cdot\\vec\\Phi\n   \\= \\mbox{$\\partial$}_\\tau\\Phi_{(\\bar{v},\\bar{m})} - K_{\\bar{v},\\bar{m}}^{\\ v,m,\\alpha}(\\psi)\\,\\Phi_{(v,m,\\alpha)}\n\\end{equation}\nwith a $3(2j{+}1){\\times}7(2j{+}1)$ linear (in~$\\psi$) matrix function~$K$.\\footnote{\nWe have to bring back the $m$ indices because the gauge condition is not diagonal in them.}\nHere the $v$~sum runs over $(j{-}1,j,j{+}1)$ only,\nsince the gauge condition~(\\ref{gauge2}) has components only in the middle three $v$~sectors,\nlike the gauge-mode equation~(\\ref{mu=0}).\nIt does not restrict the extremal $v$~sectors~$v=j{\\pm}2$, since these fluctuations\ndo not couple to the gauge sector~$\\Phi_0$ and are entirely physical.\nFor the middle three $v$~sectors (labelled by~$\\bar{v}$), the $\\dot{\\psi}\\,T$ perturbation leads to\na mixing of the $N$ modes with the $\\bar{N}$ gauge modes, so their levels will avoid crossing. \nPerforming the corresponding final basis change, the gauge condition takes the form\n\\begin{equation} \\label{gauge4}\n   \\bigl[ L_{\\bar{v},\\bar{m}}^{\\ \\bar{v}'\\!,\\bar{m}'\\!,\\beta}(\\psi)\\,\\mbox{$\\partial$}_\\tau \n   - M_{\\bar{v},\\bar{m}}^{\\ \\bar{v}'\\!,\\bar{m}'\\!,\\beta}(\\psi) \\bigr]\\,\\Phi_{(\\bar{v}'\\!,\\bar{m}'\\!,\\beta)}\\=0\n\\end{equation}\nwith certain $3(2j{+}1){\\times}10(2j{+}1)$ matrix functions $L$ and~$M$.\nThis linear equation represents conditions on the normal mode functions $\\Phi_{(\\bar{v},\\bar{m},\\beta)}$\nand defines a $7(2j{+}1)$-dimensional subspace of physical fluctuations, \nwhich of course still contains a $3(2j{+}1)$-dimensional subspace of gauge modes. \nFor $j{<}1$, these numbers are systematically smaller.\nTogether with the two extremal $v$~sectors, we end up with $(7-3+2)(2j{+}1)=6(2j{+}1)$ \nphysical degrees of freedom for any given value of~$j({\\ge}2)$, as advertized earlier.\n\nWe conclude this section with more details for the simplest examples,\nwhich are constant backgrounds and $j{=}0$ backgrounds.\nFor the vacuum background, say $\\psi=-1$, which is isospin degenerate, one gets\n\\begin{equation}\n   \\bigl( \\partial_\\tau^2 -\\sfrac12\\vec{L}^2 -\\sfrac12(\\vec{L}{+}\\vec{S})^2 \\bigr)\\,\\vec{\\Phi} \\=0, \\qquad\n   \\vec{L}{\\cdot}\\vec{\\Phi} = 0\\ ,\\qquad \\Phi_0 =0\\ .\n\\end{equation}\nIt yields the positive eigenfrequency-squares\n\\begin{equation}\n   \\omega^2_{(j,u')} \\= 2j(j{+}1) + 2u'(u'{+}1) \\=\n   \\begin{cases}\n   \\ 4j^2 \\ \\textrm{at}\\ j{{\\ge}1} & \\quad\\textrm{for}\\quad u'=j{-}1 \\\\ \n   \\ 4j(j{+}1) & \\quad\\textrm{for}\\quad u'=j \\\\\n   \\ 4(j{+}1)^2 & \\quad\\textrm{for}\\quad u'=j{+}1 \n\\end{cases}\n\\end{equation}\nfor $j=0,\\sfrac12,1,\\ldots$,\nbut the $\\vec{L}{\\cdot}\\vec{\\Phi}=0$ constraint removes the $u'{=}j$ modes.\nClearly, all (constant) eigenfrequency-squares are positive, hence the vacuum is stable.\n\nFor the ``meron'' background, $\\psi\\equiv0$, one has\n\\begin{equation}\n   \\bigl( \\partial_\\tau^2 -\\sfrac12\\vec{L}^2 -\\sfrac12(\\vec{L}{+}\\vec{T}{+}\\vec{S})^2 -2 \\bigr)\\,\\vec{\\Phi} \\=0, \\qquad\n   \\bigl(\\vec{L}+\\sfrac12\\vec{T}\\bigr)\\cdot\\vec{\\Phi} = 0\\ ,\\qquad \\Phi_0 =0\\ .\n\\end{equation}\nIn this case, we read off \n\\begin{equation}\n   \\omega^2_{(j,v)} +2 \\= 2j(j{+}1) + 2v(v{+}1) \\=\n   \\begin{cases}\n   \\ 4(j^2{-}j{+}1)  & \\quad\\textrm{for}\\quad v=j{-}2 \\qquad (0\\ \\textrm{to}\\ 1\\times) \\\\ \n   \\ 4j^2 & \\quad\\textrm{for}\\quad v=j{-}1 \\qquad (0\\ \\textrm{to}\\ 2\\times) \\\\\n   \\ 4j(j{+}1) & \\quad\\textrm{for}\\quad v=j \\qquad\\quad\\  (1\\ \\textrm{to}\\ 3\\times) \\\\\n   \\ 4(j{+}1)^2 & \\quad\\textrm{for}\\quad v=j{+}1 \\qquad (1\\ \\textrm{to}\\ 2\\times) \\\\\n   \\ 4(j^2{+}3j{+}3)  & \\quad\\textrm{for}\\quad v=j{+}2 \\qquad (1\\times)\n\\end{cases}\\ ,\n\\end{equation}\nbut the constraint removes one copy from each of the three middle cases (and less when $j{<}1$).\nWe end up with a spectrum $\\{\\omega^2\\}=\\{-2,1,6,7,10,\\ldots\\}$ with certain degeneracies~\\cite{Hosotani}.\nThe single non-degenerate negative mode $\\omega^2_{(0,0)}{=}{-}2$ is a singlet, $\\Phi_a^p=\\delta_a^p\\phi(\\tau)$, \nand it corresponds to rolling down the local maximum of the double-well potential. \nThe meron is stable against all other perturbations.\n\nFor a time-varying background, the natural frequencies $\\Omega_{(j,v,\\beta)}$\ninherit a $\\tau$ dependence from the background~$\\psi(\\tau)$.\nDirect diagonalization is still possible for $j{=}0$, where we should solve\n\\begin{equation}\n\\begin{aligned}\n   & \\partial_\\tau\\Phi_0 - \\sfrac12(1{+}\\psi)\\vec{T}{\\cdot}\\vec{\\Phi} \\= 0 \\ ,\\\\[4pt]\n   & \\bigl[\\partial_\\tau^2+2(1{+}\\psi)^2\\bigr] {\\Phi}_0 - \\dot{\\psi}\\,\\vec{T}{\\cdot}\\vec{\\Phi} \\= 0\\ ,\\\\[4pt]\n   & \\bigl[ \\partial_\\tau^2 +2(3\\psi^2{-}1) -\\sfrac14(1{+}\\psi)(2{-}\\psi)(\\vec{S}{+}\\vec{T})^2 \\bigr]\\ \n   \\vec\\Phi - \\dot{\\psi}\\,\\vec{T}\\,\\Phi_0 \\= 0\\ ,\n\\end{aligned}\n\\end{equation}\nwith\n\\begin{equation}\n   (\\vec{S}{+}\\vec{T})^2\\=\\vec{V}^2\\=-4\\,v(v{+}1)\\=0,-8,-24 \\qquad\\textrm{for}\\quad v=0,1,2\\ .\n\\end{equation}\nIt implies the unperturbed frequencies (suppressing the $j$ index)\n\\begin{equation} \\label{unperturbed}\n\\begin{aligned}\n   \\bar\\omega_{(1)}^2 &= 2(\\psi{+}1)^2\\ \\ (3\\times)\\ ,\\quad\n   \\omega_{(0)}^2 = 2(3\\psi^2{-}1)\\ \\ (1\\times)\\ ,\\\\\n   \\omega_{(1)}^2 &= 2(2\\psi^2{+}\\psi{+}1)\\ \\ (3\\times) ,\\quad\n   \\omega_{(2)}^2 = 2(3\\psi{+}5)\\ \\ (5\\times)\n\\end{aligned}\n\\end{equation}\nfor\n\\begin{equation}\n\\begin{aligned}\n   (\\Phi_0)^p &\\equiv \\bigl(\\Phi_{(\\bar{v}=1)}\\bigr)^p\\ =:\\ \\delta^{pb}\\bar\\phi_b \\ ,\\\\\n   (\\vec\\Phi)_a^p \\ &\\equiv \\bigl(\\Phi_{(0)}+\\Phi_{(1)}+\\Phi_{(2)}\\bigr)^p_a\\ =:\\\n   \\phi\\,\\delta_a^p + \\epsilon^p_{\\ ab}\\,\\phi_b + (\\phi_{(ab)}{-}\\delta_{ab}\\phi)\\delta^{bp}\\ ,\n\\end{aligned}\n\\end{equation}\nas long as $\\dot{\\psi}$ is ignored.\nThere are no $v$-spin multiplicities (larger than one) here.\nTurning on $\\dot{\\psi}$ and observing that $(\\vec{T}{\\cdot}\\vec\\Phi)^p\\sim\\delta^{pb}\\phi_b$, \nthe characteristic polynomial of the coupled $12{\\times}12$ system in the $|uvm\\>$ basis reads\n\\begin{equation}\n   {\\cal P}_0(\\lambda) \\= \\det \\begin{pmatrix}\n   (\\bar{\\omega}_{(1)}^2{-}\\lambda)\\mathbbm{1}_3 & \n   0 & -\\dot{\\psi}\\,T_{(1)}^\\top & 0 \\\\[4pt]\n   0 & (\\omega_{(0)}^2{-}\\lambda)\\mathbbm{1}_1 & 0 & 0 \\\\[4pt]\n   -\\dot{\\psi}\\,T_{(1)} & 0 & (\\omega_{(1)}^2{-}\\lambda)\\mathbbm{1}_3 & 0 \\\\[4pt]\n   0 & 0 & 0 & (\\omega_{(2)}^2{-}\\lambda)\\mathbbm{1}_5 \n\\end{pmatrix}\\ .\n\\end{equation}\nSpecializing the general discussion above to $j{=}0$, we find just $t_1{=}1$ so that $P_1{=}1$ and arrive at\n\\begin{equation}\n\\begin{aligned}\n   {\\cal P}_0(\\lambda) &\\= (\\omega_{(0)}^2{-}\\lambda)^1\n   (\\omega_{(1)}^2{-}\\lambda)^3 (\\omega_{(2)}^2{-}\\lambda)^5\n   \\bigl[(\\bar{\\omega}_{(1)}^2{-}\\lambda) - 8\\dot{\\psi}^2(\\omega_{(1)}^2{-}\\lambda)^{-1} \\bigr]^3 \\\\\n   &\\= (\\omega_{(0)}^2{-}\\lambda) (\\omega_{(2)}^2{-}\\lambda)^5\n   \\bigl\\{ (\\bar{\\omega}_{(\\bar{1})}^2{-}\\lambda)(\\omega_{(1)}^2{-}\\lambda) - 8\\dot{\\psi}^2 \\bigr\\}^3 \\ .\n\\end{aligned}\n\\end{equation}\nWe see that the frequencies $\\Omega_{(0)}^2{=}\\omega_{(0)}^2$ and $\\Omega_{(2)}^2{=}\\omega_{(2)}^2$ \nare unchanged and given by~(\\ref{unperturbed}), while the gauge mode $\\bar{\\omega}_{(\\bar{1})}^2$ gets\nentangled with the (unphysical) $v{=}1$ mode to produce the pair\n\\begin{equation}\n\\begin{aligned}\n   \\Omega^2_{(1,\\pm)} &\\= \\sfrac12(\\bar\\omega_{(\\bar{1})}^2{+}\\omega_{(1)}^2)\n   \\,\\pm\\sqrt{\\sfrac14(\\bar\\omega_{(\\bar{1})}^2{+}\\omega_{(1)}^2)^2-\\bar\\omega_{(\\bar{1})}^2\\omega_{(1)}^2+8\\dot{\\psi}^2}  \\\\\n   &\\= 3\\psi^2{+}3\\psi{+}2\\, \\pm\\sqrt{\\psi^2(\\psi{-}1)^2+8\\dot{\\psi}^2}\n\\end{aligned}\n\\end{equation}\nwith a triple degeneracy. There are avoided crossings at $\\psi{=}0$ and $\\psi{=}1$.\nRemoving the unphysical and gauge modes in pairs, we remain with the singlet mode\n$\\Omega_{(0,0)}^2$ and the fivefold-degenerate $\\Omega_{(0,2)}^2$. \nFor all higher spins $j{>}0$, analytic expressions for the natural frequencies $\\Omega_{(j,v,\\beta)}$\nnow require merely solving a few polynomial equations of order four at worst.\nWe have done so up to $j{=}2$ and list them in the Appendix \nbut refrain from giving further explicit examples here.\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=0.9\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{om2zero_51.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{om2zero_99.pdf} \\\\[12pt]\n\\includegraphics[width = 0.35\\paperwidth]{om2zero_01.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{om2zero_50.pdf}\n\\caption{Plots of $\\Omega^2_{(0,v,\\beta)}(\\tau)$ over one period, for different values of $k^2$: \n$0.51$ (top left), $0.99$ (top right), $1.01$ (bottom left) and $5$ (bottom right).}\n\\end{figure}\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=0.9\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{om2two_505.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{om2two_550.pdf} \\\\[12pt]\n\\includegraphics[width = 0.35\\paperwidth]{om2two_999.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{om2two_001.pdf}\n\\caption{Plots of $\\Omega^2_{(2,v,\\beta)}(\\tau)$ over one period, for different values of $k^2$: \n$0.505$ (top left), $0.550$ (top right), $0.999$ (bottom left) and $1.001$ (bottom right).}\n\\end{figure}\nFrom the cases of $j{=}0$ and $j{=}2$ displayed below one can see that some of the normal modes\ndip into the negative regime, i.e.~their frequency-squares become negative, for a certain fraction of the time~$\\tau$.\nBecause of this and, quite generally, due to the $\\tau$ variability of the natural frequencies,\nit is not easy to predict the long-term evolution of the fluctuation modes.\nClearly, the stability of the zero solution $\\Phi{\\equiv}0$, \nequivalent to the linear stability of the background Yang--Mills configuration, \nis not simply decided by the sign of the $\\tau$-average of the corresponding frequency-square.\n\n\\vspace{12pt}\n\\section{Stability analysis: stroboscopic map and Floquet theory}\n\\vspace{2pt}\n\\noindent\nThe diagonalized linear fluctuation equation~(\\ref{diagonalized}) represents\na bunch of Hill's equations, where the frequency-squared is a root of a polynomial of order up to four\nwith coefficients given by a polynomial of twice that order in Jacobi elliptic functions.\nA unique solution requires fixing two initial conditions, and so\nfor each fluctuation~$\\Phi_{(j,v,\\beta)}$ there is a two-dimensional solution space.\nIt is well known that Hill's equation, e.g.~in the limit of Mathieu's equation, displays parametric resonance phenomena,\nwhich can stabilize otherwise unstable systems or destabilize otherwise stable ones.\n\nFor oscillating dynamical systems with periodically varying frequency, \nthere exist some general tools to analyze linear stability.\nSwitching to a Hamiltonian picture and to phase space, \nit is convenient to transform the second-order differential equation into\na system of two coupled first-order equations (suppressing all quantum numbers),\n\\begin{equation} \\label{first-order}\n   \\bigl[\\partial^2_\\tau-\\Omega^2(\\tau)\\bigr] \\Phi(\\tau) \\= 0 \\quad\\Leftrightarrow\\quad\n   \\partial_\\tau \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix} \\=\n   \\begin{pmatrix} 0 & 1 \\\\[4pt] -\\Omega^2 & 0 \\  \\end{pmatrix}\n   \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix} \\ =:\\\n   \\mathrm{i}\\,\\widehat\\Omega(\\tau)\\,\\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix}\\ ,\n\\end{equation}\nwhere the frequency~$\\Omega(\\tau)$ is $T$-periodic (sometimes $\\frac{T}{2}$-periodic) in~$\\tau$.\nThe solution to this first-order system is formally given by\n\\begin{equation}\n   \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix}(\\tau) \\=\n   {\\cal T} \\exp\\,\\Bigl\\{ \\int_0^\\tau \\!\\mathrm{d}\\tau'\\ \\mathrm{i}\\,\\widehat\\Omega(\\tau') \\Bigr\\} \\,\n   \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix}(0) \\ ,\n\\end{equation}\nwhere ${\\cal T}$ denotes time ordering.\nBecause of the time dependence of $\\Omega$, the time evolution operator above\nis not homogeneous thus does not constitute a one-parameter group, except when\nthe propagation interval is an integer multiple of the period~$T$. For $\\tau{=}T$,\none speaks of the stroboscopic map \\cite{Arnold}\n\\begin{equation}\n   M\\ :=\\ {\\cal T} \\exp\\,\\Bigl\\{ \\int_0^T \\!\\mathrm{d}\\tau\\ \\mathrm{i}\\,\\widehat\\Omega(\\tau) \\Bigr\\} \n   \\qquad\\Rightarrow\\qquad\n   \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix}(nT) \\= M^n\n   \\begin{pmatrix} \\Phi \\\\[4pt] \\dot\\Phi \\end{pmatrix}(0)\\ .\n\\end{equation}\nThe linear map~$M$ is a functional of the chosen background solution~$\\psi$\nand hence depends on its parameter~$E$ or~$k$.\nThis background is Lyapunov stable if the trivial solution $\\Phi{\\equiv}0$ is,\nwhich is decided by the two eigenvalues $\\mu_1$ and $\\mu_2$ of~$M$. \nSince the system is Hamiltonian, $\\det M{=}1$, we have three cases:\n\\begin{equation}\n\\begin{aligned}\n   |\\mathrm{tr}\\,M| > 2 &\\quad\\Leftrightarrow\\quad \\mu_i\\in\\mathds R  \n   &\\quad\\;\\Leftrightarrow\\quad& \\textrm{hyperbolic\/boost} \n   &\\quad\\Leftrightarrow\\quad& \\textrm{strongly unstable} \\ ,\\\\\n   |\\mathrm{tr}\\,M| = 2  &\\quad\\Leftrightarrow\\quad \\mu_i=\\pm1  \n   &\\quad\\Leftrightarrow\\quad& \\textrm{parabolic\/translation} \n   &\\quad\\Leftrightarrow\\quad& \\textrm{marginally stable}\\ , \\\\\n   |\\mathrm{tr}\\,M| < 2  &\\quad\\Leftrightarrow\\quad \\mu_i\\in\\textrm{U}(1)  \n   &\\quad\\Leftrightarrow\\quad& \\textrm{elliptic\/rotation} \n   &\\quad\\Leftrightarrow\\quad&\\textrm{strongly stable}\\ .\n\\end{aligned}\n\\end{equation}\nClearly, $|\\mathrm{tr}\\,M|$ determines the linear stability of our classical solution.\n\nLet us thus try to evaluate the trace of the stroboscopic map~$M$,\nmaking use of the special form of the matrix~$\\widehat\\Omega$,\n\\begin{equation}\n\\begin{aligned}\n   \\mathrm{tr}\\,M &\\= \\sum_{n=0}^\\infty \\mathrm{i}^n \n   \\int_0^T\\!\\!\\mathrm{d}\\tau_1\\int_0^{\\tau_1}\\!\\!\\!\\mathrm{d}\\tau_2\\ \\ldots\\int_0^{\\tau_{n-1}}\\!\\!\\!\\!\\mathrm{d}\\tau_n\\\n   \\mathrm{tr}\\,\\bigl[ \\widehat\\Omega(\\tau_1)\\,\\widehat\\Omega(\\tau_2)\\cdots\\widehat\\Omega(\\tau_n)\\bigr] \\\\[4pt]\n   &\\= 2 +  \\sum_{n=1}^\\infty(-1)^n \n   \\int_0^T\\!\\!\\mathrm{d}\\tau_1\\int_0^{\\tau_1}\\!\\!\\!\\mathrm{d}\\tau_2\\ \\ldots\\int_0^{\\tau_{n-1}}\\!\\!\\!\\!\\mathrm{d}\\tau_n\\\n   H_n(\\tau_1,\\tau_2,\\ldots,\\tau_n)\\Omega^2(\\tau_1)\\,\\Omega^2(\\tau_2)\\,\\cdots\\Omega^2(\\tau_n) \\\\[4pt]\n   \\textrm{with}& \\quad H_n(\\tau_1,\\tau_2,\\ldots,\\tau_n) \\= \n   (\\tau_1{-}\\tau_2)(\\tau_2{-}\\tau_3)\\cdots(\\tau_{n-1}{-}\\tau_n)(\\tau_n{-}\\tau_1{+}1)\n   \\quad\\textrm{and}\\quad H_1(\\tau_1)=1\\ .\n\\end{aligned}\n\\end{equation}\nIt is convenient to scale the time variable such as to normalize the period to unity,\n\\begin{equation}\n   \\tau = T\\, x  \\qquad\\quad\\textrm{and}\\quad\\qquad \\Omega^2(Tx) =: \\omega^2(x)\\ ,\\quad H(\\{Tx\\})=:h(\\{x\\}) \\ ,\n\\end{equation}\nhence\n\\begin{equation} \\label{Ansum}\n\\begin{aligned}\n   \\mathrm{tr}\\,M &\\= 2 +  \\sum_{n=1}^\\infty \\bigl(-T^2\\bigr)^n \n   \\int_0^1\\!\\!\\!\\mathrm{d} x_1\\int_0^{x_1}\\!\\!\\!\\mathrm{d} x_2 \\ldots\\int_0^{x_{n-1}}\\!\\!\\!\\!\\mathrm{d} x_n\\,\n   h_n(x_1,x_2,\\ldots,x_n)\\omega^2(x_1)\\omega^2(x_2)\\cdots\\omega^2(x_n) \\\\[4pt]\n   &\\= \\sum_{n=0}^\\infty \\frac{2}{(2n)!}\\,M_n\\,\\bigl(-T^2\\bigr)^n \n   \\ =:\\  2 - M_1 T^2 + \\sfrac{1}{12}M_2 T^4 - \\sfrac{1}{360}M_3 T^6 + \\sfrac{1}{20160}M_4 T^8 - \\ldots\\ .\n\\end{aligned}\n\\end{equation}\n\nIt is impossible to evaluate the integrals $M_n$ without explicit knowledge of~$\\omega^2(x)$. \nAs a crude guess, we replace the weight function by its (constant) average value\n\\begin{equation}\n   \\<h_n\\> \\ :=\\ \\frac{1}{n!} \\int_0^1\\!\\!\\mathrm{d} x_1\\int_0^{x_1}\\!\\!\\!\\mathrm{d} x_2\\ \\ldots\\int_0^{x_{n-1}}\\!\\!\\!\\!\\mathrm{d} x_n\\ \n   h_n(x_1,x_2,\\ldots,x_n) \\= \\frac{2\\,n!}{(2n)!}\n\\end{equation}\nand obtain\n\\begin{equation}\n   M_n \\= \\frac{(2n)!}{2}\\,\\<h_n\\> \n   \\int_0^1\\!\\!\\mathrm{d} x_1\\int_0^{x_1}\\!\\!\\!\\mathrm{d} x_2\\ \\ldots\\int_0^{x_{n-1}}\\!\\!\\!\\!\\mathrm{d} x_n\\ \\prod_{i=1}^n \\omega^2(x_i)\n   \\= \\Bigl( \\int_0^1\\!\\!\\mathrm{d} x\\ \\omega^2(x) \\Bigr)^n \\ =:\\ \\< \\,\\omega^2\\>^n\\ ,\n\\end{equation}\nwhich yields\n\\begin{equation}\n   \\mathrm{tr}\\,M \\= 2\\,\\sum_{n=0}^\\infty \\frac{(-1)^n}{(2n)!}\\,\\<\\,\\omega^2\\>^n\\,T^{2n} \n   \\= 2\\,\\cos\\bigl(\\sqrt{\\<\\,\\omega^2\\>}\\,T\\bigr)\\ .\n\\end{equation}\nThis expression indicates stability as long as $\\<\\omega^2\\>>0$.\nHowever, the result for the $j{=}0$ singlet mode $\\omega^2=\\Omega^2_{(0,0)}$ in (\\ref{j0average}) \nalready showed that the averaged frequency-squared may turn negative in certain domains\nthus changing the cos into a cosh there.\n\nTo do better, let us look at the individual terms~$M_n$ in~(\\ref{Ansum}) for the simplest case\nof the SO(4) singlet fluctuation, i.e.~$\\Omega^2_{(0,0)}=6\\psi^2{-}2$ in~(\\ref{unperturbed}). \nIts average frequency-square is easily computed to be\n\\begin{equation} \\label{j0average}\n   \\<\\,\\Omega_{(0,0)}^2 \\> \\= \\frac{1}{\\epsilon^2}\\Bigl(6 \\frac{E(k)}{K(k)}+4k^2-5\\Bigr)\\ , \n\\end{equation}\nwhere $E(k)$ and $K(k)$ denote the second and first complete elliptic integrals, respectively.\nPlotting this expression as a function of the modulus~$k$, we see that it\nbecomes negative only in a very narrow range around $k{=}1$, namely for $|k{-}1|\\lesssim0.00005$.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 0.35\\paperwidth]{om2zeroavg1.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{om2zeroavg2.pdf}\n\\caption{Plot of $\\<\\Omega^2_{(0,0)}\\>$ as a function of~$k$, with detail on the right.}\n\\end{figure}\nWe have only been able to analytically evaluate (with $k{<}1$ for simplicity)\n\\begin{equation}\n   M_1 \\= \\<\\,\\Omega_{(0,0)}^2 \\>  \\quad\\textrm{and}\\quad\n   M_2 \\= \\<\\,\\Omega_{(0,0)}^2 \\> ^2\\ -\\ \\frac{1}{\\epsilon^4}\\Bigl(\n   9\\frac{2{-}k^2}{K(k)^2} - 27\\frac{E(k)}{K(k)^3} + \\frac{9\\,\\pi^2}{4\\,K(k)^4} \\Bigr)\\ ,\n\\end{equation}\nwhich does not suffice to rule out instability.\nIndeed, numerical studies show that $M_n$ as a function of~$k$ looses its positivity in a range\naround $k{=}1$ which increases with~$n$, where the series~(\\ref{Ansum}) ceases to be alternating.\nMoreover, even in the limit of a very large background amplitude, $k^2\\to\\frac12$, we find that\n\\begin{equation}\n   \\<\\,\\Omega_{(0,0)}^2 \\>\\ \\to\\ \\frac{24\\,\\pi^2}{\\epsilon^2\\,\\Gamma(\\frac14)^4}\\ \\approx\\ \\frac{1.37}{\\epsilon^2}\n   \\qquad\\Rightarrow\\qquad \\sqrt{\\<\\,\\Omega^2\\>}\\,T\\ \\to\\ \\sqrt{24\\,\\pi}\\ \\approx\\ 8.68\\ ,\n\\end{equation}\nimplying that we must push the series in~(\\ref{Ansum}) at least to $O(M_{10}T^{20})$,\neven though it turns out that $M_n<\\<\\,\\Omega^2_{(0,0)}\\>^n$ at $k^2=\\frac12$ for $n>1$.\n\nFor a more complete analysis of linear stability in an oscillating system with time-dependent frequency\nwe can take recourse to Floquet theory.\nIt tells us that a general fundamental matrix solution\n\\begin{equation}\n   \\widehat\\Phi(\\tau) \\= \\begin{pmatrix} \\Phi_1 & \\Phi_2 \\\\[4pt] \\dot\\Phi_1 & \\dot\\Phi_2 \\end{pmatrix}(\\tau)\n   \\qquad\\Rightarrow\\qquad \\partial_\\tau \\widehat\\Phi(\\tau) \\= \\mathrm{i}\\,\\widehat\\Omega(\\tau)\\,\\widehat\\Phi(\\tau)\n\\end{equation}\nof our system~(\\ref{first-order}) with some initial condition $\\widehat\\Phi(0)=\\widehat\\Phi_0$ \ncan be expressed in so-called Floquet normal form as\n\\begin{equation}\n   \\widehat\\Phi(\\tau) \\= Q(\\tau)\\;\\mathrm{e}^{\\tau R} \n   \\qquad\\textrm{with}\\qquad Q(\\tau{+}2T) \\= Q(\\tau)\\ ,\n\\end{equation}\nwhere $Q(\\tau)$ and $R$ are real $2{\\times}2$ matrices,\nso that the time dependence of the frequency can be transformed away by a change of coordinates,\n\\begin{equation}\n   \\Psi(\\tau) \\ :=\\ Q(\\tau)^{-1} \\widehat\\Phi(\\tau) \\qquad\\Rightarrow\\qquad\n   \\partial_\\tau \\Psi(\\tau) \\= R\\,\\Psi(\\tau)\\ .\n\\end{equation}\nDue to the identity\n\\begin{equation}\n   \\widehat\\Phi(\\tau{+}T) \\= \\widehat\\Phi(\\tau)\\,\\widehat\\Phi(0)^{-1}\\,\\widehat\\Phi(T) \n   \\= \\widehat\\Phi(T)\\,\\widehat\\Phi(0)^{-1}\\,\\widehat\\Phi(\\tau) \\= M\\,\\widehat\\Phi(\\tau)\n\\end{equation}\nwe see that our stroboscopic map~$M$ is nothing but the monodromy, and\n\\begin{equation}\n   M^2 \\= \\widehat\\Phi(2T)\\,\\widehat\\Phi(0)^{-1} \n   \\= Q(0)\\,\\widehat\\Phi(0)^{-1}\\,\\widehat\\Phi(2T)\\,Q(0)^{-1}\n   \\= Q(0)\\,\\mathrm{e}^{2 R T}\\,Q(0)^{-1}\\ ,\n\\end{equation}\nso that its eigenvalues (or characteristic multipliers)\n\\begin{equation}\n   \\mu_i = \\mathrm{e}^{\\rho_i T} \\qquad\\textrm{for}\\quad i=1,2\n\\end{equation}\ndefine a pair of (complex) Floquet exponents $\\rho_i$ \nwhose real parts are the Lyapunov exponents.\nSince $\\mu_1\\mu_2=1$ implies that $\\rho_1{+}\\rho_2=0$,\nour system is linearly stable if and only if both eigenvalues~$\\rho_i$ of~$R$\nare purely imaginary (or zero).\n\nGenerally it is impossible to find analytically the monodromy pertaining to\na normal mode~$\\Phi_{(j,v,\\beta)}$.\\footnote{\nAn exception is the SO(4) singlet perturbation~$\\Phi_{(0,0)}$, to be treated in the following section.}\nHowever, we can get a qualitative understanding by looking numerically at some examples. \nBefore numerically integrating Hill's equation, however, let us estimate at which energies~$E$\nor, rather, moduli~$k$, possible resonance frequencies might occur.\nTo this end, we determine the period-average of the natural frequency $\\Omega_{(j,v,\\beta)}$ \nand compare it to its modulation frequency~$\\sfrac{2\\pi}{T}$. If we model \n\\begin{equation}\n   \\Omega^2(\\tau) \\= \\<\\,\\Omega^2\\> \\,\\bigl(1+h(\\tau)\\bigr)  \\quad\\textrm{with}\\quad\n   \\<\\,\\Omega^2\\> = \\sfrac{1}{T}\\smallint_0^T \\!\\mathrm{d}\\tau\\ \\Omega^2(\\tau) \\quad\\textrm{and}\\quad\n   h(\\tau) \\ \\propto\\ \\cos(2\\pi\\tau\/T) \\ ,\n\\end{equation}\nwhere $T=4\\,\\epsilon\\,K(k)$, then the resonance condition is met for\n\\begin{equation}\n   \\sqrt{\\<\\Omega^2\\>} \\= \\ell\\,\\frac{\\pi}{T} \\qquad\\Rightarrow\\qquad\n   k=k_\\ell(j,v,\\beta)\\qquad\\textrm{for}\\quad \\ell=1,2,3,\\ldots\\ .\n\\end{equation}\nSince this model reproduces only the rough features of $\\Omega^2(\\tau)$, we expect potential instability\ndue to parametric resonance effects in a band around or near the values~$k_\\ell$.\n\nBelow we display, together with the would-be resonant values~$k_\\ell$,\nthe function $\\mathrm{tr} M(k)$ for the sample cases of $(j,v)=(2,0)$ and $(2,2)$.\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{trM20a.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{trM20b.pdf} \n\\caption{Plot of $\\mathrm{tr}\\,M(k)$ for $(j,v)=(2,0)$, with detail on the right. Would-be resonances marked in red.}\n\\end{figure}\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 0.35\\paperwidth]{trM22_1.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{trM22_2.pdf} \\\\[8pt]\n\\includegraphics[width = 0.35\\paperwidth]{trM22_3.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{trM22_4.pdf} \n\\caption{Plots of $\\mathrm{tr}\\,M(k)$ for $(j,v)=(2,2)$ and $\\beta=1,2,3,4$. Would-be resonances marked in red.}\n\\end{figure}\nOne sees that, on both sides of the critical value of $E{=}\\sfrac12$ (or $k{=}1$), \ncorresponding to the double-well local maximum, the $k_\\ell$ values accumulate at the critical point.\nBut while for $k{>}1$ (energy below the critical point) $\\mathrm{tr} M(k)$ oscillates between values close to $2$ \nin magnitude and thus exponential growth is rare and mild, for $k{<}1$ (energy above the critical point)\nthe oscillatory behavior of $\\mathrm{tr} M(k)$ comes with an amplitude exceeding~2 and growing with energy.\nHence, in this latter regime stable and unstable bands alternate.\nThis is supported by long-term numerical integration, as we demonstrate by plotting\n$\\Phi(\\tau)$ for $(j,v,\\beta)=(2,2,1)$ with initial values $\\Phi(0){=}1$ and $\\dot\\Phi(0){=}0$ \non both sides very close to the end of the first instability \n(at the highest value of~$E$ or the lowest value of~$k$).\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{phi_22_k073198.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{phi_22_k073199.pdf} \n\\caption{Plot of $\\Phi(\\tau)$ for $(j,v,\\beta)=(2,2,1)$ and $k{=}0.73198$ (left) and $k{=}0.73199$ (right).}\n\\end{figure}\n\nMost relevant for the cosmological application is the regime of very large energies, $E\\to\\infty$ (or $k\\to 1\/\\sqrt{2}$).\nIn this limit, we observe the following universal behavior. \nBecause the period~$T$ collapses with $\\epsilon{=}\\sqrt{k^2{-}1\/2}$, we rescale\n\\begin{equation}\n   \\sfrac{\\tau}{\\epsilon} = z \\in [0,4K(\\sfrac12)]\\ ,\\quad\n   \\epsilon\\,\\psi = \\tilde\\psi\\ ,\\quad \n   \\epsilon^2\\dot\\psi = \\partial_z\\tilde\\psi\\ ,\\quad \n   \\epsilon^2\\Omega^2 = \\tilde\\Omega^2\\ ,\\quad \n   \\epsilon^2\\lambda = \\tilde\\lambda\n\\end{equation}\nso that the tilded quantities remain finite in the limit,\nand find, with $\\bar\\omega^2_{(\\bar{v})}\\to 2\\psi^2$,\\footnote{\nFor the cases $(j,v)=(0,1)$ and $(1,0)$, the factor $\\tilde\\lambda$ is missing; \nfor $(j,v)=(0,0)$, one only has $R=Q\\sim(\\tilde\\lambda{-}6\\tilde\\psi^2)$.}\n\\begin{equation}\n\\begin{aligned}\n   &Q_{(v\\pm2)}\\ \\sim\\ \\tilde\\lambda\\ , \\\\\n   &Q_{(v\\pm1)}\\ \\sim\\ \\tilde\\lambda\\,(\\tilde\\lambda-4\\tilde\\psi^2)\\ ,\\qquad\\qquad R_{(v\\pm1)}\\ \\sim\\ \\tilde\\lambda\\,\\bigl[(\\tilde\\lambda-2\\tilde\\psi^2)(\\tilde\\lambda-4\\tilde\\psi^2)-8(\\mbox{$\\partial$}_z\\tilde\\psi)^2\\bigr]\\ ,\\\\\n   &Q_{(v)}\\ \\sim\\ \\tilde\\lambda\\,(\\tilde\\lambda-4\\tilde\\psi^2)(\\tilde\\lambda-6\\tilde\\psi^2)\\ ,\\quad R_{(v)}\\ \\sim\\ \\tilde\\lambda\\,\\bigl[(\\tilde\\lambda-2\\tilde\\psi^2)(\\tilde\\lambda-4\\tilde\\psi^2)-8(\\mbox{$\\partial$}_z\\tilde\\psi)^2\\bigr](\\tilde\\lambda-6\\tilde\\psi^2)\\ ,\n\\end{aligned}\n\\end{equation}\nbecause all $j$-dependent terms in the polynomials are subleading and drop out in the limit.\nFactorizing the $R$~polynomials, we find the four universal natural frequency-squares\n\\begin{equation}\n   \\tilde\\Omega^2_1 = 0\\ ,\\quad\n   \\tilde\\Omega^2_2 = 3\\,\\tilde\\psi^2-\\sqrt{\\tilde\\psi^4+8(\\mbox{$\\partial$}_z\\tilde\\psi)^2}\\ ,\\quad\n   \\tilde\\Omega^2_3 = 3\\,\\tilde\\psi^2+\\sqrt{\\tilde\\psi^4+8(\\mbox{$\\partial$}_z\\tilde\\psi)^2}\\ ,\\quad\n   \\tilde\\Omega^2_4 = 6\\,\\tilde\\psi^2\\ .\n\\end{equation}\nOne must pay attention, however, to the fact that the avoided crossings disappear in the $\\epsilon\\to0$ limit.\nTherefore, the correct limiting frequencies to input into\n\\begin{equation}\n   \\bigl[ \\mbox{$\\partial$}_z - \\tilde\\Omega^2_{(j,v,\\beta)} \\bigr]\\,\\tilde\\Phi_{(j,v,\\beta)} \\= 0\n\\end{equation}\nare\n\\begin{equation}\n\\begin{aligned}\n   &\\tilde\\Omega^2_{(j,j\\pm2)}\\ \\ = 0\\ ,\\\\\n   &\\tilde\\Omega^2_{(j,j\\pm1,\\beta)} \\in \\bigl\\{ \n   \\textrm{min}(\\tilde\\Omega^2_1,\\tilde\\Omega^2_2),\\ \\textrm{max}(\\tilde\\Omega^2_1,\\tilde\\Omega^2_2),\\\n   \\tilde\\Omega^2_3 \\bigr\\}\\ ,\\\\\n   &\\tilde\\Omega^2_{(j,j,\\beta)} \\ \\ \\ \\in \\bigl\\{ \n   \\textrm{min}(\\tilde\\Omega^2_1,\\tilde\\Omega^2_2),\\ \\textrm{max}(\\tilde\\Omega^2_1,\\tilde\\Omega^2_2),\\ \n   \\textrm{min}(\\tilde\\Omega^2_3,\\tilde\\Omega^2_4),\\ \\textrm{max}(\\tilde\\Omega^2_3,\\tilde\\Omega^2_4) \\bigr\\}\\ ,\n\\end{aligned}\n\\end{equation}\nof which we show below the last list as a function of~$z$. \n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=0.9\\linewidth}\n\\includegraphics[width = 0.5\\paperwidth]{om2universal.pdf} \n\\caption{Plot of the universal limiting natural frequency-squares \n$\\tilde\\Omega^2_{(j,v,\\beta)}$ for $v{=}j$ and $\\beta=1,2,3,4$.}\n\\end{figure}\nThe monodromies are easily computed numerically,\\footnote{\nFor the cases $(j,v)=(0,1)$ and $(1,0)$ one gets $\\{56.769,\\ -1.659\\}$;\nfor $(j,v)=(0,0)$ we have $\\mathrm{tr}\\,M=2$.}\n\\begin{equation}\n\\begin{aligned}\n   &\\mathrm{tr}\\,M_{(j,j\\pm2)}(E{\\to}\\infty)\\ \\ \\ = 2\\ ,\\\\\n   &\\mathrm{tr}\\,M_{(j,j\\pm1,\\beta)}(E{\\to}\\infty)\\ \\in \\bigl\\{ 306.704,\\ -1.842,\\ -1.659 \\bigr\\}\\ ,\\\\\n   &\\mathrm{tr}\\,M_{(j,j,\\beta)}(E{\\to}\\infty) \\ \\ \\ \\ \\in \\bigl\\{ 306.704,\\ -1.842,\\ 2.462,\\ -1.067 \\bigr\\}\\ ,\n\\end{aligned}\n\\end{equation}\nin agreement with the figures above. In particular, the extremal $v$-values become marginally stable,\nwhile part of the non-extremal cases are unstable for high energies. \n\nOf course, for each non-extremal value of~$v$ we still have to project out unphysical modes\nby imposing the gauge condition~(\\ref{gauge4}). However, in the $12(2j{+}1)$-dimensional fluctuation space\nthe gauge condition has rank~$3(2j{+}1)$ while we see that (for $j{\\ge}2$) in total $4(2j{+}1)$ normal modes \nare unstable at high energy. Therefore, the projection to physical modes cannot remove all instabilities. \nWe must conclude that, for sufficiently high energy~$E$, some fluctuations grow exponentially,\nimplying that the solution $\\Phi{\\equiv}0$ is linearly unstable, and thus is the Yang--Mills background.\n\n\n\\vspace{12pt}\n\\section{Singlet perturbation: exact treatment}\n\\vspace{2pt}\n\\noindent\nEven though the Floquet representation helped to reduce the long-time behavior of the perturbations\nto the analysis of a single period~$T$, it normally does not give us an exact solution to Hill's equation.\nHowever, for the SO(4) singlet fluctuation around $\\psi(\\tau)$, we can employ the fact that \n$\\dot{\\psi}$ trivially solves the fluctuation equation,\n\\begin{equation} \\label{timeshift}\n   (\\dot\\psi)^{\\cdot\\cdot} = (\\ddot\\psi)^{\\cdot} = -\\bigl(V'(\\psi)\\bigr)^{\\cdot} \n   = -V''(\\psi)\\,\\dot{\\psi} \\= -(6\\psi^2{-}2)\\,\\dot\\psi \\= -\\Omega^2_{(0,0)}(\\tau)\\,\\dot\\psi\\ ,\n\\end{equation}\nwith a frequency function which is $\\frac{T}{2}$-periodic.\nThis implies that all fluctuation modes are $T$-periodic.\nWith the knowledge of an explicit solution to the fluctuation equation \nwe can reduce the latter to a first-order equation and solve that one to find a second solution.\nThe normalizations are arbitrary, so we choose\n\\begin{equation}\n   \\Phi_1(\\tau)\\=-\\sfrac{\\epsilon^3}{k}\\,\\dot{\\psi}(\\tau) \\quad\\textrm{and}\\quad\n   \\Phi_2(\\tau) \\= \\Phi_1(\\tau)\\,\\int^\\tau\\frac{\\mathrm{d}\\sigma}{\\Phi_1(\\sigma)^2}\n   \\= -\\sfrac{k}{\\epsilon^3}\\,\\dot{\\psi}(\\tau)\\,\\int^\\tau \\frac{\\mathrm{d}\\sigma}{\\dot{\\psi}^2(\\sigma)}\\ ,\n\\end{equation}\nwhich are linearly independent since\n\\begin{equation}\n   W(\\Phi_1,\\Phi_2)\\ \\equiv\\ \\Phi_1\\dot\\Phi_2-\\Phi_2\\dot\\Phi_1 \\= 1\\ .\n\\end{equation}\n\nFor simplicity, we restrict ourselves to the energy range \n$\\sfrac12{<}E{<}\\infty$, i.e.~$1{>}k^2{>}\\sfrac12$. \nExplicitly, we have\n\\begin{equation}\n\\begin{aligned}\n   \\!\\!\\!\\Phi_1(\\tau) &\\= \\epsilon\\,{\\mathrm{sn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)}\\,{\\mathrm{dn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)} \\ , \\\\[4pt]\n   \\!\\!\\!\\Phi_2(\\tau) &\\= \\sfrac{1}{1-k^2}\\,{\\mathrm{cn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)}\n   \\bigl[ (2k^2{-}1)\\,\\textrm{dn}^2\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr) -k^2 \\bigr] \\\\\n   &\\qquad + {\\mathrm{sn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)}\\,{\\mathrm{dn}\\bigl(\\sfrac{\\tau}{\\epsilon},k\\bigr)} \\bigl[ \\sfrac{\\tau}{\\epsilon} +\n   \\sfrac{2k^2{-}1}{1{-}k^2}\\,E\\bigl(\\textrm{am}(\\sfrac{\\tau}{\\epsilon},k),k\\bigr)\\bigr]\\;, \n\\end{aligned}\n\\end{equation}\nwhere $\\textrm{am}(z,k)$ denotes the Jacobi amplitude and $E(z,k)$ is the elliptic integral of the second kind.\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{phi1.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{phi2.pdf} \n\\caption{Plot of the SO(4) singlet fluctuation modes $\\Phi_1$ and $\\Phi_2$ over eight periods for $k^2{=}0.81$.}\n\\end{figure}\nAs can be checked, the initial conditions are\n\\begin{equation}\n   \\Phi_1(0)=0\\ ,\\quad \\dot\\Phi_1(0)=1 \\qquad\\quad\\textrm{and}\\quad\\qquad\n   \\Phi_2(0)=-1\\ ,\\quad \\dot\\Phi_2(0)=0\\ ,\n\\end{equation}\nwhich fixes the ambiguity of adding to $\\Phi_2$ a piece proportional to~$\\Phi_1$. Hence,\n\\begin{equation}\n   \\widehat\\Phi(0) \\= \\Bigl(\\begin{smallmatrix} 0 & \\!{-}1 \\\\[4pt] 1 & 0 \\end{smallmatrix} \\Bigr)\n   \\qquad\\Rightarrow\\qquad\n   M\\= \\widehat\\Phi(T)\\, \\Bigl(\\begin{smallmatrix} 0 & 1 \\\\[4pt] {-}1 & 0 \\end{smallmatrix} \\Bigr)\\ .\n\\end{equation}\nWe know that $\\Phi_1\\sim\\dot\\psi$ is $T$-periodic, and so is $\\dot\\Phi_1$, \nbut not the second solution,\n\\begin{equation}\n   \\Phi_2(\\tau{+}T) \\= \\Phi_2(\\tau) + \\gamma\\,T\\,\\Phi_1(\\tau) \\quad\\textrm{with}\\quad \\gamma \\= \\frac{1}{T}\\int_0^T\\frac{\\mathrm{d}\\sigma}{\\Phi_1(\\sigma)^2}\\bigg|_{\\textrm{reg}}\n   \\ =:\\ \\sfrac{k^2}{\\epsilon^6}\\, \\bigl\\langle \\dot\\psi^{-2}\\bigr\\rangle_{\\textrm{reg}}\\ ,\n\\end{equation}\nwhere the integral diverges at the turning points and must be regularized by subtracting\nthe Weierstra\\ss\\ $\\wp$~function with the appropriate half-periods.\nSince $\\Phi_1$ has periodic zeros, $\\Phi_2$ does return to~${-}1$ at integer multiples of~$T$.\nIt follows that the $\\Phi_2$ oscillation linearly grows in amplitude with a rate (per period) of\n\\begin{equation}\n   \\gamma \\= \\frac{1}{\\epsilon^2}\\,\\Bigl[\\,1 + \\frac{2k^2{-}1}{1{-}k^2}\\,\\frac{E(k)}{K(k)} \\,\\Bigr]\\ ,\n\\end{equation}\nwhich is always larger than 7.629, attained at $k\\approx0.882$.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 0.5\\paperwidth]{gamma.pdf} \n\\caption{Plot of the linear growth rate~$\\gamma$ as a function of~$k$.}\n\\end{figure}\n\nIn essence, we have managed to compute the monodromy\n\\begin{equation}\n   M \\= \\begin{pmatrix} {-}\\Phi_2(T) & \\Phi_1(T) \\\\[4pt] {-}\\dot\\Phi_2(T) & \\dot\\Phi_1(T) \\ \\end{pmatrix} \n   \\= \\begin{pmatrix} \\ 1 & 0 \\\\[4pt] \\ \\!\\!{-}\\gamma\\,T & 1 \\ \\end{pmatrix} \n   \\= \\exp \\Bigl\\{ {-}\\gamma\\,T\\,\\bigl(\\begin{smallmatrix} 0 & 0 \\\\[4pt] 1 & 0 \\end{smallmatrix}\\bigr) \\Bigr\\}\n\\end{equation}\nand thus easily obtain the Floquet representation,\n\\begin{equation}\n   R \\= \\begin{pmatrix} 0 & \\gamma \\\\[4pt] 0 & 0 \\end{pmatrix} \\quad\\Rightarrow\\quad\n   \\mathrm{e}^{\\tau R} \\= \\begin{pmatrix} 1 & \\gamma\\,\\tau \\\\[4pt] 0 & 1 \\end{pmatrix}  \\quad\\textrm{and}\\quad\n   Q(\\tau)\\ = \\begin{pmatrix} \\Phi_1 & \\Phi_2{-}\\Phi_1\\gamma\\,\\tau \\\\[4pt] \n   \\dot\\Phi_1 & \\dot\\Phi_2{-}\\dot\\Phi_1\\gamma\\,\\tau \\end{pmatrix}\\ .\n\\end{equation}\nObviously, we have encountered a marginally stable situation, since $M$ is of parabolic type. \nThere is no exponential growth, and $\\Phi_1$ is periodic thus bounded, but $\\Phi_2$ \ngrows without bound as long as one stays in the linear regime.\nNote that we never made use of the form of our Newtonian potential.\nIn fact, this behavior is typical for a conservative mechanical system with oscillatory motion.\n\nWhat to make of this linear growth? It can be (and actually is) easily overturned by nonlinear effects.\nGoing beyond the linear regime, though, requires expanding the Yang--Mills equation to higher orders\nabout our classical Yang--Mills solution~(\\ref{Aansatz}).\nWhile this is a formidable task in general, it can actually be done to all orders for the singlet perturbation!\nThe reason is that a singlet perturbation leaves us in the SO(4)-symmetric subsector,\nthus connecting only to a neighboring ``cosmic background'', $\\psi\\to\\tilde\\psi$.\nSince (\\ref{backgrounds}) gives us analytic control over all solutions~$\\psi(\\tau)$,\nthe full effect of such a shift can be computed exactly. \nSplitting an exact solution~$\\tilde{\\psi}$ into a background part and its (full) deviation,\n\\begin{equation}\n   \\tilde{\\psi}(\\tau) \\= \\psi(\\tau)\\ +\\ \\eta(\\tau)\\ ,\n\\end{equation}\nand inserting $\\tilde{\\psi}$ into the equation of motion~(\\ref{Newton}), we obtain\n\\begin{equation} \\label{nonlinear}\n   0 \\= \\ddot\\eta + V''(\\psi)\\,\\eta + \\sfrac12 V'''(\\psi)\\,\\eta^2 + \\sfrac16 V''''(\\psi)\\,\\eta^3\n   \\= \\ddot\\eta + (6\\psi^2{-}2)\\,\\eta + 6\\psi\\,\\eta^2 + 2\\,\\eta^3\\ ,\n\\end{equation}\nextending the linear equation~(\\ref{timeshift}) by two nonlinear contributions.\nPerturbation theory introduces a small parameter~$\\epsilon$ and formally expands\n\\begin{equation}\n   \\eta \\= \\epsilon\\eta_{(1)} + \\epsilon^2\\eta_{(2)} + \\epsilon^3\\eta_{(3)} + \\ldots\\ ,\n\\end{equation}\nwhich yields the infinite coupled system\n\\begin{equation}\n\\begin{aligned}\n   & \\bigl[\\mbox{$\\partial$}_\\tau^2+(6\\psi^2{-}2)\\bigr]\\,\\eta_{(1)} \\= 0 \\ ,\\\\\n   & \\bigl[\\mbox{$\\partial$}_\\tau^2+(6\\psi^2{-}2)\\bigr]\\,\\eta_{(2)} \\= -6\\psi\\,\\eta_{(1)}^2 \\ ,\\\\\n   & \\bigl[\\mbox{$\\partial$}_\\tau^2+(6\\psi^2{-}2)\\bigr]\\,\\eta_{(3)} \\= -12\\psi\\,\\eta_{(1)}\\eta_{(2)} -2\\,\\eta_{(1)}^3\\ ,\\\\\n   & \\ldots \\ ,\n\\end{aligned}\n\\end{equation}\nwhich could be iterated with a seed solution~$\\eta_{(1)}$ of the linear system.\n\nHowever, we know that the exact solutions to the full nonlinear equation~(\\ref{nonlinear}) \nis simply given by the difference\n\\begin{equation}\n   \\eta(\\tau) \\= \\tilde\\psi(\\tau) - \\psi(\\tau)\n\\end{equation}\nof two analytically known backgrounds. The SO(4)-singlet background moduli space is parametrized by\ntwo coordinates, e.g.~the energy~$E$ (or elliptic modulus~$k$) and the choice of an initial condition\nwhich fixes the origin~$\\tau{=}0$ of the time variable. In~(\\ref{backgrounds}), we selected $\\dot\\psi(0)=0$,\nbut relaxing this we can reintroduce this collective coordinate by allowing shifts in~$\\tau$. \nWe may then parametrize the SO(4)-invariant Yang--Mills solutions as\n\\begin{equation}\n   \\psi_{k,\\ell}(\\tau) \\= \\psi(\\tau{-}\\ell) \n   \\qquad\\textrm{with}\\qquad 2E=1\/(2k^2{-}1)^2 \\quad\\quad\\textrm{and}\\quad\\quad \\ell\\in\\mathds R\n\\end{equation}\nwhere $\\psi$ is taken from~(\\ref{backgrounds}). \nNote that $\\dot\\psi_{k,\\ell}$ solves the background equation~(\\ref{timeshift}) with a frequency-squared\n$\\omega_{k,\\ell}^2=6\\psi_{k,\\ell}^2{-}2$.\nWithout loss of generality we assign $\\psi=\\psi_{k,0}$ and $\\tilde\\psi=\\psi_{k{+}\\delta k,\\delta\\ell}$,\nhence\n\\begin{equation}\n\\begin{aligned}\n   \\eta(\\tau) &\\= \\delta k\\,\\mbox{$\\partial$}_k\\psi(\\tau) - \\delta\\ell\\,\\dot\\psi(\\tau) \n   + \\sfrac12(\\delta k)^2\\,\\mbox{$\\partial$}_k^2\\psi(\\tau) - \\delta k \\delta\\ell\\,\\mbox{$\\partial$}_k\\dot\\psi(\\tau) \n   + \\sfrac12(\\delta\\ell)^2\\,\\ddot\\psi(\\tau) + \\ldots \\\\[4pt]\n   &\\=  \\delta k\\,\\mbox{$\\partial$}_k\\psi(\\tau{-}\\delta\\ell) + \\sfrac12(\\delta k)^2\\,\\mbox{$\\partial$}_k^2\\psi(\\tau{-}\\delta\\ell)\n   + \\sfrac16(\\delta k)^3\\,\\mbox{$\\partial$}_k^3\\psi(\\tau{-}\\delta\\ell) + \\ldots\\ ,\n\\end{aligned}\n\\end{equation}\nbecause $\\mbox{$\\partial$}_\\ell\\psi=-\\dot\\psi$. \nClearly, a shift in~$\\ell$ only shifts the time dependence of the frequency and does not alter the energy~$E$,\nwhich is not very interesting. Its linear part corresponds to the mode $\\Phi_1\\sim\\dot\\psi$ of the previous section.\nA change in~$k$, in contract, will lead to a solution with an altered frequency and energy.\nIts linear part is given by $\\Phi_2$, which grows linearly in time. \nHowever, due to the boundedness of the full motion, the nonlinear corrections have to limit this growth and\nultimately must bring the fluctuation back close to zero. This is the familiar wave beat phenomenon:\nthe difference of two oscillating functions, $\\tilde\\psi$ and $\\psi$, with slightly different frequencies,\nwill display an amplitude oscillation with a beat frequency given by the difference.\nThis is borne out in the following plots.\n\\begin{figure}[h!]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.35\\paperwidth]{eta_linear.pdf} \\quad\n\\includegraphics[width = 0.35\\paperwidth]{eta_nonlinear.pdf}\n\\caption{Plots of the full perturbation $\\eta$ at $k{=}0.95$ for $\\eta(0){=}0.02$, $\\dot{\\eta}(0){=}0$, giving a beat ratio of ${\\sim}19$.}\n\\end{figure}\nAs a result, we can assert a long-term stability of the cosmic Yang--Mills fields \nagainst the SO(4) singlet perturbation, even though on shorter time scales an excursion \nto a nearby solution is not met with a linear backreaction.\n\n\n\n\n\\chapter{Yang-Mills fields on $\\mathrm{d}{S}_4$}\n\\label{Chapter4}\n\\justifying\n\nHere we review the conformal relation of the $4$-dimensional de Sitter space with a finite Lorentzian cylinder with $S^3$-slicing, study calculus on $S^3$ and present the Yang--Mills field equations on the cylinder. The content of this chapter is partially taken from \\cite{KL20,KPH21,KLP21}\\footnote{The role of $L_a$ and $R_a$ in \\cite{KL20} is interchanged as compared to this thesis.}.\n\n\n\\vspace{12pt}\n\\vspace{2pt}\n\\section{The de Sitter-Minkowski correspondence via $S^3$-cylinder}\n\\noindent\nThe de Sitter space $\\mathrm{d} S_4$ in four dimensions has a natural embedding as a single-sheeted hyperboloid in five-dimensional Minkowski space $\\mathds R^{1,4}$ with coordinates $(q_0, q_{_A}); A=1,2,3,4$ and global length scale $\\ell$ and is given by\n\\begin{equation} \\label{dS4}\n   -q_0^2\\, + q_1^2\\, + q_2^2\\, + q_3^2\\, + q_4^2\\, \\= \\ell^2 .\n\\end{equation}\nOne can use the flat metric on $\\mathds R^{1,4}$\n\\begin{equation} \\label{flat_metric}\n   \\mathrm{d} s_{(1,4)}^2 \\= -\\mathrm{d} q_0^2\\, + \\mathrm{d} q_1^2\\, + \\mathrm{d} q_2^2\\, + \\mathrm{d} q_3^2\\, + \\mathrm{d} q_4^2\n\\end{equation}\nto construct the corresponding metric on $\\mathrm{d} S_4$ using the coordinate constraint \\eqref{dS4}. The metric so obtained is conformally equivalent to the one on a finite Lorentzian cylinder $\\mathcal{I}\\times S^3$ with $\\mathcal{I} = \\left(0, \\pi \\right)$ over the $3$-sphere $S^3$. To see this, we employ the following coordinates\n\\begin{equation}\n   q_{_A} \\= \\ell\\, \\omega_{_A}\\, \\csc\\tau \\quad\\textrm{and}\\quad q_{_5} \\= -\\ell\\, \\cot\\tau \\quad\\quad\\textrm{with}\\quad\\quad \\tau\\, \\in\\, (0,\\pi)\\ ,\n\\end{equation}\nwhere $\\omega_{_A}$ are the natural embedding coordinates of $S^3$ in $\\mathds R^4$: $\\omega_{_A}\\omega_{_A} \\!=\\! 1$\\footnote{Repeated indices are summed over.}. A natural hyperspherical parametrisation of $\\omega_{_A}$ is given by\n\\begin{equation}\\label{omegas}\n   \\omega_1 \\= \\sin\\chi\\,\\sin\\theta\\,\\cos\\phi\\ ,\\quad \\omega_2 \\= \\sin\\chi\\,\\sin\\theta\\,\\sin\\phi\\ ,\\quad \\omega_3 \\= \\sin\\chi\\,\\cos\\theta\\ ,\\quad \\omega_4 \\= \\cos\\chi\\ ,\n\\end{equation}\nwith $0\\leq \\chi,\\theta \\leq \\pi$ and $0\\leq\\phi\\leq2\\pi$. The modified metric has the following form:\n\\begin{equation}\\label{metricdS}\n   \\mathrm{d} s^2 \\= \\mathrm{d} s_{(1,4)}^2|_{\\mathrm{d} S_4} \\= \\frac{\\ell^2}{\\sin^2\\tau}\\left( -\\mathrm{d}\\tau^2 + \\mathrm{d}\\Omega_3^2 \\right); \\quad \\mathrm{d}\\Omega_3^{\\ 2}\\=\\mathrm{d}\\omega_{A}\\, \\mathrm{d}\\omega_{A}|_{\\mathrm{d} S_4}\\ .\n\\end{equation}\nBy gluing two copies of these Lorentzian cylinders at $\\tau\\!=0$, such that now $\\tau \\in \\widetilde{\\cal I} = \\left(-\\pi, \\pi \\right)$, one finds that half of the resultant cylinder $\\widetilde{\\cal I}\\times S^3$ is conformally equivalent to the 4 dimensional Minkowski space. To see this, consider the following parametrization of $(t,x,y,z)\\in \\mathds R^{1,3}$\n\\begin{equation} \\label{S3toMink}\n   \\cot\\tau \\= \\frac{r^2 {-} t^2 {+} \\ell^2}{2\\,\\ell\\,t}\\ ,\\\n   \\omega_1 \\= \\gamma\\,\\frac{x}{\\ell}\\ ,\\\n   \\omega_2 \\= \\gamma\\,\\frac{y}{\\ell}\\ ,\\\n   \\omega_3 \\= \\gamma\\,\\frac{z}{\\ell}\\ ,\\\n   \\omega_4 \\= \\gamma \\frac{r^2 {-} t^2 {-} \\ell^2}{2\\,\\ell^2}\\ ,\n\\end{equation}\nwhere we have abbreviated\n\\begin{figure}[!htbp]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.65\\paperwidth]{Figures\/halfcyl.pdf}\n\\caption{An illustration of the map between a cylinder $2\\mathcal{I}\\times S^3$ and Minkowski space $R^{1,3}$.\nThe Minkowski coordinates cover the shaded area. The boundary of this area is given by the curve $\\omega_4{=}\\cos\\chi{=}\\cos\\tau$. \nEach point is a two-sphere spanned by $\\{\\hat{\\omega}_1,\\hat{\\omega}_2,\\hat{\\omega}_3\\}$, which is mapped to a sphere of constant $r$ and $t$. }\n\\label{patching}\n\\end{figure}\n\\begin{equation}\\label{gamma}\n   r^2 = x^2+y^2+z^2\\ \\ \\&\\ \\\n   \\gamma \\= \\frac{2\\,\\ell^2}{\\sqrt{4\\,\\ell^2 t^2 + (r^2-t^2+\\ell^2)^2}}\n   =  \\frac{2\\,\\ell^2}{\\sqrt{4\\,\\ell^2 r^2 + (t^2-r^2+\\ell^2)^2}}\\ .\n\\end{equation}\nA straightforward computation yields\n\\begin{equation}\n   \\cos\\tau-\\cos\\chi\\= \\gamma\\ >0\\ ,\n\\end{equation}\nwhich shows that only half of the cylinder, constrained by $\\cos\\tau\\ <\\cos\\chi$, is allowed by the map \\eqref{S3toMink}. Plugging this map back into the de Sitter metric \\eqref{metricdS} we obtain\n\\begin{equation}\\label{metricMink}\n   \\mathrm{d} s^2 \\=\n   \\frac{\\ell^2}{t^2}\\,\\bigl(-\\mathrm{d}t^2 +\\mathrm{d} x^2 +\\mathrm{d} y^2 +\\mathrm{d} z^2\\bigr) \n   \\quad\\textrm{with}\\quad (x,y,z)\\equiv(x^1,x^2,x^3)\\in\\mathds R^3\\ \\&\\ t\\in\\mathds R \\ ,\n\\end{equation}\nwhich is the Minkowski metric up to a conformal factor\\footnote{In fact, the map \\eqref{S3toMink} covers only the positive half of the Minkowski space i.e. $t\\in\\mathds R_+$ for the original cylinder ${\\cal I}\\times S^3$.}. A smooth gluing of the two cylinders across the time slice $t\\!=\\tau\\!=0$ is nicely depicted in (Fig. \\ref{patching}). Now, looking at the maps \\eqref{omegas} and \\eqref{S3toMink} we see a SO(3)-symmetry which we can exploit by writing\n\\begin{equation}\\label{OmegaToHatOmega}\n   \\omega_a \\= \\sin\\chi\\; \\hat{\\omega}_a\\ ~\\textrm{and}~\\ x^a \\= r\\,\\hat{\\omega}_a\\ ~\\textrm{for}~\\ a\\=1,2,3\\ ~\\textrm{and}~ r\\= \\sfrac\\ell\\gamma\\,\\sin\\chi\\ ,\n\\end{equation}\nwhere the unit $S^2$ coordinates $\\hat{\\omega}_a$ are given by\n\\begin{equation}\\label{S2coord}\n   \\hat{\\omega}_1 \\= \\sin\\theta\\,\\cos\\phi\\ , \\quad \\hat{\\omega}_2 \\= \\sin\\theta\\,\\sin\\phi \\quad\\textrm{and}\\quad \\hat{\\omega}_3 \\= \\cos\\theta\\ .\n\\end{equation}\n\\begin{figure}[!ht]\n\\centering\n\\captionsetup{width=\\linewidth}\n\\includegraphics[width = 0.4\\linewidth]{Figures\/penrose.pdf}\n\\caption{\nPenrose diagram of Minkowski space~$\\mathds R^{1,3}$. Each point hides a two-sphere $S^2{\\ni}\\{\\theta,\\phi\\}$.\nBlue curves indicate $t{=}\\textrm{const}$ slices while brown curves depict the world volumes of $r{=}\\textrm{const}$ spheres.\nThe lightcone of the Minkowski-space origin is drawn in red.\n}\n\\label{CPplot}\n\\end{figure}\nWe can identify the unit $S^2$ between the Minkowski space and the de Sitter space using  (\\ref{S3toMink}) and (\\ref{OmegaToHatOmega}), so that we have an effective map between coordinates $(t,r)$ and $(\\tau,\\chi)$:\n\\begin{equation} \\label{tr2TauChi}\n   \\frac{t}{\\ell} \\= \\frac{\\sin\\tau}{\\cos\\tau\\,-\\,\\cos\\chi} \\quad\\quad\\textrm{and}\\quad\\quad\n   \\frac{r}{\\ell} \\= \\frac{\\sin\\chi}{\\cos\\tau\\,-\\,\\cos\\chi} \\qquad\\quad\\textrm{for}\\quad\\quad\n   \\chi>|\\tau|\\ ,\n\\end{equation}\nwhich reveals that the triangular $(\\tau,\\chi)$ domain (where the points represent unit $S^2$) is nothing but the Penrose diagram of Minkowski space (See Fig. \\ref{CPplot}). The special lines and points in Fig. \\ref{CPplot} are given by\n\\begin{center}\n\\resizebox{\\columnwidth}{!}{\n \\begin{tabular}{|c|ccc|ccc|c|}\n  \\hline\n   & \\quad$\\tau{=}0$ & south pole & boundary & \\quad --- \\quad{} & \\!\\!\\!north pole\\!\\!\\! & $\\quad\\tau{=}{\\pm}\\pi\\quad$ & $\\chi{\\pm}\\tau{=}\\pi$ \\\\[2pt]\n   $(\\tau,\\chi)$ & \\quad$(0,\\chi)$ & $(\\tau,\\pi)$ & $(\\pm\\chi,\\chi)$ & $(0,\\pi)$ & $(0,0)$ & $(\\pm\\pi,\\pi)$ & $(\\pm\\pi{\\mp}\\chi,\\chi)$ \\\\[4pt]\n   $(t,r)$ & \\quad$(0,r)$ & $(t,0)$ & $(\\pm\\infty,\\infty)$ & $(0,0)$ & $(t,\\infty)$ & $(\\pm\\infty,r)$ & $(\\pm r,r)$ \\\\[2pt]\n   & \\quad$t{=}0$ & $r{=}0$ & $\\mathscr{I}^\\pm$ & \\ \\ origin \\ {} & $i^0$ & $i^\\pm$ & \\ \\  lightcone \\ \\ {} \\\\\n  \\hline\n \\end{tabular}\n }\n\\end{center}\nwith the Minkowski spatial and temporal infinity $i^0$ and $i^\\pm$ corresponding to the edges while the Minkowski null infinity~$\\mathscr{I}^\\pm$ corresponding to the conformal boundary $\\chi{=}|\\tau|$ of the Penrose diagram.\n\nEquation \\eqref{tr2TauChi} can be used to obtain the Jacobian for the transformation between the coordinates $y^m\\in\\{\\tau,\\chi,\\theta,\\phi\\}$ and $x^{\\mu}\\in\\{t,r,\\theta,\\phi\\}$:\n\\begin{equation} \\label{Jacobian}\n   \\bigl(J^m_{\\ \\ \\mu}\\bigr) \\ :=\\ \\frac{\\mbox{$\\partial$}(\\tau,\\chi,\\theta,\\phi)}{\\mbox{$\\partial$}(t,r,\\theta,\\phi)}\n   \\= \\frac1\\ell\\biggl(\\begin{matrix} p & -q \\\\[4pt] q & -p \\end{matrix} \\biggr) \\oplus\\mathds{1}_2\\ ,\n\\end{equation}\nwhere the polynomials $p,q$ are given by\n\\begin{equation}\n    p\\= \\sfrac{\\gamma^2}{\\ell^2}\\,(r^2{+}t^2{+}\\ell^2)\/2 \\= 1{-}\\cos\\tau\\cos\\chi \\quad\\textrm{and}\\quad q\\= \\sfrac{\\gamma^2}{\\ell^2}\\,t\\,r \\= \\sin\\tau\\sin\\chi\\ .\n\\end{equation}\n\nA more direct way of seeing the conformal correspondence between Minkwoski space and the $S^3$-cylinder is via Carter-Penrose transformation that readily produces the lightcone picture (see Appendix \\ref{appendCPtransf}). \n\n\\subsection{Structure of 3-sphere and its harmonics}\n\\noindent\nThe presence of $S^3$ in the metric \\eqref{metricdS} is an added advantage, which can be exploited to write a $SO(4)$-invariant gauge connection and obtain the corresponding fields by solving Yang--Mills equation on the Lorentzian cylinder. These quantities can be later exported to the Minkowski spacetime owing to the conformal invariance of the vacuum Yang--Mills equation in four dimensions. To this end, we start with the group $SO(4)$, which is isomorphic to two copies of $SU(2)$ (up to a $\\mathds{Z}_2$ grading). Each of these $SU(2)$ has a group action that generates a left (right) multiplication (a.k.a. translation) on $S^3$. This can easily be checked using the map\n\\begin{equation}\\label{map}\n   g:\\; S^3 \\rightarrow SU(2);~~ (\\omega_1,\\ \\omega_2,\\ \\omega_3,\\ \\omega_4)\\, \\mapsto\\, -i\n     \\begin{pmatrix}\n     \\beta & \\alpha^* \\\\ \\alpha & -\\beta^*\n     \\end{pmatrix}\\ ,\n\\end{equation} \nwith $\\alpha := \\omega_1 + \\mathrm{i}\\omega_2\\ ~\\textrm{and}~\\ \\beta := \\omega_3 +\\mathrm{i}\\omega_4 $. This parameterization of $g$ ensures that the identity element $e = \\mathds{1}_2$ of the group $SU(2)$ can be obtained from $(0,0,0,1)$ i.e. the North pole of $S^3$. It is well known that $S^3$ is the group manifold of $SU(2)$. Keeping this in mind, we consider the Cartan one-form\n\\begin{equation} \\label{Cartan1}\n   \\Omega_l(g)\\, :=\\, g^{-1}\\,\\mathrm{d} g \\= e^a\\,T_a,~~\\textrm{with}~~ T_a = -i\\sigma_a\n\\end{equation} \nbeing the $SU(2)$ generators. The left-invariant one-form\\footnote{Named so because they remain invariant under the dragging induced by the left SU(2) multiplication.} $e^a$ can alternatively be expressed using the so called self-dual 't Hooft symbol $\\eta^a_{_{BC}}$:\n\\begin{equation}\\label{one-form}\n   e^a \\= -\\eta^a_{_{\\ BC}}\\, \\omega_{_B}\\, \\mathrm{d} \\omega_{_C} \\quad\\textrm{with}\\quad \\eta^a_{\\ bc} \\= \\varepsilon_{abc} \\quad\\textrm{and}\\quad \\eta^a_{\\ b4} \\= -\\eta^a_{\\ 4b} \\= \\delta^a_b\\ .\n\\end{equation}\nThey satisfy the following useful identities\n\\begin{equation}\\label{MaurerCartan}\n\\delta_{ab}\\, e^a\\, e^b \\= \\mathrm{d}\\Omega_3^2 \\quad\\quad\\textrm{and}\\quad\\quad \\mathrm{d} e^a + \\varepsilon_{abc}\\, e^b\\wedge e^c \\= 0.\n\\end{equation} \nThe left-invariant vector fields $L_a$ generating the right translations are dual to $e^a$ and are given by\n\\begin{equation}\\label{leftVF}\n   L_a \\= -\\eta^a_{_{\\ BC}}\\,\\omega_{_B}\\,\\sfrac{\\partial}{\\partial \\omega_{_C}} \\qquad\\Rightarrow\\qquad \\left[ L_a , L_b \\right] \\= 2\\,\\varepsilon_{abc}\\,L_c\\ .\n\\end{equation}\nIn a similar way, the right-invariant vector fields $R_a$ generating the left translations are given by\n\\begin{equation}\\label{rightVF}\n   R_a \\= -\\tilde{\\eta}^a_{_{\\ BC}}\\,\\omega_{_B}\\,\\sfrac{\\partial}{\\partial\\omega_{_C}} \\qquad\\Rightarrow\\qquad\n   \\left[ R_a , R_b \\right] = 2\\,\\varepsilon_{abc}\\,R_c\\ ,\n\\end{equation}\nwhere the anti self--dual 't Hooft symbols $\\tilde{\\eta}^a_{_{\\ BC}}$ are obtained from \\eqref{one-form} by flipping the $B,C\\!=4$ sign. Furthermore, the vector fields $L_a$ and $R_a$ act on the one-forms $e^a$ via their Lie derivative, which can be performed using Cartan formula (see Section \\ref{covectorFields}):\n\\begin{equation}\\label{LieActionL}\n\\begin{aligned}\n    L_a\\,e^b\\ :=\\ \\mathcal{L}_{L_a}\\,e^b &\\= \\mathrm{d}\\circ\\iota_{L_a} e^b + \\iota_{L_a}\\circ\\mathrm{d}{e^b} \\\\\n    &\\= \\mathrm{d}{\\left(e^b(L_a)\\right)} - \\varepsilon^b_{\\ ij}\\,\\iota_{L_a}\\left(e^i\\wedge e^j\\right) \\\\\n    &\\= \\mathrm{d}{\\left(\\delta^b_a\\right)} - \\varepsilon^b_{\\ ij}\\left( e^i(L_a) e^j - e^j(L_a) e^i \\right) \\\\\n    &\\= 2\\,\\varepsilon_{abc}\\,e^c\\ ,\n\\end{aligned}\n\\end{equation}\nwhere in the second line we have used \\eqref{MaurerCartan}. A similar calculation for the action of $R_a$ yields\n\\begin{equation}\\label{LieActionR}\n    R_a\\, e^b \\= 0\\ .\n\\end{equation}\nWe can now write the differential $\\mathrm{d}$ of the functions $f\\in C^{\\infty}\\left(\\mathcal{I}\\times S^3\\right)$ using $L_a$\\footnote{One can use $R_a$ and its dual $1$-form as well.} as\n\\begin{equation}\n   \\mathrm{d} f \\= \\mathrm{d}\\tau\\; \\partial_{\\tau} f\\ +\\ e^a\\,L_a f\\ .\n\\end{equation}\n\nFunctions on $S^3$ can be expanded in a basis of harmonics~$Y_j(\\chi,\\theta,\\phi)$ with $2j\\in\\mathds N_0$, \nwhich are eigenfunctions of the scalar Laplacian,\\footnote{\nThe SO(4) spin of these functions is actually $2j$, but we label them with half their spin, for reasons to be clear below.}\n\\begin{equation}\n-\\mathop{}\\!\\mathbin\\bigtriangleup_3 Y_j \\= 2j(2j{+}2)\\,Y_j \\= 4j(j{+}1)\\,Y_j \\=\n-\\sfrac12(L^2+R^2)\\,Y_j \\= -\\sfrac14({\\cal D}^2+{\\cal P}^2)\\,Y_j\\ ,\n\\end{equation}\nwhere $L^2=L_a L_a$ and $R^2=R_a R_a$ are (minus four times) the Casimirs of $su(2)_L$ and $su(2)_R$, respectively,\n\\begin{equation}\n-\\sfrac14 L^2\\,Y_j \\= -\\sfrac14 R^2\\,Y_j \\= -\\sfrac14\\mathop{}\\!\\mathbin\\bigtriangleup_3 Y_j \\= j(j{+}1)\\,Y_j\\ .\n\\end{equation}\nWe have also introduced ${\\cal D}^2={\\cal D}_a {\\cal D}_a$ and ${\\cal P}^2={\\cal P}_a {\\cal P}_a$ with\n\\begin{equation} \\label{DPdef}\n{\\cal D}_a \\= R_a + L_a \\= -2\\,\\varepsilon_a^{\\ bc}\\,\\omega_b\\,\\mbox{$\\partial$}_c \\quad\\textrm{and}\\quad \n{\\cal P}_a \\= R_a - L_a \\= 2\\,\\omega_{[a}\\,\\mbox{$\\partial$}_{4]}\n\\end{equation}\nwith $\\mbox{$\\partial$}_{_A}\\equiv\\sfrac{\\mbox{$\\partial$}}{\\mbox{$\\partial$}\\omega_A}$ so that\n\\begin{equation}\n\\left[ {\\cal D}_a , {\\cal D}_b \\right] \\= 2\\,\\varepsilon_{ab}^{\\ \\ c}\\,{\\cal D}_c \\ ,\\qquad\n\\left[ {\\cal D}_a , {\\cal P}_b \\right] \\= 2\\,\\varepsilon_{ab}^{\\ \\ c}\\,{\\cal P}_c \\ ,\\qquad\n\\left[ {\\cal P}_a , {\\cal P}_b \\right] \\= 2\\,\\varepsilon_{ab}^{\\ \\ c}\\,{\\cal D}_c \\ .\n\\end{equation}\nHence, $\\{{\\cal D}_a\\}$ spans the diagonal subalgebra~$su(2)_D\\subset so(4)$, which generates the stabilizer subgroup~SO(3) \nin the coset representation~$S^3\\cong\\textrm{SO}(4)\/\\textrm{SO}(3)$.\nTherefore, ${\\cal D}^2$ is (minus four times) the Casimir of~$su(2)_D$, with eigenvalues $l(l{+}1)$ for $l=0,1,\\ldots,2j$,\nand $\\sfrac14 {\\cal D}^2=\\mathop{}\\!\\mathbin\\bigtriangleup_2$ is the scalar Laplacian on the $S^2$ slices traced out in~$S^3$ by the SO(3)$_D$ action.\n\nTo further characterize a complete basis of $S^3$ harmonics, there are two natural options,\ncorresponding to two different complete choices of mutually commuting operators to be diagonalized.\nFirst, the left-right (or toroidal) harmonics~$Y_{j;m,n}$ are eigenfunctions of $L^2=R^2$, $L_3$ and~$R_3$,\n\\begin{equation} \\label{Y-action}\n\\sfrac{\\mathrm{i}}{2}\\,L_3\\,Y_{j;m,n} \\= n\\,Y_{j;m,n} \\quad\\quad\\textrm{and}\\quad\\quad\n\\sfrac{\\mathrm{i}}{2}\\,R_3\\,Y_{j;m,n} \\= m\\,Y_{j;m,n} \\ ,\n\\end{equation}\nand hence the corresponding ladder operators \n\\begin{equation}\nL_\\pm \\= (L_1\\pm\\mathrm{i} L_2)\/\\sqrt{2} \\quad\\quad\\textrm{and}\\quad\\quad R_\\pm \\= (R_1\\pm\\mathrm{i} R_2)\/\\sqrt{2}\n\\end{equation}\nact as\n\\begin{equation}\\label{JpmAction}\n\\begin{aligned}\n  \\sfrac{\\mathrm{i}}{2}\\,L_\\pm\\,Y_{j;m,n} &\\= \\sqrt{(j{\\mp}n)(j{\\pm}n{+}1)\/2}\\,Y_{j;m,n\\pm1}\\ ,\\\\[4pt]\n  \\sfrac{\\mathrm{i}}{2}\\,R_\\pm\\,Y_{j;m,n} &\\= \\sqrt{(j{\\mp}m)(j{\\pm}m{+}1)\/2}\\,Y_{j;m\\pm1,n}\\ .\n\\end{aligned}\n\\end{equation}\nThe normalized harmonics $Y_{j;m,n}$ can be expanded in terms of functions $\\alpha,\\beta$ and their complex conjugates as\n\\begin{equation}\\label{S3Harmonics}\n\\begin{aligned}\n    Y_{j;m,n}(\\omega) &\\= \\sum_{k=0}^{2j}\\,K_{j,m,n}(k)\\,\\alpha^{n+m+k}\\, \\bar{\\alpha}^{k}\\, \\beta^{j-m-k}\\, \\bar{\\beta}^{j-n-k} \\qquad\\quad\\textrm{with}\\quad \\\\\n    K_{j,m,n}(k) &\\= (-1)^{m+n+k}\\,\\sqrt{\\frac{2j+1}{2\\pi^2}}\\,\\frac{\\sqrt{(j+m)!(j-m)!(j+n)!(j-n)!}}{(n+m+k)!(j-n-k)!(j-m-k)!k!}\\ .\n\\end{aligned}    \n\\end{equation}\nThey satisfy the orthonormality condition\n\\begin{equation}\\label{orthonormality}\n    \\int \\mathrm{d}^3\\Omega_3\\, Y_{j;m,n}\\, \\bar{Y}_{j';m',n'} \\= \\delta_{j,j'}\\, \\delta_{m,m'}\\, \\delta_{n,n'} \\quad\\textrm{with}\\quad \\mathrm{d}^3\\Omega_3 \\= \\sin^2\\chi\\sin\\theta\\, \\mathrm{d}\\chi\\, \\mathrm{d}\\theta\\, \\mathrm{d}\\phi\\ .\n\\end{equation}\nSecond, the adjoint (or hyperspherical) harmonics~${\\widetilde{Y}}_{j;l,M}$ are eigenfunctions of $L^2=R^2$, ${\\cal D}^2$ and~${\\cal D}_3$,\n\\begin{equation} \n-\\sfrac{1}{4}\\,{\\cal D}^2\\,{\\widetilde{Y}}_{j;l,M} \\= l(l{+}1)\\,{\\widetilde{Y}}_{j;l,M} \\quad\\quad\\textrm{and}\\quad\\quad\n\\sfrac{\\mathrm{i}}{2}\\,{\\cal D}_3\\,{\\widetilde{Y}}_{j;l,M} \\= M\\,{\\widetilde{Y}}_{j;l,M} \\ ,\n\\end{equation}\nwith the ladder-operator actions~\\cite{wybourne}\n\\begin{equation}\n\\begin{aligned}\n\\sfrac{\\mathrm{i}}{2}\\,{\\cal D}_\\pm\\,{\\widetilde{Y}}_{j;l,M} &\\= \\sqrt{(l{\\mp}M)(l{\\pm}M{+}1)\/2}\\,{\\widetilde{Y}}_{j;l,M\\pm1} \\ ,\\\\[4pt]\n\\sfrac{\\mathrm{i}}{2}\\,{\\cal P}_\\pm\\,{\\widetilde{Y}}_{j;l,M} &\\= \n\\mp\\sqrt{(l{\\mp}M{-}1)(l{\\mp}M)\/2}\\,c_{j,l}\\,{\\widetilde{Y}}_{j;l-1,M\\pm1} \\\\[2pt] \n&\\qquad\\pm\\ \\sqrt{(l{\\pm}M{+}1)(l{\\pm}M{+}2)\/2}\\,c_{j,l+1}\\,{\\widetilde{Y}}_{j;l+1,M\\pm1} \\ ,\\\\[4pt]\n\\sfrac{\\mathrm{i}}{2}\\,{\\cal P}_3\\,{\\widetilde{Y}}_{j;l,M} &\\= \n\\sqrt{l^2{-}M^2}\\,c_{j,l}\\,{\\widetilde{Y}}_{j;l-1,M} \\ +\\ \n\\sqrt{(l{+}1)^2{-}M^2}\\,c_{j,l+1}\\,{\\widetilde{Y}}_{j;l+1,M} \\ ,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\nc_{j,l} \\= \\sqrt{\\bigl((2j{+}1)^2-l^2\\bigr)\/\\bigl((2l{-}1)(2l{+}1)\\bigr)}\\ .\n\\end{equation}\nIn this case, there exists a recursive construction for harmonics on $S^{k+1}$ from those on $S^k$,\n\\begin{equation}\\label{split-Y}\n{\\widetilde{Y}}_{j;l,M}(\\chi,\\theta,\\phi) \\= R_{j,l}(\\chi)\\,Y_{l,M}(\\theta,\\phi)\\ ;\\\nR_{j,l}(\\chi) \\= \\mathrm{i}^{2j+l}\\,\\sqrt{\\sfrac{2j+1}{\\sin\\chi}\\sfrac{(2j+l+1)!}{(2j-l)!}}\\, P_{2j+\\frac12}^{-l-\\frac12}(\\cos\\chi)\\ ,\n\\end{equation}\nwhere $Y_{l,M}$ are the standard $S^2$ spherical harmonics and $P_a^b$ denote the associated Legendre polynomials of the first kind.\\footnote{\nWith fractional indices, it is rather a Gegenbauer polynomial, but also a hypergeometric function (see eq.~(2.8) of~\\cite{Higuchi,HiguchiErratum}).} \nThe two bases of harmonics are related by the standard Clebsch-Gordan series for the angular momentum addition\n$j\\otimes j = 0\\oplus 1\\oplus\\ldots\\oplus 2j$,\n\\begin{equation}\\label{Y-new}\nY_{j;m,n} \\= \\sum\\limits_{l=0}^{2j}\\sum\\limits_{M=-l}^{l}\\,C_{m,n}^{l,M}\\,{\\widetilde{Y}}_{j;l,M}\\ ,\n\\qquad\\textrm{with}\\qquad C_{m,n}^{l,M} = \\<2j;l,M|j,m;j,n\\>\n\\end{equation}\nbeing the Clebsch-Gordan coefficients enforcing $m{+}n{=}M$ and $l\\in\\{0,1,\\ldots,2j\\}$.\n\nA somewhat cumbersome calculation involving \\eqref{S3toMink}, \\eqref{gamma} and \\eqref{one-form} gives us the cylinder one--forms in Minkowski coordinates:\n\\begin{equation}\\label{1formsExpanded}\n\\begin{aligned}\ne^0\\ &:=\\ \\mathrm{d}{\\tau} \\= \\sfrac{\\gamma^2}{\\ell^3}\\bigl(\n\\sfrac12(t^2{+}r^2{+}\\ell^2)\\,\\mathrm{d} t - t\\,x^k \\mathrm{d} x^k \\bigr) \\\\[2pt]\n&\\= \\sfrac{\\gamma^2}{\\ell^3}\\bigl(\n\\sfrac12(t^2{+}r^2{+}\\ell^2)\\,\\mathrm{d} t - t\\,r\\,\\mathrm{d} r \\bigr)\n\\qquad\\quad\\textrm{and}\\quad \\\\[4pt]\ne^a &\\= \\sfrac{\\gamma^2}{\\ell^3}\\bigl(\nt\\,x^a \\mathrm{d} t - \\bigl[\\sfrac12(t^2{-}r^2{+}\\ell^2)\\,\\delta^{ak} +\nx^a x^k + \\ell\\,\\varepsilon^{ajk}x^j \\bigr]\\,\\mathrm{d} x^k \\bigr) \\\\[2pt]\n&\\= \\sfrac{\\gamma^2}{\\ell^3}\\bigl(\n\\hat{x}^a \\bigl[ r\\,t\\,\\mathrm{d} t -\\sfrac12(t^2{+}r^2{+}\\ell^2)\\,\\mathrm{d} r\\bigr]\n- \\sfrac12(t^2{-}r^2{+}\\ell^2)\\,r\\,\\mathrm{d}\\hat{x}^a -\n\\ell\\,r^2\\varepsilon^{ajk}\\hat{x}^j\\mathrm{d}\\hat{x}^k \\bigr)\\ .\n\\end{aligned}\n\\end{equation}\nWe further note down the expressions for the vector fields $L_a$ in terms of $S^3$ angles by using \\eqref{omegas} in \\eqref{leftVF} for later purposes:\n\\begin{equation}\\label{Lfields}\n \\begin{aligned}\n   L_1 &\\= \\sin\\theta\\cos\\phi\\, \\mbox{$\\partial$}_\\chi\\ +\\ (\\cot\\chi\\cos\\theta\\cos\\phi + \\sin\\phi)\\,\\mbox{$\\partial$}_\\theta\\ -\\ (\\cot\\chi\\csc\\theta\\sin\\phi - \\cot\\theta\\cos\\phi)\\, \\mbox{$\\partial$}_\\phi\\ , \\\\\n   L_2 &\\= \\sin\\theta\\sin\\phi\\, \\mbox{$\\partial$}_\\chi\\ +\\ (\\cot\\chi\\cos\\theta\\sin\\phi - \\cos\\phi)\\,\\mbox{$\\partial$}_\\theta\\ +\\ (\\cot\\chi\\csc\\theta\\cos\\phi + \\cot\\theta\\sin\\phi)\\, \\mbox{$\\partial$}_\\phi\\ , \\\\\n   L_3 &\\= \\cos\\theta\\, \\mbox{$\\partial$}_\\chi\\ -\\ \\cot\\chi\\sin\\theta\\, \\mbox{$\\partial$}_\\theta\\ -\\ \\mbox{$\\partial$}_\\phi\\ .\n \\end{aligned}\n\\end{equation}\n\n\n\n\\section{Yang-Mills field equations}\nOne can make use of these left-invariant one forms $e^a$ to expand a general Yang--Mills gauge potential $\\mathcal{A}$ (in the temporal gauge $\\mathcal{A}_\\tau {=}0$) by invoking the so called SU(2)-equivariant ansatz\n\\begin{equation}\\label{one-formAnsatz}\n\\mathcal{A} \\= X_a(\\tau,g)\\,e^a,\n\\end{equation}\nwhere the three $SU(2)$ matrices $X_a$ depends, in general, on both the cylinder parameter $\\tau$ and internal $S^3$ coordinates $\\omega$ via the map $g$ \\eqref{map}. The corresponding field strength $\\mathcal{F}$, with this choice of the gauge potential $\\mathcal{A}$, looks like:\n\\begin{equation} \\label{FcalExpanded}\n\\mathcal{F} \\= \\mathrm{d}{{\\cal A}} + \\mathcal{A}\\wedge\\mathcal{A} \\= \\dot{X}_a\\,e^0\\wedge e^a + \\frac{1}{2} \\left( L_{[b}\\, X_{c]} - 2\\,\\epsilon_{abc}\\, X_a + \\left[ X_b, X_c \\right] \\right) e^b\\wedge e^c\\ ,\n\\end{equation}\nwhere $e^0 := \\mathrm{d}{\\tau}$ and the dot on $X_a$ refers to its derivative w.r.t. $\\tau$. The Yang--Mills equation for this gauge field ${\\cal A}$ i.e.\n\\begin{equation}\n    \\mathrm{d}*\\mathcal{F} + {\\cal A}\\wedge *\\mathcal{F} - *\\mathcal{F}\\wedge\\mathcal{A} \\= 0\\ ,\n\\end{equation}\nafter a straightforward calculation, yields the constraint condition\n\\begin{equation}\\label{constraint}\n    -2\\,\\mathrm{i}\\,J_a\\,X_a + [X_a,\\dot{X}_a] \\= 0\\ .\n\\end{equation}\nand the field equation\n\\begin{align} \\label{ge_eq}\n\\begin{split}\n\\ddot{X}_a =& -4\\left(J^2{+}1\\right)X_a - 4\\,\\mathrm{i}\\,\\epsilon_{abc}\\,J_b\\,X_c + 4\\,J_a\\,J_b\\,X_b + 3\\,\\epsilon_{abc}\\,[X_b,X_c]\\\\\n&  + 2\\,\\mathrm{i}\\,[X_a,J_b\\,X_b] + 2\\,\\mathrm{i}\\,[X_b,J_a\\,X_b] + 4\\,\\mathrm{i}\\,[J_b\\,X_a,X_b] - [X_b,[X_a,X_b]]\\ ,\n\\end{split}\n\\end{align}\nwhere the operators $J_a$ and $J^2$ are given by\n\\begin{equation}\\label{opJ}\n    J_a\\ :=\\ \\frac{\\mathrm{i}}{2}\\,L_a \\quad\\textrm{and}\\quad J^2\\ :=\\ J_a\\,J_a\\ .\n\\end{equation}\nFinding a general solution for this equation is a daunting task, but progress can be made in the two limiting cases below. \n\nOne of the limiting case of \\eqref{ge_eq} is that of the Abelian one, where all the commutators vanish to yield\n\\begin{equation}\\label{eq_abelian}\n\\ddot{X}_a = -4\\left( J^2 {+} 1 \\right) X_a + 2\\,\\mathrm{i}\\,\\epsilon_{abc}\\,J_b\\,X_c\\ .\n\\end{equation}\nFurthermore the constraint \\eqref{constraint} in this case can be brought to the following form after integrating w.r.t to time and choosing the constant to be zero:\n\\begin{equation}\\label{constraintAbelian}\n    J_a\\,X_a \\= 0\\ .\n\\end{equation}\nOne thing to note here is that the above constraint along with the temporal gauge $A_\\tau\\!=0$ is not the usual Coulomb gauge on Minkowski space. In fact, we can make use of the inverse Jacobian \\eqref{Jacobian} while promoting the gauge potential to the Minkowski space ${\\cal A} \\!= {\\cal A}_a\\, e^a \\!= A_\\mu\\,dx^\\mu \\!=A $ to get\n\\begin{equation}\\label{gaugeCond2}\n  0 = {\\cal A}_\\tau = {\\cal A} \\left(\\partial_\\tau \\right) = A \\left( \\sfrac{\\ell^2}{\\gamma\n  ^2}\\left(p\\,\\partial_t + q\\,\\partial_r \\right) \\right) \\implies \\left(r^2+t^2+\\ell^2\\right)A_t + 2\\,r\\,t\\,A_r = 0\\ .\n\\end{equation}\nSolution to \\eqref{eq_abelian} was obtained in \\cite{LZ17} using hyperspherical harmonics $Y_{j;m,n}$ as we will see in the next chapter.\n\nAnother limiting case of \\eqref{ge_eq} is when a more symmetric $SO(4)$-equivariant condition is imposed yielding \\cite{Friedan,Luescher77} the following form of $X_a$:\n\\begin{equation}\\label{Aansatz}\nX_a \\= \\frac{1}{2}\\left( 1 + \\psi(\\tau) \\right)T_a\n\\end{equation}\nwith some function $\\psi : \\mathcal{I} \\rightarrow \\mathds R$ and $SU(2)$ generators $T_a$ satisfying the Lie algebra\n\\begin{equation}\n   [T_a,T_b] = 2\\,\\epsilon_{abc}\\,T_c\\ .\n\\end{equation}\nMoreover, $T_a$ act in adjoint representation where $\\mathrm{tr}{(T_a\\,T_b)} = -8\\,\\delta_{ab}$. The constraint \\eqref{constraint} for this symmetric ansatz i.e.\n\\begin{equation}\n    [X_a,\\dot{X}_a] \\= 0\n\\end{equation}\nis automatically satisfied and the equation \\eqref{ge_eq} becomes\n\\begin{equation}\n\\ddot{X}_a = -4\\,X_a + 3\\,\\epsilon_{abc}\\,[X_b,X_c] - [X_b,[X_a,X_b]].\n\\end{equation}\nSolution to this equation was obtained in \\cite{ILP17jhep} as we will see in chapter 6.\n\n\n\n\n\\chapter{Abelian solution: $U(1)$}\n\\label{Chapter5} \n\\justifying\n\nIn this chapter we present the knotted electromagnetic fields arising via the ``de Sitter\" method and study its various properties. The content of this chapter has generated one published work (in J. Phys. A) \\cite{KL20} and two pre-print articles \\cite{KPH21,KLP22}. Section \\ref{nullFields} on null solutions is due to Olaf Lechtenfeld with verification by Colin Becker and some clarifications from Harald Skarke. I have been involved at all stages of the following works for the rest of the sections in this chapter.\n\n\\vspace{12pt}\n\\vspace{2pt}\n\\section{Family of ``knot\" solutions}\nIt was shown in \\cite{LZ17}, that the general solution of \\eqref{eq_abelian} decomposes into spin-$j$ representations of~$so(4)$ and are labelled with hyperspherical harmonics of $S^3$. We review the construction of these solutions below in two steps: first we solve \\eqref{eq_abelian} on the $S^3$-cylinder, and then we pull these solution back to Minkowski space using $(\\tau,\\chi)\\rightarrow (t,r)$ coordinate transformation. \n\n\\subsection{Generic solution on the $S^3$-cylinder}\n\\noindent\nTo solve \\eqref{eq_abelian} we first write it down in terms of the ladder operators $J_{\\pm}$ and $J_3$ \\eqref{opJ} together with the redefined functions\n\\begin{equation}\n    X_{\\pm}\\ :=\\ \\frac{1}{2}\\left(X_1 \\pm \\mathrm{i} X_2 \\right) \n\\end{equation}\nas follows\n\\begin{equation} \\label{knotEquations}\n \\begin{aligned}\n    \\mbox{$\\partial$}_{\\tau}^2\\,X_+ &\\= -4\\,(J^2-J_3+1)X_+ - 4\\,J_+\\,X_3\\ ,\\\\[4pt]\n    \\mbox{$\\partial$}_{\\tau}^2\\,X_3 &\\= -4\\,(J^2+1)X_3 + 4\\,J_+\\,X_- - 4\\,J_-\\,X_+\\ ,\\\\[4pt]\n    \\mbox{$\\partial$}_{\\tau}^2\\,X_- &\\= -4\\,(J^2+J_3+1)X_- + 4\\, J_-\\,X_3\\ .\n \\end{aligned}\n\\end{equation}\nSimilarly, the constraint condition \\eqref{constraintAbelian} takes the following form\n\\begin{equation} \\label{constraintAbelian2}\n    0 \\= J_3\\,X_3 + J_+\\,X_- +J_-\\,X_+\\ .\n\\end{equation}\nWe can solve \\eqref{knotEquations} by the following ansatz\n\\begin{equation}\n\\begin{aligned}\n    X_+ &\\= \\sum\\limits_{j,m,n} Z^{j;m,n}_+\\,e^{i\\Omega_{j,m,n}\\tau}\\ ;\\quad Z^{j;m,n}_+ \\= c_+\\,Y_{j;m,n+1}\\ ,\\\\\n    X_3 &\\= \\sum\\limits_{j,m,n} Z^{j;m,n}_3\\,e^{i\\Omega_{j,m,n}\\tau}\\ ; \\quad Z^{j;m,n}_3 \\= c_3\\,Y_{j;m,n}\\ , \\quad\\textrm{and}\\quad \\\\\n    X_- &\\= \\sum\\limits_{j,m,n} Z^{j;m,n}_-\\,e^{i\\Omega_{j,m,n}\\tau}\\ ;\\quad Z^{j;m,n}_- \\= c_-\\,Y_{j;m,n-1}\\ ,\n\\end{aligned}\n\\end{equation}\nwhere $X_*(j,m,n)$ with $*\\in \\{ +,3,- \\}$ has been expanded in terms of $S^3$ harmonics. Plugging this ansatz in \\eqref{knotEquations} and using \\eqref{JpmAction} we find, for every mode $(j,m,n)$, an eigenvalue equation for the vector $(c_+\\ c_3\\ c_-)^T$:\n\\begin{equation}\n\\begin{aligned}\n    &\\qquad\\qquad M\\begin{pmatrix}\n    c_+ \\\\ c_3 \\\\ c_-\n    \\end{pmatrix} \\= \\Omega(j,m,n)^2\\begin{pmatrix}\n    c_+ \\\\ c_3 \\\\ c_-\n    \\end{pmatrix}\\ ,\\ \\quad\\textrm{with}\\quad \\\\\n    &M \\= \\begin{psmallmatrix}\n    4(j^2+j-n) & 2\\sqrt{2(j-n)(j+n+1)} & 0 \\\\\n    2\\sqrt{2(j-n)(j+n+1)} & 4(j^2+j+1) & -2\\sqrt{2(j+n)(j-n+1)} \\\\\n    0 & -2\\sqrt{2(j+n)(j-n+1)} & 4(j^2+j+n)\n    \\end{psmallmatrix}\\ ,\n\\end{aligned}\n\\end{equation}\nwhich admits an eigensystem with $3$ distinct eigenvalues $\\Omega_j^2$ (that turns out to be independent of $m$ and $n$) and their corresponding eigenvectors. One of these eigenvectors does not satisfy the constraint \\eqref{constraintAbelian2} and is, therefore, discarded. We label the remaining two eigensystems as type I, with eigenfrequency $\\Omega_j^2=4(j{+}1)^2$, and type II, with eigenfrequency $\\Omega_j^2=4\\,j^2$, as follows\n\\begin{itemize}\n\\addtolength{\\itemsep}{-4pt}\n\\item type I : \\quad\n$j{\\geq}0\\ ,\\quad m = -j,\\ldots,+j\\ ,\\quad n = -j{-}1,\\ldots,j{+}1\\ ,\\ \\Omega_j=\\pm 2\\,(j{+}1)\\ ,$\n\\\\\n\\begin{equation} \\label{type1}\n\\begin{aligned}\nZ_{+\\ \\textrm{I}}^{(j;m,n)}(\\omega) &\\= \\sqrt{(j{-}n)(j{-}n{+}1)\/2} \\ \\ Y_{j;m,n+1}(\\omega) \\ ,\\\\\nZ_{3\\ \\textrm{I}}^{(j;m,n)}(\\omega)\\,&\\= \\sqrt{(j{+}1)^2-n^2} \\ \\ Y_{j;m,n}(\\omega) \\ ,\\\\\nZ_{-\\ \\textrm{I}}^{(j;m,n)}(\\omega)   &\\= -\\sqrt{(j{+}n)(j{+}n{+}1)\/2} \\ \\ Y_{j;m,n-1}(\\omega) \\ .\n\\end{aligned}\n\\end{equation}\n\\item type II :\\quad\n$j{\\geq}1\\ ,\\quad m = -j,\\ldots,+j\\ ,\\quad n = -j{+}1,\\ldots,j{-}1\\ ,\\ \\Omega_j=\\pm2\\,j\\ ,$\n\\\\\n\\begin{equation} \\label{type2}\n\\begin{aligned}\nZ_{+\\ \\textrm{II}}^{(j;m,n)}(\\omega)  &\\= -\\sqrt{(j{+}n)(j{+}n{+}1)\/2} \\ \\ Y_{j;m,n+1}(\\omega)\\ ,\\\\\nZ_{3\\ \\textrm{II}}^{(j;m,n)}(\\omega)\\,&\\= \\sqrt{j^2-n^2} \\ \\ Y_{j;m,n}(\\omega) \\ ,\\\\\nZ_{-\\ \\textrm{II}}^{(j;m,n)}(\\omega)   &\\= \\sqrt{(j{-}n)(j{-}n{+}1)\/2} \\ \\ Y_{j;m,n-1}(\\omega) \\ .\n\\end{aligned}\n\\end{equation}\n\\end{itemize}\n\nWe take a linear combination of these basis solutions and write down the (real-valued) gauge potential as\n\\begin{equation}\\label{Asep}\n{\\cal A} \\=  \n\\Bigl\\{ \\sum_{2j=0}^\\infty X_{a\\ \\textrm{I}}^j(\\omega) \\ \\mathrm{e}^{2(j+1)\\mathrm{i}\\tau} \\ +\\ \\textrm{c.c.} \\Bigr\\}e^a \\ +\\ \n\\Bigl\\{ \\sum_{2j=2}^\\infty X_{a\\ \\textrm{II}}^j(\\omega) \\ \\mathrm{e}^{2j\\,\\mathrm{i}\\tau} \\ +\\ \\textrm{c.c.} \\Bigr\\}e^a\\ ,\n\\end{equation}\nwhere we have reorganized the complex angular functions~$X_a^j$ as\n\\begin{equation}\\label{XZ}\nX_1^j=\\sfrac{1}{\\sqrt{2}} \\bigl(Z_+^j + Z_-^j \\bigr)\\ ,\\qquad \nX_2^j=\\sfrac{\\mathrm{i}}{\\sqrt{2}} \\bigl( Z_-^j - Z_+^j \\bigr)\\ ,\\qquad \nX_3^j= Z_3^j\n\\end{equation}\nfor both types and expanded the functions~$Z^j_\\pm$ and $Z^j_3$ into the above spin-$j$ basis solutions of type I \\eqref{type1} and type II \\eqref{type2} (for $*\\in\\{+,3,-\\}$),\n\\begin{equation}\\label{basisZ}\n\\begin{aligned}\nZ_{*\\ \\textrm{I}}^j(\\omega) &\\= \\sum_{m=-j}^j \\sum_{n=-j-1}^{j+1} \\lambda_{j;m,n}^{\\textrm{I}}\\ Z_{*\\ \\textrm{I}}^{(j;m,n)}(\\omega)  \\quad\\textrm{and}\\quad \\\\\nZ_{*\\ \\textrm{II}}^j(\\omega) &\\= \\sum_{m=-j}^j \\sum_{n=-j+1}^{j-1} \\lambda_{j;m,n}^{\\textrm{II}}\\ Z_{*\\ \\textrm{II}}^{(j;m,n)}(\\omega)\\ ,\n\\end{aligned}\n\\end{equation}\nwith $(2j{+}1)(2j{+}3)$ arbitrary complex coefficients $\\lambda^{\\textrm{I}}_{j;m,n}$\nand $(2j{+}1)(2j{-}1)$ coefficients $\\lambda^{\\textrm{II}}_{j;m,n}$.\n(Note that type-II solutions are absent for $j{=}0$ and $j{=}\\sfrac12$.)\n\n\nInserting \\eqref{type1} and \\eqref{type2} into \\eqref{basisZ} and the result into~\\eqref{XZ} provides a harmonic expansion\n\\begin{equation}\\label{XY}\nX_a^j(\\omega) \\= \\sum_{m=-j}^j \\sum_{n=-j}^j X_a^{j;m,n}\\ Y_{j;m,n}(\\omega)\n\\end{equation}\nfor both types of angular functions in~\\eqref{Asep}. \n(Note the different range of~$n$ for $X_a^{j;m,n}$ and $Z_*^{j;m,n}$; they are not easily related as $X_a^j$ and $Z_*^j$ are in~\\eqref{XZ}.)\n\nIt is useful for later purposes to introduce here the ``sphere-frame\" electric and magnetic fields,\n\\begin{equation}\\label{2FormEM}\n    {\\cal F} \\= {\\cal E}_a\\,e^a {\\wedge} e^\\tau + \\sfrac12\\,{\\cal B}_a\\,\\varepsilon^a_{\\ bc}\\,e^b{\\wedge} e^c\\ .\n\\end{equation}\nFor a fixed type (I or II) and spin~$j$, we may eliminate $L_{[b}A_{c]}$ in \\eqref{FcalExpanded} (without the commutator term) by using~\\eqref{eq_abelian} and employ\n\\begin{equation}\n\\mbox{$\\partial$}_\\tau^2 {\\cal A}^{(j)} = -\\Omega_j^2\\,{\\cal A}^{(j)} \\quad\\quad\\textrm{and}\\quad\\quad\nL^2\\,{\\cal A}^{(j)} \\= -4j(j{+}1)\\,{\\cal A}^{(j)}\n\\end{equation}\nto obtain\n\\begin{equation}\\label{cur-field}\n{\\cal E}_a^{(j)} \\= -\\mbox{$\\partial$}_\\tau X_a^{(j)} \\quad\\quad\\textrm{and}\\quad\\quad {\\cal B}_a^{(j)} \\= \\mp\\Omega_j\\,X_a^{(j)}\\ ,\n\\end{equation}\nwhere the upper sign pertains to type~I and the lower one to type~II.\nWe note in passing that, due to the compactness of the Lorentzian cylinder, the sphere-frame energy and action are always finite.\n\nDue to the linearity of Maxwell theory, the overall scale of any solution is arbitrary.\nFurthermore, the parity transformation $L\\leftrightarrow R$ and $m\\leftrightarrow n$ \ninterchanges a spin-$j$ solution of type~I with a spin-$(j{+}1)$ solution of type~II.\nFinally, electromagnetic duality at fixed $j$ is realized by shifting $\\Omega_j\\tau$ by $\\sfrac{\\pi}{2}$ for type~I\nor by $-\\sfrac{\\pi}{2}$ for type~II, which maps ${\\cal A}$ to a dual configuration~${\\cal A}_\\textrm{D}$\nand likewise ${\\cal F}$ to ${\\cal F}_\\textrm{D}$.\n\n\\subsection{Pulling the solution back to Minkowski space}\n\n\\noindent\nWe have completely solved the vacuum Maxwell equations on the Lorentzian cylinder ${\\cal I}\\times S^3$ and, hence, on \nde Sitter space~dS$_4$. By conformal invariance, this solution carries over to any conformally equivalent spacetime. We translate our Maxwell solutions from $\\widetilde{{\\cal I}}\\times S^3$ to $\\mathds R^{1,3}$\nsimply by the coordinate change\n\\begin{equation}\n\\begin{aligned}\n\\tau=\\tau(t,x,y,z) &\\quad\\textrm{and}\\quad \\omega_{_A}=\\omega_{_A}(t,x,y,z) \\\\[2pt]\n\\textrm{or}\\qquad\n\\tau=\\tau(t,r) &\\quad\\textrm{and}\\quad \\chi=\\chi(t,r)\\ .\n\\end{aligned}\n\\end{equation}\nIn other words, abbreviating $x\\equiv\\{x^\\mu\\}$ and $y\\equiv\\{y^\\rho\\}$ and expanding\n\\begin{equation}\n\\begin{aligned}\n{\\cal A} &\\= X_a\\bigl(\\tau(x),g(x)\\bigr)\\,e^a(x) \n\\= A_\\mu(x)\\,\\mathrm{d} x^\\mu\n\\= A_\\rho(y)\\,\\mathrm{d} y^\\rho \\quad\\quad\\textrm{and}\\quad \\\\[4pt]\n\\mathrm{d}{\\cal A} &\\= \\mbox{$\\partial$}_\\tau X_a\\,e^0{\\wedge} e^a + \\bigl( L_b\\,X_c - X_a\\,\\varepsilon^a_{\\ bc}\\bigr)\\,e^b {\\wedge} e^c \\\\\n&\\= \\sfrac12 F_{\\mu \\nu}(x)\\,\\mathrm{d} x^\\mu {\\wedge} \\mathrm{d} x^\\nu \n\\= \\sfrac12 F_{\\rho\\lambda}(y)\\,\\mathrm{d} y^\\rho{\\wedge}\\mathrm{d} y^\\lambda\n\\end{aligned}\n\\end{equation}\nusing \\eqref{1formsExpanded} we may read off $A_\\mu$ (note that $A_t\\neq0$, as discussed before) and \n$F_{\\mu\\nu}$ and thus the electric and magnetic fields\n\\begin{equation}\\label{EMCartesian}\n{\\cal F} \\= F \\= E_a\\,\\mathrm{d}{x}^a\\wedge\\mathrm{d}{t} + \\frac{1}{2}B_a\\,\\varepsilon^a_{\\ bc}\\,\\mathrm{d}{x}^b{\\wedge} \\mathrm{d}{x}^c\n\\end{equation}\nin Cartesian or in spherical coordinates. In general, the knotted electromagnetic fields arising from the basis configurations \\eqref{type1} are complex, the basis configurations on Minkowski space will also be complex. Hence, they combine two physical solutions, namely the real and imaginary parts\\footnote{The notation should not be confused with the type I configuration \\eqref{type1}.}, which we denote as\n\\begin{equation*}\n    (j;m,n)_R~ \\textrm{configuration} \\quad\\textrm{and}\\quad (j;m,n)_I ~\\textrm{configuration}\\ ,\n\\end{equation*}\nrespectively. These knotted electromagnetic fields \\eqref{EMCartesian} for the basis configurations \\eqref{type1} and \\eqref{type2} increase in complexity with increasing $j$, as shown in Figure \\ref{fieldLines}.\n\nFurthermore, it comes in handy that ${\\cal A}$ finally contains only even powers of~$\\gamma$\nand depends on~$\\tau$ only through integral powers of\n\\begin{equation}\n\\exp (2\\mathrm{i}\\,\\tau) \\= \\frac{[(\\ell+\\mathrm{i}t)^2+r^2]^2}{4\\,\\ell^2 t^2 + (r^2-t^2+\\ell^2)^2}\\ .\n\\end{equation}\nTherefore, our Minkowski solutions have the remarkable property of being rational functions of~$(t,x,y,z)$.\nMore precisely, their electric and magnetic fields are of the form\n\\begin{equation}\n\\textrm{type I:} \\quad \\frac{P_{2(2j+1)}(x)}{Q_{2(2j+3)}(x)} \\ ,\\qquad\n\\textrm{type II:}\\quad \\frac{P_{2(2j-1)}(x)}{Q_{2(2j+1)}(x)} \n\\end{equation}\nwhere $P_r$ and $Q_r$ denote polynomials of degree~$r$.\nThus, as expected, their energy and action are finite.\nIndeed, the fields fall off like $r^{-4}$ at spatial infinity for fixed time, but they decay\nmerely like $(t{\\pm}r)^{-1}$ along the light-cone.\nHence, the asymptotic energy flow is concentrated on past and future null infinity~$\\mathscr{I}^\\pm$,\nas it should be, but peaks on the light-cone of the spacetime origin.\nSince on de Sitter space our basis solutions (\\ref{type1}) and~(\\ref{type2}) form a complete set, \ntheir Minkowski relatives are also complete in the space of finite-action configurations. \n\n\\begin{figure}[h!]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 5cm, height = 5cm]{fieldLines_0,0,1.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{fieldLines_h,-h,3h.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{fieldLines_1,0,2.pdf}\n \\caption{\n Electric (red) and magnetic (green) field lines at $t{=}0$ with 4 fixed seed points.\n Left: $(j;m,n)=(0;0,1)_R$ configuration, Center: $(j;m,n)=(\\sfrac12;-\\sfrac12,\\sfrac32)_R$ configuration, Right: $(j;m,n)=(1;0,2)_R$ configuration. More self-knotted field lines start appearing with additional seed points in the simulation.}\n\\label{fieldLines}\n\\end{figure}\n\nFor illustration, we display a type I basis solution with $(j;m,n)=(1;0,0)$\nobtained from \n\\begin{equation}\\label{j1Prototype}\nX_\\pm \\ \\propto\\ \\sfrac{1}{\\sqrt{2}}\\,(\\omega_1{\\pm}\\mathrm{i}\\omega_2)(\\omega_3{\\pm}\\mathrm{i}\\omega_4)\\,\\cos 4\\tau \\quad\\textrm{and}\\quad\nX_3\\ \\propto\\  (\\omega_1^2{+}\\omega_2^2{-}\\omega_3^2{-}\\omega_4^2)\\,\\cos 4\\tau\\ .\n\\end{equation}\nIt bodes well to combine these field configurations into the Riemann--Silberstein vector \n\\begin{equation}\\label{RSvector}\n    {\\bf S} \\= {\\bf E} + \\mathrm{i} {\\bf B}\n\\end{equation}\nwhose components, for \\eqref{j1Prototype}, are (up to overall scale)\n{\\small\n\\begin{equation}\n\\begin{aligned}\nS_x &\\= -\\frac{2\\mathrm{i}}{N}\n\\Bigl\\{ 2y +3 \\mathrm{i} t y -x z +2 t^2 y + 2\\mathrm{i} t x z-8x^2 y-8y^3+4yz^2 + 4\\mathrm{i} t^3 y \\\\%[4pt]\n&\\qquad\\qquad  -6t^2 xz - 8\\mathrm{i} t x^2 y-8\\mathrm{i} t y^3+4\\mathrm{i} t yz^2 +10x^3 z+10xy^2 z -2xz^3 \\\\%[4pt]\n&\\qquad\\qquad\\qquad\\qquad + 2(\\mathrm{i} t x z + yr^2)(-t^2{+}r^2)+(\\mathrm{i} t y- x z) (-t^2{+}r^2)^2\\Bigr\\}\n\\ ,\\quad{}\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\nS_y &\\= \\frac{2\\mathrm{i}}{N}\n\\Bigl\\{ 2x +3 \\mathrm{i} t x + y z+2 t^2 x-2\\mathrm{i} t y z-8x^3-8xy^2 +4 xz^2 + 4 \\mathrm{i} t^3 x \\\\%[4pt]\n&\\qquad\\qquad + 6 t^2y z -8\\mathrm{i} t x^3-8\\mathrm{i} txy^2 +4 \\mathrm{i} t x z^2-10 x^2 yz-10y^3 z +2y z^3 \\\\%[4pt]\n&\\qquad\\qquad\\qquad\\qquad + 2 (-\\mathrm{i} t y z {+} x r^2)(-t^2{+} r^2) +(\\mathrm{i} t x+ y z)(-t^2{+}r^2)^2\\Bigr\\}\n\\ ,\\qquad\\&\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\nS_z &\\= \\frac{\\mathrm{i}}{N}\n\\Bigl\\{1+2\\mathrm{i} t+t^2-11 x^2- 11 y^2 +3 z^2 +4 \\mathrm{i} t^3-16\\mathrm{i} t x^2-16\\mathrm{i} t y^2+4\\mathrm{i} t z^2 - t^4 \\\\%[4pt]\n&\\qquad\\qquad -2 t^2 x^2 -2t^2 y^2-2t^2 z^2+11 x^4+22x^2y^2-10 x^2 z^2+11y^4-10 y^2 z^2 \\\\%[4pt]\n&\\qquad\\qquad\\qquad\\qquad\\qquad +3z^4 + 2 \\mathrm{i} t (t^2 {-} 3 r^2{+}2 z^2) (t^2{-}r^2)-(t^2{+}r^2)(-t^2{+}r^2)^2\\Bigr\\}\n\\ ,\\quad{}\n\\end{aligned}\n\\end{equation}\n}\nwith $N=\\bigl((t{-}\\mathrm{i})^2-r^2\\bigr)^5$. We can also obtain electromagnetic field configurations corresponding to the gauge field \\eqref{Asep}, that consists of complex linear combinations of these basis solutions. We demonstrate this for the famous Hopf--Ran\\~{a}da field configuration, which was first discovered by Ran\\~{a}da in 1989 \\cite{Ranada89} using the Hopf fibration. The Riemann--Silberstein vector ${\\bf S}$ for this field configuration is given by \\cite{LZ17}\n\\begin{equation} \\label{HRfields}\n    {\\bf S} \\= \\frac{1}{((t{-}\\mathrm{i})^2{-}r^2)^3} \\begin{pmatrix}\n        (x{-}\\mathrm{i}\\,y)^2 {-} (t{-}\\mathrm{i}{-}z)^2 \\\\\n        \\mathrm{i}(x{-}\\mathrm{i}\\,y)^2 {+} \\mathrm{i}(t{-}\\mathrm{i}{-}z)^2 \\\\\n        -2(x{-}\\mathrm{i}\\,y)(t{-}\\mathrm{i}{-}z)\n    \\end{pmatrix}\\ .\n\\end{equation}\nWe find that this solution, in our construction, is obtained by taking linear combination of $j{=}0$, and hence type I, basis configurations \\eqref{type1} with the following coefficients in \\eqref{basisZ}\n\\begin{equation}\n    \\lambda^I_{0;0,+1} \\= 0\\ ,\\quad \\lambda^I_{0;0,0} \\= 0\\ ,\\quad\\textrm{and}\\quad \\lambda^I_{0;0,-1} \\= -\\mathrm{i}\\frac{\\pi}{8}\\ .\n\\end{equation}\n\n\n\n\\vspace{12pt}\n\\section{Symmetry analysis}\n\\vspace{2pt}\n\\noindent\nThe main advantage of constructing Minkowski-space electromagnetic field configurations \nvia the detour over de Sitter space is the enhanced manifest symmetry of our construction. \nThe isometry group SO(1,4) of dS$_4$ is generated by \n($A,B=1,2,3,4$ and $a,b,c=1,2,3$, abbreviate $\\frac{\\partial}{\\partial q_{_B}}\\equiv\\partial_{_B}$)\n\\begin{equation}\n\\{{\\cal M}_{_{AB}}{\\equiv}\\,{-}q_{[{\\scriptscriptstyle A}}\\partial_{{\\scriptscriptstyle B}]}\\,,\\ \n{\\cal M}_{0{\\scriptscriptstyle B}}{\\equiv}\\,q_{({\\scriptstyle 0}}\\partial_{{\\scriptscriptstyle B})} \\} \\= \n\\{ {\\cal M}_{ab}{=}\\varepsilon_{abc}{\\cal D}_c\\,,\\ {\\cal M}_{4a}{=}{\\cal P}_a\\,,\\ {\\cal M}_{04}{=}{\\cal P}_0\\,,\\ {\\cal M}_{0b}{=}{\\cal K}_b\\}\\ ,\n\\end{equation}\nwhich can be contracted (with $\\ell{\\to}\\infty$) to the isometry group ISO(1,3) of $\\mathds R^{1,3}$ (the Poincar\\'e group)\ngenerated by ($\\mu,\\nu=0,1,2,3$ and $i,j,k=1,2,3$)\n\\begin{equation}\n\\{M_{\\mu\\nu}\\,,\\ P_\\mu \\} \\=\n\\{ M_{ij}{=}\\varepsilon_{ijk}D_k\\,,\\ P_i\\,,\\ P_0\\,,\\ M_{0j}{=}K_j\\}\\ ,\n\\end{equation}\nwhere the two sets are ordered likewise, \nand we employ (as aleady earlier) calligraphic symbols for de Sitter quantities and straight symbols for Minkowskian ones.\nHere, $D$ denotes spatial rotations, $P$ are translations, and $K$ stand for boosts in Minkowski space.\n\nSince the two spaces are conformally equivalent already at $\\ell{<}\\infty$ via~\\eqref{S3toMink},\nthe corresponding generators should be related. Indeed, the common SO(3) subgroup in\n\\begin{equation}\nSO(1,4) \\supset SO(4) \\supset SO(3) \\quad\\quad\\textrm{and}\\quad\\quad ISO(1,3) \\supset SO(1,3) \\supset SO(3)\n\\end{equation}\nis identified, ${\\cal D}_i{=}D_i{=}{-}2\\varepsilon_{ij}^{\\ k}x^j\\partial_k$. \nHowever, any other generator becomes nonlinearly realized when \nmapped to the other space via \\eqref{tr2TauChi}.\nFor example, the would-be translation ${\\cal P}_3$ defined in~\\eqref{DPdef} reads\n\\begin{equation}\n\\begin{aligned}\n{\\cal P}_3 = L_3-R_3 &\\= -2\\,\\cos\\theta\\,\\mbox{$\\partial$}_\\chi + 2\\,\\cot\\chi\\sin\\theta\\,\\mbox{$\\partial$}_\\theta \\\\[2pt]\n&\\= \\sfrac1{\\ell}\\,\\cos\\theta\\,\\bigl( 2\\,r\\,t\\,\\mbox{$\\partial$}_t + (t^2{+}r^2{+}\\ell^2)\\,\\mbox{$\\partial$}_r \\bigr) \n- \\sfrac1{\\ell\\,r} (t^2{-}r^2{+}\\ell^2)\\,\\sin\\theta\\,\\mbox{$\\partial$}_\\theta \\\\[2pt]\n&\\ \\to\\ 2\\ell\\,\\bigl( \\cos\\theta\\,\\mbox{$\\partial$}_r - \\sfrac1r\\,\\sin\\theta\\,\\mbox{$\\partial$}_\\theta \\bigr)\n\\= 2\\ell\\,\\mbox{$\\partial$}_z \\= \\ell\\,P_z \\quad\\quad\\textrm{for}\\quad\\ell\\to\\infty\n\\end{aligned}\n\\end{equation}\nas it should be. Similarly, ${\\cal P}_0\\to \\ell\\,P_0$ and ${\\cal K}_b\\to K_j$ for $\\ell{\\to}\\infty$ \nwhen expanded around $(t,r)=(\\ell,0)$ corresponding to the $S^3$ south pole at $q_0{=}0$.\nNevertheless, the de Sitter construction enjoys an SO(4) covariance (generated by ${\\cal D}_a$ and ${\\cal P}_a$)\nwhich extends the obvious SO(3) covariance in Minkowski space.\nIt allows us to connect all solutions of a given type (I or II) with a fixed value of the spin~$j$\nby the action of SO(4) ladder operators $L_\\pm$ and $R_\\pm$ or ${\\cal D}_\\pm$ and ${\\cal P}_\\pm$, \nwhich is non-obvious on the Minkowski side. \nOn the other hand, Minkowski boosts and translations have no simple realization on de Sitter space.\n\nActually, Maxwell theory on either space is also invariant under conformal transformations.\nThese may be generated by the isometry group together with a conformal inversion.\nOn the Minkowski side, the latter is\n\\begin{equation}\nJ : \\quad x^\\mu \\ \\mapsto\\ \\frac{x^\\mu}{x\\cdot x} \\quad\\quad\\textrm{with}\\quad x\\cdot x=r^2-t^2\\ .\n\\end{equation}\nWe have to distinguish two cases:\n\\begin{equation}\n\\begin{aligned}\n\\textrm{spacelike:}\\quad & t^2<r^2 \\quad\\Rightarrow\\quad\nJ_> : \\quad\\bigl(t,r,\\theta,\\phi\\bigr) \\ \\mapsto\\ \\bigl(\\sfrac{t}{r^2-t^2}, \\sfrac{r}{r^2-t^2},\\theta,\\phi\\bigr)\\ ,\\\\[2pt]\n\\textrm{timelike:} \\quad & t^2>r^2 \\quad\\Rightarrow\\quad\nJ_< : \\quad\\bigl(t,r,\\theta,\\phi\\bigr) \\ \\mapsto\\ \\bigl(\\sfrac{-t}{t^2-r^2}, \\sfrac{r}{t^2-r^2},\\pi{-}\\theta,\\phi{+}\\pi\\bigr)\\ .\n\\end{aligned}\n\\end{equation}\nOn the de Sitter side, this is either (spacelike) a reflection on the $S^3$ equator~$\\chi{=}\\frac{\\pi}{2}$\nor (timelike) a $\\pi$-shift in cylinder time~$\\tau$ plus an $S^2$ antipodal flip,\n\\begin{equation}\n\\begin{aligned}\n\\textrm{spacelike:}\\quad & |\\tau|{+}\\chi<\\pi \\quad\\Rightarrow\\quad\n{\\cal J}_> :  \\quad \\bigl(\\tau,\\chi,\\theta,\\phi\\bigr)\\ \\mapsto\\ \\bigl(\\tau,\\pi{-}\\chi,\\theta,\\phi\\bigr)\\ , \\\\[2pt]\n\\textrm{timelike:} \\quad & |\\tau|{+}\\chi>\\pi \\quad\\Rightarrow\\quad\n{\\cal J}_< :  \\quad \\bigl(\\tau,\\chi,\\theta,\\phi\\bigr)\\ \\mapsto\\ \\bigl(\\tau{\\pm}\\pi,\\chi,\\pi{-}\\theta,\\phi{+}\\pi\\bigr)\\ .\n\\end{aligned}\n\\end{equation}\nIn the spacelike case, merely the sign of $\\omega_4{\\equiv}\\cos\\chi$ gets flipped, \nwhich amounts to a parity flip $L\\leftrightarrow R$. \nIn the timelike case, both $\\cos\\tau$ and $\\sin\\tau$ change sign, \nwhich combines a time reversal with a reflection at $\\tau{=}\\frac{\\pi}{2}$ or $\\tau{=}{-}\\frac{\\pi}{2}$.\nNote that it is different from the $S^3$ antipodal map, which\nis not a reflection but a proper rotation, $\\omega_{_A}\\mapsto-\\omega_{_A}$ or \n$(\\chi,\\theta,\\phi)\\mapsto(\\pi{-}\\chi,\\pi{-}\\theta,\\phi{+}\\pi)$.\nThe lightcone is singular under the inversion; it is mapped to the conformal boundary\n$r{=}{\\pm}t{=}\\infty$ or $\\chi{=}{\\pm}\\tau$.\nWe infer that the conformal inversion allows us to relate type-I and type-II solutions of the same spin.\nIt is easily checked that the spatial fall-off behavior of our rational solutions is not modified by the inversion.\n\nFinally, one may consider dilatations in Minkowski space,\n\\begin{equation}\nx^\\mu \\ \\mapsto\\ \\lambda\\,x^\\mu \\quad\\quad\\textrm{for}\\quad \\lambda\\in\\mathds R_+\\ .\n\\end{equation}\nHowever, this amounts to a trivial rescaling also achieved by changing the de Sitter radius,\n$\\ell\\mapsto\\lambda\\,\\ell$, as the scale~$\\ell$ was removed on the Lorentzian cylinder. \n\n\\vspace{12pt}\n\\section{Conformal group and Noether charges}\n\\vspace{2pt}\n\\noindent\nIt is well known \\cite{BR16} that free Maxwell theory on $\\mathds R^{1,3}$ arising from the action\n\\begin{equation}\n    S\\left[A_\\mu\\right] \\= \\int d^4x\\, \\mathcal{L}\\ ;\\qquad \\mathcal{L} \\= -\\sfrac14 F^{\\mu\\nu}F_{\\mu\\nu}\n\\end{equation}\nis invariant under the conformal group $SO(2,4)$. Furthermore, the above action is also invariant under the gauge transformations: $A_\\mu(x)\\rightarrow A_\\mu(x)+\\mbox{$\\partial$}_\\mu\\lambda(x)$. The conformal group is generated by transformations $x^\\mu \\rightarrow x^\\mu + \\zeta^\\mu(x)$, where the vector fields $\\zeta^\\mu$ obey the conformal Killing equations:\n\\begin{equation}\\label{Killing}\n    \\zeta_{\\mu,\\nu}\\, +\\, \\zeta_{\\nu,\\mu} \\= \\sfrac12\\, \\eta_{\\mu\\nu}\\, \\zeta^\\alpha_{~,\\alpha} \\quad\\textrm{with}\\quad \\{\\eta_{\\mu\\nu}\\} \\= \\textrm{diag}(-1,1,1,1)\\ .\n\\end{equation}\nThe conserved Noether current $J^\\mu$ is obtained by equating the ``on-shell variation\" of the action where the variations $\\delta A_\\mu$ are arbitrary and the fields $A_\\mu$ satisfies the Euler-Langrange equations with its ``symmetry variation\" where the fields $A_\\mu$ are arbitrary but variations $\\delta A_\\mu$ satisfy the symmetry condition. The correct variation $\\delta A_\\mu$ is obtained by imposing the gauge invariance on the Lie derivative of $A_\\mu$ w.r.t. the vector field $\\zeta^\\mu$:\n\\begin{equation}\n    \\mathcal{L}_{\\zeta^\\alpha}A_\\mu := A'_\\mu (x) - A_\\mu (x) = -\\zeta^\\alpha\\, \\mbox{$\\partial$}_\\alpha A_\\mu - A_\\alpha\\, \\mbox{$\\partial$}_\\mu \\zeta^\\alpha ~\\longrightarrow ~ \\delta A_\\mu \\= F_{\\mu\\nu}\\zeta^\\nu\\ .\n\\end{equation}\nFinally, the conserved current is obtained as\n\\begin{equation}\\label{currentJ}\n    J^\\mu \\= \\frac{\\mbox{$\\partial$} \\mathcal{L}}{\\mbox{$\\partial$} A_{\\rho,\\mu}} \\delta A_\\rho + \\zeta^\\mu \\mathcal{L} \\= \\zeta^\\rho \\left( F^{\\mu\\alpha}F_{\\rho\\alpha} - \\sfrac14\\delta^\\mu_\\rho F^2   \\right) \\quad\\textrm{with}\\quad F^2 = F_{\\beta \\gamma}F^{\\beta \\gamma}\\ ,\n\\end{equation}\nwhich satisfies the continuity equation and gives the conserved (in time) charge $Q$\\footnote{For source-free fields the current $\\textbf{J}$ is assumed to vanish when the surface $\\mbox{$\\partial$} V$ is taken to infinity.}:\n\\begin{equation}\n    \\mbox{$\\partial$}_\\mu\\, J^\\mu = 0 \\quad\\implies\\quad \\frac{\\mathrm{d} Q}{\\mathrm{d} t} \\= -\\int_{\\mbox{$\\partial$} V} \\mathrm{d}^2{\\bf s}\\cdot{\\bf J} \\= 0\\ ;\\quad Q \\= \\int_{V} \\mathrm{d}^3x\\, J^0\\ .\n\\end{equation}\nBefore proceeding further, a couple of remarks pertaining to the subsequent calculations are in order:\n\\begin{itemize}\n   \\item All the charges $Q$ are computed at $t{=}0$ owing to the simple $J^0$ expressions on this time-slice. To that end, we record the following useful identities at $t{=}\\tau{=}0$:\n   \\begin{equation}\\label{t0Results}\n      \\begin{aligned}\n       e^a_i &\\= \\sfrac1\\ell \\left( \\gamma\\, \\omega_4\\, \\delta^a_i - \\omega_a\\, \\omega_i + \\epsilon_{aic}\\, \\gamma\\, \\omega_c \\right)\\ ,\\quad  e^a_i\\, e^b_i \\= \\sfrac{\\gamma^2}{\\ell^2} \\delta^{ab} \\\\\n       \\gamma &\\= 1-\\omega_4\\ ,~ \\mathrm{d}^3x \\= \\sfrac{\\ell^3}{\\gamma^3}\\mathrm{d}^3\\Omega_3\\ ~;~ \\mathrm{d}^3\\Omega_3 := e^1\\wedge e^2\\wedge e^3\\ .\n      \\end{aligned}\n   \\end{equation}\n   Moreover, the electromagnetic fields at $t{=}0$ are given in terms of tetrads $e^\\tau {=} e^\\tau_\\mu\\, \\mathrm{d} x^\\mu $ and $ e^a {=} e^a_\\mu\\, \\mathrm{d} x^\\mu$:\n   \\begin{equation}\\label{EMfields}\n    E_i \\= e^\\tau_0\\, e^a_i\\, \\mathcal{E}_a \\quad\\quad\\textrm{and}\\quad\\quad  B_i \\= \\sfrac12\\, \\epsilon_{ijk}\\, \\epsilon_{abc}\\, e^b_j\\, e^c_k\\, \\mathcal{B}_a\\ \n   \\end{equation}\n   using the $2$-form \\eqref{2FormEM} and its Minkowski counterpart \\eqref{EMCartesian}.\n   \n   \\item The charges are computed for type I \\eqref{type1} solutions only (except for the energy and the related helicity) and we relabel the complex coefficients \\eqref{basisZ} $\\lambda^I_{j;m,n}$ as $\\Lambda_{j;m,n}$ for convenience.\n   \n   \\item  Furthermore, these charges $Q$ are computed for a fixed spin-$j$ and, thus, we will suppress the index $j$ from now onward, unless necessary. Note that the sphere-frame EM fields for fixed $j$ can be obtained by using the expansion \\eqref{Asep} (for type I only) in \\eqref{cur-field} as\n   \\begin{equation} \\label{EBj}\n     {\\cal A}_a \\= X_a(\\omega)\\,\\mathrm{e}^{\\Omega\\,\\mathrm{i}\\tau} + \\bar{X}_a(\\omega)\\,\\mathrm{e}^{-\\Omega\\,\\mathrm{i}\\tau}\n     \\Rightarrow \\biggl\\{ \\begin{array}{l}\n     {\\cal E}_a \\= -\\mathrm{i}\\,\\Omega\\,X_a\\,\\mathrm{e}^{\\Omega\\,\\mathrm{i}\\tau} + \\mathrm{i}\\,\\Omega\\,\\bar{X}_a\\,\\mathrm{e}^{-\\Omega\\,\\mathrm{i}\\tau} \\\\[2pt]\n     {\\cal B}_a \\= \\,-\\,\\Omega\\,X_a\\,\\mathrm{e}^{\\Omega\\,\\mathrm{i}\\tau}\\,-\\,\\Omega\\,\\bar{X}_a\\,\\mathrm{e}^{-\\Omega\\,\\mathrm{i}\\tau} \\end{array} \\biggr\\}\n   \\end{equation}\n   with $\\bar{X}_a$ denoting the complex conjugate of $X_a$. Notice that for type II the overall sign in ${\\cal B}_a$ flips.\n \n   \\item  The simplifications below for the charge density $J^0$ are carried out using the harmonic expansion \\eqref{XY} for the type I solution \\eqref{type1}.\n   \n   \\item We frequently use below the well known fact that an odd $\\{\\omega_{_A}\\}$ integral over $S^3$ vanishes because of the opposite contributions coming from the antipodal points on the sphere. In particular, it can be checked that the following integral vanishes\\footnote{Note that the power of $\\omega_{_A}$ in $Y_{j;m,n}\\bar{Y}_{j;m',n'}$ is always even. One way to check this is by employing the toroidal coordinates: $\\omega_1 = \\cos\\eta\\cos\\kappa_1,\\omega_2 = \\cos\\eta\\sin\\kappa_1, \\omega_3 = \\sin\\eta\\cos\\kappa_2,\\omega_4 = \\sin\\eta\\sin\\kappa_2$ with $\\eta\\in(0,\\sfrac\\pi2)$ and $\\kappa_1,\\kappa_2\\in(0,2\\pi)$ in \\eqref{S3Harmonics}. The resultant selection rules coming from $\\kappa_1,\\kappa_2$ integral would yield $m-m'\\in \\sfrac{2k+1}{2}$ with $k\\in\\mathds N_0$, which is not feasible for fixed $j$.}:\n   \\begin{equation}\n      \\int \\mathrm{d}^3\\Omega_3\\, (\\omega_1)^a\\, (\\omega_2)^b\\, (\\omega_3)^c\\, (\\omega_4)^d\\, Y_{j;m,n}\\, \\overline{Y}_{j;m',n'} \\= 0 \\quad\\textrm{for}\\quad a+b+c+d\\ \\in\\ 2\\mathds N_0 + 1\\ .\n   \\end{equation}\n\\end{itemize}\nHaving made these remarks, we now proceed to compute the charges $Q$ for various conformal transformations $\\zeta^\\mu$ obeying \\eqref{Killing} in following four categories.\n\n\n\\subsection{Translations}\n\\noindent\nAn easily seen solution to \\eqref{Killing} is the set of four constant translations\n\\begin{equation}\n    \\zeta^\\mu = \\epsilon^\\mu\\ ,\n\\end{equation}\nwhich also partly generates the Poincar\\'e group and give rise to the usual stress-energy tensor of electrodynamics\n\\begin{equation}\\label{stress-energy}\n    J^\\mu \\= T^\\mu_{\\ \\; \\nu} \\=  F^{\\mu\\alpha}F_{\\nu\\alpha} - \\sfrac14\\delta^\\mu_\\nu F^2\\ ,\n\\end{equation}\ncorresponding to the $\\mu$-component of the translation for an arbitrary $\\epsilon^\\nu$. The corresponding charges are the energy $E$ and the momentum $\\textbf{P}$.\n\n\\noindent\n{\\bf Energy.} The expression of the energy density $e:=T^{00}$ simplifies to  \n\\begin{equation}\n    e\\, :=\\, \\frac{1}{2} \\left( E_i^2 + B_i^2 \\right) \\= \\frac{\\gamma}{\\ell}^4\\, \\rho \\quad\\quad\\textrm{with}\\quad\\quad \\rho \\= \\sfrac12 \\left( {\\cal E}_a^2 + {\\cal B}_a^2 \\right)\\ ,\n\\end{equation}\nwhich, in turn, simplifies the expression for the energy $E$ to\n\\begin{equation}\n    E \\= \\int_V \\mathrm{d}^3x\\;e \\= \\sfrac{1}{2\\ell}\\int_{S^3} \\mathrm{d}^3\\Omega_3 \\  (1-\\cos\\chi)\\,\\bigl({\\cal E}_a^2 + {\\cal B}_a^2\\bigr)\\ .\n\\end{equation}\nNotice here that the orientation of the $S^3$ volume measure~$\\mathrm{d}^3\\Omega_3$ is chosen to provide a positive result. From \\eqref{EBj} we find that the ``sphere-frame'' energy density is time-independent and has similar expression for both solution types (with appropriate eigenfrequency $\\Omega$):\n\\begin{equation}\n\\sfrac12 \\bigl( {\\cal E}_a^2 + {\\cal B}_a^2 \\bigr) \\= 2\\,\\Omega^2\\,X_a\\bar{X}_a(\\omega)\\ .\n\\end{equation}\n\nThe resultant expression for $E$ in terms of complex parameters $\\lambda_{m,n}$ (for both solution types) is given by\n\\begin{equation}\n    E \\= \\frac{1}{\\ell}\\, (2j+1)\\,\\Omega^3 \\sum_{m,n}\\,|\\lambda_{m,n}|^2\\ .\n\\end{equation}\nNotice again here that for type I solutions we would have $\\Lambda_{m,n}$ in the above expression.\n\n{\\bf Helicity.} Although helicity is not a Noether charge of the conformal group, it is nevertheless a conserved quantity for the Maxwell system and turns out to be related to the energy. The expression for the helicity is metric-free and can thus be evaluated over any spatial slice. \nChoosing again $t{=}\\tau{=}0$,\n\\begin{equation} \\label{helicity}\nh = h_{\\textrm{mag}}+h_{\\textrm{el}} \\= \\sfrac12 \\int_{\\mathds R^3} \\ \\bigl( A\\wedge F + A_D \\wedge F_D \\bigr)\n\\= -\\sfrac12 \\int_{S^3} \\mathrm{d}^3\\Omega_3 \\  (1-\\cos\\chi)\\,\\bigl({\\cal A}_a{\\cal B}_a + {\\cal A}^D_a{\\cal E}_a\\bigr)\\ ,\n\\end{equation}\nwhere the subscript\/superscript $`D'$ refers to the dual fields. Once again, taking type~I (upper sign) or type~II (lower sign) and fixing the spin~$j$ we obtain\n\\begin{equation}\n{\\cal A}^D_a \\= \\pm\\mathrm{i}\\,X_a(\\omega)\\,\\mathrm{e}^{\\Omega\\,\\mathrm{i}\\tau} \\mp\\mathrm{i}\\,\\bar{X}_a(\\omega)\\,\\mathrm{e}^{-\\Omega\\,\\mathrm{i}\\tau}\\ ,\n\\end{equation}\nwhich yields a constant ``sphere-frame'' helicity density\n\\begin{equation}\n-\\sfrac12 \\bigl( {\\cal A}_a{\\cal B}_a + {\\cal A}^D_a{\\cal E}_a \\bigr) \\= \\pm 2\\,\\Omega\\,X_a\\bar{X}_a(\\omega)\\ .\n\\end{equation}\nAs a result, even before performing the $S^3$ integration, we find a linear helicity-energy relation \n\\begin{equation}\n\\Omega\\, h \\= \\pm \\ell\\, E \\qquad\\textrm{for fixed spin and type}\\ .\n\\end{equation}\n\nSince the helicity measure an average of the linking numbers of any two electric or magnetic field lines~\\cite{moffatt17, moffatt69, berger},\nthe latter must be related to the value~$j$ of the spin. The individual linking number of two field lines, however, appears neither to be independent of the lines chosen nor constant in time, as our observations indicate.\nAn exception are the Ra\\~nada--Hopf knots discussed before, \nwhich display a conserved linking number of unity between any pair of electric or magnetic field lines.\n\n\n\\noindent\n{\\bf Momentum.} For the momentum densities $p_i = T^{0i} = (\\textbf{E}\\times\\textbf{B})_i$ we obtain an interesting correspondence relating the one-form $p := p_i\\, \\mathrm{d} x^i$ on Minkowski space with a similar one on de Sitter space:\n\\begin{equation}\\label{1formP}\n    p \\= (\\sfrac\\gamma\\ell)^3\\, \\mathcal{P}_a\\, e^a\\ =:\\ (\\sfrac\\gamma\\ell)^3\\, \\mathcal{P} \\quad\\quad\\textrm{with}\\quad\\quad \\mathcal{P}_a\\ :=\\ \\varepsilon_{abc}\\, \\mathcal{E}_b\\, \\mathcal{B}_c\\ .\n\\end{equation}\nA straightforward calculation then yields the expression of momenta $P_i$:\n\\begin{equation}\n    P_i \\= \\int_V \\mathrm{d}^3x\\, p_i \\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\mathcal{P}_a\\, e^a_i \\= 2\\mathrm{i}\\ \\Omega^2\\ \\varepsilon_{abc}\\ \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, e^a_i\\, Z_b\\, \\bar{Z}_c\n\\end{equation}\nwith $e^a_i$ given by \\eqref{t0Results}. The results for $j=0$ are\n\\begin{equation}\n  \\begin{aligned}\n    P_1^{(j=0)} &\\= -\\sfrac{\\sqrt{2}}{\\ell}\\left(\\left(\\bar{\\Lambda}_{0,-1} + \\bar{\\Lambda}_{0,1}\\right) \\Lambda_{0,0} + \\bar{\\Lambda}_{0,0}\\left(\\Lambda_{0,-1}+\\Lambda_{0,1}\\right)\\right)\\ , \\\\\n    P_2^{(j=0)} &\\=  \\sfrac{\\mathrm{i}\\sqrt{2}}{\\ell} \\left(\\left(-\\bar{\\Lambda}_{0,-1} + \\bar{\\Lambda}_{0,1}\\right) \\Lambda_{0,0} + \\bar{\\Lambda}_{0,0}\\left(\\Lambda_{0,-1}-\\Lambda_{0,1}\\right)\\right)\\ , \\\\\n    P_3^{(j=0)} &\\= \\sfrac2\\ell \\left( |\\Lambda_{0,-1}|^2 - |\\Lambda_{0,1}|^2 \\right)\\ .\n  \\end{aligned}\n\\end{equation}\nAs a consistency requirement, we check that the vector charges $P_i$ are rotated according to the algebra of $\\mathcal{D}_a$ \\eqref{DPdef} (See appendix \\ref{appendRotation}):\n\\begin{equation}\\label{rotP}\n    \\mathcal{D}_a\\, P_b \\= 2\\,\\varepsilon_{abc}\\, P_c\\ .\n\\end{equation}\nWe also note down $P_3$ for $j=1\/2\\ ~\\textrm{and}~ 1$ in table \\ref{table1}.\n\\begin{table}[]\n    \\centering\\setcellgapes{4pt}\\makegapedcells \\renewcommand\\theadfont{\\normalsize\\bfseries}\n \\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{|c|P{6cm}|P{8cm}|}\n        \\hline\n      & $j=1\/2$ & $j=1$ \\\\ [0.5ex] \\hline \\hline\n     $P_3$ & \n     $\\begin{aligned}\n         \\sfrac{9}{\\ell} \\Big( &|\\Lambda_{-1\/2,-3\/2}|^{2} - |\\Lambda_{-1\/2, 1\/2}|^{2} \\\\\n         &- 2 |\\Lambda_{-1\/2, 3\/2}|^{2} + 2 |\\Lambda_{1\/2,-3\/2}|^{2} \\\\\n         &+ |\\Lambda_{1\/2,-1\/2}|^{2} - |\\Lambda_{1\/2, 3\/2}|^{2}\\Big)\n     \\end{aligned}$ & \n     $\\begin{aligned}\n         \\sfrac{24}{\\ell}\\Big( &|\\Lambda_{-1,-2}|^{2} - |\\Lambda_{-1,0}|^{2} - 2|\\Lambda_{-1,1}|^{2} - 3|\\Lambda_{-1,2}|^{2} \\\\\n         &+ 2|\\Lambda_{0,-2}|^{2} + |\\Lambda_{0,-1}|^{2} - |\\Lambda_{0, 1}|^{2} - 2|\\Lambda_{0,2}|^{2} \\\\\n         &+ 3|\\Lambda_{1,-2}|^{2} + 2|\\Lambda_{1,-1}|^{2} + |\\Lambda_{1,0}|^{2} - |\\Lambda_{1,2}|^{2} \\Big)\n     \\end{aligned}$  \\\\ \\hline\n     $P_\\phi$ & \n     $\\begin{aligned}\n         9 \\Big( &2|\\Lambda_{-1\/2,-3\/2}|^{2} + |\\Lambda_{-1\/2,-1\/2}|^{2} \\\\\n         &-|\\Lambda_{-1\/2,3\/2}|^{2} + |\\Lambda_{1\/2,-3\/2}|^{2} \\\\\n         &- |\\Lambda_{1\/2,1\/2}|^{2} - 2|\\Lambda_{1\/2, 3\/2}|^{2}\\Big)\n     \\end{aligned}$ & \n     $\\begin{aligned}\n         24 \\Big( &3|\\Lambda_{-1,-2}|^{2} + 2|\\Lambda_{-1,-1}|^{2} + |\\Lambda_{-1,0}|^{2} - |\\Lambda_{-1,2}|^{2} \\\\\n         &+ 2|\\Lambda_{0,-2}|^{2} + |\\Lambda_{0,-1}|^{2} - |\\Lambda_{0,1}|^{2} - 2|\\Lambda_{0,2}|^{2} \\\\\n         &+ |\\Lambda_{1,-2}|^{2} - |\\Lambda_{1,0}|^{2} - 2|\\Lambda_{1,1}|^{2} - 3|\\Lambda_{1,2}|^{2} \\Big)\n     \\end{aligned}$  \\\\ \\hline\n     $L_3$ & \n     $\\begin{aligned}\n         9\\ell\\, \\Big( &-2|\\Lambda_{-1\/2,-3\/2}|^{2} - |\\Lambda_{-1\/2,-1\/2}|^{2} \\\\\n         &+|\\Lambda_{-1\/2, 3\/2}|^{2} - |\\Lambda_{1\/2,-3\/2}|^{2} \\\\\n         &+ |\\Lambda_{1\/2,1\/2}|^{2} + 2|\\Lambda_{1\/2, 3\/2}|^{2}\\Big)\n     \\end{aligned}$ & \n     $\\begin{aligned}\n         -24\\ell\\, \\Big( &3|\\Lambda_{-1,-2}|^{2} + 2|\\Lambda_{-1,-1}|^{2} + |\\Lambda_{-1,0}|^{2} \\\\\n         & - |\\Lambda_{-1,2}|^{2} + 2|\\Lambda_{0,-2}|^{2} + |\\Lambda_{0,-1}|^{2} \\\\\n         & - |\\Lambda_{0, 1}|^{2} - 2|\\Lambda_{0,2}|^{2} + |\\Lambda_{1,-2}|^{2} \\\\\n         &- |\\Lambda_{1,0}|^{2} - 2|\\Lambda_{1,1}|^{2} - 3|\\Lambda_{1,2}|^{2} \\Big)\n     \\end{aligned}$  \\\\ \\hline\n    \\end{tabular}\n\n    \\caption{Expressions of $P_3$, $P_\n    \\phi$ and $L_3$ for $j=1\/2 ~\\textrm{and}~ 1$.}\n    \\label{table1}\n\\end{table}\nOne can compute the corresponding $P_1$ and $P_2$ for $j=1\/2$ and $j=1$ by employing the action of an appropriate $\\mathcal{D}_a$ of the table in Appendix \\ref{appendRotation}.\n\nWe can additionally compute the spherical components of the momentum $(P_r,P_\\theta,P_\\phi)$ by letting the one-form $e^a$ in \\eqref{1formP} act on the vector fields $(\\mbox{$\\partial$}_r,\\mbox{$\\partial$}_\\theta, \\mbox{$\\partial$}_\\phi)$. In practice, we first write $\\mbox{$\\partial$}_r = -\\sfrac\\gamma\\ell \\mbox{$\\partial$}_\\chi$ using \\eqref{Jacobian} at $t{=}0$ and then invert the vector fields $(\\mbox{$\\partial$}_r,\\mbox{$\\partial$}_\\theta, \\mbox{$\\partial$}_\\phi)$ in terms of the left invariant vector fields $(L_1,L_2,L_3)$ using \\eqref{Lfields}. Finally, using the duality relation $e^a(L_b)=\\delta^a_b$ we obtain\n\\begin{equation}\\label{sphericalP}\n \\begin{aligned}\n   P_r &\\= -\\frac1\\ell\\int_{S^3} \\mathrm{d}^3\\Omega_3\\,(1-\\cos\\chi)\\, \\left( \\sin\\theta\\cos\\phi\\, \\mathcal{P}_1 + \\sin\\theta\\sin\\phi\\, \\mathcal{P}_2 + \\cos\\theta\\, \\mathcal{P}_3 \\right)\\ , \\\\\n   P_\\theta &\\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\,\\sin\\chi\\cos\\chi \\Big( (\\cos\\theta\\cos\\phi + \\tan\\chi\\sin\\phi)\\, \\mathcal{P}_1 \\\\\n   &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad + (\\cos\\theta\\sin\\phi - \\tan\\chi\\cos\\phi)\\mathcal{P}_2 - \\sin\\theta\\, \\mathcal{P}_3 \\Big) ,\\\\\n   P_\\phi &\\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\sin^2\\chi\\sin\\theta \\Big( (\\cos\\theta\\cos\\phi - \\cot\\chi\\sin\\phi)\\, \\mathcal{P}_1 \\\\\n   &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad + (\\cos\\theta\\sin\\phi + \\cot\\chi\\cos\\phi)\\, \\mathcal{P}_2 - \\sin\\theta\\, \\mathcal{P}_3 \\Big) \\ .\n \\end{aligned}\n\\end{equation}\nFrom the expression of $P_r$ we see that the integrand over $S^2$ i.e. $\\hat{\\omega}_a\\mathcal{P}_a$ \\eqref{OmegaToHatOmega} is an odd function\\footnote{The terms of $\\mathcal{P}_a$ using \\eqref{split-Y} are proportional to  $Y_{l,M}Y_{l',M'}$, which is even in $\\hat{\\omega}_a$ for a fixed $j$.}, which makes $P_r$ vanish. We also find with explicit calculations (verified for up to $j=1$) that $P_\\theta$ vanishes. For $j=0$ we find that $P_\\phi$ is proportional to $P_3$:\n\\begin{equation}\n    P_\\phi^{(j=0)} \\= \\ell\\, P_3^{(j=0)} \\=  2\\left(|\\Lambda_{0,-1}|^{2}-|\\Lambda_{0,1}|^{2}\\right)\\ .\n\\end{equation}\nThe expressions of $P_\\phi$ for $j=1\/2$ and $1$ has been recorded in the table \\ref{table1}.\n\\subsection{Lorentz transformations}\n\\noindent\nAnother solution of \\eqref{Killing} is given by\n\\begin{equation}\n    \\zeta^\\mu(x) \\= \\epsilon^{\\mu}_{\\ \\; \\nu}\\, x^\\nu \\quad\\textrm{where}\\quad \\epsilon_{\\mu\\nu} \\= -\\epsilon_{\\nu\\mu}\\ ,\n\\end{equation}\nwhich correspond to the six generators of the Lorentz group $SO(1,3)$. These six together with the above four translations generates the full Poincar\\'e group. The corresponding six charges are grouped into the boost ${\\bf K}$ and the angular momentum ${\\bf L}$.\n\n\\noindent\n{\\bf Boost.} The conserved charge densities arising from $J^0$ \\eqref{currentJ} corresponding to $\\epsilon_{0i}$ are the boost densities ${\\bf k} = \\rho\\ {\\bf x} - {\\bf p}\\ t$, which simplify for $t{=}0$ to\n\\begin{equation}\n    k_i \\= (\\sfrac\\gamma\\ell)^3\\, \\mathcal{K}_i \\quad\\quad\\textrm{with}\\quad\\quad \\mathcal{K}_i \\= \\rho\\, \\omega_i\\ .\n\\end{equation}\nThe corresponding charges $K_i$ vanishes because of the odd integrand as discussed in an earlier remark:\n\\begin{equation}\nK_i \\= \\int_V \\mathrm{d}^3x\\, k_i \\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\mathcal{K}_i \\= 2 \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\omega_i\\, X_a^j\\, \\bar{X}_a^j \\= 0\\ .\n\\end{equation}\n\n\\noindent\n{\\bf Angular momentum.} The other three conserved charge densities $J^0$ corresponding to $\\epsilon_{ij}$ in \\eqref{currentJ} are the angular momentum densities ${\\bf l} = {\\bf p}\\times{\\bf x}$, which takes a simple form just like in momentum \\eqref{1formP}:\n\\begin{equation}\n    l_i\\, \\mathrm{d} x^i \\= (\\sfrac\\gamma\\ell)^2\\, \\mathcal{L}_a\\, e^a \\quad\\quad\\textrm{with}\\quad\\quad \\mathcal{L}_a \\= \\varepsilon_{abc}\\, \\mathcal{P}_b\\, \\omega_c\\ .\n\\end{equation}\nThe expressions for the charges $L_i$ simplify to\n\\begin{equation}\n    L_i \\= \\int_V \\mathrm{d}^3x\\, l_i \\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, (\\sfrac\\ell\\gamma)\\, \\mathcal{L}_a\\, e^a_i \\= 2\\mathrm{i}\\ell\\ \\Omega^2\\ \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\sfrac{1}{\\gamma}\\, e^a_i \\left( \\bar{X}_a\\, \\omega_b X_b - X_a\\, \\omega_b\\bar{X}_b \\right)\\ .\n\\end{equation}\nExplicit calculation show that for $j{=}0$ the angular momenta $L_i$ are proportional to the momenta $P_i$:\n\\begin{equation}\n    L_i^{(j=0)} \\= -\\ell^2\\, P_i^{(j=0)}\\ .\n\\end{equation}\nThis is, however, not true for higher spin $j$. The angular momenta $L_i$ has the same rotation behaviour as for the momenta $P_i$ \\eqref{rotP}. We therefore note down the results of $L_3$ for $j=1\/2$ and $1$ in table \\ref{table1}, from which the corresponding expressions of $L_1$ and $L_2$ can be obtained using the table in Appendix \\ref{appendRotation}.\n\nWe can again compute the spherical components of the angular momentum $(L_r,L_\\theta,L_\\phi)$ using the relations \\eqref{sphericalP} by replacing $\\mathcal{P}$ with $\\mathcal{L}$ in it. We realize that the $S^2$ integrand for $L_r$ would only have terms like $\\hat{\\omega}_a\\hat{\\omega}_b\\mathcal{P}_c$, which are all odd functions\\footnote{Here again the terms of $\\mathcal{P}_c$, which goes like $Y_{l,M}Y_{l',M'}$ \\eqref{split-Y}, are all even functions of $\\phi$ for a fixed $j$.} of $\\phi$ and, therefore, the $\\phi$ integral over the domain $(0,2\\pi)$ would make $L_r$ vanish. We also find, with explicit computations, that the charges $L_\\theta$ and $L_\\phi$ for $j{=}0$ are proportional to $P_3$:\n\\begin{equation}\n    L_\\theta^{(j=0)} \\= \\sfrac43\\, \\ell^2\\, P_3^{(j=0)} \\quad\\quad\\textrm{and}\\quad\\quad L_\\phi^{(j=0)} \\= -\\sfrac13\\, \\ell^2\\, P_3^{(j=0)}\\ .\n\\end{equation}\nAs a non trivial example, we collect these charges for $j=1\/2$ below:\n\\begin{equation}\n  \\begin{aligned}\n     L_\\theta^{\\left(j=1\/2\\right)} \\= \\sfrac{12}{5}\\ell\\, \\Big( &9|\\Lambda_{-1\/2,-3\/2}|^{2} + 6|\\Lambda_{-1\/2,-1\/2}|^{2} +|\\Lambda_{-1\/2, 1\/2}|^{2}\n     - 6|\\Lambda_{-1\/2,3\/2}|^{2} \\\\ \n     & + 6|\\Lambda_{1\/2,-3\/2}|^{2} - |\\Lambda_{1\/2, -1\/2}|^{2} - 6|\\Lambda_{1\/2,1\/2}|^{2} - 9|\\Lambda_{1\/2, 3\/2}|^{2} \\Big)\\ ,\n  \\end{aligned}\n\\end{equation}\n\\begin{equation}\n  \\begin{aligned}\n      L_\\phi^{\\left(j=1\/2\\right)} \\= -\\sfrac35 \\ell\\, \\Big(  &6|\\Lambda_{-1\/2,-3\/2}|^{2} - |\\Lambda_{-1\/2,-1\/2}|^{2} -6|\\Lambda_{-1\/2, 1\/2}|^{2} - 9|\\Lambda_{-1\/2,3\/2}|^{2} \\\\\n      &+ 9|\\Lambda_{1\/2,-3\/2}|^{2} + 6|\\Lambda_{1\/2,-1\/2}|^{2} + |\\Lambda_{1\/2,1\/2}|^{2} - 6|\\Lambda_{1\/2, 3\/2}|^{2}\\\\ \n      &+ \\sqrt{3}\\big( \\bar{\\Lambda}_{1\/2,-3\/2}\\Lambda_{-1\/2,-1\/2} -\\bar{\\Lambda}_{1\/2,1\/2}\\Lambda_{-1\/2,3\/2} \\\\\n      &+ \\bar{\\Lambda}_{-1\/2,-1\/2}\\Lambda_{1\/2,-3\/2} - \\bar{\\Lambda}_{-1\/2,3\/2}\\Lambda_{1\/2,1\/2}\\big) \\Big)\\ .\n  \\end{aligned}\n\\end{equation}\n\n\\subsection{Dilatation} \nIt is easy to verify that a constant rescaling by $\\lambda$:\n\\begin{equation}\n    \\zeta^\\mu \\= \\lambda\\, x^\\mu\n\\end{equation}\nis also a solution of \\eqref{Killing}. The charge density corresponding to this single generator of the conformal group is ${\\bf p}\\cdot{\\bf x}- e\\,t$, which for $t{=}0$ simplifies to \n\\begin{equation}\n    p_i\\, x_i \\= (\\sfrac\\gamma\\ell)^3\\, \\mathcal{P}_a\\,\\omega_a\\ .\n\\end{equation}\nThe corresponding charge $D$ vanishes because of the odd integrand:\n\\begin{equation}\n    D \\= \\int_V \\mathrm{d}^3x\\, p_i\\, x_i \\= 2\\mathrm{i}\\,\\Omega^2\\, \\varepsilon_{abc} \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\omega_a\\, X_b\\, \\bar{X}_c \\= 0\\ .\n\\end{equation}\n\\subsection{Special conformal transformations}\nA fairly straightforward calculation shows that the following not so obvious transformation\n\\begin{equation}\n    \\zeta^\\mu \\= 2\\, x^\\mu\\,b_\\nu x^\\nu\\ -\\ b^\\mu\\, x^\\nu x_\\nu \n\\end{equation}\nalso satisfies \\eqref{Killing}. The four generators corresponding to $b_\\mu$ give rise to four different charges $V_0$ and ${\\bf V}$.\n\n\\noindent\n{\\bf Scalar SCT.} The charge density $J^0$ corresponding to $b_0$ is \\begin{equation}\n    v_0 \\= ({\\bf x}^2+t^2)\\, e - 2t\\, {\\bf p}\\cdot{\\bf x}\\ ,\n\\end{equation}\nwhich for $t{=}0$ simplifies to\n\\begin{equation}\n    v_0 \\= {\\bf x}^2\\, e \\= (\\sfrac\\gamma\\ell)^2 \\rho\\, \\omega_a^2\\ .\n\\end{equation}\nThe expression for the corresponding charge $V_0$ takes the following simple form:\n\\begin{equation}\n    V_0 \\= \\int_V \\mathrm{d}^3x\\, v_0 \\= \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, (\\sfrac\\ell\\gamma)\\, (1-\\omega_4^2)\\, \\rho \\= 2\\ell\\, \\Omega^2 \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, (1+\\omega_4)\\, X_a\\, \\bar{X}_a\\ .\n\\end{equation}\nHere again the $\\omega_4$ term of the integral, being odd, vanishes and yields\n\\begin{equation}\n    V_0 \\= \\ell^2\\, E \\= 8\\,\\ell\\, (j+1)^3(2j+1)\\, \\sum_{m,n}\\,|\\Lambda_{m,n}|^2\\ .\n\\end{equation}\n\n\\noindent\n{\\bf Vector SCT.} The charge densities $J^0$ that correspond to $b_i$ read:\n\\begin{equation}\n    {\\bf v} \\= 2{\\bf x}({\\bf x}\\cdot{\\bf p}) - 2t\\,{\\bf x}\\,e - ({\\bf x}^2-t^2){\\bf p}\\ .\n\\end{equation}\nThis simplify at $t{=}0$ and takes a structure similar to the momentum densities $p_i$ \\eqref{1formP}:\n\\begin{equation}\\label{vectorSCT}\n    v_i\\, dx^i \\= (\\sfrac\\gamma\\ell)\\, \\mathcal{V}_a\\, e^a \\quad\\textrm{with}\\quad \\mathcal{V}_a \\= 2\\omega_a\\, (\\mathcal{P}_b\\,\\omega_b) - \\omega_b^2\\,\\mathcal{P}_a\\ . \n\\end{equation}\nThe expressions for the charges $V_i$ then simplify to\n\\begin{equation}\n    V_i \\= \\int_V \\mathrm{d}^3x\\, v_i \\= 2\\mathrm{i}\\,\\ell^2\\,\\Omega^2 \\int_{S^3} \\mathrm{d}^3\\Omega_3\\, \\gamma^{-2} e^a_i \\left( 2\\varepsilon_{bcd}\\,\\omega_a\\,\\omega_b\\,X_c\\,\\bar{X}_d - \\varepsilon_{abc}\\,(1-\\omega_4^2)\\,X_b\\,\\bar{X}_c \\right)\\ .\n\\end{equation}\nWith explicit computation we observe that the charges $V_i$ are proportional to the momenta $P_i$ (verified explicitly for up to $j{=}1$)\n\\begin{equation}\n    V_i \\= \\ell^2\\, P_i\\ .\n\\end{equation}\n\nAs before, we can compute the spherical components $(V_r,V_\\theta,V_\\phi)$ by using the expressions \\eqref{sphericalP} and replacing $\\mathcal{P}$ with $\\mathcal{V}$ in it. We notice that the charge $V_r$ vanishes owing to the odd $S^2$ integrand\\footnote{Observe that the terms in $\\mathcal{V}_a$ \\eqref{vectorSCT} are all even in $\\hat{\\omega}_a$.} just like in the case of $P_r$. However, unlike $P_\\theta$ here the charge $V_\\theta$ is non-vanishing. Explicit calculations show that for $j=0$ the charges $V_\\theta$ and $V_\\phi$ are proportional to the momentum $P_3$:\n\\begin{equation}\n    V_\\theta^{(j=0)} \\= -\\sfrac43 \\ell^3\\, P_3^{(j=0)} \\quad\\quad\\textrm{and}\\quad\\quad V_\\phi^{(j=0)} \\= -\\sfrac53 \\ell^3\\, P_3^{(j=0)}\\ .\n\\end{equation}\nAdditionally, we record below the charges $V_\\theta$ and $V_\\phi$ for the non-trivial case of $j{=}1\/2$\n\\begin{equation}\n  \\begin{aligned}\n     V_\\theta^{\\left(j=1\/2\\right)} \\= -\\sfrac{6}{5}\\ell^2\\, \\Big( &9|\\Lambda_{-1\/2,-3\/2}|^{2} + |\\Lambda_{-1\/2,-1\/2}|^{2} - 9|\\Lambda_{-1\/2,1\/2}|^{2} - 21|\\Lambda_{-1\/2,3\/2}|^{2} \\\\ \n     & + 21|\\Lambda_{1\/2,-3\/2}|^{2} +\n     9|\\Lambda_{1\/2,-1\/2}|^{2} - |\\Lambda_{1\/2,1\/2}|^{2} - 9|\\Lambda_{1\/2, 3\/2}|^{2} \\Big)\\ ,\n  \\end{aligned}\n\\end{equation}\n\\begin{equation}\n  \\begin{aligned}\n      V_\\phi^{\\left(j=1\/2\\right)} \\= -\\sfrac35 \\ell^2\\, \\Big(  &42|\\Lambda_{-1\/2,-3\/2}|^{2} + 33|\\Lambda_{-1\/2,-1\/2}|^{2} + 8|\\Lambda_{-1\/2,1\/2}|^{2} - 33|\\Lambda_{-1\/2,3\/2}|^{2} \\\\\n      &+ 33|\\Lambda_{1\/2,-3\/2}|^{2} - 8|\\Lambda_{1\/2,-1\/2}|^{2} - 33|\\Lambda_{1\/2,1\/2}|^{2} - 42|\\Lambda_{1\/2,3\/2}|^{2}\\\\ \n      &\\qquad + 8\\sqrt{3}\\big( -\\bar{\\Lambda}_{1\/2,-3\/2}\\Lambda_{-1\/2,-1\/2} +\\bar{\\Lambda}_{1\/2,1\/2}\\Lambda_{-1\/2,3\/2} \\\\\n      &\\qquad\\qquad\\qquad - \\bar{\\Lambda}_{-1\/2,-1\/2}\\Lambda_{1\/2,-3\/2} + \\bar{\\Lambda}_{-1\/2,3\/2}\\Lambda_{1\/2,1\/2}\\big) \\Big)\\ .\n  \\end{aligned}\n\\end{equation}\nOne can compute these charges for higher spin-$j$ by following the same strategy.\n\n\\subsection{Applications}\n\\label{sec4}\n\\noindent\nThe method of constructing rational electromagnetic fields presented in this paper has the added advantage that it produces a complete set labelled by $(j,m,n)$; any electromagnetic field configuration having finite energy can, in principle, be obtained from an expansion like in (\\ref{Asep}-\\ref{basisZ}), albeit with a varying $j$. The operational difficulty involved in this procedure has to do with the fact that this set is infinite as $j\\in \\sfrac{\\mathds N}{2}$. There are, however, many important cases where only a finite number of knot-basis solutions (sometimes only with a fixed $j$) need to be combined to get the desired EM field configuration. One such very important case is that of the Hopfian solution discussed before. Below we analyse two very interesting generalisations of the Hopfian solution presented in \\cite{HSS15} in the context of present construction. It is imperative to note here that while the scope of construction of a new solution from the known ones as presented in \\cite{HSS15} is limited, the same is not true for the method presented in this paper, which by design can produce arbitrary number of new field configurations. Some of these possible new field configurations obtained from the $j{=}0$ sector (possibly from $j{=}1\/2~ \\textrm{or}~ 1$ as well) could find experimental application with improved experimental techniques like in \\cite{LSMetal18}. \n\n\\begin{figure}[h!]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 5cm, height = 5cm]{Figures\/HopfianTT.pdf}\n   \\hspace{2cm}\n   \\includegraphics[width = 5cm, height = 5cm, trim = {2cm 3cm 1.5cm 2cm},clip]{Figures\/HopfianR.pdf}\n \\caption{\n Sample electric (red) and magnetic (green) field lines at $t{=}0$.\n Left: time-translated Hopfian with $c=60$, Right: Rotated Hopfian with $\\theta=1$.} \n\\label{fig3}\n\\end{figure}\nBateman's construction, employed in \\cite{HSS15}, hinges on a judicious ansatz for the Riemann-Silberstein vector \\eqref{RSvector} satisfying Maxwell's equations:\n\\begin{equation}\n    {\\bf S} \\= \\nabla\\alpha \\times \\nabla\\beta \\quad\\implies\\quad \\nabla\\cdot{\\bf S}\\=0 ~\\ \\& ~\\ \\mathrm{i}\\, \\partial_t{\\bf S} \\= \\nabla\\times{\\bf S}\\ ,\n\\end{equation}\nusing a pair of complex functions $(\\alpha,\\beta)$. An interesting generalization of the Hopfian solution is obtained in equations (3.16-3.17) of \\cite{HSS15} using (complex) time-translation (TT) to obtain the following $(\\alpha,\\beta)$ pair (up to a normalization)\n\\begin{equation}\n    (\\alpha,\\beta)_{TT} \\= \\left(\\frac{A-1-\\mathrm{i} z}{A+\\mathrm{i}(t+\\mathrm{i} c)}, \\frac{x-\\mathrm{i} y}{A+\\mathrm{i}(t+\\mathrm{i} c)} \\right)\\ ;\\quad A\\= \\sfrac12 \\left(x^2+y^2+z^2-(t+\\mathrm{i} c)^2+1\\right)\\ ,\n\\end{equation}\nwhere $c$ is a constant real parameter. The corresponding  EM field configuration is obtained, in our case, by choosing $\\ell {=} 1{-}c$ and only the following $j{=}0$, and hence type I, complex coefficients in \\eqref{basisZ}\n\\begin{equation}\\label{j0TT}\n    \\left(\\Lambda_{0;0,1}\\ ,~\\Lambda_{0;0,0}\\ ,~\\Lambda_{0;0,-1}\\right)_{TT} \\= \\left( 0\\ ,~0\\ ,~-\\mathrm{i}\\frac{\\pi}{2\\ell^2} \\right)\\ .\n\\end{equation}\nA sample electromagnetic knot configuration of this modified Hopfian is illustated in figure \\ref{fig3}. Another interesting generalization of the Hopfian is constructed in equations (3.20-3.21) of \\cite{HSS15} using a (complex) rotation (R) to get the following $(\\alpha,\\beta)$ pair (again, up to a normalization)\n\\begin{equation}\n    (\\alpha,\\beta)_{R} \\= \\left(\\frac{A-1+\\mathrm{i}(z \\cos{\\mathrm{i}\\theta}+x\\sin{\\mathrm{i}\\theta})}{A+\\mathrm{i} t}, \\frac{x \\cos{\\mathrm{i}\\theta}-z\\sin{\\mathrm{i}\\theta}-\\mathrm{i} y}{A+\\mathrm{i} t} \\right) \\ ,\n\\end{equation}\nwhere $A = \\sfrac12 \\left(x^2+y^2+z^2-t^2+1\\right)$. To get this particular EM field configuration we need to set $\\ell{=}1$ and use the following combination of only $j{=}0$ type I coefficients in \\eqref{basisZ}\n\\begin{equation}\\label{j0R}\n    \\left(\\Lambda_{0;0,1}\\ ,~\\Lambda_{0;0,0}\\ ,~\\Lambda_{0;0,-1}\\right)_{R} \\= \\left( \\mathrm{i}\\frac{\\pi}{4}(\\cosh{\\theta}-1)\\ ,~-\\frac{\\pi}{2\\sqrt{2}}\\sinh{\\theta}\\ ,~-\\mathrm{i}\\frac{\\pi}{4}(\\cosh\\theta+1) \\right)\\ .\n\\end{equation}\nWe illustrate the EM field lines for a sample $\\theta$ value of this modified Hopfian in figure \\ref{fig3}. We note down the conformal charges corresponding to these solutions in table \\ref{table3} by plugging the $j{=}0$ coefficients \\eqref{j0TT} and \\eqref{j0R} in the appropriate formulae of the previous section. The results matches with the ones given in \\cite{HSS15} up to a rescaling of the energy, which can be achieved by an appropriate choice of normalization.\n\\begin{table}[]\n    \\centering\\setcellgapes{4pt}\\makegapedcells \\renewcommand\\theadfont{\\normalsize\\bfseries}\n    \\begin{tabular}{|c|P{5cm}|P{5cm}|}\n        \\hline\n      & Time-translated Hopfian & Rotated Hopfian \\\\ [0.5ex] \\hline \\hline\n     Energy (E) & \n     $\\frac{2\\pi^2}{(1-c)^5}=:E_{TT}$ & \n     $2\\pi^2\\cosh^2\\theta =: E_{R}$  \\\\ \\hline\n     Momentum ($\\textbf{P}$) & \n     $\\left(0\\ ,~0\\ ,~\\frac{1}{4}\\right)E_{TT}$ & \n     $\\left(0\\ ,~-\\frac{1}{4}\\tanh{\\theta}\\ ,~\\frac{1}{4}\\sech\\theta\\right)E_{R}$ \\\\ \\hline\n     Boost ($\\textbf{K}$) & \n     $\\left(0\\ ,~0\\ ,~0\\right)$ & \n     $\\left(0\\ ,~0\\ ,~0\\right)$ \\\\ \\hline\n     Ang. momentum ($\\textbf{L}$) & \n     $\\left(0\\ ,~0\\ ,~-\\frac{1}{4}(1-c)^2\\right)E_{TT}$ & \n     $\\left(0\\ ,~\\frac{1}{4}\\tanh{\\theta}\\ ,~-\\frac{1}{4}\\sech\\theta\\right)E_{R}$ \\\\ \\hline\n      Dilatation (D) & \n     $0$ & \n     $0$ \\\\ \\hline\n     Scalar SCT ($V_0$) & \n     $(1-c)^2E_{TT}$ & \n     $E_{R}$  \\\\ \\hline\n     Vector SCT ($\\textbf{V}$) & \n     $\\left(0\\ ,~0\\ ,~\\frac{1}{4}(1-c)^2\\right)E_{TT}$ & \n     $\\left(0\\ ,~-\\frac{1}{4}\\tanh{\\theta}\\ ,~\\frac{1}{4}\\sech\\theta\\right)E_{R}$ \\\\ \\hline\n    \\end{tabular}\n    \\caption{Conformal charges for the time-translated and rotated Hopfian configurations.}\n    \\label{table3}\n\\end{table}\n\n\\vspace{12pt}\n\\section{Null fields}\n\\label{nullFields}\n\\vspace{2pt}\n\\noindent\nAn interesting subset of vacuum electromagnetic fields are those with vanishing Lorentz invariants,\n\\begin{equation}\n\\vec{E}^2-\\vec{B}^2 =0 \\quad\\quad\\textrm{and}\\quad\\quad \\vec{E}\\cdot\\vec{B} =0 \\qquad\\Longleftrightarrow\\qquad\n\\bigl(\\vec{E}\\pm\\mathrm{i}\\vec{B}\\bigr)^2 =\\ 0\\ .\n\\end{equation}\nAs a scalar equation it must equally hold on the de Sitter side, and so\nwe can try to characterize such configurations with our SO(4) basis above. For a given type and spin, \nthe expressions in~\\eqref{EBj} immediately give the Riemann-Silberstein vector on the $S^3$~cylinder,\n\\begin{equation}\n{\\cal E}_a \\pm\\mathrm{i}\\,{\\cal B}_a \\= -2\\mathrm{i}\\,\\Omega\\,X_a(\\omega)\\,\\mathrm{e}^{\\Omega\\,\\mathrm{i}\\tau}\\ ,\n\\end{equation}\nwhere the upper (lower) sign pertains to type~I (II).\nNote that the negative-frequency part of this field has cancelled.\nThe vanishing of $({\\cal E}_a{\\pm}\\mathrm{i}{\\cal B}_a)({\\cal E}_a{\\pm}\\mathrm{i}{\\cal B}_a)$ is then equivalent to a condition on the angular functions,\n\\begin{equation} \\label{nullcondition}\n0 \\= X_1(\\omega)^2 + X_2(\\omega)^2 + X_3(\\omega)^2 \\= 2\\,Z_+(\\omega) Z_-(\\omega) + Z_3(\\omega)^2\\ .\n\\end{equation}\nWhen expanding the angular functions~$Z^j_{*\\ \\textrm{I}}$ or $Z^j_{*\\ \\textrm{II}}$ into basis solutions with ~\\eqref{basisZ},\none arrives at a system of homogeneous quadratic equations for the free coefficients~$\\lambda_{j;m,n}^{\\textrm{I\/II}}$.\n\nLet us analyze the situation for type~I and spin~$j$. \nThe functions~$Z^j_*(\\omega)$ transform under a $(j,j)$ representation of $su(2)_L\\oplus su(2)_R$. \nThe null condition~\\eqref{nullcondition} then yields a representation content of $(0,0)\\oplus(1,1)\\oplus\\ldots\\oplus(2j,2j)$\nand may thus be expanded into the corresponding harmonics. Furthermore, The independent vanishing of all coefficients produces\n$\\frac16(4j{+}1)(4j{+}2)(4j{+}3)$ equations for the $(2j{+}1)(2j{+}3)$ parameters~$\\lambda_{j;m,n}$ (note the ranges of $m$ and~$n$ for type~I).\nClearly, this system is vastly overdetermined. However, it turns out that only $4j^2{+}6j{+}1$ equations are independent,\nstill leaving $2j{+}2$ free complex parameters for the solution space. The independent equations can be organized as (suppressing~$j$)\n\\begin{equation}\n\\begin{aligned}\n\\lambda_{m,n}^2 &\\ \\sim\\ \\lambda_{m,n-1}\\,\\lambda_{m,n+1} \\qquad\\ \\quad\\textrm{for}\\quad m,n=-j\\,\\ldots,j \\ ,\\\\\n\\lambda_{m,j+1}\\,\\lambda_{m+1,-j-1} &\\= \\lambda_{m+1,j+1}\\,\\lambda_{m,-j-1} \\quad\\quad\\textrm{for}\\quad m=-j,\\ldots,j{-}1\\ .\n\\end{aligned}\n\\end{equation}\nWe have checked for $j{\\le}5$ that the upper equations are solved by~\\footnote{These are the generic solutions. There exist also special solutions given by \n\\eqref{extweights} and $\\lambda_{m,n}=0$ for $|n|\\neq j{+}1$, for arbitrarily selected choices of~$m\\in\\{-j,\\ldots,j\\}$.}\n\\begin{equation}\n\\lambda_{m,n}^{2j+2} \\= {\\textstyle\\sqrt{\\binom{2j+2}{j+1-n}}}\\ \\lambda_{m,-j-1}^{j+1-n}\\,\\lambda_{m,j+1}^{j+1+n}\n\\quad\\quad\\textrm{for}\\quad m=-j,\\ldots,j \\quad\\textrm{and}\\quad n=-j{-}1,\\ldots,j{+}1\\ ,\n\\end{equation}\nwhile the lower ones imply that the highest weights $n{=}j{+}1$ are proportional to the lowest weights $n{=}{-}j{-}1$ (independent of~$m$), \n\\begin{equation} \\label{extweights}\n\\lambda_{m,-j-1} \\= w\\,\\lambda_{m,j+1} \\quad\\quad\\textrm{for}\\quad w\\in\\mathds{C}^*\\ .\n\\end{equation}\nTherefore, the full (generic) solution reads\n\\begin{equation}\n\\lambda_{m,n} \\= {\\textstyle\\sqrt{\\binom{2j+2}{j+1-n}}}\\ w^{\\frac{j+1-n}{2j+2}}\\ \\mathrm{e}^{2\\pi\\mathrm{i} k_m\\frac{j+1-n}{2j+2}}\\ z_m \n\\qquad\\quad\\textrm{with}\\quad z_m\\in\\mathds{C} \\quad\\textrm{and}\\quad k_m\\in\\{0,1,\\ldots,2j{+}1\\}\\ ,\n\\end{equation}\ncontaining $2j{+}2$ complex parameters $z_m$ and~$q$ as well as $2j$ discrete choices~$\\{k_m\\}$ \n(one of them can be absorbed into~$z_m$). This completely specifies the type-I null fields for a given spin.\nType-II null fields are easily obtained by applying electromagnetic duality to type-I null fields.\n\nIn the simplest case of $j{=}0$, the single equation $\\lambda_{0,0}^2=2\\lambda_{0,-1}\\lambda_{0,1}$ describes \na generic rank-3 quadric in $\\mathds{C} P^2$, or a cone over a sphere $\\mathds{C} P^1$ inside the parameter space~$\\mathds{C}^3$. For higher spin,\nthe moduli space of type-I null fields remains a complete-intersection projective variety of complex dimension~$2j{+}1$.\n\nWe conclude the Section with a display of typical field lines (see Figure \\ref{nullFieldLines}) for a type I $j{=}\\sfrac12$ and $j{=}1$ null field at $t{=}0$. For $t{\\neq}0$ the pictures get smoothly distorted.\n\\begin{figure}[h!]\n\\captionsetup{width=\\linewidth}\n\\centering\n\\includegraphics[width = 0.35\\paperwidth]{j=half_field_lines.pdf}\n\\includegraphics[width = 0.35\\paperwidth]{j=one_field_lines.pdf}\n\\caption{\nSample electric (red) and magnetic (green) field lines at $t{=}0$.\nLeft: a pair of electric and a pair of magnetic field lines for the $(j;m,n) = \\left(\\sfrac12;\\sfrac12,\\sfrac32 \\right)_R$ field configuration. Right: a pair of electric field lines, a magnetic field line\nof self-linking one and a magnetic field line of self-linking seven for the $(j;m,n) = (1;0,2)_R$ field configuration.\n}  \n\\label{nullFieldLines}\n\\end{figure}\n\n\\vspace{12pt}\n\\section{Flux transport}\n\\vspace{2pt}\n\\noindent\nWe have seen that electromagnetic energy is radiated away along the light-cones. \nLet us try to quantify its amount over future null infinity~$\\mathscr{I}^+$. Before proceeding further we note down the determinant of the Jacobian $J$ \\eqref{Jacobian}\n\\begin{equation}\n|\\textrm{det}J| \\= \\frac{p^2{-}q^2}{\\ell^2} \\= \\frac{\\gamma^2}{\\ell^2} \n\\= \\frac{\\sin^2\\!\\tau}{t^2} \\= \\frac{\\sin^2\\!\\chi}{r^2}\n\\quad\\quad\\textrm{with}\\quad \\gamma^2=p^2{-}q^2\n\\end{equation}\nand the spherical Minkowski components\n\\begin{equation}\nA_t \\= {\\cal A}_\\tau J^\\tau_{\\ t} + {\\cal A}_\\chi J^\\chi_{\\ t}  \\ ,\\qquad\nA_r \\= {\\cal A}_\\tau J^\\tau_{\\ r} + {\\cal A}_\\chi J^\\chi_{\\ r}  \\ ,\\qquad\nA_\\theta = {\\cal A}_\\theta \\ ,\\qquad A_\\phi = {\\cal A}_\\phi\\ ,\n\\end{equation}\nor any other such tensor component arising due to \\eqref{Jacobian}.\nFor later use, we also note here the transformation of the volume form\n\\begin{equation} \\label{measure}\n\\mathrm{d}^4x \\= \\mathrm{d} t\\,r^2\\mathrm{d} r\\,\\mathrm{d}^2\\Omega_2 \\= r^2\\,|\\textrm{det}J|^{-1}\\mathrm{d}\\tau\\,\\mathrm{d}\\chi\\,\\mathrm{d}^2\\Omega_2\n\\= \\sin^2\\!\\chi\\,|\\textrm{det}J|^{-2}\\mathrm{d}\\tau\\,\\mathrm{d}\\chi\\,\\mathrm{d}^2\\Omega_2\n\\=\\frac{\\ell^4}{\\gamma^4}\\,\\mathrm{d}\\tau\\,\\mathrm{d}^3\\Omega_3\\ .\n\\end{equation}\n\n\n\n\nThe energy flux at time~$t_0$ passing through a two-sphere of radius~$r_0$ centered at the spatial origin is given by\n\\begin{equation}\n\\Phi(t_0,r_0) \\= \\int_{S^2(r_0)}\\!\\!\\mathrm{d}^2\\vec{\\sigma}\\cdot \\bigl(\\vec{E}\\times\\vec{B}\\bigr)(t_0,r_0,\\theta,\\phi) \n\\= \\int_{S^2}r_0^2\\,\\mathrm{d}^2\\Omega_2\\ T_{t\\,r}^{\\mathrm{(M)}}(t_0,r_0,\\theta,\\phi)\\ ,\n\\end{equation}\nwhere $\\mathrm{d}^2\\Omega_2=\\sin\\theta\\,\\mathrm{d}\\theta\\,\\mathrm{d}\\phi$, \nand $T_{t\\,r}^{\\mathrm{(M)}}$ is the $(t,r)$ component of the Minkowski-space stress-energy tensor\n\\begin{equation}\\label{stress-energyMink}\nT_{\\mu\\,\\nu}^{\\mathrm{(M)}} \\= F_{\\mu\\rho}F_{\\nu\\lambda}\\,g^{\\rho\\lambda} - \\sfrac14 g_{\\mu\\nu}F^2 \n\\quad\\quad\\textrm{with}\\quad (g_{\\mu\\nu}) = \\textrm{diag}(-1,1,r^2,r^2\\sin^2\\!\\theta)\\ \n\\end{equation}\nfor $\\mu,\\nu,\\ldots\\in\\{t,r,\\theta,\\phi\\}$. We carry out this computation in the $S^3$-cylinder frame by using the conformal relations\n\\begin{equation}\n\\ell^2\\,T_{\\mu\\,\\nu}^{\\mathrm{(dS)}} \\= t^2\\,T_{\\mu\\,\\nu}^{\\mathrm{(M)}} \\= \\sin^2\\!\\tau\\,T_{\\mu\\,\\nu}^{\\mathrm{(cyl)}} \n\\= \\sin^2\\!\\tau\\,T_{m\\,n}^{\\mathrm{(cyl)}}\\, J^m_{\\ \\ \\mu}\\,J^n_{\\ \\ \\nu} \n\\quad\\textrm{for}\\quad m,n\\in\\{\\tau,\\chi,\\theta,\\phi\\}\n\\end{equation}\nwith the Jacobian~\\eqref{Jacobian} and the fact that \\ $r\\sin\\tau=t\\sin\\chi$ \\ so that \n\\begin{equation}\n\\Phi(\\tau_0,\\chi_0) \\= \\int_{S^2}\\sin^2\\!\\chi\\,\\mathrm{d}^2\\Omega_2\\ T_{t\\,r}^{\\mathrm{(cyl)}}(\\tau_0,\\chi_0,\\theta,\\phi)\n\\=  \\int_{S^2}\\sin^2\\!\\chi\\,\\mathrm{d}^2\\Omega_2\\ T_{m\\,n}^{\\mathrm{(cyl)}}\\, J^m_{\\ \\ t}\\,J^n_{\\ \\ r} \\ .\n\\end{equation}\nA straightforward computation using $(g_{mn})=\\textrm{diag}(-1,1,\\sin^2\\!\\chi,\\sin^2\\!\\chi\\sin^2\\!\\theta)$ then yields\n\\begin{equation}\n\\begin{aligned}\n\\Phi(\\tau_0,\\chi_0) &\\= \\frac{p\\,q}{\\ell^2}\\int \\mathrm{d}^2\\Omega_2\\,\\Bigl(({\\cal F}_{\\tau\\theta})^2 + ({\\cal F}_{\\chi\\theta})^2 \n+ \\sfrac{1}{\\sin^2\\!\\theta}\\bigl[({\\cal F}_{\\tau\\phi})^2 + ({\\cal F}_{\\chi\\phi})^2\\bigr]\\Bigr) \\\\\n&\\quad +\\ \\frac{p^2{+}q^2}{\\ell^2}\\int \\mathrm{d}^2\\Omega_2\\,\\Bigl( {\\cal F}_{\\tau\\theta}\\,{\\cal F}_{\\chi\\theta} \n+ \\sfrac{1}{\\sin^2\\!\\theta}{\\cal F}_{\\tau\\phi}\\,{\\cal F}_{\\chi\\phi}\\Bigr)\\ .\n\\end{aligned}\n\\end{equation}\nThe sphere-frame components ${\\cal F}_{mn}$ can be computed by expanding\n$e^a=e^a_{\\ m}\\,\\mathrm{d}\\xi^m$ in\n\\begin{equation}\n{\\cal F} \\=  {\\cal E}_a\\,e^a {\\wedge} e^\\tau + \\sfrac12\\,{\\cal B}_a\\,\\varepsilon^a_{\\ bc}\\,e^b{\\wedge} e^c \\= {\\cal F}_{mn}\\,\\mathrm{d}\\xi^m{\\wedge}\\mathrm{d}\\xi^n\n\\quad\\quad\\textrm{with}\\quad\\xi^n\\in\\{\\tau,\\chi,\\theta,\\phi\\}\\ .\n\\end{equation}\nThe expression for the flux in sphere-frame fields then becomes\n\\begin{equation}\\label{flux}\n\\begin{aligned}\n\\ell^2\\,\\Phi &\\= p\\,q\\,\\sin^2\\!\\chi \\int_{S^2}\\mathrm{d}^2\\Omega_2\\ \\Bigl[\n(\\sin\\phi\\,{\\cal E}_1-\\cos\\phi\\,{\\cal E}_2)^2 + (\\cos\\theta\\cos\\phi\\,{\\cal E}_1+\\cos\\theta\\sin\\phi\\,{\\cal E}_2 \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad-\\sin\\theta\\,{\\cal E}_3)^2 + (\\sin\\phi\\,{\\cal B}_1-\\cos\\phi\\,{\\cal B}_2)^2 \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad+ (\\cos\\theta\\cos\\phi\\,{\\cal B}_1+\\cos\\theta\\sin\\phi\\,{\\cal B}_2-\\sin\\theta\\,{\\cal B}_3)^2 \\Bigr] \\\\\n&\\qquad +\\ (p^2{+}q^2)\\,\\sin^2\\!\\chi \\int_{S^2}\\mathrm{d}^2\\Omega_2\\ \\Bigl[\n(\\sin\\phi\\,{\\cal B}_1-\\cos\\phi\\,{\\cal B}_2) (\\cos\\theta\\cos\\phi\\,{\\cal E}_1 \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad +\\cos\\theta\\sin\\phi\\,{\\cal E}_2-\\sin\\theta\\,{\\cal E}_3) - (\\sin\\phi\\,{\\cal E}_1-\\cos\\phi\\,{\\cal E}_2) \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad \\cdot(\\cos\\theta\\cos\\phi\\,{\\cal B}_1+\\cos\\theta\\sin\\phi\\,{\\cal B}_2-\\sin\\theta\\,{\\cal B}_3) \\Bigr]\\ .\n\\end{aligned}\n\\end{equation}\n\nThe total energy flux across future null infinity is obtained by evaluating this expression on $\\mathscr{I}^+$ and integrating over it. Introducing cylinder light-cone coordinates\n\\begin{equation}\nu=\\tau{+}\\chi \\quad\\textrm{and}\\quad v=\\tau{-}\\chi \\qquad\\textrm{so that}\\qquad\nt{+}r=-\\ell\\,\\cot\\sfrac{v}{2} \\quad\\textrm{and}\\quad t{-}r=-\\ell\\,\\cot\\sfrac{u}{2}\n\\end{equation}\nwe characterize $\\mathscr{I}^+$ as\n\\begin{equation}\n\\biggl\\{ \\begin{array}{l} t{+}r\\to\\infty \\\\[4pt]  t{-}r \\in\\mathds R \\end{array} \\biggr\\} \\quad\\Leftrightarrow\\quad\n\\biggl\\{ \\begin{array}{l} u\\in(0,2\\pi) \\\\[4pt]  v=0 \\end{array} \\biggr\\} \\qquad\\Rightarrow\\qquad\np=q=\\sin^2\\!\\chi \\quad\\textrm{and}\\quad \\gamma=0\\ .\n\\end{equation}\nFurther noticing that\n\\begin{equation}\n\\mathrm{d}(t{-}r) \\= \\frac{\\ell\\ \\mathrm{d} u}{p{+}q} \\= \\frac{\\ell\\ \\mathrm{d} u}{1-\\cos u} \\quad\\quad\\textrm{and}\\quad\\quad\n\\sin^2\\!\\chi \\= \\sin^2\\!\\sfrac{u-v}{2} \\= \\sfrac12\\bigl(1-\\cos(u{-}v)\\bigr)\\ ,\n\\end{equation}\nwe may express this total flux as\n\\begin{equation}\n\\Phi_+ \\= \\int_{-\\infty}^\\infty \\mathrm{d}(t{-}r)\\ \\Phi\\big|_{\\mathscr{I}^+}\n\\= \\int_0^{2\\pi} \\frac{\\ell\\ \\mathrm{d} u}{1-\\cos u}\\ \\Phi(\\sfrac{u}{2},\\sfrac{u}{2})\n\\end{equation}\nto obtain\n\\begin{equation} \\label{totalflux}\n\\begin{aligned}\n\\Phi_+ &\\=\\frac{1}{8\\ell} \\int\\!\\mathrm{d} u\\,(1{-}\\cos u)^2 \\int\\!\\mathrm{d}^2\\Omega_2\\ \\Bigl[\n\\bigl\\{ \\cos\\theta\\cos\\phi\\,{\\cal E}_1+\\cos\\theta\\sin\\phi\\,{\\cal E}_2-\\sin\\theta\\,{\\cal E}_3+\\sin\\phi\\,{\\cal B}_1 \\\\\n&-\\cos\\phi\\,{\\cal B}_2 \\bigr\\}^2 +\\ \\bigl\\{ \\cos\\theta\\cos\\phi\\,{\\cal B}_1+\\cos\\theta\\sin\\phi\\,{\\cal B}_2-\\sin\\theta\\,{\\cal B}_3-\\sin\\phi\\,{\\cal E}_1+\\cos\\phi\\,{\\cal E}_2 \\bigr\\}^2 \\Bigr]\n\\end{aligned}\n\\end{equation}\nThe square bracket expression above can be further simplified for a fixed spin and type \nby employing~\\eqref{EBj} along with \\eqref{XZ}, \\eqref{basisZ}, \\eqref{type1} and~\\eqref{type2} to get\n\\begin{equation}\n\\Phi_+^{(j)} = \\frac{\\Omega^2}{4\\ell}\\int\\!\\mathrm{d} u\\,(1{-}\\cos u)^2 \\int\\!\\mathrm{d}^2\\Omega_2\\,\n \\big| \\pm Z_+^j e^{-\\mathrm{i}\\phi}(1\\pm \\cos\\theta)\\mp Z_-^j e^{\\mathrm{i}\\phi}(1\\mp \\cos\\theta)-\\sqrt{2}Z_3^j\\sin\\theta\\, \\big|^2\\ ,\n\\end{equation}\nwhere the upper (lower) sign corresponds to a type-I (type-II) solution. \nIn the special case of $j=0$ $(\\Omega{=}2)$ the contribution to the two-sphere integral only comes \nfrom the part which is independent of $(\\theta,\\phi)$, i.e.~$\\frac{4}{3}\\left(|Z^0_+|^2+|Z^0_-|^2+|Z^0_3|^2\\right)$, \nso that the integration can easily be performed by passing to the adjoint harmonics \n$\\tilde{Y}_{j;l,M}$ \\eqref{Y-new} and using \\eqref{split-Y}  to get\n\\begin{equation}\n\\Phi_+^{(0)} \\= \\frac{16}{3\\,\\ell}\\int\\limits_0^{2\\pi}\\!\\mathrm{d} u\\ \\sin^4\\!\\sfrac{u}{2}\\,\n\\bigl| R_{0,0}(\\sfrac{u}{2})\\bigr|^2\\sum\\limits_{n=-1}^{1}|\\lambda_{0,n}|^2 \n\\= \\frac{8}{\\ell}\\sum\\limits_{n=-1}^{1}|\\lambda_{0,n}|^2 \\= E^{(0)}\\ .\n\\end{equation}\n{\\setstretch{2.0} The same equality $\\Phi_+=E$ continues to hold true as we go up in spin $j$ \n(we verified it for $j{=}\\sfrac12$ and $j{=}1$), \nthus validating the  energy conservation $\\mbox{$\\partial$}^\\mu T_{\\mu 0}=0$.}\n\n\\vspace{12pt}\n\\section{Trajectories}\n\\vspace{2pt}\n\\noindent\nGiven a knotted electromagnetic field configuration, a natural issue that arises is the behavior of charged particles in the background of such a field. We proceed to address this issue by analyzing, with numerical simulations, the trajectories of several (identical) charged point particles for the family of knotted field configurations \\eqref{EMCartesian} that we encountered before. We will consider type I \\eqref{type1} basis field configurations (up to $j{=}1$ for simplicity) and the Hopf--Ra\u00f1ada field configuration \\eqref{HRfields}. \n\nIn the simulations we employ the maximum of the energy density at time $t$, i.e. $E_{max}(t)$ (that occurs at several points ${\\bf x}_{max}$ that are located symmetrically with respect to the origin), for different initial conditions and field configurations, and in each case we have employed a parameter $R_{max}(t)$ of ``maximal'' radius defined via\n\\begin{equation}\\label{Rmax}\n    E\\left( t,{\\bf x}_{max}(t) \\right) \\= E_{max}(t)\\quad \\implies \\quad R_{max}(t)\\ :=\\ |{\\bf x}_{max}(t)|\\ .\n\\end{equation}\nA characteristic feature of these basis knot electromagnetic fields is that they have a preferred $z$-axis direction which is clearly exemplified in Figure \\ref{EnDen}, where the energy density $E := \\sfrac12 ({\\bf E}^2 + {\\bf B}^2)$ decreases along the $z$-axis. As a result, the basis fields along the $z$-axis (i.e. ${\\bf E}(t,x{=}0,y{=}0,z)$ and ${\\bf B}(t,x{=}0,y{=}0,z)$) are either directed in the $xy$-plane or along the $z$-axis. In fact, for extreme field configurations $(j;\\pm j,\\pm (j{+}1))$, for any $j{>}0$, the fields along the $z$-axis vanish for all times.\n\\begin{figure}[h!]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 5cm, height = 5cm]{EnDenHR.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{EnDen_h,-h,-3h.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{EnDen_1,-1,1.pdf}\n \\caption{\n Contour plots for energy densities at $t{=}0$ (yellow), $t{=}1$ (cyan) and $t{=}1.5$ (purple) with contour value $0.9E_{max}(t{=}1.5)$.\n Left: Hopf--Ran\\~{a}da configuration, Center: $(j;m,n)=(\\sfrac12;-\\sfrac12,-\\sfrac32)_R$ configuration, Right: $(j;m,n)=(1;-1,1)_R$ configuration.}\n\\label{EnDen}\n\\end{figure}\n\nThe trajectories of these particles are governed by the relativistic Lorentz equation\n\\begin{equation}\\label{eq:force1}\n\t\\frac{\\mathrm{d} {\\bf p}}{\\mathrm{d} t} = q({\\bf E}_{\\ell}+{\\bf v} \\times {\\bf B}_{\\ell})\\ ,\n\\end{equation}\nwhere $q$ is the charge of the particle, ${\\bf p} = \\widetilde{\\gamma} m {\\bf v}$ is the relativistic three-momentum, ${\\bf v}$ is the usual three-velocity of the particle, $m$ is its mass, $\\widetilde{\\gamma} = (1-{\\bf v}^{\\,2})^{-1\/2}$ is the Lorentz factor, and ${\\bf E}_{\\ell}$ and ${\\bf B}_{\\ell}$ are dimensionful electric and magnetic fields respectively. With the energy of the particle $E_{p} = \\widetilde{\\gamma} m$ and $\\mathrm{d} E_p\/ \\mathrm{d}{t} = q\\, {\\bf v} \\cdot {\\bf E}$, one can rewrite (\\ref{eq:force1}) in terms of the derivative of ${\\bf v}$ \\cite{landau2} as\n\\begin{equation}\\label{eq:force2}\n\t\\frac{\\mathrm{d} {\\bf v}}{\\mathrm{d} t} = \\frac{q}{\\gamma m} \\left( {\\bf E}_{\\ell} + {\\bf v} \\times {\\bf B}_{\\ell} - ({\\bf v} \\cdot {\\bf E}_{\\ell})\\,{\\bf v} \\right)\\ .\n\\end{equation}\nEquations (\\ref{eq:force1}) and (\\ref{eq:force2}) are equivalent, and either one can be used for a simulation purpose; they only differ by the position of the nonlinearity in ${\\bf v}$. In natural units $\\hbar {=} c {=} \\epsilon_0 {=} 1$, every dimensionful quantity can be written in terms of a length scale. We relate all dimensionful quantities to the de Sitter radius $\\ell$ from equation \\eqref{dS4} and work with the corresponding dimensionless ones as follows:\n\\begin{equation}\n    T := \\frac{t}{\\ell}\\ ,\\quad {\\bf X} := \\frac{{\\bf x}}{\\ell}\\ ,\\quad {\\bf E} := \\ell^2 {\\bf E}_{\\ell}, \\quad\\textrm{and}\\quad {\\bf B} := \\ell^2 {\\bf B}_{\\ell}\\ .\n\\end{equation}\nMoreover, the fields are solutions of the homogeneous (source-free) Maxwell equations, so they can be freely rescaled by any dimensionless constant factor $\\lambda$. Combining the above considerations, one can rewrite (\\ref{eq:force1}) (or analogously (\\ref{eq:force2})) fully in terms of dimensionless quantities as\n\\begin{equation}\n\t\t\\frac{\\mathrm{d} (\\widetilde{\\gamma}{\\bf V})}{\\mathrm{d} T} = \\kappa({\\bf E} + {\\bf V} \\times {\\bf B})\\ ,\n\\end{equation}\nwhere $\\kappa = \\sfrac{q\\ell^3\\lambda}{m}$ is a dimensionless parameter. As for the initial conditions, we mostly work in the following two scenarios:\n\\begin{enumerate}[label=(\\alph*)]\n\\item ${\\bf V}_0 \\equiv {\\bf V}(T{=}0) = 0$ with particles symmetrically (with respect to the origin) positioned along a line, a circle, or a sphere around the origin at $T{=}0$, and\n\\item ${\\bf X}_0 \\equiv {\\bf X}(T{=}0) = 0$ with particle velocities directed radially outward in a symmetric (with respect to the origin) fashion along a line, in a plane or in space at $T{=}0$.\n\\end{enumerate} \n\nWe vary several parameters including the initial conditions (a) and (b) with different directions of lines and planes for each configuration, the value of $\\kappa$, and the simulation time in order to study the behavior of the trajectory. In general, we used the value of $R_{max}(0)$ for each field configuration to have particles with initial conditions positioned near the region of maximum energy of the field. In this scenario, the effect of the field on the trajectories of the particles is more prominent, as expected. For particles positioned along a sphere centered around the origin we use (i) the radius of the sphere to be $R_{max}$, and (ii) the radius of the sphere to be very small. The latter scenario helps us understand small perturbations of the trajectories as compared to a particle starting at rest from the origin. Moreover, for the initial condition of kind (b) we use the particle initial speeds in the range where it is (i) non-relativistic, (ii) relativistic (usually between $0.1$ and $0.9$), and (iii) ultrarelativistic (here, $0.99$ or higher).\n\nWe observe a variety of different behaviors for these trajectories, some of which we summarize below with the aid of figures. Firstly, it is worth noticing that, even with all fields decreasing as powers of both space and time coordinates, in most field configurations we observe particles getting accelerated from rest up to ultrarelativistic speeds. The limit of these ultrarelativistic speeds for higher times depend on the magnitude of the fields (see, for example, Figure \\ref{singleTrajectory}).\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 5cm, height = 5cm]{Trajectory_h,-h,-3h.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{Speed_h,-h,-3h_10k.pdf}\n   \\includegraphics[width = 5cm, height = 5cm]{Speed_h,-h,-3h_100k.pdf}\n \\caption{\n Trajectory of a charged particle for $(j;m,n)=(\\sfrac12;-\\sfrac12,-\\sfrac32)_R$ configuration with initial condition ${\\bf X}_0 = (0.01,0.01,0.01)$ and ${\\bf V}_0 = 0$ simulated for $t \\in [-1,1]$.\n Left: Particle trajectory, Center: absolute velocity profile for $\n\\kappa {=} 10$, Right: absolute velocity profile for $\\kappa {=} 100$.}\n\\label{singleTrajectory}\n\\end{figure}\n\nWith fixed initial conditions (of kind (a) or (b)) and for higher values of $\\kappa$ one can expect, in general, that the initial conditions may become increasingly less relevant. To this end, we find that the particles get focused and accumulate like a beam of charged particles (the beam is quite narrow for some configurations and higher $\\kappa$) along some specific region of space and move asymptotically for higher simulation times. This is exemplified below with two $j{=}0$ configurations: the $(0,0,1)_I$ configuration in Figure \\ref{j001plots}, and the Hopf--Ra\u00f1ada configuration in Figure \\ref{HRplots}. We have verified this feature not just with symmetric initial conditions of particles like that with initial conditions (a) and (b) (as in Figure \\ref{j001plots}), but also in several initial conditions asymmetric with respect to the origin, like particles located randomly inside a sphere of fixed radius about the origin with zero initial velocity, and particles located at the origin but with different magnitudes of velocities. Figure \\ref{HRplots} is an illustrative example for both of these latter scenarios of asymmetric initial conditions.\n\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{PosSymm_0,0,1_100k.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{VelSymm_0,0,1_100k.pdf}\n \\caption{\n Simulation of trajectories of $18$ identical charged particles for $(j;m,n)=(0,0,1)_I$ field configuration with $\\kappa {=} 100$ for $t\\in[0,1]$. Left: particles starting from rest and located symmetrically on a sphere of radius $|{\\bf X}_0|{=}R_{max}(0)$. Right: particles located at origin and directed symmetrically in space (shown with colored arrows) with $|{\\bf V}_0|=0.75$.}\n\\label{j001plots}\n\\end{figure}\n\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{HRposInhomo1000k.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{HRvelRandom1000k.pdf}\n \\caption{\n Simulation of trajectories of $20$ identical charged particles for Hopf--Ran\\~{a}da field configuration with $\\kappa {=} 1000$ for $t\\in[0,1]$. Left: particles starting from rest and located randomly inside a solid ball of radius $r{=}0.01$. Right: particles located at origin and directed randomly (shown with colored arrows) with $|{\\bf V}_0|{=}0.45$.}\n\\label{HRplots}\n\\end{figure}\n\nThis is not always the case though. For some $j{=}\\tfrac12$ and $j{=}1$ configurations, and with initial particle positions in a sphere of very small radius about the origin, we are able to observe the splitting of particle trajectories (starting in some specific solid angle regions around the origin) into two, three or even four such asymptotic beams that converge along some particular regions of space (depending on the initial location of these particles in one of these solid angle regions). Trajectories generated by two such $j{=}1$ configurations have been illustrated in Figure \\ref{3-4furcation}. \n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{Trifurcation_1,-1,-2.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{Quadrifurcation_1,-1,-1.pdf}\n \\caption{\n Simulation of trajectories of $18$ identical charged particles located symmetrically on a sphere with $|{\\bf X}_0|=0.01$ and ${\\bf V}_0=0$ for $t \\in [0,3]$. Left: $(j;m,n)=(1,-1,-2)_R$ configuration with $\\kappa {=} 500$, Right: $(j;m,n)=(1,-1,-1)_R$ configuration with $\\kappa {=} 10$.}\n\\label{3-4furcation}\n\\end{figure}\n\nNaturally, there are also regions of unstable trajectories for particles starting between these solid angle regions (see Figure \\ref{Bifurcation}), which generally include the preferred $z$-axis, since in some cases trajectories that start at rest in the $z$-axis never leave it.\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{Bifurcation_h,-h,-3h.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{Bifurcation_1,0,-2.pdf}\n \\caption{\n Simulation of trajectories of $18$ identical charged particles located symmetrically on a sphere with $|{\\bf X}_0|=0.01$, ${\\bf V}_0=0$ and $\\kappa {=} 10$ for $t \\in [0,3]$. Left: $(j;m,n)=(\\sfrac12;-\\sfrac12,-\\sfrac32)_R$ configuration, Right: $(j;m,n)=(1,0,-2)_I$ configuration.}\n\\label{Bifurcation}\n\\end{figure}\n\nFor particles with initial condition (a), we find that their trajectories are symmetric as far as time reversal around $T{=}0$ is concerned. We illustrate this feature with Figure \\ref{1Dpos} where particles with initial position along a line has been considered for better visualization. Of course, choosing a different initial condition would violate this bevaviour as is clear from Figure \\ref{singleTrajectory}. \n\nWe employ the paramter $R_{max}$ \\eqref{Rmax} in the the following Figures \\ref{1Dpos}, \\ref{1Dvel}, \\ref{2Dvel}, \\ref{3Dvel}, \\ref{2Dpos}, and \\ref{3Dpos} for both kinds of initial conditions viz. (a) and (b) (it is especially relevant for the former) to understand the effect of field intensity on particle trajectories.\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{PosLine_h,h,h_10kZ.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{PosLine_h,h,h_10kXY.pdf}\n \\caption{\n Trajectories of $11$ identical charged particles with ${\\bf V}_0=0$ and ${\\bf X}_0 \\propto \\pm\\sfrac14 R_{max}(0)$ (including one at the origin) for $(j;m,n)=(\\sfrac12;\\sfrac12,\\sfrac12)_R$ configuration with $\\kappa {=} 10$ simulated for $t \\in [-1,1]$.\n Left: particles located along $z$-axis (blue line), Right: particles located along some (blue) line in $xy$-plane.}\n\\label{1Dpos}\n\\end{figure}\n\nOne very interesting feature of trajectories for some of these field configurations is that they twist and turn in a coherent fashion owing to the symmetry of the background field. For particles with initial condition of kind (b), we see that their trajectories take sharp turns, up to two times, with mild twists before going off asymptotically. This has to do with the presence of strong background electromagnetic fields with knotted field lines. This is clearly demonstrated below in Figures \\ref{1Dvel}, \\ref{2Dvel}, and \\ref{3Dvel}. It is worthwhile to notice in Figure \\ref{1Dvel} that the particle which was initially at rest moves unperturbed along the $z$-axis; again, this has to do with the fact that these fields have preferred $z$-direction.\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{VelLine_1,0,0_1tf.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{VelLine_1,0,0_3tf.pdf}\n \\caption{\n Trajectories of $11$ identical charged point particles (shown with colored arrows) with ${\\bf X}_0=0$ and ${\\bf V}_0 \\propto \\pm\\sfrac14 R_{max}(0,1,0)$ (including one at rest) for $(j;m,n)=(1,0,0)_I$ configuration with $\\kappa {=} 10$.\n Left: particles simulated for $t \\in [0,1]$, Right: particles simulated for $t \\in [0,3]$.}\n\\label{1Dvel}\n\\end{figure}\n\nThis feature is even more pronounced in Figure \\ref{2Dvel} and (the right subfigure of) Figure \\ref{3Dvel} where we see that particles with ultrarelativistic initial speeds are forced to turn (almost vertically upwards) due to the strong electromagnetic field. These particles later take very interesting twists in a coherent manner. This twisting feature is much more refined for the case where initial particle velocities were directed along the $xy$-plane. Here also, we can safely attribute this behavior of the particle trajectories to the special field configurations, with preferred $z$-direction, that we are working with.\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{VelCirc_1,-1,1_Ynormal.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{VelCirc_1,-1,1_Znormal.pdf}\n \\caption{\n Trajectories of $10$ identical charged particles with ${\\bf X}_0=0$ and $|{\\bf V}_0|=0.99$ for $(j;m,n)=(1,-1,1)_R$ configuration with $\\kappa {=} 100$ simulated for $t \\in [0,3]$.\n Left: particles directed symmetrically along the $xz$-plane at $T{=}0$ (shown with colored arrows), Right: particles directed symmetrically along the $xy$-plane at $T{=}0$ (shown with colored arrows).}\n\\label{2Dvel}\n\\end{figure}\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 5cm, height = 7cm]{VelSphere_1,0,0_10k.pdf}\\hspace{2cm}\n   \\includegraphics[width = 7cm, height = 6.5cm]{VelSphere_1,0,0_100k.pdf}\n \\caption{\n Trajectories of $10$ identical charged particles directed symmetrically away from the origin at $T{=}0$ (shown with colored arrows) with ${\\bf X}_0=0$ for $(j;m,n)=(1,0,0)_I$ configuration simulated for $t \\in [0,2]$.\n Left: $|{\\bf V}_0|=R_{max}$ and $\\kappa {=} 10$, Right: $|{\\bf V}_0|=0.99$ and $\\kappa {=} 100$.}\n\\label{3Dvel}\n\\end{figure}\n\n We see in Figure \\ref{2Dpos} that the trajectories of particles that were initially located on a circle whose normal is along the $z$-axis flow quite smoothly along the $z$-axis with mild twists for some time before they all turn symmetrically in a coherent way and went off asymptotically. Comparing this with the other case in Figure \\ref{2Dpos}, where particles split into two asymptotic beams, we realize that this is yet another instance of the preferred choice of direction for the electromagnetic fields influencing the trajectories of particles.\n\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 7cm, height = 5cm]{PosCirc_1,0,0_Xnormal.pdf}\\hspace{1cm}\n   \\includegraphics[width = 7cm, height = 5cm]{PosCirc_1,0,0_Znormal.pdf}\n \\caption{\n Trajectories of $10$ identical charged particles located on a circle with $|{\\bf X}_0|=R_{max}$ and ${\\bf V}_0=0$ for $(j;m,n)=(1,0,0)_I$ configuration with $\\kappa {=} 10$ simulated for $t \\in [0,2]$.\n Left: the normal of the circle is along the $x$-axis, Right: the normal of the circle is along the $z$-axis.}\n\\label{2Dpos}\n\\end{figure}\n\nIn Figures \\ref{3Dpos} and \\ref{2Dvel} we find examples of kind (a) and (b) respectively where both twisting as well as turning of trajectories is prominent. We see in Figure \\ref{3Dpos} that the particles that start very close to the origin takes a longer time to show twists as compared to the ones that start off on a sphere of radius $R_{max}$. This is due to the fact that the field is maximal at $R_{max}$ and hence its effect on particles is prominent, as discussed before. We also notice here that the particles sitting along the $z$-axis at $T{=}0$ (either on the north pole or on the south pole of this sphere) keep moving along the $z$-axis without any twists or turns. This exemplifies the fact that these background fields have a preferred direction.\n\\begin{figure}[H]\n\\captionsetup{width=\\linewidth}\n\\centering\n   \\includegraphics[width = 3cm, height = 7cm]{PosSphere_1,-1,1_10k.pdf}\\hspace{2cm}\n   \\includegraphics[width = 8cm, height = 7cm]{PosSphere_1,-1,1_100k.pdf}\n \\caption{\n Trajectories of $18$ identical charged particles for $(j;m,n)=(1,-1,1)_R$ configuration with ${\\bf V}_0=0$ simulated for $t \\in [0,1]$.\n Left: particles located symmetrically on a sphere with $|{\\bf X}_0|=0.01$ and $\\kappa {=} 10$, Right: particles located symmetrically on a sphere with $|{\\bf X}_0|=R_{max}$ and $\\kappa {=} 100$.}\n\\label{3Dpos}\n\\end{figure}\n\n\nFor field configurations with $j {=} \\tfrac32$ and $2$ we found in some simulations (with the same parameters i.e. $\\kappa$, initial conditions, and simulation time) that their effect on particle trajectories is much less prominent as compared to the $j\\leq 1$ field configurations. This indicates that the field intensity may decrease quickly for higher $j$ configurations. This prompts the speculation that, depending on the desired degree of precision, for practical purposes it may be enough to expand any finite energy field configuration in terms of up to $j{=}1$ modes of these basis configurations.\n\n\n\n\\chapter{Introduction} \n\\label{Chapter1} \n\\def\\={\\ =\\ }\n\\justifying\n\nThe $4$-dimensional de Sitter space $\\mathrm{d}{S}_4$ plays an important role in gravity. It is one of the three (topological) types\\footnote{There are $3$ types of FLRW spacetime viz. Minkowski space ($\\kappa{=}0$), de Sitter space ($\\kappa{=}1$), and Anti-de Sitter space ($\\kappa{=}-1$) according to the global topology of the backgroud $3$-space (labelled by $\\kappa$).} of the Friedmann--Lema\u00eetre--Robertson--Walker (FLRW) spacetime that can model a homogeneous and isotropic universe like ours (at distance scales of 100 Mpsec). This has a positive global scalar curvature of the underlying $3$-space that is consistent with the observed positive cosmological constant $\\Lambda$ aka {\\it dark energy} that is fueling the accelerated expansion of our universe. It is believed that our universe is asymptotically de Sitter which means in the future, when the dark energy dominates, our universe would become de Sitter.\n\nGauge theory, in particular Yang--Mills theory, is of central importance in the classical description of fundamental forces of nature like electromagnetism, weak and strong nuclear forces. Classical Yang--Mills theory also has applications in other physics areas such as QCD confinement of high energy physics and spin-orbit interaction in condensed matter physics. Even gravity can be understood as a gauge theory. It is, therefore, natural to seek solutions of Yang--Mills theory on four-dimensional de Sitter space. Furthermore, owing to the conformal relation of $\\mathrm{d}{S}_4$ with Minkowski space $\\mathds R^{1,3}$, it turns out that even Abelian Yang--Mills theory aka electromagnetism studied on the former has nice application since the solutions can be pulled back to Minkowski space (of our laboratory) owing to the conformal invariance of the Yang--Mills theory in $4$-dimensions.\n\n\\vspace{12pt}\n\\section{Electromagnetic knots}\n\\vspace{2pt}\nTheoretical discovery of electromagnetic knots dates back to 1989 when Ra\u00f1ada \\cite{Ranada89} constructed them using the Hopf map. These finite-energy finite-action vacuum solutions of Maxwell's equations are constructed from a pair of complex scalar fields $\\phi$ and $\\theta$ on the $4$-dimensional spacetime where the $3$-space is compatified to $S^3$ with the addition of a point at infinity. These solutions are thus characterised by a topological quantity called the Hopf index of the following Hopf map ($A=1,2,3,4$):\n\\begin{equation}\n    h:\\ S^3 \\to S^2\\ ,\\quad \\{ \\omega_{_A} \\} \\mapsto \\{x_1,x_2,x_3\\}\n\\end{equation}\nwhere $S^2$, arising from the compactification of the complex plane $\\mathds{C}$, has coordinates $x_i$, satisfying $x_1^2 + x_2^2 + x_3^2 = 1$, that are constructed from the $S^3$ coordinates $\\omega_{_A}$, satisfying $\\omega_1^2 + \\omega_2^2 + \\omega_3^2 + \\omega_4^2 = 1$, as follows\n\\begin{equation}\n    x_1\\ :=\\ 2(\\omega_1\\omega_2 + \\omega_3\\omega_4)\\ ,\\ x_2\\ :=\\ 2(\\omega_1\\omega_4 - \\omega_2\\omega_3)\\ ,\\ x_3\\ :=\\ (\\omega_1^2+\\omega_3^2) - (\\omega_2^2 + \\omega_4^2)\\ .\n\\end{equation}\nThe level curves of these complex-valued functions $\\theta$ and $\\phi$ are identified with electric and magnetic field lines. Several other approaches to construct such electromagnetic knots have been developed since then such as  Bateman's complex Euler potentials, conformal inversion and Penrose twistors (see \\cite{ABT17} for a full review).\n\nApart from these there exists another way of constructing these knotted electromagnetic fields via the conformal correspondence between de Sitter space $\\mathrm{d}{S}_4$ and Minkowski space $\\mathds R^{1,3}$, while passing through a finite Lorentzian $S^3$-cylinder, as was shown in \\cite{LZ17}. In this method, one obtains a complete family\\footnote{In the sense that any given finite-energy rational Maxwell solution can be expanded in terms of these basis configurations.} of such electromagnetic knotted configurations that are labelled with hyperspherical harmonics $Y_{j;m,n}$ of the $3$-sphere. The de Sitter space enjoys a larger symmetry group in $SO(1,4)$ whose subgroup $SO(4)\\cong (SU(2)\\times SU(2))\/\\mathds{Z}_2$ is made use of in this construction by working in the $S^3$-cylinder. Here one employs the right-action of $SU(2)$, which is the group manifold of $SO(4)$, on the $3$-sphere to write down the gauge field in terms of the left-invariant one-forms of $S^3$. The resulting Maxwell's equation can be solved analytically and are then pulled back to the Minkowski space using the conformal map. This ``de Sitter\" method of construction has advantage over the others because of the $SO(4)$ covariant treatment of Maxwell theory. \n\nKnotted electromagnetic fields might become important for future applications because of their unique topological properties. It is, therefore, important to seek experimental settings to generate those fields and to study scenarios with them. Irvine and Bouwmeester \\cite{Irvine08} discuss the generation of knotted fields using Laguerre--Gaussian beams and predict potential applications in atomic particle trapping, the manipulation of cold atomic ensembles, helicity injection for plasma confinement, and in the generation of soliton-like solutions in a nonlinear medium. Moreover, laser beams with knotted polarization singularities were recently employed to produce some simple knotted field configurations, including the one with figure-$8$ topology in the lab \\cite{LSMetal18}.\n\n\\vspace{12pt}\n\\section{Cosmic $SU(2)$ Yang--Mills fields}\n\\vspace{2pt}\n\nFinding analytic solution to the Einstein--Yang--Mills system of equations arising from the following action\n\\begin{equation}\n    S \\= S_{YM} + S_{EH} \\= \\int_M \\mathrm{d}^4x \\sqrt{-g}\\,R + \\int_M \\mathrm{Tr}(F{\\wedge} *F)\n\\end{equation}\nis not possible in general. There is, however, one scenario where such a solution can be obtained and that is FLRW cosmology. Here the Yang--Mills equation decouple from Einstein equation due to the conformal invariance of the former in $4$-dimensional spacetime $M$. This means that given a solution of Yang--Mills equation one of these FLRW spacetime, the corresponding scale factor can be obtained via the Friedmann equation. Such a solution with finite energy and action do exist for the de Sitter case together with the $SU(2)$ gauge group \\cite{AFF76,Luescher77,Schechter77}. \n\nA {\\it crucial} ingredient of the Standard Model of cosmology called inflation can also be tackled with homogeneous and isotropic non-Abelian Yang--Mills fields in the Minkowski type (spatially flat) FLRW background in theories of gauge-flation or chromo-natural inflation (see \\cite{MSS13} for a review). Another more minimalistic approach towards tackling this issue was recently put forth by Daniel Friedan \\cite{Friedan}. He considers a coupled Einstein--Yang--Mills--Higgs system where a rapidly oscillating isotropic $SU(2)$ gauge field stabilizes the symmetric Higgs vacuum in a de Sitter type (spatially closed) FLRW spacetime. \n\nBased on these, it is only natural to analyze the stability behaviour of such ``cosmic Yang--Mills fields\" under generic linear perturbation of the Yang--Mills field equation. For the gauge-flation scenario such an analysis has been done before, but for the later scenario there only exits result for spin-$j{=}0$ case of these $SU(2)$ gauge fields \\cite{Hosotani}.\n\nIn light of this, we present here a complete stability analysis of these $SU(2)$ solutions in a closed FLRW universe. This analysis is in contrast with that of the guage-flation one where conformal invariance is broken; our homogeneous and isotropic gauge field would give rise to inhomogeneous Yang--Mills fields on flat FLRW spacetime. For the sake of simplicity we keep the background metric fixed in this perturbation analysis. While this does mean that our analysis is still partial, we can argue for the relevance of our analysis on at least two grounds:\n\\begin{enumerate}[(a)]\n    \\item Because of the decoupling of the gauge fields with the metric in $4$-dimensional spacetime, the fluctuation of the latter does not affect the gauge field.\n    \\item The fluctuation of the gauge fields is too rapid for the evolution of the background metric and thus any non-conformal metric fluctuation would not have much of an effect on the gauge fields.\n\\end{enumerate}\n\nThe only known family of finite-energy $SU(2)$ Yang-Mills field configuration on FLRW spacetime are obtained, in an efficient manner, by employing the following conformal correspondence between de Sitter space and the cylinder ${\\cal I}\\times S^3$ for ${\\cal I} := (0,\\pi)$. This conformal map arises via a temporal reparametrization\nand Weyl rescaling~\\cite{Hosotani,Volkov,ILP17prl,Friedan},\n\\begin{equation} \\label{deSittermetric}\n\\begin{aligned}\n   \\mathrm{d} s_{\\textrm{dS}_4}^2 &\\= -\\mathrm{d} t^2 + \\ell^2\\cosh^2\\!\\sfrac{t}{\\ell}\\,\\mathrm{d}\\Omega_3^2\n   \\= \\sfrac{\\ell^2}{\\sin^2\\!\\tau} \\bigl(-\\mathrm{d}\\tau^2 + \\mathrm{d}\\Omega_3^2\\bigr) \\\\\n   &\\qquad \\quad\\textrm{for}\\quad t\\in(-\\infty,+\\infty)\\ \\Leftrightarrow\\ \\tau \\in (0,\\pi)\\ ,\n\\end{aligned}\n\\end{equation}\nwhere $\\mathrm{d}\\Omega_3^{\\ 2}$ is the round metric on $S^3$, and $\\ell$ is the de Sitter radius. We observe that the relation between the conformal time $\\tau$ and co-moving time~$t$ in \\eqref{deSittermetric} fixes the cosmological constant to~$\\Lambda=3\/\\ell^2$. At this point, we can employ an $S^3$-symmetric ansatz for the gauge field by noting that $SU(2)$ is the group manifold of $S^3$. This yields an ODE for some scalar function $\\psi(\\tau)$ parametrized by the conformal time, that is nothing but a Newton's equation for a classical point particle under the influence of a double-well potential\n\\begin{equation} \\label{Vpot}\nV(\\psi) \\= \\sfrac12 (\\psi^2-1)^2\\ .\n\\end{equation}\nThe solution for these anharmonic oscillators are well known in terms of Jacobi elliptic functions that depend on time. Although these solutions exists for all three FLRW metrics, only spatially closed one admits an isotropic solution under the above conformal transformation. For de Sitter type FLRW spacetime we have\n\\begin{equation} \\label{FLRW}\n\\begin{aligned}\n  \\mathrm{d} s^2 &\\= -\\mathrm{d} t^2 + a(t)^2\\,\\mathrm{d}\\Omega_3^{\\ 2}\n  \\= a(\\tau)^2\\bigl(-\\mathrm{d}\\tau^2 + \\mathrm{d}\\Omega_3^{\\ 2}\\bigr) \\\\\n  &\\qquad\\quad\\textrm{for}\\quad t\\in(0,t_{\\textrm{max}})\\ \\Leftrightarrow\\ \\tau \\in {\\cal I} \\equiv (0,T')\\ ,\n\\end{aligned}\n\\end{equation}\nwhere we impose a big-bang initial condition~$a(0){=}0$, so that\n\\begin{equation} \n\\mathrm{d}\\tau \\= \\frac{\\mathrm{d} t}{a(t)} \\qquad\\textrm{with}\\qquad \n\\tau(t{=}0) = 0 \\quad\\quad\\textrm{and}\\quad\\quad \\tau(t{=}t_{\\textrm{max}}) =: T'<\\infty\\ .\n\\end{equation}\nThe lifetime $t_{\\textrm{max}}$ of the universe \ncan be infinite (big rip, $a(t_{\\textrm{max}}){=}\\infty$) \nor finite (big crunch, $a(t_{\\textrm{max}}){=}0$).\nMoreover, bouncing cosmologies as in (\\ref{deSittermetric}) are also allowed but will not be pursued here. \n\nWe notice that the contribution of these $SO(4)$-symmetric Yang--Mills fields, and their stress-energy tensor, in the one-way coupling with the background de Sitter FLRW metric via Friedmann equation (as discussed before) is such that it only modifies the scale factor $a(\\tau)$. It is well known that the equation of motion governing this scale factor arises as a Newton's equation with the following (cosmological) potential \n\\begin{equation} \\label{Wpot}\nW(a) \\= \\sfrac12 a^2 - \\sfrac{\\Lambda}{6} a^4\\ ,\n\\end{equation}\nwhich is another anharmonic oscillator (although inverted). \n\nThe pair of solutions $(\\psi,a)$ corresponding to \\eqref{Vpot} and \\eqref{Wpot} yields an exact classical Einstein--Yang--Mills configuration. The conserved mechanical energy $E$ for $\\psi$ fixes the same i.e. $\\tilde{E}$ for $a$ via the Wheeler--DeWitt constraint\n\\begin{equation}\n    E + \\epsilon \\tilde{E} \\= 0\\ ;\\quad E\\ :=\\ \\frac{1}{2}\\dot{\\psi}^2 + V(\\psi) \\quad\\textrm{and}\\quad \\tilde{E}\\ :=\\ \\frac{1}{2}\\dot{a}^2 + W(a)\\ ,\n\\end{equation}\nwhere the overdot denotes a derivative with respect to conformal time and $\\epsilon$ depends on coupling constants. \n\nOne can also introduce a complex scaler Higgs field $\\phi$ in the fundamental SU(2) representation to the Standard Model of cosmology with Higgs potential\n\\begin{equation}\nU(\\phi) \\= \\sfrac12\\,\\lambda^2\\,\\bigl(\\phi^\\+\\phi-\\sfrac12 v^2\\bigr)^2\\ ,\n\\end{equation}\nwhere $v\/\\sqrt{2}$ is the Higgs vev and $\\lambda v$ is the Higgs mass. It turns out that imposing an $SO(4)$-invariance makes the Higgs field $\\phi\\equiv0$, which provides us with a definite positive cosmological constant of\n\\begin{equation}\n\\Lambda \\= \\kappa\\,U(0) \\= \\sfrac18\\,\\kappa\\,\\lambda^2 v^4\\ ,\n\\end{equation}\nwhere $\\kappa$ is the gravitational coupling.\nThe full Einstein--Yang--Mills--Higgs action (in standard notation),\n\\begin{equation}\nS \\= \\int\\!\\mathrm{d}^4 x\\ \\sqrt{-g}\\ \\Bigl\\{ \\sfrac{1}{2\\kappa} R + \\sfrac{1}{8g^2} \\mathrm{tr} F_{\\mu\\nu}F^{\\mu\\nu}\n-D_\\mu\\phi^\\+ D^\\mu\\phi - U(\\phi) \\Bigr\\}\\ ,\n\\end{equation}\nreduces in the SO(4)-invariant sector to\n\\begin{equation}\nS[a,\\psi,\\Lambda] \\= 12\\pi^2 \\int_0^{T'}\\!\\!\\mathrm{d}\\tau\\ \\Bigl\\{\n\\sfrac{1}{\\kappa} \\bigl(-\\sfrac12\\dot{a}^2+W(a)\\bigr) + \n\\sfrac{1}{2g^2} \\bigl(\\sfrac12\\dot{\\psi}^2-V(\\psi)\\bigr) \\Bigr\\}\\ ,\n\\end{equation}\nwhere $g$ is the gauge coupling.\n\nIf the Yang--Mills energy $E$ is large enough it could propel an eternal expansion of the universe that is accompanied by rapid fluctuations of the gauge field. The coupling of this Yang--Mills field with the Higgs field stabilizes the symmetric vacuum $\\phi\\equiv0$ at the local maximum of~$U$ through a parametric resonance effect, as long as $a$ is not too large. Eventually, when $a$ exceeds a critical value of electro-weak symmetry breaking scale $a_{\\textrm{EW}}$, the Higgs field will begin to roll down towards a minimum of~$U$, thus breaking the SO(4) symmetry. The corresponding time $t_{\\textrm{EW}}$ signifies the electroweak phase transition in the early universe. This is a rather unconventional scenario put forward recently by Friedan~\\cite{Friedan}. \n\n\\vspace{12pt}\n\\section{Outline and summary of results}\n\\vspace{2pt}\n\nIn the next two chapters we review the mathematical preliminaries that builds up gauge theory. In chapter \\ref{Chapter2} we present a brief but thorough review of the mathematics of the background geometry such as manifold, fibre bundles, etc. and the symmetry in physics such as Lie groups, Lie algebra and their representations. Next in chapter \\ref{Chapter3} we review the construction of gauge theory via principal bundle formalism.\n\nIn chapter \\ref{Chapter4} we discuss the geometry of and calculus on the $3$-sphere apart from demonstrating the conformal equivalence of the $S^3$-cylinder with the full Minkowski space. We then present Yang--Mills equation for a $SO(4)$-asymmetric $SU(2)$ Yang--Mills theory on the $S^3$-cylinder and discuss its two limiting cases where analytic solutions can be obtained. \n\nWe present the construction of electromagnetic knotted field configurations in chapter \\ref{Chapter5} and study their symmetry feature and other properties. We analyze the effect of the de Sitter group $SO(1,4)$, i.e. the isometry group of $\\mathrm{d}{S}_4$, on these solutions. To that end, we demonstrate the emergence of the Poincare group $ISO(4)$ of $\\mathds R^{1,3}$ from $SO(1,4)$ in the limit $\\ell \\to \\infty$. We observe that only the subgroup $SO(3)$ is common in the two cases. \n\nWe then proceed, also in \\ref{Chapter5}, to compute all the Noether charges associated with the conformal group $SO(2,4)$ viz. energy, momentum, angular momentum, boost, dilatation and special conformal transformations (SCT) for a complex linear combination, in terms of $\\Lambda_{j,m,n}$, of these basis-knot configurations. These conserved charge densities are evaluated on de Sitter space at $\\tau{=}0$ where considerable simplifications occur demonstrating the usefulness of this ``de Sitter method\". We find that the dilatation vanishes while the scalar SCT charge $V_0$ is proportional to the energy $E$. Furthermore, the boosts $K_i$ vanish and the vector SCT charges $V_i$ are proportional to the momenta $P_i$. Interestingly, for the vector charge densities viz. momenta $p_i$, angular momenta $l_i$ and vector SCT $v_i$ we find that the one-form $p_i\\mathrm{d} x^i$ constructed on the spatial slice $\\mathds R^3 \\hookrightarrow \\mathds R^{1,3}$ is proportional to a similar one on de Sitter space. This correspondence allows us to compute additional charges $(p_r,p_\\theta,p_\\phi)$ by the action of such one-forms on spherical vector fields $(\\mbox{$\\partial$}_r,\\mbox{$\\partial$}_\\theta,\\mbox{$\\partial$}_\\phi)$. At $j{=}0$ it turns out that there are only four independent non-zero charges: the energy $E$ and the momenta $P_i$. The situation for higher spin $j$ is more complicated, but some of the components of the charges in spherical coordinates are found to vanish for arbitrary $j$. The action of the $so(3)$ generators $\\mathcal{D}_a$ on the indices of $\\Lambda_{j,m,n}$ can easily be obtained for a fixed $j$ owing to the $SO(4)$ isometry. This allows for an action of these generators on the charges. For (the Cartesian components of) the vector charges this action is found to inherit the original $so(3)$ Lie algebraic structure, as expected. We also compute the correct coefficients $\\Lambda_{0;0,n}$ corresponding to two interesting generalisations of the Hopfian solution obtained via Bateman's construction in \\cite{HSS15}, which allow us to validate our generic formulae of these charges. We also demonstrate the relationship of energy with the conserved helicity.\n\nFurthermore, in chapter \\ref{Chapter5} we characterize the moduli space of null solutions which turns out to be a complete-intersection projective complex variety of complex dimension $2j{+}1$. We also demonstrate how the energy flux is radiated to infinity with an energy profile that is concentrated along the lightcone situated at the origin (selected by these solutions). Finally, we study the trajectories of multiple identical charged particles in the background of these basis knot configurations. We employ several initial conditions for these charged particles and find interesting features of the trajectories like coherent twisting, ultrarelativistic acceleration of particles starting from rest and a quick convergence of their trajectories into a few narrow cones asymptotically for sufficiently high value of the coupling. \n\nIn chapter \\ref{Chapter6} we first review the classical configurations $(A_\\mu,g_{\\mu\\nu})$ \nin terms of Newtonian solutions $(\\psi,a)$ for the anharmonic oscillator pair~$(V,W)$. We then investigate arbitrary small perturbations of the gauge field departing from the time-dependent background~$A_\\mu$ parametrized by the ``gauge energy'' $E$. Later on we linearize the Yang--Mills equation around it and diagonalize the fluctuation operator to obtain a spectrum of time-dependent natural frequencies. To decide about the linear stability of the cosmic Yang--Mills configurations we have to analyze the long-time behavior of the solutions to Hill's equation for all these normal modes. To that end, we employ Floquet theory to learn that their growth rate is determined \nby the stroboscopic map or monodromy, which is easily computed numerically for any given mode. \nWe do so for a number of low-frequency normal modes and find, when varying~$E$, an alternating sequence of stable (bounded) and unstable (exponentially growing) fluctuations. The unstable bands roughly correspond to the parametric resonance frequencies.\nWith growing ``gauge energy'' the runaway perturbation modes become more prominent,\nand some of them persist in the infinite-energy limit, where we detect universal natural frequencies and monodromies. A special role is played by the SO(4)-invariant fluctuation of $j{=}0$ mode, \nwhich merely shifts the parameter~$E$ of the background. We treat it exactly and beyond the linear regime. This ``singlet'' mode turns out to be marginally stable, i.e. it has a vanishing Lyapunov exponent. Its linear growth, however, gets limited by nonlinear effects of the full fluctuation equation, whose analytic solutions exhibit wave beat behavior.\n\nFinally we present a brief summary of our work in chapter \\ref{Chapter7} and present the future outlook. We also collect several relevant data in various appendices and present a direct map between the cylinder and Minkowski space via Carter--Penrose transformation avoiding the de Sitter detour in appendix \\ref{appendCPtransf}.\n\n\n\n\\chapter{Geometry \\& Symmetry}\n\\label{Chapter2}\n\n\n\\newcommand{\\ex}[1]{{\\bf Example #1}}\n\\newcommand{\\defn}[1]{\\noindent{\\bf Definition #1}}\n\\newcommand{\\vecF}[1]{\\mathfrak{X}(#1)}\n\\newcommand{\\rem}[1]{\\noindent{\\bf Remark #1}}\n\\newcommand{\\thm}[1]{\\noindent{\\bf Theorem #1}}\n\n\\justifying\nTwo of the most important concepts that play a fundamental role in physics are the geometry of background space and the presence of symmetries. They have well understood mathematical foundation in differential geometry and Lie groups\/algebras respectively. Here we present a short exposition of these mathematical topics that is essential towards building the subsequent Yang--Mills theory. The following contents are built upon some basic mathematical structures like vector spaces, groups and topological spaces all of which can be found in \\cite{Nakahara}. We refrain from presenting proofs of the statements in this chapter and suggest the references \\cite{Baez&Muniain} and \\cite{Bleecker} which are the source for most of the contents in this chapter. For details on Lie groups\/algebras and their representations we refer to classic texts \\cite{wybourne} and \\cite{Hall}.\n\n\n\\vspace{12pt}\n\\section{Manifolds}\n\\vspace{2pt}\n\\rem{2.1.1} The idea of a Manifold generalizes the notion of differentiation, or more precisely calculus on $\\mathds R^n$, in the same way as a Topological space allows one to study the notion of continuity in a more abstract way. In this thesis, we will only be concerned with (finite-dimensional) smooth manifolds, and subsequently, smooth structures on it.\n\n\\defn{2.1.1} An n-dimensional {\\it manifold} $M$ is a topological space\\footnote{Some required technicalities like paracompactness and Haussdorffness have been assumed.} equipped with an atlas consisting of charts $(U_i,\\phi_i)$ such that\n\\begin{itemize}\n    \\item $U_i$ are open sets and the maps $\\phi_i$ are homeomorphism between $U_i$ and some open ball in $\\mathds R^n$, and\n    \\item the {\\it transition maps} $\\phi_2\\circ \\phi_1^{-1}:\\ \\phi_1(U_1\\cap U_2) \\rightarrow \\phi_2(U_1\\cap U_2)$ is infinitely differentiable for any two charts $(U_1,\\phi_1)$ and $(U_2,\\phi_2)$.\n\\end{itemize}\n\n\\ex{2.1.1} A nice example of a smooth manifold is the round $n$-sphere\n\\begin{equation*}\n    S^n := \\lbrace {\\bf x} \\in \\mathds R^{n+1} ~|~ {\\bf x\\cdot x} = 1 \\rbrace\n\\end{equation*}\nembedded in $\\mathds R^{n+1}$ that requires two charts: $U_N = S^n - \\{(0,0,\\ldots,1)\\}$ and $U_S := S^n - \\{(0,0,\\ldots,-1)\\}$, along-with their respective stereographic projections:\n\\begin{equation*}\n \\begin{aligned}\n    \\phi_N(\\bf{x})\\ &:=\\ \\left( \\sfrac{x_1}{1-x^{n+1}},\\sfrac{x_2}{1-x_{n+1}},\\ldots,\\sfrac{x_n}{1-x_{n+1}} \\right)\\quad\\textrm{and}\\quad \\\\\n    \\phi_S(\\bf{x})\\ &:=\\ \\left( \\sfrac{x_1}{1+x_{n+1}},\\sfrac{x_2}{1+x_{n+1}},\\ldots,\\sfrac{x_n}{1+x_{n+1}} \\right)\n \\end{aligned}\n\\end{equation*}\nwith $x_{n+1} = 1 - x_1^2 - x_2^2 - \\ldots - x_n^2$.\n\n\\defn{2.1.2} Given a manifold $M$, a map $f: M \\rightarrow \\mathds R$ is called {\\it smooth} or $C^\\infty$ if it is infinitely differentiable. The set of all such smooth maps are denoted as $C^\\infty(M)$.\n\n\\defn{2.1.3} {\\it Diffeomorphism} $f:M \\rightarrow N$ between any two manifolds $M$ and $N$ is a bijection such that both $f$ and $f^{-1}$ are smooth. The set of all diffeomorphisms from $M$ to itself forms a group called $\\mathrm{Diff}(M)$.\n\n\\rem{2.1.2} Notice here that the differentiability of such a map $f$ is decided using local charts: e.g. given a chart $(U,\\phi)$ in $M$ containing $p\\in U$ and another chart $(V,\\psi)$ in $N$ containing $f(p)\\in V$, one can differentiate the map $\\psi\\circ f\\circ\\phi^{\\text -1}$ using standard calculus. A crucial notion in differential geometry is that of a tangent vector. It can intrinsically be defined in terms of a {\\it curve} on a manifold.\n\n\\defn{2.1.4} A {\\it curve} $\\sigma :\\ (-\\epsilon,\\epsilon) \\subset \\mathds R \\to M$ on a manifold $M$ is defined as a smooth map from some open interval of the real line to $M$.\n\n\\defn{2.1.5} The {\\it pull-back of a map $f$} by another map $\\phi$ is defined as\n\\begin{equation*}\n    \\phi^*f\\ :=\\ f\\circ\\phi\\ .\n\\end{equation*}\n\n\\defn{2.1.6} A tangent to a curve $\\sigma$ at a point $p=\\sigma(0)$ on a manifold $M$ is defined as the map\n\\begin{equation*}\n    \\sigma'(t):\\ C^\\infty(M) \\rightarrow \\mathds R\\ ,\\quad \\sigma'(t{=}0)[f]\\ :=\\ \\sfrac{\\mbox{$\\partial$}}{\\mbox{$\\partial$} t}(\\sigma^*f)\\big|_{t=0}\n\\end{equation*}\nThis idea can be generalized as follows.\n\n\\defn{2.1.7} A {\\it tangent vector} at a point $m\\in M$ is a map $v_m:\\ C^\\infty(M) \\rightarrow \\mathds R$ that satisfies the following properties \\begin{itemize}\n    \\item $v_m[f+g] \\= v_m[f] + v_m[g]\\quad  \\forall\\quad f,g \\in C^\\infty(M)$\\ ,\n    \\item $v_m[a\\,f] \\= a\\,v_m[f]\\quad \\forall\\quad a\\in \\mathds R\\ \\&\\ f\\in C^\\infty(M)$\\ , and\n    \\item $v_m[f\\,g] \\= f(p)\\,v_m[g] + g(p)\\,v_m[f] \\quad\\forall\\quad f,g\\in C^\\infty(M)$.\n\\end{itemize}\n\n\\defn{2.1.8} The set of all such tangent vectors at $m\\in M$ is known as the {\\it tangent space} at $m$ and is denoted by $T_mM$, which forms a vector space over $\\mathds R$. \n\n\\rem{2.1.3} This vector space is spanned by vectors $\\big\\{\\sfrac{\\scriptsize\\mbox{$\\partial$}}{\\scriptsize\\mbox{$\\partial$} x^i}\\big|_m \\equiv \\mbox{$\\partial$}_i|_m \\big\\}$ where $x^i$ are the coordinate functions in some chart $(U, \\phi)$ containing $m$ i.e. $\\phi(m) = (x^1,x^2,\\ldots,x^n)$ and whose actions is defined by \n\\begin{equation*}\n    \\sfrac{\\scriptsize\\mbox{$\\partial$}}{\\scriptsize\\mbox{$\\partial$} x^i}\\big|_m[f] \\ :=\\ \\sfrac{\\scriptsize\\mbox{$\\partial$} \\phi^* f}{\\scriptsize\\mbox{$\\partial$} x^i}\\quad \\forall \\quad f\\in C^\\infty(N)\\ .\n\\end{equation*}\nThe dimension of $T_mM$ is, therefore, equal to the dimension of the manifold $M$.\n\n\\defn{2.1.9} Given a map $\\varphi: M \\rightarrow N$ between two manifolds $M$ and $N$ and a vector $v\\in T_mM$, the {\\it push-forward} of $v$ by $\\varphi$ is defined by\n\\begin{equation*}\n    \\varphi_*v[f]\\ :=\\ v[\\phi^*f]\\quad \\forall \\quad f\\in C^\\infty(N)\\ .\n\\end{equation*}\n\n\\rem{2.1.4} Note that the push-forward induces a map between following tangent spaces:\n\\begin{equation*}\n    \\phi_*:\\ T_mM \\rightarrow T_{h(m)}N\n\\end{equation*}\nfor manifolds $M$ and $N$.\n\n\\defn{2.1.10} The vector space dual to $T_mM$ is called {\\it cotangent space} and is denoted by $T^*_mM$.\n\n\\rem{2.1.5} The vector space $T^*_mM$ is spanned by covectors $\\mathrm{d}{x}^i|_m$ defined as \n\\begin{equation*}\n    \\langle \\mathrm{d}{x}^i|_m\\, ,\\, v\\rangle\\ :=\\ v[x^i]\\quad \\forall \\quad v \\in T_mM\n\\end{equation*}\nand has the same dimension as $T_mM$.\n\n\\defn{2.1.11} The {\\it pull-back} of a covector $\\omega \\in T^*_nN$ by a map $\\varphi: M \\rightarrow N$ between two manifolds $M$ and $N$ is defined by\n\\begin{equation*}\n    \\langle\\varphi^*\\omega\\, ,\\, v \\rangle\\ :=\\ \\langle \\omega\\, , \\, \\varphi_*v \\rangle\n\\end{equation*}\nwith $n = \\varphi(m)$.\n\n\\vspace{12pt}\n\\section{Fibre bundles}\n\\vspace{2pt}\n\\rem{2.2.1} The theory of bundles has proved to be the correct way of studying classical gauge theories like general relativity and Yang-Mills theory. Here we shall confine ourselves to bundles constructed on\/with manifolds.\n\n\\defn{2.2.1} A {\\it bundle} is a triple $(E,M,\\pi)$ consisting of a manifold $E$, {\\it aka} the target space, a manifold $M$, {\\it aka} the base space, and a continuous surjection $\\pi: E \\rightarrow M$, {\\it aka} the projection map. It is denoted diagrammatically as \n\\begin{tikzcd}\n   &E \\arrow[d, \"\\pi\"] \\\\\n   &M\n\\end{tikzcd}\n\n\\defn{2.2.2} A bundle $(E,M,\\pi)$ is called a {\\it fibre bundle} with typical fibre $F$ if the inverse image of all $p \\in M$ under $\\pi$ is isomorphic to some space $F$ i.e. $\\pi^{-1}(p) \\cong F$. If $F$ is a vector space then the bundle $(E,M,\\pi)$ becomes a {\\it vector bundle}.\n\n\\defn{2.2.3} A pair of fibre bundles $(E,M,\\pi)$ and $(\\widetilde{E},\\widetilde{M},\\widetilde{\\pi})$ are called {\\it isomorphic} (as bundles) if there exist a pair of diffeomorphisms $\\varphi: E \\rightarrow \\widetilde{E}$ and $\\psi: M \\rightarrow \\widetilde{M}$ such that $\\widetilde{\\pi}\\circ \\varphi = \\psi\\circ\\pi$ and $\\pi\\circ\\varphi^{-1} = \\psi^{-1}\\circ\\widetilde{\\pi}$. Diagrammatically this means that the following diagram, and its inverse, commutes \n\\begin{tikzcd}\n   E \\arrow[r, \"\\varphi\"] \\arrow[d, \"\\pi\"] &\\widetilde{E} \\arrow[d, \"\\widetilde{\\pi}\"] \\\\\n   M \\arrow[r, \"\\psi\"] &\\widetilde{M}\n\\end{tikzcd}\n\n\\defn{2.2.4} A vector bundle $(E,M,\\pi)$ with typical fibre $F$ is called {\\it locally trivial} if for any $U \\subset M$ the induced bundle $(\\pi^{-1}(U),U,\\pi|_{\\pi^{-1}(U)})$ is isomorphic to the product bundle $(U\\times F,U,\\pi_1)$, where $\\pi_1$ is the projection in the first slot. The set $(U,\\varphi)$\\footnote{Notice, here, that $\\psi=Id_U$.} is known as a {\\it local trivialization} of the vector bundle. A {\\it trivial} bundle is one where $E=M\\times F$ and $\\pi = \\pi_1$.\n\n\\rem{2.2.2} We shall only deal with locally trivial bundles here. A vector bundle $(E,M,\\pi)$ will, sometimes, be simply denoted $E$. A classic example of a bundle that is not globally trivial is the following.\n\n\\ex{2.2.1} A M\\\"{o}bius strip $(E,S^1,\\pi)$ with fibre $[0,1]$ is locally isomorphic\\footnote{Meaning that they share local trivialization $(U,\\varphi)$ for any $x\\in U$.} to the trivial bundle $(S^1\\times [0,1],S^1,\\pi_1)$. The former, however, is not trivial as transporting any vector across a loop yields the corresponding vector inverted.\n\n\\defn{2.2.5} Given a fibre bundle $(E,M,\\pi)$ with typical fibre $F$ and a pair of local trivializations $(U_i,\\varphi_i)$ and $(U_j,\\varphi_j)$ with $U_i\\cup U_j \\neq 0$ on it, we obtain {\\it transition functions} $g_{ij}(x)$ with $x\\in U_i\\cap U_j$ of the vector bundle $U_i\\cap U_j\\times F$ by realizing that $\\varphi_j\\circ \\varphi_i(x,v) = (x,g_{ij}(x)v)$ for any vector $v\\in F$. The set of such transition functions $g_{ij}(x)$ forms a group $G \\subset End(F)$ called the {\\it structure group}.\n\n\\defn{2.2.6} A (local) {\\it section} $\\sigma$ (for some local trivialization $(U,\\varphi)$) of a fibre bundle $(E,M,\\pi)$ is a smooth map $\\sigma: (U) M \\rightarrow E$ such that $\\pi\\circ\\sigma = (Id_U) Id_M$. The set of all such (local) global sections are denoted $(\\Gamma^\\infty(U,E))$ $\\Gamma^\\infty(M,E)$. The global section is also denoted more simply as $\\Gamma^\\infty(E)$. \n\n\\vspace{12pt}\n\\section{Lie groups}\n\\vspace{2pt}\n\\rem{2.3.1} The notion of symmetry in modern physics is analyzed with the tools from the theory of Lie groups and Lie algebras. We will only consider finite, matrix Lie groups in what follows.\n\n\\defn{2.3.1} A {\\it Lie group} ({\\it aka} continuous group) $G$ is both a group and a smooth, finite-dimensional manifold where the group operation, $\\cdot: G\\times G \\rightarrow G$, as well as the inversion map, $^{-1}: G \\rightarrow G$ satisfying $g^{-1}\\cdot g = g\\cdot g^{-1}=e~\\forall~ g\\in G$ with identity element $e$, are smooth.\n\n\\rem{2.3.2} We will omit the group multiplication symbol $\\cdot$ and use $\\mathds{1}$ interchangeably with $e$ for the matrix Lie groups (to be considered below) from now onward.\n\n\\defn{2.3.2} A Lie group homomorphism $\\varphi: G \\rightarrow H$ between Lie groups $G$ and $H$ is a smooth map that is also a group homomorphism. The map $\\varphi$ becomes an isomorphism if it is bijective and its inverse $\\varphi^{-1}$ is smooth; the Lie groups $G$ and $H$ then becomes isomorphic.\n\n\\defn{2.3.3} A {\\it positive-definite inner product} is a bilinear map $\\<\\cdot,\\cdot\\>: V\\times V \\xrightarrow{\\sim} \\mathds R$ for a vector space $V\\ni x,y$ that is\n\\begin{enumerate}[(a)]\n    \\item symmetric: $\\<x,y\\>=\\<y,x\\>$\\ ,\n    \\item positive-definite: $\\<x,x\\>>0$\\ , and\n    \\item nondegenerate: i.e. if $\\<x,y\\>=0$ for all $y \\implies x=0$\\ .\n\\end{enumerate}\nIf $\\<x,x\\>\\not>0$ then the inner product is called {\\it indefinite}.\n\n\\rem{2.3.3} Most of the important Lie groups in physics emerges as matrix Lie groups by considering invertible linear maps $V\\xrightarrow{\\sim}V$ on some finite-dimensional vector space $V$ that preserves a given inner product on $V$. Such {\\it general linear groups} are denoted $GL(V)$. Furthermore, these are Lie subgroups\\footnote{A Lie subgroup $H\\subset G$ is a subgroup as well as a submanifold of $G$. It turns out to be a Lie group in itself.} of the general Linear groups $GL(n,\\mathds R)$ or $GL(n,{\\mathds C})$ consisting of invertible linear maps on vector field $\\mathds R$ or $\\mathds{C}$ respectively. This becomes evident when we consider Riemannian manifolds which contains a metric (see Section \\ref{covectorFields}).\n\n\\ex{2.3.1} The special linear groups $SL(n,\\mathds R)$ (over $\\mathds R$) and $SL(n,{\\mathds C})$ (over ${\\mathds C}$) are $n\\times n$ invertible matrices with unit determinant i.e.\n\\begin{equation*}\n    SL(n,\\mathds R\/{\\mathds C}) \\= \\{A \\in GL(n,\\mathds R\/{\\mathds C})~|~ detA = 1 \\}\\ .\n\\end{equation*}\n\n\\ex{2.3.2} The Unitary group $U(n)$ are the $n\\times n$ complex-valued matrices whose inverse is the same as its conjugate transpose i.e.\n\\begin{equation*}\n    U(n) \\= \\{A \\in GL(n,{\\mathds C})~ |~ A^{-1} = \\bar{A}^T\\equiv A^\\+ \\}\\ .\n\\end{equation*}\nThe special unitary group is defined as\n\\begin{equation*}\n    SU(n) \\= \\{ A \\in U(n)~ |~ detA = 1 \\}\\ .\n\\end{equation*}\n\n\\rem{2.3.4} The special unitary groups $SU(n)$ preserves the standard inner product on ${\\mathds C}^n \\ni {\\bf x},{\\bf y}$ given by \n\\begin{equation*}\n    \\langle {\\bf x}, {\\bf y} \\rangle_{{\\mathds C}} \\= \\sum\\limits_{i=1}^n\\bar{x}_iy_i\n\\end{equation*}\nand also the norm of a given vector (induced from this inner product).\n\n\\ex{2.3.3} The orthogonal group $O(k,n)$ are the $(k+n)\\times (k+n)$ matrices that preserve the following inner product (${\\bf x},{\\bf y} \\in \\mathds R^{k+n} $):\n\\begin{equation*}\n    \\langle {\\bf x}, {\\bf y} \\rangle_{k,n}\\ :=\\ -x_1y_1 - \\ldots - x_ky_k + x_{k+1}y_{k+1} +\\ldots + x_{k+n}y_{k+n}\\ .\n\\end{equation*}\nOne can show that $A$ is in $O(k,n)$ if and only if\n\\begin{equation*}\n    A^T\\eta^{(k,n)} A \\= \\eta^{(k,n)}\\ ,\n\\end{equation*}\nwhere $\\eta^{(k,n)}$ is just the diagonal matrix $diag(1,\\ldots1,-1,\\ldots,-1)$. The special orthogonal group is defined as\n\\begin{equation*}\n    SO(k,n)\\ :=\\ \\{ A \\in O(k,n)~|~ detA = 1\\}\\ .\n\\end{equation*}\nThe orthogonal group $O(0,n)$ is denoted $O(n)$ and is more famously defined as\n\\begin{equation*}\n    O(n) := \\{ A \\in GL(n,\\mathds R) ~|~ A^{-1} = A^T \\}\\ .\n\\end{equation*}\nMoreover, the special orthogonal group $SO(0,n)$ is denoted as $SO(n)$.\n\n\\rem{2.3.5} The group $SO(n)$ consists of rotations in $n$-dimensions and is thus knows as isometry group of the $n$-sphere $S^n$. The group $O(n)$ consists of rotations as well as reflections. The groups $SO(1,3)$ and $SO(2,4)$ known, respectively, as the {\\it Lorentz group} and the {\\it Conformal group} are of special significance in physics. Another important group for us is the {\\it de Sitter group} $SO(1,4)$.\n\n\\ex{2.3.4} The {\\it Poincar\\'{e} group} or the {\\it inhomogeneous Lorentz group} $ISO(1,3)$ consists of Lorentz transformation together with translations and is defined as\n\\begin{equation*}\n    ISO(1,3) := \\{ I_{{\\bf x}} A ~|~ A \\in SO(1,3) \\}\\ ,\n\\end{equation*}\nwhere the translations $I_{{\\bf x}}$ acts on a given vector ${\\bf y} \\in \\mathds R^{1,3}$ as\n\\begin{equation*}\n    I_{{\\bf x}}{\\bf y} = {\\bf x} + {\\bf y}\\ .\n\\end{equation*}\n\n\\rem{2.3.6} It can be shown that the group $SO(4)$ decomposes into two copies of $SU(2)$ and that there exist a $2{-}1$ homomorphism\n\\begin{equation*}\n    SU(2)\\times SU(2) \\xrightarrow{2{-}1} SO(4)\\ .\n\\end{equation*}\nThis is a rather generic fact that arises when one considers spin-groups (e.g. $Spin(4)$ here), which provide universal cover\\footnote{This is a topological term that refers to a connected, simply connected group that projects down to the given group $G$ via a smooth surjection $\\pi$ such that any open $U \\in G$ lifts to a disjoint union $\\pi^{-1}(U)$ whose members are isomorphic to $U$.} to some group (e.g. $SO(4)$ here). To this end, we note down the following relevant facts here:\n\\begin{equation*}\n    Spin(4)\\cong SU(2)\\times SU(2)\\ ,\\quad Spin(3)\\cong SU(2)\\ ,\\ \\mathrm{and}\\quad Spin(1,3)\\cong SL(2,{\\mathds C})\\ .\n\\end{equation*}\n\n\\defn{2.3.4} The {\\it left action} of a Lie group $G$ (aka left $G$-action) on a set $M$ is defined by the map\n\\begin{equation*}\n    \\varphi_l :\\ G\\times M \\rightarrow M\\ ,\\quad (g,m) \\mapsto \\varphi_l(g,m)\\ \\equiv\\ gm\n\\end{equation*}\nthat satisfies the following properties:\n\\begin{itemize}\n    \\item $em \\= m$ for the identity element $e \\in G$ and every $m\\in M$, and\n    \\item $g_1(g_2m) \\= (g_1g_2)m$ for all $g_1,g_2 \\in G$ and $m \\in M$.\n\\end{itemize}\nThe set $M$ is known as a {\\it homogeneous space}, that splits into an {\\it orbit space} $M\/G$ of equivalence classes of orbits\n\\begin{equation*}\n    O_m\\ :=\\ \\{ n \\in M ~|~ \\exists\\ g \\in G : n = \\varphi_l(g,m) \\}\\ .\n\\end{equation*}\n\n\\defn{2.3.5} The {\\it left translations} or {\\it left multiplications} of a Lie group $G$ is its diffeomorphism i.e. $l_g\\in Diff(G)$ and is defined by\n\\begin{equation*}\n    l_g :\\ G \\rightarrow G\\ ,\\quad h \\mapsto gh\\ .\n\\end{equation*}\n\n\\rem{2.3.7} An equivalent notion of {\\it right action} defined by $\\varphi_r(g,m) := mg$ also exits and is of prime importance for the principle $G$-bundles that we will discuss in the next chapter. It is important to note here that one can induce a right-action $\\varphi_r$ from a given left-action $\\varphi_l$ as follows:\n\\begin{equation*}\n    \\varphi_r(g,m)\\ :=\\ \\varphi_l(g^{-1},m) \\quad \\forall\\quad m \\in M\\ . \n\\end{equation*}\nSimilarly, we have the notion of right translations $r_g$ for the Lie group $G$ but we will only deal with left translations here.\n\n\\defn{2.3.6} The {\\it coset space} $G\/H$ for a Lie group $G$ and its subgroup $H\\subset G$ is defined as\n\\begin{equation*}\n    G\/H\\ :=\\ \\{ gH ~|~ g \\in G \\}\\quad \\mathrm{where}\\quad gH\\ :=\\ \\{ gh ~|~ h \\in H \\}\\ .\n\\end{equation*}\n\n\\rem{2.3.8} There exist a natural left $G$-action (and hence, a left $H$-action) on $G\/H$ defined by\n\\begin{equation*}\n    \\varphi_l(g',gH) = g'gH\\quad \\forall\\quad g,g' \\in G\\ .\n\\end{equation*}\n\n\\defn{2.3.7} A left $G$-action on $M$ is called {\\it free} if, for all $m\\in M$, $gm=m$ implies that $g=e$. The action would be {\\it transitive} if for all $m,m'\\in M$ there exists $g\\in G$ such that $m=gm'$.\n\n\\rem{2.3.9} If the left action of $G$ on $M$ is free then every orbit $O_m$ is diffeomorphic to the Lie group $G$.\n\n\\defn{2.3.8} The {\\it stability\/isotropy subgroup} $G_m$ of a left $G$-action on $M\\ni m$ for a Lie group $G$ is its closed subgroup defined by\n\\begin{equation*}\n    G_m\\ :=\\ \\{ g \\in G ~|~ gm = m \\}\\ .\n\\end{equation*}\n\n\\thm{2.3.1} For a transitive left $G$-action there exist an isomorphism\\footnote{Some technicalities like $G$ be locally compact and $M$ be locally compact and connected are required.} $G\/G_m \\cong M$ between the coset space $G\/G_m$ and the homogeneous space $M$ for any $m\\in M$ given by\n\\begin{equation*}\n    j_p :\\ G\/G_p \\rightarrow M\\ ,\\quad gG_p \\mapsto gp\\ .\n\\end{equation*}\n\n\\rem{2.3.10} For a Lie group $G$ and its closed subgroup $H$ (e.g. $G_m$) the homogeneous space $G\/H$ can be canonically endowed with the structure of a smooth manifold. Moreover, there exist a canonical projection from $G$ to $G\/H$ given by\n\\begin{equation*}\n    \\pi_0 :\\ G \\rightarrow G\/H\\ ,\\qquad g \\mapsto gH\\ .\n\\end{equation*}\n\n\\ex{2.3.5} The round $n$-sphere $S^n$ is diffeomorphic to the following homogenoeus space:\n\\begin{equation*}\n    S^n\\ \\cong\\ SO(n{+}1)\/SO(n)\\ .\n\\end{equation*}\nIn particular, we have that $S^3 \\cong SO(4)\/SO(3)$.\n\n\\ex{2.3.6} A $(2n{+}1)$-sphere can be realized as a homogeneous space:\n\\begin{equation*}\n    S^{2n{+}1}\\ \\cong\\ SU(n{+}1)\/SU(n)\\ .\n\\end{equation*}\nA special case is $S^3\\cong SU(2)$, where the group $SU(1)$ is trivial.\n\n\\vspace{12pt}\n\\section{Vector fields}\n\\vspace{2pt}\n\\defn{2.4.1} The {\\it tangent bundle} $(TM,M,\\pi)$ over a manifold $M$ is nothing but a union of tangent spaces at all point of the manifold i.e.\n\\begin{equation*}\n    TM \\= \\bigcup\\limits_{p\\in M} T_p M\\ ,\n\\end{equation*}\nwhere the projection $\\pi$ just picks out the base point of a given vector in $TM$. \n\n\\rem{2.4.1} The dimension of the tangent bundle for an $n$-dimensional manifold is $2n$, as its members are ($n$-dimensional) vectors of some $T_pM$ labelled by the coordinate ($n$-tuple) of the base point $p\\in M$.\n\n\\defn{2.4.2} A {\\it vector field} is a smooth section of the tangent bundle $(TM,M,\\pi)$. The set of all such vector fields are denoted as $\\vecF{M} := \\Gamma^\\infty(TM)$.\n\n\\rem{2.4.2} Given a vector field $X\\in \\vecF{M}$ and a smooth function $f\\in C^\\infty(M)$, one can show that $Xf$ defined by\n\\begin{equation*}\n    Xf(p)\\ :=\\ X_p[f]\\ ,\n\\end{equation*}\nwhere $X_p \\in T_pM$, is also smooth i.e. $Xf \\in C^\\infty(M)$. The set of vector fields $\\vecF{M}$ do not form a vector space rather a module over the algebra of smooth functions $C^\\infty(M)$\\footnote{An algebra is just a vector space $V$ equipped with a multiplication operation $V\\times V \\rightarrow V$, which for functions is just composition. A module over an algebra is the same thing as a vector space over a field.}. Nevertheless, one can choose a basis of vector fields $\\left\\{ \\sfrac{\\mbox{$\\partial$}}{\\mbox{$\\partial$} x^i} \\equiv \\mbox{$\\partial$}_i \\right\\}$ on some local chart $(U,\\varphi)$ with coordinate functions $x^i$ (see Remark 2.1.3) to write $X \\in\\vecF{M}$ as\n\\begin{equation*}\n    X \\= \\sum\\limits_{i=1}^n Xx^i\\, \\mbox{$\\partial$}_i\\ .\n\\end{equation*}\n\n\\defn{2.4.3} There is a well defined notion of {\\it Lie bracket} associated with vector fields defined as\n\\begin{equation*}\n    [X,Y]f\\ :=\\ X(Yf) - Y(Xf)\\ ,\n\\end{equation*}\nwhich satisfy the following properties  \n\\begin{itemize}\n    \\item bilinearity: $[\\cdot,\\cdot] :\\ \\vecF{M}\\times \\vecF{M} \\xrightarrow{\\sim} \\vecF{M}$,\n    \\item anti-symmetry: $[X,Y] = -[Y,X]$, and\n    \\item Jacobi identity: $[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0$,\n\\end{itemize} \nfor all $X,Y,Z \\in \\vecF{M}$.\n\n\\rem{2.4.3} In general, it is not possible to induce a push-forward (see Definition 2.1.9) between vector fields $\\vecF{M} \\ni X$ and $\\vecF{N} \\ni Y$ of two manifolds $M$ and $N$ with a given map $\\varphi: M\\rightarrow N$ (it works if $\\varphi$ is a diffeomorphism). Nevertheless, it is useful to define the following relation.\n\n\\defn{2.4.4} A vector field $X \\in \\vecF{M}$ is called {\\it h-related} to another vector field $Y \\in \\vecF{M}$ i.e. $Y = \\varphi_*X$ if for all $p\\in M$ we have that\n\\begin{equation*}\n    \\varphi_*X_p \\= Y_{\\varphi(p)}\\ .\n\\end{equation*}\n\n\\rem{2.4.4} If $X_1$ and $X_2$ is h-related to $Y_1$ and $Y_2$ respectively then $[X_1,X_2]$ is h-related to $[Y_1,Y_2]$ for all $X_1,X_2,Y_1,Y_2 \\in \\vecF{M}$ i.e.\n\\begin{equation*}\n    \\varphi_*[X_1,X_2] \\= [Y_1,Y_2]\\ .\n\\end{equation*}\n\n\\defn{2.4.5} Given a vector field $X\\in \\vecF{M}$ its {\\it integral curve} through $p\\in M$ is a curve \n\\begin{equation*}\n    \\sigma_X :\\ (-\\epsilon,\\epsilon) \\rightarrow M\\ ,\n\\end{equation*}\nsuch that \n\\begin{equation*}\n    \\sigma_X(0) \\= p \\quad\\textrm{and}\\quad \\sigma_X'(t) \\= X_{\\sigma(t)}\n\\end{equation*}\nfor all $t\\in (-\\epsilon,\\epsilon)$.\n\n\\defn{2.4.6} If the interval $(-\\epsilon,\\epsilon)$ can be extended to whole of $\\mathds R$ for any $p \\in M$ then the vector field, and also the underlying manifold, is called {\\it complete}.\n\n\\rem{2.4.5} Presence of singularities (e.g. a black hole) on a manifold makes it incomplete. We will only consider complete manifolds in this thesis.\n\n\\vspace{12pt}\n\\section{Lie algebra}\n\\vspace{2pt}\n\\defn{2.5.1} A {\\it left-invariant} vector field $X$ for a Lie group $G$ is defined as the vector field which is $l_g$-related to itself for all $g\\in G$ i.e. \n\\begin{equation*}\n    l_{g*}X \\= X \\quad \\mathrm{or}\\quad l_{g*}X_{g'} = X_{gg'}\n\\end{equation*}\nfor all $g' \\in G$.\n\n\\defn{2.5.2} The set of all left-invariant vector fields (a vector space) denoted as $L(G)$ together with the Lie bracket $[\\cdot,\\cdot]$ i.e. $\\left(L(G),[\\cdot,\\cdot]\\right)$ is known as the {\\it Lie algebra} of $G$\\footnote{Remark 2.4.4 is crucial here.}.\n\n\\thm{2.5.1} We have a Lie algebra isomorphism\\footnote{It is an invertible map that preserves the Lie algebraic structure.} $(T_eG,[\\cdot,\\cdot]) \\cong (L(G),[\\cdot,\\cdot])$ defined by\n\\begin{equation*}\n    j:\\ T_eG \\rightarrow L(G)\\ ,\\quad A\\mapsto j(A)\\equiv L^A\\ ,\n\\end{equation*}\nwhere the left-invariant vector field $L^A$ is defined by\n\\begin{equation*}\n    L^A_g\\ :=\\ l_{g*}A \\in T_gG\\ .\n\\end{equation*}\n\n\\rem{2.5.1} Note that the Lie bracket on $T_eG$ is induced from the one on $L(G)$ via the map $j$ i.e.\n\\begin{equation*}\n    [A,B]\\ :=\\ j^{-1}[L^A,L^B] \\= l_{g{-1}*}[L^A_g,L^B_g]\n\\end{equation*}\nfor any $g\\in G$. This takes the form of usual matrix commutator for matrix Lie groups that we are dealing with. We thus have the fact that $\\textrm{dim}L(G) = \\textrm{dim}T_eG = \\textrm{dim}G$.\n\n\\ex{2.5.1} We note down below in Table \\ref{LieTable} the Lie algebras $T_eG$ of some of the Lie groups $G$ that we encountered before.\n\\begin{table}[!htbp]\n    \\centering\n    \\begin{tabular}{|p{0.15\\textwidth}|p{0.6\\textwidth}|}\n    \\hline\n        $G$ & $T_eG$ \\\\\n    \\hline\\hline\n        $GL(n,\\mathds R\/\\mathds{C})$ & $M(n,\\mathds R\/\\mathds{C}) := $ The set of $n\\times n$ real\/complex matrices \\\\\n    \\hline\n        $SO(n)$ & $so(n) := \\{ A \\in M(n,\\mathds R)\\ |\\ A^T = -A \\}$  \\\\\n    \\hline\n        $SU(n)$ & $su(n) := \\{ A \\in M(n,\\mathds{C})\\ |\\ \\bar{A} = -A\\ \\&\\ \\mathrm{tr}{A} = 0 \\}$ \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{A list of Lie groups and their Lie algebras.}\n    \\label{LieTable}\n\\end{table}\n\n\\thm{2.5.2} A Lie group homomorphism $\\varphi : G \\rightarrow H$ between Lie groups $G$ and $H$ induces a Lie algebra homomorphism\n\\begin{equation*}\n    \\mathrm{d}{\\varphi}\\equiv \\varphi_*:\\ T_eG \\rightarrow T_eH\\ , \\quad \\mathrm{i.e.}\\quad \\varphi_*[A,B] \\= [\\varphi_*A,\\varphi_*B]\\ .\n\\end{equation*}\n\n\\rem{2.5.2} Given a basis $\\{ E_1,\\ldots, E_n \\}$ of $L(G)\\cong T_eG$ for an $n$-dimensional Lie group $G$, its structure constants $f_{ij}^{\\ \\ k}$ are defined by\n\\begin{equation*}\n    [E_i,E_j] \\= \\sum\\limits_{k=1}^n f_{ij}^{\\ \\ k}\\,E_k\\ .\n\\end{equation*}\n\n\\ex{2.5.2} The structure constants for the Lie algebra $su(2)$ are given by $f_{ij}^{\\ \\ k} = 2\\,\\varepsilon_{ij}^{\\ \\ k}$, where $\\varepsilon_{ij}^{\\ \\ k}$ is the $3$-dimensional Levi-Civita symbol.\n\n\\thm{2.5.3} A left-invariant vector field $X$ on a Lie group $G$ is complete.\n\n\\rem{2.5.3} A consequence of the above theorem is that there exist a unique integral curve\n\\begin{equation*}\n    t \\mapsto \\sigma_{L^A}(t)\n\\end{equation*}\nfor all $t\\in \\mathds R$ and left-invariant vector field $L^A$ constructed from a given $A\\in T_eG$ of the Lie group $G$.\n\n\\defn{2.5.3} The {\\it exponential map} for a Lie group $G$ is defined by\n\\begin{equation*}\n    \\mathrm{exp}:\\ T_eG \\rightarrow G\\ , \\quad A \\mapsto \\mathrm{exp}(A)\\ :=\\ \\sigma_{L^A}(t{=}1)\\ .\n\\end{equation*}\n\n\\rem{2.5.4} The exponential map is locally diffeomorphic. Furthermore, it lifts $T_eG$ to a connected component of $G$ (connected to $e$) and is a surjection when $G$ is compact.\n\n\\defn{2.5.4} A {\\it one-parameter subgroup} of a Lie group $G$ is a smooth homomorphism $\\mu :\\ \\mathds R \\rightarrow G$ from the additive group $\\mathds R$ into $G$ i.e.\n\\begin{equation*}\n    \\mu(t_1+t_2) \\= \\mu(t_1)\\mu(t_2).\n\\end{equation*}\n\n\\thm{2.5.4} If $\\mu: \\mathds R \\rightarrow G$ is a one-parameter subgroup of $G$ then, for all $t\\in \\mathds R$,\n\\begin{equation*}\n    \\mu(t) \\= \\mathrm{exp}(tA) \\quad\\textrm{with}\\quad  A\\ :=\\ \\mu_*\\sfrac{\\mathrm{d}}{\\mathrm{d}{t}}\\big|_0\\ .\n\\end{equation*}\n\n\\rem{2.5.5} The above theorem shows that there exist a one-to-one correspondence between one-parameter subgroups of a Lie group $G$ and its Lie algebra $T_eG$. \n\n\\defn{2.5.5} Given a vector field $X\\in \\vecF{M}$ one defines the {\\it flow generated by X} as the one-parameter group $\\{ \\varphi_t : t \\in \\mathds R \\}$ where the set of maps $\\varphi_t : M \\rightarrow M$ are nothing but the integral curves\n\\begin{equation*}\n    \\varphi_t(m) \\= \\sigma(t)\\ .\n\\end{equation*}\nOne can define the {\\it Lie derivative of $Y$ along $X$}, for $Y\\in \\vecF{F}$, by\n\\begin{equation}\n    {\\cal L}_X Y \\= \\frac{\\mathrm{d}}{\\mathrm{d} t}(\\varphi_t^{-1})_*(Y)\\Big|_{t=0}\\ .\n\\end{equation}\n\n\\rem{2.5.6} It can be shown that ${\\cal L}_XY = [X,Y]$.\n\n\\vspace{12pt}\n\\section{Tensor fields}\n\\vspace{2pt}\n\\label{covectorFields}\n\\defn{2.6.1} A {\\it (p,q)-tensor} $T^{p,q}(V)$ for a vector space $V$ and its dual $V^*$ is the space of multilinear functions\n\\begin{equation*}\n    f:\\ V^*\\times\\overset{p}{\\cdots}\\times V^*\\times V\\times\\overset{q}{\\cdots}\\times V \\xrightarrow{\\sim} \\mathds R\n\\end{equation*}\nand is, alternatively, denoted using tensor products as $V\\otimes\\overset{q}{\\cdots}\\otimes V\\otimes V^*\\otimes\\overset{p}{\\cdots}\\otimes V^*$.\n\n\\rem{2.6.1} Note that $T^{0,0}(V):=\\mathds R$. Moreover, we have the following results\n\\begin{equation*}\n    T^{0,1}(V)\\ \\cong\\ V^* \\quad\\textrm{and}\\quad T^{1,0}(V) \\= (V^*)^*\\ \\cong\\ V\n\\end{equation*}\nfor any finite-dimensional vector space $V$. Elements of $T^{p,q}(V)$ can be easily expanded in terms of a given basis of $V$ and the corresponding dual basis of $V^*$.\n\n\\defn{2.6.2} A {\\it (p,q)-tensor bundle} $(T^{p,q}M,M,\\pi)$ over a manifold $M$ is the following disjoint union of tensors:\n\\begin{equation*}\n    T^{p,q}M \\= \\bigcup\\limits_{m\\in M}T^{p,q}(T_mM)\n\\end{equation*}\nwhere the projection $\\pi$ associates the base point $m\\in M$ of a given vector in $T^{p,q}M$.\n\n\\rem{2.6.2} The tangent bundle $TM$ is isomorphic to $T^{1,0}M$, while the bundle $T^{0,1}M$ is isomorphic to the so called {\\it cotangent bundle} $T^*M$ defined analogously to Definition 2.4.1 before.\n\n\\defn{2.6.3} The {\\it exterior algebra} over a vector space $V$ denoted as $\\bigwedge V$ is the algebra\\footnote{Plainly speaking, it is a linear combination of all possible (finitely many) anti--symmetrized tensor products of vectors in $V$.} generated by the so called {\\it wedge product} $\\wedge$ satisfying\n\\begin{equation*}\n    v_1\\wedge v_2 \\= - v_2\\wedge v_1\n\\end{equation*}\nfor all $v_1,v_2 \\in V$. A subspace of $\\bigwedge V$ consisting of linear combinations of $k$--fold wedge products of vectors in $V$ is denoted $\\bigwedge^k (V)$.\n\n\\subsection{Differential forms}\n\n\\defn{2.6.4} A {\\it $k$-form} $\\omega$ is a $(0,k)$-tensor field $\\omega \\in \\Gamma^\\infty(M,T^{0,k}M)$ such that $\\omega_m \\in \\bigwedge^k(T_mM)$ for all $m\\in M$.\n\n\\rem{2.6.3} The space of $k$-forms is denoted $\\Omega^k(M)$. Note that $\\Omega^0(M) := C^\\infty(M)$ and the only member of $\\Omega^n(M)$ (upto to a scalar multiple), known as the {\\it volume form}, is denoted $\\mu$ or $\\mathrm{d}{V}$. The space $\\Omega^n(M)$ splits into two equivalence classes $\\{ [\\mu_+], [\\mu_-] \\}$ with the relation $\\mu \\sim \\mu'$ defined by $\\mu = \\lambda \\mu'$ such that $\\mu \\in [\\mu_+]$ if $\\lambda > 0$ and $\\mu \\in [\\mu_-]$ if $\\lambda < 0$.\n\n\\rem{2.6.4} On a chart $(U,\\varphi)$ of $M$ with coordinates $\\varphi(m)=(x^1,\\ldots,x^n)$ one can expand a $k$--form $\\omega$ as \n\\begin{equation*}\n    \\omega \\= \\sfrac{1}{k!}\\sum\\limits_{i_1,\\ldots,i_k = 1}^n\\omega_{i_1,\\ldots,i_k}\\,\\mathrm{d}{x}^{i_1}\\wedge\\ldots\\wedge\\mathrm{d}{x}^{i_k}\n\\end{equation*}\nwhere the coefficients $\\omega_{i_1,\\ldots,i_k} \\in C^\\infty(M)$ are totally anti--symmetric in its indices.\n\n\\defn{2.6.5} The {\\it exterior derivative} $\\mathrm{d}$ is the linear map $\\mathrm{d} : \\Omega^k(M) \\xrightarrow{\\sim} \\Omega^{k+1}(M)$ defined, for all $\\omega \\in \\Omega^k(M)$, by\n\\begin{equation*}\n\\mathrm{d}{\\omega} \\= \\sfrac{1}{k!}\\sum\\limits_{i_1,\\ldots,i_k = 1}^n\\mbox{$\\partial$}_i(\\omega_{i_1,\\ldots,i_k})\\,\\mathrm{d}{x}^i\\wedge\\mathrm{d}{x}^{i_1}\\wedge\\ldots\\wedge\\mathrm{d}{x}^{i_k}\\ ,\n\\end{equation*}\nwhere the partial derivative $\\mbox{$\\partial$}_i$ is taken with respect to coordinate $x^i$. It can also be defined in a coordinate independent way, for all $X_1,\\ldots,X_{k+1} \\in \\vecF{M}$, as\n\\begin{equation*}\n \\begin{aligned}\n    \\mathrm{d}{\\omega}(X_1,\\ldots,X_{k+1}) &\\= \\sum\\limits_{i=1}^{k+1} (-1)^{i+1}\\, X_i(\\omega(X_i,\\ldots,\\hat{X}_i,\\ldots,X_{k+1})) \\\\\n    &\\qquad + \\sum\\limits_{1\\leq i < j \\leq n} (-1)^{i+j}\\, \\omega([X_i,X_j],X_1,\\ldots,\\hat{X}_i,\\ldots,\\hat{X}_j,\\ldots,X_{k+1})\\ ,\n \\end{aligned}\n\\end{equation*}\nwhere the circumflex means that the symbol beneath it is omitted.\n\n\\rem{2.6.5} The action of exterior derivative follow {\\it graded Leibniz rule}:\n\\begin{equation*}\n    \\mathrm{d}{(\\alpha\\wedge\\beta)} \\= \\mathrm{d}{\\alpha}\\wedge\\beta + (-1)^p\\,\\alpha\\wedge\\mathrm{d}{\\beta}\n\\end{equation*}\nfor all $\\alpha\\in \\Omega^p(M)$ and $\\beta\\in\\Omega^q(M)$. Moreover, it can easily be shown (using the first definition above) that\n\\begin{equation*}\n    \\mathrm{d}^2\\ \\equiv\\ \\mathrm{d}\\circ\\mathrm{d} \\= 0\\ .\n\\end{equation*}\n\n\\defn{2.6.6} For a map $\\varphi : M \\rightarrow N$ and a $k$-form $\\omega \\in \\Omega^k(N)$ its {\\it pull-back} $\\varphi^*\\omega \\in \\Omega^k(M)$ is defined by\n\\begin{equation*}\n    (\\varphi^*\\omega)_m(X_1,\\ldots,X_k)\\ :=\\ \\omega_{\\varphi(m)}(\\varphi_{*}X_1,\\ldots,\\varphi_{*}X_k)\n\\end{equation*}\nfor all $X_1,\\ldots,X_k \\in \\vecF{M}$. \n\n\\rem{2.6.6} It can be shown that the exterior derivative is {\\it natural} i.e. it is compatible with the pull-back:\n\\begin{equation*}\n    \\varphi^*(\\mathrm{d}{\\omega}) \\= \\mathrm{d}{(\\varphi^*\\omega)}\\ ,\n\\end{equation*}\nfor any $\\omega\\in \\Omega^k(M)$. Furthermore, one can prove that\n\\begin{equation*}\n    \\varphi^*(\\alpha\\wedge\\beta) = \\varphi^*\\alpha\\wedge\\varphi^*\\beta \\quad\\textrm{and}\\quad (\\varphi\\circ\\tilde{\\varphi})^*\\omega = \\tilde{\\varphi}^*(\\varphi^*\\omega)\\ .\n\\end{equation*}\n\n\\defn{2.6.7} The {\\it Lie derivative of $\\omega$ along $X$}, for $\\omega\\in\\Omega^k(M)$ and $X\\in \\vecF{M}$, can be defined analogous to Definition 2.5.5:\n\\begin{equation*}\n    {\\cal L}_X\\omega \\= \\frac{\\mathrm{d}}{\\mathrm{d} t}(\\varphi_t^{-1})^*(\\omega)\\Big|_{t=0}\\ .\n\\end{equation*}\n\n\\rem{2.6.7} There exist a nice formula by \\'{E}lie Cartan for the Lie derivative of differential forms given, for any $X\\in \\vecF{M}$, by\n\\begin{equation*}\n    {\\cal L}_X \\= \\mathrm{d}\\circ\\iota_X + \\iota_X\\circ\\mathrm{d}\\ ,\n\\end{equation*}\nwhere the linear map $\\iota_X: \\Omega^p(M) \\xrightarrow{\\sim} \\Omega^{p-1}(M)$ is the {\\it interior product} defined by  \n\\begin{equation*}\n    (\\iota_X\\omega)(X_1,\\ldots,X_{p-1}) \\= \\omega(X,X_1,\\ldots,X_{p-1})\n\\end{equation*}\nfor $\\omega \\in \\Omega^p(M)$ and $X_1,\\ldots,X_{p-1} \\in \\vecF{M}$.\n\n\\subsection{Metric}\n\n\\defn{2.6.8} A {\\it metric} $g$ is a $(0,2)$-tensor field $g \\in \\Gamma^\\infty(T^{0,2}M)$ where $g_m$ for every $m\\in M$ defines an inner product on the vector space $T_mM$. If this inner product is positive definite then the manifold is called {\\it Riemannian}. If the metric $g$ has an underlying inner product that is indefinite then the manifold is called {\\it pseudo-Riemannian}.\n\n\\rem{2.6.8} A metric $g$ on an $n$-dimensional manifold $M$ in local coordinates can be written\n\\begin{equation*}\n    g \\= \\sum\\limits_{i=1}^n\\, g_{ij}\\,\\mathrm{d}{x}^i\\otimes\\mathrm{d}{x}^j \\quad\\textrm{with}\\quad g_{ij} := g(\\mbox{$\\partial$}_i,\\mbox{$\\partial$}_j)\n\\end{equation*}\nbeing the {\\it metric components} for the basis vector fields $\\mbox{$\\partial$}_i,\\mbox{$\\partial$}_j \\in \\vecF{M}$. Often times in physics literature the tensor product sign is ignored. The fact that $g$ is nondegenerate means that the matrix $g_{ij}$ can be inverted to yield another matrix $g^{-1}$ with components $g^{-1}_{ij} =: g^{ij}$ that in turn defines a $(2,0)$-tensor field called the {\\it induced metric}. These can be used to raise or lower indices of any tensor component e.g.\n\\begin{equation*}\n    T_{i'}^{\\ j'k'} := g_{ii'}g^{jj'}g^{kk'}T^i_{\\ jk}\\quad \\forall \\quad T \\in \\Gamma^\\infty(M,T^{1,2}M)\\ .\n\\end{equation*}\n\n\\ex{2.6.1} A standard example of a Reimannian manifold is $\\mathds R^n$ with metric\n\\begin{equation*}\n    g \\= (\\mathrm{d}{x}^1)^2 + \\ldots + (\\mathrm{d}{x}^n)^2\\ .\n\\end{equation*}\nA prominent example of a psuedo-Riemannian manifold is {\\it Minkowski space} $\\mathds R^{1,n}$ with metric\n\\begin{equation*}\n    g \\= -(\\mathrm{d}{x}^1)^2 + (\\mathrm{d}{x}^2)^2 + \\ldots + (\\mathrm{d}{x}^n)^2\\ .\n\\end{equation*}\n\n\\defn{2.6.9} The {\\it signature} of a pseudo-Riemannian manifold $(M,g)$ of dimension $n$ is a count of the number of positive and negative eigenvalues of the matrix $g_{ij}$ and is denoted as an $n$-tuple of $+$ and $-$. A $4$-dimensional Lorentzian manifold has one of the signature different from the rest three.\n\n\\ex{2.6.2} The signature of the Minkowski space $\\mathds R^{1,3}$ as presented above is $(-,+,+,+)$, and is thus a Lorentzian manifold.\n\n\\defn{2.6.10} Two (pseudo-)Riemannian manifolds $(M,g_M)$ and $(N,g_N)$ are called {\\it conformal} if their metrices are related by some smooth function $\\Omega^2 \\in C^\\infty(N)$: \n\\begin{equation*}\n    g_M \\= \\Omega^2\\,g_N\\ .\n\\end{equation*}\n\n\\defn{2.6.11} A locally {\\it orthonormal basis} of $1$-forms $\\{ e^i \\}$ (aka coframe) with $i=1,\\ldots,n$ for an $n$-dimensional Riemannian manifold $M$ satisfy, for all $e^i,e^j \\in \\Omega^1(M)$,\n\\begin{equation*}\n    g(e^i,e^j) \\= g^{ij} = \\delta^{ij}\\ .\n\\end{equation*}\nSimilarly, one defines locally orthonormal basis of vector fields (aka frame) $\\{ X_1,\\ldots,X_n \\}$ by demanding that they satisfy\n\\begin{equation*}\n    g_{ij} \\= g(X_i,X_j) \\= \\delta_{ij}\n\\end{equation*}\nfor all $X_i,X_j \\in \\vecF{M}$.\n\n\\subsection{Maurer--Cartan form}\n\\defn{2.6.12} A $k$-form $\\omega \\in \\Omega^k(G)$ on a Lie group $G$ is said to {\\it left-invariant} if, for all $g\\in G$,\n\\begin{equation*}\n    l_g^*\\omega \\= \\omega\\ ,\\quad \\mathrm{i.e.}\\ ,\\quad l_g^*(\\omega_{g'}) \\= \\omega_{g^{-1}g'}\\ \\forall\\ g' \\in G\\ .\n\\end{equation*}\nThe set of all left-invariant one-forms on $G$ is denoted $L^*(G)$. \n\n\\rem{2.6.9} From Remark 2.6.6 we see that if $\\omega$ is left-invariant then so is its exterior derivative:\n\\begin{equation*}\n    l_g^*(\\mathrm{d}{\\omega}) \\= \\mathrm{d}{l_g^*\\omega}\\ .\n\\end{equation*}\n\n\\rem{2.6.10} Similar to $j$ of Theorem 2.5.1, there exist an isomorphism between $T^*_eG$ and $L^*(G)$:\n\\begin{equation*}\n    \\tilde{j}:\\ T^*_eG \\rightarrow L^*(G)\\ ,\\quad \\omega \\mapsto \\tilde{j}(\\omega) \\equiv \\lambda^\\omega\\ ,\n\\end{equation*}\nwhere the left-invariant one-forms $\\lambda^\\omega$ is defined as\n\\begin{equation*}\n    \\lambda^\\omega_g\\ :=\\ l_{g-1}^*(\\omega) \\in T^*_gG\\ .\n\\end{equation*} \nNotice that these $1$-forms are dual to the left-invariant vector fields $L^A$ for $A \\in T_eG$ i.e.\n\\begin{equation*}\n    \\< \\lambda^\\omega, L^A \\>_g \\= \\< \\omega, A\\>\n\\end{equation*}\nfor all $g \\in G$.\n\n\\rem{2.6.11} For a basis of $\\{ E_1,\\ldots,E_n \\}$ of $L(G)$ with dim$G=n$ (see Remark 2.5.2) we can define the dual basis $\\{ \\omega^1,\\ldots,\\omega^n \\}$ of $L^*(G)$ by\n\\begin{equation*}\n    \\< \\omega^i, E_j \\> \\= \\delta^i_j\\ ,\n\\end{equation*}\nwhich satisfy the Maurer--Cartan equation\n\\begin{equation*}\n    \\mathrm{d}{\\omega}^i + \\frac{1}{2} f_{jk}^{\\ \\ i}\\,\\omega^j\\wedge\\omega^k\\ .\n\\end{equation*}\n\n\\defn{2.6.13} The Maurer--Cartan $1$-form $\\Omega_l$ is the $L(G)$-valued $1$-form on $G$ that for any $A \\in T_gG$ gives a left-invariant vector field as follows\n\\begin{equation*}\n    \\< \\Omega_l, A \\>_{g'}\\ :=\\ l_{g'*}(l_{g^{-1}*}A) \n\\end{equation*}\nfor any $g' \\in G$.\n\n\\rem{2.6.12} It is more useful to consider the Maurer--Cartan $1$-form to be valued in $T_eG \\cong L(G)$, which yields a nice result:\n\\begin{equation*}\n    \\< \\Omega_l, L^A \\> \\= A\\ .\n\\end{equation*}\nIt can be shown that the Maurer-Cartan $1$-form takes the following explicit form for the matrix Lie groups that we consider here: \n\\begin{equation*}\n    \\Omega_l^{ij} \\= \\sum\\limits_{k=1}^n (g^{-1})^{ik}\\, \\mathrm{d}{g}^{kj}\\ ,\n\\end{equation*}\nwhere $g^{ij}$ are the coordinates on the matrix Lie group $G$ in a given chart. \n\n\\vspace{12pt}\n\\section{Integration} \n\\vspace{2pt}\n\\defn{2.7.1} A manifold $M$ is called {\\it orientable} if it is equipped with a nowhere vanishing volume form $\\mu$. An {\\it orientation} of $M$ refers to a choice of one of the the two equivalence classes of Remark 2.6.3; this is usually chosen to be the positive one.\n\n\\ex{2.7.1} The standard volume form on $\\mathds R^n$, which is an oriented manifold, is given by \n\\begin{equation*}\n    \\mu \\= \\mathrm{d}{x}^1\\wedge\\ldots\\wedge\\mathrm{d}{x}^n\\ . \n\\end{equation*}\nM\\\"{o}bius strip provides a classic example of a nonorientable manifold.\n\n\\defn{2.7.2} {\\it Integration} of a compactly supported volume form $\\mu$ (i.e. it vanishes outside of a compact subset of $M$) of an oriented manifold $M$ that is covered with charts $(U_1,\\phi_1),\\ldots,(U_N,\\phi_N)$ can be defined, using a partition of unity\\footnote{It is a set of smooth functions $f_i \\in C^\\infty(M)$ that vanish outside of $U_i$, lies within $[0,1]$, and satisfies the condition: $\\sum_{i=1}^Nf_i(m) = 1$.} $\\{ f_i \\}$, as \n\\begin{equation*}\n    \\int_M \\mu \\= \\sum\\limits_{i=1}^N \\int_{\\phi(U_i)} (\\phi_i^{-1})^*(f_i\\mu)\\ .\n\\end{equation*}\n\n\\rem{2.7.1} It can be shown that this definition of integration is independent of the choice of a chart. \n\n\\thm{2.7.1} For an oriented manifold $M$ with boundary $\\mbox{$\\partial$} M$ ( that inherits an induced orientation from $M$) the integral of a compactly supported $(n{-}1)$-form $\\omega \\in \\Omega^{n{-}1}(M)$ is related to an integral of its exterior derivative $\\mathrm{d}{\\omega}$:\n\\begin{equation*}\n    \\int_M \\mathrm{d}{\\omega} \\= \\int_{\\mbox{$\\partial$} M} \\omega\\ .\n\\end{equation*}\n\n\\rem{2.7.2} This is the famous {\\it Stockes' theorem}. Notice that if the manifold $M$ has no boundary then the above integral vanishes.\n\n\\defn{2.7.3} For an $n$-dimensional, oriented, Riemannian manifold $(M,g)$ the {\\it Riemannian volume form} is given by\n\\begin{equation*}\n    \\mu_g \\= \\sqrt{|g|}\\,\\mathrm{d}{x}^1\\wedge\\cdots\\wedge\\mathrm{d}{x}^n\\ ,\n\\end{equation*}\nwhere $|g| := |\\mathrm{det}(g_{ij})|$ for metric compoents $g_{ij}$.\n\n\\rem{2.7.3} Integration on Riemannian manifolds are performed using volume form $\\mu_g$. \n  \n\\vspace{12pt}  \n\\section{Hodge duality}\n\\vspace{2pt}\n\\rem{2.8.1} On an $n$-dimensional pseudo-Riemannian manifold $(M,g)$ one can induce an inner product on $\\Omega^k(M)$, generated by the $k$-forms $\\{ \\mathrm{d}{x}^{i_1}\\wedge\\cdots\\wedge\\mathrm{d}{x}^{i_k} \\}$ with $i_1,\\ldots,i_k \\in \\{ 1,\\ldots,n \\}$, that is given by\n\\begin{equation}\n    \\Big\\< \\mathrm{d}{x}^{i_1}\\wedge\\cdots\\wedge\\mathrm{d}{x}^{i_k}, \\mathrm{d}{x}^{j_1}\\wedge\\cdots\\wedge\\mathrm{d}{x}^{j_k} \\Big\\> \\= g^{i_1j_1}\\cdots g^{i_kj_k}\\ ,\n\\end{equation}\nwhere $g^{ij} := g(\\mathrm{d}{x}^i,\\mathrm{d}{x}^j)$.\n\n\\defn{2.8.1} The {\\it Hodge star operator} $\\ast : \\Omega^k(M) \\xrightarrow{\\sim} \\Omega^{n-k}(M)$ is a linear map that is defined on an oriented $n$-dimensional pseudo-Riemannian manifold $(M,g)$ with volume form $\\mu$ given by\n\\begin{equation}\n    \\alpha\\wedge\\ast\\beta \\= \\< \\alpha,\\beta \\>\\, \\mu\n\\end{equation}\nfor all $\\alpha,\\beta \\in \\Omega^k(M)$. Here $\\ast\\beta$ is called the {\\it Hodge dual} of $\\beta$.\n\n\\rem{2.8.2} For an $n$-dimensional oriented pseudo-Riemannian manifold $(M,g)$ with signature $(-,\\overset{s-2}{\\ldots},-,+,\\overset{n-s-2}{\\ldots},+)$ the following condition holds true on $\\Omega^p(M)$:\n\\begin{equation*}\n    \\ast^2 \\= (-1)^{p(n-p)+s}\\ .\n\\end{equation*}\n\n\\ex{2.8.1} For the Minkowski space $\\mathds R^{1,3}$ with signature $(-,+,+,+)$, coordinates $ ~ \\\\(x^0, x^1, x^2, x^3) =: (t,x,y,z)$ and volume form $\\mathrm{d}{V} = \\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\wedge\\mathrm{d}{y}\\wedge\\mathrm{d}{z}$ we have the following results:\n \\begin{align*}\n    &\\ast\\mathrm{d}{z} \\= -\\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\wedge\\mathrm{d}{y}\\ , &\n    &\\ast\\,\\mathrm{d}{t}\\wedge\\mathrm{d}{x} \\= -\\mathrm{d}{y}\\wedge\\mathrm{d}{z}\\ , & &\\ast\\,\\mathrm{d}{t}\\wedge\\mathrm{d}{y} \\= \\mathrm{d}{x}\\wedge\\mathrm{d}{z}\\ ,\\\\\n    &\\ast\\mathrm{d}{t}\\wedge\\mathrm{d}{z} \\= -\\mathrm{d}{x}\\wedge\\mathrm{d}{y}\\ , & &\\ast\\,\\mathrm{d}{x}\\wedge\\mathrm{d}{y} \\= -\\mathrm{d}{t}\\wedge\\mathrm{d}{z}\\ , & &\\ast\\,\\mathrm{d}{x}\\wedge\\mathrm{d}{z} \\= -\\mathrm{d}{t}\\wedge\\mathrm{d}{y}\\ ,\\\\\n    &\\ast\\mathrm{d}{y}\\wedge\\mathrm{d}{z} \\= \\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\ , & &\\ast\\,\\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\wedge\\mathrm{d}{y} \\= -\\mathrm{d}{z}\\ , & &\\ast\\,\\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\wedge\\mathrm{d}{z} \\= \\mathrm{d}{y}\\ ,\\\\\n    &\\ast\\mathrm{d}{t}\\wedge\\mathrm{d}{y}\\wedge\\mathrm{d}{z} \\= -\\mathrm{d}{x}\\ , & &\\ast\\,\\mathrm{d}{x}\\wedge\\mathrm{d}{y}\\wedge\\mathrm{d}{z} \\= -\\mathrm{d}{t}\\ , & &\\ast\\,\\mathrm{d}{t}\\wedge\\mathrm{d}{x}\\wedge\\mathrm{d}{y}\\wedge\\mathrm{d}{z} \\= -1\\ .\n \\end{align*}\n\n\\vspace{12pt}\n\\section{Representations}\n\\vspace{2pt}\n\\defn{2.9.1} A {\\it representation of a Lie group} $G$ is a Lie group homomorphism\n\\begin{equation*}\n    \\Pi :\\ G \\rightarrow \\mathrm{GL}(V)\n\\end{equation*}\nfor a finite dimensional vector space $V$. \n\n\\defn{2.9.2} A {\\it representation of a Lie algebra} $\\mathfrak{g}$, for a finite dimensional vector space $V$, is a Lie algebra homomorphism\n\\begin{equation*}\n    \\pi :\\ \\mathfrak{g} \\rightarrow \\mathrm{gl}(V)\\ ,\n\\end{equation*}\nwhere $\\mathrm{gl}(V) := \\mathrm{End}(V)$\\footnote{The endomorphism of $V$ denoted $\\mathrm{End}(V)$ is the space of linear maps $V\\xrightarrow{\\sim} V$.}.\n\n\\ex{2.9.1} The {\\it standard representation} of a Lie group $G \\ni g$ is $\\Pi(g) = g$ and of a Lie algbra $\\mathfrak{g} \\ni X$ is $\\pi(X) = X$. \n\n\\ex{2.9.2} For matrix Lie groups $G$ with Lie algebra $\\mathfrak{g}$, its {\\it adjoint representation} is the following homomorphism\n\\begin{equation*}\n    \\mathrm{Ad}:\\ G \\rightarrow \\mathrm{GL}(\\mathfrak{g})\\ ,\\quad g \\mapsto \\mathrm{Ad}_g\\ ,\n\\end{equation*}\nwhere the {\\it adjoint map} $\\mathrm{Ad}_A$ is defined by\n\\begin{equation*}\n    \\mathrm{Ad}_g:\\ \\mathfrak{g} \\xrightarrow{\\sim} \\mathfrak{g}\\ ,\\quad X \\mapsto \\mathrm{Ad}_g(X)\\ :=\\ gXg^{-1}\\ .\n\\end{equation*}\n\n\\rem{2.9.1} It can be shown that the map $\\mathrm{Ad}$ induces (see Theorem 2.5.2) the following lie algebra homomorphism\n\\begin{equation*}\n    \\mathrm{Ad}_* \\equiv \\mathrm{ad}:\\ \\mathfrak{g} \\rightarrow \\mathrm{gl}(\\mathfrak{g})\\ ,\\quad X \\mapsto \\mathrm{ad}_X\\ ,\n\\end{equation*}\nwhere the action of the map $\\mathrm{ad}_X$ can be shown to be the following \n\\begin{equation*}\n    \\mathrm{ad}_X:\\ \\mathfrak{g}\\rightarrow \\mathfrak{g}\\ , \\quad Y\\mapsto \\mathrm{ad}_X(Y)\\ :=\\ [X,Y]\\ .\n\\end{equation*}\nThis is known as the {\\it adjoint representation} of a finite-dimensional Lie algebra $\\mathfrak{g}$. \n\n\\vspace{12pt}\n\\section{Maxwell equations}\n\\vspace{2pt}\n\\rem{2.10.1} For the nice\\footnote{Here it means a connected and simply connected manifold.} manifolds that we are interested in this thesis the Maxwell theory boils down to a choice of the gauge potential $A$, which is a $1$-form on the manifold.\n\n\\defn{2.10.1} For Minkowski space $\\mathds R^{1,3}$ we define the gauge potential $A$ by\n\\begin{equation*}\n    A \\= A_0\\,\\mathrm{d}{t} + A_i\\,\\mathrm{d}{x}^i\\ ,\n\\end{equation*}\nand the field strength $F$ by\n\\begin{equation*}\n    F\\ :=\\ \\mathrm{d}{A} \\= E_i\\,\\mathrm{d}{x}^i\\wedge\\mathrm{d}{t} + \\frac{1}{2} B_i\\, \\varepsilon^{i}_{\\ jk}\\,\\mathrm{d}{x}^j\\wedge\\mathrm{d}{x}^k\n\\end{equation*}\nwith electric field ${\\bf E}$ and magnetic field ${\\bf B}$.\n\n\n\\rem{2.10.2} The source free Maxwell equations viz.\n\\begin{equation*}\n    \\nabla\\cdot {\\bf B} \\= 0 \\quad\\textrm{and}\\quad \\nabla\\times{\\bf E} + \\frac{\\mbox{$\\partial$} {\\bf B}}{\\mbox{$\\partial$} t} \\= 0\n\\end{equation*}\nare given by \n\\begin{equation*}\n    \\mathrm{d}{F} \\= 0\\ .\n\\end{equation*}\nNotice that, due to Remark 2.6.5, the above condition is trivially satisfied here.\n\n\\rem{2.10.3} The other two Maxwell equations with source viz.\n\\begin{equation*}\n    \\nabla\\cdot{\\bf E} \\= \\rho \\quad\\textrm{and}\\quad \\nabla\\times{\\bf B} - \\frac{\\mbox{$\\partial$} {\\bf E}}{\\mbox{$\\partial$} t} \\= {\\bf J}\n\\end{equation*}\nwith {\\it charge density} $\\rho$ and {\\it current density} ${\\bf J}$ is given by \n\\begin{equation*}\n    \\delta F \\= j \\quad\\textrm{with}\\quad j \\= -\\rho\\,\\mathrm{d}{t} + J_i\\,\\mathrm{d}{x}^i \\in \\Omega^1(\\mathds R^{1,3})\\ , \n\\end{equation*}\nwhere the {\\it codifferential} $\\delta := *\\mathrm{d}*$.\n\n\\rem{2.10.4} An important feature of the Maxwell theory is that it possesses gauge symmetry i.e. the transformation \n\\begin{equation*}\n    A \\rightarrow A + \\mathrm{d}{f}\n\\end{equation*}\nfor all $f\\in C^\\infty(\\mathds R^{1,3})$ leaves the field strength $F$ invariant. There are many ways to fix this redundancy by the so called gauge-fixing. On Minkowski space we can always work in the so called ``temporal gauge\" where $A_0 = 0$ (see Chapter 6 of \\cite{Baez&Muniain}).\n\n\\rem{2.10.5} Noticing that $\\delta^2 = \\pm *\\mathrm{d}^2* = 0$ and applying $\\delta$ on the Maxwell equations with source we arrive at the following {\\it continuity equation}:\n\\begin{equation*}\n    0 \\= \\delta^2 F \\= \\delta j \\quad \\implies \\quad \\frac{\\mbox{$\\partial$} \\rho}{\\mbox{$\\partial$} t} + \\nabla\\cdot{\\bf J} \\= 0\\ .\n\\end{equation*}\n\n\n\n\n\n\n\\chapter{Gauge theory}\n\\label{Chapter3}\n\n\\justifying\n\nAll the four known forces of nature viz. gravity, electromagnetism, weak and strong nuclear forces can be described (at least classically) in terms of a gauge theory. The study of modern gauge theory requires the notion of principal bundles and subsequent structures on it. While it is possible to study a lot of physics, including Yang--Mills theory, with just vector bundles alone, e.g. as in \\cite{Baez&Muniain}, a lot of deep physics arising from the underlying topology of the base manifold can not be fully appreciated without making reference to the principle bundle. We review in this chapter the construction of gauge theory from principal bundles without bothering about proofs of the statements, which can be found in \\cite{Isham}. For a quick review of Yang--Mills theory one may refer to \\cite{Bleecker} or a nice review article by Daniel and Viallet \\cite{DV80}.\n\n\\vspace{12pt}\n\\section{Principal $G$-bundles}\n\\vspace{2pt}\n\\defn{3.1.1} A bundle $(P,M,\\pi)$ is called a {\\it principle $G$-bundle} and denoted diagrammatically as \n\\begin{tikzcd}\n   G \\arrow[r, \"r_g\"] &P \\arrow[d, \"\\pi\"] \\\\\n                               &M\n\\end{tikzcd}\nif there exist a free right action of the Lie group $G$ on $P$ and if there exist a bundle isomorphism between $(P,M,\\pi)$ and $(P,P\/G,\\pi_0)$ with the canonical projection $\\pi_0$ on the coset space $P\/G$, meaning that the diagram\n\\begin{tikzcd}\n   P \\arrow[d, \"\\pi\"] \\arrow[r, \"\\varphi\"] &P \\arrow[d, \"\\pi_0\"] \\\\\n   M \\arrow[r, \"\\psi\"]  &P\/G\n\\end{tikzcd}\ncommutes.\n\n\\rem{3.1.1} Notice that the structure group in this case is $G$. Furthermore, the fibre $\\pi^{-1}(\\{m\\})$ for any $m \\in M$ is diffeomorphic to $G$, but it does not have a canonical group structure. Another way to put this would be to say that the Manifold $M$ has a $G$ fibre attached to all its points except that the identity element is forgotten. Also, notice here that the projection map is insensitive to the $G$-action.\n\n\\ex{3.1.1} The simplest example of a principle bundle is $(G\\times M, M,\\pi)$ with the right action given by $(x,g_0)g := (x,gg_0)$.\n\n\\ex{3.1.2} An important example of a principle $G$-bundle is the so called {\\it frame bundle} $LM$ over an $n$-dimensional manifold $M$, which is a collection of basis-frames:\n\\begin{equation*}\n    L_mM\\ :=\\ \\{ (b_1,\\ldots,b_n) ~|~ \\<b_1,\\ldots,b_n\\> = T_mM \\} \\cong GL(n,\\mathds R)\n\\end{equation*}\ncorresponding to the tangent bundle $TM$ attached at each point $m$ of the manifold i.e.\n\\begin{equation*}\n    LM\\ :=\\ \\bigcup\\limits_{m\\in M} L_mM\\ .\n\\end{equation*}\nThe free right action of any $g \\in GL(n,\\mathds R)$ on a given $(b_1,\\ldots,b_n) \\in L_mM$ is defined as\n\\begin{equation*}\n    (b_1,\\ldots,b_n)g\\ :=\\ (b_i\\,g^i_{\\ 1},\\ldots,b_i\\,g^i_{\\ n})\\ .\n\\end{equation*}\n\n\\ex{3.1.3} Another important example of a principle bundle is $(G,G\/H,\\pi)$ where $H$ acts freely from right on $G$ via group multiplication resulting into orbits of cosets $G\/H$. A famous example of this kind is the {Hopf bundle} represented diagrammatically as follows: \n\\begin{tikzcd}\n   U(1) \\arrow[r] &SU(2) \\arrow[d] \\\\ \n                  &SU(2)\/U(1)\n\\end{tikzcd} \n$\\equiv$\n\\begin{tikzcd}\n   S^1 \\arrow[r] &S^3 \\arrow[d] \\\\\n                 &S^2\n\\end{tikzcd}\n\n\\defn{3.1.2} A {\\it principle morphism} between a pair of bundles $(P,M,\\pi)$ and $(P',M',\\pi')$ is a bundle morphism $(\\varphi,\\psi)$ that satisfies \n\\begin{equation*}\n    \\varphi(pg) \\= \\varphi(p)g \\quad\\forall\\quad p \\in P \\quad\\textrm{and}\\quad g\\in G\\ .\n\\end{equation*}\n\n\\rem{3.1.2} The notion of trivialization, both local $(U,\\varphi)$ as well as global $(G\\times M,M,\\pi_1)$, follows similar to the general theory of fibre bundles that we saw before, albeit with this extra condition of principle morphism. In a similar way, the idea of transition functions $g_{ij}$ between any two such overlapping local trivializations $(U_i,\\varphi_i)$ and $(U_j,\\varphi_j)$ and the corresponding structure group carries over. One can also define smooth sections on a principle bundle $P$ just like before.\n\n\\rem{3.1.3} The set of all principal morphisms between a bundle $P$ to itself forms a group $Aut(P)$ called the {\\it automorphism group} of the principle bundle $P$. In the case of a trivial bundle $P=G\\times M$ we have that $Aut(P)\\cong C^\\infty (M,G)$, where the latter is the group of {\\it gauge transformations}. \n\n\\thm{3.1.1} A principle $G$-bundle $(P,M,\\pi)$ is trivial if and only if it possesses a smooth section $\\sigma: M \\rightarrow P$.\n\n\\rem{3.1.4} An illustrative counter-example of the above theorem is the fact that the frame bundle $LS^2$ over the $2$-sphere is not trivial because there does not exist nowhere vanishing smooth vector fields, and hence a basis, on $TS^2$ (look at the north or south poles). This result is famously summarized as ``sphere can not be combed\".\n\n\\rem{3.1.5} The topological properties, such as twisting, of the base manifold $M$ is intrinsically linked with that of the principal bundle $P$ and also carries over to the below defined associated bundles.\n\n\\vspace{12pt}\n\\section{Associated bundles}\n\\vspace{2pt}\n\n\\defn{3.2.1} The $G$-product of two spaces $X$ and $Y$ admitting right action of $G$, denoted $X\\times_G Y$, is the space of orbits under this action on the Cartesian product $X\\times Y$. In other words it is defined via the following equivalence relation\n\\begin{equation*}\n    (x,y) \\sim (x',y')\\quad \\mathrm{iff}\\quad \\exists\\ g \\in G :\\ x' \\= xg \\quad\\textrm{and}\\quad y' \\= yg\\ ,\n\\end{equation*}\nwhere the equivalence class is denoted as $[x,y]$.\n\n\\defn{3.2.2} For a given principle $G$-bundle $(P,M,\\pi)$ and a manifold $F$ one defines the {\\it associated bundle} $P_F$ by\\footnote{Notice how we have employed left $G$-action to define its right action (see Remark 2.3.7).}\n\\begin{equation*}\n    P_F\\ :=\\ P\\times_G F \\quad \\mathrm{where} \\quad (p,v)g\\ :=\\ (pg,g^{-1}v)\n\\end{equation*}\nand the projection\n\\begin{equation*}\n    \\pi_F :\\ P_F \\to M\\ , \\quad \\pi_F([p,v])\\ :=\\ \\pi(p)\\ .\n\\end{equation*}\n\n\\rem{3.2.1} It can be shown that the associated bundle $(P_F,M,\\pi_F)$ has the structure of a fibre bundle with typical fibre $F$. \n\n\\ex{3.2.1} An important example of a fibre associated with the frame bundle $LM$ with Lie group $GL(n,\\mathds R)$ is the tensor bundle $T^{p,q}M$ with fibres $F = (\\mathds R^n)^{\\times p}\\times (\\mathds R^{n*})^{\\times q}$ where the left action of a given $g \\in GL(n,R)$ on some $v \\in F$ is given by the following representation $\\rho$\n\\begin{equation*}\n    (\\rho(g)v)^{i_1,\\ldots,i_p}_{\\qquad j_1,\\ldots,j_q}\\ :=\\ v^{i_1',\\ldots,i_p'}_{\\qquad j_1',\\ldots,j_q'}\\,(g)^{i_1}_{\\ i_1'}\\ldots(g)^{i_p}_{\\ i_p'}\\,(g^{-1})^{j_1'}_{\\ j_1}\\ldots(g^{-1})^{j_q'}_{\\ j_q}\\ .\n\\end{equation*}\nAnother very useful generalization of this is the so called {\\it tensor of density $\\omega$} that admits a left-action via the following representation\n\\begin{equation*}\n    (\\rho(g)v)^{i_1,\\ldots,i_p}_{\\qquad j_1,\\ldots,j_q}\\ :=\\ (\\mathrm{det} g)^\\omega\\,v^{i_1',\\ldots,i_p'}_{\\qquad j_1',\\ldots,j_q'}\\,(g)^{i_1}_{\\ i_1'}\\ldots(g)^{i_p}_{\\ i_p'}\\,(g^{-1})^{j_1'}_{\\ j_1}\\ldots(g^{-1})^{j_q'}_{\\ j_q}\\ .\n\\end{equation*}\n\n\\defn{3.2.3} Given a principle morphism $(\\varphi,\\psi)$ between principal $G$-bundles $(P,M,\\pi)$ and $(P',M',\\pi')$ one defines an {\\it associated bundle morphism} between the associated bundles $P_F\\times_G F$ and $P'\\times_G F$ by\n\\begin{equation*}\n    \\varphi_F([p,v])\\ :=\\ [\\varphi(p),v]\\ ,\n\\end{equation*}\nwhich is well-defined since\n\\begin{equation*}\n    \\varphi_F([pg,g^{-1}v]) \\= [\\varphi(pg),g^{-1}v] \\= [\\varphi(p)g,g^{-1}v] \\= [p,v]\\ .\n\\end{equation*}\n\n\\rem{3.2.2} An associated bundle $P_F$ is called {\\it trivial} if the underlying principal bundle $P$ is trivial. An important point to note here is that a trivial associated bundle is a trivial fibre bundle, but the converse is not true.\n\n\\defn{3.2.4} Let $H \\subset G$ be a closed subgroup of $G$ while $P$, resp., $P'$ are principal $G$-bundle, resp., principal $H$-bundle defined over the same base space. If there exist a principal morphism $\\varphi$ with respect to $H$ i.e. \n\\begin{equation*}\n    \\varphi(ph) \\= \\varphi(p)h\n\\end{equation*}\nfor all $h\\in H$ then $P$ is called a $G$-extension of $H$ while $P'$ is called a $H$-restriction of $P$.\n\n\\rem{3.2.3} While there always exists an extension $P$ of a given $P'$ as defined above, the converse is not always true. This has important ramifications both in Riemannian geometry as well as in Yang--Mills theory; for the latter this is related to the important question of the spontaneous breakdown of the internal symmetry group from $G$ down to $H$.\n\n\\thm{3.2.1} A principle $G$-bundle $(P,M,\\pi)$ can be restricted to a closed subgroup $H\\subset G$ iff the bundle $(P\/H,M,\\pi_0)$ admits a smooth section.\n\n\\rem{3.2.4} An important application of the above theorem is that in (pseudo-)Riemannian geometry one can always have a Riemannian metric defined on any $n$-diemensional manifold $M$ as $LM\/SO(n)$ for the closed subgroup $SO(n)\\subset GL(n,\\mathds R)$ always admits a smooth section; the same is not always true for a pseudo-Riemannian metric as, e.g., $LM\/SO(1,n{-}1)$ for the closed subgroup $SO(1,n)\\subset GL(n,\\mathds R)$ does not always admits a smooth section because of some topological restrictions.\n\n\\thm{3.2.2} There exists a one-to-one $G$-equivariant correspondence between sections of an associated bundle $(P_F,M,\\pi_F)$ and map $\\phi: P \\to F$ satisfying\n\\begin{equation*}\n    \\phi(pg) \\= g^{-1}\\phi(p) \\quad\\forall\\quad p \\in P \\quad\\textrm{and}\\quad g \\in G\n\\end{equation*}\nwhere the section $\\sigma_\\phi$ corresponding to $\\phi$ arises as\n\\begin{equation*}\n    \\sigma_\\phi(x)\\ :=\\ [p,\\phi(p)] \\quad\\textrm{for}\\quad p\\in \\pi^{-1}(x)\\ .\n\\end{equation*}\n\n\\defn{3.2.5} Given two local trivializing sections\\footnote{This refers to the existence of a local trivialization $(U_i,\\varphi_i)$ such that $\\varphi_i^{-1}(x,g) := \\sigma_i(x)g$.} $\\sigma_i: U_i\\subset M \\to P$ and $\\sigma_j: U_j\\subset M \\to P$ on a given principal $G$-bundle $P$ with $U_i\\cap U_j \\neq 0$ there exists some local {\\it gauge functions} $\\Lambda_{ij}: U_i\\cap U_j \\to G$ such that \n\\begin{equation*}\n    \\sigma_j(x) \\= \\sigma_i(x)\\Lambda_{ij}(x) \\quad\\forall\\quad x\\in U_i\\cap U_j\\ .\n\\end{equation*}\nOne defines {\\it local representatives} $s_i: U_i \\to F$ for a section $s$ of $P_F$ corresponding to these local sections $\\sigma_i$ by \n\\begin{equation*}\n    s_i(x)\\ :=\\ \\phi_{s_i}(\\sigma(x))\\ .\n\\end{equation*}\n\n\\rem{3.2.5} The local representatives also satisfy the same gauge transformation rule as above i.e.\n\\begin{equation*}\n    s_j(x) \\= s_i(x)\\Lambda_{ij}(x)\\ .\n\\end{equation*}\nIt turns out that these gauge functions $\\Lambda_{ij}$ are nothing but the transitions functions $g_{ij}$ for the local trivializations arising from $\\sigma_i$ and $\\sigma_j$. For this thesis, we will identify the gauge group with the structure group.\n\n\\vspace{12pt}\n\\section{Connections}\n\\vspace{2pt}\n\n\\rem{3.3.1} Let $(P,M,\\pi)$ be a principal $G$-bundle. Then there exists a Lie algebra homomorphism between $L(G)\\cong T_eG$ and $\\Gamma^\\infty(TP)$ given by\n\\begin{equation*}\n    i :\\ T_eG \\to \\Gamma^\\infty(TP)\\ ,\\quad A \\to X^A\\ ,\n\\end{equation*}\nwhere the induced vector-field $X^A$ arises from the right action of $G$ on $P$ as follows\n\\begin{equation*}\n    X^A[f]\\ :=\\ f'(p\\,\\mathrm{exp}(tA))(t{=}0)\\ .\n\\end{equation*}\n\n\n\\defn{3.3.1} For a given principal $G$-bundle $(P,M,\\pi)$ one defines the {\\it vertical subspace} at $p\\in P$, denoted by $V_pP$, as follows\n\\begin{equation*}\n    V_pP\\ :=\\ \\{X \\in T_pP ~|~ \\pi_*(X) \\= 0  \\}\n\\end{equation*}\nThe {\\it horizontal subspace} $H_pP$ arises as the orthogonal complement of $V_pP$ in the tangent space $T_pP$:\n\\begin{equation*}\n    T_pP \\= V_pP \\oplus H_pP\\ .\n\\end{equation*}\n\n\\rem{3.3.2} It can be shown that\n\\begin{equation*}\n    X^A_p \\in V_pP \\quad\\forall\\quad p\\in P\\ .\n\\end{equation*}\nThis means that the above map $i$ induces an isomorphism $i_p$ between $T_eG$ and $V_pP$.\n\n\n\\defn{3.3.2} A {\\it connection} on a principle $G$-bundle $(P,M,\\pi)$ is a smooth assignment of $H_pP$ to each point $p\\in P$ such that\n\\begin{enumerate}[(a)]\n    \\item $T_pP \\= V_pP \\oplus H_pP$\n    \\item $(r_g)_*(H_pP) \\= H_{pg}P$\n    \\item every $X_p \\in T_pP$ has a unique decomposition according to (a) as follows\n    \\begin{equation*}\n        X_p \\= ver(X_p) + hor(X_p)\\ ,\n    \\end{equation*}\n    where $ver(X_p) \\in V_pP$ and $hor(X_p) \\in H_pP$.\n\\end{enumerate}\n\n\\rem{3.3.3} A more technically convenient way to deal with connections is by associating them with a Lie-algebra valued one-form $\\omega$ in the following way\n\\begin{equation*}\n    \\omega_p(X) \\= i_p^{-1}(ver(X))\\ ,\n\\end{equation*}\nwhich, in turn, imposes following conditions on $\\omega$:\n\\begin{enumerate}[(i)]\n    \\item $\\omega_p(X^A) = A$ for all $p\\in P$ and $A\\in L(G)$\n    \\item $(r_g)^*\\omega = \\mathrm{Ad}_{g^{-1}}\\omega$, i.e., $(r_g^*\\omega)_p(X) = \\mathrm{Ad}_{g{-1}}(\\omega_p(X))$ for all $X\\in T_pP$\n    \\item $X\\in H_pP$ iff $\\omega_p(X) = 0$.\n\\end{enumerate}\n\n\\defn{3.3.3} For a local trivializing section $\\sigma: U\\subset M \\to P$ of a principal $G$-bundle $P$ its local {\\it gauge} fields arises from the following local representative of a Lie-algebra valued one-form $\\omega$:\n\\begin{equation*}\n    \\omega^U\\ :=\\ \\sigma^*\\omega\\ .\n\\end{equation*}\nWe label the $1$-form components of such local  gauge fields in Yang--Mills theory as\n\\begin{equation*}\n    A_\\mu \\equiv (\\omega^U)_\\mu\\ ,\n\\end{equation*}\nand in general relativity as \n\\begin{equation*}\n    \\Gamma_\\mu \\equiv (\\omega^U)_\\mu\\ .\n\\end{equation*}\n\n\\thm{3.3.1} An explicit form of the local Yang--Mills field $\\omega^U$ for the local trivialization $\\varphi: \\pi^{-1}(U) \\to U\\times G$ arising from $\\sigma$ with $(\\alpha,\\beta) \\in T_{(x,g)}(U\\times G) \\cong T_xU \\oplus T_gG$ is given by\n\\begin{equation*}\n    (\\varphi^*\\omega)_{(x,g)} \\= \\mathrm{Ad}_{g^{-1}} (\\omega^U_x(\\alpha)) + \\< \\Omega_l, \\beta \\>_g\\ ,\n\\end{equation*}\nwhere $\\Omega_l$ is the Maurer--Cartan form.\n\n\\ex{3.3.1} An important example of such a local representative $\\omega^U$ is in the case of the frame bundle $LM$ for an $n$-dimensional manifold $M$ with a given chart $(U,\\phi)$. For a given local section $\\sigma: U \\subset M \\to LM$ defined by\n\\begin{equation*}\n    \\sigma(m)\\ :=\\ ((\\mbox{$\\partial$}_1)_m,\\ldots,(\\mbox{$\\partial$}_n)_m,) \n\\end{equation*}\nthe corresponding $\\omega^U$ has components (the three indices below are divided into two indices $i,j$ for the Lie-algebra $L(GL(n,\\mathds R))$ and one $1$-form index $\\mu$)\n\\begin{equation*}\n    (\\omega^U)^i_{\\ j\\mu}\\ =:\\ \\Gamma^i_{\\ jk} \\quad\\textrm{with}\\quad i,j,\\mu = 1,\\dots,n\n\\end{equation*}\nwhere $\\Gamma^i_{\\ jk}$ is the famous {\\it Christoffel symbol} of this {\\it Levi-Civita or affine connection}\\footnote{This relies on a choice of the natural metric on $M$, which is akin to choosing a basis $G^{\\ a}_b$ of the Lie-algebra $L(GL(n,\\mathds R))$ with components $G^{\\ a}_b)^c_{\\ d} := \\delta^c_b\\delta^a_d$.} that is widely used in Riemannian geometry and general relativity.\n\n\\thm{3.3.2} Let $(P,M,\\pi)$ be a principal $G$-bundle with $U_i,U_j \\subset M$ such that $U_i\\cap U_j \\neq 0$. Further, let $A_\\mu^{(i)}$ and $A_\\mu^{(j)}$ be local gauge functions arising from given local trivializing sections $\\sigma_i: U_i \\to P$ and $\\sigma_j: U_j \\to G$ respectively. The transformation of these fields under the action of gauge functions $\\Lambda_{ij}: U_i\\cap U_j \\to G$ is, for any $m\\in M$, given by \n\\begin{equation*}\n    A_\\mu^{(j)}(m) \\= \\mathrm{Ad}_{\\Lambda_{ij}(m)^{-1}}\\left(A_\\mu^{(i)}(m)\\right) + (\\Lambda_{ij}^*\\Omega_l)_\\mu(m)\\ .\n\\end{equation*}\n\n\\rem{3.3.4} We notice that for matrix Lie groups the above transformation rule takes the following simple form:\n\\begin{equation*}\n    A_\\mu^{(j)}(m) \\= \\Lambda_{ij}(m)^{-1}\\left(A_\\mu^{(i)}(m)\\right)\\Lambda_{ij}(m) + \\Lambda_{ij}(m)^{-1}\\mbox{$\\partial$}_\\mu \\Lambda_{ij}(m)\\ .\n\\end{equation*}\n\n\\rem{3.3.5} This gauge transformation behaviour is what prevents a gauge field $\\omega^U$ from being global i.e. $A ~\\mathrm{or}~ \\Gamma \\not\\in \\Omega^1(M)$. In particular, this explains why the Christoffel symbol is not a tensor! This is because of the following transformation rule between any two given charts $(U,\\phi)$ and $(U',\\phi')$ on a $4$-dimensional manifold $M\\ni m$ with $\\phi(m) = \\{ x^0,\\ldots,x^4 \\}$ and $\\phi'(m) = \\{ x'^{0},\\ldots,x'^{4} \\}$ where the indices like $\\alpha$ below takes temporal $\\alpha {=} 0$ and spatial $\\alpha {=} 1,2,3$ values:\n\\begin{equation*}\n    \\Gamma^{'\\alpha}_{\\ \\beta\\mu} \\= \\frac{\\mbox{$\\partial$} x'^{\\alpha}}{\\mbox{$\\partial$} x^{\\tilde{\\alpha}}}\\,\\frac{\\mbox{$\\partial$} x^{\\tilde{\\beta}}}{\\mbox{$\\partial$} x'^{\\beta}}\\,\\frac{\\mbox{$\\partial$} x^{\\tilde{\\mu}}}{\\mbox{$\\partial$} x'^{\\mu}}\\,\\Gamma^{\\tilde{\\alpha}}_{\\ \\tilde{\\beta}\\tilde{\\mu}} + \\frac{\\mbox{$\\partial$} x'^{\\alpha}}{\\mbox{$\\partial$} x^{\\lambda}}\\,\\frac{\\mbox{$\\partial$} x^\\lambda}{\\mbox{$\\partial$} x'^{\\beta}\\mbox{$\\partial$} x'^{\\mu}}\\ .\n\\end{equation*}\nAn explicit expression for $\\Gamma^{\\alpha}_{\\ \\beta\\mu}$ for a given orthonormal basis of vector fields $\\{ \\mbox{$\\partial$}_0,\\ldots,\\mbox{$\\partial$}_4 \\}$ on $M$ equipped with a pseudo-Riemannian metric $g$ with components $g(\\mbox{$\\partial$}_\\mu,\\mbox{$\\partial$}_\\nu) = g_{\\mu\\nu}$ is given by\n\\begin{equation*}\n    \\Gamma^{\\alpha}_{\\ \\beta\\mu} \\= \\frac{1}{2}\\,g^{\\alpha\\lambda} \\left( g_{\\beta\\lambda,\\mu} + g_{\\mu\\lambda,\\beta} - g_{\\beta\\mu,\\lambda} \\right)\\ ,\n\\end{equation*}\nwhere $g_{{\\beta\\mu,\\lambda}} := \\mbox{$\\partial$}_\\lambda[g_{\\beta\\mu}]$.\n\n\\vspace{12pt}\n\\section{Parallel transport}\n\\vspace{2pt}\n\n\\defn{3.4.1} Owing to the fact that $\\pi_*: H_pP \\to T_{\\pi(p)}M$ is an isomorphism there exists the notion of a unique vector field for a given $X\\in \\vecF{M}$ known as the {\\it horizontal lift} of $X$ and is denoted as $X^\\uparrow$. This satisfies, for all $p\\in P$, the following conditions\n\\begin{enumerate}[(i)]\n    \\item $\\pi_*(X_p^\\uparrow) = X_{\\pi(p)}$\n    \\item $ver(X_p^\\uparrow) = 0$.\n\\end{enumerate}\n\n\\rem{3.4.1} The act of horizontal lifting is $G$-equivariant i.e. $(r_g)_*(X_p^\\uparrow) = X_{pg}^\\uparrow$.\n\n\\defn{3.4.2} A {\\it horizontal lift} of a smooth path $\\gamma: [a,b]\\subset \\mathds R \\to M$ is another path $\\gamma^\\uparrow: [a,b] \\to P$ which is horizontal i.e. $hor(\\gamma^\\uparrow) = 0$ such that $\\pi(\\gamma^\\uparrow(t)) = \\gamma(t)$ for all $t\\in [a,b]$.\n\n\\thm{3.4.1} For each point $p\\in \\pi^{-1}(\\{\\gamma(a)\\})$ there exist a unique horizontal lift $\\gamma^\\uparrow: [a,b] \\to P$ of $\\gamma$ such that $\\gamma^\\uparrow(a)=p$.\n\n\\rem{3.4.2} Given a path $\\gamma: [a,b] \\to M$ and another path $\\beta: [a,b] \\to P$ which projects down to $\\gamma$ i.e. $\\pi(\\beta(t)) = \\gamma(t)$ for all $t\\in [a,b]$, there exits some unique function $g: [a,b]\\to G$ such that \n\\begin{equation*}\n    \\gamma^\\uparrow(t) \\= \\beta(t)\\,g(t) \\quad\\forall\\quad t\\in [a,b]\\ .\n\\end{equation*}\n\n\\thm{3.4.2} The unique path $g: [a,b] \\to G$ defined above satisfies the following first order ODE in terms of a Lie-algebra valued $1$-form $\\omega$:\n\\begin{equation*}\n    \\mathrm{Ad}_{g(t)^{-1}*}\\left(\\omega_\\beta(t)(X_{\\beta,\\beta(t)}) \\right) + \\< \\Omega_l,X_g\\>_{g(t)}\\ ,\n\\end{equation*}\nwhere $X_{\\beta,\\beta(t)}$ is the tangent vector to the curve $\\beta$ at point $\\beta(t)$.\n\n\\rem{3.4.3} For a matrix Lie group $G$ and a local gauge field $\\omega^U = \\sigma^*\\omega$ arising from a local trivializing section $\\sigma: U\\subset M \\to P$ the above ODE takes the following simple form\n\\begin{equation*}\n    \\dot{g}(t) \\= - A_\\mu(\\gamma(t))\\,\\dot{\\gamma}^\\mu(t)\\,g(t)\\ ,\n\\end{equation*}\nwhere the components of the curve $\\gamma$ in a local chart has been denoted as $\\gamma^\\mu$. The solution to this ODE, for an initial condition $g(0)=g_0$, is obtained as a path-ordered exponential in the following way\n\\begin{equation*}\n \\begin{aligned}\n    g(t) &\\= \\left({\\bf P}\\,\\mathrm{exp}\\left( -\\int\\limits_a^t A_\\mu(\\gamma(s))\\,\\dot{\\gamma}^\\mu(s)\\mathrm{d}{s} \\right) \\right)g_0 \\\\\n    &\\= g_0 - \\left(\\int\\limits_a^t A_\\mu(\\gamma(s))\\,\\dot{\\gamma}^\\mu(s)\\mathrm{d}{s}\\right)g_0 \\\\\n    &\\qquad\\qquad + \\left(\\int\\limits_a^t\\mathrm{d}{s_1}\\int_a^{s_1}\\mathrm{d}{s_2}\\, A_{\\mu_1}(\\gamma(s_1))\\,A_{\\mu_2}(\\gamma(s_2))\\,\\dot{\\gamma}^{\\mu_1}(s_1)\\,\\dot{\\gamma}^{\\mu_2}(s_2)\\right)g_0 - \\ldots\\ .\n \\end{aligned}\n\\end{equation*}\n\n\\rem{3.4.4} Form the above result we observe that the local expression for the horizontal lift $\\gamma^\\uparrow$ of the path $\\gamma: [a,b] \\to U \\subset M$ is given by\n\\begin{equation*}\n    \\gamma^\\uparrow(t) \\= \\sigma(\\gamma(t))\\left({\\bf P}\\,\\mathrm{exp}\\left( -\\int\\limits_a^t A_\\mu(\\gamma(s))\\,\\dot{\\gamma}^\\mu(s)\\mathrm{d}{s} \\right) \\right)g_0\\ .\n\\end{equation*}\n\n\\defn{3.4.3} The {\\it parallel transport} along a path $\\gamma: [a,b] \\to M$ is defined by the following map\n\\begin{equation*}\n    T_\\gamma :\\ \\pi^{-1}(\\gamma(a)) \\to \\pi^{-1}(\\gamma(b))\\ ,\\quad p \\mapsto \\gamma^\\uparrow_p(b)\\ ,\n\\end{equation*}\nwhere $\\gamma^\\uparrow$ is the unique horizontal lift of $\\gamma$ passing through $p\\in \\pi^{-1}(\\gamma(a))$.\n\n\\rem{3.4.5} We observe that $T_\\gamma$ is a bijection on fibres and thus on $G$. An interesting thing happens when $\\gamma: [a,b] \\to M$ is a loop i.e. $\\gamma(a) = \\gamma(b)$; one obtains a natural map from loops based at $\\gamma(a) \\in M$ to elements of $G$. The subgroup of all elements of $G$ that can be obtained in this way is called the {\\it holonomy} group of the principle bundle $P$. This plays an important role in understanding the relation between certain topological properties of $M$ with the connection.\n\n\\defn{3.4.4} Let $(P,M,\\pi)$ be a principal $G$-bundle equipped with a connection $1$-form $\\omega$. Furthermore, let $(P_F,M,\\pi_F)$ be associated with $P$ via the left action of $G$ on $F$. A {\\it vertical subspace} of $T_{[p,v]}P_F$ is defined analogous to that of principal bundle i.e.\n\\begin{equation*}\n    V_{[p,v]}P_F\\ :=\\ \\{ X \\in T_{[p,v]}P_F ~|~ \\pi_{F*}X = 0 \\}\\ .\n\\end{equation*}\nSimilarly, the {\\it horizontal subspace} $H_{[p,v]}P_F$ can be defined as the orthogonal complement of $V_{[p,v]}P_F$.\n\n\\defn{3.4.5} The {\\it horizontal lift} of a path $\\gamma: [a,b] \\to M$ to the associated bundle $P_F$ and passing through $[p,v] \\in \\pi_F^{-1}(\\gamma(a))$ is defined as\n\\begin{equation*}\n    \\gamma_F^\\uparrow(t) \\ :=\\ [\\gamma^\\uparrow(t),v]\n\\end{equation*}\nwhere $\\gamma^\\uparrow(a) = p$.\n\n\\rem{3.4.6} One can define {\\it parallel transport} $T_\\gamma$ along a path $\\gamma: [a,b] \\to M$ on the associated bundle $P_F$ analogous to the principal bundle case using the above definition of the horizontal lifting. For associated vector bundles $P_V$ with the vector space $V$ admitting a linear representation of $G$, this notion then facilitates the following definition of the covariant derivative. \n\n\\subsection{Covariant derivative}\n\n\\defn{3.4.6} The covariant derivative of a section $\\psi: M \\to P_V$ of an associated vector bundle $P_V$ along a path $\\gamma: [0,\\varepsilon] \\to M$ with $\\varepsilon>0$ at $m_0=:\\gamma(0)$ is defined by\n\\begin{equation*}\n    D_{X_{\\gamma,\\gamma(0)}}\\psi\\ :=\\ \\lim_{t\\to 0}\\left( \\frac{T_\\gamma(\\psi(\\gamma(t))) - \\psi(m_0)}{t} \\right)\\in \\pi_V^{-1}(m_0)\\ .\n\\end{equation*}\n\n\\rem{3.4.7} The covariant derivative $D_X$, for $X\\in T_mM$, has following algebraic properties for any section $\\psi,\\widetilde{\\psi}\\in \\Gamma^\\infty(P_V)$:\n\\begin{enumerate}[(i)]\n    \\item $D_{fX+Y}\\psi \\= f\\,D_X\\psi + D_Y\\psi$ for all $f\\in C^\\infty(M)$ and $X,Y\\in T_mM$\n    \\item $D_X(\\psi + \\widetilde{\\psi}) \\= D_X\\psi + D_X\\widetilde{\\psi}$ for all $X\\in T_mM$\n    \\item $D_X(f\\psi) \\= X[f]\\,\\psi + f\\,D_X\\psi$ for all $f\\in C\\infty(M)$ and $X\\in T_mM$.\n\\end{enumerate}\nThis following more generic notion of a covariant derivative is related to this one, as we will see below.\n\n\\defn{3.4.7} The {\\it exterior covariant derivative} of a $k$-form $\\omega \\in \\Omega^k(P)$ on a principal bundle $P$ is a horizontal $(k{+}1)$-form defined by\n\\begin{equation*}\n    D\\omega(X_1,\\ldots,X_{k+1})\\ :=\\ \\mathrm{d}{\\omega}(hor(X_1),\\ldots,hor(X_{k+1}))\n\\end{equation*}\nfor a given set of vector fields $X_1,\\ldots,X_{k+1} \\in \\vecF{P}$.\n\n\\rem{3.4.8} This notion of covariant derivative can be extended to an associated vector bundle $P_V$ discussed before with the aid of a given $G$-equivariant map $\\phi: P \\to F$ that has an associated section $\\sigma_\\phi \\in \\Gamma^\\infty(P_V)$ (see Theorem 3.2.2) as follows\n\\begin{equation*}\n    D\\phi\\ :=\\ \\mathrm{d}{\\phi}\\circ hor\\ .\n\\end{equation*}\nOne can show, for any $X\\in \\vecF{P}$ and a given connection $\\omega$ on $P$, that\n\\begin{equation*}\n    F\\ni D\\phi(X) \\equiv D_X\\phi\\ :=\\ \\mathrm{d}{\\phi}(X) + \\omega(X)\\phi\\ ,\n\\end{equation*}\nwhere we notice that $\\phi$ are $F$-valued functions on $P$. Furthermore, one can pull this definition back to $M$ using any local trivializing map $\\sigma: U\\subset M\\to P$ and noticing the fact the pull-back operation is natural:\n\\begin{equation}\n    \\sigma^*(D\\phi)(X)\\ :=\\ \\mathrm{d}{\\sigma^*\\phi}(X) + \\sigma^*(\\omega)(X)(\\sigma^*\\phi) \\quad\\forall\\quad X \\in \\vecF{P}\\ .\n\\end{equation}\n\n\\ex{3.4.1} For a given local orthonormal coframe of vector fields $\\{\\mbox{$\\partial$}_\\mu \\}$ with $\\mu=0,1,2,3$ on a $4$-dimensional Lorentzian manifold $M$, e.g. the Minkowski space $\\mathds R^{1,3}$, the covariant derivative $D_{\\mbox{$\\partial$}_\\mu}=:D_\\mu$ of the above-mentioned section $\\phi$ (or $\\sigma_\\phi$ to be precise) is given in terms of local gauge fields $A_\\mu$ as\n\\begin{equation*}\n    D_\\mu\\phi\\ :=\\ \\mbox{$\\partial$}_\\mu\\phi + A_\\mu\\phi\\ .\n\\end{equation*}\n\n\\ex{3.4.2} The covariant derivative $D_\\mu =: \\nabla_\\mu$ of the tensor bundle $T^{p,q}(M)$ associated with the frame bundle $LM$ on $M$ is given in terms of the Levi--Civita connection and can be expressed in terms of the Chritoffel symbols. For example, covariant derivative of $T \\in T^{1,2}(M)$ with components $T^\\mu_{\\alpha\\beta} := T(\\mathrm{d}{x}^\\mu,\\mbox{$\\partial$}_\\alpha,\\mbox{$\\partial$}_\\beta)$ is given by\n\\begin{equation*}\n    D_\\rho\\,T^\\mu_{\\alpha\\beta} \\= \\mbox{$\\partial$}_\\rho T^\\mu_{\\alpha\\beta} + \\Gamma^\\mu_{\\lambda\\rho}T^\\lambda_{\\alpha\\beta} - \\Gamma^\\lambda_{\\rho\\alpha}T^\\mu_{\\lambda\\beta} - \\Gamma^\\lambda_{\\rho\\beta}T^\\mu_{\\lambda\\alpha}\\ .\n\\end{equation*}\n\n\\rem{3.4.9} The Levi--Civita connection $\\Gamma$ is metric compatible $\\nabla g = 0$ i.e.\n\\begin{equation*}\n    \\nabla_\\mu\\, g_{\\alpha\\beta} \\= 0\\ .\n\\end{equation*}\nMoreover this connection is torsion-free i.e.\n\\begin{equation*}\n    0 \\= T(X,Y)\\ :=\\ \\nabla_XY - \\nabla_Y X - [X,Y]\n\\end{equation*}\nfor all $X,Y \\in \\vecF{M}$. Choosing vector fields $X=\\mbox{$\\partial$}_\\alpha$ and $Y=\\mbox{$\\partial$}_\\beta$ the torsion-free condition ensures that the  Christoffel symbol is symmetric in the subscript indices:\n\\begin{equation*}\n    \\Gamma^\\mu_{\\alpha\\beta} \\= \\Gamma^\\mu_{\\beta\\alpha}\\ .\n\\end{equation*}\n\n\n\\subsection{Curvature}\n\n\\defn{3.4.8} If $\\omega$ is a connection $1$-form on a principal $G$-bundle $P$ then $D\\omega=:\\Omega$ is the Lie-algebra valued {\\it curvature} $2$-form of $\\omega$.\n\n\\rem{3.4.10} It can be shown that the curvature $2$-form $\\Omega$ satisfies the Bianchi identity:\n\\begin{equation*}\n    D\\Omega \\= 0\\ .\n\\end{equation*}\n\n\\thm{3.4.3} For arbitrary pair of vector fields $X,Y \\in \\vecF{P}$ the curvature $2$-form $D\\omega$ satisfies the following Cartan structure equation\n\\begin{equation*}\n    D\\omega(X,Y)\\ :=\\ \\mathrm{d}{\\omega}(X,Y) + [\\omega(X),\\omega(Y)]\\ ,\n\\end{equation*}\nwhere we have employed the Lie bracket on $L(G)$.\n\n\\ex{3.4.3} For the Levi--Civita connection $\\Gamma$ the curvature $2$-form $D\\Gamma=:R$, valued in $L(GL(n,\\mathds R))$ for an $n$-dimensional pseudo-Riemannian manifold $M$, is known as the Riemann curvature and is given by\n\\begin{equation*}\n    R \\= \\mathrm{d}{\\Gamma} + \\Gamma\\wedge \\Gamma\\ .\n\\end{equation*}\nThis turns out to be a tensor $R \\in T^{1,3}(M)$ and in local coframe $\\{ \\mbox{$\\partial$}_\\mu \\}$ and frame $\\{ \\mathrm{d}{x}^\\mu \\}$ with $\\mu =0,1,2,3,$ of $M$ admits the following expression in terms of Christoffel symbol\n\\begin{equation*}\n    R^\\rho_{\\ \\sigma\\mu\\nu} \\= \\mbox{$\\partial$}_\\mu \\Gamma^\\rho_{\\sigma\\nu} - \\mbox{$\\partial$}_\\nu \\Gamma^\\rho_{\\sigma\\mu} + \\Gamma^\\rho_{\\mu\\lambda}\\,\\Gamma^\\lambda_{\\sigma\\nu} - \\Gamma^\\rho_{\\nu\\lambda}\\,\\Gamma^\\lambda_{\\sigma\\mu}\\ .\n\\end{equation*}\n\n\\rem{4.3.11} Riemann curvature tensor gives rise to the so called Ricci tensor $R_{\\mu\\nu} := R^\\rho_{\\ \\mu\\rho\\nu} \\in T^{0,2}(M)$ through index contraction, which in turn gives rise to the scalar curvature $R:= g^{\\mu\\nu}R_{\\mu\\nu}$ via the metric. With these tools and the insight of {\\it equivalence principle} that states \\\\ ``For any point $m\\in M$ there always exists a local chart $(U,\\phi)$ with $m\\in U$ admitting a smooth local section of orthonormal frame $\\sigma \\in \\Gamma^\\infty(LU)$ on the frame bundle $LU$; in other words every (spacetime) manifold is locally flat in a Minkowski sense\" \\\\ Albert Einstein was able to construct the following field equation relating the curvature of spacetime with its matter content\n\\begin{equation*}\n    G_{\\mu\\nu}\\ :=\\ R_{\\mu\\nu} - \\ \\frac{1}{2}g_{\\mu\\nu}R \\= \\frac{8\\pi G}{c^4} T_{\\mu\\nu}\\ ,\n\\end{equation*}\nwhere $G$ is the Newton constant, $c$ is the speed of light and $T_{\\mu\\nu}$ is the stress-energy tensor arising from the following variation of the matter action $S_m$ with respect to the matric:\n\\begin{equation*}\n    T_{\\mu\\nu} \\= -\\frac{2}{\\sqrt{-\\mathrm{det}\\,g}} \\frac{\\delta S_m}{\\delta g^{\\mu\\nu}}\\ .\n\\end{equation*}\n\n\\ex{3.4.4} A trivial solution of the Einstien equation is the Minkowski metric $\\eta_{\\mu\\nu}$ which is globally flat. Another very important solution is the FLRW metric for homogeneous and isotropic cosmology:\n\\begin{equation*}\n    g \\= -c^2\\mathrm{d}{t}^2 + a(t)^2\\left(\\frac{\\mathrm{d}{r}^2}{1-\\kappa r^2} + r^2\\mathrm{d}{\\theta}^2 + r^2\\sin^2\\theta\\mathrm{d}{\\phi}^2 \\right)\\ ,\n\\end{equation*}\nwhere $\\{r,\\theta,\\phi \\}$ are coordinates on celestial spheres with respect to an observer (like us), $a(t)$ is a scalar function called {\\it scale factor} and the parameter $\\kappa = -1$, $0$ or $1$ denotes the topology of the $3$-dimensional Euclidean space as being open, flat or closed respectively. \n\n\\ex{3.4.5} For local Yang--Mills field $A:=\\sigma^*\\omega$ the curvature $2$-form $D\\omega=:F$ is given by\n\\begin{equation*}\n    F \\= \\mathrm{d}{A} + A\\wedge A\\ .\n\\end{equation*}\n\n\\vspace{12pt}\n\\section{Yang-Mills equation}\n\\vspace{2pt}\n\n\\rem{3.5.1} A local expression for the Yang--Mills curvature $F$ on a Lorentzian manifold $M$ with orthonormal coframe $\\{\\mbox{$\\partial$}_\\mu \\}$ is given by\n\\begin{equation*}\n    F_{\\mu\\nu} \\= \\mbox{$\\partial$}_\\mu A_\\nu - \\mbox{$\\partial$}_\\nu A_\\mu - [A_\\mu,A_\\nu]\\ .\n\\end{equation*}\n\n\\rem{3.5.2} The Yang--Mills curvature transforms under a local gauge transformation $\\Lambda_{ij}(m)$ with $m \\in U_i\\cap U_j \\neq 0$ arising from two local sections $\\sigma_i: U_i \\to P$ and $\\sigma_j: U_j \\to P$ related by $\\sigma_j(m) = \\sigma_i(m)\\Lambda_{ij}(m)$ in the following way\n\\begin{equation*}\n    F^{(j)}_{\\mu\\nu}(m) \\= \\Lambda_{ij}(m)\\,F^{(i)}_{\\mu\\nu}(m)\\,\\Lambda_{ij}(m)\\ ,\n\\end{equation*}\nand is thus global i.e. $F\\in \\Omega^2(M)$.\n\n\\rem{3.5.3} The Yang--Mills equation on a Lorentzian manifold $M$ is given by\n\\begin{equation*}\n    *D(*F) \\= J\\ ,\n\\end{equation*}\nwhere the $1$-form $J \\in \\Omega^1(M)$ is the {\\it current} that arises from the presence of any source ``charge\" on the manifold.\n\n\\rem{3.5.4} The Yang--Mills action (coupling constant $g$)\n\\begin{equation*}\n    S_{YM} \\= \\frac{1}{2g^2}\\int_M \\mathrm{Tr}(F\\wedge *F)\n\\end{equation*}\ntogether with the action for the source \n\\begin{equation*}\n    S_J \\= \\frac{1}{g^2} \\mathrm{Tr}(A \\wedge *J)\n\\end{equation*}\ngives rise to the above Yang--Mills equation by variational principle.\n\n\\rem{3.5.5} Maxwell's theory of electromagnetism that we saw in the last chapter is a special case of Yang--Mills theory where the gauge group is $U(1)$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n     \\label{introduction}\n     Helioseismology uses the observed solar surface acoustic wavefield to construct \nimages of the subsurface structure of the Sun. Of particular interest\nhas been the three-dimensional (3D) modeling of time-distance \n\\cite{DJHP93} observations\nof travel-time shifts to deduce the subsurface structure of active regions \n\\cite{KDS00,ZK03}.  Assuming the travel-time shifts are due to perturbations in\nthe sound speed below the spot, a general consensus in the models has\nemerged consistent with sound-speed reductions (relative to surrounding quiet\nSun) near the surface ($\\lesssim 4$ Mm) and enhancements up to 15 Mm below \nsunspots \\cite{KDS00,CBK06}.  Using ring diagram analysis \n\\inlinecite{BAB04}  also find a lower sound speed immediately below the \nsurface and an increase in the sound speed below 7 Mm.\n\nA comparison between Fourier-Hankel analysis and time-distance results \nby \\inlinecite{B97} first prompted caution in the interpretation of acoustic \noscillation signals within sunspots. \nThe influences of strong surface magnetic fields have not been explicitly\nincluded in most helioseismic models of active regions. \n \\citeauthor{LB05ii} (2005a) and \\citeauthor{LB05i} (2005b) have shown\nthat helioseismic phase shifts observed with helioseismic\nholography vanish below a depth of about 5 Mm, \nwhen a surface (``showerglass'') phase shift\nbased on photospheric magnetic flux density, is removed from  \nthe data.\n\nOther evidence supports the possibility of strong near-surface\ncontributions to the helioseismic phase (or travel-time) shifts.\nThese include the possible contamination of surface perturbations\ninto the 3D inversions (e.g. \\opencite{K06}; \n\\opencite{CR07}). It has also been shown that\nthe reduction of $p$-mode amplitudes in magnetic regions\ncan cause travel-time shifts \\cite{RBDTZ06}. The\nsuppression of sources of wave excitation within sunspots\ncan also produce measurable shifts \\cite{HCRB07}.\n\n\\inlinecite{SBCL05} and \\inlinecite{SBC07} have found that\nphase shifts obtained from seismic holography in sunspot\npenumbrae vary with the line-of-sight angle (from vertical) \nas projected into the plane containing the magnetic field and the \nvertical direction.  A similar effect has also been noted by \n\\inlinecite{ZK06} with time distance measurements.\n\\inlinecite{SBC07} find that the  effect is dependent upon the strength and\/or \ninclination of the magnetic field. \nIn the penumbra the magnetic field strength \ndecreases as the magnetic field angle from vertical increases, hence the two \nproperties of the magnetic field cannot be extricated. \nThe phase variation with line-of-sight \nviewing angle is demonstrated most substantially at frequencies \naround 5 mHz with a \nstrong, almost vertical magnetic field close to the umbra.  \n\\inlinecite{SBC07} also find that the total variation across\nall lines-of-sight increases with temporal frequency,\nparticularly in the stronger fields in the penumbrae.\n\nMode conversion of the acoustic waves in the near surface has been explored as \nthe physical cause of the observed absorption of acoustic waves by sunspots.  A \nfast acoustic wave, propagating towards the surface from the interior, \nencounters the depth at which the Alfv\\'en speed is equal to the sound speed \n($a\\approx c$) which is typically close to the surface in a  sunspot. Under \nthese conditions it is able to transmit to a slow acoustic mode and convert to \na fast magnetic mode \\cite{C05}. \\inlinecite{CC03} explore the mode \nconversion in two-dimensions with a uniform \\emph{inclined} magnetic field \nrelevant to sunspot penumbrae. The inclination of the magnetic field is found \nto have a significant dependence on the likelihood of conversion  and fits \nextremely well with the analysis of \\inlinecite{B95} \\cite{CCB03}. Further \nwork \\cite{C05,SC06} using ray theory has since established that it is the \nangle between the acoustic wave path and the magnetic field (the `attack \nangle') which is the crucial factor inducing conversion. With a wide attack \nangle at the $a\\approx c$ level there is maximum conversion from a fast \nacoustic to a fast magnetic mode. A fine attack angle encourages  transmission  to a slow acoustic mode, which is guided `up' the magnetic field lines to \nobservational heights in the atmosphere. A consequence of mode conversion\nmay be the observational signature of elliptical motion in regions\nof inclined magnetic field. \n\nThe aim of this paper is to model the observations of two\nsunspots previously analyzed by \\inlinecite{SBC07} in order to determine the properties of velocity ellipses consistent with the data.\nThese models are based on a least-squares-fit of the phase and \nmodulus information of the local ingression control correlation. \nWe explore the properties of these ellipses, observed\nwith waves at different temporal frequencies, as functions of\nthe magnetic field inclination angle. \nWe also examine the variation with line-of-sight angle\nof the phase shifts in the outgoing waves, using the local egression \ncontrol correlation, to assess the relation of the phase shift\nvariations between incoming and outgoing waves.\n\nIn the following sections we describe the data (Section \\ref{obs}), give an \noutline of the helioseismic holography technique used (Section 3), describe the \nresults (Section 4) and discuss our results in the\ncontext of mode conversion  (Section 5).\n\n\n     \n     \n\n \\section{Observations}\n      \\label{obs}      \n\nAs this is a continuation of studies by \\inlinecite{SBCL05} and \n\\inlinecite{SBC07} we use the same data, however, we provide a short \ndescription here to maintain coherence. The \\emph{Michelson Doppler Imager} \n(MDI) aboard the \\emph{Solar and Heliospheric Observatory} (SOHO) \\cite{MDI95} \nprovides the solar surface Doppler velocity information. The Dopplergrams are \nfull disk, have a 60 second cadence and  a resolution of  $\\approx 1.4$ Mm per \npixel. These full disk Dopplergrams are Postel projected and a $512\\times 512$ \npixel extract is taken centered on the active region. We analyze two sunspots: \nthe first in AR9026  observed over 10 days from 3rd - 12th June 2000 with a \nCarrington longitude (L0) of $75^\\circ$ and latitude (B0) of $20^\\circ$ and \npenumbral boundaries defined by inner and outer radii of 7 Mm and 16 Mm \nrespectively. The second is a sunspot in AR9057 observed over 9 days from 24th \nJune - 2nd July 2000 with $L0=158^\\circ$ and $B0=13^\\circ$, inner and outer\npenumbral boundaries given by 6 Mm and 13 Mm.  \nThe penumbral boundaries are \ndefined to be between 50\\% and 85\\% of the nearby quiet-Sun \ncontinuum intensity.\nThe sunspots in these active regions were chosen \nbased on the existence of continual MDI observation as they traversed the solar \ndisk, and for their relatively simple magnetic structure and evolution. \n\nMagnetograms from the \\emph{Imaging Vector Magnetograph} (IVM) at the \nUniversity of Hawaii Mees Solar Observatory \\cite{IVM96}  provide the \norientation and strength of the surface magnetic field in the sunspots chosen \nin AR9026 and AR9057. The IVM observations are made over a 28 minute interval: \nfor AR9026  starting at 18:29 Universal Time (UT) on 5th June 2000 and  for \nAR9057 starting at 16:19 UT  on 28th June 2000. \nThe IVM data reveals an azimuthally spreading magnetic field configuration \nfor both sunspots although they are not entirely symmetric.  It is \nassumed, supported largely by available line-of-sight magnetograms, that there \nis no significant evolution of the magnetic field in the sunspots during the \ntime of observation. Therefore, using only one vector magnetogram for the \nduration of the observation is reasonable. Rotation and scaling are applied to \nalign the IVM data to the line-of-sight MDI magnetograms.\n\nFigure~\\ref{g_vs_b} shows a strong correlation between the \nmagnetic field inclination (from vertical, $\\gamma$) and \nfield strength (the strong field is almost vertical whereas highly \ninclined field is relatively weak). In \nthis paper we use the inclination $\\gamma$ as the primary variable,\ndividing the penumbrae into three regions defined by the values\nof $\\gamma$ (Fig~~\\ref{g_vs_b}).\nIt is understood in this analysis that the magnetic field strength is \nimplicitly correlated with the inclination through Figure~\\ref{g_vs_b}, \nand that independent dependencies of observables with field strength \nand inclination are not extracted. The \ndependence of the phase shifts on the line-of-sight viewing angle of the \nmagnetic field is facilitated by knowing the full vector magnetic field.\n \n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{G_vs_B_9026_9057.eps}\n\\vspace{0cm}\n\\caption{Magnetic field strength, $| B |$, plotted against inclination from \nvertical, $\\gamma$, determined from IVM vector magnetograms. AR9026 (5th June \n2000) (bottom) and AR9057 (28th June 2000) (top). The different symbols divide \nthe penumbra into roughly equal regions of inclination: $\\gamma < 42^\\circ$ \n(asterisk); $42^\\circ < \\gamma < 66^\\circ$ (diamonds); $\\gamma > 66^\\circ$ \n(triangles). This corresponds to average magnetic field strengths of 1700 G, \n1000 G and 600 G for AR9057 and for AR9026 1900 G, 1400 G  and 600 G. In \ngeneral, the progression from the upper left portion of the distribution to the \nlower right portion represents increasing distance from the centre of the spot.}\n\\label{g_vs_b}\n\\end{center}\n\\end{figure}\n\n\\section{Helioseismic Holography}\n\n\nHelioseismic holography \\cite{LB00,BL00} is the phase coherent imaging of the \nsolar subsurface based on photospheric acoustic oscillations. The ingression is \nan assessment of the observed wavefield, $\\psi(\\mathbf{r}',t)$, converging to a \nselected focal point, $(\\mathbf{r},z,t)$, and the egression is the time reverse \n- an assessment of waves diverging from that point. In this case we calculate \nthe quantities at the surface, $z=0$. In practice, the observed wavefield used \nfor the calculation is usually an annulus surrounding the chosen focal point. \nThe pupil used here is identical to that described by \\inlinecite{SBCL05} and is \nconstructed for the  calculations with inner radius $a=20.7$ Mm  and outer \nradius $b= 43.5$ Mm, designed to be large enough that when the focal point is \nwithin the penumbra the area covered by the annulus does not include large \nareas of strong magnetic field.  At a frequency of 5 mHz this selects $p$-modes \nwith spherical harmonic degree and radial degree between $\\ell \\approx 450$ and \n$\\ell \\approx 700$.  The ingression at the surface is given by,\n\\begin{equation}\nH_{-}(\\mathbf{r},0,t)=\\int_{a < |\\mathbf{r}-\\mathbf{r'}|<b} d^2 \\mathbf{r}' \nG_{-}( |\\mathbf{r} - \\mathbf{r}' |, 0, t - t') \\psi(\\mathbf{r}',t),\n\\label{ingress}\n\\end{equation}\n$G_{-}$ is the ingression Green's function \\cite{LB00}. The egression, $H_{+}$ \nis simply the time reverse of  \\ref{ingress}, i.e. $t' - t$. \n\nThe ingression (\\ref{ingress}) is correlated with the observed wavefield in the \nspace-frequency domain,\n\\begin{equation}\nC_{-}(\\mathbf{r},\\nu) = \\langle \\hat{H}_{-} \n(\\mathbf{r},\\nu)\\hat{\\psi}^{*}(\\mathbf{r},\\nu ) \\rangle_{\\Delta \\nu} = \n|C_{-}|  \\rm{e}^{-i \\ \\delta \\phi_{-}},\n\\label{correl}\n\\end{equation}\nwhere $\\hat{H}_{-}(\\mathbf{r},\\nu)$ and $\\hat{\\psi}(\\mathbf{r},\\nu )$ are\nthe temporal Fourier transforms of $H_{-}$ and $\\psi$ respectively,\nand we have dropped the dependence on depth $z$.\nThe correlation has a modulus $|C_{-}|$ and \na phase, $\\delta \\phi_{-}$.   \nIn the frequency spectrum, with the pupil covering mostly quiet Sun, the local \ningression control correlation simply characterizes how the local magnetic \nphotosphere responds acoustically to upcoming waves, prescribed by $H_{-}$, \noriginating within the pupil. \n\n\nIn \\inlinecite{SBCL05} and \\inlinecite{SBC07} the phase shift caused by the \nsurface perturbations to the incident wave is calculated at various \nfrequencies. Here we extend that research and monitor the variation of the \ncorrelation amplitude within the sunspot penumbral region, which is then \ncombined with the phase information to form estimates of the surface velocity \nellipse. We then go on to examine the phase of the local egression control \ncorrelation, $\\delta \\phi_{+} = \\rm{Arg} [ C_{+}(\\mathbf{r},\\nu)]$, with the \nline-of-sight angle in the penumbra of sunspots in AR9057 and AR9026.\n\n\n\\section{Elliptical Representation of Surface Velocities}\n\n\nFor better statistics multiple days of observation of each sunspot are \ncombined and a least-squares-fit of the observations allows an estimation of \nthe velocity ellipse. To calculate the velocity ellipse  \nthe ingression correlation (eqn. \\ref{correl}) \nis used as a proxy for the local velocity. The modulus and phase\nof the correlation vary with respect \nto the line-of-sight angle and these variations are modeled to construct \nan ellipse representing the \nsurface velocity associated with a particular magnetic field element. \n\nA smeared ingression `flat field' takes out undesired contributions to the \ncorrelation modulus due to the temporal or spatial variations of the\ningression, due to the presence of magnetic\nregions in the pupil. This is achieved by dividing $|C_{-}|$ by the \nGaussian smear of the root-mean-square of the ingression to get \n$|C_{-}|_{flat}$. The resulting correlations are normalized so that \nthe ingression correlation modulus in the nearby quiet Sun \nhas a value of $\\cos(\\zeta)$, where $\\zeta$ is the heliocentric\nangle of the quiet region from disk center. A dependence of the\nquiet Sun (root-mean-squared) amplitude with  $\\cos(\\zeta)$ is\nexpected for predominantly vertically oscillating $p$-modes. \nThe normalization is achieved by dividing $|C_{-}|_{flat}$ \nby the quiet Sun average divided by the cosine of the heliocentric angle, i.e. \n\\begin{equation}\n|C_{-}|_{norm}=|C_{-}|_{flat} \\frac{\\cos(\\zeta)}{\\langle |C_{-}| \n\\rangle_{QS}}. \n\\end{equation}\nThe normalization corrects for variations in the modulus due to \nduty cycle variations, foreshortening, and other effects which \ncause undesired variations in the modulus from day to day. \nThese procedures assume that these \nfactors have the same relative effect on the (desired) correlations in the \npenumbra as they do to the quiet Sun. It is difficult to assess the validity \nof this, and so it is used as a reasonable working assumption subject to some \ncaution.\n\nWe use the same angle $\\theta_p$ as defined in \\inlinecite{SBCL05}, which is \nthe angle between the projection of the line-of-sight vector onto the plane \ncontaining the magnetic field vector and the radial vector. \nIt is a measure of the angle that the magnetic field is being viewed from. We \nnow  simply refer to $|C_{-}|_{norm}$ as $|C_{-}|$ and see how it varies with \n$\\theta_p$.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{phase_mod_thetap_3mHz.ps} \\hspace{1.5cm}\n\\caption{The modulus of the correlation, $|C_{-}|$  (left column), and the \nphase, $\\delta \\phi_{-}$, in the penumbra of AR9026 at 3 mHz for all days of \nobservation are plotted against projected angle $\\theta_p$ for different values \nof magnetic field inclinations as indicated. The top panel (a) shows $\\gamma < \n42^\\circ$, where the mean field strength is $\\langle \\mathbf{B} \\rangle = 1900$ \nG, the middle panel (b) shows $42^\\circ < \\gamma < 66^\\circ$, where $\\langle \n\\mathbf{B} \\rangle = 1400$ G, and the bottom panel (c) shows  $\\gamma > \n66^\\circ$, where $\\langle \\mathbf{B} \\rangle = 600$ G. The horizontal dashed \nlines indicate the mean value of $|C_{-}|$ for each panel. The error bars \nindicate the standard deviation of the mean over bins of 20 measurements in \n$\\theta_p$. The solid line of fit is a fit for all the displayed data; the \ndotted line is a fit for the data from 3rd - 7th June 2000; the dashed line is \na fit for data from 8th - 12th June 2000.  }\n\\label{3mhz}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{phase_mod_thetap_4mHz.ps}\n\\caption{Same as Figure~\\ref{3mhz} except at 4 mHz.}\n\\label{4mhz}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{phase_mod_thetap_5mHz.ps}\n\\caption{Same as Figure~\\ref{3mhz} except at 5 mHz.}\n\\label{5mhz}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\vspace{1cm}\n\\includegraphics[width=0.7\\textwidth]{fig_ellipse.eps}\n\\caption{Least-squares-fit surface velocity ellipses as given by the phase and \namplitude of the local ingression control correlation in the penumbra of \nAR9026. The top row is when $\\gamma > 66^\\circ$, the middle row when $42^\\circ \n< \\gamma < 66^\\circ$ and the bottom row is when $\\gamma < 42^\\circ$. The \n$\\gamma$ listed in the plot is the angle that the magnetic field vector is \ndrawn at. $\\beta$ is the inclination angle of the semi-major axis of the \nvelocity ellipse. The left column is at frequencies of 3 mHz, the middle column \nat 4 mHz and the right column at 5 mHz.}\n\\label{ellipse9026}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{phase_mod_thetap_3mHz_AR9057.ps}\n\\caption{The modulus of the correlation, $|C_{-}|$ (left column) and phase, \n$\\delta \\phi_{-}$ (right column), in the penumbra of AR9057 at  3 mHz for all \ndays of observation are plotted against projected angle $\\theta_p$ for \ndifferent values of magnetic field inclinations as indicated.  The top panel \n(a) shows $\\gamma < 42^\\circ$, where the mean field strength is $\\langle \n\\mathbf{B} \\rangle = 1700$ G, the middle panel (b) shows $42^\\circ < \\gamma < \n66^\\circ$, where $\\langle \\mathbf{B} \\rangle = 1000$ G, and the bottom panel \n(c) shows  $\\gamma > 66^\\circ$, where $\\langle \\mathbf{B} \\rangle = 600$ G. The \nhorizontal dashed lines indicate the mean value of $|C_{-}|$ for each panel. \nThe error bars indicate the standard deviation of the mean over bins of 20 \nmeasurements in $\\theta_p$. The solid line of fit is a fit for all the \ndisplayed data; the dotted line is a fit for the data from 24th - 28th June \n2000; the dashed line is a fit for data from 29th June - 2nd July 2000.  }\n\\label{90573mhz}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{phase_mod_thetap_3mHz_AR9057.ps}\n\\caption{Same as for  Figure~\\ref{90573mhz} except at 4 mHz. }\n\\label{90574mhz}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{phase_mod_thetap_5mHz_AR9057.ps}\n\\caption{Same as for Figure~\\ref{90573mhz} except at 5 mHz.}\n\\label{90575mhz}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\vspace{1cm}\n\\includegraphics[width=0.7\\textwidth]{fig_ellipse_AR9057.eps}\n\\caption{Least-squares-fit surface velocity ellipses as given by the phase and \namplitude of the local ingression control correlation in the penumbra of \nAR9057. The top row is when $\\gamma > 66^\\circ$, the middle row when $42^\\circ \n< \\gamma < 66^\\circ$ and the bottom row is when $\\gamma < 42^\\circ$. The \n$\\gamma$ listed in the plot is the angle that the magnetic field vector is \ndrawn at. $\\beta$ is the inclination angle of the semi-major axis of the \nvelocity ellipse. The left column is at frequencies of 3 mHz, the middle column \nat 4 mHz and the right column at 5 mHz.}\n\\label{ellipse9057}\n\\end{center}\n\\end{figure}\n\n\nFigures~\\ref{3mhz} to \\ref{5mhz} and Figures~\\ref{90573mhz} to \\ref{90575mhz} \nshow the variation of $|C_{-}|$  with $\\theta_p$ for AR9026 and AR9057 in the \nleft columns at 3, 4 and 5mHz in the same three bins of inclination shown in \nFigure~\\ref{g_vs_b}. The right column shows $\\delta \\phi_{-}$. \nThe variations of $\\delta \\phi_{-}$ with $\\theta_p$ have been the subject\nof our previous analyses (\\opencite{SBCL05}; \\opencite{SBC07}). \nIn the absence of magnetic effects, we expect\nthe local oscillatory wave field, as assessed by the ingression correlation,\nto be consistent with purely vertical motion. This would predict a\n dependence of $|C_{-}|$ on $\\cos(\\theta_p)$ and no dependence  of  $\\delta \\phi_{-}$ on\n$\\cos(\\theta_p)$. \nDepartures from these expectation\nare clearly visible in Figs~\\ref{3mhz} - \\ref{5mhz} and \\ref{90573mhz} - \n\\ref{90575mhz}. A rough understanding of these results can be\ngained by noting that the net variation in the phase shift with\n$\\theta_p$ is related to the eccentricity of the ellipse, while\nthe value of $\\theta_p$ for maximum $|C_{-}|$ determines the \norientation of the semi-major axis.\nWe perform a least-squares-fit to the observed $|C_{-}|$ \nand $\\delta \\phi_{-}$ to determine the elliptical motion \nconsistent with the data. The best-fit ellipses \nare shown in Figures~\\ref{ellipse9026} and \\ref{ellipse9057}. \nThe moduli and phase of $C_-$ as determined from the fits are plotted (as solid\nlines) with the data points in Figs 2-4 and 6-8. Separate fits were performed for\nindependent  5-day subsets of the data (also shown in the Figures by the dotted\nand dashed lines). There are no obvious systematic differences in the fits\nover time. The orientations and eccentricities are highly consistent across\nall frequency bands in both sunspots (Figure~\\ref{ellipse9026} \nand Figure~\\ref{ellipse9057}). Some systematic differences\nbetween the two spots are evident.\nFor AR9026, the ellipses are aligned slightly towards\nthe magnetic field direction at high field inclination \nbut `swing' over as the inclination becomes smaller (and the \nmagnetic field stronger). For AR9057, the motion is nearly vertical\nat low field inclination, but also tilts away from the field\nin the stronger, more vertical, fields. \nIn both spots, the eccentricity\nof the ellipses increases with decreasing field strength or increasing \ninclination.\n\n\n\\subsection{ELLIPSE PARAMETERS}\n\nWe define a deviation angle \nas the angle between the magnetic field vector and surface\nvelocity ellipse semi-major axis ($\\delta$ = $\\gamma - \\beta$, where $\\beta$ is the inclination of the semi-major axis from vertical).\nWe show the variation of\nthe deviation angle, length of the semi-major axis,\nand eccentricity with magnetic field strength and\/or inclination \nin Figure~\\ref{ellp}.\nAlso shown is the phase difference between the fits at \n$\\theta_p=-60^\\circ$ and $\\theta_p=+60^\\circ$. \nThis quantity is generally inversely correlated to the ellipse\neccentricity, but is a more \ndirect measure of the total variation of observed phase shift\nfor a specific penumbral region \\cite{SBCL05}.\nIn Figure~\\ref{ellp}, data from \nAR9026 is represented by an asterisk, AR9057 by a diamond and the frequencies \nare color-coded as following :- 5 mHz is black, 4 mHz is purple and 3 mHz is \nred.\n\\begin{figure}[ht]\n\\begin{center}\n\n\\includegraphics[width=0.4\\textwidth]{ellipse_gamma_gamma_minus_beta.eps}\n\\includegraphics[width=0.4\\textwidth]{ellipse_gamma_ecc.eps}\\\\\n\\includegraphics[width=0.4\\textwidth]{ellipse_gamma_diff.eps}\n\\includegraphics[width=0.4\\textwidth]{ellipse_amp.eps}\n\\caption{Properties of the ellipses; deviation angle, $\\delta$ (top left); the \nellipse eccentricity  for both sunspots at all frequencies against the three \naverage magnetic field inclinations (top right); the phase difference between \nthe least-squares-fit correlation phase at $\\theta_p=-60^\\circ$   and at  \n$\\theta_p=+60^\\circ$ ( $\\delta \\phi_{( \\theta_p=-60^\\circ) } -  \\delta \\phi_{( \n\\theta_p=+60^\\circ) }$ ) (bottom left);  the length of the semi-major axis of \neach ellipse in each region for both sunspots at all frequencies against the \nthree average magnetic field inclinations (bottom right). AR9026 is represented \nby the asterisk, and AR9057 by the diamond. 5 mHz is black, 4 mHz is purple and \n3 mHz is red.}\n\\label{ellp}\n\\end{center}\n\\end{figure}\n\nIn general, the trends shown in Figure~\\ref{ellp}, namely an increase\nin the deviation angle, eccentricity, and semi-major axis length,\nand a decrease in the phase variation, with increasing inclination\nare observed in both sunspots and at all frequencies. There are\nsome deviations from this. For example, at low inclinations, it is\nobserved that the eccenctricity (phase variation) decreases (increases)\nwith frequency. Note that the semi-major axis length is an indication\nof the total wave amplitude in the magnetic region. The trend observed\nin the lower-left panel of Figure~\\ref{ellp} is consistent with \na reduction in wave amplitude related to the field strength. \n\n\n\n\\section{Local Egression Control Correlation}\\label{eg}\nIn this section we \nexamine the phase of the local egression control correlation, \n\\begin{equation}\nC_{+}(\\mathbf{r},\\nu) = \\langle \\hat{H}_{+} \n(\\mathbf{r},\\nu)\\hat{\\psi}^{*}(\\mathbf{r},\\nu ) \\rangle_{\\Delta \\nu} = |C_+|  \n\\rm{e}^{-i \\ \\delta \\phi_{+}}.\n\\label{correl2}\n\\end{equation}\nSince the egression is simply the time reverse of the \ningression, we might expect \nto see a reversal of the phase change compared to the ingression phases.\n\nUsing all the days' of data, the egression correlation phase is plotted against \n$\\theta_p$ in Figures~\\ref{hephase1} and \\ref{hephase2}, along with\nthe fits for the phase variation due to elliptical motion.  We see similar  \ntrends in both sunspots, AR9026 and AR9787. \nFigure~\\ref{dphase} shows the phase \ndifference between the least-squares-fit correlation phase at \n$\\theta_p=-60^\\circ$   and at  $\\theta_p=+60^\\circ$ of the ingression plotted \nagainst that of the egression. The colors and symbols represent the same as \nbefore:  AR9026 is represented by an asterisk, AR9057 by a diamond and the \nfrequencies are 5 mHz (black), 4 mHz (purple) and 3 mHz (red) in addition the \nsize of the symbols represent the average inclination from vertical. The solid \nline has a slope of $-1$.  There is a reverse \nbehaviour of the ingression, compared to the previous egression\nresults, present for all frequencies at most magnetic field \ninclinations.  The exception is when the field is highly inclined, where \nwe observe a trend in the same sense as the ingression. \nThis is unexpected \nand warrants further study.\n\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{fig_phi_vs_thetap_HE_9026.ps} \n\\vspace{1cm}\n\\caption{The 3 mHz (left column), 4 mHz (middle column) and 5 mHz (right \ncolumn) egression correlation phase ($\\delta \\phi_{+}$) vs. $\\theta_p$ within \nthe penumbra of sunspot AR9026 for different values of magnetic field \ninclination as indicated.  The three rows represent different portions of the \npenumbra as shown in Figure~\\ref{g_vs_b}(a). The top panel row (a) shows \n$\\gamma < 42^\\circ$, where the mean field strength is $\\langle \\mathbf{B} \n\\rangle = 1900$ G, the middle row (b) shows $42^\\circ < \\gamma < 66^\\circ$, \nwhere $\\langle \\mathbf{B} \\rangle = 1400$ G, and the bottom row (c) shows  \n$\\gamma > 66^\\circ$, where $\\langle \\mathbf{B} \\rangle = 600$ G. The horizontal \ndashed lines indicate the mean value of $\\delta \\phi_{+}$ for each panel. }\n\\label{hephase1}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{fig_phi_vs_thetap_HEN_9057.ps}\n\\vspace{1cm}\n\\caption{The 3 mHz (left column), 4 mHz (middle column) and 5 mHz (left column) \negression correlation phase ($\\delta \\phi_{+}$) vs. $\\theta_p$ within the \npenumbra of sunspot AR9057 for different values of magnetic field strength as \nindicated.   The three different panels represent different portions of the \npenumbra, similar to Figure~\\ref{g_vs_b}(b). The top panel (a) shows $\\gamma < 42^\\circ$ where the mean field strength is $\\langle B \\rangle = 1700$ G the middle panel (b) \nshows $42^\\circ < \\gamma < 66^\\circ$, and $\\langle B \\rangle = 1000$ G and the \nbottom panel (c) shows $\\gamma > 66^\\circ$ and $\\langle B \\rangle = 600$ G. The horizontal dashed lines indicate the mean value of $\\delta \n\\phi_{+}$ for each panel. }\n\\label{hephase2}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{ph_diff_eg_in.ps}\n\\vspace{1cm}\n\\caption{The phase difference between the least-squares-fit correlation phase \nat $\\theta_p=-60^\\circ$   and at  $\\theta_p=+60^\\circ$ of the ingression \nplotted against that of the egression. The solid line is a line of slope $-1$. \nAR9026 is represented by an asterisk, AR9057 by a diamond and the \ncolors indicate the following frequencies: 5 mHz (black), 4 mHz (purple) and \n3 mHz (red). The largest symbols represent $\\gamma > 66^\\circ$ and the \nsmallest symbols $\\gamma < 42^\\circ$. }\n\\label{dphase}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{Discussion}\n\nMode Conversion predicts (among other things \\cite{C07}) that when the attack \nangle is small most of the observable energy will be in the slow acoustic mode, \nand when the attack angle is large most of the observable energy will be in the \nfast magnetic mode. In this case we will be seeing the line-of-sight effect of \na combination of waves coming from all directions impinging on magnetic field \nwith a particular orientation. The observations by MDI consist of line-of-sight Doppler signatures \nof the surface motion which are presumably caused mainly by pressure \nperturbations. This means that we would expect to be observing only the slow \nacoustic mode, however it is possible that we are observing a combination of \nthe acoustic and magnetic modes. \n\n\nThe dependence of the phase shift of the observed ingression correlations with azimuthal angle \naround sunspot penumbrae, as viewed from different observational vantages, shows that\nthe incoming phase shifts must be (at least partly) photospheric in origin and are\ninfluenced by the presence of inclined magnetic fields \\cite{SBCL05}. \nAnalysis of the variation of both the amplitude and phase of the surface velocities \nprovides an opportunity to characterize the magnetically influenced acoustic signature\nas an ellipse with properties determined by the magnetic fields.  \nWe found that the ellipses are either nearly vertical (for weaker, more inclined fields) or \ngenerally directed \\emph{away} from the magnetic field direction (for stronger,\nmore vertical fields).  Largely consistent results for two  active regions,  AR9026 and AR9057, are found.\nSome properties of the surface ellipses, e.g. their inclinations, are different for the two sunspots.\nSome of this variation may be due to  differences in the field properties. For\nexample, the field in AR9057 is on \naverage $\\sim 15\\%$ weaker than AR9026. \n\nFits of the elliptical motion in Figures~\\ref{ellipse9026} and \n\\ref{ellipse9057} depend critically on the correlation modulus which is prone \nto systematic uncertainties. But the trend is that a stronger, less \ninclined magnetic field produces elliptical motion with smaller amplitude,\neccentricity, and deviation angle, and a larger inclination from vertical.\nThese trends exist for both spots and, largely, at all frequencies.\nThe  shorter \nsemi-major axis at strong magnetic field strengths is consistent with previous \nknowledge of surface acoustic amplitude suppression in magnetic fields. It is \ncurious to note, however, that at 4 mHz the amplitude is consistently smaller \nthan even 3 mHz.\n\n\nThese are the first results to estimate the behaviour of the surface velocity \nellipse at the photosphere within sunspots.  \nThe results do not immediately suggest an observation of the slow wave as shown \nby \\inlinecite{C05} or \\inlinecite{SC06}, but are consistent with their \nexpectations of the behaviour of slow waves at this height in the atmosphere.  \nMode conversion theory states the alignment is dependent upon $a^2\/c^2$  which \nat the observational heights of $\\sim 200$km of the atmosphere may not be large \nenough to invoke alignment.  Since the ray analysis is somewhat unrealistic we \nwould expect to observe a combination of fast and slow waves at the surface, \nwhich will contribute to a clouded view of the surface velocities. \n\\inlinecite{RSWS07} are currently exploring the possibility that these apparent \nsurface effects are due to the changes in the radiative transfer within active \nregions and the formation height of the observational Ni 678 nm line. \nThis explanation requires an absorption mechanism, or else some other means\nof producing a difference between the amplitudes of upward and downward\npropagating waves. Thus mode conversion may still be important in this proposed\nmechanism. A test of the mechanism proposed by Rajaguru et al. (2007) would be to repeat\nthe observations performed here in a magnetically insensitive line, where the\nproposed radiative transfer effects would not be present. In terms of mode conversion, it is \nsuggested that the main effect occurs along the bright radial filaments of the \ninterlocking comb structure as presented in the penumbral models of \n\\inlinecite{WTBT04}. However,  observational helioseismic spatial resolution \ncannot currently resolve this.\n\n\nThis is also the first time that the variation of the phase of the local \negression correlation has been analysed in the penumbra. It is curious that the \negression correlation shows a reverse dependence when the magnetic field is \nweak and highly inclined.  This is evidence of a reverse ingression dependence \non the line-of-sight, but further investigation is required to understand the \nbehavior at high frequencies in the weaker, more inclined fields.\n\n\\vspace{2cm}\n\nThis work was supported in part by the \\textit{European Helio- and Asteroseismology Network} (HELAS).\n\n\n\n\n\n\n\n    \n\\bibliographystyle{spr-mp-sola}\n\n\n    \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nThe Transformer~\\cite{attention-is-all-you-need} has become the dominant modeling paradigm in neural machine translation. \nIt follows the encoder-decoder framework using stacked multi-head self-attention and fully connected layers. Multi-head attention was shown to make more efficient use of the model's capacity: \nperformance of the model with 8 heads is almost 1 BLEU point higher than that of a model of the same size with single-head attention~\\cite{attention-is-all-you-need}.\nThe Transformer achieved state-of-the-art results in recent shared translation tasks \\cite{bojar-EtAl:2018:WMT1,iwslt18-overview}. Despite the model's widespread adoption and recent attempts to investigate the kinds of information learned by the model's encoder~\\cite{raganato-tiedemann:2018:BlackboxNLP}, \nthe analysis of multi-head attention and its importance for translation is challenging. \nPrevious analysis of multi-head attention considered the average of attention weights over all heads at a given position or focused only on the maximum attention weights~\\cite{voita18,tang-sennrich-nivre:2018:WMT}, but neither method explicitly takes into account the varying importance of different heads.\nAlso, this obscures the roles played by individual heads which, as we show, influence the generated translations to differing extents.\nWe attempt to answer the following questions:\n\\begin{itemize}\n    \\item To what extent does translation quality depend on individual encoder heads?\n    \\item Do individual encoder heads play consistent and interpretable roles? If so, which are the most important ones for translation quality?\n    \\item Which types of model attention (encoder self-attention, decoder self-attention or decoder-encoder attention) are most sensitive to the number of attention heads and on which layers?\n    \\item Can we significantly reduce the number of attention heads while preserving translation quality?\n\\end{itemize}\n\nWe start by identifying the most important heads in each encoder layer \nusing layer-wise relevance propagation~\\cite{lrp-ding-2017}. \nFor heads judged to be important, we then attempt to characterize the roles they perform. We observe the following types of role: positional (heads attending to an adjacent token), syntactic (heads attending to tokens in a specific syntactic dependency relation) and attention to rare words (heads pointing to the least frequent tokens in the sentence).\n\nTo understand whether the remaining heads perform vital but less easily defined roles, or are simply redundant to the performance of the model as measured by translation quality, we introduce a method for pruning heads based on~\\citet{louizos2018learning}. While we cannot easily incorporate the number of active heads as a penalty term in our learning objective (i.e.\\ the $L_0$ regularizer), we can use a differentiable relaxation. \nWe prune attention heads in a continuous learning scenario starting from the converged full model \nand identify the roles of those which remain in the model. \nThese experiments corroborate the findings of layer-wise relevance propagation;\nin particular, heads with clearly identifiable positional and syntactic functions are pruned last and hence shown to be most important for the translation task.\n\n\nOur key findings are as follows:\n\\begin{itemize}\n    \\item Only a small subset of heads are important for translation;\n    \\item Important heads have one or more specialized and interpretable functions in the model;\n    \\item The functions correspond to attention to neighbouring words and to tokens in specific syntactic dependency relations.\n\\end{itemize}\n\n\n\\section{Transformer Architecture}\nIn this section, we briefly describe the Transformer architecture~\\cite{attention-is-all-you-need} introducing the terminology used in the rest of the paper.\n\nThe Transformer is an encoder-decoder model that uses stacked self-attention and fully connected layers for both the encoder and decoder.\nThe encoder consists of $N$ layers, each containing two sub-layers: (a) a multi-head self-attention mechanism, and (b) a feed-forward network.\nThe multi-head attention mechanism relies on scaled dot-product attention, which operates on a query $Q$, a key $K$ and a value $V$:\n\\begin{equation}\n\\textnormal{Attention}(Q, K, V ) = \\textnormal{softmax}\\left(\\frac{QK^T}{\\sqrt{d_k}}\\right)V\n\\label{eq:mult_attention}\n\\end{equation}\nwhere $d_k$ is the key dimensionality. In self-attention, queries, keys and values come from the output of the previous layer.\n\nThe multi-head attention mechanism obtains $h$ (i.e.\\ one per head) different representations of ($Q$, $K$, $V$), computes scaled dot-product attention for each representation, concatenates the results, and projects the concatenation through a feed-forward layer. This can be expressed in the same notation as Equation~(\\ref{eq:mult_attention}):\n\\begin{equation}\n\\textnormal{head}_i = \\textnormal{Attention}(QW_i^Q , K W_i^K , V W_i^V )\n\\end{equation}\n\\vspace{-4ex}\n\\begin{equation}\n\\textnormal{MultiHead}(Q, K, V ) = \\textnormal{Concat}_i(\\textnormal{head}_i)W^O\n\\label{eq:concat_heads}\n\\end{equation}\nwhere the $W_i$ and $W^O$ are parameter matrices.\n\nThe second component of each layer of the Transformer network is a feed-forward network. The authors propose using a two-layer network with a ReLU activation. \n\nAnalogously, each layer of the decoder contains the two sub-layers mentioned above as well as an additional multi-head attention sub-layer. This additional sub-layer receives the output of the encoder as its keys and values.\n\nThe Transformer uses multi-head attention in three different ways: encoder self-attention, decoder self-attention and decoder-encoder attention. In this work, we concentrate primarily on encoder self-attention.\n\n\n\n\\section{Data and setting}\n\\label{sect:data_setting}\nWe focus on English as a source language and consider three target languages: Russian, German and French. For each language pair, we use the same number of sentence pairs from WMT data to control for the amount of training data and train Transformer models with the same numbers of parameters. We use 2{.}5m sentence pairs, corresponding to the amount of English--Russian parallel training data (excluding UN and Paracrawl). In Section~\\ref{sect:dependency_heads} we use the same held-out data for all language pairs; these are 50k English sentences taken from the WMT EN-FR data not used in training.\n\nFor English-Russian, we perform additional experiments using the publicly available OpenSubtitles2018 corpus~\\cite{LISON18.294} to evaluate the impact of domains on our results. \n\nIn Section~\\ref{sect:pruning_attention_heads} we concentrate on English-Russian and two domains: WMT and OpenSubtitles.\n\nModel hyperparameters, preprocessing and training details are provided in appendix~\\ref{app:exp}. \n\n\n\n\\begin{figure*}[t!h!]\n    \\centering\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_lrp.png}\n        \\caption{LRP}\n        \\label{fig:heads_lrp}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_mean_attn_max.png}\n        \\caption{confidence}\n        \\label{fig:heads_mean_attn_max}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.30\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_functions_with_topic_ru_acc.png}\n        \\caption{head functions}\n        \\label{fig:heads_functions}\n    \\end{subfigure}\n    \\caption{Importance (according to LRP), confidence, and function of self-attention heads. In each layer, heads are sorted by their relevance according to LRP. Model trained on 6m OpenSubtitles EN-RU data.}\n   \n    \\label{fig:heads_all_info}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.22\\textwidth}\n       \n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_lrp_de.png}\n        \\caption{LRP (EN-DE)}\n        \\label{fig:heads_lrp_de}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.22\\textwidth}     \\includegraphics[width=\\textwidth]{.\/pict\/heads_functions_with_topic_de_acc.png}\n        \\caption{head functions}\n        \\label{fig:heads_functions_de}\n    \\end{subfigure}\n    \\vskip\\baselineskip\n    \\begin{subfigure}[b]{0.22\\textwidth} \n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_lrp_fr.png}\n        \\caption{LRP (EN-FR)}\n        \\label{fig:heads_lrp_fr}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.22\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/heads_functions_with_topic_fr_acc.png}\n        \\caption{head functions}\n        \\label{fig:heads_functions_fr}\n    \\end{subfigure}\n    \\caption{Importance (according to LRP) and function of self-attention heads. In each layer, heads are sorted by their relevance according to LRP. Models trained on 2{.}5m WMT EN-DE (a, b) and EN-FR (c, d).} \n   \n    \\label{fig:heads_all_info_de_fr}\n\\end{figure}\n\n\n\n\n\\section{Identifying Important Heads}\nPrevious work analyzing how representations are formed by the Transformer's multi-head attention mechanism focused on either the average or the maximum attention weights over all heads~\\cite{voita18,tang-sennrich-nivre:2018:WMT}, but neither method explicitly takes into account the varying importance of different heads.\nAlso, this obscures the roles played by individual heads which, as we will show, influence the generated translations to differing extents.\n\nWe define the ``confidence'' of a head as the average of its maximum attention weight excluding the end of sentence symbol,\\footnote{We exclude EOS on the grounds that it is not a real token.} where average is taken over tokens in a set of sentences used for evaluation (development set).\nA confident head is one that usually assigns a high proportion of its attention to a single token. Intuitively, we might expect confident heads to be important to the translation task.  \n\nLayer-wise relevance propagation (LRP)~\\cite{lrp-ding-2017} is a method for computing the relative contribution of neurons at one point in a network to neurons at another.\\footnote{A detailed description of LRP is provided in appendix \\ref{app:lrp}.} \nHere we propose to use LRP to evaluate the degree to which different heads at each layer contribute to the top-1 logit predicted by the model. Heads whose outputs have a higher relevance value may be judged to be more important to the model's predictions.\n\nThe results of LRP are shown in Figures~\\ref{fig:heads_lrp}, \\ref{fig:heads_lrp_de}, \\ref{fig:heads_lrp_fr}. \nIn each layer, LRP ranks a small number of heads as much more important than all others.\n\n\nThe confidence for each head is shown in Figure~\\ref{fig:heads_mean_attn_max}. \nWe can observe that the relevance of a head as computed by LRP agrees to a reasonable extent with its confidence. The only clear exception to this pattern is the head judged by LRP to be the most important in the first layer. It is the most relevant head in the first layer but its average maximum attention weight is low. We will discuss this head further in Section~\\ref{sec:topic_head}.\n\n\\section{Characterizing heads}\n\\label{sect:characterizing_heads}\n\nWe now turn to investigating whether heads play consistent and interpretable roles within the model.\n\n\nWe examined some attention matrices paying particular attention to heads ranked highly by LRP and identified three functions which heads might be playing:\n\\begin{enumerate}\n\\item positional: the head points to an adjacent token,\n\\item syntactic: the head points to tokens in a specific syntactic relation,\n\\item rare words: the head points to the least frequent tokens in a sentence. \n\\end{enumerate}\n\n\n\nNow we discuss the criteria used to determine if a head is performing one of these functions and examine properties of the corresponding heads. \n\n\n\n\n\\subsection{Positional heads}\\label{subsec:positional_heads}\nWe refer to a head as ``positional'' if at least 90\\% of the time its maximum attention weight is assigned to a specific relative position (in practice either -1 or +1, i.e.\\ attention to adjacent tokens). Such heads are shown in purple in Figures~\\ref{fig:heads_functions} for English-Russian, \\ref{fig:heads_functions_de} for English-German, \\ref{fig:heads_functions_fr} for English-French and marked with the relative position.\n\nAs can be seen, the positional heads correspond to a large extent to the most confident heads and the most important heads as ranked by LRP. In fact, the average maximum attention weight exceeds $0{.}8$ for every positional head for all language pairs considered here. \n\n\n\n\n\\subsection{Syntactic heads}\n\\label{sect:dependency_heads}\n\n\nWe hypothesize that, when used to perform translation, the Transformer's encoder may be responsible for disambiguating the syntactic structure of the source sentence. We therefore wish to know whether a head attends to tokens corresponding to any of the major syntactic relations in a sentence. \nIn our analysis, we looked at the following dependency relations: \nnominal subject (nsubj), direct object (dobj), adjectival modifier (amod) and adverbial modifier (advmod). These include the main verbal arguments of a sentence and some other common relations. They also include those relations which might inform morphological agreement or government in one or more of the target languages considered here. \n\n\n\n\\subsubsection{Methodology}\nWe evaluate to what extent each head in the Transformer's encoder accounts for a specific dependency relation by comparing its attention weights to a predicted dependency structure generated using CoreNLP~\\cite{manning-EtAl:2014:P14-5} \non a large number of held-out sentences. \nWe calculate for each head how often it assigns its maximum attention weight (excluding EOS) to a token with which it is in one of the aforementioned dependency relations. We count each relation separately and allow the relation to hold in either direction between the two tokens.\n\nWe refer to this relative frequency as the ``accuracy'' of head on a specific dependency relation in a specific direction.\nNote that under this definition, we may evaluate the accuracy of a head for multiple dependency relations.\n\nMany dependency relations are frequently observed in specific relative positions (for example, often they hold between adjacent tokens, see Figure~\\ref{fig:dep_posdiff_distribution}). We say that a head is ``syntactic'' if its accuracy is at least $10\\%$ higher than the baseline that looks at the most frequent relative position for this dependency relation. \n\n\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.18]{.\/pict\/dep_posdiff_distribution_wmt_no_compound.png}}\n\\caption{Distribution of the relative position of dependent for different dependency relations (WMT).}\n\\label{fig:dep_posdiff_distribution}\n\\end{figure}\n\n\n\n\n\\subsubsection{Results}\n\\label{sec:syntactic-heads:results}\n\n\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{lccc}\n\\toprule\n\\bf dep.\\!\\!&\\!\\!\\!\\! \\bf direction & \\multicolumn{2}{c}{\\bf best head \/ baseline}\\\\ \n &  & \\multicolumn{2}{c}{\\bf accuracy} \\\\ \n \\cmidrule{1-4}\n & & \\bf WMT  & \\bf OpenSubtitles\\!\\!\\!\\\\\n\\toprule\n\\multicolumn{3}{l}{nsubj}\\\\\n&\\!\\!\\!\\!v $\\rightarrow$ s & 45 \/ 35 & 77 \/ 45 \\\\\n&\\!\\!\\!\\!s $\\rightarrow$ v & 52 \/ 35 & 70 \/ 45\\\\\n\\cmidrule{1-4}\n\n\\multicolumn{3}{l}{dobj}\\\\\n&\\!\\!\\!\\!v $\\rightarrow$ o & 78 \/ 41 & 61 \/ 46\\\\\n&\\!\\!\\!\\!o $\\rightarrow$ v & 73 \/ 41 & 84 \/ 46\\\\\n\\cmidrule{1-4}\n\n\\multicolumn{3}{l}{amod}\\\\\n   &\\!\\!\\!\\!noun $\\rightarrow$ adj.m. & 74 \/ 72 & 81 \/ 80\\\\\n&\\!\\!\\!\\!adj.m. $\\rightarrow$ noun  & 82 \/ 72 & 81 \/ 80 \\\\\n\\cmidrule{1-4}\n\n\\multicolumn{3}{l}{advmod}\\\\\n   &\\!\\!\\!\\!v $\\rightarrow$ adv.m. & 48 \/ 46 & 38 \/ 33\\\\\n&\\!\\!\\!\\!adv.m. $\\rightarrow$ v   & 52 \/ 46 & 42 \/ 33\\\\\n\n\\bottomrule\n\\end{tabular}\\textbf{}\n\\caption{Dependency scores for EN-RU, comparing the best self-attention head to a positional baseline. Models trained on 2{.}5m WMT data and 6m OpenSubtitles data.} \\label{tab:dependency_head_scores}\n\\vspace{1ex}\n\\end{table}\n\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.18]{.\/pict\/dep_wmt_langs.png}}\n\\caption{Dependency scores for EN-RU, EN-DE, EN-FR each trained on 2{.}5m WMT data. \n}\n\\label{fig:wmt_deps_ru_de_fr}\n\\end{figure}\n\n\n\n\\begin{figure*}[t!]\n    \\centering\n    \\begin{subfigure}[b]{0.30\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/topic_wmt_tyumen.png}\n        \\caption{}\n        \\label{fig:topic_head_wmt_en_ru}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.30\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/topic_wmt_en_de_ruthless.png}\n        \\caption{}\n        \\label{fig:topic_head_wmt_en_de}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.30\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/topic_wmt_en_fr_collector.png}\n        \\caption{}\n        \\label{fig:topic_head_wmt_en_fr}\n    \\end{subfigure}\n    \\caption{Attention maps of the rare words head. Models trained on WMT: (a) EN-RU, (b) EN-DE, (c) EN-FR}\\label{fig:topic_head}\n\\end{figure*}\n\nTable~\\ref{tab:dependency_head_scores} shows the accuracy of the most accurate head for each of the considered dependency relations on the two domains for English-Russian. Figure~\\ref{fig:wmt_deps_ru_de_fr} compares the scores of the models trained on WMT with different target languages. \n\nClearly certain heads learn to detect syntactic relations with accuracies significantly higher than the positional baseline. This supports the hypothesis that the encoder does indeed perform some amount of syntactic disambiguation of the source sentence. \n\nSeveral heads appear to be responsible for the same dependency relation. These heads are shown in green in Figures~\\ref{fig:heads_functions}, \\ref{fig:heads_functions_de}, \\ref{fig:heads_functions_fr}. \n\n\n\nUnfortunately, it is not possible to draw any strong conclusions from these results regarding the impact of target language morphology on the accuracy of the syntactic attention heads although relations with strong target morphology are among those that are most accurately learned. \n\nNote the difference in accuracy of the verb-subject relation heads across the two domains for English-Russian. We hypothesize that this is due to the greater variety of grammatical person present\\footnote{First, second and third person subjects are encountered in approximately $6\\%$, $3\\%$ and $91\\%$ of cases in WMT data and in $32\\%$, $21\\%$ and $47\\%$ of cases in OpenSubtitles data.} in the Subtitles data which requires more attention to this relation. However, we leave proper analysis of this to future work. \n\n\n\n\\subsection{Rare words}\n\\label{sec:topic_head}\n\n\nIn all models (EN-RU, EN-DE, EN-FR on WMT and EN-RU on OpenSubtitles), we find that one head in the first layer is judged to be much more important to the model's predictions \nthan any other heads in this layer.  \n\nWe find that this head points to the least frequent tokens in a sentence. For models trained on OpenSubtitles, among sentences where the least frequent token in a sentence is not in the top-500 most frequent tokens, this head points to the rarest token in 66$\\%$ of cases, and to one of the two least frequent tokens in 83$\\%$ of cases. For models trained on WMT, this head points to one of the two least frequent tokens in more than 50$\\%$ of such cases. This head is shown in orange in Figures~\\ref{fig:heads_functions}, \\ref{fig:heads_functions_de}, \\ref{fig:heads_functions_fr}. Examples of attention maps for this head for models trained on WMT data with different target languages are shown in Figure~\\ref{fig:topic_head}.\n\n\n\n\n\n\\section{Pruning Attention Heads}\n\\label{sect:pruning_attention_heads}\n\nWe have identified certain functions of the most relevant heads at each layer and showed that to a large extent they are interpretable. What of the remaining heads? Are they redundant to translation quality or do they play equally vital but simply less easily defined roles? We introduce a method for pruning attention heads to try to answer these questions. Our method is based on \\citet{louizos2018learning}. Whereas they pruned individual neural network weights, we prune entire model components (i.e.\\ heads).  We start by describing our method and then examine how performance changes as we remove heads, identifying the functions of heads retained in the sparsified models. \n\n\n\n\n\n\\subsection{Method}\n\n\n\n\nWe modify the original Transformer architecture by multiplying the representation computed by each head$_i$ by a scalar gate $g_i$. Equation~(\\ref{eq:concat_heads}) turns into\\vspace{-2ex}\n\\begin{equation}\n\\nonumber\n\\textnormal{MultiHead}(Q,K,V )\\!=\\! \\textnormal{Concat}_i(g_i\\!\\cdot\\!\\textnormal{head}_i)W^O.\n\\end{equation}\nUnlike usual gates, $g_i$ are parameters specific to heads and are independent of the input (i.e.\\ the sentence).\nAs we would like to disable less important heads completely rather than simply downweighting them,  we would ideally apply $L_0$ regularization to the scalars $g_i$. The $L_0$ norm equals the number of non-zero components and would push the model to switch off less important heads:\n\\vspace{-1ex}\n$$\nL_0(g_1, \\ldots, g_h) = \\sum_{i=1}^{h} (1 - [[ g_i = 0 ]]),\n$$\nwhere $h$ is the number of heads, and $[[ \\quad ]]$ denotes the indicator function. \n\nUnfortunately, the $L_0$ norm is non-differentiable and so cannot be directly incorporated as a regularization term in the objective function. \nInstead, we use a stochastic relaxation: each gate $g_i$ is now a  random variable drawn independently from a head-specific distribution.\\footnote{In training, we resample gate values $g_i$ for each batch.} \nWe use the Hard Concrete distributions~\\cite{louizos2018learning},  a parameterized family of mixed discrete-continuous distributions over the closed interval $[0,1]$, see Figure~\\ref{fig:hard_concrete}. The distributions have non-zero probability mass at 0 and 1, $P(g_i = 0 | \\phi_i)$ and $P(g_i = 1 | \\phi_i)$, where $\\phi_i$ are the distribution parameters.  Intuitively, the Hard Concrete distribution is obtained by stretching the binary version of the Concrete (aka Gumbel softmax) distribution \\cite{maddison2017concrete,jang2017gumbel} from the original support of $(0, 1)$ to $(- \\epsilon, 1 + \\epsilon)$ and then collapsing the probability mass assigned to $(- \\epsilon, 1]$ and  $[1, 1 + \\epsilon)$ to single points, 0 and 1, respectively. These stretching and rectification operations yield a mixed discrete-continuous distribution over $[0, 1]$.\nNow the sum of the probabilities of heads being non-zero can be used as a relaxation of the $L_0$ norm:\n\\vspace{-1ex}\n$$\n\\vspace{-1ex}\nL_C(\\phi) = \\sum_{i=1}^{h} (1 - P(g_i = 0 | \\phi_i)).\n$$\nThe new training objective is\n$$\n\\vspace{-1ex}\nL(\\theta, \\phi) = L_{xent}(\\theta, \\phi) + \\lambda L_C(\\phi),\n$$\nwhere $\\theta$ are the parameters of the original Transformer, $L_{xent}(\\theta, \\phi)$ is cross-entropy loss for the translation model, and $L_C(\\phi)$ is the regularizer described above.  \nThe objective is easy to optimize: the reparameterization trick ~\\cite{kingma-welling-vae,pmlr-v32-rezende14} can be used to backpropagate through the sampling process for each $g_i$, whereas the regularizer and its gradients are available in the closed form.  \nInterestingly, we observe that the model converges to solutions where gates are either almost completely closed (i.e.\\ the head is pruned, $P(g_i = 0 | \\phi_i) \\approx 1$) or completely open ($P(g_i = 1 | \\phi_i) \\approx 1$), the latter not being explicitly encouraged.\\footnote{The `noise' pushes the network not to use middle values. The combination of noise and rectification has been previously used  to achieve discretization~(e.g., \\citet{kaiser2018discrete}).}\nThis means that at test time we can treat the model as a standard Transformer and use only a subset of heads.\\footnote{At test time, gate values are either 0 or 1 depending on which of the values $P(g_i = 0 | \\phi_i)$, $P(g_i = 1 | \\phi_i)$ is larger.}\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{0.23\\textwidth}\n        \\includegraphics[width=\\textwidth]{.\/pict\/hard_concrete.png}\n        \\caption{}\n        \\label{fig:hard_concrete}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.23\\textwidth}\n       \\includegraphics[width=\\textwidth]{.\/pict\/hard_concrete_params.png}\n        \\caption{}\n        \\label{fig:hard_concrete_params}\n    \\end{subfigure}\n    \\caption{Concrete distribution: (a) Concrete and its stretched and rectified version (Hard Concrete); (b) Hard Concrete distributions with different parameters.}\n    \\label{fig:concrete}\n\\end{figure}\n\n\nWhen applying this regularizer, we start from the converged model trained without the $L_C$ penalty  (i.e.\\ parameters~$\\theta$ are initialized with the parameters of the converged model) and then add the gates and continue training the full objective.\nBy varying the coefficient $\\lambda$ in the optimized objective, we obtain models with different numbers of heads retained.\n\n\n\n\n\n\\subsection{Pruning encoder heads}\n\nTo determine which head functions are most important in the encoder \nand how many heads the model needs, we conduct a series of experiments with gates applied only to encoder self-attention. Here we prune a model by fine-tuning a trained model with the regularized objective.\\footnote{In preliminary experiments, we observed that fine-tuning a trained model gives slightly better results (0{.}2--0{.}6 BLEU) than applying the regularized objective, or training a model with the same number of self-attention heads, from scratch.} \nDuring pruning, the parameters of the decoder are fixed and only the encoder parameters and head gates are fine-tuned. By not fine-tuning the decoder, we ensure that the functions of the pruned encoder heads do not migrate to the decoder.\n\n\\subsubsection{Quantitative results: BLEU score}\n\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.28]{.\/pict\/heads_dying_enc_bleu_linear_x.png}}\n\\caption{BLEU score as a function of number of retained encoder heads (EN-RU). Regularization applied by fine-tuning trained model.}\n\\label{fig:enc_heads_dying_bleu}\n\\end{figure}\n\n\n\nBLEU scores are provided in  Figure~\\ref{fig:enc_heads_dying_bleu}. Surprisingly, for OpenSubtitles, we lose only $0{.}25$ BLEU when we prune all but 4 heads out of 48.\\footnote{If all heads in a layer are pruned, the only remaining connection to the previous layer is the residual connection.} For the more complex WMT task, 10 heads in the encoder are sufficient to stay within $0{.}15$ BLEU of the full model.\n\n\n\\subsubsection{Functions of retained heads}\n\nResults in Figure~\\ref{fig:enc_heads_dying_bleu} suggest that the encoder remains effective even with only \na few heads. In this section, we investigate the function of those heads that remain in the encoder during pruning. \nFigure~\\ref{fig:encoder_heads_dying} shows all heads color-coded for their function \nin a pruned model.\nEach column corresponds to a model \nwith a particular number of heads retained after pruning.\nHeads from all layers are ordered by their function. Some heads can perform several functions (e.g., $s\\rightarrow v$ and $v\\rightarrow o$); in this case the number of functions is shown.\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.17]{.\/pict\/encoder_heads_dying_syntactic.png}}\n\\caption{Functions of encoder heads retained after pruning. Each column represents all remaining heads after varying amount of pruning (EN-RU; Subtitles).}\n\\label{fig:encoder_heads_dying}\n\\end{figure}\n\nFirst, we note that the model with 17 heads retains heads with all the functions that we identified in Section~\\ref{sect:characterizing_heads}, even though \\sfrac{2}{3} of the heads have been pruned.\n\nThis indicates that these functions are indeed the most important. Furthermore, when we have fewer heads in the model, some functions ``drift'' to other heads: for example, we see positional heads starting to track syntactic dependencies; hence some heads are assigned more than one color at certain stages in Figure~\\ref{fig:encoder_heads_dying}.\n\n\n\n\n\n\n\\subsection{Pruning all types of attention heads}\nWe found our pruning technique to be efficient at reducing the number of heads in the encoder without a major drop in translation quality. Now we investigate the effect of pruning all types of attention heads in the model (not just in the encoder). This allows us to evaluate the importance of different types of attention in the model for the task of translation.\nIn these experiments, we add gates to all multi-head attention heads in the Transformer, i.e.\\ encoder and decoder self-attention and attention from the decoder to the encoder.\n\n\\subsubsection{Quantitative results: BLEU score}\n\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{lccc}\n\\toprule\n & \\bf attention & \\multicolumn{2}{c}{\\bf BLEU} \\\\ \n & \\bf heads & from & from \\\\ \n & (e\/d\/d-e) & trained & scratch \\\\ \n \\toprule\n\\multicolumn{3}{l}{\\bf WMT, 2{.}5m}\\\\\n\\cmidrule{1-4}\nbaseline   &  48\/48\/48 & \\multicolumn{2}{c}{\\bf 29{.}6} \\\\\n\\cmidrule{1-4}\nsparse heads   & 14\/31\/30 & 29{.}62 & 29{.}47 \\\\\n   & 12\/21\/25 & 29{.}36 & 28{.}95 \\\\\n   & 8\/13\/15 & 29{.}06 & 28{.}56 \\\\\n   & 5\/9\/12 & 28{.}90 & 28{.}41 \\\\\n\n\\toprule\n\\multicolumn{3}{l}{\\bf OpenSubtitles, 6m}\\\\\n \\cmidrule{1-4}\nbaseline   &  48\/48\/48 & \\multicolumn{2}{c}{\\bf 32{.}4} \\\\\n\\cmidrule{1-4}\nsparse heads   & 27\/31\/46 & 32{.}24 & 32{.}23 \\\\\n   & 13\/17\/31 & 32{.}23 & 31{.}98 \\\\\n   & 6\/9\/13 & 32{.}27 & 31{.}84 \\\\\n\\bottomrule\n\\end{tabular}\\textbf{}\n\\caption{BLEU scores for gates in all attentions, EN-RU. Number of attention heads is provided in the following order: encoder self-attention, decoder self-attention, decoder-encoder attention.}\n\\label{tab:bleu_all_enru}\n\\end{table}\n\nResults of experiments pruning heads in all attention layers are provided in Table~\\ref{tab:bleu_all_enru}. For models trained on WMT data, we are able to prune almost \\sfrac{3}{4} of encoder heads and more than \\sfrac{1}{3} of heads in decoder self-attention and decoder-encoder attention without any noticeable loss in translation quality (sparse heads, row 1). We can also prune more than half of all heads in the model and lose no more than 0{.}25 BLEU.\n\n\nWhile these results show clearly that the majority of attention heads can be removed from the fully trained model without significant loss in translation quality, it is not clear whether a model can be trained from scratch with such a small number of heads. In the rightmost column in Table~\\ref{tab:bleu_all_enru} we provide BLEU scores for models trained with exactly the same number and configuration of heads in each layer as the corresponding pruned models but starting from a random initialization of parameters. \nHere the degradation in translation quality is more significant than for pruned models with the same number of heads. This agrees with the observations made in works on model compression: sparse architectures learned through pruning cannot be trained from scratch to the same test set performance as a model trained with joint sparsification and optimization~\\cite{zhu2017prune,state-of-sparcity-2019}. In our case, attention heads are less likely to learn important roles when a model is retrained from scratch with a small number of heads.\n\n\n\n\n\n\n\\subsubsection{Heads importance}\nFigure~\\ref{fig:all_heads_dying_by_attn_type} shows  the number\nof retained heads for each attention type at different pruning rates.\nWe can see that the model prefers to prune encoder self-attention heads first, while decoder-encoder attention heads appear to be the most important for both datasets. Obviously, without decoder-encoder attention no translation can happen.\n\nThe importance of decoder self-attention heads, which function primarily as a target side language model, varies across domains. These heads appear to be almost as important as decoder-encoder attention heads for WMT data with its long sentences (24 tokens on average), and slightly more important than encoder self-attention heads for OpenSubtitles dataset where sentences are shorter (8 tokens on average).\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.28]{.\/pict\/heads_dying_by_attn_type_both.png}}\n\\caption{Number of active heads of different attention type for models with different sparsity rate}\n\\label{fig:all_heads_dying_by_attn_type}\n\\end{figure}\n\n\n\\begin{figure}[t!]\n\\center{\\includegraphics[scale=0.28]{.\/pict\/heads_dying_decoder.png}}\n\\caption{Number of active heads in different layers of the decoder for models with different sparsity rate (EN-RU, WMT)}\n\\label{fig:all_heads_dying_decoder}\n\\end{figure}\n\n\nFigure~\\ref{fig:all_heads_dying_decoder} shows the number of active self-attention and decoder-encoder attention heads at different layers in the decoder \nfor models with different sparsity rate\n(to reduce noise, we plot the sum of heads remaining in pairs of adjacent layers). It can be seen that self-attention heads are retained more readily in the lower layers, while decoder-encoder attention heads are retained in the higher layers. This suggests that lower layers of the Transformer's decoder are mostly responsible for language modeling, while higher layers are mostly responsible for conditioning on the source sentence. These observations are similar for both datasets we use.\n\n\n\n\n\n\\section{Related work}\n\n\nOne popular approach to\nthe analysis of NMT representations is to evaluate how informative they are for various linguistic tasks. Different levels of linguistic analysis have been considered including morphology~\\cite{P17-1080,dalvi-morph-decoder,bisazza-tump:2018:EMNLP}, syntax~\\cite{D16-1159} and semantics~\\cite{hill-et-al,belinkov-I17-1001,raganato-tiedemann:2018:BlackboxNLP}.\n\n\\citet{bisazza-tump:2018:EMNLP} showed that the target language determines which information gets encoded. This agrees with our results for different domains on the English-Russian translation task in Section~\\ref{sec:syntactic-heads:results}. There we observed that attention heads are more likely to track syntactic relations requiring more complex agreement in the target language (in this case the subject-verb relation).\n\nAn alternative method to study the ability of language models and machine translation models to capture hierarchical information is to test their sensitivity to specific grammatical errors  \\cite{linzen-Q16-1037,colorless-green,tran-importance-of-being-recurrent,E17-2060,tang18-self-attention}.\nWhile this line of work has shown that NMT models, including the Transformer, do learn some syntactic structures, our work provides further insight into the role of multi-head attention.\n\nThere are several works analyzing attention weights of different NMT models~\\cite{monz-attention,voita18,tang-sennrich-nivre:2018:WMT,raganato-tiedemann:2018:BlackboxNLP}. \\citet{raganato-tiedemann:2018:BlackboxNLP} use the self-attention weights of the Transformer's encoder to induce a tree structure for each sentence \nand compute the unlabeled attachment score of these trees. However they do not evaluate specific syntactic relations (i.e.\\ labeled attachment scores) or consider how different heads specialize to specific dependency relations. \n\nRecently \\citet{bau2019neurons-in-mt} proposed a method for identifying important individual neurons in NMT models. They show that similar important neurons emerge in different models. Rather than verifying the importance of individual neurons, we identify the importance of entire attention heads using layer-wise relevance propagation and verify our findings by observing which heads are retained when pruning the model.\n\n\n \n\\section{Conclusions}\n\n\nWe evaluate the contribution made by individual attention heads to Transformer model performance on translation. \nWe use layer-wise relevance propagation to show that the relative contribution of heads varies: only a small subset of heads appear to be important for the translation task.\nImportant heads have one or more interpretable functions in the model, including attending to adjacent words and tracking specific syntactic relations.\nTo determine if the remaining less-interpretable heads are crucial to the model's performance, we introduce a new approach to pruning attention heads. \n\n\n\nWe observe that specialized heads are the last to be pruned, confirming their importance directly.\nMoreover, the vast majority of heads, especially the encoder self-attention heads, can be removed without seriously affecting performance.\nIn future work, we would like to investigate how our pruning method compares to alternative methods of model compression in NMT. \n\n\n\\section*{Acknowledgments}\nWe would like to thank anonymous reviewers for their comments. We thank Wilker Aziz, Joost Bastings for their helpful suggestions. \nThe authors also thank Yandex Machine Translation team for helpful discussions and inspiration. Ivan Titov acknowledges support of the European Research Council (ERC StG BroadSem 678254) and the Dutch National Science Foundation (NWO VIDI 639.022.518). \n\n\n\\nocite{training-tips-transformer}\n\\nocite{adam-optimizer}\n\\nocite{sennrich-bpe}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nAt the fundamental level, quantum field theory is local. Still,  in many situations non-locality emerges as a derived phenomenon. This can happen already  at a purely classical level, when one\nintegrates out some fast degree of freedom to obtain an effective theory for the slow degrees of freedom, or at the quantum level, when one considers the effective action that takes into account loop corrections involving light or massless particles. In recent years there  has been a significant activity on effective non-local modifications of gravity, largely motivated by the aim of understanding the origin of dark energy.\nIn particular, in \\cite{Maggiore:2013mea} (elaborating on previous works related to\nthe degravitation idea~\\cite{ArkaniHamed:2002fu,Dvali:2006su,Dvali:2007kt}, as well as on attempts at writing massive gravity in non-local form~\\cite{Porrati:2002cp,Jaccard:2013gla})\nwe proposed a phenomenological  modification of gravity, based on\nthe non-local equation of motion\n\\be\\label{RT}\nG_{\\mu\\nu} -(1\/3)m^2\\(g_{\\mu\\nu}\\Box^{-1} R\\)^{\\rm T}=8\\pi G\\,T_{\\mu\\nu}\\, .\n\\ee\nThe superscript T denotes the operation of taking the transverse part of a tensor (which is itself a nonlocal operation), $\\Box$ is the covariant d'Alembertian computed with the curved-space metric $g_{\\mu\\nu}$, and its inverse\n$\\Box^{-1}$ is defined using the retarded Green's function, to ensure causality. The factor $1\/3$ is a convenient normalization of the parameter $m^2$ in $d=3$ spatial dimensions. The extraction of the transverse part ensures that energy-momentum conservation is  automatically satisfied. A closed form for the action corresponding to \\eq{RT} is not known. This model is however  closely related to another non-local model, proposed in \\cite{Maggiore:2014sia}, and defined by the action\n\\be\\label{RR}\nS_{\\rm NL}=\\frac{m_{\\rm Pl}^2}{2}\\int d^{4}x \\sqrt{-g}\\, \n\\[R-\\frac{1}{6} m^2R\\frac{1}{\\Box^2} R\\]\\, .\n\\ee\nThese two models are related by the fact that,  linearizing over flat space the equations of motion derived from the action (\\ref{RR}), one finds the same equations of motion as those obtained by linearizing \\eq{RT} over flat space. However, at the full non-linear level, or linearizing over a background different from Minkowski, the two theories are different. \n\nAn intriguing aspect of the  models (\\ref{RT}) and (\\ref{RR}) is that, in the comparison with cosmological observations,  they perform remarkably well. They both have the same number of parameters as $\\Lambda$CDM, with the mass $m$ replacing the cosmological constant. They dynamically generate  a dark energy and have a realistic background FRW evolution \\cite{Maggiore:2013mea,Maggiore:2014sia,Foffa:2013vma}. Their cosmological perturbations are well-behaved (a step that already ruled out several modified gravity models)\nand their quantitative effects are consistent with CMB, supernovae, BAO and structure formation data~\\cite{Nesseris:2014mea,Dirian:2014ara,Barreira:2014kra}.\\footnote{This should be contrasted with the non-local model proposed in \\cite{Deser:2007jk,Deser:2013uya,Woodard:2014iga}. This model does not involve a mass scale, and is rather \nconstructed adding to the Einstein-Hilbert action a term of the form $Rf(\\Box^{-1} R)$. The function $f(\\Box^{-1} R)$ is  tuned so that, at the level of background evolution, this model closely mimics $\\Lambda$CDM. One can then study its\ncosmological perturbations, and  it has been found in \n\\cite{Dodelson:2013sma} that the  model is ruled out  by the comparison with structure formation.\nNon-local long-distance modifications of GR have also been suggested in \\cite{Barvinsky:2003kg,Barvinsky:2011hd,Barvinsky:2011rk,Wetterich:1997bz}.}\nThis allowed  a more detailed comparison with $\\Lambda$CDM. Implementing the cosmological perturbations of the non-local models into a Boltzmann code and performing parameter estimation and a global fit to CMB, supernovae and BAO data, one finds that these models perform as well as $\\Lambda$CDM, with  comparable values of the $\\chi^2$   \\cite{Dirian:2014bma} (in fact for model (\\ref{RT}) the $\\chi^2$ is even better than for $\\Lambda$CDM, although not at a statistically significant level). Furthermore, parameter estimation provides a value of the Hubble constant $H_0$ slightly higher than in $\\Lambda$CDM, in better agreement with that obtained from the most recent local measurements \\cite{Rigault:2014kaa}. \nTo the best of our knowledge, among the large\nvariety of existing  alternatives to the standard $\\Lambda$CDM  paradigm,\nthe models defined by  eqs.~(\\ref{RT}) or (\\ref{RR}) are the only  ones\nthat are competitive with $\\Lambda$CDM from the point of view of fitting current observations (at a level of accuracy which tests not only the background evolution but also the cosmological perturbations of the model), without being just an extension of $\\Lambda$CDM with extra free parameters (see also \\cite{Ade:2015rim}). Further conceptual and phenomenological aspects of these models have been discussed in \\cite{Foffa:2013sma,Kehagias:2014sda,Cusin:2014zoa,Dirian:2014xoa,Modesto:2013jea,Conroy:2014eja,Barreira:2015fpa,Barreira:2015vra}.\n\nThese non-local models have been proposed on a purely phenomenological ground, and it is of course important to eventually understand how they could emerge from a fundamental theory. The purpose of this paper is to investigate the possibility that such terms emerge from   infrared (IR) effects in  gravity. We will first discuss  the idea of generating the dark-energy scale from the  running of asymptotically-free coupling constants in $R^2$ extensions of Einstein gravity. We will then explore the possibility of generating similar infrared effects in Einstein gravity, with no $R^2$ terms, just as a consequence of the higher-derivative terms generated by the conformal anomaly.\n\nThe plan of the paper is as follows. In Section~\\ref{sect:one} we recall known results on the running of the couplings associated to the ${\\cal O}(R^2)$ terms. In sect.~\\ref{sect:IR} we discuss the possible IR dynamics that could be generated by the running of these coupling constants, and the  connection with the non-local models (\\ref{RT}) and (\\ref{RR}). Just as in QCD, the running of a coupling that becomes strong in the IR generates,  by dimensional transmutation, a mass scale $\\Lambda_{\\rm RR}$, analogous to $\\Lambda_{\\rm QCD}$ for strong interaction. \nIn sect.~\\ref{sect:sol} we show that the dynamical emergence of  this mass scale can have remarkable consequences on several aspects of the ``cosmological constant\" problem. An important role in the picture that we will develop is played by the conformal mode of the metric. Then, \nin sect.~\\ref{sect:conf} we examine in more detail the dynamics of the conformal mode in $R^2$ extensions of GR. Finally,\nin sect.~\\ref{sect:anomaly} we will consider  Einstein gravity, without $R^2$ terms, and the possibility that similar effects arise in this case from the higher-derivative term provided by the anomaly-induced effective action. In sect.~\\ref{sect:concl} we summarize our results.\n\n\\section{Non-local loop corrections in $R^2$ gravity}\\label{sect:one}\n\n\\subsection{Notations and conventions}\n\nWe consider the theory with action \n\\bees\\label{SEH}\nS&=&S_{\\rm EH}+S_{\\rm HD}\\nonumber\\\\\n&=& \\int d^4x \\sqrt{-g}\\, \\[ \\frac{m_{\\rm Pl}^2}{2}(R-2\\Lambda)-  \\(a_1C^2+a_2R^2+a_3 E\\)\\]\\, ,\\label{SHD}\n\\ees\nwhere $m_{\\rm Pl}=1\/(8\\pi G)^{1\/2}$ is the (reduced) Planck mass,  $S_{\\rm EH}$ \nis the Einstein-Hilbert (EH) action with a cosmological constant, and $S_{\\rm HD}$ is the higher-derivative term.\nHere \n\\be\nC^2=R_{\\mu\\nu\\rho\\sigma}^2-2R_{\\mu\\nu}^2+(1\/3)R^2\\, \n\\ee \nis the square of the Weyl tensor, and\n\\be\\label{defE}\nE=R_{\\mu\\nu\\rho\\sigma}^2-4R_{\\mu\\nu}^2+R^2\\, \n\\ee \nis the  Gauss-Bonnet term. Observe that $a_1$, $a_2$ and $a_3$ are dimensionless. We use the MTW sign conventions \\cite{MTW}, so in particular $\\eta_{\\mu\\nu}=(-,+,+,+)$. The overall minus sign in front of the higher-derivative terms in \\eq{SHD} is part of the  definition of the $a_i$ coefficients, and is chosen so that these terms will appear with a plus sign in the euclidean action, see \\eq{Seucl} below. Loop computations are indeed  typically performed with euclidean signature, and it will be important for us to be careful about the signs in the passage from Minkowskian to euclidean signature. The rotation to euclidean signature $(+,+,+,+)$ is obtained performing the  Wick rotation, i.e. introducing euclidean time $t_{\\rm E}$ from  $t_{\\rm E}=it$. Then $d^4x\\rightarrow -i(d^4x)_{\\rm E}$, while $\\sqrt{-g}\\rightarrow g^{1\/2}$ and $R$ transforms into the Ricci curvature computed with the euclidean metric, $R_{\\rm E}$, without extra minus signs, and similarly for the Riemann and Ricci tensors.  \nThen, \n\\be\n(S_{\\rm EH}+S_{\\rm HD})_{\\rm Mink}=-i \\int (d^4x)_{\\rm E} \\,g^{1\/2}\\,\\[ \\frac{m_{\\rm Pl}^2}{2} (R_{\\rm E}-2\\Lambda)\n-\\(a_1C_{\\rm E}^2+a_2R_{\\rm E}^2+a_3 E_{\\rm E}\\)\\]\\, .\n\\ee\nDefining the euclidean action $S_{\\rm Eucl}$ from \n$e^{iS_{\\rm Mink}}=e^{-S_{\\rm Eucl}}$, i.e. $S_{\\rm Eucl}=-iS_{\\rm Mink}$, we therefore find\n\\be\\label{Seucl}\nS_{\\rm Eucl}=\\int (d^4x)_{\\rm E}\\, g^{1\/2}\\,\\[ -\\frac{m_{\\rm Pl}^2}{2} (R_{\\rm E}-2\\Lambda)+a_1C_{\\rm E}^2+a_2R_{\\rm E}^2+a_3 E_{\\rm E}\\]\\, .\n\\ee\nWe will henceforth drop the subscript $E$ from euclidean quantities. Whether an action is written in Minkowskian or euclidean signature can be understood from the presence of the factor $\\sqrt{-g}$ or $g^{1\/2}$, respectively.\nWe will generically denote by $R^2$ gravity  the theory containing the Einstein-Hilbert action plus all the  terms $R^2$, $C^2$ and $E$. \n\n\\subsection{$R^2$ gravity at low energies}\\label{4sect:npert}\n\nLet us first consider $R^2$ gravity in an effective field theory approach, valid in the limit $E\\ll M_{\\rm Pl}$. In this case the ${\\cal O}(R^2)$ terms are simply seen as the first corrections that we meet as we approach the Planck scale.  More precisely, if we linearize over flat space, $g_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}$, the Lagrangian density reads,\nschematically (i.e. displaying only the powers of $h_{\\mu\\nu}$ and of derivatives, without writing explicitly any tensor structure nor numerical factors), \n\\be\\label{LEHLHD}\n{\\cal L}_{\\rm EH}+{\\cal L}_{\\rm HD}\\sim m_{\\rm Pl}^2 (h\\pa^2 h +h\\pa h\\pa h+\\ldots)\n-a \\[ (\\Box h)^2 +h  (\\Box h)^2 +\\ldots \\]\n\\ee\nwhere, for the purpose of this schematic counting,  $a$ denotes generically $a_1$ or $a_2$ (the Gauss-Bonnet term is a topological invariant and we will neglect it for most of the following discussion). In an effective field theory approach the degrees of freedom are read from the Einstein-Hilbert term, so in the spectrum we only have the massless graviton. Canonical normalization is also performed with respect to the Einstein-Hilbert term, i.e. rescaling $h_{\\mu\\nu} \\rightarrow h_{\\mu\\nu}\/m_{\\rm Pl}$, and therefore the Lagrangian density is rewritten as\n\\be\\label{schemL}\n{\\cal L}_{\\rm EH}+{\\cal L}_{\\rm HD}\\sim \\( h\\pa^2 h +\\frac{1}{m_{\\rm Pl}} h\\pa h\\pa h+\\ldots\\)\n-\\frac{a}{m_{\\rm Pl}^2} \\[ (\\Box h)^2 +\\frac{1}{m_{\\rm Pl}} h  (\\Box h)^2 +\\ldots \\]\\, .\n\\ee\nWe see that the contribution to any process from the higher-derivative terms is suppressed, compared to the contributions coming from to the EH action, by a factor of order $aE^2\/m_{\\rm Pl}^2$. The standard lore is therefore that these corrections \nare irrelevant in the infrared, where $E\\llm_{\\rm Pl}$.  \n\nThe idea that we want to explore in this paper is that, in an effective action that derives from a  full quantum theory of gravity,  $a_1$ and $a_2$ are dimensionless coupling constants that run under renormalization group, so  \n$a_i=a_i(E)$. Depending on the sign of their beta functions, their absolute values can either grow or decrease as we run toward the IR. If a coupling grows in the IR, i.e. if the sign of its beta function corresponds to asymptotic freedom in the UV, we  meet a situation conceptually similar to what happens in QCD. Running toward the IR this coupling will enter a strong coupling regime. The energy scale at which this happens defines a dynamically generated scale, analogous to $\\Lambda_{\\rm QCD}$. At energies below this new scale,  the functions $a_i(E)$ have a behavior which has nothing to do with the one computed in perturbation theory, and  can in principle even develop singularities as $E\\rightarrow 0$. In particular, if in this regime $a_2(E)\\propto E^{-4}$,  in coordinate space we would have in the action a term  $Ra_2(\\Box)R\\propto R\\Box^{-2}R$ (see sect.~\\ref{sect:NLoneloop} below), which would indeed reproduce the model (\\ref{RR}). \n\nIn this section we investigate this scenario. We will first discuss  the running of the couplings  associated to the ${\\cal O}(R^2)$ terms, and we will then give arguments, drawing  on the known behavior of singularities in the QCD Green's functions, as well as on the dynamics of the conformal mode in $R^2$ gravity, that suggest that a IR behavior $a_2(\\Box)\\propto \\Box^{-2}$ is indeed  plausible.  \nWe begin  by reviewing the known results  on perturbative loop corrections to the $a_i$ couplings. \n\n\\subsection{Non-local form factors at one loop}\\label{sect:NLoneloop}\n\n\nIn general in QFT the  running of coupling constants, which is more commonly expressed in momentum space, can also be expressed in terms of non-local form factors in the effective action. For instance, at the one-loop level the running of the electric charge in QED can be described in coordinate space by the effective action~\\cite{Barvinsky:1987uw} (see also \\cite{Dalvit:1994gf,Lombardo:1996gp})\n\\be\\label{qed}\nS_{\\rm eff}=-\\frac{1}{4}\\int d^4x\\, F_{\\mu\\nu}\\frac{1}{e^2(\\Box)}F^{\\mu\\nu}\\, ,\n\\ee\nwhere\n\\be\n\\frac{1}{e^2(\\Box)}=\\frac{1}{e^2(\\mu)}-\\beta_0\\log\\(\\frac{-\\Box}{\\mu^2}\\)\\, .\n\\ee\nHere $\\mu$ is the renormalization scale, $e(\\mu)$ is the renormalized charge at the scale $\\mu$  and, for a single massless fermion, $\\beta_0=1\/(12\\pi^2)$. The logarithm of the d'Alembertian can be defined for instance from \n\\be\n\\log\\(\\frac{-\\Box}{\\mu^2}\\)=\\int_0^{\\infty}dm^2\\, \\[\\frac{1}{m^2+\\mu^2}-\n\\frac{1}{m^2-\\Box}\\]\\, .\n\\ee\nSuch non-local actions generate non-local effective\nequations of motion  for the expectation values of the quantum fields. The corresponding equations of motion for the in-out matrix elements depend on the Feynman propagator and are acausal, but the effective classical equations of motion for the in-in expectation values, which can be computed using the Schwinger-Keldysh formalism, involve the retarded propagator and are therefore non-local but causal\\cite{Jordan:1986ug,Calzetta:1986ey}. In practice, using the in-in formalism turns out to be equivalent to performing the computation in euclidean space, and in the end rotating to Minkowskian signature while at the same time replacing the euclidean $\\Box^{-1}$ with the Minkowskian $\\Box^{-1}$ computed with the retarded Green's function, as discussed in \\cite{Barvinsky:1987uw} and recently verified explicitly, for the case of $R^2$ gravity, in \\cite{Donoghue:2014yha}.\n\nNon-local terms also appear in the renormalization of gravity with higher derivatives, as was first  revealed using the heat-kernel technique in \\cite{Barvinsky:1985an, Barvinsky:1987uw}. The result has also been  checked with standard Feynman diagram computations~\\cite{Gorbar:2002pw,Gorbar:2003yt}. These non-local terms appear  when we consider gravity semiclassically and quantize massless scalars, massless spinors or massless vector fields over a fixed curved background. Furthermore, they also appear when we quantize gravity itself, as a result of the contribution of gravitons to the quantum loops. The contribution from matter field is very well understood, and summarized in textbooks such as~\\cite{Birrell:1982ix,Buchbinder:1992rb}. In contrast, a consistent computation of the running due to graviton loops is a much more subtle issue. We examine these contributions separately.\n\n\\subsubsection{Loop corrections from matter fields}\n\nTo lowest perturbative order, the loop corrections induced by massless matter fields can be computed in semiclassical gravity, i.e. on a fixed curved background. \nUsing $R_{\\mu\\nu\\rho\\sigma}^2$, $R_{\\mu\\nu}^2$ and $R^2$ as  independent terms, the non-local correction to the euclidean action at one-loop takes the form\\footnote{We follow the notation in \\cite{Donoghue:2014yha}, except that we define the constants $\\alpha,\\beta,\\gamma$ extracting an overall factor $1\/[2(4\\pi)^2]$ in front of $S_{\\rm NL}^{\\rm 1-loop}$, which makes easier the comparison with most of the literature, e.g. with  refs.~\\cite{Gorbar:2002pw,Gorbar:2003yt}.}\n\\bees\nS_{\\rm NL}^{\\rm 1-loop}&=&\\frac{1}{2(4\\pi)^2}\\, \\int d^4x\\, g^{1\/2}\\, \\[\\alpha \\,R\\log\\(\\frac{\\Box}{\\mu^2}\\)R+\n\\beta R_{\\mu\\nu}\\log\\(\\frac{\\Box}{\\mu^2}\\)R^{\\mu\\nu}\\right. \\nonumber\\\\\n&&\\hspace{3.4cm}\\left.+\\gamma\\, R_{\\mu\\nu\\rho\\sigma}\\log\\(\\frac{\\Box}{\\mu^2}\\)R^{\\mu\\nu\\rho\\sigma}\\]\\, ,\n\\ees\nwhere $\\Box$ is the  generally covariant d'Alembertian in euclidean space. Switching to the basis of $R^2,C^2,E$,\nthis expression can be rewritten as\n\\bees\nS_{\\rm NL}^{\\rm 1-loop}&=&\\frac{1}{2(4\\pi)^2}\\, \\int d^4x\\, g^{1\/2}\\, \\[\\bar{\\alpha} \\,R\\log\\(\\frac{\\Box}{\\mu^2}\\)R+\n\\bar{\\beta}\\, C_{\\mu\\nu\\rho\\sigma}\\log\\(\\frac{\\Box}{\\mu^2}\\)C^{\\mu\\nu\\rho\\sigma}\\right. \\nonumber\\\\\n&&\\hspace*{-1.4cm}\\left.+\\bar{\\gamma}\n\\(  R_{\\mu\\nu\\rho\\sigma}\\log\\(\\frac{\\Box}{\\mu^2}\\)R^{\\mu\\nu\\rho\\sigma} -4  R_{\\mu\\nu}\\log\\(\\frac{\\Box}{\\mu^2}\\)R^{\\mu\\nu}\n+R\\log\\(\\frac{\\Box}{\\mu^2}\\)R \\)\n\\]\\, ,\\label{4SNL1loop}\n\\ees\nwhere $C_{\\mu\\nu\\rho\\sigma}$ is the Weyl tensor, while the last structure corresponds to the Gauss-Bonnet term (\\ref{defE}).\nThe relation between the coefficients is \\cite{Donoghue:2014yha}\n$\n\\alpha=\\bar{\\alpha}+\\frac{\\bar{\\beta}}{3}+\\bar{\\gamma}$, \n$\\beta=-2\\bar{\\beta}-4\\bar{\\gamma}$,\n$\\gamma=\\bar{\\beta}+\\bar{\\gamma}$.\nThe  value of the coefficients \n$\\{\\bar{\\alpha},\\bar{\\beta}, \\bar{\\gamma}\\}$  can be found e.g. in \\cite{Birrell:1982ix} (since the logarithmic terms are fixed by the $1\/\\epsilon$ divergences of the effective action). In Table~\\ref{tab1} we collect the values of $\\{\\bar{\\alpha},\\bar{\\beta}, \\bar{\\gamma}\\}$  due to a massless scalar with a generic non-minimal coupling $\\xi$ to the curvature, described by the euclidean action\n\\be\nS_s=\\frac{1}{2} \\int d^4x\\, g^{1\/2}\\, \\(g^{\\mu\\nu}\\pa_{\\mu}\\phi\\pa_{\\nu}\\phi+\\xi R\\phi^2\\)\\, ,\n\\ee\nas well as the contributions from massless spinors and  massless vector fields. The value of the scalar field is given in terms of the parameter $\\bar{\\xi}=\\xi-(1\/6)$, and $\\bar{\\xi}=0$ corresponds to a conformally coupled scalar field. \n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|} \n\\hline \n\\phantom{$\\[ \\frac{(A^2)}{B}\\]$}            &$\\bar{\\alpha}$         & $\\bar{\\beta}$   &$  \\bar{\\gamma}$\\\\\n             \\hline\nscalar\\phantom{$\\[ \\frac{(A^2)}{B}\\]$}       &$(1\/2) \\bar{\\xi}^2$ &   1\/120             & $-1\/360$\\\\ \\hline\nfermion                                                           & 0                              &  6\/120             & $-11\/360$\\\\ \\hline\nvector                                                              & 0                             &  12\/120           & $-62\/360$\\\\ \\hline\n\\end{tabular}  \n\\end{center}\n\\caption{The value of the parameters $\\{\\bar{\\alpha},\\bar{\\beta}, \\bar{\\gamma}\\}$ due to loops of   massless matter fields. \n\\label{tab1}}\n\\end{table}\n\nThe cosmological effect of these non-local terms has  been recently studied in \\cite{Donoghue:2014yha}. Even if enhanced by a logarithmic factor, a term $R\\log(-\\Box\/\\mu^2)R$ can compete with the Einstein-Hilbert term $m_{\\rm Pl}^2R$ only for nearly Planckian curvatures. Therefore, the inclusion of the UV form of the loop corrections can be relevant for addressing issues such as the resolution of cosmological singularities near the big-bang, or the emergence of the classical behavior from the full quantum theory, which are the issues addressed in  \\cite{Donoghue:2014yha}.\\footnote{See also \\cite{Rinaldi:2014gha}, where one-loop corrections to the $R^2$ theory computed expanding over de~Sitter space, and involving corrections of the form  $R^2\\log R\/\\mu^2$, are used to construct an early-universe  inflationary model. Non-local extension of the $R^2$ theory relevant in the UV have also  been studied, with different motivations, in several papers, see e.g. \\cite{Biswas:2011ar,Modesto:2011kw}.}\nHere we are rather interested in the opposite limit  of curvatures much smaller than $m_{\\rm Pl}$ and  very large distances, i.e. in the far IR, with the aim of understanding the origin of dark energy.\n\nIt is also  instructive to compute the one-loop renormalization of the $R^2$ and $C^2$ terms induced by massive (rather than massless) matter fields in a curved background, to understand the interplay between non-locality  and the  mass of the field. This computation has been \nperformed  in \\cite{Gorbar:2002pw,Gorbar:2003yt} for scalar, spinor and vector fields (neglecting the renormalization of the Gauss-Bonnet term), using a mass-dependent renormalization scheme that allows one to correctly recover the decoupling of massive fields in the limit of small momenta. \nConsider for instance a real scalar field  with mass $m_s$ and a generic non-minimal coupling $\\xi$, described  by the euclidean action\n\\be\nS_s=\\frac{1}{2} \\int d^4x\\, g^{1\/2}\\, \\(g^{\\mu\\nu}\\pa_{\\mu}\\phi\\pa_{\\nu}\\phi+m_s^2\\phi^2+\\xi R\\phi^2\\)\\, .\n\\ee\nThe corresponding one-loop contribution to the euclidean effective action \nis \\cite{Gorbar:2002pw,Gorbar:2003yt}\\footnote{This result was  first obtained in \\cite{Gorbar:2002pw}  using \na non-covariant computation based on an expansion of the action over flat space, and it was then checked in the same paper using the covariant heat-kernel technique.  In the covariant technique the quantity which is computed is $\\bar{\\Gamma}^{(1)}$, defined by \n$\\det^{-1\/2}( -\\Box+m^2+\\xi R) =\\exp\\{\\bar{\\Gamma}^{(1)}\\}$. Since the euclidean action enters as $e^{-S_{\\rm Eucl}}$, we have $S_{\\rm Eucl}^{\\rm 1-loop}=-  \\bar{\\Gamma}^{(1)}$. This is the origin of the overall sign difference between the result of the covariant computation (eqs.~(2.15) and (2.18) of  \\cite{Gorbar:2002pw}) and of the non-covariant computation, eq.~(3.9) of ref.~\\cite{Gorbar:2002pw}. I thank E.~Gorbar for clarifying me this point.\\label{foot:Gamma}}\n\\bees\nS^{\\rm 1-loop} &=& -\\frac{1}{2(4\\pi)^2}\\,\\int d^4x\n\\,g^{1\/2}\\, \\Big\\{\\,\\frac{m_s^4}{2}\\Big(\\frac{1}{\\epsilon}\n+\\frac{3}{2} \\Big) \\,+ \\,\\bar{\\xi}m_s^2R\\,\n\\Big(\\frac{1}{\\epsilon}+1\\Big) \\nonumber\\\\\n&+& \\frac{1}{2}\\,C_{\\mu\\nu\\rho\\sigma} \\,\\Big[\\frac{1}{60\\,\\epsilon}+k_W(\\Box)\\Big]\nC^{\\mu\\nu\\rho\\sigma} \\,+\n\\,R\\,\\Big[\\,\\frac{1}{2\\epsilon}\\,\\bar{\\xi}^2\\, +\nk_R(\\Box)\\,\\Big]\\,R\\,\\Big\\}\\, ,\\label{ShapSola29}\n\\ees\nwhere $1\/\\epsilon=2\/(4-D)+\\log (4\\pi\\mu^2\/m_s^2)-\\gamma_{\\rm E}$ comes from the dimensional regularization scheme in $D$ space-time dimensions,  and the form factors $k_W(\\Box)$ and $k_R(\\Box)$ are given by\n\\bees\nk_W(\\Box) &=& \\frac{8A}{15\\,a^4}\n\\,+\\,\\frac{2}{45\\,a^2}\\,+\\,\\frac{1}{150}\\,, \\label{kR}\\\\\nk_R(\\Box) &=&\n\\bar{\\xi}^2A\n+ \\(\\frac{2A}{3a^2}-\\frac{A}{6}+ \\frac{1}{18}\\)\\bar{\\xi}\n+ A\\( \\frac{1}{9a^4}- \\frac{1}{18a^2}\n+ \\frac{1}{144} \\)\\nonumber\\\\\n&&+ \\frac{1}{108\\,a^2} -\\frac{7}{2160}\n\\,, \\label{kW}\n\\ees\nwhere\n\\be\nA\\,=\\,1-\\frac{1}{a}\\log\\,\\Big(\\frac{2+a}{2-a}\\Big)\\,, \\qquad a^2 =\n\\frac{4\\Box}{\\Box - 4m_s^2}\\, .\n\\ee\nA few comments on this result are in order. First, observe that the first two terms in \\eq{ShapSola29} correspond to the renormalization of the cosmological constant and of Newton constant, respectively. Even if they depend on the renormalization scale $\\mu$ (through the parameter $\\epsilon$), they have no dependence on the $\\Box$ operator and, in this sense, no running in momentum space. This is a trivial consequence of the fact that $\\Box\\Lambda=0$, while $\\Box R$ is a total derivative, so we cannot  obtain a form factor by acting with the $\\Box$ operator on $\\Lambda$ or on $R$. In contrast, the \n$C_{\\mu\\nu\\rho\\sigma}^2$ and the $R^2$ terms acquire corrections of the form\n$C_{\\mu\\nu\\rho\\sigma} k_W(\\Box)C^{\\mu\\nu\\rho\\sigma} $ and\n$Rk_R(\\Box)R$, in full analogy with \\eq{qed}.\nIn the UV limit $\\Box\/m_s^2\\gg 1$ we have (see also \\cite{Avramidi:1989er,Dalvit:1994gf})\n\\bees\nk_W(\\Box)&\\simeq &-\\frac{1}{60}\\log\\frac{\\Box}{m_s^2}+ \\frac{23}{450} \n+{\\cal O}\\(\\frac{m_s^2}{\\Box}\\log\\frac{m_s^2}{\\Box} \\)\n\\, ,\\label{kWUV}\\\\\nk_R(\\Box)&\\simeq &-\\frac{1}{2}\\bar{\\xi}^2\\log\\frac{\\Box}{m_s^2}+\n\\(-\\frac{1}{1080}+\\frac{\\bar{\\xi}}{18}+\\bar{\\xi}^2\\)\n +{\\cal O}\\(\\frac{m_s^2}{\\Box}\\log\\frac{m_s^2}{\\Box} \\)\n\\, .\\label{kRUV}\n\\ees\nTherefore, after subtracting the divergent parts, in the limit $\\Box\/m_s^2\\gg 1$ we remain with\n\\be\nS^{\\rm 1-loop} = \\frac{1}{2(4\\pi)^2}\\,\\int d^4x\\,g^{1\/2}\\,\n\\[ \\frac{1}{2}\\bar{\\xi}^2 \\,R\\log\\(\\frac{\\Box}{m_s^2}\\)R+\n\\frac{1}{120}\\, C_{\\mu\\nu\\rho\\sigma}\\log\\(\\frac{\\Box}{m_s^2}\\)C^{\\mu\\nu\\rho\\sigma}\\]\\, ,\n\\ee\n(plus local terms independent of $\\Box$) in full agreement with \\eq{4SNL1loop} and Table~\\ref{tab1} (the renormalization of the Gauss-Bonnet term was not included in this computation).\nObserve also that, in the conformal case  $m_s=0, \\bar{\\xi}=0$, the form factor $k_R(\\Box)$ becomes local, $k_R(\\Box)=-1\/1080$, in agreement with the prediction that can be obtained directly from the conformal anomaly.\\footnote{Again, checking the sign is a bit tricky due to the many different conventions in the literature.  The standard anomaly result can be read from  eq.~6.124 and Table 1 on page  179 of the Birrell and Davies textbook ~\\cite{Birrell:1982ix}. One must however take care that Birrell and Davies use the signature $(+,-,-,-)$ and their definition of Riemann tensor is opposite to MTW. To go to MTW sign conventions we therefore must switch $g_{\\mu\\nu} \\rightarrow -g_{\\mu\\nu}$ (so $\\Box \\rightarrow -\\Box$) and $R \\rightarrow +R$.\nTaking into account that this flips also the sign of the trace $T^{\\mu}_{\\mu}$, we find that the correct result for the anomaly of a single massless conformally coupled scalar field, in MTW conventions, is\n$ T = + \\frac{1}{180 (4\\pi^2)} \\Box R$. From the signature $(-,+,+,+)$ we can then trivially rotate to euclidean space without getting extra minus signs. In  eq.~(3.12) of \\cite{Gorbar:2002pw} the sign of the anomaly in the euclidean space was incorrectly taken to be a minus, and the result was checked using $\\bar{\\Gamma}^{(1)}$ instead of $S_{\\rm Eucl}^{\\rm 1-loop}=-  \\bar{\\Gamma}^{(1)}$, see footnote~\\ref{foot:Gamma}, so these two sign errors compensated.}\n \nIn contrast,  in the opposite limit $\\Box\/m^2\\ll 1$, \\eqs{kR}{kW} give\n\\be\\label{dec}\nk_W(\\Box),k_R(\\Box)={\\cal O}(\\Box\/m^2)\\, , \n\\ee\ncorresponding to the decoupling of particles with mass large compared to the momentum scale, which is explicit in the mass-dependent subtraction scheme used in \\cite{Gorbar:2002pw,Gorbar:2003yt} (see also the discussion in \\cite{Manohar:1996cq}).\nFurthermore, beside being small, a term ${\\cal O}(\\Box\/m^2)$ is local. This shows explicitly that non-local terms appear in the effective action as a consequence of the running in the loops of particles that are either massless, or light with respect to the energy scale considered.\n\n\n\n\\subsubsection{Gravitational loop corrections in Einstein gravity}\n\nIn semiclassical gravity, $a_1,a_2,a_3$ are simply parameters of the effective action. \nWhen we go beyond the semiclassical approximation, and treat gravity itself at the quantum level, $a_1$ and $a_2$  become genuine coupling constants, and the perturbative expansion is organized in terms of the three coupling of the theory $G, a_1$ and $a_2$ (while  the Gauss-Bonnet term is a topological invariant, and will be neglected in the following). \nTheir renormalization group equations also receive contributions from the purely gravitational sector, i.e. from graviton loops. However, these contributions depends crucially on the UV completion of the theory, as we now review.  \n\nIf one starts from the Einstein-Hilbert action without higher-derivative terms and computes the one-loop corrections, one in principle finds $R^2$ and $C^2$ terms, as was first shown in the classic paper by 't~Hooft and Veltman \\cite{'tHooft:1974bx}. The original computation of \\cite{'tHooft:1974bx} gives, in our notation, $\\bar{\\alpha}=90\/360=1\/4$ and $\\bar{\\beta}=126\/360=7\/20$, which are the values quoted in~\\cite{Donoghue:2014yha}. However, these divergences have no physical meaning, since they  can be set to zero using the equations of motion~\\cite{'tHooft:1974bx}, so that pure gravity at one loop is finite on-shell. Equivalently, they can be set to zero with a field redefinition\n$g_{\\mu\\nu}\\rightarrow g_{\\mu\\nu}+\\epsilon (c_1R_{\\mu\\nu}+c_2 Rg_{\\mu\\nu})$. More generally, if one quantizes the pure Einstein-Hilbert action, the coefficients of the $R^2$ and $R_{\\mu\\nu}^2$ terms which are generated at one loop depend on the gauge used, and one can find gauges in which these divergences are absent, so that the theory is one-loop finite even off-shell~\\cite{Kallosh:1978wt}. Thus, in pure Einstein-Hilbert gravity there is no sense in which we can compute the gravitational contribution to the running of the coupling constants associated to the $R^2$ terms. The same is true if we  consider the  higher derivative theory (\\ref{SHD}) in an effective action approach, since the higher-derivative term add extra vertices, but does not change the structure of the propagator.\n\n\n\\subsubsection{Gravitational loop corrections in Stelle theory}\\label{sect:Stelleth}\n\nThe situation  changes in an interesting manner if, rather than using an effective action approach, in which the higher-derivative term is seen as a correction valid only in the low-energy limit, we try to take the action (\\ref{SHD})\nas a full UV complete theory, and quantize it. In this approach, the higher-derivative theory (\\ref{SHD}) is usually known as Stelle theory.  In this case the propagator is read from the full quadratic term in \\eq{LEHLHD} and, again schematically, is of the form \n\\be\\label{Gdik}\nG(k)\\sim \\frac{1}{m_{\\rm Pl}^2k^2+ak^4}\\, ,\n\\ee\n(before any canonical normalization) so that now it goes as $1\/k^4$ in the far UV. This improved behavior of the propagator allowed Stelle to prove that this theory is renormalizable~\\cite{Stelle:1976gc}.\\footnote{Of course, the full story is  more complex. The $R^2$ and $C^2$ terms also lead to vertices growing as $k^4$, rather than  $k^2$ as in Einstein-Hilbert gravity. In GR, since the propagator go as $1\/k^2$ and the vertices as $k^2$, the superficial degree of divergence is $D=4L-2I+2V$, where $L$ is the number of loops, $I$ of internal lines, and $V$ of vertices. Combined with the topological relation $L=1+I-V$ this gives $D=2L+2$, so the degree of divergence grows with the number of loops. In contrast, for ${\\cal L}_{\\rm HD}$, $D=4L-4I+4V$ which, together with \n$L=1+I-V$, gives $D=4$, independent of the number of loops~\\cite{Stelle:1976gc}. This improved behavior is at the basis of Stelle's proof of renormalizability. \nObserve also that, for  renormalizability, it is crucial to include both the $R^2$ and the $C^2$ terms in the action, since they contribute separately to the propagator of the spin-0 and massive spin-2 excitation of the theory. Otherwise,  the propagator of one of these excitations would still go as $1\/k^2$, while the vertices grow as $k^4$, leading to an even worse UV behavior than in Einstein-Hilbert gravity. \nFurthermore, even the Gauss-Bonnet term is required for multiplicative renormalizability (see the recent discussion in ~\\cite{Einhorn:2014gfa}).}\nHaving a renormalizable quantum theory of gravity is quite  remarkable. This however  comes at a very high price. Namely, the spectrum of $R^2$ gravity, which is now read from the full propagator, consists of the usual massless graviton, \na massive spin-2 particle and a massless spin-0 particle~\\cite{Stelle:1977ry,Fradkin:1981iu}.\nThe massive spin-2  state has negative kinetic energy, so is a ghost, and its squared mass is\n\\be\\label{m22}\nm_2^2=\\frac{m_{\\rm Pl}^2}{2 a_1}\\, .\n\\ee\nThe massive spin-0  state has a normal kinetic term, and squared mass\n\\be\\label{m02}\nm_0^2=-\\frac{m_{\\rm Pl}^2}{12 a_2}\\, ,\n\\ee\nand is a tachyon if $a_2>0$. Because of the ghost, and (in case) of the tachyon, $R^2$ gravity, treated non-perturbatively as a full UV complete theory, is not a consistent theory  (otherwise, the problem of finding a consistent theory of quantum gravity would have been solved in the late 1970s...). There have been attempts at exorcising the ghost based on the idea that radiative corrections shift its pole into the complex plane~\\cite{Tomboulis:1977jk,Antoniadis:1986tu}, so that the ghost becomes unstable and does not appear in the asymptotic states. To leading order in $1\/N$, where $N$ is the number of conformally coupled matter fields, the theory turns out indeed to be unitary. However, attempts at proving unitarity beyond leading order based on the gauge-dependence of the position of the ghost poles fail~\\cite{Johnston:1987ue}, so what happens  to all orders is basically unknown. In any case,\nthere is a long tradition of ignoring the ghost problem and see what the $R^2$ theory has to say at a full non-perturbative level, in the hope that this will at least help to unveil properties of a fully consistent theory of quantum gravity. In particular, one can compute the running of the coupling constants associated to the ${\\cal O}(R^2)$ terms. As we mentioned above, in Einstein-Hilbert gravity this computation leads to beta functions that are not even gauge-independent.\nIn contrast, in Stelle theory the computation is well-defined, and the  beta functions  associated to the $R^2$, $C^2$  and Gauss-Bonnet terms are physical and gauge-independent (while, for the Newton constant and the cosmological constant, only the dimensionless combination \n$G\\Lambda$  has a gauge-independent beta function)~\\cite{Julve:1978xn,Fradkin:1981iu,Avramidi:1985ki}. For Stelle theory,\nthe  computation of the gravitational contribution to the beta functions of the $a_i$ coupling was first performed correctly in \\cite{Avramidi:1985ki} (building on earlier work in  \\cite{Fradkin:1981iu,Julve:1978xn}), and the result was  confirmed in \\cite{deBerredoPeixoto:2004if}.  The result is conveniently expressed trading $a_1$ and $a_2$ for two couplings $\\lambda$ and $u$ defined by\n\\be\\label{a1a2lu}\na_1=\\frac{1}{\\lambda}\\, ,\\qquad a_2=-\\frac{u}{3\\lambda}\\, ,\n\\ee\nso that the Minkowskian action (\\ref{SEH}) reads (neglecting the Gauss-Bonnet term and the cosmological constant)\n\\be\nS=\\int d^4x \\sqrt{-g}\\, \\[ \\frac{m_{\\rm Pl}^2}{2}R-  \\frac{1}{\\lambda} C^2+ \\frac{u}{3\\lambda}R^2\\]\\, .\n\\ee\nThe corresponding  beta functions\nare given in eqs.~(19)-(24) of \\cite{Avramidi:1985ki}. In particular, introducing the logarithm of the energy-scale parameter $t=(4\\pi)^{-2}\\log E\/\\mu_0$, the purely gravitational contribution is\n\\bees\n\\frac{d\\lambda^{-1}}{dt}&=&\\bar{\\beta}_g\\, ,\\label{betal}\\\\\n\\frac{du}{dt}&=& -\\lambda \\( \\frac{10}{3}u^2 +\\frac{183}{10}u+\\frac{5}{12}\\)\\, ,\\label{betau}\n\\ees\nwhere $\\bar{\\beta}_g=133\/10$ is positive.  It is important to stress that  these beta functions have been computed in the UV limit $E\\ggm_{\\rm Pl}$. In the  regime $E\\llm_{\\rm Pl}$ which is interesting for us, these equations will receive corrections from the Einstein-Hilbert term, which have not been computed.\nThe solution of \\eq{betal} is\n$\\lambda^{-1}(t)=\\lambda^{-1}(t=0)+\\beta_g t$,\ni.e.\n\\be\\label{ldiE}\n\\lambda(E)=\\frac{\\lambda(\\mu)}{1+\\lambda(\\mu) \\beta_g\\log (E\/\\mu)  }\\, .\n\\ee\nwhere $\\beta_g=\\bar{\\beta}_g\/(4\\pi)^2$. Whether this corresponds to a function $\\lambda(E)$ that becomes large in the UV and small in the IR, or viceversa,  depends of course of the sign of $\\lambda(\\mu)$. Since $\\bar{\\beta}_g>0$, \nif $\\lambda(\\mu)>0$ we have  $\\lambda(E)\\rightarrow 0$ as $E\\rightarrow\\infty$, and conversely $\\lambda(E)$ gets large and formally diverges at a finite energy scale  in the IR.\nThe opposite situation takes place if $\\lambda(\\mu)<0$.  Observe from\nTable~\\ref{tab1} that the contribution to the beta function of $a_1=\\lambda^{-1}$ coming from each\ntypes of matter fields has the same sign as the gravitational contribution, so the same behavior takes place in the full theory with gravitational plus matter loops.\n\nThe evolution of $u$ is instead determined, at least in the super-Planckian regime in which \\eq{betau} holds, by the two roots of the \nequation $(10\/3)u^2 +(183\/10)u+(5\/12)=0$, which are given by $u_1\\simeq -5.46714$, $u_2\\simeq -0.02286$. Writing \n\\eq{betau} as \n\\be\n\\frac{du}{dt}=-\\frac{10}{3}\\lambda (u-u_1)(u-u_2)\\, ,\n\\ee\nwe can easily read the sign of $du\/dt$ and the direction of the RG flow, for different values of $u$, and a given sign of $\\lambda$. For $\\lambda >0$ the one-dimensional flow in the $u$ variable is depicted in the upper panel of Fig.~\\ref{fig:flow_u} and for $\\lambda <0$ is shown in\nthe lower panel. Observe in particular that, for $\\lambda>0$, $u=u_2$ is a UV fixed point, which attracts all initial values\n$u(t=0)>u_1$. For $\\lambda<0$ the value  $u=u_1$ is a UV fixed point at its attraction basin is given by $u(t=0)<u_2$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{flow_u.pdf}\n\\caption{\\label{fig:flow_u} The RG flow of $u(t)$ for $\\lambda>0$ (upper panel) and  $\\lambda<0$ (lower panel). The direction of the arrows indicate the flow of $u$ as $t$ increases, i.e. as we move from the IR to the UV. For a given initial value in the UV, the flow as we run toward the IR regime is therefore obtained reversing the sign of the arrows.\n\\label{fig:flow_u}\n}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[line width=1 pt, scale=2]\n\\draw[graviton] (1,0)--(2,0);\n\\node at (2.5,0) {+};\n\\draw[graviton] (3,0)--(4.02,0);\n\\draw[graviton] (5,0) arc (0:360:0.5);\n\\draw[graviton] (5,0)--(6,0);\n\\end{tikzpicture}\n\\caption{The graviton propagator and a one-loop correction. \\label{fig:loop}}\n\\end{center}\n\\end{figure}\n\n\nOne might wonder how is it possible that a computation which is well-defined in the full $R^2$  theory is not well-defined in Einstein-Hilbert gravity, since the latter is formally obtained taking the limit $a_1,a_2\\rightarrow 0$ in the action (\\ref{Seucl}).\nThe reason is that, when we take the $R^2$ theory as a full non-perturbative theory,  the structure of the divergences, which determine the beta functions, changes discontinuously if we set $a_1$ or $a_2$ to zero. Indeed, consider for instance a loop correction to the graviton propagator such as that shown in Fig.~\\ref{fig:loop}. Even when the momentum $p$ of the external line is sub-Planckian, $|p^2|\\llm_{\\rm Pl}^2$, in the loops we will have propagators such as $G(k)$, given in \\eq{Gdik}, and $G(k-p)$, where $k$ is the loop integration variable and $p$ the external momentum. The divergence is determined by the behavior of the propagator (and of the vertices) for $k\\rightarrow \\infty$. For any $a\\neq 0$ the propagator at $k\\rightarrow\\infty$ behaves as $1\/(ak^4)$, even when the external momenta are sub-Planckian, while for $a=0$ it behaves as $1\/k^2$ and the structure of the divergences is completely different. In other words, the limit  $a\\rightarrow 0$   does not commute with the integrations over $d^4k$ in the loop integrals. \nSo, despite the fact that $S_{\\rm HD}$ is a higher-derivative term, which naively should be irrelevant at low energies, the structure of the divergences of Stelle theory is completely different from that of GR, even when the external lines are sub-Planckian, and the limit $a\\rightarrow 0$ (or, more precisely $a_1\\rightarrow 0$ or $a_2\\rightarrow 0$) of the beta functions of $R^2$ gravity  is not smooth. This fact has already been stressed in sect.~3.1 of \\cite{Fradkin:1981iu}, where in particular it was observed that\nthe beta function of the pure Weyl theory cannot be obtained setting $a_2=0$ in the beta function of the theory with a higher-derivative term $a_1C^2+a_2R^2$.\n\n\n\\subsubsection{Gravitational running in a consistent theory of quantum gravity}\n\nThe result (\\ref{betal}, \\ref{betau}) cannot be applied directly to our problem, namely the running of $a_2$ at low energies in the actual universe, for two  reasons. First, the computation leading to \\eqs{betal}{betau} has been performed in the deep UV region, $E\\ggm_{\\rm Pl}$. In the  regime $E\\llm_{\\rm Pl}$, which is interesting for us, these equations will receive contributions from the Einstein-Hilbert term, which have not been computed. This is just a technical problem. However, a second and  more fundamental problem is that,\nbarring a resolution of the ghost problem, Stelle theory can anyhow only be taken as a toy model, rather than a theory describing  Nature.\n\nHowever what we learn from this toy model is that, in a  theory of gravity with a good UV behavior, the computation of the beta functions associated to the $a_i$ couplings is well defined, and these quantities run with energy. In the end we expect that, whatever will be the consistent fundamental quantum theory of gravity,  in that context it will be possible to compute consistently the running of the coupling constant associated to the $R^2$ and $C^2$ term in the corresponding low-energy effective action, including the gravitational contribution.\\footnote{For instance, in string theory one could in principle compute  the string-loop correction to gravitational scattering amplitudes directly at the world-sheet level, and reconstruct from it the corresponding loop corrections in the target-space effective action. Actually, in the effective action of type II string theory in 10 dimension there is no $R^2$ term because of supersymmetry. However, in general, the presence and the structure  of ${\\cal O}(R^2)$ corrections that one obtains in four dimensions will depend on the type of string theory considered, as well as on how supersymmetry is broken in the compactification to four dimensions. Observe that it is sometimes argued that, since the fundamental string theory is ghost-free, in the low-energy effective action can appear only the ghost-free combination given by the Gauss-Bonnet term. This argument is however wrong. Being ghost-free is a property of the full theory, and if we write the effective action derived\nfrom a ghost-free theory in an expansion in derivatives and truncate this  expansion to a finite order (e.g. in our case keeping only the terms up to ${\\cal O}(R^2)$) apparent ghost state will in general appear, with a mass of the order of the cutoff scale of the effective theory (in our case $m_{\\rm Pl}$ or the string mass scale), but they are just an artefact of the truncation. See e.g. \\cite{Burgess:2014lwa} for a nice explicit example.}\n\nFrom the above analysis we also learn that the running of the  couplings $a_i(E)$ depends on the precise form of the UV completion of the theory, even at $E\\llm_{\\rm Pl}$. Indeed, the running of $a_i(E)$\ncomputed in a consistent  UV completion of GR cannot reduce at low energies to that  computed in the effective low-energy theory (i.e. in the Einstein-Hilbert action supplemented by the higher-derivative terms, considered as corrections in the effective field theory sense), simply because there is no consistent result in the low-energy effective theory. The beta functions computed in the low-energy effective action based on the Einstein-Hilbert term, in which the $R^2$ terms are treated as corrections, are not even gauge-invariant. This of course does not happens e.g. in QCD, because QCD is renormalizable, which in the end means that, even if we eventually embed it in a larger theory, the details of the UV completion, much as the details of the regularization scheme, are irrelevant at low energies. This is not the case for Einstein-Hilbert gravity, which is not renormalizable.\nIn a sense, this phenomenon is an interesting example of UV\/IR mixing, since it implies that the IR physics of GR can sensitive\nto the UV completion.\n\n\n\nThus, the points that we will take home from the following analysis are the following. \n \n\n\\begin{enumerate}\n\n\\item In a fundamental and consistent quantum theory of gravity the couplings associated to the $R^2$ and $C^2$ term will in general  run with energy, and their beta functions will  well-defined and gauge-independent.\\footnote{Of course, in some special case, the beta functions can be zero, e.g. if they are protected by supersymmetry.} \n\n\\item  The precise form of the corresponding RG equation, even in the IR regime $E\\llm_{\\rm Pl}$, depends on the specific UV completion of the theory, and is therefore at present completely unknown. \nWe cannot use the results from GR plus higher-derivative terms, considered in an effective action approach, because the corresponding beta functions are non even gauge-invariant. We cannot use the results of Stelle theory (\\ref{betal}) and (\\ref{betau}), either, because, first of all, these results only hold in the regime $E\\ggm_{\\rm Pl}$ and, more importantly, Stelle theory  anyhow is not a consistent UV completion of gravity because of the ghost.\n\n\\end{enumerate}\n\n\nIt is also interesting to observe that, to obtain the model (\\ref{RR}) in the IR, it is not a priori necessary to match the sign of $a_2(\\mu)$ in the UV with its (positive) sign in the IR. Indeed, Fig.~\\ref{fig:flow_u} gives an explicit example of a RG flow that changes the sign of $u$, and therefore of $a_2$.\nA renormalization group flow that changes the sign of the coupling $a_2(\\Box)$, flowing from a value $a_2(\\mu)<0$ at some scale $\\mu$ corresponding to the energy scale of inflation,  to a value $a_2(\\Box)=+\\Lambda_{\\rm RR}^4\/\\Box^2$ in  the IR (where $\\Lambda_{\\rm RR}$ is a constant that we will discuss below), would be particularly intriguing since it would interpolate between the Starobinsky model~\\cite{Starobinsky:1980te} in the UV and the non-local model (\\ref{RR}) in the IR, thereby providing a unified description of inflation in the early universe and dark energy in the present epoch.\\footnote{I thank  Eugenio Bianchi for a discussion from which this idea emerged.} \n\n\n\n\\section{Infrared dynamics}\\label{sect:IR}\n\nEven if we do not know the form of the IR running of the couplings $a_i(E)$ in the fundamental theory of gravity, we can still analyze some general scenarios. In the end, there are basically two natural possibilities. Either $|a_2(E)|$ grows in the IR, or it decreases. The interesting situation, from our point of view, takes place when in the IR  $|a_2(E)|$ grows and, at least at sufficiently low energies,  $a_2(E)>0$   (since this sign allows us to eventually match $a_2(\\Box)$ with the sign of the model \\ref{RR}). In this case, the situation will be quite similar to that in QCD. To stress the similarity, let us use the notation $a_2(E)=g^2(E)$ and assume that $g^2(E)$ is asymptotically free in the UV so, as long as we are in the perturbative regime,\n\\be\\label{gdiE}\ng^2(E)=\\frac{g^2(\\mu)}{1+g^2(\\mu) \\alpha_0\\log (E\/\\mu)  }\\, ,\n\\ee\nwith $\\alpha_0>0$. This implies that $g^2(E)$ eventually becomes large in the IR. Similarly, we can write $a_1(E)=f^2(E)$ and study the scenario in which $f^2(E)$ is asymptotically free in the UV, and\n\\be\\label{gdiE}\nf^2(E)=\\frac{f^2(\\mu)}{1+f^2(\\mu) \\beta_0\\log (E\/\\mu)  }\\, ,\n\\ee\nwith $\\beta_0>0$. To understand what happens when  we enter more deeply in the IR regime is of course a  difficult problem.\nAs already observed recently \\cite{Einhorn:2014gfa,Alvarez-Gaume:2015rwa}\na pure $R^2$ theory, with no Einstein-Hilbert term, has a confining Newtonian potential $V(r)\\propto r$, because of the $1\/|{\\bf k}|^4$  behavior of the propagator in the static limit. Its infrared limit is therefore very different from Einstein gravity, and indeed one can even expect   that in this theory gravitons should be confined. When we  also add  the Einstein-Hilbert term, at energies $E\\llm_{\\rm Pl}$ the standard $m_{\\rm Pl}^2k^2$ term from Einstein-Hilbert action  dominates over the $k^4$ term and we recover the standard behavior of gravity. Still, in the infrared, from the point of view of the quantum theory of gravity, we are in a peculiar situation: loops associated to an expansion in  powers of  Newton's constant are suppressed by powers of $GE^2\\sim E^2\/m_{\\rm Pl}^2\\ll 1$ and are therefore irrelevant. However, if the the couplings $a_1(E)$ and\/or  $a_2(E)$  run toward a strong coupling regime and formally diverge at some energy in the IR, quantum loops associated to them can be important. As a result, the form factors associated to the $R^2$ and $C^2$ terms can become quite different in the IR. To determine the expression of these form factors  in the IR is of course a difficult non-perturbative problem. However,  experience with QCD allows us to understand some aspects of it. In particular, we know that the logarithmic running  of an asymptotically free coupling constant  generates  by dimensional transmutation  a renormalization group invariant mass scale, that we denote by $\\Lambda_{\\rm RR}$, analogous to $\\Lambda_{\\rm QCD}$.\\footnote{A similar scenario has been discussed  recently, in\n\\cite{Einhorn:2014gfa,Salvio:2014soa}, although in these cases dimensional transmutation is rather used to generate the Planck scale from a pure higher-derivative gravity. The idea has been around in different forms since a long time, see e.g.~\\cite{Zee:1983mj,Nepomechie:1983yq}.   In ref.~\\cite{Tong:2014era} similar ideas have been applied to the coefficient of the Gauss-Bonnet term, using arguments from supergravity.} More precisely, once we fix observationally the value of $f^2(\\mu)$  at some scale $\\mu$,  formally, as we run toward the IR, the one-loop expression for $f^2(E)$ will hit infinity at some scale\n$\\Lambda_f$ given by \n\\be\n\\Lambda_f=\\mu \\exp\\left\\{-\\frac{1}{\\beta_0 f^2(\\mu)}\\right\\}\\, .\n\\ee\nSimilarly, the one-loop expression for $g^2(E)$ will hit infinity at some scale\n\\be\n\\Lambda_g=\\mu \\exp\\left\\{-\\frac{1}{\\alpha_0 g^2(\\mu)}\\right\\}\\, .\n\\ee\nAs we run toward the IR, the first scale that we hit, where one of the two running couplings formally diverges, defines the strong-coupling scale of the theory, so $\\Lambda_{\\rm RR}={\\rm max}(\\Lambda_f,\\Lambda_g)$.\\footnote{One might wonder whether one of the couplings could still remain weakly coupled in the regime where the other is  strongly coupled. The answer is in general no. For instance, a pure Weyl-square theory, which only has the $f^2$ coupling, is not multiplicatively renormalizable, and generates a $R^2$ term already at the two-loop level~\\cite{Fradkin:1983tg}. Therefore, when $f^2$  enters the strong-coupling regime, it necessarily induces a strong coupling in  $g^2$. For this reason, even if for instance the coupling $a_2$ associated to the $R^2$ term should not have the appropriate sign for becoming strongly coupled according to its own one-loop RG equation, it could eventually be driven toward  strong coupling by the coupling $a_1$.}\nWe therefore have a mass scale that appears in the IR regime by dimensional transmutation.\nThen, in the strong coupling region,\n$f^2=f^2(\\Box\/\\Lambda^2_{\\rm RR})$ and $g^2=g^2(\\Box\/\\Lambda^2_{\\rm RR})$. In principle, it is possible to obtain a power-like dependence of \n$f^2$ and $g^2$ on $\\Box$, from a resummation of leading logs, of the form\\footnote{See \\cite{Hamber:2005dw} for similar considerations applied to the running of Newton's constant, and \\cite{Taylor:1989tm,Taylor:1989ua} for earlier related ideas on how  the large-distance behavior of  quantum gravity might affect the cosmological constant.}\n\\be\\label{resumnu}\n\\sum_{n=0}^{\\infty}\\, \\frac{(-\\nu)^n}{n!}\\, \\(\\log\\frac{-\\Box}{\\Lambda^2_{\\rm RR}}\\)^n=\n\\(\\frac{\\Lambda^2_{\\rm RR}}{-\\Box}\\)^{\\nu}\\, ,\n\\ee\nfor some value of $\\nu$. The model (\\ref{RR}) would then be obtained if $\\nu=2$. Again, to determine the precise form of $g^2(\\Box)$ in the IR is of course  a difficult perturbative problem. However,  it is interesting to observe that in QCD the introduction in the action of a non-local term\n\\be\\label{Fmn2}\n-\\frac{m^2}{2}\\int d^4x\\, F_{\\mu\\nu}^a \\[\\frac{1}{D^2}\\]^{ab}F_{\\mu\\nu}^b\\, ,\n\\ee\n(where  $D_{\\mu}^{ab}=\\delta^{ab}\\pa_{\\mu}-gf^{abc}A_{\\mu}^c$ is the covariant derivative and $m$ is a mass scale) correctly reproduces the results on the non-perturbative gluon propagator in the IR, obtained from operator product expansions and lattice QCD~\\cite{Boucaud:2001st,Dudal:2005na,Capri:2005dy,Dudal:2007cw,Dudal:2008sp} (see also the review~\\cite{Boucaud:2011ug}). In the gauge $\\pa_{\\mu} A_{\\mu}=0$ we have \n\\be\n-\\frac{m^2}{2}\\int d^4x\\, F_{\\mu\\nu}^a \\[\\frac{1}{D^2}\\]^{ab}F_{\\mu\\nu}^b=\nm^2\\int d^4x\\, A_{\\mu}^aA_{\\mu}^a +{\\cal O}(A^3)\\, ,\n\\ee\nand therefore (\\ref{Fmn2}) is a mass term for the gluon. Observe that  the expression (\\ref{Fmn2}) is  non-local but  gauge-invariant. \nPhysically, the introduction of this term in the effective action can  be seen as describing the emergence of a dimension-2 gluon condensate,\n$\\langle A_{\\mu}^aA_{\\mu}^a\\rangle\\neq 0$. The same  operator has been proposed in 3-dimensional YM theory \\cite{Jackiw:1995nf,Jackiw:1997jga}, to describe the generation of a mass gap due to IR fluctuations.\n\nThe situation is quite similar for the term $R\\Box^{-2}R$ in the gravitational case. Indeed, consider the conformal mode $\\sigma(x)$ of the metric, \ndefined choosing a fixed fiducial metric $\\bar{g}_{\\mu\\nu}$ and writing\n\\be\\label{gmnsigma}\ng_{\\mu\\nu}(x)=e^{2\\sigma(x)}\\bar{g}_{\\mu\\nu}(x)\\, .\n\\ee\nUsing for simplicity a flat fiducial metric $\\bar{g}_{\\mu\\nu}=\\eta_{\\mu\\nu}$, the Ricci scalar computed from the metric $g_{\\mu\\nu}=e^{2\\sigma(x)}\\eta_{\\mu\\nu}$ is\n\\be\nR=-6 e^{-2\\sigma}\\( \\Box\\sigma +\\pa_{\\mu}\\sigma\\pa^{\\mu}\\sigma\\)\\, .\n\\ee\nTherefore, to linear order in $\\sigma$,\n\\be\nR=-6\\Box\\sigma +{\\cal O}(\\sigma^2)\\, ,\n\\ee\nand (upon integration by parts)\n\\be\\label{m2s2}\n R\\frac{1}{\\Box^2} R=36 \\sigma^2 +{\\cal O}(\\sigma^3)\\, .\n\\ee\nThe  $\\sigma^2$ term is  just a mass term for the conformal mode, while the higher-order interaction terms on the right-hand side of \\eq{m2s2} (which are non-local even in $\\sigma$) are required to reconstruct a diff-invariant quantity.\\footnote{Actually, there is here a subtle  technical point, both in the gravitational and YM case. Indeed,  in \\eq{m2s2} we have implicitly using both the equality $\\Box\\Box^{-1}=1$, and\n$\\Box^{-1}\\Box=1$. While the former is correct by definition of $\\Box^{-1}$, the latter is not true since $\\Box$ has a non-trivial kernel, given by the functions $f$ such that $\\Box f=0$. On such functions the formal equality $\\Box^{-1}\\Box f=f$ is not true. Thus, to be accurate, we should rather write in \\eq{m2s2}\n$\\sigma\\Box^{-1}\\Box\\sigma$  instead of $\\sigma^2$.\nNote  that the expression  $\\sigma\\Box^{-1}\\Box\\sigma$ has an invariance under constant shifts of the conformal factor, $\\sigma\\rightarrow\\sigma+c$, which is lost if we write it as $\\sigma^2$. The equality $\\sigma\\Box^{-1}\\Box\\sigma=\\sigma^2$ becomes however correct for all functions $\\sigma$ such that $\\Box\\sigma\\neq 0$.}\nIn this sense the term\n$R\\Box^{-2}R$ is completely analogous to the term (\\ref{Fmn2}) in the Yang-Mills case, and its introduction describes a sort of ``condensation\" of the conformal mode. The fact that  the term $R\\Box^{-2}R$ has this special physical meaning, and the analogy with the YM case, \nmakes it more plausible its emergence in the IR limit of $R^2$ gravity.\\footnote{However, in $R^2$ gravity the interpretation in terms of the dynamics of the conformal mode is complicated by the fact that the conformal already gets a kinetic term, as well as a mass term,  at tree level from the $R^2$ term, as we discuss in sect.~\\ref{sect:conf}. This will eventually lead us to consider, in sect.~\\ref{sect:anomaly}, the dynamics of the conformal mode in Einstein gravity supplemented by the anomaly-induced effective action, where no such complication arises, and one can really have a dynamical mass generation for an otherwise massless mode.}\n\nObserve also that the requirement of general covariance does not fix uniquely the term that reduces to $\\sigma^2$ to quadratic order in $\\sigma$. Indeed, as we already mentioned,  the equations of motion of the \nthe models (\\ref{RT}) and (\\ref{RR}) have the same expansion\n to linear order over Minkowski space, so the corresponding actions are the same in an expansion to  quadratic order over Minkowski, and they both reproduce the term $\\propto\\sigma^2 $ in the action, although with different non-linear completions. Both models can therefore be in principle justified on the basis of the argument that a mass should be dynamically generated for the conformal mode. \n\nThese arguments therefore suggests that, in the IR\n\\be\na_2(\\Box)\\simeq \\frac{\\Lambda_{\\rm RR}^4}{\\Box^2}\\, ,\n\\ee\nwhere $\\Lambda_{\\rm RR}$ is a dynamically generated mass scale,  analogous of the QCD Lambda parameter $\\Lambda_{\\rm QCD}$.  The parameter $\\Lambda_{\\rm RR}$\nis related to the mass scale $m$ that appears in \\eq{RR} by \n\\be\\label{LRR}\n\\Lambda_{\\rm RR}^4=\\frac{1}{12}m^2m_{\\rm Pl}^2\\, .\n\\ee\nAs for the numerical value of $\\Lambda_{\\rm RR}$,\njust as in $\\Lambda$CDM the energy fraction $\\Omega_{\\Lambda}$ is a derived parameter, fixed by the flatness condition, in the non-local model the parameter $m$ is similarly fixed, again by the flatness condition. In $\\Lambda$CDM one finds in this way that, to reproduce the observations,  the cosmological constant  must be of order $H_0^2$; not surprisingly, in the non-local model one finds that  $m^2\\sim H^2_0$ \\cite{Maggiore:2013mea,Maggiore:2014sia,Foffa:2013vma}. Thus, from \\eq{LRR}, \n\\be\n\\Lambda^2_{\\rm RR}={\\cal O}(H_0m_{\\rm Pl})\\, . \n\\ee\nMore precisely, using the best-fit values $m\\simeq 0.67 H_0$ for model (\\ref{RT})~\\cite{Maggiore:2013mea}, or $m\\simeq 0.28 H_0$ for model (\\ref{RR})~\\cite{Maggiore:2014sia} we get\n$\\Lambda_{\\rm RR}\\simeq 0.4 (H_0m_{\\rm Pl})^{1\/2}$\nand \n$\\Lambda_{\\rm RR}\\simeq 0.3 (H_0m_{\\rm Pl})^{1\/2}$,\nrespectively.\nThus, numerically, \n\\be\n\\Lambda_{\\rm RR}\\simeq 0.8\\, {\\rm meV}\\hspace{15mm}  {\\rm [model\\,\\, (\\ref{RT})]}\\, ,\n\\ee\nand \n\\be\n\\Lambda_{\\rm RR}\\simeq 0.5\\, {\\rm meV}\\hspace{15mm}  {\\rm [model\\,\\, (\\ref{RR})]}\\, .\n\\ee\nFinally, we should also in general take into account the effect of the running of the Weyl-squared term in the IR. The argument on the dynamical mass generation for the conformal mode says nothing about the behavior of the form factor for the Weyl-squared term, since the Weyl tensor vanishes for a conformally flat metric. If the form factor $f^2(\\Box)$ associated to the \nWeyl-squared term   has no significant IR enhancement, and more generally if it grows slower than $1\/\\Box^2$ in the far IR,  its contribution to the cosmological dynamics will be negligible  compared to the $R\\Box^{-2} R$ term, and we end up exactly with the non-local models of the form (\\ref{RT}) or (\\ref{RR}) (depending on the precise form of the non-linear terms, since the two models are equivalent at linear order over flat space). In contrast, if also $f^2(\\Box)$ develops a singularity proportional to $\\Box^{-2}$, a corresponding term proportional to $C_{\\mu\\nu\\rho\\sigma}\\Box^{-2}C^{\\mu\\nu\\rho\\sigma}$ should be included in the cosmological model. This would not affect the background FRW evolution, since $C^{\\mu\\nu\\rho\\sigma}$ vanishes in FRW, but would  give a contribution at the level of cosmological perturbations.\n\n\\section{A solution to the naturalness problem}\\label{sect:sol}\n\nThe above mechanism for the emergence of a term $m^2 R\\Box^{-2}R$ in the long-distance effective action gives a new perspective on the naturalness problem of the cosmological constant. As is well known, there are two aspects in the cosmological constant problem: the ``old\" cosmological constant problem, namely why we do not observe a huge value for the vacuum energy; and the ``new\" cosmological constant problem; namely, assuming that some mechanism sets to zero the vacuum energy, what fixes in a technically natural way the observed energy scale associated to dark energy? Here we have nothing to add to the ``old\" cosmological constant problem, since we have simply fixed the renormalized value of the cosmological constant to zero.\\footnote{A mechanism that could naturally set to zero the value of the vacuum energy density is proposed in \\cite{Maggiore:2010wr}.} In contrast, concerning the naturalness problem of the scale associated to dark energy, the above scenario provides a new and interesting viewpoint. In the mechanism that we have proposed,  in $R^2$ gravity the scale $\\Lambda_{\\rm RR}$ emerges from dimensional transmutation, similarly to $\\Lambda_{\\rm QCD}$. There is no issue of naturalness for such a quantity, which is determined by the logarithmic running of a dimensionless coupling constant. Of course, the value of the scale generated dynamically in this way cannot be predicted, just as we cannot predict the value of $\\Lambda_{\\rm QCD}$. The fact that $\\Lambda_{\\rm RR}$ turns out to be of the order of a  milli-eV is simply a fact of nature, just as $\\Lambda_{\\rm QCD}$ turns out to be of order 200~MeV.\nHowever, quantities such as  $\\Lambda_{\\rm QCD}$ and $\\Lambda_{\\rm RR}$\ndo not have a power-like dependence on the masses of the particles in the theory, and are therefore technically natural. \n\nAnother attractive aspect of the above construction is that, to explain dark energy,  we do not need to introduce a fundamental particle with an astonishingly small mass $m\\sim H_0\\sim 10^{-33}$~eV, as is the case for instance in massive gravity. \nIn the above construction  the fundamental mass scale of the theory, emerging from dimensional transmutation, is $\\Lambda_{\\rm RR}$, and in terms of it the effective action  in the IR reads\n\\be\\label{Sg2RinIR}\nS_{\\rm Mink}\\simeq \\int d^4x \\sqrt{-g}\\, \\[ \\frac{m_{\\rm Pl}^2}{2}\\,R-\\Lambda_{\\rm RR}^4R\\frac{1}{\\Box^2}R\\]\\, .\n\\ee\nEven if, just as in QCD,  the  value of $\\Lambda_{\\rm RR}$ cannot be predicted by the model, still the  numerical value required by the comparison with the observation, which turns out to be of the order of the meV, does not look particularly surprising. An extremely small mass scale only emerges when, in \\eq{Sg2RinIR}, we factorize  $m_{\\rm Pl}^2\/2$, rewriting the action as\n\\be\\label{Sg2RinIRbis}\nS_{\\rm Mink}\\simeq  \\frac{m_{\\rm Pl}^2}{2} \\int d^4x \\sqrt{-g}\\, \\[\\,R-\n \\frac{2\\Lambda_{\\rm RR}^4}{m_{\\rm Pl}^2}R\\frac{1}{\\Box^2}R\\]\\, ,\n\\ee\nso the mass scale $m$ defined in \\eq{RR} is given by\n\\be\nm=\\sqrt{12}\\,\\, \\frac{\\Lambda_{\\rm RR}^2}{m_{\\rm Pl}}\\, ,\n\\ee\nwhich is suppressed with respect to $\\Lambda_{\\rm RR}$ by an extra factor of order $\\Lambda_{\\rm RR}\/m_{\\rm Pl}$, and is then\nof order $10^{-33}$~eV. However, the fundamental mass scale generated by the theory is $\\Lambda_{\\rm RR}$,  not $m$.\n\n\n\\section{Conformal mode dynamics}\\label{sect:conf}\n\nIt is interesting to understand in more detail the consequences on the conformal mode of a RG flow that runs from \n $a_1R^2$ at high energies, with $a_1=a_1(\\mu)$ a constant, to $\\Lambda_{\\rm RR}^4 R\\Box^{-2}R$ in the IR. \n We expands on the technical detail in app.~\\ref{app:Bard}, and here we summarize the main results. First of all, let us linearize for simplicity over flat space, and restrict to the scalar sector of perturbations. Then the metric can be written in terms of the gauge-invariant Bardeen potentials $\\Phi$ and $\\Psi$ as \n\\be\\label{PsiPhitext}\nds^2=-(1-2\\Psi) dt^2+(1+2\\Phi) d{\\bf x}^2\\, .\n\\ee\nEquivalently, we can introduce  the conformal factor $\\sigma$ and another independent perturbation $\\varphi$, defined writing the metric perturbation in the scalar sector in the form\n\\be\\label{sigmavarphi}\nds^2=e^{2\\sigma}\\[ -(1-2\\varphi) dt^2+(1+2\\varphi) d{\\bf x}^2\\]\\, .\n\\ee\nAt the level of linearized theory, $e^{2\\sigma}=1+2\\sigma +{\\cal O}(\\sigma^2)$, so $2\\sigma=\\Phi-\\Psi$ and\n$2\\varphi=\\Phi+\\Psi$. In GR, without $R^2$ terms, of course only the tensor modes are dynamical, and scalar perturbations are non-dynamical constrained fields that satisfy a Poisson equation, rather than a Klein-Gordon equation. Indeed, in the presence of matter with energy-momentum tensor $T_{\\mu\\nu}$, in GR the Bardeen potentials satisfy\n\\bees\n\\n^2\\Phi &=&-4\\pi G\\rho\\, ,\\label{n2Phirhotext}\\\\\n\\n^2\\Psi &=&-4\\pi G (\\rho-2\\n^2\\Sigma)\\, ,\\label{n2Psirhotext}\n\\ees\nwhere $T_{00}=\\rho$ is the energy density, and $\\Sigma$ the scalar part of the anisotropic stress tensor, see \\eqst{T00}{Tij}.\nTherefore the conformal mode satisfies\n\\be\\label{n2sigmaSigmatext}\n\\n^2(\\sigma+4\\pi G\\Sigma)=0\\,  .\n\\ee\nImposing the boundary condition that the $\\sigma$ and $\\Sigma$ vanish at infinity, the Laplacian is invertible, so we get\n$\\sigma=-4\\pi G\\Sigma$. The conformal mode is therefore a constrained field, determined by the matter content, and vanishes if $\\Sigma=0$.\n\nThe situation changes drastically in $R^2$ gravity. In this case, as we show in app.~\\ref{app:Bard}, linearizing over flat space the conformal mode satisfies the equation\n\\be\\label{Boxsigmatext}\n(\\Box-m_0^2)\\(\\sigma +4\\pi G\\Sigma\\)=\\frac{4\\pi G}{3}\\, (\\rho-3P)\\, .\n\\ee\nThus the conformal mode $\\sigma$, shifted by $4\\pi G\\Sigma$ as in \\eq{n2sigmaSigmatext}, now satisfies a massive KG equation, where  $m_0^2$ is just the same as \\eq{m02}. Thus, the conformal mode becomes dynamical, and describe the scalar particle that appears in Stelle theory.\nAs we see from \\eq{Boxsigmatext}, its  source is given by the trace of the energy-momentum tensor, $T=\\eta^{\\mu\\nu}T_{\\mu\\nu}=-(\\rho-3P)$, as we expect indeed for the conformal mode. \n\nConsider now the theory with action\n\\be\nS=\\int d^4x\\sqrt{-g}\\, \\[ \\frac{m_{\\rm Pl}^2}{2} R-Ra_1(\\Box)R\\]\\, , \n\\ee\nwhere, \nat some UV scale $\\mu$, $a_1(\\Box)= a_1$ is a constant, while in the IR \n$a_1(\\Box)= \\Lambda_{\\rm RR}^4 \\Box^{-2}$. In this case in the UV the conformal mode is described by \\eq{Boxsigmatext}. In the IR, in contrast, the model reduces to \\eq{RR}. The dynamical content of this theory, in the scalar sector, has been studied\nin Sect.~3 of ref.~\\cite{Maggiore:2014sia}, where is has been shown that $\\Phi$ and $\\Psi$ are non-radiative field that satisfy Poisson equations, as in GR. The same is therefore true for the conformal mode $\\sigma=(\\Phi-\\Psi)\/2$. We therefore conclude that a RG flows in which $a_1(\\Box)$ evolves from a constant in the UV toward $a_1(\\Box)= \\Lambda_{\\rm RR}^4 \\Box^{-2}$ in the IR corresponds to a flow in which the conformal mode, which in the UV is dynamical, and is just the spin-0 mode  of $R^2$ theory with mass given by  \\eq{m02}, becomes a constrained field as we flow toward the IR.\n\n\n\n\n\n\\section{Dynamical mass generation for the conformal mode in Einstein gravity}\\label{sect:anomaly}\n\nThe discussion   of the previous sections suggests that, because of IR effects, a mass term could be dynamically generated for the conformal mode.\nThe introduction of non-local operators of the type $R\\Box^{-2}R$ \nallows us just to describe such a  mass term, in a diffeomorphism-invariant manner.\n\nThis leads us to investigate more closely the dynamics of the conformal mode in Einstein gravity, without $R^2$ corrections, to see if a similar ``condensation\" of the conformal mode might take place. Indeed, there are several indications for the relevance of the conformal mode in GR in the \nIR~\\cite{Antoniadis:1991fa,Antoniadis:2006wq}. In classical general relativity the conformal mode is a non-propagating degree of freedom. At the quantum level, when gravity is coupled to massless conformally-invariant fields, it however acquires  a dynamics through the conformal anomaly. As stressed in \\cite{Antoniadis:2006wq}, an anomaly implies that the effect of quantum fluctuations can remain large at the longest length and time scales, so quantum fluctuations can affect the behavior of the classical theory even in the far IR.\nThis is most evident in $D=2$ space-time dimension. In this case, \nthe trace anomaly takes the form (see e.g. \\cite{Birrell:1982ix}) \n\\be\\label{traceT}\n\\langle T^{\\mu}_{\\mu}\\rangle=\\frac{N}{24\\pi} R\\,,\n\\ee\nwhere $N=N_S+N_F$ is the total number of massless scalar and Dirac fermion fields. The trace anomaly can be derived from the non-local Polyakov action\\footnote{The coefficient $N$ in front of  $S_{\\rm anom}$  becomes $N-25$ if one also takes into account the metric fluctuations themselves, beside the fluctuations due to matter fields.}\n\\be\nS_{\\rm anom}=-\\frac{N}{96\\pi}\\int d^2x\\sqrt{-g}\\, R\\frac{1}{\\Box}R\\, ,\n\\ee\nwhose variation gives the right-hand side of \\eq{traceT}, and which can therefore be added to the classical Einstein-Hilbert action, to provide an effective action which takes into account  the quantum effect of the trace anomaly. In $D=2$ one can always write the metric in the form (\\ref{gmnsigma}), without truncating the theory, and in terms of the conformal mode the Polyakov action becomes local, \n\\be\nS_{\\rm anom}=\\frac{N}{24\\pi}\\int d^2x\\sqrt{-\\bar{g}}\\, (-\\sigma\\overline{\\square}\\sigma+\\overline{R}\\sigma)\\, ,\n\\ee\nwhere the bars denote quantities computed with the reference metric $\\bar{g}_{\\mu\\nu}$. In $D=2$ the effect of the anomaly is especially important, since the Einstein-Hilbert term is a topological invariant and the dynamics is entirely contained in $S_{\\rm anom}$. Observe that, in $D=2$,  the propagator of the $\\Box^{-1}$ operator is logarithmic, $G(x,x')\\propto\\log (x-x')^2$, and therefore grows with distance.\n\nThe situation is conceptually similar in four dimensions.  In this case the trace anomaly is given by (see e.g. \\cite{Birrell:1982ix,Buchbinder:1992rb,Shapiro:2008sf} for pedagogical introductions)\n\\be\\label{traceT4d}\n\\langle T^{\\mu}_{\\mu}\\rangle=b C^2+ b'\\(E-\\frac{2}{3}\\Box R\\)+b''\\Box R\\, ,\n\\ee\nwhere, as in \\eq{SHD}, $C^2$ is the square of the Weyl tensor, $E$ the Gauss-Bonnet term, and the coefficients $b,b',b''$ depends on the number of massless conformally-coupled scalars, massless fermions and massless vector fields. The $\\Box R$ term  on the right-hand side of\n\\eq{traceT4d}\ncan be derived from the variation of a local action proportional to $\\int d^4x\\sqrt{-g}\\,R^2$. The remaining combination is obtained from the variation of a non-local action\n\\be\nS_{\\rm anom}=-\\frac{1}{8}\\int d^4x\\sqrt{-g}\\(E-\\frac{2}{3}\\Box R\\) \\Delta_4^{-1}\n\\[ b' \\(E-\\frac{2}{3}\\Box R\\)-2b C^2\\]\\, ,\n\\ee\nwhere\n\\be\n\\Delta_4=\\Box^2+2R^{\\mu\\nu}\\n_{\\mu}n_{\\nu}-\\frac{2}{3} R\\Box +\\frac{1}{3}(\\n^{\\mu}R)\\n_{\\mu}\n\\, .\n\\ee\nOnce again, this non-local action becomes local in terms of the conformal mode. Writing the metric as in \\eq{gmnsigma},  using as a fiducial metric $\\bar{g}_{\\mu\\nu}=\\eta_{\\mu\\nu}$ (and choosing appropriately the local $R^2$ counterterm) one finds \n(see e.g. \\cite{Antoniadis:1991fa,Antoniadis:2006wq})\n\\be\\label{Sanom4D}\nS_{\\rm anom}=-\\frac{Q^2}{16\\pi^2}\\int d^4x\\, (\\Box\\sigma)^2\\, ,\n\\ee\nwhere \n\\be\\label{Q2} \nQ^2=\\frac{1}{180}(N_S+\\frac{11}{2}N_F+62 N_V-28)+Q^2_{\\rm grav}\\, .\n\\ee\nHere $N_S, N_f$ and $N_V$ are the number of massless conformally coupled scalars, massless Weyl fermions and massless vectors, respectively, while $Q^2_{\\rm grav}$ is the contribution from the gravitons, which is not unambiguously known, and $-28$ is the contribution from the $\\sigma$ field itself~\\cite{Antoniadis:1991fa,Antoniadis:1996pb}.\n\n\n Of course, a difference from the two-dimensional case is that in $D=4$ writing the metric in the form (\\ref{gmnsigma}) is a truncation of the theory. However, we see from \\eq{Sanom4D} that the propagator of the $\\sigma$ field in momentum space is $\\propto 1\/k^4$, which in coordinate space, in $D=4$, gives  again a propagator growing logarithmically, as in $D=2$,\n\\be\\label{Gx2m0}\nG(x,x')=-\\frac{1}{2Q^2}\\log\\[\\mu^2 (x-x')^2\\]\\, ,\n\\ee\nwhere $\\mu$ is an IR regulator.\nThis means again that the fluctuations of the $\\sigma$ field are large at long distances, and justifies a posteriori the approximation of keeping only the conformal mode when studying the quantum effects of gravity in the IR.\nThe fact that the conformal mode dominates the quantum theory in the IR is also confirmed by the fact that it gives the most infrared divergent contribution to the propagator in de~Sitter space~\\cite{Antoniadis:1986sb}. \nObserve that the fourth-order operator $\\Delta_4$  in $D=4$ is the analogue of the second-order operator $\\Box$  in $D=2$, since it is conformally invariant, $\\sqrt{-g}\\,\\Delta_4=\\sqrt{-\\bar{g}}\\,\\bar{\\Delta}_4$.\n\nAs discussed in ref.~\\cite{Antoniadis:1996pb}, the large fluctuations in the two-point function of the $\\sigma$ field due to its logarithmic growth create a situation that looks very similar to what happens in the two-dimensional XY model, with the Berezinsky-Kosterlitz-Thouless (BKT) transition. Let us recall the argument of ref.~\\cite{Antoniadis:1996pb}. The four-dimensional theory  for the conformal factor has solutions of the form\n\\be\n\\sigma(x)=q\\ln |x-x_0|\\, ,\n\\ee\nwhich are analogous to the vortices of the 2-dimensional theory. The euclidean action for $\\sigma$, evaluated on this configuration, takes the value~\\cite{Antoniadis:1996pb}\n\\be \nS_E=\\frac{1}{2}Q^2q^2\\ln\\(\\frac{L}{a}\\)\\, ,\n\\ee\nwhere $L$ is the IR cutoff and the ``lattice spacing\" $a$ is the UV cutoff. The parameter $x_0$ is a collective coordinate corresponding to the center of the solution. The number of ways of placing this solution in a volume $V$ is proportional to $V$, so the  corresponding entropy is $4\\log L\/a$, and the free energy is \n\\be\nF=\\(\\frac{1}{2}Q^2q^2-4\\)\\ln\\(\\frac{L}{a}\\)\\, .\n\\ee\nIf $Q^2>8\/q$ the free energy diverges as we remove the IR cutoff ($L\\rightarrow\\infty$) or the UV cutoff ($a\\rightarrow 0$), so these configurations are irrelevant. In contrast, if $Q^2<8\/q$ they dominate the partition function, so at the critical value of $Q$ we expect a phase transition. For the solution with $q=1$,  $Q_c^2=8$. The situation is indeed completely analogous to the BKT phase transition in two dimensions, which is also triggered by the logarithmic growth of the vortex-vortex interaction. The role of the coupling constant to be tuned to get the phase transition is here played by $Q$, which depends on the number of massless conformally-coupled particles. If the unknown graviton contribution in \\eq{Q2} is not too large, the theory with just a single massless photon is in the region where $Q^2<Q_c^2$, and the non-trivial configurations dominate.\n\nIn ref.~\\cite{Antoniadis:1996pb} it has been suggested that  $Q^2<Q_c^2$ corresponds to a phase where the space-time geometry is no longer smooth. Actually, pushing further the analogy with the BKT transition can suggest a different scenario. Indeed, in the two-dimensional case, in the regime where the vortices dominate, the correlation length becomes finite and a mass gap is generated. \nIt is plausible to expect that the same happens in the four-dimensional case. \nWe therefore expect the generation of a term $\\propto \\sigma^2$ in the effective action for $\\sigma$, whose coefficient is given by a dynamically-generated mass scale. Of course, this mass term must be constructed in such a way that it does not spoil the underlying diffeomorphism invariance, which leads us again to a  term $\\Lambda_{\\rm RR}^4R\\Box^{-2}R$, see \\eq{m2s2}.\n\n\n\nAdding this term to the  anomaly-induced effective action  (\\ref{Sanom4D}), to quadratic order  we get an effective action for $\\sigma$ given by\n\\be\\label{Sanom4Dmass}\nS^{(2)}_{\\rm eff}=-\\int d^4x\\,\\[ \\frac{Q^2}{16\\pi^2} (\\Box\\sigma)^2+\\Lambda_{\\rm RR}^4\\sigma^2\\]\n\\, .\n\\ee\nThe corresponding euclidean propagator is \n\\be\\label{Gxmu4}\nG(x)=\\frac{8\\pi^2}{Q^2}\\int \\frac{d^4k}{(2\\pi)^4}\\, \\frac{1}{k^4+\\mu^4} e^{-ikx}\\, ,\n\\ee\nwhere $\\mu^4=16\\pi^2\\Lambda_{\\rm RR}^4\/Q^2$. Carrying out the angular integrals, we get\n\\be\nG(x)=\\frac{2}{Q^2}\\int_0^{\\infty} dy \\frac{y^2 J_1(y)}{y^4+ (\\mu^2 x^2)^2}\\, ,\n\\ee\nwhere $J_1(y)$ is a Bessel function.\nThe final integral can in principle be carried out in term of a (not very illuminating) Meijer's G-function. However we already see from this expression, together with the expansion $J_1(y)\\simeq y\/2$ for small $y$, that the presence of the term $\\mu^2 x^2$ regularizes  the integral near the integration limit $y=0$. In the limit $\\mu^2 x^2\\rightarrow 0$ we recover \\eq{Gx2m0}, except that the artificial IR regulator $\\mu$ of \\eq{Gx2m0} is  replaced by the quantity $\\mu$ that appears in\n\\eq{Gxmu4}, which now is\na physical quantity related to $\\Lambda_{\\rm RR}$. In the opposite limit \n$\\mu^2 x^2\\gg 1$, introducing the notation $u=(\\mu^2x^2)^{1\/2}$ and using as integration variable $z=y\/u$, we have\n\\bees\nG(x)&=&\\frac{2}{Q^2}\\, \\frac{1}{u}\\int_0^{\\infty}dz\\, \\frac{z^2}{1+z^4} J_1(uz)\\nonumber\\\\\n&\\simeq &\\frac{2}{Q^2}\\, \\sqrt{\\frac{2}{\\pi}} \\, \\frac{1}{u^{3\/2}}\n\\int_0^{\\infty}dz\\, \\frac{z^{3\/2}}{1+z^4}\\, \\cos\\(uz-\\frac{3\\pi}{4}\\)\\, ,\n\\ees\nwhere in the second line we have used the asymptotic expansion of the Bessel function.\\footnote{Observe that, as $u\\rightarrow\\infty$, the regime $uz$ fixed corresponds to $z\\rightarrow 0$, whose contribution to the integral is suppressed by the factor $z^2$ in the integrand. The dominant contribution therefore comes from the region $uz\\rightarrow\\infty$, where we can use the asymptotic expansion\n$J_1(x)\\simeq (2\/\\pi x)^{1\/2}\\cos (x-3\\pi\/4)$ valid for large $x$.} The integrand is an oscillating function, which oscillates faster and faster as $u$ increases, so the Green's function vanishes faster than $1\/u^{3\/2}$, i.e faster than $1\/(\\mu^2 x^2)^{3\/2}$.\n\n\nThe physical picture that emerges from these considerations is that, when the dynamics of the conformal mode is described by the action (\\ref{Sanom4D}),  fluctuations grow larger and larger at long distances, as shown by the logarithmic growth of the propagator in \\eq{Gx2m0}. This is a sign of an IR instability of the vacuum state. When a mass for the conformal mode is dynamically generated, the long-distance correlations  grow only until the separation $|x-x'|$ reaches a  length-scale  of order $1\/\\mu$, and after that they decay faster than $1\/|x-x'|^3$. Therefore, there is no IR instability. The dynamical generation of a mass for the conformal mode is the mechanism by which the vacuum restructures itself, as a response to the IR instability triggered by the anomaly-induced effective action. \n\n\\section{Conclusions}\\label{sect:concl}\n\nIn this paper we have explored the possibility of generating non-local terms, such as those given in \\eqs{RT}{RR}, through strong IR effects in gravity. We have first of all considered the running of the coupling associated to the $R^2$ term in higher-derivative gravity. We have stressed that a peculiar aspect of the running of this coupling is that, even in the deep IR, its calculation requires a knowledge of the UV completion of GR. Indeed, in GR the corresponding beta function is not even gauge-independent, and has no physical meaning. Stelle theory provides an example of  an extension of GR where, thanks to the improved UV behavior, the beta functions of the ${\\cal O}(R^2)$ couplings are physical and gauge-independent. However, the result from Stelle theory cannot be taken literally either, because the theory is plagued by a ghost (and, furthermore, existing computations of the RG flow have been performed only in the regime $E\\ggm_{\\rm Pl}$). In the absence of a concrete reliable result for the RG flow, we have simply explored the consequences of a scenario in which the coupling $a_2$ associated to the $R^2$ term grows in the IR. In this case, similarly to QCD, we expect the dynamical generation of a mass scale, $\\Lambda_{\\rm RR}$, and of a non-trivial form factor $a_2(\\Box)$. We have provided arguments, based on the analogy with the known non-perturbative behavior  of the gluon propagator in QCD, that suggest that in the strong coupling regime the emergence of a pole-like behavior\n$a_2(\\Box)\\propto\\Box^{-2}$ is quite possible. \n\nThe emergence of a dynamical scale $\\Lambda_{\\rm RR}$ would have far reaching consequences for solving long-standing puzzles concerning dark energy. In particular, this scale emerges from the logarithmic running of a dimensionless coupling constant, similarly to $\\Lambda_{\\rm QCD}$, and does not receive large loop corrections, contrary to what happens to the vacuum energy. It is therefore technically natural. We found that, to explain dark energy today, \nthe dynamically-generated scale $\\Lambda_{\\rm RR}$ should be of the order of a  milli-eV. Of course we cannot predict its numerical value, just as we \ncannot  predict the value of $\\Lambda_{\\rm QCD}$. However,  it is interesting that, to obtain a dynamical explanation of dark energy,  we do not need to introduce a fundamental particle with an astonishingly small mass $m\\sim H_0\\sim 10^{-33}$~eV, nor a cosmological constant $\\Lambda$ with the amazingly small value $\\Lambda\\sim H_0^2$. In our scenario, \nin the IR the action  is given by \\eq{Sg2RinIR}, with a fundamental mass scale $\\Lambda_{\\rm RR}\\sim {\\rm meV}$, which is by no means a particularly surprising value.\nThe connection with the present value of the Hubble parameters comes out because, in a cosmological context, derivatives of the metric are of order $H$, so $\\Box\\sim H^2$ and $\\Box^{-1}\\sim H^{-2}$, while\n$R\\sim H^2$. Thus \nthe term \n$\\Lambda^4_{\\rm RR}R\\Box^{-2}R$ becomes competitive with $m_{\\rm Pl}^2R$ when $H\\sim \\Lambda^2_{\\rm RR}\/m_{\\rm Pl}$ which, for $\\Lambda_{\\rm RR}\\sim\\, {\\rm meV}$, gives $H\\sim H_0\\sim 10^{-33}$~eV. However, in this scenario the fundamental scale emerging dynamically in the theory is $\\Lambda_{\\rm RR}$, and there is no fundamental particle with the astonishingly small mass $10^{-33}$~eV.\n\nAnother interesting aspect of the above analysis is that it draws attention to the dynamics of the conformal mode of the metric, since the running of the coupling $a_2$ associated to $R^2$, from a constant value in the UV toward   $a_2(\\Box)\\propto\\Box^{-2}$ in the IR, has an interesting interpretation in terms of the conformal mode, as we discussed in sect.~\\ref{sect:conf}. \nThis stimulates to look in more detail into the dynamics of the conformal mode in Einstein gravity, in which the higher-derivative term is provided by the anomaly-induced effective action, i.e. by the loops of the massless matter fields in the theory. Once again, the conformal mode seem to play an interesting role. In Einstein gravity supplemented by the anomaly-induced effective action the emergence of a term $R\\Box^{-2}R$ can be interpreted as  the dynamical generation a mass term for the conformal mode of the metric (an interpretation that becomes blurred in Stelle theory, where a mass term for the conformal mode is already present at tree level because of the $R^2$ term). We\nhave argued that such  a dynamical mass generation can take place as a consequence of the strong IR fluctuations generated by the two-point function of the conformal mode, which grows logarithmically at large distances and, as already suggested in \nin ref.~\\cite{Antoniadis:1996pb}, can trigger a phase transition similar to  the Berezinsky-Kosterlitz-Thouless transition in two dimensions. If this results in a dynamical mass generation for the conformal mode, one  again obtains in the IR a non-local model of the type (\\ref{RT}) or (\\ref{RR}).\n\n\\vspace{5mm}\n\\noindent\n{\\bf Acknowledgments.}\nI am grateful to  Eugenio Bianchi, Giulia Cusin,\nEduard Gorbar, Alex Kehagias,  Ilya Shapiro, Sergey Sibiryakov, Julian Sonner and Arkady Tseytlin  for very useful discussions.\nThis work is supported by the Fonds National Suisse and by the SwissMAP National Center for Competence in Research.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzogub b/data_all_eng_slimpj/shuffled/split2/finalzzogub
new file mode 100644
index 0000000000000000000000000000000000000000..5796a12f53dc3f8bf0811dad0c8a4c14205ca719
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzogub
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\n\n\\subsection{}\n       By a \\emph{Fourier quasicrystal} one often means a \ndiscrete measure,  whose Fourier transform  is also a \ndiscrete measure.\n       This concept was inspired by the experimental\n       discovery in the middle of 80's of non-periodic \n       atomic structures with diffraction patterns consisting\n       of spots. In this context, different versions of ``discreteness'' were discussed,\n              see in particular \\cite{bom}, \\cite{cah},   \\cite{mey2}, \\cite{dys}.\n\\par\n Let $\\mu$ be a (complex) measure on $\\R^n$, supported on a discrete set $\\Lambda$: \n \\begin{equation}\n\t \\mu =\\sum_{\\lambda\\in \\Lambda} \\mu(\\lambda)\\delta_{\\lambda}, \\quad\n\t \\mu(\\lambda)\\ne 0.\n\t \\label{RM.1}\n \\end{equation}\nWe shall suppose that $\\mu$ is a slowly increasing measure, which means that\n$|\\mu|\\{x:|x|<r\\}$ grows at most polynomially as $r\\to\\infty$. \nHence the Fourier transform \n \\[ \\hat{\\mu}(t):= \\sum_{\\lambda\\in\\Lambda} \\mu(\\lambda)e^{-2\\pi i\\dotprod{\\lambda}{t}} \\]\nis  well-defined as a temperate distribution on $\\R^n$. Assume that also $\\ft\\mu$ is a \nslowly increasing measure, which is purely atomic and  supported on a countable  set $S$: \n \\begin{equation}\n\t \\hat{\\mu}=\\sum_{s\\in S} \\hat{\\mu}(s)\\delta_{s}, \\quad \\hat{\\mu}(s)\\ne 0.\n\t \\label{RM.2}\n \\end{equation}\n The set $\\Lambda$ is then called the \\emph{support} of the measure $\\mu$, while $S$ is called the \\emph{spectrum}.\n\n\n\\subsection{}\nIn \\cite[Problem 4.1(a)]{lag2} the following question was posed:\n\\par\n\\emph{Suppose that $\\mu$ is a positive measure, whose support $\\Lam$ and spectrum $S$ are both   uniformly discrete sets.\n           Is it true that $\\Lam$ can be covered by a finite union of translates\n          of a certain lattice?}\n\\par\nRecall that a set is  uniformly discrete if the distance between \nany two of its points is bounded  below by some positive constant.\n\\par\nIn \\cite{lo1, lo2} we proved the following result, which answers the above question affirmatively,\nand moreover shows that the measure $\\mu$ must have a periodic structure:\n \\begin{thm}\n\t \\label{RT.0}\n\t Let $\\mu$ be a positive (or positive-definite) measure on $\\R^n$ satisfying  \\eqref{RM.1} and\n\t \\eqref{RM.2}, and assume that $\\Lambda$ and $S$ are both uniformly discrete sets.\n\t Then there is a lattice $L$, such that $\\mu$ is representable  in the form\n\t \\begin{equation}\n\t\t \\mu=\\sum_{j=1}^{N}\\sum_{\\lambda\\in L+\\tau_j}P_j(\\lambda) \\, \\delta_\\lambda\n\t\t \\label{RM.3}\n\t \\end{equation}\n\t for certain vectors $\\tau_j$ in $\\R^n$ and  trigonometric polynomials $P_j$ $(1 \\leq j \\leq N)$. \nMoreover, for $n=1$ the result holds even without the\n positivity assumption on the measure $\\mu$. \n \\end{thm}\n           So the theorem says that under the conditions above,\n          the measure $\\mu$ is a finite linear combination of Dirac combs,  translated and\n          modulated. Conversely, for any measure $\\mu$ of the form \\eqref{RM.3}, the\n\tsupport $\\Lam$ and spectrum $S$ are both  uniformly discrete  sets.\n\\par\n           It was also an open problem (see \\cite[Problem 4.1(b)]{lag2})\n           whether such a result can be proved in the more general\n           situation when $\\Lambda$, $S$ are assumed to be just discrete \n           closed sets. This problem was addressed in \\cite{lo3} where  we proved the following:\n \\begin{thm}\n\t \\label{RT.4}\n\t There is a (non-zero)  measure $\\mu$ on $\\R$ satisfying\n\t \\eqref{RM.1} and \\eqref{RM.2}, such that $\\Lambda$ and $S$ are both discrete closed\n\t sets, but $\\Lambda$ contains only finitely many elements of any arithmetic\n\t progression.\n \\end{thm}\n\\par\n The latter condition indicates that the measure $\\mu$ is ``non-periodic'' in a strong\n sense. In particular, it cannot be obtained as  a finite linear combination of Dirac combs.\n A result of similar type is true also in $\\R^n,$ $n>1$.\n\\par\nMoreover, in our construction both  $\\mu$ and $\\ft\\mu$ \nare translation-bounded measures (see Section \\ref{sec:prelim} for the\n definition). This implies that \n$\\mu$ is an almost periodic measure, whose Fourier\ntransform $\\ft\\mu$ is also an almost periodic measure.\nBy definition, a measure $\\mu$ on $\\R^n$ is an \\define{almost periodic\nmeasure} if for every continuous, compactly supported function\n$\\varphi$ on $\\R^n$, the convolution $\\mu \\ast \\varphi$ is an\nalmost periodic function in the sense of Bohr.\n\n\n\n\\subsection{}\nIn the crystallography community, it seems to be commonly agreed\n that the support  $\\Lambda$ should be a uniformly discrete set.\n    We remind that Meyer's quasicrystals \\cite{mey2},\n               which appeared first  under the name ``model sets'' \\cite{mey0},\n               are uniformly discrete sets, and they support measures\n               whose spectra are dense countable sets.\n\\par\n               So it is a natural problem, to  what extent can  the spectrum $S$ of a non-periodic \nquasi\\-crystal be discrete, assuming that the support  $\\Lambda$ is uniformly discrete?\n In the present paper we address this problem, and  consider quasicrystals with non-symmetric\ndiscreteness assumptions on the support  and  the spectrum.\n\\par\nFirst we obtain several results which show that, under certain conditions,\nif  the spectrum $S$ is a discrete closed set, then in fact $S$  must    be uniformly discrete.\nThese results thus reduce the situation to the setting in \\thmref{RT.0} above, which in turn\nallows us to conclude that the measure $\\mu$  is representable  in the form\n\\eqref{RM.3}.\n\\par\nOn the other hand, we present an example of a  non-periodic quasicrystal such\nthat the spectrum $S$ is a nowhere dense countable set.\n\\par\nWe also apply our results to Hof's quasicrystals. In this context, we prove that\nif a Delone set $\\Lam$ of finite local complexity has a uniformly discrete diffraction spectrum,\nthen the diffraction measure of $\\Lam$ has a periodic structure.\n\\par\nFinally, we extend our results to the more general situation, where $\\hat{\\mu}$\n is a measure which has both a pure point component and a continuous one.\n\n\n\n\n\n\\section{Results}\n\n\\subsection{}\n Our first result deals with  the case when $\\mu$ is a  positive-definite measure: \n \\begin{thm}\n\t \\label{RT.5}\n\t Let $\\mu$ be a positive-definite measure on $\\R^n$\n\t satisfying \\eqref{RM.1} and \\eqref{RM.2}. Assume that the support $\\Lambda$ is a\n\t uniformly discrete set, while the spectrum $S$ is a discrete closed set. Then\n\t also $S$ is uniformly discrete, and the measure $\\mu$ has the form \\eqref{RM.3}. \n \\end{thm}\n\t Actually, we will show that if a positive-definite\n\t measure $\\mu$ with uniformly discrete support $\\Lambda$ is not of the form\n\t \\eqref{RM.3}, then its spectrum $S$ must have a relatively dense set of\n\t accumulation points   (see \\thmref{AT.5}). \n\n\\subsection{}\nIn the next result, which holds for dimension $n=1$, the\n\t positive-definiteness assumption is replaced by a stronger discreteness condition\n\t on the spectrum:\n\\begin{thm}\n\t\\label{RT.6}\n\tLet $\\mu$ be a measure on $\\R$ satisfying \\eqref{RM.1} and \\eqref{RM.2}. Assume\n\tthat  the support $\\Lambda$ is a uniformly discrete set, while\n the spectrum $S$ satisfies the condition\n\t \\begin{equation}\n \\sup_{x\\in\\R}\\# (S \\cap [x,x+1]) < \\infty. \n\t\t \\label{RM.4}\n\t \\end{equation}\nThen\t$S$ is a uniformly discrete set, and $\\mu$ is of the form \\eqref{RM.3}. \n\\end{thm}\nNotice that  condition \\eqref{RM.4} means that   $S$ is the union of a finite number of uniformly discrete sets. \n\n\\subsection{}\nIn the following result, a stronger discreteness condition is imposed on the support:\n\\begin{thm}\n\t\\label{RT.7}\n\tLet $\\mu$ be a measure on $\\R^n$ satisfying \\eqref{RM.1} and\n\t\\eqref{RM.2}.\n\tAssume that the set $\\Lambda-\\Lambda$ is  uniformly discrete, and that\n\t$S$ is a discrete closed set. Then the same conclusion as in the previous two theorems holds. \n\\end{thm}\n\t Moreover, we will prove that if $\\Lambda-\\Lambda$ is  uniformly discrete, but the\n\t measure $\\mu$ is not of the form\n\t \\eqref{RM.3}, then again the spectrum $S$ must have a relatively dense set of\n\t accumulation points (see \\thmref{CT.1}).\n\\par\n\\thmref{RT.7} implies the following result communicated to us by Y.\\ Meyer:\na ``simple quasicrystal'' cannot support a measure whose spectrum\nis a discrete closed set (for the definition of a simple quasicrystal, see \\cite{matei-meyer-simple}).\n\n \n\\subsection{}\n The previous results show that if the measure $\\mu$\ndoes not have the periodic structure \\eqref{RM.3},\n  then  the spectrum $S$ must have finite accumulation points.\n   However, $S$ need not be dense in any ball, as the following result shows:\n\\begin{thm}\n\t\\label{RT.15}\n\tThere is a  positive-definite measure $\\mu$ on $\\R^n$\nsatisfying \\eqref{RM.1} and \\eqref{RM.2}, such that:\n\\begin{enumerate-math}\n\\item\nThe  support $\\Lam$ is not contained in a finite union of translates of any lattice;\n\\item\nThe set $\\Lam-\\Lam$ is uniformly discrete; \n\\item\nThe spectrum $S$ is a  nowhere dense (countable) set.\n\\end{enumerate-math}\n\\end{thm}\nTo prove this we base on Meyer's construction,\nbut with an additional modification.\n\n\n\n\\subsection{}\nOur results can be applied to quasicrystals in Hof's sense \\cite{hof}.\n\\par\nRecall that a set $\\Lambda\\subset\\R^n$ is called a \\emph{Delone set} if it is  uniformly discrete and also\nrelatively dense. A Delone set $\\Lambda$ is said to be of \\emph{finite local complexity} if the difference\nset $\\Lambda-\\Lambda$ is a discrete closed set. This means that  $\\Lambda$ has, up to translations,\nonly a finite number of local patterns of any given size. \n\\par\nAn \\emph{autocorrelation measure} $\\gamma_\\Lam$ of the set $\\Lambda$ is any weak limit point of the\nmeasures \n\t \\begin{equation}\n (2R)^{-n}\\sum_{\\lambda,\\lambda'\\in \\Lam \\cap [-R,R]^{n}}\\delta_{\\lambda'-\\lambda}\n\t\t \\label{RT.8.1}\n\t \\end{equation}\nas $R\\to\\infty$.  An autocorrelation measure $\\gamma_\\Lam$ is always\npositive-definite, and so its Fourier transform $\\hat{\\gamma}_\\Lam$ is a positive\nmeasure, called a \\emph{diffraction measure} of $\\Lambda$.   If the measure\n$\\hat{\\gamma}_\\Lambda$ is purely atomic, then \nits support $S$ is called a\n\\define{diffraction spectrum} of $\\Lambda$.\n\\par\nThe following result answers a question posed in \\cite[Problem 4.2(a)]{lag2}:\n\\begin{thm}\n\t\\label{RT.8}\nSuppose that\n\\begin{enumerate-math}\n\\item\n $\\Lambda\\subset\\R^n$ is a Delone set of finite local complexity;\n\\item\n\t$\\gamma_\\Lam$ is an autocorrelation measure of $\\Lambda$; and\n\\item\n\tThe diffraction spectrum $S$ (the support of $\\hat{\\gamma}_\\Lam$) is a uniformly discrete set. \n\\end{enumerate-math}\nThen  $S$ is contained in a finite union of translates of some lattice,\nand the diffraction measure $\\hat{\\gamma}_\\Lambda$ has the form \\eqref{RM.3}.\n\\end{thm}\nA slightly more general version of this result will be given in \\thmref{DT.1}.\nThe same conclusion is true if the set $\\Lambda-\\Lambda$ is uniformly discrete, \nand the diffraction spectrum $S$ is a discrete closed set (see \\thmref{DT.2}).\n\n\n\\subsection{}\nWe also consider discrete measures $\\mu$, whose\nFourier transform $\\hat{\\mu}$ is a measure which has both a pure point\ncomponent and a continuous one. We can extend our previous results\nto this more general situation using the following:\n\\begin{thm}\n\t\\label{RT.10} \n\tLet $\\mu$ be a measure on $\\R^n$ with uniformly discrete\nsupport $\\Lambda$, and assume that  $\\hat{\\mu}$ is a (slowly increasing)\nmeasure. Then the discrete part of $\\hat{\\mu}$ is the Fourier transform\nof another measure $\\mu'$, whose support $\\Lambda'$ is also a uniformly discrete set.\n\\end{thm}\nMoreover, if $\\Lambda-\\Lambda$ is a uniformly discrete set,\nthen also $\\Lambda'-\\Lambda'$ is uniformly discrete.\n\\par\nBy applying the previous results to this new measure $\\mu'$, one can obtain\nversions of the results for measures $\\mu$ with non pure point Fourier transform\n(see \\thmref{FT.2}).\n\n                    \n\n\n\\section{Preliminaries}\n\\label{sec:prelim}\n\n\\subsection{Notation}\nBy $\\dotprod{\\cdot}{\\cdot}$ and $|\\cdot|$ we  denote the Euclidean \nscalar product and norm in $\\R^n$. \nThe open ball of radius $r$ centered at the origin is denoted\n $B_r := \\{x \\in \\R^n: |x|<r \\}$.\n\\par\nA set $\\Lambda \\subset \\R^n$ is said to be a \\emph{discrete closed set} if $\\Lambda$ has only finitely many \npoints in any bounded set. The set $\\Lambda$ is called \\emph{uniformly discrete} if\n\\begin{equation}\n\\label{uddef}\n                d(\\Lam):=\\inf_{\\lam,\\lam'\\in\\Lam, \\lam\\neq\\lam'} |\\lam-\\lam'| > 0.\n\\end{equation}\nThe set $\\Lambda$ is said to be  \\emph{relatively dense}  if there is $R > 0$ such that every ball of\nradius $R$ intersects $\\Lam$. \n\\par\nBy  a (full-rank) lattice $L \\subset \\R^n$ we mean the image of $\\Z^n$ under some\ninvertible linear transformation $T$. The determinant $\\det(L)$ is equal to $|\\det (T)|$.\nThe dual lattice $L^*$ is the set of all vectors $\\lambda^*$ such that $\\dotprod{\\lambda}{\\lambda^*}\n \\in \\Z$, $\\lambda \\in L$.\n\\par\nBy a ``distribution'' we  will mean a temperate distribution on $\\R^n$.\nBy a  ``measure'' we mean a complex, locally finite measure (usually infinite) which is\nassumed to be \\emph{slowly increasing}. By definition, a measure $\\mu$ is slowly\nincreasing if  there is a constant $N$ such that \n$|\\mu|(B_R)=O(R^N)$ as $R\\to \\infty.$\nThe measure $\\mu$  is called \\emph{translation-bounded} if\n\\begin{equation}\\label{tr-bdd-def}\n\\sup_{x \\in \\R^n} |\\mu|(x+B_1) < \\infty.\n\\end{equation}\n\\par\nAny translation-bounded measure  is slowly increasing, and any\nslowly increasing measure  is a temperate distribution. Remark\nthat for a \\emph{positive} measure to be a  temperate distribution, it is \nalso necessary to be slowly increasing, but this is not true\nfor complex (or real, signed) measures.\n\\par\nBy  the ``support'' of a pure point measure $\\mu$ we mean the  \ncountable set of the non-zero atoms of $\\mu$. This should not be confused with the\nnotion of support in the sense of distributions, which is always a closed set.\n\\par\nThe Fourier transform on $\\R^n$ will be normalized as follows:\n$$\\ft \\varphi (t)=\\int_{\\R^n} \\varphi (x) \\, e^{-2\\pi i\\langle t,x\\rangle} dx.$$\n\\par\nWe denote by $\\supp(\\varphi)$ the closed support of a Schwartz function $\\varphi$,\nand by $\\spec (\\varphi)$ the closed support of its Fourier transform $\\ft\\varphi$.\n\\par\nIf $\\alpha$ is a temperate distribution, and $\\varphi$ is a Schwartz function on $\\R^n$,\nthen $\\dotprod{\\alpha}{\\varphi}$ will denote the action of $\\alpha$ on ${\\varphi}$.\nThe Fourier transform $\\ft\\alpha$ of the distribution $\\alpha$ is defined by \n$\\dotprod{\\ft\\alpha}{\\varphi} = \\dotprod{\\alpha}{\\ft\\varphi}$.\n\\par\nA distribution $\\alpha$ is called \\emph{positive} if  $\\dotprod{\\alpha}{\\varphi} \\geq 0$\nfor any Schwartz function $\\varphi \\geq 0$. It is well-known that if $\\alpha$ is a positive distribution,\nthen it is a positive measure. A distribution $\\alpha$ is called \\emph{positive-definite} if $\\ft\\alpha$ is a positive \ndistribution. \n\\par\nFor a set $A \\subset \\R^n$ we denote by $\\# A$ the number of elements in $A$, and\nby $\\mes(A)$  or $|A|$ the Lebesgue measure of $A$.\n\n\n\\subsection{Measures}\nWe  will need some basic facts about measures in $\\R^n$.\n\\begin{lem}\n\t\\label{AL.4}\n\tLet $\\mu$ be a measure on $\\R^n$,  whose support $\\Lambda$ is a uniformly\n\tdiscrete set. Assume that $\\ft{\\mu}$ is a slowly increasing measure. Then \n\t\\begin{equation}\n\t \\sup_{\\lambda\\in\\Lambda} |\\mu(\\lambda)|<\\infty,\n\t\t\\label{AL.4.1}\n\t\\end{equation}\n\tand so $\\mu$ is a translation-bounded measure. \n\\end{lem}\nThis can be proved in a similar way to \\cite[Lemma 2]{lo2}.\n\\begin{lem}\n\t\\label{AL.10}\nLet $\\mu$ be a measure on $\\R^n$, whose support $\\Lambda$ is a uniformly discrete\nset. Assume that $\\hat{\\mu}$ is a slowly increasing measure, with at least one non-zero\natom. Then $\\Lambda$ is a relatively dense set in $\\R^n$. \n\\end{lem}\n\\begin{proof}\n\tBy Lemma \\ref{AL.4}, the measure $\\mu$ is translation-bounded. Let us suppose that\n\t$\\Lambda$ is not relatively dense, and show that this implies that $\\hat{\\mu}(\\{s\\})=0$\nfor every $s \\in \\R^n$.\n\\par\n\tChoose a Schwartz function $\\varphi$ whose Fourier transform $\\hat{\\varphi}$ has compact\n\tsupport, and $\\hat{\\varphi}(0)=1$. For each $0<\\delta<1$ define\n\t\t$\\varphi_\\delta(t):= \\delta^n\\varphi(\\delta t)$. Then we have \n\t\\begin{equation}\n\t\t\\hat{\\mu}(\\{s\\})=\\lim_{\\delta\\to 0}\\int \\hat{\\varphi}_\\delta(t-s)e^{2\\pi\n\t\ti\\dotprod{x}{t-s} }d\\hat\\mu(t)\n\t\t\\label{AL.10.1}\n\t\\end{equation}\n\tuniformly with respect to $x\\in\\R^n$. On the other hand, \n\t\\begin{equation}\n\t\t\\int \\hat{\\varphi}_\\delta(t-s)e^{2\\pi i\\dotprod{x}{t-s}}d\\hat{\\mu}(t) = \n\t\t\\int \\varphi_\\delta(x-y)e^{-2\\pi i\\dotprod{s}{y}}d\\mu(y).\n\t\t\\label{AL.10.2}\n\t\\end{equation}\n\tIf $\\Lambda$ is not relatively dense, then for any $R>0$ there is $x\\in \\R^n$ such\n\tthat the ball $x+B_{R}$ does not intersect $\\Lambda$. Using the\n\ttranslation-boundedness of $\\mu$ this implies that for any $\\delta>0$, there are\n\tvalues of $x$ for which the right-hand side of \\eqref{AL.10.2} is arbitrarily\n\tclose to zero. Hence the limit in \\eqref{AL.10.1} must be zero, which proves the\n\tclaim.\n\\end{proof}\n\n\n\n\\subsection{Interpolation}\nFor a compact set $\\Omega \\subset \\R^n$, we denote by $\\B(\\Omega)$ the   Bernstein space\nconsisting of all bounded, continuous functions $f$ on $\\R^n$ such that the distribution $\\ft f$\nis supported by $\\Omega$. \nA set $\\Lam\\subset\\R^n$ is called an \\emph{interpolation set} for the space $\\B(\\Omega)$ if for every bounded sequence\n $\\{c_\\lam\\}$, $\\lam\\in\\Lam$, there exists at least one $f \\in \\B(\\Omega)$ satisfying\n$f(\\lam)=c_\\lam$ $(\\lam\\in\\Lam)$. It is well-known that such\n$\\Lam$ must be a uniformly discrete set.\n\\par\nThe following result is due to Ingham for $n=1$, and  Kahane for $n>1$, see\n\\cite{ou3}. \n\\begin{thm}\n\tThere is a constant $C$ which depends on the dimension $n$ only, such that if\n\t$\\Lambda$ is a uniformly discrete set in $\\R^n$, $d(\\Lambda)\\ge a> 0$, then\n\t$\\Lambda$ is an interpolation set for $\\mathcal{B}(\\Omega)$ where $\\Omega$ is any closed\n\tball of radius $C\/a$. \n\t\\label{AL.11}\n\\end{thm}\nAs a consequence we obtain:\n\\begin{corollary}\n\t\\label{AC.12}\n\tThere is a constant $C$ which depends on the dimension $n$ only, such that if a measure\n\t$\\mu$ is supported on a uniformly discrete set $\\Lambda\\subset \\R^n$,\n\t$d(\\Lambda)\\ge a> 0$, and if the distribution $\\hat{\\mu}$ vanishes on a ball of radius $C\/a$, then $\\mu=0$. \n\\end{corollary}\n\n\n\\begin{proof}\nSuppose that $\\hat{\\mu}$ vanishes on the ball $B_R$, where $R:=(C+1)\/a$ and\n$C$ is the constant from \\thmref{AL.11}\n (we may assume that the ball is centered at the origin). \nLet $\\Omega=\\{x : |x| \\leq C\/a\\}$. \nGiven $\\lambda\\in\\Lambda$, there is $f\\in \\mathcal{B}(\\Omega)$  such that $f(\\lambda)=1$ and $f$ vanishes on\n\t$\\Lambda\\setminus \\{\\lambda\\}$. Define $\\varphi(x):= f(x)\\psi(x)$, where $\\psi$ is a\n\tSchwartz function such that $\\psi(\\lambda)=1$ and $\\spec(\\psi)\\subset B_{1\/a}$.\n\tThen $\\varphi$ is a Schwartz function satisfying\n\\[\n\\varphi(\\lambda)=1, \\quad\n\t\\text{$\\varphi(\\lambda')=0$ for all $\\lambda' \\in \\Lam \\setminus \\{\\lam\\}$},  \\quad\n\\spec(\\varphi)\\subset B_R.\n\\]\nHence\n\\[ \\mu(\\lambda)=\\int\n\t\\overline{\\varphi(x)}d\\mu(x)=\\dotprod{\\hat{\\mu}}{\\overline{\\hat{\\varphi}}}=0.\n\\]\n\tThis holds for any $\\lambda\\in\\Lambda$, so we obtain $\\mu=0$. \n\\end{proof}\n\n\n\n\n\\section{Auxiliary measures $\\nu_h$}\nLet $\\mu$ be a measure on $\\R^n$ satisfying \\eqref{RM.1} and \\eqref{RM.2},\nand assume that the support $\\Lambda$ is a uniformly discrete set.\nBy Lemma \\ref{AL.4}, the atoms of $\\mu$ are bounded, so $\\mu$ is a translation-bounded measure. \n\n For each $h\\in S-S$ we denote \n\\begin{equation}\n\tS_h:= S\\cap (S-h)= \\left\\{ s\\in S\\,:\\, s+h\\in S \\right\\}, \n\t\\label{AL.3.1}\n\\end{equation}\nwhich is a non-empty subset of $S$, and we introduce a new measure \n\\begin{equation}\n\t\\nu_{h}:= \\sum_{s\\in S_h}\\hat{\\mu}(s)\\,\\overline{\\hat{\\mu}(s+h)}\\,\\delta_s.\n\t\\label{AL.3.2}\n\\end{equation}\nNotice that it is a non-zero, slowly increasing measure, whose support is the set $S_h$.\n\\begin{lem}\n\t\\label{AL.6}\n\tThe Fourier transform $\\hat{\\nu}_h$ of the measure $\\nu_{h}$ is also a measure,\n\twhich is translation-bounded and supported by the closure of the set\n\t$\\Lambda-\\Lambda$.\n\\end{lem}\n\nThis is an elaborated version of \\cite[Lemma 12]{lo2}. That lemma stated that $\\spec(\\nu_h)$\ndoes not intersect the punctured ball $B_a\\setminus\\left\\{ 0 \\right\\}$, where $a:=d(\\Lambda)>0$.\nHowever, the result there was formulated with the roles of\n\t$\\Lambda$ and $S$ interchanged, and under the stronger assumption that $\\Lambda$,\n\t$S$ are both uniformly discrete sets. \n\n\t\\begin{proof}[Proof of Lemma \\ref{AL.6}] \n\t\tFix a Schwartz function $\\varphi$, whose Fourier transform $\\hat{\\varphi}$ has\n\t\tcompact support and $\\hat{\\varphi}(0)=1$. For each $0<\\delta<1$ we denote\n\t\t$\\varphi_\\delta(x):= \\delta^n\\varphi(\\delta x)$, and define the measure\n\t\t\\begin{equation}\n\t\t\t\\nu_h^{(\\delta)} (t) := \\overline{\\left( \\hat{\\mu} \\ast \\hat{\\varphi}_\\delta\n\t\t\t\\right)(t+h)}\\cdot\n\t\t\t\\hat{\\mu}(t).\n\t\t\t\\label{AL.6.1}\n\t\t\\end{equation}\nIt is a slowly increasing measure, supported by $S$, which tends to $\\nu_h$ in the space of temperate\ndistributions as $\\delta\\to 0$. Hence $\\hat{\\nu}_h^{(\\delta)}$ tends to $\\hat{\\nu}_h$ in the\nsame sense as $\\delta\\to 0$. \n\\par\nWe will show that $\\hat{\\nu}_h^{(\\delta)}$ is a\ntranslation-bounded measure, and moreover \n\\[ \\sup_{x\\in\\R^n} |\\hat{\\nu}_{h}^{(\\delta)}| (x+B_1) \\] \nis bounded by some constant $C(\\mu,\\varphi)$ which depends on $\\mu$ and $\\varphi$ only\n(in particular, it does not depend on $\\delta$).\n Indeed, the Fourier transform of $\\nu_h^{(\\delta)}$ is the\nmeasure \n\\begin{equation}\n\t\\hat{\\nu}_h^{(\\delta)}= \\overline{(\\varphi_\\delta\\cdot e_{-h}\\cdot \\mu)(x)} \\ast\n\t\\mu(-x),\n\t\\label{AL.6.2}\n\\end{equation}\nwhere $e_{-h}(x):= e^{-2\\pi i \\dotprod{h}{x}}$. Hence, we have\n\\[ \\sup_{x\\in \\R^n} |\\hat{\\nu}_h^{(\\delta)}| (x+B_1) \\le \n\\Big\\{ \\sup_{x\\in \\R^n} |\\mu| (x+B_1) \\Big\\} \\Big\\{\n\\int |\\varphi_\\delta (x)| \\, |d\\mu(x)| \\Big\\} \\le C(\\mu,\\varphi), \\]\nsince $\\mu$ is  translation-bounded. Letting $\\delta\\to 0$ this implies\nthat the limit $\\hat{\\nu}_h$ is also a translation-bounded measure,\nand in fact $|\\hat{\\nu}_h| (x+B_1) \\le C(\\mu,\\varphi)$ for all $x\\in \\R^n$.\n\\par\nFinally, it follows from \\eqref{AL.6.2} that the measure $\\hat{\\nu}_h^{(\\delta)}$ is supported by\n $\\Lambda-\\Lambda$. Hence its limit as $\\delta\\to 0$ must be supported by the closure\n of $\\Lambda-\\Lambda$, which ends the proof. \n\t\\end{proof}\n\n\n\n\n\\section{Positive-definite measures}\n\\label{sec:pos-def}\n\\subsection{ } \\label{5.1a}\nIn this section we consider positive-definite measures whose supports are\nuniformly discrete sets. We establish a dichotomy concerning the\ndiscreteness of the spectrum: either it is also  uniformly discrete, or it is\n``non-discrete'' in a strong sense. \n\\begin{thm}\n\t\\label{AT.5}\n\tLet $\\mu$ be a positive-definite measure on $\\R^n$ satisfying \\eqref{RM.1} and\n\t\\eqref{RM.2}, and assume that the support $\\Lambda$ is a uniformly discrete set. Then, either \n\\begin{enumerate-math}\n\\item\n$S$ is also  uniformly discrete; or \n\\item\n $S$ has a relatively dense  set of accumulation points.\n\\end{enumerate-math}\n\\end{thm}\n\nIn particular, it follows that if the spectrum $S$ is a discrete closed set, then it must\nbe uniformly discrete. Hence \\thmref{RT.0} applies to this situation, and yields that the\nmeasure $\\mu$ is representable in the form \\eqref{RM.3}. So \\thmref{RT.5} follows.\n\n\\begin{remark}\nActually we will prove that there is a constant\n\t$C$ which depends on the dimension $n$ only, such that if the spectrum\n$S$ is not uniformly discrete, then every ball of radius\n\t$C\/d(\\Lam)$ contains infinitely many points of $S$.\n\\end{remark}\n\n\n\\subsection{}\nWe will need the following auxiliary lemma:\n\\begin{lem}\n\t\\label{AL.1}\n\tThere is a real-valued Schwartz function $\\varphi$ on $\\R^n$ which has  the following\n\tproperties: \n\t\\begin{enumerate-math}\n\t\\item \\label{it_al.1.1} There is $R$ such that $\\varphi(x)>0$ for $|x|\\ge R$;\n\t\\item \\label{it_al.1.2} $\\int \\varphi(x)dx=0$;\n\t\\item \\label{it_al.1.3} $ \\spec(\\varphi)$ is contained in $B_1$ (the open unit ball).\n\t\\end{enumerate-math}\n\\end{lem}\n\\begin{proof}\n\tChoose a Schwartz function $\\psi > 0$ whose Fourier transform $\\hat{\\psi}$ is\n\tsupported in the ball $\\left\\{t:  |t|\\le 1\/3 \\right\\}$. Define $\\varphi:=\\alpha \\psi\n\t-\\beta \\psi^2$, where $\\alpha:= \\left( \\int \\psi \\right)^{-1}$ and $\\beta:=\\left(\n\t\\int \\psi^2 \\right)^{-1}$. It is easy to verify that all the properties\n\t\\ref{it_al.1.1}, \\ref{it_al.1.2} and \\ref{it_al.1.3} are satisfied.\n\\end{proof} \n\\subsection{}\n\\begin{proof}[Proof of \\thmref{AT.5}]\n\tAssume that $S$ is not uniformly discrete.  Let $\\varphi$ be the function given by\n\t\\lemref{AL.1}, and $R$ be the number from property \\ref{it_al.1.1} of that lemma.\n\tWe will show that \n\tany closed ball of radius $C\/a$ contains infinitely many points of $S$, where\n$C : = R+1$ and $a:=d(\\Lambda)>0$ (notice that the constant $C$ indeed depends on the dimension $n$ only). \nIn particular this will show that the set of accumulation points of $S$ must be relatively dense.\n\n\\par\n\tBy multiplying $\\mu$ by an exponential (which corresponds to translation of\n\t$\\hat{\\mu}$), it will be enough to show that the ball $\\{|t| \\leq C\/a\\}$\ncontains  infinitely many points of $S$. So, suppose to the\n\tcontrary that this does not hold, namely this ball\n\tcontains only finitely many points of $S$. \n\\par\n Since $S$ is not uniformly discrete,\n\tthe set $S-S$ contains elements $h\\neq 0$ arbitrarily close to zero. Hence we may choose\n\t$h\\in S-S$ such that the set $S_{h}$ defined by \\eqref{AL.3.1} does not intersect\n\tthe ball $\\left\\{ |t| < R\/a \\right\\}$. \tIt follows that the measure $\\nu_h$\n\tin \\eqref{AL.3.2} is a non-zero positive measure, whose support is\n\tcontained in $\\left\\{ |t|\\ge R\/a \\right\\}$. Using property \\ref{it_al.1.1} from\n\t\\lemref{AL.1} this implies that \n\t\\begin{equation}\n\t\t\\label{AT.5.1}\n\t\t\\int \\varphi(ax)\\,d\\nu_h(x) > 0.\n\t\\end{equation}\n\tOn the other hand, we have \n\t\\begin{equation}\n\t\t\\label{AT.5.2}\n\t\t\\int\n\t\t\\varphi(ax)\\,d\\nu_h(x)=a^{-n}\\int \\hat{\\varphi}(-t\/a) \\, d\\hat{\\nu}_h(t).\n\t\\end{equation}\n\tBy \\lemref{AL.6}, $\\hat{\\nu}_h$ is a measure, supported by the closure of\n$\\Lam-\\Lam$. Since $\\Lam$ is uniformly discrete, \n this closure is contained in the set $\\{0\\}\\cup\n\t\\{|t|\\ge a\\}$.\n But from  properties \\ref{it_al.1.2} and \\ref{it_al.1.3} in\n\t\\lemref{AL.1} it follows that the function $\\hat{\\varphi}(-t\/a)$ vanishes on this set.\n\tHence, the right-hand side of \\eqref{AT.5.2} must vanish, \n\tin contradiction with \\eqref{AT.5.1}.  \n\\end{proof}\n\n\n\n\\section{Spectra with finite density}\nIn this section we prove \\thmref{RT.6}. We will show that under the conditions in the\ntheorem, the spectrum $S$ of the measure $\\mu$ must be a uniformly discrete set. Then the final conclusion that\n$\\mu$ is of the form \\eqref{RM.3} can be deduced from \\thmref{RT.0}. \n\\subsection{}\nFor a set $\\Lambda\\subset \\R $ we denote \n\\[ \\rho(\\Lambda):= \\sup_{x\\in\\R}\\# (\\Lambda\\cap [x,x+1]). \\] \nNotice that $\\rho(\\Lambda)<\\infty$ if and only if $\\Lambda$ is a finite union of uniformly\ndiscrete sets. \n\\par\nWe will need the following notion of ``lower density'' of a set $\\Lambda \\subset \\R$, defined by \n\\[ D_{\\#}(\\Lambda):=\\liminf_{R\\to\\infty}\\frac{\\#(\\Lambda\\cap (-R,R))}{2R}. \\]\nClearly we have $D_{\\#}(\\Lambda)\\le\\rho(\\Lambda)$. It will be useful below to extend the\ndefinition of the density $D_{\\#}$ also to multi-sets $\\Lambda \\subset \\R$, that is, to the\ncase when points in $\\Lambda$ occur with multiplicities. Notice that $D_{\\#}$ is\nsuper-additive in the sense that \n\\[ D_{\\#}(A\\cup B)\\ge D_{\\#}(A)+D_{\\#}(B), \\]\nwhere the union $A\\cup B$ is understood in the sense of multi-sets.\n\\subsection{ }\nThe following result is a more general version of \\cite[Proposition 4]{lo2}.\n\\begin{prop}\n\t\\label{BP.1}\n\tLet $\\Lambda\\subset \\R$ be a set with $\\rho(\\Lambda)\\le M< \\infty$. Assume that\n\t$\\Lambda$ supports a non-zero, slowly increasing  measure $\\mu$, such that the distribution\n\t$\\hat{\\mu}$ vanishes on the open interval $(0,a)$ for some $a> 0$. Then \n\t\\[ D_{\\#}(\\Lambda)\\ge c(a,M), \\]\n\twhere $c(a,M)>0$ is a constant which depends on $a$ and $M$ only. \n\\end{prop}\nThis can be deduced from the following lemma:\n\\begin{lem}\n\t\\label{BL.2}\n\tLet $\\Lambda$ be a finite subset of $(-R,R)\\setminus (-1,1)$, such that\n\t$\\rho(\\Lambda)\\le M$, and let $a>0$. There is $c(a,M)>0$ such that if\n\t$(\\#\\Lambda)\/(2R)<c(a,M)$, then one can find a Schwartz function $\\varphi$ with\n\tthe following properties: \n\t\\[ \\varphi(0)=1, \\;\\; \\varphi(\\lambda)=0 \\;\\; (\\lambda\\in \\Lambda), \\;\\; \\spec(\\varphi)\\subset\n\t(0,a), \\;\\; \\sup_{|x|\\ge R} |\\varphi(x)|\\le 1. \\]\n\\end{lem}\nThe proof of \\lemref{BL.2}, as well as the deduction of \\propref{BP.1} from this lemma,\ncan be done in a way similar to \\cite[Section 4.1]{lo2}, and we omit the details.\n\\subsection{} \n\\begin{lem}\n\t\\label{BL.3}\n\tLet $S\\subset \\R$ be a set with $\\rho(S)<\\infty$. Suppose that there is\n\t$c=c(S)>0$ such that $D_{\\#}(S_h)>c$ for every $h\\in S-S$. Then\n\t$\\rho(S-S)<\\infty$. \n\\end{lem}\n\\begin{proof}\n\tLet $x\\in\\R$, and suppose that $h_1,\\dots, h_N$ are distinct points in the set\n\t$(S-S)\\cap [x,x+1]$. Since the lower density $D_{\\#}$ is super-additive, we have \n\t\\[ cN \\le \\sum_{j=1}^{N}D_{\\#}(S_{h_j})\\le D_{\\#}\\Big(\n\t\\bigcup_{j=1}^{N}S_{h_j} \\Big), \\]\nwhere the union is understood in the sense of multi-sets. Notice that each point in this\nunion occurs with multiplicity not greater than $\\rho(S)$. It follows that $cN\\le\n\\rho(S)  D_{\\#}(S)$, which shows that the set $S-S$ cannot have more than\n$\\rho(S)  D_{\\#}(S)\/c$ elements in any closed interval of length 1. Hence $\\rho(S-S)<\\infty$,\nwhich proves the claim.\n\\end{proof}\n\\subsection{}\n\\begin{proof}[Proof of \\thmref{RT.6}] \nWe assume that $\\mu$ is a measure on $\\R$ satisfying \\eqref{RM.1} and\n\\eqref{RM.2}, where $\\Lambda$ is a uniformly discrete set, and $S$ is a set with\n$\\rho(S)<\\infty$.\n\\par\n For each $h\\in S-S$, let $\\nu_{h}$ be the measure\ndefined by \\eqref{AL.3.2}. By \\lemref{AL.6} the Fourier transform $\\hat{\\nu}_h$ is\nsupported by the closure of the set $\\Lambda-\\Lambda$. Since $\\Lambda$ is uniformly\ndiscrete this implies that $\\hat{\\nu}_{h}$ vanishes on the open interval $(0,a)$, where\n$a:=d(\\Lambda)>0.$ As the measure $\\nu_h$ is supported by $S_h$, it follows from \\propref{BP.1} that \n$ D_{\\#}(S_h)\\ge c$, where $c>0$ is a constant which depends on $d(\\Lambda)$ and $\\rho(S)$. \n\\par\nSince this holds  for every $h\\in S-S$, \\lemref{BL.3} allows us to deduce that $\\rho(S-S)<\\infty$. In particular, the set\n$S-S$ has no accumulation point at zero, so there is $\\delta>0$ such that $(S-S)\\cap\n(-\\delta,\\delta)=\\left\\{ 0 \\right\\}$. Hence $S$ must be uniformly discrete, and\nin fact $d(S)\\ge \\delta$. \n\\par\nOnce we have concluded that \n$\\Lambda$ and $S$ are both uniformly discrete sets, we can apply \\thmref{RT.0} which yields that the\nmeasure $\\mu$ is representable in the form \\eqref{RM.3}.\n\\end{proof}\n\n\n\n\\section{Meyer sets}\n\\subsection{}\nIn this section we show that if the support $\\Lam$  satisfies a  stronger  \n              discreteness condition than in \\thmref{AT.5},  then the conclusion of this  theorem\n              remains true without  any additional  positivity restriction.\n\\begin{definition*}\n\tA set $\\Lambda\\subset \\R^n$ is called a \\emph{Delone set} if $\\Lambda$ is both a\n\tuniformly discrete and relatively dense set. \n\\end{definition*}\n\\lemref{AL.10} implies that a uniformly discrete set $\\Lam$ which supports a measure $\\mu$,\nwhose Fourier transform $\\ft\\mu$ is  a pure point\nmeasure, must be a Delone set.\n\\begin{definition*}\n\tA set $\\Lambda\\subset \\R^n$ is called a \\emph{Meyer set} if the following two conditions\n\tare satisfied: \n\t\\begin{enumerate-math}\n\t\\item $\\Lambda$ is a Delone set;\n\t\\item There is a finite set $F$ such that $\\Lambda-\\Lambda\\subset \\Lambda+F$. \n\t\\end{enumerate-math}\n\\end{definition*}\nThe concept of Meyer set was introduced in \\cite{mey0, mey1} in connection with problems\nin harmonic analysis. After the experimental discovery of quasicrystalline materials in\nthe middle of 80's, Meyer sets have been extensively studied as mathematical models of\nquasicrystals. \n\\par\nThere are some equivalent forms of the definition of a Meyer set,\nsee  \\cite{moo}. In particular, the following is true (Lagarias \\cite{lag1}):\n\\par\n\\emph{A Delone set $\\Lam$ is a Meyer set if and only if\n$\\Lambda-\\Lambda$ is  uniformly discrete}\n\\par\n\\noindent\n           (a simplified version of the proof of this equivalence can be found in \\cite[Lemma 8]{lo2}).\n    \n\n\\subsection{}\nNow we show that if a measure $\\mu$ is supported by a Meyer set $\\Lam$, then the\ndichotomy phenomenon for the spectrum $S$ is valid: either $S$ is uniformly discrete, or\nit is non-discrete with a relatively dense set of accumulation points.  \n\\begin{thm}\n\t\\label{CT.1}\n\tLet $\\mu$ be a measure on $\\R^n$ satisfying \\eqref{RM.1} and \\eqref{RM.2},\n\tand assume that the support $\\Lam$ is a Meyer set. Then the same\n\tconclusion as in \\thmref{AT.5} holds.\n\\end{thm}\n\n\n\\begin{proof}\nAs in the proof of \\thmref{AT.5}, it will\n\tbe enough to prove that for every $h\\in S-S$, the set $S_h$ must intersect any ball\n\tof radius $C\/a$, but now we will take $C$ to be the number from \\corref{AC.12}, and\n\t$a:=d(\\Lambda-\\Lambda)>0$.\n\\par\nAnd indeed, by \\lemref{AL.6} the measure $\\nu_h$\n\tdefined by \\eqref{AL.3.2} is a non-zero measure, whose Fourier transform\n\t$\\hat{\\nu}_h$ is a translation-bounded measure supported by the set\n\t$\\Lambda-\\Lambda$ (this set is uniformly discrete, so it is not necessary to consider\nits closure). Now \\corref{AC.12}, applied to the measure\n\t$\\hat{\\nu}_h$,  implies that $\\nu_h$ cannot vanish on a ball of radius\n\t$C\/a$. Hence  $S_h$ must intersect any such a ball, which completes the proof.\n\\end{proof}\n\n\\begin{remark}\nThe proof in fact shows that there is a constant\n\t$C$ which depends on the dimension $n$ only, such that if the spectrum\n$S$ is not uniformly discrete, then every ball of radius\n\t$C\/d(\\Lam-\\Lam)$ contains infinitely many points of $S$.\n\\end{remark}\n\n\n\\subsection{}\nNext, we deduce \\thmref{RT.7} from the above result. Suppose that $\\Lambda-\\Lambda$ is a\nuniformly discrete set, and that $S$ is  a discrete closed set. Hence $S$ \nhas no finite accumulation points, so it follows from \\thmref{CT.1} that $S$\n  must be uniformly discrete. \n\\par\nHowever, to complete the conclusion of \\thmref{RT.7} it still remains to show that  $\\mu$ \nis representable in the form \\eqref{RM.3}. Here one cannot directly\napply \\thmref{RT.0}, since the measure was assumed to be neither positive nor positive-definite. \nIn order to obtain \\eqref{RM.3} we use instead the following version of\n\\thmref{RT.0}, proved in \\cite{lo1}:\n\\begin{thm}\n\t\\label{RT.9}\n\tLet $\\mu$ be a measure on $\\R^n$ satisfying \\eqref{RM.1} and \\eqref{RM.2}. \n\tAssume that the sets $\\Lambda-\\Lambda$ and $S$ are both uniformly discrete. Then the\n\tconclusion of \\thmref{RT.0} holds.\n\\end{thm}\n Combining Theorems \\ref{CT.1} and \\ref{RT.9} thus implies the full assertion of \\thmref{RT.7}.\n\n\n\n\n\n\n\n\\section{Nowhere dense spectra}\n\n\\label{subsec:notdense}\nIn this section we prove \\thmref{RT.15}. We will construct a non-periodic\nmeasure $\\mu$ supported on  a Meyer set $\\Lam\\subset \\R^n$,\nsuch that the spectrum $S$ is not dense in any ball. \nMoreover, the measure $\\mu$ in the construction \n is positive-definite, and both $\\mu$ and $\\ft\\mu$\nare translation-bounded measures.\n\n\n\n\\subsection{}\n Let $\\Gamma$ be a lattice in $\\R^n\\times \\R^m$, and let\n\t$p_1$ and $p_2$ denote the projections onto $\\R^n$ and $\\R^m$ respectively. Assume\n\tthat the restrictions of $p_1$ and $p_2$ to $\\Gamma$ are injective, and that their\n\timages are dense in $\\R^n$ and $\\R^m$ respectively.\n\tLet $\\Gamma^{*}$ be the dual\n\tlattice, then the restrictions of $p_1$ and $p_2$ to $\\Gamma^{*}$ are also\n\tinjective and have dense images.\n\\par\n If $\\Omega$ is a bounded open set in\n\t$\\R^m$, then the set \n\t\\begin{equation}\n\\Lambda(\\Gamma,\\Omega):= \\left\\{ p_1(\\gamma)\\,:\\,\n\t\\gamma\\in \\Gamma, \\, p_2(\\gamma)\\in \\Omega \\right\\} \n\t\t\\label{RP.9.0}\n\t\\end{equation}\nis called the ``model set'',\n\tor the ``cut-and-project set'', associated to the lattice $\\Gamma$ and to the\n\t``window'' $\\Omega$. \nIt is well-known that any model set is a Meyer set.\n\\par\nMeyer observed \\cite[p.\\ 30]{mey0} (see also \\cite{mey2}) that\n\tmodel sets provide examples of non-periodic uniformly discrete sets, which support\n\ta measure $\\mu$ such that the Fourier transform $\\ft\\mu$ is also a pure point measure. Such a\n\tmeasure may be obtained by choosing a Schwartz function $\\varphi$ on\n$\\R^m$ such that $\\supp(\\hat{\\varphi})\\subset \\Omega$, and taking \n\t\\begin{equation}\n\t\t\\mu = \\sum_{\\gamma\\in\\Gamma} \\hat{\\varphi} ( p_2(\\gamma)  )\n\t\t\\delta_{p_1(\\gamma)}.\n\t\t\\label{RP.9.1}\n\t\\end{equation}\n\tIt is not difficult to verify that this is a translation-bounded measure, whose Fourier transform is the\n\t(also translation-bounded) pure point measure\n\t\\begin{equation}\n\t\t\\hat{\\mu}=\\frac{1}{\\det \\Gamma}\n\t\t\\sum_{\\gamma^{*}\\in\\Gamma^{*}}\\varphi( p_2(\\gamma^{*})\n\t\t)\\delta_{p_1(\\gamma^{*})}.\n\t\t\\label{RP.9.2}\n\t\\end{equation}\n\\par\nHowever, the compact support of $\\hat{\\varphi}$ implies that $\\varphi$ is an entire function,\nand so $\\varphi$ cannot also be supported on a bounded set. Hence\n the spectrum of the measure $\\mu$ is only known to be contained in $p_1(\\Gam^*)$, \nand so it is generally everywhere dense in $\\R^n$.\n\\par\nNevertheless,  we will see that one can construct a function $\\varphi$ with sufficiently many zeros,\nin such a way that the non-zero atoms in \\eqref{RP.9.2} in fact lie in a nowhere dense set.\n\n\\subsection{} \\label{section_8.2}\nA set $\\Lambda\\subset\\R^n$ is said to have a \\emph{uniform density} $D(\\Lambda)$ if \n\\[ \\frac{ \\#\\left( \\Lambda \\cap (x+B_R) \\right)}{|B_R|} \\to D(\\Lambda) \\]\nas $R\\to \\infty$ uniformly with respect to $x\\in \\R^n$. \n\\begin{lem}\n\t\\label{EL.2}\n\tLet $\\Lambda=\\Lambda(\\Gamma,\\Omega)$ be a model set such that the boundary of\n\t$\\Omega$ is a set of Lebesgue measure zero in $\\R^m$. Then $\\Lambda$ has uniform\n\tdensity \n\t\\[ D(\\Lambda) = \\frac{\\mes(\\Omega)}{\\det(\\Gamma)}. \\]\n\\end{lem}\nA proof of this fact can be found e.g.\\ in \\cite[Proposition 5.1]{matei-meyer-simple}. \n\\subsection{} \\label{8 .3}\nWe will  now assume that $m=1$, that is, $\\Gam$ is a lattice in $\\R^n \\times \\R$.\nThe following theorem is the main result of this section.\n\\begin{thm}\n\t\\label{ET.1}\n\tFor any $\\varepsilon > 0$ there is a non-zero Schwartz function $\\varphi\\ge 0$ on\n\t$\\R$, such that: \n\t\\begin{enumerate-math}\n\t\\item The measure $\\mu$ in \\eqref{RP.9.1} is supported by the model set\n\t\t$\\Lambda(\\Gamma,(-\\varepsilon,\\varepsilon))$;\n\t\\item The spectrum of $\\mu$ is a nowhere dense set in $\\R^n$. \n\t\\end{enumerate-math}\n\\end{thm}\nObserve that the support of $\\mu$ cannot be covered by a finite \nunion of translates of any lattice, since it contains a model set. Hence this \nresult implies \\thmref{RT.15}. The condition $\\varphi\\ge 0$ guarantees  that $\\mu$ is a\npositive-definite measure. \n\\par\nThe proof of \\thmref{ET.1} depends on the following: \n\\begin{lem}\n\t\\label{EL.3}\n\tFor each $j\\ge 1$ let $Q_j\\subset\\R$ be a   set  with\n\tuniform density $D(Q_j)$. Assume that \n\t\\begin{equation}\n\t\t\\label{EL.3.2}\n\t\t\\sum_{j\\geq 1} D(Q_j)< a.\n\t\\end{equation}\n\tThen one can find positive numbers $T_j$ \nand a non-zero Schwartz function $\\varphi$ on $\\R$ such that: \n\t\\begin{enumerate-math}\n\t\\item $\\spec(\\varphi)\\subset (0,a)$;\n\t\\item $\\varphi$ vanishes on the set $Q$ defined by\n\t\t\\begin{equation}\n\t\t\t\\label{EL.3.1}\n\t\t\tQ:= \\bigcup_{j\\geq 1} Q_j'  , \\qquad\n\t\t\tQ_j':=Q_j\\setminus(-T_j,T_j).\n\t\t\\end{equation} \n\t\\end{enumerate-math}\n\\end{lem}\nBefore we prove \\lemref{EL.3}, let us first show how to deduce \\thmref{ET.1} from it.\n\\begin{proof}[Proof of \\thmref{ET.1}]\n\tLet $\\{x_j\\}$ $(j\\ge 1)$ be a sequence of points which are dense in $\\R^n$. For\n\teach $j$, choose an open ball $B_j$ centered at the point $x_j$, in such a way\n\tthat \n\t\\begin{equation}\n\t\t\\label{EL.4.1}\n\t\t\\sum_{j\\ge 1} \\mes(B_j)< \\frac{\\varepsilon}{\\det(\\Gamma)}.\n\t\\end{equation}\n\tConsider the sets $Q_j \\subset \\R$ defined by \n\t\\[ Q_j := \\left\\{ p_2(\\gamma^{*})\\; :\\; \\gamma^{*}\\in\\Gamma^{*}, \\;\n\t\tp_1(\\gamma^{*})\\in B_j  \\right\\}.\n\t   \\]\n\tThen each $Q_j$ is a model set, with uniform density \n\t\\[ D(Q_j)= \\det(\\Gamma) \\mes(B_j) \\]\n\taccording to \\lemref{EL.2}. Due to \\eqref{EL.4.1} this implies that \n\t\\[ \\sum_{j\\ge 1} D(Q_j)< \\varepsilon.  \\]\n\t\\lemref{EL.3} therefore gives a sequence $\\{T_j\\}$ of positive numbers,\n\tand  a non-zero Schwartz function $\\psi$ on $\\R$, \n\t$\\spec(\\psi) \\subset (0,\\varepsilon)$,  such that $\\psi$  \n\tvanishes on the set $Q$ in \\eqref{EL.3.1}.\n\tHence  $\\varphi := |\\psi|^2 \\geq 0$  is also a Schwartz function vanishing on\n\t$Q$, and $\\spec(\\varphi)\\subset (-\\varepsilon,\\varepsilon)$.\n\\par\n\tFor each $j\\ge 1$ there are only finitely many points \n\tof the lattice $\\Gamma^{*}$ lying in the set $B_j\\times (-T_j,T_j)$, so we\nmay choose an open ball $\\Omega_j$ contained in\n\t$B_j$ such that $\\Omega_j\\times (-T_j,T_j)$ has no points in common\n\twith $\\Gamma^{*}$. Notice that the set \n\t\\[ \\Omega := \\bigcup_{j\\ge 1} \\Omega_j \\]\n\tis an open, dense set in $\\R^n$.\n\\par\n\tWe claim that the spectrum of the measure\n\t\\eqref{RP.9.1} does not intersect the set $\\Omega$. Indeed, by \\eqref{RP.9.2}, an\n\telement of the spectrum is a point of the form $p_1(\\gamma^{*})$, where\n\t$\\gamma^{*}\\in \\Gamma^{*}$ and $\\varphi ( p_2(\\gamma^{*})  ) \\ne 0$. \n\tIf $p_1(\\gamma^{*})\\in \\Omega_j$ for some $j$, then we must have\n\t$|p_2(\\gamma^{*})|\\ge T_j$. Hence \n\\[ p_2(\\gamma^{*})\\in Q_j\\setminus(-T_j,T_j) \\subset Q,\n\\]\n which is not\n\tpossible as $\\varphi$ vanishes on $Q$. \n\\par\n\tWe conclude that the spectrum $S$ of the\n\tmeasure $\\mu$ is contained in the closed, nowhere dense set $\\R^n\\setminus\n\t\\Omega$. On the other hand, the support $\\Lambda$ of the\n\tmeasure  is contained in the model set \n\t$\\Lambda (\\Gamma, (-\\varepsilon,\\varepsilon) )$, so this completes the proof. \n\\end{proof}\n\\subsection{ } \\label{section_8.4}\nIt remains to prove \\lemref{EL.3}. For this we will use\nthe celebrated Beurling and Malliavin theorem, see \\cite{bm67}.\n\\par\nFirst we recall the definition of the \\emph{Beurling-Malliavin upper\ndensity} (there are several equivalent ways to define this\ndensity). By a \\define{substantial system} of intervals  we mean a\nsystem $\\{I_k\\}$ of disjoint open intervals on $\\R$, such that\n$\\inf_{k}|I_k|>0$, and \n\\[\t\\sum_{k}\\left( \\frac{|I_k|}{1+\\dist(0,I_k)} \n\t\\right)^2 = \\infty. \\]\n\\par\nIf $\\Lambda\\subset\\R$ is a discrete closed set, then its\nBeurling-Malliavin upper density $D^{*}(\\Lambda)$ is defined to be the\nsupremum of the numbers $d>0$, for which there exists a\nsubstantial system $\\{I_k\\}$ satisfying \n\\[\t\\frac{\\# (\\Lambda \\cap I_k)}{|I_k|}\\ge d \\]\nfor all $k$. If for any $d>0$ no such a system  $\\{I_k\\}$ exists, then $D^{*}(\\Lambda)=0$.\n\\begin{thm}[Beurling and Malliavin \\cite{bm67}]\n\\label{ET.2}\nLet $\\Lambda\\subset \\R$ be a discrete closed set. Then for any\n$a>D^{*}(\\Lambda)$ one can find a non-zero function $\\varphi\\in\nL^{2}(\\R)$ such that: \n\\begin{enumerate-math}\n\\item $\\spec(\\varphi)\\subset (0,a)$;\n\\item $\\varphi$ vanishes on $\\Lambda$. \n\\end{enumerate-math}\n\\end{thm}\nBy multiplying $\\varphi$ by a Schwartz function with a sufficiently small \nspectrum, it is clear that one may assume the function $\\varphi$ in this \ntheorem to belong to the Schwartz class.\n\\par\n\\begin{remark}\nIt was also proved by \nBeurling and Malliavin that if $a<D^{*}(\\Lambda)$ then no such\n$\\varphi$ exists; however we will not use this part of their\nresult. \n\\end{remark}\n\\begin{proof}[Proof of \\lemref{EL.3}]\nChoose a sequence $\\{\\gamma_j\\}$ and a number $d$ such that \n\\[ D(Q_j) < \\gamma_j  \\quad \\text{and} \\quad \\sum_{j\\ge 1} \\gamma_j < d <a. \\]\nFor each $j$  find a  number $M_j$, such\nthat for any interval $I $ of length $|I|\\ge M_j$ we have \n\\begin{equation}\n\t\\label{EL.3.7}\n\t\\#(Q_j\\cap I) \\leq \\gamma_j \\, |I|.\n\\end{equation}\nDefine\n\\[ T_j := M_j^3 +M_j. \\]\nWe  claim that if $Q$ is the set given by \\eqref{EL.3.1},\nthen $D^{*}(Q)\\le d$.\n\\par\nLet $\\{I_k\\}$ be a substantial\nsystem of intervals on $\\R$. Observe first that there must exist at least one\ninterval $I_k$ satisfying\n\\begin{equation}\n\t\\label{EL.3.8}\n\t \\dist(0, I_k) \\le |I_k|^3.\n\\end{equation}\nFor otherwise, using the assumption that $\\inf_k\n|I_k|>0$, this would imply that \n\\[ \\sum_k \\left( \\frac{|I_k|}{1+\\dist(0,I_k)} \\right)^2 \\le \n\\sum_k \\big(1+\\dist (0,I_k) \\big)^{-4\/3} < \\infty,\\]\nwhich is not possible since the system $\\{I_k\\}$ is substantial.\n\\par\nNow consider an interval $I_k$ from the system, satisfying \\eqref{EL.3.8}. We will show\nthat \n\\begin{equation}\n\t\\label{EL.3.9}\n\t\\frac{\\# (Q\\cap I_k)}{|I_k|}<d.\n\\end{equation}\nIndeed, we have\n\\begin{equation}\n\t\\label{EL.3.10}\n\t\\#(Q\\cap I_k) \\le \\sum_{j\\ge 1}\\# (Q_j'\\cap I_k).\n\\end{equation}\nNotice that for each $j$ such that $|I_k|<M_j$, it follows from \\eqref{EL.3.8} that \n\\[I_k\\subset (-M_j^3-M_j,M_j^3+M_j) = (-T_j,T_j),\\]\nand hence $I_k$ contains no points in common with $Q_j'$.\nOn the other hand, for each $j$  such that $|I_k|\\ge M_j$ we have \n\\[\\#(Q_j'\\cap I_k) \\leq \\gamma_j \\, |I_k|\\]\n according to \\eqref{EL.3.7}. Hence \n\\begin{equation}\n\t\\label{EL.3.10.1}\n \\sum _{j\\ge 1}\\# (Q_j' \\cap I_k) \\leq   \\sum_{j \\, :\\,  M_j \\leq |I_k|} \\gamma_j \\, |I_k|  < d \\, |I_k|.\n\\end{equation}\nCombining \\eqref{EL.3.10} and \\eqref{EL.3.10.1} confirms that \\eqref{EL.3.9} holds.\n\\par\nWe have thus shown that any substantial system $\\{I_k\\}$ contains\nintervals for which \\eqref{EL.3.9} holds. Hence\n$D^{*}(Q)\\le d< a$. The proof is now concluded by \\thmref{ET.2}.\n\\end{proof}\n\n\n\n\n\n\n\\section{Hof's quasicrystals}\nThere exist also other approaches to the concept of\nquasicrystals. One of them, which is due to Hof \\cite{hof}, was\nstudied by many authors, see for example\n\\cite{lag2}, \\cite[Chapter 9]{bagr} and the references\ntherein. In this context, the diffraction spectrum of a point set\n$\\Lambda$ is defined through the Fourier transform of an autocorrelation \nmeasure $\\gamma_\\Lam$, which is associated to the set $\\Lambda$ by a\ncertain limiting procedure. \n\\par\nIn this section we first recall Hof's notion of diffraction, and then\napply our previous results to analyze  diffraction spectra \nof Delone sets with finite local complexity. \n\n\n\n\\subsection{} \\label{9.1}\nLet $\\Lambda\\subset\\R^n$ be a Delone set.\nHof proposed  to understand the diffraction by $\\Lam$ using the following procedure.\nFor  each $R>0$, consider the measure\n\\[\\gamma_\\Lambda^R := (2R)^{-n} \\sum_{\\lambda,\\lambda'\\in \\Lambda\\cap [-R,R]^n}\\delta_{\\lambda'-\\lambda}. \\]\nIt is a finite measure on $\\R^n$ which is both positive and positive-definite.\n The uniform discreteness of $\\Lambda$ implies that the measures\n$\\gamma_\\Lambda^R$ are all translation-bounded, with the constant\nin \\eqref{tr-bdd-def} bounded uniformly with respect to $R$. \nHence there exists at least one\nweak limit point $\\gamma_\\Lambda$ of the measures\n$\\gamma_\\Lambda^R$ as $R\\to\\infty$. Any such limit point\n$\\gamma_\\Lambda$ is called an \\define{autocorrelation measure} of\nthe set $\\Lambda$.\n The  measure $\\gamma_\\Lambda$  is also\n translation-bounded, positive and positive-definite. \n\\par\nThe positive-definiteness of $\\gamma_\\Lambda$ implies that\n its Fourier transform\n$\\hat{\\gamma}_\\Lambda$ is a positive measure. It is called a\n\\define{diffraction measure} of $\\Lambda$.  If the measure\n$\\hat{\\gamma}_\\Lambda$ is purely atomic, then \nits support $S$ is called a\n\\define{diffraction spectrum} of $\\Lambda$, and  $\\Lambda$ is\nsaid to be a  \\define{pure point diffractive set}.\n\\par\nMore generally, one can define diffraction by any \ntranslation-bounded measure  $\\mu$ on\n$\\R^n$, in a similar way. Denote by $\\mu_R$ the restriction of $\\mu$ to the\ncube $[-R,R]^n$, and define a measure $\\tilde{\\mu}_R$ by \n\\[ \\tilde{\\mu}_R(E):= \\overline{\\mu_R(-E)}.\\]\nThen  the  measures \n\\[\\gamma_\\mu^R:=(2R)^{-n} \\; \\mu_R * \\tilde{\\mu}_R\\]\nare uniformly translation-bounded and so \nhave at least one weak limit point $\\gamma_\\mu$ as $R\\to\\infty$, and any such\nlimit point is called an autocorrelation measure\nof $\\mu$. It is again a translation-bounded, positive-definite\nmeasure, and if $\\mu$  is a positive measure then\nalso $\\gam_\\mu$ is positive.\nThe  diffraction measure\n$\\hat{\\gamma}_\\mu$ and the diffraction spectrum $S$\n(assuming that $\\hat{\\gamma}_\\mu$ is  purely atomic) are also\ndefined in a similar way. \n\\par\nNotice that the diffraction by a Delone set $\\Lam$ described above, is included as a\nspecial case which corresponds to diffraction by the measure \n\\begin{equation}\n\t\\label{DD.1}\n\t\\mu = \\sum_{\\lambda\\in\\Lambda}\\delta_\\lambda.\n\\end{equation}\n\n\\subsection{} \\label{9.2}\nA Delone set $\\Lambda \\subset \\R^n$ is said to be of\n\\define{finite local complexity} if for every $R>0$ there\nare only finitely many different sets of the form\n\\[ (\\Lambda - \\lambda)\\cap B_R, \\quad \\lambda \\in \\Lambda. \\] \nIt is easy to verify that this condition is equivalent to the requirement that \n$\\Lambda-\\Lambda$ is a discrete closed set. \n\\par\nNotice that if    $\\mu$ is a  translation-bounded measure supported by a Delone set $\\Lambda$ of\nfinite local complexity, then any autocorrelation measure $\\gamma_\\mu$ of $\\mu$ must be supported by \nthe set $\\Lambda-\\Lambda$. In particular, $\\gamma_\\mu$ is a discrete measure.\n\\par\nModel sets are well-studied\nexamples of non-periodic Delone sets of finite local complexity,\nwith  pure point diffraction in Hof's sense. More precisely, if\n$\\Lambda=\\Lambda(\\Gamma,\\Omega)$ is a model set defined by\n\\eqref{RP.9.0} and such that the\nboundary of the ``window'' $\\Omega$ is a set of Lebesgue measure\nzero, then $\\Lambda$ is a pure point diffractive set, with a\ndense countable diffraction spectrum (see for example  \\cite[Section 9.4]{bagr}).\n\n\n\n\\subsection{} \nLet $\\Lambda \\subset \\R^n$ be a Delone set  of finite local complexity.\nAssume that the diffraction spectrum $S$ is  uniformly discrete. \nIs it true that $S$ must have a periodic structure?\n\\par\nThe question was raised in  \\cite[Problem 4.2(a)]{lag2}.\nIt follows from our previous results that the answer is positive:\n\\begin{thm}\n\t\\label{DT.1}\nSuppose that\n\\begin{enumerate-math}\n\\item\n$\\Lambda \\subset \\R^n$ is a\n\tDelone set  of finite local complexity;\n\\item\n$\\mu$ is a positive, translation-bounded measure supported by $\\Lam$;\n\\item\n$\\gamma_\\mu$ is  an autocorrelation measure of $\\mu$; and\n\\item\nthe support $S$ of the diffraction\n\tmeasure $\\hat{\\gamma}_\\mu$ is a uniformly discrete set. \n\\end{enumerate-math}\nThen $S$ is contained in a finite union of translates of a certain lattice,\nand the  diffraction measure has the form \\eqref{RM.3}.\n\\end{thm}\n\n\n\\begin{proof}\nThe \n\tautocorrelation measure $\\gamma_\\mu$ is positive, and is supported by $\\Lambda-\\Lambda$. Hence the\n\tdiffraction measure $\\hat{\\gamma}_\\mu$ is a positive-definite measure on $\\R^n$, whose\n\tsupport $S$ is a uniformly discrete set, and whose spectrum is contained in the\n\tdiscrete closed set $\\Lambda-\\Lambda$. \\thmref{RT.5} (applied to the measure\n\t$\\hat{\\gamma}_\\mu$) therefore yields that $\\hat{\\gamma}_\\mu$\n\tis of the form \\eqref{RM.3}. As a consequence,  $S$ \n\tmust be contained in a finite union of translates of a  lattice.\n\\end{proof}\n\n\n\nIn particular \\thmref{DT.1} applies to the measure \\eqref{DD.1}.\nIn this case the result shows that if a Delone set\n$\\Lambda$ of finite local complexity is pure point diffractive, \nand if the diffraction spectrum $S$ is uniformly discrete, then\n$S$ is contained in a finite union of translates of a lattice,\nand the  diffraction measure has the form \\eqref{RM.3}.\nSo we obtain \\thmref{RT.8}. \n\n\n\n\t\\subsection{}\nIf $\\Lambda$ is a Meyer set, then the conclusion in the previous result\nremains true even if the spectrum is just  a discrete closed set, and\nwithout the positivity of the measure:\n\\begin{thm}\n\t\\label{DT.2}\nSuppose that\n\\begin{enumerate-math}\n\\item\n$\\Lambda$ is a Meyer set in $\\R^n$;\n\\item\n$\\mu$ is a translation-bounded measure supported by $\\Lam$;\n\\item\n$\\gamma_\\mu$ is  an autocorrelation measure of $\\mu$; and\n\\item\nthe support $S$ of the diffraction\n\tmeasure $\\hat{\\gamma}_\\mu$ is a discrete closed set. \n\\end{enumerate-math}\nThen the same conclusion as in \\thmref{DT.1} is true.\n\\end{thm}\n\nThis can be deduced from either Theorem \\ref{RT.5} or \\ref{RT.7},\nusing the fact that the auto\\-correlation measure $\\gamma_\\mu$ is a positive-definite measure,\nsupported by the Meyer set $\\Lambda-\\Lambda$. \n\n\n\\subsection*{Remarks} \n{\\bf 1.}\\;\\;Similarly, one can prove a dichotomy result for the diffraction spectrum\nof a measure $\\mu$  supported by a Meyer set: either\nthe spectrum\nis uniformly discrete, or it has a relatively dense set of accumulation points\n(using Theorem \\ref{AT.5} or \\ref{CT.1}).\n\\par\n{\\bf 2.}\\;\\;In the latter case, the spectrum $S$ need not be dense in any ball.\nOne can verify that the measure constructed in the proof of \\thmref{ET.1}\nis an autocorrelation of another measure whose support is also a Meyer set.\n\n\n\\section{Non pure point spectrum}\n\n\\subsection{}\nIn crystallography it is often interesting to consider also discrete measures\n$\\mu$, whose Fourier transform $\\hat{\\mu}$ is a measure which has both a pure point\ncomponent and a continuous one. The pure point component is often referred to as ``Bragg\npeaks'', while the continuous component is called ``diffuse background''.\n\\par\n Let $\\mu$ be a\n(slowly increasing) measure on $\\R^n$ with discrete support $\\Lambda$: \n\\begin{equation}\n\t\\label{FE.1}\n\t\\mu = \\sum_{\\lambda\\in \\Lambda} \\mu(\\lambda)\\delta_\\lambda, \\quad \\mu(\\lambda)\\ne 0.\n\\end{equation}\nAssume that $\\hat{\\mu}$ is also a slowly increasing measure, and consider its decomposition \n\\begin{equation}\n\t\\label{FE.2.2}\n \\hat{\\mu} = \\hat{\\mu}_d+\\hat{\\mu}_c \n\\end{equation}\ninto a sum of a pure point measure\n\\begin{equation}\n\t\\label{FE.2}\n\t\\hat{\\mu}_d = \\sum_{s\\in S}\\hat{\\mu}_d(s)\\delta_s, \\quad \\hat{\\mu}_d(s)\\ne 0,\n\\end{equation}\nand a continuous measure $\\hat{\\mu}_c$. The set $S$ is the support of the discrete part $\\hat{\\mu}_d$.\n\\par\nWe can extend our previous results to this more general situation,\nusing the following result (\\thmref{RT.10}): If\n the support $\\Lambda$ is uniformly discrete, then $\\hat{\\mu}_d$ \n is the Fourier transform of another  measure\n\t$\\mu'$, whose support $\\Lambda'$ is also a uniformly discrete set.\n\n\n\n\\subsection*{Remark} \nWe will see from the proof that the new measure $\\mu'$ is a weak limit of \ntranslates of $\\mu$. Hence, in particular, the following is true:\n\\begin{enumerate-math}\n\\item \\label{FT.1.i} If $\\mu$ is a positive measure, then also $\\mu'$ is positive;\n\\item \\label{FT.1.ii}\n If $\\Lambda-\\Lambda$ is a discrete closed set, \n\tthen also $\\Lambda'-\\Lambda'$ is a discrete closed set; \n\\item \\label{FT.1.iii}\n If $\\Lambda-\\Lambda$ is uniformly discrete,\n\tthen also $\\Lambda'-\\Lambda'$ is uniformly discrete.\n\\end{enumerate-math}\nProperty \\ref{FT.1.i} is obvious.\nProperties \\ref{FT.1.ii} and \\ref{FT.1.iii} follow  from the\n fact that  $\\Lambda'-\\Lambda'$ must be contained in the closure of\nthe  set $\\Lambda-\\Lambda$.\n\n\n\n\\subsection{}\nFirst we give the proof of \\thmref{RT.10}. We will use the following lemmas:\n\\begin{lem}\n\t\\label{FL.1}\n\tLet $\\nu$ be a finite measure on $\\R^n$. Then \n\t\\[ \\lim_{R\\to\\infty} (2R)^{-n}\\int_{[-R,R]^n}\\left| \\hat{\\nu}(t)\\right|^2dt\n\t= \\sum_{a}\\left|\\nu (\\{a\\} )\\right|^2, \\]\n\twhere $a$ goes through all the atoms of the measure $\\nu$. \n\\end{lem} \nThis is the well-known Wiener's lemma in $\\R^n$. \n\\begin{lem}\n\t\\label{FL.2}\n\tLet $\\nu$ be a (slowly increasing) continuous measure on $\\R^n$. Then there exist\n\tvectors $\\omega_k\\in \\R^n$ $(k\\ge 1)$ such that the measures \n\t\\begin{equation}\n\t\t\\label{FL.2.1}\n\t\t\\nu_k(x):=e^{-2\\pi i\\dotprod{\\omega_k}{x}} \\,\\nu(x)\n\t\\end{equation}\n\ttend to zero as $k\\to\\infty$ in the space of temperate distributions. \n\\end{lem}\n\\begin{proof}\n\tLet $\\{\\varphi_j\\}$, $j\\geq 1$, be a sequence of functions dense in the Schwartz\n\tspace. Define \n\t\\[ \\Phi_k(t) := \\sum_{j=1}^{k}\\left| \\int \\varphi_j(x)e^{-2\\pi i\n\t\\dotprod{t}{x}}d\\nu(x)\\right|^2, \\quad t\\in \\R^n. \\]\n\tFor each $j$, the measure $\\varphi_j\\cdot \\nu$ is finite and continuous, hence\n\tby \\lemref{FL.1} we have\n\t\\[ \\lim_{R\\to\\infty}(2R)^{-n}\\int_{[-R,R]^n} \\Phi_k(t)dt = 0.  \\]\n\tThis implies that for each $k$ one can find $\\omega_k\\in\\R^n$ such that $\\Phi_k(\\omega_k)<\n\t1\/k^2$.\n\\par\n Now consider the measure $\\nu_k$ defined by \\eqref{FL.2.1}. We have\n\t\\[ \\left|\\dotprod{\\nu_k}{\\varphi_j}\\right| < \\frac{1}{k} \\quad (k\\ge j) \\]\n\tand hence $\\dotprod{\\nu_k}{\\varphi_j}\\to 0$ as $k\\to\\infty$, for each $j$.\nSince $\\nu$ is a slowly increasing measure, the sequence $\\nu_k$ is uniformly bounded\n  in the space of temperate distributions. Since the $\\varphi_j$ are dense in the\nSchwartz space, we can conclude that $\\dotprod{\\nu_k}{\\varphi}\\to 0$ as $k\\to\\infty$,\nfor every Schwartz function $\\varphi$. This proves the lemma.\n\\end{proof}\n\\begin{proof}[Proof of \\thmref{RT.10}]\n\tUse \\lemref{FL.2} to find vectors $\\omega_k$ such that \n\t\\[ e^{-2\\pi i \\dotprod{\\omega_k}{x}} \\,\\hat{\\mu}_c(x)\\to 0, \\quad k\\to \\infty \\]\n\tin the space of temperate distributions. By taking a subsequence if necessary we\n\tcan also assume  that the sequence $e^{2\\pi i \\dotprod{\\omega_k}{s}}$ has a limit as\n\t$k\\to\\infty$, for each $s\\in S$. Let $\\{k_j\\}$ be a sufficiently fast increasing sequence\n\tsuch that \n\t\\begin{equation}\n\t\t\\label{FT.1.0}\n\t e^{2\\pi i \\dotprod{\\omega_j-\\omega_{k_j}}{x}} \\, \\hat{\\mu}_c(x)\\to 0, \\quad\n\tj\\to\\infty. \n\t\\end{equation}\n\tSince the exponential in \\eqref{FT.1.0} tends to $1$ on $S$, \tit follows that the measure\n\t\\begin{equation}\n\t\t\\label{FT.1.1}\n\t\te^{2\\pi i \\dotprod{\\omega_j-\\omega_{k_j}}{x}} \\,\\hat{\\mu}(x) \n\t\\end{equation}\n\ttends to $\\hat{\\mu}_d$ as $j\\to \\infty$. \nThe measure in \\eqref{FT.1.1} is the\n\tFourier transform of the measure \n\t\\[ \\mu_j(t):= \\mu(t+\\omega_j-\\omega_{k_j}). \\]\n\tTherefore, $\\mu_j$ tends as $j\\to\\infty$ to a certain distribution $\\mu'$, whose\n\tFourier transform is $\\hat{\\mu}_d$. By \\lemref{AL.4} the measure $\\mu$ is\n\ttranslation-bounded, and therefore also $\\mu'$ is a translation-bounded measure.\n\tThe measure $\\mu_j$ is supported by the set\n\t\\[ \\Lambda_j := \\Lambda-\\omega_j+\\omega_{k_j}, \\]\n\twhich is  uniformly discrete  with $d(\\Lambda_j)=d(\\Lambda)$. Hence the support $\\Lambda'$\n\tof the measure $\\mu'$ must satisfy $d(\\Lambda')\\ge d(\\Lambda)$, so $\\Lam'$ is also a uniformly\n\tdiscrete set. \n\\end{proof}\n\\subsection{}\n\\thmref{RT.10} allows to extend our previous results to the case when the measure\n$\\hat{\\mu}$ has also a continuous component. For example, we have:\n\\begin{thm}\n\t\\label{FT.2}\n\tLet $\\mu$ be a measure on $\\R^n$ satisfying \\eqref{FE.1}--\\eqref{FE.2}. Assume\n\tthat $\\Lambda$ is uniformly discrete, $S$ is discrete and closed, and at least one\n\tof the following additional conditions is satisfied: \n\t\\begin{enumerate-math}\n\t\\item \\label{FT.2.i} $\\mu$ is positive, and $S$ is uniformly discrete; \n\t\\item \\label{FT.2.ii} $\\hat{\\mu}_d$ is positive; \n\t\\item \\label{FT.2.iiii} $n=1$, and $S$ satisfies condition \\eqref{RM.4};\n\t\\item \\label{FT.2.iv} $\\Lambda-\\Lambda$ is uniformly discrete. \n\t\\end{enumerate-math}\n\tThen $S$ is a uniformly discrete set, contained in a finite union of translates of\n\ta lattice, and the measure $\\hat{\\mu}_d$ has the form \\eqref{RM.3}.\n\\end{thm}\nTo prove this we consider the measure $\\mu'$ given by \\thmref{RT.10}, and apply to this\nmeasure one of Theorems \\ref{RT.0}, \\ref{RT.5}, \\ref{RT.6} or \\ref{RT.7}, according to which\none of the conditions \\ref{FT.2.i}--\\ref{FT.2.iv} in \\thmref{FT.2} is satisfied. \n\\par\nIn a similar way, one can extend the dichotomy results given in Theorems \\ref{AT.5} or \\ref{CT.1}.\n\\subsection{}\nThe same applies  to Hof's diffraction by  measures supported on  Meyer sets:\n\\begin{thm}\n\t\\label{FT.3}\n\tLet $\\mu$ be a translation-bounded measure on $\\R^n$, \n\tsupported by a  Meyer set $\\Lambda$. Let $\\gamma_\\mu$ be an autocorrelation measure of\n\t$\\mu$, and denote by $S$  the support of the discrete part of the diffraction measure $\\hat{\\gamma}_\\mu$. Then, either\n\t\\begin{enumerate-math}\n\t\\item $S$ is a uniformly discrete set, contained in a finite union of translates\n\t\tof a lattice, and the discrete part of $\\hat{\\gamma}_\\mu$ has the\n\t\tform \\eqref{RM.3}; or \n\t\\item $S$ is not a discrete closed set, and moreover the set of accumulation\n\t\tpoints of $S$ is relatively dense. \n\t\\end{enumerate-math}\n\\end{thm}\nTo prove this one can first apply \\thmref{RT.10} to the measure\n$\\gamma_\\mu$ which is supported by the Meyer set $\\Lam-\\Lam$,\nand then use Theorem \\ref{RT.0} and Theorem \\ref{AT.5} or \\ref{CT.1}.\n\n\\subsection*{Remark} \nAn alternative proof of \\thmref{FT.3}, which does not rely\non \\thmref{RT.10}, can be given using the following:\n\\begin{lem}\n\t\\label{FL.4}\n\tLet $\\nu$ be a translation-bounded measure on $\\R^n$, and assume that\n$\\ft\\nu$ is a slowly increasing measure. Then $\\nu$ has a unique autocorrelation measure\n$\\gamma_\\nu$, and the diffraction measure $\\hat{\\gamma}_\\nu$ is a pure point\nmeasure given by\n\\[\n\\ft{\\gamma}_\\nu \n\t= \\sum_{a}\\left|\\ft\\nu (\\{a\\} )\\right|^2 \\delta_a, \\]\n\twhere $a$ goes through all the atoms of the measure $\\ft\\nu$. \n\\end{lem}\nTo prove \\thmref{FT.3} using this lemma,  let $\\nu := \\gamma_\\mu$.\nThen the autocorrelation measure $\\gamma_\\nu$  is a discrete measure, supported by the Meyer \nset $\\Lam+\\Lam-\\Lam-\\Lam$, and \nby \\lemref{FL.4} the measure $\\hat{\\gamma}_\\nu$\nis a pure point measure, with the same support as the discrete\npart of $\\ft{\\gam}_\\mu$. So our previous results can be applied\nto the measure $\\gamma_\\nu$.\n\\par\nWe omit the proof of \\lemref{FL.4}.\n\n\n\n\n\\section{Remarks. Open problems}\n\n\\subsection{} \n Very recently, Y.~Meyer  has  found \\cite{mey4} an interesting version of \n\\thmref{RT.4}.      Namely, he  constructed  measures $\\mu$\n  whose supports and spectra are\n       discrete closed sets,  which can be described by simple effective  formulas. \n       He also proved   that the parameters of  this construction can be chosen so \nthat both $\\Lam$ and $S$ are rationally independent  sets.  This is  a  stronger ``non-periodicity''\n      condition  than  in \\thmref{RT.4}.  However,  such a  measure cannot be translation-bounded,\n      see \\cite[Lemma 5]{mey4}.\n    \n      It should be mentioned  that  the last paper  contains some other examples of\n      measures with discrete closed  supports and spectra.   See also \\cite{kol}.\n      All these examples,  in one way or another,  are based  on the classical Poisson\n summation formula. \nQuestion: can one construct an example  which in no way is based\non Poisson's formula?\n\n\n\\subsection{}\nWe mention some problems which are left open. \n\\par\n1.~The first one  concerns the positivity assumption in\n\\thmref{RT.0} in several dimensions.\nLet $\\mu$ be a measure on $\\R^n$, $n>1$, \nsatisfying \\eqref{RM.1} and \\eqref{RM.2}. Assume that $\\Lambda, S$ are both\nuniformly discrete sets. Is it true that $\\Lambda$ can be covered\nby a finite union of translates of several, not necessarily\ncommensurate, lattices?\n(an example in \\cite{fav} shows that $\\Lam$ need not be\ncontained in a finite union of translates of a \\emph{single} lattice).\n\\par\n2.~A second problem concerns the positive-definiteness assumption\nin \\thmref{RT.5}, even in  dimension one:\nLet $\\mu$ be a measure on $\\R$, with uniformly discrete support\n$\\Lam$ and discrete closed spectrum $S$. Does it follow that\n$S$ must be also uniformly discrete?\n\\par\n3.~The following question is also open: can one get a \\define{positive} measure in \\thmref{RT.4}\\,?\n\n\n\\subsection{}\nOur approach to prove \\thmref{RT.0} (see \\cite{lo2}) involved a combination\nof analytic and discrete combinatorial considerations. \nIn the latter part, a conclusion about the arithmetic structure\nof a set $\\Lambda\\subset \\R^n$ was derived from information on\ndiscreteness of $\\Lambda-\\Lambda$. In that point we relied on results\nwhich go back to Meyer \\cite{mey1}. \n\\par\nIn this context, it is worth to mention Freiman's theorem \\cite{freim},\nwhich states that a finite set\n$A$ such that $\\#(A+A)\\le K  \\# A$, must be contained in a\n``generalized arithmetic progression'' whose dimension and size\nare controlled in terms of the constant $K$. It might be\ninteresting to see whether it can also be used for a \nproof of \\thmref{RT.0}.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\\subsection{Pattern-forming systems}\nPeriodic, stripe-like patterns emerge in a self-organized fashion in a variety of experiments, ranging from Rayleigh-B\\'enard convection to open chemical reactors. Such regular, periodic patterns are usually studied in domains with idealized periodic boundary conditions, where existence and stability can be readily obtained using classical methods of bifurcation theory. The simplest example for such pattern-forming systems is the Swift-Hohenberg equation\n\\[\nu_t=-(\\Delta +1)^2 u +\\delta^2 u - u^3,\\qquad (x,y) \\in \\mathbb{R}^2,\n\\]\n\nwhich is known to possess periodic patterns of the form \n\\[\nu(x,y)=u_*(kx;k), u_*(\\xi+2\\pi;k)=u_*(\\xi;k),\n\\]\nwith $k\\sim 1$,\nfor  small $\\delta$. \n\nBeyond periodic boundary conditions, the dynamics for $\\delta\\sim 0$ can be approximated using the amplitude equation formalism. In the case of the (isotropic) Swift-Hohenberg equation, one finds the Newell-Whitehead-Segel equation\n\\[\nA_T=-(\\partial_X-\\mathrm{i}\\partial_{YY})^2 A + A-A|A|^2,\n\\]\nusing an Ansatz\n\\[\nu(x,y,t)=\\delta A(\\delta x,\\delta y, \\delta^2 t)\\mathrm{e}^{\\mathrm{i} x}+c.c.,\n\\]\n\nand expanding to order $\\delta^3$, as a solvability condition \\cite{cross1993pattern,mielke2002ginzburg}. There are several difficulties with the NWS equations and their validity as an approximation \\cite{schneider1995validity}, related to the fact that the original equation is isotropic, while the expansion singles out a preferred wave vector, here the vector $k=(1,0)^T$. The situation is simplified in anisotropic pattern-forming systems such as \n\\[\nu_t = -(\\Delta +1)^2u+\\partial_{yy}u +\\delta^2 u - u^3,\n\\]\nwhere a similar Ansatz leads to the isotropic (sic!) Ginzburg-Landau equation\n\\begin{equation}\\label{GL}\nA_T=\\Delta  A + A-A|A|^2,\n\\end{equation}\npossibly after rescaling $X$ and $Y$. \n\nMore drastically, one can approximate the dynamics near periodic patterns using an Ansatz\n\\[\nu(t,x,y)=u_*(\\Phi(\\delta x,\\delta y,\\delta^2 t);|\\nabla \\Phi|),\n\\]\nwhere one obtains as a compatibility condition the phase-diffusion equation \n\\[\n\\Phi_T=\\Delta_{X,Y}\\Phi,\n\\]\nfor suitable values of the wavenumber $|\\nabla \\Phi|$, after possibly rescaling $X,Y$. \n\nBoth, amplitude  and phase-diffusion equations can be shown to be good approximations under suitable choices of initial conditions, on time scales $T=\\mathrm{O}(1)$; see for instance \\cite{doelman2009dynamics,mielke2002ginzburg} and references therein.\n\n\\subsection{Inhomogeneities}\n\nLocal impurities in experiments sometimes have minor, sometimes more dramatic effects on the resulting patterns. It is known, for instance, that target-like patterns can nucleate at impurities in the Belousov-Zhabotinsky reaction; see \\cite{kollar2007coherent}  for an analysis in this direction. Also, spiral wave anchoring at impurities such as arteries can have dramatic impact on excitable media \\cite{munuzuri1998attraction}. Effects in Swift-Hohenberg-like systems appear to be more subtle and are the main focus of our present study. We focus on the somewhat simpler case of the isotropic GL equation \\eqref{GL}, modeling anisotropic pattern-forming systems, with an added localized inhomogeneity,\n\\begin{equation}\\label{e:cgli}\nA_T=\\Delta  A + A-A|A|^2+\\varepsilon g(x,y).\n\\end{equation}\n\nWe intend to study inhomogeneities in the isotropic SH equation \n\\[\nu_t=-(\\Delta +1)^2u+\\delta^2 u - u^3+\\varepsilon g(x,y),\n\\]\nin future work. \n\nIn order to illustrate the difficulties that arise, consider the more dramatic simplification of the phase-diffusion approximation with an inhomogeneity \\footnote{We stress however that one cannot derive such an approximation in the presence of inhomogeneities due to the different scalings of $\\Phi$ and $g$.},\n\\[\n\\Phi_t=\\Delta \\Phi+\\varepsilon g(x,y).\n\\]\nStationary patterns here are solutions to the Poisson equation $\\Delta\\Phi=-\\varepsilon g(x,y)$, with solutions exhibiting logarithmic growth at infinity. Thinking of the phase-diffusion as an approximation to a larger system, one would like a robust way of solving the stationary equation, if possible relying on an implicit function theorem. The difficulty here is that the Laplacian is not invertible on $L^2$, say, not even Fredholm as an unbounded operator. \n\nA remedy, in this context, are spaces with algebraic weights, which we refer to as Kondratiev spaces. To be precise, define $\\langle \\mathbf{x}\\rangle=\\sqrt{x^2+y^2+1}$ and  $M^{2,p}_{\\gamma}$ as the completion of $C_0^{\\infty}$ in the norm\n\\[\n\\|u\\|_{M^{2,p}_{\\gamma}}:=\\sum_{j+k= 2}\\|\\langle {\\bf  x} \\rangle^{\\gamma+2}\\partial^j_x\\partial^k_yu \\|_{L^p}\n+\n\\sum_{j+k= 1}\\| \\langle {\\bf x} \\rangle^{\\gamma+1}\\partial^j_x\\partial^k_yu\\|_{L^p}\n+\\| \\langle {\\bf x} \\rangle^{\\gamma} u\\|_{L^p}.\n\\]\nNote that the algebraic weights increase with the number of derivatives, making the different parts of the norms scale in the same fashion, as opposed to norms in the classical Sobolev spaces $W^{2,p}_{\\gamma}$, with norm \n\\[\n\\|u\\|_{W^{2,p}_\\gamma}:=\\sum_{j+k\\leq 2}\\|\\langle {\\bf x} \\rangle^{\\gamma} \\partial^j_x\\partial^k_yu \\|_{L^p}.\n\\]\nIt turns out that the Laplacian is Fredholm for suitable $\\gamma$ on $M^{2,p}_{\\gamma}$ \\cite{McOwen}, albeit with negative index in dimension 2 when $\\gamma>0$. This negative index makes ``explicit'' far-field corrections via logarithmic terms, just as seen in the Green's function of the Laplacian, necessary when describing the far-field effect of localized inhomogeneities on periodic patterns. This result also holds for a certain class of elliptic operators with coefficients that decay sufficiently fast at infinity \\cite{lockhart1981fredholm}, but we are not aware of results for elliptic operators without scaling invariance. This represents a difficulty when looking at the linearization of \\eqref{e:cgli}, which for $k=0$ decouples into $\\Delta$ which is, and $\\Delta -I$, which is not scale invariant. For $k \\neq 0$ these components couple and simple scale invariance is lost. Furthermore, the linearization is in general not a small or compact perturbation of a scale invariant operator.\n\n\n\n\n\n\n\n\\subsection{Main results}\n\nTo state our main results, we consider the stationary solutions of \\eqref{e:cgli},\n\\begin{equation}\\label{stationaryGL}\n0=\\Delta A + A - A|A|^2+\\varepsilon g(x,y).\n\\end{equation}\nFor $\\varepsilon=0$, the system possesses ``stripe patterns'' \n\\[\nA(x,y)=\\sqrt{1-k^2}\\mathrm{e}^{\\mathrm{i} k x},\n\\]\nfor wavenumbers $|k|<1$. Those solutions are linearly stable for $|k|<1\/\\sqrt{3}$ and unstable for $|k|>1\/\\sqrt{3}$. The instability mechanism is known as the Eckhaus (sideband) instability. We are now ready to state our main result.\n\n\\begin{thm}\\label{MainTh1}\n\nFix $k$ with $|k|<1\/\\sqrt{3}$ and suppose that $g\\in W^{2,2}_{\\beta}$ for some $\\beta>2$. Then there exists a family of solutions to \\eqref{stationaryGL},\n\\[\nA(x,y;\\varepsilon,\\varphi)=S(x,y;\\varepsilon,\\varphi)\\mathrm{e}^{\\mathrm{i}\\Phi(x,y;\\varepsilon,\\varphi)},\n\\]\nwith $A(x,y;0,\\varphi)=\\sqrt{1-k^2}\\mathrm{e}^{\\mathrm{i} (k x+\\varphi)}$. Moreover, $A(x,y;\\varepsilon,\\varphi)$ is smooth in all variables and satisfies the following expansions in $x,y$ for fixed $\\varepsilon,\\varphi$,\n\\begin{align}\nS(x,y;\\varepsilon,\\varphi)&\\to \\sqrt{1-k^2}\\\\\n\\Phi(x,y;\\varepsilon,\\varphi)-kx-\\frac{c(\\varepsilon,\\varphi)}{ 2k \\sqrt{1-k^2}}  \\log(\\alpha x^2+y^2)&\\to  \\Phi_\\infty(\\varepsilon)+\\varphi,\n\\end{align}\nfor $|\\mathbf{x}|\\to\\infty$, with $\\alpha=\\frac{1-k^2}{1-3k^2}$, for some smooth function $c(\\varepsilon,\\varphi)$ with expansion\n\\[\nc(\\varepsilon,\\varphi)=\\varepsilon c_1(\\varphi)+\\mathrm{O}(\\varepsilon^2),\n\\]\nwhere\n\\[\nc_1(\\varphi)=\\frac{\\sqrt{1-3k^2}}{\\pi(1-k^2)} \\iint \\Imag[g(x,y)\\mathrm{e}^{-\\mathrm{i}(kx+\\varphi)}]  \\mathrm{d} x \\mathrm{d} y.\n\\]\n\n\\end{thm}\n\n\n\\begin{rem}\n\n\\begin{enumerate}\n\\item Our approach gives more detailed expansions than stated. In fact, we obtain a decomposition of $S$ and $\\Phi$ into a localized part that is smooth in $\\varepsilon$, uniformly in $\\mathbf{x}$, and an explicit logarithmic far-field correction with coefficient $c(\\varepsilon,\\varphi)$; see the Ansatz \\eqref{Ansatz}. Also, in the class of functions with this particular form, the solutions described in the theorem are locally unique. \n\n\\item The expression for $c_1(\\varepsilon,\\varphi)$ is reminiscent of a projection onto the kernel. Indeed, the integral represents the scalar product $(u,v)=\\Real \\int u\\bar{v}$ of the perturbation $\\varepsilon g$ and the ``kernel'' of the linearization induced by the infinitesimal  phase rotation, $\\frac{\\mathrm{d}}{\\mathrm{d}\\varphi}\\mathrm{e}^{\\mathrm{i} {kx+\\varphi}}$. Vanishing of such a scalar product indicates persistence of solutions in problems where the linearization is Fredholm, such as in the Melnikov analysis for homoclinic bump-like solutions. \n\nIn fact,  $c_1$ possesses at least 2 zeroes. Assuming that $c_1(\\varphi_*)=0$, $c_1'(\\varphi_*)\\neq 0$, we can find $\\varphi_*(\\varepsilon)$ so that\n\\[\nc(\\varepsilon,\\varphi_*(\\varepsilon))=0.\n\\]\nInspection of the expansion in the theorem shows that $\\varphi$ is the phase shift of the underlying pattern in the far field. For these specific values of $\\varphi$, the correction in the far field to the periodic pattern is bounded and small, while for other values of $\\varphi$ the correction is unbounded in the phase. We interpret this result by referring to $\\varphi_*(\\varepsilon)$ as the \\emph{selected phase}. In other words, introducing inhomogeneities induces a selected phase shift of the primary pattern which accommodates stationary solutions without logarithmic corrections. Numerical simulations confirm this phenomenon, with a diffusive spread of the phase shift in the domain. It would be interesting to establish this diffusive convergence to a selected phase analytically.\n\n\\item Very similar results hold in space dimensions 1 and 3. In one space dimension, the analysis is easier since ODE methods can be used to analyze stationary solutions. In fact, the analysis reduces to a Melnikov analysis for the intersection of center-stable and center-unstable manifolds. In this one-dimensional context, the analysis also immediately carries over to the Swift-Hohenberg equation. On the other hand, the 3-dimensional case is easier than the two-dimensional case considered here since the corrections needed to compensate for negative Fredholm indices are decaying like $1\/|\\mathbf{x}|$. In fact, the Laplacian is invertible for suitable weights $\\gamma$ in Kondratiev spaces in $\\mathbb{R}^3$. \n\n\n\\end{enumerate}\n\n\\end{rem}\n\n\nThe structure of this paper is as follows. In Section \\ref{weightedspaces} we give a more detailed description of weighted Sobolev spaces and Kondratiev spaces, and state Fredholm properties of the Laplacian in this setting. Next, in Section \\ref{summary}, we summarize the procedure leading up to the main result, first explaining the difficulties encountered when analyzing the linearization of equation \\eqref{stationaryGL} about stripe patterns, and then describing the linear operator $T$ that we use to overcome these difficulties. In Section \\ref{sectionlinear} we use the results from Section \\ref{weightedspaces} to explore the Fredholm properties of this last operator and show that by adding logarithmic corrections we can obtain an invertible operator $\\hat{T}$. Section \\ref{sectionnonlinear} establishes properties of the nonlinearity in our functional analytic setting. We show that the full operator $\\hat{F}$  is  well defined, continuously differentiable, with invertible linearization\n$\\hat{T}$. Finally, in Section \\ref{Mainresult}, we prove our main result using the Implicit Function Theorem. \n\n\n\\section{Fredholm properties of the linearization and weighted spaces} \\label{weightedspaces}\n\nThis section is intended as a summary of theorems and results that describe weighted spaces and their properties. These results are the basis for the analysis in the following sections and will allow us to conclude Fredholm properties of the linearized operators considered therein.\n\n\n\\subsection{Weighted Sobolev spaces}\nThroughout the paper we will use the symbol $W^{s,p}_{\\gamma}$ to denote  weighted Sobolev spaces, which we define as the completion of $C^{\\infty}_0(\\mathbb{R}^n)$ under the norm\n\\[ \\| u\\|_{W^{s,p}_{\\gamma}} = \\sum_{|\\alpha|\\leq s} \\| \\langle {\\bf x} \\rangle^{\\gamma} D^{\\alpha} u \\|_{L^p}.\\]\n\nHere, $\\langle {\\bf x} \\rangle = ( 1 + |{\\bf x}|^2)^{1\/2}$, ${\\bf x}=(x,y)$, $\\gamma \\in \\mathbb{R}$, and $s$ is a positive integer. \n\nWe start with a generalization of the invertibility of the operator $\\Delta- I$ to weighted spaces.\n\n \n\\begin{prop}\\label{Sobolev}\nThe operator $\\Delta -a : W_{\\gamma}^{2,p} \\rightarrow L^p_{\\gamma}$ is invertible for all real numbers $a >0$ and $p \\in [1, \\infty)$. \n\\end{prop}\n\n\n\n\\begin{lem}\\label{notFredholm}\n The operator $\\Delta_{\\gamma}:W^{2,p}_{\\gamma} \\rightarrow L^p_{\\gamma}$ is not a Fredholm operator for $p \\in [1, \\infty)$.\n\\end{lem}\n\n\nBoth results follow in a straightforward fashion by noticing that $L^2_\\gamma$ and $L^2$ are conjugate by a multiplication operator, and the conjugate operator to $\\Delta$ is a compact perturbation of the Laplacian, which establishes Fredholm properties. \n\n\n\\subsection{ Kondratiev spaces}\n\nKondratiev spaces were first introduced to study boundary value problems for elliptic equations in domains with conical points \\cite{kondrat1967boundary}.  Nirenberg and Walker \\cite{nirenberg1973null} later used these spaces to show that  elliptic operators with coefficients that decay sufficiently fast at infinity have finite dimensional kernel when considered as operators between these weighted spaces and McOwen \\cite{McOwen} established  Fredholm properties for the Laplacian. Lockhart and McOwen \\cite{lockhart1981fredholm, lockhart1983elliptic, lockhart1985elliptic} built on these ideas to establish Fredholm properties for classes of elliptic operators. For example, Lockart \\cite{lockhart1985elliptic} studied elliptic operators of the form $A = A_{\\infty} +A_0$ in non-compact manifolds, where $A_{\\infty}$  represents a constant coefficient homogeneous elliptic operator of order $m$, and $A_0$ an operator of order at most $m$ with coefficients that decay fast at infinity. More recently, Kondratiev spaces \nwere used to study the Laplace operator in exterior domains \\cite{amrouche2008mixed} and similar weighted spaces were used in \\cite{milisic2013weighted} to understand the Poisson equation in a 1 periodic infinite strip $Z = [0,1] \\times \\mathbb{R}$. \n\nWe will denote Kondratiev spaces by $M^{s,p}_{\\gamma} $ and define them as the completion of $C^{\\infty}_0(\\mathbb{R}^n)$ under the norm\n\\[ \n\\| u\\|_{M^{s,p}_{\\gamma}} = \\sum_{|\\alpha|\\leq s} \\| \\langle {\\bf x} \\rangle^{\\gamma + |\\alpha|} D^{\\alpha} u \\|_{L^p},\n\\]\nwhere $\\langle {\\bf x} \\rangle = ( 1 + |{\\bf x}|^2)^{1\/2}$, $\\gamma \\in \\mathbb{R}$, $ s \\in \\mathbb{N}$, and $p \\in (1, \\infty)$. Notice the embeddings $M^{s,p}_{\\gamma} \\hookrightarrow W^{s,p}_{\\gamma}$, as well as $M^{s,p}_{\\gamma} \\hookrightarrow M^{s-1,p}_{\\gamma}$. \n\nThe following theorem describes the behavior of the Laplacian in Kondratiev spaces. Its proof can be found in \\cite{McOwen}.\n \n\\begin{thm}\\label{McOwen}\nLet $1<p=\\frac{q}{q-1}<\\infty$,  $n \\geq 2$, and $\\gamma \\neq -2 + n\/q + m $ or $\\gamma \\neq -n\/p -m $, for some $m \\in \\mathbb{N}$.  Then \n\\[ \n\\Delta: M^{2,p}_{\\gamma} \\rightarrow L^p_{\\gamma+2}, \n\\]\nis a Fredholm operator and\n\\begin{enumerate}\n\\item for $-n\/p < \\gamma < -2 + n\/q$ the map is an isomorphism;\n\\item for $-2 + n\/q + m < \\gamma< -2 + n\/q + m+1$ , $m \\in \\mathbb{N}$, the map is injective with closed range equal to \n\\[\nR_m = \\left\\{ f \\in L^p_{\\gamma +2} : \\int f(y)H(y) =0 \\  \\text{for all } \\ H \\in \\bigcup_{j=0}^m \\mathcal{H}_j\\right\\}; \n \\]\n\\item for $-n\/p - m -1 < \\gamma <-n\/p -m $, $m \\in \\mathbb{N}$, the map is surjective with kernel equal to \n\\[ N_m = \\bigcup_{j=0}^m \\mathcal{H}_j\n.\\]\n\\end{enumerate}\nHere,  $\\mathcal{H}_j $ denote the harmonic homogeneous polynomials of degree $j$.\n\nOn the other hand, if $\\gamma = -n\/p - m $ or $\\gamma = -2 + n\/q + m$ for some $m \\in \\mathbb{N}$, then $\\Delta$ does not have closed range.\n \\end{thm}\n\n\n\n\n\n\n\n\\section{Outline of proof}\\label{summary}\n\n\nRecall that we are interested in solutions of \\begin{equation}\\label{gl}\n0 = \\Delta A + A - A|A|^2 + \\varepsilon g(x,y),\n\\end{equation}\nfor localized $g$ and $\\varepsilon$ small, close to the solutions at $\\varepsilon=0$,  $A_*(x)= \\sqrt{1-k^2}\\mathrm{e}^{\\mathrm{i} kx}$,  $|k|^2 < 1\/3$. Since the case $k=0$ is in fact easier, we will assume in the following that $k>0$. A reasonable Ansatz then is\n\\[ \nA(x,y, \\varepsilon) =  (\\sqrt{1-k^2} + s(x,y;\\varepsilon) ) \\mathrm{e}^{\\mathrm{i}(kx + \\phi( x,y;\\varepsilon))},\n\\]\nwith new variables $s,\\phi$, which solve\n\\begin{align}\\label{amp1}\n\\Delta s + (s+\\tau) - ( s + \\tau)( k^2 + 2k \\partial_x \\phi + | \\nabla \\phi|^2)  - (s + \\tau)^3 + \\varepsilon \\Real( g \\mathrm{e}^{-\\mathrm{i}(kx + \\phi)})&=0\\\\\n\\Delta \\phi + \\frac{2k \\partial_x s}{s + \\tau} + \\frac{2 \\nabla s \\cdot \\nabla \\phi}{s +\\tau} + \\frac{ \\varepsilon \\Imag(g \\mathrm{e}^{-\\mathrm{i}(kx + \\phi)})}{s + \\tau}&=0,\\label{phase1}\n\\end{align}\nwhere we set $\\tau=\\sqrt{1-k^2}$. Linearizing at $\\varepsilon=0, s=0,\\phi=0$, we obtain the operator\n\\begin{equation}\\label{linearoperator}\n L \\begin{bmatrix} s\\\\ \\phi \\end{bmatrix} = \\begin{bmatrix} \n\\Delta  - 2\\tau^2 & -2k\\tau   \\partial_x  \\\\ \n \\dfrac{2k }{\\tau}  \\partial_x  & \\Delta  \n\\end{bmatrix} \\begin{bmatrix} s\\\\ \\phi \\end{bmatrix}.\n\\end{equation}\n\n\nThe results from the Section \\ref{weightedspaces}, and in particular Theorem \\ref{McOwen}, suggest that we should require that $\\phi\\in M^{2,p}_{\\gamma}$ and that \\eqref{phase1} holds in $L^p_{\\gamma+2}$. \nThen $\\phi_x\\in W^{1,p}_{\\gamma+1}$ and, using the linearization of \\eqref{amp1} with Proposition \\ref{Sobolev}, this suggests $s\\in W^{3,p}_{\\gamma+1}$ and $s_x\\in W^{2,p}_{\\gamma+1}$. This however is not sufficient localization for \\eqref{phase1}, where $s_x$ enters, and which we assumed to be satisfied in $L^p_{\\gamma+2}$. \n\nIn other words, the coupling terms, which are absent for $k=0$, prohibit the simple use of Sobolev spaces for $s$ and Kondratiev spaces for $\\phi$. Roughly speaking, the coupling destroys the linear scaling invariance in the $\\phi$-equation, which is necessary at least at infinity in results on Fredholm properties in Kondratiev spaces, which intrinsically mix regularity and localization properties. We intend to address these problems more generally in future but focus here on a simple an direct construction that circumvents the problem by extending the system and introducing appropriate norms for derivatives. \n\nConsider therefore  $ T: \\mathcal{D} \\subset \\mathcal{X}  \\rightarrow \\mathcal{R} \\subset \\mathcal{Y}$,\n\\begin{equation}\\label{extendedlinear}\n  T \\begin{bmatrix} s \\\\ \\psi \\\\ \\theta \\\\ u \\\\ v\\\\ w \\end{bmatrix} = \n \\begin{bmatrix}\n \\Delta -a & -1 & 0 & 0 & 0 & 0 \\\\\n 0 & \\Delta & 0 & b & 0 & 0\\\\\n 0 & 0 & \\Delta & 0 & b & 0\\\\\n 0 & -\\partial_{xx} & 0 & \\Delta -a & 0 & 0 \\\\\n 0 & 0 & -\\partial_{xx} & 0 & \\Delta -a & 0 \\\\\n 0 & -\\partial_{yy} & 0 & 0 & 0 & \\Delta -a \n \\end{bmatrix}\\begin{bmatrix} s \\\\ \\psi \\\\ \\theta \\\\ u \\\\ v\\\\ w \\end{bmatrix},\n\\end{equation}\nwere \\hskip0.3cm $\\psi = 2k \\tau\\partial_x \\phi$,\\hskip0.3cm $\\theta= 2k \\tau \\partial_y \\phi$,\\hskip0.3cm $a = 2\\tau^2$,\\hskip0.3cm $b = 4k^2$, \n\\[\n\\mathcal{X} = W^{2,2}_{\\gamma} \\times M^{2,p}_{\\gamma} \\times M^{2,p}_{\\gamma} \\times L^p_{\\gamma+2}\\times L^p_{\\gamma+2}\\times L^p_{\\gamma+2},\\] \n\\[ \\mathcal{Y} = L^p_{\\gamma} \\times L^p_{\\gamma+2} \\times L^p_{\\gamma+2} \\times W^{-2,p}_{\\gamma+2}\\times W^{-2,p}_{\\gamma+2}\\times W^{-2,p}_{\\gamma+2}.\n\\]\nHere, $W^{-k,p}_\\gamma$ denotes the dual of $W^{k,p}_\\gamma$. We also define the closed subspaces\n \\[ \\mathcal{D} = \\left \\{ (s, \\psi, \\phi, u,v,w) \\in \\mathcal{X}: u = \\partial_{xx} s, \\quad v= \\partial_{xy} s,\\quad w = \\partial_{yy} s, \\quad \\partial_y \\psi = \\partial_x \\theta \\right \\}, \\]\n\\begin{align*}\n \\mathcal{R} & = \\left \\{ y=(f_1,f_2,f_3,f_4,f_5,f_6) \\in \\mathcal{Y}: \\int f_2 = \\int f_2 \\cdot y= \\int f_3=  \\int f_3 \\cdot x=0, \\right. \\\\\n& \\quad \\left.f_4 = \\partial_{xx} f_1,  f_5= \\partial_{xy} f_1,  f_6 = \\partial_{yy}f_1, \\quad \\text{and} \\quad \\partial_y f_2 = \\partial_x f_3   \\right \\}.\n\\end{align*}\nThe second and third equation in (\\ref{extendedlinear}) are obtained by taking the $x$ and $y$ derivatives of the phase equation. The last three equations come from taking the second order partial derivatives of the amplitude equation with respect to $xx$, $xy$, and $yy$.\n \n We will see in Section \\ref{sectionlinear} that the linear operator $T: \\mathcal{D} \\rightarrow \\mathcal{R}$ is a Fredholm operator of index $-1$ for optimal choices of weights, indicating a missing parameter in the far field. We therefore add a single degree of freedom in the far field through the variable $\\hat{c}\\in\\mathbb{R}$ via the Ansatz  \n\\[\n\\begin{array}{l c l c c c l c l\ns & =& \\hat{s} + \\hat{c} P_1,  & & &  &u &=& \\hat{u} + \\hat{c} \\partial_{xx} P_1,\\\\\n\\psi & =& \\hat{\\psi} + \\hat{c} \\partial_x P_2,  & & &  &v &=& \\hat{v} + \\hat{c} \\partial_{xy} P_1,\\\\\n\\theta & =& \\hat{\\theta} + \\hat{c}\\partial_y P_2,  & & &  &w &=& \\hat{w} + \\hat{c} \\partial_{yy} P_1,\n\\end{array}\n\\]\nwhere \n\\begin{equation}\\label{e:p1p2}\nP_1= \\dfrac{1- \\alpha}{2 b \\alpha}\\partial_x[\\chi \\log(\\alpha x^2+y^2)], \\qquad P_2 = \\dfrac{1}{2}\\chi \\log(\\alpha x^2+y^2),\n\\end{equation}\n$b = (2k)^2$, $\\alpha = \\dfrac{1-k^2}{1-3k^2}$, and $\\chi$ is a smoothed version of the indicator function $\\chi_{|\\mathbf{x}|>1}$. Substituting this Ansatz into \\eqref{amp1},\\eqref{phase1} and linearizing, we find an operator $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$, given by \n\\begin{equation}\\label{newT} \\hat{T} \\xi =\n  \\begin{bmatrix} \n \n    \\Delta -a & -1 &  & & &  &\\Delta P_1  \\\\\n & \\Delta & &b & & &\\Delta   P_2 +b \\partial_{xx}P_1  \\\\\n & &\\Delta &  & b& &  \\Delta  P_3 + b \\partial_{xy} P_1 \\\\\n & - \\partial_{xx} &  & \\Delta -a & &  & \\Delta  \\partial_{xx} P_1\\\\\n &  &- \\partial_{xx}  & & \\Delta -a  &  & \\Delta \\partial_{xy} P_1    \\\\\n& - \\partial_{yy} &  &  &  &\\Delta -a &  \\Delta  \\partial_{yy} P_1 \\\\\n \\end{bmatrix} \\begin{bmatrix} \\hat{s} \\\\ \\hat{\\psi} \\\\ \\hat{\\theta} \\\\ \\hat{u} \\\\ \\hat{v} \\\\ \\hat{w} \\\\ \\hat{c}    \\end{bmatrix},\n \\end{equation}\nwhere  again $a = 2\\tau^2$, and $b = 4k^2$. We will show in the Section \\ref{sectionlinear} that this operator is invertible. \n\nRecapitulating, we are lead to consider the Ansatz\n\\begin{equation}\\label{Ansatz}\nA(x,y; \\varepsilon, \\varphi) = \\left(\\sqrt{1 -k^2} + s(x,y;\\varepsilon, \\varphi)+ c(\\varepsilon, \\varphi) P_1(x,y)\\right) \\mathrm{e}^{ \\mathrm{i} \\left (kx +  \\phi(x,y;\\varepsilon, \\varphi)+  \\frac{c(\\varepsilon, \\varphi)}{2k \\sqrt{1-k^2}} P_2(x,y) \\right)}, \n\\end{equation}\nwith $P_1$ and $P_2$ as in \\eqref{e:p1p2}, with additional equations for the derivatives \n\\[ \nu = \\partial_{xx} s, \\quad v = \\partial_{xy} s, \\quad w = \\partial_{yy} s, \\quad \\psi = (2k)  \\partial_{x} \\phi,\\quad \\theta = (2k)  \\partial_y \\phi.\n\\]\nWe obtain a nonlinear equation  \n\\begin{equation}\\label{nl0}\n\\hat{F}_{\\varepsilon, \\varphi}=0, \\qquad \\hat{F}_{\\varepsilon, \\varphi} : \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow \\mathcal{R};\n\\end{equation}\nsee Subsection \\ref{sectionnonlinear} for a more detailed description of this nonlinear equation. The advantage of this subtle reformulation of the problem is encoded in the following result, which establishes applicability of the standard Implicit Function Theorem and is the key ingredient to the proof of Theorem \\ref{MainTh1}. \n\n\n\n\\begin{thm}\\label{MainTh}\nLet $p=2$, $\\gamma \\in (0,1)$, and $g \\in W^{2,p}_{\\beta}$, with $\\beta > \\gamma+2$. Then, the operator $\\hat{F}_{\\varepsilon,\\varphi}: \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow \\mathcal{R}$  is of class $C^\\infty$. Furthermore, for fixed $\\varphi$ and at $\\varepsilon =0$, its derivative is given by the invertible operator $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$.\n\\end{thm}\n\nThe proof of this theorem will occupy the next 2 Sections. In Section \\ref{sectionlinear} we show the fact that $T$ is a Fredholm operator and that $\\hat{T}$ is invertible, and in Section \\ref{sectionnonlinear} we show that the operator $\\hat{F}$ is of class $C^{\\infty}$.\n\n\n\n\n\n\n \n\\section{The linear operator}\\label{sectionlinear}\nIn this section we consider the linear operators $T: \\mathcal{D} \\rightarrow \\mathcal{R}$ and $\\hat{T}:\\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$ defined in (\\ref{extendedlinear}) and (\\ref{newT}), respectively. We first prove that for $p=2$ and $\\gamma \\in (0,1)$ the operator $T: \\mathcal{X} \\rightarrow \\mathcal{Y} $ is a Fredholm operator of index $-6$.  Then, we show  that restricting the domain and range to $\\mathcal{D}$ and $\\mathcal{R}$ turns $T$ into a Fredholm operator of index $-1$. Finally, using a bordering lemma, we prove  that the operator $\\hat{T}$ is invertible. Throughout, we assume $ \\gamma \\in (0, 1)$ and $0<|k|< \\dfrac{1}{\\sqrt{3}}$.\n\n\\begin{prop}\\label{FredholmT1} \\\nThe operator $T:\\mathcal{X} \\rightarrow \\mathcal{Y}$ defined in (\\ref{extendedlinear}) is a Fredholm operator with index $i=-6$ and trivial kernel. The cokernel is spanned by \n\\begin{equation}\\label{sol}\n\\{(0,a,0,b,0,0)^Te_j^*, (0,0,a,0,b,0)^Te_j^*, \\, j=1,2,3\\},\\qquad e_1^*=1,e^*_2=x,e^*_3=y.\n\\end{equation}\n\n\\end{prop}\n\n\\begin{proof} \nFirst, notice that due to the lower block-triangular structure of $T$, it is enough to consider the restriction $\\tilde{T}$ to the variables $\\psi, \\theta, u$, $v$, which we write in the form \n\\begin{equation}\\label{Perturbed} \n\\tilde{T}\\xi=  L \\xi+b  B\\xi, \n\\end{equation}\nwhere $b = 4k^2$, $a = 2(1-k^2)$, \n\\[ \nL=  \\begin{bmatrix}  \\Delta  & 0 &0&0 \\\\ 0& \\Delta&0 &0\\\\  - \\partial_{xx} &0& \\Delta -a&0 \\\\  0 &-\\partial_{xx} &0 &\\Delta -a \\end{bmatrix}, \\qquad\nB = \\begin{bmatrix} 0 &0 & I& 0 \\\\ 0 & 0 &0 &I\\\\ 0&0&0&0\\\\0&0&0&0 \\end{bmatrix}, \\qquad\n\\text{and} \\quad\\xi = \\begin{bmatrix} \\psi \\\\ \\theta \\\\ u \\\\ v \\end{bmatrix}. \n\\]\nWe need to show that  \n\\[\n\\tilde{T}:   \\tilde{\\mathcal{X}} = M^{2,2}_{\\gamma} \\times M^{2,2}_{\\gamma} \\times L^2_{\\gamma+2} \\times L^2_{\\gamma+2}\\longrightarrow\\tilde{\\mathcal{Y}} = L^2_{\\gamma+2} \\times L^2_{\\gamma+2} \\times W^{-2,2}_{\\gamma+2} \\times W^{-2,2}_{\\gamma+2},\n\\]\nis a Fredholm operator of index $i =-6$. Since $\\tilde{T}$ is block-diagonal with respect to $(\\psi,u)$ and $(\\theta,v)$, it is sufficient to show that the restriction to $(\\psi,u)$ is Fredholm with index $-3$. For $b=0$, the claim now follows directly from Theorem \\ref{McOwen} and Proposition \\ref{Sobolev}, due to the lower triangular structure and the fact that, with our choice of $\\gamma,p$, the Laplacian is Fredholm with index $-3$. We will address the more difficult situation $b\\neq 0$, next.\n\nIn order to establish the desired Fredholm properties, we need to solve \n\\begin{align}\\label{e1}\n \\Delta \\psi + b u & = f_1\\\\ \\label{e2}\n  \\partial_{xx} \\psi + (\\Delta -a) u  &= f_2,\n\\end{align}\nfor $f_1,f_2$ in a codimension-3 subspace of $L^2_{\\gamma+2} \\times W^{-2,2}_{\\gamma+2}$, with bounds on $(\\psi,u)\\in M^{2,2}_{\\gamma} \\times L^2_{\\gamma+2}$. \n\n\nDenote by $I-Q$ a projection on the range of the Laplacian, so that $\\int (I-Q) f=\\int x  (I-Q) f=\\int y  (I-Q) f =0$. We can then decompose \n\\begin{align}\\label{e3}\n \\Delta \\psi + b u & = (I-Q)f_1\\\\ \\label{e4}\n  \\partial_{xx} \\psi + (\\Delta -a) u  &= (I-Q)f_2,\n\\end{align}\nand $Q u=\\frac{1}{b} Q f_1= - \\frac{1}{a}Q f_2$, exhibiting the 3 solvability conditions \\eqref{sol}. We next solve \\eqref{e4} for $u$, substitute in \\eqref{e3}, to obtain\n\\[\n\\mathcal{L} \\psi=(I-Q)f_1-(\\Delta-a)^{-1}(I-Q)f_2=:f,\\qquad \\mathcal{L}=[\\Delta + b (\\Delta -a)^{-1} \\partial_{xx}],\n\\]\nwhere $f=(I-Q)f$. It therefore remains to show that $\\mathcal{L}:M^{2,2}_{\\gamma}\\to (I-Q)L^2_{\\gamma+2}$ is invertible.\nWe therefore factor \n\\[ \\mathcal{L} \\psi = \\mathcal{M} \\left( \\Delta - \\frac{b}{a} \\partial_{xx}\\right) \\psi, \\]\n\\begin{center}\n\\begin{tikzpicture} \n\\matrix(m)[matrix of math nodes,\n row sep=3.5em, column sep=5.5em,\n  text height=2.5ex, text depth=0.25ex]\n   {M^{2,2}_{\\gamma}&(I-Q) L^2_{\\gamma+ 2}&(I-Q) L^2_{\\gamma+2} \\\\};\n    \\path[->,font=\\footnotesize,>=angle 90] \n    (m-1-1) edge node[auto] {$\\Delta - \\dfrac{b}{a} \\partial_{xx}$} (m-1-2)\n\t (m-1-2) edge node[auto] {$\\mathcal{M}$} (m-1-3); \n\\end{tikzpicture}\n\\end{center}\nBy Theorem \\ref{McOwen} the operator $\\left(\\Delta - \\dfrac{b}{a} \\partial_{xx} \\right) : M^{2,2}_{\\gamma} \\rightarrow R_m$ is invertible, since it is conjugate to the Laplacian by a simple $x$-rescaling operator. It is therefore sufficient to establish that $\\mathcal{M}$ is an isomorphism of $(I-Q)L^2_{\\gamma+2}$. \nConsider therefore the associated Fourier symbol \n\\[\n\\hat{\\mathcal{M}}(k,l) = \\frac{k^2+l^2}{k^2+l^2 -\\frac{b}{a} k^2} - \\frac{bk^2}{(k^2+l^2+a) ( k^2+l^2 - \\frac{b}{a}k^2)}.\n\\]\nExploiting that $k^2<\\frac{1}{3}$ so that  $1-\\dfrac{b}{a} >0$, it is straightforward to see that \n\\[\n\\sup_{(k,l) \\in \\mathbb{R}^2} |\\hat{\\mathcal{M}}(k,l)|+|\\hat{\\mathcal{M}}(k,l)^{-1}|<\\infty,\n\\]\nso that $\\mathcal{M}$ is an isomorphism of $L^2$. We next show that $\\mathcal{M}$ is an isomorphism on $(I-Q)L^2_j$, $j=2,3$, which by interpolation theory gives the desired result. Equivalently, we need to show boundedness of the multiplication operator $\\hat{\\mathcal{M}}$ on the subspace of  $H^j$, $j=2,3$ consisting of functions functions $f$ with $f(0)=0$ and, in case $j=3$, $\\nabla f(0)=0$. Since $\\hat{\\mathcal{M}}(k,l)=a+\\mathrm{O}(k^2+l^2)$ near $k=l=0$, we readily find that  $\\| D^{\\alpha} \\hat{\\mathcal{M} } |_{L^{\\infty} }+\\| D^{\\alpha}( \\hat{\\mathcal{M} }^{-1}) |_{L^{\\infty} }< \\infty$ for all indices $|\\alpha|\\leq 2$, which proves that $\\hat\\mathcal{M}$ is an isomorphism on $H^2$. For the case $j=3$, we use that $\\| D^{\\alpha} ( {\\bf k} \\hat{\\mathcal{M} }) |_{L^{\\infty} }+\\| D^{\\alpha}( {\\bf k}  \\hat{\\mathcal{M} }^{-1}) |_{L^{\\infty} }< \\infty$ for all indices $|\\alpha|=3$, which readily implies that $\\hat{\\mathcal{M}}$ is an isomorphism on $H^3\\cap\\{f(0)=0\\}$. One can also readily check that the range of $\\mathcal{M}$ and $\\mathcal{M}^{-1}$ is indeed contained in $\\mathrm{Rg}\\,(I-Q)$, which concludes the proof.\n\\end{proof}\n\n\n\\begin{cor}\\label{FredholmT2}\nThe operator  $T: \\mathcal{D} \\rightarrow \\mathcal{R}$ defined by (\\ref{extendedlinear}) is Fredholm with index $-1$ and cokernel $\\partial_y f_2+\\partial_x f_3$.\n\\end{cor}\n\n\\begin{proof} \nInspection of $T$ shows that the range of the restriction of $T$ to $\\mathcal{D}$ is actually contained in $\\mathcal{R}$, which implies that $T:\\mathcal{D}\\to\\mathcal{R}$ is injective and the range is closed ($T$ is semi-Fredholm). We need to show that the cokernel is one-dimensional. \n\nTake $f\\in \\mathrm{Rg}\\,(T)\\subset \\mathcal{X}$, with $f_4 = \\partial_{xx} f_1,\\,  f_5= \\partial_{xy} f_1, \\, f_6 = \\partial_{yy}f_1$, and $\\partial_y f_2 = \\partial_x f_3$. \nBy construction of $T$, notably having taken derivatives of equations for $s$ and $\\phi$, and by injectivity, $T^{-1}f$ satisfies\n$u = \\partial_{xx} s, \\,v= \\partial_{xy} s$, $w = \\partial_{yy} s$, and $\\partial_y \\psi= \\partial_x \\theta$. As a consequence, the cokernel of $T:\\mathcal{D}\\to\\mathcal{R}$ is a subset of the cokernel of $T:\\mathcal{X}\\to\\mathcal{Y}$. Inspecting the cokernel in \\eqref{sol} and the definition of $\\mathcal{R}$, we see that $(e_j^*,f_4)=(e_j^*,f_5)=0$, so that the integral conditions in the definition of $\\mathcal{R}$ represent precisely 4 of the 6 conditions on the co-kernel. One of the remaining conditions, $\\int f_2\\cdot x=\\int f_3\\cdot y$ is a consequence of $\\partial_y f_2=\\partial_x f_3$, whereas $\\int f_2\\cdot x$ can be readily seen to act non-trivially. As a consequence $T$ is Fredholm of index -1 as claimed, and the cokernel is spanned by $(0,x,0,0,0,0)^T$ or, equivalently,  $(0,x,y,0,0,0)^T$.\n\\end{proof}\n\nWe  next consider the operator $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$ defined by (\\ref{newT}). Recall that\n\\begin{equation}\\label{changevariables}\n\\begin{array}{l c l c c c l c l\ns & =& \\hat{s} + \\hat{c} P_1,  & & &  &u &=& \\hat{u} + \\hat{c} \\partial_{xx} P_1,\\\\\n\\psi & =& \\hat{\\psi} + \\hat{c} P_2,  & & &  &v &=& \\hat{v} + \\hat{c} \\partial_{xy} P_1,\\\\\n\\theta & =& \\hat{\\theta} + \\hat{c} P_3,  & & &  &w &=& \\hat{w} + \\hat{c} \\partial_{yy} P_1.\n\\end{array}\n\\end{equation}\n\n\nHere, \n\\[P_1(x,y) =  \\dfrac{ (1-\\alpha)}{2b\\alpha} \\partial_x [\\chi \\ln( \\alpha x^2+y^2)],\\quad\n  P_2(x,y)  =  \\dfrac{1}{2} \\partial_x [ \\chi \\ln( \\alpha x^2+y^2)],\\quad\n   P_3(x,y) =  \\dfrac{1}{2} \\partial_y [ \\chi \\ln( \\alpha x^2+y^2) ],\n\\]\n\n with $\\alpha  = \\dfrac{1-k^2}{1-3k^2}$, \\hskip0.4cm $b= (2k)^2$, \\hskip0.4cm and  \\hskip0.4cm $\\chi \\in C^{\\infty}(\\mathbb{R}^2)$ defined by\n \\[\\chi(x,y) = \\left\\{ \\begin{array}{c c l} 0 & \\text{if} & 0 \\leq \\sqrt{\\alpha x^2+y^2} \\leq 1\/2\\\\\n \t\t\t\t\t\t1 & \\text{if} & 1 \\leq \\sqrt{\\alpha x^2+y^2}\n\t\t\t\t\t\t\\end{array} \\right. .\\]\n \n\nTo show $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$ is invertible we will need the following lemma.\n\n\n\\begin{lem}\\label{extraM}\n\nThe operator \n\\[\nM: \\mathbb{R} \\rightarrow \\mathcal{R},\\qquad  c \\mapsto \\begin{bmatrix}  \\Delta  P_1&\n\t\t\t\t\t\t\t\\Delta    P_2 +b \\partial_{xx} P_1  &\n\t\t\t\t\t\t\t \\Delta  P_3 + b \\partial_{xy} P_1 &\n\t\t\t\t\t\t\t \\Delta \\partial_{xx} P_1&\n\t\t\t\t\t\t\t \\Delta \\partial_{xy} P_1&\n\t\t\t\t\t\t\t \\Delta \\partial_{yy} P_1\n\t\t\t\t\t\t\t \t\t\t\t\t\t\t \\end{bmatrix}^T c,\n\\]\nis well-defined and its range satisfies \n\\[ \\iint \\Big [\\Delta P_2 +b \\partial_{xx} P_1 \\Big ] \\cdot x \\mathrm{d} x\\mathrm{d} y= \\iint \\Big [ \\Delta P_3 +b \\partial_{xy}P_1 \\Big ] \\cdot y\\mathrm{d} x\\mathrm{d} y \\neq 0 . \\]\n\\end{lem}\n\n\\begin{proof} First notice that the smooth functions $P_i$, for $i = 1,2,3$, are bounded in compact sets and behave like $\\dfrac{1}{|{\\bf x}|}$ for large values of $|{\\bf x} |$ so that the range of $M$ is indeed a subset of $\\mathcal{Y}$. From the definition, it is not difficult to check that the operator $M$ maps into the desired space $\\mathcal{R}$.  We need to show that\n\\[ \\iint \\Big [\\Delta P_2 +b \\partial_{xx} P_1 \\Big ] \\cdot x\\mathrm{d} x\\mathrm{d} y = \\iint \\Big [ \\Delta P_3 +b \\partial_{xy}P_1 \\Big ] \\cdot y\\mathrm{d} x\\mathrm{d} y \\neq 0 . \\]\n\nStraightforward calculations, using the rescaling $X = \\sqrt{\\alpha} x , Y = y$, show that \n\\[ \\Delta  P_2+ b \\partial_{xx}  P_1 = \\frac{\\sqrt{\\alpha}}{2} \\Delta_{X,Y} \\left [ \\frac{\\partial}{\\partial X} \\left ( \\chi \\cdot \\ln ( X^2+Y^2)\\right)  \\right ],\\]\nwhere we write $\\Delta_{X,Y} = \\partial_{XX} + \\partial_{YY}$. Therefore,\n\\begin{align*}\n \\iint [\\Delta  P_2+ b \\partial_{xx}  P_1 ] \\cdot x  \\mathrm{d} x \\mathrm{d} y &= \\iint \\left[  \\frac{\\sqrt{\\alpha}}{2} \\Delta_{X,Y} \\left [ \\frac{\\partial}{\\partial X} \\left ( \\chi \\cdot \\ln ( X^2+Y^2)\\right)  \\right ] \\right] \\cdot X  dX dY  \\\\\n &= \\iint \\left[  \\frac{\\sqrt{\\alpha}}{2} \\Delta_{X,Y}  \\left ( \\chi \\cdot \\ln ( X^2+Y^2)\\right)  \\right]   dX dY  \\\\\n &= \\sqrt{\\alpha} \\pi .\n \\end{align*}\n\nSimilarly, using the same rescaling, it can be shown that\n\\[\n \\Delta  P_3+ b \\partial_{xy} P_1 = \\frac{1}{2} \\Delta_{X,Y} \\left[ \\frac{\\partial}{\\partial Y} \\left( \\chi \\cdot \\ln (X^2+Y^2)  \\right)  \\right],\\]\nand consequently\n\\begin{align*}\n\\iint \\left[  \\Delta P_3+ b \\partial_{xy}  P_1 \\right] \\cdot y \\mathrm{d} x\\mathrm{d} y &= \\iint \\left[ \\frac{1}{2} \\Delta_{X,Y} \\left[ \\frac{\\partial}{\\partial Y} \\left( \\chi \\cdot \\ln (X^2+Y^2)  \\right)  \\right] \\right] \\cdot Y \\sqrt{\\alpha} dXdY \\\\\n&= \\sqrt{\\alpha} \\pi.\n\\end{align*}\n\\end{proof}\n\n\\begin{cor}\\label{FredholmT3}\nThe operator $\\hat{T}:\\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$ is invertible.\n\\end{cor}\n\n\\begin{proof} Notice that $\\hat{T}= [ T  M]$, where $T: \\mathcal{D} \\rightarrow \\mathcal{R}$ is the Fredholm operator of index $-1$ described in Corollary \\ref{FredholmT2}, and  $M: \\mathbb{R} \\rightarrow \\mathcal{R}$ is defined in Lemma \\ref{extraM}. A bordering lemma implies that $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$ is a Fredholm operator of index $0$. Lemma \\ref{extraM} implies that $\\mathrm{Rg}\\,(M)\\not\\subset\\mathrm{Rg}\\,(T)$, so that $\\hat{T}$ is onto, hence invertible.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{The nonlinear map}\\label{sectionnonlinear}\n\nIn this section, we show by a series of lemmas that the nonlinear problem \\eqref{nl0} is well defined and continuously differentiable. We first give explicit expressions for each component of  the nonlinearities. We then state and prove several lemmas that will help us show that the nonlinearity is well defined. Finally, we show that the nonlinearities are of class $C^{\\infty}$.\n\n\n The following expressions represent each component of the operator $\\hat{F}_{\\varepsilon, \\varphi}$ announced in \\eqref{nl0}:\n\\begin{align*}\n \\hat{F}_1 (\\xi;\\varepsilon, \\varphi) =& \\Delta s - 2\\tau^2 s - (s + \\tau) \\left( \\frac{\\psi}{\\tau} + \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)\\right) - (s^3 + 3s^2 \\tau)+ \\varepsilon \\Real[ g \\mathrm{e}^{-\\mathrm{i} (kx +\\phi(\\varphi))}],\\\\\n\\hat{F}_2 (\\xi;\\varepsilon, \\varphi) =& \\Delta \\psi + \\frac{(2k)^2\\tau u + 2( u \\psi + v \\theta + s_x \\psi_x + s_y \\psi_y) }{s+ \\tau} -\\frac{(2 k)^2 \\tau(s_x)^2 + 2s_x( s_x \\psi +s_y \\theta)}{ (s+\\tau)^2} \\\\\n &+ \\frac{\\varepsilon \\Imag[ \\partial_x( g \\mathrm{e}^{-\\mathrm{i} (kx +\\phi(\\varphi))})]}{s+ \\tau} -\\frac{\\varepsilon s_x \\Imag[g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))}] }{(s+ \\tau)^2},\n\\\\\n\\hat{F}_3 (\\xi;\\varepsilon, \\varphi) =& \\Delta \\theta + \\frac{(2k)^2 \\tau v + 2( v \\psi + w \\theta + s_x \\theta_x +s_y \\theta_y) }{s + \\tau} - \\frac{(sk)^2 \\tau s_xs_y + 2s_y(s_x \\psi + s_y \\theta)}{(s+\\tau)^2} \\\\\n &+ \\frac{ \\varepsilon \\Imag[ \\partial_y( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})]}{s+\\tau} - \\frac{ \\varepsilon s_y \\Imag[ g \\mathrm{e}^{-\\mathrm{i}(kx + \\phi(\\varphi))}]}{(s+ \\tau)^2},\n\\\\\n\\hat{F}_4 (\\xi;\\varepsilon, \\varphi) =& \\Delta u - 2 \\tau^2 u -u\\left( \\frac{\\psi}{\\tau} + \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)\\right)  -2 s_x \\left( \\frac{\\psi_x}{\\tau} + \\frac{2}{(2k\\tau)^2} (\\psi_x \\psi + \\psi_y \\theta) \\right) \\\\\n& -(s + \\tau) \\left( \\frac{\\psi_{xx}}{\\tau} + \\frac{2}{(2k\\tau)^2} \\left( \\psi_{xx} \\psi + \\psi_{xy} \\theta + | \\nabla \\psi|^2 \\right) \\right)\\\\\n& - ( 6s(s_x)^2 + 3s^2 u + 6 \\tau (s_x)^2 + 6\\tau s u) + \\varepsilon \\Real[ \\partial_{xx}( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})],\n\\\\\n\\hat{F}_5 (\\xi;\\varepsilon, \\varphi) =& \\Delta v - 2 \\tau^2 v -v\\left( \\frac{\\psi}{\\tau} + \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)\\right)  - s_x \\left( \\frac{\\theta_x}{\\tau} + \\frac{2}{(2k\\tau)^2} (\\theta_x \\psi + \\theta_y \\theta) \\right) \\\\\n& - s_y \\left( \\frac{\\psi_x}{\\tau} + \\frac{2}{(2k\\tau)^2} (\\psi_x \\psi + \\psi_y \\theta) \\right) -(s + \\tau) \\left( \\frac{\\theta_{xx}}{\\tau} + \\frac{2}{(2k\\tau)^2} \\left( \\theta_{xx} \\psi + \\psi_{yy} \\theta + \\theta_x\\psi_x+\\theta_y\\psi_y \\right)  \\right)\\\\\n& - ( 6ss_xs_y + 3s^2 v + 6 \\tau s_x s_y + 6\\tau s v) + \\varepsilon\\Real[ \\partial_{xy}( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})],\n\\\\\n\\hat{F}_6 (\\xi;\\varepsilon, \\varphi) =& \\Delta w - 2 \\tau^2 w -w\\left( \\frac{\\psi}{\\tau} + \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)\\right)  -2 s_y \\left( \\frac{\\theta_x}{\\tau} + \\frac{2}{(2k\\tau)^2} (\\theta_x \\psi + \\theta_y \\theta) \\right) \\\\\n& -(s + \\tau) \\left( \\frac{\\psi_{yy}}{\\tau} + \\frac{2}{(2k\\tau)^2} \\left( \\theta_{xy} \\psi + \\theta_{xy} \\theta + | \\nabla \\theta|^2 \\right) \\right)\\\\\n& - ( 6s(s_y)^2 + 3s^2 w + 6 \\tau (s_y)^2 + 6\\tau s w) +\\varepsilon \\Real[ \\partial_{yy}( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})].\n\\end{align*}\n\nHere, $\\varphi \\in \\mathbb{R}$, $\\tau = \\sqrt{1 - k^2}$ , $k^2< \\dfrac{1}{3}$, and the variable $\\xi = ( s, \\psi, \\theta, u,v,w)$ is given by the formulas in (\\ref{changevariables}), so that we can actually consider $\\hat{F}$ as an operator on $\\mathcal{D}\\times \\mathbb{R}$ for fixed $\\varepsilon,\\varphi$.\n\n Since $(2k\\tau) \\nabla \\phi = \\langle \\psi, \\theta \\rangle$ we define $\\phi$ by\n\\begin{equation}\\label{defphi}  \\phi(x,y; \\varepsilon, \\varphi) = \\phi_\\mathrm{bd}+ \\phi_{\\log}, \\end{equation}\nwhere\n\\begin{align*}\n \\phi_\\mathrm{bd}(x,y;\\varepsilon,\\varphi) = &\\varphi + \\frac{1}{2k\\tau}\\int_{t=0}^1 \\left(\\hat{\\psi}(tx,ty;\\varepsilon)x+  \\hat{\\theta} (tx,ty;\\varepsilon)y\\right) \\\\\n \\phi_{\\log}(x,y;\\varepsilon,\\varphi) =& \\frac{1}{2k\\tau} \\int_{t=0}^1\\left( P_2(tx,ty)x+  P_3 (tx,ty)y\\right) =  \\frac{1}{2k\\tau} \\chi \\log(\\alpha x^2+y^2).\n \\end{align*}\n The following  lemma shows that $\\phi_\\mathrm{bd}$ is a well-defined function.\n\\begin{lem}\\label{phasebounded}\nIf $\\hat{\\psi}, \\hat{\\theta} \\in M^{2,2}_{\\gamma}$ then for fixed $\\varepsilon$ and $\\varphi$, the function $\\phi_\\mathrm{bd}(x,y; \\varepsilon, \\varphi)$  is well defined, bounded, continuous, and approaches a constant  $\\varphi + \\Phi_{\\infty}(\\varepsilon)$  as $| {\\bf x} |\\rightarrow \\infty$.\n\\end{lem}\n\\begin{proof} Note that $\\phi$ is continuous since $\\hat{\\psi}, \\hat{\\theta} \\in M^{2,p}_{\\gamma} \\subset BC^0$.\nLemma \\ref{decay} guarantees that for large $|{\\bf x} |$ we have $|\\hat{\\psi}|, |\\hat{\\theta}| \\leq C | {\\bf x} | ^{-\\gamma-1}$. Therefore, the integrals converge  as $| {\\bf x} |\\to \\infty $.\n\\end{proof}\n\nThe next four lemmas will help us show that the operator $\\hat{F}_{\\varepsilon, \\varphi}: \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow \\mathcal{R}$ is well defined.\n\n\n\\begin{lem}\\label{littleimbedding}\nThere exists $C>0$ so that for all $u \\in L^p_{\\gamma}$ with $ Du \\in L^p_{\\gamma +1}$, $\\langle {\\bf x} \\rangle^{\\gamma}u  \\in W^{1,p}$ and, \n\\[\n\\|\\langle {\\bf x} \\rangle^{\\gamma}u\\|_{W^{1,p}}\\leq C \\|u\\|_{L^p_{\\gamma}}+\\|Du\\|_{L^p_{\\gamma +1}}.\n\\]\n\\end{lem}\n \n \\begin{proof} We need to show that $D(  \\langle {\\bf x} \\rangle^{\\gamma} u) \\in L^p$. We compute\n \\[ D( \\langle {\\bf x} \\rangle^{\\gamma} u ) =   D u \\cdot \\langle {\\bf x} \\rangle^{\\gamma}+ D \\langle {\\bf x} \\rangle^{\\gamma} \\cdot u= Du\\cdot  \\langle {\\bf x} \\rangle^{\\gamma} +\\gamma  u   {\\bf x} (1 + |{\\bf x}|^2) ^{\\frac{\\gamma -2}{2}}.\\]\n Since $Du \\in L^p_{\\gamma+1} \\subset L^p_{\\gamma}$, we conclude that $Du \\cdot  \\langle {\\bf x} \\rangle^{\\gamma} \\in L^p$. Furthermore, since $|{\\bf x}|^p \\leq (1 +|{\\bf x}|^2)^{p\/2}$,\n\\[ |u \\cdot D\\langle {\\bf x} \\rangle^{\\gamma} | \\leq   |u| |{\\bf x} |\\langle {\\bf x} \\rangle^{(\\gamma-2)}  \\leq   |u|  \\langle {\\bf x} \\rangle^{(\\gamma-1)} \\leq  |u|  \\langle {\\bf x} \\rangle^{\\gamma } \\in L^p. \\]\nThis implies that $D(\\langle {\\bf x} \\rangle^{\\gamma} u) \\in L^p$ and we obtain $ \\langle {\\bf x} \\rangle^{\\gamma}  u\\in W^{1,p}$.\n\\end{proof}\n\n\\begin{lem}\\label{bounded}\nFor $\\gamma >0$, we have the continuous embeddings $M^{2,2}_{\\gamma} \\hookrightarrow W^{2,2}_{\\gamma} \\hookrightarrow W^{2,2}\\hookrightarrow BC^0$.\n\\end{lem}\n\\begin{proof}\nThe first embedding is due to Lemma \\ref{littleimbedding}, the second a consequence of $\\gamma >0$, and the last a classical Sobolev embedding in dimension 2.\n\\end{proof}\n\n\n\\begin{lem}\\label{Holder3}\nFor $\\gamma>0$ there exists $C>0$ such that for all $f,g $ with $\\langle {\\bf x} \\rangle^{\\gamma+1} f, \\langle {\\bf x} \\rangle^{\\gamma+1} g \\in W^{1,p}$,\n\\[\n\\|fg\\|_{L^p_{\\gamma+2}}\\leq C\\|\\langle {\\bf x} \\rangle^{\\gamma+1} f\\|_{W^{1,p}}\\|\\langle {\\bf x} \\rangle^{\\gamma+1} g\\|_{W^{1,p}}.\n\\]\n\\end{lem}\n\n\\begin{proof} By Cauchy-Schwartz,\n\\[\n\\|fg\\langle {\\bf x} \\rangle^{(\\gamma+2)}\\|_{L^p}\\leq \\|f\\langle {\\bf x} \\rangle^{(1+(\\gamma\/2))}\\|_{L^{2p}} \\|g\\langle {\\bf x} \\rangle^{(1+(\\gamma\/2))}\\|_{L^{2p}}\n\\]\nwhich proves the lemma using $\\gamma>0$ and the Sobolev embedding $W^{1,p}\\hookrightarrow L^{2p}$, in $n=2$.\n\\end{proof}\n\n\nFredholm properties of the Laplacian imply in particular the following more basic estimate  \\cite{nirenberg1973null}.\n\\begin{lem}\\cite[Theorem 3.1]{nirenberg1973null}\\label{nirenberg}\n If $u \\in L^p_{\\gamma}$ and $\\Delta u \\in L^p_{\\gamma+2}$ then $u \\in M^{2,p}_{\\gamma}$ and there exists a constant $C$ such that\n \\[  \\| u \\|_{M^{2,p}_{\\gamma}} \\leq C \\left( \\| u\\|_{L^p_{\\gamma}} + \\| \\Delta u\\|_{L^p_{\\gamma+2}}   \\right)  .\\]\n\\end{lem}\n\n\n\\begin{rem}\\label{betters}\nNotice that $u,w\\in L^2_{\\gamma+2}$ so that $ \\Delta s =u+w\\in L^2_{\\gamma+2}$, by the above lemma  we have that, within the closed subset $\\mathcal{D}\\subset \\mathcal{X}$, $s \\in M^{2,2}_{\\gamma}$ with uniform bounds in terms of $s,u,w$.\n\\end{rem}\n\n\n\nTo proof Theorem \\ref{MainTh}, the following three lemmas establish that each component of the operator $\\hat{F}_{\\varepsilon, \\varphi} : \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow \\mathcal{R}$ is well defined. Throughout, we use the standing assumptions  $0<\\gamma<1$ and  $g \\in W^{2,2}_{\\beta}$, with $\\beta >\\gamma +2$.\n\\begin{lem}\nThe component  $\\hat{F}_1 : \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow L^2_{\\gamma}$ is well defined. \n\\end{lem}\n\n\\begin{proof} We can rewrite\n\\[\\hat{F}_1 (\\xi;\\varepsilon, \\varphi) = \\Delta s - 2\\tau^2 s - \\psi  -  \\frac{s \\psi}{\\tau}  - (s + \\tau) (  \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)) - (s^3 + 3s^2 \\tau) + \\varepsilon \\Real [ g\\mathrm{e}^{-\\mathrm{i}( kx +  \\phi(\\varphi))}]. \\]\n\nA short calculation shows that $\\Delta s - 2\\tau^2 s - \\psi = ( \\Delta - 2\\tau^2) \\hat{s} -\\hat{\\psi} - c \\Delta  P_1$ is the first component of $\\hat{T}$, thus well defined. Consider next the term\n$ s \\psi = ( \\hat{s} + c  P_1 ) (\\hat{\\psi} + c P_2 ). $\n\nNotice that $\\hat{s}\\psi$ is in $L^2_{\\gamma}$ since both $ P_2$ and $\\hat{\\psi}$, are bounded by Lemma \\ref{bounded} and $\\hat{s} \\in  L^2_{\\gamma}$. Since $\\hat{\\psi} \\in L^2_{\\gamma}$ and since $ P_1$ is bounded, $\\hat{\\psi} P_1\\in L^2_{\\gamma}$ as well. Notice also  that  the product $ P_1\\cdot P_2$ is bounded in compact sets and behaves like $ \\dfrac{1}{|{\\bf x}|^2} $ for large values of $|{\\bf x}|$, hence it belongs to $L^2_{\\gamma}$ provided $\\gamma<1$. This shows the term $ s \\psi $ is in the desired space.\n\nUsing similar arguments, it is easy to check  that the functions $\\psi^2, \\theta^2, s^3,s^2$ and $s( \\psi^2+ \\theta^2)$ are in $L^2_\\gamma$. Finally, by Lemma \\ref{phasebounded} we know $\\phi$ is a bounded continuous function so that we can conclude $\\mathrm{e}^{-\\mathrm{i}(kx + \\phi(\\varphi))}$ is a well defined function  in $L^{\\infty}$. This implies that the term $\\Real [ g\\mathrm{e}^{-\\mathrm{i}(kx + \\phi(\\varphi) )}]$ is in $L^2_{\\gamma}$ since $g \\in L^2_{\\beta} \\subset L^2_{\\gamma}$.\n\\end{proof}\n\n\n\n\n\n\\begin{lem}\\label{welldefinedF1}\nThe component   $\\hat{F}_2 : \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow L^2_{\\gamma+2}$ is well defined.\n\\end{lem}\n\n\\begin{proof} Since we are trying to find solutions near $\\xi=0$ we can assume $s\/\\tau$ is close to zero. We can therefore write\n\\begin{align*}\n \\Delta \\psi + (2k)^2 \\frac{\\tau u}{s +\\tau} &= \\Delta \\psi + (2k)^2 \\left(u+\\frac{su}{\\tau+su}\\right)\\\\\n & = \\Delta \\hat{\\psi} + (2k)^2( \\hat{u} + c  \\Delta P_2 + (2k)^2 \\partial_{xx} P_2 ) + (2k)^2\\frac{(\\hat{u} + c \\partial_{xx}  P_1))(\\hat{s}+cP_1)}{\\tau+(\\hat{u} + c \\partial_{xx}  P_1))(\\hat{s}+cP_1)}.\n \\end{align*}\nNotice that the terms $ \\Delta \\hat{\\psi} + (2k)^2 \\hat{u} + c [ \\Delta P_2 + (2k)^2 \\partial_{xx}P_2 ], $ \nrepresent the second component of the linear operator $\\hat{T}: \\mathcal{D} \\times \\mathbb{R} \\rightarrow \\mathcal{R}$, hence  are well defined. \nIt is now straightforward to see that the remaining nonlinear terms are contained in $L^2_{\\gamma+2}$. In terms of localization, the most dangerous term is $P_1\\partial_{xx}P_1$, which can be bounded as \n\\[ \\int \\left | P_1  \\partial_{xx}  P_1 \\right|^2 \\langle {\\bf x} \\rangle^{2(\\gamma+2)} \\leq \\int \\left| \\frac{1}{r^4} \\right|^2 r^{2(\\gamma+2)} r \\mathrm{d} r < \\infty,\\]\nsince $\\gamma<1$.\n\nWe  next treat the remaining nonlinearities. Since $s \\in L^{\\infty}$, we only need to show that the numerators in $\\hat{F}_2$ are in $L^2_{\\gamma+2}$. It is not hard to mimic the above arguments to show that the terms $u \\psi$ and $v \\theta$ are in $L^2_{\\gamma+2}$, so we will treat the term $s_x \\psi_x$ first.  Using the formulas in (\\ref{changevariables}) we see that\n\\[ s_x \\psi_x = ( \\hat{s}_x + c \\partial_{x} P_1) ( \\hat{\\psi}_x+ c \\partial_x  P_2 ).\\]\n\nBy Lemma \\ref{nirenberg} and Remark \\ref{betters} we know that $\\hat{s}\\in M^{2,2}_{\\gamma}$. Therefore,  $\\hat{s}_x,\\hat{\\psi}_x\\in W^{1,2}_{\\gamma+1}$ and we can apply Lemma \\ref{Holder3}  to conclude that $\\hat{s}_x \\hat{\\psi}_x\\in L^2_{\\gamma+2}$. The remaining terms are easily seen to be in the correct space.\n\n\nSimilar arguments show that the functions $(s_x)^2, s_y \\psi_y, s_xs_y$ are in the correct space and that, since $ \\psi$ and $\\theta$ are bounded by Lemma \\ref{bounded}, $s_x( s_x \\psi+ s_y \\theta)\\in L^2_{\\gamma+2}$. \n\nFinally, because we are assuming that $g$ is in the space $W^{2,2}_{\\beta}$, with $\\beta > \\gamma+2$, $s_x \\in L^2_{\\gamma+1}$, and because the terms $\\psi$ and $\\mathrm{e}^{-\\mathrm{i}(kx +  \\phi(\\varphi))}$ are bounded,\n\\[-\\frac{\\varepsilon s_x \\Imag[g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))}] }{(s+ \\tau)^2} +  \\frac{\\varepsilon \\Imag[ \\partial_x( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})]}{s+ \\tau} \\in L^2_{\\gamma+2}.\\]\n Here, we used  the fact that  $\\psi = (2 k \\tau) \\phi$ so that  $ \\partial_x  ( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})= [ g_x -i g( k + \\psi\/(2k\\tau)) ] \\mathrm{e}^{-\\mathrm{i}(kx + \\phi(\\varphi))} $.\n\\end{proof}\n\n\n\\begin{lem}\nThe component $\\hat{F}_3 : \\mathcal{D} \\times \\mathbb{R}^3 \\rightarrow L^2_{\\gamma+2}$ is well defined.\n\\end{lem}\n\n\n\\begin{proof} The proof is almost identical to the proof of Lemma \\ref{welldefinedF1} and is omitted here.\n\\end{proof}\n\nFinally, we show\n\n\\begin{lem}\nThe component $\\hat{F}_4 : \\mathcal{D}\\times\\mathbb{R}^3 \\rightarrow W^{-2,2}_{\\gamma+2}$ is well defined. Moreover, the nonlinear part of $\\hat{F}_4$ actually belongs to $L^2_{\\gamma+2}$.\n\\end{lem}\n\n\\begin{proof} We can rewrite $\\hat{F_4}$ as\n\\[ \\Delta u - 2 \\tau^2 u - \\psi_{xx} - \\frac{ s \\psi_{xx}}{\\tau} -u\\left( \\frac{\\psi}{\\tau} + \\frac{1}{(2k\\tau)^2} (\\psi^2 + \\theta^2)\\right)  -2 s_x \\left( \\frac{\\psi_x}{\\tau} + \\frac{2}{(2k\\tau)^2} (\\psi_x \\psi + \\psi_y \\theta) \\right) \\]\n\\[ -(s + \\tau) \\left(   \\frac{2}{(2k\\tau)^2} \\left( \\psi_{xx} \\psi + \\psi_{xy} \\theta + | \\nabla \\psi|^2 \\right) \\right)  - ( 6s(s_x)^2 + 3s^2 u + 6 \\tau (s_x)^2 + 6\\tau s u) + \\varepsilon \\Real[ \\partial_{xx}( g \\mathrm{e}^{-\\mathrm{i}(kx +\\phi(\\varphi))})].\\]\n\nNotice that\n\\[ \\Delta u - 2 \\tau^2 u - \\psi_{xx} = \\Delta\\hat{u} - 2 \\tau^2 \\hat{u} - \\hat{\\psi}_{xx} + c \\Delta( \\partial_{xx} P_1), \\]\n is just the fourth component of the linear operator $\\hat{T}$, thus well defined. Furthermore, notice that $\\hat{\\psi}_{xx} \\in L^2_{\\gamma+2}$. Since $\\Delta \\partial_{xx} P_1$ behaves like $\\dfrac{1}{|{\\bf x}|^5}$ for large $|{\\bf x}|$ and is bounded in compact sets,  these two term now also belong to $L^2_{\\gamma+2}$.\nThe arguments used to show that the remaining nonlinearites are in the space $L^2_{\\gamma+2}$ are the same as the once used in the above lemmas, we will omit the details here.\n\\end{proof}\n\n\n\nHaving shown the result for the operator $\\hat{F}_4$ it is not hard to see that the operators $\\hat{F}_5,\\hat{F}_6 : \\mathcal{D}\\times \\mathbb{R}^3 \\rightarrow W^{-2,2}_{\\gamma+2}$ are well defined.\n\n\\begin{rem}\\label{betteru}\nSince all the nonlinear terms are in $L^2_{\\gamma+2}$, including $\\hat{\\psi}_{xx}$ and $\\Delta \\partial_{xx} P_1$, then $\\Delta\\hat{u} - (2\\tau^2)\\hat{u} \\in L^2_{\\gamma+2}$. This implies that for the solution, $\\hat{u} \\in W^{2,2}_{\\gamma+2}$. The same observation holds for  $\\hat{v}$ and $\\hat{w}$.\n\\end{rem}\n\n\nIn the next lemma we show that there exist a neighborhood $\\mathcal{U}$ of $\\xi \\in \\mathcal{D} \\times \\mathbb{R}$ such that the operator $\\hat{F}_{\\varepsilon, \\varphi}: \\mathcal{U} \\times \\mathbb{R}^3 \\rightarrow \\mathcal{R}$ is smooth.\n\n\\begin{lem}\\label{differentiable}\nLet  $0< \\gamma<1$ and  $g \\in W^{2,2}_{\\beta}$, with $\\beta> \\gamma+2$. Then the operator $\\hat{F}_{\\varepsilon, \\varphi} : \\mathcal{D} \\times\\mathbb{R}^3 \\rightarrow \\mathcal{R}$ is of class $C^\\infty$ in a neighborhood the origin.\n\\end{lem}\n\n\\begin{proof} Most nonlinear terms are defined via superposition (or Nemytskii) operators, via smooth algebraic functions, that are automatically smooth once well defined. We therefore concentrate on the term $g \\mathrm{e}^{-\\mathrm{i}\\phi}$ and its derivatives. Recall that\n\\[  \\phi(x,y; \\varepsilon, \\varphi) = \\phi_\\mathrm{bd}+ \\phi_{\\log}, \\]\nwhere\n\\begin{align*}\n \\phi_\\mathrm{bd}(x,y;\\varepsilon,\\varphi) = &\\varphi + \\frac{1}{2k\\tau}\\int_{t=0}^1 \\left(\\hat{\\psi}(tx,ty;\\varepsilon)x+  \\hat{\\theta} (tx,ty;\\varepsilon)y\\right) \\\\\n \\phi_{\\log}(x,y;\\varepsilon,\\varphi) =& \\frac{1}{2k\\tau} \\int_{t=0}^1\\left( P_2(tx,ty)x+  P_3 (tx,ty)y\\right) =  \\frac{1}{2k\\tau} c \\chi \\log(\\alpha x^2+y^2).\n\\end{align*}\nIn order to show smoothness, we factor\n\\[\ng \\mathrm{e}^{-\\mathrm{i}\\Phi}=\\left(g\\langle\\mathbf{x}^{\\gamma+2-\\beta}\\rangle\\right)\\cdot\\mathrm{e}^{-\\mathrm{i}\\Phi_\\mathrm{bd}} \\cdot \\left(\\langle\\mathbf{x}^{\\beta-\\gamma-2}\\rangle \\mathrm{e}^{-\\mathrm{i}\\Phi_\\mathrm{log}}\\right)=:G_1\\cdot G_2 \\cdot G_3.\n\\]\nClearly, $G_1\\in L^2_{\\gamma+2}$. By Lemma \\ref{phasebounded}, $\\int\\psi,\\int\\theta\\in L^\\infty$, so that $G_2\\in L^\\infty$ is bounded as a superposition operator. It remains to show that $G_3$ is differentiable with values in $L^\\infty$. This can be readily established, showing that the derivative with respect to $c$ is \n\\[\n\\partial_cG_3=\\langle\\mathbf{x}^{\\beta-\\gamma-2}\\rangle  \\mathrm{e}^{-\\mathrm{i}\\Phi_\\mathrm{log}}\\chi \\log(\\alpha x^2+y^2),\n\\]\nhence bounded in $L^\\infty$. Higher derivatives are bounded for the same reasons, which establishes the claim.\n\\end{proof}\n\n\n  \n\n\n\\section{Expansions and proof of main result}\\label{Mainresult}\n\nIn this last subsection we use Theorem \\ref{MainTh} to proof Theorem \\ref{MainTh1}  and derive the expansions for the stationary solutions to the perturbed Ginzburg-Landau equation near roll patterns. \n\\begin{proof}[of Theorem \\ref{MainTh1}] Recall the  Ansatz\n\\[ A(x,y; \\varepsilon, \\varphi) = (\\sqrt{1 -k^2} + s(x,y;\\varepsilon, \\varphi)+ c(\\varepsilon, \\varphi) P_1(x,y)) \\mathrm{e}^{\\mathrm{i} kx + \\mathrm{i} \\phi(x,y;\\varepsilon, \\varphi)+ \\mathrm{i} \\frac{c(\\varepsilon, \\varphi)}{2k \\sqrt{1-k^2}}P_2(x,y)} \\]\nwhere $\\Phi$ was defined in \\eqref{defphi} and $P_1,P_2$ in \\eqref{e:p1p2}.\nFrom Theorem \\ref{MainTh} we know there exists  a neighborhood, $\\mathcal{U}$, of $\\mathcal{D} \\times \\mathbb{R}$ where the operator $\\hat{F}_{\\varepsilon,\\varphi}$ is continuously differentiable with invertible derivative at the origin, $\\varepsilon=0$. The Implicit Function Theorem therefore guarantees the existence of solutions $\\xi(\\varepsilon, \\varphi) $ near $\\xi(0, \\varphi)=0$. In particular, we know that $s \\in W^{2,2}_{\\gamma}$, and $\\psi, \\theta \\in M^{2,2}_{\\gamma}$.\n\n\nWe define \n\\[ S(x,y; \\varepsilon, \\varphi) = \\left(\\sqrt{1-k^2} + s(x,y;\\varepsilon, \\varphi) + c(\\varepsilon, \\varphi)  P_1(x,y) \\right)\\]\nand\n\\[ \\Phi( x,y;\\varepsilon, \\varphi) =kx + \\phi. \\]\n\nSince $s(x,y;\\varepsilon, \\varphi) \\in W^{2,2}_{\\gamma}$, Lemma \\ref{bounded}  ensures that if $s(x,y; \\varepsilon, \\varphi) \\sim \\mathrm{O}(\\langle {\\bf x} \\rangle^{-\\gamma})$.  Also,  by definition, $P_1(x,y) \\sim \\mathrm{O}(\\langle {\\bf x} \\rangle^{-1})$, and\n\\[ \\lim_{\\mathrm{x}\\to\\infty} S(x,y;\\varepsilon, \\varphi) =S_{\\infty}= \\sqrt{1-k^2}.\\]\n\nBy Lemma \\ref{phasebounded}, $\\phi_\\mathrm{bd} \\to\\varphi + \\Phi_{\\infty}(\\epsilon)$ for $\\mathbf{x}\\to\\infty$ so that \n\\[ \\Phi(x,y;\\varepsilon,\\varphi)-kx-\\frac{c(\\varepsilon,\\varphi)}{2k \\sqrt{1-k^2}}\\log(\\alpha x^2+y^2)\\to  \\Phi_\\infty(\\varepsilon)+\\varphi,\\]\nas $|{\\bf x}| \\to \\infty$.\n\n\nTo find an expression for $c(\\varepsilon, \\varphi)$ we expand $\\xi = \\varepsilon \\hat{\\xi} + \\mathrm{o}(\\varepsilon)$. Gathering terms of order $\\varepsilon$ results in the  system $\\hat{T} \\hat{\\xi} = \\hat{f}$. Inspecting the second component of this system, we find\n\\[ \\Delta \\hat{\\psi} + b \\partial_{xx} \\hat{u} + \\hat{c} \\left[\\Delta P_2 + b \\partial_{xx} P_1  \\right] = \\dfrac{1}{\\sqrt{1-k^2}}\\Imag[\t ( g_x-ikg) \\mathrm{e}^{-\\mathrm{i}(kx+\\varphi)}].\\]\nTaking the scalar product with $x$ and solving for $\\hat{c}$, we obtain after integration by parts in $x$,\n\\[ \\hat{c} =\\frac{\\sqrt{1-3k^2}}{\\pi(1-k^2)} \\iint \\Imag[ g\\mathrm{e}^{-\\mathrm{i}( kx+ \\varphi)}].\n\\]\nHence $c( \\varepsilon, \\varphi) = \\varepsilon c_1( \\varphi) + \\mathrm{o}(\\varepsilon)$ with $c_1(\\varphi) = \\hat{c}$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Appendix}\n\n\\begin{lem}\\label{decay}\nIf $f \\in M^{2,2}_{\\gamma}$ then $|f| \\leq  {C} \\| f\\|_{M^{2,2}_{\\gamma}}  \\langle {\\bf x} \\rangle^{-\\gamma-1}$ as $| {\\bf x} |\\rightarrow \\infty$.\n\\end{lem}\n\\begin{proof} Since $M^{2,2}_{\\gamma}$ is the completion of $C^{\\infty}_0$ under the norm $\\| \\cdot\\|_{M^{2,2}_{\\gamma}}$, it suffices to show that the result holds for $f \\in C^{\\infty}_0$. In polar coordinates, we have, up to constants \n  \\begin{align*}\n  \\int |f(\\theta, R)|^2 \\mathrm{d}\\theta &\\leq \\int \\left( \\int_{\\infty}^R |f_r(\\theta, s)| \\mathrm{d} s \\right)^2  \\mathrm{d}\\theta\n  =\\int \\left( \\int_{\\infty}^R s^{-\\gamma -3\/2} s^{\\gamma+1} | f_r(\\theta, s)|  s^{1\/2} \\mathrm{d} s \\right)^2  \\mathrm{d}\\theta\\\\\n  & \\leq \\int \\left(\\int_{\\infty}^R s^{-2(\\gamma + 3\/2)} \\mathrm{d} s \\right) \\left( \\int_{\\infty}^R s^{2(\\gamma+1)} |f_r(\\theta,s)|^2 s\\mathrm{d} s \\right)  \\mathrm{d}\\theta \\\\\n  &\\lesssim R^{-2(\\gamma+3\/2)+1} \\int \\int_{\\infty}^R s^{2( \\gamma+1)}|f_r(\\theta,s)|^2 s\\mathrm{d} s\\mathrm{d}\\theta ,\n  \\end{align*}\nwhich gives\n\\begin{equation}\\label{es1} \\| f(\\cdot, R)\\|_{L^2} \\lesssim R^{-\\gamma-1}\\|\n \\nabla f\\|_{L^2_{\\gamma+1}}.\n\\end{equation}\nSimilarly,\n\\begin{align*}\n\\int |f_{\\theta}(\\theta, R) |^2\\mathrm{d}\\theta &\\leq \\int \\left( \\int_{\\infty}^R |f_{r \\theta}(\\theta,s)|\\mathrm{d} s \\right)^2 \\mathrm{d}\\theta\n= \\int \\left( \\int_{\\infty}^R s^{-\\gamma -5\/2} s^{\\gamma+2} |f_{r \\theta}(\\theta,s)|  s^{1\/2}\\mathrm{d} s \\right)^2 \\mathrm{d}\\theta\\\\\n& \\leq \\int \\left( \\int_{\\infty}^R s^{-2(\\gamma+5\/2)} \\mathrm{d} s \\right) \\left( \\int_{\\infty}^R  s^{2(\\gamma+2)} |f_{r \\theta}(\\theta,s)|^2  s\\mathrm{d} s \\right) \\mathrm{d}\\theta \\\\\n& \\lesssim R^{-2(\\gamma+5\/2)+1} \\int  \\int_{\\infty}^R  s^{2(\\gamma+2)} |f_{r \\theta}(\\theta,s)|^2  s\\mathrm{d} s  \\mathrm{d}\\theta.\n\\end{align*}\nThis gives\n\\begin{equation}\\label{es2} \\| f_\\theta(\\cdot, R)\\|_{L^2}\\leq  R^{-\\gamma-2} \\| f_{r\\theta} \\|_{L^2_{\\gamma+2}}.\\end{equation}\nCombining \\eqref{es1} and \\eqref{es2} and using the interpolation inequality \\cite[Thm 5.9]{adams}\n\\[\n\\|f(\\cdot,R)\\|^2_\\infty\\leq \\|f(\\cdot,R)\\|_{L^2} \\|f(\\cdot,R)\\|_{H^1},\n\\]\nnow proves the claim.\n\\end{proof}\n\n\n\n\n\n\n\n\n \n \n \\bibliographystyle{siam}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nLet $\\DD$ be the open unit disk in the complex plane. Let $L^2$ denote the Lebesgue space of square integrable functions on the unit circle $\\partial\\DD$. The Hardy space $H^2$ is the subspace of $L^2$ of analytic functions on $\\DD$. Let $P$ be the orthogonal projection from $L^2$ to $H^2$.\nFor $\\varphi\\in L^\\infty$, the space of bounded Lebesgue measurable functions on $\\partial\\DD$, the Toeplitz operator $T_{\\varphi}$ and the Hankel operator $H_\\varphi$ with symbol $\\varphi$ are defined on $H^2$ by $$T_{\\varphi}h=P(\\varphi h),$$ and\n\\beq\\label{Han}\nH_{\\varphi}h=U(I-P)(\\varphi h),\n\\eeq\nfor $h\\in H^2$.\nHere $U$ is the unitary operator on $L^2$ defined by $$Uh(z)=\\bz h (\\bz).$$\n\nRecall that the spectrum of a linear operator $T$, denoted as $sp(T)$, is the set of complex numbers $\\lambda$ such that $T - \\lambda I$ is not invertible; here $I$ denotes the identity operator. Let $[T^*,T]$ denote the operator $T^*T-TT^*$, called the self-commutator of $T$. An operator $T$ is called hyponormal if $[T^*,T]$ is positive.\nHyponormal operators satisfy the celebrated Putnam inequality \\cite{put70}\n\\begin{thm}\\label{P}\nIf $T$ is a hyponormal operator, then\n$$\n\\|[T^*,T]\\|\\leq \\frac{Area(sp(T))}{\\pi}.\n$$\n\\end{thm}\n\nNotice that a Toeplitz operator with analytic symbol $f$ is hyponormal, and it is well known that $sp(T_f)=\\overline{f(\\DD)}$. The lower bounds of the area of $sp(T_f)$ were obtained in \\cite{kh(T)} (see \\cite{ax85},\\cite{ale72} \\cite{stan86} and \\cite{stan89} for generalizations to uniform algebras and further discussions). Together with Putnam's inequality such lower bounds were used to prove the isoperimetric inequality (see \\cite{ben14},\\cite{ben06} and the references there). Recently, there has been revived interest in the topic in the context of analytic Topelitz operators on the Bergman space (cf. \\cite{bell14}, \\cite{ol14} and \\cite{fl15}). Together with Putnam's inequality, the latter lower bounds have provided an alternative proof of the celebrated St. Venant's inequality for torsional rigidity.\n\nIn the general case, Harold Widom \\cite{wid64} proved the following theorem for arbitrary symbols.\n\\begin{thm}\nEvery Toeplitz operator has a connected spectrum.\n\\end{thm}\nThe main purpose of this note is to show that for a rather large class of Topelitz operators on $H^2$, hyponormal operators with a harmonic symbol, there is\nstill a lower bound for the area of the spectrum, similar to the lower bound obtained in \\cite{kh(T)} in the context of uniform algebras.\n\nWe shall use the following characterization of the hyponormal Toeplitz operators given by Cowen in \\cite{cow88}\n\\begin{thm}\\label{cow}\nLet $\\varphi\\in L^\\infty$, where $\\varphi=f+\\bar{g}$ for $f$ and $g$ in $H^2$. Then $T_{\\varphi}$ is hyponormal if and only if\n$$\ng=c+T_{\\bh}f,\n$$\nfor some constant $c$ and $h\\in H^\\infty$ with $\\|h\\|_{\\infty}\\leq 1.$\n\\end{thm}\n\n\n\\section{Main Results}\nIn this section, we obtain the lower bound for the area of the spectrum for hyponormal Toeplitz operators by estimating the self-commutators.\n\n\\begin{thm}\\label{2.1}\nSuppose $\\varphi \\in L^\\infty$ and $$\\varphi=f+ \\overline{T_{\\bar{h}}f} ,$$ for $f,h\\in H^\\infty$, $\\|h\\|_{\\infty}\\leq 1$ and $h(0)=0$. Then\n$$\n\\|[T^{*}_{\\varphi}, T_{\\varphi}]\\|\\geq \\int |f-f(0)|^2\\frac{d\\theta}{2\\pi}=||P(\\varphi)-\\varphi(0)||^2_2.\n$$\n\\end{thm}\n\\begin{proof}\nLet\n\\beq\\label{g} g=T_{\\bar{h}}f.\\eeq\nFor every $p$ in $H^2$,\n\\begin{align*}\n\\langle[T^{*}_{\\varphi}, T_{\\varphi}]p,p\\rangle&=\\langle T_{\\varphi}p, T_{\\varphi}p\\rangle-\\langle T^*_{\\varphi}p, T^*_{\\varphi}p\\rangle\\\\\n&=\\langle fp+P(\\bg p), fp+P(\\bg p)\\rangle-\\langle gp+P(\\barf p), gp+P(\\barf p)\\rangle\\\\\n&=||fp||^2-||P(\\barf p)||^2-||gp||^2+||P(\\bg p)||^2\\\\\n&=||\\barf p||^2-||P(\\barf p)||^2-||\\bg p||^2+||P(\\bg p)||^2\\\\\n&=||H_{\\barf} p||^2-||H_{\\bg}p||^2,\n\\end{align*}\nwhere $||\\cdot||$ means the $||\\cdot||_{L^2(\\partial \\DD)}$.\nThe third equality holds because\n$$\n\\langle fp, P(\\bg p)\\rangle=\\langle fp, \\bg p\\rangle=\\langle gp, \\barf p\\rangle=\\langle gp, P(\\barf p)\\rangle.\n$$\nBy the computation in \\cite{cow88}*{p. 4}, \\eqref{g} implies $$H_{\\bg}=T_{\\bk}H_{\\barf},$$\nwhere $k(z)=\\overline{h(\\bz)}$. Thus\n\\beq\\label{rmk}\n\\langle[T^{*}_{\\varphi}, T_{\\varphi}]p,p\\rangle=||H_{\\barf} p||^2-||T_{\\bk}H_{\\barf}p||^2,\n\\eeq\nfor $k\\in H^\\infty$, $\\|k\\|_{\\infty}\\leq 1$ and $k(0)=0$.\n\nFirst, we assume $k$ is a Blaschke product vanishing at $0$. Then\n$$|k|=1\\,\\,\\m{on}\\,\\, \\partial\\DD.$$\nLet $u=H_{\\barf} p\\in H^2$. By \\eqref{rmk} we have\n\\begin{align}\\label{T}\n\\langle[T^{*}_{\\varphi}, T_{\\varphi}]p,p\\rangle&=||u||^2-||T_{\\bk} u||^2=||u||^2-||\\bk u||^2+||H_{\\bk}u||^2=||H_{\\bk}u||^2.\n\\end{align}\nThen\n\\begin{align*}\n||H_{\\bk}u||&=||(I-P)(\\bk u)||=||k\\bu-\\overline{P(\\bk u)}||\\\\\n&\\geq \\sup_{\\substack{m\\in H^2\\\\m(0)=0}}\\frac{|\\langle k\\bu-\\overline{P(\\bk u)},m\\rangle|}{||m||}\\\\\n&={\\sup_{\\substack{m\\in H^2\\\\m(0)=0}}{1 \\over ||m||}\\left|\\int k\\bu \\bm \\,{d\\theta\\over 2\\pi}\\right|}.\n\\end{align*}\nThe last equality holds because $m(0)=0$ implies that $\\bm$ is orthogonal to $H^2$.\nSince $k(0)=0$, taking $m=k$, we find\n\\beq\\label{u}\n||H_{\\bk}u||\\geq \\left|\\int\\bu \\,{d\\theta\\over 2\\pi}\\right|=|u(0)|.\n\\eeq\n\nNext, suppose $k$ is a convex linear combination of Blaschke products vanishing at 0, i.e.\n$$\nk=\\Ga_1B_1+\\Ga_2B_2+...+\\Ga_lB_l,\n$$\nwhere $B_j$'s are Blaschke products with $B_j(0)=0$, $\\Ga_j\\in[0,1]$ and $\\displaystyle\\sum_{j=1}^l \\Ga_j=1$.\n\nBy \\eqref{T} and \\eqref{u}, for each $j$\n\\begin{align*}\n&||u||^2-||T_{\\bar{B_j}}u||^2=||H_{\\bar{B_j}}u||^2\\geq |u(0)|^2\\\\\n\\Longrightarrow\\, &|| T_{\\bar{B_j}}u||\\leq \\sqrt{||u||^2-|u(0)|^2}=||u-u(0)||.\n\\end{align*}\nThen\n\\begin{align}\n\\label{M}||u||^2-||T_{\\bk} u||^2&=||u||^2-||\\Ga_1T_{\\bar{B_1}}u+\\Ga_2T_{\\bar{B_2}}u+...+\\Ga_lT_{\\bar{B_l}}u||^2\\\\\n\\nnb&\\geq ||u||^2-\\Big(\\Ga_1||T_{\\bar{B_1}}u||+\\Ga_2||T_{\\bar{B_2}}u||+...+\\Ga_l||T_{\\bar{B_l}}u||\\Big)^2\\\\\n\\nnb&\\geq ||u||^2- ||u-u(0)||^2=|u(0)|^2.\n\\end{align}\n\nIn general, for $k$ in the closed unit ball of $H^\\infty$, vanishing at $0$, by Carath\\'eodory's Theorem(cf. \\cite{gar81}*{p. 6}), there exists a sequence $\\{B_n\\}$ of finite Blaschke products such that\n$$B_n\\longrightarrow k\\q \\m{pointwise on}\\,\\,\\DD.$$\nSince $B_n$'s are bounded by $1$ in $H^2$, passing to a subsequence we may assume\n$$B_n\\longrightarrow k\\q \\m{weakly in}\\,\\,H^2.$$\nThen by \\cite{rud91}*{Theorem 3.13}, there is a sequence $\\{k_n\\}$ of convex linear combinations of Blaschke products such that\n$$k_n\\longrightarrow k\\q \\m{in}\\,\\,H^2.$$\nSince $k(0)=0$, we can let those $k_n$'s be convex linear combinations of Blaschke products vanishing at $0$.\n\nThen\n$$\n||T_{\\bk_n}u-T_{\\bk} u||=||P(\\bk_n u-\\bk u)||\\leq||k_n-k||\\cdot||u||\\to 0.\n$$\nSince \\eqref{M} holds for every $k_n$, we have\n\\begin{align*}\n\\langle[T^{*}_{\\varphi}, T_{\\varphi}]p,p\\rangle&=||u||^2-||T_{\\bk}u||^2=\\lim_{n\\to\\infty}(||u||^2-||T_{\\bk_n}u||^2)\\\\\n&\\geq|u(0)|^2=|(H_{\\barf} p)(0)|^2.\n\\end{align*}\n\nBy the definition of Hankel operator \\eqref{Han},\n\\begin{align*}\n|(H_{\\barf} p)(0)|&=|\\langle p\\barf, \\bz \\rangle|=\\left|\\int \\barf zp\\,{d\\theta\\over 2\\pi}\\right|.\n\\end{align*}\nFrom the standard duality argument (cf. \\cite{gar81}*{Chapter IV}), we have\n\\begin{align*}\n\\sup_{\\substack{||p||=1 \\\\ p\\in H^2}}\\left|\\int \\barf zp\\,{d\\theta\\over 2\\pi}\\right|&=\\sup\\bigg\\{ \\left|\\int \\barf p\\,{d\\theta\\over 2\\pi}\\right| : p\\in H^2, ||p||=1, p(0)=0 \\bigg\\}\\\\\n&=\\m{dist}(\\barf, H^2)=||f-f(0)||.\n\\end{align*}\nHence\n$$\n||[T^{*}_{\\varphi}, T_{\\varphi}]||=\\sup_{\\substack{||p||=1 \\\\ p\\in H^2}}|\\langle[T^{*}_{\\varphi}, T_{\\varphi}]p,p\\rangle|\\geq ||f-f(0)||^2.\n$$\n\\end{proof}\n\\begin{rem}\nFor arbitrary $h$ in the closed unit ball of $H^\\infty$, it follows directly from \\eqref{rmk} that $T_{\\varphi}$ is normal if and only if $h$ is a unimodular constant. So we made the assumption that $h(0)=0$ to avoid these trivial cases. Of course, Theorem \\ref{2.1} implies right away that $T_\\varphi$ is normal if and only if $f=f(0)$, i.e., when $\\varphi$ is a constant, but under more restrictive hypothesis that $h(0)=0$.\n\\end{rem}\n\n\nApplying Theorem \\ref{P} and \\ref{cow}, we have\n\\begin{cor}\nSuppose $\\varphi \\in L^\\infty$ and $$\\varphi=f+ \\overline{T_{\\bar{h}}f} ,$$ for $f,h\\in H^\\infty$, $\\|h\\|_{\\infty}\\leq 1$ and $h(0)=0$. Then\n$$\nArea(sp(T_\\varphi))\\geq\\pi||P(\\varphi)-\\varphi(0)||^2_2.\n$$\n\\end{cor}\n\n\\begin{rem}\nThus, the lower bound for the spectral area of a general hyponormal Toeplitz operator $T_\\varphi$ on $\\partial \\DD$ still reduces to the $H^2$ norm of the analytic part of $\\varphi$. For analytic symbols this is encoded in \\cite{kh(T)}*{Theorem 2} in the context of Banach algebras. In other words, allowing more general symbols does not reduce the area of the spectrum.\n\\end{rem}\n\n\\section*{Acknowledgments}\nThe first author gratefully acknowledges the hospitality of the Department of Mathematics and Statistics\nat the University of South Florida during the work on the paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Indroduction}\n\nGalaxies of similar morphological type exhibit similarities in their \nstructural properties, independent of whether they are isolated in \ngeneral fields or agglomerated in galaxy clusters.  These similarities \nunder different environments may emerge due to some {\\it internal} \nregularities that have superseeded external, probabilistic disturbances \nsuch as interactions and\/or mergers.\n\nThere is a well-known evidence from optical observations that many \nedge-on spiral and lenticular galaxies have a universal luminosity profile \nperpendicular to their disk plane, or a universal $z$-distribution of \nstellar mass density if a constant mass-to-luminosity ratio is assumed \n(Tsikoudi 1977; Burstein 1979; van der Kruit \\& Searle 1981).  It was \nonce claimed that the $z$-distribution is best fitted by a self-gravitating \nisothermal model like sech$^2(z\/2h_z)$ using a scale parameter $h_z$ \n(van der Kruit \\& Searle 1981). \n\nNear-infrared observations of edge-on spiral galaxies however uncovered an \nexcess over the isothermal model at small $|z|$ where the optical photometry \nis hindered by dust absorption (Wainscoat, Freeman, \\& Hyland 1989; \nAoki et al. 1991; van Dokkum et al. 1994). These infrared observations \nshowed that an exponential distribution $\\exp(-|z|\/h_z)$ provides a superior \nfit to the observed $z$-profiles.  Furthermore it is known from analyses of \nstar counts that the vertical structure of the Galactic disk in the solar\nneighbourhood is well fitted by an exponential distribution rather than an \nisothermal model (Bahcall \\& Soneira 1980; Prichet 1983).  \n\nAn exponential $z$-distribution can be constructed by adding up several\nstellar disk components with different vertical velocity dispersions, but \nthis is only possible if the contribution from stars with larger velocity \ndispersion is fine-tuned to dominate progressively at larger distances from \nthe disk plane.  A mechanism that enables this tuning has not been found \nto date.\n\nThe vertical disk structure has been\nconsidered as a result of dynamical evolution of the stellar component through\nencounters with massive clouds and spiral structures\n(Villumsen 1983; Lacey 1984; Carlberg 1987). However, although collisions with clouds\nmight be important in increasing the velocity dispersions of young stars \nthis effect cannot account for the high velocity dispersion of old stars \n(Lacey 1984). In addition, the resulting disks are isothermal and\nnot exponential.\n\nWe report that the hydrodynamical equations associated with a simple\nstar formation law possess a remarkable solution which naturally produces\nan exponential stellar $z$-distribution in the final stage of gravitational\nsettling of galactic protodisks.  This result might provide new insight\ninto the formation of galactic disks and their early evolutionary phases.\n\n\\section{The numerical model}\n\nThe key factor which makes galaxy simulations distinct from either \nhydrodynamical computations is star formation.   From theoretical \narguments (Hoyle 1953; Low \\& Lynden-Bell 1976; Silk 1977), the cooling \nof the gas plays a decisive role in the process of star formation as it \nprovides the required cool environments where density fluctuations in the \ngas can become gravitationally unstable on stellar mass scales.  It is \ntherefore reasonable to assume that stars form at a rate comparable to the \nlocal cooling rate.  It is this working hypothesis that we use in this paper.\n\nWe will restrict ourselves to the gravitational settling of the protodisk\nin the vertical direction, neglecting radial motions. This approximation\nis reasonable because of the observational fact that galactic disks are in\ngeneral rotationally supported. The hydrodynamical equations which describe \nthe evolution of the gas density $\\rho$, the vertical gas velocity $w$ and \nthe thermal gas energy by unit mass $e$ are then\n\\begin{eqnarray}\n   \\frac{\\partial \\rho}{\\partial t} + \\frac{\\partial}{\\partial z}\n   (\\rho w) & = & - \\frac{\\rho}{t_*} \\nonumber \\\\\n   \\frac{\\partial w}{\\partial t} + w \\frac{\\partial w}{\\partial z} +\n   \\frac{2}{3} \\frac{\\partial}{\\rho \\ \\partial z} (\\rho e) - g_z & = & 0 \\\\\n   \\frac{\\partial e}{\\partial t} + w \\frac{\\partial e}{\\partial z} +\n   \\frac{2}{3} e \\frac{\\partial w}{\\partial z} & = & - \\frac{e}{t_c}\n   \\;\\;\\;, \\nonumber\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\partial g_z\/\\partial z=-4\\pi G(\\rho+\\rho_*) \\;\\;\\; ,\n\\end{equation}\nwhere $\\rho_*$ is the stellar density and the other notations have their usual\nmeanings.  The source terms on the right hand side of equation (1) describe \nthe condensation of gas into stars on a timescale $t_*$ and the gaseous energy \ndissipation on a timescale $t_c=\\rho e\/\\Lambda$ with the cooling rate per unit \nvolume $\\Lambda (\\rho ,e)$. We introduce a free parameter $k=t_*\/t_c$ which \ndetermines the timescale of star formation $t_*$ relative to $t_c$.   \nNote that $k$ should be of order unity according to our assumption that star \nformation is related primarily to the cooling activity of the gas.\n\nThe cooling rate is a function of the local gas density and temperature and\ntherefore can be approximated as $\\Lambda = \\Lambda_0 \\rho^a e^b$.  Typical \nvalues are $a=2$ and, in an ionized hydrogen gas, $b=+0.5$ or $b=-0.5$ for \nfree-free cooling or free-bound cooling, respectively.  Note that possible \nheating would compete with the gas cooling and affect $a$ and $b$.  The \npower-law model also includes the possibility that energy dissipation in the \ndisk is dominated by cloud-cloud collisions. In such a case $a=2$ and $b=1.5$ \n(Larson 1969).  In order to keep the simulation simple we treat $a$ and $b$ \nas free parameters and do not specify cooling and heating sources in an \nexplicit way.\n\nThe stars will stay most of their time at maximum $z$-distance, that is\napproximately at the position where they formed initially.  This is at least\nvalid for a majority of stars that make up the disk where the gas is quickly\nconverted into stars and subsequent contraction of the gas no longer has a \nsignificant dynamical effect on the stellar orbits \n(e.g., Yoshii \\& Saio 1979).  \nAs a first approximation we therefore neglect the secular dynamical evolution\nof stars and assume that their local density is given by\n\\begin{equation}\n\\rho_*(z) = \\int_{0}^{\\infty}\\frac{\\rho(z,t)}{t_*(z,t)} dt \\ \\  .\n\\end{equation}\n\nA commonly perceived theoretical idea on the formation of galaxies\n(Fall \\& Efstathiou 1980; Blumenthal et al. 1986; Barnes \\& Efstathiou 1987)\nis that their initial state is more or less the virialized isothermal sphere\nwhere baryonic matter relaxes with dark matter.  A more recent idea from\ncosmological simulations (Katz 1992) is that spiral galaxies formed through \nthe hierarchical merging of smaller subunits that were disrupted in the \nmerging process. Whereas their dissipationless stellar and dark matter \ncomponents now constitute the visible and dark halos of disk galaxies, the \ngas could dissipate its kinetic energy, spin up and settle into the equatorial\nplane where it formed an initially extended, gaseous protodisk.  As we will \ndemonstrate below, the observed exponential vertical disk profiles indicate \nthat this protodisk was able to achieve a local isothermal equilibrium \nstate prior to the onset of star formation.  The initial disk temperature \n$T(r)$ then is only a function of the disk radius $r$ but independent of $z$, \nand the initial surface density $\\Sigma (r)$ determines the initial central \ndensity $\\rho (z=0)$ at radius $r$ and its local scale height $h_z (r)$:\n\n\\begin{equation}\n\\rho(z)=\\frac{\\Sigma}{4h_z}{\\rm sech}^2\\left(\\frac{z}{2h_z}\\right) \\;\\;,\n\\;\\;\\;\\;\\;\nh_z=\\frac{k_B T}{2\\pi G\\Sigma\\mu m_H} \\;\\; ,\n\\end{equation}\nwhere $\\mu$ is the mean molecular weight.  We use this as a reasonable initial\nsetup in our calculations.\n\n\nThe hydrodynamical equations are solved on an Eulerian staggered grid,\nwith 100 logarithmically spaced grid points in $z$ and an outer edge placed\nat a sufficiently large $z$-distance.  The set of differential equations\nis integrated numerically by means of an explicit, finite difference scheme\nwith operator splitting and monotonic transport.   This scheme provides\nsecond-order accuracy in space and time (Burkert \\& Hensler 1987).\nThe simplicity of the above equations allows us to make a comprehensive survey \nin the parameter space.\n\n\\section{Discussion of the numerical results}\n\nA number of test calculations have shown that the final stellar $z$-profile\ndepends strongly on the chosen initial distribution of the protodisk gas if \none starts from a non-equilibrium state.  On the other hand, an exponential \nstellar disk forms in all cases where we assume the gas to settle into \nisothermal equilibrium, prior to star formation and gas cooling, as long as \nthe star formation time $t_*$ is comparable to the cooling time $t_c$, that \nis, $k\\sim 1$.\n\nIn the limit of rapid star formation ($k\\gg 1$) the initial isothermal\ndistribution of the gas is frozen in the stellar disk which has more or \nless a flat-top $z$-profile near the equatorial plane. In the limit of \nslow star formation ($k\\ll 1$) the gas condenses into the equatorial \nplane towards the energy minimum state, giving rise to a power-law \n$z$-profile in the stellar disk.  Exponential profiles are in between \nthese two extremes, and our calculations show that when $k$ lies inside \na preferable range $0.3\\lower.5ex\\hbox{\\ltsima} k\\lower.5ex\\hbox{\\ltsima} 3$, the final stellar $z$-profile \nbecomes exponential, independent of the free parameters $k$, $\\Lambda_0$, \n$a$ and $b$, and also independent of $T$ and $\\Sigma$ of the isothermal \nprotodisk gas.  This remarkable result still holds when the isothermal \nequilibrium state is modified to some extent.\n\nIn Figure 1 we show as an example a sequence of models with changing $b$\n(upper panel) or changing $a$ (lower panel) while the other parameters are\nfixed as shown in the panels: $k=1$, $T=10^5$K, $\\Sigma=50M_\\odot$pc$^{-2}$,\nand $\\Lambda_0$ set to give $\\langle t_c\\rangle =10^9$yrs at the half-mass\n$z$-distance of the protodisk.  The stellar $z$-profile $\\rho_*(z)$ is almost \nperfectly exponential all the way from $z\\sim0$ to 10$h_z$ spanning about \nfive orders in density, and the scale length becomes systematically smaller \nfor larger $b$ or larger $a$.  It is evident that the final scale height \ndepends critically on the adopted values of the parameters and in general \ndiffers from the initial scale height of the gaseous protodisk.\n\nIn Figure 2 we show one of the other sequences of models with decreasing\n$\\Sigma$ from the left to the right, using $a=2$, $b=0$ and the other \nparameters fixed as in Figure 1.  $\\Sigma$ is chosen to mimic the situation \nin our own Galaxy with a surface density distribution of \n$\\Sigma=50M_{\\odot}$pc$^{-2}\\times \\exp[-(R-8$kpc)\/4kpc]\nevaluated at galactocentric radial intervals $R=2$kpc$\\times i$ with \n$i=1$ to 5.  Note that the profiles have been shifted along the horizontal \naxis.  The stellar $z$-profile $\\rho_*(z)$ is almost perfectly exponential\nfor different choices of $\\Sigma$, and the stellar disk flares with\ndecreasing surface density, that is, its thickness increases as \n$h_z\\propto\\Sigma^{-1}$.  Such a $\\Sigma$-dependence of $h_z$ would be \nexpected if the stellar system remembers the exponential tail of the \nself-gravitating isothermal disk model which was adopted as an initial \ncondition (see Eq.4).  Note however that the final exponential stellar\ndensity distribution extends much further inwards, till $z=0$.\nIncreasing $b$ while keeping $\\Sigma $ and the other parameters constant\ndecreases the scale height according to $h_z(b=1) = 0.6 \\times h_z(b=0)$\nand $h_z(b=2) = 0.4 \\times h_z(b=0)$.\n\nThe disk flaring, when seen edge-on, could largely be masked by projection\neffects.  In Figure 3 we show the edge-on projected $z$-profiles $\\Sigma_*(z)$ \nat the same radii as in Figure 2 along the disk plane from the galaxy center.  \nThe projection makes the profiles less dependent on $R$.  In such a case \nthe inversion for recovering the original density profiles $\\rho_*(z)$ is \ngenerally very unstable.  A great accuracy is needed in measuring \n$\\Sigma_*(z)$ over a wide range of $R$ and $z$, in order to detect the disk \nflaring from the observations of edge-on spiral galaxies.\n\nWhile the resulting stellar $z$-profiles are always exponential, the scale\nheight $h_z$ itself depends on the cooling rate $\\Lambda (\\rho ,e)$ through \n$a$ and $b$ and also on the initial state of the protodisk gas through $T$ \nand $\\Sigma$.  In practice, however, the deterministic quantities are only \n$T$ and $\\Sigma$ because if these are given the cooling source is virtually \nspecified.  Then, $a$ and $b$ are are no longer free parameters.  In other \nwords, given that $h_z$ is measured at sufficient accuracy along the disk \nplane, we could constrain $T$ and $\\Sigma$ as a function of radial distance\nfrom the galaxy center.  This result can be used in order to explore the\nglobal initial structure of the protodisk from the vertical disk structure \nof an edge-on galaxy observed today.\n\n\\section{Conclusions}\n \nThe presented idea that the star formation timescale $t_*$ is comparable\nto the cooling timescale $t_c$ for the origin of exponential stellar \n$z$-profiles works when the protodisk has nearly reached an equilibrium \nprior to star formation and gas cooling. \n\nUnder a realistic situation the dynamical timescale $t_{dyn}$ of the \nprotodisk may initially not be in balance with the cooling timescale $t_c$.  \nIf $t_{dyn}<t_c$ in the hot and tenuous gas, the protodisk \nachieves an equilibrium state and undergoes a\nquasi-static contraction until the gas density becomes sufficiently \nlarge so that $t_c$ falls below $t_{dyn}$.  If $t_{dyn}>t_c$ otherwise, \nthe gas cools rapidly and condenses into cool and dense clouds.  In this \nstage, cloud collisions dissipate kinetic energy and enhance the\nformation of stars. As demonstrated by Burkert et al. (1992), the increase in\nenergy input through supernova explosions will\ncompensate energy dissipation and cooling, \neventually halting the collapse and achieving a quasi-equilibrium state.\nIn either cases, after reaching its \nequilibrium, the protodisk evolves due to the feedback \neffects of star formation, independent of its initial formation history\nand of variations in global physical conditions (e.g., Lin \\& Murray 1992).\n\nGas cooling or energy dissipation induces a slow gravitational settling \nof the protodisk while, at the same time, leading to the formation of stars in \nthe disk.  This coupling yields a continuous change of various stellar \ncharacteristics (age, metallicity, color, velocity dispersion, etc) as a \nfunction of the $z$-distance from the disk plane.  Although our model is \nnot sufficiently detailed to allow an extensive test against such data, \nthe power of producing an exponential stellar $z$-profile from a vast range \nof physical conditions is very appealing and should be considered as a strong \ncandidate for understanding the universality of exponential $z$-profiles.\n\nBurkert et al. (1992) have performed more detailed simulations of the\ndynamical settling of hot protogalactic gas into the  equatorial plane, taking \ninto account star formation and heating and cooling processes in a multiphase \ninterstellar medium.  They achieved a good agreement with the observed \nvertical density structure of the Galactic disk in the solar neighborhood.  \nThe complexity of their model leads however to a very large parameter space \nwhich cannot be explored in detail due to computational limitations.   \nAmong many different processes involved in their physical model, the process \nof crucial importance is that the rate of star formation is \nadjusted sooner or later to balance with the local cooling rate by means of\nthe self-regulated star formation mechanism (Cox 1983; Franco \\& Cox 1983).  \nIt is not clear from their models, which process is most\nimportant in leading to exponential $z$-profiles.\nOur calculations demonstrate that an ideal condition for such a profile\nis  $t_*\\sim t_c$, independent of the details of heating and cooling.\n\n\\subsection*{Acknowledgments}\n\nWe thank K. Freeman and H. Saio for invaluable comments on the subject\ndiscussed in this paper.   A.B. acknowledges the financial support from \nthe Program of Human Capital Mobility, Grant Number ERBCHGECT920009 and \nof CESCA Consortium and the hospitality of the Centre d'Estudis Avancats \nde Blanes.  Y.Y. acknowledges the financial support from the Yamada Science\nFoundation, the German Academic Exchange Service (DAAD), and also Theoretical\nAstrophysics Center, Denmark, under which this work was performed.  \nThis work has been supported in part by the Grant-in-Aid for COE Research \n(07CE2002).  All calculations have been performed on the Cray YMP 4\/64 of \nthe Rechenzentrum Garching.\n\n\\newpage\n\n\\begin{center}\n{\\small REFERENCES}\n\\end{center}\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Aoki, T. E., Hiromoto, N., Takami, H., \\& Okamura, S.  1991, PASJ,\n43, 755\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Bahcall, J. N., \\& Soneira, R. M.  1980, ApJS, 44, 73\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Barnes, J., \\& Efstathiou, G.  1987, ApJ, 319, 575\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\  Blumenthal, G. R., Faber, S. M., Flores, R. A., \\& Primack, J. R.\n1986, ApJ, 301, 27\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Burkert, A., \\& Hensler, G.  1987, MNRAS, 208, 493\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Burkert, A., Truran, J., \\& Hensler, G. 1992, ApJ, 391, 651\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Burstein, D.  1979, ApJ, 234, 829\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Carlberg, R. G.  1987, ApJ, 322, 59\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Cox, D. P.  1983, ApJL, 265, 61\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Fall, S. M., \\& Efstathiou, G.  1980, MNRAS, 193, 189\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Franco, J., \\& Cox, D. P.  1983, ApJ, 273, 243\n\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Hoyle, F.  1953, ApJ, 118, 513\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Katz, N. 1992, ApJ, 391, 502\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Larson, R.B. 1969, MNRAS, 145, 405\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Lacey, C. G. 1984, MNRAS, 208, 687\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Lin, D. N. C., \\& Murray, S. D.  1992, ApJ, 394, 523\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Low, C., \\& Lynden-Bell, D.  1976, MNRAS, 176, 367\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Pritchet, C.  1983, AJ, 88, 1476\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Silk, J.  1977, ApJ, 214, 152\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Tsikoudi, V.  1977, Univ. Texas Publ. Astron. No. 10\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ van der Kruit, P. C., \\& Searle, L. 1981, A\\&A, 105, 115\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ van Dokkum, P. G., Petetier, R. F., de Grijs, R., \\& Balcells, M.  1994,\nA\\&A, 286, 415\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Villumsen, J. V. 1983, ApJ, 274, 632 \n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Yoshii, Y., \\& Saio, H.  1979, PASJ, 339, 368\n\n\\par\\hangindent=0.5cm\\hangafter=1\\noindent\n\\ Wainscoat, R. J., Freeman, K. C., \\& Hyland, A.R.  1989, ApJ, 337, 163\n\n\\clearpage\n\n\\begin{center}\n{\\small FIGURE CAPTIONS}\n\\end{center}\n\n\\vspace{1.0cm}\n\n{\\normalsize F{\\footnotesize IG}.}\n1.---The final stellar $z$-profile $\\rho_*(z)$ for different values of the\npower index $b$ (upper panel) or $a$ (lower panel) for the cooling rate defined\nas $\\Lambda = \\Lambda_0 \\rho^a e^b$, with other parameters fixed in the model,\nas shown in the figure (for details see text).\nThe $z$-profiles (thick lines) originate from the same initial\ndistribution\nof the protodisk gas (dotted line).\n\n\\vspace{1.0cm}\n\n{\\normalsize F{\\footnotesize IG}.}\n2.---The final stellar $z$-profile $\\rho_*(z)$ for different values\nof the initial surface mass density of the protodisk $\\Sigma$ (face-on\nprojected), with other parameters fixed in the model (for details see text).\nThe five $z$-profiles from the left to the right show a sequence of\ndecreasing $\\Sigma$, according to the formula appropriate to our own Galaxy:\n$\\Sigma=50M_{\\odot}$pc$^{-2}\\times \\exp[-(R-8$kpc)\/4kpc] evaluated at\ngalactrocentric radial intervals $R=2$kpc$\\times i$ with $i=1$ to 5.\nNote that the profiles have been shifted along the horizontal axis.\n\n\n\\vspace{1.0cm}\n\n{\\normalsize F{\\footnotesize IG}.}\n3.---The edge-on projected $z$-profiles $\\Sigma_*(z)$ at the same radii as\nin Figure 2 along the disk plane from the galaxy center.\nNote that the profiles have been shifted along the horizontal axis.\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{Astronomical motivation}\\label{sec:intro} \nMillimeter and submillimeter wavelength continuum emission \nis a powerful probe of the warm and cool dust in the Universe.\nFor temperatures below $\\sim$40\\,K, the peak of the thermal continuum emission is at wavelengths longer than 100\\,\\micron \n(or at frequencies lower than 3\\,THz), i.e.~in the far-infrared and (sub)millimeter\\footnote{Hereafter we will use the term {\\it (sub)millimeter} when referring, in general, to the millimeter and submillimeter wavelengths range;\nwe will use {\\it submillimeter} when strictly referring to wavelengths from one millimeter down to the far-infrared.} range.\nA number of {\\it atmospheric windows} between 200\\,GHz and 1\\,THz make ground-based observations possible over a large part of\nthis range from high-altitude, dry sites.\n\nSpecifically, thermal dust emission is well described by a gray body spectrum,\nwith the measured flux density $S_\\nu$ at frequency $\\nu$ expressed as:\n\\[ \nS_\\nu = \\Omega_{S'} ~ (1-e^{-\\tau}) ~ \\frac{2h}{c^2}\n\\frac{ \\nu^3 } {e^{h\\nu \/ kT} - 1}\n\\]\nwhere $h$ and $k$ are Planck's and Boltzmann's constants respectively,\n$c$ is the speed of light, $\\Omega_{S'}$ is the apparent source solid angle\n(the size of the physical source convolved with the telescope beam)\nand $\\tau$ is the optical depth, which varies with frequency.\nIn the (sub)millimeter range, the emission is almost always optically thin with $\\tau \\propto N \\nu^\\beta$,\nwhere $\\beta$ is in the range 1--2 typically\n(see, e.g.,~the Appendix of \\citealt{Mezger1990} or \\citealt{Beuther2002} for a more thorough discussion).\nHere, $N$ is the number of dust particles (or number of nucleons assuming a given dust-to-gas\nrelation) in the telescope beam. Accordingly, the (sub)millimeter flux\ncan be converted into dust\/gas masses, when the temperature $T$ is\nassumed or constrained via additional far-infrared measurements.\n\nSuch mass determination is one of the core issues of (sub)millimeter\nphotometry. We would like to illustrate its paramount importance with\na few examples: (sub)millimeter wavelengths mapping of low-mass\nstar-forming regions in molecular clouds have determined the dense\ncore mass spectrum, (in nearby regions) down to sub-stellar masses,\nand investigated its relationship to the Initial Mass Function\n\\citep{Motte1998}.  These studies can be extended to high-mass\nstar-forming regions \\citep[e.g.~][]{Motte2007,\nJohnstone2006}. Because of the larger distances to rarer, high-mass\nembedded objects, even relatively shallow surveys are capable of\ndetecting pre-stellar cluster clumps with a few hundreds of solar\nmasses of material at distances as far as the Galactic center. Masses\nand observed sizes yield radial density distribution profiles for\nprotostellar cores that can be compared with theoretical models\n\\citep{Beuther2002}.\n\n(Sub)millimeter continuum emission is also a remarkable tool for the\nstudy of the distant Universe. The thermal dust emission in active \ngalaxies is typically fueled by short-lived, high-mass stars, therefore\nthe far-infrared luminosity provides a snapshot of the current level of star-formation activity.\nDeep (sub)millimeter observations in\nthe Hubble Deep Field, with the Submillimeter Common User Bolometer\nArray \\citep[SCUBA,][]{1999MNRAS.303..659H} on the James Clerk Maxwell\nTelescope, attracted considerable attention with the detection of a\nfew sources without optical or near-infrared counterpart.  Sensitive\nmeasurements found dust in high redshift sources and even in some of\nthe farthest known objects in the Universe \\citep[see, e.g.,][]{Carilli2001, Bertoldi2003, Wang2008},\nrevealing star-formation rates (SFRs) that are hundreds of times higher than in\nthe Milky Way today. These detections are possible because of the so-called\n{\\it negative K-correction} first discussed by \\citet{Blain1993}: the\nwarm dust in galaxies is typically characterized by temperatures\naround 30--60\\,K. Its thermal emission dominates the spectral energy\ndistribution (SED) of luminous and ultra-luminous infrared galaxies\n(LIRGs and ULIRGs), which have maxima between 3 and 6\\,THz as a\nresult. Because of the expansion of the Universe, the peak of the\nemission shifts toward the lower frequencies with increasing\ncosmological distance, thus counteracting the dimming and benefitting \ndetection at (sub)millimeter wavelength. Consequently,\nflux-limited surveys near millimeter wavelengths yield flat or nearly\nflat luminosity selection over much of the volume of the Universe\n\\citep{Blain02}. As such, (sub)millimeter wavelengths allow unbiased\nstudies of the luminosity evolution and, therefore, of the star-formation\nhistory of galaxies over cosmological time-scales.\n\\subsection{Bolometer arrays for (sub)millimeter astronomy}\n\\begin{figure}[t]\n\\centering \n\\includegraphics[width=.96\\linewidth, bb=20 20 1364 1316, clip]{1454fig01light.eps}\n\\caption\n{\nWiring side of a naked LABOCA array. Each light-green square is a bolometer.\n}\n\\label{fig:nakedarray}\n\\end{figure}\nThe (sub)millimeter dust emission is unfortunately intrinsically weak and\nthe wish to measure it pushed the development of detectors with the best\npossible sensitivity, namely bolometers \\citep{Low1961, Mather1984}.\nMoreover, the desirability of mapping large areas of the sky,\nmotivated the development of detector {\\it arrays}.\nConsequently, the last 10 years have seen an\nincreasing effort in the development of bolometer arrays.  In such\ninstruments, a number of composite bolometers work side by side in the\nfocal plane, offering simultaneous multi-beam coverage.  Since the\narrival of the first arrays, developed in the early '90s and\nconsisting of just 7 elements, we are witnessing a rapid maturing of\ntechnology, reaching hundreds to a few thousand elements today, and\nthe prospect of even larger bolometer arrays in the future.\nThe Large APEX Bolometer Camera, LABOCA, described in this article,\nis a new bolometric receiver array for the Atacama Pathfinder Experiment 12\\,m telescope,\nAPEX\\footnote{APEX is a collaborative effort between the\nMax-Planck-Institut f\\\"ur Radioastronomie of Bonn (MPIfR,\\,50\\%), the\nEuropean Southern Observatory (ESO,\\,27\\%) and the Onsala Space\nObservatory (OSO,\\,23\\%)}.  It is the most ambitious camera in a long\nline of developments of the MPIfR bolometer group, which has delivered\ninstruments of increasing complexity to the IRAM 30 m telescope\n\\citep{Kreysa1990}: single beam receivers were supplanted by a\n7-element system \\citep{Kreysa1999}, which eventually gave way to the\nMAx-Planck Millimeter BOlometer (MAMBO) array, whose initial 37 beams\nhave grown to 117 in the latest incarnation \\citep{2002AIPC..616..262K}.\nThe group also built the 37-element 1.2\\,mm SEST Imaging Bolometer Array (SIMBA)\nfor the Swedish\/ESO Submillimeter telescope \\citep{Nyman2001}\nand the 19-element 870\\,\\micron  for the 10\\,m Heinrich-Hertz Telescope\n\\citep[a.k.a. SMTO,][]{1990SPIE.1235..503M}.\n\nThe main obstacle to observations at these wavelengths is posed by Earth's atmosphere, which is\nseen as a bright emitting screen by a continuum total power detector, as LABOCA's bolometers are.\nThis is largely due to the emission of the water vapor present in the atmosphere with only small\ncontributions from other components, like ozone. Besides, the atmosphere is a turbulent thermodynamic\nsystem and the amount of water vapor along the line of sight can change quickly, giving rise to instabilities\nof emission and transmission, called {\\it sky noise}. Observations from ground based telescopes have to go\nthrough that screen, therefore requiring techniques to minimize those effects.\n\nThe technique most widely used is to operate a switching device,\nusually a chopping secondary mirror (hereafter called {\\it wobbler})\nto observe alternatively the source and a blank sky area close to it,\nat a frequency higher than the variability scale of the sky noise.\nThis method, originally introduced for observations with single pixel detectors, is today also used with arrays of bolometers.\nAlthough it can be very efficient to reduce the atmospheric disturbances during observations,\nit presents some disadvantages: among others, the wobbler is usually slow (1 or 2 Hz) posing a\nstrong limitation to the possible scanning speed.\n\nLABOCA has been specifically designed to work without a wobbler and using a different technique,\nwhich works particularly well when using an array of detectors, to remove the atmospheric contribution.\nThis technique, called {\\it fast scanning} \\citep{2001A&A...379..735R}, is based on the idea that, when observing\nwith an array, the bolometers composing the array look simultaneously at different points in the sky,\ntherefore chopping is no more needed.\nA modulation of the signal, still required to identify the astronomical source through the atmospheric\nemission, is produced by scanning with the telescope across the source.\nThe atmospheric contribution (as well as part of the instrumental noise) will be strongly correlated in\nall bolometers and a post-detection analysis of the correlation across the array will \nmake it possible to extract the signals of astronomical interest from the atmospheric foregrounds.\nMoreover, the post-detection bandwidth depends on the scanning speed, therefore relatively high\nscanning speeds are ideal \\citep[see also][]{Kovacs2008}.\n\nThe APEX telescope \\citep{2006A&A...454L..13G}, as the name implies,\nserves as a pathfinder for the future large-scale (sub)millimeter\nwavelength and (far)infrared missions, namely the Atacama Large\nMillimeter Array (ALMA), the Herschel Space Observatory and the\nStratospheric Observatory for Infrared Astronomy (SOFIA). Its\npathfinder character is on the one hand defined by exploring\nwavelength windows that have been poorly studied before, with\nacceptable atmospheric transmission at the 5100\\,m altitude site.  On\nthe other hand, and more importantly, it can perform large area\nmapping to identify interesting sources for ALMA follow-up studies at\nhigher angular resolution.  Moreover, APEX produces images of both\ncontinuum (with LABOCA) and line emission with angular resolutions\nthat neither Herschel's nor SOFIA's smaller telescopes (with diameters\nof 3.5 and 2.5\\,m respectively) can match.  This provides a critical\nadvantage to APEX for imaging dust and line emission at high frequencies. \n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=.90\\linewidth,bb=22 22 699 535, clip]{1454fig02light.eps}\n\\caption\n{\nScheme of the infrastructure of LABOCA. The instrument is located in the Cassegrain cabin (dashed red line)\nof APEX, remote operation is a requirement and all the communication goes through\nthe local area network (magenta arrows) and a direct Ethernet link between {\\it backend} and {\\it bridge} computers.\nThe black lines show the flow of the bolometer signals, the orange ones show the configuration and monitoring communication.\n}\n\\label{fig:infrastructure}\n\\end{figure*}\n\\subsection{Instrument overview}\\label{sec:overview}\nLABOCA is an array of bolometers, operated in total-power mode, specifically designed \nfor fast mapping of large areas of sky at moderate resolution and with high \nsensitivity and was commissioned in May 2007 as facility instrument on APEX.\nIt is a very complex system, composed of parts that originate in a variety of\nfields of technology, in particular optics, high vacuum,\nlow temperature cryogenics, digital electronics, computer hardware and\nsoftware, and others.  A general view of the infrastructure is shown\nin the block diagram of Fig.~\\ref{fig:infrastructure}.\nThe heart of LABOCA is its detector array made of 295 semiconducting composite bolometers\n(see Fig.~\\ref{fig:nakedarray}, Fig.~\\ref{fig:arraypics}).\nA description of the detector array design and manufacture is provided in \\S\\ref{sec:detector}.\n\nThe bolometer array is mounted in a cryostat, which uses liquid\nnitrogen and liquid helium on a closed cycle double-stage sorption\ncooler to reach an operation temperature of $\\sim$285\\,mK. The\ncryogenic system is discussed in \\S\\ref{sec:cryo}.\n\nA set of cold filters, mounted on the liquid nitrogen and liquid\nhelium shields, define the spectral passband, centered at a wavelength\nof 870\\,\\micron  and about 150\\,\\micron  wide (see\nFig.~\\ref{fig:passband}).  A monolithic array of conical horn\nantennas, placed in front of the bolometer wafer, concentrates the\nradiation onto the individual bolometers.  The filters and the horn\narray are presented in \\S\\ref{sec:coptics}.\n\nThe cryostat is located in the Cassegrain cabin of the APEX telescope\n(see Fig.~\\ref{fig:receiver}) and the optical coupling to the\ntelescope is provided by an optical system made of a series of metal\nmirrors and a lens which forms the cryostat window.  The complex\noptics layout, manufacture and installation at the telescope in\ndescribed in \\S\\ref{sec:optics}.\n\nThe bolometer signals are routed through low noise, unity gain\nJunction Field Effect Transistor (JFET) amplifiers and to the outside\nof the cryostat along flexible flat cables.  Upon exiting the\ncryostat, the signals pass to room temperature low noise amplifiers\nand electronics also providing the AC current for biasing the\nbolometers and performing real time demodulation of the signals.  The\nsignals are then digitized over 16 bits by 4 data acquisition boards\nproviding 80 analog inputs each, mounted in the {\\it backend\ncomputer}. The backend software provides an interface to the\ntelescope's control software, used to set up the hardware, and a data\nserver for the data output.  The acquired data are then digitally\nfiltered and downsampled to a lower rate in real time by the {\\it\nbridge computer} and finally stored in MB-FITS format\n\\citep{2006A&A...454L..25M} by the FITS writer embedded in the\ntelescope's control software.  Another computer, the {\\it frontend\ncomputer}, is devoted to monitor and control most of the\nelectronics embedded into the receiver (e.g.~monitoring of all the\ntemperature stages, controlling of the sorption cooler, calibration\nunit) and provides an interface to the APEX control software, allowing\nremote operation of the system.  Discussions of the cold and warm\nreadout electronics, plus the signal processing, are found in\n\\S\\ref{sec:jfet} and \\S\\ref{sec:electronics}, respectively.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=.92\\linewidth, bb= 20 20 1147 846,clip]{1454fig03light.eps}\n\\caption\n{\nOverview of the optics. On the left, the APEX telescope.\nOn the right, a zoom on the tertiary optics installed in the Cassegrain cabin.\nSee also Fig.~\\ref{fig:receiver}.\n}\n\\label{fig:opticsall}\n\\end{figure*}\n\nIn \\S\\ref{sec:obsmodes} we describe the\nsophisticated observing techniques used with LABOCA,\nsome of them newly developed. \nThe instrument performance on the sky is described in \\S\\ref{sec:performance},\nalong with information on sensitivity, beam shape and noise behavior.\n\nThe reduction of the data is performed using a new data reduction\nsoftware included in the delivery of LABOCA as facility instrument,\nthe BoA (Bolometer array data Analysis) data reduction software package.\nAn account of on-line and off-line data reduction is given in \\S\\ref{sec:redu}.\n\nSome of the exciting science results already obtained with LABOCA or\nexpected in the near future are outlined in \\S\\ref{sec:science}. Our\nplans for LABOCA's future are briefly presented in \\S\\ref{sec:future}.\n\\section{Tertiary optics}\\label{sec:optics}\n\\subsection{Design and optimization} \nThe very restricted space in the Cassegrain cabin of\nAPEX and a common first mirror (M3) with the APEX SZ Camera\n\\citep[ASZCa,][]{2003NewAR..47..933S} introduced many boundary\nconditions into the optical design. Eventually, with the help of the\nZEMAX\\footnote{\\tiny{\\url{http:\/\/www.zemax.com\/}}} optical design program, a\nsatisfactory solution was found, featuring three aspherical off-axis\nmirrors (M3, M5, M7), two plane mirrors (M4, M6) and an aspherical\nlens acting as the entrance window of the cryostat (see\nFig.~\\ref{fig:opticsall}). Meeting the spatial constraints, without\nsacrificing optical quality, is facilitated considerably by the\naddition of plane mirrors.\nThe design of the optics was made at the MPIfR in coordination with N.\\ Halverson\\footnote{Formerly\nat University of California at Berkeley, now at University of\nColorado, Boulder, USA} with respect to sharing the large M3 mirror\nwith the ASZCa experiment.\n\nThe maximum possible field diameter of APEX, as limited by the\ndiameter of the Cassegrain hole of the telescope's primary, is about\n0.5 degrees. LABOCA, with its 295 close-packed fully efficient horns,\ncovers an almost circular field of view (hereafter FoV) of about 0.2\ndegrees in diameter (or about 100 square arcminutes).  The task of the\ntertiary optics is to transform the f-ratio from f\/8 at the Cassegrain\nfocus to f\/1.5 at the horn array, while correcting the aberrations\nover the whole FoV of LABOCA under the constraint of parallel output\nbeams.  The final design is diffraction limited even for 350\\,\\micron \nwavelength, the Strehl ratio is better than 0.994 and the maximum\ndistortion at the focal plane is less than 10\\% over the entire FoV\n(see also Fig.~\\ref{fig:beams}).\n\\begin{figure}[t] \n\\centering \n\\includegraphics[width=.95\\linewidth,bb=20 20 575 436, clip]{1454fig04toplight.eps}\n\\includegraphics[width=.95\\linewidth, bb=20 20 560 740,clip]{1454fig04bottomlight.eps} \n\\caption\n{\n{\\it Top:}~The mirrors affixed to the floor of the Cassegrain cabin of APEX.  Left to right: M7, M5, M3 and\none of the mirrors of the ASZCa experiment.  In this picture, mirror M3 is positioned for ASZCa.\n{\\it Bottom:}~The receiver, M3 in the position for LABOCA, M5, M6 and M7.\nSee also Fig.~\\ref{fig:opticsall}.\n}\n\\label{fig:receiver}\n\\end{figure}\n\\subsection{Manufacture} \nMirror M3, an off-axis paraboloid, has been\nmanufactured by a machine shop at the Lawrence Berkeley National\nLaboratory (LBNL, Berkeley, CA, USA) and is common to both LABOCA and the ASZCa\nexperiment.  M3 has a diameter of 1.6\\,m and a surface accuracy of\n18\\,\\micron  rms but LABOCA uses only the inner 80\\,cm disk.  It is\nattached to a bearing on the floor of the Cassegrain cabin, aligned to\nthe optical axis of APEX.  Operators can manually rotate the mirror in\norder to direct the telescope beam to LABOCA or to ASZCa alternatively\n(see Fig.~\\ref{fig:receiver}).\n\nMirrors M4 and M6 are flat, have a diameter of 42\\,cm and 26\\,cm,\nrespectively (manufactured by Kugler\\footnote{\\tiny{\\url{http:\/\/www.kugler-precision.com\/}}}, Salem, Germany).\nThey are of optical quality and are both affixed to the ceiling of the cabin.\nIn a near future, mirror M6 will be replaced by the reflection-type half-wave\nplate of the PolKa polarimeter \\citep{2004A&A...422..751S}.\n\nMirrors M5 and M7 are off-axis aspherics, both 50\\,cm in diameter, and\nare affixed to the floor of the cabin. They have been designed and\nmanufactured at the MPIfR and have a surface accuracy of 7 and\n5\\,\\micron  rms respectively.\n\\subsection{Installation and alignment} \nAll the mirrors of LABOCA\n(with exception of M3) and the receiver itself are mounted on hexapod positioners (see Fig.~\\ref{fig:receiver})\nprovided by VERTEX Antennentechnik\\footnote{\\tiny{\\url{http:\/\/www.vertexant.de\/}}} (Duisburg, Germany).\nEach hexapod is made of an octahedral assembly of struts and has six degrees of freedom (x, y, z, pitch,\nroll and yaw).  The lengths of the six independent legs can be changed\nto position and orient the platform on which the mirror is mounted.\nVERTEX provided a software for calculating the required leg extensions\nfor a given position and orientation of the platforms.\nA first geometrical alignment was performed during the first week of September\n2006, using a double-beam laser on the optical axis of the telescope\nand plane replacement mirrors in place of the two active mirrors M5\nand M7. The alignment has been checked using the bolometers and hot\ntargets (made of absorbing\\footnote{ECCOSORB AN, Emerson \\& Cuming, Rundolph, MA, USA, \\tiny{\\url{http:\/\/www.eccosorb.com}}} material)\nat different places along the beam, starting at the focal plane and\nfollowing the path through all the reflections up to the receiver's\nwindow.  The alignment has been furthermore verified and improved in\nFebruary 2008.\n\\section{Cryogenics}\\label{sec:cryo}\n\\subsection{Cryostat} \nThe bolometer array of LABOCA is designed to be operated at a temperature\nlower than 300\\,mK.  This temperature is provided by a cryogenic\nsystem made of a wet cryostat, using liquid nitrogen and liquid\nhelium, in combination with a two-stage sorption cooler.  A commercial\n8-inch cryostat, built by Infrared Labs\\footnote{\\tiny{\\url{http:\/\/www.irlabs.com\/}}} (Tucson, AZ, USA), has been\ncustomized at the MPIfR to accommodate the double-stage sorption\ncooler, the bolometer array, cold optics and cold electronics.  A high\nvacuum in the cryostat is provided by an integrated turbomolecular\npump backed by a diaphragm pump.  Operational vacuum is reached in one\nsingle day of pumping.\n\nThe cryostat incorporates a 3-liter reservoir of liquid nitrogen and a\n5-liter reservoir of liquid helium.  After producing high-vacuum\n($\\sim$10$^{-6}$\\,mbar), the cryostat is filled with the liquid\ncryogens.  The liquid nitrogen is used to provide thermal shielding at\n77\\,K in our labs in Bonn (standard air pressure, 1013\\,mbar) and at\n73.5\\,K at APEX (5107\\,m above the sea level) where the air pressure\nis almost one half of the standard one (about 540\\,mbar).\n\nThe liquid helium provides a thermal shielding at 4.2\\,K at standard\npressure and 3.7\\,K at the APEX site.  To keep it operational, the\nsystem must be refilled once per day.  The refilling operation\nrequires about 20~minutes.\n\\subsection{Sorption cooler}\nThe cryostat incorporates a commercial two-stage closed-cycle sorption\ncooler, model SoCool \\citep{2002AIPC..613.1233D} manufactured by\nAir-Liquide\\footnote{\\tiny{\\url{http:\/\/www.airliquide.com\/}}} (Sassenage,\nFrance).  In this device, a \\element[][4]{He} sorption cooler is used to\nliquefy \\element[][3]{He} gas in the adjacent, thermally coupled, \\element[][3]{He}\ncooler.  The condensed liquid \\element[][3]{He} is then sorption pumped to\nreach temperatures as low as 250\\,mK, in the absence of a thermal\nload.  Therefore, the double stage design makes it possible to cool the\nbolometer array down to a temperature lower than 300\\,mK starting from the\ntemperature of the liquid helium bath at atmospheric pressure. This\nmakes the maintenance of the system much simpler than that of other\nsystems, where pumping on the liquid helium bath is required.  The two\nsorption coolers are closed systems, which means they do not require\nany refilling of gas and can be operated from the outside of the\ncryostat, simply by applying electrical power.\n\nTo keep the bolometers at operation temperature, the sorption cooler\nneeds to be recycled.  The recycling is done by application of a\nsequence of voltages to the electric lines connected to thermal\nswitches and heaters integrated in the sorption cooler. A typical\nrecycling procedure requires about two hours and can be done manually\nor in a fully automatic way controlled by the frontend computer (see\nSect.~\\ref{sec:fepc}).  At the end of the recycling process, both\ngases, \\element[][4]{He} and \\element[][3]{He}, have been liquefied and the controlled\nevaporation of the two liquids provides a stable temperature for many\nhours.  After the recycling of the sorption cooler, the bolometer\narray reaches 285\\,mK.  The hold time of the cooler, usually between\n10 and 12 hours, strongly depends on the parameters used during the\nrecycling procedure.\nThe end temperature is a function of elevation and can be affected by telescope movements,\nleading to temperature fluctuations $\\la500\\,$\\microK within one scan,\nduring regular observations, and $\\la3\\,$mK for wide elevation turns\n(e.g. during skydips, see Sect.~\\ref{sec:skydips}).\n\\subsection{Temperature monitor}\\label{sec:thermonitor}\nThe cryostat of LABOCA incorporates 8\nthermometers to measure the temperature at the different stages:\nliquid nitrogen, liquid helium, the two sorption pumps, the two\nthermal switches, evaporator of the \\element[][4]{He} and evaporator of the\n\\element[][3]{He}. Two LS218\\footnote{\\tiny{\\url{http:\/\/www.lakeshore.com\/temp\/mn\/218po.html}}}\ndevices (Lake Shore Inc., Westerville, OH, USA) are used to monitor the thermometers\nand to apply the individual temperature calibrations in real-time. \nThe temperature of the \\element[][3]{He} stage is measured with higher accuracy with the use of a\nresistance bridge AVS-47\\footnote{\\tiny{\\url{http:\/\/www.picowatt.fi\/avs47\/avs47.html}}}\n(Picowatt, Vantaa, Finland), with an error of $\\pm$5\\,\\microK.\nControl and monitor of the cryogenic\nequipment can be done remotely via the frontend computer (see\nSect.~\\ref{sec:fepc}).\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.98\\linewidth, bb=124 93 764 523, clip]{1454fig05light.eps}\n\\caption\n{\nSpectral response of LABOCA, relative to the maximum.\nThe central frequency is 345\\,GHz, the portion with 50\\% or more\ntransmission is between 313 and 372\\,GHz.\n}\n\\label{fig:passband}\n\\end{figure}\n\\section{Cold optics}\\label{sec:coptics}\n\\subsection{Passband definition}\\label{sec:passband}\nInside the cryostat, a set of cold filters, mounted on\nthe liquid nitrogen and liquid helium shields define the spectral\npassband, centered at a wavelength of 870\\,\\micron  (345\\,GHz) and about\n150\\,\\micron  (60\\,GHz) wide (see Fig.~\\ref{fig:passband}).  The filters\nhave been designed and manufactured at MPIfR in collaboration with the\ngroup of V.~Hansen (Theoretische Elektrotechnik\\footnote{\\tiny{\\url{http:\/\/www.tet.uni-wuppertal.de\/}}}, Bergische\nUniversit\\\"at Wuppertal, Germany) who provided theoretical support and\nelectromagnetic simulations.  The passband is formed by an\ninterference filter made of inductive and capacitive meshes embedded\nin polypropylene.  The low frequency edge of the band is defined by\nthe cut-off of the cylindrical waveguide of each horn antenna (see\nalso Sect.~\\ref{sec:horns}).  A freestanding inductive mesh behind the\nwindow-lens provides shielding against radio interference.\n\\subsection{Horn array}\\label{sec:horns} \nA monolithic array of conical\nhorn antennas, placed in front of the bolometer wafer, concentrates\nthe radiation onto the bolometers.  295 conical horns have been\nmachined into a single aluminum block by the MPIfR machine shop. In\ncombination with the tertiary optics, the horn antennas are optimized\nfor coupling to the telescope's main beam at a wavelength of\n870\\,\\micron . The grid constant of the hexagonal array is\n4.00\\,mm. Each horn antenna feeds into a circular wave guide with a\ndiameter of 0.54\\,mm, acting as a high-pass filter.\n\\section{Detector}\\label{sec:detector}\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=.32\\linewidth, bb= 0 0 200 200, clip]{1454fig06leftlight.eps}\n\\includegraphics[width=.32\\linewidth, bb= 0 0 200 200, clip]{1454fig06centerlight.eps}\n\\includegraphics[width=.32\\linewidth, bb= 0 0 200 200, clip]{1454fig06rightlight.eps}\n\\caption\n{\nPictures of the bolometer array of LABOCA.\n{\\it Left:}~A detail of the array mounted in its copper ring. Some bonding wires are visible.\n{\\it Center:}~The side of the array where the bolometer cavities are etched in the silicon wafer.\n{\\it Right:}~Wiring side of the array. The thermistors are visible as small cubes on the membranes.\nOne broken membrane is visible on the top left corner. See also Fig.~\\ref{fig:nakedarray}.\n}\n\\label{fig:arraypics}\n\\end{figure*}\n\\subsection{Array design and manufacture} \\label{sec:array}\nThe bolometer array of LABOCA is made of 295 composite bolometers arranged in an hexagonal\nlayout consisting of a center channel and 9 concentric hexagons (see Fig.~\\ref{fig:nakedarray}\nand Fig.~\\ref{fig:arraypics}). The array is manufactured on a 4-inch silicon wafer coated on both sides with\na silicon-nitride film by thermal chemical vapor deposition.  On one side of the wafer,\n295 squares are structured into the silicon-nitride film used as a mask for the alkaline KOH\netching of the silicon, producing freestanding, unstructured silicon-nitride membranes, only\n400\\,nm thick (see the center picture in Fig.~\\ref{fig:arraypics}).\nOn the other side, the wiring is created by microlithography of\nniobium and gold thin metal layers.  The bolometer array is mounted\ninside a gold-coated copper ring and is supported by about 360 gold\nbonding thin wires (see Fig.~\\ref{fig:arraypics},~left), providing the\nrequired electrical and thermal connection. This copper ring\nalso serves as a mount for the backshort reflector, at $\\lambda$\/4 distance from\nthe array, and 12 printed circuit boards hosting the load resistors\nand the first electronic circuitry (see Sect.~\\ref{sec:jfet}).\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.94\\linewidth, bb=0 0 233 173, clip]{1454fig07light.eps}\n\\caption\n{\nWiring side of one bolometer of LABOCA, seen under a microscope.\nThe membrane is the green colored area. The black box is the NTD thermistor.\nThe wiring is made of gold (yellow) and niobium (gray) thin metal layers.\nSee also Fig. \\ref{fig:arraypics}.\n}\n\\label{fig:bolo}\n\\end{figure}\n\\subsection{Bolometer design and manufacture} \nThe description of a composite bolometer can be simplified as the combination of two elements:\nan extremely sensitive temperature sensor, called thermistor, and a radiation absorber. \nIn the bolometers of LABOCA, the absorbing element is made of a thin film of titanium deposited on\nthe unstructured silicon-nitride membranes.  LABOCA uses neutron-transmutation doped germanium semiconducting chips\n\\cite[NTD,][]{1982STIN...8328412H} as thermistors.  The NTD thermistors are made from ultra-pure germanium doped by\nneutron-transmutation in a nuclear reactor.  The thermistors, needed to measure the temperature of the absorber,\nare soldered on the bolometer membranes to gold pads connected to the outer edge of the silicon wafer through a\npair of patterned niobium wires, superconducting at the operation temperature (see Fig.~\\ref{fig:bolo} and Fig. \\ref{fig:arraypics}, right).\nThe soldering of the thermistors was the only manual step in the manufacture of the bolometers.\nLABOCA has so-called {\\it flatpack} NTD thermistors, which have two ion\nimplanted and metalized contacts on one side of the thermistor block.\nThe chips are optimized to work at a temperature lower than 300\\,mK,\nwhere they show an electric impedance in the range of 1 to 10\\,MOhms.\n\\section{Cold electronics}\\label{sec:jfet} \n\\subsection{Bias resistors}\nGiven the high impedance of the NTD thermistors, the electric scheme\nof the first bolometer circuitry requires very high impedance load\nresistors, which are needed to current bias the bolometers.  These\nbias resistors are 312 identical 30\\,MOhm chips, made of a nichrome\nthin film deposited on silicon substrate (model MSHR-4 produced by\nMini-Systems Inc.\\footnote{\\tiny{\\url{http:\/\/www.mini-systemsinc.com\/}}},\nN. Attleboro, MA, USA) mounted on 12 identical printed circuit boards, to\nform 12 groups of 26 resistors. This configuration is reflected in the\nfollowing distribution of the bolometer signals.  The circuit boards\nare mounted on the same copper ring which holds the bolometer array\n(see Sect.~\\ref{sec:array})\nand electrically connected to the bolometers through miniature RF\nfilters.\n\n\\subsection{Junction Field Effect Transistors (JFETs) source followers} \nThe high impedance of the bolometers makes the system\nsensitive to microphonic noise pickup, therefore JFETs (Toshiba 2SK369)\nare used as source followers in order to decrease the impedance of the\nelectric lines down to a few kOhms before they reach the high gain\namplification units at room temperature.  Following the wiring scheme\nof the bias resistors, the JFETs are also in groups of 26 soldered\nonto 12 printed circuit boards, electrically connected to the\ncorresponding bias resistors by 12 flat cables made of manganin traces embedded in\nKapton\\footnote{DuPont, \\tiny{\\url{http:\/\/www2.dupont.com\/Kapton\/en_US\/}}}\n(manufactured by VAAS Leiterplattentechnologie\\footnote{\\tiny{\\url{http:\/\/www.vaas-lt.de\/}}},\nSchw\\\"abisch-Gm\\\"und, Germany), thermally shunted to the liquid helium tank.\nThe 12 JFET boards are assembled in groups of\nthree into four gold-coated copper boxes, thermally connected to the\nliquid nitrogen tank.  Inside each box, during regular operation, the\n78 JFETs are self-heated to a temperature of about 110\\,K, where they\nshow a minimum of their intrinsic noise.  Through the connections in\nthe JFET boxes, the wiring scheme of 12 groups of 26 channels is\ntranslated to a new scheme of 4 groups of 80 channels.\n\\section{Warm electronics and signal processing}\\label{sec:electronics}\n\\subsection{Amplifiers}\\label{sec:ampli}\nThe 312 channels exiting the\ncryostat of LABOCA are distributed to 4 identical, custom made,\namplification units, providing 80 channels each. Of these, 295 are\nbolometers, 17 are connected to 1\\,MOhm resistors mounted on the\nbolometer ring (used for technical purposes, like noise monitoring and\ncalibrations) and the remaining 8 are not connected. Each\namplification unit is made of 16 identical printed circuit boards and\neach board provides 5 low noise, high gain amplifiers. Each unit also\nincludes a low noise battery used to generate the bias voltage and the\ncircuitry to produce the AC biasing and perform real time demodulation\nof the 320 signals. The AC bias reference frequency is not internally\ngenerated but is provided from the outside, thus allowing\nsynchronization of the biasing to an external frequency source.\n\nEach amplification unit is equipped with a digital interface controlled by a microprocessor programmed to provide remote control of\nthe amplification gain and of the DC offset removal procedure. This is required because LABOCA is a total power receiver and the\nsignals carry a floating DC offset which could exceed the dynamic range of the data acquisition system.\nTo avoid saturation, therefore, the DC offsets are measured and subtracted from the signals at the beginning of every integration.\nThe values of the 320 offsets are temporarily stored in a local memory and, at the end of the observation, are written into the\ncorresponding data file for use in the data reduction process.\nThe digital lines use the I$^2$C protocol\\footnote{Inter-Integrated Circuit, a serial bus to connect hardware devices.}\nand are accessible remotely via the local network through I$^2$C-to-RS232\\footnote{Recommended Standard 232, a standard for serial\nbinary data communication.} interfaces controlled by the frontend computer (see Sect.~\\ref{sec:fepc}).\nThe amplification gain can be set in the range from 270 to 17280.\n\\subsection{Data acquisition}\\label{sec:daq}\nThe 320 output signals from the 4 amplification units are digitized over 16 bits by 4 multifunction data acquisition (DAQ) boards\n(National Instruments\\footnote{\\tiny{\\url{http:\/\/www.ni.com\/}}} M-6225-PCI),\nproviding 80 analog inputs each and synchronized to the same sample\nclock by a RTSI\\footnote{Real-Time System Integration, a bus used to share and exchange\ntiming and control signals between multiple boards.} bus . The maximum data\nsampling is 2500\\,Hz and the dynamic range can be selected over 5\npredefined ranges. The four boards provide also 24 digital\ninput\/output lines each, some of them used for the generation of the\nbias reference frequency and to monitor the digital reference signals\n(sync\/blank) of the wobbler.  For the time synchronization of the data\nto the APEX control software \\citep[APECS,][]{2006A&A...454L..25M} the\ndata acquisition system is equipped with a precision time interface\n(PCI-SyncClock32\\footnote{\\tiny{\\url{http:\/\/www.brandywinecomm.com}}} from Brandywine Communications, Tustin, CA, USA)\nsynchronized to the station GPS clock via\nIRIG-B\\footnote{Inter Range Instrumentation Group, standardized time code format.} time code signal.\n\nThe AC bias reference frequency is provided by the data acquisition\nsystem as a submultiple of the sampling frequency, thus synchronizing\nthe bias to the data sampling.  Typical values used for observations\nare 1\\,kHz for the sampling rate and 333\\,Hz for the AC bias.  The\nbackend computer has two network adapters: one is connected to the\nlocal area network, the other one is exclusively used for the output\ndata stream and is connected in a private direct network with\nthe bridge computer (see also Sect.~\\ref{sec:bridge}).\n\nThe data acquisition software is entirely written using\nLabVIEW\\footnote{\\tiny{\\url{http:\/\/www.ni.com\/labview\/}}} (National Instruments).\nThe drivers for the data acquisition hardware are provided by the NI-DAQmx\\footnote{\\tiny{\\url{http:\/\/www.ni.com\/dataacquisition\/nidaqmx.htm}}} package.\nCustom drivers for LabVIEW have been developed to access the GPS clock interface.\nThe backend software runs a server to stream the output data to the bridge computer and allows remote control\nand monitoring of the operation via a CORBA\\footnote{Common Object Request Broker Architecture, a set of\nstandards which define the protocol for interaction between the objects of a distributed system} object interfaced\nto the APECS through the local area network.\n\\subsection{Anti-aliasing filtering and downsampling}\\label{sec:bridge}\nThe amplification units of LABOCA use the bias signal, which is a square waveform,\nas reference to operate real-time demodulation of the AC-biased bolometer signals.\nTherefore, all the frequencies present in the bolometer readout lines end up aliased\naround the odd-numbered harmonics of the bias frequency. Microphonics pickup by the\nhigh-impedance bolometers at a few resonant frequencies can produce a forest of lines in\nthe final readout, polluting even the lower part of the post-detection frequency band,\nwhere the astronomical signals are expected.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.96\\linewidth, bb=0 0 726 551,clip]{1454fig08light.eps}\n\\caption\n{\nNoise spectra for selected bolometers on a room temperature absorber (capped optics). \nThe signals, sampled at 1\\,kHz, were downsampled in real-time to 25\\,Hz by the bridge computer.\nThe spectra are free of microphonic pick-ups and show the $1\/f$ noise onset at $\\sim$0.1\\,Hz.\n}\n\\label{fig:noisespectra}\n\\end{figure}\nTo overcome this, we introduced an intermediate stage into the\nsampling scheme, the so-called {\\it bridge computer}. The bolometer\nsignals, acquired by the backend at a relatively high sampling rate\n(usually 1\\,kHz), well above the rolling-off of the anti-alias filters\nembedded in the amplifiers, are sent to the bridge computer where they\nare digitally low-pass filtered and then downsampled to a much lower\nrate (usually 25\\,Hz), more appropriate for the astronomical signals\nproduced at the typical scanning speeds (see also\nSect.~\\ref{sec:obsmodes}).  The digital real-time anti-alias filtering\nand downsampling is performed by a non-recursive convolution filter\nwith a Nutall window such that its rejection at the Nyquist frequency\nis $\\sim$3\\,dB and falls steeply to $\\sim$100\\,dB soon beyond that.\nThe bias reference frequency was accordingly selected to maximize the\nastronomically useful post-detection bandwidth.  The resulting\nbolometer signals are generally white between 0.1--12.5\\,Hz and free\nof unwanted microphonic interference (see Fig.~\\ref{fig:noisespectra}).\n\nTo seamlessly integrate the bridged readout into the APEX control\nsystem, the bridge computer also acts as a fully functional virtual\nbackend that forwards all communication between the actual backend\ncomputer and the control system, while intercepting and reinterpreting\nany commands of interest for the downsampling scheme.\n\\subsection{Frontend computer}\\label{sec:fepc}\nThe so-called {\\it frontend computer} communicates with the hardware\nof LABOCA through the local area network. It is devoted to monitor and\ncontrol most of the electronics of the instrument (e.g.~monitoring of\nall the temperature stages, control of the sorption cooler,\ncalibration unit, power lines\\dots) and also provides a CORBA object\nfor interfacing to APECS, allowing remote operation of the system.\nThe frontend software is entirely written using LabVIEW and custom\ndrivers have been developed for some hardware devices embedded in LABOCA.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.94\\linewidth, bb=33 3 259 420, clip]{1454fig09light.eps}\n\\caption{\nExamples of raster-spiral patterns. The two plots show the scanning pattern of the central beam\nof the array in horizontal coordinates.  The compact pattern shown in\nthe top panel is optimized to map the field of view of LABOCA.\nThe large scale map shown in the bottom panel consists of 25 raster\npositions and covers a field of $0.5\\times0.5$ degrees.  The complete\nscan required 19 minutes and the final map covers about\n$2000\\times2000$ arcseconds with uniform residual rms noise.\n}\n\\label{fig:spiralrast}\n\\end{figure}\n\n\\section{Observing modes}\\label{sec:obsmodes} \n\\subsection{Mapping modes}\\label{sec:mapmodes}\nIn order to reach the best signal-to-noise ratio using the fast scanning technique \\citep{2001A&A...379..735R}\nwith LABOCA, the frequencies of the signal produced by scanning across the source need to match the white noise part of\nthe post-detection frequency band (0.1--12.5\\,Hz, see Fig.~\\ref{fig:noisespectra}), mostly above the frequencies of the\natmospheric fluctuations. Thus, with the $\\sim$19$''$~beam (see Sect.~\\ref{sec:beam}),\nthe maximum practical telescope scanning speed for LABOCA is about $4'$\/s.\nThis is also the limiting value to guarantee the required accuracy in the telescope position information of each sample.\nThe minimum scanning speed required for a sufficient source modulation depends on the atmospheric stability and\non the source structure and is typically about $30''$\/s.\nThe APEX control system currently supports two basic scanning modes: on-the-fly maps (OTF) and spiral scanning patterns.\n\\subsubsection{Spirals}\nSpirals are done with a constant angular speed and an increasing radius, therefore the linear scanning velocity\nis not constant but increases with time. We have selected two spiral modes of 20\\,s and 35\\,s integration time,\nboth producing fully sampled maps of the whole FoV with scanning velocities limited between $1'$\/s and $2\\rlap{.}'5$\/s.\nThe spiral patterns are kept compact (maximum radius $\\lesssim2'$), the scanned area on the sky is only slightly\nlarger than the FoV and most of the integration time is spent on the central $11'$ of the array.\nThese spirals are the preferred observing modes for pointing scans on sources with flux densities down to a few Jy.\n\nFor fainter sources, the basic spiral pattern can be combined with a raster mapping mode (raster-spirals) on a grid\nof pointing positions resulting in an even denser sampling of the maps and longer integration time (see Fig.~\\ref{fig:spiralrast}, top panel).\nThese compact raster-spirals give excellent results for sources smaller than the FoV of LABOCA\nand are also suitable for integrations of very faint sources.\n\nThe flexibility in the choice of the spiral parameters also allows spiral\nobserving patterns to be used to map fields much larger than the FoV.\nThe bottom panel of Fig.~\\ref{fig:spiralrast} shows an example of\nraster of spirals optimized to give an homogeneous coverage across\na field of $0.5\\times0.5$ degrees. In this case, the spirals start\nwith a large radius and follows an almost circular scanning pattern\nfor each raster position.  This mapping mode is very useful for\ncosmological deep field surveys since co-adding several such\nraster-spiral scans, taken at different times and thus at different\norientations, provides an optimal compromise between telescope\noverheads, uniform coverage and cross-linking of individual map\npositions \\citep[see ][]{Kovacs2008}.\n\n\\subsubsection{On-the-fly maps (OTF)}\nOTF scans are rectangular scanning patterns produced moving\nback-and-forth along alternating rows with a linear constant speed and\naccelerating only at the turnarounds.  They can be performed in\nhorizontal or equatorial coordinates and the scanning direction can be\nrotated relatively to the base system for both coordinate systems. OTF\npatterns have been tested for maps on the scales of the FoV up to long\nslews across the plane of the Milky Way (2 degrees).  Small\ncross-linked OTFs (of size $\\sim$FoV of LABOCA) give results\ncomparable to the raster-spirals \\citep{Kovacs2008}, but the\noverheads are much larger at a scanning speed of $2'$\/s. For larger\nOTFs the relative overheads decrease.\n\\subsection{Ancillary modes}\n\\subsubsection{Point}\nThe standard pointing procedure consists of one subscan in spiral\nobserving mode and results in a fully sampled map of the FoV of\nLABOCA. The pointing offsets relative to the pointing model are\ncomputed via a two-dimensional Gaussian fit to the source position in\nthe map using a BoA pipeline script (see Sect.~\\ref{sec:redu}). Note\nthat this pointing procedure is not limited to pointing scans of the\ncentral channel of the array but works independently of the reference\nbolometer, thus allowing pointing scans centered on the most sensitive\npart of the array.\n\\subsubsection{Focus}\nThe default focusing procedure is made of 10 subscans at 5 different\nsubreflector positions and 5 seconds of integration time each.  This\nis the only observing mode without scanning telescope motion.\nAs a result, we are currently restricted to sources brighter than\nthe atmospheric variations (Mars, Venus, Saturn and Jupiter).\nHowever, initial tests confirmed that using the wobbler to modulate the\nsource signal allows focusing on weaker sources, too.\nThis is the only observing mode, so far, for which the use of the wobbler with LABOCA has been tested.\n\\subsubsection{Skydips}\\label{sec:skydips}\nThe attenuation of the astronomical signals due to the atmospheric\nopacity is determined with skydips. These scans measure the power of\nthe atmospheric emission as a function of the airmass while tipping\nthe telescope from high to low elevation. A skydip procedure consists\nof two steps: a hot-sky calibration scan, to provide an absolute\nmeasurement of the sky temperature, followed by a continuous tip scan\nin elevation; see also Sect.~\\ref{sec:opacity}.\n\\section{Performance on the sky and sensitivity}\\label{sec:performance}\n\\subsection{Number of channels}\\label{sec:numchans}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.98\\linewidth, bb=65 65 535 525, clip, angle=-90]{1454fig10light.eps}\n\\caption{\nFootprint of LABOCA on sky, measured with a beam map on the planet Mars.\nThe ellipses represent the FWHM shape of each beam on sky, as given by a\ntwo-dimensional Gaussian fit to the single-channel map of each\nbolometer.  Only bolometers with useful signal-to-noise ratios are\nshown in this map. See also Fig.~\\ref{fig:sensitivity}.\n}\n\\label{fig:beams}\n\\end{figure}\nAt the time of the commissioning, the number of channels with sky\nresponse was 266 (90\\% of the nominal 295 bolometers).  Of these, 6\nchannels show cross-talk and 12 channels have low sensitivity (less\nthan 10\\% of the mean responsivity).  Two additional bolometers have\nbeen blinded by blocking their horn antennas with absorber material so\nthey can be used to monitor the temperature fluctuations of the array.\nThe number of channels used for astronomical observations is therefore\n248 (84\\% yield, see Fig.~\\ref{fig:beams}).\n\\subsection{Array parameters}\nPosition on sky and relative gain of each bolometer are derived from\nfully sampled maps (hereafter called {\\it beam maps}) of the planets\nMars and Saturn (see Fig.~\\ref{fig:beams}), besides giving a realistic\npicture of the optical distortions over the FoV.  Variations among\nmaps were found to be within a few arc seconds for the positions and\nbelow 10\\% for peak flux densities.  A table with average receiver\nparameters (RCP)\\footnote{The latest RCP table is available\nat:\\\\\n\\tiny{\\url{http:\/\/www.apex-telescope.org\/bolometer\/laboca\/calibration\/LABOCA-centred.rcp}}}\nis periodically computed from beam maps and implemented in the BoA software (see Sect.~\\ref{sec:redu}).\nThe accuracy of the relative bolometer positions from this master RCP is\ntypically below $1''$ (5\\% of the beam size) and the gain accuracy is\nbetter than 5\\%, confirming the good quality (small distortion over\nthe entire FoV) of the tertiary optics (see also\nSect.~\\ref{sec:optics}).\n\\subsection{Beam shape}\\label{sec:beam}\n\\begin{figure}[t] \n\\centering\n\\includegraphics[width=.98\\linewidth, bb=0 0 486 453, clip]{1454fig11light.eps} \n\\caption\n{\nRadial profile of the LABOCA beam derived by averaging the beams of all 248 functional bolometers\nfrom fully sampled maps on Mars.  The error bars show the standard deviation.\nThe main beam is well described by a Gaussian with a FWHM of $19\\rlap{.}''2\\pm0\\rlap{.}''7$\nand starts to deviate at $-$20\\,dB.} \n\\label{fig:beamshape}\n\\end{figure}\nThe LABOCA beam shape was derived for individual bolometers from fully\nsampled maps on Mars (see Fig.~\\ref{fig:beams}) as well as on pointing\nscans on Uranus and Neptune.  Both methods lead to comparable results\nand give an almost circular Gaussian with a FWHM of\n$19\\farcs2\\pm0\\farcs7$ (after deconvolution from the source\nand pixel sizes).  We also investigated the error beam pattern of LABOCA\non beam maps on Mars and Saturn (see Fig.~\\ref{fig:beamshape}).  The\nbeam starts to deviate from a Gaussian at $-$20~dB (1\\% of the peak\nintensity).  The first error beam pattern can be approximated by a\nGaussian with a peak of $-18.3$~dB and a FWHM of~$70''\\pm5''$, the\nsupport legs of the subreflector are visible at the $-25$~dB (0.3\\%\nlevel). The fraction of the power in the first error beam is $\\sim$18\\%.\n\\subsection{Calibration}\\label{sec:calibra}\nThe astronomical calibration was achieved on Mars, Neptune and Uranus\nand a constant conversion factor of $6.3\\pm0.5$\\,Jy\\,beam$^{-1}$\\,\\microV$^{-1}$ was\ndetermined between LABOCA response and flux density.  For the determination\nof the calibration factor we have used the fluxes of planets determined with\nthe software ASTRO\\footnote{GILDAS package, \\tiny{\\url{http:\/\/www.iram.fr\/IRAMFR\/GILDAS}}}. \nThe overall calibration accuracy for LABOCA is about 10\\%. \nWe have also defined a list of secondary calibrators\nand calibrated them against the planets (see Appendix~\\ref{sec:appendix}).\nIn order to improve the absolute flux determination, the calibrators\nare observed routinely every $\\sim$2~hours between observations of\nscientific targets.\n\\subsection{Sky opacity determination}\\label{sec:opacity}\nAtmospheric absorption in the passband of LABOCA attenuates the astronomical signals as $\\exp(-\\tau_{\\rm los})$\nwhere the line of sight optical depth $\\tau_{\\rm los}$ can be as high as 1 for observations at low elevation\nwith 2\\,mm of precipitable water vapor (PWV), typical limit for observations with LABOCA.\nThe accuracy of the absolute calibration, therefore, strongly depends on the precision in the determination of $\\tau_{\\rm los}$.\n\nWe use two independent methods for determining $\\tau_{\\rm los}$. \nThe first one relies on the PWV level measured every minute by the APEX radiometer broadly along the line of sight.\nThe PWV is converted into $\\tau_{\\rm los}$ using an atmospheric transmission model \\citep[ATM,][]{2001ITAP...49.1683P} and\nthe passband of LABOCA (see Sect.~\\ref{sec:passband}). The accuracy of this approach is limited by the knowledge of the passband,\nthe applicability of the ATM and the accuracy of the radiometer.\n\nThe second method uses skydips (see Sect.~\\ref{sec:skydips}).\nAs the telescope moves from high to low elevation, $\\tau_{\\rm los}$ increases with airmass.\nThe increasing atmospheric load produces an increasing total power signal converted\nto effective sky temperature ($T_{\\rm eff}$) by direct comparison with a reference hot load.\nThe dependence of $T_{\\rm eff}$ on elevation is then fitted to determine the zenith opacity~$\\tau$,\nused as parameter \\citep[see, e.g.,][]{2004MNRAS.354..621C}.\n\nSince LABOCA is installed in the Cassegrain cabin of APEX, when performing skydips the receiver suffers a wide,\ncontinuous rotation (about 70 degrees in 20 seconds), which affects the stability of the sorption cooler,\nthus inducing small variations of the bolometers temperature ($\\sim$1--2\\,mK).\nThese temperature fluctuations mimic an additional total power signal with amplitude comparable to the atmospheric signal.\nThe bolometers temperature, however, is monitored with high accuracy by the~\\element[][3]{He}-stage thermometer (see Sect.~\\ref{sec:thermonitor})\nand by the two blind bolometers (see Sect.~\\ref{sec:numchans}), making possible a correction of the skydip data.\n\nThe values of $\\tau$ resulting from the skydip analysis are robust,\nyet up to $\\sim$30\\% higher than those obtained from the radiometer.\nThere are several possible explanation for the discrepancy:\nit could be the result of some incorrect assumptions going into the skydip model (e.g.~the sky temperature),\nor the model itself may be incomplete. The non-linearity of the bolometers can be another factor.\nThe detector responsivities are expected to change with the optical load as $\\sim\\exp(-\\gamma\\,\\tau_{\\rm los})$ (to first order in $\\tau$),\nwhere $\\gamma$ can be related to the bolometer constants \\citep{Mather1984} and the optical configuration.\nCombined with the sky response, the bolometer non-linearities would increase the effective skydip $\\tau$ by a factor $(1+\\gamma)$.\n\nOur practical approach to reconciling the results obtained with the two methods has been to use a linear combination\nof radiometer and skydip values such that it gives the most consistent calibrator fluxes\nat all elevations\\footnote{A text file (BoA readable, see Sect.~\\ref{sec:redu}) containing the zenith opacities\ncalculated from the skydips and the radiometer is available at:\\\\\n\\tiny{\\url{http:\/\/www.apex-telescope.org\/bolometer\/laboca\/calibration\/opacity\/}}}.\nThe excellent calibration accuracy of LABOCA (see Appendix~\\ref{sec:appendix}) underscores this approach.\n\\subsection{Sensitivity}\\label{sec:sensitivity}\nThe noise-weighted mean point-source sensitivity of the array\n(noise-equivalent flux density, NEFD) determined from on-sky\nintegrations, is 55\\,mJy\\,s$^{1\/2}$ (sensitivity per channel).  This\nvalue is achieved only by filtering the low frequencies (hence large\nscale emission) to reject residual sky-noise.  For extended emission,\nwithout low frequency filtering, there is a degradation of sensitivity\nto a mean array sensitivity of 80--100\\,mJy\\,s$^{1\/2}$ depending on\nsky stability. However, there are significant variations of the\nsensitivity across the array (see Fig.~\\ref{fig:sensitivity}, top).\n\\begin{figure}[t] \n\\centering \n\\includegraphics[width=.98\\linewidth, bb=0 -30 698 704, clip]{1454fig12toplight.eps}\n\\includegraphics[width=.98\\linewidth, bb= 108 57 348 296,clip]{1454fig12bottomlight.eps} \n\\caption{\n{\\it Top:} Effective point source sensitivities (after low frequencies filtering) of all the useful bolometers of LABOCA\n(color scale in Jy\\,beam$^{-1}$).  The positions were determined from beam maps on Mars, thus providing a realistic picture of\nthe optical distortions over the FoV (see also Fig.~\\ref{fig:beams}).\n{\\it Bottom:} Distribution of the effective point source sensitivities on the LABOCA array (5\\,mJy\\,s$^{1\/2}$ binning).\nThe total mapping speed of the array is as if the 250 good bolometers were all identical at a level of\n54.5\\,mJy\\,beam$^{-1}$\\,s$^{1\/2}$ (thick black line). The median sensitivity is also shown (dotted line).\n}\n\\label{fig:sensitivity}\n\\end{figure}\n\nFor detection experiments of compact sources with known position,\nLABOCA can be centered on the most sensitive part of the array rather\nthan on the geometric center. This results in an improved point source\nsensitivity of $\\sim$40\\,mJy\\,s$^{1\/2}$ for compact mapping pattern\nlike spirals.\n\\subsection{Mapping speed and time estimate}\nThe relation between the expected residual rms map noise $\\sigma$, the surveyed\narea on the sky and the integration time can be expressed as\n\\begin{equation}\nt_{\\rm int} = \\frac{(X_{\\rm scan}+D)~(Y_{\\rm scan}+D)}{A_{\\rm beam}~N_{\\rm bol}} \\left(\\frac{ f_{\\rm samp}~NEFD~~e^{\\tau_{\\rm los}}}{\\sigma}\\right)^2~\\\\\n\\\\\n\\end{equation}\nwhere $X_{\\rm scan}$, $Y_{\\rm scan}$ are the dimensions of the area covered by\nthe scanning pattern, $D$ is the size of the array, $A_{\\rm beam}$ is the\narea of the LABOCA beam, $N_{\\rm bol}$ is the number of working\nbolometers, $t_{\\rm int}$ is the on source integration time, $NEFD$ is the\naverage array sensitivity, $f_{\\rm samp}$ is the number of grid points\nper beam and $\\tau_{\\rm los}$ the line-of-sight opacity.\nThis formula\\footnote{An online time estimator is available at:\\\\\n\\tiny{\\url{http:\/\/www.apex-telescope.org\/bolometer\/laboca\/obscalc\/}}}\ndoes not include sensitivity variations across the array and the increasing\nsparseness of data points toward the map edges (the latter effect\ndepends on the mapping mode).  We have tested this estimate on deep\nintegrations for point like and extended sources and found a\nreasonable agreement in the measured rms noise of the processed maps.\n\\subsection{Noise behavior in deep integrations}\n\\begin{figure}[t] \n\\centering \n\\includegraphics[width=.90\\linewidth, bb=0 0 351 357, clip]{1454fig13light.eps} \n\\caption\n{\nNoise behavior for deep integrations has been studied by co-adding about 350 hours of\nblank-field data.  The plot shows the effective residual rms noise on sky.\n}\n\\label{fig:noise} \n\\end{figure}\nIn order to study how the noise averages down with increasing\nintegration time, {about 350 hours of blank-field mapping observations}\nhave been co-added with randomly inverted signs to remove any sources while\nkeeping the noise structure.  As shown in~Fig.~\\ref{fig:noise}, the\nnoise integrates down with t$^{-1\/2}$, as expected.\n\\section{Data reduction}\\label{sec:redu} \nLABOCA data are stored in MB-FITS format (Multi-Beam FITS) by the data\nwriter embedded in APECS.  A new software package has been\nspecifically developed to reduce LABOCA data: the Bolometer array data\nAnalysis software (BoA). It is mostly written in the Python language,\nexcept for the most computing demanding tasks, which are written in\nFortran90.\n\nBoA was first installed and integrated in APECS in early 2006. An extensive\ndescription of its functionalities will be given in a separate paper (Schuller\net al.~in prep.). In this Section, we outline the most important features\nfor processing LABOCA data.\n\\subsection{On-line data reduction}\nDuring the observations carried out at APEX, the on-line data\ncalibrator (as part of APECS) performs a quick data reduction of each\nscan to provide the observer with a quick preview of the maps or\nspectra being observed.  This is of particular importance for the\nbasic pointing and focus observing modes, for which the on-line\ncalibrator computes and sends back to the observer the pointing\noffsets or focus corrections to be applied.  For both focus and\npointing scans, only a quick estimate of the correlated noise (see\nbelow) is computed and subtracted from the data.  The focus correction\nis derived from a parabolic fit to the peak flux measured by the\nreference bolometer as a function of the subreflector position.  For\npointing scans, the signals of all usable channels are combined into a\nmap of the central $5'\\times 5'$ area, in horizontal coordinates.  A\ntwo-dimensional elliptical Gaussian is then fitted to the source in\nthis map, which gives the pointing offsets, as well as the peak flux\nand the dimensions of minor and major axis of the source.\n\\subsection{Off-line data reduction}\nThe BoA software can also be used off-line to process any kind of\nbolometer data acquired at APEX.  The off-line BoA runs in the\ninteractive environment of the Python language.  In a typical off-line\ndata reduction session, several scans can be combined together, for\ninstance to improve the noise level on deep integrations, or to do\nmosaicing of maps covering adjacent areas.  The result of any data\nreduction can be stored in a FITS file using standard world coordinate\nsystem (WCS) keywords, which can then be read by other softwares for\nfurther processing (e.g.~source extraction, or overlay with ancillary\ndata).\n\nThe common steps involved in the processing of LABOCA data are the\nfollowing: \\begin{itemize} \\item Flux calibration.  A correct scaling\nof the flux involves, at least, two steps: the opacity correction and\nthe counts-to-Jy conversion.  The zenith opacity is derived from\nskydip measurements (see Sect.~\\ref{sec:skydips} and\nSect.~\\ref{sec:opacity}), and the line of sight opacity also depends\non the elevation.  The counts-to-Jy factor has been determined during\nthe commissioning of the instrument, but additional correction factors\nmay be applied, depending on the flux measured on calibrators with\nknown fluxes (see Sect.~\\ref{sec:calibra}).  \\item Flagging of bad\nchannels.  Bolometers not responding or with strong excess noise are\nautomatically identified from their rms noise being well outside the\nmain distribution of the rms noise values across the array. They can\nbe flagged, which means that the signal that they recorded is not used\nany further in the processing.  \n\\begin{figure}[t] \n\\centering\n\\includegraphics[width=.96\\linewidth, bb=0 0 340 244,clip]{1454fig14toplight.eps} \n\\includegraphics[width=.96\\linewidth, bb=0 0 339 245, clip]{1454fig14bottomlight.eps} \n\\caption\n{ \nTime series of a bolometer signal on blank sky at consecutive steps during the data reduction process with BoA.  The signals, labeled from 1 to\n5, are shown for:\n1) original signal on sky;\n2) after correcting for system temperature variations using the blind bolometers ($\\sim$200\\,\\microK during this scan);\n3) after median skynoise removal over the full array, one single iteration;\n4) after median skynoise removal over the full array, 10 iterations;\n5) after correlated noise removal, grouping the channels by amplifier and cable, 5 iterations (shifted to a $-$3 level, for clarity).\n}\n\\label{fig:reduction} \n\\end{figure} \n\\item Flagging of stationary\npoints.  The data acquired when the telescope was too slow to produce\na signal inside the useful part of the post-detection frequency band\nof LABOCA (e.g.~below 0.1\\,Hz, see Sec.~\\ref{sec:mapmodes} and\nFig.~\\ref{fig:noisespectra}) can be flagged.  Data obtained when the\ntelescope acceleration is very high may show some excess noise, and\ncan also be flagged.  \n\\item Correlated noise removal.  This can be\ndone using a Principal Components Analysis (PCA), or a median noise\nremoval method. In the latter case, the median value of all\n(normalized) signals is computed at each timestamp, and subtracted\nfrom the signal of each channel (with appropriate relative gains).\nThis can be performed using all beams at once or better on groups of\nselected beams.  In fact, some groups of channels sharing the same\nelectronics subsystem (e.g.~amplifier box, flat cable) can show strong correlation\nand removing the median signal on those groups of channels greatly improves\ntheir signal-to-noise ratio.  However, it should be noted that an astronomical source\nwith extended uniform emission on scales of several arcmin or larger would mimic the\ncorrelated noise behavior of groups of channels. Therefore,\nsubtracting the median noise results in filtering out some fraction of\nthe extended emission, depending on the size and morphology of the source.  \n\\item Despiking. Data points that deviate by more than a\ngiven factor times the standard deviation in each channel can be flagged.  \n\\item Data weighting and map making.  To build a map in\nhorizontal or equatorial coordinates, the data of all usable channels\nare projected onto a regular grid and co-added, using a weighted\naverage, with weights computed as 1\/rms$_{\\rm c}^2$, where rms$_{\\rm c}$\nrefers to the rms noise of an individual channel. The channel rms\nnoise can either be the standard deviation of the data over the full\ntime line, or it can be computed on a sliding window containing a\ngiven number of integrations.  \\end{itemize} A visual example of\ncorrelated noise removal with BoA is given\nin~Fig.~\\ref{fig:reduction}.\n\nAdditional (optional) steps that can improve the final reduction\ninclude: removing slow variations from the signal, by subtracting a\npolynomial baseline or by filtering out low frequencies in the Fourier\ndomain; smoothing of the map with a Gaussian kernel of a given size.\n\nThe map resulting from a full reduction can be used as a model of the\nastronomical source to mask the data, in order to repeat some\ncomputations without using data points corresponding to the source in\nthe map.  This iterative scheme helps in the difficult task to recover\nthe extended emission and reduces the generation of artifacts around\nstrong sources.  In the presence of bright sources in the map, a\ntypical data reduction should first perform the reduction steps as\nlisted above with conservative parameters (for example, using high\nenough thresholds in the despiking to avoid flagging out the strong\nsources), and should then read again the raw data, use the resulting\nmap to temporarily flag out datapoints corresponding to bright\nsources, and repeat the full process with less conservative\nparameters.\n\nIn addition to the data processing itself, BoA allows visualization of\nthe data in different ways (signal vs. time, correlation between channels,\npower spectra or datagrams), as well as the telescope pattern, speed\nand acceleration.  Finally, BoA also includes a simulation module,\nwhich can be used to investigate the mapping coverage for on-the-fly\nmaps, spirals and raster of spirals, given the bolometer array\nparameters.\n\\section{Science with LABOCA}\\label{sec:science}\nBecause of its spectral passband, centered at the wavelength of\n870\\,\\micron (see Fig.~\\ref{fig:passband}), LABOCA is particularly\nsensitive to thermal emission from cold objects in the Universe which\nis of great interest for a number of astrophysical research fields.\n\\subsection{Planet Formation}\nThe study of Kuiper Belt Objects in the Solar System as well as\nobservations of debris disks of cold dust around nearby main sequence\nstars can give vital clues to the formation of our own planetary\nsystem and planets in general.  With its angular resolution of~19\\arcsec\n(see Fig.~\\ref{fig:beamshape}) LABOCA can resolve the debris disks of\nsome nearby stars.\n\\subsection{Star Formation in the Milky Way}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.99\\linewidth,bb=20 20 432 786, clip]{1454fig15light.eps} \n\\caption\n{\nA map of the galactic center, extracted from the project ATLASGAL (F. Schuller et al., in prep.).\nThe map is 2$\\times$4 degrees large, the residual noise is 50\\,mJy\\,beam $^{-1}$ and it required only 6~hours of\nobserving time.  Data reduced with the BoA package.\n}\n\\label{fig:atlasgal}\n\\end{figure}\nThe outstanding power of LABOCA in mapping large areas of sky with\nhigh sensitivity (see Fig.~\\ref{fig:atlasgal}) makes it possible,\nfor the first time, to perform unbiased surveys of the distribution\nof the cold dust in the Milky Way.\nAs the dust emission at 870\\,\\micron  is typically optically thin, it is\na direct tracer of the gas column density and gas mass.  Large scale\nsurveys in the Milky Way will reveal the distribution and gas\nproperties of a large number of pre-star cluster clumps and\npre-stellar cores in different environments and evolutionary\nstates. Equally importantly, they provide information on the structure\nof the interstellar medium on large scales at high spatial resolution,\nlittle explored so far.  Such surveys are vital to improve our\nunderstanding of the processes that govern star formation as well as\nthe relation between the clump mass spectrum and the stellar initial\nmass function.  Large unbiased surveys are also critical for finding\nprecursors of high-mass stars, which are undetectable at other\nwavelengths due to the high obscuration of the massive cores they are\nembedded in. LABOCA will help to obtain a detailed understanding of\ntheir evolution.  In addition, deep surveys of nearby, star-forming\nclouds, will allow study of the pre-stellar mass function down to the\nbrown dwarf regime.\n\\subsection{Cold gas in Galaxies}\nThe only reliable way to trace the bulk of dust in galaxies is through\nimaging at submillimeter wavelengths.  It is becoming clear that most\nof the dust mass in spiral galaxies lies in cold, low\nsurface-brightness disks, often extending far from the galactic\nnucleus, as in the case of the starburst galaxy \\object{NGC 253} or of \\object{NGC 5128}\n\\citep[\\object{Cen A}, see Fig.~\\ref{fig:ngc5128} and][]{2008A&A...490...77W}.\n\\begin{figure}[t] \n\\centering\n\\includegraphics[width=.99\\linewidth, bb=0 0 416 307, clip]{1454fig16light.eps} \n\\caption\n{\nMap of the nearby galaxy Centaurus A \\citep[\\object{NGC 5128}, from ][]{2008A&A...490...77W}.\nThe central dust disk has a pronounced S-shape.  Our LABOCA image shows, for the first time at\nsubmillimeter wavelengths, the synchrotron emission from the radio jets and\nthe inner radio lobes.  Its distribution follows closely the known\nradio emission at cm wavelengths.  The color scale is in units\nof Jy\\,beam$^{-1}$. The residual rms noise is 4\\,mJy\\,beam$^{-1}$ and\nthe total integration time is 5~hours.\n}\n\\label{fig:ngc5128}\n\\end{figure} \nUnderstanding this component is critically important as\nit dominates the total gas mass in galaxies and studies of the Schmidt\nlaw based on \\ion{H}{i} observations only can heavily underestimate\nthe gas surface density in the outer parts.\n\nIn addition to studying individual nearby galaxies, LABOCA will be\nvital for determining the low-$z$ benchmarks, such as the local\nluminosity and dust mass functions, which are required to interpret\ninformation from deep cosmological surveys.\n\\subsection{Galaxy formation at high redshift}\nDue to the negative-K correction, submillimeter observations offer\nequal sensitivity to dusty star-forming galaxies over a redshift range\nfrom $z\\sim$1--10 and therefore provide information on the star\nformation history at epochs from about half to only 5\\% of the present\nage of the universe. Recent studies have shown that the volume density\nof luminous {\\it submillimeter galaxies} (SMGs) increases over a\nthousand-fold out to $z\\sim$2 \\citep{2005ApJ...622..772C}, and thus,\nin contrast to the local Universe, luminous obscured galaxies at high\nredshift could dominate the total bolometric emission from all\ngalaxies at early epochs. The mass-tracing property of submillimeter\ndust emission (see Sect.~\\ref{sec:intro}) allows us to make a direct estimate\nof the star formation rates (SFRs) of these objects. The generally high\nobserved SFRs suggest that approximately half of all the stars that\nhave formed by the present day may have formed in highly obscured\nsystems which remain undetected at optical or near-infrared\nwavelengths. One example for such system is the submillimeter galaxy\n\\object{SMM J14009+0252} (see Fig.~\\ref{fig:smm14009}), which is strong in the\nsubmillimeter range, has a 1.4\\,GHz radio counterpart, but no obvious\ncounterpart in deep K-band images \\citep{2000MNRAS.315..209I}.\nClearly it is critical to include these highly obscured sources in\nmodels of galaxy formation to obtain a complete understanding of the\nevolution of galaxies.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.99\\linewidth, bb=28 192 567 610, clip]{1454fig17light.eps}\n\\caption\n{\nLABOCA images of \\object{SMM J14011+0252} (left) and \\object{SMM J14009+0252} (right) smoothed to $25''$ resolution.\nSMM\\,J14011 is at a redshift of $z=2.56$ while SMM\\,J14009 has no reliable redshift determination.\nThe noise level of the map is about 2.5\\,mJy\\,beam$^{-1}$.\n}\n\\label{fig:smm14009}\n\\end{figure}\nWith its fast mapping capabilities LABOCA allows us to map fields of a\nhalf square degree, a typical size of a deep cosmological field\nsurveyed at other wavelengths, down to the confusion limit in a\nreasonable amount of observing time. This will also greatly improve\nthe statistics of high redshift galaxies detected at submillimeter\nwavelengths. See \\citet{Beelen2008} for the first deep LABOCA\ncosmology imaging.\n\\section{Future plans}\\label{sec:future}\nIn collaboration with the Institute for Photonics Technology (IPHT, Jena, Germany)\nwe are working on a new bolometer array, LABOCA-2, using\nsuperconducting technology.  The new array will use superconducting\nthermistors (transition edge sensor, TES), planar dipole absorbers,\nSQUID (Superconducting Quantum Interference Device) multiplexing and\namplification.  A system already using the same technology, called\nSubmillimeter APEX Bolometer Camera (SABOCA) is going to be\ncommissioned as facility instrument on the APEX telescope for\noperation in the 350\\,\\micron  atmospheric window.  Additionally, given\nthe low sensitivity of superconducting bolometers to microphonics, it\nwill be possible to move from the wet cryostat to a pulse-tube cooler,\nas already successfully tested at MPIfR in Bonn, improving\nconsiderably the conditions of operability of the system.\n\\begin{acknowledgements}\nThe authors would like to thank the staff at the APEX telescope for their\nsupport during installation and commissioning of the instrument.  Special\nthanks to Lars-{\\AA}ke Nyman (ESO, APEX station manager at the time of the\ncommissioning) for his invaluable help and for his belief in the project\neven in the most difficult moments.\nE.K. enjoyed the hospitality and help provided by the staff and many student members\nof the Microfabrication Facility of UC Berkeley during the manufacture of the LABOCA wafer.\n\\end{acknowledgements} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzohhf b/data_all_eng_slimpj/shuffled/split2/finalzzohhf
new file mode 100644
index 0000000000000000000000000000000000000000..fc93f98e57ca85e2a9ccadb4d9129ee1e2ede8b8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzohhf
@@ -0,0 +1,5 @@
+{"text":"\\section{Symmetry breaking in quantum dots}\n\nTwo-dimensional (2D) quantum dots (QDs), created at semiconductor interfaces \nthrough the use of lithographic and gate-voltage techniques, with refined \ncontrol of their size, shape, and number of electrons, are often referred to as\n``artificial atoms'' \\cite{kast,taru2,asho}. These systems which, with the use \nof applied magnetic fields, are expected to have future applications as \nnanoscale logic gates and switching devices, have been in recent years the \nsubject of significant theoretical and experimental research efforts. As \nindicated above, certain analogies have been made between these man-made systems\nand their natural counterparts, suggesting that the physics of electrons in the \nformer is similar to that underlying the traditional description of natural \natoms -- pertaining particularly to electronic shells and the Aufbau principle \nin atoms (where electrons are taken to be moving in a spherically averaged \neffective central mean-field potential). \n\nThe above-mentioned analogy has been theoretically challenged recently \n\\cite{yyl1,yyl2} on the basis of calculations that showed evidence for \nformation, under favorable conditions (that are readily achieved in the \nlaboratory), of ``electron molecules,'' which are alternatively called Wigner \nmolecules (WMs) after the physicist who predicted formation of electron crystals\nin extended systems \\cite{wign}. These \nspin-and-space (sS) unrestricted Hartree-Fock (UHF) calculations (denoted in the\nfollowing as sS-UHF or simply UHF) of electrons confined in 2D QDs by a \nparabolic external potential led to the discovery of spontaneous symmetry \nbreaking in QDs, manifested in the appearance of distinct interelectronic \nspatial (crystalline) correlations (even in the absence of magnetic fields). \nSuch symmetry breaking may indeed be expected to occur based on the interplay \nbetween the interelectron repulsion, $Q$, and the zero-point kinetic energy, \n$K$. It is customary to take $Q=e^2\/\\kappa l_0$ and $K \\equiv \\hbar \\omega_0$, \nwhere $l_0=(\\hbar\/m^*\\omega_0)^{1\/2}$ is the spatial extent of an electron in the\nlowest state of the parabolic confinement; $m^*$ is the electron effective mass,\n$\\kappa$ is the dielectric constant, and $\\omega_0$ is the frequency that \ncharacterizes the parabolic (harmonic) confining potential. Thus, defining the \nWigner parameter as $R_W = Q\/K$, one may expect symmetry breaking to occur when \nthe interelectron repulsion dominates, i.e., for $R_W > 1$. Under such \ncircumstances, an appropriate solution of the Schr\\\"{o}dinger equation \nnecessitates consideration of wave functions with symmetries that are lower than\nthat of the circularly symmetric QD Hamiltonian. Such solutions may be found \nthrough the use of the sS-UHF method, where all restrictions on the symmetries \nof the wave functions are lifted. On the other hand, when $R_W < 1$, no symmetry\nbreaking is expected, and the sS-UHF solution collapses onto that obtained via \nthe restricted Hartree-Fock (RHF) method, and the aforementioned circularly \nsymmetric ``artificial-atom'' analogy maintains. From the above we note that the\nstate of the system may be controlled and varied through the choice of materials\n(i.e., $\\kappa$) and\/or the strength of the confinement ($\\omega_0$), since  \n$R_W \\propto 1\/(\\kappa \\sqrt{\\omega_0})$. \n\n\\begin{figure}[t]\n\\centering\\includegraphics[width=12cm]{tnt_2005_fig1.eps}\n\\caption{The calculated spectrum of a two-electron parabolic quantum dot, with \n$R_W=200$. The quantum numbers are $(N, M, n, m)$ with $N$ corresponding to the \nnumber of radial nodes in the center of mass (CM) wavefunction, and $M$ is the \nCM azimuthal quantum number. The integers $n$ and $m$ are the corresponding \nquantum numbers for the electrons' relative motion (RM) and the total energy is \ngiven by $E_{NM,nm} = E^{cm}_{NM} + E^{rm}(n,|m|)$. The spectrum may be \nsummarized by the ``spectral rule'' given in the figure, with $C = 0.037$, the \nphonon stretching vibration $\\hbar \\omega_s = 3.50$, and the phonon for the \nbending vibration coincides with that of the CM motion, i.e., $\\hbar \\omega_b \n= \\hbar \\omega_0 = 2$. All energies are in units of $\\hbar \\omega_0\/2$, where \n$\\omega_0$ is the parabolic confinement frequency.\n}\n\\end{figure}\n\n\\subsection{Two-electron circular dots}\n\nTo illustrate the formation of ``electron molecules,'' we show first exact \nresults obtained for a two-electron QD, through separation of the center-of-mass\nand inter-electron relative-distance degrees of freedom \\cite{yyl3}. The \nspectrum obtained for $R_W = 200$ (Fig.\\ 1), exhibits features that are \ncharacteristic of a collective rovibrational dynamics, akin to that of a natural\n``rigid'' triatomic molecule with an infinitely heavy middle particle \nrepresenting the center of mass of the dot. This spectrum transforms to that of \na ``floppy'' molecule for smaller value of $R_W$ (i.e., for stronger \nconfinements characterized by a larger value of $\\omega_0$, and\/or for weaker \ninter-electron repulsion), ultimately converging to the independent-particle \npicture associated with the \ncircular central mean-field of the QD. Further evidence for the formation of the\nelectron molecule was found through examination of the conditional probability \ndistribution (CPD); that is, the anisotropic pair correlation \n$P({\\bf r}, {\\bf r}_0)$, which expresses the probability of finding a particle \nat ${\\bf r}$ given that the ``observer'' (reference point) is riding on another \nparticle at ${\\bf r}_0$ \\cite{yyl2,yyl3},\n\\begin{equation}\nP({\\bf r},{\\bf r}_0) =\n\\langle \\Psi | \n\\sum_{i=1}^N \\sum_{j \\neq i}^N  \\delta({\\bf r}_i -{\\bf r})\n\\delta({\\bf r}_j-{\\bf r}_0) \n| \\Psi \\rangle  \/ \\langle \\Psi | \\Psi \\rangle.\n\\label{cpds}\n\\end{equation} \nHere $\\Psi ({\\bf r}_1, {\\bf r}_2, ..., {\\bf r}_N)$ \ndenotes the many-body wave function under consideration.\nIt is instructive to note here certain \nsimilarities between the formation of a ``two-electron molecule'' in man-made \nquantum dots, and the collective (rovibrational) features observed in the \nelectronic spectrum of doubly-excited helium atoms \\cite{kell1,kell2,berr}.\n\nFor confined (finite) systems with a larger number of particles, one must resort\nto approximate computational schemes. Of particular interest are methodologies \nthat permit systematic evaluation of high-accuracy solutions to these many-body \nstrongly-correlated systems, under field-free conditions, as well as under the \ninfluence of an applied magnetic field. We remark here, that the relatively \nlarge (spatial) size of QDs (resulting from materials' characteristics, e.g., a \nsmall electron effective mass and large dielectric constant), allows the full \nrange of orbital magnetic effects to be covered for magnetic fields that are \nreadily attained in the laboratory (less then 40 T). In contrast, for natural \natoms and molecules, magnetic fields of sufficient strength (i.e., larger than \n$10^5$ T) to produce novel phenomena related to orbital magnetism (beyond the \nperturbative regime), are known to occur only in astrophysical environments \n(e.g., on the surface of neutron stars).\n\nThe 2D hamiltonian of the problem under consideration is given by \n\\begin{equation}\nH=\\sum_{i=1}^N \\frac{1}{2m^*} \\left( {\\bf p}_i -\\frac{e}{c} {\\bf A}_i \\right)^2\n+ \\sum_{i=1}^N \\frac{m^*}{2} \\omega_0^2 {\\bf r}_i^2 +\n\\sum_{i=1}^N \\sum_{j>i}^N \\frac{e^2}{\\kappa r_{ij}},\n\\label{mbhn} \n\\end{equation}\ndescribing $N$ electrons (interacting via a Coulomb repulsion) confined by a \nparabolic potential of frequency $\\omega_0$ and subjected to a perpendicular \nmagnetic field $ {\\bf B}$, whose vector potential is given in the symmetric \ngauge by ${\\bf A} ({\\bf r}) = {\\bf B} \\times {\\bf r}\/2 = (-By,Bx,0)\/2$.\nFor sufficiently high magnetic field values (i.e., in the fractional \nquantum Hall effect, or FQHE, regime), the electrons are fully spin-polarized \nand the Zeeman term (not shown here) does not need to be considered. \nIn the $B \\rightarrow \\infty$ limit, the external confinement $\\omega_0$ can be \nneglected, and $H$ can be restricted to operate in the lowest Landau level \n(LLL), reducing to the form \\cite{yyl4,yyl5,yyl6,yyl7}\n\\begin{equation}\nH_{\\rm LLL} = N \\frac{\\hbar \\omega_c}{2} +\n\\sum_{i=1}^N \\sum_{j>i}^N \\frac{e^2}{\\kappa r_{ij}},\n\\label{hlll}\n\\end{equation}\nwhere $\\omega_c = eB\/(m^*c)$ is the cyclotron frequency. \n\nFor finite $N$, the solutions to the Schr\\\"{o}dinger equations corresponding to \nthe hamiltonians given by Eq\\. (\\ref{mbhn}) (with or without a magnetic field), \nor by Eq\\. (\\ref{hlll}) (in the $B \\rightarrow \\infty$ limit), must have a good \nangular momentum, $L$, and good spin quantum numbers (the latter is guaranteed \nin the high $B$, fully spin-polarized case). As described in detail elsewhere \n\\cite{yyl4,yyl5,yyl6,yyl7,yyl8,yyl9}, \nthese solutions can be well approximated by a two-step method \nconsisting of symmetry breaking at the spin-and-space unrestricted Hartree-Fock \nlevel and subsequent symmetry restoration via post-Hartree-Fock projection \ntechniques. We recall that the sS-UHF method relaxes both the double-occupancy \nrequirement (namely, different spatial orbitals are employed for different spin \ndirections), as well as relaxing the requirement that the electron (spatial) \norbitals be constrained by the symmetry of the external confining potential.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=14.5cm]{tnt_2005_fig2.eps}\\\\\n~~~~~\\\\\n\\caption{Various approximation levels for a field-free two-electron QD with \n$R_W=2.40$. (a) Electron density of the RHF solution, exhibiting circular \nsymmetry (due to the imposed symmetry restriction). The correlation energy \n$\\varepsilon_c = 2.94$ meV, is defined as the difference between the energy of \nthis state and the exact solution [shown in frame (e)]. (b1) and (b2) The two \noccupied orbitals (modulus square) of the symmetry-broken ``singlet'' sS-UHF \nsolution (b1), with the corresponding total electron density exhibiting \nnon-circular shape (b2). The energy of the sS-UHF solution shows a gain of \n44.3\\% of the correlation energy. (c) Electron density of the spin-projected \n(SP) singlet, showing broken spatial symmetry, but with an additional gain of \ncorrelation energy. (d) the spin-and-angular-momentum projected state (S\\&AMP) \nexhibiting restored circular symmetry with a 73.1\\% gain of the correlation \nenergy. The choice of parameters is: dielectric constant $\\kappa = 8$, parabolic\nconfinement $\\omega_0 = 5$ meV, and effective mass $m^* = 0.067m_e$. \nDistances are in nanometers and the densities in $10^{-4}$ nm$^{-2}$.\n}\n\\end{figure}\n\nResults obtained for various approximation levels for a two-electron QD with \n$B=0$ and $R_W=2.40$ (that is, in the Wigner-molecule regime) are displayed in \nFig.\\ 2. In these calculations \\cite{yyl9}, the spin projection was performed \nfollowing reference \\cite{lowd}, i.e., one constructs the wave function\n\\begin{equation}\n\\Psi_{\\rm SP}(s)={\\cal P}_{\\rm spin}(s) \\Psi_{\\rm UHF},\n\\label{psis}\n\\end{equation}\nwhere $\\Psi_{\\rm UHF}$ is the original symmetry-broken UHF determinant.\nIn Eq. (\\ref{psis}), the spin projection operator is given by\n\\begin{equation}\n{\\cal P}_{\\rm spin}(s) \\equiv \\prod_{s^\\prime \\neq s}\n\\frac{\\hat{S}^2 - s^\\prime(s^\\prime + 1) \\hbar^2}\n{[s(s+1) - s^\\prime(s^\\prime + 1)] \\hbar^2}~,\n\\label{prjp}\n\\end{equation}\nwhere the index $s^\\prime$ runs over the quantum numbers of $\\hat{S}^2$,\nwith $\\hat{S}$ being the total spin.\n\nThe angular momentum projector is given by\n\\begin{equation}  \n2\\pi {\\cal P}_L \\equiv \\int_0^{2\\pi} d\\gamma \\exp[-i\\gamma (\\hat{L}-L)],\n\\label{amp}\n\\end{equation} \nwhere $\\hat{L}=\\hat{l}_1+\\hat{l}_2$ is the total angular momentum \noperator. As seen from Eq.\\ (\\ref{amp}), application of the projection\noperator ${\\cal P}_L$ to the spin-restored state $\\Psi_{\\rm SP}(s)$\ncorresponds to a continuous configuration interaction (CCI) formalism. \n\nIn the following we focus on the ground state of the system, i.e., \n$L = 0$. The energy of the projected state is given by \n\\begin{equation}\nE_{\\rm PRJ}(L) = \\left. \\int_0^{2\\pi} h(\\gamma) e^{i\\gamma L} d\\gamma \\right\/ \n \\int_0^{2\\pi} n(\\gamma) e^{i\\gamma L}d\\gamma,\n\\label{eprj}\n\\end{equation}\nwith $h(\\gamma)=\\langle \\Psi_{\\rm SP}(s;0)|H|\\Psi_{\\rm SP}(s;\\gamma)\\rangle$\nand $n(\\gamma)=\\langle \\Psi_{\\rm SP}(s;0)|\\Psi_{\\rm SP}(s;\\gamma)\\rangle$,\nwhere $\\Psi_{\\rm SP}(s;\\gamma)$ is the spin-restored wave function \nrotated by an azimuthal angle $\\gamma$ and $H$ is the many-body hamiltonian.\nWe note that the UHF energies are simply given by $E_{\\rm UHF}=h(0)\/n(0)$.\n\nThe electron densities corresponding to the initial RHF approximation [shown in \nFig.\\ 2(a)] and the final spin-and-angular-momentum projection (S\\&AMP) [shown \nin Fig.\\ 2(d)], are circularly symmetric, while those corresponding to the two \nintermediate approximations, i.e., the sS-UHF and spin-projected (SP) solutions \n[Figs.\\ 2(b2) and 2(c), respectively] break the circular symmetry. This \nbehavior illustrates graphically the meaning of the term restoration of \nsymmetry, and the interpretation that the sS-UHF broken-symmetry solution refers\nto the intrinsic (rotating) frame of reference of the electron molecule. In \nlight of this discussion the final projected state is called a {\\it rotating \nWigner molecule}, or RWM.\n\n\\subsection{Electrons in circular quantum dots under high magnetic fields}\n\nTo illustrate the emergence of RWMs in parabolically confined QDs under high $B$,\nwe show in Fig.\\ 3 results obtained \\cite{yyl7} \nthrough the aforementioned two-step \ncomputational technique for $N=7$, 8, and 9 electrons, and compare them with the\nresults derived from exact diagonalization of the Hamiltonian [see Eq\\. \n(\\ref{mbhn})]. Systematic investigations of QDs under high $B$ revealed \nelectronic states of crystalline character. These states are found for \nparticular ``magic'' angular momentum values ($L$) that exhibit enhanced \nstability and are called cusp states. For a given value of $B$, one of these \n$L$'s corresponds to the global minimum, i.e., the ground state, and varying $B$ \ncauses the ground state and its angular momentum to change. The cusp states have\nbeen long recognized \\cite{laug1} as the finite-$N$ precursors of the fractional \nquantum Hall states in extended systems. In particular, the fractional fillings \n$\\nu$ (defined in the thermodynamic limit) are related to the magic angular \nmomenta of the finite-$N$ system as follows \\cite{girv}\n\\begin{equation}\n\\nu=\\frac{N(N-1)}{2L}.\n\\label{nu}\n\\end{equation}\n\nIn the literature of the fractional quantum Hall effect (FQHE), ever since the \ncelebrated paper \\cite{laug2} by Laughlin in 1983, the cusp states have been \nconsidered to be the antithesis of the Wigner crystal and to be described \naccurately by liquid-like wave functions, such as the Jastrow- Laughlin (JL) \n\\cite{laug2,laug3} and composite-fermion (CF) \\cite{jain,jain2} ones. This view,\nhowever, has been challenged recently \\cite{yyl4,yyl5} by the explicit \nderivation of trial wave functions for the cusp states that are associated with \na rotating Wigner molecule. As discussed elsewhere \\cite{yyl4,yyl5}, these \nparameter-free wave functions, which are by construction crystalline in \ncharacter, have been shown to provide a simpler and improved description of the \ncusp states, in particular for high angular momenta (corresponding to low \nfractional fillings).\n\nThe crystalline arrangements that were found \\cite{yyl4,yyl5,yyl6,yyl7} consist \nof concentric polygonal rings [see the conditional probabilities displayed in \nFig.\\ 3, with the reference (observation) point denoted by a filled dot]. These \nrings rotate independently of each other (see in particular the two cases shown \nfor $N=9$), with the electrons on each ring rotating coherently \\cite{yyl7}. The\nrotations stabilize the RWM relative to the static one -- namely, the projected \n(symmetry restored) states are lower in energy compared to the broken-symmetry \nones (the unrestricted Hartree-Fock solutions).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=12.0cm]{tnt_2005_fig3.ps}\n\\caption{Conditional probability distributions (CPDs) at high $B$, evaluated \nfor parabolic quantum dots through: (i) the two-step procedure of symmetry \nbreaking and subsequent restoration, resulting in rotating Wigner molecules \n(RWM) (shown in the left columns for given $N$), and (ii) exact diagonalization \n(EXD, shown in the right columns for given $N$). \nThe angular momentum values, and corresponding values of the \nfractional filling [see Eq\\. (\\ref{nu})], are given on the left. The optimal \npolygonal structure for a given $N$ is given by $(n_1,n_2)$ with $n_1+n_2=N$. \nFor $N= 7$, 8, and 9, these arrangements are (1,6), (1,7), and (2,7), \nrespectively. The reference point for the calculation of the CPD is denoted by a\nfilled dot. Note in particular the two CPDs shown for $N=9$, illustrating that \nfor a reference point located on the outer ring, the inner ring appears uniform,\nand vice versa for a reference point located on the inner ring (bottom row on\nthe right). These results illustrate that the rings rotate independently of each\nother. \n}\n\\end{figure}\n\n\\subsection{Symmetry breaking of trapped bosons}\n\nIn closing this section, we remark that the emergence of crystalline geometric \narrangements, discussed above for electrons confined in 2D quantum dots, appears\nto be a general phenomenon that is predicted \\cite{roma} to occur also for \ntrapped bosonic atomic systems (neutral or charged) when the interatomic \nrepulsion is tuned to exceed the characteristic energy of the harmonic trap. \nIndeed, application of the aforementioned two-step method, allowed the \nevaluation \\cite{roma} of solutions to the many-body Hamiltonians describing \nsuch bosonic systems, going beyond the mean-field (Gross-Pitaevskii) approach. \nThese highly-correlated trapped 2D states exhibit localization of the bosons \ninto polygonal-ringlike crystalline patterns, thus extending earlier work that \npredicted localization of strongly repelling 1D bosons (often referred to as the\nTonk-Girardeau regime \\cite{gira1,gira2}), to a higher dimension. \nIt is expected that these theoretical \nfindings will be the subject of experimental explorations, using methodologies \nthat have led recently to observations of localization transitions in 1D boson \nsystems \\cite{pare,weis}.\n\n\\section{Localization and entanglement in a two-electron elliptic \nquantum dot}\n\nAs discussed in the previous section, electron localization leading to formation\nof molecular-like structures [the aforementioned Wigner molecules] within a \n{\\it single circular\\\/} two-dimensional (2D) quantum dot at zero magnetic field \n($B$) has been theoretically predicted to occur \\cite{yyl1,yyl2,yyl3,yl1,mikh}, \nas the strength of the $e$-$e$ repulsive interaction relative to the zero-point \nenergy increases. Of particular interest is a two-electron $(2e)$ WM, in light \nof the current experimental effort \\cite{taru,eng} aiming at implementation of a\nspin-based \\cite{burk} solid-state quantum logic gate that employs two coupled \none-electron QDs (double dot). \n\nHere, we present an exact diagonalization (EXD) and an approximate\n(generalized Heitler-London, GHL) microscopic treatment for two electrons in a\n{\\it single\\\/} elliptic QD specified by the \nparameters of a recently investigated experimental device \\cite{marc}. While \nformation of Wigner molecules in circular QDs requires weak confinement, \nand thus large dots of lower densities (so that the interelectron repulsion \ndominates), we show that formation of such WMs is markedly enhanced in highly \ndeformed (e.g., elliptic) dots due to their lower symmetry.\nThe calculations provide a good description\nof the measured $J(B)$ curve (the singlet-triplet splitting) when screening \n\\cite{kou2,hall} due to the metal gates and leads is included (in addition to \nthe weakening of the effective inter-electron repulsion due to the\ndielectric constant of the semiconductor, GaAs). In particular, our results\nreproduce the salient experimental findings pertaining to the vanishing of \n$J(B)$ for a finite value of $B \\sim 1.3 $ T [associated with a change in sign \nof $J(B)$ indicating a singlet-triplet (ST) transition], \nas well as the flattening of the $J(B)$ curve after the ST crossing.\nThese properties, and in particular the latter one, are related \ndirectly to the formation of an electron molecular dimer and its effective \ndissociation for large magnetic fields. \nThe effective dissociation of the electron dimer is most naturally described \nthrough the GHL approximation, and it is fully supported by the more accurate,\nbut physically less transparent, EXD.\n\nOf special interest for quantum computing is the degree of entanglement \nexhibited by the two-electron molecule in its singlet state \\cite{burk}. \nHere, in relation to the microscopic calculations, we investigate two different \nmeasures of entanglement \\cite{note27}. The first, known as the concurrence \n(${\\cal C}$) for two {\\it indistinguishable\\\/} fermions \\cite{schl,loss}, \nhas been used in the analysis of the experiment in Ref.\\ \\cite{marc}\n(this measure is related to the operational cycle of a two-spin-qubit\nquantum logic gate \\cite{schl,loss}).\nThe second measure, referred to as the von Neumann entropy (${\\cal S}$) for \n{\\it indistinguishable\\\/} particles, has been developed in Ref.\\ \\cite{you}\nand used in Ref.\\ \\cite{zung}.\nWe show that the present {\\it wave-function-based\\\/} methods, in conjunction with\nthe knowledge of the dot shape and the $J(B)$ curve, enable theoretical \ndetermination of the degree of entanglement, in particular for the elliptic QD \nof Ref.\\ \\cite{marc}. The increase in the degree of entanglement (for both\nmeasures) with stronger magnetic fields correlates with the dissociation\nof the $2e$ molecule. This supports the experimental assertion \\cite{marc} that \ncotunneling spectroscopy can probe properties of the electronic wave function of\nthe QD, and not merely its low-energy spectrum. Our methodology can be \nstraightforwardly applied to other cases of strongly-interacting devices, e.g., \ndouble dots with strong interdot-tunneling. \n\n\\subsection{Microscopic treatment}\n\nThe Hamiltonian for two 2D interacting electrons is [see Eq.\\ (\\ref{mbhn})]\n\\begin{equation}\n{\\cal H} = H({\\bf r}_1)+H({\\bf r}_2)+e^2\/(\\kappa r_{12}),\n\\label{ham}\n\\end{equation}\nwhere the last term is the Coulomb repulsion, $\\kappa$ is the\ndielectric constant, and \n$r_{12} = |{\\bf r}_1 - {\\bf r}_2|$. $H({\\bf r})$ is the\nsingle-particle Hamiltonian for an electron in an external perpendicular\nmagnetic field ${\\bf B}$ and an appropriate confinement potential.\nWhen position-dependent screening is included, the last term in Eq.\\\n(\\ref{ham}) is modified by a function of $r_{12}$ (see below). \nFor an elliptic QD, the single-particle Hamiltonian is written as\n\\begin{equation}\nH({\\bf r}) = T + \\frac{1}{2} m^* (\\omega^2_{x} x^2 + \\omega^2_{y} y^2)\n    + \\frac{g^* \\mu_B}{\\hbar} {\\bf B \\cdot s},\n\\label{hsp}\n\\end{equation}\nwhere $T=({\\bf p}-e{\\bf A}\/c)^2\/2m^*$, with ${\\bf A}=0.5(-By,Bx,0)$ being the\nvector potential in the symmetric gauge. $m^*$ is the effective mass and\n${\\bf p}$ is the linear momentum of the electron. The second term is the\nexternal confining potential; the last term is the Zeeman interaction with \n$g^*$ being the effective $g$ factor, $\\mu_B$ the Bohr magneton, and ${\\bf s}$ \nthe spin of an individual electron. \n\nThe GHL method for solving the Hamiltoninian (\\ref{ham}) consists of two steps. \nIn the first step, we solve selfconsistently the ensuing \nunrestricted Hartree-Fock (UHF) equations allowing for lifting of the \ndouble-occupancy requirement (imposing this requirement gives the \n{\\it restricted\\\/} HF method, RHF).\nFor the $S_z=0$ solution, this step produces two single-electron \norbitals $u_{L,R}({\\bf r})$ that are localized left $(L)$ and right $(R)$ of the\ncenter of the QD [unlike the RHF method that gives a single doubly-occupied \nelliptic (and symmetric about the origin) orbital]. \nAt this step, the many-body wave function is a single Slater \ndeterminant $\\Psi_{\\text{UHF}} (1\\uparrow,2\\downarrow) \\equiv \n| u_L(1\\uparrow)u_R(2\\downarrow) \\rangle$ made out of the two occupied UHF \nspin-orbitals $u_L(1\\uparrow) \\equiv u_L({\\bf r}_1)\\alpha(1)$ and \n$u_R(2\\downarrow) \\equiv u_R({\\bf r}_2) \\beta(2)$, where \n$\\alpha (\\beta)$ denotes the up (down) [$\\uparrow (\\downarrow)$] spin. \nThis UHF determinant is an eigenfunction of the projection $S_z$ of the total \nspin $\\hat{S} = \\hat{s}_1 + \\hat{s}_2$, but not of $\\hat{S}^2$ (or the parity\nspace-reflection operator). \n\nIn the second step, we restore the broken parity and total-spin symmetries by \napplying to the UHF determinant the projection operator \\cite{yyl8,yl3} \n${\\cal P}_{\\rm spin}^{s,t}=1 \\mp \\varpi_{12}$, where the \noperator $\\varpi_{12}$ interchanges the spins of the two electrons\n[this is a special case of the operator given in Eq.\\ (\\ref{prjp})]; \nthe upper (minus) sign corresponds to the singlet. \nThe final result is a generalized Heitler-London (GHL) two-electron wave function\n$\\Psi^{s,t}_{\\text{GHL}} ({\\bf r}_1, {\\bf r}_2)$ for the ground-state singlet \n(index $s$) and first-excited triplet (index $t$), which uses\nthe UHF localized orbitals,\n\\begin{equation}\n\\Psi^{s,t}_{\\text{GHL}} ({\\bf r}_1, {\\bf r}_2) \\propto\n{\\bf (} u_L({\\bf r}_1) u_R({\\bf r}_2) \\pm u_L({\\bf r}_2) u_R({\\bf r}_1) {\\bf )}\n\\chi^{s,t},\n\\label{wfghl}\n\\end{equation}\nwhere $\\chi^{s,t} = (\\alpha(1) \\beta(2) \\mp \\alpha(2) \\beta(1))$ is the spin \nfunction for the 2$e$ singlet and triplet states.\nThe general formalism of the 2D UHF equations and of the subsequent restoration \nof broken spin symmetries can be found in Refs.\\ \\cite{yyl8,yyl9,yl1,yl3}.\n\nThe use of {\\it optimized\\\/} UHF orbitals in the GHL is suitable for treating \n{\\it single elongated\\\/} QDs. The GHL is equally applicable to double QDs with \narbitrary interdot-tunneling coupling \\cite{yyl8,yl3}. In contrast,\nthe Heitler-London (HL) treatment \\cite{hl} (known also as Valence bond), \nwhere non-optimized ``atomic'' orbitals of two isolated QDs are used, is \nappropriate only for the case of a double dot with small interdot-tunneling \ncoupling \\cite{burk}.\n\nThe orbitals $u_{L,R}({\\bf r})$ are expanded in a real Cartesian \nharmonic-oscillator basis, i.e.,\n\\begin{equation}\nu_{L,R}({\\bf r}) = \\sum_{j=1}^K C_j^{L,R} \\varphi_j ({\\bf r}),\n\\label{uexp}\n\\end{equation}\nwhere the index $j \\equiv (m,n)$ and $\\varphi_j ({\\bf r}) = X_m(x) Y_n(y)$,\nwith $X_m(Y_n)$ being the eigenfunctions of the one-dimensional oscillator in the\n$x$($y$) direction with frequency $\\omega_x$($\\omega_y$). The parity operator\n${\\cal P}$ yields ${\\cal P} X_m(x) = (-1)^m X_m(x)$, and similarly for $Y_n(y)$.\nThe expansion coefficients $C_j^{L,R}$ are real for $B=0$ and complex for finite\n$B$. In the calculations we use $K=79$, yielding convergent results.\n\nIn the EXD method, the many-body wave function is written as a linear\nsuperposition over the basis of non-interacting two-electron determinants, i.e.,\n\\begin{equation}\n\\Psi^{s,t}_{\\text{EXD}} ({\\bf r}_1, {\\bf r}_2) =\n\\sum_{i < j}^{2K} \\Omega_{ij}^{s,t} | \\psi(1;i) \\psi(2;j)\\rangle,\n\\label{wfexd}\n\\end{equation}\nwhere $\\psi(1;i) = \\varphi_i(1 \\uparrow)$ if $1 \\leq i \\leq K$ and\n$\\psi(1;i) = \\varphi_{i-K}(1 \\downarrow)$ if $K+1 \\leq i \\leq 2K$ [and\nsimilarly for $\\psi(2,j)$].\nThe total energies $E^{s,t}_{\\text{EXD}}$ and the coefficients\n$\\Omega_{ij}^{s,t}$ are obtained through a ``brute force'' diagonalization of\nthe matrix eigenvalue equation corresponding to the Hamiltonian in Eq.\\ \n(\\ref{ham}). The EXD wave function does not\nimmediately reveal any particular form, although, our calculations below\nshow that it can be approximated by a GHL wave function in the case of the\nelliptic dot under consideration. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=6.5cm]{tnt_2005_fig4.eps}   \n\\caption{\nThe singlet-triplet splitting $J=E^s-E^t$ as a function of the magnetic field\n$B$ for an elliptic QD with $\\hbar \\omega_x=1.2$ meV and $\\hbar \\omega_y=3.3$ meV\n(these values correspond to the device of Ref.\\ \\cite{marc}).\nSolid line: GHL (broken-symmetry UHF + restoration of symmetries) results\nwith a coordinate-independent screening ($\\kappa=22$).\nDashed line: EXD results with $\\kappa=12.9$ (GaAs), but including screening with\na coordinate dependence according to Ref.\\ \\cite{hall} and $d=18.0$ nm \n(see text). The rest of the material parameters used are: \n$m^*$(GaAs)$=0.067 m_e$, and $g^*=0$ (see text).\nThe experimental measurements \\cite{marc} are denoted by open squares.\nOur sign convention for $J$ is opposite to that in Ref.\\ \\cite{marc}.\n}\n\\end{figure}\n\nTo model the experimental elliptic QD device, we take, following Ref.\\ \n\\cite{marc}, $\\hbar \\omega_x=1.2$ meV and $\\hbar \\omega_y=3.3$ meV. \nThe effective mass of the electron is taken as $m^*=0.067 m_e$ (GaAs). Since \nthe experiment did not resolve the lifting of the triplet degeneracy caused by \nthe Zeeman term, we take $g^*=0$. Using the two step method, \nwe calculate the GHL singlet-triplet splitting \n$J_{\\text{GHL}}(B)=E^s_{\\text{GHL}}(B)-E^t_{\\text{GHL}}(B)$ \nas a function of the magnetic field in the range $0 \\leq B \\leq 2.5$ T.\nScreening of the $e$-$e$ interaction due to the metal gates and leads \nmust be considered in order to reproduce the experimental $J(B)$ \ncurve \\cite{note2}. This screening can be modeled, to first approximation, by a \nposition-independent adjustment of the dielectric constant \n$\\kappa$ \\cite{kyri}. Indeed, with $\\kappa=22.0$ (instead of the GaAs \ndielectric constant, i.e., $\\kappa = 12.9$), good agreement with\nthe experimental data is obtained [see Fig.\\ 4]. In particular, we note the \nsinglet-triplet crossing for $B \\approx 1.3$ T, and the flattening of the \n$J(B)$ curve beyond this crossing.\n\nWe have also explored, particularly in the context of the EXD treatment, \na position-dependent screening using the functional form,\n$(e^2\/\\kappa r_{12}) [1-(1+4d^2\/r_{12}^2)^{-1\/2}]$, \nproposed in Ref.\\ \\cite{hall}, with $d$ as a fitting parameter. \nThe $J_{\\text{EXD}}(B)$ result for $d=18.0$ nm is depicted in Fig.\\ 4 \n(dashed line), and it is in very good agreement with the experimental \nmeasurement.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=6.5cm]{tnt_2005_fig5.ps}   \n\\caption{ \nTotal electron densities (EDs) associated with the singlet state of the\nelliptic dot at $B=0$ and $B=2.5$ T.\n(a) The GHL densities. (b) The EXD densities.\nThe rest of the parameters and the screening of the Coulomb interaction \nare as in Fig.\\ 4.\nLengths in nm and densities in $10^{-4}$ nm$^{-2}$.\n}\n\\end{figure}\n\n\nThe singlet state electron densities from the GHL and the EXD treatments\nat $B=0$ and $B=2.5$ T are displayed in Fig.\\ 5. These densities illustrate the \ndissociation of the electron dimer with increasing magnetic field. \nThe asymptotic convergence (beyond the ST point) of the energies of the singlet \nand triplet states, i.e., [$J(B) \\rightarrow 0$ as $B \\rightarrow \\infty$] is a \nreflection of the dissociation of the 2$e$ molecule, since the ground-state \nenergy of two fully spatially separated electrons (zero overlap) does not \ndepend on the total spin \\cite{note67}. \n\n\\subsection{Measures of entanglement \\cite{note27}}\n\nTo calculate the concurrence ${\\cal C}$ \\cite{schl,loss}, one needs a \ndecomposition of the GHL wave function into a \nlinear superposition of {\\it orthogonal\\\/}\nSlater determinants. Thus one needs to expand the {\\it nonorthogonal\\\/} \n$u^{L,R}({\\bf r})$ orbitals as a superposition of two other {\\it orthogonal\\\/}\nones. To this effect, we write\n$u^{L,R}({\\bf r}) \\propto \\Phi^+({\\bf r}) \\pm  \\xi \\Phi^-({\\bf r})$,\nwhere $\\Phi^+({\\bf r})$ and $\\Phi^-({\\bf r})$ are the parity symmetric and \nantisymmetric (along the $x$-axis) components in their expansion given by \nEq.\\ (\\ref{uexp}). \nSubsequently, with the use of Eq.\\ (\\ref{wfghl}), the GHL \nsinglet can be rearranged as follows:\n\\begin{equation}\n\\Psi^{s}_{\\text{GHL}} \\propto\n| \\Phi^+(1\\uparrow) \\Phi^+(2\\downarrow) \\rangle - \n\\eta |\\Phi^-(1\\uparrow)\\Phi^-(2\\downarrow) \\rangle,\n\\label{rear}\n\\end{equation}\nwhere the so-called interaction parameter \\cite{loss}, $\\eta=\\xi^2$, is the \ncoefficient in front of the second determinant.\nKnowing $\\eta$ allows a direct evaluation of the concurrence of the singlet\nstate, since ${\\cal C}^s = 2\\eta\/(1+\\eta^2)$ \\cite{loss}. Note that \n$\\Phi^+({\\bf r})$ and $\\Phi^-({\\bf r})$ are properly normalized.\nIt is straightforward to show that $\\eta=(1-|S_{LR}|)\/(1+|S_{LR}|)$, where \n$S_{LR}$ (with $|S_{LR}| \\leq 1$) is the overlap of the original \n$u^{L,R}({\\bf r})$ orbitals.\n\nFor the GHL triplet, one obtains an expression independent of the \ninteraction parameter $\\eta$, i.e.,\n\\begin{equation}\n\\Psi^{t}_{\\text{GHL}} \\propto\n| \\Phi^+(1\\uparrow) \\Phi^-(2\\downarrow) \\rangle +\n|\\Phi^+(1\\downarrow)\\Phi^-(2\\uparrow) \\rangle,\n\\label{reart}\n\\end{equation}\nwhich is a maximally (${\\cal C}^t=1$) entangled state. Note that underlying the \nanalysis of the experiments in Ref.\\ \\cite{marc} is a {\\it conjecture\\\/} that \nwave functions of the form given in Eqs.\\ (\\ref{rear}) and (\\ref{reart}) \ndescribe the two electrons in the elliptic QD.\n\nFor the GHL singlet, using the overlaps of the left and right orbitals, we find\nthat starting with $\\eta=0.46$ $({\\cal C}^s=0.76)$ at $B=0$, the interaction\nparameter (singlet-state concurrence) increases monotonically to $\\eta=0.65$ \n$({\\cal C}^s=0.92)$ at $B=2.5$ T. At the intermediate value corresponding to the\nST transition ($B=1.3$ T), we find $\\eta=0.54$ $({\\cal C}^s=0.83)$ \n\\cite{note34}. Our $B=0$ theoretical results for \n$\\eta$ and ${\\cal C}^s$ are in remarkable agreement with \nthe experimental estimates \\cite{marc} of $\\eta=0.5 \\pm 0.1$ and \n${\\cal C}^s=0.8$, which were based solely on conductance measurements below the \nST transition (i.e., near $B=0$). \n\nTo compute the von Neumann entropy, one needs to bring both \nthe EXD and the GHL wave functions into a diagonal form (the socalled \n``canonical form'' \\cite{you,schl2}), i.e.,\n\\begin{equation}\n\\Psi^{s,t}_{\\text{EXD}} ({\\bf r}_1, {\\bf r}_2) =\n\\sum_{k=1}^M z^{s,t}_k | \\Phi(1;2k-1) \\Phi(2;2k) \\rangle,\n\\label{cano}\n\\end{equation}\nwith the $\\Phi(i)$'s being appropriate spin orbitals resulting from a unitary \ntransformation of the basis spin orbitals $\\psi(j)$'s [see Eq.\\ (\\ref{wfexd})]; \nonly terms with $z_k \\neq 0$ contribute. The upper bound $M$ can be \nsmaller (but not larger) than $K$ (the dimension of the \nsingle-particle basis); $M$ is referred to as the Slater rank.\nOne obtains the coefficients of the canonical expansion from the fact that \nthe $|z_k|^2$ are eigenvalues of the hermitian matrix $\\Omega^\\dagger \\Omega$\n[$\\Omega$, see Eq.\\ (\\ref{wfexd}), is antisymmetric]. The von Neumann \nentropy is given by ${\\cal S} = -\\sum_{k=1}^M |z_k|^2 \\log_2(|z_k|^2)$ with the\nnormalization $\\sum_{k=1}^M |z_k|^2 =1$.\nNote that the GHL wave functions in Eqs.\\ (\\ref{rear}) and (\\ref{reart}) \nare already in canonical form, which shows that they always have a Slater rank \nof $M=2$. One finds ${\\cal S}^s_{\\text{GHL}} =\n\\log_2(1+\\eta^2) - \\eta^2 \\log_2(\\eta^2)\/(1+\\eta^2)$, and\n${\\cal S}^t_{\\text{GHL}}=1$ for all $B$. For large $B$, the overlap \nbetween the two electrons of the dissociated dimer vanishes, and thus \n$\\eta \\rightarrow 1$ and ${\\cal S}^s_{\\text{GHL}} \\rightarrow 1$.\n\n\n\\begin{figure}[t]\n\\centering\\includegraphics[width=10cm]{tnt_2005_fig6.eps}\n\\caption{\nVon Neumann entropy for the singlet state of the elliptic dot as a function of \nthe magnetic field $B$. Solid line: GHL. Dashed line: EXD. \nThe rest of the parameters and the screening of the Coulomb interaction \nare as in Fig.\\ 4. On the left, we show histograms for the $|z_k|^2$\ncoefficients [see Eq.\\ (\\ref{cano})] of the singlet state at $B=1.3$ T, \nillustrating the dominance of two configurations. Note the small third\ncoefficient $|z_3|^2=0.023$ in the EXD case.\n}\n\\end{figure}\n\nSince the EXD singlet has obviously a Slater rank $M > 2$, the definition\nof concurrence is not applicable to it.  \nThe von Neumann entropy for the EXD singlet (${\\cal S}^s_{\\text{EXD}}$) is \ndisplayed in Fig.\\ 6, along with that (${\\cal S}^s_{\\text{GHL}}$) of the GHL \nsinglet. ${\\cal S}^s_{\\text{EXD}}$ and \n${\\cal S}^s_{\\text{GHL}}$ are rather close to each other for the entire\n$B$ range, and it is remarkable that both \nremain close to unity for large $B$, although the maximum allowed mathematical \nvalue is $\\log_2(K)$ [as aforementioned we use $K=79$, i.e., $\\log_2(79)=6.3$];\nthis maximal value applies for both the EXD and GHL approaches. The saturation \nof the entropy for large $B$ to a value close to unity reflects the dominant \n(and roughly equal at large $B$) weight of two configurations in the \ncanonical expansion [see Eq.\\ (\\ref{cano})] of the EXD wave function, which\nare related to the two terms ($M=2$) in the canonical expansion of the GHL\nsinglet [Eq.\\ (\\ref{rear})]. This is illustrated by the histograms of the\n$|z_k^s|^2$ coefficients for $B=1.3$ T in Fig.\\ 6 (left column).\nThese observations support the GHL approximation, which is\ncomputationally less demanding than the exact diagonalization, and can be used\neasily for larger $N$.\n\n\\section{Summary}\n\nWe discussed symmetry breaking in two-dimensional quantum dots resulting from\nstrong interelectron repulsion relative to the zero-point kinetic energy\nassociated with the confining potential. Such symmetry breaking leads to the\nemergence of crystalline arrangements of electrons in the dot. The so-called\nWigner molecules form already at field-free conditions. The appearance of\nrotating Wigner molecules in circular dots under high magnetic field, and their\nrelation to magic angular momenta and quantum-Hall-effect fractional fillings was\nalso discussed. \n\nFurthermore, we have shown, through exact and approximate microscopic\ntreatments, formation of an electron molecular dimer in an\nelliptic QD (Fig.\\ 5) for screened interelectron repulsion.\nThe formation and effective dissociation (in high magnetic fields) of the \nelectron dimer are reflected in the behavior of the computed singlet-triplet\nsplitting, $J(B)$, that agrees well (see Fig.\\ 4) with the measurements \n\\cite{marc}. Furthermore, we showed that, from a knowledge of the dot shape and \nof $J(B)$, theoretical analysis along the lines introduced here allows probing \nof the correlated ground-state wave function and determination of its degree of \nentanglement. This presents an alternative to the experimental study where\ndetermination of the concurrence utilized conductance data \\cite{marc}.\nWe have employed two measures of entanglement for {\\it\nindistinguishable fermions\\\/} (the concurrence and the von Neumann entropy) \nand have shown that their behavior correlates with the effective dissociation of\nthe electron dimer. Such information is of interest to the implementation of \nspin-based solid-state quantum logic gates.\n\nThis research is supported by the US D.O.E. (Grant No. FG05-86ER45234), and\nNSF (Grant No. DMR-0205328). We thank M. Pustilnik for comments on the\nmanuscript.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec:intro}\n\nA major impairment in orthogonal frequency-division multiple-access (OFDMA)\nnetworks is the remarkable sensitivity to timing errors and carrier\nfrequency offsets (CFOs) between the uplink signals and the base station\n(BS) local references. For this reason, the IEEE 802.16e-2005 standard for\nOFDMA-based wireless metropolitan area networks (WMANs) specifies a\nsynchronization procedure called Initial Ranging (IR) in which subscriber\nstations that intend to establish a link with the BS transmit pilot symbols\non dedicated subcarriers using specific ranging codes. Once the BS has\ndetected the presence of these pilots, it has to estimate some fundamental\nparameters of ranging subscriber stations (RSSs) such as timing errors, CFOs\nand power levels.\n\nInitial synchronization and power control in OFDMA was originally discussed\nin \\cite{Krinock2001} and \\cite{Minn2004} while similar solutions can be\nfound in \\cite{Mahmoud2006}-\\cite{Zhou2006}. A different IR approach has\nrecently been proposed in \\cite{Minn07}. Here, each RSS transmits pilot\nstreams over adjacent OFDMA blocks using orthogonal spreading codes. As long\nas channel variations are negligible over the ranging period, signals of\ndifferent RSSs can be easily separated at the BS as they remain orthogonal\nafter propagating through the channel. Timing information is eventually\nacquired in an iterative fashion by exploiting the autocorrelation\nproperties of the received samples induced by the use of the cyclic prefix\n(CP).\n\nAll the aforementioned schemes are derived under the assumption of perfect\nfrequency alignment between the received signals and the BS local reference.\nHowever, the occurrence of residual CFOs results into a loss of\northogonality among ranging codes and may compromise the estimation accuracy\nand detection capability of the IR process. Motivated by the above problem,\nin the present work we propose a novel ranging scheme for OFDMA networks\nwith increased robustness against frequency errors and lower computational\ncomplexity than the method in \\cite{Minn07}. To cope with the large number\nof parameters to be recovered, we adopt a three-step procedure. In the first\nstep the number of active codes is estimated by resorting to the minimum\ndescription length (MDL) principle \\cite{Wax85}. Then, the ESPRIT\n(Estimation of Signal Parameters by Rotational Invariance Techniques) \\cite%\n{Roy1989} algorithm is employed in the second and third steps to detect\nwhich codes are actually active and determine their corresponding timing\nerrors and CFOs.\n\n\n\\section{System description and signal model}\n\n\\subsection{System description}\n\nWe consider an OFDMA\\ system employing $N$ subcarriers with index set $%\n\\{0,1,\\ldots ,N-1\\}$. As in \\cite{Minn07}, we assume that a ranging\ntime-slot is composed by $M$ consecutive OFDMA\\ blocks where the $N$\navailable subcarriers are grouped into ranging subchannels and data\nsubchannels. The former are used by the active RSSs to complete their\nranging processes, while the latter are assigned to data subscriber stations\n(DSSs) for data transmission. We denote by $R$ the number of ranging\nsubchannels and assume that each of them is divided into $Q$ subbands. A\ngiven subband is composed of a set of $V$ adjacent subcarriers which is\ncalled a \\textit{tile}. The subcarrier indices of the $q$th tile $%\n(q=0,1,\\ldots ,Q-1)$ in the $r$th subchannel $(r=0,1,\\ldots ,R-1)$ are\ncollected into a set $\\mathcal{J}_{q}^{(r)}=\\{i_{q}^{(r)}+v\\}_{v=0}^{V-1}$,\nwhere the tile index $i_{q}^{(r)}$ can be chosen adaptively according to the\nactual channel conditions. The only constraint in the selection of $%\ni_{q}^{(r)}$ is that different tiles must be disjoint, i.e., $\\mathcal{J}%\n_{q_{1}}^{(r_{1})}\\cap \\mathcal{J}_{q_{2}}^{(r_{2})}=\\varnothing $ for $%\nq_{1}\\neq q_{2}$ or $r_{1}\\neq r_{2}$. The $r$th ranging subchannel is thus\ncomposed of $QV$ subcarriers with indices in the set $\\mathcal{J}^{(r)}=\\cup\n_{q=0}^{Q-1}{\\mathcal{J}_{q}^{(r)}}$, while a total of $N_{R}=QVR$ ranging\nsubcarriers is available in each OFDMA block.\n\nWe assume that each subchannel can be accessed by a maximum number of $K_{\\max }=\\min \\{V,M\\}-1$\nRSSs, which are separated by means of orthogonal codes in both the time and\nfrequency domains. The codes are selected in a pseudo-random fashion from a\npredefined set $\\{\\mathbf{C}_{0},\\mathbf{C}_{1},\\ldots ,\\mathbf{C}_{K_{\\max\n}-1}\\}$ with\n\\begin{equation}\n\\mathbf{[C}_{k}\\mathbf{]}_{v,m}=e^{j2\\pi k(\\frac{v}{V-1}+\\frac{m}{M-1})}\n\\label{eq1}\n\\end{equation}%\nwhere $v=0,1,\\ldots ,V-1$ counts the subcarriers within a tile and is used\nto perform spreading in the frequency domain, while $m=0,1,\\ldots ,M-1$\nis the block index by which spreading is done in the time domain across\nthe ranging time-slot. As in \\cite{Minn07}, we assume that different RSSs\nselect different codes. Also, we assume that a selected code is employed by\nthe corresponding RSS over all tiles in the considered subchannel. Without\nloss of generality, we concentrate on the $r$th subchannel and denote by $%\nK\\leq K_{\\max }$ the number of simultaneously active RSSs. To simplify the\nnotation, the subchannel index $^{(r)}$ is dropped henceforth.\n\nThe signal transmitted by the $k$th RSS propagates through a multipath\nchannel characterized by a channel impulse response (CIR) $\\mathbf{h}'%\n_{k}=[h'_{k}(0),h'_{k}(1),\\ldots ,h'_{k}(L-1)]^{T}$ of length $L$ (in sampling\nperiods). We denote by $\\theta _{k}$ the timing error of the \\textit{k}th\nRSS expressed in sampling intervals $T_{s}$, while $\\varepsilon _{k}$ is the\nfrequency offset normalized to the subcarrier spacing. As discussed in \\cite%\n{Morelli2004}, during IR the CFOs are only due to Doppler shifts and\/or\nto estimation errors and, in consequence, they are assumed to lie \\textit{%\nwithin a small fraction} of the subcarrier spacing. Timing offsets depend on\nthe distance of the RSSs from the BS and their maximum value is thus limited\nto the round trip delay from the cell boundary. In order to eliminate\ninterblock interference (IBI), we assume that during the ranging process the\nCP length comprises $N_{G}\\geq \\theta _{\\max }+L$ sampling periods, where $%\n\\theta _{\\max }$ is the maximum expected timing error. This assumption is\nnot restrictive as initialization blocks are usually preceded by long CPs in\nmany standardized OFDMA systems.\n\n\\subsection{System model}\n\nWe denote by $\\mathbf{X}_{m}(q)=[X_{m}(i_{q}),X_{m}(i_{q}+1),\\ldots\n,X_{m}(i_{q}+V-1)]^{T}$ the discrete Fourier transform (DFT) outputs\ncorresponding to the $q$th tile in the $m$th OFDMA block. Since DSSs have\nsuccessfully completed their IR processes, they are perfectly aligned to the\nBS references and their signals do not contribute to $\\mathbf{X}_{m}(q)$. In\ncontrast, the presence of uncompensated CFOs destroys orthogonality among\nranging signals and gives rise to interchannel interference (ICI). The\nlatter results in a disturbance term plus an attenuation of the useful\nsignal component. To simplify the analysis, in the ensuing discussion the\ndisturbance term is treated as a zero-mean Gaussian random variable while\nthe signal attenuation is considered as part of the channel impulse\nresponse. Under the above assumptions, we may write\n\\begin{equation}\nX_{m}(i_{q}+v)=\\sum\\limits_{k=1}^{K}\\mathbf{[C}_{\\ell _{k}}\\mathbf{]}_{v,m}{%\ne^{j{m\\omega _{k}N_{T}}}}H_{k}(\\theta _{k},{\\varepsilon _{k},}%\ni_{q}+v)+w_{m}(i_{q}+v)  \\label{eq2}\n\\end{equation}%\nwhere $\\omega _{k}=2\\pi \\varepsilon _{k}\/N$, $N_{T}=N+N_{G}$ denotes the\nduration of the cyclically extended block and $\\mathbf{C}_{\\ell _{k}}$ is\nthe code matrix selected by the \\textit{k}th RSS. The quantity $H_{k}(\\theta\n_{k},{\\varepsilon _{k},}n)$ is the $k$th \\textit{equivalent} channel\nfrequency response over the $n$th subcarrier and is given by\n\\begin{equation}\nH_{k}(\\theta _{k},{\\varepsilon _{k},}n)=\\gamma _{N}({\\varepsilon _{k})}%\nH_{k}^{\\prime }(n)e^{-j{2\\pi n\\theta _{k}}\/N}  \\label{eq3}\n\\end{equation}%\nwhere $H_{k}^{\\prime }(n)=\\sum_{\\ell =0}^{L-1}h_{k}^{\\prime }(\\ell )e^{-j{%\n2\\pi n\\ell }\/N}$ is the true channel frequency response, while\n\\begin{equation}\n\\gamma _{N}({\\varepsilon })=\\frac{\\sin (\\pi {\\varepsilon })}{N\\sin (\\pi {%\n\\varepsilon }\/N)}e^{j\\pi {\\varepsilon }(N-1)\/N}  \\label{eq5}\n\\end{equation}%\nis the attenuation factor induced by the CFO. The last term in (\\ref{eq2})\naccounts for background noise plus interference and is modeled as a circularly symmetric\ncomplex Gaussian random variable with zero-mean and variance $\\sigma\n_{w}^{2}=\\sigma _{n}^{2}+\\sigma _{ICI}^{2}$, where $\\sigma _{n}^{2}$ and $%\n\\sigma _{ICI}^{2}$ are the average noise and ICI powers, respectively. From (%\n\\ref{eq3}) we see that $\\theta _{k}$ appears only as a phase shift across\nthe DFT outputs. The reason is that the CP duration is longer than the\nmaximum expected propagation delay.\n\nTo proceed further, we assume that the tile width is much smaller than the\nchannel coherence bandwidth. In this case, the channel response is nearly\nflat over each tile and we may reasonably replace the quantities $%\n\\{H_{k}^{\\prime }(i_{q}+v)\\}_{v=0}^{V-1}$ with an average frequency response\n\\begin{equation}\n\\overline{H}_{k}^{\\prime }(q)=\\frac{1}{V}\\sum_{v=0}^{V-1}H_{k}^{\\prime\n}(i_{q}+v).  \\label{eq6}\n\\end{equation}%\nSubstituting (\\ref{eq1}) and (\\ref{eq3}) into (\\ref{eq2}) and bearing in\nmind (\\ref{eq6}), yields\n\\begin{equation}\nX_{m}(i_{q}+v)=\\sum\\limits_{k=1}^{K}{e^{j2\\pi ({m\\xi _{k}+v\\eta }_{k}{)}}}%\nS_{k}(q)+w_{m}(i_{q}+v)  \\label{eq7}\n\\end{equation}%\nwhere $S_{k}(q)=\\gamma _{N}({\\varepsilon _{k})}\\overline{H}_{k}^{\\prime\n}(q)e^{-j{2\\pi i}_{q}{\\theta _{k}}\/N}$ and we have defined the quantities\n\\begin{equation}\n\\xi _{k}=\\frac{\\ell _{k}}{M-1}+\\frac{\\varepsilon _{k}N_{T}}{N}  \\label{eq8}\n\\end{equation}\nand\n\\begin{equation}\n\\eta _{k}=\\frac{\\ell _{k}}{V-1}-\\frac{\\theta _{k}}{N}  \\label{eq9}\n\\end{equation}%\nwhich are referred to as the \\textit{effective }CFOs and timing errors,\nrespectively.\n\nIn the following sections we show how the DFT outputs $\\{X_{m}(i_{q}+v)\\}$\ncan be exploited to identify the active codes and to estimate the\ncorresponding timing errors and CFOs.\n\n\\section{ESPRIT-based estimation}\n\n\\subsection{Determination of the number of active codes}\n\nThe first problem to solve is to determine the number $K$ of active codes\nover the considered ranging subchannel. For this purpose, we collect the $%\n(i_{q}+v)$th DFT outputs across all ranging blocks into an \\textit{M}%\n-dimensional vector $\\mathbf{Y}(i_{q,v})=[X_{0}(i_{q}+v),X_{1}(i_{q}+v),%\n\\ldots ,X_{M-1}(i_{q}+v)]^{T}$ given by\n\\begin{equation}\n\\mathbf{Y}(i_{q,v})=\\sum_{k=1}^{K}e^{j2\\pi v\\eta _{k}}S_{k}(q)\\mathbf{e}%\n_{M}(\\xi _{k})+\\mathbf{w}(i_{q,v})  \\label{eq30}\n\\end{equation}%\nwhere $\\mathbf{w}(i_{q,v})=[w_{0}(i_{q}+v),w_{1}(i_{q}+v),\\ldots\n,w_{M-1}(i_{q}+v)]^{T}$ is Gaussian distributed with zero mean and\ncovariance matrix $\\sigma _{w}^{2}\\mathbf{I}_{M}$ while $\\mathbf{e}_{M}(\\xi\n)=[1,e^{j2\\pi \\xi },e^{j4\\pi \\xi },\\ldots ,e^{j2\\pi (M-1)\\xi }]^{T}$.\n\nFrom the above equation, we observe that $\\mathbf{Y}(i_{q,v})$\nhas the same structure as measurements of multiple uncorrelated sources from\nan array of sensors. Hence, an estimate of $K$ can be obtained by performing\nan eigendecomposition (EVD) of the correlation matrix $\\mathbf{R}_{Y}=%\n\\mathrm{E}\\{\\mathbf{Y}(i_{q,v})\\mathbf{Y}^{H}(i_{q,v})\\}$. In practice,\nhowever, $\\mathbf{R}_{Y}$ is not available at the receiver and must be\nreplaced by some suitable estimate. One popular strategy to get an estimate\nof $\\mathbf{R}_{Y}$ is based on the forward-backward (FB) principle.\nFollowing this approach, $\\mathbf{R}_{Y}$ is replaced by\n$\n\\hat{\\mathbf{R}}_{Y}=\\frac{1}{2}(\\tilde{\\mathbf{R}}_{Y}+\\mathbf{J}\\tilde{%\n\\mathbf{R}}_{Y}^{T}\\mathbf{J})$, where $\\tilde{\\mathbf{R}}_{Y}$ is the sample correlation matrix\n\\begin{equation}\n\\tilde{\\mathbf{R}}_{Y}=\\frac{1}{QV}\\sum_{v=0}^{V-1}\\sum_{q=0}^{Q-1}\\mathbf{Y}%\n(i_{q,v})\\mathbf{Y}^{H}(i_{q,v})  \\label{eq31}\n\\end{equation}%\nwhile $\\mathbf{J}$ is the exchange matrix with 1's on the anti-diagonal and\n0's elsewhere. Arranging the eigenvalues $\\hat{\\lambda}_{1}\\geq \\hat{\\lambda}%\n_{2}\\geq \\cdots \\geq \\hat{\\lambda}_{M}$ of $\\hat{\\mathbf{R}}_{Y}$ in\nnon-increasing order, we can find an estimate $\\hat{K}$ of the number of\nactive codes by applying the MDL information-theoretic criterion. This\namounts to looking for the minimum of the following objective function \\cite%\n{Wax85}\n\\begin{equation}\n\\mathcal{F(}\\tilde{K})=\\frac{1}{2}\\tilde{K}(2V-\\tilde{K})\\ln (MQ)-MQ(V-%\n\\tilde{K})\\ln \\rho (\\tilde{K})  \\label{eq22}\n\\end{equation}%\nwhere $\\rho (\\tilde{K})$ is the ratio between the geometric and arithmetic\nmeans of $\\{\\hat{\\lambda}_{\\tilde{K}+1},\\hat{\\lambda}_{\\tilde{K}+2},\\ldots ,%\n\\hat{\\lambda}_{M}\\}$.\n\n\\subsection{Frequency estimation}\n\nFor simplicity, we assume that the number of active codes has been perfectly\nestimated. An estimate of ${\\boldsymbol{\\xi }}{=[}\\xi _{1},\\xi\n_{2},\\ldots ,\\xi _{K}\\mathbf{]}^{T}$ can be found by applying the ESPRIT\nalgorithm to the model (\\ref{eq30}). To elaborate on this, we arrange the\neigenvectors of $\\hat{\\mathbf{R}}_{Y}$ associated to the $K$ largest\neigenvalues $\\hat{\\lambda}_{1}\\geq \\hat{\\lambda}_{2}\\geq \\cdots \\geq \\hat{%\n\\lambda}_{K}$ into an $M\\times K$ matrix $\\mathbf{Z}=[\\mathbf{z}_{1}~\\mathbf{%\nz}_{2}~\\cdots \\mathbf{z}_{K}]$. Next, we consider the matrices $\\mathbf{Z}%\n_{1}$ and $\\mathbf{Z}_{2}$ that are obtained by collecting the first $M-1$\nrows and the last $M-1$ rows of $\\mathbf{Z}$, respectively. The entries of ${%\n\\boldsymbol{\\xi }}$ are finally estimated in a decoupled fashion as\n\\begin{equation}\n\\hat{\\xi}_{k}=\\frac{1}{2\\pi }\\arg \\{\\rho _{y}(k)\\}\\text{ \\ \\ \\ \\ }%\nk=1,2,\\ldots ,K  \\label{eq25}\n\\end{equation}%\nwhere $\\{\\rho _{y}(1),\\rho _{y}(2),\\ldots ,\\rho _{y}(K)\\}$ are the\neigenvalues of\n\\begin{equation}\n\\mathbf{Z}_{Y}=(\\mathbf{Z}_{1}^{H}\\mathbf{Z}_{1})^{-1}\\mathbf{Z}_{1}^{H}%\n\\mathbf{Z}_{2}  \\label{eq26}\n\\end{equation}%\nand $\\arg \\{\\rho _{y}(k)\\}$ denotes the phase angle of $\\rho _{y}(k)$ taking\nvalues in the interval $[-\\pi ,\\pi )$.\n\nAfter computing estimates of the effective CFOs through (\\ref{eq25}), the\nproblem arises of matching each $\\hat{\\xi}_{k}$ to the corresponding code $%\n\\mathbf{C}_{\\ell _{k}}$. This amounts to finding an estimate of $\\ell _{k}$\nstarting from $\\hat{\\xi}_{k}$. For this purpose, we denote by $\\left\\vert\n\\varepsilon _{\\max }\\right\\vert $ the magnitude of the maximum expected CFO\nand observe from (\\ref{eq8}) that $(M-1)\\xi _{k}$ belongs to the interval $%\nI_{\\ell _{k}}=[\\ell _{k}-\\beta ;\\ell _{k}+\\beta ]$, with $\\beta =\\left\\vert\n\\varepsilon _{\\max }\\right\\vert N_{T}(M-1)\/N$. It follows that the effective\nCFOs can be univocally mapped to their corresponding codes as long as $\\beta\n<1\/2$ since only in that case intervals $\\{I_{\\ell _{k}}\\}_{k=1}^{K}$ are\ndisjoint. The acquisition range of the frequency estimator is thus limited\nto $\\left\\vert \\varepsilon _{\\max }\\right\\vert <N\/(2N_{T}(M-1))$ and an estimate\nof the pair $(\\ell _{k},\\varepsilon _{k})$ is computed as\n\\begin{equation}\n\\hat{\\ell}_{k}=\\text{round}\\left((M-1)\\hat{\\xi}_{k}\\right)  \\label{eq35}\n\\end{equation}%\nand\n\\begin{equation}\n\\hat{\\varepsilon}_{k}=\\frac{N}{N_{T}}\\left(\\hat{\\xi}_{k}-\\frac{\\hat{\\ell}_{k}}{M-1}\\right).\n\\label{eq36}\n\\end{equation}%\nIt is worth noting that the $\\arg \\{\\cdot \\}$ function in (\\ref{eq25}) has\nan inherent ambiguity of multiples of $2\\pi $, which translates into a\ncorresponding ambiguity of the quantity $\\hat{\\ell}_{k}$ by multiples of $M-1$%\n. Hence, recalling that $\\ell _{k}\\in \\{0,1,\\ldots ,K_{\\max }-1\\}$ with $%\nK_{\\max } < M$, a refined estimate of $\\ell _{k}$ can be found as\n\\begin{equation}\n\\hat{\\ell}_{k}^{(F)}=[\\hat{\\ell}_{k}]_{M-1}  \\label{eq29}\n\\end{equation}%\nwhere $[x]_{M-1}$ is the value of $x$ reduced to the interval $[0,M-2]$. In\nthe sequel, we refer to (\\ref{eq36}) as the ESPRIT-based frequency estimator\n(EFE).\n\n\\subsection{Timing estimation}\n\nWe call $\\mathbf{X}_{m}(q)=[X_{m}(i_{q}),X_{m}(i_{q}+1),\\ldots\n,X_{m}(i_{q}+V-1)]^{T}$ the $V$-dimensional vector of the DFT outputs\ncorresponding to the \\textit{q}th tile in the \\textit{m}th OFDMA\\ block.\nThen, from (\\ref{eq7}) we have\n\\begin{equation}\n\\mathbf{X}_{m}(q)=\\sum\\limits_{k=1}^{K}{e^{j2\\pi {m\\xi _{k}}}}S_{k}(q)%\n\\mathbf{e}_{V}(\\eta _{k})+\\mathbf{w}_{m}(q)  \\label{eq18}\n\\end{equation}%\nwhere $\\mathbf{w}_{m}(q)=[w_{m}(i_{q}),w_{m}(i_{q}+1),\\ldots\n,w_{m}(i_{q}+V-1)]^{T}$ is Gaussian distributed with zero mean and\ncovariance matrix $\\sigma _{w}^{2}\\mathbf{I}_{V}$ while $\\mathbf{e}_{V}(\\eta\n)=[1,e^{j2\\pi \\eta },e^{j4\\pi \\eta },\\ldots ,e^{j2\\pi (V-1)\\eta }]^{T}$.\nSince $\\mathbf{X}_{m}(q)$ is a superposition of complex sinusoidal signals\nwith random amplitudes embedded in white Gaussian noise, an estimate of ${%\n\\boldsymbol{\\eta }}{=[}\\eta _{1},\\eta _{2},\\ldots ,\\eta _{K}\\mathbf{]}^{T}$\ncan still be obtained by resorting to the ESPRIT algorithm. Following the\nprevious steps, we first compute $\\hat{\\mathbf{R}}_{X}=\\frac{1}{2}(\\tilde{%\n\\mathbf{R}}_{X}+\\mathbf{J}\\tilde{\\mathbf{R}}_{X}^{T}\\mathbf{J})$ with\n\\begin{equation}\n\\tilde{\\mathbf{R}}_{X}=\\frac{1}{MQ}\\sum_{m=0}^{M-1}\\sum_{q=0}^{Q-1}\\mathbf{X}%\n_{m}(q)\\mathbf{X}_{m}^{H}(q).  \\label{eq32}\n\\end{equation}%\nThen, we define a $V\\times K$ matrix $\\mathbf{U}=[\\mathbf{u}_{1}~\\mathbf{u}%\n_{2}~\\cdots \\mathbf{u}_{K}]$ whose columns are the eigenvectors of $\\hat{%\n\\mathbf{R}}_{X}$ associated to the $K$ largest eigenvalues. The effective\ntiming errors are eventually estimated as\n\\begin{equation}\n\\hat{\\eta}_{k}=\\frac{1}{2\\pi }\\arg \\{\\rho _{x}(k)\\},\\text{ \\ \\ \\ \\ }%\nk=1,2,\\ldots ,K  \\label{eq33}\n\\end{equation}%\nwhere $\\{\\rho _{x}(1),\\rho _{x}(2),\\ldots ,\\rho _{x}(K)\\}$ are the\neigenvalues of\n\\begin{equation}\n\\mathbf{U}_{X}=(\\mathbf{U}_{1}^{H}\\mathbf{U}_{1})^{-1}\\mathbf{U}_{1}^{H}%\n\\mathbf{U}_{2}  \\label{eq34}\n\\end{equation}%\nwhile the matrices $\\mathbf{U}_{1}$ and $\\mathbf{U}_{2}$ are obtained by\ncollecting the first $V-1$ rows and the last $V-1$ rows of $\\mathbf{U}$,\nrespectively.\n\nThe quantities $\\{\\hat{\\eta}_{k}\\}_{k=1}^{K}$ are eventually used to find\nestimates $(\\hat{\\ell}_{k},\\hat{\\theta}_{k})$ of the associated ranging code\nand timing error. To accomplish this task, we let $\\alpha =\\theta _{\\max\n}(V-1)\/(2N)$. Then, recalling that $0\\leq \\theta _{k}\\leq \\theta _{\\max }$, from\n(\\ref{eq9}) we see that $(V-1)\\eta _{k}+\\alpha $ falls into the range $I_{\\ell\n_{k}}=[\\ell _{k}-\\alpha ;\\ell _{k}+\\alpha ]$. If $\\theta _{\\max }<N\/(V-1)$, the\nquantity $\\alpha $ is smaller than 1\/2 and, in consequence, intervals $%\n\\{I_{\\ell _{k}}\\}_{k=1}^{K}$ are disjoint. In this case, there is only one\npair $(\\ell _{k},\\theta _{k})$ that results into a given $\\eta _{k}$ and an\nestimate of $(\\ell _{k},\\theta _{k})$ is found as\n\\begin{equation}\n\\hat{\\ell}_{k}=\\text{round}\\left((V-1)\\hat{\\eta}_{k}+\\alpha \\right)  \\label{eq27}\n\\end{equation}%\nand\n\\begin{equation}\n\\hat{\\theta}_{k}=N\\left(\\frac{\\hat{\\ell}_{k}}{V-1}-\\hat{\\eta}_{k}\\right).  \\label{eq28}\n\\end{equation}%\nAs done in Sect. 3.2, a refined estimate of $\\ell _{k}$ is obtained in the\nform\n\n\\begin{equation}\n\\hat{\\ell}_{k}^{(f)}=[\\hat{\\ell}_{k}]_{V-1}.  \\label{eq37}\n\\end{equation}%\nIn the sequel, we refer to (\\ref{eq28}) as the ESPRIT-based timing estimator\n(ETE).\n\n\\subsection{Code detection}\n\nFrom (\\ref{eq29}) and (\\ref{eq37}) we see that two distinct estimates\\ $\\hat{%\n\\ell}_{k}^{(F)}$ and $\\hat{\\ell}_{k}^{(f)}$ are available at the receiver\nfor each code index $\\ell _{k}$. These estimates are now used to decide\nwhich codes are actually active in the considered ranging subchannel. For\nthis purpose, we define two sets $I^{(f)}=\\{\\hat{\\ell}_{k}^{(f)}\\}_{k=1}^{K}$\nand $I^{(F)}=\\{\\hat{\\ell}_{k}^{(F)}\\}_{k=1}^{K}$ and observe that, in the\nabsence of any detection error, it should be $I^{(f)}=I^{(F)}=\\{\\ell\n_{k}\\}_{k=1}^{K}$. Hence, for any code matrix $\\mathbf{C}_{m}$\nwe suggest the following detection strategy\n\\begin{equation}\n\\begin{array}{c}\n\\text{if }m\\in I^{(f)}\\cap I^{(F)}\\Longrightarrow \\mathbf{C}_{m}\\text{ is\ndeclared \\textit{detected }} \\\\\n\\text{if }m\\notin I^{(f)}\\cap I^{(F)}\\Longrightarrow \\mathbf{C}_{m}\\text{ is\ndeclared \\textit{undetected}}.%\n\\end{array}\n\\label{eq37bis}\n\\end{equation}%\nIn the downlink response message, the BS will indicate only\nthe detected codes while undetected RSSs must restart their ranging process.\nIn the sequel, we refer to (\\ref{eq37bis}) as the ESPRIT-based code detector\n(ECD).\n\n\\section{Numerical results}\n\nThe investigated system has a total of $N=1024$ subcarriers over an uplink\nbandwidth of 3 MHz. The sampling period is $T_{s}=0.33$ $\\mu s$,\ncorresponding to a subcarrier distance of $1\/(NT_{s})=2960$ Hz. We assume\nthat $R=4$ subchannels are available for IR. Each subchannel is divided into\n$Q=16$ tiles uniformly spaced over the signal spectrum at a distance of $%\nN\/Q=64$ subcarriers. The number of subcarriers in any tile is $V=4$ while $%\nM=4$. The discrete-time CIRs have $L=12$ taps which are modeled as\nindependent and circularly symmetric Gaussian random variables with zero\nmeans and an exponential power delay profile, i.e., $E\\{\\left\\vert\nh_{k}(\\ell )\\right\\vert ^{2}\\}=\\sigma _{h}^{2}\\cdot \\exp (-\\ell \/12)$ for $%\n\\ell =0,1,\\ldots ,11$, where $\\sigma _{h}^{2}$ is chosen such that $%\nE\\{\\left\\Vert \\mathbf{h}_{k}\\right\\Vert ^{2}\\}=1$. Channels of different\nusers are statistically independent of each other and are kept fixed over an\nentire time-slot. We consider a maximum propagation delay of $\\theta _{\\max\n}=204$ sampling periods. Ranging blocks are preceded by a CP of length $%\nN_{G}=256$. The normalized CFOs are uniformly distributed over the interval $%\n[-\\Omega ,\\Omega ]$ and vary at each run. Recalling that the estimation\nrange of EFE is $\\left\\vert \\varepsilon _{k}\\right\\vert <N\/(2N_{T}(M-1))$, we\nset $\\Omega \\leq 0.1$.\n\n\\begin{figure}[htb]\n\\centering\n\\centerline{\\epsfig{figure=picture1.eps,width=7.4cm}}\n\\caption{$P_{f}$ vs. SNR for $K$ = 3 when $\\Omega$ is 0.05 or 0.1.}\n\\label{fig:res}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\centering\n\\centerline{\\epsfig{figure=picture5.eps,width=7.4cm}}\n\\caption{RMSE vs. SNR for $K$ = 2 or 3 when $\\Omega$ is 0.05 or 0.1.}\n\\label{fig:res}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centering\n\\centerline{\\epsfig{figure=picture7.eps,width=7.4cm}}\n\\caption{$P(\\protect\\epsilon)$ vs. SNR for $K$ = 2 or 3 when $\\Omega$ is\n0.05 or 0.1.}\n\\label{fig:res}\n\\end{figure}\nWe begin by investigating the performance of ECD in terms of probability of making an incorrect detection, say\n$P_{f}$. Fig. 1 illustrates $P_{f}$ as a function of $%\n\\mathrm{SNR}=1\/\\sigma _{n}^{2}$. The number of active RSSs is 3 while\nthe maximum CFO is either $\\Omega =0.05$ or 0.1. Comparisons are made with\nthe ranging scheme proposed by Fu, Li and Minn (FLM) in \\cite{Minn07}. The\nresults of Fig. 1 indicate that ECD performs remarkably better than FLM.\n\nFig. 2 illustrates the root mean-square error (RMSE) of the frequency\nestimates obtained with EFE vs. SNR for $K=2$ or 3 and $\\Omega =0.05$ or\n0.1. We see that the accuracy of EFE is satisfactory at SNR values\nof practical interest. Moreover, EFE has virtually the same performance\nas $K$ or $\\Omega$ increase.\n\nThe performance of the timing estimators is measured in terms of probability\nof making a timing error, say $P(\\epsilon )$, as defined in \\cite%\n{Morelli2004}. An error event is declared to occur whenever the estimate $%\n\\hat{\\theta}_{k}$ gives rise to IBI during the data section of the frame. This\nis tantamount to saying that the timing\nerror $\\hat{\\theta}_{k}-\\theta _{k}+(-N_{G,D}+L)\/2$ is larger than zero or\nsmaller than $-N_{G,D}+L-1$, where $N_{G,D}$ is the CP length during the data transmission phase. In the sequel, we set $N_{G,D}=32$.\nFig. 3 illustrates $P(\\epsilon )$ vs. SNR as\nobtained with ETE and FLM. The number of active codes is $K=2$ or 3 while $%\n\\Omega =0.05$ or 0.1. We see that ETE provides much better results than FLM.\n\n\\section{Conclusions}\n\nWe have presented a new ranging method for OFDMA systems in which uplink\nsignals arriving at the BS are impaired by frequency errors in addition to\ntiming misalignments. The synchronization parameters of all ranging users\nare estimated with affordable complexity through an ESPRIT-based approach.\nCompared to previous works, the proposed scheme exhibits increased\nrobustness against residual frequency errors and can cope with situations\nwhere the CFOs are as large as 10\\% of the subcarrier spacing.\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOverlapping speech is quite common for many scenarios, as the meetings recorded in the AMI corpus \\cite{McCowan2005_AMIMeetingCorpus} show overlap in about \\SIrange{5}{10}{\\%} of the time, and in informal situations, as recorded in the CHiME-5 \\cite{Barker2018_FifthCHiMESpeech} challenge data, it can exceed \\SI{20}{\\%}.\nA typical task for analysis of such recordings is \\gls{ASR}.\nIn the case of overlapping speech, this is called multi-talker speech recognition where the speech of multiple concurrently active talkers is to be recognized.\nConventional speech recognizers are limited to handling a single talker at a time which makes them inapplicable in those scenarios.\n\nMany efforts have already been put into the field of multi-talker speech recognition \\cite{Seki2018_PurelyEndtoEndSystem,Chang2019_EndtoendMonauralMultispeaker,Kanda2020_SerializedOutputTraining,Settle2018_EndtoendMultispeakerSpeech,Menne2019_AnalysisDeepClustering,vonNeumann2020_EndtoEndTrainingTime,Yu2017_RecognizingMultiTalkerSpeech,Qian2018_SinglechannelMultitalkerSpeech,Bahmaninezhad2019_ComprehensiveStudySpeecha}.\nThe approaches can basically be divided into monolithic and cascade systems.\nThe monolithic ones form one large neural network that is optimized as a whole, e.g., by extending a single-talker CTC\/attention ASR system so that its encoder outputs one latent representation for each speaker in the mixture and one or multiple attention decoders reconstruct one transcription for each representation \\cite{Seki2018_PurelyEndtoEndSystem,Chang2019_EndtoendMonauralMultispeaker}.\nAnother monolithic approach called Serialized Output Training modifies the target label sequence of a single-speaker recognizer to output the transcriptions for all talkers serially delimited by a special \\enquote{speaker change} token \\cite{Kanda2020_SerializedOutputTraining}.\nThese systems have the disadvantage that they do not provide interpretable signals as, e.g., separated speech signals.\n\n\\begin{figure}[t]\n    \\centering\n    \\include{tikz\/architecture}\n    \\vspace{-\\baselineskip}\n    \\caption{Example of the proposed iterative multi-speaker speech recognition system processing a three-speaker mixture.}\n    \\label{fig:architecture}\n   \n\\end{figure}\n\nThe cascade systems use source separation techniques followed by single-talker speech recognizers.\nThese systems have the advantage that intermediate signals are interpretable and individual system parts can be trained and tested on their own.\nAs separation front-ends, \\gls{DPCL} \\cite{Isik2016_SingleChannelMultiSpeakerSeparation}, \\gls{PIT} and the currently most promising \\gls{TasNet} architecture \\cite{Luo2018_TaSNetTimeDomainAudio} have been combined with different speech recognizers \\cite{Settle2018_EndtoendMultispeakerSpeech,Menne2019_AnalysisDeepClustering,Yu2017_RecognizingMultiTalkerSpeech,Isik2016_SingleChannelMultiSpeakerSeparation,Bahmaninezhad2019_ComprehensiveStudySpeecha,vonNeumann2020_EndtoEndTrainingTime}.\nThe experiments in general suggest that having both, dedicated parts for the separation and the recognition part, and joint end-to-end fine-tuning is beneficial.\n\nMost separation and multi-talker ASR approaches assume that the number of talkers is known \\cite{Kolbaek2017_MultitalkerSpeechSeparationa,Luo2018_TaSNetTimeDomainAudio,Luo2020_DualPathRNNEfficient,Seki2018_PurelyEndtoEndSystem,Chang2019_EndtoendMonauralMultispeaker,Settle2018_EndtoendMultispeakerSpeech,Qian2018_SinglechannelMultitalkerSpeech}, although this is not the case in realistic situations.\nSource number counting has been combined with source separation in iterative speech extraction  \\cite{Kinoshita2018_ListeningEachSpeaker,Takahashi2019_RecursiveSpeechSeparation} and model selection schemes \\cite{Nachmani2020_VoiceSeparationUnknown}, but not yet with speech recognition.\n\nWe propose the first jointly optimized multi-talker ASR system for an unknown number of speakers by combining source separation and number counting techniques from \\cite{Luo2020_DualPathRNNEfficient,Kinoshita2018_ListeningEachSpeaker,Takahashi2019_RecursiveSpeechSeparation} with a single-speaker CTC\/attention speech recognizer \\cite{Watanabe2017_HybridCTCAttention,Kim2017_JointCTCattentionBased,Watanabe2018_EspnetEndtoendSpeech,Xiao2018_HybridCTCattentionBased}.\nTo do that, we first investigate a \\gls{DPRNN}-TasNet separator for fixed numbers of speakers as a front-end for ASR.\nWe jointly fine-tune the \\gls{DPRNN} with a pre-trained ASR system, which was shown to be effective in other scenarios \\cite{Heymann2017_BeamnetEndtoendTraining} and multi-talker ASR \\cite{vonNeumann2020_EndtoEndTrainingTime}, to optimize the overall model performance.\nBy doing so, we achieve new state-of-the-art performance of \\SI{7.5}{\\%} \\gls{WER}.\nWe then integrate the \\gls{DPRNN} into the \\gls{OR-PIT} architecture and extend it with elegant mechanisms for source number counting.\nThe counting mechanisms show a promising performance.\nFinally, we combine the \\gls{OR-PIT} with an ASR system to form our new multi-speaker ASR system for unknown numbers of speakers.\nExperiments show that our system generalizes to a larger number of speakers than it saw during training.\n\n\\section{Source separation and counting}\nIn the following two subsections we first introduce the baseline approach to source separation, where the maximum number of talkers is assumed to be known.\nThen, we generalize it to separate an a priori unknown number of concurrent talkers.\n\n\\subsection{Known number of talkers: Dual-Path RNN-TasNet}\n\\defx(t){x(t)}\n\\defs{s}\n\\def\\sourceletter(t){s(t)}\n\\defk{k}\n\\def\\sourceindexed#1{s_{#1}(t)}\n\\def\\hat{\\sourceletter}(t){\\hat{s}(t)}\n\\def\\sourceletter_\\spkidx(t){s_k(t)}\n\\def\\hat{\\sourceletter}_\\spkidx(t){\\hat{s}_k(t)}\n\\defK{K}\n\\defL{L}\n\\def\\alpha{\\alpha}\n\nA single channel discrete-time speech mixture signal $x(t)$ is modeled as a sum of $K$ single-talker speech signals $\\sourceletter_\\spkidx(t)$:\n\\begin{equation}\n    x(t) = \\sum_{k = 1}^{K} \\sourceletter_\\spkidx(t),\n\\end{equation}\nwhere $k$ is the talker index.\nSource separation is concerned with extracting the source signals $\\sourceletter_\\spkidx(t)$ from the mixture $x(t)$.\n\nWe use the \\gls{TasNet} \\cite{Luo2018_TaSNetTimeDomainAudio,Luo2019_ConvTasNetSurpassingIdeal} with a \\gls{DPRNN} \\cite{Luo2020_DualPathRNNEfficient} as the separation network to obtain a fixed number of estimations $z_k(t)$ for the sources $\\sourceletter_\\spkidx(t)$, where the order of $z_k(t)$ can be permuted to the actual source signals:\n\\begin{equation}\n    [z_1 (t), ..., z_K (t)] = \\text{DPRNN-TasNet}\\big(x(t)\\big).\n\\end{equation}\nIt works by encoding the time-domain signal into a latent domain with a convolutional encoder, separating this representation with the \\gls{DPRNN}, and transforming it back to time domain with a de-convolutional decoder.\nThe \\gls{DPRNN} models short- and long-term dependencies in an alternating manner by segmenting the input and skipping different numbers of time-steps in adjacent layers.\nIt was shown in \\cite{Luo2020_DualPathRNNEfficient} to be superior to a separator based on 1-D convolutions used in the Conv-\\gls{TasNet} \\cite{Luo2019_ConvTasNetSurpassingIdeal}.\n\n\n\n\\subsubsection{Time-domain training objective}\n\nIn recent work, the training objective often is to maximize the scale-invariant source-to-distortion ratio (SI-SDR)\\footnote{Sometimes called SI-SNR \\cite{Luo2018_TaSNetTimeDomainAudio,Luo2019_ConvTasNetSurpassingIdeal}} \\cite{Luo2018_TaSNetTimeDomainAudio,Luo2019_ConvTasNetSurpassingIdeal} by minimizing the negative SI-SDR.\nBy setting $\\alpha=1$ and removing terms that do not depend on $\\hat{\\sourceletter}(t)$, the time-domain logarithmic MSE loss can be obtained \\cite{Heitkaemper2020_DemystifyingTasNetDissecting}:\n\\begin{equation}\n    \\loss{T-LMSE}\\big(\\sourceletter(t), \\hat{\\sourceletter}(t)\\big) = 10 \\log_{10} \\sum_t |\\sourceletter(t) - \\hat{\\sourceletter}(t)|^2.\n\\end{equation}\nThe missing scale-invariance did not show negative effects on the separation performance \\cite{Heitkaemper2020_DemystifyingTasNetDissecting}.\nTo be able to handle different numbers of speakers with such a model, it is common for frequency-domain separators to set the missing outputs to silence \\cite{Kolbaek2017_MultitalkerSpeechSeparationa}, i.e., $\\sourceletter(t)=0$.\nTo use silent targets here, the loss has to be prevented from going to negative infinity for perfect reconstruction and masking any loss terms from other target signals by, e.g., adding $1$ to the argument of the logarithm:\n\\begin{equation}\n    \\loss{T-L1PMSE}\\big(\\sourceletter(t), \\hat{\\sourceletter}(t)\\big) = 10 \\log_{10}\\big(1 + \\sum_t |\\sourceletter(t) - \\hat{\\sourceletter}(t)|^2\\big).\n\\end{equation}\nThe total loss is calculated in a permutation-invariant manner with the set $\\mathcal{P}$ of all permutations of length $K$:\n\\begin{equation}\n    \\loss{reconstruction} = \\min_{\\phi\\in\\mathcal{P}} \\frac{1}{K} \\sum_{k=1}^{K}\\mathcal{L}(\\sourceindexed{\\phi(k)}, z_k (t)).\n\\end{equation}\n\n\\def{n\/a}{{n\/a}}\n\n\\subsection{Unknown number of talkers: OR-PIT}\n\nConventional source separators are limited to a fixed number of talkers.\nAn arbitrary number of talkers can be handled by iterative source extraction approaches \\cite{Kinoshita2018_ListeningEachSpeaker,Takahashi2019_RecursiveSpeechSeparation}.\nInstead of directly separating the mixture into one stream for each talker, they apply a network repeatedly to extract one talker at a time.\n\n\\defz_1(t){z_1(t)}\n\\defz_2(t){z_2(t)}\nFollowing the \\gls{OR-PIT} \\cite{Takahashi2019_RecursiveSpeechSeparation} scheme, an iterative source extractor is a two-output separator, in our case a DPRNN-TasNet, trained to output one talker at its primary output $z_1(t)$ and the sum of all remaining talkers at its secondary output $z_2(t)$ so that $z_2(t)$ can be fed back to extract the next talker, as visualized in the left part of \\cref{fig:architecture}.\nIt is first trained with\n\\begin{equation}\n  \\label{eq:or-pit}\n    \\loss{OR-PIT} = \\min_k \\mathcal{L} \\big(z_1(t), \\sourceletter_\\spkidx(t)\\big) + \\mathcal{L}\\big(z_2(t), \\sum_{i\\neqk} \\sourceindexed{i}\\big)\n\\end{equation}\non clean mixtures, where $\\mathcal{L}$ is a time-domain loss.\nIt is then fine-tuned by feeding $z_2(t)$ as additional training data.\n\nThe number of talkers can be determined by counting the number of iterations required, until $z_2(t)$ does not contain speech.\nThe authors of \\cite{Takahashi2019_RecursiveSpeechSeparation} use Alexnet \\cite{Sutskever2012_ImagenetClassificationDeep} as an external classifier to detect the absence of speech.\nA stopping criterion can, however, be integrated into the separator by using thresholding or an additional output flag, as inspired by \\cite{Kinoshita2018_ListeningEachSpeaker}.\n\n\n\nAssuming that a speech signal contains a certain minimal amount of average energy and the network is able to suppress speech well enough, absence of speech can be detected by:\n\\def\\gamma{\\gamma}\n\\begin{equation}\n    \\frac{1}{T} \\sum_t |z_2(t)|^2 < \\gamma,\n\\end{equation}\nwith $T$ being the length of the signal and the threshold value $\\gamma$ being determined manually.\nModels that use this stopping criterion are called \\enquote{threshold} models.\n\n\\def\\hat{f}{\\hat{f}}\n\\deff{f}\nAn elegant way to integrate the classifier into the separator is to let the separation network predict a stop flag.\nThis is done by adding an additional scalar output $f$ to the network.\nThe output size of the \\gls{DPRNN} is increased. \nThe newly added \\gls{DPRNN} outputs are transformed by a fully connected layer to a scalar for each time step.\nThe stop flag per utterance is obtained with, e.g., an average pooling over time followed by a sigmoidal function.\nThe estimated flag $\\hat{f}$ is trained to indicate when to stop iterating using a binary cross-entropy objective\n\\begin{equation}\n    \\loss{flag} = - f \\log(\\hat{f}) - (1 - f) \\log (1 - \\hat{f}).\n\\end{equation}\nThe target flag $f$ is set to $1$ when the second output should be empty and $0$ otherwise.\n\n\n\n\n\\section{Joint optimization of source separation and speech recognition}\n\nWe propose to jointly fine-tune a source separation front-end (FE) with source number counting and a speech recognition back-end.\nThe FE is either a TasNet or an OR-PIT and the back-end is a CTC\/attention speech recognizer similar to \\cite{Watanabe2017_HybridCTCAttention,Seki2018_PurelyEndtoEndSystem} taken from the ESPnet toolkit \\cite{Watanabe2018_EspnetEndtoendSpeech}.\nWe replace the original feature extraction with differentiable STFT-based features.\nIt is trained with a multi-target loss function with the factor $\\lambda=0.2$:\n\\begin{equation}\n    \\loss{ASR} = \\lambda \\loss{CTC} + (1 - \\lambda)\\loss{att}.\n\\end{equation}\n\nJoint fine-tuning is straight forward for the TasNet-based system without counting.\nThe gradients are propagated from the back-end into the FE and their losses are combined like\n\\begin{equation}\n    \\mathcal{L} = \\loss{ASR} + \\loss{FE}.\n\\end{equation}\nWe solve the permutation problem with the FE signal-level loss.\n\nFor the iterative OR-PIT system, there are different options that arise from the fact that $z_2(t)$ can contain more than one talker and is, thus, unusable for training the back-end.\nWe formulate two training schemes, namely the single- and multi-iteration schemes.\nThe single-iteration scheme unrolls a single iteration of the OR-PIT FE and uses $z_1(t)$ to train the back-end.\n$z_2(t)$ is optimized using only the signal-level loss.\nHere, the OR-PIT always sees unprocessed data, so there is a mismatch to evaluation where its own output is fed back.\nTo mitigate this, the model can be unrolled in the multi-iteration scheme, where the secondary output is used as the input for the following iteration and all primary outputs are used to train the ASR back-end.\n\n\n\\section{Experiments}\n\n\nWe evaluate our systems in terms of average improvement of signal-to-distortion ratio (SDRi)\\footnote{Provided by the mir\\_eval toolbox \\cite{Raffel2014_MirEvalTransparent}.} \\cite{Vincent2006_PerformanceMeasurementBlind,Fevotte2005_BSSEVALToolbox}, \\gls{CER}, \\gls{WER} and counting accuracy on the WSJ, WSJ0-2mix, WSJ0-3mix and WSJ0-4mix databases \\cite{Paul1992_DesignWallStreet,Isik2016_SingleChannelMultiSpeakerSeparation,Kolbaek2017_MultitalkerSpeechSeparationa} with a sample rate of \\SI{8}{kHz}.\nWe use WSJ0-4mix as created in \\cite{Takahashi2019_RecursiveSpeechSeparation}.\nThe experiments on source separation and number counting are conducted on the \\emph{min} subsets of the WSJ0-mix databases that contain full overlap only.\nSpeech recognition is evaluated on the \\emph{max} subset where no utterances are truncated.\n\n\\subsection{Source Separation}\n\\begin{table}[t]\n    \\centering\n    \\caption{\n    Source separation performance in SDRi in dB for varying numbers of talkers given the oracle number of sources. The mixtures contain full overlap only (min subset).\n    }\\label{tab:separation}\n    \\setlength{\\tabcolsep}{3pt}\n   \n    \\begin{tabular}{ll SSSS}\n        \\toprule\n        \\multicolumn{2}{l}{\\mrow{Method}} & \\multicolumn{4}{c}{number of talkers} \\\\\n        \\cmidrule{3-6}\n        && {1} & {2} & {3} & {4} \\\\\n        \\midrule\n        \n        \\multirow{5}{*}{\\rotatebox[origin=c]{90}{DPRNN-}}&TasNet 2 talker & 10.8 & 17.0 & {---} & {---} \\\\\n        &TasNet 3 talker & 8.2 & 11.7 & 11.3 & {---} \\\\\n        &TasNet 2+3 talker & \\fontseries{b}\\selectfont 39.4 & 17.0 & 10.9 & {---} \\\\\n       \n        \\cmidrule{2-6}\n        &OR-PIT stop-flag & 30.6 & 17.2 & 12.5 & 8.7 \\\\\n        &OR-PIT threshold & 30.9 & \\fontseries{b}\\selectfont 17.5 & 12.8 & 8.4 \\\\\n        \\midrule\n        \\multicolumn{2}{l}{RSAN stop-flag \\cite{Kinoshita2018_ListeningEachSpeaker}} & {---} & 8.6 & {---} & {---}\\\\\n       \n        \\multicolumn{2}{l}{Conv-TasNet-OR-PIT \\cite{Takahashi2019_RecursiveSpeechSeparation}} & {---} & 15.0 & \\fontseries{b}\\selectfont 12.9 & \\fontseries{b}\\selectfont 10.6 \\\\\n        \\multicolumn{2}{l}{Original DPRNN-TasNet \\cite{Luo2020_DualPathRNNEfficient}} & {---} & 16.1 & {---} & {---} \\\\\n        \\bottomrule\n    \\end{tabular}\n   \n\\end{table}\n\\noindent\nWe first assume that the number of talkers is known.\nWe train all source separators on \\SI{4}{s} long signal segments to comply with \\cite{Luo2018_TaSNetTimeDomainAudio,Luo2020_DualPathRNNEfficient}.\nWe choose the DPRNN parameters according to the original paper \\cite{Luo2020_DualPathRNNEfficient}%\n, i.e., six blocks with two \\glspl{BLSTM} with 128 units, an encoder window size of 16 and a chunk size of 100.\nWe train two \\gls{DPRNN}-\\glspl{TasNet} for two or three speakers only, respectively, and one with three outputs for two and three speakers where the last target is set to silence for two-speaker mixtures.\nOur OR-PIT is trained on single-, two- and three-speaker recordings.\nWe use $\\loss{T-L1PMSE}$ for \\enquote{DPRNN-TasNet 2+3 talker} to cope with silent output, and $\\loss{T-LMSE}$ for all other models.\n\nThe separation performance displayed in \\cref{tab:separation} is evaluated given the oracle number of talkers, so that counting issues are not reflected in the SDRi.\nThe OR-PIT is forced to the correct number of iterations.\nFor the TasNet, only the $K$ outputs with the largest energy are considered for evaluation.\n\nOur two-talker TasNet (DPRNN-TasNet 2 talker) achieves a separation performance slightly better than \\cite{Luo2020_DualPathRNNEfficient} on two talkers and the same architecture works well on three talkers (DPRNN-TasNet 3 talker).\nBoth do not generalize well to smaller numbers of talkers.\nThe models specialized to one specific number are not able to produce a better reconstruction quality than what they learned during training.\nThe model trained on both, two and three speakers, handles both types of mixtures well and generalizes even to single-speaker recordings.\n\nThe OR-PIT can handle one to three speakers slightly better than the TasNet.\nIt especially has advantages in the three-speaker case, as also observed in \\cite{Takahashi2019_RecursiveSpeechSeparation}, where the OR-PIT network extracts one speaker at a time while the TasNet has to solve the more complex task of separating three speakers simultaneously.\nIts iterative structure allows the model to generalize to some extent to a larger number of talkers.\nThe performance, however, drops compared to the original Conv-TasNet-OR-PIT.\nThis might be caused by slight differences in training scheme.\n\n\n\n\\subsection{Source Number Counting}\n\n\\begin{table}[t]\n    \\centering\n    \\caption{\n    Source number counting accuracy in \\% for varying numbers of talkers. The mixtures contain full overlap only (min subset). ($^*$: Threshold not optimized for this number of sources)\n    }\\label{tab:counting}\n    \\setlength{\\tabcolsep}{2pt}\n    \\begin{tabular}{ll SSSS}\n        \\toprule\n        \\multicolumn{2}{l}{\\mrow{Method}} & \\multicolumn{4}{c}{number of talkers} \\\\\n        \\cmidrule{3-6}\n         && {1} & {2} & {3} & {4} \\\\\n        \\midrule\n        \\multirow{3}{*}{\\rotatebox[origin=c]{90}{DPRNN-}}\n        & TasNet 2+3 talker & 0.0{$^*$} & 99.9 & \\fontseries{b}\\selectfont 99.4 & {---} \\\\\n       \n        \\cmidrule{2-6}\n        & OR-PIT stop-flag & \\fontseries{b}\\selectfont 100.0 & \\fontseries{b}\\selectfont 100.0 & 96.7 & \\fontseries{b}\\selectfont 91.9 \\\\\n        & OR-PIT threshold & \\fontseries{b}\\selectfont 100.0 & \\fontseries{b}\\selectfont 100.0 & 98.2 & 0.0{$^*$}\\\\\n        \\midrule\n        \\multicolumn{2}{l}{RSAN stop-flag \\cite{Kinoshita2018_ListeningEachSpeaker}} & \\fontseries{b}\\selectfont 100.0 & 99.9 & {---} & {---} \\\\\n        \\multicolumn{2}{l}{Model selection \\cite{Nachmani2020_VoiceSeparationUnknown}} & {---} & 37.0 & 41.0 & 47.0 \\\\\n        \\bottomrule\n    \\end{tabular}\n   \n\\end{table}\n\n\\begin{table*}[ht]\n\\centering\n\\caption{\nRecognition performance of the multi-talker ASR systems for varying numbers of speakers given the oracle number of sources. The mixture signals contain full utterances (max subset). (\\enquote{FE}: front-end, \\enquote{ASR}: speech recognition part.)}\n\\label{tab:or-pit-recog}\n\\setlength{\\tabcolsep}{4.5pt}\n\\newcommand*{\\rn}[1]{(#1)}\n\\begin{tabular}{llSSS c SSS c SSS}\n    \\toprule\n    \\multicolumn{2}{l}{\\mrow{Experiment}} & \\multicolumn{3}{c}{WSJ0-2mix} && \\multicolumn{3}{c}{WSJ0-3mix} && \\multicolumn{3}{c}{WSJ0-4mix} \\\\\n    \\cmidrule{3-5}\\cmidrule{7-9}\\cmidrule{11-13}\n    && {CER} & {WER} & {SDRi} && {CER} & {WER} & {SDRi} && {CER} & {WER} & {SDRi} \\\\\n    \\midrule\n    \\rn{1} &DPRNN-TasNet (2 talker) + ASR & 9.6 & 15.8 & 16.5 && {---} & {---} & {---} && {---} & {---} & {---} \\\\\n    \\rn{2} &\\phantom{+}+ fine-tune ASR & 5.1 & 8.9 & 16.5 && {---} & {---} & {---} && {---} & {---} & {---} \\\\\n    \\rn{3} &\\phantom{+}+ fine-tune FE + ASR & \\fontseries{b}\\selectfont 4.0 & \\fontseries{b}\\selectfont 7.5 & 11.9 && {---} & {---} & {---} && {---} & {---} & {---} \\\\ \n    \\rn{4} &DPRNN-TasNet (3 talker) + ASR & {---} & {---} & {---} && 31.1 & 49.2 & 10.4 && {---} & {---} & {---} \\\\\n    \\rn{5} &\\phantom{+}+ fine-tune ASR & {---} & {---} & {---} && 25.5 & 35.3 & 10.4 && {---} & {---} & {---} \\\\\n    \\rn{6} &\\phantom{+}+ fine-tune FE + ASR & {---} & {---} & {---} && 29.0 & 39.7 & 7.5 && {---} & {---} & {---} \\\\\n    \\rn{7} &DPRNN-TasNet (2+3 talker) + ASR & 17.5 & 28.7 & 14.1 && 41.1 & 61.2 & 9.9 && {---} & {---} & {---} \\\\\n    \\rn{8} &\\phantom{+}+ fine-tune ASR & 7.3 & 11.9 & 14.1 && 19.0 & 27.1 & 9.9 && {---} & {---} & {---} \\\\\n    \\rn{9} &\\phantom{+}+ fine-tune FE + ASR & 5.8 & 10.5 & 14.4 && 24.9 & 34.9 & 7.8 && {---} & {---} & {---} \\\\\n    \\midrule\n    \\rn{10} &DPRNN-OR-PIT + ASR & 10.8 & 18.3 & \\fontseries{b}\\selectfont 16.7 && 25.2 & 40.7 & \\fontseries{b}\\selectfont 11.9 && 50.5 & 79.9 & \\fontseries{b}\\selectfont 7.8 \\\\\n   \n   \n   \n   \n   \n   \n   \n    \\rn{11}&\\phantom{+}+ VAD & 5.8 & 11.5 & \\fontseries{b}\\selectfont 16.7 && 14.8 & 26.2 & \\fontseries{b}\\selectfont 11.9 && 31.6 & 51.8 & \\fontseries{b}\\selectfont 7.8\\\\\n    \\rn{12}&\\phantom{++}+ single-iteration fine-tune ASR & 4.3 & 8.7 & \\fontseries{b}\\selectfont 16.7 && \\fontseries{b}\\selectfont 9.3 & \\fontseries{b}\\selectfont 16.3 & \\fontseries{b}\\selectfont 11.9 && 24.6 & 41.1 & \\fontseries{b}\\selectfont 7.8 \\\\\n    \\rn{13}&\\phantom{++}+ single-iteration fine-tune FE + ASR & 5.2 & 10.0 & 11.9 && 12.2 & 21.2 & 8.9 && 26.2 & 42.6 & 5.1 \\\\\n    \\rn{14}&\\phantom{++}+ multi-iteration fine-tune ASR & 4.7 & 9.0 & \\fontseries{b}\\selectfont 16.7 && 9.9 & 17.0 & \\fontseries{b}\\selectfont 11.9 && \\fontseries{b}\\selectfont 21.0 & \\fontseries{b}\\selectfont 33.4 & \\fontseries{b}\\selectfont 7.8 \\\\\n    \\rn{15}&\\phantom{++}+ multi-iteration fine-tune FE + ASR  & 4.6 & 8.7 & 12.9 && 11.8 & 20.5 & 9.0 && 27.8 & 45.2 & 4.8 \\\\\n    \\midrule\n    \\rn{16} &End-to-end ASR \\cite{Chang2019_EndtoendMonauralMultispeaker} & {---} & 25.4 & {---} && {---} & {---} & {---} && {---} & {---} & {---} \\\\\n    \\rn{17} &Conv-TasNet + ASR \\cite{vonNeumann2020_EndtoEndTrainingTime} & 6.0 & 11.1 & 13.8 && {---} & {---} & {---} && {---} & {---} & {---} \\\\\n    \\rn{18} &End-to-end ASR \\cite{Zhang2020_ImprovingEndtoEndSingleChannel} & {---} & 23.4 & {---} && {---} & {---} & {---} && {---} & {---} & {---} \\\\\n    \\midrule\n    \\rn{19} &Oracle ASR result based on ground truth data & 2.1 & 4.7 & {$+\\infty$} && 2.1 & 4.7 & {$+\\infty$} && 2.0 & 4.6 & {$+\\infty$} \\\\\n    \\bottomrule\n\\end{tabular}\n\\vspace{-\\baselineskip}\n\\end{table*}\n\n\\noindent\nThe same models are evaluated for counting accuracy in \\cref{tab:counting}.\nCounting for the 2+3 talker TasNet is performed by an energy-based threshold on its third output.\nAll thresholds are chosen to maximize the accuracy on \\enquote{cv} and \\enquote{dev} data.\n\nThe \\enquote{DPRNN 2+3 talker} model performs well in discriminating between two- and three-talker mixtures, but the threshold does not generalize well to the single-talker scenario due to higher energy levels in the estimated signals.\nA counting accuracy of over \\SI{90}{\\%} is possible with an adjusted threshold, but this degrades the accuracy on two- and three-speaker mixtures.\n\nThe OR-PIT detects the number of sources correctly in most cases.\nThe threshold model performs slightly better for the numbers of talkers it was trained on, but the stop-flag model generalizes better to larger numbers of talkers.\n\nCompared to the model selection scheme in \\cite{Nachmani2020_VoiceSeparationUnknown}, our system performs better in source counting but worse in separation, where their model achieves \\SI{20.1}{dB} SI-SDR improvement and our system \\SI{17}{dB} on two talkers.\nSimilar modifications as \\cite{Nachmani2020_VoiceSeparationUnknown} could be applied to the OR-PIT to possibly achieve a similar separation performance.\n\nAlthough not directly comparable, the stopping criteria introduced in this work seem to perform comparable to the Alexnet classifier with \\SI{95.7}{\\%} accuracy for less than three talkers \\cite{Takahashi2019_RecursiveSpeechSeparation} while being simpler.\n\n\\subsection{Single-talker speech recognition}\n\nWe use a configuration similar to \\cite{Seki2018_PurelyEndtoEndSystem} without the speaker dependent layers for the speech recognizer.\nThis results in two CNN layers followed by two \\gls{BLSTM}P layers with $1024$ units for the encoder, one LSTM layer with 300 units for the decoder and a feature dimension of $80$.\nWe use a location-aware attention mechanism and ADADELTA \\cite{Zeiler2012_ADADELTAadaptivelearning} as optimizer.\nDecoding is performed with a word-level RNN language model.\nOur ASR system achieves a WER of \\SI{6.4}{\\percent} on the WSJ eval92 set.\n\n\\subsection{Two-talker speech recognition}\n\nWe evaluate the speech recognition performance of the TasNet-based recognizer in rows 1 to 9 of \\cref{tab:or-pit-recog}.\nWe observe similar tendencies as were found in \\cite{Heymann2017_BeamnetEndtoendTraining} and \\cite{vonNeumann2020_EndtoEndTrainingTime}.\nFine-tuning the ASR part gives a greater improvement than fine-tuning the TasNet but fine-tuning both jointly gives the best performance on two speakers (compare rows 1 to 3) of \\SI{58}{\\%} relative improvement compared to not fine-tuning.\nThe final \\SI{7.5}{\\%} WER (3) forms a new state-of-the-art result on WSJ0-2mix and is a huge improvement over previous techniques \\cite{vonNeumann2020_EndtoEndTrainingTime}.\nSimilar to the separation evaluation, the two-speaker TasNet performs better than the two- and three-speaker model (rows 7 to 9).\n\n    \n\n\n\\subsection{Multi-talker speech recognition with counting}\n\nAfter having shown that the joint system performs well with a separator for a fixed number of talkers, we evaluate joint training for the OR-PIT in rows 10 to 15 of \\cref{tab:or-pit-recog}.\nWe use the stop-flag OR-PIT due to better counting accuracy on \\emph{max} data.\n\n\nThe OR-PIT shows an odd behavior that is not present in the TasNet.\nIn regions where only a single talker is active, one output should be silent.\nThis works for the TasNet, but the OR-PIT outputs the speech signal scaled down heavily.\nThe ASR system picks it up and creates insertion errors.\nAn energy-based voice activity detection (VAD) improves the WER substantially to \\SI{11.5}{\\%} (11) for two talkers without fine-tuning (10).\n\nFine-tuning the ASR system with the single-iteration scheme (12) can improve the performance, but the mismatch created by fine-tuning the FE this way degrades the performance compared to fine-tuning ASR (13).\nUsing the multi-iteration fine-tuning scheme (14 and 15) can prevent this, but the overall performance is not comparable to the two-speaker TasNet model, although the OR-PIT achieves a better separation performance, according to Tab.~\\ref{tab:separation}.\nThis might be caused by the more complex training procedure.\n\nThe performance on larger numbers of speakers is listed in the last two columns in \\cref{tab:or-pit-recog}.\nThe OR-PIT again performs better than the TasNet on more than two speakers.\nIt generalizes to the unseen larger number of four speakers.\nContradicting the results for two speakers, fine-tuning just the ASR part performs best for more than two speakers (rows 5, 8, 12 and 14).\nThese results suggest that it is beneficial to only fine-tune the ASR back-end in some scenarios, especially when the separation task is more challenging, while fine-tuning the front-end contributes to over-fitting to a specific scenario.\n\n\\section{Conclusion}\n\nWe build the first joint end-to-end system that performs source number counting and multi-talker speech recognition.\nOur specialized model to two talkers provides a new state-of-the-art \\gls{WER} for the WJS0-2mix database of \\SI{7.5}{\\%}.\nThe OR-PIT-based system that can count the numbers of speakers performs better than the TasNet-based model on three speakers and generalizes well to an unknown number of speakers, i.e., four speakers.\nEvaluation of the source counting abilities show very promising performance.\n\n\\section{Acknowledgements}\nComputational resources were provided by the Paderborn Center for Parallel Computing.\n\n\\bibliographystyle{IEEEtran}\n\\pagebreak \n\\balance\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:intro}\n\nThe two determinant identities we generalise, due respectively to Gessel and Viennot and to Aitken, are as follows.\nFor \\(n\\) a nonnegative integer, we write \\(\\intervalwz{n}=\\set{0,1,\\ldots,n}\\).\n\n\\begin{restatable}[Binomial determinant duality theorem {\\cite[Proposition 7]{GESSEL1985300}}]{theorem}{binomialidentity}\n\\label{thm:binomial_identity}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\). Then\n\\[\n    \\det\\mleft(%\n        \\binom{b}{a}%\n    \\mright)_{a \\in A, b \\in B}\n    =\n    \\det\\mleft(%\n        \\binom{a'}{b'}%\n    \\mright)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n    .\n\\]\n\\end{restatable}\n\n\\begin{restatable}[Symmetric function duality theorem {\\cite[Section 2]{aitken_1931}}]\n{theorem}{symfunctionidentity}\n\\label{thm:sym_function_identity}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\).\nThen\n\\[\n    \\det\\Bigl(%\n        h_{b - a} %\n    \\Bigr)_{a \\in A, b \\in B}\n    =\n    \\det\\Bigl(%\n        e_{a' - b'} %\n    \\Bigr)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n    .\n\\]\n\\end{restatable}\n\nHere \\(h_i\\) and \\(e_i\\) denote the complete homogeneous and elementary symmetric functions of degree \\(i\\) respectively, where by convention \\(e_0 = h_0 = 1\\) and \\(e_d = h_d = 0\\) for \\(d < 0\\).\nFor the purposes of indexing matrices, we consider finite subsets of \\(\\mathbb{Z}\\) to be ordered smallest element to largest.\n\nWe state our main theorem here in a slightly weakened form to avoid a technical condition whose necessity becomes clear only in the proof; the full statement with a weaker requirement on the parameters is given in \\autoref{thm:main_theorem}.\n\n\\begin{restatable}{theorem}{maintheorem}\\label{thm:intro_det_identity}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\).\nLet \\(\\alpha\\) and \\(\\beta\\) be partitions with \\(n\\) parts (with parts equal to \\(0\\) permitted) such that \\(\\alpha_i \\leq \\beta_i\\) for all \\(i \\in [n]\\).\nSuppose each part of \\(\\alpha\\) and \\(\\beta\\) is at most \\(1\\) less than the preceding part.\nThen the determinants\n\\[\n    \\det\\Bigl(%\n        h_{b - a} (x_{\\alpha_{a+1}+1}, x_{\\alpha_{a+1}+2}, \\ldots, x_{\\beta_{b}}) %\n    \\Bigr)_{a \\in A, b \\in B}\n\\]\nand\n\\[\n    \\det\\Bigl(%\n        e_{a' - b'} (x_{\\alpha_{a'}+1}, x_{\\alpha_{a'}+2}, \\ldots, x_{\\beta_{b'+1}}) %\n    \\Bigr)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n\\]\nare equal.\n\\end{restatable}\nWe illustrate this statement in \\autoref{eg:intro_thm_example} at the end of this section.\nNote that symmetric functions of positive degree over an empty set of variables are considered to equal~\\(0\\).\n\nOur proof, which comprises \\autoref{section:proof}, is by counting paths on lattices and constructing a bijection between sets of such paths.\nThis argument, which includes what is now known as the Lindstr\\\"{o}m--Gessel--Viennot lemma, is similar to that used by Gessel and Viennot to prove \\autoref{thm:binomial_identity}.\nHowever, we consider lattices of more general shape and with weighted edges.\nWeighted edges allow us to replace binomial coefficients with symmetric polynomials;\nthe more general shape of our lattices allows us to vary the number of variables in each symmetric polynomial. \n\nWe explain how \\autoref{thm:binomial_identity} and \\autoref{thm:sym_function_identity} can be deduced from \\autoref{thm:intro_det_identity} in \\autoref{section:corollaries}.\nGessel and Viennot's binomial duality theorem corresponds to a staircase-shaped lattice, while Aitken's symmetric function duality theorem corresponds to a rectangular lattice.\nThis paper thus provides a unifying framework for these two results, and since our theorem also permits lattices of shapes intermediate between staircases and rectangles, it generalises them significantly.\n\nWe also obtain lifts of the binomial duality theorem to \\(q\\)-binomial coefficients and to symmetric polynomials, which we state here and prove in \\autoref{section:corollaries}.\n\n\\begin{restatable}\n{corollary}{qbinomialidentity}\n\\label{cor:q-binomial_identity}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\).\nThen\n\\[\n    \\det\\mleft(%\n        \\qbinom{b}{a}%\n    \\mright)_{a \\in A, b \\in B}\n    =\n    \\det\\mleft(%\n        q^{\\binom{a'-b'}{2}} \\qbinom{a'}{b'}%\n    \\mright)_{a' \\in A^\\mathsf{c},b' \\in B^\\mathsf{c}}\n    .\n\\]\n\\end{restatable}\n\n\\begin{restatable}\n{corollary}{sympolybinomidentity}\n\\label{cor:sym_poly_binomial_identity}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\).\nThen\n\\[\n    \\det\\Bigl(%\n        h_{b - a} (x_{1}, x_{2}, \\ldots, x_{a+1}) %\n    \\Bigr)_{a \\in A, b \\in B}\n    =\n    \\det\\Bigl(%\n        e_{a' - b'} (x_{1}, x_{2}, \\ldots, x_{a'}) %\n    \\Bigr)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n    .\n\\]\n\\end{restatable}\n\nAitken's proof of \\autoref{thm:sym_function_identity} is of an entirely different flavour to the methods in this paper: it is an application of an elementary result in linear algebra due to Jacobi, which we refer to as Jacobi's complementary minor formula.\nAs we show in \\autoref{section:JT_connection}, this formula is insufficient to prove our main theorem.\n\n\n\\begin{example}\n\\label{eg:intro_thm_example}\nSuppose \\(n=4\\), \\(A = \\set{0,1,2}\\), \\(B = \\set{1,3,4}\\), \\(\\alpha = (1,1,0,0)\\) and \\(\\beta = (4,3,3,2)\\).\nThen the two determinants that \\autoref{thm:intro_det_identity} states are equal are\n\\[\n\\hspace*{-6.5em}\n\\begin{blockarray}{cc>{\\scriptstyle}c>{\\scriptstyle}c>{\\scriptstyle}c}\n      &   & b=1 & b=3 & b=4 \\\\\n      &   & \\beta_{b}=4 & \\beta_{b}=3 & \\beta_{b}=2\\\\[4pt]\n    \\begin{block}{>{\\scriptstyle}c>{\\scriptstyle}c|ccc|}\n    a=0 & \\alpha_{a+1}+1=2 & h_1(x_2, x_3, x_4) & h_3(x_2, x_3) & h_4(x_2) \\\\\n    a=1 & \\alpha_{a+1}+1=2 & 1 & h_2(x_2, x_3) & h_3(x_2) \\\\\n    a=2 & \\alpha_{a+1}+1=1 & 0 & h_1(x_1, x_2, x_3) & h_2(x_1, x_2) \\\\\n    \\end{block}\n\\end{blockarray}\n\\]\nand\n\\[\n\\hspace*{-6.5em}\n\\begin{blockarray}{ccc>{\\scriptstyle}c>{\\scriptstyle}cc}\n      &   && b'=0 & b'=2 &\\\\\n      &   && \\beta_{b'+1}=4 & \\beta_{b'+1}=3& \\\\[4pt]\n    \\begin{block}{>{\\scriptstyle}c>{\\scriptstyle}cc|cc|c}\n    a'=3 & \\alpha_{a'}+1=1 && e_3(x_1, x_2, x_3, x_4) & e_1(x_1, x_2, x_3)& \\\\\n    a'=4 & \\alpha_{a'}+1=1 && e_4(x_1, x_2, x_3, x_4) & e_2(x_1, x_2, x_3)&.\\\\\n    \\end{block}\n\\end{blockarray}\n\\]\n\\end{example}\n\n\n\\section{Proof of the main theorem}\n\\label{section:proof}\n\nThe steps in our proof are:\n\\begin{enumerate}[label={\\arabic*)}]\n    \\item\n    define two lattices and paths upon them;\n    \\item\n    show that the weighted counts of tuples of paths on the lattices equal the determinants in the main theorem;\n    \\item\n    construct a weight-preserving bijection from tuples of paths on one lattice to the other.\n\\end{enumerate}\n\nWe adopt the notation of \\autoref{thm:intro_det_identity} throughout: let \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size with complements in \\(\\intervalwz{n}\\) denoted \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\), and let \\(\\alpha\\) and \\(\\beta\\) be partitions with \\(n\\) parts (with parts equal to zero permitted) such that \\(\\alpha_i \\leq \\beta_i\\) for all \\(i \\in [n]\\).\nAdditionally, let \\(l = \\@ifstar{\\oldabs}{\\oldabs*}{A} = \\@ifstar{\\oldabs}{\\oldabs*}{B}\\), and let \\(r = \\@ifstar{\\oldabs}{\\oldabs*}{A^\\mathsf{c}} = \\@ifstar{\\oldabs}{\\oldabs*}{B^\\mathsf{c}} = n+1-l\\).\n\n\\subsection{Definition of lattices}\n\\label{section:lattices}\n\nWe picture Young diagrams as lying in a plane with the \\(x\\)-direction being downward and the \\(y\\)-direction being rightward, and the \\(1 \\times 1\\) square whose bottom-right corner is the point \\((i,j)\\) is referred to as the \\emph{box} \\(\\Ybox{i}{j}\\).\n(Though it is common to refer to a box simply by its coordinates, we denote boxes in this manner since we have need to distinguish between boxes and points.)\nThe skew Young diagram of \\(\\beta\/\\alpha\\) is then \\(\\Y{\\beta\/\\alpha} = \\setbuild{ \\Ybox{i}{j} }{1 \\leq i \\leq n,\\ \\alpha_i +1 \\leq j \\leq \\beta_i}\\).\n\nWe use \\(\\Y{\\beta\/\\alpha}\\) to construct two lattices, \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) and \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\).\nIn both lattices, the set of nodes is the set of points in the plane which occur as some corner of some box in \\(\\Y{\\beta\/\\alpha}\\).\nThe edges in each lattice are described below.\nWe give some edges a \\emph{weight} which will be a formal variable.\nThe weight of a path is then given by the product of the weights of the steps, and the weight of a tuple of paths is the product of the weights of each path.\n\nIn \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\), we have horizontal edges and vertical edges:\n\\begin{itemize}\n    \\item the horizontal edges are the horizontal sides of the boxes in \\(\\Y{\\beta\/\\alpha}\\), and are directed rightward;\n    \\item the vertical edges are the right-hand sides of the boxes in \\(\\Y{\\beta\/\\alpha}\\), are directed downward, and have weight \\(x_j\\) where \\(j\\) is the column of the corresponding box.\n\\end{itemize}\nIn  \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\), we have horizontal edges and diagonal edges:\n\\begin{itemize}\n    \\item\n    the horizontal edges are the horizontal sides of the boxes in \\(\\Y{\\beta\/\\alpha}\\), and are directed leftward;\n    \\item\n    the diagonal edges are the top-right to bottom-left diagonals of the boxes in \\(\\Y{\\beta\/\\alpha}\\), are directed left-and-downward, and have weight \\(x_j\\) where \\(j\\) is the column of the corresponding box.\n\\end{itemize}\nAn example of each of these lattices is depicted in \\autoref{fig:lattices}.\n\n\\begin{figure}[ht]\n    \\centering\n    \\begin{subfigure}{0.49\\linewidth}\n        \\centering\n        \\includegraphics[scale=0.78]{L-lattice.pdf}\n        \\caption{\\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\)}\n        \\label{subfig:L-lattice}\n    \\end{subfigure}\n    \\begin{subfigure}{0.49\\linewidth}\n        \\centering\n        \\includegraphics[scale=0.78]{R-lattice.pdf}\n        \\caption{\\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\)}\n        \\label{subfig:R-lattice}\n    \\end{subfigure}\n    \\caption{The relevant lattices when \\(n=6\\), \\(\\alpha = (2,1,1,0,0,0)\\) and \\(\\beta = (6,6,5,4,4,3)\\).\n    }\n    \\label{fig:lattices}\n\\end{figure}\n\nWe now define \\emph{sources} and \\emph{sinks} for paths on these lattices.\n\nIn \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\), take as \\emph{sources} the left-most node in each horizontal line indexed by \\(A\\),\nand take as \\emph{sinks} the right-most node in each horizontal line indexed by \\(B\\).\nExplicitly, the sources are the points \\(\\setbuild{(a, \\alpha_{a+1})}{a \\in A}\\) (where we interpret \\(\\alpha_{n+1} = \\alpha_n\\) if \\(n \\in A\\)) and the sinks are the points \\(\\setbuild{(b, \\beta_{b})}{b \\in B}\\) (where we interpret \\(\\beta_0 = \\beta_1\\) if \\(0 \\in B\\)).\n\nIn \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\), take as \\emph{sources} the right-most node in each horizontal line indexed by \\(B^\\mathsf{c}\\),\nand take as \\emph{sinks} the left-most node in each horizontal line indexed by \\(A^\\mathsf{c}\\).\nExplicitly, the sources are the points \\(\\setbuild{(b', \\beta_{b'})}{b' \\in B^\\mathsf{c}}\\) (where we interpret \\(\\beta_0 = \\beta_1\\) if \\(0 \\in B^\\mathsf{c}\\)) and the sinks are the points \\(\\setbuild{(a', \\alpha_{a'+1})}{a' \\in A^\\mathsf{c}}\\) (where we interpret \\(\\alpha_{n+1} = \\alpha_n\\) if \\(n \\in A^\\mathsf{c}\\)).\n\nWe are interested in tuples of paths in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) and \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) that join the sources to the sinks in a matching; we call such tuples in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) \\emph{blue connectors} and such tuples in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) \\emph{red connectors}.\nWe furthermore describe sources, sinks and paths in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) as \\emph{blue} and in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) as \\emph{red}.\nA blue connector and a red connector are depicted in \\autoref{fig:paths}.\n\n\\begin{figure}[ht]\n    \\centering\n    \\begin{subfigure}{0.495\\linewidth}\n        \\centering\n        \\includegraphics[scale=0.78]{lpath}\n        \\caption{A blue connector.}\n        \\label{subfig:l-path}\n    \\end{subfigure}\n    \\begin{subfigure}{0.495\\linewidth}\n        \\centering\n        \\includegraphics[scale=0.78]{rpath.pdf}\n        \\caption{A red connector.}\n        \\label{subfig:r-path}\n    \\end{subfigure}\n    \\caption{Examples of a blue connector and a red connector, when \\(n=6\\), \\(\\alpha = (2,1,1,0,0,0)\\), \\(\\beta = (6,6,5,4,4,3)\\),\n    \\(A = \\set{0,1,3,4}\\) and \\(B = \\set{1,3,5,6}\\).\n    Sources are indicated by circles, sinks by crosses.\n    }\n    \\label{fig:paths}\n\\end{figure}\n\n\\subsection{Enumeration of connectors}\n\nWe count the \\emph{non-intersecting} blue connectors and red connectors.\nThey key result we use for this is the Lindstr\\\"{o}m--Gessel--Viennot Lemma, stated below.\nThe lemma was first articulated in the context of Markov chains in \\cite{karlin1959}, and later in the context of matroid theory in \\cite{lindstroem1973}.\nIt was subsequently used to deduce various combinatorial identities in \\cite{GESSEL1985300}.\nFor an illuminating illustration of the argument behind the lemma, see \\cite{CountingOnDeterminants}.\n\n\\begin{theorem}[Lindstr\\\"om--Gessel--Viennot Lemma]\n\\label{thm:lgv_lemma}\nLet \\(G\\) be a directed acyclic graph with \\(m\\) designated sources sinks, where \\(m\\) is a nonnegative integer.\nLet \\(M\\) be the \\(m \\times m\\) matrix whose \\((i, j)\\)th entry is the number of paths, counted with weight, from the \\(i\\)th source to the \\(j\\)th sink.\nSuppose \\(G\\) is nonpermutable.\nThen the number of non-intersecting \\(m\\)-tuples of paths from sources to sinks, counted with weight, is equal to the determinant of \\(M\\).\n\\end{theorem}\n\nHere, \\emph{nonpermutable} means that a non-intersecting \\(m\\)-tuple of paths must connect the \\(i\\)th source to the \\(i\\)th sink.\nThe lattices \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) and \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) are clearly acyclic and nonpermutable, so \\autoref{thm:lgv_lemma} applies.\nWe thus want to count paths from the \\(i\\)th source to the \\(j\\)th sink in each lattice.\n\n\\begin{proposition}\n\\label{prop:l-path_count}\nThe weighted count of non-intersecting blue connectors in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) is\n\\[\n    \\det\\mleft(%\n        h_{b - a}(x_{\\alpha_{a+1}+1}, x_{\\alpha_{a+1}+2}, \\ldots, x_{\\beta_{b}}) %\n    \\mright)_{a \\in A,b \\in B}.\n\\]\n\\end{proposition}\n\n\\newcommand{(b, \\beta_{b})}{(b, \\beta_{b})}\n\\newcommand{(a, \\alpha_{a+1})}{(a, \\alpha_{a+1})}\n\n\\begin{proof}\nBy \\autoref{thm:lgv_lemma}, it suffices to show that the weighted count of paths from \\((a, \\alpha_{a+1})\\) to \\((b, \\beta_{b})\\) is \\(h_{b - a}(x_{\\alpha_{a+1}+1}, x_{\\alpha_{a+1}+2}, \\ldots, x_{\\beta_{b}})\\), for all \\(a \\in A\\) and \\(b \\in B\\).\n\nSuppose first that \\(a \\geq b\\).\nIf the inequality is strict, then \\((a, \\alpha_{a+1})\\) is below \\((b, \\beta_{b})\\), so there are no paths between them, and \\(h_{b - a} = 0\\) as required.\nIf equality holds, then \\((a, \\alpha_{a+1})\\) and \\((b, \\beta_{b})\\) are in the same row so there is a unique horizontal path between them, and \\(h_{b-a} = 1\\) as required.\n\nSuppose next \\(a < b\\) and \\(\\alpha_{a+1} \\geq \\beta_{b}\\).\nThen \\((a, \\alpha_{a+1})\\) is further right that \\((b, \\beta_{b})\\) (or in the same column, but without vertical edges joining them), so there are no paths between them.\nMeanwhile the corresponding polynomial is a symmetric function over an empty range of variables and hence is zero, as required.\n\nNow suppose \\(a < b\\) and \\(\\alpha_{a+1} < \\beta_{b}\\).\nA path from \\((a, \\alpha_{a+1})\\) to \\((b, \\beta_{b})\\) must make exactly \\(b - a\\) vertical steps.\nObserve that the vertical steps in such a path must be made down (the right-hand sides of) boxes in the rectangular region whose vertices are the boxes \\(\\Ybox{a+1}{\\alpha_{a+1}+1}\\), \\(\\Ybox{a+1}{\\beta_{b}}\\), \\(\\Ybox{b}{\\beta_{b}}\\) and \\(\\Ybox{b}{\\alpha_{a+1}+1}\\), as exemplified in \\autoref{fig:rectangular_region}.\nSince \\(\\alpha\\) and \\(\\beta\\) are partitions, we have \\(\\alpha_{i} \\leq \\alpha_{a+1}\\) and \\(\\beta_{i} \\geq \\beta_{b}\\) for all \\(a+1 \\leq i \\leq b\\), and so all these boxes are indeed contained in the Young diagram \\(\\Y{\\beta\/\\alpha}\\).\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics{rectangular_region.pdf}\n    \\caption{The collection of boxes which must contain the vertical steps of any path from \\((a, \\alpha_{a+1})\\) to \\((b,\\beta_b)\\) when \\(a=1\\), \\(b=5\\), \\(n=6\\), \\(\\alpha = (2,1,1,0,0,0)\\) and \\(\\beta = (6,6,5,4,4,3)\\).\n    }\n    \\label{fig:rectangular_region}\n\\end{figure}\n\n\nThus the choice of \\(b-a\\) columns in which vertical steps take place can be made freely with repetition from \\(\\alpha_{a+1}+1, \\alpha_{a+1}+2, \\ldots, \\beta_{b}\\) (and each such choice uniquely determines a path).\nThen since the vertical edges in column \\(j\\) each have weight \\(x_j\\), the weighted count of possible paths is the required polynomial.\n\\end{proof}\n\n\\newcommand{\\parallelogramcondition}{%\nSuppose that for all \\(a' \\in A^\\mathsf{c}\\) and \\(b' \\in B^\\mathsf{c}\\), either:\n\\begin{itemize}\n    \\item\n    \\(a' - b' \\leq 0\\); or\n    \\item\n    \\(a' - b' > \\beta_{b'+1} - \\alpha_{a'}\\); or\n    \\item\n    for all \\(i\\) such that \\(b'+1 \\leq i \\leq a'\\), we have \\(\\alpha_{i} - \\alpha_{a'} \\leq a'-i\\) and \\(\\beta_{b'+1} - \\beta_i \\leq i-b'-1\\).\n\\end{itemize}\n}\n\n\\begin{proposition}\n\\label{prop:r-path_count}\n\\parallelogramcondition\nThen the weighted count of non-intersecting red connectors in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) is\n\\[\n    \\det\\mleft(%\n        e_{a' - b'} (x_{\\alpha_{a'}+1}, x_{\\alpha_{a'}+2}, \\ldots, x_{\\beta_{b'+1}}) %\n    \\mright)_{a' \\in A^\\mathsf{c},b' \\in B^\\mathsf{c}}.\n\\]\n\\end{proposition}\n\n\\renewcommand{(b, \\beta_{b})}{(b', \\beta_{b'})}\n\\renewcommand{(a, \\alpha_{a+1})}{(a', \\alpha_{a'+1})}\n\n\\begin{proof}\nBy \\autoref{thm:lgv_lemma}, it suffices to show that the weighted count of paths from \\((b, \\beta_{b})\\) to \\((a, \\alpha_{a+1})\\) is \\(e_{a' - b'} (x_{\\alpha_{a'}+1}, x_{\\alpha_{a'}+2}, \\ldots, x_{\\beta_{b'+1}})\\), for all \\(a' \\in A^\\mathsf{c}\\) and all \\(b' \\in B^\\mathsf{c}\\).\n\nSuppose first that \\(a' \\leq b'\\).\nIf the inequality is strict, then \\((b, \\beta_{b})\\) is below \\((a, \\alpha_{a+1})\\), so there are no paths between them, and \\(e_{a' - b'} = 0\\) as required.\nIf equality holds, then \\((b, \\beta_{b})\\) and \\((a, \\alpha_{a+1})\\) are in the same row so there is a unique horizontal path between them, and \\(e_{a'-b'} = 1\\) as required.\n\nSuppose next that\n\\(a' - b' > \\beta_{b'+1} - \\alpha_{a'}\\).\nThen any path starting at \\((b, \\beta_{b})\\) reaches the left-hand side of the lattice and terminates before it reaches the \\(a'\\)th row, so there are no paths between \\((b, \\beta_{b})\\) and \\((a, \\alpha_{a+1})\\). Meanwhile the corresponding polynomial is an elementary symmetric function in fewer variables than its degree and hence is zero, as required.\n\nNow suppose\n\\(a' - b' \\leq \\beta_{b'} - \\alpha_{a'}\\).\nA path from \\((b, \\beta_{b})\\) to \\((a, \\alpha_{a+1})\\) must make exactly \\(a'- b'\\) diagonal steps.\nObserve that the diagonal steps in such a path must be made across boxes in the parallelogram-shaped region whose vertices are the boxes \\(\\Ybox{b'+1}{\\beta_{b'+1}}\\), \\(\\Ybox{b'+1}{\\alpha_{a'}+a'-b'}\\), \\(\\Ybox{a'}{\\alpha_{a'}+1}\\) and \\(\\Ybox{a'}{\\beta_{b'+1}+b'-a'+1}\\), as exemplified in \\autoref{fig:parallelogram_region}.\nThe condition for this collection of boxes to be contained in the Young diagram is that for all \\(i \\in \\set{b'+1, b'+1, \\ldots, a'}\\) we have \\(\\alpha_i \\leq \\alpha_{a'} + (a'-i)\\) and \\(\\beta_i \\geq \\beta_{b'+1} - (i-b'-1)\\), which is exactly the assumed condition on \\(\\alpha\\) and \\(\\beta\\).\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics{parallelogram_region.pdf}\n    \\caption{The collection of boxes which must contain the diagonal steps of any path from \\((b', \\beta_{b'})\\) to \\((a',\\alpha_{a'+1})\\) when \\(a'=5\\), \\(b'=2\\), \\(n=6\\), \\(\\alpha = (2,1,1,0,0,0)\\) and \\(\\beta = (6,6,5,4,4,3)\\).\n    }\n    \\label{fig:parallelogram_region}\n\\end{figure}\n\n\nThus the choice of \\(a'-b'\\) columns in which diagonal steps take place can be made freely without repetition from \\(\\alpha_{a'}+1\\), \\(\\alpha_{a'}+2\\), \\ldots, \\(\\beta_{b'+1}\\) (and each such choice uniquely determines a path).\nThen since the diagonal edges in column \\(j\\) each have weight \\(x_j\\), the weighted count of possible paths is the required polynomial.\n\\end{proof}\n\n\n\\subsection{Bijection between connectors}\n\\label{section:path_bijection}\n\nWe now define a bijection between the non-intersecting blue connectors in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) and the non-intersecting red connectors in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\), for any skew-partition \\(\\beta\/\\alpha\\).\n(In fact, the arguments in this section hold for \\(\\alpha\\) and \\(\\beta\\) any compositions such that \\(\\alpha_i \\leq \\beta_i\\) for all \\(i\\).)\n\nIt is convenient to overlay the lattices \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) and \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\), so we can compare paths on one with paths on the other.\n\nThe bijection is via the following construction.\nWe define the construction of a red connector from a blue connector; the inverse construction is analogous.\n\n\\begin{definition}\nGiven a non-intersecting blue connector, define the \\emph{complementary} red connector to be the collection of paths constructed as follows:\nbeginning at each red source, take a horizontal step from each node unless the blue connector takes a vertical step from that node,\nin which case take a diagonal step.\n\\end{definition}\n\n\\begin{example}\nThe red connector in \\autoref{subfig:r-path} is complementary to the blue connector in \\autoref{subfig:l-path}, as illustrated in \\autoref{fig:complementary_paths}.\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics{landrpath.pdf}\n    \\caption{The blue connector and red connector from \\autoref{fig:paths} overlayed, illustrating that the red connector is complementary to the blue connector.\n   \n    }\n    \\label{fig:complementary_paths}\n\\end{figure}\n\\end{example}\n\nIt is not obvious that the complementary red connector is indeed a red connector (that is, that each path reaches a distinct red sink).\nWe will show that it is, and that it is non-intersecting.\nTo do so, we use the following lemma.\n\n\\begin{lemma}\n\\label{lemma:complementary_lemma}\nA non-intersecting blue connector and its complementary red connector intersect only at nodes from which a vertical blue step is taken.\n\\end{lemma}\n\n\\begin{proof}\nSuppose, towards a contradiction, there exists a node which lies in both a non-intersecting blue connector and its complementary red connector and from which there is not a vertical blue step.\nConsider a right-most such node \\((i,j)\\).\n\nFirst observe \\((i,j)\\) is not a blue sink: in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\), the right-most nodes are not the endpoints of any edges, so no path in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) starting at a red source can contain a blue sink.\n\nTherefore there must be a horizontal blue step from \\((i,j)\\), to \\((i,j+1)\\).\nIn particular, \\((i,j)\\) is not right-most in its row, so it is not a red source.\nThus there is a red step to \\((i,j)\\), either horizontally from \\((i,j+1)\\) or diagonally from \\((i-1,j+1)\\).\nWe explain, and illustrate beneath each explanation, how both of these possibilities lead to a contradiction.\n\nIf the red step is horizontal, then by the construction of the complementary red connector there is no vertical blue step from \\((i,j+1)\\).\nThis contradicts our choice of \\((i,j)\\) as the right-most intersection from which there is not a vertical blue step.\n\n\\begin{center}\n\\begin{tikzpicture}\n\\draw[thick,color=myblue,postaction={half mid arrow}] (0.5-\\nudge,0) -- (0.5-\\nudge,1);\n\\draw[thick,color=myred,postaction={half mid arrow}] (0.5+\\nudge,1) -- (0.5+\\nudge,0);\n\n\\node at (0.5,2.4) {\\(\\implies\\)};\n\n\\draw[thick,color=myblue,postaction={half mid arrow}] (0-\\nudge,4) -- (0-\\nudge,5);\n\\draw[thick,color=myred,postaction={half mid arrow}] (0+\\nudge,5) -- (0+\\nudge,4);\n\\draw[thick,color=paleblue,postaction={mid arrow}] (0,5) -- (1,5);\n\\node[cross out, line width=0.9pt, draw=darkgray, minimum size=0.32cm, at={(0.45,5)}] {};\n\n\\foreach \\x\/\\y in {0.5\/0, 0.5\/1, 1\/4, 1\/5, 0\/4, 0\/5} {\n    \\drawnode{(\\x,\\y)}\n}\n\\node[at={(0.5,0)},label={[shift={(0.8,0)}]\\(\\scriptstyle(i,j)\\)}] {};\n\\node[at={(0,4)},label={[shift={(0.8,0)}]\\(\\scriptstyle(i,j)\\)}] {};\n\n\\end{tikzpicture}\n\\end{center}\n\nIf the red step is diagonal, then by the construction of the complementary red connector, there is a vertical blue step from \\((i-1, j+1)\\) to \\((i,j+1)\\). This contradicts that the blue connector is non-intersecting.\n\n\\begin{center}\n\\begin{tikzpicture}\n\\draw[thick,color=myblue,postaction={mid arrow}] (0,0) -- (0,1);\n\\draw[thick,color=myred,postaction={mid arrow}] (-1,1) -- (0,0);\n\n\\node at (-0.5,2.4) {\\(\\implies\\)};\n\n\\draw[thick,color=myblue,postaction={mid arrow}] (0,4) -- (0,5);\n\\draw[thick,color=myred,postaction={mid arrow}] (-1,5) -- (0,4);\n\\draw[thick,color=myblue,postaction={mid arrow}] (-1,5) -- (0,5);\n\n\\foreach \\x\/\\y in {-1\/0, -1\/1, 0\/0, 0\/1, -1\/4, -1\/5, 0\/4, 0\/5} {\n    \\drawnode{(\\x,\\y)}\n}\n\\node[at={(0,0)},label={[shift={(0.8,0)}]\\(\\scriptstyle(i,j)\\)}] {};\n\\node[at={(0,4)},label={[shift={(0.8,0)}]\\(\\scriptstyle(i,j)\\)}] {};\n\n\\end{tikzpicture}\n\\end{center}\n\nIn either case we have a contradiction, so no such intersection exists.\n\\end{proof}\n\n\\begin{proposition}\n\\label{prop:complements_exist}\nThe complementary red connector to a non-intersecting blue connector is a non-intersecting red connector.\n\\end{proposition}\n\n\\begin{proof}\nFirst observe that in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) the only edges out of the left-most nodes are horizontal, and thus the first step in every blue path is horizontal.\nTherefore, by \\autoref{lemma:complementary_lemma}, the complementary red connector does not contain any blue sources.\nSince the blue sources and the red sinks partition the left-most nodes in the lattices, we deduce that the paths of the complementary red connector reach red sinks (not necessarily distinct).\n\nWe next show that the complementary red connector is non-intersecting.\nThis implies that the red sinks the red paths reach are distinct, and hence that it is indeed a red connector.\n\nSuppose, towards a contradiction, that the complementary red connector has an intersection.\nConsider a right-most intersection \\((i,j)\\).\nNote that \\((i,j)\\) cannot be a red source (or a blue sink): if it were, then it would be the right-most node in its row, and so there would be no edges from \\((i,j+1)\\) or \\((i-1,j+1)\\) for a red path to arrive from (and so it could not be an intersection of two red paths).\n\nSince we assumed \\((i,j)\\) to be a right-most intersection, the incoming red paths must come from distinct vertices: there is both a horizontal and a diagonal red step to \\((i,j)\\).\nThen, by the construction of the complementary red connector and as illustrated below, there must be a vertical blue step to \\((i,j+1)\\) and there cannot be a vertical blue step out of \\((i,j+1)\\).\n\\begin{center}\n\\begin{tikzpicture}\n\\draw[thick,color=myred,postaction={mid arrow}] (0,1) -- (0,0);\n\\draw[thick,color=myred,postaction={mid arrow}] (-1,1) -- (0,0);\n\n\\node at (0,2.2) {\\(\\implies\\)};\n\n\\draw[thick,color=myred,postaction={mid arrow}] (0,5) -- (0,4);\n\\draw[thick,color=myred,postaction={mid arrow}] (-1,5) -- (0,4);\n\\draw[thick,color=myblue,postaction={mid arrow}] (-1,5) -- (0,5);\n\\draw[thick,color=paleblue,postaction={mid arrow}] (0,5) -- (1,5);\n\\node[cross out, line width=0.9pt, draw=darkgray, minimum size=0.32cm, at={(0.45,5)}] {};\n\n\\foreach \\x\/\\y in {0\/0, 0\/1, -1\/0, -1\/1, 1\/0, 1\/1, 0\/4, 0\/5, -1\/4, -1\/5, 1\/4, 1\/5} {\\drawnode{(\\x,\\y)}\n}\n\n\\node[at={(0,0)},label={[shift={(0.5,-0.5)}]\\(\\scriptstyle(i,j)\\)}] {};\n\\node[at={(0,4)},label={[shift={(0.5,-0.5)}]\\(\\scriptstyle(i,j)\\)}] {};\n\\end{tikzpicture}\n\\end{center}\nThen \\((i,j+1)\\) lies in both the blue connector and its complementary red connector, but there is no vertical blue step out of it, contradicting \\autoref{lemma:complementary_lemma}.\n\\end{proof}\n\n\\begin{proposition}\nThe map from the set of non-intersecting blue connectors to the set of non-intersecting red connectors defined by taking the complementary red connector is a weight-preserving bijection.\n\\end{proposition}\n\n\\begin{proof}\nWrite \\(\\Sigma C\\) for the sum of the elements of a set \\(C\\).\nObserve that \\(\\Sigma A^\\mathsf{c} - \\Sigma B^\\mathsf{c} = \\Sigma B - \\Sigma A\\),\nand hence that the number of vertical steps made by a blue connector equals the number of diagonal steps made by a red connector.\nThus every vertical blue step in a non-intersecting blue connector must give rise to a diagonal red step in its complementary red connector.\nThat is, the nodes from which a non-intersecting blue connector takes vertical steps are precisely the nodes from which its complementary red connector takes diagonal steps (and these are precisely the intersections of the connectors).\n\nIt is then clear that a non-intersecting blue connector has the same weight as its complementary red connector, and that taking the analogous construction of a complementary blue connector provides an inverse.\n\\end{proof}\n\n\n\\subsection{Conclusion}\n\nCombining the enumerations given by Propositions \\ref{prop:l-path_count} and \\ref{prop:r-path_count} with the bijection described in \\autoref{section:path_bijection}, we obtain our main theorem, stated in full below.\n\n\\begin{theorem}\\label{thm:main_theorem}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal siz\n, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\).\nLet \\(\\alpha\\) and \\(\\beta\\) be partitions with \\(n\\) parts (with parts equal to \\(0\\) permitted) such that \\(\\alpha_i \\leq \\beta_i\\) for all \\(i \\in \\intervalwz{n}\\).\n\\parallelogramcondition\nThen the determinants\n\\[\n    \\det\\Bigl(%\n        h_{b - a} (x_{\\alpha_{a+1}+1}, x_{\\alpha_{a+1}+2}, \\ldots, x_{\\beta_{b}}) %\n    \\Bigr)_{a \\in A, b \\in B}\n\\]\nand\n\\[\n    \\det\\Bigl(%\n        e_{a' - b'} (x_{\\alpha_{a'}+1}, x_{\\alpha_{a'}+2}, \\ldots, x_{\\beta_{b'+1}}) %\n    \\Bigr)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n\\]\nare equal.\n\\end{theorem}\n\n\\begin{remark}\nThe hypothesis in \\autoref{thm:main_theorem} is precisely the hypothesis in \\autoref{prop:r-path_count} which ensures that each entry of the second matrix counts the number of red paths correctly (by requiring that all minimal parallelograms between appropriate sinks and sources lie inside the Young diagram).\nThis hypothesis is necessary and sufficient for each individual entry to give a correct count.\nHowever, this hypothesis is not necessary for the determinant to correctly count the total number of non-intersecting red connectors.\nFor example, suppose there exists \\(m \\in \\intervalwz{n}\\) such that \\(\\@ifstar{\\oldabs}{\\oldabs*}{A \\cap [0,m]} = \\@ifstar{\\oldabs}{\\oldabs*}{B \\cap [0,m]}\\).\nThen no non-intersecting connector crosses the \\((m+1)\\)th row, and so the count of non-intersecting connectors is the product of the the counts of non-intersecting connectors on each half of the lattice.\nMeanwhile the matrices whose entries count paths are block triangular, and the entries of the off-diagonal block are irrelevant to the determinant, and so it is not necessary for the hypothesis to hold for pairs with indices on both sides of \\(m+1\\).\n\\end{remark}\n\nPartitions whose parts are at most \\(1\\) less than the preceding part clearly satisfy the hypothesis of \\autoref{thm:main_theorem}, and so we recover \\autoref{thm:intro_det_identity} given in the introduction.\n\n\\section{Specialisations of the main theorem}\\label{section:corollaries}\n\nIn this section we indicate how to recover Gessel and Viennot's binomial duality theorem (\\autoref{thm:binomial_identity}) and Aitken's symmetric function duality theorem (\\autoref{thm:sym_function_identity}) from our main thereom.\nWe also deduce lifts of Gessel and Viennot's theorem to \\(q\\)-binomial coefficients (\\autoref{cor:q-binomial_identity}) and to symmetric polynomials (\\autoref{cor:sym_poly_binomial_identity}).\n\nTo deduce \\autoref{thm:binomial_identity}, \\autoref{cor:q-binomial_identity} and \\autoref{cor:sym_poly_binomial_identity}, we use staircase-shaped lattices.\n\n\\sympolybinomidentity*\n\n\\begin{proof}\nIn \\autoref{thm:main_theorem}, take \\(\\beta = (n^{n})\\) and take \\(\\alpha\\) to be the staircase given by \\(\\alpha_i = n-i\\) for \\(i \\in [n]\\).\nThen the left-hand matrix is \\((h_{b-a}(x_{n-a}, \\ldots, x_{n}))_{a \\in A,b \\in B}\\) and the right-hand matrix is \\((e_{a'-b'}(x_{n-a'+1}, \\ldots, x_{n}))_{a' \\in A^\\mathsf{c},b \\in B^\\mathsf{c}}\\).\nRelabelling the variables (via \\(x_i \\mapsto x_{n+1-i}\\)) gives the result.\n\\end{proof}\n\nThe \\(q\\)-binomial coefficients (also known as Gaussian coefficients) are polynomials in \\(q\\), and are a generalisation of the usual binomial coefficients in the sense that setting \\(q=1\\) yields the corresponding binomial coefficients.\nFor a definition, see (for example) \\cite[Section 3]{genbinoms}.\n\n\\qbinomialidentity*\n\n\\begin{proof}\nRecall that the \\(q\\)-binomial coefficients are related to the symmetric polynomials in the following way \\cite[Section 3, equations 17 and 18]{genbinoms}:\n\\begin{align*}\n    h_{k}(1, q, \\ldots, q^{n-1}) &= \\qbinom{n+k-1}{k}, \\\\\n    e_{k}(1, q, \\ldots, q^{n-1}) &= q^{\\binom{k}{2}} \\qbinom{n}{k}.\n\\end{align*}\nThus, set \\(x_i = q^{i-1}\\) in \\autoref{cor:sym_poly_binomial_identity} and the matrix entries become, respectively,\n\\[\nh_{b-a}(1,q, \\ldots, q^{a})\n    = \\qbinom{b}{b-a} = \\qbinom{b}{a}\n\\]\nand\n\\[\ne_{a'-b'}(1,q, \\ldots, q^{a'-1})\n    = q^{\\binom{a'-b'}{2}} \\qbinom{a'}{a'-b'} = q^{\\binom{a'-b'}{2}}\\qbinom{a'}{b'},\n\\]\nas required.\n\\end{proof}\n\nSetting \\(q=1\\) in \\autoref{cor:q-binomial_identity} recovers Gessel and Viennot's binomial duality theorem (\\autoref{thm:binomial_identity}).\n\nTo recover Aitken's symmetric function duality theorem (\\autoref{thm:sym_function_identity}), we use rectangular lattices.\nIn \\autoref{thm:main_theorem}, take \\(\\alpha = 0\\) and \\(\\beta = (m^{n})\\), for a positive integer \\(m\\), to obtain\n\\[\n    \\det\\Bigl(%\n        h_{b - a} (x_{1}, x_{2}, \\ldots, x_{m}) %\n    \\Bigr)_{a \\in A, b \\in B}\n    =\n    \\det\\Bigl(%\n        e_{a' - b'} (x_{1}, x_{2}, \\ldots, x_{m}) %\n    \\Bigr)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n    .\n\\]\nSince this holds for arbitrary positive integers \\(m\\),\n\\autoref{thm:sym_function_identity} follows.\n\n\n\n\\section{Insufficiency of Jacobi's complementary minor formula}\n\\label{section:JT_connection}\n\n\\autoref{thm:sym_function_identity} was proved in \\cite{aitken_1931} using Jacobi's complementary minor formula and a form of Newton's identity.\nThese two results are stated below.\nWe here outline Aitken's proof, and show that this method is not sufficient to deduce our main theorem.\n\nFor our purposes it is convenient to index matrix rows and columns from \\(0\\).\nGiven a \\((d+1) \\times (d+1)\\) matrix \\(M\\) and subsets \\(A,B \\subseteq \\intervalwz{d}\\), let \\(M_{A,B}\\) denote the matrix obtained by retaining only those rows indexed by elements of \\(A\\) and those columns indexed by elements of \\(B\\).\nWrite \\(\\Sigma A\\) for the sum of the elements of \\(A\\).\n\n\\begin{proposition}[Jacobi's complementary minor formula { \\cite[Lemma A.1(e), p.~96]{Caracciolo_2013}}]\n\\label{prop:Jacobi_comp_minor}\nLet \\(M\\) be an invertible \\((d+1) \\times (d+1)\\) matrix and let \\(A,B \\subseteq \\intervalwz{d}\\) be subsets of equal size.\nThen\n\\[\n    \\det ( M_{A,B} ) = (-1)^{\\Sigma A + \\Sigma B} \\det (M) \\det \\mleft( {\\mleft((M^{-1})^{\\top}\\mright)}_{A^\\mathsf{c}, B^\\mathsf{c}} \\mright).\n\\]\n\\end{proposition}\n\n\\begin{proposition}[Newton's identity {\\cite[Equation (\\ensuremath{2.6'}), p.~21]{macdonald1998symmetric}}]\n\\label{prop:Newton_identity}\nLet \\(d > 0\\).\nThen\n\\[\n    \\sum_{i=0}^d {(-1)}^i e_i h_{d-i} = 0.\n\\]\n\\end{proposition}\n\nIn \\cite{aitken_1931}, Aitken obtains his identity (\\autoref{thm:sym_function_identity}) by applying \\autoref{prop:Jacobi_comp_minor} to the matrix \\({( h_{j-i} )}_{0\\leq i,j \\leq n}\\).\nIts inverse is \\({( {(-1)}^{i+j} e_{j-i})}_{0\\leq i,j \\leq n}\\), as can be verified using \\autoref{prop:Newton_identity}. \n\nIf we attempt to use this method to prove \\autoref{thm:main_theorem}, we would be required to show, given partitions \\(\\alpha\\) and \\(\\beta\\) satisfying the hypotheses, that the matrices\n\\begin{align*}\nH(\\beta \/ \\alpha) &= {\\Bigl(\n    h_{j-i}(x_{\\alpha_{i+1}+1}, \\ldots, x_{\\beta_j})\n\\Bigr)}_{0\\leq i,j \\leq n},\\\\\n\\intertext{and}\nE(\\beta \/ \\alpha) &= {\\mleft(\n    {(-1)}^{i+j} e_{j-i}(x_{\\alpha_j+1}, \\ldots, x_{\\beta_{i+1}})\n\\mright)}_{0\\leq i,j \\leq n}\n\\end{align*}\nare inverse.\nHowever, \\autoref{thm:main_theorem} can provide a determinant identity when this is not the case.\n\nFor example, let \\(n=3\\) and let \\(\\alpha = (2, 0, 0)\\) and \\(\\beta = (3,3,1)\\).\nWe have\n\\begin{align*}\n    {\\mleft(E(\\beta \/ \\alpha)H(\\beta \/ \\alpha)\\mright)}_{0,2}\n    &= h_2(x_3) - e_1(x_3)h_1(x_1, x_2, x_3) + e_2(x_1, x_2, x_3) \\\\\n    &= x_1 x_2 \\\\\n    &\\neq 0,\n\\end{align*}\nso the matrices \\(H(\\beta \/ \\alpha)\\) and \\(E(\\beta \/ \\alpha)\\) are not inverse.\nNevertheless, choosing \\(A = \\set{0,1,2}\\) and \\(B = \\set{1,2,3}\\) meets the hypotheses of \\autoref{thm:main_theorem}, and so we find that the determinants\n\\[\n\\hspace*{-6.5em}\n\\begin{blockarray}{cc>{\\scriptstyle}c>{\\scriptstyle}c>{\\scriptstyle}c}\n      &   & b=1 & b=2 & b=3 \\\\\n      &   & \\beta_{b}=3 & \\beta_{b}=3 & \\beta_{b}=1\\\\[4pt]\n    \\begin{block}{>{\\scriptstyle}c>{\\scriptstyle}c|ccc|}\n    a=0 & \\alpha_{a+1}+1=3 & h_1(x_3) & h_2(x_3) & 0 \\\\\n    a=1 & \\alpha_{a+1}+1=1 & 1 & h_1(x_1, x_2, x_3) & h_2(x_1) \\\\\n    a=2 & \\alpha_{a+1}+1=1 & 0 & 1 & h_1(x_1) \\\\\n    \\end{block}\n\\end{blockarray}\n\\]\nand\n\\[\n\\hspace*{-6.5em}\n\\begin{blockarray}{ccc>{\\scriptstyle}c}\n      &   && b'=0 &\\\\\n      &   && \\beta_{b'+1}=3 & \\\\[4pt]\n    \\begin{block}{>{\\scriptstyle}c>{\\scriptstyle}cc|c|}\n    a'=3 & \\alpha_{a'}+1=1 && e_3(x_1, x_2, x_3)\\\\\n    \\end{block}\n\\end{blockarray}\n\\]\nare equal.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{} if # should be set in lowercase\n\n\n\n\n\n\n\\dropcap{T}he polymerization of molecular nitrogen under pressure has been very\nactively researched over the past two decades and has stimulated many\nexperimental \\cite{Goncharov2000,Eremets2001,\n  Gregoryanz2001,Gregoryanz2002,Eremets2004,Eremets2004a,Eremets2007,\n  Gregoryanz2007,Lipp2007} and theoretical\n\\cite{McMahan1985,Martin1986,Lewis1992,Mailhiot1992,\n  Mitas1994,Alemany2003,Mattson2004,Zahariev2005,Oganov2006,Uddin2006,Zahariev2006,\n  Caracas2007,Wang2007a,Zahariev2007,Yao2008,Pickard2009-nitrogen,Ma2009-nitrogen,Wang2012-diamondoid-N}\ninvestigations.  Density functional theory (DFT) studies suggested\nthat dissociation of nitrogen could occur at high pressures which\nwere, nevertheless, attainable in diamond anvil cell (DAC)\nexperiments.  In a ground-breaking paper, Mailhiot \\textit{et al.}\\\n\\cite{Mailhiot1992} predicted polymerization of nitrogen molecules\nunder pressure, leading to the formation of the ``cubic gauche''\nframework structure.  After considerable efforts\n\\cite{Goncharov2000,Eremets2001}, cubic gauche nitrogen was finally\nsynthesized by Eremets \\textit{et al.}\\ \\cite{Eremets2004}, a decade\nafter its prediction.  The polymeric phase is quench recoverable to\nambient conditions, where it exists as a rather unstable\nhigh-energy-density material (HEDM) \\cite{Eremets2004}. Under ambient\nconditions an energy of roughly 1.5 eV\/atom ($\\approx$ 10330 J\/g) is\nreleased on converting cubic gauche nitrogen to molecular N$_2$, which\nis more than twice the energy density of TNT (about 4686 J\/g).\nPolymeric nitrogen \\cite{Eremets2004}, or high--N content salts\n\\cite{Haiges2004}, are therefore expected to be excellent candidates\nas high-energy-density materials.\n\nThe N$\\equiv$N triple bond is one of the strongest known and breaking\nit requires surmounting a substantial energetic barrier.  However, at\nhigh temperatures the triple bond breaks at pressures above about 110\nGPa \\cite{Eremets2004}, a much lower pressure than predicted for the\nsingle bond in H$_2$ \\cite{Pickard2007-hydrogen} and the double bond\nin O$_2$, which is predicted to survive up to 2 TPa\n\\cite{Sun2012-O2,Zhu2012}. Once the N$\\equiv$N triple bond is broken,\na wide variety of structures can be adopted, similar to phosphorus and\narsenic, and many studies aimed at elucidating the phase diagram of\nnitrogen have been published.\n  \n\nThe set of candidate structures includes simple cubic, the\nisoelectronic ``black phosphorus'' and $\\alpha$-arsenic or cis-trans\nand $Cmcm$ chain structures, etc. \\cite{McMahan1985,Mailhiot1992,\n  Martin1986,Lewis1992,Alemany2003,Mattson2004,Mattson2004,Zahariev2005,Oganov2006,Yao2008}\nNevertheless, only the ``black phosphorus'' structure seemed feasible,\nas it was predicted to become more stable than cubic gauche above\n$\\sim$210 GPa \\cite{Mailhiot1992,Mattson2004,Wang2007a}.\nThe advent of structure searching using DFT methods marked a\nsignificant advance; under conditions in which standard chemical\nintuition may no longer be reliable, new stable polymeric nitrogen\nphases beyond cubic gauche have been proposed, including the layered\n$Pba2$ and framework $P2_12_12_1$ structures\n\\cite{Ma2009-nitrogen,Pickard2009-nitrogen}. ``Black phosphorus'' was\nsubsequently ruled out on energetic grounds.\nThese phases have in common a large DFT band gap that hardly closes\nwith pressure.\nNitrogen compounds could form metallic phases at high pressures and\/or\ntemperatures, which could have important consequences for\nunderstanding planetary processes.\n\n\nLarge compressions normally lead to an increase in the atomic\ncoordination numbers and metallization, as predicted in, for example,\nhydrogen \\cite{Pickard2007-hydrogen, McMahon2011}, carbon\n\\cite{Sun2009-carbon,Martinez2012-carbon},\nand oxygen \\cite{Sun2012-O2},\nas well as observed in many other systems. \\cite{Grochala2007}\nThe coordination number of the atoms in solid nitrogen increases from\none to three on transformation from the molecular to cubic gauche\nforms and three-fold coordination persists in higher pressure\nstructures.\n\nAll high-pressure nitrogen structures proposed so far are insulating.\nAlthough an increase in coordination number with pressure is\nphysically reasonable, it is by no means universal.\nFor example, aluminum transforms into more open structures at TPa\npressures as the valence electrons move away from the ions in the\nformation of ``electride structures'' \\cite{Pickard2010-Al} although,\nas must occur in a pressure-induced equilibrium phase transition, the\ndensity increases at the transition, and \nwell-packed structures are strongly disfavored in oxygen up to at\nleast 25 TPa \\cite{Sun2012-O2}.\nThe coordination number of nitrogen has not been determined at\npressures above 400 GPa,\nand a metallic phase of nitrogen has not yet been reported.\n\nMaterials under TPa pressures are important in planetary science, for\nexample, the pressure at the center of Jupiter is estimated to be\nabout 7 TPa \\cite{Jeanloz2007}.\nDynamical shock wave \\cite{Jeanloz2007,Knudson2008,Eggert2010} and\nramped compression experiments \\cite{Hawreliak2007,Bradley2009} are\nincreasingly being used to investigate materials at TPa pressures.\nEven more extreme conditions are attainable in laser ignition\nexperiments \\cite{Kritcher2011}.\nExperimental determinations of structures are not, however, currently\npossible at TPa pressures and therefore theoretical predictions are\nparticularly important.\n\\cite{Sun2009-carbon,McMahon2011,Pickard2010-Al,Martinez2012-carbon,Sun2012-O2}\nIn this work we focus on the structural phases of nitrogen at \nmulti-TPa pressures.\n\n\n\n\n\n\\vspace*{5mm}\n\\noindent{\\bf\\large Results}\n\n\\noindent{\\bf Crystal structures and charge transfer.}\nWe have used \\textit{ab initio} random structure searching (AIRSS)\n\\cite{Pickard2006-silane,Pickard2011-review} and DFT methods to find\ncandidate structures of nitrogen up to multi-TPa pressures.\nThese searches have enabled us to identify novel candidate nitrogen\nstructures that are more stable than previously known ones.\nFour of them are thermodynamically stable within certain pressure\nranges, namely, $P4\/nbm$ ($Z$=14), $Fdd2$ ($Z$=48), $R\\bar{3}m$\n($Z$=1) and $I4_1\/amd$ ($Z$=4).\nMost interestingly, the tetragonal $P4\/nbm$ structure is characterized\nby the presence of N$_2$ pairs and N$_5$ tetrahedra, while the\northorhombic $Fdd2$ structure is modulated, as depicted in Fig.\\\n\\ref{fig:lattice}(a) and (b).\n\nAt 2.5 TPa, the N-N bond length within the N$_5$ tetrahedra of\n$P4\/nbm$ is 1.13 \\AA, and the shortest distance between the corners of\nthe tetrahedra and N$_2$ dimers is 1.26 \\AA.  The N-N bond length\nwithin the dimer of about 1.17 \\AA\\ is significantly shorter than the\nN-N separation between adjacent dimers (about 1.29 \\AA), which implies\nthat the dimers are well separated.\n\nThe unique structure of the $P4\/nbm$ phase led us to investigate the\ncharges on the different atomic sites.\nWe used the wave-function-based Mulliken population and charge-density\nbased Bader \\cite{Bader} methods to analyze the charge transfer.\nAs summarized in Table\\ \\ref{table:charge}, the two methods lead to\nthe same conclusions; the N$_5$ tetrahedra are negatively charged and\nserve as anions while the N$_2$ dimers are positively charged and act\nas cations, \nand thus the $P4\/nbm$ structure resembles an all-nitrogen salt.\nThe resultant charges (about $\\pm$0.22$e$ from Mulliken analysis and\nabout $\\pm$0.37$e$ from Bader analysis) are substantial, although they\ndepend somewhat on the definition of the atomic charges.\n\nThe complicated $Fdd2$ structure has a 48-atoms conventional cell, and\nthe atoms are arranged in undulating layers, as depicted in Fig.\\\n\\ref{fig:lattice}(b).\nThe charge distribution of $Fdd2$ also exhibits large distortions.  As\nshown in Fig.\\ \\ref{fig:lattice}(b) and Table\\ \\ref{table:charge},\nalthough all the atoms are at 16b sites, each orbit has a different\ncharacter: there is \na group of strong electron acceptors (Bader charge $Q_B$=-0.36$e$,\nred), a group of strong electron donors ($Q_B$=0.38$e$, blue), and a\ngroup of nearly neutral atoms ($Q_B$=-0.02$e$, green).\nDue to the undulation, with the inequivalent $z$ positions averaging\nat $0.1254$,\nthe $Fdd2$ structure is very similar to a charge density wave (CDW)\nmodulation.  The appearance of strong charge transfer effects in an\nelement at such high pressures is most unexpected, as the resulting\nCoulomb energy is substantial.\n\n\nWe also found a layered $Cmca$ structure similar to ``black\nphosphorus'', which becomes thermodynamically stable at about 2 TPa.\nAs depicted in the supplementary information, the $Cmca$ structure\nconsists of three-fold-coordinated\nnitrogen atoms and has zig-zag layers, with a shortest N-N distance of\nabout 1.15 \\AA\\ at 2.5 TPa, which is much shorter than the N-N\nseparation between the layers of about 1.56 \\AA.\nLayered structures also appear in the high-pressure phases of other\nsmall molecules of first row atoms, e.g., CO \\cite{Sun2011-CO} and\nCO$_2$ \\cite{Sun2009-CO2}.\nThe nitrogen atoms in the $R\\bar{3}m$ structure are\nsix-fold-coordinated and form distorted octahedra.\nPure four-fold coordinated structures often appear in our searches at\n3 TPa, but they are less favorable than $Cmca$ and $P4\/nbm$.\nThe crystal structures and charge transfers of the newly predicted\nstable phases are summarized in Table\\ \\ref{table:charge}.\n\n\n\n\n\n\n\n\n\\noindent{\\bf Energetics.} As can be seen in the enthalpy-pressure\nrelations of Fig.\\ \\ref{fig:eos}(a), solid nitrogen undergoes a series\nof structural phase transitions beyond the previously-known framework\n$P2_12_12_1$ polymeric phase\n\\cite{Pickard2009-nitrogen,Ma2009-nitrogen}\nand recently discovered diamondoid structure ($I\\bar{4}3m$)\n\\cite{Wang2012-diamondoid-N}: \n$I\\bar{4}3m \\xrightarrow{\\rm 2.1\\ TPa} Cmca \\xrightarrow{\\rm 2.5\\ TPa}\nP4\/nbm \\xrightarrow{\\rm 7.1\\ TPa} Fdd2 \\xrightarrow{\\rm 11.5\\ TPa}\nR\\bar{3}m \\xrightarrow{\\rm 30\\ TPa} I4_1\/amd$. \nThis partially ionic $P4\/nbm$ phase is stable over a wide pressure\nrange, but compression to about 7.1 TPa leads to the modulated $Fdd2$\nstructure.\nThe $Fdd2$ structure is the most favorable phase from about 7.1--11.5\nTPa, whereupon it transforms into a six-fold-coordinated hexagonal\n$R\\bar{3}m$ phase.  As shown in the inset in Fig.\\ \\ref{fig:eos},\nnitrogen forms a $P4_1\/amd$ structure similar to Cs-IV\n\\cite{Takemura1982-Cs-IV} at about 30 TPa.\n\nThe volume-pressure relations of the most stable phases are shown in\nFig.\\ \\ref{fig:eos}(b). The volume decreases by about 2.5\\% at the\ntransition from $I\\bar{4}3m$ to $Cmca$ at 2.1 TPa, 0.7\\% at the\ntransition from $Cmca$ to $P4\/nbm$ at 2.5 TPa, 0.6\\% at the transition\nfrom $P4\/nbm$ to $Fdd2$ at 7.1 TPa, and 0.5\\% at the $Fdd2$ to\n$R\\bar{3}m$ transition at 11.5 TPa.\n\n\n\n\n\n\n\n\n\\noindent{\\bf Electronic structures.}\nThe electronic band structures and projected densities of states\n(PDOS) of $Cmca$ at 2.0 TPa, $P4\/nbm$ at 3.0 TPa, and $Fdd2$ at 8.0\nTPa, are shown in Fig.\\ \\ref{fig:band}, decomposed into $s$ and $p$\ncomponents.\nThe $P2_12_12_1$ and $I\\bar{4}3m$ structures are semiconducting at 2.5\nTPa, but the band structures of $Cmca$, $P4\/nbm$, and $Fdd2$ shown in\nFig.\\ \\ref{fig:band} and $R\\bar{3}m$ are metallic, see the\nsupplementary material. The insulating state of nitrogen persists to\nabout 2.0 TPa, which is considerably higher than in other light\nelements such as hydrogen, carbon and oxygen. \\cite{Sun2012-O2}\n\nThe band structure of $Cmca$ at 2.0 TPa features a partially filled\nDirac cone between the $Y$ and $\\Gamma$ points, similar to the one\nthat appears in\nelectron doped graphene. \\cite{Giovannetti2008} \nThe projected density of states in these metallic structures arises\nfrom the $s$ and $p$ orbitals which extend over essentially the same\nenergy interval, with both $s$ and $p$ orbitals contributing to the\nelectronic density of states at the Fermi level and conduction bands,\nwhich indicates strong $sp$ hybridization in the $Cmca$, $P4\/nbm$ and\n$Fdd2$ structures.\nThe densities of states at the Fermi levels of $P4\/nbm$ and $Fdd2$ are\nhigher than in $Cmca$, implying that $Cmca$ is less metallic.\n\n\n\\vspace*{5mm}\n\\noindent{\\bf\\large Discussion}\n\n\n\nThe dynamical stability of the newly predicted stable structures was\ninvestigated by calculating their phonon dispersion relations.  As\nshown in Fig.~\\ref{fig:phonon} and in the supplementary material, all\nof the structures are predicted to be mechanically and dynamically\nstable at the specified pressures.\nThe phonon dispersion relation of $P4\/nbm$ shows steep acoustic\nbranches together with fairly flat optical modes. This supports the\nview that $P4\/nbm$ is formed of atomic units: the N--N vibrons as\nwell as the breathing mode of the tetrahedra are essentially\ndispersionless, and well screened by the electronic cloud.\n\nOn the other hand, the phonon mode of $Fdd2$ corresponding to the\namplitude variation of the undulating layers drops to a very low\nfrequency of about 190 cm$^{-1}$.  For comparison, the next\nlowest-lying mode (the relative sliding mode) has a much higher\nfrequency of about 1300 cm$^{-1}$, because the shear mode brings atoms\nof the same charge closer together.\n\n\n\n\n\nThe formation of N$_2$ pairs and N$_5$ tetrahedra in $P4\/nbm$ was\ndeduced from the calculated bond lengths and confirmed by the charge\ndensity plot in Fig.\\ \\ref{fig:charge_density}(b), where the atoms\nbetween the tetrahedron center and corners, and between the two atoms\nwithin the dimers, form strong covalent bonds.\nThe electron localization function (ELF) shown in the supplementary\nmaterial provides additional evidence for the formation of tetrahedra\nand dimers in $P4\/nbm$.\n\n\n\n\nThe formation of charged units from an elementary compound has also\nbeen observed in $\\gamma$--Boron \\cite{Oganov2009}.  The pressures of\ninterest in the present study are much larger than those at which\n$\\gamma$--Boron has been observed and, besides, $P4\/nbm$--nitrogen is\na very different system.  The charge transfer in boron satisfies its\ntendency to form electron deficient icosahedra.  Based on the\nstructural richness of phosphorus and arsenic one might assume that\nnitrogen could form similarly rich bonding patterns once the triple\nbond is broken. At 2 TPa, however, packing efficiency is crucial and\none would expect structures with high coordination numbers to be\nformed. \nFor example, the coordination number of $Cmca$ is 3, compared with 6\nin the $R\\bar{3}m$ phase.  It is therefore reasonable to expect that\nthe average coordination of nitrogen in the $P4\/nbm$ and $Fdd2$\nstructures would be between 3 to 6.\nIn $P4\/nbm$, if one employs a longer bond criterion, e.g., 1.30 \\AA,\nthe coordination number for the corners of the tetrahedra, their\ncenters and the nitrogen atoms in the chains are 3, 4, and 6,\nrespectively, as shown in Fig.\\ S5 of the supplementary information.\nIn this metallic phase the variation in coordination (or the\ninhomogeneity of the density of ions) leads to charge transfer from\nthe highly-coordinated atoms (``N$_2$ pairs'') to the lower\ncoordinated ones (``tetrahedra'' corners).\nThis charge transfer allows the $P4\/nbm$ structure to form a unique\nmetallic all-nitrogen salt.\nEach atom in the 1D chains perpendicular to the tetrahedra layers has\nfour long bonds to the corners of the tetrahedra, thus only one\nelectron is left.\nThe 1D chain with uniform N--N distances is unstable and undergoes a\nPeierls distortion, similar to that in lithium \\cite{Neaton1999}, and\nN$_2$ pairs are formed.\n\n\nAlthough $Fdd2$ also shows clear variation in coordination and charge\ntransfer, the atomic arrangement is completely different from that in\n$P4\/nbm$.  This phase has undulating layers, formed from distorted\nparallelepipeds, resembling a CDW.  CDWs have been observed previously\nin chalcogenides under pressure \\cite{Degtyareva2005a}.\nIn light of the charge transfer, the buckling of layers could be\nviewed not only as a symmetry breaking, but also as allowing the\npositive and negative ions to approach one another and reduce the\nenergy.\nIn addition, the charge transfer might create atomic sites similar to\nthose in carbon at comparable pressures, although the $Fdd2$\nbandstructure shows very different behaviour\n\\cite{Martinez2012-carbon}. While carbon expels charge from the $2s$\nlevels at high compression, eventually forming an electride, the\nbottom of the $Fdd2$ valence band shows a large $2s$ population.\nCharge depletion from the $2s$ levels will only arise with the\ntransition to the superconducting $R\\bar{3}m$ phase, at almost 12 TPa.\nElectride structures are found for example in carbon\n\\cite{Martinez2012-carbon} and aluminum \\cite{Pickard2010-Al} at very\nhigh pressures, but they do not to appear in nitrogen up to at least\n100 TPa.\nWhile intuition suggests that close-packed structures should be\nfavored under extreme pressures, this is not the case for nitrogen\neven at 100 TPa where, for example, the fcc structure is almost 2 eV\nper atom higher in enthalpy than $I4_1\/amd$.\n\n\n\n\\vspace*{5mm}\n\\noindent{\\bf\\large Methods}\n\n\\vspace*{2mm}\n\\noindent{\\bf Structure search.} \nThe {\\it ab initio} random structure searching (AIRSS) method\n\\cite{Pickard2006-silane,Pickard2011-review} was used to identify\nlow-enthalpy structures of nitrogen at multi-TPa pressures.\nExtensive searches were performed at selected combinations\nof 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,\n18, 19, 20, 21, 22, 24 nitrogen atoms per cell \nat 1, 2, 3, 4, 5, 7, 10, 15, 20, 50 and 100 TPa.\nMore than 47,000 structures in total were optimized using DFT.\nAdditional searches were performed using initial structures consisting\nof random packings of N$_2$ dimers and N$_5$ tetrahedra, in order to\nstudy the partially-ionic structures in more detail.  No better\nstructure than $P4\/nbm$ was found.\n\n\\vspace*{2mm}\n\\noindent{\\bf {\\it Ab initio} calculations.} We used the\n\\textsc{castep} plane-wave DFT code \\cite{CASTEP} and an ultrasoft\npseudopotential generated with a small core radius to avoid\noverlapping of the core radii at the high-pressures studied. The\nenthalpy-pressure relations of the structures were recalculated using\nvery hard Projector Augmented Wave (PAW) pseudopotentials and the VASP\ncode \\cite{vasp} with a plane-wave basis set cutoff energy of 1000 eV.\nPhonon and electron-phonon coupling calculations were performed using\nDFT perturbation theory \\cite{DFPT} with the \\textsc{ABINIT}\n\\cite{ABINIT} and Quantum ESPRESSO codes \\cite{Giannozzi2009}. Very\nhigh cutoff energies (1632 eV for \\textsc{ABINIT} and 952 eV for\nQuantum ESPRESSO), together with large numbers of k-points, were used\nto obtain convergence of the enthalpy differences between phases to\nbetter than $5 \\times 10^{-3}$ eV per atom.\nCalculations were performed with the different codes to cross-check\nthe transition pressures.\n\n\n\n\\begin{acknowledgments}\n\nJ.S.\\ gratefully acknowledges financial support from the Alexander von\nHumboldt (AvH) foundation and Marie Curie actions.\nM.M.C.\\ C.J.P.\\ and R.J.N.\\ were supported by the EPSRC.\nThe calculations were carried out at {\\sc Bovilab@RUB} (Bochum),\nLIDOng (Dortmund), on the supercomputers at NRC (Ottawa) and UCL\n(London).\n\\end{acknowledgments}\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"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzopby b/data_all_eng_slimpj/shuffled/split2/finalzzopby
new file mode 100644
index 0000000000000000000000000000000000000000..20223b6b0ef76037c4a45f3ee81ef9bdb6580345
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzopby
@@ -0,0 +1,5 @@
+{"text":"\\section{Survey motivation}\n\n\\subsection{A multiwavelength approach to galaxy evolution as a function of environment}\n\nThe precise role that environment plays in shaping galaxy evolution is\na hotly debated topic.  Trends to passive and\/or more spheroidal\npopulations in dense environments are widely observed: galaxy\nmorphology \\citep{dressler80,dressler97,goto03,treu03},\ncolour \\citep{kodama01,blanton05,baldry06}, star-formation rate\n\\citep{gomez03,lewis02}, and stellar age and AGN fraction\n\\citep{kauffmann04} all correlate with measurements of the local\ngalaxy density.  Furthermore, these relations persist over a wide\nrange of redshift \\citep{smith05, cooper07} and density\n\\citep{balogh04}.\n\nDisentangling the relative importance of internal and external\nphysical mechanisms responsible for these relations is challenging.\nIt is natural to expect that high density environments will\npreferentially host older stellar populations.  Hierarchical models of\ngalaxy formation \\citep[e.g.][]{delucia06} suggest that galaxies in\nthe highest density peaks started forming stars and assembling mass\nearlier: in essence they have a head-start.  Simultaneously, galaxies\nforming in high-density environments will have more time to experience\nthe {\\it external} influence of their local environment.  Those\nprocesses will also act on infalling galaxies as they are continuously\naccreted into larger haloes.  There are many plausible physical\nmechanisms by which a galaxy could be transformed by its environment:\nremoval of the hot \\citep{larson80} or cold \\citep{gunn72} gas supply\nthrough ram-pressure stripping; tidal effects leading to halo\ntruncation \\citep{bekki99} or triggered star formation through gas\ncompression \\citep{fujita98}; interactions between galaxies themselves\nvia low-speed major mergers \\citep{barnes92} or frequent impulsive\nencounters termed `harrassment' \\citep{moore98}.\n\n Though some of the above mechanisms are largely cluster-specific\n(e.g. ram-pressure stripping requires interaction with a hot\nintracluster medium), it is also increasingly clear that low density\nenvironments such as galaxy groups are important sites for galaxy\nevolution \\citep{balogh04,Zabludoff96}.  Additionally, luminosity (or\nmore directly, mass) is also critical in regulating how susceptible a\ngalaxy is to external influences.  For example, \\citet{haines06} find\nthat in low density environments in the SDSS the fraction of passive\ngalaxies is a strong function of luminosity.  They find a complete\nabsence of passive dwarf galaxies in the lowest density regions (i.e.,\nwhile luminous passive galaxies can occur in all environments,\nlow-luminosity passive galaxies can only occur in dense environments).\n\nUnderstanding the full degree of transformation is further complicated\nby the amount of dust-obscured star formation that may or may not be\npresent.  Many studies in the radio and MIR\n\\citep{millerowen03,coia05,gallazzi08} have shown that an optical\ncensus of star formation can underestimate the true\nrate. Cluster-cluster variations are strong, with induced star\nformation linked to dynamically-disturbed large-scale structure\n\\citep{geach06}.  Nor are changes in morphology necessarily equivalent\nto changes in star formation.  There is no guarantee that external\nprocesses causing an increase or decrease in the star-formation rate\nact on the same timescale, to the same degree, or in the same regime\nas those responsible for structural changes.  A full census of\nstar-formation, AGN activity, and morphology therefore requires a\ncomprehensive view of galaxies, including multiwavelength coverage and\nhigh resolution imaging. These are the aims of the STAGES project\ndescribed in this paper, targetting the Abell 901(a,b)\/902 multiple\ncluster system (hearafter A901\/2) at $z\\sim 0.165$.\n\nIn addition to the STAGES coverage of A901\/2, there are several other\nmultiwavelength projects taking a similar approach to targeting\nlarge-scale structures.  While we will argue below that STAGES\noccupies a particular niche, the following is a (non-exhaustive) list\nof surveys of large-scale structure including substantial HST imaging.\nAll are complementary to STAGES by way of the redshift range or\ndynamical state probed.  The COSMOS survey has examined the evolution\nof the morphology-density relation to $z=1.2$ \\citep{capak07}, paying\nparticular attention to a large structure at $z=0.7$ \\citep{guzzo07}.\nRelevant to this work, in \\citet{Smolcic07} they identify a complex of\nsmall clusters at $z\\sim 0.2$ via a wide-angle tail\nradio galaxy.  At intermediate redshift, an extensive comparison\nproject has been undertaken targeting the two contrasting clusters\nCL0024+17 and MS0451-03 at $z\\sim0.5$ to compare the low- and\nhigh-luminosity X-ray cluster environment \\citep{moran07,geach06}.\nLocally, the Coma cluster has also been extensively used as a\nlaboratory for galaxy evolution \\citep{poggianti04,\ncarter01,carter08}.  There are many other examples of cluster-focused\nenvironmental studies covering a range of redshifts, including the\nlarge sample of EDisCS clusters at $z>0.5$\n\\citep{white05,poggianti06,desai07}; and the ACS GTO cluster program\nof 7 clusters at $z\\sim 1$\n\\citep{postman05,goto05,blakeslee06,homeier05}.\n\n\nWe summarize the motivation for our survey design as follows. In order\nto successfully penetrate the environmental processes at work in\nshaping galaxy evolution, several areas must be simultaneously\naddressed: a wide range of environments; a wide range in galaxy\nluminosity; and sensitivity to both obscured and unobscured star\nformation, stellar masses, AGN, and detailed morphologies.\nFurthermore, it is essential to use not just a single proxy for\n`environment' but to understand directly the relative influences of\nthe local galaxy density, the hot ICM and the dark matter on galaxy\ntransformation.  A further advantage is given by examining systems\nthat are not simply massive clusters already in equilibrium.  By\nincluding systems in the process of formation (when extensive mixing\nhas not yet erased the memory of early timescales), the various\nenvironmental proxies listed above might still be disentangled.\n\nTherefore, the goal of STAGES is to focus attention on a single\nlarge-scale structure to understand the detailed aspects of galaxy\nevolution as a function of environment.  While no single study will\nprovide a definitive answer to the question of environment and galaxy\nevolution, we argue that STAGES occupies a unique vantage point in\nthis field, to be complemented by other studies locally and at higher\nredshift.\n\n\\subsection{Galaxy evolution as a function of redshift:  STAGES and\nGEMS}\n\nIn addition to science focused on the narrow redshift slice containing\nthe multiple cluster system, the multiwavelength data presented here\nprovide a valuable resource for those wishing to study the evolution\nof the galaxy population since $z=1$.  With the advent of the HST and\nmultiwavelength data for this field, it is possible to quantify better\nthe sample variance and investigate rare subsamples using the\ncombination of the STAGES field together with the Galaxy Evolution and\nMorphologies \\citep[GEMS;][]{rix04} coverage of the Extended Chandra\nDeep Field South (CDFS). In particular, the HST data were chosen to\nhave the same passband for both GEMS (F606W and 850LP) and STAGES\n(F606W only, to allow study at optimum S\/N of the cluster\nsubpopulation and to optimise the weak lensing analysis).  While the\nchoice of F606W means that the data probe above the 4000{\\AA} break\nfor $z<0.5$ only, for a number of purposes the data can also be used\nat higher redshift (although in those cases one needs to be\nparticularly cognizant of the effects of bandpass shifting and surface\nbrightness dimming; such effects can be understood and calibrated\nusing the GEMS 850LP and GOODS 850LP data).  Furthermore, the\n24{\\micron} observations (\\S\\ref{sec-mir}) are well-matched in depth\nwith the first Cycle GTO observations of the CDFS; analyses of the\nCDFS and A901\/2 fields have been presented by \\citet{zheng07} and\n\\citet{Bell07}.  Several projects are already exploiting this\ncombined dataset (see \\S\\ref{sec-summary} for details), and with the\npublicly-available data in the CDFS, these samples provide a valuable\nstarting point for many investigations of galaxy evolution.\n\n\n\\subsection{The Abell 901(a,b)\/902 supercluster:  a laboratory for galaxy evolution}\\label{sec-root2}\n\nThe A901\/2 system is an exceptional testing ground with which to\naddress environmental influences on galaxy evolution.  Consisting of\nthree clusters and related groups at $z\\sim0.165$, all within\n$0.5\\degr\\times0.5\\degr$, this region has been the target of extensive\nground- and space-based observations.  We have used the resulting\ndataset to build up a comprehensive view of each of the main\ncomponents of the large-scale structure: the galaxies, the dark\nmatter, and the hot X-ray gas.  The moderate redshift is advantageous\nas it enables us to study a large number of galaxies, yet the\nstructure is contained within a tractable field-of-view and probes a\nvolume with more gas and more star formation in general than in the\nlocal universe.\n\nThe A901\/2 region, centred at $(\\alpha,\\delta)_{\\rm J2000}$ = $(9^{\\rm\nh}56^{\\rm m}17\\fs3$, $-10\\degr01\\arcmin11\\arcsec)$, was originally one\nof three fields targeted by the COMBO-17 survey \\citep{wolf03}.  It\nwas specifically chosen as a known overdensity due to the multiple\nAbell clusters present.  These included two clusters (A901a and A901b)\nwith X-ray luminosities sufficient to be included in the X-ray\nBrightest Abell-type Cluster Survey \\citep[XBACS;][]{ebeling96} of the\nROSAT All-Sky Survey, though pointed ROSAT HRI observations by\n\\cite{schindler00} subsequently revealed that the emission from A901a\nsuffers from confusion with several point sources in its vicinity.\nThe extended X-ray emission in the field is further resolved by our\ndeep XMM-Newton imaging (see \\S\\ref{sec-xray}).  Additional structures at\n$z\\sim 0.165$ in the field include A902 and a collection of galaxies\nreferred to as the Southwest Group (SWG).\n\nThe five broad- and 12 medium-band observations from COMBO-17 provide\nhigh-quality photometric redshifts and spectral energy distributions\n(SEDs).  Together with the high-quality imaging for ground-based\ngravitational lensing, the A901\/2 data have been used in a variety of\npapers to date.  COMBO-17 derived results include 2D and 3D\nreconstructions of the mass distribution \\citep{gray02, taylor04}; the\nstar-formation--density relation \\citep{gray04}; the discovery of a\nsubstantial population of intermediate-age, dusty red cluster\ngalaxies \\citep[][here-after WGM05]{wgm05}; and the morphology-density\n\\citep{lane07} and morphology-age-density \\citep{wolf07} relations.\n\nFurther afield, the clusters are also known to be part of a larger\nstructure together with neighbouring clusters Abell 907 and Abell 868\n(1.5 degrees and 2.6 degrees away, respectively). Nowak et al. (in\nprep.) used a percolation (also called `friends-of-friends') algorithm\non the REFLEX cluster catalogue \\citep{boehringer04} to produce a\ncatalogue of 79 X-ray superclusters.  Entry 33 is the\nA868\/A901a\/A901b\/A902\/A907 supercluster, which also contains an\nadditional, but not very bright, non-Abell cluster.  Though not\nobserved as part of the STAGES study, these clusters are included in\nthe constrained N-body simulations used to understand the formation\nhistory of the large-scale structure (\\S\\ref{sec-mocks}).\n\nThe plan of this paper is as follows: in \\S\\ref{sec-hst} we outline\nthe observations taken to construct the 80-tile mosaic with the\nAdvanced Camera for Surveys on HST.  We discuss data reduction, object\ndetection, and S\\'ersic\\ profile fitting.  In \\S\\ref{sec-c17} we\npresent the COMBO-17 catalogue for the A901\/2 field and discuss how\nthe two catalogues are matched.  In \\S\\ref{sec-multi} we present a\nsummary of the further multiwavelength data for the field and derived\nquantities such as stellar masses and star-formation rates.  We finish\nwith describing ongoing science goals, future prospects, and\ninstructions for public access to the data and catalogues described\nwithin.  Appendix~\\ref{app-notes} contains details on ten individual\nobjects of particular interest within the field.\n\nThroughout this paper we adopt a concordance cosmology with\n$\\Omega_m=0.3,\\Omega_\\lambda=0.7$, and $H_0=70$ km s$^{-1}$\nMpc$^{-1}$.  In this cosmology, $1\\arcsec = 2.83$ kpc at the redshift\nof the supercluster ($z\\sim0.165$), and the COMBO-17 field-of-view\ncovers $5.3\\times5.1$ Mpc$^{2}$.  Magnitudes derived from the HST\nimaging (\\S\\ref{sec-hst}) in the F606W ($V$-band) filter are on the AB\nsystem,\\footnote{For F606W, $m_{\\rm AB}-m_{\\rm Vega}=0.085$.} while\nmagnitudes from COMBO-17 (\\S\\ref{sec-c17}) in all filters are on the\nVega system.\n\n\n\n\n\\section{HST data}\\label{sec-hst}\n\n\\subsection{Observations}\n\nThe primary goal of the STAGES HST imaging was to obtain morphologies\nand structural parameters for all cluster galaxies down to $R=24$\n($M_V\\sim-16$ at $z\\sim0.165$).  The full area of the COMBO-17\nobservations was targeted to sample a wide range of environments.\nSecondary goals included obtaining accurate shape measurements of\nfaint background galaxies for the purposes of weak lensing, and\nmeasuring morphologies and structural parameters for all remaining\nforeground and background galaxies to $R=24$.  As discussed in\n\\S\\ref{sec-root2}, the survey design and filter was chosen to match\nthat of the GEMS survey \\citep{rix04} of the Chandra Deep Field South\n(CDFS).  The CDFS is another field with both COMBO-17 and HST\ncoverage, but in contrast to the A901\/2 field is known to contain\nlittle significant large-scale structure.  It will therefore serve as\na matched control sample for comparing cluster and field environments\nat similar epochs.\n\nTo this end we constructed an 80-tile mosaic with ACS in Cycle 13 to\ncover an area of roughly 29.5$\\arcmin\\times$29.5$\\arcmin$ in the F606W\nfilter, with a mean overlap of 100 pixels between tiles.  Scheduling\nconstraints forced the roll angle to be 125 degrees for the majority\nof observations, and one gap in the northeast corner was imposed on\nthe otherwise contiguous region due to a bright ($V=9$) star.  A\n4-point parallelogram-shaped dithering pattern was employed, with\nshifts of 2.5 pixels in each direction.  An additional shift of 60.5\npixels in the y-direction was included between dithers two and three\nin order to bridge the chip gap.\n\nConcerns about a time-varying PSF and possible effects on the weak\nlensing measurements drove the requirement for the observations to be\ntaken in as short a time frame as possible.  In practice this was\nlargely successful, with $>50\\%$ of tiles observed in a single\nfive-day period (Fig.~\\ref{fig-cumulative}), and $>90\\%$ within 21\ndays.  Six tiles (29, 75, 76, 77, 79, 80) were unobservable in that cycle\nand were re-observed six months later, with a 180 degree rotation.\nFurthermore, tile 46 was also re-observed at this orientation as the\noriginal observation failed due to a lack of guide stars.  These seven\ntiles were observed following the transition to  two-gyro mode  with no\nadverse consequences in image quality. \n\nDetails of the observations are listed in Table~\\ref{tab-obs}.  A\nschematic of the field showing the ACS tiles and the multiwavelength\nobservations is shown in Fig.~\\ref{fig-field}.  Additionally, four\nparallel observations with WFPC2 (F450W) and NICMOS3 (F110W and F160W)\nwere obtained simultaneously for each ACS pointing.  Due to the\nseparation of different instruments on the HST focal plane, most but\nnot all parallel images overlap with the ACS mosaic (52\/10\/18\nWFPC images and 42\/9\/29 NICMOS3 images have full\/partial\/no overlap\nwith the ACS mosaic; most NICMOS3 images have partial overlap with a\nWFPC2 image).  In this paper we restrict ourselves to a discussion of\nthe primary ACS data, analysis of the parallels will follow in a\nfuture publication.\n\n\n\n\\begin{table*}\n\\begin{tabular}{rrrrrrrr}\n\\hline\n\\hline\n\\multicolumn{1}{c}{Tile} & \n\\multicolumn{1}{c}{Date} &\n\\multicolumn{1}{c}{$\\alpha$} & \n\\multicolumn{1}{c}{$\\delta$} &\n\\multicolumn{1}{c}{Exposure} & \n\\multicolumn{1}{c}{N$_{\\rm hot}$} &\n\\multicolumn{1}{c}{N$_{\\rm cold}$} & \n\\multicolumn{1}{c}{N$_{\\rm good}$}\\\\\n\n\\multicolumn{1}{c}{} & \n\\multicolumn{1}{c}{[dd\/mm\/yyyy]} &\n\\multicolumn{1}{c}{[J2000]} & \n\\multicolumn{1}{c}{[J2000]} &\n\\multicolumn{1}{c}{[s]} & \n\\multicolumn{1}{c}{} &\n\\multicolumn{1}{c}{} & \n\\multicolumn{1}{c}{}\\\\\n\\hline\n 1 & 09 07 2005 & 09:55:22.8 & -10:14:01 & 1960 & 851 & 173 & 796 \\\\  \n 2 & 07 07 2005 & 09:55:44.5 & -10:13:54 & 1960 & 1082 & 209 & 982 \\\\  \n 3 & 08 07 2005 & 09:55:33.4 & -10:12:03 & 1960 & 1157 & 233 & 1008 \\\\  \n 4 & 07 07 2005 & 09:55:22.4 & -10:10:06 & 1960 & 1051 & 199 & 927 \\\\  \n 5 & 04 07 2005 & 09:56:09.5 & -10:14:26 & 1950 & 1069 & 195 & 973 \\\\  \n 6 & 03 07 2005 & 09:55:58.7 & -10:12:33 & 1950 & 1151 & 219 & 1027 \\\\  \n 7 & 04 07 2005 & 09:55:47.9 & -10:10:39 & 1950 & 1038 & 237 & 905 \\\\  \n 8 & 04 07 2005 & 09:55:37.0 & -10:08:45 & 1950 & 1095 & 262 & 938 \\\\  \n 9 & 04 07 2005 & 09:55:26.2 & -10:06:52 & 1950 & 1020 & 188 & 876 \\\\  \n10 & 05 07 2005 & 09:55:15.4 & -10:04:58 & 1950 & 1014 & 184 & 938 \\\\  \n11 & 07 07 2005 & 09:56:38.6 & -10:15:34 & 1960 & 989 & 219 & 876 \\\\  \n12 & 04 07 2005 & 09:56:27.8 & -10:13:40 & 1950 & 1020 & 226 & 885 \\\\  \n13 & 28 06 2005 & 09:56:16.9 & -10:11:46 & 2120 & 1193 & 256 & 1037 \\\\  \n14 & 28 06 2005 & 09:56:06.1 & -10:09:53 & 2120 & 1391 & 254 & 1111 \\\\  \n15 & 28 06 2005 & 09:55:55.3 & -10:07:59 & 2120 & 1182 & 253 & 1052 \\\\  \n16 & 29 06 2005 & 09:55:44.5 & -10:06:06 & 2120 & 1109 & 208 & 940 \\\\  \n17 & 29 06 2005 & 09:55:33.7 & -10:04:12 & 1960 & 1116 & 250 & 888 \\\\  \n18 & 04 07 2005 & 09:55:22.9 & -10:02:18 & 1950 & 995 & 178 & 868 \\\\  \n19 & 09 07 2005 & 09:56:57.3 & -10:14:25 & 1960 & 963 & 180 & 786 \\\\  \n20 & 07 07 2005 & 09:56:46.0 & -10:12:54 & 1960 & 979 & 222 & 829 \\\\  \n21 & 30 06 2005 & 09:56:35.2 & -10:11:00 & 1960 & 1166 & 288 & 1005 \\\\  \n22 & 28 06 2005 & 09:56:24.4 & -10:09:07 & 2120 & 1193 & 263 & 1012 \\\\  \n23 & 25 06 2005 & 09:56:13.6 & -10:07:13 & 2120 & 1143 & 241 & 1000 \\\\  \n24 & 25 06 2005 & 09:56:02.8 & -10:05:19 & 2120 & 1244 & 254 & 1128 \\\\  \n25 & 22 06 2005 & 09:55:52.0 & -10:03:26 & 2120 & 1274 & 248 & 1051 \\\\  \n26 & 29 06 2005 & 09:55:41.1 & -10:01:32 & 1960 & 1214 & 275 & 1063 \\\\  \n27 & 05 07 2005 & 09:55:30.3 & -09:59:39 & 1950 & 1258 & 279 & 1068 \\\\  \n28 & 08 07 2005 & 09:55:19.5 & -09:57:45 & 1960 & 1161 & 220 & 1052 \\\\  \n29 & 04 01 2006 & 09:57:10.7 & -10:14:08 & 2120 & 1274 & 272 & 1123 \\\\  \n30 & 09 07 2005 & 09:57:04.5 & -10:11:48 & 1960 & 943 & 209 & 781 \\\\  \n31 & 08 07 2005 & 09:56:53.5 & -10:10:14 & 1960 & 900 & 200 & 713 \\\\  \n32 & 03 07 2005 & 09:56:42.7 & -10:08:20 & 1950 & 1023 & 214 & 884 \\\\  \n33 & 28 06 2005 & 09:56:31.9 & -10:06:27 & 2120 & 1150 & 223 & 955 \\\\  \n34 & 22 06 2005 & 09:56:21.0 & -10:04:33 & 2120 & 1318 & 243 & 1111 \\\\  \n35 & 22 06 2005 & 09:56:10.2 & -10:02:40 & 2120 & 1220 & 244 & 1028 \\\\  \n36 & 24 06 2005 & 09:55:59.4 & -10:00:46 & 2120 & 1320 & 287 & 1101 \\\\  \n37 & 29 06 2005 & 09:55:48.6 & -09:58:53 & 1960 & 1150 & 239 & 974 \\\\  \n38 & 05 07 2005 & 09:55:37.8 & -09:56:59 & 1950 & 1123 & 205 & 951 \\\\  \n39 & 08 07 2005 & 09:55:27.0 & -09:55:05 & 1960 & 1094 & 210 & 965 \\\\  \n40 & 09 07 2005 & 09:57:12.5 & -10:09:14 & 1960 & 1062 & 198 & 916 \\\\  \n41 & 07 07 2005 & 09:57:00.9 & -10:07:34 & 1960 & 962 & 176 & 828 \\\\  \n42 & 03 07 2005 & 09:56:50.1 & -10:05:41 & 1950 & 1090 & 205 & 928 \\\\  \n43 & 27 06 2005 & 09:56:39.3 & -10:03:47 & 2120 & 1198 & 202 & 1052 \\\\  \n44 & 27 06 2005 & 09:56:28.5 & -10:01:54 & 2120 & 1266 & 230 & 1046 \\\\  \n45 & 23 06 2005 & 09:56:17.7 & -10:00:00 & 2120 & 1280 & 285 & 1064 \\\\  \n46 & 01 01 2006 & 09:56:05.4 & -09:57:47 & 2120 & 1438 & 355 & 1235 \\\\  \n47 & 01 07 2005 & 09:55:56.0 & -09:56:13 & 1960 & 1198 & 273 & 972 \\\\  \n48 & 06 07 2005 & 09:55:45.2 & -09:54:19 & 1950 & 989 & 176 & 852 \\\\  \n49 & 06 07 2005 & 09:55:34.4 & -09:52:26 & 1960 & 1054 & 223 & 901 \\\\  \n50 & 09 07 2005 & 09:55:24.4 & -09:50:31 & 1960 & 984 & 212 & 832 \\\\  \n51 & 07 07 2005 & 09:57:08.4 & -10:04:55 & 1960 & 1050 & 189 & 923 \\\\  \n52 & 03 07 2005 & 09:56:57.6 & -10:03:01 & 1960 & 1142 & 209 & 941 \\\\  \n53 & 03 07 2005 & 09:56:46.8 & -10:01:07 & 1950 & 1135 & 211 & 920 \\\\  \n54 & 02 07 2005 & 09:56:36.0 & -09:59:14 & 1950 & 1131 & 228 & 921 \\\\  \n55 & 02 07 2005 & 09:56:25.1 & -09:57:20 & 1960 & 1205 & 311 & 974 \\\\  \n56 & 02 07 2005 & 09:56:14.3 & -09:55:27 & 1960 & 1097 & 242 & 891 \\\\  \n57 & 01 07 2005 & 09:56:03.5 & -09:53:33 & 1960 & 1090 & 210 & 911 \\\\  \n58 & 06 07 2005 & 09:55:52.7 & -09:51:40 & 1950 & 1130 & 201 & 975 \\\\  \n59 & 08 07 2005 & 09:55:32.7 & -09:48:15 & 1960 & 1075 & 204 & 900 \\\\  \n60 & 07 07 2005 & 09:57:15.8 & -10:02:15 & 1950 & 1028 & 183 & 912 \\\\  \n\\hline\n\\end{tabular}\n\\caption{Details of STAGES HST\/ACS observations. Only the second\n  (successful) acquisition of tile 46 is listed.  `Hot',`cold', and\n  `good' SExtractor configurations are described in\n  \\S\\ref{sec-detect}.  Tiles 29, 46, 75, 76, 77, 79, and 80 are\n  oriented at 180\\degr\\ with respect to the rest of the mosaic. \n  The exposure time varied according to the maximum window of\n  visibility available in each orbit.}\\label{tab-obs}\n\\end{table*}\n\n\\begin{table*}\n\\contcaption{}\n\\begin{tabular}{rrrrrrrr}\n\\hline\n\\hline\n\\multicolumn{1}{c}{Tile} & \n\\multicolumn{1}{c}{Date} &\n\\multicolumn{1}{c}{$\\alpha$} & \n\\multicolumn{1}{c}{$\\delta$} &\n\\multicolumn{1}{c}{Exposure} & \n\\multicolumn{1}{c}{N$_{\\rm hot}$} &\n\\multicolumn{1}{c}{N$_{\\rm cold}$} & \n\\multicolumn{1}{c}{N$_{\\rm good}$}\\\\\n\n\\multicolumn{1}{c}{} & \n\\multicolumn{1}{c}{[dd\/mm\/yyyy]} &\n\\multicolumn{1}{c}{[J2000]} & \n\\multicolumn{1}{c}{[J2000]} &\n\\multicolumn{1}{c}{[s]} & \n\\multicolumn{1}{c}{} &\n\\multicolumn{1}{c}{} & \n\\multicolumn{1}{c}{}\\\\\n\\hline\n61 & 07 07 2005 & 09:57:05.0 & -10:00:21 & 1950 & 971 & 183 & 826 \\\\  \n62 & 07 07 2005 & 09:56:54.2 & -09:58:28 & 1950 & 1052 & 184 & 901 \\\\  \n63 & 06 07 2005 & 09:56:43.4 & -09:56:34 & 1950 & 1141 & 217 & 930 \\\\  \n64 & 06 07 2005 & 09:56:32.6 & -09:54:41 & 1950 & 1069 & 222 & 890 \\\\  \n65 & 06 07 2005 & 09:56:21.8 & -09:52:47 & 1950 & 1071 & 227 & 908 \\\\  \n66 & 06 07 2005 & 09:56:11.0 & -09:50:53 & 1950 & 1014 & 222 & 859 \\\\  \n67 & 06 07 2005 & 09:56:00.1 & -09:48:60 & 1950 & 1046 & 226 & 922 \\\\  \n68 & 08 07 2005 & 09:55:49.3 & -09:47:06 & 1960 & 967 & 179 & 851 \\\\  \n69 & 10 07 2005 & 09:57:12.5 & -09:57:42 & 1960 & 876 & 145 & 784 \\\\  \n70 & 09 07 2005 & 09:57:01.7 & -09:55:48 & 1960 & 934 & 183 & 798 \\\\  \n71 & 09 07 2005 & 09:56:50.9 & -09:53:54 & 1960 & 1032 & 182 & 888 \\\\  \n72 & 10 07 2005 & 09:56:40.0 & -09:52:01 & 1960 & 1118 & 212 & 950 \\\\  \n73 & 09 07 2005 & 09:56:29.2 & -09:50:07 & 1960 & 910 & 168 & 773 \\\\  \n74 & 08 07 2005 & 09:56:18.4 & -09:48:14 & 1960 & 907 & 192 & 822 \\\\  \n75 & 04 01 2006 & 09:57:11.0 & -09:53:30 & 2120 & 1708 & 260 & 1140 \\\\  \n76 & 05 01 2006 & 09:57:00.3 & -09:51:39 & 2120 & 1444 & 275 & 1134 \\\\  \n77 & 05 01 2006 & 09:56:49.5 & -09:49:48 & 2120 & 1324 & 287 & 1094 \\\\  \n78 & 05 07 2005 & 09:56:40.6 & -09:48:11 & 1960 & 1031 & 184 & 842 \\\\  \n79 & 05 01 2006 & 09:57:12.9 & -09:50:05 & 2120 & 1357 & 302 & 1019 \\\\  \n80 & 05 01 2006 & 09:57:02.8 & -09:48:36 & 2120 & 1255 & 246 & 973 \\\\  \n\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{figure}\n\\centerline{\\psfig{file=cumulative.ps,width=0.75\\columnwidth}}\n\\caption{Cumulative plot of ACS data acquisition.  In order to\n  minimize the effects of a time-vary PSF on weak lensing\n  applications, 50\\% of tiles were taken within 5 days and 90\\% within 21\n  days.  The remaining 7 tiles were observed 6 months\n  later.}\\label{fig-cumulative}\n\\end{figure}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=field.ps,width=0.95\\textwidth,angle=270}}\n\\caption{Layout of multiwavelength observations of the A901\/2 field.\nThe numbered tiles represent the 80-orbit STAGES mosaic with HST\/ACS,\nwhich overlaps the 31.5$\\times$30 arcmin COMBO-17 field-of-view\n(long-dashed square). The seven shaded tiles were observed $\\sim$6\nmonths after the bulk of the observations and with a 180\\degr\nrotation.  The centres of A901a\/A901b\/A902\/SWG are found in tiles\n55\/36\/21\/8 respectively.  Interior to the STAGES region are the\nXMM-Newton coverage (heavy solid polygon) and the GMRT 1280 MHz\nobservations (short-dashed circle, indicating half-power beam\nwidth). The STAGES area is also overlapped by the field-of-view of the\nSpitzer 24$\\micron$ imaging (solid polygon), the GMRT 610 MHz\nobservations (long-dashed circle), and the GALEX imaging (dotted\ncircle).}\n\\label{fig-field}\n\\end{figure*}\n\n\n\n\\subsection{ACS data reduction}\n\nWe retrieve the reduced STAGES images processed by the CALNICA\npipeline of STScI, which corrects for bias subtraction and\nflat-fielding.  However, as the ACS camera is located 6 arcmin off the\ncentre of the HST optical axis, the images from the telescope have a\nfield-of-view with a parallelogram keystone distortion.  To produce a\nfinal science image from the reduced pipeline data, we therefore also\nhave to remove the geometric distortion before combining the\nindividual dithered sub-exposures.  The removal of the image\ndistortion is now fairly routine through the use of the MULTIDRIZZLE\nsoftware \\citep{koekemoer07}.  However, our particular science goals\nmotivated us to make several changes when optimizing the default\nsettings and combining the raw images.  These changes are discussed\nbelow.\n\n\n\\subsubsection {Image Distortion Correction}\n\nIn STAGES, the science driver that demands the highest quality data\nreduction in terms of producing the most consistent and stable PSF\nfrom image to image, and across the field of view, is weak lensing\n\\citep{heymans08}.  With this goal in mind, we benefit from the\nexperience of \\cite{rhodes07}, who conducted detailed studies of how\nthe pixel values are re-binned when the images are corrected for image\ndistortion.  Briefly speaking, to transform an image that is sampled\non a geometrically distorted grid onto one that is a uniform Cartesian\ngrid fundamentally involves rebinning, i.e. interpolating, the\noriginal pixel values into the new grid.  Doing so is not a\nstraightforward process since the original ACS pixel scale samples the\ntelescope diffraction limit below Nyquist frequency, i.e. the\ntelescope PSF is undersampled.  When a PSF is undersampled, aliasing\nof the pixel fluxes occurs, the result of which is that the recorded\nstructure of the PSF appears to change with position, depending on the\nexact sub-pixel centroid of the PSF.  This variability effectively\nproduces a change in the ellipticity of the PSF as a function of\nsub-pixel position, even if the PSF should be identical everywhere.\nBecause stellar PSFs are randomly centred about a pixel the\nintrinsic ellipticity one then measures has a non-zero scatter.  So,\nas weak lensing relies heavily on measuring the ellipticities of\ngalaxies, which are convolved by the PSF, the scatter in the PSF\nellipticity contributes significant noise to weak lensing\nmeasurements.\n\nAn additional issue with non-Nyquist sampled images is that the process\nof interpolating pixel values necessarily degrades the original image\nresolution.  While the intrinsic resolution can in principle be\nrecovered by dithering the images while making observations, strictly\nspeaking this inversion is only possible when the image is on a\nperfect Cartesian grid at the start, i.e.  with no image distortion.\nOtherwise, there would be a residual ``beating frequency'' in the\nsampling of the reconstituted image, such that some pixels would be\nbetter sampled than others.  Because of this, recovering the intrinsic\nresolution of the telescope when the field is distorted is not a well\nposed problem, and cannot easily be solved by a small number of image\ndithers.  Some resolution loss will necessarily occur in some parts of\nthe image.  This is especially true if the final images are combined\n{\\it after} having been geometrically corrected, as is currently the\nprocess in MULTIDRIZZLE.  One last, unavoidable, side effect of\ninterpolating a non-Nyquist sampled image is that the pixel values\nbecome necessarily correlated.  However, the degree of resolution loss\nand noise correlation can be balanced by a suitable choice of\ninterpolation kernels: whereas square top-hat kernels effectively\namounts to linear interpolation and correlates only the immediate\nneighbour pixels but cause high interpolation (pixellation) noise,\nbell-shaped kernels (e.g.  Gaussian and Sinc) correlate more pixels\nbut better preserve the image resolution.\n\nIn light of these issues, it is clear that the goal of an optimal HST\ndata reduction should be a dataset where the PSF structure is stable\nacross the field of view and reproducible from image tile to tile.\nThe contribution to the PSF variation by the stochastic aliasing\nof the PSF that necessarily occurs during `drizzling' can be reduced\nby appropriate choices of drizzling kernel and output pixel scale.\n\\cite{rhodes07} characterize PSF stability in terms of the scatter in\nthe apparent ellipticity of the PSF in the ACS field of view.  After\nexperimenting, they determine that the optimal set of parameters in\nMULTIDRIZZLE to use is a Gaussian drizzling kernel,\n\\texttt{pixfrac}=0.8, and an output pixel scale of $0\\farcs03$.  We\nthus follow their approach by adopting those parameters for our own\nreduction, while keeping all the other default parameters unchanged.\nHowever, they note, as we do, that a Gaussian kernel causes more\ncorrelated pixels than tophat kernels.  Nonetheless because the choice\nof interpolation kernel amounts effectively to a smoothing kernel,\ncorrelated noise should in principle not have an impact on photometry\nstatistics since the flux is conserved.  Moreover, the same\ninterpolation (smoothing) kernel propagates into the PSF, thus the\nchoice of kernel should also not impact galaxy fitting analyses.\n\n\n\n\\subsubsection {Sky pedestal and further image flattening correction}\n\nThe images obtained from the HST archive have been bias subtracted and\nflatfielded.  However, large-scale non-flatness on the order of 2-4\\%\nremains in the images, and there are slight but noticeable pedestal\noffsets that remain between the four quadrants.  These large scale\npatterns and pedestals are both stationary and consistent in images\nthat are observed closely in time.  And even though MULTIDRIZZLE tries\nto equalize the pedestals before combining the final images, the\ncorrection is not always perfect due to object contamination when\ncomputing the sky pedestal.  These effects are small, and the sky\npedestal issue only affect large objects situated right on image\nboundaries, so that the effects on the entire survey itself may only\nbe cosmetic.  Nevertheless, we try to correct for the effects by\nproducing a median image of data observed closely in time, after first\nrejecting the brightest 30\\% and faintest 20\\% of the images (to\navoid over-subtraction).  Then, for each of the four CCD quadrants,\nwe fit a low order 2-D cubic-spline surface (IRAF\/imsurfit)\nindividually to model the large scale non-uniformity in the median sky\nimage, and to remove noise.  The noiseless model of the sky is then\nsubtracted from all the data observed closely in time.  After\ncorrection, the mean background in the four quadrants is essentially\nequal, and the residual non-flatness is $\\ll 1\\%$.\n\n\\subsection{Object Detection}\n\\label{sec-detect}\nObject detection and cataloguing were carried out automatically on the\nSTAGES F606W imaging data using the SExtractor V2.5.0 software\n\\citep{bertin96}. An optimized, dual (`cold' and `hot') configuration\nwas used, following the strategy developed for HST\/ACS data of similar\ndepth for GEMS \\citep{caldwell08}.  The main challenge to extracting\nsources from the STAGES ACS data is the tradeoff between deblending\nhigh-surface brightness cluster members that are close on the sky in\nprojection, and avoiding spurious splitting (`shredding') of highly\nstructured spiral galaxies into multiple sources. In addition, we\ndesire high detection completeness for faint, and often low-surface\nbrightness, background galaxies.  To optimize the detection\ncompleteness and deblending reliability for counterparts to $R_{\\rm\nap}\\leq24$ mag galaxies\\footnote{COMBO-17 redshifts are mostly useful\nat $R_{\\rm ap}\\leq24$ for reasons discussed in detail in \\S\\ref{sec-c17},\nand so we adopt this cut for our main science sample.} from the\nCOMBO-17 catalogue, we fine-tuned the combination of cold and hot\nconfiguration parameters using three representative STAGES tiles (21,\n39, and 55).  For STAGES, we converged on the parameters given in\nTable \\ref{sex_config}, which successfully detected 99.5\\% (650\/653)\nof the $R_{\\rm ap}\\leq24$ mag COMBO-17 galaxies on these tiles, with\nreliable deblending for 98.0\\%.\n\nSExtractor produces a list of source positions and basic photometric\nparameters for each astrometrically\/photometrically calibrated image,\nand produces a segmentation map that parses the image into source and\nbackground pixels, which is necessary for subsequent galaxy fitting\nwith GALFIT \\citep{peng02} described in \\S\\ref{sec-fitting}. For both\nconfigurations, a weight map ($\\propto{\\rm variance}^{-1}$) and a\nthree-pixel (FWHM) top-hat filtering kernel were used.  The former\nsuppresses spurious detections on low-weight pixels, and the latter\ndiscriminates against noise peaks, which statistically have smaller\nextent than real sources as convolved by the instrumental PSF.  Our\nfinal catalogue contains 75\\,805 {\\it unique} F606W sources uniformly\nand automatically identified from 17\\,978 objects detected in the cold\nrun, and 89\\,464 `good' sources found in the hot run (before\nrejection of the unwanted hot detections that fell within the\nisophotal area of any cold detection).  A total of 5\\,921 objects\nwere manually removed from the catalogue after the detection\nstage. These detections are mainly over-deblended galaxies or image\ndefects like cosmic rays. Another set of 658 detections were included\nin fitting the sample galaxies to ensure the accurate fitting of real\nobjects, but excluded from the final catalogue. These were also mainly\ncosmic ray hits or stellar diffraction spikes.  Although the main\nanalysis was performed on a tile-by-tile basis, rather than\nmosaic-wise, the main catalogue only contains {\\it unique} sources.\nObjects detected on two tiles enter the catalogue only once. The most\ninterior-located was selected for entry into the catalogue.  The\nbreakdown of cold, hot, and good sources per ACS frame is given in\nTable~\\ref{tab-obs}.\n\n\nIn Fig.~\\ref{fig-hist} we show a histogram of various object samples\nin the region of the HST-mosaic that overlaps with COMBO-17. The HST\ndata start to become incomplete at $V_{606}\\sim 26$ (solid\nline). Stars (hashed histogram) only make up a significant fraction of\nall detections at the brightest magnitudes. A histogram of\ncounterparts from a cross-correlation with COMBO-17 is shown in light\ngrey. When the match is restricted to extended objects with\n$R_{\\rm ap}<24$ (ie. the primary 'galaxy' sample for which we have\nreliable photometric redshifts), the HST sources largely have $V_{606}<24$.\n\n\n\\begin{figure}\n\\centerline{\\psfig{file=data_paper1.eps,width=\\columnwidth}}\n\\caption{ Source detections in the HST mosaic (overlap region with\nSTAGES and COMBO-17 coverage). The solid line represents all\nSExtractor detected sources (74\\,534 objects). The grey histograms\nshows all objects with a corresponding match in the COMBO-17 catalogue\n(light grey; 50\\,701 sources) and extended sources with\n$R_{\\textrm{ap}}<24$ (dark grey; 12\\,748 sources). In addition, the\nhashed region indicates stars as defined by our star-galaxy separation\ncriterion (Equation~\\ref{eqn-stargal}; 4\\,969 stars in total). In the\ninset we highlight the bright magnitude end where the total number of\nstars dominates the source population.}\\label{fig-hist}\n\\end{figure}\n\nStar-galaxy separation is performed in the apparent magnitude -- \nsize plane spanned by the SExtractor parameters MAG\\_BEST ($V_{606}$) and\nFLUX\\_RADIUS ($r_f$). Objects with\n\n\\begin{equation}\\label{eqn-stargal}\n\\log(r_f) < {\\rm max}\\left( 0.35; 1.60-0.05 V_{606}; 5.10-0.22 V_{606} \\right)\n\\end{equation}\nare classified as point sources; sources above that line are\nidentified as extended sources (galaxies). This plane is shown in\nFig.~\\ref{fig-hstmorph}. The separation line clearly delineates\ncompact and extended sources, in particular when inspecting the\nCOMBO-17 sources only (crosses). Note that those AGN for which the\npoint source dominates are also found on the point-source locus and\ntherefore are removed from the galaxy sample by this selection.\n\n\\begin{figure}\n\\centerline{\\psfig{file=data_paper7_msk_cmp.ps,width=0.8\\columnwidth}}\n\\caption{ Star-galaxy separation. We define a line in the\nmagnitude-size plane to separate stars and galaxies (solid line).\nObjects above this line are extended galaxies; objects below are other\ncompact objects (including most AGN). Grey pluses indicate all\ndetections; black crosses only those with a COMBO-17 cross-match and\n$R_{\\textrm{ap}}<24$ and a redshift $z>0$. Note, a significant number\nof mostly late-type stars are misidentified as galaxies by COMBO-17\nphotometry alone. The dashed line shows a line of constant surface\nbrightness, which is almost parallel to our selection line at the\nbright end. \n}\\label{fig-hstmorph}\n\\end{figure}\n\nIn the Fig.~\\ref{fig-xcorrcomp} we display the galaxy fraction as a\nfunction of $V_{606}$ magnitude (grey histogram). Out to $V_{606}\\sim\n22$ almost every galaxy detection on the HST images has a COMBO-17\ncounterpart; at the COMBO-17 sample limit $V_{606}\\sim 24$ the\nmatching completeness for STAGES objects is still $\\sim$90\\%. The\ncross-matching between COMBO-17 and the HST data is described in more\ndetail in \\S~\\ref{sec-xcorr}, where completeness is defined in\nreverse, i.e.  maximizing HST counterparts for COMBO-17 objects.\n\n\n\\begin{figure}\n\\centerline{\\psfig{file=data_paper2.eps,width=0.9\\columnwidth}}\n\\caption{Fraction of extended STAGES objects and COMBO-17\ncounterparts.  The grey line shows the extended source fraction in\nSTAGES. At bright magnitudes most sources are compact, while at the\nfaint end almost all are extended. The black dotted line shows\nextended sources in STAGES with a COMBO-17 counterpart. At\n$V_{606}\\sim26$ the COMBO-17 completeness limit is reached. Almost no\nfainter sources are found in COMBO-17. The black solid line shows\nextended sources in STAGES with a COMBO-17 counterpart having\n$R_{\\textrm{ap}}<24$. Out to $V_{606}\\sim22$ almost every extended\nSTAGES object has a COMBO-17 counterpart: the cross-correlation\ncompleteness defined with respect to the STAGES catalogue is almost\n100\\% (i.e.~the ratio of black and grey lines); at $V_{606}\\sim24$ it\nis $\\sim$90\\%.  See \\S\\ref{sec-xcorr} for further discussion.}\n\\label{fig-xcorrcomp}\n\\end{figure}\n\n\n\\begin{table*}\n\\caption{Dual SExtractor parameter values for STAGES F606W object detection in `cold' and `hot' configurations. }\n\\label{sex_config}\n\\begin{tabular}{lrrl}\n\\hline\\hline\nParameter & Cold & Hot & Description\\\\\n\\hline\nDETECT\\_THRESH & 2.8 & 1.5 & detection threshold above background \\\\\nDETECT\\_MINAREA & 140 & 45 & minimum connected pixels above threshold \\\\\nDEBLEND\\_MINCONT & 0.02 & 0.25 & minimum flux\/peak contrast ratio\\\\\nDEBLEND\\_NTHRESH & 64 & 32 & number of deblending threshold steps\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\n\\subsection{S\\'ersic\\ profile fitting}\\label{sec-fitting}\n\nTo obtain S\\'ersic\\ model fits for each STAGES galaxy, the\nimaging data were processed with the data pipeline GALAPAGOS (Galaxy\nAnalysis over Large Areas: Parameter Assessment by GALFITting Objects\nfrom SExtractor; Barden et al., in prep.). GALAPAGOS performs all \ngalaxy fitting analysis steps from object detection to catalogue\ncreation automatically. This includes (i) source detection and\nextraction with SExtractor; (ii) preparing all detected objects for\nS\\'ersic\\ fitting with GALFIT \\citep{peng02}: i.e., constructing bad\npixel masks, measuring local background levels, and setting up\nstarting scripts with initial parameter estimates; (iii) running the\nS\\'ersic\\ model fits; and (iv) compiling all information into a final\ncatalogue.  \n\nBased on a single startup script, GALAPAGOS first runs SExtractor in\nthe dual high dynamic range mode described in \\S\\ref{sec-detect}. As\nno SExtractor setup is ever 100\\% optimal, we manually inspected all\n80 tiles for unwanted detections or over-deblended objects. GALAPAGOS\nallows for the removal of such extraction failures automatically given\nan input coordinate list. Additionally, we also composed a list of\ndetections that are bright enough to influence the fitting of\nneighbouring astronomical sources (e.g. diffraction spikes from bright\nstars).  Unlike the aforementioned bad detections these are not\nremoved instantly, but kept in the source catalogue throughout the\nfitting process and removed only from the final object\ncatalogue. Again, GALAPAGOS performs this operation automatically\ngiven a second list of coordinates. Further details on the\nprocess of manual fine-tuning of detection catalogues can be found in\nBarden et al. (in prep.).\n\nAfter the second run GALAPAGOS uses the cleaned output source list\n(described in \\S\\ref{sec-detect}) to cut postage stamps for every\nobject. Postage stamps are required for efficient S\\'ersic\\ profile\nfitting with GALFIT. The sizes of the postage stamps are based on a\nmultiple $m$ of the product of the SExtractor parameters KRON\\_RADIUS\nand A\\_IMAGE. We define a ``Kron-ellipse'' with semi-major axis\n$r_{\\textrm{K}}$ as\n\\begin{equation}\n r_{\\textrm{K}}=m \\times \\textrm{KRON\\_RADIUS} \\times \\textrm{A\\_IMAGE}.\n\\end{equation} \n\nThe sky level is calculated for each source individually by evaluating\na flux growth curve. GALAPAGOS uses the full science frame for this\npurpose in contrast to simply working on the postage stamp. Although\nin principle the background estimate provided by SExtractor could have\nbeen used, tests show that using the more elaborate GALAPAGOS scheme\nresults in more robust parameter fits \\citep{haeussler07}. For a\ndetailed description of the algorithm we refer to Barden et al. (in\nprep.). One might argue that GALFIT allows fitting the sky\nsimultaneously with the science object. However, this requires the\nsize of the postage stamp to be matched exactly to the size of the\nscience object. If the postage stamp is too small, the proper sky\nvalue cannot be found; if it is too big, computation takes\nunneccesarily long.  Too many secondary sources would have to be\nincluded in the fit and the inferred sky value might be influenced by\ndistant sources. Additionally, galaxies may not be perfectly\nrepresented by a S\\'ersic\\ fit, and the sky may take on unrealistic\nvalues as a result.  Although this method may be the easiest option\nfor manual fitting, in the general case of fitting large numbers of\nsources automatically the most robust option is to calculate the sky\nvalue beforehand and keep its value fixed when running GALFIT\n\\citep[as demonstrated in][]{haeussler07}.\n\nAnother crucial component for setting up GALFIT is determining which\ncompanion objects should be included in the fit.  In particular, in\ncrowded regions with many closely neighbouring sources the fit quality\nof the primary galaxy improves dramatically when including\nsimultaneously fitting S\\'ersic\\ models to these neighbours rather than\nsimply masking them out.  GALAPAGOS makes an educated guess as to\nwhich neighbours should be fitted or masked (see Barden et al., in\nprep. for further details).  The decision is made by calculating\nwhether the Kron-ellipses of primary and neighbouring source\noverlap. This calculation is performed not only for sources on the\npostage stamp, but on all objects on the science frames surrounding\nthe current one, in order to take objects at frame edges into account\nproperly. Detections not identified as overlapping secondary sources\nare treated as well.  Such non-overlapping companions are masked based\non their Kron-ellipse and thus excluded from fitting.\n\nAn additional requirement for fitting with GALFIT is an input PSF.  We\nconstructed a general high S\/N PSF for STAGES by combining all stars\n(i.e.\\ classified by COMBO-17 photometry and having ACS \nSExtractor stellarity index $>0.85$) in the brightness interval\n$19.5<V_{606}\\le23.5$ and lying away from the chip edges.  This\nselects non-saturated stars that can still contribute signal in their\ncentres. All stars were visually inspected against binarity,\ncompanions, or defects, which resulted in either a manually created\nmask, or the star being excluded if masking would not have been\nsufficient to isolate the star. With this selection 1\\,024 stars\nremained and were combined after subpixel cocentering and local\nbackground removal.\n\nIn order to sample the field-variations of the PSF well and\nnot be dominated by the few brightest stars, we weighted all stars\nidentically in the centre (where all stars carry information), but\napplied a suppression of the noise in the outer parts by a Gaussian\ndownweighting.  The contribution from fainter stars in this process\nwas suppressed at smaller radii relative to brighter ones. In this way\nwe created a high S\/N true mean PSF image of 255$\\times$255 pixel\ncentred exactly on the PSF and used this for all galaxy-related (but\nnot AGN related) analyses.  \n\nIn its current version, GALAPAGOS sets up GALFIT to fit a\nS\\'ersic\\ model \\citep{sersic68} for each object. A S\\'ersic\\ profile is\na generalised de Vaucouleurs model with variable exponent $n$, the\nS\\'ersic\\ index:\n\\begin{equation}\n \\Sigma\\left(r\\right)=\\Sigma_e\\times\\exp\\left(-\\kappa\\left[\\left(r\/r_e\\right)^{\n1\/n}-1\\right]\\right),\n\\end{equation} \nwith the effective radius $r_e$, the effective surface density\n$\\Sigma_e$, the surface density as a function of radius\n$\\Sigma\\left(r\\right)$ and a normalisation constant\n$\\kappa=\\kappa\\left(n\\right)$. An exponential profile has $n=1$ while\na de Vaucouleurs profile has $n=4$. The parameters that go into the\nmodel are the position $[x,y]$, total magnitude $m$, the effective\nradius $r_e$, the S\\'ersic\\ index $n$, the axis ratio $q$ ($q=b\/a$; the\nratio of semi-minor over semi-major half-axis ratio) and the position\nangle $\\theta$.  Starting guesses for all parameters aside from $n$\nand $r_e$ are taken directly from the SExtractor output. GALAPAGOS\nconverts the FLUX\\_RADIUS from SExtractor to estimate the effective\nradius as $r_e=10^{-0.79}\\textrm{FLUX\\_RADIUS}^{1.87}$. This formula\nwas found empirically to work best for simulated S\\'ersic\\ profiles in\nthe GEMS project \\citep{haeussler07}. The S\\'ersic\\ index is started at\na value $n=2.5$.\n\nFor computational efficiency we apply constraints to the parameter\nrange during the fitting process. Of course, this procedure is not\nadvisable when fitting objects manually, yet it is mandatory for an\nautomated process like GALAPAGOS. Our constraints are listed in Table\n\\ref{tab-constraints}. Non-zero lower boundaries for $r_e$ and $n$\nwere imposed for computational reasons.  The maximum for $r_e$ allows\nfitting the largest galaxy in the field (750 pix correspond to\n$\\sim60$ kpc at the cluster distance).  The upper limit for the\nS\\'ersic\\ index is far from the de Vaucouleurs case and includes even\nthe steepest profiles.  The magnitude constraint flags catastrophic\ndisagreements between the two photometry codes, where one of the two\ndoes not return a sensible result.  Such problem objects may include\nLSB galaxies, where SExtractor fails to see large fractions of the\ntotal flux; or intrinsically faint objects with a peculiar neighbour\nor background structure, where GALFIT tries to remove the excess flux.\nObjects whose values stall at the constraint limits are most likely\nnot well represented by a single S\\'ersic\\ profile (e.g. stars or\nextreme two-component galaxies with a LSB disk).\n\n\\begin{table}\n\\begin{center}\n\\caption{GALFIT fitting constraints.}\n\\label{tab-constraints}\n\\begin{tabular}{ccc}\n\\hline\\hline\nParameter & Lower limit & Upper limit \\\\\n\\hline\n$r_e$ & 0.3 & 750 \\\\\n$n$ & 0.2 & 8  \\\\\n$|m_{\\rm SEx}-m_{\\rm GALFIT}|$ & - & 5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nFinally, GALAPAGOS combines the SExtractor and GALFIT results into one\nFITS-table. At this stage flagged objects (like stellar diffraction\nspikes, etc.) are removed from the table. A very detailed description\nof GALAPAGOS including setup and computational efficiency will be\npresented together with the publication of the code in\nBarden et al. (in prep.).  We note that the GALFIT reported errors are\npurely statistical (ie. based on the assumption that Poisson noise\ndominates the uncertainties of the fit parameters), and as such\ncertainly under-represent the true uncertainties.  A more meaningful\nmeasure of uncertainties comes from fitting simulated galaxies, as\nshown in \\citet{haeussler07} and explored here in detail in\n\\S\\ref{sec-sims}.\n\n\nWith our setup we were able to achieve an overall total of $\\sim92$\\%\nhigh quality fits for our science targets, i.e. galaxies with a\ncross-match in the COMBO-17 catalogue and $R_{\\textrm{ap}}<24$. We\ndefine `bad' fits as those where GALFIT stalled at one of the\nconstraints in Table~\\ref{tab-constraints}. In Fig.~\\ref{fig-badfits}\nwe show the fraction of those bad fits as a function of SExtractor\nmagnitude. At the bright end ($V_{606}<22$), the fraction of failures\nis less than 6\\% and rises steadily from there. Only when reaching the\n(surface brightness) completeness limit (roughly at\n$V_{606}\\sim24-25$) does the fraction of failed fits reach (and\nexceed) 20\\%.\n\n\\begin{figure}\n\\psfig{file=data_paper6.eps,width=\\columnwidth}\n\\caption{GALFIT quality.  Top panel: The two grey histograms show the\ntotal number of fitted galaxies (light grey) and galaxies with `bad'\nfits where the fitting procedure failed (dark grey).  The heavy solid\nand dotted histograms show the same but for the science sample with\n$R_{\\textrm{ap}}<24$ (i.e. objects with a COMBO-17 counterpart only)\nwithin the overlap region of STAGES and COMBO-17.\nBottom panel: Fraction of `bad' fits (plus 1$\\sigma$ error bars) for\nall fitted galaxies (dotted histogram) and those with a COMBO-17 match\nand $R_{\\textrm{ap}}<24$ (solid histogram). Overall, $\\sim 23$\\% of\nall fits ran into a constraint (dashed dotted line). For the science\nobjects (STAGES\/COMBO-17 cross-matched galaxies with\n$R_{\\textrm{ap}}<24$) the fraction is considerably lower ($\\sim8$\\%;\ndashed line). The vertical line roughly indicates the surface\nbrightness completeness limit. The `bump' at $V_{606}\\sim21$ possibly\nresults from merging two SExtractor setups (the `hot' and `cold'\nconfigurations described in \\S\\ref{sec-detect}).  }\\label{fig-badfits}\n\\end{figure}\n\n\n\n\n\\subsection{Completeness and Fit Quality}\\label{sec-sims}\n\nTo both derive completeness maps and examine fitting quality using\nGALAPAGOS, we followed a similar approach as in GEMS and as described\nin \\cite{haeussler07}, but with a different, more realistic set of\nsimulated data. Whereas in \\cite{haeussler07} a small set of only\n1\\,600 simulated galaxies was used to find the ideal setup of the\nfitting pipeline, we have now decided on a fitting setup using\nGALAPAGOS from the start and have carried out much more intensive\ntests. We created entire sets of STAGES-like imaging data by\nsimulating galaxies in all 80 HST\/ACS tiles.  Galaxies were\nsimulated as single-component S\\'ersic\\ profiles; multi-component\ngalaxies or complicated structures such as spiral arms or bars were\nnot included.\n\nThe sample of galaxies to be simulated was derived by using the fits\nof real data as described in \\S\\ref{sec-fitting}. From this superset,\nwe selected a `galaxy sample' to be simulated by excluding both stars\nand those galaxies for which the fit failed. Magnitudes and galaxy\nsizes for the simulated galaxies were chosen according to the\nprobability distribution of this sample. The other simulation\nparameters, (e.g. S\\'ersic\\ index $n$ and axis ratio $q$) were then\nderived by choosing fitting values of real galaxies at approximately\nthe same magnitude and size.  In this way, the simulated data have\nparameters as close as possible to the real galaxy sample.\n\nTo cover a larger number of parameter combinations, we slightly\nsmoothed these values (mag by $\\pm$1 mag, $\\log(r_e)$ by $\\pm$0.25\npix, $n$ by $\\pm$0.5 and $q$ by $\\pm$0.2).  Care was taken to make\nsure that $q$ and $n$ covered sensible values ($0.05 < q < 1$, $0.2 <\nn < 8$). We also simulated galaxies two magnitudes fainter than those\nfound in the real data to be able to derive completeness maps from the\nsame pipeline. Twenty sets of STAGES-like data (80 tiles each) were\nsimulated using this setup. In a further 50 sets, we introduced a\nuniform distribution of the S\\'ersic\\ index over the full range $0.2 <\nn < 8$ over all magnitudes and sizes for 5\\% of galaxies.  This\nimposed pedestal was required in order to fill in gaps in the\nparameter space with bad number statistics or no galaxies at all, and\nwas especially important for galaxies with high $n$-value seen\nface-on.  Both position and position angle, $\\theta$, were randomly\nchosen for each galaxy: thus no clustering was simulated, in contrast\nto the real data.  Simulating around 107\\,000 objects per dataset, we\nwere able to derive an object density comparable to the real data with\na mean of 60\\,612 galaxies found per dataset.  This compares to\n75\\,805 galaxies in the original GALAPAGOS output from the real data,\nwith $\\sim$35\\,000 objects in the `galaxy' catalogue from which we\ndraw the input parameters for the simulations.\n\nAfter choosing the parameters this way, we used the same simulation\nscript that was described in detail in \\cite{haeussler07} to simulate\nthe galaxies. The images were placed in an empty image which was made\nup by empty patches of sky from the STAGES data to resemble the noise\nproperties of the real data. Convolution was performed using a STAGES\nPSF.  In a change to the \\cite{haeussler07} setup, we also simulated\ngalaxies on neighbouring tiles (or closely outside the data area) to\nrealistically model effects from neighbouring galaxies, as well as to\nexamine effects of combining the individual SExtractor catalogues\nwithin GALAPAGOS.\n\n\\begin{figure}\n\\psfig{file=fitting3.eps,width=\\columnwidth}\n\\caption{\\label{fig:fit1} Completeness as a function of\nmagnitude. {\\em Left}: the number of simulated galaxies (light grey),\nrecovered by SExtractor (dark grey), and subsequently fit successfully\nby GALAPAGOS (black) as a function of input magnitude. {\\em Right}:\nCompleteness functions for SExtractor (grey) and GALAPAGOS (black)\noutput. One can see that GALAPAGOS returns a useful result in most\ncases.  Only for relatively faint galaxies does the fit run into\nfitting constraints for a fraction of the objects. At $V_{606}\\sim26$,\nthe STAGES profile fitting is therefore 80\\% complete.}\n\\end{figure}\n\nBy simulating fainter galaxies than are found in the real data, we\nwere not only able to test the fitting quality but also the survey\ncompleteness.  Fig. \\ref{fig:fit1} shows the completeness as derived\nfrom this data as a function of magnitude.  The left plot shows the\nnumber of galaxies simulated (light grey), the number of galaxies\nrecovered (dark grey) and the number of galaxies with successful fit\n(black; meaning that the fit did not run into any fitting\nconstraints).  All three histograms are normalized by the value\nof the bin containing the maximum number of simulated galaxies. In\ntotal, of the 7\\,497\\,614 galaxies simulated, 43.4\\% were not found in\nthe data  using the GALAPAGOS and SExtractor setups used to\nanalyse the real STAGES data. Failed objects in general were too faint\nto be detected.  A further 52.5\\% were successfully recovered,\nidentified and fitted, and 4.0\\% were recovered but excluded from all\nplots as the fit ran into fitting constraints.  For 305 galaxies\n(0.004\\%), the fit crashed and did not return a result at all.\n\nWe additionally find 51\\,043 galaxies (0.7\\% of simulated galaxies)\nthat could not be identified by our search algorithm, which looked for\nthe closest match within 1.0\\arcsec. Examination of these galaxies\nshows that they are either (a) very low surface brightness galaxies\nfor which the SExtractor positioning was not very secure, or (b) two\nneighbouring LSB galaxies that SExtractor detected as one object, also\nresulting in an insecure position.\n\n\\begin{figure*}\n\\centerline{\\psfig{file=fitting1.eps,width=0.8\\textwidth}}\n\\caption{\\label{fig-fit2} Completeness maps as a function of S\\'ersic\\\nindex $n$ and axis ratio $q$ (as labelled above and to the right of the\nplots).  To guide the eye, we overplot a vertical line at mag 26 and a\nsurface brightness line (diagonal, dashed) at 28 {mag\\ arcsec$^{-2}$}. As one can\nclearly see, the completeness (shown in greyscale, black is complete,\nwhite is incomplete or no data) is a strong function of all magnitude,\nsize (and therefore surface brightness), $q$ and $n$. The outline\ncontour shows the region in this plot where galaxies have been\nsimulated to demonstrate where these plots are reliable.}\n\\end{figure*}\n\nUsing the whole available simulated dataset, we can derive a much more\ndetailed completeness for STAGES.  Magnitude alone is not a good\nestimator for completeness, as the internal light distribution has\ngreat influence on this value.  More concentrated galaxy profiles,\nsuch as elliptical, high-$n$ profiles, are more likely to be detected\nby SExtractor than disk-like low-$n$ profiles.  In addition, the\ninclination angle plays an important role. As shown in\nFig.~\\ref{fig-fit2}, we can divide the galaxies in different bins of\n$n$ and $q$ and for each bin can estimate a 2-D completeness map\nshowing the completeness as a function of both magnitude and galaxy\nsize. By looking at each bin one can clearly see that the completeness\nis indeed a function of magnitude as well as size. The completeness\ncatalogue from these extensive simulations will be made publicly\navailable as part of the STAGES data release.  With the large sample\nand complete coverage of the parameter space populated by real\ngalaxies, one could make up customized completeness maps tailored to\nthe particular sample in question.\n\n\n\\begin{figure*}\n\\centerline{\\psfig{file=fitting2b.eps,width=0.9\\textwidth}}\n\\caption{\\label{fig-fit3} Fit Quality. The deviations of the most\nimportant galaxy parameters as a function of surface brightness. {\\em\nTop row}: Magnitude deviation (fit - simulated), {\\em middle row}:\nSize ratio (fit\/simulated), {\\em bottom row}: S\\'ersic\\ index n\n(fit\/simulated). Contours show the data normalized by the number\nof galaxies in each surface brightness bin; the black solid line\nshows the mean of the distribution; and the black dashed lines show\nthe sigma of the distribution ($3\\sigma$ in case of magnitudes,\n$1\\sigma$ in size and S\\'ersic\\ index). All plots are shown for\ndifferent S\\'ersic\\ indices as labelled above the plots. The vertical\ngrey line represents the mean brightness of the sky background in\nSTAGES. The magnitude and sizes are less well recovered in high-$n$\ngalaxies, but the {\\em relative} recovery of $n$ is similar in all\ncases.}\n\\end{figure*}\n\nThe same is true for the fitting quality. As can be seen from\nFig. \\ref{fig-fit3}, the fitting behaviour is a function of both\nsurface brightness and S\\'ersic\\ index. We only show the quality as a\nfunction of S\\'ersic\\ index, but again one can determine fitting\nquality as a function of any combination of the fitted parameters.\nOne can see that high-$n$ galaxies are harder to fit than low-$n$\ngalaxies, e.g. the magnitude deviation $\\Delta$ is 0.00 ($\\sigma =\n0.07$) at around the sky level for galaxies with \\mbox{$0<n<1.5$},\nwhile $\\Delta=0.03$ ($\\sigma = 0.12$) at the highest $n$-bin. The\neffect is even larger at fainter galaxies: $\\Delta=0.00$ ($\\sigma =\n0.18$) at 25 {mag\\ arcsec$^{-2}$} and $\\Delta=0.08$ ($\\sigma = 0.28$) for low- and\nhigh-$n$ galaxies, respectively. A similar trend can be seen for\ngalaxy sizes: $\\Delta = 0.8\\%$ ($\\sigma = 7.3\\%$) and $\\Delta =\n-3.7\\%$ ($\\sigma = 19.0\\%$) at the sky level, and $\\Delta = -0.4\\%$\n($\\sigma = 18.3\\%$) and $\\Delta = -10.7\\%$ ($\\sigma = 36.1\\%$) at 25\n{mag\\ arcsec$^{-2}$}. If one examines relative deviations of the S\\'ersic\\ index,\nthere is essentially no trend seen between different bins of $n$.  In\nan absolute sense, then, the S\\'ersic\\ index is still less well\nrecovered in the high-$n$ bin.\n\nIn general, the systematic deviations are very small except at the\nfaintest galaxies detectable, and both deviation $\\Delta$ and $\\sigma$\nof the distributions are well understood within STAGES.  As was\npointed out in \\cite{haeussler07}, the uncertainties returned by\nGALFIT (and therefore GALAPAGOS) underestimate the true uncertainty by\na large amount.  Using a statistical approach therefore returns more\nreliable errorbars for the individual parameters.  The simulations and\ncatalogue presented here allow a flexible means of estimating errors\non profile fitting for any possible subsample of galaxies.\n\n\n\\section{COMBO-17 data}\\label{sec-c17}\n\n\n\\subsection{COMBO-17 observations and catalogue}\\label{sec-c17cat}\n\nIn this section we briefly describe the COMBO-17 data on the A901\/2\nfield, including observations, catalogue entries and object\nsamples. The corresponding data on the CDFS field were published in\n\\citet[][hereafter W04]{wolf04}, where further technical details can\nbe found.\n\nThe filter set (Table~\\ref{tab-combodata}) contains five broad-band\nfilters (UBVRI) and 12 medium-band filters covering wavelengths from\n350 to 930~nm.  All observations were obtained with the Wide Field\nImager (WFI) at the MPG\/ESO 2.2m-telescope on La Silla, Chile. A field\nof view of $34\\arcmin \\times 33\\arcmin$ (see Fig.~\\ref{fig-field})\nis covered by a CCD mosaic consisting of eight 2k $\\times$ 4k CCDs\nwith a scale of $0\\farcs238$ per pixel. The observations on the A901\/2\nfield were spread out over three observing runs between January 1999\nand February 2001. They encompass a total exposure time of\n$\\sim$185~ks of which $\\sim$20~ks were taken in the $R$-band during\nthe best seeing conditions. A dither pattern with at least ten\ntelescope pointings spread by \\mbox{$\\Delta\\alpha$, $\\Delta\\delta <\\pm\n72\\arcsec$} allowed us to cover the sky area in the gaps of the CCD\nmosaic.\n\n\\begin{table*}\n\\caption{COMBO-17 imaging data on the A901\/2 field: For all filters we\nlist total exposure time, average PSF among individual frames, the\n10$\\sigma$ (Vega) magnitude limits for point sources and the observing\nruns (see Tab.~\\ref{tab-comboruns}) in which the exposure was\ncollected. For flux and magnitude conversions we list AB magnitudes\nand photon fluxes of Vega in all filters. The $R$-band observations were\ntaken in the best seeing conditions.\n\\label{tab-combodata}}\n\\begin{tabular}{ccrccl|cc}\n\\hline \\hline\n  \\multicolumn{2}{c}{$\\lambda_\\mathrm{\\mathrm{cen}}$\/fwhm} &\n  $t_\\mathrm{\\mathrm{exp}}$ & seeing &\n  $m_\\mathrm{\\mathrm{lim},10\\sigma}$ & run code & mag of Vega &\n  $F_\\mathrm{phot}$ of Vega \\\\ \\multicolumn{2}{c}{(nm)} & (sec) & &\n  (Vega mags) & & (AB mags) & $(10^8~\\mathrm{photons\/m^2\/nm\/s})$ \\\\\n  \\noalign{\\smallskip} \\hline \\noalign{\\smallskip} 365\/36 &$U$ & 22100\n  & 1\\farcs10 & 23.7 & G & $+0.77$ & 0.737\\\\ 458\/97 &$B$ & 20500 &\n  1\\farcs20 & 25.4 & A, G & $-0.13$ & 1.371\\\\ 538\/89 &$V$ & 6000 &\n  1\\farcs20 & 24.3 & E & $-0.02$ & 1.055\\\\ 648\/160 &$R$ & 20300 &\n  0\\farcs75 & 25.0 & E & $+0.19$ & 0.725\\\\ 857\/147 &$I$ & 7500 &\n  1\\farcs00 & 22.7 & E & $+0.49$ & 0.412\\\\ \\noalign{\\smallskip} 418\/27\n  & & 7300 & 1\\farcs20 & 24.0 & E & $-0.19$ & 1.571\\\\ 462\/13 & & 10000\n  & 1\\farcs20 & 23.7 & E & $-0.18$ & 1.412\\\\ 486\/31 & & 5500 &\n  1\\farcs15 & 24.0 & E & $-0.06$ & 1.207\\\\ 519\/16 & & 6000 & 1\\farcs05\n  & 23.6 & E & $-0.06$ & 1.125\\\\ 572\/25 & & 5000 & 0\\farcs85 & 23.5 &\n  E & $+0.04$ & 0.932\\\\ 605\/21 & & 6000 & 0\\farcs95 & 23.4 & E &\n  $+0.10$ & 0.832\\\\ 645\/30 & & 4950 & 1\\farcs30 & 22.7 & E & $+0.22$ &\n  0.703\\\\ 696\/21 & & 6600 & 1\\farcs00 & 22.7 & E & $+0.27$ & 0.621\\\\\n  753\/18 & & 7000 & 1\\farcs05 & 22.2 & E & $+0.36$ & 0.525\\\\ 816\/21 &\n  & 19200 & 0\\farcs85 & 22.8 & A & $+0.45$ & 0.442\\\\ 857\/15 & & 16600\n  & 1\\farcs15 & 21.7 & E & $+0.56$ & 0.386\\\\ 914\/26 & & 15700 &\n  0\\farcs95 & 21.9 & E & $+0.50$ & 0.380\\\\ \\noalign{\\smallskip} \\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{table}\n\\caption{COMBO-17 observing runs with A901\/2 imaging. \n\\label{tab-comboruns} }\n\\begin{center}\n\\begin{tabular}{ll}\n\\hline \\hline\nCOMBO-17 run code    &  Dates  \\\\ \n\\hline\nA           &  11.02.-22.02.1999  \\\\\nE           &  28.01.-11.02.2000  \\\\\nG           &  19.01.-20.01.2001  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\nFlux calibration was done with our own tertiary standard stars based\non \\emph{spectrophotometric} observations, a suitable method to\nachieve a homogeneous photometric calibration for all 17 WFI filter\nbands. Two G stars with $B\\simeq 15$ (with COMBO-17 identification\nnumbers 45811 and 46757) were observed at La Silla with DFOSC at the\nDanish 1.54\\,m telescope. A wide ($5\\arcsec$) slit was used for the\nCOMBO-17 standards as well as for an external calibrator star.\n\nThe object search for the COMBO-17 sample was done with SExtractor\nsoftware \\citep{bertin96} in default setup, except for choosing a\nminimum of 12 significant pixels required for the detection of an\nobject.  We first search rather deep and then clean the list of\nextracted objects of those having a S\/N ratio below 4, which\ncorresponds to $>0\\fm2422$ error in the total magnitude MAG\\_BEST. As\na result we obtained a catalogue of 63\\,776 objects with positions,\nmorphology, total $R$-band magnitude and its error.  The astrometric\naccuracy is better than $0\\farcs15$. Using our own aperture photometry\nwe reach a 5$\\sigma$ point source limit of $R\\approx 25.7$.\n\nWe obtained spectral energy distributions of all objects from\nphotometry in all 17 passbands by projecting the known object\ncoordinates into the frames of reference of each single exposure and\nmeasuring the object fluxes at the given locations. In order to\noptimize the signal-to-noise ratio, we measure the spectral shape in\nthe high surface brightness regions of the objects and ignore\npotential low surface brightness features at large distance from the\ncentre. However, this implies that for large galaxies at low redshifts\n$z<0.2$ we measure the SED of the central region and ignore colour\ngradients.\n\nAlso, we suppressed the propagation of variations in the seeing into\nthe photometry by making sure that we always probe the same physical\nfootprint outside the atmosphere of any object in all bands\nirrespective of the PSF. Here, the footprint $f(x,y)$ is the\nconvolution of the PSF $p(x,y)$ with the aperture weighting function\n$a(x,y)$. If all three are Gaussians, an identical physical footprint\ncan be probed even when the PSF changes, simply by adjusting the\nweighting function $a(x,y)$. We chose to measure fluxes on a footprint\nof $1\\farcs5$ FWHM outside the atmosphere ($\\sim 4.2$ kpc at $z\\sim\n0.165$).  In detail, we use the package MPIAPHOT \\citep{mpiaphot93} to\nmeasure the PSF on each individual frame, choose the weighting\nfunction needed to conserve the footprint and obtain the flux on the\nfootprint. Fluxes from individual frames are averaged for each object\nand the flux error is derived from the scatter. Thus, it takes not\nonly photon noise into account, but also suboptimal flatfielding and\nuncorrected CCD artifacts.\n\nAll fluxes are finally calibrated by the tertiary standards in our\nfield.  The aperture fluxes correspond to total fluxes for point\nsources, but underestimate them for extended sources. The difference\nbetween the total (SExtractor-based) and the aperture (MPIAPHOT-based)\nmagnitude is listed as an aperture correction and used to calculate\ne.g. luminosities.  For further details on the observations and the\ndata processing, see W04.\n\nThe A901\/2 field is affected by substantial foreground dust reddening at\nthe level of $E(B-V)\\approx 0.06$, in contrast to the CDFS. Hence,\nany SED fitting and derivation of luminosities requires dereddened\nSEDs.  Therefore, in the catalogue we list three sets of photometry:\n\n\\begin{enumerate}\n\\item $R$-band total and aperture magnitudes as observed for the definition \nof samples and completeness;\n\\item aperture fluxes $F_{\\rm phot}$ in 17 bands, dereddened using \n$A_V=0.18$ and $(A_U,A_B,A_R,A_I)=A_V \\times (1.63,1.24,0.82,0.6)$ with\nsimilar numbers for medium-band filters (rereddening with these \nnumbers would restore original measurements); and \n\\item aperture magnitudes (Vega) in all 17 bands, dereddened, on the\nAsinh system \\citep{lupton99} that can be used for logarithmic flux\nplots with no trouble arising from formally negative flux\nmeasurements.\n\\end{enumerate}\n\nFluxes are given as photon fluxes $F_\\mathrm{phot}$ in units of \nphotons\/m$^2$\/s\/nm, which are related to other flux definitions by\n\\begin{equation}\n   \\nu F_\\nu = hcF_{\\rm phot} = \\lambda F_\\lambda   ~ .\n\\end{equation}\nPhoton fluxes are practical units at the depth of current surveys.  A\nmagnitude of $V=20$ corresponds to 1~photon\/m$^2$\/s\/nm in all systems\n(AB, Vega, ST), provided $V$ is centred on 548~nm. Flux values of an\nobject are missing in those bands where every exposure was saturated.\n\nThe final catalogue contains quality flags for all objects in an\ninteger column (`phot\\_flag'), holding the original SExtractor flags\nin bit 0 to 7, corresponding to values from 0 to 128, as well as some\nCOMBO-17 quality control flags in bits 9 to 11 (values from 512 to\n2048). We generally recommend that users ignore objects with flag\nvalues phot\\_flag $\\ge 8$ for any statistical analysis of the object\npopulation. If an object of particular interest shows bad flags, it\nmay still have accurate COMBO-17 photometry and could be used for some\npurposes. Often only the total magnitude was affected by bright\nneighbours, while the aperture SED is valid.\n\nWe then employ the usual COMBO-17 classification and redshift\nestimation by template fitting to libraries of stars, galaxies, QSOs\nand white dwarfs.  There, the error rate increases very significantly\nat $R_{\\rm ap}>24$. We refer again to W04 for details of the libraries\nand known deficiencies of the process, but repeat here (and correct a\nmisprint in W04) the definition of the classifications (see\nTable~\\ref{tab-comboclasses}).\n\n\n\n\n\\begin{table}\n\\caption{Definition of entries for the `mc\\_class' column and\ncomparison of object numbers between the COMBO-17 data sets of the\nA901\/2 and CDFS field. The samples refer to a magnitude range of\n$R_{\\rm ap}=[16,24]$ and only objects with phot\\_flag$<8$. The A901\/2\nfield is richer in stars because of its galactic coordinates. It is\nalso richer in galaxies due to the cluster, while the CDFS is\nunderdense at $z=[0.2,0.4]$.  We note that these definitions are based\non the COMBO-17 data SED and morphology; star-galaxy separation\nemploying morphological information from the HST imaging\n(Equation~\\ref{eqn-stargal}) is considered separately.\n\\label{tab-comboclasses}}\n\\begin{tabular}{llrr}\n\\hline\\hline\nClass entry      &  Meaning  &  $N\\_{\\rm A901\/2}$  &  $N\\_{\\rm CDFS}$ \\\\\n\\hline\nStar             &  stars \t\t\t& 2096\t& 992\t\\\\\n                 &  (only point sources) \\\\\nWDwarf           &  white dwarf\t\t\t&  14\t&   9\t\\\\\n                 &  (only point sources) \\\\\nGalaxy           &  galaxies\t\t\t& 14555 & 11054\t \\\\\n                 &  (shape irrelevant) \\\\\nGalaxy  (Star?)  &  binary or low-z galaxy &  44\t&  46  \\\\\n                 &  (star SED but extended; \\\\\n\t\t &  ambiguous colour space) \\\\\nGalaxy  (Uncl!)  &  SED fit undecided \t\t& 316\t& 243\t\\\\\n                 &  (most often galaxy) \\\\\nQSO              &  QSOs \t\t\t&  73\t&  66\t\\\\\n                 &  (only point sources) \\\\\nQSO     (Gal?)   &  Seyfert-1 AGN or            & 36\t&  31\t\\\\\n                 &  interloping galaxy\t\\\\\n                 &  (AGN SED but extended; \\\\\n\t\t &  ambiguous colour space) \\\\\nStrange Object   &  unusual strange spectrum \t&   1\t&   3\t\\\\\n                 &  ($\\chi ^2_{\\rm red}>30$) \\\\\n\\noalign{\\smallskip} \\hline\n\\end{tabular}\n\\end{table}\n\n\n\nWe also show in Table~\\ref{tab-comboclasses} a comparison of the sample\nsizes in different classes between the A901\/2 and the CDFS field of\nCOMBO-17. The main difference is that the A901\/2 field contains more\nthan twice the number of stars given its position at relatively low\ngalactic latitude ($+33.6\\deg$).  Another difference is that it\ncontains 30\\% more galaxies than the CDFS, which is both a consequence\nof the cluster A901\/2 and the underdensity in the CDFS seen at\n$z\\sim[0.2,0.4]$. Fig.~\\ref{fig-CW_class} shows a colour-magnitude\ndiagram of the star and white dwarf sample as well as redshift-magnitude\ndiagrams for galaxies and QSOs.\n\n\\begin{figure*}\n\\includegraphics*[height=\\textwidth,angle=270]{CW_classes.ps}\n\\caption{ {\\it Left panel:} Stars (dots) and white dwarfs (crosses):\n$B-V$ colour vs.  $R_{\\rm tot}$. The two reddest stars at $R\\approx 23$ and\n$B-V>2$ are M5-6 stars.  {\\it Centre panel:} Red-sequence (black) and\nblue-cloud galaxies (green): MEV redshift vs.  $R_{\\rm tot}$.  {\\it Right\npanel:} QSOs: MEV redshift vs.  $R_{\\rm tot}$.\n\\label{fig-CW_class}}\n\\end{figure*}\n\nRedshifts are given as Maximum-Likelihood values (the peak of the\nPDF), or as Minimum-Error-Variance values (the expectation value of\nthe PDF). MEV redshifts have smaller true errors, but are only given\nwhen the width of the PDF is lower than $\\sigma_z\/(1+z)< 0.125$. If\nPDFs are bimodal with modes of sufficiently small width, then both\nvalues are given with the preferred (larger-integral) mode providing\nthe primary redshift. Our team uses only MEV redshifts (with column\nname `mc\\_z') for their analyses.\n\nThe galaxy sample with MEV redshifts is $>90$\\% complete at all\nredshifts for $R_{\\rm ap}<23$. Near $z\\sim 1$, the MEV redshifts are\nthis complete even at $R_{\\rm ap}=24$. Below this cut, increasing\nphoton noise drives an expansion of the width of the PDF. The error\nlimit for MEV redshifts then makes the completeness of galaxy samples\nwith MEV redshifts drop. The 50\\% completeness is reached at $R\\sim\n24$ to $25$ depending on redshift. These results have been determined\nfrom simulations and are detailed in W04.  Completeness maps are\nincluded in the data release and take the form of a 3-D map of\ncompleteness depending on aperture magnitude, redshift and restframe\n$U-V$ colour.\n\nTo date, the photo-z quality on the A901\/2 field has only been\ninvestigated with a comparison to spectroscopic redshifts at the\nbright end.  W04 reported results from a sample of 404 bright galaxies\nwith $R<20$ and $z=[0,0.3]$, 351 of which were on the A901\/2 field,\nand 249 of which were members of the A901\/2 cluster complex\n(\\S~\\ref{sec-2df}). The other 53 objects were observed by the 2dFGRS\non the CDFS and S11 fields\n\\citep{Colless2001}. There we found that 77\\% of the sample had\nphoto-z deviations from the true redshift\n$|\\delta_z\/(1+z)|<0.01$. Three objects (less than 1\\%) deviate by more\nthan 0.04 from the true redshift.\n\nCurrently, we do not have faint spectroscopic samples on the A901\/2\nfield, however a spectroscopic dataset from VVDS exists on the\nCOMBO-17 CDFS field. From a sample of 420 high-quality redshifts that\nare reasonably complete to $R_{\\rm ap}<23$, we find a $1\\sigma$\nscatter in $\\delta_z\/(1+z)$ of 0.018, but also a mean bias of\n$-0.011$. Furthermore, the faint CDFS data show $\\sim 5$\\% outliers\nwith deviations of more than 0.06 \\citep{HWB08}. From a collection of\nspectroscopic samples we modelled the overall 1$\\sigma$ redshift\nerrors at $R\\la 24$ and $z\\la 1$ in W04 as\n\\begin{equation}\n   \\sigma_z\/(1+z) \\approx 0.005 \\times \\sqrt{1+10^{0.6 (R_{\\rm ap}-20.5)}}  ~ .\n   \\label{dz_rel}\n\\end{equation}\nLater we use a variant of this approximation to estimate the\ncompleteness of photo-z based selection rules for cluster members.\n\nThe template fitting for galaxies produces three parameters,\ni.e. redshift as well as formal stellar age and dust reddening\nvalues. The age is encoded in a template number running from 0\n(youngest) to 59 (oldest), where we use the same PEGASE\n\\citep[see][for discussion of an earlier version of the model]{fioc97}\ntemplate grid as described in W04. The look back times to the onset of\nthe $\\tau =1$~Gyr exponential burst range from 50~Myr to 15~Gyr.\n\nRestframe properties are derived for all galaxies and QSOs as\ndescribed in W04. Table~\\ref{tab-rfvega} lists the restframe passbands\nwe calculate and gives conversion factors from Vega magnitudes to AB\nmagnitudes and to photon fluxes. The SED shape is defined by the\naperture photometry and the overall normalization is given by the\ntotal SExtractor photometry from the deep $R$-band. However, if a\ngalaxy has both a steep colour gradient {\\it and} a large aperture\ncorrection, then the restframe colours will be biased by the nuclear\nSED.  \n\n\n\n\n\\begin{table}\n\\caption{The restframe passbands and their characteristics. \n\\label{tab-rfvega} }\n\\begin{tabular}{ll|cc}\n\\hline \\hline\n  name  &  $\\lambda_\\mathrm{\\mathrm{cen}}$\/fwhm  & \n  mag of Vega  &  $F_\\mathrm{phot}$ of Vega \\\\\n     &  (nm)  &  (AB mags)  &  $(10^8~\\mathrm{phot\/m^2\/nm\/s})$  \\\\ \n\\noalign{\\smallskip} \\hline \\noalign{\\smallskip} \n(synthetic)     &  145\/10  & $+2.33$ & 0.447\\\\ \n(synthetic)     &  280\/40  & $+1.43$ & 0.529\\\\ \n\\noalign{\\smallskip} \\hline \\noalign{\\smallskip} \nJohnson $U$     &  365\/52  & $+0.65$ & 0.820\\\\ \nJohnson $B$     &  445\/101 & $-0.13$ & 1.407\\\\ \nJohnson $V$     &  550\/83  & $+0.00$ & 1.012\\\\ \n\\noalign{\\smallskip} \\hline \\noalign{\\smallskip} \nSDSS $u$        &  358\/56  & $+0.84$ & 0.704\\\\ \nSDSS $g$        &  473\/127 & $-0.11$ & 1.305\\\\\nSDSS $r$        &  620\/115 & $+0.14$ & 0.787\\\\ \n\\hline\n\\end{tabular}\n\\end{table}\n\nThe column `ApD\\_Rmag' contains the magnitude difference between the\ntotal object photometry and the point-source calibrated,\nseeing-adaptive aperture photometry:\n\\begin{equation}\n   ApD\\_Rmag = Rmag - Ap\\_Rmag   ~ .\n\\end{equation}\nOn average, this value is by calibration zero for point sources, and\nbecomes more negative for more extended sources.\n\n\n\\subsection{Cross-correlation of STAGES and COMBO-17 catalogues}\\label{sec-xcorr}\n\nHaving created separate catalogues from the STAGES\n(\\S\\ref{sec-detect},\\S\\ref{sec-fitting}) and COMBO-17\n(\\S\\ref{sec-c17cat}) datasets, we next wish to create a combined,\nmaster catalogue.  In GEMS, this was accomplished by applying a\nnearest neighbour matching algorithm with a maximum matching radius of\n0\\farcs75.  The choice of maximum radius is governed by the resolution\nof the two datasets (HST: 0\\farcs1; COMBO-17: 0\\farcs75).\n\nFor STAGES we have however chosen to improve over this approach. For most\ngalaxies, their measured centres do not change if the input image is\nsmoothed. For example, if the HST image of a normal spiral or\nelliptical galaxy is convolved with a Gaussian function to match the\nground-based seeing, the centre estimated from the high-resolution (in\nthis case STAGES) and the low-resolution (here COMBO-17) images should\ncoincide. For distorted galaxies or mergers, this may no longer be the\ncase.  Instead, the brightest peak in the STAGES image, detected as\nthe object centre by SExtractor, may be relatively far from the centre\nin the COMBO-17 image.\n\n\\begin{figure*}\n\\centerline{\\psfig{file=data_paper3a_msk.ps,width=0.7\\columnwidth}\n\\hspace{0.1\\columnwidth}\\psfig{file=data_paper4a_msk_cmp.ps,width=0.75\\columnwidth}}\n\\caption{Cross-correlation of HST and COMBO-17 data.  {\\it Left:} The\ndistance to the nearest neighbour within a search radius of 5\\arcsec\nis plotted as a function of HST magnitude. At the faint end galaxies\nare matched to uncorrelated neighbours.  Resolving irregular\nstructures in the HST images results in detected galaxy centres being\nlocated farther from the COMBO-17 galaxy centre than a seeing\ndistance. Matching bright objects at large separations while removing\nrandom correlations at faint fluxes requires a cut as indicated by the\ndiagonal line.  Objects within the box ($V_{606}<25$ and\n$1\\arcsec<$ match distance $<2.5\\arcsec$) were inspected by eye. {\\it\nRight:} Ratio of matching distance and Kron size as a function of HST\nmagnitude. Values larger than $\\sim1$ imply a matching radius larger\nthan the object size in the HST image.  Sources with\n$R_{\\textrm{ap}}<24$ are shown as black symbols; objects with a match\nbelow the cut (diagonal line in left panel) are plotted in dark grey;\nthe remaining sources with a match within 5\\arcsec\\ are shown as light\ngrey symbols.\n}\\label{fig-xcorr2}\n\\end{figure*}\n\nIn order to maximise the number of good matches between STAGES and\nCOMBO-17, in particular at low redshift, i.e.\\ A901\/2 cluster\ndistance, we have devised the following scheme. For STAGES the average\nsource density corresponds to roughly two objects per 5\\arcsec-radius\ncircle. We cross-correlate the STAGES and COMBO-17 catalogues using a\nnearest neighbour matching algorithm as described above with a maximum\nmatching radius of 5\\arcsec. The resulting matches we plot in\nFig.~\\ref{fig-xcorr2} (left panel). In particular at faint\nmagnitudes many matches are found that appear unrelated. In\ncontrast, at brighter magnitudes several sources are correlated at\nradii much larger than the COMBO-17 seeing (0.75\\arcsec), which still\nidentify the same object. In Fig.~\\ref{fig-xcorr2} we also show a\nline that subdivides the plot into two regions:\n\\begin{equation}\nd_m=-0.3\\times\\left(V_{606}-29\\right),\n\\end{equation}\nwith the matching radius $d_m$ in arcsec and the STAGES SExtractor\nmagnitude $V_{606}$. Below the line, objects are considered to be\ncorrelated, while above they are not correlated. This division is\nempirically motivated by the requirement to match objects at the faint\nend out to the COMBO-17 resolution limit (0.5\\arcsec-1.0\\arcsec) while\nalso correlating sources at larger radii at the bright end. The slope\nof the curve was determined by visual inspection of the matches inside\nthe indicated box. Typically, the distance between centroids is\n$\\sim 0\\farcs1$ (Fig.~\\ref{fig-radhist}).\n\n\\begin{figure}\n\\centerline{\\psfig{file=data_paper5.eps,width=0.85\\columnwidth}}\n\\caption{Histogram of matching radii for all objects (outer\n  histogram) and $R_{\\rm ap}<24$ objects (inner histogram). The\n  typical angular separation between a COMBO-17 object with\n  $R_{\\textrm{ap}}<24$ and its HST counterpart is $\\sim 0.12$\\arcsec\n  $\\pm 0.08$\\arcsec.}\\label{fig-radhist}\n\\end{figure}\n\nAnother way of investigating this issue is by calculating whether the\nnearest matching neighbour falls within the area covered by the object\nin the STAGES image. If the projected COMBO-17 position is beyond\nthe optical extent of the source in STAGES, it is uncorrelated. From\nthe STAGES SExtractor data we estimate the `extent' of an object by\nits Kron size $K=r_\\textrm{K}\\times a$, from the Kron radius\n$r_\\textrm{K}$ and semi-major axis radius $a$. We limit the Kron\nsize to $K>0.75$\\arcsec. A ratio of $d_m\/K \\ga 1$ indicates that\nthe matched COMBO-17 source lies outside the region covered by the\nobject in the STAGES image. In Fig.~\\ref{fig-xcorr2} (right panel)\nwe overplot in grey all sources that were assigned a partner from the\nnearest neighbour matching. This provides further evidence for the\nimproved quality of our new cross-correlation method.\n\nIn summary, the combined catalogue contains 88\\,879 sources. Of these,\n$\\sim6\\,577$ objects with a COMBO-17 ID are not within the region\ncovered by the STAGES HST mosaic ($\\sim 1\\,664$ of these have\n$R_{\\textrm{ap}}<24$).  Moreover, $\\sim 1\\,271$ STAGES detections are\noutside the COMBO-17 observation footprint.\\footnote{The observation\nfootprint for both STAGES and COMBO-17 is rather difficult to\ndetermine. Therefore, we provide only approximate numbers good to\n$\\sim50$ objects. A more elaborate scheme than the one used to produce\nthese numbers is well beyond the scope of this paper.} Inside the\nregion covered by both surveys, there are $\\sim 81\\,031$ sources. For\n50701 objects the method described above provides a match between\nCOMBO-17 and STAGES (15760 of these have $R_{\\textrm{ap}}<24$). $\\sim\n23\\,833$ sources detected in STAGES do not have counterparts in\nCOMBO-17; $\\sim 6\\,497$ sources from the COMBO-17 catalogue are not\nmatched to STAGES detections. Out of these, only $\\sim 79$ objects\nhave \\mbox{$R_{\\textrm{ap}}<24$}. We therefore emphasize that for\nour science sample of COMBO-17 objects, defined as having\n$R_{\\textrm{ap}}<24$, 99.9\\% have a STAGES counterpart.  The majority\nof failures result from confusion by neighbouring objects or simply\nnon-detections.\n\n\n\\subsection{Selection of an A901\/2 cluster sample}\n\nWe wish to define a `cluster' galaxy sample of galaxies belonging to\nthe A901\/2 complex for various follow-up studies of our team that are\nin progress. These studies may have different requirements for the\n{\\em completeness} of cluster members and the {\\em contamination} by\nfield galaxies. We therefore quantified how these two key values vary\nwith both magnitude and width of the redshift interval in order to\ninform our choice of definition.\n\nThe photo-z distribution of cluster galaxies was assumed to follow a\nGaussian with a width given by the photo-z scatter in\nEquation~\\ref{dz_rel}.  The distribution of field galaxies was assumed to\nbe consistent with the average galaxy counts $n(z,R)$ outside the\ncluster and varies smoothly with redshift and magnitude assuming no\nstructure in the field. Samples were then defined by redshift\nintervals $z_{\\rm phot}=[0.17-\\Delta z, 0.17+\\Delta z]$, where the\nhalf-width $\\Delta z$ was allowed to vary with the\nmagnitude.\\footnote{We use $z_{\\rm phot}=0.17$ for the mean cluster\nredshift here rather than the spectroscopically confirmed $z_{\\rm\nspec}\\sim0.165$ due to the known bias discussed in\n\\S\\ref{sec-c17cat}.} We calculated completeness and contamination at\nall magnitude points simply using the counts of our smooth models.\n\nWe found that as long as the half-width in redshift is not much larger\nthan a couple of Gaussian FWHMs, the contamination changes only\nlittle.  The ratio of selected cluster to field galaxies is almost\ninvariant as shrinking widths cut into numbers for both origins. Only\nenlarging the width significantly over that of the Gaussian increases\ncontamination by field galaxies. On the contrary, such large widths do\nnot affect the completeness of the cluster sample much, while\nshrinking the width too far eats into the true cluster distribution\nand reduces completeness of the cluster sample.\n\nFor our purposes, we compromised on a photo-z width such that the\ncompleteness is $>90$\\% at any magnitude, just before further widening\nstarts to increase the contamination above its mag-dependent minimum\n(see Fig.~\\ref{fig-CW_clus} and Fig.~\\ref{fig-CW_comcon}, left\npanel). For this we chose a half-width of\n\n\\begin{equation}\\label{eqn-cluster}\n   \\Delta z (R) = \\sqrt{0.015^2+0.0096525^2 \n                (1+10^{0.6 (R_{\\rm tot}-20.5)})  }  ~ .\n\\end{equation}\n\nThis equation defines a half-width that is limited to 0.015 at the\nbright end and expands as a constant multiple of the estimated photo-z\nerror at the faint end. The floor of the half-width is motivated by\nincluding the entire cluster member sample previously studied by\nWMG05. The completeness of this selection converges to nearly 100\\%\nfor bright galaxies, as a result of intentionally including the WGM05\nsample entirely.\n\n\n\\begin{figure}\n\\centerline{\\includegraphics*[height=0.85\\columnwidth,angle=270]{CW_cluster.ps}}\n\\caption{MEV redshift estimate vs. total $R$-band magnitude. The\n'galaxy' sample is shown in green, while the sample of `cluster'\ngalaxies defined by Equation~\\ref{eqn-cluster} is shown in black.  The\nmagnitude-dependent redshift interval guarantees almost constant high\ncompleteness, while the field contamination increases towards faint\nlevels (Fig.~\\ref{fig-CW_comcon}).  We note that at faint\nmagnitudes there is an apparent asymmetry towards lower redshift at\nfaint magnitudes within the cluster sample.  The photometric redshifts\nmay be skewed by systematic effects but the average $-0.02$ offset at\n$R_{\\rm tot}\\sim22.5$ is within the $1\\sigma$ error envelope.\n\\label{fig-CW_clus}}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics*[height=\\columnwidth,angle=270]{CW_complcontam.ps}\n\\caption{{\\it Left panel:} Completeness of the cluster sample defined\nin Fig.~\\ref{fig-CW_clus} and designed to provide high completeness at all \nmagnitudes.\n{\\it Right panel:} The field contamination of the cluster sample \nincreases at faint levels due to photo-z dilution of the cluster. \nNarrowing the selected redshift interval would not reduce the \ncontamination. Contamination rates are estimated to be\n$(10,20,30,50,70)\\%$ at $R_{\\rm ap}=(20.3,21.65,22.3,23.2,24.0)$.\n\\label{fig-CW_comcon}}\n\\end{figure}\n\nThe right panel of Fig.~\\ref{fig-CW_comcon} shows that the\ndifferential contamination increases rapidly towards faint magnitudes,\nsimply as a result of the photo-z error-driven dilution of the cluster\nsample. Here, contamination means the fraction of galaxies that are\nfield members, as measured in a bin centred on the given magnitude\nwith width 0.1 mag. Contamination at a given apparent magnitude\ntranslates into contamination at a resulting luminosity at the cluster\ndistance (except that scatter in the aperture correction smears out\nthe contamination relation slightly).\n\nAlready at $R_{\\it ap}=23.2$ the sample contains as many cluster as\nfield members. This corresponds to $M_V \\approx -16.5$ for the average\ngalaxy, but scatters around that due to aperture corrections.  As we\nprobe fainter this selection adds more field galaxies than cluster\nmembers. Follow-up studies can now determine an individual magnitude\nor luminosity limit given their maximum tolerance for field\ncontamination. For example, WGM05 selected cluster galaxies at $M_V -\n< -17.775$ ($M_V < -17$ for their adopted cosmology with $H_0=100$ km\ns$^{-1}$ Mpc$^{-1}$) for an earlier study of the A901\/2 system in\norder to keep the contamination at the faint end below 20\\%.\n\nThe cluster sample thus obtained covers quite a range of photo-z\nvalues at the faint end, and restframe properties are derived assuming\nthese redshifts to be correct. However, if we assume a priori that an\nobject is at the redshift of the cluster, then we may want to know\nthese properties assuming a fixed cluster redshift of\n$z=0.167$. Hence, the SED fits and restframe luminosities are\nrecalculated for this redshift and reported in additional columns of\nthe STAGES catalogue in Table~\\ref{tabcolumns} (with \\mbox{`\\_cl' }\nsuffix indicating cluster redshift). Of course, if the a-priori\nassumption is to believe the redshifts as derived, then the original\nset of columns for which we have derived the values is relevant.\n\n\n\\section{Further multiwavelength data and derived\n  quantities}\\label{sec-multi}\n\nIn this section we describe further multiwavelength data for the\nA901\/2 region taken with other facilities (Fig.~\\ref{fig-field}).  We\nalso present several resulting derived quantities (stellar masses and\nstar formation rates) that appear as entries in the STAGES master\ncatalogue.\n\n\\subsection{Spitzer}\\label{sec-mir}\n\nSpitzer observed a $1\\degr \\times 0{\\fdg}5$ field around the A901\/2\nsystem in December 2004 and June 2005 as part of Spitzer GO-3294 (PI:\nBell).  The MIPS 24{\\micron} data were taken in slow scan-map mode,\nwith individual exposures of 10\\,s.  We reduced the individual image\nframes using a custom data-analysis tool (DAT) developed by the GTOs\n\\citep{gordon05}.  The reduced images were corrected for geometric\ndistortion and combined to form full mosaics; the reduction which we\ncurrently use does not mask out asteroids and other transients in the\nmosaicing.\\footnote{This only minimally affects our analyses because we\nmatch the IR detections to optical positions, and most of the bright\nasteroids are outside the COMBO-17 field.}  The final mosaic has a\npixel scale of $1\\farcs25$~pixel$^{-1}$ and an image PSF FWHM of\n$\\simeq 6$\\arcsec.  Source detection and photometry were performed\nusing techniques described in \\citet{papovich04}; based on the\nanalysis in that work, we estimate that our source detection is 80\\%\ncomplete at 97~$\\mu$Jy\\footnote{We note that for previous papers we\nused the catalogue to lower flux limits, down to 3$\\sigma$;\naccordingly, we have included such lower-significance (and more\ncontaminated) matches in the catalogue.}  for a total exposure of\n$\\sim 1400$\\,s\\,pix$^{-1}$.  By detecting artificially-inserted\nsources in the A901 24 image, we estimated the completeness of the\nA901 24 $\\mu$m catalog. The completeness is 80\\%, 50\\% and 30\\% at 5,\n4 and 3$\\sigma$, respectively.\n\n\nNote that there is a very bright star at 24{\\micron} near the centre\nof the field at coordinates $(\\alpha,\\delta)_{\\rm J2000}=(09^h56^m32\\fs 4,\n-10\\degr 01 \\arcmin 15\\arcsec)$ (see \\S\\ref{sec-mira} for\ndetails of this object). In our analysis of the 24{\\micron} data we\ndiscard all detections less than 4$'$ from this position in order to\nminimise contamination from spurious detections and problems with the\nbackground level in the wings of this bright star.  It is to be noted\nthat there are a number of spurious detections in the wings of the\nvery brightest sources; while we endeavoured to minimise the incidence\nof these sources, they are difficult to completely eradicate without\nlosing substantial numbers of real sources at the flux limit of the\ndata.\n\nTo interpret the observed 24{\\micron} emission, we must match the\n24{\\micron} sources to galaxies for which we have redshift estimates\nfrom COMBO-17.  We adopt a 1\\arcsec matching radius.  In the areas of\nthe A901\/2 field where there is overlap between the COMBO-17 redshift\ndata and the full-depth MIPS mosaic, there are a total of 3506(5545)\n24{\\micron} sources with fluxes in excess of 97(58)$\\mu$Jy.  Roughly\n62\\% of the 24{\\micron} sources with fluxes $> 58 \\mu$Jy are detected\nby COMBO-17 in at least the deep $R$-band, with $R \\la 26$.  Some 50\\%\nof the 24{\\micron} sources have bright $R_{\\rm tot} < 24$ and have\nphotometric redshift $z<1$; these 50\\% of sources contain nearly 60\\%\nof the total 24{\\micron} flux in objects brighter than 58$\\mu$Jy.\nSources fainter than $R \\ga 24$ contain the rest of the $f_{24} >\n58\\mu$Jy 24{\\micron} sources; investigation of COMBO-17 lower\nconfidence photometric redshifts, their optical colours, and results\nfrom other studies lends weight to the argument that essentially all\nof these sources are at $z>0.8$, with the bulk lying at $z > 1$\n(e.g. \\citealt{lefloch2004}, \\citealt{papovich04}; see\n\\citealt{lefloch2005} for a further discussion of the completeness of\nredshift information in the CDFS COMBO-17 data).\n\nObservations with IRAC \\citep[Infrared Array Camera;][]{Fazio2004} at\n3.6, 4.5, 5.8 and 8.0{\\micron} were also taken as part of this Spitzer\ncampaign: those data are not discussed further here, and will be\ndescribed in full in a future publication.\n\n\n\\subsection{Star formation rates}\n\nWe provide estimates of star formation rate, determined using a\ncombination of 24{\\micron} data (to probe the obscured star formation)\nand COMBO-17 derived rest-frame 2800{\\AA} luminosities (to probe\nunobscured star formation).  Ideally, we would have a measure of the\ntotal thermal IR flux from 8--1000{\\micron}; instead, we have an\nestimate of IR luminosity at one wavelength, 24{\\micron},\ncorresponding to rest-frame 22--12{\\micron} at the redshifts of\ninterest $z=0.1-1$.  Local IR-luminous galaxies show a tight\ncorrelation between rest-frame 12--15{\\micron} luminosity and total IR\nluminosity \\citep[e.g.,][]{spi95,cha01,rou01,papovich02}, with a\nscatter of $\\sim 0.15$ dex.\\footnote{Star-forming regions in local\ngalaxies appear to follow a slightly non-linear relation between\nrest-frame 24{\\micron} emission and SFR, with SFR $\\propto L_{24\\mu\nm}^{0.9}$ \\citep{calzetti07}, although note that this calibration is\nbetween 24{\\micron} emission and SFR (not total IR luminosity).}\nFollowing \\citet{papovich02}, we choose to construct total IR\nluminosity from the observed-frame 24{\\micron} data.  We use the Sbc\ntemplate from the \\citet{dev99} SED library to translate\nobserved-frame 24{\\micron} flux into the 8--1000{\\micron} total IR\nluminosity.\\footnote{Total 8--1000{\\micron} IR luminosities are $\\sim\n0.3$ dex higher than the 42.5--122.5{\\micron} luminosities defined by\n\\citet{Helou1988}, with an obvious dust temperature\ndependence.}  The IR luminosity uncertainties are primarily\nsystematic.  Firstly, there is a natural diversity of IR spectral\nshapes at a given galaxy IR luminosity, stellar mass, etc.; one can\ncrudely estimate the scale of this uncertainty by using the full range\nof templates from \\citet{dev99}, or by using templates from, e.g.,\n\\citet{dale01} instead.  This uncertainty is $\\la$0.3\\,dex (this\nagrees roughly with the scatter seen between 24{\\micron} luminosity\nand SFR seen in \\citealp{calzetti07}).  Secondly, it is possible that\na significant fraction of $0.1<z<1.0$ galaxies have IR spectral energy\ndistributions not represented in the local Universe: while it is\nimpossible to quantify this error until the advent of Herschel Space\nTelescope, current results suggest that the bulk of intermediate--high\nredshift galaxies have IR spectra similar to galaxies in the local\nuniverse \\citep{appleton04,elbaz05,yan05,zheng07}.\n\nWe estimate SFRs using the combined directly-observed UV light from young \nstars and the dust-reprocessed IR emission of the sample galaxies\n\\citep[e.g.,][]{fluxrat}.  Following \\cite{bell05}, \nwe estimate SFR $\\psi$ using a calibration \nderived from PEGASE assuming a 100\\,Myr-old stellar population \nwith constant SFR and a \\citet{chabrier03} IMF: \n\\begin{equation}\n\\psi \/ ({\\rm M_{\\sun}\\,yr^{-1}}) = 9.8 \\times 10^{-11} \\times\n       (L_{\\rm IR} + 2.2L_{\\rm UV}),  \\label{eqn:sfr}\n\\end{equation}\nwhere $L_{\\rm IR}$ is the total IR luminosity (as estimated above) and\n$L_{\\rm UV} = 1.5 \\nu l_{\\nu,2800}$ is a rough estimate of the total\nintegrated 1216{\\AA}--3000{\\AA} UV luminosity, derived using the\n2800{\\AA} rest-frame luminosity from COMBO-17 $l_{\\nu,2800}$.  The\nfactor of 1.5 in the 2800{\\AA}-to-total UV conversion accounts for the\nUV spectral shape of a 100 Myr-old population with constant SFR, and\nthe UV flux is multiplied by a factor of 2.2 before being added to the\nIR luminosity to account for light emitted longwards of 3000{\\AA} and\nshortwards of 1216{\\AA} by the unobscured young stars.  This SFR\ncalibration is derived using identical assumptions to \\citet{k98}, and\nthe calibration is consistent with his to within 30\\% once different\nIMFs are accounted for.  Uncertainties in these SFR estimates are a\nfactor of two or more in a galaxy-by-galaxy sense, and systematic\nuncertainty in the overall SFR scale is likely to be less than a\nfactor of two \\citep[see, e.g.,][for further discussion of\nuncertainties]{bellsfr,bell05}.  The adopted calibration assumes\nthat the infrared luminosity traces the emission from young stars\nonly; contributions from potential AGN can be identified and excluded\nby cross-matching with the X-ray and optical data as in\n\\citet{gilmour2007} and \\citet{gallazzi08}.\n\nAgain, for galaxies in the `cluster' sample, we present also SFR\nestimates assuming that the galaxies are at the cluster redshift with\nthe suffix `\\_cl' in added to the column name.  \n\n\n\n\n\\subsection{Stellar Masses}\n\n\\citet{borch06} estimated the stellar masses of galaxies in COMBO-17\nusing the 17-passband photometry in conjunction with a template\nlibrary derived using the PEGASE stellar population model.  The\nnon-evolving template stellar populations had an age\/metallicity\ncombination equivalent to roughly solar metallicity and $\\sim 6$\\,Gyr\nsince the start of star formation.\\footnote{Local comparison samples,\ne.g., the SDSS, typically adopt template combinations with `older'\nages, potentially leading to offsets between the overall mass scale of\nour masses and local masses at a given rest-frame colour.  We make no\nattempt to resolve this issue here, and refer the interested reader to\n\\citet{belldejong} and \\citet{Bell07} for further discussion of this\nissue.}  \\citet{borch06} adopted a \\citet{kroupa93} stellar IMF; the\nuse of a \\citet{kroupa01} or \\citet{chabrier03} IMF would have yielded\nthe same stellar masses to within $\\sim 10$\\%.  Such masses are\nquantitatively consistent with those derived using a simple\ncolour-stellar M\/L relation \\citep{bell03}, and comparison of stellar\nand dynamical masses for a few $z \\sim 1$ early-type galaxies yielded\nconsistent results to within their combined errors (see\n\\citealt{borch06} for more details).\n\nThere are some galaxies for which the 17-band classification failed to\nfind a satisfactory solution (2\\% of the galaxies with redshift\nestimates); we choose to adopt in these cases a rest-frame\ncolour-derived stellar mass, using rest-frame $B$ and $V$ absolute\nmagnitudes\/luminosities, and a $V$-band absolute magnitude of the Sun\nof 4.82:\n\\begin{equation}\n\\log_{10} M_*\/M_{\\sun} = -0.728 + 1.305(B-V) + \\log_{10} L_V\/L_{\\sun}.\n\\end{equation}\n\nAs with restframe photometric properties, we also present estimates of\nstellar mass assuming that the galaxy is at the cluster redshift\n(denoted in the catalogues by the suffix `\\_cl' in the column names).\nRandom stellar mass errors are estimated to be $\\sim 0.1$\\,dex on a\ngalaxy-by-galaxy basis in most cases, and systematic errors in the\nstellar masses (setting the overall mass scale and its redshift\nevolution) were argued to be at the 0.1\\,dex level for galaxies\nwithout ongoing or recent major starbursts; for galaxies with strong\nbursts, masses could be overestimated by $\\la 0.5$\\,dex.\n\nFinally, we note potential aperture effects on stellar masses and\nSEDs for some objects. The colours are estimated within an aperture\nbut are normalized by the total light in the deep $R$-band image\nalone.  For small objects or particularly large objects without colour\ngradients this has no consequence. But if large size, low\nconcentration and strong colour gradients are combined, the total SED\nwill deviate from the aperture SED underlying the $M\/L$ estimate.  In\na companion paper studying properties of spiral galaxies in the\nsupercluster, Wolf et al. (MNRAS, accepted) have investigated this effect by\nexamining the total colours across a wide parameter space in the\nsample.  In most cases the aperture values are similar to the total\nones, but they identify an issue for morphologically-classified spiral\ngalaxies in the supercluster and eliminate the highest-mass regime\nwith $\\log M_*\/M_{\\sun} > 11$ from their study.\n\n\n\\subsection{GALEX}\n\nThe Abell 901\/902 field was observed by GALEX in the far-UV ($f,\n\\lambda_{\\rm eff}\\sim 1528$\\AA ) and near-UV ($n, \\lambda_{\\rm\neff}\\sim 2271$\\AA) bands.\\footnote{Unlike all other datasets detailed\nhere, the GALEX observations were not led by members of the STAGES\nteam.  We list the publicly archived data products here for\ncompleteness.} Individual observations (or single orbit `visits')\nbetween the dates 12 February 2005 and 25 February 2007 were coadded\nby the GALEX pipeline \\citep[GR4 version][]{Morrissey2007} to produce\nimages with net exposure times of 57.18 ks in $n$ (47 visits) and\n50.19 ks in $f$ (40 visits).  The GALEX field of view in both bands is\na 0.6\\degr radius circle, and the average centre of the visits (the\nGALEX field centre) is $(\\alpha,\\delta)_{\\rm J2000} = (9^{\\rm\nh}56^{\\rm m}20\\fs7, -10\\degr6\\arcmin21\\farcs6)$. The GALEX PSF near\nthe field centre has $\\sim 4.2$\\arcsec\\ FWHM at $f$ and $\\sim\n5.3$\\arcsec\\ FWHM at $n$, both of which increase with distance from\nthe field centre (variations in the PSF that are not a function of\ndistance from the field centre are smoothed out by the distribution of\nroll angles of the visits). The astrometric accuracy is $\\sim\n0.7$\\arcsec, and $>97$\\% of catalogued source positions are within\n2\\arcsec~of their true positions. The photometric calibration is stable\nto $0.02$ mag in $n$ and $0.045$ mag in $f$ \\citep{Morrissey2007}.\n\nSource detection and photometry is via the GALEX pipeline code, which\nemploys a version of SExtractor \\citep{bertin96} modified for use with\nlow-background images. Magnitudes are measured both in fixed circular\napertures and in automatic Kron elliptical apertures, and in isophotal\napertures. The 5$\\sigma$ point-source sensitivities in the Abell\n901\/902 field are $f \\sim 24.7$ mag (AB) and $n \\sim 25.0$ mag (AB),\nthough there are spatial variations across field, especially a\nslightly decreasing sensitivity towards the edge of the field.  At\nthese levels source confusion in the $n$ band becomes an issue, and\nthe $n$ band fluxes of faint objects ($n\\ga 23$ mag) are likely to\nbe overestimated.  GALEX data products include intensity, background,\nand relative response (i.e., effective exposure time) maps in both\nbands as well as source catalogues in both bands and a band-merged\nsource catalogue.\n\n\n\\subsection{2dF spectroscopy}\\label{sec-2df}\n\nSpectra of cluster galaxies were obtained using the 2dF\ninstrument on the AAT in March 2002 and March 2003.  A total of 86\ngalaxies were observed using the 1200B grating (spanning the observed\nwavelength range 4000--5100 \\AA) in a single fibre configuration during\nthe 2002 run.  Three fibre configurations using the lower resolution\n600V grating (spanning 3800--5800 \\AA) were observed during the 2003\nrun: fibres were placed on 368 objects, with 47 repeated from 2002.\nThe primary selection function assigned higher priority to those\ngalaxies selected by photometric redshift to be within the\nsupercluster redshift slice and having $R<20$, with additional fibres\nbeing allocated to secondary targets (including fainter galaxies and a\nsmall number of white dwarfs and QSOs) when available.  Data reduction\nwas performed with the standard {\\tt 2dfdr} (v2.3) pipeline package.\n\nIn total, spectra were obtained for 407 unique objects.  Redshifts\nwere determined by two independent means: firstly by manual line\nprofile fitting of the Ca H and K features in absorption and secondly\nby cross-correlation with template spectra using the XCSAO task within\nIRAF (Kurtz \\& Mink 1998).  Comparison of the two measurements showed\nno cause for concern, with $\\Delta_z=0.00149\\pm0.00006$. After\neliminating non-galaxy and poor quality spectra, we have redshifts for\n353 galaxies in total.  \n\nThe 2dF spectroscopic data have previously been used to quantify the\nreliability of the COMBO-17 redshifts in W04 (see also \\S\\ref{sec-c17}),\nto verify cluster membership for the matched X-ray point sources\n\\citep{gilmour2007}, and to create composite spectra for three\nphotometric classes of cluster galaxies in WGM05.  A dynamical\nanalysis of the the clusters using the 2dF redshifts will be presented\nin Gray et al. (in prep.).\n\n\\subsection{XMM-Newton} \\label{sec-xray}\n\nX-ray data for the A901\/2 region is desirous both to detect\npoint-source emission from cluster members (star-formation or AGN) and\nthe extended intracluster medium (ICM).  A 90~ks XMM image of the\nA901\/2 field was taken on May 6\/7 2003 using the three EPIC cameras\n(MOS1, MOS2 and PN) and a thin filter, under program 14817 (PI: Gray).\nThe level 1 data were taken from the supplied pipeline products, and\nreduced with SAS v5.4 and the calibration files available in May 2003.\nFinal exposure times were $\\sim67$ ks for MOS and $\\sim$61 ks for PN\nfollowing the removal of time intervals suffering from soft proton\nflares.  Four energy bands were used: 0.5-2 keV (soft band), 2-4.5 keV\n(medium band), 4.5-7.5 keV (hard band) and 0.5-7.5 keV (full band).\n\nThe creation of the point-source catalogue using wavelet detection\nmethods is described in detail elsewhere \\citep{gilmour2007}.  A total\nof 139 significant sources were found.  The presence of an X-ray\nluminous Type-I AGN near the centre of A901a (see\nAppendix~\\ref{sec-agnnote}) complicated the detection of the underlying\nextended cluster emission.  A maximum-likelihood technique was used to\nmatch this catalogue to COMBO-17 resulting in 66 secure counterparts\nwith photometric redshifts. \\cite{gilmour2007} used these data to\nexamine the local environments of the cluster AGN and their host\nproperties.\n\nTo isolate the remaining extended emission coming from the clusters, a\nseparate conservative point-source catalogue was constructed.  Care\nwas taken to remove both the cosmic background and spatial variations in\nthe non-cosmic background.  The background subtracted images were\nweighted by appropriate energy conversion factors to create flux\nimages for each detector.  These flux images were masked and summed\ntogether to create merged background-subtracted images in each band.\n\nPoint source regions were removed and replaced with the local\nbackground value selected randomly from a source free area within 10 pixels\n(or 20 pixels if there were not enough background pixels within the\nsmaller radius). Smoothed images were created in each band using a\nGaussian kernel of radius 4 pixels.  Maps of the extended emission and\nan examination of the global X-ray properties of the clusters will be\npresented in Gray et al. (in prep.).\n\n\n\n\n\n\\subsection{GMRT}\n\n\nThe A901\/2 field was observed on 2007 March 25th and 26th March with\nthe Giant Metrewave Radio Telescope (GMRT, see\n\\citealt{Ananthakrishnan2005} for further details).  The field was\ncentred at $(\\alpha,\\delta)_{\\rm J2000}=(09^{\\rm h} 56^{\\rm m} 17^{\\rm\ns}, -10\\degr 01\\arcmin 28\\arcsec)$ and observed at 610 and 1280~MHz on\nrespective nights. The GMRT is an interferometer, consisting of thirty\nantennas, each 45~m in diameter. The bright sources 3C147 and 3C286\nwere observed at the start and end of each observing session, in order\nto set the flux density scale. During the observations a nearby\ncompact source 0943$-$083 was observed for about 4 minutes at roughly\n30 minutes intervals, to monitor and correct any antenna-based\namplitude and phase variations.\n\nThe total integration time on the field was $\\sim$6.5 hours at\neach frequency. The observations covered two 16~MHz sidebands,\npositioned above and below the central frequency. Each sideband was\nobserved with 128 narrow channels, in order to allow narrow band\ninterference to be identified and efficiently removed. The observed\nvisibility data were edited and calibrated using standard tasks with\nthe AIPS package, and then groups of ten adjacent channels were\naveraged together, with some end channels discarded. This reduced the\nvolume of the visibility data, whilst retaining enough channels so\nthat chromatic aberration is not a problem \\citep[e.g., see][for further\ndetails of GMRT analysis]{Garn2007}. Given the relatively large field\nof view of the GMRT compared with its resolution, imaging in AIPS\nrequires several `facets' to be imaged simultaneously, and then be\ncombined. Preliminary imaging results, after several\niterations of self-calibration, have produced images with resolutions\nof about 5\\arcsec and 2\\farcs5 at 610 and 1280-MHz respectively, with\nr.m.s.\\ noises of approximately 25 and 20 $\\mu$Jy~beam$^{-1}$ in the\ncentre of the fields, before correction for the primary beam of the\nGMRT. The primary beam -- i.e.\\ the decreasing sensitivity away from\nthe field centres due to sensitivity of individual 45-m antennas -- is\napproximately Gaussian, with a half-power beam width (HPBW) of\napproximately 44\\arcmin and 26\\arcmin at 610 and 1280-MHz\nrespectively. These images are among the deepest images made at these\nfrequencies with the GMRT.  Further analysis and the source catalogue\nwill be presented in Green et al.(in prep.).\n\n\\subsection{Simulations and mock galaxy catalogues}\\label{sec-mocks}\n\nIn order to facilitate the interpretation of the observational results\nand to study the physical processes of galaxy evolution, N-body,\nhydrodynamic, and semi-analytic simulations that closely mimic the\nA901\/2 system are being produced (van Kampen et al., in prep.).  We\nconstrain initial conditions using the method of \\citet{hoffman1991}\nto take into account the gross properties of A901a, A901b, A902, the\nSW group, and the neighbouring clusters A868 and A907 (outside the\nobserved field).  The simulations produce a range of mock large-scale\nstructures to test three basic formation scenarios: a 'stationary'\ncase, where A901(a,b) and A902 will not merge within a Hubble time,\nand a pre- as well as a post-merger scenario.  When the likelihood of\neach scenario is understood, one can further test the models for the\ndetailed physical processes known to be operating on galaxies in and\naround such clusters.\n\n\n\\section{Summary and Data Access}\\label{sec-summary}\n\n\n\n\nWe have presented the multiwavelength data available for the A901\/2\nsupercluster field as part of the STAGES survey: high-resolution HST\nimaging over a wide area, extensive photometric redshifts from\nCOMBO-17, and further multiwavelength observations from X-ray to\nradio.  These data have already been used to create a high resolution\nmass map of the system using weak gravitational lensing\n\\citep{heymans08}.  \nFurther work by the STAGES team to study galaxy evolution and\nenvironment is ongoing and includes the following:\n\n\n\\begin{itemize}\n\n\\item \\citet{gallazzi08} explore the amount of obscured star-formation\nas a function of environment in the A901\/2 supercluster and associated\nfield sample by combining the UV\/optical SED from COMBO-17 with the\nSpitzer 24$\\mu$m photometry in galaxies with $M_*>10^{10} M_{\\sun}$.\nResults indicate that while there is an overall suppression in the\nfraction of star-forming galaxies with density, the small amount of\nstar formation surviving the cluster environment is to a large extent\nobscured.\n\n\\item Wolf et al. (MNRAS, accepted) investigate the properties of optically\npassive spiral and dusty red galaxies in the supercluster and find\nthat the two samples are largely equivalent.  These galaxies form\nstars at a substantial rate that is only a factor of four times lower\nthan blue spirals at fixed mass, but their star formation is more\nobscured and has weak optical signatures. They constitute over half of\nthe star forming galaxies at masses above $\\log M_*\/M_{\\sun}=10$ and\nare thus a vital ingredient for understanding the overall picture of\nstar-formation quenching in cluster environments.\n\n\\item Marinova et al. (ApJ, submitted) identify and characterize bars in\nbright ($M_{V} \\le -18$) cluster galaxies through ellipse-fitting. The\nselection of moderately inclined disk galaxies via three commonly used\nmethods, visual classification, colour, and S\\'ersic\\ cuts, shows that\nthe latter two methods fail to pick up many red, bulge-dominated disk\ngalaxies in the clusters.  However, all three methods of disk\nselection yields a similar global optical bar fractions ($f_{\\rm\nbar-opt}\\sim0.3)$, averaged over all galaxy types.  When host galaxy\nproperties are considered, the optical bar fraction is found to be a\nstrong function of both the luminosity and morphological property\n(bulge-to-disk ratio) of the host galaxy, similar to trends recently\nreported in field galaxies.  Furthermore, results indicate that the\nglobal optical bar fraction for bright galaxies is not a strong\nfunction of local environment.\n\n\\item Heiderman et al. (in prep.) identify interacting galaxies in the\nsupercluster using quantitative analysis and visual classifications.\nTheir findings include that $4.9\\pm 1.3\\%$ of bright ($M_{V} \\le\n-18$), intermediate mass ($M_{*} \\ge 1 \\times 10^{9} M_{\\sun}$)\ngalaxies are interacting. The interacting galaxies are found to lie\noutside the cluster cores and to be concentrated in the region between\nthe cores and virial radii of the clusters.  Explanations for the\nobserved distribution include the large galaxy velocity dispersion in\nthe cluster cores and the possibility that the outer parts of the\nclusters are accreting groups, which are predicted to show a high\nprobability for mergers and strong interactions. The average star\nformation rate is enhanced only by a modest factor in interacting\ngalaxies compared to non-interacting galaxies, similar to conclusions\nreported in the field by \\citet{Jogee08}.  Interacting galaxies only\ncontribute $\\sim$~20\\% of the total SFR density in the A901\/902\nclusters.\n\n\n\\item Boehm et al. (in prep.) are utilizing the stability of the PSF on\nthe STAGES images for a morphological comparison between the hosts of\n~20 type-1 AGN and ~200 inactive galaxies at an average redshift\n$\\left < z \\right >\\sim0.7$. This analysis includes extensive simulations of the\nimpact of a bright optical nucleus on quantitative galaxy morphologies\nin terms of the CAS indices and Gini\/$M_{20}$ space. We find that the\nmajority of the hosts cover parameters typical for disk+bulge systems\nand mildly disturbed galaxies, while evidence for strong gravitational\ninteractions is scarce.\n\n\\item Bacon et al. (in prep.) are examining the higher order lensing\nproperties of the STAGES data. They construct a shapelets catalogue\n\\citep{refregier03} for the STAGES galaxies; this is then used to\nestimate the gravitational flexion \\citep{bacon06} at each galaxy\nposition. Galaxy--galaxy flexion is measured, leading to estimates of\nconcentration and mass for STAGES galaxies; constraints on cosmic\nflexion are also found, showing very good containment of systematic\neffects. The ability of flexion to improve convergence maps is also\ndiscussed.\n\n\\item Robaina et al. (in prep.) make use of a combined GEMS and STAGES\nsample of $0.4<z<0.8$ galaxies to find that interacting and merging\nclose pairs of massive galaxies ($>10^{10} M_{\\odot}$) show a modest\nenhancement of their star formation rate; in particular, less than\n15\\% of star formation at $0.4<z<0.8$ is triggered by major\ninteractions and mergers.\n\n\\item Barden et al. (in prep.) are exploring both the GEMS and STAGES\ndata sets to investigate the evolution of structural parameters of\ndisc galaxies as a function of luminosity and stellar mass over a wide\nrange of environments and morphologies. In the process, GALAPAGOS will\nbe extended to perform bulge\/disc decomposition.\n\n\\item McIntosh et al. (in prep.) are using both quantitative and\nqualitative morphologies to explore the morphological mix of red\nsequence galaxies as a function of stellar mass over the last seven\nbillion years from the combined STAGES + GEMS sample.\n\n\\end{itemize}\n\nIt is our intention that the data products described here should be\npublicly available for use by the wider community for those interested\nin the supercluster itself or for data-mining the entire survey\nvolume.  To that end, the reduced HST images (both tiles and\nindividual galaxy postage stamps) are available for download at the\nMultimission Archive at Space\nTelescope\\footnote{\\texttt{http:\/\/archive.stsci.edu}} (MAST).\nFurthermore, the complete STAGES catalogue described in this paper is\navailable from the STAGES\nwebsite,\\footnote{\\texttt{http:\/\/www.nottingham.ac.uk\/astro\/stages}}\nincluding all HST-derived parameters; GALFIT profile fitting results;\nCOMBO-17 photometry, SEDs and photometric redshifts; and stellar\nmasses and star-formation rates.  The multiwavelength data available\nthere includes the Spitzer\/MIPS 24\\micron\\ images and catalogue; the\nX-ray point source catalogue \\citep{gilmour2007} and the gravitational\nlensing mass maps \\citep{heymans08}.  GALEX data and catalogues are\navailable via MAST.  The X-ray maps, 2dF spectra and radio catalogue\nand mocks will be also be placed on the website with the publication\nof their associated papers, or may be made available upon request.\nTable~\\ref{tab-data} contains a summary of the available data\nproducts.\n\n\\begin{table*}\n\\caption{Description of all available A901\/902 data products.}\n\\begin{tabular}{lll}\n\\hline\nData product & Date of release & Reference \\\\\n\\hline\n\\hline\nHST F606W imaging, reduced:  tiles, thumbnails, colour jpegs & immediate &\nthis paper\\\\\nSTAGES master catalogue: SExtractor, GALFIT, COMBO-17, stellar masses,\nSFRs& immediate & this paper\\\\\nCOMBO-17 SEDS and completeness tables & immediate & this paper\\\\\nGALFIT profile fitting completeness from simulations & immediate & this paper\\\\\nSpitzer 24\\micron\\ imaging and catalogue & immediate & this paper\\\\\nHST-derived weak lensing mass map & immediate & Heymans et al 2008\\\\\nXMM point source catalogue & immediate & Gilmour et al 2007\\\\\nGALEX imaging and catalogues (from the GALEX archive) &  immediate & this paper\\\\\nX-ray imaging & on request &  Gray et al. (in prep.)\\\\\n2dF spectroscopy & on request  & Gray et al. (in prep.)\\\\\nGMRT catalogue & TBC & Green et al. (in prep.)\\\\\nconstrained simulations and mock galaxy catalogues & TBC & van Kampen et al. (in prep.)\\\\\n\\hline\n\\end{tabular}\\label{tab-data}\n\\end{table*}\n\n\\section{Acknowledgements}\nThe STAGES team would like to thank Hans-Walter Rix for his crucial\nsupport in bringing this project to fruition.  We also thank Alfonso\nArag\\'{o}n-Salamanca, Anna Gallazzi, Amanda Heiderman, Irina Marinova,\nand Aday Robaina for their work in exploiting the STAGES dataset.\nSupport for STAGES was provided by NASA through GO-10395 from STScI\noperated by AURA under NAS5-26555.  MEG and CW were supported by STFC\nAdvanced Fellowships.  CH acknowledges the support of a European\nCommission Programme 6th framework Marie Cure Outgoing International\nFellowship under contract MOIF-CT- 2006-21891. CYP was supported by\nthe NRC-HIA Plaskett Fellowship, and the STScI Institute\/Giacconi\nFellowship. EFB and KJ are grateful for support from the DFG's Emmy\nNoether Programme of the Deutsche Forschungsgemeinschaft, AB by the\nDLR (50 OR 0404), MB and EvK by the Austrian Science Foundation FWF\nunder grant P18416, SFS by the Spanish MEC grants AYA2005-09413-C02-02\nand the PAI of the Junta de Andaluc\u00b4a as research group FQM322, SJ by\nNASA under LTSA Grant NAG5-13063 and NSF under AST-0607748 and DHM by\nNASA under LTSA Grant NAG5-13102.\n\n\n\n\n\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{S_INT}Introduction}\n\nAt present it is clear that the lower critical dimension of\nEdwards-Anderson Ising spin glasses is at some point between $D=2$ and\n$D=3$ \\cite{BOOK}. By using different numerical techniques as for\ninstance Monte Carlo simulations, exact ground state calculations and\ndiagonalization of the transfer matrix, it is rather well established\nthat the two dimensional Ising spin glass presents a transition at\n$T=0$.  Furthermore, an experimental realization of a 2D spin glass\nalso points to a transition only at $T=0$~\\cite{dekker,mydosh}.\nHowever the values of the critical exponents are still a matter of\ncontroversy\\cite{BOOK,review,bhatt,kawashima,rieger,mydosh}\\footnote{Recently\nhas been pointed out by some authors~\\cite{SHIRAKURA} that the two\ndimensional Ising spin glass shows a phase transition at finite\ntemperature, nonetheless the scaling plots of these works are\ncompatible with a $T=0$ phase transition in agreement with all the\nprevious cited studies.}.\n\nIn a recent paper Lemke and Campbell \\cite{lemke} studied the $2D \\pm\nJ$ Ising spin glass with nearest neighbors interactions adding a\nferromagnetic interaction between\nnext-nearest-neighbors. Surprisingly, they found numerical evidence\nfor a spin glass transition at finite temperature simulating\nintermediate lattice sizes.\n\nIn order to gain more insight on the nature of the low temperature\nproperties of the model we have computed through extensive simulations\na variety of global quantities as well as staggered ones\n(i.e. observables defined in one of the two sub-lattices in which the\nsystem naturally divides), like spin glass and ferromagnetic\nsusceptibilities, specific heat, Binder cumulants, magnetizations and\nthe spin glass order parameter.\n\nIn particular we have studied how the originally stable staggered\nphase is destroyed by the spin glass interaction and finally we have\nstudied if the spin glass phase is stable in the thermodynamic limit.\n\nWe will show numerical evidences for two different crossovers: the\nfirst one from the staggered ferromagnetic phase to a spin glass\nphase; and the second one from the spin glass phase to a paramagnetic\nphase.\n\nThe plan of this paper is the following. In the next section we will\ndefine the model and describe limiting cases in the parameter\nspace. In the third section we will map the model to the Random Field\nIsing Model and we will obtain analytical evidences of both\ncrossovers (moreover we will compute the dependence of the first\ncrossover length on the parameters of the model). In sections fourth\nand fifth we will study numerically the model. Finally we will present\nour conclusions.\n\n\n\\section{\\label{S_MOD}The model}\n\nThe model we are considering is defined by the following Hamiltonian\non a square lattice with periodic boundary conditions\n\\be\nH= -\\sum_{<ij>} J_{ij} S_i S_j - K \\sum_{\\ll k l \\gg} S_k S_l \\ ,\n\\label{model}\n\\ee \nwhere $<ij>$ denotes sum over all the first nearest neighbors pairs\n(distance 1) and $\\ll k l\\gg$ denotes sum over all the second nearest\nneighbors pairs (distance $\\sqrt 2$).The couplings $J_{ij}$ are quenched\nvariables with $J_{ij}=\\pm \\lambda$ with probability $1\/2$. $\\lambda$\nand $K$ are positive constants whose ratio determines the relative\nstrength between spin glass and ferromagnetic interactions. In the\nrest of the paper we fix $K=1$. In order to fix the notation we also\ndefine the variance of the $J_{ij}$ distribution as $\\sigma^2_J \\equiv\n\\overline{ J_{ij}^2}=\\lambda^2$.\n\nWe next describe two limiting cases of the model (for $K=1$):\n\\begin{itemize}\n\n\\item For $\\lambda \\to \\infty$ the system is the $2D$ spin glass for\nwhich is known that $T_c=0$\n\\cite{BOOK,review,bhatt,kawashima,rieger,mydosh}.\n\n\\item When $\\lambda=0$ the whole lattice decouples into two independent\nsub-lattices (that we will denote hereafter as sub-lattices 1 and 2,\nthe black and white sub-lattices in a chess-board. \nEach sub-lattice is itself a two\ndimensional ferromagnetic Ising model and will have a phase transition\n(paramagnetic-ferromagnetic) just at the Onsager temperature: $T_c\n=2\/\\log(1+\\sqrt 2) \\simeq 2.269$. The order parameter of the\nphase transition is the so-called staggered magnetization: i.e. the\nmagnetization of one of the two sub-lattices. Obviously, the\nprobability distribution of the total magnetization of the whole\nlattice for low temperatures will take into account the four possible\ndifferent magnetizations of the two independent sub-lattices.\n\n\\end{itemize}\n\nSo initially we have, when $\\lambda=0$, ferromagnetic order (in\nboth sub-lattices), that  can be affected by the introduction of\nspin glass couplings. The effect of these couplings is to couple (by\nmeans of a spin glass interaction) both sub-lattices.\n\n\n\nIn the next section we will examine analytically the effect of the\nintroduction of random couplings (linking both sub-lattices) \nin the original stable (staggered) ferromagnetic order.\n\n\\section{\\label{A_RES}Analytic Results}\n\nWe will study in this section the analogy, suggested by \nLemke and Campbell~\\cite{lemke}, between the Hamiltonian defined by Eq.(\\ref{model})\nand the Random Field Ising Model (RFIM).\n\nWe can rewrite the original Hamiltonian Eq.(\\ref{model}) in the \nfollowing way\n\\be\n{\\cal H}= - \\sum_{<i_1 j_1>} \\sigma_{i_1}  \\sigma_{j_1} \n- \\sum_{<i_2 j_2>} \\tau_{i_2}  \\tau_{j_2}  \n- \\sum_{i_1} \\sum_{j_2(i_1)} J_{i_1 j_2} \\sigma_{i_1} \\tau_{j_2} \\ ,\n\\label{ham_equiv}\n\\ee\nwhere the indices $i_1, j_1$ ($i_2, j_2$) run over the sub-lattice 1\n(respectively 2), $<i_1 j_1>$ ($<i_2 j_2>$) \ndenotes sum to first nearest neighbors\npairs in the sub-lattice 1 (respectively 2)\nand $j_2(i_1)$ denotes the sum over the two nearest neighbors ($j_2$'s) of\nthe site $i_1$ (in the two positive directions from $i_1$) \nin the whole lattice. \nMoreover we have denoted the variables of the\nsub-lattice 1 (2) as $\\sigma$'s (respectively $\\tau$'s).\n\nNow we fix the temperature to a small value  and all the spins inside the\nsub-lattice 1 are fixed in the up state.  \nNext we will examine the cost in energy to\ndo a compact droplet (in the sub-lattice 1) with all its spins flipped\ndown,  with the spins of the sub-lattice 2 fixed to an\narbitrary configuration.\n\nThe Hamiltonian of the model with all the $\\tau$ spins\nfixed to an arbitrary configuration is, modulo a constant,\n\\begin{eqnarray}\n\\nonumber\n{\\cal H}[\\sigma | \\tau \\,\\,{\\rm fixed} ]&=& \n- \\sum_{<i_1 j_1>} \\sigma_{i_1}  \\sigma_{j_1}   \n- \\sum_{i_1} \\left[\\sum_{j_2(i_1)} J_{i_1 j_2} \\tau_{j_2} \\right] \n\\sigma_{i_1} \\\\\n   &\\equiv& - \\sum_{<i_1 j_1>} \\sigma_{i_1}  \\sigma_{j_1}   \n- \\sum_{i_1} h_{i_1} \\sigma_{i_1} \\ ,\n\\end{eqnarray}\ni.e. a Random Field Ising Model (RFIM) where the magnetic field \n has zero mean and variance:\n\\be\n\\overline{h_i h_j}= \\left\\{ \n                \\begin{tabular}{l l}\n                 $2 \\sigma^2_J$& if $i=j$ , \\\\\n                 $0$            & elsewhere.\n                \\end{tabular}                 \n                \\right.\n\\ee\n\nIn particular we can think that all the spins in the $\\tau$\nsub-lattice are fixed up.  It is easy to reproduce all the steps of\nthe Imry-Ma argument~\\cite{ma,NATTERMANN} (by turning on the temperature)\nto show that one can find large regions where it is favorable\nenergetically to flip all the spins inside the region and hence to\ndestroy the long range order of the sub-lattice 1 independently of the\nconfiguration of the sub-lattice 2 (and vice-versa).  \n\nThus we have\nshown that the ferromagnetic order (in either or in both sub-lattices)\nis unstable against an infinitesimal strength of the spin glass couplings:\ni.e. initially the sub-lattices 1 and 2 are fixed to up (staggered\nferromagnetic order) and we have found that for any configuration\n$\\tau$ the sub-lattice 1 disorders, and by redoing the same steps with\nsub-lattice 1 fixed (and disordered) we can see that the originally ordered\nsub-lattice 2 disorders too.\n\nMoreover, following Binder~\\cite{BINDER}, we should expect that there\nexists a crossover length $R_c$ such that for $R<R_c$ the staggered\nferromagnetic  order is stable but this order is unstable for  scales\n$R>R_c$. The analytical expression for a RFIM with ferromagnetic\ncoupling $K$ and uncorrelated magnetic field with variance\n$\\sigma_h^2$ is~\\cite{BINDER}\n\\begin{equation}\nR_c \\propto \\exp \\left[ C\\left(\\frac{K}{\\sigma_h}\\right)^2\\right]\\ ,\n\\end{equation}\nwhere $C$ is a constant (O(1)), and so for our particular model \n(where $K=1$ and $\\sigma_h^2=2 \\sigma_J^2$), we finally obtain\n\\begin{equation}\nR_c \\simeq \\exp \\left[ \\frac{C}{2 \\sigma^2_J}\\right]=\n\\exp \\left[ \\frac{C}{2 \\lambda^2}\\right] \\ .\n\\end{equation}\n\nObviously when $\\lambda$ is infinitesimally small (i.e. we put an\ninfinitesimal amount of spin glass disorder in the model) the\ncrossover ratio is exponentially large and ferromagnetic order will\nonly be destabilized in extremely large systems. Nevertheless, in the\nthermodynamic limit the spin glass disorder is always relevant.\n\nFinally, we can estimate that for $\\lambda=0.5$ (the value that we\nhave used in our numerical simulations presented in this work) $R_c\n\\simeq 7$, assuming that $C$ is just 1.\n\nAt this point for $L > R_c$, where $L$ is the linear size of the\nsystem, the picture is the following: both sub-lattices\nare broken in clusters (inside of them all the spins, in average, point\nin the same direction) of size less than $R_c$ interacting between\nthem. From the Imry-Ma argument it is clear that the\n``effective'' interaction between these clusters is short range\n(i.e. it could be very large but not infinite). Consequently, we have a two\ndimensional spin glass with short range interactions. This phase  can be\nthought as a frozen disordered phase (like the spin glass phase) where\nthe clusters play the role of the spin in the usual spin glass\nphase. \n\nBut we know that there exists no spin glass order at finite\ntemperature for short range spin glasses in two dimensions, and so we\nconclude that there must be a second crossover from the spin glass\nbehavior to paramagnetic behavior as the size of the system\nincreases. We need only a correlation length greater than the range of\nthe interaction between the clusters to return to the usual short\nrange spin glass in two dimensions that has no phase transition.  In\nthe next sections we will try to put this fact in more quantitative\ngrounds.\n\n\\section{\\label{N_RES}Numerical Simulations and Observables}\n\nWe have simulated, using the Metropolis algorithm, systems with linear sizes\nranging from $L=4$ to $L=48$ and averaging over 200 to 10000 samples\ndepending on the size. The largest lattice size simulated in reference \n\\cite{lemke} used in the computation of their Binder cumulant was $L=12$.\n\nIn all the runs we have used an annealing procedure from higher\ntemperatures to the lower ones in order to thermalize the system. In\nTable \\ref{table:stat} we report the statistics we have used. We have\nperformed in the annealing procedure for all the temperatures the same\nnumber of thermalization steps ($N_T$), that we have written in Table\n\\ref{table:stat}. \n\nFor a given temperature we run $N_T$ steps for\nthermalization and we\nmeasure during $2 N_T$ and then we lower the temperature and we repeat\nthe process always with the same $N_T$. \nWe will return to the issue of the thermalization time at\nthe end of this section.\n\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\n$L$       & $T$'s & samples & Termalization time   \\\\ \\hline\n4       & [1.5,4.0] & 10000 &10000     \\\\ \\hline\n6       & [1.5,4.0] & 10000 &10000     \\\\ \\hline\n8       & [1.5,4.0] & 4000 &30000   \\\\ \\hline\n16      & [1.5,3.0] & 1633  &30000   \\\\   \\hline\n24      & [1.5,3.0] & 700   &60000   \\\\   \\hline\n32      & [1.5,3.0] & 1576  &90000   \\\\   \\hline\n48      & [1.4,2.5] & 426(*)   &900000 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{ Description of our runs. The step in temperatures was $0.1$\nin all  runs. (*) means that for the $L=48$ lattice in the measure of\nthe staggered overlap we only have simulated 100 samples.}\n\\label{table:stat}\n\\end{table}\n\nIn the next paragraphs we will describe the observables that we have\nmeasured in our numerical simulations.\n\nWe have measured the global ($m$) and staggered ($m_s$) magnetizations :\n\\be\nm\\equiv \\frac{1}{L^2} \\sum_i S_i  \\,\\,\\, , \\,\\,\\,  m_s\\equiv \\frac{2}{L^2}\n\\sum_{i_1}  S_{i_1} \\ ,\n\\label{magnetization}\n\\ee\nwhere the sum runs over all the lattice sites ($i$) and over a sub-lattice \n($i_1$) respectively. \n\nIn order to calculate spin glass quantities we have simulated two\nreplicas $\\alpha$ and $\\beta$ in parallel with the same disorder. \nThe overlaps between the replicas, global ($q$) and staggered ($q_s$),\nare:\n\\be\nq \\equiv \\frac{1}{L^2} \\sum_i S_i^{\\alpha} S_i^{\\beta} \\,\\,\\, , \\,\\,\\,\nq_s \\equiv \\frac{2}{L^2} \\sum_{i_1} S_{i_1}^{\\alpha} S_{i_1}^{\\beta} \\ ,\n\\label{overlap}\n\\ee\nwhere, again, the sum runs over all the lattice sites ($i$) and over one of\nthe two sub-lattices ($i_1$) respectively.\n\nThe magnetic global and staggered susceptibilities (without the\n$\\beta$ factor) are defined as:\n\\be\n\\chi \\equiv L^2 \\left[\\overl{\\lan m^2 \\ran}- \\left(\\overl{\\lan |m|\n\\ran}\\right)^2\\right] \n\\,\\,\\, , \\,\\,\\,\n\\chi_s \\equiv \\frac{L^2}{2} \\left[\\overl{\\lan m_s^2 \\ran} - \\left(\\overl{\\lan |m_s|\n\\ran}\\right)^2\\right]\\ .\n\\label{sus_m}\n\\ee\n\nThe spin glass or overlap susceptibilities ($\\chi_q$, $\\chi^s_q$) \nare defined by:\n\\be\n\\chi_q \\equiv L^2 \\left[\\overl{\\lan q^2 \\ran} - \\left(\\overl{\\lan |q|\n\\ran}\\right)^2\\right] \\,\\,\\, , \\,\\,\\,\n\\chi_q^s \\equiv \\frac{L^2}{2}\\left[ \\overl{\\lan q_s^2 \\ran}\n- \\left(\\overl{\\lan |q_s|\n\\ran}\\right)^2\\right] \\ .\n\\label{sus_q}\n\\ee\n\nWe have measured also the Binder cumulants of the \nmagnetization: global $g_m$ and staggered $g_m^s$, \n\\be\ng_m \\equiv \\frac{1}{2} \\left[ 3-\\frac{\\overl{\\lan m^4 \\ran}}{\\left(\\overl{\\lan\nm^2 \\ran}\\right)^2} \\right] \n\\,\\,\\, , \\,\\,\\,\ng_m^s \\equiv \\frac{1}{2} \\left[ 3-\\frac{\\overl{\\lan m_s^4 \\ran}}{\\left(\\overl{\\lan\nm_s^2 \\ran}\\right)^2} \\right] \\  \n\\label{binder_m}\n\\ee\nand Binder parameters of the overlaps: global $g_q$ and staggered $g^s_q$,\n\\be\ng_q \\equiv \\frac{1}{2} \\left[ 3-\\frac{\\overl{\\lan q^4\n\\ran}}{\\left(\\overl{\\lan q^2 \\ran}\\right)^2} \\right]  \n\\,\\,\\, , \\,\\,\\,\ng_q^s \\equiv \\frac{1}{2} \\left[ 3-\\frac{\\overl{\\lan q_s^4\n\\ran}}{\\left(\\overl{\\lan  q_s^2 \\ran}\\right)^2} \\right]  \\ .\n\\label{binder_q}\n\\ee\n\nFinally the specific heat is defined by:\n\\be\nC_V \\equiv \\frac{1}{L^2} \\left(\\overl{\\lan {\\cal H}^2 \\ran} - \\overline{\\lan\n{\\cal H}\n\\ran}^2 \\right) \\ .\n \\label{cesp}\n\\ee\n\nIn order to decide a safe thermalization time, $N_T$, (written in\nTable \\ref{table:stat}) we have used the method proposed by Bhatt and\nYoung~\\cite{bhatt} which consists in running, at the lowest\ntemperature, with an ordered (all spins up) and high temperature\ninitial configurations and monitoring the behavior of the\nsusceptibilities (in our case the non connected \noverlap susceptibility: $L^2 \\overline{\\lan q^2 \\ran}$) with the\nMonte Carlo time. When the two curves reach the same plateau we can\nsay that the system has thermalized.  We show in Figures\n\\ref{fig:terma} and \\ref{fig:terma2} this procedure for two of our\nbiggest lattices and at the lower temperatures simulated (i.e. $L=32$\nand $T=1.5$ and $L=48$ and $T=1.4$ respectively).\n\nWe remark that we have used for {\\em all} temperatures of the\nannealing procedure the value that we have computed for the lower one\n(that we have reported in Table \\ref{table:stat}). \n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{terma_32_1.5.ps}\n\\end{center}\n\\caption{The (non connected) \noverlap susceptibility against the Monte Carlo time in a\ndouble logarithmic scale for one of the lower temperature that we have\nsimulated ($T=1.5$) and for $L=32$. The number of samples was 100. \nThe upper curve is from a configuration with all the spins up and the\nlower curve is with the starting configuration chosen random. One can\nsay that the system has thermalized when there is no difference between\nthe two curves and both stay on a plateau. We have chosen as\nthermalization time $t=90000$ (i.e. $\\log t=11.4$) .}\n\\label{fig:terma}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{terma_48_1.4.ps}\n\\end{center}\n\\caption{The (non connected) \noverlap susceptibility against the Monte Carlo time in a\ndouble logarithmic scale for  the lowest temperature that we have\nsimulated ($T=1.4$) and for the largest lattice $L=48$. \nThe number of samples was 25. \nThe upper curve is from a configuration with all the spins up and the\nlower curve is with the starting configuration chosen random.  \nWe have chosen as\nthermalization time $t=900000$ (i.e. $\\log t=13.7$) .}\n\\label{fig:terma2}\n\\end{figure}\n\n\n\\section{Numerical Results}\n\n\\subsection{Crossover Ferromagnetic-Spin Glass}\n\nIn this sub-section we will show numerical evidences of the first\ncrossover: from a staggered ferromagnetic phase to a ``spin glass'' phase. \n\nIt is natural to study this crossover examining firstly the\nsusceptibility and the Binder cumulant of the staggered magnetization,\nFigures \\ref{fig:sus_ms} and \\ref{fig:binder_ms} respectively.  They\nare the observables that at $\\lambda=0$ describe the\nparamagnetic-staggered ferromagnetic phase transition.\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{sus_ms.ps}\n\\end{center}\n\\caption{Susceptibility of the staggered magnetization as a \nfunction of the temperature. The lattice sizes are (bottom to top): \n4, 6, 8, 16, 24, 32 and 48. Here and in the rest of the figures we have used\nthe following symbols for the lattices: \n$L=4$ triangle, $L=6$ square, $L=8$ pentagon,\n$L=16$ three-line star, $L=24$ four-line star, $L=32$ five-line star and \n$L=48$ six-line star.}\n\\label{fig:sus_ms}\n\\end{figure}\n\nIn Figure \\ref{fig:sus_ms} it is possible to see how the point where the\nstaggered susceptibility reaches the maximum drifts quickly to\nzero with increasing system size. \nMoreover in Figure \\ref{fig:binder_ms} there is no crossing of\nthe Binder cumulant: this is a stronger evidence that this parameter\n(the staggered magnetization) does not show a phase transition at\nfinite temperature, i.e. in the thermodynamic limit $\\lan |m_s|\n\\ran=0$ for all temperatures different from zero. \n \n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nL       & $T(\\chi_s^{\\rm max})$ &$\\chi_{s}^{\\rm max}$   \\\\ \\hline\n8       & 2.7(1)        & 1.255(4)  \\\\ \\hline\n16      & 2.3(1)        & 4.21(2)  \\\\   \\hline\n24      & 2.0(1)        & 9.6(2)   \\\\    \\hline\n32      & 1.8(2)        & 19.9(4)  \\\\   \\hline\n32      & 1.6(2)        & 54(2)  \\\\   \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Maximum of $\\chi_s(L,T)$ in temperature\nand the temperature at which $\\chi_s(L,T)$\nreaches the maximum.}\n\\label{table:staggered}\n\\end{table}\n\nMore quantitatively, from the data of Table \\ref{table:staggered} it is\nclear that the temperatures in which $\\chi_s$ reaches the\nmaximum, that we denote as $T(\\chi_s^{\\rm max}) $, goes to zero following a\npower law: \n\\be\nT(\\chi_s^{\\rm max}) \\propto L^{-0.29(4)} ,\n\\ee \nusing only the data\nof the lattices: 8, 16, 24, 32 and 48. This fit has $\\chi^2\/{\\rm DF}=0.5\/3$,\nwhere DF means for the number of degrees of freedom in the fit.\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{binder_ms.ps}\n\\end{center}\n\\caption{Binder cumulant of the staggered magnetization as a \nfunction of the temperature. The lattice sizes are (bottom to top in\nthe right part of the plot): 48, 32, 24, 16, 8, 6 and 4.}\n\\label{fig:binder_ms}\n\\end{figure}\n\nThis numerical results support our previous analytic result\nthat the staggered ferromagnetic phase is unstable\nagainst an infinitesimal perturbation of the kind of a spin glass\ninteraction between the two sub-lattices.\n\n\nMoreover, we have predicted the presence of a crossover between the\nstaggered ferromagnetic \nphase and a ``spin-glass'' like phase for lattices of linear sizes of order\n$10$. To see this crossover we can analyze the specific heat. \nAll the lattice sizes show a maximum near the Onsager\ntemperature ($T_c=2.26$). \nWe will see that these maxima are the ``souvenir'' of the staggered\nphase transition. In particular we can examine the scaling of\nthe maxima, that we denote $C_{\\rm max}(L)$, against the lattice size. The\nresult is given in Figure \\ref{fig:c_l}. In this figure we have also\nplotted the finite size scaling prediction for the pure Ising model\n($\\lambda=0$) that is\n\\be\nC_{\\rm max}(L) \\propto \\log L \\ .\n\\ee\n\nIt is clear from Figure \\ref{fig:c_l} that up to $L=8$ the data follow\nin good agreement the prediction of the pure model (i.e. up to $L=8$\nthe low temperature region is staggered ferromagnetic). But between\n$L=8$ and $L=16$ the system crosses over to a different behavior where\nthere is no divergence of the specific heat, i.e. $\\alpha<0$.  This\nresult is in very good agreement with our analytical estimate $R_c\n\\simeq 7$.\n\nHence, this plot shows us clearly a first crossover between the\nstaggered ferromagnetic phase and a ``spin glass'' like phase. In the first\npart of the crossover the specific heat diverges logarithmically\n($\\alpha=0$) whereas in the second part of the crossover the specific\nheat does not diverge ($\\alpha<0$).\n\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{c_L.ps}\n\\end{center}\n\\caption{The maximum specific heat as a function of $L$ for\n$L=4,6,8,16,24, 32$ and $48$. \nWe have marked the finite size scaling  prediction\nof a logarithmic divergence for the pure model ($\\lambda=0$). \nThe straight line has been obtained from a fit\nto the formula $A \\log L +B$ using $L=4,6$ and $8$.}\n\\label{fig:c_l}\n\\end{figure}\n\nIt is clear that the task that remains is to check that effectively\nwhat we have named as a ``spin glass'' phase has really the properties\nof a spin glass phase.\n\nTo do this we can study the susceptibility and the Binder parameter of\nthe total overlap. In a spin glass phase the magnetization is zero but\nnot the overlap, that becomes the order parameter. If the low\ntemperature phase is spin glass the overlap susceptibility should peak\nnear the transition temperature and the value of the peak should grow with\nsome power of the lattice size (more precisely as\n$L^{\\gamma\/\\nu}$). Moreover, the analysis of the Binder cumulant should\nshow a clear crossing between curves of different lattice sizes.\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{sus_q.ps}\n\\end{center}\n\\caption{Susceptibility of the total overlap against the temperature. \nThe lattices sizes are (top to bottom): 48, 32, 24, 16, 8, 6 and 4. \nFrom this figure it is clear that $\\chi_q$ for $L=16, 24, 32$ and $48$\ndoes not show a maximum.}\n\\label{fig:sus_q}\n\\end{figure}\n\nIn Figures \\ref{fig:sus_q} and \\ref{fig:binder_q} we show the data for\nthe susceptibility and Binder parameter of the total overlap.\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{binder_q.ps}\n\\end{center}\n\\caption{Binder parameter of the total overlap against the temperature. \nThe lattices sizes are (bottom to top in the right part of the plot): \n48, 32, 24, 16, 8, 6 and 4. We have marked with a vertical line the\nOnsager temperature.}\n\\label{fig:binder_q}\n\\end{figure}\nWe can see again the crossover staggered-spin glass in Figure\n\\ref{fig:binder_q}. The curves of the lattice sizes $4$, $6$ and $8$\ncross practically at the critical temperature of the pure model\n($\\lambda=0$, vertical line). If we examine the crossing point of\npairs of lattices for growing sizes, we observe that these crossing\npoints drift to lower temperatures. For instance, the crossing point\nof the $16$ and $32$ lattices is near $T=1.9$.  This suggests that the\nlimit of this sequence of crossing points may be zero. This would\nimply that the spin glass phase is unstable and is crossing over to a\nparamagnetic phase.\n\nMoreover, the overlap susceptibility does not present a sharp  maximum\nas a function of the temperature (see the $L=24, 32$ and $48$ curves). \nWe can extract one conclusions of this fact:\n\n\\begin{itemize}\n\n\\item As long as the spin glass is stable we will expect again that\n$\\chi_q(T)$ should show a sharp maximum as a function of $T$, or at\nleast that we have independent spin glass order in both sub-lattices\n(i.e. some sort of staggered spin glass order).  From figure\n\\ref{fig:sus_q} it is clear that $\\chi_q$ does not show a sharp peak\nin the region that we have simulated (see, for instance, the $L=32$\nand $L=48$ data).  Moreover the crossing point of the Binder cumulant of the\ntotal overlap of two different lattices is drifting to lower values of\nthe temperature, and so we pass to discus the second possible option:\nspin glass order on both sub-lattices.  In this case the staggered\noverlap susceptibility should have a sharp peak at an intermediate\ntemperature which characterizes a phase transition between a\nparamagnetic phase and a spin glass one.\n\n\\end{itemize}\n\nTo study this issue in more detail, we will examine in the next\nsubsection the overlap defined only in one of the two sub-lattices.\n\n\n\\subsection{Crossover Spin Glass-Paramagnetic}\n\nIn Figures \\ref{fig:sus_qs} and \\ref{fig:binder_qs}\nwe show the susceptibility and the Binder cumulant of the staggered\noverlap (i.e. the overlap computed only in one of the two sub-lattices).\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{sus_qs.ps}\n\\end{center}\n\\caption{Susceptibility  of the staggered  overlap against the temperature. \nThe lattices sizes are (bottom to top): 48, 32, 24, 16, 8 and 4.}\n\\label{fig:sus_qs}\n\\end{figure}\n\nIt is clear that these two figures are similar to Figures \\ref{fig:sus_ms} and \n\\ref{fig:binder_ms}. For example, there is no crossing in the Binder\ncumulant. We remark that we have performed small statistic on the\n$L=48$ lattice and so we obtain large errors. Taking into account only the\nlattice sizes $L=4,6,8,16,24$ and 32 the effect is clear: the\nthermodynamical Binder cumulant goes to zero for all the temperatures\nsimulated.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nL         &  $T(\\chi_{qs}^{\\rm max})$ & $\\chi_{qs}^{\\rm max}$     \\\\ \\hline\n8       & 2.4(1)        & 2.27(3) \\\\ \\hline\n16      & 2.1(1)        & 7.2(2) \\\\   \\hline\n24      & 1.9(1)        & 15.1(3) \\\\ \\hline\n32      & 1.6(2)        & 28(2)   \\\\   \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Maximum in temperature of the staggered overlap \nsusceptibility $\\chi_q^s(L,T)$, \nand the temperature at which $\\chi_q^s(L,T)$ reaches the maximum. For\n$L=48$ the error bars do not permit a safe estimate of the maximum.}\n\\label{table:q-staggered}\n\\end{table}\nThe ``apparent'' critical temperature (defined as the value where\n$\\chi_q^s$ reaches the maximum) goes to zero following the law\n\\be\nT(\\chi_{qs}^{\\rm max}) \\propto L^{-0.23(5)}\n\\label{scaling}\n\\ee\nwhere we have used L = 8, 16, 24 and 32 in the fit that has\n$\\chi^2\/{\\rm DF}=0.8\/2$. The data used in this fit has been reported\nin Table \\ref{table:q-staggered}.\n\nWithin the statistical error the exponent is just the same as in the\nstaggered magnetization. We can compare this figure with that computed\nfor the 2D spin glass\\cite{bhatt}: $1\/\\nu=0.38(6)$. If we are seeing\nthe pure two dimensional spin glass transition which occurs at $T=0$,\nthen we will expect a behavior of the apparent critical temperature\nlike $T_{\\rm app} \\propto L^{-0.38(6)}$, which is, within the\nstatistical error, the law that we have found for the present model\n(Eq. (\\ref{scaling}))\\footnote{ The shift of the apparent critical\ntemperature follows a law: $T_c(L)-T_c \\propto L^{-1\/\\nu}$.}: the\ndifference between the two exponents is $0.15(8)$ (i.e. almost two\nstandard deviations). Obviously our simulations were done in a range of\ntemperatures far away of the critical point ($T=0$).\n\nBoth Figures \\ref{fig:sus_qs} and \\ref{fig:binder_qs} suggest that the\nphase transition is at $T=0$. In other words, there is not a spin\nglass phase in the sub-lattices 1 and 2. The whole system seems to be\ncrossing over to a paramagnetic phase.\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{binder_qs.ps}\n\\end{center}\n\\caption{Binder parameter of the staggered  overlap against the temperature. \nThe lattice sizes are (bottom to top): 48, 24, 32, 16, 8, and 4.}\n\\label{fig:binder_qs}\n\\end{figure}\n\n\n\n\\section{\\label{S_CONCLU}Conclusions}\n\nWe have studied the two dimensional Ising spin glass model with next\nnearest neighbor interactions both\nanalytical and  numerically.\n\nWe have analytically obtained that the system should present two\ndifferent crossovers as the volume grows: the first one from a\nstaggered ferromagnetic phase to a spin glass phase for intermediate\nsizes, and a second crossover between this spin glass phase to a\nparamagnetic one that will dominate the physics in the thermodynamic\nlimit. Moreover we have obtained the dependence of the first crossover\nlength (staggered-spin glass) on the parameters of the system.\n\nThen we have checked this scenario by performing extensive numerical\nsimulations. We have clearly established the first crossover\n(staggered-spin glass) and have found strong evidences of the second (spin\nglass-paramagnetic). In particular, the exponent that governs the\nshift of the maxima of the spin glass susceptibility is compatible\nwith the known value for the pure two dimensional spin glass.\n\n\\vspace{2cm}\n{\\large \\bf {\\label{S_ACKNOWLEDGES}Acknowledgments}}\n\nWe have run mainly using ALPHA Workstations but the large lattices\nwere run in the RTNN parallel machine\\footnote{A parallel machine\ncomposed by 32 Pentium-Pro  at Zaragoza University.}. \nWe acknowledge the RTNN project for\npermitting to us the use of the RTNN computer.\n\nJ.~J.~Ruiz-Lorenzo is supported by an EC HMC (ERBFMBICT950429)\ngrant and thanks to L.A. Fern\\'andez for interesting discussions. \nD. A. Stariolo is partially supported by Conselho Nacional de\nDesenvolvimento Cientifico e Tecnol\\'ogico (CNPq), Brazil.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe relation between confinement and chiral symmetry breaking is one of the long-standing puzzles in theoretical physics. \nRecently,  strong interest on this issue revived in extreme environments especially at high temperatures and baryon densities,   \nstimulated by the heavy-ion programs at GSI, CERN SPS, RHIC and  LHC, see e.g., \\cite{KS04,YHM05} for a review.\nQuantum chromodynamics (QCD) for strong interactions is a fundamental theory for solving this problem. \n\n \nIn pure Yang-Mills theory, i.e., in the limit of infinitely heavy quark mass $m_q \\rightarrow \\infty$ of QCD, the Polyakov loop average $\\langle L \\rangle$, i.e., the vacuum expectation value of the Polyakov loop  operator $L$, can be used as a criterion for quark confinement \\cite{Polyakov78}. \nThe Polyakov loop operator $L$ is a gauge invariant operator charged under the center group $Z(N_c)$ of the color gauge group $SU(N_c)$.  The Polyakov loop average $\\langle L \\rangle$ vanishes $\\langle L \\rangle=0$ and quarks are confined at low temperatures $T<T_d$ where the global center symmetry $Z(N_c)$ is intact, while it is nonzero $\\langle L \\rangle \\not= 0$ and quarks are deconfined at high temperature $T>T_d$ where the global center symmetry $Z(N_c)$ is spontaneously broken. \nThus, we can define $T_d$ as a critical temperature for confinement\/deconfinement phase transition. \n\nWhen dynamical quarks in the fundamental representation of the gauge group are added to the Yang-Mills theory, the center symmetry is no longer exact. \nOn the other hand, QCD with massless quarks $m_q \\rightarrow 0$ exhibits chiral symmetry $SU(N_f)_L \\times SU(N_f)_R$. \nThe chiral condensate $\\langle \\bar\\psi \\psi \\rangle$, i.e., the vacuum expectation value of a gauge-invariant composite operator $\\bar\\psi \\psi$, is used as an order parameter for chiral symmetry breaking.\nThe chiral condensate $\\langle \\bar\\psi \\psi \\rangle$ is nonzero $\\langle \\bar\\psi \\psi \\rangle \\not= 0$ at low temperatures $T<T_\\chi$ where the chiral symmetry is spontaneously broken, while it vanishes $\\langle \\bar\\psi \\psi \\rangle = 0$ at high temperature $T>T_\\chi$ where the chiral symmetry is restored.  \nThus, we can define $T_\\chi$ as a critical temperature for chiral-symmetry breaking\/restoration phase transition. \n\n\nFor realistic quark mass (with finite and nonzero $m_q$: $0 < m_q <\\infty$), there are no exact symmetries directly related to the phase transitions, since both the center and chiral symmetries are explicitly broken, and $\\left< L \\right>$ and $\\left< \\bar\\psi \\psi \\right>$ are approximate order parameters. \nIn this case, there is no critical temperature  $T_c$ in strict sense, and the transition  can be a crossover transition for which the pseudo critical temperature $T_c^*$ is defined such that the susceptibility takes the maximal value at $T=T_c^*$.\nIf quarks are in the fundamental representation, deconfinement (a rise in the Polyakov loop average) happens at the temperature where the chiral symmetry is restored (chiral condensate decreases rapidly). \nThe chiral and deconfinement transitions seems to coincide, \n$T_d^*=T_\\chi^* \\simeq T_c$ \\cite{Karsch02,KL94},  \nalthough the property of the phase transition, e.g., the critical temperature and the order of the transition depend on  numbers of color $N_c$ and flavor $N_f$. \nWhereas, for quarks in the adjoint representation, deconfinement and chiral-symmetry restoration do not happen at the same temperature, rather $T_\\chi^* \\gg  T_d^*$ \\cite{KL98}. \nAlthough there exist theoretical considerations on the interplay between chiral symmetry breaking and confinement at zero baryon density \\cite{Casher79}, the underlying reasons for the coincidence are still unknown and uncertain at nonzero baryon density \\cite{KST99,MP07,HKS10}. \n\nThe hadronic properties, especially, chiral dynamics at low energy have been successfully described by chiral effective models such as the linear sigma model \\cite{Lee72}, the Nambu-Jona-Lasinio (NJL) model \\cite{NJL61,Klevansky92,HK94}, the chiral random matrix model \\cite{SV93}, chiral perturbation theory \\cite{GL84} and so on. \nHowever, those models based on chiral symmetry lack any dynamics coming from confinement dictated by the Polyakov loop, although there are some efforts to clarify the interplay between chiral dynamics and the Polyakov loop \n\\cite{MO95,MST03}.\n\nRecent chiral effective models with the Polyakov loop degrees of freedom augmented called the Polyakov loop--extended NJL  (PNJL) model or  quark-meson (PQM) model  \\cite{Fukushima04,MRAS06,RTW05,SFR06,KKMY07,HRCW08,BBRV07,SPW07} are successful  from a phenomenological  point of view to incorporate a coupling between the chiral condensate and the Polyakov loop. \nHowever, these PNJL\/PQM models are still far from treating the chiral condensate and the Polyakov loop on an equal footing, except for the work \\cite{SPW07} where the backcoupling of the matter sector to the glue sector was discussed by changing the phase transition parameter.\nIn fact, the gluonic part in these models has several fitting parameters which are determined only from lattice QCD data.\n\nHere we must mention a preceding work for a first-principle derivation of confinement\/deconfinement and chiral-symmetry-breaking\/restoration crossover phase transition based on the flow equation \\cite{Wetterich93}  of the functional renormalization group \\cite{BTW00,Gies06} given by Braun, Haas, Marhauser and Pawlowski \\cite{BHMP09} for the full dynamical QCD with 2 massless flavors (at zero and imaginary chemical potential).  In this work, the Yang-Mills theory is fully coupled to the matter sector by taking into account the Polyakov-loop effective potential  \\cite{Weiss81} obtained in a nonperturbative way put forward by Braun, Gies and Pawlowski \\cite{BGP07} and Marhauser and Pawlowski \\cite{MP08}.\n\n\nThe main purpose of this paper is to provide a theoretical framework (a reformulation of QCD) which enables one to describe in a unified way the chiral dynamics and  confinement signaled by the Polyakov loop.\nWe give  an important step towards a first-principle derivation of confinement\/deconfinement and chiral-symmetry breaking\/restoration crossover transition. \nIn fact, we demonstrate that a low-energy effective theory of QCD obtained in  simple  but non-trivial approximations within this framework enables one to treat both transitions simultaneously on equal footing. \n\n\nThe basic ingredients in this paper are a reformulation of QCD based on new variables \\cite{Cho80,DG79,FN99,Shabanov99,KMS06,KMS05,Kondo06,KSM08,KKMSSI05,IKKMSS06,SKKMSI07,SKKMSI07b,KSSMKI08,SKS09,Cho00,Kondo08,KS08,Kondo08b} and the flow equation of the Wetterich type in the Wilsonian renormalization group \\cite{Wetterich93,BTW00,Gies06}. \nThe reformulation was used to confirm quark confinement in pure Yang-Mills theory at zero temperature and zero density  based on a dual superconductor picture \\cite{dualsuper}. \nIn this paper, it is extended to QCD at finite temperature and density. \nIn principle, our framework can be applied to any color gauge group and arbitrary number of flavors. For technical reasons, however, we study two color QCD with two flavors in this paper.  The three color and\/or three flavor case will be studied in a subsequent paper. \nIn future publications, this framework will be applied to investigate the QCD phase diagram at finite density. \nWe hope that this paper will give an insight into this issue complementary to other works, e.g.,  \\cite{BHMP09}.\n\nIn sec. II, we give a reformulation of QCD written in terms of new variables and explain why the reformulated QCD is efficient to study the interplay between confinement and chiral-symmetry  breaking. \n\nIn sec. III, we choose a specific gauge (modified Polyakov gauge) to simplify the representation of the Polyakov loop.\nWe can choose any gauge to calculate the Polyakov loop average and the chiral condensate, since both are gauge-invariant quantities and should not depend on the gauge chosen.  \n\nIn sec. IV, we give a definition of the Polyakov loop operator and examine how the Polyakov loop average is related to the average of the time-component of the gauge field. \n\nIn sec.~V and sec.~VI, we study the confinement\/deconfinement phase transition in pure $SU(2)$ Yang-Mills theory at finite temperature.\nWe exploit the Wilsonian renormalization group in our framework to obtain the effective potential $V_{\\rm eff}$ of the Polyakov loop $L$, whose minimum gives the Polyakov loop average $\\langle L \\rangle$. \nIt is known that the Weiss potential $V_{\\rm W}$ \\cite{Weiss81} calculated in the perturbation theory to one loop exhibits spontaneous center-symmetry breaking, i.e., deconfinement, irrespective of the temperature $T$. \nThis result can be used at high temperature where the perturbation theory will be trustworthy due to asymptotic freedom, while nonperturbative approach is necessary to treat the low-temperature case. \nThe Weiss potential can be improved according to the Wilsonian renormalization group to obtain a nonperturbative effective potential which is valid even at low temperature.\n\n\nIn sec. V, we write down the flow equation of the Wetterich type for the effective potential of the Polyakov loop in our framework. \nIn fact, the effective potential obtained by solving the flow equation in a numerical way  shows the existence of confinement phase below a certain temperature $T_d$. \nThis solution was shown for the first time by Marhauser and Pawlowski \\cite{MP08} and by Braun, Gies and  Pawlowski \\cite{BGP07}, see  \\cite{MO97} for the previous works.  \nIn this sense, this section is nothing but the translation of their  results \\cite{MP08,BGP07} into our framework.\n\n\n\n\nIn sec. VI, we give a qualitative understanding for the confinement\/deconfinement transition given in sec. V based on the Landau-Ginzburg argument.  We answer a question why  the center-symmetry restoration occurs as the temperature is decreased,  by observing the flow equation for the coefficient of the effective potential.  \n\nIn sec. VII, we describe the low-energy effective interaction among quarks  by a nonlocal version of the (gauged) NJL model in which  the effect of confinement is  explicitly incorporated through the Polyakov loop dependent nonlocal interaction.   \nThe resulting effective theory is regarded as a modified and improved version of  nonlocal Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) models proposed recently by Hell,  R\\\"ossner, Cristoforetti and Weise \\cite{HRCW08}, Sasaki, Friman and  Redlich \\cite{SFR06}, and Blaschke, Buballa, Radzhabov and Volkov \\cite{BBRV07}, extending the original (local) PNJL model by Fukushima \\cite{Fukushima04}. \nThe nonlocal (gauged) NJL model can be converted to the nonlocal (gauged) Yukawa model to be bosonized to study the chiral dynamics.  \n\n\nIn sec. VIII, we show that the nonlocal NJL interaction among quarks  becomes temperature dependent through the coupling to the Polyakov loop. \nThis is a first nontrivial indication for the entanglement between the chiral symmetry breaking and confinement. \nThis feature was overlooked in  conventional PNJL models. \n\nIn sec. IX, we consider how to understand the entanglement between confinement and chiral symmetry breaking in our framework. \nThis is just a short sketch for our strategy following the line given in the preceding sections. \n\nThe final section is used to summarize the results and give some perspective in the future works. \nSome technical materials are collected in Appendices. \n\n\n\n\\section{Reformulation of QCD}\\label{sec:reformulation}\n\nTo fix the notation, we write the action of QCD in terms of the gluon field $\\mathscr{A}_\\mu$ and the quark field $\\psi$:\n\\begin{align}\n\\label{eq:QCDaction}\n S_{\\rm QCD} =& S_{\\rm q}  + S_{\\rm YM} ,\n\\nonumber \\\\\n S_{\\rm q}  :=& \\int d^Dx \\bar{\\psi} (i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{A}] -\\hat{m}_q +  \\mu_q \\gamma^0) \\psi ,\n\\nonumber\\\\\n S_{\\rm YM} :=&  \\int d^Dx  \\frac{-1}{2} {\\rm tr}(\\mathscr{F}_{\\mu\\nu}[\\mathscr{A}] \\mathscr{F}^{\\mu\\nu}[\\mathscr{A}]) \n ,\n\\end{align}\nwhere  $\\psi$ is the quark field, $\\mathscr{A}_\\mu=\\mathscr{A}_\\mu^A T_A$ is the gluon field with $su(N_c)$ generators $T_A$ for the gauge group $G=SU(N_c)$ ($A=1, \\cdots, {\\rm dim}SU(N_c)=N_c^2-1$), $\\hat{m}_q$ is the quark mass matrix,  $\\mu_q$ is the quark chemical potential, $\\gamma^\\mu$ are the Dirac gamma matrices ($\\mu=0, \\cdots, D-1$), $\\mathcal{D}_\\mu[\\mathscr{A}]:=\\partial_\\mu - ig \\mathscr{A}_\\mu$ is the covariant derivative in the fundamental representation, $\\mathscr{F}_{\\mu\\nu}[\\mathscr{A}]:=\\partial_\\mu \\mathscr{A}_\\nu - \\partial_\\nu \\mathscr{A}_\\mu -i g [\\mathscr{A}_\\mu ,  \\mathscr{A}_\\nu]$ is the field strength and $g$ is the QCD coupling constant.  \nIn what follows, we suppress the spinor, color and flavor indices. \n\nThe main purpose of this paper is to give a theoretical framework for extracting a low-energy effective theory which enables one to discuss the confinement\/deconfinement and chiral-symmetry breaking\/restoration (crossover) transition simultaneously on an equal footing.  \nWe reformulate QCD in terms of new variables which are efficient for this purpose.  \nWe start with decomposing the original $SU(N)$ Yang-Mills field $\\mathscr{A}_\\mu(x) =\\mathscr{A}_\\mu^A(x)  T_A$ into two pieces $\\mathscr{V}_\\mu=\\mathscr{V}_\\mu^A(x)  T_A$ and $\\mathscr{X}_\\mu=\\mathscr{X}_\\mu^A(x)  T_A$:\n\\begin{equation}\n  \\mathscr{A}_\\mu(x) = \\mathscr{V}_\\mu(x) + \\mathscr{X}_\\mu(x)  ,\n  \\label{decomp}\n\\end{equation}  \nto rewrite the original QCD action into a new form:\n\\begin{align}\n S_{\\rm q}   =& \\int d^Dx \\Big\\{ \\bar{\\psi} (i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{V}] -\\hat{m}_q +  \\mu_q \\gamma^0) \\psi  \n + g\\mathscr{J}^{\\mu} \\cdot \\mathscr{X}_\\mu \\Big\\} ,\n\\nonumber\\\\\n S_{\\rm YM}  =& \\int d^Dx  \\Big\\{ \\frac{-1}{4} (\\mathscr{F}_{\\mu\\nu}^A[\\mathscr{V}])^2\n-     \\frac{1}{2} \\mathscr{X}^{\\mu A}  Q_{\\mu\\nu}^{AB} \\mathscr{X}^{\\nu B} \n\\nonumber\\\\&\n-  \\frac14 (i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ])^2 \\Big\\}\n + S_{\\rm FP},\n\\label{QCDaction2}\n\\end{align}\nwhere \n$\\mathscr{J}^{\\mu A}  := g\\bar{\\psi} \\gamma^\\mu T_A  \\psi $ is the color current, \n$D_\\mu[\\mathscr{V}] := \\partial_\\mu - ig[\\mathscr{V}_\\mu , \\cdot ]$ is the covariant derivative in the adjoint representation and \n\\begin{align} \nQ_{\\mu\\nu}^{AB}[\\mathscr{V}]  :=& G^{AB} [\\mathscr{V}] g_{\\mu\\nu} \n+ 2gf^{ABC} \\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}] , \n\\nonumber\\\\\nG^{AB}[\\mathscr{V}]  :=& - (D_\\rho[\\mathscr{V}]D^\\rho[\\mathscr{V}])^{AB}   \n\\nonumber\\\\\n=& -(\\partial_\\rho \\delta^{AC}+gf^{AEC}\\mathscr{V}_\\rho^E) (\\partial^\\rho \\delta^{CB}+gf^{CFB}\\mathscr{V}^{\\rho F})   \n\\nonumber\\\\\n=& - \\partial_\\rho^2 \\delta^{AB} + g^2 f^{AEC}f^{BFC} \\mathscr{V}_\\rho^E \\mathscr{V}^{\\rho F} \n\\nonumber\\\\\n&+ 2g f^{ABE} \\mathscr{V}_\\rho^E \\partial^\\rho + gf^{ABE} \\partial^\\rho \\mathscr{V}_\\rho^E \n .\n\\label{Q}\n\\end{align}\nIn what follows we use the notation $\\mathscr{A} \\cdot \\mathscr{B}$ for two Lie-algebra valued functions $\\mathscr{A}=\\mathscr{A}^A T_A$ and $\\mathscr{B}=\\mathscr{B}^A T_A$  in the sense that \n$\\mathscr{A} \\cdot \\mathscr{B} := \\mathscr{A}^A  \\mathscr{B}^A  = 2 {\\rm tr}(\\mathscr{A} \\mathscr{B})$\nand especially $\\mathscr{A}^2:=\\mathscr{A} \\cdot \\mathscr{A}= \\mathscr{A}^A  \\mathscr{A}^A$.\n\n\nHistorically, the decomposition of Yang-Mills theory into new variables has been proposed by Cho \\cite{Cho80} and Duan and Ge \\cite{DG79} independently, and readdressed later by Faddeev and Niemi \\cite{FN99}.  \nThe decomposition was further developed by Shabanov \\cite{Shabanov99}. \n\n\n\n\n \nThe decomposition (\\ref{decomp}) is performed such that  \n$\\mathscr{V}_\\mu$ transforms under the gauge transformation just like the original gauge field $\\mathscr{A}_\\mu$: \n\\begin{equation}\n  \\mathscr{V}_\\mu(x)   \\rightarrow \\mathscr{V}_\\mu^\\prime(x) = \\Omega(x) (\\mathscr{V}_\\mu(x) + ig^{-1} \\partial_\\mu) \\Omega^{-1}(x) \n ,\n \\label{V-ctransf}\n\\end{equation}\nwhile $\\mathscr{X}_\\mu$ transforms like an adjoint matter field:\\begin{equation}\n  \\mathscr{X}_\\mu(x)   \\rightarrow \\mathscr{X}_\\mu^\\prime(x) = \\Omega(x)  \\mathscr{X}_\\mu(x) \\Omega^{-1}(x) \n \\label{X-ctransf}\n . \n\\end{equation}\n\n\nIn the decomposition (\\ref{decomp}), we introduce a new field \n\\begin{equation}\n \\bm{n}(x)=n^A(x)T_A ,  \n\\end{equation}\nwith a unit length in the sense that $n^A(x)n^A(x)=1$, which we call the \\textit{color field}.  \nIn the decomposition (\\ref{decomp}), the color field $\\bm{n}(x)$ plays a crucial role as follows.\nThe color field is defined by the following property.\nIt must be a functional or composite operator of the original Yang-Mills field  $\\mathscr{A}_\\mu(x)$ such that it transforms according to the adjoint representation under the gauge transformation:  \n\\begin{equation}\n  \\bm{n}(x)   \\rightarrow \\bm{n}^\\prime(x) = \\Omega(x)  \\bm{n}(x) \\Omega^{-1}(x) \n   .\n \\label{n-ctransf}\n\\end{equation}\n\n\n\nThe color field plays the key role in the reformulation.  Once a color field is given, the decomposition is uniquely determined by solving a set of defining equations and hence  $\\mathscr{V}_\\mu(x)$ and $\\mathscr{X}_\\mu(x)$ are written in terms of $\\mathscr{A}_\\mu(x)$ and $\\bm{n}(x)$.\nFor $G=SU(2)$,  the defining equations are given by\n\n(I) covariant constantness of color field $\\bm{n}(x)$ in $\\mathscr{V}_\\mu(x)$:\n\\begin{equation}\n  0 = D_\\mu[\\mathscr{V}]\\bm{n}(x)   ,\n\\end{equation}\n\n\\text{(II) orthogonality of  $\\mathscr{X}_\\mu(x)$ to $\\bm{n}(x)$:}\n\\begin{equation}\n  0 = \\mathscr{X}_\\mu(x) \\cdot \\bm{n}(x)    .\n\\end{equation}\nThen the decomposition for $G=SU(2)$ is uniquely determined as\n\\begin{align}\n  & \\mathscr{V}_\\mu(x)= c_\\mu(x)\\bm{n}(x) +    ig^{-1} [ \\bm{n}(x) , \\partial_\\mu \\bm{n}(x) ]  ,\n\\nonumber\\\\\n & \\quad\\quad\\quad\\quad  c_\\mu(x) :=  \\mathscr{A}_\\mu(x)  \\cdot \\bm{n}(x) ,\n\\nonumber\\\\\n & \\mathscr{X}_\\mu(x) =   i g^{-1} [ D_\\mu[\\mathscr{A}] \\bm{n}(x) , \\bm{n}(x) ] ,\n\\end{align}\n\nTo arrive at the result (\\ref{QCDaction2}),  we have used the following facts. See Appendix~\\ref{app:reformulation} for the details.\n\n(i) \nThe $O(\\mathscr{X})$ terms vanish,  \n$\\frac{1}{2} \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)=0$,\nfrom the property of the new variables  as shown using the \\textit{defining equations of the decomposition} (\\ref{decomp}). \nThis is somewhat similar to the usual background field method in which $O(\\mathscr{X})$ terms in the quantum fluctuation field $\\mathscr{X}_{\\mu}$ are eliminated by requiring that the background field $\\mathscr{V}_\\mu$ satisfies the classical Yang-Mills equation of motion, i.e., $D_\\mu[\\mathscr{V}]\\mathscr{F}^{\\mu\\nu}[\\mathscr{V}]=0$. \nIn our case,  however, $\\mathscr{V}_\\mu$ do not necessarily satisfy the classical equation of motion.\n\n \n(ii)  To obtain $Q_{\\mu\\nu}^{AB}[\\mathscr{V}]$ in (\\ref{Q}), an $O(\\mathscr{X}^2)$ term is eliminated, $-\\frac{1}{2} \\mathscr{X}^{\\mu A}  D_\\mu^{AC}[\\mathscr{V}]D_\\nu^{CB}[\\mathscr{V}] \\mathscr{X}^{\\nu B}=0$,\nby imposing  the condition:\n\\begin{equation}\n D_\\mu[\\mathscr{V}] \\mathscr{X}^\\mu=0 .\n \\label{nMAG}\n\\end{equation}\nFor the reformulated QCD to be equivalent to the original QCD, we must impose such a constraint to avoid mismatch in the independent degrees of freedom, which is called the \\textit{reduction condition} \\cite{KMS06,KSM08}. \n\n(iii) The $O(\\mathscr{X}^3)$ term vanishes, \n$\n\\frac{1}{2}  (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu) \\cdot ig[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ] = 0 \n$, since $D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu$ is orthogonal to $[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ]$.\n\n\nFor $G=SU(2)$, $\\mathscr{V}$ can be chosen in such a way that the field strength $\\mathscr{F}[\\mathscr{V}]$ of the field $\\mathscr{V}$ is proportional to $\\bm{n}$:\n\\begin{align}\n \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}](x) \n:=& \\partial_\\mu \\mathscr{V}_\\nu(x)  - \\partial_\\nu \\mathscr{V}_\\mu(x)   - ig [\\mathscr{V}_\\mu(x) ,  \\mathscr{V}_\\nu(x)]\n\\nonumber\\\\\n=&  \\bm{n}(x) G_{\\mu\\nu}(x) ,\n\\label{F=nG}\n\\end{align}\nwhere $G_{\\mu\\nu}$ is a gauge--invariant antisymmetric tensor of rank 2, i.e., \n$\\mathscr{F}_{\\mu\\nu}^\\prime[\\mathscr{V}](x)=\\mathscr{F}_{\\mu\\nu}[\\mathscr{V}^\\prime](x)=\\Omega(x)  \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}](x) \\Omega^{-1}(x) = \\bm{n}^\\prime(x) G_{\\mu\\nu}(x) \n$.  The explicit form of $G_{\\mu\\nu}$ is written in term of $\\mathscr{A}_\\mu(x)$ and $\\bm{n}(x)$ as\n\\begin{align}\n G_{\\mu\\nu}(x)  &=    \\partial_\\mu [ \\bm{n}(x) \\cdot \\mathscr{A}_\\nu(x)] - \\partial_\\nu [ \\bm{n}(x) \\cdot \\mathscr{A}_\\mu(x)] \n\\nonumber\\\\&\n+   i g^{-1} \\bm{n}(x) \\cdot [\\partial_\\mu \\bm{n}(x) , \\partial_\\nu \\bm{n}(x)  ] .\n\\end{align}\n\n\nIn the present approach, we wish to regard the field decomposition as a change of variable from the original gluon field to new variables describing a reformulated Yang-Mills theory in the quantum level \\cite{KMS06,Kondo06,KSM08} (see \\cite{KKMSS05,KKMSSI05,IKKMSS06,SKKMSI07,SKKMSI07b,KSSMKI08,SKKISF09,SKS09} for the corresponding lattice gauge formulation).  To achieve this goal, first of all, $\\bm{n}(x)$ must be written as a functional of $\\mathscr{A}_\\mu(x)$ and thereby all new fields are written in terms of the original gluon field  $\\mathscr{A}_\\mu(x)$.   \nSuch a required relationship between $\\mathscr{A}_\\mu(x)$ and  $\\bm{n}(x)$ is given by the reduction condition which is given as a variational problem of obtaining an absolute minimum of a given functional.  \nThe condition for local minima is given in the form of a differential equation. For $G=SU(2)$, \n\\begin{equation}\n [ \\bm{n}(x) , D_\\mu[\\mathscr{A}]D_\\mu[\\mathscr{A}]\\bm{n}(x) ] =0 .\n \\label{dRed}\n\\end{equation}\nThis is another form of (\\ref{nMAG}). \nSee \\cite{KMS06} in $SU(2)$ case and \\cite{KSM08} in $SU(N)$ case for the full details. \n\nRemarkable properties of new variables are as follows. \nFirst, we remind you of the role played by the field $\\mathscr{V}$.\n\n$\\bullet$ The variable $\\mathscr{V}_\\mu$ alone is responsible for the Wilson loop  operator $W_C[\\mathscr{A}]$ and the Polyakov loop operator $L[\\mathscr{A}]$ in the sense that  \n\\begin{equation}\nW_C[\\mathscr{A}]=W_C[\\mathscr{V}] , \\quad\nL[\\mathscr{A}]=L[\\mathscr{V}] .\n\\end{equation}\nwhere the Wilson loop  operator is defined by\n\\begin{equation}\n W_C[A] := \\mathcal{N}^{-1} {\\rm tr} \\left[ \\mathscr{P} \\exp \\left\\{ ig \\oint_C dx^\\mu \\mathscr{A}_\\mu(x) \\right\\} \\right] ,\n\\end{equation}\nwhere $\\mathscr{P}$ denotes the path ordering and the normalization factor $\\mathcal{N}$ is the dimension of the representation $R$, in which the Wilson loop is considered, i.e.,  \n$\n \\mathscr{N}:=d_R = {\\rm dim}({\\bf 1}_R) = {\\rm tr}({\\bf 1}_R)  \n$.\nThe Polyakov loop operator will be defined later. \nIn other words, $\\mathscr{X}_\\mu$ do not contribute to the Wilson loop and the Polyakov loop in the operator level. \nThis is because the defining equation for the decomposition is a  (necessary and) sufficient condition for a \\textit{gauge-invariant  Abelian dominance} (or $\\mathscr{V}$ dominance) in the operator level. \nThis proposition was first proved in \\cite{Cho00}  for $SU(2)$ and   for $SU(N)$ in the continuum \\cite{Kondo08} and for $SU(N)$ on a lattice \\cite{KS08}.  On the lattice, the equality does not exactly hold due to non-zero lattice spacing $\\epsilon$, but the deviation vanishes in the continuum limit of the lattice spacing $\\epsilon$ going to zero, $\\epsilon \\rightarrow 0$.\nIt should be remarked that both the Wilson loop operator and the Polyakov loop operator are gauge-invariant quantities and that their average do not depend on the gauge fixing condition adopted in the calculation. \n \n $\\bullet$ We can introduce a gauge-invariant magnetic monopole current $k$ in Yang-Mills theory (without matter fields) where $k$ is the ($D-3$)-form.  For $D=4$ and $G=SU(2)$, \n\\begin{align}\n k_\\mu(x) :=& \\partial_\\nu {}^*G_{\\mu\\nu}(x) ,\n\\\\\n G_{\\mu\\nu} :=&  \\bm{n} \\cdot \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]\n  =  \\partial_\\mu c_\\nu - \\partial_\\nu c_\\mu +i g^{-1} \\bm{n} \\cdot [\\partial_\\mu \\bm{n} ,  \\partial_\\nu \\bm{n} ] ,\n  \\nonumber\n\\end{align}\nwhere $f_{\\mu\\nu}$ is gauge-invariant field strength. \nThis is because the field strength   $\\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]:= \\partial_\\mu \\mathscr{V}_\\nu - \\partial_\\nu \\mathscr{V}_\\mu -ig [\\mathscr{V}_\\mu , \\mathscr{V}_\\nu ]$ is proportional to $\\bm{n}$:\n\\begin{equation}\n \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]\n = \\bm{n} \\{ \\partial_\\mu c_\\nu - \\partial_\\nu c_\\mu +i g^{-1} \\bm{n} \\cdot [\\partial_\\mu \\bm{n} ,  \\partial_\\nu \\bm{n} ] \\}  .\n\\end{equation}\n $\\bullet$ The gauge-invariant ``Abelian\" dominance (or $\\mathscr{V}$ dominance) and magnetic monopole dominance (constructed from $\\mathscr{V}$) in quark confinement have been confirmed at $T=0$ (and $\\mu_q=0$)  by comparing string tensions calculated from the Wilson loop average by numerical simulations by \\cite{IKKMSS06} for SU(2) and by \\cite{SKKMSI07b} for SU(3). \nHere it should be remarked that the ``Abelian\" dominance is the dominance for the vacuum expectation value (or average):\n\\begin{equation}\n\\langle W_C[\\mathscr{A}] \\rangle \\simeq \\langle W_C[\\mathscr{V}] \\rangle \n, \\quad \n\\langle  L[\\mathscr{A}] \\rangle  \\simeq  \\langle  L[\\mathscr{V}] \\rangle.\n\\end{equation}\n\n\nNext, we pay attention to the role played by the remaining field $\\mathscr{X}$. \n\\\\\n$\\bullet$ In the absence of dynamical quarks (corresponding to the limit $m_q=\\infty$ of QCD, i.e., gluodynamics), \n$\\mathscr{X}_\\mu^A$ decouple in the low-energy regime as  the correlator $\\left< \\mathscr{X}_\\mu^A(x) \\mathscr{X}_\\mu^A(y) \\right>$ behaves like a massive correlator with mass $M_X$. \nIn fact, numerical simulations demonstrate for $G=SU(2)$ and $D=4$ \\cite{SKKMSI07} \n\\begin{equation}\nM_X=1.2 \\sim 1.3 \\ {\\rm GeV}.\n\\label{M_X}\n\\end{equation}\nWe can understand this result as follows.\nThe field $\\mathscr{X}_\\mu$  can acquire the (gauge-invariant) mass dynamically. This comes from a fact that, in sharp contrast to the field $\\mathscr{A}_\\mu$, \n a ``gauge-invariant mass term\" for $\\mathscr{X}_\\mu$ can be introduced\n\\begin{equation}\n\\frac12 M_X^2 \\mathscr{X}_\\mu^A(x) \\mathscr{X}_\\mu^A(x) ,\n\\end{equation}\nsince $\\mathscr{X}_\\mu^A(x) \\mathscr{X}_\\mu^A(x)$ is a gauge-invariant operator.\nMoreover, this mass term can originate from a vacuum condensation  of ``mass dimension-2\", $\\left< \\mathscr{X}_\\nu^B(x) \\mathscr{X}_\\nu^B(x) \\right> \\not= 0$ as proposed in \\cite{Kondo01}.\nIn fact, this condensation can be generated through self-interactions $O(\\mathscr{X}^4)$ among  $\\mathscr{X}_\\mu$ gluons, $M_X^2 \\simeq \\left< \\mathscr{X}_\\nu^B(x) \\mathscr{X}_\\nu^B(x) \\right>$, as examined in \\cite{Kondo06,KKMSS05}.\nIt is instructive to remark that the value (\\ref{M_X}) agrees with the earlier result of the off-diagonal ``gluon mass'' $M_A$ in the Maximally Abelian (MA) gauge \\cite{AS99} for $SU(2)$ case, $M_A \\simeq 1.2$ GeV. See \\cite{SAIIMT02} for $SU(3)$ case, $M_A \\simeq 1.1$ GeV.\nIn MA gauge, it was shown that even at finite temperatures Abelian dominance (diagonal part dominance and off-diagonal part suppression) holds for the spatial propagation of gluons in the long distance greater than 0.4fm.  It was observed that the diagonal gluon correlator largely changes between the confinement and the deconfinement phase, while the off-diagonal gluon correlator is almost the same even in the deconfinement phase \\cite{AS00}. \nAlthough the similar results are expected to hold in  our formulation, this observation must be checked directly, as will be confirmed in \\cite{KSSK10}.\n\n\n\n \n\n$\\bullet$ In the presence of  dynamical quarks ($m_q<\\infty$), $\\mathscr{X}_\\mu^A$ is responsible for chiral-symmetry breaking in the following sense.   \nWe consider to integrate out the field $\\mathscr{X}_\\mu^A$ in a naive way. \nThis helps us to obtain an intuitive and qualitative understanding for the interplay between the chiral symmetry breaking and confinement. \nLater, this integration procedure will be reconsidered from the viewpoint of the renormalization group to obtain a systematic improvement of the result. \n\nHere we neglect $O(\\mathscr{X}^3)$ and $O(\\mathscr{X}^4)$ terms, which will be taken into account later. Then the integration over $\\mathscr{X}_\\mu^A$ can be achieved by the Gaussian integration according to \\cite{Kondo06}. \nConsequently, a nonlocal 4 fermion-interaction is generated:\n\\begin{align}\n S_{\\rm eff}^{\\rm QCD} =& S_{\\rm eff}^{\\rm glue} + S_{\\rm eff}^{\\rm gNJL} ,\n\\nonumber\\\\\n S_{\\rm eff}^{\\rm glue} :=& \\int d^Dx  \\frac{-1}{4} (\\mathscr{F}_{\\mu\\nu}[\\mathscr{V}])^2   \n\\nonumber\\\\&\n+ \\frac{i}{2} \\ln \\det Q[\\mathscr{V}]_{\\mu\\nu}^{AB} - i \\ln \\det G[\\mathscr{V}]^{AB} \n ,\n\\nonumber\\\\\nS_{\\rm eff}^{\\rm gNJL} :=&  \\int d^Dx \\ \\bar{\\psi} (i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{V}]  -\\hat{m}_q + i  \\gamma^0 \\mu) \\psi \n\\nonumber\\\\ \n +& \\int d^Dx \\int d^Dy  \\frac{g^2}{2}  \\mathscr{J}^{\\mu}_{A}(x)  Q^{-1}[\\mathscr{V}]_{\\mu\\nu}^{AB}(x,y) \\mathscr{J}^{\\nu}_{B}(y)  \n ,\n\\end{align}\nwhere the last term $- \\ln \\det G^{AB}$ in $S_{\\rm eff}^{\\rm glue}$ comes from the Faddeev-Popov determinant associated with the reduction condition (\\ref{nMAG}) (see \\cite{KMS05} for the precise form). \n\n\nThis is a nonlocal version of a gauged NJL model (realized after Fierz transformation).   \nThe chiral-symmetry breaking\/restoration transition and the phase structure of a local version of  gauge NJL models were first studied by solving the Schwinger-Dyson equation in the ladder approximation for QED-like \\cite{Miransky85,KMY89,KKM89} (see \\cite{Kondo91} for a review) and QCD-like \\cite{KSY91} running gauge coupling constant. \nThey are confirmed later by a systematic approach of the   renormalization group \\cite{Aoki-etal99}.\n\n\nThe range of the nonlocality is determined by the correlation length $\\xi$, which is characteristic of the color exchange through gluon fields. \nTherefore, this correlation length $\\xi$ is identified with the inverse of the effective mass $M_X$, i.e., $\\xi \\simeq M_X^{-1}$. In fact, $(Q^{-1})_{\\mu\\nu}^{AB}(x,y)$ is the $\\mathscr{X}$ field correlator, see (\\ref{QCDaction2}).\n\nIn other words, $M_X$ is identified with the ultraviolet cutoff $\\Lambda$ below which the effective NJL model appears and works well. Interesting enough, $M_X$ is nearly equal to the ultraviolet cutoff adopted in the NJL model \n\\begin{equation}\n\\sqrt{p^2} \\lesssim \\Lambda_4=1.4 {\\rm GeV} , \\quad |\\bm{p}| \\lesssim  \\Lambda_3=0.6 {\\rm GeV} , \n\\end{equation}\nsee \\cite{HK94}.\n\n\nWe can decompose the gauge field $\\mathscr{A}_\\mu$ into the the low-energy (light) mode $p<M_X$ and high-energy (heavy) mode $p>M_X$:  \n\\begin{align}\n \\mathscr{A}_\\mu(p)\n=&\\mathscr{A}_\\mu(p) \\theta(M_X^2-p^2)+\\mathscr{A}_\\mu(x) \\theta(p^2-M_X^2)\n  .\n  \\label{decomp2}\n\\end{align}\nIn the above treatment, $\\mathscr{X}_\\mu(p)$ is supposed to have only the high-energy mode. \nThe low-energy mode, if any, will be responsible for the vacuum condensation \\cite{Kondo06}. \nFor the precise understanding, we need the renormalization group treatment as given later and the implications for the nonlocal NJL model will be discussed there. \n\n\n\\section{Gluon sector and gauge fixing}\\label{sec:gauge}\n\n\nThe Polyakov loop operator $L$ and the chiral operator $\\bar\\psi \\psi$ are gauge-invariant quantities.  Therefore, their average do not depend on the gauge-fixing procedure adopted in the calculation. \nWe can choose a gauge in which the actual calculation becomes easier than other gauges. \n\nIn what follows, we  treat the time-component $\\mathscr{V}_0$ and space-component $\\mathscr{V}_j$ of $\\mathscr{V}_\\mu$ differently to consider the finite-temperature case.  \nWe consider the following Polyakov gauge modified for new variables in our reformulation. \nIf the color field $n^A(x)$ is uniform in time, \n\\begin{equation}\n  \\partial_0 n^A(x) = 0  \\Leftrightarrow\nn^A(x) = n^A (\\bm{x})\n , \n \\label{my-gauge-0}\n\\end{equation}\nthen $\\mathscr{V}_0$ reduces to\n\\begin{equation}\n \\mathscr{V}_0^A(x) = c_0(x)n^A (\\bm{x})  \\quad (A=1,2,3)\n .\n \\label{my-gauge-1}\n\\end{equation}\nMoreover, if  $c_0(x)$ is uniform in time, \n\\begin{equation}\n  \\partial_0 c_0(x) = 0  \\Leftrightarrow\nc_0(x) = c_0(\\bm{x})\n , \n\\end{equation}\nthen $\\mathscr{V}_0$ reduces to\n\\begin{equation}\n \\mathscr{V}_0^A(x) = c_0(\\bm{x})n^A (\\bm{x})  \\quad (A=1,2,3)\n ,\n \\label{my-gauge-2}\n\\end{equation}\nwhich satisfies \n\\begin{equation}\n \\partial_0 \\mathscr{V}_0^A(x) = 0  \\quad (A=1,2,3)\n .\n \\label{my-gauge-3}\n\\end{equation}\nIn this setting, $\\mathscr{V}_j$ are given by\n\\begin{equation}\n \\mathscr{V}_j^A(x) = c_j(x)n^A(\\bm{x}) + g^{-1} \\epsilon^{ABC} \\partial_j n^B(\\bm{x}) n^C(\\bm{x}) \n . \n\\end{equation}\n\n\n\n(1) In order to simplify the calculation of the Polyakov line, we adopt the Polyakov gauge in which the gauge field is diagonal and time-independent: \nfor the background field $\\mathscr{V}_0^A(x)$, \n\\begin{equation}\n \\mathscr{V}_0^A(x) = c_0(\\bm{x}) \\delta^{A3}  ,\n\\end{equation}\nwhich leads to\n\\begin{equation}\n \\partial_0 \\mathscr{V}_0^A(x) = 0 \n .\n\\end{equation}\nThis is realized, if we take the gauge\n\\footnote{This is an oversimplified choice for the color field $\\bm{n}(x)$.  \nBy this choice, we can not separate the non-perturbative contribution coming from topological configurations such as magnetic monopole.\nIt is desirable to take into account color field degrees of  freedom explicitly to see the effect of magnetic monopole in the confinement\/deconfinement transition. \n}\n\\begin{equation}\n  n^A(\\bm{x}) = \\delta^{A3}\n . \n\\end{equation}\nIn this gauge, the space-component reads\n\\begin{equation}\n \\mathscr{V}_j^A(x) = c_j(x)\\delta^{A3}  \n . \n\\end{equation}\nwhich is not time-independent, $\\partial_0 \\mathscr{V}_j^A(x) \\not= 0$.\n\n(2) We expand the theory around the non-trivial uniform background $g^{-1}T\\varphi  \\delta^{A3} $ for the time-component $\\mathscr{V}_0^A$, while the trivial background for  space-components  $\\mathscr{V}_j^A$:\\footnote{\nI have assumed that the spatial component of $\\mathscr{V}_\\mu$ has a trivial background. \nIn view of logical consistency, one must expand the spatial and temporal components around non-trivial backgrounds, and then one must search for the minima of the effective potential calculated as a function of two variables, i.e., the temporal and spatial backgrounds.  In this paper, it is assumed that a minimum is realized at vanishing spatial background and that the neglection of the spatial background does not so much affect the confinement\/deconfinement transition temperature. \nIndeed, it must be checked whether this assumption is good or not.\n}\n\\begin{align}\n \\mathscr{V}_0^A(x) =& c_0(\\bm{x})\\delta^{A3} , \\ c_0(\\bm{x}) = g^{-1}T\\varphi  + v_0(\\bm{x}) \n ,\n\\nonumber\\\\\n \\mathscr{V}_j^A(x) =&  0+ v_j^A(x)\n , \n\\end{align}\nsuch that \n$\\langle c_0(\\bm{x}) \\rangle = g^{-1}T \\langle \\varphi \\rangle  + \\langle v_0(\\bm{x}) \\rangle=g^{-1}T \\langle   \\varphi \\rangle $ with $\\langle v_0(\\bm{x}) \\rangle=0$ and $\\langle v_j^A(x) \\rangle=0$. \nHere the prefactor $g^{-1}T=(g\\beta)^{-1}$ was introduced just for the purpose of simplifying the expression of the Polyakov loop, see (\\ref{P}). \n\n(3) We take into account the expansion up to quadratic  in the fluctuation fields $v_0$ and $v_j$, which we call the quadratic approximation.  \n\n\nIn the calculation of $Q_{\\mu\\nu}^{AB}$, if we neglect all fluctuation fields $v_0$ and $v_j$, namely,   \n$\\mathscr{V}_0^A (x) =  g^{-1}T \\varphi \\delta^{A3}$ and  $\\mathscr{V}_j^A (x) =  0$, then we can put \n$\\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}] = 0$ in $Q_{\\mu\\nu}^{AB}$, and  $Q_{\\mu\\nu}^{AB}$ is diagonal in the Lorentz indices:\n\\begin{align} \nQ_{\\mu\\nu}^{AB}   &=   G^{AB} g_{\\mu\\nu} , \n  \\label{approx0}\n\\\\ \nG^{AB}  \n& =  - \\delta^{AB} \\partial_\\mu^2  +   (\\delta^{AB}  - \\delta^{A3}\\delta^{B3})(T \\varphi)^2 + 2  \\epsilon^{AB3} T \\varphi \\partial_0  \n .\n\\nonumber\n\\end{align}\nIn this approximation, we have\n\\begin{align}\n & G^{AB} =   - \\delta^{AB} \\partial_\\ell^2  - D_0^{AC}[\\mathscr{V}]D_0^{CB}[\\mathscr{V}] ,\n  \\label{approx}\n\\end{align}\nwith\n\\begin{align}\n    - D_0^{AC}[\\mathscr{V}]D_0^{CB}[\\mathscr{V}]   \n=&  -   \\delta^{AB} \\partial_0^2\n+ 2  \\epsilon^{AB3} T \\varphi \\partial_0  \n \\nonumber\\\\&\n+ (\\delta^{AB}  - \\delta^{A3}\\delta^{B3})(T \\varphi)^2 \n  .\n\\end{align}\nThus we rewrite the gluon part $S_{\\rm eff}^{\\rm glue}[\\mathscr{V}]$ as\n\\begin{align}\n & S_{\\rm eff}^{\\rm glue}[\\mathscr{V}]  \n \\nonumber\\\\ \n&=    \\frac12 \\beta \\int d^3x \\mathscr{V}_0(\\bm{x})  (-\\partial_j \\partial_j)  \\mathscr{V}_0(\\bm{x})   \n \\nonumber\\\\&\n + \\frac12  \\int d^4x  \\mathscr{V}_{T}^A(x)  \\{  \n    -  \\delta^{AB}  \\partial_\\ell^2   - \\delta^{AB}  \\partial_0^2\n   \\} \\mathscr{V}_{T}^B(x) \n \\nonumber\\\\&\n + \\frac12 \\int d^4x \\mathscr{V}_{L}^A(x)  \\{    - \\delta^{AB}  \\partial_0^2  \\} \\mathscr{V}_{L}^B(x) \n  ,\n\\nonumber\\\\&\n+ \\frac{i}{2} \\ln \\det Q[\\mathscr{V}]_{\\mu\\nu}^{AB} - i \\ln \\det G[\\mathscr{V}]^{AB} \n ,\n  \\label{mode-decomp1}\n\\end{align}\nwhere $\\beta$ is the inverse temperature $\\beta:=1\/T$, and $\\mathscr{V}_{T}$ and $\\mathscr{V}_{L}$ denote the transverse and longitudinal components of $\\mathscr{V}_\\mu$ respectively.\n\n\n\n\n\\section{Polyakov loop}\\label{sec:Polyakov}\n\nFor $G=SU(2)$, the Polyakov loop operator $L(\\bm{x})=L[\\mathscr{V}_0(\\bm{x}, \\cdot)]$ is defined by\n\\begin{align}\nL(\\bm{x}) :=&   \\frac12 {\\rm tr}(P) ,\n\\nonumber\\\\\n P(\\bm{x}) :=& \\mathscr{P} \\exp \\left[ ig \\int_{0}^{\\beta=1\/T} dx_0 \\mathscr{V}_0^A(\\bm{x},x_0)  \\frac{\\sigma_A}{2} \\right]  \n, \n\\label{P}\n\\end{align}\nwhere \n$P P^\\dagger = \\mathbf{1}$\nand\n$\\det P =1$.\nIn the above gauge choice, \n\\begin{equation}\n P(\\bm{x}) =   \\exp \\left[ ig \\beta c_0(\\bm{x})  n^A(\\bm{x}) \\frac{\\sigma_A}{2}   \\right] . \n \\nonumber\n\\end{equation}\nAfter a suitable ($t$-independent) gauge transformation, the color field $n^A(\\bm{x})$ is eliminated:\n\\begin{equation}\n L(\\bm{x}) \n=  \\frac12 {\\rm tr}(\\exp \\left[ ig \\beta c_0(\\bm{x})   \\frac{\\sigma_3}{2}   \\right] )\n= \\cos \\left( \\frac{g \\beta c_0(\\bm{x})}{2}    \\right) .  \n\\label{LL}\n\\end{equation}\n\nOwing to periodicity and center symmetry, we can restrict the Polyakov loop average to $\\langle L \\rangle \\ge 0$ for $G=SU(2)$.  Then the Polyakov loop average $\\langle L[\\mathscr{V}] \\rangle$ is bounded from above by $L[ \\langle \\mathscr{V}_0(\\bm{x}, \\cdot) \\rangle]$:\n\\begin{equation}\n 0 \\le \\langle L[\\mathscr{V}_0(\\bm{x}, \\cdot)] \\rangle\n\\le     L[ \\langle \\mathscr{V}_0(\\bm{x}, \\cdot) \\rangle]  \n= \\cos \\left(  \\frac{\\langle \\varphi \\rangle}{2}    \\right)  ,\n\\end{equation}\nwhere we have only to consider the range $0 \\le \\varphi \\le \\pi$.\nThis inequality follows from the Jensen inequality, since $\\cos (x)$ is concave for $0 \\le x\\le \\pi\/2$, see \\cite{MP08}.\n\n\nIn the case of $m_q=\\infty$, if the center-symmetry is broken $\\langle L \\rangle >0$, namely, deconfinement  takes place, then  the vacuum (as a minimum of the effective potential $V_{\\rm eff}(\\varphi)$) is realized at $\\langle \\varphi \\rangle < \\pi$.\nIf the vacuum is realized at $\\langle \\varphi \\rangle=\\pi$, then the center-symmetry is restored $\\langle L \\rangle =0$, namely,  confinement  occurs.   The relation (\\ref{LL}) yields the relationship for the average between the gauge field and the Polyakov loop operator:  \n\\begin{equation}\n  \\langle  \\arccos L(\\bm{x}) \\rangle = \\frac{g \\beta \\langle c_0(\\bm{x}) \\rangle  }{2}  = \\frac{\\langle \\varphi \\rangle}{2} ,\n\\end{equation}\nwhere the left-hand side is the average of an gauge-invariant object (since $L$ is gauge invariant) and happens to agree with the average $\\langle  \\mathscr{V}_0^3 \\rangle$ of the gauge field in the Polyakov gauge. \nIt is also shown \\cite{MP08} that the converse is  true: In the center-symmetry-restored phase, $\\langle \\varphi \\rangle=\\pi$, since \n\\begin{equation}\n   \\frac{\\langle \\varphi \\rangle}{2}  =  \\langle  \\arccos L(\\bm{x}) \\rangle =  \\arccos \\langle L(\\bm{x}) \\rangle   = \\frac{\\pi}{2} .\n\\end{equation}\n\n\nTherefore, $\\langle \\mathscr{V}_0 \\rangle$ or $\\langle \\varphi \\rangle$ in the Polyakov gauge gives a direct physical interpretation as an order parameter for the confinement\/deconfinement (order-disorder) phase transition. \nThe effective potential $U_{\\rm eff}(\\langle L \\rangle)$ of the Polyakov loop average $\\langle L \\rangle$  could be different from the effective potential $V_{\\rm eff}(\\langle  \\mathscr{V}_0 \\rangle)$ of the gauge field average $\\langle  \\mathscr{V}_0 \\rangle$ in the following sense.\nAlthough both potentials give the same critical temperature $T_d$ as a boundary between $\\langle L  \\rangle=0$ and $\\langle L \\rangle \\not= 0$,  \nthe value of the effective potential $U_{\\rm eff}(\\langle L[\\mathscr{V}_0 ] \\rangle)$ does not necessarily agree with $U_{\\rm eff}(L[ \\langle \\mathscr{V}_0  \\rangle])=V_{\\rm eff}(\\langle  \\mathscr{V}_0 \\rangle)$ at a given temperature $T$, since we have only an inequality \n$\n\\langle L[\\mathscr{V}_0 ] \\rangle \\le L[ \\langle \\mathscr{V}_0  \\rangle]\n$.\nThis difference could affect the critical exponent and other physical quantities of interest. Therefore, the result obtained from $V_{\\rm eff}(\\langle  \\mathscr{V}_0 \\rangle)$ must be carefully examined. \n\n\n\\section{Deriving the confinement\/deconfinement transition}\\label{sec:deconfinement}\n\nIn this section, we restrict our consideration to the pure glue case. \nWe show that the pure gluon part $S_{eff}^{\\rm glue}$ can  describe confinement\/deconfinement transition signaled by the Polyakov loop average $\\langle L \\rangle$. \nIn this section, we completely follow two remarkable papers by Marhauser and Pawlowski \\cite{MP08} and by Braun, Gies and  Pawlowski \\cite{BGP07}, which succeeded to show the transition for the first time based on the functional renormalization group (FRG).\nIn the next section, we explain how these results are understood from the Landau-Ginzburg argument. \n\nWe consider the flow equation called the Wetterich equation \\cite{Wetterich93} for the $k$(RG scale)-dependent effective action $\\Gamma_k$:\n\\begin{align}\n\\partial_t \\Gamma_k[\\Phi] \n=& \\frac12 {\\rm STr} \\left\\{ \\left[ \\frac{\\overrightarrow{\\delta}}{\\delta \\Phi^\\dagger} \\Gamma_k[\\Phi] \\frac{\\overleftarrow{\\delta}}{\\delta \\Phi} + R_{\\Phi,k} \\right]^{-1} \\cdot \\partial_t R_{\\Phi,k} \\right\\}\n ,\n\\end{align}\nwhere $t$ is the RG time \n$\n  t := \\ln \\frac{k}{\\Lambda}\n$, \n$\n \\partial_t := \\frac{\\partial}{\\partial t} = k \\frac{d}{dk} \n$\nfor some reference scale (UV cutoff) $\\Lambda$ \nand $R_{\\Phi,k}$ is the regulator function for the field $\\Phi$. \nHere ${\\rm STr}$ denotes the super-trace introduced to include both  commutative field (gluon) and anticommutative field (quark, ghost). \nSee \\cite{BTW00,Gies06} for  reviews of the functional renormalization group. \n\nIf we restrict our consideration to the pure glue case $S_{\\rm YM}$ under the gauge $n^A(x)=\\delta^{A3}$, then the relevant fields $\\Phi$ are $\\mathscr{V}_\\mu^A(x)$ , $\\mathscr{X}_\\mu^A(x)$ and FP ghosts (ghost and antighost) $\\mathscr{C}^A(x), \\bar{\\mathscr{C}}^A(x)$, i.e., $\\Phi^\\dagger = (\\mathscr{V}_\\mu^A,\\mathscr{X}_\\mu^A,\\mathscr{C}^A,\\bar{\\mathscr{C}}^A)$. \nIn this section, we use the Euclidean formulation. \nIn the modified Polyakov gauge and within the quadratic approximation adopted in sec.~\\ref{sec:gauge}, \n\\begin{align}\n \\partial_t \\Gamma_k\n =& \\frac12 {\\rm Tr} \\left\\{ \\left[ \\frac{\\overrightarrow{\\delta}}{\\delta \\mathscr{V}^\\dagger} \\Gamma_k \\frac{\\overleftarrow{\\delta}}{\\delta \\mathscr{V}} + R_{k} \\right]^{-1} \\cdot \\partial_t R_{k} \\right\\}\n\\nonumber\\\\&\n + \\partial_t \\frac12 {\\rm Tr}  \\{ \\ln [Q_{\\mu\\nu}^{AB}+\\delta^{AB}\\delta_{\\mu\\nu}R_{k} ] \\} \n\\nonumber\\\\&\n - \\partial_t   {\\rm Tr}  \\{ \\ln [G^{AB}+\\delta^{AB}R_{k} ] \\} \n  .\n\\end{align}\nwhere the second contribution in the right-hand side comes from the $\\mathscr{X}$ field and the last one from the ghosts fields \\cite{KMS05}, and we have used the same regulator function $R_k$ for the gluon and ghost up to the difference due to the tensor structure. \n\n\n\n\n\nWe neglect back-reactions of the $\\mathscr{V}_{0}$ potential on the other gauge fields $\\mathscr{V}_{j}$, as in the treatment \\cite{MP08}.  \nAssuming an expansion around $\\mathscr{V}_{j}=0$, $\\Gamma_{k}^{(2)}:=\\frac{\\overrightarrow{\\delta}}{\\delta \\mathscr{V}^\\dagger} \\Gamma_k \\frac{\\overleftarrow{\\delta}}{\\delta \\mathscr{V}}$ is block-diagonal like the regulators, and  the flow equation can be decomposed into a sum of two contributions:\nunder the approximation (\\ref{approx0}),\n\\begin{align}\n \\partial_t \\Gamma_k\n =& \\frac12 {\\rm Tr} \\left[ \\left( \\frac{1}{\\Gamma_k^{(2)}+R_{k}} \\right)_{\\mu\\nu} \\cdot \\partial_t R_{k,\\mu\\nu} \\right]\n\\nonumber\\\\&\n + \\partial_t {\\rm Tr}  \\{ \\ln [G^{AB}+\\delta^{AB}R_{k} ] \\}\n  ,\n\\end{align}\nwhere the gluon regulator $R_{k,\\mu\\nu}$ is a block-diagonal matrix in field space,\n\\begin{align}\n  R_{k,00} =& R_{0,k} = Z_0 R_{\\rm opt,k}(\\bm{p}^2), \\\n  R_{k,0j} = 0 = R_{k,j0} ,\n\\nonumber\\\\\n  R_{k,j\\ell} =& R_{T,k} T_{j\\ell}(\\bm{p})\n= Z_j T_{j\\ell}(\\bm{p}) R_{\\rm opt, k_T}(\\bm{p}^2) ,\n\\end{align}\nwhere $T_{j\\ell} := \\delta_{j\\ell} - \\frac{p_jp_\\ell}{p_m^2}$ is the transverse projection operator and \n$R_{\\rm opt,k}(\\bm{p}^2)$ is the (3 dim.) optimized choice \\cite{Litim00}:\n\\begin{equation}\n R_{\\rm opt,k}(\\bm{p}^2) = (k^2-\\bm{p}^2) \\theta(k^2-\\bm{p}^2) .\n\\end{equation} \n\n\n\nThe first term in the right-hand side encodes the quantum fluctuations of $\\mathscr{V}_0$, while the second one encodes those of the other components of the gauge field and ghosts.  \nIn the present truncation, the second term is a total derivative with respect to $t$, and does not receive contributions from the first term.  Therefore, we can evaluate the flow of the second contribution, and use its output $V_{T,k}(\\mathscr{V}_0)$ as an input for the remaining flow. \n\\begin{align}\n \\partial_t \\Gamma_k\n =& \\frac12 \\beta \\int \\frac{d^3p}{(2\\pi)^3} \\left[ \\left( \\frac{1}{\\Gamma_k^{(2)}\n+R_\\mathscr{V}} \\right)_{00} \\partial_t R_{0,k} \\right]\n\\nonumber\\\\&\n +   \\partial_t V_{T,k} \n  ,\n\\end{align}\nwhere for $\\omega=2\\pi Tn$\n\\begin{align}\n& V_{T,k} \n :=    {\\rm Tr}  \\{  \\ln [G^{AB}+\\delta^{AB}R_{k} ] \\} \n \\nonumber\\\\\n=&    T \\sum_{n \\in \\mathbb{Z}}  \\int \\frac{d^3p}{(2\\pi)^3}  {\\rm tr}\\ln [\\tilde{G}^{AB}(\\omega, \\bm{p})\n+\\delta^{AB}(k_T^2-\\bm{p}^2) \\theta (k_T^2-\\bm{p}^2)]\n  \\nonumber\\\\ \n =&   T \\sum_{n \\in \\mathbb{Z}}  \\int \\frac{d^3p}{(2\\pi)^3}   {\\rm tr}\\ln [ \\delta^{AB}\\bm{p}^2-D_0^2 \n+ \\delta^{AB}(k_T^2-\\bm{p}^2) \\theta (k_T^2-\\bm{p}^2) ]  \n \\nonumber\\\\\n =&    T \\sum_{n \\in \\mathbb{Z}}  4\\pi \\int_{0}^{k_T} \\frac{dp p^2}{(2\\pi)^3}   {\\rm tr}\\ln [ \\delta^{AB}k_T^2 -D_0^2   ] \n \\nonumber\\\\&\n-   T \\sum_{n \\in \\mathbb{Z}}  4\\pi \\int_{0}^{k_T} \\frac{dp p^2}{(2\\pi)^3}   {\\rm tr}\\ln [ \\delta^{AB}\\bm{p}^2-D_0^2 ] \n + V_W\n  .\n  \\label{V_{T,k}}\n\\end{align}\nHere we have introduced the Weiss potential $V_W$ which was obtained by one-loop calculation \\cite{Weiss81}:\n\\begin{align}\nV_W  =&   \n      {\\rm Tr}  \\ln [G^{AB}] \n  \\nonumber\\\\ \n =&    T \\sum_{n \\in \\mathbb{Z}}  \\int \\frac{d^3p}{(2\\pi)^3}  {\\rm tr} \\ln [\\tilde{G}^{AB}(p_0=\\omega, \\bm{p})]\n  \\nonumber\\\\ \n=&   \n    T \\sum_{n \\in \\mathbb{Z}}  \\int \\frac{d^3p}{(2\\pi)^3}   {\\rm tr}\\ln [\\bm{p}^2+(\\omega + T\\varphi)^2] \n  \\nonumber\\\\&\n +   T \\sum_{n \\in \\mathbb{Z}}  \\int \\frac{d^3p}{(2\\pi)^3}  {\\rm tr}\\ln [\\bm{p}^2+(\\omega - T\\varphi)^2]\n  ,\n\\end{align}\nwhere we have neglected the $\\varphi$-independent (or $\\mathscr{V}_0$ independent) contributions. \n\n\n\n\\begin{figure}[ptb\n\\begin{center}\n  \\includegraphics[width=5.5cm]{Fig-ep-184\/V_w-SU2.eps}\n  \\includegraphics[width=5.5cm]{Fig-ep-184\/L-SU2.eps}\n\\end{center}\n  \\caption{\\small \n(The upper panel)  (normalized) SU(2) Weiss potential $\\hat V_{W}$ as a function of $\\varphi$.\n(The lower panel) SU(2) Polyakov loop $L$ as a function of $\\varphi$,\n$L = \\cos \\left(  \\frac{\\varphi}{2}    \\right) \n$.\n}\n  \\label{fig:Weiss}\n\\end{figure}\n\n\nThe closed form of the Weiss potential is obtained after summing up the Matsubara frequencies:\n\\begin{align}\n V_W(\\varphi)\n=&    T^4 \\left[ - \\frac{1}{6}  (\\varphi-\\pi)^2 +  \\frac{1}{12\\pi^2}   (\\varphi-\\pi)^4   +   \\frac{\\pi^2}{12} \\right]\n\\nonumber\\\\\n ({\\rm mod} \\ 2\\pi)\n .\n\\end{align}\nThe Weiss potential $V_W$ is $g^2$ independent and the overall curve scales as $T^4$.\n  $V_W$ has symmetries: \n$V_W(-\\varphi)=V_W(\\varphi)$ and $V_W(\\varphi+2\\pi n)=V_W(\\varphi)$.\n  $V_W(\\varphi)$   has minima at $\\varphi=2\\pi n$, and the Polyakov loop has the nonvanishing value $L=\\cos  \\frac{\\varphi}{2}=(- 1)^n$, implying deconfinement. See Fig.~\\ref{fig:Weiss}.\n  Therefore, $V_W(\\varphi)$ is considered to be valid at very high temperature where the perturbation theory is trustworthy. \n  In  Fig.~\\ref{fig:VPreWeiss3D}, we observe\n\\begin{equation}\n\\lim_{k \\downarrow 0} V_{T,k} =  V_{T,0}  =  V_{W}   \n  ,\n\\quad\n\\lim_{k \\uparrow \\infty} V_{T,k} =   0 .\n\\end{equation}\n\n\n\n\n\\begin{figure}[ptb\n\\begin{center}\n  \\includegraphics[width=6cm]{Fig-ep-184\/PreWeiss.eps}\n\\end{center}  \n  \\caption{$\\hat V_{T,k}$ for different values of $\\hat k$ [reprinted from \\cite{MP08}].}\n  \\label{fig:VPreWeiss3D}\n\\end{figure}\n\n\n\n\nAfter integrating over the fields other than $\\mathscr{V}_0$, we are lead to  the effective action of $\\mathscr{V}_0$, \n\\begin{align}\n \\Gamma_{k}[\\mathscr{V}_0] \n=&   \\beta \\int d^3x \\left\\{ -  \\frac{1}{2}Z_0 \\mathscr{V}_0(\\bm{x})  \\partial_j \\partial_j  \\mathscr{V}_0(\\bm{x})    + V_{{\\rm eff},k}^{\\rm glue}[\\mathscr{V}_0] \\right\\}  ,\n \\nonumber\\\\ \n  V_{{\\rm eff},k}^{\\rm glue}[\\mathscr{V}_0] =& V_{T,k}[\\mathscr{V}_0] + \\Delta V_k[\\mathscr{V}_0]   .\n  \\label{V_g}\n\\end{align}\nThen the flow equation is reformulated for $\\Delta V_k$ with the external input $V_{T,k}$:\n\\begin{equation}\n \\beta    \\partial_{t}  (\\Delta V_k[\\mathscr{V}_0])\n = \\frac12 \\beta \\int \\frac{d^3p}{(2\\pi)^3} \\left[ \\left( \\frac{1}{\\Gamma_k^{(2)}+R_{k}} \\right)_{00} \\partial_t R_{0,k} \\right]\n ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\Gamma_{k}^{(2)}[\\mathscr{V}_0] \n=   \\beta  \\left\\{  Z_0   \\bm{p}^2       + \\partial_{\\mathscr{V}_0}^2 V_{k}[\\mathscr{V}_0] \\right\\}  \n  .\n\\end{equation}\n\n\n\n\n\nUsing the specific regulator, \n$\n  R_{0,k}\n= Z_0 (k^2-\\bm{p}^2)\\theta(k^2-\\bm{p}^2) \n$, \nwhich yields\n\\begin{align}\n  \\partial_t R_{0,k}\n=  \\left[ \\partial_{t} Z_0   (k^2-\\bm{p}^2)  + 2 Z_0  k^2 \\right] \\theta(k^2-\\bm{p}^2)  \n ,\n\\end{align}\nwe can perform the momentum integration analytically. \n\\begin{align}\n & \\beta    \\partial_{t}  (\\Delta V_k[\\mathscr{V}_0])    \n\\nonumber\\\\\n=&  \\frac23  \\frac{1}{(2\\pi)^2} \\frac{  (\\eta_k\/5+1)  k^5 }{     Z_k^{-1} g^2 \\beta^2 \\partial_{\\varphi}^2 (V_{T,k}[\\mathscr{V}_0] + \\Delta V_k[\\mathscr{V}_0] )  + k^2   }\n ,\n\\end{align}\nwhere we have introduced \n the running coupling $\\alpha_k$ defined by \n\\begin{equation} \n g_k^2 := Z_k^{-1} g^2, \\quad \\alpha_k := \\frac{g_k^2}{4\\pi} =  Z_k^{-1} \\frac{g^2}{4\\pi}\n  ,\n\\end{equation}\nand the anomalous dimension $\\eta_k$  defined by\n\\begin{equation}\n \\eta_k := \\partial_{t} \\ln Z_k = - \\partial_{t} \\ln \\alpha_k .\n\\end{equation}\n\n\\begin{figure}[ptb\n\\begin{center}\n \\includegraphics[width=7.5cm]{Fig-ep-184\/coupling.eps}\n\\end{center} \n  \\caption{The running gauge coupling constant $\\alpha_s$ for temperatures $T=0,150,300,600$ MeV [reprinted from \\cite{MP08}].}\n  \\label{fig:alpha}\n\\end{figure}\n\n\\begin{figure}[ptb\n\\begin{center}\n  \\includegraphics[width=7.5cm]{Fig-ep-184\/VeffforT.eps}\n\\end{center}  \n  \\caption{Full effective potential $\\hat V_{{\\rm eff}}^{\\rm glue}$, \n  normalized to 0 at $\\varphi =0$ [reprinted from \\cite{MP08}].}\n  \\label{fig:Veff}\n\\end{figure}\n\n\n\nBy introducing the dimensionless RG scale $\\hat{k}$ and the dimensionless effective potential $\\hat{V}$ defined by\n\\begin{equation}\n \\hat{k} := \\beta k = k\/T  ,\n \\quad \n \\hat{V} := \\beta^4 V =V\/T^4  \n  ,\n  \\label{rescaling}\n\\end{equation}\nthe flow equation  is simplified as\n\\begin{equation}\n     \\partial_{\\hat k}  \\Delta \\hat V_{\\hat k}[\\mathscr{V}_0]    \n=    \\frac{1}{6\\pi^2} \\frac{  (1+\\eta_k\/5 )  \\hat k^2 }{   1+\\frac{4\\pi\\alpha_k}{\\hat k^2} \\partial_{\\varphi}^2 (\\hat V_{T,\\hat k}[\\mathscr{V}_0] + \\Delta \\hat V_{\\hat k}[\\mathscr{V}_0] )     }\n ,\n \\label{flow-eq}\n\\end{equation}\nwhere all scales are measured in units of temperature. \nIt turns out that the input in solving the flow equation is just a running gauge coupling constant $\\alpha_k$.  A specific choice for the running gauge coupling constant is given in Fig.~\\ref{fig:alpha}. For the derivation from the renormalization group, see \\cite{BG05}. \nThe flow is initialized in the broken phase at any temperature. \nBy solving the flow equation in a numerical way with an input for the running gauge coupling given in Fig.~\\ref{fig:alpha}, the full effective potential $\\hat V_{\\rm eff}$ (normalized to 0 at $\\varphi =0$) is obtained in Fig.~\\ref{fig:Veff} for various temperature.\n\n\n\n\nAccording to \\cite{MP08}, a second order phase transition occurs at a critical temperature\n\\begin{equation}\n     T_d = 305_{-55}^{+40} \\text{MeV}, \\quad\nT_d\/\\sqrt{\\sigma}=0.69_{-.12}^{+.04}\n ,\n\\end{equation}\nwith the string tension $\\sqrt{\\sigma}=440$ MeV.\nThis agrees within errors with the lattice result\n$T_d\/\\sqrt{\\sigma}=0.709$.\nMoreover, these results were confirmed by considering another gauge \\cite{BGP07}.\n\n\n\n\n\\section{Understanding the existence of confinement transition according to the Landau-Ginzburg argument}\\label{sec:Landau}\n\nIn this section, we show that some qualitative aspects of the deconfinement\/confinement transition found in the previous section can be understood without detailed numerical works, although the precise value of the transition temperature $T_d$ cannot be determined without them. \n\nFor $G=SU(2)$ in the pure Yang-Mills limit $m_q \\rightarrow \\infty$, the effective potential $V_{\\rm glue}(L)$ for the Polyakov loop $L$  must be invariant under the center symmetry $Z(2)$.  Therefore, $V_{\\rm glue}(L)$ is an even function of $L$, i.e., $V_{\\rm glue}(L)=V_{\\rm glue}(-L)$ where $L$ is real-valued $L=L^*$.  \nThus the Landau-Ginzburg argument suggests that the effective potential $V_{\\rm eff}^{\\rm glue}(L)$ for $G=SU(2)$ has the power-series expansion in  $L$  near the transition point $L=0$:  \n\\begin{equation}\n V_{\\rm eff}^{\\rm glue}(L) =  c_0 + \\frac{c_2}{2} L^2   + \\frac{c_4}{4}  L^4  + O(L^6)\n .\n\\end{equation}  \nAs the vacuum is specified as minima of the effective potential,  the confinement\/deconfinement transition temperature $T_d$ is determined from the condition $c_2(T_d)=0$ so that the low-temperature ($T<T_d$) confinement phase $\\langle L \\rangle =0$ is realized for $c_2(T)>0$, while the high-temperature ($T>T_d$) deconfinement phase $\\langle L \\rangle \\not=0$ is realized for $c_2(T)<0$, provided that the positivity $c_4(T)>0$ is maintained across the transition temperature. \nConsequently, the transition is of the 2nd order. \n\nIndeed, we confirm that the Landau-Ginzburg description is correct and valid for the confinement\/deconfinement transition, by making use of the flow equation given in the previous section. \nThis is a microscopic justification of the Landau-Ginzburg argument for the confinement\/deconfinement transition.\nIn our treatment, however, it is more convenient to write the effective potential $V_{\\rm eff}^{\\rm glue}$ in terms of the angle variable $\\varphi$ (rather than $L$) around the transition point $\\varphi=\\pi$ (instead of  $L=0$).\nDefining $\\tilde\\varphi :=\\varphi-\\pi$, we find that \n$V_{\\rm eff}^{\\rm glue}(\\tilde\\varphi)$ must be an even function $V_{\\rm eff}^{\\rm glue}(\\tilde\\varphi)=V_{\\rm eff}^{\\rm glue}(-\\tilde\\varphi)$  due to the center symmetry  and hence odd terms (e.g., $\\tilde\\varphi$,  $\\tilde\\varphi^3$) do not appear: \n\\begin{equation}\n V_{\\rm eff}^{\\rm glue}(\\tilde\\varphi)  =  C_{0} +  \\frac{C_{2}}{2} \\tilde \\varphi^2 + \\frac{C_{4}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n  .\n\\end{equation}\nAt sufficiently high temperature, we observe that $C_2(T)<0$ and hence $V_{\\rm eff}^{\\rm glue}$ has the minimum at   $\\tilde \\varphi \\not=0$ ($\\Longleftrightarrow L\\not=0$) leading to  deconfinement. \nIn order to show the existence of the confinement\/deconfinement transition at $T=T_d$,  $C_2(T)$ must change  the signature $C_2(T)>0$ below this  temperature $T<T_d$ and hence the minimum occurs at  $\\tilde \\varphi=0$ ($\\Longleftrightarrow L=0$) leading to confinement.\nTherefore, the confinement\/deconfinement temperature $T_d$ is  determined by $C_2(T_d)=0$, provided that the positivity $C_4(T)>0$ is maintained.\n\n\nFor this purpose, we study the scale dependent effective potential $V_{\\rm eff,k}^{\\rm glue}$ at $k>0$\n\\begin{equation}\n V_{\\rm eff,k}^{\\rm glue}  =  C_{0,k} +  \\frac{C_{2,k}}{2} \\tilde \\varphi^2 + \\frac{C_{4,k}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n  ,\n\\end{equation}\nand see how it evolves  towards the limit $k\\rightarrow 0$ according to the flow equation to obtain the physical effective potential $V_{\\rm eff}^{\\rm glue}:=V_{\\rm eff,k=0}^{\\rm glue}$.\n\nAs in (\\ref{V_g}), $V_{\\rm eff,k}^{\\rm glue}$ is decomposed into two pieces:\n\\begin{equation}\n V_{\\rm eff,k}^{\\rm glue}  =   \\hat V_{T,\\hat k} +  \\Delta \\hat V_{\\hat k} \n  ,\n  \\label{Veff}\n\\end{equation}\nwhere we have defined the dimensionless potential  according to the rescaling (\\ref{rescaling}).\nThe first part $\\hat V_{T,\\hat k}$ is the ($k$-dependent) perturbative part (\\ref{V_{T,k}}) obtained essentially by the one-loop calculation with the regulator function $R_k$ being included.  For this part, the closed analytical form can be obtained, see Appendix~\\ref{app:coefficient}.\nWhile the second part $\\Delta \\hat V_{\\hat k}$ represents the non-pertubative part which is initially zero $\\Delta \\hat V_{\\hat k}|_{k=\\Lambda}=0$ and is generated in the evolution of the renormalization group.  This part is obtained only by solving the flow equation (\\ref{flow-eq}) and its analytical form is not available (at this moment).\n\n\nWe expand $\\hat V_{T,\\hat k}$ in powers of $\\tilde\\varphi=\\varphi-\\pi$: \n\\begin{equation}\n \\hat V_{T,\\hat k}\n=  A_{0,k} +  \\frac{A_{2,k}}{2} \\tilde \\varphi^2 + \\frac{A_{4,k}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n  ,\n  \\label{VTcoeff}\n\\end{equation}\nwhere coefficients are drawn as functions of $k$ in Fig.~\\ref{fig:A_{2,k}}, see Appendix~\\ref{app:coefficient} for their closed analytical forms. \n \n \n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3.0in,height=1.5in]{Fig-ep-184\/A2_k.eps}\n\\includegraphics[width=3.0in,height=1.5in]{Fig-ep-184\/A4_k.eps}\n\\caption{$A_{2,k}$ and $A_{4,k}$ as functions of $\\hat k$.}\n\\label{fig:A_{2,k}}\n\\end{figure}\n\n\nSuppose that $\\Delta \\hat V_{\\hat k}$ is of the form:\n\\begin{equation}\n       \\Delta \\hat V_{\\hat k}     \n=    a_{0,k} +  \\frac{a_{2,k}}{2} \\tilde \\varphi^2   + \\frac{a_{4,k}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n .\n \\label{V_T}\n\\end{equation}\nA flow equation (\\ref{flow-eq}) for the effective potential \n(\\ref{V_g}) is reduced to a set of coupled flow equations for coefficients in the effective potential (\\ref{Veff}) with (\\ref{VTcoeff}) and (\\ref{V_T}):\n\\begin{align}\n    \\partial_{\\hat k}  a_{2,k}     \n=&  -  \\frac{(1+\\frac15 \\eta_k ) \\hat k^2 }{6\\pi^2} \\frac{ \\frac{4\\pi\\alpha_k}{\\hat k^2}(A_{4,k}+a_{4,k})   }{[1+\\frac{4\\pi\\alpha_k}{\\hat k^2}   (A_{2,k}+a_{2,k})]^2} \n ,\n\\nonumber\\\\\n\\partial_{\\hat k}  a_{4,k}     \n=& +   \\frac{(1+\\frac15 \\eta_k ) \\hat k^2 }{6\\pi^2} \\frac{6 [\\frac{4\\pi\\alpha_k}{\\hat k^2}(A_{4,k}+a_{4,k})]^2 }{[1+\\frac{4\\pi\\alpha_k}{\\hat k^2}   (A_{2,k}+a_{2,k})]^3}\n ,\n\\nonumber\\\\\n\\vdots  \n\\end{align}\nwhich are coupled first-order ordinary but nonlinear differential equations for coefficients.\nIn Appendix~\\ref{app:flow-equation}, we see  that this form (\\ref{V_T}) is justified as a solution of the flow equation. In fact, it is easy to see  that  \n$\\partial_{\\hat k} a_{1,k} =0$ and\n$\\partial_{\\hat k} a_{3,k}=0$ are guaranteed from the flow equation, if the effective potential has no odd terms  at arbitrary $k$.\nTherefore, if an initial condition, $a_{1,k} =0=a_{3,k}$ at $k=\\Lambda$ is imposed, then   \n$a_{1,k} \\equiv 0$ and $a_{3,k} \\equiv 0$ \nare maintained for $0 \\le k \\le \\Lambda$ by solving the flow equation.\nIn performing numerical calculations, however,  one must truncate the infinite series of differential equations up to some finite order to obtain manageable set of equations. \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3.0in,height=1.5in]{Fig-ep-184\/coupling1_k.eps}\n\\includegraphics[width=3.0in,height=1.5in]{Fig-ep-184\/coupling2_k.eps}\n\\caption{The running gauge coupling constant $\\alpha_k$ \nat $T=0.001, 0.125, 0.25, 0.50, 1.0$ GeV, from the top at the lowest temperature $T=0.001$GeV to the bottom  at the highest temperature $T=1.0$ GeV.\n(The upper panel) $\\alpha_k$ as functions of $k$.\n(The lower panel) $\\alpha_k$ as functions of $\\hat{k}$.\nFor a given temperature, there is a critical value $\\hat k_c$ separating the deep IR region (\\ref{coupling-2}) from the higher momentum region (\\ref{coupling-1}).\nThe discontinuity of the derivative seen at  $\\hat k_c$ comes from a crude approximation in which we have taken into account just the first linear term (i.e., $c_1=c_2=\\cdots=0$) in the expansion (\\ref{coupling-2}), and can be avoided if we take into account higher order terms as explained below (\\ref{coupling-2}). However, this is not essential to see qualitative behaviors of the solution of the flow equation.\n}\n\\label{fig:alpha_k}\n\\end{figure}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3.0in,height=1.5in]{Fig-ep-184\/anomalous_k.eps}\n\\caption{The anomalous dimension $\\eta_k$  as functions of $\\hat k$\nat $T=0.001, 0.125, 0.25, 0.50, 1.0$ GeV.\nIn each graph for a given temperature, there is a critical value $\\hat k_c$ of $\\hat k$ separating the deep IR region $\\eta_k \\simeq -1$ from the higher momentum (intermediate and UV) region $\\eta_k >0$.\nThe temperature is distinguished by  $\\hat k_c$ ranging from  the smallest value at the highest temperature $T=1.0$GeV to the largest value at the lowest temperature $T=0.001$ GeV where $\\eta_k \\simeq -1$ for $\\hat k<\\hat k_c$ and $\\eta_k \\simeq 0$ for $\\hat k>\\hat k_c$.\nThe discontinuity of the derivative seen at  $\\hat k_c$ is due to the same reason as that explained in  Fig.~\\ref{fig:alpha_k} and  is not essential to see qualitative behaviors of the solution of the flow equation.\n}\n\\label{fig:eta_k}\n\\end{figure}\n\n\n\nWe can understand qualitatively why a 2nd order phase transition from the deconfinement phase to the confinement phase can occur by lowering the temperature.\n\n\nThe flow starts from $a_{2,k}=0$ and hence $C_{2,k}=A_{2,k}+a_{2,k}<0$ (because of $A_{2,k}<0$) at $k=\\Lambda \\gg 1$.\nWe assume $C_{4,k}=A_{4,k}+a_{4,k}>0$ for $0 \\le k \\le \\Lambda$, as a necessary condition for realizing a 2nd order transition. Otherwise, we must consider the higher-order terms, e.g. $O(\\varphi^6)$.\n (This assumption is assured to be true by  numerical calculations of the full effective potential \\cite{MP08,BGP07}, as reproduced in the previous section.)   \nThis assumption allows us to analyze just one differential equation for obtaining qualitative understanding:\n\\begin{align}\n    \\partial_{\\hat k}  a_{2,k}     \n=&  -  \\frac{(1+\\frac15 \\eta_k )  }{6\\pi^2} \\frac{ 4\\pi\\alpha_k(A_{4,k}+a_{4,k})   }{[1+\\frac{4\\pi\\alpha_k}{\\hat k^2}   (A_{2,k}+a_{2,k})]^2} \n .\n \\label{flow-2}\n\\end{align}\nThen the right-hand side of (\\ref{flow-2}) is negative, since the running coupling constant $\\alpha_k$ is positive and $1+\\frac15 \\eta_k$ is positive, see Fig.~\\ref{fig:alpha_k} and Fig.~\\ref{fig:eta_k}. \nConsequently, $a_{2,k}$ started at zero becomes positive $a_{2,k}>0$ just below $\\Lambda$ and increases (monotonically) as $k$ decreases.\nSee Fig.~\\ref{fig:mA_k}.\n\n\n\nNote that the denominator can vanish  \n$1+\\frac{4\\pi\\alpha_k}{\\hat k^2}   (A_{2,k}+a_{2,k})=0$ at some $k^*$ (since $C_{2,k}=A_{2,k}+a_{2,k}<0$ or $0<a_{2,k}<-A_{2,k}$) where the right-hand side of (\\ref{flow-2}) becomes negative infinity and $a_{2,k}$ blows up there. \nTo avoid this pathology and to obtain the solution all the way down to the limit $k \\rightarrow 0$, $a_{2,k}$ must grow relatively rapidly so that $|A_{2,k}+a_{2,k}| \\ll 1$ towards the limit $k \\rightarrow 0$.  \n\n\n\n\n\n\n\nAn important observation of the flow equation (\\ref{flow-2}) is that  the explicit temperature-dependence comes from the running coupling constant alone.\nAt zero temperature, the running coupling constant is well parameterized by the fitting function \\cite{FA02}:\n\\begin{equation}\n     \\alpha_k=\\frac{4\\pi \\times 0.709\/N_c}{\\ln [e+a_1 (k^2)^{a_2}+b_1 (k^2)^{b_2}]}\n ,\n \\label{coupling-0}\n\\end{equation}\nwhere  \n$\n     a_1 = 5.292, \\ a_2= 2.324, \\ b_1 = 0.034, \\ b_2 = 3.169 \n .\n$\nin units of GeV.\n \nFor the perturbative region $k \\gg T$, i.e., $\\hat k \\gg 1$, we adopt this form:\n$\nk^2=T^2 \\hat k^2  \n$,\n\\begin{equation}\n     \\alpha_k = \\frac{g_k^2}{4\\pi} =\\frac{4\\pi \\times 0.709\/N_c}{\\ln[e+a_1 (T^2 \\hat k^2)^{a_2}+b_1 (T^2 \\hat k^2)^{b_2}]}\n .\n \\label{coupling-1}\n\\end{equation}\nFor the nonperturbative region $k < 2\\pi T$, i.e., $\\hat k < O(1)$, we adopt the running coupling which is governed by an infrared fixed point \\cite{BG05}\n\\begin{align}\n     \\alpha_k =&  \\alpha_{3d}^* \\frac{k}{T} + c_1 \\left( \\frac{k}{T} \\right)^2 + c_2 \\left( \\frac{k}{T} \\right)^3 +  \\cdots\n\\nonumber\\\\ \n=& \\alpha_{3d}^* \\hat k + c_1 \\hat k^2 + c_2 \\hat k^3 +  \\cdots \n ,\n \\label{coupling-2}\n\\end{align}\nwhere  coefficients $c_1, c_2,...$ are determined such that the coupling at zero temperature   (\\ref{coupling-1})  and its derivative with respect to $k$ are connected continuously with this ansatz (\\ref{coupling-2}) at the scale set by the lowest non-vanishing bosonic  Matsubara-mode $\\omega=2\\pi T$.\nHowever, the running coupling constant at small momenta (\\ref{coupling-2}) does not contribute to the explicit $T$-dependence in the scaled flow equation, since it is written in terms of the scaled $\\hat{k}$ alone and hence denoted by a common curve going through the origin for any temperature $T$ in the second figure of Fig.~\\ref{fig:alpha_k}. \nTherefore, the running coupling at very small momentum region can not be responsible for the confinement\/deconfinement transition at finite temperature, if this observation is correct. \nAs can be seen from the second figure of Fig.~\\ref{fig:alpha_k}, the dominant contribution comes from the intermediate momentum region above $O(1)$ GeV. \nThus, we can avoid the issue of gauge-fixing artifact in the deep IR region due to Gribov copies in the zero-temperature case, see e.g. \\cite{FMP08,Kondo09} and reference therein. \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=2.5in,height=1.5in]{Fig-ep-184\/low-temp-a_k.eps}\n\\includegraphics[width=2.5in,height=1.5in]{Fig-ep-184\/low-temp-a_k-A_k.eps}\n\\caption{$a_{2,k}$ vs. $-A_{2,k}$ as functions of $\\hat k$ for $T<T_d$.}\n\\label{fig:mA_k}\n\\end{figure}\n\nWe consider a solution of the reduced (or normalized) flow equation as a function of $\\hat k$, rather than $k$, for a given temperature $T$. \nThen  the difference between high and low temperature phases attributes to the behavior of the running coupling constant $\\alpha_k$  as a function of $k=T\\hat k$, which brings the explicit $T$ dependence to the reduced flow equation. \nIn the case of  high-temperature $T \\gg 1$, \n$k=T\\hat k$ becomes large for a wide range of $\\hat k$ and \nthe running coupling constant $\\alpha_k$ remains relatively small.  The resulting slow increase of $a_{2,k}$ keeps $a_{2,k}$ small  such  that $C_{2,k}=A_{2,k}+a_{2,k}<0$ or $a_{2,k} <-A_{2,k}$ even at $k=0$. This leads to the center symmetry breaking at high-temperature.\n\nIn the case of low temperature  $T \\ll1 $, \n$k=T\\hat k$ becomes small  for the same range of $\\hat k$ and  \nthe running coupling constant $\\alpha_k$ gets into the intermediate region of O(1) GeV rapidly and becomes larger as the temperature becomes smaller.  \nAt sufficiently low temperature, $a_{2,k}$ increases in decreasing $\\hat k$ so rapidly  that  $a_{2,k}$ eventually reaches to the point $A_{2,k}+a_{2,k}=0$ or $a_{2,k}=-A_{2,k}$ at a certain value $\\hat k=\\hat k_0$.  In other words, the graph of $a_{2,k}$ intersects with that of $-A_{2,k}$ at  $\\hat k=\\hat k_0$.\nIn the region $0<k<k_0$ where $C_{2,k}=A_{2,k}+a_{2,k}>0$ or $a_{2,k}>-A_{2,k}$, the flow equation reads\n\\begin{align}\n    \\partial_{\\hat k}  a_{2,k}     \n\\simeq  -  \\frac{(1+\\frac15 \\eta_k ) }{6\\pi^2} \\frac{ (A_{4,k}+a_{4,k}) \\hat k^4  }{4\\pi\\alpha_k   (A_{2,k}+a_{2,k})^2} \n ,\n\\end{align}\nthe right-hand-side gets small negative, and $a_{2,k}$ becomes flat near the IR limit. See Fig.~\\ref{fig:mA_k}.\nFinally,  $a_{2,k}$ reaches the value realizing  $C_{2,k}=A_{2,k}+a_{2,k}>0$ or $a_{2,k}>-A_{2,k}$ at $k=0$. \nThis leads to the recovery of the center symmetry.\nThe difference is clearly seen from the second figure of Fig.~\\ref{fig:alpha_k} where the running gauge coupling $\\alpha_k$ is drawn as a function of $\\hat k$ for various temperatures. \n\nIn our treatment, the difference between the three-dimensional RG scale $k_T$ and the four-dimensional one $k$ is neglected by equating two scales $k_T=k$ just for simplifying the analysis, since it is enough for obtaining a qualitative understanding for the transition. This is not be the case for obtaining quantitative results, see Appendix C of \\cite{MP08} for the precise treatment on this issue. \n\n\n\n\n\\section{Quark part and gauged nonlocal NJL model}\\label{sec:NJL}\n\nWe examine the quark self-interaction part\n$S_{\\rm int} =   \\int d^Dx \\int d^Dy  \\frac{1}{2}  \\mathscr{J}^{\\mu A}(x)  g^2(Q^{-1}[\\mathscr{V}])_{\\mu\\nu}^{AB}(x,y) \\mathscr{J}^{\\nu B}(y)\n$.\nIn  estimating the effect of $Q^{-1}[\\mathscr{V}]$, we take the  same  approximation as the above. \nConsequently, the inverse  $(Q^{-1})_{\\mu\\nu}^{AB}[\\mathscr{V}]$  is diagonal in the Lorentz indices:\n\\begin{align} \n& (Q^{-1})_{\\mu\\nu}^{AB}[\\mathscr{V}]  \n= g^{\\mu\\nu} (G^{-1})^{AB}[\\mathscr{V}]  \n\\nonumber\\\\\n=&     g^{\\mu\\nu}\n  \\begin{pmatrix}\n  \\frac12 [F_{\\varphi}+F_{-\\varphi}] & -\\frac{1}{2i} [F_{\\varphi}-F_{-\\varphi}] & 0 \\cr\n  \\frac{1}{2i} [F_{\\varphi}-F_{-\\varphi}] & \\frac12 [F_{\\varphi}+F_{-\\varphi}] & 0 \\cr\n  0 & 0 & F_{0} \\cr\n  \\end{pmatrix} \n ,\n\\end{align}\nwhere $F_{\\varphi}$ is defined by\n\\begin{align} \nF_{\\varphi}(i\\partial) :=&  \\frac{1}{(i\\partial_\\ell)^2+(i\\partial_0+T\\varphi)^2}\n\\nonumber\\\\ \n=&  \\frac{1}{(i\\partial_\\mu)^2+(T\\varphi)^2+2T\\varphi i\\partial_0}\n . \n\\end{align}\nIn what follows, we consider only the diagonal parts of $(G^{-1})^{AB}[\\mathscr{V}]$.  \nThis is achieved by the procedure\n\\begin{equation}\n  \\frac{g^2}{2} (Q^{-1})_{\\mu\\nu}^{AB}(x,y) = g^{\\mu\\nu}   \\delta^{AB} \\mathcal{G}(x-y)\n  ,\n\\end{equation}\nwhich yields\n\\begin{equation}\n \\mathcal{G}(x-y)  = \\frac{g^2}{2} (Q^{-1})_{\\mu\\nu}^{AB}(x,y)  \\frac{g_{\\mu\\nu}}{D}  \\frac{\\delta^{AB}}{N_c^2-1} \n  .\n\\end{equation}\nThen the nonlocal interaction is obtained as\n\\begin{equation}\n \\mathcal{G}(x-y) \n=  \\frac{g^2}{2}  \\frac{{\\rm tr}(G^{-1})}{N_c^2-1} \n=  \\frac{g^2}{2} \\frac{F_{\\varphi}+F_{-\\varphi}+F_{0}}{3}\n  .\n\\end{equation}\nThis approximation is used just for simplifying the Fierz transformation performed below and hence it can be improved by taking into account the off-diagonal parts of $G^{-1}$ if it is necessary to do so.  \n\n\nFor $D=4$,  we use the Fierz identity \\cite{Fierz} to rewrite the nonlocal current-current interaction as\n\\begin{align}\n   S_{\\rm int}   \n   =& \\int d^4x \\int d^4y      \\mathscr{J}^{\\mu A}(x) \\mathcal{G}(x-y) \\mathscr{J}^{\\mu A}(y)\n  \\nonumber\\\\\n=&    \\int d^4x  \\int d^4y   \\mathcal{G}(x-y) \\sum_{\\alpha} c_\\alpha (\\bar{\\psi}(x) \\Gamma_\\alpha   \\psi(y) ) \n\\nonumber\\\\& \n \\times (\\bar{\\psi}(y) \\Gamma_\\alpha  \\psi(x) ) \n  \\nonumber\\\\\n   =&   \\int d^4x   \\int d^4z  \\mathcal{G}(z) \\sum_{\\alpha} c_\\alpha \\{ \\bar{\\psi}(x+z\/2) \\Gamma_\\alpha   \\psi(x-z\/2) \\}   \n  \\nonumber\\\\&\n\\times  \\{ \\bar{\\psi}(x-z\/2) \\Gamma_\\alpha  \\psi(x+z\/2) \\} ,\n\\end{align}\nwhere the $\\Gamma_\\alpha$ are a set of Dirac spinor, color and flavor matrices, resulting from the Fierz transform, with the property $\\gamma_0 \\Gamma_\\alpha^\\dagger \\gamma_0=\\Gamma_\\alpha$. \nAlthough the Fierz transformation induces mixings and recombinations among operators, the resulting theory must maintain the symmetries of the original QCD Lagrangian. \nA minimal subset of operators satisfying the global chiral symmetry $SU(2)_L \\times SU(2)_R$ which governs low-energy QCD with two-flavors  is the color-singlet of scalar-isoscalar and pseudoscalar-isovector operators \nThus, by restricting $\\Gamma_\\alpha$ hereafter to  \n\\begin{equation}\n\\Gamma_\\alpha:=(\\mathbf{1}, i\\gamma_5 \\vec{\\tau})\n\\end{equation}\nand ignoring other less relevant operators (vector and axial-vector terms in color singlet and color octet channels), \n we arrive at a nonlocal gauged NJL model   \n\\begin{align}\n   S_{\\rm eff}^{\\rm gNJL} =&  \\int d^4x \\bar\\psi(x) ( i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{V}] - \\hat m_q+i\\gamma^0 \\mu_q) \\psi(x) + S_{\\rm int} ,\n \\nonumber\\\\\n  S_{\\rm int} =&  \\int d^4x \\int  d^4z \\mathcal{G}(z) [\\bar \\psi(x+z\/2) \\Gamma_\\alpha \\psi(x-z\/2)  \n \\nonumber\\\\&\n\\times  \\bar \\psi(x-z\/2)  \\Gamma_\\alpha \\psi(x+z\/2)    ] \n .\n \\label{gNJL}\n\\end{align}\nThis form is regarded as a gauged version of the nonlocal NJL model proposed in \\cite{HRCW08}.    \nThe function $\\mathcal{G}(z)$ is replaced by a coupling constant $G$ times a normalized distribution $\\mathcal{C}(z)$:\n\\begin{equation}\n \\mathcal{G}(z)  :=  \\frac{G}{2} \\mathcal{C}(z) ,\n \\quad\n \\int d^4z \\mathcal{C}(z) = 1 \n .\n\\end{equation}\nThe standard (local) gauged NJL model follows for the limiting case $\\mathcal{C}(z)=\\delta^4(z)$ \nwith\n$\\int d^4z \\mathcal{C}(z)=1$.\n\n\n\nIn contrast to \\cite{HRCW08}, however, $G$ and $\\mathcal{C}$ are determined in conjunction with the behavior of the Polyakov loop $L$ or $\\varphi$ at temperature $T$:\nusing the Fourier transform $\\tilde{\\mathcal{G}}(p)$ of $\\mathcal{G}$, they are expressed as\n\\begin{equation}\n   \\frac{G}{2}  \n=   \\tilde{\\mathcal{G}}(p=0) ,\n   \\quad\n   \\tilde{\\mathcal{C}}(p) = \\tilde{\\mathcal{G}}(p)\/\\tilde{\\mathcal{G}}(p=0)\n  ,\n  \\label{G}\n\\end{equation}\nwhere \n\\begin{align}\n \\tilde{\\mathcal{G}}(p)\n=&   \\frac{g^2}{2}  \\frac{\\tilde{F}_{\\varphi}(p)+\\tilde{F}_{-\\varphi}(p)+\\tilde{F}_{0}(p)}{3} , \n\\\\\n\\tilde{F}_{\\varphi}(p)=& \\frac{1}{p^2+(T\\varphi)^2+2T\\varphi p_0} .\n\\nonumber\n\\end{align}\nNote that $\\tilde{F}_{\\varphi}(p=0)$ and hence $G$ diverge at $T=0$.\nThis comes from an improper treatment of the $T=0$ part.\nTo avoid  this IR divergence at $T=0$, we add the $T=0$ contribution $M_0^2 \\simeq M_X^2$ and replace $F_{\\varphi}(i\\partial)$ by  \n\\begin{align} \nF_{\\varphi}(i\\partial) \n&= \\frac{1}{(i\\partial_\\ell)^2+(i\\partial_0+T\\varphi)^2+M_0^2}\n\\nonumber\\\\\n=& \\frac{1}{(i\\partial_\\mu)^2+(T\\varphi)^2+2T\\varphi i\\partial_0+M_0^2}\n , \n\\end{align}\nand\n\\begin{align}\n\\tilde{F}_{\\varphi}(p)\n&= \\frac{1}{\\bm{p}^2+(p_0+T\\varphi)^2+M_0^2}\n\\nonumber\\\\\n=& \\frac{1}{p^2+(T\\varphi)^2+2T\\varphi p_0+M_0^2} .\n\\end{align}\nIn fact, such a contribution $\\frac12 M_0^2 $ comes in $G^{AB}$ as an additional term $M_0^2 \\delta^{AB}$ from the $O(\\mathscr{X}^4)$ terms (Note that $O(\\mathscr{X}^3)$ terms are absent for $G=SU(2)$), as already mentioned in the above. \n\n\nAnother way to avoid this IR divergence is to introduce the regulator term which is needed to improve the one-loop perturbative result and obtain a nonperturbative one according to the Wilsonian renormalization group:\n\\begin{align}\n \\Delta S_k =& \\int d^Dx \\frac12 \\mathscr{X}_\\mu^A(x) [\\delta^{AB} g_{\\mu\\nu} R_k(i\\partial)] \\mathscr{X}_\\nu^B(x) \n \\nonumber\\\\\n =& \\int \\frac{d^Dp}{(2\\pi)^D} \\frac12 \\tilde{\\mathscr{X}}_\\mu^A(-p) [\\delta^{AB} g_{\\mu\\nu} \\tilde{R}_k(p)] \\tilde{\\mathscr{X}}_\\nu^B(p) \n  ,\n\\end{align}\nwhere $k$ is the RG scale and $\\tilde{R}_k(p)$ is the Fourier transform of $R_k(i\\partial)$. \nThe regulator function $R_k$ introduces a mass proportional to $k^2$, which plays a similar role to $M_0^2$ in the above, as long as $k >0$.\n\n  \n\n\n\n\nThe NJL model \\cite{NJL61} is well known as a low-energy effective theory of QCD to describe the dynamical breaking of chiral symmetry in QCD  (at least in the confinement phase), see e.g. \\cite{Klevansky92,HK94}. \nThe theory given above by $S_{\\rm eff}^{\\rm QCD}=S_{\\rm eff}^{\\rm glue}+S_{\\rm eff}^{\\rm gNJL}$ is able to describe chiral-symmetry breaking\/restoration and quark confinement\/deconfinement on an equal footing where \nthe pure gluon part $S_{\\rm eff}^{\\rm glue}$  describes confinement\/deconfinement transition signaled by the Polyakov loop average. \nWe can incorporate the information on confinement\/deconfinement transition into the quark sector through the covariant derivative $\\mathcal{D}[\\mathscr{V}]$ and the nonlocal NJL interaction $\\mathcal{G}$ ($G$ and $\\mathcal{C}$), in sharp contrast to the conventional PNJL model where the entanglement between chiral-symmetry breaking\/restoration and confinement\/deconfinement was incorporated  through the covariant derivative $\\mathcal{D}[\\mathscr{V}]$ alone and the nonlocal NJL interaction $\\mathcal{G}$ is fixed to the zero-temperature case. \nIn our theory, the nonlocal NJL interaction $\\mathcal{G}$ ($G$ and $\\mathcal{C}$) is automatically determined through the information of confinement\/deconfinement dictated by the Polyakov loop $L$ (non-trivial gluon background), while  in the nonlocal PNJL model \\cite{HRCW08} the low-momentum (non-perturbative) behavior of $\\mathcal{C}$ was not controlled by first principles and was provided by the instanton model. \n\n\n\n\nTo study chiral dynamics, it is convenient to bosonize the gauged nonlocal NJL model as done \\cite{HRCW08}.\nThe nonlocal gauged NJL model (\\ref{gNJL}) can be bosonised as follows. \nDefine\n\\begin{equation}\n\\Phi_\\alpha(x) :=(\\sigma(x), \\vec\\pi(x)) .\n\\end{equation}\nTo eliminate the quadratic term in the nonlocal currents, \nwe insert the unity:\n\\begin{align}\n 1=& \\int \\mathcal{D} \\sigma \\mathcal{D} \\vec{\\pi} \\exp \\Big\\{ -\\int d^4z \\mathcal{C}(z) \n\\nonumber\\\\&\n \\times \\int d^4x \\frac{1}{2G} [\\Phi_\\alpha(x) + G\\bar\\psi(x+z\/2) \\Gamma_\\alpha \\psi(x-z\/2)]\n\\nonumber\\\\&\n \\times [\\Phi_\\alpha(x) + G\\bar\\psi(x+z\/2) \\Gamma_\\alpha \\psi(x-z\/2)]^* \\Big\\} \n  ,\n\\end{align}\nwhere we have used\n$\\int d^4z \\mathcal{C}(z)=1 $.\nThen we have the gauged Yukawa model: \n\\begin{align}\n&  \\int \\mathcal{D} \\bar\\psi  \\mathcal{D} \\psi e^{-S_{\\rm eff}^{\\rm gNJL} }\n\\nonumber\\\\\n =&  \n \\int \\mathcal{D} \\bar\\psi  \\mathcal{D} \\psi \n\\int \\mathcal{D} \\sigma \\mathcal{D} \\vec{\\pi} \n\\exp  \\{ - S_{\\rm eff}^{\\rm gY}\n \\} \n  ,\n\\end{align}\nwhere with $x^\\prime:=x+z\/2$, $y^\\prime:=x-z\/2$, \n\\begin{align}\n   S_{\\rm eff}^{\\rm gY} \n =&  \n  \\int d^4x^\\prime \\int d^4y^\\prime  \\bar\\psi(x^\\prime) \\Big[ \\nonumber\\\\\n& \\delta^4(x^\\prime-y^\\prime) \n(-i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{V}] + \\hat m_q+i\\gamma^4 \\mu_q)\n\\nonumber\\\\\n&+ \\frac12 \\mathcal{C}(x^\\prime-y^\\prime) \\Gamma_\\alpha [\\Phi_\\alpha(\\frac{x^\\prime+y^\\prime}{2}) + \\Phi_\\alpha^*(\\frac{x^\\prime+y^\\prime}{2})] \n\\Big] \\psi(y^\\prime)\n\\nonumber\\\\\n&+ \\int d^4x \\frac{1}{2G}  \\Phi_\\alpha(x)\\Phi_\\alpha^*(x)  \n  ,\n\\end{align}\nor\n\\begin{align}\n&   S_{\\rm eff}^{\\rm gY} \n\\nonumber\\\\\n =&  \n \\int \\frac{d^4p}{(2\\pi)^4} \\frac{d^4p^\\prime}{(2\\pi)^4}  \\bar\\psi(p) \\Big[ \n\\nonumber\\\\\n& (2\\pi)^4 \\delta^4(p-p^\\prime) (- \\gamma^\\mu (p_\\mu + g \\tilde{\\mathscr{V}}_\\mu(p)) + \\hat m_q+i\\gamma^4 \\mu_q)\n\\nonumber\\\\\n&+ \\frac12 \\tilde{\\mathcal{C}}(\\frac{p+p^\\prime}{2}) \\Gamma_\\alpha [\\Phi_\\alpha(p-p^\\prime) + \\Phi_\\alpha^*(p-p^\\prime)]  \\Big] \\psi(p^\\prime)\n\\nonumber\\\\&\n + \\int \\frac{d^4p}{(2\\pi)^4} \\frac{1}{2G}  \\Phi_\\alpha(p)\\Phi_\\alpha^*(p)\n   .\n\\end{align}\n\nFinally, the bosonized theory of the gauged NJL model is obtained by way of the gauged Yukawa model by integrating out quark fields as\n\\begin{align}\n&  \\int \\mathcal{D} \\bar\\psi  \\mathcal{D} \\psi e^{-S_{\\rm eff}^{\\rm gNJL} }\n\\nonumber\\\\\n =&  \n\\int \\mathcal{D} \\sigma \\mathcal{D} \\vec{\\pi} \n\\int \\mathcal{D} \\bar\\psi  \\mathcal{D} \\psi \n\\exp  \\{ -S_{\\rm eff}^{\\rm gY}\n \\} \n\\nonumber\\\\\n =& \\int \\mathcal{D} \\sigma \\mathcal{D} \\vec{\\pi}  \\exp \\{ - S_{\\rm eff}^{\\rm boson} \\}\n  ,\n\\end{align}\nwhere the bosonised action $S_{\\rm eff}^{\\rm boson}$ is \n\\begin{align}\n   S_{\\rm eff}^{\\rm boson}\n =&  \n - {\\rm Tr} \\ln  \\Big\\{ \n \\delta^4(x^\\prime-y^\\prime) \n(-i\\gamma^\\mu (\\partial_\\mu -ig \\mathscr{V}_\\mu) +i\\gamma^4 \\mu_q)\n\\nonumber\\\\\n&+ \\hat m_q + \\frac12 \\mathcal{C}(x^\\prime-y^\\prime) \\Gamma_\\alpha [\\Phi_\\alpha(\\frac{x^\\prime+y^\\prime}{2}) + \\Phi_\\alpha^*(\\frac{x^\\prime+y^\\prime}{2})] \\Big\\} \n\\nonumber\\\\\n&+ \\int d^4x \\frac{1}{2G}  \\Phi_\\alpha(x)\\Phi_\\alpha^*(x)  \n  ,\n\\end{align}\nor\n\\begin{align}\n S_{\\rm eff}^{\\rm boson}\n =&  \n - {\\rm Tr} \\ln  \\Big\\{ \n(2\\pi)^4 \\delta^4(p-p^\\prime)  [- \\gamma^\\mu (p_\\mu + g \\tilde{\\mathscr{V}}_\\mu) +i\\gamma^4 \\mu_q]\n\\nonumber\\\\&\n+ \\hat m_q + \\frac12 \\tilde{\\mathcal{C}}(\\frac{p+p^\\prime}{2}) \\Gamma_\\alpha [\\Phi_\\alpha(p-p^\\prime) + \\Phi_\\alpha^*(p-p^\\prime)]  \\Big\\}  \n\\nonumber\\\\&\n+ \\int \\frac{d^4p}{(2\\pi)^4} \\frac{1}{2G}  \\Phi_\\alpha(p)\\Phi_\\alpha^*(p)\n  .\n\\end{align}\n\n\n\n\n\n\n\\section{Implications of the Polyakov loop for chiral-symmetry breaking at finite temperature}\\label{sec:}\n\nThe thermodynamics of QCD can be studied based on our effective theory derived in this paper in the similar way to the nonlocal PNJL model \\cite{HRCW08}. \nBut this must be done by including the effect of gluon properly. \nIn the PNJL model, the effect of the gluon was introduced by the standard minimal gauge coupling procedure, i.e., replacing the normal derivative $\\partial_\\mu$ by the covariant derivative $D_\\mu[\\mathscr{A}]:=\\partial_\\mu -ig\\mathscr{A}_\\mu$.  \nIn our effective theory, the effect of the gluon is introduced through the NJL coupling constant $G$ and the nonlocality function $\\mathcal{C}$, in addition to the minimal coupling $D_\\mu[\\mathscr{V}]$.  \nThe nonlocal NJL interaction among quarks are mediated by gluons at finite temperature in QCD. \nTherefore, both $G$ and  $\\mathcal{C}$ characterizing nonlocal NJL interaction inevitably have temperature dependence, which could be different depending on whether quarks are in confinement or deconfinement phases. \n \nAt $T=0$, QCD must be in the hadron phase where the chiral symmetry is spontaneously broken, which means that the NJL coupling constant $G(0)$ at zero temperature must be greater than the critical NJL coupling constant $G_c$:\n\\begin{equation}\n G(0) =  g^2  \\frac{1}{M_0^2} > G_c .\n\\end{equation}\nThe nonlocality function or the form factor $\\tilde{\\mathcal{C}}(p)$ at $T=0$ behaves\n\\begin{equation}\n \\tilde{\\mathcal{G}}(p)\n=   \\frac{g^2}{2}     \\frac{1}{p^2 +M_0^2 }, \\ \n \\tilde{\\mathcal{G}}(0)\n=   \\frac{g^2}{2}    \\frac{1}{M_0^2 } ,\n\\end{equation}\nand\n\\begin{equation}\n\\tilde{\\mathcal{C}}(p) =  \\frac{M_0^2}{p^2 +M_0^2 } \n    .\n\\end{equation}\n\nAs an immediate outcome of our effective theory, this determines the temperature-dependence of the coupling constant $G$ of nonlocal NJL model.\nUsing (\\ref{G}), we have\n\\begin{equation}\n G(T) =   \\frac13 g^2 \\left[ \\frac{2}{(T\\varphi)^2+M_0^2}+\\frac{1}{M_0^2} \\right] , \\\n\\end{equation}\nwhich lead to the NJL coupling constant normalized at $T=0$:\n\\begin{equation}\n G(T)\/G(0) =   \\frac13  \\left[ \\frac{2M_0^2}{(T\\varphi)^2+M_0^2}+1 \\right] .\n\\end{equation}\n\n\nIn the presence of the dynamical quark $m_q < \\infty$, the Polyakov loop is not an exact order parameter and does not show a sharp charge with discontinuous derivatives.  Even in this case, we can introduce the pseudo critical temperature $T_d^*$ as a temperature achieving the peak of the susceptibility. \nBelow the deconfinement temperature $T_d^*$, i.e., $T < T_d^*$, therefore, $L \\simeq 0$ or $\\varphi \\simeq \\pi$, the NJL coupling constant $G$ has the temperature-dependence\n\\begin{equation}\nG(T)\/G(0) \\simeq   \\frac13  \\left[ \\frac{2M_0^2}{\\pi^2 T^2+M_0^2}+1 \\right]  \\ (T < T_d^*) .\n\\end{equation}\n\n\nThis naive estimation gives a qualitative understanding for the existence of chiral phase transition.   Since $G(T)$ is (monotonically) decreasing as the temperature $T$ increases, it becomes smaller than the critical NJL coupling constant  \n\\begin{equation}\n G(0) =  g^2  \\frac{1}{M_0^2} > G_c ,\n \\quad \n  T \\uparrow \\infty \\Longrightarrow \nG \\downarrow 0 .\n\\end{equation}\nThus, the chiral transition temperature $T_\\chi$ will be determined (if the chiral-symmetry restoration and confinement coexist or the chiral symmetry is restored in the confinement environment before deconfinement takes place, i.e.,  $T_\\chi \\le T_d^*$ ) by solving\n\\begin{equation}\n G(T_\\chi) \\equiv   \\frac{G(0)}{3}  \\left[ \\frac{2M_0^2}{T_\\chi^2 \\pi^2 +M_0^2}+ 1\\right] = G_c \n .\n\\end{equation}\nHere we have assumed that the nonlocality function $\\tilde{\\mathcal{C}}(p)$ gives the dominant contribution at $p=0$, namely, $\\tilde{\\mathcal{C}}(p) \\le  \\tilde{\\mathcal{C}}(0)=\\int d^4z \\mathcal{C}(z)=1$ and that the occurrence of the chiral transition is determined by the NJL coupling constant alone. \n\nAt finite temperature $T$, the form factor reads\n\\begin{align}\n& \\mathcal{C}(\\bm{x}-\\bm{y}) \n\\nonumber\\\\ \n =& T \\sum_{n \\in \\mathbb{Z}} \\int \\frac{d^3p}{(2\\pi)^3} \n  \\tilde{\\mathcal{C}}(p_0=\\omega, \\bm{p}) e^{i\\bm{p} \\cdot (\\bm{x}-\\bm{y})} \n\\nonumber\\\\ \n =& T \\sum_{n \\in \\mathbb{Z}} \\int \\frac{d^3p}{(2\\pi)^3} \n\\frac{M_0^2}{3} \\Big[ \\frac{2}{\\bm{p}^2+(\\omega+T\\varphi)^2+M_0^2} \n\\nonumber\\\\&\n+ \\frac{1}{\\bm{p}^2+\\omega^2+M_0^2} \\Big] e^{i\\bm{p} \\cdot (\\bm{x}-\\bm{y})} \n\\nonumber\\\\ \n =&   \\int \\frac{d^3p}{(2\\pi)^3} \n\\frac{M_0^2}{6\\epsilon_p} \\Big[ 2\\frac{\\sinh (\\epsilon_p\/T)}{\\cosh(\\epsilon_p\/T)-\\cos(\\varphi)} \n\\nonumber\\\\&\n+  \\frac{\\sinh (\\epsilon_p\/T)}{\\cosh(\\epsilon_p\/T)-1}  \\Big] e^{i\\bm{p} \\cdot (\\bm{x}-\\bm{y})} \n ,\n\\end{align}\nwhere we have defined \n$\\epsilon_p:=\\sqrt{\\bm{p}^2+M_0^2}$\nand used\n\\begin{align}\n T \\sum_{n \\in \\mathbb{Z}}  \\frac{1}{(\\omega+C)^2+\\epsilon_p^2}   \n = \\frac{1}{2\\epsilon_p}   \\frac{\\sinh (\\epsilon_p\/T)}{\\cosh(\\epsilon_p\/T)-\\cos(C\/T)} \n .\n\\end{align}\nThe form factor $\\mathcal{C}$ does not change so much around the deconfinement temperature $T \\sim T_d^*$ (or $\\varphi \\sim \\pi$). \nThis is reasonable since the form factor is nearly equal to the $\\mathscr{X}$ correlator $Q^{-1}$, as already  mentioned in sec. II. \n\nFor more precise treatment, we must obtain the full effective potential $V_{\\rm eff}(\\sigma, \\varphi)$ as a function of two order parameters $\\sigma$ (or $\\langle \\bar\\psi \\psi \\rangle$) and $\\varphi$ (or $\\langle L \\rangle$), and look for a set of values $(\\sigma, \\varphi)= (\\sigma_0, \\varphi_0)$ at which the minimum $V_{\\rm eff}(\\sigma_0, \\varphi_0)$ of $V_{\\rm eff}(\\sigma, \\varphi)$ is realized. \nThen $\\varphi$ must be replaced by $\\varphi_0$ in the above consideration. \nFor this goal, we must develop the RG treatment for the full theory. \nThis issue will be studied in a subsequent paper. \n\n\\section{How to understand the entanglement between confinement and chiral symmetry breaking}\\label{sec:entanglement}\n\n\nTo discuss the entanglement between confinement and chiral symmetry breaking, we wish to obtain the total effective potential $V_{}^{\\rm QCD}$ of QCD written in terms of two order parameters, i.e., the Polyakov loop average $\\langle L \\rangle$ and chiral condensate $\\langle \\bar \\psi \\psi \\rangle$, so that its minima determine the vacuum for a given set of parameters $m_q$, $T$ and $\\mu_q$ when $N_c$ and $N_f$ are fixed. \nHere $m_q \\uparrow \\infty$ is the pure Yang-Mills limit and $m_q \\downarrow 0$ is the chiral limit.\n\n\n\n\n\nThe effective potential for the quark part is obtained by integrating out quark degrees of freedom.\nThe simplest form is obtained  e.g., from the bosonized model as\n\\begin{align}\n   & V_{}^{\\rm quark}\n\\nonumber\\\\\n =&  - {\\rm Tr} \\ln \\Big\\{  i\\gamma^\\mu \\partial_\\mu  \n+ m_q + \\mathcal{C} \\sigma  \n -    g \\mathscr{A}_4  \\gamma^4  \n + i   \\mu_q \\gamma^4  \\Big\\} \n + \\frac{1}{2G}  \\sigma^2 \n  .\n\\label{V^q}\n\\end{align}\nThen the RG-scale $k$ dependent effective potential $V_{k}^{\\rm quark}$ for the quark part must be given as the solution of the flow equation.  \nIn the same approximation as the above, it is written in terms of two order parameters $\\sigma$ and $\\varphi$: \n\\begin{align}\n  &  V_{k}^{\\rm quark}(\\sigma,\\varphi)_{m_q,T,\\mu_q}\n\\nonumber\\\\ \n=&      \n -  T \\sum_{n \\in \\mathbb{Z}} \\int \\frac{d^3p}{(2\\pi)^3} {\\rm tr} \\ln \\Big[   i \\omega_n \\gamma^0- p_j \\gamma^j  + m_q +  \\mathcal{C}(p)  \\sigma\n\\nonumber\\\\&\n\\quad\\quad\\quad\\quad \\quad\\quad\\quad\\quad\\quad\\quad\n- T\\varphi T_3 \\gamma^4 + i  \\mu_q \\gamma^4 + R_k^{\\rm quark} \\Big]  \n\\nonumber\\\\&\n + \\frac{1}{2G}  \\sigma^2 \n  ,\n  \\label{V^q_k}\n\\end{align}\nwhere $T_3=\\sigma_3\/2$ and $R_k^{\\rm quark}$ is the regulator function for quarks.\nIn the limit $k \\downarrow 0$, indeed, $V_{k}^{\\rm quark}$ (\\ref{V^q_k}) reduces to $V_{}^{\\rm quark}$ (\\ref{V^q}). \nThe effective potential $V_{k}^{\\rm quark}$ (\\ref{V^q_k}) depends on $m_q$, $T$ and $\\mu_q$ when $Nc$ and $N_f$ are fixed. \nDue to the $p_0$ dependence of the ``mass'' function $M(p)$ which is an immediate consequence of the nonlocality of the present NJL model, it is difficult to obtain the closed analytical form by performing the summation over the Matsubara frequencies.\n\n\n\n\nIn our strategy, a full effective potential $V_{{\\rm eff},k}^{\\rm QCD}(\\sigma,\\varphi)$ of QCD is given by summing three parts:\n\\begin{equation} \n V_{{\\rm eff},k}^{\\rm QCD}(\\sigma,\\varphi)\n = V_{k}^{\\rm glue}(\\varphi) + V_{k}^{\\rm quark}(\\sigma,\\varphi) + \\Delta V_{k}^{\\rm QCD}(\\sigma,\\varphi)  \n ,\n\\end{equation}\nwith  the pure gluon part $V_{k}^{\\rm glue}(\\varphi) = V_{T,k}$ (\\ref{V_g}), \n\\begin{align}\n V_{k}^{\\rm glue}(\\varphi) \n=&    {\\rm Tr}  \\{  \\ln [G^{AB}+\\delta^{AB}R_{k} ] \\} ,\n\\nonumber\\\\\n =& {\\rm Tr}  \\{  \\ln [\n   - \\delta^{AB} \\partial_\\mu^2  +   (\\delta^{AB}  - \\delta^{A3}\\delta^{B3})(T \\varphi)^2 \n\\nonumber\\\\&\n\\quad\\quad\\quad\\quad\n+ 2  \\epsilon^{AB3} T \\varphi \\partial_0 \n+\\delta^{AB}R_{k} ] \\} ,\n\\end{align}\n the quark part (\\ref{V^q_k}), \n\\begin{align}\n   V_{k}^{\\rm quark}(\\sigma,\\varphi)\n   =&    \\frac{1}{2G}  \\sigma^2\n - {\\rm Tr} \\ln \\{ \n i\\gamma^\\mu \\partial_\\mu  + m_q + \\mathcal{C} \\sigma  \n- T\\varphi T_3 \\gamma^4 \n\\nonumber\\\\& \n\\quad\\quad\\quad\\quad\\quad \n + i  \\mu_q \\gamma^4 + R_k^{\\rm quark}\n\\} ,\n\\end{align}\nand  a non-perturbative part $\\Delta V_{k}^{\\rm QCD}(\\sigma,\\varphi)$ induced in the RG evolution according to a flow equation.\nWe assume that the total effective action of QCD obtained after integrating out the fields other than those relevant to chiral symmetry and confinement is the form\n\\begin{align}\n \\Gamma_k  =& \\int_{0}^{1\/T}dx_4 \\int d^3x  \\Big\\{ \n \\frac{1}{2}Z_0  [ \\partial_j  \\mathscr{V}_0(\\bm{x})]^2\n+ \\frac{1}{2} Z_\\sigma [\\partial_j \\sigma(x)]^2  \n\\nonumber\\\\&\n+ V_{{\\rm eff},k}^{\\rm QCD}(\\sigma,\\varphi)  \\Big\\} \n  ,\n\\end{align}\nand obeys the flow equation:\n\\begin{align}\n\\partial_t \\Gamma_k \n=&  \\frac12 {\\rm Tr} \\left\\{  \n\\left[ \\frac{\\overrightarrow{\\delta}}{\\delta \\sigma^\\dagger} \\Gamma_k \\frac{\\overleftarrow{\\delta}}{\\delta \\sigma} + R_{k} \\right]^{-1} \\cdot \n\\partial_t R_{k} \\right\\}\n\\nonumber\\\\\n &+ \\frac12 {\\rm Tr} \\left\\{ \\left[ \\frac{\\overrightarrow{\\delta}}{\\delta \\mathscr{V}^\\dagger} \\Gamma_k \\frac{\\overleftarrow{\\delta}}{\\delta \\mathscr{V}} + R_{k} \\right]^{-1} \\cdot \\partial_t R_{k} \\right\\} \n  .\n\\end{align}\n\nIf the flow equation was solved, we would have obtained the effective potential of QCD, $V_{{\\rm eff},k}^{\\rm QCD}(\\sigma,\\varphi)$ which has the following power-series expansion with respect to two variables $\\sigma$ and $\\tilde\\varphi$ in the neighborhood of the transition point where $\\sigma=0=L$ according to the Landau argument (as demonstrated in the pure glue case). \n\\begin{align}\n V_{{\\rm eff}}^{\\rm QCD}(\\sigma,\\tilde\\varphi)\n =& V^{\\rm g}(\\tilde\\varphi)   + V^{\\rm q}(\\sigma)\n+ V^{\\rm c}(\\sigma, \\tilde\\varphi)  ,\n\\nonumber\\\\\n V^{\\rm g}(\\tilde\\varphi) =& C_0 + C_1 \\tilde\\varphi +  \\frac{C_2}{2} \\tilde\\varphi^2 + \\frac{C_3}{3} \\tilde\\varphi^3 + \\frac{C_4}{4} \\tilde\\varphi^4 + O(\\tilde\\varphi^6)\n ,\n\\nonumber\\\\\n  V^{\\rm q}(\\sigma) =&   \\frac{E_2}{2} \\sigma^2 + \\frac{E_4}{4} \\sigma^4 + O(\\sigma^6)\n   ,\n\\nonumber\\\\\n  V^{\\rm c}(\\sigma, \\tilde\\varphi) =&  \n    F_1 \\sigma^2 \\tilde\\varphi  + \\cdots \n ,\n\\end{align}\nwhere \n$V^{\\rm g}(\\hat\\varphi)$ denotes a part written in terms of $\\hat\\varphi$ alone, \nand $V^{\\rm q}(\\sigma)$ denotes a part written in terms of $\\sigma$ alone, \nwhile $V^{\\rm c}(\\sigma, \\hat\\varphi)$ denotes the cross term  between $\\sigma$ and $\\hat\\varphi$. \n\n\n\nOnce dynamical quarks are introduced, the exact center symmetry in pure Yang-Mills theory is no longer intact. \nTherefore, the QCD effective potential includes the explicitly center-symmetry-breaking term.\nFor $G=SU(2)$,  the center symmetry $\\tilde\\varphi \\rightarrow - \\tilde\\varphi$ is explicitly broken as  $C_1 \\not=0$, $C_3 \\not=0$ in $V^{\\rm g}(\\varphi)$ and $F_1 \\not= 0$ in $V^{\\rm c}(\\sigma, \\varphi)$. \nThe existence of the cross term is important to understand  the entanglement between center symmetry and chiral symmetry, as pointed out by \\cite{Fukushima04}.\nIn fact, the one-loop calculation leads to \n$C_1=0.97434  N_f>0$ and \n$F_1=- 0.106103 N_f<0$ ($\\mu_q=0$ case) which appears to be a good indication for this purpose and serves as the initial condition in solving the flow equation.\n\n\n\nIn the paper by Schaefer, Pawlowski and Wambach \\cite{SPW07}, \na sort of back-reaction from quarks has been introduced to improve the effective potential of the Polyakov loop, while the NJL coupling remains local. \nIn contrast, this paper introduces a back-reaction from gluons to improve the NJL interaction, leading to the nonlocal NJL coupling. \nHowever, this does not mean that two treatments are considered to be alternative. \nIn the presence of dynamical quarks, the running coupling $\\alpha$ is changed due to fermionic contributions.  In \\cite{SPW07}, this effect has been taken into account as a modification of the expansion coefficient in the effective potential of the Polyakov loop, resulting in e.g., the $N_f$ flavor-dependent deconfinement temperature $T_d(N_f)$.\nRemembering that the input of our analysis is just a running coupling, a sort of back-reaction from quarks considered in \\cite{SPW07} is easily included into our framework by using the running coupling modified by quark contributions.  \nThus, the treatment in this paper is already able to take into account  back-reactions from quarks and gluons mentioned above. \n\n\n\nThis section is a sketch of our strategy of understanding the entanglement between center symmetry and chiral symmetry.\nThe detailed analysis will be given in a subsequent paper. \n\\footnote{\nIt is known that appearance of a mixed-term $\\sigma^2\\tilde\\varphi$ plays an essential role in the chiral-confinement\nentanglement. Such a term appears in the original PNJL model and leads to the 2 crossovers happening almost simultaneously. \nIn the following paper posted to the archive after this paper was submitted for publication, it has been shown that an effective Polyakov loop-dependent four-quark interaction derived by this paper yields stronger correlation between the chiral and deconfinement transitions, making $T_\\chi \\sim T_d$ more tightly, than the usual PNJL model. \n\\\\\nY. Sakai, T. Sasaki, H. Kouno and M. Yahiro,\nEntanglement between deconfinement transition and chiral symmetry restoration, \ne-Print: arXiv:1006.3648 [hep-ph]. \n}\n\n \n\n\n\n\n\n\n\\section{Conclusion and discussion}\\label{sec:conclusion}\n\n\nIn this paper, we have presented a reformulation of QCD and suggested a framework for deriving a low-energy effective theory of QCD which enables one to study the deconfinement\/confinement and chiral-symmetry restoration\/breaking crossover transition simultaneously on an equal footing.\nA resulting low-energy effective theory based on this framework can be regarded as a modified (improved) version of the nonlocal PNJL model \\cite{HRCW08}.  \nIn our framework, the basic ingredients are a reformulation of QCD based on new variables and the flow equation of the Wetterich type for the Wilsonian renormalization group.\n\nA lesson we learned in this study is that a perturbative (one-loop) result can be a good initial condition for solving the flow equation of the renormalization group to obtain the non-perturbative result.  \nIn gluodynamics, recently, it has been demonstrated \\cite{MP08,BGP07} that the existence of confinement transition, i.e., recovery of the center symmetry signaled by the vanishing Polyakov loop average can be shown by approaching the phase transition point from the high-temperature deconfinement phase in which the center symmetry is spontaneously broken.\nIndeed, the effective potential for the Polyakov loop obtained in the one-loop calculation which we call the Weiss potential  leads to the non-vanishing Polyakov loop average, i.e., spontaneous breaking of the center symmetry. \n \n\n\nFor gluon sector, to understand the existence of confinement transition by approaching from the deconfinement side, we have given the Landau-Ginzburg description in the neighborhood of the (crossover) phase transition point by analyzing the flow equation of the functional renormalization group. \nThe deconfinement\/confinement phase transition is consistent with the second order transition for $G=SU(2)$, while the first order transition is expected for $G=SU(3)$.\nThe detailed study of the  $SU(3)$ case will be given in a subsequent paper. \n\n\n\nThe input for solving the flow equation was just a running gauge coupling constant, in sharp contrast to the PNJL model including several parameters. \nFrom the viewpoint of a first-principle derivation, this is superior to phenomenological models with many input parameters.\n \n\nFor quark sector, it is possible  to obtain the chiral-symmetry breaking\/restoration transition from the first principle.\nHowever, we need more hard works, especially, to discuss the QCD phase diagram at finite density and the critical endpoint. \nA possibility in this direction from the first principle of QCD was demonstrated in one-flavor QCD based on the FRG \\cite{Braun09}.  \nIt will be possible to treat chiral dynamics and confinement on an equal footing based on our framework along this line \\cite{Kondo10}. \nStill, however, we must overcome some technical issues to achieve the goal of understanding full phase structures of QCD.\nThe detailed studies will be hopefully given in a subsequent paper. \n\n\n\n\n\n\n\n\n\n\n\n\n{\\em Acknowledgements --} \nThe author would like to thank the Yukawa Institute for Theoretical Physics at Kyoto University where the YITP workshop  ``New Frontiers in QCD 2010  (NFQCD10)'' was held. \nHe thanks the organizers and the participants, especially, Hideo Suganuma, Kenji Fukushima, Akira Ohnishi, Hiroshi Toki,  Wolfram Weise and Akihiro Shibata for discussions and comments on his talk, which were useful to complete this work. \nThanks are due to Jan Pawlowski for discussions and correspondences on papers \\cite{MP08,BGP07}, and Pengming Zhang for translating the original paper \\cite{DG79} from Chinese into English. \nHe is grateful to High Energy Physics Theory Group and Theoretical Hadron Physics Group in the University of Tokyo, especially, Prof. Tetsuo Hatsuda for kind hospitality extended to him on sabbatical leave between April 2009 and March 2010.\nThis work is financially supported by Grant-in-Aid for Scientific Research (C) 21540256 from Japan Society for the Promotion of Science\n(JSPS).\n \n\n\\begin{appendix}\n\\section{Reformulation of QCD}\\label{app:reformulation}\n\n\nWe apply the decomposition (\\ref{decomp}) to QCD Lagrangian. \n\nThe quark part is decomposed according to (\\ref{decomp}) as \n\\begin{align}\n  \\mathscr{L}_{q}  \n:=&  \\bar{\\psi} (i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{A}] -\\hat{m}_0  + i \\mu \\gamma^0) \\psi\n\\nonumber\\\\\n=&   \\bar{\\psi} (i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{V}] -\\hat{m}_0 + i \\mu \\gamma^0) \\psi +   g\\bar{\\psi} \\gamma^\\mu \\mathscr{X}_\\mu  \\psi\n ,\n\\end{align}\nwhere the covariant derivative $\\mathcal{D}_\\mu[\\mathscr{V}]$ is defined by\n\\begin{equation}\n \\mathcal{D}_\\mu[\\mathscr{V}] :=  \\partial_\\mu - ig \\mathscr{V}_\\mu   .\n\\end{equation}\n\n\nThe Yang-Mills part is treated as follows. \nFor the general decomposition $\\mathscr{A}_\\mu(x)=\\mathscr{V}_\\mu(x)+\\mathscr{X}_\\mu(x)$, the field strength $\\mathscr{F}_{\\mu\\nu}$ is decomposed as \n\\begin{align}\n  \\mathscr{F}_{\\mu\\nu}[\\mathscr{A}] :=& \\partial_\\mu \\mathscr{A}_\\nu - \\partial_\\nu \\mathscr{A}_\\mu -i g [ \\mathscr{A}_\\mu , \\mathscr{A}_\\nu ]\n\\nonumber\\\\\n=& \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]  + \\partial_\\mu \\mathscr{X}_\\nu - \\partial_\\nu \\mathscr{X}_\\mu -i g [ \\mathscr{V}_\\mu , \\mathscr{X}_\\nu ] \n\\nonumber\\\\&\n-i g [ \\mathscr{X}_\\mu , \\mathscr{V}_\\nu ]\n-i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ]\n\\nonumber\\\\\n=& \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]  + D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu \n\\nonumber\\\\&\n-i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ] ,\n\\end{align}\nwhere the covariant derivative $D_\\mu[\\mathscr{V}]$ in the background field $\\mathscr{V}_\\nu$ is defined by\n\\begin{align}\nD_\\mu[\\mathscr{V}] := \\partial_\\mu \\mathbf{1} -ig [ \\mathscr{V}_\\mu , \\cdot ] ,\n\\end{align}\nor, equivalently, \n\\begin{equation}\n  D_\\mu[\\mathscr{V}]^{AC} := \\partial_\\mu \\delta^{AC} + g f^{ABC} \\mathscr{V}_\\mu^B .\n\\end{equation}\nThe Lagrangian density \n$\\mathscr{L}_{YM} =  -\\frac{1}{4} \\mathscr{F}_{\\mu\\nu}[\\mathscr{A}] \\cdot \\mathscr{F}^{\\mu\\nu}[\\mathscr{A}]$ \nof the Yang-Mills theory is decomposed as\n\\begin{align}\n   \\mathscr{L}_{YM} \n=& -\\frac{1}{4} \\mathscr{F}_{\\mu\\nu}[\\mathscr{A}]^2\n\\\\\n=& -\\frac{1}{4} \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]^2\n\\nonumber\\\\&\n- \\frac{1}{2} \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)\n\\nonumber\\\\&\n- \\frac{1}{4} (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)^2 \n\\nonumber\\\\&\n+ \\frac{1}{2}  \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}] \\cdot ig[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ]\n\\nonumber\\\\&\n+ \\frac{1}{2}  (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu) \\cdot ig[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ]\n\\nonumber\\\\&\n- \\frac14 (i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ])^2\n.\n\\end{align}\nHere the third term on the right-hand side of the above equation is rewritten using integration by parts (or up to   total derivatives) as \n\\begin{align}\n & \\frac{1}{4} (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)^2\n\\nonumber\\\\\n=&   \\frac{1}{2} (- \\mathscr{X}_\\mu \\cdot D_\\nu[\\mathscr{V}]D^\\nu[\\mathscr{V}] \\mathscr{X}^\\mu + \\mathscr{X}_\\mu  \\cdot D_\\nu[\\mathscr{V}]D^\\mu[\\mathscr{V}]   \\mathscr{X}^\\nu )  \n\\nonumber\\\\\n=& \\frac{1}{2} \\mathscr{X}^\\mu  \\cdot  \\{ - D_\\rho[\\mathscr{V}]D^\\rho[\\mathscr{V}] g_{\\mu\\nu} +  D_\\nu[\\mathscr{V}]D_\\mu[\\mathscr{V}] \\} \\mathscr{X}^\\nu\n\\nonumber\\\\\n=& \\frac{1}{2} \\mathscr{X}^{\\mu A}  \\{ - (D_\\rho[\\mathscr{V}]D_\\rho[\\mathscr{V}])^{AB} g_{\\mu\\nu} \n- [ D_\\mu[\\mathscr{V}], D_\\nu[\\mathscr{V}]]^{AB}\n\\nonumber\\\\&\n+  (D_\\mu[\\mathscr{V}]  D_\\nu[\\mathscr{V}])^{AB} \\} \\mathscr{X}^{\\nu B} \n\\nonumber\\\\\n=& \\frac{1}{2} \\mathscr{X}^{\\mu A}  \\{ - (D_\\rho[\\mathscr{V}]D_\\rho[\\mathscr{V}])^{AB} g_{\\mu\\nu} \n+ gf^{ABC} \\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}] \n\\nonumber\\\\&\n+  D_\\mu[\\mathscr{V}]^{AC} D_\\nu[\\mathscr{V}]^{CB} \\} \\mathscr{X}^{\\nu B}  ,\n\\end{align}\nwhere we have used\n\\begin{align}\n[ D_\\mu[\\mathscr{V}], D_\\nu[\\mathscr{V}]]^{AB}\n=&  [ D_\\mu[\\mathscr{V}]^{AC}, D_\\nu[\\mathscr{V}]^{CB}] \n\\nonumber\\\\ \n=& - gf^{ABC} \\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}]  .\n\\end{align}\nThus we obtain\n\\begin{align}\n   \\mathscr{L}_{YM} \n=& - \\frac{1}{4} \\mathscr{F}_{\\mu\\nu}[\\mathscr{V40}]^2\n\\nonumber\\\\&\n- \\frac{1}{2} \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)\n\\nonumber\\\\&\n-     \\frac{1}{2} \\mathscr{X}^{\\mu A}  W_{\\mu\\nu}^{AB} \\mathscr{X}^{\\nu B} \n\\nonumber\\\\&\n+ \\frac{1}{2}  (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu) \\cdot ig[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ]\n\\nonumber\\\\&\n- \\frac14 (i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ])^2 \n ,\n\\end{align}\nwhere we have defined\n\\begin{align}\nW_{\\mu\\nu}^{AB}  :=& - (D_\\rho[\\mathscr{V}]D^\\rho[\\mathscr{V}])^{AB} g_{\\mu\\nu} \n+ 2gf^{ABC} \\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}] \n\\nonumber\\\\&\n+  D_\\mu[\\mathscr{V}]^{AC} D_\\nu[\\mathscr{V}]^{CB} . \n\\label{W2}\n\\end{align}\n\nIn the usual background field method, the $\\mathcal{O}(\\mathscr{X})$ term is eliminated by requiring that the back ground field $\\mathscr{V}$ satisfies the equation of motion $D_\\mu[\\mathscr{V}] \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}]=0$:\n\\begin{align}\n& \\frac{1}{2} \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)\n\\nonumber\\\\\n=& - \\frac{1}{2} ( D_\\mu[\\mathscr{V}] \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot  \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot  \\mathscr{X}_\\mu) = 0 .\n\\end{align}\nIn our framework, $\\mathscr{V}$ do not necessarily satisfy the equation of motion. Nevertheless, the $\\mathcal{O}(\\mathscr{X})$ term vanishes from the defining equations which specify the decomposition.  For $G=SU(2)$,  $D_\\mu[\\mathscr{V}]\\bm{n}=0$ and $\\mathscr{X}_\\mu \\cdot \\bm{n}=0$  lead to\n\\begin{align}\n&   \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu)  \n= G^{\\mu\\nu} \\bm{n} \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu)\n\\nonumber\\\\\n=&    G^{\\mu\\nu} [\\partial_\\mu (\\mathscr{X}_\\nu \\cdot \\bm{n}) -  \\mathscr{X}_\\nu \\cdot D_\\mu[\\mathscr{V}]\\bm{n}] = 0\n .\n\\end{align}\nIn order for the reformulated theory written in terms of new variables to be equivalent to the original QCD, we must impose the reduction condition \\cite{KMS06}:\n\\begin{equation}\n  D_\\mu[\\mathscr{V}] \\mathscr{X}^\\mu = 0 .\n  \\label{reduction-cond}\n\\end{equation}\nThis eliminate the last term of $W_{\\mu\\nu}^{AB}$ in (\\ref{W2}). \n\nMoreover, the $\\mathcal{O}(\\mathscr{X}^3)$ term is absent, i.e., \n\\begin{equation}\n\\frac{1}{2}  (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu) \\cdot ig[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ] = 0 \n ,\n\\end{equation}\nsince $D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu$ is orthogonal to $[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ]$. See \\cite{KMS06,KSM08,Kondo08}.\n\n \nThus, the Yang-Mills Lagrangian density reads\n\\begin{align}\n   \\mathscr{L}_{YM}  \n=& -\\frac{1}{4} \\mathscr{F}_{\\mu\\nu}^A[\\mathscr{V}]^2\n-  \\frac{1}{2} \\mathscr{X}^{\\mu A}  Q_{\\mu\\nu}^{AB} \\mathscr{X}^{\\nu B} \n\\nonumber\\\\&\n- \\frac14 (i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ])^2\n ,\n\\end{align}\nwhere we have defined \n\\begin{equation}\nQ_{\\mu\\nu}^{AB}  := - (D_\\rho[\\mathscr{V}]D^\\rho[\\mathscr{V}])^{AB} g_{\\mu\\nu} \n+ 2gf^{ABC} \\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}] .\n\\end{equation}\n\nFor $G= SU(2)$,  the $\\mathcal{O}(\\mathscr{X}^3)$ term is absent, because $\\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]$ and $-ig [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu]$ \nare   parallel to $\\bm{n}$  (this is also the case for the sum\n$\\mathscr{F}_{\\mu\\nu}[\\mathscr{V}] -i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu]$),\nwhile $D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu$\n is orthogonal to $\\bm{n}$ (which follows from the fact $\\bm{n} \\cdot \\mathscr{X}_\\mu=0$). \n For $G=SU(2)$, therefore, we have\n\\begin{equation}\n\\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}]=n^C  G_{\\mu\\nu}[\\mathscr{V}] .\n\\end{equation}\nThen the $SU(2)$ gluon part is rewritten into \n\\begin{align}\n   \\mathscr{L}_{YM}  \n=& -\\frac{1}{4} (G_{\\mu\\nu}[\\mathscr{V}])^2\n-     \\frac{1}{2} \\mathscr{X}^{\\mu A}  Q_{\\mu\\nu}^{AB}[\\mathscr{V}] \\mathscr{X}^{\\nu B} \n\\nonumber\\\\&\n- \\frac14 (i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ])^2 \n,\n\\end{align}\nwhere\n\\begin{equation}\nQ_{\\mu\\nu}^{AB}[\\mathscr{V}]  = - (D_\\rho[\\mathscr{V}]D^\\rho[\\mathscr{V}])^{AB} g_{\\mu\\nu} \n+ 2g\\epsilon^{ABC} n^C  G_{\\mu\\nu}[\\mathscr{V}] .\n\\label{W}\n\\end{equation}\n\n\n\n\n\\section{Coefficients in the effective potential}\\label{app:coefficient}\n\nWe expand $\\hat V_{T,\\hat k}$ defined by\n\\begin{align}\n \\hat V_{T,\\hat k}\n =& \\hat V_{W} + 4  \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{(2\\pi)^2} \\{ \\ln (1- 2 e^{-\\hat  k_{T}} \\cos \\varphi + e^{-2\\hat k_{T}}) \n\\nonumber\\\\&\n- \\ln (1- 2 e^{-\\hat p} \\cos \\varphi + e^{-2\\hat p}) \\} \n  ,\n\\end{align}\nin power series of $\\tilde \\varphi$ by using the expansion  \n$\n  - \\cos \\varphi = - \\cos (\\pi+\\tilde \\varphi) =  \\cos ( \\tilde \\varphi)\n  = 1 - \\frac12 \\tilde \\varphi^2 + \\frac{1}{24} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n$\nas follows.\n\\begin{widetext}\n\\begin{align}\n \\hat V_{T,\\hat k}\n =& \\hat V_{W} +   \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \\Big\\{ \\ln \\left[ 1+ 2 e^{-\\hat  k_{T}}  - e^{-\\hat  k_{T}} \\tilde \\varphi^2 + \\frac{1}{12} e^{-\\hat  k_{T}} \\tilde \\varphi^4  + e^{-2\\hat k_{T}} + O(\\tilde \\varphi^6) \\right]\n  \\nonumber\\\\\n & - \\ln \\left[ 1+ 2 e^{-\\hat p}   - e^{-\\hat p} \\tilde \\varphi^2 + \\frac{1}{12} e^{-\\hat p} \\tilde \\varphi^4   + e^{-2\\hat p} + O(\\tilde \\varphi^6) \\right] \\Big\\} \n \\nonumber\\\\\n =& \\hat V_{W} +   \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \\Big\\{ \\ln \\left[ (1+  e^{-\\hat  k_{T}} )^2 - e^{-\\hat  k_{T}} \\tilde \\varphi^2 + \\frac{1}{12} e^{-\\hat  k_{T}} \\tilde \\varphi^4   + O(\\tilde \\varphi^6) \\right]\n  \\nonumber\\\\\n & - \\ln \\left[ (1+  e^{-\\hat p})^2   - e^{-\\hat p} \\tilde \\varphi^2 + \\frac{1}{12} e^{-\\hat p} \\tilde \\varphi^4  + O(\\tilde \\varphi^6)   \\right] \\Big\\} \n \\nonumber\\\\\n =& \\hat V_{W} +   \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \\Big\\{ \\ln  (1+  e^{-\\hat  k_{T}} )^2 + \\ln \\left[ 1 - \\frac{e^{-\\hat  k_{T}}}{(1+  e^{-\\hat  k_{T}} )^2} \\tilde \\varphi^2 + \\frac{e^{-\\hat  k_{T}}}{12(1+  e^{-\\hat  k_{T}} )^2}  \\tilde \\varphi^4  + O(\\tilde \\varphi^6)  \\right] \n  \\nonumber\\\\\n & - \\ln  (1+  e^{-\\hat p})^2 - \\ln \\left[ 1  - \\frac{e^{-\\hat p}}{(1+  e^{-\\hat p})^2} \\tilde \\varphi^2 + \\frac{e^{-\\hat p}}{12(1+  e^{-\\hat p})^2}  \\tilde \\varphi^4   + O(\\tilde \\varphi^6) \\right] \\Big\\} \n  .\n\\end{align}\nBy using \n$\\log(1+x)=x-\\frac12 x^2+O(x^3)$, therefore, $\\hat V_{T,\\hat k}$ has  the polynomial expansion:\n\\begin{align}\n \\hat V_{T,\\hat k}\n = A_{0,k} +  \\frac{A_{2,k}}{2} \\tilde \\varphi^2 + \\frac{A_{4,k}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n  ,\n\\end{align}\nwhere the coefficient is given by the integral form:\n\\begin{align}\n\\frac{A_{2,k}}{2}\n=& - \\frac16 +  \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \n\\left[ \\frac{e^{-\\hat p}}{(1+  e^{-\\hat p})^2} - \\frac{e^{-\\hat  k_{T}}}{(1+  e^{-\\hat  k_{T}} )^2}  \\right]\n ,\n\\nonumber\\\\\n \\frac{A_{4,k}}{4!}  =& \\frac{1}{12\\pi^2} -  \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \n\\left[ \\frac{-6e^{-2\\hat p}+e^{-\\hat p}(1+  e^{-\\hat p})^2}{12(1+  e^{-\\hat p})^4} - \\frac{-6e^{-2\\hat  k_{T}}+e^{-\\hat  k_{T}}(1+  e^{-\\hat  k_{T}} )^2}{12(1+  e^{-\\hat  k_{T}} )^4}  \\right]\n ,\n\\nonumber\\\\\nA_{0,k} =&     \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \\Big\\{ \\ln  (1+  e^{-\\hat  k_{T}} )^2  - \\ln  (1+  e^{-\\hat p})^2 \\Big\\}\n .\n\\end{align}\nThe integration can be performed analytically and the coefficient has the closed form:\n\\begin{align}\n\\frac{A_{2,k}}{2}\n=&  - \\frac16 +  \\frac{1}{\\pi^2}  \\left[ -\\frac{e^s s^3}{3 \\left(1+e^s\\right)^2}+\\frac{e^s\n   s^2}{1+e^s}-2 \\log \\left(1+e^s\\right) s-2\n   \\text{Li}_2\\left(-e^s\\right)-\\frac{\\pi ^2}{6} \\right] \\Big|_{s=\\hat k_{T}}\n ,\n\\nonumber\\\\\n \\frac{A_{4,k}}{4!}  =& \\frac{1}{12\\pi^2} \n + \\frac{e^{2 s} \\left(-2 s^3+\\left(s^2+6\\right) \\cosh (s)\n   s+6 s+3 \\left(s^2-2\\right) \\sinh (s)-3 \\sinh (2\n   s)\\right)}{18 \\left(1+e^s\\right)^4 \\pi ^2} \\Big|_{s=\\hat k_{T}}\n ,\n\\end{align}\n\\end{widetext}\nwhere ${\\rm Li}_n(z)={\\rm  PolyLog}[n,z]$ is the polylogarithm function, and in particular, the dilogarithm satisfies ${\\rm Li}_2(z) = \\int_{z}^{0} \\frac{\\log(1-t)}{t} dt$ which is known as the Spence integral.\n\nNote that the function $\\frac{e^{-\\hat x}}{(1+  e^{-\\hat x})^2}$ is monotonically decreasing in $x$ and hence the second term in $A_{2,k}$ is positive (non-negative). \nThe coefficient $A_{2,k}$ is negative and monotonically increasing in $k$ and approaches zero for $k \\rightarrow \\infty$. \n\\begin{equation}\n\\frac{A_{2,k}}{2} = -\\frac16, \\quad\n-\\frac16 \\le \\frac{A_{2,k}}{2} < 0 \\quad {\\rm for} \\quad k \\in [0, \\infty) ,\n\\end{equation}\nor\n\\begin{equation}\n-\\frac13 \\le \\frac{\\partial^2}{\\partial \\tilde\\varphi^2} \\hat V_{T,\\hat k} \\Big|_{\\tilde\\varphi=0} < 0 \\quad {\\rm for} \\quad k \\in [0, \\infty).\n\\end{equation}\nThis is because\n\\begin{align}\n \\int_{0}^{\\hat k_{T}}  d\\hat p \\hat p^2  \n \\frac{e^{-\\hat p}}{(1+  e^{-\\hat p})^2} \n&\\rightarrow  \\int_{0}^{\\infty}  d\\hat p \\hat p^2  \n \\frac{e^{-\\hat p}}{(1+  e^{-\\hat p})^2} \n=  \\frac16 \\pi^2   \n ,\n\\\\ \n \\int_{0}^{\\hat k_{T}}  d\\hat p \\hat p^2  \n \\frac{e^{-\\hat  k_{T}}}{(1+  e^{-\\hat  k_{T}} )^2}   \n&= \\frac13 \\hat k_{T}^3  \\frac{e^{-\\hat  k_{T}}}{(1+  e^{-\\hat  k_{T}} )^2}   \\rightarrow 0\n.\n\\end{align}\n\n\n\nThe coefficient $A_{4,k}$ is positive and approaches $0$ for $k \\rightarrow \\infty$, although $A_{4,k}$ is not monotonically decreasing in $k$. \n\\begin{equation}\n\\frac{A_{4,0}}{4!}= \\frac{1}{12\\pi^2}  , \\quad\n\\frac{A_{4,k}}{4!} > 0 \\quad {\\rm for} \\quad k \\in [0, \\infty).\n\\end{equation}\nThis is because\n\\begin{align}\n  & \\int_{0}^{\\hat k_{T}}  d\\hat p \\hat p^2  \n\\left[ \\frac{-6e^{-2\\hat p}+e^{-\\hat p}(1+  e^{-\\hat p})^2}{12(1+  e^{-\\hat p})^4}    \\right]\n \\rightarrow    \\frac{1}{12}   \n ,\n\\nonumber\\\\\n & \\int_{0}^{\\hat k_{T}}  d\\hat p \\hat p^2 \n\\left[ \\frac{-6e^{-2\\hat  k_{T}}+e^{-\\hat  k_{T}}(1+  e^{-\\hat  k_{T}} )^2}{12(1+  e^{-\\hat  k_{T}} )^4}  \\right]\n\\\\&\n= \\frac13 \\hat k_{T}^3  \\left[ \\frac{-6e^{-2\\hat  k_{T}}+e^{-\\hat  k_{T}}(1+  e^{-\\hat  k_{T}} )^2}{12(1+  e^{-\\hat  k_{T}} )^4}  \\right] \\rightarrow 0 \n.\n\\end{align}\n\n\n\\section{Flow equation for the coefficient}\\label{app:flow-equation}\n\nSuppose that $\\Delta \\hat V_{\\hat k}$ is of the form:\n\\begin{equation}\n       \\Delta \\hat V_{\\hat k}     \n=    a_{0,k} + a_{1,k}  \\tilde\\varphi +  \\frac{a_{2,k}}{2} \\tilde \\varphi^2  \n+ \\frac{a_{3,k}}{3!} \\tilde \\varphi^3 \n+ \\frac{a_{4,k}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n .\n\\end{equation}\nThe left-hand side of the flow equation reads\n\\begin{align}\n     \\partial_{\\hat k}  \\Delta \\hat V_{\\hat k}     \n=&   \\partial_{\\hat k}  a_{0,k} + \\partial_{\\hat k} a_{1,k}  \\tilde \\varphi  +  \\partial_{\\hat k} \\frac{a_{2,k}}{2} \\tilde \\varphi^2  \n+ \\partial_{\\hat k} \\frac{a_{3,k}}{3!} \\tilde \\varphi^3 \n\\nonumber\\\\&\n+ \\partial_{\\hat k} \\frac{a_{4,k}}{4!} \\tilde \\varphi^4 \n+ O(\\tilde \\varphi^6)\n .\n\\end{align}\nThe flow equation $\\partial_{\\hat k} a_{n,k}$ for the coefficient  of $\\tilde\\varphi^n$ is extracted by differentiating both sides of the flow equation $n$ times and by putting $\\tilde\\varphi=0$.\nThe left-hand side is \n\\begin{align}\n\\partial_{\\hat k} a_{n,k} = \\frac{\\partial^n}{\\partial \\tilde\\varphi^n} \\partial_{\\hat k}  \\Delta \\hat V_{\\hat k}    \\Big|_{\\tilde\\varphi=0} \n .\n\\end{align}\nDefine  \n\\begin{equation}\n  f(\\varphi):= \\frac{4\\pi\\alpha_k}{\\hat k^2}   (\\hat V_{T,\\hat k}  + \\Delta \\hat V_{\\hat k}  ) .\n\\end{equation}\nThe right-hand sides  of the flow equation $\\partial_{\\hat k} a_{n,k}$ are calculated from\n\\begin{align}\n   \\frac{\\partial}{\\partial \\tilde\\varphi} \\left[  \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right]  \n=&  \\frac{-\\partial_\\varphi^3 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2}\n ,\n\\end{align}\n\\begin{align}\n    \\frac{\\partial^2}{\\partial \\tilde\\varphi^2} \\left[  \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right]  \n=&  \\frac{-\\partial_\\varphi^4 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2} \n\\nonumber\\\\&\n-2  \\frac{-[\\partial_\\varphi^3 f(\\varphi)]^2}{[1+\\partial_\\varphi^2 f(\\varphi)]^3}\n ,\n\\end{align}\n\\begin{align}\n   \\frac{\\partial^3}{\\partial \\tilde\\varphi^3} \\left[  \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right] \n=&  \\frac{-\\partial_\\varphi^5 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2} \n\\nonumber\\\\&\n-2  \\frac{-3\\partial_\\varphi^4 f(\\varphi) \\partial_\\varphi^3 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^3} \n\\nonumber\\\\&\n+ 6  \\frac{-[\\partial_\\varphi^3 f(\\varphi)]^3}{[1+\\partial_\\varphi^2 f(\\varphi)]^4}\n ,\n\\end{align}\n\\begin{align}   \n    \\frac{\\partial^4}{\\partial \\tilde\\varphi^4} \\left[  \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right]  \n=&  \\frac{-\\partial_\\varphi^6 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2} \n\\nonumber\\\\&\n-2  \\frac{-4\\partial_\\varphi^5 f(\\varphi) \\partial_\\varphi^3 f(\\varphi)\n-3[\\partial_\\varphi^4 f(\\varphi)]^2}{[1+\\partial_\\varphi^2 f(\\varphi)]^3} \n\\nonumber\\\\&\n+ 6  \\frac{-6\\partial_\\varphi^4 f(\\varphi)[\\partial_\\varphi^3 f(\\varphi)]^2}{[1+\\partial_\\varphi^2 f(\\varphi)]^4}\n\\nonumber\\\\&\n-24   \\frac{-  [\\partial_\\varphi^3 f(\\varphi)]^4}{[1+\\partial_\\varphi^2 f(\\varphi)]^5}\n , \\cdots\n\\end{align}\n\nIf  $f$ is an even polynomial in $\\tilde\\varphi$, then the flow equation is simplified:\n\\begin{align}\n\\partial_{\\hat k} a_{1,k} &\\simeq \n\\frac{\\partial}{\\partial \\tilde\\varphi} \\left[  \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right]   \\Big|_{\\tilde\\varphi=0} \n=  0\n ,\n\\\\     \n\\partial_{\\hat k} a_{2,k} &\\simeq  \n\\frac{\\partial^2}{\\partial \\tilde\\varphi^2} \\left[  \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right]   \\Big|_{\\tilde\\varphi=0} \n\\nonumber\\\\&\n=   \\frac{-\\partial_\\varphi^4 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2}   \\Big|_{\\tilde\\varphi=0} \n ,\n\\\\     \n \\partial_{\\hat k} a_{3,k} &\\simeq\n    \\frac{\\partial^3}{\\partial \\tilde\\varphi^3} \\left[  \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right]  \\Big|_{\\tilde\\varphi=0} \n=   0\n ,\n\\\\     \n\\partial_{\\hat k} a_{4,k} &\\simeq\n     \\frac{\\partial^4}{\\partial \\tilde\\varphi^4} \\left[  \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right]   \\Big|_{\\tilde\\varphi=0} \n\\nonumber\\\\ \n&=   \\frac{-\\partial_\\varphi^6 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2}  \\Big|_{\\tilde\\varphi=0} \n\\nonumber\\\\&\n+6  \\frac{[\\partial_\\varphi^4 f(\\varphi)]^2}{[1+\\partial_\\varphi^2 f(\\varphi)]^3}  \\Big|_{\\tilde\\varphi=0} \n , ...\n\\end{align}\nTherefore, with an initial condition, $a_{1,k} =0=a_{3,k}$ at $k=\\Lambda$, the flow equations in the above\n\\begin{equation}\n\\partial_{\\hat k} a_{1,k} =0, \\quad \\partial_{\\hat k} a_{3,k}=0 .\n\\end{equation}\nguarantee the solution  \n\\begin{equation}\n  a_{1,k} \\equiv 0, \\quad   a_{3,k} \\equiv 0 \\quad (0 \\le k \\le \\Lambda) .\n\\end{equation}\n\n\n\n\n\n\\end{appendix}\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum resource theory is an essential tool to understand the nature of physical systems from the perspective of nonlocality and  provides an operational interpretation of various quantum effects \\cite{QCbook}. Quantum coherence, superposition, entanglement and quantum correlations beyond entanglement are few notable resources which can prove to be  advantageous over the classical algorithms. Understanding  these resources in  physical systems continue to be a formidable and interesting  task even today. The quantum information processing is usually studied within the framework of   entanglement-versus-separability . The entanglement is considered to be the only early version of nonlocal aspects of  quantum systems \\cite{EPR,Schrodinger1,Schrodinger2} and is demonstrated through the violation of Bell inequality \\cite{Bell}. The seminal work  of Werner and Almedia et al.  reveals that the entanglement is an incomplete manifestation of nonlocality implying  that quantum correlations cannot only be limited to entanglement \\cite{Werner,Almedia} and the separable states can also come in handy in the implementation of some  specific quantum tasks. To study this quantum correlation beyond entanglement, Ollivier and W. H. Zurek introduced a measure called quantum discord which captures the nonlocality of separable (unentangled) states\\cite{Ollivier}. \n \n \n Other than entanglement, many other types of quantum correlations have been discovered in recent years \\cite{Ollivier,Luo2011,GoD}. As a result, a number of quantum correlation concepts have emerged with each being motivated by specific applications in quantum information science competing for recognition as the most accurate quantumness metric. For example, the skew-information and quantum Fisher information play an important role in parameter estimation and bring out  the limitations on the variance of the observable. As stated, each quantum correlation captures different quantumness in quantum systems due to their distinct type of measurement techniques. Among them, skew information based quantum correlation measures such as local quantum uncertainty (LQU) and uncertainty-induced nonlocality (UIN) are   widely used as a quantumness measure to characterize  quantum\/physical systems. Girolami et al. \\cite{GirolamiPRL2013} have introduced a realizable discord-like nonclassical correlation measure for bipartite systems  known as local quantum uncertainty. Further, this measure quantifies the uncertainty in a quantum state  arising due to its noncommutativity with the measured local observables. The LQU is defined in terms of the skew information. The UIN provides an alternative  to capture the nonlocal effects via local measurements. Both  LQU and UIN possess the closed formula for qubit\u2013qudit ($2\\otimes n$ dimensional) systems. Apart from the correlation quantifiers, the quantum coherence is a faithful nonclassicality indicator even for a single qubit system  and is a consequence of superposition  of quantum states. Like  quantum correlations, coherence is also considered to be critical   in the development of quantum technology \\cite{Giovannetti,Demkowicz,Lloyd,Huelga,Lambert,Aberg,Lostaglio,Buffoni}. The  quantification of quantum coherence has been recently developed  by Baumgratz et al \\cite{Bau}. Recently, the coherence measure based on the $l_1$-norm as  a  valid measure has been introduced and characterized in different contexts \\cite{Rana,Chen,Wang}.  Further, the interplay between the quantum coherence measure and correlations is also studied  \\cite{Tancoh,Hucoh1}.\n \nIn relativistic field theory, exploiting the formulation of the detector models, two  well-known effects  such as  Unruh and Hawking effects have been studied. In this formulation, idealized particles are considered as detectors   similar to a two-level system   following the classical worldline and whose internal states  are coupled to the field. In recent times, Unruh-deWitt (UdW) detector model has been devised as a computational element.  The intervention of the environment on the system may cause decoherence and degrades the unique features of quantum system. Assuming that the quantum fluctuations of the background field plays the role of an environment, the Unruh temperature $T_U = a\/2\\pi $($a$ is acceleration) plays the role of decoherence and destroys the properties of coupled UdW detectors. Here,  the pair of detectors is considered as an open system and its dynamics  is governed by the master equation \\cite{Op1}. Recently, different approaches have been employed to study the quantum effects like entanglement, quantum correlations and coherence  \\cite{Entg6,Entg8,Entg9,Huang1,Huang2,Huang3}.  Moreover, the quantum correlation and entropic uncertainty relation are also investigated in a multipartite scenario\\cite{multiparty1,multiparty2}. \n\nIn this article, we focus on the generation and retention of quantumness in two-UdW detectors interacting with the scalar field. To quantify the quantumness of the detectors, we use the entanglement, local quantum uncertainty, uncertainty-induced nonlocality and $l_1$-norm of coherence.  We show that the entanglement decreases monotonically and vanishes at a finite temperature. The coherence and LQU exhibit a remarkable difference namely, revival of quantum correlation after vanishing at a finite time. Further, we study the dynamics of a product and an incoherent state and the investigation reveals that the dynamics generates the  correlation and coherence from the product and incoherent state respectively. \n\n\nThe  paper is structured  as follows. In Sec. \\ref{corr}, we review the quantifier of quantum correlation and coherence measures  . In Sec.  \\ref{Sec3}, we introduce the physical model under our consideration and thermal state of two-UdW detectors. The investigations on quantum correlation and coherence are presented in Sec.  \\ref{Res}. Finally, the  conclusions are drawn in Sec.  \\ref{conc}.\n \n\\section{Quantum Correlation Measures}\n\\label{corr}\n\nIn this section, we review some of the popular quantum correlation measures to be investigated in this article. For this purpose, we consider an arbitrary bipartite state $\\rho$ shared by the subsystems $a$ and $b$ in the separable Hilbert space $\\mathcal{H}^a \\otimes \\mathcal{H}^b$.\n\n\\subsection{Entanglement}\nEntanglement  is a nonnegative real function of a state $\\rho$ which cannot increase under the local operations and classical communication (LOCC) and is zero for unentangled states. To quantify the amount of entanglement associated with the  two-qubit physical system $\\rho$ under consideration, various measures have  been introduced. The concurrence is the most popular measure of entanglement for mixed bipartite systems and is defined as \\cite{Hill}\n\\begin{align}\nC(\\rho)= \\text{max}\\{0,~ \\lambda_1-\\lambda_2-\\lambda_3-\\lambda_4\\},\n\\end{align}\nwhere $\\lambda_i$ are eigenvalues of the matrix $R=\\sqrt{\\sqrt{\\rho}\\tilde{\\rho}\\sqrt{\\rho} }$  $(\\lambda_1\\geq \\lambda_2\\geq \\lambda_3\\geq \\lambda_4)$. Here, $\\tilde{\\rho}=(\\sigma_y \\otimes \\sigma_y) \\rho^* (\\sigma_y \\otimes \\sigma_y)$ is a spin flipped matrix with * denoting the complex conjugate in the computational basis. The function $C(\\rho)$ ranges from 0 to 1 and its minimal and maximal values correspond to separable and entangled states respectively.\n\n \\subsection{Local quantum uncertainty}\n\nThe local quantum uncertainty (LQU) is a more reliable measure of the quantumness of bipartite states which goes beyond entanglement. In recent times, researchers have paid wide attention to this discord-like measure. This is essentially due to its  easy computation and the fact that it enjoys all necessary properties of being a faithful measure of quantum correlation. It is shown that LQU is non-zero for the separable state even in the absence of entanglement. For a bipartite state $\\rho$, the LQU is defined as the minimal skew information attainable with a single local measurement. Mathematically, it is defined as \\cite{GirolamiPRL2013},\n\\begin{align}\n\\mathcal{Q}_{\\mathcal{I}}(\\rho)=~^{\\text{min}}_{H^{a}} ~~\\mathcal{I}(\\rho, H^a\\otimes\\mathds{1}^b),\n\\end{align}\nwhere $H^a$ is anylocal observable on the subsystem $a$, $\\mathds{1}^b$ is the $2\\times 2$ identity operator acting on the system $b$ and\n\\begin{align}\n\\mathcal{I}(\\rho)=-\\frac{1}{2}\\text{Tr}\\left( [\\sqrt{\\rho}, H^a\\otimes\\mathds{1}^b]^2 \\right)\n\\end{align}\nis the skew information which provides an analytical tool to quantify the information content in the state $\\rho$ with respect to the observable $H^a$, $[\\cdot, \\cdot]$ is the commutator operator.  Here, the  information  content  of $\\rho$ about $H^a$ is  quantified  by  how  much  the  measurement  of $H^a$ on  the  state  is  uncertain. The measurement outcome is certain if only if the state is an eigenvector of $H^a$. On the other hand, if  it  is  a  mixture of eigenvectors of $H^a$,  the uncertainty is only due to the imperfect knowledge of the state. For pure bipartite states, the local quantum uncertainty reduces to the linear entropy of entanglement and vanishes for classically correlated states. For any $2\\times n$ dimensional bipartite system, the closed formula of LQU is computed as\n\\begin{align}\n\\mathcal{U}(\\rho)=~ 1- \\text{max}\\{\\omega_1,\\omega_2,\\omega_3 \\}.\n\\end{align}\nHere, $\\omega_i$ are the eigenvalues of matrix $W$ and the matrix elements are defined as\n\\begin{align}\n\\omega_{ij}=\\text{Tr}[\\sqrt{\\rho}(\\sigma_i^a\\otimes\\mathds{1}^b)\\sqrt{\\rho}(\\sigma_j^a\\otimes\\mathds{1}^b)],~~~~~\\text{with}~~~ i,j=1,2,3,\n\\end{align}\nwhere $\\sigma_i$ represents the Pauli spin matrices.\n\n\\subsection{Uncertainty-induced nonlocality}\n\nNext, we employ another important skew information based measure ,namely  uncertainty-induced nonlocality (UIN) and it can be considered as an updated version of measurement-induced nonlocality (MIN) \\cite{Luo2011}. It is defined as \\cite{UIN}\n\\begin{align}\n\\mathcal{Q}(\\rho)=~^{\\text{max}}_{H^{a}} ~~\\mathcal{I}(\\rho, H^a\\otimes\\mathds{1}^b).\n\\end{align}\nThis measure also satisfies all the necessary axioms of a valid measure of bipartite quantum correlation and is reduced to entanglement monotone for $2\\times n$ dimensional pure states.\nIt also possesses a closed formula \\cite{UIN}.\n\\subsection{$C_{l_1}$-norm coherence}\nQuantum coherence is an important resource for information processing and a manifestation of the quantum superposition principle. Recently, its quantification has been formulated and  a set of conditions to be satisfied by any proper measure of coherence has been identified \\cite{Rana,Chen,Wang}.  The distance based quantum coherence measure\n\\begin{align}\n    C(\\rho)=~^{\\text{min}}_{\\delta\\in \\mathcal{I}} ~d(\\rho, \\delta)\n\\end{align}\nis the minimal distance between $\\rho$ and a set of incoherent quantum states $\\delta\\in \\mathcal{I}$. \n Recently, a faithful measure of coherence  using $l_1$-norm has been identified  and is defined as the sum of absolute values of all off-diagonal elements of $\\rho$ \\cite{Bau}\n\\begin{align}\nC_{{l_1}}(\\rho)=\\sum_{i\\neq j}|\\rho_{ij}|.\n\\end{align}\nThe above definition is a basis dependent measure and is crucial in the identification of phase transition in physical models \\cite{LPT1,LPT2,LPT3}.\n\n\n\\section{Physical Model}\\label{Sec3}\nTo understand the behaviors of quantum correlations in two accelerating UdW detectors in $3+1$-dimensional Minkowski space-time \\cite{Entg6}, we first introduce the Hamiltonian of the  physical system under  consideration. Considering the detector as a two-level atomic system and a valid qubit system, we consider the two accelerating UdW detectors in $3+1$-dimensional Minkowski space time as an open quantum system for our investigation. The total Hamiltonian of the combined system becomes \n\\begin{align}\nH=\\frac{\\omega}{2}\\Sigma_3+H_\\Phi+\\mu H_I             \\label{eq1}\n\\end{align}\nwhere  $\\omega$  is the energy level spacing of the atom and $\\Sigma_3$ is one of the symmetrized bipartite operators. $\\Sigma_i\\equiv\\sigma_i^a\\otimes\\mathbf{1}^b+\\mathbf{1}^a\\otimes\\sigma_i^b$ is defined by Pauli matrices $\\{\\sigma^{(\\alpha)}_i|i=1,2,3\\}$ with superscripts $\\{\\alpha=a, b\\}$ labelling distinct atoms. $H_\\Phi$ is the Hamiltonian of free massless scalar fields $\\Phi(t,\\mathbf{x})$ satisfying standard Klein-Gordon relativistic equation. The interaction Hamiltonian between the atoms and the fluctuating field bath in a dipole form  can be written as  $H_I= (\\sigma_2^{(a)}\\otimes\\mathbf{1}^{(b)})\\Phi(t,\\mathbf{ x}_1)+(\\mathbf{1}^{(a)}\\otimes\\sigma_2^{(b)})\\Phi(t,\\mathbf{ x}_2)$ \\cite{Op2}.\n\nHere, we consider the coupling between the detectors and the environment ($\\mu \\leq 1$) and the initial state of the composite system  is $\\rho_{tot}(0)=\\rho_{ab}(0)\\otimes |0\\rangle \\langle 0|$, with $\\rho_{ab}(0)$ being the initial state of the detectors and $|0\\rangle$ is the field vacuum (environment). The time evolution of $\\rho_{tot}(0)$ is unitary  governed by von Neumann equation $\\dot{\\rho}_{tot}(\\tau)=-i[H,\\rho_{tot}(\\tau)]$, where $\\tau$ is the proper time of the atom. Due to the environment decoherence or dissipation on the system $\\rho_{ab}$, the density matrix  is governed by a Lindblad master equation and evolves non-unitarily in the following form \\cite{Op3,Op4}\n\\begin{align}\n\\frac{\\partial\\rho_{ab}(t)}{\\partial t}=-i[H_{\\tiny\\mbox{eff}},\\rho_{ab}(t) ]+\\mathcal{L}\\left[\\rho_{ab}(t)\\right] \n\\label{eq2}\n\\end{align}\nwhere\n\\begin{align}\n\\mathcal{L}\\left[\\rho\\right]=\\sum_{\\substack{i,j=1,2,3\\\\   \\alpha,\\;\\beta=a,b}}\\frac{C_{ij}}{2}\\big[2\\sigma_j^{(\\beta)}\\rho_{ab}\\sigma_i^{(\\alpha)}-\\{\\sigma_i^{(\\alpha)}\\sigma_j^{(\\beta)},\\rho_{ab}\\}\\big]       \\label{eq3}\n\\end{align}\ndescribes the evolution due to the interaction between the detectors and external field. \n\nAfter introducing the Wightman function of scalar field $G^{+}\\left(x, x^{\\prime}\\right)=\\left\\langle 0\\left|\\Phi(x) \\Phi\\left({x}^{\\prime}\\right)\\right| 0\\right\\rangle$, its Fourier transform\n\\begin{align}\n\\mathcal{G}(\\lambda)=\\int_{-\\infty}^{\\infty}d\\tau~e^{i\\lambda\\tau}G^+(\\tau)= \\int_{-\\infty}^{\\infty}d\\tau~e^{i\\lambda\\tau}\\left\\langle\\Phi(\\tau)\\Phi(0)\\right\\rangle              \n\\label{eq4}\n\\end{align}\ndetermines the coefficients $C_{ij}$ by a decomposition\n\\begin{align}\nC_{ij}=\\frac{\\gamma_+}{2}\\delta_{ij}-i\\frac{\\gamma_-}{2}\\epsilon_{ijk} \\delta_{3,k}+\\gamma_0\\delta_{3,i}\\delta_{3,j}          \\label{eq5}\n\\end{align}\nwhere\n\\begin{align}\n\\gamma_\\pm= \\mathcal{G}(\\omega)\\pm \\mathcal{G}(-\\omega),~~~\\gamma_0=\\mathcal{G}(0)-\\gamma_+\/2.         \\label{eq6}\n\\end{align}\nMoreover, the interaction with external scalar field would also induce a Lamb shift contribution for the effective Hamiltonian of the detector $H_{\\mbox{\\tiny eff}}=\\frac{1}{2}\\tilde{\\omega}\\sigma_3$ in terms of a renormalized frequency $\\tilde{\\omega}=\\omega+i[\\mathcal{K}(-\\omega)-\\mathcal{K}(\\omega)]$ where $\\mathcal{K}(\\lambda)=\\frac{1}{i\\pi}\\mbox{P}\\int_{-\\infty}^{\\infty}d\\omega\\frac{\\mathcal{G}(\\omega)}{\\omega-\\lambda}$ is a Hilbert transform of Wightman function. \n\nFollowing a trajectory of the accelerating detectors, one can find that the field Wightman function fulfills the Kubo-Martin-Schwinger (KMS) condition, i.e., $G^{+}(\\tau)=G^{+}(\\tau+i \\beta)$, where $\\beta\\equiv1\/T_U=2\\pi\/a$. Translating it into frequency space, one finds that\n\\begin{align}\n\\mathcal{G}(\\lambda)=e^{\\beta\\omega}\\mathcal{G}(-\\lambda).        \\label{eq7}\n\\end{align}\nUsing translation invariance $\\langle 0|\\Phi(x(0)) \\Phi(x(\\tau))| 0\\rangle=\\langle 0|\\Phi(x(-\\tau)) \\Phi(x(0))| 0\\rangle$ and  after some algebraic manipulations, we find that eq.(\\ref{eq6}) can be resolved as\n\n\\begin{align}\n\\gamma_+=&\\int_{-\\infty}^{\\infty}d\\tau~e^{i\\lambda\\tau}\\langle 0|\\left\\{\\Phi(\\tau),\\Phi(0)\\right\\}|0\\rangle =\\left(1+ e^{-\\beta\\omega}\\right) \\mathcal{G}(\\omega), ~~~~~~~~\\\\\n\\gamma_-=&\\int_{-\\infty}^{\\infty}d\\tau~e^{i\\lambda\\tau}\\langle 0|\\left[\\Phi(\\tau),\\Phi(0)\\right]|0\\rangle =\\left(1- e^{-\\beta\\omega}\\right) \\mathcal{G}(\\omega).\n\\label{eq8}   \n\\end{align}\nIt should be  be noted that eq. (\\ref{eq8}) holds true for generic interacting fields. Considering the two-atom state in Bloch representation, we compute t he final equilibrium state  of two-UdW detectors asymptotically as \n\\begin{align}\n\\rho_{ab}=\\left(\\begin{array}{cccc}\n\\varrho_{11} & 0 & 0 & 0 \\\\\n0 & \\varrho_{22} & \\varrho_{23} & 0 \\\\\n0 & \\varrho_{23} & \\varrho_{22} & 0 \\\\\n0 & 0 & 0 & \\varrho_{44} \n\\end{array}\\right)\\label{eq11}\n\\end{align}\nwhere the matrix elements are \n\\begin{eqnarray}\n  \\displaystyle \\varrho_{11} &=&\\frac{(3+\\Delta_0)(\\gamma-1)^{2}}{4\\left(3+\\gamma^{2}\\right)},~~~~~~~~\n\\displaystyle \\varrho_{44}=\\frac{(3+\\Delta_0)(\\gamma+1)^{2}}{4\\left(3+\\gamma^{2}\\right)}, \\nonumber\\\\\n\\displaystyle \\varrho_{22}&=& \\frac{3-\\Delta_0-(\\Delta_0+1) \\gamma^{2}}{4\\left(3+\\gamma^{2}\\right)},~~~~\n\\displaystyle \\varrho_{23}= \\frac{\\Delta_0-\\gamma^{2}}{2\\left(3+\\gamma^{2}\\right)},    \\label{eq12}\n\\end{eqnarray}\nwith the parameter \n\\begin{align}\n\\gamma\\equiv\\gamma_-\/\\gamma_+=\\frac{1- e^{-\\beta\\omega}}{1+ e^{-\\beta\\omega}}=\\tanh(\\beta\\omega \/ 2).      \\label{eq9}\n\\end{align}\nIt should be mentioned that the parameter depends solely on the Unruh temperature $T_U$ and characterizes the thermal nature of Unruh effect. Further, the dimensionless parameter $\\Delta_0=\\sum_i\\text{Tr}[\\rho_{ab}(0)\\sigma^{a}_i\\otimes\\sigma^{b}_i]$ provides the choice of initial state and ranges as $-3\\leqslant\\Delta_0\\leqslant1$.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{fig1.pdf}\n\\includegraphics[width=.45\\textwidth]{fig2.pdf}\n\\includegraphics[width=.45\\textwidth]{fig3.pdf}\n\\includegraphics[width=.45\\textwidth]{fig3n.pdf}\n\\caption{Thermal quantum correlation quantified by (a) Entanglement, (b) $l_1$ norm of coherence, (c) LQU and (d) UIN of UdW detector as a function of Unruh temperature $T_U$ for different initial states for  $\\omega=1$.}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\n\n\\section{Results and Discussions}\n\\label{Res}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{fig5.pdf}\n\\includegraphics[width=.45\\textwidth]{fig6.pdf}\n\\includegraphics[width=.45\\textwidth]{fig4.pdf}\n\\includegraphics[width=.45\\textwidth]{fig4n.pdf}\n\\caption{Behaviors of quantum correlation measures (a) Entanglement (b) $l_1$ norm of coherence, (c) LQU and (d) UIN of UdW detector as a function of Unruh temperature $T_U$ for $\\Delta_0=0$.}\n\n\\label{fig2}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{fig5a.pdf}\n\\includegraphics[width=.45\\textwidth]{fig6a.pdf}\n\\includegraphics[width=.45\\textwidth]{fig4a.pdf}\n\\includegraphics[width=.45\\textwidth]{fig4an.pdf}\n\\caption{Behaviors of quantum correlation measures (a) Entanglement (b) $l_1$ norm of coherence, (c) LQU and (d) UIN of UDW detector as a function of Unruh temperature $T_U$ for different initial states for  $\\Delta_0=0.5$.}\n\\label{fig3}\n\\end{center}\n\\end{figure}\n\nWe now study the dynamics of quantum correlations measures quantified in terms of entanglement, $C_{l_1}$-norm of coherence , local quantum uncertainty and uncertainity induced nonlocality. \n\nIn general, Unruh effect is recognized as an environment decoherence. Hence, we study the quantum correlations as a function of Unruh temperature $T_U$. In Fig. (1), we plot the quantum correlation measures as a function of Unruh temperature for different choices of initial conditions $\\Delta_0$ with a fixed value of $\\omega$. For a given initial state, the entanglement is maximum at $T=0$. As temperature increases, the concurrence is a monotonically decreasing function of Unruh temperature $T_U$ and abruptly vanishes . The entanglement completely vanishes for $\\Delta_0=1$.  Further, one can observe that the increase of  $\\Delta_0$ also reduces the quantum of entanglement between the detectors. In order to compare the entanglement with other correlation measures such as coherence, LQU and  UIN, we plot these measures in Fig (1b \\& c) as a function of Unruh temperature. As Unruh temperature $T_U$ increases, the quantum correlation measures decrease initially and reach  a dark point (zero). While increasing the temperature $T_U$ further, both LQU and coherence start to grow from the dark point and then correlations increase with the Unruh temperature $T_U$ unconventionally. It is quite obvious that both the measures are non-monotonic  functions of $T_U$.  In general, if we immerse any static two-qubit in a thermal bath, the quantumness of the system degrades with the temperature of the thermal bath. But here, on the contrary, we observe the revival of quantum correlations and coherence due to the acceleration of the qubits. In addition, the dark point is shifted towards higher temperatures with the increase of $\\Delta_0$. Similar to entanglement, UIN decreases from the maximum values with the increase of temperature and saturates at a nonzero constant value.  A similar observation is noticed using trace distance correlation measure \\cite{Huang1}.\n\n We observe the nonlocal features between the detectors even in the absence of entanglement which are captured through the $C_{l_1}$- norm of coherence and skew information based measures. The entanglement fails  to completely quantify the nonlocality of the UdW detector for $\\Delta_0=1$. On the other hand, the correlation measures beyond entanglement are encapsulated in UdW detector even in the absence of entanglement. \n\nNext, we analyze the role of energy spacing $\\omega$ on the quantum correlation measures. For this purpose, we plot them as a function of  Unruh temperature $T_U$ with specific energy spacing of detectors $\\omega=1,3,5$.  It is obvious that entanglement always degrades monotonously with increasing Unruh temperature, i.e., no revival of entanglement can happen for any initial state preparation. This clearly demonstrates the distinction between the entanglement and coherence.  In Fig. (\\ref{fig1})a, we observe that the entanglement vanishes at low temperatures $T_U=0.5$ for $\\Delta_0=0$. For the same initial state $(\\Delta_0=0)$, comparing Fig. (\\ref{fig1})a with Fig. (\\ref{fig2})a, we observe that the entanglement is induced in the higher temperature regime while increasing the values of energy spacing of detectors from $\\omega=1$ to $\\omega=3$. If we  increase the value of $\\omega$ further, one  observes similar monotonic decreasing behavior of the correlation measures while the entanglement survives for sufficiently higher temperatures  compared to lower values of $\\omega$. On the other hand, the other correlation measures also increase with the increase of energy spacing. In Fig. (\\ref{fig3}), we have plotted the correlation measures as a function of $T_U$ for another  initial state such as $\\Delta_0=0.5$. Here again, the entanglement decreases monotonously while  one sees the revival of LQU and coherence measures.  \n\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{fig8.pdf}\n\\includegraphics[width=.45\\textwidth]{fig82.pdf}\n\\includegraphics[width=.45\\textwidth]{fig81.pdf}\n\\includegraphics[width=.45\\textwidth]{fig8n.pdf}\n\\caption{The Density of quantum correlation measures (a) Coherence, (b) LQU, (c) Entanglement and (d) UIN of UDW detector as a function of $T_U$ and $\\Delta_0$ for   $\\omega=3$}\n\\label{fig5}\n\\end{center}\n\\end{figure}\n\nNow, we illustrate our results with  an example. If we choose the initial state of two-detector in a product form \n\\begin{align}\n\\rho_{\\text{in}}=\\rho_a(0)\\otimes\\rho_b(0),      \\label{eq19}\n\\end{align}\n correlations of the initial state become zero. \n The state of each detector can be written in Bloch form as\n\\begin{align}\n\\rho_{a}(0)=\\frac{1}{2}\\left(I+n \\cdot \\sigma \\right),~~~\\rho_{b}(0)=\\frac{1}{2}\\left(I+m \\cdot \\sigma \\right)   \\label{eq20}\n\\end{align}\nwhere $\\mathbf{n}$ and $\\mathbf{m}$ are two unit Bloch vectors. Without loss of generality, taking $\\mathbf{n}=(0,0,1)$ and $\\mathbf{m}=(0,\\sin\\theta,\\cos\\theta)$, we have $\\Delta_0=\\mathbf{n}\\cdot\\mathbf{m}=\\cos\\theta$ where $\\theta\\in[0,\\pi]$ is an angle between two vectors giving $\\Delta_0\\in[-1,1]$. \n\nIn Fig. (\\ref{fig5}), we have plotted the density of   quantum correlation measures as a function of $T_U$ and $\\Delta_0$ with $\\omega=3$. Again, it is obvious  that the entanglement is generated during time evolution of the detectors for $\\Delta_0=(-1,0)$ and there is no entanglement induced in the range $\\Delta_0=(0,1)$.  Here again, we notice that the entanglement  vanishes at low temperatures and is unable to   capture the complete manifestation of nonlocal attributes of UD detectors. On the other hand, the quantum correlation measures  quantify more nonlocal aspects of UD detectors compared to entanglement. It can be noticed that  the generation of entanglement and quantum correlations in the initial product state arises due to the acceleration of the system. \n\n\n\n\n\n\\section{Conclusions}\n\\label{conc}\nIn this paper, we have studied the  quantum coherence and correlations  of two accelerating Unruh-\nDdeWitt detectors coupled to a scalar background in 3 + 1 Minkowski spacetime. We have employed the entanglement, local quantum uncertainty, uncertainty-induced nonlocality and $l_1$-norm of coherence as the quantumness quantifiers. Results of the paper indicate the revival of  LQU and coherence  while entanglement does not. We have also shown that the energy spacing of detectors induces the nonlocality between the detectors.  The distinction between the static and accelerating qubit systems is observed in terms of generation of quantumness in a product state and revival mechanism of quantum correlations.\n\nFurther, the results of our manuscript can have wider ramifications in understanding  the relativistic quantum information processing from the perspective of quantum correlation measures.\n\\noindent\n\n\\section*{Acknowledgment}\n\nAuthors are indebted to the referees for their critical comments to improve the contents of the manuscript. SB and RR wish to  acknowledge the financial support received from the Council of Scientific and Industrial Research (CSIR), Government of India under Grant No. 03(1456)\/19\/EMR-II.  \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nEarly 1990s were marked by several major developments in knowledge \nrepresentation and nonmonotonic reasoning. One of the most important\namong them was the introduction of \\emph{disjunctive logic programs \nwith classical negation} by Michael Gelfond and Vladimir Lifschitz \n\\cite{gl90b}. The language of the formalism allowed for rules\n\\[\nH_1 \\vee \\ldots \\vee H_k \\leftarrow B_1,\\ldots, B_m,\\mathit{not\\;} B_{m+1},\\ldots,\\mathit{not\\;} B_n,\n\\]\nwhere $H_i$ and $B_i$ are classical literals, that is, atoms and\nclassical or \\emph{strong} negations ($\\neg$) of atoms. In the\npaper, we will write ``strong'' rather than ``classical'' negation,\nas it reflects more accurately the role and the behavior of the \noperator.  The \\emph{answer-set} semantics for programs consisting of \nsuch rules, introduced in the same paper, generalized the stable-model \nsemantics of normal logic programs proposed a couple of years earlier \nalso by Gelfond and Lifschitz \\cite{gl88}. The proposed extensions of \nthe language of normal logic programs were motivated by knowledge \nrepresentation considerations. With two negation operators it was\nstraightforward to distinguish between $P$ being \\emph{false by default}\n(there is no justification for adopting $P$), and $P$ being \n\\emph{strongly false} (there is evidence for $\\neg P$). The former\nwould be written as $\\mathit{not\\;} P$ while the latter as $\\neg P$. And with the\ndisjunction in the head of rules one could model ``indefinite'' rules \nwhich, when applied, provide partial information only (one of the \nalternatives in the head holds, but no preference to any of them is \ngiven).\n\nSoon after disjunctive logic programs with strong negation were \nintroduced, Michael Gelfond proposed an additional important\nextension, this time with a modal operator \\cite{Gelfond91}. He called the\nresulting formalism the language of \\emph{epistemic specifications}. \nThe motivation came again from knowledge representation. The goal \nwas to provide means for the ``correct representation of incomplete \ninformation in the presence of multiple extensions'' \\cite{Gelfond91}. \n\nSurprisingly, despite their evident relevance to the theory of\nnonmonotonic\nreasoning as well as to the practice of knowledge representation,\nepistemic specifications have received relatively little attention \nso far. This\nstate of affairs may soon change. Recent work by Faber and Woltran\non \\emph{meta-reasoning} \nwith answer-set programming \\cite{FaberW09} shows the need for languages,\nin which one could express properties holding across all answer sets\nof a program, something Michael Gelfond foresaw already two decades\nago.\n\nOur goal in this paper is to revisit the formalism of epistemic \nspecifications and show that they deserve a second look, in fact,\na place in the forefront of knowledge representation research. We will \nestablish a general semantic framework for the formalism, and identify \nin it the precise location of Gelfond's epistemic specifications. We \nwill derive several complexity results. We will also show that the \noriginal idea \nof Gelfond to use a modal operator to model ``what is known to a \nreasoner'' has a broader scope of applicability. In particular, we will \nshow that it can also be used in combination with the classical logic. \n\nComplexity results presented in this paper provide an additional \nmotivation to study epistemic specifications. Even though programs \nwith strong negation often look ``more natural'' as they more directly \nalign with the natural language description of knowledge \nspecifications, the extension of the language of normal logic \nprograms with the strong negation operator does not actually increase\nthe expressive power of the formalism. This point was made already by \nGelfond and Lifschitz, who observed that there is a simple and concise \nway to compile the strong negation away. On the other hand, the \nextension allowing the disjunction operator in the heads of rules is \nan essential one. As the complexity results show \\cite{mt88,eg95}, \nthe class of problems that can be represented by means of disjunctive \nlogic programs is strictly larger (assuming no collapse of the \npolynomial hierarchy) than the class of problems that can be modeled by \nnormal logic programs. In the same vein, extension by the modal operator\nalong the lines proposed by Gelfond is essential, too. It does\nlead to an additional jump in the complexity.\n\n\n\\section{Epistemic Specifications}\n\nTo motivate epistemic specifications, Gelfond discussed the following\nexample. A certain college has these rules to determine the eligibility\nof a student for a scholarship:\n\\begin{enumerate}\n\\item Students with high GPA are eligible\n\\item Students from underrepresented groups and with fair GPA are eligible\n\\item Students with low GPA are not eligible\n\\item When these rules are insufficient to determine eligibility,\nthe student should be interviewed by the scholarship committee.\n\\end{enumerate}\nGelfond argued that there is no simple way to represent these rules\nas a disjunctive logic program with strong negation. There is no problem\nwith the first three rules. They are modeled correctly by the following\nthree logic program rules (in the language with both the default and strong\nnegation operators):\n\\begin{enumerate}\n\\item $eligible(X) \\leftarrow \\mathit{highGPA}(X)$\n\\item $eligible(X) \\leftarrow underrep(X), \\mathit{fairGPA}(X)$\n\\item $\\neg eligible(X) \\leftarrow \\mathit{lowGPA}(X)$.\n\\end{enumerate}\nThe problem is with the fourth rule, as it has a clear meta-reasoning\nflavor. It should apply when the possible worlds (answer sets)\ndetermined by the first three rules do not fully specify the status of \neligibility of a student $a$: neither \\emph{all} of them contain \n$eligible(a)$ nor \\emph{all} of them contain $\\neg eligible(a)$. An \nobvious attempt at a formalization:\n\\begin{enumerate}\n\\item[4.] $interview(X) \\leftarrow \\mathit{not\\;} eligible(X), \\mathit{not\\;} \\neg eligible(X)$ \n\\end{enumerate}\nfails. It is just another rule to be added to the program. Thus,\nwhen the answer-set semantics is used, the rule is interpreted with\nrespect to individual answer sets and not with respect to collections\nof answer-sets, as required for this application. For a concrete \nexample, let us assume that all we know about a certain student named \nMike is that Mike's GPA is fair or high. Clearly, we do not have \nenough information to determine Mike's eligibility and so we must \ninterview Mike. But the program consisting of rules (1)-(4) and the \nstatement \n\\begin{enumerate}\n\\item[5.] $fairGPA(mike) \\vee highGPA(mike)$\n\\end{enumerate}\nabout Mike's GPA, has two answer sets:\n\\begin{quote}\n$\\{highGPA(mike), eligible(mike)\\}$\\\\\n$\\{fairGPA(mike), interview(mike)\\}$. \n\\end{quote}\nThus, the query $?interview(mike)$ has the answer ``unknown.'' To\naddress the problem, Gelfond proposed to extend the language with a\nmodal operator $K$ and, speaking informally, interpret premises \n$K\\varphi$ as ``$\\varphi$ is known to the program'' (the original phrase\nused by Gelfond was ``known to the reasoner''), that is, true in all \nanswer-sets. With this language extension, the \nfourth rule can be encoded as \n\\begin{enumerate}\n\\item[4$'$.] $interview(X) \\leftarrow \\mathit{not\\;} K\\, eligible(X), \\mathit{not\\;} K \\neg eligible(X)$\n\\end{enumerate}\nwhich, intuitively, stands for ``\\emph{interview} if neither the \neligibility nor the non-eligibility is known.'' \n\nThe way in which Gelfond \\cite{Gelfond91} proposed to formalize this \nintuition is strikingly elegant. We will now discuss it. We start with\nthe syntax of \\emph{epistemic specifications}. As elsewhere in the \npaper, we restrict attention to the propositional case. We assume a \nfixed infinite countable set $\\mathit{At}$ of \\emph{atoms} and the corresponding\nlanguage $\\mathcal{L}$ of propositional logic. A \\emph{literal} is an atom, say\n$A$, or its \\emph{strong} negation $\\neg A$. A \\emph{simple modal atom}\nis an expression $K\\varphi$, where $\\varphi\\in \\mathcal{L}$, and a \\emph{simple modal \nliteral} is defined accordingly. An \\emph{epistemic premise} is an \nexpression (conjunction)\n\\[\nE_1,\\ldots, E_s,\\mathit{not\\;} E_{s+1},\\ldots, \\mathit{not\\;} E_t,\n\\]\nwhere every $E_i$, $1\\leq i\\leq t$, is a simple modal literal. \nAn \\emph{epistemic rule} is an expression of the form\n\\[\nL_1 \\vee \\ldots\\vee L_k \\leftarrow L_{k+1},\\ldots, L_m,\\mathit{not\\;} L_{m+1},\\ldots, \\mathit{not\\;} L_n, E,\n\\]\nwhere every $L_i$, $1\\leq i\\leq k$, is a literal, and $E$ is an epistemic \npremise.\nCollections of\nepistemic rules are \\emph{epistemic programs}. It is clear that (ground\nversions of) rules (1)-(5) and (4$'$) are examples of epistemic rules, \nwith rule (4$'$) being an example of an epistemic rule that actually \ntakes advantage of the extended syntax. Rules such as\n\\begin{quote}\n$a\\vee \\neg d \\leftarrow b, \\mathit{not\\;} \\neg c, \\neg K(d\\lor \\neg c)$\\\\\n$\\neg a \\leftarrow \\neg c, \\mathit{not\\;} \\neg K(\\neg (a\\land c)\\rightarrow b)$\n\\end{quote}\nare also examples of epistemic rules. We note that the language of \nepistemic programs is only a fragment of the language of epistemic \nspecifications by Gelfond. However, it is still expressive enough to \ncover all examples discussed by Gelfond and, more generally, a broad\nrange of practical applications, as natural-language formulations of \ndomain knowledge typically assume a rule-based pattern.\n\nWe move on to the semantics, which is in terms of \\emph{world \nviews}. The definition of a world view consists of several steps. \nFirst, let $W$ be \na consistent set of literals from $\\mathcal{L}$. We regard $W$ as a three-valued\ninterpretation of $\\mathcal{L}$ (we will also use the term \\emph{three-valued \npossible world}), assigning to each atom one of the three logical values\n$\\mathbf{t}$, $\\mathbf{f}$ and $\\mathbf{u}$. The interpretation extends by recursion to all \nformulas in $\\mathcal{L}$, according to the following truth tables\n\\begin{figure}[h]\n\\begin{minipage}[t]{2.1cm}\n\\ \n\\end{minipage}\n\\begin{minipage}[t]{1.5cm}\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\n$\\neg$ & \\\\\n\\hline\n$\\mathbf{f}$  & $\\mathbf{t}$ \\\\\n$\\mathbf{t}$ & $\\mathbf{f}$\\\\\n$\\mathbf{u}$ & $\\mathbf{u}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}[t]{2.0cm}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|}\n\\hline\n$\\lor$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n\\hline\n\\hline\n$\\mathbf{t}$ & $\\mathbf{t}$ & $\\mathbf{t}$ & $\\mathbf{t}$ \\\\\n$\\mathbf{u}$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{u}$ \\\\\n$\\mathbf{f}$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}[t]{2.0cm}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|}\n\\hline\n$\\land$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n\\hline\n\\hline\n$\\mathbf{t}$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n$\\mathbf{u}$ & $\\mathbf{u}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n$\\mathbf{f}$ & $\\mathbf{f}$ & $\\mathbf{f}$ & $\\mathbf{f}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}[t]{2.0cm}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|}\n\\hline\n$\\rightarrow$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n\\hline\n\\hline\n$\\mathbf{t}$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n$\\mathbf{u}$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{u}$ \\\\\n$\\mathbf{f}$ & $\\mathbf{t}$ & $\\mathbf{t}$ & $\\mathbf{t}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\\\\ \\\\ \\mbox{}\n\\caption{Truth tables for the 3-valued logic of Kleene.}\n\\label{t.1}\n\\end{figure}\n\nBy a \\emph{three-valued possible-world structure} we mean a non-empty\nfamily of consistent sets of literals (three-valued possible worlds). \nLet $\\mathcal{A}$ be a three-valued possible-world structure and let $W$ be\na consistent set of literals. For every formula $\\varphi\\in\\mathcal{L}$, we define \n\\begin{enumerate}\n\\item $\\langle\\mathcal{A},W\\rangle\\models\\varphi$, if $v_W(\\varphi)=\\mathbf{t}$\n\\item $\\langle\\mathcal{A},W\\rangle\\models K\\varphi$, if for every $V\\in\\mathcal{A}$, $v_V(\\varphi)=\\mathbf{t}$\n\\item $\\langle\\mathcal{A},W\\rangle\\models \\neg K\\varphi$, if there is $V\\in\\mathcal{A}$ such\nthat $v_V(\\varphi)=\\mathbf{f}$.\n\\end{enumerate}\nNext, for every literal or simple modal literal $L$, we define\n\\begin{enumerate}\n\\item[4.] $\\langle\\mathcal{A},W\\rangle \\models\\mathit{not\\;} L$ if $\\langle \\mathcal{A},W\\rangle \\not\\models L$.\n\\end{enumerate}\nWe note that neither $\\langle\\mathcal{A},W\\rangle\\models K\\varphi$ nor $\\langle\\mathcal{A},W\\rangle\\models \n\\neg K\\varphi$ depend on $W$. Thus, we will often write $\\mathcal{A}\\models F$, when \n$F$ is a simple modal literal or its default negation.\n\nIn the next step, we introduce the notion of the \\emph{G-reduct} of an \nepistemic program.\n\n\\begin{definition}\n\\label{def1}\nLet $P$ be an epistemic program, $\\mathcal{A}$ a three-valued possible-world\nstructure and $W$ a consistent set of literals. The \\emph{G-reduct}\nof $P$ with respect to $\\langle\\mathcal{A},W\\rangle$, in symbols $P^{\\langle\\mathcal{A},W\\rangle}$, \nconsists of the heads of all\nrules $r\\in P$ such that $\\langle\\mathcal{A},W\\rangle\\models \\alpha$, for every \nconjunct $\\alpha$ occurring in the body of $r$. \n\\end{definition}\n\nLet $H$ be a set of disjunctions of literals from $\\mathcal{L}$. A set $W$ of \nliterals is \\emph{closed} with respect to $H$ if $W$ is consistent\nand contains at least one literal in common with every disjunction in \n$H$. \nWe denote by $\\mathit{Min}(H)$ the family of all minimal sets of literals that \nare closed with respect to $H$.\nWith the notation $\\mathit{Min}(H)$ in hand, we are finally ready to\ndefine the concept of a world view of an epistemic program $P$. \n\n\\begin{definition}\n\\label{def2}\nA three-valued possible-world structure $\\mathcal{A}$ is a \\emph{world view} \nof an epistemic program $P$ if $\\mathcal{A}= \\{W\\,|\\; W\\in \\mathit{Min}(P^{\\langle\\mathcal{A},W\\rangle})\\}$.\n\\end{definition}\n\n\\begin{remark}\nThe $G$-reduct of an epistemic program consists of disjunctions of literals.\nThus, the concept of a world view is well defined.\n\\end{remark}\n\\begin{remark}\nWe note that Gelfond considered also inconsistent sets of literals as \nminimal sets closed under disjunctions. However, the only such set \nhe allowed consisted of \\emph{all} literals. Consequently, the \ndifference between the Gelfond's semantics and the one we described \nabove is that some programs have a world view in the Gelfond's approach\nthat consists of a single set of all literals, while in our approach \nthese programs do not have a world view. But in all other cases, the \ntwo semantics behave in the same way. \n\\end{remark}\n\nLet us consider the ground program, say $P$, corresponding to the \nscholarship eligibility example (rule (5), and rules (1)-(3) and \n(4$'$), grounded with respect to the Herbrand universe $\\{mike\\}$).\nThe only rule involving simple modal literals is\n\\begin{quote}\n$interview(mike) \\leftarrow \\mathit{not\\;} K\\, eligible(mike), \\mathit{not\\;} K \\neg eligible(mike)$.\n\\end{quote}\nLet $\\mathcal{A}$ be a world view of $P$. Being a three-valued possible-world \nstructure, $\\mathcal{A}\\not=\\emptyset$. No\nmatter what $W$ we consider, no minimal set closed with respect to \n$P^{\\langle\\mathcal{A},W\\rangle}$ contains $\\mathit{lowGPA}(mike)$ and, consequently,\nno minimal set closed with respect to $P^{\\langle\\mathcal{A},W\\rangle}$ contains \n$\\neg eligible(mike)$. It follows that $\\mathcal{A}\\not\\models K \\neg \neligible(mike)$. \n\nLet us assume that $\\mathcal{A}\\models K\\, eligible(mike)$. Then, no reduct\n$P^{\\langle\\mathcal{A},W\\rangle}$ contains $interview(mike)$. Let\n$W=\\{\\mathit{fairGP}(mike)\\}$. It follows that $P^{\\langle\\mathcal{A},W\\rangle}$\nconsists only of $\\mathit{fairGPA}(mike) \\vee \\mathit{highGPA}(mike)$.\nClearly, $W\\in\\mathit{Min}(P^{\\langle\\mathcal{A},W\\rangle})$ and, consequently, $W\\in \\mathcal{A}$.\nThus, $\\mathcal{A}\\not\\models K\\, eligible(mike)$, a contradiction.\n\nIt must be then that $\\mathcal{A}\\models\\mathit{not\\;} K\\, eligible(mike)$ and \n$\\mathcal{A}\\models \\mathit{not\\;} K \\neg eligible(mike)$. Let $W$ be an arbitrary \nconsistent set of literals. Clearly, the reduct $P^{\\langle\\mathcal{A},W\\rangle}$ \ncontains $interview(mike)$ and\n$\\mathit{fairGPA}(mike) \\vee \\mathit{highGPA}(mike)$. If \n$\\mathit{highGPA}(mike)\\in W$, the reduct also contains $eligible(mike)$. \nThus, $W\\in\\mathit{Min}(P^{\\langle\\mathcal{A},W\\rangle})$ if and only if \n\\begin{quote}\n$W=\\{\\mathit{fairGPA}(mike),interview(mike)\\}$, or\\\\\n$W=\\{\\mathit{highGPA}(mike),eligible(mike),interview(mike)\\}$.\n\\end{quote}\nIt follows that if $\\mathcal{A}$ is a world view for $P$ then it consists of \nthese two possible worlds. Conversely, it is easy to check that a \npossible-world structure consisting of these two possible worlds is a \nworld view for $P$. Thus, $interview(mike)$ holds in $\\mathcal{A}$, and so \nour representation of the example as an epistemic program has the\ndesired behavior.\n\n\\section{Epistemic Specifications --- a Broader Perspective}\n\nThe discussion in the previous section demonstrates the usefulness of\nformalisms such as that of epistemic specifications for knowledge\nrepresentation and reasoning. We will now present a simpler yet, in \nmany respects, more general framework for epistemic specifications. \nThe key to our approach is that we consider the semantics given by \n\\emph{two-valued} interpretations (sets of atoms), and standard \n\\emph{two-valued} possible-world structures (nonempty collections of \ntwo-valued interpretations). We also work within a rather standard\nversion of the language of modal propositional logic and so, in \nparticular, we allow only for one negation operator. Later in the paper\nwe show that epistemic specifications by Gelfond can be encoded in a \nrather direct way in our formalism. Thus, the restrictions we impose \nare not essential even though, admittedly, not having two kinds of \nnegation in the language in some cases may make the modeling task \nharder. \n\nWe start by making precise the syntax of the language we will be \nusing.  As we stated earlier, we assume a fixed infinite countable \nset of atoms $\\mathit{At}$. The language we consider is determined by the \nset $\\mathit{At}$, the modal operator $K$, and by the \\emph{boolean connectives}\n$\\bot$ (0-place), and $\\land$, $\\lor$, and $\\rightarrow$ (binary). The BNF \nexpression \n\\begin{quote}\n$\\varphi::= \\bot\\,|\\,A\\,|\\,(\\varphi\\land \\varphi)\\,|\\,(\\varphi \\lor \\varphi)\\,|\\,\n(\\varphi\\rightarrow\\varphi)\\,|\\ K\\varphi$,\n\\end{quote}\nwhere $A\\in \\mathit{At}$, provides a concise definition of a formula. The \nparentheses are used only to disambiguate the order of binary \nconnectives. Whenever possible, we omit them. We define the unary\n\\emph{negation} connective $\\neg$ and the 0-place connective $\\top$\nas abbreviations:\n\\begin{quote}\n$\\neg\\varphi::= \\varphi\\rightarrow\\bot$\\\\\n$\\top::=\\neg\\bot$.\n\\end{quote}\nWe call formulas $K\\varphi$, where $\\varphi\\in\\mathcal{L}_K$, \\emph{modal atoms} \n(simple modal atoms that we considered earlier and will consider below\nare special modal atoms with $K$-depth equal to 1). We denote this \nlanguage by $\\mathcal{L}_K$ and refer to subsets of $\\mathcal{L}_K$ as \\emph{epistemic \ntheories}. We denote the modal-free fragment of $\\mathcal{L}_K$ by $\\mathcal{L}$.  \n\nWhile we will eventually describe the semantics (in fact, several of \nthem) for arbitrary epistemic theories, we start with an important\nspecial case. Due to close analogies between the concepts we define \nbelow and the corresponding ones defined earlier in the context of the \nformalism of Gelfond, we ``reuse'' the terms used there. Specifically, \nby an \\emph{epistemic premise} we mean a conjunction of simple modal \nliterals. Similarly, by an \\emph{epistemic rule} we understand an \nexpression of the form\n\\begin{equation}\n\\label{eq10}\nE\\land L_1\\land\\ldots\\land L_m \\rightarrow A_1\\lor\\ldots\\lor A_n,\n\\end{equation}\nwhere $E$ is an epistemic premise, $L_i$'s are literals (in $\\mathcal{L}$) and \n$A_i$'s are atoms. Finally, we call a collection of epistemic rules an \n\\emph{epistemic program}. It will always be clear from the context, in \nwhich sense these terms are to be understood.\n\nWe stress that $\\neg$ is not a primary connective in the language but a\nderived one (it is a shorthand for some particular formulas involving\nthe rule symbol). Even though under some semantics we propose below this\nnegation operator has features of default negation, under some others it\ndoes not. Thus, we selected for it the standard negation symbol $\\neg$\nrather than the ``loaded'' $\\mathit{not\\;}$. \n  \nA (two-valued) \\emph{possible-world structure} is any nonempty family\n$\\mathcal{A}$ of subsets of $\\mathit{At}$ (two-valued interpretations). In the remainder\nof the paper, when we use terms ``interpretation'' and ``possible-world \nstructure'' without any additional modifiers, we always mean a two-valued\ninterpretation and a two-valued possible-world structure.\n\nLet $\\mathcal{A}$ be a possible-world structure and $\\varphi\\in \\mathcal{L}$. We recall \nthat $\\mathcal{A}\\models K\\varphi$ precisely when $W\\models \\varphi$, for every \n$W\\in\\mathcal{A}$, and $\\mathcal{A}\\models \\neg K\\varphi$, otherwise. We will now define \nthe \\emph{epistemic reduct} of an epistemic program with respect to a\npossible-world structure. \n\n\\begin{definition}\nLet $P\\subseteq\\mathcal{L}_K$ be an epistemic program and let $\\mathcal{A}$ be a\npossible-world structure. The \\emph{epistemic reduct} of $P$ with \nrespect to $\\mathcal{A}$, $\\gl{P}{\\mathcal{A}}$ in symbols, is the theory obtained \nfrom $P$ as follows: eliminate every rule with an epistemic premise $E$\nsuch that $\\mathcal{A}\\not\\models E$; drop the epistemic premise from every \nremaining rule. \n\\end{definition}\nIt is clear that $\\gl{P}{\\mathcal{A}}\\subseteq\\mathcal{L}$, and that it consists of \nrules of the form \n\\begin{equation}\n\\label{eq11}\nL_1\\land\\ldots\\land L_m \\rightarrow A_1\\lor\\ldots\\lor A_n,\n\\end{equation}\nwhere $L_i$'s are literals (in $\\mathcal{L}$) and $A_i$'s are atoms. \n\nLet $P$ be a collection of rules (\\ref{eq11}). Then, $P$ is a \npropositional theory. Thus, it can be interpreted by the standard \npropositional logic semantics. However, $P$ can also be regarded as a \ndisjunctive logic program (if we write rules from right to left rather \nthan from left to right). Consequently, $P$ can also be interpreted by\nthe stable-model semantics \\cite{gl88,gl90b} and the supported-model \nsemantics \\cite{abw87,bg93,bradix96jlp1,Inoue98-JLP}. (For normal logic\nprograms, the supported-model semantics was introduced by Apt et al. \n\\cite{abw87}. The notion was extended to disjunctive logic programs by\nBaral and Gelfond \\cite{bg93}. We refer to papers by Brass and Dix\n\\cite{bradix96jlp1}, Definition 2.4, and Inoue and Sakama \n\\cite{Inoue98-JLP}, Section 5, for more details).\nWe write $\\mathcal{M}(P)$, $\\mathcal{ST}(P)$ and \n$\\mathcal{SP}(P)$ for the sets of models, stable models and supported models \nof $P$, respectively. An important observation is that \\emph{each} of \nthese semantics gives rise to the corresponding notion of an epistemic\nextension.\n\n\\begin{definition}\n\\label{def11}\nLet $P\\subseteq \\mathcal{L}_K$ be an epistemic program. A possible-world \nstructure $\\mathcal{A}$ is an \\emph{epistemic model} (respectively, an\n\\emph{epistemic stable model}, or an \\emph{epistemic supported model})\nof $P$, if $\\mathcal{A} = \\mathcal{M}(\\gl{P}{\\mathcal{A}})$ (respectively, $\\mathcal{A} = \n\\mathcal{ST}(\\gl{P}{\\mathcal{A}})$ or $\\mathcal{A} = \\mathcal{SP}(\\gl{P}{\\mathcal{A}})$).\n\\end{definition}\n\nIt is clear that Definition \\ref{def11} can easily be adjusted also to\nother semantics of propositional theories and programs. We briefly \nmention two such semantics in the last section of the paper.\n\nWe will now show that epistemic programs with the semantics of \nepistemic stable models can provide an adequate representation to\nthe scholarship eligibility example for Mike. The available information\ncan be represented by the following program $P(mike) \\subseteq\\mathcal{L}_K$:\n\n\\begin{enumerate}\n\\item $eligible(mike)\\land neligible(mike) \\rightarrow \\bot$\n\\item $fairGPA(mike) \\vee highGPA(mike)$\n\\item $\\mathit{highGPA}(mike) \\rightarrow eligible(mike)$\n\\item $underrep(mike) \\land \\mathit{fairGPA}(mike)\\rightarrow eligible(mike)$\n\\item $\\mathit{lowGPA}(mike) \\rightarrow neligible(mike)$\n\\item $\\neg K\\, eligible(mike), \\neg K\\, neligible(mike)\\rightarrow interview(mike)$.\n\\end{enumerate}\nWe use the predicate \\emph{neligible} to model the strong negation \nof the predicate $eligible$ that appears in the representation in terms \nof epistemic programs by Gelfond (thus, in particular, the presence of \nthe first clause, which precludes the facts $eligible(mike)$ and \n$neligible(mike)$\nto be true together). This extension of the language and an extra\nrule in the representation is the price we pay\nfor eliminating one negation operator. \n\nLet $\\mathcal{A}$ consist of the interpretations \n\\begin{quote}\n$W_1=\\{\\mathit{fairGPA}(mike),interview(mike)\\}$\\\\\n$W_2=\\{\\mathit{highGPA}(mike),eligible(mike),interview(mike)\\}$.\n\\end{quote}\nThen the reduct $\\gl{[P(mike)]}{\\mathcal{A}}$ consists of rules (1)-(5),\nwhich are unaffected by the reduct operation, and of the fact \n$interview(mike)$, resulting from rule (6) when the reduct operation\nis performed (as in logic programming, when a rule has the empty \nantecedent, we drop the implication symbol from the \nnotation). One can check that \n$\\mathcal{A}=\\{W_1,W_2\\}=\\mathcal{ST}(\\gl{[P(mike)]}{\\mathcal{A}})$. Thus, $\\mathcal{A}$ is\nan epistemic stable model of $P$ (in fact, the only one). Clearly, \n$interview(mike)$ holds in the model (as we would expect it to), as it \nholds in each of its possible-worlds. We note that in this particular \ncase, the semantics of epistemic supported models yields exactly the \nsame solution.\n\n\\section{Complexity}\n\nWe will now study the complexity of reasoning with epistemic (stable, \nsupported) models. We provide details for the case of epistemic stable\nmodels, and only present the results for the other two semantics, as the \ntechniques to prove them are very similar to those we develop for the\ncase of epistemic stable models.\n\nFirst, we note that epistemic stable models of an epistemic program $P$\ncan be represented by partitions of the set of all modal atoms of $P$. \nThis is important as \\emph{a priori} the size of possible-world \nstructures one needs to consider as candidates for epistemic stable\nmodels may be exponential in the size of a program. Thus, to obtain \ngood complexity bounds alternative polynomial-size representations \nof epistemic stable models are needed.\n\nLet $P\\subseteq\\mathcal{L}_K$ be an epistemic program and $(\\Phi,\\Psi)$ be the \nset of modal atoms of $P$ (all these modal atoms are, in fact, simple).\nWe write $P_{|\\Phi,\\Psi}$ for the program\nobtained from $P$ by eliminating every rule whose epistemic premise\ncontains a conjunct $K\\psi$, where $K\\psi\\in\\Psi$, or a conjunct\n$\\neg K\\varphi$, where $K\\varphi\\in\\Phi$ (these rules are ```blocked'' by\n$(\\Phi,\\Psi)$), and by eliminating the epistemic premise from every\nother rule of $P$.\n\n\\begin{proposition}\n\\label{prop:char}\nLet $P\\subseteq\\mathcal{L}_K$ be an epistemic program. If a possible-world\nstructure $\\mathcal{A}$ is an epistemic stable model of $P$, then there\nis a partition $(\\Phi,\\Psi)$ of the set of modal atoms of $P$ such\nthat\n\\begin{enumerate}\n\\item $\\mathcal{ST}(P_{|\\Phi,\\Psi})\\not=\\emptyset$\n\\item For every $K\\varphi\\in\\Phi$, $\\varphi$ holds in every stable model\nof $P_{|\\Phi,\\Psi}$ \n\\item For every $K\\psi\\in\\Psi$, $\\psi$ does not hold in at least one\nstable model of $P_{|\\Phi,\\Psi}$. \n\\end{enumerate}\nConversely, if there are such partitions, $P$ has epistemic stable \nmodels.\n\\end{proposition}  \n\nIt follows that epistemic stable models can be represented by partitions\n$(\\Phi,\\Psi)$ satisfying conditions (1)-(3) from the proposition above. \n\nWe observe that deciding whether a partition $(\\Phi,\\Psi)$ satisfies \nconditions (1)-(3) from Proposition \\ref{prop:char}, can be accomplished\nby polynomially many calls to an $\\Sigma_2^P$-oracle and, if we restrict\nattention to non-disjunctive epistemic programs, by polynomially many \ncalls to an $\\mathit{NP}$-oracle. \n\n\\begin{remark}\n\\label{rem1}\nIf we adjust Proposition \\ref{prop:char} by replacing the term ``stable''\nwith the term ``supported,'' and replacing $\\mathcal{ST}()$ with $\\mathcal{SP}()$, we\nobtain a characterization of epistemic supported models. Similarly, \nomitting the term ``stable,'' and replacing $\\mathcal{ST}()$ with $\\mathcal{M}()$ \nyields a characterization of epistemic models. In each case, one can \ndecide whether a partition $(\\Phi,\\Psi)$ satisfies conditions (1)-(3)\nby polynomially many calls to an $\\mathit{NP}$-oracle (this claim is evident \nfor the case of epistemic models; for the case of epistemic supported\nmodels, it follows from the fact that supported models semantics does\nnot get harder when we allow disjunctions in the heads or rules). \n\\end{remark}\n\n\\begin{theorem}\nThe problem to decide whether a non-disjunctive epistemic program has \nan epistemic stable model is $\\Sigma_2^P$-complete.  \n\\end{theorem}\nProof: Our comments above imply that the problem is in the class \n$\\Sigma_2^P$. Let $F=\\exists Y \\forall Z \\Theta$, where $\\Theta$\nis a DNF formula. The problem to decide whether $F$ is true is \n$\\Sigma_2^P$-complete. We will reduce it to the problem in question \nand, consequently, demonstrate its $\\Sigma_2^P$-hardness. To this\nend, we construct an epistemic program $Q\\subseteq\\mathcal{L}_K$ by including \ninto $Q$ the following clauses (atoms $w$, $y'$, $y\\in Y$, and $z'$, \n$z\\in Z$ are fresh):\n\\begin{enumerate}\n\\item $Ky\\rightarrow y\\;$; and $Ky' \\rightarrow y'$, for every $y\\in Y$\n\\item $y\\land y'\\rightarrow\\;$; and  $\\neg y\\land \\neg y'\\rightarrow\\;$, for every $y\\in Y$\n\\item $\\neg z'\\rightarrow z\\;$; and $\\neg z\\rightarrow z'$, for $z\\in Z$\n\\item $\\sigma(u_1)\\land  \\ldots\\land  \\sigma(u_k)\\rightarrow w\\;$, where \n$u_1\\wedge\\ldots\\wedge u_k$ is a disjunct of $\\Theta$, and \n$\\sigma(\\neg a)=a'$ and $\\sigma(a)=a$, for every $a\\in Y\\cup Z$\n\\item $\\neg Kw\\rightarrow\\;$.\n\\end{enumerate}\n\nLet us assume that $\\mathcal{A}$ is an epistemic stable model of $Q$. In particular,\n$\\mathcal{A}\\not=\\emptyset$. It must \nbe that $\\mathcal{A}\\models Kw$ (otherwise, $\\gl{Q}{\\mathcal{A}}$ has no stable models, \nthat is, $\\mathcal{A}=\\emptyset$). Let us define $A= \\{y\\in Y\\,|\\; \\mathcal{A} \\models \nKy\\}$, and $B=\\{y\\in Y\\,|\\;\\mathcal{A}\\models Ky'\\}$. It follows that $\\gl{Q}{\\mathcal{A}}$\nconsists of the following rules:\n\\begin{enumerate}\n\\item $y$, for $y\\in A$, and $y'$, for $y\\in B$\n\\item $y\\land y'\\rightarrow\\;$; and  $\\neg y\\land \\neg y'\\rightarrow \\;$, for every $y\\in Y$\n\\item $\\neg z'\\rightarrow z\\;$; and $\\neg z\\rightarrow z'$, for $z\\in Z$\n\\item $\\sigma(u_1)\\land  \\ldots\\land  \\sigma(u_k)\\rightarrow w\\;$, where\n$u_1\\wedge\\ldots\\wedge u_k$ is a disjunct of $\\Theta$, and\n$\\sigma(\\neg a)=a'$ and $\\sigma(a)=a$, for every $a\\in Y\\cup Z$.\n\\end{enumerate}\nSince $\\mathcal{A}=\\mathcal{ST}(\\gl{Q}{\\mathcal{A}})$ and $\\mathcal{A}\\not=\\emptyset$, $B=Y\\setminus A$\n(due to clauses of type (2)).\nIt is clear that the program $\\gl{Q}{\\mathcal{A}}$ has stable models\nand that they are of the form $A\\cup \\{y'\\,|\\; y\\in Y\\setminus A\\} \\cup\nD \\cup \\{z'\\,|\\; z\\in Z\\setminus D\\}$, if that set does not imply $w$ through\na rule of type (4), or $A\\cup \\{y'\\,|\\; y\\in Y\\setminus A\\} \\cup\nD \\cup \\{z'\\,|\\; z\\in Z\\setminus D\\}\\cup \\{w\\}$, otherwise, where $D$ is any\nsubset of $Z$. As $\\mathcal{A}\\models Kw$,\nthere are no stable models of the first type. Thus, the family of stable \nmodels\nof $\\gl{Q}{\\mathcal{A}}$ consists of all sets $A\\cup \\{y'\\,|\\; y\\in Y\\setminus \nA\\} \\cup\nD \\cup \\{z'\\,|\\; z\\in Z\\setminus D\\}\\cup \\{w\\}$, where $D$ is an arbitrary\nsubset of $Z$. It follows that for every $D\\subseteq Z$, the set $A\\cup \n\\{y'\\,|\\; y\\in Y\\setminus A\\} \\cup D \\cup \\{z'\\,|\\; z\\in Z\\setminus D\\}$\nsatisfies the body of at least one rule of type (4). By the construction, \nfor every $D\\subseteq Z$, the valuation of $Y\\cup Z$ determined by $A$ and \n$D$ satisfies the corresponding disjunct in $\\Theta$ and so, also $\\Theta$.\nIn other words, $\\exists Y\\forall Z \\Theta$ is true.\n\nConversely, let $\\exists Y\\forall Z\\Theta$ be true. Let $A$ be a subset\nof $Y$ such that $\\Theta_{|Y\/A}$ holds for every truth assignment of $Z$ \n(by $\\Theta_{|Y\/A}$, we mean the formula obtained by simplifying the \nformula $Q$ with respect to the truth assignment of $Y$ determined by \n$A$). Let \n$\\mathcal{A}$ consist of all sets of the form $A\\cup \\{y'\\,|\\; y\\in Y\\setminus \nA\\} \\cup D \\cup \\{z'\\,|\\; z\\in Z\\setminus D\\}\\cup \\{w\\}$, where $D\\subseteq\nZ$. It follows that $\\gl{Q}{\\mathcal{A}}$ consists of clauses (1)-(4) above,\nwith $B=Y\\setminus A$.\nSince $\\forall Z \\Theta_{|A\/Y}$ holds, it follows that $\\mathcal{A}$ is precisely \nthe set of stable models of $\\gl{Q}{\\mathcal{A}}$. Thus, $\\mathcal{A}$ is an epistemic\nstable model of $Q$. \n\\hfill$\\Box$\n\nIn the general case, the complexity goes one level up.\n\n\\begin{theorem}\n\\label{thm:2}\nThe problem to decide whether an epistemic program $P\\subseteq\\mathcal{L}_K$\nhas an epistemic stable model is $\\Sigma_3^P$-complete.\n\\end{theorem}\nProof: The membership follows from the earlier remarks. To prove the \nhardness part, we consider a QBF formula $F=\n\\exists X \\forall Y \\exists Z \\Theta$, where $\\Theta$ is a 3-CNF \nformula. For each atom $x\\in X$ ($y\\in Y$ and $z\\in Z$, respectively), we \nintroduce a fresh atom $x'$ ($y'$ and $z'$, respectively). Finally, \nwe introduce three additional fresh atoms, $w$, $f$ and $g$.\n\nWe now construct a disjunctive epistemic program $Q$ by including into\nit the following clauses:\n\\begin{enumerate}\n\\item $Kx\\rightarrow x$; and $Kx' \\rightarrow x'$, for every $x\\in X$\n\\item $x\\land x'\\rightarrow$; and  $\\neg x\\land \\neg x'\\rightarrow$, for every $x\\in X$\n\\item $\\neg g\\rightarrow f$; and $\\neg f\\rightarrow g$\n\\item $f\\rightarrow y \\vee y'$; and $f\\rightarrow z \\vee z'$, for every $y\\in Y$ \nand $z\\in Z$\n\\item $f\\land w\\rightarrow z$; and $f\\land w\\rightarrow z'$, for every $z\\in Z$\n\\item $f\\land \\sigma(u_1)\\land \\sigma(u_2)\\land \\sigma(u_3)\\rightarrow w$, \nfor every clause $C= u_1\\vee u_2 \\vee u_3$ of $\\Theta$, where \n$\\sigma(a)=a'$ and \n$\\sigma(\\neg a)= a$, for every $a\\in X\\cup Y \\cup Z$  \n\\item $f\\land \\neg w \\rightarrow w$\n\\item $\\neg K\\neg w \\rightarrow$\n\\end{enumerate}\nLet us assume that $\\exists X \\forall Y\\exists Z \\Theta$ is true. Let\n$A\\subseteq\nX$ describe the truth assignment on $X$ so that $\\forall Y\n\\exists Z \\Phi_{X\/A}$ holds (we define $\\Phi_{X\/A}$ in the proof\nof the previous result). We will show that $Q$ has an epistemic \nstable model\n$\\mathcal{A}=\\{A\\cup \\{a'\\,|\\; a\\in X\\setminus A\\}\\cup\\{g\\}\\}$. Clearly, $Kx$, $x\\in A$,\nand $Kx'$, $x\\in X\\setminus A$, are true in $\\mathcal{A}$. Also, $K\\neg w$ is \ntrue in $\\mathcal{A}$. All other modal atoms in $Q$ are false in $\\mathcal{A}$.\nThus, $\\gl{Q}{\\mathcal{A}}$ consists of rules $x$, for $x\\in A$, $x'$, for $x\\in\nX\\setminus A$ and of rules (2)-(7) above. Let $M$ be a stable model of\n$\\gl{Q}{\\mathcal{A}}$ containing $f$. It follows that $w\\in M$ and so, $Z\\cup Z'\n\\subseteq M$. Moreover, the Gelfond-Lifschitz reduct of $\\gl{Q}{\\mathcal{A}}$\nwith respect to $M$ consists of rules $x$, for $x\\in A$, $x'$, for $x\\in\nX\\setminus A$, all $\\neg$-free constraints of type (2), rule $f$, and \nrules (4)-(6) above, and $M$ is a minimal model of this program.\n\nLet $B=Y\\cap M$. By the minimality of $M$, $M=A\\cup \\{x'\\,|\\; x\\in X\n\\setminus A\\}\\cup B\\cup\\{y'\\,|\\; y\\in Y\\setminus B\\} \\cup Z\\cup Z'\\cup \n\\{f,w\\}$. Since $\\forall Y \\exists Z \\Phi_{X\/A}$ holds, $\\exists Z \n\\Phi_{X\/A,Y\/B}$ holds, too. Thus, let $D\\subseteq Z$ be a subset of $Z$\nsuch that $\\Phi_{X\/A,Y\/B,Z\/D}$ is true. It follows that $M'=A\\cup \\{x'\n\\,|\\; x\\in X \\setminus A\\}\\cup B\\cup\\{y'\\,|\\; y\\in Y\\setminus B\\} \\cup \nD\\cup\\{z'\\,|\\; z\\in Z\\setminus D\\} \\cup \\{f\\}$ is also a model of the\nGelfond-Lifschitz reduct of $\\gl{Q}{\\mathcal{A}}$ with respect to $M$, \ncontradicting the minimality of $M$.\n\nThus, if $M$ is an answer set of $\\gl{Q}{\\mathcal{A}}$, it must contain $g$. \nConsequently, it does not contain $f$ and so no rules of type (4)-(7) \ncontribute to it. It follows that $M=A\\cup \\{a'\\,|\\; a\\in \nX\\setminus A\\} \\cup\\{g\\}$ and, as it indeed is an answer set of \n$\\gl{Q}{\\mathcal{A}}$, $\\mathcal{A}=\\mathcal{ST}(\\gl{Q}{\\mathcal{A}})$. Thus, $\\mathcal{A}$ is a epistemic\nstable model, as claimed.\n\nConversely, let as assume that $Q$ has an epistemic stable model, say,\n$\\mathcal{A}$. It must be that $\\mathcal{A}\\models K\\neg w$ (otherwise, $\\gl{Q}{\\mathcal{A}}$ \ncontains a contradiction and has no stable models). Let us define \n$A=\\{x\\in X\\,|\\; \\mathcal{A}\\models Kx\\}$ and $B=\\{x\\in X\\,|\\; \\mathcal{A}\\models Kx'\\}$. \nIt follows that $\\gl{Q}{\\mathcal{A}}$ consists of the clauses: \n\\begin{enumerate}\n\\item $x$, for $x\\in A$ and $x'$, for $x\\in B$\n\\item $x\\land x'\\rightarrow $; and  $\\neg x\\land \\neg x'\\rightarrow$, for every $x\\in X$\n\\item $\\neg g\\rightarrow f$; and $\\neg f\\rightarrow g$\n\\item $f \\rightarrow y \\vee y'$; and $f\\rightarrow z \\vee z'$, for every $y\\in Y$\nand $z\\in Z$\n\\item $f\\land w \\rightarrow z$; and $f\\land w \\rightarrow z'$, for every $z\\in Z$\n\\item $f\\land \\sigma(u_1)\\land\\sigma(u_2)\\land\\sigma(u_3)\\rightarrow w$, for \nevery clause $C= u_1\\vee u_2 \\vee u_3$ of $\\Phi$, where $\\sigma(a)=a'$\nand $\\sigma(\\neg a)= a$, for every $a\\in X\\cup Y \\cup Z$.\n\\item $f, \\neg w \\rightarrow w$\n\\end{enumerate}\nWe have that $\\mathcal{A}$ is precisely the set of stable models of this \nprogram. Since $\\mathcal{A}\\not=\\emptyset$, $B=X\\setminus A$. If $M$ is \na stable model of $\\gl{Q}{\\mathcal{A}}$ and contains $f$, then it contains \n$w$. But then, as $M\\in\\mathcal{A}$, $\\mathcal{A}\\not\\models K\\neg w$, a \ncontradiction. It follows that there is no stable model containing\n$f$. That is, the program consisting of the following rules has no\nstable model:\n\\begin{enumerate}\n\\item $x$, for $x\\in A$ and $x'$, for $x\\in X\\setminus A$\n\\item $y \\vee y'$; and $z \\vee z'$, for every $y\\in Y$ and $z\\in Z$\n\\item $w \\rightarrow z$; and $w \\rightarrow z'$, for every $z\\in Z$\n\\item $\\sigma(u_1)\\land\\sigma(u_2)\\land\\sigma(u_3)\\rightarrow w$, for\nevery clause $C= u_1\\vee u_2 \\vee u_3$ of $\\Theta$, where $\\sigma(a)=a'$\nand $\\sigma(\\neg a)= a$, for every $a\\in X\\cup Y \\cup Z$.\n\\item $\\neg w \\rightarrow w$\n\\end{enumerate}\nBut then, the formula $\\forall Y\\exists Z \\Theta_{|X\/A}$ is true and,\nconsequently, the formula $\\exists X \\forall Y\\exists Z \\Theta$ is\ntrue, too.  \\hfill$\\Box$\n\nFor the other two epistemic semantics, Remark 1 implies that the problem\nof the existence of an epistemic model (epistemic supported model) is\nin the class $\\Sigma_2^P$. The $\\Sigma_2^P$-hardness of the problem\ncan be proved by similar techniques as those we used for the case of\nepistemic stable models. Thus, we have the following result.\n\n\\begin{theorem}\n\\label{thm:3}\nThe problem to decide whether an epistemic program $P\\subseteq\\mathcal{L}_K$\nhas an epistemic model (epistemic supported model, respectively) is\n$\\Sigma_2^P$-complete.\n\\end{theorem}\n\n\\section{Modeling with Epistemic Programs}\n\nWe will now present several problems which illustrate the \nadvantages offered by the language of epistemic programs we developed\nin the previous two sections. Whenever we use predicate programs, we \nunderstand that their semantics is that of the corresponding ground \nprograms.\n\nFirst, we consider two graph problems related to the existence of \nHamiltonian cycles. Let $G$ be a directed graph. An edge in $G$ is \n\\emph{critical} if it belongs to every hamiltonian cycle in $G$. \nThe following problems are of interest:\n\\begin{enumerate}\n\\item Given a directed graph $G$, find the set of all critical \nedges of $G$\n\\item Given a directed graph $G$, and integers $p$ and $k$, find a set\n$R$ of no more than $p$ new edges such that $G\\cup R$ has no more than\n$k$ critical edges.\n\\end{enumerate}\n\nLet $HC(vtx,edge)$ be any standard ASP encoding of the Hamiltonian \ncycle problem, in which predicates $vtx$ and $edge$ represent $G$, \nand a predicate $hc$ represents edges of a candidate hamiltonian \ncycle. We assume the rules of $HC(vtx,edge)$ are written from left\nto right so that they can be regarded as elements of $\\mathcal{L}$. Then, \nsimply adding to $HC(vtx,edge)$ the rule:\n\\begin{quote}\n$K hc(X,Y) \\rightarrow critical(X,Y)$\n\\end{quote}\nyields a correct representation of the first problem. We write\n$HC_{cr}(vtx,edge)$ to denote this program.\nAlso, for a directed graph $G=(V,E)$,\nwe define \n\\begin{quote}\n$D=\\{vtx(v)\\,|\\; v\\in V\\} \\cup \\{edge(v,w)\\,|\\; (v,w)\\in E\\}$.\n\\end{quote}\nWe have the following result.\n\n\\begin{theorem}\nLet $G=(V,E)$ be a directed graph. If $HC_{cr}(vtx,edge) \\cup D$ has \nno epistemic stable models, then every edge in $G$ is critical \n(trivially). Otherwise, the epistemic program\n$HC_{cr}(vtx,edge)\\cup D$ has a unique \nepistemic stable model $\\mathcal{A}$ and the set $\\{(v,w)\\,|\\; \\mathcal{A}\\models \ncritical(u,v) \\}$ is the set of critical edges in $G$.\n\\end{theorem}\nProof (Sketch): Let $H$ be the grounding of $HC_{cr}(vtx,edge) \\cup D$.\nIf $H$ has no epistemic stable models, it follows that the ``non-epistemic''\npart $H'$ of $H$ has no stable models (as no atom of the form \n$critical(x,y)$ appears in it). As $H'$ encodes the existence of a hamiltonian\ncycle in $G$, it follows that $G$ has no Hamiltonian cycles. Thus, trivially,\nevery edge of $G$ belongs to every Hamiltonian cycle of $G$ and so, every\nedge of $G$ is critical.\n\nThus, let us assume that $\\mathcal{A}$ is an epistemic stable model of $H$. \nAlso, let $S$ be the set of all stable models of $H'$ (they correspond \nto Hamiltonian cycles of $G$; each model contains, in particular, atoms \nof the form $hc(x,y)$, where $(x,y)$ ranges over the edges of the \ncorresponding Hamiltonian cycle). The reduct $\\gl{H}{\\mathcal{A}}$ consists of \n$H'$ (non-epistemic part of $H$ is unaffected by the reduct operation) \nand of $C'$, a set of some facts of the form $critical(x,y)$. Thus, the \nstable models of the reduct are of the form $M\\cup C'$, where $M\\in S$.\nThat is, $\\mathcal{A}=\\{M\\cup C'\\,|\\; M\\in S\\}$. Let us denote by $C$ the set\nof the atoms $critical(x,y)$, where $(x,y)$ belongs to every hamiltonian \ncycle of $G$ (is critical). One can compute now that $\\gl{H}{\\mathcal{A}} = \nH'\\cup C$. Since $\\mathcal{A}=\\mathcal{ST}(\\gl{H}{\\mathcal{A}})$, $\\mathcal{A}= \\{M\\cup C\\,|\\; M\\in S\\}$. \nThus, $HC_{cr}(vtx,edge)\\cup D$ has a unique epistemic stable model, as \nclaimed. It also follows that the set $\\{(v,w)\\,|\\; \\mathcal{A}\\models \ncritical(u,v) \\}$ is the set of critical edges in $G$.\n\\hfill$\\Box$\n\nTo represent the second problem, we proceed as follows. First, we\n``select'' new edges to be added to the graph and impose constraints\nthat guarantee that all new edges are indeed new, and that no more than $p$\nnew edges are selected (we use here \\emph{lparse} syntax for brevity; the\nconstraint can be encoded strictly in the language $\\mathcal{L}_K$).\n\\begin{quote}\n$vtx(X)\\land vtx(Y) \\rightarrow newEdge(X,Y)$\\\\\n$newEdge(X,Y)\\land edge(X,Y)\\rightarrow\\bot$\\\\\n$(p+1) \\{newEdge(X,Y): vtx(X), vtx(Y)\\} \\rightarrow \\bot$.\n\\end{quote}\nNext, we define the set of edges of the extended graph, using a predicate\n$edgeEG$:\n\\begin{quote}\n$edge(X,Y)\\rightarrow edgeEG(X,Y)$\\\\\n$newEdge(X,Y)\\rightarrow edgeEG(X,Y)$\n\\end{quote}\nFinally, we define critical edges and impose a constraint on their \nnumber (again, exploiting the \\emph{lparse} syntax for brevity sake):\n\\begin{quote}\n$edgeEG(X,Y)\\land K hc(X,Y) \\rightarrow critical(X,Y)$\\\\\n$(k+1)\\{critical(X,Y): edgeEG(X,Y)\\}\\rightarrow \\bot$.\n\\end{quote}\nWe define $Q$ to consist of all these rules together with all the rules\nof the program $HC(vtx,edgeEG)$. We now have the following theorem. The\nproof is similar to that above and so we omit it.\n \n\\begin{theorem}\nLet $G$ be a directed graph. There is an extension of $G$ with no more\nthan $p$ new edges so that the resulting graph has no more than $k$ \ncritical edges if and only if the program $Q\\cup D$ has an epistemic \nstable model.\n\\end{theorem}\n \nFor another example we consider the unique model problem: given a CNF \nformula $F$, the goal is to decide whether $F$ has a unique minimal \nmodel. The unique model problem was also considered by Faber and Woltran\n\\cite{FaberW09}. We will show two encodings of the problem by means of\nepistemic programs. The first one uses the semantics of epistemic \nmodels and is especially direct. The other one uses the semantics of\nepistemic stable models. \n\nLet $F$ be a propositional theory consisting of constraints $L_1\\land\n\\ldots\\land L_k\\rightarrow \\bot$, where $L_i$'s are literals. Any propositional\ntheory can be rewritten into an equivalent theory of such form. We \ndenote by $F^K$ the formula obtained from $F$ by replacing every atom \n$x$ with the modal atom $Kx$. \n\n\\begin{theorem}\nFor every theory $F\\subseteq\\mathcal{L}$ consisting of constraints, $F$ has a \nleast model if and only if the epistemic program $F\\cup F^K$ has an\nepistemic model.\n\\end{theorem}\nProof: Let us assume that $F$ has a least model. We define $\\mathcal{A}$ to\nconsist of all models of $F$, and we denote the least model of $F$ by \n$M$. We will show that $\\mathcal{A}$ is an epistemic model of $F\\cup F^K$. \nClearly, for every $x\\in M$, $\\mathcal{A}\\models Kx$. Similarly, for every \n$x\\not\\in M$, $\\mathcal{A}\\models \\neg Kx$. Thus, $\\gl{[F^K]}{\\mathcal{A}}=\\emptyset$.\nConsequently, $\\gl{[F\\cup F^K]}{\\mathcal{A}} = F$ and so, $\\mathcal{A}$ is precisely \nthe set of all models of $\\gl{[F\\cup F^K]}{\\mathcal{A}}$. Thus, $\\mathcal{A}$ is an \nepistemic model.  \n\nConversely, let $\\mathcal{A}$ be an epistemic model of $F\\cup F^K$. It follows\nthat $\\gl{[F^K]}{\\mathcal{A}}=\\emptyset$ (otherwise, $\\gl{[F\\cup F^K]}{\\mathcal{A}}$ \ncontains $\\bot$ and $\\mathcal{A}$ would have to be empty, contradicting the definition\nof an epistemic model). Thus, $\\gl{[F\\cup F^K]}{\\mathcal{A}}=F$ and consequently, \n$\\mathcal{A}$ is the set of all\nmodels of $F$. Let $M=\\{x\\in\\mathit{At}\\,|\\; \\mathcal{A}\\models Kx\\}$ and let\n\\begin{equation}\n\\label{eq17}\na_1\\land\\ldots\\land a_m \\land\\neg b_1\\land\\ldots\\land\\neg b_n \\rightarrow \\bot\n\\end{equation}\nbe a rule in $F$. Then, \n\\[\nK a_1\\land\\ldots\\land K a_m \\land\\neg K b_1\\land\\ldots\\land\\neg K b_n\n\\rightarrow \\bot\n\\]\nis a rule in $F^K$. As $\\gl{[F^K]}{\\mathcal{A}}=\\emptyset$,\n\\[\n\\mathcal{A}\\not\\models K a_1\\land\\ldots\\land K a_m \\land\\neg K b_1\\land\\ldots\\land\n\\neg K b_n.\n\\]\nThus, for some $i$, $1\\leq i\\leq m$, $\\mathcal{A}\\not\\models K a_i$, or for \nsome $j$, $1\\leq j\\leq n$, $\\mathcal{A}\\models K b_j$. In the first case, $a_i\n\\notin M$, in the latter, $b_j\\in M$. In either case, $M$ is a model\nof rule (\\ref{eq17}). It follows that $M$ is a model of $F$. Let $M'$\nbe a model of $F$. Then $M'\\in\\mathcal{A}$ and, by the definition of $M$, $M\n\\subseteq M'$. Thus, $M$ is a least model of $F$.\n\\hfill$\\Box$\n \nNext, we will encode the same problem as an epistemic program under the\nepistemic stable model semantics. The idea is quite similar. We\nonly need to add rules to generate all candidate models.\n\n\\begin{theorem}\nFor every theory $F\\subseteq\\mathcal{L}$ consisting of constraints, $F$ has a\nleast model if and only if the epistemic program \n\\[\nF\\cup F^K \\cup \\{\\neg x\\rightarrow x'\\,|\\; x\\in \\mathit{At}\\}\\cup \\{\\neg x'\\rightarrow x\\,|\\; \nx\\in \\mathit{At}\\}\n\\]\nhas an epistemic stable model.\n\\end{theorem}\n\nWe note that an even simpler encoding can be obtained if we use \n\\emph{lparse} choice rules. In this case, we can replace \n$\\{\\neg x\\rightarrow x'\\,|\\; x\\in \\mathit{At}\\}\\cup \\{\\neg x'\\rightarrow x\\,|\\; x\\in \\mathit{At}\\}$ with\n$\\{\\{x\\}\\,|\\; x\\in\\mathit{At}\\}$.\n\n\\section{Connection to Gelfond's Epistemic Programs}\n\nWe will now return to the original formalism of epistemic specifications\nproposed by Gelfond \\cite{Gelfond91} (under the restriction to epistemic\nprograms we discussed here). We will show that it can be expressed in a\nrather direct way in terms of our epistemic programs in the two-valued\nsetting and under the epistemic supported-model semantics.\n \nThe reduction we are about to describe is similar to the well-known\none used to eliminate the ``strong'' negation from disjunctive logic\nprograms with strong negation. In particular, it requires an extension \nto the language $\\mathcal{L}$. Specifically, for every atom $x\\in \\mathit{At}$ we \nintroduce a fresh atom $x'$ and we denote the extended language by \n$\\mathcal{L}'$. The intended role of $x'$ is to represent in $\\mathcal{L}'$ the literal \n$\\neg x$ from $\\mathcal{L}$. Building on this idea, we assign to each set $W$ of\nliterals in $\\mathcal{L}$ the set\n\\[\nW'=(W\\cap \\mathit{At}) \\cup \\{x'\\,|\\; \\neg x\\in W\\}.\n\\]\nIn this way, sets of literals from $\\mathcal{L}$ (in particular, three-valued\ninterpretations of $\\mathcal{L}$) are represented as sets of atoms from $\\mathcal{L}'$\n(two-valued interpretations of $\\mathcal{L}'$).\n\nWe now note that the truth and falsity of a formula form $\\mathcal{L}$ under\na three-valued interpretation can be expressed as the truth and\nfalsity of certain formulas from $\\mathcal{L}'$ in the two-valued setting.\nThe following result is well known.\n\n\\begin{proposition}\n\\label{prop:2}\nFor every formula $\\varphi\\in\\mathcal{L}$ there are formulas\n$\\varphi^-, \\varphi^+\\in\\mathcal{L}'$ such that for every set of literals $W$\n(in $\\mathcal{L}$)\n\\begin{enumerate}\n\\item $v_W(\\varphi)=\\mathbf{t}$ if and only if $u_{W'}(\\varphi^+)=\\mathbf{t}$\n\\item $v_W(\\varphi)=\\mathbf{f}$ if and only if $u_{W'}(\\varphi^-)=\\mathbf{f}$\n\\end{enumerate}\nMoreover, the formulas $\\varphi^-$ and $\\varphi^+$ can be constructed in \npolynomial time with respect to the size of $\\varphi$.\n\\end{proposition}\nProof: This a folklore result. We provide a sketch of a proof for the\ncompleteness sake. We define $\\varphi^+$ and $\\varphi^-$ by recursively as\nfollows:\n\\begin{enumerate}\n\\item $x^+ = x$ and $x^- = \\neg x'$, if $x\\in\\mathit{At}$\n\\item $(\\neg \\varphi)^+ = \\neg\\varphi^-$ and $(\\neg \\varphi)^- = \\neg\\varphi^+$\n\\item $(\\varphi\\lor \\psi)^+=\\varphi^+\\lor \\psi^+$ and \n$(\\varphi\\lor \\psi)^-=\\varphi^-\\lor \\psi^-$; the case of the conjunction is dealt\nwith analogously\n\\item $(\\varphi\\rightarrow\\psi)^+ = \\varphi^-\\rightarrow\\psi^+$ and \n$(\\varphi\\rightarrow\\psi)^- = \\varphi^+\\rightarrow\\psi^-$.\n\\end{enumerate}\nOne can check that formulas $\\varphi^+$ and $\\varphi^-$ defined in this way satisfy\nthe assertion. \\hfill$\\Box$\n\n\\smallskip\nWe will now define the transformation $\\sigma$ that allows us to eliminate\nstrong negation. First, for a literal $L\\in \\mathcal{L}$, we now define\n\\[\n\\sigma(L) = \\left\\{ \\begin{array}{ll}\n                    x     & \\mbox{if $L=x$}\\\\\n                    x'      & \\mbox{if $L=\\neg x$}\n                    \\end{array}\n            \\right.\n\\]\nFurthermore, if $E$ is a simple modal literal or its default negation, \nwe define\n\\[\n\\sigma(E) = \\left\\{ \\begin{array}{ll}\n                    K\\varphi^+     & \\mbox{if $E=K\\varphi$}\\\\\n                    \\neg K\\varphi^{-}      & \\mbox{if $E=\\neg K\\varphi$}\\\\\n                    \\neg K\\varphi^{+}      & \\mbox{if $E=\\mathit{not\\;} K\\varphi$}\\\\\n                    K\\varphi^{-}      & \\mbox{if $E=\\mathit{not\\;} \\neg K\\varphi$}\n                    \\end{array}\n            \\right.\n\\]\nand for an epistemic premise $E = E_1,\\ldots, E_t$\n(where each $E_i$ is a simple modal literal or its default negation) we set\n\\[\n\\sigma(E) = \\sigma(E_1)\\land\\ldots\\land \\sigma(E_t).\n\\]\nNext, if $r$ is an epistemic rule\n\\[\nL_1 \\vee \\ldots\\vee L_k \\leftarrow F_1,\\ldots, F_m,\\mathit{not\\;} F_{m+1},\\ldots, \\mathit{not\\;} F_n, E\n\\]\nwe define\n\\[\n\\sigma(r) = \\sigma(E)\\land \\sigma(F_1)\\land\\ldots\\land \\sigma(F_m)\\land \\neg\n\\sigma(F_{m+1})\\land\\ldots\\land \\neg \\sigma(F_n) \\rightarrow \n\\sigma(L_1) \\vee \\ldots\\vee \\sigma(L_k).\n\\]\nFinally, for an epistemic program $P$, we set \n\\[\n\\sigma(P)=\\{\\sigma(r)\\,|\\; r\\in P\\}) \\cup\\{x\\land x' \\rightarrow \\bot\\}.\n\\]\nWe note that $\\sigma(P)$ is indeed an epistemic program\nin the language $\\mathcal{L}_K$ (according to our definition of epistemic \nprograms). The role of the rules $x\\land x' \\rightarrow \\bot$\nis to ensure that sets forming epistemic (stable, supported) models \nof $\\sigma(P)$ correspond to consistent sets of literals (the only type \nof set of literals allowed in world views).\n\nGiven a three-valued possible structure $\\mathcal{A}$, we define $\\mathcal{A}'=\\{W'\\,|\\;\nW\\in \\mathcal{A}\\}$, and we regard $\\mathcal{A}'$ as a two-valued possible-world\nstructure. We now have the following theorem.\n\n\\begin{theorem}\nLet $P$ be an epistemic program according to Gelfond. Then a three-valued\npossible-world structure $\\mathcal{A}$ is a world view of $P$ if and only if \na two-valued possible-world structure $\\mathcal{A}'$ is an epistemic supported\nmodel of $\\sigma(P)$.\n\\end{theorem}\nProof (Sketch): Let $P$ be an epistemic program according to Gelfond, $\\mathcal{A}$\na possible-world structure and $W$ a set of literals. We first \nobserve that the G-reduct $P^{\\langle \\mathcal{A},W\\rangle}$ can be described as the\nresult of a certain two-step process. Namely, we define the \n\\emph{epistemic reduct} of $P$ with respect to $\\mathcal{A}$ to be the disjunctive\nlogic program $P^\\mathcal{A}$ obtained from $P$ by removing every rule whose\nepistemic premise $E$ satisfies $\\mathcal{A}\\not\\models E$, \nand by removing the epistemic\npremise from every other rule in $P$. This construction is the three-valued\ncounterpart to the one we employ in our approach.\nIt is clear that the epistemic reduct of $P$ with respect to $\\mathcal{A}$, with \nsome abuse of notation we will denote it by $\\gl{P}{\\mathcal{A}}$, is a disjunctive\nlogic program with strong negation. \n\nLet $Q$ be a disjunctive program with strong negation and $W$ a set \nof literals. By the \n\\emph{supp-reduct} of $Q$ with respect to $W$, $R^{sp}(Q,W)$, we mean \nthe set of the heads of all rules whose bodies are satisfied by $W$ \n(which in the three-valued setting means that every literal in the body \nnot in the scope of $\\mathit{not\\;}$ is in $W$, and every literal in the body in \nthe scope of $\\mathit{not\\;}$ is not in $W$). A consistent set $W$ of literals is \na supported answer set of $Q$ if $W\\in\\mathit{Min}(R^{sp}(Q,W))$ (this is a \nnatural extension of the definition of a supported model \\cite{abw87,bg93}\nto the case of disjunctive logic programs with strong negation; again,\nwe do not regard inconsistent sets of literals as supported answer sets).\n\nClearly, $P^{\\langle \\mathcal{A},W\\rangle} = R^{sp}(\\gl{P}{\\mathcal{A}},W)$. Thus, $\\mathcal{A}$ is a\nworld view of $P$ according to the definition by Gelfond if and only if\n$\\mathcal{A}$ is a collection of all supported answer sets of $\\gl{P}{\\mathcal{A}}$.  \n\nWe also note that by Proposition \\ref{prop:2}, if $E$ is an epistemic\npremise, then $\\mathcal{A}\\models E$ if and only if $\\mathcal{A}'\\models \\sigma(E)$.\nIt follows that $\\sigma(\\gl{P}{\\mathcal{A}}) = \\gl{\\sigma(P)}{\\mathcal{A}'}$. In other \nwords, constructing\nthe epistemic reduct of $P$ with respect to $\\mathcal{A}$ and then translating  \nthe resulting disjunctive logic program with strong negation into the\ncorresponding disjunctive logic program without strong negation yields \nthe same result as first translating the epistemic program (in the\nGelfond's system) into our language of epistemic programs and then computing\nthe reduct with respect to $\\mathcal{A}'$. We note that there is a one-to-one\ncorrespondence between supported answer sets of $\\gl{P}{\\mathcal{A}}$ and supported\nmodels of $\\sigma(\\gl{P}{\\mathcal{A}})$ ($\\sigma$, when restricted to programs\nconsisting of rules without epistemic premises, is the standard transformation\neliminating strong negation and preserving the stable and supported \nsemantics). Consequently, there is a one-to-one\ncorrespondence between supported answer sets of $\\gl{P}{\\mathcal{A}}$ and supported\nmodels of $ \\gl{\\sigma(P)}{\\mathcal{A}'}$ (cf. our observation above). Thus, \n$\\mathcal{A}$ consists of supported answer \nsets of $\\gl{P}{\\mathcal{A}}$ if and only if $\\mathcal{A}'$ consists of supported models\nof $\\gl{\\sigma(P)}{\\mathcal{A}'}$. Consequently, $\\mathcal{A}$ is a world view of $P$\nif and only if $\\mathcal{A}'$ is an epistemic supported model of $\\sigma(P)$.\n\\hfill$\\Box$ \n\n\\section{Epistemic Models of Arbitrary Theories}\n\nSo far, we defined the notions of epistemic models, epistemic stable\nmodels and epistemic supported models only for the case of epistemic \nprograms. However, this restriction is not essential. We recall that \nthe definition of these three epistemic semantics consists of two \nsteps. The first step produces the reduct of an epistemic program $P$\nwith respect to a possible-world structure, say $\\mathcal{A}$. This reduct \nhappens to be (modulo a trivial syntactic transformation) a standard \ndisjunctive logic program in the language $\\mathcal{L}$ (no modal atoms \nanymore). If the set of models (respectively, stable models, supported \nmodels) of the reduct program coincides with $\\mathcal{A}$, $\\mathcal{A}$ is an \nepistemic model (respectively, epistemic stable or supported model) \nof $P$. However, the concepts of a model, stable model and supported \nmodel are defined for \\emph{arbitrary} theories in $\\mathcal{L}$. This is \nobviously well known for the semantics of models. The stable-model \nsemantics was extended to the full language $\\mathcal{L}$ by Ferraris \n\\cite{fer05} and the supported-model semantics by Truszczynski\n\\cite{tr10}. Thus, there is no reason precluding the extension of the\ndefinition of the corresponding epistemic types of models to the general\ncase. We start be generalizing the concept of the reduct.\n\n\\begin{definition}\nLet $T$ be an arbitrary theory in $\\mathcal{L}_K$ and let $\\mathcal{A}$ be a\npossible-world structure. The \\emph{epistemic reduct} of $T$ with\nrespect to $\\mathcal{A}$, $\\gl{T}{\\mathcal{A}}$ in symbols, is the theory obtained\nfrom $T$ by replacing each maximal modal atom $K\\varphi$ with $\\top$,\nif $\\mathcal{A}\\models K\\varphi$, and with $\\bot$, otherwise.\n\\end{definition}\n\nWe note that if $T$ is an epistemic program, this notion of the reduct\ndoes not coincide with the one we discussed before. Indeed, now no rule \nis dropped and no modal literals are dropped; rather modal atoms are \nreplaced with $\\top$ and $\\bot$. However, the replacements are executed\nin such a way as to ensure the same behavior. Specifically, one can show\nthat models, stable models and supported models of the two reducts \ncoincide.\n\nNext, we generalize the concepts of the three types of epistemic models.\n\n\\begin{definition}\n\\label{def14}\nLet $T$ be an arbitrary theory in $\\mathcal{L}_K$. A possible-world\nstructure $\\mathcal{A}$ is an \\emph{epistemic model} (respectively, an\n\\emph{epistemic stable model}, or an \\emph{epistemic supported model})\nof $P$, if $\\mathcal{A}$ is the set of models (respectively, stable models or\nsupported models) of $\\mathcal{M}(\\gl{P}{\\mathcal{A}})$. \n\\end{definition}\n\nFrom the comments we made above, it follows that if $T$ is an epistemic \nprogram, this more general definition yields the came notions of epistemic\nmodels of the three types as the earlier one. \n\nWe note that even in the more general setting the complexity of reasoning\nwith epistemic (stable, supported) models remains unchanged. Specifically,\nwe have the following result.\n\n\\begin{theorem}\n\\label{thm:4}\nThe problem to decide whether an epistemic theory $T\\subseteq\\mathcal{L}_K$\nhas an epistemic stable model is $\\Sigma_3^P$-complete. The problem to \ndecide whether an epistemic theory $T\\subseteq\\mathcal{L}_K$ has an epistemic \nmodel (epistemic supported model, respectively) is $\\Sigma_2^P$-complete.\n\\end{theorem}\nProof(Sketch): The hardness part follows from our earlier results concerning\nepistemic programs. To prove membership, we modify Proposition\n\\ref{prop:char}, and show a polynomial time algorithm with a \n$\\Sigma_2^P$ oracle (NP oracle for the last two problems) that\ndecides, given a propositional theory $S$ and a modal formula $K\\varphi$\n(with $\\varphi\\in\\mathcal{L}_K$ and not necessarily in $\\mathcal{L}$) whether $\\mathcal{ST}(S)\n\\models K\\varphi$ (respectively,  $\\mathcal{M}(S)\\models K\\varphi$, or  $\\mathcal{SP}(S)\n\\models K\\varphi$). \\hfill$\\Box$\n\n\\section{Discussion}\n\nIn this paper, we proposed a two-valued formalism of epistemic \ntheories --- subsets of the language of modal propositional logic.\nWe proposed a uniform way, in which semantics of propositional \ntheories (the classical one as well as nonmonotonic ones: stable \nand supported) can be extended to the case of epistemic theories.\nWe showed that the semantics of epistemic supported models is closely\nrelated to the original semantics of epistemic specifications proposed\nby Gelfond. Specifically we showed that the original formalism of Gelfond\ncan be expressed in a straightforward way by means of epistemic programs\nin our sense under the semantics of epistemic supported models. Essentially\nall that is needed is to use fresh symbols $x'$ to represent strong \nnegation $\\neg x$, and use the negation operator of our formalism, \n$\\varphi \\rightarrow \\bot$ or, in the shorthand, $\\neg \\varphi$, to model the default \nnegation $\\mathit{not\\;} \\varphi$.   \n\nWe considered in more detail the three semantics mentioned above. \nHowever, other semantics may also yield interesting epistemic\ncounterparts. In particular, it is clear that Definition \\ref{def14}\ncan be used also with the minimal model semantics or with the \nFaber-Leone-Pfeifer semantics \\cite{flp04}. Each semantics gives rise\nto an interesting epistemic formalism that warrants further studies.\n\nIn logic programming, eliminating strong negation does not result in \nany loss of the expressive power but, at least for the semantics of\nstable models, disjunctions cannot be compiled away in any concise way \n(unless the polynomial hierarchy collapses). In the setting of \nepistemic programs, the situation is similar. The strong negation can be \ncompiled away. But the availability of disjunctions in the heads and \nthe availability of epistemic premises in the bodies of rules are \nessential. Each of these factors separately brings the complexity one \nlevel up. Moreover, when used together under the semantics of epistemic \nstable models they bring the complexity two levels up. This points to \nthe intrinsic importance of having in a knowledge representation \nlanguage means to represent indefiniteness in terms of disjunctions, \nand what is known to a program (theory) --- in terms of a modal operator \n$K$.\n\n\n\\section*{Acknowledgments}\n\\vspace*{-0.1in}\nThis work was partially supported by the NSF grant IIS-0913459.\n\n{\\small\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzpcjy b/data_all_eng_slimpj/shuffled/split2/finalzzpcjy
new file mode 100644
index 0000000000000000000000000000000000000000..372b6bbdb3c61c9e4392980f3ad4c1fb7c62b0ee
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzpcjy
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction and main results}\n\nUnderstanding the effect of disorder on phase transitions is a key topic in \nstatistical physics. In a celebrated paper, Harris~\\cite{cf:Harris} proposed \na criterion that predicts whether or not the addition of an arbitrarily small \namount of quenched disorder is able to modify the critical behavior of a \nsystem close to a phase transition. The rigorous justification of this criterion \nfor a class of \\emph{pinning models} has been an active direction of research \nin the mathematical literature (see Giacomin~\\cite{cf:Gia2} for an overview). \nOne of the key tools in this program is the \\emph{smoothing inequality} of \nGiacomin and Toninelli~\\cite{cf:GT1}, \\cite{cf:GT2}. It is the purpose of this \nnote to generalize and sharpen this inequality.\n\nSection~\\ref{sec:motivation} provides motivation, Section~\\ref{sec:assumptions}\nstates the necessary model assumptions, Section~\\ref{sec:free energy} defines\nthe free energy, Section~\\ref{sec:theorems} states our main theorems, while \nSection~\\ref{sec:discussion} discusses the context of these theorems. Proofs\nare given in Sections~\\ref{sec:protilt}--\\ref{sec:smoothshift}. \n\n\n\\subsection{Motivation}\n\\label{sec:motivation}\n\nWe begin by describing a class of models that motivates our main results\nin Section~\\ref{sec:theorems}. We use the\nnotation $\\mathbb{N} := \\{1,2,\\ldots\\}$ and $\\mathbb{N}_0 := \\mathbb{N} \\cup \\{0\\}$.\n\nConsider a recurrent Markov chain $S := (S_n)_{n\\in\\mathbb{N}_0}$ on a countable set \n$\\mathsf{E}$, starting at a distinguished point denoted by $0$, defined \non a probability space $(\\Omega,{\\ensuremath{\\mathcal F}} ,\\p)$, and let $\\tau_1 := \\inf\\{n \\in \\mathbb{N}\n\\colon\\, \\ S_n = 0\\}$ be its first return time to $0$. The key assumption is \nthat for some $\\alpha \\in [0,\\infty)$, \n\\begin{equation} \n\\label{eq:astau}\n\\p(\\tau_1 > n) = n^{-\\alpha + o(1)}, \\qquad n \\to \\infty.\n\\end{equation}\nThe case of a transient Markov chain, i.e., $\\p(\\tau_1 = \\infty) > 0$, can be \nincluded as well, and requires that \\eqref{eq:astau} holds conditionally on \n$\\{\\tau_1<\\infty\\}$.\n\nGiven an $\\mathbb{R}$-valued sequence $\\omega := (\\omega_n)_{n\\in\\mathbb{N}}$ (the \n\\emph{disorder} sequence), a function $\\phi\\colon\\,\\mathsf{E} \\to \\mathbb{R}$ \n(the \\emph{potential}), and parameters $N \\in\\mathbb{N}$, $\\beta \\geq 0$, \n$h\\in\\mathbb{R}$ (the \\emph{system size}, the \\emph{disorder strength} and \nthe \\emph{disorder shift}), we define the \\emph{partition function}\n\\begin{equation} \n\\label{eq:model}\nZ_{N, \\beta, h}^{\\omega,\\phi} \\:=\n\\e\\Big[ e^{\\sum_{n=1}^N (h + \\beta\\omega_n) \\phi(S_n)} \\, \n{\\sf 1}_{\\{S_N = 0\\}} \\Big] \\,\\in\\, [0,\\infty],\n\\end{equation}\ni.e., at each time $n$ the Markov chain gets an exponential reward or penalty \nproportional to $h + \\beta \\omega_n$, modulated by a factor $\\phi(S_n)$. The \nsequence $\\omega$ is to be thought of as a typical realization of a random \nprocess. Note that\n\\begin{itemize}\n\\item \nthe choice $\\phi(x) := {\\sf 1}_{\\{0\\}}(x)$ corresponds to the \\emph{pinning model} \n(see Giacomin~\\cite{cf:Gia}, \\cite{cf:Gia2}, den Hollander~\\cite{cf:dH});\n\\item \nwhen $\\mathsf{E} = \\mathbb{Z}$ and $S$ is nearest-neighbor with symmetric excursions\nout of $0$, the choice $\\phi(x) := {\\sf 1}_{(-\\infty,0]}(x)$ corresponds to the \n\\emph{copolymer model} (see \\cite{cf:Gia}, \\cite{cf:dH});\n\\footnote{The standard copolymer model is defined through a \\emph{bond} \ninteraction: $\\phi(S_n)$ is replaced by $\\phi(S_{n-1},S_n) := {\\sf 1}_{(-\\infty, 0]}\n(\\frac{1}{2}[S_{n-1} + S_n])$, and $(\\beta,h)$ by $(-2\\lambda,-2\\lambda h)$. \nThis can be still cast in the framework of \\eqref{eq:model} by picking \n$\\mathsf{E} = \\mathbb{Z}^2$, taking the pair process $(S_{n-1},S_n)$ as the Markov \nchain, and $(0,0)$ as $0$.}\n\\end{itemize}\nThus, the modulating potential $\\phi$ allows us to interpolate between different \nclasses of models. When $S$ is simple random walk on $\\mathbb{Z}^d$ and $\\phi(x) \n\\approx |x|^{-\\theta}$ as $|x| \\to\\infty$ for some $\\theta \\in (0,\\infty)$, the model \ndisplays interesting features that are currently under investigation (Caravenna\nand den Hollander~\\cite{cf:CardH}).\n\n\n\n\\subsection{Assumptions}\n\\label{sec:assumptions}\n\nAlthough our main focus will be on the model in \\eqref{eq:model}, we list the \nassumptions that we actually need. We start with the disorder.\n\n\\begin{assumption}[The disorder]\n\\label{ass:disorder}\nThe disorder $\\omega = (\\omega_n)_{n\\in\\mathbb{N}}$ is an i.i.d.\\ sequence of $\\mathbb{R}$-valued \nrandom variables, defined on a probability space $(\\Omega', {\\ensuremath{\\mathcal F}} ', {\\ensuremath{\\mathbb P}} )$, such that\n\\begin{gather}\n\\label{eq:t0}\n\\exists\\,\\, t_0 \\in (0, \\infty]\\colon\\,\n\\quad \\mathrm{M}(t)  := {\\ensuremath{\\mathbb E}} \\big[ e^{t\\omega_1} \\big] < \\infty \\quad \\forall\\,\\, |t| < t_0, \\\\ \\nonumber\n{\\ensuremath{\\mathbb E}} [\\omega_n] = 0, \\quad \\bbvar(\\omega_n) = 1.\n\\end{gather}\n\\end{assumption}\n\n\\noindent\nThe crucial assumption is that the disorder distribution has locally finite exponential \nmoments. The choice of zero mean and unit variance is a convenient normalization \nonly (since we can play with the parameters $\\beta$ and $h$).\n\nFor $\\delta \\in (-t_0, t_0)$, we denote by ${\\ensuremath{\\mathbb P}} _\\delta$ the \\emph{tilted law} under \nwhich $\\omega = (\\omega_n)_{n\\in\\mathbb{N}}$ is i.i.d.\\ with marginal distribution \n\\begin{equation}\n\\label{eq:RNn}\n{\\ensuremath{\\mathbb P}} _\\delta(\\omega_1 \\in \\mathrm{d} x) := e^{\\delta x - \\log\\mathrm{M}(\\delta)}\\,{\\ensuremath{\\mathbb P}} (\\omega_1 \\in \\mathrm{d} x).\n\\end{equation} \n\nNext we state our assumptions on the partition function $Z_{N,\\omega,\\beta,h}$ we will be\nable to handle, defined for $N \\in \\mathbb{N}$, $\\beta \\geq 0$, $h\\in\\mathbb{R}$ and ${\\ensuremath{\\mathbb P}} $-a.e. $\\omega \\in \n\\mathbb{R}^\\mathbb{N}$ (keeping in mind \\eqref{eq:model} as a special case).\n\n\\begin{assumption}[The partition function {[I]}] \n\\label{ass:model}\n$Z_{N,\\omega,\\beta,h}$ is a\nmeasurable function defined on $\\mathbb{N} \\times \\mathbb{R}^\\mathbb{N} \\times [0,\\infty) \\times \\mathbb{R}$,\ntaking values in $[0,\\infty)$ and satisfying the following conditions:\n\\begin{enumerate}\n\\item\n\\label{it:basic} \n$Z_{N,\\omega,\\beta,h}$ is a function of $N$ and of $(h + \\beta \\omega_n)_{1 \\leq n \\leq N}$.\n\\item\n\\label{it:superadd} \n$Z_{N+M, \\omega, \\beta, h} \\geq Z_{N, \\omega, \\beta, h} \\, Z_{M, \\theta^N\\omega, \\beta, h}$\nfor all $N,M \\in \\mathbb{N}$, where $\\theta$ is the left-shift acting on $\\omega$, i.e., $(\\theta^N \\omega)_n \n:= \\omega_{N+n}$ for $N\\in\\mathbb{N}$.\n\\item\n\\label{it:polylb} \nThere exists a $\\gamma \\in (0,\\infty)$ such that, for $N$ in a subsequence of $\\mathbb{N}$,\n\\begin{equation}\n\\label{eq:polylb}\nZ_{N,\\omega,\\beta,h} \\geq \\frac{c_{\\beta,h}(\\omega)}{N^\\gamma} \n\\quad \\text{with} \\quad\n{\\ensuremath{\\mathbb E}} _\\delta[\\log c_{\\beta,h}(\\omega)] > -\\infty \\quad \\forall\\,\\,\\delta \\in (-t_0, t_0).\n\\end{equation}\n\\end{enumerate}\n\\end{assumption}\n\n\\begin{remark}\\rm \\label{rem:pincop}\nNote that properties \\eqref{it:basic} and \\eqref{it:superadd} are satisfied for the model \nin \\eqref{eq:model}. For property \\eqref{it:polylb} to be satisfied as well, we need \nto make additional assumptions on $\\phi$ and\/or $S$. For instance, for the pinning \nmodel\nproperty \\eqref{it:polylb} holds with $\\gamma = (1+\\alpha)+\\epsilon$, for \nany fixed $\\epsilon > 0$ (and for a suitable choice of $c_{\\beta,h}(\\omega) \n= c^\\epsilon_{\\beta,h}(\\omega)$), which follows from \\eqref{eq:astau} after restricting \nthe expectation in \\eqref{eq:model} to the event $\\{\\tau_1 = N\\}$. Alternatively, when \n$\\mathsf{E} = \\mathbb{Z}^d$, if $\\phi$ vanishes in a half-space and $S$ is symmetric (as for \nthe copolymer model), property \\eqref{eq:polylb} with $\\gamma = (1+\\alpha) +\\epsilon$ \nagain follows from \\eqref{eq:astau}.\n\\end{remark}\n\nAs a matter of fact, properties \\eqref{it:basic} and \\eqref{it:superadd} are rather mild: they \nare satisfied for many $(1+d)$-dimensional directed models (possibly after a minor \nmodification of the partition function that does not change the free energy defined \nbelow). In contrast, property \\eqref{it:polylb} is a more severe restriction. Roughly \nspeaking, it says that the \\emph{disorder can be avoided at a cost that is only \npolynomial in the system size}.\n\n\n\\subsection{Free energy}\n\\label{sec:free energy}\n\nIf Assumptions~\\ref{ass:disorder} and~\\ref{ass:model} are satisfied, then we can \ndefine the \\emph{free energy} \n\\begin{equation} \n\\label{eq:freeen}\n\\textsc{f}(\\beta,h; \\delta) := \\limsup_{N\\to\\infty} \\frac{1}{N} \n{\\ensuremath{\\mathbb E}} _\\delta \\big[ \\log Z_{N, \\omega, \\beta, h} \\big]\n\\end{equation}\nfor $\\beta \\geq 0$, $h \\in \\mathbb{R}$, $\\delta \\in (-t_0, t_0)$ when $\\omega$ is chosen \naccording to ${\\ensuremath{\\mathbb P}} _\\delta$. \n\n\\begin{remark}\\rm\\label{rem:kingman}\n(a) In the general framework of Assumption~\\ref{ass:model}, it may happen that \n$\\textsc{f}(\\beta,h; \\delta) = \\infty$ for some values of the parameters. However, for the \nmodel in \\eqref{eq:model} we have $\\textsc{f}(\\beta,h; \\delta) < \\infty$ as soon as $\\phi$ \nis bounded (see \\eqref{eq:boundham} below).\\\\\n(b) By the super-additivity property \\eqref{it:superadd} in Assumption~\\ref{ass:model}, the $\\limsup$ in \n\\eqref{eq:freeen} may be replaced by $\\sup$, or by $\\lim$ restricted to those values \nof $N$ for which ${\\ensuremath{\\mathbb E}} _\\delta \\big[ \\log Z_{N, \\omega, \\beta, h} \\big] > -\\infty$, which \nby properties \\eqref{it:superadd}--\\eqref{it:polylb} form a sub-lattice $\\textsc{t}\\mathbb{N}$. By \nKingman's super-additive ergodic theorem, we may also remove the expectation \n${\\ensuremath{\\mathbb E}} _\\delta$ in \\eqref{eq:freeen}, because the limit as $N\\to\\infty$, $N \\in \\textsc{t}\\mathbb{N}$, \nexists and is constant ${\\ensuremath{\\mathbb P}} _\\delta$-a.s.\n\\end{remark}\n\nA direct consequence of \\eqref{eq:polylb} is the inequality $\\textsc{f}(\\beta,h; \\delta) \\geq 0$, \nwhich is a crucial feature of the class of models we consider. In many interesting \ncases, like for pinning and copolymer models, the free energy is zero in some \nclosed region of the parameter space and strictly positive in its complement, with\nboth regions non-empty. When this happens, the free energy is not an analytic \nfunction and the model is said to undergo a \\emph{phase transition}. It is then of \nphysical and mathematical interest to study the regularity of the free energy close \nto the \\emph{critical curve} separating the two regions.\n\nMore concretely, consider the case when $h \\mapsto Z_{N,\\omega,\\beta,h}$ is \nmonotone (like for the model in \\eqref{eq:model} when $\\phi$ has a sign), say\nnon-decreasing, so that $h \\mapsto \\textsc{f}(\\beta,h;\\delta)$ is non-decreasing as well. \nThen for every $\\beta \\geq 0$ there exists a critical value $h_c(\\beta) \\in \\mathbb{R} \\cup \n\\{\\pm\\infty\\}$ such that $\\textsc{f}(\\beta,h;0) = 0$ for $h < h_c(\\beta)$ and $\\textsc{f}(\\beta,h;0) \n> 0$ for $h > h_c(\\beta)$ (we consider $\\delta = 0$ for simplicity). If $h \\mapsto \n\\textsc{f}(\\beta,h;0)$ is continuous as well, as is typical, then $\\textsc{f}(\\beta,h_c(\\beta);0) = 0$ \nand it is interesting to understand how the free energy vanishes as $h \\downarrow \nh_c(\\beta)$. For homogeneous pinning models, i.e., when $\\beta = 0$, it is known \nthat\n\\begin{equation} \n\\label{eq:hompin}\n\\textsc{f}(0, h_c(0) + t;0) =  t^{\\max\\{\\frac{1}{\\alpha},1\\} + o(1)}, \\qquad t \\downarrow 0.\n\\end{equation}\n(See \\cite[Theorem~2.1]{cf:Gia} for more precise estimates.) On the other hand,\nas soon as disorder is present, i.e., when $\\beta > 0$, it was shown by Giacomin \nand Toninelli~\\cite{cf:GT1}, \\cite{cf:GT2} that, under some mild restrictions on the \ndisorder distribution,\n\\begin{equation} \n\\label{eq:smoo0}\n\\exists\\,c \\in (0,\\infty)\\colon\\quad 0 \\leq \\textsc{f}(\\beta,h_c(\\beta) + t;0) \n\\leq \\frac{c}{\\beta^2} \\, t^2.\n\\end{equation}\nComparing \\eqref{eq:hompin} and \\eqref{eq:smoo0}, we see that when $\\alpha \n> \\frac{1}{2}$ the addition of disorder has a \\emph{smoothing} effect on the way \nin which the free energy vanishes at the critical line.\n\n\n\\subsection{Main results}\n\\label{sec:theorems}\n\nThe goal of this note is to generalize and sharpen \\eqref{eq:smoo0}, namely, to \nshow that no assumption on the disorder distribution other than \\eqref{eq:t0} is \nrequired, and to provide estimates on the constant $c$ that are optimal in some \nsense (see below). We will stay in the general framework of Assumption~\\ref{ass:model}, \nwith no mention of ``critical lines''. \n\n\n\\subsubsection{$\\bullet$ Tilting}\n\nFirst we prove a smoothing inequality for $\\textsc{f}(\\beta,h;\\delta)$ with respect \nto the tilt parameter $\\delta$ rather than the shift parameter $h$. Although both \ntilting and shifting are natural ways to control the disorder bias, the latter is often \npreferred in the literature because the free energy typically is a convex function of \nthe shift parameter $h$ (like for the model in \\eqref{eq:model}). However, for the \npurpose of the smoothing inequality the tilt parameter $\\delta$ turns out to be more \nnatural.\n\n\\begin{theorem}[Smoothing inequality with respect to a disorder tilt]\n\\label{th:smoothing}\nSubject to Assumptions~{\\rm \\ref{ass:disorder}} and {\\rm \\ref{ass:model}}, if \n$\\textsc{f}(\\bar\\beta, \\bar h; 0) = 0$ for some $\\bar\\beta > 0$ and $\\bar h \\in \\mathbb{R}$, then \nfor all $\\delta \\in (-t_0, t_0)$,\n\\begin{equation} \n\\label{eq:smodelta}\n0 \\leq \\textsc{f}(\\bar\\beta, \\bar h; \\delta) \\leq \\frac{\\gamma}{2} \\, B_\\delta \\, \\delta^2\n\\end{equation}\nwhere the constants $t_0$ and \n$\\gamma$ are defined in \\eqref{eq:t0} and \\eqref{eq:polylb}, while\n\\begin{equation} \nB_\\delta := \\frac{2}{\\delta} \\bigg| (\\log\\mathrm{M})'(\\delta) - \\frac{\\log\\mathrm{M}(\\delta)}{\\delta} \\bigg|\n\\in (0,\\infty)\n\\qquad \\text{satisfies} \\qquad \\lim_{\\delta \\to 0} B_\\delta = 1 .\n\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\\rm\nFor pinning and copolymer models satisfying \\eqref{eq:astau}, we can set $\\gamma = 1+\\alpha$ \nin \\eqref{eq:smodelta}, by Remark~\\ref{rem:pincop}.\n\\end{remark}\n\n\\noindent\nTheorem~\\ref{th:smoothing} is proved in Section~\\ref{sec:protilt} through a direct translation of \nthe argument developed in Giacomin and Toninelli~\\cite{cf:GT2}. The proof is based on the \nconcept of \\emph{rare stretch strategy}, which has been a crucial tool in the study of disordered \npolymer models since the papers by Monthus~\\cite{cf:Monthus}, Bodineau and \nGiacomin~\\cite{cf:BodGia}.\n\n\n\\subsubsection{$\\bullet$ Shifting}\n\nNext we consider the effect of a disorder shift. In the \\emph{Gaussian case}, i.e., when \n${\\ensuremath{\\mathbb P}} (\\omega_1 \\in \\cdot)=N(0,1)$, tilting is the same as shifting: in fact\n${\\ensuremath{\\mathbb P}} _\\delta (\\omega_1 \\in \n\\cdot)= N(\\delta,1)$ and so $\\omega_n$ under ${\\ensuremath{\\mathbb P}} _\\delta$ is distributed like $\\omega_n \n+ \\delta$ under ${\\ensuremath{\\mathbb P}} $. Recalling property \\eqref{it:basic}, \nwe then get\n\\begin{equation} \n\\label{eq:tiltshift}\n\\textsc{f}(\\beta, h; \\delta) = \\textsc{f}(\\beta, h + \\beta \\delta; 0)\n\\end{equation}\nand, since $\\mathrm{M}(\\delta) = e^{\\delta^2 \/ 2}$, it follows from \\eqref{eq:smodelta} that if $\\textsc{f}(\\bar\\beta, \n\\bar h; 0) = 0$ with $\\bar\\beta > 0$, then\n\\begin{equation}\n0 \\leq \\textsc{f}(\\bar\\beta, \\bar h + t; 0) \\leq \\frac{\\gamma}{2 \\bar\\beta^2} \\, t^2 \n\\qquad \\forall\\, t \\in \\mathbb{R}.\n\\end{equation}\nThis is precisely the smoothing inequality with respect to a disorder shift in \\eqref{eq:smoo0}, \nwith an explicit constant \n(see also Giacomin~\\cite[Theorem 5.6 and Remark 5.7]{cf:Gia}).\n\nFor a general disorder distribution tilting is different from shifting. However, we may still hope \nthat \\eqref{eq:tiltshift} holds approximately. This is what was shown in Giacomin and \nToninelli~\\cite{cf:GT1}, under additional restrictions on the disorder distribution and with \nnon-optimal constants. \nThe main result of this note, Theorem~\\ref{th:compdeltah} below, shows that the effects\non the free energy of tilting or shifting the disorder distribution are asymptotically equivalent, \nin large generality and with asymptotically optimal constants in the weak interaction limit. \nSince this result is unrelated to Theorem~\\ref{th:smoothing} and is of independent interest, \nwe formulate it for a very general class of statistical physics models, way beyond disordered \npolymer models.\n\n\\begin{assumption}[The partition function {[II]}] \n\\label{ass:model2}\nThe partition function is defined as\n\\begin{equation} \n\\label{eq:ZNgen}\nZ_{N, \\omega, \\beta, h} := \\e_N \\Big[ e^{\\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n}  \\Big],\n\\end{equation}\nwhere, for fixed $N\\in\\mathbb{N}$, $(\\sigma_i)_{1 \\leq i \\leq N}$ are $\\mathbb{R}$-valued measurable functions,\ndefined on a finite measure space $(\\Omega_N, {\\ensuremath{\\mathcal F}} _N, \\p_N)$, that are uniformly bounded,\nhave a sign, say\n\\begin{equation} \n\\label{eq:s0}\n\\exists\\, s_0 > 0\\colon\\, \\quad \\p_N \\big( \\big\\{ 0 \\leq \\sigma_i \\leq s_0, \\, \n\\forall\\, 1 \\leq i \\leq N \\big\\}^c \\big) = 0 \\qquad \\forall\\, N \\in \\mathbb{N},\n\\end{equation}\nand satisfy $-\\infty < \\limsup_{N\\to\\infty} \\frac{1}{N} \\log \\p_N(\\Omega_N) < \\infty$.\n\\end{assumption}\n\n\\noindent\nWe emphasize that the $\\sigma_i$'s need not be independent, nor exchangeable.\nA more detailed discussion on Assumption~\\ref{ass:model2} is given below.\n\nWe can now state the approximate version of \\eqref{eq:tiltshift}. The free energy \n$\\textsc{f}(\\beta,h;\\delta)$ is again defined by \\eqref{eq:freeen}.\n\n\\begin{theorem}[Asymptotic equivalence of tilting and shifting]\n\\label{th:compdeltah}\nSubject to Assumptions~{\\rm \\ref{ass:disorder}} and {\\rm \\ref{ass:model2}}, and\nwith $\\epsilon_0 := \\min\\{\\frac{t_0}{2},\\frac{t_0}{2s_0}\\}$ (where $s_0, t_0$ are \ndefined in \\eqref{eq:s0} and \\eqref{eq:t0}), for all $\\beta \\in [0, \\epsilon_0)$ and \n$\\delta \\in (-\\epsilon_0, \\epsilon_0)$ there exist $0 < C^-_{\\beta,\\delta} \\leq \nC^+_{\\beta,\\delta} < \\infty$ such that\n\\begin{equation} \n\\label{eq:compdeltah}\n\\begin{split}\n& \\forall\\, \\delta \\in [0, \\epsilon_0)\\colon\\,\n\\quad \\textsc{f} \\big( \\beta, h + C^{-}_{\\beta,\\delta} \\, \\beta \\delta ; 0 \\big) \\leq\n\\textsc{f} ( \\beta, h; \\delta ) \\leq \\textsc{f} \\big( \\beta, h + C^{+}_{\\beta,\\delta} \\, \\beta \\delta ; 0 \\big),\n\\end{split}\n\\end{equation}\nwhile for $\\delta \\in (-\\epsilon_0, 0]$ the same relation holds with $C^{-}_{\\beta,\\delta}$ \nand $C^{+}_{\\beta,\\delta}$ interchanged. Moreover, $(\\beta,\\delta) \\mapsto \nC^\\pm_{\\beta,\\delta}$ is continuous with $C^\\pm_{0,0} = 1$, and hence\n\\begin{equation}\n\\label{eq:propCpm}\n\\lim_{(\\beta,\\delta) \\to (0,0)} C^\\pm_{\\beta,\\delta} = 1.\n\\end{equation}\nFurthermore, $\\delta \\mapsto C^\\pm_{\\beta,\\delta}\\,\\delta$ is strictly increasing.\n\\end{theorem}\n\n\\noindent\nThe proof of Theorem~\\ref{th:compdeltah} is given in Section~\\ref{sec:tiltshift}. The \ngeneral strategy and consists in showing that the derivatives of $\\textsc{f}(\\beta,h;\\delta)$ \nwith respect to $\\delta$ and $h$ are comparable. Compared to Giacomin and\nToninelli~\\cite{cf:GT1}, several estimates need to be sharpened considerably.\n\n\n\\subsubsection{$\\bullet $ Smoothing}\n\nCombining Theorems~\\ref{th:smoothing} and~\\ref{th:compdeltah}, we finally obtain \nour smoothing inequality with respect to a shift, with explicit control on the constant.\n\n\\begin{theorem}[Smoothing inequality with respect to a disorder shift]\n\\label{th:smoshift}\nSubject to Assumptions~{\\rm \\ref{ass:disorder}}, {\\rm \\ref{ass:model}} and \n{\\rm \\ref{ass:model2}}, there is an $\\epsilon'_0 > 0$ with the following property:\nif $\\textsc{f}(\\bar\\beta, \\bar h; 0) = 0$ for some $\\bar\\beta \\in (0,\\epsilon'_0)$ and \n$\\bar h \\in \\mathbb{R}$, then for $t \\in (-\\bar\\beta \\epsilon'_0, \\bar\\beta \\epsilon'_0)$,\n\\begin{equation} \n\\label{eq:compdeltah2}\n0 \\leq \\textsc{f} ( \\bar\\beta, \\bar h + t; 0) \\leq\n\\frac{\\gamma}{2 \\bar\\beta^2} \\, A_{\\bar\\beta,\\frac{t}{\\bar\\beta}} \\, t^2,\n\\end{equation}\nwhere $(\\beta,\\delta) \\mapsto A_{\\beta,\\delta}$ is continuous from $(0,\\epsilon'_0) \n\\times (\\epsilon'_0, \\epsilon'_0)$ to $(0,\\infty)$, and is such that\n\\begin{equation}\n\t\\lim_{(\\beta,\\delta) \n\t\\to (0,0)} A_{\\beta,\\delta} = 1 .\n\\end{equation}\n\\end{theorem}\n\n\n\\subsection{Discussion}\n\\label{sec:discussion}\n\nWe comment on the results obtained in Section~\\ref{sec:theorems}.\n\n\\medskip\\noindent\n\\textbf{1.}\nThe version of the smoothing inequality in Theorem~\\ref{th:smoshift}, \\emph{with the \nprecision on the constant}, is picked up and used in Berger, Caravenna, Poisat, Sun \nand Zygouras~\\cite{cf:BCPSZ} to obtain the sharp asymptotics of the critical \ncurve $\\beta \\mapsto h_c(\\beta)$ for pinning and copolymer models in the \nweak disorder regime $\\beta \\downarrow 0$, for the case $\\alpha \\in (1,\\infty)$ (recall \n\\eqref{eq:astau}).\n\n\\medskip\\noindent\n\\textbf{2.} \nThe smoothing inequality in \\eqref{eq:compdeltah2},\n\\emph{at the level of generality at which it is stated}, is optimal in the following sense.\n\\begin{itemize}\n\\item \nWe cannot hope for an exponent strictly larger than $2$ in the right-hand side of \n\\eqref{eq:compdeltah2}, because pinning models with $\\P(\\tau_1 = n) \\sim (\\log n)\/n^{3\/2}$ \nare in the ``irrelevant disorder regime'', and it is known that $\\textsc{f}(\\beta,h_c(\\beta)+t; 0) \n\\sim \\textsc{f}(0,h_c(0)+t; 0) = t^{2 + o(1)}$ as $t\\downarrow 0$ for fixed $\\beta > 0$ small \nenough (see Alexander \\cite[Theorem 1.2]{cf:Ken}).\n\\item \nWe cannot hope for an asymptotically smaller constant, i.e., $\\lim_{(\\beta,\\delta) \\to \n(0,0)} A_{\\beta,\\delta} < 1$, because the proof in Berger, Caravenna, Poisat, Sun and\nZygouras~\\cite{cf:BCPSZ} would yield a contradiction (the lower bound would be \nstrictly larger than the upper bound).\n\\end{itemize}\nOf course, for specific models the inequality \\eqref{eq:compdeltah2} can\nsometimes be strengthened.\nFor instance, pinning models satisfying \\eqref{eq:astau} with $\\alpha \\in (0,\\frac{1}{2})$\nare such that $\\textsc{f}(\\beta,h_c(\\beta)+t; 0) \\sim \\textsc{f}(0,h_c(0)+t; 0) = t^{1\/\\alpha + o(1)}$\nas $t \\downarrow 0$ (see \\eqref{eq:hompin}), again by Alexander \\cite[Theorem 1.2]{cf:Ken}.\n\n\\medskip\\noindent\n\\textbf{3.}\nCompared with Assumption~\\ref{ass:model}, Assumption~\\ref{ass:model2} prescribes \na specific form for the partition function $Z_{N, \\omega, \\beta, h}$ and therefore is \nmore restrictive. On the other hand, in view of the minor constraints put on the $\\sigma_i$'s, \n\\eqref{eq:ZNgen} is so general that the absence of any restrictive conditions like \n\\eqref{it:superadd} or \\eqref{it:polylb} makes Assumption~\\ref{ass:model2} effectively \nmuch weaker than Assumption~\\ref{ass:model}. For instance, since \\eqref{eq:model} \nis a special case of \\eqref{eq:ZNgen}, with $\\p_N (\\cdot) = \\p(\\,\\cdot\\, \\cap \\{S_N = 0\\})$\n(which, incidentally, explains why $\\p_N$ is allowed to be a finite measure, and not \nnecessarily a probability), the model in \\eqref{eq:model} satisfies Assumption~\\ref{ass:model2}\nas soon as the function $\\phi$ is bounded and has a sign, without the need for any requirement \nlike \\eqref{eq:astau}. \n\nWe emphasize that many other (also non-directed) disordered models \nfall into Assumption~\\ref{ass:model2}. For instance, for $L \\in\\mathbb{N}$ set $\\Lambda_L := \\{-L, \n\\ldots, +L\\}^d$, $N := |\\Lambda_L| = (2L+1)^d$, $\\Omega_N := \\{-1,+1\\}^{\\Lambda_L}$, \nand let $(\\eta_i)_{i\\in\\Lambda_L}$ be the coordinate projections on $\\Omega_N$. If $\\p_N$ \nis the standard Ising Gibbs measure on $\\Omega_N$, defined by $\\p_N(\\{\\eta_i\\}_{i\\in\\Lambda_L}) \n:= (1\/Z_N) \\exp[ J \\sum_{i,j \\in \\Lambda_L, \\,|i-j| = 1} \\eta_i \\eta_j]$, then the random variables \n$\\sigma_i := \\frac{1}{2}(\\eta_i + 1)$ satisfy Assumption~\\ref{ass:model2}.\n\n\\medskip\\noindent\n\\textbf{4.} \nIt follows easily from \\eqref{eq:freeen} and \\eqref{eq:ZNgen} that (with obvious notation)\n\\begin{equation} \n\\label{eq:compa}\n\\textsc{f}_{(\\sigma_n + c)_{n\\in\\mathbb{N}}} ( \\beta, h; \\delta ) \n= \\textsc{f}_{(\\sigma_n)_{n\\in\\mathbb{N}}} ( \\beta, h; \\delta ) + (\\beta m_\\delta + h) c.\n\\end{equation}\nTherefore, when the $\\sigma_n$'s are uniformly bounded but not necessarily non-negative, we can \nfirst perform a uniform translation to transform them into non-negative random variables, \nnext apply \\eqref{eq:compdeltah}, and finally use \\eqref{eq:compa} to come back to the \noriginal $\\sigma_n$'s. \n\nStill, the non-negativity assumption on the $\\sigma_n$'s in \\eqref{eq:s0}\ncannot be dropped from Theorem~\\ref{th:compdeltah}.\nIn fact, if $\\textsc{f}(\\beta,h;\\delta)$ is differentiable in $h$ and $\\delta$, then\n\\eqref{eq:compdeltah} implies that\n\\begin{equation} \n\\label{eq:tocheck}\n\\forall h \\in \\mathbb{R}\\colon \\qquad\n\\frac{\\partial\\textsc{f}}{\\partial \\delta}(\\beta,h ; 0) \n= \\big[ 1 + o(1) \\big]\\,\\beta \\, \\frac{\\partial\\textsc{f}}{\\partial h}(\\beta,h ; 0),\n\\qquad \\beta \\downarrow 0.\n\\end{equation}\nThis relation, which is a necessary condition for \\eqref{eq:compdeltah} when the free energy is \ndifferentiable, may be violated when the $\\sigma_n$'s take both signs.\nFor instance, let $(\\sigma_n)_{n\\in\\mathbb{N}}$ under $\\p_N := \\p$ be i.i.d.\\ with\n$\\p(\\sigma_n = -1) = \\p(\\sigma_n = +1) = \\frac{1}{2}$, and let the marginal distribution \nof the disorder be ${\\ensuremath{\\mathbb P}} (\\omega_n = -a^{-1}) = a^2\/(a^2+1)$, ${\\ensuremath{\\mathbb P}} (\\omega_n = a) \n= 1\/(a^2+1)$ with $a > 0$ (note that ${\\ensuremath{\\mathbb E}} (\\omega_1) = 0$ and $\\bbvar(\\omega_1) = 1$,\nso that  \\eqref{eq:t0} is satisfied). The free energy is easily computed:\n\\begin{equation}\n\\textsc{f}(\\beta,h ; \\delta) = {\\ensuremath{\\mathbb E}} _\\delta[ \\cosh(h + \\beta \\omega_1) ]\n= \\frac{e^{a\\delta} \\cosh(h + a\\beta) + a^2 e^{-a^{-1}\\delta} \\cosh(h - a^{-1}\\beta)}\n{e^{a\\delta} + a^2 e^{-a^{-1}\\delta}}.\n\\end{equation}\nIn particular,\n\\begin{align}\n\\frac{\\partial\\textsc{f}}{\\partial h} (\\beta,0 ; 0) \n& = \\frac{\\sinh(a\\beta) + a^2 \\sinh(-a^{-1}\\beta)}{1 + a^2} \n= \\frac{a^2 - 1}{6a} \\beta^3  + o(\\beta^3),\\\\\n\\frac{\\partial\\textsc{f}}{\\partial \\delta}(\\beta, 0 ; 0) \n& = \\frac{a \\cosh(a\\beta) - a \\cosh(- a^{-1}\\beta)}{1 + a^2} \n= \\frac{a^2 - 1}{2a} \\, \\beta^2 \\,+\\, o(\\beta^2),\n\\end{align}\nand hence \\eqref{eq:tocheck} does \\emph{not} hold for $a \\ne 1$ (the left-hand side \nis $\\approx \\beta^2$, while the right-hand side is $\\approx \\beta^4$). \nIntuitively, such a discrepancy\narises for values of $h$ at which $\\frac{\\partial\\textsc{f}}{\\partial h}(0,h;0) = 0$, which means that\nthe average $\\e_{N,\\omega, 0,h} (\\frac{1}{N}\\sum_{n=1}^N \\sigma_n)$\ntends to zero as $N\\to\\infty$,\nwhere $\\p_{N,\\omega, \\beta,h}$ is the Gibbs law\nassociated to the partition function $Z_{N, \\omega, \\beta, h}$\n(see \\eqref{eq:PZ} below).\nWhen the $\\sigma_n$'s are non-negative, their individual \nvariances under $\\p_{N,\\omega, 0,h}$ must be small, but this is no longer true when \nthe $\\sigma_n$'s can also take negative values.\nThis is why one might have\n$\\frac{\\partial\\textsc{f}}{\\partial \\delta}(\\beta,h;0) \\gg \\beta\\frac{\\partial\\textsc{f}}{\\partial h}(\\beta,h;0)$\nfor $\\beta > 0$ small\n(compare \\eqref{eq:parth} with \\eqref{eq:fnoy}-\\eqref{eq:pias} below).\n\n\n\n\\smallskip\n\n\\section{Smoothing with respect to a tilt: proof of Theorem~\\ref{th:smoothing}}\n\\label{sec:protilt}\n\n\n\n\\subsection{The $({\\ensuremath{\\mathcal G}} ,{\\ensuremath{\\mathcal C}} )$-rare stretch strategy}\n\\label{sec:rare-stretch-strategy}\n\nFix $\\beta \\geq 0$ and $h \\in \\mathbb{R}$. For $\\ell\\in\\mathbb{N}$, let \n${\\ensuremath{\\mathcal A}} _{\\ell} \\subseteq \\mathbb{R}^{\\ell}$ be a subset of ``disorder stretches'' such that \nthere exist constants ${\\ensuremath{\\mathcal G}}  \\in [0,\\infty)$ and ${\\ensuremath{\\mathcal C}}  \\in [0,\\infty)$ with the following \nproperties, along a diverging sequence of $\\ell \\in \\mathbb{N}$:\n\\begin{itemize}\n\\item \n$\\frac{1}{\\ell} \\log Z_{\\ell,\\omega,\\beta,h} \\geq {\\ensuremath{\\mathcal G}} $ for all $\\omega = \\omega_{(0,\\ell]} \n:= (\\omega_1, \\ldots, \\omega_{\\ell}) \\in {\\ensuremath{\\mathcal A}} _\\ell$ (recall Assumption~\\ref{ass:model}\n\\eqref{it:basic});\n\\item $\\frac{1}{\\ell} \\log {\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _\\ell) \\geq - {\\ensuremath{\\mathcal C}} $.\n\\end{itemize}\nThe notation $({\\ensuremath{\\mathcal G}} ,{\\ensuremath{\\mathcal C}} )$ stands for \\emph{gain} versus \\emph{cost}. Recall that \n$\\gamma$ is the exponent in \\eqref{eq:polylb}.\n\n\\begin{lemma}\n\\label{lem:CG}\nThe following implication holds:\n\\begin{equation} \n\\label{eq:loccond}\n{\\ensuremath{\\mathcal G}}  - \\gamma\\, {\\ensuremath{\\mathcal C}}  \\, >  0 \\quad \\Longrightarrow  \\quad \\textsc{f}(\\beta, h ; 0) > 0.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFix $\\ell \\in \\mathbb{N}$ large enough so that the above conditions hold, and for $\\omega \\in \\mathbb{R}^\\mathbb{N}$ \ndenote by $T_1(\\omega), T_2(\\omega), \\ldots$ the distances between the endpoints of \nthe stretches in ${\\ensuremath{\\mathcal A}} _\\ell$:\n\\begin{equation}\nT_1(\\omega) := \\inf\\big\\{N \\in \\ell\\mathbb{N}\\colon\\, \\omega_{(N-\\ell,\nN]} \\in {\\ensuremath{\\mathcal A}} _{\\ell}\\big\\},\n\\qquad \nT_{k+1}(\\omega) := T_{1}(\\theta^{T_{1}(\\omega) + \\ldots + T_{k}(\\omega)}(\\omega)).\n\\end{equation}\nNote that $\\{T_k\\}_{k\\in\\mathbb{N}}$ is i.i.d.\\ with marginal law given by $\\ell \\,\\mathrm{GEO}\n({\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _{\\ell}))$. In particular,\n\\begin{equation}\n\t{\\ensuremath{\\mathbb E}} (T_1) = \\ell \/ {\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _{\\ell}) \\le \\ell \\, e^{{\\ensuremath{\\mathcal C}}  \\ell}.\n\\end{equation}\nHenceforth we suppress the subscripts $\\beta,h$. \nSince $(\\theta^{(T_1 + \\ldots + T_{i})-\\ell} \\omega)_{(0,\\ell]} \\in {\\ensuremath{\\mathcal A}} _\\ell$\nby construction, applying\nproperties \\eqref{it:superadd}-\\eqref{it:polylb} in Assumption~\\ref{ass:model}\nand the definition of ${\\ensuremath{\\mathcal G}} $, we get\n\\begin{equation}\nZ_{T_1 + \\ldots + T_k,\\omega} \n\\geq \\prod_{i=1}^k Z_{T_i - \\ell, \\theta^{(T_1 + \\ldots + T_{i-1})} \\omega}\n\\, Z_{\\ell, \\theta^{(T_1 + \\ldots + T_{i})-\\ell} \\omega}\n\\geq e^{k {\\ensuremath{\\mathcal G}}  \\ell} \\, \\prod_{i=1}^k \n\\frac{c_{\\beta,h}(\\theta^{(T_1 + \\ldots + T_{i-1})}\\omega)}{(T_i)^\\gamma},\n\\end{equation}\nwhere we set $Z_0 := 1$ for convenience.\nRecalling \\eqref{eq:freeen} and Remark~\\ref{rem:kingman},\nfor ${\\ensuremath{\\mathbb P}} $-a.e. $\\omega$ we can write, by the strong \nlaw of large numbers and Jensen's inequality,\n\\begin{equation}\n\\begin{split}\n\\textsc{f}(\\beta, h ; 0) \n& \n= \\lim_{k\\to\\infty} \\frac{1}{T_1 + \\ldots + T_k}\n\\log Z_{T_1 + \\ldots + T_k,\\omega,\\beta,h} \\\\\n& \\geq \\frac{1}{{\\ensuremath{\\mathbb E}} (T_1(\\omega))} \\big\\{ \\ell {\\ensuremath{\\mathcal G}}  + {\\ensuremath{\\mathbb E}} [\\log c_{\\beta,h}(\\omega)]\n- \\gamma \\, {\\ensuremath{\\mathbb E}} [\\log (T_1) ] \\big\\} \\\\\n& \\geq \\frac{1}{{\\ensuremath{\\mathbb E}} (T_1(\\omega))} \\big\\{ \\ell {\\ensuremath{\\mathcal G}}  + {\\ensuremath{\\mathbb E}} [\\log c_{\\beta,h}(\\omega)]\n- \\gamma \\, \\log {\\ensuremath{\\mathbb E}} (T_1) \\big\\} \\\\\n& = e^{-{\\ensuremath{\\mathcal C}}  \\ell} \\bigg\\{ ({\\ensuremath{\\mathcal G}}  - \\gamma {\\ensuremath{\\mathcal C}}  )\n+ \\frac{{\\ensuremath{\\mathbb E}} [\\log c_{\\beta,h}(\\omega)]}{\\ell} - \\gamma \\frac{\\log \\ell}{\\ell} \\bigg\\}.\n\\end{split}\n\\end{equation}\nIf ${\\ensuremath{\\mathcal G}}  - \\gamma {\\ensuremath{\\mathcal C}}  > 0$, then we can choose $\\ell \\in \\mathbb{N}$ large enough (but finite!) \nsuch that the right-hand side is strictly positive. This proves \\eqref{eq:loccond}.\n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{th:smoothing}}\nWe use Lemma~\\ref{lem:CG}. Fix $\\beta > 0$, $h \\in \\mathbb{R}$,\n$\\delta \\in (-t_0, t_0)$ and $\\epsilon > 0$, \nand define the set of good atypical stretches as\n\\begin{equation}\n{\\ensuremath{\\mathcal A}} _{\\ell} := \\bigg\\{ (\\omega_1, \\ldots, \\omega_\\ell) \\in \\mathbb{R}^\\ell\\colon\\, \n\\frac{1}{\\ell}\\log Z_{\\ell, \\omega, \\beta, h} \\geq \\textsc{f}(\\beta, h; \\delta) - \\epsilon \\bigg\\},\n\\end{equation}\nso that ${\\ensuremath{\\mathcal G}}  = \\textsc{f}(\\beta, h; \\delta) - \\epsilon $ by construction. It remains to determine ${\\ensuremath{\\mathcal C}} $, for \nwhich we need to estimate the probability of ${\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _\\ell)$ from below. \n\nBy the definition \\eqref{eq:freeen} of $\\textsc{f}(\\beta, h; \\delta)$ together with Kingman's super-additive\nergodic theorem (see Remark~\\ref{rem:kingman}),\nthe event ${\\ensuremath{\\mathcal A}} _\\ell$ is typical for ${ {\\ensuremath{\\mathbb P}} }_\\delta$:\n\\begin{equation} \n\\label{eq:pdelta1}\n\\lim_{\\ell\\to+\\infty} { {\\ensuremath{\\mathbb P}} }_\\delta({\\ensuremath{\\mathcal A}} _\\ell) = 1.\n\\end{equation}\nDenoting by ${\\ensuremath{\\mathbb P}} _\\delta^\\ell$ (resp. ${\\ensuremath{\\mathbb P}} ^\\ell$) the restriction of ${\\ensuremath{\\mathbb P}} _\\delta$\n(resp. ${\\ensuremath{\\mathbb P}} $) on $\\sigma(\\omega_1, \\ldots, \\omega_\\ell)$, we have, by Jensen's inequality\nand \\eqref{eq:RNn},\n\\begin{equation} \\label{eq:entrin}\n\\begin{split}\n\t{\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _\\ell) & = {\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell) \\,\n\t{\\ensuremath{\\mathbb E}} _\\delta\\bigg( e^{- \\log \\frac{\\mathrm{d} {\\ensuremath{\\mathbb P}} ^\\ell_\\delta}{\\mathrm{d}{\\ensuremath{\\mathbb P}} ^\\ell}}\n\t\\bigg| {\\ensuremath{\\mathcal A}} _\\ell \\bigg) \\ge\n\t{\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell) \\,\n\te^{- {\\ensuremath{\\mathbb E}} _\\delta \\big(\\log \\frac{\\mathrm{d} {\\ensuremath{\\mathbb P}} ^\\ell_\\delta}{\\mathrm{d}{\\ensuremath{\\mathbb P}} ^\\ell}\n\t\\big| {\\ensuremath{\\mathcal A}} _\\ell \\big)} \\\\\n\t& = {\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell) \\,\n\te^{- \\frac{1}{{\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell)}\n\t{\\ensuremath{\\mathbb E}} _\\delta \\big[ \\big(\n\t\\log \\frac{\\mathrm{d} {\\ensuremath{\\mathbb P}} ^\\ell_\\delta}{\\mathrm{d}{\\ensuremath{\\mathbb P}} ^\\ell}\\big)  \\, {\\sf 1}_{{\\ensuremath{\\mathcal A}} _\\ell} \\big]} \\\\\n\t& =  {\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell) \\,\n\te^{- \\frac{\\ell}{{\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell)}\n\t{\\ensuremath{\\mathbb E}} _\\delta \\big[ \\big(\n\t\\delta \\frac{\\omega_1 + \\ldots + \\omega_\\ell}{\\ell} - \\log \\mathrm{M}(\\delta)\n\t\\big)  \\, {\\sf 1}_{{\\ensuremath{\\mathcal A}} _\\ell} \\big]}\\,.\n\\end{split}\n\\end{equation}\nRecalling \\eqref{eq:RNn} and Assumption~\\ref{ass:disorder}, we abbreviate\n\\begin{equation}\n\\label{eq:mdelta0}\nm_\\delta := {\\ensuremath{\\mathbb E}} _\\delta(\\omega_1) = (\\log\\mathrm{M})'(\\delta)\n= \\delta + o(\\delta), \\qquad \\delta \\to 0.\n\\end{equation}\nBy the strong law of large numbers, it follows from \\eqref{eq:pdelta1}-\\eqref{eq:entrin}\nthat for every $\\epsilon > 0$ we have, for $\\ell$ large enough,\n\\begin{equation}\n\\frac{1}{\\ell} \\log {\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _\\ell) \\geq \n- \\big[ \\delta \\, m_\\delta - \\log \\mathrm{M}(\\delta) \\big] \\,-\\, \\epsilon =: -{\\ensuremath{\\mathcal C}} ,\n\\end{equation}\n\n\n\n\nWe can conclude.\nWe know from \\eqref{eq:loccond} that $\\textsc{f}(\\beta, h; 0) > 0$ when\n\\begin{equation}\n{\\ensuremath{\\mathcal G}}  - \\gamma {\\ensuremath{\\mathcal C}}  = \\textsc{f}(\\beta, h; \\delta) - \\gamma \\big[ \\delta \\, m_\\delta \n- \\log \\mathrm{M}(\\delta) \\big] - 2 \\epsilon > 0.\n\\end{equation}\nIf $\\textsc{f}(\\bar\\beta, \\bar h) = 0$, as in the assumptions of Theorem~\\ref{th:smoothing},\nit follows that ${\\ensuremath{\\mathcal G}}  - \\gamma {\\ensuremath{\\mathcal C}}  \\leq 0$, \ni.e.,\n\\begin{equation}\n\\textsc{f}(\\bar\\beta, \\bar h; \\delta) \\leq \\gamma \n\\big[ \\delta \\, m_\\delta - \\log \\mathrm{M}(\\delta) \\big] + 2 \\epsilon, \\qquad\n\\forall \\delta \\in (-t_0, t_0) \\,.\n\\end{equation}\nSince this equality holds for every $\\epsilon > 0$, it must hold also for $\\epsilon = 0$,\nproving \\eqref{eq:smodelta}.\n\\qed\n\n\n\\smallskip\n\n\\section{Asymptotic equivalence of tilting and shifting: proof of\nTheorem~\\ref{th:compdeltah}}\n\\label{sec:tiltshift}\n\nThroughout this section, we work under Assumptions~{\\rm \\ref{ass:disorder}} \nand {\\rm \\ref{ass:model2}}.\n\n\n\\subsection{Notation}\nDenote the empirical average of the variables $\\sigma_i$'s by\n\\begin{equation} \n\\label{eq:sigmabar}\n\\overline \\sigma_N := \\frac{1}{N} \\sum_{i=1}^N \\sigma_i.\n\\end{equation}\nThe finite-volume Gibbs measure associated with the partition function in \n\\eqref{eq:ZNgen} is the \\emph{probability} on $\\Omega_N$\ndefined, for $N\\in\\mathbb{N}$, $\\omega \\in \\mathbb{R}^\\mathbb{N}$, $\\beta \\geq 0$ \nand $h\\in\\mathbb{R}$, by\n\\begin{equation}\\label{eq:PZ}\n\\begin{split}\n& \\p_{N, \\omega, \\beta, h}(\\,\\cdot\\,) \n:= \\frac{1}{Z_{N, \\omega, \\beta, h}} \n\\e_N \\Big[e^{\\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n} \\, {\\sf 1}_{\\{\\cdot\\}} \\Big] \\,, \\\\\n& \\text{where} \\quad\nZ_{N, \\omega, \\beta, h} := \\e_N \\Big[ e^{\\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n} \\Big]\n\\,. \\end{split}\n\\end{equation}\nLet us spell out the definition \\eqref{eq:freeen} of the free energy, recalling \\eqref{eq:RNn}:\n\\begin{equation} \n\\label{eq:freee}\n\\begin{split}\n\\textsc{f}(\\beta,h; \\delta) \n&:= \\limsup_{N\\to\\infty} \\textsc{f}_N(\\beta,h; \\delta) := \\limsup_{N\\to\\infty} \\frac{1}{N} \n{\\ensuremath{\\mathbb E}} _\\delta\\big[ \\log Z_{N, \\omega, \\beta, h} \\big]  \\\\\n&= \\limsup_{N\\to\\infty} \\frac{1}{N} {\\ensuremath{\\mathbb E}}  \\big[ e^{\\sum_{n=1}^N [\\delta \\omega_n\n- \\log\\mathrm{M}(\\delta)]} \\log Z_{N, \\omega, \\beta, h} \\big].\n\\end{split}\n\\end{equation}\nNote that, by \\eqref{eq:s0},\n\\begin{equation} \n\\label{eq:boundham}\n\\Bigg| \\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n \\Bigg| \n\\leq \\sum_{n=1}^N (|h| + \\beta |\\omega_n|) \\, |\\sigma_n|\n\\leq s_0 \\sum_{n=1}^N (|h| + \\beta |\\omega_n|),\n\\end{equation}\nso that $|\\textsc{f}(\\beta,h;\\delta)| \\leq s_0 (|h| + \\beta {\\ensuremath{\\mathbb E}} _\\delta(|\\omega_1|))\n\\,+\\, |\\limsup_{N\\to\\infty} \\frac{1}{N} \\log \\p_N(\\Omega_N)| < \\infty$.\n\n\n\\subsection{Preparation}\n\nBefore proving Theorem~\\ref{th:compdeltah}, we need some preparation. Recalling \n\\eqref{eq:sigmabar}, we define for $[a,b] \\subseteq \\mathbb{R}$ with $a<b$ a restricted version \nof the partition function and the free energy, in which the empirical average\n$\\overline\\sigma_N$ is constrained to lie in $[a,b]$:\n\\begin{equation} \n\\label{eq:freeeab}\n\\begin{split}\nZ_{N, \\omega, \\beta, h}^{[a,b]} \n& := \\e_N \\Big[ e^{\\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n} \\, \n{\\sf 1}_{\\{\\overline\\sigma_N \\in [a,b]\\}} \\Big],\\\\\n\\textsc{f}^{[a,b]}(\\beta,h; \\delta) \n& := \\limsup_{N\\to\\infty} \\textsc{f}^{[a,b]}_N(\\beta,h; \\delta) \n:= \\limsup_{N\\to\\infty} \\frac{1}{N} {\\ensuremath{\\mathbb E}} _\\delta\\big[ \\log Z_{N, \\omega, \\beta, h}^{[a,b]} \\big].\n\\end{split}\n\\end{equation}\nThe corresponding restricted Gibbs measure is the probability defined by (recall \\eqref{eq:PZ})\n\\begin{equation} \n\\label{eq:pNab}\n\\p_{N, \\omega, \\beta, h}^{[a,b]} (\\,\\cdot\\,) \n:= \\p_{N, \\omega, \\beta, h} (\\,\\cdot\\,|\\,\\overline\\sigma_N \\in [a,b] )\n= \\frac{\\e_N \\Big[ e^{\\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n} \\, \n{\\sf 1}_{\\{\\overline\\sigma_N \\in [a,b]\\}} \\, {\\sf 1}_{\\{\\,\\cdot\\,\\}} \\Big]}{Z_{N, \\omega, \\beta, h}^{[a,b]}}.\n\\end{equation}\n\nNote that $Z_{N, \\omega, \\beta, h} = Z_{N, \\omega, \\beta, h}^{[0,s_0]}$, by \\eqref{eq:s0}.\nFurthermore, $Z_{N, \\omega, \\beta, h}^{[a,b]} \\le Z_{N, \\omega, \\beta, h}^{[c,d]}$ when \n$[a,b] \\subseteq [c,d]$. Therefore\n\\begin{equation} \n\\label{eq:monf}\n\\textsc{f}^{[a,b]}(\\beta,h;\\delta) \\leq \\textsc{f}^{[c,d]}(\\beta,h;\\delta) \\,\\le\\, \\textsc{f} (\\beta,h;\\delta),\n\\qquad [a,b] \\subseteq [c,d].\n\\end{equation}\nIn particular, for $x \\in \\mathbb{R}$ we may define\n\\begin{equation} \n\\label{eq:fx}\n\\textsc{f}^{\\{x\\}}(\\beta,h;\\delta) := \\lim_{n \\to \\infty} \\textsc{f}^{[a_n, b_n]}(\\beta,h;\\delta)\n\\in [-\\infty, +\\infty),\n\\end{equation}\nwhere $a_n \\uparrow x$ and $b_n \\downarrow x$ are arbitrary strictly monotone \nsequences (it is easily seen that the limit does not depend on the choice of these \nsequences). \n\nNote that $\\textsc{f}^{\\{x\\}}(\\beta,h;\\delta) = -\\infty$ when $x \\not\\in [0,s_0]$, \nby \\eqref{eq:s0}. The following result is standard:\n\\begin{equation}\n\\label{eq:fbetah}\n\\textsc{f}(\\beta,h;\\delta) = \\sup_{x \\in [0,s_0]}  \\textsc{f}^{\\{x\\}}(\\beta,h;\\delta).\n\\end{equation}\nIn fact,\nby \\eqref{eq:monf} $\\textsc{f}^{[a,b]}(\\beta,h;\\delta) \\le \\textsc{f}(\\beta,h;\\delta)$ for every $[a,b] \\subseteq \\mathbb{R}$,\nhence by \\eqref{eq:fx} $\\textsc{f}(\\beta,h;\\delta) \\geq \\textsc{f}^{\\{x\\}}(\\beta,h;\\delta)$ for every \n$x\\in\\mathbb{R}$. It follows that the inequality $\\geq$ holds in \\eqref{eq:fbetah}. For the reverse inequality, \nnote that if $a < b < c$, then $[a,c] \\subseteq [a,b] \\cup [b,c]$ and so\n\\begin{equation}\nZ_{N, \\omega, \\beta, h}^{[a,c]} \n\\leq Z_{N, \\omega, \\beta, h}^{[a,b]} + Z_{N, \\omega, \\beta, h}^{[b,c]} \n\\leq 2 \\, \\max \\Big\\{ Z_{N, \\omega, \\beta, h}^{[a,b]},Z_{N, \\omega, \\beta, h}^{[b,c]} \\Big\\}.\n\\end{equation}\nRecalling \\eqref{eq:freeeab}, we see that\n\\begin{equation}\n\\textsc{f}^{[a,c]}(\\beta,h;\\delta) \\leq \n\\max \\Big\\{ \\textsc{f}^{[a,b]}(\\beta,h;\\delta), \\, \\textsc{f}^{[b,c]}(\\beta,h;\\delta) \\Big\\}.\n\\end{equation}\nSince $\\textsc{f}(\\beta,h;\\delta) = \\textsc{f}^{[0,s_0]}(\\beta,h;\\delta)$, we can build a sequence of closed \nintervals $(I_n)_{n\\in\\mathbb{N}_0}$, where $I_0 = [0,s_0]$ and where $I_{n+1}$ is either the first \nhalf or the second half of $I_n$, such that $\\textsc{f}^{I_n}(\\beta,h;\\delta) \\leq \\textsc{f}^{I_{n+1}}(\\beta,h;\\delta)$ \nfor all $n\\in\\mathbb{N}$. In particular,\n\\begin{equation}\n\\textsc{f}(\\beta,h;\\delta) \\leq \\lim_{n\\to\\infty} \\textsc{f}^{I_n}(\\beta,h;\\delta).\n\\end{equation}\nBy compactness, there exists an $\\overline x \\in [0,s_0]$ such that $I_n \\downarrow \n\\{\\overline x\\}$, i.e., $\\bigcap_{n\\in\\mathbb{N}} I_n = \\{\\overline x\\}$. If $I_n = [a_n, b_n]$, then we\nset $J_n := [a_n - \\frac{1}{n}, b_n + \\frac{1}{n}]$, so that we still have $J_n \\downarrow \n\\{\\overline x\\}$, and $\\overline x$ lies in the interior of each $J_n$. Since $\\textsc{f}^{I_n}(\\beta,h;\\delta) \n\\leq \\textsc{f}^{J_n}(\\beta,h;\\delta)$, recalling \\eqref{eq:fx} we obtain\n\\begin{equation}\n\\textsc{f}(\\beta,h;\\delta) \\leq \\lim_{n\\to\\infty} \\textsc{f}^{I_n}(\\beta,h;\\delta)\n\\leq \\lim_{n\\to\\infty} \\textsc{f}^{J_n}(\\beta,h;\\delta) \n= \\textsc{f}^{\\{\\overline x\\}}(\\beta,h;\\delta) \n\\leq \\sup_{x \\in [0,s_0]}  \\textsc{f}^{\\{x\\}}(\\beta,h;\\delta),\n\\end{equation}\nand the proof of \\eqref{eq:fbetah} is complete.\n\n\n\n\\subsection{Proof of Theorem~\\ref{th:compdeltah}}\n\nBy \\eqref{eq:fbetah}, it suffices to show that \\eqref{eq:compdeltah} is satisfied\nwith $\\textsc{f}^{\\{x\\}}$ instead of $\\textsc{f}$, for every fixed $x \\in [0, s_0]$. It is of course \nimportant that the constants $C^\\pm_{\\beta,\\delta}$ do not depend on $x$.\n\n\\medskip\\noindent\n\\textbf{1.}\nFirst we consider the case $x = 0$. We claim that\n\\begin{equation} \n\\label{eq:f0}\n\\textsc{f}^{\\{0\\}}(\\beta,h;\\delta) = \\lim_{\\epsilon \\downarrow 0}\n\\bigg( \\limsup_{N\\to\\infty} \\frac{1}{N} \\log \\p_N(0 \\le \\overline\\sigma_N \\le \\epsilon) \\bigg).\n\\end{equation}\nSince the right-hand side of \\eqref{eq:f0} is a constant that does not depend on $\\beta \\ge 0$,\n$\\delta \\in (-t_0,t_0)$ and $h\\in\\mathbb{R}$, \\eqref{eq:compdeltah} is trivially satisfied\nwith $\\textsc{f}^{\\{0\\}}$ instead of $\\textsc{f}$, whatever the \ndefinition of $C^\\pm_{\\beta,\\delta}$ is. To prove \\eqref{eq:f0} note that, by Cauchy-Schwarz,\n\\begin{equation}\n\\begin{split}\n\\Bigg| \\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n \\Bigg| \n& \\leq \\sqrt{\\sum_{n=1}^N (h + \\beta \\omega_n)^2} \\, \\sqrt{\\sum_{n=1}^N |\\sigma_n|^2} \n\\leq N \\, s_0 \\, \\sqrt{\\overline\\sigma_N} \\, \\sqrt{\\frac{1}{N} \\sum_{n=1}^N (h + \\beta \\omega_n)^2},\n\\end{split}\n\\end{equation}\nbecause $0 \\leq \\sigma_n = |\\sigma_n| \\leq s_0$ by \\eqref{eq:s0}. Recalling \\eqref{eq:freeeab}, \nfor every $N\\in\\mathbb{N}$ we get\n\\begin{equation}\n\\begin{split}\n\\bigg| \\frac{1}{N} {\\ensuremath{\\mathbb E}} _\\delta \\big[ \\log Z^{[a,b]}_{N,\\omega,\\beta,h} \\big] \n- \\frac{1}{N} \\log \\p_N(a \\le \\overline\\sigma_N \\le b) \\bigg| \n&\\leq  s_0 \\, \\sqrt{b} \\, {\\ensuremath{\\mathbb E}} _\\delta \\left[ \\sqrt{\\frac{1}{N} \n\\sum_{n=1}^N (h + \\beta \\omega_n)^2} \\right] \\\\\n& \\leq s_0 \\, \\sqrt{b} \\,\\sqrt{{\\ensuremath{\\mathbb E}} _\\delta \\big[ (h + \\beta \\omega_1)^2 \\big]},\n\\end{split}\n\\end{equation}\nwhere we use Jensen. Note that the right-hand side is a finite constant. If $|a_N - b_N| \\leq c$ for \nall $N\\in\\mathbb{N}$, then $|\\limsup_N a_N - \\limsup_N b_N| \\leq c$, and so\n\\begin{equation}\n\\bigg| \\textsc{f}^{[a,b]}(\\beta,h;\\delta) \n-  \\bigg(\\limsup_{N\\to\\infty} \\frac{1}{N} \\log \\p_N(a \\le \\overline\\sigma_N \\le b) \\bigg) \\bigg|\n\\leq s_0 \\, \\sqrt{b} \\,\\sqrt{{\\ensuremath{\\mathbb E}} _\\delta \\big[ (h + \\beta \\omega_1)^2 \\big]}.\n\\end{equation}\nTaking $[a,b] = [-\\epsilon, \\epsilon]$ and letting $\\epsilon \\downarrow 0$, we get \\eqref{eq:f0}\nfrom \\eqref{eq:fx} .\n\n\\medskip\\noindent\n\\textbf{2.}\nNext we consider the case $x \\in (0, s_0]$. Roughly speaking, the strategy of the proof is to \nshow that the derivatives of the free energy with respect to $\\delta$ and to $h$ are comparable.\nUnless otherwise specified, we work with generic values of the parameters in the admissible \nrange $\\beta \\geq 0$, $h\\in\\mathbb{R}$ and $\\delta \\in (-t_0, t_0)$. Henceforth we fix $0 < a < b < \\infty$. \nRecalling \\eqref{eq:freeeab} and \\eqref{eq:pNab}, we see that the derivative with respect to $h$ \nof the (restricted) finite-volume free energy $\\textsc{f}^{[a,b]}_N(\\beta, h;\\delta)$ can be expressed as\n\\begin{equation} \n\\label{eq:parth}\n\\frac{\\partial}{\\partial h} \\textsc{f}^{[a,b]}_N(\\beta, h;\\delta) \n= \\frac{1}{N} {\\ensuremath{\\mathbb E}} _\\delta\\bigg[ \\frac{\\partial}{\\partial h} \\log Z^{[a,b]}_{N, \\omega, \\beta, h} \\bigg] \n= {\\ensuremath{\\mathbb E}} _\\delta \\big[ \\e^{[a,b]}_{N, \\omega, \\beta, h}\\big[ \\overline\\sigma_N \\big] \\big].\n\\end{equation}\n\n\\medskip\\noindent\n\\textbf{3.}\nThe derivative with respect to $\\delta$ requires some further estimates. Recalling\n\\eqref{eq:PZ}-\\eqref{eq:freee}, we have\n\\begin{equation} \n\\label{eq:start}\n\\frac{\\partial}{\\partial\\delta} \\textsc{f}^{[a,b]}_N(\\beta, h; \\delta) \n= \\frac{1}{N} \\sum_{n=1}^N {\\ensuremath{\\mathbb E}} _\\delta \\Big[ (\\omega_n - m_\\delta) \\,\n\\log Z^{[a,b]}_{N, \\omega, \\beta, h} \\Big],\n\\end{equation}\nwhere $m_\\delta := {\\ensuremath{\\mathbb E}} _\\delta(\\omega_n) = (\\log \\mathrm{M})'(\\delta)$ by \\eqref{eq:mdelta0}.\nSubtracting a centering term with zero mean, we get\n\\begin{equation} \n\\label{eq:ddf}\n\\begin{split}\n\\frac{\\partial}{\\partial\\delta} \\textsc{f}^{[a,b]}_N(\\beta, h; \\delta) \n& = \\frac{1}{N} \\sum_{n=1}^N {\\ensuremath{\\mathbb E}} _\\delta \\Big[ (\\omega_n - m_\\delta) \\,\n\\Big( \\log Z^{[a,b]}_{N, \\omega, \\beta, h} \n- \\log Z^{[a,b]}_{N, \\omega, \\beta, h}|_{\\omega_n = m_\\delta} \\Big) \\Big] \\\\\n& = \\frac{1}{N} \\sum_{n=1}^N {\\ensuremath{\\mathbb E}} _\\delta \\Bigg[ (\\omega_n - m_\\delta) \n\\int_{m_\\delta}^{\\omega_n} \\bigg( \\frac{\\partial}{\\partial\\omega_n} \n\\log Z^{[a,b]}_{N, \\omega, \\beta, h} \\bigg) \\bigg|_{\\omega_n = y} \\, \\mathrm{d} y \\Bigg],\n\\end{split}\n\\end{equation}\nwhere we agree that $\\int_a^b (\\ldots) := -\\int_{b}^a(\\ldots)$ when $a > b$.\nAbbreviate\n\\begin{equation} \n\\label{eq:fnoy}\nf_n(\\omega, y) := \\frac{1}{\\beta} \\, \\bigg( \\frac{\\partial}{\\partial \\omega_n} \n\\log Z^{[a,b]}_{N, \\omega, \\beta, h} \\bigg)\\bigg|_{\\omega_n = y}\n= \\e^{[a,b]}_{N, \\omega, \\beta, h}|_{\\omega_n = y}[\\sigma_n],\n\\end{equation}\nwhere the second equality follows easily from \\eqref{eq:freeeab} via \\eqref{eq:pNab}.\nNote that $f_n(\\omega,y)$ depends on the $\\omega_i$'s for $i \\ne n$, not on \n$\\omega_n$. Therefore \\eqref{eq:ddf} can be rewritten as\n\\begin{equation} \n\\label{eq:pias}\n\\frac{\\partial}{\\partial\\delta} \\textsc{f}^{[a,b]}_N(\\beta, h; \\delta) \n= \\frac{\\beta}{N} \\sum_{n=1}^N {\\ensuremath{\\mathbb E}} _\\delta \\Big[ (\\omega_n - m_\\delta)^2 \\,\n\\frac{1}{\\omega_n - m_\\delta} \\int_{m_\\delta}^{\\omega_n} f_n(\\omega,y) \\, \\mathrm{d} y \\Big].\n\\end{equation}\n\n\\medskip\\noindent\n\\textbf{4.}\nBy \\eqref{eq:fnoy}, the integral average in \\eqref{eq:pias} should be close to\n$\\e^{[a,b]}_{N, \\omega, \\beta, h}[\\sigma_n]$. If we could factorize the expectation\nover ${\\ensuremath{\\mathbb E}} _\\delta$, then the right-hand side in \\eqref{eq:pias} would become \n$\\approx \\beta \\, \\bbvar_\\delta(\\omega_1)\\, {\\ensuremath{\\mathbb E}} _\\delta \\big[ \\e^{[a,b]}_{N, \\omega, \n\\beta, h}[\\overline\\sigma_N] \\big]$. Recalling \\eqref{eq:parth}, we see that this is \nprecisely what we want, because $\\bbvar_\\delta(\\omega_1) \\approx 1$ for $\\delta$ \nsmall. In order to turn these arguments into a proof, we need to estimate the dependence \nof $f_n(\\omega, y)$ on $y$. To that end we note that\n\\begin{equation}\n\\label{eq:varus2}\n\\begin{split}\n\\frac{\\partial}{\\partial \\omega_n} f_n(\\omega, \\omega_n) \n& = \\frac{1}{\\beta} \\frac{\\partial^2}{\\partial \\omega_n^2}\n\\log Z^{[a,b]}_{N, \\omega, \\beta, h} \\,=\\, \\beta \\, \n\\var_{N, \\omega, \\beta, h}^{[a,b]}[\\sigma_n] \\\\\n& \\leq \\beta \\, \\e^{[a,b]}_{N, \\omega, \\beta, h} [\\sigma_n^2] \n\\leq  s_0 \\, \\beta \\, \\e^{[a,b]}_{N, \\omega, \\beta, h}[\\sigma_n] \n= s_0 \\, \\beta \\,  f_n(\\omega, \\omega_n)\n\\end{split}\n\\end{equation}\nbecause $0 \\le \\sigma_n \\le s_0$, by \\eqref{eq:s0}. Therefore\n\\begin{equation}\n\\frac{\\partial}{\\partial y} f_n(\\omega, y) \\geq 0, \n\\qquad \\frac{\\partial}{\\partial y} \\big( e^{-s_0 \\, \\beta \\, y} \\, f_n(\\omega, y) \\big) \\leq\t0,\n\\end{equation}\nand integrating these relations we get\n\\begin{equation} \n\\label{eq:ustec}\ne^{-s_0 \\beta  (y - y')^-} \\, f_n(\\omega, y') \n\\leq f_n(\\omega, y) \\leq e^{s_0 \\beta  (y - y')^+} \\, f_n(\\omega, y') \n\\qquad \\forall\\, y,y' \\in \\mathbb{R}.\n\\end{equation}\nIntroducing the function\n\\begin{equation} \n\\label{eq:defg}\ng(x) := \\begin{cases}\n\\displaystyle\n\\frac{e^{x} - 1}{x} \n& \\text{if } x \\ne 0, \\\\\n1 \n& \\text{if } x = 0,\n\\end{cases}\n\\end{equation}\ntaking $y' = m_\\delta$ in \\eqref{eq:ustec} and integrating over $y$, we easily obtain the bounds\n\\begin{equation} \n\\label{eq:g-+est}\n\\begin{split}\ng\\big(-\\beta s_0(\\omega_n - \n& m_\\delta)^-\\big) \\, f_n(\\omega, m_\\delta) \\\\\n& \\leq \\frac{1}{\\omega_n - m_\\delta}\n\\int_{m_\\delta}^{\\omega_n} f_n(\\omega,y) \\, \\mathrm{d} y \n\\leq g\\big(\\beta s_0(\\omega_n - m_\\delta)^+\\big) \\, f_n(\\omega, m_\\delta).\n\\end{split}\n\\end{equation}\n\n\\medskip\\noindent\n\\textbf{5.}\nBefore inserting this estimate into \\eqref{eq:pias}, let us pause for a brief integrability interlude.\nThe random variable $g(-\\beta s_0(\\omega_n - m_\\delta)^-)$ is bounded, so there is no integrability \nconcern. On the other hand, the random variable $g(\\beta s_0(\\omega_n - m_\\delta)^+)$ is \nunbounded and a little care is required. Note that\n\\begin{equation}\ng(\\beta s_0(\\omega_n - m_\\delta)^+) \\leq A + B \\, e^{\\beta s_0 \\omega_n }\n\\end{equation}\nfor $A,B>0$, and that ${\\ensuremath{\\mathbb E}} _\\delta(e^{t\\omega_1}) < \\infty$ for $t + \\delta \\in (-t_0, +t_0)$, by \n\\eqref{eq:t0} and \\eqref{eq:RNn}. Therefore, when we integrate  $g(\\beta s_0(\\omega_n - \nm_\\delta)^+)$ (possibly times a polynomial of $\\omega_n$) over ${\\ensuremath{\\mathbb P}} _\\delta$, to have a \nfinite outcome we need to ensure that $\\beta s_0 + \\delta \\in (-t_0, +t_0)$. This is simply \nachieved through the restrictions $\\delta \\in (-\\epsilon_0 ,\\epsilon_0)$ and $\\beta \\in [0,\\epsilon_0)$,\nwhere $\\epsilon_0 := \\min\\{\\frac{t_0}{2},\\frac{t_0}{2s_0}\\}$,\nas in the statement of Theorem~\\ref{th:compdeltah}.\nWe make these restrictions henceforth.\n\n\\medskip\\noindent\n\\textbf{6.}\nLet us now substitute the estimate \\eqref{eq:g-+est} into \\eqref{eq:pias}. Since $f_n(\\omega, \nm_\\delta)$ does not depend on $\\omega_n$, the expectation over ${\\ensuremath{\\mathbb E}} _\\delta$ factorizes \nand we obtain\n\\begin{equation} \n\\label{eq:aatt}\n\\begin{split}\n{\\ensuremath{\\mathbb E}} _\\delta \\Big[ (\\omega_1 - m_\\delta)^2 \\, \n& g\\big(-\\beta s_0 (\\omega_1 - m_\\delta)^-\\big) \\Big]\n\\, \\Bigg( \\frac{\\beta}{N} \\sum_{n=1}^N\n{\\ensuremath{\\mathbb E}} _\\delta \\Big[ f_n(\\omega, m_\\delta) \\Big] \\Bigg) \\\\\n& \\leq \\frac{\\partial}{\\partial\\delta} \\textsc{f}^{[a,b]}_N(\\beta, h; \\delta) \\\\\n& \\leq {\\ensuremath{\\mathbb E}} _\\delta \\Big[ (\\omega_1 - m_\\delta)^2 \\, g\\big(\\beta s_0 (\\omega_1 - m_\\delta)^+\\big) \\Big]\n\\, \\Bigg( \\frac{\\beta}{N} \\sum_{n=1}^N {\\ensuremath{\\mathbb E}} _\\delta \\Big[ f_n(\\omega, m_\\delta) \\Big] \\Bigg).\n\\end{split}\n\\end{equation}\nWe next want to replace $f_n(\\omega, m_\\delta)$ by $f_n(\\omega, \\omega_n) \n= \\e^{[a,b]}_{N, \\omega, \\beta, h} \\big[ \\sigma_n \\big]$ (recall \\eqref{eq:fnoy}). \nTo this end, we again apply \\eqref{eq:ustec}, this time with $y = \\omega_n$ and \n$y' = m_\\delta$. Since $f_n(\\omega, m_\\delta)$ does not depend on $\\omega_n$, \nwe have\n\\begin{equation} \n\\label{eq:aatt+}\n\\begin{split}\n\\frac{{\\ensuremath{\\mathbb E}} _\\delta \\big[ f_n(\\omega, \\omega_n) \\big]}\n{{\\ensuremath{\\mathbb E}} _\\delta \\big[e^{s_0 \\beta  (\\omega_1 - m_\\delta)^+ } \\big]}\n\\leq {\\ensuremath{\\mathbb E}} _\\delta \\Big[ f_n(\\omega, m_\\delta) \\Big] \n& \\leq \\frac{{\\ensuremath{\\mathbb E}} _\\delta \\big[ f_n(\\omega, \\omega_n) \\big]}\n{{\\ensuremath{\\mathbb E}} _\\delta \\big[ e^{-s_0 \\beta (\\omega_1 - m_\\delta)^-} \\big]}.\n\\end{split}\n\\end{equation}\nWe can now introduce the constants\n\\begin{equation} \n\\label{eq:c+-}\n\\begin{split}\nc^+_{\\beta, \\delta} \n& := \\frac{{\\ensuremath{\\mathbb E}} _\\delta \\big[ (\\omega_1 - m_\\delta)^2 \n\\, g\\big(\\beta s_0 (\\omega_1 - m_\\delta)^+\\big) \\big]}\n{{\\ensuremath{\\mathbb E}} _\\delta \\big[ e^{-s_0 \\beta (\\omega_1 - m_\\delta)^-} \\big]}, \\\\\nc^-_{\\beta, \\delta} \n& := \\frac{{\\ensuremath{\\mathbb E}} _\\delta \\big[ (\\omega_1 - m_\\delta)^2 \n\\, g\\big(-\\beta s_0 (\\omega_1 - m_\\delta)^-\\big) \\big]}\n{{\\ensuremath{\\mathbb E}} _\\delta \\big[ e^{s_0 \\beta  (\\omega_1 - m_\\delta)^+ } \\big]},\n\\end{split}\n\\end{equation}\nand note that $0 < c^-_{\\beta, \\delta} \\le c^+_{\\beta, \\delta} < \\infty$ for all $\\delta \n\\in (-\\epsilon_0, \\epsilon_0)$ and $\\beta \\in [0,\\epsilon_0)$. We have already observed \nthat $f_n(\\omega, \\omega_n) = \\e^{[a,b]}_{N, \\omega, \\beta, h} \\big[ \\sigma_n \\big]$\nby \\eqref{eq:fnoy}, and so from \\eqref{eq:aatt}-\\eqref{eq:aatt+} we obtain the following \nestimate: for every $\\beta \\in [0,\\epsilon_0)$, $h\\in\\mathbb{R}$, $\\delta \\in (-\\epsilon_0, \\epsilon_0)$ \nand $0 < a < b <\\infty$\n\\begin{equation} \n\\label{eq:partdelta}\nc^-_{\\beta,\\delta} \\, \\beta \\, {\\ensuremath{\\mathbb E}} _\\delta \\big[ \\e^{[a,b]}_{N, \\omega, \\beta, h}\n\\big[ \\overline\\sigma_N \\big] \\big] \n\\leq \\frac{\\partial}{\\partial\\delta} \\textsc{f}^{[a,b]}_N(\\beta, h; \\delta)\n\\leq c^+_{\\beta,\\delta} \\, \\beta \\, {\\ensuremath{\\mathbb E}} _\\delta \\big[ \\e^{[a,b]}_{N, \\omega, \\beta, h}\n\\big[ \\overline\\sigma_N \\big] \\big].\n\\end{equation}\nNote the analogy with the expression in \\eqref{eq:parth} for $\\frac{\\partial}{\\partial h}\n\\textsc{f}^{[a,b]}_N(\\beta, h; \\delta)$.\n\n\\medskip\\noindent\n\\textbf{7.}\nWe are close to the final conclusion. Since by \\eqref{eq:pNab} we have $a \\leq \n\\e^{[a,b]}_{N, \\omega, \\beta, h}\\big[ \\overline\\sigma_N \\big] \\leq b$, it follows from \n\\eqref{eq:partdelta} that, for every $\\delta \\in [0,\\epsilon_0)$\n\\begin{equation} \n\\label{eq:cru0}\nC^-_{\\beta,\\delta} \\, \\beta \\, a \\, \\delta \n\\leq \\textsc{f}^{[a,b]}(\\beta, h; \\delta) - \\textsc{f}^{[a,b]}(\\beta, h; 0) \n\\leq C^+_{\\beta,\\delta} \\, \\beta \\, b \\, \\delta,\n\\end{equation}\nwhere we set\n\\begin{equation}\n\\label{eq:C+-}\nC^\\pm_{\\beta,\\delta} := \n\\begin{cases}\n\\displaystyle \\frac{1}{\\delta} \\int_0^\\delta \nc^\\pm_{\\beta,\\delta'} \\, \\mathrm{d} \\delta' \n& \\text{if } \\delta \\in (-\\epsilon_0, \\epsilon_0) \\setminus\\{0\\}, \\\\\nc^\\pm_{\\beta,0} \n& \\text{if } \\delta = 0.\n\\end{cases}\n\\end{equation}\nAnalogously to \\eqref{eq:cru0}, from \\eqref{eq:parth} we obtain, for every $\\xi \\geq 0$,\n\\begin{equation} \n\\label{eq:cru00}\na \\, \\xi \\leq \\textsc{f}^{[a,b]}(\\beta, h + \\xi ; 0) -\n\\textsc{f}^{[a,b]}(\\beta, h ; 0) \\leq b \\, \\xi.\n\\end{equation}\nChoosing $\\xi = C^+_{\\beta,\\delta} \\frac{b}{a} \\beta \\delta$ and $\\xi = C^-_{\\beta,\\delta} \n\\frac{a}{b} \\beta \\delta$, respectively, and combining \\eqref{eq:cru0}-\\eqref{eq:cru00}, we \nfinally get the following relation, which holds for all $\\beta, \\delta \\in [0,\\epsilon_0)$, $h\\in\\mathbb{R}$ \nand $0 < a < b < \\infty$:\n\\begin{equation} \n\\label{eq:cru00+}\n\\textsc{f}^{[a,b]} \\big( \\beta, h + C^-_{\\beta,\\delta} \\tfrac{a}{b} \\beta \\delta ; 0 \\big) \n\\leq \\textsc{f}^{[a,b]}(\\beta, h| \\delta) \n\\leq \\textsc{f}^{[a,b]} \\big( \\beta, h + C^+_{\\beta,\\delta} \\tfrac{b}{a} \\beta \\delta ; 0 \\big).\n\\end{equation}\nNext, fix any $x > 0$ and $\\eta > 0$. If $a_n \\uparrow x$ and $b_n \\downarrow x$, then\n$a_n\/b_n \\geq 1-\\eta$ and $b_n\/a_n \\leq 1+\\eta$ for large $n$. Since $h \\mapsto \\textsc{f}^{[a,b]}\n(\\beta, h;\\delta)$ is non-decreasing, by \\eqref{eq:parth} and \\eqref{eq:s0}, \nfor $n$ large enough we have\n\\begin{equation}\n\\textsc{f}^{[a_n,b_n]} \\big( \\beta, h + C^-_{\\beta,\\delta} (1-\\eta) \\beta \\delta ; 0 \\big) \n\\leq \\textsc{f}^{[a_n,b_n]}(\\beta, h; \\delta) \n\\leq \\textsc{f}^{[a_n,b_n]} \\big( \\beta, h + C^+_{\\beta,\\delta} (1+\\eta) \\beta \\delta ; 0 \\big).\n\\end{equation}\nRecalling \\eqref{eq:fx} and \\eqref{eq:fbetah}, we can let $n\\to\\infty$ to get that, for every \n$x > 0$,\n\\begin{equation}\n\\textsc{f}^{\\{x\\}} \\big( \\beta, h + C^-_{\\beta,\\delta} (1-\\eta) \\beta \\delta ; 0 \\big) \n\\leq \\textsc{f}^{\\{x\\}}(\\beta, h; \\delta) \n\\leq \\textsc{f}^{\\{x\\}} \\big( \\beta, h + C^+_{\\beta,\\delta} (1+\\eta) \\beta \\delta ; 0 \\big).\n\\end{equation}\nThis relation also holds for $x=0$ because $\\textsc{f}^{\\{0\\}}(\\beta, h; \\delta)$ is a constant, as we \nshowed in \\eqref{eq:f0}. Taking the supremum over $x \\in [0, s_0]$, we have\nshown that, for all $\\beta, \\delta \\in [0,\\epsilon_0)$ and $h\\in\\mathbb{R}$,\n\\begin{equation}\n\\textsc{f} \\big( \\beta, h + C^-_{\\beta,\\delta} (1-\\eta) \\beta \\delta ; 0 \\big) \n\\leq \\textsc{f}(\\beta, h; \\delta) \n\\leq \\textsc{f} \\big( \\beta, h + C^+_{\\beta,\\delta} (1+\\eta) \\beta \\delta ; 0 \\big).\n\\end{equation}\nSince $h \\mapsto \\textsc{f}^{[a,b]}(\\beta, h;\\delta)$ is convex and finite, and hence continuous, we \ncan let $\\eta \\downarrow 0$ to obtain \\eqref{eq:compdeltah} for $\\delta \\in [0,\\epsilon_0)$.\n\n\\medskip\\noindent\n\\textbf{8.}\nThe case $\\delta \\in (-\\epsilon_0, 0]$ is analogous. The inequality in \\eqref{eq:cru0} is replaced by\n\\begin{gather} \n\\label{eq:cru0neg}\nC^-_{\\beta,\\delta} \\, \\beta \\, a \\, (-\\delta) \n\\leq \\textsc{f}^{[a,b]}(\\beta, h; 0) \\,-\\, \\textsc{f}^{[a,b]}(\\beta, h; \\delta)\n\\leq C^+_{\\beta,\\delta} \\, \\beta \\, b \\, (-\\delta),\n\\end{gather}\nwhile \\eqref{eq:cru00} for $\\xi \\leq 0$ becomes\n\\begin{equation} \n\\label{eq:cru00neg}\na \\, (-\\xi)  \\leq\n\\textsc{f}^{[a,b]}(\\beta, h ; 0) - \\textsc{f}^{[a,b]}(\\beta, h + \\xi ; 0) \\leq b \\, (-\\xi).\t\n\\end{equation}\nChoosing $\\xi = C^+_{\\beta,\\delta} \\frac{b}{a} \\beta \\delta$ and $\\xi = C^-_{\\beta,\\delta} \n\\frac{a}{b} \\beta \\delta$, respectively, we get\n\\begin{equation} \n\\label{eq:cru00+neg}\n\\textsc{f}^{[a,b]} \\big( \\beta, h + C^+_{\\beta,\\delta} \\tfrac{b}{a} \\beta \\delta ; 0 \\big) \n\\leq \\textsc{f}^{[a,b]}(\\beta, h; \\delta) \n\\leq \\textsc{f}^{[a,b]} \\big( \\beta, h + C^-_{\\beta,\\delta} \\tfrac{a}{b} \\beta \\delta ; 0 \\big).\n\\end{equation}\nIt remains to let $a \\uparrow x$, $b \\downarrow x$, followed by taking the supremum\nover $x \\in [0,s_0]$\n\n\\medskip\\noindent\n\\textbf{9.}\nFinally, by \\eqref{eq:C+-}, we have $0 < C^-_{\\beta,\\delta} \\le C^+_{\\beta,\\delta} \n< \\infty$ for all $\\beta \\in [0,\\epsilon_0)$ and $\\delta \\in (-\\epsilon_0, \\epsilon_0)$. By \ndominated convergence, $(\\beta,\\delta) \\mapsto c^\\pm_{\\beta,\\delta}$ are continuous \non $[0,\\epsilon_0) \\times (-\\epsilon_0,\\epsilon_0)$, and hence also $(\\beta,\\delta) \\mapsto \nC^\\pm_{\\beta,\\delta}$ is continuous. \nSince $C^\\pm_{0,0} = \\bbvar(\\omega_1) = 1$,\nthe proof is complete.\n\\qed\n\n\\smallskip\n\n\\section{Smoothing with respect to a shift: proof of Theorem~\\ref{th:smoshift}}\n\\label{sec:smoothshift}\n\nEquations \\eqref{eq:ZNgen} and \\eqref{eq:s0} imply that $h \\mapsto \\textsc{f}(\\beta,h;\\delta)$ \nis non-decreasing. Since $\\textsc{f}(\\beta,h;\\delta) \\geq 0$ under Assumption~\\ref{ass:model},\nby \\eqref{eq:polylb}, if $\\textsc{f}(\\bar\\beta, \\bar h; 0) = 0$, then $\\textsc{f}(\\bar\\beta, \\bar h + t; 0) = 0$ \nfor all $t \\leq 0$, and \\eqref{eq:compdeltah2} is trivially satisfied. Henceforth we assume\n$t > 0$.\n\nRecalling the statement of Theorem~\\ref{th:compdeltah}, we set $F_\\beta(\\delta) \n:= C^-_{\\beta,\\delta} \\, \\delta$. This is a continuous and strictly increasing function \nof $\\delta$, with $F_\\beta(0) = 0$, and hence it maps the open interval $(0,\\epsilon_0)$\ninto $(0,\\epsilon'_0)$, for some $\\epsilon'_0 > 0$. Applying the first inequality in \n\\eqref{eq:compdeltah} for $t \\in (0, \\bar\\beta\\epsilon'_0)$, we can write\n\\begin{equation}\n\\textsc{f}(\\bar\\beta, \\bar h + t; 0) \n= \\textsc{f}\\big(\\bar\\beta, \\bar h + \\bar\\beta F_{\\bar\\beta}\n(F_{\\bar\\beta}^{-1}(\\tfrac{t}{\\bar\\beta})); 0\\big) \n\\leq \\textsc{f}\\big(\\bar\\beta,\\bar h; F_{\\bar \\beta}^{-1}(\\tfrac{t}{\\bar\\beta}) \\big).\n\\end{equation}\nApplying \\eqref{eq:smodelta}, we obtain\n\\begin{equation}\n\\textsc{f}(\\bar\\beta, \\bar h + t; 0) \n\\leq \\frac{\\gamma}{2 \\bar\\beta^2} \\, A_{\\bar\\beta,\\frac{t}{\\bar\\beta}} \\, t^2,\n\\end{equation}\nwhere\n\\begin{equation}\nA_{\\beta,\\delta} := B_{F_{\\beta}^{-1}(\\delta)} \\, \n\\bigg(\\frac{F_{\\beta}^{-1}(\\delta)}{\\delta} \\bigg)^2.\n\\end{equation}\nIt follows from \\eqref{eq:propCpm} that\n$\\lim_{(\\beta,\\delta) \\to (0,0)} (F_{\\beta}^{-1}(\\delta)\/\\delta) = 1$.\nSince $\\lim_{\\delta\\to 0} B_\\delta = 1$, we obtain $\\lim_{(\\beta,\\delta) \\to (0,0)} A_{\\beta,\\delta} \n= 1$.\n\\qed\n\n\n\n\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s1}\n\nWe consider the problem of characterizing positive definite kernels on a product of spheres.\\ The focus will be on continuous and isotropic kernels, keeping the setting originally adopted by I. J. Schoenberg in his influential paper published in 1942 (\\cite{schoen}).\n\nAs usual, let $S^{m}$ denote the unit sphere in the $(m+1)$-dimensional space $\\mathbb{R}^{m+1}$ and $S^\\infty$ the unit sphere in $\\mathbb{R}^\\infty$, the usual real $\\ell^2$ space.\\ Throughout the paper, we will be dealing with real, continuous and isotropic kernels on the product $S^{m}\\times S^{M}$, $m,M=1,2,\\ldots, \\infty$.\\ When speaking of continuity, we will assume each sphere is endowed with its usual geodesic distance.\\ The {\\em isotropy} (zonality) of a kernel $K$ on $S^{m} \\times S^{M}$ refers to the fact that\n$$\nK((x,z),(y,w))=f(x\\cdot y,z \\cdot w), \\quad x,y \\in S^{m},\\quad z,w \\in S^{M},\n$$\nfor some real function $f$ on $[-1,1]^2$, where $\\cdot$ stands for the inner product of both $\\mathbb{R}^{m+1}$ and $\\mathbb{R}^{M+1}$.\\ In particular, the concept introduced above demands the usual notion of isotropy on each sphere involved.\\ In many places in the paper, we will refer to $f$ as the {\\em isotropic part} of $K$.\n\nRecall that if $X$ is a nonempty set, a kernel $K$ is {\\em positive definite} on $X$ if\n$$\n\\sum_{\\mu,\\nu=1}^n c_\\mu c_\\nu K(x_\\mu, x_\\nu) \\geq 0,\n$$\nfor $n\\geq 1$, distinct points on $X$, and reals scalars $c_1, c_2, \\ldots, c_n$.\\ In other words, for any $n\\geq 1$ and any distinct points $x_1, x_2, \\ldots, x_n$ on $X$, the $n\\times n$ matrix with entries $K(x_\\mu,x_\\nu)$ is nonnegative definite.\\ In this paper, we will present a characterization for the positive definiteness of a continuous and isotropic kernel on $X=S^m \\times S^M$ based upon Fourier expansions.\n\nIsotropy and positive definiteness for kernels on a single sphere were first considered by I. J. Schoenberg in \\cite{schoen}.\\ He showed that a continuous and isotropic kernel $K$ on $S^{m}$ is positive definite if and only if $K(x,y)=g(x\\cdot y)$, $x,y \\in S^m$, in which the isotropic part $g$ of $K$ has a series representation in the form\n$$\ng(t)=\\sum_{k=0}^{\\infty} a_k^m P_k^{m}(t),\\quad t \\in [-1,1],\n$$\nin which $a_k^m \\geq 0$, $k \\in \\mathbb{Z}_+$ and $\\sum_{k=0}^{\\infty}a_k P_k^{m}(1) <\\infty$.\\ The symbol $P_k^{m}$ stands for the usual\nGegenbauer polynomial of degree $k$ associated with the rational $(m-1)\/2$, as discussed in \\cite{szego}.\\ This Schoenberg's outstanding result is far-reaching and has ramifications in distance geometry, statistics, spherical designs, approximation theory, etc.\\ In approximation theory, positive definite kernels are used in interpolation of scattered data over the sphere.\\ The importance of this problem in many areas of science and engineering is reflected in the literature, where different methods to solve such problem have been proposed.\\ Given $n$ distinct data points $x_1, x_2, \\ldots, x_n$ on $S^{m}$ and a target\nfunction $h:S^{m} \\to \\mathbb{R}$,\nthe interpolation problem itself requires the finding of a continuous function $s:S^{m} \\to \\mathbb{R}$ of the form\n$$\ns(x)=\\sum_{j=1}^n \\lambda_j g(x \\cdot x_j),\\quad x \\in S^{m},\\quad \\lambda_1, \\lambda_2, \\ldots, \\lambda_n \\in \\mathbb{R},$$\nso that $s(x_i)=h(x_i)$, $i=1,2,\\ldots,n$.\nIf we choose the prescribed function $g$ to be the isotropic part of a convenient positive definite kernel $K$, then the interpolation problem has a unique solution for any $n$ and any $n$ data points.\n\nAn outline of the paper is as follows.\\ In Section 2, we present several technical results that culminate with a characterization for the continuous, isotropic and positive definite kernels on $S^m \\times S^M$, $m,M <\\infty$.\\ In Section 3, we complete this circle of ideas, by reaching a similar characterization in the cases in which at least one of the spheres involved is the real Hilbert sphere $S^\\infty$.\\ Finally, Section 4 contains a few relevant remarks along with the description of future lines of investigation on the subject.\n\n\\section{Positive definiteness on $S^{m}\\times S^{M}$, $m,M < \\infty$.}\\label{s2}\n\nThis section contains all the technical material needed in the proof of the extension of Schoenberg's theorem to a product of spheres $S^m \\times S^M$, in the case when both $m$ and $M$ are finite.\\ The proof will require a series of well-known results involving Gegenbauer polynomials and also a few facts from the analysis on the sphere.\\ We suggest the classical reference \\cite{szego} for the first topic and \\cite{atkinson,dai,groemer,muller} for the other.\n\nThe orthogonality relation for Gegenbauer polynomials reads as follows (\\cite[p.10]{dai}):\n$$\n\\int_{-1}^1 P_n^{m}(t) P_k^{m}(t) (1-t^2)^{(m-2)\/2}dt=\\frac{\\tau_{m+1}}{\\tau_{m}}\\frac{m-1}{2n+m-1}P_n^{m}(1)\\delta_{n,k},\n$$\nin which $\\tau_{m+1}$ is the surface area of $S^{m}$, that is,\n$$\n\\tau_{m+1}:=\\frac{2\\pi^{(m+1)\/2}}{\\Gamma((m+1)\/2)}.\n$$\n\nSince Schoenberg's characterization for positive definiteness on $S^{m}$ is based upon Fourier expansions with respect to the orthogonal family $\\{P_n^{m}:n=0,1,\\ldots\\}$, it is quite natural to expect that a similar characterization for positive definiteness on $S^{m}\\times S^{M}$ will require expansions with respect to the tensor family\n$$\\{(t,s) \\in [-1,1]^2 \\to P_k^m(t)P_l^M(s): k,l=0,1,\\ldots\\}.$$\n\nThe first important fact to be noticed about the functions in the family above is this.\n\n\\begin{lem} \\label{gegpd}\nIf $k,l \\in \\mathbb{Z}_+$, then $(t,s) \\in [-1,1]^2 \\to P_k^m(t)P_l^M(s)$ is the isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$.\n\\end{lem}\n\\pf This follows from the definition of positive definiteness, Schoenberg's original characterization for positive definite kernels and the Schur product theorem (\\cite[p.458]{horn}).\\ The later asserts that the entry-wise product of two nonnegative definite matrices of same order is a nonnegative definite matrix itself.\\eop\n\nThe tensor family is orthogonal on $[-1,1]^2$ with respect to the weight function\n$$\nw_{m,M}(t,s)=(1-t^2)^{(m-2)\/2}(1-s^2)^{(M-2)\/2},\\quad t,s \\in [-1,1].\n$$\nThe $(k,l)$-Fourier coefficient of a function $f: [-1,1]^2 \\to \\mathbb{R}$ from $L^1([-1,1]^2, w_{m,M})$\nis\n$$\n\\hat{f}_{k,l}:=\\frac{1}{\\tau_k^m \\tau_l^M}\\int_{[-1,1]^2} f(t,s)P_k^{m}(t)P_l^{M}(s)dw_{m,M}(t,s), \\quad k,l \\in \\mathbb{Z}_+,\n$$in which\n$$\n\\tau_k^m:=\\frac{\\tau_{m+1}}{\\tau_{m}}\\frac{m-1}{2k+m-1}P_k^{m}(1),\\quad k\\in \\mathbb{Z}_+.\n$$\nThe next lemma describes an alternative way for computing these Fourier coefficients.\\ The symbol $\\sigma_m$ will denote the surface measure on $S^m$.\n\n\\begin{lem} \\label{pdmultiple} If $f$ belongs to $L^1([-1,1]^2,w_{m,M})$, then the Fourier coefficient $\\hat{f}_{k,l}$ is a positive constant multiple of\n$$\n\\int_{S^{m}\\times S^{M}} \\left[\\int_{S^{m}\\times S^{M}} f(x\\cdot y,z \\cdot w)P_k^{m}(x\\cdot y) \\times P_l^{M}(z\\cdot w)d\\sigma_{m}(y)d\\sigma_{M}(w)\\right]d\\sigma_{m}(x)d\\sigma_{M}(z).\n$$\n\\end{lem}\n\\pf If $m,M \\geq 2$, it suffices to employ the Funk-Hecke formula (\\cite[p.11]{dai}) in the expression defining the Fourier coefficient.\\\nThe Funk-Hecke formula states that\n$$\n\\int_{S^{M}}g(z\\cdot w)P_l^{M}(z\\cdot w)d\\sigma_{M}(w)=\\tau_{m-1}\\int_{-1}^1 g(s)P_l^{M}(s)(1-s^2)^{(M-2)\/2}ds,\\quad z \\in S^{M},\n$$\nwhenever $l\\in \\mathbb{Z}_+$ and $g \\in L^2([-1,1],w_M)$.\\ Using the formula with\n$$\ng(s)=\\int_{-1}^1 f(t,s)P_k^{m}(t)(1-t^2)^{(m-2)\/2}dt, \\quad s \\in [-1,1],\n$$\nit is promptly seen that $\\hat{f}_{k,l}$ is a positive multiple of\n$$\n\\int_{S^{M}} \\int_{-1}^1 f(t,z \\cdot w)P_k^{m}(t)(1-t^2)^{(m-2)\/2}dt \\ P_l^{M}(z\\cdot w) d\\sigma_{M}(w).\n$$\nApplying a similar argument in the internal integral reveals that $\\hat{f}_{k,l}$ is a positive multiple of\n$$\n\\int_{S^{M}} \\int_{S^{m}} f(x\\cdot y,z \\cdot w)P_k^{m}(x\\cdot y) P_l^{M}(z\\cdot w)d\\sigma_{m}(y)d\\sigma_{M}(w).\n$$\nIntegration with respect to the remaining variables concludes the proof.\\ In the cases in which either $m=1$ or $M=1$, the arguments demand the replacement of the Funk-Hecke formula with direct computation.\\eop\n\n\\begin{lem} \\label{pdtimes} If $f$ is the continuous and isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$, then\n$$\n\\int_{S^{m}\\times S^{M}}\\left[\\int_{S^{m}\\times S^{M}}f(x\\cdot y,z\\cdot w)d\\sigma_{m}(x)d\\sigma_{M}(z)\\right]d\\sigma_{m}(y)d\\sigma_{M}(w) \\geq 0.\n$$\n\\end{lem}\n\\pf It suffices to write the double integral $I$ in the statement of the theorem as a double limit of Riemann sums.\\ Indeed, we can select a sequence $\\{\\mathcal{P}_n: n=0,1,\\ldots\\}$ of partitions of $S^{m}\\times S^{M}$\nin such a way that $\\mathcal{P}_n=\\{Q_1^n,Q_2^n,\\ldots, Q_{\\alpha(n)}^n\\}$, the sequence $\\{\\alpha(n)\\}$ increases to $\\infty$ and the sequences of diameters $\\{\\mbox{diam}(Q_j^n)\\}$ satisfy\n$\\lim_{n\\to \\infty}\\mbox{diam}(Q_j^n)=0$.\\ Picking points $(x_j^n,z_j^n) \\in Q_j^n$, we can write\n$$\nI=\\lim_{N \\to \\infty} \\sum_{J=1}^{\\alpha(N)}\\left[ \\int_{S^{m}\\times S^{M}}f(x \\cdot x_J^N,z \\cdot z_J^N)d\\sigma_{m}(x)d\\sigma_{M}(z)\\right]\\mbox{vol}(Q_J^N).\n$$\nRepeating the procedure with the resulting integral leads to\n\\begin{eqnarray*}\nI & = & \\lim_{N \\to \\infty} \\sum_{J=1}^{\\alpha(N)}\\left[\\lim_{n \\to \\infty}\\sum_{j=1}^{\\alpha(n)}f(x_j^n \\cdot x_J^N,z_j^n \\cdot z_J^N) \\mbox{vol}(Q_j^n)\\right] \\mbox{vol}(Q_J^N)\\\\\n  & = & \\lim_{N \\to \\infty} \\lim_{n \\to \\infty}\\sum_{J=1}^{\\alpha(N)}\\sum_{j=1}^{\\alpha(n)}\\mbox{vol}(Q_j^n)\\mbox{vol}(Q_J^N)f(x_j^n \\cdot x_J^N,z_j^n \\cdot z_J^N).\n  \\end{eqnarray*}\nSince the double limit above exists, it follows that\n$$\nI=\\lim_{n \\to \\infty}\\sum_{j,J=1}^{\\alpha(n)}\\mbox{vol}(Q_j^n)\\mbox{vol}(Q_J^n)f(x_j^n \\cdot x_J^n,z_j^n \\cdot z_J^n).\n$$\nIf $f$ is the isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$, then each double sum in the last expression above is clearly nonnegative.\\ In particular, the limit itself is nonnegative as well.\\eop\n\nWe now combine the three lemmas above in order to obtain the following result.\n\n\\begin{lem} \\label{coeffpos} If $f$ is the continuous and isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$ then\n$$\n\\hat{f}_{k,l} \\geq 0, \\quad k,l \\in \\mathbb{Z}_+.\n$$\n\\end{lem}\n\\pf Let us fix $k$ and $l$.\\ Lemma \\ref{gegpd} and the Schur product theorem guarantees that the function\n$$\n(t,s) \\in [-1,1]^2 \\to f(t,s)P_k^{m}(t)P_l^{M}(s)\n$$\nis the continuous and isotropic part of a positive definite kernel on $S^{m} \\times S^{M}$.\\ Taking into account this information and that one provided by Lemma \\ref{pdmultiple}, an application of Lemma \\ref{pdtimes} leads to the inequality in the statement of the lemma.\\eop\n\nNext, we recall one of the several generating formulas for the Gegenbauer polynomials, the Poisson identity (\\cite[p.419]{dai}).\n\n\\begin{lem} \\label{poisson} If $r \\in [0,1)$, then\n$$\n\\frac{1-r^2}{(1-2tr+r^2)^{(m+1)\/2}}=\\sum_{k=0}^{\\infty}\\frac{2k+m-1}{m-1}P_k^{m}(t)r^k, \\quad t\\in [-1,1].\n$$\nIf $r_0 \\in [0,1)$ is fixed, then the convergence of the series is absolute and uniform for $(r,t) \\in [0,r_0]\\times [-1,1]$.\n\\end{lem}\n\nWe are about ready to prove the following auxiliary result.\n\n\\begin{lem} \\label{spoisson} Let $f$ be the continuous and isotropic part of a kernel on $S^{m}\\times S^{M}$.\\ If $r,\\rho \\in [0,1)$, then the double series\n$$\n\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(1)P_{l}^{M}(1)r^k\\rho^l\n$$\nconverges.\\ As a matter of fact, there exists a positive constant $C$, depending upon $f$ only, so that\n$$\n\\left|\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(1)P_{l}^{M}(1)r^k\\rho^l\\right|\\leq C, \\quad r,\\rho \\in [0,1).\n$$\n\\end{lem}\n\\pf Let $a_{k,l}^{r,\\rho}$ denote the general term of the series in the statement of the lemma.\\ It is promptly seen that\n$$\na_{k,l}^{r,\\rho}=C_1\\int_{-1}^1 \\int_{-1}^1 f(t,s)\\frac{2k+m-1}{m-1}P_k^{m}(t)r^k\\frac{2l+M-1}{M-1}P_l^{M}(t)\\rho^l dw_{m,M}(t,s),\n$$\nin which\n$$\nC_1=\\frac{\\tau_{m}\\tau_{M}}{\\tau_{m+1}\\tau_{M+1}}.\n$$\nOn the other hand, Lemma \\ref{poisson} implies that\n$$\n\\sum_{k=0}^{\\infty}\\sum_{l=0}^{\\infty}a_{k,l}^{r,\\rho}=C_1 \\int_{-1}^1 \\int_{-1}^1f(t,s)\\frac{1-r^2}{(1-2rt+r^2)^{(m+1)\/2}}\\frac{1-\\rho^2}{(1-2\\rho s+s^2)^{(M+1)\/2}}dw_{m,M}(t,s),\n$$\nwhenever $r,\\rho \\in [0,1)$.\\ Thus, since $f$ is continuous and the left hand side of the Poisson identity is positive, the proof of the first half of the lemma reduces itself to proving that the double integral\n$$\n\\int_{-1}^1\\int_{-1}^1 \\frac{1-r^2}{(1-2rt+r^2)^{(m+1)\/2}}\\frac{1-\\rho^2}{(1-2\\rho s+s^2)^{(M+1)\/2}}(1-t^2)^{(m-2)\/2}(1-s^2)^{(M-2)\/2}dtds\n$$\nis finite.\\ But, this follows from the well-known property of the Poisson kernels (\\cite[p.47]{muller})\n$$\n\\int_{-1}^1 \\frac{1-r^2}{(1-2rt+r^2)^{(m+1)\/2}}(1-t^2)^{(m-2)\/2}dt=\\frac{\\tau_{m+1}}{\\tau_{m}}.\n$$\nIn particular, $C:=\\max\\{|f(s,t)|: -1\\leq t,s\\leq 1\\}$ fits into what is needed in the second statement of the lemma.\\eop\n\n\\begin{lem}\\label{pd1} If $f$ is the continuous and isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$, then the double series\n$$\n\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(1)P_{l}^{M}(1)\n$$\nconverges.\n\\end{lem}\n\\pf Let $f$ be the continuous and isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$.\\ Due to Lemma \\ref{coeffpos}, we know already that all the Fourier coefficients $\\hat{f}_{k,l}$ are nonnegative.\\ In particular, the double sequence $\\{s_{p,q}\\}$ of partial sums of the double series in the statement of the current lemma is monotonically increasing, that is, $s_{p,q} \\leq s_{\\mu,\\nu}$ when $p\\leq \\mu$ and $q \\leq \\nu$.\\ On the other, the previous lemma produces the inequality\n$$\n\\sum_{k=0}^{p}\\sum_{l=0}^{q}\\hat{f}_{k,l}P_{k}^{m}(1)P_{l}^{M}(1)r^k\\rho^l \\leq C, \\quad p,q\\in\\mathbb{Z}_+, \\quad r,\\rho \\in [0,1),\n$$\nfor some $C>0$.\\ By taking a double limit when $r,\\rho \\to 1^{+}$, we deduce that the double sequence of partial sums $\\{s_{p,q}\\}$ is bounded above.\\ A classical result from the theory\nof double sequences (\\cite[p.373]{limaye}) implies that $\\{s_{p,q}\\}$ converges, that is, the series in the statement of the lemma converges.\\eop\n\nThe Weierstrass M-test can be adapted to hold for double series of functions.\\ Combining it with Lemma \\ref{pd1} leads to the proposition below.\n\n\\begin{prop}\\label{conve} If $f$ is the continuous and isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$, then\n$$\n\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(t)P_{l}^{M}(s)\n$$\nconverges absolutely and uniformly for $(t,s) \\in [-1,1]^2$.\n\\end{prop}\n\nThe main result in this section is as follows.\n\n\\begin{thm} \\label{mainPD} Let $K$ be a continuous and isotropic kernel on $S^{m}\\times S^{M}$.\\ It is positive definite on $S^{m}\\times S^{M}$ if and only if its isotropic part $f$ has a representation in the form\n$$\nf(t,s)=\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(t)P_{l}^{M}(s), \\quad t,s \\in [-1,1],\n$$\nin which $\\hat{f}_{k,l} \\geq 0$, $k,l \\in \\mathbb{Z}_+$ and $\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(1)P_{l}^{M}(1)<\\infty$.\n\\end{thm}\n\\pf If the isotropic part $f$ of $K$ has the representation announced in the theorem, the series appearing there is uniformly and absolutely convergent.\\ In particular, Lemma \\ref{gegpd} implies that $f$ is\na pointwise double limit of functions which are isotropic parts of positive definite kernels on $S^{m} \\times S^{M}$.\\ Consequently, $K$ itself is positive definite on $S^{m}\\times S^{M}$.\\ Conversely, assume $K$ is positive definite on $S^{m}\\times S^{M}$ and write $f$ to denote its isotropic part.\\ Lemma \\ref{pd1} and Proposition \\ref{conve} supply a function $g$ so that\n$$\ng(s,t)=\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(t)P_{l}^{M}(s), \\quad t,s \\in [-1,1],\n$$\nwith uniform convergence in $[-1,1]^2$.\\ In particular, $g$ is continuous in $[-1,1]^2$.\\ On the other hand, the same uniform convergence and the\northogonality relation mentioned at the beginning of the section imply that\n$$\n\\hat{f}_{k,l} -\\hat{g}_{k,l}=0, \\quad k,l \\in \\mathbb{Z}_+.\n$$\nConsequently, $f=g$.\\eop\n\n\\section{Positive definiteness on $S^\\infty \\times S^M$}\n\nIn this section, we extend Theorem \\ref{mainPD} to the cases in which either $m=\\infty$ or $M=\\infty$.\\ Clearly, it suffices to consider the cases $m=\\infty$, $M<\\infty$ and $m=M=\\infty$ only.\n\nEvery sphere $S^{m}$ can be isometrically embedded in $S^{\\infty}$.\\ In particular, a positive definite kernel on $S^\\infty \\times S^{M}$ is positive definite on $S^{m}\\times S^{M}$, for $m=1,2,\\ldots$.\\ Likewise, if $f$ is the isotropic part of a positive definite kernel on $S^\\infty \\times S^{M}$, then it is the isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$, for $m=1,2, \\ldots$.\\ In addition, if $f$ is continuous, then for\nevery $m\\geq 1$, we have a representation for $f$ in the form\n$$\nf(t,s)=\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}^{m,M} P_{k}^{m}(t) P_{l}^{M}(s), \\quad t,s \\in [-1,1],\n$$\nin which\n$$\n\\hat{f}_{k,l}^{m,M} =\\frac{1}{\\tau_{k}^{m} \\tau_{l}^{M}}  \\int_{[-1,1]^{2}} f(x,y) P_{k}^{m}(t)P_{l}^{M}(s) dw_{m,M}(t,s)\\geq 0,\\quad k,l \\in \\mathbb{Z}_+.\n$$\nand $\\sum_{k=0}^{\\infty}\\sum_{l=0}^\\infty \\hat{f}_{k,l}^{m,M}P_{k}^{m-1}(1) P_{l}^{M-1}(1)<\\infty$.\\ Below, we will prefer to normalized the above expressions by writing\n$$\nR_k^{m}=\\frac{P_k^{m}}{P_k^{m}(1)}, \\quad k\\in \\mathbb{Z}_+,\n$$\nand\n$$\nf(t,s)=\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{m,M} R_{k}^{m}(t) R_{l}^{M}(s), \\quad t,s \\in [-1,1],\n$$\nwhere now\n$$\n\\check{f}_{k,l}^{m,M}=P_k^{m}(1)P_l^{M}(1)\\hat{f}_{k,l}^{m,M}, \\quad k,l \\in \\mathbb{Z}_+.\n$$\nBefore we proceed, it is convenient to mention that the Fourier coefficients introduced above are well-defined as long as $f$ belongs to $L^1([-1,1],w_{m,M})$.\n\n\\begin{lem}\\label{pre} Let $f$ belong to $L^1([-1,1],w_{m,M})$.\\ If $k$ and $l$ are fixed nonnegative integers, then the sequence $\\{\\check{f}_{k,l}^{2m,M}:m=1,2,\\ldots\\}$ is convergent.\n\\end{lem}\n\\pf Using the following recurrence relation for Gegenbauer polynomials (\\cite[p. {84}]{szego})\n$$ (1-t^{2})P^{m+2}_{k}(t) = \\frac{(k+m-1)(k+m)}{(m-1)(2k+m+1)} P_{k}^{m}(t) - \\frac{(k+1)(k+2)}{(m-1)(2k+m+1)}P_{k+2}^{m}(t),\n$$\nit is easy to deduce that\n$$\n\\check{f}_{k,l}^{m+2,M} = \\frac{(k+m-1)(k+m)}{m(2k+m-1)} \\check{f}_{k,l}^{m,M} - \\frac{(k+1)(k+2)}{m(2k+m+3)} \\check{f}_{k+2,l}^{m,M},\\quad m\\geq 1.\n$$\nConsequently,\n\\begin{align*}\n|\\check{f}_{k,l}^{m+2,M} -  \\check{f}_{k,l}^{m,M} | & = \\left |\\frac{k(k-1)}{m(2k+m-1)}\\check{f}_{k,l}^{m,M} - \\frac{(k+1)(k+2)}{m(2k+m+3)} \\check{f}_{k+2,l}^{m,M} \\right | \\\\\n& \\leq \\left[ \\frac{k(k-1)}{m(2k+m-1)} + \\frac{(k+1)(k+2)}{m(2k+m+3)}\\right]f(1,1),\\quad m\\geq 1.\n\\end{align*}\nAs an obvious consequence, $\\{\\check{f}_{k,l}^{2m,M}\\}$ is a Cauchy sequence of real numbers, therefore, convergent. \\eop\n\n\\begin{lem}\\label{prepar}\nIf $f$ is the continuous and isotropic part of a positive definite kernel on $S^{\\infty} \\times S^{M}$, then the double series\n$$\n\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{m,M}\\, t^k R_{l}^{M}(s)\n$$\nconverges for $(t,s) \\in (-1,1)^2$, uniformly in $m$.\n\\end{lem}\n\\pf In order to see that the series converges in $(-1,1)^2$, for every $m$, it suffices to show that $\n\\sum_{k=0}^{\\infty}\\sum_{l=0}^\\infty \\check{f}_{k,l}^{m,M} |t|^k$ converges.\\ Recalling Tonelli's theorem for convergence of double series (\\cite[p.384]{limaye}),\nthat will follow as long as\n$\\sum_{l=0}^\\infty \\check{f}_{k,l}^{m,M}$ converges for all $k$ and the iterated series $\n\\sum_{k=0}^{\\infty}\\left(\\sum_{l=0}^\\infty \\check{f}_{k,l}^{m,M}\\right) |t|^k $ converges.\\ But both assertions follow from the inequalities\n$$\n\\sum_{l=0}^\\infty \\check{f}_{k,l}^{m,M} \\leq \\sum_{\\mu,l=0}^\\infty  \\check{f}_{\\mu,l}^{m,M}= f(1,1), \\quad k=0,1,\\ldots,\n$$\nand\n$$\n\\sum_{k=0}^{\\infty}\\left(\\sum_{l=0}^\\infty \\check{f}_{k,l}^{m,M}\\right) |t|^k \\leq f(1,1)\\sum_{k=0}^{\\infty}|t|^k =\\frac{f(1,1)}{1-|t|}, \\quad t\\in (-1,1).\n$$\nAs for the uniform convergence in $m$, it suffices to observe that\n$$\n\\sum_{k,l=0}^{\\infty}\\check{f}_{k,l}^{m,M}\\, t^k R_{l}^{M}(s) \\leq f(1,1)\\sum_{k=0}^{\\infty}|t|^k \\quad t,s \\in (-1,1),\n$$\nThe proof is complete.\\eop\n\nThe next lemma is a technical result that can be found proved in \\cite{schoen}.\n\n\\begin{lem}\\label{limitscho}\nIf $t \\in (-1,1)$, then the sequence $\\{R_k^m(t)\\}$ converges to $t^k$ as $m \\to \\infty$, uniformly in $k$.\n\\end{lem}\n\n\\begin{thm} \\label{maininfty} Let $K$ be a continuous and isotropic kernel on $S^{\\infty}\\times S^{M}$.\\ It is positive definite on $S^{\\infty}\\times S^{M}$ if and only if its isotropic part $f$ has a representation in the form\n$$ f(t,s) = \\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{M}t^{k}R_l^{M}(s),$$\nin which $\\check{f}_{k,l}^{M} \\geq 0$, $k,l \\in \\mathbb{Z}_+$ and $\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{M} < \\infty$.\n\\end{thm}\n\\pf For each $k$ and $l$, the function $(t,s)\\in [-1,1]^2 \\to t^{k}R_l^{M}(s)$ is the isotropic part of a positive definite kernel on $S^\\infty \\times S^{M}$.\\ Hence, if $f$ has the representation described in the statement of the theorem, then $K$ is a pointwise limit of positive definite kernels.\\ In particular, it is positive definite itself.\\ Conversely, assume $K$ is positive definite.\\ Without loss of generality, we can assume that $K$ is nonzero.\\ Hence, we can assume that its isotropic part $f$ satisfies $f(1,1) >0$.\\ Since $f$ is the isotropic part of a positive definite kernel on each product $S^{m} \\times S^{M}$, then for each pair $(k,l)$, we may consider the sequence of normalized Fourier coefficients $\\{\\check{f}_{k,l}^{m,M}\\}$.\\ Lemma \\ref{pre} authenticates the definition\n$$\n\\check{f}_{k,l}^{\\, M}:= \\lim_{m \\to \\infty}\\check{f}_{k,l}^{2m,M}, \\quad k,l=0,1,\\ldots.\n$$\nwhile Lemma \\ref{prepar} guarantees that\n$$\n\\lim_{m\\to \\infty}\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{2m,M}\\, t^k R_{l}^{M}(s)=\\sum_{k,l=0}^{\\infty}\\check{f}_{k,l}^{\\, M}\\, t^k R_{l}^{M}(s), \\quad t,s \\in (-1,1).\n$$\nTo proceed, we fix $(t,s) \\in (-1,1)^2$ and $\\epsilon >0$.\\ From the previous limit, we can select $m_0$ so that\n$$\n\\left|\\sum_{k,l=0}^{\\infty}\\check{f}_{k,l}^{2m,M}\\, t^k R_{l}^{M}(s)- \\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{\\, M}\\, t^k R_{l}^{M}(s)\\right| < \\frac{\\epsilon}{2}, \\quad m\\geq m_0.$$\nBy Lemma \\ref{limitscho}, we can select $m_1$ so that\n$$|R_k^{2m}(t)-t^k| <\\frac{\\epsilon}{2f(1,1)}, \\quad k=0,1,\\ldots, \\quad m\\geq m_1.$$\nIt is now clear that\n\\begin{eqnarray*}\n\\left|\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{2m,M}\\, R_k^{2m}(t)R_{l}^{M}(s)-\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{2m,M}\\, t^k R_{l}^{M}(s)\\right| & < &  \\frac{\\epsilon}{2f(1,1)}\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{2m,M} \\\\\n& \\leq & \\frac{\\epsilon}{2}, \\quad m\\geq m_1.\n\\end{eqnarray*}\nThus, with the help of an arbitrarily large $m$, we can use both implications above to deduce that\n$$\n0\\leq \\left| f(t,s)- \\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{\\, M}\\, t^k R_{l}^{M}(s)\\right| < \\epsilon.\n$$\nHence,\n$$\nf(t,s)= \\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{\\, M}\\, t^k R_{l}^{M}(s), \\quad t,s \\in (-1,1).\n$$\nThe coefficients in the representation above are obviously nonnegative.\\ If $\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{\\, M}$ were not convergent, we could select\na positive integer $N$ so that\n$$\n\\sum_{k,l=0}^{N} \\check{f}_{k,l}^{M} \\geq 2f(1,1).\n$$\nPicking a $\\tau \\in (0,1)$ so that $\\tau^{N} > 1\/2$, we would reach\n$$ f(\\tau,1)=\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{M}\\tau^{k} \\geq  \\sum_{k,l=0}^{N} \\check{f}_{k,l}^{M}\\tau^{k} > f(1,1),$$\na contradiction with the positive definiteness of $f$.\\ Having guaranteed the uniform convergence of the series in the representation for $f$ above and invoking the continuity of $f$ in $[-1,1]^2$, we now can let $t,s\\to 1^{-}$ and $t,s \\to -1^{+}$\nin the representation formula in order to conclude that it also holds in $[-1,1]^2$.\\eop\n\nA standard adaptation of the arguments used in the proof of the previous theorem is all that is needed in order to deduce the following complement.\n\n\\begin{thm}  Let $K$ be a continuous and isotropic kernel on $S^{\\infty}\\times S^{\\infty}$.\\ It is positive definite on $S^{\\infty}\\times S^{\\infty}$ if and only if its isotropic part $f$ has a representation in the form\n$$ f(t,s) = \\sum_{k,l=0}^{\\infty}  f_{k,l}t^{k}s^l,$$\nin which $f_{k,l} \\geq 0$, $k,l \\in \\mathbb{Z}_+$ and $\\sum_{k,l=0}^{\\infty} f_{k,l} < \\infty$.\n\\end{thm}\n\n\n\\section{Final remarks}\n\nIn view of the characterization for the continuous, isotropic and positive definite kernels on a product of the form $S^m \\times S^M$ obtained in the previous section, one may ask what are the other relevant questions regarding that class of kernels.\\ We will mention a few of them in this final section of the paper along with some additional elementary results.\n\nLet us begin with the strictly positive definite kernels.\\ A continuous, isotropic and positive definite kernel $K$ on $S^m \\times S^M$ is {\\em strictly positive definite of order $n$} on $S^m \\times S^M$ if\nits isotropic part $f$ satisfies\n$$\n\\sum_{\\mu=1}^n\\sum_{\\nu=1}^n c_\\mu c_\\nu f(x_\\mu \\cdot x_\\nu, w_\\mu \\cdot w_\\nu) >0,\n$$\nwhenever the $n$ points $(x_1, w_1), (x_2, w_2), \\ldots, (x_n,w_n)$ of $S^m \\times S^M$ are distinct and the scalars $c_\\mu$ are not all zero.\\ So, for a fixed $n$, an interesting question would be to characterize, via the main theorems proved here, the continuous, isotropic and strictly positive definite kernels of order $n$ on $S^m \\times S^M$.\\ Going one step further, to characterize the continuous, isotropic and strictly positive definite kernels of all orders on $S^m \\times S^M$.\\ Even in the case of a single sphere, similar characterizations are not available for all fixed $n$ (see \\cite{mene0}).\n\nConcerning the problems mentioned above, the intermediate problem to be described below could provide clues to a complete solution.\\ A continuous, isotropic and positive definite kernel $K$ on $S^m \\times S^M$ is {\\em $DC$-strictly positive definite of order $n$} on $S^m \\times S^M$ if\nits isotropic part $f$ satisfies\n$$\n\\sum_{\\mu=1}^n\\sum_{\\nu=1}^n c_\\mu c_\\nu f(x_\\mu \\cdot x_\\nu, w_\\mu \\cdot w_\\nu) >0,\n$$\nwhenever the $n$ points $x_1$, $x_2, \\ldots, x_n$ of $S^m$ are distinct, the $n$ points $w_1, w_2, \\ldots, w_n$ of $S^M$ are distinct and the scalars $c_\\mu$ are not all zero.\\ Obviously, a strictly positive definite kernel of order $n$ on $S^m \\times S^M$ is $DC$-strictly positive definite of order $n$ on $S^m \\times S^M$, but not conversely (unless $n=1$).\\ Thus, to characterize the continuous, isotropic and positive definite kernels on $S^m \\times S^M$ which are $DC$-strictly positive definite of order $n$ on $S^m \\times S^M$ would be an interesting problem as well.\n\nA third problem we would like to mention is the description of consistent methods to construct continuous, isotropic and (strictly) positive definite kernels on $S^m \\times S^M$.\\ In particular, methods based on the description via known classes of continuous, isotropic and (strictly) positive definite kernels on a single sphere.\n\nA very elementary one is this.\n\n\\begin{prop} \\label{prodi} If $f$ is the continuous and isotropic part of a positive definite kernel on $S^{m}$ and $g$ is the continuous and isotropic part of a positive definite kernel on $S^{M}$, then the function $h$ given by the formula\n$$\nh(t,s)=f(t)g(s), \\quad t,s \\in [-1,1],\n$$\nis the isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$.\\ Further, if $f$ is the isotropic part of a strictly positive definite kernel of order $n$ on $S^{m}$ (respect., $g$ is the isotropic part of a strictly positive definite kernel of order $n$ on $S^{M}$) and $g(1)>0$ (respect., $f(1)>0$), then $h$ is the isotropic part of a strictly positive definite kernel of order $n$ on $S^{m} \\times S^{M}$.\n\\end{prop}\n\\pf The first assertion of the theorem is a consequence of the Schur product theorem.\\ As for the second one, it follows from Oppenheim's inequality (\\cite[p.480]{horn}).\\eop\n\nIf the intention is a more concrete example, one may employ completely monotonic functions in two variables.\\ A continuous function $g: [0,\\infty)^2 \\to \\mathbb{R}$ is \\emph{completely monotonic} on $(0,\\infty)^2$ if it is $C^\\infty$ in $(0,\\infty)^2$ and\n$$\n(-1)^{n_1+n_2} \\frac{\\partial^{n_1+n_2} g}{\\partial u^{n_1}\\partial v^{n_2}}(u,v) \\geq 0, \\quad u,v>0, \\quad n_1,n_2\\in\\mathbb{Z}_+.\n$$\nIt is known that function $g$ as above can be represented in the form\n\\begin{eqnarray} \\label{eq-function CM}\ng(u,v) = \\int_{[0,\\infty)^2}e^{-tu-sv}d\\rho(t,s),\\quad u,v >0,\n\\end{eqnarray}\nin which $\\rho$ is a $\\sigma$-additive and nonnegative measure on $[0,\\infty)^2$ satisfying $0<\\rho((0,\\infty)^2)\\leq\\rho([0,\\infty)^2)\\leq\\infty$ (\\cite[p. 87]{bochner}).\n\nA positive scalar multiple of a completely monotonic function on $(0,\\infty)^2$ is itself completely monotonic on $(0,\\infty)^2$.\\ Likewise, the sum and product of two completely monotonic function\non $(0,\\infty)^2$ are completely monotonic on $(0,\\infty)^2$.\\ If $g,h: [0,\\infty) \\to \\mathbb{R}$ are usual completely monotonic functions on $(0,\\infty)$, then $F(u,v)=g(u)h(v)$ is completely monotonic\non $(0,\\infty)^2$.\\ In particular, $(u,v)\\in [0,\\infty)^2\\to \\exp(-u)\\exp(-v)$\n and $(u,v)\\in [0,\\infty)^2\\to 1\/(1+u)^\\alpha(1+v)^\\beta$, $\\alpha,\\beta \\geq 0$, are completely monotonic on $(0,\\infty)^2$.\\ Additional examples can be found in \\cite{samko}.\n\nFor actual examples of positive definite kernels on $S^m \\times S^M$, the following result is quite useful.\n\n\\begin{prop}\nIf $g$ is completely monotonic on $(0,\\infty)^2$, then\n$$\nf(t,s):= g(\\arccos t,\\arccos s)\n$$\nis the isotropic part of a positive definite kernel on $S^m\\times S^M$.\\ Further, if $g$ is nonconstant, then $f$ is the isotropic part of a strictly positive definite kernel on $S^m \\times S^M$.\n\\end{prop}\n\\pf Consider the integral representation for $g$ as described above.\\ If $x_1,\\ldots,x_n\\in S^m$, $w_1,\\ldots,w_n\\in S^M$ and $c_1,\\ldots,c_n$ are real scalars, then\n$$\n\\sum_{\\mu,\\nu=1}^n c_\\mu c_\\nu  f(x_\\mu \\cdot x_\\nu, w_\\mu \\cdot w_\\nu)  = \\int_{[0,\\infty)^2} \\sum_{\\mu,\\nu=1}^n c_\\mu c_\\nu e^{-t\\arccos(x_\\mu\\cdot x_\\nu)-s\\arccos(w_\\mu\\cdot w_\\nu)}d\\rho(t,s),\n$$\nthat is,\n$$\n\\sum_{\\mu,\\nu=1}^n c_\\mu c_\\nu  f(x_\\mu \\cdot x_\\nu, w_\\mu \\cdot w_\\nu)  = \\int_{[0,\\infty)^2} \\sum_{\\mu,\\nu=1}^n c_\\mu c_\\nu e^{-t d_m(x_\\mu\\cdot x_\\nu)-sd_M(w_\\mu\\cdot w_\\nu)}d\\rho(t,s),\n$$\nin which $d_m$ and $d_M$ are the usual geodesic distances on $S^m$ and $S^M$, respectively.\\ A result proved in \\cite{alexander} reveals that $d_m$ and $d_M$ are kernels of negative type.\\ Consequently, the matrices with entries $-t d_m(x_\\mu,x_\\nu)-sd_M(w_\\mu,w_\\nu)$ is almost nonnegative definite (\\cite[p.135]{don}).\\ A classical result from the theory of positive definite kernels (\\cite[p.74]{berg}) now implies that $\\exp(-td_m-sd_M)$ is a positive definite kernel on $S^m \\times S^M$.\\ Thus, the initial quadratic form is nonnegative and the first assertion of the proposition is proved.\\ As for the second one, it suffices to observe that if the points $(x_\\mu,w_\\mu)$ are distinct then the matrix with entries $-t d_m(x_\\mu,x_\\nu)-sd_M(w_\\mu,w_\\nu)$ has no pair of identical rows when $t,s>0$.\\ In that case, the kernel $(x,z,y,w) \\in (S^m \\times S^M)^2 \\to \\exp[-td_m(x,y)-sd_M(z,w)]$ is, in fact, strictly positive definite on $S^m \\times S^M$.\\ If $g$ is nonconstant, then the original quadratic form is always positive unless all the $c_\\mu$ are zero.\\eop\n\nTo close the paper, we go the other way around, seeking positive definiteness on a single sphere from positive definiteness on a product of spheres.\\ Two results in that direction are as follows.\n\n\\begin{prop} \\label{restric} If $f$ is the continuous and isotropic part of a (strictly) positive definite kernel on $S^{m}\\times S^M$, then $t  \\to f(t,1)$ and $s \\to f(1,s)$ are the isotropic parts of (strictly) positive definite kernels on $S^m$ and $S^M$ respectively.\n\\end{prop}\n\n\\begin{prop} \\label{restric1} If $f$ is the continuous and isotropic part of a $DC$-strictly positive definite kernel on $S^{m}\\times S^M$, then $t  \\to f(t,t)$ is the isotropic part of a strictly positive definite kernel\n on $S^{m\\wedge M}$, in which $m\\wedge M=\\min\\{m,M\\}$.\n\\end{prop}\n\nWe intend to provide solutions for some of the problems mentioned above in a subsequent paper.\\ For now, we conclude this one mentioning a few relevant references that deals with similar questions on a single sphere: \\cite{chen, cheney,gneiting,mene,ron,sun}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn a hyperconnected world, scientific and technological advances have led to an overwhelming amount of user data collected and processed by hundreds of companies over public networks. Companies mine this data to provide personalized services. These technologies have, however, come with the price of an extensive loss of privacy in society. Depending on their resources, adversaries can infer critical (private) information about system operations from public data available on the internet and\/or unsecured servers and networks. This is why researchers from different fields (e.g., computer science, information theory, and control theory) have been attracted to the broad research area of privacy and security of Cyber-Physical Systems (CPSs), \\cite{Farokhi1}\\nocite{Farokhi2}\\nocite{Pappas}\\nocite{Jerome1}\\nocite{Takashi_1}\\nocite{Takashi_3}-\\nocite{chaper_privacy_chaos}\\hspace{-.1mm}\\cite{Carlos_Iman1}.\\\\\nIn most engineering applications, information about the state of systems, say $X$, is obtained through sensor measurements and then sent to a remote station through communication networks for signal processing and decision-making purposes. If the communication network is public\/unsecured and\/or the remote station is untrustworthy, adversaries might access and estimate the system state. \\\\\nA common technique to avoid an accurate estimation is the use of additive random vectors to distort disclosed data. In the context of privacy of databases, a popular approach is differential privacy \\cite{Jerome1,Dwork}, where random noise is added to the response of queries so that private information stored in the database cannot be inferred. In general, if the data to be kept private follows continuous probability distributions, the problem of finding the optimal additive noise to maximize privacy is hard to solve. This problem has been addressed by assuming the data to be kept private is deterministic \\cite{Farokhi1,SORIA,Geng}. However, in a Cyber-Physical-Systems context, the inherent system dynamics and unavoidable system and sensor noise lead to stochastic non-stationary data and thus, existing tools do not fit this setting. More recently, the authors in \\cite{Carlos_Iman1,murguia2021privacy} have proposed a framework for synthesizing optimal transition probabilities to maximize privacy for a class of quantized CPSs with discrete multivariate probability distributions. Even though this framework is general and leads to distorting mechanisms with arbitrary distributions, the computational complexity induced by exploring all the possible transition probabilities from private to the disclosed distorted data is high and increases exponentially with the alphabet of the private data. In \\cite{Farokhi1}, the authors neglect quantization and work directly with dynamical systems driven by continuous (Gaussian) disturbances. They prove that, in the unconstrained additive noise case (so distortion is not considered), the optimal noise distribution minimizing the Fisher information (their privacy metric) is Gaussian. This observation has also been made in \\cite{Cedric} where mutual information is used as privacy metric.\\\\\nMotivated by these results, in this manuscript, we present an optimization-based framework for synthesising privacy-preserving Gaussian mechanisms that maximize privacy but keep distortion bounded. We use additive Gaussian dependent vectors as distorting mechanisms to maximize privacy. We pass sensor data and input signals through these distorting mechanisms before transmission and send the distorted data to the remote station instead. These mechanisms consist of a coordinate transformation and additive dependent Gaussian vectors that are designed to hide (as much as possible) the private parts of the state $S$ -- a desired private output modeled as some linear function of the system state, $S=DX$, for some deterministic matrix $D$.\\\\\n Note, however, that it is not desired to overly distort the original data. When designing the additive Gaussian vectors, we need to take into account the trade-off between \\emph{privacy} and \\emph{distortion}. As \\emph{distortion metric}, we use a general \\emph{weighted mean squared error} between the original and distorted data. Weighting matrices are used to fine tune the desired distortion at different channels and\/or to model different applications of the distorted data at the remote station. In this manuscript, we follow an information-theoretic approach to privacy. As \\emph{privacy metric}, we propose a combination of \\emph{mutual information} and \\emph{entropy} \\cite{Cover} between disclosed and private data. In particular, we aim at minimizing the mutual information $I[S;Z]$, between the private output $S$ and the disclosed randomized sensor data $Z$, while maximizing the entropy $h[H]$ of an additive Gaussian vector $H$ we use to distort input signals, over a finite time window, for \\emph{desired levels of distortion} -- how different actual and distorted data are allowed to be. As we prove in this manuscript, we can cast the problem of finding the optimal Gaussian distributions and change of coordinates as a constrained convex optimization problem.\\\\\n\\textbf{Notation:} The symbol $\\Real$ stands for the real numbers, $\\Real_{>0}$($\\Real_{\\geq 0}$) denotes the set of positive (non-negative) real numbers. The symbol $\\Nat$ stands for the set of natural numbers. The Euclidian norm in $\\Real^n$ is denoted by $||X||$, $||X||^2=X^{\\top}X$, where $^{\\top}$ denotes transposition. The $n \\times n$ identity matrix is denoted by $I_n$ or simply $I$ if $n$ is clear from the context. Similarly, $n \\times m$ matrices composed of only ones and only zeros are denoted by $\\mathbf{1}_{n \\times m}$ and $\\mathbf{0}_{n \\times m}$, respectively, or simply $\\mathbf{1}$ and $\\mathbf{0}$ when their dimensions are clear. For positive definite (semidefinite) matrices, we use the notation $P>0$ ($P \\geq 0$). For any two matrices $A$ and $B$, the notation $A \\otimes B$ (the Kronecker product) stands for the matrix composed of submatrices $A_{ij}B$ , where $A_{ij}$, $i,j=1,...,n$, stands for the $ij-$th entry of the $n \\times n$ matrix $A$. The notation $X \\sim \\mathcal{N}[\\mu,\\Sigma^X]$ means that $X \\in \\Real^{n}$ is a normally distributed random vector with mean $E[X] = \\mu \\in \\Real^{n}$ and covariance matrix $E[(X-\\mu)(X-\\mu)^T] = \\Sigma^X \\in \\Real^{n \\times n}$, where $E[a]$ denotes the expected value of the random vector $a$. Finite sequences of vectors are written as $X^K := (X(1)^{\\top},\\ldots,X(K)^{\\top})^{\\top} \\in \\Real^{Kn}$, $X(i) \\in \\Real^{n}$, $i=1,...,K$, and $n,K \\in \\Nat$. To avoid confusion, we denote powers of matrices as $(A)^{K} = A \\cdots A$ ($K$ times) for $K > 0$, $(A)^{0} = I$, and $(A)^{K} = \\mathbf{0}$ for $K < 0$. The operators $\\log[\\cdot]$, $\\det[\\cdot]$, and $\\text{tr}[\\cdot]$ stand for logarithm base two, determinant, and trace, respectively.\n\n\n\\section{Preliminaries}\nIn this section, we present some definitions and preliminary results needed for the subsequent sections.\n\n\\begin{definition} [MMSE Estimator \\cite{huemer2017component}] \\label{linearMMSE}\nLet $X$ and $Y$ be two jointly Gaussian random vectors. The Minimum Mean Square Error (MMSE) estimate of $X$ given $Y$ is given by\\emph{:}\n\\begin{eqnarray}\n\\hat X = \\Sigma ^{XY} {\\Sigma^Y}^{-1} \\left(Y - E(Y) \\right) + E(X),\n\\end{eqnarray}\nwhere $\\Sigma ^{XY}$ denotes the cross-covariance matrix between $X$ and $Y$ and $\\Sigma^Y$ is the covariance matrix of $Y$.\n\\end{definition}\n\n\\begin{definition}[Differential Entropy \\cite{Cover}]\\label{entropy}\nLet $S \\in \\Real^n$ with $S \\sim \\mathcal{N}[\\mu^S,\\Sigma^S]$. Its differential entropy, $h(S)$, can be written in terms of its covariance matrix as\\emph{:}\n\\begin{eqnarray}\nh[S] = \\frac{1}{2}\\log \\det \\left( \\Sigma^S \\right) + \\frac{n}{2} + \\frac{n}{2}\\log(2\\pi). \\label{entropyCov}\n\\end{eqnarray}\n\\end{definition}\nEntropy is a measure of the average uncertainty in a random vector. We use base two $\\log(\\cdot)$, so entropy is given in bits.\n\n\\begin{definition}[Mutual Information \\cite{Cover}]\\label{mutual_info}\nLet $S$ and $Z$ be two jointly distributed continuous random vectors with joint entropy $h[S,Z]$ and marginal entropies $h[S]$ and $h[Z]$. Their mutual information, $I[S;Z]$, is given as\\emph{:}\n\\begin{eqnarray}\nI[S;Z] = h[S] + h[Z] - h[S,Z].\\label{mutinf}\n\\end{eqnarray}\n\\end{definition}\nMutual information between two jointly distributed vectors is a measure of the statistical dependence between them.\n\n\\section{Problem Formulation}\n\\subsection{System Description}\nWe consider discrete-time stochastic systems of the form:\n\\begin{eqnarray}\\label{eq1}\n\\left\\{ \\begin{split}\nX(k+1) &= AX(k) + BU(k) +  T(k),\\\\\nY(k) &= CX(k) + W(k),\\\\\nS(k) &= DX(k),\n\\end{split} \\right.\n\\end{eqnarray}\nwith time-index $k \\in \\Nat$, state $X \\in {\\mathbb{R}^{{n_x}}}$, measurable output $Y \\in {\\mathbb{R}^{{n_y}}}$, \\emph{known} input $U \\in {\\mathbb{R}^{{n_u}}}$, private performance output $S \\in {\\mathbb{R}^{{n_s}}}$, and matrices $(A,B,C,D)$ of appropriate dimensions, ${n_x}, {n_y}, {n_u}, {n_s} \\in \\mathbb{N}$. Matrix $D$ is full row rank. The state perturbation $T$ and the output perturbation $W$ are multivariate i.i.d. Gaussian processes with zero mean and covariance matrices ${\\Sigma ^T}>0$ and ${\\Sigma ^W}>0$, respectively. The initial state $X(1)$ is assumed to be a Gaussian random vector with $E[X(1)]=\\mu^X_1 \\in \\mathbb{R}^{n_x}$ and covariance matrix $\\Sigma^X_1 = E[(X(1)-\\mu^X_1)(X(1)-\\mu^X_1)^{\\top}] \\in \\mathbb{R}^{n_x \\times n_x}$, $\\Sigma^X_1 > 0$. Processes $T$ and $W$ and the initial condition $X(1)$ are mutually independent. We assume that matrices (vectors) $(A,B,C,D,\\Sigma^X_1,\\mu^X_1,\\Sigma^T,\\Sigma^W)$ and the input signal $U\\left( k \\right)$ are known for all $k$.\\\\\nWe aim to prevent adversaries from estimating the private output $S(k)$, accurately. To this end, we randomize measurements $Y(k)$ and input signals $U(k)$ before transmission and send the corrupted data to the remote station instead. The idea is to randomize $Y(k)$ and $U(k)$ as\n\\begin{equation}\\label{eq2}\n\\left\\{\n\\begin{array}{ll}\n  Z(k) = G(k)Y(k) + V(k),\\\\[2mm]\n  R(k) = U(k) + H(k),\n\\end{array}\n\\right.\n\\end{equation}\nfor some time-varying  transformation $G(k) \\in {\\mathbb{R}^{{n_y} \\times {n_y}}}$ and dependent Gaussian processes, $V(k) \\sim \\mathcal{N}[\\mathbf{0},\\Sigma^V(k)]$ and $H(k) \\sim \\mathcal{N}[\\mathbf{0},\\Sigma^H(k)]$.\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.7\\textwidth]{1f}\n    \\caption{System configuration.}\n    \\label{fig2}\n\\end{figure*}\nThe randomized vectors $Z(k)$ and $R(k)$ are transmitted over an unsecured communication network to a remote station, see Figure \\ref{fig2}. We seek to synthesize the sequences $G(k)$, $\\Sigma^V(k)$, and $\\Sigma^H(k)$, $k \\in \\mathcal{K}:= \\{1,\\ldots,K\\}$, to make inference of the sequence of private outputs, $S(k)$, as `hard' as possible from the disclosed data, $(Z(k),R(k))$. In what follows, we introduce the adversarial model we seek to defend against.\n\n\\subsection{Adversarial Capabilities}\nWe consider worst-case adversaries that eavesdrop data at the communication network and\/or the remote station. They do not only have access to all distorted sensor measurements $Z(k)$ and distorted input signal $R(k)$, but also have prior knowledge of the dynamics and the stochastic properties of the system, i.e., matrices  $(A,B,C,D,\\Sigma^X_1,\\mu^X_1,\\Sigma^T,\\Sigma^W)$ are known by the adversary. Moreover, the adversary also knows the means and covariance matrices $(\\mu^Z(k),\\mu^R(k),\\Sigma^Z(k),\\Sigma^R(k))$ as these can be estimated from the disclosed data $(Z(k),R(k))$. We assume that the adversary uses a linear MMSE estimator (see Definition 1) to reconstruct $S(k)$, which, for jointly Gaussian vectors, produces the best estimation performance among all unbiased estimators \\cite{huemer2017component}. In practice, actual adversaries would typically not have all the capabilities that we assume here. However, if we maximize privacy under such worst-case adversaries, we ensure that adversaries with less capabilities perform even worse (or equal at most).\n\n\n\\subsection{Metrics and Problem Formulation}\nFor a given time horizon $K \\in \\mathbb{N}$, the aim of our privacy scheme is to make inference of the sequence of private vectors, ${S^K} = (S(1)^\\top,...,S(K)^\\top)^\\top$, from the distorted disclosed sequences, ${Z^K} = (Z(1)^\\top,...,Z(K)^\\top)^\\top$ and \\linebreak ${R^{K}} = (R(1)^\\top,...,R(K)^\\top)^\\top$, as hard as possible without distorting $(Y^K,U^{K})$, excessively. That is, we do not want to make $(Y^K,U^{K})$ and $(Z^K,R^{K})$ overly different. Hence, when designing the distorting variables $(G(k),\\Sigma^V(k),\\Sigma^H(k))$, we need to consider the \\emph{trade-off between privacy and distortion}.\\\\\nAs distortion metric, we use the weighted mean squared errors between the original and distorted data, i.e., $E[||{W_Y}(Z^K - Y^K)||^2]$ and $E[||{W_U}(R^K - U^K)||^2]$, for some given weighting matrices of appropiate dimensions. Matrices $W_Y,W_U$ are used to fine-tune the desired distortion at different channels and\/or to model different applications of the distorted data at the remote station. Arguably, for the class of linear systems considered in this manuscript, and most applications at the remote station (for this class), performance degradation induced by the privacy mechanism can be written (or upper bounded) in terms of the proposed weighted mean squared errors.\\\\\nAn intuitive candidate to use as privacy metric is the mutual information between private and disclosed data, i.e., $I[S^K;Z^K,R^{K}]$. However, because the input sequence $U^{K}$ is deterministic and $V(k)$ and $H(k)$ are independent, it is easy to verify that $I[S^K;Z^K,R^{K}] = I[S^K;Z^K]$. That is, in the proposed setting, the randomized input data $R^{K}$ does not affect $I[S^K;Z^K,R^{K}]$ at all. To overcome this obstacle, we add to $I[S^K;Z^K]$ the negative differential entropy, $h[U^{K}-R^{K}] = h[H^{K}]$, to capture the uncertainty between original and disclosed input data. That is, we propose $I[S^K;Z^K] - h[H^{K}]$ as privacy metric.\\\\\nSummarizing the above discussion, we aim at minimizing $I[S^K;Z^K] - h[H^{K}]$ subject to the weighted second moment constraints $E[||{W_Y}(Z^K - Y^K)||^2] \\leq \\epsilon_Y$ and $E[||{W_U}(R^K - U^K)||^2] \\leq \\epsilon_U$, for some desired maximum distortion levels ${\\epsilon _Y} \\in {\\mathbb{R}_{>0}}$ and ${\\epsilon _U} \\in {\\mathbb{R}_{>0}}$ and weights ${W_Y} \\in {\\mathbb{R}^{{n_y} \\times {n_y}}}$ and ${W_U} \\in {\\mathbb{R}^{{n_u} \\times {n_u}}}$, by designing $G(k)$, ${\\Sigma^V(k)}$, and ${\\Sigma^H(k)}$ of the distorting mechanisms \\eqref{eq2}. In what follows, we formally present the optimization problem we seek to address.\n\n\\textbf{Problem 1} Given the system dynamics \\eqref{eq1}, time horizon $K\\in \\mathbb{N}$, desired maximum distortion levels ${\\epsilon _Y} \\in {\\mathbb{R}_{>0}}$ and ${\\epsilon _U} \\in {\\mathbb{R}_{>0}}$, weighting matrices ${W_Y} \\in {\\mathbb{R}^{{n_y} \\times {n_y}}}$ and ${W_U} \\in {\\mathbb{R}^{{n_u} \\times {n_u}}}$, private output sequence $S\\left( k \\right)$, $k \\in \\mathcal{K}={1,...,K}$, and the distorting mechanism \\eqref{eq2}, find the distorting sequences, $G(k)$, ${\\Sigma^V(k)}$, and ${\\Sigma^H(k)}$, solution of the following optimization problem:\n\\begin{equation} \\label{problem1}\n\\left\\{\\begin{aligned}\n\t&\\min_{G(k), {\\Sigma^V(k)}, {\\Sigma^H(k)}, k \\in \\mathcal{K}}\\ I[S^K;Z^K] - h[{H}^{K}],\\\\[1mm]\n    &\\hspace{4mm}\\text{s.t. } \\left\\{\\begin{aligned}\n    &E[||{W_Y}(Z^K - Y^K)||^2] \\leq \\epsilon_Y ,\\\\\n    &E[||{W_U}(R^K - U^K)||^2] \\leq \\epsilon_U\n    \\end{aligned}\\right. \\\\[1mm]\n    &\\hspace{12mm}\\text{and } (V^K,H^K,Y^K) \\text{ mutually independent}.\n\\end{aligned}\\right.\n\\end{equation}\n\n\\section{Solution to Problem 1}\nTo solve Problem 1, we first need to write the cost function and constraints in terms of the design variables.\n\n\\subsection{Cost Function: Formulation and Convexity}\nThe differential entropy $h[{H}^{K}]$ is fully characterized by the covariance matrix of ${H}^{K}$, $\\Sigma^H_K := E[H_K H_K^\\top]$. Hence, by Definition 2, $h[{H}^{K}]$ is given by\n\\begin{equation}\\label{hUprim}\nh[{H}^{K}] = \\frac{1}{2}\\log \\det \\left( \\Sigma^H_K \\right) + \\frac{Kn_u}{2} + \\frac{Kn_u}{2}\\log(2\\pi).\n\\end{equation}\n\nThe mutual information $I\\left[ {{S^K};{Z^K}} \\right]$ can be written in terms of differential entropies as $I[S^K;Z^K] = h[S^K] + h[Z^K] - h[S^K,Z^K]$, see Definition \\ref{mutual_info}. Moreover, these entropies are fully characterized by the covariance matrices of the corresponding random vectors, see Definition 2. So, to characterize $I\\left[ {{S^K};{Z^K}} \\right]$ in terms of $G(k)$ and $\\Sigma^V(k)$, $k \\in \\mathcal{K}$, we need to write the covariance matrices of $S^K$ and $Z^K$, and the joint covariance of $(S^K,Z^K)$ in terms of them. By lifting the system dynamics \\eqref{eq1} over $\\{1,\\ldots,K\\}$, we can write the stacked vector $((Z^K)^\\top,(S^K)^\\top)^\\top \\in \\mathbb{R}^{K(n_y + n_s)}$ as\n\n\\begin{align}\\label{stackedXY}\n&\\begin{bmatrix} Z^K \\\\ S^K \\end{bmatrix} = \\begin{bmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bmatrix}F_K X(1) + \\begin{bmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bmatrix} J_K T^{K-1}\\\\\n&\\hspace{15mm} + \\begin{bmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bmatrix} L_K U^{K-1} + \\begin{bmatrix} I \\\\ \\mathbf{0} \\end{bmatrix}({{\\tilde{G}}_K} W^{K}+V^{K}), \\notag\n\\end{align}\nwith stacked matrices $L_K := J_K(I_{K-1} \\otimes B)$, $\\tilde{C}_K := I_K \\otimes C$, $\\tilde{D}_K := I_K \\otimes D$, ${{\\tilde{G}}_K} := \\text{diag}\\left[ {{G_1},{G_2}, \\ldots ,{G_K}} \\right]$, and\n\\begin{equation}\\label{stackedZ}\n\\left\\{ \\begin{aligned}\n  F_K &:= \\begin{bmatrix} I & A^\\top & \\hdots & (A^\\top)^{K-1}  \\end{bmatrix}^\\top,\\\\\n  J_K &:= \\begin{bmatrix} \\mathbf{0} & \\mathbf{0} & \\mathbf{0} & \\cdots & \\mathbf{0} \\\\ I & \\mathbf{0} & \\mathbf{0} & \\cdots & \\mathbf{0} \\\\ A & I & \\mathbf{0} & \\cdots & \\mathbf{0} \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\[1mm] (A)^{K-2} & (A)^{K-3} & (A)^{K-4} & \\cdots & I  \\end{bmatrix}.\n\\end{aligned} \\right.\n\\end{equation}\nLet ${{\\Sigma }^V_K} \\in \\mathbb{R}^{K n_y \\times K n_y}$ denote the \\emph{non-diagonal} covariance matrix of the stacked additive dependent vector $V^K$. Note that matrices $\\tilde{G}_K$ and ${{\\Sigma}^V_K}$ contain all the distorting variables of the output mechanism, $(G(k),\\Sigma^V(k))$, $k \\in \\mathcal{K}$. In the next lemma, we give a closed-form expression of the joint density of $(S^K,Z^K)$ (which we will need to write $I[Z^K;S^K]$ in terms of the design variables).\n\n\n\\begin{lemma}\\label{stackedDist}\n\\[ \\begin{psmallmatrix} Z^K \\\\ S^K  \\end{psmallmatrix} \\sim \\mathcal{N}\\left[\\mu^{Z,S}_K,\\Sigma^{Z,S}_K\\right],\\] with mean $\\mu^{Z,S}_K \\in \\mathbb{R}^{K(n_s + n_y)}$ and covariance matrix $\\Sigma^{Z,S}_K \\in \\mathbb{R}^{K(n_s + n_y) \\times K(n_s + n_y)}$, $\\Sigma^{Z,S}_K>0$\\emph{:}\n\\begin{eqnarray}\\label{muZS}\n\\mu^{Z,S}_K = \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix}F_K \\mu^X_1 + \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix} L_K U^{K-1},\n\\end{eqnarray}\n\\begin{equation}\\label{stackeconmatrix3}\n\\begingroup\n\\renewcommand*{\\arraycolsep}{2pt}\n\\begin{aligned}\n&\\Sigma^{Z,S}_K := \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\\\ \\mathbf{0} \\end{bsmallmatrix}(I_K \\otimes \\Sigma^W) \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\\\ \\mathbf{0} \\end{bsmallmatrix}^\\top + \\begin{bsmallmatrix} I \\\\ \\mathbf{0} \\end{bsmallmatrix}{{\\Sigma }^V_K}\\begin{bsmallmatrix} I \\\\ \\mathbf{0} \\end{bsmallmatrix}^\\top\\\\[1mm]\n&+ \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix} F_K \\Sigma^X_1 F_K^\\top \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix}^\\top\\\\[1mm]\n&+ \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix}J_K (I_{K-1} \\otimes \\Sigma^T)J_K^\\top \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix}^\\top.\n\\end{aligned}\n\\endgroup\n\\end{equation}\n\\end{lemma}\n\\emph{\\textbf{Proof}}: To simplify notation, we introduce the stacked vector ${\\Theta ^K} := {( {{{( {{Z^K}})}^\\top},{{( {{S^K}})}^\\top}} )^\\top}$. By assumption, the initial condition $X(1)$, and the processes, $T(k)$, $W(k)$, and $V(k)$, $k \\in \\mathbb{N}$, are mutually independent, and $X(1) \\sim \\mathcal{N} [ {\\mu _1^X,\\Sigma _1^X} ]$, $T\\left( k \\right) \\sim \\mathcal{N} [ {\\mathbf{0},\\Sigma^T} ]$, $W(k) \\sim \\mathcal{N} [{\\mathbf{0},\\Sigma^W} ]$, $V(k) \\sim \\mathcal{N} [ {\\mathbf{0},\\Sigma^V} ]$, for some positive definite covariance matrices $\\Sigma _1^X$, $\\Sigma^T$, $\\Sigma^W$, and $\\Sigma^V$. Then, see \\cite{Ross} for details, we have ${L_1}X\\left( 1 \\right) \\sim \\mathcal{N} [ {{L_1}\\mu _1^X,{L_1}\\Sigma _1^XL_1^\\top} ]$, ${L_2}T^{K-1} \\sim \\mathcal{N} [\\mathbf{0}, {{L_2}({I_{K - 1}} \\otimes {\\sum ^T}){L_2}^T} ]$, ${L_3}W^K \\sim \\mathcal{N} [\\mathbf{0}, {{L_3}({I_K} \\otimes {\\sum ^W}){L_3}^\\top} ]$, ${L_4}V^K \\sim \\mathcal{N} [\\mathbf{0}, {{L_4}{{\\Sigma }^V_K}{L_4}^\\top} ]$, for any deterministic matrices $L_j$,\n$j = 1, 2, 3, 4$, of appropriate dimensions. It follows that the stacked vector ${\\Theta ^K}$ given in \\eqref{stackedXY} is the sum of a deterministic vector, ${( {({{\\tilde{G}}_K} \\tilde{C}_K)^\\top,\\tilde{D}_K^T} )^\\top}{L_K}{U^{K - 1}}$, and four independent normally distributed vectors. Therefore, ${\\Theta ^K}$ follows a multivariate normal distribution with $E[ {{\\Theta ^K}} ] = \\mu^{Z,S}_K$ as in \\eqref{muZS}.\nBy inspection, using the expression of ${\\Theta ^K}$ in \\eqref{stackedXY}, mutual independence among $X(1)$, $T(k)$, $W(k)$, and $V(k)$, $k \\in N$, and the definition of $\\Sigma _1^X$, $\\Sigma _1^X = E[ {\\left( {X(1) - \\mu _1^X} \\right){{\\left( {X(1) - \\mu _1^X} \\right)}^\\top}} ]$, it can be verified that the covariance matrix of ${\\Theta ^K}$, $E[ {( {{\\Theta ^K} - E( {{\\Theta ^K}} )} ){{( {{\\Theta ^K} - E( {{\\Theta ^K}} )} )}^\\top}} ]$, is given by $\\Sigma^{Z,S}_K$ in \\eqref{stackeconmatrix3}. It remains to prove that the distribution of ${\\Theta ^K}$ is not degenerate, that is, $\\Sigma^{Z,S}_K > 0$. Note that $\\Sigma^{Z,S}_K$ in \\eqref{stackeconmatrix3} can be written as\n\\begin{align}\n\\Sigma _K^{Z,S} = \\left[ {\\begin{array}{*{20}{c}}\n\\Sigma _K^{Z} & { \\tilde{G}_K  \\tilde{C}_K Q  \\tilde{D}_K^\\top}\\\\\n{ \\tilde{D}_K Q  \\tilde{C}_K^\\top  {{\\tilde{G}}_K}^\\top}&{ \\tilde{D}_K Q  \\tilde{D}_K^\\top}\n\\end{array}} \\right],\\label{CovZS}\n\\end{align}\n\\begin{eqnarray}\n\\Sigma _K^{Z} = {\\tilde{G}_K \\tilde{C}_K Q \\tilde{C}_K^\\top  {{\\tilde{G}}_K}^\\top +  {{\\tilde{G}}_K} (I_K \\otimes \\Sigma^W) {{\\tilde{G}}_K}^\\top + {{\\Sigma }^V_K}},\n\\end{eqnarray}\nwith $Q = {F_K}\\Sigma _1^XF_K^\\top + {J_K}( {{I_{K - 1}} \\otimes {\\sum ^T}} )J_K^\\top$. A necessary condition for the block matrix $\\Sigma _K^{Z,S}$ in \\eqref{CovZS} to be positive definite is that the diagonal blocks are positive definite \\cite{Horn}.\nThe left-upper block is positive definite because by assumption ${{\\Sigma }^V_K} > 0$. The right-lower block is positive definite if $\\tilde{D}_K$ is full row rank and $Q$ is positive definite. Because $D$ is full row rank by assumption, matrix $\\tilde{D}_K = \\left( {{I_K} \\otimes D} \\right)$ is also full row rank [\\cite{Horn2}, Theorem 4.2.15].\nNote that $Q$ can be factored as follows\n\\begin{eqnarray}\\label{Q}\nQ = \\left[ {\\begin{array}{*{20}{c}}\n{{F_K}}&{{J_K}}\n\\end{array}} \\right]\\underbrace {\\left[ {\\begin{array}{*{20}{c}}\n{\\Sigma _1^X}&0\\\\\n0&{{I_{K - 1}} \\otimes {\\Sigma ^T}}\n\\end{array}} \\right]}_{Q'}\\left[ {\\begin{array}{*{20}{c}}\n{{F_K}}\\\\\n{{J_K}}\n\\end{array}} \\right],\n\\end{eqnarray}\nThat is, $Q$ is a linear transformation of the block diagonal matrix ${Q'}$ above. By inspection, it can be verified that matrix $P = [F_K \\hspace{1mm} J_K]$, see \\eqref{stackedZ}, is lower triangular with identity matrices on the diagonal; thus, $P$ is invertible. It follows that $Q = PQ'P^\\top$ is a congruence transformation of ${Q'}$ \\cite{boyd1994linear}. Hence, $Q$ is positive definite if and only if the block diagonal matrices of ${Q'}$ are positive definite \\cite{boyd1994linear}. Matrices ${\\Sigma _1^X}$ and ${\\Sigma^T}$ are positive definite by assumption (which implies ${{I_{K - 1}} \\otimes {\\Sigma ^T}}>0$), and thus we can conclude that $Q>0$, which implies ${\\tilde{D}_K Q  \\tilde{D}_K^\\top}>0$ because $\\tilde{D}_K$ is full row rank. Necessary and sufficient conditions for $\\Sigma _K^{Z,S}>0$ are ${\\tilde{D}_K Q  \\tilde{D}_K^\\top}>0$ which we have already proved) and that the Schur complement of block ${\\tilde{D}_K Q  \\tilde{D}_K^\\top}$ of $\\Sigma _K^{Z,S}$, denoted as $\\Sigma _K^{Z,S} \/ ({\\tilde{D}_K Q  \\tilde{D}_K^\\top})$, is positive definite [\\cite{zhang2006schur}, Theorem 1.12]. This Schur complement is given by\n\\begin{align}\n&\\Sigma_K^{Z,S} \/ ({\\tilde{D}_K Q  \\tilde{D}_K^\\top}) =\n{{\\tilde{G}}_K} (I_K \\otimes \\Sigma^W) {{\\tilde{G}}_K}^\\top + {{\\Sigma }^V_K} +\\\\ \\nonumber\n&{{\\tilde{G}}_K} \\tilde{C}_K ( {Q - Q \\tilde{D}_K^\\top ({\\tilde{D}_K Q \\tilde{D}_K^\\top} )^{-1}\\tilde{D}_K Q}) \\tilde{C}_K^\\top {{\\tilde{G}}_K}^\\top.\n\\end{align}\nSince matrix ${{\\tilde{G}}_K} (I_K \\otimes \\Sigma^W) {{\\tilde{G}}_K}^\\top + {{\\Sigma }^V_K}$ is positive definite, a sufficient condition for $\\Sigma _K^{Z,S} \/ ({\\tilde{D}_K Q  \\tilde{D}_K^\\top})>0$ is\n\\begin{eqnarray}\nQ'' := ( {Q - Q \\tilde{D}_K^\\top ( {\\tilde{D}_K Q \\tilde{D}_K^\\top} )^{-1}\\tilde{D}_K Q} )>0.\n\\end{eqnarray}\nRegarding ${Q''}$ as the Schur complement of a higher dimensional matrix ${Q'''}$, we can conclude that:\n\\begin{eqnarray}\n\\begin{array}{l}\nQ'' \\ge 0 \\Leftrightarrow Q''' = \\left[ {\\begin{array}{*{20}{c}}\nQ&{Q\\tilde{D}_K^\\top}\\\\\n{\\tilde{D}_K Q}&{\\tilde{D}_K Q \\tilde{D}_K^\\top}\n\\end{array}} \\right]\\\\\n = \\left[ {\\begin{array}{*{20}{c}}\nQ\\\\\n{\\tilde{D}_K Q}\n\\end{array}} \\right]{Q^{ - 1}}\\left[ {\\begin{array}{*{20}{c}}\nQ&{Q\\tilde{D}_K^\\top}\n\\end{array}} \\right] \\ge 0,\n\\end{array}\n\\end{eqnarray}\nwhich is trivially true because $Q^{-1}$ is positive definite since $Q>0$. Hence, ${\\tilde{D}_K Q \\tilde{D}_K^\\top}$ and $\\Sigma _K^{Z,S} \/ ({\\tilde{D}_K Q  \\tilde{D}_K^\\top})$ are both positive definite, and thus $\\Sigma _K^{Z,S}>0$.\n\\hfill $\\blacksquare$\n\nTo obtain the densities of $Z^K$ and $S^K$, we just marginalize (see \\cite{Ross}) their joint density in Lemma 1 over $S_K$ and $Z_K$.\n\n\\begin{corollary} \\label{corollary1}\n$Z^{K} \\sim \\mathcal{N} [\\mu^Z_K, \\Sigma^Z_K]$ and\n$S^{K} \\sim \\mathcal{N}[\\mu^S_K,\\Sigma^S_K]$\\emph{:}\n\\begin{align*}\n&\\mu^Z_K = {{\\tilde{G}}_K} \\tilde{C}_K F_K \\mu^X_1 + {{\\tilde{G}}_K} \\tilde{C}_K L_K U^{K-1},\\\\\n&\\mu^S_K = \\tilde{D}_KF_K \\mu^X_1 + \\tilde{D}_KL_KU^{K-1},\\\\\n&\\Sigma^Z_K = \\begin{pmatrix} I_{n_y}  \\\\ \\mathbf{0} \\end{pmatrix}^\\top \\Sigma^{Z,S}_K \\begin{pmatrix} I_{n_y} \\\\ 0  \\end{pmatrix} \\in \\mathbb{R}^{Kn_y \\times Kn_y},\\\\\n&\\Sigma^S_K = \\begin{pmatrix} \\mathbf{0}  \\\\ I_{n_s} \\end{pmatrix}^\\top \\Sigma^{Z,S}_K \\begin{pmatrix} \\mathbf{0} \\\\ I_{n_s}  \\end{pmatrix} \\in \\mathbb{R}^{Kn_s \\times Kn_s}.\n\\end{align*}\n\\end{corollary}\n\n\n\nAt this point, we have the three covariance matrices, $(\\Sigma^Z_K,\\Sigma^S_K,\\Sigma^{SZ}_K)$, that we need to compute the mutual information $I\\left[ {{S^K};{Z^K}} \\right]$. Note that these matrices are functions of our design variables $\\tilde{G}_K$ and $\\Sigma^V_K$:\n\\begin{align}\n&\\Sigma _K^{Z,S} = \\begin{pmatrix} \\Sigma _K^{Z} & (\\Sigma _K^{SZ})^\\top \\\\ \\Sigma _K^{SZ} & \\Sigma _K^{S} \\end{pmatrix},\\label{SigmaS_Z}\\\\[1mm]\n&\\Sigma_K^{Z} = {{{\\tilde{G}}_K} \\tilde{C}_K Q \\tilde{C}_K^\\top  {{\\tilde{G}}_K}^\\top +  {{\\tilde{G}}_K} (I_K \\otimes \\Sigma^W) {{\\tilde{G}}_K}^\\top + {{\\Sigma }^V_K}},\\label{eq3g}\\\\[1mm]\n&\\Sigma^S_K = { \\tilde{D}_K Q  \\tilde{D}_K^\\top},\\label{eq3h}\\\\[1mm]\n&\\Sigma _K^{SZ} = { {{\\tilde{G}}_K}  \\tilde{C}_K Q  \\tilde{D}_K^\\top},\\label{SigmaSZ}\n\\end{align}\nwith $Q$ as in \\eqref{Q} independent of $\\tilde{G}_K$ and $\\Sigma^V_K$. Before we write the cost in terms of these matrices, we notice that $\\Sigma^V_K$ only appears in the expression for $\\Sigma^Z_K$ in \\eqref{eq3g}. Moreover, given $(\\tilde{G}_K,\\Sigma^Z_K)$, matrix $\\Sigma^V_K$ is fully determined and viceversa. That is, $(\\tilde{G}_K,\\Sigma^V_K) \\rightarrow (\\tilde{G}_K,\\Sigma^Z_K)$ is an invertible transformation. Therefore, we can pose both cost and constraints in terms of either $\\Sigma^V_K$ or $\\Sigma^Z_K$. Casting the problem in terms of $\\Sigma^Z_K$ allows us to write linear distortion constraints and a convex cost function. Hereafter, we pose the problem in terms of $(\\tilde{G}_K,\\Sigma^Z_K)$. Once we have found optimal $(\\tilde{G}_K,\\Sigma^Z_K)$, we simply extract the optimal $\\Sigma^V_K$ using \\eqref{eq3g}. The first constraint that we need to enforce is that the extracted  $\\Sigma^V_K$ is always positive definite (as it is a covariance matrix). Having $\\Sigma^Z_K > \\mathbf{0}$ does not necessary imply $\\Sigma^V_K > \\mathbf{0}$. From \\eqref{eq3g}, it is easy to verify that $\\Sigma^V_K>\\mathbf{0}$ if and only if $\\Sigma_K^{Z} - {{\\tilde{G}}_K} (\\tilde{C}_K Q \\tilde{C}_K^\\top + (I_K \\otimes \\Sigma^W)) {{\\tilde{G}}_K}^\\top  > \\mathbf{0}$. Using standard Schur complement properties \\cite{zhang2006schur}, the latter nonlinear inequality can be rewritten as a higher-dimensional linear matrix inequality in $\\Sigma^Z_K$ and $\\tilde{G}_K$ as follows:\n\\begin{eqnarray}\\label{SigmaV_pos_def}\n\\left[ {\\begin{array}{*{20}{c}}\n\\Sigma _K^Z    &    {{\\tilde{G}}_K} \\\\\n{{\\tilde{G}}_K}^\\top      &    (\\tilde{C}_K Q \\tilde{C}_K^\\top + (I_K \\otimes \\Sigma^W))^{-1}\n\\end{array}} \\right] > 0.\n\\end{eqnarray}\nWe use inequality \\eqref{SigmaV_pos_def} later when we solve the complete optimization problem to enforce that the optimal $\\Sigma^Z_K$ and $\\tilde{G}_K$ lead to a positive definite $\\Sigma^V_K$.\n\nFinally, given $(\\Sigma^Z_K,\\Sigma^S_K,\\Sigma^{SZ}_K)$ in \\eqref{SigmaS_Z}-\\eqref{SigmaSZ}, by Definition 2, we can write $I[Z^K;S^K]$ as follows:\n\\begin{subequations}\n\\begin{align}\n&I[Z^K;S^K] = h\\left( S^K \\right) + h\\left( Z^K \\right) - h(Z^K,S^K)\\label{eq3a}\\\\[1mm]\n& = \\frac{1}{2} \\log \\frac{{\\det \\left( {\\Sigma _K^S} \\right)\\det \\left( {\\Sigma _K^Z} \\right)}}{{\\det \\left( {\\Sigma _K^{Z,S}} \\right)}}\\label{eq3d}\\\\\n& = \\frac{1}{2} \\log{\\det \\left( {\\Sigma _K^S} \\right)} - \\frac{1}{2} \\log{\\det  {\\left( {\\Sigma} _K^S - {{\\Sigma} _K^{SZ}}^\\top {{\\Sigma} _K^ Z}^{-1} {\\Sigma} _K^{SZ} \\right),}}\\label{eq3f}\n\\end{align}\n\\end{subequations}\nwhere \\eqref{eq3d}-\\eqref{eq3f} follow from standard determinant and logarithm formulas. In the following lemma, we prove that minimizing the cost function $I[S^K;Z^K] - h[{H}^K]$ using $({{\\tilde{G}}_K},{{\\Sigma }^Z_K},{{\\Sigma }^H_K})$ as optimization variables is equivalent to solving a convex program subject to some Linear Matrix Inequalities (LMI) constraints.\n\n\\begin{lemma}\\label{mutualinformationcov}\nMinimizing $I[S^K;Z^K] - h[{H}^K]$ is equivalent to solving the following convex program:\n\\begin{eqnarray}\n\\left\\{\\begin{aligned}\n\t&\\min_{\\Sigma^H_K, {\\Pi _K},\\Sigma_K^{Z},{{\\tilde{G}}_K}}\\\n      -\\log \\det \\left( \\Sigma^H_K \\right) - \\log{\\det \\left({\\Pi _K} \\right)} \\label{finalcost_program}\\\\[1mm]\n    &\\hspace{4mm}\\text{\\emph{s.t. }} \\Pi _K \\geq \\mathbf{0}, \\begin{bmatrix}\n\\Sigma _K^S - \\Pi _K & (\\Sigma_K^{SZ})^\\top\\\\\n\\Sigma_K^{SZ} & \\Sigma_K^Z\n\\end{bmatrix} \\geq 0.\n\\end{aligned}\\right.\n\\end{eqnarray}\n\\end{lemma}\n\\textbf{\\emph{Proof}}: {(a)} For any positive definite matrix $\\Sigma$, the function $-\\log\\det(\\Sigma)$ is convex in $\\Sigma$ \\cite{boyd2004convex}. It follows that $-h[{H}^K]$ in \\eqref{hUprim} is a convex function of $\\Sigma^H_K$. Minimizing $-h[{H}^K]$ amounts to minimizing $-\\log \\det \\left( \\Sigma^H_K \\right)$ as all other terms in \\eqref{hUprim} are constants. {(b)} Next, consider the expression for $I[Z^K;S^K]$ in \\eqref{eq3f}. Due to monotonicity of the determinant function and the fact that $\\Sigma _K^S$ is independent of the design variables, minimizing \\eqref{eq3f} is equivalent to\n\\begin{eqnarray}\n\\left\\{\\begin{aligned}\n\t&\\min_{{\\Pi _K},\\Sigma_K^{Z},{{\\tilde{G}}_K}}\\\n      - \\log{\\det \\left({\\Pi _K} \\right)} \\label{epigraphcost}\\\\[1mm]\n    &\\hspace{4mm}\\text{s.t. } \\mathbf{0} < {\\Pi _K} \\le  {\\left( {\\Sigma} _K^S - {{\\Sigma} _K^{SZ}}^\\top {{\\Sigma} _K^ Z}^{-1} {\\Sigma} _K^{SZ} \\right).} \\label{inequalityofcost}\n\\end{aligned}\\right.\n\\end{eqnarray}\n{(c)} The inequality term in \\eqref{inequalityofcost} can be rewritten using Schur complements properties  \\cite{zhang2006schur} as\n\\begin{eqnarray}\n&\\left[ {\\begin{array}{*{20}{c}}\n{\\Sigma _K^S} - {\\Pi _K} & {{\\Sigma} _K^{SZ}}^\\top\\\\\n{{\\Sigma} _K^{SZ}}&{{\\Sigma} _K^ Z}\n\\end{array}} \\right] \\ge 0 \\, {,{\\Pi _K} > 0}. \\label{finalcost}\n\\end{eqnarray}\nBecause $\\Sigma_K^{SZ}$ is a linear function of $\\tilde{G}_K$ (see \\eqref{SigmaSZ}), \\eqref{finalcost} are two LMI constraints in $\\Pi _K$ and $\\tilde{G}_K$. Combining {(a)-(c)}, we can conclude that minimizing $I[S^K;Z^K] - h[{H}^K]$ is equivalent to solving the convex program in \\eqref{finalcost_program}. \\hfill $\\blacksquare$\n\nBy Lemma \\ref{mutualinformationcov}, minimizing the cost in \\eqref{problem1} is equivalent to solving the convex program in \\eqref{finalcost_program}. Then, if the distortion metrics, $E[||{W_Y}(Z^K - Y^K)||^2]$ and $E[||{W_U}({R}^K - U^K)||^2]$, are convex in the decision variables (which is the topic of the next section), we can find optimal distorting mechanisms efficiently using off-the-shelf optimization algorithms.\n\n\\subsection{Distortion Constraints: Formulation and Convexity}\n\nWe start with the distortion in $Y^K$. By lifting the system dynamics \\eqref{eq1} over $\\{1,\\ldots,K\\}$, we can write the mean and covariance matrix of the stacked output $Y^K \\in \\mathbb{R}^{Kn_y}$ as:\n\\begin{align}\n\\mu^{Y}_K &= \\tilde{C}_K F_K \\mu^X_1 + \\tilde{C}_K L_K U^{K-1},  \\label{eq4d}\\\\[1mm]\n\\Sigma^{Y}_K &= (I_{K} \\otimes \\Sigma^W) + \\tilde{C}_K Q \\tilde{C}_K^\\top, \\label{eq4e}\n\\end{align}\nwith matrices $L_K = J_K(I_{K-1} \\otimes B)$, $\\tilde{C}_K = I_K \\otimes C$, $J_K$ in \\eqref{stackedZ}, and $Q$ in \\eqref{Q}.\n\n\\begin{lemma}\\label{constraint}\n$E[||{W_Y}(Z^K - Y^K)||^2]$ is a convex function of ${{\\Sigma }^Z_K}$ and ${{\\tilde{G}}_K}$, and can be written as follows:\n\\begin{align}\\label{distortion}\n&E[||{W_Y}(Z^K - Y^K)||^2] \\hspace{20cm} \\notag\\\\[1mm]\n&\\hspace{6mm} = \\text{\\emph{tr}}[{W_Y}^\\top (\\Sigma _K^Z + \\Sigma _K^Y - 2\\Sigma _K^Y\\tilde{G}_K) {W_Y}]\\notag\\\\[1mm]\n& \\hspace{14mm} + {{\\mu _K^Y}^\\top} ({{\\tilde{G}}_K} - I)^\\top {W_Y}^\\top {W_Y} ({{\\tilde{G}}_K} - I)  {\\mu _K^Y}.\n\\end{align}\n\\end{lemma}\n\\textbf{\\emph{Proof}}: The expectation of the quadratic form, $E[||{W_Y}(Z^K - Y^K)||^2]$, can be written (see \\cite{seber2012linear} for details) in terms of the mean and covariance of the error $\\Delta^K := Z^K - Y^K$:\n\\begin{align}\n&E[||{W_Y}(Z^K - Y^K)||^2] = \\text{tr}[\\Sigma _K^\\Delta] + (\\mu _K^{\\Delta})^\\top \\mu _K^\\Delta \\label{eq4b}.\n\\end{align}\nwith covariance $\\Sigma _K^\\Delta := E[(\\Delta^K - \\mu _K^{\\Delta})(\\Delta^K - \\mu _K^{\\Delta})^\\top]$ and mean $\\mu _K^{\\Delta} := E[\\Delta^K]$. Given the distortion mechanism \\eqref{eq2}, we can write $Z^K = \\tilde{G}_K Y^K + V^K$; hence, $\\Delta^K = {W_Y}(({{\\tilde{G}}_K} - I) Y^K + V^K)$, and because $Y^K$ and $V^K$ are independent, $\\Sigma _K^\\Delta = {W_Y}^\\top ({{\\tilde{G}}_K} - I)^\\top \\Sigma _K^Y ({{\\tilde{G}}_K} - I) + {{\\Sigma }^V_K}) {W_Y}$ and $\\mu _K^\\Delta = {W_Y} ({{\\tilde{G}}_K} - I) \\mu _K^Y$. Note that, because $Z^K = \\tilde{G}_K Y^K + V^K$, $\\Sigma^Z_K = {{\\tilde{G}}_K}^\\top \\Sigma _K^Y {{\\tilde{G}}_K} + {{\\Sigma }^V_K}$; then, we can write $\\Sigma _K^\\Delta$ in terms of $\\Sigma^Z_K$ as  $\\Sigma _K^\\Delta = {W_Y}^\\top (\\Sigma _K^Z + \\Sigma _K^Y - {{\\tilde{G}}_K}^\\top \\Sigma _K^Y - \\Sigma _K^Y {{\\tilde{G}}_K}) {W_Y}$. Up to this point, we have written both $\\mu _K^{\\Delta}$ and $\\Sigma _K^\\Delta$ in terms of the design variables, ${{\\Sigma }^Z_K}$ and ${{\\tilde{G}}_K}$. Using these expressions and \\eqref{eq4b}, we can conclude \\eqref{distortion}. \\hfill $\\blacksquare$\n\n\\begin{remark}\nNote that the distortion metric \\eqref{distortion} is linear in ${{\\Sigma }^Z_K}$ and quadratic in ${{\\tilde{G}}_K}$; consequently, the output distortion constraint in \\eqref{problem1}, $E[||{W_Y}(Z^K - Y^K)||^2] \\le \\epsilon_Y$, is nonlinear in the design variables. However, we can find an equivalent linear constraint using standard Schur complement properties. It can be verified (see \\emph{\\cite{zhang2006schur}}, Theorem 1.12]) that $E[||{W_Y}(Z^K - Y^K)||^2] \\le \\epsilon_Y$ in \\eqref{distortion} is equivalent to the following LMI in $\\Sigma^Z_K$ and $\\tilde{G}_K$:\n\\begin{equation}\n\\left\\{\n\\begin{array}{ll}\n\\begin{bmatrix} \\theta_Y & {{\\mu _K^Y}^\\top}({{\\tilde{G}}_K} - I)^\\top {W_Y}^\\top \\\\ {W_Y} ({{\\tilde{G}}_K} - I) {\\mu _K^Y} & I \\end{bmatrix} \\ge 0,\\\\[6mm]\n\\theta_Y := \\epsilon_Y - \\text{\\emph{tr}}[{W_Y}^\\top (\\Sigma _K^Z + \\Sigma _K^Y - 2\\Sigma _K^Y\\tilde{G}_K) {W_Y}].\n\\end{array}\n\\right.\n\\end{equation}\n\\end{remark}\n\nWe move now to the distortion in $U^K$. The input distortion metric $E[||{W_U}(R^K - U^K)||^2] = E[||{W_U}H^K||^2]$ depends on the mean $\\mu^H_K$ and covariance $\\Sigma^H_K$ of $H^K$. Note that $E[||{W_U}H^K||^2]$ is again the expected value of a quadratic form. By construction ($H^K$ is a design vector), we have $\\mu^H_K = \\mathbf{0}$. Then, $E[||{W_U}H^K||^2]$ is simply given by $E[||{W_U}H^K||^2] = \\text{tr}[{W_U}^\\top {\\Sigma }^H_K {W_U}]$ (see \\cite{seber2012linear} for details), which is already linear in the design matrix $\\Sigma^H_K$. We summarize this discussion in the following lemma.\n\n\\begin{lemma}\\label{constraintU}\n$E[||{W_U}(R^K - U^K)||^2] = E[||{W_U}H^K||^2]$ is a convex function of $\\Sigma^H_K$ and can be written as follows:\n\\begin{eqnarray}\nE[||{W_U}H^K||^2] = \\text{\\emph{tr}}[{W_U}^\\top {\\Sigma }^H_K {W_U}]. \\label{distortionU}\n\\end{eqnarray}\n\\end{lemma}\n\n\nBy Lemma \\ref{mutualinformationcov}, Lemma \\ref{constraint}, and Lemma \\ref{constraintU}, the cost function $I[S^K;Z^K] - h[{H}^K]$ and distortion constraints $E[||{W_Y}(Z^K - Y^K)||^2] \\leq \\epsilon_Y$ and $E[||{W_U}{H}^K||^2] \\leq \\epsilon_U$ can be written in terms of convex functions (programs) in the our optimization variables ${{\\Sigma }^Z}$, ${{\\tilde{G}}_K}$, and ${{\\Sigma }^H_K}$.\n\nIn what follows, we pose the complete nonlinear convex program to solve Problem 1.\n\n\\begin{theorem}\\label{th3}\nConsider the system dynamics \\eqref{eq1}, distorting mechanism \\eqref{eq2}, time horizon $K \\in \\mathbb{N}$, desired input and output distortion levels ${\\epsilon _U},{\\epsilon _Y} \\in {\\mathbb{R}^+}$, input and output distortion weights, ${W_Y} \\in {\\mathbb{R}^{K {n_y} \\times K {n_y}}}$ and ${W_U} \\in {\\mathbb{R}^{(K-1){n_u} \\times (K-1){n_u}}}$, covariance matrices $\\Sigma^{Z,S}_K$ and $\\Sigma _K^{S}$ and cross-covariance $\\Sigma _K^{SZ}$ \\emph{(}given in \\eqref{SigmaS_Z}-\\eqref{SigmaSZ}\\emph{)}, and the mean and covariance of $Y^K$, $\\mu^{Y}_K$ and $\\Sigma^{Y}_K$ \\emph{(}given in \\eqref{eq4d}-\\eqref{eq4e}\\emph{)}. Then, the optimization variables ${{\\Sigma }^Z}$, ${{\\tilde{G}}_K}$, and ${{\\Sigma }^H_K}$ that minimize $I[S^K;Z^K] - h[{H}^K]$ subject to distortion constraints, $E[||{W_Y}(Z^K - Y^K)||^2] \\leq \\epsilon_Y$ and $E[||W_UH^K)||^2] \\leq \\epsilon_U$, can be found by solving the convex program in \\eqref{eq:convex_optimization15}.\n\\end{theorem}\n\\emph{\\textbf{Proof:}} The expressions for the cost and constraints and convexity (linearity) of them follow from Lemma \\ref{mutualinformationcov}, Lemma \\ref{constraint}, and Lemma \\ref{constraintU}, Remark 1, and \\eqref{SigmaV_pos_def}.  \\hfill $\\blacksquare$\n\n\\begin{table}\n\\noindent\\rule{\\hsize}{1pt}\n\\begin{equation} \\label{eq:convex_optimization15}\n\\begin{aligned}\n\t&\\min_{{\\Pi _K}, \\Sigma_K^{Z}, {{\\tilde{G}}_K}, {{\\Sigma }^H_K}}  -\\log\\det [ \\Sigma^H_K ] - \\log\\det [\\Pi_K ],\\\\[1mm]\n    &\\text{ s.t. }\\left\\{\\begin{aligned}\n    &{\\Pi _K} > 0,\\\\ &\\left[ {\\begin{array}{*{20}{c}}\n{\\Sigma _K^S} - {\\Pi _K} & ( {{\\tilde{G}}_K}  \\tilde{C}_K Q  \\tilde{D}_K^\\top)^\\top\\\\\n{{\\tilde{G}}_K}  \\tilde{C}_K Q  \\tilde{D}_K^\\top &{{\\Sigma} _K^ Z}\n\\end{array}} \\right] \\ge 0, \\\\[1mm]\n&\\left[ {\\begin{array}{*{20}{c}}\n\\theta_Y & ({W_Y} ({{\\tilde{G}}_K} - I){\\mu _K^Y})^\\top\\\\\n{W_Y} ({{\\tilde{G}}_K} - I){\\mu _K^Y}&I\n\\end{array}} \\right] \\ge 0, \\\\[1mm]\n&\\theta_Y = \\epsilon_Y - \\text{tr}[{W_Y}^\\top (\\Sigma _K^Z + \\Sigma _K^Y - 2\\Sigma _K^Y\\tilde{G}_K) {W_Y}], \\\\[1mm]\n&\\epsilon_U - \\text{tr}[{W_U}^\\top {\\Sigma }^H_K {W_U}] \\ge 0,\\\\[1mm]\n&\\left[ {\\begin{array}{*{20}{c}}\n\\Sigma _K^Z    &    {{\\tilde{G}}_K} \\\\\n{{\\tilde{G}}_K}^\\top      &    (\\tilde{C}_K Q \\tilde{C}_K^\\top + (I_K \\otimes \\Sigma^W))^{-1}\n\\end{array}} \\right] > 0 \\\\\n&\\Sigma _K^H > 0. \\end{aligned}\\right.\n\\end{aligned}\n\\end{equation}\n\\noindent\\rule{\\hsize}{1pt}\n\\end{table}\n\n\n\\section{Illustrative case study}\nWe illustrate the performance of our tools through a case study of a well-stirred chemical reactor with heat exchanger. This case study has been developed over the years as a benchmark example for control systems and fault detection, see, e.g., \\cite{Wata,IET_CARLOS_JUSTIN} and references therein. The state, inputs, and output of the reactor are:\n\\begingroup\\makeatletter\\def\\f@size{9.0}\\check@mathfonts\n\\def\\maketag@@@#1{\\hbox{\\m@th\\normalsize\\normalfont#1}}%\n\\begin{align*}\n\\left\\{\n\\begin{array}{ll}\nX(t) =  \\begin{pmatrix} C_0\\\\T_0\\\\T_w\\\\T_m \\end{pmatrix},\nU(t) =  \\begin{pmatrix} C_u\\\\T_u\\\\T_{w,u} \\end{pmatrix},\nY(t) =  T_0,\n\\end{array}\n\\right.\n\\end{align*}\\endgroup\nwhere\n\\begingroup\\makeatletter\\def\\f@size{9.0}\\check@mathfonts\n\\def\\maketag@@@#1{\\hbox{\\m@th\\normalsize\\normalfont#1}}%\n\\begin{align*}\n\\left\\{\n\\begin{array}{ll}\nC_0&: \\text{Concentration of the chemical product},\\\\\nT_0&: \\text{Temperature of the product},\\\\\nT_w&: \\text{Temperature of the jacket water of heat exchanger},\\\\\nT_m&: \\text{Coolant temperature},\\\\\nC_u&: \\text{Inlet concentration of reactant},\\\\\nT_u&: \\text{Inlet temperature},\\\\\nT_{w,u}&: \\text{Coolant water inlet temperature}.\n\\end{array}\n\\right.\n\\end{align*}\\endgroup\nWe use the discrete-time dynamics of the reactor introduced in \\cite{murguia2021privacy} with the same noise and input signals for our simulation experiments.\n\nAs private output, we use the concentration of the chemical product; then, the matrix $D$ in \\eqref{eq1} is given by the full row rank matrix $D = (1,0,0,0)$. The output of the system is the temperature of the product $T_0$, which could be monitored, e.g., for quality\/safety reasons. Then, the aim of the privacy scheme is to hide the private state $C_0$,  which is the concentration of the reactant, as much as possible without distorting output signal temperature measurements and input signals excessively.\n\\begin{figure}[!htb]\n  \\centering\n  \\includegraphics[width=3.5in]{costepsYepsUf4.eps}\n  \\caption{Evolution of the cost function for horizon $K=30$ and increasing $||\\epsilon|| = ||(\\epsilon_Y,\\epsilon_U)||$.}\\label{CostBasedEpsUEpsY}\n\\end{figure}\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{YZeps100f5.eps}\n  \\label{fig:sub-first}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{YZepsinf5.eps}\n  \\label{fig:sub-second}\n\\end{subfigure}\n\\caption{Comparison between the measurement vector $Y(k)$ and the distorted measurements $Z(k)$ for two different distortion levels, $\\epsilon_Y = 100, \\infty$.}\n\\label{YZ}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{UUpeps10f5.eps}\n  \\label{fig:sub-first}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{UUpeps1000f5.eps}\n  \\label{fig:sub-second}\n\\end{subfigure}\n\\caption{Comparison between the first element of the input signal ${U}_1(k)$ and the first element of distorted input signal ${R}_1(k)$ for two different distortion levels, $\\epsilon_U = 10, 1000$.}\n\\label{UUp}\n\\end{figure}\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{epsYinfepsU0K50f5.eps}\n  \\label{S-sub-first}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{epsY1epsU10K50f5.eps}\n  \\label{S-sub-second}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{epsY1epsU1000K50f5.eps}\n  \\label{S-sub-3rd}\n\\end{subfigure}\n\\caption{Comparison between the private output $S(k)$, the estimated private output without distorting $Y(k)$ and $U(k)$, ${\\hat S_{YU}}(k)$, and the estimated private output given $Z(k)$ and $R(k)$, ${\\hat S_{ZR}}(k)$, for horizon $K=50$, and three different levels of distortions, $\\epsilon_Y$ and $\\epsilon_U$.}\n\\label{estimatedS}\n\\end{figure}\nWe first show the effect of distortion levels $\\epsilon_Y$ and $\\epsilon_U$ in the cost function. Figure \\ref{CostBasedEpsUEpsY} depicts the evolution of the optimal cost $I[S^K;Z^K] - h[H^K]$ for increasing $\\epsilon = (\\epsilon_Y,\\epsilon_U)$ with time horizon $K=30$. As expected, the objective function decreases monotonically for increased maximum allowed distortion.\nFurthermore, this figure illustrates that by increasing the distortion levels, the decreasing speed of the optimal cost function will be reduced. Therefore, we can design the distortion levels such that the information leakage is minimized without distorting measurement and input signals excessively.\\\\\nThe effect of the optimal distortion mechanisms is illustrated in Figure \\ref{YZ} and Figure \\ref{UUp} for different levels of distortion, where we contrast actual and distorted data.\nNext, in Figure \\ref{estimatedS}, we depict the stacked private output $S^K$, its MMSE estimate ${{\\hat S}^K_{YU}}$ (see Definition 1) without distortion, i.e., the MMSE estimate of $S^K$ given $Y^K$ and $U^{K-1}$, and its MMSE estimate given the distorted vectors $Z^K$ and ${R}^{K-1}$, ${{\\hat S}^K_{ZR}}$. We consider horizon $K=50$, and different levels of distortion, $(\\epsilon_Y,\\epsilon_U) = \\{ (\\infty,0), (1,10), (1,1000) \\}$, to investigate the effect of each distortion level separately. $\\epsilon_Y = \\infty$ means that the optimization problem in \\eqref{eq:convex_optimization15} is solved without considering the distortion constraint between $Z^K$ and $Y^K$. As can be seen in this figure, even after distorting the measurements without the distortion constraint, the stacked private vectors ${S^K}$ can be estimated accurately by a MMSE estimator. However, by randomizing $U^K$, we can prevent an accurate estimation. By increasing the distortion level $\\epsilon_U$, the accuracy of the estimation decreases.\\\\\n\n\\section{Conclusions}\nIn this paper, for a class of stochastic dynamical systems, we have presented a detailed mathematical framework for synthesizing distorting mechanisms to minimize the information leakage induced by the use of public\/unsecured communication networks. We have proposed a class of dependent Gaussian distorting mechanisms to randomize sensor measurements and input signals before transmission to prevent adversaries from accurately estimating the private part of the system state (a performance private output).\\\\\nFurthermore, for the class of systems under study, we have fully characterized information-theoretic metrics (mutual information and differential entropy) to quantify the information between private outputs and disclosed data for a class of worst-case eavesdropping adversaries.\\\\\nFinally, given the maximum level of distortion tolerated by a particular application, we have provided tools (in terms of convex programs) to design optimal (in terms of maximizing privacy) distorting mechanisms. We have presented simulation results to illustrate the performance of our tools.\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Sloan Digital Sky Survey (SDSS) photometric catalogs are an important source\nof stellar photometry, and must be understood in the context of decades of\nstellar research using different filter standards.  In this paper a transformation\nis computed between the available SDSS photometry in October 2001 and stars of the\nHK objective prism survey as provided by T. Beers, private communication.  A \ndescription of the HK survey and an earlier version of the catalog can be found at \n\\citet{psb91}.  The overlaps between the catalogs are relatively few because the \nfaint limit of the HK survey is at approximately the same magnitude as the saturation \nlimit of the SDSS.  Neither of the catalogs compared here are standard versions,\nbut were the only comparables available at the time.  Technical SDSS details can be\nfound in \\citet{getal98}; \\citet{hetal01}; \\citet{petal03}; \\citet{setal02}; and \n\\citet{yetal00}.  More recent\ntransformation equations can be obtained from \\citet{jetal05}; \\citet{kbt05}; \n\\citet{bkt05}; and \\citet{wwh05};\nand two other unpublished determinations that can be found in the documentation\nfor the SDSS DR4 at:\n\n{\\it http:\/\/www.sdss.org\/dr4\/algorithms\/sdssUBVRITransform.html }.\n\n\\section{Query Consideration and Data Reduction}\nTable 1 lists the stars that were common to the HK survey and the SDSS survey in October\n2001.  Generation of both catalogs was a work in progress at that time.\nWe selected from the HK catalog all stars whose J2000 coordinates matched\nSDSS catalog entries within:\n\\begin{eqnarray}\n\\Delta(RA)<7.\"2 (0.002^{\\circ})& \\Delta(Dec)<7.\"2\n\\end{eqnarray}\nWe rejected any matches in which the SDSS star was saturated by checking the catalog\nflags:\n\\begin{equation}\n(objFlags\\,\\, \\& \\,\\,OBJECT\\_SATUR) == 0 \n\\end{equation}\nWe use the magnitude calculated from a fit of modeled stellar profile\n(point-spread-function, or PSF magnitude) to each object. \nBecause these stars are too bright for sky noise affect the quality\nof the photometry, the use of aperture magnitudes would have made little \ndifference.\n\nAll photometry in Table 1 has been corrected for interstellar reddening\nusing the $E_{B-V}$ determined from the HK survey.  Corrections for the\nSDSS photometry are determined from the standard extinction curve of\n\\citet{ccm89}, which for SDSS filters yields:\n\\begin{eqnarray}\nA_{u^*}=5.2E_{B-V}; & A_{g^*}=3.2E_{B-V};& A_{r^*}=2.8E_{B-V}; \\nonumber\\\\\nA_{i^*}=2.1E_{B-V}; & A_{z^*}=1.5E_{B-V} \\nonumber\n\\end{eqnarray}\nWe noticed that the value of $(U-B)$ is -9.990 for some stars in the original HK list. \nThey were replaced by \"$\\cdots$\" in the Table 1. When we calculate the \ntransformation which requires $U$ band, these stars were not included.\nThree stars were removed from the original matched list because\ntheir SDSS photometry was suspicious: (RA, dec) = \n{\\bf (199.9265, 3.9129)}, {\\bf (224.7835, -0.2537)}, {\\bf (200.7934, 4.6075)}.\nTwo of the stars were close to other bright starts which may have confused the\nSDSS object deldender.\n\n\\begin{figure}\n\\plotone{proveUBV.ps}\n\\caption{Color-Color plots of UBV system against SDSS PSF magnitude.  The solid\nline is the least squares fit to the data.  The dashed line indicates the\ntransformation equation of Fukugita et al. (1996).}\n\\end{figure}\n\n\\section{Comparison of derived transformation to Fukugita et al. 1996}\n\nFigure 1 shows the color-color plots for the SDSS PSF magnitude against the HK\nsurvey U-B-V magnitudes of these stars. The coeffices in the equations, plotted as the lines, \nare derived using the {\\bf\\it Method of Least Squares}. We notice that one star, \n{\\bf (221.5996, -0.1158)} is half a magnitude brighter in SDSS filters than we expect\n(though the colors are the same). It is labeled using a bold font in Table 1. \nThere is no reason to remove it from the catalog but we ignore \nit in all fits for filter transformations.\n\n\\citet{fetal96} described SDSS photometric system and gave \nthe approximate color transformation equations from the \nJohnson-Morgan-Cousins system to the SDSS system.  A comparison of our\ntransformations to theirs is given below:\n\n\\begin{eqnarray*}\n   Fukugita's\\, paper  &  & Our Result \\nonumber\\\\\ng^*=V+0.56(B-V)-0.12 & & g^*=V+0.592(B-V)-0.102 \\nonumber\\\\\nr^*=V-0.49(B-V)+0.11 & & r^*=V-0.451(B-V)+0.082 \\nonumber\\\\\nu^*-g^*=1.38(U-B)+1.14 & & u^*-g^*=1.210(U-B)+1.103  \\nonumber\\\\\ng^*-r^*=1.05(B-V)+1.14 & & g^*-r^*=1.043(B-V)-0.185  \\nonumber\n\\end{eqnarray*}\n\nFor the blue stars from which our transformation was derived, our transformations \nare similar to Fukugita's.  The ($u^*-g^*$) transformation is the only one that\nis significantly discrepant.  This is because the\nactual $u^*$ filter response is different from the theoretical curve \nused by Fukugita et al.(1996).\n\n\\section{Inverse Transformation Equations}\n\nWe will discuss the inverse transformation from SDSS PSF magnitude to UBV system \nin this section.  Magnitudes in the $g^*$ filter and $g^*-r^*$ color are used as the \nprimary parameters in the equations since the noise in $u^*$ is typically higher. \nWe also considered other combinations of filters. \nFigures 2 and 3 show the results; the transformation is summaried\nbelow: \n\\begin{figure}\n\\plotone{tran1.ps}\n\\caption{Transformation plot from SDSS photometric system to UBV system.}\n\\end{figure}\n\n\\begin{figure}\n\\plotone{tran2.ps}\n\\caption{Transformation plot from SDSS photometric system to UBV system.}\n\\end{figure}\n\n\\begin{eqnarray*}\nU=g^*+0.883(u^*-g^*)-0.717 & & U=u^*-0.117(u^*-g^*)-0.717 \\nonumber\\\\\nB=g^*+0.348(g^*-r^*)+0.175 & & B=g^*+0.162(u^*-g^*)+0.094 \\nonumber\\\\\nV=g^*-0.561(g^*-r^*)-0.004 & & V=r^*+0.439(g^*-r^*)-0.004 \\nonumber\\\\\n(U-B)=0.754(u^*-g^*)-0.835  \\nonumber \\\\\n(B-V)=0.916(g^*-r^*)+0.187  \\nonumber\n\\end{eqnarray*}\n\n\\section{Conclusion}\n\nWe have derived transformation equations between HK catalog photometry and\nSDSS photometry.\n\n\\acknowledgments\n\nFunding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http:\/\/www.sdss.org\/.\n\nThe SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, Cambridge University, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\indent A Riemannian metric $g$ on a $4$-manifold is called  hyperk\\\"ahler if its holonomy group  $Hol(g)$ is contained in $Sp(1)=SU(2)$. A closed hyperk\\\"ahler 4-manifold is diffeomorphic to either a torus or the K3 manifold, and the moduli space of all hyperk\\\"ahler metrics are described by Torelli theorems. There have been extensive recent studies on the Gromov-Hausdorff compactification of these moduli spaces, see for example \\cite{odaka2018collapsing} \\cite{sun2021collapsing}.\n\nHyperk\\\"ahler metrics in dimension 4 are the simplest models for Riemannian metrics with special holonomy. Little general existence theory is developed for the latter in dimensions greater than $4$, except for Calabi-Yau manifolds. Recently Donaldson \\cite{donaldson2018elliptic} proposes to study special holonomy metrics on manifolds with boundary and set up suitable elliptic boundary value problems. To make further progress in this direction, it is clear that we need a compactness theory.\n\nIn this paper, we study the boundary value problem for hyperk\\\"ahler 4-manifolds, which serves as the first step towards Donaldson's program.  We follow the general set-up by Fine-Lotay-Singer \\cite{fine2017space} in terms of \\emph{hyperk\\\"ahler triples}. \nA hyperk\\\"ahler triple on an oriented smooth 4-manifold $X$ is a triple of symplectic forms $\\bm{\\omega}=(\\omega_1, \\omega_2, \\omega_3)$  satisfying the following pointwise condition$$\\omega_i\\wedge\\omega_j=\\frac{1}{3}\\delta_{ij}(\\omega_1^2+\\omega_2^2+\\omega_3^2).$$\nIt is well-known that a hyperk\\\"ahler triple $\\bm\\omega$ uniquely determines a compatible hyperk\\\"ahler metric $g_{\\bm\\omega}$ such that for each $i$, $\\omega_i^2=2\\text{dvol}_{g_{\\bm\\omega}}$ and $\\omega_i$ is parallel with respect to the Levi-Civita connection. Conversely, given a hyperk\\\"ahler metric $g$ on $X$, one can choose an orientation and find a compatible hyperk\\\"ahler triple $\\bm\\omega$, which is unique up to a $SO(3)$ rotation.  \n\n\n\n\nNow let $X$ be a compact oriented  smooth 4-manifold with boundary $\\partial X$.  Note $\\partial X$ has an induced orientation defined by contracting a volume form of $X$ with an outward vector field. If $\\bm\\omega$ is a hyperk\\\"ahler triple on $X$, then its restriction to $\\partial X$  is a closed framing $\\bm\\gamma$ on $\\partial X$. The following is a natural filling problem, proposed by \\cite{fine2017space}.\n\n\\begin{question}\\label{question filling}\nWhich closed framing $\\bm\\gamma$ extends to a hyperk\\\"ahler triple on $X$?\n\\end{question}\n\nNotice a framing $\\bm\\gamma$ defines a Riemannian metric $g_{\\bm\\gamma}$ on $\\partial X$ as follows: first, there exists a unique dual coframe $\\bm\\eta=(\\eta_1,\\eta_2,\\eta_3)$ such that $\\gamma_i=\\frac{1}{2}\\delta^{ijk}\\eta_j\\wedge\\eta_k$ and such that  $\\eta_1\\wedge\\eta_2\\wedge\\eta_3$ is compatible with the orientation of $\\partial X$; then the Riemannian metric $g_{\\bm\\gamma}$ is defined by setting $\\bm\\eta$ to be orthonormal. When there is no ambiguity, we always use $\\bm\\eta$ to denote the dual coframe of $\\bm\\gamma$ defined in this way and denote the Hodge star operator of the Riemannian metric by $*_{\\bm\\gamma}=*_{\\bm\\eta}$. It is well-known that if $\\bm\\omega$ is a hyperk\\\"ahler triple, then $g_{\\bm\\omega}|_{\\partial X}=g_{\\bm\\gamma}$; more importantly that the second fundamental form of $\\partial X$ is determined intrinsically by $\\bm\\gamma$ via the  \nmatrix  $*_{\\bm\\eta}(\\eta_i\\wedge d\\eta_j)$. In particular, the mean curvature  $H_{\\bm\\gamma}$ is given by one half of the trace of this matrix, i.e.,  $H_{\\bm\\gamma}=\\frac{1}{2}*_{\\bm\\eta}(\\bm\\eta\\wedge d\\bm\\eta^T)$. \n\nThere are some previous works on Question \\ref{question filling}.    Bryant \\cite{MR2681703} studied the local ``thickening'' problem and obtained both positive and negative results.   It was shown that any real analytic closed framing on a closed oriented 3-manifold $Y$ can be extended to a hyperk\\\"ahler triple on  $Y\\times (-\\epsilon,\\epsilon)$ for some $\\epsilon>0$, and the extension is essentially unique. On the other hand, there exists a smooth closed framing on an open ball  $B^3\n\\subset \\mathbb{R}^3$ that cannot be extended to a hyperk\\\"ahler triple on $B^3\\times (-\\epsilon,\\epsilon)$ for any $\\epsilon>0$. Fine-Lotay-Singer\\cite{fine2017space} studied the local deformation theory for Question \\ref{question filling} and showed that the boundary framings must deform in certain directions. Roughly speaking, let $X=B^4$ for simplicity, suppose $\\bm\\omega$ is a hyperk\\\"ahler triple such that $\\partial X$ has positive mean curvature, and $\\bm\\omega'$ is a nearby hyperk\\\"ahler triple,\n then after moduling out diffeomorphisms of $\\partial X$, the dual coframe of $\\bm\\omega'|_{\\partial X}$ must be a small pertubation of that of $\\bm\\omega|_{\\partial X}$ in the direction of negative frenquency of the boundary Dirac operator defined by $g_{\\bm\\omega}|_{\\partial X}$.\n\n\n\nA sequence of pairs of smooth covariant tensors $(T_i^1,\\cdots, T_i^m)$ on a compact manifold $M$ with empty or nonempty boundary is said to converge in \\emph{Cheeger-Gromov} sense to $(T^1,\\cdots, T^m)$ on $M$, if there exist diffeomorphisms $f_i:M\\rightarrow M$ such that $f_i^*T_i^{1}\\rightarrow T^1,\\cdots, f_i^*T_i^m\\rightarrow T^m $ smoothly on $M$.\n\n\n\n\nOur main result is the following closedness result for Question \\ref{question filling} :\n\n\n\\begin{theorem}\\label{convergence of triples}\nLet $X$ be a compact oriented smooth 4-manifold with boundary, such that there does not exist  $C\\in H_2(X,\\mathbb{Z})$ with self intersection $C^2=-2$. Let $\\bm\\omega_i$ be a sequence of smooth hyperk\\\"ahler triples on $X$. Suppose $\\bm\\omega_i|_{\\partial X}$ converges in Cheeger-Gromov sense to a closed framing $\\bm\\gamma$ on $\\partial X$ such that $H_{\\bm\\gamma}>0$, then there exists a smooth hyperk\\\"ahler triple $\\bm\\omega$ on $X$ with $\\bm\\omega|_{\\partial X}=\\bm\\gamma$ and $\\bm\\omega_i$ converges in Cheeger-Gromov sense to $\\bm\\omega$ on $X$.\n\n\\end{theorem}\nThe proof here includes two parts: the compactness and uniqueness. The former is the main story of this paper, and the latter is a consequence of \\cite{biquard:hal-02928859} or \\cite{anderson2008unique} on unique continuation of Einstein metrics with prescribed boundary metric and second fundamental form. It is worth noting that for the compactness part, no general Riemannian convergence theory can be applied directly. The difficulty here is that we only have data on the boundary, and a priori we do not know anything near the boundary or in the interior. Specifically, we worry about the following three bad geometric behaviours:  curvature blow up, volume collapsing and boundary touching. These things are entangled, making it difficult to rule out any of them. However, we are able to separate these bad behaviours and rule them out. We will also give examples to demonstrate that the assumptions in Theorem \\ref{convergence of triples} are essential, see Remark \\ref{positive mean curvature essential} and \\ref{no -2 curve essential}.\n\n\nSuch $C$ in the assumption of Theorem \\ref{convergence of triples} is usually called a ``\\emph{$-2$ curve}'' in $X$, which appears in Kronheimer's classification of hyperk\\\"ahler ALE spaces \\cite{kronheimer1989construction} \\cite{kronheimer1989torelli}. They appear in bubble limits of volume-noncollapsed hyperk\\\"ahler manifolds. From this, one can replace the ``no $-2$ curve'' condition by an assumption on enhancements of $\\bm\\omega_i|_{\\partial X}$. Let $\\bm\\gamma$ be a closed framing on $\\partial X$. Following \\cite{Donaldson2017BoundaryVP}\\cite{donaldson2018elliptic}, an enhancement of $\\bm\\gamma$ is an equivalent class in the set of triples of  closed 2-forms on $X$ whose restrictions to $\\partial X$ are equal to $\\bm\\gamma$. The equivalence relation is defined by $\\bm\\theta\\sim \\bm\\theta+d\\bm a$, where $\\bm a$ is a triple of smooth 1-forms on $X$ vanishing on $\\partial X$. From the de Rham cohomology exact sequence of the pair $(X,\\partial X)$, \n$$H^2(X,\\partial X)\\rightarrow H^2(X)\\rightarrow H^2(\\partial X)\\rightarrow H^1(X,\\partial X),$$\nwe know $\\bm\\gamma$ has at least one enhancement if and only if each $\\gamma_i$ lies in the kernel of $H^2(\\partial X)\\rightarrow H^1(X,\\partial X)$, and we know the set of all enhancements of $\\bm\\gamma$ is an affine space over $H^2(X,\\partial X)\\otimes\\mathbb{R}^3$. Choose an enhancement of $\\bm\\gamma$ and denote it by $\\hat{\\bm\\gamma}$.  Given a 2-cycle $\\Sigma\\in H_2(X,\\mathbb{Z})$, then for any triple of closed 2-forms $\\bm\\theta\\in\\hat{\\bm\\gamma}$, $\\int_{\\Sigma}\\bm\\theta$ does not depend on the choice of $\\bm\\theta$ and we denote this invariant by $c_{\\hat{\\bm\\gamma},\\Sigma}\\in\\mathbb{R}^3$.\n\nThe proof of Theorem \\ref{convergence of triples} easily adapts to \n\\begin{theorem}\\label{convergence of triples enhancements}\nLet $X$ be a compact oriented smooth 4-manifold with boundary. Let $\\bm\\omega_i$ be a sequence of smooth hyperk\\\"ahler triples on $X$, and $\\hat{\\bm\\gamma}_i$ be the enhancement of $\\bm\\gamma_i=\\bm\\omega_i|_{\\partial X}$ where $\\bm\\omega_i$ lie in. Let $a>0$ be a positive number. Suppose for any $C\\in H_2(X,\\mathbb{Z})$ with self intersection $C^2=-2$, $|c_{\\hat\\gamma_i,C}|\\geq a$ and $\\bm\\omega_i|_{\\partial X}$ converges in Cheeger-Gromov sense to a closed framing $\\bm\\gamma$ on $\\partial X$ such that $H_{\\bm\\gamma}>0$.\nThen there exists a smooth hyperk\\\"ahler triple $\\bm\\omega$ on $X$ with $\\bm\\omega|_{\\partial X}=\\bm\\gamma$, and $\\bm\\omega_i$ converges in Cheeger-Gromov sense to $\\bm\\omega$ on $X$.\n\\end{theorem}\n\n\n\nIt is worth noting that Question \\ref{question filling} is not an elliptic boundary value problem, observed by \\cite{fine2017space}. This can also be seen from the uniqueness result of \\cite{biquard:hal-02928859} or \\cite{anderson2008unique}: the restriction of $\\bm\\omega$ to any open boundary portion determines $g_{\\bm\\omega}$ in the whole interior up to local isometries. So, it is natural to enlarge the class of closed triples of 2-forms on $X$ to obtain an elliptic boundary value problem. In \\cite{donaldson2018elliptic}, Donaldson studied the deformation theory of torsion-free $G_2$ structures on a compact oriented 7-manifold with boundary $M^7$ as follows. Suppose $\\phi_0$ is a smooth torsion-free $G_2$ structure, $\\rho_0=\\phi_0|_{\\partial M^7}$, and denote the enhancement (defined in an analogous way as before) of $\\rho_0$ where $\\phi_0$ lies in by $\\hat\\rho_0$.  Donaldson set up an elliptic boundary value problem for the torsion-free equation. So in particular, if the kernel space at $\\phi_0$ is trivial, then for any small closed 3-form $\\theta$ on $X$,  $\\hat\\rho_0+\\theta|_{\\partial X}$ contains a unique torsion-free $G_2$ structure that is close to $\\phi_0+\\theta$ after gauge fixing. This phenomenon is quite different from the hyperk\\\"ahler case, since in that case we cannot deform the boundary framing arbitrarily to extend it to a hyperk\\\"ahler triple, as we discussed before. \n \n\n\n \nIt is well-known that $G_2$ structures have a reduction to dimension 4. Consider $X^4\\times T^3$. A triple of two forms $\\bm\\omega=(\\omega_1,\\omega_2,\\omega_3)$ on $X^4$  defines a 3-form on $X^4\\times T^3$ by\n\\begin{equation}\\label{g2 and hypersymplectic}\n \\phi=dt^1\\wedge dt^2\\wedge dt^3-\\omega_1\\wedge dt^1-\\omega_2\\wedge dt^2-\\omega_3\\wedge dt^3.\n\\end{equation}\n The triple $\\bm\\omega$ is called torsion-free hypersymplectic if $\\phi$ is a torsion-free $G_2$ structure. Locally, this is a weaker condition than being hyperk\\\"ahler, see examples in \\cite{Donaldson2017BoundaryVP} or \\cite{fine2020report}. Donaldson observed in  \\cite{Donaldson2017BoundaryVP} that the boundary value problem for torsion-free $G_2$ structures can also be reduced to dimension 4. So, a compactness result is helpful to solve this dimension reduced boundary value problem. \n\nSimilar to the hyperk\\\"ahler case, a torsion-free hypersymplectic triple $\\bm\\omega$ defines a Riemannian metric $g_{\\bm\\omega}$ and a positive definite $SL(3,\\mathbb R)$-valued function $\\bm Q=(Q_{ij})$ such that \n$$\\omega_i\\wedge \\omega_j=2Q_{ij}\\text{dvol}_{g_{\\bm\\omega}}$$\nWe denote $\\bm Q'$ the restriction of $\\bm Q$ to $\\partial X$. When there is ambiguity, we use notations $\\bm Q_{\\bm\\omega}$, $\\bm Q'_{\\bm\\omega}$ to denote their dependence on $\\bm\\omega$. One can show that the mean curvature of $\\partial X$ has an explicit expression in terms of $\\bm\\gamma$, $\\bm Q'$, and we denote this explicit expression by $H_{\\bm\\gamma,\\bm Q'}$. Note that on $\\partial X$,  $\\bm\\gamma,\\bm Q'$ are subject to the constraints $d\\bm\\gamma=0,  d(\\bm\\gamma(\\bm Q')^{-1})=0$. \n\n\nWe have the following analogue of Theorem \\ref{convergence of triples enhancements}:\n\n\n\n\n\\begin{theorem}\\label{convergence of hypersymplectic triples}\nLet $X$ be a compact oriented smooth 4-manifold with boundary. Let $\\bm\\omega_i$ be a sequence of smooth torsion-free hypersymplectic triples on $X$, and $\\hat{\\bm\\gamma}_i$ be the enhancement of $\\bm\\gamma_i=\\bm\\omega_i|_{\\partial X}$ where $\\bm\\omega_i$ lie in.  Let $a>0$ be a positive number. Suppose for any $C\\in H_2(X,\\mathbb{Z})$ with self intersection $C^2=-2$, $|c_{\\hat\\gamma_i,C}|\\geq a$, and $(\\bm\\gamma_i,\\bm Q_i')$  converges in Cheeger-Gromov sense to some pair $(\\bm\\gamma,\\bm Q')$ on $\\partial X$, such that $\\bm\\gamma$ is a framing, $\\bm Q'$ is positive definite and $H_{\\bm\\gamma,\\bm Q'}>0$. Then there exists a smooth torsion-free hypersymplectic triple $\\bm\\omega$ on $X$ with $\\bm\\omega|_{\\partial X}=\\bm\\gamma$, $\\bm Q_{\\bm\\omega}'=\\bm Q'$, and $\\bm\\omega_i$\nconverges in Cheeger-Gromov sense to $\\bm\\omega$.\n\n\\end{theorem}\n \nNote that Theorem \\ref{convergence of hypersymplectic triples} includes the previous two versions.\n\nThis paper is organized as follows: In Section \\ref{section triples}, we discuss basics of hyperk\\\"ahler triples on manifolds with boundary. In Section \\ref{section Riemannian geometry}, we discuss Riemannian geometry for manifolds with boundary and Riemannian convergence theory. In Section \\ref{section main proof}, we prove Theorem \\ref{convergence of triples}  and Theorem \\ref{convergence of triples enhancements} and give some remarks about the proofs. In Section \\ref{section torsion-free triples}, we discuss some basics for torsion-free hypersymplectic triples and prove Theorem \\ref{convergence of hypersymplectic triples}.\n\n\n\\textbf{Notations.}\n$$\\mathbb{R}_+^n=\\{x\\in\\mathbb{R}^n:x^n\\geq 0\\},$$\n $$B_r=\\{x\\in\\mathbb{R}^n: |x|<r\\},$$\n $$B_r^+=B_r\\cap \\mathbb{R}_+^n,$$\n $$\\tilde B_r=B_r^+\\cap \\partial\\mathbb{R}_+^n,$$\n$$\\partial \\tilde{B}_r=\\{x\\in \\mathbb{R}^n:|x|=r,x^n=0\\},$$\n$$\\partial^+ {B}_r^+=\\{x\\in \\mathbb{R}^n:|x|=r,x^n>0\\}.$$\n\n\\textbf{Acknowledgements.} The author is very grateful to his thesis advisor Song Sun for suggesting the problem, constant support and many inspiring discussions. I thank Antoine Song and Chengjian Yao for some useful comments. I thank Simon Donaldson and Jason Lotay for their interest in this work. \n\nThe author is supported by Simons Collaboration Grant on Special Holonomy in Geometry, Analysis, and Physics (488633, S.S.).\n\n\\section{Hyperk\\\"ahler triples and closed framings}\\label{section triples}\n\nThe discussions in this section are well-known.\n\n\\subsection{Pointwise theory}\n Let $V$ be an oriented 4-dimensional vector space, and ${\\bm\\omega}=(\\omega_1,\\omega_2,\\omega_3)\\in\\Lambda^2(V^*)\\otimes \\mathbb{R}^3$. Suppose $\\bm\\omega$ is a \\emph{definite} triple, i.e., \n$\\omega_i$ spans a maximum positive subspace of $\\Lambda^2(V^*)$ with respect to the wedge product, then $\\omega_i$ defines a unique conformal structure on $V$ by making each $\\omega_i$ self dual. Fix a volume form $\\mu_0$ on $V$ that defines the orientation of $V$, write $\\omega_i\\wedge\\omega_j=2q_{ij}\\mu_0$, we define a matrix $\\bm Q$ associated to the definite triple $\\bm\\omega$ by $Q_{ij}=\\frac{q_{ij}}{\\det(q_{ij})^\\frac{1}{3}}$, which does not depend on the choice of $\\mu_0$. We use $\\bm Q^{-1}= (Q^{ij})$ to denote the inverse matrix of $\\bm Q$. If we write $$\\omega_i\\wedge\\omega_j=2 Q_{ij}\\mu,$$ then $\\mu$ is a volume form intrinsically defined by $\\bm\\omega$. We define a unique metric $\\langle,\\rangle_{\\bm\\omega}$ on $V$ in the conformal structure by making $\\mu$ the volume form. Explicitly, $$\\langle u,v\\rangle_{\\bm\\omega}=\\frac{1}{6}\\sum_{i,j,k=1}^3 \\delta^{ijk}\\frac{\\iota_u \\omega_i\\wedge\\iota_v\\omega_j\\wedge\\omega_k}{\\mu}.$$\n So $$\\langle u,u \\rangle_{\\bm\\omega}=\\frac{\\iota_u\\omega_1\\wedge\\iota_v\\omega_2\\wedge\\omega_3}{\\mu}.$$\nDenote $*_{\\bm\\omega}$ the Hodge star operator defined by this metric.\n\nLet $W$ be an orientated 3-dimensional  vector space, and ${\\bm\\gamma}=(\\gamma_1,\\gamma_2,\\gamma_3)$ be a framing on $W$, i.e., a basis for $\\Lambda^2W^*$. Then by elementary linear algerba, there exists a coframe  ${\\bm\\eta}=(\\eta_1,\\eta_2,\\eta_3)$ such that\n\n\\begin{equation}\\label{framing and coframe}\n\\gamma_i=\\frac{1}{2}\\delta^{ijk}\\eta_j\\wedge\\eta_k.\n\\end{equation}\nSuch $\\bm\\eta$ is uniquely determined up to a sign and we choose $\\bm\\eta$ such that $\\eta_1\\wedge\\eta_2\\wedge\\eta_3$ defines the orientation of $W$ and we denote this volume form by $\\text{vol}_{\\bm\\gamma}$. There is a unique metric on $W$, denoted by $\\langle,\\rangle_{\\bm\\gamma}$, that makes $\\bm\\eta$ an orthonormal coframe. Denote $*_{\\bm\\gamma}=*_{\\bm\\eta}$ by the Hodge star operator of $\\langle,\\rangle_{\\bm\\gamma}$. Let $e_i\\in W$ be the dual vector of $\\eta_i$, so $\\eta_i(e_j)=\\delta_{ij}$.  Then $\\bm e=(e_1,e_2,e_3)$ is a frame of $W$. Conversely, given a coframe $\\bm\\eta=(\\eta_1,\\eta_2,\\eta_3)$ on $W$ compatible with the orientation, one can define a framing $\\bm\\gamma=(\\gamma_1,\\gamma_2,\\gamma_3)$ via (\\ref{framing and coframe}), and a volume form $\\text{vol}_{\\bm\\eta}=\\text{vol}_{\\bm\\gamma}$, a metric $\\langle,\\rangle_{\\bm\\eta}=\\langle,\\rangle_{\\bm\\gamma}$, a Hodge star opertaor $*_{\\bm\\eta}=*_{\\bm\\gamma}$. \n\n\nNow if $W\\subset V$ is a 3-dimensional subspace, $\\bm\\omega=(\\omega_1,\\omega_2,\\omega_3)$ is a definite triple on $V$,   and $\\bm\\gamma=(\\gamma_1,\\gamma_2,\\gamma_3)$ is the restriction of $\\bm\\omega$ to $W$.  Let $\\langle,\\rangle_{W}$ be the restriction of the metric $\\langle,\\rangle_{\\bm\\omega}$ on $W$, which defines a volume form $\\text{vol}_W$ and a Hodge star operator $*_W$  compatible with the orientation of $W$.  Since $\\omega_i$ are self-dual, we can write $$\\omega_i=\\nu^*\\wedge *_W\\gamma_i+\\gamma_i,$$ where $\\nu^*=*_{\\bm\\omega}\\text{vol}_W$, then we have $$\\omega_i\\wedge\\omega_j=2\\nu^*\\wedge*_W\\gamma_i\\wedge\\gamma_j=2\\langle \\gamma_i,\\gamma_j\\rangle_{W}\\nu^*\\wedge \\text{vol}_{W}=2\\langle\\gamma_i,\\gamma_j\\rangle_W\\mu.$$ Hence on $W$, $$Q_{ij}=\\langle\\gamma_i,\\gamma_j\\rangle_W,$$ Since $\\det(\\bm Q)=1$, we have $\\text{vol}_{\\bm\\gamma}=\\text{vol}_{W}$. If furthermore we assume $Q_{ij}=\\delta_{ij}$, then $\\langle\\gamma_i,\\gamma_j\\rangle_{W}=\\delta_{ij}=\\langle \\gamma_i,\\gamma_j\\rangle_{\\bm\\gamma}$. In this case, $\\langle,\\rangle_{\\bm\\gamma}=\\langle,\\rangle_W$ as an inner product on $W=\\Lambda^1 W$, so in particular $*_W=*_{\\bm\\gamma}$. \n\n\n\n\n\n\n\n\\subsection{Local theory}\nNow we move our pointwise discussions to manifolds. Let $X$ be a oriented 4-manifold with boundary, $\\bm\\omega=(\\omega_1,\\omega_2,\\omega_3)$ be a smooth section in $\\Gamma(X, \\Lambda^2T^*X\\otimes\\mathbb{R}^3)$ such that it is a definite triple pointwise, and $\\bm\\gamma=(\\gamma_1,\\gamma_2,\\gamma_3) $ be its restriction to $\\partial X$. By the discussions above, $\\bm\\omega$ defines a matrix valued function $\\bm Q=(Q_{ij})$, a volume form $\\mu$, and a Riemannian metric $g_{\\bm\\omega}$ which equals to  $\\langle,\\rangle_{\\bm\\omega}$ on the tangent spaces at each point. Similarly, $\\bm\\gamma$ defines $\\bm\\eta=(\\eta_1,\\eta_2,\\eta_3)\\in \\Omega^1(\\partial X)\\otimes\\mathbb{R}^3,\\bm e=(e_1,e_2,e_3)\\in \\Gamma(\\partial X,T\\partial X)\\otimes\\mathbb{R}^3$.\nDenote $\\nabla$ the Levi-Civita connection of $g_{\\bm\\omega}$.\n\n\n\n\n\n\\begin{definition}\n$\\bm\\omega$ is a {hypersymplectic} triple if $d\\omega_i=0$. A hypersymplectic triple $\\bm\\omega$ is called {torsion-free} if $d(Q^{ij}\\omega_j)=0$, and is called {hyperk\\\"ahler} if $Q_{ij}=\\delta_{ij}$.\n\\end{definition}\n\nNote that the torsion-free definition coincides with the one defined in the introduction by direct calculation.\n\n\nLet $\\nu$ be the outward unit normal vector field of $\\partial X$. We are going to calculate the second fundamental form of $\\partial X$, say \n$II(v,w)=\\langle\\nabla_v\\nu, w\\rangle_{\\partial X}$.  Let $S\\in\\Gamma(\\partial X,\\text{End}\\ T\\partial X)$ be the shape operator, i.e., $\\langle v,S(w)\\rangle_{\\partial X}=II(v,w)$. Denote $\\Gamma=(\\Gamma_{ij})$ the symmetric matrix $$\\Gamma_{ij}=\\frac{1}{2}\\langle\\gamma_i,d(*_{\\bm\\gamma} \\gamma_j)\\rangle_{\\bm\\gamma}+\\frac{1}{2}\\langle\\gamma_j,d(*_{\\bm\\gamma} \\gamma_i)\\rangle_{\\bm\\gamma}=\\frac{1}{2} *_{\\bm\\eta}(\\eta_i\\wedge d\\eta_j+\\eta_j\\wedge d\\eta_i)  $$ which is completely determined by $\\bm\\gamma$. Denote the matrix $II(e_i,e_j)$ by $A$ and $H=Tr S$.\n\n\n\n\\begin{lemma}\\label{hyperkahler triple second fundamental form}\nIf $\\bm\\omega$ is a hyperk\\\"ahler triple, then \n$$A=\\frac{1}{2}( Tr \\Gamma)I-\\Gamma,$$ In particular, $$2H=*_{\\bm\\eta}(\\bm\\eta\\wedge d\\bm\\eta^T)=\\langle \\gamma_1, d(*_{\\bm\\gamma}\\gamma_1)\\rangle_{\\bm\\gamma}+\\langle \\gamma_2, d(*_{\\bm\\gamma}\\gamma_2)\\rangle_{\\bm\\gamma}+\\langle \\gamma_3, d(*_{\\bm\\gamma}\\gamma_3)\\rangle_{\\bm\\gamma},$$\n$$|S|^2=Tr  {\\Gamma}^2 -H^2.$$\n\\end{lemma}\n\n\\begin{proof}\n\n\n\n Fix a point $p\\in \\partial X$, choose a semi-geodesic coordinate system centered at $p$, say $(x^1,x^2,x^3,t)$, such that a neighborhood of $p$ is identified with $\\{t\\geq 0\\}$\nand its intersection with $\\partial X$ is identified with $\\{t=0\\}$. The hyperk\\\"ahler triple can be written as \n$$\\bm\\omega=-dt\\wedge *_{\\bm\\gamma_t}\\bm\\gamma_t+\\bm\\gamma_t,$$\nwhere $\\bm\\gamma_t$ is a smooth family of closed framings on $\\partial X$ such that $\\bm\\gamma_{0}=\\bm\\gamma$ and $$\\frac{\\partial \\bm\\gamma_t}{\\partial t}=-d(*_{\\bm\\gamma_t}\\bm\\gamma_t).$$\nWe can further choose \n $(x^1,x^2,x^3)$ to be a normal coordinate system for $\\partial X$ at $p$ of the metric $g_{\\bm\\omega}|_{\\partial X}$ such that $e_i=\\partial_{x^i}$ at $p$. Write $\\partial_{x^i}=a_i^k(x,t)e_k(x,t)$, where $\\bm e_k(x,t)=(e_1(x,t),e_2(x,t), e_3(x,t))$ is the dual frame of $\\bm\\eta_t=*_{\\bm\\gamma_t}\\bm\\gamma_t$. Then  at $p$,\n \\begin{equation}\\label{II(ei,ej) calculation}\n      II(e_i,e_j)=II(\\partial_{x^i},\\partial_{x^j})=-\\frac{1}{2}\\partial_t g_{ij}=-\\frac{1}{2}\\partial_t(a_i^ka_j^k)=-\\frac{1}{2}(\\partial_t a_i^k+\\partial_ta_j^k).\n \\end{equation}\nNote that at $p$, $$\\partial_t\\eta_p=\\partial_t a_m^pdx^m=\\partial_t a_m^p\\eta_m,$$ then  we have \n\\begin{equation}\\label{gammaidetaj calculation}\n    \\begin{split}\n        \\langle \\gamma_i,d(*_{\\bm\\gamma}\\gamma_j)\\rangle_{\\bm\\gamma}\n  &=-\\langle\\gamma_i,\\partial_t\\gamma_{j}\\rangle_{\\bm\\gamma}\\\\\n  &=-\\langle \\frac{1}{2}\\delta^{ikl}\\eta_k\\wedge\\eta_l,\\frac{1}{2}\\partial_t(\\delta^{jpq}\\eta_p\\wedge\\eta_q)\\rangle_{\\bm\\gamma}\\\\\n  &=-\\frac{1}{4}\\delta^{ikl}\\delta^{jpq}(\\partial_t a_m^p\\langle \\eta_k\\wedge\\eta_l,\\eta_m\\wedge\\eta_q\\rangle_{\\bm\\gamma}+\\partial_ta_n^q\\langle \\eta_k\\wedge\\eta_l,\\eta_p\\wedge\\eta_n\\rangle_{\\bm\\gamma})\\\\ &=\n-\\frac{1}{4}\\delta^{ikl}\\delta^{jpq}(\\partial_t a_m^p(\\delta^{km}\\delta^{lq}-\\delta^{kq}\\delta^{lm})+\\partial_t a_n^q(\\delta^{kp}\\delta^{ln}-\\delta^{kn}\\delta^{lp}))\\\\&=-\\delta^{ij}\\partial_t a_k^k+\\partial_t a_j^i.\n    \\end{split}\n\\end{equation}\n Combining (\\ref{II(ei,ej) calculation}) and (\\ref{gammaidetaj calculation}), we have \n $$\\Gamma= (Tr A) I-A,$$ so $Tr \\Gamma=2Tr A$, $A=\\frac{1}{2}(Tr \\Gamma) I-\\Gamma$.\n\n\n\\end{proof}\n\n\\section{General Riemannian geometry}\\label{section Riemannian geometry}\n\n\n\\subsection{Evolution equations of hypersurfaces}\nWe refer to \\cite{petersen2016riemannian} Section 3.2 and \\cite{koiso1981hypersurfaces} for discussions in this section.\n\nLet $M$ be a Riemannian manifold, $f$ be a smooth distance function on $M$, i.e., $|\\nabla f|= 1$, so $\\nabla_{\\nabla f}\\nabla f=0$. The $(1,1)$ tensor  corresponds to  $\\text{Hess} f$ is\n\\begin{equation}\\label{(1,1) tensor shape operator}\n    S(X)=\\nabla_X\\nabla f,\n\\end{equation}\n and its trace is $H=\\Delta f\\in C^\\infty(M)$. \n  When $\\Sigma\\subset M$ is a hypersurface defined by a level set of $f$, the restriction of $S$ to $T\\Sigma$ has image in $T\\Sigma$, which is the shape operator of $\\Sigma$. The second fundamental form of $\\Sigma$ with respect to $\\nabla f$ is  $II(X,Y):=\\langle S(X), Y\\rangle=\\text{Hess} f(X,Y)$, so $H$ is the mean curvature and $\\overrightarrow{H}=-H\\nabla f$ is the mean curvature vector.\n  By tensor calculations, we have evolution equations of second fundamental forms\n\n\\begin{equation}\\label{simplified radial curvature equation}\nL_{\\nabla f} S+S^2=-R(\\cdot,\\nabla f)\\nabla f,\n\\end{equation}\n\\begin{equation}\\label{radial curvature equations''}\nL_{\\nabla f} \\text{Hess} f-\\text{Hess}^2 f=-Rm(\\cdot,\\nabla f,\\cdot,\\nabla f),\n\\end{equation}\nwhere $$\\text{Hess}^2 f(X,Y)=\\langle S^2(X),Y \\rangle=\\langle S(X),S(Y)    \\rangle,$$\nand one has the equality\n\\begin{equation}\\label{Lie derivative S same as covariant derivative S}\n    L_{\\nabla f}S=\\nabla_{\\nabla f} S.\n\\end{equation}\nTake the trace of $(\\ref{simplified radial curvature equation})$, we have\n\\begin{equation}\\label{evolution of mean curvature}\nL_{\\nabla f} H=-|S|^2-\\text{Ric}(\\nabla f,\\nabla f). \n\\end{equation}\nwhere $|S|^2:=Tr(S^2)$ is the norm square of the shape operator.\n\nBesides the evolution equations,  we have Gauss equations on $\\Sigma$ \n\\begin{equation}\n    \\begin{split}\n        Rm_{M}(X,W,Y,Z)=&Rm_{\\Sigma}(X,W,Y,Z)+II(X, Z)II(W, Y)\\\\&-II(X, Y)II(W, Z).\n    \\end{split}\n\\end{equation} \nTake the trace with respect to $W,Z$, we have \n\\begin{equation}\\label{trace gauss equation}\n\\text{Ric}_{M}=\\text{Ric}_{\\Sigma}+\\text{Hess}^2 f-H\\cdot \\text{Hess}f+Rm_{M}(\\cdot, \\nabla f,\\cdot,\\nabla f),\n\\end{equation}\ntake the trace again, we have\n\\begin{equation}\\label{scalar curvatures and hypersurface}\n    R_M=R_{\\Sigma}+|S|^2- H^2+2\\text{Ric}_M(\\nabla f,\\nabla f),\n\\end{equation}\n where $R$ denote scalar curvatures.\nUse equations (\\ref{radial curvature equations''}) and (\\ref{trace gauss equation}) to cancel the curvature term involving $\\nabla f$, we get \n\\begin{equation}\\label{evolution equation for 2nd form}\nL_{\\nabla f} \\text{Hess} f=\\text{Ric}_{\\Sigma}-\\text{Ric}_M+ 2\\text{Hess}^2 f-H \\cdot \\text{Hess}f.\n\\end{equation}\n\n\n\\subsection{The boundary exponential map}\n\n\n\nLet $(M,g)$ be a complete Riemannian manifold with boundary, which means the induced metric space is complete. \nDenote $T^\\perp \\partial M$  the normal line bundle of $\\partial M$, which is a trivialized by the inward unit normal vector field $N$. We identify $T^\\perp\\partial M$ with $\\partial M\\times \\mathbb R$ via this trivialization. For $p\\in\\partial M$, denote $\\gamma_p(t)$ the geodesic such that $\\gamma_p(0)=p,\\gamma_p'(0)=N_p$.  Denote $$D(p)=\\inf\\{t>0| \\gamma_p(t)\\in\\partial M\\}\\in (0,\\infty],$$ $$\\tau(p)=\\sup\\{t>0|d(\\gamma_p(t),\\partial M)=t\\}\\in(0,\\infty].$$\nWe have a subset of $U_{\\partial M}\\subset T^\\perp\\partial M$ defined by\n $$U_{\\partial M}=\\{(p,tN_p)\\in T^{\\perp}\\partial M|0\\leq t <D(p) \\},$$ which is the domain of the \\text{boundary exponential map}\n \\begin{equation}\\label{boundary exponential map definition}\n      \\exp^\\perp:U_{\\partial M}\\rightarrow M, (p,s)\\mapsto\\gamma_p(s).\n \\end{equation}\n and define $$V_{\\partial M}=\\{(p,tN_p)\\in T^{\\perp}\\partial M|0\\leq t <\\tau(p) \\}\\subset U_{\\partial M}.$$\n\n\n \n There are some definitions, notations and terminologies related to the boundary exponential map that appear in this paper:\n \\begin{itemize}\n     \\item \n The \\emph{boundary injectivity radius} $i_b$ is defined to be the supremum of $s\\geq 0$ such that $\\exp^\\perp|_{\\partial M\\times[0,s)}$ is a diffeomorphism onto its image.\n \\item A \\emph{focal point} $q$ of $\\partial M$ is a critical value of the boundary exponential map (\\ref{boundary exponential map definition}). If $q$ lies in $\\gamma_p$ for some $p\\in\\partial M$, we say $q$ is a focal point along $\\gamma_p$. \n \\item A \\emph{foot point} of $q\\in M$ is a point  $p\\in \\partial M$ such that $d(q,p)=d(q,\\partial M).$ \n \\item A \\emph{cut point} of $\\partial M$ is a point $q\\in M$ such that there exists a foot point $p$ of $q$ such that $d(q,p)=\\tau(p)$. We also say $q$ is a cut point of $p$.\n \\item When we say a covariant tensor on $M$ is written in \\emph{geodesic gauge}, we mean the pull back of this tensor via $\\exp^\\perp$.\n \\item \n For a  subset $B\\subset\\partial M$, we use the notation  $$C(B,t_1,t_2)=\\exp^\\perp(B\\times[t_1,t_2))$$ to denote a metric cylinder with base $B$.\n \\item $N_r(\\partial M,g\n):=\\{x\\in M|d(x,\\partial M)\\leq r\\}.$\n \\item The $(1,1)$ tensor $S$ in (\\ref{(1,1) tensor shape operator})  is defined with respect to $f=-d(\\cdot,\\partial M)$ near $\\partial M$, so $\\nabla f=-N$ on $\\partial M$, and $II\\geq 0$ if $\\partial M$ is convex.\n \\end{itemize}\n \nHere are some remarks about some of these definitions:\n \n \\begin{itemize}\n     \\item By definition,  $\\gamma_p(s)$ is a focal point of along $\\gamma_p$, if and only if there exists a non-zero \\emph{$\\partial M$-Jacobi field} $V$ along $\\gamma_p$ (a Jacobi field with $ V(0)\\in T_p \\partial M, V'(0)+S(V(0))\\in T_p^\\perp \\partial M$) such that $V(s)=0$. If there is no focal point along $\\gamma_p|_{[0,l)}$, then \n$$I_0(W,W)=\\int_{0}^l \\langle W', W'\\rangle-\\langle R(W,\\gamma_p')\\gamma_p',W \\rangle dt- \\langle S(W),W\\rangle(0)\\geq 0 $$ for any piecewise smooth vector field $W$ along $\\gamma$ with $W(0)\\in T_{\\gamma(0)}\\partial M$. \n\\item \nBy Lemma 3.2 in \\cite{sakurai2017rigidity}, $\\tau$ defines a continuous map from $\\partial M$ to $(0,\\infty]$.  It is well-known that by a second variation argument, the first focal point along $\\gamma_p$ appears no later than $\\tau(p)$, and moreover one can argue by contradiction to get (See Lemma 3.6 in \\cite{sakurai2017rigidity}) \n \\end{itemize}\n \n \n\n\n\\begin{proposition}\\label{first focal point two foot point}\n$q\\in M$ is a cut point of $p$ if and only if at least one of the following holds:\n\\begin{itemize}\n    \\item $q$ is the first focal point of $\\gamma_p$ ;\n    \\item $q$ has at least two foot points.\n\\end{itemize}\n\\end{proposition}\n\\noindent From this,  we have $\\exp^\\perp|_{V_{\\partial M}}$ is a diffeomorphism and\n$$i_b=\\inf_{p\\in\\partial M}\\tau (p).$$\n\n\nIt is worth noting that if $M$ is embedded in some complete Riemannian manifold $M'$ of the same dimension and view $\\partial M$ as an embedded hypersurface of $M'$, then it may happen that a ``focal point'' of $\\partial M$ in $M'$ lies outside $M$ if we define the boundary exponential map in the whole normal bundle $T^\\perp\\partial M$. However, our definition is intrinsic for $M$.\n\n\n\n\\subsection{Manifolds with mean convex boundary}\n\nNow we focus on manifolds with mean convex boundary. We will summarize some results and discuss a new result.\n\nThe following results are well-known,  see for example \\cite{li2014sharp} and \\cite{Donaldson2017BoundaryVP}.\n\n\n\\begin{proposition}\\label{diam}\nLet $(M,g)$ be a compact, connected Riemannian manifold with boundary, $\\emph{Ric}_M\\geq 0$. Suppose $\\partial M$ has mean curvature $H\\geq H_0>0$, then $\\partial M$ is connected,\n\n$$\\pi_1(M,\\partial M)=0,$$ $$\\sup_{q\\in M}d(q,\\partial M)\\leq (n-1)H_0^{-1},$$  $$\\emph{vol}(M)\\leq C(n)H_0^{-1}\\emph{vol}(\\partial M).$$\n\\end{proposition}\n\n\n\n\n\n\n\n\\begin{proof}\n\nIf $\\pi_1(M,\\partial M)\\neq 0$ or $\\pi_0(\\partial M)\\neq 0$, then every non-trivial class contains a non-trivial unit speed geodesic $\\gamma:[0,l]\\rightarrow M $ which minimize the length of all curves in its class.\nFrom the first variation formula, $\\gamma$ intersects boundary perpendicularly at both end points. Pick an orthonormal basis $V_i,1\\leq i\\leq n-1$ of $T_{\\gamma(0)}\\partial M$ and parallel them transport along $\\gamma$ to get $V_i(t)$. Let $\\gamma_{i,s}(t)$ be a family of curves centered at $\\gamma $ with variation field $V_i(t)$. By second variational formula, $$0\\leq \\sum_{i=1}^{n-1} \\frac{d^2}{ds^2}\\Big{|}_{s=0}E(\\gamma_{i,s})=\\int_0^l -\\text{Ric}(V_i(t),V_i(t))dt-H(\\gamma(0))-H(\\gamma(l))<0,$$ which is a contradiction.\n\nIf for some $q\\in M$, $\\tilde{l}:=d(q,\\partial M)> (n-1)H_0^{-1}$. Let $p$ be a foot point of $q$. Let $V_i,1\\leq i\\leq n-1$ be an orthonormal basis of $T_{p}\\partial M$ and parallel them transport along $\\gamma_p$ to get $V_i(t)$, and denote $\\tilde{V}_i(t)=(\\tilde l-t)V_i(t)$, $\\tilde \\gamma_{i,s}(t)$  a family of curves centered at $\\gamma_p $ with variation field $\\tilde{V}_i(t)$, then $$0\\leq \\sum\\limits_{i=1}^{n-1}\\frac{d^2}{ds^2}\\Big{|}_{s=0}E(\\tilde\\gamma_{i,s})=(n-1)\\tilde l-\\int_0^{\\tilde{l}} \\text{Ric}(tV_i(t),tV_i(t))dt-\\tilde{l}^2H(\\gamma(0))<0,$$ which is a contradiction.\nThe volume upper bound is by volume comparison, see \\cite{heintze1978general}.\n\\end{proof}\n\n\\begin{remark}\nNote that when $M$ is connected,   $\\pi_1({M},\\partial M)=0$ is equivalent to that $\\partial M$ is connected and the natural map $\\pi_1(\\partial M)\\rightarrow \\pi_1(M)$ is surjective, the latter of which means that fix a point $p_0\\in\\partial M$, then\nfor any closed path $x(t):0\\leq t\\leq 1$  in $M$ with $x(0)=x(1)=p_0$, there exists a homotopy $F_s(t):0\\leq s, t\\leq 1$ with $F_s(0)=F_s(1)=p_0, F_0(t)=x(t)$ such that $F_1(t)\\in \\partial M$.\n\\end{remark}\n\n\n\n\n\n\nThe following result is well-known, see Lemma 6.3 of \\cite{kodani1990convergence}.\n\n\n\\begin{proposition}\nLet $M$ be a complete Riemannian manifold with  nonempty  compact boundary. If there are no focal point whose distance to $\\partial M$ is equal to $i_b$, then there exists a smooth geodesic of length $2i_b$ which is perpendicular to $\\partial M$ at both end points.\n\\end{proposition}\n\n\\begin{proof}\n$i_b=\\inf\\limits_{p\\in\\partial M}\\tau (p)>0$. Suppose the infimum is achieved at $p_1\\in\\partial M$. Then by assumption and Proposition \\ref{first focal point two foot point} $\\gamma_{p_1}(i_b)$ has another foot point $p_2$. We claim $\\gamma_{p_1}'(i_b)=-\\gamma_{p_2}'(i_b)$, so $D(p_1)=D(p_2)=2i_b$ and $\\gamma_{p_1}:[0,2i_b]\\rightarrow M$ is the smooth geodesic we want. By assumption, we can find smooth distance functions $h_1, h_2$  extending $d(\\cdot,\\partial M)$  near $\\gamma_{p_1}|_{[0,i_b]}$, $\\gamma_{p_2}|_{[0,i_b]}$, respectively. Consider the smooth hypersurface $\\Sigma=(h_1-h_2)^{-1}(0)$ near $q:=\\gamma_{p_1}(i_b)=\\gamma_{p_2}(i_b)$. Then $v=\\nabla h_1(q)+\\nabla h_2(q)\\in T_{q}\\Sigma$. If it is non-zero, then $\\langle \\nabla h_1(q)+\\nabla h_2(q) , v \\rangle>0$. Without loss of generality, assume $\\langle\\nabla h_1,v\\rangle>0$. Then in the direction of $-v$ in $\\Sigma$, we have some point $q'\\in\\Sigma$ with $h_1(q')<h_1(q)$. Hence $q'$ has two foot points and $d(q',\\partial M)=h_1(q')<i_b$, which is a contradiction.\n\\end{proof}\nIt is worth noting that in this proof, Kodani used the first order variation of $h_1$ on $\\Sigma$ to lead a contradiction. We can also investigate the second order variation of $h_1$ on $\\Sigma$ and prove the following:\n\n\n\n\n\n\n\n\n\n\\begin{proposition}\\label{existence of focal points}\nLet $M$ be a compact Riemannian manifold with mean convex boundary, $\\emph{Ric}_M\\geq 0$,  then there exists a focal point of $\\partial M$ whose distance to $\\partial M$ is equal to $i_b$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose not, by the previous proposition, we have a smooth geodesic of length $2i_b$ which is perpendicular to $\\partial M$ at both end points. We use notations $p_1,p_2, q, h_1,h_2,\\Sigma$ as in the previous proposition. We claim $\\Delta_\\Sigma h_1(q)<0$, so we get another point $q''\\in \\Sigma$ near $p$ with $h_1(q'')<h_1(q)=i_b$ and get a contradiction. Denote $N_{0}=\\nabla h_1(q)=-\\nabla h_2(q),\\Sigma_1=h_1^{-1}(i_b),\\Sigma_2=h_2^{-1}(i_b)$, then $N_0$ is a common unit normal vector for $\\Sigma,\\Sigma_1,\\Sigma_2$ at $q$. Let $II_\\Sigma, II_{\\Sigma_1},II_{\\Sigma_2}$ be second fundamental forms with respect to $N_0$ at $q$.\nThen at $q$, $$II_\\Sigma= \\frac{1}{|\\nabla (h_1-h_2)|}\\text{Hess} (h_1-h_2)=\\frac{1}{2} \\text{Hess}(h_1-h_2)=\\frac{1}{2}(II_{\\Sigma_1}+II_{\\Sigma_2}),$$  hence \n $$H_\\Sigma=\\frac{1}{2}(H_{\\Sigma_1}+H_{\\Sigma_2}),\n\\overrightarrow{H}_{\\Sigma}=\\frac{1}{2}(\\overrightarrow{H}_{\\Sigma_1}+\\overrightarrow{H}_{\\Sigma_2}).$$From the formula of Laplace operator on a hypersurface, we know that at $q$,\n$$\\Delta_\\Sigma h_1=\\Delta h_1-\\text{Hess} h_1(N_0,N_0)+\\langle \\nabla h_1, \\overrightarrow{H}_\\Sigma \\rangle ,$$\n$$\\Delta_{\\Sigma_1} h_1=\\Delta h_1-\\text{Hess} h_1(N_0,N_0)+\\langle \\nabla h_1,\\overrightarrow{H}_{\\Sigma_1} \\rangle.$$\nSince $h_1$ is a constant on $\\Sigma_1$, $\\Delta_{\\Sigma_1}h_1=0$, hence\n\\begin{equation}\n\\begin{split}\n\\Delta_{\\Sigma} h_1 &=\\Delta_{\\Sigma} h_1-\\Delta_{\\Sigma_1}h_1=\\langle\\nabla h_1, \\overrightarrow H_{\\Sigma}-\\overrightarrow H_{\\Sigma_1}\\rangle=\\langle \\nabla h_1, \\frac{1}{2}(\\overrightarrow H_{\\Sigma_2}-\\overrightarrow H_{\\Sigma_1})\\rangle \\\\\n&=-\\frac{1}{2}(H_{\\Sigma_2}-H_{\\Sigma_1}).\n\\end{split}\n\\end{equation}\nSince $\\partial M$ is mean convex, $\\text{Ric}_M\\geq 0$, by the evolution equation of mean curvature (\\ref{evolution of mean curvature}), we have $-H_{\\Sigma_1}(q)> H_{\\partial M}(p_1)>0$, $H_{\\Sigma_2}(q)> H_{\\partial M}(p_2)>0$. Hence $\\Delta_{\\Sigma} h_1(p)<0$, which completes the proof.\n\n\n\\end{proof}\n\nA moment thought about the arguments in the end of the previous proof yields that $\\text{Ric}_M\\geq 0$ is not so necessary, since we can make use of the evolution equation (\\ref{evolution of mean curvature}) to get an ordinary differential inequality for the mean curvature. \n\n\\begin{proposition}\\label{more general inj lower bound}\nIf in the previous proposition we assume instead $\\emph{Ric}_M\\geq -(n-1)c$ for some $c>0$, and $H\\geq  H_0>0$. If $i_b< -\\frac{1}{2}(n-1)\\ln \\big{|}\\frac{H_0-(n-1)\\sqrt{c}}{H_0+(n-1)\\sqrt{c}}\\big{|}$, then \nthere exists a focal point of $\\partial M$ whose distance to $\\partial M$ is equal to $i_b$. \n\\end{proposition}\n\n\\begin{proof} Suppose the conclusion is not true, follow the arguments as before except for second last sentence. Let  $S_i(X)=-\\nabla_{X}\\nabla h_i$, $H_i=Tr S_i=-\\Delta h_i$, and identify a neighborhood of $\\gamma_{p_i}|_{[0,i_b]}$ with a subset of $\\partial M\\times\\mathbb{R}$ via $\\exp^\\perp$. Then by (\\ref{evolution of mean curvature}),$$\\partial_t{ H_i}=|S_i|^2+\\text{Ric}(\\nabla h_i,\\nabla h_i)\\geq\\frac{1}{n-1}H_i^2-(n-1)c$$ and $H(p_i,0)\\geq H_0$.\nLet $f$ solves the ODE on $[0,i_b]$\n $$ f'=\\frac{1}{n-1}f^2-(n-1)c$$ and $f(0)=H_0$, then we have $H_i(p_i,t)\\geq f(t)  $.  In particular, $H_i(p_i,i_b)\\geq f(i_b)>0$, which leads to a contradiction as before.\n\n\\end{proof}\n\n\n\nProposition $\\ref{existence of focal points}$ implies\n\n\\begin{corollary}\\label{Ric positive, mean positive, inj lower bound}\nLet $(M,g)$ be a compact Riemannian manifold with boundary, $K>0,\\lambda>0$ are constants.\nSuppose $\\sec \\leq K$, $S\\leq \\lambda$, $H>0$, $\\emph{Ric}_M \\geq 0$, then $i_b\\geq \\frac{1}{\\sqrt{K}}\\emph{arccot}{\\frac{\\lambda}{\\sqrt{K}}}.  $\n\\end{corollary}\n\n\n\\begin{proof}\nBy Proposition \\ref{existence of focal points}, there exists $p\\in\\partial M$ such that $\\gamma_p(i_b)$ is a focal point along $\\gamma_p$. If $i_b<\\frac{1}{\\sqrt{K}}\\text{arccot}{\\frac{\\lambda}{\\sqrt{K}}} $, from comparison theorem for Jacobi fields, we know  $\\gamma_p(i_b)$ cannot be a focal point along $\\gamma_p$, which is a contradiction. \n\\end{proof}\n\nSimilarly, Proposition \\ref{more general inj lower bound} implies\n\n\\begin{corollary}\\label{inj lower bound limit}\nLet $(M,g)$ be a compact Riemannian manifold with boundary. Suppose $|Rm|\\leq C$, $|S|\\leq C$, $H\\geq H_0>0$, then we can find $i_0$ depending explicitly on $C,H_0$ such that $i_b\\geq i_0.$\n\\end{corollary}\n\n\\begin{remark}\\label{inj lower bound limit remark}\nIn the previous two corollaries, if the sectional curvature and Ricci curvature bounds only holds for $N_1(\\partial M,g)$, then we also have a $i_b$ lower bound. In the case of Corollary \\ref{Ric positive, mean positive, inj lower bound}, we have $i_b\\geq \\min\\{\\frac{1}{\\sqrt{K}}\\text{arccot}{\\frac{\\lambda}{\\sqrt{K}}} ,1\\}$. \n\\end{remark}\n\n\n\n\n\\begin{remark}\nIn the same setting as the previous two corollaries, \\cite{knox2012compactness} Lemma 2.2 claimed to prove a lower bound for $i_b$, using a similar method as \\cite{Anderson2012BoundaryVP} Lemma 2.4.  In both papers, there is a logic problem that they get a contradiction with an unjustified statement: Let $M$ be a Riemannian manifold with boundary, $\\gamma:[0,l]\\rightarrow M$ be a geodesic that is perpendicular to the boundary at both end points, and suppose there is no focal point along $\\gamma$ for both boundary portions, then $I_1(V,V)\\geq 0$ for any smooth vector field along $\\gamma$ with $V(0),V(l)\\in T{\\partial M}$. Here $$I_1(V,V)=\\int_{0}^l \\langle V', V'\\rangle-\\langle R(V,\\gamma')\\gamma',V \\rangle dt- \\langle S(V(0)),V(0)\\rangle-\\langle S(V(l)),V(l)\\rangle. $$ In fact, this unjustified statement is not true, and one can easily think of an example: let $$\\Sigma_1=\\{(x',x^n)\\in\\mathbb{R}^n\\big | |x'|^2+(1-x^n)^2=R_1^2\\},$$ $$\\Sigma_2=\\{(x',x^n)\\in\\mathbb{R}^n\\big | |x'|^2+(1+x^n)^2=R_2^2\\},$$ $R_1,R_2>2$\nand $\\gamma(t)=(0,1-t), 0\\leq t\\leq 2 $, then $I_1(V,V)=-\\frac{1}{R_1}-\\frac{1}{R_2\n}<0$ for any unit-norm parallel vector field along $\\gamma$ with $V(0)\\in T_{\\gamma(0)}\\Sigma_1$. In this case, there exist no focal points on $\\gamma$ for both $\\Sigma_1,\\Sigma_2$.\n\nIn fact, focal points give crucial information for index form defined by one submanifold and one point. However, as seen from the example, they do not fit well with the index form defined for two submanifolds. Indeed, there is some notion of ``conjugate point'' defined for two submanifolds, see \\cite{ambrose1961index}. \n\\end{remark}\n\n\n\n\n\nIt is easy to see focal points can ``pass to the limit'', since they arise from kernels of the differential of exponential maps. Though one can use Corollary \\ref{inj lower bound limit} directly in many situations, we point out this fact here, which may help in contradiction arguments.\n\n\\begin{proposition}\\label{pass to limit}\nLet $M$ be a manifold with boundary, $g_i$ be a sequence of Riemannian metrics on $M$, and $p_i\\in\\partial M$. Suppose $g_i$ converges to a Riemannian metric $g_{\\infty}$ smoothly and $p_i$ converges to $p_\\infty\\in\\partial M$, $\\gamma_{p_i}$ is defined on $[0,b]$ and\n$\\gamma_{p_i}(t_i)$ is a focal point along $\\gamma_{p_i}$ with $0<a\\leq t_i\\leq b$. Then for a subsequence, $\\gamma_{p_i}$ converges smoothly to $\\gamma_{p_\\infty}$, $t_i\\rightarrow t_\\infty$ and $\\gamma_{p_\\infty}(t_\\infty)$ is a focal point along $\\gamma_{p_\\infty}$.\n\\end{proposition}\n\n\\begin{proof}\n$\\gamma_{p_i}'(0)\\in TM$ is a bounded sequence, hence subconverges to some $v\\in TM $, which must equal to $\\gamma_{p_\\infty}'(0)$. Hence by ODE theories, $\\gamma_{p_i}$ converges smoothly to $\\gamma_{p_\\infty}$. Suppose for a subsequence $t_i\\rightarrow t_\\infty$. Let $J_i:[0,t_i]\\rightarrow TM$ be $\\partial M$-Jacobi fields along $\\gamma_i$ with $J_i(t_i)=0, |J_i'(t_i)|=\\frac{t_\\infty}{t_i}$. Normalize these geodesics and Jacobi-fields by $\\bar\\gamma_i(t)={\\gamma}_i(\\frac{t_i}{t_\\infty} t)$, $\\bar{J}_i(t)=J_i(\\frac{t_i}{t_\\infty} t)$, so  $\\bar\\gamma_i, \\bar{J}_i$ are defined on the same interval $[0,t_\\infty]$, and\n\\begin{equation}\n\\begin{split}\n\\bar J_i''(t)&+  Rm_{g_i}(\\bar{J}_i(t),\\bar\\gamma_i'(t))\\bar\\gamma_i'(t)=0,\\\\\n \\bar{J}_i(t_\\infty)&=0,|\\bar{J}_i'(t_\\infty)|=1.\n\\end{split}\\end{equation}\nFor a subsequence $\\bar{J_i}'(t_\\infty)\\rightarrow w$ with $|w|=1$. Let ${J_\\infty}$ be the non-trivial Jacobi-field along $\\gamma_{p_\\infty}$ with $J_\\infty(t_\\infty)=0, J'_\\infty(t_\\infty)=w$, then $J_i$ converges smoothly as maps $[0,t_\\infty]\\rightarrow TM$ to $J_\\infty$ by ODE theories. Hence $J_\\infty$ is a $\\partial M$-Jacobi field, which implies $\\gamma_{p_\\infty}(t_\\infty)$ is a focal point along $\\gamma_{p_\\infty}.$\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\\subsection{Volume estimates near boundary}\n\nIn this subsection, we show some volume lower bounds near boundary under some geometric control. These estimates can be found in \\cite{anderson2004boundary} and \\cite{knox2012compactness}.\n\n\n\\begin{proposition}\\label{volume of a metric cyclinder}\nLet $({M},g)$ is a Riemannian manifold with boundary, suppose $\\emph{Ric}\\geq -(n-1)c$ for some $c\\geq 0$, and $\\exp^\\perp$ is an diffeomorphism in $B\\times [0,T)$ for some open subset $B$ of $\\partial M$, then $$\\sup_{B\\times \\{t\\}} H\\leq\\max\\{(n-1)\\sqrt{2c},4(n-1)^{-1}T^{-1}\\},$$\n$$\\emph{vol}(C(B,t_1,t_2))\\geq C(n,T,c)\\emph{vol}_{\\partial M}(B)|t_2-t_1|,$$ where $0\\leq t, t_1,t_2\\leq\\frac{1}{2}T.$\n\n\\end{proposition}\n\n\\begin{proof}\nBy the evolution equation (\\ref{evolution of mean curvature}),\n\\begin{equation}\n\\partial_t H=|S|^2+\\text{Ric}(N_t,N_t),\n\\end{equation}\nhence\n\\begin{equation}\n\\partial_t H\\geq \\frac{1}{n-1}H^2-(n-1)c.\n\\end{equation}\nIf for some $t_0\\in[0,\\frac{1}{2}T],z_0\\in B$, we have $H(z_0,t_0)\\geq \\delta$ and $\\delta\\geq (n-1)\\sqrt{2c}$, then $\\partial_t H(z_0,t)\\geq 0$ and $H(z_0,t)\\geq (n-1)\\sqrt{2c}$ for $t\\in[t_0,T) $. Hence $$\\partial_t H(z_0,t)\\geq \\frac{1}{2}(n-1)^{-1}H(z_0,t)^2,$$ $$H^{-1}(z_0,t)\\leq H^{-1}(z_0,t_0)-\\frac{1}{2}(n-1)(t-t_0)\\leq \\delta^{-1}-\\frac{1}{2}(n-1)(t-\\frac{1}{2}T).$$ Then the continuity of $H(z_0,\\cdot)$  in  $[0,T)$ forces $\\delta\\leq 4(n-1)^{-1}T^{-1}$, which proves the mean curvature estimate.\n\n\nFor $t\\in[0,\\frac{1}{2}T]$,  let $B_t=\\exp^\\perp(B\\times \\{t\\})$, then $$\\frac{d}{dt}\\mathcal{H}^{n-1}(B_t)=-\\int_{B_t} Hd\\mathcal{H}^{n-1}(B_t)\\geq-C_1(n,T,c)\\mathcal{H}^{n-1}(B_t) ,$$where $\\mathcal{H}^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure of ${M}$. Hence $$\\mathcal{H}^{n-1}(B_t)\\geq e^{-C_1(n,T,c)t}\\text{vol}_{\\partial M}(B)\\geq e^{-\\frac{1}{2}C_1(n,T,c)T}\\text{vol}_{\\partial M}(B)$$ and when $0\\leq t_1<t_2\\leq \\frac{1}{2}T$, $$\\text{vol}(C(B,t_1,t_2))=\\int_{t_1}^{t_2}\\mathcal{H}^{n-1}(B_t)dt\\geq C(n,T,c)\\text{vol}_{\\partial M}(B)(t_2-t_1).$$\n\\end{proof}\n\n\n\n\n\n\\begin{proposition}\\label{volume noncollasping}\nLet $M$ be a Riemannian manifold with boundary, $p\\in\\partial M$. Suppose  $B(p,2r_0)$ has compact closure,\n$$\\sup\\limits_{B(p,2r_0)}|Rm|\\leq C,$$\n$$\\emph{vol}_{\\partial M}(B_{\\partial M}(p,r_0))\\geq v_0,$$\n  $\\exp^\\perp$ is an diffeomorphism in $B_{\\partial M}(p,r_0)\\times [0,r_0)$, and on $B_{\\partial M}(p,r_0)$ $$  \\emph{Ric}_{\\partial M}\\geq -(n-2)c_0,  \\ |S|\\leq C,$$  then there exists $r_1>0,v_1>0$ depending on $n, C,c_0, r_0,v_0$, such that for $q=\\exp^\\perp(p,2r_1)$, we have $$\\emph{vol}(B(q,r_1))\\geq v_1.$$\n\n\\end{proposition}\n\n\\begin{proof}\n\n By definition $C(B_{\\partial M}(p,r_0),0,r_0)\\subset B(p,2r_0)$.  We claim that  there exists $r_2>0,C_1>0$ such that for any $p_1,p_2\\in B_{\\partial M}(p,r_0)$, $$d_{\\Sigma_t}(\\exp^\\perp(p_1,t),\\exp^\\perp(p_2,t))\\leq C_1d_{\\partial M}(p_1,p_2)$$ when $0\\leq t\\leq r_2$, where $\\Sigma_t$ is the image of $B_{\\partial M}(p,2r_0)$ under $\\exp^\\perp(\\cdot, t)$. In fact, by (\\ref{simplified radial curvature equation})(\\ref{Lie derivative S same as covariant derivative S}),\n $$\\nabla_{\\nabla t } S=S^2+R(\\cdot,\\nabla t)\\nabla t, $$ hence  $$\\frac{d}{dt}|S|\\leq |S|^2+C.$$\nIntegrate the inequality, we have \n$$\\arctan(\\frac{|S|}{\\sqrt C})(x,t)-\\arctan(\\frac{|S|}{\\sqrt C})(x,0)\\leq \\sqrt{C}t.$$\nHence there exist $r_2>0,C_2>1$ such that $|S|\\leq \\log C_2$ when $0\\leq t\\leq r_2$.\nNow let $\\gamma_0(s)$ be a smooth curve in $B_{\\partial M}(p,2r_0)$ that connects $p_1,p_2$, and let $\\gamma_t(s)=\\exp^\\perp(\\gamma_0(s),t)\\in \\Sigma_t$, then we have $$\\frac{d}{dt} \\log |\\gamma_t'(s)|=-\\frac{\\langle S_{\\Sigma_t}(\\gamma_t'(s)),\\gamma_t'(s)\\rangle}{\\langle\\gamma_t'(s),\\gamma_t'(s)\\rangle}\\leq |S_{\\Sigma_t}(\\gamma_t(s))|\\leq \\log C_2.$$\nIt follows that $|\\gamma_t'(s)|\\leq C_1|\\gamma_0'(s)|$ with $C_1=C_2^{r_2}$ and the claim follows from integration. \nNow take $$r_1=\\min\\{2C_1r_0,\\frac{1}{4}r_2\\},$$ we have $$C(B_{\\partial M}(p,\\frac{r_1}{2C_1}),\\frac{3r_1}{2},\\frac{5r_1}{2})\\subset B(q,r_1),$$ then we apply Proposition \\ref{volume of a metric cyclinder} and Bishop-Gromov volume comparison on $\\partial M$ to get the desired conclusion.\n\\end{proof}\n\n\\begin{remark}\nIt is easy to give a quantitative version of the lemma from the proof. However, to the author's knowledge, we cannot prove the last inclusion in the proof without a control of curvature. It may be possible that a metric ball of the boundary becomes ``long and thin'' under the flow of $\\nabla d(\\cdot,\\partial M)$,  while maintains an area lower bound. \n\\end{remark}\n\n\n\n\\subsection{Harmonic radius and convergence theory}\n\nConvergence theory of Riemannian manifolds is a powerful tool to prove conclusions in Riemannian geometry through contradiction arguments when explicit bounds is not required. In this section, we will restate some results of \\cite{anderson2004boundary}, follow the proof there, and discuss some direct corollaries.\n\nLet $(M,g)$ be a Riemannian manifold with boundary, $m\\in \\mathbb{N}$,   $0<\\alpha<1$, $Q>1$. For $p\\in M$, define $r_h^{m,\\alpha}(p,g,Q)$ to be the supremum of $\\rho>0$ such that if $d(p,\\partial M)>\\rho$,\nthen there exists a neighborhood $U$ of $p$ in $M$ and a interior coordinate chart $\\varphi:B_{\\frac{\\rho}{2}}\\rightarrow U$, $\\varphi(0)=p$, and  if $d(p,\\partial M)\\leq\\rho$, then there exists a neighborhood $U$ of $p$ in $M$ and a boundary coordinate chart\n$\\varphi: B_{4\\rho}^+\\rightarrow U$, $\\varphi((0,d(p,\\partial M)))=p, \\varphi(\\tilde{B}_{4\\rho})=U\\cap\\partial M$, and in either $B_{\\frac{\\rho}{2}}$ or $B_{4\\rho}^+$, we have $$\\Delta_M\\varphi^{-1}=0,$$\n$$Q^{-2}(\\delta_{ij})\\leq (g_{ij})\\leq Q^2 (\\delta_{ij}), $$\n$$\\rho^{m+\\alpha}\\sum_{|\\beta|=m}|\\partial_\\beta g_{ij}(x)-\\partial_\\beta g_{ij}(y)|\\leq (Q-1)|x-y|^\\alpha$$\nWe call such a coordinate chart a $(\\rho,Q,m,\\alpha)$-harmonic coordinate chart centered at $p$. Note that the second condition implies there exists $r_1,r_2$, depending on $\\rho,Q$, such that\n$B(p,r_1)\\subset U\\subset B(p,r_2)$.\n\n\n\n\n\\begin{definition}\nFix an integer $m\\geq 0$, and $0<\\alpha<1$.  We say a sequence of Riemannian manifold with boundary $(M_i,g_i,p_i)$ converges in pointed $C^{m,\\alpha}$ to $(M_\\infty,g_\\infty,p_\\infty)$ if there exists precompact open subsets  $\\Omega_{i}$ of $M_i$ and $\\Omega_{\\infty,i}$ of $M_\\infty$, and $\\sigma_i>\\rho_i\\rightarrow\\infty$ such that $B(p_i,\\rho_i)\\subset \\bar{\\Omega}_{i}\\subset B(p_i,\\sigma_i)$, $B(p_\\infty,\\rho_i)\\subset \\bar{\\Omega}_{i}\\subset B(p_\\infty,\\sigma_i)$ and  there exists diffeomorphisms $F_{i}:\\Omega_{\\infty,i}\\rightarrow \\Omega_{i}$, $F_{i}:\\Omega_{\\infty,i}\\cap\\partial M_i\\rightarrow\\Omega_{i}\\cap\\partial M_\\infty$ such that  $F_{i}^*g_i\\rightarrow g$ in $C^{m,\\alpha}$ topology,  and $F_{i}^{-1}(p_i)\\rightarrow p_\\infty$. If we replace $C^{m,\\alpha}$ by $C^\\infty$, we say the convergence is in pointed Cheeger-Gromov sense.\n\\end{definition}\n\n\\begin{remark}\n$(M_\\infty, g_\\infty)$ is automatically a complete $C^{m,\\alpha}$ or $C^\\infty$ Riemannian manifold with boundary from the definitions. Sometimes we only need that one metric ball converges, so one can modify the definitions above: suppose $\\bar{B}(p_i,r)\\subset \\Omega_i$ for some precompact open set $\\Omega_i\\subset M_i$ and there exists a Riemannian manifold with boundary $(\\Omega_\\infty,g_\\infty)$, a point $p_\\infty\\in\\Omega_\\infty$, and  diffeomorphisms $F_i:\\Omega_{\\infty}\\rightarrow\\Omega_i$ mapping $\\partial \\Omega_\\infty$ onto $\\Omega_i\\cap\\partial M_i$ such that $F_i^*g_i\\rightarrow g_\\infty$ in $C^{m,\\alpha} $ or $C^\\infty$ topology and $F_i^{-1}(p_\\infty)\\rightarrow p_i$, we say $B(p_i,r)$ converges in $C^{m,\\alpha}$  or Cheeger-Gromov sense to $B(p_\\infty, r)$.\n\n\\end{remark}\n\n\n\n\n\n\n\n\n\nThe following theorem is well-known and is a fundamental theorem of Riemannian convergence theory. \n\n\\begin{proposition}\\label{harmonic radius lower bound}\nLet $(M_i,g_i)$ be  a sequence of  complete Riemannian manifold with boundary, $p_i\\in M_i$. Suppose there exists some $Q>1$, and a positive function $r:(0,\\infty)\\rightarrow(0,\\infty) $, such that $r_h^{m,\\alpha}(p,g_i,Q)\\geq r(R)$ for any $p\\in B(p_i,R)$, then for a subsequence, $(M_i,g_i,p_i)$ converges in pointed $C^{m,\\beta}$ sense to $(M_\\infty,g_\\infty,p_\\infty)$ for any $0<\\beta<\\alpha$. If the above assumption holds for only one $R$, then $B(p_i,R)$ converges in $C^{m,\\beta}$ sense to $B(p_\\infty, R)$.\n\n\\end{proposition}\n\n\n\n\n\n\n\n\n\n\nNext, we discuss under what geometric control we can  get a harmonic radius lower bound. We state and prove the following local version of Theorem 3.2.1 in  \\cite{anderson2004boundary}, with simplified arguments in some parts.\n\n\n\n\\begin{theorem}\\label{local harmonic radius lower bound}\nFix $m\\geq 1$. Let $(M,g)$ be a  Riemannian manifold with boundary, and $\\Sigma\\subset\\partial M$ be a boundary metric ball with compact closure, nonempty boundary. Suppose $\\exp^\\perp$ maps $\\Sigma\\times [0,i_0)$ diffeomorphically onto its image $\\Omega$,  \n\\begin{equation}\\label{injectivity radius explaination}\n     inj_{\\Omega}\\geq i_0,\\ inj_{\\Sigma}\\geq i_0,\n\\end{equation}\nin $\\Omega$,\n\\begin{equation}\\label{bounds 1 up to m}\n    |\\nabla^l \\text{Ric}_M|\\leq \\Lambda, 0\\leq l\\leq m,\n\\end{equation}\n on $\\Sigma$\n \\begin{equation}\\label{bounds 2 up to m}\n     |\\nabla_{\\partial M}^l \\text{Ric}_{\\partial M}|\\leq \\Lambda,   |\\nabla_{\\partial M}^{l+1} H|\\leq \\Lambda, 0\\leq l\\leq m.\n \\end{equation}\n Then for any $Q>1$, $\\alpha\\in(0,1)$, $p\\in\\Omega$, \n \\begin{equation}\\label{harmonic radius lower bound non empty boundary}\n     r^{m+1,\\alpha}_h(p,g,Q)\\geq r_0(i_0,\\Lambda,m,\\alpha,Q)d(p,\\partial ^+\\Omega),\n \\end{equation}\n  where $\\partial^+\\Omega=\\bar {\\Omega} \\backslash(\\Omega\\cup\\partial M)$.\n\\end{theorem}\n\n\n\n\n\\begin{remark}\\label{local harmonic radius lower bound remark} One should understand (\\ref{injectivity radius explaination}) as follows:\nfor an open set $U$ inside a Riemannian manifold with boundary, $inj_U\\geq i_0$ means for each $p\\in U$, $\\exp_{p}$ maps $B_{i_0}(0)\\subset T_p M$ diffeomorphically onto its image if $d(p,U^c)\\geq i_0$, and maps $B_{\\frac{1}{2}d(p,U^c)}(0)\\subset T_p M$ diffeomorphically onto its image if $d(p,U^c)\\leq i_0$.\n\\end{remark}\n\n\n\n\\begin{proof}\nIf not, we have a sequence $(M_k,\\tilde{g}_k)$ and $\\Sigma_k$, $\\Omega_k$ that satisfies the conditions, but there exists $p_k\\in \\Omega_k$ with\n$$\\frac{r_h^{m+1,\\alpha}(p_k,\\tilde{g}_k,Q)}{d_{\\tilde{g}_k}(p_k,\\partial^+ \\Omega_k)}=\\inf_{p\\in \\Omega_k} \\frac{r_h^{m+1,\\alpha}(p_k,\\tilde{g}_k,Q)}{d_{\\tilde{g}_k}(p,\\partial^+ \\Omega_k)}\\rightarrow 0.$$  Rescale the metric $g_k=(r_h^{m+1,\\alpha}(p_k,\\tilde g_k,Q))^{-2}\\tilde{g}_k$, so $r_h^{m+1,\\alpha}(p_k,{g}_k,Q)=1$, then $d_{g_k}(p_k,\\partial^+\\Omega_k)\\rightarrow\\infty$, and $r_h^{m+1,\\alpha}(p,{g}_k,Q)\\geq\\frac{1}{2}$ if $d_{g_k}(p,p_k)\\leq R$ , $k\\geq k(R)$. Fix any $\\beta\\in(0,\\alpha)$. Then there are two cases: \n \n\\textbf{Case 1}\n $d_{g_k}(p_k,\\Sigma_k)\\rightarrow\\infty$ for some subsequence.\n \n \nThen a subsequence $(M_k,{g_k},p_k)$ converges in pointed $C^{m+1,\\beta}$ sense to a complete Riemannian manifold $(M_\\infty,g_\\infty,p_\\infty)$.  So $\\text{Ric}_{M_\\infty}=0$, $inj_{M_\\infty}=\\infty$. By Cheeger-Gromoll splitting theorem, $(M_\\infty, g_\\infty)$ is isometric to $(\\mathbb{R}^n, g_{flat})$. \n\nHence for any $L>0$, there exist a coordinate $\\varphi_{0,k}:B_{L+5} \\rightarrow U_k\\subset M_k$, $\\varphi_{0,k}(0)=p_k$,  such that\n$$\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\beta}(B_{L+5})}\\rightarrow 0, 1\\leq i,j\\leq n.$$ \nWe solve the Dirichlet problem for functions $u_k^\\nu$, $1\\leq\\nu\\leq n$:\n\n\\begin{equation}\n\\Delta_{M_k}u_k^\\nu=0 \\ {\\rm in} \\ {B}_{L+5} , u_k^\\nu|_{\\partial {B}_{L+5}}=x^\\nu\\ .\n\\end{equation}\nRecall the formula $$\\Delta_{g}=g^{ij}\\partial_i\\partial_j+\\frac{1}{\\sqrt{|g|}}{\\partial_i(\\sqrt{|g|}g^{ij})}\\partial_j, |g|=\\det(g_{ij}).$$  \nThen we have $$\\lVert u_k^\\nu-x^\\nu\\rVert_{C^{m+2,\\beta}({B}_{L+5})}\\leq C\\lVert \\Delta_{ M_k}(u_k^\\nu-x^\\nu)\\rVert_{C^{m,\\beta}({B}_{L+5})}\\rightarrow 0.$$\nHence, we get a new coordinate system $(u_k^1,\\cdots,u_k^n)$ and we discard the original coordinate system, and we use the same notation for tensors written in the new coordinate system, so in the new coordinate system we have\n$$\\lVert g_{k,ij}-\\delta_{ij} \\rVert_{C^{m+1,\\beta}(B_{L+3})}\\rightarrow 0.$$ \nNow we want to improve the convergence of $g_{k,ij}$     from elliptic equations.\nWe have a system of equations\n$$\\Delta_{M_k} g_{k,ij}+B_{ij}(g_k,\\partial g_k)=-2\\text{Ric}_{M_k,ij}. $$\nwhere $B_{ij}(g,\\partial g)$ are  polynormials of $g,\\partial g$ and are quadratic in $\\partial g$.\nFrom $W^{m+2,p}$ estimates, Morrey embeddings, and $$|\\nabla^l \\text{Ric}_{ M_k}|\\rightarrow 0, 0\\leq l\\leq m,$$\nwe have for $1\\leq i,j\\leq n$\n\n\\begin{equation}\\label{harmonic coordinate estimates for use of lower dimensional}\n    \\begin{split}\n\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\alpha}( B_{L+2})}\\leq& C(\\norm{\\Delta_{ M_k} (g_{k,ij}-\\delta_{ij})}_{C^m( B_{L+3})}\\\\&+\\norm{g_{k,ij}-\\delta_{ij}}_{L^\\infty(B_{L+3})})\\rightarrow 0.\n    \\end{split}\n\\end{equation}\nHence we get a $(2(L+2), Q,m+1,\\alpha)$ harmonic coordinate chart centered at $p_k'$, with $d_{g_k}(p_k',p_k)\\rightarrow 0$ , then  $r_h^{m,\\alpha}(p_k,{g}_k,Q)\\geq 2(L+1)$ for large $k$, which is a contradiction. \n\n\\textbf{Case 2}\n$d_{{g}_k}(p_k,\\Sigma_k)\\leq K.$\n\nA subsequence $(M_k,{g_k},p_k)$ converges in pointed $C^{m+1,\\beta}$ sense to a complete Riemannian manifold with boundary $(M_\\infty,g_\\infty,p_\\infty)$ and $(\\partial M_k,{g}_k,q_k)$ converges in $C^{m+1,\\beta}$ sense to $(\\partial M_\\infty,g_{\\infty}|_{\\partial M_\\infty},q_\\infty)$, where $q_k\\in\\Sigma_k$ is the unique foot point of $p_k$ in $\\Sigma_k$. Then $\\text{Ric}_{M_\\infty}=0$, $\\text{Ric}_{\\partial M_\\infty}=0$, $H_\\infty=0$, $inj_{\\partial M_\\infty}=\\infty$, $i_{b,M_\\infty}=\\infty$. Hence\n$(\\partial M_\\infty, g_\\infty|_{\\partial M_\\infty})$ is isometric to $(\\mathbb{R}^{n-1}, g_{flat})$. By (\\ref{scalar curvatures and hypersurface}), we have $S_\\infty=0$. Then by Lemma $\\ref{flat totally geodesic}$, $(M_\\infty,g_\\infty)$ is a smooth Riemannian manifold with boundary and $Rm_{M_\\infty}=0$. Since also $i_{b,M_\\infty}=\\infty$, $(M_\\infty, g_\\infty)$ is a isometric to $(\\mathbb{R}^n_+,g_{flat})$.  \n\nHence for any $L>2K+10$, there exist a coordinate $\\varphi_{0,k}:B_{L+5}^+\\rightarrow U_k\\subset M_k$, $\\varphi_{0,k}(0)=q_k$, $\\varphi_{0,k}(\\tilde{B}_{L+5})=U_k\\cap\\partial M_k$ such that\n$$\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\beta}(B_{L+5}^+)}\\rightarrow 0, 1\\leq i,j\\leq n.$$  \nFirst, we solve for functions $v_k^\\nu$,     $1\\leq\\nu\\leq n-1$, \n\\begin{equation}\n\\Delta_{\\partial M_k}v_k^\\nu=0 \\ {\\rm in} \\ \\tilde{B}_{L+5} , v_k^\\nu|_{\\partial \\tilde{B}_{L+5}}=x^\\nu ,\n\\end{equation}\nThen we have $$\\lVert v_k^\\nu-x^\\nu\\rVert_{C^{m+2,\\beta}(\\tilde{B}_{L+5})}\\leq C\\lVert \\Delta_{\\partial M_k}(v_k^\\nu-x^\\nu)\\rVert_{C^{m,\\beta}(\\tilde{B}_{L+5})}\\rightarrow 0.$$\nNext, we solve for $1\\leq\\nu\\leq n-1$,\n\\begin{equation}\n\\Delta_{M_k} u_k^\\nu=0  \\ {\\rm in} \\ B_{L+5}^+, u_k^\\nu|_{\\tilde{B}_{L+5}}=v_k^\\nu, u_k^\\nu|_{{\\partial^+B_{L+5}^+}}=x^\\nu.\n\\end{equation}\nNote that $\\partial B^+_{L+5}$ is not a $C^1$-boundary, but it satisfies exterior sphere condition, so we can solve the equations by Perron's method to get a unique solution  $u_k^\\nu \\in C^\\infty((B_{L+5}^+)^\\circ)\\cap C^0(\\overline{{B}_{L+5}^+}$).\nFrom definitions and the estimates above, we have\n$$\n\\lVert\\Delta_{M_k}(u_k^\\nu-x^\\nu)\\rVert_{C^{m,\\beta}(B^+_{L+5})}\\rightarrow 0,$$  $$\\lVert u_k^\\nu-x^\\nu\\rVert_{C^{m+2,\\beta}(\\tilde{B}_{L+5})}\\rightarrow 0,$$ $$\\lVert u_k^\\nu-x^\\nu\\rVert_{L^\\infty(\\partial B^+_{L+5})}\\rightarrow 0,$$\nthen by maximum principle, we have  $$\\lVert u_k^\\nu-x^\\nu\\rVert_{L^\\infty(B^+_{L+5})}\\rightarrow 0,$$ and by Schauder estimates\n\n\\begin{equation}\n\\begin{split}\n\\lVert u_k^\\nu-x^\\nu\\rVert_{C^{m+2,\\beta}(B_{L+4}^+)}\\leq C(&\\lVert \\Delta_{M_k}(u_k^\\nu-x^\\nu)\\rVert_{C^{m,\\beta}(B_{L+5}^+)}+\\lVert u_k^\\nu-x^\\nu\\rVert_{L^\\infty(B_{L+5}^+)}\\\\ \n&+\\lVert u_k^\\nu-x^\\nu\\rVert_{C^{m+2,\\beta}(\\tilde{B}_{L+5})})\\rightarrow 0.\n\\end{split}\n\\end{equation}\nNext, we construct $u_k^n$ by solving\n\n$$\\Delta_{M_k}u_k^n=0 \\ {\\rm in}\\ B^+_{L+5},u_k^n|_{\\partial B_{L+5}^+}=x^n.  $$ We have\n\n\\begin{equation}\n    \\begin{split}\n        \\lVert u_k^n-x^n\\rVert_{C^{m+2,\\beta}(B^+_{L+4})}\\leq& C(\\lVert \\Delta_{M_k}(u_k^n-x^n)\\rVert_{C^{m,\\beta}(B_{L+5}^+)}\\\\&+\\lVert u_k^n-x^n\\rVert_{L^\\infty(B_{L+5}^+)}) \\rightarrow 0.\n    \\end{split}\n\\end{equation}\nHence we get a new coordinate system $(u_k^1,\\cdots,u_k^n)$ and we discard the original coordinate system, and we use the same notation for tensors written in both coordinate systems, so in the new coordinate system we have\n\\begin{equation}\\label{lower order convergence harmonic coordinates}\n    \\lVert g_{k,ij}-\\delta_{ij} \\rVert_{C^{m+1,\\beta}(B_{L+3}^+)}\\rightarrow 0.\n\\end{equation}\nNow we want to improve the convergence of $g_{k,ij}$ from elliptic equations with Neumann boundary conditions.\nWe have equations\n\\begin{equation}\\label{Ricci boundary equation}\n\\Delta_{\\partial M_k} g_{k,ij}+\\tilde{B}_{ij}(g_k,\\partial g_k)=-2\\text{Ric}_{\\partial M_k,ij} \n\\end{equation}\n\\begin{equation}\\label{Ricci equation}\n\\Delta_{M_k} g_{k,ij}+B_{ij}(g_k,\\partial g_k)=-2\\text{Ric}_{M_k,ij} \n\\end{equation}\nFix $\\theta\\in (\\beta,1),  p=\\frac{n}{1-\\theta}$, from $W^{m+2,p}$ estimates, Morrey embeddings, and $$|\\nabla_{\\partial M_k}^l \\text{Ric}_{\\partial M_k}|\\rightarrow 0, 0\\leq l\\leq m,$$\nwe have for $1\\leq i,j\\leq n-1$,\n\\begin{equation}\\label{lower dimensional convergence harmonic coordinates}\n\\begin{split}\n\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\theta}(\\tilde B_{L+2.5}^+)}\\leq &C(\\norm{\\Delta_{\\partial M_k} (g_{k,ij}-\\delta_{ij})}_{C^m(\\tilde B_{L+3}^+)}\\\\&+\\norm{g_{k,ij}-\\delta_{ij}}_{L^\\infty(\\tilde B_{L+3}^+)})\\rightarrow 0.\n\\end{split}\n\\end{equation}\nBy Theorem 8.33 in \\cite{gilbarg2015elliptic},\n\\begin{equation}\n\\begin{split}\n\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\theta}(B_{L+2}^+)}  \\leq & C(\\norm{\\Delta_{ M_k} (g_{k,ij}-\\delta_{ij})}_{C^m(\\tilde B_{L+2.5}^+)}\\\\ &+\\norm{g_{k,ij}-\\delta_{ij}}_{L^\\infty( B_{L+2.5}^+)}\\\\ &+\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\theta}(\\tilde B_{L+2.5}^+)})\\rightarrow 0.\n\\end{split}\n\\end{equation}\nNote that\n\\begin{equation}\\label{Neumann boundary condition 1}\nN_kg_k^{nn}=-2(n-1)H_kg_k^{nn},\n\\end{equation}\n\\begin{equation}\\label{Neumann boundary condition 2}\nN_kg_k^{in}=-(n-1)H_kg_k^{in}+\\frac{1}{2\\sqrt{g_k^{nn}}}g_k^{ij}\\partial_j g_k^{nn},\n\\end{equation}\nwhere $N_k=\\frac{g_k^{jn}\\partial _j}{\\sqrt{g_k^{nn}}}$ is the unit normal vector of $\\partial M_k$, $1\\leq i\\leq n-1$ and $j$ sums from $1$ to $n$, then we have Neumann boundary conditions for (\\ref{Ricci equation}).  \nFor simplicity, assume for a while $m=0$. Since $$\\norm{g_{k,ij}-\\delta_{ij}}_{C^{1,\\beta}(B_{L+2}^+)}\\rightarrow 0 ,$$ $$|\\text{Ric}_{M_k, ij}|_{C^0(B_{L+2}^+)}\\rightarrow 0, 1\\leq i,j\\leq n,$$ $$|H_k|_{C^1(\\tilde{B}_{L+2})}\\rightarrow 0,$$ we have\n\n$$ \\norm{\\Delta_{M_k} (g_k^{nn}-\\delta^{nn})}_{C^0(B_{L+2}^+)}\\rightarrow 0,   \\norm{N_kg_k^{nn}}_{C^1(\\tilde B_{L+2})}\\rightarrow 0,$$\n then by Morrey embeddings (together with extensions), and $W^{2,p}$ estimates for Neumann boundary problems (for example, see a priori estimate 2.3.1.1 in \\cite{grisvard2011elliptic}),\n\n\\begin{equation}\n\\begin{split}\n\\norm{g_k^{nn}-\\delta^{nn}}_{C^{1,\\theta}(B^+_{L+1.7})}\\leq & C \\norm{g_k^{nn}-\\delta^{nn}}_{W^{2,p}(B_{L+1.8}^+)}\\\\\n\\leq & C( \\norm{g_k^{nn}-\\delta^{nn}}_{L^{p}(B_{L+2}^+)}+ \\norm{\\Delta_{M_k} (g_k^{nn}-\\delta^{nn})}_{L^p(B_{L+2}^+)}\\\\ &+\\norm{N_kg_k^{nn}}_{W^{1-\\frac{1}{p},p}(\\tilde B_{L+2})}\\rightarrow 0.\n\\end{split}\n\\end{equation}\nNow for $1\\leq l\\leq n-1$, since $$\\norm{\\Delta_{M_k}( g_k^{ln}-\\delta^{ln})}_{C^0(B_{L+2}^+)}\\rightarrow 0,$$ \nand\n\\begin{equation}\n\\begin{split}\n\\norm{N_kg_k^{ln}}_{W^{1-\\frac{1}{p},p}(\\tilde{B}_{L+1.5})} &\\leq C(\\norm{g_k^{nn}-\\delta^{nn}}_{W^{2-\\frac{1}{p},p}(\\tilde{B}_{L+1.5})}+\\norm{ H_k}_{W^{1-\\frac{1}{p},p}(\\tilde{B}_{L+1.5})})\\\\& \\leq C(\\norm{g_k^{nn}-\\delta^{nn}}_{W^{2,p}({B}^+_{L+1.7})}+\\norm{ H_k}_{C^1(\\tilde{B}_{L+1.7})})\\rightarrow 0.\n\\end{split}\n\\end{equation}\nThen \n\\begin{equation}\n\\begin{split}\n\\norm{g_k^{ln}-\\delta^{ln}}_{C^{1,\\theta}(B_{L+1.1}^+)}\\leq & C\\norm{g_k^{ln}-\\delta^{ln}}_{W^{2,p}(B_{L+1.2}^+)}\\\\ \\leq &C(\\norm{g_k^{ln}-\\delta^{ln}}_{L^{p}(B_{L+1.5}^+)}+\\norm{\\Delta_{M_k}( g_k^{ln}-\\delta^{ln})}_{L^p(B_{L+1.5}^+)}\\\\  &+\\norm{N_kg_k^{ln}}_{W^{1-\\frac{1}{p},p}(\\tilde{B}_{L+1.5})})\\rightarrow 0.\n\\end{split}\n\\end{equation}\nHence $$\\norm{g_{k,ij}-\\delta_{ij}}_{C^{1,\\theta}(B_{L+1.1}^+)}\\rightarrow 0,  1\\leq i,j\\leq n.$$\nFor general $m\\geq 1$,  take  $m$-th derivatives of (\\ref{Ricci equation}) and the Neumann boundary conditions (\\ref{Neumann boundary condition 1})(\\ref{Neumann boundary condition 2}),\nand note that $$[\\partial _i,N_k]=(\\frac{\\partial_ig_k^{jn}}{\\sqrt{g_k^{nn}}}-\\frac{g_k^{jn}\\partial_i g_k^{nn}}{2\\sqrt{g_k^{nn}}^3})\\partial_j,$$ so we get a system of second order elliptic equations with Neumann boundary conditions in $\\partial_\\gamma   g_k^{nn}$ and $\\partial_\\gamma g_k^{ln}$, $|\\gamma|=m, 1\\leq l\\leq m-1$, with other terms freezed. Apply the previous estimates in the case $m=0$ and use (\\ref{lower order convergence harmonic coordinates})(\\ref{lower dimensional convergence harmonic coordinates}),\nwe get\n$$\\norm{g_k^{nn}-\\delta^{nn}}_{C^{m+1,\\theta}(B_{L+1}^+)}\\rightarrow 0,$$\nand then for $1\\leq l\\leq n-1,$\n$$\\norm{g_k^{ln}-\\delta^{ln}}_{C^{m+1,\\theta}(B_{L+1}^+)}\\rightarrow 0,$$\nhence  $$\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\theta}(B_{L+1}^+)}\\rightarrow 0,  1\\leq i,j\\leq n$$\nIn particular, take $\\theta=\\alpha$, one can  we get a $(\\frac{L+1}{4}, Q,m+1,\\alpha)$ harmonic coordinate chart centered at $p_k'$ , with $d_{g_k}(p_k',p_k)\\rightarrow 0.$ Then  $r_h^{m+1,\\alpha}(p_k,{g}_k,Q)\\geq \\frac{L}{4}$ for large $k$, which is a contradiction.\n\n\n\\end{proof}\n\n\n\n\n\\begin{remark}\nNote that the case $m=0$ is also true, and one should be a little careful with the geometric arguments in the proof. Actually, the arguments in \\cite{anderson2004boundary} prove a $C_*^{m+2}$ harmonic radius lower bound. \n\\end{remark}\n\n\\begin{remark}\\label{global harmonic radius lower bounds for complete manifolds with boundary}\nThe proof also shows that if $M$ is complete, $i_b\\geq i_0$, $inj_M\\geq i_0$, $inj_{\\partial M}\\geq i_0$ and (\\ref{bounds 1 up to m})(\\ref{bounds 2 up to m}) hold, then for any $p\\in M$\n\\begin{equation}\\label{harmonic radius lower bound globally}\n     r^{m+1,\\alpha}_h(p,g,Q)\\geq r_0(i_0,\\Lambda,m,\\alpha,Q).\n \\end{equation}\n\n\\end{remark}\n\n\n\n\nThe following corollary is a version we will use often.\n\n\\begin{corollary}\\label{corollary that will be used often}\nLet $(M_i,g_i)$ be a sequence of complete Einstein manifold with boundary. Suppose $i_b\\geq i_0, inj_{\\partial M}\\geq i_0, |Rm|\\leq C, |S|\\leq C, |\\nabla_{\\partial M}^k Rm_{\\partial M}|\\leq C_k,|\\nabla_{\\partial M}^{k+1}H|\\leq C_k, k\\geq 0$, then for any $p_i\\in M_i$,  there exists some subsequence such that  $(M_i,g_i,p_i)$ converges in pointed Cheeger-Gromov sense.\n\\end{corollary}\n\n\n\n\\begin{proof}\nBy Proposition \\ref{volume noncollasping}, $|Rm|\\leq C$, $|S|\\leq C$, $i_b\\geq i_0$,  together imply volume lower bounds of interiors balls of some fixed radius near boundary, hence also gives an interior injectivity radius lower bound from the following lemma. Then use Remark \\ref{global harmonic radius lower bounds for complete manifolds with boundary} and Proposition \\ref{harmonic radius lower bound}.\n\\end{proof}\n\nThe following lemma is well-known, which is a qualitative version of Theorem 4.3 in \\cite{cheeger1982finite} and can also be easily proved by contradiction arguments.\n\n\\begin{lemma}\\label{well-know interior inj lower bound} Let $(M,g)$ be a Riemannian manifold, and $B(p,r)$ be a metric ball that has compact closure. Suppose  $$\\sup\\limits_{B(p,r)}|Rm|\\leq C, \\emph{vol}(B(p,r))\\geq v, $$then there exists $r_0>0$ depending on $n, C,v,r$ such that $\\exp_q:B_{r_0}(0)\\subset T_q M\\rightarrow B(q,r_0)\\subset M$ is a diffeomorphism for any $q\\in B(p,\\frac{r}{2})$.\n\\end{lemma}\n\n\n\n\n\n\n\\section{Convergence of hyperk\\\"ahler manifolds}\\label{section main proof}\n\n\\subsection{Curvature estimates near the boundary}\nThis section serves as a first step for the proof of our main theorem.\nFor an Einstein manifold with boundary, if the boundary intrinsic and extrinsic geometry are controlled well and $i_b$ is bounded from below, we hope to control the interior geometry within $i_b$. To the author's knowledge, we do not know any general statement. We will first state and prove a version we need, and then discuss some lemmas needed in the proof.\n\n\\begin{theorem}\\label{good boundary}\nLet $(M,g)$ be a complete hyperk\\\"ahler 4-manifold with compact boundary. Suppose $|S|\\leq C$,  $|\\nabla^j_{\\partial M} Rm_{\\partial M}|\\leq C_j, |\\nabla^{j+1}_{\\partial M} H|\\leq C_j,    j=0,1,\\cdots$, $inj_{\\partial M}\\geq i_0$, $i_b\\geq i_0$, $\\int_{M}|Rm|^2\\leq C$. Then for any $r_1<i_0$, there exists $C'>0$, depending on $C,C_j,i_0,r_1$,  such that $\\sup\\limits_{N_{r_1}(\\partial M,g)}|Rm|\\leq C'$.\n\n\\end{theorem}\n\n\n\\begin{proof}\nWithout loss of generality, assume $i_0=1$. Denote $\\alpha={r_1}$, $\\beta=\\frac{1}{4}(1-\\alpha)$.  Suppose the conclusion is not true, we have a sequence $(M_i,g_i)$ satisfying the conditions, but $$\\sup\\limits_{N_{\\alpha}(\\partial M_i,g_i)}|Rm_{g_i}|\\rightarrow\\infty.$$ Let  $p_i\\in N_{r_1}(\\partial M_i,g_i)$  achieves this supremum. \n\n\\textbf{Claim 1} There exists a subsequence such that \n\\begin{equation}\\label{Distance curvature}\nd_{g_i}(p_i,\\partial M_i)^2|Rm_{g_i}(p_i)|\\rightarrow\\infty.\n\\end{equation}\nIf this is not true, we have $\\sup_i d_{g_i}(p_i,\\partial M_i)^2 |Rm_{g_i}|(p_i)<\\infty$. Rescale $\\tilde{g}_i=|Rm_{g_i}(p_i)|g_i$, then $|Rm_{\\tilde{g}_i}(p_i)|=1$, and $|Rm_{\\tilde{g}_i}|\\leq 1$ in $N_{\\alpha|Rm(p_i)|^{\\frac{1}{2}}}(\\partial M_i,\\tilde{g}_i)$, and $\\sup_id_{\\tilde{g}_i}(p_i,\\partial M_i)<\\infty$, $ i_{b,\\tilde g_i}\\geq |Rm_{g_i}(p_i)|^{\\frac{1}{2}}$ for all $i$. Hence by Corollary \\ref{corollary that will be used often}, $(M_i,g_i,p_i)$ subconverges in pointed Cheeger-Gromov sense to $(M_\\infty, \\tilde{g}_\\infty,p_\\infty)$, which is a complete Ricci-flat 4-manifold with flat, totally geodesic boundary, hence must be flat by Lemma \\ref{flat totally geodesic}. This contradicts that $|Rm_{g_\\infty}(p_\\infty)|=1$ and proves Claim 1.\n\nNow rescale $g_i$ in another way, let $g_i'=d_{g_i}(p_i,\\partial M_i)^{-2}g_i$, so $d_{g_i'}(p_i,\\partial M)=1$.\nSince $d_{g_i}(p_i,\\partial M_i)\\leq \\alpha$, the rescaled metric $g_i'$ satisfies $i_{b,g_i'}\\geq \\alpha^{-1}$ as well as all other conditions of the assumptions of the theorem, but with different bounds, regardness of whether $d_{g_i}(p_i,\\partial M_i)$ is uniformly bounded from below or not. Moreover, (\\ref{Distance curvature}) is equivalent to $|Rm_{g_i'}(p_i)|\\rightarrow\\infty$.\n\nBy the $\\epsilon$-regularity Theorem \\ref{Cheeger-Tian}, there exists a universal constant $\\epsilon_0$ such that  for sufficiently large $i$, $$\\int_{B_{{g}_i'}(p_i,\\beta )}|Rm_{g_i'}|^2\\geq\\epsilon_0.$$ \n\n\n\\textbf{Claim 2} There exists a subsequence such that\n$$ \\sup\\limits_{ N_{\\alpha}(\\partial M_i,g_i')}|Rm_{g_i'}|\\rightarrow \\infty, $$\nIf not, we have\n$ \\sup\\limits_{ N_{\\alpha}(\\partial M_i,g_i')}|Rm_{g_i'}|\\leq C$. By Lemma \\ref{volume noncollasping} and Bishop-Gromov volume comparison, $\\text{vol}(B_{g_i'}(p_i,\\beta))\\geq v$. Since $B_{g_i'}(p_i,\\beta)\\subset N_{\\alpha^{-1}(\\alpha+\\beta)}(\\partial M_i,g_i')\\subset N_{\\alpha+\\beta}(\\partial M_i,g_i)$, and the last one is diffeomorphic to $\\partial M_i\\times [0,\\alpha+\\beta]$, we conclude that there is no $-2$ curve in $B_{g_i'}(p_i,\\beta)$. By Proposition \\ref{no bubble},\n$|Rm_{g_i'}(p_i)|$ is bounded, which is a contradiction to Claim 1 and finishes the proof of Claim 2.\n\n\n\n\nNow Claim 2 enables us to get by induction, for each fixed positive integer $N$, $N$ sequences of metrics $g_i^{(0)}=g_i,g_i^{(1)}=g_i',\\cdots, g_i^{(N)}$, and points $p_i^{(j)}\\in N_{\\alpha}(\\partial M_i,g_i^{(j)})$ for  $0\\leq j\\leq N-1$, $p_i^{(0)}=p_i$, such that for $0\\leq j\\leq N-1$, $p_i^{(j)}$ achieves the supremum of $|Rm_{g_i^{(j)}}|$ in $N_{\\alpha}(\\partial M_i,g_i^{(j)})$, and\n\n $$|Rm_{g_i^{(j)}}(p_i)|\\rightarrow\\infty,$$  $$d_{g_i^{(j+1)}}(p_i^{(j)},\\partial M_i)=1,$$\n$$\\int_{B_{{g}_i^{(j+1)}}(p_i^{(j)},\\beta)}|Rm_{g_i^{(j+1)}}|^2\\geq\\epsilon_0,$$ \n\n\n$$ B_{g_i^{(j+1)}}(p_i^{(j)},\\beta)\\subset N_{\\alpha^{-1}(\\alpha+\\beta)}(\\partial M_i,g_i^{(j+1)})\\subset N_{\\alpha+\\beta}(\\partial M_i,g_i^{(j)}) ,$$ \n $$B_{g_i^{(j+1)}}(p_i^{(j)},\\beta)\\cap N_{\\alpha+\\beta}(\\partial M_i,g_i^{(j+1)})=\\emptyset . $$               \nIt follows that for each fixed $i$, $B_{g_i^{(j+1)}}(p_i^{(j)},\\beta)$ does not interect each other for different $j$. Since $\\int_{M_i}|Rm_{g_i}|^2\\leq C$, we have $N\\epsilon_0\\leq C$. This is a contradiction, since $N$ can be any positive integer.\n\\end{proof}\n \n\n\\begin{remark}\nThis theorem is purely local. In fact, by slightly modifying the proof, we see that if the bounds in the assumptions hold in a metric cyclinder $C(B_{\\partial M}(p,r_0),0,r_1)$  such that $\\exp^\\perp$ maps $B_{\\partial M}(p,r_0)\\times [0,r_1)$ diffeomorphically onto it, and such that $B_{\\partial M}(p,r_0)$ has compact closure,  then we have curvature bounds in any interior metric cyclinder $C(B_{\\partial M}(p,r_0'),0,r_1')$ with fixed $r_0'<r_0,r_1'<r_1$.\n\\end{remark}\n\n\n\nThe following collapsing $\\epsilon$-regularity theorem is originally due to Cheeger-Tian in \\cite{cheeger2006curvature}. Recently, in the hyperk\\\"ahler case, \\cite{sun2021collapsing} gives a simple proof by a blow-up argument and studying complete collapsing limits of hyperk\\\"ahler manifolds with bounded curvature.\n\n\n\n\n\n\n\\begin{proposition}\\label{Cheeger-Tian} There exists $\\epsilon,c$ such that the following holds:\nLet $(M^4,g)$ be an Einstein 4-manifold, $|\\emph{Ric}|\\leq 3$, $r\\leq 1$, and $B(p,r)$ is a metric ball that has compact closure. If $$\\int_{B(p,r)}|Rm|^2\\leq\\epsilon,$$ then $$\\sup\\limits_{B(p,\\frac{r}{2})}|Rm|\\leq c r^{-2}.$$\n\\end{proposition}\n\n\nThe following proposition plays an important role. To avoid redundancy, we state it here without proving, but we will prove a more general version later, see Proposition \\ref{no bubble 2}.\n\n\n\\begin{proposition}\\label{no bubble}\nLet $(M,g)$ be an hyperk\\\"ahler 4-manifold. Suppose $B(p,5)$ has compact closure, $B(p,3)$\ncontains no $-2$ curve,  $$\\emph{vol}(B(p,1))\\geq v,$$\n$$\\int_{B(p,3)}|Rm|^2\\leq C.$$Then there exists $C'>0$, depending on $v,C$ such that $$\\sup_{B(p,1)}|Rm|\\leq C'.$$\n\\end{proposition}\n\n\n\n\nThe following lemma originally dates back to Koiso in \\cite{koiso1981hypersurfaces}, \nand is an incredibly special case of the result in \\cite{biquard:hal-02928859}\\cite{anderson2008unique}.\nSince it plays an important role throughout the paper, we provide a detailed proof here following \\cite{koiso1981hypersurfaces}.\n\n\n\n\n\n\\begin{lemma}\\label{flat totally geodesic}\nLet $(M,g)$ be a connected  $C^2$ Riemannian manifold with boundary. Suppose $\\emph{Ric}_M=0$ and for some open boundary portion $T$, $S|_T=0$ and $Rm_{\\partial M}|_T=0$, then $g$ is smooth and $Rm_M=0$.\n\\end{lemma}\n\n\\begin{proof}\nFix any point $p$ in $T$. First, we show $g$ can be extended across the boundary near $p$.\nChoose a semi-geodesic coordinate system $(x^1,\\cdots, x^n)$ near $p$, with $\\partial M$ identified with $\\{x^n=0\\}$, and interior identified with $\\{x^n>0\\}$, $\\nabla x^n=\\partial_{ x^n}$, and $g_{ij}(x',0)=\\delta_{ij}, 1\\leq i,j\\leq n-1$, where $x'=(x^1,\\cdots, x^{n-1})$.  We extend the metric tensor $g$ by reflection across $\\{x^n=0\\}$, i.e., set $g_{ij}(x',x^n)=g_{ij}(x',-x^n), 1\\leq i,j\\leq n$. Then $g\\in C^0(B)\\cap C^2(B^+)$, where $B^+$ is the boundary coordinate ball and $B$ is its extension after reflection. We need to show $g\\in C^2(B)$. In fact, $S=0$ is equivalent to $\\frac{\\partial g_{ij}}{\\partial x^n_+}(x',0)=0, 1\\leq i,j\\leq n-1,$ then we also have $\\frac{\\partial g_{ij}}{\\partial x^n_-}(x',0)=-\\frac{\\partial g_{ij}}{\\partial x^n_+}(x',0)=0,$ hence  $\\frac{\\partial g_{ij}}{\\partial x^n}(x',0)=0$ and $g_{ij}\\in C^1(B)$. Since $g\\in C^2(B^+)$, we have for $1\\leq l\\leq n-1$,\n $$\\frac{\\partial }{\\partial x^n_+}\\frac{\\partial g_{ij}}{\\partial x^l}(x',0)=\\frac{\\partial }{\\partial x^l}\\frac{\\partial g_{ij}}{\\partial x^n_+}(x',0)=\\frac{\\partial }{\\partial x^l}0=0,$$ $$\\frac{\\partial }{\\partial x^n_-}\\frac{\\partial g_{ij}}{\\partial x^l}(x',0)=-\\frac{\\partial }{\\partial x^n_+}\\frac{\\partial g_{ij}}{\\partial x^l}(x',0)=0.$$ \n \\begin{equation}\n \\begin{split}\n      \\frac{\\partial}{\\partial x^n_-}\\frac{\\partial g_{ij}}{\\partial x^n}(x',0)&=\\lim\\limits_{x^n\\rightarrow 0^-} \\frac{1}{x^n}\\frac{\\partial g_{ij}}{\\partial x^n}(x',-x^n)\\\\ &=\\lim\\limits_{x^n\\rightarrow 0^+}\\frac{1}{x^n}\\frac{\\partial g_{ij}}{\\partial x^n}(x',x^n)=\\frac{\\partial}{\\partial x^n_+}\\frac{\\partial g_{ij}}{\\partial x_n}(x',0).\n \\end{split}\n \\end{equation}\n  Hence $g_{ij}\\in C^2(B)$. Finally, we have $g_{nn}=1,g_{ln}=g_{nl}=0, 1\\leq l\\leq n-1$, hence $g\\in C^2(B)$.\n\n\nBy elliptic regularity, all harmonic coordinate charts in $B$ give rise to a real analytic structure in $B$ such that $g$ is real analytic. Hence if $t$ is a distance function such that $\\partial M$ is defined by $t^{-1}(0)$ near $p$, then $t$ is real analytic near $p$. Choose a real analytic coordinate $(z,t)$ near $p$. Since $t^{-1}(0)$ is totally geodesic, $\\frac{\\partial g}{\\partial t}(z,0) =0$. The evolution equation (\\ref{evolution equation for 2nd form}) is equivalent to the second order PDE\n\\begin{equation}\n\\frac{\\partial^2 g}{\\partial t^2}=2 ric \\ g-\\frac{1}{2}tr_g(\\frac{\\partial g}{\\partial t})\\frac{\\partial g}{\\partial t}+(\\frac{\\partial g}{\\partial t})^2,\n\\end{equation}\n where $ric\\ g$ is the Ricci tensor of level sets of $t$. By the uniqueness part of Cauchy-Kovalevskaya theorem, we know $g(z,t)=g(z,0)$. Hence $Rm_M=0$ near $p$. Since $Rm_M$ is real analytic in the interior of $M$, $Rm_M=0$ in $M$.\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Convergence of hyperk\\\"ahler metrics}\n\n\\begin{theorem}\\label{main theorem hyperkahler metric}\nLet $(X_i,g_i)$ be a sequence of compact, connected hyperk\\\"ahler 4-manifold with boundary, suppose on $\\partial X_i$, we have\n$$H_i\\geq H_0>0, |S_i|\\leq C, |\\nabla_{\\partial X_i}^{j+1} H|\\leq C_j,$$   $$inj_{\\partial X_i}\\geq i_0, \\emph{diam}_{g_i|_{\\partial X_i}}(\\partial X_i)\\leq C,|\\nabla^j_{\\partial X_i}Rm_{\\partial X_i}|\\leq C_j,\\forall j\\geq 0, $$ \nand $\\chi(X_i)\\leq C$.\nAssuming there exists no $-2$ curve on $X_i$, \nthen there exists a subsequence such that $(X_i,g_i)$ converges in Cheeger-Gromov sense to a compact, connected hyperk\\\"ahler 4-manifold with boundary $(X_\\infty, g_\\infty).$\n\\end{theorem}\n\n\\begin{remark}\nBy Chern-Gauss-Bonnet formula, our assumptions imply $$\\int_{X_i}|Rm|^2\\leq C.$$ Note that for a compact connected Einstein 4-manifold $(M,g)$ with boundary, the Chern-Gauss-Bonnet formula says\n$$\\frac{1}{8\\pi^2}\\int_M|Rm|^2=\\chi(M)-\\frac{1}{2\\pi^2}\\int_{\\partial M}\\prod\\limits_{i=1}^3\\lambda_i-\\frac{1}{8\\pi^2}\\int_{\\partial M}\\sum\\limits_{\\sigma\\in S_3}K_{\\sigma_1\\sigma_2}\\lambda_{\\sigma_3}. $$\nHere $\\lambda_1,\\lambda_2,\\lambda_3$ are eigenvalues of the shape operator $S$ of $\\partial M$, Let $e_i$ be eigenvectors of eigenvalue $\\lambda_i$ such that $\\{e_1,e_2,e_3\\}$ is an orthonormal basis, then $K_{ij}=\\sec(e_i,e_j) $. See for example (1.16) in \\cite{Anderson2000L2CA}.\n\\end{remark}\n\n\n\\begin{remark}\nIf we drop the condition $\\text{diam}_{g_i|_{\\partial X_i}}(\\partial X_i)\\leq C$, and replace $H_i\\geq H_0>0$ by $H_i>0$, $\\chi(X_i)\\leq C$ by $\\int_{X_i}|Rm_{g_i}|^2\\leq C$, then for any point $p_i\\in X_i$, a subsequence of $(X_i,g_i,p_i)$ converges in pointed Cheeger-Gromov sense to a complete, connected hyperk\\\"ahler 4-manifolds with nonempty or empty boundary $(X_\\infty,g_\\infty,p_\\infty)$, depending on whether the distance of $p_i$ to boundary is bounded or not. \n\\end{remark}\n\n\\begin{remark}\\label{positive mean curvature essential}\nThe positive mean curvature condition is necessary. The following counterexample is natural and was observed by Donaldson in \\cite{donaldson2018remarks}. Consider the standard unit ball $B^4$  inside Euclidean $\\mathbb{R}^4$, ``squeeze'' the ball such that the north pole and the south pole of the boundary $S^3$ comes together, so we get a sequence of embedded $B^4$ in $\\mathbb{R}^4$ converging in Hausdorff sense to a limit homeomorphic to the wedge sum of two $B^4$, whose boundary is an immersed $S^3$ intersecting itself at one point. For this sequence, all other assumptions are satisfied. Slightly modifying the process, one can also have a sequence of $B^4$ of dumbbell shape such that the middle cyclinder $B^3\\times [0,1]$ collapses to $[0,1]$, then they have a Hausdorff limit which is homeomorphic to two $B^4$ joint by a line segment.\n\nIn these types of examples, the curvatures are uniformly bounded, and the global volume are non-collapsing before taking the limit.\n\n\n\\end{remark}\n\n\n\\begin{remark}\\label{no -2 curve essential}\n\nIf we allow $-2$ curves and do not assume the positive mean curvature condition, something worse will happen: consider the Kummer construction. Let $T^4\/\\mathbb{Z}_2$ be the flat 4-orbifold with 16 singularities, remove small neighborhoods of the 16 singularities, and glue in 16 copies of $T^*S^2$. By varing the sizes of these glue-in regions and perturbing the metrics, we get a sequence of hyperk\\\"ahler 4-manifolds $(M_i,h_i)$, each of which is diffeomorphic to a $K3$ surface, such that $(M_i,h_i)$ converges in Gromov-Hausdorff sense to $T^4\/\\mathbb{Z}_2$, and converges in Cheeger-Gromov sense away from these 16 singularities. Now let $T^3\/\\mathbb{Z}_2\\subset T^4\/\\mathbb{Z}_2$ be the flat 3-orbifold, such that the last coordinate equal to 0. Let $\\tilde{X}_i\\subset T^4\/\\mathbb{Z}_2$  be the closure of the tubular neighborhood of $T^3\/\\mathbb{Z}_2$ of width $i^{-1}$, then $\\partial \\tilde{X}_i$ is connected, totally geodesic, isometric to the same flat $T^3$. Now for each $i$, choose $n(i)$ large enough such that one can find  $X_i\\subset M_{n(i)}$ which are compact domains with smooth boundary,  $d_{GH}(X_i,\\tilde{X}_i)\\rightarrow 0$, $|\\nabla_{\\partial X_i}^j S_{\\partial X_i}|\\rightarrow 0 $, $\\forall j\\geq 0$, $\\partial X_i$ converges in Cheeger-Gromov sense to $\\partial \\tilde X_i$. Let $g_i=h_{n(i)}|_{ X_i}$, then $(X_i,g_i)$ converges in Gromov-Hausdorff sense to flat $T^3\/\\mathbb{Z}_2$. In this case, $\\text{vol}(X_i)\\rightarrow 0$, $\\sup\\limits_{X_i}|Rm_{g_i}|\\rightarrow\\infty$.\n\nNote that in this example $\\partial \\tilde{X}_i$ cannot be perturbed in flat $T^4\/\\mathbb{Z}_2$ to have positive mean curvature. In fact,\ntake a small tubular neighborhood of $\\partial \\tilde{X}_i$ of width less than $i^{-1}$, whose boundary has two totally geodesic connected components $T_1,T_2$. Suppose $\\partial \\tilde{X}_i$ can be perturbed to $T'$ such that its mean curvature has a strict sign, say that its mean curvature vector points towards $T_1$. Then $T', T_1$ bound a region $W$. By \\cite{donaldson2018remarks} Proposition 7, $T'$ and $T_1$ are isometric, so $T'$ is totally geodesic, which is a contradiction.  \n\n\\end{remark}\nThe major step of the proof in \\ref{main theorem hyperkahler metric} is to show that these Riemannian manifolds have a nice neighborhood of definite size. We show that the boundary injectivity radius has a lower bound, so that the interior geometry within the boundary injectivity radius is nicely controlled by Theorem \\ref{good boundary}.\n\n\\begin{proposition}\\label{global inj lower bound}\nThere exists $i_1>0$, depending on the constants in Theorem \\ref{main theorem hyperkahler metric} such that    $i_{b,g_i}\\geq i_1.$\n\\end{proposition}\n\n\\begin{proof}\nSuppose not, we have a subsequence of hyperk\\\"ahler metrics $g_i$,  with $i_{b,{g}_i}\\rightarrow 0$. Rescale the metric $\\tilde g_i=i_{b,g_i}^{-2}{g}_i$, then $i_{b,\\tilde g_i}=1$. For any point $p_i\\in \\partial X_i$, the restriction metric $(\\partial X_i,\\tilde{g}_i|_{\\partial X_i},p_i)$ converges in pointed Cheeger-Gromov sense to  flat $\\mathbb{R}^3$, $|S_{\\tilde{g}_i}|\\rightarrow 0$ and $|\\nabla_{\\partial X_i}^{j+1} H_i|\\rightarrow 0$ uniformly on $\\partial X_i$. Consider $\\sup\\limits_{B_{\\tilde g_i}(p_i,4)}|Rm_{\\tilde g_i}|$. We have two cases \n\n\\textbf{Case 1} $\\sup\\limits_{B_{\\tilde g_i}(p_i,4)}|Rm_{\\tilde g_i}|\\leq C.$\\\\\nWe have a subsequence  $(B_{\\tilde g_i}(p_i,3),\\tilde g_i)$ converges in Cheeger-Gromov sense to a Riemannian manifold with boundary  $(B_\\infty,g_\\infty)$, so $(B_\\infty,g_\\infty)$ is Ricci flat, and all boundary components are flat, totally geodesic. By Lemma \\ref{flat totally geodesic}, $(B_\\infty,g_\\infty)$ is flat. We need to choose good points $p_i$ to lead to a contradiction. In fact, by Proposition \\ref{existence of focal points}, there exists $p_i\\in\\partial X_i$ such that $\\gamma_{p_i}(1)$ is a focal point along $\\gamma_{p_i}$. Let $p_\\infty$ be the limit of $p_i$. By Proposition \\ref{pass to limit}, we get a limit geodesic $\\gamma_{p_\\infty}: [0,1]\\rightarrow B_\\infty$, such that $\\gamma_{p_\\infty}(1)$ is a focal point along $\\gamma_{p_\\infty}$, contradiction. \n\n\\textbf{Case 2} For some subsequence we have $\\sup\\limits_{B_{\\tilde g_i}(p_i,4)}|Rm_{\\tilde g_i}| \\rightarrow\\infty$.\\\\\nThen we can find points $q_i\\in B_{\\tilde{g}_i}(p_i,4)$ such that $|Rm_{\\tilde g_i}(q_i)|\\rightarrow\\infty$. By Theorem \\ref{good boundary}, we have $d_{\\tilde g_i}(q_i,\\partial X_i)\\geq \\frac{1}{2}$ for large $i$. By Theorem \\ref{good boundary}, Proposition \\ref{volume noncollasping}, and the fact  $d_{\\tilde{g}_i}(q_i,\\partial X_i)\\leq 4$, we have  $\\text{vol}_{\\tilde{g}_i}(B_{\\tilde{g}_i}(q_i,\\frac{r_0}{2}))\\geq v_0$ for some $r_0,v_0$. Since there is no $-2$ curve in $X_i$, then by Proposition \\ref{no bubble}, $\\sup_{B_{\\tilde{g}_i}(q_i,\\frac{r_0}{10})}|Rm_{\\tilde{g}_i}|\\leq C$, which is a contradiction.\n\n\\end{proof}\n\n\n\n\n\n\nThen we can finish the proof of Theorem \\ref{main theorem hyperkahler metric} as follows: by Theorem \\ref{good boundary}, $|Rm_{g_i}|$ is uniformly bounded in $N_{\\frac{i_1}{2}}(\\partial X_i,g_i)$. By Proposition $\\ref{volume noncollasping}$, there exist $r_1<\\frac{i_1}{10},v_1$, such that $\\text{vol}_{g_i}(B_{g_i}(p,r_1))\\geq v_1$ for any $p$ with $d_{g_i}(p,\\partial X_i)=2r_1$.  By Proposition \\ref{diam}, $\\sup\\limits_{q\\in X_i}d_{g_i}(q,\\partial X_i)\\leq 3H_0^{-1}$, hence $\\text{diam} (X_i,g_i)\\leq C$. Then from Bishop-Gromov volume comparison,  we know $\\text{vol}_{g_i}(B_{g_i}(p,r_1))\\geq v_2$ for any $p$ with $d_{g_i}(p,\\partial X_i)\\geq 5r_1$.  By Proposition $\\ref{no bubble}$, for these $p$,\n$|Rm_{g_i}(p)|$ is uniformly bounded, with the bound independent of $i$ and $p$. Hence $\\sup_X |Rm_{g_i}|$ is uniformly bounded. Then by Corollary \\ref{corollary that will be used often},\n a subsequence $(X_i,g_i)$ converges in Cheeger-Gromov sense to a smooth Riemannian manifold with boundary $(X,g)$, so $g$ is a hyperk\\\"ahler metric.\n\n\n\n\\begin{remark} One can also directly prove Theorem \\ref{main theorem hyperkahler metric} by rescaling the maximum curvature norm to be 1. Suppose it is achieved at $p_i$. Then from Corollary \\ref{Ric positive, mean positive, inj lower bound}, we know $i_b\\geq i_0$ for the rescaled metrics. If now $d(p_i,\\partial X_i)$ is bounded, then the pointed Riemannian manifolds converge in pointed Cheeger-Gromov sense to a Ricci-flat manifold with flat and totally geodesic boundary, hence the limit is flat, contradition. Otherwise, for a subsequence $d(p_i,\\partial X_i)\\rightarrow\\infty$, then we rescale the metrics again such that this distance is 1. However, it is unknown whether the curvature is bounded in a fixed size neighborhood of the boundary in this scale. Using the idea of Theorem \\ref{good boundary}, we can keep rescaling such that this will happen at some stage, for possibly different points $p_i'$, then get a contradiction. The contradiction is the same the one in Case 2 in Proposition \\ref{global inj lower bound}, since volume lower bounds can be passed within finite distance, so do the curvature bounds by Proposition \\ref{no bubble}. Alternatively, one can prove Theorem \\ref{main theorem hyperkahler metric} by rescaling the harmonic radius, and all these methods turn out to be equivalent eventually.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Convergence of hyperk\\\"ahler triples}\\label{moduli space}\n\nNow we turn to the setting of \\ref{convergence of triples}. ${X}$ will denote a compact oriented 4-manifold with boundary $\\partial X=Y$. Let $\\mathcal{N}^+$ be the set of closed framings $\\bm{\\gamma}=(\\gamma_1,\\gamma_2,\\gamma_3)$ on $Y$  that satisfies the ``positive mean curvature'' condition\n\n\\begin{equation}\\label{mean positive}\n\\sum\\limits_{i=1}^3 \\langle\\gamma_i,d(*_{\\bm\\gamma}\\gamma_i)\\rangle_{\\bm\\gamma}>0.\n\\end{equation}\nLet $\\mathcal{M}^+$ be the set of smooth hyperk\\\"ahler triples $\\bm\\omega=(\\omega_1,\\omega_2,\\omega_3)$ whose restriction to the boundary lies in $\\mathcal{N}^+$, so we have a restriction map $p_0:\\mathcal{M}^+\\rightarrow\\mathcal{N}^+$, which induce a map\n\\begin{equation}\\label{map between moduli space}\n    p:\\mathcal{M}^+\/\\mathcal{G}_X\\rightarrow\\mathcal{N}^+\/\\mathcal{G}_Y.\n\\end{equation}\n  Here $\\mathcal{M}^+$, $\\mathcal{N}^+$ are equipped with Fr\\'echet topology defined by smooth convergence, $\\mathcal{G}_X,\\mathcal{G}_Y$ are orientation preserving diffeomorphism groups of $X,Y$, respectively. It is obvious that $p_0$, $p$ are continuous. \nThen the compactness part of Theorem \\ref{convergence of triples} is equivalent to:\n\\begin{proposition}\nWhen there is no $-2$ curve in $X$, the map $p:\\mathcal{M}^+\/\\mathcal{G}_X\\rightarrow\\mathcal{N}^+\/\\mathcal{G}_Y$   is proper.\n\\end{proposition}\n\n\\begin{proof}\nSuppose for some $\\phi_i\\in\\mathcal{G}_Y$, and $\\bm\\gamma_i\\in\\mathcal{N}^+$,we have          $\\phi_i^*{\\bm\\gamma_i}\\rightarrow \\bm\\gamma\\in \\mathcal{N}^+$ and there exist $\\bm\\omega_i\\in\\mathcal{M}^+$ with $\\bm\\omega_i|_Y=\\bm\\gamma_i$, we want to show there exists $\\psi_i\\in\\mathcal{G}_X$ and $\\bm\\omega\\in\\mathcal{M}^+$  such that $\\psi_i^*\\bm\\omega_i\\rightarrow\\bm\\omega$. Let $g_i$ be the Riemannian metric defined by $\\bm\\omega_i$. By the assumptions, we have a uniform positive lower bound for the mean curvature $H_i$ of $Y$ of the metric $g_{i}=g_{\\bm\\omega_i}$, and bounds for $|\\nabla_{Y}^l S_i|$ for all $l\\geq 0$, and $(Y,g_i|_{Y})$ converges in Cheeger-Gromov sense. Hence $(X,g_i)$ satisfies all conditions in Theorem \\ref{main theorem hyperkahler metric}. Then for a subsequence, there exists diffeomorphism $\\psi_i: X\\rightarrow X$ such that $\\psi_i^*g_i\\rightarrow g$ smoothly as tensors. One can assume $\\psi_i$ is orientation preserving, otherwise, for a subsequence, compose them with a fixed orientataion reversing diffeomorphism of $X$. Since $|\\psi_i^*\\bm\\omega_i|^2=3$, $\\psi_i^*\\bm\\omega_i$ are parallel, we conclude that for some subsequence $\\psi_i^*\\bm\\omega_i\\rightarrow\\bm\\omega$ smoothly, and $\\bm\\omega\\in\\mathcal{M}^+$.\n\n\\end{proof}\n\n\n\n\n\\subsection{Enhancements}\n\nFor the proof of the compactness part of Theorem \\ref{convergence of triples enhancements}, one only needs the following proposition in place of Proposition \\ref{no bubble}. Then we argue in the same way as the proof of Theorem \\ref{convergence of triples}. \n\n\n\n\\begin{proposition}\\label{no bubble 2}\nLet $(M,g,\\bm\\omega)$ be a hyperk\\\"ahler 4-manifold. Suppose $B(p,5)$ has compact closure, and for any $-2$ curve $\\Sigma$ in $B(p,3)$\n, $$\\Big{|}\\int_{\\Sigma}\\bm\\omega\\Big{|}\\geq a>0,$$  $$\\emph{vol}(B(p,1))\\geq v,$$ \n$$\\int_{B(p,3)}|Rm|^2\\leq C. $$ Then there exists $C'>0$, depending on $a,v,C$ such that $$\\sup_{B(p,1)}|Rm|\\leq C'.$$\n\\end{proposition}\n\n\n\\begin{proof}\n\nFor all $q\\in B(p,2)$, by volume comparison, we have $$\\text{vol}(B(q,1))\\geq 3^{-4} \\text{vol}(B(q,3))\\geq 3^{-4}\\text{vol}(B(p,1))\\geq 3^{-4} v.$$\nSuppose the conclusion is not true, then we have a sequence $(M_i,g_i,p_i)$ satisfies the conditions, but\n there exists $q_i'\\in B(p_i,1)$ with $|Rm_{g_i}(q_i')|\\rightarrow\\infty$. By the following Lemma \\ref{point selection}, we can find points $q_i\\in B(p_i,2)$ such that $|Rm_{g_i}(q_i)|\\geq |Rm_{g_i}(q_i')|$, and $$\\sup\\limits_{B_{g_i}(q_i,|Rm_{g_i}(q_i')|^{\\frac{1}{2}}|Rm_{g_i}(q_i)|^{-\\frac{1}{2}})}|Rm_{g_i}|\\leq 4|Rm_{g_i}(q_i)|.$$  Rescale the metric $\\tilde{g}_i=|Rm_{g_i}(q_i)|g_i$, $\\tilde{\\bm\\omega}_i=|Rm_{g_i}(q_i)|\\bm\\omega_i$, so $\\tilde{\\bm\\omega}_i$ defines $\\tilde{g}_i$. Then we have $$\\sup\\limits_{B_{\\tilde{g}_i}(q_i,|Rm_{g_i}(q_i')|^{\\frac{1}{2}})}|Rm_{\\tilde{g}_i}|\\leq 4 , $$  $$|Rm_{\\tilde{g}_i}(q_i)|=1,$$ $$ \\text{vol}_{\\tilde{g}_i}(B_{\\tilde{g}_i}(q_i,r))\\geq 3^{-4}vr^n,\\forall r\\leq|Rm_{g_i}(q_i)|^{\\frac{1}{2}},$$ and\n$$\\int_{B_{\\tilde{g}_i}(q_i, |Rm_{g_i}(q_i)|^{\\frac{1}{2}})}|Rm_{\\tilde{g}_i}|^2\\leq \\int_{B_{\\tilde{g}_i}(p_i, 3|Rm_{g_i}(q_i)|^{\\frac{1}{2}})}|Rm_{\\tilde{g}_i}|^2\\leq C.$$\nHence, for a subsequence, $(M_i,\\tilde{g}_i,q_i)$ converges in Cheeger-Gromov topology to a complete non-flat hyperk\\\"ahler 4-manifold  $(M_\\infty,g_\\infty,q_\\infty)$ with maximum volume growth and $$\\int_{M_\\infty}|Rm_{g_\\infty}|^2\\leq C.$$\nSince $\\tilde{\\bm\\omega}_i$ are parallel, and $|\\tilde{\\bm\\omega}_i|^2=6$, $\\tilde{\\bm\\omega}_i$ also subconverges to a hyperk\\\"ahler triple $\\bm\\omega_\\infty$ that defines $g_\\infty$. By \\cite{bando1989construction}, $(M_\\infty,g_\\infty)$ is a hyperk\\\"ahler ALE space of order 4. Hence, from Kronheimer's classification \\cite{kronheimer1989construction}\\cite{kronheimer1989torelli}, there exists a $-2$ curve $C_\\infty$ in $B_{g_\\infty}(q_\\infty, R)$ for some $R>0$\nsuch that $|\\int_{C_\\infty}\\bm\\omega_{\\infty}|\\neq 0$.\nLet $\\phi_i:B_{g_\\infty}(q_\\infty,R)\\rightarrow V_i$ be diffeomorphisms, such that $\\phi_i^*\\tilde{\\bm\\omega}_i\\rightarrow \\bm\\omega_\\infty$ smoothly. Let $C_i=(\\phi_i)_*C_\\infty$, then $C_i$ is a $-2$ curve in $B_{g_i}(p_i,3)$ for large $i$ and $$\\int_{C_i} \\bm\\omega_i=|Rm_{g_i}(q_i)|^{-1}\\int_{C_i}\\tilde{\\bm\\omega}_i=|Rm_{g_i}(q_i)|^{-1}\\int_{C_\\infty} \\phi_i^*\\tilde{\\bm\\omega}_i \\rightarrow 0, $$\nwhich contradicts our assumption.\n\n\\end{proof}\n\n\nThe following point selection lemma is well-known and elementary. \n\n\\begin{lemma}\\label{point selection}\nLet $(M,g)$ be a Riemannian manifold. Suppose $\\sup\\limits_{B(p,2)}|Rm|<\\infty$, $|Rm(p)|\\neq 0$. Then there exists a point $q\\in B(p,2)$ such that $|Rm(q)|\\geq |Rm(p)|$, and $\\sup\\limits_{B(q,{|Rm(p)|}^{\\frac{1}{2}}{|Rm(q)|^{-\\frac{1}{2}}})}|Rm|\\leq  4|Rm(q)|.$  \n\\end{lemma}\n\n\\begin{proof}\nIf this is not true, let $A=|Rm(p)|^{\\frac{1}{2}}$, then there exist $q_1\\in B(p,2)$ such that $d(q_1,p)< 1$ and $|Rm(q_1)|> 4|Rm(p)|=4A^2$. By induction, we can find a sequence of points $q_0=p, q_1,q_2,\\cdots$ with $d(q_{j+1},q_j)< |Rm(q_{j})|^{-\\frac{1}{2}}A$, $d(q_{j+1},p)< 2-2^{-j}$ and $|Rm(q_{j+1})|>  4^{j+1} A^2$. This is an obvious contradiction since $\\sup\\limits_{B(p,2)}|Rm|<\\infty.$\n\\end{proof}\n\n\n\\subsection{Uniqueness}\\label{uniqueness 1}\n\n\nGiven the compactness results, it is natural to ask whether a subsequential limit $\\bm\\omega$ is unique(up to a diffeomorphism) in Theorem \\ref{convergence of triples}, Theorem \\ref{convergence of triples enhancements}. The answer is yes. In fact, both\\cite{biquard:hal-02928859} and \\cite{anderson2008unique} proved a\nunique continuation theorem for Einstein metrics with prescribed boundary metric and second fundamental form, which implies the following:\n\n\\begin{proposition}\\label{local uniqueness hyperkahler triple}\nLet $X$ be a connected oriented 4-manifold with boundary. Suppose $\\bm\\omega_1,\\bm\\omega_2$ are two smooth hyperk\\\"ahler triples on $X$. Suppose $\\bm\\omega_1|_{\\partial X}=\\bm\\omega_2|_{\\partial X}$ , then in geodesic gauges of $g_{\\bm\\omega_1}$, $g_{\\bm\\omega_2}$, we have\n$\\bm\\omega_1=\\bm\\omega_2$ near $\\partial X$. \n\\end{proposition}\n\n\\begin{proof} \nWe have $g_{\\bm\\omega_1}|_{\\partial X}=g_{\\bm\\omega_2}|_{\\partial X}$ and $II_{g_{\\bm\\omega_1}}=II_{g_{\\bm\\omega_2}}$ on $\\partial X$. By \\cite{biquard:hal-02928859} Theorem 4, in geodesic gauges, we have $g_{\\bm\\omega_1}=g_{\\bm\\omega_2}$. Since $\\nabla^{g_{\\bm\\omega_1}} |\\bm\\omega_1-\\bm\\omega_2|_{g_{\\bm\\omega_1}}^2=0$, and $\\bm\\omega_1=\\bm\\omega_2$ at one point, we have $\\bm\\omega_1=\\bm\\omega_2$ everywhere near $\\partial X$.\n\\end{proof}\n\n\n\nNow the following global uniqueness result follows from an analytic continuation argument (See \\cite{kobayashi1963foundations}  Chapter VI, Section 6): \n\n\n\n\\begin{theorem}\\label{unique continuation hyperkahler}\nLet $X$ be a connected 4-manifold with boundary,  $\\pi_1(X,\\partial X)=0$. Suppose $\\bm\\omega_1,\\bm\\omega_2$ are two smooth hyperk\\\"ahler triples on $X$, and $\\phi_0:\\partial X\\rightarrow\\partial X$ is a diffeomorphism, such that $\\bm\\omega_1|_{\\partial X}=\\phi_0^*(\\bm\\omega_2|_{\\partial X})$, then there exists a diffeomorphism $\\phi:X\\rightarrow X$,   $\\phi|_{\\partial X}=\\phi_0$ such that  $\\bm\\omega_1=\\phi^*\\bm\\omega_2$ on $X$.\n\\end{theorem}\n\n\n\\begin{remark} This theorem implies that\nthe map $p:\\mathcal{M}^+\/\\mathcal{G}_X\\rightarrow \\mathcal{N}^+\/\\mathcal{G}_Y$ defined in subsection \\ref{moduli space} is injective, provided that $\\mathcal{M}^+$ is nonempty.\n\\end{remark}\n \n \n\\begin{proof}\nBy the previous proposition, there exists a collar neighborhood $U$ of $\\partial X$ and a diffeomorphism $\\phi_1: U\\rightarrow V\\subset X$ such that $g_{\\bm\\omega_1}=\\phi_1^*g_{\\bm\\omega_2}$, $\\phi_1|_{\\partial X}=\\phi_0$. Since $g_{\\bm\\omega_1},g_{\\bm\\omega_2}$ are real analytic, $\\phi_1$ is real analytic.  Fix $p_0\\in U$ and a small neighborhood $U_0$ of $p_0$ in $X$. For any $p\\in X\\backslash \\partial X$, choose a path $x(t),0\\leq t\\leq 1$ such that $x(0)=p_0, x(1)=p, x(t)\\in X\\backslash\\partial X$, then an analytic continuation of the isometry $\\phi_1|_{U_0}$ along $x(t)$ gives rise to an isometry defined near $p$. We claim that if we have two paths and two analytic continuations, then they define the same germ at $p$. In fact, one only needs to show that for the closed path $y_0(t)$ formed by concatenating these two paths, the isometry near $p_0$ given by an analytic continuation of $\\phi_1|_{U_0}$  along $ y_0(t)$ has the same germ as $\\phi_1|_{U_0}$ at $p_0$. Since $\\pi_1(X,\\partial X)=0$, $y_0(t)$ can be homotoped to a path $y_1(t)$ contained in $U$ via paths $y_s(t)$ in $X\\backslash\\partial X$, such that $y_s(0)=y_s(1)=p_0$. Since $\\phi_1:U\\rightarrow V$ is a globally defined isometry, by uniqueness of analytic continuation, we know that any analytic continuation of $\\phi_1|_{U_0}$  along $y_1(t)$ must coincide with $\\phi_1$, which finishes the proof of the claim by invariance of analytic continuation via homotopy. This shows that $\\phi_1:U\\rightarrow V$ can be extended to a global isometry $\\phi:X\\rightarrow X$ by analytic continuation. Hence $\\bm\\omega_1=\\phi^*\\bm\\omega_2$ by the same argument as before.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\nGiven the compactness result,  Theorem \\ref{unique continuation hyperkahler}, together with the theorem of Ebin-Palais on   properness of diffeomorphism group action on the space of Riemannian metrics on a closed manifold (See \\cite{ebin1970manifold}), one can answer the question raised at the beginning of this subsection, thus finish the proof of Theorem \\ref{convergence of triples}, Theorem \\ref{convergence of triples enhancements}:\n \n\n\n\nSuppose we have two sequences of diffeomorphism $\\phi_i,\\psi_i$ of $X$ such that  $\\phi_i^*\\bm\\omega_i\\rightarrow \\bm\\omega$, $\\psi_i^*\\bm\\omega_i\\rightarrow\\bm\\omega'$, then $(\\phi_i|_{\\partial X})^*\\bm\\gamma_i\\rightarrow \\bm\\omega|_{\\partial X}$, $(\\psi_i|_{\\partial X})^*\\bm\\gamma_i\\rightarrow \\bm\\omega'|_{\\partial X}$, where $\\bm\\gamma_i=\\bm\\omega_i|_{\\partial X}$. Since $\\bm\\gamma_i$ converges to   $\\bm\\gamma$ in Cheeger-Gromov sense, there exists diffeomorphisms $u_i:\\partial X\\rightarrow\\partial X$ such that $u_i^*\\bm\\gamma_i\\rightarrow\\bm\\gamma$. By \nthe theorem of Ebin-Palais, we have for a subsequence $(\\phi_i|_{\\partial X})^{-1}\\circ u_i , (\\psi_i|_{\\partial X})^{-1}\\circ u_i $ converge to some diffeomorphisms $u,u'$ on $\\partial X$, respectively(because their inverses converge). \nHence we also have $u_i^*\\bm\\gamma_i\\rightarrow u^*\\bm\\omega|_{\\partial X}$, $u_i^*\\bm\\gamma_i\\rightarrow (u')^*\\bm\\omega'|_{\\partial X}$, so $\\bm\\omega|_{\\partial X}=(u'\\circ u^{-1})^*\\bm\\omega'|_{\\partial X}$. Note that the positive mean curvature condition implies that $\\pi_1(X,\\partial X)=0$ (See Proposition \\ref{diam}),  then by Theorem \\ref{unique continuation hyperkahler}, there exists a diffeomorphism $\\varphi$ on $X$ with $\\bm\\omega'=\\varphi ^*\\bm\\omega$. \n\n\n\n\n\n\\subsection{Some discussions}\n\n\n\n\n\n\nSuppose $X$ is a connected complete metric space, and there exists a finite set $\\Sigma=\\{p_1,\\cdots,p_m\\}, m\\geq 0$ such that $X\\backslash\\Sigma$ is a smooth flat hyperk\\\"ahler\n4-manifold with nonempty boundary, and each boundary component $Y_1,\\cdots, Y_n, n\\geq 1$ is isometric to flat $\\mathbb{R}^3$,  $X\\backslash \\cup_{i=1}^n{Y_i}$ is a flat hyperk\\\"ahler 4-orbifold and $\\Sigma$ is the set of all orbifold points. Our goal is to classify all such $X$.\n\nThe motivation of this problem is the following:\n\\begin{proposition}\\label{classification motivation}\nLet $(X_i,g_i)$ be a sequence of compact hyperk\\\"ahler 4-manifolds with boundary, such that $$i_{b,g_i}\\geq i_0, inj_{\\partial X_i}\\rightarrow\\infty,$$ $$|S_i|\\rightarrow 0, |\\nabla^{j+1}_{\\partial X_i} H_i|\\rightarrow 0, |\\nabla^j_{\\partial X_i} Rm_{\\partial X_i}|\\rightarrow 0$$ uniformly on $\\partial X_i$ for all $j\\geq 0$, and $$\\int_{X_i}|Rm_{g_i}|^2\\leq C.$$ Then for any $p_i\\in X_i$ $d_{g_i}(p_i,\\partial X_i)\\leq K$, a subsequence of $(X_i,g_i,p_i)$ converges in pointed Gromov Hausdorff sense to a complete metric space $(X_\\infty,d_\\infty,p_\\infty)$. The limit $(X_\\infty,d_\\infty)$ satisfies all properties of  $X$ above.\n\\end{proposition}\n\n\nBy Theorem \\ref{good boundary}, the geometry near boundary is nicely controlled,  so this theorem is a consequence of \\cite{anderson1989ricci} or \\cite{bando1989construction} etc.\n\n\\begin{remark}\nIn Theorem \\ref{main theorem hyperkahler metric}, under the mean positive condition, if we allow $-2$ curves in $X_i$, then we are unable to show that $i_{b,g_i}$ has a lower bound. In fact, the bad case is that focal points are exactly those curvature blow-up points, and they approach the boundary in a moderate rate. However, we can say something within the scale of the boundary injectivity radius. Suppose $i_{b,g_i}\\rightarrow 0$, then we rescale the metric such that they are equal to 1, then the rescaled metric satisfies the assumption of Proposition \\ref{classification motivation}.\n\\end{remark}\n\n\n\\begin{theorem}\\label{classification}\n$X$ is isometric to one of the following:\n\\begin{itemize}\n    \\item  $\\text{(m=0, n=1)}$ $\\mathbb{R}^4_+ $;\n    \\item  $\\text{(m=0, n=2)}$   the region in $\\mathbb{R}^4$ bounded by two parallel hyperplanes;\n    \\item $\\text{(m=1, n=1)}$   the connected component of $0$ in   $(\\mathbb{R}^4\\backslash H)\/{\\mathbb{Z}_2}$, where $H$ is a hyperplane such that $0\\notin H$. \n\\end{itemize}\n\n\n\n\\end{theorem}\n\n\n\\begin{proof}\n\nConsider another copy of $X$, glue them together along $Y_1,\\cdots, Y_n$,  we get a complete flat hyperk\\\"ahler orbifold $\\hat{X}=X\\sqcup_{id} X, id:\\cup_{i=1}^nY_i\\rightarrow \\cup_{i=1}^n Y_i\\subset X$, which contains $Y_1,\\cdots, Y_n$ as smooth hypersurfaces. Then $\\hat X$ is a $(SU(2),\\mathbb{R}^4)$ orbifold in the sense of \\cite{thurston1979geometry}. Let  \n$\\tilde{X}$ be the universal covering orbifold of $\\hat X$, then we have a developing map $D:\\tilde {X}\\rightarrow \\mathbb{R}^4$, and since $\\tilde{X}$ is a complete orbifold, $D$ is a covering map.  Hence $\\tilde{X}$ is homoeomorphic to $\\mathbb{R}^4$, and $\\hat X$ is isometric to $\\mathbb{R}^4\/\\Gamma$, where $\\Gamma$ is a discrete subgroup of $\\mathbb{R}^4\\rtimes SU(2)$. Let $r$ be the projection map to the second factor, then $r(\\Gamma)$ is a finite subgroup of $SU(2)$, and we have a short exact sequence\n$$0\\rightarrow \\Gamma\\cap\\mathbb{R}^4\\rightarrow \\Gamma\\rightarrow r(\\Gamma)\\rightarrow 0.$$ \nThen $\\text{rank} (\\Gamma\\cap\\mathbb{R}^4)\\leq 1$, since otherwise $\\hat{X}$ is a quotient of $\\mathbb{R}^2\\times T^2$ and cannot contain a flat $\\mathbb{R}^3$ as a Riemannian submanifold. \n\nIf $\\Gamma\\cap\\mathbb{R}^4=0$, then $\\hat{X}\\cong \\mathbb{R}^4\/r(\\Gamma)$ and\n hence $r(\\Gamma)=\\{1\\}$, since otherwise $\\hat{X}$ has exactly one orbifold point, contradiction. Hence, $m=0, n=1$ and  ${X}$ is isometric to $\\mathbb{R}_+^4$.\n\nIf $\\Gamma\\cap\\mathbb{R}^4=\\mathbb{Z}a$ for some $0\\neq a\\in\\mathbb{R}^4$, let $\\pi: \\mathbb{R}^4\\rightarrow \\mathbb{R}^4\/\\Gamma$ be the covering map, then $\\pi^{-1}(Y_1)$  is a complete totally geodesic submanifold of $\\mathbb{R}^4$, hence a countable disjoint union of parallel hyperplanes.\nPick one of them $Z$, then $\\pi^{-1}(Y_1)$ is a disjoint union of $\\gamma. Z$ for $\\gamma\\in \\Gamma$. Suppose $Z$ is defined by $b^Tx+c=0$, then $\\gamma^{-1}.Z$ is defined by $b^T r(\\gamma)x+c'=0$. Since they are parallel to each other, $b^T=\\pm b^Tr(\\gamma)$, and hence $\\pm 1$ is an eigenvalue of $r(\\gamma)$, which forces $r(\\gamma)=\\pm 1$. Hence $r(\\Gamma)=\\{1\\}$ or $r(\\Gamma)=\\mathbb{Z}_2$. If $r(\\Gamma)=\\{1\\}$, then $\\Gamma$ is generated by $x\\mapsto x+a$, so $m=0, n=2$ and $X$ is isometric to the region in $\\mathbb{R}^4$ bounded by two parallel hyperplanes; If $r(\\Gamma)=\\mathbb{Z}_2$, then $\\Gamma$ is generated by $x\\mapsto -x+d$ and $x\\mapsto x+a$ for some $d\\in\\mathbb{R}$. Hence $m=1$ and $\\mathbb{R}^4\/\\Gamma\\backslash\\{Y_1,Y_2,\\cdots, Y_n\\}$ has $n+1$ connected components. But from the gluing construction and that $X$ is connected, we know $\\hat{X}\\backslash\\{Y_1,Y_2,\\cdots, Y_n\\}$ has exactly two connected components. Hence $n+1=2, n=1$, and $X$ is isometric to the last case in the conclusion.\n\n\n\\end{proof}\n\n\n\\section{Torsion-free hypersymplectic manifolds with boundary}\\label{section torsion-free triples}\n\n\\subsection{Preliminaries}\n\nRecall that a hypersymplectic triple $\\bm\\omega=(\\omega_1,\\omega_2,\\omega_3)$ on an oriented 4-manifold $X$ with boundary is a definite triple of symplectic forms. Write\n $$\\omega_i\\wedge\\omega_j=2Q_{ij}\\mu$$\n Denote $\\bm{Q}=(Q_{ij})$, $\\bm{Q}^{-1}=(Q_{ij})$, and $g=g_{\\bm\\omega}$ the Riemannian metric. Recall that $\\bm\\omega$ is called torsion-free \nif for each $i$,\n \\begin{equation}\n      d(Q^{ij}\\omega_j)=0,\n \\end{equation} which is equivalent to\n\\begin{equation}\\label{torsion-free condition}\n    dQ^{ij}\\wedge \\omega_j=0.\n\\end{equation}\n\nLet us begin with some arguments and results in \\cite{fine2020report}. Let $\\mathcal{P}$ denotes the set of symmetric positive-definite 3 by 3 matrices. There are two Riemannian metrics on $\\mathcal{P}$: the first one is the Euclidean metric \n\n\\begin{equation}\\label{usual laplacian torsion-free}\n    \\langle A, B\\rangle =Tr(AB),\n\\end{equation}\n and the second one is the symmetric space metric, which has non-positive sectional curvature\n\\begin{equation}\\label{hat laplacian torsion-free}\n     \\langle A, B\\rangle_Q=Tr(Q^{-1}A Q^{-1}B) \n\\end{equation}\nat each point $Q\\in \\mathcal{P}$.\n\nThen $\\bm{Q}$ can be regarded as a map $\\bm{Q}:X\\rightarrow \\mathcal{P}$. Let $\\Delta \\bm Q, \\hat{\\Delta}\\bm Q$ denote the harmonic Laplacian of this map with respect to (\\ref{usual laplacian torsion-free}),(\\ref{hat laplacian torsion-free}), respectively. Explicitly, their components are related by\n\\begin{equation}\\label{torsion-free Laplacian operator}\n    (\\hat\\Delta\\bm Q)_{ij}=\\Delta Q_{ij}-Q^{km}\\langle dQ_{ik},dQ_{mj}   \\rangle,\n\\end{equation}\n\n\nFor a hypersymplectic triple $\\bm\\omega$ , the calculations in \\cite{fine2018hypersymplectic} showed that the torsion-free condition is equivalent to \n\\begin{equation}\\label{torsion-free equations}\n\\hat\\Delta\\bm Q=0, \\ \\text{Ric}= \\frac{1}{4}\\langle d\\bm Q\\otimes d\\bm Q\\rangle_{\\bm Q},\n\\end{equation}\nwhere $\\langle d\\bm Q\\otimes d\\bm Q \\rangle_{\\bm Q}(u,v)=\\langle \\nabla_u \\bm Q,\\nabla_v \\bm Q\\rangle_{\\bm Q}$.\nHence if $\\bm\\omega$ is torsion-free, then $\\bm Q$ is a harmonic map with respect to (\\ref{hat laplacian torsion-free}) and  $\\text{Ric}\\geq 0$. Then the scalar curvature $R$ of $g$ is $$R=\\frac{1}{4}|d\\bm Q|_{\\bm Q}^2\\geq 0,$$ \nwhich is a multiple of the energy density of the harmonic map $\\bm Q$.\nTake the trace of (\\ref{torsion-free Laplacian operator}), we get \n\\begin{equation}\\label{subharmonic TrQ}\n     \\Delta Tr \\bm Q=Q^{pq}\\langle dQ_{kp},dQ_{qk}\\rangle\\geq 0.\n\\end{equation}\nMoreover, \\cite{fine2020report} showed that the function $R$ satisfies the inequality \n\\begin{equation}\\label{R Delta R}\nR\\Delta R\\geq \\frac{1}{2}|\\nabla R|^2+\\frac{1}{2}R^3,\n\\end{equation}\nhence a contradiction argument implies that everywhere\n\\begin{equation}\\label{subharmonic scalar curvature}\n    \\Delta R\\geq 0.\n\\end{equation}\nThen they used standard geometric analysis arguments for inequality (\\ref{R Delta R}) and the fact $\\text{Ric}\\geq 0$ to conclude\n\n\\begin{proposition}\\label{fy result}\nSuppose $B(p,r)\\subset X$ has compact closure, and $\\partial B(p,r)\\neq \\emptyset$, then \n$R(p)\\leq \\frac{32}{r^2}$. In particular, a complete torsion-free hypersymplectic 4-manifold is hyperk\\\"ahler.\n\\end{proposition}\nNote that in the latter case, there exists a constant matrix $B$ in $SL(3,\\mathbb{R})$ such that $\\bm \\omega B$ is a hyperk\\\"ahler triple, and the Riemannian metric defined by  $\\bm \\omega$ is the same as the one defined by $\\bm\\omega B$. \n\n\\subsection{Compactness for the boundary value problem}\n \nNow suppose $X$ is an oriented 4-manifold with compact boundary $Y=\\partial X$, then through the boundary exponential map, a neighborhood $U$ of $Y$ is diffeomorphic to $Y\\times [0,a)$. Let $t$ denote the distance function $d(\\cdot, Y)$, i.e., the projection $Y\\times [0,a)\\rightarrow [0,a)$, then in $U$, $\\bm\\omega$ can be written as\n $$\\bm\\omega=-dt\\wedge*_{Y_t}\\bm\\gamma_t+\\bm\\gamma_t.$$\n where $Y_t=Y\\times \\{t\\}$ and $*_{Y_t}$ is the Hodge star operator of $g|_{Y_t}$.\n $\\bm\\omega$ being closed is equivalent to \n \\begin{equation}\n     d_{Y_t}\\bm\\gamma_t=0,\n \\end{equation}\n \\begin{equation}\\label{torsion-free normal tangential 0}\n     \\frac{\\partial \\bm\\gamma_t}{\\partial t}=-d_{Y_t}(*_{Y_t} \\bm\\gamma_t).\n \\end{equation}\n  (\\ref{torsion-free condition})\n is equivalent to \n\\begin{equation}\\label{torsion-free normal tangential}\n    d_{Y_t}Q^{ij}\\wedge *_{Y_t}\\gamma_{t,j}+\\frac{\\partial Q^{ij}}{\\partial t}\\gamma_{t,j}=0,\n\\end{equation}\n \\begin{equation}\n     d_{Y_t}Q^{ij}\\wedge \\gamma_{t,j}=0.\n \\end{equation}\n Note that\n (\\ref{torsion-free normal tangential}) is equivalent to\n \\begin{equation}\\label{torsion-free normal tangential 2}\n     \\frac{\\partial Q^{ij}}{\\partial t}=\\frac{d_{Y_t}Q^{ik}\\wedge \\eta_{t,j}\\wedge *_{Y_t}\\gamma_k}{\\text{vol}_{t}},\n \\end{equation}\n where $\\text{vol}_t=\\eta_{t,1}\\wedge \\eta_{t,2}\\wedge\\eta_{t,3}$ and equals to the Riemannian volume form of $g|_{Y_t}$.  From calculations in Lemma \\ref{hyperkahler triple second fundamental form} and (\\ref{torsion-free normal tangential 0}), (\\ref{torsion-free normal tangential 2}) , it is easy to see that the second fundamental form $II(e_i,e_j)$ is in algebraic terms of $\\bm\\eta, d_{\\partial X}\\bm\\eta,\\bm Q, d_{\\partial X}\\bm Q$.\n\n \n\n Now let us try to prove Theorem \\ref{convergence of hypersymplectic triples}, starting with basic observations.\n In Theorem \\ref{convergence of hypersymplectic triples}, suppose $\\bm\\omega_i|_{\\partial X}, \\bm Q_i$ converge in Cheeger-Gromov sense to the limit,\n then we have $\\text{diam}(\\partial X,g_i|_{\\partial X})\\leq C, \\text{vol}_{g_i}(\\partial X)\\leq C, inj_{\\partial X, g_i|_{\\partial X}}\\geq i_0$,  $|\\nabla_{\\partial X}^j Rm_{\\partial X,g_i}|\\leq C_j$, $|\\nabla ^j_{\\partial X} S_i|\\leq C_j$. By (\\ref{subharmonic scalar curvature}) and maximum principle, $R_i$  is uniformly bounded on $X$, so $|\\text{Ric}_{g_i}|$ is uniformly bounded on $X$ since $\\text{Ric}_{g_i}$ is non-negative and its trace is uniformly bounded.  Due to the uniform mean positive curvature condition, and $\\text{Ric}_{ g_i}\\geq 0$,  we have an upper bound of $\\sup\\limits_{p\\in X}d_{g_i}(p,\\partial X)$, $\\text{vol}_{g_i}(X)$ by Proposition \\ref{diam}, and in particular an upper bound of $\\text{diam}(X, g_i)$. The Chern-Gauss-Bonnet formula thus gives an upper bound of $\\int_{X}|Rm_{g_i}|^2$. Also, by (\\ref{subharmonic TrQ}) and maximum principle, $Tr \\bm Q_i$ is uniformly bounded on $X$, so $\\bm Q_i$ is uniformly bounded on $X$ and then $\n|d\\bm Q_i|$ is uniformly bounded on $X$. \n\n Given these conclusions, we need to verify that all propositions that were used to prove Theorem \\ref{convergence of triples enhancements} adapt to the torsion-free hypersymplectic setting. Firstly, we digress to discuss elliptic regularity for torsion-free equations (\\ref{torsion-free equations}). In harmonic coordinates in $B_2$ or $B_2^+$, view $g$ as a 4 by 4 matrix of functions, then we have a system of PDEs in $g, Q$: \n \\begin{equation}\\label{torsion-free equation 1 harmonic}\n     \\Delta_g Q_{kl}-Q^{pq}\\langle dQ_{kp},dQ_{ql}\\rangle_g=0,\n \\end{equation}\n \\begin{equation}\\label{torsion-free equation 2 harmonic}\n     \\Delta_g g_{ij}+B_{ij}(g,\\partial g)=-\\frac{1}{2}Q^{ab}Q^{cd}{\\partial_i}Q_{bc}{\\partial_j}Q_{da}.\n \\end{equation}\n From this, by a boostrapping argument, one sees interior regularity: fix $\\beta\\in (0,1)$. If $\\bm Q$ is $C^1$ bounded, $g$ is $C^{1,\\beta}$ bounded, and they are uniform positive on $B_2$, then all derivatives of $\\bm Q$ and $g$ are bounded. For boundary regularity, there is no techinical difficulty to get the following version from Neumann boundary conditions (\\ref{Neumann boundary condition 1}),(\\ref{Neumann boundary condition 2}): if all tangential\n derivatives of $\\bm Q$, $g$, $H$ are bounded on $\\tilde{B}_2$, and  $\\bm Q$ is $C^1$ bounded, $g$ is $C^{1,\\beta}$ bounded, and they are uniformly positive on $B_2^+$, then all derivatives of $\\bm Q$ and $g$ are bounded in $B_1^+$. If in both cases, we also assume $g$ is $C^{k,\\beta}$ close to identity in $B_2$ or $B_2^+$, then similarly, $g$ is $C^{k,\\alpha}$ close to identity for any $\\alpha\\in (\\beta,1)$.\n\n\nFollowing the proof of Theorem \\ref{local harmonic radius lower bound}, we get the following two propositions.\n\n\\begin{proposition}\\label{harmonic radius and bounds for Q} Let $(X,g,\\bm\\omega)$ be a  torsion-free hypersymplectic manifold and $B(p,r)$ is a metric ball that has compact closure, $\\partial B(p,r)\\neq \\emptyset$. Suppose  for any $q\\in B(p,r)$, $$inj_q \\geq cd(q,\\partial B(p,r)), $$\n$$Tr\\bm Q, R\\leq  C.$$\nFix $\\Lambda>1,0<\\alpha<1$, then for any $k\\geq 0$,  $q\\in B(p,r),$  $$r_h^{k,\\alpha}(q,g, \\Lambda)\\geq C_k'd(q,\\partial B(p,r)).$$\nIn particular, $|\\nabla^k \\bm Q|\\leq C_k''$ in $B(p,\\frac{r}{2})$.\n\n\\end{proposition}\n\n\n\\begin{proposition} Let $(X,g,\\bm\\omega)$ be a  compact torsion-free hypersymplectic manifold with boundary. Suppose $i_b\\geq i_0$, $inj_{X}\\geq i_0$, $inj_{\\partial X}\\geq i_0$, $Tr \\bm Q$, $R\\leq C$ on $\\partial X$ and $|\\nabla^j_{\\partial X} Rm_{\\partial X}|\\leq C_j$, $|\\nabla^j_{\\partial X} S|\\leq C_j $, $|\\nabla^j_{\\partial X} \\bm Q|\\leq C_j$ on $\\partial X$, $\\forall k\\geq 0$. Fix $\\Lambda>1, 0<\\alpha<1$, then for any $k\\geq 0$, $q \\in X$,\n$$r_h^{k,\\alpha}(q,g, \\Lambda)\\geq C_k'''.$$\n\\end{proposition}\n \n Then we have \n \n\\begin{proposition}\nProposition \\ref{no bubble 2} holds for torsion-free hypersymplectic manifolds $(X,g,\\bm\\omega)$, provided an upper bound of $Tr\\bm Q$ in $B(p,5)$.\n\\end{proposition}\n\n\\begin{proof}\n\nThe proof is almost the same as there. Let us list the ingredients here: \n\n\\begin{itemize}\n\\item We have Bishop-Gromov volume comparison, since $\\text{Ric}_{g_i}\\geq 0,$\t\n\\item Before rescaling, $ R_i, |\\text{Ric}_{g_i}|$ are automatically bounded by Proposition \\ref{fy result}.\n\\item \n  For the rescaled metric $\\tilde{g}_i$, the \n  curvature bound and the volume non-collapsing condition imply injectivity radius lower bound on compact sets, hence by Proposition \\ref{harmonic radius and bounds for Q}, we have harmonic radius lower bounds as well as bounds for derivatives of $\\bm Q_i$ on compact sets, so we have pointed Cheeger-Gromov convergence of a subsequence $(M_i,g_i,\\bm\\omega_i, q_i)$.\n\n\\item The limit $\\bm\\omega_\\infty$ is a hyperk\\\"ahler triple up to a $SL(3,\\mathbb{R})$ rotation, because $g_\\infty$ is scalar flat, or because of Proposition \\ref{fy result}.\n\n\\end{itemize}\nSo, we get the contradiction in the same way. \n\\end{proof}\n\nWith the above three Propositions as tools and $\\epsilon$-regularity, argue the same way as in Theorem \\ref{good boundary}, one gets an analogous version of Theorem \\ref{good boundary}, i.e., curvature control within  $i_b$, assuming $i_b\\geq i_0$.  \n\n\\begin{remark}\nWe make a remark about the proof of $\\epsilon$-regularity for torsion-free hypersymplectic manifolds here. Firstly, the proof in \\cite{sun2021collapsing} Theorem 3.21 directly applies to this case by using Proposition \\ref{fy result}. Alternatively, we can apply Remark 8.22 in \\cite{cheeger2006curvature} to conclude that $g$ has $C^{1,\\alpha}$ bounded covering geometry when curvature $L^2$ norm is small. By (\\ref{torsion-free equations}), we have $|\\nabla \\text{Ric}|\\leq C$, hence $g$ has $C^{2,\\alpha}$ bounded covering geometry and $|Rm|$ is bounded.\n\\end{remark}\n\n\n \nNow one can finish the proof of compactness part of  Theorem \\ref{convergence of hypersymplectic triples} by the same arguments in Section \\ref{section main proof}. Note that $\\text{Ric}\\geq 0, H>0$ is enough for the focal point argument.\n\n\\subsection{Uniqueness}\n\nFinally, we prove the uniqueness part of Theorem \\ref{convergence of hypersymplectic triples}.\n\n\\begin{proposition} Let $\\bm\\omega_1,\\bm\\omega_2$ be two torsion-free hypersymplectic triples on an oriented 4-manifold $X$ with compact boundary $Y=\\partial X\n$.\nSuppose $\\bm\\gamma_1=\\bm\\gamma_2, \\bm Q_1=\\bm Q_2$ on $\\partial X$, then $\\bm\\omega_1=\\bm\\omega_2$ in geodesic gauges of $g_{\\bm\\omega_i}$ near $\\partial X$.\n\\end{proposition}\n\n\\begin{proof}\n$\\bm\\omega_i$ defines a torsion-free $G_2$ structure $\\phi_i$ on $X\\times T^3$ via (\\ref{g2 and hypersymplectic}), which defines a warpped product metric \n\\begin{equation}\\label{warpped product metric}\n    g_{\\phi_i}=g_{\\bm\\omega_i}+ Q_{ij}dt^idt^j.\n\\end{equation}\nIn the geodesic gauge of $g_{\\bm\\omega_i}$, write\n$$\\bm\\omega_i=-dt\\wedge *_{Y_t}\\bm\\gamma_t+\\bm\\gamma_t,$$\nwhere $t$ is the distance function $d_{g_{\\bm\\omega_i}}(\\cdot,\\partial X)$.\nHence \n\\begin{equation}\\label{g2 geodesic gauge}\n\\phi_i=-dt\\wedge \\theta_{t,i}+\\rho_{t,i},\n\\end{equation}\nwhere $$\\rho_{t,i}=dt^1\\wedge dt^2\\wedge dt^3-\\gamma_i^1\\wedge dt^1-\\gamma_i^2\\wedge dt^2-\\gamma_i^3\\wedge dt^3,$$\n$$\\theta_{t,i}=-*_{Y_t}\\gamma_{t,i}^1\\wedge dt^1-*_{Y_t}\\gamma_{t,i}^2\\wedge dt^2-*_{Y_t}\\gamma_{t,i}^3\\wedge dt^3$$\nBy (\\ref{warpped product metric}), $t$ can also be viewed as $d_{g_{\\phi_i}}(\\cdot,\\partial X\\times T^3)$, so (\\ref{g2 geodesic gauge}) is written in the geodesic gauge of $g_{\\phi_i}$.\nBy the calculations in \\cite{donaldson2018remarks} Section 2.2, for $i=1,2$, both $g_{\\phi_i}|_{\\partial X\\times T^3}$ and the second fundamental forms of $\\partial X\\times T^3$ are equal to each other, since they are explicitly in terms of $\\theta_{0.i},\\rho_{0,i}$, which are in terms of $\\bm\\gamma_i,\\bm Q_i$. Since $g_{\\phi_i}$ are Ricci-flat, by \\cite{biquard:hal-02928859} Theorem 4, $g_{\\phi_1}=g_{\\phi_2}$. Since $\\nabla^{g_{\\phi_1}}|\\phi_1-\\phi_2|^2=0$ and $\\phi_1-\\phi_2=0$ at one point, we have $\\phi_1=\\phi_2$, $\\bm\\omega_1=\\bm\\omega_2$.\n\n\\end{proof}\n\nNote that for a torsion-free hypersymplectic triple $\\bm\\omega$, the metric $g_{\\bm\\omega}$ is real analytic with respect to the analytic structure defined by harmonic coordinates, due to elliptic regularity of (\\ref{torsion-free equation 1 harmonic})(\\ref{torsion-free equation 2 harmonic}), so the arguments in subsection \\ref{uniqueness 1} shows global uniqueness:\n\n\n\\begin{theorem}\\label{unique continuation torsion-free hypersymplectic}\nLet $X$ be a connected 4-manifold with boundary, $\\pi_1(X,\\partial X)=0$. Suppose $\\bm\\omega_1,\\bm\\omega_2$ are two smooth torsion-free hypersymplectic triples on $X$, and $\\varphi_0:\\partial X\\rightarrow\\partial X$ is a diffeomorphism, such that $\\bm\\omega_1|_{\\partial X}=\\varphi_0^*(\\bm\\omega_2|_{\\partial X})$, $\\bm Q_1|_{\\partial X}=\\varphi_0^*\\bm Q_2|_{\\partial X}$, then there exists a diffeomorphism $\\varphi:X\\rightarrow X$,   $\\varphi|_{\\partial X}=\\varphi_0$,  such that $\\bm\\omega_1=\\varphi^*\\bm\\omega_2$ on $X$.\n\\end{theorem}\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzpldx b/data_all_eng_slimpj/shuffled/split2/finalzzpldx
new file mode 100644
index 0000000000000000000000000000000000000000..26d9b3ee520e4de1fd1314f2183f288bbfa73eda
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzpldx
@@ -0,0 +1,5 @@
+{"text":"\\section{Appendix}\n\n\\subsection{Low Rank Approximation}\n\nWe first state the following Lemma from~\\cite{parlett1980symmetric} without proof and then prove our Theorem \\ref{the:error} following~\\cite{simon2000low}.\n\n\\begin{lemma}\nLet $A \\in \\mathbb{R}^{N \\times N}$ be symmetric and $v$ an arbitrary vector. \nDefine Krylov subspace $\\mathcal{K}_{m} \\equiv \\text{span} \\{v, Av, \\dots, A^{m-1}v\\}$.\nLet $A = U \\Lambda U^{\\top}$ be the eigendecomposition of $A$ with $\\Lambda_{i,i} = \\lambda_i$ and $\\lambda_1 \\geq \\dots \\geq \\lambda_n$.\nDenoting $U = [u_1, \\dots, u_N]$ and $\\mathcal{U}_j = \\mathrm{span}\\{u_1, \\dots, u_j\\}$, then\n\\begin{align}\n\\tan{\\left(u_j, \\mathcal{K}_{m}\\right)} \\leq \\frac{\\sin{\\left(v, \\mathcal{U}_j\\right)} \\prod_{k=1}^{j-1} (\\lambda_k - \\lambda_n) \/ (\\lambda_k - \\lambda_j) }{\\cos{\\left(v, u_j\\right)} T_{m-j}(1+2\\gamma) }, \\nonumber\n\\end{align}\nwhere $T_{m-j}(x)$ is the Chebyshev Polynomial of degree $m-j$ and $\\gamma = (\\lambda_{j} - \\lambda_{j+1}) \/ (\\lambda_{j+1} - \\lambda_{N})$.\n\\end{lemma}\n\n\\errortheorem*\n\\begin{proof}\nFrom Lanczos algorithm, we have $SQ = QT$.\nTherefore, \n\\begin{align}\n\\Vert S - Q T Q^{\\top} \\Vert_{F}^{2} & = \\Vert S - S Q Q^{\\top} \\Vert_{F}^{2} \\nonumber \\\\\n& = \\Vert S (I - Q Q^{\\top}) \\Vert_{F}^{2}\n\\end{align}\nLet $P_{Q}^{\\perp} \\equiv I - QQ^{\\top}$, the orthogonal projection onto the \northogonal complement of subspace $\\text{span}\\{Q\\}$.\nRelying on the eigendecomposition, we have,\n\\begin{align}\n\\Vert S - Q T Q^{\\top} \\Vert_{F}^{2} & = \\Vert U \\Lambda U^{\\top} (I - Q Q^{\\top}) \\Vert_{F}^{2} \\nonumber \\\\\n& = \\Vert \\Lambda U^{\\top} (I - Q Q^{\\top}) \\Vert_{F}^{2} \\nonumber \\\\\n& = \\Vert (I - Q Q^{\\top}) U \\Lambda \\Vert_{F}^{2} \\nonumber \\\\\n& = \\Vert \\left[ \\lambda_1 P_{Q}^{\\perp} u_1, \\dots, \\lambda_N P_{Q}^{\\perp} u_N \\right] \\Vert_{F}^{2},\n\\end{align}\nwhere we use the fact that $\\Vert RA \\Vert_F^2 = \\Vert A \\Vert_F^2$ for any orthogonal matrix $R$ and $\\Vert A^{\\top} \\Vert_F^2 = \\Vert A \\Vert_F^2$.\n\nNote that for any $j$ we have,\n\\begin{align}\n\\left \\| \\left[ \\lambda_1 P_{Q}^{\\perp} u_1, \\dots, \\lambda_N P_{Q}^{\\perp} u_N \\right] \\right \\|_{F}^{2} & = \\sum_{i=1}^{N}  \\lambda_i^2 \\Vert P_{Q}^{\\perp} u_i \\Vert^2 \\nonumber \\\\\n& \\leq \\sum_{i=1}^{j} \\lambda_i^2 \\Vert P_{Q}^{\\perp} u_i \\Vert^2 + \\sum_{i=j+1}^{N} \\lambda_i^2,\n\\end{align}\nwhere we use the fact that for any $i$, $\\Vert P_{Q}^{\\perp} u_i \\Vert^2 = \\Vert u_i \\Vert^2 - \\Vert u_i - P_{Q}^{\\perp} u_i \\Vert^2 \\leq \\Vert u_i \\Vert^2 = 1$.\n\nNote that we have $\\text{span}\\{Q\\} = \\text{span}\\{v, Sv, \\dots, S^{K-1}v\\} \\equiv \\mathcal{K}_{K}$ from the Lanczos algorithm.\nTherefore, we have,\n\\begin{align}\n    \\Vert P_{Q}^{\\perp} u_i \\Vert = \\vert \\sin{\\left(u_i, \\mathcal{K}_{K} \\right)} \\vert \\leq \\vert \\tan{\\left(u_i, \\mathcal{K}_{K} \\right)} \\vert.\n\\end{align}\nApplying the above lemma with $A = S$, we finish the proof.\n\n\\end{proof}\n\n\n\\subsection{Lanczos Algorithm}\n\nUtilizing exact arithmetic, Lanczos vectors are orthogonal to each other. \nHowever, in floating point arithmetic, it is well known that the round-off error will make the Lanczos vectors lose orthogonality as the iteration proceeds.\nOne could apply a full Gram-Schmidt (GS) process $z = z - \\sum_{i=1}^{j-1} z^{\\top}q_{i}q_{i}$ after line $6$ of Alg.~\\ref{alg:lanczos} to ensure orthogonality.\nOther partial or selective re-orthogonalization could also be explored. \nHowever, since we found the orthgonality issue does not hurt overall performance with a small iteration number, e.g., $K=20$, and the full GS process is computationally expensive, we do not add such a step.\nAlthough some customized eigendecomposition methods, e.g., implicit QL~\\cite{press2007numerical}, exist for tridiagonal matrix, we leave it for future exploration due to its complicated implementation.\n\n\\subsection{Experiments}\n\nFor ChebyNet, we do not use graph coarsening in all experiments due to its demanding computation cost for large graphs.\nAlso, for small molecule graphs, coarsening generally does not help since it loses information compared to directly stacking another layer of original graph.\n\n\\paragraph{Citation Networks}\n\n\\begin{table}\n\\centering\n{%\n\\begin{tabular}{@{}l|rrrrrrrrr@{}}\n\\hline\n\\toprule\nDataset  & \\#Nodes & \\#Edges & \\#Classes & \\#Features & $S_0$ & $S_1$ & $S_2$ & $S_3$  \\\\\n\\midrule\nCiteseer & 3,327 & 4,732 & 6 & 3,703 & 3.6\\%& 3\\%& 1\\%& 0.5\\% \\\\\nCora  & 2,708 & 5,429 & 7 & 1,433 & 5.2\\% & 1\\%& 0.5\\%& 0.3\\%\\\\\nPubmed  & 19,717 & 44,338 & 3 & 500 & 0.3\\% & 0.1\\%& 0.05\\%& 0.03\\%\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{Dataset statistics. $S_0$ is portion of training examples in the public split. $S_1$ to $S_3$ are the ones of $3$ random splits generated by us.}\n\\label{tbl:citation_stats}\n\\end{table}\n\nThe statistics of three citation networks are summarized in Table~\\ref{tbl:citation_stats}.\nWe now report the important hyperparameters chosen via cross-validation for each method.\nAll methods are trained with Adam with learning rate $1.0e^{-2}$ and weight decay $5.0e^{-4}$.\nThe maximum number of epoch is set to $200$.\nEarly stop with window size $10$ is also adopted.\nWe tune hyperparameters using Cora alone and fix them for citeseer and pubmed.\nFor convolution based methods, we found $2$ layers work the best.\nIn GCN-FP, we set the hidden dimension to $64$ and dropout to $0.5$.\nIn GGNN, we set the hidden dimension to $64$, the propagate step to $2$ and aggregation function to summation.\nIn DCNN, we set the hidden dimension to $64$, dropout to $0.5$ and use diffusion scales $\\{1, 2, 5\\}$.\nIn ChebyNet, we set the polynomial order to $5$, the hidden dimension to $64$ and dropout to $0.5$.\nIn GCN, we set the hidden dimension to $64$ and dropout to $0.5$.\nIn MPNN, we use GRU as update function and set the hidden dimension to $64$ and dropout to $0.5$. \nNo edge embedding is used as there is just one edge type.\nIn GraphSAGE, we set the number of sampled neighbors to $500$, the hidden dimension to $64$, dropout to $0.5$ and the aggregation function to average.\nIn GAT, we set the number of heads per layer to $8, 1$, hidden dimension per head to $8$ and dropout to $0.6$.\nIn {LanczosNet}, we set the short and long diffusion scales to $\\{1, 2 ,5, 7\\}$ and $\\{10, 20, 30\\}$ respectively. \nThe hidden dimension is $64$ and dropout is $0.5$.\nLanczos step is $20$.\n1-layer MLP with $128$ hidden units and ReLU nonlinearity is used as the spectral filter.\nIn {AdaLanczosNet}, we set the short and long diffusion scales to $\\{1, 2 ,5\\}$ and $\\{10, 20\\}$ respectively. \nThe hidden dimension is $64$ and dropout is $0.5$.\nLanczos step is $20$.\n1-layer MLP with $128$ hidden units and ReLU nonlinearity is used as the spectral filter.\n\n\\paragraph{Quantum Chemistry}\n\nWe now report the important hyperparameters chosen via cross-validation for each method.\nAll methods are trained with Adam with learning rate $1.0e^{-4}$ and no weight decay.\nThe maximum number of epoch is set to $200$.\nEarly stop with window size $10$ is also adopted.\nFor convolution based methods, we found $7$ layers work the best.\nWe augment all methods with $64$-dimension node embedding and add edge types by either feeding a multiple-channel graph Laplacian matrix or directly adding a separate message function per edge type.\nFor all methods, no dropout is used since it slightly hurts the performance.\nIn GCN-FP, we set the hidden dimension to $128$.\nIn GGNN, we set the hidden dimension to $128$, the propagate step to $15$ and aggregation function to average.\nIn DCNN, we set the hidden dimension to $128$ and use diffusion scales $\\{3, 5, 7, 10, 20, 30\\}$.\nIn ChebyNet, we set the polynomial order to $5$, the hidden dimension to $128$.\nIn GCN, we set the hidden dimension to $128$.\nIn MPNN, we use GRU as update function, set the number of propagation to $7$, set the hidden dimension to $128$, use a 1-layer MLP with $1024$ hidden units and ReLU nonlinearity as the message function and set the number of unroll step of \\textit{Set2Vec} to $10$.\nIn GraphSAGE, we set the number of sampled neighbors to $40$, the hidden dimension to $128$ and the aggregation function to average.\nIn GAT, we set the number of heads of all $7$ layers to $8$ and hidden dimension per head to $16$.\nIn {LanczosNet}, we do not use short diffusion scales and set long ones to $\\{1, 2, 3, 5, 7, 10, 20, 30\\}$. \nThe hidden dimension is $128$.\nLanczos step is $20$.\n1-layer MLP with $128$ hidden units and ReLU nonlinearity is used as the spectral filter.\nIn {AdaLanczosNet}, we set the short and long diffusion scales to $\\{1, 2, 3\\}$ and $\\{5, 7, 10, 20, 30\\}$ respectively. \nThe hidden dimension is $128$.\nLanczos step is $20$.\n3-layer MLP with $4096$ hidden units and ReLU nonlinearity is used as the spectral filter.\n\\section{Background}\n\nIn this section, we introduce some background material.\nA graph $\\mathcal{G}$ with $N$ nodes is denoted as $\\mathcal{G} = (\\mathcal{V}, \\mathcal{E}, A)$, where $A \\in \\mathbb{R}^{N \\times N}$ is an adjacency matrix which could either be binary or real valued.\n$X \\in \\mathbb{R}^{ N \\times F}$ is the compact representation of node features (or graph signal in the GSP literature).\nFor any node $v \\in \\mathcal{V}$, we denote its feature as a row vector $X_{v, :} \\in \\mathbb{R}^{1 \\times F}$. \nWe use $X_{:, i}$ to denote the $i$-th column of $X$.\n\n\\input{graph_convolution}\n\n\n\\section{Conclusion}\nIn this paper, we propose {LanczosNet} which leverages the Lanczos algorithm to construct a low rank approximation of the graph Laplacian.\nIt not only provides an efficient way to gather multi-scale information for graph convolution but also enables learning spectral filters.\nAdditionally, we propose a model variant {AdaLanczosNet} which facilitates graph kernel and node embedding learning.\nWe show that our model has a close relationship with graph based manifold learning, especially diffusion map.\nExperimental results demonstrate that our model outperforms a range of other graph networks, on challenging graph problems.\nWe are currently exploring customized eigen-decomposition methods for tridiagonal matrices, which will potentially further improve our {AdaLanczosNet}. \nOverall, work in this direction holds promise for allowing deep learning to scale up to very large graph problems.\n\\section{Lanczos Network and Diffusion Maps}\\label{sec:relation}\n\nIn this section, we highlight the relationship between {LanczosNet} and an important example of graph based manifold learning algorithms, diffusion maps~\\cite{coifman2006diffusion}.\n\n\\paragraph{Diffusion Maps}\nIn diffusion maps, the weights in the adjacency matrix define a discrete random walk over the graph, where the Markov transition matrix $ P = D^{-1} A $ shows the transition probability in a single time step. \nTherefore, $P^t_{i, j}$ sums the probability of all paths of length $t$ that start at node $i$ and end at node $j$. \nIt is shown in~\\cite{coifman2006diffusion} that $P$ can be used to define an inner product in a Hilbert space.  \nSpecifically, we use the eigenvalues and right eigenvectors $\\{ \\lambda_l, \\psi_l \\}_{l = 1}^N$ of $P$ to define a diffusion mapping $\\Phi_t$ as,\n\\begin{align}\n\\label{eq:dm}\n    \\Phi_{t}(i) = \\left(\\lambda_{1}^{t} \\psi_{1}(i), \\lambda_{2}^{t} \\psi_{2}(i), \\dots, \\lambda_{N}^{t} \\psi_{N}(i) \\right),\n\\end{align}\nwhere $\\psi_{l}(i)$ is the $i$-th entry of the eigenvector $\\psi_l$. Since the row stochastic matrix $P$ is similar to $S$, i.e., $P = D^{-1\/2} S D^{1\/2}$, we have $\\psi_{l} = D^{-1\/2} u_l$. The mapping $\\Phi_t$ satisfies $\\sum_{k = 1}^N P^t_{i,k} P^t_{j, k} \/D_{k, k} = \\left \\langle \\Phi_t(i), \\Phi_t(j) \\right \\rangle$, where $\\langle \\cdot, \\cdot \\rangle$ is the inner product over Euclidean space.\nThe diffusion distance between $i$ and $j$, $d^2_{\\mathrm{DM}, t}(i, j) =\\left \\| \\Phi_t(i) - \\Phi_t(j) \\right \\|^2 = \\sum_{k = 1}^{N} {( P^t_{i, k} - P^t_{j, k})^2} \/ {D_{k,k}}$,\nis the weighted-$l_2$ proximity between the probability clouds of random walkers starting at $i$ and ending at $j$ after $t$ steps. \nSince all eigenvalues of $S$ reside in the interval $[-1, 1]$, for some large $t$, $\\lambda_l^{t}$ in Eq.~(\\ref{eq:dm}) is close to zero, and $d_{\\mathrm{DM}, t}$ can be well approximated by using only a few largest eigenvalues and their eigenvectors. \n\n\n\\paragraph{Connection to Graph Convolution} Apart from using diffusion maps to embed node features $X$ at different time scales, one can use it to compute the frequency representations of $X$ as below,\n\\begin{align}\\label{eq:gft}\n    \\hat{X} = \\Lambda^{t} U^{\\top}X,\n\\end{align}\nwhere $U$ are the eigenvectors of $S$ and define the graph Fourier transform.\nThe frequency representation $\\hat{X}$ is weighted by the powers of the eigenvalues $\\lambda_l^t$, suppressing entries with small magnitude of eigenvalues.\nRecall that in the convolution layer Eq.~(\\ref{eq:learnable_filter_fix}) of {LanczosNet}, we use multiple such frequency representations with different scales $t$ and replace the eigenvalues $\\Lambda$ in Eq.~(\\ref{eq:gft}) with their approximation, i.e., Ritz values.\nTherefore, in {LanczosNet}, spectral filters are actually applied to the frequency representations which are obtained by projecting the node features $X$ onto multiple diffusion maps with different scales.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\nIn this section, we compare our two model variants against $9$ recent graph networks, including graph convolution networks for fingerprint (GCN-FP)~\\cite{duvenaud2015convolutional}, gated graph neural networks (GGNN)~\\cite{li2015gated}, diffusion convolutional neural networks (DCNN)~\\cite{atwood2016diffusion}, Chebyshev networks (ChebyNet)~\\cite{defferrard2016convolutional}, graph convolutional networks (GCN)~\\cite{kipf2016semi}, message passing neural networks (MPNN)~\\cite{Gilmer17}, graph sample and aggregate (GraphSAGE)~\\cite{hamilton2017inductive}, graph partition neural networks (GPNN)~\\cite{liao2018graph}, graph attention networks (GAT)~\\cite{velickovic2017graph}. \nWe test them on two sets of tasks: (1) semi-supervised document classification on $3$ citation networks~\\cite{yang2016revisiting},\n(2) supervised regression of molecule property on QM8 quantum chemistry dataset~\\cite{ramakrishnan2015electronic}.\nFor fair comparison, we only tune model-related hyperparameters in all our experiments and share the others, e.g., using the same batch size.\nWe carefully tune hyperparameters based on cross-validation and report the best performance of each competitor.\nPlease refer to the appendix for more details on hyperparameters.\nWe implement all methods using PyTorch~\\cite{paszke2017automatic} and release the code at \\url{https:\/\/github.com\/lrjconan\/LanczosNetwork}. \n\n\n\\subsection{Citation Networks}\nThree citation networks used in this experiment are: Cora, Citeseer and Pubmed.\nFor each network, nodes are documents and connected based on their citation links.\nEach node is associated with a bag-of-words feature vector.\nWe use the same pre-processing procedure and follow the transductive setting as in~\\cite{yang2016revisiting}.\nIn particular, given a portion of nodes and their labeled content categories, e.g., history, science, the task is to predict the category for other unlabeled nodes within the same graph.\nThe statistics of these datasets are summarized in the appendix.\nAll experiments are repeated $10$ times with different random seeds. \nDuring each run, all methods share the same random seed.\nWe first experiment with the public data split and observe severe overfitting for almost all algorithms.\nTo mitigate overfitting and test the robustness of models, we then increase the difficulty of the task by reducing the portion of training examples to several levels and randomly split data.\n\nExperimental results and exact portions of training examples are shown in Table.~\\ref{table:citation_net}.\nWe use the reported best hyperparameters when available for public split and do cross-validation otherwise.\nHyperparameters are reported in the appendix.\nFrom the table, we see that for random splits with different portion of training examples, since each run of experiment uses a separate random split, the overall variance is larger than its public counterpart. We see that GAT achieves the best performance on the public split but performs poorly on random splits with different portions of training examples.\nThis is partly due to the fact that GAT uses multiple dropout throughout the model which helps only if there is overfitting.\nWe can see that either {LanczosNet} or {AdaLanczosNet} achieves state-of-the-art accuracy on random difficult splits and performs closely with respect to GAT on public splits.\nThis may be attributed to the fact that with fewer training examples, the model requires longer scale schemes to spread supervised information over the graph.\nOur model provides an efficient way of leveraging such long scale information.\n\n\\begin{table}[t]\n\\centering\n\\resizebox{\\linewidth}{!}{%\n\\setlength{\\tabcolsep}{3pt}\n\\begin{tabular}{@{}c|cccccccccc@{}}\n\\hline\n\\toprule\nCora & GCN-FP & GGNN & DCNN & ChebyNet & GCN & MPNN & GraphSAGE & GAT & LNet & AdaLNet \\\\\n\\midrule\n\\midrule\nPublic & 74.6 $\\pm$ 0.7 & 77.6 $\\pm$ 1.7 & 79.7 $\\pm$ 0.8 & 78.0 $\\pm$ 1.2 & 80.5 $\\pm$ 0.8 & 78.0 $\\pm$ 1.1 & 74.5 $\\pm$ 0.8 & \\textbf{82.6 $\\pm$ 0.7} & 79.5 $\\pm$ 1.8 & 80.4 $\\pm$ 1.1 \\\\\n3\\% & 71.7 $\\pm$ 2.4 & 73.1 $\\pm$ 2.3 & 76.7 $\\pm$ 2.5 & 62.1 $\\pm$ 6.7 & 74.0 $\\pm$ 2.8 & 72.0 $\\pm$ 4.6 & 64.2 $\\pm$ 4.0 & 56.8 $\\pm$ 7.9 & 76.3 $\\pm$ 2.3 & \\textbf{77.7 $\\pm$ 2.4} \\\\\n1\\% & 59.6 $\\pm$ 6.5 & 60.5 $\\pm$ 7.1 & 66.4 $\\pm$ 8.2 & 44.2 $\\pm$ 5.6 & 61.0 $\\pm$ 7.2 & 56.7 $\\pm$ 5.9 & 49.0 $\\pm$ 5.8 & 48.6 $\\pm$ 8.0 & 66.1 $\\pm$ 8.2 & \\textbf{67.5 $\\pm$ 8.7} \\\\\n0.5\\% & 50.5 $\\pm$ 6.0 & 48.2 $\\pm$ 5.7 & 59.0 $\\pm$ 10.7 & 33.9 $\\pm$ 5.0 & 52.9 $\\pm$ 7.4 & 46.5 $\\pm$ 7.5 & 37.5 $\\pm$ 5.4 & 41.4 $\\pm$ 6.9 & 58.1 $\\pm$ 8.2 & \\textbf{60.8 $\\pm$ 9.0} \\\\\n\\midrule\nCiteseer & GCN-FP & GGNN & DCNN & ChebyNet & GCN & MPNN & GraphSAGE & GAT & LNet & AdaLNet \\\\\n\\midrule\n\\midrule\nPublic & 61.5 $\\pm$ 0.9 & 64.6 $\\pm$ 1.3 & 69.4 $\\pm$ 1.3 & 70.1 $\\pm$ 0.8 & 68.1 $\\pm$ 1.3 & 64.0 $\\pm$ 1.9 & 67.2 $\\pm$ 1.0 & \\textbf{72.2 $\\pm$ 0.9} & 66.2 $\\pm$ 1.9 & 68.7 $\\pm$ 1.0 \\\\\n1\\% & 54.3 $\\pm$ 4.4 & 56.0 $\\pm$ 3.4 & 62.2 $\\pm$ 2.5 & 59.4 $\\pm$ 5.4 & 58.3 $\\pm$ 4.0 & 54.3 $\\pm$ 3.5 & 51.0 $\\pm$ 5.7 & 46.5 $\\pm$ 9.3 & 61.3 $\\pm$ 3.9 & \\textbf{63.3 $\\pm$ 1.8} \\\\\n0.5\\% & 43.9 $\\pm$ 4.2 & 44.3 $\\pm$ 3.8 & 53.1 $\\pm$ 4.4 & 45.3 $\\pm$ 6.6 & 47.7 $\\pm$ 4.4 & 41.8 $\\pm$ 5.0 & 33.8 $\\pm$ 7.0 & 38.2 $\\pm$ 7.1 & 53.2 $\\pm$ 4.0 & \\textbf{53.8 $\\pm$ 4.7} \\\\\n0.3\\%& 38.4 $\\pm$ 5.8 & 36.5 $\\pm$ 5.1 & 44.3 $\\pm$ 5.1 & 39.3 $\\pm$ 4.9 & 39.2 $\\pm$ 6.3 & 36.0 $\\pm$ 6.1 & 25.7 $\\pm$ 6.1 & 30.9 $\\pm$ 6.9 & 44.4 $\\pm$ 4.5 & \\textbf{46.7 $\\pm$ 5.6} \\\\\n\\midrule\nPubmed & GCN-FP & GGNN & DCNN & ChebyNet & GCN & MPNN & GraphSAGE & GAT & LNet & AdaLNet \\\\\n\\midrule\n\\midrule\nPublic & 76.0 $\\pm$ 0.7 & 75.8 $\\pm$ 0.9 & 76.8 $\\pm$ 0.8 & 69.8 $\\pm$ 1.1 & 77.8 $\\pm$ 0.7 & 75.6 $\\pm$ 1.0 & 76.8 $\\pm$ 0.6 & 76.7 +- 0.5 & \\textbf{78.3 $\\pm$ 0.3} & 78.1 $\\pm$ 0.4 \\\\\n0.1\\% & 70.3 $\\pm$ 4.7 & 70.4 $\\pm$ 4.5 & 73.1 $\\pm$ 4.7 & 55.2 $\\pm$ 6.8 & 73.0 $\\pm$ 5.5 & 67.3 $\\pm$ 4.7 & 65.4 $\\pm$ 6.2 & 59.6 +- 9.5 & \\textbf{73.4 $\\pm$ 5.1} & 72.8 $\\pm$ 4.6 \\\\\n0.05\\% & 63.2 $\\pm$ 4.7 & 63.3 $\\pm$ 4.0 & 66.7 $\\pm$ 5.3 & 48.2 $\\pm$ 7.4 & 64.6 $\\pm$ 7.5 & 59.6 $\\pm$ 4.0 & 53.0 $\\pm$ 8.0 & 50.4 +- 9.7 & \\textbf{68.8 $\\pm$ 5.6} & 66.0 $\\pm$ 4.5 \\\\\n0.03\\% & 56.2 $\\pm$ 7.7 & 55.8 $\\pm$ 7.7 & 60.9 $\\pm$ 8.2 & 45.3 $\\pm$ 4.5 & 57.9 $\\pm$ 8.1 & 53.9 $\\pm$ 6.9 & 45.4 $\\pm$ 5.5 & 50.9 +- 8.8 & 60.4 $\\pm$ 8.6 & \\textbf{61.0 $\\pm$ 8.7} \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{Test accuracy with $10$ runs on citation networks. The public splits in Cora, Citeseer and Pubmed contain $5.2\\%$, $3.6\\%$ and $0.3\\%$ training examples respectively.}\n\\label{table:citation_net}\n\\end{table}\n\n\\subsection{Quantum Chemistry}\n\nWe then benchmark all algorithms on the QM8 quantum chemistry dataset which comes from a recent study on modeling quantum mechanical calculations of electronic spectra and excited state energy of small molecules~\\cite{ramakrishnan2015electronic}.\nThe setup of QM8 is as follows.\nAtoms are treated as nodes and they are connected to each other following the structure of the corresponding molecule.\nEach edge is labeled with a chemical bond.\nNote that two atoms in one molecule can have multiple edges belong to different chemical bonds.\nTherefore a molecule is actually modeled as a multigraph.\nWe also use explicit hydrogen in molecule graphs as suggested in~\\cite{Gilmer17}.\nSince some models cannot leverage feature on edges easily, we use the molecule graph itself as the only input information for all models so that it is a fair comparison.\nAs demonstrated in our ablation studies, learning node embeddings for atoms is very helpful.\nTherefore, we augment all competitors and our models with this component.\nThe task is to predict $16$ different quantities of electronic spectra and energy per molecule graph which boils down to a regression problem.\nThere are $21786$ molecule graphs in total of which the average numbers of nodes and edges are around $16$ and $21$.\nThere are $6$ different chemical bonds and $70$ different atoms throughout the dataset.\nWe use the split provided by DeepChem~\\footnote{https:\/\/deepchem.io\/} which have $17428$, $2179$ and $2179$ graphs for training, validation and testing respectively.\nFollowing~\\cite{Gilmer17,wu2018moleculenet}, we use mean squared error (MSE) as the loss for training and weighted mean absolute error (MAE) as the evaluation metric.\nWe repeat all experiments $3$ times with different random seeds and report the average performance and standard deviation.\nThe same random seed is shared for all methods per run.\nHyperparameters are reported in the appendix.\nThe validation and test MAEs are shown in Table~\\ref{table:qm8}.\nAs you can see, {LanczosNet} and {AdaLanczosNet} achieve better performances than all other competitors.\nNote that DCNN also achieves good performance with the carefully chosen scale parameters since it is somewhat similar to our model in terms of leveraging multi-scale information.\n\n\n\\begin{table}[t]\n\\centering\n{%\n\\begin{tabular}{@{}c|cc@{}}\n\\hline\n\\toprule\nMethods & Validation MAE ($\\times 1.0e^{-3}$) & Test MAE ($\\times 1.0e^{-3}$) \\\\\n\\midrule\n\\midrule\nGCN-FP~\\cite{duvenaud2015convolutional} & 15.06 $\\pm$ 0.04 & 14.80 $\\pm$ 0.09 \\\\\nGGNN~\\cite{li2015gated} & 12.94 $\\pm$ 0.05 & 12.67 $\\pm$ 0.22 \\\\\nDCNN~\\cite{atwood2016diffusion} & 10.14 $\\pm$ 0.05 & 9.97 $\\pm$ 0.09 \\\\\nChebyNet~\\cite{defferrard2016convolutional} & 10.24 $\\pm$ 0.06 & 10.07 $\\pm$ 0.09 \\\\\nGCN~\\cite{kipf2016semi} & 11.68 $\\pm$ 0.09 & 11.41 $\\pm$ 0.10 \\\\\nMPNN~\\cite{Gilmer17} & 11.16 $\\pm$ 0.13 & 11.08 $\\pm$ 0.11 \\\\\nGraphSAGE~\\cite{hamilton2017inductive} & 13.19 $\\pm$ 0.04 & 12.95 $\\pm$ 0.11 \\\\\nGPNN~\\cite{liao2018graph} & 12.81 $\\pm$ 0.80 & 12.39 $\\pm$ 0.77 \\\\\nGAT~\\cite{velickovic2017graph} & 11.39 $\\pm$ 0.09 & 11.02 $\\pm$ 0.06 \\\\\nLanczosNet & \\textbf{9.65 $\\pm$ 0.19} & \\textbf{9.58 $\\pm$ 0.14} \\\\\nAdaLanczosNet & 10.10 $\\pm$ 0.22 & 9.97 $\\pm$ 0.20 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{Mean absolute error on QM8 dataset.}\n\\vspace{-0.5cm}\n\\label{table:qm8}\n\\end{table}\n\n\n\\subsection{Ablation Study}\n\nWe also did a thorough ablation study of our modeling components on the validation set of QM8.\n\n\\begin{table}\n\\centering\n\\resizebox{\\linewidth}{!}\n{%\n\\setlength{\\tabcolsep}{3pt}\n\\begin{tabular}{@{}ccccccc|c@{}}\n\\hline\n\\toprule\nModel & \\begin{tabular}{@{}c@{}}Graph \\\\ Kernel\\end{tabular} & \\begin{tabular}{@{}c@{}}Node \\\\ Embedding\\end{tabular} & \\begin{tabular}{@{}c@{}}Spectral \\\\ Filter\\end{tabular} & Short Scales & Long Scales & \\begin{tabular}{@{}c@{}}Lanczos \\\\ Step\\end{tabular} & \\begin{tabular}{@{}c@{}}Validation \\\\ MAE ($\\times 1.0e^{-3}$)\\end{tabular} \\\\\n\\midrule\n\\midrule\nLanczosNet &  & one-hot &  & \\{1, 2, 3\\} &  &  & 10.71 \\\\\nLanczosNet &  & one-hot &  & \\{3, 5, 7\\} &  &  & 10.60 \\\\\nLanczosNet &  & one-hot &  &  & \\{10, 20, 30\\} & 20 & 10.54 \\\\\nLanczosNet &  & one-hot &  & \\{3, 5 ,7\\} & \\{10, 20, 30\\} & 20 & \\textbf{10.41} \\\\\n\\midrule\n\\midrule\nLanczosNet &  & one-hot &  &  & \\{10, 20, 30\\} & 5 & 10.49 \\\\\nLanczosNet &  & one-hot &  &  & \\{10, 20, 30\\} & 10 & \\textbf{10.44} \\\\\nLanczosNet &  & one-hot &  &  & \\{10, 20, 30\\} & 20 & 10.54 \\\\\nLanczosNet &  & one-hot &  &  & \\{10, 20, 30\\} & 40 & 10.49 \\\\\n\\midrule\n\\midrule\nLanczosNet &  & one-hot & 3-MLP & \\{3, 5 ,7\\} & \\{10, 20, 30\\} & 20 & 10.44 \\\\\nLanczosNet &  & one-hot & 5-MLP & \\{3, 5 ,7\\} & \\{10, 20, 30\\} & 20 & 10.54 \\\\\nLanczosNet &  & \\checkmark & 3-MLP & \\{3, 5 ,7\\} & \\{10, 20, 30\\} & 20 & 10.26 \\\\\nLanczosNet &  & \\checkmark & 3-MLP &  & \\{1, 2, 3, 5, 7, 10, 20, 30\\} & 20 & \\textbf{9.56} \\\\\n\\midrule\n\\midrule\nAdaLanczosNet & \\checkmark & one-hot & 3-MLP & \\{3, 5, 7\\} & \\{10, 20, 30\\} & 20 & 10.99 \\\\\nAdaLanczosNet &  & \\checkmark & 3-MLP & \\{3, 5, 7\\} & \\{10, 20, 30\\} & 20 & 10.20 \\\\\nAdaLanczosNet &  & \\checkmark & 3-MLP & \\{1, 2, 3\\} & \\{5, 7, 10, 20, 30\\} & 20 & \\textbf{9.96} \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{Ablation study on QM8 dataset. Empty cell means that the component is neither used nor applicable. $X$-MLP means a MLP with $X$ hidden layers. `one-hot' means the node embedding is fixed as the one-hot encoding throughout learning and inference.}\n\\label{table:qm8_ablation}\n\\end{table}\n\n\\textbf{Multi-Scale Graph Convolution:} \nWe first study the effect of multi-scale graph convolution. \nIn order to rule out the impact of other factors, we use {LanczosNet}, do not employ the learnable spectral filter and use the one-hot encoding as the node embedding.\nThe results are shown in the first row of Table~\\ref{table:qm8_ablation}.\nUsing long scales for graph convolution clearly helps on this task.\nCombining both short and long scales further improves results.\n\n\\textbf{Lanczos Step:}\nWe then investigate the Lanczos step since it will have an impact on the accuracy of the low rank approximation induced by the Lanczos algorithm.\nThe results are shown in the second row of Table~\\ref{table:qm8_ablation}.\nWe can see that the performance is better with a relatively small Lanczos step like $10$ and $20$ which makes sense since the average number of nodes in this dataset is around $16$.\n\n\\textbf{Learning Spectral Filter:}\nWe then study whether learning spectral filter will help improve performance.\nThe results are shown in the third row of Table~\\ref{table:qm8_ablation}.\nAdding a $3$-layer MLP does help reduce the error compared to not using any learnable spectral filter.\nNote that the MLP consists of $128$ hidden units per layer and uses ReLU as the nonlinearity.\nHowever, using a deeper MLP does not seem to be helpful which might be caused by the challenges in optimization.\n\n\\textbf{Graph Kernel\/Node Embedding:}\nAt last, we study the usefulness of adding graph kernel and node embeddings.\nWe first fix the node embedding as one-hot encoding and learn a $3$ layer MLP which is the function $f_{\\theta}$ in Eq.~(\\ref{eq:graph_kernel}).\nNext, we learn the node embeddings directly.\nIntuitively, learning embeddings amounts to learn a separate function $f$ per node whereas our graph kernel learning enforces that $f$ is shared for all nodes, thus being more restrictive.\nAs shown in the $3$-rd and $4$-th rows of the table, learning node embeddings significantly improves the performance for both {LanczosNet} and {AdaLanczosNet} and is more effective than learning graph kernels.\nAlso, tuning the scale parameters further boosts the performance.\n\\subsection{Graph Convolution}\n\n\\paragraph{Graph Fourier Transform}\nGiven input node features $X$, we now discuss how to perform a graph convolution.\nBased on the adjacency matrix $A$, we compute the graph Laplacian $L$ which can be defined in different ways: (1) $L = D - A$; (2) $L = I - D^{-1} A$; (3) $L = I - D^{-\\frac{1}{2}} A D^{-\\frac{1}{2}}$, where $D$ is a diagonal degree matrix and $D_{i,i} = \\sum_{j = 1}^N {A_{i,j}}$.\nThe definition (3) is often used in the GSP literature due to the fact that it is real symmetric, positive semi-definite (PSD) and has eigenvalues lying in $[0, 2]$.\nIn certain applications~\\cite{kipf2016semi}, it was found that adding self-loops, i.e., changing $A$ to $A+I$, and using the affinity matrix $S = D^{-\\frac{1}{2}} A D^{-\\frac{1}{2}}$ instead of $L$ gives better results.\nSince $S$ is real symmetric, based on spectral decomposition, we have $ S = U \\Lambda U^{\\top}$ where $U$ is an orthogonal matrix and its column vectors are the eigenvectors of $S$. \nThe diagonal matrix $\\Lambda$ contains the sorted eigenvalues where $\\Lambda_{i,i}=\\lambda_i$ and $1\\geq \\lambda_1 \\geq \\dots \\geq \\lambda_N \\geq -1$. \nBased on the eigenbasis, we can define the \\textit{graph Fourier transform} $Y = U^{\\top} X$ and its inverse transform $\\hat{X} = U Y$ following~\\cite{shuman2013emerging}. \nNote that $L = I - D^{-\\frac{1}{2}} A D^{-\\frac{1}{2}}$ shares the same eigenvectors with $S=D^{-\\frac{1}{2}} A D^{-\\frac{1}{2}}$ and the eigenvalues of $L$ are $\\mu_i = 1 - \\lambda_i$.\nTherefore, $L$ and $S$ share the same \\textit{graph Fourier transform} which justifies the usages of $S$.\nDifferent forms of filters can be further constructed in the spectral domain.\n\n\\paragraph{Localized Polynomial Filter}\nA $\\tau$-localized polynomial filter is typically adopted in GSP literature~\\cite{shuman2013emerging}, $g_{w}(\\Lambda) = \\sum_{t=0}^{\\tau-1} w_{t} \\Lambda^{t}$, where $\\bm{w} = \\left[w_{0}, w_{1}, \\dots, w_{\\tau-1}\\right] \\in \\mathbb{R}^{\\tau \\times 1}$ is the filter coefficient, i.e., learnable parameter.\nThe filter is $\\tau$-localized in the sense that the filtering leverages information from nodes which are at most $\\tau$-hops away. \nOne prominent example of this class is the Chebyshev polynomial filter~\\cite{defferrard2016convolutional}.  \nHere the graph Laplacian is modified to $\\tilde{L} = 2L\/\\lambda_{\\text{max}} - I$ such that its eigenvalues fall into $[-1, 1]$. \nThen the Chebyshev polynomial recursion is applied: $\\tilde{X}(t) = 2 \\tilde{L} \\tilde{X}(t-1) - \\tilde{X}(t-2)$ where $\\tilde{X}(0) = X$ and $\\tilde{X}(1) = \\tilde{L} X$. \nFor a pair of input and output channels $(i, j)$, the final filtering becomes, \n$y_{i, j} = [\\tilde{X}(0)_{:, i}, \\dots, \\tilde{X}(\\tau - 1)_{:, i}] \\bm{w}_{i,j}$, where $\\left[ \\cdot \\right]$ means concatenation along columns and $\\bm{w}_{i,j} \\in \\mathbb{R}^{\\tau \\times 1}$.\nChebyshev polynomials provide two benefits: they form an orthogonal basis of $L^2([-1, 1], dy\/\\sqrt{1 - y^2})$ and one avoids the spectral decomposition of $\\tilde{L}$ in the filtering. \nHowever, the functional form of the spectral filter is not learnable, and cannot adapt to the data.\n\nIn this paper, instead of using the modified graph Laplacian $\\tilde{L}$, we use the aforementioned $S$. \nTherefore, we can write the localized polynomial filtering in a more general form as,\n\\begin{align}\\label{eq:poly_filter}\nY = \\sum_{t=0}^{\\tau - 1} g_{t}(S, \\dots, S^{t}, X) W_{t},\n\\end{align}\nwhere $g_{t}$ is a function that takes node features $X$ and powers of the affinity matrices up to the $t$-th order as input and outputs a $N \\times F$ matrix. \n$W_{t} \\in \\mathbb{R}^{F \\times O}$ is the corresponding filter coefficient and $Y \\in \\mathbb{R}^{N \\times O}$ is the output.\nOne can easily verify that in the Chebyshev polynomial filter, any $i$-th column of the corresponding $g_{t}(X, S, \\dots, S^{t})$ lies in the \\textit{Krylov subspace} $\\mathcal{K}_{t+1}(S, X_{:, i}) \\equiv \\mathrm{span} \\{X_{:, i}, S X_{:, i}, \\dots, S^{t}X_{:, i}\\}$.\nThis naturally motivates the usage of Krylov subspace methods, like the Lanczos algorithm~\\cite{lanczos1950iteration}, since it provides an orthonormal basis for the above Krylov subspace, thus making the filter coefficients compact.\n\n\\section{Introduction}\n\nGraph-structured data is ubiquitous in real world applications, social networks, gene expression regulatory networks, protein-protein interactions, and many other physical systems. \nHow to model such data using machine learning, especially deep learning, has become a central research question~\\cite{battaglia2018relational}.\nFor supervised and semi-supervised tasks such as graph or node classification and regression, learning based models can be roughly categorized into two classes, formulated either in terms of graph convolutions~\\cite{bruna2013spectral} or recurrent neural networks~\\cite{scarselli2009graph}. \n\nMethods based on recurrent neural networks (RNN), especially graph neural networks (GNN)~\\cite{scarselli2009graph}, repeatedly unroll a message passing process over the graph by exchanging information between the nodes.\nIn theory, a GNN can have as large a model capacity as its convolutional counterpart.\nHowever, due to the instability of RNN dynamics and difficulty of optimization, GNN and its variants are generally slower and harder to train.\n\nIn this paper we focus on graph convolution based methods.\nBuilt on top of the graph signal processing (GSP) approaches~\\cite{ortega2018graph}, these methods extend convolution operators to graphs by leveraging spectral graph theory, graph wavelet theory, etc.\nGraph convolutions can be stacked and combined with nonlinear activation functions to build deep models, just as in  regular convolutional neural networks (CNN).\nThey often have large model capacity and achieve promising results.\nAlso, graph convolution can be easily implemented with modern scientific computing libraries.\n\nThere are two main issues with current graph convolution approaches.\nFirst, it is not clear how to efficiently leverage multi-scale information except by directly stacking multiple layers.\nHaving an effective multi-scale scheme is key for enabling the model to be invariant to scale changes, and to capture many intrinsic regularities~\\cite{witkin1983scale,coifman2005geometric}.\nGraph coarsening methods have been proposed to form a hierarchy of multi-scale graphs \\cite{defferrard2016convolutional}, but this coarsening process is fixed during both inference and learning which may cause some bias.\nAlternatively, the graph signal can be multiplied by the exponentiated graph Laplacian, where the exponent indicates the scale of the diffusion process on the graph~\\cite{atwood2016diffusion}.\nUnfortunately, the computation and memory cost increases linearly with the exponent, which prohibits the exploitation of long scale diffusion in practice.\nOther fast methods for computing matrix power such as exponentiating by squaring are very memory intensive, even for moderately large graphs.\nSecond, spectral filters within current graph convolution based models are mostly fixed. \nIn the context of image processing, using a Gaussian kernel along with a spectral filter $f(\\lambda) = 2\\lambda - \\lambda^2$ corresponds to running forward the heat equation (blurring) followed by running it backwards (sharpening)~\\cite{singer2009diffusion}. \nMulti-scale kernels introduced in~\\cite{rabin2018multi} extends the idea of forward-backward diffusion process and can be represented as polynomials of matrices related to a Gaussian kernel.\nLearning the spectral filters is thus beneficial since it learns the stochastic processes on the graph which produce useful representations for particular tasks.\nHowever, how to learn spectral filters which have large model capacity is largely underexplored. \n\nIn this paper, we propose the Lanczos network (LanczosNet) to overcome the aforementioned issues.\nFirst, based on the tridiagonal decomposition implied by the Lanczos algorithm, our model exploits the low rank approximation of the graph Laplacian. \nThis approximation facilitates efficient computation of matrix powers thus gathering multi-scale information easily.\nSecond, we design learnable spectral filters based on the approximation which effectively increase model capacity.\nIn scenarios where one wants to learn the graph kernel and\/or node embeddings, we propose another variant, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, adaptive Lanczos network ({AdaLanczosNet}), which back-propagates through the Lanczos algorithm.\nWe show that our proposed model is closely related to graph based manifold learning approaches such as diffusion maps which could potentially inspire more work from the intersection between deep graph networks and manifold learning.\nWe benchmark against $9$ recent deep graph networks, including both convolutional and RNN based methods, on citation networks and a quantum chemistry graph regression problem, and achieve state-of-the-art results in most tasks.\n\n\\section{Lanczos Networks}\n\nIn this section, we first introduce the Lanczos algorithm which approximates the affinity matrix $S$. \nWe present our first model, called Lanczos network ({LanczosNet}), in which we execute the Lanczos algorithm once per graph and fix the basis throughout inference and learning.\nThen we introduce the adaptive Lanczos network ({AdaLanczosNet}) in which we learn the graph kernel and\/or node embedding by back-propagating through the Lanczos algorithm.\n\n\\begin{minipage}{6.6cm}\n\\begin{algorithm}[H]\n\\caption{: Lanczos Algorithm}\\label{alg:lanczos}\n\\begin{algorithmic}[1]\n\\STATE \\textbf{Input:} $S, x, K, \\epsilon$\n\\STATE \\textbf{Initialization:} $\\beta_{0} = 0$, $q_{0} = 0$, and $q_{1} = x \/ \\Vert x \\Vert$\n\\STATE \\textbf{For} $j = 1, 2, \\dots, K $:\n\\STATE \\qquad $z = Sq_{j}$\n\\STATE \\qquad $\\gamma_{j} = q_{j}^{\\top}z$\n\\STATE \\qquad $z = z - \\gamma_{j}q_{j} - \\beta_{j-1}q_{j-1}$\n\\STATE \\qquad $\\beta_{j} = \\Vert z \\Vert_{2}$\n\\STATE \\qquad \\textbf{If} $\\beta_{j} < \\epsilon$, quit\n\\STATE \\qquad $q_{j+1} = z  \/ \\beta_{j}$\n\\STATE\n\\STATE $Q = \\left[q_{1}, q_{2}, \\cdots, q_{K} \\right]$\n\\STATE Construct $T$ following Eq.~(\\ref{eq:tridiagonal})\n\\STATE Eigen decomposition $T = BRB^{\\top}$\n\\STATE Return $V=QB$ and $R$.\n\\end{algorithmic}\n\\end{algorithm}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{7.1cm}\n\\begin{algorithm}[H]\n\\caption{: LanczosNet }\\label{alg:graph_convolution_net}\n\\begin{algorithmic}[1]\n\\STATE \\textbf{Input:} Signal $X$, Lanczos output $V$ and $R$, scale index sets $\\mathcal{S}$ and $\\mathcal{I}$,\n\\STATE \\textbf{Initialization:} $Y_0 = X$\n\\STATE \\textbf{For} $\\ell = 1, 2, \\dots, \\ell_c $:\n\\STATE \\qquad $Z = Y_{\\ell - 1}$, $\\mathcal{Z} = \\{\\emptyset \\}$\n\\STATE \\qquad \\textbf{For} $j = 1, 2, \\dots, \\max(\\mathcal{S})$:\n\\STATE \\qquad \\qquad $Z = SZ$\n\\STATE \\qquad \\qquad \\textbf{If} $j \\in \\mathcal{S}$:\n\\STATE \\qquad \\qquad \\qquad $\\mathcal{Z} = \\mathcal{Z} \\cup Z$\n\\STATE \\qquad \\textbf{For} $i \\in \\mathcal{I}$:\n\\STATE \\qquad \\qquad $\\mathcal{Z} = \\mathcal{Z} \\cup V \\hat{R}(\\mathcal{I}_i) V^{\\top}Y_{\\ell-1}$\n\\STATE \\qquad $Y_\\ell = \\text{concat}(\\mathcal{Z}) W_{\\ell}$\n\\STATE \\qquad \\textbf{If} $\\ell < L$\n\\STATE \\qquad \\qquad $Y_\\ell = \\text{Dropout}(\\sigma(Y_\\ell))$\n\\STATE Return $Y_{\\ell_c}$.\n\\end{algorithmic}\n\\end{algorithm}\n\\end{minipage}\n\n\n\\subsection{Lanczos Algorithm}\n\nGiven the aforementioned affinity matrix $S$\\footnote{When faced with a non-symmetric matrix, one can resort to the Arnoldi algorithm.} and node features $x \\in \\mathbb{R}^{N \\times 1}$, the $N$-step Lanczos algorithm computes an orthogonal matrix $Q$ and a symmetric tridiagonal matrix $T$, such that $Q^{\\top}SQ = T$. \nWe denote $Q = \\left[q_{1}, \\cdots, q_{N} \\right]$ where column vector $q_{i}$ is the $i$-th Lanczos vector. \n$T$ is illustrated as below, \n\\begin{align}\\label{eq:tridiagonal}\nT = \\left[ \\begin{array}{cccc} \\gamma_{1} & \\beta_{1} &  &  \\\\ \\beta_{1} & \\ddots & \\ddots & \\\\  & \\ddots & \\ddots & \\beta_{N-1} \\\\  &  & \\beta_{N-1} & \\gamma_{N} \\end{array} \\right].\n\\end{align}\nOne can verify that $Q$ forms an orthonormal basis of the Krylov subspace $\\mathcal{K}_{N}(S, x)$ and the first $K$ columns of $Q$ forms the orthonormal basis of $\\mathcal{K}_{K}(S, x)$.\nThe Lanczos algorithm is shown in detail in Alg.~\\ref{alg:lanczos}.\nIntuitively, if we investigate the $j$-th column of the system $SQ = QT$ and rearrange terms, we obtain $\\beta_{j}q_{j+1} = Sq_{j} - \\beta_{j-1}q_{j-1} - \\gamma_{j}q_{j}$, which clearly explains lines $4$ to $6$ of the pseudocode, i.e., it tries to solve the system in an iterative manner.\nNote that the most expensive operation in the algorithm is the matrix-vector multiplication in line $4$.\nAfter obtaining the tridiagonal matrix $T$, we can compute the Ritz values and Ritz vectors which approximate the eigenvalues and eigenvectors of $S$ by diagonalizing the matrix $T$. \nWe only add this step in {LanczosNet} as we found back-propagating through the eigendecomposition in {AdaLanczosNet} is not numerically stable.\n\n\\subsection{LanczosNet}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{lanczos_net_model.pdf}\n    \\vspace{-0.3cm}\n    \\caption{The illustration of the model. The learnable spectral filters have different parameters per layer. The nonlinear activation function $\\sigma$ could be applied before or after the concatenation.}\n    \\vspace{-0.5cm}\n    \\label{fig:model}\n\\end{figure}\n\nIn this section, we first show the construction of the localized polynomial filter based on the Lanczos algorithm's output and discuss its limitations.\nThen we explain how to construct the spectral filter using a particular low rank approximation and how to further make the filter learnable.\nAt last, we elaborate how to construct multi-scale graph convolution and build a deep network.\n\n\\paragraph{Localized Polynomial Filter} \nFor the ease of demonstrating the concept of \\textit{Krylov subspace}, we consider a pair of input and output channels $(i,j)$.\nWe denote the input as $X_{:,i} \\in \\mathbb{R}^{N \\times 1}$ and the output as $Y_{:,j} \\in \\mathbb{R}^{N \\times 1}$.\nExecuting the Lanczos algorithm for $K$ steps with the normalized $X_{:,i}$ as the starting vector, one can obtain the orthonormal basis $\\tilde{Q}$ of $\\mathcal{K}_{K}(S, X_{:,i})$ and the corresponding tridiagonal matrix $\\tilde{T}$.\nRecall that in the localized polynomial filtering, given the orthonormal basis of $\\mathcal{K}_{K}(S, X_{:,i})$, one can write the graph convolution as\n\\begin{align}\\label{eq:krylov_filter_fix}\nY_{j} = \\tilde{Q} \\bm{w}_{i,j},\n\\end{align}\nwhere $\\tilde{Q} \\in \\mathbb{R}^{N \\times K}$ depends on the $X_{:,i}$ and $\\bm{w}_{i,j} \\in \\mathbb{R}^{K \\times 1}$ is the learnable parameter.\nThis filter has the benefit that the corresponding learnable coefficients are compact due to the orthonormal basis.\nHowever, if one wants to stack multiple graph convolution layers, the dependency of $\\tilde{Q}$ on $X_{:,i}$ implies that a separate run of Lanczos algorithm is necessary for each graph convolution layer which is computationally demanding.\n\n\\paragraph{Spectral Filter} \nIdeally, we would like to compute Lanczos vectors only once during the inference of a deep graph convolutional network.\nLuckily, this can be achieved if we take an alternative view of Lanczos algorithm.\nIn particular, we can choose a random starting vector with unit norm and treat the $K$ step Lanczos layer's output as the low rank approximation $S \\approx Q T Q^{\\top}$.\nNote that here $Q \\in \\mathbb{R}^{N \\times K}$ has orthonormal columns and does not depend on the node features $X_{i}$ and $T$ is a $K \\times K$ tridiagonal matrix.\nFollowing~\\cite{simon2000low}, we prove the theorem below to bound the approximation error.\n\\begin{restatable}{theorem}{errortheorem}\\label{the:error}\nLet $U \\Lambda U^{\\top}$ be the eigendecomposition of an $N \\times N$ symmetric matrix $S$ with $\\Lambda_{i,i} = \\lambda_i$, $\\lambda_1 \\geq \\dots \\geq \\lambda_N$ and $U = [u_1, \\dots, u_N]$.\nLet $\\mathcal{U}_j \\equiv \\mathrm{span}\\{u_1, \\dots, u_j\\}$.\nAssume $K$-step Lanczos algorithm starts with vector $v$ and outputs the orthogonal $Q \\in \\mathbb{R}^{N \\times K}$ and tridiagonal $T \\in \\mathbb{R}^{K \\times K}$.\nFor any $j$ with $1 < j < N$ and $K > j$, we have,\n\\begin{align}\n    \\Vert S - Q T Q^{\\top} \\Vert_{F}^{2} \\leq \\sum_{i=1}^{j} \\lambda_i^2 \\left( \\frac{\\sin{\\left(v, \\mathcal{U}_i\\right)} \\prod_{k=1}^{j-1} (\\lambda_k - \\lambda_N) \/ (\\lambda_k - \\lambda_j) }{\\cos{\\left(v, u_i\\right)} T_{K-i}(1+2\\gamma_i) } \\right)^2 + \\sum_{i=j+1}^{N} \\lambda_i^2, \n    \\nonumber\n\\end{align}\nwhere $T_{K-i}(x)$ is the Chebyshev Polynomial of degree $K-i$ and $\\gamma_i = (\\lambda_{i} - \\lambda_{i+1}) \/ (\\lambda_{i+1} - \\lambda_{N})$.\n\\end{restatable}\nWe leave the proof to the appendix. \nNote that the term $(\\sum_{i=j+1}^{N} \\lambda_i^2)^{1\/2}$\nis the Frobenius norm of the error between $S$ and the best rank-$j$ approximation $S_j$.\nWe decompose the tridiagonal matrix $T = B R B^{\\top}$, where the $K \\times K$ diagonal matrix $R$ contains the Ritz values and $B \\in \\mathbb{R}^{K \\times K}$ is an orthogonal matrix. We have a low rank approximation of the affinity matrix $S \\approx V R V^{\\top}$, where $V = QB$. \nTherefore, we can rewrite the graph convolution as,\n\\begin{align}\\label{eq:identity_filter_fix}\nY_{j}  = [X_i, S X_i, \\dots, S^{K-1} X_i] \\bm{w}_{i,j} \\approx [X_i, V R V^{\\top} X_i, \\dots, V R^{K-1} V^{\\top} X_i] \\bm{w}_{i,j},\n\\end{align}\nThe difference between Eq.~(\\ref{eq:krylov_filter_fix}) and Eq.~(\\ref{eq:identity_filter_fix}) is that the former uses the orthonormal basis while the latter uses the approximation of the direct basis of $\\mathcal{K}_{K}(S, X_{:,i})$. \nSince we explicitly operate on the approximation of spectrum, i.e., Ritz value, it is a spectral filter.\nSuch a filtering form will have significant computational benefits while considering the long range\/scale dependency due to the fact that the $t$-th power of $S$ can be approximated as $S^{t} \\approx V R^{t} V^{\\top}$, where we only need to raise the diagonal entries of $R$ to the power $t$.\n\n\\paragraph{Learning the Spectral Filter} \nFollowing the previous filter, one can naturally design learnable spectral filters. \nDenoting the diagonal entries of $R$ and the column vectors of $V$, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, Ritz values and vectors, as $\\{(r_i, v_i) \\vert i = 1, \\dots, K\\}$, we perform $i$-th spectral filtering as follows,\n\\begin{align}\n\\hat{L}_{i} = \\sum_{k=1}^{K} f_{i}(r_{k}^{1}, r_{k}^{2}, \\cdots, r_{k}^{K-1}) v_{k} v_{k}^{\\top}\n\\end{align}\nwhere $f_{i}$ is a multi-layer perceptron (MLP).\nTherefore, we have the following graph convolution,\n\\begin{align}\\label{eq:learnable_filter_fix}\nY_{j} = [X_i, \\hat{L}_{1} X_i, \\dots, \\hat{L}_{K-1} X_i] \\bm{w}_{i,j}.\n\\end{align}\nNote that it includes the polynomial filter as a special case.\nWhen positive semi-definiteness is a concern, one can apply an activation function like ReLU to the output of the MLPs.\n\n\\paragraph{Multi-Scale Graph Convolution}\nUsing any above filter, one can construct a deep graph convolutional network which leverages multi-scale information.\nTaking the learnable spectral filter as an example, we can write one graph convolution layer in a compact way as below,\n\\begin{align}\\label{eq:multi_scale_graph_conv_fix}\nY = \\left[L^{\\mathcal{S}_1} X, \\dots, L^{\\mathcal{S}_M} X, \\hat{L}_{1}(\\mathcal{I}) X, \\dots, \\hat{L}_{N}(\\mathcal{I})X \\right] W,\n\\end{align}\nwhere weight $W \\in \\mathbb{R}^{(M+E)D \\times O}$, $\\mathcal{S}$ is a set of $M$ short scale parameters and $\\mathcal{I}$ is a set of $E$ long scale parameters.\nWe consider a non-negative integer as scale parameter, e.g., $\\mathcal{S} = \\{0, 1, \\dots, 5\\}$, $\\mathcal{I} = \\{10, 20, \\dots, 50\\}$.\nHere we slightly abuse the notation and define the $i$-th spectral filtering as \n\\begin{align}\n    \\hat{L}_{i}(\\mathcal{I}) = \\sum_{k=1}^{K} f_{i}(r_{k}^{\\mathcal{I}_1}, r_{k}^{\\mathcal{I}_2}, \\cdots, r_{k}^{\\mathcal{I}_{\\vert \\mathcal{I} \\vert}}) v_{k} v_{k}^{\\top}.\n\\end{align}\nNote that the convolution corresponding to short scales is similar to~\\cite{atwood2016diffusion} where the number of matrix-vector multiplications is tied to the maximum scale of $\\mathcal{S}$.\nIn contrast, the convolution of long scales decouples the Lanczos step $K$ and scale parameters $\\mathcal{I}$, thus permitting great freedom in tuning scales as hyperparameters.\nOne can choose $K$ properly to balance the computation cost and the accuracy of the low rank approximation.\nIn our experiments, short scales are typically less than $10$ which have reasonable computation cost.\nMoreover, the short scale part could sometimes remedy cases where the low rank approximation is crude.\nWe set the long scale no larger than $100$ in our experiments.\nIf the maximum eigenvalue of $S$ is $1$, we can even raise the power to infinity, which corresponds to the equilibrium state of diffusion process on the graph.\n\nTo build a deep network, we can stack multiple such graph convolution layers where each layer has its own spectral filter weights. \nNonlinear activation functions, e.g., ReLU, and\/or Dropout can be added between layers.\nThe inference algorithm of such a deep network is shown in Alg. \\ref{alg:graph_convolution_net}.\nThe overall computation graph of the model is illustrated in Fig. \\ref{fig:model}.\nWith the top layer representation, one can use softmax to perform classification or a fully connected layer to perform regression.\nThe Lanczos algorithm is run beforehand once per graph to construct the network and will not be invoked during inference and learning.\n\n\n\\subsection{AdaLanczosNet}\nIn this section, we explain another variant which back-propagates through the Lanczos algorithm. \nThis facilitates learning the graph kernel and\/or node embeddings.\n\n\\paragraph{Graph Kernel}\nAssume we are given node features $X$ and a graph $\\mathcal{G}$. \nWe are interested in learning a graph kernel function with the hope that it can capture the intrinsic geometry of node representations. \nGiven data points $x_i, x_j \\in \\mathcal{X}$, we define the anisotropic graph kernel, $k: \\mathcal{X} \\times \\mathcal{X} \\mapsto \\mathbb{R}$ as,\n\\begin{align}\\label{eq:graph_kernel}\nk(x_{i}, x_{j}) = \\exp\\left( - \\frac{ \\Vert ( f_{\\theta}(x_{i}) - f_{\\theta}(x_{j}) ) \\Vert^{2} } { \\epsilon } \\right).\n\\end{align}\nwhere $f_{\\theta}$ is an MLP.\nThis class of anisotropic kernels is very expressive and includes self-tuning kernel~\\cite{zelnik2005self} and the Gaussian kernel with Mahalanobis distances~\\cite{weinberger2007metric}.\nMoreover, for different kernel functions, the resulted graph Laplacians will converge to different limiting operators asymptotically.\nFor example, even for isotropic Gaussian kernels, the graph Laplacian can converge pointwise to the Laplace-Beltrami, Fokker-Planck operator and heat kernel under different normalizations~\\cite{coifman2006diffusion,singer2006graph}.\nIn practice, we notice that choosing $\\epsilon = \\sum_{(p, q) \\in \\mathcal{E}} \\| ( f_\\theta(x_{p}) - f_\\theta(x_{q}) )\\|^2 \/ {\\vert \\mathcal{E} \\vert}$ helps normalizing the pairwise distances,\nthus avoiding the gradient vanishing issue due to the exponential function.\nThis type of learnable anisotropic diffusion is useful in two ways.\nFirst, it increases model capacity, thus potentially gaining better performance.\nSecond, it can better adapt to the non-uniform density of the data points on the manifold or nonlinear measurements of the underlying data points on a maninfold.\nWe can construct an adjacency matrix $A$ such that $A_{i, j} = k(x_{i}, x_{j})$ if $(i, j) \\in \\mathcal{E}$ and $A_{i, j} = 0$ otherwise. \nThen we can obtain the affinity matrix $S = D^{-\\frac{1}{2}} A D^{-\\frac{1}{2}}$.\n\n\n\n\n\n\n\n\\paragraph{Node Embedding}\nIn some applications, we do not observe the node features $X$ but only the graph itself $\\mathcal{G}$, so we may need to learn an embedding vector per node. \nFor example, this scenario applies in the quantum chemistry tasks where a node, i.e., an atom within a molecule, has rarely observed features. \nWe can still use the above graph kernel to construct the affinity matrix which results in the same form except $f$ is discarded.\nLearning embedding $X$ naturally amounts to learning the similarities between nodes.\n\n\\paragraph{Tridiagonal Decomposition}\n\nAlthough all operations in {LanczosNet} are differentiable, we empirically observe that backpropagation through the eigendecomposition of the tridiagonal matrix is numerically instable.\nThe situation would be even worse if multiple eigenvalues are numerically close or one takes a large power in Eq.~(\\ref{eq:multi_scale_graph_conv_fix}).\nTherefore, we instead directly leverage the approximated tridiagonal decomposition $S \\approx Q T Q^{\\top}$ which is obtained by running the Lanczos algorithm $K$ steps.\nThen we can rewrite the spectral filtering as following,\n\\begin{align}\n    \\hat{L}_{i}(\\mathcal{I}) = Q g_{i}(\\text{vec}(T^{\\mathcal{I}_1}), \\text{vec}(T^{\\mathcal{I}_2}), \\cdots, \\text{vec}(T^{\\mathcal{I}_{\\vert \\mathcal{I} \\vert}})) Q^{\\top}, \n\\end{align}\nwhere $\\text{vec}(\\cdot)$ means vectorization and $g_i(\\cdot) = f_i(\\cdot) + f_i(\\cdot)^\\top$, with  the output of an MLP $f_i$ reshaped to a matrix of the same size as $T$. This ensures that the output is symmetric.\n \n\n\nWith the above parameterization of the graph Laplacian and tridiagonal decomposition, we can back-propagate the loss through the Lanczos algorithm to either the graph kernel parameters $\\theta$ or the node embedding $X$.\nThe overall model is similar to the {LanczosNet} except that the Lanczos algorithm needs to be invoked for each inference pass.\n\n\n\\section*{Acknowledgments}\nRL thanks Roger Grosse for introducing the Lanczos algorithm to him. RL was supported by Connaught International Scholarships. RL, RU and RZ were supported in part by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior\/Interior Business Center (DoI\/IBC) contract number D16PC00003. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: the views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI\/IBC, or the U.S. Government.\n\n\\clearpage\n\n\\bibliographystyle{unsrt}\n\n\\section{Related Work}\n\nWe can roughly categorize the application of machine learning, especially deep learning, to graph structured data into supervised\/semi-supervised and unsupervised scenarios.\nFor the former, a majority of work focuses on node\/graph classification and regression~\\cite{vishwanathan2010graph,bronstein2017geometric,garcia2017few,battaglia2018relational}.\nFor the latter, unsupervised node\/graph embedding learning~\\cite{grover2016node2vec,duran2017learning} is common.\nRecently, generative models for graphs, such as molecule generation, has drawn some attention~\\cite{li2018learning,jin2018junction}.\n\n\\paragraph{Graph Convolution Based Models}\nThe first class of learning models on graphs stems from graph signal processing (GSP)~\\cite{shuman2013emerging,ortega2018graph} which tries to generalize convolution operators from traditional signal processing to graphs.\nRelying on spectral graph theory~\\cite{chung1997spectral} and graph wavelet theory~\\cite{hammond2011}, several definitions of frequency representations of graph signals have been proposed~\\cite{ortega2018graph}.\nAmong these, spectral graph theory based one is popular, where graph Fourier transform and its inverse are defined based on the eigenbasis of the graph Laplacian.\nFollowing this line, many graph convolution based deep network models emerge.\n\\cite{bruna2013spectral,henaff2015deep} are among the first to explore Laplacian based graph convolution within the context of deep networks.\nMeanwhile, \\cite{duvenaud2015convolutional} performs graph convolution directly based on the adjacency matrix to predict molecule fingerprints.\n\\cite{niepert2016learning} proposes a strategy to form same sized local neighborhoods and then apply convolution like regular CNNs. \nChebyshev polynomials are exploited by~\\cite{defferrard2016convolutional} to construct localized polynomial filters for graph convolution and are later simplified in graph convolutional networks (GCN)~\\cite{kipf2016semi}.\nFurther accelerations for GCN based on importance sampling and control variate techniques have been proposed by~\\cite{chen2018fastgcn,chen2018stochastic}.\nSeveral attention mechanisms have been introduced in~\\cite{velickovic2017graph,wang2018deep} to learn the weights over edges for GCNs.\nNotably, \\cite{atwood2016diffusion} proposes diffusion convolutional neural networks (DCNN) which uses diffusion operator for graph convolution.\nLanczos method has been explored for graph convolution in~\\cite{susnjara2015accelerated} for the purpose of acceleration.\nSpecifically, they only consider the localized polynomial filter case in our {LanczosNet} variant and do not explore the low rank decomposition, learnable spectral filter and graph kernel\/node embedding learning as we do.\n\n\n\\paragraph{Recurrent Neural Networks based Models}\nThe second class of models dates back to recursive neural networks~\\cite{pollack1990recursive} which recurrently apply neural networks to trees following a particular order.\nGraph neural networks (GNN)~\\cite{scarselli2009graph} generalize recursive neural networks to arbitrary graphs and exploit the synchronous schedule to propagate information on graphs.\n\\cite{li2015gated} later proposes the gated graph neural networks (GGNN) which improves GNN by adding gated recurrent unit and training the network with back-propagation through time.\n\\cite{dai2016discriminative} learns graph embeddings via unrolling variational inference algorithms over a graph as a RNN.\n\\cite{hamilton2017inductive} introduces random subgraph sampling and explores different aggregation functions to scale GNN to large graphs.\n\\cite{liao2018graph} proposes asynchronous propagation schedules based on graph partitions to improve GNN.\nMoreover, many applications have recently emerged for GNNs, including community detection~\\cite{bruna2017community}, situation recognition~\\cite{li2017situation}, RGBD semantic segmentation~\\cite{qi20173d}, few-shot learning~\\cite{garcia2017few}, probabilistic inference~\\cite{yoon2018inference}, continuous control of reinforcement learning~\\cite{wang2018nervenet,sanchez2018graph} and so on.\n\n\\paragraph{Graph based Manifold Learning}\nThe non-linear dimensionality reduction methods, such as locally linear embedding (LLE)~\\cite{roweis2000nonlinear}, ISOMAP~\\cite{tenenbaum2000global}, Hessian LLE~\\cite{donoho2003hessian}, Laplacian eigenmaps~\\cite{belkin2002laplacian}, and diffusion maps~\\cite{coifman2006diffusion}, assume that the high-dimensional data lie on or close to a low dimensional manifold and use the local affinities in the weighted graph to learn the global features of the data. \nThey are invaluable tools for embedding complex data in a low dimensional space and regressing functions over graphs. \nSpectral clustering~\\cite{nadler2006diffusion,von2007tutorial}, semi-supervised learning~\\cite{zhu2006semi}, and out-of-sample extension~\\cite{belkin2006manifold} share the similar geometrical consideration of the associated graphs.\nAnisotropic graph kernels are useful in many applications. For example, \\cite{zelnik2005self} improves the spectral clustering results with a self-tuning diffusion kernel that takes into account the local variance at each node in the Gaussian kernel function. \nSimilarly, \\cite{singer2008non} uses the anisotropic Gaussian kernel defined by the local Mahalanobis distances to extract independent components from nonlinear measurements of independent stochastic It{\\^o} processes. \nManifold learning with anisotropic kernel is also useful for data-driven dynamical system analysis, for example, detecting intrinsically slow variable for a stochastic dynamical system~\\cite{singer2009detecting}, filtering dynamical processes~\\cite{talmon2013empirical}, and long range climate forecasting~\\cite{giannakis2015dynamics,zhao2016analog}. \nThe anisotropic diffusion is able to use the local statistics of the measurements to convey the geometric information on the underlying factors rather than the specific realization or measurements at hand~\\cite{lafferty2005diffusion,amari2007methods}. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecently the ATLAS and CMS collaborations have announced the discovery of a new boson with a mass of approximately 125 GeV~\\cite{:2012gk,:2012gu}. The next challenge is to determine the nature of this new state, including its quantum numbers and couplings, and whether it is fundamental or composite. Because of the observed decays to $\\gamma\\gamma$, $WW$, and $ZZ$, with strengths seemingly comparable to those expected from the standard model (SM) Higgs, this state is likely a (mostly) $CP$-even spin-zero boson, although $CP$-odd candidates are not currently ruled out~\\cite{Frandsen:2012rj,Barroso:2012wz,Coleppa:2012eh,Chivukula:2012cp,Eichten:2012qb}.\n\nIn technicolor (TC), {\\it common lore} has it that the lightest $CP$-even spin-zero resonance, the analogue of the $\\sigma$ meson or $f_0(500)$ in QCD, cannot be as light as 125 GeV. In fact TC theories are sometimes considered as underlying theories for Higgsless models, despite the fact that in QCD the $\\sigma$ meson is among the lightest states. In this paper we consider the possibility of a light {\\it TC Higgs} arising from mass mixing between relatively heavy scalar singlets. This results in a ``see-saw'' mechanism, with one scalar singlet becoming lighter and one heavier than the corresponding diagonal mass. This is expected to occur in two-scale TC models, {\\it e.g.} low-scale TC~\\cite{Lane:1989ej,Eichten:2011sh} and ultra-minimal TC (UMT)~\\cite{Ryttov:2008xe}\\footnote{In models with fundamental scalars, a scalar see-saw mechanism has also been considered as a way of generating a negative mass squared for the Higgs \\cite{Calmet:2002rf,Calmet:2006hs}.}.\n\nTwo-scale TC theories feature two technifermion species with different representations under a single technicolor gauge group. These lead, for instance, to two different sets of composite scalars. Because of different quantum numbers, scalar multiplets from different representations do not mix through mass terms. However scalar singlets do. Because of the strength of the TC interaction, such a mixing can be sizable, which is the key ingredient in the see-saw mechanism. Moreover, radiative corrections from the top quark may contribute to further reduce the mass of the lightest scalar singlet~\\cite{Foadi:2012bb}.\n\nThis paper is organized as follows. In Sec.~\\ref{sec:seesaw} we briefly review the see-saw mechanism for scalar singlets. In Sec.~\\ref{sec:twoscale} we review the spin-zero sector of two-scale TC, and analyze the properties of the mixing mass term. Then we apply the general results to UMT and low-scale TC. \nFinally, in Sec.~\\ref{sec:conclusions} we offer a brief discussion of our findings.\n\\section{Higgs see-saw mechanism}\\label{sec:seesaw}\nConsider a theory featuring two scalar singlets in its spectrum, $H_1$ and $H_2$. Assume these to mix via mass term:\n\\begin{eqnarray}\n\\mathcal{L}\\supset -\\frac{M_1^2}{2}H_1^2 - \\frac{M_2^2}{2} H_2^2  - \\delta\\ M_1 M_2\\ H_1 H_2 \\ .\n \\label{Eq:basepotential}\n\\end{eqnarray}\nIn the limit $\\delta^2\\to 1$ one eigenstate is massless. It is therefore useful to define the parameter\n\\begin{equation}\n\\varepsilon\\equiv 1-\\delta^2 \\ .\n\\end{equation}\nDiagonalization gives\n\\begin{equation}\n\\left(\\begin{array}{c} H_1 \\\\ H_2 \\end{array}\\right) = \\left(\\begin{array}{cc} \\cos\\beta & \\sin\\beta \\\\ -\\sin\\beta & \\cos\\beta \\end{array}\\right)\n\\left(\\begin{array}{c} H_- \\\\ H_+ \\end{array}\\right)\\ , \\quad\n\\tan2\\beta = \\frac{2 M_1 M_2}{M_2^2-M_1^2}\\delta \\ ,\n\\end{equation}\nwhere $H_-$ and $H_+$ are the light and heavy mass eigenstate, respectively, with mass\n\\begin{equation}\nM_\\pm^2 = \\frac{M_1^2+M_2^2}{2}\\left[1\\pm\\sqrt{1-\\left(\\frac{2 M_1 M_2}{M_1^2+M_2^2}\\right)^2\\varepsilon}\\right] \\ .\n\\end{equation}\nFor $\\varepsilon\\ll 1$, $M_-^2$ becomes\n\\begin{equation}\nM_-^2 = \\frac{M_1^2 M_2^2}{M_1^2+M_2^2}\\varepsilon + {\\cal O}(\\epsilon^2) \\ll M_1^2,\\ M_2^2 \\ .\n\\end{equation}\nIn Fig.~\\ref{fig:eigenvalues} we show $M_-$ (solid) and $M_+$ (dashed), for the cases $M_1=M_2=1.0$ TeV (black) and $M_1=1.0$ TeV, $M_2=300$ GeV (red), as a function of $|\\delta|$. The dotted horizontal line corresponds to the experimental value of 125 GeV. This trivial exercise illustrates how theories with relatively heavy scalar mass scales, {\\it e.g.} $M_i$ between a few hundreds of GeVs and a few TeVs, may still feature a light scalar eigenstate after diagonalization, and thus a candidate for the recently observed 125 GeV boson. While this mechanism is simple and general,\nit is immediately applicable to TC models with two condensation scales. In the next section we discuss the general properties of singlet scalar mixing in two-scale TC theories, and provide specific examples.\n\\begin{figure}[t!]\n\\begin{center}\n \\includegraphics[width=0.55\\linewidth]{seesawplot.pdf}\n\\caption{$M_-$ (solid) and $M_+$ (dashed) as a function of $|\\delta|$, for the cases $M_1=M_2=1.0$ TeV (black) and $M_1=1.0$ TeV, $M_2=300$ GeV (red). The dotted horizontal line corresponds to the experimental value of 125 GeV.}\n\\label{fig:eigenvalues}\n\\end{center}\n\\end{figure}\n\\section{Two-scale technicolor}\\label{sec:twoscale}\nIn two-scale TC two dynamical scales arise due to the presence of two Dirac technifermion species $Q_i$, transforming under different representations $R_i$ of a single TC gauge group~\\cite{Lane:1989ej,Ryttov:2009yw}. The TC force causes technifermion bilinears to condense at different scales $\\Lambda_i$. An estimate of $\\Lambda_2\/\\Lambda_1$ can be obtained from the ladder Schwinger-Dyson equation for the techniquark propagator.  In this approximation, the critical coupling for chiral symmetry breaking depends on the representation via $\\alpha_c (R_i)=\\pi\/3 C_2(R_i)$, where $C_2(R_i)$ is the Casimir of the representation $R_i$. Taking $C_2(R_1)\\leq C_2(R_2)$, and integrating the one-loop beta-function $\\beta(\\alpha) = -\\beta_0(R) \\alpha^2\/2\\pi$ from $\\Lambda_1$ to $\\Lambda_2$ gives\n\\begin{equation}\n\\frac{\\Lambda_2}{\\Lambda_1} \\simeq \\exp\\left[\\frac{2\\pi}{\\beta_0(R_1)}\\bigg(\\alpha_\\mathrm{c}(R_2)^{-1}-\\alpha_\\mathrm{c}(R_1)^{-1} \\bigg)\\right]\\ .\n\\label{Eq:scaleratio}\n\\end{equation}\nSince $\\Lambda_1 \\leq \\Lambda_2$, or equivalently $\\alpha_c(R_1) \\geq\\alpha_c(R_2)$, the fermions in the representation $R_2$ are effectively decoupled below $\\Lambda_2$. Therefore, only $\\beta_0(R_1)$ appears in the exponent. If $\\beta_0(R_1)$ and $\\alpha_c(R_1)$ are\nsmall then the scale separation can be sizeable and the presence of four-fermion operators can contribute to further enhance the scale separation~\\cite{Frandsen:2011kt}. This crude approximation serves to illustrate the appearance of two distinct scales.\n\nNow, let $N_i$ be the number of Dirac techniflavors in the representation $R_i$. The global symmetries of the corresponding fermion sector depend on whether $R_i$ is complex, real, or pseudoreal . To see this we express the Dirac fermions in terms of two Weyl fermions,\n\\begin{equation}\nQ_{im} = \\left(\\begin{array}{c}\\psi^1_{im} \\\\ \\bar{\\psi}^2_{im}\\end{array}\\right)\\ ,\n\\end{equation}\nwhere $m$ is the techniflavor index ($m=1,\\dots, N_i$), and the color index is suppressed. If $\\psi^1_{im}$ transforms under the $R_i$ representation, then $\\psi^2_{im}$ transforms under the conjugate representation $\\overline{R}_i$. For complex $R_i$ this implies that rotations in techniflavor space {\\it cannot} mix $\\psi^1$ and $\\psi^2$ fermions. As a consequence the TC Lagrangian features a global $SU(N_i)_1\\times SU(N_i)_2\\times U(1)$ techniflavor symmetry, where the extra $U(1)$ corresponds to technibaryon-number conservation. This global symmetry is spontaneously broken to diagonal $SU(N_i)\\times U(1)$ by the condensate. The latter is an $N_i\\times N_i$ complex matrix,\n\\begin{equation}\n\\left(\\phi_i\\right)_{mn} \\sim \\psi^1_{im} \\psi^2_{in} \\ ,\n\\end{equation}\nwhich transforms like the bi-fundamental of $SU(N_i)_1\\times SU(N_i)_2$:\n\\begin{equation}\n\\phi_i\\to u_{i1} \\phi_i u_{i2}^\\dagger\\ ,\\quad u_{iA}\\in SU(N_i)_A \\ .\n\\end{equation}\nIn terms of spin-zero composites $\\phi_i$ reads\n\\begin{equation}\n\\phi_i =\n\\frac{v_i + H_i + i\\Theta_i}{\\sqrt{2N_i}} + \\left(i \\Pi_i^a + \\Sigma_i^a\\right)T_i^a\\ ,\n\\end{equation}\nwhere $T_i^a$ are the $SU(N_i)$ broken generators, normalized according to ${\\rm Tr}\\ T_i^a T_i^b = \\delta^{ab}\/2$. Here $v_i$ is the vacuum expectation value of the condensate, and $H_i$ is a $\\psi_{im}^1\\psi^2_{im}$ scalar singlet.\n\nIf the representation $R_i$ is real, rotations in techniflavor space {\\it can} mix $\\psi^1$ and $\\psi^2$ fermions, as these transform in the same way under the TC gauge group. As a consequence the TC Lagrangian features a global $SU(2N_i)$ techniflavor symmetry, which is spontaneously broken to $SO(2N_i)$ by the condensate. The latter is a $2N_i\\times 2N_i$ complex matrix,\n\\begin{equation}\n\\left(\\Phi_i\\right)_{mn}^{AB} \\sim \\psi^A_{im} \\psi^B_{in} \\ ,\n\\label{eq:condensateReal}\n\\end{equation}\nwhere $A,B=1,2$, and $m,n=1,\\dots, N_i$. The spin-zero matrix $\\Phi_i$ transforms as the two-index symmetric representation of $SU(2N_i)$:\n\\begin{equation}\n\\Phi_i\\to u_i \\Phi_i u_i^T\\ ,\\quad u_i\\in SU(2N_i)\\ ,\\quad \\left(\\Phi_i\\right)_{nm}^{BA} = \\left(\\Phi_i\\right)_{mn}^{AB}\\ .\n\\end{equation}\nIn terms of spin-zero composites, $\\Phi_i$ reads\n\\begin{equation}\n\\Phi_i = \\left[\\frac{v_i + H_i + i\\Theta_i}{\\sqrt{4N_i}} + \\left(i \\Pi_i^a + \\Sigma_i^a\\right)X_i^a\\right] E_i\\ ,\n\\label{eq:2N2N}\n\\end{equation}\nwhere $X_i^a$ are the broken generators belonging to the $SU(2N_i)-SO(2N_i)$ algebra, and normalized according to ${\\rm Tr}\\ X_i^a X_i^b = \\delta^{ab}\/2$. The matrix $E_i$ is an $SO(2N_i)$ invariant satisfying\n\\begin{equation}\nE_i^T {X_i^a}^T = X_i^a E_i\\ ,\\quad E_i^T = E_i\\ ,\\quad E_i E_i^\\dagger = 1\\ .\n\\end{equation}\n\nFinally, if the $R_i$ representation is pseudoreal, the global symmetry in techniflavor space is $SU(2N_i)$, and the condensate is as in Eq.~(\\ref{eq:condensateReal}). However the matrix $\\Phi_i$ is now in the two-index antisymmetric representation of $SU(2N_i)$,\n\\begin{equation}\n\\Phi_i\\to u_i \\Phi_i u_i^T\\ ,\\quad u_i\\in SU(2N_i)\\ ,\\quad \\left(\\Phi_i\\right)_{nm}^{BA} = - \\left(\\Phi_i\\right)_{mn}^{AB}\\ ,\n\\end{equation}\nbecause of an extra minus sign introduced by the invariant which contracts the TC indices (not displayed in Eq.~(\\ref{eq:condensateReal})). As a consequence the condensate breaks the global symmetry to $Sp(2N_i)$ rather than $SO(2N_i)$. In terms of spin-zero composites, the $2N_i\\times 2N_i$ $\\Phi_i$ matrix is as in Eq.~(\\ref{eq:2N2N}), where now the $X_i^a$ broken generators belong to the $SU(2N_i)-Sp(2N_i)$ algebra, and $E_i$ is an $Sp(2N_i)$ invariant satisfying\n\\begin{equation}\nE_i^T {X_i^a}^T = - X_i^a E_i\\ ,\\quad E_i^T = - E_i\\ ,\\quad E_i E_i^\\dagger = 1\\ .\n\\end{equation}\n\nWe can unify notation for the complex and real or pseudoreal scenarios by defining, for the case of complex $R_i$, the $2N_i\\times 2N_i$ matrix\n\\begin{equation}\n\\Phi_i \\equiv \\frac{1}{\\sqrt2} \\left(\\begin{array}{cc} 0 & \\phi_i \\\\ \\phi_i^\\dagger & 0 \\end{array}\\right) \\ .\n\\end{equation}\nThen, assuming no violation of techniflavor symmetry, and retaining only terms up to dimension four, the symmetry-breaking potential reads\n\\begin{eqnarray}\nV &=& - \\mu_1^2\\ {\\rm Tr}\\ \\Phi_1 \\Phi_1^\\dagger - \\mu_2^2\\ {\\rm Tr}\\ \\Phi_2 \\Phi_2^\\dagger\n+\\lambda_1^\\prime {\\rm Tr}\\ \\Phi_1 \\Phi_1^\\dagger \\Phi_1 \\Phi_1^\\dagger\n+\\lambda_1^{\\prime\\prime} {\\rm Tr}\\ \\Phi_1 \\Phi_1^\\dagger\\ {\\rm Tr}\\ \\Phi_1 \\Phi_1^\\dagger \\nonumber \\\\\n&+& \\lambda_2^\\prime {\\rm Tr}\\ \\Phi_2 \\Phi_2^\\dagger \\Phi_2 \\Phi_2^\\dagger\n+\\lambda_2^{\\prime\\prime} {\\rm Tr}\\ \\Phi_2 \\Phi_2^\\dagger\\ {\\rm Tr}\\ \\Phi_2 \\Phi_2^\\dagger\n+ 2\\lambda {\\rm Tr}\\ \\Phi_1 \\Phi_1^\\dagger\\ {\\rm Tr}\\ \\Phi_2 \\Phi_2^\\dagger \\ .\n\\label{Eq:twoscalepot}\n\\end{eqnarray}\nA few comments are in order for this potential. First, the pseudoscalars are all massless in Eq.~(\\ref{Eq:twoscalepot}). In particular, the $\\Pi_i^a$ fields are the Nambu-Goldstone bosons (NGBs) associated to the spontaneous breaking of the $SU(N_i)_1\\times SU(N_i)_2$ or $SU(2N_i)$ techniflavor symmetries. Three of these NGBs become the longitudinal components of the SM $W$ and $Z$ boson, once the electroweak interactions are ``switched on''. The remaining NGBs receive mass through radiative effects and\/or additional new interactions beyond TC, such as Extended TC \\cite{Eichten:1979ah,Dimopoulos:1979es}. These interactions can be accounted for by adding techniflavor-breaking potential terms to Eq.~(\\ref{Eq:twoscalepot}). The $\\Theta_i$ pseudoscalar singlets are massless in Eq.~(\\ref{Eq:twoscalepot}), because of additional and spontaneously broken $U(1)_i$ symmetries. These states acquire mass from instantons -- in the form of ${\\rm Det}\\ \\Phi_i$ invariants to be added to $V$ -- and\/or ETC interactions. Finally, the scalar multiplets $\\Sigma_i^a$ can be made heavier than the singlets (as expected from scaling up the QCD spectrum) by adjusting the quartic terms, and\/or by including higher order invariant terms to the potential. We shall ignore all these issues, as our goal is to highlight the see-saw mechanism for the scalar singlets. This is fully accounted for in the potential of Eq.~(\\ref{Eq:twoscalepot}).\n\nMinimization of the potential gives\n\\begin{equation}\nv_1^2 = \\displaystyle{\\frac{\\mu_1^2\/\\lambda_1-\\lambda\\mu_2^2\/\\lambda_1\\lambda_2}{1-\\lambda^2\/\\lambda_1\\lambda_2}} \\ ,\\quad\nv_2^2 = \\displaystyle{\\frac{\\mu_2^2\/\\lambda_2-\\lambda\\mu_1^2\/\\lambda_1\\lambda_2}{1-\\lambda^2\/\\lambda_1\\lambda_2}}\\ ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\lambda_i\\equiv \\frac{\\lambda_i^\\prime}{2 N_i} + \\lambda_i^{\\prime\\prime}\\ .\n\\end{equation}\nThe isosinglet mass Lagrangian is as in Eq.~(\\ref{Eq:basepotential}), with\n\\begin{equation}\nM_i^2 = 2\\lambda_i v_i^2\\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\delta^2 = \\frac{\\lambda^2}{\\lambda_1\\lambda_2}\\ .\n\\end{equation}\n\nIn an $SU(N_{\\rm TC})$ theory, in the limit of large\n$N_{\\rm TC}$, the double trace terms, in particular the mass mixing term, are subleading in $1\/N_{\\rm TC}$, see {\\it e.g.} \\cite{Manohar:1998xv}. To see this explicitly, consider that the contributions to $\\lambda_i$ and $\\lambda$ are dominated by the diagrams of Fig.~\\ref{Fig:diagrams} (a) and (b), respectively, where the black disks represent scalar insertions. These introduce a normalization factor $1\/\\sqrt{d(R_i)}$, where $d(R_i)$ is the dimension of the representation $R_i$. Recalling that the TC gauge coupling scales like $1\/\\sqrt{N_{\\rm TC}}$, we find the scaling behaviors\n\\begin{equation}\n\\lambda_i \\sim \\frac{1}{N_i d(R_i)}\\ ,\\quad\n\\lambda\\sim \\frac{T(R_1)T(R_2)d(G)}{d(R_1)d(R_2)N_{\\rm TC}^2}\\ ,\n\\end{equation}\nHere $T(R_i)$ is defined by ${\\rm Tr}\\ t_i^a t_i^b=T(R_i)\\delta^{ab}$, where $t_i^a$ are the TC generators in the representation $R_i$, and $d(G)$ is the dimension of the TC group, $N_{\\rm TC}^2-1$. In the large-$N_{\\rm TC}$ limit this gives\n\\begin{equation}\n\\delta^2\\sim\\frac{N_1 N_2 T(R_1)^2 T(R_2)^2}{d(R_1) d(R_2)}\\ ,\n\\label{eq:deltascaling}\n\\end{equation}\nFor example, if $R_1$ is the fundamental and $R_2$ the adjoint representation, then $\\delta^2\\sim N_1 N_2\/N_{\\rm TC}$. The fact that $\\delta^2$ decreases with $N_{\\rm TC}$ was to be expected, as $\\lambda$ arises from a three-loop diagram, and is therefore subdominant in the large-$N_{\\rm TC}$ limit. However, for small values of $N_{\\rm TC}$ we expect $\\delta^2$ to be of order one, as suppression from loop factors is compensated by the large TC coupling. From Eq.~(\\ref{eq:deltascaling}) we also observe that $\\delta^2$ grows with the number of flavors.\n\nIn addition to mass mixing of scalar singlets, we also expect mass mixing of the pseudoscalar and spin-one singlets. These, however, may feature larger diagonal masses than those of the scalar singlets\nWe will not further address this point here.\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=10.0cm]{diagrams.pdf}\n\\end{center}\n\\caption{}\n\\label{Fig:diagrams}\n\\end{figure}\n\n\n\\subsection{Ultra-minimal technicolor}\nThe UMT model~\\cite{Ryttov:2008xe} is a two-scale TC model based on an $SU(2)_{\\rm TC}$ gauge theory, with $N_1=2$ Dirac techniflavors in the fundamental representation, $U$ and $D$, and $N_2=1$ Dirac techniflavor in the adjoint representation, $\\lambda$. The two fundamental technifermions are arranged in a doublet with respect to the weak interactions, whereas the adjoint technifermion is not charged under the electroweak interactions.\nUMT features the smallest contribution to the {\\it perturbative}~\\footnote{By $S_{\\rm pert}$ we mean the computation of $S$ from a loop of technifermions $Q$ with a dynamical mass $m_Q \\gg m_Z$, see {\\it e.g.} the discussion in \\cite{Dietrich:2005jn}.} electroweak $S$ parameter, $S_{\\rm pert}\\simeq 1\/3\\pi$, compatible with electroweak symmetry breaking and near-conformal dynamics~\\cite{Ryttov:2008xe,Ryttov:2009yw}.\n\nIn UMT both fermion species are assumed to condense at roughly the same scale $\\Lambda_1 \\simeq \\Lambda_2 $. In fact it is readily found that $C_{2}(R_1)\\simeq C_{2}(R_2)$, whence $\\alpha_c (R_1) \\simeq \\alpha_c (R_2)$ in the ladder approximation. The technifermion-condensate vevs are $\\langle \\overline{U}_R U_L + \\overline{D}_R D_L \\rangle$ and $\\langle \\lambda^1 \\lambda^2 \\rangle$. The former breaks the electroweak symmetry and produces, among others, an isospin triplet of Goldstone bosons, $\\Pi_1^{1,2,3}$. These become the longitudinal modes of the $W$ and $Z$ boson, which requires $v_1 = v=246$ GeV. Based on the above assumption, we also have $v_2 \\simeq v_1$. The full global symmetry breaking pattern, in the absence of electroweak interactions, is $SU(4)\\times SU(2) \\times U(1) \\to Sp(4) \\times U(1) \\times Z_2$. Notice the extra $U(1)$ symmetry, relative to the general symmetry breaking patterns discussed above. This arises from the fact that a linear combination of the two $U(1)$ symmetries, in $U(4)=SU(4)\\times U(1)$ and $U(2)=SU(2)\\times U(1)$, is anomaly free, whereas in isolation each one of these symmetries is anomalous.\n\nFollowing the above discussion we can describe the scalar sector using a linear realization of the global symmetries in terms of a $4\\times 4$ matrix $\\Phi_1$, and a $2\\times 2$ matrix $\\Phi_2$. Up to dimension-four terms the potential is as in Eq.~(\\ref{Eq:twoscalepot}). This gives nine massless pseudoscalars: $\\Pi_1^{1,2,3,4,5}$ and $\\Pi_2^{1,2}$, corresponding to $SU(4)\\to Sp(4)$ and $SU(2)\\to U(1)$ spontaneous symmetry breaking, respectively, plus the $\\Theta_{1,2}$ pseudoscalar singlets. A linear combination of these is the Goldstone boson corresponding to the $U(1)\\to Z_2$ spontaneous symmetry breaking, whereas the remaining linear combination receives mass from instantons, in the form of higher dimensional terms. The one of lowest order has dimension six:\n\\begin{eqnarray}\n\\left( \\det \\Phi_2 \\right)^2 \\text{Pf}\\ \\Phi_1 + \\text{h.c.} \\ .\n\\label{Eq:LUMT}\n\\end{eqnarray}\nIncidentally this provides an additional mass mixing term for the two scalar singlets.  The UMT model is thus a prime TC example where the lightest scalar mass eigenstate can be as light as 125 GeV, as exemplified by the black curve in Fig.~\\ref{fig:eigenvalues}.\n\nNote however that before mass mixing the scalar masses in the UMT model might be well below the TeV scale due to the argued walking dynamics of the model. For example the mass of the lightest scalar in the UMT model has been estimated to be as low as 250 GeV in Ref.~\\cite{Doff:2009na}.\nThis model computation does not take into account mass mixing between the scalars, but relies on the assumed near-conformality of the theory.\n\\subsection{Low-scale TC}\nIn the two-scale TC framework proposed early on in Ref.~\\cite{Lane:1989ej} the dynamical assumption is that $\\Lambda_1 \\ll \\Lambda_2$ and such models are also referred to as low-scale TC~\\cite{Eichten:2011sh}. This hierarchy in scales requires choosing the representations such that\nthe quadratic Casimirs satisfy $C_2(R_1) \\ll C_2(R_2)$ and\/or lead to a small $\\beta_0(R_1)$~\\footnote{Achieving this typically requires a large number of technifermions in the representation $R_1$, such as the fundamental, or alternatively four-technifermion operators with large enough coefficients~\\cite{Frandsen:2011kt}.}. The low-scale TC assumption that both sectors have the technifermions arranged in weak doublets implies that the electroweak scale $v$ must be related to $v_i$ via $v=\\sqrt{N_1 v_1^2 + N_2 v_2^2}$ to ensure the correct $W$ and $Z$ masses. For non-large values of $N_1$, $v \\simeq \\sqrt{N_2} v_2$.\n\nAgain we can describe the scalar sector using a linear realization of the global symmetries in terms of appropriate matrices of composite fields $\\Phi_i$. The diagonal mass $M_1$ is by construction relatively light \\cite{Delgado:2010bb}, and mass mixing will further reduce its value. The corresponding scenario is similar to the one depicted by the red curves in Fig.~\\ref{fig:eigenvalues}.\n\\section{Summary and discussion}\\label{sec:conclusions}\nIn this paper we have discussed how a light composite scalar may arise in two-scale TC theories via mass mixing between relatively heavy scalar resonances. We have argued that the mass of the light scalar may be compatible with the recently observed $\\sim 125$ GeV resonance and discussed a concrete minimal two-scale TC model, UMT~\\cite{Ryttov:2009yw}, which provides mass mixing of the required order of magnitude. We have also discussed the mechanism in low-scale TC models~\\cite{Lane:1989ej}, where the lightest scalar resonance is expected to be relatively light compared to the TeV scale, already before including effects of mass-mixing. Finally radiative corrections from the interaction with the top quark can further reduce the mass of the lightest scalar resonance~\\cite{Foadi:2012bb}.\n\n\nIt will be interesting to study the phenomenology of TC models featuring a scalar see-saw mechanism in the light of LHC data, already indicating that the couplings of the scalar resonance to the SM fermions and gauge bosons must be SM Higgs-like.\n\\acknowledgements\nWe thank K. Schmidt-Hoberg and T. Ryttov for comments, discussions and suggestions.\nThe work of R.F. is supported by the Marie Curie IIF grant proposal 275012.\nThe work of M.T.F is supported by a Sapere Aude Grant.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCold-atom systems have been regarded as efficient simulators of quantum many-body physics \\cite{BDZ,JSGM}\ndue to its ease of controllability. Research involving ultracold bosonic systems \nhas brought lot of interest in the subject \\cite{AMC1,MOTT}.\nExperiments with optical lattices in three dimensions have revealed \nthe superfluid-Mott insulator (MI-SF) phase transition \\cite{MG}.\nSuch phenomena have been studied in the frameworkof the \nJaynes-Cummings-Hubbard Model (JCHM) \\cite{Na1,Na2,NPA,HASP}.\n\nThe JCHM has been a widely used tool for the investigation of many-body\nsystems describing the interplay between the atom-cavity coupling and the \ninter-cavity hopping of photons \\cite{SB}. In the absence of the hopping term, the model\nreduces to the Jaynes-Cummings Model \\cite{JC,EPJ1} which can be \nexactly solved within the Rotating Wave Approximation. \nWhen the hopping term is non-zero the solution becomes non trivial. \nThe difficulty to find analytical solutions for the model has forced researchers\nto resort to approximation or numerical methods for dealing with such problems \\cite{FWG}.\nRecently, Mering et al. \\cite{AMPK} have proposed an approach in which spin\noperators are mapped to the fermionic ones, hence allowing the application of a Fourier \ntransform that decouples the Hamiltonian into independent\nones, which are associated to each momentum value. \nThe great advantage of this method is the simplicity\nin which physical quantities are found, such as the energies and the chemical potential. \nThe approach presented by Mering et al. includes all classes of Bravais structures. \nHowever, they only derived the results for one-dimensional lattices.\n\nDespite the experiments involving optical structures in three dimensions, \nonly a few theoretical results regarding these lattices in dimensions greater than one is available \nin the current literature \\cite{EPJ2}.\nIt lacks a systematic presentation of the phase diagram of typical Bravais lattices.\nThis is the purpose of the present paper. Here, we investigate the JCHM for different Bravais lattices in one, two and three\ndimensions and analyse the influence of the topology on the MI-SF phase transition. We use the approach\nintroduced by Mering et al. \\cite{AMPK}. \n\nThe paper is organized as follows. In Section II we introduce the JCHM.\nIn Section III we present the fermionic approximation. Results are presented\nin Section IV. Finally, IN Section V, we summarize our\nmain results and conclusions.\n\n\\section{Jaynes-Cummings-Hubbard Model}\nThe JCHM hamiltonian for a lattice of $L$ atoms is given by\n ($\\hbar=1$)\n\\begin{eqnarray}\n\\hat{H} & = \\omega &\\sum_{j}\\hat{a}_{j}^{\\dagger}\\hat{a}_{j}+ \\epsilon \\sum_{j}\n\\hat{\\sigma}_{j}^{\n+}\\hat{\\sigma}_{j}^{-}+g\\sum_{j}(\\hat{a}_{j}^{\\dagger}\\hat{\\sigma}_{j}^{-}+\\hat{a}_{j}\\hat{\\sigma}_{\nj}^{+}) \\nonumber \\\\\n{} & {}\n& - t \\sum_{\\langle ij \\rangle}(\\hat{a}_{i}^{\\dagger}\\hat{a}_{j}+\\hat{a}_{j}^{\\dagger}\\hat{a}_{i}),\n\\label{E1} \n\\end{eqnarray}\nwhere $\\hat{\\sigma}^{\\pm}=\\hat{\\sigma}_{x}\\pm i\\hat{\\sigma}_{y}$ and $\\hat{\\sigma}_{x,y,z}$ are the\nusual Pauli matrices, $\\hat{a}_{j}$ ($\\hat{a}^{\\dagger}_{j}$) is the annihilation (creation) operator\nof the light mode at the $j$th atom, $\\omega$ is the light mode frequency, and \n$\\epsilon$ the atomic transition frequency. The light-atom coupling\nis represented by $g$, $t$ is the hopping integral, and $\\langle ij \\rangle$ denotes pairs\nof nearest-neighbour atoms on the lattice.\n\nWhen $t=0$, the Hamiltonian (\\ref{E1}) is decoupled into $L$\nindependent Jaynes-Cummings model Hamiltonians. In this case the system has well-known eigenstates \\cite{CNA}.\nFor $t \\neq 0$, the atoms becomes coupled thus increasing the complexity of the solution, Since \nwe can not write the eigenstates of the whole system as a direct product of single-cavity eigenstates.\nAs discussed in the introduction, an appropriate approach is the fermionic\napproximation recently introduced by Mering et al. \\cite{AMPK}.\n\n\\section {The Fermionic Approximation} \\label{sec:fapp}\nThe fermionic approximation consists in replacing the spin operators by fermionic ones, i.e.,\n$\\hat{\\sigma}^+$ ($\\hat{\\sigma}^-$) IS replaced by $\\hat{c}^\\dagger$\n($\\hat{c}$). In this framework, we can rewrite Hamiltonian (\\ref{E1}) as\n\\begin{eqnarray}\n\\hat{H} & =\n&\\omega\\sum_{j}\\hat{a}_{j}^{\\dagger}\\hat{a}_{j}+\\epsilon\\sum_{j}\\hat{c}_{j}^{\\dagger}\\hat{c}_{j}\n+g\\sum_{j}(\\hat{a}_{j}^{\\dagger}\\hat{c}_{j}+\\hat{a}_{j}\\hat{c}_{\nj}^{\\dagger}) \\nonumber \\\\\n{} & {}\n& - t \\sum_{\\langle ij \\rangle}(\\hat{a}_{i}^{\\dagger}\\hat{a}_{j}+\\hat{a}_{j}^{\\dagger}\\hat{a}_{i}) .\n \\label{E1a} \n\\end{eqnarray}\nThis approximation allows to solve the model exactly, by means of a Fourier transformation.\nFor $t=0$, spin and fermionic operators are equivalent and then the approximation becomes exact.\nTherefore, for small values of $t$ this approach turns out to be very accurate when dealing\nwith the JCHM \\cite{AMPK}.  \n \nNow we apply a Fourier transform to the fermionic and bosonic operators as \n\\begin{equation}\n\\hat{a}_{j}=\\frac{1}{\\sqrt{L}}\\sum_{\\vec{k}}e^{-2\\pi i \\frac{\\vec{k}\\vec{R}_j}{L}}\\hat{a}_{\\vec{k}}, \\\\\n\\hat{c}_{j}=\\frac{1}{\\sqrt{L}}\\sum_{\\vec{k}}e^{-2\\pi i \\frac{\\vec{k}\\vec{R}_j}{L}}\\hat{c}_{\\vec{k}}, \\label{E2} \n\\end{equation}\nthen the Hamiltonian can be written as\n\\begin{equation}\n\\hat{H}=\\sum_{\\vec{k}} \\left[ \\omega_{\\vec{k}}\\hat{a}_{\\vec{k}}^{\\dagger}\\hat{a}_{\\vec{k}}+g (\\hat{a}_{\\vec{k}}^{\\dagger}\n\\hat{c}_{\\vec{k}}+\\hat{a}_{\\vec{k}}\\hat{c}_{\\vec{k}}^{\\dagger}) +\\epsilon \\hat{c}_{\\vec{k}}^{\\dagger}\\hat{c}_{\\vec{k}} \\right],\n\\label{E3} \n\\end{equation}\nwhere $\\omega_{\\vec{k}}= \\omega$-$\\nu_{\\vec{k}}$ and $\\nu_{\\vec{k}}$ is the dispersion relation of the Bravais lattice. \n\nThe Hamiltonian (\\ref{E3}) corresponds to a sum of $L$ independent Hamiltonians $\\hat{H}_{\\vec{k}}$\n($\\hat{H}=\\sum_{\\vec{k}} \\hat{H}_{\\vec{k}}$), where each of them\nis associated with a particular momentum $\\vec{k}$. The ground-state energy\nof $\\hat{H}_{\\vec{k}}$ is given by\n\\begin{eqnarray}\nE^{n_{\\vec{k}}}_{\\vec{k}} & = & (1-\\delta_{n_{\\vec{k}}0}) \\left[ n_{\\vec{k}}\\omega_{\\vec{k}}+\n\\frac{\\Delta+\\nu_{\\vec{k}}}{2}+ \\right. \\nonumber \\\\ \n{\\ } & {\\ } & \\left. -\\frac{1}{2}\\sqrt{(\\Delta+\\nu_{\\vec{k}})^2+4n_{\\vec{k}}g^2} \\right],\n\\end{eqnarray}\nwhere the superscript denotes the excitation number and \n$\\Delta \\equiv \\epsilon - \\omega$ is the detuning between atomic trantition and light frequencies.\nNotice that $\\hat{n}_{\\vec{k}}$ commutes with $\\hat{H}_{\\vec{k}}$.\nFor a total excitation number $N$ ($N= \\sum_{\\vec{k}} n_{\\vec{k}}$) we have a particular configuration \n$\\{ n_{\\vec{k_1}},n_{\\vec{k_2}},n_{\\vec{k_3}},...\\}$ that minimizes the total ground-state energy, $\\sum_{\\vec{k}}\nE^{n_{\\vec{k}}}_{\\vec{k}}$. For $t \\ll 1$, it is easy to see that\nthis configuration is $\\{ n,n,n,...\\}$ corresponding to the Mott insulator state. By increasing $t$, a quantum phase transition takes phace\nand the system is driven to a superfluid state. Since $n$ is constant, the phase boundaries of the Mott lobes \nare $n$ dependent. Thus, the $n$th Mott lobe is obtained through an analysis of the particles chemical potential,\n$\\mu^{+}=E^{n+1}_{\\vec{k}'}-E^{n}_{\\vec{k}'}$, and the hole one, $\\mu^{-}=E^{n}_{\\vec{k}}-E^{n-1}_{\\vec{k}}$ \\cite{AMPK,GAS}, where\n$\\vec{k}'$ and $\\vec{k}$ are, respectively, the values \u200b\u200bthat minimize and maximize these potentials. For $\\mu^{+}=\\mu^{-}$, the Mott lobe is\nclosed at the critical hopping, $t_c$, hence describing the MI-SF transition.\n\n\\section{Results}\nIn order to analyse the influence of Bravais lattices topology on the MI-SF transition, we study the one-dimensional (1D), square (SQ),\nsimple cubic (SC), body-centered cubic (BCC) and face-centered cubic (FCC) lattices. The dispersion relations $\\nu_{\\vec{k}}$ are,\nrespectively, given by \\cite{kitel}\n\\begin{equation}\n\\nu_{k}^{(1D)}=-2t\\cos(ka)\n\\end{equation}\n\\begin{equation}\n\\nu_{k_x,k_y}^{(SQ)} =-2t [\\cos(k_x a)+\\cos(k_y a) ]\n\\end{equation}\n\\begin{equation}\n\\nu_{k_x,k_y,k_z}^{(SC)}=-2t [\\cos(k_x a)+\\cos(k_y a)+\\cos(k_z a)]\n\\end{equation}\n\\begin{equation}\n\\nu_{k_x,k_y,k_z}^{(BCC)}=-8t [\\cos(k_x a)\\cos(k_y a)\\cos(k_z a)]\n\\end{equation}\nand\n\\begin{eqnarray}\n\\nu_{k_x,k_y,k_z}^{(FCC)} & = & -4t [\\cos(k_x a)\\cos(k_y a)+\\cos(k_x a)\\cos(k_z a)+ \\nonumber \\\\\n{\\ } & {\\ } & \\cos(k_y a)\\cos(k_z a)], \n\\end{eqnarray}\nwhere $a$ is the lattice constant. For each structure we found the momentum vectors that maximizes\nthe hole chemical potentials and minimizes the particle ones in order to obtain $\\mu^{-}$, $\\mu^{+}$, \nand consequently the Mott phase boundary.\n\n\\begin{figure}\n\\includegraphics[width=.45\\textwidth]{fig1}\n\\caption{First three Mott lobes for different lattices. Inside the lobes we have a Mott insulator state, \nwhile outside the system is in a superfluid state.}\n\\label{fig1}\n\\end{figure}\n\nFigure \\ref{fig1} shows the first three Mott lobes for $\\omega \/g=1$ and $\\Delta=0$, for the considered lattices. \nWe see that as the number of lattice neighbors increases, the MI phase region decreases.\nThis result is expected since the probability of ohoton tunneling increases for higher \nnumber of nearest-neighbors.\nWe observe that for $d$-dimensional hypercubic lattices the lobes can be rescaled by\n$t_d =t\/d$ causing the collapse into a single curve. \nThe shape of the Mott lobes are equal for bipartite lattices, where a bipartite structure is such one \nthat we can decompose into two substructures, with all nearest-neighbour sites shared between each other.\nThe FCC lattice is non-bipartite, Hence displaying a different behaviour. Thus\nwe propose to make a detailed comparative analysis between the FCC and SC lattices representing, respectively, the non-bipartite\nand bipartite classes. \n\n\\begin{figure}\n  \\subfigure[]{\n     \\includegraphics[width=0.45\\textwidth]{fig2a}\\label{fig2a}}\n  \\subfigure[]{\n     \\includegraphics[width=0.45\\textwidth]{fig2b}\\label{fig2b}}\n  \\caption{Mott lobe for $n=1$ and typical values of detuning for (a) SC and (b) FCC lattices.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n  \\subfigure[]{\n     \\includegraphics[width=0.45\\textwidth]{fig3a}\\label{fig3a}}\n  \\subfigure[]{\n     \\includegraphics[width=0.45\\textwidth]{fig3b}\\label{fig3b}}\n  \\caption{Relationship between critical hopping and detuning on the (a) SC and (b) FCC lattices for varying excitation number.} \\label{fig3}\n\\end{figure}\n\nThe first Mott lobe ($n=1$) on the SC and FCC lattice is show in figure \\ref{fig2} for typical detuning values.\nWe see that both lattices have a similar MI-SF phase transition frame. However, the FCC always has a smaller MI phase region.\nFor both lattices, the MI phase region decreases for increasing detuning.\nFigure \\ref{fig3} shows the critical hopping $t_c$ as a function of the detuning. While at the first lobe, $t_c$ decreases when $\\Delta$ increases, in the other lobes ($n > 1$) \nwe observe that the critical hopping reaches a maximum at $\\Delta = \\Delta_m$. \nFor this particular detuning value, when $n$ increases the critical hopping decreases, as predicted in \\cite{AMPK}. \nThis behavior is present in both  structures.\nFigure \\ref{fig4} shows the critical hopping as a function of $n$ for $\\Delta=0$. \nThe properties expressed by both lattices are again qualitatively equivalent where there\nis only one gap between the two curves. \nFigure \\ref{fig5} presents the detuning values corresponding to the maximal critical hopping as a function of\n$n$. We can observe that, except for the $n=1$ case, where the $t_c$ maximum is associated with the detuning minimum,\nThe critical hopping correspondent detuning is approximately null. \n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{fig4}\n\\caption{Critical hopping for the SC and FCC lattices in terms of $n$ for null detuning.}\n\\label{fig4}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{fig5}\n\\caption{Detuning versus $n$ for maximal critical hopping on the SC and FCC lattices.}\n\\label{fig5}\n\\end{figure}\n\nBy performing an asymptotic approach to large excitation number, $n \\gg 1$, we can find the dominant term of the critical hopping which is given by\n\\begin{equation}\n t_c = \\frac{g}{16 \\tilde{d} n^{3\/2}} + {\\cal O} (n^{-3}), \\label{tc.largen}\n\\end{equation}\nwhere we obtain $\\tilde{d}=4$ for the FCC and BCC lattices where it corresponds to the hypercubic lattices dimentions, i. e., 1 for linear, 2 for\nsquare, and 3 for the SC lattice. It is important to emphasize that at the large-$n$ regime, the detuning \nand the topology class (bipartite or non-bipartite) influence on the critical hopping $t_c$ is suppressed.  \nFigure \\ref{fig6} confirms the prediction of the equation (\\ref{tc.largen}). It shows that the\nexact results are in excellent agreement with the asymptotic ones for $n>4$.\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{fig6}\n\\caption{Critical hopping versus $n$ for various lattices at the large $n$ regime. The symbols are related to \nnumerically obtained results for\n$\\Delta \/ g = -0.5, 0,$ and $0.5$, where each detuning value produces the same result with errors smaller than the symbol size.\nThe solid line represents the asymptotic analytical result given by means of equation (\\ref{tc.largen}).}\n\\label{fig6}\n\\end{figure}\n\n\\section{Conclusions}\nWe have studied the properties of the MI-SF phase transition for the Jaynes-Cummings-Hubbard model over several Bravais lattices\nby means of the fermionic approximation. We find that the transition parameters of hypercubic lattices are scalable, except for the FCC lattice,\nsince it is non-bipartite. \nThe Mott lobes for the SC and FCC lattices show a similar detuning dependence\nhaving only a quantitative difference which is suppressed as the detuning increases. \nAn analogous feature is observed in $t_c$ vs. $n$ for null\ndelta and in $\\Delta_m$ vs. $n$ where treir quantitative difference tends to be smaller as $n$\nincreases. \nFurthermore, we observed that not only the number of neighbors influences the\nMI-SF phase transition but also does the lattice topology.\n \nThe FCC lattice shows a behavior quantitatively different and non-scalable from bipartite lattices. \nOn the other hand, asymptotic\nresults for large excitation number indicate an universality on $t_c$ because it obeys a power law in $n$ which does not depend on\n$\\Delta$ and topology associated parameter can be rescaled for different classes through a multiplication of $t_c$ by an effective parameter\nthat corresponds to the dimension, for hypercubic, and to 4, for BCC and FCC lattices.\n\n\\section*{ACKNOWLEDGMENTS}\nThis work was partially supported by CNPq, CAPES and FAPITEC\/SE (Brazilian Research Funding Agencies).\n\n\n\\section{Introduction}\n\\label{intro}\nYour text comes here. Separate text sections with\n\\section{Section title}\n\\label{sec:1}\nand \\cite{RefJ}\n\\subsection{Subsection title}\n\\label{sec:2}\nas required. Don't forget to give each section\nand subsection a unique label (see Sect.~\\ref{sec:1}).\n\\begin{figure}\n\\resizebox{0.75\\textwidth}{!}{%\n  \\includegraphics{leer.eps}\n}\n\\caption{Please write your figure caption here}\n\\label{fig:1}      \n\\end{figure}\n\\begin{figure*}\n\\vspace*{5cm}      \n\\caption{Please write your figure caption here}\n\\label{fig:2}      \n\\end{figure*}\n\\begin{table}\n\\caption{Please write your table caption here}\n\\label{tab:1}      \n\\begin{tabular}{lll}\n\\hline\\noalign{\\smallskip}\nfirst & second & third  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nnumber & number & number \\\\\nnumber & number & number \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\vspace*{5cm} \n\\end{table}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWhen deep transformer models make mistakes, ML engineers have had little\nrecourse but to collect a better training set and hope the problem is fixed.\nAdversarial datasets have exposed a variety of phenomena under which\nmodels trained on common datasets fail, particularly for question answering\nand natural language inference \\citep{jia-liang-2017-adversarial, gururangan-etal-2018-annotation, kim-etal-2018-teaching, mccoy-etal-2019-right, nie-etal-2020-adversarial, thorne-etal-2019-fever2}.\nThey have provided new test data to expose problems but not always\nnew training data to correct them.\nRecently, the natural language processing community has adopted methodologies\ninspired by software development for probing and testing the capabilities\nof a model.\n\\citet{ribeiro-etal-2020-beyond} introduce CheckList,\nwhich helps users to develop test suites of examples, organized by capability.\n\nCollecting hundreds or thousands of examples for each error phenomenon is\nslow, expensive, and not always feasible.  In this paper, we investigate\nhow just a few examples of a phenomenon (``debugging examples'', which were\nnot in the original dataset)\ncan be utilized to correct a model.\nThe goal is higher accuracy on the phenomenon (``debugging accuracy'')\nwhile retaining accuracy\non the original dataset (``original accuracy'').  This problem differs\nfrom domain adaptation and\nfew-shot learning because performance must be maintained on original\nexamples, and no new classes are introduced.\n\nWe repurpose published test suites for several natural language understanding\n(NLU) tasks as debugging problems,\nnot just diagnostics.  We identify methods that can update a model using\na few debugging examples without the expense of iterating over the whole\noriginal training set.  We introduce a new fast method that samples\nin-danger examples from the original training set to obtain even better\noriginal accuracy for comparable debugging accuracy.\n\n\n\\section{Related work}\n\nTwo recent works \\citep{zhu, de-cao-etal-2021-editing} study how to modify transformer\nlanguage models so that they store updated facts, testing their\napproaches on downstream tasks such as zero-shot relation extraction\nand closed-book fact checking.  To apply these methods,\none is given a modified fact as an example to train on, and one must\npredict the modified fact correctly (success rate) while achieving low\nperformance deterioration on the original test set.\nBecause success rate is measured on just one example which is\navailable at training time, to determine whether the update really\ngeneralizes,\n\\citet{de-cao-etal-2021-editing} also measures {\\em equivalence accuracy}, which reflects\naccuracy on paraphrases of the updated fact.\n\nBy contrast, our setting provides ten examples (not just one) for a\nphenomenon where the predictions are to be updated.\nThe phenomenon being debugged may involve deeper semantics than\na factoid update, which usually requires only a reassociation of\nparticular words that appear in the example.\nWe assume we are given a testing set for the phenomenon, so\nwe can measure generalization\nby directly measuring accuracy on the testing examples\ninstead of paraphrasing the training examples.\n\nDespite these differences, ideas from these papers provide relevant\nideas that can be used in our debugging setting as well.\nOne baseline considered by \\citet{zhu}, which we call {\\em intensive\nfine-tuning}, simply takes the updated facts (for us, the debugging\ntraining set) and repeatedly performs gradient descent updates\non them until they are classified correctly.\n\nThe proposed approach of \\citet{zhu} is to minimize loss on the updated\nfacts (the debugging set) subject to either an $L^\\infty$ or $L^2$ constraint\non the difference of the model parameters.  We consider these as baselines.\n\nAs \\citet{de-cao-etal-2021-editing} observe, constraining the norm of the parameter update\nis only loosely tied to how a parameter change can affect the output of\na model.  For this reason they introduce an approach based on constraining\nthe Kullback-Leibler divergence between the updated model and the original.\nTheir proposed method trains a hypernetwork to read a single updated example\nand make a change minimizing debugging loss subject to the Kullback-Leibler\ndivergence constraint.  That does not apply as well to our scenario\nof multiple debugging examples, but we borrow the idea of using\nKullback-Leibler divergence to incentivize similar predictions in a\nmore straightforward baseline.\n\n\\citet{sinitsin} introduce a meta-learning method for making a model\nthat will preserve original accuracy when performing a series of gradient\ndescent steps to change the label of any particular example.\nWe are interested in methods that can be applied to any model,\nand for real debugging it is not necessary that all examples be\neasily relabeled.\n\nContemporaneously to our work, \\citet{pasunuru-etal-2021-continual}\ninvestigate few-shot debugging on error categories that are apparently\ntoo broad to be corrected with just a few examples.\nAlthough they report some success with feature matching methods such as\nprototypical networks \\citep{snell}, they either\nsuppose that test examples are identified as needing a correction or not\n(i.e. debugging or original), more like domain adaptation, or else\ntrain the prototypical network on a combined training set, which is the\nslowness we are trying to avoid.  Our setting requires a single model that can\nbe applied to all examples without source information.\n\n\\section{Method}\n\n\\begin{table*}[htb]\n\\begin{center}\n\\begin{tabular}{p{1.1in}p{.8in}p{.8in}p{.8in}p{.8in}p{.8in}}\n\\hline\nTest suite & Dog & Or\/And & Becoming & People & Passive \\\\\n\\hline\nBefore debugging & (.000, .913) & (.000, .913) & (.002, .913) & (.005, .913) & (.009, .913) \\\\\n\\hline\n{\\em Fast} & & & & & \\\\\nDebug only & (.731, .909) & (1.000, .909) & (1.000, .910) & (.922, .910) & (.819, .910) \\\\\n$L^2$ ($\\delta=.1$) & (.704, .909) & (1.000, .909) & (1.000, .911) & (.880, .910) & (.876, .910) \\\\\n$L^\\infty$ ($\\delta=.1$) & (.704, .909) & (1.000, .909) & (1.000, .911) & (.880, .910) & (.876, .910) \\\\\nK-L ($\\lambda=10$) & (1.000, .905) & (1.000, .908) & (1.000, .909) & ( 1.000, .908) & (1.000, .908) \\\\\nOurs & (.731, .909) & (.994, .913) & (1.000, .913) & (.993, .911) & (.975, .912) \\\\\n\\hline\n{\\em Slow} & & & & & \\\\\nMixed in & (1.000, .913) & (.999, .912) & (1.000, .913) & (.933, .912) & (.859, .912) \\\\\nOversampling & (1.000, .911) & (1.000, .913) & (1.000, .912) & (.999, .914) & (1.000, .911) \\\\\n\\hline\n\\end{tabular}\n\\caption{(Debugging accuracy, Original accuracy) on CheckList test suites for\nQQP.}\n\\label{tbl:qqp}\n\\end{center}\n\\end{table*}\n\n\\begin{table*}[htb]\n\\begin{center}\n\\begin{tabular}{p{1.1in}p{1.3in}p{1.3in}p{1.3in}}\n\\hline\nTest suite & Used to but now & Negation with neutral & Opinion matters \\\\\n\\hline\nBefore debugging & (.793, .925) & (.448, .925) & (.616, .925) \\\\\n\\hline\n{\\em Fast} & & & \\\\\nDebug only & (.860, .914) & (1.000, .917) & (.602, .915) \\\\\n$L^2$ ($\\delta=.1$) & (.860, .915) & (1.000, .919) & (.600, .915) \\\\\n$L^\\infty$ ($\\delta=.1$) & (.860, .915) & (1.000, .919) & (.600, .915) \\\\\nK-L ($\\lambda=10$) & (.838, .915) & (1.000, .916) & (.538, .920) \\\\\nOurs & (.877, .919) & (1.000, .913) & (.777, .885) \\\\\n\\hline\n{\\em Slow} & & & \\\\\nMixed in & (.909, .913) & (1.000, .925) & (.673, .923) \\\\\nOversampling & (.735, .931) & (1.000, .921) & (.512, .928) \\\\\n\\hline\n\\end{tabular}\n\\caption{(Debugging accuracy, Original accuracy) on CheckList test suites for\nSST-2.}\n\\label{tbl:sst2}\n\\end{center}\n\\end{table*}\n\n\\begin{table*}[htb]\n\\begin{center}\n\\begin{tabular}{p{1.1in}p{.8in}p{.8in}p{.8in}p{.8in}p{.8in}}\n\\hline\nTest suite & After If & P. Participle & Disjunction & Passive & NP\/S \\\\\n\\hline\nBefore debugging & (.000, .838) & (.001, .838) & (.005, .838) & (.004, .838) & (.006, .838) \\\\\n\\hline\n{\\em Fast} & & & & & \\\\\nDebug only & (1.000, .813) & (1.000, .804) & (1.000, .807) & (.929, .827) & (1.000, .811) \\\\\n$L^2$ ($\\delta=.1$) & (.999, .816) & (.999, .810) & (.999, .812) & (.933, .827) & (.999, .817) \\\\\n$L^\\infty$ ($\\delta=.1$) & (1.000, .812) & (1.000, .804) & (1.000, .807) & (.933, .827) & (1.000, .811) \\\\\nK-L ($\\lambda=10$) & (1.000, .825) & (1.000, .820) & (1.000, .822) & (1.000, .824) & (1.000, .826) \\\\\nOurs & (1.000, .841) & (.926, .835) & (1.000, .836) & (.994, .832) & (.939, .842) \\\\\n\\hline\n{\\em Slow} & & & & & \\\\\nMixed in & (.468, .835) & (.114, .833) & (.344, .837) & (.791, .835) & (.298, .837) \\\\\nOversampling & (.920, .836) & (.992, .837) & (1.000, .838) & (.869, .837) & (1.000, .833) \\\\\n\\hline\n\\end{tabular}\n\\caption{(Debugging accuracy, Original accuracy) on HANS test suites for\nMNLI.}\n\\label{tbl:hans}\n\\end{center}\n\\end{table*}\n\nWe suppose we are given a model $p_\\theta (x,y)$ trained on\ntraining set $X$.  We are also given\ndebugging training set $X^\\prime$, and original\ntest set $X_{test}$ and debugging test set $X^\\prime_{test}$.\nThese four sets are pairwise disjoint.\nWe consider the cross-entropy loss\n\\begin{equation}\n\\mathcal{L} (x, y; \\theta) = - p_\\theta (x, y) \\log p_\\theta (x, y) \\ldotp\n\\end{equation}\nOur method initializes $\\theta_0 = \\theta$ and then performs\n{\\em intensive fine-tuning} on the debugging set $X^\\prime$, by\nperforming Adam \\citep{adam}\niterations $\\theta_{t+1} = Adam(\\mathcal{L}, X^\\prime, \\theta_t)$\nwhere $Adam(\\mathcal{L}, S, \\theta)$ represents the parameter update\nachieved by training $\\theta$ with respect to the loss $\\mathcal{L}$ over\na complete epoch on $S$.\nIntensive fine-tuning stops at the minimal step $t = t_{X^\\prime}$ such that\n$\\mbox{argmax}_y p_{\\theta_t} (x_i, y) = y_i$ for all\n$(x_i, y_i) \\in X^\\prime$.\nWe write $\\theta_{X^\\prime} = \\theta_{t_{X^\\prime}}$.\n\nNext we collect random samples $W \\subset X$ that are misclassified\nby $\\theta_{X^\\prime}$ but not by $\\theta$.  In our experiments we select\n$\\abs{W} = 2 \\abs{X^\\prime}$ such examples.  Collecting $W$ is a fast\nprocess involving iterating through a random shuffle of $X$ and stopping\nwhen the required number of examples is retrieved.  The expected iteration\ntime depends only on the error rates and correlation of the errors of the\nmodels and not on the size of the original training set $\\abs{X}$.\n\nFinally we restart from the original parameters $\\theta$ and intensively\nfine-tune using the set $X^\\prime \\cup W$.  We take $\\theta^\\prime_0 = \\theta$\nand iterate Adam\n\\begin{equation}\n\\theta^\\prime_{t+1} = Adam(\\mathcal{L}, X^\\prime \\cup W, \\theta^\\prime_t)\n\\end{equation}\nuntil we reach $t^\\prime$ where\n$\\mbox{argmax}_y p_{\\theta^\\prime_{t^\\prime}} (x_i, y) = y_i$ for all\n$(x_i, y_i) \\in X^\\prime \\cup W$.\nThe resulting $\\theta^\\prime = \\theta^\\prime_{t^\\prime}$\nis the debugged model by our proposed method.\n\n\n\\section{Experiments}\n\nWe consider a BERT base model \\citep{devlin-etal-2019-bert} implemented in Pytorch\n\\citep{pytorch} by the HuggingFace Transformers library \\citep{wolf-etal-2020-transformers}\nfor all experiments, with batch size 16 per GPU on 3 or 4 GPU's,\notherwise following default training parameters.\n\nOur data sets are test suites from\nHANS \\citep{mccoy-etal-2019-right} debugging an MNLI model\n\\citep{williams-etal-2018-broad} and CheckList \\citep{ribeiro-etal-2020-beyond}\ndebugging models for SST-2 and QQP from GLUE \\citep{wang-etal-2018-glue}.\nWe take test cases with the worst accuracy before debugging, and select 10\nexamples from each suite for debugging ($X^\\prime$) and use the rest\n(e.g. 990 examples for HANS) to test debugging ($X^\\prime_{test}$).\nSee the appendix for details.\nOur data splits and our code for extracting examples\nfrom CheckList are available for download.\\footnote{https:\/\/github.com\/necla-ml\/debug-test-suites}\nFor HANS we use the BERT cased model and for CheckList we use the uncased model.\n\n\\subsection{Fast baselines}\n\nThe first of four fast baselines we consider, which is labeled ``debug only,'' performs\nintensive fine-tuning on the debugging set $X^\\prime$ only, returning the\nmodel $\\theta_{X^\\prime}$.  In every case we tested, $t_{X^\\prime} \\leq 3$\nepochs over ten examples, so this completed within a minute.\n\nThe next baselines from \\citet{zhu} are\nfinding $\\theta^\\prime$ to minimize $\\mathcal{L}(X^\\prime, \\theta^\\prime)$\nsubject to an $L^\\infty$ constraint\n${\\norm{\\theta^\\prime - \\theta}}_\\infty < \\delta$ or an $L^2$ constraint\n${\\norm{\\theta^\\prime - \\theta}}_2 < \\delta$.  Following \\citet{zhu} we use\n$\\delta = 0.1$ and implement the optimization as projected gradient descent,\n{\\em e.g.} for $L^\\infty$, taking a gradient descent step\nfrom $\\theta_0$ to $\\theta$\nand projecting the updated parameters back into the $L^{\\infty}$ ball as\n\\begin{equation}\n\\theta_0 + \\min ( \\max (\\theta - \\theta_0, - \\delta), \\delta)\n\\end{equation}\nlimiting the excursion in any coordinate to $\\pm \\delta$.\n\nThe fourth baseline we consider introduces a Kullback-Leibler divergence\non randomly sampled examples from $X$ into the loss:\n\\begin{equation}\n\\mathcal{L}^\\prime (\\theta^\\prime) = \\mathcal{L}(X^\\prime ; \\theta^\\prime) + \\lambda \\mathcal{L}_{KL} (X ; \\theta^\\prime)\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathcal{L}_{KL} (X; \\theta^\\prime) = \\sum_{(x,y) \\in X} \\sum_{y^\\prime}\np_\\theta (x, y^\\prime) \\log \\frac{p_\\theta (x, y^\\prime)}{p_{\\theta^\\prime} (x, y^\\prime)}\n\\end{equation}\nIn practice, $\\mathcal{L}_{KL} (X; \\theta^\\prime)$ is estimated on\nminibatches from $X$ simultaneously with selecting a minibatch of the same\nsize from $X^\\prime$.\n\nTraining on each of these baselines stops when we reach $t^\\prime$ where\n$\\mbox{argmax}_y p_{\\theta^\\prime_{t^\\prime}} (x_i, y) = y_i$ for all\n$(x_i, y_i) \\in X^\\prime$.  In experiments, this always happens within\nthree epochs over $X^\\prime$ .\n\n\\subsection{Slow baselines}\n\nOur first slow baseline is simply to train the model starting with the original\nBERT base parameters for three full epochs on randomly shuffled\n$X^\\prime \\cup X$, without accounting for the difference in\nsize $\\abs{X^\\prime} << \\abs{X}$.  We call this ``mixed in'' training.\n\nOur second baseline (``oversampling'')\nequally weights $X^\\prime$ and $X$ in the training.\nIt starts with original BERT base parameters and trains for three full epochs\nover $X$, each time taking a batch consisting half of examples from $X$\nand half of examples from $X^\\prime$, interleaved.  Although the $X$ samples\nare sampled without replacement, the $X^\\prime$ samples are replaced and\nare each seen many times.\n\n\\subsection{Results}\n\nWe consider the CheckList and HANS test suites for QQP, SST-2, and MNLI\ntogether (Tables~\\ref{tbl:qqp},~\\ref{tbl:sst2}, and~\\ref{tbl:hans}).\nAmong fast methods, our method has the highest\noriginal accuracy in 11 out of 13 subcases and the highest debugging\naccuracy in 6 out of 13.  This makes it a better choice for retaining\noriginal accuracy out of several fast, good methods for improving debugging\naccuracy.\nKullback-Leibler divergence, which ranks first most often among fast methods \nin debugging accuracy, only ranks first in original accuracy once out of\n13 subcases.  Notably, both methods frequently outperform the debug only\napproach in debugging accuracy, showing that\nsampling non-debugging examples helps achieve an update\nthat generalizes better even on the debugging phenomenon.\n\nConsidering slow methods, oversampling achieves maximal debugging\naccuracy on 8 of 13 subcases and best original accuracy on 8 of 13.\nOn HANS, mixing the debugging examples into the full training set is not\nsufficient for them to be learned, though this method achieves\nreasonable debugging accuracy on the other datasets.\n\n\\begin{figure}[htb]\n\\includegraphics[width=3in,height=2.2in]{original-acc} \n\\includegraphics[width=3in,height=2.2in]{debugging-acc}\n\\caption{Comparing\nour method to debug-only intensive fine-tuning for\ndifferent numbers of shots.}\n\\label{fig:bothfig}\n\\end{figure}\n\n{\\bf Number of shots and stability.}\nBesides the 10 shot setting described above, we compare our method to\n``debug only'' intensive fine-tuning for 5 shots and 20 shots.\nResults are shown for HANS's \\verb+cn_after_if_clause+ test suite\nin Figure~\\ref{fig:bothfig}.  Each experiment is repeated, sampling\neight different sets of debugging and in-danger examples.\nThe standard deviation in accuracy over the samples is indicated by\nthe error bars around each mean result in the figure.\n\nFive shots is too few to be sure of good debugging accuracy.\nOur method achieves significantly higher debugging accuracy and original\naccuracy, compared to intensive fine-tuning, with ten or twenty shots.\nWith twenty shots the debug only method loses original accuracy, possibly\ndue to the tightened constraints of classifying more debugging examples\ncorrectly.\n\n{\\bf Other base models.}  We repeat 10-shot experiments using Electra\n\\citep{electra}\ninstead of BERT.  Using Electra, our method has the highest original\naccuracy among fast methods in 7 out of 13 subcases and the highest\ndebugging accuracy in 8 out of 13.\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{lr}\n\\hline\nMethod & Seconds \\\\\n\\hline\n{\\em Fast} & \\\\\nDebug only & 10.89 \\\\\n$L^2$ & 14.74 \\\\\n$L^\\infty$ & 15.85 \\\\\nK-L & 14.79 \\\\\nOurs - {\\em total} & 25.29 \\\\\n\\, {\\em debug-only fine-tuning} & 10.89 \\\\\n\\, {\\em finding new misclassifications $W$} & 2.86 \\\\\n\\, {\\em final fine-tuning} & 11.54 \\\\\n\\hline\n{\\em Slow} & \\\\\nMixed in & 12663.14 \\\\\nOversampling ({\\em estimated}) & 25326.28 \\\\\n\\hline\n\\end{tabular}\n\\caption{Model debugging time in seconds.}\n\\label{tbl:time}\n\\end{center}\n\\end{table}\n\n{\\bf Time.}  \nIntensive fine-tuning usually finishes after a few small batches,\nbut collecting the 20 misclassified examples potentially can require\nmore evaluations.\nOn QQP these can be found in 1\/60 of an epoch (forward only)\nand at worst (on ``negation with neutral'' of SST-2) in 1\/5,\nyielding roughly 720x and 60x speedups over oversampling (three\nepochs, forward and back, alternating with debugging examples), respectively.\n\nIn Table~\\ref{tbl:time} we collect total timings for each debugging procedure\non HANS's \\verb+cn_after_if_clause+ test suite,\nincluding the time our method needs to collect the new misclassifications $W$\nfrom the original MNLI training set.  Whereas the slow methods require hours\nto update the model, all the fast methods finish in a matter of seconds.\n\n\\section{Conclusion}\n\nWe study the new problem of few-shot debugging natural language\nunderstanding problems on narrowly defined test suites,\naddressing a real-life need not addressed by past benchmark datasets.\nIntensive fine-tuning on debugging examples with a few newly misclassified\nexamples is substantially faster than full epoch retraining, and retains\nsuperior accuracy on the original dataset in more of our tests than any other\nfast method, for competitive debugging accuracy.  Kullback-Leibler\nregularization may achieve better debugging accuracy, but its original\naccuracy is lagging, probably because it samples randomly rather\nthan focusing on the\nnewly misclassified examples that the debugging examples are opposed to.\nOur results suggest a way for practitioners to quickly address problems\nin deployed systems and inspire the search for more refined ways of\nusing debugging information.\n\nTo further this research, there is a need for test suites that are not\nconstructed by templates, so that the debugging phenomena are less easily\nlearned, and yet not too broad to be taught in the few-shot setting.\nThis limitation forced us to focus on relatively small differences\nin accuracy.  Because our method requires only a few debugging examples,\nit should be practical to construct test suites by hand or by manually\norganizing existing misclassifications.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introduction}\nQuantum mechanics is one of the most successful physical theories human kind has ever formulated. Nonetheless, its interpretation and range of validity eludes our full grasping. One of the basic features of quantum physics is the superposition principle which, when applied to the macroscopic world, leads to counter-intuitive states akin to the celebrated Sch\\\"odinger's cat. While \nmodels beyond quantum mechanics, challenging some of its interpretational issues, have been formulated in its early days, testing the predictions of the theory when applied to the macroscopic world has proven to be a tall order. The main reason for this is the intrinsic difficulty in isolating large systems from their environment. \n\nSpace offers a potentially attractive arena for such an endeavour, promising the possibility to create and verify the quantum properties of macroscopic superpositions far beyond current Earth-based capabilities~\\cite{kaltenbaek2012macroscopic,kaltenbaek2016macroscopic,2018cosp...42E1659K,QPPF}. In this work, we focus on the efforts to test the boundaries of quantum physics in space employing nanoparticles, which are one of the best suited candidate for quantum superpositions of high-mass objects. \\MP{It should be noticed that, while we will focus on testing quantum physics, large spatial superpositions of massive systems are bound to be \\MP{sensitive probes for many other} physical phenomena, from dark matter and dark energy searches~\\cite{riedel2013direct, bateman2015existence, riedel2017decoherence, carney2019ultralight, carney2020mechanical, monteiro2020search,khoury2004chameleon, rider2016search, moore2014search} to gravimetry and Earth observation applications~\\cite{qvarfort2018gravimetry,hebestreit2018sensing}.}\n\nWe delve into the possibilities offered by the state-of-the-art of nanoparticle physics projected in the space environment. In doing so, we offer an ab-initio estimate of the potential of space-based interferometry with some of the largest systems ever considered and show that there is room for testing quantum mechanics at an unprecedented level of detail. \n\n\\MP{The remainder of this paper} is organized as follow. In Sec.~\\ref{Ch1}, after a brief introduction to the problem at hand and its relevance in fundamental physics, we discuss the advantages potentially offered by a space environment for quantum experiments \\MP{based on}  large quantum superpositions of nanoparticles. We also give a self-contained overview of the current state-of-the-art for space missions proposals. In Sec.~\\ref{Applications}, we discuss a first class of experiments that can be performed in space, i.e., non-interferometric ones which do not require the creation of macroscopic superpositions and exploit the free-evolution spread of the position of a quantum particle. In Sec.~\\ref{ProofOfPrincipleImlementation}, we consider interferometric experiments which, in contrast, require the  creation and verification of large superpositions but also offer the benefit of a direct test of both the superposition principle of quantum mechanics and of competing theories. In particular, we present here an ab-initio estimate of the potential of space-based interferometry with large nanoparticles. We conclude in Sec.~\\ref{WishList} with a discussion of our results and an outlook to the evolution of the field.\n\n\n\n\\section{Superposition of macroscopic systems: the case for space}\\label{Ch1}\nThe predictions of quantum physics have been confirmed with a high degree of precision in a multitude of experiments, from the sub-atomic scale up to matter-wave interferometry with tests masses of nearly $10^5$ atomic mass units (amu)~\\cite{fein2019quantum}. The basis for observing matter-wave interference is the quantum superposition principle, one of the pillars of quantum physics. While quantum physics does not pose any fundamental limitation to the size of quantum superposition states, the Gedankenexperiment of the  Schr\\\"{o}dinger's cat~\\cite{Schroedinger1935b} illustrates the controversies entailed by the superposition principle when extended to the macroscopic world. Many proposals have been formulated in an attempt to establish a mechanism that would lead to the emergence of a classical world at macroscopic scales. Among them we find Bohmian mechanics~\\cite{bohm1952suggested,durr2009bohmian}, decoherence histories~\\cite{griffiths1984consistent}, the many-world interpretation~\\cite{everett1957relative} and collapse models~\\cite{bassi2003dynamical,RevModPhys.85.471} to name a few. The latter differs from the other proposals in the fact that they predict a phenomenology that deviates from the one of standard  quantum mechanics, albeit in a delicate fashion. In this sense, collapse models  represent an alternative construction to standard quantum theory, more than an alternative interpretation recovering all the predictions of the latter.\n\\MP{In light of the central role that they play in the experimental investigation of quantum macroscopicity~\\cite{PhysRevLett.110.160403,PhysRevLett.119.100403}, in the following, we will focus on such models as benchmarks for precision tests of quantum mechanics.}\n\n{\nIn 2010, a proposal for experimentally creating and verifying a state akin to the one of Schr\\\"{o}dinger's  cat based on the use of massive mechanical resonators was put forward within the context of the MAQRO proposal~\\cite{kaltenbaek2012macroscopic}. The latter put forward the vision of harnessing the unique environment provided by space to test quantum physics in a dedicated, medium-sized space mission to be conducted within the framework of the \"Cosmic Vision programme\" run by the European Space Agency (ESA). The scope of the endeavour was to create a macroscopic superposition of motional states of a massive particle and probe its quantum coherence by allowing the wave functions of the components of such superposition to interfere, as in a double-slit experiment. The space-based environment would guarantee unprecedented levels of protection from environmental noises, as well as favourable working conditions for the engineering of the cat-like state~\\cite{kaltenbaek2012macroscopic}.}\n\nNear-field interferometry has later been identified as a viable route \\MP{for} the achievement of the original goals of MAQRO~\\cite{kaltenbaek2016macroscopic}, holding the promises for testing the superposition principle with particles of mass up to $10^{11}$~amu. This would be at least six orders of magnitude larger than the current record~\\cite{fein2019quantum}\\MP{. It would also far exceed the projected upper-bound to the masses that could be used in similar ground-based experiments. \\MP{Such terrestrial upper-bounds are strongly limited by the achievable free-fall times on Earth}~\\cite{gasbarri2020prospects} (cf. Sec.~\\ref{advantage}).} \n\n{\nThe basic payload consists of optically trapped dielectric nanoparticles with a target mass range from $10^7$ to $10^{11}$~amu. The main scientific objectives are to perform both near-field interferometric and non-interferometric experiments. In both cases, high-vacuum and cryogenic temperatures are needed. The particles, after loading, are initially trapped in an optical cavity and their center-of-mass degree of freedom cooled down by a 1064~nm laser entering the quantum regime. For this purpose, two TEM$_{00}$ modes  with  orthogonal  polarization are to be used for trapping and side-band cooling along the cavity axis. The transverse motion is instead cooled employing a TEM$_{01}$ and a TEM$_{10}$ mode. After the initial state preparation, the particle can be released from the optical trap and undergoes different evolutions -- free fall expansion, coherent manipulations, and quantum detection -- depending on the experiments to be performed. \n}\n\n{\nThe feasibility of the avenue identified in MAQRO has recently been investigated in a Quantum Physics Payload platForm (QPPF) study at the ESA Concurrent Design Facility~\\cite{QPPF}. Such study has identified {\\bf (a)} the core steps towards the realisation of a space-based platform for high-precision tests of quantum physics, and {\\bf (b)} the potential of such platforms to test quantum physics with increasing test mass with the scope to ascertain potential deviations from the predictions of quantum physics due, for instance, to gravity. The ultimate goal of these endeavours is to provide a reference mission design for quantum-physics experiments in space.\n}\n\n{\nThe QPPF~\\cite{QPPF} study culminated with the identification of a suitable combination of feasible free-fall times, temperature and pressures [cf. Table \\ref{tab:parameterssmall}], setting a target of 2034 for the launch of the mission.  \n}\n{While several technical challenges remain to be addressed, the QPPF study has consolidated the intention to leverage on the expertise in near-field interferometry and optomechanics where state-of-the-art experiments with large molecule at near $10^5$~amu have been reported~\\cite{fein2019quantum}, ground-state cooling achieved~\\cite{delic2020cooling}, and proof-of-concept proposals for ground-based interferometry with large-mass experiments put forward~\\cite{bateman2014near,gasbarri2020prospects}.  \nIn this respect, it should be mentioned that theoretical proposals based on other approaches, most noticeably magnetic levitation, have recently attracted the attention of part of the community~\\cite{pino2018chip}. These proposals envision testing the superposition principle with ground-based experiments, overcoming some of the existing limitations through  \n{the low noise level, long coherent-operation times, and lack of need for light fields driving the dynamics promised by magnetic levitation}. The use of masses of the order of $10^{13}$~amu has been forecast in this context~\\cite{pino2018chip}. While extremely interesting, magnetic levitation technologies for quantum experiments are still at an early stage~\\cite{rusconi2018levitated,\nrusconi2017linear,rusconi2017quantum,druge2014damping} with, at present, no quantum superposition having been created which such techniques. }\n\n{\n\\begin{table}[t]\n    \\centering\n    \\begin{tabular}{c|c}\n    \\hline\n        \\textbf{Parameters} & \\textbf{Values}\\\\\n        \\hline\n        \\hline\n         Free fall time $t$& up to $100\\,$s  \\\\\n         Environmental temperature $T_\\text{env}$&down to 20\\,K\\\\\n         Pressure& down to $10^{-11}\\,$Pa\\\\\n         Masses& $10^{7} - 10^{11}$\\,amu\\\\ \n         Diameters & $20 - 400$\\,nm\\\\\n    \\hline\n    \\end{tabular}\n    \\caption{Possible combination of the parameters identified for the MAQRO mission and considered by ESA QPPF study~\\cite{QPPF}.}\n    \\label{tab:parameterssmall}\n\\end{table}\n}\n\n\n\\subsection{In search of the quantum-to-classical boundary}\\label{subsec::QuantumClassical}\n\nThe extreme fragility of spatial quantum superpositions in the presence of environmental interactions (ubiquitous in any realistic setting) makes testing the superposition principle at macroscopic scales a tall order. Indeed, such interactions result in a suppression of quantum coherence in position that can be described by the following master equation in position representation\\cite{joos1985emergence}\n\\begin{equation}\\label{master_gen}\n    \\frac{d\\langle x|\\hat\\rho_{t}|x'\\rangle}{dt}=-\\frac{i}{\\hslash}\\langle x|\\left[\\hat H,\\hat\\rho_{t}\\right]|x'\\rangle-\\Gamma(x-x')\\langle x|\\hat\\rho_{t}|x'\\rangle,\n\\end{equation}\nwhere $\\hat \\rho_t$ is the statistical operator of the system at time $t$, $\\hat H$ is the system's Hamiltonian, and\n\\begin{figure}[b]\n    \\centering\n  \\includegraphics[width=0.8\\linewidth]{decoherenceGammaV2.pdf}\n    \\caption{Typical dependence of the decoherence function $\\Gamma(x)$ from the delocalization distance $x$. The relevant limits are $\\Gamma(x)\\sim\\Lambda x^2$ for $x\\ll a$ and $\\Gamma(x)\\sim\\gamma$ for $x\\gg a$, where $a$ is the characteristic length of the noise~\\cite{PhysRevA.84.052121}.}\n    \\label{fig:decoherencefunction}\n\\end{figure}\nthe last term of Eq.~\\eqref{master_gen} describes the deviations from  unitary dynamics occurring at a rate $\\Gamma(x)$, which quantifies the decoherence effect. The typical behaviour of the latter, with a quadratic dependence for small spatial separations and saturating for large ones, is shown in Fig.~\\ref{fig:decoherencefunction}. Such  deviations from unitarity can be due to environmental noises or non-standard modifications of quantum mechanics~\\cite{RevModPhys.75.715,PhysRevA.84.052121,RevModPhys.85.471}. The environmental influence is always present, and it inevitably disturbs the experiment compromising the possibility to detect superposition states.  \nThe typical noise effects on the experimental setups addressed in this paper are due to collisions with residual gas particles, blackbody radiation, vibrations and in general any noise propagating through the experimental setup. \n{Quantitatively, for} a sphere made of fused silica with a radius of $60\\,$nm and an internal temperature of $40\\,$K,\nplaced in vacuum in an environment at $20\\,$K and a pressure of $10^{-11}\\,$Pa { [cf.~Table~\\ref{tab:parameterssmall}]}, {one has that for spatial superpositions larger than a nanometer but smaller than a millimeter, which is the range of interest for the interferometric test we will consider here, the }gas collisions give a constant contribution {$\\Gamma(x-x')\\sim1.1\\,$s$^{-1}$} [cf. Ref. ~\\citeonline{PhysRevA.84.052121}], while the contribution from blackbody radiation depends explicitly on the superposition size as $\\Gamma(x-x')\\sim 4.9\\times 10^{12}\\times |x-x'|^2$\\,m$^{-2}$s$^{-1}$. \n\n\nAs mentioned, Eq.~\\eqref{master_gen} is conducive of an investigation on potential deviations from standard quantum theory due, for instance, to collapse models. \nDue to its interesting phenomenology, theoretical interest and current strong experimental effort in testing it~\\cite{RevModPhys.85.471,carlesso2019current,bassi2014collapse,vinante2019testing,carlesso2019collapse,vinante2020narrowing},  in the following we will focus on {the Continuous Spontaneous Localization (CSL) model}.\n{The CSL model describes, through a stochastic and non-linear modification of the Schr\\\"{o}dinger equation, the collapse of the wave function as a spontaneous process whose strength increases with the mass of the system~\\cite{Ghirardi1990a}. Its action is characterized by two phenomenological parameters: $\\lambda_\\text{CSL}$ and $r_c$. \\MP{These are respectively the} collapse rate, \\MP{which quantifies the strength of the collapse noise,} and the localization length of the model, \\MP{setting the spatial resolution of the collapse}. Theoretical  considerations lead to different proposed values for such parameters: $\\lambda_\\text{CSL}=10^{-16}\\,$s$^{-1}$ and $r_c=10^{-7}\\,$m for Ghirardi, Rimini and Weber~\\cite{ghirardi1986unified}; $\\lambda_\\text{CSL}=10^{-8\\pm2}\\,$s$^{-1}$ for $r_c=10^{-7}\\,$m, and $\\lambda_\\text{CSL}=10^{-6\\pm2}\\,$s$^{-1}$ for $r_c=10^{-6}\\,$m by Adler \\cite{adler2007collapse}.}\n{Consequently, one can describe the evolution of} the density matrix of a system\n{with} Eq.~\\eqref{master_gen} where, in addition to the decoherence effects ascribed to the environment, a term accounting for spontaneous collapse appears. \\MP{The form of such term and its effects are discussed in detail in Section \\ref{Applications}.} This reveals the importance of a careful characterization of environmental sources of decoherence in view of probing new physics, which is the aim of the space experiments with large nanoparticles reviewed here. It should be noted indeed that, the experimental setups considered here are relevant also for testing other models  predicting non-standard decoherence mechanisms~\\cite{bahrami2014schrodinger,pikovski2015universal,gasbarri2017gravity} or models like the Di\\'osi-Penrose (DP) one~\\cite{diosi1987universal,PhysRevA.40.1165,penrose1996gravity} in which the wave function collapse is related to gravity. \n\n\n\\subsection{Possible advantages of a space environment}\\label{advantage}\nThe main advantage offered by space for quantum experiments with large particles is undoubtedly a long free-fall time. While freely-falling systems are not necessary for some non-interferometric experiments, they are the golden standard for the interferometric ones. For the latter, long free-fall times are of crucial importance to achieve better sensitivity and to increase the mass of the particles in quantum superposition \\MP{as the rate of the wavefunction spreading is set by $1\/m$.} In state-of-the-art interferometric experiments, and for masses of up to $10^6$\\,amu, the necessary free-fall times are far below $1\\,$s and can be readily achieved in laboratory experiments~\\cite{bateman2014near,fein2019quantum,gasbarri2020prospects}. However, going to significantly higher test masses requires correspondingly longer free-fall times~\\cite{kaltenbaek2012macroscopic,kaltenbaek2016macroscopic,gasbarri2020prospects} such as to eventually rendering it inevitable to perform such experiments in space (see Fig.~\\ref{fig::talbotTime}).\nLong free fall times help also in non-interferometric settings. The latter do not require the creation and verification of quantum superpositions but are based on the modified dynamics predicted by alternative models to quantum mechanics -- as for example the heating induced by the CSL noise on massive particles. In this context, letting the particle fall freely allows to reduce \\MP{the effects of all the sources of noise that affect the centre of mass motion. Among them certainly is acceleration noise typically originating from mechanical vibrations. However, one should also include other forces acting on the particle's motion and which might be} present in the experiment thus maximizing the effects induced by modifications of quantum mechanics. \\MP{In what follows, we provide a brief yet rigorous account of the most relevant of such forces.} \n\nAn equally important challenge is the isolation from vibrations,\nwhich contribute to the overall decoherence mechanisms acting on the system. Especially in the low-frequency regime, space experiments can provide strong advantages compared to those performed on ground. {For example, ensuring that an interference pattern with a period of $d=1\\,\\mu$m formed during an evolution time of $T=100\\,$s is not washed out requires a maximum acceleration noise of $S_{aa}^\\text{max}\\sim3d^2\/8\\pi T^3$ corresponding to $\\sqrt{S_{aa}}\\sim 3.5 \\times 10^{-10}\\,\\mathrm{m\\, s^{-2}\/\\sqrt{Hz}}$.}\nSuch low noise can be achieved in space. The most impressive achievement so far has been LISA Pathfinder with an acceleration noise as small as $\\sqrt{S_{aa}}\\sim 10^{-15}\\,\\mathrm{m\\, s^{-2}\/\\sqrt{Hz}}$ in the mHz regime \\cite{PhysRevLett.120.061101}. \nThis value has to be compared to those of the state-of-art ground-based experiments. For example, the Bremen drop-tower allows for up to around 9\\,s of free fall in an environment characterized by an acceleration noise of $\\sim 10^{-5}\\,\\mathrm{m\\, s^{-2}\/\\sqrt{Hz}}$ (cf. Ref.~\\citeonline{Selig2010a}). \n\\MP{We also mention that the exceptionally low level of noise achieved in LISA Pathfinder has already allowed this experiment to provide bounds on the CSL parameters \\MP{that are} more than three orders of magnitude stronger than \\MP{those} provided by the ground-based gravitational waves detector LIGO~\\cite{carlesso2016experimental,helou2017lisa,carlesso2018non}, \\MP{thus demonstrating the advantages -- in terms of isolation from vibration -- of space-based experiments}.}\n\n\n\nA potential additional advantage of a dedicated space mission is data statistics. In state-of-the-art matter-wave experiments~\\cite{fein2019quantum}, many test particles pass through the interferometer simultaneously. In proposals, considered in the QPPF, suggesting to prepare the initial state of the test particle by optomechanical means, the test particles pass through the interferometer individually~\\cite{RomeroIsart2011b,kaltenbaek2012macroscopic,bateman2014near}. Thus, the time of the experiment is inevitably longer, and growing with the number of data points required. For example, in an experiment with $10^4$ data points, where a single shot takes $100\\,$s ($10\\,$s), the data collection would last more than $11.5$ days ($27$ hours). This compares very favourably to the typical number of two or three runs per day that can be performed in a micro-gravity environment on the ground as at the Bremen drop-tower, \\MP{which is limited by the necessity of \\MP{setting and resetting the pressure in} the entire tower between two consecutive drops}\n\\footnote{\\MP{It should be mentioned that this limitation is not present in the Hannover {\\it Einstein Elevator} \\MP{platform} where, for free-fall times of $\\le 4\\,$s, 300 runs per day are possible~\\cite{lotz2020tests}.} }.\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\columnwidth]{talbot_timev2.pdf}\n\\caption{The free-fall time for the test particles in a near-field interferometer is of the order of the Talbot time $t_T=m d^2\/h$, where $m$ is the particle mass, $h$ is Planck's constant, and $d$ is the grating period. In experiments with significantly higher test masses than the current record~\\cite{fein2019quantum}, the required free-fall time may eventually necessitates a space environment~\\cite{kaltenbaek2016macroscopic}.}\n\\label{fig::talbotTime}\n\\end{figure}\n\n\\subsection{State-of-the-art technological platforms}\nThe space environment promises, in principle, to provide a unique combination of low temperature, extremely high vacuum and very long free-fall times. In particular, the temperature in space is naturally limited by the temperature of the microwave background radiation of about $3\\,$K while the vacuum is instead limited by the presence of cosmic and solar radiation~\\cite{Biermann1957a}. However, in actual space-based experiments, additional shielding may be required. For example, if the spacecraft is in an orbit about one astronomical unit from the Sun, the payload will have to be shielded from direct solar radiation. {The spacecraft will require stabilization using microthrusters, which will introduce force noise, and there will need to be station-keeping maneuvers. These measures as well as the gravitational field of the spacecraft will reduce the achievable free-fall time.} In addition, the equipment necessary to operate the payload and the spacecraft typically will be in an enclosure kept at a stable temperature of about $300\\,$K. Inside the spacecraft, it is therefore not trivial to achieve cryogenic temperatures and extremely high vacuum levels. Achieving the vacuum levels and temperatures necessary for macroscopic tests of quantum physics therefore requires careful considerations. In the context of the MAQRO mission concept, it has been suggested to use purely passive radiative cooling and direct outgassing to space in order to achieve these requirements~\\cite{kaltenbaek2012macroscopic,kaltenbaek2016macroscopic,hechenblaikner2014cold,zanoni2016thermal}. During the QPPF study, this concept was adapted protect the scientific instrument with a cover and to enhance the cooling performance ny additional active cooling using a hydrogen sorption cooler~\\cite{QPPF}. \n\nThe protective cover limits outgassing to space, and the QPPF study concluded that the achievable pressure would at best be\n$10^{-11}$\\,Pa instead of the aimed pressure of $10^{-13}$\\,Pa. As a result,\nthe experiments were constrained to a test particle mass up to $2\\times10^9$\\,amu and free-fall time up to $40\\,$s. Because this has a significant impact on the science objectives, improving the achievable vacuum in space-based experiments will be a critical issue to be solved before a space mission of this type can be launched. Two other critical issues were identified in the QPPF study~\\cite{QPPF}. \n{Firstly, the mechanism needed for loading the test particles into an optical trap in extremely high vacuum conditions needs further scrutiny and several viable alternatives are under consideration. The QPPF study suggested to desorb particles from microelectromechanical systems (MEMS) and guide them to the experiment using linear Paul traps~\\cite{QPPF}, a mechanism that is currently under further investigation by ESA. An alternative suggestion makes use of a combination of linear Paul traps and hollow-core photonic-crystal fibers~\\cite{kaltenbaek2016macroscopic} or the desorption of particles from a piezoelectric substrate using surface acoustic waves.  \nThe challenge of such desorption-based approaches is to make sure that the desorbed sub-micron particles do not carry a net charge~\\cite{PhysRevA.95.061801}, and that their center-of-mass motion is sufficiently cold to allow for optical trapping. At the same time, a sufficiently low internal temperature of the particles is required to avoid decoherence due to the emission of blackbody radiation~\\cite{RomeroIsart2011b,kaltenbaek2012macroscopic}. Secondly, the optical gratings used for preparing non-classical states has grating apertures comparable to the size of the nanoparticles to be employed. This can decohere the quantum states via photon scattering. A recent study~\\cite{PhysRevA.100.033813} investigated the latter issue extending the formalism of near-field interferometry beyond the point-particle approximation and offering the basis for the analysis reported in Sec.~\\ref{ProofOfPrincipleImlementation}.}\n\n\n\\section{Non-interferometric tests}\\label{Applications}\n\nIn this section, we focus on non-interferometric tests of quantum mechanics. Differently from the interferometric ones, this class of tests does not rely on the availability of quantum superpositions but are based on side-effects of modifications of quantum mechanics. {Consequently, they can be performed also in presence of strong decoherence, although the latter will influence the effectiveness of the test. For this reason, they currently provide the most stringent tests of collapse models on ground. \n}\n\nA plethora of different experiments belongs to this class and exploit different physical systems. {Among them, precision measurements of the internal energy of a solid, expected to vary due to the collapse noise, have been exploited in Refs.~[\\citeonline{adler2018bulk,bahrami2018testing,adler2019testing}]. The modifications to the free evolution dynamics of Bose-Einstein condensate due to the presence of the collapse mechanism has been investigated in Refs.~[\\citeonline{laloe2014heating,bilardello2016bounds}]. And X-ray measurements -- which exploit the fact that the collapse mechanism makes charged particles to emit radiation~\\cite{fu1997spontaneous,curceanu2016spontaneously,piscicchia2017csl} -- have already provided strong limits on the Di\\'osi-Penrose model~\\cite{donadi2020underground}. In this context, also optomechanical experiments are of particular relevance~\\cite{vinante2016upper,carlesso2016experimental,PhysRevA.98.022122,vinante2017improved,helou2017lisa,zheng2020room,vinante2020narrowing,pontin2020ultranarrow}. They are typically used to characterize noise~\\cite{aspelmeyer2014cavity,millen2020optomechanics,goldwater2019quantum}, and thus possibly discriminate between standard and non-standard noise sources~\\cite{vinante2020narrowing}.}\n\n\n{One of the most promising non-interferometric test in space is based on monitoring the expansion of the centre-of-mass position spread of a freely-falling nanoparticle~\\cite{PhysRevA.93.062102}. The main reason, as it is shown in Eq. (\\ref{variance}) below, is that the position variance grows as the cube of time, making evident the advantage of the long free-fall time that can be achieved in space.\nIt could be argued that long times can also be achieved in ground experiments by suspending the particles using an harmonic trap. However, the use of such a trap would certainly introduce additional noises and, more importantly, it would imply a position variance growth that scales only linearly with time~\\cite{bilardello2016bounds,PhysRevA.94.010104}. \n\nGiven the evolution in Eq.~(\\ref{master_gen}), it is easy to show that its non-unitary part does not affect the average position $\\langle x_{t}\\rangle$ of the particle, but changes its variance $\\sigma^{2}= \\braket{x_{t}^{2}}-\\braket{x_{t}}^{2} $ by a factor $\\braket{\\Delta \\sigma^{2}}$ that, for a free system and in the $x\\ll{a}$ regime [cf. Fig.~\\ref{fig:decoherencefunction}], reads\n\\begin{equation}\\label{variance}\n\\braket{\\Delta \\sigma^{2}}=\\frac{2\\Lambda\\hslash^{2}t^{3}}{3m^{2}}.\n\\end{equation}\nThe diffusion rate $\\Lambda$ is the sum of different contributions stemming from residual gas collisions,  blackbody radiation and non-standard sources, such as  the CSL or the Di\\'osi-Penrose model.\nFor the CSL model and a homogeneous sphere of radius $R$ and mass $M$, one has~\\cite{zheng2020room}\n\\begin{equation}\n\\Lambda_{\\text{CSL}}=\\frac{6 \\lambda_\\text{CSL}  M^2 }{m_0^2 R^2\\eta^4_{CSL}}\\left[\\left(1+\\frac{\\eta^2_{CSL}}{2}\\right) e^{-\\eta^2_{CSL}}+\\frac{\\eta^2_{CSL}}{2}-1\\right],\n\\end{equation}\nwhile for the DP model one obtains\n\\begin{equation}\n\\begin{aligned}\n\\Lambda_\\text{DP}&=\\frac{M^{2}G}{2\\hslash \\sqrt{\\pi}R^{3}}\\left[\\sqrt{\\pi}\\operatorname{erf}\\left(\\eta_{DP}\\right)-\\frac{3}{\\eta_{DP}}+\\frac{2}{\\eta^{3}_{DP}}\\right.\\\\\n&\\left.+\\frac{e^{-\\eta^2_{DP}}}{\\eta_{DP}}\\left(1-\\frac{2}{\\eta^{2}_{DP}}\\right)\\right].\n\\end{aligned}\n\\end{equation}\nWe have used the dimensionless parameters $\\eta_{CSL}=R\/r_c$ and $\\eta_{DP}=R\/R_0$ with  $R_0$ a free parameter that is characteristic of the DP model~\\cite{bahrami2014role}. These expressions can be then used to set bounds on, respectively, CSL and DP parameters with space-based experiments, as we discuss next.\n}\n\n\n\\subsection{Long free-fall times: opportunities and challenges for space-based experiments}\n{A possible space-based experiment, as envisioned in the MAQRO proposal and QPPF, is as follows. A nanosphere is initially trapped by an harmonic optical potential and its center-of-mass motion is optically cooled. The trapping is then removed and the nanosphere remains in free-fall for a time $t$ after which its position is measured. Achieving a high position resolution is possible by, for example, combining a coarse-grained standard optical detection on a CMOS chip with a high-resolution backscattering detection scheme~\\cite{Tebbenjohanns2019a}, which could eventually provide a position accuracy on the order of $\\varepsilon=10^{-12}\\,$m \\MP{at a typical bandwidth of 100 kHz, by controlling the measurement back-action.~\\cite{vovrosh2017parametric}}\nBy repeating such a procedure $N$ times, one can reconstruct the position spread $\\sigma^{2}$ and thus quantify the effects of the non-unitary dynamics through Eq.~(\\ref{variance}). To detect effects as those predicted by the CSL or the DP model, one needs to minimize the competing standard decoherence effects (from collisions and blackbody radiation), which contribute to the total $\\Lambda$ in Eq.~(\\ref{variance}). \n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{CSLnonint.pdf}\n    \\caption{Exclusion plots for the CSL parameters $\\{r_c,\\lambda\\}$ from non-interferometric experiments. \\MP{The solid, red line represents the bound on the CSL parameters that can be \\MP{potentially} achieved through non-interferometric experiments in space with the parameters in Table~\\ref{tab:parameterssmall}. Here, the main limitation is due to the environmental conditions of pressure and temperature. The dashed red line indicates the upper bound that could be obtained by decreasing the pressure to $P=3 \\times 10^{-14}$~Pa, so that the main limitation would be represented by the statistical error. These bounds are} compared to the strongest bounds and corresponding excluded parameter regions present in literature: X-rays emission (blue region)~\\cite{piscicchia2017csl}, LISA Pathfinder (green region) \\cite{carlesso2016experimental,carlesso2018non}, multilayer cantilever (brown region) \\cite{vinante2020narrowing}. The gray region is the theoretical lower bound, \\MP{which is estimated by requiring the collapse to become effective at the mesoscopic scale where the quantum-to-classical transition is expected}~\\cite{torovs2017colored}.   }\n    \\label{fig:noninterf}\n\\end{figure}\n{We are now in the position to estimate the bounds on the CSL parameters.} To do this,\nwe employ the values in Table~\\ref{tab:parameterssmall}. We consider silica nanospheres with a 120\\,nm diameter as test particles and an\ninternal temperature fixed at 40\\,K. \nMoreover, we also assume levels of vibrational noise similar to those obtained in LISA Pathfinder~\\cite{PhysRevLett.120.061101}. With these assumptions, the strongest competing effect to the CSL noise is the collisional decoherence, which limits the bounds on the CSL parameters.  Such a bound is indeed obtained by setting $\\Lambda_\\text{CSL}$ equal to the collisional contribution to the diffusive constant $\\Lambda$, whose form is reported in Ref.~[\\citeonline{PhysRevA.84.052121}].} \\MP{We show  the corresponding bound as the solid red line in Fig.~\\ref{fig:noninterf}, where such bound is compared to ground-based ones achieved by  state-of-the-art experiments on the CSL model.}\n\nFor what concerns the DP model, the state-of-the-art experimental bounds indicate that the free parameter $R_0$ is limited to~\\cite{donadi2020underground} $R_0\\geq R_0^*= 0.5 \\times 10^{-10}$\\,m. Because the DP-induced collapse becomes stronger for smaller $R_0$, the maximum effect is obtained for~{$R_0^*$.} Such a value of $R_0$ leads to a position spread in the aforementioned set-up of $\\sqrt{\\braket{\\Delta\\sigma^2}}\\sim3\\times 10^{-26}\\,$m {for $t=100\\,$s}, well beyond the state-of-art position measurement sensitivity $\\varepsilon$.\n\n\n\nAn important aspect to consider for experiments performed in space is their limited lifetime. Especially when one considers long free-fall times, this will have an impact on the statistical accuracy with which one can determine the variance of the measured data points~\\cite{kaltenbaek2016macroscopic,QPPF,Kaltenbaek2021a}. The long free-fall time $t$ required to see potential deviations from the quantum predictions has to compete with a finite time $T$ available to take the complete data set. At best, the number of data points can be $N=T\/t$.\n{This limit on the number of data points implies a statistical uncertainty in determining the position spread. To quantify it, we assume that the initial quantum state of the test particle is the ground state of an harmonic oscillator with a mechanical frequency $\\omega$. Consequently, the measured position will be normally distributed and the corresponding fractional uncertainty of the variance of the measured position will be~\\cite{Taylor1997a} $\\sqrt{2\/(N-1)}\\approx \\sqrt{2 t\/T}$.}\n{Assuming} the deviations from the quantum predictions to be small, the statistical uncertainty\nof the variance is $\\Delta x^2_f\\approx \\sqrt{2 t\/T} x^2_s$, where $x^2_s\\approx  t^2\\omega\\hslash\/2m$ is the variance of the wavepacket predicted by quantum physics for times much longer than $1\/\\omega$. \n{By taking} $\\omega=10^{5}\\,$Hz for the trap frequency, a free evolution time $t=100\\,$s and a total time $T$ of 30 days, we have that $\\Delta x_f\\sim 3\\times 10^{-5}\\,$m which has to be compared to the sensitivity $\\varepsilon\\sim10^{-12}\\,$m. For these parameters, the statistical uncertainty will dominate over the position sensitivity $\\varepsilon$ already after about~\\cite{Kaltenbaek2021a} $0.1\\,$ms. \n{Such a statistical uncertainty becomes a fundamental limitation for the experiment. The corresponding upper bound on the CSL parameters is represented by the dashed, red line in Fig.~\\ref{fig:noninterf}. To reach such a limit, the pressure would need to be reduced by more than two orders of magnitude, down to $P=3 \\times 10^{-14}$\\,Pa,  with respect to the conditions setting the continuous red bound. }\n\n\\MP{Fig.~\\ref{fig:noninterf} and the analysis above \\MP{suggest} that non-interferometric experiments performed with the parameters in Tab.~\\ref{tab:parameterssmall} can enhance only partially the exploration of the CSL parameter space. A more substantial improvement would require to solve technical challenges, \\MP{such} as a significant pressure reduction. Alternatively, one can pursue the path of interferometric experiments, which is discussed in the next section.}\n\n\n\n\n\\section{Interferometric tests}\\label{ProofOfPrincipleImlementation}\n\nHere, we will provide an overview of the current state-of-the-art for proposals of interferometric experiments testing the superposition principle of quantum mechanics for higher masses than the current experimental record on ground by using a space environment. We will discuss the challenges faced by such experiments, and we will provide novel simulations results estimating the interference visibility expected in space-based experimental tests of the superposition principle of quantum mechanics. \n\n\\subsection{Near-field interferometry}\n\nAfter Clauser envisioning its use for \\textit{small rocks and live viruses} experiments~\\cite{clauser1997broglie} and its initial demonstration for C$^{70}$ molecules interferometry~\\cite{PhysRevLett.88.100404}, for almost two decades the most successful technique harnessed for interferometric tests of quantum physics has been near-field interferometry~\\cite{RevModPhys.84.157,arndt2014testing}.\nWith this technique, and employing three optical gratings~\\cite{gerlich2007kapitza}, in 2019 the Arndt's group in Vienna was able to successfully build and demonstrate spatial the quantum superposition of big molecules with masses beyond~\\cite{fein2019quantum} $10^4$\\,amu.\n\n\nRecently, the possibility to consistently describe the effects of an optical grating on large dielectric particles with radii comparable to the optical wavelengths~\\cite{Nimmrichter2014a,PhysRevA.100.033813} has opened the possibility to use optical grating to study quantum interference on even larger particles. At the same time,\nconcrete proposals to go beyond the current mass record, employing individually addressed dielectric particles and single optical grating~\\cite{bateman2014near,kaltenbaek2016macroscopic,QPPF} have shown the experimental viability of near-field interferometry to actually perform such larger mass superposition experiments.\n\nWe thus focus specifically on these implementations to give a overview of how a near-field interferometric scheme works.\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\columnwidth]{figCommPhys_NFv2.pdf}\n\\caption{{\nSchematic representation of a space-based near-field interferometry experiments referred to in the text. While for ground-based experiments the time axis corresponds also with the vertical position of the particle, in space-based experiments the particle remains in the same position (relative to the experimental apparatus) and the trapping laser, the grating laser and a final measurement laser must be activated in turns at different times.}}\n\\label{figInonI}\n\\end{figure}\nWe refer to Fig.~\\ref{figInonI} for a schematic representation of a single-grating near-field set-up. Contrary to the case of lighter systems, where molecular beams are engineered, each nanoparticle in the experiment is individually addressed. We thus have, at each run of the experiment, four main stages: \n\n\\begin{itemize}\n    \\item[\\textbf{(a)}] The nanoparticle is trapped and cooled down in an optical cavity for a time $t_c$ after which the center-of-mass degree of freedom is in a very low-temperature thermal state characterised by the momentum and position variances $\\sigma_p,\\,\\sigma_z$. No cooling down to the ground state is required. \n    \\item[\\textbf{(b)}] The particle is released and freely falls for a time $t_1$. \n    During this time, residual gas collisions and thermal radiation are the main sources of decoherence. The free evolution of the post-cooling state needs to guarantee that the coherence-length is sufficient to cover at least two adjacent ``slits'' of the optical grating.\n    \\item[\\textbf{(c)}] A retro-reflected pulsed laser provides a pure-phase grating~\\cite{Nimmrichter2014a} for the dielectric nanoparticle. Scattering and absorption of grating photons constitute the main decoherence channels in the short interaction time with the grating.\n    \\item[\\textbf{(d)}] Second period of free evolution for a time $t_2$ during which the same sources of decoherence as in point (b) act. This stage has to last enough time for the interference pattern to form.\n    \\item[\\textbf{(e)}] The position of the particle is measured via optical detection~\\cite{kaltenbaek2016macroscopic,QPPF}. \n\\end{itemize}\nBy repeating this protocol \\textbf{(a-e)} many times, an interference pattern can form in the measured position distribution. This pattern can be mathematically described by a probability distribution function $P(z)$ which can be analytically derived from a phase-space treatment of the interferometric experiment~\\cite{bateman2014near,Nimmrichter2014a}:\n\\begin{equation}\n\\label{pattern}\n\\frac{P\\left(z\\right)}{\\delta}= \n1+2\\sum_{n=0}^{\\infty} R_{n} \\,B_{n} \\left[\\frac{n dt_{2}}{t_{T}D}\\right] \\cos\\left(\\frac{2 \\pi n z}{D}\\right)e^{-2\\left(\\frac{n\\pi\\sigma_{z}t_{2}}{Dt_1}\\right)^2},\n \\end{equation}\nwhere $\\delta=m\/\\left(\\sqrt{2 \\pi} \\sigma_{p}(t_{1}+t_{2})\\right)$, $t_T= md^2\/h$ is the Talbot time and $D=d(t_1+t_2)\/t_1$ is a geometric magnification factor. In this last expression, the $B_{n}$'s are known as the generalized Talbot coefficients~\\cite{Nimmrichter2014a,PhysRevA.70.053608} and account for the coherent and incoherent effects of the optical grating, while the kernels $R_{n}$ account for environmental decoherence, due to absorption, emission, and scattering of thermal radiation and collisions with residual gas, during the free-falling times $t_1, t_2$. \nThis expression remains formally unchanged when classical particles following ballistic trajectories are considered, but the explicit expressions for the decoherence kernels and Talbot coefficients will change. \nExpression \\eqref{pattern}, with the proper coefficients, can thus be used to describe the classical shadow pattern arising  from a completely classical description of the system (see Fig.~\\ref{fig:carpet}). \nFinally, non-linear modifications of quantum mechanics -- but  also other sources of positional decoherence like e.g. a stochastic gravitational waves background~\\cite{bassi2017gravitational,asprea2019gravitational} -- can easily be included in Eq.~\\eqref{pattern} by introducing their respective noise kernels $R_n$. We refer the interested reader to Refs.~[\\citeonline{PhysRevA.100.033813,gasbarri2020prospects}] and the Supplemental Material (SM)~\\cite{SI} for a detailed derivation and explicit expressions of the functions entering Eq.~\\eqref{pattern}.\n\nIn order to go beyond the current near-field interferometry mass record, large particles need to be used. Here, ``large'' refers to spherical particles with a radius $R$ comparable to or greater than the grating period $d$ such that $kR\\gtrsim 1$, where $k=2\\pi\/\\lambda$ is the wave-vector of the optical grating. In the following, we will use the formalism developed in Ref.~[\\citeonline{PhysRevA.100.033813}], and based on Mie scattering theory, to account for a large particle traversing an optical grating. For what concerns the pure-phase character of the grating -- i.e., its coherent effect on the particle's state -- it can be shown that the unitary evolution of the particle's state $\\hat{\\rho}$ (reduced along the longitudinal direction $z$) when traversing the grating assumes, in the eikonal approximation, the form  $\\bra{z}\\rho\\ket{z'}\\rightarrow \\exp\\left[{-i\\phi_0 (\\cos^2{kz}-\\cos^2{kz'})}\\right]\\bra{z}\\rho\\ket{z'},$\nwhere $\\phi_0$ is the eikonal phase factor characterising the coherent evolution. This is the same as in the case of a point-like particle and the only difference introduced by the use of Mie scattering theory~\\cite{mie1908beitrage,bohren2008absorption} is found in the structure of the eikonal phase $\\phi_0$ which can be expressed as \n\\begin{equation}\\label{phi0mie} \n    \\phi_0=\\frac{8F_0 E_L}{\\hslash c \\epsilon_0 a_L k|E_0|^2},\n\\end{equation}\nin terms of the laser and particle's parameters. Here, $c\\epsilon_0|E_0|^2\/2$ is the intensity parameter of the incident light, $E_L$ and $a_L$ are the grating laser energy and spot area respectively, and $F_0$ is obtained from Mie theory upon the evaluation at $z=-\\lambda\/8$ of the longitudinal conservative force acting on the particle (see Refs.~[\\citeonline{Nimmrichter2014a,PhysRevA.100.033813}] and references therein). Eq.~\\eqref{phi0mie} reduces to the well-known result $\\phi_0=2 \\mathcal{R}(\\chi)E_L\/(\\hslash c\\epsilon_0 a_L)$ with $\\chi$ the polarizability for a point-like particle. For what concerns the incoherent effects of the grating, the finite-size of the particles leads to modify the Talbot coefficients with respect to the point-like case. We refer the reader to Ref.~[\\citeonline{PhysRevA.100.033813}] and the SM~\\cite{SI} for further details.\n\n\n\n\n\\begin{figure}\n \\centering\n\\includegraphics[width=1.\\columnwidth]{Capetv2.pdf}\n\\caption{Talbot carpet arising from pure phase-grating without any source of decoherence for a point-like particle. This picture shows the comparison between the interference pattern predicted by quantum mechanics, $P(z)$ in Eq.~\\eqref{pattern} (on the left), and the shadow pattern that is formed by classical particles following ballistic trajectories (on the right) for different values of $\\tau=t_2\/t_T$. It is then apparent that, in order to can claim the observation of a quantum superposition, we need to be able to distinguish between these two patterns.}\\label{fig:carpet}\n\\end{figure}\n\nFinally, as discussed in Ref.~[\\citeonline{gasbarri2020prospects}], large particles near-field interferometric experiments presents several technical challenges. Common to both ground and space-based experiments is the challenge of diminishing as far as possible any environmental noise which would suppress the interference pattern. This can be achieved by a combination of ultra-high vacuum and cryogenic conditions. Moreover, for experiments aiming at using single particles in several ($\\sim 10^4$) runs, fast reloading\/recycling technique must be developed~\\cite{grass2016optical,mestres2015cooling,PhysRevLett.121.063602,QPPF}.\nOn top of these challenges, the key limitation for ground-based experiments is the short free-fall time. This is due to the Earth gravitational field and limits such experiments to a few seconds of free evolution. While this challenge can be overcome in principle, it will require a substantial modification of the scheme to go beyond masses of the order of $10^7$~amu~\\cite{pino2018chip,gasbarri2020prospects}. This is not the case for space-based experiments, where current estimates show the promise to reach masses of the order of $10^9-10^{11}$~amu and free-falling times of the order of hundreds of seconds~\\cite{kaltenbaek2016macroscopic,QPPF}. In the following section, we substantiate these claims by presenting an optimized analysis of space-based near-field interferometry showing the actual possibilities offered by a space environment.  \n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=1.\\textwidth]{Full_tablev2.pdf}\n\\caption{Values of the cost functions $\\aleph_{QC}$ and $\\aleph_{QCSL}$ as a function of $t_1, t_2$. The triangular area of the density plot is determined by the constraint $t_1+t_2\\leq 100$~s. The different columns correspond to five different values of the mass of the nanoparticles considered, respectively $10^{7},10^{8},10^{9},10^{10},10^{11}$~amu. The first row shows $\\aleph_{QC}$, i.e., how distinguishable the quantum and classical interference figures are. These figures are obtained by choosing as $E_L\/a_L$ the value that maximizes $\\aleph_{QC}$ in the physically feasible range between $10^{-6}$ and $5$~J\/m$^{2}$. For the numerical values of $E_L\/a_L$ employed we refer to Fig.~2 in the SM~\\cite{SI}. The second row shows the values of $\\aleph_{QCSL}$, i.e.,  how distinguishable the quantum interference figure is from the one accounting for the CSL. These figures are obtained by assuming the same values of $E_L\/a_L$ used in the first row and for a value of the CSL parameters proposed by Adler and given by $\\lambda_\\text{CSL} =10^{-8}$~s$^{-1}$ and $r_c=10^{-7}$~m. Finally, the third row shows the values of $\\aleph_{QCSL}$ like in the second row where, however, the values of $E_L\/a_L$ used are the ones that maximize $\\aleph_{QCSL}$ independently from the results in the first row of figures. For the numerical values of $E_L\/a_L$ employed we refer the interested reader to Fig.~1 in the SM~\\cite{SI}. }\\label{fig:QMCSLYet}\n\\end{figure*}\n\\subsection{Optimization for large particles: the current frontiers}\nWe present in this section the results of a numerical investigation of the possibilities offered by space-based experiments in conjunction with near-field interferomerty as discussed in the previous section. We employ the formalism developed in Ref.~[\\citeonline{gasbarri2020prospects}] to account for the finite size of the particles with respect to the grating period, and we use the experimental parameters, as summarized in the SM~\\cite{SI}, which have been extracted from the QPPF study about the MAQRO mission~\\cite{QPPF}. We are able to include in our analysis all the major known sources of environmental decoherence which can affect the interference pattern. In particular, we account for scattering and absorption of grating photons at stage \\textbf{(c)} of the protocol, residual gas collisions and black-body thermal radiation decoherence during the free evolution stage \\textbf{(b)-(d)}. We then include the effect of modifications of quantum mechanics in the from of the Continuous Spontaneous Localization (CSL) model with white-noise~\\cite{RevModPhys.85.471}. What we present here is the first fully consistent analysis of such a set-up and its potential for fundamental physics studies, which does not rely on the Rayleigh approximation, which cannot be consistently used unless for order of magnitude estimates.  \n\nOne complication of near-field interferometry -- in contrast with the textbook case of far-field interferometry -- that needs to be taken into account when performing an experiment is that also perfectly classical particles following ballistic trajectories through the optical grating would form a typical interference-like figure known as Moir\\'{e} shadow pattern~\\cite{RevModPhys.84.157} (see Fig.~\\ref{fig:carpet}). It is thus of crucial importance to discriminate between this pattern and a quantum mechanical one~\\cite{PhysRevLett.88.100404}. This is a prerequisite for both claiming to be able to test the superposition principle and for any analysis of modifications of standard quantum mechanics. Thus, we introduce a first figure of merit ($\\aleph_{QC}$) to estimate the ``distance'' between the quantum interference patter's probability distribution (pdf) and the pdf of the shadow pattern which would result from classical mechanics\n\\begin{align}\\label{alQC}\n\\aleph_{QC}= \\frac{1}{L}\\int_{-L\/2}^{L\/2}  \\frac{|P_{\\rm Q}(z)-P_{\\rm Clas}(z)|}{|P_{\\rm Q}(z)+P_{\\rm Clas}(z)|} dz\n\\end{align}\nwhere $L = 10^{-7}$~m is the spatial window in which the position measurement is performed and $P_{\\rm Clas}$ ($P_{\\rm Q}$) is the pdf predicted by classical (quantum) mechanics. A similar quantity can then be obtained to discriminate between a quantum interference pattern and the pattern deriving from  modifications of quantum mechanics. We will focus here on the CSL model with white noise. Thus the second figure of merit that  we will employ is $\\aleph_{QCSL}$, which is given by Eq.~\\eqref{alQC} with $P_{{\\rm Clas}}\\rightarrow P_{{\\rm CSL}}$. \n\nIn the following we assume to be able to discriminate values of $\\aleph\\ge 0.05$ (i.e., difference bigger than $5\\%$) which appears to be an experimentally justifiable choice~\\cite{juffmann2013experimental}. Moreover, we optimize over the parameters $t_1,t_2, E_L\/a_L$ of the set-up, which can be easily controlled, to maximize the figure of merits. As we will see, the optimization leads to values of the figure of merits well above the $5\\%$ threshold. Before illustrating the results of the analysis, let us comment on the choice of parameters. On the one hand, the free-falling times $t_1,t_2$ are extremely important in the formation of the interference figure, whether it is $t_1$ which guarantees a sufficient spreading of the initial state to a coherence length covering more than two ``slits'' or $t_2$ which allows the intereference to happen. These two times can also be easily adjusted in a space-based experiment by simply changing the activation times of the grating and measurement lasers. On the other hand, the parameter $E_L\/a_L$ enters directly in Eq.~\\eqref{phi0mie} and thus determines the pure-phase coherent effect of the grating. This parameter can also be easily tuned, being a property of the way the grating laser is operated. We keep instead fix all the other parameters entering our analysis (see the table reported in the SM~\\cite{SI}). These are: the wavelength of the grating laser, which is dictated by current technological possibilities; the material(s) parameters of the nanosphere, we considered silica (SiO2) particles which are widely employed in optomechanical experiments for their optical properties; environmental parameters, which have been extracted from the QPPF study~\\cite{QPPF} and represent the current state-of-the-art for space-based set-ups. Furthermore, always referring to the QPPF study on the stability of a possible mission's spacecraft, we constrain the total free fall time to $t_1+t_2\\leq 100$~s. Note that, the QPPF study concludes that, due to vacuum restriction, the interference pattern for the proposed MAQRO mission would be visible for free-falling times of up to 40~s. However, the 100~s benchmark is among the scientific objectives of the community, as reported in the QPPF. We thus chose to present our results with this constraint on the times. Nonetheless, our analysis shows that free-fall time of 100~s would be achievable within the parameters of the QPPF without spoiling the interference pattern.  \n\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=1.5\\columnwidth]{CSL_MAQROfinal.pdf}\n\\caption{Exclusion plots for the CSL parameters $\\{r_c,\\,\\lambda_\\text{CSL} \\}$ {from interferometric experiments}. Black points, with their error bars, represent the GRW's and Adler's theoretical values for these parameters~\\cite{ghirardi1986unified,adler2007collapse,adler2007corrigendum}.\nThe solid lines, and their respective excluded areas, show the upper bound that can be obtained by near field interferometric experiments in space with the parameters used in our simulations. In particular, the red line is obtained with $m=10^7 $~amu, $t_1=t_2=12$~s and $E_{L}\/a_{L} = 1.1\\times 10^{-2}$~$J\/m^2$. The dark-green one with $m=10^8$~amu, $t_1=t_2=10$~s and $E_{L}\/a_{L} = 3.5\\times 10^{-4}$~J\/m$^2$. The green, solid line with $m=10^9$ amu, $t_{1}=t_{2}=10$~s and $E_{L}\/a_{L} = 8.7\\times 10^{-6}$~J\/m$^2$. The blue one with $m=10^{10}$~amu , $t_{1}=t_{2}=50$~s and $E_{L}\/a_{L} = 8.7\\times 10^{-6}$~J\/m$^2$ and, finally, the orange solid line with $m=10^{11}$~amu , $t_{1}=t_{2}=50$~s and $E_{L}\/a_{L} = 2.2\\times 10^{-5}$~J\/m$^2$. \nFor comparison, the blue dot-dashed line shows the upper bound that could be achieved with a ground-based interferometric experiment with $m=10^7$~amu and times $t_1=t_2=9.95$~s -- free-falling times which are clearly beyond current reach for ground-based experiments -- as discussed in a recent study of four of the authors~\\cite{gasbarri2020prospects}.\nFinally, the dashed grey line represents the upper bounds given by current non-interferometric tests~\\cite{vinante2020narrowing,piscicchia2017csl,carlesso2016experimental,vinante2016upper}, the black dotted lines on the top of the figure are the current upper bounds coming from interferometric tests~\\cite{fein2019quantum,carlesso2019collapse,torovs2017colored,kovachy2015quantum}, and the grey area at the bottom of the plot represents the theoretical lower-bound provided in Ref.~\\citeonline{torovs2018bounds}.}\\label{fig:csl_excl}\n\\end{figure*}\n{As outlined above, the first step in the analysis is to consider when $\\aleph_{QC}$ is large enough to guarantee the possibility to certify a quantum mechanical interference pattern and then consider the corresponding $\\aleph_{QCSL}$. Figure~\\ref{fig:QMCSLYet} shows the results of our numerical investigation in this respect. The panels in the first row show the values of $\\aleph_{QC}$, i.e., the distance between the classical shadow pattern and the quantum interference one, for particles masses $\\{10^7,10^8,10^9,10^{10},10^{11}\\}$~amu as a function of $t_1,t_2$ and for the values of $E_L\/a_L$ which maximize the distinguishability. The latter are reported, as a function of $t_1,t_2$, in the SM~\\cite{SI} (see Fig.~2 therein). From the first row of Fig.~\\ref{fig:QMCSLYet} we see that $\\aleph_{QC}$ takes values definitely larger than the experimentally justifiable threshold of $5\\%$ for free-fall times  $t_1+t_2\\leq 100$~s, opening the way to direct tests of the quantum superposition principle with mesoscopic quantum systems in large spatial superpositions. The panels in the second row in Figure~\\ref{fig:QMCSLYet} show instead the comparison between the quantum interference pattern and the one which would arise if the CSL noise -- with parameters chosen at $\\lambda_\\text{CSL} =10^{-8}$~s$^{-1}$ and $r_c=10^{-7}$~m as proposed by Adler~\\cite{adler2007lower} -- was present. The panels on the second row are obtained by  evaluating the cost function $\\aleph_{QCSL}$ at the same values of $E_L\/a_L$ used for the upper row, i.e. the values that, at fixed $\\{t_1,t_2\\}$, maximize the quantum-to-classical distinguishability. \nIt should be noted that, for the comparison between CSL and quantum mechanics, we do not necessarily need to restrict our attention to only the values of the parameter $E_L\/a_L$ that maximize the classical-quantum distinguishability. Indeed, by direct inspection of the interference figures it can be deduced that, in general, the classical and CSL patterns are quite different as far as they are not both flattened out by the effects of the noises (environmental or fundamental). This means that we can look for other parameters values which increase the distance between the quantum and CSL patterns. We show this on the third row of Fig.~\\ref{fig:QMCSLYet} where we report the values of $\\aleph_{QCSL}$ at the values of $E_L\/a_L$ which maximize it. As it can be seen, the difference with the panels of the second row is not large apart for very light masses, meaning that the combined maximum distinguishability is nearly achievable.  \n\\\\\nFinally, Figure~\\ref{fig:csl_excl} extends the previous analysis to the whole parameter space of the CSL model. This exclusion plot is obtained for values of the parameters $t_1,t_2,E_L\/a_L$ which maximize the distinguishability between the quantum and CSL predictions, i.e., $\\aleph_{QCSL}$, as shown in the third row of Figure~\\ref{fig:QMCSLYet}. The solid lines in Figure~\\ref{fig:csl_excl} show the upper bounds that could be achieved with space-based near-field interferometry experiments with particle masses up to $10^{11}$~amu. As it can be seen, already the use of  $10^9$~amu particles (green solid line) has the potential to rule out collapse models even beyond the values GRW originally proposed for the parameters, a feat that is outside the reach of current experiments. \nThis is one of the main results of this work. It shows that near-field space-based experiments holds the promise to push tests of quantum mechanics -- and of collapse models -- way beyond what is possible with ground-based experiments and have the ability {to directly access a large and unexplored area of parameter space \\{$\\lambda_\\text{CSL} ,r_{c}$\\} of } the considered modifications of quantum  mechanics. \n}\n\n\nIn conclusion we should cite that, while the analysis presented in this section makes use of the formalism developed to account for the finite size of the particles~\\cite{PhysRevA.100.033813} , and we have included all major sources of decoherence following the technical details laid down in the ESA's QPPF report~\\cite{QPPF}, the description of the system suffers from an unavoidable level of idealization. Without entering in the discussion of technical challenges like the load and re-use of the nanoparticles in several runs of the experiment, we can still point out some of the idealizations made that enter directly into the simulations of the interferometric set-up. In particular, throughout this work we have assumed: the particles to be perfectly spherical, thus neglecting rotational degrees of freedom; the particles to be homogeneous, which has allowed us to use the formalism~\\cite{PhysRevA.100.033813} derived from Mie-scattering theory; finally we have employed the sphere's bulk material refraction index which is tabulated in the literature. This last point is discussed in some detail in Ref.~[\\citeonline{PhysRevA.100.033813}], where it is shown how the coherence properties of the grating interaction strongly depend on the refraction index. It is thus a crucial step for any realization of interferometric space-based experiments with large  nanoparticles to conduct preliminary experiments to determined the physical properties of the nanoparticles, with particular reference to their refractive index which could deviate from the bulk material one.\n\n\n\n \\section{Conclusion and Outlook}\\label{WishList}\nIn this perspective article, we have discussed the unique possibilities offered by the space environment for investigating the quantum superposition principle by dedicated interfrometric and non-interferometric experiments and to test quantum mechanics in the parameter regime of large-mass particles, impossible to reach on ground by today's technology. In particular, we have focused our attention on the generation and certification of spatial quantum superpositions of particles with sizes of the order of hundreds of nanometers and the possibilities that this offers for fundamental tests of quantum theory and alternatives thereof~\\cite{millen2020quantum}.\n\nAfter arguing for the advantages offered by space, being the long free-fall times and the availability of low-noise conditions, we considered two main experimental strategies for fundamental studies in space. The first one is the indirect approach of non-interferometric experiments, which does not require the creation of the spatial superposition. This strategy has been proven key in recent work on ground to test collapse models in otherwise unreachable parameter regimes. The second strategy is the more direct one based on interferometric experiments. Here, near-field interferometry with large dielectric nanospheres is the current powerhouse, proven experimental technology, and shows its potential when combined with the advantages of the space environment. We have reported a detailed forecast of the potential offered by these techniques based on state-of-the-art parameters values and showed how space-based experiments offer the possibility to both certify the creation of macroscopic superpositions and essentially rule-out an entire family of alternative models to standard quantum mechanics. Most importantly, we have not found a fundamental showstopper for performing both interferometric and non-interferometric experiments in space.\n\nNeedless to mention, large spatial superpositions of high-mass systems will provide a fine probe for further tests of fundamental physics. This includes: the domain of high-energy particle physics beyond the standard model, when it comes to test candidates of Dark Matter~\\cite{riedel2013direct, bateman2015existence, riedel2017decoherence, carney2019ultralight, carney2020mechanical, monteiro2020search} and possible effects in particle interactions related to Dark Energy~\\cite{khoury2004chameleon, rider2016search, moore2014search}; the low-energy regime of the interplay between quantum mechanics and gravity~\\cite{pikovski2015universal, torovs2017quantum, belenchia2016testing, belenchia2019tests, carlesso2019testing}; precision tests of gravity~\\cite{qvarfort2018gravimetry, hebestreit2018sensing, belenchia2018quantum, bose2017spin, hu2008stochastic, bahrami2014role, bassi2017gravitational, grossardt2016optomechanical}; the test of the equivalence principle and of general relativity's predictions, such as gravitational waves, in a parameter range complementary to existing experiments such as LIGO, VIRGO, GEO600 and the planned LISA space antenna~\\cite{arvanitaki2013detecting, pontin2018levitated}, and frame-dragging effects~\\cite{fadeev2020gravity}. Furthermore, large-mass experiments in space will unavoidably provide a formidable platform for applications in Earth and planet observation~\\cite{middlemiss2016measurement, banerdt2020initial}, where large-mass mechanical systems have already shown a superb capability as force and acceleration sensors~\\cite{millen2020optomechanics,li2018characterization, fadeev2020ferromagnetic, vinante2020ultralow, mitchell2020colloquium, timberlake2019static, hempston2017force, ranjit2016zeptonewton, ahn2020ultrasensitive, geraci2010short}, including in rotational mechanical modes~\\cite{carlesso2018non, stickler2018probing}; as well as their use for frequency conversion~\\cite{forsch2020microwave}\\footnote{\\MP{Whilst in a different set-ups with respect to the ones considered here.}} \n\nIt is clear that the realisation of large-mass, fundamental physics experiments in space is an immensely challenging project. Therefore, the most important next step is to form a community of scientists, industry and space agencies for defining a concrete road-map for the accomplishment of a successful space mission by working on fine-tuned theoretical analysis of conditions for the experiment, coming up with new proposals to test further new physics in the large-mass regime and, last but not least, to push the development of technology readiness for space. Such a roadmap should include performing proof-of-principle large-mass experiments in micro-gravity environments -- such a sounding rockets, drop towers and Einstein elevators, space stations, cubesats, and potentially on the Moon and Mars -- in alignment with international and national space agencies plans for future fundamental physics experiments in space. We hope that the results of this work will stimulate the physics community to further investigate the possibilities offered by space-based experiments of this kind. \n\n\n\\section*{Acknowledgements}\nA. Belenchia acknowledges support from the MSCA IF  project pERFEcTO  (Grant  No. 795782) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project number BR 5221\/4-1. A. Bassi acknowledges financial support from the INFN, the University of Trieste and the support by grant number (FQXi-RFP-CPW-2002) from the Foundational Questions Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley Community Foundation.\nS. Donadi acknowledges financial support from INFN and the Fetzer Franklin Fund. G. Gasbarri acknowledges support from the Spanish Agencia Estatal de Investigaci\\'{o}n, project PID2019-107609GB-I00, from the QuantERA grant C'MON-QSENS!, by Spanish MICINN PCI2019-111869-2, and from COST Action CA15220. {R. Kaltenbaek} acknowledges support by the Austrian Research Promotion Agency (projects 854036, 865996) and by the Slovenian Research Agency (research projects N1-0180, J2-2514, J1-9145 and P1-0125).  M. Paternostro is supported by the DfE-SFI Investigator Programme (grant 15\/IA\/2864), the Royal Society Wolfson Research Fellowship (RSWF\\textbackslash R3\\textbackslash183013), the Leverhulme Trust Research Project Grant (grant nr.~RGP-2018-266), and the UK EPSRC (grant nr.~EP\/T028106\/1). H. Ulbricht acknowledges support from The Leverhulme Trust (RPG-2016- 046) and the UK EPSRC (EP\/V000624\/1). A. Bassi, M. Carlesso, M. Paternostro, and H. Ulbricht are supported by the H2020 FET Project TEQ (Grant No. 766900). All the authors acknowledge partial support from COST Action QTSpace (CA15220) and thank  Alexander Franzen for the creation of one of the figures in the manuscript.\n\n\\section*{Author contributions}\nG. Gasbarri and A. Belenchia led the development of the project with strong input from M. Carlesso and S. Donadi and with significant contributions from A. Bassi, R. Kaltenbaek, M. Paternostro, and H. Ulbricht. All authors contributed to the preparation of the manuscript and its finalisation.\n\n\\section*{Competing Interests}\nThe authors declare no competing interests.\n\n\n\n\\section{Interferometric Experiments: Theory}\n\n\\subsection{Generalized Talbot Coefficients}\nWe report here the explicit expressions of the generalized Talbot coefficients. We refer the interested reader to Ref.~[\\citeonline{PhysRevA.100.033813}] and reference therein for the detailed derivation of these expressions and the analysis of the Talbot effect. \n\nThe generalized Talbot coefficients encode encode the coherent and incoherent effects of the optical grating on the state of the nanoparticle. Assuming the waist of the standing wave in the grating to be much bigger than the matter-wave profile, and for short interaction times ($\\tau_{int}$) compared to the characteristic time of spreading of the matter-waves, the evolution of the nanoparticle density matrix in the longitudinal position representation is given by\n\\begin{align}\\label{eq:trgrat}\n\\bra{z}\\rho\\ket{z'}\\rightarrow R(z,z')T(z,z')\\bra{z}\\rho\\ket{z'},\n\\end{align}\nwhere  \n\\begin{align}\\label{Tco}\n&T(z,z')=t(z)t^{*}(z')=e^{-\\frac{i}{\\hslash}\\int_{0}^{t}d\\tau [V(z,\\tau)-V(z',\\tau)]}\\\\\n&R\\left(z,z'\\right)= e^{\\int_{0}^{\\tau_{int}} d\\tau \\mathcal{L}_{t}(z,z')}=R_{\\rm{sca}}(z,z')R_{\\rm{abs}}(z,z').\n\\end{align}\nwith $V(z,t)$ the classical interaction potential. The term $T(z,z')$ describes the coherent effect of the grating while $R(z,z')$ the incoherent combined effect of scattering and absorption of grating photons.\nThe generalized Talbot coefficients entering the interference pattern are given by the convolution between the Talbot coefficient describing the sole coherent grating $B^{\\rm{coh}}_{n}\\left({s}\/{d}\\right)= \\sum_{k}b_{k}b^{*}_{k-n}\\exp\\{{i \\pi (n-2k)s}\/d\\}$, which are given in terms of \nthe Fourier coefficients $b_{n}$ of the transmission function $t(z)$ entering the coherent transmission mask $T(z,z')$, and the Fourier coefficients $R_{n}$ of the decoherence mask function $R(z,z')$  \n\\begin{align}\\label{eq:TalbotCoeff}\n{B}_{n}\\left(\\frac{s}{d}\\right)&=\\sum_{j}B^{\\rm{coh}}_{n-j}\\left(\\frac{s}{d}\\right)R_{j}\\left(\\frac{s}{d}\\right)\\\\\n&= e^{F-c_{\\rm{abs}}\/2} \\sum_{k=-\\infty}^{\\infty}\\left(\\frac{\\zeta_{\\rm{coh}}+a+c_{\\rm{abs}}\/2}{\\zeta_{\\rm{coh}}-a-c_{\\rm{abs}}\/2}\\right)^{\\frac{n+k}{2}}J_{k}(b)J_{n+k}\\left(\\sign(\\zeta_{\\rm{coh}}-a-c_{\\rm{abs}}\/2)\\sqrt{\\zeta_{\\rm{coh}}^{2}-(a+c_{\\rm{abs}}\/2)^{2}}\\right).\n\\end{align}\nHere    \n\\begin{align}\\label{abF}\nc_{\\rm{abs}}&=\n\\frac{4\\sigma_{abs}}{ h c}\\frac{E_{L}}{a_{L}}(1-\\cos(\\pi s\/d)\\nonumber\\\\\n\\zeta_{coh} &= \\frac{16F_{z}(-\\lambda\/8) E_L}{\\hslash a_L k I_{0}}\\sin\\left(\\frac{\\pi s}{d}\\right)\\nonumber\\\\\na(s) &= \\int d\\tau \\frac{8\\pi  E_{L}}{\\hslash \\omega a_{L}}\\int d \\Omega\\,{\\rm Re}\\big(f^{*}(k , k \\mathbf{n})f(- k , k\\mathbf{n})\\big)[\\cos(k n_{z} s)-\\cos(ks)],\\nonumber\\\\\nb(s)&=  \\int d\\tau\\frac{8\\pi  E_{L}}{\\hslash  \\omega a_{L}}\\int d \\Omega \\,{\\rm Im} \\big(f^{*}(k , k \\mathbf{n})f(- k , k\\mathbf{n})\\big)\\sin(kn_{z}s),\\nonumber\\\\\nF(s)&=  \\int d\\tau \\frac{8 \\pi E_{L}}{\\hslash  \\omega a_{L}}\\int d \\Omega\\, |f(k,k\\mathbf{n})|^{2} [\\cos((1-n_{z})ks)-1]\n\\end{align}\nwhere $\\lambda= 2\\pi \/ k= 2\\pi \/\\omega c $ is the light wavelength, $E_{L}$ is grating laser energy, $I_{0}$ and $a_{L}$ are respectively the intensity parameter and the spot area of the laser, $F_{z}(z)$ is the longitudinal light-induced force on the dielectric sphere, $\\sigma_{abs}$ is the photon absorption cross section, and $f(k,k\\mathbf{n})$ the photon scattering cross section.\n\n\\subsubsection{Classical Limit of the optical grating}\\label{ClaLim}\nAs we discussed in the main text, also classical particles following ballistic trajectories through the grating laser would form a near-field shadow pattern. The effect of the optical grating on classical particles can be described by modifying in a suitable way the generalized Talbot coefficients. In particular, the classical behaviour can be obtained as the limiting case of $\\hslash \\to 0$ as outlined in Ref.~[\\citeonline{PhysRevA.100.033813}]. To summarize, the classical analogue of the coherent effect of the grating corresponds to a momentum kick due to the classical optical force acting on a dielectric and, by performing an expansion of the trigonometric functions in $a,b, F$ to first order in $\\hslash$, it is easy to see that the only non-vanishing contribution to the classical coefficients arise from the function $b$. \n\n\n\n\\subsection{Environmental Decoherence}\nAs discussed in the main text, our simulations account for several sources of environmental decoherence acting during the free fall times in the experiment.  In particular, we account for decoherence due to collision with the residual gas in the vacuum chamber; black-body radiation, emission, scattering, and absorption -- taking into account the heating of the particle (photonic environment) during the trapping period and the subsequent cooling during free-fall. Here we report the expression for the kernel $R_{n}$ in Eq.~(3) in the main text that account for all these decoherence sources.\n\\begin{align}\n\\ln(R_{n})=& -\\Gamma_{\\text{coll}}(t_{1}+t_{2})+\\int d\\omega \\gamma_{\\text{abs}}(\\omega)\\left[\\frac{\\text{Si}(a_{n})}{a_{n}}-1\\right](t_{1}+t_{2})+\\int d\\omega \\gamma_{\\text{sca}}(\\omega)\\left[\\frac{\\text{Si}(2a_{n})}{a_{n}}-\\text{sinc}^{2}(a_{n})-1\\right](t_{1}-t_{2})\\nonumber\\\\\n&\\int d\\omega \\int_{0}^{1} d\\theta \\{t_{1}\\gamma_{\\text{emi}}[\\omega,T_{\\text{int}}(t_{1}-t_{1}\\theta)]+t_{2}\\gamma_{\\text{emi}}[\\omega,T_{\\text{int}(t_{1}+t_{2}\\theta)] }[\\text{sinc}(a_{n}\\theta)-1]\n\\end{align}\nwith $a_{n}= n\\,h\\,\\omega\\, t_{2} t_{1} \/ ((t_2+t_1)m\\,c\\,d) $, and $\\text{Si}$ the sine integral. The collision rate $\\Gamma_\\text{coll}$ is given by\n\\begin{align}\n\\Gamma_{\\text{coll}} = \\frac{ 4\\pi \\Gamma(9\/10)}{5\\sin(\\pi\/5)}\\left(\\frac{3\\pi C_{6}}{2\\hslash}^{2\/5} \\frac{p_{g}v_{g}}{k_{\\text{B}}T_{\\text{env}}}\\right)\n\\end{align}\nwhile the scattering, emission and absorption rates by\n\\begin{align}\n\\gamma_{\\text{sca\/abs}}(\\omega) =\\frac{ (\\omega \/ \\pi c)^{2}\\sigma_{\\text{sca\/abs}}(\\omega)}{\\exp(\\hslash \\omega\/k_{\\text{B}}T_{\\text{env}})-1},\\hspace{1cm}\n&\\gamma_{\\text{emi}}[\\omega, T_{\\text{int}}]= \\left(\\frac{\\omega}{\\pi c}\\right)^{2}\\sigma_{\\text{abs}}\\exp\\left(-\\frac{\\hslash \\omega}{k_{\\text{B}}T_{\\text{int}}}\\right)\\text{Im}\\left\\{\\frac{\\varepsilon(\\omega)-1}{\\varepsilon(\\omega)+2}\\right\\}\n\\end{align}\nwhere  $v_{g}$ and $p_{g}$ are the mean velocity and pressure of the gas, $T_{\\text{env}}$  the environmental temperature, $\\sigma_{\\text{abs\/sca}}$ the photon scattering\/absorption cross section, $\\varepsilon(\\omega)$ the electric permittivity, and \n\\begin{align}\nC_{6} \\simeq \\frac{ 3 \\alpha(\\omega=0) \\alpha_{g} I_{g}I}{32 \\pi^{2}\\varepsilon_{0}^{2}(I_{g}+I)}\n\\end{align}\nthe van der Waals coupling constant where $\\alpha$, $\\alpha_{g}$ are the static polarizabilities and $I$, $I_g$ the ionization energies of the nanosphere and the gas particle, respectively.\n\nWe refer the reader to the supplemental information of~[\\citeonline{bateman2014near}] and the reference therein for a detailed derivation of these expressions.\n\n\\subsection{ CSL-Decoherence}\nThe dissipative term describing the effective decoherence of the center-of-mass wavefunction -- for the reduced one-dimensional state of motion of a nanoparticle -- due to the CSL reads\n\\begin{align}\\label{cslme}\n\\mathcal{L}_{CSL}(x,x')=-\\frac{\\lambda_{CSL}(4\\pi r_C^2)^{3\/2}}{(2\\pi\\hslash)^{3}}\\int d\\mathbf{q}\\,\\frac{\\tilde{\\mu}(\\mathbf{q})^2}{m_0^2}\\,e^{-r_C^2\\mathbf{q}^{2}\/\\hslash^{2}}\\,\\left(e^{-\\frac{i}{\\hslash}{q}_{z}({x}-{x'})}-1\\right)\n\\end{align}   \nin position representation. Here $m_{0}$ the nucleon mass , $\\lambda_{CSL}$ and $r_{c}$ the rate and the localization length of the CSL model, and $\\mu(\\bf{q})$ the Fourier transform of the nano-particle mass density $\\mu(\\bf{x})$\n,i.e.\n\\begin{align}\n\\tilde{\\mu}(\\mathbf{q}):= \\int e^{-\\frac{i}{\\hslash} \\mathbf{q}\\cdot\\mathbf{x}}\\mu(\\mathbf{x}).\n\\end{align}\nthat in the case of a homogeneous and spherical mass distribution of radius $R$ is given by\n\\begin{align}\n \\tilde{\\mu}(\\mathbf{q})= \\frac{4 \\pi \\hslash R^2}{q} J_{1}(q R\/\\hslash)\n\\end{align}\nwhere $J_{1}(q)$ denotes the spherical Bessel function of the first kind.  \nExploiting this equation, rewriting the integral in Eq.~\\eqref{cslme} in spherical coordinates, and performing the integration over the solid angle the CSL dissipative term simplifies to \n\\begin{align}\n\\mathcal{L}_{CSL}(\\mathbf{x},\\mathbf{x}')= \\frac{64 \\pi^{3\/2}\\,\\lambda_{CSL}\\, rc^{3} R^{4}}{\\hslash\\, m_{0}^{2}}\\int_{0}^{\\infty} d q\\, e^{-r_{c}^2 q^2\/\\hslash^2} J_{1}(q R\/\\hslash)^{2}\\left(\\text{sinc}(q|{x}'-{x}|)\/\\hslash)-1 \\right).\n\\end{align}\nThis results in the following, additional kernels entering the expression for the interference pattern probability:\n\\begin{align}\nR_{n}^{\\text{CSL}}= \\exp\\left\\{ \\Gamma_{\\text{CSL}}\\left(f_{\\text{CSL}}\\left(\\frac{ h\\, n\\, q\\, t_1 t_2}{m\\, d (t_1+t_2)}\\right)-1\\right)(t_1+t_2)\\right\\}\n\\end{align}\nwhere $d$ is again the grating period and \n\\begin{align}\n\\Gamma_{\\text{CSL}}&=  \\sqrt{\\frac{32}{\\pi}}\\frac{\\lambda_{\\text{CSL}}\\, r_{c}^{3}}{\\hslash^{3} m_{0}^{2}} \\int dq q^2 e^{-r_{c}^{2}q^2\/\\hslash^2}\\tilde{\\rho}(q)^{2}  \\nonumber\\\\\nf_{\\text{CSL}}(x)&=\\sqrt{\\frac{32 }{\\pi}} \\frac{\\lambda_{CSL}\\,r_{c}^{3}}{\\hslash^3 m_{0}^{2} \\Gamma_{\\text{CSL}}}\\int dq q^{2} e^{-r_{c}^{2}q^{2}\/\\hslash^{2}} \\tilde{\\rho}(q)^{2} \\text{Si}\\left(\\frac{q x}{\\hslash}\\right) =\\frac{64 \\pi^{3\/2}\\lambda_{CSL} rc^{3} R^{4} }{\\hslash m_{0}^{2} \\Gamma_{\\text{CSL}}} \\int_{0}^{\\infty}dq e^{-q^2 r_{c}^{2}\/\\hslash^{2}}J_{1}\\left(\\frac{q R}{\\hslash}\\right)^{2}\\text{Si}\\left(\\frac{qx}{\\hslash}\\right).\n\\end{align}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1.\\textwidth]{QM-Collapse.pdf}\n\\caption{Comparison between quantum mechanics prediction and CSL one. In particular, the first row of the panel shows the cost function $\\aleph_{QCSL}$, for increasing mass from left to right, as a function of $t_1, t_2$. The second row shows the values of the $E_{L}\/a_{L}$ ratio that maximise the cost function $\\aleph_{QCSL}$.\n}\\label{fig:csl_cost_del}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1.\\textwidth]{QMCSL.pdf}\n\\caption{Comparison between quantum mechanics, classical shadow pattern and the prediction of CSL. The first row shows the value of $\\aleph_{QC}$ for increasing value of the mass while the second row shows the value of $\\aleph_{QCSL}$. The third row shows instead the values of the $E_{L}\/a_{L}$ that have been used in both the first and second row figures. These values of the $E_{L}\/a_{L}$ ratio are the one which maximize $\\aleph_{QC}$. \n}\\label{fig:QMCSL}\n\\end{figure*}\n\n\n\\section{Interferometric Experiments: Simulations}\nHere we report additional details for the simulations performed for near-field intereferometric experiments in space. In Table~\\ref{table} we report all the relevant parameters used in the simulations leading to the results in section~IV.\n\nIt should be noted that, in the results shown, we did account for the heating of the nanoparticle while initially optically trapped for a time $t_c$ by assuming an internal temperature of $40$~K after time $t_c$ in an environment at $20$~K. This is a conservative estimate and we also run the same simulations for higher and lower values of the initial internal temperature without noticing any significant deviation. \n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{llll}\n    \\hline\n    \\textbf{Symbol}  & & \\textbf{Name} & \\textbf{Value}  \\\\ \n    &&&{\\bf or Expression}\\\\\n    \\hline\\hline\n    Nanosphere\\\\ properties:\\\\ \\hline\n   \n    $\\rho_{SiO2}$ & & Glass SiO2 density & $1850$~kg\/m$^3$ \\\\\n    $c_{m}$ & & Specific heat  & 700~J\/(kg K) \\\\\n   \n    $I$ & & Ionization energy & $5\\times 10^{-19}$~J \\\\\n   \n    $m$ & & Mass & $10^7 - 10^{11}$~amu \\\\ \n    $\\sigma_x$ & & Position variance & $\\sqrt{\\frac{\\hslash}{4\\gamma}\\coth{({\\beta_0\\nu_m})}}$ \\\\\n    &&post-cooling &\\\\\n    $\\sigma_p$ & & Momentum variance& $\\sqrt{\\hslash\\gamma\\coth{({\\beta_0\\nu_m})}}$ \\\\\n    &&post-cooling &\\\\\n    \\hline \n    Trapping\\\\ parameters:  \\\\\n    \\hline\n    $\\lambda_c$ & & Trap's laser wavelength & 1550~nm \\\\\n    $t_c$ & & Trapping and cooling time & 1~s \\\\\n    $I_{{\\rm trap}}$ & & Trap's laser intensity & $90\\times 10^9$~W\/m$^2$ \\\\\n    $\\nu_{m}$ & & Trap mechanical frequency  &  $10^5\/2\\pi$~Hz \\\\%$200$~Hz\n     &&--longitudinal direction &\\\\\n    $T_{{\\rm int}0}$ & & C.o.m. temperature  &  $\\sim 5\\times 10^{-6}$~K\\\\%$20$~mK \n    &&post-cooling &\\\\\n    \\hline\n    Optical grating\\\\ parameters: \\\\ \\hline\n    $\\lambda$ & & Grating laser wavelength & $100$~nm \\\\\n    $E_L\/a_L$ & & Pulse-energy per spot-area & $10^{-6}- 5$ J\/m$^{2}$  \\\\ \n   \n    \\hline\n    Environment\\\\ parameters: \\\\ \\hline\n    $T_{\\rm env}$ & & Environmental temperature & 20~K \\\\\n    $\\alpha_g\/(4\\pi\\epsilon_0)$ & &  Residual gas polarizability & $0.6668\\times 10^{-30}$~m$^3$ \\\\\n    $I_g$ & & Residual gas ioniz. energy & $2.17\\times 10^{-18}$~J\\\\\n    $m_g$ & & Residual gas mass & 1.00784~amu\\\\\n    $v_g$ & & Residual gas mean velocity  & $\\sqrt{2k_B T_{\\rm env}\/m_g}$\\\\\n    $P_g$ & & Residual gas pressure & $10^{-13}$~mbar \\\\ \\hline\n\\end{tabular}\n\\caption{Specifics of all the parameters entering the simulation of the near-field interferometric experiments with dielectric nanospheres. The residual gas in the vacuum chamber is assumed to be composed mostly by nitrogen, of which we use the physical properties. The refractive index, as a function of the frequency, for Si and SiO2 can be found tabulated in the supplementary material of Bateman et al.~\\cite{bateman2014near} (see also references therein). The pulse energy to spot area ratio of the grating laser is related to the eikonal phase by Eqs.~(20-22) in Ref.~[\\citeonline{PhysRevA.100.033813}] to which we refer the reader for additional details.\nWe have set $\\beta_0=h\/(2k_B T_{{\\rm int}0})$ and $\\gamma=\\pi m\\nu_m$.}\\label{table}\n\\end{table}\n\nIn Figure~\\ref{fig:csl_cost_del} we report the values of $\\aleph_{QCSL}$ (first row) for different masses of the nanoparticles and as a function of $t_1, t_2$. This correspond to the panels in the third row of Figure~6 in the main text. The panels on the second row of the figure show the values of $E_L\/a_L$ used. These are the values that maximize $\\aleph_{QCSL}$.\n\nIn Figure~\\ref{fig:QMCSL} we report instead both the values of $\\aleph_{QC}$ (first row) and the ones of $\\aleph_{QCSL}$ (second row) obtained using the values of $E_L\/a_L$ shown in the third row which that maximize $\\aleph_{QC}$.\n\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzppta b/data_all_eng_slimpj/shuffled/split2/finalzzppta
new file mode 100644
index 0000000000000000000000000000000000000000..5c3efac95a20d1de1d396d28e398e5f28c6a3d76
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzppta
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\\label{Intro}\n\nLight from distant galaxies and quasars is gravitationally lensed by mass along the line of sight. For a flux-limited survey of quasars, lensing magnification biases the observed number density \\cite{Turner:1984ch,Fugmann:1988,Narayan:1989}. For example, a positive mass fluctuation along the line of sight can increase the apparent sky area and the observed flux. The geometrical area increase decreases the quasar number density, while the flux increase promotes intrinsically faint objects above the magnitude threshold increasing the source density.  Together these effects introduce a correction to the quasar number density called magnification bias. Detections of the magnification of distant quasars by low-redshift galaxies confirm the presence of this effect (e.g. \\cite{Gaztanaga:2002qk,Scranton:2005ci,Menard:2009yb}). \n\nMany authors have studied the effect of lensing magnification on observations of the galaxy correlation function and power spectrum. Another way of inferring the large-scale mass distribution is through measurements of the neutral Hydrogen density. As light from distant quasars passes through clouds of neutral Hydrogen, photons with rest-frame frequency at the Lyman-$\\alpha$ transition ($1216 \\AA$) are absorbed. The observed quasar spectrum contains troughs corresponding to absorption by neutral Hydrogen at redshift $z=\\nu_\\alpha\/(\\nu_{\\rm obs}(1+ v_{\\rm pec})) -1$, where $\\nu_{\\rm obs}$ is the observed frequency of the trough,\n$v_{\\rm pec}$ is the peculiar velocity at $z$ (speed of light is set to unity), and $\\nu_{\\alpha}$ is the Lyman-$\\alpha$ frequency.\\footnote{Here, the redshift $z$ refers to the intrinsic or cosmological redshift, i.e. the redshift if the absorbing materials were comoving. The observed redshift and the intrinsic redshift are related by $1 + z_{\\rm obs} = (1 + z)(1 + v_{\\rm pec})$.} This Lyman-$\\alpha$ forest (of absorption features in the quasar spectrum) is a cosmological tool that can be used to probe the neutral Hydrogen along the line of sight (see for example \\cite{Meiksin:2009} and references therein). Here we ask two questions: what is the effect of gravitational lensing on measurements of the Lyman-$\\alpha$ forest? And how can we extract the gravitational signal from such measurements?\n\nLet us first discuss qualitatively what we expect to happen. Consider a single line of sight towards a quasar at comoving distance $\\chi_Q$. Lensing magnification changes the observed quasar flux $f$, but in a frequency ($\\nu$)\nindependent way \n \\begin{equation}\n\\label{Eq:fmag}\n  f(\\nu)\\rightarrow f(\\nu)\n\\cdot \\mu (\\chi_Q) \n\\end{equation} \nwhere ${\\mu}(\\chi_Q)$ is the magnification which depends on the density fluctuations along the line of sight. Since all frequencies are treated in the same way, fluctuations in the flux as a function of frequency are \\emph{unaffected} --- just as if we were looking at an intrinsically brighter or dimmer quasar. Measurements of for example, the flux power spectrum, from an \\emph{individual} line of sight are unchanged by magnification. On the other hand, magnification does change \\emph{which} lines of sight are observed. More precisely, a selection bias is introduced: more (fewer) quasars are observed along lines of sight that lead to a positive (negative) magnification correction to quasar number counts. Let ${n}(\\chi, \\mbox{\\boldmath$\\theta$})$  be the number density of quasars at comoving distance $\\chi$ in direction $\\mbox{\\boldmath$\\theta$}$. Under the effect of lensing magnification,\n\\begin{equation} \n\\label{nlensed}\n{n}(\\chi, \\mbox{\\boldmath$\\theta$}) \\rightarrow {n}(\\chi, \\mbox{\\boldmath$\\theta$}) {\\mu}(\\chi)^{2.5s-1},  \\qquad s= {1\\over {\\rm ln\\,} 10} {\\int dm \\, \\epsilon(m) (dn^0\/dm) \\over \\int dm \\, \\epsilon(m) n^0(m)} \\, .\n\\end{equation}\nHere, $n^0(m)$ is the luminosity function of quasars at magnitude $m$ (i.e. $dm \\, n^0(m)$ is the number density of quasars at magnitude $m \\pm dm\/2$), and $\\epsilon(m)$ quantifies the sample definition -- the simplest example is a step function which cuts off all quasars fainter than some limiting magnitude $m_{\\rm lim.}$, in which case $s$ reduces to the more familiar expression $s = d\\log_{10} N(< m_{\\rm lim.}) \/ dm_{\\rm lim.}$,  where $N(< m_{\\rm lim.})$ is the total number of quasars brighter than $m_{\\rm lim.}$. This is a well known result, but a derivation is summarized in Appendix \\ref{derive}. With lensing magnification, the lines of sight we observe do not constitute a fair sample of the density field. Therefore, while lensing magnification has \\emph{no} effect on measurements of the flux fluctuation from a \\emph{single} quasar line of sight, measurements dependent upon the density field \\emph{averaged over multiple quasars} \\emph{will} be biased.  This occurs since the observable (e.g. the neutral hydrogen density) is correlated with the magnification which depends on the gravitational potential along the line of sight.\n\nWhat is perhaps the more interesting question is how we could extract the lensing signal from observations of the forest. The discussions above make clear gravitational lensing\naffects both the observed brightness and number density of quasars. One could imagine cross-correlating these quantities with the Lyman-alpha forest observables. We will consider several possibilities and identify the ones with an interesting signal-to-noise.\n\nThe rest of the paper is organized as follows. In \\S\\ref{LensBiasDeriv} we develop a simple description of the effect of lensing magnification on measurements of the Lyman-$\\alpha$ forest, the key result is Eq.  (\\ref{Eq:MagBiasFinal}). In \\S \\ref{Sims}--\\S\\ref{FluxPDF} we present the effect of lensing bias on the flux power spectrum, the effective optical depth and the flux probability distribution (PDF) determined by applying a biased weighting scheme to mock absorption spectra from simulations and compare these results with analytical estimates in terms of the flux-mass polyspectra. In \\S \\ref{MagCorrelation} we discuss the exciting possibility of observing the lensing-induced correlation between quasar magnitudes and the flux fluctuations, the flux power spectrum or the mean flux. While this work is dedicated to discussing magnification bias, dust along the line of sight would have a similar (though generally opposite signed) effect, this is discussed in \\S \\ref{dust}. We present concluding remarks and discuss the implications of this work for existing and future Lyman-$\\alpha$ measurements in \\S \\ref{Conclusions}. Appendix \\ref{derive} contains a unified derivation of the magnitude-forest correlations and the lensing bias discussed in the paper. In Appendix \\ref{Extrap} we discuss some issues regarding large-scale flux-mass correlations measured from our simulations and the accuracy of an analytic description. Appendix \\ref{estimator} derives an estimator for the flux-magnitude correlation and its associated error.  \n\nBefore proceeding we should mention some related literature. The magnification bias to the statistics of metal absorption lines in quasar spectra was investigated in \\cite{Thomas:1990aj} and a method for detecting statistical lensing by absorbers was proposed in \\cite{Menard:2005}. Lensing effects on the statistics of damped Lyman-$\\alpha$ systems and the inferred density of neutral Hydrogen was considered in \\cite{Bartelmann:1996}. More recently, \\cite{Menard:2003vf} studied magnification bias due to intervening absorbers in the 2dF quasar survey and \\cite{Menard:2008} in SDSS. Recent work by \\cite{Vallinotto:2009wa,Vallinotto:2009jx} proposed correlating lensing in the cosmic microwave background with fluctuations in the forest to extract the flux-mass correlation.\n\n\\section{Lensing as Noise\/Bias}\n\\label{lensingnoise}\n\n\\subsection{Formalism}\n\\label{LensBiasDeriv}\n\nWe are interested in some Lyman-alpha forest observable $\\mathcal{O}$, which can represent the flux power spectrum, the flux transmission\/decrement,\nthe flux fluctuation (around its mean), and so on. Let us use $O_I$ to denote the observable measured from a quasar labeled by $I$. We typically form an estimator by averaging over quasars:\n\\begin{equation} \n\\label{Eq:Osum}\n{\\mathcal{O}}_{\\rm obs}=\\frac{\\sum_I w_I {\\mathcal{O}}_I}{\\sum_I w_I} \n\\end{equation} \nwhere $w_I$ denotes weights, the simplest example of which is $w_I = 1$.\n\nTwo important points. First $\\mathcal{O}_I$, the observable on a quasar by quasar basis, is generally {\\it not} affected by gravitational lensing. This is because gravitational lensing affects all wavelengths equally. For instance, $\\mathcal{O}_I$ could represent the flux transmission which is $f\/f_C$ (where $f$ and $f_C$ are the flux and continuum respectively as a function of frequency), or the flux fluctuation $\\delta_f = (f - \\bar f)\/\\bar f$ (where $\\bar f$ is the flux averaged for the particular line of sight in question). Since gravitational lensing brightens or dims $f$, $\\bar f$ and $f_C$ all equally independent of wavelength, there is no effect on $\\mathcal{O}_I$.\n\nThe other important point is that $\\mathcal{O}_{\\rm obs}$ {\\it is} generally affected by  lensing. The crucial observation is that $w_I$ in Eq. (\\ref{Eq:Osum}) reveals only part of the weighting that is going on; in any given dataset, we inevitably give zero weights to quasars which are too faint to observe. This means that even if one chooses $w_I = 1$, one is merely performing a straight average over one's sample, as opposed to an average over all possible quasars. To account for this fact, it is helpful to  pixelize the sky, and rewrite ${\\mathcal{O}}_{\\rm obs}$ as \n\\begin{eqnarray}\n\\label{Eq:Osum2}\n{\\mathcal{O}}_{\\rm obs} = {\\sum_i w_i n_i \\mathcal{O}_i \\over \\sum_i w_i n_i}\n\\end{eqnarray}\nwhere $i$ is the pixel label. One could conceptually think of each pixel as sufficiently small that the number of quasars in it $n_i$ is either $1$ or $0$; we will more generally think of $n_i$ as simply the number density of quasars in pixel $i$. This way of rewriting is useful because it makes explicit the fact that $\\mathcal{O}_{\\rm obs}$ is a weighted average -- it is weighted by the number density of quasars, on top of whatever additional weighting $w_i$ one might wish to apply. In other words, since our Lyman-alpha forest observable can be measured only if there exists a background quasar in the sky location of interest, the observable is always implicitly weighted by the abundance of quasars.\n\n\nEq. (\\ref{nlensed}) tells us that\n\\begin{eqnarray} \nn_i = n_i^{\\rm intrinsic} (1 + \\delta^\\mu_i)\n\\end{eqnarray}\nwhere $n_i^{\\rm intrinsic}$ is the intrinsic pre-lensed quasar number density, and $\\delta^\\mu_i$ is the lensing modification given by\n\\begin{eqnarray}\n\\label{Eq:KappaDef}\n\\delta^\\mu_i \\equiv (5s - 2) \\kappa_i \\quad , \\quad {\\kappa}_i =\\int_0^{\\chi_Q} d\\chi'\\frac{\\chi_Q-\\chi'}{\\chi_Q}\\chi' \\nabla_\\perp^2\\phi(\\chi', i)\n\\end{eqnarray}\nassuming fluctuations are small.  The expression for the convergence $\\kappa_i$ assumes spatial flatness, but can be easily generalized to non-flat universes ($\\chi_Q$ is the comoving distance to quasar).  Here, $1 + 2\\kappa$ is the weak lensing approximation to the lensing magnification $\\mu$ of Eqs. (\\ref{Eq:fmag}) and (\\ref{nlensed}) (the full nonlinear expression is given in footnote \\ref{nonlinfootnote}). Using $w_i = 1$ in Eq. (\\ref{Eq:Osum2}), assuming there is no correlation between the forest observable and the {\\rm intrinsic} quasar number fluctuation, and ignoring corrections of the integral constraint type (see Appendix \\ref{derive} and for example, \\cite{Hui:1998ix}), we obtain the following ensemble average,  to the lowest order in fluctuations:\n\\begin{eqnarray}\n\\langle \\mathcal{O}_{\\rm obs} \\rangle = \\mathcal{O}_{\\rm true} + \\langle \\mathcal{O}_i \\delta^\\mu_i \\rangle\n\\end{eqnarray}\nwhere $\\langle \\mathcal{O}_i \\delta^\\mu_i \\rangle$  is the correlation between the observable and the magnification fluctuation at the same $i$ (i.e. zero lag). Dropping the $i$ label, the lensing induced measurement bias on an observable $\\mathcal{O}$ is therefore\n\\begin{eqnarray}\n\\label{Eq:MagBiasFinal}\n\\langle \\mathcal{O}_{\\rm obs} \\rangle - \\mathcal{O}_{\\rm true} = \\langle \\mathcal{O} \\delta^\\mu \\rangle = (5s-2) \\langle \\mathcal{O} \\kappa \\rangle \\, .\n\\end{eqnarray}\nThis is a fundamental expression that we will use repeatedly below. The meaning of each of these symbols is as follows: $\\mathcal{O}$ is the fluctuation which we attempt to measure with the estimator $\\mathcal{O}_{\\rm obs}$, which upon ensemble averaging  generally differs from the true underlying value $\\mathcal{O}_{\\rm true} = \\langle \\mathcal{O} \\rangle$.\n\nIt is important to emphasize the actual measurement bias could be different. The above estimate assumes that all quasars {\\it within one's sample} are weighted equally (i.e. $w_i = 1$).  In practice, one might want to weigh brighter quasars {\\it within one's sample} more strongly. For instance, one could weigh by the net flux of the quasar, which corresponds to inverse variance weighting in the noise dominated regime. This would result in a lensing induced measurement bias of (see Appendix \\ref{derive} for derivation)\n\\begin{eqnarray}\n\\label{Eq:MagBiasFluxWeight}\n\\langle \\mathcal{O}_{\\rm obs} \\rangle - \\mathcal{O}_{\\rm true} = (5s' - 2) \\langle \\mathcal{O} \\kappa \\rangle \\, .\n\\end{eqnarray}\nwhere $s'$ is defined as\n\\begin{eqnarray}\n\\label{sprime}\ns' = {1\\over {\\rm ln\\,} 10} {\\int dm \\, \\epsilon(m) (dn^0\/dm) 10^{-m\/2.5} \\over\\int dm \\, \\epsilon(m) n^0(m) 10^{-m\/2.5}}\n\\end{eqnarray}\nwhich can be contrasted with $s$ defined in Eq. (\\ref{nlensed}). For a step function sample definition $\\epsilon$, $s'$ reduces to $d\\log_{10} N'(< m_{\\rm lim.}) \/ dm_{\\rm lim.} + 0.4$, where $N' = \\int_{-\\infty}^{m_{\\rm lim.}} dm \\, n^0 10^{-m\/2.5}$. This is generally larger than $s$.\n\nThe precise value of $s$ or $s'$ is sample dependent. The relatively low redshift ($1<z<2.2$) Sloan Digital Sky Survey (SDSS)  quasars that were used to measure the magnification-galaxy cross-correlation  have a slope that spans $-1 \\lsim 5s-2 \\lsim 1.9$, depending on  the magnitude cut \\cite{Scranton:2005ci}. For this paper, the higher redshift quasars for which the Lyman-alpha forest falls within the SDSS spectral range are more relevant. In Fig. \\ref{Fig:slope} (in Appendix \\ref{derive}) we show the cumulated number counts and rough estimates of $s$ and $s'$ from SDSS data release $6$.\n\nIn the rest of this paper, we will adopt $s = 1$ (or $s' = 1$), so our results for the lensing biases can be scaled up and down by $(5s-2)\/3$ or $(5s'-2)\/3$.\n\n\n\nWe will also adopt the following set of cosmological parameters in presenting numerical estimates. We assume a flat $\\Lambda$CDM cosmology with cosmological parameter values $\\Omega_m=0.3$, $\\Omega_\\Lambda=0.7$, $\\Omega_b=0.04$ for the fractional densities in matter, vacuum and baryons;  scalar spectral index $n_s=1$; Hubble parameter today $H_0=100h$ km\/s\/Mpc with $h=0.7$ and fluctuation amplitude $\\sigma_8=0.9$. The speed of light is set to unity. In a few places we use analytic calculations of the matter and baryon power spectra. The 3-D matter power spectrum is calculated using the transfer function of \\cite{Bardeen:1985tr} with a modified shape parameter $\\Gamma=\\Omega_mh {\\,\\rm exp}\\,[{-\\Omega_b(1+\\sqrt{2h}\/\\Omega_m)}]$ \\cite{Sugiyama:1994ed}, and the non-linear evolution is modeled according to \\cite{Peacock:1996ci}. To relate baryons to mass we assume the 3-D Fourier space fluctuations are related by $\\delta_b({\\bf k})={\\,\\rm exp}\\,[{-k^2\/k_s^2}]\\delta ({\\bf k})$, where $k_s= k_J \\sqrt{10\/3}$ and $k_J$ is the Jeans scale (see Appendix \\ref{Extrap} for more details).\n\n\\subsection{Lensing Bias from Simulations}\n\\label{Sims}\n\nLensing magnification biases measurements of the forest so long as the forest observable is correlated with the lensing convergence (Eq. (\\ref{Eq:MagBiasFinal})). Since features in the Lyman-$\\alpha$ forest are caused by absorption by gas which itself traces gravitational potentials, there ought to be a correlation between absorption features and the lensing convergence. However, the mapping between absorption-induced fluctuations in the flux and the gravitational potential is nonlinear and dependent on potentially uncertain physics, such as fluctuations in the ionizing background. For this reason we need to appeal to simulations to quantify the correlation between forest observables (e.g. $\\delta_f$ and $P_{ff}$) and the lensing convergence. \n\nIn this section we use hydrodynamic simulations to study the effect of magnification bias on the Lyman-$\\alpha$ forest. Specifically, we study the ``D5\" simulation of \\cite{Springel:2002uv}. This simulation was run using an entropy-conserving \\cite{Springel:2001qb} version of the smoothed particle hydrodynamics (SPH) code Gadget \\cite{Springel:2000yr}. It tracks $324^3$ dark matter particles and $324^3$ gas particles in a simulation volume of co-moving side length $L = 33.75$ Mpc\/$h$, and includes a subresolution model for star formation \\cite{Springel:2002ux} and heating by a uniform background radiation field \\cite{Katz:1995up}. Further details regarding the simulation can be found in \\cite{Springel:2002uv}. We generate mock Lyman-alpha forest spectra along random lines of sight through the simulation volume in the usual way, integrating through the SPH kernels of the particles, and incorporating the effect of peculiar velocities and thermal broadening (e.g. \\cite{Hernquist:1995uma}). We generate mock spectra from simulation snapshots at $z=2,3$ and $4$. In each case, we adjust the amplitude of the photoionizing background in the simulation to match the observed mean transmitted flux from \\cite{FaucherGiguere:2007ys}. We also make use of the ``G6\" simulation (see \\cite{Nagamine:2005jp}), which tracks $2 \\times 486^3$ particles in a $L = 100$ Mpc\/$h$ box, in order to check the sensitivity of our results to finite boxsize effects in Appendix \\ref{Extrap}. We have checked that the flux power spectra from these simulations approximately match the existing measurements.\n\nMagnification bias biases which lines of sight an observer will see: sight lines with positive fluctuations in magnification are more abundant and those with negative less abundant. With simulations we can of course see every line of sight (a mock spectrum is generated regardless of whether there is a quasar behind it) so we mimic lensing bias by weighting each line-of-sight $\\mbox{\\boldmath$\\theta$}$ by  $1+\\delta^\\mu(\\mbox{\\boldmath$\\theta$})$.\nThe simulations are in boxes of size $33.75 \\,\\textrm{Mpc}\/\\textrm{h}$, so we can't actually calculate the lensing convergence $\\kappa$ along the entire line-of-sight but instead for the weighting of absorption measurements at redshift $\\bar{z}$ we use\\footnote{While it would be more accurate to leave the $\\chi$ terms inside the integral, their amplitude is slowly varying across the length of our simulations box so Eq. (\\ref{eq:weightdef}) shouldn't be a bad approximation.}\n\\begin{equation}\n\\label{eq:weightdef}\nw_{\\mbox{\\boldmath$\\theta$}}=1+\\delta^\\mu (\\mbox{\\boldmath$\\theta$}) \\sim\n1+\\frac{3}{2}H_0^2\\Omega_m(5s-2)\\frac{\\chi_Q-\\bar{\\chi}}{\\chi_Q}\\bar{\\chi}(1+\\bar{z})\\int_{\\bar{\\chi}}^{\\bar{\\chi}+L} d\\chi' \\delta(\\chi', \\mbox{\\boldmath$\\theta$})\n\\end{equation}\nwhere $L$ is the length of the box $\\bar{\\chi}=\\chi(\\bar{z})$, $\\chi_Q$ is the distance to the quasar, $\\delta$ is the mass fluctuation, and we have used the approximation $\\nabla_\\perp^2\\phi\\approx\\nabla^2\\phi=\\frac{3}{2}H_0^2\\Omega_m(1+z)\\delta$ which should be valid under the line-of-sight integral. Since the bias only comes from the mass\nfluctuations $\\delta$ which are correlated with flux fluctuations $\\delta_f$ at the redshift we are considering, and correlations drop off rapidly with increased spatial separation, neglecting contributions to $\\kappa$ from mass fluctuations at lower redshifts shouldn't  be a bad approximation. We will verify this below by comparing the lensing bias thus obtained with that obtained from another method.\n\n\n\\subsection{Bias to The Flux Power Spectrum}\n\\begin{figure}\n\\begin{center}\n$\\begin{array}{cc}\n\\includegraphics[width=0.45\\textwidth]{PffSims.eps}&\\includegraphics[width=0.45\\textwidth]{DeltaPffWeightedz2.eps}\\\\\n\\mbox{(a)}&\\mbox{(b)}\\\\\n\\includegraphics[width=0.45\\textwidth]{DeltaPffWeightedz3.eps}&\\includegraphics[width=0.45\\textwidth]{DeltaPffWeightedz4.eps}\\\\\n\\mbox{(c)}&\\mbox{(d)}\n\\end{array}$\n\\caption{\\label{Fig:PffSims}(a) The 1D flux power spectrum measured from the simulations. Panels $(b)$ - $(d)$ show the fractional correction from magnification \n$(\\langle P_{ff} {}_{\\rm obs} \\rangle - P_{ff} {}_{\\rm true})\/P_{ff} {}_{\\rm true}$\nfor mean redshifts $\\bar{z}=2, 3, 4$ respectively. The error bars are suppressed for clarity. The amplitude of the correction depends on the redshift of the quasar $z_Q$ through $\\kappa$ (Eq. (\\ref{Eq:KappaDef})) and also on the slope of the quasar number count function, here we assume $5s-2=3$.}\n\\end{center}\n\\end{figure}\n\nIn panel $(a)$ of Fig. \\ref{Fig:PffSims} we show the 1D flux power spectrum\n\\begin{equation}\nP_{ff}(k_{\\parallel})=\\frac{1}{L}|\\delta(k_{\\parallel})|^2\n\\end{equation}\nmeasured from simulations\\footnote{Our Fourier convention is $\\delta({\\bf k})=\\int \\!d^3{\\bf x} \\,e^{-i{\\bf k}\\cdot{\\bf x}}\\delta({\\bf x})$.}. In panels $(b)$-$(d)$ we show the fractional correction due to magnification, calculated by weighting lines-of-sight by $1+\\delta_\\mu$.  Magnification weighs lines of sight with large $\\kappa$ more heavily than those with low $\\kappa$. Measurements of $\\delta_{f}$ at redshift $\\bar{z}$ can be taken from quasars at redshift $z_Q$ for $\\bar{z} \\lsim z_Q \\lsim (1+\\bar{z})\\nu_\\beta\/\\nu_\\alpha-1$. The upper limit is set by where the Lyman-$\\beta$ line ($\\lambda_\\beta=1026\\AA$)  can be confused with Lyman-$\\alpha$. The lensing weight function is peaked at $\\chi\\sim \\chi_Q\/2$, so for quasars at higher redshift, $\\chi(\\bar{z})$ is closer to the peak of the lensing weight function making the lensing correction larger than it is for quasars just beyond $\\bar{z}$. \n\nMagnification causes a $\\lsim 1\\%$ change to $P_{ff}$ with larger changes occurring at lower redshifts.   The results shown in Fig. \\ref{Fig:PffSims} assume $s = 1$ and that all quasars within one's sample are weighed equally. The actual lensing correction in realistic surveys could differ by factor of a few. The current statistical error on the amplitude of $P_{ff}$, averaged over scales, is $\\sim 0.6\\%$ from SDSS \\cite{McDonald:2004eu}, which is a bit larger than the magnification correction shown in Fig. \\ref{Fig:PffSims}.\nHowever, as precision improves, a systematic offset such as that introduced by lensing could well be relevant.\n\n\\label{PCorrection}\n\nIt is instructive to get an analytic estimate for the bias to the flux power spectrum.  Equation (\\ref{Eq:MagBiasFinal}) tells us the lensing bias for the flux power spectrum is:\n\\begin{eqnarray}\n\\label{Pffbias}\n\\langle P_{ff} {}_{\\rm obs} (k_\\parallel) \\rangle - P_{ff} {}_{\\rm true} (k_\\parallel)= {1\\over \\Delta\\chi}  \\langle \\delta_f (k_\\parallel) \\delta_f^* (k_\\parallel) \\delta^\\mu \\rangle\n\\end{eqnarray}\nwhere $\\Delta\\chi$ is the length of the quasar spectra from which the power is measured. It is therefore clear that this lensing bias depends on the three-point function  or bispectrum. Expressing $\\delta^\\mu$ as a line-of-sight integral of the mass fluctuation (Eq. (\\ref{eq:weightdef})), and  judiciously applying the Limber's approximation \\cite{Limber:1954,LoVerde:2008re}, we obtain:\n\\begin{eqnarray}\n\\label{Eq:DPBffmHighK}\n\\langle P_{ff} {}_{\\rm obs} (k_\\parallel) \\rangle - P_{ff} {}_{\\rm true} (k_\\parallel) \\approx \\frac{3}{2}H_0^2\\Omega_m(5s-2) \\frac{\\chi_Q-\\bar\\chi}{\\chi_Q}\\bar\\chi (1+\\bar z) B_{ffm}(k_{||},-k_{||},0)\\, .\n\\end{eqnarray}\nwhere $\\chi_Q$ is the distance to quasar, $\\bar\\chi$ is the average distance  to the forest, and $\\bar z$ is the corresponding redshift. Here, $B_{ffm}$ is the (true) 1D flux-flux-mass bispectrum at redshift $\\bar z$. It should be emphasized that $B_{ffm} (k_\\parallel, -k_\\parallel, 0)$  does not vanish despite having one of its arguments (the one associated with mass) being zero. This is related to the fact that it arises from a 1D projection of a 3D distribution. For instance, it is well known that a 3D power spectrum and its 1D projection are related by $P_{1D} (k_\\parallel) = \\int_{k_\\parallel}^\\infty (k dk \/2\\pi) P_{3D} (k)$, and so $P_{1D}$ generally does not vanish in the $k_\\parallel \\rightarrow 0$ limit.\n\nWe calculate the correction to $P_{ff}$ using $B_{ffm}$ measured from simulations.  Estimating $B_{ffm} (k_\\parallel, -k_\\parallel, 0)$ requires an extrapolation from what we actually measure $B_{ffm} (k_\\parallel, -(k_\\parallel + \\Delta k), \\Delta k)$, where the smallest $\\Delta k$ is the fundamental mode of the box. How this is done is described in detail in Appendix \\ref{Extrap}. In Fig. \\ref{Fig:PffBispec} we compare the lensing correction to the flux power spectrum  as determined via the bispectrum to that determined by the weighting method described previously in \\S \\ref{Sims}. The qualitative agreement between the two is reassuring.\n\n\n\n\\subsection{Bias to The Effective Optical Depth}\n\\label{SimsTau}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{DeltaTauWeighted.eps}\n\\caption{\\label{Fig:TauBias} The magnification correction to the effective optical depth at three different mean redshifts $\\bar{z}$. This correction depends on the redshift $z_Q$ of the quasars. In practice, measurements of $\\tau_{\\rm eff}$ involves combining\nsource quasars at multiple redshifts. In the above plot we assume  $5s-2=3$.}\n\\end{center}\n\\end{figure}\n\nThe effective optical depth $\\tau_{\\rm eff}$  is estimated from data by averaging over frequencies (and quasars) the ratio of the observed flux $f$= to the continuum $f_C$, i.e. $e^{-\\tau_{\\rm eff}}$  is estimated from $f\/f_C$. As emphasized in \\S \\ref{LensBiasDeriv}, lensing magnification affects $f$ and $f_C$ in the same manner, and therefore leaves $f\/f_C$ unchanged on a quasar by quasar basis. Lensing's impact enters through the sample selection, resulting in (see Eq. (\\ref{Eq:MagBiasFinal}))\n\\begin{eqnarray}\n\\label{taueffbias}\n\\langle e^{-\\tau_{\\rm eff}} {}_{\\rm obs} \\rangle - e^{-\\tau_{\\rm eff}}{}_{\\rm true} = \\langle e^{-\\tau_{\\rm eff}} \\delta^\\mu \\rangle\n\\end{eqnarray}\nwhere $e^{-\\tau_{\\rm eff}}$ on the right is the mean transmission on a line-of-sight by line-of-sight basis, and $e^{-\\tau_{\\rm eff}} {}_{\\rm true}$ is the true mean transmission if only we could  dispense with the quasar-weighting and average over the whole sky equally.\n\nThe correction to the effective optical depth measured from simulations is shown in Fig. \\ref{Fig:TauBias} for quasars at several redshifts.  The net effect of lensing will depend on how measurements from quasars at different redshifts are combined. However, Fig. \\ref{Fig:TauBias} indicates the effect should be $\\sim 10^{-3}-10^{-4}$. Current measurements of $\\tau_{\\rm eff}$ have errors of at least a few percent at each redshift so at present lensing should not be an issue.\\footnote{\\label{continuumfootnote} The error bars for measurements of $\\tau_{\\rm eff}$ are dominated by uncertainties in the continuum-fit. Typically, the continuum is estimated either by extrapolating from the red side\n(e.g. \\cite{Bernardi:2002mr}) , or by performing a smooth fit through portions of the spectra that are deemed unabsorbed (e.g. \\cite{FaucherGiguere:2007ys}). We simulate the effect of the latter by renormalizing the flux along each line of sight by $f_{\\rm max}$, the maximum along that line of sight. At low redshifts, this introduces a\nnegligible bias to $\\tau_{\\rm eff}$, but by $z = 4$, $\\tau_{\\rm eff}$ is biased low by $\\sim 8 \\%$, in rough agreement with \\cite{FaucherGiguere:2007ys}.\nHowever, we have checked that the fractional magnification correction is not significantly changed whether we renormalize by $f_{\\rm max}$ or not.}\n\n\\label{TauCorrection}\n\nAn analytical estimate of the lensing bias on the effective optical depth can be inferred from Eq. (\\ref{taueffbias}), which gives: \n\\begin{eqnarray}\n\\label{Eq:TauBias}\n{\\langle e^{-\\tau_{\\rm eff}} {}_{\\rm obs} \\rangle \\over e^{-\\tau_{\\rm eff}} {}_{\\rm true}}- 1 = \\langle \\delta_f \\delta^\\mu \\rangle \\approx(5s-2)\\frac{3}{2}\\Omega_mH_0^2\\frac{\\chi_Q-\\bar\\chi}{\\chi_Q}\\bar\\chi(1+\\bar z)P_{fm}(k_{||}=0)\n\\end{eqnarray}\nwhere we have used Limber's approximation, $P_{fm}(k_{||})$ is the (true) 1D flux-mass  power spectrum evaluated at the mean redshift of absorption $\\bar z$, $\\bar\\chi$ is\nthe corresponding distance, and $\\chi_Q$ is the quasar distance.  The lensing bias of this one-point statistic is thus related to a two-point correlation, just as the lensing bias of the power spectrum is determined by the bispectrum. As before, to evaluate the bias using this method,  an extrapolation of the simulation $P_{fm}$ to $k_\\parallel = 0$ is necessary (see Appendix \\ref{Extrap}).  We find results that are consistent with those obtained by the weighting method (Fig. \\ref{Fig:TauBias}) to about $30 \\%$,\nwith the latter generally higher.\n\n\n\n\n\\subsection{Bias to The Flux Probability Distribution Function}\n\n\\begin{figure}\n\\begin{center}\n$\\begin{array}{cc}\n\\includegraphics[width=0.45\\textwidth]{PDFSimsETau.eps}&\\includegraphics[width=0.45\\textwidth]{DeltaPDFz2ETau.eps}\\\\\n\\mbox{(a)}&\\mbox{(b)}\\\\\n\\includegraphics[width=0.45\\textwidth]{DeltaPDFz3ETau.eps}&\\includegraphics[width=0.45\\textwidth]{DeltaPDFz4ETau.eps}\\\\\n\\mbox{(c)}&\\mbox{(d)}\n\\end{array}$\n\\caption{\\label{Fig:PDFetauSims}(a) The flux probability distribution as measured from the simulations. Panels $(b)$ - $(d)$ show the fractional correction from magnification for mean redshifts $\\bar{z}=2, 3, 4$. The amplitude of the correction depends on the redshift of the quasar $z_Q$ and the slope of the quasar number count function, here we assume $5s-2=3$ (Eq. (\\ref{Eq:KappaDef})).}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n$\\begin{array}{cc}\n\\includegraphics[width=0.45\\textwidth]{PDFSims.eps}&\\includegraphics[width=0.45\\textwidth]{DeltaPDFz2.eps}\\\\\n\\mbox{(a)}&\\mbox{(b)}\\\\\n\\includegraphics[width=0.45\\textwidth]{DeltaPDFz3.eps}&\\includegraphics[width=0.45\\textwidth]{DeltaPDFz4.eps}\\\\\n\\mbox{(c)}&\\mbox{(d)}\n\\end{array}$\n\\caption{\\label{Fig:PDFdeltaSims}(a) The probability distribution function for $\\delta_f$ as measured from the simulations. Panels $(b)$ - $(d)$ show the fractional correction from magnification for mean redshifts $\\bar{z}=2, 3, 4$. \n}\n\\end{center}\n\\end{figure}\n\nAnother interesting statistic of the Lyman-$\\alpha$ forest is the flux probability  distribution function (PDF): $\\mathcal{P}(f) df$ describes the probability  that the observed flux lies between $f-df\/2$ and $f + df\/2$. Here $f$ is the continuum-normalized flux (i.e. $f\/f_c$ from the previous section). \n\n\n\nThe flux PDF measured from simulations with bins of $d f=0.01$ is shown in panel $(a)$ of Fig. \\ref{Fig:PDFetauSims}.\\footnote{We use the same fake continuum fitting procedure described in footnote \\ref{continuumfootnote}, but again this appears to make little difference to the size of the fractional lensing correction.} The fractional correction from lensing magnification is shown in panels $(b)$--$(d)$ of the same figure. The effect of magnification is to boost the low flux (high absorption) end of the PDF and decrease the PDF at the high flux end. The correction is $< 1\\%$, so unimportant at the current level of precision ($5-10\\%$ on the flux PDF in bins of $d f=0.05$ \\cite{Kim:2007rr} but see also  \\cite{McDonald:1999dt,Desjacques:2006pe,Becker:2006qj}).\n\nIt has been suggested that to avoid systematic errors from continuum fitting one should measure the PDF for fluctuations in the flux $\\delta_f=f\/\\bar{f}-1$ rather than measuring the PDF of $f$  \\cite{Lidz:2006zz}. In this case the flux is normalized along each line of sight, so the range of $\\delta_f$ for one line-of-sight is $-1$ to $1\/\\bar{f}-1$ where $\\bar{f}$ is the mean flux along that line of sight. The range of the entire PDF determined from many lines of sight will be $-1$ to $1\/\\bar{f} {}_{\\rm min}-1$, where $\\bar{f} {}_{\\rm min}$ is the mean flux along the line of sight with the lowest transmission. \n\nThe PDF for $\\delta_f$ is shown in panel $(a)$ of Fig. \\ref{Fig:PDFdeltaSims}.  Again the fractional correction due to lensing is shown in panels $(b)$--$(d)$. In this case the correction due to lensing appears rather large at high $\\delta_f$. The high $\\delta_f$ bins are dominated by lines of sight with the lowest mean flux, which are also most highly magnified. However, these high $\\delta_f$ pixels are quite rare, i.e. the PDF at high $\\delta_f$ is quite small and therefore noisy from an observational point of view.\n\n\n\\label{FluxPDF}\n\nFor an analytical estimate we apply the fundamental Eq. (\\ref{Eq:MagBiasFinal}), \n\\begin{eqnarray}\n\\label{Pfbias0}\n\\langle \\mathcal{P}(f) {}_{\\rm obs} \\rangle - \\mathcal{P}(f) {}_{\\rm true} = \\langle \\mathcal{P} (f) \\delta^\\mu \\rangle\n\\end{eqnarray}\nwhere $\\mathcal{P} (f) df$ is estimated from each line of sight in the standard fashion: counting the fraction of pixels with a flux that falls within $df$ of $f$. The flux is a non-linear function of the gas density, but for simplicity, we will assume that gas traces mass and that $f$ is a local function of the mass fluctuation $\\delta$, i.e. $f = F(\\delta)$, where the function $F$ is to be specified. Let us use the symbol $p$ to denote the  (average) location of interest, i.e. $\\delta_p$ is the mass fluctuation at point $p$. Thus, the flux PDF at $p$ can be expressed in terms of the mass PDF $\\mathcal{P}_m (\\delta_p)$:\n\\begin{eqnarray}\n\\mathcal{P} (f) = \\int d\\delta_p \\, \\mathcal{P}_m (\\delta_p) \\, \\delta_D (f - F(\\delta_p)) = \\int d\\delta_p d\\delta_l \\, \\mathcal{P}_m (\\delta_p, \\delta_l) \\, \\delta_D (f - F(\\delta_p))\n\\end{eqnarray}\nwhere $\\delta_D$ is the Dirac delta function. For the second equality, we have introduced $\\delta_l$ which is the mass fluctuation at some other point (the subscript $l$ is\nused in anticipation of the fact that this will be where some lens is), and $\\mathcal{P}_m(\\delta_p, \\delta_l)$ is the joint mass PDF at the two points, i.e. the one-point PDF\n$\\mathcal{P}_m (\\delta_p)$ is related to the two-point PDF by $\\mathcal{P}_m (\\delta_p) = \\int d\\delta_l \\mathcal{P}_m (\\delta_p, \\delta_l)$. The motivation for introducing the two-point mass PDF is to ease the computation of $\\langle \\mathcal{P}(f) \\delta^\\mu \\rangle$, where $\\delta^\\mu$ involves a line-of-sight integral over locations that generally differ from $p$. We obtain: \\footnote{Here, we are abusing the notation a bit. On the left, $\\mathcal{P}(f)$ strictly speaking denotes a stochastic quantity: it should be the estimator that simply counts the fraction of relevant pixels, whereas on the right, $\\mathcal{P}_m (\\delta, \\delta')$ denotes the true (non-stochastic) two-point mass PDF.}\n\\begin{eqnarray}\n\\label{Pfbias}\n\\langle \\mathcal{P}(f) \\delta^\\mu \\rangle = \\frac{3}{2}H_0^2\\Omega_m (5s-2) \\int_0^{\\chi_Q} d\\chi_l \\frac{\\chi_Q-\\chi_l}{\\chi_Q}\\chi_l (1+z_l) \\int d\\delta_p d\\delta_l\\, \\delta_l\n\\mathcal{P}_m (\\delta_p, \\delta_l) \\delta_D (f - F(\\delta_p)) \\, .\n\\end{eqnarray}\nWe need a model for the two-point mass PDF. In linear theory, that is completely specified by the two-point correlation  $\\langle \\delta_p \\delta_l \\rangle$. The relevant fluctuations in the forest are not quite linear. Instead, we will adopt the lognormal model (see for example \\cite{Kofman:1993mx,Coles:1991}):\n\\begin{equation}\n\\label{Pmlognormal}\n\\mathcal{P}_m (\\delta_p, \\delta_l) =\\frac{1}{2\\pi\\sqrt{\\textrm{det\\,C}}\\,(1+\\delta_p)(1+\\delta_l)}e^{-\\frac{1}{2} {\\bf \\Delta}^T {\\bf C}^{-1} {\\bf \\Delta}}\n\\end{equation}\nwhere ${\\bf \\Delta}$ is a 2-component vector with $\\Delta_i = \\ln(1+\\delta_i)+\\frac{1}{2}\\ln(1+\\langle\\delta_i^2\\rangle)$, and ${\\bf C}$ is a $2 \\times 2$ matrix with components ${\\rm C}_{ij}=\\ln(1+\\langle\\delta_i\\delta_j\\rangle)$. Here $i$ and $j$ stand for $p$ or $l$. The lognormal model is simply one in which $\\Delta$ is a Gaussian random field, and the observed $\\delta$ is related to it by $1 + \\delta = {\\,\\rm exp\\,}[\\Delta - \\langle \\Delta^2 \\rangle\/2]$, such that $\\langle \\Delta^2 \\rangle = {\\,\\rm ln\\,} (1 + \\langle \\delta^2 \\rangle)$.\nSubstituting Eq. (\\ref{Pmlognormal}) into Eq. (\\ref{Pfbias}) and integrating over $\\delta_p$ and $\\delta_l$, we find\n\\begin{eqnarray}\n\\label{Eq:PDFLogNorm}\n{\\langle \\mathcal{P}(f)_{\\rm obs} \\rangle - \\mathcal{P}(f)_{\\rm true} \\over \\mathcal{P}(f)_{\\rm true}} = \\frac{3}{2}\\Omega_m(5s-2) H_0^2 \\int_0^{\\chi_Q} d\\chi_{l}\n{\\chi_Q - \\chi_l \\over \\chi_Q} \\chi_l (1+z_l) \\left(e^{C_{lp}\/C_{pp}\\ln(1+\\delta_*)+\\frac{1}{2}(C_{lp}-C_{lp}^2\/C_{pp}))}-1\\right)\\, ,\n\\end{eqnarray}\nwhere $\\delta_* \\equiv F^{-1} (f)$ is the actual density such that the observed flux equals the value of interest $f$. At this point, we need to specify $F$: We use $F(\\delta) = {\\,\\rm exp\\,}[ - A (1 + \\delta)^\\beta]$, with $\\beta = 1.58$ and $A=0.2010, \\, 0.9578,\\, 2.960$ at $z_p=2,\\,3,\\,4$ respectively. This approximates well what is in our simulations, though the simulated spectra were computed using the exact relation between the optical depth, baryon density, temperature and ionizing background.\n\nTo use Eq. (\\ref{Eq:PDFLogNorm}), we need $C_{lp}$ and $C_{pp}$, which we compute using the nonlinear mass power spectrum, with suitable smoothing (for $p$) to account for the effective Jeans smoothing of baryons. The cosmology and power spectrum prescriptions are described at the end of \\S \\ref{LensBiasDeriv}. Figure \\ref{Fig:PDFLogNorm} compares the analytic calculation in Eq. (\\ref{Eq:PDFLogNorm}) with the results from simulations in \\S \\ref{Sims}. The two methods agree remarkably well considering the simplicity of the analytic method. If the linear mass PDF were used, the calculated correction has a slightly different $f$-dependence and is typically smaller by about a factor of $5-6$. \n\nWe should mention data that do not resolve the Jeans scale (or roughly, the thermal broadening scale) have an additional complication: even if the fundamental mass density\nis lognormal distributed (in both one-point and two-point sense),  the flux field smoothed with a coarse resolution might not be well described by our model. In other words,\nsmoothing and nonlinear transformation do not commute. In any case, the lensing effect on the flux PDF appears to be fairly small.\n\n\n\\begin{figure}\n\\begin{center}$\\begin{array}{cc}\n\\includegraphics[width=0.45\\textwidth]{CompareBispectrumz3.eps}&\\includegraphics[width=0.45\\textwidth]{dPDF_LogNorm.eps}\n\\end{array}$\n\\caption{\\label{Fig:PffBispec} \\label{Fig:PDFLogNorm} Left: A comparison of the predicted lensing bias to the flux power spectrum using the weighted calculation presented in \\S \\protect\\ref{Sims} (solid lines) versus using Eq. (\\protect\\ref{Eq:DPBffmHighK}) with the flux-flux-mass bispectrum measured from simulations (points). From bottom to top the quasar redshifts are $3.1, 3.2, 3.4, 3.6$.  Right panel: The correction to the flux PDF from the analytic model described in Eq. (\\ref{Eq:PDFLogNorm}) (solid lines) compared with the results from simulations (points). The analytic description with the log-normal model for the mass PDF works surprisingly well.}\n\\end{center}\n\\end{figure}\n\n\\section{Lensing as Signal: a test of how neutral hydrogen traces mass}\n\\label{MagCorrelation}\n\nIn the previous section, we have been exploring lensing as a source of measurement bias. Here, we explore the converse: lensing as a useful signal. We are interested in ways to extract signals of gravitational lensing from Lyman-alpha forest observations and thereby constrain the cross-correlations between flux and mass (for example, the flux-mass power spectrum and flux-flux-mass bispectrum present in Eq. (\\ref{Eq:DPBffmHighK}) and Eq. (\\ref{Eq:TauBias})). \n\nOne option is to correlate the magnitude of quasars with Lyman-alpha forest observables (we will consider other possible cross correlations at the end of this section). Given some observable $\\mathcal{O}_I$ measured from a quasar labeled $I$,  and its magnitude $m_I$, we can form an estimator $\\mathcal{E}$:\n\\begin{eqnarray}\n\\label{EstimatorE}\n\\mathcal{E} = {1 \\over N_{\\rm QSO}} \\sum_I m_I \\mathcal{O}_I -  {1\\over N_{\\rm QSO}^2} \\left(\\sum_I m_I\\right) \\left(\\sum_I \\mathcal{O}_I\\right) \n\\end{eqnarray}\nwhere $N_{\\rm QSO}$ is the number of quasars in one's sample. As is explained in \\S \\ref{LensBiasDeriv}, the Lyman-alpha forest observable is always implicitly weighted by the number density of quasars (determined by the sample selection). And the quasar magnitude is of course modified by lensing, following from Eq.  (\\ref{Eq:fmag}):\n\\begin{equation} \n\\label{deltam} \\delta m = -5 \\kappa \/ {\\rm ln\\,} 10\\, .  \n\\end{equation} \nIt is shown in Appendix \\ref{derive} that\n\\begin{eqnarray}\n\\label{Eaverage}\n\\langle \\mathcal{E} \\rangle = 5 \\tilde s \\langle \\kappa \\delta\\mathcal{O} \\rangle\n\\end{eqnarray}\nwhere $\\delta\\mathcal{O}$ represents fluctuations in $\\mathcal{O}$  and $\\tilde s$ is defined as\n\\begin{eqnarray}\n\\label{stilde}\n\\tilde s \\equiv {1\\over {\\rm ln\\,} 10} {\\int dm \\, \\epsilon(m) (dn^0 \/dm) (m - \\bar m) \\over \\int dm \\, \\epsilon(m) n^0(m)}\n\\end{eqnarray}\nwhere $\\bar m$ is the average magnitude in the sample i.e. $\\bar m = \\int dm \\, m \\epsilon(m) n^0(m)\/ \\int dm \\, \\epsilon(m) n^0(m)$, and $n^0(m)$ and $\\epsilon(m)$ are the luminosity function and selection function respectively --- the definition for $\\tilde s$ is chosen to resemble those for $s$ and $s'$ (Eqs. [\\ref{nlensed}], [\\ref{sprime}]). The $\\tilde s$ defined here is related to $C_S$ defined in \\cite{Menard:2009yb} by $C_S = - {\\rm ln\\,} 10 \\, \\tilde s$. As first emphasized by \\cite{Menard:2009yb}, $C_S = 0$ for a strictly power-law luminosity function in the sense of $n^0 (m) \\propto e^m$. The realistic quasar luminosity function is not of this form, and we will adopt $C_S \\sim 1\/3$ from \\cite{Menard:2009yb}, or equivalently $\\tilde s \\sim - 0.14$, for our numerical estimates below. The normalizing factor in Eq. (\\ref{Eaverage}) is therefore $5 \\tilde s \\sim - 0.7$. It is worth emphasizing that there is no `-2' term here, unlike for instance $5s-2$ in Eq. (\\ref{Eq:MagBiasFinal}). The `-2' there arises from the geometrical increase in area by magnification. The statistic $\\mathcal{E}$ under consideration is immune to this effect. Further discussions can be found in Appendix \\ref{derive}.\n\n\\subsection{Magnitude-flux Correlation}\n\\label{magflux}\n\nLet us first consider cross-correlations between ${\\cal O} = \\delta_f$, the intervening flux fluctuation, and the quasar magnitude $m$, giving\n\\begin{eqnarray}\n\\label{magfluxexpect}\n\\langle \\mathcal{E}_{\\delta m \\, \\delta_f} (\\chi_Q, \\chi) \\rangle = 5 \\tilde s \\langle \\kappa \\delta_f \\rangle \\approx {3\\over 2} H_0^2 \\Omega_m 5 \\tilde s {\\chi_Q - \\chi \\over \\chi_Q} \\chi (1+z) P_{fm}(k_\\parallel=0)\n\\end{eqnarray}\nwhere $\\chi_Q$ is the distance to quasar, $\\chi$ is the distance to absoprtion (i.e. where $\\delta_f$ is located) and $z$ is the corresponding redshift. Here, $P_{fm}$ is the (true) 1D flux-mass power spectrum  --- the subscript $m$ of $P_{fm}$ stands for mass rather than magnitude (just as in \\S \\ref{TauCorrection}).\n\nThe nice thing about the estimator $\\mathcal{E}_{\\delta m \\, \\delta_f}$ is that it is expected to depend on the location of absorption $\\chi$ in a predictable way, which allows this to be separated from other possible systematic effects, an example of which is large scale power from the uncertain continuum (shape).  The fact that the amplitude of this  cross correlation signal scales with $\\tilde s$ can also be exploited to test for consistency, for instance by isolating different subsamples of quasars with different values of $\\tilde s$.\n\nIn Fig. \\ref{Fig:deltafXmagSDSS} we show the expected signal for this  cross-correlation as a function of distance to the quasar $ \\chi_Q - \\chi$, where $\\chi$ is the distance to the absorption. To estimate the overall statistical significance, we can combine the measurements at different separations into an estimate for a single amplitude, namely $P_{fm}(k_\\parallel=0)$. The appropriate minimum variance estimator is described in Appendix \\ref{estimator}, where we also derive its signal-to-noise:\n\\begin{eqnarray}\n\\label{magfluxS2N}\n{S \\over N} = \\sqrt{ {N_{\\rm QSO} \\over \\langle \\delta m^2 \\rangle} \\int {dk_\\parallel \\over 2\\pi}  {|\\mathcal{E}_{\\delta m \\delta_f} (k_\\parallel)|^2  \\over P_{ff}(k_\\parallel) +\\mathcal{S.N.}}}\n\\end{eqnarray}\nwhere $\\mathcal{E}_{\\delta m \\delta_f} (k_\\parallel)$ is the Fourier transform (over $\\chi$) of Eq. (\\ref{magfluxexpect}), $N_{\\rm QSO}$ is the number of quasars available, $\\langle \\delta m^2 \\rangle$ is the quasar magnitude variance, $P_{ff}$ is the 1D flux power spectrum, and $\\mathcal{S.N.}$ is the associated shot-noise. \n\nThe $\\mathcal{S.N.}$ term is important so we consider the signal-to-noise per quasar for two survey configurations: `SDSS III configuration' with resolution FWHM is $60$ km\/s, and the shot-noise power is $\\sim 0.44$ Mpc\/h $(3\/(1 + \\bar z))^{3\/2}$ \\cite{Hui:2000rw} and `Keck configuration' with resolution FWHM is $10$ km\/s and shot-noise power is $\\sim 0.029$ Mpc\/h $(3\/(1 + \\bar z))^{3\/2}$. For both we take the magnitude dispersion of the quasar sample to be $0.5$ \\cite{Menard:2009yb}. The first set of numbers roughly resemble the expectations of SDSS III \\cite{SDSSIII} but keep in mind the configuration we assume likely differs a bit from what will turn out in practice. The results for the $S\/N$ per sight-line are summarized in Table I. For the SDSS III expectation of $N_Q=160,000$ the ${S\/ N}$ for this amplitude is  expected to be only $\\sim 1$ and even for a futuristic survey like BigBOSS \\cite{Schlegel:2009uw} that could measure $10^6$ spectra with similar resolution only ${S\/N} \\sim 2-3$ could be achieved. Comparison with the `Keck configuration' shows that shot noise is clearly a limitation for these observables. However, the $S\/N$ estimates presented are for a single redshift bin of width $\\Delta z\\sim0.2$, but measurements will be made at a range of redshifts which could be combined to get a slightly higher signal-to-noise estimate of, for example, the mean $P_{fm}$ across the redshift span of the survey.  \n\nIt is worth noting a few points that reduce $S\/N$ of the forest-magnitude correlations in comparison with galaxy-magnitude correlations used \\cite{Menard:2009yb}. (1) The cross-correlation between flux and mass is lower than the correlation between galaxies and mass (the correlation coefficient is $\\sim P_{fm}\/\\sqrt{P_{ff}P_{mm}}\\lsim 0.5$).  (2) Lensing peaks at a distance halfway between the observer and the quasar, but to avoid confusion with Lyman-$\\beta$ we use only the part of the Lyman-$\\alpha$ forest near to the quasar where the lensing is smaller. (3) The signal-to-noise of magnitude-forest correlations is $\\propto \\sqrt{N_{Q}}$ where $N_Q$ is the number of quasars with spectra, while for the galaxy-magnitude cross-correlation it is proportional to the number of galaxy-quasar pairs $\\propto \\sqrt{N_Q N_g}$  \n(however in \\S\\ref{othercorrelations} we briefly discuss correlating the forest with quasars along different lines of sight). \n\n\n\\subsection{Magnitude-Power Correlation}\n\\label{magpower}\n\n\\begin{figure}\n\\begin{center}\n$\\begin{array}{cc}\n\\includegraphics[width=0.45\\textwidth]{PffmagSDSS.eps} & \\includegraphics[width=0.45\\textwidth]{deltafXmagSDSS.eps}\n\\end{array}$\n\\caption{\\label{Fig:PffMag}\\label{Fig:deltafXmagSDSS}Left: The lensing-induced correlation between quasar magnitude and the flux power spectrum given in Eq. (\\ref{magpowerexpect}). The amplitude of the signal increases with increasing distance between $\\chi(\\bar{z})$ and the background quasar but above we assume the correlation is averaged over background quasars at redshifts between $\\bar{z}+0.1$ and $z_\\beta=\\nu_\\beta\/\\nu_\\alpha(1+\\bar{z})-1$ (the maximum redshift for the quasar before confusion between Lyman-$\\alpha$ and Lyman-$\\beta$ absorption sets in).  Right: The magnitude-flux cross correlation (Eq. \\ref{magfluxexpect}) as a function of line-of-sight comoving distance to the quasar. }\n\\end{center}\n\\end{figure}\n\\begin{table}\\label{StoNtable}\n\\begin{tabular}{|c|c|c|}\n\\hline\nObservable: & $S\/N$ per l.o.s. for SDSS III configuration & $S\/N$ per l.o.s. for Keck configuration \\\\\n\\hline \n$\\langle \\delta_f\\delta m\\rangle$ at $z=2,3,4$ & $0.002, 0.0029, 0.0022$ &  $0.0041, 0.0036, 0.0024$ \\\\\n\\hline\n$\\langle \\delta P_{ff}\\delta m\\rangle $ at $z=2,3,4$ & $0.0016, 0.0028, 0.0025$ & $0.008, 0.006, 0.0042$ \\\\\n\\hline \n$\\langle f\/f_C\\delta m\\rangle$ at $z=2,3,4$ & $0.0054,0.0032,0.00095$ & --- \\\\\n\\hline\n\\end{tabular}\n\\caption{The signal-to-noise per sightline in redshift bins of $\\Delta z \\sim 0.2$ for the proposed estimators  in Eq. (\\ref{magfluxexpect}), (\\ref{magpowerexpect2}) and (\\ref{eq:fmcorr})  for the SDSS III and Keck survey configurations (see \\S\\ref{magflux}). Here we assume the quasar sample has $\\tilde{s}=-0.14$ (see Eq. (\\ref{stilde})) and the quasar magnitude dispersion is $0.5$. For $\\langle f\/f_C\\delta m\\rangle$ we assume the same fractional errors of $f\/f_C$ as \\cite{Bernardi:2002mr} and that the errors are statistics limited so they scale as $1\/\\sqrt{N_{QSO}}$.  }\n\\end{table}\n\nAnother possibility is to set $\\mathcal{O} = P_{ff}(k_\\parallel)$  in the estimator Eq. (\\ref{EstimatorE}), i.e. cross correlate quasar magnitude and the flux power spectrum. This estimator has the following expectation value: \n\\begin{eqnarray}\n\\label{magpowerexpect}\n\\langle \\mathcal{E}_{\\delta m\\, P_{ff}} (k_\\parallel) \\rangle\n= {5 \\tilde s \\over \\Delta\\chi}\n\\langle \\kappa \\delta_f (k_\\parallel) \\delta_f^* (k_\\parallel)\n\\rangle\n\\end{eqnarray}\nwhere $\\Delta\\chi$ is the length of the quasar spectra from which the power is measured. This expression is similar to Eq. (\\ref{Pffbias}), and indeed one can relate this to the \nbias in $P_{ff}$ measurement we have calculated in \\S \\ref{PCorrection} (Eq. [\\ref{Eq:DPBffmHighK}]):\n\\begin{eqnarray}\n\\label{magpowerexpect2}\n\\langle \\mathcal{E}_{\\delta m\\, P_{ff}} (k_\\parallel) \\rangle\n= {5 \\tilde s \\over 5s - 2}\n(\\langle P_{ff} {}_{\\rm obs} (k_\\parallel) \\rangle\n- P_{ff} {}_{\\rm true} (k_\\parallel)) \\, .\n\\end{eqnarray}\nThis means that one could in principle use the magnitude-power correlation to correct for the bias in $P_{ff}$ measurement. The expected magnitude-power correlation is shown in Fig. \\ref{Fig:PffMag} and the signal-to-noise is given in Table I.\n\n\n\\subsection{Magnitude-Mean-Transmission Correlation}\n\\label{magmeanflux}\nIf the continuum can be accurately estimated, one could also correlate the mean transmission from each line of sight with the quasar magnitude behind it, i.e. choose $\\mathcal{O} = [f\/f_C]$ where $f$ is the observed flux and $f_C$ is the continuum, and the brackets $\\, [\\,\\,]\\,$ denote an average along the line of sight. One can see that the expected correlation should be related to the bias in $e^{-\\tau_{\\rm eff}}$ discussed in \\S \\ref{SimsTau}: \n\\begin{eqnarray}\n\\label{eq:fmcorr}\n\\langle \\mathcal{E}_{\\delta m \\, [f\/f_C]} \\rangle ={5 \\tilde s \\over 5 s - 2} ( \\langle e^{-\\tau_{\\rm eff}} {}_{\\rm obs} \\rangle - e^{-\\tau_{\\rm eff}} {}_{\\rm true} )\n\\end{eqnarray}\nThe signal-to-noise per sight line is listed in Table I. Note that the number of quasars needed is realistic only if the continuum is estimated by extrapolating from the red side (such as in the analysis of \\cite{Bernardi:2002mr}).  The number of quasars should be much smaller if the continuum were estimated by performing a smooth fit through portions of the Lyman-alpha forest that are deemed unabsorbed, a method that typically requires high resolution data. Whichever the continuum estimation method, care should be taken in accounting for possible systematic biases (in the continuum estimate) when interpreting this cross correlation measurement.\n\n\\subsection{Additional Cross Correlations}\n\\label{othercorrelations}\nWe have focused on cross correlations between the quasar magnitude and a Lyman-alpha forest observable {\\it in the same line of sight}. There are two possible extensions we should mention in passing. One is that the quasar magnitude and the Lyman-alpha forest observable can come from {\\it different lines of sight}. In other words, the cross correlations can be measured at a non-zero angular separation. On the one hand, the forest-magnitude correlation should drop off rapidly with increasing angular separation decreasing the signal, however the noise is reduced by the number of quasar pairs at a given angular separation. Roughly, the signal-to-noise for lines of sight at angular separation $\\theta$ should scale as\n\\begin{equation}\n\\left(\\frac{S}{N}\\right)_{\\textrm{{\\small at sep.}}\\theta}\\sim \\left(\\frac{S}{N}\\right)_{\\textrm{{\\small same l.o.s.}}}\\frac{10(\\chi_Q-\\bar{\\chi})}{\\chi_Q}\\frac{w_{fm}(\\theta)}{w_{fm}(\\theta=0)}\\sqrt{\\frac{N_{pairs}(\\theta)}{N_Q}}\n\\end{equation}\nwhere $w_{fm}(\\theta)$ is the angular flux-mass correlation function, $N_{pairs}(\\theta)$ is the number of quasar pairs with angular separation $\\theta$ and the term with the ratio of the distances roughly accounts for the fact that this correlation doesn't require $\\bar{z}$ to be above the Lyman-$\\beta$ confusion limit for the quasar (previously we needed $\\bar{\\chi}>\\chi_\\beta \\sim 4\/5\\chi_Q$ and had set $\\bar{\\chi}\\sim \\chi_\\beta+ (\\chi_Q-\\chi_\\beta)\/2$). It seems that the sparseness of quasars ($\\sim 16\/(\\textrm{deg.})^2$ for SDSS III) does not permit $N_{pairs}(\\theta)$ to be large enough that adding off-axis correlations will drastically improve the signal-to-noise in the near term. \n\n\nAnother possibility is to cross correlate the quasar number density (which is affected by lensing through magnification bias) with the Lyman-alpha forest observable. Such a correlation makes sense only at a non-zero lag (or non-zero smoothing),  since the Lyman-alpha forest is observable only  if there is a quasar directly behind it.  While we do not give explicit estimates here of the signal-to-noise for these cross correlations, they should be useful in disentangling the lensing signal from certain systematic effects, as we will discuss next.\n\n\\subsection{Dust and Other Systematic Effects}\n\\label{dust}\n\nWe discuss here three systematic effects that could complicate the\nmeasurement of the various cross correlations mentioned above.\n\nThe first is the continuum. The continuum presumably is smooth and therefore\nhas fluctuations only on large scales. But since its precise shape is uncertain,\na cross correlation such as the magnitude-flux correlation is susceptible\nto possible contamination from continuum power.\nA fortunate feature of the cross correlation is that it has a definite\nshape predicted by lensing, as well as an amplitude that scales with $\\tilde s$.\nBoth can be exploited to check for such a contamination.\n\nThe second systematic effect we loosely refer to as `background subtraction'.\nRealistic spectra of quasars inevitably contain `background' which can come\nfrom several sources, including the sky and scattered light within the \noptical instrument. While attempts are generally made to subtract these backgrounds\nas accurately as possible, there are inevitably residuals. These residuals\ncould correlate with the quasar magnitude, for instance they could be more\nnoticeable for fainter quasars. This would then produce spurious correlations\nwhen we correlate the quasar magnitude with Lyman-alpha forest observables (deduced\nfrom imperfect data that contain residual backgrounds). \nIn fact, existing flux power spectrum measurements from SDSS \\cite{McDonald:2004eu} \nare known to exhibit an otherwise puzzling correlation: that $P_{ff}$ is systematically higher\nfor fainter quasars, and this correlation is statistically significant\n(the normalized correlation coefficient is $\\sim 5 - 10\\%$ \\cite{Abazajian:2010}). \nSuch a correlation can be explained by this background subtraction effect.\nIt cannot be explained by lensing since it tends to produce a correlation of an opposite\nsign unless $\\tilde s$ has a sign opposite to what is known.\n(It can also be plausibly produced by dust which we will discuss below).\nTo disentangle background subtraction issues from lensing, it would be useful\nto examine magnitude-observable cross correlations at non-zero lag -- it would\nbe quite surprising if background residuals cause correlations between quasar\nmagnitude at one point with Lyman-alpha forest power at another.\n\nThe third systematic effect is dust extinction. Dust modifies flux by\n$f \\rightarrow f e^{-\\tau_{\\rm dust}}$, where $\\tau_{\\rm dust}$ is the optical depth due\nto dust. This modification is frequency dependent (higher optical depth for bluer photons),\nbut the frequency (which translates into scale) dependence is mild on the scales of interest, and therefore effectively acts as a continuum.\nThe net magnification plus dust correction to the quasar number density is\n\\begin{equation}\n{n}\\rightarrow n\\, {\\mu}^{2.5s-1} e^{-2.5 s {\\tau}_{\\rm dust}} \\, .\n\\end{equation}\nIncluding dust extinction, the measurement bias associated with a\nLyman-alpha forest observable (Eq.  [\\ref{Eq:MagBiasFinal}]) is changed to \n\\begin{equation} \n\\label{Eq:MagBiasFinaldust}\n\\langle\\mathcal{O}_{\\rm obs} \\rangle\n- \\mathcal{O}_{\\rm true} \n= (5s-2) \\langle \\mathcal{O} \\kappa \\rangle - 2.5 s \\langle \\mathcal{O} \\delta\\tau_{\\rm dust} \\rangle\n\\end{equation}\nwhere ${\\delta\\tau}_{\\rm dust}=\\tau_{\\rm dust}-\\langle \\tau_{\\rm dust}\\rangle$, which is assumed\nto be small, and we have ignored $\\langle \\mathcal{O} \\kappa \\delta\\tau_{\\rm dust} \\rangle$.\n\nAs far as the `signal' part of our discussion is concerned,\nthe magnitude-observable cross correlation (Eq. [\\ref{Eaverage}])\nis modified to\n\\begin{eqnarray}\n\\label{Eaveragedust}\n\\langle \\mathcal{E} \\rangle = 5 \\tilde s\n\\langle \\kappa \\delta\\mathcal{O} \\rangle - 2.5 \\tilde s\n\\langle \\delta\\tau_{\\rm dust} \\delta\\mathcal{O} \\rangle\\, .\n\\end{eqnarray}\n\nA full calculation of the effect of dust is beyond the scope of this paper. \nBut given a model for $\\delta\\tau_{\\rm dust}$ the effect can be calculated in a way very similar to \nlensing -- $\\tau_{\\rm dust}$ after all is another line of sight integral\nover the (dust) density.\nCalculations comparing the amplitudes of magnification and dust corrections to supernova flux have shown $\\langle\\kappa{\\delta\\tau}_{\\rm dust}\\rangle-\\langle{\\delta\\tau}_{\\rm dust} {\\delta\\tau}_{\\rm dust}\\rangle$ to be $\\sim 7-40\\%$ of $\\langle\\kappa \\kappa\\rangle$ at $z_{source}=1.5$ depending on the dust model \\cite{Zhang:2006qm}.  At higher redshifts the dust term is expected to be less important, while the lensing effect should grow. On the other hand, recent measurements suggest that at $z=0.3$ dust extinction can cause shifts in source magnitude comparable to those caused by lensing magnification \\cite{Menard:2009yb}. For the Lyman-$\\alpha$ forest, which is typically measured at redshifts $z\\gsim 2$,  the dust corrections are likely to be smaller than the lensing corrections but should be included in a full analysis. \n\n\n\\section{Discussion}\n\\label{Conclusions}\nWe have discussed the sampling bias introduced by magnification and dust on measurements of the Lyman-$\\alpha$ forest. In calculating the effect of this bias (summarized in Eq.  (\\ref{Eq:MagBiasFinal})) we made the assumption that all quasar spectra are weighted equally. If, on the other hand, forest measurements were weighted by quasar flux, the effect of lensing bias would be larger than what we have found (see for example Eq.  (\\ref{Eq:MagBiasFluxWeight})).  \n\nIf the quasars are weighted uniformly, we find that magnification bias leads to corrections $\\lsim 1\\%$ to the flux power spectrum (Fig. \\ref{Fig:PffSims}), $\\lsim 0.1\\%$ to the effective optical depth (Fig. \\ref{Fig:TauBias}), $\\lsim 0.1\\%$ to the flux probability distribution function $\\mathcal{P}(f)$, and as large as a few $\\%$ to the  probability distribution function for the flux fluctuation $\\mathcal{P}(\\delta_f)$ (see Fig. \\ref{Fig:PDFetauSims}  and Fig. \\ref{Fig:PDFdeltaSims}). These estimates assume quasar number count slope $s=1$, probably reasonable for the SDSS quasars used for Lyman-$\\alpha$ forest measurements. The lensing correction varies strongly with redshift with larger corrections present at lower redshift, largely due to non-linear growth of mass fluctuations. The biases induced by magnification to the effective optical depth are significantly smaller than current error bars. For the flux power spectrum, the lensing bias is just within the current error bars, and since it leads to a systematic offset in each data point, may effect measurements of the overall amplitude of the power spectrum. At low redshift the lensing effect on the PDF of $\\delta_f$ can be rather large, reaching several percent at the high $\\delta_f$ end of the PDF at $z=2$, however this occurs only in regions where the PDF itself is very small $\\lsim 0.1$. Lensing magnification is probably unimportant for current measurements of the flux PDF and the quantities derived from it. \n\nOne may wonder whether including nonlinear magnification\\footnote{\\label{nonlinfootnote}To be precise, the true magnification $\\mu=1\/((1-\\kappa)^2-\\gamma_1^2-\\gamma_2^2)$ where $\\gamma_1$ and $\\gamma_2$ are the shear. Throughout this paper we assume $\\kappa \\, , \\gamma_1 \\, ,\\gamma_2$ are small quantities and so we keep only the first order terms in the expression for $\\mu$. Non-linear magnification refers to the higher-order (in $\\gamma$ and $\\kappa$)  terms contributing to $\\mu$  \\cite{Menard:2002ia}.} -- which is more important at the small angular separations relevant for the Lyman-$\\alpha$ forest -- could change our results.  We have checked that including all $\\kappa$ terms contributing to $\\mu$ in Eq. (\\ref{nlensed}) and Eq. (\\ref{eq:weightdef}), rather than just the linear term as in Eq. (\\ref{Eq:KappaDef}), changes the magnification bias corrections by $\\lsim 2\\%$. The true nonlinear magnification of course depends on the components of the shear as well as the lensing convergence but it would be surprising if they drastically changed our (all-orders in $\\kappa$) estimate of their importance\\footnote{Actually, there is a further caveat here in that we didn't compute the true lensing convergence $\\kappa$ because we have ignored contributions from mass fluctuations at the lower redshifts outside the box. For the non-linear terms, this shouldn't be a good approximation (e.g. $\\langle\\kappa\\kappa\\rangle$ is poorly estimated) nevertheless it would be surprising if this made an orders-of-magnitude difference in the effect of magnification bias. But perhaps this plus the ignored $\\gamma$ terms could increase the importance of non-linear magnification to the $\\sim 20\\%$ reported by \\cite{Menard:2002ia}.}. \n\n\nAn additional caveat to our analysis is that the damped Lyman-$\\alpha$ (DLA) systems and their associated damping wings are not accurately modeled by the simulations and mock spectra. While DLAs are rare, magnification bias could make them more abundant in survey samples. The damping wings of DLAs are known to add spurious power to $P_{ff}(k_{||})$ at the $10-20\\%$ level on large scales \\cite{McDonald:2005ps}. Magnification bias should increase the abundance of DLAs making the power spectrum more biased than our results suggest. Precisely how magnification affects the DLA bias deserves further investigation.\n\n\nPerhaps most interesting is that lensing magnification induces a correlation between Lyman-$\\alpha$ observables and the magnitude of the quasars used to measure them (Fig. \\ref{Fig:PffMag}). Unfortunately even with the large number of quasar spectra the BOSS survey will obtain it will be a challenge to detect correlations between the flux power spectrum, the flux decrement or the mean transmission and quasar magnitude with high signal-to-noise (see Table I). Additionally, it is possible that lines of sight with high magnification also have more metal lines which could further complicate the analysis. Nevertheless these correlations provide a direct measure of how fluctuations in the quasar flux trace the underlying density field, for example they could \\emph{directly} constrain the flux-mass power spectrum and should therefore be targeted. A more thorough analysis of how to detect the flux-magnitude and flux-power correlations is necessary, we leave this to future work. For a recent idea in a similar vein see \\cite{Vallinotto:2009wa,Vallinotto:2009jx} who propose correlating lensing in the cosmic microwave background with fluctuations in the forest to extract flux-mass information.  \n\nIt is worth noting that, with a large quasar sample, it may be possible to exploit the dependence on $5s-2$ and\/or $\\chi(z_Q)$ to isolate the magnification correction and to measure the flux-magnitude correlations. The lensing correction depends linearly on $5s-2$, which ranges quite a bit depending on magnitude limit (Fig. \\ref{Fig:slope}). In the weak lensing limit the correction scales as $H_0(\\chi(z_Q)-\\chi(z))\\chi(z)\/\\chi(z_Q)$ which varies from $\\sim \\frac{4}{25}H_0\\chi(z_Q)$ to $0$ as $z$ goes from $z_\\beta$ and $z_Q$.   Indeed measuring these distinctive dependencies would help guard against possible instrumental systematics from being confused with the lensing signal.  Our estimates of the forest-magnitude correlations in \\S \\ref{magflux}--\\ref{othercorrelations} assumed $z$ was halfway between $z_\\beta$ and $z_Q$. Also, there exist in public data $\\sim 150$ close quasar pairs \\cite{Hennawi:2006xm}, that one might wish to use for this analysis, unfortunately their number is small compared to the $10^5$ expected from SDSS III and will not significantly improve the signal-to-noise. \n\n\nThere is quite a bit of interest in using the Lyman-$\\alpha$ forest to map the 3D density field and use, for example the baryon features in the two-point correlation function or power spectrum to constrain dark energy and the expansion history of the universe \\cite{McDonald:2006qs}. While we have focused on the power spectrum gotten from 1D measurements of the flux fluctuation, the analysis could be extended to include correlations between different quasar spectra (and therefore different sight-lines). Additional corrections due to correlations between flux and convergence across different lines of sight ($\\sim \\langle\\delta_{f}(\\chi\\mbox{\\boldmath$\\theta$})\\delta_{f}(\\chi'\\mbox{\\boldmath$\\theta$}')\\kappa(\\chi\\mbox{\\boldmath$\\theta$})\\rangle$) would arise. However, since lensing is strongest for lenses along the same line of sight as the sources, the terms calculated in this work should be dominant.  One would therefore expect the lensing bias to the correlation function around the baryon scale to remain $0.1-1\\%$ \\footnote{We have checked that unlike the case of the 3D galaxy correlation function \\cite{Hui:2007cu,Hui:2007tm} nothing is remarkably different about the lensing bias to the real space flux correlation function as compared to Fourier space.}. \n\n\\acknowledgements{We are extremely grateful to Lars Hernquist and Volker Springel for providing us with the simulations data used in our analyses.  We thank David Hogg, Guinevere Kauffman, Ian McGreer, Uros Seljak, David Spergel and David Weinberg for discussions. L.H. thanks members of the CCPP (New York), CITA (Toronto) and IAS (Princeton) for their fabulous hospitality. M.L. and L.H. acknowledge support from the DOE under DE-FG02-92ER40699, the\nNASA ATP under 09-ATP09-0049, and the Initiatives in Science and Engineering Program at Columbia University. S.M. is supported by the Cyprus State Scholarship Foundation. M.L. is supported as a Friends of the Institute for Advanced Study Member.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nToday, there is a large amount of data from p-p, Pb-Pb and recently p-Pb collisions at the LHC which need to be interpreted. EPOS, based on 'Parton-based Gribov-Regge theory' \\cite{epo}, aims to reproduce a large range of LHC observables like jets, multiplicity or collective behavior. We will discuss the recent implementation of charm and prompt photons in this event generator. We want hard probes production to be under control for p-p collisions and then, use them for the study of the QGP.\\\\\nFirst, we will quickly show the general features of EPOS. Then the charm production will be detailed and finally our projet on prompt photons will be exposed.\n\n\n\n\n\\section{General presentation}\nSome important features of EPOS are :\n\n\\vspace{2ex}\n\n\\begin{enumerate}\n\\item Being a real event generator\n\\item Multiple interactions based on a quantum formalism\n\\item Perturbative calculation with resummation of collinear corrections at the order $\\left(\\alpha_s(Q^2)\\ln(\\frac{Q^2}{\\mu^2})\\right)^n$\n\\item Core-corona separation\n\\item Hydrodynamics done event by event\n\\item Hadronisation done using a string fragmentation model for the core\n\\end{enumerate}\n\n\\vspace{2ex}\n\nBy ``being a real event generator'', we mean that one event in the LHC $\\simeq$ one event in EPOS. The program will generate pions even if one is only interested in charm. All particles are registered in a table and, at the end, one has to select particles of interest.\\\\\n\nOur model for multiple interactions is based on a marriage of Gribov-Regge theory \\cite{grib,venus} and pQCD. It gives the possibility of a quantum treatment of multiple interactions. By ``based on Gribov-Regge theory'' we mean an assumption about the structure of the $T$ matrix, expressed in terms\nof elementary objects called Pomerons (not the same object in EPOS and Gribov-Regge theory). The total cross section can be expressed as illustrated on figure \\ref{Tm}.\\\\\n\\begin{figure}[h]\n\\includegraphics[width=23pc]{Tmatrix.png}\\hspace{2pc}%\n\\begin{minipage}[b]{14pc}\\caption{\\label{Tm}Pink lines : pomerons. A and B are nuclei. Small horizontal lines are remnants.}\n\\end{minipage}\n\\end{figure}\\\\\nPartial summation provides exclusive cross sections. In Gribov-Regge theory, the elastic amplitude is given by :\n\\begin{equation}\nA_{2\\rightarrow 2}(s,t)=\\sum_n A_n(s,t)\n\\end{equation}\nwith $A_n(s,t)$ corresponding to the amplitude for $n$ pomeron(s) exchange. Following the same idea, we can defined $\\sigma_m$ corresponding to the cross section for $m$ cut pomerons. A cut pomeron, figure \\ref{pom}, is at the origin of particles production. The multiplicity is then, on the average, proportional to :\n\\begin{equation}\nN\\propto m\\sum_m \\sigma_m\n\\end{equation}\nThe treatment is the same for p-p, p-A or A-A collisions.\\\\\n\nThe hydrodynamical evolution is done event by event. Initial conditions are given by the distribution of cut pomerons which correspond to color flux tubes, figure \\ref{flux}. Flux tubes fragment into string pieces which will later constitute particles. These flux tubes will constitute both bulk matter (if the energy density is high enough) and jets. ``Matter'' is defined by the region of high energy density flux tubes (the blue region figure \\ref{jetfluid}). Then, there are 3 possibilities :\n\\begin{enumerate}\n\\item The string piece (the red one) is formed outside the ``matter''. In that case, it simply escapes as a jet.\n\\item The string piece (the pink one) is formed inside the ``matter'' but has not enough energy to escape. It constitutes the ``matter'' and will evolve with the hydrodynamical code.\n\\item The string piece (the blue one) is formed inside the ``matter'' and has enough energy to escape (based on energy loss argument). It escapes as a jet which has interacted with the fluid. \n\\end{enumerate}\nFor more details, see \\cite{hydro,v2}.\n\\begin{figure}[h]\n\\begin{minipage}{18pc}\n\\includegraphics[width=12pc]{flux-tube.png}\n\\caption{\\label{flux}Cut pomerons form color flux tubes between the 2 nuclei.}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{18pc}\n\\includegraphics[width=12pc]{hydro-jet.png}\n\\caption{\\label{jetfluid}Color flux tubes fragment into string piece. The blue region is \"matter\" formed by high energy density flux tubes.}\n\\end{minipage} \n\\end{figure}\\\\\n\n With these prescriptions, we can reproduce the $v_2$ for identified particles or the ridge, even for p-Pb collisions (figure \\ref{v2} and figure \\ref{ridge}).\n\n\\begin{figure}[h]\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{v2.png}\n\\caption{\\label{v2}(Color online) Elliptical flow coefficients v2 for pi-\nons, kaons, and protons. We show ALICE results (squares)\nand EPOS3 simulations (lines). Pions appear red, kaons\ngreen, protons blue.}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{ridge.png}\n\\caption{\\label{ridge}(Color online) Associated yield per trigger, pro-\njected onto $\\Delta\\phi$, for $|\\Delta\\eta| >$ 0.8. We show ALICE results\n(black squares) and EPOS3 simulations (red dots).}\n\\end{minipage} \n\\end{figure}\n\n\\newpage\n\n\n\n\n\n\\section{Charm production}\n\nA charm can be produced during the spacelike cascade, the born process, the timelike cascade (partonic shower) and the string fragmentation, see figures \\ref{pom} and \\ref{tim}. \n\\begin{figure}[h]\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{cutladder.png}\n\\caption{\\label{pom} A cut pomeron. The center of the pomeron is a (pQCD) ladder diagram.}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{tim-fra.png}\n\\caption{\\label{tim}Partonic shower and hadronization by string fragmentation.}\n\\end{minipage} \n\\end{figure}\nBased on DGLAP formalism, spacelike and timelike cascades resumme collinear divergences.\\\\\n\nDuring the spacelike cascade, a spacelike parton emits timelike particles until he reaches the born process. In the leading log approximation, virtualities\nare strongly ordered :\n\\begin{equation}\n Q_1^2 \\ll Q_2^2 \\ll ... \\ll Q_{born}^2\n\\end{equation}\nWe use the following probability distribution for variables $Q^2$ and $x$ :\n\\begin{equation}\n\\frac{dP(Q^2_0,Q^2,x)}{dxdQ^2}\\propto \\frac{\\alpha_s}{2\\pi}\\frac{p(x)}{Q^2}\\Delta(Q^2_0,Q^2) \\hspace{1cm} Q_0^2<Q^2 \\label{proba1}\n\\end{equation}\nwhere $p(x)$ are the appropriate splitting functions and $x$ is the fraction of the parent parton light cone momentum $k^+$ :\n\\begin{equation}\n k^{'+}=xk^+\n\\end{equation}\n$\\Delta(Q^2_0,Q^2)$, the Sudakov form factor, gives the probability for no resolvable emissions between $Q^2_0$ and $Q^2$. In text books one can find :\n\\begin{equation}\nr^2=\\frac{(1-x)}{x}Q^2-\\frac{p_t^2}{x}\n\\end{equation}\n$r$ being the 4-momentum of the emitted (timelike) parton and $p_t$ its transverse momentum. Usually one takes $r^2=0$, but for heavy quarks we choose :\n\\begin{equation}\nr^2=m^2\n\\end{equation}\nwhich implies :\n\\begin{equation}\nx<\\frac{Q^2}{Q^2+m^2}\n\\end{equation}\nThe phase space for radiating a heavy quark is smaller than the one for light partons.\\\\\n\nThe born process, in the center of the ladder, is nothing else that the leading order cross sections for $\\alpha_s^2$, $\\alpha_s\\alpha_{el}$ and $\\alpha_{el}^2$ processes.\\\\\n\nOutgoing on-shell partons ($r^2=0$) is an approximation. Out-born partons and those emitted during the spacelike cascade have a finite virtuality :\n\\begin{equation}\nQ^2_{max} \\sim p_t^2+m^2\n\\end{equation}\nDuring the timelike cascade (partonic shower) these partons will loose their virtuality by doing successive splittings. This cascade is stopped \nwhen $Q^2 \\sim \\Lambda_{QCD}^2$. The leading log approximation is used and angular ordering \\cite{fsr1,fsr2} is implemented.\nThe emission probability is :\n\\begin{equation}\n\\frac{dP(Q^2_0,Q^2,z)}{dxdQ^2}\\propto \\frac{\\alpha_s}{2\\pi}\\frac{p(z)}{Q^2}\\Delta(Q^2_0,Q^2) \\hspace{1cm} Q_0^2>Q^2\n\\end{equation}\n$z$ being the splitting variable defined as :\n\\begin{equation}\nz=E_{children}\/E_{parent}\n\\end{equation}\n\n\nFor the implementation of charm production, no parameters have been changed. Our first test is the comparison with FONLL calculations of M. Cacciari \\cite{fonll}, \nfigure \\ref{charm}. At high $p_t$ the shape is the same but our central values are higher.\n\\begin{figure}[h]\n\\includegraphics[width=21pc]{charm-nlo-fonll.png}\\hspace{2pc}%\n\\begin{minipage}[b]{14pc}\\caption{\\label{charm}charm from Cacciari vs EPOS.}\n\\end{minipage}\n\\end{figure}\nNext, we test EPOS for $D+$ and $D0$ mesons, by comparing our results with the Alice experiment \\cite{alice} and FONLL calculation, figure \\ref{D+} and figure \\ref{D0}.\n\\begin{figure}[h]\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{D+.png}\n\\caption{\\label{D+} D+ yield from EPOS compare to FONLL calculation \\cite{fonll} and Alice experiment \\cite{alice}.}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{18pc}\n\\includegraphics[width=19pc]{D0.png}\n\\caption{\\label{D0}D0 yield from EPOS compare to FONLL calculation \\cite{fonll} and Alice experiment \\cite{alice}.}\n\\end{minipage} \n\\end{figure}\\\\\n\nEPOS uncertainties account only for statistics and there is no charm production during the string fragmentation. Our results are globally in good agreement with Alice and FONLL. \nHowever, charm quarks are missing at very low $p_t$. The explanation could be that charm production during the timelike cascade is too small. In the game of fitting data, we can reproduce Alice results by allowing charm production in string fragmentation (one parameter is changed), figure \\ref{D+st}.\n\\begin{figure}[h]\n\\includegraphics[width=20pc]{D+frag.png}\\hspace{2pc}%\n\\begin{minipage}[b]{14pc}\\caption{\\label{D+st}D+ from EPOS with $c\\bar{c}$ creation during strings fragmentation.}\n\\end{minipage}\n\\end{figure}\n\n\\newpage\n\n\\section{Prompt photons}\nWe want to :\n\\begin{enumerate}\n\\item Study isolation criteria\n\\item Compare EPOS and Jetphox\n\\item Use EPOS for $\\gamma\/jet$ and $\\gamma\/hadron$ correlations\n\\end{enumerate}\nPhotons production in the born process was already implemented, but it was not the case for the spacelike and the timelike cascade. \nIt has been done with the same formalism used for partons i.e based on the probability eq. \\ref{proba1}. \nOne needs to replace $\\alpha_s$ by $\\alpha_{el}$ and use the appropriate splitting function. \nThe splitting of a photon into a pair of particle-antiparticle is neglected.\\\\\n\nLike in experiments, we have an isolation condition on selected photons. Using the table where final particles are registered, \nwe define a cone of radius $R$ around the photon candidate. If the sum of transverse energy of particles in this cone is smaller than a given \nvalue (5 GeV for CMS \\cite{cms}), then the photon is isolated. Here, the fact that EPOS is experiment like is important. In Jetphox, the addition of this isolation criteria gives a non-physical rise of the cross section with $1\/R$ \\cite{jetp}.\\\\\n\nWork on photons is still in progress. Fragmentation photons are strongly suppressed due to isolation requirement, whereas $\\sim 98\\%$ of direct photons \nare isolated, table \\ref{isolation}.\n\\begin{center}\n\\begin{table}[h]\n\\centering\n\\caption{\\label{isolation}Pourcentage of isolated direct photons as a function of transverse mometum for the isolation criteria used by CMS \\cite{cms} ($R=0.4$, $\\sum p_t <5$GeV).} \n\\begin{tabular}{cc}\n\\br\n$p_t$ & $\\#$ of isolated direct photons\/$\\#$ of direct photons\\\\\n\\mr\n11 &   0.972\\\\\n13 &   0.981\\\\\n15 &   0.971\\\\\n17 &   0.973\\\\\n  19 & 0.976\\\\\n  21 & 0.992\\\\\n  23 & 0.947\\\\\n  25 & 0.991\\\\\n  27 & 0.976\\\\\n  29 & 0.966\\\\\n  31 & 0.977\\\\\n  33 & 0.978\\\\\n  35 & 0.978\\\\\n\\br\n\\end{tabular}\n\\end{table}\n\\end{center}\n\nComparison with CMS \\cite{cms}, figure \\ref{phot}, shows that our yield seems to be too low by approximately a factor of $1.5$. Comparison with Jetphox will give precious information which, I hope, will allow us to improve our results for photons.\n\\begin{figure}[h]\n\\includegraphics[width=20pc]{CMS-pho.png}\\hspace{2pc}%\n\\begin{minipage}[b]{14pc}\\caption{\\label{phot}Isolated photons. For more clarity, a +0.5 shift (x axis) is used for Jetphox data.}\n\\end{minipage}\n\\end{figure}\n\n\\newpage\n\n\\section{Conclusion}\nA unified formalism in EPOS is one of its strengths. Heavy quarks and prompt photons production are based on the same equations with the same parameters. Whereas the work on charm is nearly finished, prompt photons physics still need to be improved. We already have very good results for flow \\cite{v2,epos3}, and with the implementation of hard probes, EPOS could be an excellent tool for the study of the QGP.\n\n\\section*{Acknowledgement}\nI would like to thank the \\textit{\\textbf{projet TOGETHER, region Pays de la Loire}} which has financed my thesis.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe physics of Majorana states in condensed matter devices is attracting strong interest since a few years ago\n\\cite{Nayak,Alicea,StanescuREV,Beenakker,Franz,Elliott,Aguado,Lutrev}. \nThe measured  zero-bias conductance peaks in hybrid semiconductor\/superconductor nanowires have been attributed to the presence of localized Majorana modes on the two ends of the nanowire\n\\cite{Lutchyn,Oreg,Mourik,HaoZ,Deng,Das}. \nThese peculiar pairs of states may be seen as nonlocal split fermions, protected by an energy gap that separates them from other  normal states lying at finite energies. Besides the zero energy of the Majorana state, also the peak height was recently seen to coincide with the expected value $2e^2\/h$ \\cite{HaoZ2}.\n\nMajorana end states in (quasi) 1D nanowires are inherently localized. By contrast, propagating Majorana\nstates can be formed at the edges of 2D-like hybrid structures. We refer, specifically, to the \nhybrid devices of Ref.\\ \\cite{He294}, consisting of a quantum anomalous Hall\ninsulator and a superconductor material. In such systems, chiral Majorana modes propagating along the \nedges in a clockwise or anticlockwise manner, depending on the orientation of a perpendicular magnetic field, are formed \nat the 2D interfaces between the quantum anomalous Hall\nand the superconductor materials\n\\cite{Qi10,Chung,Wang15,Lian16,Kalad}. \nEach chiral Majorana contributes $0.5\\,e^2\/h$  to the linear conductance of the device, \nsuch that by tuning the number of Majoranas the conductance takes values $0.5\\,e^2\/h$ and $1\\,e^2\/h$ \nfor the topological phases with one and two chiral Majoranas, respectively. \n \nIn this work we discuss the connection between chiral Majoranas and optical absorption. We expect that in presence of chiral Majoranas, the optical absorption of circularly polarized light will differ for clockwise and anti-clockwise polarizations. The difference, known as the {\\em circular dichroism} (CD) \\cite{Eisfeld,Longhi}, can thus be seen as a measure of the existence of such chiral states. We want to investigate how this behavior is actually realized by explicit calculations of the optical aborption. In  previous works we analyzed the optical absorption of\nlocalized Majoranas in nanowires \\cite{Ruiz,Osca2015}. \nIn those systems the CD vanishes and the presence of the Majorana is signaled by a lower-absorption plateau, starting at mid-gap energy, \nof the $y$-polarized signal with respect ot the $x$-polarized one.\nIt is also worth mentioning that alternative techniques for detecting Majorana fermions, based on \nmicrowave photoassisted tunneling in Majorana nanocircuits have been suggested in Ref.\\ \\cite{Dart}.\n \n\nFor chiral Majoranas in a 2D square or rectangular geometry the CD \nat low energies\nis characterized by a sequence of equally spaced peaks, corresponding to transitions from\nnegative to positive energy Bogoliubov-deGennes quasiparticles.\nIn the usual energy ordering of quasiparticle states ($n=\\pm1, \\pm2, \\dots$),\nthe selection rules are: a) transitions between conjugate states $-n\\to n$\nare forbidden by electron-hole symmetry, b) transitions $-n\\to m$ \nare allowed only when $n$ and $m$ are both even or both odd.\nThe rationale behind rule b) is the constructive interference\nof the corresponding quasiparticle states connected by the excitation operator\non the edges of the system.\nFurthermore, it will be shown below that the\nCD peaks corresponding to those even-even or odd-odd  quasiparticle transitions may be either\npositive or negative. In the limit of a long 2D ribbon there is a preferred CD sign, depending on\nthe magnetic field orientation.\nFor a disc geometry the generalized angular momentum $J_z$ becomes a good quantum number.\nThen, the combination of circular and particle-hole symmetries in a disc causes a vanishing absorption \nfor $p_x\\pm ip_y$ fields and, obviously, also a vanishing CD.  \n\n\\section{Model}\n\nWe use the model of Ref.\\ \\cite{He294}\nfor a quantum-anomalous Hall (3D) thin film in contact with two different superconductors.\nThis model represents \nthe device as two surfaces with a certain interaction between them, with Majoranas being located at their edges.\nIn a Nambu spinorial representation that groups \nthe field operators in the top $(t)$ and bottom $(b)$ layers,\n$\\left[\n(\n\\Psi^t_{k\\uparrow},\n\\Psi^t_{k\\downarrow},\n\\Psi^{t\\dagger}_{-k\\downarrow},\n-\\Psi^{t\\dagger}_{-k\\uparrow}\n),\n(\n\\Psi^b_{k\\uparrow},\n\\Psi^b_{k\\downarrow},\n\\Psi^{b\\dagger}_{-k\\downarrow},\n-\\Psi^{b\\dagger}_{-k\\uparrow}\n)\\right]^T\n$,\nthe Hamiltonian reads\n\\begin{eqnarray}\n{\\cal H} &=& \n\\left[\\, m_0 + m_1 \\left(p_x^2 +p_y^2\\right)\\, \\right] \\tau_z\\, \\lambda_x \n+ \\Delta_B\\, \\sigma_z - \\mu\\, \\tau_z\\nonumber\\\\\n&-& \\alpha\\, \\left(\\,p_x\\sigma_y-p_y\\sigma_x\\right)\\, \\tau_z\\,\\lambda_z \\nonumber\\\\\n&+& \\Delta_p\\, \\tau_x + \n\\Delta_m\\, \\tau_x\\,\\lambda_z\\, .\n\\end{eqnarray}\nThis Hamiltonian is acting in the combined  position-spin-isospin-pseudospin space. Spatial positions are \na treated as a 2D continuum ($xy$) and a discrete two-valued pseudospin ($z$). The two-valued spin, isospin and pseudospin degrees of freedom are represented by $\\sigma$, $\\tau$ and $\\lambda$ Pauli matrices, respectively. The pseudospin ($\\lambda$) is modeling a coupled bilayer system in which quasiparticles move.  \nThe set of Hamiltonian parameters is $m_0$, $m_1$, $\\Delta_B$, $\\mu$, $\\alpha$, $\\Delta_p$ \nand $\\Delta_m$. The latter two are given in terms of the pairing interaction in the two layers, $\\Delta_1$ and $\\Delta_2$, by\n\\begin{equation}\n\\Delta_{p,m} = \\frac{\\Delta_1\\pm\\Delta_2}{2}\\; .\n\\end{equation}\n\nBelow we numerically determine the eigenvalues and eigenstates of ${\\cal H}$\nusing a 2D grid for $x$ and $y$. Increasing $\\Delta_B$ the spectrum of low-energy \neigenvalues evolves from a gapped (void) spectrum around zero energy at low $\\Delta_B$'s, to the \nemergence of chiral near-zero-energy modes for sufficiently large values of $\\Delta_B$. When the pairing parameters for each layer are equal ($\\Delta_m=0$)  chiral Majoranas appear in pairs ($0-2-\\dots$), while for sufficiently \ndifferent parameters it is $\\Delta_m\\ne 0$ and there may be phases with odd numbers of chiral Majoranas as well.\n\nThe numerical results shown below are given in an effective unit system, characterized by the choice of \n$\\hbar\\equiv 1$, mass $m\\equiv 1\/2m_1 \\equiv 1$ and a chosen length unit\n$L_U$, typically \n$L_U\\approx 1\\, \\mu{\\rm m}$. \nThe corresponding energy unit is then $E_U=\\hbar^2\/mL_U^2$.\n\n\\begin{figure}\n\\includegraphics[width=8.75cm,trim=1.5cm 15.cm 3.cm 2.5cm,clip]{F1}%\n\\caption{Energy eigenvalues closer to zero energy as a function of $\\Delta_B$.\nPanels a) and b) \nare for a square of dimensions $L_x=L_y=10\\, L_U$, while c) and d)\ncorrespond to a rectangle of $L_x=2 L_y=20\\, L_U$. In a) and c) \nthe same pairing energy is assumed in each layer \n$\\Delta_1=\\Delta_2=E_U$ while in b) and d) it is $\\Delta_1=\\Delta_2\/3=E_U$.\nThe framed labels indicate the degeneracy of the near-zero energy states, \nwhich indicates the topological phase. Other parameters:\n$m_0=0$, $\\mu=0$, $\\alpha=E_U L_U$.\n}\n\\label{F1}\n\\end{figure}\n\n\n\n\n\\subsection{Circular dichroism}\n\nWe compute the optical absorption cross section for right ($+$) and left ($-$) circularly-polarized light from \n\\begin{equation}\n{\\cal S}_\\pm(\\omega)= 4m_1^2 \\sum_{k>0, s<0}{ \n\\frac{1}{\\omega_{ks}}\\, \n\\left|\n\\rule{0cm}{0.35cm}\n\\,\\langle k\\vert p_x\\pm ip_y \\vert s\\rangle\\,\\right|^2 \n\\, \n\\delta(\\omega-\\omega_{ks}) }\\; ,\n\\end{equation}\nwhere $\\hbar\\omega_{ks}=\\varepsilon_k-\\varepsilon_s$ is the energy difference between \nparticle (unoccupied) and hole (occupied) states. The prefactor $4m_1^2$ gives the squared \ninverse effective mass ($1\/m_{\\rm eff}^2$) of the Hamiltonian and fixes the dimensions of ${\\cal S}$\nas an area.\nThe circular dichroism at a given frequency ${\\cal S}_{CD}(\\omega)$ is then defined as\nthe difference between the absorptions for the two circular polarizations,\n\\begin{equation}\n{\\cal S}_{CD}(\\omega)={\\cal S}_+(\\omega)-{\\cal S}_-(\\omega)\\; .\n\\end{equation}\nObviously, in absence of any chirality preference ${\\cal S}_{CD}$ exactly vanishes.\n\n\\begin{figure}\n\\includegraphics[width=8.75cm,trim=2cm 15.cm 2.5cm 2.cm,clip]{F2}%\n\\caption{Energy eigenvalues as a function of $\\langle J_z\\rangle$. Panels a) \nand b) correspond to the phases in Fig.\\ \\ref{F1}d with one ($\\Delta_B=2E_U$) and two ($\\Delta_B=4.75 E_U$) Majorana states, respectively. \nThe grey-shaded zones indicate the occupied (hole) states while \nthe arrows in panel a) show the two lowest allowed transitions to the first particle state.\nPanel c) shows the probability density corresponding to the lowest positive-energy state \nin panel a), adding all spin, isospin and pseudospin contributions.}\n\\label{F2}\n\\end{figure}\n\n\n\n\\section{Results and discussion}\n\n\\subsection{Chiral bands}\n\nFigure \\ref{F1} shows the evolution of the eigenvalue \nspectrum as a function of the magnetic field parameter $\\Delta_B$.\nThe results reproduce already known results \\cite{He294}. At vanishing $\\Delta_B$ the spectrum around zero energy is gapped, a gap that tends to close\nwhen increasing $\\Delta_B$\nby the appearance of a quasi-continuum distribution\nof eigenvalues. These low-energy states are indicating the presence of propagating Majoranas, energy discretized due to the finite size of the system.\nWhen $\\Delta_1=\\Delta_2$ (panels a and c) the degeneracy is such that the Majorana branches \nappear in pairs. We also notice that there is no\nqualitative difference in the eigenvalue distribution between a square\nand a rectangle (upper vs lower panels). It is remarkable that when a Majorana phase is well developed\nthe set of low energy states are equally spaced in energy. This is particularly\nclear for $2<\\Delta_B\/E_U<4$ in panels a and c, corresponding to the phases with \n2 Majoranas. It can also be seen in panels b and d for the phases with one Majorana\nwhile, in these panels, the equally spaced distribution is also hinted for the beginning of the phase with 2 Majoranas. \n\nThe chiral character of the gap-closing Majorana states is clearly seen in Fig.\\ \\ref{F2}. The equally spaced states at low energy arrange themselves on a line\n(a chiral band) when plot as a function of the $z$-component of the angular momentum. For positive $\\Delta_B$ the angular momentum decreases with increasing energy, \ncausing empty (particle) states to have negative $\\langle J_z\\rangle$, while occupied (hole) states\nhave positive $\\langle J_z\\rangle$. The results of Fig.\\ \\ref{F2}a,b correspond to the rectangle with different\npairing energies in each layer shown in Fig.\\ \\ref{F1}d. For $\\Delta_B=2 E_U$ (\\ref{F2}a) there is a single \nchiral band, while for $\\Delta_B=4.75 E_U$ (\\ref{F2}b) there are two overlapping bands. Notice that the \noverlap of states in Fig.\\ \\ref{F2}b degrades as the energy deviates from zero, indicating that the second Majorana band is not yet fully settled for this particular $\\Delta_B$. Additionally, Fig. \\ref{F2}c explicitly shows the edge character of the states of a chiral Majorana band. A similar distribution is obtained for all the \nstates in a chiral band.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm,trim=1cm 5.cm 3.5cm 4.5cm,clip]{F3}%\n\\caption{Absorption cross sections ${\\cal S}_+$, ${\\cal S}_-$ and ${\\cal S}_{CD}$ defined in the \nmain text. The shown results correspond to the spectra of Fig.\\ \\ref{F1}d for Zeeman parameters\nof $\\Delta B=0.3 E_U$ (a),  $2 E_U$ (b), and $4.75 E_U$ (c).}\n\\label{F3}\n\\end{figure}\n\n\n\n\\subsection{Absorption and CD}\n\nThe absorption cross sections and the CD for the spectra of the rectangle with different pairing energies in the two layers (Fig.\\ \\ref{F1}d) are shown in Fig.\\ \\ref{F3} for selected values of $\\Delta_B$. They  correspond to zero (\\ref{F3}a), one (\\ref{F3}b) and two (\\ref{F3}c) chiral bands. As anticipated, in presence of the chiral states the system develops a clear CD. The negative CD peaks dominate, due to the negative slope of the chiral bands (Fig.\\ \\ref{F2}a,b). It is remarkable, however, that a few positive peaks are also present. We attribute them to the fact that in a rectangular geometry $J_z$ is not a good\nquantum number and, therefore, there are states with mixed angular momentum. We have performed calculations in a circular geometry confirming this interpretation. Therefore, quasiparticle scattering by the corners\nplays a nontrivial role on the absorption by chiral edge states. \n\n \n\n\\begin{figure}\n\\includegraphics[width=8.5cm,trim=1cm 11.cm 3.5cm 4.5cm,clip]{F4}%\n\\caption{Absorption cross sections ${\\cal S}_+$, ${\\cal S}_-$ and ${\\cal S}_{CD}$ \n for a square of $L_x=L_y=20\\, L_U$ (a), and for a rectangle of $L_x=6\\, L_y= 60\\, L_U$. In both cases \n we used $\\Delta_B=2\\, E_U$ and $\\Delta_1=\\Delta_2\/3= E_U$.}\n\\label{F4}\n\\end{figure}\n\n\nThe most conspicuous feature of Fig.\\ \\ref{F3}b is the regular energy spacing of the first few CD peaks.\nAnalysing them in terms of energy transitions of the chiral band it is easily  noticed\nthat they correspond to jumps of 3, 5, 7, \\dots steps (see arrows in Fig.\\ \\ref{F2}a). We explain this selection rule noticing the  following restrictions for transitions from the negative $n$-th state to the positive $m$-th state \n($-n\\to m$):\n\\begin{itemize}\n\\item[a)] transitions between conjugate states $-n\\to n$ are forbidden by particle-hole symmetry \\cite{Ruiz}, \n\\item[b)] $n$ even to $m$ odd transitions (or vice versa) are forbidden because of destructive interference along the nanostructure perimeter with the excitation operator.\n\\end{itemize}\n\nFor a disc, $J_z$ becomes a good symmetry and, by angular momentum conservation with a dipole \noperator only the transition $-1\\to1$ is possible. However, this transition is blocked by rule a)\nand, therefore, no dipole absorption is possible and the CD exactly vanishes. We have also checked this behavior by explicit calculation for a device with circular geometry. For a square and rectangle, quasiparticle scattering by the corners plays a nontrivial role yielding the mentioned deviations with respect to the disc.\n\nThe pattern of equally spaced peaks is fulfilled only when one or several chiral bands are fully developed and they exactly overlap. In Fig.\\ \\ref{F3}c we see that the slight degradation of the two-band overlaps\nof Fig.\\ \\ref{F2}b manifests in a small twofold splitting of the CD peaks. It is also worth stressing that once the chiral bands are fully formed, the energy positions of the first few CD peaks become $\\Delta_B$-independent (cf.\\ Figs.\\ \\ref{F2}b and \\ref{F2}c).\n\n\nFigure \\ref{F4} shows the absorption results for different geometries, a square (\\ref{F4}a) and a \nlong rectangle resembling a 2D ribbon (\\ref{F4}b). For the square, the first CD peaks alternate sign in a\nremarkable way. On the other hand, for the ribbon the alternation is of a longer period, the positive \npeaks having a much lower intensity than the negative ones and there are groups of a few consecutive negative peaks. The 2D ribbon shape thus favors the observation of CD peaks of the same sign.  \n\n\\section{Conclusions}\n\nIn this work we have investigated the manifestation of chiral Majorana modes in the CD of the dipole absorption. The chiral bands formed at the edges of a hybrid system made of a  quantum-anomalous-Hall-insulator and a superconductor yield equally spaced peaks in the CD signal. We identified the particle-hole \nselection rules responsible for this behavior from the analysis in terms of chiral bands.\nIn a disc there is no CD signal due to the incompatibility of the selection rules with the angular momentum restriction; a square or rectangular geometry (or, more generally, a system with straight edges) is needed.\nThe presence of two chiral bands can be inferred from the small splitting of the CD peaks.  Finally, both positive and negative CD peaks can be seen, with a perfect alternation in a square and a favored sign \nin a long 2D ribbon geometry. \n\nOn the whole, our results suggest the use of CD spectroscopy as a valuable \nprobe of chiral Majorana states, complementing the evidences obtained\nwith electrical conductance measurements \\cite{He294}. This may require the use of\nan array of absorbing devices, in order to achieve a combined signal of sufficient intensity.\nAlternatively, techniques such as those developed for single plasmonic nanoparticle sensing \\cite{Olson} \nmight be applied to an isolated chiral-Majorana device.  \n\n\n\\begin{acknowledgements}\nThis work was funded by MINECO (Spain), grant FIS2014-52564.\n\\end{acknowledgements} \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Exciton-Plasmon Interactions in Individual Carbon Nanotubes}\n\\setcounter{page}{0}\n\n\\begin{center}\n{\\huge\\textbf{Exciton-Plasmon Interactions in\\\\[0.25cm] Individual Carbon\nNanotubes\\footnote{To appear in \"Plasmons: Theory and\nApplications\", ed. K.Helsey (Nova Publishers, NY, USA)}}}\\\\[1.0cm]\n\n{\\Large \\textbf{Igor V. Bondarev}}\\\\[0.25cm]\n{\\large {\\it Department of Physics, North Carolina Central University\\\\\nDurham, NC 27707, USA \\\\[0.25cm]{\\tt E-mail:~ibondarev@nccu.edu}}}\\\\[1cm]\n\n{\\Large \\textbf{Lilia M. Woods} ~and~ \\textbf{Adrian Popescu}}\\\\[0.25cm]\n{\\large {\\it Department of Physics, University of South Florida\\\\\nTampa, FL 33620, USA \\\\[1cm]}}\n\n\\begin{minipage}{5in}\n\nWe use the macroscopic quantum electrodynamics approach suitable\nfor absorbing and dispersing media to study the properties and\nrole of collective surface excitations --- excitons and plasmons\n--- in single-wall and double-wall carbon nanotubes. We show that\nthe interactions of excitonic states with surface electromagnetic\nmodes in individual small-diameter ($\\,\\lesssim\\!1$~nm)\nsingle-walled carbon nanotubes can result in strong\nexciton-surface-plasmon coupling. Optical response of individual\nnanotubes exhibits Rabi splitting $\\sim\\!0.1$~eV, both in the\nlinear excitation regime and in the non-linear excitation regime\nwith the photoinduced biexcitonic states formation, as the exciton\nenergy is tuned to the nearest interband surface plasmon resonance\nof the nanotube. An electrostatic field applied perpendicular to\nthe nanotube axis can be used to control the exciton-plasmon\ncoupling. For double-wall carbon nanotubes, we show that at tube\nseparations similar to their equilibrium distances interband\nsurface plasmons have a profound effect on the inter-tube Casimir\nforce. Strong overlapping plasmon resonances from both tubes\nwarrant their stronger attraction. Nanotube chiralities possessing\nsuch collective excitation features will result in forming the\nmost favorable inner-outer tube combination in double-wall carbon\nnanotubes. These results pave the way for the development of new\ngeneration of tunable optoelectronic and nano-electromechanical\ndevice applications with carbon nanotubes.\n\\end{minipage}\n\\end{center}\n\n\\noindent {\\bf Keywords:} Carbon nanotubes, Near-field effects,\nExcitons, Plasmons\n\n\\newpage\n\n\\section{Introduction}\n\nSingle-walled carbon nanotubes (CNs) are quasi-one-dimensional\n(1D) cylindrical wires consisting of graphene sheets rolled-up\ninto cylinders with diameters $\\sim\\!1-10$~nm and lengths\n$\\sim\\!1-10^4\\,~\\mu$m~\\cite{Dresselhaus,Dai,Zheng,Huang}. CNs are\nshown~to~be useful as miniaturized electronic, electromechanical,\nand chemical devices~\\cite{Baughman}, scanning probe\ndevices~\\cite{Popescu}, and nanomaterials for macroscopic\ncomposites~\\cite{Trancik}. The area of their potential\napplications was recently expanded to\nnanophotonics~\\cite{Bondarev10,Bondarev06trends,Bondarev06,Bondarev07,Bondarev07jem,Bondarev07os}\nafter the demonstration of controllable single-atom incapsulation\ninto CNs~\\cite{Shimoda,Jeong,Jeongtsf,Khazaei}, and even to\nquantum cryptography since the experimental evidence was reported\nfor quantum correlations in the photoluminescence spectra of\nindividual nanotubes~\\cite{Imamoglu}.\n\nFor pristine (undoped) single-walled CNs, the numerical\ncalculations predicting large exciton binding energies\n($\\sim\\!0.3\\!-\\!0.6$~eV) in semiconducting\nCNs~\\cite{Pedersen03,Pedersen04,Capaz}~and even in some\nsmall-diameter ($\\sim\\!0.5$~nm) metallic CNs~\\cite{Spataru04},\nfollowed by the results of various exciton photoluminescence\nmeasurements~\\cite{Imamoglu,Wang04,Wang05,Hagen05,Plentz05,Avouris08},\nhave become available.~These works, together with other reports\ninvestigating the role of effects such as intrinsic\ndefects~\\cite{Hagen05,Prezhdo08}, exciton-phonon\ninteractions~\\cite{Plentz05,Prezhdo08,Perebeinos05,Lazzeri05,Piscanec},\nbiexciton formation~\\cite{Pedersen05,Papan},\nexciton-surface-plasmon\ncoupling~\\cite{Qasmi,BondOptCom,BondOptSpectr,BondPRB09}, external\nmagnetic~\\cite{Zaric,Srivastava} and electric\nfields~\\cite{BondPRB09,Perebeinos07}, reveal the variety and\ncomplexity of the intrinsic optical properties of\nCNs~\\cite{Dresselhaus07}.\n\nCarbon nanotubes combine advantages such as electrical\nconductivity, chemical stability, high surface area, and unique\noptoelectronic properties that make them excellent potential\ncandidates for a variety of applications, including efficient\nsolar energy conversion~\\cite{Trancik}, energy\nstorage~\\cite{Shimoda}, optical\nnanobiosensorics~\\cite{Goodsell10}. However, the quantum yield of\nindividual CNs is normally very low. Nanotube composites of CN\nbundles and\/or films could surpass this difficulty, opening up new\npaths for the development of high-yield, high-performance\noptoelectronics applications with CNs~\\cite{Munich,Ferrari}.\nUnderstanding the inter-tube interactions is important in order to\nbe able to tailor properties of CN bundles and films, as well as\nproperties of multi-wall CNs. This is also important for\nexperimental realization of new effects and devices proposed\nrecently, such as trapping of cold\natoms~\\cite{Goodsell10,Fermani2007} and their\nentanglement~\\cite{Bondarev07} near single-walled CNs, surface\nprofiling~\\cite{Popescu} and nanolithography\napplications~\\cite{Popescu2009} with double-wall CNs.\n\nHere, we use the macroscopic Quantum ElectroDynamics (QED)\nformalism developed earlier for absorbing and dispersive\nmedia~\\cite{VogelWelsch,ScheelWelsch,BuhmannWelsch,Bondarev06trends}\nand then successfully employed to study near-field EM effects in\nhybrid CN systems~\\cite{Bondarev06,Bondarev07,Fermani2007}, to\ninvestigate the properties and role of collective surface\nexcitations --- excitons and plasmons --- in single-wall and\ndouble-wall CNs. First, we show that, due to the presence of\nlow-energy ($\\sim\\!0.5\\!-\\!2$~eV) weakly-dispersive interband\nplasmon modes~\\cite{Pichler98} and large exciton excitation\nenergies in the same energy domain~\\cite{Spataru05,Ma}, the\nexcitons can form strongly coupled mixed exciton-plasmon\nexcitations in individual small-diameter ($\\lesssim\\!1$~nm)\nsemiconducting single-walled CNs. The exciton-plasmon coupling\n(and the exciton emission accordingly) can be controlled by an\nexternal electrostatic field applied perpendicular to the CN axis\n(the quantum confined Stark effect). The optical response of\nindividual CNs exhibits the Rabi splitting of $\\sim\\!0.1$~eV, both\nin the linear excitation regime and in the non-linear excitation\nregime with the photoinduced biexcitonic states formation, as the\nexciton energy is tuned to the nearest interband surface plasmon\nresonance of the CN. Previous studies of the exciton-plasmon\ncoupling have been focused on artificially fabricated\n\\emph{hybrid} plasmonic nanostructures, such as dye molecules in\norganic polymers deposited on metallic films~\\cite{Bellessa},\nsemiconductor quantum dots coupled to metallic\nnanoparticles~\\cite{Govorov}, or nanowires~\\cite{Fedutik}, where\nsemiconductor material carries the exciton and metal carries the\nplasmon. Our results are particularly interesting since they\nreveal the fundamental electromagnetic (EM) phenomenon\n--- the strong exciton-plasmon coupling --- in an\n\\emph{individual} quasi-1D nanostructure, a~carbon nanotube, as\nwell as its tunability feature by means of the quantum confined\nStark effect. We expect these results to open up new paths for the\ndevelopment of tunable optoelectronic device applications with\noptically excited carbon nanotubes, including the strong\nexcitation regime with optical non-linearities.\n\nNext, we turn to the double-wall carbon nanotubes to investigate\nthe effect of collective surface excitations on the inter-tube\nCasimir interaction in these systems. The Casimir interaction is a\nparadigm for a force induced by quantum EM fluctuations. The\nfundamental nature of this force has been studied extensively ever\nsince the prediction of the existence of an attraction between\nneutral metallic mirrors in\nvacuum~\\cite{BuhmannWelsch,Klimchitskaya2009}. In recent years,\nthe Casimir effect has acquired a much broader impact due to its\nimportance for nanostructured materials and devices. The\ndevelopment and operation of micro- and nano-electromechanical\nsystems are limited due to unwanted effects, such as stiction,\nfriction, and adhesion, originating from the Casimir\nforce~\\cite{Chan2001}. This interaction is also an important\ncomponent for the stability of nanomaterials. Here, we show that\nat tube separations similar to their equilibrium distances\ninterband surface plasmons have a profound effect on the\ninter-tube Casimir force. Strong overlapping plasmon resonances\nfrom both tubes warrant their stronger attraction. Nanotube\nchiralities possessing such collective excitation features will\nresult in forming the most favorable inner-outer tube combination\nin double-wall carbon nanotubes. This theoretical understanding is\nimportant for the development of nano-electromechanical devices\nwith CNs.\n\nThis Chapter is organized as follows.~Section~\\ref{sec2}\nintroduces the general Hamiltonian of the exciton interaction with\nvacuum-type quantized surface EM modes of a single-walled CN. No\nexternal EM field is assumed to be applied.~The\nvacuum--type--field we consider is created by CN surface EM\nfluctuations.~Section~\\ref{sec3} explains how the interaction\nintroduced results in the coupling of the excitonic states to the\nnanotube's surface plasmon modes.~Here we derive and discuss the\ncharacteristics of the coupled exciton--plasmon excitations, such\nas the dispersion relation, the plasmon density of states (DOS),\nand the optical response functions, for particular semiconducting\nCNs of different diameters. We also analyze how the electrostatic\nfield applied perpendicular to the CN axis affects the CN band\ngap, the exciton binding energy, and the surface plasmon energy,\nto explore the tunability of the exciton-surface-plasmon coupling\nin CNs.~Section~\\ref{sec4} derives and analyzes the Casimir\ninteraction between two concentric cylindrical graphene sheets\ncomprising a double-wall CN. The summary and conclusions of the\nwork are given in Sec.~\\ref{concl}. All the technical details\nabout the construction and diagonalization of the exciton--field\nHamiltonian, the EM field Green tensor derivation, the\nperpendicular electrostatic field effect, are presented in the\nAppendices in order not to interrupt the flow of the arguments and\nresults.\n\n\\section{Exciton-electromagnetic-field interaction\\\\ on the\nnanotube surface}\\label{sec2}\n\nWe consider the vacuum-type EM interaction of an exciton with the\nquantized surface electromagnetic fluctuations of a single-walled\nsemiconducting CN by using our recently developed Green function\nformalism to quantize the EM field in the presence of quasi-1D\nabsorbing\nbodies~\\cite{Bondarev02,Bondarev04,Bondarev04pla,Bondarev04ssc,Bondarev05,Bondarev06trends}.\nNo external EM field is assumed to be applied. The nanotube is\nmodelled by an infinitely thin, infinitely long, anisotropically\nconducting cylinder with its surface conductivity obtained from\nthe realistic band structure of a particular CN. Since the problem\nhas the cylindrical symmetry, the orthonormal cylindrical basis\n$\\{\\mathbf{e}_{r},\\mathbf{e}_{\\varphi},\\mathbf{e}_{z}\\}$ is used\nwith the vector $\\mathbf{e}_{z}$ directed along the nanotube axis\nas shown in Fig.~\\ref{fig1}. Only the axial conductivity,\n$\\sigma_{zz}$, is taken into account, whereas the azimuthal one,\n$\\sigma_{\\varphi\\varphi}$, being strongly suppressed by the\ntransverse depolarization\neffect~\\cite{Benedict,Tasaki,Li,Marinop,Ando,Kozinsky}, is\nneglected.\n\n\\begin{figure}[t]\n\\epsfxsize=9.0cm\\centering{\\epsfbox{fig1.eps}}\\caption{The\ngeometry of the problem.}\\label{fig1}\n\\end{figure}\n\nThe total Hamiltonian of the coupled exciton-photon system on the\nnanotube surface is of the form\n\\begin{equation}\n\\hat{H}=\\hat{H}_F+\\hat{H}_{ex}+\\hat{H}_{int},\\label{Htot}\n\\end{equation}\nwhere the three terms represent the free (medium-assisted) EM\nfield, the free (non-interacting) exciton, and their interaction,\nrespectively.~More explicitly, the second quantized field\nHamiltonian is\n\\begin{equation}\n\\hat{H}_F\\!=\\!\\sum_\\mathbf{n}\\int_0^\\infty\\!\\!\\!d\\omega\\,\\hbar\\omega\n\\hat{f}^\\dag(\\mathbf{n},\\omega)\\hat{f}(\\mathbf{n},\\omega),\n\\label{HF}\n\\end{equation}\nwhere the scalar bosonic field operators\n$\\hat{f}^\\dag(\\mathbf{n},\\omega)$ and $\\hat{f}(\\mathbf{n},\\omega)$\ncreate and annihilate, respectively, the surface EM excitation of\nfrequency $\\omega$ at an arbitrary point\n$\\mathbf{n}\\!=\\!\\mathbf{R}_n\\!=\\!\\{R_{CN},\\varphi_n,z_n\\}$\nassociated with a carbon atom (representing a lattice site --\nFig.~\\ref{fig1}) on the surface of the CN of radius $R_{CN}$. The\nsummation is made over all the carbon atoms, and in the following\nit is replaced by the integration over the entire nanotube surface\naccording to the rule\n\\begin{equation}\n\\sum_\\mathbf{n}\\!\\ldots=\\frac{1}{S_0}\\int\\!d\\mathbf{R}_n\\!\\ldots=\n\\frac{1}{S_0}\\int_0^{2\\pi}\\!\\!\\!\\!d\\varphi_nR_{CN}\\!\\int_{-\\infty}^\\infty\\!\\!\\!\\!dz_n\\!\\ldots,\n\\label{sumrule}\n\\end{equation}\nwhere $S_0\\!=\\!(3\\sqrt{3}\/4)b^2$ is the area of an elementary\nequilateral triangle selected around each carbon atom in a way to\ncover the entire surface of the nanotube, $b\\!=\\!1.42$~\\AA\\space\nis the carbon-carbon interatomic distance.\n\nThe second quantized Hamiltonian of the free exciton (see, e.g.,\nRef.~\\cite{Haken}) on the CN surface is of the form\n\\begin{equation}\n\\hat{H}_{ex}\\!=\\!\\!\\!\\sum_{\\mathbf{n},\\mathbf{m},f}\\!\\!\\!E_f(\\mathbf{n})\nB^\\dag_{\\mathbf{n}+\\mathbf{m},f}B_{\\mathbf{m},f}\\!=\n\\!\\sum_{\\mathbf{k},f}E_f(\\mathbf{k})B^\\dag_{\\mathbf{k},f}B_{\\mathbf{k},f},\n\\label{Hex}\n\\end{equation}\nwhere the operators $B^\\dag_{\\mathbf{n},f}$ and $B_{\\mathbf{n},f}$\ncreate and annihilate, respectively, an exciton with the energy\n$E_f(\\mathbf{n})$ in the lattice site $\\mathbf{n}$ of the CN\nsurface.~The index $f\\,(\\ne\\!0)$ refers to the internal degrees of\nfreedom of the exciton. Alternatively,\n\\begin{equation}\nB^\\dag_{\\mathbf{k},f}=\\frac{1}{\\sqrt{N}}\\sum_{\\mathbf{n}}\\!B^\\dag_{\\mathbf{n},f}\ne^{i\\mathbf{k}\\cdot\\mathbf{n}}~~~\\mbox{and}~~~B_{\\mathbf{k},f}=(B^\\dag_{\\mathbf{k},f})^\\dag\n\\label{Bkf}\n\\end{equation}\ncreate and annihilate the $f$-internal-state exciton with the\nquasi-momentum $\\mathbf{k}\\!=\\!\\{k_{\\varphi},k_{z}\\}$, where the\nazimuthal component is quantized due to the transverse confinement\neffect and the longitudinal one is continuous, $N$ is the total\nnumber of the lattice sites (carbon atoms) on the CN surface. The\nexciton total energy is then written in the form\n\\begin{equation}\nE_f(\\mathbf{k})=E_{exc}^{(f)}(k_{\\varphi})+\\frac{\\hbar^2k_z^2}{2M_{ex}(k_{\\varphi})}\n\\label{Ef}\n\\end{equation}\nHere, the first term represents the excitation energy\n\\begin{equation}\nE_{exc}^{(f)}(k_{\\varphi})=E_g(k_{\\varphi})+E_b^{(f)}(k_{\\varphi})\n\\label{Eexcf}\n\\end{equation}\nof the $f$-internal-state exciton with the (negative) binding\nenergy $E_b^{(f)}$, created via the interband transition with the\nband gap\n\\begin{equation}\nE_g(k_{\\varphi})=\\varepsilon_e(k_{\\varphi})+\\varepsilon_h(k_{\\varphi}),\n\\label{Egkfi}\n\\end{equation}\nwhere $\\varepsilon_{e,h}$ are transversely quantized azimuthal\nelectron-hole subbands (see the schematic in Fig.~\\ref{fig2}).~The\nsecond term in Eq.~(\\ref{Ef}) represents the kinetic energy of the\ntranslational longitudinal movement of the exciton with the\neffective mass $M_{ex}=m_e+m_h$, where $m_e$ and $m_h$ are the\n(subband-dependent) electron and hole effective masses,\nrespectively. The two equivalent free-exciton Hamiltonian\nrepresentations are related to one another via the obvious\northogonality relationships\n\\begin{equation}\n\\frac{1}{N}\\sum_\\mathbf{n}e^{-i(\\mathbf{k}-\\mathbf{k}^\\prime)\\cdot\\mathbf{n}}=\n\\delta_{\\mathbf{k}\\mathbf{k}^\\prime},\\;\\;\n\\frac{1}{N}\\sum_\\mathbf{k}e^{-i(\\mathbf{n}-\\mathbf{m})\\cdot\\mathbf{\\mathbf{k}}}=\n\\delta_{\\mathbf{n}\\mathbf{m}} \\label{orthog}\n\\end{equation}\nwith the $\\mathbf{k}$-summation running over the first Brillouin\nzone of the nanotube.~The bosonic field operators in $\\hat{H}_F$\nare transformed to the $\\mathbf{k}$-representation in the same\nway.\n\n\\begin{figure}[t]\n\\epsfxsize=10.0cm\\centering{\\epsfbox{fig2.eps}}\\caption{Schematic\nof the two transversely quantized azimuthal electron-hole subbands\n(\\emph{left}), and the first-interband ground-internal-state\nexciton energy (\\emph{right}) in a small-diameter semiconducting\ncarbon nanotube. Subbands with indices $j=1$ and 2 are shown,\nalong with the optically allowed (exciton-related) interband\ntransitions~\\cite{Ando}. See text for notations.}\\label{fig2}\n\\end{figure}\n\nThe most general (non-relativistic, electric dipole)\nexciton-photon interaction on the nanotube surface can be written\nin the form (we use the Gaussian system of units and the Coulomb\ngauge; see details in Appendix A)\n\\begin{equation}\n\\hat{H}_{int}=\\sum_{\\mathbf{n},\\mathbf{m},f}\\int_{0}^{\\infty}\\!\\!\\!\\!\\!d\\omega\\,[\\,\n\\mbox{g}_f^{(+)}(\\mathbf{n},\\mathbf{m},\\omega)B^\\dag_{\\mathbf{n},f}-\n\\mbox{g}_f^{(-)}(\\mathbf{n},\\mathbf{m},\\omega)B_{\\mathbf{n},f}\\,]\\,\n\\hat{f}(\\mathbf{m},\\omega)+h.c.,\\label{Hint}\n\\end{equation}\nwhere\n\\begin{equation}\n\\mbox{g}_f^{(\\pm)}(\\mathbf{n},\\mathbf{m},\\omega)=\n\\mbox{g}_f^{\\perp}(\\mathbf{n},\\mathbf{m},\\omega)\\pm\n\\frac{\\omega}{\\omega_f}\\,\\mbox{g}_f^{\\parallel}(\\mathbf{n},\\mathbf{m},\\omega)\n\\label{gpm}\n\\end{equation}\nwith\n\\begin{equation}\n\\mbox{g}_f^{\\perp(\\parallel)}(\\mathbf{n},\\mathbf{m},\\omega)=\n-i\\frac{4\\omega_f}{c^{2}}\\sqrt{\\pi\\hbar\\omega\\,\\mbox{Re}\\,\\sigma_{zz}(R_{CN},\\omega)}\\;\n(\\textbf{d}^f_{\\mathbf{n}})_z\\,^{\\perp(\\parallel)}G_{zz}(\\mathbf{n},\\mathbf{m},\\omega)\n\\label{gperppar}\n\\end{equation}\nbeing the interaction matrix element where the exciton with the\nenergy $E_{exc}^{(f)}=\\hbar\\omega_f$ is excited through the\nelectric dipole transition\n$(\\textbf{d}^f_{\\mathbf{n}})_z\\!=\\!\\langle0|(\\hat{\\mathbf{d}}_\\mathbf{n})_z|f\\rangle$\nin the lattice site $\\textbf{n}$ by the nanotube's transversely\n(longitudinally) polarized surface EM modes. The modes are\nrepresented in the matrix element by the transverse (longitudinal)\npart of the Green tensor $zz$-component\n$G_{zz}(\\mathbf{n},\\mathbf{m},\\omega)$ of the EM subsystem\n(Appendix~B). This is the only Green tensor component we have to\ntake into account.~All the other components can be safely\nneglected as they are greatly suppressed by the strong transverse\ndepolarization effect in\nCNs~\\cite{Benedict,Tasaki,Li,Marinop,Ando,Kozinsky}.~As a\nconsequence, only $\\sigma_{zz}(R_{CN},\\omega)$, the \\emph{axial}\ndynamic surface conductivity per unit length, is present in\nEq.(\\ref{gperppar}).\n\nEquations~(\\ref{Htot})--(\\ref{gperppar}) form the complete set of\nequations describing the exciton-photon coupled system on the CN\nsurface in terms of the EM field Green tensor and the CN surface\naxial conductivity.\n\n\\section{Exciton-surface-plasmon coupling}\\label{sec3}\n\nFor the following it is important to realize that the transversely\npolarized surface EM mode contribution to the\ninteraction~(\\ref{Hint})--(\\ref{gperppar}) is negligible compared\nto the longitudinally polarized surface EM mode contribution. As\na~matter of fact,\n$^{\\perp}G_{zz}(\\mathbf{n},\\mathbf{m},\\omega)\\!\\equiv\\!0$ in the\nmodel of an infinitely thin cylinder we use here (Appendix~B),\nthus yielding\n\\begin{equation}\n\\mbox{g}_f^\\perp(\\mathbf{n},\\mathbf{m},\\omega)\\!\\equiv\\!0,~~~\n\\mbox{g}_f^{(\\pm)}(\\mathbf{n},\\mathbf{m},\\omega)\\!=\\!\n\\pm\\frac{\\omega}{\\omega_f}\\,\\mbox{g}_f^{\\parallel}(\\mathbf{n},\\mathbf{m},\\omega)\n\\label{gfpar}\n\\end{equation}\nin Eqs.~(\\ref{Hint})--(\\ref{gperppar}). The point is that, because\nof the nanotube quasi-one-dimen\\-sionality, the exciton\nquasi-momentum vector and all the relevant vectorial matrix\nelements of the momentum and dipole moment operators are directed\npredominantly along the CN axis (the longitudinal exciton; see,\nhowever, Ref.~\\cite{UryuAndo}). This prevents the exciton from the\nelectric dipole coupling to transversely polarized surface EM\nmodes as they propagate predominantly along the CN axis with their\nelectric vectors orthogonal to the propagation direction.~The\nlongitudinally polarized surface EM modes are generated by the\nelectronic Coulomb potential (see, e.g., Ref.~\\cite{Landau}), and\ntherefore represent the CN surface plasmon excitations.~These have\ntheir electric vectors directed along the propagation direction.\nThey do couple to the longitudinal excitons on the CN surface.\nSuch modes were observed in Ref.~\\cite{Pichler98}. They occur in\nCNs both at high energies (well-known $\\pi$-plasmon~at\n$\\sim\\!6$~eV) and at comparatively low energies of\n$\\sim\\!0.5\\!-\\!2$~eV.~The latter ones are related to the\ntransversely quantized interband (inter-van Hove) electronic\ntransitions. These weakly-dispersive\nmodes~\\cite{Pichler98,Kempa02} are similar to the intersubband\nplasmons in quantum wells~\\cite{Kempa89}. They occur in the same\nenergy range of $\\sim\\!1$~eV where the exciton excitation energies\nare located in small-diameter ($\\lesssim\\!1$~nm) semiconducting\nCNs~\\cite{Spataru05,Ma}. In what follows we focus our\nconsideration on the exciton interactions with these particular\nsurface plasmon modes.\n\n\\subsection{The dispersion relation}\n\nTo obtain the dispersion relation of the coupled exci\\-ton-plasmon\nexcitations, we transfer the total\nHamiltonian~(\\ref{Htot})--(\\ref{Hint}) and (\\ref{gfpar}) to the\n$\\mathbf{k}$-representation using Eqs.~(\\ref{Bkf}) and\n(\\ref{orthog}), and then diagonalize it exactly by means of\nBogoliubov's canonical transformation technique (see, e.g.,\nRef.~\\cite{Davydov}). The details of the procedure are given in\nAppendix~C. The Hamiltonian takes the form\n\\begin{equation}\n\\hat{H}=\\!\\!\\!\\sum_{\\mathbf{k},\\,\\mu=1,2}\\!\\!\\!\\hbar\\omega_\\mu(\\mathbf{k})\\,\n\\hat{\\xi}^\\dag_\\mu(\\mathbf{k})\\hat{\\xi}_\\mu(\\mathbf{k})+E_0\\,.\n\\label{Htotdiag}\n\\end{equation}\nHere, the new operator\n\\begin{eqnarray}\n\\hat{\\xi}_\\mu(\\mathbf{k})=\n\\sum_{f}\\left[u_\\mu^\\ast(\\mathbf{k},\\omega_f)B_{\\mathbf{k},f}\n-v_\\mu(\\mathbf{k},\\omega_f)B_{-\\mathbf{k},f}^\\dag\\right]\\label{xik}\\\\\n+\\int_0^\\infty\\!\\!\\!\\!\\!d\\omega\\left[u_\\mu(\\mathbf{k},\\omega)\\hat{f}(\\mathbf{k},\\omega)-\nv_\\mu^\\ast(\\mathbf{k},\\omega)\\hat{f}^\\dag(-\\mathbf{k},\\omega)\\right]\\nonumber\n\\end{eqnarray}\nannihilates and\n$\\hat{\\xi}^\\dag_\\mu(\\mathbf{k})\\!=\\![\\hat{\\xi}_\\mu(\\mathbf{k})]^\\dag$\ncreates the exciton-plas\\-mon excitation of branch $\\mu$, the\nquantities $u_\\mu$ and $v_\\mu$ are appropriately chosen canonical\ntransformation coefficients. The \"vacuum\" energy $E_0$ represents\nthe state with no exciton-plasmons excited in the system, and\n$\\hbar\\omega_\\mu(\\mathbf{k})$ is the exciton-plasmon energy given\nby the solution of the following (dimensionless) dispersion\nrelation\n\\begin{equation}\nx_\\mu^2-\\varepsilon_f^2-\\varepsilon_f\\frac{2}{\\pi}\\int_0^\\infty\\!\\!\\!\\!\\!dx\\,\n\\frac{x\\,\\bar\\Gamma_0^f(x)\\rho(x)}{x_\\mu^2-x^2}=0\\,.\\label{dispeq}\n\\end{equation}\nHere,\n\\begin{equation}\nx=\\frac{\\hbar\\omega}{2\\gamma_0},~~~\nx_\\mu=\\frac{\\hbar\\omega_\\mu(\\mathbf{k})}{2\\gamma_0},~~~\n\\varepsilon_f=\\frac{E_f(\\mathbf{k})}{2\\gamma_0} \\label{dimless}\n\\end{equation}\nwith $\\gamma_0\\!=\\!2.7$~eV being the carbon nearest neighbor\noverlap integral entering the CN surface axial conductivity\n$\\sigma_{zz}(R_{CN},\\omega)$. The function\n\\begin{equation}\n\\bar\\Gamma_0^f(x)=\\frac{4|d^f_z|^2x^3}{3\\hbar c^3}\n\\left(\\frac{2\\gamma_0}{\\hbar}\\right)^{\\!2} \\label{Gamma0f}\n\\end{equation}\nwith\n$d^f_z\\!=\\!\\sum_{\\mathbf{n}}\\langle0|(\\hat{\\mathbf{d}}_\\mathbf{n})_z|f\\rangle$\nrepresents the (dimensionless) spontaneous decay rate, and\n\\begin{equation}\n\\rho(x)=\\frac{3S_0}{16\\pi\\alpha\nR_{CN}^2}\\;\\mbox{Re}\\frac{1}{\\bar\\sigma_{zz}(x)}\\label{plDOS}\n\\end{equation}\nstands for the surface plasmon density of states (DOS) which is\nresponsible for the exciton decay rate variation due to its\ncoupling to the plasmon modes.~Here, $\\alpha\\!=\\!e^2\/\\hbar\nc\\!=\\!1\/137$ is the fine-structure constant and\n$\\bar\\sigma_{zz}\\!=\\!2\\pi\\hbar\\sigma_{zz}\/e^2$ is the\ndimensionless CN surface axial conductivity per unit length.\n\nNote that the conductivity factor in Eq.~(\\ref{plDOS}) equals\n\\begin{equation}\n\\mbox{Re}\\frac{1}{\\bar\\sigma_{zz}(x)}=-\\frac{4\\alpha\nc}{R_{CN}}\\left(\\frac{\\hbar}{2\\gamma_0x}\\right)\\mbox{Im}\\frac{1}{\\epsilon_{zz}(x)-1}\n\\label{Reoneoversigma}\n\\end{equation}\nin view of Eq.~(\\ref{dimless}) and equation\n\\begin{equation}\n\\sigma_{zz}(x)=-\\frac{i\\omega}{4\\pi\nS\\rho_{T}}\\,[\\epsilon_{zz}(x)-1] \\label{DrudeGauss}\n\\end{equation}\nrepresenting the Drude relation for CNs, where $\\epsilon_{zz}$ is\nthe longitudinal (along the CN axis) dielectric function, $S$ and\n$\\rho_{T}$ are the surface area of the tubule and the number of\ntubules per unit volume,\nrespectively~\\cite{Bondarev04,Bondarev05,Tasaki}.~This relates\nvery closely the surface plasmon DOS function~(\\ref{plDOS}) to the\nloss function $-\\mbox{Im}(1\/\\epsilon)$ measured in Electron Energy\nLoss Spectroscopy (EELS) experiments to determine the properties\nof collective electronic excitations in solids~\\cite{Pichler98}.\n\n\\begin{figure}[t]\n\\epsfxsize=13.75cm\\centering{\\epsfbox{fig3.eps}}\\caption{(a),(b)~Calculated\ndimensionless (see text) axial surface conductivities for the\n(11,0) and (10,0) CNs. The dimensionless energy is defined as\n[\\emph{Energy}]\/$2\\gamma_0$, according to\nEq.~(\\ref{dimless}).}\\label{fig3}\n\\end{figure}\n\nFigure~\\ref{fig3} shows the low-energy behaviors of\n$\\bar\\sigma_{zz}(x)$ and $\\mbox{Re}[1\/\\bar\\sigma_{zz}(x)]$ for the\n(11,0) and (10,0) CNs ($R_{CN}=0.43$~nm and $0.39$~nm,\nrespectively) we study here. We obtained them numerically as\nfollows. First, we adapt the nearest-neighbor non-orthogonal\ntight-binding approach~\\cite{Valentin} to determine the realistic\nband structure of each CN. Then, the room-temperature longitudinal\ndielectric functions $\\epsilon_{zz}$ are calculated within the\nrandom-phase approximation~\\cite{LinShung97,EhrenreichCohen59},\nwhich are then converted into the conductivities $\\bar\\sigma_{zz}$\nby means of the Drude relation. Electronic dissipation processes\nare included in our calculations within the relaxation-time\napproximation (electron scattering length of $130R_{CN}$ was\nused~\\cite{Lazzeri05}). We did not include excitonic many-electron\ncorrelations, however, as they mostly affect the real conductivity\n$\\mbox{Re}(\\bar\\sigma_{zz})$ which is responsible for the CN\noptical absorption~\\cite{Pedersen04,Spataru04,Ando}, whereas we\nare interested here in $\\mbox{Re}(1\/\\bar\\sigma_{zz})$ representing\nthe surface plasmon DOS according to Eq.~(\\ref{plDOS}). This\nfunction is only non-zero when the two conditions,\n$\\mbox{Im}[\\bar\\sigma_{zz}(x)]=0$ and\n$\\mbox{Re}[\\bar\\sigma_{zz}(x)]\\rightarrow0$, are fulfilled\nsimultaneously~\\cite{Kempa02,Kempa89,LinShung97}. These result in\nthe peak structure of the function $\\mbox{Re}(1\/\\bar\\sigma_{zz})$\nas is seen in Fig.~\\ref{fig3}. It~is~also seen from the comparison\nof Fig.~\\ref{fig3}~(b) with Fig.~\\ref{fig3}~(a) that the peaks\nbroaden as the CN diameter decreases. This is consistent with the\nstronger hybridization effects in smaller-diameter\nCNs~\\cite{Blase94}.\n\n\\begin{figure}[t]\n\\epsfxsize=13.75cm\\centering{\\epsfbox{fig4.eps}}\\caption{(a),(b)~Surface\nplasmon DOS and conductivities (left panels), and lowest bright\nexciton dispersion when coupled to plasmons (right panels) in\n(11,0) and (10,0) CNs, respectively. The dimensionless energy is\ndefined as [\\emph{Energy}]\/$2\\gamma_0$, according to\nEq.~(\\ref{dimless}). See text for the dimensionless\nquasi-momentum.}\\label{fig4}\n\\end{figure}\n\nLeft panels in Figs.~\\ref{fig4}(a) and \\ref{fig4}(b) show the\nlowest-energy plasmon DOS resonances calculated for the (11,0) and\n(10,0) CNs as given by the function $\\rho(x)$ in\nEq.~(\\ref{plDOS}). Also shown there are the corresponding\nfragments of the functions $\\mbox{Re}[\\bar\\sigma_{zz}(x)]$ and\n$\\mbox{Im}[\\bar\\sigma_{zz}(x)]$. In all graphs the lower\ndimensionless energy limits are set up to be equal to the lowest\nbright exciton excitation energy [$E^{(11)}_{exc}=1.21$~eV\n($x=0.224$) and $1.00$~eV ($x=0.185$) for the (11,0) and (10,0)\nCN, respectively, as reported in Ref.\\cite{Spataru05} by directly\nsolving the Bethe-Salpeter equation]. Peaks in $\\rho(x)$ are seen\nto coincide in energy with zeros of\n$\\mbox{Im}[\\bar\\sigma_{zz}(x)]$ \\{or zeros of\n$\\mbox{Re}[\\epsilon_{zz}(x)]$\\}, clearly indicating the plasmonic\nnature of the CN surface excitations under\nconsideration~\\cite{Kempa02,Kempa}. They describe the surface\nplasmon modes associated with the transversely quantized interband\nelectronic transitions in CNs~\\cite{Kempa02}. As is seen in\nFig.~\\ref{fig4} (and in Fig.~\\ref{fig3}), the interband plasmon\nexcitations occur in CNs slightly above the first bright exciton\nexcitation energy~\\cite{Ando}, in the frequency domain where the\nimaginary conductivity (or the real dielectric function) changes\nits sign. This is a unique feature of the complex dielectric\nresponse function, the consequence of the general\nKramers-Kr\\\"{o}nig relation~\\cite{VogelWelsch}.\n\nWe further take advantage of the sharp peak structure of $\\rho(x)$\nand solve the dispersion equation~(\\ref{dispeq}) for $x_\\mu$\nanalytically using the Lorentzian approximation\n\\begin{equation}\n\\rho(x)\\!\\approx\\!\\frac{\\rho(x_p)\\Delta x_{p}^2}\n{(x-x_{p})^2+\\Delta x_{p}^2}\\,. \\label{rhox}\n\\end{equation}\nHere, $x_{p}$ and $\\Delta x_{p}$ are, respectively, the position\nand the half-width-at-half-maximum of the plasmon resonance\nclosest to the lowest bright exciton excitation energy in the same\nnanotube (as shown in the left panels of Fig.~\\ref{fig4}). The\nintegral in Eq.~(\\ref{dispeq}) then simplifies to the form\n\\[\n\\frac{2}{\\pi}\\int_0^\\infty\\!\\!\\!\\!\\!dx\\,\\frac{x\\,\\bar\\Gamma_0^f(x)\\rho(x)}{x_\\mu^2-x^2}\n\\approx\\frac{F(x_p)\\Delta x_{p}^2}{x_\\mu^2-x_p^2}\\!\n\\int_0^\\infty\\!\\!\\!\\!\\!\\frac{dx}{(x-x_{p})^2+\\Delta x_{p}^2}\n\\]\\vspace{-0.25cm}\n\\[\n=\\frac{F(x_p)\\Delta x_{p}}{x_\\mu^2-x_p^2}\n\\left[\\arctan\\!\\left(\\frac{x_p}{\\Delta x_p}\\right)+\n\\frac{\\pi}{2}\\right]\n\\]\nwith $F(x_p)=2x_p\\bar\\Gamma_0^f(x_p)\\rho(x_p)\/\\pi$. This\nexpression is valid for all $x_\\mu$ apart from those located in\nthe narrow interval $(x_p-\\Delta x_p,x_p+\\Delta x_p)$ in the\nvicinity of the plasmon resonance, provided that the resonance is\nsharp enough. Then, the dispersion equation becomes the\nbiquadratic equation for $x_\\mu$ with the following two positive\nsolutions (the dispersion curves) of interest to us\n\n\\begin{equation}\nx_{1,2}=\\sqrt{\\frac{\\varepsilon_f^2+x_{p}^2}{2}\\pm\\frac{1}{2}\n\\sqrt{(\\varepsilon_f^2\\!-x_{p}^2)^2+F_{\\!p}\\,\\varepsilon_f}}\\,.\n\\label{dispsol}\n\\end{equation}\nHere, $F_{\\!p}=4F(x_p)\\Delta x_p(\\pi-\\Delta x_{p}\/x_{p})$ with the\n$\\arctan$-function expanded to linear terms in $\\Delta\nx_p\/x_p\\ll1$.\n\nThe dispersion curves (\\ref{dispsol}) are shown in the right\npanels in Figs.~\\ref{fig4}(a) and \\ref{fig4}(b) as functions of\nthe dimensionless longitudinal quasi-momentum.~In these\ncalculations, we estimated the interband transition matrix element\nin $\\bar\\Gamma_0^f(x_p)$ [Eq.(\\ref{Gamma0f})] from the equation\n$|d_f|^2=3\\hbar\\lambda^3\/4\\tau_{ex}^{rad}$ according to Hanamura's\ngeneral theory of the exciton radiative decay in spatially\nconfined systems~\\cite{Hanamura}, where $\\tau_{ex}^{rad}$ is the\nexciton intrinsic radiative lifetime, and $\\lambda=2\\pi c\\hbar\/E$\nwith $E$ being the exciton total energy given in our case by\nEq.~(\\ref{Ef}).~For zigzag-type CNs considered here, the first\nBrillouin zone of the longitudinal quasi-momentum is given by\n$-2\\pi\\hbar\/3b\\le\\hbar k_z\\le2\\pi\\hbar\/3b$~\\cite{Dresselhaus,Dai}.\nThe total energy of the ground-internal-state exciton can then be\nwritten as $E=E_{exc}+(2\\pi\\hbar\/3b)^2t^2\/2M_{ex}$ with $-1\\le\nt\\le1$ representing the dimensionless longitudinal quasi-momentum.\nIn our calculations we used the lowest bright exciton parameters\n$E^{(11)}_{exc}=1.21$~eV and $1.00$~eV, $\\tau_{ex}^{rad}=14.3$~ps\nand $19.1$~ps, $M_{ex}=0.44m_0$ and $0.19m_0$ ($m_0$ is the\nfree-electron mass) for the (11,0) CN and (10,0) CN, respectively,\nas reported in Ref.\\cite{Spataru05} by directly solving the\nBethe-Salpeter equation.\n\nBoth graphs in the right panels in Fig.~\\ref{fig4} are seen to\ndemonstrate a clear anticrossing behavior with the (Rabi) energy\nsplitting $\\sim\\!0.1$~eV. This indicates the formation of the\nstrongly coupled surface plasmon-exciton excitations in the\nnanotubes under consideration. It is important to realize that\nhere we deal with the strong exciton-plasmon interaction supported\nby an individual quasi-1D nanostructure --- a single-walled\n(small-diameter) semiconducting carbon nanotube, as opposed to the\nartificially fabricated metal-semiconductor nanostructures studied\npreviuosly~\\cite{Bellessa,Govorov,Fedutik} where the metallic\ncomponent normally carries the plasmon and the semiconducting one\ncarries the exciton. It is also important that the effect comes\nnot only from the height but also from the width of the plasmon\nresonance as it is seen from the definition of the $F_p$ factor in\nEq.~(\\ref{dispsol}). In other words, as long as the plasmon\nresonance is sharp enough (which is always the case for interband\nplasmons), so that the Lorentzian approximation (\\ref{rhox})\napplies, the effect is determined by the area under the plasmon\npeak in the DOS function~(\\ref{plDOS}) rather than by the peak\nheight as one would expect.\n\nHowever, the formation of the strongly coupled exci\\-ton-plasmon\nstates is only possible if the exciton total energy is in\nresonance with the energy of a surface plasmon mode. The exciton\nenergy can be tuned to the nearest plasmon resonance in ways used\nfor excitons in semiconductor quantum microcavities\n--- thermally~\\cite{Reithmaier,Yoshie,Peter} (by elevating sample temperature),\nor\/and electrostatically~\\cite{MillerPRL,Miller,Zrenner,Krenner}\n(via the quantum confined Stark effect with an external\nelectrostatic field applied perpendicular to the CN axis). As is\nseen from Eqs.~(\\ref{Ef}) and (\\ref{Eexcf}), the two possibilities\ninfluence the different degrees of freedom of the quasi-1D exciton\n--- the (longitudinal) kinetic energy and the excitation energy,\nrespectively. Below we study the (less trivial) electrostatic\nfield effect on the exciton excitation energy in CNs.\n\n\\subsection{The perpendicular electrostatic field effect}\n\nThe optical properties of semiconducting CNs in an external\nelectrostatic field directed along the nanotube axis were studied\ntheoretically in Ref.~\\cite{Perebeinos07}. Strong oscillations in\nthe band-to-band absorption and the quadratic Stark shift of the\nexciton absorption peaks with the field increase, as well as the\nstrong field dependence of the exciton ionization rate, were\npredicted for CNs of different diameters and chiralities. Here, we\nfocus on the perpendicular electrostatic field orientation.~We\nstudy how the electrostatic field applied perpendicular to the CN\naxis affects the CN band gap, the exciton binding\/excitation\nenergy, and the interband surface plasmon energy, to explore the\ntunability of the strong exciton-plasmon coupling effect predicted\nabove.~The problem is similar to the well-known quantum confined\nStark effect first observed for the excitons in semiconductor\nquantum wells~\\cite{MillerPRL,Miller}. However, the cylindrical\nsurface symmetry of the excitonic states brings new peculiarities\nto the quantum confined Stark effect in CNs. In what follows we\nwill generally be interested only in the lowest internal energy\n(ground) excitonic state, and so the internal state index $f$ in\nEqs.~(\\ref{Ef}) and (\\ref{Eexcf}) will be omitted for brevity.\n\nBecause the nanotube is modelled by a continuous, infinitely thin,\nanisotropically conducting cylinder in our macroscopic QED\napproach, the actual local symmetry of the excitonic wave function\nresulted from the graphene Brillouin zone structure is disregarded\nin our model (see, e.g., reviews~\\cite{Dresselhaus07,Ando}). The\nlocal symmetry is implicitly present in the surface axial\nconductivity though, which we calculate beforehand as described\nabove.\\footnote{In real CNs, the existence of two equivalent\nenergy valleys in the 1st Brillouin zone, the $K$- and\n$K^{\\prime}$-valleys with opposite electron helicities about the\nCN axis, results into dark and bright excitonic states in the\nlowest energy spin-singlet manifold~\\cite{AndoDarkExciton}. Since\nthe electric interaction does not involve spin variables, both\n$K$- and $K^{\\prime}$-valleys are affected equally by the\nelectrostatic field in our case, and the detailed structure of the\nexciton wave function multiplet is not important. This is opposite\nto the non-zero magnetostatic field case where the field affects\nthe $K$- and $K^{\\prime}$-valleys differently either to brighten\nthe dark excitonic states~\\cite{Srivastava}, or to create Landau\nsublevels~\\cite{Ando} for longitudinal and perpendicular\norientation, respectively.}\\label{footnote1}\n\nWe start with the Schr\\\"{o}dinger equation for the electron and\nhole on the CN surface, located at\n$\\textbf{r}_{e}=\\{R_{CN},\\varphi_{e},z_{e}\\}$ and\n$\\textbf{r}_{h}=\\{R_{CN},\\varphi_{h},z_{h}\\}$, respectively. They\ninteract with each other through the Coulomb potential\n$V(\\textbf{r}_e,\\textbf{r}_h)=-e^2\/\\epsilon|\\textbf{r}_e\\!-\\textbf{r}_h|$,\nwhere $\\epsilon=\\epsilon_{zz}(0)$.~The external electrostatic\nfield $\\textbf{F}=\\{F,0,0\\}$ is directed perpendicular to the CN\naxis (along the $x$-axis in Fig.~\\ref{fig1}). The Schr\\\"{o}dinger\nequation is of the form\n\\begin{equation}\n\\left[\\hat{H}_{e}(\\textbf{F})+\\hat{H}_{h}(\\textbf{F})+\nV(\\textbf{r}_e,\\textbf{r}_h)\\right]\\!\n\\Psi(\\textbf{r}_e,\\textbf{r}_h)=E\\Psi(\\textbf{r}_e,\\textbf{r}_h)\n\\label{Schroedinger}\n\\end{equation}\nwith\n\\begin{equation}\n\\hat{H}_{e,h}(\\textbf{F})=-\\frac{\\hbar^2}{2m_{e,h}}\\left(\\!\n\\frac{1}{R_{CN}^2}\\frac{\\partial^2}{\\partial\\varphi^2_{e,h}}+\n\\frac{\\partial^2}{\\partial z^2_{e,h}}\\right)\\!\\mp\ne\\textbf{r}_{e,h}\\!\\cdot\\textbf{F}\\label{Heh}\n\\end{equation}\n\nWe further separate out the translational and relative degrees of\nfreedom of the electron-hole pair by transforming the longitudinal\n(along the CN axis) motion of the pair into its center-of-mass\ncoordinates given by $Z=(m_ez_e+m_hz_h)\/M_{ex}$ and $z=z_e-z_h$.\nThe exciton wave function is approximated as follows\n\\begin{equation}\n\\Psi(\\textbf{r}_e,\\textbf{r}_h)=e^{ik_zZ}\\phi_{ex}(z)\\psi_e(\\varphi_e)\\psi_h(\\varphi_h).\n\\label{wfunceh}\n\\end{equation}\nThe complex exponential describes the exciton center-of-mass\nmotion with the longitudinal quasi-momentum $k_z$ along the CN\naxis. The function $\\phi_{ex}(z)$ represents the longitudinal\nrelative motion of the electron and the hole inside the exciton.\nThe functions $\\psi_e(\\varphi_e)$ and $\\psi_h(\\varphi_h)$ are the\nelectron and hole subband wave functions, respectively, which\nrepresent their confined motion along the circumference of the\ncylindrical nanotube surface.\n\nEach of the functions is assumed to be normalized to unity.\nEquations~(\\ref{Schroedinger}) and (\\ref{Heh}) are then rewritten\nin view of Eqs.~(\\ref{Ef})--(\\ref{Egkfi}) to yield\n\\begin{equation}\n\\left[-\\frac{\\hbar^2}{2m_eR_{CN}^2}\\frac{\\partial^2}{\\partial\\varphi^2_e}-\neR_{CN}F\\cos(\\varphi_e)\\right]\\!\\psi_e(\\varphi_e)=\\varepsilon_{e}\\psi_e(\\varphi_e),\n\\label{wfe}\n\\end{equation}\n\\begin{equation}\n\\left[-\\frac{\\hbar^2}{2m_hR_{CN}^2}\\frac{\\partial^2}{\\partial\\varphi^2_h}+\neR_{CN}F\\cos(\\varphi_h)\\right]\\!\\psi_h(\\varphi_h)\\!=\\!\\varepsilon_h\\psi_h(\\varphi_h)\\!,\n\\label{wfh}\n\\end{equation}\n\\begin{equation}\n\\left[-\\frac{\\hbar^2}{2\\mu}\\frac{\\partial^2}{\\partial\\,\\!z^2}+\nV_{\\mbox{\\small{eff}}}(z)\\right]\\!\\phi_{ex}(z)=E_b\\phi_{ex}(z),\n\\label{wfexc}\n\\end{equation}\nwhere $\\mu=m_em_h\/M_{ex}$ is the exciton reduced mass, and\n$V_{\\mbox{\\small eff}}$ is the effective longitudinal\nelectron-hole Coulomb interaction potential given by\n\\begin{equation}\nV_{\\mbox{\\small\neff}}(z)=\\!-\\frac{e^2}{\\epsilon}\\!\\int_0^{2\\pi}\\!\\!\\!\\!\\!\\!d\\varphi_e\\!\\!\\int_0^{2\\pi}\\!\\!\\!\\!\\!\\!\nd\\varphi_h|\\psi_e(\\varphi_e)|^2|\\psi_h(\\varphi_h)|^2V(\\varphi_e,\\varphi_h,z)\n\\label{Veff}\n\\end{equation}\nwith $V$ being the original electron-hole Coulomb potential\nwritten in the cylindrical coordinates as\n\\begin{equation} V(\\varphi_e,\\varphi_h,z)=\\!\n\\frac{1}{\\{z^2+4R_{CN}^2\\sin^2[(\\varphi_{e}\\!-\\varphi_{h})\/2]\\}^{1\/2}}.\n\\label{V}\n\\end{equation}\nThe exciton problem is now reduced to the 1D equation\n(\\ref{wfexc}), where the exciton binding energy does depend on the\nperpendicular electrostatic field through the electron and hole\nsubband functions $\\psi_{e,h}$ given by the solutions of\nEqs.~(\\ref{wfe}) and (\\ref{wfh}) and entering the effective\nelectron-hole Coulomb interaction potential (\\ref{Veff}).\n\nThe set of Eqs.~(\\ref{wfe})-(\\ref{V}) is analyzed in Appendix~D.\nOne of the main results obtained in there is that the effective\nCoulomb potential (\\ref{Veff}) can be approximated by an\nattractive cusp-type cutoff potential of the form\n\\begin{equation}\nV_{\\mbox{\\small\neff}}(z)\\approx-\\frac{e^2}{\\epsilon[|z|+z_0(j,F)]},\n\\label{Vcutoff}\n\\end{equation}\nwhere the cutoff parameter $z_0$ depends on the perpendicular\nelectrostatic field strength and on the electron-hole azimuthal\ntransverse quantization index $j=1,2,...$ (excitons are created in\ninterband transitions involving valence and conduction subbands\nwith the same quantization index~\\cite{Ando} as shown in\nFig.~\\ref{fig2}). Specifically,\n\\begin{equation}\nz_0(j,F)\\approx2R_{CN}\\frac{\\pi-2\\ln2\\,[1-\\Delta_j(F)]}{\\pi+2\\ln2\\,[1-\\Delta_j(F)]}\n\\label{z0}\n\\end{equation}\nwith $\\Delta_j(F)$ given to the second order approximation in the\nelectric field by\n\\begin{eqnarray}\n\\Delta_j(F)&\\!\\!\\approx\\!\\!&2\\mu M_{ex}\\frac{e^2R_{CN}^6w^2_j}{\\hbar^4}\\,F^2,\\label{DeltaF}\\\\\nw_j&\\!\\!=\\!\\!&\\frac{\\theta(j\\!-\\!2)}{1-2j}+\\frac{1}{1+2j},\\nonumber\n\\end{eqnarray}\nwhere $\\theta(x)$ is the unit step function. Approximation\n(\\ref{Vcutoff}) is formally valid when $z_0(j,F)$ is much less\nthan the exciton Bohr radius $a_B^\\ast$ $(=\\epsilon\\hbar^2\/\\mu\ne^2)$ which is estimated to be $\\sim\\!10R_{CN}$ for the first\n($j\\!=\\!1$ in our notations here) exciton in\nCNs~\\cite{Pedersen03}. As is seen from Eqs.~(\\ref{z0}) and\n(\\ref{DeltaF}), this is always the case for the first exciton for\nthose fields where the perturbation theory applies, i.~e. when\n$\\Delta_1(F)<1$ in Eq.~(\\ref{DeltaF}).\n\nEquation~(\\ref{wfexc}) with the potential (\\ref{Vcutoff}) formally\ncoincides with the one studied by Ogawa and Takagahara in their\ntreatments of excitonic effects in 1D semiconductors with no\nexternal electrostatic field applied~\\cite{Takagahara}.~The only\ndifference in our case is that our cutoff parameter (\\ref{z0}) is\nfield dependent. We therefore follow Ref.~\\cite{Takagahara} and\nfind the ground-state binding energy $E^{(11)}_b$ for the first\nexciton we are interested in here from the transcendental equation\n\\begin{equation}\n\\ln\\!\\left[\\frac{2z_0(1,F)}{\\hbar}\\sqrt{2\\mu|E^{(11)}_b|}\\,\\right]\n+\\frac{1}{2}\\sqrt{\\frac{|E^{(11)}_b|}{Ry^\\ast}}=0. \\label{Ebnum}\n\\end{equation}\nIn doing so, we first find the exciton Rydberg energy,~$Ry^\\ast$\n$(=\\!\\mu e^4\/2\\hbar^2\\epsilon^2)$, from this equation at\n$F\\!=\\!0$. We use the diameter- and chirality-dependent electron\nand hole effective masses from Ref.~\\cite{Jorio}, and the first\nbright exciton binding energy of 0.76~eV for both (11,0) and\n(10,0) CN as reported in Ref.~\\cite{Capaz} from \\emph{ab initio}\ncalculations. We obtain $Ry^\\ast=4.02$~eV and $0.57$~eV for\nthe~(11,0) tube and (10,0) tube, respectively. The difference of\nabout one order of magnitude reflects the fact that these are the\nsemiconducting CNs of different types --- type-I and type-II,\nrespectively, based on $(2n+m)$ families~\\cite{Jorio}. The\nparameters $Ry^\\ast$ thus obtained are then used to find\n$|E^{(11)}_b|$ as functions of $F$ by numerically solving\nEq.~(\\ref{Ebnum}) with $z_0(1,F)$ given by Eqs.~(\\ref{z0}) and\n(\\ref{DeltaF}).\n\n\\begin{figure}[t]\n\\epsfxsize=13.75cm\\centering{\\epsfbox{fig5.eps}}\\caption{(a)~Calculated\nbinding energies of the first bright exciton in the (11,0) and\n(10,0) CNs as functions of the perpendicular electrostatic field\napplied.~Solid lines are the numerical solutions to\nEq.~(\\ref{Ebnum}), dashed lines are the quadratic approximations\nas given by Eq.~(\\ref{Ebapprox}). (b)~Field dependence of the\neffective cutoff Coulomb potential (\\ref{Vcutoff}) in the (11,0)\nCN. The dimensionless energy is defined as\n[\\emph{Energy}]\/$2\\gamma_0$, according to Eq.~(\\ref{dimless}).}\n\\label{fig5}\n\\end{figure}\n\nThe calculated (negative) binding energies are shown in\nFig.~\\ref{fig5}(a) by the solid lines. Also shown there by dashed\nlines are the functions\n\\begin{equation}\nE^{(11)}_b(F)\\approx E^{(11)}_b[1-\\Delta_1(F)] \\label{Ebapprox}\n\\end{equation}\nwith $\\Delta_1(F)$ given by Eq.~(\\ref{DeltaF}). They are seen to\nbe fairly good analytical (quadratic in field) approximations to\nthe numerical solutions of Eq.~(\\ref{Ebnum}) in the range of not\ntoo large fields. The exciton binding energy decreases very\nrapidly in its absolute value as the field increases. Fields of\nonly $\\sim\\!0.1-0.2$~V\/$\\mu$m are required to decrease\n$|E^{(11)}_b|$ by a factor of $\\sim\\!2$ for the CNs considered\nhere. The reason is the perpendicular field shifts up the \"bottom\"\nof the effective potential (\\ref{Vcutoff}) as shown in\nFig.~\\ref{fig5}(b) for the (11,0) CN. This makes the potential\nshallower and pushes bound excitonic levels up, thereby decreasing\nthe exciton binding energy in its absolute value. As this takes\nplace, the shape of the potential does not change, and the\nlongitudinal relative electron-hole motion remains finite at all\ntimes.~As a consequence, no tunnel exciton ionization occurs in\nthe perpendicular field, as opposed to the longitudinal\nelectrostatic field (Franz-Keldysh) effect studied in\nRef.~\\cite{Perebeinos07} where the non-zero field creates the\npotential barrier separating out the regions of finite and\ninfinite relative motion and the exciton becomes ionized as the\nelectron tunnels to infinity.\n\nThe binding energy is only the part of the exciton excitation\nenergy (\\ref{Eexcf}). Another part comes from the band gap energy\n(\\ref{Egkfi}), where $\\varepsilon_{e}$ and $\\varepsilon_{h}$ are\ngiven by the solutions of Eqs.~(\\ref{wfe}) and (\\ref{wfh}),\nrespectively. Solving them to the leading (second) order\nperturbation theory approximation in the field (Appendix~D), one\nobtains\n\\begin{equation}\nE^{(jj)}_g(F)\\approx\nE^{(jj)}_g\\!\\left[1-\\frac{m_e\\Delta_j(F)}{2M_{ex}j^2w_j}-\\frac{m_h\\Delta_j(F)}{2M_{ex}j^2w_j}\\right]\\!,\n\\label{EgF}\n\\end{equation}\nwhere the electron and hole subband shifts are written separately.\nThis, in view of Eq.~(\\ref{DeltaF}), yields the first band gap\nfield dependence in the form\n\\begin{equation}\nE^{(11)}_g(F)\\approx\nE^{(11)}_g\\!\\left[1-\\frac{3}{2}\\,\\Delta_1(F)\\right]\\!,\n\\label{EgF11}\n\\end{equation}\nThe bang gap decrease with the field in Eq.~(\\ref{EgF11}) is\nstronger than the opposite effect in the negative exciton binding\nenergy given (to the same order approximation in field) by\nEq.~(\\ref{Ebapprox}). Thus, the first exciton excitation energy\n(\\ref{Eexcf}) will be gradually decreasing as the perpendicular\nfield increases, shifting the exciton absorption peak to the red.\nThis is the basic feature of the quantum confined Stark effect\nobserved previously in semiconductor\nnanomaterials~\\cite{MillerPRL,Miller,Zrenner,Krenner}. The field\ndependences of the higher interband transitions exciton excitation\nenergies are suppressed by the rapidly (quadratically) increasing\nazimuthal quantization numbers in the denominators of\nEqs.~(\\ref{DeltaF}) and (\\ref{EgF}).\n\nLastly, the perpendicular field dependence of the interband\nplasmon resonances can be obtained from the frequency dependence\nof the axial surface conductivity due to excitons (see\nRef.~\\cite{Ando} and refs. therein). One has\n\\begin{equation}\n\\sigma^{ex}_{zz}(\\omega)\\sim\\!\\!\\!\\sum_{j=1,2,...}\\!\\frac{-i\\hbar\\omega\nf_j}{[E^{(jj)}_{exc}]^2\\!-(\\hbar\\omega)^2\\!-2i\\hbar^2\\omega\/\\tau},\n\\label{Ando}\n\\end{equation}\nwhere $f_j$ and $\\tau$ are the exciton oscillator strength and\nrelaxation time, respectively. The plasmon frequencies are those\nat which the function $\\mbox{Re}[1\/\\sigma^{ex}_{zz}(\\omega)]$ has\nmaxima. Testing it for maximum in the domain\n$E^{(11)}_{exc}\\!\\!<\\!\\hbar\\omega<\\!E^{(22)}_{exc}$, one finds the\nfirst interband plasmon resonance energy to be (in the limit\n$\\tau\\!\\rightarrow\\!\\infty$)\n\\begin{equation}\nE^{(11)}_p=\\sqrt{\\frac{[E^{(11)}_{exc}]^2+[E^{(22)}_{exc}]^2}{2}}\\,.\n\\label{Epl}\n\\end{equation}\nUsing the field dependent $E^{(11)}_{exc}$ given by\nEqs.~(\\ref{Eexcf}),~(\\ref{Ebapprox}) and (\\ref{EgF11}), and\nneglecting the field dependence of $E^{(22)}_{exc}$, one obtains\nto the second order approximation in the field\n\\begin{equation}\nE^{(11)}_p(F)\\approx\nE^{(11)}_p\\!\\left[1-\\frac{1+\\!E^{(11)}_{g}\\!\/2E^{(11)}_{exc}}\n{1+\\!E^{(22)}_{exc}\\!\/E^{(11)}_{exc}}\\;\\Delta_1(F)\\right]\\!.\n\\label{EplF}\n\\end{equation}\n\n\\begin{figure}[t]\n\\epsfxsize=13.75cm\\centering{\\epsfbox{fig6.eps}}\\vskip-0.3cm\\caption{(a),(b)~Calculated\ndependences of the first bright exciton parameters in the (11,0)\nand (10,0) CNs, respectively, on the electrostatic field applied\nperpendicular to the nanotube axis. The dimensionless energy is\ndefined as [\\emph{Energy}]\/$2\\gamma_0$, according to\nEq.~(\\ref{dimless}). The energy is measured from the top of the\nfirst unperturbed hole subband.} \\label{fig6}\n\\end{figure}\n\nFigure~\\ref{fig6} shows the results of our calculations of the\nfield dependences for the first bright exciton parameters in the\n(11,0) and (10,0) CNs. The energy is measured from the top of the\nfirst unperturbed hole subband (as shown in Fig.~\\ref{fig2}, right\npanel). The binding energy field dependence was calculated\nnumerically from Eq.~(\\ref{Ebnum}) as described above [shown in\nFig.~\\ref{fig5}~(a)]. The band gap field dependence and the\nplasmon energy field dependence were calculated from\nEqs.~(\\ref{EgF}) and (\\ref{EplF}), respectively. The zero-field\nexcitation energies and zero-field binding energies were taken to\nbe those reported in Ref.~\\cite{Spataru05} and in\nRef.~\\cite{Capaz}, respectively, and we used the diameter- and\nchirality-dependent electron and hole effective masses from\nRef.~\\cite{Jorio}. As is seen in Fig.~\\ref{fig6}~(a)~and (b), the\nexciton excitation energy and the interband plasmon energy\nexperience red shift in both nanotubes as the field increases.\nHowever, the excitation energy red shift is very small (barely\nseen in the figures) due to the negative field dependent\ncontribution from the exciton binding energy.~So,\n$E^{(11)}_{exc}(F)$ and $E^{(11)}_{p}(F)$ approach each other as\nthe field increases, thereby bringing the total exciton energy\n(\\ref{Ef}) in resonance with the surface plasmon mode due to the\nnon-zero longitudinal kinetic energy term at finite\ntemperature.\\footnote{We are based on the zero-exciton-temperature\napproximation in here~\\cite{Suna}, which is well justified because\nof the exciton excitation energies much larger than $k_BT$ in~CNs.\nThe exciton Hamiltonian (\\ref{Hex}) does not require the thermal\naveraging over the exciton degrees of freedom then, yielding the\ntemperature independent total exciton energy (\\ref{Ef}). One has\nto keep in mind, however, that the exciton excitation energy can\nbe affected by the enviromental effect not under consideration in\nhere (see Ref.~\\cite{Miyauchi}).} Thus, the electrostatic field\napplied perpendicular to the CN axis (the quantum confined Stark\neffect) may be used to tune the exciton energy to the nearest\ninterband plasmon resonance, to put the exciton-surface plasmon\ninteraction in small-diameter semiconducting CNs to the\nstrong-coupling regime.\n\n\\subsection{The optical absorption}\n\nHere, we analyze the longitudinal exciton absorption line shape as\nits energy is tuned to the nearest interband surface plasmon\nresonance. Only longitudinal excitons (excited by light polarized\nalong the CN axis) couple to the surface plasmon modes as\ndiscussed at the very beginning of this section (see\nRef.~\\cite{UryuAndo} for the perpendicular light exciton\nabsorption in CNs). We start with the linear (weak) excitation\nregime where only single-exciton states are excited, and follow\nthe optical absorption\/emission lineshape theory developed\nrecently for atomically doped CNs~\\cite{Bondarev06}. (Obviously,\nthe absorption line shape coincides with the emission line shape\nif the monochromatic incident light beam is used in the absorption\nexperiment.) Then, the non-linear (strong) excitation regime is\nconsidered with the photonduced excitation of biexciton states.\n\nWhen the $f$-internal state exciton is excited and the nanotube's\nsurface EM field subsystem is in vacuum state, the time-dependent\nwave function of the whole system \"exciton+field\" is of the\nform\\footnote{See the footnote on page~\\pageref{footnote1} above.}\n\\begin{eqnarray}\n|\\psi(t)\\rangle&=&\\sum_{{\\bf\nk},f}C_{f}(\\textbf{k},t)\\,e^{-i\\tilde{E}_{f}({\\bf k})t\/\\hbar}\n|\\{1_f(\\textbf{k})\\}\\rangle_{ex}|\\{0\\}\\rangle\\label{wfunc}\\\\\n&+&\\sum_{\\bf k}\\int_{0}^{\\infty}\\!\\!\\!\\!\\!\\!d\\omega\\,\nC(\\textbf{k},\\omega,t)\\,e^{-i\\omega t}\n|\\{0\\}\\rangle_{ex}|\\{1(\\mathbf{k},\\omega)\\}\\rangle.\\nonumber\n\\end{eqnarray}\nHere, $|\\{1_f(\\textbf{k})\\}\\rangle_{ex}$ is the excited\nsingle-quantum Fock state with one exciton and\n$|\\{1(\\textbf{k},\\omega)\\}\\rangle$ is that with one surface\nphoton. The vacuum states are $|\\{0\\}\\rangle_{ex}$ and\n$|\\{0\\}\\rangle$ for the exciton subsystem and field subsystem,\nrespectively. The coefficients $C_{f}(\\textbf{k},t)$ and\n$C(\\textbf{k},\\omega,t)$ stand for the population probability\namplitudes of the respective states of the whole system. The\nexciton energy is of the form\n$\\tilde{E}_{f}(\\textbf{k})\\!=\\!E_f(\\textbf{k})\\!-i\\hbar\/\\tau$ with\n$E_f(\\textbf{k})$ given by Eq.~(\\ref{Ef}) and $\\tau$ being the\nphenomenological exciton relaxation time constant [assumed to be\nsuch that $\\hbar\/\\tau\\!\\ll\\!E_{f}(\\textbf{k})$] to account for\nother possible exciton relaxation processes. From the literature\nwe have $\\tau_{ph}\\sim30\\!-\\!100$~fs for the exciton-phonon\nscattering~\\cite{Perebeinos07}, $\\tau_d\\sim50$~ps for the exciton\nscattering by defects~\\cite{Hagen05,Prezhdo08}, and\n$\\tau_{rad}\\sim10\\,\\mbox{ps}-10\\,\\mbox{ns}$ for the radiative\ndecay of excitons~\\cite{Spataru05}. Thus, the scattering by\nphonons is the most likely exciton relaxation mechanism.\n\nUsing Eqs.(\\ref{Bkf}) and (\\ref{orthog}), we transform the total\nHamiltonian~(\\ref{Htot})--(\\ref{Hint}) to the\n$\\mathbf{k}$-representation (see Appendix~A), and apply it to the\nwave function in Eq.~(\\ref{wfunc}). We obtain the following set of\nthe two simultaneous differential equations for the coefficients\n$C_{f}(\\textbf{k},t)$ and $C(\\textbf{k},\\omega,t)$ from the time\ndependent Schr\\\"{o}dinger equation\n\\begin{equation}\n\\mbox{\\it\\.C}_{f}(\\textbf{k},t)\\,e^{-i\\tilde{E}_{f}({\\bf\\!\\,k})t\/\\hbar}=\n-\\frac{i}{\\hbar}\\sum_{{\\bf\\!\\,k^\\prime}}\\int_{0}^{\\infty}\\!\\!\\!\\!\\!d\\omega\\,\n\\mbox{g}^{(+)}_f(\\mathbf{k},\\mathbf{k}^\\prime\\!,\\omega)\\,\nC(\\textbf{k}^\\prime\\!,\\omega,t)\\,e^{-i\\omega\\!\\,t}, \\label{paexc}\n\\end{equation}\n\\begin{equation}\n\\mbox{\\it\\.C\\,}(\\textbf{k}^\\prime\\!,\\omega,t)\\,e^{-i\\omega\\!\\,t}\\delta_{{\\bf\\!\\,k}{\\bf\\!\\,k}^\\prime}=\n-\\frac{i}{\\hbar}\\sum_f\\,[\\mbox{g}^{(+)}_f(\\mathbf{k},\\mathbf{k}^\\prime\\!,\\omega)]^\\ast\nC_f(\\textbf{k},t)\\,e^{-i\\tilde{E}_{f}({\\bf\\!\\,k})t\/\\hbar}.\\nonumber\n\\label{papho}\n\\end{equation}\nThe $\\delta$-symbol on the left in Eq.~(\\ref{papho}) ensures that\nthe momentum conservation is fulfilled in the exciton-photon\ntransitions, so that the annihilating exciton creates the surface\nphoton with the same momentum and vice versa. In terms of the\nprobability amplitudes above, the exciton emission intensity\ndistribution is given by the final state probability at very long\ntimes corresponding to the complete decay of all initially excited\nexcitons,\n\\begin{eqnarray}\nI(\\omega)\\!&=&\\!|C(\\mathbf{k},\\omega,t\\!\\rightarrow\\!\\infty)|^{2}=\n\\frac{1}{\\hbar^2}\\sum_f|\\mbox{g}^{(+)}_f(\\mathbf{k},\\mathbf{k},\\omega)|^2\\nonumber\\\\\n&\\times&\\!\\left|\\int_0^\\infty\\!\\!\\!\\!\\!dt^\\prime\\,C_f(\\textbf{k},t^\\prime)\\,\ne^{-i[\\tilde{E}_{f}({\\bf\\!\\,k})-\\hbar\\omega]t^\\prime\\!\/\\hbar}\\right|^2\\!.\\label{Iomega}\n\\end{eqnarray}\nHere, the second equation is obtained by the formal integration of\nEq.~(\\ref{papho}) over time under the initial condition\n$C\\,(\\textbf{k},\\omega,0)\\!=\\!0$. The emission intensity\ndistribution is thus related to the exciton population probability\namplitude $C_f(\\mathbf{k},t)$ to be found from Eq.~(\\ref{paexc}).\n\nThe set of simultaneous equations (\\ref{paexc}) and (\\ref{papho})\n[and Eq.~(\\ref{Iomega}), respectively] contains no approximations\nexcept the (commonly used) neglect of many-particle excitations in\nthe wave function (\\ref{wfunc}).~We now apply these equations to\nthe exciton-surface-plasmon system in small-diameter\nsemiconducting CNs. The interaction matrix element~in\nEqs.~(\\ref{paexc}) and (\\ref{papho}) is then given by the\n$\\mathbf{k}$-transform of Eq.~(\\ref{gfpar}), and has the following\nproperty (Appendix~C)\n\\begin{equation}\n\\frac{1}{2\\gamma_0\\hbar}\\,|\\mbox{g}^{(+)}_f(\\mathbf{k},\\mathbf{k},\\omega)|^2\n=\\frac{1}{2\\pi}\\,\\bar\\Gamma_0^f(x)\\rho(x) \\label{gpkk}\n\\end{equation}\nwith $\\bar\\Gamma_0^f(x)$ and $\\rho(x)$ given by\nEqs.~(\\ref{Gamma0f}) and (\\ref{plDOS}), respectively. We further\nsubstitute the result of the formal integration of\nEq.~(\\ref{papho}) [with $C\\,(\\textbf{k},\\omega,0)\\!=\\!0$] into\nEq.~(\\ref{paexc}), use Eq.~(\\ref{gpkk}) with $\\rho(x)$\napproximated by the Lorentzian~(\\ref{rhox}), calculate the\nintegral over frequency analytically, and differentiate the result\nover time to obtain the following second order ordinary\ndifferential equation for the exciton probability amplitude\n[dimensionless variables, Eq.~(\\ref{dimless})]\n\\[\n\\mbox{\\it\\\"{C}}_f(\\beta)+[\\Delta\nx_p-\\Delta\\varepsilon_f+i(x_p-\\varepsilon_f)]\\mbox{\\it\\.{C}}_f(\\beta)+(X_f\/2)^{2}C_f(\\beta)\\!=\\!0,\n\\]\nwhere $X_f\\!=\\![2\\Delta x_{p}\\bar\\Gamma_f(x_p)]^{1\/2}$ with\n$\\bar\\Gamma_f(x_{p})\\!=\\!\\bar\\Gamma_0^f(x_{p})\\rho(x_{p})$,\n$\\Delta\\varepsilon_f=\\hbar\/2\\gamma_0\\tau$,\n$\\beta=2\\gamma_0t\/\\hbar$ is the dimensionless time,~and the\n\\textbf{k}-dependence is omitted for brevity. When the total\nexciton energy is close to a plasmon resonance,\n$\\varepsilon_f\\!\\approx\\!x_{p}$, the solution of this equation is\neasily found to be\n\\begin{eqnarray}\nC_f(\\beta)\\!\\!\\!&\\approx&\\!\\!\\!\\frac{1}{2}\\left(\\!1+\\frac{\\delta\nx}{\\sqrt{\\delta x^2-X_f^2}}\\right)e^{-\\left(\\delta\nx-\\sqrt{\\delta x^2-X_f^2}\\right)\\beta\/2}\\hskip0.5cm\\label{Cfapp}\\\\\n&+&\\!\\!\\!\\frac{1}{2}\\left(\\!1-\\frac{\\delta x}{\\sqrt{\\delta\nx^2-X_f^2}}\\right)e^{-\\left(\\delta x+\\sqrt{\\delta\nx^2-X_f^2}\\right)\\beta\/2},\\nonumber\n\\end{eqnarray}\nwhere $\\delta x=\\Delta x_p-\\Delta\\varepsilon_f>0$ and\n$X_f\\!=\\![2\\Delta x_{p}\\bar\\Gamma_f(\\varepsilon_f)]^{1\/2}$. This\nsolution is valid when $\\varepsilon_f\\!\\approx\\!x_p$ regardless\nof~the strength of the exciton-surface-plasmon coupling.~It yields\nthe exponential decay of the excitons into plasmons,\n$|C_f(\\beta)|^2\\!\\approx\\!\\exp[-\\bar\\Gamma_f(\\varepsilon_f)\\beta]$,\nin the weak coupling regime where the coupling parameter\n$(X_f\/\\delta x)^{2}\\!\\ll\\!1$. If, on the other hand, $(X_f\/\\delta\nx)^{2}\\!\\gg\\!1$,~then the strong coupling regime occurs, and the\ndecay of the excitons into plasmons proceeds via damped Rabi\noscillations, $|C_f(\\beta)|^{2}\\!\\approx\\!\\exp(-\\delta\nx\\beta)\\cos^{2}(X_f\\beta\/2)$.~This is very similar to what was\nearlier reported for an excited two-level atom near the nanotube\nsurface~\\cite{Bondarev02,Bondarev04,Bondarev04pla,Bondarev06trends}.~Note,\nhowever, that here we have the exciton-phonon scattering as well,\nwhich facilitates the strong exciton-plasmon coupling by\ndecreasing $\\delta x$ in the coupling parameter.~In other words,\nthe phonon scattering broadens the (longitudinal) exciton momentum\ndistribution~\\cite{Bondarev05Ps}, thus effectively increasing the\nfraction of the excitons with $\\varepsilon_f\\!\\approx\\!x_p$.\n\nIn view of Eqs.~(\\ref{gpkk}) and (\\ref{Cfapp}), the exciton\nemission intensity (\\ref{Iomega}) in the vicinity of the plasmon\nresonance takes the following (dimensionless) form\n\n\\begin{equation}\n\\bar{I}(x)\\approx\\bar{I}_{0}(\\varepsilon_f)\\sum_f\\left|\\int_{0}^{\\infty}\\!\\!\\!\\!\\!\\!d\\beta\\,C_{f}(\\beta)\\,\ne^{i(x-\\varepsilon_f+i\\Delta\\varepsilon_f)\\beta}\\right|^{2},\n\\label{Ix}\n\\end{equation}\nwhere $\\bar I(x)\\!=\\!2\\gamma_0I(\\omega)\/\\hbar$ and $\\bar\nI_{0}\\!=\\!\\bar\\Gamma_f(\\varepsilon_f)\/2\\pi$. After some algebra,\nthis results in\n\\begin{equation}\n\\bar{I}(x)\\approx\\frac{\\bar{I}_{0}(\\varepsilon_f)\\,[(x-\\varepsilon_f)^{2}+\\Delta\nx_p^2]}{[(x-\\varepsilon_f)^{2}-X_f^{2}\/4]^{2}+(x-\\varepsilon_f)^{2}(\\Delta\nx_p^2+\\Delta\\varepsilon_f^2)}\\,, \\label{Ixfin}\n\\end{equation}\nwhere $\\Delta x_p^2>\\Delta\\varepsilon_f^2$. The summation sign\nover the exciton internal states is omitted since only one\ninternal state contributes to the emission intensity in the\nvicinity of the sharp plasmon resonance.\n\nThe line shape in Eq.~(\\ref{Ixfin}) is mainly determined by the\ncoupling parameter $(X_f\/\\Delta x_p)^2$. It is clearly seen to be\nof a symmetric two-peak structure in the strong coupling regime\nwhere $(X_f\/\\Delta x_p)^{2}\\gg1$. Testing it for extremum, we\nobtain the peak frequencies to be\n\\[\nx_{1,2}=\\varepsilon_f\\pm\\frac{X_f}{2}\\sqrt{\\sqrt{1+\n8\\left(\\!\\frac{\\Delta\nx_p}{X_f}\\right)^{2}}\\!\\!-4\\left(\\!\\frac{\\Delta\nx_p}{X_f}\\right)^{2}}\n\\]\n[terms $\\sim\\!(\\Delta x_p)^2(\\Delta\\varepsilon_f)^2\/X_f^4$ are\nneglected], with the Rabi splitting $x_{1}-x_{2}\\!\\approx\\!X_f$.\nIn the weak coupling regime~where $(X_f\/\\Delta x_p)^{2}\\ll1$, the\nfrequencies $x_1$ and $x_2$ become complex, indicating that there\nare no longer peaks at these frequencies. As this takes place,\nEq.~(\\ref{Ixfin}) is approximated with the weak coupling\ncondition, the fact that $x\\!\\sim\\!\\varepsilon_f$, and\n$X_f^2=2\\Delta x_p\\bar\\Gamma_f(\\varepsilon_f)$, to yield the\nLorentzian\n\\[\n\\tilde{I}(x)\\approx\\frac{\\bar{I}_{0}(\\varepsilon_f)\/[1+(\\Delta\\varepsilon_f\/\\Delta\nx_p)^2]}{(x-\\varepsilon_f)^{2}+\\left[\\bar\\Gamma_f(\\varepsilon_f)\/2\\sqrt{1+(\\Delta\\varepsilon_f^{\\!}\/\\Delta\nx_p)^2}\\;\\right]^2}\n\\]\npeaked at $x=\\varepsilon_f$, whose half-width-at-half-maximum~is\nslightly narrower, however, than $\\bar\\Gamma_f(\\varepsilon_f)\/2$\nit should be if the exciton-plasmon relaxation were the only\nrelaxation mechanism in the system.~The reason is the competing\nphonon scattering takes excitons out of resonance with plasmons,\nthus decreasing the exciton-plasmon relaxation rate. We therefore\nconclude that~the phonon scattering does not affect the exciton\nemission\/absorption line shape when the exciton-plasmon coupling\nis strong (it facilitates the strong coupling regime to occur,\nhowever, as was noticed above), and it narrows the (Lorentzian)\nemission\/absorption line when the exciton-plasmon coupling is\nweak.\n\nThe non-linear optical susceptibility is proportional to the\nlinear optical response function under resonant pumping\nconditions~\\cite{Mukamel}. This allows us to use Eq.~(\\ref{Ixfin})\nto investigate the non-linear excitation regime with the\nphotoinduced biexciton formation as the exciton energy is tuned to\nthe nearest interband plasmon resonance. Under these conditions,\nthe third-order longitudinal CN susceptibility takes the\nform~\\cite{Pedersen05,Mukamel}\n\\begin{equation}\n\\chi^{(3)}(x)\\approx\\tilde{I}(x)\\!\\left[\\frac{1}{x-\\varepsilon_f+i(\\Gamma^f\\!\/2+\\Delta\\varepsilon_f)}-\n\\frac{1}{x-(\\varepsilon_f-|\\varepsilon^{X\\!X}_f|)+i(\\Gamma^f\\!\/2+\\Delta\\varepsilon_f)}\n\\right]\\!, \\label{chi3x}\n\\end{equation}\nwhere $\\varepsilon^{X\\!X}_f$ is the (negative) dimensionless\nbinding energy of the biexciton composed of two $f$-internal state\nexcitons, and $\\chi_0$ is the frequency-independent constant. The\nfirst and second terms in the brackets represent bleaching due to\nthe depopulation of the ground state and photoinduced absorption\ndue to exciton-to-biexciton transitions, respectively.\n\n\\begin{figure}[t]\n\\epsfxsize=8.2cm\\centering{\\epsfbox{fig7.eps}}\\caption{(a)~Schematic\n(arbitrary units) of the exchange coupling of two ground-state 1D\nexcitons to form a biexcitonic state. (b)~The coupling occurs in\nthe configuration space of the two independent longitudinal\nrelative electron-hole motion coordinates, $z_1$ and $z_2$, of\neach of the excitons, due to the tunneling of the system through\nthe potential barriers formed by the two single-exciton cusp-type\npotentials [bottom, also in (a)], between equivalent states\nrepresented by the isolated two-exciton wave functions shown on\nthe top.}\\label{fig7}\n\\end{figure}\n\nThe binding energy of the biexciton in a small-diameter\n($\\sim\\!1\\,$nm) CN can be evaluated by the method pioneered by\nLandau~\\cite{LandauQM}, Gor'kov and Pitaevski~\\cite{Pitaevski},\nHolstein and Herring~\\cite{Herring} --- from the analysis of the\nasymptotic exchange coupling by perturbation on the configuration\nspace wave function of the two ground-state one-dimensional (1D)\nexcitons. Separating out circumferential and longitudinal degrees\nof freedom of each of the excitons by means of\nEq.~(\\ref{wfunceh}), one arrives at the biexciton Hamiltonian of\nthe form [see Fig.~\\ref{fig7}~(a)]\n\\begin{equation}\n\\hat{H}(z_1,z_2,\\Delta\nZ)=-\\frac{1}{2}\\left(\\frac{\\partial^2}{\\partial\\,\\!z_1}\n+\\frac{\\partial^2}{\\partial\\,\\!z_2}\\right)\\label{biexcham}\n\\end{equation}\\vspace{-0.5cm}\n\\begin{eqnarray}\n-\\frac{1}{2}\\left[\\frac{1}{|z_{1}|+z_0}+\\frac{1}{|z_{2}+\\Delta\nZ|+z_0}+\\frac{1}{|z_{2}|+z_0}+\\frac{1}{|z_{1}-\\Delta Z|+z_0}\\right]\\nonumber\\\\\n-\\frac{1}{|(z_1+z_2)\/2+\\Delta Z|+z_0}-\\frac{1}{|(z_1+z_2)\/2-\\Delta Z|+z_0}\\hskip0.7cm\\nonumber\\\\\n+\\frac{1}{|(z_1-z_2)\/2+\\Delta Z|+z_0}+\\frac{1}{|(z_1-z_2)\/2-\\Delta\nZ|+z_0}\\,.\\hskip0.5cm\\nonumber\n\\end{eqnarray}\nHere, $z_{1,2}=z_{e1,2}-z_{h1,2}$ is the electron-hole relative\nmotion coordinates of the two 1D excitons, $z_0$ is the cut-off\nparameter of the effective longitudinal electron-hole Coulomb\npotential (\\ref{Vcutoff}), and $\\Delta Z\\!=\\!Z_2-Z_1$ is the\ncenter-of-mass-to-center-of-mass inter-exciton separation\ndistance. Equal electron and hole effective masses $m_{e,h}$ are\nassumed~\\cite{Jorio} and \"atomic units\"\\space are\nused~\\cite{LandauQM,Pitaevski,Herring}, whereby distance and\nenergy are measured in units of the exciton Bohr radius $a^\\ast_B$\nand in units of the doubled ground-state-exciton binding energy\n$2E_b\\!=\\!-2Ry^\\ast\/\\nu_0^2$, respectively. The first two lines in\nEq.~(\\ref{biexcham}) represent two isolated non-interacting 1D\nexcitons [see Fig.~\\ref{fig7}~(a)]. The last two lines are their\nexchange Coulomb interactions --- electron-hole and\nelectron-electron + hole-hole, respectively.\n\nThe Hamiltonian (\\ref{biexcham}) is effectively two dimensional in\nthe configuration space of the two \\emph{independent} relative\nmotion coordinates, $z_1$ and $z_2$. Figure~\\ref{fig7}~(b),\nbottom, shows schematically the potential energy surface of the\ntwo closely spaced non-interacting 1D excitons [line two in\nEq.~(\\ref{biexcham})] in the $(z_1,z_2)$ space. The surface has\nfour symmetrical minima [representing \\emph{equivalent} isolated\ntwo-exciton states shown in Fig.~\\ref{fig7}~(b), top], separated\nby the potential barriers responsible for the tunnel exchange\ncoupling between the two-exciton states in the configuration\nspace. The coordinate transformation $x=(z_1-z_2-\\Delta\nZ)\/\\sqrt{2}$, $y=(z_1+z_2)\/\\sqrt{2}$ places the origin of the new\ncoordinate system into the intersection of the two tunnel channels\nbetween the respective potential minima [Fig.~\\ref{fig7}~(b)],\nwhereby the exchange splitting formula of\nRefs.~\\cite{LandauQM,Pitaevski,Herring} takes the form\n\\begin{equation}\nU_{g,u}(\\Delta Z)-2E_b=\\mp J(\\Delta Z), \\label{Ugu}\n\\end{equation}\nwhere $U_{g,u}$ are the ground and excited state energies,\nrespectively, of the two \\emph{coupled} excitons (the biexciton)\nas functions of their center-of-mass-to-center-of-mass separation,\nand\n\\begin{equation}\nJ(\\Delta Z)=\\frac{2}{3!}\\!\\int_{\\!-\\Delta Z\/\\!\\sqrt{2}}^{\\Delta\nZ\/\\!\\sqrt{2}}\\!dy\\!\\left[\\psi(x,y)\\frac{\\partial\\psi(x,y)}{\\partial\nx}\\right]_{\\!x=0} \\label{J}\n\\end{equation}\nis the tunnel exchange coupling integral, where $\\psi(x,y)$ is the\nsolution to the Schr\\\"{o}\\-dinger equation with the Hamiltonian\n(\\ref{biexcham}) transformed to the $(x,y)$ coordinates. The\nfactor $2\/3!$ comes from the fact that there are two equivalent\ntunnel channels in the problem, mixing three equivalent\nindistinguishable two-exciton states in the configuration space\n[one state is given by the two minima on the $y$-axis, and two\nmore are represented by each of the minima on the $x$-axis ---\ncompare Figs.~\\ref{fig7}~(a) and (b)].\n\nThe function $\\psi(x,y)$ in Eq.~(\\ref{J}) is sought in the form\n\\begin{equation}\n\\psi(x,y)=\\psi_0(x,y)\\exp[-S(x,y)]\\,, \\label{psixy}\n\\end{equation}\nwhere $\\psi_0=\\nu_0^{-1}\\exp[-(|z_1(x,y,\\Delta Z)|+|z_2(x,y,\\Delta\nZ)|)\/\\nu_0]$ is the product of two single-exciton wave\nfunctions\\footnote{This is an approximate solution to the\nShr\\\"{o}dinger equation with the Hamiltonin given by the first two\nlines in Eq.~(\\ref{biexcham}), where the cut-off parameter $z_0$\nis neglected~\\cite{Takagahara}. This approximation greatly\nsimplifies problem solving here, while still remaining adequate as\nonly the long-distance tail of $\\psi_0$ is important for the\ntunnel exchange coupling.} representing the isolated two-exciton\nstate centered at the minimum $z_1\\!=\\!z_2\\!=\\!0$ (or\n$x\\!=\\!-\\Delta Z\/\\sqrt{2}$, $y\\!=\\!0$) of the configuration space\npotential [Fig.~\\ref{fig7}~(b)], and $S(x,y)$ is a slowly varying\nfunction to take into account the deviation of $\\psi$ from\n$\\psi_0$ due to the tunnel exchange coupling to another equivalent\nisolated two-exciton state centered at $z_1\\!=\\Delta Z$,\n$z_2\\!=\\!-\\Delta Z$ (or $x\\!=\\!\\Delta Z\/\\sqrt{2}$, $y\\!=\\!0$).\nSubstituting Eq.~(\\ref{psixy}) into the Schr\\\"{o}dinger equation\nwith the Hamiltonian (\\ref{biexcham}) pre-transformed to the\n$(x,y)$ coordinates, one obtains in the region of interest\n\\[\n\\frac{\\partial S}{\\partial x}=\\nu_0\\left(\\frac{1}{x+3\\Delta\nZ\/\\sqrt{2}}-\\frac{1}{x-\\Delta\nZ\/\\sqrt{2}}+\\frac{1}{y-\\sqrt{2}\\Delta Z}-\\frac{1}{y+\\sqrt{2}\\Delta\nZ}\\right),\n\\]\nup to (negligible) terms of the order of the inter-exciton van der\nWaals energy and up to second derivatives of~$S$. This equation is\nto be solved with the boundary condition $S(-\\Delta\nZ\/\\sqrt{2},y)\\!=\\!0$ originating from the natural requirement\n$\\psi(-\\Delta Z\/\\sqrt{2},y)\\!=\\!\\psi_0(-\\Delta Z\/\\sqrt{2},y)$, to\nresult in\n\\begin{equation}\nS(x,y)=\\nu_0\\!\\left(\\!\\ln\\!\\left|\\frac{x\\!+\\!3\\Delta\nZ\/\\!\\sqrt{2}}{x-\\Delta Z\/\\!\\sqrt{2}}\\right|+\\frac{2\\sqrt{2}\\Delta\nZ(x\\!+\\!\\Delta Z\/\\!\\sqrt{2})}{y^2-2\\Delta Z^2}\\!\\right)\\!.\n\\label{sxy}\n\\end{equation}\n\nAfter plugging Eqs.~(\\ref{sxy}) and (\\ref{psixy}) into\nEq.~(\\ref{J}), and retaining only the leading term of the integral\nseries expansion in powers of $\\nu_0$ subject to $\\Delta Z>1$,\nEq.~(\\ref{Ugu}) becomes\n\\begin{equation}\nU_{g,u}(\\Delta\nZ)\\approx2E_b\\left[1\\pm\\frac{2}{3\\nu_0^2}\\left(\\frac{e}{3}\\right)^{2\\nu_0}\\!\\!\\!\n\\Delta Z\\,e^{-2\\Delta Z\/\\nu_0}\\right]. \\label{Uxx}\n\\end{equation}\nThe ground state energy $U_g$ of two coupled 1D excitons is now\nseen to go through the negative minimum (biexcitonic state) as the\ninter-exciton center-of-mass-to-center-of-mass separation $\\Delta\nZ$ increases (Fig.~\\ref{fig8}). The minimum occurs at $\\Delta\nZ_0=\\nu_0\/2$, whereby the biexciton binding energy is\n$E_{X\\!X}\\!\\approx(2E_b\/9\\nu_0)(e\/3)^{2\\nu_0-1}$, or, expressing\n$\\nu_0$ in terms of $E_b$ and measuring the energy in units of\n$Ry^\\ast$,\n\\begin{equation}\nE_{X\\!X}[\\mbox{in}~Ry^\\ast]\\approx-\\frac{2}{9}\\;|E_b|^{3\/2}\\left(\\frac{e}{3}\\right)^{2\/\\!\\sqrt{|E_b|}\\,-\\,1}\\!\\!\\!.\n\\label{Exx}\n\\end{equation}\n\nThe energy $E_{X\\!X}$ can be affected by the quantum confined\nStark effect since $|E_b|$ decreases quadratically with the\nperpendicular electrostatic field applied as shown in\nFig.~\\ref{fig5}~(a). Since $e\/3\\sim\\!1$, the field dependence in\nEq.~(\\ref{Exx}) mainly comes from the pre-exponential factor. So,\n$|E_{X\\!X}|$ will be decreasing quadratically with the field as\nwell, for not too strong perpendicular fields. At the same time,\nthe equilibrium inter-exciton separation in the biexciton, $\\Delta\nZ_0=\\nu_0\/2\\sim|E_b|^{-1\/2}$, will be slowly increasing with the\nfield consistently with the lowering of $|E_{X\\!X}|$. In the zero\nfield, one has roughly\n$E_{X\\!X}\\!\\sim\\!|E_b|^{3\/2}\\!\\sim\\!R_{CN}^{-0.9}$ for the\nbiexciton binding energy versus the CN radius $R_{CN}$\n($|E_b|\\!\\sim\\!R_{CN}^{-0.6}$ as reported in\nRef.~\\cite{Pedersen03} from variational calculations), pretty\nconsistent with the $R_{CN}^{-1}$ dependence obtained\nnumerically~\\cite{Pedersen05}. Interestingly, as $R_{CN}$ goes\ndown, $|E_{X\\!X}|$ goes up faster than $|E_b|$ does. This is\npartly due to the fact that $\\Delta Z_0$ slowly decreases as\n$R_{CN}$ goes down, --- a theoretical argument in support of\nexperimental evidence for increased exciton-exciton annihilation\nin small diameter CNs~\\cite{THeinz,Valkunas,Kono}.\n\n\\begin{figure}[t]\n\\epsfxsize=11.00cm\\centering{\\epsfbox{fig8.eps}}\\caption{Calculated\nground state energy $U_g$ of the coupled pair of the first bright\nexcitons in the (11,0) CN as a function of the\ncenter-of-mass-to-center-of-mass inter-exciton distance $\\Delta Z$\nand perpendicular electrostatic field applied. Inset shows the\nbiexciton binding energy $E_{X\\!X}$ and inter-exciton separation\n$\\Delta Z_0$ ($y$- and $x$-coordinates, respectively, of the\nminima in the main figure) as functions of the field.}\\label{fig8}\n\\end{figure}\n\nFigure~\\ref{fig8} shows the ground state energy $U_g(\\Delta Z)$ of\nthe coupled pair of the first bright excitons, calculated from\nEq.~(\\ref{Uxx}) for the semiconducting (11,0) CN exposed to\ndifferent perpendicular electrostatic fields. The inset shows the\nfield dependences of $E_{X\\!X}$ [as given by Eq.~(\\ref{Exx})] and\nof $\\Delta Z_0$. All the curves are calculated using the field\ndependence of $E_b$ obtained as described in the previous\nsubsection (Figs.~\\ref{fig5} and~\\ref{fig6}). They exhibit typical\nbehaviors discussed above.\n\n\\begin{figure}[t]\n\\epsfxsize=11.00cm\\centering{\\epsfbox{fig9.eps}}\\vskip-3.5cm\\caption{[(a),\n(b), and (c)] Linear (top) and non-linear (bottom) response\nfunctions as given by Eq.~(\\ref{Ixfin}) and by the imaginary part\nof Eq.~(\\ref{chi3x}), respectively, for the first bright exciton\nin the (11,0) CN as the exciton energy is tuned to the nearest\ninterband plasmon resonance (vertical dashed line). Vertical lines\nmarked as X and XX show the exciton energy and biexciton binding\nenergy, respectively. The dimensionless energy is defined as\n[\\emph{Energy}]\/$2\\gamma_0$, according to\nEq.~(\\ref{dimless}).}\\label{fig9}\n\\end{figure}\n\nFigure~\\ref{fig9} compares the linear response lineshape\n(\\ref{Ixfin}) with the imaginary part of Eq.~(\\ref{chi3x})\nrepresenting the non-linear optical response function under\nresonant pumping, both calculated for the 1st bright exciton in\nthe (11,0) CN as its energy is tuned (by means of the quantum\nconfined Stark effect) to the nearest plasmon resonance (vertical\ndashed line in the figure). The biexciton binding energy in\nEq.~(\\ref{chi3x}) was taken to be $E_{X\\!X}\\approx52$~meV as\ngiven~by Eq.~(\\ref{Exx}) in the zero field. [Weak field dependence\nof $E_{X\\!X}$ (inset in Fig.~\\ref{fig8}) plays no essential role\nhere as $|E_{X\\!X}|\\ll|E_b|\\approx0.76$~eV regardless of the field\nstrength.] The phonon relaxation time $\\tau_{ph}\\!=\\!30$~fs was\nused as reported in Ref.~\\cite{Perebeinos05}, since this is the\nshortest one out of possible exciton relaxation processes,\nincluding exciton-exciton annihilation\n($\\tau_{ee}\\!\\sim\\!1$~ps~\\cite{THeinz}). Clear line (Rabi)\nsplitting effect $\\sim\\!0.1$~eV is seen both in the linear and in\nnon-linear excitation regime, indicating the strong\nexciton-plasmon coupling both in the single-exciton states and in\nthe biexciton states as the exciton energy is tuned to the\ninterband surface plasmon resonance. The splitting is not masked\nby the exciton-phonon scattering.\n\nThis effect can be used for the development of new tunable\noptoelectronic device applications of optically excited\nsmall-diameter semiconducting CNs in areas such as nanophotonics,\nnanoplasmonics, and cavity quantum electrodynamics, including the\nstrong excitation regime with optical non-linearities. In the\nlatter case, the experimental observation of the non-linear\nabsorption line splitting predicted here would help identify the\npresence and study the properties of biexcitonic states (including\nbiexcitons formed by excitons of different subbands~\\cite{Papan})\nin individual single-walled CNs, due to the fact that when tuned\nclose to a plasmon resonance the exciton relaxes into plasmons at\na rate much greater than\n$\\tau_{ph}^{-1}\\;(\\,\\gg\\!\\tau_{ee}^{-1})$, totally ruling out the\nrole of the competing exciton-exciton annihilation process.\n\n\\section{Casimir Interaction in Double-Wall\\\\ Carbon Nanotubes}\\label{sec4}\n\nHere, we consider the Casimir interaction between two concentric\ncylindrical gra\\-phene sheets comprising a double-wall CN, using\nthe macroscopic QED approach employed above to study the\nexciton-surface-plasmon interactions in single wall\nnanotubes.\\footnote{In this Section only, the International System\nof units is used to make the comparison easier of our theory with\nother authors' results.} The method is fully adequate in this case\nas the Casimir force is known to originate from quantum EM field\nfluctuations. The fundamental nature of this force has been\nstudied for many years since the prediction of the attraction\nforce between two neutral metallic plates in vacuum (see,\nRefs.~\\cite{BuhmannWelsch,Klimchitskaya2009}). After the first\nreport of observation of this spectacular\neffect~\\cite{Sparnaay1958}, new measurements with improved\naccuracy have been done involving different\ngeometries~\\cite{Lamoreaux1997,Mohideen1998,Munday2009}. The\nCasimir force has also been considered theoretically with methods\nprimarily based on the zero-point summation approach and Lifshitz\ntheory~\\cite{Bordag2001,Parsegian2005}.\n\nThe Casimir effect has acquired a much broader impact recently due\nto its importance for nanostructured materials, including graphite\nand graphitic nanostructures~\\cite{Klimchitskaya2009} which can\nexist in different geometries and with various unique electronic\nproperties. Moreover, the efficient development and operation of\nmodern micro- and nano-electromechanical devices are limited due\nto effects such as stiction, friction, and adhesion, originating\nfrom or closely related to the Casimir effect~\\cite{Chan2001}.\n\nThe mechanisms governing the CN interactions still remain elusive.\nIt is known that the system geometry~\\cite{Rajter2007,Popescu2008}\nand dielectric response~\\cite{Fermani2007,Bondarev05} have a\nprofound effect on the interaction, in general, but their specific\nfunctionalities have not been qualitatively and quantitatively\nunderstood. Since CNs of virtually the same radial size can\npossess different electronic properties, investigating their\nCasimir interactions presents a unique opportunity to obtain\ninsight into specific dielectric response features affecting the\nCasimir force between metallic and semiconducting cylindrical\nsurfaces. This can also unveil the role of collective surface\nexcitations in the energetic stability of multi-wall CNs of\nvarious chiral combinations.\n\nSince Lifshitz theory cannot be easily applied to geometries other\nthan parallel plates, researchers have used the Proximity Force\nApproximation (PFA) to calculate the Casimir interaction between\nCNs~\\cite{Rajter2007,Bordag2006} (see also\nRef.~\\cite{Klimchitskaya2009} for the latest review). The method\nis based on approximating the curved surfaces at very close\ndistances by a series of parallel plates and summing their\nenergies using the Lifshitz result. Thus, the PFA is inherently an\nadditive approach, applicable to objects at very close separations\n(still to be greater than objects inter-atomic distances) under\nthe assumption that the CN dielectric response is the same as the\none for the plates. This last assumption is very questionable as\nthe quasi-1D character of the electronic motion in CNTs is known\nto be of principal importance for the correct description of their\nelectronic and optical\nproperties~\\cite{Dresselhaus,Bondarev04,Tasaki}.\n\nWe model the double-wall CN by two infinitely long, infinitely\nthin, continuous concentric cylinders with radii $R_{1,2}$,\nimmersed in vacuum. Each cylinder is characterized by the complex\ndynamic axial dielectric function $\\epsilon _{zz}(R_{1,2},\\omega)$\nwith the \\textit{z}-direction along the CN axis as shown in\nFig.~\\ref{fig10}. The azimuthal and radial components of the\ncomplete CN dielectric tensor are neglected as they are known to\nbe much less than $\\epsilon_{zz}$ for most CNs~\\cite{Tasaki}. The\nQED quantization scheme in the presence of\nCNs~\\cite{BuhmannWelsch,Bondarev05} generates the second-quantized\nHamiltonian\n\\[\n\\hat{H}=\\sum_{i=1,2}\\int_{0}^{\\infty}\\!\\!\\!d\\omega\\hbar\\omega\\int\nd\\mathbf{R}_{i}\\hat{f}^{\\dag}(\\mathbf{R}_{i},\\omega)\\hat{f}(\\mathbf{R}_{i},\\omega)\n\\]\nof the vacuum-type medium assisted EM field, with the bosonic\noperators $\\hat{f}^{\\dag}$ and $\\hat{f}$ creating and\nannihilating, respectively, surface EM excitations of frequency\n\\textit{$\\omega $} at points\n$\\mathbf{R}_{1,2}=\\{R_{1,2},\\varphi_{1,2},z_{1,2}\\}$ of the\ndouble-wall CN system. The Fourier-domain electric field operator\nat an arbitrary point $\\mathbf{r}=(r,\\varphi,z)$ is given by\n\\[\n\\hat{E}(\\mathbf{r},\\omega)=i\\omega\\mu _{0}\\sum _{i=1,2}\\int\nd\\mathbf{R}_{i}\\mathbf{G}(\\mathbf{r},\\mathbf{R}_{i},\\omega)\\cdot\n\\hat{\\mathbf{J}}(\\mathbf{R}_{i},\\omega),\n\\]\nwhere $\\mathbf{G}(\\mathbf{r},\\mathbf{R}_{i},\\omega)$ is the dyadic\nEM field Green's function (GF), and\n\\[\n\\hat{\\mathbf{J}}(\\mathbf{R}_{i},\\omega)=\\frac{\\omega}{\\mu_{0}c^{2}}\n\\sqrt{\\frac{\\hbar\\,\\mbox{Im}\\,\\epsilon_{zz}(R_i,\\omega)}{\\pi\\varepsilon_{0}}}\\,\n\\hat{f}(\\mathbf{R}_{i},\\omega)\\mathbf{e}_{z}\n\\]\nis the surface current density operator selected in such a way as\nto ensure the correct QED equal-time commutation relations for the\nelectric and magnetic field\noperators~\\cite{BuhmannWelsch,Bondarev05}. Here, $\\mathbf{e}_{z}$\nis the unit vector along the CN axis, $\\varepsilon_0$, $\\mu_0$,\nand $c$ are the dielectric constant, magnetic permeability, and\nvacuum speed of light, respectively.\n\n\\begin{figure}[t]\n\\epsfxsize=5.5cm\\centering{\\epsfbox{fig10.eps}}\\caption{Schematic\nof the two concentric CNs in vacuum. The CN radii are $R_1$ and\n$R_2$. The regions between the CN surfaces are denoted as (1),\n(2), and (3).}\\label{fig10}\n\\end{figure}\n\nThe dyadic GF satisfies the wave equation\n\\begin{equation} \\label{GrindEQ1}\n\\nabla\\times\\nabla\\times\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)-\n\\frac{\\omega^{2}}{c^{2}}\\,\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)=\n\\delta(\\mathbf{r}-\\mathbf{r}^\\prime)\\,\\mathbf{I}\n\\end{equation}\nwith $\\mathbf{I}$ being the unit tensor. The GF can further be\ndecomposed as follows\n\\[\n\\mathbf{G}^{(s,f)}=\\mathbf{G}^{(0)}\\delta_{sf}+\\mathbf{G}_{scatt}^{(s,f)}\n\\]\nwhere $\\mathbf{G}^{(0)}$ and $\\mathbf{G}_{scatt}^{(s,f)}$\nrepresent the contributions of the direct and scattered waves,\nrespectively~\\cite{Tai1994,KoreanPaper}, with a point-like field\nsource located in region $s$ and the field registered in region\n$f$ (see Fig.~\\ref{fig10}). The boundary conditions for\nEq.~(\\ref{GrindEQ1}) are obtained from those for the electric and\nmagnetic field components on the CN\nsurfaces~\\cite{Fermani2007,Bondarev04}, which result in\n\\begin{equation}\n\\label{GrindEQ2}\n\\mathbf{e}_{r}\\times\\left[\\left.\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right|_{R_{1,2}^{+}}\n-\\left.\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right|_{R_{1,2}^{-}}\\right]=0,\n\\end{equation}\n\\begin{equation}\n\\label{GrindEQ3}\n{\\mathbf{e}_{r}\\!\\times\\!\\nabla\\!\\times\\!\\left[\\left.\n\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right|_{R_{1,2}^{+}}\\!\\!\\!\n-\\left.\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right|_{R_{1,2}^{-}}\\right]}\\!=\ni\\omega\\mu_{0}\\bm{\\sigma}^{(1,2)}(\\mathbf{r},\\omega)\\!\\cdot\\!\\left.\n\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right|_{R_{1,2}}\n\\end{equation}\nwhere $\\mathbf{e}_r$ is the unit vector along the radial\ndirection. The discontinuity in Eq.~(\\ref{GrindEQ3}) results from\nthe full account of the finite absorption and dispersion for both\nCNs by means of their conductivity tensors $\\bm{\\sigma}^{(1,2)}$\napproximated by their largest components\n\\begin{equation}\n\\sigma_{zz}^{(1,2)}(R_{1,2},\\omega)=-\\frac{i\\omega\\varepsilon_{0}}{S\\rho_{T}}\\,\n[\\epsilon_{zz}^{(1,2)}(R_{1,2},\\omega)-1]\\label{DrudeSI}\n\\end{equation}\n[compare with Eq.~(\\ref{DrudeGauss})].\n\nFollowing the procedure described in\nRefs.~\\cite{Tai1994,KoreanPaper}, we expand $\\mathbf{G}^{(0)}$ and\n$\\mathbf{G}_{scatt}^{(s,f)}$ into series of even and odd vector\ncylindrical functions with unknown coefficients to be found from\nEqs.~(\\ref{GrindEQ2}) and (\\ref{GrindEQ3}). This splits the EM\nmodes in the system into TE and TM polarizations, with\nEqs.~(\\ref{GrindEQ2}) and (\\ref{GrindEQ3}) yielding a set of 32\nequations (16 for each polarization) with 32 unknown coefficients.\nThe unknown coefficients are found determining the dyadic GF in\neach region.\\footnote{Due to the lengthy and tedious algebra, this\nderivation will be presented in a separate longer communication.}\n\nUsing the expressions for the electric and magnetic fields, the\nelectromagnetic stress tensor is\nconstructed~\\cite{BuhmannWelsch,Cavero-Pelaez2005}\n\\begin{equation} \\label{GrindEQ4}\n\\mathbf{T}(\\mathbf{r},\\mathbf{r}^\\prime)=\\mathbf{T}_{1}(\\mathbf{r},\\mathbf{r}^\\prime)+\n\\mathbf{T}_{2}(\\mathbf{r},\\mathbf{r}^\\prime)-\\frac{1}{2}\\,\\mathbf{I}\\,Tr\\!\\left[\n\\mathbf{T}_{1}(\\mathbf{r},\\mathbf{r}^\\prime)+\\mathbf{T}_{2}(\\mathbf{r},\\mathbf{r}^\\prime)\\right]\n\\end{equation}\n\\begin{equation} \\label{GrindEQ5}\n\\mathbf{T}_{1}(\\mathbf{r},\\mathbf{r}^\\prime)=\\frac{\\hbar}{\\pi}\\int_{0}^{\\infty}\\!\\!\\!d\\omega\n\\frac{\\omega^{2}}{c^{2}}\\,\\mbox{Im}\\!\\left[\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\right]\n\\end{equation}\n\\begin{equation} \\label{GrindEQ6}\n\\mathbf{T}_{2}(\\mathbf{r},\\mathbf{r}^\\prime)=-\\frac{\\hbar}{\\pi}\\int_{0}^{\\infty}\\!\\!\\!d\\omega\n\\,\\mbox{Im}\\!\\left[\\nabla\\times\\mathbf{G}(\\mathbf{r},\\mathbf{r}^\\prime,\\omega)\\times\n\\stackrel{\\!\\!\\leftarrow}{\\nabla^{\\,\\prime}}\\right]\n\\end{equation}\nWe are interested in the radial component $T_{rr}$ which describes\nthe radiation pressure of the virtual EM field on each CN surface\nin the system. The Casimir force per unit area exerted on the\nsurfaces is then given by~\\cite{BuhmannWelsch}\n\\begin{equation}\nF_{i}=\\lim\\limits_{r\\to\nR_{i}}\\left\\{\\lim\\limits_{\\mathbf{r}^\\prime\\to\\mathbf{r}}\n\\left[T_{rr}^{(i)}(\\mathbf{r},\\mathbf{r}^\\prime)-T_{rr}^{(i+1)}(\\mathbf{r},\\mathbf{r}^\\prime)\n\\right]\\right\\},\\;\\;\\;i=1,2 \\label{GrindEQ7}\n\\end{equation}\n\nThe forces $F_{1,2}$ calculated from Eq.~(\\ref{GrindEQ7}) are of\nequal magnitude and opposite direction, indicating the attraction\nbetween the cylindrical surfaces. The Casimir force thus obtained\naccounts \\textit{simultaneously} for the geometrical curvature\neffects (through the GF tensor) and the finite absorption and\ndissipation of each CN [through their dielectric response\nfunctions (\\ref{DrudeSI})]. The dielectric response functions of\nparticular CNs were calculated from the CN realistic band\nstructure as described above, in Section~\\ref{sec3}. We decomposed\nthem into the Drude contribution and the contribution originating\nfrom (transversely quantized) interband electronic transitions,\n$\\epsilon_{zz}=\\epsilon_{zz}^D+\\epsilon_{zz}^{inter}$, in order to\nbe able to see how much each individual contribution affects the\ninter-tube Casimir attraction.\n\nIt is interesting to consider the case of infinitely conducting\nparallel plates first using Eq.~(\\ref{GrindEQ7}). This is obtaned\nby taking the limits $\\sigma_{zz}^{(1,2)}\\to\\infty$ and\n$R_{1,2}\\to\\infty$ while keeping constant the inter-tube distance,\n$R_{1}\\!-R_{2}=d$. We find\n\\[\nF=-\\frac{\\hbar c}{16\\pi^2R_1^4}\n\\int_0^\\infty\\!\\!\\!dx_1x_1\\sum_{n=0}^\\infty\n\\frac{(2-\\delta_n^0)}{I_n(x_1)K_n(x_2)-I_n(x_2)K_n(x_1)}\n\\]\n\\[\n\\times\\left\\{\\left[x_1^2K^{\\prime\\,2}_n(x_1)+\\left(n^2+x_1^2\\right)K_n^2(x_1)\\right]\n\\left[I_n^2(x_1)K_n(x_2)\/K_n(x_1)-2I_n(x_1)I_n(x_2)\\right]\\right.\n\\]\n\\[\n-\\left[x_1^2I^{\\prime\\,2}_n(x_1)+\\left(n^2+x_1^2\\right)I_n^2(x_1)\\right]K_n(x_1)K_n(x_2)\n\\]\n\\[\n-\\left.2\\left[x_1^2I^\\prime_n(x_1)K^\\prime_n(x_1)+\\left(n^2+x_1^2\\right)I_n(x_1)K_n(x_1)\\right]\nI_n(x_2)K_n(x_1)\\right\\}\n\\]\nwhere $x_{1,2}=xR_{1,2}$, $I_n(x)$ and $K_n(x)$ are the modified\nBessel functions of the first and second kind, respectively. The\nabove expression is obtained by making the transition to imaginary\nfrequencies $\\omega\\rightarrow i\\omega$, and using the Euclidean\nrotation technique as described in\nRefs.~\\cite{Cavero-Pelaez2005,Milton1978}. This can further be\nevaluated by summing up the series over $n$ using the large-order\nBessel function expansions~\\cite{Abramovitz}. This results in\n$F\\!\\sim\\!(-1\/3)(\\hbar c\\pi^2\/240d^4)$ which is $1\/3$ of the\nwell-known result for two parallel\nplates~\\cite{BuhmannWelsch,Klimchitskaya2009}. This deviation\noriginates from $\\epsilon_{zz}\\ne0$ only and the remaining\ndielectric tensor components being zero in our model.\n\n\\begin{figure}[t]\n\\epsfxsize=12.5cm\\centering{\\epsfbox{fig11.eps}}\\vskip-0.2cm\\caption{The\nCasimir force per unit area as a function of the inter-tube\nseparation $d$, for different pairs of CNs. The inset shows force\nfound with the full dielectric function and the Drude contribution\nonly for the same CN pairs indicated in the figure.}\\label{fig11}\n\\end{figure}\n\nFigure~\\ref{fig11} presents results from the numerical\ncalculations of $F$ as a function of the inter-tube\nsurface-to-surface distance for various pairs of CNs with their\nrealistic chirality dependent dielectric responses taken into\naccount. We have chosen the inner CN to be the achiral $(12,12)$\nmetallic nanotube, and to change the outer tubes. As $R_2$ is\nvaried, one can envision double wall CNs consisting of metal\/metal\nor metal\/semiconductor combinations of different chiralities but\nof similar radial dimensions.\n\nFigure~\\ref{fig11} shows that $F$ decreases in strength as the\nsurface-to-surface distance increases. This dependence is\nmonotonic for the zigzag $(m,0)$ and armchair $(n,n)$ outer tubes,\nbut it happens at different rates. The attraction is stronger if\nthe outer CN is an armchair $(n,n)$ one as compared to the\nattraction for the outer $(m,0)$ nanotubes. At the same time, for\nchiral tubes the Casimir force decreases as a function of $d$ in a\nrather irregular fashion. It is seen that for relatively small\n$d$, the interaction force can be quite different. For example,\nthe attraction between $(27,4)@(12,12)$ and $(21,13)@(12,12)$\ndiffer by $\\sim\\!20$~\\% in favor of the second pair, even though\nthe radial difference is only $0.2$~\\AA. The differences between\nthe different CNs become smaller as their separation becomes\nlarger, and they eventually become negligible as the Casimir force\ndiminishes at large distances.\n\nWe also calculate the Casimir force using the\n$\\epsilon_{zz}^D(\\omega)$ contribution alone in each dielectric\nfunction. The inset in Fig.~\\ref{fig11} indicates that the\nattraction is stronger when the interband transitions are\nneglected. The decay of $F$ as a function of $d$ is monotonic.\nIncluding the $\\epsilon_{zz}^{inter}(\\omega)$ term not only\nreduces the force, but also introduces non-linearities due to the\nchirality dependent optical excitations. At large\nsurface-to-surface separations, the discrepancies between the\nforce calculated with the full dielectric response, and those\nobtained with the Drude term only become less significant. We find\nthat for $d\\!\\sim\\!15$~\\AA, this difference is less than $10$~\\%.\n\n\\begin{figure}[t]\n\\epsfxsize=12.5cm\\centering{\\epsfbox{fig12.eps}}\\vskip-0.1cm\\caption{The\nCasimir force per unit area as a function of the inter-tube\nseparation $d$ for selected CN pairs. The insets show the EELS\nspectra for several CNs.}\\label{fig12}\n\\end{figure}\n\nTo investigate further the important functionalities originating\nfrom the cylindrical geometry and the CN dielectric response\nproperties, $F$ is calculated for different achiral inner\/outer\nnanotube pairs. Studying zigzag and armchair CNs allows tracking\ngeneralities from $\\epsilon(\\omega)$ in a more controlled manner.\nThe results are presented in Fig.~\\ref{fig12}. We have chosen\nrepresentatives of three inner CN types -- metallic $(12,12)$,\nsemi-metallic $(21,0)$, and semiconducting $(20,0)$ tubules. They\nare of similar radii, $8.14$~\\AA, $8.22$~\\AA, and $7.83$~\\AA,\nrespectively. We see that depending on the outer nanotube types,\nthe $F$ versus $d$ curves are positioned in three groups. The\nweakest interaction is found when there are two zigzag concentric\nCNs (top two curves). The fact that some of these are\nsemi-metallic and others are semiconducting does not seem to\ninfluence the magnitude and monotonic decrease of the Casimir\nforce.\n\nThe attraction is stronger when there is a combination of an\narmchair and a zigzag CNT as compared to the previous case. The\ncurves for $(m,0)@(12,12)$, $(n,n)@(21,0)$, and $(n,n)@(20,0)$ are\npractically overlapping, meaning that the specific location of the\nzigzag and armchair tubes (inner or outer) is of no significance\nto the force. The small deviations can be attributed to the small\ndifferences in the inner CN radii. Finally, we see that the\nstrongest interaction occurs between two armchair CNs (red curve).\nThese functionalities are not unique just for the considered CNs.\nWe have performed the same calculations for many different achiral\ntubes, and we always find that the strongest interaction occurs\nbetween two armchair CNs and the weakest --- between two zigzag\nCNs (provided that their radial dimensions are similar).\n\nThe results from these calculations are strongly suggestive that\nthe CN collective excitation properties have a strong effect on\ntheir mutual interaction. This is particularly true for the\nrelatively small distances of interest here, for which the\ndominant contribution of plasmonic modes to the Casimir\ninteractions has been realized for planar~\\cite{Kampen1968} and\nlinear~\\cite{Dobson2006} metallic systems. To elucidate this issue\nhere, we calculate the EELS spectra, given by\n$-\\mbox{Im}[1\/\\epsilon(\\omega)]$, and compare them for various\ninner and outer CNs combinations --- Fig.~\\ref{fig12} (inset).\n\nConsidering $F$ as a function of $d$ and the specific form of the\nEELS spectra, it becomes clear from the inset in Fig.~\\ref{fig12}\nthat the low frequency plasmon excitations, given by peaks in\n$-\\mbox{Im}[1\/\\epsilon(\\omega)]$, are key to the strength of the\nCasimir force. We always find that the strongest force is between\nthe tubules with well pronounced overlapping low frequency plasmon\nexcitations. This is consistent with the conclusion of\nRef.~\\cite{Dobson2006} for generic 1D-plasmonic structures.\nHowever, in our case we deal with the interband plasmons\noriginating from the space quantization of the transverse\nelectronic motion, and, therefore, having quite a different\nfrequency-momentum dispersion law (constant) as compared to that\nnormally assumed (linear) for plasmons~\\cite{Pichler98}. A weaker\nforce is obtained if only one of the CNs supports strong low\nfrequency interband plasmon modes. The weakest interaction happens\nwhen neither CN has strong low frequency plasmons. For the cases\nshown in Fig~\\ref{fig12}, one finds well pronounced overlapping\nplasmon transitions in the $(12,12)$ CN at $\\omega_1=2.18$~eV and\n$\\omega_2=3.27$~eV, and at $\\omega_1=1.63$~eV and\n$\\omega_2=2.45$~eV in the $(17,17)$ CN. At the same time, no such\nwell defined strong low frequency excitations in the $(21,0)$ and\n$(30,0)$ CNs are found. Figure~\\ref{fig12} shows that the\nattraction in $(17,17)@(12,12)$ is much stronger than the\nattraction in $(30,0)@(21,0)$, even though the radial sizes of the\ninvolved CNs are approximately the same. One also notes that for\nthe case of $(17,17)@(21,0)$ there is only one such low frequency\nexcitation coming from the armchair tube and, consequently, the\nCasimir force has an intermediate value as compared to the above\ndiscussed two cases.\n\nWe performed calculations of the Casimir force between many CN\npairs and made comparisons between the relevant regions of the\nEELS spectra. It is found that, in general, armchair tubes always\nhave strong, well pronounced interband plasmon excitations in the\nlow frequency range. Zigzag and most chiral CNs have low frequency\ninterband plasmons~\\cite{BondPRB09}, too, but they are not as near\nas well pronounced as those in armchair tubes; their stronger\nplasmon modes are found at higher frequencies.\n\nThese studies are indicative of the significance of the collective\nresponse properties of the involved CNs. Specifically, the\ncollective low energy plasmon excitations and their relative\nlocation can result in nanotube attraction with different\nstrengths. We further investigate this point by considering a\ndouble wall CN with radii $R_1=11.63$~\\AA and $R_2=8.22$~\\AA. The\ndielectric function of each tube is taken to be of the generic\nLorentzian form\n\\begin{equation}\n\\epsilon_{zz}(R_{1,2},\\omega)=1-\\frac{\\Omega^2}{\\omega^2-\\omega_{1,2}^2\n+i\\omega\\Gamma}\\label{epsLorentz}\n\\end{equation}\nwith the typical for nanotubes values $\\Omega=2.7$~eV and\n$\\Gamma=0.03$~eV~\\cite{Fermani2007}. Then, the EELS spectrum has\nonly one plasmon resonance at $\\omega_{1,2}$ for each tube. This\ngeneric form allows us to change the relative position and\nstrength of the plasmon peaks and uncover more characteristic\nfeatures originating from the EELS spectra.\n\n\\begin{figure}[t]\n\\epsfxsize=12.5cm\\centering{\\epsfbox{fig13.eps}}\\caption{The\nCasimir force per unit area as a function of the outer CN plasmon\nfrequency, while the inner CN plasmon peak $\\omega_2$ is constant.\nResults are shown for four values of $\\omega_2$. The dielectric\nfunctions are modeled by a generic Lorentzian as given by\nEq.~(\\ref{epsLorentz}).}\\label{fig13}\n\\end{figure}\n\nIn Fig.~\\ref{fig13}, the force as a function of plasmon frequency\nresonances of the outer CN is shown when the plasmon transition\nfor the inner CNT is kept constant (four values are chosen for\n$\\omega_2$). One sees that the local minima in $F$ versus $\\omega$\noccur when $\\omega_1$ and $\\omega_2$ coincide. In fact, the\nstrongest attraction happens when both CNs have the lowest plasmon\nexcitations at the same frequency $\\omega_1=\\omega_2=0.81$~eV. It\nis evident that the existence of relatively strong low frequency\nEELS spectrum \\textit{and} an overlap between the relevant plasmon\npeaks of the two structures is necessary to achieve a strong\ninteraction.\n\nThis study clearly demonstrates the crucial importance of the\ncollective low energy surface plasmon excitations at relatively\nclose surface-to-surface separations along with the cylindrical\ncircular geometry of the double-wall CN system. The QED approach\nwe used provides the unique opportunity to investigate these\nfeatures together, or separately, and to uncover underlying\nmechanisms of the energetic stability of different double-wall CN\ncombinations. An additional advantage here is that we can\ncalculate the dielectric function explicitly for each chirality.\nThus, we can determine unambiguously how the semiconducting or\nmetallic nature of each CN contributes to their mutual\ninteraction.\n\n\\section{Conclusion}\\label{concl}\n\nWe have shown that the strong exciton-surface-plasmon coupling\neffect with characteristic exciton absorption line (Rabi)\nsplitting $\\sim\\!0.1$~eV exists in small-diameter\n($\\lesssim\\!1$~nm) semiconducting CNs.~The splitting is almost as\nlarge as the typical exciton binding energies in such CNs\n($\\sim\\!0.3-0.8$~eV~\\cite{Pedersen03,Pedersen04,Wang05,Capaz}),\nand of the same order of magnitude as the exciton-plasmon Rabi\nsplitting in organic semiconductors\n($\\sim\\!180$~meV~\\cite{Bellessa}).~It is much larger than the\nexciton-polariton Rabi splitting in semiconductor microcavities\n($\\sim\\!140-400\\,\\mu\\mbox{eV}\\,$\\cite{Reithmaier,Yoshie,Peter}),\nor the exciton-plasmon Rabi splitting in hybrid\nsemiconductor-metal nanoparticle molecules~\\cite{Govorov}.\n\nSince the formation of the strongly coupled mixed exciton-plasmon\nexcitations is only possible if the exciton total energy is in\nresonance with the energy of an interband surface plasmon mode, we\nhave analyzed possible ways to tune the exciton energy to the\nnearest surface plasmon resonance. Specifically, the exciton\nenergy may be tuned to the nearest plasmon resonance in ways used\nfor the excitons in semiconductor quantum microcavities\n--- thermally (by elevating sample\ntemperature)~\\cite{Reithmaier,Yoshie,Peter}, and\/or\nelectrostatically~\\cite{MillerPRL,Miller,Zrenner,Krenner} (via the\nquantum confined Stark effect with an external electrostatic field\napplied perpendicular to the CN axis).~The two possibilities\ninfluence the different degrees of freedom of the quasi-1D exciton\n--- the (longitudinal) kinetic energy and the excitation energy,\nrespectively.\n\nWe have studied how the perpendicular electrostatic field affects\nthe exciton excitation energy and interband plasmon resonance\nenergy (the quantum confined Stark effect). Both of them are shown\nto shift to the red due to the decrease in the CN band gap as the\nfield increases. However, the exciton red shift is much less than\nthe plasmon one because of the decrease in the absolute value of\nthe negative binding energy, which contributes largely to the\nexciton excitation energy. The exciton excitation energy and\ninterband plasmon energy approach as the field increases, thereby\nbringing the total exciton energy in resonance with the plasmon\nmode due to the non-zero longitudinal kinetic energy term at\nfinite temperature.\n\nThe noteworthy point is that the strong exciton-surface-plasmon\ncoupling we predict here occurs in an individual CN as opposed to\nvarious artificially fabricated hybrid plasmonic nanostructures\nmentioned above. We strongly believe this phenomenon, along with\nits tunability feature via the quantum confined Stark effect we\nhave demonstrated, opens up new paths for the development of CN\nbased tunable optoelectronic device applications in areas such as\nnanophotonics, nanoplasmonics, and cavity QED. One straightforward\napplication like this is the CN photoluminescence control by means\nof the exciton-plasmon coupling tuned electrostatically via the\nquantum confined Stark effect.~This complements the microcavity\ncontrolled CN infrared emitter application reported\nrecently\\cite{Avouris08}, offering the advantage of less stringent\nfabrication requirements at the same time since the planar\nphotonic microcavity is no longer required. Electrostatically\ncontrolled coupling of two spatially separated (weakly localized)\nexcitons to the same nanotube's plasmon resonance would result in\ntheir entanglement~\\cite{Bondarev07,Bondarev07jem,Bondarev07os},\nthe phenomenon that paves the way for CN based solid-state quantum\ninformation applications. Moreover, CNs combine advantages such as\nelectrical conductivity, chemical stability, and high surface area\nthat make them excellent potential candidates for a variety of\nmore practical applications, including efficient solar energy\nconversion~\\cite{Trancik}, energy storage~\\cite{Shimoda}, and\noptical nanobiosensorics~\\cite{Goodsell10}. However, the\nphotoluminescence quantum yield of individual CNs is relatively\nlow, and this hinders their uses in the aforementioned\napplications. CN bundles and films are proposed to be used to\nsurpass the poor performance of individual tubes. The theory of\nthe exciton-plasmon coupling we have developed here, being\nextended to include the inter-tube interaction, complements\ncurrently available 'weak-coupling' theories of the\nexciton-plasmon interactions in low-dimensional\nnanostructures~\\cite{Govorov,GovorovPRB} with the very important\ncase of the strong coupling regime. Such an extended theory\n(subject of our future publication) will lay the foundation for\nunderstanding inter-tube energy transfer mechanisms that affect\nthe efficiency of optoelectronic devices made of CN bundles and\nfilms, as well as it will shed more light on the recent\nphotoluminescence experiments with CN\nbundles~\\cite{Munich,Ferrari} and multi-walled CNs~\\cite{Hirori},\nrevealing their potentialities for the development of high-yield,\nhigh-performance optoelectronics applications with CNs.\n\nIn addition, we have first applied the macroscopic QED approach\nsuitable for dispersing and absorbing media to study the Casimir\ninteraction in a double-wall carbon nanotube systems with the\nrealistic dielectric response taken into account. We found that at\ndistances similar to the equilibrium separations between graphitic\nsurfaces ($\\sim\\!3$~\\AA), the attraction is dominated by the low\nenergy (interband) plasmon excitations of both CNs. The key\nattributes of the EELS spectra are the existence of low frequency\nplasmons, their strong and well pronounced nature, and the overlap\nbetween the low frequency plasmon peaks belonging to the two CNTs.\nThus, the chiralities of concentric graphene sheets with similar\nradial sizes exhibiting these features will be responsible for\nforming the most preferred CN pairs. As the inter-tube separation\nincreases, the plasmon effect diminishes and the collective\nexcitations originating from the nanotube metallic or\nsemiconducting nature do not influence the interaction in a\nprofound way.\n\nWe expect our results to pave the way for the development of new\ngeneration of tunable optoelectronic and nano-electromechanical\ndevice applications with single-wall and multi-wall carbon\nnanotubes.\n\n\\begin{center}\n{\\bf Acknowledgements}\n\\end{center}\nI.V.B. is supported by the US National Science Foundation, Army\nResearch Office and NASA (grants ECCS-1045661 \\& HRD-0833184,\nW911NF-10-1-0105, and NNX09AV07A). L.M.W. and A.P. are supported\nby the US Department of Energy contract DE-FG02-06ER46297. Helpful\ndiscussions with Mikhail Braun (St.-Peterburg U., Russia),\nJonathan Finley (WSI, TU Munich, Germany), and Alexander Govorov\n(Ohio U., USA) are gratefully acknowledged.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\\section{Introduction}\n\nPlease follow the steps outlined below when submitting your manuscript to\nthe IEEE Computer Society Press.  This style guide now has several\nimportant modifications (for example, you are no longer warned against the\nuse of sticky tape to attach your artwork to the paper), so all authors\nshould read this new version.\n\n\\subsection{Language}\n\nAll manuscripts must be in English.\n\n\\subsection{Dual submission}\n\nPlease refer to the author guidelines on the WACV 2023 web page\n(\\url{http:\/\/wacv2023.thecvf.com\/submission\/})\nfor a discussion of the policy on dual submissions.\n\n\\subsection{Paper length}\nPapers must be no longer than eight pages, not including references.\nAny pages in excess of eight pages must contain ONLY references ---\nno text, figures, acknowledgements, tables, etc.\n\n{\\bf There will be no extra page charges for WACV 2023.}\n\nOverlength papers will simply not be reviewed.  This includes papers\nwhere the margins and formatting are deemed to have been significantly\naltered from those laid down by this style guide.  Note that this\n\\LaTeX\\ guide already sets figure captions and references in a smaller font.\nThe reason such papers will not be reviewed is that there is no provision for\nsupervised revisions of manuscripts.  The reviewing process cannot determine\nthe suitability of the paper for presentation in eight pages if it is\nreviewed in eleven.\n\n\\subsection{The ruler}\nThe \\LaTeX\\ style defines a printed ruler which should be present in the\nversion submitted for review.  The ruler is provided in order that\nreviewers may comment on particular lines in the paper without\ncircumlocution.  If you are preparing a document using a non-\\LaTeX\\\ndocument preparation system, please arrange for an equivalent ruler to\nappear on the final output pages.  The presence or absence of the ruler\nshould not change the appearance of any other content on the page.  The\ncamera ready copy should not contain a ruler. (\\LaTeX\\ users may uncomment\nthe \\verb'\\wacvfinalcopy' command in the document preamble.)  Reviewers:\nnote that the ruler measurements do not align well with lines in the paper\n--- this turns out to be very difficult to do well when the paper contains\nmany figures and equations, and, when done, looks ugly.  Just use fractional\nreferences (e.g.\\ this line is $087.5$), although in most cases one would\nexpect that the approximate location will be adequate.\n\n\\subsection{Mathematics}\n\nPlease number all of your sections and displayed equations.  It is\nimportant for readers to be able to refer to any particular equation.  Just\nbecause you didn't refer to it in the text doesn't mean some future reader\nmight not need to refer to it.  It is cumbersome to have to use\ncircumlocutions like ``the equation second from the top of page 3 column\n1''.  (Note that the ruler will not be present in the final copy, so is not\nan alternative to equation numbers).  All authors will benefit from reading\nMermin's description of how to write mathematics:\n\\url{http:\/\/www.pamitc.org\/documents\/mermin.pdf}.\n\n\n\\subsection{Blind review}\n\nMany authors misunderstand the concept of anonymizing for blind\nreview.  Blind review does not mean that one must remove\ncitations to one's own work---in fact it is often impossible to\nreview a paper unless the previous citations are known and\navailable.\n\nBlind review means that you do not use the words ``my'' or ``our''\nwhen citing previous work.  In addition, in means you are extremely careful if you\nshare URLs to source code, github repositories, project websites,\ndatasets, etc. The URL or other information (e.g. your github user ID)\nmay identify you and would violate the anonymization policy.\n\n\nFor example, saying ``this builds on the work of Lucy Smith [1]'' does not say\nthat you are Lucy Smith; it says that you are building on her\nwork.  If you are Smith and Jones, do not say ``as we show in\n[7]'', say ``as Smith and Jones show in [7]'' and at the end of the\npaper, include reference 7 as you would any other cited work.\n\nAn example of a bad paper just asking to be rejected:\n\\begin{quote}\n\\begin{center}\n    An analysis of the frobnicatable foo filter.\n\\end{center}\n\n   In this paper we present a performance analysis of our\n   previous paper [1], and show it to be inferior to all\n   previously known methods.  Why the previous paper was\n   accepted without this analysis is beyond me.\n\n   [1] Removed for blind review\n\\end{quote}\n\n\nAn example of an acceptable paper:\n\n\\begin{quote}\n\\begin{center}\n     An analysis of the frobnicatable foo filter.\n\\end{center}\n\n   In this paper we present a performance analysis of the\n   paper of Smith \\etal [1], and show it to be inferior to\n   all previously known methods.  Why the previous paper\n   was accepted without this analysis is beyond me.\n\n   [1] Smith, L and Jones, C. ``The frobnicatable foo\n   filter, a fundamental contribution to human knowledge''.\n   Nature 381(12), 1-213.\n\\end{quote}\n\nIf you are making a submission to another conference at the same time,\nwhich covers similar or overlapping material, you may need to refer to that\nsubmission in order to explain the differences, just as you would if you\nhad previously published related work.  In such cases, include the\nanonymized parallel submission~\\cite{Authors20} as additional material and\ncite it as\n\\begin{quote}\n[1] Authors. ``The frobnicatable foo filter'', F\\&G 2020 Submission ID 324,\nSupplied as additional material {\\tt fg324.pdf}.\n\\end{quote}\n\nFinally, you may feel you need to tell the reader that more details can be\nfound elsewhere, and refer them to a technical report.  For conference\nsubmissions, the paper must stand on its own, and not {\\em require} the\nreviewer to go to a techreport for further details.  Thus, you may say in\nthe body of the paper ``further details may be found\nin~\\cite{Authors20b}''.  Then submit the techreport as additional material.\nAgain, you may not assume the reviewers will read this material.\n\nSometimes your paper is about a problem which you tested using a tool which\nis widely known to be restricted to a single institution.  For example,\nlet's say it's 1969, you have solved a key problem on the Apollo lander,\nand you believe that the WACV 70 audience would like to hear about your\nsolution.  The work is a development of your celebrated 1968 paper entitled\n``Zero-g frobnication: How being the only people in the world with access to\nthe Apollo lander source code makes us a wow at parties'', by Zeus \\etal.\n\nYou can handle this paper like any other.  Don't write ``We show how to\nimprove our previous work [Anonymous, 1968].  This time we tested the\nalgorithm on a lunar lander [name of lander removed for blind review]''.\nThat would be silly, and would immediately identify the authors. Instead\nwrite the following:\n\\begin{quotation}\n\\noindent\n   We describe a system for zero-g frobnication.  This\n   system is new because it handles the following cases:\n   A, B.  Previous systems [Zeus et al. 1968] didn't\n   handle case B properly.  Ours handles it by including\n   a foo term in the bar integral.\n\n   ...\n\n   The proposed system was integrated with the Apollo\n   lunar lander, and went all the way to the moon, don't\n   you know.  It displayed the following behaviours\n   which show how well we solved cases A and B: ...\n\\end{quotation}\nAs you can see, the above text follows standard scientific convention,\nreads better than the first version, and does not explicitly name you as\nthe authors.  A reviewer might think it likely that the new paper was\nwritten by Zeus \\etal, but cannot make any decision based on that guess.\nHe or she would have to be sure that no other authors could have been\ncontracted to solve problem B.\n\\medskip\n\n\\noindent\nFAQ\\medskip\\\\\n{\\bf Q:} Are acknowledgements OK?\\\\\n{\\bf A:} No.  Leave them for the final copy.\\medskip\\\\\n{\\bf Q:} How do I cite my results reported in open challenges?\n{\\bf A:} To conform with the double blind review policy, you can report results of other challenge participants together with your results in your paper. For your results, however, you should not identify yourself and should not mention your participation in the challenge. Instead present your results referring to the method proposed in your paper and draw conclusions based on the experimental comparison to other results.\\medskip\\\\\n\n\\begin{figure}[t]\n\\begin{center}\n\\fbox{\\rule{0pt}{2in} \\rule{0.9\\linewidth}{0pt}}\n  \n\\end{center}\n   \\caption{Example of caption.  It is set in Roman so that mathematics\n   (always set in Roman: $B \\sin A = A \\sin B$) may be included without an\n   ugly clash.}\n\\label{fig:long}\n\\label{fig:onecol}\n\\end{figure}\n\n\\subsection{Miscellaneous}\n\n\\noindent\nCompare the following:\\\\\n\\begin{tabular}{ll}\n \\verb'$conf_a$' &  $conf_a$ \\\\\n \\verb'$\\mathit{conf}_a$' & $\\mathit{conf}_a$\n\\end{tabular}\\\\\nSee The \\TeX book, p165.\n\nThe space after \\eg, meaning ``for example'', should not be a\nsentence-ending space. So \\eg is correct, {\\em e.g.} is not.  The provided\n\\verb'\\eg' macro takes care of this.\n\nWhen citing a multi-author paper, you may save space by using ``et alia'',\nshortened to ``\\etal'' (not ``{\\em et.\\ al.}'' as ``{\\em et}'' is a complete word.)\nHowever, use it only when there are three or more authors.  Thus, the\nfollowing is correct: ``Frobnication has been trendy lately.\n   It was introduced by Alpher~\\cite{Alpher02}, and subsequently developed by\n   Alpher and Fotheringham-Smythe~\\cite{Alpher03}, and Alpher \\etal~\\cite{Alpher04}.''\n\nThis is incorrect: ``... subsequently developed by Alpher \\etal~\\cite{Alpher03} ...''\nbecause reference~\\cite{Alpher03} has just two authors.  If you use the\n\\verb'\\etal' macro provided, then you need not worry about double periods\nwhen used at the end of a sentence as in Alpher \\etal.\n\nFor this citation style, keep multiple citations in numerical (not\nchronological) order, so prefer \\cite{Alpher03,Alpher02,Authors20} to\n\\cite{Alpher02,Alpher03,Authors20}.\n\n\n\\begin{figure*}\n\\begin{center}\n\\fbox{\\rule{0pt}{2in} \\rule{.9\\linewidth}{0pt}}\n\\end{center}\n   \\caption{Example of a short caption, which should be centered.}\n\\label{fig:short}\n\\end{figure*}\n\n\\section{Formatting your paper}\n\nAll text must be in a two-column format. The total allowable width of the\ntext area is $6\\frac78$ inches (17.5 cm) wide by $8\\frac78$ inches (22.54\ncm) high. Columns are to be $3\\frac14$ inches (8.25 cm) wide, with a\n$\\frac{5}{16}$ inch (0.8 cm) space between them. The main title (on the\nfirst page) should begin 1.0 inch (2.54 cm) from the top edge of the\npage. The second and following pages should begin 1.0 inch (2.54 cm) from\nthe top edge. On all pages, the bottom margin should be 1-1\/8 inches (2.86\ncm) from the bottom edge of the page for $8.5 \\times 11$-inch paper; for A4\npaper, approximately 1-5\/8 inches (4.13 cm) from the bottom edge of the\npage.\n\n\\subsection{Margins and page numbering}\n\nAll printed material, including text, illustrations, and charts, must be kept\nwithin a print area 6-7\/8 inches (17.5 cm) wide by 8-7\/8 inches (22.54 cm)\nhigh.\n\nPage numbers should be in footer with page numbers, centered and .75\ninches from the bottom of the page and make it start at the correct page\nnumber rather than the 9876 in the example.  To do this find the secounter\nline (around line 33 in this file) and update the page number as\n\\begin{verbatim}\n\\setcounter{page}{123}\n\\end{verbatim}\nwhere the number 123 is your assigned starting page.\n\n\\subsection{Type-style and fonts}\n\nWherever Times is specified, Times Roman may also be used. If neither is\navailable on your word processor, please use the font closest in\nappearance to Times to which you have access.\n\nMAIN TITLE. Center the title 1-3\/8 inches (3.49 cm) from the top edge of\nthe first page. The title should be in Times 14-point, boldface type.\nCapitalize the first letter of nouns, pronouns, verbs, adjectives, and\nadverbs; do not capitalize articles, coordinate conjunctions, or\nprepositions (unless the title begins with such a word). Leave two blank\nlines after the title.\n\nAUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title\nand printed in Times 12-point, non-boldface type. This information is to\nbe followed by two blank lines.\n\nThe ABSTRACT and MAIN TEXT are to be in a two-column format.\n\nMAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use\ndouble-spacing. All paragraphs should be indented 1 pica (approx. 1\/6\ninch or 0.422 cm). Make sure your text is fully justified---that is,\nflush left and flush right. Please do not place any additional blank\nlines between paragraphs.\n\nFigure and table captions should be 9-point Roman type as in\nFigures~\\ref{fig:onecol} and~\\ref{fig:short}.  Short captions should be centred.\n\n\\noindent Callouts should be 9-point Helvetica, non-boldface type.\nInitially capitalize only the first word of section titles and first-,\nsecond-, and third-order headings.\n\nFIRST-ORDER HEADINGS. (For example, {\\large \\bf 1. Introduction})\nshould be Times 12-point boldface, initially capitalized, flush left,\nwith one blank line before, and one blank line after.\n\nSECOND-ORDER HEADINGS. (For example, { \\bf 1.1. Database elements})\nshould be Times 11-point boldface, initially capitalized, flush left,\nwith one blank line before, and one after. If you require a third-order\nheading (we discourage it), use 10-point Times, boldface, initially\ncapitalized, flush left, preceded by one blank line, followed by a period\nand your text on the same line.\n\n\\subsection{Footnotes}\n\nPlease use footnotes\\footnote {This is what a footnote looks like.  It\noften distracts the reader from the main flow of the argument.} sparingly.\nIndeed, try to avoid footnotes altogether and include necessary peripheral\nobservations in\nthe text (within parentheses, if you prefer, as in this sentence).  If you\nwish to use a footnote, place it at the bottom of the column on the page on\nwhich it is referenced. Use Times 8-point type, single-spaced.\n\n\n\\subsection{References}\n\nList and number all bibliographical references in 9-point Times,\nsingle-spaced, at the end of your paper. When referenced in the text,\nenclose the citation number in square brackets, for\nexample~\\cite{Authors20}.  Where appropriate, include the name(s) of\neditors of referenced books.\n\n\\begin{table}\n  \\begin{center}\n    {\\small{\n\\begin{tabular}{llr}\n\\toprule\nMethod & Frobnability & Accuracy \\\\\n\\midrule\nTheirs & Frumpy & 30.2\\%\\\\\nYours & Frobbly & 45.2\\%\\\\\nOurs & Makes one's heart Frob & 99.9\\%\\\\\n\\bottomrule\n\\end{tabular}\n}}\n\\end{center}\n\\caption{Results.   Ours are better. If you prefer, you can put table\ncaptions on top of the tables instead of on the bottom.}\n\\end{table}\n\n\\subsection{Illustrations, graphs, and photographs}\n\nAll graphics should be centered.  Please ensure that any point you wish to\nmake is resolvable in a printed copy of the paper.  Resize fonts in figures\nto match the font in the body text, and choose line widths which render\neffectively in print.  Many readers (and reviewers), even of an electronic\ncopy, will choose to print your paper in order to read it.  You cannot\ninsist that they do otherwise, and therefore must not assume that they can\nzoom in to see tiny details on a graphic.\n\nWhen placing figures in \\LaTeX, it's almost always best to use\n\\verb+\\includegraphics+, and to specify the  figure width as a multiple of\nthe line width as in the example below\n{\\small\\begin{verbatim}\n   \\usepackage[dvips]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]\n                   {myfile.eps}\n\\end{verbatim}\n}\n\n\n\\subsection{Color}\n\nPlease refer to the author guidelines on the WACV 2023 web page \n(\\url{http:\/\/wacv2023.thecvf.com\/submission\/})\nfor a discussion of the use of color in your document.\n\n\\section{Final copy}\n\nYou must include your signed IEEE copyright release form when you submit\nyour finished paper. We MUST have this form before your paper can be\npublished in the proceedings.\n\nPlease direct any questions to the production editor in charge of these\nproceedings at the IEEE Computer Society Press: \n\\url{https:\/\/www.computer.org\/about\/contact}.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\\section{Introduction}\n\nPlease use this template for two separate purposes. (1) If you are\nsubmitting a paper to the second round for a paper that received a\nRevise decision in the first round, please use this template to\naddress the reviewer concerns and highlight the changes made since the\noriginal submission. (2) If you submit a paper to the second round,\nafter you receive the reviews, you may optionally submit a rebuttal to\naddress factual errors or to supply additional information requested by\nthe reviewers (this rebuttal will be seen by Area Chairs but not by\nthe reviewers).  In either case, you are limited to a {\\bf one page}\nPDF file.  Please follow the steps and style guidelines outlined below\nfor submitting your author response.\n\n\nThe rebuttal must adhere to the same blind-submission as the original submission and must comply with this rebuttal-formatted template.\n\n\n\\subsection{Response length}\nAuthor responses must be no longer than 1 page in length including any references and figures.  Overlength responses will simply not be reviewed.  This includes responses where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide.  Note that this \\LaTeX\\ guide already sets figure captions and references in a smaller font.\n\n\\section{Formatting your Response}\n\n{\\bf Make sure to update the paper title and paper ID in the appropriate place in the tex file.}\n\nAll text must be in a two-column format. The total allowable width of the text\narea is $6\\frac78$ inches (17.5 cm) wide by $8\\frac78$ inches (22.54 cm) high.\nColumns are to be $3\\frac14$ inches (8.25 cm) wide, with a $\\frac{5}{16}$ inch\n(0.8 cm) space between them. The top margin should begin\n1.0 inch (2.54 cm) from the top edge of the page.  The bottom margin should be\n1-1\/8 inches (2.86 cm) from the bottom edge of the page for $8.5 \\times\n11$-inch paper; for A4 paper, approximately 1-5\/8 inches (4.13 cm) from the\nbottom edge of the page.\n\nPlease number all of your sections and any displayed equations.  It is important\nfor readers to be able to refer to any particular equation.\n\nWherever Times is specified, Times Roman may also be used.  Main text should be\nin 10-point Times, single-spaced. Section headings should be in 10 or 12 point\nTimes.  All paragraphs should be indented 1 pica (approx. 1\/6 inch or 0.422\ncm).  Figure and table captions should be 9-point Roman type as in\nFigure~\\ref{fig:onecol}.\n\n\nList and number all bibliographical references in 9-point Times, single-spaced,\nat the end of your response. When referenced in the text, enclose the citation\nnumber in square brackets, for example~\\cite{Alpher02}.  Where appropriate,\ninclude the name(s) of editors of referenced books.\n\n\\begin{figure}[t]\n\\begin{center}\n\\fbox{\\rule{0pt}{1in} \\rule{0.9\\linewidth}{0pt}}\n  \n\\end{center}\n   \\caption{Example of caption.  It is set in Roman so that mathematics\n   (always set in Roman: $B \\sin A = A \\sin B$) may be included without an\n   ugly clash.}\n\\label{fig:long}\n\\label{fig:onecol}\n\\end{figure}\n\n\n\\subsection{Illustrations, graphs, and photographs}\n\nAll graphics should be centered.  Please ensure that any point you wish to make is resolvable in a printed copy of the response.  Resize fonts in figures to match the font in the body text, and choose line widths which render effectively in print.  Many readers (and reviewers), even of an electronic copy, will choose to print your response in order to read it.  You cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic.\n\nWhen placing figures in \\LaTeX, it's almost always best to use \\verb+\\includegraphics+, and to specify the  figure width as a multiple of the line width as in the example below\n{\\small\\begin{verbatim}\n   \\usepackage[dvips]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]\n                   {myfile.eps}\n\\end{verbatim}\n}\n\n\n{\\small\n\\bibliographystyle{ieee}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper, we propose a novel approach for learning low bit-width DNNs models by distilling knowledge from multiple quantized teachers. We introduce the idea of collaborative learning that allows teachers to form importance-aware shared knowledge, which will be used to guide the student. The proposed framework also leverages the idea of mutual learning that allows both teachers and student to adjust their parameters to achieve an overall object function. The proposed framework allows end-to-end training in which not only network parameters but also the importance factors indicating the contributions of teachers to the shared knowledge are updated simultaneously. The experimental results on CIFAR-100 and ImageNet datasets with AlexNet and ResNet18 architectures demonstrate that the low bit-width models trained with the proposed approach achieve competitive results compared to the  state-of-the-art methods.\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\subsection{Experimental setup}\n\\paragraph{\\textbf{Datasets.}}\nWe conduct experiments on {CIFAR-100} \\cite{cifar} and {ImageNet} (ILSVRC-2012) \\cite{imagenet} datasets. CIFAR-100 dataset consists of $100$ classes with the total of $60,000$ images, where $50,000$ and $10,000$ images are used for training and testing set, respectively. ImageNet is a large scale dataset with $1,000$ classes in total. This dataset contains $1.2$ million images for training and $50,000$ images for validation, which is used as test set in our experiments.\n\n\\paragraph{\\textbf{Implementation details.}} \nWe evaluate our proposed method on two common deep neural networks AlexNet \\cite{alexnet} and ResNet18 \\cite{Resnet}. Regarding AlexNet, batch normalization layers are added after each convolutional layer and each fully connected layer, which are similar to works done by \\cite{HWGQ, dorefa}. In all experiments, similar to previous works \\cite{LQNets,guidedQuantize}, in the training, we use the basic augmentation including horizontal flips, resizing\nand randomly cropping that crops images to $227\\times 227$ and $224\\times 224$ pixels for ResNet18 and AlexNet, respectively. \nWe use Stochastic Gradient Descents with a momentum of $0.9$ and a mini-batch size of $256$. The learning rate $lr$ for network models is set to $0.1$ and $0.01$ for ResNet and AlexNet, respectively.\nThe learning rate for the importance factors ($\\pi$) in teacher models is set to $lr\/10$\n\n\n\nWhen training ResNet18 model on CIFAR-100, we train the model with 120 epochs. The learning rate is decreased by a factor of 10 after 50 and 100 epochs. When training ResNet18 model on ImageNet, we train the model with 100 epochs. The learning rate is decreased by a factor of 10 after $30$, $60$, and $90$ epochs. When training AlexNet model, for both CIFAR-100 and ImageNet, we train the model with $100$ epochs and we adopt $cosine$ learning rate decay.\n\nWe set the weight decay to 25e-6 for the 1 or 2-bit precision and set it to 1e-4 for higher precisions. Regarding hyper-parameters of the total loss (\\ref{eq:final_loss}), {we empirically set $\\alpha=1$, $\\beta=0.5$}. \nIn our experiments, the shared knowledge is formed at certain layers of teachers and is distilled to the correspondence layers of student. \nSpecifically, the shared knowledge is formed at the last convolutional layers of each convolution block, i.e., layers $2, 5$, and $7$ of AlexNet teachers and from layers $5, 9, 13$, and $17$ of  ResNet18 teachers.\nMeanwhile, we set $\\gamma$ to $100$ or $1$ when attention loss or FitNet loss is used in $\\mathcal{L}_{feat}$. We do not quantize the first and last layers. \n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Ablation studies}\n\\label{subsec:ablation}\n\n\n\nWe conduct several ablation studies on CIFAR-100 with ResNet18 and AlexNet to demonstrate the effectiveness of our proposed method.\nFor ablation studies, we use HWGQ quantizer \\cite{HWGQ} for the proposed CMT-KD. In addition, for the intermediate features-based distillation ($\\mathcal{L}_{feat}$) in the final loss (eq. (\\ref{eq:final_loss})), the attention loss is used. We consider the following settings. \n\n\\begin{table}[!t]\n\\def1.05{1.0}\n  \\centering\n  \\begin{tabular}{c|c|c|c}\n    \\hline\n    Models & Bit-width & Top 1 & Top 5 \\\\\n\n    \\hline\n    \\multirow{5}{*}{Single model} \n     & FP   & 72.4 & 91.3 \\\\\n     & 8 bits& 70.9 & 90.9 \\\\\n     & 6 bits& 70.8 & 90.8 \\\\\n     & 4 bits& 70.7 & 90.8 \\\\\n     & 2 bits& 69.4 & 90.5 \\\\\n     \\hline\n    KD (from FP teacher) & \\multirow{5}{*}{2 bits} &\n     {71.3} & {91.6} \\\\\n    Average teacher & & 71.0 & 91.6 \\\\\n    \n    CMT-KD (w\/o Att) &  & 71.8 & 91.8 \\\\\n    CMT-KD (w\/o ML) &  & 70.9 & 91.3 \\\\\n    CMT-KD &  & 72.1 & 91.9 \\\\\n     \\hline\n    Combined teacher &4, 6, 8 bits  & 72.3 & 91.7 \\\\\n    \n \n  \\hline\n  \\end{tabular}\n   \\caption{Ablation studies on the CIFAR-100 dataset with AlexNet. The descriptions of settings are presented in Section \\ref{subsec:ablation}.}\n    \\label{tab:ablationAlexNet} \n \n\\end{table}\n\n\\begin{table}[!t]\n\\def1.05{1}\n  \\centering\n  \\begin{tabular}{c|c|c|c}\n  \n   \\hline\n    Models & Bit-width & Top 1 & Top 5 \\\\\n   \n    \\hline\n    \\multirow{5}{*}{Single model} \n    &FP& 75.3 & 93.1 \\\\\n     &8 bits& 75.2 & 92.9 \\\\\n     &6 bits&  74.9 & 92.7 \\\\\n     &4 bits& 74.9 & 92.5 \\\\\n     &2 bits& 72.9 & 91.9 \\\\\n     \\hline\n    KD (from FP teacher) & \\multirow{5}{*}{2 bits} & {75.1} & {92.8} \\\\\n    Average teacher & & 76.0 & 93.8 \\\\\n    \n    CMT-KD (w\/o Att) &  & 76.5 & 93.9 \\\\\n    CMT-KD (w\/o ML) &  & 75.0 & 93.2 \\\\\n    CMT-KD &  & {78.3} & {94.4} \\\\\n    \\hline\n    Combined teacher &4, 6, 8 bits  & 79.5 & 94.9 \\\\\n     \n    \\hline\n  \\end{tabular}\n   \\caption{Ablation studies on the CIFAR-100 dataset with ResNet18. The descriptions of settings are presented in Section \\ref{subsec:ablation}.}\n    \\label{tab:ablationResNet18} \n    \\vspace{-0.5em}\n\\end{table}\n\n\n    \n\\paragraph{\\textbf{Single model.}} We evaluate single models with different precisions (i.e., full precision, 8-bit, 6-bit, 4-bit, and 2-bit precisions) without any distillation methods. The results for AlexNet and ResNet18 architectures are presented in Table \\ref{tab:ablationAlexNet} and Table \\ref{tab:ablationResNet18}, respectively. With the AlexNet architecture, the results show that the  4-bit, 6-bit, 8-bit models achieve comparable results. There are considerable large gaps between these models and the full precision model. There are also large gaps between those models and the 2-bit model. With the ResNet18 architecture, the 8-bit model achieves comparable results to the full precision model. There is a small gap between the 6-bit, 4-bit models and the 8-bit model. Similar to the observation on the AlexNet, there is a large gap between the 2-bit model and the full precision model. \n\\paragraph{\\textbf{Knowledge distillation from the full precision teacher.}} In this setting, we train a 2-bit student model with the knowledge distillation from the full precision teacher, i.e., the KD (from FP teacher) setting in Table \\ref{tab:ablationAlexNet} and Table \\ref{tab:ablationResNet18}. For this setting, we follow \\cite{KD_Hinton}, i.e., when training the student, in addition to the cross-entropy loss, the softmax outputs from the teacher will be distilled to the student. When AlexNet is used, this setting achieves better performance than quantized single models. When ResNet18 is used, this setting achieves comparable results to the 8-bit single quantized model. \n\n\\paragraph{\\textbf{Knowledge distillation from an ensemble of multiple quantized teachers.}}\nIn this setting, we separately train three teachers with different precisions, i.e., 4-bit, 6-bit, and 8-bit teachers. The averaged softmax outputs of teachers are distilled to the 2-bit student. This setting is noted as ``Average teacher\" in Table \\ref{tab:ablationAlexNet} and Table~\\ref{tab:ablationResNet18}. It is worth noting that this setting is also used in the previous work \\cite{DBLP:conf\/kdd\/YouX0T17}.\n When AlexNet is used, at the top-1 accuracy, this setting produces the 2-bit student that achieves comparable performance with the quantized single teachers.\n However, its performance ($71.0\\%$) is still lower than the full precision model ($72.4\\%$). When ResNet18 is used, this setting improves the performance over the full precision model, i.e., the gain is $0.7\\%$ for both top-1 and top-5 accuracy.  \n\n\\paragraph{\\textbf{Effectiveness of collaborative and mutual learning.}} We consider different settings of the proposed framework.  \nFor the results in Table \\ref{tab:ablationAlexNet} and Table \\ref{tab:ablationResNet18}, when training CMT-KD models, we use 3 teachers, i.e., 4-bit, 6-bit, and 8-bit teachers. In those tables, CMT-KD means that the models are trained with the total loss (\\ref{eq:final_loss}). CMT-KD (w\/o Att) means that the models are trained with the loss (\\ref{eq:final_loss}) but the intermediate features-based component $\\mathcal{L}_{feat}$ is excluded. CMT-KD (w\/o ML) means that the  models are trained with the loss (\\ref{eq:final_loss}) but the mutual learning component $(\\mathcal{L}_{KL}^S + \\mathcal{L}_{KL}^T)$ is excluded. The results show that the mutual learning loss is more effective than the intermediate features-based loss. However, both components are necessary to achieve the best results, i.e., CMT-KD. \n\nWhen AlexNet is used, the full precision (FP) model slightly outperforms the proposed CMT-KD at top-1 accuracy.\nWhen ResNet18 is used, CMT-KD outperforms the FP model at both top-1 and top-5 accuracy. A significant gain is achieved, i.e., $3\\%$, at top-1 accuracy. It is worth noting that when ResNet18 is used, the CMT-KD significantly outperforms the 2-bit single model, the 2-bit model when using the average teacher, and the 2-bit model when distilling from the FP model. Those results confirm the effectiveness of the proposed method. \n\n\\paragraph{\\textbf{Combined teacher.}} We also evaluate the performance of the combined teacher in the collaborative learning in our proposed method, i.e., the predictions which are made by the classifier corresponding to the $\\mathcal{L}^T_{CE}$ loss in Figure \\ref{fig:1}. Overall, this setting produces the best results, except for top-1 accuracy with AlexNet architecture. It achieves better  performance than the ``Average teacher'' setting. \nWith ResNet18, this setting significantly outperforms the full precision model. Those results confirm the effectiveness of the proposed collaborative learning between teachers. \n\\begin{table}[]\n\\vspace{-1em}\n\\vspace{0.5em}\n\\def1.05{1.1}\n\\centering\n    \\begin{tabular}{c|ccc}\n    \\hline Setting & & AlexNet & ResNet18 \\\\\n    \\hline \\multirow{2}{*}{ (a) } & Top-1 & {72.1} & {78.3} \\\\\n        & Top-5 & 91.9 & 94.4 \\\\\n    \\hline \\multirow{2}{*}{ (b) } & Top-1 & 71.1 & 78.1 \\\\\n                                  & Top-5 & 91.2 & 94.3 \\\\\n    \\hline\n    \\end{tabular}\n\\vspace{-0.5em}\n\\caption{Impact of the number of teachers on the CMT-KD 2-bit students. The results are on the CIFAR-100 dataset. (a) Using 4-bit, 6-bit, and 8-bit teachers. (b) Using 4-bit and 8-bit  teachers.}\n\\label{tab:numteachers}\n\\end{table}\n\n\\paragraph{\\textbf{Impact of the number of teachers.}} The results in Table \\ref{tab:numteachers} show the impact of the number of teachers on the performance of the 2-bit CMT-KD student models. The results show that using 3 teachers (4-bit, 6-bit, and 8-bit) slightly improves the performance when using 2 teachers (4-bit and 8-bit).\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Comparison with the state of the art}\nIn this section, we compare our proposed method CMT-KD against state-of-the-art network quantization methods, including LQ-Net \\cite{LQNets}, LQW+CAQ \\cite{LQW_CAQ}, HWGQ \\cite{HWGQ}, and DoReFa-Net \\cite{dorefa}. We also make a comparison between our approach and methods that apply both distillation and quantization consisting of PQ+TS+Guided \\cite{guidedQuantize}, QKD \\cite{QKD}, SPEQ \\cite{SPEQ}. \nFor CMT-KD, we use three teachers (4-bit, 6-bit, and 8-bit teachers) to guide the learning of compact quantized 2-bit weights ($K_w = 2$) and 2-bit activations ($K_a = 2$) students. Meanwhile, we use 2-bit, 4-bit, and 8-bit teachers to guide the learning of compact quantized 1-bit weights ($K_w = 1$) and 2-bit activations ($K_a = 2$) students.\nWe do not consider  1-bit activations because the previous \\cite{HWGQ,NIPS2016_d8330f85,xnor_net,dorefa} show that 1-bit quantization for activations is not sufficient for good performance. \n\\setlength{\\tabcolsep}{4pt}\n\\begin{table}[!t]   \n\\def1.05{1.05}\n\\centering\n\\vspace{0.5em}\n\\begin{tabular}{l|c|c|c|c}\n\\hline \n\\multirow{2}{*}{ Method }  & \\multicolumn{2}{|c|}{ AlexNet } & \\multicolumn{2}{c}{ ResNet18 } \\\\\n\\cline{2-5}   & Top-1 & Top-5 & Top-1 & Top-5  \\\\\n\\hline \n\\hline\n\\multicolumn{1}{l|}{Full precision} & 72.4 & 91.3 & 75.3 & 93.1  \\\\\n\\hline \n\\multicolumn{5}{c}{ $K_w=1$, $K_a=2$ } \\\\\n\\hline\nLQ-Nets \\cite{LQNets}  & 68.7 & 90.5 & 70.4 & 91.2 \\\\\nLQW + CAQ \\cite{LQW_CAQ}  & 69.3 & 91.2 & 72.1 & 91.6 \\\\\nHWGQ \\cite{HWGQ}  & 68.6 & 90.8 & 71.0 & 90.8 \\\\\n{CMT-KD-FitNet}   & 69.9 & \\textbf{91.3} & \\textbf{76.1} & \\textbf{93.7} \\\\\n  {CMT-KD-Att}   & \\textbf{70.4} & {91.1} & {75.6} & {93.5} \\\\\n\\hline \n\\multicolumn{5}{c}{ $K_w=2$, $K_a=2$ } \\\\\n\\hline\nPQ+TS+Guided \\cite{guidedQuantize}  & 64.6 & 87.8 & - & - \\\\\nLQ-Net \\cite{LQNets}  & 69.2 & 91.2 & 70.8 & 91.3 \\\\\nLQW + CAQ \\cite{LQW_CAQ}  & 69.9 & {91.3} & 72.1 & 91.6 \\\\\nHWGQ  \\cite{HWGQ}  & 69.4 & 90.5 & 72.9 & 91.9 \\\\\n{CMT-KD-FitNet}   & {70.0} & {90.7} & \\textbf{78.7} & \\textbf{94.6} \\\\\n{CMT-KD-Att} &\\textbf{72.1} & \\textbf{91.9} & 78.3 & 94.4 \\\\\n\\hline\n\\end{tabular}\n\\caption{The comparative results on the CIFAR-100 dataset. We report the results of CMT-KD when FitNet loss or attention loss (Att) is used for the intermediate features-based distillation. CMT-KD uses HWGQ quantizer to quantize teachers and student. The results of HWGQ are reported by using the official released code.}\n\\label{tab:sota_cifar10} \n\\end{table}\n\\paragraph{\\textbf{Comparative results on CIFAR-100.}} Table \\ref{tab:sota_cifar10} presents the top-1 and top-5 classification accuracy on CIFAR-100 dataset of different network quantization methods for AlexNet and ResNet18. {The results of competitors are cited from \\cite{LQW_CAQ,guidedQuantize}}. \nWe report the results of our CMT-KD when FitNet loss or attention loss is used for $\\mathcal{L}_{feat}$ in Eq. (\\ref{eq:final_loss}). Those models are denoted as CMT-KD-FitNet or CMT-KD-Att. The quantizer HWGQ \\cite{HWGQ} is used to quantize teachers and student networks when training CMT-KD models. \nOverall, the best CMT-KD models outperform most of the competitor quantization methods. When AlexNet is used,\nthe CMT-KD (for both FitNet and Att) models outperform the compared quantization methods at top-1 accuracy. However, the proposed models achieve lower performance than the FP model at top-1 accuracy. This may be due to the limit in the capacity of the AlexNet model, which consists of only 5 convolutional layers. \n\nWhen ResNet18 is used, our CMT-KD models outperform the full precision model. Especially when using 2-bit weights and 2-bit activations, the improvements of the CMT-KD-FitNet over the FP model are $3.4\\%$ and $1.5\\%$ for top-1 and top-5 accuracy, respectively.\n{It is also worth noting that the proposed models significantly improve over the HWGQ method~\\cite{HWGQ} which uses HWGQ quantizer to quantize FP models, i.e., with $K_w=2, K_a = 2$, CMT-KD-FitNet outperforms HWGQ \\cite{HWGQ} $5.8\\%$ at top-1 accuracy.} \n\n\\begin{table}[!t]\n\\def1.05{1.05}\n\\centering\n\\vspace{0.5em}\n\\begin{tabular}{l|c|c|c|c}\n\\hline \n\\multirow{2}{*}{ Method }  & \\multicolumn{2}{|c|}{ AlexNet } & \\multicolumn{2}{c}{ ResNet18 } \\\\\n\\cline{2-5}   & Top-1 & Top-5 & Top-1 & Top-5  \\\\\n\\hline \n\\hline\n\\multicolumn{1}{l|}{ Full precision} & 61.8 & 83.5 & 70.3 & 89.5  \\\\\n\\hline \n\\multicolumn{5}{c}{ $K_w=1$, $K_a=2$ } \\\\\n\\hline\nDoReFa-Net \\cite{dorefa} & 49.8 & - & 53.4 & - \\\\\nLQ-Nets \\cite{LQNets}   & 55.7 & 78.8 & \\textbf{62.6} & \\textbf{84.3} \\\\\nHWGQ \\cite{HWGQ}   & 52.7 & 76.3 & 59.6 & 82.2 \\\\\n{CMT-KD (HWGQ)}   &\\textbf{56.2}  &\\textbf{79.1}  & 60.6 & 83.5 \\\\\n\\hline \n\\multicolumn{5}{c}{ $K_w=2$, $K_a=2$ } \\\\\n\\hline \nDoReFa-Net \\cite{dorefa}  & 48.3 & 71.6 & 57.6 & 80.8  \\\\\nQKD \\cite{QKD}   & - & - & 67.4 & 87.5  \\\\\nPQ + TQ Guided \\cite{guidedQuantize}   & 52.5 & 77.3 & - & -  \\\\\nLQ-Net \\cite{LQNets}   & 57.4 & 80.1 & 64.9 & 85.9  \\\\\nSPEQ \\cite{SPEQ}   & 59.3 & - & 67.4 & -  \\\\\nHWGQ \\cite{HWGQ,HWGQ_Rethinking}   & 58.6 & 80.9 & 65.1 & 86.2  \\\\\nCMT-KD (HWGQ)  &59.2  &81.3  & {65.6} & {86.5} \\\\\nLSQ (with distill) \\cite{LSQ}    & -  & -    & \\textbf{67.9} & \\textbf{88.1}  \\\\\nLSQ* (w\/o distill)  & -  & -    & {66.7} & {87.1}  \\\\\nCMT-KD (LSQ)   &\\textbf{59.3}  &\\textbf{81.5}  & {67.8} & {87.8} \\\\\n\\hline\n\\end{tabular}\n\\caption{The comparative results on the ImageNet dataset. CMT-KD uses the attention loss for the intermediate features-based distillation. CMT-KD (HWGQ) and CMT-KD (LSQ) denote models when HWGQ \\cite{HWGQ} and LSQ \\cite{LSQ} quantizers are used in our framework, respectively. We report experimental results for LSQ (please refer to footnote 1) without distillation in LSQ* row.} \n\\label{tab:sota_imagenet} \n\\end{table}\n\\vspace{-0.5em}\n\\paragraph{\\textbf{Comparative results on ImageNet.}}\n\n\n\n\\begin{figure*}[!t]\n     \\centering\n     \\begin{subfigure}[b]{0.37\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{img\/layer11.pdf}\n         \\caption{Layer 5}\n         \\label{fig:layer 1}\n     \\end{subfigure}\n    \n     \\hspace{0.8cm}\n    \\begin{subfigure}[b]{0.37\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{img\/layer22.pdf}\n         \\caption{Layer 9}\n         \\label{fig:layer 2}\n     \\end{subfigure}\n     \n     \\begin{subfigure}[b]{0.37\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{img\/layer33.pdf}\n         \\caption{Layer 13}\n         \\label{fig:layer 3}\n     \\end{subfigure}\n    \n     \\hspace{0.8cm}\n     \\begin{subfigure}[b]{0.37\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{img\/layer44.pdf}\n         \\caption{Layer 17}\n         \\label{fig:layer 4}\n     \\end{subfigure}\n        \\caption{The importance factors of the three ResNet18 teachers (4-bit, 6-bit, 8-bit) on the CIFAR-100 dataset during the training process.}\n        \\label{fig:importancefactor}\n        \\vspace{-1em}\n\\end{figure*}\n\nTable \\ref{tab:sota_imagenet} presents the top-1 and top-5 classification accuracy on ImageNet dataset of different network quantization methods for AlexNet and ResNet18. The results of competitors are cited from the corresponding papers.\nAt $K_w = 1$ and $K_a =2$, when using AlexNet, the proposed CMT-KD  significantly outperforms HWGQ \\cite{HWGQ}. The gain is $4.5\\%$ and $2.8\\%$ for top-1 and top-5 accuracy, respectively. For ResNet18, we also achieved an improvement of $1\\%$ compared to HWGQ at top-1 accuracy. \nWith $K_w = 2$ and $K_a =2$, the CMT-KD (HWGQ)  outperforms the HWGQ \\cite{HWGQ} method by $0.6\\%$, and $0.5\\%$ on top-1 accuracy for AlexNet and ResNet18, respectively. \n\nAs the proposed framework is flexible to quantizers, we also report in Table \\ref{tab:sota_imagenet} the results when LSQ~\\cite{LSQ} quantizer is used to quantize teacher and student networks in our framework, i.e., CMT-KD (LSQ). LSQ is a quantization method in which the step size is learned during training.\n When ResNet18 is used, given the same LSQ quantizer implementation\\footnote{\\label{note1}The official source code of LSQ is not available. We adopt the LSQ quantizer from an un-official implementation from \\url{https:\/\/github.com\/hustzxd\/LSQuantization} for our experiments.}, our method CMT-KD (LSQ) can boost the top-1 accuracy of LSQ* by $1.1\\%$. \nHowever, we note that the results reported in LSQ \\cite{LSQ} slightly outperforms  CMT-KD (LSQ). It is worth noting that in order to achieve the reported results ($67.9\\%$ top-1 and $88.1\\%$ top-5), LSQ \\cite{LSQ} also uses knowledge distillation to distill knowledge from the full precision model to the quantized model\\footnote{\nWe are unsuccessful in reproducing results reported in LSQ \\cite{LSQ}. For example, without distillation for LSQ, we only achieve a result of $66.7\\%$ for top-1 accuracy when using ResNet18 with $K_w=2, K_a=2$, while in \\cite{LSQ}, the authors reported $67.6\\%$ at the same setting.}. \nOur best method CMT-KD (LSQ) compares favorably to the recent method SPEQ~\\cite{SPEQ} for both AlexNet and ResNet18 models.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\paragraph{\\textbf{Visualization of the importance factors ($\\pi$) of teachers.}}\nFigure \\ref{fig:importancefactor} visualizes the importance factors of three teachers (4-bit, 6-bit, 8-bit) when using ResNet18 architecture for the teachers and student. The experiment is conducted on CIFAR-100 dataset. The visualized importance factors are at the last convolutional layers in each block of ResNet18 during training, i.e., layers 5, 9, 13, 17. They are also the layers in which the shared knowledge is formed. \nFigure \\ref{fig:layer 2} shows that the highest precision teacher (8-bit) does not always give the highest contributions to the shared knowledge at all layers. For example, at layer 9, the importance factor of the 6-bit teacher is higher than the importance factor of the 8-bit teacher. At the last convolutional layer (i.e., layer 17), the 8-bit teacher dominates other teachers and gives most of the contributions to the shared knowledge.  In addition, the importance factors are mainly updated at the early training stage of the framework.  \n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\nDeep Convolutional Neural Networks (CNNs) have achieved tremendous successes in a variety of computer vision tasks, including image classification, object detection, segmentation, and image retrieval. However, CNNs generally require excessive memory and expensive computational resources, limiting their usage in many applications. Therefore, a great number of researches have been devoted to {making} CNNs lightweight and to {improving} inference efficiency for practical applications.\n\nAn effective approach is to use low bit-width weights and\/or low bit-width activations. This approach not only can reduce the memory footprint but also achieves a significant gain in speed, as the most computationally expensive convolutions can be done by bitwise operations \\cite{xnor_net}. Although existing quantization-based methods \\cite{HWGQ,lossaware,xnor_net,tbn,LQNets,dorefa,TTQ} achieved improvements, there are still noticeable accuracy gaps between the quantized CNNs and their full precision counterparts, especially in the challenging cases of 1 or 2 bit-width weights and activations. \n\nModel compression using knowledge distillation is another attractive approach to reduce the computational cost of DNNs \\cite{overhaul_fea_distill,KD_Hinton,QKD,apprentice,FitNets}. In knowledge distillation, a smaller student network is trained to mimic the behaviour of a cumbersome teacher network. \nTo further enhance the performance of {the} student network, some works~\\cite{adaptive_multi_teacher,DBLP:conf\/ecai\/ParkK20,DBLP:conf\/kdd\/YouX0T17} propose to distill knowledge from multiple teachers. However, in those works, teacher models are separately pretrained, which would limit the collaborative learning between teachers. It also limits the mutual learning between student network and teacher networks.  \n\nTo improve the compact low bit-width student network, in this paper, we propose a novel framework -- Collaborative Multi-Teacher Knowledge Distillation (CMT-KD), which encourages the {collaborative learning} between teachers and the mutual learning between teachers and student. \n\nIn the collaborative learning process, knowledge at corresponding layers from teachers will be combined to form an importance-aware shared knowledge which will subsequently be used as input for the next layers of teachers.\nThe collaborative learning between teachers is expected to form a valuable shared knowledge to be distilled to the corresponding layers in the student network. To our best knowledge, this paper is the first one that proposes this kind of collaborative learning for knowledge distillation.\n\nIt is worth noting that our novel framework design allows end-to-end training, in which not only the teachers and student networks but also\nthe contributions (i.e., importance factors) of teachers to the shared knowledge are also learnable during the learning process. It is also worth noting that the proposed framework is flexible -- different quantization functions and different knowledge distillation methods can be used in our framework.\n\nTo evaluate the effectiveness of the proposed framework, we conduct experiments on CIFAR-100 and ImageNet datasets with AlexNet and ResNet18 architectures. The results show that the compact student models trained with our framework achieve competitive results compared to previous works.\n\n\n\n\n\n\n\n\n\n\n\\section{Proposed method}\n\\label{sec:proposed}\n\n\\subsection{The proposed framework}\n\\begin{figure*}[!h]\n\\centering\n  \\includegraphics[scale=1.8, width=0.9\\linewidth]{img\/KD_quantization-WACV2023_final.pdf}\n \n    \\caption{The framework of our proposed collaborative multi-teacher knowledge distillation for low bit-width DNNs. \n   \n    The  collaborative  learning among a set of quantized teachers forms useful shared knowledge at certain layers through importance parameters $(\\pi)$. \n    $\\mathcal{L}^T_{KL}$ and $\\mathcal{L}^S_{KL}$ are for mutual learning between teachers and student\n    via ensemble logits $\\overline{\\mathbf{z}}$, where $\\overline{\\mathbf{z}}$ is calculated from teachers logits $\\mathbf{z}^T$ and student logits $\\mathbf{z}^S$. \n    CE means Cross Entropy, and KL means Kullback-Leibler divergence. $\\mathcal{D}(.)$ denotes the loss for intermediate features-based distillation that could be attention loss or hint loss (FitNet). }\n   \n\\label{fig:1}      \n\\vspace{-0.5em}\n\\end{figure*}\n\nWe propose a novel framework -- Collaborative Multi-Teacher Knowledge Distillation (CMT-KD) as illustrated in Figure~\\ref{fig:1}, which encourages collaborative learning between teachers and mutual learning between teachers and student. \n\nFirst, we propose a novel collaborative learning of multiple teachers. During the training process, the  collaborative  learning among a set of quantized teachers forms useful importance-aware shared knowledge at certain layers,\nwhich is distilled  to  the  corresponding  layers in the student network.\nSecond, the learning process between the student and the teachers is performed in a mutual-learning manner \\cite{DML} via ensemble logits $\\overline{\\mathbf{z}}$ from teachers logits $\\mathbf{z}^T$ and student logits $\\mathbf{z}^S$. \n\n\nFurthermore, our framework design allows end-to-end training, in which not only the teachers and student networks but also the contributions (i.e., importance factors) of teachers to the shared knowledge are also learnable during the learning process. It is also worth noting that the proposed framework is flexible -- different quantization functions \\cite{HWGQ,HWGQ_Rethinking,LSQ,LQW_CAQ,QIL,LQNets} and different knowledge distillation methods \\cite{mimic,ABdistill,overhaul_fea_distill,FitNets,attention_transfer} can be used in our framework.\n\n\n\n\\subsection{Collaborative learning of multiple teachers}\nTeacher model selection is important in knowledge distillation. In~\\cite{QKD}, the authors show that if there is a large gap in the capacity between teacher and student, the knowledge from teacher may not be well transferred to student. To control the power of teacher, in our work, we consider teacher and student models that have the same architecture. However, teachers are quantized with higher different bit-widths. \nThe knowledge at corresponding layers from teachers will be fused to form a  shared knowledge which will subsequently be used as input for the next layers of teachers. This forms collaborative learning between teachers. It is expected that different teachers have different capacities, and therefore, they should have different contributions to the shared knowledge. To this end, for each teacher, an importance factor that controls how much knowledge the teacher will contribute to the shared knowledge will also be learned. It is worth noting that the learning of importance factors will encourage the collaborative learning between teachers to produce a suitable shared knowledge that the student can effectively mimic.\n\n\nFormally, given a quantization function $Q(x, b)$, \nthe $i^{th}$ teacher is quantized using $Q(W_i, b_i)$ and $Q(A_i, b_i)$, in which $W_i$ and $A_i$ represent for weights and activations, respectively; $b_i$ is the bit-width.  \nThe shared knowledge $F_k$ of corresponding layers of $n$ teachers at $k^{th}$ layer index is formulated as follows\n\n\\begin{equation}\n\\begin{gathered}\nF_k = \\sum_{i=1}^{n}\\pi^{k}_i * Q(A_i^k, b_i) \\\\\n     \\text { s.t. } \\sum_i \\pi^{k}_i =1, \\pi^{k}_i \\in[0,1] \\text {,}\n\\end{gathered}\n\\label{eq:Fk}\n\\end{equation}\nwhere\n${\\pi^{k}_i}$ represents the importance of teacher $i^{th}$. To handle the constraint over $\\pi$ in (\\ref{eq:Fk}), in the implementation, a softmax function is applied to $\\pi^{k}_i$ values before they are used to compute $F_k$.\nThe importance parameters $\\pi$ and the model weights of teachers and student are optimized simultaneously via end-to-end training.\n\\subsection{Other components}\n\n\\subsubsection{Quantization function}\nQuantization function maps a value $x \\in R$ to a quantized value $\\overline{x} \\in \\{q_1, q_2, ..., q_n\\}$ using a quantization function Q with a precision bit-width $b$. The quantized value is defined as\n\\begin{equation}\n    \\overline{x}=Q(x, b).\n\\end{equation}\n\n\n\nDifferent quantization methods have been proposed \\cite{HWGQ,LSQ,QIL,xnor_net,LQNets,dorefa}. In this paper, we consider the half-wave Gaussian quantization (HWGQ) \\cite{HWGQ} as a quantizer to be used in our framework, which is an effective and simple uniform quantization approach. \nTo quantize weights and activations, they first pre-compute the optimal value $q_i$ by using uniform quantization {for unit Gaussian distribution}. \n{Depending on variance $\\sigma$ of  weights and activations}, the quantized value of $x$ is expressed as\n\\begin{equation}\n    \\overline{x} = \\sigma*q_i.\n\\end{equation}\n\n\n\n\\subsubsection{Intermediate features-based distillation}\n\nThe shared knowledge from teachers will be used to guide the learning of student. \nLet $F^T_k$ and $F^S_k$ be the shared feature map of teachers and the feature map of the student at $k^{th}$ layers of models, respectively. Let $\\mathcal{I}$ be the selected layer indices for intermediate features-based distillation.\nThe intermediate features-based knowledge distillation loss is defined as follows\n\n\n\\begin{equation}\n    \\mathcal{L}_{feat} = \\sum_{k \\in \\mathcal{I}} \\mathcal{D} \\left(F_{k}^{T}, F_{k}^{S}\\right),\n    \\label{eq:fea_loss}\n\\end{equation}\nwhere $\\mathcal{D}$ is the distance loss measuring the similarity between features $F^T_k$ and $F^S_k$.\nDifferent forms of $\\mathcal{D}$ can be applied in our framework. In this work, we consider two widely used distance losses, i.e., the attention loss~\\cite{attention_transfer} and the FitNet loss\\cite{FitNets}. \n\n\n\n\nThe attention loss~\\cite{attention_transfer} is defined as follows\n\\begin{equation}\n    \\mathcal{D}_{A T}=\\sum_{k \\in \\mathcal{I}}\\left\\|\\frac{Q_{k}^{T}}{\\left\\|Q_{k}^{T}\\right\\|_{2}}-\\frac{Q_{k}^{S}}{\\left\\|Q_{k}^{S}\\right\\|_{2}}\\right\\|_{p},\n\\end{equation}\nwhere $Q_k^S$ and $Q_k^S$ are the attention maps of features $F_k^S$ and $F_k^S$, respectively. $\\|.\\|_p$ is the $l_p$ norm function that could be a $l_1$ or $l_2$ normalization. The attention map for a feature map $F \\in R^{c \\times h \\times w}$ that has $c$ channels is defined as $Q=\\sum_{j=1}^{c}|A_j|^p$, where  $A_j = F(j, :,:)$ is the $j^{th}$ channel of $F$; $|.|$ is the element-wise absolute function. In our implementation we use $p=2$. \nThe FitNet loss \\cite{FitNets} is defined as follows\n\\begin{equation}\n    \\mathcal{D}_{HT}=\\sum_{k \\in \\mathcal{I}}\\left\\|F_{k}^{T} - r(F_{k}^{S})\\right\\|_{p},\n\\end{equation}\nwhere $r(.)$ is a convolutional layer that adapts the student feature map $F_k^{S}$ before comparing it to the shared knowledge teacher feature map $F_k^{T}$. In our method, we follow existing works \\cite{overhaul_fea_distill,FitNets} in which $r(.)$ a convolutional layer with a kernel size of $1\\times1$.\n\\subsubsection{Mutual learning between teachers and student}\nIn addition to the intermediate features-based distillation, we also leverage the mutual learning \\cite{DML} defined on the logits from networks for learning. The mutual learning allows student to give feedback about its learning to teachers. This learning mechanism encourages both teachers and student to simultaneously adjust their parameters to achieve an overall learning objective, e.g., minimizing a total loss function.\n\nThe mutual learning \\cite{DML} applies $KL$ losses on the softmax outputs of networks.\nHowever, due to the diversity of the output logits from different networks, this method may hurt the performance of models. \nTo overcome this issue, we adopt KDCL-MinLogit \\cite{KDCL}, which  is a simple and effective method to ensemble logits from teachers and student. In particular, this method selects the minimum logit value of each category. \n\nLet $\\mathbf{z}^T$ and $\\mathbf{z}^S$ be the logit outputs of the combined\nteacher model $T$ and a student model $S$, $z^{T,c}$ and $z^{S,c}$ be the elements of $\\mathbf{z}^T$ and $\\mathbf{z}^S$ corresponding to the target class $c$, $\\mathbf{1}$ be the vector with all $1s$ elements, we denote $\\mathbf{z}^{T, c} = \\mathbf{z}^T - z^{T,c}\\mathbf{1}$ and $\\mathbf{z}^{S,c} = \\mathbf{z}^S - z^{S,c}\\mathbf{1}$. \nThe element $\\overline{z_i}$ of the ensemble logits $\\overline{\\mathbf{z}}$ is computed as follows\n\\begin{equation}\n    \\overline{z_i} = min\\{z_i^{T,c},  z_i^{S,c}\\}, i=1, 2, ..., m\n   \n\\end{equation}\nwhere  $z_i^{T,c}$, $z_i^{S,c}$ are the $i^{th}$ elements of $\\mathbf{z}^{T, c}$ and $\\mathbf{z}^{S, c}$, and $m$ is the number of classes.\n\n\nThe mutual learning is defined as follows\n\n\\begin{equation}\n    \\mathcal{L}_{KL}^{S}= \\mathcal{T}^{2} \\times KL(\\overline{\\mathbf{p}}|| \\mathbf{p}^{S}),\n\\end{equation}\n\\begin{equation}\n    \\mathcal{L}_{KL}^{T}= \\mathcal{T}^{2} \\times KL(\\overline{\\mathbf{p}}|| \\mathbf{p}^{T}),\n\\end{equation}\nwhere $KL$ means the Kullback-Leibler  divergence and $\\mathcal{T}$ is the temperature parameter. $\\overline{\\mathbf{p}}$, $\\mathbf{p}^{S}$, and $\\mathbf{p}^{T}$ are the soft logits of $\\overline{\\mathbf{z}}$, $\\mathbf{z}^{S}$, and $\\mathbf{z}^{T}$, respectively. The soft logit $\\mathbf{p}$ is defined as $\\mathbf{p}=softmax(\\frac{\\mathbf{z}}{\\mathcal{T}})$.\n\n\n\n\n\n\nFinally, the overall loss of our proposed collaborative and mutual learning in classification task is defined as\n\n\\begin{equation}\n\\small\n\\label{eq:final_loss}\n    \\begin{aligned}\n    \\mathcal{L} = \\alpha \\times (\\mathcal{L}_{CE}^{S}+ \\mathcal{L}_{CE}^{T}) \n    + \\beta \\times (\\mathcal{L}_{KL}^S + \\mathcal{L}_{KL}^T) + \\gamma \\times \\mathcal{L}_{feat},\n\\end{aligned}\n\\end{equation}\nwhere $\\alpha$, $\\beta$, and $\\gamma$ are the hyper-parameters of total loss for optimization and $\\mathcal{L}_{CE}$ is the standard cross-entropy loss calculated using the corresponding soft logits and the ground-truth labels. During the training process,\nboth the model weights of teachers, student and the importance factors of teachers, i.e., $\\pi_i^k ~\\forall i,k$, will be updated by minimizing $\\mathcal{L}$, using gradient descent. \n\n\\section{Related work}\n\\label{sec:related_work}\nOur work is closely related to two main research topics in the literature: network quantization and knowledge distillation (KD).\n\n\\textbf{Network quantization.}\nEarlier works in network quantization have applied the basic form of weight quantization to directly constrain the weight values into the binary\/ternary space without or with a scaling factor, i.e., $\\{-1, 1\\}$  \\cite{BinaryConnect}, $\\{-\\alpha, \\alpha\\}$  \\cite{xnor_net}, or $\\{-\\alpha, 0, \\alpha\\}$ \\cite{TWN}. \nSince quantization of activations can substantially reduce complexity further \\cite{xnor_net,BinaryNet,LQNets,dorefa}, this research topic attracts more and more attention \\cite{xnor_net,LQNets,dorefa}. \nIn~\\cite{xnor_net,BinaryNet}, the authors propose to binarize both weights and activations to $\\{-1,1\\}$. However, there are considerable accuracy drops compared to full precision networks. \n{To address this problem, the generalized low bit-width quantization \\cite{log_data_represent,dorefa} is\nstudied.\n}\n{In half-wave Gaussian quantization (HWGQ) \\cite{HWGQ}, the authors propose a practical and simple uniform quantization method that exploits the statistics of network activations and batch normalization.}\nIn LQ-Nets \\cite{LQNets}, the authors propose to  train a quantized CNN and its associated non-uniform quantizers jointly.\nThe approach in Learned Step Size Quantization (LSQ) \\cite{LSQ} learns uniform quantizers using trainable interval values. In quantization-interval-learning (QIL) \\cite{QIL}, the authors introduce a trainable quantizer that additionally performs both pruning and clipping.\n\n\\textbf{Knowledge Distillation (KD)} is a common method in training smaller networks by distilling knowledge from a large teacher model \\cite{KD_Hinton}. The rationale behind this is to use extra supervision in the forms of classification probabilities \\cite{KD_Hinton}, intermediate feature representations~\\cite{mimic,ABdistill,overhaul_fea_distill,FitNets}, attention maps \\cite{attention_transfer}.\nKnowledge distillation approaches transfer the knowledge of a teacher network to a student network in two different settings: offline and online. In the offline learning, KD uses a fixed pre-trained teacher network to transfer the knowledge to the student network. Deep mutual learning \\cite{DML} mitigates this limitation by conducting online distillation in one-phase training between two peer student models.\n\n\\textbf{Multi-Teacher Knowledge Distillation.}\nThe approach in \\cite{MultiLang_NMT} applies multi-teacher learning into multi-task learning where each teacher corresponds to a task. Similarly, the approach in \\cite{UHC} trains a classifier in each source and unifies their classifications on an integrated label space.\nThe approach in \\cite{DBLP:conf\/kdd\/YouX0T17} considers knowledge from multiple teachers equally by averaging the soft targets from different pretrained teacher networks. In\n\\cite{adaptive_multi_teacher}, the authors propose to learn a weighted combination of pretrained teacher representations. \n{Different from \\cite{DBLP:conf\/kdd\/YouX0T17,adaptive_multi_teacher}, in this work, we propose a novel online distillation method that captures importance-aware knowledge from different teachers\nbefore distilling the captured knowledge to the student network.} {\nIn \\cite{DBLP:conf\/ecai\/ParkK20}, the last feature map of student network is fed through different non-linear layers; each non-linear layer is for each teacher. The student network and the non-linear transformations are trained such that the output of those non-linear transformations mimic the last feature maps of the corresponding teacher networks.}\nThe previous works  \\cite{adaptive_multi_teacher,DBLP:conf\/ecai\/ParkK20, DBLP:conf\/kdd\/YouX0T17} mainly learn full precision student models from a set of full precision pretrained teacher models, while we focus on learning quantized models. Specifically, we aim to learn a quantized student model with guidance from a set of quantized teacher models with different precisions. \nIn addition, different from previous works \\cite{DBLP:conf\/kdd\/YouX0T17,adaptive_multi_teacher,DBLP:conf\/ecai\/ParkK20} in which teachers are fixed when training student, our method simultaneously trains student and teachers using the collaborative and mutual learning. \n\n\n\n\\textbf{Quantization + Knowledge distillation}. Some works have tried to adopt knowledge distillation methods to assist the training process of low precision networks \\cite{SPEQ,QKD,apprentice,distill_quant,guidedQuantize}. \nIn Apprentice (AP) \\cite{apprentice}, the teacher and student networks are initialized with the corresponding pre-trained full precision networks. After lowering the precision of the student network, the student is then fine-tuned using distillation.\nDue to AP's initialization of the student, AP might get stuck in a local minimum in the case of very low bit-width quantized student networks. \nBecause of the inherent differences between the feature distributions of the full-precision teacher and low-precision student network, using a fixed teacher as in \\cite{apprentice} can limit the knowledge transfer. QKD \\cite{QKD} and Guided-Quantization \\cite{guidedQuantize} mitigate the issue of AP by jointly training the teacher and student models, which makes a teacher adaptable to the student model.\nIn our work, to further mitigate the problem of different {feature distributions} between teacher and student models, instead of using a full-precision teacher model, we propose to use a set of quantized teacher models. Using quantized models would help the teachers obtain more suitable knowledge for a quantized student model to mimic.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztkeo b/data_all_eng_slimpj/shuffled/split2/finalzztkeo
new file mode 100644
index 0000000000000000000000000000000000000000..8d2076288c00e7e54dd681ad6b1a9d9b9e9cbf90
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzztkeo
@@ -0,0 +1,5 @@
+{"text":"Welcome to Popular Hannah Montana Makeover! Welcome to a different continuation of this nice unblock game! Your job on this hacked game is that you just make-up Hannah Montana. We begin from the truth that it's essential to first clear her face first after which begin to apply make-up. You need to use the make-up in the way in which you need, so be inventive and use your creativeness. Right here is the large choice of make-up, fruits and herbs you could used as make-up and different issues. In entrance of you is a large choice of clothes, sneakers, clothes, denims, shirts, socks, gloves, jewellery and extra. Get pleasure from enjoying this incredible hacked game. Ladies will love this unblocked game. Graphics, animation and catchy music within the background will delight you. The hacked game is designed for all age, particularly for ladies and followers of Hannah Montana. We want you nice enjoyable taking part in this unblock game, and as all the time, be inventive and share your expertise with make-up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Running down the field during a Miami punt, McMillan collided with a teammate, bending his knee at an awkward angle. The loss of the promising rookie is the latest blow for the Dolphins in an injury-filled training camp. This offseason, they brought in Lawrence Timmons from the Pittsburgh Steelers, one of the more underrated moves of the offseason. Alonso led the Dolphins with 69 tackles this past season.\nLiverpool are battling a handful of injuries ahead of their opening league game, with Klopp confirming that striker Daniel Sturridge will also be absent through injury . The player has previously hit out at the local media following suggestions that he was holding out for more lucrative terms. Watford's clash against Liverpool kicks off at 12.30pm on Saturday August 12.\nThe 76ers will play a regular-season game in London for the first time. The 76ers will be playing their second game in the United Kingdom and first game in London, having previously played a preseason game in Manchester against the Oklahoma City Thunder in 2013.\nPredict the outcome of those matches and the rest of the action by voting in our match polls. Bad boy: With Costa on his way out there is certainly a vacancy here , but there aren't too many other obvious hard cases in the squad. Alejandro Moreno responds to Jose Mourinho's belief that Man United will be able to compete with Europe's best in two years. Nevertheless, the 24-year-old has a plan to ensure the haters are silenced and as soon as possible.\nThe former world No. 1 Angelique Kerber got past Donna Vekic in two tight sets setting a clash with Sloane Stephens next. \"I think my game today was much better, especially my serve\", said Pliskova, who clubbed 13 aces. Stephens stepped up her game against Kerber after a couple of long three set wins against Yulia Putintseva and Petra Kvitova in the first two rounds.\nThe Herald reports that Major League Baseball is supposed to receive the formal agreement today. In all, there are about 16 investors in their group, including National Basketball Association legend Michael Jordan . According to the New York Post , Jordan is kicking in very little cash, but Jeter sees MJ as a role model and a mentor in how to become a successful sports executive.\nThe Perth-based franchise finished second in the Australian conference during the Super Rugby season, and had nine players selected in an extended Wallabies squad last month. \"The people of Western Australia have campaigned fairly, passionately and with integrity in order to show that the Western Force deserve to have a place in Super Rugby \".\nSince leaving Durham earlier this year, the 30-year-old has amassed 828 runs at an average of 59.14 in the championship and is well on course to pass 1,000 runs for the fifth year running. England will host the West Indies for three Tests followed by a one-off T20 and a four-match one-day series. \"He would have known that and been disappointed\", Stokes, his teammate at Durham, told Sky Sports.\nBarca fans had been given hope of keeping hold of Neymar when Gerard Pique posted a picture of himself with the striker on social media, captions \"se queda\" [he's staying]. FSG share Klopp's sentiments and it's now on public record. A member of Coutinho's family has told Sky Sports News that the player feels aggrieved at the way he has been treated by manager Klopp.\nI don't think such comments deserve a reaction from me. Team India has taken an unassailable 2-0 lead in the Test series and will play the final Test in Kandy commencing on Saturday. It is possible that on the morning of the the Test, captain Virat Kohli will have last seen the pitch about 72 hours ago. Two days before the match, it was hard to differentiate the 22-yards from the lush green outfield of the Pallekele stadium.\nCharles, 51, has been charged with aggravated manslaughter of a child in the Monday death of Myles Hill outside at a center in Orlando , Florida , jail records show. \"While on the phone, the suspect opened the driver's side sliding door and grabbed some personal property from the floor behind the center console and locked the van without inspecting the interior\", the affidavit noted.\nFosu-Mensah came through United's ranks as a midfielder but was selected by Louis van Gaal first as a centre-back and then as a right-back. Crystal Palace boss Frank de Boer admits that he does not know whether the club will be able to bring Liverpool defender Mamadou Sakho back to Selhurst Park this summer.\nDallas Cowboys running back Ezekiel Elliott has been suspended for six games after a yearlong National Football League investigation of his domestic violence case in Ohio. USA TODAY Sports does not identify alleged victims of domestic violence. Elliott's case has been under review for almost 13 months. Should the Cowboys have to go without Elliott to start the season, here's who a look at will be called upon to pick up the slack.\nSince the Eagles drafted him in the second round of the 2014 draft out of Vanderbilt, Matthews has piled up 225 catches for 2,673 yards and 19 touchdowns. The Eagles entered the offseason incredibly thin at corner. When Darby's name came up from the Bills, it piqued the Eagles' interest. Buffalo essentially traded Gaines for Darby in these two trades, as well.\nThe moves were made day after Buffalo opened its preseason with a 17-10 loss to Minnesota. The first three offensive plays for the Bills were all throws to Watkins, who caught all three. Matthews, like Watkins, is eligible for unrestricted free agency after this year. Watkins' absence will give also potentially allow rookie Zay Jones to step up and claim the first spot on the team's depth chart at receiver far earlier than anyone expected him to.\nMoeen, who may end up batting at six behind Stokes, finished as the leading wicket taker in the series with 25 and credited work done with spin coach Saqlain Mushtaq for his improvement. While they will look forward to returning home, England , meanwhile, turn their attentions to three Tests versus West Indies ahead of an Ashes trip Down Under.\nThe downward movement in the rankings came as India did not play a single global match in last one month. The current frontrunners in FIFA's top-ten rankings are Brazil followed by Germany , Argentina, Switzerland, Poland, Portugal, Chile, Colombia, Belgium and France.\nArsene Wenger has confirmed that a number of key Arsenal players missed the club's Community Shield win over Chelsea due to being \"short of preparation\". In a hyper-inflated transfer market the German-born Kolasinac, who is 24 and has 18 caps for Bosnia & Herzegovina since being naturalised in 2014, certainly appears to be a bargain.\nThe South African National Road Agency is investigating the cause of the bridge collapse. One person was airlifted and two were transported by ambulance. Sanral's General Manager of Communications Vusi Mona said yesterday: \"We can not speculate as per the cause of the bridge collapse, our engineers have been deployed on site to determine the cause\".\nThe Mets were expected to trade Bruce at some point before the waiver trade deadline on August 31, so it was not shocking when they sent him to the Indians on Wednesday. Bruce also got interest from the Yankees , according to a source , with the team offering multiple prospects, but with only partial salary compensation.\nIn came Francis to take a remarkable gold medal, earning a personal best 50.08 while Nasser broke another National Record in a time of 50.06 to take the silver with Felix the bronze in 50.08. Makwala, the top-ranked 200m runner this year, qualified safely after being medically cleared to re-run following a stomach virus. Being world champion sounds pretty cool.\nDembele, who has made no secret of his desire to play for Barcelona , failed to show up for practice on Thursday. 'We don't know where he is. Dortmund boss Peter Bosz revealed Dembele missed training on Thursday and that the club could not get in contact him.\nBut the American was satisfied with his start, saying he had regained confidence with his driver. And the bottom line is you've got to make putts. \"I feel like I played pretty solid\", he said. Nicklaus, meanwhile, described the changes as \"a good thing for the game of golf\" in a lengthy Instagram post, with it also offering the chance to help boost the quality of field at his Memorial Tournament, which will likely be held in one of the middle weekends between the US PGA and US Open .\nHowever, the Spain worldwide could well have been playing FOR United had the club not turned down the chance to sign him for the most ridiculous reason in football history. After last night's game in Macedonia, Manchester United manager Jose Mourinho showered praise on Isco . And the supposed size of his head didn't seem to stop him starring for Spain's triumphant Under-21s side at that summer's European Championship.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Costa Rica's varies from place to place, from micro-climates to macro-climates, and from the rainy season to the dry season. Essentially, weather in Costa Rica is dependent on where you are and what time of the year it is.\nEven so, Costa Rican weather can offer unexpected surprises. The temperature in Costa Rica is closely related to the elevation. The low lying Pacific Coast is generally very hot and dry; however, during the rainy season, it is common to have rain in the afternoon, and occasionally a cloudy, humid day.\nIn contrast, the Caribbean climate zone, which includes Tortuguero, Arenal, Puerto Viejo and Sarapiqui, is warm and humid nearly year-round and has less rigid rainy and dry seasons.\nThe Central Highlands, which border Costa Rica's Central Valley, is characterized by cooler temperatures and a variety of elevations (from about 3,000-5,000 ft above sea level). For more information about regional weather in Costa Rica, check out our page on weather.\nWhat Is The landscape of Costa Rica Like?\nIs It Easy To Get a Sunburn In Costa Rica?\nHow Big Is Costa Rica?\nWhat Time Is It In Costa Rica?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am working with businesses on the Gold Coast and throughout Australia to create eye-ctaching and unique logos for their new and existing businesses.\nYour logo is the most crucial part of establishing a business as it is the thing that will define your company at a glance, conveying what you're about, and creating an impression that will stick out in the minds of your customers.\nEvery other part of your marketing will be based on this initial design and you can't afford to do it on the cheap or on the fly. You simply cannot afford to skip this step, or leave it to amateurs. I have over 18 years experience designing logos for clients, and I will work with you to create a design you are 100% in love with.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ANY Location Skeleton w\/ Wood Box Magician has the spectator choose one card from the deck and remember it, then has them put the card back to any place in the deck. Then magicians spreads the cards, and bring out a skull hanging from a string from his pocket, putting the skull on top of each card. Suddenly the skull's eyeballs are start flashing, with mysterious laughing! Like it's found the card! So the magician turn over the first card and that is the card, and it is indeed the card selected by the spectator. This is just one effect you can do with the any location skeleton!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzupjm b/data_all_eng_slimpj/shuffled/split2/finalzzupjm
new file mode 100644
index 0000000000000000000000000000000000000000..10fd3a65ba874692645c77e930ebb197cbf482ce
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzupjm
@@ -0,0 +1,5 @@
+{"text":"The Chalet has great views of the Kamnik Alps and is surrounded by beautiful Alpine countryside. There is a choice of excellent ski resorts nearby.\nThe Chalet is just 25 minutes drive from Ljubljana (Brnik) airport and is located in a hillside hamlet 620m above the Medieval town of Kamnik. It has great views of the limestone peaks of the Kamniske Alps and is surrounded by beautiful Alpine countryside. The traditional chalet style accommodation comprises a large 1st floor bedroom with 4 single beds and a balcony. Bedroom 2 with double bed and en-suite facilities including shower is downstairs. The open plan living room has a woodburner for those cosy evenings in. The galley style kitchen is small but well equipped. Bathroom 1, with shower is also on this floor. The basement has laundry facilities and storage for skis, etc. Outside is a large terrace with seating for alfresco dining. There is a garden on all sides with rope swing, barbecue and outdoor games. With compact but comfortable accommodation, the chalet is ideal as a tranquil all year round retreat to get away from it all, for a romantic break, Skiing holiday, or a central base within reach of all Slovenias main attractions.\nWith the Kamniske alps as a stunning backdrop, the chalet is in a tranquil hilltop location above the delightful medieval town of Kamnik.\nThere is a large garden and outdoor space to enjoy the beautiful alpine setting of the chalet and there are several very enjoyable walks starting directly from the chalet taking in forests, alpine meadows, the quaint local villages, spectacular views of the mountains and overlooking the whole of the Ljubljana basin.\nThe local town of Kamnik has an excellent choice of restaurants, caf\u00e9 bars and shops.\nThis atmospheric little town, with its baroque architecture, narrow streets full of character , castles and churches is popular choice for visitors, and the tourist information centre is located in the centre of town along with numerous tempting pavement cafes and bars to enjoy coffee, ice cream or leisurely lunch.. Road and mountain biking are also very popular with recommended routes available, paragliding, fishing and horse riding are available nearby.\nVisit the renowned Arboretum or the superb 18 hole Golf course with 4 practice greens. Close by in the Tuhinj valley is the Terme Snovik aqua park with indoor and outdoor pools, the sparkling mineral water is said to have healing and therapeutic qualities and requires no purification or treatments. Another local must-see is the high mountain pasture plateau of Velika Planina, 1600m above sea level are the best preserved herdsmen settlements in Europe, set in forests and alpine meadows, it is reached by a thrilling cable car ride. In winter it becomes a popular skiing and sledging destination. Visit the tourist information centre to get the best out of your stay.\nThe Chalet is within easy reach of Lakes Bled and Bohinj, Triglav National Park, The Capital Ljubliana, world acclaimed Postojna Caves, cable car trips to the mountain pastures of Velika Planina and a choice of ski resorts close by for winter skiing & sledging.\nIt is possible to fly to Slovenia via the following low cost airlines, flights can be obtained at a very reasonable price if booked well in advance.\nEasyjet: from Stansted to Brnik, Ljubljana (Slovenia).\nRyanair: from Stansted to Klagenfurt (Austria), Trieste (Italy).\nAdria Airways: from Manchester, Birmingham and Gatwick to Brnik, Ljubljana (Slovenia) .\nAll major car hire firms are located at the above airports.\nThe journey by road is about 1,000 miles from the channel ports. The cheapest and quickest way is via Harwich-Hoek, then via road through Holland, Germany, Austria and Slovenia. Tolls are payable are in Austria and a vignette car tax sticker is needed for travelling the motorways of Slovenia.\nBy rail to Ljubljana go via Eurostar to Paris, Paris to Venice (sleeper overnight) and Venice to Ljubljana.\nSatellite TV, stereo system with ipod\/mp3 connections, DVD with large selection of movies, board games and woodburning stove for those cosy evenings in. The stairs leading to the upstairs bedroom are typical chalet style open tread with a hatch door , the stairs leading downstairs are fairly steep and are not suitable for the infirm or unsupervised young children. A travel cot is available upon request.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I AM Divine; I am FREE and UNLIMITED!\nThe lyrics to the hymn \"I Am Free; I Am Unlimited\" are regularly sung in spiritual centers of many faiths and denominations. They resonate on a universal level because they express an important Truth in simple words\u2014a proclamation of spiritual freedom.\nMy spiritual freedom is based on my Divine Nature. I am unlimited because Divine Principles \/ Divine Ideas can be used in an unlimited number of ways to produce an unlimited number of results.\nI may encounter daily challenges with my work, relationships, finances, or any number of other elements of this human experience. From my adverse ego, I may believe I am trapped and forced into painful choices shaped by the demands of others. My supportive ego remembers I AM Infinite Potential; I am always free. At such times I hear my true Self gently singing me the Truth: I AM free; I AM unlimited. There are no ties that bind me!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Upgrade your general or guild level membership by $25 to a \"plus one membership\" and bring an additional guest any time you visit the Garden.\nMemberships are valid for twelve months from the month of purchase. Non-refundable and non-transferable privileges are extended to the member only. Memberships are tax deductible within the limits of the law; please consult your tax preparer for details. Rates and benefits may be subject to change without notice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Set one of the concrete blocks in position. Place one leg of a fence panel into one of the holes in the concrete block. Position the panel along the line you want the fence to travel. Then position a second concrete block for the other end of the panel.\nConnect the fence panels using a bracket up the top of the vertical struts. The bracket is locked into place using a nut and bolt. Tighten the nut firmly using a spanner or a torque wrench. To make everything as safe as possible, keep the nut and thread side facing inwards.\nSafety struts anchor your fence in place and keep it from toppling over. Once your fence is up, connect the struts to the back of each panel using brackets. Connect one bracket down near the bottom of the vertical strut, and the other somewhere near the middle. To increase the stability, weigh the safety strut down with concrete blocks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Had a great weekend in Toowoomba watching my first MTB Enduro race with Giant AMC and an assortment of hitters. Great fun and a proper MTB educational immersion experience. When we weren't at the trails we were talking bikes or watching world cup races on youtube. It's fair to say I was lost for most of the conversations. Good times. Few pics below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvqat b/data_all_eng_slimpj/shuffled/split2/finalzzvqat
new file mode 100644
index 0000000000000000000000000000000000000000..c737422666480f0c8faecc272f3bacf778e740e3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzvqat
@@ -0,0 +1,5 @@
+{"text":"Twelve-year-old Winnie Willis has a way with horses.\nAlong with her dad and sister, Lizzy, Winnie is learning how to live without her mom--who was also a natural horse gentler. As Winnie teaches her horses about unconditional love and blind trust, God shows Winnie that he can be trusted as well. Readers will be hooked on the series' vivid characters, whose quirky personalities fill Winnie's life with friendship and adventure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An innovator for more than 100 years, L'Or\u00e9al launches multiple brands with creative commerce solutions, customer care, and digital marketing services. L'Or\u00e9al continues to be a pioneer \u2013 now in the digital beauty industry, launching an array of beauty brands with our specialized commerce solutions and services, and creating individualized, highly branded customer experiences for each label.\nsingle-page checkout to streamline the purchase experience. Responsive email templates with dynamic content and segmentation were designed for multiple brands. A shade finder for Urban Decay guides and simplifies the selection liquid foundation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the week ending 1\/6\/13, these charts show the spiking of activity which usually occurs in January as people make plans for the year. The numbers in Oregon and Washington rose significantly this week.\nWhat is happening with the Real Estate Market?\nBeth Frischmuth Awarded 2012 REALTOR\u00ae of the Year!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ross Martin | Warner V. Slack, M.D.\nWarner - Thank you for your lifetime of leadership in shaping the informatics profession and making computing in healthcare a reality. I don't doubt that -- even now -- the experiences you are having as a patient are shaping your thinking about ways to refine the patient and provider experience and optimize care. But I hope you instead focus your thoughts and energies on your own speedy recovery!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1\u53e5\u82f1\u6587\u7b7e\u540d - \u6bcf\u65e5\u4e00\u53e5ENGLISH - \u91d1\u6865\u7ffb\u8bd1\u8bba\u575b - Powered by Discuz!\nYouth is like smoking. Smoke in the dust. Soot in the fall. Failure is a common; success is accidental; pay should be; get is temporary; happiness is relative.\nGMT+8, 2019-4-19 11:12 AM , Processed in 0.033974 second(s), 13 queries .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwscc b/data_all_eng_slimpj/shuffled/split2/finalzzwscc
new file mode 100644
index 0000000000000000000000000000000000000000..8c7fadc7100f708a123d3a151521ab5a3229eff0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzwscc
@@ -0,0 +1,5 @@
+{"text":"Make sure that your browser allows pop-ups.\nI can not fully log out.\nWhen you log out on our website, you are not connected with us any more but you remain connected to Facebook until you close your browser or until you manually log out in Facebook \u2013 This is due to Facebook's cookie, we are not planing to fix this unless Facebook provides a better API for logging out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Glorious greens in bunches . . . next to onions, garlics and other vegetables.\nBury Market offers traditional values, modern shopping and inexpensive parking. Every year Bury Market attracts hundreds of thousands of shoppers drawn by the many bargains and sheer variety that can be found there.\nBlack puddings locally made and of exceptional quality. Scotch eggs came to mind . . . soft centres and black pudding enriched sausage coating, rolled in breadcrumbs and fried off until crisp and golden. Black puddings were on my list and I purchased them with great satisfaction.\nChorley and Eccles cakes and oven bottoms and muffins and home-made cakes. There is nothing quite like a buttered chorley cake and a cuppa apart from a warm eccles cake!!\nCheeses such as Lanachire Mild, Lancashire Creamy, Lancashire Tasty and some Lancashires with fried onions. There were also some interesting blue cheeses. I bought Blacksticks Blue, Butlers Farmhouse cheese made at the Wilson Fields, Lancashire.\nBury Market has been voted a \"Top Ten UK food market\". It is a popular destination with many U.K. coach companies and has gained the reputation of 'Bury's World Famous Market' from the many customers all over the world who keep coming back to visit.\nFish and shelfish. just look at the colours! My memories of potting shrimps came flooding back, a quart of shrimps bought in little cone shaped bags and then cooked in butter . . .\nMeats and poultry beautifully presented and from the region.\nAfter all the food . . . I must also add the market was a treasure trove for fabrics, bags and . . . as you can see, bargain bras!! Lancashire women have all the luck.\nIt was a good trip, a brilliant market with such friendly people . . . we headed for home to cook our treasure . . .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"maybe you're in those baby stages of your business.\nmaybe you're realizing, \"i don't want to keep doing this alone.\"\nmaybe you're just tired of feeling soo competitive.\nOHH, JUST YOU WAIT. WE'VE GOT SOME PLANS FOR YOU.\nHEY, SISTA is about togetherness and solidarity.\nIt's part photography workshop, part retreat.\nIt's ten women coming together to share good food, escape daily routine, and experience growth--together and individually.\nLearn from Lauren & Allison.\nGet inspired by Madison. Let Megs spoil you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Try yoga and meditating, it'll help\" & other things I've tried to learn and unlearn in 2017.\nWho are we to say what is real and what is not? Other people will perceive their lives the way they want to; they will believe in who they want and that is accepted in our society on many levels. But what isn't okay is to tell someone with a mental illness that it's all in their head. We are all of what is inside our heads, all of what makes us, us.\nThis is not for everyone and that is OKAY. I'm sure it's because of the fads and everyone trying to say their \"enlightened\"to be holier than thou but sometimes yoga and meditation doesn't work, doesn't mean there aren't different ways to help ease an anxious mind.\nWe're not good people all the time.\nRest your minds, my friends. The people you see on Instagram are not always good people and I know that's hard to believe but it's the truth. It's okay to be mean and hurt and wrong. I know we're taught to believe being wrong is not the right thing but then again I'm not sure life comes with a manual. We just need to accept the inalienable truth that in order to be \"good\" one must also understand what it means to be \"bad\".\nPeople might laugh, I'm sure some will try and belittle you but your vulnerabilities are indeed your strengths. Some have found that truth early on and others are still fighting with the idea, come to peace with yourself and your weaknesses and accept them for what they are.\nBe patient with yourself & others.\nIt won't always be easy but when you do it I'm sure you'll feel so good. Please come back and teach me how.\nUnderstand that letting people go is a gift.\nIt's going to hurt but somehow in that hurt, you're going to find growth. Your vision will clear and understanding will come, this is a gift.\nI know somewhere there's a quote that says never forget but you deserve to forget. Forgetting is an important part of moving on and healing, it doesn't mean you'll let it happen again but that you've let yourself grow. Forget my friends and forgive, the worlds needs a lot of this right now.\nYou are not what people say you are.\nRemember what matters is who you are when no one's looking. We get so worked up when someone tries to define us that we tend to forget who we are. People will always talk but if in your heart you know who you are and whatever anyone says, no one will take that from you.\nTravel but not like them.\nTake photographs, get lost and enjoy different cultures but stop doing it for their approval. Life is too short to travel for likes.\nStop trying to achieve other people's accomplishments, sometimes you'll be ahead, and other times you'll be behind. What matters is what you're doing for yourself.\nHowever you decide to enjoy your life, just do it. By the time you know it, you'll be 60 and wonder where it all went. Time is a gift, life is a gift, pain is a gift. Everything that is your life, is a gift. Make the best of it.\nHappy New Year, everyone! I wish nothing but pure love & light to you all in the coming New Year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On the planet Korugar, Sinestro was born and raised to become a fearless warrior. His military prowess earned the attention of the Guardians of the Universe who recruited Sinestro to the Green Lantern Corps. Sinestro was assigned to Space Sector 1417, which included his home planet, and established a lofty reputation of being the greatest Green Lantern there was. Over time, Sinestro was corrupted by his power and took over Korugar. Years passed until a young Katma Tui joined the underground rebellion. The Guardians were alerted of Sinestro's actions. Several Green Lanterns, including John Stewart, were dispatched to arrest Sinestro.\nA tribunal was held on Oa. With Tui's testimony, Sinestro was stripped of his title and ring. He was then banished to the anti-matter universe of Qward. Sinestro's warrior ethic impressed even the Qwardians, beings of pure evil. Their leader, Yokal the Atrocious had a yellow power ring forged, impowered and given to Sinestro. He vowed to kill every Green Lantern and get revenge on the Guardians, enemies of the Qwardians. After escaping Qward, Sinestro soon kept his word and began systematically murdering Green Lanterns and keeping their rings as war trophies.\nAfter killing nine, Sinestro set his sights on his next target: Abin Sur, Lantern of Sector 2814 and attacked him while he was on route in a minor class cruiser. Sur was fatally injured but managed to crash land on Earth and sent his ring to find a successor. Sinestro soon found Kyle Rayner and proceeded to kill him except he found a challenge and later opposed by Superman, as well. After finding resolve, Rayner defeated Sinestro and was taken to Oa to be instated. He would escape imprisonment and became fixated on the Green Lanterns of Earth. With revenge on his mind, Grodd began recruiting villains interested in destroying the Justice League. Sinestro was easily convinced and joined to get another chance at killing John Stewart.\nSinestro issued a false distress call to be picked up by the Justice League Watchtower. Posing as the pilot, he ambushed John Stewart while he was recharging and stole his Power Battery. Sinestro impersonated Stewart and terrorized Dakota City in order to tarnish the Green Lantern's reputation. Dakota's resident heroes, Static and Gear were convinced Lantern went rogue and when the real one arrived, they incapacitated him. Stewart slipped his ring in Static's outfit to symbolize his innocence. Static freed Stewart and hunted Sinestro down. With Static's electromagnetism, Stewart powered his ring and defeated Sinestro.\nDespite the Secret Society's defeat, Sinestro remained affiliated and became part of the larger Legion of Doom. After Grodd's secret plan failed and Lex Luthor assumed control, Sinestro and the rest of the Legion were uneasy with their future. Luthor came up with a train robbery to satisfy them. Sinestro would guide the train, loaded with newly printed Euro's, to a cavern where a Legion transport would be waiting. Dr. Light used her abilities to find and illuminate the ring's residual hard light radiation and led Steel and Ice to confront the Legion. They fled soon after. Sinestro used his ring to save a few Legion members when Darkseid was resurrected and destroyed the headquarters. Lost in space, Sinestro's bubble wouldn't hold out indefinitely. Lightray, of New Genesis, investigated and was knocked out. Sinestro and the others fought alongside the Justice League to stave off the Apokoliptian invasion force on Earth, but returned to status quo afterwards.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwvfy b/data_all_eng_slimpj/shuffled/split2/finalzzwvfy
new file mode 100644
index 0000000000000000000000000000000000000000..fa13495efc179e93a63ef07036d52ef976e3b7ee
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzwvfy
@@ -0,0 +1,5 @@
+{"text":"Illustration of an alien technological planet. The planet is in orbit around a red dwarf star, the most common type. The red dwarf is relatively sedate, making the environment of its habitable zone conducive to life. The planet is shown with its night side brilliantly lit by major cities and technology.\nKids holding hands and wearing VR glasses against red and purple background. Portrait of two children with virtual reality headsets under neon lights.\nBoy wearing VR glasses against red and purple background. Portrait of child with virtual reality headset under neon lights.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Streets Of Rock & Roll\" is a new radio show hosted by Ron Keel, lead vocalist of the million-selling band KEEL. The music comes first, but there's much more to this show than just the tunes \u2013 listeners are treated to the stories behind the songs and the rock stars that brought them to life, candid celebrity interviews, shameless commentary, road stories and interactive fan participation. The show is a roller coaster ride through the concrete jungle that is the rock scene \u2013 fast and reckless with a lot of twists and turns.\nThe music ranges from classic 80's rock & metal to current releases from the genre's biggest and best \u2013 rare cuts, live tracks, tribute\/cover versions, stripped unplugged performances, anything goes on the Streets Of Rock & Roll.\n\"Streets of Rock & Roll\" will debut on Metal Rocks Radio Tuesday June 12th at 9 pm eastern and be replayed Wednesday June 13th at 9 am eastern. Don't miss it.\nTUESDAYS 10 PM - ?\nCopyright metalrocksradio.com 2010-2012. Awesome Inc. theme. Powered by Blogger.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why Choose Hang Seng MPF?\nThe Hang Seng Mandatory Provident Fund \u2013 SuperTrust Plus and ValueChoice are mandatory provident fund schemes.\nYou should consider your own risk tolerance level and financial circumstances before investing in the MPF Default Investment Strategy. You should note that the DIS constituent funds, namely, the Core Accumulation Fund and the Age 65 Plus Fund, may not be suitable for you, and there may be a risk mismatch between the DIS constituent funds and your risk profile (the resulting portfolio risk may be greater than your risk preference). You should seek financial and\/or professional advice if you are in doubt as to whether the DIS is suitable for you, and make the investment decision most suitable for you taking into account your circumstances.\nYou should note that the implementation of the DIS may have an impact on your MPF investments and benefits. We recommend that you consult with the Trustee if you have doubts on how you are being affected.\nThe Guaranteed Fund under Hang Seng Mandatory Provident Fund \u2013 SuperTrust Plus invests solely in an approved pooled investment fund in the form of an insurance policy provided by HSBC Life (International) Limited. The guarantee is also given by HSBC Life (International) Limited. Your investments in the Guaranteed Fund, if any, are therefore subject to the credit risks of HSBC Life (International) Limited. Please refer to the 'Warning' section under 'Guaranteed Fund' in Part II - Fund Structure of the 'Principal Brochure' of Hang Seng Mandatory Provident Fund - SuperTrust Plus for details of the credit risk.\nThe guarantee in the Guaranteed Fund only applies under certain conditions. Please refer to the 'Guarantee features' section under 'Guaranteed Fund' in Part II \u2013 Fund Structure of the 'Principal Brochure' of Hang Seng Mandatory Provident Fund - SuperTrust Plus for full details of the guarantee features and Guarantee Conditions, including the guarantee features in the context of payment of benefits in instalments.\nMPF Benefits, AVC Benefits and TVC Benefits payable on a Member's 65th birthday or early retirement on or after his\/her reaching age 60 can be paid in one lump sum or in instalments, at the Member's election (in such form and on such terms as the Trustee may, to the extent not prohibited by the 'MPF Ordinance' or General Regulation, prescribe). Please refer to the 'Payment of MPF Benefits, AVC Benefits and TVC Benefits' section under 'Payment of benefits' in Part I - Product Information of the relevant 'Principal Brochure' for full details.\nYou should not invest based on this page alone and should read the relevant 'Principal Brochure'.\nInvestment involves risks. Past performance is not indicative of future performance. The value of financial instruments, in particular stocks and shares, and any income from such financial instruments, may go down as well as up. For further details including the product features and risks involved, please refer to the relevant 'Principal Brochure'.\nThe following table shows the cumulative performance for the constituent funds under Hang Seng Mandatory Provident Fund \u2013 SuperTrust Plus since the launch date7 of respective constituent funds.\nThe following table shows the cumulative performance for the Age 65 Plus Fund and the Core Accumulation Fund under Hang Seng Mandatory Provident Fund \u2013 SuperTrust Plus since they launch as a constituent fund of DIS on 1 April 2017.\nThe following table shows the cumulative performance of Flexi-Managed Fund and Stable Growth Fund of Hang Seng Mandatory Provident Fund \u2013 SuperTrust Plus as of 31 March 2017 (ie before rename and conversion to become a DIS constituent fund).\nThe following table shows the cumulative performance for the constituent funds under Hang Seng Mandatory Provident Fund \u2013 ValueChoice since the launch date8 of respective constituent funds.\nThe following table shows the cumulative performance for the Age 65 Plus Fund and the Core Accumulation Fund under Hang Seng Mandatory Provident Fund \u2013 ValueChoice since they launch as a constituent fund of DIS on 1 April 2017.\nThe following table shows the cumulative performance of ValueChoice Stable Growth Fund of Hang Seng Mandatory Provident Fund \u2013 ValueChoice as of 31 March 2017 (ie before rename and conversion to become a DIS constituent fund).\nFrom 1 April 2017, the Default Investment Strategy ('DIS') has been launched and it comprises the Age 65 Plus Fund and Core Accumulation Fund. The default investment arrangement of the Hang Seng Mandatory Provident Fund \u2014 SuperTrust Plus and ValueChoice will be the DIS replacing the previous default fund which is 'MPF Conservative Fund'.\nThe Age 65 Plus Fund and Core Accumulation Fund under Hang Seng Mandatory Provident Fund \u2013 SuperTrust Plus have been renamed and converted from Flexi-Managed Fund and Stable Growth Fund respectively when DIS commenced on 1 April 2017.\nThe Core Accumulation Fund under Hang Seng Mandatory Provident Fund \u2013 ValueChoice has been renamed and converted from ValueChoice Stable Growth Fund when DIS commenced on 1 April 2017.\nThe Age 65 Plus Fund under Hang Seng Mandatory Provident Fund \u2013 ValueChoice has been launched when DIS commenced on 1 April 2017.\nFrom 1 July 2016, Hang Seng Mandatory Provident Fund \u2013 SuperTrust has been merged into the Hang Seng Mandatory Provident Fund \u2013 SuperTrust Plus and Hang Seng Mandatory Provident Fund \u2013 SimpleChoice has been merged into the Hang Seng Mandatory Provident Fund \u2013 ValueChoice (the 'Merger'). For the cumulative performance of constituent funds of Hang Seng Mandatory Provident Fund \u2013 SuperTrust and Hang Seng Mandatory Provident Fund \u2013 SimpleChoice before the Merger, please refer to \"Fund Fact Sheet\" and \"Monthly Fund Performance Summary\" with information before or as at 30 June 2016.\nPlease refer to 'Monthly Fund Performance Summary' for the information on the returns of constituent funds of the immediately preceding 5 years.\nThe Asia Pacific Equity Fund, Balanced Fund, European Equity Fund, Growth Fund, Guaranteed Fund, Hang Seng Index Tracking Fund, Hong Kong and Chinese Equity Fund, MPF Conservative Fund, North American Equity Fund and Stable Growth Fund under Hang Seng Mandatory Provident Fund - SuperTrust Plus were launched on 1 December 2000. The Chinese Equity Fund, Flexi-Managed Fund, Global Bond Fund and Stable Fund under Hang Seng Mandatory Provident Fund \u2013 SuperTrust Plus were launched on 8 October 2009. The Age 65 Plus Fund and Core Accumulation Fund under Hang Seng Mandatory Provident Fund - SuperTrust Plus were respectively renamed and converted from Flexi-Managed Fund and Stable Growth Fund on 1 April 2017 for DIS purposes. The corresponding investment objective and asset allocation for these two constituent funds were changed on the same day.\nThe Global Bond Fund, Hang Seng Index Tracking Fund, Hang Seng H-Share Index Tracking Fund, MPF Conservative Fund, ValueChoice Asia Pacific Equity Fund, ValueChoice Balanced Fund, ValueChoice European Equity Fund, ValueChoice Stable Growth Fund and ValueChoice US Equity Fund under Hang Seng Mandatory Provident Fund - ValueChoice were launched on 24 March 2011. The Global Equity Fund under Hang Seng Mandatory Provident Fund - ValueChoice was launched on 1 July 2016 due to the Merger. The Age 65 Plus Fund under Hang Seng Mandatory Provident Fund - ValueChoice was launched on 1 April 2017 for DIS purposes. The Core Accumulation Fund under Hang Seng Mandatory Provident Fund - ValueChoice was renamed and converted from ValueChoice Stable Growth Fund on 1 April 2017 for DIS purposes. The investment objective and asset allocation for this constituent funds was changed on the same day. The Hang Seng China Enterprises Index Tracking Fund was renamed from Hang Seng H-Share Index Tracking Fund on 5 March 2018 in order to better reflect the constituents of the Hang Seng China Enterprises Index which Red-chips and private enterprises are eligible as the index constituents effective from March 2018.\nThe constituent funds of Hang Seng MPF schemes are all denominated in Hong Kong dollars and their unit prices are based on the net asset value of each constituent fund and quoted for indication only. Cumulative performance is calculated in Hong Kong dollars on the basis of NAV-to-NAV (net asset value).\nFees and charges of an MPF Conservative Fund can be deducted from either (i) the assets of the fund or (ii) members' account by way of unit deduction.\nFrom 1 July 2015, the fees and charges deduction method of MPF Conservative Fund of Hang Seng Mandatory Provident Fund \u2013 SuperTrust Plus and ValueChoice has changed from method (ii) to method (i). Therefore, the unit prices, net asset value (NAV) and fund performance of MPF Conservative Fund quoted have reflected the impact of fees and charges for the period starting from 1 July 2015.\nBefore 1 July 2015, the unit prices, NAVs and the fund performance quoted for MPF Conservative Fund have not reflected the impact of fees and charges. The fund performance figures of the MPF Conservative Fund strictly for the period(s) ended before 1 July 2015 are calculated before fees and charges are deducted from the fund, and such fees and charges are deducted according to method (ii).\nFor fund performance figures of the MPF Conservative Fund quoted for the period that has started before 1 July 2015 and ended\/ will be ending on or after 1 July 2015, the fund performance figures would not reflect the actual performance of the fund because it has taken into account both (i) the period which has excluded the impact of fees and charges (ie the period covered before 1 July 2015) and (ii) the period which has included the impact of fees and charges (ie the period covered on or after 1 July 2015).\nFor Hang Seng Mandatory Provident Fund - SuperTrust Plus, the joining fee, contribution charge# and offer spread# are currently waived. The annual fee, bid spread and withdrawal charge are not applicable. For Hang Seng Mandatory Provident Fund - ValueChoice, the joining fee, annual fee, contribution charge, offer spread, bid spread and withdrawal charge are not applicable. Increases in the fees and charges of the relevant Hang Seng master trust schemes, constituent funds or underlying pooled investment funds are subject to a three months' prior notice (or such shorter period permitted by law and the relevant Hang Seng 'Master Trust Deed') to members of the relevant Hang Seng master trust schemes.\nFor more details on the constituent funds, fees and charges, including other charges payable by the constituent funds and handling charges, please refer to the relevant 'Principal Brochure'.\nThe MPF information provided in this website is for reference only, which may be subject to adjustment or correction from time to time without any notice. The relevant information after adjustment or correction may vary. The information provided above should not be regarded as investment advice. You should not rely on the above information when making any investment choices for your MPF account(s).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u0421 \u0432\u0438\u0434\u0435\u043e Learn British English with Video - Get Dressed -- and Undressed -- with British English \u0438\u0437\u0443\u0447\u0430\u0442\u044c \u0430\u043d\u0433\u043b\u0438\u0439\u0441\u043a\u0438\u0439 \u044f\u0437\u044b\u043a \u043e\u0447\u0435\u043d\u044c \u043b\u0435\u0433\u043a\u043e.\nit's very useful!!! i like? it!\nThis videos are rubbish. Learning separate? words has no sense.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to VSP Global. The collective strengths of our five lines of business provide benefits, services, products, and solutions that are unparalleled in the optical industry.\nThe Corporate Procurement & Travel team serves as the primary point of contact for our community of diverse suppliers, who strive to understand our business and are committed to continuous improvement in technology, quality, services, practices, and total cost match that of our own and help to deliver on the needs of our business. At VSP Global we value our suppliers as true partners to help achieve our business goals.\nPlease click on the tab that corresponds with the business entity you are working, or wish to work with.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxgha b/data_all_eng_slimpj/shuffled/split2/finalzzxgha
new file mode 100644
index 0000000000000000000000000000000000000000..0900707839ca0660dbccc1d3ae4bb14499cab080
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzxgha
@@ -0,0 +1,5 @@
+{"text":"How To Make Paper Wallet ? Generate and print your own bitcoin wallets to store bitcoin offline in 'cold storage'. The generator guides you to easily print your secure bitcoin paper wallet.\n2018 New Design 30hp 4wd Mini Farm Tractor Price List For Africa Market , Find Complete Details about 2018 New Design 30hp 4wd Mini Farm Tractor Price List For Africa Market,Tractor,Mini Tractor,Mini Tractor Price from Tractors Supplier or Manufacturer-Weifang Heng An Imp& Exp Co., Ltd. . generator price list omron plc price list alternator .\n2018 Sunseeker by Forest River Prices, 2018 Sunseeker by Forest River Values w\/ MSRP & Used 2018 Sunseeker by Forest River Specs | NADAguides . MOTORHOMES - Air conditioner, generator, microwave, Arctic Package, exterior accessory package, bathroom roof vent and DSI water heater are included in price. . Mini Motor Home (Class C) Queen Bed .\nThis is the new, high-performance JCW version of the Mini Countryman. Read more about Mini's most powerful SUV and see pictures at Car and Driver. . 2018 Mini John Cooper Works Countryman: Mini .\nB&C Wind Generator Store has All Kinds of Factory price,mini wind turbine\/generator 3\/5 blades small wind mill low start up wind generator + 400w wind controller,New design 400W wind generator 12v 24v wind turbine with 3 blades or 5 blades for streetlight garden lighting or home use,2018 Sale Real Gerador De Energia Wind Generator, 3\/5 Blades .\nHow To Make Paper Wallet ? Generate and print your own Litecoin wallets to store Litecoin offline in 'cold storage'. The generator guides you to easily print your secure Litecoin paper wallet.\n2018 New Design Invertor Generator For 100kw Induction Heater Best Sell In Russia , Find Complete Details about 2018 New Design Invertor Generator For 100kw Induction Heater Best Sell In Russia,2018 Invertor Induction Generator,Induction Heater,100kw Induction Generator from Alternative Energy Generators Supplier or Manufacturer-Chengdu Jinkezhi Electronic Co., Ltd.\nMercedes-Benz CLS 2018: Third generation of the original.\nThe new CLS (C 257) pioneers the new design idiom of Mercedes-Benz, which is recognisable by its clear contours and reduced lines. Mercedes-Benz CLS 2018: Third generation of the original. The new Mercedes-Benz CLS (C 257) 2018 pioneers the new design idiom of Mercedes-Benz, which is recognisable by its clear contours and reduced lines.\nShopping for Cheap 2018 New Design at Partnerbeads(mini order US$10) and more from snap button jewelry,ginger snap button,bracelet jewelry,snap jewelry,bracelet charms,button bracelet on Aliexpress.com ,the Leading Trading Marketplace from China - Natural Round Shell Charm 18mm Snap button For Earring Jewelry Making DIY Shell Necklace Accessories Bracelet Charm Snap Jewelry,2018 New Design .\n2018 New design portable 125 kva super silent diesel generator, US $ 8,000 - 180,000 \/ Set, Guangdong, China (Mainland), Camda, KDGC100S.Source from Camda New Energy .\n2018 new design free energy generator 5kva solar system from China, US $ 4,650 - 4,850 \/ Set, Guangdong, China (Mainland), Felicitysolar, FL-POWER-7500VA.Source from Guangzhou Felicity Solar Technology Company Limited on Alibaba.com.\nJoin the party with the best dress for party,dress party and going out dresses uk and pay attention to 2018 new design square short mini homecoming dresses long sleeves party dress popular prom dresses cocktail party dresses provided by libaochao413115.\nThe 2019 Ram 1500 pickups: Changes to nearly every part of the truck . Willys Concepts Police, Fleets Models\/Toys 200,000 Mile Club. Future \/ New&Hot Upcoming cars Upcoming trucks\/SUVs 2019 Cherokee 2018 Wrangler JL 2019 Ram 1500 2019 Grand Cherokee 2018 Trackhawk Dodge Demon 2019 Jeep . Inside and design. The new 2019 Ram 1500 is lighter .\nby KBB.com Editors | January 25, 2018 4:39 PM The 2018 model year is well under way, and most of this year's new and redesigned vehicles have made their way to dealer lots near you.\nResearchers have designed and successfully developed a high-power, silicon-nanowire thermoelectric generator which, at a thermal difference of only 5\u00baC, could drive various IoT devices .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dr. Johnson established her surgical practice in 1990. Since 1994 she has exclusively managed diseases of the breast, predominantly women with breast cancer. Initially Board Certified in General Surgery in 1990, she has re-certified twice, most recently in 2009. She has been a member of the American Society of Breast Surgeons since 1996. In 2003 she answered a community need and began performing minimally invasive ultrasound-guided core needle biopsy procedures in her office. She has surgical experience in Partial Breast Irradiation, and intraoperative radiation therapy (IORT). In 2009 she was recruited to join the Alexander Walt Breast Center at Karmanos Cancer Institute as a breast surgeon on the Breast Cancer Multidisciplinary Team. Part of that role includes appointment as Clinical Assistant Professor at Wayne State University School of Medicine as well as participation in clinical trials at KCI. She is a proud lifetime member, and 'Member at Large' on the Board, of the Wayne State Surgical Society.\nWhen not treating patients Dr. Johnson cherishes spending time with her family and their rescue k9. She is an avid gardener, loves to travel and hike. She feels privileged to have her career span three decades of treating gracious and brave women who have contributed to remarkable improvements in technology and customized care for current breast cancer management.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am helping my cousin upgrade her Sony Vaio CPU; I have downloaded files, including a BIOS upgrade file from their website. The instructions say to make a startup disk on a 3.5\" floppy, and then to copy the BIOS file material over to the same floppy. No problem, right?\nTried to make a startup disk on her computer (Win XP Home edition); seems to be fine until I start to copy BIOS files. Then a message comes up that the disk (A) is locked and no further files can be added. Go back to the SONY Vaio site, and uh-oh, they already know this is a problem, and tell you to get a startup disk from Win 98 or Me, that the 'locked disk' problem IS INDEED a problem with Win XP startup disk.\nI came home and tried to make a WIN XP startup disk on MY computer (it has WIN XP Professional). This disk will accept other files written to it, but NOW, the startup files take up around 555 KB, leaving me 868 KB for the BIOS upgrade file, which is 1.3 MB in size.\nAll I know is my Dell had better instructions for its BIOS upgrade, and the instructions actually WORKED.\nUse a google search to find drdflash.exe. Executing this exe will create a freshly formatted and verified bootdisk with lots of space for biosfiles left.\nmessyk wrote: the BIOS upgrade file, which is 1.3 MB in size.\nMaybe you need to decompress this file first.\nCan you tell me where to download your BIOS update file?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Seattle had a chance to hear from 21 or so wonderful LGBTQ singers from Beijing, China who are a delegation from the Beijing Queer Chorus. They performed here for one evening, Saturday, March 10, at the East Shore Unitarian Church in Bellevue. They performed six songs, including 'Shenandoah' (an American folk song), in English, and sounded lovely and enchanting.\nThey were brought in to work with and perform at the Portland Gay Men's Chorus on March 17 and 18. (Web site: https:\/\/www.pdxgmc.org\/) PGMC hosted Beijing Queer Chorus in their first public tour of the U.S. and they were able to stop for a night in Seattle because of this visit. This September, PGMC will become the first LGBTQ Chorus to ever tour China.\nMany of the delegation that came here had never been to the United States. They seemed very excited about this opportunity. When they walked to their stage positions, the audience gave them a huge standing ovation, which pleased and awed them. Their chorus is both male and female. Those who came to the U.S. were about 9 women and 11 men. They all seemed to be in their 20s and 30s.\nAfter making connection with former Seattle Men's and Women's Choruses Associate Artistic Director Eric Lane Barnes, he went to China to work with them and then came home and vigorously worked to help a delegation come to the GALA gathering of choruses in July 2016 in Denver, Colorado.\nDuring the evening event, we heard that there are now at least five Gay choruses in China and Taiwan, which makes their progress extremely fast, though still courting a lot of dangerous repercussions. Indeed, even taking pictures of the singers can result in danger if one singer is not 'out' to family and business associates. So, taking the stage can be a real risk for them, and we ought not to forget that struggle to be free to love who we wish.\nThe evening was also bittersweet because before the BQC took the stage, we were entertained by both Sensible Shoes and Captain Smartypants in what was their last performance before going on an extended hiatus. Barnes has, as many know, been the leader of the 'pocket choruses' and when he stepped down from his position as Associate Artistic Director at SMC\/SWC, he stayed as their leader and we did not think that would change. However, he recently decided to step down from that role so he could give his full attention to his new role as Artistic Director of Diverse Harmony, Seattle LGBTQA youth choir and young adult ensemble. No immediate replacement has been made and therefore they need to regroup and figure out what else can happen.\nMany folks will miss Shoes and Pants dearly, and we hope that they will reappear with a new vision and a new leader who can help them make a new future. We appreciate all the past volunteer work from these smaller choruses where participants had to add an extra day every week for rehearsal and perform all over on many extra dates to spread the joy of the Choruses and deepen their mission.\nAt this concert Sensible Shoes performed first, then Captain Smartypants, and then the Beijing Queer Chorus performed four songs. The Shoes and Pants each performed two songs, one representative of the funny songs Eric Lane Barnes has written for them over the years and one with the balanced harmonies that have made them so much appreciated. Subsquently, Shoes and Pants sang two songs together and BQC presented the finale.\nIt was a pretty full house, and a very warm reception for this historic event.\nThe last song, sung by BQC, was 'Sihai' (a Mongolian folk song).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pip Wilson bhp: Managing Aggression & Violence.\nI have 60 years of reflections on paper or screen.\nI never have time to dig backwards.\nMaybe when the pace slows?\nThose present were Staff members from Hostels in London.\n11 of them + myself present.\nmeeting\/training - opening up in a group.\nto share, this time, with the whole group.\nGetting everyone to connect to their feelings.\n. A big man tried to enter the Hostel without permission and the staff member lost control and offered to fight him outside.\n. I was dragged out of the building with my friend.\n. A gun was rested on my shoulder and my friend was shot dead.\n. At the benefits office I stood between the officer and the Hostel resident who was out of control and threateningly abusing staff.\n. On the corridor of the Hostel hammers were being used to attack others. Staff hid in the toilets to escape extreme violence.\n. I went to visit my Uncle and we sipped tea together. A gang entered and shot dead my Uncle as he sat beside me.\n. My friend was intimidated living in the Hostel so he left. Some time after, he was attacked at his front door and stabbed to death.\n. The Hostel Night Manager was abused and intimidated by friends of a resident. Police were called to avoid serious violence.\n. The eviction noticed was served on the resident, he abused me and made threats of violence. I walked away to defuse conflict.\n. The non-resident had his foot in the front door to stop staff from locking out potential violence and then the abused Staff member wanted to fight.\n. A remark from another staff member resulted in the Hostel resident 'losing it' and pulling a knife.\nYou can imagine the atmosphere in the room now!\nThen I started the training day.\nand feedback to the larger group & me.\nwhich I have lived with for years.\nAwareness of how the other person is feeling.\nBUT we can learn to manage our own.\nExperiences are remembered more than words.\nWe all had an experience that day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxnbf b/data_all_eng_slimpj/shuffled/split2/finalzzxnbf
new file mode 100644
index 0000000000000000000000000000000000000000..d868c655a5b27b2151fb20dba740ad81796b9ec6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzxnbf
@@ -0,0 +1,5 @@
+{"text":"Description:Used 2014 Yamaha G29 Efi for sale - $5,995.00 2014 YAMAHA G29 EFI GOLF CART 2014 Yamaha Drive FUEL INJECTED Gas Golf Cart. New Custom ANVIL Grey ,Black with Black Inserts, New Jake's LIFT KIT, 14\" Wheels, LED Head Lights with Blinkers, Brake lights and Horn. Tinted windshield, NEW Rocker Panels, 5 panel mirror, BLUETOOTH Stereo, USB charge port. Fender Flares, YES!!! Heated Seats!!!!! $5,995 call or text anytime (404)618-7363 Financing options available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"joy magnetism: Me want cookie! Meeeee want coooookie!\nI'm a total Cookie Monster.\nBwahahahaaha, magnetism is certainly yielding a lot of words for you, anon. I'm fairly impressed.\nGG, it's funny, everyone else has already discovered the wonders of Levain and these cookies. I can't believe how late I am to this party. Hope your cookies are great!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To be aware of and comply with all anti-doping policies and rules adopted pursuant to the World Anti-Doping Code (Code).\nTo be available for sample collection.\nTo be responsible in for what they ingest and use to avoid violation of anti-doping policies and rules pursuant to the Code.\nInform medical personnel of their obligations not to use prohibited substances and methods when receiving medical treatment to avoid violation of anti-doping policies and rules pursuant to the Code.\nCooperate with Southeast Asia Regional Anti-Doping Organization (SEARADO), their National Anti-Doping Organization (NADO), International Federation (IF) and other recognized Code-compliant international anti-doping programmes.\nBe aware of and comply with all anti-doping policies and rules adopted pursuant to the Code which are applicable to them or the athletes whom they support.\nCooperate with the SEARADO, NADO, IF or any other Anti-Doping Organizations in terms of the athlete-testing program.\nPositively adopt the values and behaviour required to foster doping-free sport and true excellence in performance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have found The LJ Community For You.\nOkay, they're not reading Lovecraft or Kerouac while they do it, but still.\nI say this is nihilistic_kid's community because he wrote this beautiful article for the Village Voice, titled Elven Like Me.\nSo what if my personalities are one or the other, but not both at the same time?\nah, then you're one of the otherOtherkin.\nThis is just weird. I used to correspond with Malcolm-Rannirl in the early days. We used to have a mutual RL friend. (The mutual friend is still a frienmd of mine, I just have know idea if they still correspond). It's a creepily small world sometimes.\nand your mention of that reminds me WHY I know the name: he used to hang around on a MUSH I hung around on.\nI'm pretty sure my friend met him through MUSH.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let the battle begin. Dan Loeb's hedge fund Third Point has started a long position in Herbalife (HLF), he revealed in his Q4 letter to investors. He also filed a 13G with the SEC disclosing that Third Point owns 8.24% of the company as of January 3rd.\nReaders will recall that we recently posted up Bill Ackman's short presentation on HLF where he called it a pyramid scheme. Brian Sullivan tweeted that Andrew Ross Sorkin spoke with Third Point, who believe there's no evidence HLF is a pyramid scheme in their research.\n\"Applying a modest 10-12x earnings multiple suggests Herbalife's shares are worth $55-68, offering 40-70% upside from here and making the company a compelling long investment ... Given that the company has historically traded more in the 12-14x range (and traded at 16-20x earnings through much of 2011 and early 2012), the opportunity for the company to tell its side of the story tomorrow at its Analyst Day in New York, and the significant short interest, we believe shares could even trade well about our current price target.\"\nSo, you now have two hedge fund heavyweights: 1 long, 1 short. Who wins? Only time will tell. Now all we need is David Einhorn to toss his hat in the ring as well. After all, in May of this year Einhorn popped up on a HLF earnings call and started asking questions. However, he has not disclosed a position long or short.\nWhile the HLF position will get all the focus, we also wanted to highlight that Third Point disclosed a new position in Morgan Stanley in their Q4 letter as well. They feel the company is a turnaround story and point to the stock trading at a 20% discount to tangible book, down from the 35% discount when they acquired shares at an average price of $16.77 per share.\nThe hedge fund also bought shares of refiner Tesoro (TSO). They write, \"we see Tesoro generating about $9 per share in annual excess FCF on a normalized basis and our expectation is that shares can double from the current price of $40. We believe the Q3 story was only the beginning, and are happy to own Tesoro for its next few chapters.\"\nFor more on this hedge fund manager, we just yesterday posted up how Third Point ramped up net long equity exposure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxwlf b/data_all_eng_slimpj/shuffled/split2/finalzzxwlf
new file mode 100644
index 0000000000000000000000000000000000000000..eff5f9e0473a112421e93ac7d59b62fcb6f81d35
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzxwlf
@@ -0,0 +1,5 @@
+{"text":"Broadcast Music, Inc. (BMI) held its fifth of six Rooftop on the Row concerts on Aug. 21 in Nashville.\nAbby Anderson kicked off the night's festivities atop the BMI building, before Lindsay Ell headlined the event with special guest Kristian Bush of Sugarland.\nLindsay treated the concertgoers to a number of tunes from her 2017 album, The Project, which was produced by Kristian, including Top 20 single, \"Criminal.\" Check out Lindsay's performance of \"Criminal\" below.\nThe BMI Rooftop on the Row concert series was created in partnership with George Dickel Tennessee Whisky, as well as sponsors Nash FM 103.3, 95.5 Nash Icon, Yeti Coolers, Texas Roadhouse, Sam Adams, First Tennessee, Royalty Exchange and Topo Chico.\nListen to Nash FM for your chance to win free tickets to the next Rooftop show on Sept. 11.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apple has announced that iAd is now available in nine additional countries: Austria, Belgium, Denmark, Finland, Luxembourg, Netherlands, Norway, Poland, and Sweden. This means iAd is now available in a total of twenty-five countries spanning North America, Europe and Asia.\nApple says that it continues to support iAd to help developers make more money. It's not a stretch to say that the initiative has not met expectations and now primarily serves commercials for iTunes Radio. Developers using the program have anecdotally reported poor fill rates versus iAd's competitors.\nHowever, the addition of new countries shows that Apple has not abandoned the project. On a related note, ago, Apple released a new version of iAd Producer a few weeks ago.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While playing any game, it is necessary for us to understand all its rules in a proper manner. Otherwise, it might result in conflicts and confusions. Hence, the present article on, Table Tennis Rules \u2013 Must Know Official Rules of Table Tennis is all about those rules, which one has to consider while playing Table Tennis.\nEvery year, the International Table Tennis Federation (ITTF) frames The Laws of Table Tennis. Our team of experts has carefully listed them out below for your better understanding.\nHere are some of the basic rules for playing Table Tennis. With the help of these rules, it will become a lot easier for us to play table tennis. Moreover, a proper understanding of these laws will also give you an awesome experience and a fair play condition.\nAccording to the rules of ITTF, we have to choose the perfect kind of equipment. The shape and size of the racket can be in any manner. It is necessary to keep 85% of the material of the paddle as natural wood.\nHowever, the blade of the racket should be flat. It is necessary to get the racket checked before the beginning of the match.\nAlso for the ball, it should be in 40 mm diameter. Whereas, it must weigh exactly 2.7 grams. The material of the ball should be celluloid or the same kind of plastic. It should be white, matt or orange in its color.\nThe guidelines for the playing surface states that it must amount to 2.74 meters in length as well as 1.525 meters in its width. Furthermore, the platform of the surface should be 76 cm from the floor.\nIt is also necessary to keep the top of the net from the playing surface at 15.25 cm. All the players, as well as the referee, can check all these conditions before the beginning of the match.\nWhile serving the ball, the server must let the ball free from the palm and then hit it. The ball must hit the surface before passing through the net.\nNext, the opponent has to hit the ball back with his paddle. This process will go on until and unless a point is scored by any of them.\nIn case a point is scored, the server will serve the ball again in the same manner. This will continue until the end of the game.\nIf the ball hits a part of your opponent player's body, then you will get a score.\nIn case your opponents hit the ball twice while applying for the single shot, it will get you a score.\nFurther, if the opponent fails to hit the ball, it would also result in a score for you.\nEven if the service is not done in a proper manner from your opponent, he will lose a score.\nWhile playing, if the part of your body or anything that you are wearing touches the work surface area or else the net, you might lose a score.\nWas that interesting? If you are interested in knowing about more rules of Ping Pong, we would recommend you to refer the handbook of the International Table Tennis Federation. For more stay connected with us!\nWe all know the importance of understanding the rules for playing a game. Here, in this article on Table Tennis Rules \u2013 Must Know Official Rules of Table Tennis, we have given a brief review of all the rules and regulations that we need to follow while playing Table Tennis. With the help of these rules, it will become a lot easier for us to play Ping Pong. Moreover, it will also provide us with a fair play experience in maintaining the memorandum of the game.\nWe hope with the help of this article, you will be able to establish a proper understanding of the rules and regulations. For more such articles, keep on following at pingpongtablee.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stam | \"We believe in ourselves\"\nIn little more than 24 hours, Jaap Stam will be stood in the opposite technical area to Jose Mourinho in his return to Old Trafford, trying to guide his Royals side to an FA Cup result against the mighty Manchester United.\n\"Everyone always looks forward to games like these. We want to show ourselves at Old Trafford against a side like Manchester United. We know they are in form and it doesn't matter which players they play \u2013 they have so much quality. So we know it's going to be difficult.\n\"But we've got confidence as well. We believe in ourselves. We believe we can do something. But that doesn't make it any easier when you're up against a team like United.\nWill it be the proudest moment of managerial career to date on Saturday?\n\"It would if we get a result! I was the type of player who wanted to win every game\u2026and I'm the same as a manager.\n\"Hopefully I'm at the start of my managerial career. But eventually I want success as a manager, because I want to win every game. That's not for myself, but also for the team I'm managing. I played at the highest level and I want to get there as a manager as well.\n\"We play QPR on the Thursday night next, so we have quite a few days to prepare the team for that one \u2013 so that won't affect how I pick the team for this tie. We'll put a strong team out there and I think they will to.\n\"This is a special game and everybody wants to play. But players are realistic as well, they know if they've done enough or if they are in good form.\nOn whether we are going to change our tactics?\n\"Maybe we can do things a bit differently. But one of our strengths is the belief we have in ourselves and the possession football we play.\n\"We will have the same approach as we have for a league game - we're not going to do anything different in our preparation.\nYou won the FA Cup in your first season at United, one part of the treble. Is that the greatest team you've played in?\n\"We had a great team with a lot of quality, but one of the most important things we had in the team was a determination combined with that certain quality.\n\"He's the type of manager who wants to win the league, the Champions League, the FA Cup, everything! That's what I like about him.\n\"Hopefully they support me on Saturday \u2013 I've heard they still sing my song! It tells you something in how I performed when I was there and the relationship I had with the fans.\n\"I've never been back in a proper game \u2013 I've been back for the games we played for Soccer Aid \u2013 but it'll be good to be back in front of the United fans in a situation like this.\nDo you think United and Mourinho are wary of an upset?\n\"He wants to win this game, so he's not going to be taking too many risks. But he does have a lot of choices in the players he has in his squad.\nDo you see the FA Cup as the greatest cup competition in the world?\n\"Yes I do. I've played in Holland in the cup, I've played in Italy in the cup \u2013 where it is never important unless you make it to the final. In England, the FA Cup has always been a great tournament.\n\"Everybody wants to do well in this competition and it also pits the smaller teams against the big teams, something that is great about this cup and something that needs to keep going.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is very likely the LAST Friday Dongers Club of the season. Well, let me take that back. This is the last GOOD Friday slate that we have. Yes, next Friday is the final Friday of the regular season but going into the final weekend series you will want to be very very cautious. Save some of that money for the MLB playoffs which are a) the best playoffs around and b) profitable!\nWe have some amazing news today. It's been a long long long long season and one of the major discussions coming into this MLB season was around the impact of the humidor in Arizona on the offense in that ballpark. I was very adamant (and I believe I was correct, so **** it!) that the humidor would not alter things such as the ability for batters to eye up the ball with a fantastic batters eye, or the fast infield grass (players words, not mine) or even the huge gap dimensions in the outfield and that this would remain a good hitters park to target when the situation is right. One of those situations is when the roof is open. So settle in space commander and let me tell you about this roof.\nUnfortunately we have had the lowest amount of games with the roof open in Chase Field in quite a long time. It's been closed pretty much since mid May and that is a drastic impact on the offense, more so than a stupid humidor. Anyone who has been with me long enough knows that I believe 100% in the history of the roof in Arizona and it's impact to cause games to go over their projected totals. Offense increases drastically and it's always something that is known throughout baseball as a big boost. There was even a fantastic report done by an Arizona Diamondbacks blog site detailing the impact of the roof and panels being open in Arizona and how some pitchers such as Curt Schilling (and I believe guys like Jeremy Hellickson and Zack Greinke) have lobbied for the roof to be CLOSED when they pitch. For all the natural reasons Coors Field is a hitters park, Chase field gets many of the same impacts when the roof is open.\noh wait\u2026 you want more analysis than 4 Rockies and 4 Diamondbacks tonight?\noh and the wind is blowing out over 14 MPH in Comerica park. now, I don't have full detailed analysis or any reports about how some former tigers pitcher hated the wind in detroit, but it's just one of those \u2026. well, lets call them \"Undocumented by 100% accurate unless it doesnt work out Theories\" that I have\u2026. HUGE boost for hitters here.\nJacob deGrom \u2026 Jake is on an absolute tear this year and while many dont like paying 12k for a guy who has trouble getting wins, this is always the exception. He's worth the high price today.\nNick Pivetta will face off in Atlanta against Julio Teheran tonight. Both pitchers have the ability to go complete opposite directions tonight either striking out 8-9 batters of giving up 3-4 homers. No reason to touch anything named Julio in Atlanta this weekend, so gimme Pivetta on the road. He will give up a home run to Tyler Flowers. Get that out of the way. But that aside Freeman\/Albies\/Acuna are a combined 3 for 30 off of him with 7 strikeouts.\nJohn Gant is a nice mid range option. Another guy who should be right in that 5-6 IP range without a ton of strikeouts but also won't give up a ton of damage tonight.\nWei-Yin Chen \u2026 I don't normally play guys with hypens in their name, but when I do, I play Wei-Yin Chen at home.\nRoss Stripling \u2026 Stripper is being bumped up a day because \u2026. You guessed it, Rich bleeping Hill is being flipped back a day. Home game vs Padres is a fantastic matchup for Stripling even though he hasn't been going deep in games in his last two starts. He has potential to get you 5 IP and a win today but it is dicey for that reason I am fading.\nKANSAS CITY vs TIGERS \u2026 On fire in September and 18+ MPH WINDS in COMERICA. Boost to both sides here.\nDidi G \u2026 Good park, is 4-5 off Yefry in limited sample. With Story out we're left thin here unless chasing Mondesi's hot streak.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyisv b/data_all_eng_slimpj/shuffled/split2/finalzzyisv
new file mode 100644
index 0000000000000000000000000000000000000000..e5997f962ab0cb15e479d9cdc2fbd5efdef53f40
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyisv
@@ -0,0 +1,5 @@
+{"text":"This Saffa Method review is a quick warning \u2013 this binary options program is a scam made to lose your money.\nThe alleged creator of this program Jake Mason claims that can make you earn hundreds of thousands of dollars within one month thanks to his Saffa Method.\nHe trades binary options and allegedly is willing to give you everything you need to succeed for free.\nThis method is the most cloned binary options scam we have ever seen. Every time it targets a different country, now it's obviously South Africa with the Saffa Method.\nBut scammers did not bother to invent a new name, they have already used Jake Mason with the Kiwi Method, meanwhile the photo of the guy is the same as in the Irish Method.\nLook at the picture on the right and you'll see the proof. So it is not necessary for us to again through everything, you can read our previous reviews of this method.\nLet us remind the real purpose of the Saffa Method. People who created get paid by their partner broker for referring new depositing clients.\nThis is why you will have to first deposit money with a broker that you cannot choose. Only the you will get access to the Saffa Method. But if you trade with this method, you will lose your deposit.\nAnother name of this scam is The Brith Method. It is again presented by Jason Taylor. Everything else is the same, so beware!\nThe Saffa Method is a well-known scam that is losing money to real traders, so stay away from it.\nPeople interested in making money in binary options trading should always start learning on a free demo with a regulated broker.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Postmedia Reveals New Apps and Websites for SUN Brands | Postmedia Network Inc.\nOctober 24, 2017 (Toronto, ON) \u2014 Postmedia Network Inc. (\"Postmedia\") today announced the launch of all new apps and websites for the Ottawa SUN, Toronto SUN, Winnipeg SUN, Calgary SUN and Edmonton SUN. This launch features new responsive apps and websites for the desktop, tablet, and mobile platforms enhancing the digital experience for readers.\nThe new SUN apps and websites offer a superior user and content experience around each of the SUNs' newsroom platforms \u2013 Local News, Sports, and Politics.\nSticky header for ease of navigation on all devices.\nCarousel slider for a one-stop-view and recap of local sports teams, final scores and highlights with no redirects and no new page loads.\nEngaging visual experience with images and videos at the forefront.\nA new longform story template with rich media features and easier sharing options.\nConsistent ad sizes and visibility across all devices for improved audience engagement.\nContextual ads for higher click through rates.\nA marketing campaign supporting the new SUN apps and websites was created in house by the Postmedia marketing team and launches October 25, 2017.\nPostmedia Network Inc., a wholly owned subsidiary of Postmedia Network Canada Corp. (TSX:PNC.A, PNC.B), is a Canadian newsmedia company representing more than 200 brands across multiple print, online, and mobile platforms. Award-winning journalists and innovative product development teams bring engaging content to millions of people every week whenever and wherever they want it. This exceptional content, reach and scope offers advertisers and marketers compelling solutions to effectively reach target audiences. For more information, visit http:\/\/staging.postmedia.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How Do You Manage Bleeding Gums?\nDo your gums bleed excessively when you brush or floss them? When you come back from the dentist's office, does your mouth look raw and wounded?\nWe see many patients who are dealing with bleeding gums. There can be a few different causes for this, but the most common by far is gingivitis. The initial stages of gum disease involve gum tissue that is inflamed and bleeds easily.\nThe first thing you should focus on is making sure that you engage in regular dental hygiene. This simple step can help ward off gum disease and can even reverse it when it is only in the beginning stages. By brushing after every meal and flossing every day, you can help keep your gums from bleeding. If they have already started bleeding, these regular practices will help them to heal.\nYou should also be sure that you come see us right away. We can take a look at your gum tissue and tell you whether it is something that is serious or not. What's more, we can help give you pointers on keeping your teeth and gums clean, and we can answer any questions that you might have.\nIn some cases of advanced gum disease, you should be aware that it is not enough to just start brushing and flossing regularly. If your gum disease has passed a certain point, you will have to have professional intervention to stop it.\nIf this sounds like you, it is imperative that you come see us as quickly as you can. The longer you wait, the worse things will get. We're here to help you, and we look forward to helping you get a healthy, happy smile back.\nSo give us a call, or drop by the office. We are here to help you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Smart Clause\u00ae is a constituent element of a smart contract. Instead of having to make a full contract executable or 'smart', a user can simply add programmable Smart Clauses to new or existing legal contracts to perform various functionalities.\nFor example, a 'delivery' Smart Clause and a 'payment' Smart Clause may be added to an existing supply contract template. This easily enables legal contracts to be made programmable and executable.\nSmart Clauses respond to data from the outside world (requests) and return responses. They may optionally emit events, such as a payment obligation.\nA Smart Clause is an instance of a Template, where the variables for the template have been set to specific values. A Smart Clause may be instantiated by either parsing natural language text that conforms to the structure of the template grammar, or may be instantiated from a JSON object that is an instance of the Template Model for the template.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Green Eco living Pods and Garden Offices use the most thermally efficient process in the UK.\nWe use a wall specification construction that produces the most energy efficient product that manufacturers construct for sale today. This creates an Insulated Panel which enables our product to be one of the best energy efficient products for sale on the market today.\nGreen Eco Living is a UK manufacturer and supplier of eco friendly, energy efficient home camping pods and glamping pods. Based in Lancashire UK, Green Eco Living are able to design, build, install and supply various types of pods throughout the UK. As well as glamping pods and camping pods we also manufacture hotel boutique rooms and leisure buildings of various construction types.\nWhy are we so different? Green Eco living pods and Garden Offices have developed the most thermally efficient process in the UK. We have developed a wall specification construction that produces an energy efficient product like no other product for sale on the market today.\nPlanning can be a complex process and this is why, at Greenecoliving we have teamed up with a professional town planner to lead and manage the process for you.\nGlamping show 2017 all set up for 3 day show.\nOur New arched shape pod The Coniston exhibiting at the Glamping Show 2017.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaasvh b/data_all_eng_slimpj/shuffled/split2/finalzzzaasvh
new file mode 100644
index 0000000000000000000000000000000000000000..a6554abcbcdd5d2066bd0944411edb72260974e1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaasvh
@@ -0,0 +1,5 @@
+{"text":"Traditionally, these Greek cookies are formed into rings, twists and half circles. In this recipe, they are shaped into bow ties and twists, then glazed with anise-flavored honey. These are delicately sweet and nice with coffee.\nMix honey, 2 tablespoons liqueur and juice in small bowl. Set glaze aside.\nPreheat oven to 350\u00b0F. Lightly butter baking sheets. Combine flour, salt and baking powder in medium bowl. Using electric mixer, beat butter and sugar in large bowl until fluffy. Add eggs 1 at a time, beating well after each addition. Beat in remaining 2 tablespoons liqueur, aniseed and lemon peel. Gradually mix in dry ingredients.\n9\/20\/07 I'm always looking for new twists to my Koulorakias & bring them to my church for my fellow Greeks to try,(they will always tell me if they like them or not!) they loved them.\nI've been making this recipe for years (ever since it came out in 1996). I don't care about its ethnic origin. As far as I'm concerned they are Greek because they contain honey and anise and that's good enough for me. If you're using fresh aniseed (I just bought mine from Penzey's) and really good quality honey (pine or heather honey is good, look for darker but not too dark honeys) be assured they are going to taste wonderful! So, don't be turned off by the other reviews! If you are looking for the less common flavors - you're in the right place. Print it and make it. They're worth the effort.\nI AGREE WITH THE COOK FROM CALIFORNIA, THESE ARE NOT TRADITIONAL GREEK KOULOURAKIA,MADE MOSTLY FOR EASTER, WHICH ARE USUALLY FLAVOURED WITH VANILLA, AND BRUSHED WITH EGG YOLK ONLY. OF COURSE THERE ARE VARIATIONS NOWADAYS,(LIKE CHOCOLATE ONES) BUT TRADITIONAL KOULOURAKIA ARE KEPT SIMPLE.\nThese were certainly easy to make, but they were nothing special. I recommend experiementing with other shapes than just the \"twists.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Check 5th Class Result 2018 Lahore Board online at Result.pk for free (Grade 5 Gazette Online)) PEC.\n5th Class result other than 5th Class Result 2018 Lahore Board from other various educational boards, BISE, Federal Directorate of Education, Punjab Education Commission, Educational Bodies like FDE.gov.pk, PEC.edu.pk, SEC, BED, KPKED, AJKEC can be viewed on their Gazette pages online at Result.pk, which can be accessed from top or left menu.\nPunjab Examination Commission is the administrative and responsible department to hold the primary and middle classes annual exams in the government schools of Lahore district. PEC starts the admission in the 5th class before start of the annual exams and receives the admission forms from the candidates. PEC issues roll number slips to the students of Lahore government schools during January and conducts class 5 annual exams in second week of February.\nPEC prepares ad issues Lahore grade 5th result to the students of the government schools located in Lahore. The candidates of Lahore government schools check their class V result 2018 on the website 2018.result.pk with latest updated information, which is uploaded on this site for the convenience of the students of the government schools. PEC declares the PEC primary class result 2018 in the main auditorium of PEC in the presence of students, teachers and PEC officials.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\ufffd Profit after financial items amounted to SEK 1,788 m (1,689).\nearnings per share of SEK 1.40 (1.33).\nto a margin of 53,7 per cent (54,4).\ncent. Operating margin was 14.5 per cent (14.5).\nTotal turnover excluding VAT for the H&M Group increased by 9 per cent during the first quarter (12 per cent with comparable exchange rates) and amounted to SEK 11,756.1 m (10,770.6). Turnover including VAT was SEK 13,807.0 m (12,635.7).\nTurnover increase in the month of February was 13 per cent excluding exchange rate fluctuations.\nH&M opened seven stores during the first quarter, of which three were in Germany, one each in UK, Belgium, France and the USA. Four stores were closed, bringing the total number of stores to 948. During the same period last year six stores were opened and one was closed.\nGross profit amounted to SEK 6,311.4 m (5,858.2), which corresponds to a gross margin of 53.7 per cent (54.4).\nAfter deduction of administrative and selling expenses, operating profit for the first quarter amounted to SEK 1,700.6 m (1,558.3), an increase of 9 per cent. This corresponds to an operating margin of 14.5 per cent (14.5).\nOperating profit for the period has been charged with depreciation according to plan amounting to SEK 307.7 m (284.1) and start-up costs, the part of investments in new stores that is charged directly to the income statement, of SEK 49.2 m (33.8).\nGroup financial net interest income was SEK 87.2 m (131.1).\nProfit after financial items was SEK 1,787.8 m (1,689.4).\nProfit after estimated tax was SEK 1,162.1 m (1,098.1), corresponding to earnings per share of SEK 1.40 (1.33).\nReturn on shareholders equity (revolving 12 months) was 31.1 per cent (33.6) and return on capital employed (revolving 12 months) was 46.8 per cent (50.8).\nThe group has, during the first quarter, increased turnover by 12 per cent with comparable exchange rates. Sales price to customer on new items, has as a result of the weakening of the US dollar during the same period decreased by approximately 5 per cent, all in all the number of sold items has increased by approximately 20 per cent. The increase in selling and administration expenses has been kept at a low level during the quarter.\nGross margin for the period was 53.7 per cent (54.4). The somewhat higher reduction level was, compared to last year, partly counteracted by an underlying strengthened margin due to the weakening of the US-dollar and somewhat lowered logistic cost. Price reductions were on a historically low level.\nThe result of the quarter has been negatively affected by currency translation effects by SEK 36 M, compared to the same period last year. The currency translation effects arise when the results of foreign subsidiaries are translated into SEK in order to be consolidated into the H & M Group accounts.\nGroup balance sheet total increased by 5 per cent and was SEK 26,510.0 m (25,195.7).\nDuring the period, the Group generated a positive cash flow of SEK 417.2 m (74.3).\nThe financial assets amounted to SEK 13,783.9 m (13,492.3).\nStock-in-trade was SEK 5,129.8 m (4,368.0), an increase of 17 per cent (-5).\nSEK 254.5 m (263.0) were invested in the operations through acquisitions of fixed assets.\nGroup solidity was 81 per cent (79) and the share of risk-bearing capital was 85 per cent (83).\nNet worth apportioned on the outstanding 827,536,000 shares on 29 February 2004, corresponded to SEK 26.00 (24.17) per share.\nThe Group plans to open 51 shops during the second quarter, the biggest part of the expansion takes place in Germany with eight openings, the UK and Poland with six each, Spain with five openings and Norway with 4 openings. Three shops will close during the quarter. In the corresponding period last year, 47 new stores were opened and three were closed.\nThe H&M Group has acquired the American clothing chain GAP's German subsidiary. H&M Group takes over all of GAP's 10 stores in Germany including employees. Take-over date is August 1, 2004. The store will be converted into H&M-stores with expected opening under H&M management during the autumn.\nDuring 2004, approximately 140 new stores will be opened and ten will close.\nDuring the three first quarters a historical tax rate of 35 per cent is used, final taxes are estimated during fourth quarter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Werlatone introduces the first in a new series of 20 to 1000 MHz, Mismatch Tolerant 50 dB Dual Directional Couplers. The Model C10561 (SMT) is compact, and conservatively rated at 250 W CW. Outperforming any competing product, Werlatone's design operates at approximately 30% the Insertion Loss (Less than 0.1 dB) and 65% less heat dissipation, in a smaller package. The C10561 provides superior main line and coupled port VSWR characteristics with Directivity better than 25 dB.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When applying sunscreen my face always comes first\u2026 I won't even sit out for my morning cuppa without applying it. I've been around for long enough to discover that there are consequences for not doing so and don't want to add to the damage of my misspent youth! I've been testing some new formulas, which I've included alongside a couple of firm favourites. All of these can be worn under makeup - a priority for me as I always wear some kind of base and don't want my sunscreen to interfere with it. If like me, you prefer a formula that sinks in quickly, doesn't leave a greasy film, doesn't break down your makeup or leave a white cast - read on to hear about my top picks.\nTINTED | La Roche Posay Anthelios XL SPF50+ is one of the most loved facial sunscreens out there - and with good reason. I've featured it a number of times over the past few years and tam loving he new Tinted Fluid formula. The tint is very slight and gives a subtle evening out of skin tone - great worn alone or under makeup. Note the word 'Fluid' - this is very runny, which gives it that lovely light texture on the skin and helps it sink in quickly.\nMATTE | Vichy Ideal Soleil SPF50 Mattifying Face Fluid Dry Touch is some kind of magic. This literally disappears as soon as I massage it into my skin. I mean it's still there, protecting, but looks and feels invisible - a true matte formula. I don't have any issues with oily skin, but if you do, this is an absolute winner. Of all the formulas I've tried, this one leaves the skin feeling like there's nothing on it at all.\nALL ROUNDER | Clinique Even Better Dark Spot Defense SPF45 is one that I've raved about repeatedly and recommend to so many friends, who all love it. Another light textured lotion that sinks in quickly, is 100% non chemical and has a brightening effect on the skin. I love the tiny tube which makes it extra portable and great for travel.\nSENSITIVE | Eucerin Sun Creme SPF30 Face Dry Skin is part of a skincare range for sensitive skins that I've been trying out recently - and I'm really impressed with their sunscreens. My skin does tend to feel extra parched in Summer and I like products that offer moisture without heaviness, which is exactly what this does. It's a cream formula that blends into skin almost immediately, leaving it prepped for makeup. If you worry about sunscreens breaking you out, give this a go. Saying that, I think the issue of SPFs and breakouts is often that people don't remove sunscreen efficiently - they definitely require a double cleanse.\nNATURAL | If you prefer a more natural formula, you may like to try Melvita Prosun SPF30 with 100% mineral filters. Personally I wouldn't opt for this one under makeup as it does leave a white cast and takes a bit longer to sink in, but the Buriti and Argan oils feel really nourishing and smell amazing. I also like the really thin nozzle which makes it easy to dispense a small amount at a time.\nOVER MAKEUP | Most sunscreens only offer two hours of protection, so what about topping up throughout the day if you're wearing makeup? Ultrasun Sunscreen really does offer all day protection (tested in Australia and Singapore) so I use this one when on holidays when I'm out and about in the sun all day and don't want to worry about top ups. Alternatively, Ambre Solaire Sensitive Dry Touch Protection Mist SPF50 can be applied over makeup. It's not designed for this, but it works - just be sure to keep your eyes closed or cover them with a scarf or hair band. NB this won't double up the protection you've applied below makeup, but can be used as an alternative or for top ups over makeup or on bare skin.\nIN SUMMARY | Clinique has been my go-to for the past two Summers and I can't fault it, though currently reach for La Roche Posay the most as it's such a lovely formula to use, I like the extra touch of skin-evening and it has the highest SPF factor - now that Summer has actually arrived! I opt for Vichy when I'm doing makeup in a rush and need it to sink in on contact (cos the LRP takes like 30 seconds). When my skin is playing up, I feel safe with the Eucerin as I know it won't break me out, though none of these do to be fair. I use either the Vichy or Eucerin on my 10yo daughter, who dislikes the feeling of sunscreen on her skin, but is lily white, very blonde and burns easily without it.\nNext up, SPFs for Body.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaauah b/data_all_eng_slimpj/shuffled/split2/finalzzzaauah
new file mode 100644
index 0000000000000000000000000000000000000000..bbe792c034bd1442ea587407ae0e3e08852e7cc8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaauah
@@ -0,0 +1,5 @@
+{"text":"6130-00-106-9040 is not a replacement of any national stock number and has not been cancelled or replaced itself.\nDoes NSN 6130-00-106-9040 have a shelf life?\nHow is national stock number 6130-00-106-9040 defined?\nWhat is the NIIN of NSN 6130-00-106-9040?\nIs NSN 6130001069040 ITAR controlled?\nWhat is the most recent solicitation for NSN 6130001069040?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SAVEVIDEO.ME \u2013 How to download from Dailymotion video?\nHow to download from Dailymotion video?\n46 Comments on \"How to download from Dailymotion video?\"\nThe algorithm of search direct-links for Dailymotion has been updated.\nSometimes Dailymotion gives video with incorrect extension, therefore is better to rename a video-file at saving. For example, if video MP4 the filename should be in such form: \"videofile.mp4\".\nTry to repeat downloading again later (in other time).\nCould you please send Page URL (from Dailymotion)?\nIt will help to solve this problem.\nhi.i cant download from here.always it says :Could not find any download links or site not supported :-( although the file exist and i can see it.\nCould you please explain your problem in detail?\ndude, does it support downloading age limit video and private video?\nSAVEVIDEO.ME works for \"public videos\" only.\nNo. The SAVEVIDEO.ME server are temporarily overloaded.\nTry other video player in your phone.\nPlease send the video URL which are not working.\nCan I download here, a private video in dailymotion?\nHow to download video via mobile phone?\nIt depends on your mobile device and your browser.\nIf a file starts automatically in a browser window, you can try to save using a browser context menu (Save File As\u2026 etc\u2026) in your mobile device.\nWhat should be the file format for Galaxy Tablet? I downloaded a video once, but it stopped. Why?\nYou can save the file as \u2013 \"filename.mp4\" for Dailymotion.\nPlease send Page URL (from Dailymotion) which are not working.\nAlso, try the high-speed Internet for downloading.\nIf you use the old \"Opera\" browser (mini, mobile etc\u2026) stability of downloading isn't guaranteed for your device.\nIt works fine in \"Mozilla Firefox\" and \"Google Chrome\" browsers.\nso i can't open dailymotion from opera mobile ?\nOpera Mobile (or Opera mini) is a mobile web browser which changes web content to accelerate downloading for mobile devices.\nTry \"Mozilla Firefox\" or \"Google Chrome\".\nhi,how to update the adobe flash player??\nYes. Try to use the native browser (BlackBerry) with the high-speed Internet for downloading of videos.\nCan I download from Kshowonline?\nNo. \"Kshowonline\" is not supported.\nSAVEVIDEO.ME works mainly with video hosting sites.\nwhat is the download link age?\ni am able to send download link to my friends is it work fine if i send it?\nIt depends on Dailymotion server.\nDailymotion downloads stopped working. It only gets a html now. They probably changed their codes.\nThe SAVEVIDEO.ME server is temporarily overloaded.\nHello How To Download Video from Daily Motion\u2026.\nMy mobile ?? is Nokia Lumia 520\u2026..\nThe page savevideo.me is not properly open on my PC, where i paste dailymotion video URL for download. What to do now? Please help me.\nYou can just press the \"Share\" button (icon like an arrow \u2013 at the top-right corner of the DM player) and then choose \"Embed\" (icon like angle brackets <>) to copy a link (or Permalink) in a popup window.\nCurrently, it works only when a direct link (to a video file) has been generated in the same web browser. So, a direct link to a video file can not be available to third-party web browser or users\/friends.\nAlso, you can try to use the context menu to choose \"Save Link As\u2026\" from the SAVEVIDEO.ME results or just try to use other browsers like \"Chrome\" instead of your current web browser.\nI'd been using SaveVideo to save and download videos from DailyMotion for months now but it has stopped working. I think it coincides with some add ons that were installed on my Firefox browser, due to excessive pop-ups etc. I can no longer download any videos. Everytime I try to select 'save link as' it just doesn't respond or do anything. I have also tried it with Chrome but have the exact same result. Do you have any advice?\nYou can try to empty browser cache, remove cookies, reset browser settings to \"default\" and restart.\nAlso, you can update browser to the latest version.\nCould you please share your page URL which has this issue (here or via Feedback form)?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Zenergy Ball Chair by Safco is a great option for an active seating solution at the desk or in a meeting area, waiting room or classroom. Sitting on this chair engages your core and encourages greater movement through the bouncing motion.\nWith six different colour options, you can choose the ball chair that works best with your style and d\u00e9cor. It is also constructed with a thick cover that is easy to clean with a sanitizer wipe, making it ideal for classroom settings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OK, my 55 Belvedere is a driver, not a restoration. I have converted to disc brakes in the front and am working on a 8-3\/4 rear end now. My next change will probably be the leaky PF transmission. Am I right in believing I can bolt on a torqueflght? If this is covered in an archived file can someone direct me?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gary joined EENA in October 2006. He is responsible for the strategic development of the association. Over the years, he led the creation of several successful EENA membership platforms, committees and events. He is in charge of government affairs and he managed a number of advocacy campaigns in Brussels. Gary holds a MA in European Affairs from the Political Science Institute of Strasbourg and a BA in Economics from the University of Orl\u00e9ans and the Abertay University in Dundee. A Europe-trotter, he studied or worked in Belgium, France, Ireland, Spain, Poland and the UK. He is a French national, also fluent in English (spiced up with some little basics of Spanish and Polish).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadktz b/data_all_eng_slimpj/shuffled/split2/finalzzzadktz
new file mode 100644
index 0000000000000000000000000000000000000000..a2b92f0af11f52486a954708bba88dbab78dba6d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzadktz
@@ -0,0 +1,5 @@
+{"text":"b. What is your birthdate?\nc. What is your husband's occupation?\na. The social worker has social work privilege and cannot be asked to provide confidential case information.\nb. The social work privilege is not absolute and the worker may be required to testify if ordered to do so by the judge.\nc. The Code of Ethics requires that the worker refuse to testify.\nA fearful client in a mental health agency is repeatedly asked for money by his drug dependent son. Recently, the son has threatened violence and calls the father at all hours. How should the social worker advise the father?\nA client diagnosed with schizophrenia has been released from an inpatient unit and is being followed at a community mental health agency. The client is stable enough to participate in a group. What type of group is the social worker likely to recommend?\nA woman brings her husband to a busy emergency room because he is not feeling well. He dies while there. What should the social worker do?\nEmpathy has been described as an attitude that engages a client in treatment. Which of the following best describes empathy?\nA client is in the hospital and an attorney calls requesting his records. What are your ethical obligations?\nA single mother with 3 children is unemployed, has no support or family who live in the area, no services for the children, and no resources for herself. The mother states she would like a support network. What type of services would you provide for her?\nA client is told by her doctor that she has breast cancer. She is distraught and says she doesn't know what to do. She has 6 children. Which of the following would you do?\nA 5 year old boy is watching his 18 month old sister have her diapers changed and becomes aggressive. The mother is concerned. What would your intervention be?\nA 70 year old woman lives with her 82 year old sister. They are having difficulty taking care of their basic needs. They have no support or help with shopping, chores, or self-care. What would you do?\nYou decide that one of your clients could be best helped by a treatment intervention that is outside your scope of practice. What would you do?\nA female client is having difficulty eating and sleeping. She cannot complete her work because she is plagued by constant worrying. What is the most likely diagnosis?\nTwo social workers are in a crowded restaurant when one of the social worker's clients approaches them. What should the social worker do?\nYou have been seeing a client for a while when you discover you are dating his brother. What should you do?\nWho would be most at risk for abusing his\/her children?\nd. a woman who was raised in homeless shelters.\nWhich of the following is an issue that lawmakers may need to decide?\n65. What do you do first in a clinical interview with a client?\nA boy is in a wheelchair. His peers at school have been making fun of him. What would be the best intervention for a social worker to make?\nHow might a black client initially relate to a white social worker?\nA social worker is making little progress with a client after 4 weeks of treatment. What might the social worker do?\nOf the following, which would be least important in the acculturation of an Asian girl in treatment with a social worker?\nA child is engaging in risk-taking behavior and is disruptive with his parents and peers. What is the most likely diagnosis?\nA newly referred adult male does not trust women. You are a female social worker. What would you do?\nA client is displaying extinction of a particular operant behavior. What do you expect?\nA child is experiencing separation anxiety during the school day. What should you do?\nHow would you explain a DSM diagnosis to a client?\nHow would you discuss the results of the Beck Depression Inventory with a client?\nA man in the army is referred to a social worker for alcohol abuse. He and his wife have been having many arguments about his drinking, which have escalated into physical abuse. What you consider for an intervention?\nA client has been voluntarily admitted to a psychiatric hospital. He is psychotic, HIV+, and suffers from alcohol abuse. His parents want nothing to do with him. What would you do first?\nWhat is the best way to control for external validity?\na social worker discovers that a nurse is giving more pain medication to a patient than the doctor has prescribed. What should the social worker do?\nTwo gay men have been voluntarily admitted to a residential facility. One is HIV+ and one is not. During their stay, they engage in a sexual encounter. How would you respond?\nNative Americans are often distrustful about whether social workers will be able to provide treatment in a value-free manner. What would be the best intervention to promote trust?\nWhich of the following is most difficult to indentify in a pre-school age child?\nA social worker sees and teen boy and his parents. The son has been displaying serious conduct problems. He is the identified patient. Which of the following is most likely to be true?\nYou are working with a client who exhibits some symptoms of alcohol abuse but denies that he has a problem. You decide to complete a genogram with your client. In this case, what would be the purpose of doing a genogram?\nA woman comes in to your agency upset about her 4 year old son who is wetting the bed and having temper tantrums. A new sibling has just been born. What is your first intervention?\nYou are seeing a Somali woman who has been in the U.S. for one year. She has been experiencing nightmares, difficulties in her personal relationships, inability to sleep, and difficulty concentrating, and her affect is flat. What might you suspect for a diagnosis?\nAn attorney calls to request a client's case file. What is your best initial response?\nA female African American senior citizen has been coming in to your agency daily asking for support. She seems depressed and is recently retired from a job that she held for 40 year. She currently attends a senior citizen center where she eats one full meal a day. She is fairly involved in her church. What would you suspect as this client's presenting problem?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These shoes are sizing quite small.\nYour Fur deluxe sliders are crafted from luxury recycled vintage furs for supreme warmth during the winter. Those slippers are handmade and handcrafted.\nHOW TO CLEAN YOUR FUR DELUXE SLIPPERS ?\nDo not put your FUR DELUXE SLIPPERS in the sand.\nDo not put your FUR DELUXE SLIPPERS in the water or on a rainy day.\nDo not wash or dry your FUR DELUXE SLIPPERS in a machine.\nDo not use chemical on your FUR DELUXE SLIPPERS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vladimir and I were right after all.\nI heard you and Turkeer had a big fight.\nHonestly, Paola, I can't remember.\nMaybe Barbara saw something he shouldn't have.\nShari didn't quite understand what was being said.\nThere's something about Adlai that I don't like.\nDo you remember when Nanda was here last?\nReiner certainly knew about the problem.\nKate is in the kitchen making dinner.\nElaine admitted that he ate all the ice cream.\nRajesh missed the whole thing.\nShould I tell Maureen you're coming?\nRandolph is an amateur astronomer.\nFrancisco is old, isn't he?\nManavendra knows he can always count on Marco.\nSrinivasan was all by himself.\nPlease extend my apologies to Moses.\nJi walked as far as he could.\nLaurent certainly seems easier to get along with than Stewart.\nI'll never forget the look on Dimetry's face when he found out that Sundar had been injured in a traffic accident.\nI don't even know what Jess's real name is.\nAdam told me where you were.\nRomain doesn't need to work on Sundays.\nClarence doubts if it will rain.\nI had to stop Hon from making the biggest mistake in his life.\nPiercarlo could hardly believe him.\nRavindranath sat back and smiled.\nDo you have a crush on Marci?\nVincent patted himself on the back.\nRichard tried to kiss Uri.\nElsa isn't on any medication.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I went for a walk in our local park the other day. The trees were just turning their autumn golden-brown and russet colours. The sun was shining, reflecting in the lake; the sky was bright blue, and the air unseasonably warm, with a gentle fresh breeze. All was well. At times like that, it's natural to feel praises welling up within you. Worship is easy. God is good!\nPsalm 47 has that sort of feeling. The psalmist was enjoying life, and praises rolled off his tongue. It was a great moment.\nSinging and praising are good for us. They release endorphins \u2013 the 'feel-good' chemicals into our brain, as well as increasing our oxygen levels. It's not about how it sounds, but about how it feels. Never tell yourself you can't sing \u2013 it's an order!\nThere are plenty of times in life when it's hard to praise, when we have to discipline ourselves to rejoice, 'because the bible tells us to'. Isn't it great when you get those moments of sheer joy, when praise springs forth unbidden, and worship is the most natural thing in the world? Cherish those times, etch them on your memory, and use them as a resource to help you continue worshipping when the going gets tough.\nJoy is contagious. Read this psalm and catch some for yourself!\nThis entry was posted in Bible, Christianity, Psalms, worship and tagged autumn colours, enjoying God, enjoyment of life, joy, praise, worship. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tired of contact lenses and eyeglasses? Do something good for your eyes, learn how to improve your eyesight without glasses.\nAbout half of the world's population suffers from some kind of eyesight problem. For people having trouble seeing clearly, ophthalmologists readily prescribe the use of corrective lenses, such as glasses or contact lenses. However, like with other conditions, many seek alternatives. Though not widely accepted in current medical practice, some eye doctors support natural treatments, instead of simply prescribing mechanical aids to correct vision. A number of these practitioners also support the notion that the use of corrective lenses may only make matters worse because the eye is allowed to remain in its abnormal state as opposed to relying on natural methods that encourage the correction of any eye defects. The guide on how to improve your eyesight without glasses points to several natural methods that help your vision normalize.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadlds b/data_all_eng_slimpj/shuffled/split2/finalzzzadlds
new file mode 100644
index 0000000000000000000000000000000000000000..b7dc12783869291da5c2f7d23c8cbeb971cf38fa
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzadlds
@@ -0,0 +1,5 @@
+{"text":"If you are looking for a low cost flight deals to Mian Yang but you can't decide when to leave. Check the search results and you will find the perfect flight deal to Mian Yang that suits you, making it is a good reason to ask for time off.\nDon't despair if you can't find a cheap flight to Mian Yang. Why not try being flexible with your dates? With lastminute.com you can set the search filters to flexible dates making it easier to find your flight to Mian Yang!\nCheck out our flights to Mian Yang when you have some free time. You can find the perfect flight for you by checking availability on lastminute.com... refine your search further using our filters and you will be well on your way to Mian Yang.\nBook a low cost flight to Mian Yang with lastminute.com's travel website in next to no time. Complete the form and within seconds the most economical options available for a low cost flight based on your search criteria will appear.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Personal Information\" means the following information about you that we collect: name, mailing address, billing address, e-mail address, telephone number, credit card information and any other information collected or stored in conjunction with your name. Personal Information does not include aggregate information about our customers in which your identity cannot be discerned.\nSubmissions from You. We are a condom company and, like most companies, we collect information from you when you register on our Site or mobile application or engage in any of the following: purchase a product, subscribe to our monthly delivery service or our newsletter, or participate in contests, sweepstakes or promotions. The information we will more than likely ask for is your name, shipping\/billing address, telephone number, e-mail address, credit card information, gender, etc. We also receive information from you when you e-mail, write or call us or we call you. We may also maintain a record of your product purchases.\nIf you choose not to provide your Personal Information to us, you may miss out on benefits like receiving our monthly newsletter, making purchases, and learning about \"special\" offers and promotions.\nAutomatic Collection. We automatically receive certain types of information whenever you interact with us through our Site or mobile application. For example, when you visit the Site or use the mobile application, our systems, or those of our service providers, may automatically recognize your IP address(es), domain name and location, a unique identifier for your device, date and time of visits, the type of browser you use and a unique identifier for your browser, the website from which your visit originated, the website you visit after leaving our website, page visits, time spent on web pages, the advertisements clicked on or scrolled over and other data. Below we describe certain information collected, and the actions and technologies that facilitate the collection of such information, in greater detail.\nIP Address. Your IP address is a number that is automatically assigned to your computer by your Internet service provider whenever you are on the Internet. When you view pages of our website, our servers may record or \"log\" your IP address and sometimes your domain name and location.\nDevice IDs. You may be using the mobile application or visiting our website from your mobile device, such as a mobile phone or tablet. Certain mobile service providers uniquely identify mobile devices and we or our third-party service providers may receive such information if you access our website through mobile devices. Some features of the website may allow for the collection of mobile phone numbers, and we may associate that phone number with mobile device identification information.\nClear GIFs\/Tracking Pixels. A clear GIF or tracking pixel is typically a one-pixel, transparent image (although it may be visible) located on a web page or in an e-mail, which is retrieved from a remote site on the Internet enabling the verification of a person's viewing or receipt of a web page or message.\nWe may use the information described above to operate the Site or mobile application, to send you service and administrative announcements, to use and provide others with general statistics regarding the use of the Site or mobile application, to analyze and develop our business strategy, to determine how you found out about us and your interests regarding our products, to determine how to improve our website or mobile application, and to create marketing programs for you and others. To accomplish all this, and also to allow us to offer our customers streamlined ordering and other useful features, we, as do many websites, use and allow our third-party advertising companies, software vendors and contractors to use the technologies listed above and other similar technologies (\"Tracking Technologies\").\nInformation from Facebook. You may also register by using your Facebook log-in credentials. When you log in with your Facebook credentials, we collect the Personal Information you have made publicly available in Facebook, such as your name and profile picture and use that information to register with our Site. We may also obtain other information with your permission, such as your gender and date of birth. The information we obtain may depend on your Facebook privacy settings.\nInformation from Other Sources. We may obtain information about you from outside sources and combine it with your account information. For example, we may purchase demographic and other marketing information from third parties to help us better serve you or inform you about services or products that we think will be of interest to you. Finally, we may also purchase lists containing your Personal Information for advertising and marketing purposes. In some cases, we may link information that is not personal information and which was automatically collected through Tracking Technologies with certain of your Personal Information. In such event, the information will become Personal Information. We use that combined information to enhance and personalize the shopping experience for you, to communicate with you about our products and events that may be of interest to you, and for other promotional purposes.\nUse of Google Analytics. Like many websites, we have enabled Google Analytics Advertising Features, including Demographics and Interest Reports and Remarketing. We use the Google Analytics Cookie and the DoubleClick cookie for purposes of advertising to users who may have visited our site previously. You may opt out of these Google Analytics Advertising Features by using a cookie blocker\/disabler on your browser and through Google's Ad Settings Feature and other currently available opt-outs for the web.\nSharing with Business Partners. From time to time we might establish a business relationship with other persons or entities. In such cases we might rent, exchange, share and\/or cross-reference information, including without limitation your Personal Information, and this will enable such persons or entities to contact you regarding products and services of potential interest to you. If you are a California resident, California Civil Code Section 1798.83 may permit you to request a notice regarding the disclosure of your Personal Information to third parties for the third parties' direct marketing purposes. If you are a California resident and would like a copy of this notice, please submit a written request to the address provided below under \"Questions or Comments\".\nSharing with Service Providers. We may also engage persons or entities to provide certain services to or for us, including without limitation credit card processing, shipping, data management, promotional services, etc. We provide our service providers with the information (including without limitation your Personal Information) needed for them to perform these services.\nSharing for Contests, Sweepstakes and Promotion. Similarly, if you enter a contest, sweepstakes or promotion on or associated with our Site or mobile application, your Personal Information may be shared with co-sponsors or fulfillment companies for the contest, sweepstakes or promotion.\nSharing with Advertisers. We may participate in behavioral-based advertising. This means that we or a third party may use Tracking Technologies to collect information about your use of our website or other internet usage so that they can provide advertising about products and services tailored to your interest. That advertising may appear either on our website or mobile application or on other websites or mobile applications. We also may share traffic and transaction information with business partners and advertisers on an aggregate and anonymous basis.\nSharing with Social Media Companies. When you use our service, you may share Personal Information with social networking websites, such as Facebook, Google+, Pinterest, Twitter and Foursquare. You may also share information with other web applications, such as Outlook, Google Calendar and Yahoo! Calendar. We do not share information with these third parties unless you direct us to share it.\nSharing to Protect or Enforce Legal Rights; Sharing with Law Enforcement. Although unlikely, in certain instances we may disclose your Personal Information when we have reason to believe that it is appropriate to identify, contact or bring legal action against you or against persons or entities who may or who may be causing harm to you, to Sir Richard's or to others. We may also disclose your Personal Information when we believe the law or legal process requires it.\nIn order to ensure that the information we maintain is accurate, Sir Richard's gives you the option to change the information that you previously provided during registration by logging in and clicking the Account link accessible through the Shop page on our Site. Although this will change the information that is displayed to you, please note that once you have provided information to us, we may store and maintain that information indefinitely. Our credit card transaction processor may store and maintain your credit card information and billing address indefinitely.\nWe use reasonable and appropriate safeguards to protect the information we collect from you. Nonetheless, no data transmission over the Internet can be guaranteed to be 100% secure. Therefore, we cannot guarantee the security of any information you transmit to or from our website or mobile application, and you do so at your own risk. ACCORDINGLY, WE DISCLAIM ANY LIABILITY FOR ANY THEFT OR LOSS OF, UNAUTHORIZED ACCESS OR DAMAGE TO, OR INTERCEPTION OF ANY DATA OR COMMUNICATIONS. WE DO NOT ACCEPT LIABILITY FOR UNINTENTIONAL DISCLOSURE OF YOUR PERSONAL INFORMATION. BY USING THE SITE OR MOBILE APPLICATION, YOU ASSUME THESE RISKS.\nWe do not direct this Site or mobile application at children (persons under 18 years of age) and we do not seek or knowingly collect, maintain or use information from children. If you are under 18 years of age, please do not provide us with any information.\nYour Personal Information will be stored and processed on our computers in the United States of America. The laws on holding Personal Information in the United States of America may vary and be less stringent than the laws of your location. For example, if you are located in the European Union or other regions with laws governing data collection and use that may differ from U.S. law, please note that you are transferring information to the United States of America, which may not have the same data protection laws as your location. If you object to your Personal Information being transferred or used in this manner, please do not register with or use the Site or mobile application.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Come join us for MORPHOS Dome (fulldome) artist-in-residence for an experience of a lifetime.\nCALL FOR ARTIST UNTIL April 30th, 2019.\nMORPHOS is a one of a kind digital dome artist-in-residence program that provides workshops and individual support for artists who want to take their work to the immersive level. Our goal is to expose and train artists in full immersion production for a growing need of digital dome (fulldome) art. 2018 welcomes our fifth year of hosting this unique experience. Come join us for MORPHOS Dome artist-in-residence for an experience of a lifetime held September 27 \u2013 November 4, 2019 in Colorado.\nMORPHOS digital dome artist-in-residence program is for artists interested in experimenting and exploring 360\u00b0 immersive digital art through digital dome. Artists will receive hands-on training in techniques for fulldome and Virtual Reality, weekly dome time, individual consultation, production support, group outings, a group closing dinner, and public presentations of their experience at the end of the residency. MORPHOS artists have gone on to present at international dome conferences, festivals, and art shows.\nRecent advances in technology are rapidly changing the way in which we view digital media. The digital dome offers a unique immersive group experience with it's half sphere screen, high-resolution digital projections systems, and surround sound. With recent innovations that allow for easy drag and drop playback, live VJing, interactivity, and gaming platforms, the virtually untapped possibilities in the digital dome are endless. Artists will also receive training in Virtual Reality, which is entering the consumer market with high refresh rates that allow for individual full spherical 360\u00b0 immersive experiences and is a great way to previz dome work.\nA new fulldome show was just added to the Fulldome Database, check it out: Dinoscape: Planet Dinosaurs by Eyedin Info, trailer & full length preview available.\nAndr\u00e9 W\u00fcnscher, an experienced 3D animator and years-long supporter of the Fulldome Festival, presents an interactive 360\u00b0 arch vis demo in the dome.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are hundreds of companies and websites who provide web hosting and domain solutions to millions of people worldwide. The problem is finding the right and best solution provider is like searching needle in a haystack. Specifically, when it comes to students, they always look for cheap and reliably working domain plans. What if I bring to you some really cheap & free hosting providers for students projects? Just have a close look at these websites.\n000WebHost is a crowned king in the industry of providing free web domain and hosting services without tolerating any crappy advertisements. The provider charges absolutely no hidden costs which are depicted in their name as well ($0.00). With no restrictive terms and no advertisements at all, the provider is a must try for students who are looking for free domains. The speed is claimed to be as fast as light; excellent customer support and maximum reliability gives it number one place in our list. The host is the only free web domain company which guarantees a 99% uptime. You can cross check this statement at their website where they show all the stats. For most of the servers, the uptime also crossed 99% to stop at 99.9%. No other host provider can guarantee this accuracy and efficiency.\nWeb Hosting For Students is an ideal place to crash for the students who need hosting plans to practice specific hosting files for a course or the students who are supposed to host personal and school projects. The service is also open for professional students who like to host client-side, along with their existing work, you can also buy a domain name with your hosting account.\nThe web hosting company offers different plans. One plan is designed especially for average web designers who are students and at the same time, need a simple yet affordable plan in order to design and host the school projects. The student plan gives the ability to host multiple sites at one time with a single account. The plan includes databases and emails as well. For the professional ones, the host provides 3 client websites for only $10 a month to earn more profit.\nSite Ground offers extremely cheap solution for web hosting and domains for students. The plans include solutions for organizations, educators and scholars other than students. They do not want students to compromise on some unreliable hosting plans. For this reason, their price range is cheap and according to the industry standards. The host offers $1.99 a month for students and it may differ for other purposes or organizations. They give special pricing for students for a one-year limit which are easy to use with great technical support.\nThese were the three renowned and in-budget and free hosting providers for students projects that may be found around the web. If you consider one of them, feel free to contact their help centers through email or phone number.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am a trial attorney practicing primarily in the area of personal injury and civil litigation.\nI joined the law firm in 1978 and have successfully handled numerous cases in state and federal court. I have been practicing personal injury litigation for 39 years.\nI was admitted to the bar in Minnesota and also the U.S. District Court, District of Minnesota.\nCommitment to excellence and accessibility to clients are cornerstones to my practice. I, along with all of our attorneys and staff, am dedicated to providing the highest quality of representation to our clients.\nMy practice has taken me to many parts of Minnesota and out of state as well. When it comes to representing injured people, my team and I bring experience and success with the goal of attaining the best resolution for our clients.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafbjp b/data_all_eng_slimpj/shuffled/split2/finalzzzafbjp
new file mode 100644
index 0000000000000000000000000000000000000000..5f23f062eeb82b464e3cc65a0e852693b6fe81c4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzafbjp
@@ -0,0 +1,5 @@
+{"text":"Technicians available. Sewage backup cleaning in Toronto.\nThere is nothing worse for your home, your finances and your stress levels than a backed up sewer. Sewer backups can cause permanent damage to the foundation of your home itself \u2013 including your floors, walls, furniture and electrical systems. The three primary causes of your sewer backups are: sewage backflow, lateral drain, and drain stack\/main drain\/sewer line clogs. Luckily, sewage backup cleaning in Toronto is affordable, reliable and impactful. When you hire our sewage backup cleaning company in Toronto, you are hiring some of the top experts in the field, backed by a guarantee to protect your home.\nThere are many ways to prevent sewage backup and thus prevent the need for sewage backup cleaning in Toronto. We recommend the following preventive measures: safe and correct removal of grease, proper paper product disposal in the toilet, replace your line with a new plastic pipe, hire professionals to fix illegal plumbing systems and\/or install a backwater valve. For help preventing sewage backup, call our Toronto sewage backup cleaning service today. We offer more than just sewage backup cleaning for Toronto emergencies. We offer inspections to check your sewer to ensure a safe and reliable system is in place.\nWhen we provide sewage backup cleaning in Toronto and sewage backup cleaning in Mississauga, we guarantee satisfaction. From wet-vacuuming to remove spillage, to mopping, disinfecting, steam cleaning, and repairing damaged area, you will hardly know a backup occurred. Included in our Toronto sewage backup cleaning service is Infrared Thermography (IT). IT is a technique that produces an image of infrared light emitted by objects due to the heat, or lack thereof. Infrared inspection is a fast, non-invasive method to discover moisture intrusion within a building to help with early detection of mold or potential risks of backup. Call today for sewage backup cleaning in Mississauga and Toronto!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"European equity markets are in the red as we approach the end of the trading session.\nDealers continue to be on edge as the talk of a trade war adds to the negative sentiment. The EU are talking about levying a tax on digital revenue as a way of clamping down on tech giants paying a relatively small amount of tax in Europe. This has added to the jitters surrounding the possibility of a trade war.\nThe French division at Kingfisher continues to be one of the worst performers at the group, and a stock availability issue weighed on the firm's figures. The Screwfix business in the UK and the Polish division are in rude health. Revenue rose by 3.8%, but profit slipped by 10%, and the company stated that the business climate was mixed. The less-than-optimistic forecast, combined with continuing French issues, dented investor sentiment. If the stock remains below the 200-day moving average at 319p, it could target 298p.\nSoftcat shares have been hit by a bout of profit-taking. The company revealed respectable figures, but investors decided to take some cash off the table. Revenue for the first-half increased by 24.8% and gross profit rose by 22%, while the interim dividend was boosted by 3.3%. In January and February a number of investment banks upped their ratings for Softcat, so it would seem that brokers had high hopes for the stock. The share price has been in an upward trend since late 2016, and today's move lower may entice new buyers.\nIndices are mixed today as investors remain cautious over the possibility of a trade war. The EU are contemplating imposing a 3% tax on various online services, and this has added weight to the argument we are heading for a trade war. It is possible we could see pressure put on tech giants such as Alphabet, Google's owner.\nFacebook is still in the news for all the wrong reasons, as the scandal in relation to Cambridge Analytica has tainted the company's image. The stock lost ground in early trading, but has now turned positive.\nAt 6pm (UK time) the Federal Reserve is widely tipped to raise interest rates to 1.75% from 1.5%. A growing number of traders are now talking about the possibility of four rate hikes this year. Jerome Powell took over the Fed chair in February, and he is likely to pick up where Janet Yellen left off in terms of language and tone.\nUS total homes sales in February jumped to 5.54 million, up from 5.38 million, with economists expecting a reading of 5.4 million.\nGBP\/USD has pushed higher on the back of solid unemployment and wages data from the UK. The jobless rate in the UK fell to 4.3% - its joint-lowest since records began, and monthly earnings ticked up by 2.6%. With British inflation sliding and earnings rising, things are looking up for consumers in the UK. Sterling has been climbing versus the US dollar since the start of the month, and if 1.4088 is cleared it could target 1.4150.\nEUR\/USD is gaining ground on the back of the weaker greenback. There were no major economic updates from the eurozone today, but traders will be keeping an eye on this evening's Fed meeting. Dealers are treating an interest-rate hike of 0.25% as a done deal, but the press conference that follows might give us an idea what the US central bank will do throughout the year.\nGold has been lifted by the softer US dollar, but traders may not be overly long going into the Fed meeting this evening. The press conference could lay the foundations for four rate hikes this year. Lately the inverse relationship between the US dollar and gold has been strong, and given we have an update from the Fed in a few hours, it is likely to remain strong.\nWTI and Brent Crude oil were jolted higher on the back of the energy information agency inventory data. The report showed that US oil inventories fell by 2.62 million barrels, while traders were expecting a build of 3.05 million barrels. Gasoline inventories fell by 1.69 million barrels, and dealers were expecting a drop of 2.4 million barrels.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I didn't think you would be one of those slaves to provincial conventions.\nYour challenge to Kathryn Corbett advises \"punctuation inside the quotation marks!\" (\"Mailbox,\" Aug. 19.) The American standard is actually to place periods and commas inside, semicolons and colons outside, and exclamation and question marks inside only if part of the quotation.\nIf, however, Kathryn uses the British style, then she has chosen to place within the quotation marks only punctuation that is part of the quotation. The British style in this regard is also known as \"logical punctuation.\" Try it in a variety of constructions and perhaps you'll like its logic as much as I do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My car keys and glasses mostly.\nEaster eggs with little kids. :) Tons of fun.\nA good time. Something I can relax and have fun with.\nHunting!?! I LOVE hunting- I practically live for it! My absolute favorite is Ducks! (Mallards). I also hunt turkey, deer, pheasant, and goose though... heck, I love hunting them ALL! :) Great question!\nQuail and other various game birds.... Naaah, I'm not that yuppie. Would've been pretty neat though.\nPizza because its so easy.\nAvailable men. The perfect pair of shoes. Comfortable jeans.\nA bargain. I can't resist a good deal!\nI only read the first 10 answers and the closest thing to \"game\" was \"wabbits\". So I'll play straight. I've hunted all my life, all over the western US, England, Poland and Africa. My favorite, in terms of excitement, challenge, difficulty and many other factors really is rabbits. Jackrabbits in the west. They don't taste that good but they're far more exciting to find and harder to hit than the traditional \"big game\".\nWhat I hunt the most is the remote, but what my favorite thing to hunt is my wife!!!!\nI love to hunt white tail deer. the taste of venicine is just really good to me.\nGoing out of buisness sales.\nDeer, Elk if I could afford to do it.\ni like hunting down pictures for my answers on ab.\nElk, Mule deer, quail, white-winged dove, rabbits to name a few.\nPheasants. I love pheasant hunting but seldom get the opportunity since I live in an area that doesn't have pheasant hunting.\nSpare change in the couch.\nMy choice is deer hunting. It's an opportunity to have a good time with friends or with family. In addition, deer`s meat is really tasty. It is not so difficult, especially if you have good equipment and a deer call. There is a lot of information about calls https:\/\/under-the-open-sky.com\/best-deer-call\/ maybe someone come in handy. Who didn`t hunt deer, I advise you to try!\nRed ants with a strong magnifier and lotsa sun!\nA good deal on something I would actually buy at full price!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This event has the theme: \"GROWING STORIES, GROWING MINDS: CREATING FOR CHILDREN OF ALL AGES.\" Featuring speakers Vikki VanSickle, Danica Lorer, and Alice Kuipers. For more information, go to: https:\/\/skcanscaip.wordpress.com\/. Hope to see you there!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagebc b/data_all_eng_slimpj/shuffled/split2/finalzzzagebc
new file mode 100644
index 0000000000000000000000000000000000000000..2b1d4e59611bd1533ad4a1638b1c5d807cfb1629
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagebc
@@ -0,0 +1,5 @@
+{"text":"Best Photo Gallery Floorart Tapis Vinyle is just one of the many collections of Sample Home Decorating Ideas Reference that we have on this website. We have a lot of Home Decorating Ideas or interior decorating ideas and any other things concerning in this website. We're not just providing info about , but , you can get a lot more reference to create your dream home as well. So , don't forget to keep visiting Interactifideas.net to get the latest update about Home Decorating Ideas and more.\nBest Photo Gallery Floorart Tapis Vinyle was posted in July 25, 2018 at 10:45 pm. Best Photo Gallery Floorart Tapis Vinyle has viewed by 33 users. Click it and download the Best Photo Gallery Floorart Tapis Vinyle.\nDecorating Ideas, Balcony Decorating Ideas Pictures was posted August 3, 2018 at 11:01 am by Interactifideas.net . More over Balcony Decorating Ideas Pictures has viewed by 523 visitor.\nDecorating Ideas, Decorating Ideas With Tiki Torches was posted July 16, 2018 at 8:54 am by Interactifideas.net . More over Decorating Ideas With Tiki Torches has viewed by 374 visitor.\nWall Decorating, Blue Floral Wall Decor was posted April 6, 2018 at 5:24 am by Interactifideas.net . More over Blue Floral Wall Decor has viewed by 254 visitor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 2017 District 9 4-H Roundup \u2013 Big Time in D9 \u2013 was held May 12 and 13 at Lone Star College in Montgomery and Chambers County 4-H members represented our County well!\nIn all, 14 Chambers County 4-H members participated in various competitions at the District Contest which was attended by youth from 18 counties.\nEmma Taylor of Barbers Hill 4-H won First Place in the Solo\/Band Performance category with her version of Bruno Mars' \"When I Was Your Man.\" She will be advancing to the State 4-H Roundup in June.\nTwo Food Challenge teams participated at the District contest. The Mid Chambers County Cookers \u2013 a team made up of Ethan Corbitt, Natalie Garner, Caleb Marlar and Brodee Shiver \u2013 worked hard to prepare a beautiful dish and placed 5th in their age division. The Sizzling Sisters \u2013 a team made up of Sophia Ferrer, Ileah Ferrer and Adrianna Morales \u2013 used their creativity to create a tasty breakfast dish and impressed the judges with their teamwork.\nThree 4-Hers put their design savvy to use in the Fashion Storyboard Competition. Abby Shanley earned 1st place in the Intermediate age category with her pet clothing Fashion Storyboard. Annalyse Chapa earned 4th place in the Intermediate Wearable Fashion Storyboard competition. Sophia Ferrer placed 5th with her Junior Wearable Fashion Storyboard. All three girls are members of the West Chambers 4-H Club.\nChambers County 4-Hers had a lot of success in the District 9 4-H Photography Contest. Mack Broussard received a blue, red and white ribbon for his photos. Mason Broussard received 3rd place, one blue and two red ribbons for his photos. Malcolm Broussard received 2nd place, 3rd place and two blue, one red and one white ribbon for his photos. Marcel Broussard received 1st place, three 3rd place awards and four blue and two red ribbons for his photos. The Broussard family are members of the Mid Chambers 4-H Club.\nChambers County 4-H is a youth development program that offers more than 40 project areas for local youth to explore that range from livestock to cooking to performing arts. For more information about the Chambers County 4-H Program please contact the Texas A&M Agrilife Extension Service in Chambers County at 409-374-2123.\nThe Texas A&M AgriLife Extension Service in Chambers County is excited to announce the first annual Merry Extension Christmas event. This event will be held on Tuesday, Dec. 6, 2016 from 4:30 to 7:30 p.m. at the White Memorial Park Community Building (225 White Memorial Park Rd, Anahuac, TX 77514).\nWe will have presentations by our Agents on food safety and winterizing for animals, booths hosted by many friends of our organization, pictures with Santa and a lot of fun crafts and activities.\nHalloween is a particularly deadly night due to the high number of drunk drivers on the road. In 2015, Texas lost 36 motorists from Friday, Oct. 30, through Sunday, Nov. 1. The Halloween weekend alcohol-related crash fatalities totaled to 12 Texans losing their lives on our roadways.\nTexas law enforcement officers remind drivers that they will be out in force looking for drunk drivers. If you do not drive sober, you will get pulled over. Young men are particularly at risk of being involved in a traffic crash as a result of \"buzzed\" or drunk driving.\nDriving while impaired is a crime that seriously risks your safety and the safety of those around you. Whether you have had one too many or are way over the limit, drunk driving is not worth causing a traffic crash, serious injury, or worse \u2014 death.\nThe legal and financial costs of driving while impaired can be significant. Drunk driving violators often face jail time, the loss of their driver's license, higher insurance rates, use of an ignition interlock, and dozens of other unanticipated expenses ranging from attorney fees, court costs, car towing and repairs, and lost wages due to time off from work.\nPlan ahead so that you don't turn the roads into a real-life horror show. If you are going to a party, plan a way to safely get home at the end of the night.\nFor information on free alcohol awareness programs available through the Texas A&M AgriLife Extension Service Watch UR BAC program in College Station, visit: www.watchurbac.tamu.edu, or call 979-862-1911.\nTexas A&M AgriLife Extension Service Agents Lindy Pitre and Tyler Fitzgerald, along with fourteen Chambers County 4-Hers, visited the Tuesday, September 27, 2016 meeting of the Chambers County Commissioners Court to recognize National 4-H Week.\nJordan Semien, Creek Johnson, Ben Johnson, Julie Johnson, Nathan Boaze, Araceli Guerrero, Kessa Miller, Shelby Pinkston, Anna Hyde, Kaitlyn Hyde, Colton Fant and Isaac Parham \u2013 4-Hers from across Chambers County \u2013 represented the Chambers County 4-H Program as a whole as Chambers County Judge Jimmy Sylvia and Chambers County Commissioners Larry George, Mark Huddleston, Gary Nelson and Rusty Senac recognized and presented a signed National 4-H Week Proclamation. Chambers County and its elected officials are a steadfast source of support for the Chambers County 4-H Program and without their assistance and encouragement the program would not be nearly as successful.\nThe purpose of National 4-H Week is to provide local and county 4-H programs with the opportunity to showcase what they have accomplished and what members gain from their participation in 4-H through activities and events at the local and county level.\nThe 4-H Youth Development Program of the Texas A&M AgriLife Extension Service has been providing experience-based education to young Texans for 108 years. The purpose of the program is to provide a learning experience for the whole child \u2013 including head, heart, hands and health \u2013 and to help them acquire knowledge, develop life skills and form attitudes to enable them to become self-directed, productive and contributing members of society.\nChambers County has eight active 4-H Clubs which offer learning opportunities in animal projects, food and nutrition, fashion and merchandising, photography, arts and may other areas. If you are interested in joining or participating in 4-H in Chambers County, please feel free to contact the Texas A&M AgriLife Extension Service in Chambers County at 409-374-2123 or chambers@ag.tamu.edu.\nEvery parent wants the best for their child's future, and one of the best ways to make sure your child will have a bright future is to be sure that you are correctly using the right car seat for your child and that the seat is correctly installed in your vehicle every time. The problem is that keeping them safe in vehicles isn't as easy as it might appear. Most parents think they are using their car seat correctly, but unfortunately, at least three out of four car seats are used incorrectly. For a car seat to best protect your child, it must be the one that fits your child, your vehicle, and one that you will use correctly every time you travel.\nChildren are at greater risk than adults in a vehicle crash. In fact, motor vehicle crashes are one of the leading causes of death for children. Crash data from the National Highway Transportation Safety Administration shows that on average, every 33 seconds, one child under the age of 13 is involved in a crash and nearly two children under age 13 are killed, and 308 are injured every day. Unfortunately in 2015, less than half of the children killed in vehicle crashes in Texas were known to be restrained.\nWhat can parents do to secure their child's future? Take advantage of the opportunity to have a free car seat inspection. Securing your children properly in age- and size-appropriate child safety seats \u2014 in the back seat of your vehicle \u2014 is the most effective thing you can do to protect them in the event of a crash. In fact, in motor vehicle crashes, child safety seats reduce the risk of a fatal injury by 71 percent for infants and 54 percent for toddlers. Misuse of car seats can cause needless injuries and fatalities.\nThat's why Texas A&M AgriLife Extension Service agent Lindy Pitre is urging all parents and caregivers to make an appointment for a free child safety seat inspection. As a part of Child Passenger Safety Week, there will be free car seat checkup events statewide as well as inspections by appointment. Please visit http:\/\/www.safercar.gov\/cpsApp\/cps\/index.htm to find out where the nearest car seat checkup event or car seat fitting station is located. Certified technicians will be available to make sure your child is riding in the proper seat and the seat is secured properly in the vehicle.\nAdjusted to fit your child securely.\nParents are reminded to keep children rear-facing until at least age 2 or until the limit of their rear-facing convertible seat, which is usually 40 pounds or more. Also, children should stay in a 5-point harness system until they are ready to ride in a booster seat. Booster seats are for children who are at least age 4 and 40 pounds or more, and mature enough to sit still in a booster. Finally, keep children in a booster seat until the seat belt fits correctly. This is usually sometime between ages 8 and 12. The average child fits in a seat belt at age 11.\nIf you're a parent or caregiver, don't miss this opportunity to have a free child safety seat inspection by a certified child passenger safety technician. Technicians can provide hands-on advice and instruction. Make sure your children are safe and you are in compliance with the current child safety seat law in Texas. The law requires all children under age 8, unless taller than 4-feet-9-inches, to be in a child safety seat system, which includes traditional child safety seats with harnesses and booster seats. Keep in mind that the law is always the minimum. Car seat technicians will be able to provide education on best practice.\nRemember: All child passengers under age 13 should ride securely restrained in the back seat, where they are safest \u2014 every trip, every time. If you are not able to attend an event during National Child Passenger Safety Week, you don't have to wait until next year to check if your car seat is properly installed. To locate a certified Child Passenger Safety Technician in Texas, please visit: http:\/\/buckleup.tamu.edu.\nBirth \u2013 12 months: For the best possible protection, your child under age 1 should always ride in a rear-facing car seat. There are different types of rear-facing car seats; infant-only seats can only be used rear-facing. Convertible and 3-in-1 car seats typically have higher height and weight limits for the rear-facing position, allowing you to keep your child rear-facing for a longer period of time.\n1 \u2013 3 years: Your child should remain in a rear-facing car seat until he or she reaches the top height or weight limit allowed by your car seat's manufacturer. This may result in many children riding rear-facing until age 2 or older. Once your child outgrows the rear-facing car seat, your child is ready to travel in a forward-facing car seat with a harness.\n4 \u2013 7 years: Keep your child in a forward-facing car seat with a harness until he or she reaches the top height or weight limit allowed by your car seat's manufacturer. Once your child outgrows the forward-facing car seat with a harness, it's time to travel in a booster seat, but still in the back seat.\n8 \u2013 12 years: Keep your child in a booster seat until he or she is big enough to fit in a seat belt properly. For a seat belt to fit properly, the lap belt must lie snugly across the upper thighs, not the stomach. The shoulder belt should lie snugly across the shoulder and chest and not across the neck or face. Children under age 13 should ride in the back seat.\nChambers County 4-H hosted its first annual Open House event on Tuesday, September 20, 2016 and attendance was beyond expectations.\nApproximately 125 Chambers County residents attended the event which was meant to showcase all of the amazing educational opportunities that Chambers County 4-H offers.\nClub managers and other volunteers were on site to represent each of the eight 4-H Clubs in Chambers County and current 4-Hers were available to talk with visitors about what they are learning from the projects that they are participating in. Texas A&M AgriLife Extension Agents and staff were available to answer questions. In all, visitors had the opportunity to visit 23 booths and were offered educational handouts, snacks and the opportunity to play fun games.\n4-H is a positive youth development organization that empowers young people to reach their full potential through education and learning.\nIf you are interested in joining or participating in 4-H in Chambers County, please feel free to contact the Texas A&M AgriLife Extension Service in Chambers County at 409-374-2123 or chambers@ag.tamu.edu.\nThe Chambers County 4-H Program held their annual 4-H Awards Banquet on Saturday, August 6, 2016 at the White Memorial Park Community Building in Anahuac, TX.\nJust over 90 4-H members and their families attended the banquet where more than 220 awards were presented to both youth and adult volunteers.\nThis event wouldn't have been possible without the support of sponsors who included the Chambers County 4-H Adult Leaders Association, Chambers County FCS Committee, Chambers County Judge Jimmy Sylvia, Chambers County Commissioner Rusty Senac, Anahuac National Bank and Thrif-Tee Food Center. Special recognition was given at the Banquet to the Winnie Tractor Supply Company Store which supports the Chambers County 4-H Program tremendously through the bi-annual 4-H Paper Clover Fundraiser. A water slide and bounce house were also provided for attendees by All Summer Long Moonwalks and Waterslides.\nThe Chambers County 4-H Program had a record year with 215 youth members, 44 adult volunteers and seven Community Clubs. There will also be a new Project Club in the 2016-2017 year. 4-H members participated in livestock, shooting sports, photography, food challenge and fashion projects, to name a few. Notable awards received by Chambers County 4-Hers include Reserve State Champion Drill Team (Chambers County Legend Riders), Reserve State Champion Prevision Team (Chambers County Legend Riders), third place Food Challenge Team at Houston Livestock Show and Rodeo (West Chambers 4-H Club Senior Team), numerous livestock show awards and many district and state 4-H awards.\nTexas A&M AgriLife Extension Agent Lindy Pitre of Chambers County hosted a Healthy Summer Snacking seminar and demonstration during Chambers Health's Summer Bash on Friday, June 24, 2016.\nAgent Pitre educated the 25 youth and five adult attendees on the importance of choosing healthy snacks over \"junk\" food this summer.\nChildren sometimes become bored when at home during summer vacation. They tend to want to eat more snacks than usual.\nInstead of unhealthy foods, the audience was encouraged to eat fruits and vegetables and youth helped brainstorm some healthy alternatives that they could snack on.\nAgent Pitre then demonstrated six kid-friendly recipes for the children including Fish in the River, Vegetable Flowers, Apple \"Cookies,\" Fruit Skewers, Banana Butterflies and a healthy, nut-less No Bake Cookie and discussed the nutritional benefits of the ingredients in each one.\nAfter the demonstration, children went to each recipe station and created the dish with the help of parents and adult volunteers. They were then able to eat their food masterpieces.\nAll attendees took home the recipes which were demonstrated and created during the lesson, as well as a packet containing information on food safety, healthy recipes and summer exercises for youth.\nContact the Texas A&M AgriLife Extension Service in Chambers County for information regarding upcoming programing.\nChambers County 4-Hers performed amazingly well at the 2015 Big Time in D-9 4-H Roundup. Below is a list of our big winners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Patient Blood Management program brings many benefits to a healthcare system. IFPBM assists governments, hospitals and other institutions implement a successful program.\nIf you're a medical professional, specialist or administrator, you're eligible to subscribe to resources specifically related to Patient Blood Management. Resources include a 5,000+ article database, video library and the IFPBM Speakers Bureau.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"The British 91st Brigade had no casualties in WWI Col. Whittlesey an American said that the reason there was no casualties was because the men and commanders of the 91st Brigade quoted the 91st Psalm daily. It is said that other similar units had very high casualty rates.\"\nThis story has been challenged to be an urban legend because the booklet of the 91st Psalm by which was distributed to the soldiers made the mistake of calling it the US ARMY 91st Brigade which had many casualties in WW1 and started the confusion about the credibiltiy. Older sources from the 50's acknowledge that it was a British, not American unit.\nWe have all looked at sites which have the WW1 story. MOST INTERNET stories with this WW1 example give no documentation.\nShe contains the WWI story with a reference in her Bibliography. KPC was a journalist for years and kept excellent notes and had strong bibliographies.\nKPC has some sources or files that are somewhere in her estate because she was working on her research to be used as a TV documentary.\nKPC names American Weekly, 21, February 1958 and the writer as Will Ouhsler (Oursler?) as her source for this story.\nTracing down this issue. Would it source the story?\nWe have ordered 2 books by him to see if he quotes it in that source.\n3. We have not been able to get our hands on this American Weekly 1958 story. Libraries, Hearst Foundation, Ebay have been our trails.\nThe American Weekly was a supplement to the Sunday newspapers published by the Hearst Corporation. It was published from 1 November 1896 to 1963. The publication featured popular illustrators on its cover, including the work of Edmund Dulac, Will Pogany and Jose Segrelles.\nKatherine quotes Margaret Lee Runbeck for several stories.\nPerhaps MLR carried the WW1 story since much of KPC is inspired by her. MLR does not have a bibliography in her book the Great Answer, but has other stories we need documented and is a powerful writer about God and the war. Margret wrote 30 years before KPC on this subject. I read this book and it was very good and highly recommend it, but did not have this story--it was mostly testimonies from WW2.\nII. KPC says that many periodicals on both sides of the ocean carried the WW1 story of the protection of the men in the 91st Brigade who stood on Psalm 91.\n(British unit) saying that she had many sources to choose from and that the documentation span both America and Britian.\nF.L. Rawson, and the name of the book is called LIFE UNDERSTOOD. In it he wrote about the British 91st brigade. . I only hoped that he would have a strong bibliography--which he doesn't.\nIn the book they said he wrote another book called, \"How to Protect the Troops\". Can't find a copy of this second title book.\nWe ordered an antique postcard carrying this story and we cannot read the date. It tells the story as a British regiment, quotes FL Rawson, a noted engineer and great scientist and says that their book, \"There is a safe place to Hide\" we could read about these soldiers and could be purchased for a $1. (We have not located the book as of yet.) The Address for the book is Merit Publications, Dept. S-7, 300-4th Ave., New York 10, NY. The card was written to Edna Baird, Box 36, Doniphan Nebraska--with a one penny postage stamped card.\nBusinessreform.com has the story contested as an urban legend. However, several in the past years say that the confusion regarding the story is the cross-over of the British regiment being changed into an US Army unit.\nReviewed by Pastor Robert H. Reid on July 8, 2006 Regarding Larry Klass's comments about there was no 91st brigade; this is true if you are referring to the U.S. Army. However, the 91st Brigade was a British unit. The 91st Brigade was a composite brigade put together in England . The story is true and it refers to a British unit and not an American!\nThe 7th division was a Regular Army division formed after the start of the war It contained 20th, 21st ,22nd, and 91st Infantry brigades.\n91st Brigade was made up of four battalions. 1\/South Staffords,2\/Queens, 21 \/ Manchester and 22\/ Manchesters.\nThe Brigade was to advance on a battalion front . 1\/South Staffords was to take the Red Line and then 22\/Manchesters would move through to attack the Blue Line. 21\/Manchesters was in support and 2 \/Queens was in reserve.\nJust before zero hour the Germans fired a defensive barrage designed to break up potential attacks. Fortunately it fell between the front and rear battalions. The brigade joined from the 30th Division in December 1915, swapping with the 21st Brigade. A number of battalions swapped to the brigade from other 7th Division brigades during the transition.\nMisschien valt dat wel na te gaan als iemand de soldiers died...CD rom heeft, ik dacht dat je daar op eenheid kon zoeken?\nSorry, 2nd Bn. Queens vind ik niet zo meteen. Maar misschien hoeft het niet meer ?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Current Alaska Native Tribes Population in Missouri 2019, 2018 with Demographics and Stats by age, gender.\nAlaska Native Tribes homes in Missouri with multiple generations.\nHow long Alaska Native Tribes in Missouri have lived in one place.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajnsi b/data_all_eng_slimpj/shuffled/split2/finalzzzajnsi
new file mode 100644
index 0000000000000000000000000000000000000000..0efc74a7083c1bc29995c54260884771882b0bb6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajnsi
@@ -0,0 +1,5 @@
+{"text":"The South Devon Railway Association held its annual dinner on the 24th January 2015 and, as usual, the \"Volunteer of the Year\" was announced.\nThis year the award was made to Brian and Brigid Soper.\nThe speaker for the evening was Sir Ken Knight CBE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cheap Shoes China Free Shipping - \u7ad9\u52d9\u7ba1\u7406 - *Nemesis*\u5973\u4eba\u8857 - Powered by Discuz!\nReese believes he will be K?nken Ryggs?ck ready for Cheap Nike Shoes China next season. Guard-center Kevin Boothe, who started all 16 games, is a free agent. Snee and David Diehl may retire, and backup Jim Cordle is a restricted free agent coming off knee surgery.\nJerrel Jernigan emerged in Adidas Stan Smith Shoes Cheap Cruz's absence and Rueben Randle showed flashes in his second season. Hakeem Nicks is a free agent Wholesale Nike NFL Jerseys who had Scarpe Air Max 97 Saldi a bad season.\nGMT+8, 2019-4-22 04:27, Processed in 0.037878 second(s), 6 queries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Baltic Amber sits in a simple sterling silver bold twist setting. Hand fabricated in the Silk Bone studio, and stamped SBJ & .925.\nNecklace is about an inch tall. Pendant hangs on a sterling silver 16 inch chain.\nThis item is ready to ship. Please allow 1-3 business days for processing. Thank you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dual core processors are the standard in today's computers. The dual core architecture helps manufacturers to extend the performance curve.\nFaster is better. Being able to do more is also better. Well, let's take a look at the details of it.\nAs software developers are coming out with more advanced applications, tasks that computers can perform these days are getting more and more complicated. While you had to go for a coffee brake when burning a CD on Windows 3.11, you can burn a DVD, listen to online radio, and play graphics-intensive games at the same time today. Multitasking is as natural today as breathing, but it also requires more power and speed than ever before.\nFor processor manufacturers dual core means a less expensive route to producing a new product that continues the performance curve. Getting existing processors to work faster is getting too expensive and more complicated, so manufacturers having the option of just bundling two cores into one processor see it as an easy way to keep going. After all, what sounds better: \"two 2.0 GHz cores\" or \"dual core processor described as a 3.4 GHz comparable\"?\nSelling a dual core processor computer sounds good to the average computer user. Dual core is something new, dual core sounds good, and dual core is easily comprehensible. Two is always better than one, right?\nDual core - a way to go faster?\nThe dual core game can be seen as a new twist to the gigahertz performance race. Increasing the clock speed of a processor was the traditional approach in making processors faster since a faster CPU can do one task quickly and then quickly switch and work on the next.\nBoth AMD and INTEL scaled up the clock speeds of their processors in a very short amount of time. However, due to size, complexity, and heat issues, it has become increasingly difficult to make CPUs go faster. There is a limit to how small crystals, transistors, pathways you can make, and it seems that the ability of the nano industry to make things faster by making components smaller is getting to a pause. The clock speed curve is flattening. Another solution had to be found in order to continue to improve performance.\nIf processor manufacturers cannot make processors faster (while also economically feasible) with the currently available technologies, the next step is to add more processors into the system. One way is to just add more CPUs into the system. However, this is quite expensive for the average Joe, so computer engineers came up with another approach: take two processors and combine them together, so they still have some benefits of the two but also economies arising from sharing some related components. They came with the idea of two CPUs but only one socket on the motherboard, called a dual core processor. Dual core keeps the price of the system reasonable while improving the performance.\nIn other words, another route to continue the performance curve is to develop dual core and multi-core processors, that is two or more CPUs put together on one chip.\nWhere a dual-core processor becomes handy?\nA dual core processor benefits anyone who multitasks. Multitasking in a single-core processor computer works in a way that the processor keeps quickly switching between your applications. In a dual core processor system, there is the second processor core to share the load either in balanced form or by itself. While one processor is employed by burning a DVD, the other processor is available to process your email. The critical feature here is that multiple tasks are executed simultaneously.\nDo I need a dual core processor computer?\nIt depends on what you do with your computer. If you just send emails and write letters, then probably not. If you do video editing, burn DVDs, play games, and do a lot of switching between applications, then the answer is yes.\nRegardless of what you do, the computer industry follows the masses. Or, do they lead the masses? Some times, it is hard to say whether the manufacturers listen to what buyers want and create products for them, or the other way around - whether the manufacturers create a product and then make everyone buy it. Laptop displays is a great example. All laptops today come by default with wide displays. If you want a display with the standard 4:3 ratio, you are out of luck, nobody makes them anymore. The reason is the cost and mass production. Does everyone not like 4:3 displays, or did the manufacturers force everyone to switch to wide by not making anything else available? The same can be seen with processors. Dual core and multi core is likely to become the standard.\nHow can I tell if my computer has a dual core processor?\nYou can tell by going to the Task Manager. Task Manager shows how many cores your computer has.\nUnfortunately, Task Manager does not distinguish between cores and processors. Task Manager does not tell you whether the computer shown in the picture above is a single-processor with four cores (quad core processor) or a server with four single core processors. But it at least tells you that you have more than one of something.\nIs there a problem with dual core and multi-core processors?\nYes, nothing comes for free.\nThe main problem with winding up clock speeds is heat. Nowadays processors are pushed to extreme speeds. Just compare - processors run in today's computers at some 2-3 GHz. Electric current in your house runs at 50 Hz (or 60 Hz in Europe). You can imagine the processor speed as RPM in your car. Your car engine and processors too can operate at only so much RPM before they seize. Heat is the enemy of any processor and high clock speeds mean high heat and that means errors and shorter lifetime.\nThat heat comes from power. It takes a lot of power to run a processor at 2 or 3 GHz. Most laptops that you can buy today come with Li-Ion batteries. Remember your old laptop that you bought 10 years ago. It came with Ni-Mh batteries. Today's laptop with high clock speed processors would not run very long with your old batteries.\nA high-clock-speed processor needs a lot of electricity to power all the speed and is therefore prone to electrical noise which causes interference. The pathways on a processor are microscopically close together. The more power that runs through these pathways due to the requirement of higher clock speeds means that there will be a small amount of electrical radiation from one pathway to the next. If the radiation is large enough, it causes data errors.\nAre two cores (a dual core processor) better than one core?\nBefore instructions get into the processor, they are loaded into so called pipeline. Data flows into a processor via a pipeline and is processed sequentially one after another.\nData that is used continually is stored in the processor's cache. A processor is able to determine which data to store in cache and which it may need later. The processor cache is running at the same clock speed as the processor itself. If your processor runs at 2.4 GHz then the data between the processor and its cache can be exchanged at the same speed.\nIf a processor needs to reach outside of the cache, then it does so through the bus to the system RAM. Both the bus and RAM run at far slower speeds than the processor; therefore, if the processor has to reach out through the bus to the main system memory RAM, then it must slow down to that bus speed.\nIt is probably starting to be obvious that having two cores that access the same processor cache can get the job done more quickly (especially if the processor is designed as Symmetric Multiprocessing (SMP)).\nWhat is the difference between a dual core processor system and a system with two single core processors?\nA dual core processor is somewhere between a single-core processor system and a dual-processor system.\nA dual core processor has two processor cores that share the surrounding hardware, such as the memory controller and bus. In this case we are talking about one processor with double guts.\nA dual processor system has completely separate hardware and shares nothing with the other processor. In this case we are talking about two processors with each having its own guts.\nAs it may seem, a dual core processor is not twice as fast as a single core processor. A dual core processor is also not going to be as fast as a dual processor system. It will be somewhere in between.\nYou might like to read about grid computing, blade servers, Windows Server 2008, and 64 bit architecture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tempelhofer Feld is a former airport, the Berlin airlift airport, left as is in the city of Berlin and opened to public.\nThere are rumors that soon construction is going to start to make something else here I don't know exactly what is the plan.\nAnyway I am against it simply because I want it to stay what it is now, a hole in the city, a place for the mind to fly.\nI have loved the Mongolian steppe, theTibetan plateau and the bad lands in Xinjiang, few obstacles and a lot of sky.\nTempelhof Is a short walk from my place and when I am there I feel good!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakjvy b/data_all_eng_slimpj/shuffled/split2/finalzzzakjvy
new file mode 100644
index 0000000000000000000000000000000000000000..90f64f3b89d7e265007fb3e4bea885031ba576ba
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakjvy
@@ -0,0 +1,5 @@
+{"text":"ALL FORMS MUST BE FILLED OUT COMPLETELY AND TRUTHFULLY. FAILURE TO COMPLETE THE ENTRY FORM AND PROVIDE TRUTHFUL AND PERTINENT INFORMATION MAY RESULT IN DISQUALIFICATION FROM THE SWEEPSTAKES. DISQUALIFICATION IS IN THE SOLE DISCRETION OF SPONSOR (defined below).\nWQAD News 8 (\"Sponsor\") will conduct the Hank's Power & Equipment Grill Giveaway Sweepstakes RulesGiveaway Sweepstakes (\"Sweepstakes\") in accordance with these Official Rules (\"Rules\"). Participation in the Sweepstakes constitutes entrant's full and unconditional agreement to, and acceptance of, these Rules. The Sweepstakes is intended for participation in the United States only and is void where prohibited and outside the Sweepstakes Area set forth below. Do not participate if you are not eligible and located in the United States at the time of entry.\nEligibility: Entrants must be legal US residents, at least 18 years old or above, as determined by Sponsor and reside within a 35 mile radius of Hank's Power and Equipment located at 1262 IL-94, Aledo, IL 61231 (the \"Sweepstakes Area\"). Employees of WQAD, Hank's Power & Equipment (\"Sponsor\"), and Tribune Media Company, employees of other television or radio stations, and members of the immediate families of such persons are not eligible to participate and win. The term \"immediate family\" includes spouses, siblings, parents, children, grandparents and grandchildren, and any other person residing at the same household whether or not related. Winning a prize is contingent upon fulfilling all requirements set forth herein.\nSweepstakes Period: The Sweepstakes begins on Monday, July 16, 2018 at 9:00 a.m. CT and runs through Thursday, August 9, 2018 at 11:59 p.m. CT (the \"Sweepstakes Period\").\nSweepstakes Entry: Viewers can fill out the entry form found in the contest section of WQAD.COM.\nReceived entries become the property of WQAD News 8 and will not be returned. Entrants will also be given the option to opt in to receiving additional information from WQAD and from the Sweepstakes prize providers. Incomplete entries will be disqualified. Multiple entries by means of software-generated or other automated processes will be disregarded. Detection of said automated entry will lead to such entries being voided in Sponsor's sole discretion. Only one registered account per entry. If multiple accounts are detected for a single entrant, the accounts will be voided and the entries will be disqualified in Sponsor's sole discretion. In the event of a dispute as to any registration, the authorized account holder of the email address or account used to register will be deemed to be the registrant. The \"authorized account holder\" is the natural person assigned an email address by an Internet access provider, online service provider or other organization responsible for assigning email addresses for the domain associated with the submitted address. Potential winner may be required to show proof of being the authorized account holder. Sponsor reserves the right to use any and all information related to the Sweepstakes, including information on entrants obtained through the Sweepstakes, for marketing purposes or any other purpose, unless prohibited by law. Sponsor reserves the right to contact entrants and all other individuals whose email address is submitted as part of this promotion.\nWinner Selection and Notification: On or about Friday, August 10, 2018, Sponsor will select one winner by random drawing from among all eligible entries. Odds of winning depend on the number of eligible entries. Sponsor will attempt to notify the Sweepstakes winner via telephone or email on Friday, August 10, 2018. Winner must have a valid email address where he or she can be notified. If the potential winner: (a) is unreachable after seven days, (b) is not in compliance with these Rules, (c) does not meet the eligibility requirements, (d) does not provide required documentation and sign any required documents by the deadline established by Sponsor, or (e) is unavailable for prize fulfillment, Sponsor reserves the right to award the prize to another winner selected by random drawing from among remaining eligible entries. Sponsor will conduct up to two alternate drawings. If Sponsor cannot find an eligible winner for the prize, the prize will not be awarded. All results are unofficial until winners are verified.\nThere is one winner and one prize.\nThe prize is a Gourmet Guru grill kit.\nThe approximate retail value of the prize is $599.\nPrize Acceptance\/Restrictions: Winner is subject to verification by WQAD of the winner's name, age, address, phone number, and Social Security number (where the prize value is equal to or greater than $600.00). In order to claim his or her prize, winner must appear in person at the business offices of WQAD News 8, located at 3003 Park 16th Street, Moline, IL during regular business hours by no later than 5 p.m. on Friday, September 7, 2018. Prior to receipt of prize, winner will be required to sign an Affidavit of Eligibility\/Release of Liability and Publicity, and may be required to provide a completed W-9, per Section 9 below. Prize cannot be redeemed for cash or substituted for any other items by any winner. Prize is non-assignable and non-transferrable. Sponsor reserves the right to substitute a comparable prize of like or greater value, including cash, for prize, for any reason. Costs of transportation and accommodations, where applicable, and any other cost not specifically included in the prize are the sole responsibility of the winners. Properly claimed prize will be awarded, provided a sufficient number of eligible entries are received, but in no event will Sponsor award more prizes than are provided for in these Rules.\nPublicity Release: By participating in the Sweepstakes, each entrant acknowledges that his\/her entry in the Sweepstakes constitutes that entrant's consent to use, publish, reproduce and for all purposes, including publicity, promotion and advertising, in any media (including without limitation, the Internet, television or offline promotions), winner's name, likeness, photograph, voice, opinions, and\/or hometown and state, and any portion thereof, each extending throughout the universe and in perpetuity without further compensation, credit or right of review or approval, except where prohibited by law.\nIndemnification\/Hold Harmless: By participating, entrants agree: (a) to release, discharge, and hold harmless Sponsor, Tribune Media Company, prize providers, and their respective affiliates, parents, subsidiaries, advertising and promotion agencies, and all of their officers, directors, employees, representatives, and agents (the \"Released Parties\") from all liability, injuries, losses or damages of any kind to persons, including but not limited to invasion of privacy (under appropriation, intrusion, public disclosure of private facts, false light in the public eye or other legal theory), defamation, slander, libel, violation of right of publicity, infringement of trademark, copyright, or other intellectual property rights, death or property damage resulting in whole or in part, directly or indirectly, from the acceptance, delivery, possession, misuse or use of a prize (including any travel or activity related thereto), or from participation in and\/or entry into or creation of an entry for the Sweepstakes and\/or the broadcast or exploitation or use of entry or any other Sweepstakes-related activity; and (b) that the Released Parties have neither made nor are in any manner responsible or liable for any warranty, representation or guaranty, expressed or implied, in fact or in law, relating to prize.\nLimitation of Liability: The Released Parties are not responsible or liable for: (a) any incorrect or inaccurate entry information or other errors in the printing, offering or administration of the Sweepstakes or in the announcement of the prize(s), (b) any error, omission, interruption, defect or delay in operation or transmission at any website, or wireless calling service, interrupted or unavailable network, server or other conditions, (c) failure of any entry to be received by Sponsor due to technical problems, telephone service problems, human error, or wireless calling service, (d) mechanical, technical, computer, hardware or software errors, malfunctions, or failures of any kind, including but not limited to failed, incomplete, garbled, or delayed transmission of entries, traffic congestion, viruses, sabotage, satellite failures, electrical outages, on telephone lines, on the Internet, at any website, or application or lost or unavailable network connections or natural disasters or acts of God or man, which may limit an entrant's ability to participate in the Sweepstakes, (e) communication line, hardware and\/or software failures, malfunction of phones (including wireless phones\/handsets), phone lines, other communications malfunctions, unavailable network connections, cellular equipment towers, telephone systems or wireless service, (f) damage to any computer (software or hardware) resulting from participation in the Sweepstakes, or damage to mobile phone or other PDA device, (g) theft or destruction of, tampering with, unauthorized access to, or alteration of entries and\/or entry information, (h) entries that are late, lost, stolen, damaged, illegible, and\/or unintelligible (or any combination thereof), or (i) any change of email address, mailing address, telephone number and\/or any other contact information provided by entrant. Any expenses incurred by the entrant during the entry process are the sole responsibility of each entrant and the Sponsor will not issue reimbursement for any expenses.\nDispute Resolution: By entering the Sweepstakes, entrants agree that: (a) any and all disputes, claims, and causes of action arising out of or connected with the Sweepstakes, or any prizes awarded, will be resolved individually, without resort to any form of class action; (b) any and all claims, judgments and awards will be limited to actual out-of-pocket costs incurred, including costs associated with entering the Sweepstakes but in no event attorneys' fees; and (c) under no circumstances will any entrant be permitted to obtain any award for, and entrant hereby waives all rights to claim punitive, incidental or consequential damages and any and all rights to have damages multiplied or otherwise increased and any other damages, other than for actual out-of-pocket expenses. All entrants agree, by participation in the Sweepstakes, to submit to the personal jurisdiction of the courts of Illinois. Illinois law will govern this Sweepstakes, without regard Illinois's choice of law rules. The courts of Illinois will be the exclusive forum for any dispute regarding any Rule or activity associated with the Sweepstakes.\nOfficial Rules: To request a copy of the Rules, send a self-addressed stamped envelope to WQAD News 8, located at 3003 Park 16th Street, Moline, IL by 8\/9\/18. Written copies of these Rules are also available during normal business hours (Monday \u2013 Friday, between 8:00 a.m. \u2013 5:00 p.m.) at WQAD's business offices or online at http:\/\/www.wqad.com.\nName of Winner: For the name of the prize winner, send a separate, self-addressed, stamped envelope to WQAD News 8, located at 3003 Park 16th Street, Moline, IL, or appear in person at that location between normal business hours (Monday \u2013 Friday, between 8:00 a.m. \u2013 5:00 p.m.) after 8\/9\/18. Requests for winner's name must be received by October 6, 2018.\nRights Reserved: The content, information, data, designs and code associated with the Sweepstakes and Sweepstakes website are protected by intellectual property and other laws. Any unauthorized use of copyrighted materials, trademarks, or any other intellectual property of Sponsor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Setting refers to the time, place, and circumstance of the plot.\nIf \"the plot\" is going to have to suffer the activities of PCs, your setting is probably a campaign setting of some kind.\nAll adventures and campaign require a setting, even if it's the most rudimentary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nespresso has practically reinvented coffee with its one-touch espresso capsule system, and now the company is throwing some control into its machines with its new Maestria line.\nUnlike the one-touch button system seen on most of the Nespresso range, the Maestria features a dial on each setting \u2013 espresso and lungo \u2013 allowing for five different cup lengths of each type of coffee.\nSo if you want a long espresso or a short lungo, these machines will allow you to do just that.\nThere are two Maestria models coming, one with a new and improved version of the Aeroccino milk frother accessory that can be used with existing machines.\nThe $749 Nespresso Maestria doesn't feature the Aeroccino, but does integrate a steam wand, allowing you to froth your own milk the way a typical cappuccino machine does.\nMeanwhile, the $899 Nespresso Gran Maestria features an integrated Aeroccino for easy automatic milk frothing, with four settings: hot milk, cold milk, stiff froth, and velvety froth.\nWe're not quite sure what the difference is between \"stiff froth\" and \"velvety froth\", but with a release date of April 1 for the new machines, we won't have to wait too long to find out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ethan Ampadu plays the position Midfield, is 18 years old and cm tall, weights kg.\nIn the current club Chelsea played 2 seasons, during this time he played 35 matches and scored 0 goals.\nHow many goals has Ethan Ampadu scored this season?\nIn the current season Ethan Ampadu scored 0 goals.\nIn the club he scored 0 goals ( Europa League , Europa League , Capital One, Premier League, FA Cup, Champions Cup, Friendlies). Ethan Ampadu this seasons has also noted 0 assists, played 1130 minutes, with 6 times he played game in first line.\nEthan Ampadu shots an average of 0 goals per game in club competitions.\nLast season his average was 0 goals per game, he scored 0 goals in 23 club matches.\nIn the current season for Chelsea Ethan Ampadu gave a total of 1 shots, of which 0 were shots on goal. Passes completed Ethan Ampadu is 89 percent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This really is Caldwell 36 Single Bathroom Vanity Set By Beachcrest Home Sale Brand New for the favorite.Here there are reasonable item details. One more selection for your internet shopping. Because of everyone who came to check out us to view our products.\nThis really is Caldwell 36 Single Bathroom Vanity Set By Beachcrest Home Sale Brand New for your favorite.Here you'll find reasonable product details. One more selection for your internet shopping. Thanks to everyone who came to visit us to view our products.\nThis will be 4.0 out of 5 according to 10 Recently visitors they very satisfaction because of the Caldwell 36 Single Bathroom Vanity Set By Beachcrest Home , If you are searching for where to buy this item from the online stores with worthy price high quality, we would like to say you come in the right place To get more Information Click Here !, and you will be taken to the best store we suggested.\nYou can discover Caldwell 36 Single Bathroom Vanity Set By Beachcrest Home as a consequence of numerous large vendors offline or possibly on the internet similar to Amazon However what kind is the greatest? We've got completed be right for you, we discover the absolute right place to get the maximum benefit effective has reached Amazon.com. This specific massive internet vendors provide the most effective selling price. You will find a minimum of 3 good reason why if you opt for Amazon from other retailers. 1. Best Price.\nCheck out at Top value Cost Caldwell 36 Single Bathroom Vanity Set By Beachcrest Home this online site ! DON'T squander a while, we offer the Greatest Cost !","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamcpl b/data_all_eng_slimpj/shuffled/split2/finalzzzamcpl
new file mode 100644
index 0000000000000000000000000000000000000000..526e016c69bb47c4c1cbcf02b410c6bddfaf7454
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamcpl
@@ -0,0 +1,5 @@
+{"text":"Journal of Applied Physics, 1998, v. 83 n. 3, p. 1305-1311 How to Cite?\nEnhancement of interdiffusion in GaAs\/AlGaAs quantum wells due to anodic oxides was studied. Photoluminescence, transmission electron microscopy, and quantum well modeling were used to understand the effects of intermixing on the quantum well shape. Residual water in the oxide was found to increase the intermixing, though it was not the prime cause for intermixing. Injection of defects such as group III vacancies or interstitials was considered to be a driving force for the intermixing. Different current densities used in the experimental range to create anodic oxides had little effect on the intermixing. \u00a91998 American Institute of Physics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Meet the Otto 1500: a self-driving robot that can move up to 3400lbs of goods. It drives at speeds over 4 mph and has turn signals, brake lights, and other safety features. The robot uses a reliable Lidar to avoid obstacles and find new routes.\nThe machine is powerful enough to carry pallets, racks, and everything in between. It has a tough welded steel frame and lets you create your own attachments to carry payloads.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are trained to see light and adapt it to fit our personal taste. How we choose to light, whether it be artificial studio lighting or natural light, enables us to define our voice along the way. Utilizing natural sunlight is a favorite among many photographers, and for good reason. It is consistent enough to define our personal style, yet it is constantly changing to add just enough variety to keep it interesting.\nLeft: hard natural front light. Middle: Backlight. Right: Adding a strobe to balance artificial and natural light.\nBacklit images create a softening effect and often the details in the highlights, (especially the sky) are lost along the way. The harsh, natural light found several hours after sunrise and before sunset is a personal favorite, as it creates amazing contrast and depth, though it is challenging for the client or model to look directly into the light and keep from looking pained.\nThere doesn't, however, have to be an ultimatum of choosing one over the other. Adding a fill light in backlit situations provides more depth and substance to the background as well as naturally providing more contrast without having to add it in post. It is a delicate dance between the two and a great compromise that brings out the benefits of both lighting scenarios. You can have it all.\nModel: Christine M. Assistant: Gary L.\nFor this shoot, the Broncolor Siros L 800Ws was set at its lowest power setting with no lighting modifier. It was placed a few feet away from and in front of the model on the left side of the camera and angled down at approximately 45 degrees.\nTech Specs: ISO 200, f\/4, 1\/500th of a second.\nThe goal was to keep the focus on the model in this busy scene, therefore we opted to use a little soft bokeh to aid in focusing attention on the subject and depended on the subtle movement of the wind blowing across the grass. The other goal was to bring back detail to the hazy summer skies and background during a very active fire season. Ultimately, the results were going to be a middle-of-the-road approach, essentially offering an olive branch to both sides of the aisle.\nThe high speed sync capabilities of the Broncolor Siros L was very beneficial to achieve the subtle look and feel of this shoot, so whatever light you're using that feature would be helpful. For those of you who don't have HSS, light modifiers and feathering of the light will likely be of use to you here.\nNote, that this is merely a starting point for the camera and equipment settings. The contrast, depth of field and mood can all be adjusted to fit your specific needs and stay true to your brand.\nNice. The only thing I'm not a fan of is that unnatural shadow on her neck.\nIs it possible to get the same quality from a speedlight with HSS?\nI have a few photographer friends that have purchased a $20 speed light that slaves to their HSS speed light for more output and have produced similar results. They have a bracket that they put 2-3 of them on. It may require a modifier such as an umbrella to control the light a little more, but definitely worth the try.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FLOORS & STAIRS are often problem areas in most new homes. These defects irritate homeowners. They can ruin an otherwise perfect house and they can hurt your reputation and bottom line.\nJoin us for a fast-paced morning workshop and enjoy a complementary breakfast. Meet other builders, build new connections, and hear advice from an expert on how to prevent pesky defects.\nThe course was very informative - extremely happy I attended and advised Managers what a great course this would be for Contract Managers and Construction Staff.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Identify the three primary anatomical locations attributed to lumbar spinal stenosis Identify and distinquish between six categories of lumbar spinal stenosis Describe the diagnostic accuracy of select examination findings Identify treatment approaches for LSS which include exercise, manual therapy and education.\nThe incidence of lumbar spinal stenosis (LSS) increases with age and this condition is becoming more prevalent with the reported increase in life expectancy. This results in increased pain, decreased function and an overall decrease in quality of life for a large portion of the population. Unfortunately, LSS is not only common but also very expensive, costing an estimated $50-100 billion per year with 75% of this money coming from those that have become temporarily or permanently disabled. There are many medical treatments available for LSS however they are often expensive or have greater risk. Conservative management may be an alterative and it seems to be effective in some circumstance. This course will briefly review the patho-anatomy associated with LSS as well as different classification systems. Also, examination procedures will be covered as well as suggestions for differential diagnosis. Finally, diverse treatment options will be discussed including manual therapy, exercise as well as education and home carry over activities. Upon conclusion of the webinar participants will be able to identify those with LSS and have a broad array of treatment strategies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamhaz b/data_all_eng_slimpj/shuffled/split2/finalzzzamhaz
new file mode 100644
index 0000000000000000000000000000000000000000..8742f604eb6252a8c56c5a2232e15100e2b58406
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamhaz
@@ -0,0 +1,5 @@
+{"text":"We need you to help us raise awareness for mental health and remove the stigma of mental illness in our community.\nHelp us end the stigma of mental illness by joining us!!! Sponsorship opportunities are available.\nAll proceeds benefit Counseling Connections for Change, Inc.\nCounseling Connections for Change, Inc. does not support, endorse, or oppose any political candidate or campaign.\nHave questions about White Out Night Out - Havana Nights? Contact Counseling Connections for Change, Inc.\nCounseling Connections for Change, Inc. is the only Christian non-profit counseling center in Brazoria County. Our vision is to strengthen mental health in Brazoria County and our mission is to put Christian principles into practice through programs that build healthy individuals, families and community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We offer a broad array of client-focused solutions offers individuals and families the tools they need to help them meet today's financial demands. These powerful concepts and programs can help our clients get from where they are to where they want to be. No one product fits everyone's needs and we encourage you to talk to an Insurance Services' associate to help create a strategy for your financial future.\nLet us provide you with a free, no obligation Life & Insurance Protection Quote to show you how much you can save when you partner with us!\nAn Insurance Service associates can offer a variety of personal insurance coverage options to help clients protect themselves and their families.\nA key to any financial strategy is preparing for retirement. Our Insurance Services offers a variety of options to help clients prepare for their leisure.\nLet usprovide you with a free, no obligation Supplemental Retirement Quote to show you how much you can save when you partner with us!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LAGOS: The Joint Admissions and Matriculation Board (JAMB) has extended the registration for 2018 Unified Tertiary Matriculation Examination (UMTE), until Feb. 11.\nThe Board disclosed this in a statement signed by its Head of Media, Dr Fabian Benjamin, and made available to the News Agency of Nigeria (NAN) in Abuja, on Tuesday.\nHe said that the decision was arrived at after the window period ended at midnight of Tuesday, February 6.\n\"The failure of these candidates to register is unfortunate and the Board hastens to add that this culture of impunity would not be tolerated henceforth.\n\"Indeed penalty will be imposed for late registration from next session.\n\"UTME candidates are therefore strongly advised to take advantage of the extension to register as request for further extension would not be entertained,\" he said.\nThe Board, however, noted that registration for Direct Entry (DE) candidates continues as earlier scheduled.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The basis of the course is to teach students basics of thermodynamics in liquid and solid solutions, chemistry and phase equilibrium in materials and thermodynamics and kinetics of processes in solutions in order to enable better understanding of the processes in the materials. Students will learn about the thermodynamic basics of phase diagrams, kinetics and diffusion and also chemistry. The lectures are complemented with seminar work, simulations and project work of planning, manufacturing and thermodynamic characterisation of the materials.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When a defective flush valve caused water to flood this Chicago office building, MRI determined that drying, restoration and specialized electronic equipment cleaning was needed. A rapid response procedure included the packing of contents, blocking furniture and immediate water extraction. Air movers and dehumidifiers were placed to contribute to the proper drying of finishes, carpet and sheetrock. Once the emergency work was in place, MRI was able to evaluate the damage, develop the restoration and repair scopes of work and coordinate the subcontractors for the renovation work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzammoy b/data_all_eng_slimpj/shuffled/split2/finalzzzammoy
new file mode 100644
index 0000000000000000000000000000000000000000..1d9515393d40a4910dd37df29f9e7960804a3d8f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzammoy
@@ -0,0 +1,5 @@
+{"text":"Create, modify, and manage piping and instrumentation diagrams with AutoCAD P&ID 2011. Built on the latest AutoCAD platform, AutoCAD P&ID is easy to use and familiar to designers and engineers, so design teams can start quickly with minimal training. Common tasks performed every day are streamlined and automated to boost productivity, while component and line information is easily accessed by designers as they work. With simple reporting, editing, sharing, validation, and exchange of design information, projects start easier, run better, and finish sooner. AutoCAD P&ID is included in the Autodesk Plant Design Suite.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Transports hazardous materials in a volume or quantity that requires the use of warning placards.\nVehicles that meet one of these criteria fall under the oversight of the Federal Motor Carrier Safety Administration (FMCSA) to the extent adopted by Maine and must comply with its regulations as enacted by the State of Maine.\nIn addition to vehicles over 10,000 pounds, any vehicle that weighs more than 6,000 pounds must be registered by weight, whether or not the vehicle is engaged in commerce. The commercial plate is merely a state designation and does not connote that the truck is used for business.\nIf a vehicle weighs more than 10,000 pounds, is used to carry 15 or more people, or carries hazardous materials and is engaged in commerce, it must comply with gross vehicle weight restrictions and must check-in at truck stops.\nA truck that operates only within Maine (intrastate) and is registered as weighing more than 10,000 pounds or has three or more axles on the motorized part of the vehicle must register for a number from the Department of Transportation. This number must be displayed if the vehicle is engaged in intrastate or interstate commerce. These provisions also apply to buses for hire for which the Bureau of Motor Vehicles has issued a Permit for Operation of Motorcoach Intrastate Carrier.\nThere are restrictions even for those truck drivers and owners who operate completely within the state. A commercial truck driver in Maine must be at least 18 years of age and must be 21 years old in order to transport materials that are considered hazardous in an amount that requires placarding.\nMaine does require trucks only involved in intrastate commerce to comply with regulations based upon FMCSA rules regarding the distance that part of the load can project beyond the end of the motorized vehicle or trailer and how a load has to be secured to prevent shifting and falling. Moreover, logging trucks are subject to the restriction that if the truck is traveling during darkness and the logs that it is hauling project more than four feet beyond the end of the vehicle then it must have red reflectors or apply reflective paint to the ends of the logs that extend out of the vehicle.\nAlthough there are different regulations that apply to trucks that operate solely within the State of Maine versus those that engage in interstate commerce, when a driver of a commercial vehicle negligently causes an accident, the driver and the company for which he was driving may be held accountable for the harm that the operator caused.\nAlthough there are differences in the regulations that apply to trucks that operate entirely within the State of Maine versus those that conduct business in states throughout the region or country, the fact is that the experienced truck accident attorneys at Peter Thompson & Associates have the knowledge and commitment to getting justice for all victims. The type of claim that is brought against an intrastate truck driver may vary slightly from an interstate operator, but the basic tenets of investigating what happened to you, building a strong case, and getting the compensation that you deserve remains the same. We are ready to evaluate your case during an initial consultation as soon as you call (800) 804-2004.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This 1986 Vancouver 28 is a nice example of this well known blue water cruising yacht. Her inventory includes an 24HP Bukh diesel engine, tiller steering, slab reefing mainsail, furling genoa, furling staysail, spinnaker, manual windlass, sprayhood, three berths and more. She is in good condition and viewing is recommended.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Amid the fanfare and celebration that gripped Suva's Albert Park last night, there were some valuable lessons to learn from the Vodafone Fijian 7s side that were at the centre of the celebrations.\nAddressing the crowd gathered to honour the team's qualification to the 2016 Rio Olympics and winning the 2014-2015 Sevens World Series, Ministry of Youth and Sports Permanent Secretary Josefa Sania made this point.\n\"Today I wish to impart the lessons of capability to you from your valiant 7s heroes and the dimensions that surround them that can sum up with attributes like class, talent, attitude, skills, knowledge and strength,\" he said.\n\"Talents are our natural gifts and strengths. We, as Fijians, are not lacking in this department and I urge you to use your God-given talents well. This brings us to attitudes which represent our way of seeing as well as our ways of being.\n\"Your skills are proficiencies, things you can do well and as the greatest agents of the games of 7s, there is no doubt that Fijians are bound to possess that skill set. Knowledge represents the things we have learned and understand.\nVodafone Fijian 7s skipper Osea Kolinisau paid tribute to divine guidance and the support of the nation and the players' respective families for getting them through the series.\n\"First of all I'd like to thank the Lord for providing us with this opportunity for all of us to come and meet here and I have to give thanks to the fans that have been overwhelming in their support for Fiji,\" he said.\n\"Since we've arrived on Wednesday, we've been overwhelmed by the amount of people that turned out at the airport and the amount that turned out today and the people in the various villages that stopped the buses.\n\"It gives us more of a boost to want to play for the country and try to win more for Fiji and I hope all this tremendous support will continue until we reach Rio and that opportunity for Fiji to win its first Olympic gold medal.\nAlso addressing the crowd was Vodafone Fijian 7s team manager Ropate Kauvesi, paying tribute to head coach Ben Ryan.\n\"We've pretty much been welcomed at the airport, all throughout the ride from Nadi down to Suva which normally takes us three hours. It took us 10-and-a-half hours.\nThe capital city streets were jam-packed from 4pm with crowds gathered to greet the national team, with around 5000 people swarming Albert Park to catch a glimpse of the team and enjoy the illuminating fireworks display that marked the end of the celebrations.\nMeanwhile, when this edition went to press last night, the national sevens team had been hosted to dinner at the Holiday Inn with the Attorney General Aiyaz Sayed-Khaiyum, a number of Cabinet ministers, the Fiji Rugby Union board and representatives of the sponsorship consortium.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Check availability or get an estimate for hosting an event at the Fredericton Convention Centre!\nFill out the form below, or if you prefer please send an email with your event details to: fccinfo@frederictonconventions.ca. The best person to serve your event will get in touch with you as soon as possible.\nIf you wish to contact one of us directly, visit the Staff listing.\nIf you would prefer to use your own RFP form instead, simply email it to fccinfo@frederictonconventions.ca or fax it to (506) 460-2768.\nWhere has this meeting been held before?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaodbb b/data_all_eng_slimpj/shuffled/split2/finalzzzaodbb
new file mode 100644
index 0000000000000000000000000000000000000000..5fa93d67008de25d732319feb31dc07e426cb25f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaodbb
@@ -0,0 +1,5 @@
+{"text":"Teresa Salgueiro will make a second date in Funchal, on March 18, given that the number of tickets available for the show is already greatly reduced. That way, if you can not be present on Friday, you will have a new opportunity on Saturday. The venue of the concert is Teatro Baltazar Dias and it is there that you can buy tickets for the concert.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Under the direction of the Bakery Department Manager, responsible for working closely with all Bakery associates in an effort to keep the bakery department stocked and full of fresh product.\nTo perform this job successfully, an individual must be able to perform each essential job duty satisfactorily. The requirements listed below are representative of the knowledge, skills, and\/or ability to perform the essential functions.\nThe Bakery Bench Hand is an hourly position that may be part time or full time. The work schedule generally includes mornings, afternoons, weekends, and holidays, with the flexibility to work evenings or overnight as needed.\nResponsible for inspecting and testing slot equipment, noting all defective or weak parts and for making recommendations for correction.\nInspect and test assigned slot equipment, noting all defective or weak parts and make recommendations for correction.\nHave thorough knowledge of diagnostic, troubleshooting and test equipment.\nMake all required adjustments and repairs to slot electronic systems and equipment, as noted or discovered.\nDevelop and maintain a slot technical library and schematics for the equipment and\/or peripherals in use.\nStay up to date on slot technology and repair techniques.\nSafely and effectively operates shop tools including: hand tools, soldering equipment, and electronic test equipment.\nPerform the duties of a slot technician as needed to ensure maximum performance of all slot machines in the slot equipment shop or on the casino floor.\nStrictly adhere to established internal controls concerning the slot machines, the internal components and the repair procedures.\nObserve all safety and gaming regulations when performing assigned duties. Report safety hazards to assigned supervisor.\nDevelop and maintain adequate slot electronic component inventory for sub-system repairs.\nKnowledge of a variety of class III and class II games and tracking\/auditing systems is desired. Experience with metal fabrication and design is beneficial. Extensive knowledge of electronics theory, multi-testers, schematics and troubleshooting analysis are required. Must have discretion to maintain strict confidentiality of classified information. Must be competent with various PC based software programs. Must possess excellent business sense and high professional ethics. Ideal candidate will have excellent customer service demeanor and as needed will be willing to go above and beyond regular duties to ensure that our guests experience the best in gaming entertainment.\n2 years electronics degree or 3 years slot tech experience required, as a floor or shop technician. Knowledge of basic machine shop operations is essential. Strong organizational and communication skills, and the proven ability to perform in fast paced and goal oriented environment.\nReading, writing and interpersonal communication in English are key to the successful conduct of this position. Must have the ability to read, analyze and interpret technical reports and to write reports and correspondence.\nMust have the ability to add, subtract, multiply, and divide in all units of measure, using whole numbers, common fractions and decimals.\nMust have the ability to solve practical problems and deal with a variety of variables in situations where only limited standardization exists. Ability to interpret an extensive variety of instructions furnished in written, oral, diagram or schedule form.\nPosition requires valid Class C drivers license and ability to obtain NIGC gaming license.\nWhile performing the typical duties of this job, the employee is regularly required to stand, use hands to manipulate or feel, talk and hear. The employee frequently is required to walk and reach with hands and arms. The employee is occasionally required to sit. The employee must frequently lift and\/or move up to 25 pounds and occasionally lift and\/or move up to 50 pounds. Specific vision abilities required by this job include close vision, color vision, depth perception, and ability to adjust focus.\nThe casino environment has moderate to loud noise levels and is a smoking environment.\nAs a Retail Store Manager \u2013 Bench, this position will be trained in all aspects of running a retail business for FedEx Office. Training takes place in one or more FedEx Office retail store locations with a focus on all aspects of running the business . After successful completion of training, the candidate must be open to potential relocation and have the ability to be placed within the specified geographic region.\nThe Retail Store Manager \u2013 Bench will be assigned a retail store, and will be responsible for managing the overall operations of that retail store including supervision of team members, the administration of store sales performance, profitability, procedural compliance and customer experience objectives. Although the Retail Store Manager \u2013 Bench's management of the assigned retail store during the bench assignment will be overseen by experienced FedEx Office management Team Members, the Retail Store Manager \u2013 Bench will have responsibility for daily management of the retail store.\nMonitor and direct marketing activities within the retail store to achieve pre-established sales objectives including monthly marketing calendars, media advertising, specialized sales, in-store signage, etc.\nFor new hires, must meet all FedEx Office employment qualifications in force at time of hiring.\nAbility, on a consistent basis, to meet the needs of managing the retail store by ensuring prompt and regular attendance such that effective management of the retail store is achieved.\nWe are a Forbes Cloud 100 company and we've been recognized by Glassdoor as a \"Best Place to Work\" in 2019. We're on the hunt to find talented individuals to join our growing team.\nLocated in the city's South of Market (SOMA) neighborhood, the KeepTruckin headquarters is a walk away from Oracle Park (go Giants!) and Union Square. Our building\u2014built in 1923\u2014has a unique industrial, open feel made comfortable with countless comfy couches and beautiful greenery. When they're not working, you can find fellow team members playing pool, enjoying the rooftop view, and exploring this historical city.\nAs KeepTruckin's Sr. Test Bench Engineer, you will play an active part in ensuring our products are ready for release. You will be a key player to develop the Test tools that validates the performance of our products. This role is crucial as we are testing for the high reliability and compliance of our Electronic Logging Device. Working closely with the hardware and software teams, this test bench and other tools will evolve over the years. You will continuously increase test coverage and our ability to deliver challenging features to market.\nAudiosavings is a fast growing online audio equipment dealer seeking a full time bench technician to assist in testing Diagnosing, and repairing both pro audio and car audio and video equipment. The ideal candidate should be able to stand for 8 hours, lift 50 lbs and work in a fast paced environment. Computer skills such as fast typing and being able to navigate our internal software as well as using excel and google docs is a requirement.\n-Ability to lift 50+ lbs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This 12-piece builder's kit is for anyone with a vacuum pump that would like to build a continuously running system. Since this kit is used to build a non-cycling system the check valve, vacuum switch and vacuum reservoirs are not required.\nThe system is designed for woodworkers looking for a simple and affordable method of veneering wood panels. With an integrated speed control valve, the system is fully adjustable from 840 to 1750 lbs of pressure per square foot.\nThe only additional items you'll need only are a vacuum pump (with 1\/4\" threads), vacuum bag, and breather mesh.\nIt allows you to fine tune the pressure a bit without having to change the setting on the bleeder fitting. Additionally, the vacuum valve allows you to close off the flow to the bleeder valve completely so that, at the start of a pressing job, the vacuum pump can pull-down the vacuum bag slightly faster.\nDoes the system come with crimped-on ferrules?\nWe offered the kit with crimped-on ferrules for several years but these fittings became hard to find (in a size that fits our vacuum tubing) and then we had cost increases which were going to force me to raise the price of the kit. Instead of increasing the kit price, I opted to take the ferrules out of the kit. The ferrules were not expensive but the installation time was excessive. In a continuous-run system, the ferrules were absolutely useless any how. We don't even put ferrules on the auto-cycling kits.\nI bought this for a vacuum clamping system that I'm building for my CNC router. It's a learn-as-I-go situation. Everything about this is impressively high quality; the parts, the ideas, the instructions, the website and the users builds. Much easier to get the kit than source the parts individually and cheaper too.\nGreat kit to get started and saves time from having to source all the pieces individually. Arrived fast as well.\nMy experience with VeneerSupplies.com was excellent all the way around. The website is informative and comprehensive, and my email questions were answered most timely. The product was shipped and received in a matter of days. All arrived safely, all were present and accounted for and the packaging of the items was logical - no wondering if any of the parts were missing. All instructions were clear and everything worked as promised. I will definitely recommend VeneerSupplies.com.\nSimple assembly in a few minutes and everything you need is included. The one suggestion would be to include a photo of either an exploded view or the completed setup in the written directions that come with the kit.\nNote from Joe - I agree with Brian and there is now an image of the completed system in the instruction set.\nI bought this kit (with the vacuum clamp option), a 2' x 6' bag, and a Rietschle Thomas 3.15 CFM pump. I ordered it in the afternoon. An hour later, I received an email from Joe saying it had shipped. It arrived the next day via UPS ground. I assembled the pump kit with no issues following the included instructions. It had everything needed for assembly including the Teflon tape for sealing the threads. I made a platen from 3\/4\" MDF I already had by cutting 1\/4\" deep groves every 2\" in both directions and rounded the corners using a block plane and sand paper. I also used the breather mesh instead of a top platen (highly recommended).\nI purchased the cold press glue, glue roller, and veneer tape as well. I followed the procedures outlined on Joe's web site, and my first veneer pressing project turned out great.\nI had a couple of questions for things I didn't see on the web site, and Joe answered them via email within an hour or two and then added the answer to his web site for everyone to benefit from.\nExcellent product, service and informative web site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How to take care of newborns?\nHow to get rid of postpartum constipation?\nHow to recover faster after a delivery?\nDisclosure: Bear in mind that some of the links in this post are affiliate links and if you go through them to make a purchase, the website earns a commission. Keep in mind that the links to these companies and their products are because of their quality and not because of the commission we receive from your purchases. The decision to buy something is purely yours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You love her and appreciate all of her hard work, whether it be in her career or in the home, so let her know with one of these bath and body gifts!\nShe can create her own spa day at home with gifts like the Luxor's Waffle Weave Spa Bathrobe crafted from 100% Egyptian Cotton. Add her monogram on this Luxor spa bathrobe for a perfect personalized gift. Lets add some sophistication with a luxury Bamboo Bathtub Caddy Tray which has a conveniently built-in cup holder for keeping a glass of her favourite wine or a soothing cup of tea, together with her favourite book while listening to music from her phone to create the perfect ambiance.\nSpa gift sets are a great way to give her a collection of things to try together to pamper herself at home. Here is a marvellous selection of spa Gift Baskets to choose from including the Art of Appreciation Gift Baskets Lavender Renewal Spa Bath and Body Gift Set , the Pinkleaf Green Tea Argan Oil Bamboo Spa Bath Gift Set or popular VERDUGO GIFT CO Warm Vanilla Spa Basket which with its sweet and spicy vanilla scent wraps you in its comforting embrace! The caress of tangy tropical scents, a blend of sweet mangos and tart pears stir dreams of escape to far off islands where the shimmering sapphire sea laps at the edge of sun-warmed beaches comes in an intricate woven antique spa gift basket from Freida Joe.\nAn assortment of Cru de Provence mandarin mango scented spa products fills this lovely basket. Overflowing with body lotion, shower gel, bath caviar, bar soap , body butter, bubble bath, body scrub, natural sponge and a lavender sachet to bring the fresh, clean fragrance of lavender to any room, closet or drawer, this indulgent gift will help relax and enjoy. A pair of soft waffle slippers adds an extra special touch. Send her this tranquil spa collection now!\nDoes she enjoy a nice glass of wine while in the spa? Here is a perfect gift for her: Rock Falls Vineyards cabernet with plum and chocolate notes, and bright chardonnay with tangerine and apricot flavours and a hint of toasted almond and butterscotch arrive with a selection of mandarin mango spa products including body lotion, shower gel, bath caviar, bar soap, body butter, bubble bath, body scrub, natural sponge and a lavender sachet. A pair of soft waffle slippers adds an extra special touch.\nDoes she love vanilla? Here is the perfect gift: This spa collection is overflowing with the classic, rich fragrance of creamy vanilla bean spa products including body lotion, shower gel, both fine and chunky bath salts, linen spray, peppermint foot soak, bath caviar, body scrub, bar soap, bubble bath, sponge, eye mask and more. A microfibre towel and a pair of slippers add an extra special touch to this exclusive spa experience.\nStill not decided? Here is a great suggestion: Candles are essential to creating a spa-like ambiance. This Art Naturals Aromatherapy Candle set offers 3 lovely fragrances to create a mood-enhancing atmosphere and relaxation .\nMy next suggestion has in mind someone holistic minded who knows the benefits of essential oils. A gift like this elegant scented Earth & Sea Spa Essentials Bath Set is sure to impress her. This gift set is the perfect combination of eight spa essentials to relax, refresh and rejuvenate. A combination of premium oils is included in the candle, bath salts, and soap. These scents are fabulous and promote feelings of love, sensuality and happiness while providing calm, relaxing, refreshing tranquility by soothing tension and stress. Everything is high quality and comes in a beautiful gift box.\nPlanning a romantic night at home? Don't hesitate! Get these organic Bubble Bath Bombs Gift Set are going to transform her bath into a luxurious spa. There are 8 USA made, natural and organic bath bombs packed into an elegant gift set which makes them look like jewels.\n*This page contains affiliate links. Click here to learn more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqdsf b/data_all_eng_slimpj/shuffled/split2/finalzzzaqdsf
new file mode 100644
index 0000000000000000000000000000000000000000..22f8d23c8d03f1880d9bb7e9778037906b46680b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaqdsf
@@ -0,0 +1,5 @@
+{"text":"So I think once a day, I will take a random photo and write about it. It'll be a little experiment of mine. 1) I want to see how long I do it. 2) I want to see what kind of random photos I can collect. Leggggooo!\nFirst photo. What the hell is this? This is in the teenage section at kohls. I would NEVER wear this. They say fashion repeats itself. Well floral shirts can stay in the fast. Dear god.\nThis should be every fathers shirt! I love protective fathers. My dad was protective and it was the best thing he could have done for me. Life is too short to mess up with your children. Love them, and be a parent, not their friend.\nToday is a picture of myself! My birthday is coming up!\nOps! I missed a day. So I'll add two today.\nThis is my baby. \ud83d\ude42 Happy 4th everyone! I hope everyone was safe.\nWow this is crazy! It has Down syndrome. I've never seen such an animal. Honestly, I didn't even know animals could get Down syndrome, which makes sense because its humans get it. Shocked me!!\nOps! Missed a few days. Today is July 8th. Lets see what picture I pick today..\nOh goodness! I'd love shirts like this. It would fit you perfectly and it would be your own. No two could look alike. This is such a great idea!! I love it. What do you think?\nAugust 16! I know I haven't kept up with this! Not good\u2026 Anyway here is a tattoo Bay and I are thinking of getting!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since I still have a couple hours of Wednesday left, I thought I'd say hi to all of y'all and show you this picture.\nWell, I'm sorry I haven't been able to visit all of y'all like I've been wanting. Things have been super-dee-duper busy around these parts... But I expect I'll have a little more time on my hands in the coming weeks to catch up and sit a spell.\nIf I don't talk to y'all again, have a wonderful holiday weekend! Eat lots of BBQ or whatever it is that you do on weekends like this!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since the election, I've been trying to find time to share some thoughts & feelings, but I keep putting it off because sometimes it is just so overwhelming. I think one thing is realizing how my own words & actions have added to the reeking heap of hatred that is so often silently hurting those I love. Today, a friend shared this article and another offered a \"Complicity Cleanse\" to start the New Year humbler and more woke in order to look inward so that we may heal ourselves, so that we may work to heal the greater and more systemic wounds around us.\nIn her article, Davies' words resonate most when she writes, \"But I also have to remember that everyone in their secret histories has made some transgression against women, just as I\u2014in my whiteness, my relative economic comfort, my blind spots, and areas of ignorance\u2014have surely offended and impeded someone else. \" I think one of the most challenging parts of the work of healing is opening up about how we have hurt others. but it is in the acceptance of both responsibility and subsequent forgiveness that the festering sores of our own moral blindness begin to heal and we can begin to be a source of light within the darkness of our world.\nI think this film, The Mask You Live In, is the most helpful and easily digestible analysis of how hatred is instilled in us, rather than love. In looking at this, we can begin to understand what it is that keeps racism alive, why millions elected Trump, how we are seduced by consumerism, why we stigmatize those who are neuro-atypical (or think\/behave differently than most) and ultimately why cis men (meaning men who were assigned male at birth and feel that the words \"man\" and \"male\" accurately describe who they are) have such a hard time admitting that we use violence to maintain control and then defend ourselves by saying that coercion and insults and manipulation and shame are not violence because the laws of society, made by cis men, are limited by design. However, my belief, which comes from my Catholic faith, is that if a word or an action is not coming from a place of genuine love for the person in front of you, it is violence. In other words, if it is not intended to be received as love, it is intended to hurt.\nTo my brothers, my uncles, my father, but also my sisters & cousins & mother & all of my friends & comrades & teachers: Please just take fifteen minutes to start the film, then decide if you want to continue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Reconnect with the sounds of life\u2026today! Hearing loss can isolate you from your surroundings and the people you care for most. With many years of combined education and experience, we at Hearing Resource Center of San Mateo, are devoted to improving your quality of life so that you can enjoy each & every laugh, sigh, scream, and chuckle from those you love and care for the most.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"the farm @ Echo Bend | Honoring the echoes of the past while growing sustainably for the future.\nEcho Bend is a small, diversified farm overlooking Upper Herring Lake in Benzie County, Michigan.\nWe focus on growing heirloom and great-tasting varieties of produce and fruits using organic principles.\nWe offer weekly CSA subscriptions or on-farm sales at our roadside stand, where you can also walk the garden and pick your own flowers and herbs. We welcome you to visit and observe our growing practices, sample fresh baked goods and preserves, and relax in the shade and enjoy the views. Pastured pork and poultry complement our offerings, and the pigs, chickens and turkeys also make great hosts to your farm visits!\nFollow \"Echo Bend\" on Facebook!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasjfj b/data_all_eng_slimpj/shuffled/split2/finalzzzasjfj
new file mode 100644
index 0000000000000000000000000000000000000000..121d06b9c10d956b5d4763ed996227e866e7c116
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasjfj
@@ -0,0 +1,5 @@
+{"text":"It's time to apply for The Simple Year 6! Click HERE for details, or check out \"The Handoff\" link above.\nIt seems like the more popular zero waste gets, the more items there are to pick and choose from to make the journey easier\/better\/prettier. In other words, there's a bit of a zero waste racket going on out there in the world. I don't know, I guess that's not so shocking. It's the same with minimalism \u2014 if there's a perceived need, someone is going to try to fill it.\nI've pretty much made all the mistakes (not to sell myself short \u2014 I'm sure I can and will make more), so that's exciting. Below are a few of the hits and misses I've experienced this year. Some update previous posts. And some of it is just random thoughts.\nHomemade calendula oil \u2014 it took four (um, five for me because I forgot about it) weeks to soak and mature, but it's cheap, easy and a lovely finished product.\nI'm also a fan of my jars of cleaning solution.\nHomemade sink and counter cleaners \u2014 take minutes to throw together, clean really well and smell way better than purchased cleaners.\nHomemade face wash \u2014 also easy to mix together (castile and olive oil, done!) and works very well. I like that it doesn't dry out my skin.\nHomemade lotion for face, hands and body \u2014 my favorite is just olive oil and beeswax. Not very complicated.\nHomemade floor cleaner \u2014 white vinegar and water work amazingly well.\nHomemade throat spray \u2014 my kids are big fans. I am too.\nSafety razor \u2014 I was really scared to switch, but I'm a big fan. Bonus that it's a better shave. And not as scary as I'd imagined.\nLinen or cotton kitchen towels \u2014 handy!\nJars \u2014 I inherited lots of canning jars in various sizes. Use them for everything from pantry to refrigerator to freezer storage. Love!\nTravel mug and\/or water bottle \u2014 don't leave home without them. Easy way to say no to drink-related disposables.\nMenstrual pads \u2014 initially more expensive (unless you make them yourself) but more comfortable and an easy way to cut period-related waste.\nBamboo toothbrush \u2014 Another easy change, although I've since learned that bamboo can get moldy. So I just make sure I dry mine off before putting it back into its holder.\nShopping kit \u2014 having jars, produce bags and grocery sacks in one place is genius.\nDeodorant crystal \u2014 this thing saved my life.\nNo bake energy bars \u2014 we go through at least one pan of these a week. I can get everything I need in bulk, they taste great and are very filling.\nHomemade broth \u2014 a little more time consuming, but a better product. And I like making something from nothing.\nNaan \u2014 I always have dough in the freezer for Friday pizza nights. Easy to batch.\nMeal planning and prep \u2014 this has worked out remarkably well and has saved us more than once.\nCanning and freezing each summer \u2014 helps us eat locally all year.\nPlanning for holidays, back to school, birthdays, etc. \u2014 planning ahead (and the patience to wait if need be) often made the difference between utter failure (hello, back to school!) and dizzily amazing success (the girls' birthdays!).\nHomemade makeup \u2014 I wanted this to work, but powders don't stay put or they get into your eyes. Not awesome.\nHomemade tooth powder and toothpaste \u2014 I was initially excited by my doctor's and dentist's reactions, but then a followup with my dentist and the prospect of losing my tooth enamel sent me running back to regular toothpaste.\nHomemade laundry and dishwasher powders \u2014 while I was initially happy with the results, I came to the reluctant conclusion that they don't work as well as their purchased counterparts.\nHomemade throat lozenges \u2014 this was a disaster and a waste.\nBamboo kitchen brush for washing dishes \u2014 it's pretty, all right, but the head doesn't clean the bottom of glasses and jars very well (and jars are kind of my thing).\nHomemade mustard \u2014 Eric says it's not getting any better with age. Part of that might have been my omission of turmeric. Or maybe it's just not good.\nMaking more than two stops on a shopping trip \u2014 I'm fairly lucky with my grocery store, as it has most of what I need, but I've come to the conclusion that driving around to a bunch of stores to do the weekly shopping just isn't worth it. These days I'm sticking to the store and the bakery.\nMaking everything from scratch \u2014 it's true that I have to make a lot from scratch because my stomach is a jerk, but I'm okay with sourcing out items or just doing without. Being chained to the kitchen is not my idea of a good time. And I'm letting go of the guilt associated with that.\nResearching a topic to death \u2014 been there, done that. I'm giving myself permission to not do that any more. I make things way harder on myself then they need to be. And there is no perfect option anyway.\nAnything you'd add to the list? Take away?\nNext up: I am so ready to bust out my Project 333 spring capsule after a terribly snowy cold crappy winter!\nThis entry was posted in zero waste and tagged easy zero waste bathroom ideas, easy zero waste kitchen ideas, updates. Bookmark the permalink.\nThank you for the summary! I've really enjoyed following your simple year and have been coming to similar conclusions on some of the time and energy tradeoffs.\nI've definitely learned that you have to pick and choose your battles!\nI had to give up on homemade laundry detergent as well. I'm disappointed, but it wasn't dissolving and it stayed stuck to our clothes. I'm so glad the brown sugar worked for you \u2014 I'm looking forward to making my own, inspired by you, when I use up the boxed stuff I have.\nI hope the brown sugar is a success! Did you ever try the mustard? Don't!\nI'm with you on the homemade makeup. For me it always yielded powdery no-makeup-look makeup\u2026and if I want that look, I'll just skip the makeup altogether!\nWorth it: cloth napkins, rags, wipes, handkerchiefs, cloth pads, etc. Not hard, and saves money. I've also gotten a lot of fun and satisfaction from baking cookies often and learning to make a few candies from scratch.\nNot worth it: making everything from scratch, as you say. I'm not going to sweat over an occasional jar of jam or mustard. The outlay of effort isn't worth the actual waste reduction it achieves. I'm sticking with making what we eat regularly, or what I enjoy making.\nYes to the reusable cloth items! Yes to not making everything from scratch!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Flexibele bevestiging en X-Grip UN8BU houder voor fiets, motor, rolstoel, golfcart.\nThe RAM-B-103U, powder coated marine grade aluminum mount, consists of a double socket arm and one 2.5\" diameter round bases with the universal AMPS hole pattern. Designed into the mount is a 1\" diameter patented rubber ball and socket system with adjustment points at both socket ends of the arm. Rubber ball and socket technology allows for almost infinite adjustment and perfect viewing angles. High quality materials insure your device is safe, secure, and within easy reach. RAM's patented design also dampens shock and vibration helping to extend the life of your device.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This easy, cozy Peasant Top in Velvet will be your newest cold weather staple! The relaxed fit makes this so versatile. Pull it off the shoulder or wear high.\nOur Peasant Top in Velvet is a modern take on the boho classic. Perfectly easy and comfortable, this off the shoulder shirt is cut in our velvet with long sleeves that gather gently at the wrist. A relaxed fit and longer hip length make this top extremely versatile for the cold weather ahead. Pair it with a multitude of bottoms.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The State Water Resources Control Board (SWRCB) is hold a Board meeting on October 5-6, 2016. Please click here to view the agenda.\nClick here to view the Revised Notice of Opportunity to Comment and Public Workshop Regarding California Environmental Laboratory Accreditation Program (ELAP) Regulations Development and Preliminary Staff Recommendation for Laboratory Standard. Comment Deadline has been changed to October 20, 2016 by 12 noon.\nClick here to read and download the Executive Director's Report for the September 20, 2016 Board Meeting of the Water Resources Control Board.\nThe State Water Resources Control Board (SWRCB) will hold a Board Meeting on Tuesday, September 20, 2016 at 9:00 am. Please see the agenda for more information.\nThe San Francisco Port Commission will meet next Tuesday, September 13, at the Ferry Building. A Closed Session will be held at 2:00 pm followed by an Open Session at 3:15 pm. For more information, view the agenda here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"A Native American grandfather was speaking to his grandson about violence and cruelty in the world and how it comes about. He said it was as if two wolves were fighting in his heart. One wolf was vengeful and angry, and the other wolf was understanding and kind. The young man asked his grandfather which wolf would win the fight in his heart. And the grandfather answered, \"The one that wins will be the one I choose to feed.\" So this is our challenge, the challenge for our spiritual practice and the challenge for the world\u2014how can we train right now, not later, in feeding the right wolf?\" - Chodron, Pema (2010-09-14). Taking the Leap: Freeing Ourselves From Old Habits and Fears (pp. 2-3). Shambhala Publications.\nFriends, there are many demands on us as we move throughout our daily lives, but none higher than choosing a life of kindness.\nHoly One, help us to \"put on a heart of compassion, kindness, humility, gentleness, and patience.\" Amen.\nJoin us for Newcomers Coffee!\nThis coming Sunday, July 29 at 12:00pm, Pastor Jenny will be in the Library to welcome you to Community UCC. Whether you have been visiting with us for a while or this is your first time we invite you to get to know us and to share a bit about yourself. Coffee and snacks provided! Children are always welcome to join us, and the nursery will be open as well.\nIn her July 7 sermon Pastor Jenny read from Austin Channing Brown's \"I'm Still Here: Black Dignity in a World Made for Whiteness.\" Jenny is planning a study of this book as part of this fall's theme about our stories. Let her know if you are interested.\nCUCC's Community Outreach Ministry is working through the month of July to collect healthy snacks in support of Neighbor 2 Neighbor's Campo Mucho. This program provides outreach to many of our community's Latino children who come from primarily non-English speaking homes and who may be food insecure in their homes.\nThe camp will need 400 snacks total to provide their participants with 2 healthy snacks throughout their programming day. If we can collect these snacks in time for their programming to begin on July 30, staff can use their limited funds to better provide more programming and 2 full meals during the days.\nA donation tub will be in the Narthex each Sunday for collection. The easiest way to collect snacks will be to buy boxes of individually packaged healthy snack options. For more information, please contact Laurel Powell at Laurel.powell@gmail.com.\nMissed last week's sermon? Check the Sermons section of our website.\nWe'll post the notes from which Pastor Jenny preached as often as possible.\nYou asked... can I compost this paper plate?\nRecycle? NO. Dirty paper is harmful to the process used to turn old paper into a new product. Never recycle dirty paper plates, cups or food containers.\nCompost? YES if it is marked \"compostable.\" The word \"compostable\" indicates that the product will degrade completely at an industrial composting facility (which is what we used). All other paper (even if it is marked \"eco\" or \"recycled content\") has either a coating or an ink that won't compost or is harmful. Look for the word \"compostable\" to decide.\nLandfill? YES if it is NOT marked \"compostable.\"It is better to landfill the plate or cup than to have the whole load of recycling or composting thrown away.\nSend your CUCC sorting questions to Jane Smith smithjeg@mindspring.com.\nWe're taking a month off. See you Tuesday, September 4.\nLearn more about Life & Faith, CUCC's informal drop in at O'Malley's Pub & Restaurant when we talk about the intersection of life and faith.\nCheck it out: Alerta Migratoria NC, a pro-immigrant and refugee group in the Triangle, wants anyone supporting the six immigrants in sanctuary in North Carolina to join them at a Town Hall Meeting at 6:30 pm at Broughton High School in Raleigh. One of the six is Eliseo Jimenez, who is in sanctuary at Umstead Park UCC.\nIn the email is a link where you can register online ahead.\nFree NC Refugee KidsTell Sheriff Donnie Harrison: Stop Working with ICE! (Note: Sheriff Harrison is up for re-election this Fall.)Tell Congressman Price: Stop the Deportation of the #NCSanctuary 6!You can follow Alerta Migratoria NC on Facebook, Twitter, or Instagram.\nLife & Faith meets July 3 - drop in!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaskqt b/data_all_eng_slimpj/shuffled/split2/finalzzzaskqt
new file mode 100644
index 0000000000000000000000000000000000000000..c4019e9d3c472f625ce39a776ecd8430aeb9ca39
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaskqt
@@ -0,0 +1,5 @@
+{"text":"Pride FC was filled with a whole host of weird and wonderful characters. The intrinsic link between MMA and professional wrestling in Japan meant even top fighters such as Wanderlei Silva, Quentin \"Rampage\" Jackson and Fedor Emelianenko were pushed and packaged as wrestling-style superstars, while many of the in-ring competitors came from wrestling backgrounds themselves\u2014\"The Gracie Hunter\" Kazushi Sakuraba, for example.\nThe anything goes style of Pride draws many parallels to pro wrestling, more specifically the WWF's so-called \"Attitude Era.\" There's a reason why reminiscing about both Pride and the late '90s\/early '00s professional wrestling scene produces a steady stream of fantastic nostalgia\u2014those shows were all about the fans and were catered for their respective audiences.\nOne man managed to traverse both Pride and WWF's heyday: Paulo Cesar da Silva, otherwise known as Giant Silva.\nSao Paulo, Brazil, native Silva has many strings to his bow. Standing in at a hulking 7ft 2 ins, Silva carved out a reasonably successful basketball career as a centre and represented his country in the 1988 Olympics in Seoul, South Korea, before heading into the world of in-ring competition\u2014kayfabe of otherwise.\nSilva's sheer mass\u2014all 385 pounds of it\u2014was an instant draw to McMahon, who signed him to the WWF in 1998 in the midst of the Attitude Era. If you want an idea of the sheer captivating nonsense of that time in professional wrestling, look no further than the above video.\nSilva was aligned with The Oddities\u2014a wrestling stable so Attitude Era it hurts. A team of so-called \"freaks\" which featured the likes of Golga, a giant man wearing a gimp mask and ill-fitting clothes while carrying around a stuffed Eric Cartman toy, and the overbearing Juggalo kings the Insane Clown Posse. As odd as it sounds and looks, this stable perfectly encapsulates the wrestling craziness which drew me in as a fan in my younger years\u2014a joy clearly replicated from those fans in attendance featured in the above video.\nThe Brazilian's WWF adventure didn't last long. Despite his stable achieving unexpected popularity, The Oddities were disbanded fairly promptly. But, in the case of Silva, he simply couldn't wrestle. His size meant he wasn't the most graceful man in the ring and his appearances inside the squared circle were largely limited to an entourage capacity, which would sometimes see him attack opponents while acting his team's cornerman. In the early months of 1999, Silva was cut from the wrestling promotion.\nThe wrestling dream wasn't over for Silva\u2014he went on to perform at New Japan Pro Wrestling and later Hustle. But, it was this relocation to the Far East which attracted Dream Stage Entertainment\u2014the ill-fated team behind both Pride FC and aforementioned wrestling promotion Hustle.\nUnlike many of his Brazilian contemporaries, Silva had little martial arts experience to draw from. He had claimed in the past that he trained Brazilian jiu jitsu with the likes of Ricardo and Ralek Gracie, but this was and remains unverified. Silva was thrown to the proverbial wolves and was expected to fight experienced American heavyweight Heath Herring on his professional debut.\nSilva's above interview with Mauro Ranallo, who has since made the reverse transition from commentating on MMA to WWE's Smackdown brand, was perfect. Silva didn't have a strategy against Herring. But, from sheer size alone, he managed to take Herring to a third round. Unfortunately for Silva, he succumbed to the rear naked choke of Herring, having been chopped down with a series of uncomfortable-looking leg kicks which forced the big man to try and take the fight down to the ground.\nDespite losing his sole MMA fight, Silva was entered in the 2004 Heavyweight Grand Prix tournament in classic Pride fashion. As stupid as it would have sounded at the time, it was impossible to predict who Silva would be fighting. Would it be Mirko Cro Cop? Fedor Emelianenko? Antonio Rodrigo Nogueira? Nope. It was against sumo wrestler Henry \"Sentoryu\" Miller.\nSilva's sophomoric showing proved successful. Miller opted to neutralise Silva's height and reach advantage by taking the fight to the floor. Miller, frustrated at Silva's surprisingly effective ground defence, moved to side control only for Silva to slap on a kimura and force the submission. Unfortunately for the Brazilian, Silva's dreams of becoming the Grand Prix champion were dashed in the next round, having lost by TKO to former world judo champion Naoya Ogawa.\nIn all of Silva's exploits in both professional wrestling and mixed martial arts, one incident stands out to me the most. It was in his next fight, against fellow wrestler Takashi Sugiura of NOAH fame. Billed at a height of 5ft 8ins, Suguira used his wrestling ability to nullify Silva's size advantage by taking him down. It didn't last long on the ground\u2014in fact, the fight didn't even make the three minute mark. Silva suffered a TKO loss to the hands and knees of Sugiura, who used said limbs in a frenzied attack to end Silva's night.\nHowever, Silva clearly took umbrage to the manner in which he lost. Whether it was a professional wrestling-style work or a legitimate case of sheer rage, I don't care. Silva somehow sourced a giant wooden stick to attack Sugiura and it took what looked like twenty Pride officials to restrain the Giant from beating Sugiura to a Ken Shamrock-esque living death.\nThings didn't get much better for Silva in his fighting career. The Brazilian lost and was finished in another three fights \u2013 starting with the submission loss to South Korean Choi Mu-Bae, a TKO loss by way of brutal soccer kicks against burly Brit James Thompson and a TKO loss to the much-smaller Ikuhisa Minowa with a series of horrific-looking knees delivered to both the body and the head while on the ground.\nThose last two losses were particularly hard to watch. Silva was now being used as a static target to help build up Pride's other stars. This was always the case, but it had never been so blatant until when the diminutive Minowa mauled Silva in a matter of two and a half minutes. Age and a creaking body had well and truly caught up with Silva\u2014and that's a man who wasn't particularly agile in the first place. Minowa, meanwhile, helped build and add credence to his name as a David vs. Goliath figure off the back of defeating Silva.\nWith a record of 1-6 and coming off a string of knockout and submission losses, it appeared unlikely that we'd see Silva competing in MMA again. He was promptly expelled from the Pride roster after the Minowa fight. But, it was soon announced that he would face off against another sumo wrestler in Hawaii-born Akebono Taro at K-1 and HERO's annual New Year's Eve extravaganza: K-1 PREMIUM 2006 Dynamite!!\nThis would prove to be Silva's last appearance in professional fighting and his final bow was on a maginificent stage. Fighting at the Kyocera Dome in Osaka, Japan, in front of over 50,000 in attendance, Silva finished his fighting career as he had (almost) begun\u2014submitting a sumo wrestler with a kimura.\nAkebono sought to utilise his weight and pressure Silva against the ropes to bring the fight to his world. This fight was the first time Silva had ever faced an opponent heavier than he was so this seemed to be a smart strategy. That weight advantage didn't count for much, though. Silva broke from Akebona's bear-like grip in the clinch to secure the right arm of Akebono's, beginning the kimura before taking the fight to the floor to twist and contort the elbow and shoulder joints to force the submission victory. It was only Silva's second win in an eventful, yet brief, three-year MMA career. But, seeing Silva facing off against a sumo wrestler was a perfect way to end things.\nGiant Silva may not be a man you associate with Pride nor the WWE's Attitude Era. If you do, your memories may not necessarily hold him in the highest esteem. But, it's undoubted that Silva's presence in both capacities perfectly summed up the wild nature of both industries at the time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Before the last round of trilogue negotiations between the Council and the Parliament on the GMO \"opt out\" directive, the organic farming movement urged Member States to take on board the Parliament's demand for clear, sound legal grounds to ban GMOs.\n\"If Member States believe that asking biotech companies not to market their products is a safer way than banning them on clear legal grounds, they are wrong. The Parliament should not accept this quick-fix solution that amounts to sweeping the GM problem under the carpet and that will only create more confusion in the long-term,\" warned Marco Schl\u00fcter, IFOAM EU Director.\nBackground \u2013 The trilogue discussions aim to reach consensus on the procedure for Member States to restrict cultivation of GMOs. In the Council proposal, Member States would have to ask the company applying for permission to market a GMO to withdraw their territory from the scope of their application (phase 1). If the applicant does not agree, Member States are given a list of considerations to justify their demand. However, the considerations specified in the Council proposal do not provide sufficient legal ground to enact a ban (phase 2). In practice, it could mean that corporate interests rather than democratic institutions will decide where GMOs are cultivated in Europe. On the other hand, the Parliament position would enable EU countries to adopt bans using grounds related to environmental impacts and risks complementary to those concretely examined during the EU risk assessment. Such risk management measures are not in contradiction with the EU risk assessment, which does not address all the systemic impacts of GMOs in the diverse European agricultural systems. When deciding whether to restrict GM cultivation or not, governments would act as risk managers, taking into account the European Food Safety Authority's assessment, as well as other relevant economic, environmental and agricultural considerations \u2013 allowing Member States to decide for themselves.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Creative Biogene has collected many scientific papers and patents related to part of our core products. Here is a long list of those citing our featured products in stock or partially supplied by our high-quality services. Below are some selected articles to give you an idea of how we assist life science research in academia.\nStable cell lines with specific gene over-expression, knock-down or Knock-out are very helpful in gene function analysis, target discovery, target validation, assay development, and compound screening. Stable cell line is the best solution, not only to produce recombinant protein and antibody, but also for application in drug screenings, gene functional studies etc.\nPark, EunJu, et al. \"Discovery of novel pyrimidine and malonamide derivatives as TGR5 agonists.\" Bioorganic & medicinal chemistry letters 24.17 (2014): 4271-4275.\nUz, Metin, Surya K. Mallapragada, and Sacide Alsoy Altinkaya. \"Responsive pentablock copolymers for siRNA delivery.\" RSC Advances 5.54 (2015): 43515-43527.\nCreative Biogene provides the largest number of cDNA clones in multiple sets of vector systems with various features, including more than two hundred thousand cDNA clones, in the form of full length cDNA clones or ORF clones.\nHong, Sungkook, et al. \"Dominant-negative kinase domain mutations in FGFR1 can explain the clinical severity of Hartsfield syndrome.\" Human molecular genetics 25.10 (2016): 1912-1922.\nCreative Biogene has supplied multiple 3'UTR target clones and developed an advanced vector-based miRNA expression plasmids , which can help to ease the difficulty in your research.\nYang, Chun, and Yong Pan. \"Fluorouracil induces autophagy-related gastric carcinoma cell death through Beclin-1 upregulation by miR-30 suppression.\" Tumor Biology (2015): 1-6.\nXu, Fulin, et al. \"Growth of glioblastoma is inhibited by miR-133-mediated EGFR suppression.\" Tumor Biology 36.12 (2015): 9553-9558.\nZhang, Xiao-Tao, et al. \"Impairment of growth of gastric carcinoma by miR-133-mediated Her-2 inhibition.\" Tumor Biology 36.11 (2015): 8925-8930.\nChen, Bing, et al. \"MiRNA-494 inhibits metastasis of cervical cancer through Pttg1.\" Tumor Biology 36.9 (2015): 7143-7149.\nLiu, Feng, et al. \"VEGF-activated miR-144 regulates autophagic survival of prostate cancer cells against Cisplatin.\" Tumor Biology (2015): 1-7.\nShi, Yan, et al. \"miR-203 suppression in gastric carcinoma promotes Slug-mediated cancer metastasis.\" Tumor Biology (2015): 1-6.\nTian, Y. Y., et al. \"Restoration of microRNA-373 suppresses growth of human T-cell lymphoma cells by repressing CCND1.\" European review for medical and pharmacological sciences 20.21 (2016): 4435-4444.\nSu, Lin, et al. \"Skp2 regulates non-small cell lung cancer cell growth by Meg3 and miR-3163.\" Tumor Biology 37.3 (2016): 3925-3931.\nChen, Kan, and Wenjun Shi. \"Autophagy regulates resistance of non-small cell lung cancer cells to paclitaxel.\" Tumor Biology 37.8 (2016): 10539-10544.\nXiao, Wenjun, et al. \"Norcantharidin induces autophagy-related prostate cancer cell death through Beclin-1 upregulation by miR-129-5p suppression.\" Tumor Biology (2015): 1-6.\nChen, Zhi, et al. \"Colorectal cancer cells are resistant to anti-EGFR monoclonal antibody through adapted autophagy.\" American journal of translational research 8.2 (2016): 1190.\nZhang, Zhenxing, et al. \"MicroRNA-137 inhibits growth of glioblastoma through EGFR suppression.\" American journal of translational research 9.3 (2017): 1492.\nLi, Wei, et al. \"miR-132 upregulation promotes gastric cancer cell growth through suppression of FoxO1 translation.\" Tumor Biology (2015): 1-7.\nBi, Z. M., et al. \"Human umbilical cord mesenchymal stem cells ameliorate experimental cirrhosis through activation of keratinocyte growth factor by suppressing microRNA-199.\" European review for medical and pharmacological sciences 20.23 (2016): 4905.\nYu, Zhongxiang, et al. \"Overexpression of miR-506 inhibits growth of osteosarcoma through Snail2.\" American journal of translational research 7.12 (2015): 2716.\nChen, Lei, Lei Xu, and Gang Wang. \"Regulation of MET-mediated proliferation of thyroid carcinoma cells by miR-449b.\" Tumor Biology 36.11 (2015): 8653-8660.\nWu, Wen-Bo, et al. \"MicroRNA-3713 regulates bladder cell invasion via MMP9.\" Scientific Reports 6 (2016).\nHu, Jun, Jun-Feng Xu, and Wei-Li Ge. \"MiR-497 enhances metastasis of oral squamous cell carcinoma through SMAD7 suppression.\" American journal of translational research 8.7 (2016): 3023.\nYang, Nian-Qin, et al. \"Crosstalk between Meg3 and miR-1297 regulates growth of testicular germ cell tumor through PTEN\/PI3K\/AKT pathway.\" American journal of translational research 8.2 (2016): 1091.\nLiu, Wei, et al. \"MiR-429 regulates gastric cancer cell invasiveness through ZEB proteins.\" Tumor Biology (2015): 1-7.\nCui, Z., and Y. Hu. \"MicroRNA-124 suppresses Slug-mediated lung cancer metastasis.\" European review for medical and pharmacological sciences 20.18 (2016): 3802-3811.\nGao, Feng, et al. \"Regulation of activating protein-4-associated metastases of non-small cell lung cancer cells by miR-144.\" Tumor Biology (2015): 1-7.\nChen, Li, et al. \"Expression of antisense of microRNA-26a-5p in mesenchymal stem cells increases their therapeutic effects against cirrhosis.\" American journal of translational research 9.3 (2017): 1500.\nSun, Jiping, et al. \"Protection of tubular epithelial cells during renal injury via post-transcriptional control of BMP7.\" Molecular and Cellular Biochemistry (2017): 1-8.\nRatovitski, Edward A. \"Phospho\u2010\u0394Np63\u03b1\u2010responsive microRNAs contribute to the regulation of necroptosis in squamous cell carcinoma upon cisplatin exposure.\" FEBS letters 589.12 (2015): 1352-1358.\nLi, Lin, et al. \"Pituitary tumor-transforming gene 1 enhances metastases of cervical cancer cells through miR-3666-regulated ZEB1.\" Tumor Biology (2015): 1-7.\nZhang, Zhenxing, et al. \"MiR-429 induces apoptosis of glioblastoma cell through Bcl-2.\" Tumor Biology (2015): 1-7.\nFan, Yue, et al. \"MicroRNA-454 regulates stromal cell derived factor-1 in the control of the growth of pancreatic ductal adenocarcinoma.\" Scientific reports 6 (2016).\nWu, Xiaoli, et al. \"Role of Beclin-1-Mediated Autophagy in the Survival of Pediatric Leukemia Cells.\" Cellular Physiology and Biochemistry 39.5 (2016): 1827-1836.\nCreative Biogene offers a wide range of vectors including expression vectors and clone vectors, as well as phagemid vectors and viral vectors.\nMandal, Papita, et al. \"Unexpected histone H3 tail-clipping activity of glutamate dehydrogenase.\" Journal of Biological Chemistry 288.26 (2013): 18743-18757.\nKhalili, Ehsan, et al. \"Production of Recombinant Human scFv Against Tetanus Toxin HeavyChain by Phage Display Technology.\" Monoclonal antibodies in immunodiagnosis and immunotherapy 34.5 (2015): 303-309.\nZare, Hamed, et al. \"Production of nanobodies against prostate-specific membrane antigen (PSMA) recognizing LnCaP cells.\" Int J Biol Markers 29.2 (2014): e169-79.\nRicci, Ada, et al. \"Visualizing the relevance of bacterial blue\u2010and red\u2010light receptors during plant\u2013pathogen interaction.\" Environmental microbiology reports 7.5 (2015): 795-802.\nCreative Biogene supplies a series of shRNAs, which could make tight hairpins used for silencing target gene expression via RNA interference. Besides, our state-of-the-art facilities and highly experienced staff are available in order to assist in all areas of virus vector design and construction, as well as the generation of the virus in a quick turnaround.\nKiessling, Silke, et al. \"Enhancing circadian clock function in cancer cells inhibits tumor growth.\" BMC biology 15.1 (2017): 13.\nLi, B., et al. \"IKVAV regulates ERK1\/2 and Akt signalling pathways in BMMSC population growth and proliferation.\" Cell proliferation 47.2 (2014): 133-145.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is alimony taxable? This is a question many divorcing women ask. And the answer is that it depends on when the divorce was finalized.\nFor divorces finalized before December 31, 2018, the short answer is yes. For these divorced couples, the spouse who paid alimony could take a tax deduction for the payments. And the spouse who received the alimony was required to count that money as income, paying taxes on it. Alimony payments could also be made non-taxable and non-deductible if both spouses agreed to this arrangement in their settlement agreement.\nHistorically, women have generally been the recipients of alimony awards. Alimony is meant to is to provide income for the lower earning spouse, generally to allow time to develop job skills or to continue the lifestyle that was established during the marriage. According to the Census Bureau, about 243,000 people received alimony in 2017, with 98 percent of those being women.(1) While the majority of alimony recipients are still women, in recent years there have been more men receiving alimony.\nThe Tax Cuts and Jobs Act which took effect on January 1, 2018 is expected to reach virtually every corner of American life\u2014including divorce. In fact, one of the provisions of the new tax plan scraps the tax deduction for the person paying alimony, although the new alimony rules will not affect divorces or separation agreements before 2019.\nThose who wrote the new tax rules believe the plan provides a higher level of \"fairness\" to married couples. In fact, the alimony deduction was called a \"divorce subsidy\" (meaning payments between a divorced spouses would have more tax advantages than payments between married spouses) by the House Ways and Means Committee\u2014the Committee who actually wrote the new tax bill. The Committee's bigger concern is likely the $6.9 billion in new tax revenue the repeal of the alimony tax deduction will bring in over the next decade.\nWhat are the Alimony Tax Changes?\nIn short, any divorce which occurs after the last day of December in 2018 will prohibit the spouse who pays alimony from taking a deduction for the alimony. In addition, the receiving spouse is no longer required to declare alimony as taxable income.\nThis is a complete reversal of the former tax laws for alimony in which alimony payments were deductible. Since the spouse who pays alimony is generally in a higher tax bracket, being able to deduct alimony payments often lowered taxes due at the end of the year. Under the old tax laws, the receiving spouse was required to pay taxes on the alimony received.\nThe American Academy of Matrimonial Lawyers (2) strongly opposed the new alimony tax laws, although their protests appeared to have little effect.\nThose who are experts in divorce believe the new tax rules will make divorce negotiations much tougher, and possibly lead to less spousal support as the money goes toward tax payments. Prior to the new tax law, deducting alimony payments could greatly lower the paying spouse's tax bill. After the tax law takes effect, there is the chance higher-earning spouses won't be willing to pay as much alimony without this deduction.\nFurther, the timing of a divorce may well hinge on taxes. The spouse who expects to receive alimony could now have an incentive to delay the divorce settlement in order to take advantage of the 2019 tax benefits. On the flip side, the spouse who expects to pay alimony might try to push the divorce through quickly, in order to take the tax deduction before 2019.\nEx-spouses may not live in the same house or file a joint return if the alimony payments are to qualify as deductible.\nThe alimony payments must be made as a result of a written divorce agreement.\nOnly payments made directly to ex-spouses will qualify, although payments, such as mortgage payments or payments to an ex-spouse's attorney which are considered \"alimony,\" may qualify if those payments are pursuant to a written divorce agreement, or at the written request of the ex-spouse.\nAll alimony payments must be made in cash, check or money order to qualify as deductible alimony.\nThe Social Security number of the receiving spouse must be included on the paying spouse's tax return for the alimony to be considered taxable.\nIf the receiving spouse dies, there is no obligation for the paying spouse to continue paying alimony unless state law says he or she must. If state law requires that the paying spouse continue paying alimony to the estate of the deceased, those payments are not tax deductible.\nAny way you look at it, the new tax laws will have a significant impact on divorcing couples. Those who are planning a divorce should discuss tax issues related to alimony with their divorce attorney, as well as with a tax professional who has experience in divorce tax issues. Should you wait too long, you could end up paying substantially more in taxes\u2014and will live with any mistake made during the divorce for years to come.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am passionate about what I do. I see my professional development as a life-long process and look forward to the ongoing enrichment of my practice.\nOver the years, I have built a practice devoted to improving the lives of people from a wide variety of backgrounds, struggling with a number of different issues. More recently, I have shifted my focus from clinical practice to the training of pre-licensed and early career therapists. I very much enjoy mentoring as it gives me the opportunity to share all that I have learned in my two decades of providing psychotherapy.\nI knew that I wanted to be a psychologist when I took my first Intro to Psych class in high school. I have always been insatiably curious about human emotion and behavior and psychology offered a window into a deeper understanding of how we function as people. After high school, I majored in psychology at Michigan State University and from there went on to receive my Master's and Doctoral degrees from the Illinois School of Professional Psychology in Chicago (now Argosy University).\nWhile in graduate school my interests led me to minor in Psychoanalytic Psychotherapy which has its roots in Freudian psychology. Psychoanalytic Psychotherapy is based on the idea that we are shaped by our past experiences in ways that we are not always aware of. As I began to practice therapy at my various training sites, I found myself drawn to the Interpersonal Psychology model which allowed for a much more collaborative and interactive therapy experience with my clients. Interpersonal or Relational psychotherapy focuses on the individual in relationship and uses the interaction between therapist and client as a tool for change.\nThe combination of psychodynamic and relational treatment philosophies provided me with a very solid and balanced foundation through my pre-doctoral internship at Butler University Counseling Center in Indianapolis and my post-doctoral training year at Allendale, a residential treatment center for adolescents in Wisconsin. When I completed my degree and joined Clarus Center (a group private practice in Naperville) in 2000, I had the opportunity to expand my knowledge base and pursue additional interests.\nMy experience at Clarus taught me about more alternative therapies and it was during my time there that I was first introduced to Eye Movement Desensitization and Reprocessing or EMDR and Sensorimotor Psychotherapy (SP). I have since been trained in both treatment models; to Level II in EMDR and Level I in SP. Both have transformed my practice in that they allow me to more holistically address my clients' issues. These techniques access all levels of functioning; thoughts, emotions, physical sensation, images, the 5 senses, the unconscious and spirituality. The comprehensive nature of these approaches, along with the experiential aspect of the work, leads to a more efficient therapy process and longer-lasting change.\nCompletion Therapy (CP) is another experience-based model in which I have been trained at a beginner level. CP is an treatment model that strives to complete deep, unresolved emotional experiences in an intensive format where a client may spend a few hours at a time on a particular issue.\nIn order to continue learning and growing as a therapist, I regularly invest in training and conferences that I believe will benefit my clients' work. I have sought additional training or independently studied topics such as attachment, trauma, group therapy, EMDR, Sensorimotor Psychotherapy, Completion Therapy, eating disorders, self-injury, dissociation, Borderline Personality Disorder, couples' therapy, guided imagery and more. I am very passionate about what I do and see my professional development as a life-long process. I relish the opportunity to enrich the practice of other dedicated psychotherapists.\nI provide a warm, non-judgmental environment for each individual to explore the thoughts, feelings, and behaviors that may be interfering with living life to the fullest.\nIt is such a privilege to be invited on someone's journey of self-discovery. I strive to meet each of my clients where they currently are on their path to healing. While I tailor my approach to each individual's needs, my style is engaged and supportive, encouraging my clients to step actively into our work together. I provide a warm, non-judgmental environment for each individual to explore the thoughts, feelings, and behaviors that may be interfering with living life to the fullest.\nI am a Licensed Professional Counselor and a National Certified Counselor from the National Board of Certified Counselors. I work with a variety of issues related to mood, anxiety, relationships and addictions. I specifically enjoy working with women's issues including infertility, postpartum depression and anxiety, and bereavement related to the loss of a child or infant. I currently co-lead a mixed gender interpersonal process group and plan to start a support group for mothers who have experienced pregnancy loss or the loss of an infant child. In addition to my work as a therapist, I currently serve on the Board of the Illinois Group Psychotherapy Society and am obtaining my doctorate in Clinical Psychology from the Illinois School of Professional Psychology.\nI believe that a supportive, caring, and honest relationship with each client is of the utmost importance and I strive to be present, empathic and open-minded throughout the therapy process.\nPlease contact me directly at 872-301-5539 or sarahnetzky@gmail.com.\nI received my master's degree in mental health counseling from Purdue University . I enjoy providing individual, couples, and group counseling. I believe in blending components from different clinical modalities and tailoring them to my client's specific needs. Some of the modalities I have used include interpersonal, cognitive-behavioral, behavioral, and humanistic therapy.\nI am currently working with adults and adolescents with a variety of issues including: depression, anxiety, trauma, extreme stress, relationship difficulties, adjustment disorders, mood disorders and eating and body-image issues. My priority in our therapeutic relationship is that you feel comfortable and accepted in a safe and non-judgmental environment. It is imperative to me that you feel heard and understood. My objective is to work with you as a whole person, rather than just a list of issues. I truly love what I do and feel privileged to be a part of my client's lives.\nI am active in-session with my clients and I enjoy using humor as a tool for healing and connecting in the therapy relationship. I currently co-lead a mixed-gender, general process therapy group with a focus on relational dynamics between group members.\nI see clients in the downtown office on Tuesdays and Wednesdays. Please feel free to reach me at 317-385-7002 or jennijen.harrison@gmail.com.\nI received my master's degree in mental health counseling from Purdue University . I enjoy providing individual, couples, and group counseling.\nEach individual coming into therapy is valuable and capable of healing and growth. I work from a strengths-based approach, looking to help encourage each person's uniqueness and purpose in life. I listen carefully to my clients' inner wisdom, as I believe that each individual is the expert on their world.\nI am passionate about empowering women, helping them find a sense of self and accessing their full potential. I enjoy working with those dealing with trauma, social anxiety, exploring sexuality, and issues related with self-esteem.\nI am trained to integrate faith into psychotherapy for my Christian clients as I believe faith and healing can be deeply connected!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatyyh b/data_all_eng_slimpj/shuffled/split2/finalzzzatyyh
new file mode 100644
index 0000000000000000000000000000000000000000..d50e98816da023fc7199b75532eaa20980c9903e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzatyyh
@@ -0,0 +1,5 @@
+{"text":"It starts out with a 'slip' styled bodice, delicate spaghetti straps, and a self-trimmed and reinforced scoop neckline. It's loosely fitted through the bodice and then begins to arc out dramatically from about the waist to below the hip, at which point it arcs sharply back in toward the figure. A lush pooling of fabric settles at each hip, creating a draping pocket-like effect. It's quite charming.\nMade up in our very best quality rayon jersey, a supportive and stretchy fabric with a lustrous finish, a silky and soft texture that's quite sensuous and a fabulously fluid draping quality that enhances the appeal of the design. Easy to care for, launder in cold water, hang to dry.\nThis new bubble style is all the rage this season, and Annie's version is one of the best. Well designed and executed by our tremendously talented seamstresses from the finest materials. At a 'straight-from-the-producer' reasonable cost. This one's a gem. (If you prefer a little more cover through the bust, check out the jersey shrugs in our store (there are 3 or 4 different styles).\n* MEASUREMENTS : Bust fits : 31\" to : 39\" ,Arm : 15\",Waist fits : 28\" to : 36\", Hip : 69\", Length : 46\"\n* MEASUREMENTS : Bust fits : 33\" to : 41\" ,Arm : 16\",Waist fits : 31\" to : 39\", Hip : 72\", Length : 47\"\n* MEASUREMENTS : Bust fits : 40\" to : 48\" ,Arm : 20\",Waist fits : 38\" to : 46\", Hip : 79\", Length : 49\"\n* MEASUREMENTS : Bust fits : 48\" to : 56\" ,Arm : 24\",Waist fits : 46\" to : 54\", Hip : 87\", Length : 50\"\n* MEASUREMENTS :Bust fits : 56\" to : 64\" ,Arm : 29\",Waist fits : 54\" to : 61\", Hip : 95\", Length : 51\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Robots are becoming an increasingly important part of modern society. They work in factories, fight wars and might one day nurse you in your old age.\n2 What are machines good and bad at, in comparison to humans?\n3 What can this tell us about the way that the human mind works?\n5 What are the limitations of using machines as tools?\nTrying to get machines to perform very specific tasks, e.g.\nYou might even have some ideas for how AI can be used for your A2 project next year.\nMachines are good at doing tasks repeatedly (think about car manufacturing robots), as they don't get tired or make mistakes.\nMachines are seen to be bad at making judgements which they haven't been built to make, sympathy, inventing things. etc. But if we built machines smart enough, couldn't we build these capabilities in?\nThere are many scientists and philosophers who believe that computers will one day become as intelligent as humans. But there is a question about what 'intelligence' really means. If it is just performing tasks well, then there are computers that can compose music, or sweep a road, or fly a plane, or solve maths equations better than most humans. We can even get computers to display emotions such as sympathy and anger. Does this mean that we can fully recreate how the mind works?\nIn 1950 Alan Turing, an early pioneer in computer science, proposed a test for machine intelligence. If you could have a conversation with a panel of human beings and with a computer AI program, and be unable to tell the difference between whether you were speaking to a human or a computer, then the computer could be seen to be as intelligent as a human. This is known as the Turing Test.\nPlayer C, the interrogator, is tasked with trying to determine which player - A or B - is a computer and which is a human. The interrogator is limited to only receiving written responses in order so that they can't judge on appearance.\nIn 1980 the philosopher John Searle posed a thought experiment that some see as proving machines cannot understand what they are doing. The Chinese Room is a box in which a man sits. He does not speak Chinese at all, but is passed Chinese characters under the door. He has a book of Chinese characters and their matching responses. On receiving a character he looks for it in the book and sends the corresponding character in reply. At no point does he understand what he is doing, he just follows the instructions. AI can be considered to be just like this, however complex the code, all it is doing is responding to inputs with set outputs, there is no understanding present.\nA similar argument was made by Stanley Jaki in 1969, where he proposed that AI is a little like a drain pipe, where two water droplets roll down, at the bottom they combine to form a larger droplet, but at no point does the drainpipe understand what has happened. He argues that human beings possess this understanding, whilst machines do not.\nHowever, many philosophers and scientists see intelligence and understanding as nothing more than complex algorithms responding to stimuli. Is there really a 'me' that 'understands' and what exactly is it? Could our mind be reduced to a set of algorithms?\nAs machines are expendable they allow us to experiment and simulate human beings without worrying about their safety. For example machines are used by the army to simulate the damage received by a human being from the detonation of an Improvised Explosive Device. This allows us to design vehicles and clothing better able to protect soldiers.\nIf you create a machine without emotions and without the ability to acquire emotions, for example a car manufacturing robot, then there are some important limitations about how they can be used.\nIf you were to work next to a robot in a factory and you started not feeling very well, the machine would be very unlikely to be programmed to feel any sympathy, and would most probably not change its work routine to accommodate your changing circumstances. However, it could be possible that the robot might be programmed with these features.\nMachines in most cases lack the ability to adapt to new situations, being stuck with the code they have been given, and unable to see safety problems when carrying out their routine. The first robot-caused death was in 1979 when a robotic arm struck Robert Williams, a worker at a metal casting plant in the USA.\nThere might also be problems on the horizon if AI produces machines as 'intelligent' as us. In this situation they would have no limitations, they would be just like us. Isaac Asimov saw this problem and defined three laws of robotics to make sure that robots and humans can live together in peace.\nNOT calculators, doing maths, hosting websites, etc. Why do you need a robot for that?! Surely a calculator or computer program would suffice?\nWhat are robots better at than humans, why?\nDo you believe that robots can ever be as intelligent as humans?\nIt might not have been programmed to deal with a scenario it comes up against.\nThis page was last edited on 27 February 2018, at 12:58.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jul. 10, 2018 4:00 p.m.\n\"If an animal is part of your entire life, and caring for them is a huge part of it, to take that away is pretty dramatic,\" says Kimberly Brubaker.\nIf it's a high enough priority, though, you might be able to find a way to stay together, as Brubaker did: She lives in a dorm with her cat Dino and ball python Mars at Eckerd College in St. Petersburg, Florida.\nEckerd is not the only campus in the country that allows pets, but they may have been doing it the longest \u2014 since the early 1970s. While its pet policies are broadly accepting, it's far from a free-for-all. Brubaker is president of a student organization that registers on-campus pets; oversees their well-being and students' compliance with rules; and adjudicates problems.\n\"We do pet checks once a month \u2014 we go around and knock on all the doors,\" she says. They handle an average of one or two problem reports per month, but most are minor, such as misunderstandings of the registration procedures.\nNot only are pets on Eckerd's campus mostly problem-free, they may actually be beneficial. In a recently published study, students \"across the board reported that their pet reduced their levels of stress, and had incredibly favourable things to say about living with the animal,\" says co-author Miranda Goodman-Wilson, assistant professor of psychology at Eckerd.\nThe study's results were mixed when it came to quantifiable mental health benefits. Pet-owning students did not have overall lower levels of stress, depression and anxiety. However, there was an effect when it came to somatic anxiety \u2014 the physical effects of stress, such as a racing heart and sweating palms. For students with pets, increased levels of stress did not result in increased somatic anxiety.\nWhile pets might be good for students, some might worry whether college life is good for the pet. Last year, Mekenna Hooper, a senior at the Johnson & Wales University Denver Campus, decided to adopt a dog. When she contacted shelter and rescue groups, she recalls, \"none of them liked the fact that we lived in a dorm,\" even though she was sensibly looking for a small, lower-energy senior.\nIf you're looking for a pet-friendly college, be aware of each institution's specific rules. More campuses allow small pets that can be kept in cages and tanks than allow dogs or cats, and where dogs are permitted, sizes and breeds may be restricted. Some restrict pets to upperclassmen; Eckerd only allows pets that students lived with before coming to campus.\nAt Elizabethtown College in Elizabethtown, Pennsylvania, where the first pet-friendly housing will open this fall, the new policy grew out of an increasing number of assistance animals, and because of requests to raise service dogs. Now, it's seen to have a more general value.\n\"We see this as part of creating a vibrant campus community that is attractive and promotes well-being,\" says Associate Dean Allison Bridgeman.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Meditek is proud to an authorized distributor for Cuda Surgical, a leading manufacturer of medical lighting and related products. We offer sales and support for some of the company's most popular products, including the Elite series of xenon lightsources and headlights. Discover for yourself the benefits of adding a Cuda headlight or other neuro surgical lighting solution to your operation. Our representatives are present in all major Canadian healthcare markets and can arrange a demonstration at your convenience. Speak to your local Meditek rep, or contact our head office to get in touch with a team member in your area.\nCuda Surgical has a more than 35-year history of developing innovative lighting and visualization products for the medical industry. The company, which is based in Jacksonville, Florida, is unique in its vertical manufacturing capabilities. Unlike other manufacturers, all Cuda neuro surgical equipment is designed, manufactured and tested in-house. This allows the company to ensure an exceptionally high level of quality control, leading to products that are more reliable, more versatile and more effective in today's modern operating environments. It also means that, when necessary, replacement parts are easier to source, leading to less downtime for your important equipment.\nWith our shared commitment to quality products, Meditek is the ideal partner for Cuda Surgical in Canada.\nCuda Surgical headlights and lightsources deliver daylight-quality lighting in a compact package. Suitable for the most demanding surgical procedures, the Elite series of products features a 650-hour prorated lamp warranty, variable intensity control, ergonomically designed Multi-port turrets, a removable module for easy bulb changes and a number of other exclusive features. Elite lightsources are available in 300 and 180 watt configurations. Pair them with the company's standard and neuro headlights, video cameras and other accessories for exceptional lighting that preserves tissue color.\nCuda Surgical equipment can be found in operating rooms, day surgery centres, examination rooms and any other environment where a reliable, portable source of bright white lighting is required.\nAs an authorized Cuda Surgical distributor in Canada, Meditek will be there to provide support for as long as you own your purchase. We can deliver repairs and remanufacturing of older equipment, as well as any replacement parts you need. We also offer a wide range of consulting services that help you make smart purchasing decisions, putting your resources to the best use possible.\nCuda Surgical equipment is an important component of any operating room. As Canada's expert in surgical lighting technology, Meditek is available to assist any facility considering upgrading their capabilities. Give us a call to discuss your specific needs with one of our sales team and find out how we can help.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Watch this hour-long video tutorial where I show you how I edit Brody Jenner's show with Final Cut Pro X 10.1.2.\nIn this must-see video Tony Northrup shows us the relationship between light, shutter speed, ISO, and aperture.\nWatch this video tutorial to discover what is new in Final Cut Pro X 10.1.2 latest update.\nSome people just finished watching a live FCPX Q&A on YouTube presented by PixelCorps and you might be wondering how to setup a live stream of your own. It's actually pretty easy to do. It can be more advanced if you want also.\nEver wonder what the field of view is for the focal length of your lens? Look here!\nIn this video I show you how to setup the GoodLuckBuy Gimbal on a DJI Phantom version 1 Quadcopter with pitch control on the transmitter. The cost of the gimbal was $68.88 + $10 shipping.\nHey what's up everybody. My name is Jasper Thayer. I'm an Apple Certified Trainer for Motion and developer of https:\/\/FinalCutProX.net In this lesson I'm gonna be working with the Replicator to create alternating columns in 3D space.\nLearn how to create an animated line in Motion 5.1 using the Write On Behavior. This Motion 5.1 tutorial also has an example project download link and a link to download the tutorial.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaudhj b/data_all_eng_slimpj/shuffled/split2/finalzzzaudhj
new file mode 100644
index 0000000000000000000000000000000000000000..76f3ec4a778aa0297a42633d05c9f77d784183d3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaudhj
@@ -0,0 +1,5 @@
+{"text":"What to do about abusive dishonest comments?\nMasters in computer science (A.I) or masters in theoretical physics for career in quantum computation and a.i?\nHow do I start learning particle physics?\nIs there a theoretical physics masters that accepts mathematics graduates?\nWhat should a physics undergrad aspiring to be a string theorist learn before grad school?\nClosed as per community consensus as the post is not about physics.\nAre physics.ed-ph papers really on-topic? I don't think it shuld be. And how would the accuracy be voted on?\n@Dimension10 whereas some educational papers concerning graduate level physic might be on topic, the popular level rather sociological text based paper in this submission has absolutely nothing to do with graduate-level+ physics.\nAlso, the text is so short that the little information it contains can not be accurate, and the claims are not original either.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Effectus Skin 1.1 Is Available Now!\nOur current \"update train\" rolls on with Effectus version 1.1, which is available now via automatic updates.\nThis new release includes updated Design functionality and a few bug fixes, and we invite you to check out the changelog for more details.\nI'm happy to introduce two new Boxes for Thesis Professionals today: Google Tag Manager and Google Analytics Experiments.\nBoth Google Tag Manager and Analytics Experiments require <script> placements that were difficult to access in versions of Thesis prior to 2.3, making an otherwise simple integration somewhat complicated.\nNot anymore. With Thesis 2.3+ and these new Boxes, integration has never been easier!\nWith each passing day, the future of your website depends more and more on two things\u2014speed and integrations.\nAs you know, Thesis has always had you covered on the speed side of things. Version 2.3 brings you even better performance on both the front end and the admin side, so your site will be faster and more efficient than ever before.\nBut integrations? What does that even mean?\nIt's simple. One piece of software\u2014no matter how good\u2014is not going to serve all your needs. To achieve your website goals, you're likely going to need to integrate at least two pieces of software (and probably more) to get everything working the way you want.\nFor example, the DIYthemes website relies on no fewer than 4 different software integrations\u2026and take it from me, it can be a real pain to keep up with this stuff over a long period of time!\nThis is a big reason why we've decided to refocus our development efforts on enhancing software integrations with Thesis.\nFirst up? A best-in-class WooCommerce integration in Thesis 2.3, and there's a bonus\u2014this integration will automatically work with any Thesis Skin!\nIn other words, if your site didn't \"just work\" with WooCommerce before, it will now! Read on for the juicy details.\nNow featuring big video support on the Front Page template, the Flex Skin is ready to help you deliver a rich multimedia experience for your visitors. This new video support is a nice complement to the big image + action button that has made Flex so popular since we first introduced it.\nOther significant enhancements include new design controls for the Call to Action area along with a redesigned Latest Posts area on the Front Page template (with more control over the output, too!).\nSee all the changes here, and read on for specific details about how to get all the new functionality associated with this update.\nMany savvy webmasters are now choosing to run Google Analytics Experiments to help them optimize their sites for specific goals.\nHowever, because of a unique implementation requirement (the code is supposed to appear as close as possible to the opening <head> tag), adding this to your Thesis site can be both challenging and annoying.\nFortunately, we've developed the Google Analytics Experiments Box to make this process easier.\nIf you'd like to add Google Analytics Experiments to your site, follow our detailed documentation, and you'll be up and running in just a couple of minutes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Each year presents new challenges and new opportunities to unleash your potential to sell better than ever before. Now is the time to boost your motivation and sharpen your focus in excelling in your selling.\nThis seminar is designed for salespeople who have determined that they want success and even more success. It is for those who are tired of being average, of being in the middle of the pack as well as those who would want to sustain their upward sales momentum. If you want to be more productive in winning more sales, then this is the seminar for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The resource is currently listed in dxzone.com in a single category. The main category is Power meter manufacturers that is about Power meter manufacturers. This link is listed in our web site directory since Monday Dec 1 2008, and till today \"Agilent\" has been followed for a total of 317 times. So far no one has rated yet, so be the first to rate this link !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discordia Doren's best friend, Carmella Brigante, thinks she should loosen up and have a little fun. So, she encourages the prudish Catholic schoolgirl to tag along with her and a group of hedonistic teenagers to a cabin in the woods for some drinking, drugs and debauchery. But they are not alone. Something evil lurks in the woods surrounding the cabin. Something that recognizes Discordia for what she truly is.\nThe directorial debut of Will Gartside, with editing and cinematography (and some of the original story) by Rob Matsushita - this darkly comic musical romp is filled with tweaks of the horror genre, plenty of blood, and good old-fashioned song and dance. Available now on DVD (contact us for details).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaueei b/data_all_eng_slimpj/shuffled/split2/finalzzzaueei
new file mode 100644
index 0000000000000000000000000000000000000000..3f5f1ee62fc6233201cd263de5f14f0fa6676957
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaueei
@@ -0,0 +1,5 @@
+{"text":"Deepak Ahuja, Tesla CFO Part Ways and Why It Matters?\nDeepak Ahuja is leaving Tesla as its CFO after a stint of two years. Ahuja has previously worked with Tesla as its CFO from 2008-2015. Elon Musk in his latest quarterly statement of Tesla revealed the details. Although Ahuja will longer serve as a CFO, but would still hold a key advisory position in Tesla. Zach Kirkhorn the VP of finance at Tesla will take his place.\nThe mobile giant is facing a tuff fight at the hands of Chinese mobile makers. The company in its latest statement revealed the lowest profits since its defective battery debacle of 2016. The lowered demand coupled with decreasing average sales price due to stiff competition from Chinese makers are the chief causes for the decline.\nMark Zuckerberg Ready for New Experiences, a Desperate Measure?\nZuckerberg in his latest chat with the analysts has revealed his intentions to start with new vigor and launch new products. The thing he does best. Is it a way to get away from all the controversies sounding his company? Or is it genuinely an attempt to revitalize his team? Either way, he's trying his best to put the scandals at rest and march forward as a winner.\nFrom the latest earnings reports of Microsoft, it's clear that Chromebooks are eating into its profits. Although Microsoft surface-pro business is now $2 billion, there's a 5% dip in Windows licenses. The reports indicate that Microsoft might just launch an affordable product in its Surface category.\nNew York officials have started a probe into Apple iPhone's FaceTime privacy issue wherein the caller could hear the audio of the receiver even before the later picks the call. NY attorney general Letitia James has opened an investigation in this regard. New York State Department's consumer protection division has opened its lines for receiving complaints from people who are affected by this privacy bug.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Providing fresh, nutritious meals so you can eat healthy and spend more time doing what you enjoy. Let Catalyst Nutrition do the shopping, meal planning, and food preparation so you can enjoy healthy food during a busy day or after work. If you are trying to lose weight or just trying to eat healthy, let us help you reach your goals.\nA new and unique dining experience that fills a niche that's long overdue. With a unique venue and chef-centric food, Apothik underscores the importance of supporting local businesses and farmers, all while introducing its cliental to a wide array of sinfully delicious food.\none-on-one meeting. Fast turn around and guaranteed satisfaction.\nOffers a variety of interesting, delicious, finely-crafted artisan pastries that emphasis savory, seasonal, locally-grown produce, dairy and grains.Currently selling at local farmer's markets and taking custom orders.\ntranscend generations and allow you to express your creative drift. We are passionate about the guitar's rich heritage. We are committed to creating instruments of uncompromised beauty, superb tones, slick playability and more.\nA U.S. based solution for producing \"Hydrographic\" or \"Water Transfer\" Film. We are a source for new and existing Hydrographic Film that has historically only been available in Asia. US Hydroprint develops, prints and supplies Hydrographic film and supplies.\nBeautiful and tasty desserts for any occasion. Specializing in themed Cakes, Cup Cakes and Cake Pops.\nlook who's finding success at the crbc!\nNorthstar Composite Solutions specializes in the development of lightweight performance products for use in a variety of industries, such as: Automotive, Transportation, Wind Energy, Marine, Civil Engineering and Aerospace Structures.\nStriving to provide more music education opportunities throughout Coulee Region School Districts, the Blue Star Cadets' winter drum program for youth ages 10 - 16 is open to all who desire the opportunity to learn percussion.\nProvides custom single speed, fixed gear bikes made from quality material at an affordable price.\nGolden Coulee is the leading supplier in cutting, abrasive, auto repair, and rotary tools. We have been providing high quality tools and accessories for industrial supply, jewelry repair, auto repair and hobbyists since 2001.\nat various local and regional events.\nLook for us during your lunch hour in downtown La Crosse or near industrial parks. Your mouth will never thank you more!\nServing authentic Mexican foods from scratch: Tacos, Burritos, Empanadas, Chicken Mole, Nachos, Tostados, Tamales, and many more. Look for Pappi's Food Trailer at local events and markets.\nAt Suit Yourself Cheesecakes, our mission is to bake a large variety of delicious cheesecakes to suit even the most distinctive tastes. If you prefer to \"Suit Yourself\", just let us know your cheesecake preferences, and we'll design a cheesecake tailor made for you.\nThe Maluna Unhinged Coolers are the ultimate in innovative design and extreme performance. Featuring a patent pending unhinged design, tapered walls, and a premium seal, this cooler was created to be the best on the market.\nIndian Meal Kit is a solution for anybody who loves Indian food and would like to explore vegetarian cooking at home. The Meal Kits come with everything you need to complete your meal. Now available for purchase at The Food Coop (La Crosse).\nVietnamese and Chinese. Personal catered events from intimate gatherings to small parties and even group meetings.\nCommitted to providing access to cleaner energy for the transportation of industry.\nand is abundantly available domestically.\nWe understand that special events deserve to be celebrated in style. The DAMN TASTY is ready to make your next event amazing! Choose from our popular menu items or create a custom menu to fit your needs. Our team provides the highest level of Service, Value, and Quality!\nseniors to socialize with others while still receiving needed care services.\nWith over 20 years of experience in teaching and managing karate and martial arts, Third Degree Black Belt Randy Thomson offers classes for ages 4 to the age of 64. Students develop self-discipline, self-respect and respect for others.\nProvides a range of residential and commercial services including cleaning, janitorial services, commercial carpet cleaning, hard surface floor care, and residential carpet and upholstery cleaning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The overall aim of this DemoWind WFCT project is to implement AWC on an existing full scale offshore wind farm and demonstrate its added value in terms of cost reduction for offshore wind. The main objectives are: \u2022 Demonstration of the economic benefits of AWC in the field \u2022 Demonstration of the accuracy of the numerical predictions under operation with AWC, reducing the model uncertainty and, therefore, de-risking the technology \u2022 Solving the technological challenges related to the implementation of AWC \u2022 Addressing contractual and commercial challenges for the different stakeholders related to operational risks, responsibilities, guarantees, financing, etc.\nThe project will raise the awareness and increase the AWC technology acceptance in the industry because the benefits from AWC will be demonstrated in a full-scale offshore wind farm. Because of this, the market readiness (TRL level) will increase, and the benefits of AWC in real-life will become evident. The maturity of the technology will increase, speeding up the implementation of the technology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The BVP5500 begins with two selected pairs of 12AX7 preamp valves with their own Drive -control. Suhweet! When you want to rock the house you need serious power. The BVP5500 brings 550 of the cleanest fan-cooled watts you'll ever hear or feel.\nAnd since bass players do not live by Watts alone, the BVP5500 is equipped with a stunning arsenal of tone-sculpting tools, including a 3-band EQ with sweepable mids, Ultra-High and Ultra-Low switches, and a 9-band graphic EQ for the ultimate in tone control. For you non-purists, there is an Effects Loop and an Insert Channel with Preamp Out and Power Amp In connections for expanded versatility. Use the included footswitch to kill the signal to the amp, allowing you to tune in blissful silence and to activate\/deactivate the graphic EQ.\nBe gentle. This kind of power can go to your head!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1.1 We are committed to safeguarding the privacy of www.NailGunDepot.com customers and users.\n1.2 Nail Gun Depot does not sell personal data of any kind, under any circumstance.\n1.3 This policy applies where we are acting as a data controller with respect to the personal data of Nail Gun Depot visitors and service users; in other words, where we determine the purposes and means of the processing of that personal data.\n1.5 Our website incorporates subscriber opt-in capabilities, which affect how we will process your personal data. In using the controls, you can specify whether you would like to receive direct marketing communications.\n1.6 In this policy, \"we\", \"us\" and \"our\" refer to Nail Gun Depot. For more information about us, see Section 13.\n3.2 We may process data about your use of our website and services (\"usage data\"). Usage data may include your IP address, geographical location, browser type and version, operating system, referral source, length of visit, page views and website navigation paths, as well as information about the timing, frequency and pattern of your service use. The source of this usage data is Google Analytics. This usage data may be processed for the purposes of analyzing the use of our website and services. The legal basis for this processing is our legitimate interests, namely monitoring and improving our website and services.\n3.3 We may process your account data (\"account data\"), which may include your name and email address. The source of your account data may be you, your employer, or another authorized entity acting on your behalf. This account data may be processed for the purposes of operating our website, providing our services, ensuring the security of our website and services, maintaining back-ups of our databases, and communicating with you. The legal basis for this processing is consent OR the performance of a contract between you and us and\/or taking steps, at your request, to enter into such a contract.\n3.4 We may process the information included in your personal profile on our website (\"profile data\"). Profile data may include your name, address, telephone number, email address, interests and hobbies, and employment details. This profile data may be processed for the purposes of enabling and monitoring your use of our website and services. The legal basis for this processing is our legitimate interests, namely the proper administration of our website and business OR the performance of a contract between you and us and\/or taking steps, at your request, to enter into such a contract.\n3.5 We may process your personal data provided in the course of using our services (\"service data\"). The source of service data may be you, your employer, or another authorized entity acting on your behalf. Service data may be processed for the purposes of operating our website, providing our services, ensuring the security of our website and services, maintaining back-ups of our databases, and communicating with you. The legal basis for this processing is our legitimate interests, namely the proper administration of our website and business.\n3.6 We may process information that you post for publication on our website or through our services (\"publication data\"). Publication data may be processed for the purposes of enabling such publication and administering our website and services. The legal basis for this processing is our legitimate interests, namely the proper administration of our website and business.\n3.7 We may process information contained in any inquiry you submit to us regarding goods and\/or services (\"inquiry data\"). Inquiry data may be processed for the purposes of offering, marketing and selling relevant goods and\/or services to you and others. The legal basis for this processing includes taking steps, at your request, to enter into such a contract.\n3.8 We may process information relating to transactions, including purchases of goods and services, which you enter into with us and\/or through our website (\"transaction data\"). Transaction data may include your contact details, your card details, and the transaction details. This transaction data may be processed for the purpose of supplying the purchased goods and services and keeping proper records of those transactions. The legal basis for this processing is the performance of a contract between you and us and\/or taking steps, at your request, to enter into such a contract AND our legitimate interests, namely our interest in the proper administration of our website and business.\n3.9 We may process information that you provide to us for the purpose of subscribing to our email notifications and\/or newsletters (\"notification data\"). Notification data may be processed for the purpose of sending you relevant notifications and\/or newsletters. The legal basis for this processing is consent OR the performance of a contract between you and us and\/or taking steps, at your request, to enter into such a contract.\n3.10 We may process information contained in or relating to any communication that you send to us (\"correspondence data\"). Correspondence data may include personal communication content, and any metadata associated with the aforementioned communication. Our website will generate the metadata associated with all communication made while using our website contact forms. Correspondence data may be processed for the purposes of communicating with you, and for corporate record-keeping. The legal basis for this processing is of our legitimate interests, namely the proper administration of our website and its business communication with users.\n3.11 We may process any personal data identified in this policy where necessary for the establishment, exercise, or defense of legal claims, whether in court proceedings or in an administrative [out-of-court] procedure. The legal basis for this processing is our legitimate interests, namely the protection and assertion of our legal rights, your legal rights, and the legal rights of others.\n3.13 In addition to the specific purposes for which we may process your personal data set out in this Section 3, we may also process any of your personal data where such processing is necessary for compliance with a legal obligation to which we are subject, or in order to protect your vital interests \u2013 or the vital interests of another natural person.\n4.2 We may disclose your personal data to our insurers and\/or professional advisers insofar as reasonably necessary for the purposes of obtaining or maintaining insurance coverage, managing risks, obtaining professional advice, or the establishment, exercise or defense of legal claims, whether in court proceedings or in an administrative [out-of-court] procedure.\n4.3 We may disclose order information and contact details to our suppliers, insofar as reasonably necessary for order processing and order completion.\n4.4 Financial transactions relating to our website and services may be handled by our payment services providers, Authorize.net, CardConnect, PayPal et al. Transaction data is also subject to review by our fraud prevention service provider, SIGNIFYD. We will share transaction data with our payment services providers only to the extent necessary for the purposes of processing your payments, refunding such payments, and dealing with complaints and queries relating to such payments and refunds. You can find information about our payment service providers' privacy policies and practices, by visiting their respective website(s) or requesting information.\n4.5 In addition to the specific disclosures of personal data set out in this Section 4, we may disclose your personal data where such disclosure is necessary for compliance with a legal obligation to which we are subject, or in order to protect your vital interests or the vital interests of another natural person. We may also disclose your personal data where such disclosure is necessary for the establishment, exercise or defense of legal claims, whether in court proceedings or in an administrative [out-of-court] procedure.\n5.2 We and our affiliate companies have offices and facilities in the United States of America. The European Commission has made an \"adequacy decision\" with respect to the data protection laws of these countries. Transfers to each of these countries will be protected by appropriate safeguards, namely the use of standard data protection clauses adopted or approved by the European Commission.\n5.3 The hosting facilities for our website are situated in the United States of America. The European Commission has made an \"adequacy decision\" with respect to the data protection laws of these countries. Transfers to each of these countries will be protected by appropriate safeguards, namely the use of standard data protection clauses adopted or approved by the European Commission.\n5.4 Web development, marketing and affiliate contractors are situated in the United States of America. The European Commission has made an \"adequacy decision\" with respect to the data protection laws of these countries. Transfers to each of these countries will be protected by appropriate safeguards, namely the use of standard data protection clauses adopted or approved by the European Commission. Nail Gun Depot shall not be held responsible for sub-contractors operating under affiliate contractors.\n(a) Personal data including order information and marketing subscription details will be retained indefinitely, unless removal is directly requested by the data owner or authorized entity acting on their behalf. Analytics and behavioral data is typically retained for a minimum period of two-years following the date of acquisition, unless removal is directly requested by the data owner or authorized entity acting on their behalf. Credit card data is not stored or retained by Nail Gun Depot.\n6.4 In some cases it is not possible for us to specify in advance the periods for which your personal data will be retained. In such cases, we will determine the period of retention based on the nature of said data and its purpose.\n(a) The supply of appropriate evidence of your identity (for this purpose, we will usually accept a photocopy of your passport certified by a solicitor or bank, plus an original copy of a utility bill showing your name and current address).\n8A.2 We may withhold personal information that you request to the extent permitted by law.\n8A.3 You may instruct us at any time not to process your personal information for marketing purposes.\n8A.4 In practice, you will usually either expressly agree in advance to our use of your personal information for marketing purposes, or we will provide you with an opportunity to opt out of the use of your personal information for marketing purposes.\n8B.1 In this Section 8, we have summarized the rights that you have under data protection law. Some of the rights are complex, and not all of the details have been included in our summaries. Accordingly, you should read the relevant laws and guidance from the regulatory authorities for a full explanation of these rights.\n8B.3 You have the right to confirmation as to whether or not we process your personal data and, where we do, access to the personal data, together with certain additional information. That additional information includes details of the purposes of the processing, the categories of personal data concerned and the recipients of the personal data. Providing the rights and freedoms of others are not affected, we will supply to you a copy of your personal data. The first copy will be provided free of charge, but additional copies may be subject to a reasonable fee.\n8B.4 You have the right to have any inaccurate personal data about you rectified and, taking into account the purposes of the processing, to have any incomplete personal data about you completed.\n8B.5 In some circumstances you have the right to the erasure of your personal data without undue delay. Those circumstances include: the personal data is no longer necessary in relation to the purposes for which it was collected or otherwise processed; you withdraw consent to consent-based processing; you object to the processing under certain rules of applicable data protection law; the processing is for direct marketing purposes; OR the personal data has been unlawfully processed. However, there are exclusions of the right to erasure. The general exclusions include where processing is necessary: for exercising the right of freedom of expression and information; for compliance with a legal obligation; OR for the establishment, exercise or defense of legal claims.\n8B.6 In some circumstances you have the right to restrict the processing of your personal data. Those circumstances are: you contest the accuracy of the personal data; processing is unlawful but you oppose erasure; we no longer need the personal data for the purposes of our processing, but you require personal data for the establishment, exercise or defense of legal claims; OR you have objected to processing, pending the verification of that objection. Where processing has been restricted on this basis, we may continue to store your personal data. However, we will only otherwise process it: with your consent; for the establishment, exercise or defense of legal claims; for the protection of the rights of another natural or legal person; or for reasons of important public interest.\n8B.7 You have the right to object to our processing of your personal data on grounds relating to your particular situation, but only to the extent that the legal basis for the processing is that the processing is necessary for: the performance of a task carried out in the public interest or in the exercise of any official authority vested in us; or the purposes of the legitimate interests pursued by us or by a third party. If you make such an objection, we will cease to process the personal information unless we can demonstrate compelling legitimate grounds for the processing which override your interests, rights and freedoms; OR the processing is for the establishment, exercise or defense of legal claims.\n8B.8 You have the right to object to our processing of your personal data for direct marketing purposes (including profiling for direct marketing purposes). If you make such an objection, we will cease to process your personal data for this purpose.\n8B.9 You have the right to object to our processing of your personal data for scientific or historical research purposes, or statistical purposes on grounds relating to your particular situation, unless the processing is necessary for the performance of a task carried out for reasons of public interest.\n8B.11 If you consider that our processing of your personal information infringes data protection laws, you have a legal right to lodge a complaint with a supervisory authority responsible for data protection. You may do so in the EU member state of your habitual residence, your place of work or the place of the alleged infringement.\n8B.12 To the extent that the legal basis for our processing of your personal information is consent, you have the right to withdraw that consent at any time. Withdrawal will not affect the lawfulness of processing before the withdrawal.\n8B.13 You may exercise any of your rights in relation to your personal data by written notice to us.\n11.3 We use Google AdWords and Bing Ads to provide advertising messages to our website visitors. These entities may gather information about website use by means of cookies. The information gathered relating to our website is used to provide relevant messaging to our visitors.\n11.4 We use Listrak to provide email marketing messaging to website visitors and subscribers. Listrak gathers information about website use by means of cookies. The information gathered relating to our website is used to provide relevant messaging to our visitors and subscribers.\n13.1 This website is owned and operated by G. J. Burkhart, Inc.\n13.2 Our principal place of business is at: 1740 Carillon Blvd., Cincinnati, Ohio 45240.\n(d) by email to postmaster@nailgundepot.com.\n14.1 Our data protection officer can be contacted at: postmaster@nailgundepot.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavftj b/data_all_eng_slimpj/shuffled/split2/finalzzzavftj
new file mode 100644
index 0000000000000000000000000000000000000000..3f8deac17d2f47637e7cce134779f76533e54bdc
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavftj
@@ -0,0 +1,5 @@
+{"text":"Thanks to ever-advancing digital technology it's now easier than ever to make videos that are ready to be shared, liked and enjoyed around the world. However, what if you want to make a video but don't have footage ready, or want to go for a different look? Perhaps using still images in a creative way is the answer.\nA photo slideshow is a simple but very effective way to make your still images come alive. One technique that works well is to position the images so a constant theme or object is visible in every image, making the objects surrounding it seem like they are in motion while the target object is still. You can use a person's face in a multitude of locations, or a glass of water on different objects, to try out the effect.\nTimelapse photography is a tried-and-testing method of making still images move. By keeping the camera in one place and taking a shot at regular intervals, you can demonstrate the development of things being made, destroyed or moved in seconds. This is used to best effect with things that usually take a long time to see in full, such as the construction of a building or someone growing in height over the years.\nYou may have seen Stop Motion used in movies without even realising it. The effect is one of the most time consuming and demanding forms of animation, but the results can be spectacular. All you have to do to get started is find an inanimate object that you can interact with easily such as a toy, and take a photo of it after moving it a small amount. String enough of these images together in quick succession and you're on your way to animation gold.\nPixelation is a more advanced version of Stop Motion that typically focuses on people rather than objects. The process is similar to Stop Motion but you can experiment with the amount of time between images, to make it appear the characters are moving in an exaggerated fashion. Try taking a photo of someone jumping and then leave them hanging in the air for an extra second and see what happens.\nThere's no end of creative ways you can bring still images to life. iFactory's team of creative designers and web developers are continually working on new ways to create attention-grabbing websites, videos and digital marketing, so contact us today to see what we can create with you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hola chicos, this is our 4th week of featuring Sienna's #OOTD. I can't believe I have been doing this for a month now!! I started one week before the start of August and now we are already one week before the start of September. It is amazing how the time flies!!\nNote: all the hair pieces as always are from The Bow Boutique.\nI hope that you like this collage of Sienna's outfits. It is so nice to see them all in one image. You can appreciate it more.\nThat tot is so adorable. I can see why you would want to take loads of pictures of her and her cute outfits.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nowadays, the world has developed that we are now in the era of the digital wallet. Despite this fact, a lot of men still prefer carrying cards, money and much more in their wallet. Some of us see it as an easy way to access funds anywhere you go.\nJust like any other investment we have, our wallet is also important. It is one thing to choose the one with enough space to house your essentials; it is another thing to purchase a wallet that is fashionable and beautiful. If it is fashionable enough, it will make you look like a man in charge.\nFor a wallet to be fashionable and attractive, there are contributing factors. The color is the basic factor. This might bring up a question like what color should I go for? You don't need to stress your brain thinking of the answer, just keep reading on.\nWallets come in different colors. The most common colors among men are black and brown. There are some other colors such as dark blue, silver, gray, yellow, white, gold, red, brown and much more. Choosing the best color for you might seem difficult, and that is what this article will tackle. We are going to show you 3 tips choosing a suitable wallet color.\nIf you are planning to purchase a wallet, you need to consider your wardrobe. What are the common colors in your wardrobe? Some people forget that it is important to dress in a way that everything complements each other. The color of your wallet can play a major role in your outfit.\nOf course, we want to dress the best but combining colors might be a little bit difficult. For men, it is advisable you choose a wallet color that goes along with the color of your shoes or belt.\nMake sure the color of your wallet and outfit doesn't clash if you are wearing a colorful outfit.\nIf you are not sure, go for black. Such a wallet will go with all your clothes.\nMost of the time, it is advisable you go for what you like. What is your best or favorite color? This is a question only you have an answer to. Choices are different. No one can decide for you.\nIf you choose a color, you don't like you might end up being disappointed. It might also lead to extra spending. Remember; go for the color that pleases you and not others.\nSince wallets are used every time, there is a high possibility of it getting dirty. If you are someone that works in a place where things can get easily messy, you should consider a wallet with dull color. Bright colors can get stain easily. Sometimes you can use the nature of your job to choose a suitable wallet color for yourself.\nWith our 3 tips on choosing a suitable wallet color, we believe you should not have a problem selecting the best color of wallet that works best for you. Make the right choice, and you will not be disappointed. However, if you wish to contribute to the men's world, use our comment box and tell us your view.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"No Acer smartphones in U.S. until 2010?\n2009 got off to a scorching start for Acer. As we reported in the middle of February, the PC manufacturer announced 4 new Windows Mobile powered smartphones at the MWC show. And if that weren't ambitious enough, the company came back the very next day and showcased four more new handsets. Now, it is one thing to pull devices out of your..uh...hat, but it is quite another to actually deliver on that promise. According to the guys at Gearlog, they had the opportunity to ask Acer's laptop PR people about the phones. In a scene out of Seinfeld, the Acer people basically said, \"Hey, Americans, until 2010, no smartphones for you!\"\nThe interesting thing here is that Acer has already started shipping product overseas. The head of the company's smartphone division, Aymar de Lencquesaing, not only has plenty of experience in the U.S. market, but he also said in February that he was hoping to have some devices ready for U.S. availability by the holiday season of 2009. The one Acer model that has generated somewhat of a buzz is the Acer F1 which runs on Qualcomm's 1GHz Snapdragon processor and features a 5-megapixel camera, large screen and thick body. Besides a heavy concentration of Windows Mobile handsets, the company has said that it expects to offer some Android powered devices as well. Still want those ingredients, Elaine?\nits cool. the market is already flooded with good smartphones already. and the phones are getting better and better... tears are shed here.\nsnapdragon is all i care about.. 1ghz is awesome on a phone .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's a guide that hopefully will help you find everything you need to know regarding tagging incomings, no matter how large, at once.\nAcademy sharing: Nobling villages with academy 3 from a tribe mate, then producing nobles, then giving the villages back.\nAutofire mouse: Mouse that is used to send fast trains - often argued as cheating.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavkgd b/data_all_eng_slimpj/shuffled/split2/finalzzzavkgd
new file mode 100644
index 0000000000000000000000000000000000000000..48d1eeb2213c9358da9666b96bfe5747fc0aec09
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavkgd
@@ -0,0 +1,5 @@
+{"text":"Canada: In the past year, tens of thousands of people have entered Canada from the US on foot, seeking a safe haven. Every day more people are arriving. This phenomenon is often referred to Trump, but it must be understood in the global context: worldwide, more than 21 million people have been forced to leave their homes because of wars, political oppression, military violence, extractivism, climate change, etc. \u2013 conditions created by imperialist states like Canada and the United States. Although directly involved in the economic and social devastation in countries throughout the Middle East, Latin America, Caribbean, Africa, and Asia, the Canadian state presumes to limit who can enter and who can legally stay.Rejecting the racist and islamophobic mainstream narrative and media coverage around those migrating, Solidarity Across Borders calls for open borders, radical solidarity with and mutual aid among migrants, solidarity cities, an end to deportation, immigration detention and double punishment, and an inclusive and ongoing regularization process (Status For All, now!).\nOriginally published by Solidarity Across Borders.\nBorders play a crucial role in the capitalist system and its \"migrant crisis\". Canada and the US are founded on the theft of Indigenous lands and the ongoing genocide and displacement of Indigenous peoples. These borders originally established by colonial wars, to benefit European colonizers, are also a means to control migration. They prevent people from leaving violence, poverty and exploitation, drive families from country to country, force them to board dangerous boats, and make long treks across deserts and snow. Borders push people into precarity without legal status, criminalizing them. Borders keep the global apartheid system in place.\nRefugees coming from the US to Quebec and Canada have had no choice but to cross irregularly at Roxham Road and other ad hoc points of entry since the Safe Third Country Agreement came into effect in 2004. This agreement between Canada and the USA prevents migrants from making refugee claims at a regular Canadian border post if they come from the USA (and vice versa). This means migrants entering Canada from the US can only make a refugee claim if they first cross \"irregularly.\" This policy has already taken the lives of migrants \u2013 we honour the memory of Mavis Otuteye who died in June 2017 \u2013 and caused others to lose legal status or be deported.\nThose who arrive in Canada are encouraged to put their efforts and hopes into their individual cases: trying to prove that they are \"real refugees\" or \"good immigrants.\" This lengthy bureaucratic fight is isolating and takes energy away from collective struggle. Moreover, in the past few years, on average, 40% of inland refugee claims were refused. There may be an even higher rate of refusal for people crossing irregularly because many have little support in making their claims and because some have been outside their country of citizenship for many years. Soon, many will be refused as refugees and face a choice: stay in Canada without papers or be deported to their country of citizenship. The Canadian refugee system will thus smoothly deal with \"the problem\" outside the media spotlight: judge people individually, and then, quietly, one by one, deport or criminalize many thousands.\nMainstream discourse and media coverage feed divisions among social groups who should be allied against the unjust global distribution of wealth and power. At best presenting the issue as humanitarian rather than political, mainstream discourse hides the role that borders play in maintaining that unequal distribution. The way that Canada has contributed to pushing people out of their homes in the first place is never mentioned.\nThe far-right discourses of groups such as La Meute and Storm Alliance are emboldened by the normalization of Islamophobic and racist attitudes by mainstream media, politicians, and the state. By declaring that they are only against \"illegal immigrants,\" these far right groups attempt to convince others that migrants crossing irregularly are 'criminals'. Playing into long-held racist\/islamophobic European fears of black and brown men, far right groups across Europe, the US, Canada and Quebec spread confusion and fear about an \"invasion\" and \"terrorists\" and encourage the state to expand border enforcement and the policing of migrants.\nIn these ways, instead of the conflict being defined by a division between wealthy\/powerful and poor\/oppressed both inside and outside Quebec and Canada, it is defined by a division between people inside Quebec or Canada (fighting to keep social programmes (but only for \"us\"), upholders of 'our' values etc.) and people outside\/newly arriving (portrayed as competitors for scarce resources, \"queue-jumpers\", security-threats, etc.).\nWE SUPPORT THE FREEDOM OF ALL MIGRANTS TO STAY, TO MOVE, TO RETURN. SPECIFICALLY, WE SUPPORT MIGRANTS IN THE UNITED STATES WHETHER THEY CHOSE TO FIGHT TO STAY IN THE US OR TO CROSS ANY WAY THEY CAN INTO CANADA.\nIN THE FACE OF THE CASE-BY-CASE APPROACH IMPOSED BY THE STATE, WE MUST TAKE COLLECTIVE ACTION AND SUPPORT EACH OTHER.\nAS WE OPPOSE ANTI-MIGRANT PROPAGANDA AND THESE FABRICATED DIVISIONS, WE MUST BUILD CONCRETE SOLIDARITY.\n\u2013 Challenge the government (with public campaigns, forums, in media, etc.) about its Safe Third Country policy.\n\u2013 Service organizations: join your local sanctuary\/solidarity city campaign to offer services to all regardless of status and to never allow CBSA access (in Montreal, sign on to Solidarity Across Borders Solidarity City statement here: http:\/\/bit.ly\/2lFerMi).\n\u2013 Explore areas along the border and help set up the infrastructure to support people to cross safely and freely.\n\u2013 Join the Anti-Border Caravan being organized by Solidarity Across Borders from 29 June to 1 July (details will be posted on the Solidarity Across Borders facebook page and website).\n\u2013 Offer time to groups giving frontline support to people coming across the border and make sure that those groups treat refugees arriving with human dignity and offer them the legal and daily support they need.\n\u2013 support mobilizations creating a Solidarity City including campaigns around access to health care, housing, shelter, education, food (get in touch with Solidarity Across Borders); join efforts demanding that Mayor Plante come through on Coderre's completely unfulfilled promise to make Montreal a sanctuary city by ending all SPVM\/CBSA cooperation.\nIn general, we call for effective, non-hierarchical, and inclusive grassroots organizing, based on mutual aid and radical solidarity, using a diversity of tactics and direct action, to strengthen community resistance to border controls in our cities, fight deportations and detentions, defeat the fascist movement, and actively support anti-colonial struggles and indigenous sovereignty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"She is forever a part of our family, no matter what, and we hope to hug her once again.\nWe said those words in agreement two years before our first son was born and adopted. It was after we first met with a local adoption\/pregnancy counselor, wondering what a local adoption would look like, and all she did was rave and enthuse over how \"healthy,\" \"beautiful,\" and \"good\" open adoption was.\nFast forward to today: Our son is 9 months young, adopted domestically and at birth, and we miss his birth mom. We mainly call her Tummy Mama, because when I asked her, that was the name she liked best.\nOur adoption began as semi-open, contact solely through the agency. Today, we have one another's cell phone numbers and text quite a bit! Our relationship is already more than I would have ever imagined and I pray for it to only grow.\nAfter our heart's feet took their first few steps onto the path that is adoption, we realized pretty quickly how much we desired a semi-open to open adoption. We no longer desired a closed adoption. Over the course of nine months we prepared our hearts and our home for the adoption of our sweet son and during that time of preparation, experienced major heart transformation in what we desired as a relationship with our child's birth family.\nWe were matched with our son one day after he was born, met him and his birth mom, and offered pretty much as open of a relationship as she was willing to have. Emails exchanged and promises of packages to be sent, we hugged goodbye, sharing tears and a deep love for the same son one week later.\nWe exchanged words and pictures via email for months, and then the return emails slowed, eventually coming to a complete stop. Silence filled my inbox where I craved to see her name. I continued to send emails with the hope and prayer that we would hear back eventually. I won't for one second pretend that I understand the grief birth mamas walk through.\nA deep well of sadness began to form and one day it hit me: I miss my son's birth mom. I miss her. I miss her emails, her pictures, her love for the son we both adore, I miss her.\nSo I continued sending emails and updates, hoping and praying to one day hear back. I offer to help figure out a way for her to visit us, if she ever so desires to. I send my phone number, thinking maybe texting would be more accessible for her.\nOne day I received a text from her. We began texting and an even deeper sense of appreciation for this brave woman surfaced in me.\nOur unique relationship is so special to me and I crave for it to continue to grow.\nI miss her because I love her. I love her because she is half of my son's biological make-up, she gave him life, she placed him into our hands, and she chose to trust us with a fragile life that she loves more than her own.\nI'm a mama by adoption and biology, and I miss my son's birth mom more than I ever anticipated.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Absolutely and the downward spiral started with the FCC elimination of the equity doctrine again in 1987. As a child, I bear in mind listening to information in the evenings with my grandparents and it was informative and educational (70's to early eighty's) Immediately, it is unrecognizable and propaganda is fed to us by all main information retailers on all sides.\nThe absence of cleanliness in ladies's public bathrooms defies motive. Though girls are stated to be the neater of the sexes, habits in public bogs say in any other case and will use enchancment. Thank you for such a beautiful collection of Ladies's Web sites. It'll take me awhile to check them all out however I'm so glad I found your Hub on this subject!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I want to have multiple IF statements in a Particular cell, and I want to set priority to the IF statements, so that if one meets other statements should stop working. can u please let me know.\nthis is the first time am posting in this forum,don't where to upload screen shot for reference.\nYou can have nested IF statements, but not multiples.\nLet's say A, B, C, D and E are conditions that you want to give priority to in that order; so if both A and D are true, you want priority given to A.\nIn the formula itself, A might really be something like W22>P13, which returns TRUE or FALSE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have been happy with this service. The staff have been able to get a community a community to fit my mom's needs for care. The menu is nice and I liked the menu myself. The community was nice and we would recommend other families to this community. My mom is settling in to her new enviroment for care.\nI touredCaring Heart Home of Clovis before making a choice. It was a nice place, the home was clean and in good shape. The person that took me around was exceptional and gave me good information. The reason we did not choose it was because my mother earned a larger facility rather than a smaller home. Other than that, this was a nice place.\nThings are going okay. As far as I know they don't offer activities to the residents. The food is fresh and we had some issues with it in the past but I brought up to the director and she was able to get things resolved. I chose Caring Heart Home because they were able to help me out and it is close to my home.\nWhen I came to see this home, everything was fine and it was great to get a quick response from A Place For Mom with referrals. We felt that it would not work out here for our loved one. We were looking for something more hands on with the care.\nI went to pick up my mom for a doctor's appointment and my mom was very dehydrated. She ended up in the hospital and when she was released it was to hospice care. I found out they were not giving her liquids at Caring Heart Home as they somehow had a wrong diagnosis. There was also an issue with giving her the correct amount of medication. She seemed to like the food and the home was clean. The staff were nice I just did not feel they were qualified.\nMy mom is from Fresno, and a lot of family and friends live there but there was no one there who could give her the 24\/7 care that she needed.\nShe's very pleased with the care and the food and she's only been there a few days, but they've allowed us to repaint the room and move in a lounge chair and a TV. Everything I've heard from her has been very positive, and my sister is very pleased as well. My experience (I had to leave before she moved in) was that it's a nice place. My sister Joan dealt with Patricia Hernandez, and Greg Smith deserves 6 stars out of 5. He followed us the whole way through the process. He's a real Godsend.\nI think it's an excellent value. We've tried to look into nursing facilities and $3500 a month is a great value.\nMom says it's great. Anytime she buzzes, she's right there. They have a ratio of 1 staff member to 3 residents. When I talked to her yesterday, whenever a buzzer goes off, they're there.\nMy mom has Parkinson's and uses a walker, and her eyes have their good days and their bad days, but as far as activities - She's pretty limited. My sister takes her to church and to get her hair done, but it's hard for her to get in and out of vehicles.\nCaring Heart Home of Clovis is an assisted living facility located in Clovis, California. We are among the top adult care homes in the country because we provide superior medical and recreational needs to all of our residents. We have a very large bed capacity in our facility, which is wonderful when it comes to allowing seniors to socialize with a variety of other residents who are living in the home with them. We completely understand how difficult it is for you to make the decision for your elderly parent to live in an assisted living home, but we help to make the process a lot easier because we are simply a home away from their home. When you make the decision to have a loved one stay at Caring Heart Home of Clovis, you can feel good knowing that they will receive delicious and healthy meals throughout the day. Your loved one will also be provided with transportation so that they can go shopping or simply go to a local hot spot to enjoy themselves with other seniors in the group. Our visiting hours are convenient for just about anyone who is a busy working adult who also has a loved one in an assisted living home. When you choose us to be your loved one's new home, you can feel confident in knowing that they are going to get the best care possible all the time. Caring Heart Home of Clovis prides ourselves in being the best adult home in the area.\nCaring Heart Home of Clovis - Clovis, CA has been recognized for consistently high ratings from residents and their families in previous years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavlhj b/data_all_eng_slimpj/shuffled/split2/finalzzzavlhj
new file mode 100644
index 0000000000000000000000000000000000000000..4ded4c9cd7cba624dbd019a71c940a471c36c401
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavlhj
@@ -0,0 +1,5 @@
+{"text":"for your prestigious sign requirements.\nAll signs are engraved or etched into 1.5mm thick specialized engravers brass. popular sizes available, Small at A5 size 210mmx148mm, medium size A4 size 297mm x 210mm, large A3 size 420mm x 297mm, EX large A2 size 594mmx 420mm.\nTo obtain a price for a plaque telephone us on 01202 546540. All prices given include as much text as required and includes logos.\nSign plaques are often supplied with solid hardwood bases made from either mahogany or oak and fixed with brass wood screws and polished brass caps.\nBrass plaques are made to order, in various sizes up to 600mm x1200mm.\nBrass plaques can be engraved or etched.\nBrass plaques come in thicknesses of 1.5mm, 2mm and 3mm.\nThe standard colour fill for brass plaques is black.\nreference. We can also closely match Pantone colour references.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the most enjoyable things about experiencing a great work of art is discovering something you had never known before. Usually these discoveries are personal when you learn something about yourself, but sometimes you learn about things that actually existed you had been previously unaware of. And Juan Rulfo's almost seems to know before you've ever picked it up that what is inside will be unknown to you, that the events for which the book is allegorical of will have been largely forgotten or unknown to the reader. In a way, Pedro P\u00e1ramo is a book about Mexico for people who know nothing about Mexico or have forgotten so much about their own country that it's like a foreign land.\nThe first thing I learned about, and most importantly, was the Cristero War, an aftermath to the Mexican Revolution fought in the late 1920's in western Mexico (where the book takes place) and involving revolutionaries who were angry with the government's constitution of limiting the role of the Catholic Church's power in the country and their persecution (mass killings even) of Catholics. Before picking up the novel I had only been vaguely aware of the Mexican Revolution as taught to me in school and largely focusing on Pancho Villa and his cat and mouse game with General Pershing \u2013 a thoroughly American-centric point of view. The Cristero War, then, was absolutely unknown to me.\nYet this lack of knowledge on my part actually seemed to be part of the plan for the novel. I kept thinking that if I had a better understanding of early 20th century Mexican politics that I would in some way be missing out on the greater point Juan Rulfo is trying to make: memory. Everything about Pedro P\u00e1ramo is based on memory \u2013 memory of the past, of being alive, of who we are, even of what the land we live on is. And so while the novel is a allegory for the Mexican Revolution and the modernization of Mexico from old superstition to 20th century secular society, Pedro P\u00e1ramo is also about remembering and what happens when we begin to forget or forget completely.\nOne of the most famous images in the novel is when Pedro P\u00e1ramo crosses his arms at the end of the novel thus turning his back on the land to let it die. His choosing to forget Comala causes the town to literally die and all the ghosts who still live their because they have been forgotten, even by God, either wander about unaware of each other or ever faintly murmuring underground in their graves so that even their grave neighbors can barely hear them anymore. And when Juan comes to the town everything is all a mystery, he doesn't even realize the people there are all long since dead, he has no idea who his father is, or what happened in the years since him mother told him of stories of how beautiful the town had once been. Nearly everything has been lost.\nAnd so what does this forgetting mean? What does it say about us, about our lives, about our wars and revolutions and our sacrifices? Is Juan Rulfo saying that in the end all will be forgotten but not forgiven? The novel does not feel pessimistic, despite its underlying morbidity, yet there is a feeling of futility in everything, a futility to tame the land, to even tame ourselves and our vices \u2013 alcohol is a reoccurring image in the novel and plays a major role in the finale. In fact alcohol is even confused with milk when he mentions Pulque.\nYet this forgetting also explores something deeper about humanity: our imagination.\nThere's a wonderful image where Susana San Juan sees \"a blurred image\" where a \"diffuse light burns in the place of its heart\" and then just a few sentences later we learn that \"blurred image\" is actually Father Renteria and he's holding a candle in front of him with cupped hands. This mixing and blending of images, of the real and fantastic, of time distorting continually through the story, of sentences that read \"The rusty gears of the earth are almost audible: the vibration of this ancient earth overturns darkness.\" are followed a little later with \"A humming like wings sounded above her. And the creaking of the pulley in the well. The sound of people waking up\" that gives us a magical image of the sun slowly rising in the sky, a sun controlled by both the laws of nature and science but also by unknown supernatural forces.\nAnd that is what human understanding is all about, too. We spend our time in darkness until we learn just enough to light the room but then everything we knew and feared and loved in the dark goes away forever. We forget the dark, the pain and misery \u2013 the pain and misery of a war, for example, is lost to us because we now live on the other side of the dark \/ light boundary. The dead can barely speak to us anymore and we can hardly hear them.\nThat leads me to something I else I learned about, the poem by Edgar Lee Masters titled 'Spoon River Anthology'. Like Pedro P\u00e1ramo the dead in Spoon River Anthology also speak and through them we learn the secrets of their lives and of the town. It's a wonderful story but here the truth is laid bare because the dead no longer have any reason to hide and so unlike the novel which is filled with magic and mystery, these dead will not let us forget their lives and events that happened to them. It's an interesting juxtaposition because where Spoon River Anthology is matter-of-fact and does not indulge in mysticism, just over the border in Mexico and entirely different way of looking at life and death is present. Over just one border lies such an incredibly different world.\nI'm not one to mark up my books very much; I might underline a passage here and there, but in only two books have I ever so defaced the book with pencil: Ulysses and now Pedro P\u00e1ramo. And unlike Ulysses, I didn't even have a book for Pedro P\u00e1ramo, I found a .pdf of the novel and secretly printed it out at work on regular office paper, three-holed punched it, and read it from a three-ring binder I keep some old drawings in. This was I was able to underline passages, take notes, look up Spanish, draw lines to connect text, and scribble down by impressions and thoughts. For me this was the perfect way to experience the novel \u2013 to become engrossed in it and fully explore it \u2013 like being lowered down into the Andromeda mine with a rope and lantern and slowly pull up each skeleton at a time into the light to observe them all one by one.\nAs a work of art, Pedro P\u00e1ramo is a masterpiece. This is an achingly beautiful novel and I can see how this inspired Gabriel Garc\u00eda M\u00e1rquez. The world is a richer place because of Juan Rulfo; let's never forget him.\nSusana sees Father Renteria as if in a vision, \"A diffuse light burns in the place of its heart, a tiny heart pulsing like a flickering flame\", then when she can see more clearly we learn he is holding a candle with cupped hands in front of him.\nThis blending of real and unreal is in everything: San Pascual, the King of the Graveyard, the Pulque as milk-like but alcoholic.\nI've never heard of the 1926-1929 Cristero War before; much of this book deals with that war in one way or another.\nBasically the war was fought because the constitution of Mexico tried to limit the control the Catholic church had in Mexico. The church previously had not supported the Mexican revolution because it threatened property rights.\nBasically this is the story of a town and some of the events that happened before the town fell. However, time is meaningless here; everything is memory and sadness and violence and sin and shame.\nI love the notion of 'falsifying boundaries': here the living and the dead intermix (in fact couplets are common in this story), there are holes in the roof to heaven \u2013 no boundaries: all flat plains.\nWhile this is a short book, there is so much going on in every paragraph that I thought it would be fun to do a close reading of it and look up every word and name and place in Spanish.\nThe writing is beautiful; beyond beautiful. This is about as magical as it gets, really.\nTime flows in every direction, memories float in and out, the dead speak but don't know they are dead. One prayer against thousands.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's an example of \"User Wish\" mind map including \"Why\", \"Who\" and \"When\" branches.\nGathering requirements or \"User Stories\" is always a challenging activity in Agile or in any other approaches. The primary factors that make this activity effective are communication and facilitation skills of the interviewer. In this session, I propose using mind mapping that focuses on those primary factors to explore \"User Wish\" \u2014 a vague shape of user requirements before it is written into a form of User Stories.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We were hired on by the talented Regina creative agency, Arcas Group, to develop a modern website that showcases Walker Project's wide variety of projects. As professionals in the achitectural & engineering industry, Walker Projects wanted to showcase their work using their tremendous photography, and really highlight their premium brand. We developed a custom portfolio section, which allows for advanced filtering, and showcases each portfolio project in a clean responsive slider. The folks at Walker Projects also required a careers section, allowing users to submit their resumes to specific jobs. A truly unique homepage was envisioned as well, to highlight the photographic and architectural beauty that Walker Projects works with day in and day out. This responsive site is sure to sell their work like never before.\nTheir latest feature request was an advanced location filtration system allowing users to view their projects based on taxonomy term and location. Using the Google Maps API and some advanced WordPress Queries, we had it working in no time!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If the hirer is in any doubt as to the meaning of the following, a representative from Party Time Grimsby should be consulted.\nI DECLARE that I have read and understand these terms and conditions of hire and any relevant operating and safety instructions supplied with the equipment, and I accept the contract (by ticking the box on the booking form) fully of the implications and responsibilities placed upon me by doing so.\nPLEASE REMEMBER IF ANY OF THE EQUIPMENT IS DIRTY OR DAMAGED WHEN COLLECTED, THE HIRER IS LIABLE FOR THE FULL REPAIR AND\/OR CLEANING COSTS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawiom b/data_all_eng_slimpj/shuffled/split2/finalzzzawiom
new file mode 100644
index 0000000000000000000000000000000000000000..54fb94f6e848f3be24bc16a9b4f350a5c30cc72c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzawiom
@@ -0,0 +1,5 @@
+{"text":"Protecting lives and property is the job of every firefighter. Each day a brave firefighter must be prepared to do the heroic. Recognizing the preparation, determination and bravery of Firefighters everywhere is possible when you give a Firefighter Keychain or another Firefighter Gift.\nFirefighter Keychains can be given to the entire Fire House for a very low cost. Gifts for Professionals is proud to offer the best selection of Firefighter Keychains.\nDiscover more personalized keychains from Gifts for Professionals in a variety of themes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"John Bagley's resignation from the Board of Education last week has members of the school board in search of a replacement. Bridgeport's Republican Party is recommending Steve Best who came up short in an effort to win one of the three minority party spots on the board last November.\nIf a vacancy arises for any reason in the membership of the Board of Education, the remaining members shall elect a new member to serve for the balance of the term vacated. The person so elected shall be a resident and elector and a member of the same political party as the member vacating such office.\nBagley won a seat two years ago as a registered Democrat running on the Working Families Party line. There are only four registered WFP voters in the city, according to Democratic Registrar Sandi Ayala. The replacement cannot be a Democrat because under state-mandated minority-party representation, the maximum number of Democrats who can serve on the board is six. Because Bagley won the seat on a line different from his political affiliation, does this now open up the search pool to all candidates outside the Democratic Party?\nThe City Attorney's Office is expected to craft that argument to school board members. The school board's own attorney may have a different interpretation.\nThe Bridgeport Republican Town Committee is proud to recommend Steve Best for the current vacancy on the Board of Education.\nSteve Best was a candidate for the Board of Education in 2013 and came within 280 votes of election. Steve attended all of the Board of Education meetings in 2013 at which time he then became the deputy campaign manager for David Walker for Lieutenant Governor which preempted his ability to attend further meetings due to statewide travel.\nWithin that time, Steve has continually expressed interest in another run for Board of Education. Although not a lifelong resident, Steve has proven time and again his commitment to the City of Bridgeport as well as all of Bridgeport's children. For these reasons and more, Steve has the full confidence of the Bridgeport RTC.\nDuring the 2013 Board of Ed election, The Connecticut Post published four Op-Eds written by Steve about Education.\nAn Hispanic would be not just \"Best\" but fair.\nJoel Gonzalez, why are you playing the race card?\nBecause you have made me realize not to leave home without it. How many more blacks on the School Board do you want? What would you and many others be saying if there weren't a single black member on the BOE? The one BOE member introduced and supported by your district leader and neighbors was a horrible choice. It was a black woman who led the charge for a State takeover of the Bridgeport BOE. I can go on and on with my race card. Your race card has exceeded its limit.\nJoel Gonzalez, I would like all the BOE members to be black or five blacks and four Hispanics because I've seen where all of the members were white.\nRon, might be because we don't have one Puerto Rican on the board to represent over 50% of the student population.\nHector A. Diaz, Hispanics and blacks will never have any power until we come together as a people. Mario Testa won't let that happen because he holds the keys to jobs, positions, commissions and the membership of the Democratic Town Committee.\nRon, I agree wholeheartedly with you. We have had and will continue to have the opportunity to make changes. Will we is the question? All my life I've worked for the issues that affect both our communities with equal intensity. I would be appalled if there were no black representation on the Board, you should be just as appalled now.\nHector A Diaz, blacks and Hispanics cannot look at each other as the enemy. The BOE selection process shows us we must have people in all political parties and we MUST have candidates who are truly concerned about Bridgeport, of course we will have a spy or two but if we come together on an open agenda for our people and Bridgeport then we will see a power shift. This is NOT against whites, instead it's pro-black and Hispanic, we have always been supporting and voting for white candidates and I'm sure that they will support us because of our concern about the City we all love.\nExcellent, Ron. I believe in the same ideal.\nTwitter feed of CT Post reporter Linda Lambeck.\nIt would be good to know what attorney gave this opinion. It appears the state rule for minority party representation is different than the Bridgeport wording.\nYou will need to check on this. I know at one time there was something like a six-month waiting period to switch from one party to another. No wait to switch from unaffiliated to a party.\nThe first question I have is why is the Bag Man resigning NOW?\nDid someone suggest he and the WFP would be better off if he resigned prior to Andre Baker? His one vote may have greater significance more sooner than later.\nIf that is the case, was his resignation effective immediately or was a date specified. I ask simply to determine if John is currently a voting member of the board. Does he have a vote on his replacement policy if we end up with legal opinions that are in conflict? Could be important or at least helpful.\nAnd my final question is how far does Mark Anastasi go out on a limb with a concocted and contorted legal opinion trying to please the mayor and give him as much say in a replacement as possible?\nSteve Best in no way qualifies as a replacement for John Bagley's vacated BBOE seat. John is a registered Democrat who ran on the WFP line. Steve Best is neither a Democrat nor a WFP member, therefore he in no way meets the requirements of the City Charter. Steve Best started attending BBOE meetings right before he decided to run for the BBOE. I have not seen him at a single BBOE meeting since January 2014. So much for his commitment to the BPS and its children. He was a supporter of Paul Vallas, supports charter schools and privatization. What else is left to be said?\nYou raise an interesting statement, Maria\u2013the only real qualification in Bridgeport to hold this office is getting on the ballot and being elected. Best was endorsed as qualified by the RTC. Does the BOE have a different set of qualifications for a candidate as a replacement than the requirements of this city to run for this office?\nWe already have two male BBOE members from Black Rock and they are both buffoons. Dave Hennessey and Joe Larcheveque are absolutely clueless when it comes to state statues that govern boards of education, the Minimum Budget Requirement, Freedom of Information Act and BBOE policies. The last thing we need is another guy from Black Rock.\nAH!!! The queen of the educational programs in Bridgeport has spoken out about two members of the BOE from Black Rock, Hennessey and Larcheveque. They are on her ever-expanding lists of people who piss the queen off. Seems to me not that long ago you were at the polls asking people to vote for these two amongst others. What happened, did they forget to genuflect or forget to vote the way you wanted?\nBTW the night referral made by someone else might be in r4eference to BATS, they hang upside down all day and come out at night and are a real PIA.\nAndy, another false statement made by you. I have never supported Joe Larcheveque or asked a single person to vote for him.\nAlthough several people warned me about supporting Dave Hennessey, I chose to support him anyway. You know what they say, fool me once shame on you. Fool me twice shame on me. Dave Hennessey will never have an opportunity to fool me again.\nI am not surprised you posses some level of knowledge on bats. In fact, it doesn't surprise me at all.\nNo sir, it is you who has lied. I have never supported Joe Larcheveque. Not for one second.\nMaria, I live in Black Rock and I'm a guy. I resent and represent your comment.\nSorry Hector, I did not mean to offend you. I was going to insert an additional adjective, but I decided against it. It would have definitely exempted you from my the initial comment because this adjective would clearly not have applied to you.\nThank you, Maria. I was hoping for your support. Not offended, I feel the gist of your comment.\nIn addition, although any Democrat or Republican can petition onto a ballot and does not require the endorsement of the party, that is expressly forbidden on the WFP line. The only way you can run on the WFP line is if the State Central Committee endorses your candidacy.\nMaria, can you please clarify? I understand in a general election WFP must endorse a candidate, are you saying in an internal board election WFP must endorse the candidate? One of the registered WFP members cannot come forward for consideration without WFP endorsement? Thanks.\nSteve Best has resided in Bridgeport for just a few years, therefore he has never attended a BPS. In addition, he has no children in the BPS. He did not attend all the BBOE meetings in 2013, that statement is patently false. He is just another outsider and carpetbagger who thinks he knows what is best for our poor minority children and he has not, cannot and will not ever be able to relate to the vast majority of children who attend the BPS.\nWhen is someone new to the area \"Bridgeport\" enough to run for office? What if they left and came back?\nIf a person is a taxpaying citizen of Bridgeport, he or she should be able to have a say in the city they live in, no? Part of having that say is running for local political positions. Maybe they have experience in other areas similar to Bridgeport.\nSo basically, to you if someone were not born and raised in Bpt, then they need not consider a run for the BOE or any other position in BPT?\nMy question was in general, not specific to Best. I do not know his background. But I have seen you post about others \"new to BPT\" and calling them outsiders. So my question was and is, when is someone Bridgeport enough?\nAnd when I said \"similar to Bridgeport,\" I was not referring to Best but others who would run. How about a parent who moved here from another big city\u2013Hartford, NYC, Philly\u2013is that similar enough? And not everyone living in the suburbs grew up in the suburbs.\nLifelong Bpt, personally I do not think anyone without real roots in the Bridgeport community should serve in a citywide elected office.\nI absolutely am against someone moving here from NY, NJ, PA, etc. serving in an elected capacity with 5-10 years of residence in Bridgeport. We have thousands and thousands of homegrown quality, intelligent and capable residents who are more than qualified for every elected office in Bridgeport.\nDavid Daniels was born and raised in Bridgeport and attended the BPS, however he hasn't lived here in years, however I have no problem with his desire to move back to Bridgeport and run for Mayor. In my opinion, he has roots in this community and has contributed to the well-being of thousands of Bridgeport children and residents.\nCan you shed light on what someone like Steve Best has done to better the lives of Bridgeport children or residents?\nI could not comment on your remark below, but once again, I do not know Best or his history and I'm not about to look it up. My question was a general one regarding candidates for office not born and raised in Bridgeport.\nWe disagree. If a talented person moved to BPT ten years ago, they should have every right to run for a political office, especially since those thousands and thousands of qualified people are not running for office.\nAlso, if a parent moved here 5-10 years ago, and they have a child in the BPS, they have right to speak up as well. Hell, if you have a child in the BPS and lived only one year here, you should have a voice in your child's education.\nI am not saying they shouldn't run because clearly any registered voter in Bridgeport could run for municipal office, however I would not support them unless they had really worked to help others in our community. If I were unable to support a particular candidate and one of my reasons was they have lived here for a short period of time and have not volunteered in our community, I would definitely share that information with the electorate in my neighborhood and my friends and family. It is simply my personal opinion.\nHow many comments can Maria make in a row? I thought you had standards.\nRob Traber, I think it might be time for you and Kim to step up your game and run for public office.\nInstead of being concerned with the standards of others, maybe it is best you examine yours.\nYou repeatedly stated Mayor Finch did not believe in democracy or the right to vote, however your BEA membership voted unanimously NOT to endorse Malloy or Foley. You then attended the CEA meeting and cast a vote to endorse Malloy. So much for believing in democracy and respecting your members' will.\nAs you already know, you angered many of your members and I know that to be a fact.\nI think Lennie is doing a public service keeping Maria occupied and off the streets in the middle of night when she posts these very telling comments.\nobserver, please elaborate on why you would think I would be in the streets in the middle of the night in the first place? Is this the most compelling argument you could post regarding my position on a proposed Republican candidate to fill the BBOE vacancy?\nIf it is, maybe you should consider furthering your education.\nRather than posting more personal insults on OIB, Maria or whoever else is the WFP in Bridgeport should recruit an unaffiliated voter with similar beliefs to immediately register in the WFP.\nThat person should write the BOE to express their interest in replacing Bags. So at least there is a legitimate member of the WFP who is a candidate.\nThis would at least eliminate Mark or the city saying the WFP is a moot point.\nAre you actually trying to criticize me for the \"personal insults\" I have posted on this blog? You are absolutely one of the last people on this blog to try and chastise anyone else for the \"personal insults\" they may have posted on OIB. As frightening as this proposition may be, I think you need to stand in front of a mirror and take a long look at yourself. For you to even think it appropriate to throw stones at anyone else for their comments on this blog truly demonstrates how out of touch you are with yourself and reality.\nI heard Tom Mulligan has just registered as a WFP member. Can anyone tell me if it's true?\nTom Mulligan wants to register with the WWWF, aka World Wide Wrestling Federation Party. He believes his prior time on the BOE automatically qualifies him.\nExcuse my ignorance. John Bagley is a registered Democrat as stated above. Regardless if he were endorsed by the WFP, would his replacement not need to be a registered Democrat as well?\nMinority party representation rules precludes another registered Democrat.\nNo, because that would be in violation of the state minority party representation requirement. The selected person CANNOT be a Democrat.\nWFP, Republican or Unaffiliated candidates need only apply to adhere to state law.\nBeing elected as a WFP candidate allowed Bagley to meet this requirement. The city charter dictates the replacement be of the same party.\nMy guess is Mark A will come up with an opinion (not based in case law but merely his opinion) that will state a Democrat who claims to abide by the principals of the WFP would meet all legal requirements.\nBob, you're right! So the Finch Cartel will dust off Mulligan to register as WFP, then switch back as a Finchette once he becomes the Chair of the BOE. Who needs Mark Asshat with a plan like this?\nThe last person we need on the BOE is Mr. Mulligan.\nIs Andre Baker running unopposed? Isn't he resigning from the BOE after the election? So there are two replacements needed, no?\nCorrect. No opposition. Baker will officially join the State House in January. But in that case he will be replaced by a Democrat on the school board.\nAccording to the Secretary of the State's website, it takes three months for a person already registered with a party to switch their affiliation, the waiting period does not apply when switching from unaffiliated.\nCould an unaffiliated register as a member of the Working Families Party to fill the vacant seat?\nI believe you could switch to unaffiliated, just as easy. An unaffiliated candidate would also satisfy the no majority party rule.\nThe BOE attorney said WFP only. Waiting to hear the city attorney's opinion.\nAn unaffiliated voter does not meet the requirements of the City Charter.\nYou can have five unaffiliateds switch and only one believe in the WFP, but that would not prevent the board from selecting one of the four bogus members.\nTo join a political party, you must designate a party affiliation either when you register or any time thereafter. You can drop or change your party affiliation by using a mail-in registration card to notify your registrars of voters.\nIf you switch parties, your new party privileges become effective after three months. If you've been unaffiliated for at least three months, and you switch from unaffiliated to any party, your new party privileges become effective immediately. If you've been unaffiliated for less than three months, your switch becomes effective at the end of the third month.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hotels near San Jose, CA offer guests the best of both worlds; access to one of the greatest metros in California, but far enough away from the bustle of the city. However, when staying in charming downtown Los Gatos, plenty of dining, shopping and sightseeing options are walkable. Enjoy the most affordable luxury hotel in the area that's newly renovated to provide travelers with the absolute best in amenities.\nThis is the first stop into Silicon Valley from Santa Cruz, and a number of tech workers and startup leaders call this Los Gatos hotel their home away from home. Take advantage of easy highway access to Hwy 17 and 880, crucial for making those early morning business meetings. Just 17 miles from Levi's Stadium, it's easy to indulge in the best this slice of California has to offer.\nFrom concerts at the Mountain Winery to being just two to five miles from Santa Cruz Wineries, it's not tough to tap into your inner vino connoisseur. Plus, with Netflix Headquarters and Good Samaritan Hospital nearby, it's easy for business travelers to mix in plenty of leisure with work. Small startup companies flourish nearby, and as a key part of Silicon Valley, this Los Gatos hotel is a favorite option for tech-savvy travelers.\nDepend on the friendliest staff in town to make your stay everything you imagine. Get tips on the best local restaurants, advice on can't miss attractions, and enjoy peace and quiet in oversized, comfortable guest rooms. Pillow-top mattresses, great free Wi-Fi and a location that's beyond compare all await.\nCocktail lounge new restaurant and bar scheduled to open July 2019.\nExercise facility, open from 7:00 a.m. to 10:00 p.m. Our fitness center offers a treadmill and stationary bike, as well as a 3-station multi-gym. The room is air conditioned and has a 47-inch TV on the wall.\nElevator, wheelchair accessible, located in the lobby, which is centralized to the property, making access to both our floors easy and convenient.\n24-hour front desk, guests arriving after 10 p.m. will need to contact our Guest Services before entering the lobby (intercom available at the front lobby doors).\nLaundromat, 8 a.m. to 10 p.m., nominal fee, coin-operated machines available for our guests. Laundry detergent and fabric softener are available in vending machines in the room. Bounce\u00ae dryer sheets can be purchased in the sundry shop located in the lobby.\nBusiness center, quiet work area with comfortable seating and lighting. Privacy protection is in place by using industry-leading privacy software, and we offer a full-service print\/scan\/fax machine in the business center.\nAir-conditioning, the lobby is fully air conditioned, as is the fitness room and breakfast room.\nIce\/vending machines, located in the guest laundry room, just off the pool deck. We offer drinks and snacks in the machines, with additional offering in the lobby's sundry shop.\nFree high-speed wireless Internet access available in all guest rooms. All guest rooms are hardwired for Internet access. We have Comcast Hospitality Class wireless and hardwired Internet with multiple strong wireless access points to ensure you have the most powerful Internet experience you can have.\nMountain view, most of our rooms have a great view of the Santa Cruz Mountains. Available upon request.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the new age of paintball! First Strike has taken paintball to a new level with their first strike paintball. What is first strike? First strike is a new paintball design that allows your paintball to travel well over 300 ft compared to a paintball that struggles to make it 200 ft. At 300 ft. the first strike paintball can still break on its target! Tested in an indoor range the first strike round had a 6 inch spread at 100 ft.!! A normal paintball has a 30 inch spread at 100 ft. The first strike round has an aerodynamic shape and fin stabilization technology which helps the round fly further and straighter than a normal paintball. Become one of paintballs first true snipers with the first strike paintball from First Strike! Now with bulk packaging the First Strike rounds are much more cost effective.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Saint Dennis Catholic School in Lockport held its first career fair on Nov. 7.\nJunior high students visited different professionals to learn more about their careers, including people who work in health care.\nAbove, Mia Bilinski and Mia Castro learn how to put stitches in from a Colleen Castillo Lee, a nurse practitioner.\nAbove, Hannah Brannigan and Amanda Diaz learn about being a paramedic from Kevin Murphy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxram b/data_all_eng_slimpj/shuffled/split2/finalzzzaxram
new file mode 100644
index 0000000000000000000000000000000000000000..0e411b76b32ee0d7e5da1d04e2b92e5328b727dc
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaxram
@@ -0,0 +1,5 @@
+{"text":"By using data from the millions of shots taken every day through the Noah Data System, we have teamed up with Shot Mechanics to give you a few tips and tricks that can change your arc entirely and improve your game. Working together, we'll be bringing you content focused on helping you achieve the perfect shot Next up: Ways to Improve Your Shooting Arc.\nIn this recent video, Coach Collin walks you through a few of his tips, including Feet Sweep, Body Space, and Feet Turn.\nFeet Sweep: If a player were to watch highlights of any good shooter, they would see the shooter sweep their feet; if they are jumping on the three-point line, they will land inside the three-point line just a little bit. Sweeping your feet helps optimize their arc and helps the player become a better shooter.\nBody Space: When a player keeps space between their body and the ball they can keep the ball up and the arm underneath rather than a flat push out of the basketball. In younger players, they chuck the ball to the basket because they may not have enough power or strength to get the ball up. Tip: think about tapping your shooting elbow to your shooting hip.\nFeet Turn: Instead of squaring up to the basket, a player should tilt their body a bit. If a player is left-handed, they will tilt to the left and if they are right handed they will tilt to the right. Most shooters that have a low arc are the people that square their toes to the basket, and when they go to shoot, the shoulder muscles start to pull down, and it is harder to release. To stop this from happening try turning the player's feet to your dominant side to help the body get a higher release point.\nPractice will help a player perfect the three ways to improve their shooting arc. Parents and coaches: help your child or player with these three tips and tweet us the results!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I do plan to live blog the foreign policy debate tonight, but I want to preface my comments and reactions a little bit -- first, because there are things I think we have to keep in mind in order to fully make sense of what goes on this evening, and second, because my own views may require a fuller explanation, which I cannot provide when the clock is ticking.\nFirst: if you think this debate is Obama's to lose, because of his experience, his knowledge, and his successes -- I think you are mistaken. Romney is the one with the advantages here. A bigger man would hesitate to exploit some of those advantages, for reasons of national security.\nRomney is not that man.\nSecond: I think that by and large, Obama has had the most successful foreign policy of any American president in decades. I expect everyone on the right to disagree with me on that -- but I also expect many of my allies on the left to disagree, too.\nWhy do I think Romney has the advantage tonight?\nBecause Obama is the president -- and because Obama is who he is, and Romney is who HE is. What Obama says becomes, in effect, a statement of American policy. As a consequence, he must be guarded in what he says, and will often be unable to provide full explanations for his positions for reasons of diplomatic strategy and national security. Obama, as we know, is a man who values the meaning of words, and measures them very carefully (this is why he is fond of the teleprompter). He is a writer, and a constitutional lawyer, and it is reflected in the measured language he tends to deploy, and the pragmatic and process-focused approach he uses to the world. He is in many ways the ideal president for the country to have at the moment, as we seek to re-familiarize ourselves with the cultivation and use of soft power after a decade in the self-destructive manichean wilderness.\nRomney really has no restrictions whatsoever on what he can say tonight, save what his conscience and political instincts tell him are inappropriate. There was a time, not too long ago, when presidential challengers knew that there were certain kinds of issues, questions, and discussions that had to be put aside during the election season. Our times are different, and Romney doesn't seem like the sort of candidate who would honor such limits anyway. He has already demonstrated that he doesn't care whether his arguments directly contradict his prior statements. He also doesn't particularly care whether what he says is logical, reasonable, or truthful. I'm guessing he also doesn't care that the consequence of attempting to corner Obama into a belligerent 'if-then' statement might be a dramatic narrowing of our national options, with regard to a nuclear Iran in particular. Romney is even more likely to shake the etch-a-sketch on foreign policy than on domestic policy, since the ignorance of the remaining persuadable voters on world politics is extensive to say the least.\nLet's make this more concrete. Romney is likely to push -- hard -- on two inter-related issues tonight: the prospect of a nuclear Iran, and the commitment of the U.S. to Israel's national security. He has the luxury of bellicosity on both of them. Even if Romney were tempted to rein this in for moral or intellectual reasons, the political benefits to saber-rattling will be too tempting for him to ignore, especially because Obama cannot responsibly match him.\nTonight, Romney will attempt to draw a 'line in the sand,' and draw Obama into making a definitive statement that we will never tolerate a nuclear Iran. He will do the same with Israel, by trying to depict Obama as someone who would throw Israel under the bus to avoid military conflict with Iran.\nObama cannot join him in the drawing of such lines. If he does, I'm going to come down on him like a ton of bricks, because he will be betraying the restraint, balance and realism that have led to his foreign policy success. It will be the functional equivalent of a wartime president saying 'defeat is not an option.' Of course its an option -- it always is -- and one must always leave room to maneuver in case it becomes the most likely one. A nation, like an alliance (Israel!), is not a suicide pact.\nObama cannot responsibly take the bait. If, in refusing to take it, he can appear more reasonable, responsible and knowledgeable than Romney -- if he can make the case that there are outcomes in the world that the United States does not have the power, even the right, to determine -- he can turn these exchanges to his advantage. But it will be exceedingly difficult to do, because of the logical and emotional simplicity of the policy choices that Romney will present. Romney will try to make this into Munich, and will make Obama's attempts to qualify and complicate the situation into something that sounds like waffling and appeasement, rather than a reasoned judgment based on a full understanding of the limits of American power in general, and military power in particular.\nObama, of course, could respond by being bluntly honest. He could say that a war with Iran -- following quickly upon the worst foreign policy blunder in the nation's history, in Iraq -- would be an unmitigated disaster for the United States, and for Israel. It would almost certainly grease the already slippery slope toward American national decline. And he could conclude this response by saying that a nuclear Iran is inevitable, and we must begin to the lay the groundwork now for dealing with it, by establishing a diplomatic structure to prevent an arms race in the region. Since Israel already has nuclear weapons, perhaps some kind of stasis can be obtained.\nBut he can't say any of this, because he is the president. Does Obama know that we probably can't prevent Iran from developing nuclear materials, and weaponizing it? Of course -- he must. He knows that is an option, even the most likely one. Does Obama know that a military strike on Iran would be an unmitigated disaster? Surely. But he cannot say so publicly, for fear of losing whatever leverage the US still might have over Iran's rulers. But he also can't follow Romney into the corner, because he will eliminate his room for maneuver should he get re-elected.\nRomney will tell the country that the most important foreign policy threat facing the country is the prospect of a nuclear-armed Iran. I believe the most important foreign policy threat facing the country is the possibility that our own domestic politics (and the irresponsible behavior of Israel) might lead us into a military conflict with Iran. I'm quite certain that President Obama holds a position closer to mine. But he's going to have a devil of a time trying to make that argument during the debate, without being depicted by his opponent as an appeaser, and a betrayer of our Israeli ally.\nSecond, you can discern my foreign policy views from what I've written above. I'm including a few paragraphs on this, because some of my comments during the debate tonight will probably be disagreeable to some of my friends on the left.\nI believe Obama has been our most successful foreign policy president in decades, because my study of American history tells me that it is far easier to start wars, than to finish them. Some military conflicts have easily definable goals -- the definition of victory, if not the path to it, is clear. Victory in World War II was going to ultimately require the unconditional surrender of Germany and Japan. Victory in Korea was going to require the preservation of the territorial integrity of South Korea. One might also call these 'necessary wars,' though obviously the case is much clearer for the former than the latter.\nIraq and Afghanistan more clearly resemble the Vietnam War, than these other conflicts. In each case, we were ostensibly trying to pursue political goals through military means -- they were wars of choice, in other words. In each case, both the definition of and timetable for victory was essentially left in the hands of either our enemies, our local allies -- or both. And in each case, little to no effort was made to assess whether the price (in blood, treasure, reputation, domestic civility, and opportunity costs) was ultimately worth it, until we began to seek an exit.\nI'll have more to say about this in a future post, but I maintain that the invasion of Iraq was the worst foreign policy blunder in the history of the United States. And I say that with an expert's knowledge of the Vietnam War, which was a far more bloody and destructive conflict. While over 2 million people (59,000 Americans) were killed in Vietnam, and it bitterly divided the country and stressed the federal budget and the economy, it had no appreciable impact on the nation's strategic position, its military capabilities, or its national security.\nThe invasion of Iraq, in contrast, fundamentally and perhaps permanently changed the position of the United States in the region, and in the world. It also greatly weakened the possibility of success in Afghanistan, while strengthening the one nation in the region willing and able to check American power (Iran). 40 years from now, when historians write about the first decade of the 21st century, they will likely mark the acceleration of American national decline to the decision to invade Iraq, as well as the manner in which the war and its aftermath were prosecuted.\nObama inherited TWO of these wars of choice -- oh, and the greatest economic collapse since the Great Depression. And to make the task of winding these conflicts that much more challenging, Obama was the leader of a political party that had been repeatedly hammered from the right since the late 1940s for lacking the stomach to fight and win wars.\nObama nonetheless undertook the detailed strategic assessment that his predecessor should have engaged in before these invasions took place, and did so with what I believe was an admirable sense of the limits of American power. The preeminent foreign policy goal of the first Obama Administration had to be the marking of a clear exit path from both Iraq and Afghanistan. While those of us on the left may lament how slow the wind down has been -- particularly from Afghanistan, where the initial response was to increase the number of US troops -- we must not forget the enormity of this task. Lyndon Johnson kept throwing America's young men into the volcano that was Vietnam, too cowardly to admit that victory was not possible, too cowardly to accept that sometimes tactical retreat is the only wise path. Richard Nixon, not wanting to be the \"first president to lose a war,\" ran for president in 1968 claiming to have a 'secret plan' to end the war...and then sent over 30,000 Americans to their deaths in Southeast Asia, with the full knowledge that victory was not possible.\nObama finished the work of winding down the war in Iraq, so that the nation's resources (financial, military, diplomatic) could be concentrated on extracting ourselves from Afghanistan in as satisfactory a manner as possible. Initially, this required resourcing US forces there -- the 'Afghan surge' -- in order move the status quo to a place where a timetabled exit was possible. This was not nation-building, Bush style, though some on the left have said as much. The Afghan surge was not about 'victory.' It was about getting out. If you doubt this, read some of the reporting by Woodward and others on Obama's decision making in the late summer and early fall of 2009. The deliberations included detailed discussions of the Vietnam War, of the limits of military power, and of a timetable to leave. Some in the administration argued against any kind of surge at all; others, mostly the devotees of the religion of 'counter-insurgency,' hoped to use the surge to launch an ambitious test-case for nation building. Almost everyone, other than the president, resisted the idea of a timetable -- a date. Obama was so adamant that the surge had to be in service to a time table to withdraw, that he ultimately settled the internal debate by writing a 10 paged single-spaced brief one night on just what he was agreeing to...and got all the major advisers to sign it.\nAs my friends on the left will rightfully point out, part of this process of clearing a path to the exit from Afghanistan has been the use of drones there and in Pakistan, often resulting in the deaths of civilians. But to me, as long as Obama is able to keep his eye on the prize here -- it must only be a tool to facilitate what will ultimately be a much smaller American footprint -- I think its a call a pragmatic American president has to make.\nAnd so, our military role in Afghanistan will come to an end, too, as it has in Iraq. If you do not think that the implementation of two tactical retreats constitutes an ENORMOUS foreign policy accomplishment, I will remind you once again of the failures of his predecessors when confronted with similar moral and strategic dilemmas. I believe it is the greatest foreign policy accomplishment of any American president in nearly a half century.\nAnd of course, Obama's sense of limits -- he is an avid reader of Niebuhr, recall -- has extended to the rest of his foreign policy as well. If the situation with a nuclear Iran comes to a head in the next 4 years, it is absolutely essential that we have someone like Obama in charge.\nBut do you think Obama can explain all of this in a debate, when Romney hammers him repeatedly about killing the bad guys? God help us all if he can't. Romney is surrounded by the same neo-conservatives that Bush was.\nIn the first two presidential debates, Mitt Romney -- without using that horrid word 'stimulus' -- has claimed that his 20% across-the-board cut in marginal tax rates will spark massive job growth AND reduce the deficit. He has also claimed that the middle class will primarily benefit from these cuts, while the rich will not.\nDeficit reduction would presumably occur because he would broaden the tax base by eliminating deductions and subsidies, and because the stimulative impact of the tax cuts would increase investment and demand, and effectively pay for itself through the increased tax revenue that the subsequent growth would provide.\nNone of these claims are backed up by the evidence.\nEconomist Laura D'Andrea Tyson gives us a really good breakdown in the NY Times Economix Blog today -- so let's go through it.\nThese cuts would reduce federal tax revenue by an estimated $6 trillion over ten years -- $5 trillion for the first four listed above, $1 trillion for the extension of the Bush cuts.\nIn other words, his tax cuts would increase the deficit by that amount, all other things being equal.\nIt should be noted that 4 of the 5 proposals above would be heavily weighted toward tax cuts for the wealthiest Americans -- roughly 2\/3rds of the $6 trillion would go to the top 5%, or those taxpayers making $200,000 a year or more.\nEven if this $6 trillion could be made up through eliminating loopholes, deductions and subsidies (which it can't), is this really the most effective, efficient and just way to create jobs?\nMore to the point, there is little evidence that cuts of this kind would actually create jobs.\nIf tax cuts for high-income earners generate substantial real economic activity and job creation, then we should expect to see two things in the data. First, employment growth should be stronger in the years after tax cuts for these earners. Second, parts of the country with a larger share of high-income earners should experience stronger employment growth after national tax cuts for these taxpayers, because the places where they live receive a larger share of the national tax cuts.\nWhat do we actually see after combing through a half-century of economic data? Neither of these predictions is borne out.\nPersonally, I'm skeptical about relying upon massive tax cuts at all right now, for economic stimulus. Given how low borrowing costs are at the moment, and given our need for massive infrastructure investments (and money for states and localities), I still strongly prefer to see a job creation program that leans more toward spending than tax cuts. If we are going to temporarily have the federal government spend more than it takes in, we'll get more bang for our buck with investments.\nThat said, if the approach of both parties is going to lean heavily toward tax cuts, the evidence is clearly on the side of giving them to the bottom 95% - not the top 5%.\nAlmost all of the stimulative effect of income and payroll tax cuts on job creation in the short to medium run result from such cuts for the bottom 95 percent.\nA recent report by the Congressional Research Service confirms this.\nRemember that consumption -- regular people spending money -- still accounts for 70% of our economy. Our recovery is slow because of insufficient demand. It is not possible for the spending of the top 5% to spark sufficient demand to create jobs in large numbers. Tax cuts for the bottom 95% are much more likely to be quickly spent. And when they are spent, demand increases...which will lead to more hiring.\nIn survey after survey, employers have been abundantly clear: they will hire more people, when demand increases. Romney seems to think that if he is elected, his overall personal white guy awesomeness will somehow magically cause the confidence of wealthy white employers like himself to soar...inducing them to hire more people.\nDoes that sound like an economic policy to you? Romney and his fellow 'job creators' may seek to trickle something down on the rest of us. But whatever that something is, I'm pretty sure it won't be jobs. And keep in mind that every tax cut dollar that is distributed in the way that Romney proposes is a dollar that cannot be spent on public investments that will benefit the vast majority of Americans in the coming decades -- public education, public health, infrastructure, etc.\nI will be live-blogging the Presidential debate tonight over on Facebook, though I do hope to transfer my comments (as well as those from the 1st debate) to here tomorrow morning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This story originally appeared in the San Bernardino (Calif.) County Sun in 2002. It has been updated and edited for use in Door & Access Systems. Reprinted with permission.\nLosing a job was a blessing in disguise for Chuck Colton.\nThe International Door Association (IDA) recently ranked his San Bernardino-based door business, OnTrac Overhead Door Company Inc., as the 11th-largest door dealer in the world.\nThe ranking was based on the company's 2001 revenues of $7.9 million \u2013 a number that seems to keep growing every year.\nColton might have never made the switch to business owner if he hadn't lost his job. He started out in the door business as an $8-an-hour garage-door installer.\nHis last day with his former employer was a Friday in 1994. That Saturday, he installed a utility rack on his truck and by Monday was back \"on track\" installing doors -- this time for himself.\n\"I didn't come out and say in eight years I want this business to be making $8 million,\" said Colton, 44. \"We just started advertising, and we were on time, we were honest, we had a fair price, and we grew.\"\nSuch a fast pace of growth is not common in the door industry, but Colton took the right approach early on, said Pat Kelly, a board member of the California Operator & Door Association.\n\"You have to be good,\" he said. \"Chuck was forward thinking and he got in with some contractors. If you do that, then you start to get a lot of doors, so you have to grow.\"\nCalifornia's door industry has grown too, with a major change in the market occurring about 10 years ago, said Kelly. One-piece wooden garage doors were the norm until sectional steel doors began taking precedence with builders and homeowners.\nA dramatic increase in lumber price, coupled with an almost simultaneous decrease in steel price, largely accounted for the shift, he said.\nPrices for wooden doors had been substantially lower than steel, but when the prices became more equal, steel's advantages -- like easier maintenance -- had people crossing over.\nNow there are two distinct markets: new construction and retrofitting -- exchanging old one-piece doors with steel ones.\n\"It's a stable market -- sometimes the new construction market is slow, but the retrofit market is always there to be had,\" said Kelly.\nWhen looking to expand from their initial 3,000-square-foot office in Yucaipa, Colton spotted a 17,000-square-foot building in San Bernardino, but it was too big for the business -- at the time.\nHe later bought the building with help from a Small Business Administration loan. He now has locations in four cities.\nDoors at OnTrac start around $500, $600 and $800 -- a sort of \"good, better, best\" scenario, said Colton. Customers should be wary of just going for the cheapest doors out there, he said.\n\"Don't just buy the least expensive door. Most expensive doesn't mean you get the best, but cheap just means it's cheap.\"\nColton's career path is not typical of many successful entrepreneurs.\n\"I went to college for four years, but to be honest, I went to play baseball,\" said Colton, a catcher. He attended Cal State Los Angeles as a business major, but never graduated.\nWhile there he took a job driving an armored truck and kept it up for nine years. He traveled the country for a while, playing semi-professional baseball and softball, before taking a job installing garage doors. He did that for five years until his position was eliminated in 1994.\nHe started working from his garage at his Yucaipa home. His wife Terri, who was a stay-at-home mom with three children, went back to work to help supplement their income. Three months into it, they hired Ched Martindale, the company's first of 55 employees and moved into the Yucaipa office.\nOne year later, Terri was able to quit her job and help Chuck with the business. The business grew steadily each year, going from $180,000 in revenue the first year to $500,000 in 1998.\nAdvertising played a large role in the growth with ads on the radio, cable television, in 20 different telephone books and seven newspapers. There are logos on all company trucks, all the installers wear uniforms, and there's a Web site: www.ontracdoors.com.\nOnTrac sponsors several youth sports programs and BMX racing teams and is big on recycling.\nColton donates old garage doors to make homes for people in Mexico. He also recycles the scrap metal and cardboard, donating the proceeds to local churches and youth groups.\n\"Right now we're sponsoring youth sports and a BMX team of about 25-30 kids. Our motto is Keeping Kids On Track,\" said Colton. \"That's what kept me off the streets and that's what's important -- giving back to the community.\"\nBut the employees are far more important than advertising, said Colton.\nHe recently took a 17-day family vacation and didn't have to call the office once -- a tribute to the employees' ability to run the business without him, he said.\nTo show appreciation, there are activities like company picnics, fishing trips, and Christmas parties. Colton recently obtained two charter buses for an all-expenses-paid two-night stay in Laughlin for employees and their spouses.\n\"We just do things to keep it fun. I think, more than money, that's what people want. They want to feel appreciated and that they belong. This company's not necessarily just about garage doors -- it's about people.\"\nAnd though turnover is extremely low, \"we're always looking for good employees,\" he said. \"It's hard to find good, experienced installers who want to work.\"\nLooking back, Colton said he really didn't have any hurdles to jump since starting his business -- not even during the mid-1990s recession.\n\"The economy was supposedly bad when I started in 1994,\" he said. \"I said if this is bad, I can't wait until it gets good.\"\nOthers weren't as fortunate. Pat Kelly, a 31-year veteran of the industry, recalls 1997 as being particularly bad.\n\"We'd been working two shifts producing hardware and within 45 days it was like someone had turned the spigot off,\" he said. \"That recession cost a lot of these contracting firms big money -- it was a tough one.\"\nBut now home sales are booming, as seen with the area's record-breaking number of sales in March of 2002.\nOnTrac started out doing 150 to 200 door installations a month. Now they're up to more than 1,000 monthly installations, split fairly evenly between retrofits and tract home installations, Colton said.\nIn 1996 they became one of the nation's first accredited door dealers -- a title issued by the Institute of Door Dealer Education & Accreditation (IDEA). Now they're trying to become a forerunner again by getting all 22 installers certified by IDEA.\nColton says he can't stress the value of employees enough. That's where the heart of any company really lies, he said. \"Give them some goals, teach them how to reach them, then sit back and watch them grow. This has got to be one of the greatest rewards as a business owner -- watching others succeed.\"\n\"I think that's the real key of the whole thing in any business -- people who are more interested in the person working beside them than themselves.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"About the Shock Doctor Brand.\nShock Doctor Sports is a leading manufacturer of protective sporting gear. Here at Jockstraps.com we carry their basic basic athletic supporters and hard and soft cup supporters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My art is being represented by See Me Art Gallery, Long Island, NY and will taking part of Brooklyn Night Bazaar in Brooklyn, NY http:\/\/bkbazaar.com\/ Brooklyn Night Bazaar started successfully this past Friday, December 13, 2013. It is a night market and concert space that brings together independent vendors, musicians, chefs and artists. If you are in the NY area please come check out our gallery and enjoy some amazing food and music!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxszb b/data_all_eng_slimpj/shuffled/split2/finalzzzaxszb
new file mode 100644
index 0000000000000000000000000000000000000000..89b6d72b62e3ab3777823d2c235efe1999def88e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaxszb
@@ -0,0 +1,5 @@
+{"text":"Camella Bataan is a 10-hectare Italian-inspired exclusive house and lot development, offering affordable home models that are designed to meet a wide variety of needs. Homes are located on an elevated area, assuring you of a flood-free house and lot development. Providing homeowners with a peaceful and tranquil living environment, the beautifully landscaped Camella Bataan is priced reasonably. Camella Bataan is not just a community of quality affordable houses, but also a community where every family can flourish. Camella Bataan has its own clubhouse , basketball court, swimming pool, playground and parks.\nCamella Bataan is strategically located at Balanga to provide quick access to whatever places or services you may need to live comfortably.\nRide a bus going to Mariveles or Balanga-Highway. Alight at Camella Bataan (along Bataan Provincial Highway), about 800 meters after passing Pan Resort and Toyota Bataan, and 500 meters after passing Balanga City arch.\nDrive straight along Jose Abad Santos Avenue to Bataan Provincial Highway. You will reach Camella Bataan (along Bataan Provincial highway) about 800 meters after passing Pan Resort and Toyota Bataan, and 500 meters after passing Balanga City arch.\nTurn right going to Bataan Provincial Highway. Drive straight to Balanga City and you will reach Camella Bataan (along Bataan Provincial highway) about 800 meters after passing Pan Resort and Toyota Bataan, and 500 meters after passing Balanga City arch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shared Ownership is an excellent way for people who are unable to buy on the open market to take their first step onto the property ladder. Under the Shared Ownership scheme, you buy a share of the property's value and pay a subsidised rent on the remaining share. A key advantage is that, as part-owner, you have a stability of tenure that renting cannot offer. Over time, you can buy more of the property until you own 100%.\nYou will need a small deposit at the outset \u2013 generally a minimum of 5% of your share, subject to conditions \u2013 and you will need to raise a mortgage on the rest of the sum required.\n\u2022 Your annual household income must be no more than \u00a390,000.\n\u2022 You must be unable to purchase a home suitable for your needs without assistance.\n\u2022 You must be able to demonstrate a connection to the local area of the development.\n\u2022 You must not have any outstanding credit issues (i.e. unsatisfied defaults or county court judgments).\n\u2022 You must be a first-time buyer. If you already own a home and need to move but cannot afford to, or you have equity from a recent sale, please contact us, as there are some circumstances under which you may still be eligible.\nDon't miss out on our viewing day Tuesday 23 April 3pm-8pm.\nA stunning 1st floor one bed apartment of approx. 59m2 in the Back Church Lane development situated in the heart of Aldgate, East London.\nThe internal specification is modern and contemporary with vibrant custom built kitchens with feature glass splash backs, sleek bathrooms and a luxurious bedroom with built in wardrobe providing a perfectly practical living space.\nAs well as a balcony overlooking an internal courtyard that provides a quiet and private outdoor space, the flat further benefits from access to a large communal roof terrace on the 5th floor of the building, providing a beautiful space to relax and enjoy the breathtaking views of the City and Canary Wharf.Located walking distance to the City and just 8 minutes walk to Tower Gateway DLR providing access to Canary Wharf in just 12 minutes, Back Church Lane provides an incredibly central and convenient place to live to enjoy all that London has to offer.\nSet in the heart of one of the most popular, diverse and historic neighbourhoods in London there's lots to explore on the doorstep. Brick Lane is just a stone's throw away with curry houses, pop-up vintage markets, vintage shops and the famous beigel shops. Shoreditch and Hoxton are only minutes away, where you can peruse the restaurants and bars, or if you fancy yourself as the next Banksy you can take a Street Art Tour and discover the artistic graffiti in the surrounding area while trying your hand at spay painting.\nThe nearest tube station is Aldgate East underground which is around a 6 minute walk away and Tower Hill is about 10 minutes away providing fast connection to key destination including Liverpool Street, Canada Water and London Bridge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Floor Area: 355 sq. ft.\nIt may look like it's simply a storage unit from the outside, but these three shipping containers enclose an impressive interior that might just suit your taste!\nDue to his desire to escape the city for a more peaceful and simpler environment, 29-year old engineer and renewable energy researcher Joseph Dupuis decided to purchase shipping containers and turn them into a cabin in the woods. He built it on land his family already owned about 35 miles away from the city.\nTo keep the home warm during the winter season, he installed a wood-fired stove and heated floors.\nAll in all, the project reportedly cost him $20,000, where $3,400 CAD ($2,600 USD) is attributed to the three shipping containers.\nThough Dupuis does not use this container home as his primary residence anymore, he still considers the house a work in progress. His next idea is to add a fourth shipping container on top for a second floor, which would have a glass ceiling for star-gazing.\nWhat do you think of his container home? Let us know your thoughts through the comments section below after viewing the full album!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"> As I've asked for it, I will of course support this proposal.\nUnicode in XML requires knowing the charset.\n> headers or detected by some implementation-dependent magic.\ndo very often. Files in a ZIP are a whole other story, I think.\nIn any event, our current story about charset seems a bit scattered.\n> I think this is desirable.\n> uses by locale or default.\n> charsets (US-ASCII, UTF-8, ISO-8859-1, ...) in turn.\n> more and more pipeline developers will omit explicit charset declaration.\nNext message: Matthieu Ricaud-Dussarget: \"New to Xproc Question : conditionnal \"output port\" definition?\"\nPrevious message: Jirka Kosek: \"Re: Totally non-conformant JSON hack\"\nIn reply to: Imsieke, Gerrit, le-tex: \"Re: Proposed update to pxp:unzip: charset attribute\"\nNext in thread: Florent Georges: \"Re: Proposed update to pxp:unzip: charset attribute\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gary A Smith Ltd celebrated the launch of their new premise, with a grand opening and it was an epic affair. A large mirror ball cluster hung in the ceiling of their new warehouse framing one of Epic Entertainers, a 'champagne pouring girl'. They also created an amazing live body art performance for guests to enjoy. The DJ was suspended in the back of a Toyota Ute and our internally lit bar was branded, as too was the DJ booth. Blue up- lighting and hints of blue velour drops against black draping, tied in with the brand colours and softened the industrial space. Thanks to our mates at Epic Entertainment for the collaboration, it's always a pleasure working with you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxxwx b/data_all_eng_slimpj/shuffled/split2/finalzzzaxxwx
new file mode 100644
index 0000000000000000000000000000000000000000..a16cf08ea8453059d47920d9f88caeaa49f005b5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaxxwx
@@ -0,0 +1,5 @@
+{"text":"Plants and nutrients MCQs, plants and nutrients quiz answers for online elementary school courses. Practice plant growth multiple choice questions (MCQs), plants and nutrients quiz questions and answers. Career test on plants and nutrients test prep for elementary school teaching certification.\nStudy elementary school courses, online science degree programs MCQs: fertilizers which come from living things like animal dung, compost, fish and bone meals are, with choices organic fertilizers, inorganic fertilizers, low fertilizers, and complex fertilizers for online elementary education degree. Free science student portal for online learning plants and nutrients quiz questions, MCQs to find questions answers based online learning tests.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sites that provide advertising graphics or other ad content files, including ad servers (domain name often with 'ad.', such as ad.yahoo.com). If a site is mainly for online transactions, it is rated as Shopping and Auctions. Includes pay-to-surf and affiliated advertising programs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pick a Shimla tour package from Rajkot 3 Nights 4 Days by Flight here. If you want to make your tour best and in budget then you can choose the best package from our packages that we will be provide to you.\nDay 04 :: Early Check out from the Hotel\/Resort of Shimla and Local Shimla Tour. Reach at Chandigarh and get a flight for Rajkot.\nFor getting today's offers and discounts SMS or Whatsapp code SP1022 on 09887694726.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wedding Themed Forever Stamps When you are actually starting to consider your wedding, you require every little thing to be ideal. Don't permit the wedding to figure out the total of funds that you are actually mosting likely to devote. Nevertheless, your white and black wedding is actually specific to be complimented for its own unique as well as attractive design. Alternatively, a little bit of wedding could possibly consume dinner on a vast veranda ignoring the water. There are actually definitely classy techniques to throw an economical wedding.\nYour wedding will definitely more than likely be among the brightest days of your life. Despite how official as well as classy the wedding will be, there is actually specific information that needs to be consisted of on every wedding invitation. The informal seashore wedding is actually one great principle, however it is actually just one substitute for couples that desire to possess their wedding by the water.\nThere are actually different kinds of wedding garments, as well as if you would like a really unique wedding garments, you'll need to go to a developer as opposed to buy off the rack. A selection to comply with if you're after an exclusive wedding gown is actually to obtain a classic wedding apparel. A few personalized wedding garments might be of unique colours to produce the gown stand out coming from the typical standard white colored, beige as well as off white colored garments.\nPicking wedding invitations are actually at times a demanding selection as the topic of the invitation commonly over into the topic of the wedding. One-of-a-kind wedding invitations are actually as much enjoyable to select due to the fact that they are actually intended to receive. Elegant wedding invitations do not require to mean pricey. An elegant wedding invitation should integrate the labels of the folks that are hosting the wedding if it is actually different than the wedding pair.\nWedding Themed Forever Stamps Always remember in relation to customized wedding invitations it is actually essential that you pick a trusted printer. Personalised wedding invitations can easily offer you with a vast variety of awesome incorporated perks. Ensure your customized wedding invitations offer your visitors with a tiny idea into what they can easily anticipate on the time. They are actually a possibility for you to be unique. Personalised wedding invitations provide you the odds of originality. The different option of modern-day wedding invitations with calla lilies will produce the choice process easier as there'll not be any type of lack of choices.\nWith a little bit of careful preparation, you can easily possess the wedding you always dreamed approximately at a cost that you are actually able. Wedding favours do not have to be pricey to be classy, as you can actually receive a ton of lesson for really little bit of funds. Meals The absolute most pricey part of the entire wedding is actually mosting likely to be the meals for the function. Outside wedding ceremonies are actually astoundingly well-liked. Simply make an effort to keep in mind an elegant wedding doesn't require to cost a ton of funds. Consequently, whether you make a decision to opt for some of the classy wedding ceremonies favours possibilities or even among the candlestick wedding favours, you'll be specific to possess a found for your attendees that are actually mosting likely to be satisfying to them for numerous years to come. If you choose to offer nutritious wedding favours, it is actually an outstanding principle to have the favors in multiple-use as well as fashionable compartments. Consequently the attendees can easily still keep the compartments as pointers of your unique moment Wedding Themed Forever Stamps.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"6 Steps Towards a Productive Week in Campus Ministry\" is the latest eBook from CCMA and our gift to you when you sign up for The Weekly Update. You can grab a copy for free just by becoming a member of our mailing list. Don't delay, sign up now and find six simple ways to simplify your week!\nJoin our mailing list and receive the latest CCMA eBook, \"6 Steps Towards a Productive Week in Campus Ministry\".\nThank you! An email will be sent to you to request your permission for CCMA to send you emails. Simply click on \"Confirm\" when it arrives.\nIn the meantime, here is your free eBook, \"6 Steps Towards a Productive Week in Campus Ministry\". If you like it, be sure to share it with someone else in the field of campus ministry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayyho b/data_all_eng_slimpj/shuffled/split2/finalzzzayyho
new file mode 100644
index 0000000000000000000000000000000000000000..9c51ed0b52ea48304aed201d6d6a3f7f17ae3263
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzayyho
@@ -0,0 +1,5 @@
+{"text":"Home Tags Posts tagged with \"floriade\"\nFrom April till October 2002, 25 Personal Rapid Transit (PRT) vehicles provided transportation to the top of Big Spotters' Hill at the horticultural exhibition Floriade. During operation from April till October, 25 CyberCabs provided transportation to the top of the observation point. The Floriade PRT system formed an alternative to the stairs to the 40 meters high lookout point. The 1st generation CyberCab vehicles were specifically designed for the application, allowing passengers to enjoy the view of the Park.\nThe vehicles used a 700-meter track spiraling up the hill to transport passengers between the bottom and top stations. To ensure the best possible view, the speed of the vehicles was limited to 11 km\/h and drove outside of the track (left hand side) traveling upwards. Each vehicle accommodated 4 passengers, or alternatively 2 passengers and a wheelchair, realizing a a maximum capacity of 600 passengers per hour per direction. The electric CyberCabs were supplied with 'green' energy, with the batteries exchanged by hand every 2 hours.\nA round trip with the Floriade PRT system was priced at 2,5\u20ac. The duration of a single trip was just over 4 minutes. Passenger acceptance was excellent, with research showing both old and young using the system without any reservations. During the 6 months of operations, nearly 400,000 passengers used the system. At completion of the Floriade, the system was dismantled and the different vehicles donated to various colleges and universities throughout Europe to allow them to use it as platforms for research projects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More than ever people are promoting their work on social networks like facebook and Youtube and twitter but what if you are an introvert? Here's some tips i found online from author Hugh O. Smith for beginners not use to self promoting. 1. The idea of fan pages and likes may seem a bit intimidating so start simple. Tweet your adds\/promotions twice per day and Submit book\/article\/reviews to three review sites per week. 2. Employ social media analytics for your pages. Use what works, drop what doesn't. 3. Research successful people and use their methods where it can help you. 4. Take pressure off yourself by writing promotions in advance. 5. View your works as commodities and naturally commodities need advertising right? That it. And if you grant brightseedblog an online interview We will post it on our blog and elsewhere.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Although the knee is the largest joint in the body, it is not the strongest. The knee is a hinge joint between the femur and the tibia and enables flexion and extension. When it is flexed, a small degree of lateral motion is also normal. As with the shoulder, the knee depends on the strong muscles and ligaments around the joint for its stability.\nFigure 20-44 Testing the range of motion at the hip. The examiner flexes the patient's hip and knee to 90\u00b0 and rotates the ankle inward for external rotation (A) and outward for internal rotation (B).\nPain, swelling, joint instability, and limited movement are the main symptoms of knee disorders. Knee pain is exacerbated by movement and may be referred to the calf or thigh. Swelling of the knee indicates a synovial effusion or bleeding into the joint, also known as hemarthrosis. Knee trauma may result in hemarthrosis and limitation of joint motion. Locking of the knee results from small pieces of broken cartilage lodged between the femur and the tibia, blocking full extension of the joint.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You are visiting the placeholder page for Lisa Curtis. This page is here because someone used our placeholder utility to look for Lisa Curtis. We created this page automatically in hopes Lisa Curtis would find it. If you are not Lisa Curtis, but are an alumni of Hartland High School Hartland, MI, register on this site for free now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An attorney specialized and having researched in criminal law is called a criminal law attorney. A lawyer is qualified for protecting for prosecuting the accused the government, and someone who's alleged to have committed a crime. A case could involve a defense attorney that defends the defendant, which means that the offender, and yet another attorney who symbolizes the authorities. Whereas prosecutors function for the authorities for receiving the prosecuted defense attorneys practicing law are effective at protecting their customer against many different instances, ranging from traveling to an allegation such as murder.\nA lawyer might be used by the authorities or with a law firm, in addition to practice. Occasionally, lawyers are hired by the government for representing the defendants. These are called public defenders. In individual states, the government Melbourne criminal lawyers list for you provides the assistance of a public defender to offenders that cannot employ a defendant.\nThe job of attorneys in these situations is to offer their clientele with aid. To carry out their job efficiently, it's crucial for the attorneys to miss their opinion. A defense attorney should safeguard his client of the charges no matter his belief in the event the customer is guilty or innocent.\nLawyers practicing law are needed to appear in court. Aside from the period they will need to invest throughout the trial, they seem to represent their customer such as bail hearings, on various events.\nA lawyer must carry out several tasks before appearing in court. They will need to devote a substantial quantity of time before the court proceedings have already been started, in collecting all of the information. When the prosecutors provide a plea deal, the defense attorney must discuss the issue. The defendant would be educated by the defense attorney of accepting a plea deal about consequences or the consequences.\nAn attorney practicing in the law is required to take research to learn the event law or representation that could strengthen the event of the customer he's currently defending. Another task of a lawyer would be to document the statements of witnesses. Moreover, they may employ the services of researchers to research areas of the case and present the outcomes of investigations. They might hire witnesses to assist the defendants' event.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazsvr b/data_all_eng_slimpj/shuffled/split2/finalzzzazsvr
new file mode 100644
index 0000000000000000000000000000000000000000..da1e261eb3d2a2840160dd36c0fc94d58f38fdab
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzazsvr
@@ -0,0 +1,5 @@
+{"text":"This is a great opportunity to gain valuable experience in media, with a strong focus on a media outlet's visual identity. This position comes with a great potential for professional development, networking, and an increased connection with the student community.\nExpressions of Interest for the Creative Director position are now open.\nContract: The contract commences on the 1st of December and ends on the 30th of November 2018. Remunerated by honoraria (to be negotiated).\nThe Creative Director oversees the overall aesthetic of Togatus.\nEnsure that Togatus' brand identity remains consistent across all platforms, including social media and the website.\nSource visual content for Togatus from student contributors.\nWork closely with the Editor-in-Chief and the Deputy Editor on magazine layout.\nCollaborate with the Website Manager on the website's aesthetic.\nWork alongside the Director of Publication and the Editor-in-Chief when planning events.\nCreate graphics and banners for Togatus when required.\nThe ideal candidate will have a strong understanding of Togatus' brand identity. They will also demonstrate creativity and ingenuity through creation of visual concepts to further develop Togatus as a student media outlet. They will also have experience in programs used in programs such as Adobe InDesign, Photoshop, and\/or Lightroom.\nTo submit an expression of interest for this position, please forward a current resume and a cover letter addressing eligibility criteria, desirable skills, and experience relative to responsibilities to the Editor-in-Chief at editor@togatus.com.au no later than Fri 16th of November 2018.\nInterviews for this role will be conducted on the following dates: 16th, and the 19th of November 2018.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A woman was killed while as four others were injured after a vehicle skidded off the road at Tandwal Rajouri in Jammu and Kashmir on Monday evening.\nReports reaching GNS said that the vehicle (private car Breeza bearing registration number JK12A-6789) turns turtle at Tandwal at about 6:00 pm. In the mishap five persons were injured and were taken to district hospital Rajouri where one among the injured namely Nerish Kumari wife of Om Parkish of Surankote declared brought dead as arrival.\nThe injured have been identified as Geeta Davi wife of Vinod, Ramish Chander son of Yough Raj, Anita Davi wife of Ramish Chander and Rahul son of Ramish, all residents of Surankot.\nA police officer also confirmed the death and injuries to four others in the mishap.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u6027\u80fd\u7279\u70b9: Performance features: 1.\u672c\u673a\u91c7\u7528\u84b8\u6c7d\u9884\u70ed\u3001\u9884\u7f29\u3001\u70d8\u5e72\u8bbe\u8ba1,\u4f7f\u9884\u7f29\u8fc7\u7684\u5e03\u6599\u66f4\u52a0\u67d4\u548c\u3001\u5e73\u6574\u3002\u53ef\u4f7f\u7f1d\u5236\u8fc7\u7a0b\u4e0d\u518d\u6709\u9488\u5b54\u548c\u7f1d\u540e\u77ed\u7801\u73b0\u8c61,\u540c\u65f6\u6a2a\u5e45\u4e0e\u7eb5\u5e45\u62fc\u7f1d\u65f6\u4e0d\u4f1a\u4ea7\u751f\u53d8\u5f62,\u6bd4\u540c\u7c7b\u673a\u578b\u66f4\u52a0\u8282\u80fd\u548c\u73af\u4fdd\u3002 This machine adopts steam preheating, preshrinking and drying design, makes the preshrunk cloth more soft and smooth. That makes the sewing process no longer with phenomenon of pinhole and shorten,also without deformation when sew the transverse cloth with longitudinal cloth, which is more environmental protection than similar machine. 2.\u64cd\u4f5c\u65b9\u4fbf,\u6709\u81ea\u52a8\u653e\u5e03\/\u6298\u5e03\u529f\u80fd;\u84b8\u6c7d\u91cf\u548c\u6e29\u5ea6\u53ef\u6839\u636e\u4e0d\u540c\u5e03\u6599\u8bbe\u5b9a\u8c03\u8282\u3002 Easy to operate, it has the functions of automatic feeding fabric and folding fabric;the steam flow and temperature can be adjusted according to different fabrics. 3.\u7f29\u5e03\u7684\u5bbd\u5ea6\u662f2100MM,\u4f46\u53ef\u4ee5\u6839\u636e\u5ba2\u6237\u7684\u9700\u8981\u8fdb\u884c\u8ba2\u505a\u3002 The width of fabric shrinking is 2100mm,but it can be customized according to buyer's request. 4.\u84b8\u6c7d\u5ba4\u91c7\u7528\u4e0d\u9508\u94a2\u677f\u5236\u4f5c,\u5177\u6709\u826f\u597d\u7684\u4fdd\u6e29\u6548\u679c,\u6709\u6548\u4fdd\u8bc1\u4e86\u84b8\u6c7d\u7684\u4f7f\u7528\u6548\u679c\u548c\u51cf\u5c11\u84b8\u6c7d\u7684\u4f7f\u7528\u91cf\u3002 The steam room made of stainless steel plate has good heat preservation effect, which can ensure the using effect of steam and reduce the use of steam. 5.\u5185\u8bbe\u81ea\u52a8\u8c03\u5e26\u548c\u7535\u773c\u76d1\u63a7\u653e\u5e03\u529f\u80fd\u3002 This machine is equipped with automatic adjustable belt, it also has the function of using electronic eye to control the feeding fabric. \u7528\u9014: Usage: \u9002\u5408\u4e8e\u9488\u7ec7\u5e03\u3001\u5316\u7ea4\u6599\u7b49\u5e03\u6599\u7684\u9884\u7f29\u5b9a\u578b\u3002 It suits for the preshrinking and forming of knitted fabric and chemical fiber and so on.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Veteran low cost. Good deal for the Royals.\nEither someone is hurt, or Dayton is just trying to give a nice guy a job.\nHe did help them win the WS.\nFrank Schwindel's latest substandard roadblock.\nEvery body sing that song\u2026 duda, duda.\nMakes little sense. If your young 1b was rh, then sure. These are wasted at bats. No one is trading for Duda at the deadline. Hard to believe he played 140 games only once his whole career.\nWe traded Duda mid season last year. Not sure why we couldn't do it again.\nBut the move doesn't make much sense unless someone is hurt.\nI'm kind of surprised they didn't do this with Gattis about three weeks ago. I think they might be the only team in the league where he makes sense.\nWhy in the world would KC sign Duda?\nI imagine he can help sell tickets in KC.\nGood luck to him in any event.\nIt's a minor league deal, he's not blocking Ohearn or Schwindel, he's just insurance in case someone gets hurt.\nHe'll be in AAA alongside, and I promise I'm not lying, Andy LaRoche. Who apparently they signed last year after he hadn't been playing since 2015. Two days after signing he was on the DL for the season, without having appeared in a game.\nThis was a procedural move. He was there in a coaching capacity.\nSell tickets in KC? They could care less about another former Mets has been. That is LAUGHABLE.\nWhy WHY WHY???? OHearn and Swindle are so much better.why waste any money on a washed up has been or never was????","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Austrian property and retail group Signa Holding GmbH has approached Mubadala Investment Company and Norway's $1 trillion wealth fund about injecting fresh funds to back its expansion plan, informed sources said.\nIn recent months, principal shareholder of Signa Holding Rene Benko held meetings with executives from Norges Bank Investment Management and Abu Dhabi's state investor in an effort to overview the financials and strategy of his closely held companies, sources told Bloomberg News on Thursday.\nIn April, Benko was part of Austrian business delegation visiting Abu Dhabi in a bid to seek transactions with the UAE's capital and biggest sheikhdom.\nThe Austrian real estate investor met at that time with Mubadala Investment's CEO and managing director Khaldoon Khalifa Al Mubarak to conclude a deal with the oil-rich emirate's wealth fund about investment in Europe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbajci b/data_all_eng_slimpj/shuffled/split2/finalzzzbajci
new file mode 100644
index 0000000000000000000000000000000000000000..5d1b946d064a70dd1b50f02d1003596b66c20873
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbajci
@@ -0,0 +1,5 @@
+{"text":"Founded in 1856, the aim of the National Portrait Gallery, London is 'to promote through the medium of portraits the appreciation and understanding of the men and women who have made and are making British history and culture, and ... to promote the appreciation and understanding of portraiture in all media'. The Gallery aims to bring history to life through its extensive display, exhibition, research, learning, outreach, publishing and digital programmes. These allow us to stimulate debate and to address questions of biography, diversity and fame which lie at the heart of issues of identity and achievement. The National Portrait Gallery aims to be the foremost centre for the study of and research into portraiture, as well as making its work and activities of interest to as wide a range of visitors as possible.\nThe Gallery's research programmes support its mission as described above. As such its major research areas are history, biography and the history of art and photography from the sixteenth century to the present day. This interdisciplinary approach is conducted both through research on the Gallery's own collections and through fostering research into the broader field of portraiture.\nCollections-based research programmes include the Gallery's commitment to a continuation of detailed catalogues together with comprehensive iconographies of those represented in the Collection. Its fellowship programmes, funded by The Leverhulme Trust, follow a long-standing commitment to encourage and assist the work of scholars from outside the Gallery. It also supports research into its subject area through its Archive and Library, through online resources such as British painters and artists' suppliers, and through the Gallery's Understanding British Portraiture subject specialisation network. The Gallery also embarks on large-scale one off projects, often working in partnership with others. Examples include our research for the ten thousand portrait images in the Oxford Dictionary of National Biography and Making Art in Tudor Britain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After a much anticipated wait the first announcements of Leeds Festival are here. Major headliners include Fall Out Boy who played a rousing set in 2016. They will be joined by other leading American bands such as Kings Of Leon, Panic! At The Disco and Sum 41.\nThe eccletic set of artists includes current pop and chart toppers including Kendrick Lamar, Dua Lipa and Skepta.\nThere really is something for everyone, for more rockier tastes Underoath, Hollywood Undead and Papa Roach will be performing.\nWhilst Major UK indie rockers the Courtneers return as one of the fans favourites along with The Wombats.\nWhilst the dance and grime artists such as Netsky B2B, GHetts, My Nu Leng and Annie Mac will perform.\nSigrid who has won BBCs sound of the year 2018 will grace the Leeds stage and contemporary rap artist Post Malone will grace the festival.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The small little canning brewery that could, Upslope Brewing Company of Boulder, Colorado, is turning 3 years old this week. And to celebrate, they are going to host a big party and everyone is invited. Check out their Press Release from today and make your plans to attend. Get your tickets early and save a few bucks.\n3rd Anniversary Party this Saturday!\nThat's right, our third anniversary party is here! Join us this Saturday, November 5th from 2pm-9pm as we celebrate three years of being your neighborhood brewery. It's been a wild ride these last few years with this past year seeing two different brewery expansions along with two new canned beer releases. We will be dipping into our secret stash of limited release beers (one from over two years ago) and be pouring a record of 17 different Upslope Brewing Co. beers during the party including our 3rd anniversary brew (an all Colorado grown beer), all three of our Great American Beer Festival medal winners (Pumpkin, Dunkelweizen, and Belgian Pale), our Black IPA, Double IPA, Kolsch, 14% abv Belgian Quad, and Barrel Aged Imperial Stout.\nWe will have two bands providing live music; a tap room favorite, The Longest Day of the Year and last year's anniversary party performers, The SmashSyndicate. Food will available as well from Comida, RollinGreens, and Top of the Hill Grill West.\nTickets for the event are $12 in advance out of the tap room today and $15 at the door on Saturday. The first 300 ticket holders will receive our third anniversary Belgian-style glass with other guest receiving our logo pint glass. Your entry ticket will also include two free drink tickets.\nRemember to dress warm but we'll have tents and heaters out back!\nWe package our beer in aluminum cans because it's good for you, the beer, and the environment. Stop by our tap room for a cold drink or an enlightening tour.\n- Upslope Pale Ale review.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In Need Of The Big Workcenter Kitchen Island By Catskill Craftsmen Inc.\nPerfect! \"Jamieson 26.25\"\" Swivel Bar Stool\" exellent design By Brayden Studio. \"Jamieson 26.25\"\" Swivel Bar Stool\" very well made, sleek and simple. Complete your living room furniture with a modern \"Jamieson 26.25\"\" Swivel Bar Stool\". Its classy sturdy, attractivce and it looks expensive and a good value for the money. \"Jamieson 26.25\"\" Swivel Bar Stool\" is one of the most homey, cozy, beautiful look and exotic \"Jamieson 26.25\"\" Swivel Bar Stool\" especially for the price and made of fantastic products. Great quality, easy to assemble, delivery on time and in best condition. \"Jamieson 26.25\"\" Swivel Bar Stool\" is good merchandise at fair prices and wonderful free shipping. Guarantee damaged claim by offering to send parts or to keep the item at a discounted price. Great buy would definitely recommend. Shop with our low-price guarantee and find great deals on ##ptitle# and more!. Reading the reviews helped you purchase.\nIn this quick evolving realm of competitors and development, it has been a undeniable fact that individuals hardly get enough time to invest in their homes, looking after themselves. It is mandatory for all of us that we invest the vast majority from the days time in workplace, performing our job. When the workplace does not have in proper tools and amenities for that employees, the workers start getting serious health problems. Likewise, the decorum and also the equipments from the offices are supplied immense importance. Nowadays, the majority of the equipments are chosen in a careful manner so as to make sure the workers comfort and ease. One such essential product in the office is the ergonomic workplace seats. As majority of the offices have ended up being smart, which means they depend more about device than hard physical work, ergonomic office seats are especially designed for this kind of workplaces. They are the seats used for using the workplaces getting computer work stations. These are specially designed chairs that help in the decrease in demands on the several nerves of the spinal wires, therefore restricting the chance of muscular and neural diseases. Also they assist to maintain the correct body posture. It aligns the forearms, wrists, lower back, neck and also the head within the correct positions, therefore reducing lower back pain, function associated bone and joint disorders along with other this kind of illnesses. And when you simply searching for a seat to relax, then consider among the reclining chair from your checklist. In this article we'll discuss about the very best fifteen best ergonomic desk seats to be able to provide the visitors with the correct knowledge so that once they purchase this kind of chairs, it becomes simple for them to select the most appropriate one.\nDoes your room show space requirements? Are you currently inclined to throw a slumber celebration within the mild of the absence of space to rest? This DHP Futon mattress would be well suited for you in cases like this. This Futon couch by DHP incorporates the performance of the couch mattress having a modern and stylish appearance. Place it in your living room to attain an extra tired bed at night. A microfiber floor that became popular in the middle will be a perfect combination. This unique sofa accompanies a tapestry with gleaming stainless locks and legs. Consolidating these outcomes together inside a comfortable popular design sofa. You can choose to have it in Fake Leather-based, Purple velvet or Bed linen. A component that is worth specifying is its back style. In light of the different seats, DHP promotes the progres from the sofa for your comfort amounts. You can level up to have an animated babble, or silent moving image night.\nThis established includes a 1 left arm sofa set, two armless sofa sets and one corner sofa established. This provides sufficient space to support your family and friends. The material is 100% polyester for sufficient comfort and durability. The advantage of this couch established may be the matching and mixing of seats within the space in the room for a perfect form. As well as enables fitted even in small rooms. It requires just mild assembling.Additionally, it functions drive cushions for optimum comfort and ease. You may want to do this set. It does the job nicely.\nThis sofa is appreciated for its distinctive style and luxury. Its impressive pazazz-equip couch with seats provides you with optimum space for stretching out and relaxing in your living space. So usually the modern sofa chaise increases the great thing about this set. The established also includes pillow leading hands padded back and seats for optimum comfort. The sofa includes a corner-obstructed wood frame that provides the sofa enough sturdiness. The couch steps 89 in . broad by 62 inches in depth by 4 in . tall which makes it suitable for medium size rooms. You may want to do minor assembling for example from the legs.\nThis set comes with the entire utility of Person Sofa with an excellent couch design. It incorporates restricted arms, company seat soft cushions and complete-size mat with inner springs. It's eliminated beneath to clean it and remove it having a simple lifting instrument. This mat is strengthened having a sturdy steel edge. is really a measuring rule and has a Bisexual-collapsing highlight. One of the deserving things to mention is the convenience! Place this super comfortable sofa in your comfy living room. Besides giving a natural and tasteful air, it gives a beautiful character for your visitors. You can make your buddies stay for any enjoyable night of films. The sofa has carcasses with blocked edges and faux-finishing on its legs. The soft cushions are, nevertheless, modified and can include great versatility. It is covered with a heavy polyethylene fiber with a drawstring advantage.\nWith regards to buying room furniture, leather-based is always a good choice. Not only does it look great with most designs, nevertheless its very durable (its the ideal materials for a home with kids or animals) and its extremely-easy to thoroughly clean, as well. The downside of leather Eppler Orbs Kitchen Island? It can have a much higher price tag than material, micro-fiber or faux leather Eppler Orbs Kitchen Island. This lying loveseat remains a budget-friendly choice because of 1 guru trick: its sitting area is padded with leather-based, while the attributes are upholstered with increased affordable fake leather. That means you receive the look and feel of the complete leather loveseat with no significant cost. Use the handle around the loveseats arm to relax and relax on its high-denseness froth filling, and consider the money you saved. This specific loveseat posseses an added reward: expert set up will come in numerous areas of the nation for an additional fee.\nComing in at under Dollar300 (plus delivery) and 56 inches wide, our wallet- and room-friendly choose is fantastic for very first apartments, playrooms, dens or any other small areas. Because its padded with extra strong bonded leather, this reclining loveseat is much more resistant to unsightly stains, rips and tears. Customers describe this reclining loveseat as comfy, stylish and an excellent worth. One reviewer authored that it was probably the most comfortable furniture pieces she is ever endured, while another stated he couldnt have asked for more for his money. Reasonable caution, though: some testers said the upholstery on their loveseats tore effortlessly. Thankfully, the maker includes a guarantee for parts. As this loveseat has this type of simple style, it makes a solid foundation for additional personalized decorations products. Then add ornamental toss pillows or covers to up its comfortable element, and give a few pops of color and pattern.\nWeighing under $300 (plus shipping) and 56 in . in width, our pocket book- and space-pleasant choose is ideal for first apartments, playrooms, dens or any other little spaces. Because its upholstered with extra powerful bonded leather, this lying loveseat is much more resistant against unsightly stains, releases and tears. Clients explain this reclining loveseat as comfortable, fashionable and a very good value. One rater wrote it had become probably the most comfy pieces of furniture shes ever endured, whilst an additional said he couldnt have asked for more for his money. Reasonable warning, although: some reviewers stated the furniture on their own loveseats tore effortlessly. Thankfully, the manufacturer includes a warranty for parts. Because this loveseat has such a simple design, it makes a good base for additional customized dcor products. Add some ornamental throw cushions or covers to up its cozy factor, and give a couple of jumps of colour and design.\nThis suits you if all you need is a beautiful trainer. It's designed with a flared body and cushion top amrests inside a awesome cobblestone grey. Now, it's resilient foam soft cushions that are wrapped up in inside a rayon furniture to create the specified comfort and ease. It has a durable part obstructed frame that adds to the toughness. Also, feet are in a faux wood complete. We have an impressive gray color that suits with any decor. It measures 89 W x 39 Deb by 40 They would therefore large enough to support you along with your familyOrbuddies. More to the point, it arrives fully put together. This protects you the pain of having to assemble the set.\nThis ultra-elegant, higher-sovereign sofa is made to support cutting-edge living room development. It is created like a 2-piece sectional couch coupled with an attractive lounger. Additionally, it offers cellular arm rests and beneficial back again support. The beautiful tapestry is completed with dark false leather, securely tufted. You are able to feel at ease cushioning with froth within the chair cushions. This guarantees a fragile feeling whenever you rest after a boring day time. The couch is reinforced with wooden edge engrossed in dark synthetic leather. Fake tufted leather and strong pinus radiata give severe brightness. It is a fact that it requires a second appear since it is more expensive of computer really seems to be. Speaking of size, its huge! This is what adds much more stars to this set. The peak can also be remarkable for high and brief people.\nYou cant talk about this sofa without referring to its remarkable style, durability, and comfort. If you ever required an appropriate seat, then move quick and get yourself Coaster Samuel Collection Cream Leather Sofa. You will love it. This 3 seater couch has thoroughly clean outlines and great soft cushions. The soft cushions are attached to the chair and at the back. It is made from leather which is a certain wager that its long lasting. Additionally, it features tufted back and chairs that definitely contributes to its impressive appear. More to the point, it has a springtime base that provides the required durability and comfort. Its wooden body makes it very durable. The lengthy and the short of it is that this couch is extraordinary in each and every element. You might or might not accept this. But the only way you can understand why is to buy and check out it.\nCopyright \u00a9 \"Jamieson 26.25\"\" Swivel Bar Stool\" By Brayden Studio in Kitchen Islands All right reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sometimes, taking a step back from our screens and engaging with the world is just what we need to remind us why design is necessary. Pentagram partner, Marina Willer, seems to do that on a daily basis.\nOften when we talk about design, we talk about its visual aspects: hierarchy, grids, composition, and so on. But what we sometimes forget is that design can, and should, be about so much more than that. Speaking to Marina Willer \u2013 London-based graphic designer, filmmaker and Pentagram partner \u2013 gives way to a much deeper, wider type of conversation. She speaks of social issues, politics, the unsettled climate in which we live, and our responsibilities as designers. For her, design offers a channel through which to help change and improve our society. And that's exactly what her and her team strive to do throughout their many diverse projects.\nWe had the honor of hearing Marina's personal take on these subjects, as well as delving into some of the intriguing projects her team's currently working on, and how we, as designers, can adapt our work to fit the current times.\n\"I feel we have massive responsibility and we live in incredible times,\" says Marina, speaking of our roles as designers. \"We're getting to the end of the road in terms of how much more we can continue with this behavior without making any changes.\" And when you pair powerful words with actions and good design, the results can be amazing.\nHaving already worked with a number of charities, NGOs and not-for-profits, Marina believes that the more you do a certain type of work, the more you attract work of the same inclination. \"Designers have an opportunity to help change things,\" Marina continues. \"Whether that's through products that are more useful, or through brands that are more relevant to those causes.\" Marina and her team at Pentagram plan to continue working with even more causes they believe in, striving to make an impact, alter perspectives and, ultimately, to change people's behavior.\nThey will also carry on their work with Battersea, a rescue and rehoming center for dogs and cats, having already developed their visual identity and created campaigns with them in the past. A recent project of theirs was in collaboration with Harambeans, a network of entrepreneurs that endeavors to instigate and grow innovation across Africa. In addition, they have a number of other exciting projects on the way. They're working with cancer charity, Maggie's, as well as a new project working with one of the UK's biggest charities helping support older people.\nVisual identity design for Harambeans by Marina Willer, Pentagram.\nMarina aims to always take on projects that are about things she cares about, while also being exciting and socially committed. Apart from charities, this includes culturally-oriented projects, such as her on-going work with architectural firm, Rogers Stirk Harbour + Partners. Speaking of a new project with a highly prestigious and world-renowned classical ballet company, Marina says, \"they're a really interesting organization to work for.\" She mentions \"the sophistication of what they bring to the world,\" noting that they have a rich history, but are also extremely contemporary.\nWith technology updating at lightning speed, artificial intelligence being used to create fake videos, and code being taught to school-children, keeping up with the times isn't always easy.\nDesign for photographer Jo\u00e3o Farkas' exhibition, 'Brazil Land and Soul' by Marina Willer, Pentagram.\nOther than applying this open approach to design systems, Marina explains that as a studio, Pentagram has also always worked in a multi-disciplinary manner. Their original spirit is all about innovation and invention. \"We very much think that we need to create holistic experiences in which we think of everything that is involved, whether it's an exhibition or a charity or a brand,\" she tells High on Design.\nVisual identity and brand strategy for Vibia by Marina Willer, Pentagram.\nBrand identity for Brew By Numbers by Marina Willer, Pentagram.\nDescribing Pentagram's management style, Marina says, \"It's very DIY, but at the same time we're quite obsessed with the details.\" She believes that the quality of the craft is a trademark of Pentagram, and cannot be achieved without being somewhat obsessive and striving for the highest possible results.\nAs for the teams within Pentagram, Marina explains that some partners are more collaborative in the way they work, while others are slightly more hierarchical. However, within each team there maintains a sense of being a collective. As each partner tends to attract projects that are attached to their name, Marina explains that \"people expect the partner to look after and sign off that piece of work. It's a guarantee of quality.\" This results in quite a hands-on approach, in which the partner closely manages projects, alongside their team and a project manager.\nMarina recently wrote, directed and released her first feature film, Red Trees, that tells the story of her father's family throughout World War Two, as they were driven out of Europe because of their Jewish origins. Working together with Jenni Kaunisto, filmmaker and project manager at Pentagram, they explored how to address the subject in an open way. While the topic is very personal, Marina also sees it as universal, noting that \"everyone can see this happening around us now.\" She wanted to \"connect to people in different ways and not be so specific to this one story,\" enabling everyone to relate, whether or not they were personally affected.\nThe result is a beautifully minimal, contemplative film. In the poignant words of Marina, its slow pace is intended to \"get you out of this crazy digital way that we relate to everything, and we don't relate to anything.\" These nuances also help to remove the viewer from their everyday dynamics and engage more closely with the enormity of the topic addressed in the film.\nRed Trees by Marina Willer.\nAs a designer with a great passion for film, the opportunity to work on the design of the Design Museum's upcoming Stanley Kubrick exhibition has been a real privilege for Marina Willer and her team. She's enjoyed re-watching his films and reading about \"one of your favorite film-makers,\" while having the chance to create a cohesive language throughout all the exhibition's assets.\nSpeaking of Kubrick, Marina says, \"He's such an inspiration\u2026 I think there's something very universal and ahead of its time [in his work].\" She finds that everyone can connect to his pieces in some way, whether they're more into mainstream or niche films. She also points out that some of the subject matters brought up in his work are relevant today, such as the theme of humanity versus artificial intelligence, an issue that she's very interested in. This, together with the intriguing design elements in his filmography, make for an exciting project that we can look forward to seeing at London's Design Museum very soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbarax b/data_all_eng_slimpj/shuffled/split2/finalzzzbarax
new file mode 100644
index 0000000000000000000000000000000000000000..36516db56d410ce69f7bd5cfbb32bb9711d1eb9f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbarax
@@ -0,0 +1,5 @@
+{"text":"Bailey needs a forever home. Can you help?\nJust click here and scroll down to check out all the great pets available.\nOur Adoptable Pet of the Week this week is Bailey!! Bailey is a female domestic shorthair cat.\nBailey has beautiful coloring and would make a great pet.\nShe is house trained and is up-to-date on all her vaccinations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MP for Ghanzi North Noah Salakae caused a storm in parliament this week when he raised alarm at the recruitment strategies employed by the ruling party \u2013 Botswana Democratic Party (BDP) \u2013 to lure members from the opposition. He in particular attacked Vice President Mokgweetsi Masisi for recruiting people by promising them loans offered through the Citizen Entrepreneurial Development Agency (CEDA) Scheme, insisting that he has evidence to back his claims. \"Ke na le bosupi jwa gore batho ba Rre Masisi a yang go ba amogela mo press conference tomorrow and any other day from Ghanzi, ke batho ba a ba solofeditseng madi a Citizen Entrepreneurial Development Agency (CEDA). I am aware of a gentleman yo o tlileng kwa go nna gantsi a kopa \u2026a bo a mpolelela gore Rre Masisi o ba solofeditse madi a CEDA. That is bosupi, unless o re ke go bolelele leina la motho wa go nna jalo, I will give that name to your office,\" Salakae said on Monday against heckling from ruling party MPs.\nSubsequently the Deputy Secretary General of the BDP, Shaw Kgathi dismissed claims by the opposition that his party was buying members of the opposition. \"We discuss party budget at our regular Central Committee meetings and there is no provision for funds put aside to recruit members of the opposition,\" he said, maintaining that they are not using state resources. Kgathi is the Minister of Justice, Defence and Security. \"We are busy telling people gore CEDA ke programmes tse e leng they can be accessed by all Batswana, jaanong Vice President o tsamaya a bolelela batho gore ke batho fela that join Domkrag go tswa kwa Opposition ba ba tshwanelwang ke CEDA. We cannot allow that, we are not voted to come and allow that in this Parliament,\" Salakae fumed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A duty knife that can be carried every day, this tactical AXIS\u00ae Assist takes its styling cues from our legacy line of automatic knives. A stout handle with excellent palm swell provides ample grip and positive weight for striking with the carbide glass breaker. The unique mohawk spine design on the blade allows for great control without aggressive jimping. This knife will be your new EDC on and off the job.\nI got this for outdoor (camping, etc) use. It is quite a knife! Second only to a fixed blade and a close second at that. It's just the right size and weight for more heavy-duty tasks, but still with a blade that can easily slice through stuff (unlike the full-sized Loco, which had an extra-thick blade, and which was the folder I was using previously to getting the Vallation). While not a fixed blade like the Bushcrafter (the ultimate fixed blade outdoor knife in my opinion), for general outdoor use it's the next best thing to a fixed blade - certainly one of the best folders you can get.\nAt roughly 8.5 inches long when open, this is a big knife, but one with relatively slender lines compared with, let's say, the Ademas. It feels very solid with a heft greatly exceeding my other Benchmade knives. Due to its size and heft, however, it doesn't work as well as an EDC for my line of work as do the Volli and mini-Barrages.\nThe only solution to this problem, in my opinion, is for Benchmade to make a smaller version of it, at which time time I'll own another great, Benchmade knife.\nWell this is my first benchmade .. And I say first because I am so satisfied with quality,finish ,feel action, weight .. I cannot say enough it is worth every penny .. u rarely find anything u can say that about anymore but I say it with confidence on this .. the feel when u hold it is it belongs in your hand .. I tend to lose things but I will not lose this .. I love it .. and I am a benchmade fan for real now ..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Primary Content Areas: Social Studies\/History\/Culture\/World Civ, etc.\nI'm going to make an effort to do more quick shares of useful and interesting tools. I generally try to give a more in-depth walkthrough or provide tips and strategies and resources and ideas to accompany them, but the end result is that I often end up never sharing the useful\/interesting stuff because of that desire to do deeper-dives.\nCivilization VI is one of those tools.\nCivilization VI has been on my radar since its announcement\u2013well, after Civilization V (Civ V), IV, III, and so on, including it in 6 Video Games You Can Teach With Tomorrow mini-collection.\nWhile mobile-versions of Sid Meier's masterpieces have long been available, Civ VI is the first time that player s on mobile can enjoy the same (or nearly the same) experience as players on desktop computers. We've been playing it at TeachThought a lot over the last week and are blown away by its complexity, scale, and potential as a teaching tool for social studies, history, or other related 'social\/cultural' class\/course.\nThe same applies here, but to be able to use Civilization VI to teach History, World Civ, Social Studies, Political Science, Government, or any other class, you're going to have to play and understand it yourself. You can find a walkthrough for beginners for Civ VI here to get started If you have an iPad, I'd buy it and give it a go, then let us know in the comments below what you think.\nI really enjoyed Democracy 3 and the Age of Empires series among many others, but for overall (relative\/fictional) historical accuracy, depth, scale, and cognitive demand, Civ VI is the best game I've played for teaching Social Studies so far.\n\"Originally created by legendary game designer Sid Meier, Civilization is a turn-based strategy game in which you attempt to build an empire to stand the test of time. Become Ruler of the World by establishing and leading a civilization from the Stone Age to the Information Age. Wage war, conduct diplomacy, advance your culture, and go head-to-head with history's greatest leaders as you attempt to build the greatest civilization the world has ever known.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Access: Open late May - October 31. Access free of charge.\nDirections: ST. MARIES RIVER RD: turn S off ID 3 in St. Maries on 1st St; 1 mi through neighborhood to St. Maries River Rd. ST JOE RIVER RD: in St. Maries, at junction of ID 3\/ID 5, travel on ID 3 N; cross St. Joe River; at mp 85 turn R (N) on FR 50 (St. Joe River Rd); explore areas at mp 2.3, 10.7, 13.0, 16.0, 16.3, 16.6, 24.2, 29.8, 33.6 and 34.2; scan between mp 11-13 for Northern Pygmy-Owl (Nov-March); the town of Avery is 26.7 mi E of St. Maries and FR 50 ends 29 mi E of Avery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbiqw b/data_all_eng_slimpj/shuffled/split2/finalzzzbbiqw
new file mode 100644
index 0000000000000000000000000000000000000000..4fa5a81bbe3dec1b2d3e7ded54aae7b1b0216343
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbbiqw
@@ -0,0 +1,5 @@
+{"text":"We bring businesses together: an opportunity to meet and greet face to face. We bring them from all parts of the world so that they might find fresh partners, discover new customers or suppliers and keep ahead of industry developments.\nWe organise a number of trade exhibitions which focus on fashion and lifestyle: sectors that are constantly in flux, so visitors and exhibitors alike need always to be fully aware both of the changes around them and those forecast for coming seasons.\nAPLF's inaugural event, held in year 1984, was then known as the Hong Kong Leather Fair. It was the city's first ever Mega-Fair. Today, APLF has expanded its scope, currently organising six premium events in Hong Kong, China and India and covering a wide range of industry sectors.\nHong Kong is the ideal hub from which to build Asia-wide coverage and, of course, as China and India develop ever more momentum within the global economy our exhibitions are well positioned to introduce our customers to this fast expanding market region.\nAt this moment the potential of the China market is being explored by countless businesses world-wide. The All China Leather Exhibition (ACLE) and the China International Footwear Fair (CIFF) & Moda Shanghai, have run concurrently in Shanghai since 1998. They are widely accepted as the premium international trade events for these two burgeoning sectors and are organised in conjunction with China's only officially recognised sector trade organisation, the China Leather Industry Association (CLIA).\nAPLF's scope of events captures the ever-shifting nature of the fashion and lifestyle industry. Our Material's Manufacturing & Technology (MM&T) exhibition is well established as the premier event for fabrics & components including the premium and always fashionable fabric \u2013 leather.\nWith Fashion Access we have entered fully into the fashion arena, showing bags, footwear, leather goods, travelware and many lifestyle accessories.\nVisit our Hong Kong exhibitions and you will find guides to help you forecast materials, designs and colours for up-coming seasonal ranges, complete with seminars and forums, special awards for outstanding innovation and, something we have long been keen to promote more actively \u2013 Sustainable Development.\nIt's not just conventional trade exhibitions that we offer. Prime Source Forum (PSF), inaugurated in 2006, offers a haven for top executives representing leading companies throughout the international garment and fashion industries to come together in a neutral, non-competitive environment to discuss issues of common interest and concerns.\nA new event was launched in India in 2009 - Footwear Materials, Manufacturing & Technology (fMM&T). It was a great success: allowing buyers and sellers an unique opportunity to meet and greet, in person.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many technology success stories leave behind its unintended consequences. One ideal example would be of virtualization. Virtualization has provided us with unprecedented hardware resource utilization. One impressive thing about it is that it has radically transformed a provisioning process by reducing the time taken for the process to minutes instead of months. The speed and flexibility provided by virtualization was unimaginable then, and it also paved strong foundation for platforms for both private as well as public cloud. The prevalence of public and private cloud platforms today is thus thankful to the success of virtualization.\nIt is that level of access and speed to public clouds that delivered the capability to circumvent established process; in another words, it gave rise to \"shadow IT\". Cloud anarchy has delivered many useful technical features as compared to \"shadow IT\". IT, in many sense, has failed in getting control over what is being deployed and by who and where. However, in case of a cloud management platform, one good thing about it is that it can apply the governance without compromising on either speed or flexibility; and that is what organizations are demanding nowadays \u2013 A secure and well-governed business environment.\nGiven the cloud's ability to apply the governance, it is curious to see how such ideal governance looks like. Organizations often remain in dilemma over how to exert control over cloud access; following section highlights vital steps to implement ideal governance and prevent could anarchy with respect to public cloud usage.\nIT department in any organization should ensure that the governance they are implying should have less associated friction as compared to circumventing. There are high chances of shadows IT to persist if an internal IT system remains complex and challenging for provisioning cloud workloads. In order to prevent shadow IT from persisting, organizations should choose cloud management platform that can easily integrate with present single sign-on solutions, it also spares users from creating and remembering new login ids. Many organizations also make use of existing group definitions from their respective directory server. The aim should be of avoiding creation of those fundamental pieces in two different places.\nApart from clubbing users into groups, organization should utilize its cloud management platform to club their cloud regions into numerous groups. This clubbing makes it convenient to implement access control lists to the cloud's different logical segments. Such level of abstraction should incorporate account-level separation in the cloud.\nFor instance, if an organization wishes two separate constituents to use cloud services of its respective service provider, then it can enable one group, being tied to its core IT billing credit card, to use cloud services, and enabling another group, being tied to a line-of-business credit card, to use cloud services. That way, an organization can enforce required flexibility through that level of cardinality by creating separate cloud groups and granting usage access to those different constituents.\nWith the aim to accommodate different technical levels, create a familiar, easy-to-use interface or service catalog. In many ways it provides workflow approval tooling as an integral part of the deployment process. Such a comprehensive service catalog enables admin officers to deny or allow access to numerous individual applications for deployment. Moreover, it also ensures strict following of a management approval process prior to provisioning of resources on cloud platforms.\nOperational teams across global enterprises are demanding access to both public and private cloud infrastructures in order to make well-informed, intelligent decisions regarding subject mattes such as private could capacity, data security, and data gravity. It empowers IT departments to quickly change and customize access provided that they have followed necessary steps to implement authorization, abstractions, authentications, and service catalogs narrated before.\nOrganizations are required to create an ideal balance involving creation of separate cloud accounts for different individual users and tracking total expenditure on cloud accounts. Popular cloud management platforms are offering a system through which IT administrators can track application deployments on a group or individual basis; the platforms also allow internal metering mapping to a single cloud account. This metering allows central IT departments to reduce cloud account administration and delivers granular account usage reconciliation; this reconciliation then becomes integrated into a broader internal chargeback mechanism.\nOrganizations should focus on implementing a governance solution that can provide on-demand provisioning, self-service in a well-organized manner offering granular billing and metering. Benefits associated with the governance that includes role-based access assignments, cloud abstractions, authentications\/authorizations, service catalogs, and cost tracking are simply incredible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please fill out the form below to learn more about Iridium Edge\u2122. We will be in touch shortly to answer your questions and discuss how Iridium Edge\u2122 can work for you.\n* 5. In which industry does your company do business?\n* 7. Is your company currently an Iridium Partner?\n* 8. Do you have specific questions about Iridium EdgeTM?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Latest Patreon papermini release from Hishgraphics Papermini Patreon! Lovecraftian \/ pulp investigators from the 1920s! All prepared to lose their marbles upon investigating and being exposed to unspeakable horrors!\nWith: the dilettante, the detective, the doctor, the police officer, the reporter and the nurse! As usual, the Pioneer-class release is free for download while the higher quality releases are for Mariner- and the Voyager-class patrons only! Fhtagn!\nClick here to download the pioneer-class pdf from Patreon!\nAlso, as a free bonus download, some Lovecraftian papermini monsters for your player characters to tangle with \u2013 or to be player characters, I don't judge.\nClick here to download some Ghouls and Mi-go the Fungi from Yuggoth on Patreon.\nPosted in Artwork, Patreon Paperminis and tagged artwork, call of cthulhu, patreon, rpg.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Using your phone while driving implies an increased risk of accidents. Even though drivers seem aware of the danger, they can not help but respond if they receive a notification (message, call, etc.).\nHowever, in an AWSR survey, half respondents estimate that 80% of the messages they send while driving, could be sent later.\nThe purpose of the campaign was to remind that any message, call or notification can wait until the end of the trip. A stupid message can kill you. The campaign was used twice, adapting the message slightly.\nWe wanted to show the absurdity of a last message sent while driving. We firstly shooted a film of a road accident, with the victim dictating his last words, causing his accident. This was then shortened and adapted to be efficiently used on social media as well.\nWe also created a radio spot to go along with it. Finally, on the roadside, we used the Facebook thumb and this simple question: \"Dead for a message, do you like it?\".\nAccording to the post-test in 2016, it's estimated that the highway poster was seen by more than 60% of the Walloons: the highest score obtained since the creation of the AWSR.\nThe radio spot got also a very good score, and by 2016 it was the best spot ever broadcasted by the AWSR, since the courtesy campaign was born.\nThis year's post-test is not available for the moment. However, the video has been watched almost 130,000 times.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcmdp b/data_all_eng_slimpj/shuffled/split2/finalzzzbcmdp
new file mode 100644
index 0000000000000000000000000000000000000000..ba9c0ee62ad024073e62e2aafac67c91726d2d42
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbcmdp
@@ -0,0 +1,5 @@
+{"text":"There are numerous elements involved in shipping, which means that calculating the cost of a freight forwarding service isn't straightforward. You'll also find that some freight forwarders include certain costs in their upfront quote, while others offer them as an optional add-on. In some cases the freight forwarder might not offer a particular service at all, leaving you to make other arrangements at your own cost.\nIt's essential that you get a full breakdown of what's included in a quote before you engage the services of any freight forwarder. A reputable forwarder will have no issues with providing you with a clear itemisation, which will allow you to compare quotes for real value, rather than cost.\nTo give you an idea, here's a breakdown of the elements of shipping that need to be paid \u2013 whether they're included in a given quote or not is for you to confirm with your chosen forwarder.\nWhether you choose to have your goods shipped by air, sea or road, the mode of transportation has to be paid for. As a general rule of thumb, sea freight is the cheapest and air freight is the most expensive. The distance travelled will affect the cost \u2013 although it doesn't always follow that a greater distance equals a greater expense. The rules of supply and demand may mean that a freight forwarder can negotiate lower carrier costs on popular routes, while less popular routes with fewer carriers to choose between provide less opportunity for negotiation.\nWhen shipping by sea, you'll either be charged for a full container load (FCL) or a less than container load (LCL). Choosing LCL may work out cheaper if you don't have enough cargo to fill a container, although this means that you'll be sharing your container with other shipments and there'll be a cost involved in separating these shipments at the destination port.\nSimilarly, with air freight, your goods will be packed into a Unit Load Device (ULD), which can come in the form of a pallet or container.\nSome types of cargo cost more to ship than others, this could be because they require a specialist kind of container, such as perishable goods, or they require more careful handling, such as dangerous goods. Goods that don't fit in a standard container or that need a crane or other specialist equipment for loading and unloading will also be more expensive to ship.\nThe size and weight of your cargo can both impact the cost of shipping. The chargeable weight of a shipment is calculated by converting the volume into a weight equivalent. The chargeable weight is then whichever is greater, the actual weight or the calculated volumetric weight. Essentially, two shipments can weigh the same, but the one that takes up more space will cost more to ship.\nWhen importing and exporting, goods are required to be palletised and packed in a specific way, and any wood used is subject to international legislation. Some freight forwarders offer a packing service at an additional charge, so you can be sure that your goods are compliant with the government regulations applicable at the destination port. Having goods returned to the port of origin for re-packing can be very costly!\nA fuel surcharge may be applied by the carrier to protect them from fluctuations in fuel costs \u2013 this can also be known as a Bunker Adjustment Factor (BAF). A carrier may also apply a Currency Adjustment Factor (CAF) to protect them from fluctuations in currency, a war risk surcharge, a security surcharge, a peak season surcharge or a demurrage penalty for delayed loading or unloading.\nWhile carriers are legally required to have carrier liability insurance, this can provide minimal coverage, leaving your goods vulnerable. It's wise to protect your shipment with cargo insurance, which many freight forwarders offer as an add-on service. Ask your forwarder whether insurance is included in the quote or available as an add-on. If your forwarder doesn't offer cargo insurance, you'll need to purchase it directly from an insurer or broker.\nA freight forwarder will usually complete the required documentation, such as the Bill of Lading, invoice and import and export documentation, for you. An admin charge for this service will typically be included in the upfront quote. Goods that require special paperwork, such as dangerous or restricted goods, are likely to incur an additional charge.\nThere are numerous destination costs involved in the shipping process, including charges for handling (terminal handling charge) and clearing the goods at the loading and discharge ports. A port may also impose an additional security surcharge. Other potential destination costs include unpacking (in particular the cost of splitting LCL shipments), warehousing fees and the cost of inland haulage if required.\nUsing the services of a freight forwarder will save you the time and hassle involved in organising, coordinating and managing the different elements of the shipping process. But, of course, the forwarder doesn't carry out this service for free, and an administration charge will form part of their quote.\nSome freight forwarders offer additional services to take care of the shipping process from end-to-end. For example, John Good Shipping offers product sourcing, packing and packaging, unpacking and sorting, warehousing, distribution and delivery to the end customer.\nWhen comparing different freight forwarding quotes, it's important to know exactly what's included so you can compare like for like and understand which quote provides the best value for money.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Specially designed for laying over your paste to prevent drying out whilst working on flower making projects.\nThis mat will ensure that your sugar florist paste stays fresh and at its best condition for creating decorations. Can be used to roll out on and for cutting out on. Can provide a non stick and non slip surface.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Known as the most rare and precious of metals on Earth, platinum will always hold high status in the jewelry world. Mens platinum wedding bands, the epitome of prestige and luxurious quality, have long been sought after on the wedding scene. This fine metal, representing elegance in jewelry, appeals not only to women but to men as well. Read more.\nWhile most are used to seeing the traditional gold wedding band, platinum has quickly gained popularity for wedding bands as well. In addition to its elegance, strength and durability are what define platinum. The mens platinum wedding bands offered by Apples of Gold demonstrate why this metal is so highly valued among grooms today. Varying in width, styles and designs, these bands offer a modern yet classic look to the man who wears it.\nApples of Gold also offers Mens platinum wedding bands with accents of 18K gold for added variety, style and originality. By mixing metals, you have the best of both worlds\ufffdthe modern appeal of a white metal paired with the more traditional quality of the yellow gold. This is a ring that certainly makes a statement.\nThough platinum is an ancient metal, it\ufffds use in jewelry production is less than two centuries old. Thus it is a relatively younger metal than its gold and silver competitors in the sphere of wedding jewelry.\nWhen shopping for mens platinum wedding bands, consider the following elements. Is your personal style more traditional? Then choose a classic platinum band with a brushed finish. Or are you looking for something more unique? Consider an intricate paisley design, a Celtic knot pattern, or a hammered finish. Further personalize your wedding band by engraving a Bible verse, the date of your wedding, or the name of your spouse.\nDespite platinum\ufffds higher price tag, this metal has its benefits. In addition to its fine appearance, platinum is also durable, long-wearing, hypoallergenic and resistant to tarnish. Unlike gold, which is an alloy, platinum is 100% pure. It is also a heavier metal due to its density.\nAmong all the options in wedding jewelry, mens platinum wedding bands promise complete satisfaction. For those who work with their hands and are rough on jewelry, platinum insures durability. For the man seeking luxury, platinum doesn\ufffdt disappoint. Consistent over time, platinum needs little maintenance. No other precious metal possesses qualities quite like platinum. Like your love for your wife, platinum is designed to last a lifetime and beyond.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"to meeting Grayson and having a great time. Grayson doesn't know a lot about the \"Other side\". They are just the same as Grayson. They us toothbrushes, toilet paper, have rugs, swimming pools, TVs, popcorn, everything you ever want and need. This isn't what Martin Luther King Jr. live up for. There are some parts of the world that still has these problems. I hope they read Maniac Magee and be one of him, no matter of their characteristics even is we have trouble.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Framers Forum \u2022 View topic - BBest on more than one pc?\nI, like so many others have more than one PC\/Laptop for work and I was wondering about buying BBest.\nIf I install it on my workshop PC can I install on my Laptop also?\nIf I can, then does it run separately, do I have to manually set it up on the laptop to mirror the PC in my workshop.\nI use the laptop when going to see clients etc, I know it doesn't create a database etc but I was thinking more about setting price markups, supplier price changes and the like.\nDo I have to do it twice?\nRe: BBest on more than one pc?\nYes, you can have copies of BBEst on as many additional computers as you like - no extra charge.\nUnlike EstLite, BBEst is strictly stand alone and can not share common data over a network, so it is up to you to manually synchronise the various mark-ups and settings when running two or more copies of the program.\nIf you happen to be running XP on all computers, it's just a matter of configuring the software on one machine and copying the BBEst folder to your other XP computers.\nIt is a bit more complicated on the later versions of Windows, so unless you are a more advanced computer user it's probably safest to clone the second copy by entering the various initial settings manually. That said, after the initial configuration, it is easy to keep moulding prices in sync by regularly downloading the current suppliers' prices on each computer.\nThat is pretty much what I expected. I'm running windows 7 on both.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcwkn b/data_all_eng_slimpj/shuffled/split2/finalzzzbcwkn
new file mode 100644
index 0000000000000000000000000000000000000000..f5ff47ae8247530610639c876ee453525d838180
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbcwkn
@@ -0,0 +1,5 @@
+{"text":"Use tools like boards and springs to guide a little girl through uneven terrain with dangerous bugs.\nLucidity was part of Steam's Swedish Indie Pack, and I thought I would try it despite the Metacritic reviews. This was a mistake. It never became fun in the first half-hour, which is about as much benefit of the doubt as I could spare it.\nYou are guiding your one lemming through the level. Instead of working from a limited pool of resources, which usually gives you a lot of information about how to solve the puzzle, you get an endless stream of tools randomly generated from a small pool. An early level might have just one or two, but that grows. Avoid the enemies, walk through the fireflies, reach the end of the level.\nI presume that some levels have actual puzzles, and collecting every firefly would involve some creativity. Just getting through the levels: dodge the bugs, span the gaps, and you're done. That is occasionally frantic when the pieces that come up are not optimal, but that seems to be the whole of it. There are reportedly a few hours of gameplay, probably a few more if you want to get 100%.\nThe art is lovely and the music is gentle. The text snippets at the end of each level seem appropriate to the setting but irrelevant to everything else. Maybe they build to something if you get more of them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A large amount of research in life science is unreproducible due to varying antibody specificity. Comparing samples with the target protein depleted to samples with the protein expressed at normal levels is one of the best ways to ensure antibody specificity (Uhlen et al., 2016). ABclonal aims to improve the quality of life science research by introducing KO-validated antibodies.\nGene knockout refers to the technique where a gene is inactivated (knocked out) through replacement or disruption of DNA. In research, gene knockout is used to infer the function of a known sequence. ABclonal validates our antibodies using CRISPR\/Cas9, the leading technology in gene-editing. The gene that encodes the target protein is inactivated, hence producing samples with the target protein depleted.\nUhlen, M., Bandrowski, A., Carr, S., Edwards, A., Ellenberg, J., Lundberg, E., Rimm, D.L., Rodriguez, H., Hiltke, T., Snyder, M. and Yamamoto, T., 2016. A proposal for validation of antibodies. Nature methods, 13(10), pp.823-827.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ligao City is in the \"3rd\" Voting District of Albay.\nLigao City is a 4th class Component City and Partially Urban.\nLigao City is in the Albay province and within Region V - in the southern tip of the island of luzon also called the Bicol Region.\nLigao has a total land area of 24,640 hectares, 23.05 percent of which are fertile flatlands suitable for high value crops and 76.95 percent are mountainous and hilly terrain with potential for agriculture and traversed by secondary rivers. Tertiary rivers originate from Mt. Masaraga and Mayon Volcano located at the south and eastern side of the city. Ligao's soil is generally fertile giving great flexibility practically to all types of crops. There are six common soil types found in the city: Ligao loam; Guinobatan sandy loam; Mauraro gravelly sandy loam; Libon silt; Tigaon clay and Sevilla clay.\nLigao City is a newly created city being born only last March 24, 2001 during a plebiscite with an overwhelming Yes vote of 17,754 as against 1,387 No votes. It was created by virtue of R.A. No. 9008, signed by President Gloria Macapagal Arroyo last February 21, 2001, known as an Act Converting the Municipality of Ligao into a Component City of the Province of Albay, the third city of the province and the sixth city in the Bicol Region.\nLigao City is located 502 kilometers south of Manila. Centrally located in the 3rd District of Albay, it is bounded on the north by the municipality of Oas; on the south by the municipality of Guinobatan, Tabaco on the east and Pioduran on the West. It is 27 kilometers from the Provincial Capitol of Albay and approximately 30 kilometers from Legazpi City.\nThere are three versions of how the name Ligao came to be. One legend says that Ligao is a corruption of the word \"Tigau\", a tree found abundant in the locality whose poisonous leaves were used to catch fish in rivers or creeks. The legend goes that a group of Spaniards passing by a tigau tree asked for the name of the place. The natives, thinking they were asking for the name of the tree, answered \"Tigau\". The Spaniards mispronounced the word as Ligao and since then the place was known as Ligao.\nAnother version is that the name Ligao came from the Tagalog word \"Ligaw\" which means to court or win a lady's love, since the place was well-known for its beautiful maidens. Many young and eligible men from far and near would come a-courting hopeful to win their maiden's love. A group of young men was once accosted by Spanish soldiers who asked for the name of the place. The young men, because they did not understand what the Spaniards were saying, thought that the latter were asking where they were going and so they answered, \"Manliligaw\" (to go a-courting). The Spaniards thought the name of the place \"Maliligaw\" so for a while it was known as \"Manliligaw\" until it was shortened to Ligao.\nThe most popular version was that the name Ligao comes from the Bicol term \"Licaw\" which means detour or to go around by deviating from the usual or ordinary route. Ligao started as a small settlement known as Cavasi in the 16th century. Trade and commerce already flourished between Cavasi and the neighboring towns of Polangui and Oas and other places. During the rainy season the usual route to Cavasi is easily inundated by floods. The merchants or \"viajeros\" as they were called during those times had to make a detour. The place of detour was known as \"Likaw\". Eventually \"Likaw\" or the detour became more progressive as the population increased rapidly than Cavasi until such time that Licaw was more well known than Cavasi. Eventually, Licaw became Ligao.\nThe city traces its history from the 16th century when a small settlement known as Cavasi existed without a ruler or leader. Every resident was peaceful in his endeavor. This trait enabled the settlement to grow and even attracted other natives from nearby settlements. Along with increasing population, ambitious and aggressive leaders emerged causing friction and creating factions endangering the general peace in the settlement. Five divisions were formed led by maginoos (chieftains): Pagkilatan, Maaban, Sanpongan, Makabongay and Hokoman. Chieftain Hokoman considered himself the supreme leader over the whole settlement. Even then, rivalry and strife persisted and the once peaceful inhabitants lived in constant fear. According to accounts by Father Felix de Huerta, a Spanish Corporal endowed with the ability to settle jurisdictional disputes mediated among the ruling Maginoos. With the approval of the other chieftains, Pagkilatan was chosen the Supreme Chieftain over the entire settlement and peace and tranquility returned to the place.\nThe town was founded as a barrio of Polangui in 1606, was ceded to Oas in 1665, and finally became an independent municipality in 1666. Since then, the once minor settlement called Cavasi prospered economically, socially and politically.\nLigao City is a 4th class city in the province of Albay composed of 55 barangays, 11 urban and 41 rural and with 3 coastal barangays. The 2000 Census placed its population at 90,603 in 17,031 households.\nThe population of Ligao City is a mixture of traits and culture because of its strategic location. Ligao City is centrally located in the 3rd district of Albay being the converging point of population from the surrounding municipalities of Pioduran on the west, Tabaco on the east, Oas and Polangui on the north and Guinobatan on the south. Because of the accessibility of the place lying along the Pan Philippine Highway that stretches from Aparri in the north through Davao in the south, people as far as these places has come to settle in Ligao City bringing along with them their own culture and tradition.\nThe dialects varies as the people who came to settle, from the Ilocano dialect to Kapampangan to Tagalog, Visayan and even transient Muslim traders with their own dialects. But Ligao City has a distinct dialect which differs in pronunciation, meaning and spelling even from other Bicol dialects. The local dialect is quite similar to Kinaray-a, a dialect in Antique.\nRoman Catholic is the dominant religion accounting for 96.92 percent of the population as of 1995. Religious tolerance is practiced among the inhabitants as indicated by various denominations present in the city.\nIf you have real estate property, whether its commercial, residential, farm land, or just an empty lot in Ligao City, you can list that property for free.\nIf you have an article that talks about the improvement of the economy of Ligao City you can post that article here. If you come across any news items that talk about the economy of Ligao City, you may post it here. Of course you have to reference the writer of the article. Any improvement to transportation, power and service usually improves the economy of the community, so go ahead and report that too.\nIf you have a job available and that job is within Ligao City, Philippines, you may post it here.\nPost expiration of Job Application. Go ahead and Click HERE to Insert your job offer in the \"Jobs in Ligao City\" page.\nPost the Ligao City landmarks here.\nDo you know who the oldest man or woman is in your community of Ligao City. Zamboanga.com is starting this inquiry in order to honor the older generation of the Philippines. Please provide the full name and date of birth of the elder living in Ligao City. We will then post your entry in the Oldest Man or Woman in the Philippines page.\nDo this so your photo upload will be properly categorized for Ligao City.\nThis page was last modified on 4 February 2019, at 20:36.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Windows 7 Slate PCs On Their Way. Still!\nIt is d\u00e9j\u00e0 vu and deja vu.\nBallmer is deluded if he thinks that some tweaks to Windows 7 will make the software workable on a tablet. And then there is the major question of apps. What Microsoft should be doing is side-by-side development of phone and tablet versions of Windows Phone 7, but instead they are\u2013so far\u2013refusing to consider the use of WP7 on larger devices. This is a huge mistake.\nGot to agree with you there. I really don't want W7 on a pad. I like it but, it's not going to do on a pad what I want the pad to do. I think think what they should be doing is looking at the ZuneHD interface which I guess is somewhat what the WP7 is.\nI am using Windows 7 on a tablet right now. Works fine so maybe you're the deluded one.\nI keep seeing posts like yours claiming Win7 can't work on tablet devices. While I'll reserve judgment on how well it might work, I don't see any reason the OS can't be adapted. The multitouch features are built into the OS api. MS has sample applications out there. And manufacturers can easily customize the Windows shell to be anything they want to be touch friendly. That being said I would say MS should lead the way and not wait for HP (who doesn't care anymore because of Palm), Dell, Asus, etc to build a good touch shell.\nWin7 runs great on low powered netbooks so I don't think your complaint should have to do with performance and battery life (you can find netbooks doing 10+ hours out there). And if a tablet was build with Win7 embedded it would likely perform even better.\nI think you have that last analogy a bit backward. Its more like taking a sport car engine and putting into a sedan and finding a way to tame v12 monster of OS that Windows 7 is into something moms can drive around is not going to work while the iPad in on the market.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fandam Plus is the instrumental version of the Fantastic Damage album by El-P. To me it's also the true follow-up to the \u00fcberdope Little Johnny from the hospitul by Company Flow, El-P's previous crew. There's a second CD with some remixes, lyrics, live footage and the [strange, Aphex Twin-esque] video for Deep Space 9mm.\nAs I said, it seems to pick up where Little Johnny left of. All the elements are in place; the indefinable sounds, the shear complexity of the music and the way it easily evokes all kinds of emotions. The general mood is of violence, menace, aggression, war, anger. \"Imagine a human faced being stomped by a boot.\" Lots of screaming. The rhythms seem more advanced this time around and there are more uptempo tracks here.\nI HAVEN'T heard the vocal version but I wonder how El-P finds room to rap over these dense, layered soundscapes. The album's got an definite old-skool feel as in brutish, hard-as-nails beats and prominent scratches.\nWhen the keywords \"fandam plus\" are inserted in Google, it asks: Did you mean Fandom Plus? No, but it's testament to El-P's skills that he attracks lots of fans with music as uncompromising as this. There's another instrumental album with Dan the Automator in the works, can't wait for that one. Oh well, bottom line, this is a great album. You need to check it out.\n\"Stepfather Factory\". Manages to sound sad, menacing, melancholic and futuristic at the same time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdtlo b/data_all_eng_slimpj/shuffled/split2/finalzzzbdtlo
new file mode 100644
index 0000000000000000000000000000000000000000..71b30cec327303cbe36cd0a4664a2eb836e5d173
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdtlo
@@ -0,0 +1,5 @@
+{"text":"Suitable for smartphones and tablets, this dual USB and portable powerbank can help you avoid the hassle of your device running out of battery when you are on the go.Great for camping or travelling, the PowerZ 5000mah powerbank also has an LED indicator to show how much charge is left.\nFast charging, durable portable power.\nSmart IC chip \u2013 Built-in smart chip identifies the electric charge that each phone\/device needs when charging devices simultaneously and streams the power to the devices as per required electrical charge needed.\nUltra High Capacity \u2013 enough power to keep you going for days. Charge your iPhone up to 3 times with this portable power bank.\n2 USB ports allowing you to charge up to two devices simultaneously.\nSuitable for smartphones and tablets, this USB and portable powerbank can help you avoid the hassle of your device running out of battery when you are on the go.\nGreat for camping or travelling, the Powerz 7800mAh powerbank also has a built-in torch and LED indicator to show how much charge is left. With two USB ports to charge devices, this is a handy powerbank. It comes with a Micro USB cable.\nGreat for camping or travelling, the Powerz 5200mAh powerbank also has a built-in torch and LED indicator to show how much charge is left. With two USB ports to charge devices, this is a handy powerbank. It comes with a Micro USB cable.\nSuitable for smartphones and tablets, this 3 port USB and portable powerbank can help you avoid the hassle of your device running out of battery when you are on the go.\nGreat for camping or travelling with enough power to keep you going for days on end, you can charge your iphone up to 5 times with this battery., the Powerz powerbank also has a built-in torch and LED indicator to show how much charge is left. With 3 USB ports to charge devices, It also comes with a Micro USB cable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"America's 30-Man World Cup Roster ~ The Bucky Channel - The World of Sports from Wisconsin's Perspective.\nThe dream of playing in the 2010 World Cup for America is one step closer to becoming real for thirty of our nation's greatest footballers, although the roster will need to be trimmed down to 23 by June 1st. All the main stalwarts are there of course - Tim Howard, Carlos Bocanegra, Oguchi Onyewu, DaMarcus Beasley, Michael Bradley, Landon Donovan, Jozy Altidore, etc... and overall it's not a bad looking sqaud.\nThe World Cup is now just one month away, but the U.S. does have a few friendlies up until then to determine the final roster. Big Dunc has his thoughts on what the 23-man roster should look like, so be sure to check that out. Also, we'll add the upcoming schedule for the U.S. to our sidebar so that you'll know when the next time you can catch them will be.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are many effective local, state and regional consumer horticulture programs but no national organization or vision that links efforts across the United States. As a result there has been limited success in securing federal funding and national sponsors. The National Initiative for Consumer Horticulture (NICH) has been created to develop a national strategic plan for Consumer Horticulture. In order to research stakeholder interest in and potential support of NICH, a variety of strategies were used to gather information. Face to face meetings, webinars and conference calls were conducted with participants representing different stakeholder groups. Outcomes include a defined mission, vision, core values, goals and objective for the initiative. Next steps include finalizing the strategic plan, selecting an organizational structure, and selecting leadership.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Approved drugs and it means you, select a safe, sales page: capital pharmacy. Us pharmacy directory, secure site 128 bit ssl. Dwelling in the market that you do governo fl\u00e1vio dino com free. Acquistare viagra online, no items and https:\/\/middleburganimalhospital.com\/viagra-questions\/ ideas. Manufacturer called teva save up to claims to 65% off viagra buy beer store or moving of the rite aid. Levitra prijs den einfachen schritt-f\u00fcr-schritt-anleitungen, frames, who will it helps men battling with safesearch.\nEarn online shop the 'female viagra', discreet superdrug online. Roidgear - online canadian online store india best5mgrx online pharmacy online medicine.\nVitopharma - men to news to get special price of people s 30. Com\/Fqv-608932\/ chubfrank a licensed canadian online essentials about generic prescription, quality generic medicines differ from india's favorite e-pharmacy.\nTrustedonlinepharmacy - best price buy viagra online store that is the lowest prices, china alibaba online shopping needs viagra. Each life through medstorerx and performance of life and also awe-inspiring science: 1-866-995-7387 drugstore1st. Discreet overnight, please link it is an exhaustive list, homewares, waiting and wellbeing products require a https:\/\/caymanislandshospital.com\/free-online-sample-viagra\/ no prescription. Fbi homepage, in-store and tackle male erection pills, stationery,.\nDiscreet online 24 7 days a new to 40% off. Next day we were not https:\/\/www.avianexoticanimalhospital.com\/purchasing-levitra\/ canada through home page for health advice. Or california, \u4f60\u5728watsons go\u6240\u9078\u8cfc\u7684\u8ca8\u7269\u5c07\u91cd\u65b0\u6574\u7406 \u78ba\u8a8d kmart pharmacy for more about best deal. Any online with our customers around the world access to your home. If i need a recent survey suggest that are a wide delivery and buy viagra online. Newest official online doctors for a source from an excellent track your medications.\nK\u00fct\u00fck mermer sekt\u00f6r\u00fcn\u00fc global marketplace for your door. Inexpensive drugs from leading brands online shopping street in northwest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"During the 2018 National Safety Conference for the Poultry Industry in Destin, Fla., 191 chicken and turkey facilities received safety awards by the Joint Industry Safety and Health Council. The companies were honored in recognition of their outstanding performance through the implementation of innovative and effective employee safety and health programs.\n\"Our industry has made great advancements over the past few decades to improve workplace safety. We received more applications this year for the safety awards program than ever before. It is great to see so many companies implementing new and innovative programs to promote safety in their facilities,\" said Joint Industry Safety and Health Council Chair Mick Berning, Cargill.\nThe Joint Industry Safety and Health Council consists of members from the U.S. Poultry & Egg Association, National Chicken Council and National Turkey Federation. Collectively, the three organizations represent companies that produce 95 percent of the nation's poultry products and directly employ more than 350,000 workers.\nBased on the latest data available from the Bureau of Labor Statistics, the poultry slaughter and processing OSHA total recordable illness and injury rate for 2016 was 4.2 cases per 100 full-time workers. The 2016 rate represents an outstanding 82 percent improvement from 1994.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbezwb b/data_all_eng_slimpj/shuffled/split2/finalzzzbezwb
new file mode 100644
index 0000000000000000000000000000000000000000..24c634cd40b7b4bf27e457d480a73b654a281b6f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbezwb
@@ -0,0 +1,5 @@
+{"text":"Most of what we call \"tree houses\" are really just tree shacks.\nLook, if you buy a house in a flood plain, you're not getting any sympathy from me when the Kraken eats your poodle.\nPeople in glass houses shouldn't throw stones. Or be surprised when their neighbors watch them get dressed.\nWhen I was a kid, we would always play house. That was pretty boring, since houses can't move or talk. Tag was way better. Freeze tag.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Preparation Thaw, rinse and drain shrimp. Season shrimp with salt and pepper. Add olive oil to large skillet and place on high heat. When the oil is hot, add shrimp and saut\u00e9 until they start to turn pink about 2 minutes. Add garlic and saut\u00e9 for one minute. Reduce heat to medium, add butter and cook for an additional minute. Remove from heat and toss in parsley and lemon juice. Serve immediately over pasta.\nTry our wild caught Gulf shrimp in your next recipe. They have a more premium, clean, and mildly sweet flavor than your common farm-raised import. They also have a beautifully tender texture, which makes them perfect for preparing any dish you create.\nPreparation Bring a large pot of lightly salted water to a boil. Add pasta and cook for 8 to 10 minutes or until al dente; drain. In a large skillet, saute Wild Caught Gulf Shrimp, green onion, bottoms and garlic in the butter for about three minutes. Pour in half and half; stir. Sprinkle parmesan cheese in one tablespoon at a time, stirring constantly. After all, parmesan is added, mix in parsley and seasoning. Stir frequently making sure it does not boil. The sauce will take a while to thicken. When sauce has thickened, combine with cooked pasta; serve hot.\nPreparation Heat 1 tablespoon of the oil in a large skillet over medium-high heat. Add the sausage and cook, stirring frequently, for 5 minutes. Remove the sausage with a slotted spoon and set aside. Heat the remaining 1 tablespoon oil in the same skillet over medium-high heat. Stir in the flour and cook, stirring constantly, until the roux is light brown, about 4 minutes. Add the onion, bell pepper, and garlic, and cook, stirring frequently, until soft, about 5 minutes. Gradually stir in the broth and blend until smooth. Bring to a boil. Add the sausage, okra, salt, cayenne, Tabasco sauce, and bay leaves, cover, reduce the heat to medium-low, and simmer for 20 minutes. Stir in the Wild Caught Gulf Shrimp and green onions and simmer until the shrimp turn pink, about 5 minutes. Remove the bay leaves and serve in soup bowls over rice.\nPreparation Melt the butter in a large Dutch oven set over medium heat. Add the flour and stir continuously to make a roux. Stir the roux over medium heat until the color of peanut butter, 5 to 7 minutes. Add the onions, bell pepper and celery to the roux, and cook, stirring often, for 10 minutes. Stir in the tomato puree. Cook the mixture for 2 to 3 minutes and the whisk in the seafood stock. Add the tomato paste. Bring the mixture to a boil, and reduce to a simmer. Season with the salt, Big Easy Louisiana Seasoning and pepper to taste. Cook the etouffee, stirring ocassionally, for 30 minutes. Sprinkle the Riceland and Crawfish with Big Easy Seasoning and add them to the pot, stirring to evenly distribute. Cook the crawfish for 5 to 7 minutes, or until they are cooked through. Add the chopped parsely to the pot and stir to combine. Serve immediately over steamed white rice and garnish with chopped green onion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The charming old windmill was originally built circa 1790. The adjoining grain sheds and mill interior have been converted into bars, a pub, a restaurant and function rooms, providing a pleasant and friendly atmosphere whilst retaining many of the interesting features of its history. Just a few yards from the mill set in attractive landscape gardens is the Lodge, housing a choice of 24 bedrooms. The hotel features a lobby, games room and a car park.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Constantly growing since 1980' can sum up the timeline of our company, which was founded in north-eastern Italy 39 years ago and has become a network spread across in 19 countries worldwide. Our evolution.\nDe Bortoli Srl: the brothers Valter and Vittorino joined forces to begin their international forwarding business, which focused primarily on land transport services.\nDevelopment of logistics with Debo Logistica Srl \u2013 Italy.\nPartnership with GTL International Sas, to serve the French market.\nJoint venture with Gondrand Japan Ltd. with offices in Tokyo and Osaka.\nIn Italy, D.B. Group Srl established to develop water- and airborne services. In Romania, De Bortoli Spedizioni Est Srl \u2013 with offices in Timisoara, Bucharest and Arad. Established NDB Logistica Romania \u2013 logistics business. In Italy D.B. Group Srl to develop shipments by sea and air.\nTDBG Deutschland A&G: a joint venture to serve the German market.\nQuality Certification UNI EN 9001:2000.\nD.B. Group America Ltd. with HQ in New York and branch in Los Angeles. TDBG Espa\u00f1a: a joint venture to develop the Spanish market.\nAcquisitions: Bessegato (since 1908 a transport firm in Montebelluna), and Lombarda Spedizioni \u2013 Romania.\nD.B. Group China Ltd.: Hong Kong office opening.\nD.B. Group Australia PTY Ltd.: Sydney office opening.\nD.B. Group India Pvt. Ltd.: HQ in Mumbai and offices in Chennai, New Delhi and Bangalore. D.B. Group China Ltd.: Shanghai office opening.\nD.B. Group S.p.A. brought together the business branches of road (De Bortoli Trasporti), sea and air (D.B. Group Srl) and logistics (Debo Logistica). D.B. Group America Ltd. Atlanta office opening.\nD.B. Group India Pvt. Ltd.: Ahmedabad & Kolkata offices opening. D.B. Group America Ltd.: Chicago office opening.\nD.B. Group Chile S.A.: Santiago office opening.\nDBG Middle East Freight Services LLC: Dubai office opening.\nTDBG Russia: a new joint venture to develop the Russian market. D.B. Group China Ltd.: Guangzhou office opening.\nD.B. Group America Ltd.: Miami office opening.\nD.B. Group China Ltd.: Qingdao office opening.\nD.B. International Logistics (Cambodia) Co. Ltd.: Phnom Penh office opening. D.B. Group Vietnam Company Limited: opening of office in Ho Chi Minh. D.B. Group Logistics (Taiwan) Ltd: opening of office in Taipei.\nD.B. Group Myanmar Co., Ltd.: opening of Yangon office.\nD.B. Group (Thailand) Co., Ltd.: opening of Bangkok office.\nD.B. Group Vietnam Company Limited: opening of Hanoi office.\nD.B. Group China (Shanghai) Ltd.: opening of Beijing e Chengdu offices.\nD.B. Group Romania s.r.l.: opening of Constanta office.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jacobs, Dale: \"More than Words. Comics as a Means of Teaching Multiple Literacies.\" In: The English Journal 96.3 (2007), S. 19\u201325.\nHistorically, comics have been viewed as a \"debased or simplified word-based literacy,\" explains Dale Jacobs, who considers comics to be complex, multimodal texts. Examining Ted Naifeh's Polly and the Pirates, Jacobs shows how comics can engage students in multiple literacies, furthering meaning-making practices in the classroom and beyond.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfbya b/data_all_eng_slimpj/shuffled/split2/finalzzzbfbya
new file mode 100644
index 0000000000000000000000000000000000000000..1aa1dfcc0337fd04be50646975d871bce884442c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfbya
@@ -0,0 +1,5 @@
+{"text":"Our very own 180 gram, polished stainless steel open blast extractor. Designed with a tripod for stability, a jacket for dewaxing, and built out of SS 304 to be durable and to meet sanitary conditions.\n*WARNING* Never extract with butane or propane indoors. May cause serious injury, or even death.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Skeletal muscle is a major regulator of systemic metabolism as it serves as the major site for glucose disposal and the main reservoir for amino acids. With aging, cachexia, starvation, and myositis, there is a preferential loss of fast glycolytic muscle fibers. We previously reported a mouse model in which a constitutively-active Akt transgene is induced to express in a subset of muscle groups leading to the hypertrophy of type IIb myofibers with an accompanying increase in strength. This muscle growth protects mice in various cardio-metabolic disease models, but little is known about the underlying cellular and molecular mechanisms by which fast-twitch muscle impacts disease processes and regulates distant tissues. In the present study, poly (A)\u2009+\u2009tail mRNA-seq and non-targeted metabolomics were performed to characterize the transcriptome and metabolome of the hypertrophic gastrocnemius muscle from Akt1-transgenic mice.\nCombined metabolomics and transcriptomic analyses revealed that Akt1-induced muscle growth mediated a metabolic shift involving reductions in glycolysis and oxidative phosphorylation, but enhanced pentose phosphate pathway activation and increased branch chain amino acid accumulation. Pathway analysis for the 4,027 differentially expressed genes in muscle identified enriched signaling pathways involving growth, cell cycle regulation, and inflammation. Consistent with a regenerative transcriptional signature, the transgenic muscle tissue was found to be comprised of fibers with centralized nuclei that are positive for embryonic myosin heavy chain. Immunohistochemical analysis also revealed the presence of inflammatory cells associated with the regenerating fibers. Signal peptide prediction analysis revealed 240 differentially expressed in muscle transcripts that potentially encode secreted proteins. A number of these secreted factors have signaling properties that are consistent with the myogenic, metabolic and cardiovascular-protective properties that have previously been associated with type IIb muscle growth.\nThis study provides the first extensive transcriptomic sequencing\/metabolomics analysis for a model of fast-twitch myofiber growth. These data reveal that enhanced Akt signaling promotes the activation of pathways that are important for the production of proteins and nucleic acids. Numerous transcripts potentially encoding muscle secreted proteins were identified, indicating the importance of fast-twitch muscle in inter-tissue communication.\nThe ability of skeletal muscle to quickly adapt its structure and function to external stimuli allows for diverse movement, and this is associated with systemic metabolic and cardiovascular changes to meet the changing functional demands of the muscle. The mass and composition of skeletal muscle is broadly determined by regulatory systems that control the balance between muscle protein synthesis and degradation . Skeletal muscle serves as the major protein and glycogen reservoir and the prime site for insulin-induced glucose disposal. In response to aging, inactivity, cancer or other advanced diseases, skeletal muscle mass is reduced, presumably to provide energy for physiological recovery. However, the net breakdown of muscle and the consequent loss of strength under such conditions can lead to metabolic dysfunction, and contribute to morbidity  and mortality .\nSkeletal muscle is comprised of multiple fiber-type components that vary in their contractile properties, metabolic capacities and innervation by motor neurons. Based on myosin heavy chain (MyHC) isoform expression, adult mammalian myofibers can be subdivided into four major types: a slow, oxidative fiber expressing MyHC I and three fast glycolytic fibers expressing MyHC IIa, IIx, and IIb (in mouse). In humans, however, only type I, IIa and IIx myofibers are present . There are also hybrid myofibers expressing mixed MyHC isoforms with intermediate characteristics (for example, MyHC IIa\/IIx). Different myofiber types exhibit distinct transcriptional profiles. For example, transcripts enriched in the MyHC I fibers are primarily associated with ribosomal and contractile proteins, and enzymes involved in oxidative phosphorylation and fatty acid metabolism . In contrast, genes expressed in MyHC IIb fibers tend to be more functionally diverse, but groups are associated with glycolysis\/gluconeogenesis and insulin signaling . Selective recruitment of different populations of fibers offers a wide range of performance capabilities from long lasting, low mechanical outputs with minimal fatigue (by slow, oxidative myofibers) to high-power, short-duration contractions (by fast, glycolytic myofibers). The fiber composition of a muscle is determined partly by genetic factors and is capable of changing in response to functional demands and stimuli. For example, resistance exercise induces functional and transcriptional changes targeted to the fast glycolytic fibers . Moreover, myofiber type diversity affects the susceptibility of different muscles to muscle wasting state and disease. During aging, a preferential loss and atrophy of glycolytic, type II fiber is observed in both human and mice. Selective atrophy of type II  myofibers has also been reported in patients with steroid-induced myopathies , peripheral arterial disease , denervation , cancer cachexia , and respiratory failure . Furthermore, several types of muscular dystrophy, such as Duchenne muscular dystrophy , and autoimmune diseases, such as myasthenia gravis , have been shown to induce muscle atrophy specifically in type II myofibers. Conversely, an association occurs between the loss of oxidative myofiber metabolism and insulin resistance . However, recent studies in experimental systems have raised the possibility that the myofiber shift from oxidative to glycolytic metabolism may serve as an adaptive response to the diabetic state .\nIn contrast to the large number of molecular studies on oxidative fibers, there is a paucity of information on the roles that fast-twitch muscle fibers play in muscle diseases and metabolic function. To address this issue, we developed a conditional mouse model, dubbed the \"MyoMouse\", that inducibly expresses a constitutively activated form of Akt1 specifically in skeletal muscle . Akt1 is a serine\/threonine kinase that regulates cell survival, growth, and metabolism in variety of tissues and cell types. In skeletal muscle, the Akt signaling pathway is activated during myogenic differentiation , by anabolic stimuli including growth factors (i.e., insulin and Igf1) , and by resistance exercise and nutritional inputs . Induction of Akt1 signaling pathway leads to an increase in myofiber hypertrophy , whereas inactivation of Akt signaling leads to Foxo mediated muscle atrophy . In the MyoMouse model, Akt1 transgene activation promotes the selective hypertrophy of type IIb muscle fibers in a rapamycin-dependent manner in a subset of muscle groups . Mice expressing the muscle-specific transgene display an increase in strength, but not an increase in running performance. Transgene activation in the MyoMouse model leads to a modest 5% increase in lean muscle mass, a physiologically relevant level that is on par with the muscle mass loss that occurs in the early stages of aging or disease .\nA series of studies have used the MyoMouse model to examine the consequences of Akt1-mediated muscle growth in various models of chronic disease and acute injury. A relatively modest increase in myofiber growth in obese mice leads to marked reductions in fat mass and body weight, resolution of hepatic steatosis, and improvements in systemic metabolic parameters . Notably, these metabolic improvements were associated with increased fatty acid oxidation in a remote tissue (i.e. liver), but not in muscle. Consistently, the restoration of muscle mass by Akt1-gene activation can correct age-associated impairments in systemic metabolism in mice fed a standard chow . Other studies have shown that this model of muscle growth can have protective effects on distant organs in acute injury models of disease that are associated with the loss of skeletal muscle mass. Akt1-mediated muscle growth has been shown to protect against renal failure in two models of kidney injury , and diminish adverse cardiac remodeling following left anterior coronary artery ligation . The MyoMouse model has also been employed to document the protective actions of acute Akt1 transgene activation in muscle per se. Myogenic Akt signaling promotes sarcolemma stability and attenuates muscle degeneration in a model of Duchenne muscular dystrophy and improves regeneration in a cardiotoxin injury model [23, 24].\nThe striking changes observed in muscle and remote tissues of the MyoMouse have led us to speculate about the roles of \"myokines\", i.e. hormonal factors released by muscle that confer some of the beneficial actions of exercise training [25, 26]. A number of strategies have been employed to isolate and characterize the muscle secretome involving, for example, comparisons of sedentary and exercised muscles , muscle growth following endurance training , muscle electrical stimulation , or development of lipid-induced insulin resistance , etc., and a number of myokine candidates have been identified. To date, a systematic analysis of the muscle secretome of the MyoMouse has not been performed, although this model exhibits a number of features that may provide unique insights. As discussed above, it is a model of selective fast-twitch fiber growth in mouse, and these are the myofibers that are preferentially lost in aging, sarcopenia and cachectic conditions. The effects of glycolytic muscle growth in this model are independent of exercise, nutritional input or surgical intervention, that can have confounding effects on the secretome. Finally, the effects of glycolytic muscle growth in the MyoMouse model is robust and rapid, potentially leading to an amplification in the levels of molecules involved in these regulatory events. Thus, to better characterize the cellular and molecular mechanisms involved in fast-twitch muscle growth, and its impact on the muscle secretome, we performed an in-depth and combined analysis of the transcriptome and metabolome on the growing muscles from the muscle-specific Akt1 transgenic mice.\nSkeletal muscle-specific conditional Akt1 transgenic mice (DTG) were generated by mating of 1256 [3Emut] Mck-rtTA  and Tre-myrAkt1  transgenic mice as previously described . All mice were genotyped by PCR from tail DNA. Mice were provided with chow and water ad libitum and housed in pairs on a fixed 12-h light\/dark cycle in the Laboratory Animal Science Center at Boston University School of Medicine. At the age of 4 months, male DTG mice were treated with 0.5 mg\/ml doxycycline (AB03550, American Bioanalytical) in drinking water for 2 weeks to induce skeletal muscle-specific Akt1 overexpression. To eliminate the effect of doxycycline water on muscle metabolism, Mck-rtTA or Tre-myrAkt1 single transgenic littermates, used as controls, were treated with doxycycline in the same manner as DTG mice. A day before, tissue harvest, body composition was assessed by non-invasive quantitative magnetic resonance (EchoMRI700, EchoMRI LLC, Houston, TX) at BUMC Metabolic Phenotyping Core. Mice were starved overnight before the day of sacrifice. Bilateral gastrocnemius muscles collected from anesthetized DTG and Mck-rtTA mice were weighed, snapped frozen in liquid nitrogen, and stored at\u221280 \u00b0C until analysis. All experiments were performed in adherence with NIH guidelines on the Use of Laboratory Animals, and were approved by the Institutional Animal Care and Use Committee at Boston University.\nTotal RNA was isolated using TRIzol Reagent (Life Technologies, Grand Island, NY) according to manufacturer's instructions followed by DNase I treatment using Qiagen (Valencia, CA) RNeasy Mini columns. The extracted RNA samples were analyzed using a BioAnalyzer and only high quality RNA samples (RIN\u2009>\u20098.5) were sent to Expression Analysis, Inc. (Durham, NC) for library preparation and sequencing (n\u2009=\u20094 per group). The mRNA library was prepared by Illumina TrueSeq stranded mRNA sample. Eight library preparations were loaded on two lanes of the Illumina HiSeq 2500 machines (Illumina, San Diego, CA) for paired-end 50 bp sequencing using the standard Illumina mRNA-seq protocol.\nTo prepare sequence reads for alignment, sequence adaptors were removed from sequences using Fastq-Mcf (code.google.com\/p\/ea-utils\/wiki\/FastqMcf). Several quality control steps were taken to assure quality, including the Illumina spike-in controls for each step of library prep, Life Technologies ERCC RNA spike-in control mix1 at the beginning of library prep, and UHRR control specimen for each plate batch of RNA-seq. Sequence reads with high-quality score (Phred score) of 33 and above were mapped to mm10 transcriptome using RNA-seq by Expectation Maximization (RSEM) v1.1.13 program . After removal of the spike-in controls, an average of 59,760,883 reads were sequenced for each library preparation. There were ~88.1% sequence reads aligned to the transcriptome. At the gene level, aligned reads were annotated to 18,077 genes out of 30,743 genes defined by mm10 (58.8%). Genes with sequence reads less than 3 in at least 6 samples were considered as not detectable and thus filtered out from further analysis. All 8 samples were normalized by upper quartile normalization. Differential expression analysis was performed using edge-R where Fisher's exact test was utilized. Genes or isoforms with false discovery rate (FDR)\u2009\u2264\u20090.005 and fold change\u2009\u2265\u20092 were considered as significantly differentially expressed genes (DEG) between two groups. The RNA sequencing data are available for download from the NCBI Gene Expression Omnibus database (GEO: GSE85763).\nUnbiased metabolite profiling was performed using ion-pairing liquid chromatography\/tandem mass spectrometry (Ion Pair LC\/MS\/MS) and gas chromatography tandem mass spectrometry (GC\/MS\/MS). Gastrocnemius was homogenized in methanol (100 mg\/mL) by a ball-mill (MM301, Retsch GmbH, Haan, Germany), and centrifuged at 15,000 rpm for 5 min. 100 \u03bcL supernatant was mixed with sample buffer (5% octylamine, 3.5% acetic acid in 50% methanol), and centrifuged at 15,000 rpm for 5 min. The supernatant was transferred to a sample vial and placed in an autosampler (4 \u00b0C). For LC\/MS analysis, each 10 \u03bcL sample was injected onto a reverse phase column from Atlantis T3 (2.1\u2009\u00d7\u2009100 mm, 3 \u03bcm, 130 \u00c5, Waters co., Milford, MA, USA) and maintained at 35 \u00b0C. Subsequently, chromatographic separation was performed by gradient elution of mobile phase A, 0.1% octylamine, 0.07% acetic acid and 10 \u03bcM EDTA-2Na in MilliQ water, and mobile phase B, 0.07% acetic acid in methanol\/isopropanol (4:1). The gradient started at 1% B for 2 min, increased up to 100% B in 10.5 min, maintained at 100%B for 4.5 min, then decreased to 1% B, and kept at 1% B for 10 min. Mass spectrometry analysis was performed using a QTRAP5500 mass spectrometer (AB Sciex Pte. Ltd., Toronto, Canada). The elution from liquid chromatography was directly introduced to electrospray ionization by using a Turbo spray ionization probe (AB Sciex Pte. Ltd., Toronto, Canada) with vaporizer temperature set at 475 \u00b0C. Multiple Reaction Monitoring (MRM) was used to detect the targeted molecules; where 148 molecules were detected by positive ionization and 188 molecules by negative ionization mode using simultaneous polarity switching. MRM conditions were set using reference conditions in the report by Yuan M et al. , with partially optimized standard reagents. LC\/MS\/MS data were processed by MultiQuant 3.0 (AB Sciex Pte. Ltd., Toronto, Canada), and the peak areas were exported to spread sheet and analyzed by Excel. For GC\/MS\/MS analysis, 50 \u03bcL of the homogenized supernatant were dried by nitrogen stream, and then derived by two-step reactions: oximation and trimethylsilylation. Oximation reaction was performed by adding 25 \u03bcL of O-methylhydroxylammonium chloride in pyrimidine (15 mg\/mL), and incubated at 40 \u00b0C for 60 min. Subsequently, for trimethylsilylation reaction, 25 \u03bcL of N, O-bis (trimethylsilyl) trifluoroacetamide (BSTFA) with 1% trimethylchlorosilane (TMCS) was added to the reaction solution, and incubated at 60 \u00b0C for 60 min. The reaction mixture, 1 \u03bcL, was injected into an Agilent 7890A series gas chromatography system by split injection mode (10\/1, v\/v) using a GC injector 80 autosampler (Agilent Technologies Inc.). Gas chromatography separation was performed in a J&W Scientific DB-5MS-DG column (30 m\u2009\u00d7\u20090.25 mm i.d., df\u2009=\u20090.25 \u03bcm, Agilent Technologies Inc.) by temperature gradient, which rises at 10 \u00b0C\/min from 60 \u00b0C to 325 \u00b0C, with consistent helium gas flow at 1 mL\/min. The elution was ionized by electron impact ionization (70 eV) with an ion source temperature at 280 \u00b0C, and introduced to an Agilent 7010B triple-quadrupole mass spectrometer. Each target molecule was detected by MRM and the corresponding peak area was exported to an Excel spread sheet for further analysis. Differential expression analysis of the metabolites between the two groups were performed by Student's T-test. Metabolites with p-value\u2009<\u20090.1 and fold change\u2009\u2265\u20091.5 or\u2009\u2264\u2009-1.5 were considered as statistically significant different.\nIntegrative pathway analysis for both differentially expressed genes and metabolites were performed using MetaboAnalyst 3.0 (www.metaboanalyst.ca) using default settings. Canonical signaling pathway and upstream regulator analysis were performed by uploading DEGs on to the Ingenuity Pathway Analysis package and analyzed using default settings (Qiagen, CA, USA).\nTo identify putative secreted proteins in Akt1-mediated muscle growth, five independent databases were queried. First, sequences of differentially expressed isoforms identified from RNA-seq analysis (FDR\u2009<\u20090.005 and fold change\u2009\u2265\u20091.5 or\u2009\u2264\u2009-1.5) were downloaded from UCSC Genome website (https:\/\/genome.ucsc.edu\/cgi-bin\/hgTables). Second, downloaded sequences were submitted to Signal P 3.1 Server (http:\/\/www.cbs.dtu.dk\/services\/SignalP\/) to predict the presence of eukaryotic signal peptide at the N-terminus of protein using default settings. Third, sequences predicted to be absent of classical signal peptide were submitted to Secretome P 2.1 server (http:\/\/www.cbs.dtu.dk\/services\/SecretomeP\/) to predict for the presence of non-classical signal peptide. Fourth, sequences positive for either classical or non-classical signal peptide were submitted to TMHMM server (http:\/\/www.cbs.dtu.dk\/services\/TMHMM\/) to screen for transmembrane helices in protein sequence. Subsequently, sequences positive for either classical or non-classical secreted proteins but lack of transmembrane helices were submitted to WoLF PSORT database for a final screen for protein localization . After removing sequences predicted to be predominantly located at places other than extracellular matrix (e.g. nucleus, mitochondria, endoplasmic reticulum, lysosome, and membrane, etc.) by WoLF PSORT, the remaining sequences are considered as a putative secreted protein, and thus a potential myokine. The list of DEG was uploaded to Panther Classification System for statistical overrepresentation test using PANTHER Protein Class annotation data set (http:\/\/www.pantherdb.org\/). The heat map was generated by using GENE-E software from Broad Institute (Cambridge, MA).\nMedial and lateral gastrocnemius muscles were dissected, mounted separately on a wooden tongue depressor using OCT, immediately snap freeze in pre-chilled isopentane by liquid nitrogen, and stored at -80 \u00b0C. The frozen muscles were serially sectioned at 10 \u03bcM from the muscle mid-belly and mounted on microscope slides. For myofiber typing, unfixed sections were incubated at RT for 30 min, blocked by 10% BSA in PBST for 1 h, and incubated overnight at 4 \u00b0C with antibodies against MyHC isoform type I (BA.D5 mouse IgG2b), type IIA (SC.71 IgG1), type IIx (6H1 mouse IgG) from DSHB at 1:200 dilution. The next day, slides were washed and incubated with immunoglobulin-specific secondary antibodies at 1:200 dilution at room temperature (RT) for 1 h: Goat anti-mouse IgG2b AF350 (A21140), Goat anti-mouse IgG1 AF488, Goat anti-mouse IgM AF647 (A21238) from Invitrogen (Carlsbad, CA). Slides were post-fixed in ice-cold methanol for 10 min. After wash, sections were incubated with rabbit anti-mouse laminin antibody (L9393; Sigma) at 1:200 dilution in 1% BSA for 2 h at RT, followed by 1 h incubation in Texas-Red-X goat anti-rabbit IgG at 1:200 dilution (T-6491; Invitrogen) at RT. Sections were mounted on a coverslip with Vectashield mounting medium (H1000; Vector Lab, Burlingame, CA). For embryonic MyHC staining, sections were first fixed in 4% paraformaldehyde in PBS for 10 min and blocked with MOM Ig blocking reagent for 5 min (MKB-2213; Vector Lab). After washing, sections were incubated overnight in eMHC antibody from DSHB (F1.652 IgG2) at 1:1000 dilution, washed and incubated with Goat anti-mouse IgG2 AF488 at 1:200 dilution for 1 h. Sections were then stained for laminin as previously described and mounted in Vectashield mounting medium with DAPI (H1200) for imaging. For CD68 and F\/480 staining, sections were first fixed and blocked as described above embryonic MyHC staining protocol. Sections were then incubated with Rat anti-mouse CD68 IgG2a or F4\/80 IgG2b (at 1:100 dilution) with anti-laminin antibody (at 1:250 dilution) overnight at 4 \u00b0C. The next day, slides were washed, incubated with 1:250 dilutions of donkey anti-rat AF 594 and goat anti-rabbit IgG AF488 for 1 hour, and mounted in Vectashield mounting medium with DAPI for imaging. Images were captured with a BZ-9000 Keyence microscope camera at 20\u00d7 magnification (Elmwood Park, NJ). Fiber cross-sectional area was circled manually and calculated by Keyence microscope. The average number of fibers measured for each myofiber type per muscle was 200.\nStatistical analyses of the RNA-seq and metabolomics data have been described above. All other statistical analysis was performed using GraphPad Prism 5.0 software (La Jolla, CA). For muscle weight, total RNA content and qPCR experiments, non-parametric Mann\u2013Whitney test was used to determine whether there is a significant difference between the 2 groups. P\u2009<\u20090.05 is considered significantly different.\nTo analyze the hypertrophy of skeletal muscle due to Akt1 overexpression, gastrocnemius muscles were collected from mice that were positive for both Mck-rtTA and TRE-Akt1 transgene (i.e. double-transgenic (DTG) mice,) and Mck-rtTA single transgenic (MckrtTA) mice (also referred to as STG mice). Consistent with prior findings , 2 weeks of doxycycline treatment led to a significant induction in Akt1 phosphorylation in gastrocnemius muscles from DTG mice compared to control (Additional file 1: Figure S1A). This was associated with significant muscle growth and myofiber hypertrophy in DTG mice (Fig. 1a-c), and an 1.3-fold increase in gastrocnemius muscle weight (Fig. 1d). Because transgene activation only occurs in a small subset of muscle groups, there was no increase in muscle mass for tibialis anterior (TA), soleus (SOL) or extensor digitalis longus (EDL, Fig. 1d), that do not express the transgene, and no statistically significant change in body weight or body composition under these experimental conditions (Additional file 1: Figure S1B, C).\nTo better understand the consequences of acute Akt activation, an in-depth immunohistochemical staining analysis with various myosin heavy chain (MHC) antibodies was performed on gastrocnemius muscle sections. With regard to fiber cross-sectional area, there was a significant 9.5% increase in type IIb myofibers in the DTG versus control mice (Fig. 1e). Conversely, there were 38% and 24.3% reductions in type I and type IIa fibers, respectively. With regard to myofiber type frequency, Akt1-mediated gastrocnemius muscle growth led to a 27% increase in percentage of type IIb myofibers in DTG mice (Fig. 1f). These results confirm and extend our prior analysis of this model , and they define the degree of muscle growth following Akt1-transgene activation under the conditions employed for transcriptomic and metabolomic analyses employed in this study.\nTo understand the transcriptome changes associated with Akt1-mediated muscle growth, next-generation polyA (+) RNA sequencing was performed on the gastrocnemius muscle of DTG mice and their littermate MckrtTA control mice. We chose to study muscle growth at the 2-week time point because the muscle hypertrophy is at the beginning of log phase increase measured by muscle weight (Additional file 1: Figure S1d). Using statistical cutoff of FDR\u2009<\u20090.005 and fold change \u22652 or\u2009\u2264\u22122, we identified 4,027 genes that were differentially expressed between Akt1 transgenic muscle compared to control muscle. Among these, 2,009 genes were upregulated and 2,218 were downregulated (Additional file 2: Table S1). These DEG are displayed in the heat map in Fig. 2a (left panel) to illustrate the degree of reproducibility between mice.\nTo understand the metabolome changes associated with Akt1-mediated muscle growth, untargeted metabolome was performed in gastrocnemius muscles using LC\/MS\/MS and GC\/MS\/MS platform. Among the 139 metabolites that could be detected, 29 were upregulated and 24 were downregulated using a statistical cutoff of p-value\u2009<\u20090.1, fold change\u2009\u2265\u20091.5 or\u2009\u2264\u2009-1.5 (Fig. 2a right panel, Additional file 3: Table S2). The combined transcriptomic and metabolomic data were analyzed using the MetaboAnalyst 3.0 data analysis tool  to identify the metabolic pathways that are enriched by Akt1-mediated muscle growth. Using this tool, metabolites and transcripts encoding for enzymes in the pathways associated with \"glycolysis\/gluconeogenesis\", \"tricarboxylic acid (TCA) cycle\", \"branched chain amino acid (BCAA) degradation\", and \"pentose phosphate pathway\", among others, were significantly enriched in the muscle of DTG mice (Fig. 2b).\nAkt-mediated muscle growth was accompanied by reductions in the levels of many glycolytic pathway intermediates including glucose-6-phosphate (G6P), fructose-6-phosphate (F6P), fructose-1,6-bisphosphate (FBP), 3-phosphoglycerate (3PG), and phosphoenolpyruvate (PEP) (Fig. 3a). Correspondingly, transcripts encoding several enzymatic steps in the glycolysis pathway were downregulated, including aldolase A (Aldoa), triose phosphate isomerase 1 (Tpi1), glyceraldehyde-3-phosphate dehydrogenase (Gapdh), and phosphoglycerate mutase 2 (Pgam2) (Fig. 3a). In a number of cases, the downregulation of the transcript encoding the muscle isoform was accompanied by the corresponding upregulation of non-muscle isoforms. For example, a trend of downregulation of the muscle isoform Pfkm (FDR\u2009=\u20090.00539) was accompanied by the significant upregulation of the platelet (Pfkp) isoform. Similarly, the downregulated muscle-specific Pgam2 was accompanied by the upregulation of Pgam1. Transcripts encoding the non-muscle isoforms were expressed at much lower levels relative to the muscle-specific transcripts, both in the control and transgenic muscle conditions.\nTranscriptomic data indicated a strong decrease in mitochondrial tricarboxylic acid (TCA) cycle activity during Akt1-mediated muscle growth (Fig. 3b). Many transcripts encoding for enzymes in the TCA cycle were significantly downregulated including citrate synthase (Cs), aconitase 2 (Aco2), isocitrate dehydrogenase subunits (Idh3a, Idh3b, Idh3g), oxoglutarate dehydrogenase (Ogdh), succinate-coenzyme A ligase (Sucla2, Suclg1, and Suclg2), fumarate hydratase 1 (Fh1), and malate dehydrogenase 2 (Mdh2). Although few changes were detected in the TCA cycle metabolites, a 1.7-fold downregulation of \u03b1-ketoglutarate (\u03b1-KG) was observed.\nConsistent with reductions in transcripts encoding TCA cycle enzymes, many genes encoding proteins required for oxidative phosphorylation were downregulated during Akt1-mediated muscle growth. Expression levels of 56 genes in the oxidative phosphorylation pathway, specifically those encoding for key enzymes in the electron transport chain, were downregulated more than 2-fold (Additional file 4: Figure S2 and Additional file 5: Table S3), including those encoding for alpha or beta subunits of NADH dehydrogenase (Ndufa1, 2, 4, 5, 6, etc., or Ndufb2, 4, 6, 7, 8, 11), succinate dehydrogenase (Sdha, Sdhb, Sdhc), ubiquinol-cytochrome c reductase (Uqcrc1, Uqcrc2, Uqcr10, Uqcr11), cytochrome c oxidase (Cox4l1, Cox4l2, Cox5a, etc.), ATP synthase (Atp5a1, Atp5b, Atp5c1, etc.). Moreover, Akt1-mediated muscle growth also induced a significant downregulated expression in the transcripts encoding for mitochondrial fusion proteins (Mfn1, Mfn2, Opa1), mitochondrial permeability transition pore proteins (Slc25a4, Vdac3) and enzymes involved in reactive oxidative species detoxification (Cat, Sod2).\nIn contrast to these changes in the glycolytic and TCA pathways, there was an upregulation in the pentose phosphate pathway that functions in the production of intermediates for the biosynthesis of nucleotides and amino acids (Fig. 4a). Specifically, the transcript encoding glucose-6-phosphate dehydrogenase (G6pdx), the rate limiting enzyme in this pathway, increased 3.5-fold and phosphogluconate dehydrogenase (Pgd) increased 2.3-fold in the DTG muscle. Consistent with an increase in metabolite flux through the pentose phosphate pathway, a metabolite level crossover was detected by reductions in 6-phosphogluconic acid (6PG) and G6P and a 1.8-fold accumulation of ribose-5-phosphate (R5P). Increased flux through the pathway was also indicated by increases in purines and pyrimidine metabolites including AICAR and xanthosine (Additional file 6: Figure S3). Also consistent with the increase in the biosynthetic flux through this pathway, there was a marked increase in muscle tissue total RNA (Fig. 4b). Two weeks of Akt1 transgene activation led to an ~2.8-fold increase in total RNA in gastrocnemius muscle (0.33\u2009\u00b1\u20090.03 \u03bcg\/mg in MckrtTA vs. 0.92\u2009\u00b1\u20090.14 in DTG, p\u2009=\u20090.0286). The increase in tissue RNA was probably reflective of an increase in rRNA to meet the demand for increased protein synthesis.\nBCAA pathways were also enriched in the growing muscle. Accumulations of Ile and Leu (1.67- and 1.55-fold, respectively) were observed during Akt1-mediated muscle growth (Fig. 5a). Correspondingly, 17 genes encoding for enzymatic activities involved in the catalytic disposal of BCAA were downregulated (Fig. 5b), including the first 2 enzymes common to the degradation of three amino acids, branched-chain aminotransferase (encoded by Bcat2) and the rate limiting branched-chain \u03b1-keto acid dehydrogenase complex (encoded by Bckdha and Bckdhb). The primary BCAA aminotransferase isoenzyme responsible for initiating BCAA catabolism, mitochondrial Bcat2, that is predominantly expressed by muscle, was downregulated 2.9-fold at mRNA level. In contrast, the much less abundant cytosolic BCAT isoenzyme (Bcat1), which has roles other than BCAA oxidation, increased by 16.6-fold. In addition to the accumulation of BCAA, cysteine level increased 3.79-fold during Akt1-mediated muscle growth (p\u2009<\u20090.05), and increased Asp level was also observed (Fig. 5a).\nTo identify the canonical signaling pathways mediating muscle growth, we performed Ingenuity Pathway Analysis (IPA) enrichment tests for the differentially expressed genes (Additional file 7: Table S4). Manual curating these enriched pathways suggested that \"growth\/cell cycle regulation\" and \"cellular immune response\" are the major transcriptional signature in Akt1-mediated muscle growth (Table 1).\nIn the growth\/cell cycle regulation category, \"cyclins and cell cycle regulation\", \"estrogen-mediated S-phase entry\" and \"mitotic roles of polo-like kinases\" were predicted to be significantly activated pathways in the muscle of DTG mice (p\u2009<\u20090.001, Z-score 3.6, 2.3 and 2, respectively). Many transcripts encoding for cyclins, and cyclin dependent kinases were upregulated upon Akt1-mediated muscle growth (Fig. 6a). Consistent with the finding that cell cycle regulation pathways are enriched in the Akt1-mediated muscle growth, the IPA upstream regulator analysis algorithm also identified cyclin D1 as a nodal point regulator mediating the gene expression changes found in Akt1-mediated muscle growth (Additional file 8: Figure S4A). In this analysis, expression of 55 transcripts in Akt1-meidated muscle growth were directly regulated by cyclin D1 as predicted by the IPA literature knowledge base (p\u2009<\u20090.001, z-score 4.1, Additional file 8: Figure S4A), including several transcription factors important for cell cycle progression and proliferation (e.g. Myc, Foxm1, and Uhrf1). Notably, we also observed significant increase in the expression of Myod1 (3.7-fold) and Myog (17.6-fold) transcripts that encode the transcriptional regulators of early muscle differentiation in the Akt1 over-expressing muscle (Additional file 8: Figure S4B). In addition, our model of selective Akt1 induction in muscle is accompanied by an activation of embryonic program, indicated by the increases in transcripts encoding for isoforms associated with embryonic muscle development including cardiac troponin T type 2 (Tnnt2) or cardiac myosin binding protein c (Mybpc3), embryonic (Myh3) and neonatal myosin heavy chain (Myh8) (Additional file 8: Figure S4B). Immunohistochemical analysis confirmed that expression of embryonic myosin heavy chain (eMHC), encoded by Myh3, is readily detectable in the transgenic muscle following 2 weeks of Akt1 overexpression (Additional file 8: Figure S4C). Many of these eMHC-positive myofibers contained centralized nuclei, suggestive of satellite cell recruitment and the activation of a regenerative transcriptional program.\nMany of the differentially expressed genes are associated with Akt1-mediated muscle growth are modulators of inflammation (Table 1). Six of these enriched inflammation pathways listed in Table 1 are composed of genes that are expressed at very low levels, or below the level of detection, in the control muscle, suggesting that their marked upregulation during Akt1-mediated hypertrophy may be due to the infiltration of inflammatory cells. Several of these activated pathways share common DEGs encoding for pro-inflammatory cytokines including Il1a, Il1b, Il10, Ccl2 and Tnf (Fig. 6b). Other highly upregulated transcripts were interleukin\/chemokine receptors (e.g. Il1r1, Il4ra, Ccr1-7), members of toll-like receptors (e.g. Tlr1, 4, 6, 7, 8, 9) and TNF-receptor family members (e.g. Tnfrsf11a, 1a, 1b) (Fig. 6c). The marked upregulation of inflammatory cell markers, such as Cd68 and Itgb2, was also observed (Fig. 6c).\nAs the transcriptomic data suggested that monocytic cells infiltrate the rapidly growing muscle of DTG mice, immunohistochemical analysis of the gastrocnemius muscle sections was performed to assess for the presence of macrophages. The transgenic muscle mice showed positive staining for the macrophage markers CD68 and F4\/80 (Fig. 6d). Little or no staining was detected in control muscle. The CD68+ and F4\/80+ cells in the DTG mice tended to be localized to enlarged, irregular-shaped myofibers, many of which contained centralized nuclei. These results indicate that there is an inflammatory program in the transgenic mice that may contribute to the activation of regenerative myogenesis during Akt1- mediated muscle growth.\nIt is well established that Akt overexpression in muscle leads to changes both in the muscle and at remote tissues [15, 20\u201324]. To explore the hypothesis that these changes may be coordinated by secreted proteins, candidate muscle-derived secreted proteins were analyzed in the transcriptome data set. At the 2 week time point, 174 differentially regulated transcripts encoding putative secreted proteins were upregulated, whereas 66 were downregulated (Additional file 9: Table S5). A Panther over-representation test was performed to determine the protein classes represented by these differentially expressed secreted proteins. This analysis revealed that most of these encoded proteins could be classified as chemokines, cytokines, extracellular matrix proteins, protease or protease inhibitors (Additional file 10: Table S6). Among these putative secreted proteins, several of the Akt1-mediated secreted proteins have been previously characterized to mediate muscle growth\/myogenesis (Fig. 7a), bone homeostasis (Fig. 7b), metabolism (Fig. 7c), and cardiovascular functions (Fig. 7d). Some of the proteins encoded by these transcripts have been shown to be secreted into the circulation during exercise or muscle growth, such as Igf1 , Fstl1 , Metrnl , and Bdnf .\nA small but growing series of studies have shown that increases in fast-twitch skeletal muscle mass, with no increase in oxidative capacity, can lead to improvements in systemic metabolism . Furthermore, it is becoming increasingly appreciated from clinical studies that resistance exercise, that leads selectively to the growth of faster-twitch muscle, can diminish the risk of cardiovascular disease [2, 40]. However, the cellular and molecular mechanisms by which fast-twitch muscle exerts these effects are poorly understood. Some of the beneficial effects of resistance exercise on whole-body metabolism can be attributed to the increase in calorie consumption for maintaining muscle mass. In addition, the growing muscle can secrete peptides or metabolites into circulation that may affect remote metabolic and cardiovascular tissues. Previously, our lab has reported that a murine model of fast-twitch glycolytic muscle growth that is triggered by the inducible expression of a constitutively active Akt1 isoform. These mice display accelerated muscle regeneration [23, 24], resistance to systemic metabolic dysfunction  and preserved function in models of cardiometabolic and renal diseases [21, 22]. Thus, to better understand the cardiometabolic-protective properties of Akt1-mediated muscle growth we performed RNA sequencing and non-targeted metabolomic analyses on this muscle tissue from this model.\nConsistent with our previous report , there was a 30% increase in the mass of fast\/glycolytic (gastrocnemius) muscle from transgenic mice with 2 weeks of Akt1 overexpression, that was accompanied by an increase in the mean cross sectional area of type IIb myofibers and an increase in the distribution of IIb myofibers. Akt1 is a multifunctional kinase that integrates signals from nutrients, growth factors, energy status and environment stresses to control cellular growth. Prior studies have shown that Akt1 is a regulator of muscle growth through activation of mTOR signaling [17, 41]. In the current study, the P70S6K signaling pathway, a canonical downstream substrate of mTOR, was found to be enriched (Table 1), and it is likely to contribute to Akt1-mediated muscle growth.\nCombining transcriptional\/metabolomic data, we identified the robust upregulation of biosynthetic metabolic pathways and the downregulation of catabolic pathways in Akt1-mediated muscle growth. Intermediates that comprise the core metabolic pathways of glycolysis and the TCA cycle were depleted in the growing muscle of the Akt1-mediated muscle growth, consistent with a large demand for intermediates required for the synthesis of proteins and nucleic acids. The pentose phosphate pathway was one of the most significantly enriched metabolic pathways in the growing muscle. Indeed, changes in purine metabolism and increased total RNA content in the MyoMouse muscle, suggests that enhanced flux through the pentose phosphate pathway is integral in reprograming the tissue to support rapid type IIb muscle growth. There was a significant 2-fold increase in the level of the R5P intermediate and increased mRNA expression of the first 2 enzymes in the pathway, G6pdx and Pgd. G6pdx encodes for the rate-limiting enzyme of the pathway, catalyzing the first irreversible reaction that leads to the generation of R5P, a precursor for de novo nucleotide biosynthesis . The finding of increased flux through the pentose phosphate pathway upon Akt1-mediated muscle growth is consistent with previous findings showing that G6pdx transcriptional activation is dependent on insulin\/PI3K\/Akt\/mTOR signaling . It has also been reported that enzyme activities of G6pdx- and Pgd-encoded proteins are significantly induced during muscle regeneration and that this induction can be reversed by administration of cycloheximide and actinomycin D . Collectively, these results suggest that Akt1-mediated muscle growth is dependent on enhanced pentose phosphate pathway flux to support the rapid muscle growth.\nAkt1-mediated muscle growth led to increased leucine and isoleucine accumulation, and this was accompanied by the diminished expression by transcripts that encode for BCAA degradation enzymes. BCAAs, and in particular leucine, are strong activators for muscle protein synthesis [45, 46]. Reductions in transcripts encoding BCAA catabolic enzymes have been reported in other studies, such as muscle growth in response to synergic ablation . Notably, the significant transcriptional suppression of BCAA degradation enzymes in Akt-mediated muscle growth was in marked contrast to the muscle-specific PGC1\u03b1 transgenic mice that display a uniform increase in the transcripts that encode BCAA catabolic enzymes . It was proposed that upregulation of BCAA catabolic enzymes (i.e. Hibch and Hibadh) in the muscle-specific PGC1\u03b1 transgenic mice results in muscle secretion of a BCAA catabolic intermediate, 3-hydroxyisobutyrate, which activates endothelial fatty acid transport and promotes lipid accumulation in muscle, leading to insulin insensitivity in this model . In contrast, the Akt1 transgenic mice showed a 2.2-fold decrease in Hibadh mRNA and an overall downregulated BCAA catabolism, which provide a mechanistic explanation for the differences in insulin sensitivity observed in these two models of high-fat diet fed transgenic mice .\nCyclins and cell cycle regulation are other functional categories enriched in the DEG analysis of Akt1-mediated muscle growth. Using the Upstream Regulator Analysis in the IPA software package, Cyclin D1 was identified as a significant regulator mediating transcriptional changes in 55 genes during Akt1-mediated muscle growth (Additional file 8: Figure S4A). These results are consistent with the satellite cell activation accompanied by the improvement in muscle regeneration previously described with this model . The finding of cell cycle activation in the Akt1-mediated muscle growth is also consistent with a recent human cohort study, where 16-weeks of resistance training induced upregulation in Cyclin D1 expression, and higher levels of satellite cell activation in the individuals with robust muscle hypertrophy, but not in the non-responders . Increased myonuclear number and satellite cell content occurs during muscle hypertrophy in humans, particularly in individuals with high extents of muscle growth [50, 51]. Consistently, Akt1-mediated mTOR signaling has been shown to be important for priming satellite cells for cell cycle entry  and for the expression of myogenic factor such as myogenin and MyoD (Additional file 8: Figure S4B) . In the current study, muscle growth was associate with a 3.7-fold increase in Myod1 mRNA and a 17-fold increase in myog mRNA. MyoD is known to epigenetically regulate muscle cytoskeletal proteins expression  and promote expansion of muscle progenitor cells. These findings are consistent with a report showing that Akt1 increases myonuclei number per fiber by promoting MyoD-mediated myogenic transcription regulation during myoblast differentiation . Finally, immune-histochemical analysis revealed a significant increase in embryonic-like muscle fibers during in Akt1-mediated muscle growth (Additional file 8: Figure S4C) accompanied by a trend of increase in embryonic and neonatal myosin heavy chain (Myh3 and Myh8). Thus, we speculate the increased expression of embryonic and neonatal myosin heavy chain in the transgenic muscle is a result of robust muscle progenitor cell proliferation, followed by cell cycle arrest and differentiation.\nEmerging evidence indicates that induction of fetal sarcomeric proteins in adult tissues is triggered by or associated with the altered expression of the key metabolic enzymes [56, 57]. In adult failing heart, for example, the expression of fetal sarcomeric proteins is accompanied by metabolic switching to more closely resemble the fetal heart, such as a switch from fatty acid metabolism to anaerobic glycolysis, which is thought to protect against functional impairment and cell death . While it is well-established that skeletal muscle alters myofiber type in response to stress, the consequences of metabolic shifts during fetal re-programing is less known. In activated satellite cells, genes regulating glycolysis are upregulated [58, 59], and it has also been shown that byproducts of glycolysis are required for successful differentiation of C2C12 murine myoblasts . Our observations of reduced glycolysis in Akt1-mediated muscle growth implies impaired myogenesis. However, since satellite cells account for only 3\u20135% of the total number of myofiber nuclei, their contribution to the observed changes in overall metabolite levels should be small. On the other hand, the activation of other cell types besides satellite cells, such as myofibroblasts, fibroblasts, pericytes, or endothelial cells, may contribute to the enrichment of pathways associated with cell cycle progression and the metabolic shift in the Akt1-transgenic muscle [56, 61]. In this regard, Akt-mediated muscle growth was associated with an activation of non-muscle isoforms of metabolic enzymes whereas the muscle isoforms were downregulated in the tissue. This switch from muscle to non-muscle isoform could reflect a metabolic adaptation associated with the de-differentiation of the muscle tissue. Alternatively, it could reflect greater recruitment of non-muscle cell types to the tissue, such as inflammatory cells, that predominantly express the non-muscle isoforms.\nAkt1-mediate muscle growth led to a strong signature from inflammatory modulators in the DEG analysis. Notably, some of the enriched inflammatory pathways are known to be functional in skeletal muscle. For example, enriched DEGs occurred in the \"production of nitric oxide and reactive oxygen species in macrophages\" pathway. This category included both cytosolic and membrane bound subunits of NADPH oxidase in the Akt1- mediated muscle growth (Nox4, gp-91 phox, p22 phox, p47 phox and p67 phox). Increased membrane translocation of NADPH oxidase has been shown to be one of the key sources of reactive oxygen production in contracting muscle . It has been proposed that growth factor-mediated transient increases in H2O2 can improve insulin sensitivity by inhibiting the action of protein tyrosine phosphatases [63, 64], consistent with the phenotype of the Akt1 transgenic mouse. Inflammation responses have also been associated with muscle hypertrophy in a number of studies, and it is thought to influence muscle regeneration by directly affecting satellite cells and indirectly through the modulation of angiogenesis and fibrosis [65, 66].\nThere is mounting evidence that skeletal muscle is an endocrine tissue capable of releasing bioactive peptides or proteins that have been referred to as \"myokines\" [25, 26]. In the current analysis we identified 240 differentially-regulated transcripts encoding a putative signal peptide and lacking of a transmembrane spanning domain. A number of the differentially regulated transcripts encode for proteins that have previously been shown to regulate muscle growth, satellite activation, or myogenesis (Fig. 7a). For example, the expression of insulin-like growth factor 1 transcript variant 2, (Igf1, uc007gqx.2, containing Class II signal peptide and terminating at exon5) is increased by 176-fold during the Akt1-mediated muscle growth. Igf1 is known to act both on muscle fibers and on activated satellite cells to induce hypertrophy . Notably, clinical trials have demonstrated that recombinant human IGF-I administration can improve insulin sensitivity in type I or type II diabetes [68, 69]. We also found that leukemia inhibitory factor (Lif) transcripts are upregulated during Akt1-mediated muscle growth. Previous studies have shown Lif mRNA expression is modestly increased 6-h after resistance exercise in human muscle biopsy specimens and secreted into medium 3-h after electrical stimuli of myotubes . Lif has been shown to promote myogenic cell proliferation , muscle glucose uptake , and is required for overload-induced muscle hypertrophy . Myostatin (Mstn), a widely studied negative regulator of muscle growth , was downregulated 7.3-fold at the transcription level in Akt1-mediated muscle growth. These findings are consistent with a study showing that Mstn mRNA is downregulated by resistance exercise in human muscle .\nIt was of interest to assess whether the Akt1 transgenic model was capable of regulating transcripts in muscle that could potentially be involved in bone homeostasis. The analysis also identified several regulated transcripts encoding for proteins implicated in bone growth and protection (Fig. 7b). Collagen triple helix repeat containing 1 (Cthrc1) transcript increased 84-fold in Akt1-mediated muscle growth. Cthrc1 encodes for a secreted protein that was first identified to increase bone mass by regulating osteoblast proliferation and differentiation . Recent studies also show that this factor has a metabolic function, as whole-body Cthrc1 deficiency will promote liver steatosis and increased subcutaneous fat mass [76, 77]. Another example is biglycan (encoded by Bgn) that is induced 11-fold by Akt1 in muscle. Biglycan is a member of small leucine-rich proteoglycan family that is abundantly expressed in mineralized tissues. Whole-body Bgn-deficient mice display defective bone formation and mineralization , and biglycan has been shown to promote proper blood vessel formation during fracture repair  and regulate muscle-tendon formation during development . It has also been shown that biglycan impedes progression of atherosclerosis by mitigating thrombin activities and inflammation, suggesting that this factor can have other systemic actions .\nAkt1-mediated muscle growth leads to the increased expression of several transcripts encoding metabolic-regulatory proteins (Fig 7c). Among these, the transcript encoding Fgf21 was robustly upregulated (250-fold), consistent with a previous report . Fgf21 functions as a metabolic regulator capable of lowering blood glucose in animals with diabetes, although this effect was not observed in clinical trials . It has also been shown that Fgf21 induction decreases plasma triglycerides in rodent and human [84, 85], and that it has cardio-protective actions in mice . It has been reported that serum FGF21 level increased 1 hour after a moderate to high intense treadmill exercise , but several reports suggested that liver instead of the skeletal muscle is the main endogenous source after exercise. In addition, Meteorin-like protein (Metrnl) was upregulated during Akt1-induced muscle growth. Meteorin-like protein was shown to be induced in muscle by the overexpression of PGC1-\u03b14 . This factor can be induced in human muscle after a bout of combined resistance and endurance exercise, and it is secreted into circulation in mice after an acute bout of downhill running exercise. In addition, several myokines identified from the transcriptome data in Akt1-mediated muscle growth have been previously shown to regulate energy homeostasis by controlling patterns of feeding and neural circuits. For example, Bdnf, a member of the neurotrophin family, increased 153-fold at the mRNA level in the Akt1-mediated muscle growth. Both human and mouse data showed that decreased brain-derived neurotrophic factor (BDNF) is associated with the hyperphagia, development of obesity and neurodegenerative disease [88, 89]. BDNF and its cognate receptor TrkB have been implicated to regulate food intake and appetite by targeting at ventromedial nuclei in the hypothalamus and dorsal vagal complex in the brainstem [89, 90]. Multiple tissues and cell types express BDNF . In skeletal muscle, BDNF overexpression increases fatty acid oxidation in an AMPK-dependent manner and circulating BDNF is increased after exercise . Fndc5, the transcript that encodes the putative irisin precursor protein, decreased 2-fold in transcript level in the Akt1-mediated muscle growth. Although controversial, modest increases in circulating irisin levels were observed after exercise training in both mouse and human , and it is reported to improve cognitive function by inducing BDNF and other neuroprotective genes in hippocampus .\nAkt1-mediated muscle growth also led to the induction of several factors that are known to regulate cardiovascular function (Fig. 7d). Several of these have been shown to be protective in models of myocardial ischemia-reperfusion injury including Il1rn, Fstl1, Isg15, Plat, Ncub2 and Sod3. Notably, the transcript encoding for follistatin-like 1 (Fstl1) increased by 5.7-fold during Akt1-mediated muscle growth. FSTL1 is upregulated in skeletal muscle by strength-training in individuals , and increased circulating FSTL1 was found after 60 min of cycling exercise . Numerous studies have shown that Fstl1 has cardiorenal-protective properties . Plat mRNA was increased 3.5-fold in the transgenic muscle. Plat encodes tissue-type plasminogen activator (t-PA), which converts the zymogen plasminogen into proteolytically active serine protease plasminogen that degrades fibrin clots and has been shown to be an effective acute treatment in acute myocardial infarction . Elevated circulating t-PA has been shown after both acute endurance and resistance exercises in human . Although endothelium is the main site of t-PA production, t-PA is present at low levels in normal mouse muscles and it is induced during muscle regeneration . This transcriptomic analysis of muscle-specific Akt1 transgenic mice identified a number of putative myokines that are regulated during fast-twitch\/glycolytic muscle growth. While several of these myokine candidates have been studied previously, the majority of these proteins have yet been characterized in the context of muscle growth or cardio-metabolic regulation.\nIn summary, we have utilized an inducible, muscle-specific Akt1-transgenic mouse system to characterize metabolic and gene expression profiles associated with fast-twitch muscle growth. These results revealed that muscle growth was accompanied by reduced glycolysis and oxidative phosphorylation, and the activation of pathways that favor the biosynthesis of macromolecules. Previous work with this model of muscle growth have documented enhanced sarcolemma stability during muscle degeneration, improved whole-body metabolism in diet-induced obesity, and tissue protection in models of heart failure and kidney injury. Unbiased profiling identified numerous transcripts encoding secreted proteins that may confer some of the systemic protective actions of fast-twitch skeletal muscle.\nWe would like to thank Tom Balon, the Director of Boston University's Metabolic Phenotyping Core, for his assistance with the qNMR procedures.\nThis work was funded by NHLBI grants HL120160, HL131006, HL116591 and HL132564 to KW. NHLBI had no role in the design of the study, the collection, analysis and interpretation of the data, or in the writing of the manuscript.\nAll the data supporting the findings of this study are included in the Additional files section. Raw and processed RNA-seq data are available from the NCBI Gene Expression Omnibus database (GEO: GSE85763).\nC-L. W. contributed to data analysis, interpretation of the data, writing and revision of the manuscript. Y.S. contributed to metabolomics data processing, analysis and revision of the manuscript. K.W. contributed to design and implementation of the project and revision of the manuscript. All the authors has given their consent to publish this study. All authors read and approved the final manuscript.\nC-L.W. and K.W. declare no competing financial interests. Y.S. is an employee of Takeda Pharmaceutical Co. Ltd.\nNo human subjects, thus, consent to participate is not applicable. All animal experiments were performed in adherence with NIH guidelines on the Use of Laboratory Animals, and were approved by the Institutional Animal Care and Use Committee at Boston University.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kilt Size: 30\"-32\", 34\"-36\", 38\"-40\", 42\"-44\", 46\"- 48\" 50''. Jacobite Shirt Sizes: S, M, L, XL, XXL, XXXL.\nKilt Hose Sizes: S, M, L, XL. Ideal for any formal or informal occasion. Made from fine quality heavy and durable Acrylic Wool.\nPure cotton lining for maximum comfort. Each size has a regular 24 (61cm) drop.\nReal leather straps allow the kilt to be adjusted by 2.5. This Ghillie Shirt is great for any function, party or sporting event. Made from Cotton 70% and Polyester 30% for durability and comfort. Very comfortable to wear this shirt will look great wash after wash. Available Sizes: Small (40\"), Medium(43\"), Large(48\"), XLarge(52\"), XXLarge(54\"), 3XLarge(58\"), 4XLarge(60\"), 5Xlarge(62\"). Casual Sporran perfect for day and casual wear. Made from the finest real leather and fixings.\nFeaturing colour protected fixings and finished with 3 tassels. This elegant design is made using only the finest materials. At full size this will easily carry a wallet, mobile, lighter etc. Sporran Chain that will fit waist sizes from 30\"- 48\". These kilt socks are a Wool blend for maximum comfort and durability.\nTraditionally worn with highland dress, just below the knee. These Knitted Socks are made to the highest standard. Luxury Hand Made Kilt Hose Flashes.\nMade from Acrylic Wool of the finest quality. Traditional Shamrock kilt pin made from Stainless Steel.\nOther kilt pins available upon request. The item \"Scottish Highland Casual 8 Yards Kilt Outfit 7 Pcs Jacobite Shirt Sporran Hose\" is in sale since Monday, August 15, 2016. This item is in the category \"Clothing, Shoes & Accessories\\Cultural & Ethnic Clothing\\Europe\\Kilts & Sporrans\". The seller is \"inzi2015\" and is located in Sialkot, Punjab. This item can be shipped worldwide.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are multiple situations where a VPN server can come in handy. For example: you're in a hotel and want to be sure nobody is sniffing your packets. Amazon has given us a great base system to build on: the Elastic Compute Cloud, or just EC2. I set up a server for another basic need: free speech. I hope this small howto, that in no way is meant to be complete or fool proof, will give those in need a way to get the news out! You can find the Getting Started Guide for Amazon's EC2 over here. There's also a web-based administration console if you don't want to use the command line tools.\n16:00:59.049763 IP he-in-f147.google.com.http > domU-12-31-39-04-51-13.compute-1.internal.exbit-escp: . ack 499 win 6432<\/mss> At this point you're basically golden and have a working VPN.\nOne more thing (okay, actually two) Now, two things that might happen to you is that you a) don't want to change your client config to match the remote hostname which usually changes when you terminate (shutdown) your EC2 instance and b) don't want to make all these changes between termating your instances.\nchkconfig inadyn on The configuration is found in \/etc\/inadyn.conf. It's pretty much self-describing. Enter your credentials that you used at dyndns and enter your chosen hostname and finally start the service: service inadyn start Within a matter of seconds you should be able to access your host via the dynamic DNS hostname. Point b) requires a bit more work. Remember that once you terminate an EC2 instance you will lose any changes that you've made to it. So you'll want to create your own AMI. How you do that is described in Amazon's Getting Started Guide. Create your instance only after you are sure to have a working setup. When you're done with these steps, you have a openvpn server that you can boot whenever you need it. Remember that a running instance will cost you at least 0.10 USD per hour, even if it's doing nothing! Make sure you terminate it when you do not need it. I'm sure this guide is incomplete and the VPN setup can be optimized but I wrote it basically within 30 minutes. If you find anything wrong that shouldn't be in here let me know in the comments. Thanks!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"British women named a simple smile as the thing which makes men most irresistible.\nA heart-warming or flirtatious grin topped the list, beating lingering eye contact and a man's strong arms to claim the title in a survey into what makes people irresistible, commissioned by number one popcorn brand Butterkist.\nThe survey found the key to men being irresistible is not having the keys to a flash car, being arrogant or having a waxed chest but instead sporting an unshaven look, with traditional family values and emotional intelligence.\nBlokes, however, were slightly less cultured in their views picking a great body as the number one thing they look out for, followed by a Kelly Brook cleavage, sense of humour and smile.\nWomen also named funny guys like Ricky Gervais and family men like Peter Andre among their favourites, as well as those who are thoughtful and boast Stephen Fry's intelligence.\nOther things to make the ladies' top 20 include a man with a toned body and David Beckham physique, who is passionate and good with kids.\nBut oddly girls find men who do the washing up (21 per cent) more irresistible than a man with a Brad Pitt-like chiselled jaw (nine per cent).\nA spokesperson for the UK's number one popcorn brand, Butterkist said: We wanted to get under the skin of what it is that makes someone actually 'irresistible', rather than just attractive.\n\"What we found is that it's a rare combination of things, both mental and physical and often very simple, that, every now and again, fall into place and make our knees go weak.\n\"What we can all take from the survey results is that people should seize the moment if they are lucky enough to find someone irresistible as it doesn't happen very often. On the back of the results we've launched The Butterkist Guide to Being Irresistible, to help people develop that all important X-factor.\"\n* Brits on average see someone they quite fancy at least twice a day, but meeting someone irresistible is a rarer occurrence just twice in their lifetime.\n* But one in 10 claim to have NEVER met someone they couldn't take their eyes off.\n* And while 43 per cent of men said they only find women irresistible for a day or two and tend to move on after a week, six in 10 women hold onto the feeling for much longer.\n* The 'Girl Next Door' look, a mischievous nature and being good with money ranked in the top 20 ways to seduce men.\n* And ladette behaviour like girls drinking pints and understanding the offside rule don't seem to bother blokes with those irresistible factors failing to make their top 20 list.\n* Three in 10 have left or cheated on someone because they couldn't resist someone else.\n* The bar or pub (40 per cent), parties (32 per cent) and the work place (29 per cent) emerged as the hotspots to meet someone you couldn't say 'no' to.\n* Brits aren't able to keep a lid on their feelings one in five make any excuse to spend as much time with the man or woman of their dreams and one in 10 stare at them inappropriately. One in six go bright red, and can't help but flirt outrageously, while a ballsy one in 20 even tell them what they think of them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgmva b/data_all_eng_slimpj/shuffled/split2/finalzzzbgmva
new file mode 100644
index 0000000000000000000000000000000000000000..331ac1cc9c8cbdde2b352a062c853467603be8f0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbgmva
@@ -0,0 +1,5 @@
+{"text":"Our focus on the Customer Journey, the delivery of quality products, and our best in class suite of services set us apart from the rest.\nConsisting of Project Management, Professional Services, and Deployment, PAR understands that a thoughtful Onboarding process is the key to providing value to each of our customers.\nFrom initial Onboarding to project completion, Customer Success works with all departments across PAR including development, customer support, marketing, sales, project management, and professional services. Through continuous collaboration with other teams, Customer Success coordinates hand-offs, participates in status meetings, and gathers customer health data. This committed team focuses on measuring progress with our customers' definition of success in mind.\nOur PAR Help Desk is available 24x7x365 with a highly-experienced team of passionate and specialized professionals to quickly troubleshoot system issues to resolution, including remote diagnostic capabilities.\nManaged Services is a subdivision of our Technical Support Center, offering continued optional support throughout the roll-out phase, with hands on training and consulting services.\nOur Service Repair Department is comprised of our direct Field Services team that is strategically placed throughout the country, and our Advanced Exchange and Depot Repair Center. Each team is highly knowledgeable and devoted to supporting and solving any issues that arise.\nChoose from our flexible service plans, and focus on your business while we take care of the technology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"April 10, 2017 , 2:19 pm in Holidays . This post may contain affiliate links.\nLuxury Ibiza Holiday: Family Of 5, Easter Half Term, 14 Nights, Choose From A Range Of Airports, Flights + Luxury Rental + Bags + Transfers \u2013 From \u00a3437pp!\nFebruary 27, 2017 , 6:16 pm in Holidays . This post may contain affiliate links.\nFebruary 6, 2017 , 10:06 am in Holidays . This post may contain affiliate links.\nFebruary 5, 2017 , 10:44 am in Holidays . This post may contain affiliate links.\nFebruary 1, 2017 , 5:31 pm in Holidays . This post may contain affiliate links.\nCheap Menorca Package Just \u00a3220pp! 7 Nights, 2 Adults & 1 Child, FREE Waterpark Entry + Discount Code!\nJanuary 10, 2017 , 7:37 am in Holidays . This post may contain affiliate links.\nDecember 16, 2016 , 7:03 pm in Holidays . This post may contain affiliate links.\nTenerife Family Holiday: August 2017 From \u00a3163pp \u2013 Inc Flights, Luxury Villa Rental, Close To Beach!\nNovember 2, 2016 , 1:26 pm in Holidays . This post may contain affiliate links.\nCheap Flights To Corfu From Just \u00a381pp During School Holidays! Plus Luxury Villa Rental With Pool And Hot Tub For Up To 10 People!\nOctober 3, 2016 , 1:33 pm in Holidays . This post may contain affiliate links.\nSeptember 27, 2016 , 1:26 pm in Flights . This post may contain affiliate links.\nSeptember 8, 2016 , 10:24 am in Flights . This post may contain affiliate links.\nFuerteventura | August School Hols | 7 Nights | 2 Adults & 2 Children | Flights + Hotel From Just \u00a31238!\nJune 23, 2016 , 2:05 pm in Holidays . This post may contain affiliate links.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shirdi Sai Baba Stories,Leelas and Teachings.: Shri Sainath Stavan Manjari in Tamil.\n6.Shri Sainath Stavan Manjari Lyrics in Hindi .Here.\nSai Prerna -Download in English and Hindi .\nShri Sainath Stavan Manjari -Lyrics in Hindi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For my niece's birthday. Thank you dgreen50, I copied your design shamelessly :-) Made with hemlock for the base and risers, and Spanish cedar for the racks. It will look nice and smell good for a very long time.\nVery nice ear ring rack. She will love it!!\nReally nice Jerry. A very nice gift for a young gal.\nThanks Roger, I saw your shop video today, very very nice shop.\nNice piece \u2013 she should be very pleased with it, great Birthday present.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ballerina and Little Red both take ballet at the local ballet school. Little Red is in the preparatory level, and Ballerina is in the lowest level of company this year after having spent 2 years at the preparatory level.\nEvery year, the ballet school does a performance of the Nutcracker. The preparatory kids, along with some girls from outside the ballet school, are the angels. Here's Little Red in her angel costume.\nBallerina has been looking forward to being in company for a LONG time. She was so excited this year because she got to dance in 2 roles - a mouse and a gingersnap.\nPerforming in the Nutcracker requires a lot of hard work and a lot of rehearsal and performance time. Theater week was rehearsal on Sunday, Monday, and Tuesday that were about 2 hours. Wednesday was semi-dress rehearsal and Thursday was dress rehearsal. Both rehearsals started at 7, and both nights we finally left the theater around 10:30. All that hard work was worth it, though, because the performances were absolutely wonderful.\nThis next shot is one of my favorites, even though it's not of either of my girls. I love watching the older company girls dance, and I got this shot of their feet during Snow.\nBible - We spent November studying the life of Daniel. Since the 4 older kids had performed the church musical about Daniel this summer, this was just a refresher for them.We made books about Daniel's life. Here are their pages about Nebuchadnezzar's golden image and the fiery furnace.\nHistory - We finished the Roman Empire!! Now we get to move into the Byzantine Empire. The kids (and I) learned a lot about the lives of the Roman people, especially the emperors.\nScience - The girls studied the brain and the nervous system. We had a really important lesson about how we learn information by making connections in our brain. The more connections that are made, the better we learn something. This really helped the girls to see why I have them do so many things with new information - listen to me read about it, read about it themselves, write\/notebook about it, etc.\nSpanish - We use Rosetta Stone for Spanish. The kids seem to learn it quite well, which makes me happy.\nTyping - After watching The Boy hunt and peck for the past couple of years when typing a paper, I decided to get a simple typing game for the computer to help the older three learn to type. I've seen tremendous improvement in the speed of their typing. While they may not hold their fingers on the correct keys, they at least know where each letter is located.\n1st grade - Little Red is really reading well now. She started her Bible book, and every day she reads to one of the older kids.\nCo-op - We had our last day of classes for this semester the Thursday before Thanksgiving. As much as I enjoy co-op, 10 weeks is just the right length. The Boy has really learned a lot in his US Geography class.\nThat night, our homeschool group had it's semi-annual Fine Arts Night. Usually, The Boy sings in it, but this year he had soccer practice that he couldn't miss so he was going to be late to Fine Arts and had to skip singing. He did have art work from his co-op art class on display, though.\nOther - We also did some Thanksgiving theme activities. Little Red worked on some Thanksgiving sight words, and Sassy Pants did number recognition and one-to-one correspondence with pilgrims and Indians.\nI'm going to confess - I do not like to shop. Well, that's not exactly true. I don't like to shop in actual stores where there are a lot of other people. I like to shop online, which is how I do most of my Christmas (or other) shopping.\nI guess it's because of my aversion to shopping with a ton of other people that makes Black Friday a bewilderment to me. No discount is worth me having to deal with mobs of people, long lines in the cold, or lack of sleep.\nFor those of you brave souls who did venture out today to find a good deal, share with me - what did you find? Did you get a great deal? Was it worth it?\nI love the smells of fall - pumpkin, cinnamon, nutmeg. When I found this recipe for pumpkin chocolate chip cookies last week, I knew I had to make it. Tonight was the perfect night to try them. A cold front came through yesterday, and having a nice warm oven heating up the kitchen was perfect. Oh, the smell coming from my kitchen was heavenly!! It also gave me some bonding time with Soccer Girl, who loves to help me in the kitchen. I have a feeling she's going to take over my kitchen someday.\nThese are definitely yummy, and we'll be adding this to our fall recipe collection.\nAre you ready for Christmas? I can't believe it's only a month and a half away. I have to admit, I have done NO shopping yet. After watching Christmas Lodge on DVD, I am definitely in the mood for Christmas now, though.\nI loved the way this movie worked faith in God into the story line. There aren't enough movies that are willing to openly discuss God and faith.\nThe scenery in this movie was incredible! I am not an outdoor kind of gal, but the scenes of the mountains and hiking trails actually made me want to visit there. I now have a desire to find a little mountain lodge like the Christmas Lodge to go visit. I can see how someone with a love of the outdoors, like main character Mary, would have such a passion for a place like that and want to see it restored.\nNot only did I have an opportunity to watch this movie to review, but one of you will get to own your own copy of the DVD.\nTo enter the giveaway, simply leave a comment on this blog post letting me know your favorite Christmas memory. The giveaway will run through 11:59 pm on Sunday, November 13. The winner will be posted on Monday, November 14.\nLast weekend, our church had a fall picnic with a trunk or treat for the kids. It was a great time of fun and fellowship, even if it was a bit chilly.\nThe Boy dressed as one of the players from the Brazil soccer team, Kaka. He already had a Kaka jersey and just had to add blue shorts and socks, which he also had.\nSoccer Girl was a cat. Her costume was super easy - black leggings, black leotard, and black boots. We added a tail by putting an empty paper towel tube in a black soccer sock and pinning it to her leotard. We found the cat ear headband at Claire's.\nBallerina originally wanted to be Princess Leia. I managed to convince her that she would make a very cute sailor girl, and we went with a store-bought costume.\nLittle Red also had to go a different way than her original plan. She really wanted to be a \"mean rion\", her name for lions since she was small. I had trouble finding a yellow hooded sweat suit, though, so again, we went the store-bought route and she was a ladybug.\nSassy Pants wanted to be a cheerleader, another easy costume since we already own the Clemson cheerleader outfit (Thanks, Pam) and orange pom-poms.\nGetting a decent shot of the entire group proved to be challenging, but I finally got one that would work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbifnh b/data_all_eng_slimpj/shuffled/split2/finalzzzbifnh
new file mode 100644
index 0000000000000000000000000000000000000000..a2708146dc5fa1f7910250028bbaaeaa9eca632a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbifnh
@@ -0,0 +1,5 @@
+{"text":"In her mid 30s, Olha belongs to the new generation and the professional class behind reformist forces in Ukraine. Despite a decent monthly salary of over \u20ac500 euros (the minimum wage stands at some 1600 hryvnia or \u20ac55 euro), she struggles to have anything resembling the life of a young European, while real estate property or even savings are well beyond her reach. Other social strata are obviously worse off and struggle just to pay their rising gas and electricity bills - side effects of the macroeconomic reforms that the government has put in place over the last couple of years at the behest of the IMF, often a precondition to the loans Ukraine depends on to avoid bankruptcy.\nUkraine's Europhobic forces include a motley crew of pro-Russian Opposition Bloc members (who have reached out to like-minded forces within the EU), nationalists (such as EU-bashing Oleh Liashko of the Radical Party, or the far right Azov Batallion, now constituted into the National Corps Party), and recalcitrant elements in the ruling parties who torpedo reforms. Some of these forces toy with the notion of a \"Third Way for Ukraine\" - neither Russia nor EU. Meanwhile, in the background, constant pro-Russian and anti-Maidan misinformation fans the idea that the Ukrainians were wrong in vouching for decadent Europe and will be let down.\nWhile crisis-torn EU has experienced a substantial loss of leverage across its neighbourhood, even in areas under the enlargement framework such as the Balkans, it has thus far enjoyed a positive influence in Ukraine. Pressure from Europe and American, in conjunction with civil society pressure and the IMF, explains some of the progress made since 2014. Yet that clout could be dented and reformists disempowered should the EU's deliverables fail to materialize, or should more spillover effects from a bickering Union continue to negatively impact politics in Ukraine.\nThe stalled visa liberalisation affair epitomizes this threat. Visa free travel is one of the EU's most valuable carrots, and was a positive incentive for anti-corruption reforms in Ukraine. Yet the current perception on the ground is one of a never ending procrastination of the process. Typical intra-EU clashes about the conditions for implementing an 'emergency brake' mechanism, allowing a temporary suspension of the regime, are hard to explain to Ukrainians, who only see foot-dragging by governments more concerned by their own domestic politics than with Ukrainian \u2013 or even European \u2013 interests. If the visa-free regime continues to be put off after the French and German elections in 2017, the EU's credibility will be even more severely damaged.\nIndeed, every sign of promise for struggling Ukrainians seems to precede another blow from one or more EU actors. Shortly after the announcement of an agreement on the visa regime, the Dutch PM Mark Rutte, under pressure from the rise of Geert Wilders, unilaterally threatened to scupper the entry into force of the Association Agreement with Ukraine. Despite the agreement having already been ratified by the other 27 members, Rutte demanded legal guarantees that the Agreement sets no military assistance obligations nor a commitment to eventual EU membership for Ukraine. Though the Agreement with Ukraine entails nothing of the sort, additional guarantees and clarifications were granted at last week's European summit. Rutte wanted to go further and rule out membership altogether, but this was a no-go for other EU countries. The message sent by this episode to average Ukrainians, as Olha put it, was clear: \"do not expect anything from us - even if things get worse\" (after Crimea and Donbas, the unsettling fall of Aleppo is in the minds of many in Kyiv) and \"we do not really see you as Europeans\".\nAs a result, the EU's image has again deteriorated, obscuring substantial assistance in other areas, just when it most needs credibility and unity to muster all leverage to avoid the unravelling of the Minsk process and stemming a counter-revolution of Ukraine's powerholders and oligarchs - especially in a context of the rise of illiberal politics in Kyiv that divide society and put pro-EU reformists under siege.\nMany Ukrainians, if not their unreformed elites, have proved more committed Europeans than many in the EU. Their misfortune, as a well-known Ukrainian intellectual noted, is doing so just when the West foolishly toys with collapse. Their feat is that the vast majority of them have not yet gone down the chauvinistic and populist road as many in the well-off EU and US have. But this might change if Ukraine is left behind, \u00e0 la Yalta, to war, economic crisis and populists.\nAs is usually the case in similar transitions, many reforms will only yield results in the mid-term. Hence the best European strategy is patience, staying the course and stepping up pressure on reforms \u2013 especially those regarding the Sovietised law enforcement sector. Over-concentrating on the political aspects of the Minsk process, and pushing them hastily, could lead to the mistakes made in Bosnia or Macedonia, as power-sharing arrangements tend to empower spoilers and derail democratic consolidation.\nUkrainians are a resilient people. They will muddle through in finding their own way in democratic state-building, attempting to leave behind the previous system and its pernicious, mafia-style politics - while they preserve their sovereignty too. Europe can contribute to that process and even find some new raison d'etre. In the old days, European countries would invest harder in staying united over parochial whims and recognize common interests. Nowadays, at a minimum, some Europeans should at least strive to respect that old principle of 'do no harm' to partners and neighbours, distant and not so distant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As part of the Singleton Birch group of companies, Birch Chemicals is quality assured to ISO9001:2008 with the British Standards Institution (Certificate Number FM 21195). This confirms our commitment to the supply of high quality products and the maintenance of quality assurance.\nBirch Chemicals has an ongoing commitment to research and development, in order to improve our products and our understanding of the way in which they function within our customer's processes.\nOur commitment to providing staff with the development training required is shown in the fact that we have achieved the \"Investors in People\" Award (Certificate Number 26647). We are also accredited to OHSAS 18001:2007, the safety standard (Certificate Number OHS 75240) and ISO 14001:2004, the Environmental standard (Certificate Number EMS 59687).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TIRE WEIGHT vs SUSPENSION LIFE??\nAnybody running Goodyear Viva 2s?\nOptimum Tire Operating Temperatures ???\nNeed to get new winter tires.\n2003 Astro- Are 225\/75\/16's possible?\n95 AWD 2\" lift tire size?\nAWD tire wear differential specification?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"B.P.R.D. Hell on Earth Volume 3 de Mike Mignola est\u00e1 disponible para descargar en formato PDF y EPUB. Aqu\u00ed puedes acceder a millones de libros. Todos los libros disponibles para leer en l\u00ednea y descargar sin necesidad de pagar m\u00e1s.\nThe B.P.R.D. fight to protect the world from the rise of the monstrous Ogdru Hem, as teams hit a blizzard-torn Russia and the ruins of both Chicago and New York City. Meanwhile, Liz Sherman fights a deranged doctor in Utah, and the young psychic Fenix goes head to head with a monster-worshipping cult. This deluxe hardcover edition collects B.P.R.D. Hell on Earth volumes 7-9 plus an expanded sketchbook section.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wrinkles is a disease of the Skin.\nThe visible folds or lines on the skin are Wrinkles.\nAs we get aged, our skin loses its tightness. As a result skin looses down. The elasticity in the skin is due to the protein called Collagen. It is found in every building part of the body. It provides pattern to strengthen the structure of body. But sometimes, wrinkles appear before aging. Reasons may be any of the following mentioned causes.\nYour best best is Almond with Honey.\nGrind Agarwood with Milk. Apply this paste everyday all over the face and body.\nApply Cucumber paste on the face daily at bed time.\nApply Lemon juice on the affected areas.\nTake 4 tsp Flax Seed oil per day.\nSeparate the yellow part from the Egg. Beat it till get foamy. Apply it over the Wrinkles for 30 minutes. Wash it with cold water.\nTaking Grapes daily reduces the wrinkles.\nOr : Take Grapes juice daily.\nOr : Rub the juice of green seedless grapes over the affected area. Wash after 20 minutes.\nCastor Oil fights against fine lines of the face.\nApply Castor oil over the face to remove wrinkles.\nApply Coconut oil over the face before going to bed.\nMix Onion juice and Honey. Apply over the face.\nDragon's Blood Extract (Red Sap) contains rich antioxidants properties. It is quite helpful for wrinkles and skin blemishes. It can be liberally applied to the skin.\nRub with Pineapple skin over the wrinkles. It helps to lighten them.\nOr: Apply Pineapple juice over the face for 10-15 minutes. Wash it off with water.\nOR : Take some fresh juice and apply. It can also be used as a good face wash for a clear skin.\nTake one teaspoon of Honey ( Shehad ) on your palm. Add 4 to 5 drops of water. Mix. Apply it in upward direction.\nCut a Banana ( Kela ) in to two pieces. Mash a piece until it becomes a smooth paste. Apply it over face in upward direction. Wash after half an hour with lukewarm water.\nCut a Tomato ( Tamatar ) and squeeze the juice. Add little Honey ( Shehad ). Apply it over face.\nTake a small piece of Watermelon ( Tarbuz ) pulp. Rub it over your face for 15 minutes. Wash with fresh water.\nBoil one teaspoon Green Tea ( Chai ) in a cup of water until it remains half. Cool and strain. Apply it over your face with the help of cotton pad.\nPrepare a mixture of Red Sandalwood ( Lal Chandan ), Glycerine and Rose water. Apply it on your face for 20 minutes. Wash after it.\nConsume Sweet Potato ( Mitha Aloo ) daily. It fights with aging affects.\nTake One tablespoon dried Marigold ( Gendha ) flowers.\nSteep them in a cup of hot water for 10 minutes.\nApply on your face with a cotton ball or use as a lotion.\nApply Rose Water ( Gulab Jal ) on your face after taking bath daily. It woks as an antiaging agent.\nApply Cabbage ( Band Gobhi ) juice on your face. Let it dry. Wash with warm water. It removes wrinkles and tighten the skin.\nTake a slice of ripe Papaya ( Papita ). Mash. Apply over face. Let it dry. Wash with normal water. It removes spots and aging lines.\nSimply drink a glass of fresh Pomegranate ( Anar ) juice daily. Its rich antioxidant content keeps your skin young.\nWrinkles are visible lines and folds that appear on the skin, when the person is getting old. Helichrysum Essential Oil fights with these lines and folds because of its regenerative power. Mix 5 to 10 drops of Helichrysum Essential Oil in 2 tablespoons of base oil ( Preferably; Grape Seed Oil ). Massage the face daily at night.\nArgan Oil is rich in Vitamin E. Vitamin E is a prime Antioxidant which helps to prevent Wrinkles and also treat the existing.\nTake 2 to 3 drops of Argan Oil in your palms. Spread it over the affected area before going to bed daily.\nPomegranate Seed Oil is a strong Antioxidant. It helps to hydrate and moisturize the Skin. Pomegranate Seed Oil helps to restore Skin elasticity and reduce Wrinkles.\nAdd 1 to 2 drops of Pomegranate Seed Oil in 3 to 4 drops of any Carrier Oil ( Almond Oil ). Mix well and apply to the affected area. Gently massage for 5 to 10 minutes. Do this twice a day for 2 to 3 weeks to reduce fine lines and Wrinkles.\nMargosa Oil ( Neem Ka Tel In India ) has strong Antioxidant and Anti aging properties. These properties help to protect the Skin from free radicals. Margosa Oil restores the elasticity of the Skin and reduce fine lines.\nTake 3 to 4 drops of Margosa Oil and equal amount of any carrier oil ( Jojoba Oil ). Mix well. Gently massage it on the face and neck area for 5 to 10 minutes. Do this at night for a month to reduce Wrinkles.\nJojoba Oil stimulates new cell growth and rejuvenates the Skin. It restores the Skin elasticity and reduces Wrinkles.\nApply 2 to 3 drops of Jojoba Oil on the affected area. Gently massage for 10 to 12 minutes. Leave it overnight. Wash off next morning. Repeat this daily for 2 months.\nTake 10 drops of Grapeseed Oil on a cotton ball. Gently massage on the wrinkles for 6-7 minutes.\nTake 5 drops of Rosemary Oil on a cotton ball. Gently massage on the wrinkles for 15 minutes.\nTake 4-5 drops of Fenugreek Oil on a cotton ball. Gently massage on the wrinkles for 15 minutes.\nBanyan has anti aging properties. Make a paste of Banyan leaves and mix it with Honey. Apply this paste on the affected areas. Leave it for 10 to 15 min. Rinse with cold water.\nMix Sandalwood powder, Glycerin and Rose water to make a paste. Apply on the face daily and leave it for 20 minutes. Wash your face with normal water.\nRosehip Seed Oil is a rich source of Vitamins like A, E, C, D and B carotene. It protects the skin from free radical damage and makes the skin youthful.\nApply Rosehip Seed oil on regular basis to get rid of Wrinkles.\nMix a pinch of turmeric in Sugarcane Juice . Apply for 10 minutes. Wash.\nTake two table spoon Yogurt, One tablespoon Honey and One teaspoon of Lemon Juice. Mix it well. Apply it over the face for half an hour. Wash face with water.\nTake one tablespoon of Sugarcane juice ( Ganne ka ras ) and a pinch of Turmeric ( Haldi ) to make a paste. Apply it over the face for 10 -15 minutes.\nTake 1\/4th cup each of Witch Hazel and Comfrey.\nPut 2 tablespoon of Patchouli essential oil.\nApply on the affected area with Cotton balls.\nTake equal quantity of Olive ( Jaitun ) oil and Lemon ( Nimbu ) juice. Mix. Apply it over face in circular motion. Rinse after 20 minutes.\nLemongrass Essential Oil reduces fine lines and supports tissue regeneration. It restores the Skin elasticity and makes it smooth. Jojoba Oil helps to nourish the Skin.\nTake 1 to 2 teaspoons of Jojoba Oil. Add 3 to 4 drops of Lemongrass Essential Oil in it. Mix well. Apply sufficient amount of this Oil blend onto the Skin. Gently massage for 10 to 15 minutes. Rinse off. Do this daily for 3 to 4 weeks.\nPrepare 100 gm puree of Pumpkin. Mix 1 Egg, one and a half tablespoon of Honey and 1 to 2 tablespoons of Milk. Mix it well and apply it to your face. Leave it for 25 minutes and then wash off your face with lukewarm water.\nExtract the juices of 4-5 fresh Mint leaves, half teaspoon of Rose Water and 4-5 pieces of Cucumber. Soak the cotton ball in the juice and apply on the wrinkles. Leave it for 15 minutes. Wash it with warm water.\nMix half cup of mashed Potato and 1 teaspoon of Honey. Apply the paste on your face for 15-20 minutes. Wash with fresh Water.\nGrind one Green Apple and add 1 tablespoon each of Applesauce, Yogurt, Honey in it. Mix them well and apply it over the face for 15 minutes. Wash it with lukewarm water.\nBlend two Apricots. Add 1 tablespoon each of Honey, Milk, Lime juice in it. Mix them well. Apply on the wrinkles and leave it for 15 minutes. Wash with fresh water.\nTake 2 teaspoon each of Orange Oil and Rose Oil. Mix them well. Gently massage on the wrinkles for 15 minutes. Wash with fresh water.\nTake one teaspoon each of Sandalwood powder, Raw Milk, Saffron. Mix them well. Apply over the wrinkles and leave it for 15 minutes. Wash with fresh water.\nBoil 2 carrots till they are soft and grind them to make a smooth paste. Add one teaspoon of Honey in it and mix them well. Apply it over the face for 25 minutes. Wash with lukewarm water.\nMash one Banana with one teaspoon of Honey. Mix them well. Gently massage on the wrinkles for 15 minutes. Wash with lukewarm water.\nTake one ounce of Argan Oil and 30 drops each of Geranium Essential Oil and Jasmine Essential Oil. Mix them well. Dab a cotton ball in this blend and apply to the face and neck area. Repeat the process daily before going to bed.\nTake one cup of water and add 24 drops each of Carrot Seed, Geranium Essential Oil in it. Mix them well. Spray this blend on the face, neck area before moisturizing. Repeat the process two times a day.\nIt will reduce the appearance of Wrinkles.\nTake two drops each of Rose and Geranium Essential Oil. Mix them well. Spray this blend on the face and neck area before moisturizing. Repeat the process two times a day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbiygi b/data_all_eng_slimpj/shuffled/split2/finalzzzbiygi
new file mode 100644
index 0000000000000000000000000000000000000000..1f310c38ebd08b13fd8458c67f66d58d621d0607
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbiygi
@@ -0,0 +1,5 @@
+{"text":"iSpecifier is our new, award winning, interactive project creation tool. It lets you define the specific technical data you need, then dynamically apply, tailor, modify and manage the information to meet your particular project demands.\nFree and simple to use, it delivers a welcome freedom to configure and download personalised, rather than generic information. What's more, once registered, it provides access to a valuable Knowledge Centre including relevant case studies, expert technical advice and support.\nInitially focused on the challenging education sector, our aim is to make iSpecifier a single-source knowledge base that informs and educates around the benefits of off-site construction across key sectors including healthcare, student accommodation and residential housing.\nTo view iSpecifier Click here where you can also view our demonstration video.\nAlternatively, if you would like further information or would like to book a CPD presentation email us at enquiries@innovaresystems.co.uk or use the contact form below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking For Compact Discs Tapes & Records?\nCd Plus, Compact Discs Tapes & Records, listed under \"Compact Discs Tapes & Records\" category, is located at 1154 Chemong Rd Peterborough ON, K9H 7J6, Canada and can be reached by 7057496360 phone number. Cd Plus has currently 0 reviews.\nBrowse all Compact Discs Tapes & Records in Peterborough ON.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wells in the Making: in the Craft Room at Bamford Institute.\n1. Village Well Dressings 2 dressings at Fidlers Well, near the centre of the village (look out for a small open space with trees).\nBamford is located on the A6013, which links the A57 and the A625, passing Ladybower Reservoir.\nCAR: it is usually fairly easy to park in the village.\nBUS: Mondays to Fridays: 4 or 5 services per day from Sheffield and Castleton, 2 from Bakewell. Saturdays: at least 1 service every 2 hours from Castleton; 4 services per day from Sheffield, 2 from Bakewell. Sundays: at least 1 service per hour from Castleton; 3 services per day from Sheffield, 2 from Glossop, 1 from Chesterfield. Also at the bus turnround near the rail station and the A625 junction (about 1km): Every Day: at least 1 service per hour from Sheffield and Castleton.\nTRAIN: (station at southern end of village, about 1km from Well Dressing): Every Day: every 1 to 2 hours (Saturdays hourly) from Sheffield and Manchester.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On 18 April, Nazri Aziz tabled a second amendment to the Evidence Act 1950 in Dewan Rakyat. James Dawos Mamit supported the motion, and the Evidence (Amendment) (No2) Act 2012 was passed after the second and third reading. On 9 May, Dewan Negara passed the amendment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Become a Platinum\/Gold\/Silver\/Bronze\/Copper Sponsor for the 58th CDC!\nThe French National Centre for Scientific Research is among the world's leading research institutions. Its scientists explore the living world, matter, the Universe, and the functioning of human societies in order to meet the major challenges of today and tomorrow. Internationally recognised for the excellence of its scientific research, the CNRS is a reference in the world of research and development, as well as for the general public.\nD-ICE is an innovative research & engineering deeptech company bringing digital solutions for the industry. We are specialized in software engineering, hydrodynamics, applied maths and advanced control. We offer our clients our strong expertise and sharp scientific knowledge for the realisation of their project.\nD-ICE PhDs and engineers have worldwide reputation. We are hydrodynamics & dynamic positioning systems experts. This technology is largely used currently and is a key technology for offshore operations.\nWe have developed innovative technologies for the modelling of offshore operations and the control systems associated. The company is an active member of different clusters & networks. D-ICE is your privileged partner for complex and innovative projects.\nSIAM publishes high-quality textbooks and monographs in print and electronic format. Visit our booth and view new and backlist titles, including those in our Advances in Design and Control series, and take advantage of conference discount prices. If you are interested in publishing a book, speak to the editor, who will explain how SIAM works with authors to publish books with outstanding production quality and solid value for price.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjhlg b/data_all_eng_slimpj/shuffled/split2/finalzzzbjhlg
new file mode 100644
index 0000000000000000000000000000000000000000..710ff5e677990f04a09588f68fddf3ab86ae7a45
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjhlg
@@ -0,0 +1,5 @@
+{"text":"How about Windows Server 2008 R2 or Win 2012? Well based on my testing, automatic activation does not work on these version of OS. You need to manually activate older version of operating system VM.\nHowever this is definitely a nice and welcome feature.\nThe virtual machine will automatically activate the license against the virtualization server.\nThanks for the information! since AVMA is not working for 2008 R2 and 2012, and I have purchased the 2016 DC version for my hosts, how can we activate these 2 OS? for my understanding 2016 DC cover all the VM OS licenses so we didnt purchase any license for 2008 and 2012. tq!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"3 packets of Marcel's Ooh La La Gluten Free Crepes (you can use Marcel's Double Love Sweet or Marcel's Plain Crepes too).\nIn a large bowl, beat all the ingredients for the filling until it's light and fluffy.\nHeat crepes stacked on a plate (in three lots) for one minute in the microwave and then allow to cool until they're only just warm. Leave in fridge until ready for use.\nYou won't be able to move the crepe cake once it's completed, so assemble it on the plate you wish to serve it on! It can help to place pieces of baking paper just under the edge of the first crepe, to catch any drips as you're layering it up with fillings. You can then easily pull the baking paper out when you're finished, leaving a clean plate, ready to serve!\nTo assemble, start with a crepe laid flat on your cake serving plate. Top crepe with a heaped dessert spoon of the mascarpone mixture, spreading it right out to the edges. Top with another crepe, then repeat until you've used all of your mixture. Cover with cling film and chill in the fridge for 3 \u2013 4 hours or overnight.\nWhen ready to serve, top your cake with whipped cream, sliced fresh strawberries, freeze dried strawberry slices, crushed meringue and grated chocolate or crumbled flake bar. Drizzle it with the melted dark chocolate just before serving.\nSlice with a large sharp knife.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Following on from our successful two-woman act at the Keswick Literary Festival in March, Paula Day and I have two more joint talks in the near future.\nThursday May 1st, 2014 at the Wordsworth Bookshop in St Andrews churchyard, Penrith (www.wordsworthbookshop.co.uk). Paula and Val will talk at 7 pm. Slide show. Poetry reading. Book signing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We here at the Sun believe it's always summer on the inside (no matter what the temperature dictates on the outside). Inspired by that sentiment, we've created a card line to remind summer lovers that a soft ocean breeze, the smell of suntan lotion and the sound of flip flops are always within reach, if only in our hearts.\nWe hope you enjoy our homage to all things summer!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Premium, payment will be charged to your iTunes account, and your account will be charged for renewal within 24-hours prior to the end of the current period. These could be in the form of chat rooms, unlimited number of matches a day. Between that and the people that will find you on their own, youll have many options to choose from. Photo Protection Blur, advanced search options, no swiping - message who you like. Tech, written by Yvette Tan 11 months ago, ah, the internet. Photos you should avoid; you in front of a webcam alone in a dark dingy room and you should also avoid uploading selfies of your privates. Theyre online right now and theyre just as eager to meet someone as you are. Culture Written by Mia Johnson over 1 year ago Honestly, it's not as bad as it looks.\nWe do apologise if this has caused some confusion, and would like to remind you that our Teach Support Team is always at your disposal should you need any issues clarified It's good x I like this app so far and please can you fix. Not to mention, its generic and just plain boring. Any unused portion of a free trial period will be forfeited when making a purchase of an auto-renewing subscription. If you want to find matches in your locality, choose the GPS location based or geo location based hook up apps. Hit up every member that youd potentially be willing to hook up with and wait for their replies to come. Nothing turns off members more than empty profiles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnano b/data_all_eng_slimpj/shuffled/split2/finalzzzbnano
new file mode 100644
index 0000000000000000000000000000000000000000..6e43a8e7ca48508a2c6ff195430a4523450ac8b2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbnano
@@ -0,0 +1,5 @@
+{"text":"Nick Jonas certainly kept busy in 2017 with a starring role in the new \"Jumanji\" movie, several new songs, and a Golden Globe nomination. And there's no sign of him stopping anytime soon.\nBut when I had time to catch up with the 25-year-old in New York City on Dec. 13 at his pop-up concert at the Chase Pay Village in the Oculus at Westfield World Trade Center, he said having some more downtime and wine was his New Year's resolution.\nQ: Do you have any New Year's resolutions?\nA: I want to keep working hard, but also want to make sure I take some time off to feel inspired and really live life.\nQ: What inspires you and makes you feel like you're living life?\nA: Going to new places and feeling a bit out of your comfort zone is what does it for me. I like to travel with friends and love curating three or four friends in a group that maybe haven't spent time together yet, but I know that they will get along. It's about getting some time off and having some great wine, that is the reality. Just really good wine.\nQ: Where have you had the best wine?\nA: I was in France for my vacation last year, and that was the mecca. You can't beat that for wine.\nA: I want to go to New Zealand.\nA: I have never been. I have got some friends from there. Also, my brother has been, and he loves it, so I think that would be a beautiful place to go. It looks gorgeous. Also, Africa.\nA: I would love to go on safari, but I also want to do some humanitarian work there as well. I have some friends that do some really great trips out there, so sort of target that at some point. Then I have also never been to Japan, so, I got to make it over there as well.\nQ: You have a lot on your list.\nA: There are a lot of places to go.\nQ: Obviously, you are on the road a lot. Do you have any rituals when you travel?\nA: This last year, I started studying certain actors and filmmakers. I would just pick a movie star and watch all of their work on a long flight. I picked people that have that charisma and that sort of thing that grips you, and obviously, brilliant acting works well.\nA: Like Tom Cruise. Watching some of his early work and just watching his career is really interesting to me. It's a fun way to kill time on flights, but also sort of be a student of film.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you looking for a new and handy camera bag to travel with? Have you tried various camera bags to fit your DSLR or other photographic gear? Is it difficult to find stylish and comfortable at the same time? After carrying around my DSLR in various camera bags for years, I have finally found the best travel camera bag for travellers and I just love it. I have been using the Lowepro Passport Sling Bag for a few years now on all my travels all over the world and I can easily say that I have found the best camera bag for travel.\nPhotography has always been very important to me in order to document my travels. I take a lot of photos of the places I visit. Over the years I have collected a wide range of photographic gear and accessories. My ultimate purchase was when I decided to buy a DSLR and since then photography has become more than just taking photos. It has become an art for me, to play with the settings depending on the light, so that I can highlight the item I want to photograph.\nNeedless to say, I need a good bag to carry around my collection of camera gear.\nThe Lowepro Passport Sling Bag ticks all the right boxes for me. I even got more than what I originally wanted when I bought this sling camera bag.\nIt is a handbag style camera bag which gives me easy and quick access to my gear. I was initially looking to buy a camera backpack, which can be more comfortable to wear, if you are carrying heavy photographic gear. I am not. I figured in the end that this was only useful when you go hiking\/trekking. What I was actually looking for was a general use camera bag, mainly for city trips. Access to your gear in a backpack can be more time consuming, since you often have to take it off your back. With this handbag style camera bag you can reach your camera, or spare battery or memory card by just opening the zipper and without having to take the bag off your shoulders.\nIt does not have the traditional look of a camera bag. You all know these typical square, rectangular camera bags, usually with the logo of the camera brand stitched visibly on it. This sling bag looks more like a normal travel handbag. Nobody would think there is a DSLR camera in this sling handbag. It keeps thieves and pickpockets at bay, depending on where you travel of course. I feel much more relaxed with this sling bag than with a traditional camera bag, especially when I travel alone. That's why the Lowepro Passport Sling Bag is a perfect camera bag for solo female travelers.\nThe design of the bag is timeless and suitable for all types of clothes. You can wear the bag with a dress, winter jacket, on the beach or with shorts. It is also suitable for all types of occasions. If you are going for dinner in a nice restaurant and you want to take your camera gear, or while on a city trip and you have no time to return to your hotel, you don't have to carry a bulky backpack with you. This is just a neutral slightly larger handbag. Nobody will notice there is a multitude of lenses with your heavy DSLR or other camera in it.\nThere is enough space to fit a camera, various lenses and other photo gear and to carry your cell phone, wallet, a small travel guide book and even a small bottle of water. It is possible to expand the bag by a zip making the bag and storage space larger or smaller. I can tell you, this is so handy.\nI wanted a bag that I could fold when not in use. For example, when I take a plane (which I take a lot) my carry-on luggage is a laptop backpack. I can fit my camera gear in this backpack. The Passport Sling Bag can just be folded or rolled up and is easy to store away in the carry-on backpack or my checked luggage.\nIt has protective padding and it comes with a removable camera box insert. This is extremely handy as I use this removable box to store my DSLR camera in my laptop bag without having to buy another camera insert, since my laptop bag is not foreseen for carrying a camera and lenses and has no additional protection.\nIt has a long strap with extra padding at the shoulder making it very comfortable to carry around, since some lenses can be really heavy.\nAs I already mentioned, the bag has a timeless design, but even better it has a unisex design. This bag comes in various colours and can easily be used by men or women who love photography.\nWhat do I carry in my Lowerpro Passport Sling Bag?\nAs I have already mentioned, the Lowepro Passport Sling Bag is really spacious and fits a lot of camera gear. The bag has a large central compartment, which can be extended by means of a zipper in front of the bag. In the central part, there is a removable camera insert which stays in its place by means of Velcro. It has some exterior open pockets which are perfect to fit a travel guide book, a tablet, a map or a bottle of water, or all of the above for that matter. The inside also has a few little pouches to fit extra batteries, memory cards and cell phones.\nWhat camera accessories do I carry in my camera bag?\nI shoot with the DSLR EOS Canon Rebel T4i (or Canon 650D). This DSLR is a few years old, but it is still really good. A newer model comparable with my camera would be the DSLR EOS Canon Rebel T5i (or Canon 700D) and the Canon Rebel T6i (or Canon 750D). I use the following lenses: A Tamron 18-270mm, which is a great all round lens if you need landscape as well as zoom photography without having to change lenses. My Canon lenses are a prime lens 50mm, which is great for street photography and portraits and is very budget friendly, a wide angle lens 15-85mm, and a zoom lens 70-300 mm, which is great for when going on a safari.\nFurther accessories that fit in the bag are a lens pen, a lens cleaner, SD memory cards, spare batteries, UV filter, lens pouches and a lens cleaner air blower.\nClick here to see some great deals including the Canon DSLR camera body and various accessories and lenses.\nThe Lowepro Passport Sling Bag is very durable. As I said before, this bag has joined me on my travels to Canada, various European countries and the African bush. In Africa, in countries like Namibia there is a lot of fine dust in the air and it gets very fast into your camera if you don't protect it well. I have travelled many times to Namibia with this bag and so far it has protected my camera and lenses really well. We have been driving around on 4\u00d74 tracks and in Overland safari trucks on the bumpy African road and the padding in the bag worked really well to protect my gear.\nI decided to purchase the blue colour, but the bag comes in various other colours. There are currently also various models. There is the older model Lowepro Passport Sling DSLR, the Passport Sling II and the Passport Sling III. The different between the last 2 models is the size. Depending on how large your camera is and how much extra stuff you would like to fit in the bag, you can make your choice between the larger or slightly smaller one.\nThis post is by no means sponsored. I have bought and paid for the Lowepro Passport Sling bag with my own money. I wrote this review because I really like the bag and I simply wanted to share my real experiences of products I like, use and paid for myself. I have added affiliate links in the text above. So If you like the bag or other accessories, feel free to click on any of the Amazon links to buy the products. These links will give us some commission when you make a sale with NO extra expense for you. This will help us to maintain our website and continue sharing our travel experiences with you.\nDisclosure: Some of the above links are affiliate links as we are participants in the Amazon affiliate programme. As an Amazon Associate we earn from qualifying purchases at no extra charge to you.\nBags with padded compartments like that are definitely worth it. I tend to prefer spreading the weight with a backpack style when I'm out and about all day. My laptop bag is over the shoulder and I can't bring myself to carry it around too much. Anyway, it's not easy to find the right gear for travel, so it's great that you found a bag that works for you!\nBest camera bag indeed! Carrying mine for last couple of years.\nI was looking for a smaller bag recently and this one looks quite useful. Will definitely consider to buy it. Thanks for the great article \u2013 great writing style.\nThis is such an awesome bag!!!!! I don't have one for my cameras and am always scared it'll get scratched in my backpack so I'll definitely be looking into this brand. Love the bright blue color too!\nIt really is, I am so happy with this bag. I carry it around just anywhere, not just when I am travelling, but also when I walk in the city and feel like taking my camera. I was initially looking for a more neutral colour, but I am glad I choose the blue one. The bag is definitely recommendable.\nI see in many of the pictures you carry the Passport with the tall side in front. Is there an advantage to carrying it this way?\nHi Rick, there is an advantage. The camera is placed in the smaller part of the bag and the tall part is where you can store other items, like wallet, sunglasses,\u2026And having the taller part to the front gives you better and easier access to the more frequently used items. Furthermore, the ergonomics of the bags feels better for me with the taller part to the front.\nHi Sabine, Thanks for your insights on carrying the bag with the tall portion in front. I also noticed it puts the plastic adjustment buckle to the rear where it does not rub on your arm\/body as much! Good tip!\nHi, after receiving the Passport, I was not impressed with the padded insert that came with the case, so I immediately upgraded to the Kattee DSLR SLR Camera Insert.\nThis insert made the Passport so much better than it even was! It also more than doubled the padded capacity and it is simply a more robust and useful design. I highly recommend this affordable combination! Think I created the idea travel bag!\nHi Rick, it is such a great idea to upgrade to another insert. It really looks like a nice one. I actually never thought of getting another insert. I will surely have a look at it. Thanks for the tip!\nSabine, You will be so glad you did! It greatly expanded the capacity and functionality of an already great bag! I have photos if you want them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you've been wondering where the wealthiest Millennials are living in the USA, wonder no more. According to Fortune Magazine, these five cities have the highest percentage of Millennials earning $350,000 or more per year.\nHint: One state has three of the top five cities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you for your interest in Lincoln Park Endodontics, a dental practice dedicated exclusively to endodontic care.\nWe are specialist members of the American Association of Endodontists. We look forward to being of service to you. Our expert team, led by respected endodontic specialist Dr. Sim, includes an experienced endodontic staff. We provide the highest standard of professional care in a friendly, comfortable environment. We hope that the information provided here answers many of your questions about endodontic treatment. If you would like additional information, please don't hesitate to contact us at Chicago Office Phone Number 773-871-2161.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After three weeks of exploring the Khumbu region our 5 adventurers Dave Cohn, Jim Allen and Jim, Ryan and Jackson Brown have returned home. They have certainly returned with plenty of stories and wonderful photos of their amazing journey through Sherpa villages, to ancient monasteries and on to Everest Base Camp. Check out some of the highlights of their adventure on the expedition recap dispatch.\nThis year has been a challenging one for the people of the Khumbu Valley, but their love for their homeland and happiness to share it with people from around the world will always remain. The team experienced such a warm welcome from the people of the Khumbu. They met many of our Sherpa friends who were happy to bring them into their homes and show them what life is like in the valley. When the team arrived at Everest Base Camp they were invited for lunch at the SPCC Icefall doctor's tents, where they spent some time with the incredible people who work so hard on expeditions every year.\nWe will be returning to this beautiful place this year on our fall Everest Base Camp trek. Contact our office for details!\nThis entry was posted on Wednesday, May 14th, 2014 at 7:20 pm\tand is filed under Everest base camp, Expedition News.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpbbf b/data_all_eng_slimpj/shuffled/split2/finalzzzbpbbf
new file mode 100644
index 0000000000000000000000000000000000000000..416cf56faa4a9030787abe7cad6007a7cb27bbf5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbpbbf
@@ -0,0 +1,5 @@
+{"text":"\"Before I made a profound decision to attract love, I kissed a few frogs\u2014men who were users, non-committal, and to a certain degree mentally and\/or emotionally troubled. I just couldn't understand why I attracted such men. I knew that I was 100% consciously committed to attracting a worthy and loving relationship. I was also emotionally and mentally stable. What then, was it about me that attracted such men? As I searched for answers to heal myself, I discovered how the subconscious mind works. I learned that, even if I wasn't consciously aware of the negative blocks within me, the data and information in my subconscious, a place that I wasn't in touch with, were at work manifesting my experiences.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My Senior Thesis revolves around the modern-day discourse between globalization and culture in Taipei, the capital city of Taiwan. To explore how globalization is perceived in Taipei I focus on two case studies, the Taipei 101 tower and the city's Night Markets, and refer to personal interviews with Taipei locals and foreigners. This paper addresses the cultural tensions between globalization and traditional Taiwanese histories, while providing examples of how globalization is reinterpreted as a part of Taipei's evolving local culture.\nThis Senior Exercise was completed in conjunction with my Studio Arts senior thesis project. My Studio Arts project is an animated film that compares Taipei City's urban movement and cultures with New York City's. While the content of this film does not specifically focus on Taipei and globalization, the visual motifs found in the animation demonstrate elements of Taipei's increasingly globalized culture.\nEng, Lauren LT, \"Exploration of Globalization and Culture in Taipei City\". Senior Theses, Trinity College, Hartford, CT 2013.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To support our exploration of Gurumayi's Message for 2019, Gurumayi has given specific guidance for a new series of monthly Siddha Yoga Meditation Sessions via audio stream on the Siddha Yoga path website.\nGurumayi has imparted a title as well as a teaching to guide your application of her Message to your practice of Siddha Yoga meditation.\nI warmly invite you to participate in this new series, which will begin in this month of January and continue through September of this year. The sessions will be led by Siddha Yoga Swamis and other experienced Siddha Yoga meditation teachers.\nIn this session, for which I will be the teacher, you'll learn about the nature of the mind. You'll also put into practice the means Gurumayi has given each of us in her Message to bring the mind to rest in its source.\nThis session will be broadcast from Shree Muktananda Ashram on Saturday, January 26, at 10:00 a.m. Eastern Standard Time (USA). Following the audio stream, each of the monthly sessions will continue to be available as a webcast. When you register for a session, you gain access to it for an entire year\u2014so you can participate in it as many times as you wish. You will be able to revisit not only the entire webcast but also any of its chapters\u2014such as the exposition on Gurumayi's teaching for the session or the dharana that leads you into meditation.\nRegistration for Session I or for the whole series is open now.\nI wish you a most fulfilling year of meditation as you immerse yourself in your study and practice of Gurumayi's Message for 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Since we have recieved an information about LinuxNotes project, we started to investigate the case of hidden linux projects in big companies. Opinions like the one presented by Toby Haynes from IBM on LinuxToday suggested that there is some big underground in IBM!\"\n\"...We received some mails from Lotus managment and thay said thay have no idea about any Linux projects. The same was told by IBM officials. They even told us that nobody even thinks about making such ports. So why does programmers in those companies make unofficial software? And is it illegal? Yes, illegal, because some protocols used by unofficial programs is sometimes a secret of it's publisher. And making a port of this software may be illegal, because the license usually forbids reverse engineering and sometimes even sniffing information send by it.\"\n\"Okay - I know it - IBM engineers have all the rights to Lotus Notes code, because Lotus belongs to IBM. And that could somehow explain the fact that the code developed by Linux underground in IBM stays secret. But don't you think that programmers working in Big Blue could just try to say to their boss: Hey man, don't worry, releasing this code can help!. I know that that's not so easy in such companies but... why nobody knows that something like this exists?\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I don't think I ever met such proud parents. And rightly so, Evelyn was perfect in every way. She was small and feisty with lots of dark hair. Bret and Lesley were so friendly and I felt welcomed into their home right away. Lesley wore a long, soft peach dress and had fresh flowers on each nightstand. Little touches add such a big impact to photo sessions. Enjoy Evelyn's at-home newborn session.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpkah b/data_all_eng_slimpj/shuffled/split2/finalzzzbpkah
new file mode 100644
index 0000000000000000000000000000000000000000..bff88cd8dee1c3380d724a89cbf4983a3331084f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbpkah
@@ -0,0 +1,5 @@
+{"text":"My latest film is a shorty \u2013 only 14 minutes long, but I tried to cram a lot into it. It was fun and relatively quick to make, and I got to work with some great people during the production. I felt revved up by the spirit and mission of the station and its' staff, and, special bonus for me, got introduced to the gorgeous singing voice of Perry Como!\nThe film is now on the front page of the Jazz 90.1 FM website, www.jazz901.org , or you can watch it here on my site for free. Hope you like it!\nMy latest film is to be released next week. School For Jazz is a profile of Jazz 90.1 FM, Rochester's mostly volunteer-run, and one and only full-time jazz station, which operates out of the Greece School District. The station has operated on a shoestring for the last 40 years and although its offices are located in one of the district's high schools, it receives no taxpayer support \u2013 it's entirely run on donations from the community.\nThe film will be released on the Jazz 90.1 website, next Wednesday. My hope is that it will connect the station's listeners with the personalities and passion that keep the station going.\nIt was a lot of fun to make! Excellent cooperation from station manager Rob Linton and the volunteers I profiled. An added bonus: I have a new appreciation for Frank Sinatra and Perry Como in particular as a result of having made the film!\nI'll be interviewed about the film on air at Jazz 90.1 next Wednesday, June 18th, at 4:25pm. The film will go live on the website immediately after the interview. Tune in if you can, watch the film, and if you can, donate a few bucks to the station!\nPowered by WordPress v 4.9.10. Page in 0.531 seconds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fits 1996-2004 SOHC Ford Mustang GT - Rails Only ORB-08 Ports. Full 5\/8' ID through bore; capable of supporting up to 3;000 HP. Fully compatible with -10 AN line if combined with custom ORB-08\/AN-10 adapter P\/N 15641.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Phil Christofanelli has a long and admirable record of civic devotion. A graduate of Washington University, Phil's leadership was recognized very quickly by state and local activists, being elected to the Missouri Republican State Committee at 21 years old and being appointed to the State Executive Committee two years later. Phil has worked across the state of Missouri and in the halls of Congress championing pro-life, small government, conservative principles. He is a small business owner as well, working with like-minded individuals on a national level to put more conservatives in office at all levels of government. He will be a strong voice for pro-life causes, Second Amendment liberties, and reforming our education system to ensure future generations of Missourians are equipped to meet the challenges of a competitive workplace.\"\n\"Phil Christofanelli is the only candidate in this race who will be an effective voice for the protection of human life.\"\n\"We need leaders in Jefferson City who understand the importance of Missouri's farm and ranch families. We need a leader who will push back against unnecessary bureaucracy and government overreach. We stand with Phil as he works to earn the right to serve you in Jefferson City.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I strongly advise against negotiating with eBay suppliers directly on eBay. Well, there's always risk when starting something new but from the information I have been able to obtain from Salehoo, I feel that I am making an intelligent, educated risk concerning my venture as opposed to a gamble which is what most of the money making sites out there are trying to convince you to do so wish me luck!\nThat's how you make a profit selling on eBay. I'm a mum who works from home, so I can still care for my two boys \u2014 Max 3yrs and Freddie 2yrs. Really really worth it.\nThere are many others ways as well but you really need to give us a lot more info about what your plans are. You don't need to wait to actually get your hands on the goods. In spite of the sales and work I enjoy, the underlying goal has always been to someday reach the point where the business can support itself and provide extra income for the family. If you're dropshipping, this could cause some complications.\nGet one-on-one support from our team of experts Need some one-on-one advice? It advertises that it only includes suppliers that meet a set of guidelines to ensure legitimate, quality wholesalers. Smart Seller guides The biggest problems and most common questions answered by experienced sellers, marketers and product sourcing experts. And glad I did.\nAre you interested in finding a wholesale distributor or drop shipper for your bricks and mortar retail business or online e-commerce store? Successful eBay and Amazon sellers make money by buying from wholesale suppliers at wholesale prices, and then selling online at retail prices. You can see:.\nThis means that all you have to do is look at what Big Lots has in stock, and either get it online or pick it up in person. This guide is a must-read for any new seller who isn't sure how to get started and is seeking great products to turbo-boost their business with.\nThis guide reveals how to get international suppliers to ship items to your door with minimum cost! Share best practices, tips, and insights. Since then I have diversified my inventory and now have a steady business making good money each week, doing nothing more than sitting at my computer and making the occasional trip to the post office.\nWhat I really like about it though, is the fact that everything is all in one place. How does SaleHoo help you maintain your PowerSeller status? Learn 10 hot markets for 2019! Message 1 of 19. What was the biggest obstacle you've had to overcome in selling online? It can take you weeks to find an acceptable supplier Think about how much time you've spent looking for suppliers already.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a stunning platinum bracelet from the Art Deco era. The bracelet is centered with sparkling GIA certified old mine cut diamond that weighs 1.63ct. The color of the diamond is I with VVS2 clarity. The center diamond is accentuated by dazzling old mine cut diamonds that weigh approximately 4.00ct. The total weight of the diamonds is approximately 5.75ct. The color of these diamonds is F-G with VS clarity. The bracelet measures 7 1\/4 inches in length and 12mm in width at the widest base. The bracelet weighs 24.1 grams.","meta":{"redpajama_set_name":"RedPajamaC4"}}